LMFDB - Embedded newform 51.7.b.a.35.17

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [51,7,Mod(35,51)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(51, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 7, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("51.35"); S:= CuspForms(chi, 7); N := Newforms(S);

Level:	$ N $	$=$	$ 51 = 3 \cdot 17 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	51.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$11.7327582646$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			35.17
Character	$\chi$	$=$	51.35
Dual form			51.7.b.a.35.16

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.461474i q^{2} +(22.0416 + 15.5939i) q^{3} +63.7870 q^{4} +70.0200i q^{5} +(-7.19617 + 10.1716i) q^{6} +296.369 q^{7} +58.9705i q^{8} +(242.663 + 687.427i) q^{9} -32.3124 q^{10} -1247.91i q^{11} +(1405.97 + 994.687i) q^{12} -2897.94 q^{13} +136.767i q^{14} +(-1091.88 + 1543.35i) q^{15} +4055.16 q^{16} +1191.58i q^{17} +(-317.230 + 111.983i) q^{18} +3451.27 q^{19} +4466.37i q^{20} +(6532.43 + 4621.53i) q^{21} +575.879 q^{22} +20903.9i q^{23} +(-919.577 + 1299.80i) q^{24} +10722.2 q^{25} -1337.32i q^{26} +(-5370.97 + 18936.0i) q^{27} +18904.5 q^{28} -43812.9i q^{29} +(-712.217 - 503.876i) q^{30} -15663.7 q^{31} +5645.46i q^{32} +(19459.8 - 27505.9i) q^{33} -549.883 q^{34} +20751.7i q^{35} +(15478.7 + 43848.9i) q^{36} -52053.1 q^{37} +1592.67i q^{38} +(-63875.1 - 45190.0i) q^{39} -4129.11 q^{40} -13556.7i q^{41} +(-2132.72 + 3014.55i) q^{42} +19115.8 q^{43} -79600.6i q^{44} +(-48133.6 + 16991.2i) q^{45} -9646.59 q^{46} -138157. i q^{47} +(89382.1 + 63235.6i) q^{48} -29814.7 q^{49} +4948.02i q^{50} +(-18581.3 + 26264.3i) q^{51} -184851. q^{52} -259727. i q^{53} +(-8738.49 - 2478.57i) q^{54} +87378.7 q^{55} +17477.0i q^{56} +(76071.4 + 53818.6i) q^{57} +20218.5 q^{58} -26169.3i q^{59} +(-69647.9 + 98445.8i) q^{60} +115739. q^{61} -7228.42i q^{62} +(71917.6 + 203732. i) q^{63} +256925. q^{64} -202913. i q^{65} +(12693.3 + 8980.19i) q^{66} -447018. q^{67} +76007.2i q^{68} +(-325972. + 460754. i) q^{69} -9576.39 q^{70} -84440.1i q^{71} +(-40537.9 + 14309.9i) q^{72} +32980.2 q^{73} -24021.2i q^{74} +(236334. + 167201. i) q^{75} +220146. q^{76} -369842. i q^{77} +(20854.1 - 29476.7i) q^{78} +837140. q^{79} +283942. i q^{80} +(-413671. + 333626. i) q^{81} +6256.07 q^{82} +485801. i q^{83} +(416684. + 294794. i) q^{84} -83434.2 q^{85} +8821.44i q^{86} +(683212. - 965705. i) q^{87} +73589.9 q^{88} -278589. i q^{89} +(-7841.02 - 22212.4i) q^{90} -858857. q^{91} +1.33339e6i q^{92} +(-345254. - 244258. i) q^{93} +63756.0 q^{94} +241658. i q^{95} +(-88034.6 + 124435. i) q^{96} +231790. q^{97} -13758.7i q^{98} +(857848. - 302822. i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 32 q^{3} - 1024 q^{4} + 286 q^{6} + 568 q^{7} - 912 q^{9} - 744 q^{10} + 194 q^{12} - 2312 q^{13} - 6240 q^{15} + 13208 q^{16} + 2936 q^{18} + 7936 q^{19} - 21688 q^{21} + 13176 q^{22} + 18282 q^{24}+ \cdots + 1619864 q^{99}+O(q^{100}) $ 32 * q + 32 * q^3 - 1024 * q^4 + 286 * q^6 + 568 * q^7 - 912 * q^9 - 744 * q^10 + 194 * q^12 - 2312 * q^13 - 6240 * q^15 + 13208 * q^16 + 2936 * q^18 + 7936 * q^19 - 21688 * q^21 + 13176 * q^22 + 18282 * q^24 - 76792 * q^25 + 87056 * q^27 + 45712 * q^28 - 45620 * q^30 - 36728 * q^31 + 73456 * q^33 - 52272 * q^36 + 133336 * q^37 - 48720 * q^39 - 81084 * q^40 - 305352 * q^42 - 412832 * q^43 - 9936 * q^45 + 283044 * q^46 + 233266 * q^48 + 342960 * q^49 + 50152 * q^52 + 196798 * q^54 + 7104 * q^55 - 22576 * q^57 + 167484 * q^58 - 58564 * q^60 - 235496 * q^61 - 331312 * q^63 + 299888 * q^64 + 485816 * q^66 - 774272 * q^67 + 553800 * q^69 - 1318656 * q^70 + 447312 * q^72 - 33152 * q^73 + 86872 * q^75 + 1045000 * q^76 - 8652 * q^78 + 1789576 * q^79 - 2391192 * q^81 - 178380 * q^82 - 751024 * q^84 + 235824 * q^85 - 155232 * q^87 - 3091500 * q^88 - 728056 * q^90 + 2080448 * q^91 + 1012816 * q^93 + 3173160 * q^94 + 422058 * q^96 - 1502960 * q^97 + 1619864 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/51\mathbb{Z}\right)^\times$.

$n$	$35$	$37$
$\chi(n)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$			0.461474i			0.0576843i	0.999584	+	0.0288422i	$0.00918202\pi$
$2$			0.461474i			0.0576843i	−0.999584	+	0.0288422i	$0.990818\pi$
$3$	22.0416	+	15.5939i	0.816355	+	0.577551i
$3$	22.0416	+	15.5939i	0.816355	+	0.577551i
$4$	63.7870			0.996673
$5$			70.0200i			0.560160i	0.959977	+	0.280080i	$0.0903609\pi$
$5$			70.0200i			0.560160i	−0.959977	+	0.280080i	$0.909639\pi$
$6$	−7.19617	+	10.1716i	−0.0333156	+	0.0470909i
$7$	296.369			0.864048			0.432024	−	0.901862i	$-0.357800\pi$
$7$	296.369			0.864048			0.432024	+	0.901862i	$0.357800\pi$
$8$			58.9705i			0.115177i
$9$	242.663	+	687.427i	0.332871	+	0.942972i
$10$	−32.3124			−0.0323124
$11$		−	1247.91i		−	0.937574i	−0.883311	−	0.468787i	$-0.844691\pi$
$11$		−	1247.91i		−	0.937574i	0.883311	−	0.468787i	$-0.155309\pi$
$12$	1405.97	+	994.687i	0.813639	+	0.575629i
$13$	−2897.94			−1.31904			−0.659521	−	0.751686i	$-0.729241\pi$
$13$	−2897.94			−1.31904			−0.659521	+	0.751686i	$0.729241\pi$
$14$			136.767i			0.0498420i
$15$	−1091.88	+	1543.35i	−0.323521	+	0.457289i
$16$	4055.16			0.990029
$17$			1191.58i			0.242536i
$17$			1191.58i			0.242536i
$18$	−317.230	+	111.983i	−0.0543947	+	0.0192014i
$19$	3451.27			0.503173			0.251587	−	0.967835i	$-0.419048\pi$
$19$	3451.27			0.503173			0.251587	+	0.967835i	$0.419048\pi$
$20$			4466.37i			0.558296i
$21$	6532.43	+	4621.53i	0.705370	+	0.499031i
$22$	575.879			0.0540833
$23$			20903.9i			1.71808i	0.511910	+	0.859039i	$0.328938\pi$
$23$			20903.9i			1.71808i	−0.511910	+	0.859039i	$0.671062\pi$
$24$	−919.577	+	1299.80i	−0.0665204	+	0.0940250i
$25$	10722.2			0.686221
$26$		−	1337.32i		−	0.0760881i
$27$	−5370.97	+	18936.0i	−0.272874	+	0.962050i
$28$	18904.5			0.861173
$29$		−	43812.9i		−	1.79642i	−0.439567	−	0.898210i	$-0.644868\pi$
$29$		−	43812.9i		−	1.79642i	0.439567	−	0.898210i	$-0.355132\pi$
$30$	−712.217	−	503.876i	−0.0263784	−	0.0186621i
$31$	−15663.7			−0.525788			−0.262894	−	0.964825i	$-0.584677\pi$
$31$	−15663.7			−0.525788			−0.262894	+	0.964825i	$0.584677\pi$
$32$			5645.46i			0.172286i
$33$	19459.8	−	27505.9i	0.541497	−	0.765393i
$34$	−549.883			−0.0139905
$35$			20751.7i			0.484005i
$36$	15478.7	+	43848.9i	0.331763	+	0.939835i
$37$	−52053.1			−1.02764			−0.513821	−	0.857898i	$-0.671770\pi$
$37$	−52053.1			−1.02764			−0.513821	+	0.857898i	$0.671770\pi$
$38$			1592.67i			0.0290252i
$39$	−63875.1	−	45190.0i	−1.07681	−	0.761814i
$40$	−4129.11			−0.0645173
$41$		−	13556.7i		−	0.196699i	−0.995152	−	0.0983495i	$-0.968644\pi$
$41$		−	13556.7i		−	0.196699i	0.995152	−	0.0983495i	$-0.0313563\pi$
$42$	−2132.72	+	3014.55i	−0.0287863	+	0.0406888i
$43$	19115.8			0.240429			0.120214	−	0.992748i	$-0.461642\pi$
$43$	19115.8			0.240429			0.120214	+	0.992748i	$0.461642\pi$
$44$		−	79600.6i		−	0.934455i
$45$	−48133.6	+	16991.2i	−0.528215	+	0.186461i
$46$	−9646.59			−0.0991061
$47$		−	138157.i		−	1.33070i	−0.746532	−	0.665350i	$-0.768282\pi$
$47$		−	138157.i		−	1.33070i	0.746532	−	0.665350i	$-0.231718\pi$
$48$	89382.1	+	63235.6i	0.808215	+	0.571792i
$49$	−29814.7			−0.253421
$50$			4948.02i			0.0395842i
$51$	−18581.3	+	26264.3i	−0.140077	+	0.197995i
$52$	−184851.			−1.31465
$53$		−	259727.i		−	1.74457i	−0.488994	−	0.872287i	$-0.662636\pi$
$53$		−	259727.i		−	1.74457i	0.488994	−	0.872287i	$-0.337364\pi$
$54$	−8738.49	−	2478.57i	−0.0554952	−	0.0157405i
$55$	87378.7			0.525191
$56$			17477.0i			0.0995182i
$57$	76071.4	+	53818.6i	0.410768	+	0.290608i
$58$	20218.5			0.103625
$59$		−	26169.3i		−	0.127420i	−0.997968	−	0.0637099i	$-0.979707\pi$
$59$		−	26169.3i		−	0.127420i	0.997968	−	0.0637099i	$-0.0202932\pi$
$60$	−69647.9	+	98445.8i	−0.322444	+	0.455767i
$61$	115739.			0.509906			0.254953	−	0.966954i	$-0.417940\pi$
$61$	115739.			0.509906			0.254953	+	0.966954i	$0.417940\pi$
$62$		−	7228.42i		−	0.0303297i
$63$	71917.6	+	203732.i	0.287616	+	0.814774i
$64$	256925.			0.980090
$65$		−	202913.i		−	0.738875i
$66$	12693.3	+	8980.19i	0.0441512	+	0.0312359i
$67$	−447018.			−1.48628			−0.743140	−	0.669136i	$-0.766665\pi$
$67$	−447018.			−1.48628			−0.743140	+	0.669136i	$0.766665\pi$
$68$			76007.2i			0.241729i
$69$	−325972.	+	460754.i	−0.992277	+	1.40256i
$70$	−9576.39			−0.0279195
$71$		−	84440.1i		−	0.235925i	−0.993018	−	0.117962i	$-0.962364\pi$
$71$		−	84440.1i		−	0.235925i	0.993018	−	0.117962i	$-0.0376362\pi$
$72$	−40537.9	+	14309.9i	−0.108608	+	0.0383389i
$73$	32980.2			0.0847784			0.0423892	−	0.999101i	$-0.486503\pi$
$73$	32980.2			0.0847784			0.0423892	+	0.999101i	$0.486503\pi$
$74$		−	24021.2i		−	0.0592788i
$75$	236334.	+	167201.i	0.560200	+	0.396327i
$76$	220146.			0.501499
$77$		−	369842.i		−	0.810109i
$78$	20854.1	−	29476.7i	0.0439447	−	0.0621149i
$79$	837140.			1.69792			0.848960	−	0.528458i	$-0.177230\pi$
$79$	837140.			1.69792			0.848960	+	0.528458i	$0.177230\pi$
$80$			283942.i			0.554574i
$81$	−413671.	+	333626.i	−0.778394	+	0.627776i
$82$	6256.07			0.0113464
$83$			485801.i			0.849619i	0.905283	+	0.424809i	$0.139659\pi$
$83$			485801.i			0.849619i	−0.905283	+	0.424809i	$0.860341\pi$
$84$	416684.	+	294794.i	0.703023	+	0.497371i
$85$	−83434.2			−0.135859
$86$			8821.44i			0.0138690i
$87$	683212.	−	965705.i	1.03752	−	1.46652i
$88$	73589.9			0.107987
$89$		−	278589.i		−	0.395178i	−0.980285	−	0.197589i	$-0.936689\pi$
$89$		−	278589.i		−	0.395178i	0.980285	−	0.197589i	$-0.0633112\pi$
$90$	−7841.02	−	22212.4i	−0.0107559	−	0.0304697i
$91$	−858857.			−1.13972
$92$			1.33339e6i			1.71236i
$93$	−345254.	−	244258.i	−0.429229	−	0.303669i
$94$	63756.0			0.0767605
$95$			241658.i			0.281857i
$96$	−88034.6	+	124435.i	−0.0995038	+	0.140646i
$97$	231790.			0.253968			0.126984	−	0.991905i	$-0.459470\pi$
$97$	231790.			0.253968			0.126984	+	0.991905i	$0.459470\pi$
$98$		−	13758.7i		−	0.0146184i
$99$	857848.	−	302822.i	0.884107	−	0.312091i
$100$	683938.			0.683938
$101$		−	285545.i		−	0.277148i	−0.990352	−	0.138574i	$-0.955748\pi$
$101$		−	285545.i		−	0.277148i	0.990352	−	0.138574i	$-0.0442518\pi$
$102$	−12120.3	−	8574.80i	−0.0114212	−	0.00808022i
$103$	−518690.			−0.474675			−0.237337	−	0.971427i	$-0.576275\pi$
$103$	−518690.			−0.474675			−0.237337	+	0.971427i	$0.576275\pi$
$104$		−	170893.i		−	0.151923i
$105$	−323599.	+	457401.i	−0.279537	+	0.395120i
$106$	119857.			0.100635
$107$		−	806170.i		−	0.658075i	−0.944317	−	0.329037i	$-0.893276\pi$
$107$		−	806170.i		−	0.658075i	0.944317	−	0.329037i	$-0.106724\pi$
$108$	−342598.	+	1.20787e6i	−0.271966	+	0.958849i
$109$	−2.09868e6			−1.62056			−0.810282	−	0.586040i	$-0.800686\pi$
$109$	−2.09868e6			−1.62056			−0.810282	+	0.586040i	$0.800686\pi$
$110$			40323.0i			0.0302953i
$111$	−1.14733e6	−	811709.i	−0.838920	−	0.593515i
$112$	1.20182e6			0.855432
$113$		−	1.61609e6i		−	1.12003i	−0.828483	−	0.560014i	$-0.810796\pi$
$113$		−	1.61609e6i		−	1.12003i	0.828483	−	0.560014i	$-0.189204\pi$
$114$	−24835.9	+	35105.0i	−0.0167635	+	0.0236949i
$115$	−1.46369e6			−0.962398
$116$		−	2.79469e6i		−	1.79044i
$117$	−703221.	−	1.99212e6i	−0.439071	−	1.24382i
$118$	12076.5			0.00735012
$119$			353146.i			0.209562i
$120$	−91012.1	−	64388.8i	−0.0526690	−	0.0372620i
$121$	214278.			0.120954
$122$			53410.5i			0.0294136i
$123$	211401.	−	298811.i	0.113604	−	0.160576i
$124$	−999144.			−0.524038
$125$			1.84483e6i			0.944553i
$126$	−94017.0	+	33188.1i	−0.0469997	+	0.0165909i
$127$	−52204.8			−0.0254859			−0.0127429	−	0.999919i	$-0.504056\pi$
$127$	−52204.8			−0.0254859			−0.0127429	+	0.999919i	$0.504056\pi$
$128$			479874.i			0.228822i
$129$	421342.	+	298089.i	0.196275	+	0.138860i
$130$	93639.4			0.0426215
$131$		−	82813.1i		−	0.0368371i	−0.999830	−	0.0184185i	$-0.994137\pi$
$131$		−	82813.1i		−	0.0368371i	0.999830	−	0.0184185i	$-0.00586313\pi$
$132$	1.24128e6	−	1.75452e6i	0.539695	−	0.762847i
$133$	1.02285e6			0.434766
$134$		−	206288.i		−	0.0857351i
$135$	−1.32590e6	−	376075.i	−0.538902	−	0.152853i
$136$	−70267.9			−0.0279344
$137$			3.84081e6i			1.49369i	0.664996	+	0.746847i	$0.268433\pi$
$137$			3.84081e6i			1.49369i	−0.664996	+	0.746847i	$0.731567\pi$
$138$	−212626.	−	150428.i	−0.0809058	−	0.0572388i
$139$	2.83496e6			1.05561			0.527803	−	0.849367i	$-0.323016\pi$
$139$	2.83496e6			1.05561			0.527803	+	0.849367i	$0.323016\pi$
$140$			1.32369e6i			0.482394i
$141$	2.15441e6	−	3.04520e6i	0.768546	−	1.08632i
$142$	38967.0			0.0136092
$143$			3.61637e6i			1.23670i
$144$	984036.	+	2.78762e6i	0.329552	+	0.933570i
$145$	3.06778e6			1.00628
$146$			15219.5i			0.00489038i
$147$	−657163.	−	464926.i	−0.206881	−	0.146363i
$148$	−3.32031e6			−1.02422
$149$		−	83515.1i		−	0.0252468i	−0.999920	−	0.0126234i	$-0.995982\pi$
$149$		−	83515.1i		−	0.0252468i	0.999920	−	0.0126234i	$-0.00401826\pi$
$150$	−77158.8	+	109062.i	−0.0228619	+	0.0323148i
$151$	2.50988e6			0.728991			0.364495	−	0.931205i	$-0.381242\pi$
$151$	2.50988e6			0.728991			0.364495	+	0.931205i	$0.381242\pi$
$152$			203523.i			0.0579538i
$153$	−819122.	+	289151.i	−0.228704	+	0.0807330i
$154$	170673.			0.0467306
$155$		−	1.09677e6i		−	0.294525i
$156$	−4.07441e6	−	2.88254e6i	−1.07322	−	0.759279i
$157$	−5.30217e6			−1.37011			−0.685054	−	0.728492i	$-0.740221\pi$
$157$	−5.30217e6			−1.37011			−0.685054	+	0.728492i	$0.740221\pi$
$158$			386319.i			0.0979433i
$159$	4.05015e6	−	5.72479e6i	1.00758	−	1.42419i
$160$	−395295.			−0.0965076
$161$			6.19524e6i			1.48450i
$162$	−153960.	−	190898.i	−0.0362128	−	0.0449011i
$163$	−1.39663e6			−0.322491			−0.161246	−	0.986914i	$-0.551551\pi$
$163$	−1.39663e6			−0.322491			−0.161246	+	0.986914i	$0.551551\pi$
$164$		−	864741.i		−	0.196044i
$165$	1.92597e6	+	1.36257e6i	0.428743	+	0.303325i
$166$	−224185.			−0.0490097
$167$			5.53783e6i			1.18902i	0.804087	+	0.594511i	$0.202654\pi$
$167$			5.53783e6i			1.18902i	−0.804087	+	0.594511i	$0.797346\pi$
$168$	−272534.	+	385220.i	−0.0574768	+	0.0812422i
$169$	3.57123e6			0.739874
$170$		−	38502.8i		−	0.00783691i
$171$	837494.	+	2.37249e6i	0.167492	+	0.474479i
$172$	1.21934e6			0.239629
$173$			6.73423e6i			1.30062i	0.759670	+	0.650309i	$0.225361\pi$
$173$			6.73423e6i			1.30062i	−0.759670	+	0.650309i	$0.774639\pi$
$174$	445648.	+	315285.i	0.0845950	+	0.0598488i
$175$	3.17772e6			0.592928
$176$		−	5.06048e6i		−	0.928226i
$177$	408081.	−	576813.i	0.0735913	−	0.104020i
$178$	128562.			0.0227956
$179$			805353.i			0.140419i	0.997532	+	0.0702097i	$0.0223669\pi$
$179$			805353.i			0.140419i	−0.997532	+	0.0702097i	$0.977633\pi$
$180$	−3.07030e6	+	1.08382e6i	−0.526458	+	0.185840i
$181$	−2.28618e6			−0.385545			−0.192773	−	0.981243i	$-0.561748\pi$
$181$	−2.28618e6			−0.385545			−0.192773	+	0.981243i	$0.561748\pi$
$182$		−	396341.i		−	0.0657438i
$183$	2.55107e6	+	1.80482e6i	0.416264	+	0.294496i
$184$	−1.23271e6			−0.197882
$185$		−	3.64476e6i		−	0.575643i
$186$	112719.	−	159326.i	0.0175169	−	0.0247598i
$187$	1.48698e6			0.227395
$188$		−	8.81264e6i		−	1.32627i
$189$	−1.59179e6	+	5.61204e6i	−0.235776	+	0.831257i
$190$	−111519.			−0.0162588
$191$			1.53400e6i			0.220154i	0.993923	+	0.110077i	$0.0351097\pi$
$191$			1.53400e6i			0.220154i	−0.993923	+	0.110077i	$0.964890\pi$
$192$	5.66303e6	+	4.00645e6i	0.800102	+	0.566052i
$193$	7.83643e6			1.09005			0.545025	−	0.838420i	$-0.316520\pi$
$193$	7.83643e6			1.09005			0.545025	+	0.838420i	$0.316520\pi$
$194$			106965.i			0.0146500i
$195$	3.16420e6	−	4.47253e6i	0.426737	−	0.603184i
$196$	−1.90179e6			−0.252578
$197$			2.93457e6i			0.383836i	0.981411	+	0.191918i	$0.0614707\pi$
$197$			2.93457e6i			0.383836i	−0.981411	+	0.191918i	$0.938529\pi$
$198$	139744.	+	395875.i	0.0180028	+	0.0509991i
$199$	6.55724e6			0.832074			0.416037	−	0.909348i	$-0.363419\pi$
$199$	6.55724e6			0.832074			0.416037	+	0.909348i	$0.363419\pi$
$200$			632293.i			0.0790367i
$201$	−9.85299e6	−	6.97074e6i	−1.21333	−	0.858402i
$202$	131772.			0.0159871
$203$		−	1.29848e7i		−	1.55219i
$204$	−1.18525e6	+	1.67532e6i	−0.139610	+	0.197336i
$205$	949239.			0.110183
$206$		−	239362.i		−	0.0273813i
$207$	−1.43699e7	+	5.07259e6i	−1.62010	+	0.571898i
$208$	−1.17516e7			−1.30589
$209$		−	4.30687e6i		−	0.471762i
$210$	−211079.	−	149333.i	−0.0227922	−	0.0161249i
$211$	−1.58477e7			−1.68701			−0.843506	−	0.537120i	$-0.819512\pi$
$211$	−1.58477e7			−1.68701			−0.843506	+	0.537120i	$0.819512\pi$
$212$		−	1.65672e7i		−	1.73877i
$213$	1.31675e6	−	1.86119e6i	0.136259	−	0.192598i
$214$	372027.			0.0379606
$215$			1.33849e6i			0.134679i
$216$	−1.11667e6	−	316729.i	−0.110806	−	0.0314287i
$217$	−4.64224e6			−0.454306
$218$		−	968486.i		−	0.0934811i
$219$	726937.	+	514289.i	0.0692093	+	0.0489638i
$220$	5.57363e6			0.523444
$221$		−	3.45312e6i		−	0.319915i
$222$	374583.	−	529465.i	0.0342365	−	0.0483925i
$223$	−1.78123e7			−1.60622			−0.803109	−	0.595833i	$-0.796822\pi$
$223$	−1.78123e7			−1.60622			−0.803109	+	0.595833i	$0.796822\pi$
$224$			1.67314e6i			0.148863i
$225$	2.60188e6	+	7.37073e6i	0.228423	+	0.647088i
$226$	745783.			0.0646081
$227$		−	7.29499e6i		−	0.623659i	−0.950138	−	0.311829i	$-0.899058\pi$
$227$		−	7.29499e6i		−	0.623659i	0.950138	−	0.311829i	$-0.100942\pi$
$228$	4.85237e6	+	3.43293e6i	0.409401	+	0.289641i
$229$	3.49337e6			0.290896			0.145448	−	0.989366i	$-0.453538\pi$
$229$	3.49337e6			0.290896			0.145448	+	0.989366i	$0.453538\pi$
$230$		−	675454.i		−	0.0555153i
$231$	5.76726e6	−	8.15190e6i	0.467879	−	0.661337i
$232$	2.58367e6			0.206906
$233$		−	3.42027e6i		−	0.270391i	−0.990819	−	0.135195i	$-0.956834\pi$
$233$		−	3.42027e6i		−	0.270391i	0.990819	−	0.135195i	$-0.0431662\pi$
$234$	919313.	−	324519.i	0.0717490	−	0.0253275i
$235$	9.67376e6			0.745404
$236$		−	1.66926e6i		−	0.126996i
$237$	1.84519e7	+	1.30543e7i	1.38610	+	0.980634i
$238$	−162968.			−0.0120885
$239$			443887.i			0.0325147i	0.999868	+	0.0162573i	$0.00517510\pi$
$239$			443887.i			0.0325147i	−0.999868	+	0.0162573i	$0.994825\pi$
$240$	−4.42775e6	+	6.25853e6i	−0.320295	+	0.452729i
$241$	4.64356e6			0.331742			0.165871	−	0.986147i	$-0.446957\pi$
$241$	4.64356e6			0.331742			0.165871	+	0.986147i	$0.446957\pi$
$242$			98883.7i			0.00697716i
$243$	−1.43205e7	+	902917.i	−0.998018	+	0.0629259i
$244$	7.38264e6			0.508209
$245$		−	2.08762e6i		−	0.141956i
$246$	137894.	+	97556.3i	0.00926273	+	0.00655315i
$247$	−1.00016e7			−0.663707
$248$		−	923698.i		−	0.0605585i
$249$	−7.57552e6	+	1.07078e7i	−0.490698	+	0.693591i
$250$	−851342.			−0.0544859
$251$			2.17244e7i			1.37381i	0.726749	+	0.686903i	$0.241030\pi$
$251$			2.17244e7i			1.37381i	−0.726749	+	0.686903i	$0.758970\pi$
$252$	4.58741e6	+	1.29954e7i	0.286659	+	0.812062i
$253$	2.60862e7			1.61083
$254$		−	24091.2i		−	0.00147013i
$255$	−1.83902e6	−	1.30106e6i	−0.110909	−	0.0784653i
$256$	1.62217e7			0.966891
$257$			3.17561e7i			1.87080i	0.353586	+	0.935402i	$0.384962\pi$
$257$			3.17561e7i			1.87080i	−0.353586	+	0.935402i	$0.615038\pi$
$258$	−137560.	+	194439.i	−0.00801003	+	0.0113220i
$259$	−1.54269e7			−0.887931
$260$		−	1.29432e7i		−	0.736416i
$261$	3.01182e7	−	1.06318e7i	1.69397	−	0.597975i
$262$	38216.1			0.00212492
$263$		−	2.64452e7i		−	1.45372i	−0.686788	−	0.726858i	$-0.740980\pi$
$263$		−	2.64452e7i		−	1.45372i	0.686788	−	0.726858i	$-0.259020\pi$
$264$	1.62204e6	+	1.14755e6i	0.0881555	+	0.0623678i
$265$	1.81861e7			0.977240
$266$			472018.i			0.0250792i
$267$	4.34427e6	−	6.14053e6i	0.228236	−	0.322606i
$268$	−2.85140e7			−1.48134
$269$			6.52151e6i			0.335036i	0.985869	+	0.167518i	$0.0535752\pi$
$269$			6.52151e6i			0.335036i	−0.985869	+	0.167518i	$0.946425\pi$
$270$	173549.	−	611869.i	0.00881721	−	0.0310862i
$271$	−3.29788e7			−1.65702			−0.828508	−	0.559977i	$-0.810810\pi$
$271$	−3.29788e7			−1.65702			−0.828508	+	0.559977i	$0.810810\pi$
$272$			4.83203e6i			0.240117i
$273$	−1.89306e7	−	1.33929e7i	−0.930413	−	0.658244i
$274$	−1.77244e6			−0.0861627
$275$		−	1.33804e7i		−	0.643383i
$276$	−2.07928e7	+	2.93901e7i	−0.988975	+	1.39789i
$277$	3.32681e7			1.56527			0.782634	−	0.622482i	$-0.213876\pi$
$277$	3.32681e7			1.56527			0.782634	+	0.622482i	$0.213876\pi$
$278$			1.30826e6i			0.0608919i
$279$	−3.80101e6	−	1.07677e7i	−0.175019	−	0.495803i
$280$	−1.22374e6			−0.0557461
$281$			2.93917e7i			1.32466i	0.749210	+	0.662332i	$0.230433\pi$
$281$			2.93917e7i			1.32466i	−0.749210	+	0.662332i	$0.769567\pi$
$282$	1.40528e6	+	994203.i	0.0626638	+	0.0443331i
$283$	2.02195e7			0.892096			0.446048	−	0.895009i	$-0.352831\pi$
$283$	2.02195e7			0.892096			0.446048	+	0.895009i	$0.352831\pi$
$284$		−	5.38618e6i		−	0.235140i
$285$	−3.76837e6	+	5.32651e6i	−0.162787	+	0.230096i
$286$	−1.66886e6			−0.0713382
$287$		−	4.01778e6i		−	0.169957i
$288$	−3.88084e6	+	1.36994e6i	−0.162461	+	0.0573489i
$289$	−1.41986e6			−0.0588235
$290$			1.41570e6i			0.0580467i
$291$	5.10901e6	+	3.61450e6i	0.207328	+	0.146679i
$292$	2.10371e6			0.0844963
$293$			4.57915e7i			1.82046i	0.414101	+	0.910231i	$0.364096\pi$
$293$			4.57915e7i			1.82046i	−0.414101	+	0.910231i	$0.635904\pi$
$294$	214552.	−	303264.i	0.00844287	−	0.0119338i
$295$	1.83238e6			0.0713754
$296$		−	3.06960e6i		−	0.118360i
$297$	2.36305e7	+	6.70250e6i	0.901993	+	0.255839i
$298$	38540.1			0.00145634
$299$		−	6.05780e7i		−	2.26622i
$300$	1.50751e7	+	1.06652e7i	0.558336	+	0.395009i
$301$	5.66531e6			0.207742
$302$			1.15825e6i			0.0420513i
$303$	4.45276e6	−	6.29387e6i	0.160067	−	0.226251i
$304$	1.39954e7			0.498156
$305$			8.10403e6i			0.285629i
$306$	−133436.	−	378004.i	−0.00465703	−	0.0131927i
$307$	−1.94476e7			−0.672126			−0.336063	−	0.941840i	$-0.609095\pi$
$307$	−1.94476e7			−0.672126			−0.336063	+	0.941840i	$0.609095\pi$
$308$		−	2.35911e7i		−	0.807414i
$309$	−1.14327e7	−	8.08838e6i	−0.387503	−	0.274149i
$310$	506133.			0.0169895
$311$			2.67104e7i			0.887971i	0.896034	+	0.443986i	$0.146436\pi$
$311$			2.67104e7i			0.887971i	−0.896034	+	0.443986i	$0.853564\pi$
$312$	2.66488e6	−	3.76675e6i	0.0877432	−	0.124023i
$313$	−3.11441e7			−1.01565			−0.507824	−	0.861461i	$-0.669550\pi$
$313$	−3.11441e7			−1.01565			−0.507824	+	0.861461i	$0.669550\pi$
$314$		−	2.44682e6i		−	0.0790337i
$315$	−1.42653e7	+	5.03567e6i	−0.456403	+	0.161111i
$316$	5.33987e7			1.69227
$317$		−	3.85453e7i		−	1.21002i	−0.796217	−	0.605012i	$-0.793169\pi$
$317$		−	3.85453e7i		−	1.21002i	0.796217	−	0.605012i	$-0.206831\pi$
$318$	2.64185e6	+	1.86904e6i	0.0821535	+	0.0581216i
$319$	−5.46746e7			−1.68428
$320$			1.79899e7i			0.549007i
$321$	1.25713e7	−	1.77693e7i	0.380071	−	0.537223i
$322$	−2.85895e6			−0.0856325
$323$			4.11245e6i			0.122037i
$324$	−2.63868e7	+	2.12810e7i	−0.775804	+	0.625687i
$325$	−3.10723e7			−0.905155
$326$		−	644508.i		−	0.0186027i
$327$	−4.62582e7	−	3.27265e7i	−1.32296	−	0.935958i
$328$	799444.			0.0226551
$329$		−	4.09455e7i		−	1.14979i
$330$	−628792.	+	888784.i	−0.0174971	+	0.0247317i
$331$	4.61079e7			1.27143			0.635713	−	0.771925i	$-0.280706\pi$
$331$	4.61079e7			1.27143			0.635713	+	0.771925i	$0.280706\pi$
$332$			3.09878e7i			0.846792i
$333$	−1.26313e7	−	3.57827e7i	−0.342072	−	0.969037i
$334$	−2.55557e6			−0.0685879
$335$		−	3.13002e7i		−	0.832555i
$336$	2.64900e7	+	1.87410e7i	0.698336	+	0.494055i
$337$	3.00242e7			0.784478			0.392239	−	0.919863i	$-0.371701\pi$
$337$	3.00242e7			0.784478			0.392239	+	0.919863i	$0.371701\pi$
$338$			1.64803e6i			0.0426791i
$339$	2.52010e7	−	3.56211e7i	0.646873	−	0.914341i
$340$	−5.32202e6			−0.135407
$341$			1.95470e7i			0.492965i
$342$	−1.09485e6	+	386482.i	−0.0273700	+	0.00966164i
$343$	−4.37036e7			−1.08302
$344$			1.12727e6i			0.0276918i
$345$	−3.22620e7	−	2.28245e7i	−0.785658	−	0.555833i
$346$	−3.10768e6			−0.0750252
$347$			6.67109e7i			1.59664i	0.602231	+	0.798322i	$0.294279\pi$
$347$			6.67109e7i			1.59664i	−0.602231	+	0.798322i	$0.705721\pi$
$348$	4.35801e7	−	6.15995e7i	1.03407	−	1.46164i
$349$	−6.05549e7			−1.42453			−0.712267	−	0.701908i	$-0.752332\pi$
$349$	−6.05549e7			−1.42453			−0.712267	+	0.701908i	$0.752332\pi$
$350$			1.46644e6i			0.0342026i
$351$	1.55647e7	−	5.48754e7i	0.359932	−	1.26899i
$352$	7.04504e6			0.161531
$353$			2.65777e7i			0.604217i	0.953274	+	0.302109i	$0.0976905\pi$
$353$			2.65777e7i			0.604217i	−0.953274	+	0.302109i	$0.902309\pi$
$354$	266185.	+	188319.i	0.00600031	+	0.00424506i
$355$	5.91249e6			0.132156
$356$		−	1.77703e7i		−	0.393864i
$357$	−5.50691e6	+	7.78390e6i	−0.121033	+	0.171077i
$358$	−371650.			−0.00810000
$359$			3.35235e7i			0.724545i	0.932072	+	0.362273i	$0.117999\pi$
$359$			3.35235e7i			0.724545i	−0.932072	+	0.362273i	$0.882001\pi$
$360$	−1.00198e6	−	2.83846e6i	−0.0214759	−	0.0608381i
$361$	−3.51346e7			−0.746817
$362$		−	1.05502e6i		−	0.0222399i
$363$	4.72302e6	+	3.34142e6i	0.0987415	+	0.0698572i
$364$	−5.47840e7			−1.13592
$365$			2.30928e6i			0.0474894i
$366$	−832877.	+	1.17725e6i	−0.0169878	+	0.0240119i
$367$	5.03687e7			1.01897			0.509486	−	0.860479i	$-0.329835\pi$
$367$	5.03687e7			1.01897			0.509486	+	0.860479i	$0.329835\pi$
$368$			8.47684e7i			1.70095i
$369$	9.31924e6	−	3.28970e6i	0.185482	−	0.0654753i
$370$	1.68196e6			0.0332056
$371$		−	7.69749e7i		−	1.50740i
$372$	−2.20227e7	−	1.55805e7i	−0.427801	−	0.302659i
$373$	3.98528e7			0.767950			0.383975	−	0.923344i	$-0.374555\pi$
$373$	3.98528e7			0.767950			0.383975	+	0.923344i	$0.374555\pi$
$374$			686205.i			0.0131171i
$375$	−2.87680e7	+	4.06630e7i	−0.545527	+	0.771091i
$376$	8.14720e6			0.153266
$377$			1.26967e8i			2.36955i
$378$	−2.58981e6	−	734569.i	−0.0479505	−	0.0136006i
$379$	4.41790e7			0.811518			0.405759	−	0.913980i	$-0.367007\pi$
$379$	4.41790e7			0.811518			0.405759	+	0.913980i	$0.367007\pi$
$380$			1.54146e7i			0.280920i
$381$	−1.15068e6	−	814075.i	−0.0208055	−	0.0147194i
$382$	−707903.			−0.0126994
$383$		−	8.14578e7i		−	1.44989i	−0.688805	−	0.724947i	$-0.741864\pi$
$383$		−	8.14578e7i		−	1.44989i	0.688805	−	0.724947i	$-0.258136\pi$
$384$	−7.48309e6	+	1.05772e7i	−0.132156	+	0.186800i
$385$	2.58963e7			0.453791
$386$			3.61631e6i			0.0628788i
$387$	4.63869e6	+	1.31407e7i	0.0800317	+	0.226718i
$388$	1.47852e7			0.253123
$389$		−	4.02407e7i		−	0.683624i	−0.939768	−	0.341812i	$-0.888959\pi$
$389$		−	4.02407e7i		−	0.683624i	0.939768	−	0.341812i	$-0.111041\pi$
$390$	2.06396e6	+	1.46020e6i	0.0347943	+	0.0246161i
$391$	−2.49086e7			−0.416695
$392$		−	1.75819e6i		−	0.0291882i
$393$	1.29138e6	−	1.82533e6i	0.0212753	−	0.0300721i
$394$	−1.35423e6			−0.0221413
$395$			5.86165e7i			0.951106i
$396$	5.47196e7	−	1.93161e7i	0.881165	−	0.311053i
$397$	1.08106e8			1.72774			0.863870	−	0.503716i	$-0.168034\pi$
$397$	1.08106e8			1.72774			0.863870	+	0.503716i	$0.168034\pi$
$398$			3.02600e6i			0.0479976i
$399$	2.25452e7	+	1.59501e7i	0.354923	+	0.251099i
$400$	4.34802e7			0.679379
$401$			8.11812e7i			1.25899i	0.777004	+	0.629495i	$0.216738\pi$
$401$			8.11812e7i			1.25899i	−0.777004	+	0.629495i	$0.783262\pi$
$402$	3.21682e6	−	4.54690e6i	0.0495163	−	0.0699903i
$403$	4.53925e7			0.693536
$404$		−	1.82141e7i		−	0.276225i
$405$	−2.33605e7	−	2.89652e7i	−0.351655	−	0.436025i
$406$	5.99213e6			0.0895372
$407$			6.49577e7i			0.963490i
$408$	−1.54882e6	−	1.09575e6i	−0.0228044	−	0.0161336i
$409$	−3.63151e7			−0.530783			−0.265392	−	0.964141i	$-0.585501\pi$
$409$	−3.63151e7			−0.530783			−0.265392	+	0.964141i	$0.585501\pi$
$410$			438050.i			0.00635582i
$411$	−5.98931e7	+	8.46576e7i	−0.862684	+	1.21938i
$412$	−3.30857e7			−0.473095
$413$		−	7.75577e6i		−	0.110097i
$414$	−2.34087e6	−	6.63133e6i	−0.0329895	−	0.0934544i
$415$	−3.40158e7			−0.475922
$416$		−	1.63602e7i		−	0.227252i
$417$	6.24869e7	+	4.42079e7i	0.861750	+	0.609666i
$418$	1.98751e6			0.0272133
$419$		−	1.31682e8i		−	1.79013i	−0.445933	−	0.895067i	$-0.647128\pi$
$419$		−	1.31682e8i		−	1.79013i	0.445933	−	0.895067i	$-0.352872\pi$
$420$	−2.06414e7	+	2.91762e7i	−0.278607	+	0.393805i
$421$	−5.43363e7			−0.728189			−0.364094	−	0.931362i	$-0.618621\pi$
$421$	−5.43363e7			−0.728189			−0.364094	+	0.931362i	$0.618621\pi$
$422$		−	7.31330e6i		−	0.0973141i
$423$	9.49730e7	−	3.35256e7i	1.25481	−	0.442951i
$424$	1.53162e7			0.200934
$425$			1.27763e7i			0.166433i
$426$	858893.	+	607645.i	0.0111099	+	0.00785998i
$427$	3.43014e7			0.440583
$428$		−	5.14232e7i		−	0.655885i
$429$	−5.63932e7	+	7.97105e7i	−0.714257	+	1.00959i
$430$	−617677.			−0.00776884
$431$		−	1.20951e8i		−	1.51070i	−0.655320	−	0.755351i	$-0.727466\pi$
$431$		−	1.20951e8i		−	1.51070i	0.655320	−	0.755351i	$-0.272534\pi$
$432$	−2.17801e7	+	7.67886e7i	−0.270153	+	0.952457i
$433$	−1.57476e7			−0.193977			−0.0969885	−	0.995286i	$-0.530921\pi$
$433$	−1.57476e7			−0.193977			−0.0969885	+	0.995286i	$0.530921\pi$
$434$		−	2.14228e6i		−	0.0262063i
$435$	6.76186e7	+	4.78385e7i	0.821483	+	0.581179i
$436$	−1.33868e8			−1.61517
$437$			7.21448e7i			0.864491i
$438$	−237331.	+	335463.i	−0.00282444	+	0.00399229i
$439$	−9.19618e6			−0.108696			−0.0543480	−	0.998522i	$-0.517308\pi$
$439$	−9.19618e6			−0.108696			−0.0543480	+	0.998522i	$0.517308\pi$
$440$			5.15276e6i			0.0604898i
$441$	−7.23492e6	−	2.04954e7i	−0.0843564	−	0.238969i
$442$	1.59353e6			0.0184541
$443$			1.07781e8i			1.23974i	0.784705	+	0.619870i	$0.212815\pi$
$443$			1.07781e8i			1.23974i	−0.784705	+	0.619870i	$0.787185\pi$
$444$	−7.31850e7	−	5.17765e7i	−0.836128	−	0.591540i
$445$	1.95068e7			0.221363
$446$		−	8.21990e6i		−	0.0926535i
$447$	1.30232e6	−	1.84081e6i	0.0145813	−	0.0206103i
$448$	7.61444e7			0.846845
$449$			3.55140e7i			0.392338i	0.980570	+	0.196169i	$0.0628501\pi$
$449$			3.55140e7i			0.392338i	−0.980570	+	0.196169i	$0.937150\pi$
$450$	−3.40141e6	+	1.20070e6i	−0.0373268	+	0.0131764i
$451$	−1.69176e7			−0.184420
$452$		−	1.03085e8i		−	1.11630i
$453$	5.53217e7	+	3.91387e7i	0.595115	+	0.421029i
$454$	3.36645e6			0.0359753
$455$		−	6.01372e7i		−	0.638423i
$456$	−3.17371e6	+	4.48596e6i	−0.0334713	+	0.0473109i
$457$	4.19450e7			0.439472			0.219736	−	0.975559i	$-0.429480\pi$
$457$	4.19450e7			0.439472			0.219736	+	0.975559i	$0.429480\pi$
$458$			1.61210e6i			0.0167801i
$459$	−2.25637e7	−	6.39993e6i	−0.233331	−	0.0661816i
$460$	−9.33643e7			−0.959195
$461$		−	7.59633e7i		−	0.775356i	−0.921795	−	0.387678i	$-0.873277\pi$
$461$		−	7.59633e7i		−	0.775356i	0.921795	−	0.387678i	$-0.126723\pi$
$462$	3.76189e6	+	2.66144e6i	0.0381488	+	0.0269893i
$463$	−9.48911e7			−0.956054			−0.478027	−	0.878345i	$-0.658648\pi$
$463$	−9.48911e7			−0.956054			−0.478027	+	0.878345i	$0.658648\pi$
$464$		−	1.77668e8i		−	1.77851i
$465$	1.71030e7	−	2.41746e7i	0.170103	−	0.240437i
$466$	1.57837e6			0.0155973
$467$			1.00276e8i			0.984569i	0.870434	+	0.492284i	$0.163838\pi$
$467$			1.00276e8i			0.984569i	−0.870434	+	0.492284i	$0.836162\pi$
$468$	−4.48564e7	−	1.27071e8i	−0.437610	−	1.23968i
$469$	−1.32482e8			−1.28422
$470$			4.46420e6i			0.0429981i
$471$	−1.16868e8	−	8.26813e7i	−1.11849	−	0.791306i
$472$	1.54322e6			0.0146758
$473$		−	2.38548e7i		−	0.225420i
$474$	−6.02420e6	+	8.51508e6i	−0.0565672	+	0.0799565i
$475$	3.70052e7			0.345288
$476$			2.25261e7i			0.208865i
$477$	1.78543e8	−	6.30261e7i	1.64509	−	0.580718i
$478$	−204843.			−0.00187559
$479$		−	9.22244e7i		−	0.839150i	−0.907721	−	0.419575i	$-0.862179\pi$
$479$		−	9.22244e7i		−	0.839150i	0.907721	−	0.419575i	$-0.137821\pi$
$480$	−8.71293e6	−	6.16418e6i	−0.0787844	−	0.0557380i
$481$	1.50847e8			1.35550
$482$			2.14288e6i			0.0191363i
$483$	−9.66078e7	+	1.36553e8i	−0.857375	+	1.21188i
$484$	1.36681e7			0.120552
$485$			1.62299e7i			0.142263i
$486$	−416673.	−	6.60853e6i	−0.00362983	−	0.0575700i
$487$	8.11697e7			0.702760			0.351380	−	0.936233i	$-0.385713\pi$
$487$	8.11697e7			0.702760			0.351380	+	0.936233i	$0.385713\pi$
$488$			6.82517e6i			0.0587292i
$489$	−3.07839e7	−	2.17788e7i	−0.263267	−	0.186255i
$490$	963385.			0.00818864
$491$		−	1.39129e8i		−	1.17536i	−0.809093	−	0.587681i	$-0.800041\pi$
$491$		−	1.39129e8i		−	1.17536i	0.809093	−	0.587681i	$-0.199959\pi$
$492$	1.34847e7	−	1.90603e7i	0.113226	−	0.160042i
$493$	5.22064e7			0.435696
$494$		−	4.61546e6i		−	0.0382855i
$495$	2.12036e7	+	6.00665e7i	0.174821	+	0.495241i
$496$	−6.35189e7			−0.520545
$497$		−	2.50254e7i		−	0.203850i
$498$	−4.94139e6	−	3.49591e6i	−0.0400093	−	0.0283056i
$499$	2.83836e7			0.228437			0.114218	−	0.993456i	$-0.463564\pi$
$499$	2.83836e7			0.228437			0.114218	+	0.993456i	$0.463564\pi$
$500$			1.17676e8i			0.941410i
$501$	−8.63561e7	+	1.22062e8i	−0.686720	+	0.970664i
$502$	−1.00252e7			−0.0792471
$503$			2.41667e7i			0.189895i	0.995482	+	0.0949475i	$0.0302683\pi$
$503$			2.41667e7i			0.189895i	−0.995482	+	0.0949475i	$0.969732\pi$
$504$	−1.20142e7	+	4.24101e6i	−0.0938429	+	0.0331267i
$505$	1.99939e7			0.155247
$506$			1.20381e7i			0.0929194i
$507$	7.87156e7	+	5.56893e7i	0.604000	+	0.427315i
$508$	−3.32999e6			−0.0254011
$509$		−	2.43493e8i		−	1.84643i	−0.384279	−	0.923217i	$-0.625550\pi$
$509$		−	2.43493e8i		−	1.84643i	0.384279	−	0.923217i	$-0.374450\pi$
$510$	600407.	−	848662.i	0.00452621	−	0.00639770i
$511$	9.77431e6			0.0732526
$512$			3.81978e7i			0.284596i
$513$	−1.85367e7	+	6.53533e7i	−0.137303	+	0.484078i
$514$	−1.46546e7			−0.107916
$515$		−	3.63187e7i		−	0.265894i
$516$	2.68762e7	+	1.90142e7i	0.195622	+	0.138398i
$517$	−1.72408e8			−1.24763
$518$		−	7.11912e6i		−	0.0512197i
$519$	−1.05013e8	+	1.48433e8i	−0.751173	+	1.06177i
$520$	1.19659e7			0.0851011
$521$			6.49017e7i			0.458926i	0.973317	+	0.229463i	$0.0736970\pi$
$521$			6.49017e7i			0.458926i	−0.973317	+	0.229463i	$0.926303\pi$
$522$	4.90628e6	+	1.38988e7i	0.0344938	+	0.0977157i
$523$	8.71718e7			0.609356			0.304678	−	0.952455i	$-0.401451\pi$
$523$	8.71718e7			0.609356			0.304678	+	0.952455i	$0.401451\pi$
$524$		−	5.28240e6i		−	0.0367145i
$525$	7.00421e7	+	4.95530e7i	0.484040	+	0.342446i
$526$	1.22038e7			0.0838566
$527$		−	1.86646e7i		−	0.127522i
$528$	7.89124e7	−	1.11541e8i	0.536097	−	0.757761i
$529$	−2.88935e8			−1.95179
$530$			8.39241e6i			0.0563714i
$531$	1.79895e7	−	6.35032e6i	0.120153	−	0.0424143i
$532$	6.52444e7			0.433319
$533$			3.92864e7i			0.259454i
$534$	2.83370e6	+	2.00477e6i	0.0186093	+	0.0131656i
$535$	5.64480e7			0.368627
$536$		−	2.63609e7i		−	0.171185i
$537$	−1.25586e7	+	1.77513e7i	−0.0810993	+	0.114632i
$538$	−3.00951e6			−0.0193263
$539$			3.72061e7i			0.237601i
$540$	−8.45752e7	−	2.39887e7i	−0.537108	−	0.152344i
$541$	3.58122e7			0.226172			0.113086	−	0.993585i	$-0.463926\pi$
$541$	3.58122e7			0.226172			0.113086	+	0.993585i	$0.463926\pi$
$542$		−	1.52189e7i		−	0.0955839i
$543$	−5.03911e7	−	3.56504e7i	−0.314742	−	0.222672i
$544$	−6.72700e6			−0.0417854
$545$		−	1.46949e8i		−	0.907775i
$546$	6.18048e6	−	8.73598e6i	0.0379703	−	0.0536702i
$547$	−7.55148e7			−0.461392			−0.230696	−	0.973026i	$-0.574100\pi$
$547$	−7.55148e7			−0.461392			−0.230696	+	0.973026i	$0.574100\pi$
$548$			2.44994e8i			1.48872i
$549$	2.80855e7	+	7.95620e7i	0.169733	+	0.480827i
$550$	6.17470e6			0.0371131
$551$		−	1.51210e8i		−	0.903911i
$552$	−2.71709e7	−	1.92227e7i	−0.161542	−	0.114287i
$553$	2.48102e8			1.46708
$554$			1.53524e7i			0.0902914i
$555$	5.68358e7	−	8.03362e7i	0.332463	−	0.469929i
$556$	1.80834e8			1.05209
$557$			1.20303e8i			0.696165i	0.937464	+	0.348083i	$0.113167\pi$
$557$			1.20303e8i			0.696165i	−0.937464	+	0.348083i	$0.886833\pi$
$558$	4.96901e6	−	1.75407e6i	0.0286001	−	0.0100959i
$559$	−5.53963e7			−0.317136
$560$			8.41515e7i			0.479179i
$561$	3.27755e7	+	2.31878e7i	0.185635	+	0.131332i
$562$	−1.35635e7			−0.0764124
$563$			1.86833e7i			0.104696i	0.998629	+	0.0523478i	$0.0166704\pi$
$563$			1.86833e7i			0.104696i	−0.998629	+	0.0523478i	$0.983330\pi$
$564$	1.37423e8	−	1.94245e8i	0.765989	−	1.08271i
$565$	1.13158e8			0.627395
$566$			9.33080e6i			0.0514600i
$567$	−1.22599e8	+	9.88762e7i	−0.672570	+	0.542429i
$568$	4.97947e6			0.0271730
$569$		−	6.50221e7i		−	0.352959i	−0.984304	−	0.176480i	$-0.943529\pi$
$569$		−	6.50221e7i		−	0.352959i	0.984304	−	0.176480i	$-0.0564710\pi$
$570$	−2.45805e6	−	1.73901e6i	−0.0132729	−	0.00939025i
$571$	1.51488e7			0.0813708			0.0406854	−	0.999172i	$-0.487046\pi$
$571$	1.51488e7			0.0813708			0.0406854	+	0.999172i	$0.487046\pi$
$572$			2.30678e8i			1.23259i
$573$	−2.39210e7	+	3.38118e7i	−0.127150	+	0.179723i
$574$	1.85410e6			0.00980388
$575$			2.24135e8i			1.17898i
$576$	6.23461e7	+	1.76617e8i	0.326243	+	0.924198i
$577$	1.42134e8			0.739898			0.369949	−	0.929052i	$-0.379375\pi$
$577$	1.42134e8			0.739898			0.369949	+	0.929052i	$0.379375\pi$
$578$		−	655228.i		−	0.00339319i
$579$	1.72727e8	+	1.22200e8i	0.889868	+	0.629559i
$580$	1.95684e8			1.00293
$581$			1.43976e8i			0.734112i
$582$	−1.66800e6	+	2.35768e6i	−0.00846109	+	0.0119596i
$583$	−3.24116e8			−1.63567
$584$			1.94486e6i			0.00976450i
$585$	1.39488e8	−	4.92395e7i	0.696738	−	0.245950i
$586$	−2.11316e7			−0.105012
$587$		−	7.49243e7i		−	0.370432i	−0.982698	−	0.185216i	$-0.940702\pi$
$587$		−	7.49243e7i		−	0.370432i	0.982698	−	0.185216i	$-0.0592985\pi$
$588$	−4.19185e7	−	2.96563e7i	−0.206193	−	0.145876i
$589$	−5.40597e7			−0.264562
$590$			845595.i			0.00411724i
$591$	−4.57612e7	+	6.46825e7i	−0.221684	+	0.313346i
$592$	−2.11084e8			−1.01739
$593$		−	6.29987e7i		−	0.302112i	−0.988525	−	0.151056i	$-0.951733\pi$
$593$		−	6.29987e7i		−	0.302112i	0.988525	−	0.151056i	$-0.0482673\pi$
$594$	−3.09303e6	+	1.09049e7i	−0.0147579	+	0.0520309i
$595$	−2.47273e7			−0.117388
$596$		−	5.32718e6i		−	0.0251628i
$597$	1.44532e8	+	1.02253e8i	0.679267	+	0.480565i
$598$	2.79552e7			0.130725
$599$		−	1.96156e8i		−	0.912685i	−0.889804	−	0.456343i	$-0.849159\pi$
$599$		−	1.96156e8i		−	0.912685i	0.889804	−	0.456343i	$-0.150841\pi$
$600$	−9.85990e6	+	1.39367e7i	−0.0456477	+	0.0645220i
$601$	1.40293e8			0.646270			0.323135	−	0.946353i	$-0.395263\pi$
$601$	1.40293e8			0.646270			0.323135	+	0.946353i	$0.395263\pi$
$602$			2.61440e6i			0.0119835i
$603$	−1.08475e8	−	3.07292e8i	−0.494739	−	1.40152i
$604$	1.60098e8			0.726565
$605$			1.50037e7i			0.0677537i
$606$	2.90446e6	+	2.05483e6i	0.0130511	+	0.00923334i
$607$	4.87910e7			0.218159			0.109080	−	0.994033i	$-0.465210\pi$
$607$	4.87910e7			0.218159			0.109080	+	0.994033i	$0.465210\pi$
$608$			1.94840e7i			0.0866896i
$609$	2.02483e8	−	2.86205e8i	0.896470	−	1.26714i
$610$	−3.73980e6			−0.0164763
$611$			4.00371e8i			1.75525i
$612$	−5.22494e7	+	1.84441e7i	−0.227943	+	0.0804644i
$613$	−4.16948e8			−1.81009			−0.905045	−	0.425316i	$-0.860163\pi$
$613$	−4.16948e8			−1.81009			−0.905045	+	0.425316i	$0.860163\pi$
$614$		−	8.97456e6i		−	0.0387711i
$615$	2.09227e7	+	1.48023e7i	0.0899483	+	0.0636362i
$616$	2.18097e7			0.0933057
$617$			5.95345e7i			0.253462i	0.991937	+	0.126731i	$0.0404485\pi$
$617$			5.95345e7i			0.253462i	−0.991937	+	0.126731i	$0.959551\pi$
$618$	3.73258e6	−	5.27592e6i	0.0158141	−	0.0223529i
$619$	3.28361e8			1.38446			0.692229	−	0.721678i	$-0.256629\pi$
$619$	3.28361e8			1.38446			0.692229	+	0.721678i	$0.256629\pi$
$620$		−	6.99600e7i		−	0.293545i
$621$	−3.95836e8	−	1.12274e8i	−1.65288	−	0.468818i
$622$	−1.23262e7			−0.0512220
$623$		−	8.25649e7i		−	0.341453i
$624$	−2.59024e8	−	1.83253e8i	−1.06607	−	0.754218i
$625$	3.83595e7			0.157121
$626$		−	1.43722e7i		−	0.0585870i
$627$	6.71608e7	−	9.49303e7i	0.272467	−	0.385126i
$628$	−3.38210e8			−1.36555
$629$		−	6.20253e7i		−	0.249240i
$630$	−2.32383e6	−	6.58307e6i	−0.00929358	−	0.0263273i
$631$	−5.06091e7			−0.201438			−0.100719	−	0.994915i	$-0.532114\pi$
$631$	−5.06091e7			−0.201438			−0.100719	+	0.994915i	$0.532114\pi$
$632$			4.93665e7i			0.195561i
$633$	−3.49308e8	−	2.47127e8i	−1.37720	−	0.974335i
$634$	1.77877e7			0.0697993
$635$		−	3.65538e6i		−	0.0142762i
$636$	2.58347e8	−	3.65168e8i	1.00423	−	1.41945i
$637$	8.64011e7			0.334273
$638$		−	2.52309e7i		−	0.0971564i
$639$	5.80464e7	−	2.04905e7i	0.222471	−	0.0785325i
$640$	−3.36007e7			−0.128177
$641$		−	2.04729e8i		−	0.777329i	−0.921379	−	0.388665i	$-0.872937\pi$
$641$		−	2.04729e8i		−	0.777329i	0.921379	−	0.388665i	$-0.127063\pi$
$642$	8.20006e6	+	5.80134e6i	0.0309893	+	0.0219242i
$643$	−4.39417e8			−1.65289			−0.826445	−	0.563017i	$-0.809640\pi$
$643$	−4.39417e8			−1.65289			−0.826445	+	0.563017i	$0.809640\pi$
$644$			3.95176e8i			1.47956i
$645$	−2.08722e7	+	2.95023e7i	−0.0777837	+	0.109945i
$646$	−1.89779e6			−0.00703965
$647$			1.68755e8i			0.623081i	0.950233	+	0.311541i	$0.100845\pi$
$647$			1.68755e8i			0.623081i	−0.950233	+	0.311541i	$0.899155\pi$
$648$	−1.96741e7	−	2.43943e7i	−0.0723051	−	0.0896529i
$649$	−3.26570e7			−0.119465
$650$		−	1.43391e7i		−	0.0522132i
$651$	−1.02322e8	−	7.23905e7i	−0.370875	−	0.262385i
$652$	−8.90867e7			−0.321418
$653$		−	3.69047e8i		−	1.32539i	−0.748891	−	0.662693i	$-0.769413\pi$
$653$		−	3.69047e8i		−	1.32539i	0.748891	−	0.662693i	$-0.230587\pi$
$654$	1.51024e7	−	2.13470e7i	0.0539901	−	0.0763138i
$655$	5.79857e6			0.0206346
$656$		−	5.49745e7i		−	0.194738i
$657$	8.00308e6	+	2.26715e7i	0.0282202	+	0.0799437i
$658$	1.88953e7			0.0663248
$659$		−	1.87282e8i		−	0.654393i	−0.944956	−	0.327197i	$-0.893896\pi$
$659$		−	1.87282e8i		−	0.654393i	0.944956	−	0.327197i	$-0.106104\pi$
$660$	1.22852e8	+	8.69144e7i	0.427316	+	0.302315i
$661$	−5.53588e8			−1.91683			−0.958413	−	0.285386i	$-0.907878\pi$
$661$	−5.53588e8			−1.91683			−0.958413	+	0.285386i	$0.907878\pi$
$662$			2.12776e7i			0.0733413i
$663$	5.38474e7	−	7.61122e7i	0.184767	−	0.261164i
$664$	−2.86479e7			−0.0978563
$665$			7.16197e7i			0.243538i
$666$	1.65128e7	−	5.82905e6i	0.0558983	−	0.0197322i
$667$	9.15858e8			3.08639
$668$			3.53242e8i			1.18507i
$669$	−3.92610e8	−	2.77762e8i	−1.31124	−	0.927672i
$670$	1.44442e7			0.0480253
$671$		−	1.44432e8i		−	0.478074i
$672$	−2.60907e7	+	3.68786e7i	−0.0859760	+	0.121525i
$673$	2.35122e8			0.771344			0.385672	−	0.922636i	$-0.373970\pi$
$673$	2.35122e8			0.771344			0.385672	+	0.922636i	$0.373970\pi$
$674$			1.38554e7i			0.0452521i
$675$	−5.75887e7	+	2.03036e8i	−0.187252	+	0.660179i
$676$	2.27798e8			0.737412
$677$			1.46128e7i			0.0470942i	0.999723	+	0.0235471i	$0.00749598\pi$
$677$			1.46128e7i			0.0470942i	−0.999723	+	0.0235471i	$0.992504\pi$
$678$	1.64382e7	+	1.16296e7i	0.0527431	+	0.0373144i
$679$	6.86952e7			0.219440
$680$		−	4.92015e6i		−	0.0156478i
$681$	1.13757e8	−	1.60793e8i	0.360194	−	0.509127i
$682$	−9.02042e6			−0.0284364
$683$		−	4.11294e8i		−	1.29089i	−0.763805	−	0.645447i	$-0.776671\pi$
$683$		−	4.11294e8i		−	1.29089i	0.763805	−	0.645447i	$-0.223329\pi$
$684$	5.34212e7	+	1.51334e8i	0.166934	+	0.472900i
$685$	−2.68934e8			−0.836707
$686$		−	2.01681e7i		−	0.0624730i
$687$	7.69993e7	+	5.44751e7i	0.237474	+	0.168007i
$688$	7.75175e7			0.238031
$689$			7.52673e8i			2.30117i
$690$	1.05329e7	−	1.48881e7i	0.0320629	−	0.0453202i
$691$	−3.11379e8			−0.943745			−0.471873	−	0.881667i	$-0.656422\pi$
$691$	−3.11379e8			−0.943745			−0.471873	+	0.881667i	$0.656422\pi$
$692$			4.29557e8i			1.29629i
$693$	2.54239e8	−	8.97468e7i	0.763911	−	0.269662i
$694$	−3.07854e7			−0.0921013
$695$			1.98504e8i			0.591308i
$696$	5.69481e7	+	4.02893e7i	0.168908	+	0.119498i
$697$	1.61538e7			0.0477065
$698$		−	2.79445e7i		−	0.0821733i
$699$	5.33352e7	−	7.53881e7i	0.156164	−	0.220735i
$700$	2.02698e8			0.590955
$701$		−	2.56296e8i		−	0.744024i	−0.928228	−	0.372012i	$-0.878668\pi$
$701$		−	2.56296e8i		−	0.744024i	0.928228	−	0.372012i	$-0.121332\pi$
$702$	2.53236e7	+	7.18273e6i	0.0732005	+	0.0207624i
$703$	−1.79649e8			−0.517082
$704$		−	3.20619e8i		−	0.918908i
$705$	2.13225e8	+	1.50851e8i	0.608514	+	0.430509i
$706$	−1.22649e7			−0.0348539
$707$		−	8.46267e7i		−	0.239469i
$708$	2.60303e7	−	3.67932e7i	0.0733464	−	0.103674i
$709$	4.14469e8			1.16293			0.581465	−	0.813571i	$-0.302480\pi$
$709$	4.14469e8			1.16293			0.581465	+	0.813571i	$0.302480\pi$
$710$			2.72846e6i			0.00762330i
$711$	2.03143e8	+	5.75473e8i	0.565187	+	1.60109i
$712$	1.64285e7			0.0455153
$713$		−	3.27432e8i		−	0.903344i
$714$	−3.59207e6	−	2.54130e6i	−0.00986848	−	0.00698170i
$715$	−2.53218e8			−0.692750
$716$			5.13711e7i			0.139952i
$717$	−6.92192e6	+	9.78398e6i	−0.0187789	+	0.0265435i
$718$	−1.54702e7			−0.0417949
$719$			5.85161e8i			1.57430i	0.616759	+	0.787152i	$0.288445\pi$
$719$			5.85161e8i			1.57430i	−0.616759	+	0.787152i	$0.711555\pi$
$720$	−1.95189e8	+	6.89021e7i	−0.522948	+	0.184601i
$721$	−1.53723e8			−0.410142
$722$		−	1.62137e7i		−	0.0430796i
$723$	1.02351e8	+	7.24111e7i	0.270819	+	0.191598i
$724$	−1.45829e8			−0.384262
$725$		−	4.69771e8i		−	1.23274i
$726$	−1.54198e6	+	2.17955e6i	−0.00402966	+	0.00569584i
$727$	−2.14459e8			−0.558138			−0.279069	−	0.960271i	$-0.590026\pi$
$727$	−2.14459e8			−0.558138			−0.279069	+	0.960271i	$0.590026\pi$
$728$		−	5.06472e7i		−	0.131269i
$729$	−3.29726e8	−	2.03410e8i	−0.851080	−	0.525036i
$730$	−1.06567e6			−0.00273940
$731$			2.27779e7i			0.0583126i
$732$	1.62725e8	+	1.15124e8i	0.414879	+	0.293516i
$733$	6.89906e8			1.75177			0.875886	−	0.482518i	$-0.160278\pi$
$733$	6.89906e8			1.75177			0.875886	+	0.482518i	$0.160278\pi$
$734$			2.32439e7i			0.0587788i
$735$	3.25541e7	−	4.60145e7i	0.0819868	−	0.115887i
$736$	−1.18012e8			−0.296000
$737$			5.57839e8i			1.39350i
$738$	1.51811e6	+	4.30059e6i	0.00377690	+	0.0106994i
$739$	−1.55438e8			−0.385144			−0.192572	−	0.981283i	$-0.561683\pi$
$739$	−1.55438e8			−0.385144			−0.192572	+	0.981283i	$0.561683\pi$
$740$		−	2.32488e8i		−	0.573728i
$741$	−2.20450e8	−	1.55963e8i	−0.541821	−	0.383324i
$742$	3.55220e7			0.0869531
$743$		−	6.83431e8i		−	1.66621i	−0.553119	−	0.833103i	$-0.686562\pi$
$743$		−	6.83431e8i		−	1.66621i	0.553119	−	0.833103i	$-0.313438\pi$
$744$	1.44040e7	−	2.03598e7i	0.0349756	−	0.0494372i
$745$	5.84772e6			0.0141422
$746$			1.83911e7i			0.0442986i
$747$	−3.33953e8	+	1.17886e8i	−0.801167	+	0.282813i
$748$	9.48503e7			0.226639
$749$		−	2.38923e8i		−	0.568608i
$750$	−1.87649e7	−	1.32757e7i	−0.0444798	−	0.0314684i
$751$	2.63962e8			0.623192			0.311596	−	0.950215i	$-0.399136\pi$
$751$	2.63962e8			0.623192			0.311596	+	0.950215i	$0.399136\pi$
$752$		−	5.60249e8i		−	1.31743i
$753$	−3.38767e8	+	4.78839e8i	−0.793443	+	1.12151i
$754$	−5.85920e7			−0.136686
$755$			1.75742e8i			0.408351i
$756$	−1.01535e8	+	3.57976e8i	−0.234991	+	0.828491i
$757$	2.95588e7			0.0681396			0.0340698	−	0.999419i	$-0.489153\pi$
$757$	2.95588e7			0.0681396			0.0340698	+	0.999419i	$0.489153\pi$
$758$			2.03875e7i			0.0468119i
$759$	5.74980e8	+	4.06784e8i	1.31501	+	0.930333i
$760$	−1.42507e7			−0.0324634
$761$			4.79775e8i			1.08864i	0.838878	+	0.544319i	$0.183212\pi$
$761$			4.79775e8i			1.08864i	−0.838878	+	0.544319i	$0.816788\pi$
$762$	375675.	−	531008.i	0.000849077	−	0.00120015i
$763$	−6.21982e8			−1.40025
$764$			9.78494e7i			0.219421i
$765$	−2.02464e7	−	5.73549e7i	−0.0452234	−	0.128111i
$766$	3.75907e7			0.0836361
$767$			7.58371e7i			0.168072i
$768$	3.57553e8	+	2.52960e8i	0.789326	+	0.558428i
$769$	4.90886e8			1.07945			0.539724	−	0.841842i	$-0.318529\pi$
$769$	4.90886e8			1.07945			0.539724	+	0.841842i	$0.318529\pi$
$770$			1.19505e7i			0.0261766i
$771$	−4.95201e8	+	6.99956e8i	−1.08048	+	1.52724i
$772$	4.99863e8			1.08642
$773$			5.86642e8i			1.27009i	0.772475	+	0.635045i	$0.219019\pi$
$773$			5.86642e8i			1.27009i	−0.772475	+	0.635045i	$0.780981\pi$
$774$	−6.06410e6	+	2.14064e6i	−0.0130781	+	0.00461657i
$775$	−1.67950e8			−0.360807
$776$			1.36687e7i			0.0292512i
$777$	−3.40033e8	−	2.40565e8i	−0.724867	−	0.512825i
$778$	1.85701e7			0.0394344
$779$		−	4.67878e7i		−	0.0989737i
$780$	2.01835e8	−	2.85290e8i	0.425317	−	0.601177i
$781$	−1.05374e8			−0.221197
$782$		−	1.14947e7i		−	0.0240368i
$783$	8.29642e8	+	2.35318e8i	1.72825	+	0.490196i
$784$	−1.20903e8			−0.250894
$785$		−	3.71258e8i		−	0.767479i
$786$	842344.	+	595937.i	0.00173469	+	0.00122725i
$787$	8.14246e8			1.67044			0.835221	−	0.549914i	$-0.185340\pi$
$787$	8.14246e8			1.67044			0.835221	+	0.549914i	$0.185340\pi$
$788$			1.87187e8i			0.382558i
$789$	4.12383e8	−	5.82894e8i	0.839594	−	1.18675i
$790$	−2.70500e7			−0.0548639
$791$		−	4.78957e8i		−	0.967759i
$792$	1.78575e7	+	5.05877e7i	0.0359456	+	0.101828i
$793$	−3.35404e8			−0.672587
$794$			4.98881e7i			0.0996634i
$795$	4.00850e8	+	2.83591e8i	0.797775	+	0.564406i
$796$	4.18267e8			0.829305
$797$			6.26765e8i			1.23803i	0.785381	+	0.619013i	$0.212467\pi$
$797$			6.26765e8i			1.23803i	−0.785381	+	0.619013i	$0.787533\pi$
$798$	−7.36058e6	+	1.04040e7i	−0.0144845	+	0.0204735i
$799$	1.64625e8			0.322742
$800$			6.05318e7i			0.118226i
$801$	1.91509e8	−	6.76031e7i	0.372642	−	0.131543i
$802$	−3.74631e7			−0.0726240
$803$		−	4.11564e7i		−	0.0794861i
$804$	−6.28493e8	−	4.44643e8i	−1.20930	−	0.855546i
$805$	−4.33791e8			−0.831558
$806$			2.09475e7i			0.0400062i
$807$	−1.01696e8	+	1.43744e8i	−0.193500	+	0.273508i
$808$	1.68387e7			0.0319209
$809$			4.49278e8i			0.848534i	0.905537	+	0.424267i	$0.139468\pi$
$809$			4.49278e8i			0.848534i	−0.905537	+	0.424267i	$0.860532\pi$
$810$	1.33667e7	−	1.07803e7i	0.0251518	−	0.0202850i
$811$	4.58634e8			0.859813			0.429906	−	0.902874i	$-0.358547\pi$
$811$	4.58634e8			0.859813			0.429906	+	0.902874i	$0.358547\pi$
$812$		−	8.28259e8i		−	1.54703i
$813$	−7.26905e8	−	5.14267e8i	−1.35271	−	0.957011i
$814$	−2.99763e7			−0.0555783
$815$		−	9.77918e7i		−	0.180647i
$816$	−7.53501e7	+	1.06506e8i	−0.138680	+	0.196021i
$817$	6.59736e7			0.120977
$818$		−	1.67585e7i		−	0.0306179i
$819$	−2.08413e8	−	5.90402e8i	−0.379378	−	1.07472i
$820$	6.05491e7			0.109816
$821$		−	8.33343e8i		−	1.50589i	−0.658082	−	0.752946i	$-0.728632\pi$
$821$		−	8.33343e8i		−	1.50589i	0.658082	−	0.752946i	$-0.271368\pi$
$822$	−3.90673e7	−	2.76392e7i	−0.0703394	−	0.0497633i
$823$	−1.35502e8			−0.243078			−0.121539	−	0.992587i	$-0.538783\pi$
$823$	−1.35502e8			−0.243078			−0.121539	+	0.992587i	$0.538783\pi$
$824$		−	3.05874e7i		−	0.0546715i
$825$	2.08652e8	−	2.94924e8i	0.371586	−	0.525229i
$826$	3.57909e6			0.00635086
$827$			1.91581e8i			0.338717i	0.985555	+	0.169358i	$0.0541695\pi$
$827$			1.91581e8i			0.338717i	−0.985555	+	0.169358i	$0.945830\pi$
$828$	−9.16612e8	+	3.23565e8i	−1.61471	+	0.569995i
$829$	−5.01002e8			−0.879378			−0.439689	−	0.898150i	$-0.644911\pi$
$829$	−5.01002e8			−0.879378			−0.439689	+	0.898150i	$0.644911\pi$
$830$		−	1.56974e7i		−	0.0274532i
$831$	7.33281e8	+	5.18778e8i	1.27781	+	0.904021i
$832$	−7.44552e8			−1.29278
$833$		−	3.55265e7i		−	0.0614636i
$834$	−2.04008e7	+	2.88361e7i	−0.0351682	+	0.0497094i
$835$	−3.87758e8			−0.666042
$836$		−	2.74723e8i		−	0.470193i
$837$	8.41295e7	−	2.96609e8i	0.143474	−	0.505834i
$838$	6.07680e7			0.103263
$839$		−	3.18478e8i		−	0.539254i	−0.962965	−	0.269627i	$-0.913100\pi$
$839$		−	3.18478e8i		−	0.539254i	0.962965	−	0.269627i	$-0.0869003\pi$
$840$	−2.69731e7	−	1.90828e7i	−0.0455086	−	0.0321962i
$841$	−1.32474e9			−2.22712
$842$		−	2.50748e7i		−	0.0420051i
$843$	−4.58330e8	+	6.47840e8i	−0.765061	+	1.08140i
$844$	−1.01088e9			−1.68140
$845$			2.50057e8i			0.414448i
$846$	1.54712e7	+	4.38276e7i	0.0255513	+	0.0723830i
$847$	6.35052e7			0.104510
$848$		−	1.05323e9i		−	1.72718i
$849$	4.45671e8	+	3.15301e8i	0.728267	+	0.515231i
$850$	−5.89595e6			−0.00960058
$851$		−	1.08811e9i		−	1.76557i
$852$	8.39914e7	−	1.18720e8i	0.135805	−	0.191958i
$853$	9.97381e8			1.60699			0.803497	−	0.595309i	$-0.202970\pi$
$853$	9.97381e8			1.60699			0.803497	+	0.595309i	$0.202970\pi$
$854$			1.58292e7i			0.0254147i
$855$	−1.66122e8	+	5.86413e7i	−0.265784	+	0.0938221i
$856$	4.75402e7			0.0757949
$857$			5.63488e8i			0.895245i	0.894223	+	0.447623i	$0.147729\pi$
$857$			5.63488e8i			0.895245i	−0.894223	+	0.447623i	$0.852271\pi$
$858$	−3.67844e7	−	2.60240e7i	−0.0582373	−	0.0412014i
$859$	7.29664e8			1.15118			0.575591	−	0.817738i	$-0.304772\pi$
$859$	7.29664e8			1.15118			0.575591	+	0.817738i	$0.304772\pi$
$860$			8.53780e7i			0.134230i
$861$	6.26527e7	−	8.85582e7i	0.0981590	−	0.138746i
$862$	5.58160e7			0.0871438
$863$		−	6.23185e8i		−	0.969582i	−0.874630	−	0.484791i	$-0.838896\pi$
$863$		−	6.23185e8i		−	0.969582i	0.874630	−	0.484791i	$-0.161104\pi$
$864$	−1.06903e8	−	3.03216e7i	−0.165748	−	0.0470123i
$865$	−4.71531e8			−0.728554
$866$		−	7.26711e6i		−	0.0111894i
$867$	−3.12959e7	−	2.21411e7i	−0.0480209	−	0.0339736i
$868$	−2.96115e8			−0.452794
$869$		−	1.04468e9i		−	1.59193i
$870$	−2.20762e7	+	3.12043e7i	−0.0335249	+	0.0473867i
$871$	1.29543e9			1.96047
$872$		−	1.23760e8i		−	0.186651i
$873$	5.62467e7	+	1.59338e8i	0.0845385	+	0.239485i
$874$	−3.32930e7			−0.0498676
$875$			5.46750e8i			0.816139i
$876$	4.63691e7	+	3.28050e7i	0.0689790	+	0.0488009i
$877$	−9.29480e8			−1.37798			−0.688988	−	0.724773i	$-0.741945\pi$
$877$	−9.29480e8			−1.37798			−0.688988	+	0.724773i	$0.741945\pi$
$878$		−	4.24380e6i		−	0.00627006i
$879$	−7.14066e8	+	1.00932e9i	−1.05141	+	1.48614i
$880$	3.54334e8			0.519955
$881$		−	1.16497e8i		−	0.170367i	−0.996365	−	0.0851836i	$-0.972852\pi$
$881$		−	1.16497e8i		−	0.170367i	0.996365	−	0.0851836i	$-0.0271477\pi$
$882$	9.45812e6	−	3.33873e6i	0.0137848	−	0.00486604i
$883$	4.44164e8			0.645151			0.322575	−	0.946544i	$-0.395451\pi$
$883$	4.44164e8			0.645151			0.322575	+	0.946544i	$0.395451\pi$
$884$		−	2.20264e8i		−	0.318850i
$885$	4.03885e7	+	2.85738e7i	0.0582676	+	0.0412229i
$886$	−4.97381e7			−0.0715135
$887$			9.93887e8i			1.42418i	0.702086	+	0.712092i	$0.252252\pi$
$887$			9.93887e8i			1.42418i	−0.702086	+	0.712092i	$0.747748\pi$
$888$	4.78669e7	−	6.76588e7i	0.0683591	−	0.0966240i
$889$	−1.54719e7			−0.0220210
$890$			9.00187e6i			0.0127692i
$891$	4.16335e8	+	5.16224e8i	0.588587	+	0.729803i
$892$	−1.13619e9			−1.60087
$893$		−	4.76817e8i		−	0.669573i
$894$	849485.	+	600989.i	0.00118889	+	0.000841112i
$895$	−5.63908e7			−0.0786573
$896$			1.42219e8i			0.197713i
$897$	9.44646e8	−	1.33524e9i	1.30886	−	1.85004i
$898$	−1.63888e7			−0.0226317
$899$			6.86274e8i			0.944535i
$900$	1.65966e8	+	4.70157e8i	0.227663	+	0.644934i
$901$	3.09485e8			0.423121
$902$		−	7.80702e6i		−	0.0106381i
$903$	1.24872e8	+	8.83441e7i	0.169591	+	0.119982i
$904$	9.53013e7			0.129001
$905$		−	1.60078e8i		−	0.215967i
$906$	−1.80615e7	+	2.55296e7i	−0.0242868	+	0.0343288i
$907$	1.58430e8			0.212333			0.106166	−	0.994348i	$-0.466142\pi$
$907$	1.58430e8			0.212333			0.106166	+	0.994348i	$0.466142\pi$
$908$		−	4.65326e8i		−	0.621583i
$909$	1.96292e8	−	6.92912e7i	0.261342	−	0.0922543i
$910$	2.77518e7			0.0368270
$911$			9.78306e8i			1.29396i	0.762509	+	0.646978i	$0.223967\pi$
$911$			9.78306e8i			1.29396i	−0.762509	+	0.646978i	$0.776033\pi$
$912$	3.08481e8	+	2.18243e8i	0.406672	+	0.287710i
$913$	6.06237e8			0.796581
$914$			1.93565e7i			0.0253507i
$915$	−1.26373e8	+	1.78626e8i	−0.164965	+	0.233174i
$916$	2.22832e8			0.289928
$917$		−	2.45432e7i		−	0.0318290i
$918$	2.95340e6	−	1.04126e7i	0.00381764	−	0.0134596i
$919$	5.13655e8			0.661797			0.330899	−	0.943666i	$-0.392648\pi$
$919$	5.13655e8			0.661797			0.330899	+	0.943666i	$0.392648\pi$
$920$		−	8.63143e7i		−	0.110846i
$921$	−4.28655e8	−	3.03263e8i	−0.548693	−	0.388187i
$922$	3.50551e7			0.0447259
$923$			2.44702e8i			0.311195i
$924$	3.67877e8	−	5.19985e8i	0.466322	−	0.659136i
$925$	−5.58124e8			−0.705189
$926$		−	4.37898e7i		−	0.0551493i
$927$	−1.25867e8	−	3.56561e8i	−0.158005	−	0.447605i
$928$	2.47344e8			0.309498
$929$			8.32885e8i			1.03881i	0.854527	+	0.519407i	$0.173847\pi$
$929$			8.32885e8i			1.03881i	−0.854527	+	0.519407i	$0.826153\pi$
$930$	1.11560e7	+	7.89258e6i	0.0138694	+	0.00981228i
$931$	−1.02898e8			−0.127515
$932$		−	2.18169e8i		−	0.269491i
$933$	−4.16518e8	+	5.88739e8i	−0.512848	+	0.724900i
$934$	−4.62748e7			−0.0567942
$935$			1.04119e8i			0.127378i
$936$	1.17476e8	−	4.14693e7i	0.143259	−	0.0505707i
$937$	7.27480e8			0.884305			0.442152	−	0.896940i	$-0.354215\pi$
$937$	7.27480e8			0.884305			0.442152	+	0.896940i	$0.354215\pi$
$938$		−	6.11371e7i		−	0.0740792i
$939$	−6.86466e8	−	4.85658e8i	−0.829129	−	0.586588i
$940$	6.17061e8			0.742924
$941$			1.36069e9i			1.63302i	0.577335	+	0.816508i	$0.304093\pi$
$941$			1.36069e9i			1.63302i	−0.577335	+	0.816508i	$0.695907\pi$
$942$	3.81553e7	−	5.39317e7i	0.0456460	−	0.0645196i
$943$	2.83387e8			0.337944
$944$		−	1.06121e8i		−	0.126149i
$945$	−3.92955e8	−	1.11457e8i	−0.465637	−	0.132072i
$946$	1.10084e7			0.0130032
$947$			7.33753e8i			0.863973i	0.901880	+	0.431986i	$0.142187\pi$
$947$			7.33753e8i			0.863973i	−0.901880	+	0.431986i	$0.857813\pi$
$948$	1.17699e9	+	8.32692e8i	1.38149	+	0.977371i
$949$	−9.55747e7			−0.111826
$950$			1.70769e7i			0.0199177i
$951$	6.01070e8	−	8.49599e8i	0.698849	−	0.987808i
$952$	−2.08252e7			−0.0241367
$953$			9.00794e8i			1.04075i	0.853938	+	0.520375i	$0.174208\pi$
$953$			9.00794e8i			1.04075i	−0.853938	+	0.520375i	$0.825792\pi$
$954$	2.90849e7	+	8.23932e7i	0.0334983	+	0.0948956i
$955$	−1.07411e8			−0.123321
$956$			2.83143e7i			0.0324065i
$957$	−1.20511e9	−	8.52588e8i	−1.37497	−	0.972755i
$958$	4.25592e7			0.0484058
$959$			1.13830e9i			1.29062i
$960$	−2.80532e8	+	3.96525e8i	−0.317079	+	0.448185i
$961$	−6.42151e8			−0.723547
$962$			6.96119e7i			0.0781912i
$963$	5.54183e8	−	1.95627e8i	0.620546	−	0.219054i
$964$	2.96199e8			0.330638
$965$			5.48707e8i			0.610602i
$966$	−6.30157e7	−	4.45820e7i	−0.0699065	−	0.0494571i
$967$	−7.73547e8			−0.855475			−0.427738	−	0.903903i	$-0.640689\pi$
$967$	−7.73547e8			−0.855475			−0.427738	+	0.903903i	$0.640689\pi$
$968$			1.26361e7i			0.0139311i
$969$	−6.41290e7	+	9.06449e7i	−0.0704828	+	0.0996259i
$970$	−7.48969e6			−0.00820632
$971$		−	1.22350e9i		−	1.33643i	−0.743967	−	0.668216i	$-0.767058\pi$
$971$		−	1.22350e9i		−	1.33643i	0.743967	−	0.668216i	$-0.232942\pi$
$972$	−9.13460e8	+	5.75944e7i	−0.994697	+	0.0627165i
$973$	8.40192e8			0.912095
$974$			3.74578e7i			0.0405382i
$975$	−6.84882e8	−	4.84537e8i	−0.738928	−	0.522773i
$976$	4.69339e8			0.504821
$977$		−	6.11546e8i		−	0.655761i	−0.944719	−	0.327880i	$-0.893666\pi$
$977$		−	6.11546e8i		−	0.655761i	0.944719	−	0.327880i	$-0.106334\pi$
$978$	1.00504e7	−	1.42060e7i	0.0107440	−	0.0151864i
$979$	−3.47654e8			−0.370509
$980$		−	1.33163e8i		−	0.141484i
$981$	−5.09271e8	−	1.44269e9i	−0.539438	−	1.52815i
$982$	6.42043e7			0.0678000
$983$			1.16297e9i			1.22435i	0.790722	+	0.612176i	$0.209705\pi$
$983$			1.16297e9i			1.22435i	−0.790722	+	0.612176i	$0.790295\pi$
$984$	1.76210e7	+	1.24664e7i	0.0184946	+	0.0130845i
$985$	−2.05478e8			−0.215009
$986$			2.40919e7i			0.0251328i
$987$	6.38498e8	−	9.02503e8i	0.664061	−	0.938636i
$988$	−6.37969e8			−0.661499
$989$			3.99593e8i			0.413075i
$990$	−2.77192e7	+	9.78490e6i	−0.0285676	+	0.0100844i
$991$	6.40493e8			0.658102			0.329051	−	0.944312i	$-0.393271\pi$
$991$	6.40493e8			0.658102			0.329051	+	0.944312i	$0.393271\pi$
$992$		−	8.84290e7i		−	0.0905857i
$993$	1.01629e9	+	7.19000e8i	1.03793	+	0.734313i
$994$	1.15486e7			0.0117590
$995$			4.59138e8i			0.466094i
$996$	−4.83220e8	+	6.83020e8i	−0.489065	+	0.691283i
$997$	1.15040e8			0.116082			0.0580409	−	0.998314i	$-0.481515\pi$
$997$	1.15040e8			0.116082			0.0580409	+	0.998314i	$0.481515\pi$
$998$			1.30983e7i			0.0131772i
$999$	2.79576e8	−	9.85679e8i	0.280416	−	0.988642i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	51.7.b.a.35.17	yes	32
3.2	odd	2	inner	51.7.b.a.35.16	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.7.b.a.35.16	✓	32	3.2	odd	2	inner
51.7.b.a.35.17	yes	32	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 51 = 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	51.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.461474i q^{2} +(22.0416 + 15.5939i) q^{3} +63.7870 q^{4} +70.0200i q^{5} +(-7.19617 + 10.1716i) q^{6} +296.369 q^{7} +58.9705i q^{8} +(242.663 + 687.427i) q^{9} -32.3124 q^{10} -1247.91i q^{11} +(1405.97 + 994.687i) q^{12} -2897.94 q^{13} +136.767i q^{14} +(-1091.88 + 1543.35i) q^{15} +4055.16 q^{16} +1191.58i q^{17} +(-317.230 + 111.983i) q^{18} +3451.27 q^{19} +4466.37i q^{20} +(6532.43 + 4621.53i) q^{21} +575.879 q^{22} +20903.9i q^{23} +(-919.577 + 1299.80i) q^{24} +10722.2 q^{25} -1337.32i q^{26} +(-5370.97 + 18936.0i) q^{27} +18904.5 q^{28} -43812.9i q^{29} +(-712.217 - 503.876i) q^{30} -15663.7 q^{31} +5645.46i q^{32} +(19459.8 - 27505.9i) q^{33} -549.883 q^{34} +20751.7i q^{35} +(15478.7 + 43848.9i) q^{36} -52053.1 q^{37} +1592.67i q^{38} +(-63875.1 - 45190.0i) q^{39} -4129.11 q^{40} -13556.7i q^{41} +(-2132.72 + 3014.55i) q^{42} +19115.8 q^{43} -79600.6i q^{44} +(-48133.6 + 16991.2i) q^{45} -9646.59 q^{46} -138157. i q^{47} +(89382.1 + 63235.6i) q^{48} -29814.7 q^{49} +4948.02i q^{50} +(-18581.3 + 26264.3i) q^{51} -184851. q^{52} -259727. i q^{53} +(-8738.49 - 2478.57i) q^{54} +87378.7 q^{55} +17477.0i q^{56} +(76071.4 + 53818.6i) q^{57} +20218.5 q^{58} -26169.3i q^{59} +(-69647.9 + 98445.8i) q^{60} +115739. q^{61} -7228.42i q^{62} +(71917.6 + 203732. i) q^{63} +256925. q^{64} -202913. i q^{65} +(12693.3 + 8980.19i) q^{66} -447018. q^{67} +76007.2i q^{68} +(-325972. + 460754. i) q^{69} -9576.39 q^{70} -84440.1i q^{71} +(-40537.9 + 14309.9i) q^{72} +32980.2 q^{73} -24021.2i q^{74} +(236334. + 167201. i) q^{75} +220146. q^{76} -369842. i q^{77} +(20854.1 - 29476.7i) q^{78} +837140. q^{79} +283942. i q^{80} +(-413671. + 333626. i) q^{81} +6256.07 q^{82} +485801. i q^{83} +(416684. + 294794. i) q^{84} -83434.2 q^{85} +8821.44i q^{86} +(683212. - 965705. i) q^{87} +73589.9 q^{88} -278589. i q^{89} +(-7841.02 - 22212.4i) q^{90} -858857. q^{91} +1.33339e6i q^{92} +(-345254. - 244258. i) q^{93} +63756.0 q^{94} +241658. i q^{95} +(-88034.6 + 124435. i) q^{96} +231790. q^{97} -13758.7i q^{98} +(857848. - 302822. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 32 q^{3} - 1024 q^{4} + 286 q^{6} + 568 q^{7} - 912 q^{9} - 744 q^{10} + 194 q^{12} - 2312 q^{13} - 6240 q^{15} + 13208 q^{16} + 2936 q^{18} + 7936 q^{19} - 21688 q^{21} + 13176 q^{22} + 18282 q^{24}+ \cdots + 1619864 q^{99}+O(q^{100}) \) 32 * q + 32 * q^3 - 1024 * q^4 + 286 * q^6 + 568 * q^7 - 912 * q^9 - 744 * q^10 + 194 * q^12 - 2312 * q^13 - 6240 * q^15 + 13208 * q^16 + 2936 * q^18 + 7936 * q^19 - 21688 * q^21 + 13176 * q^22 + 18282 * q^24 - 76792 * q^25 + 87056 * q^27 + 45712 * q^28 - 45620 * q^30 - 36728 * q^31 + 73456 * q^33 - 52272 * q^36 + 133336 * q^37 - 48720 * q^39 - 81084 * q^40 - 305352 * q^42 - 412832 * q^43 - 9936 * q^45 + 283044 * q^46 + 233266 * q^48 + 342960 * q^49 + 50152 * q^52 + 196798 * q^54 + 7104 * q^55 - 22576 * q^57 + 167484 * q^58 - 58564 * q^60 - 235496 * q^61 - 331312 * q^63 + 299888 * q^64 + 485816 * q^66 - 774272 * q^67 + 553800 * q^69 - 1318656 * q^70 + 447312 * q^72 - 33152 * q^73 + 86872 * q^75 + 1045000 * q^76 - 8652 * q^78 + 1789576 * q^79 - 2391192 * q^81 - 178380 * q^82 - 751024 * q^84 + 235824 * q^85 - 155232 * q^87 - 3091500 * q^88 - 728056 * q^90 + 2080448 * q^91 + 1012816 * q^93 + 3173160 * q^94 + 422058 * q^96 - 1502960 * q^97 + 1619864 * q^99

Introduction

L-functions

Modular forms

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