LMFDB - Embedded newform 51.7.b.a.35.13

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			35.13
Character	$\chi$	$=$	51.35
Dual form			51.7.b.a.35.20

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.81455i q^{2} +(26.1801 + 6.60322i) q^{3} +40.8201 q^{4} -186.576i q^{5} +(31.7916 - 126.045i) q^{6} -455.638 q^{7} -504.662i q^{8} +(641.795 + 345.746i) q^{9} -898.281 q^{10} -546.662i q^{11} +(1068.67 + 269.544i) q^{12} +1273.85 q^{13} +2193.69i q^{14} +(1232.01 - 4884.59i) q^{15} +182.767 q^{16} -1191.58i q^{17} +(1664.61 - 3089.95i) q^{18} -3937.85 q^{19} -7616.07i q^{20} +(-11928.6 - 3008.68i) q^{21} -2631.93 q^{22} +6468.05i q^{23} +(3332.39 - 13212.1i) q^{24} -19185.7 q^{25} -6133.00i q^{26} +(14519.2 + 13289.6i) q^{27} -18599.2 q^{28} -28253.9i q^{29} +(-23517.1 - 5931.55i) q^{30} +45232.4 q^{31} -33178.3i q^{32} +(3609.73 - 14311.7i) q^{33} -5736.91 q^{34} +85011.2i q^{35} +(26198.1 + 14113.4i) q^{36} +86189.5 q^{37} +18959.0i q^{38} +(33349.4 + 8411.49i) q^{39} -94158.0 q^{40} +93994.5i q^{41} +(-14485.4 + 57431.0i) q^{42} -91153.1 q^{43} -22314.8i q^{44} +(64508.1 - 119744. i) q^{45} +31140.8 q^{46} +12654.7i q^{47} +(4784.86 + 1206.85i) q^{48} +89956.7 q^{49} +92370.8i q^{50} +(7868.25 - 31195.6i) q^{51} +51998.5 q^{52} +146501. i q^{53} +(63983.4 - 69903.5i) q^{54} -101994. q^{55} +229943. i q^{56} +(-103093. - 26002.5i) q^{57} -136030. q^{58} +323762. i q^{59} +(50290.6 - 199389. i) q^{60} +132481. q^{61} -217773. i q^{62} +(-292426. - 157535. i) q^{63} -148041. q^{64} -237670. i q^{65} +(-68904.2 - 17379.2i) q^{66} -53858.1 q^{67} -48640.3i q^{68} +(-42710.0 + 169334. i) q^{69} +409291. q^{70} +449162. i q^{71} +(174485. - 323889. i) q^{72} +416903. q^{73} -414964. i q^{74} +(-502285. - 126688. i) q^{75} -160744. q^{76} +249080. i q^{77} +(40497.6 - 160562. i) q^{78} +293630. q^{79} -34100.0i q^{80} +(292360. + 443796. i) q^{81} +452541. q^{82} -952000. i q^{83} +(-486928. - 122815. i) q^{84} -222320. q^{85} +438861. i q^{86} +(186567. - 739689. i) q^{87} -275879. q^{88} -843153. i q^{89} +(-576512. - 310577. i) q^{90} -580412. q^{91} +264027. i q^{92} +(1.18419e6 + 298679. i) q^{93} +60926.8 q^{94} +734710. i q^{95} +(219084. - 868611. i) q^{96} -348695. q^{97} -433101. i q^{98} +(189006. - 350845. 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 $

$2$		−	4.81455i		−	0.601819i	−0.953653	−	0.300909i	$-0.902710\pi$
$2$		−	4.81455i		−	0.601819i	0.953653	−	0.300909i	$-0.0972902\pi$
$3$	26.1801	+	6.60322i	0.969633	+	0.244564i
$3$	26.1801	+	6.60322i	0.969633	+	0.244564i
$4$	40.8201			0.637814
$5$		−	186.576i		−	1.49261i	−0.665604	−	0.746306i	$-0.731826\pi$
$5$		−	186.576i		−	1.49261i	0.665604	−	0.746306i	$-0.268174\pi$
$6$	31.7916	−	126.045i	0.147183	−	0.583544i
$7$	−455.638			−1.32839			−0.664195	−	0.747560i	$-0.731225\pi$
$7$	−455.638			−1.32839			−0.664195	+	0.747560i	$0.731225\pi$
$8$		−	504.662i		−	0.985667i
$9$	641.795	+	345.746i	0.880377	+	0.474274i
$10$	−898.281			−0.898281
$11$		−	546.662i		−	0.410715i	−0.978687	−	0.205358i	$-0.934164\pi$
$11$		−	546.662i		−	0.410715i	0.978687	−	0.205358i	$-0.0658358\pi$
$12$	1068.67	+	269.544i	0.618446	+	0.155986i
$13$	1273.85			0.579812			0.289906	−	0.957055i	$-0.406376\pi$
$13$	1273.85			0.579812			0.289906	+	0.957055i	$0.406376\pi$
$14$			2193.69i			0.799450i
$15$	1232.01	−	4884.59i	0.365039	−	1.44729i
$16$	182.767			0.0446209
$17$		−	1191.58i		−	0.242536i
$17$		−	1191.58i		−	0.242536i
$18$	1664.61	−	3089.95i	0.285427	−	0.529827i
$19$	−3937.85			−0.574115			−0.287057	−	0.957913i	$-0.592677\pi$
$19$	−3937.85			−0.574115			−0.287057	+	0.957913i	$0.592677\pi$
$20$		−	7616.07i		−	0.952008i
$21$	−11928.6	−	3008.68i	−1.28805	−	0.324876i
$22$	−2631.93			−0.247176
$23$			6468.05i			0.531606i	0.964027	+	0.265803i	$0.0856371\pi$
$23$			6468.05i			0.531606i	−0.964027	+	0.265803i	$0.914363\pi$
$24$	3332.39	−	13212.1i	0.241059	−	0.955736i
$25$	−19185.7			−1.22789
$26$		−	6133.00i		−	0.348942i
$27$	14519.2	+	13289.6i	0.737652	+	0.675181i
$28$	−18599.2			−0.847266
$29$		−	28253.9i		−	1.15847i	−0.815161	−	0.579234i	$-0.803352\pi$
$29$		−	28253.9i		−	1.15847i	0.815161	−	0.579234i	$-0.196648\pi$
$30$	−23517.1	−	5931.55i	−0.871004	−	0.219687i
$31$	45232.4			1.51832			0.759161	−	0.650902i	$-0.225609\pi$
$31$	45232.4			1.51832			0.759161	+	0.650902i	$0.225609\pi$
$32$		−	33178.3i		−	1.01252i
$33$	3609.73	−	14311.7i	0.100446	−	0.398243i
$34$	−5736.91			−0.145963
$35$			85011.2i			1.98277i
$36$	26198.1	+	14113.4i	0.561517	+	0.302499i
$37$	86189.5			1.70157			0.850784	−	0.525515i	$-0.176127\pi$
$37$	86189.5			1.70157			0.850784	+	0.525515i	$0.176127\pi$
$38$			18959.0i			0.345513i
$39$	33349.4	+	8411.49i	0.562205	+	0.141801i
$40$	−94158.0			−1.47122
$41$			93994.5i			1.36380i	0.731445	+	0.681900i	$0.238846\pi$
$41$			93994.5i			1.36380i	−0.731445	+	0.681900i	$0.761154\pi$
$42$	−14485.4	+	57431.0i	−0.195517	+	0.775173i
$43$	−91153.1			−1.14648			−0.573239	−	0.819388i	$-0.694314\pi$
$43$	−91153.1			−1.14648			−0.573239	+	0.819388i	$0.694314\pi$
$44$		−	22314.8i		−	0.261960i
$45$	64508.1	−	119744.i	0.707907	−	1.31406i
$46$	31140.8			0.319931
$47$			12654.7i			0.121888i	0.998141	+	0.0609438i	$0.0194110\pi$
$47$			12654.7i			0.121888i	−0.998141	+	0.0609438i	$0.980589\pi$
$48$	4784.86	+	1206.85i	0.0432659	+	0.0109127i
$49$	89956.7			0.764619
$50$			92370.8i			0.738966i
$51$	7868.25	−	31195.6i	0.0593154	−	0.235171i
$52$	51998.5			0.369812
$53$			146501.i			0.984041i	0.870584	+	0.492021i	$0.163742\pi$
$53$			146501.i			0.984041i	−0.870584	+	0.492021i	$0.836258\pi$
$54$	63983.4	−	69903.5i	0.406336	−	0.443933i
$55$	−101994.			−0.613038
$56$			229943.i			1.30935i
$57$	−103093.	−	26002.5i	−0.556681	−	0.140408i
$58$	−136030.			−0.697188
$59$			323762.i			1.57641i	0.615411	+	0.788206i	$0.288990\pi$
$59$			323762.i			1.57641i	−0.615411	+	0.788206i	$0.711010\pi$
$60$	50290.6	−	199389.i	0.232827	−	0.923099i
$61$	132481.			0.583665			0.291833	−	0.956469i	$-0.405735\pi$
$61$	132481.			0.583665			0.291833	+	0.956469i	$0.405735\pi$
$62$		−	217773.i		−	0.913755i
$63$	−292426.	−	157535.i	−1.16948	−	0.630021i
$64$	−148041.			−0.564733
$65$		−	237670.i		−	0.865433i
$66$	−68904.2	−	17379.2i	−0.239670	−	0.0604504i
$67$	−53858.1			−0.179072			−0.0895358	−	0.995984i	$-0.528538\pi$
$67$	−53858.1			−0.179072			−0.0895358	+	0.995984i	$0.528538\pi$
$68$		−	48640.3i		−	0.154693i
$69$	−42710.0	+	169334.i	−0.130012	+	0.515463i
$70$	409291.			1.19327
$71$			449162.i			1.25496i	0.778635	+	0.627478i	$0.215913\pi$
$71$			449162.i			1.25496i	−0.778635	+	0.627478i	$0.784087\pi$
$72$	174485.	−	323889.i	0.467477	−	0.867759i
$73$	416903.			1.07168			0.535842	−	0.844318i	$-0.319994\pi$
$73$	416903.			1.07168			0.535842	+	0.844318i	$0.319994\pi$
$74$		−	414964.i		−	1.02404i
$75$	−502285.	−	126688.i	−1.19060	−	0.300297i
$76$	−160744.			−0.366179
$77$			249080.i			0.545590i
$78$	40497.6	−	160562.i	0.0853385	−	0.338345i
$79$	293630.			0.595550			0.297775	−	0.954636i	$-0.403755\pi$
$79$	293630.			0.595550			0.297775	+	0.954636i	$0.403755\pi$
$80$		−	34100.0i		−	0.0666016i
$81$	292360.	+	443796.i	0.550127	+	0.835081i
$82$	452541.			0.820761
$83$		−	952000.i		−	1.66496i	−0.554058	−	0.832478i	$-0.686921\pi$
$83$		−	952000.i		−	1.66496i	0.554058	−	0.832478i	$-0.313079\pi$
$84$	−486928.	−	122815.i	−0.821537	−	0.207211i
$85$	−222320.			−0.362011
$86$			438861.i			0.689973i
$87$	186567.	−	739689.i	0.283319	−	1.12329i
$88$	−275879.			−0.404829
$89$		−	843153.i		−	1.19601i	−0.801491	−	0.598007i	$-0.795959\pi$
$89$		−	843153.i		−	1.19601i	0.801491	−	0.598007i	$-0.204041\pi$
$90$	−576512.	−	310577.i	−0.790826	−	0.426032i
$91$	−580412.			−0.770216
$92$			264027.i			0.339066i
$93$	1.18419e6	+	298679.i	1.47222	+	0.371327i
$94$	60926.8			0.0733542
$95$			734710.i			0.856930i
$96$	219084.	−	868611.i	0.247626	−	0.981774i
$97$	−348695.			−0.382059			−0.191030	−	0.981584i	$-0.561183\pi$
$97$	−348695.			−0.382059			−0.191030	+	0.981584i	$0.561183\pi$
$98$		−	433101.i		−	0.460162i
$99$	189006.	−	350845.i	0.194792	−	0.361584i
$100$	−783164.			−0.783164
$101$			1.35407e6i			1.31425i	0.753783	+	0.657123i	$0.228227\pi$
$101$			1.35407e6i			1.31425i	−0.753783	+	0.657123i	$0.771773\pi$
$102$	−150193.	−	37882.1i	−0.141530	−	0.0356972i
$103$	−1.81464e6			−1.66065			−0.830326	−	0.557278i	$-0.811846\pi$
$103$	−1.81464e6			−1.66065			−0.830326	+	0.557278i	$0.811846\pi$
$104$		−	642862.i		−	0.571502i
$105$	−561348.	+	2.22560e6i	−0.484914	+	1.92256i
$106$	705337.			0.592214
$107$			162152.i			0.132365i	0.997808	+	0.0661823i	$0.0210819\pi$
$107$			162152.i			0.132365i	−0.997808	+	0.0661823i	$0.978918\pi$
$108$	592676.	+	542482.i	0.470485	+	0.430640i
$109$	−126859.			−0.0979582			−0.0489791	−	0.998800i	$-0.515597\pi$
$109$	−126859.			−0.0979582			−0.0489791	+	0.998800i	$0.515597\pi$
$110$			491056.i			0.368938i
$111$	2.25645e6	+	569129.i	1.64990	+	0.416142i
$112$	−83275.6			−0.0592739
$113$			1.64169e6i			1.13777i	0.822417	+	0.568886i	$0.192625\pi$
$113$			1.64169e6i			1.13777i	−0.822417	+	0.568886i	$0.807375\pi$
$114$	−125190.	+	496348.i	−0.0845000	+	0.335021i
$115$	1.20679e6			0.793482
$116$		−	1.15333e6i		−	0.738887i
$117$	817548.	+	440427.i	0.510453	+	0.274990i
$118$	1.55877e6			0.948715
$119$			542928.i			0.322182i
$120$	−2.46506e6	−	621746.i	−1.42654	−	0.359807i
$121$	1.47272e6			0.831313
$122$		−	637836.i		−	0.351261i
$123$	−620667.	+	2.46079e6i	−0.333536	+	1.32239i
$124$	1.84639e6			0.968408
$125$			664351.i			0.340148i
$126$	−758460.	+	1.40790e6i	−0.379159	+	0.703817i
$127$	−2.56147e6			−1.25048			−0.625242	−	0.780431i	$-0.715000\pi$
$127$	−2.56147e6			−1.25048			−0.625242	+	0.780431i	$0.715000\pi$
$128$		−	1.41066e6i		−	0.672654i
$129$	−2.38640e6	−	601904.i	−1.11166	−	0.280387i
$130$	−1.14427e6			−0.520834
$131$		−	1.30524e6i		−	0.580601i	−0.956936	−	0.290300i	$-0.906245\pi$
$131$		−	1.30524e6i		−	0.580601i	0.956936	−	0.290300i	$-0.0937553\pi$
$132$	147350.	−	584204.i	0.0640659	−	0.254005i
$133$	1.79423e6			0.762648
$134$			259303.i			0.107769i
$135$	2.47952e6	−	2.70894e6i	1.00778	−	1.10103i
$136$	−601344.			−0.239059
$137$		−	1.59134e6i		−	0.618872i	−0.950920	−	0.309436i	$-0.899860\pi$
$137$		−	1.59134e6i		−	0.618872i	0.950920	−	0.309436i	$-0.100140\pi$
$138$	815269.	+	205630.i	0.310215	+	0.0782435i
$139$	−2.20404e6			−0.820681			−0.410340	−	0.911932i	$-0.634590\pi$
$139$	−2.20404e6			−0.820681			−0.410340	+	0.911932i	$0.634590\pi$
$140$			3.47017e6i			1.26464i
$141$	−83562.0	+	331302.i	−0.0298093	+	0.118186i
$142$	2.16251e6			0.755256
$143$		−	696364.i		−	0.238138i
$144$	117299.	+	63191.0i	0.0392832	+	0.0211625i
$145$	−5.27151e6			−1.72914
$146$		−	2.00720e6i		−	0.644960i
$147$	2.35507e6	+	594004.i	0.741400	+	0.186998i
$148$	3.51827e6			1.08528
$149$		−	5.05689e6i		−	1.52871i	−0.644797	−	0.764354i	$-0.723058\pi$
$149$		−	5.05689e6i		−	1.52871i	0.644797	−	0.764354i	$-0.276942\pi$
$150$	−609945.	+	2.41828e6i	−0.180724	+	0.716526i
$151$	2.99775e6			0.870692			0.435346	−	0.900263i	$-0.356626\pi$
$151$	2.99775e6			0.870692			0.435346	+	0.900263i	$0.356626\pi$
$152$			1.98728e6i			0.565886i
$153$	411983.	−	764748.i	0.115028	−	0.213523i
$154$	1.19921e6			0.328346
$155$		−	8.43929e6i		−	2.26627i
$156$	1.36133e6	+	343358.i	0.358582	+	0.0904427i
$157$	−1.00692e6			−0.260193			−0.130096	−	0.991501i	$-0.541529\pi$
$157$	−1.00692e6			−0.260193			−0.130096	+	0.991501i	$0.541529\pi$
$158$		−	1.41369e6i		−	0.358413i
$159$	−967379.	+	3.83541e6i	−0.240661	+	0.954159i
$160$	−6.19029e6			−1.51130
$161$		−	2.94709e6i		−	0.706180i
$162$	2.13668e6	−	1.40758e6i	0.502567	−	0.331077i
$163$	1.78774e6			0.412802			0.206401	−	0.978467i	$-0.433825\pi$
$163$	1.78774e6			0.412802			0.206401	+	0.978467i	$0.433825\pi$
$164$			3.83687e6i			0.869851i
$165$	−2.67022e6	−	673491.i	−0.594422	−	0.149927i
$166$	−4.58345e6			−1.00200
$167$			4.29618e6i			0.922428i	0.887289	+	0.461214i	$0.152586\pi$
$167$			4.29618e6i			0.922428i	−0.887289	+	0.461214i	$0.847414\pi$
$168$	−1.51836e6	+	6.01993e6i	−0.320220	+	1.26959i
$169$	−3.20412e6			−0.663818
$170$			1.07037e6i			0.217865i
$171$	−2.52729e6	−	1.36150e6i	−0.505438	−	0.272288i
$172$	−3.72088e6			−0.731240
$173$		−	4.98994e6i		−	0.963734i	−0.876244	−	0.481867i	$-0.839959\pi$
$173$		−	4.98994e6i		−	0.963734i	0.876244	−	0.481867i	$-0.160041\pi$
$174$	−3.56127e6	−	898235.i	−0.676017	−	0.170507i
$175$	8.74175e6			1.63111
$176$		−	99911.8i		−	0.0183265i
$177$	−2.13787e6	+	8.47612e6i	−0.385534	+	1.52854i
$178$	−4.05940e6			−0.719784
$179$		−	6.94812e6i		−	1.21146i	−0.795671	−	0.605729i	$-0.792882\pi$
$179$		−	6.94812e6i		−	1.21146i	0.795671	−	0.605729i	$-0.207118\pi$
$180$	2.63323e6	−	4.88795e6i	0.451513	−	0.838126i
$181$	6.39241e6			1.07802			0.539012	−	0.842298i	$-0.318798\pi$
$181$	6.39241e6			1.07802			0.539012	+	0.842298i	$0.318798\pi$
$182$			2.79443e6i			0.463530i
$183$	3.46836e6	+	874801.i	0.565941	+	0.142743i
$184$	3.26418e6			0.523987
$185$		−	1.60809e7i		−	2.53978i
$186$	1.43801e6	−	5.70133e6i	0.223472	−	0.886007i
$187$	−651390.			−0.0996131
$188$			516567.i			0.0777416i
$189$	−6.61550e6	−	6.05523e6i	−0.979890	−	0.896903i
$190$	3.53730e6			0.515717
$191$			1.06477e7i			1.52811i	0.645149	+	0.764057i	$0.276795\pi$
$191$			1.06477e7i			1.52811i	−0.645149	+	0.764057i	$0.723205\pi$
$192$	−3.87574e6	−	977551.i	−0.547584	−	0.138113i
$193$	4.57188e6			0.635949			0.317975	−	0.948099i	$-0.396997\pi$
$193$	4.57188e6			0.635949			0.317975	+	0.948099i	$0.396997\pi$
$194$			1.67881e6i			0.229931i
$195$	1.56939e6	−	6.22221e6i	0.211654	−	0.839153i
$196$	3.67204e6			0.487685
$197$			2.80943e6i			0.367469i	0.982976	+	0.183734i	$0.0588186\pi$
$197$			2.80943e6i			0.367469i	−0.982976	+	0.183734i	$0.941181\pi$
$198$	−1.68916e6	−	909980.i	−0.217608	−	0.117229i
$199$	−1.07107e7			−1.35913			−0.679563	−	0.733617i	$-0.737831\pi$
$199$	−1.07107e7			−1.35913			−0.679563	+	0.733617i	$0.737831\pi$
$200$			9.68231e6i			1.21029i
$201$	−1.41001e6	−	355637.i	−0.173634	−	0.0437944i
$202$	6.51923e6			0.790938
$203$			1.28735e7i			1.53890i
$204$	321183.	−	1.27341e6i	0.0378322	−	0.149995i
$205$	1.75372e7			2.03562
$206$			8.73667e6i			0.999412i
$207$	−2.23630e6	+	4.15116e6i	−0.252127	+	0.468014i
$208$	232817.			0.0258717
$209$			2.15268e6i			0.235798i
$210$	1.07153e7	+	2.70264e6i	1.15703	+	0.291830i
$211$	38902.1			0.00414119			0.00207060	−	0.999998i	$-0.499341\pi$
$211$	38902.1			0.00414119			0.00207060	+	0.999998i	$0.499341\pi$
$212$			5.98019e6i			0.627635i
$213$	−2.96592e6	+	1.17591e7i	−0.306917	+	1.21685i
$214$	780691.			0.0796595
$215$			1.70070e7i			1.71125i
$216$	6.70674e6	−	7.32729e6i	0.665504	−	0.727080i
$217$	−2.06096e7			−2.01692
$218$			610768.i			0.0589531i
$219$	1.09146e7	+	2.75291e6i	1.03914	+	0.262095i
$220$	−4.16341e6			−0.391004
$221$		−	1.51789e6i		−	0.140625i
$222$	2.74010e6	−	1.08638e7i	0.250442	−	0.992939i
$223$	−1.51238e7			−1.36378			−0.681891	−	0.731453i	$-0.738842\pi$
$223$	−1.51238e7			−1.36378			−0.681891	+	0.731453i	$0.738842\pi$
$224$			1.51173e7i			1.34502i
$225$	−1.23133e7	−	6.63340e6i	−1.08100	−	0.582356i
$226$	7.90399e6			0.684732
$227$		−	1.78502e7i		−	1.52604i	−0.646375	−	0.763019i	$-0.723716\pi$
$227$		−	1.78502e7i		−	1.52604i	0.646375	−	0.763019i	$-0.276284\pi$
$228$	−4.20828e6	−	1.06143e6i	−0.355059	−	0.0895540i
$229$	1.82281e7			1.51788			0.758938	−	0.651163i	$-0.225719\pi$
$229$	1.82281e7			1.51788			0.758938	+	0.651163i	$0.225719\pi$
$230$		−	5.81013e6i		−	0.477532i
$231$	−1.64473e6	+	6.52093e6i	−0.133432	+	0.529022i
$232$	−1.42586e7			−1.14186
$233$		−	1.16678e7i		−	0.922407i	−0.887294	−	0.461203i	$-0.847418\pi$
$233$		−	1.16678e7i		−	0.922407i	0.887294	−	0.461203i	$-0.152582\pi$
$234$	2.12046e6	−	3.93613e6i	0.165494	−	0.307200i
$235$	2.36107e6			0.181931
$236$			1.32160e7i			1.00546i
$237$	7.68725e6	+	1.93890e6i	0.577465	+	0.145650i
$238$	2.61395e6			0.193895
$239$			7.69522e6i			0.563673i	0.959462	+	0.281837i	$0.0909436\pi$
$239$			7.69522e6i			0.563673i	−0.959462	+	0.281837i	$0.909056\pi$
$240$	225170.	−	892742.i	0.0162883	−	0.0645791i
$241$	−1.55728e7			−1.11254			−0.556271	−	0.831001i	$-0.687768\pi$
$241$	−1.55728e7			−1.11254			−0.556271	+	0.831001i	$0.687768\pi$
$242$		−	7.09049e6i		−	0.500300i
$243$	4.72354e6	+	1.35491e7i	0.329191	+	0.944263i
$244$	5.40788e6			0.372270
$245$		−	1.67838e7i		−	1.14128i
$246$	1.18476e7	+	2.98823e6i	0.795837	+	0.200728i
$247$	−5.01622e6			−0.332879
$248$		−	2.28270e7i		−	1.49656i
$249$	6.28627e6	−	2.49235e7i	0.407188	−	1.61440i
$250$	3.19855e6			0.204707
$251$			5.81607e6i			0.367797i	0.982945	+	0.183899i	$0.0588718\pi$
$251$			5.81607e6i			0.367797i	−0.982945	+	0.183899i	$0.941128\pi$
$252$	−1.19369e7	−	6.43059e6i	−0.745913	−	0.401836i
$253$	3.53584e6			0.218339
$254$			1.23323e7i			0.752565i
$255$	−5.82036e6	−	1.46803e6i	−0.351018	−	0.0885349i
$256$	−1.62663e7			−0.969549
$257$			1.78039e7i			1.04886i	0.851455	+	0.524428i	$0.175721\pi$
$257$			1.78039e7i			1.04886i	−0.851455	+	0.524428i	$0.824279\pi$
$258$	−2.89790e6	+	1.14894e7i	−0.168742	+	0.669020i
$259$	−3.92712e7			−2.26035
$260$		−	9.70170e6i		−	0.551986i
$261$	9.76867e6	−	1.81332e7i	0.549432	−	1.01989i
$262$	−6.28416e6			−0.349417
$263$			1.17717e7i			0.647101i	0.946211	+	0.323550i	$0.104877\pi$
$263$			1.17717e7i			0.647101i	−0.946211	+	0.323550i	$0.895123\pi$
$264$	−7.22255e6	−	1.82169e6i	−0.392535	−	0.0990065i
$265$	2.73336e7			1.46879
$266$		−	8.63843e6i		−	0.458976i
$267$	5.56753e6	−	2.20738e7i	0.292502	−	1.15969i
$268$	−2.19849e6			−0.114214
$269$		−	2.64835e7i		−	1.36056i	−0.732952	−	0.680281i	$-0.761858\pi$
$269$		−	2.64835e7i		−	1.36056i	0.732952	−	0.680281i	$-0.238142\pi$
$270$	−1.30423e7	−	1.19378e7i	−0.662620	−	0.606502i
$271$	2.28992e7			1.15057			0.575285	−	0.817953i	$-0.304891\pi$
$271$	2.28992e7			1.15057			0.575285	+	0.817953i	$0.304891\pi$
$272$		−	217781.i		−	0.0108222i
$273$	−1.51953e7	−	3.83259e6i	−0.746827	−	0.188367i
$274$	−7.66158e6			−0.372449
$275$			1.04881e7i			0.504312i
$276$	−1.74343e6	+	6.91224e6i	−0.0829233	+	0.328770i
$277$	−2.85856e7			−1.34496			−0.672478	−	0.740118i	$-0.734770\pi$
$277$	−2.85856e7			−1.34496			−0.672478	+	0.740118i	$0.734770\pi$
$278$			1.06114e7i			0.493901i
$279$	2.90299e7	+	1.56389e7i	1.33670	+	0.720102i
$280$	4.29019e7			1.95435
$281$			1.86108e7i			0.838776i	0.907807	+	0.419388i	$0.137755\pi$
$281$			1.86108e7i			0.838776i	−0.907807	+	0.419388i	$0.862245\pi$
$282$	1.59507e6	+	402314.i	0.0711267	+	0.0179398i
$283$	1.24389e7			0.548812			0.274406	−	0.961614i	$-0.411519\pi$
$283$	1.24389e7			0.548812			0.274406	+	0.961614i	$0.411519\pi$
$284$			1.83349e7i			0.800428i
$285$	−4.85146e6	+	1.92348e7i	−0.209574	+	0.830908i
$286$	−3.35268e6			−0.143316
$287$		−	4.28274e7i		−	1.81166i
$288$	1.14713e7	−	2.12937e7i	0.480213	−	0.891400i
$289$	−1.41986e6			−0.0588235
$290$			2.53799e7i			1.04063i
$291$	−9.12888e6	−	2.30251e6i	−0.370458	−	0.0934379i
$292$	1.70180e7			0.683535
$293$			3.20615e6i			0.127462i	0.997967	+	0.0637311i	$0.0203000\pi$
$293$			3.20615e6i			0.127462i	−0.997967	+	0.0637311i	$0.979700\pi$
$294$	2.85986e6	−	1.13386e7i	0.112539	−	0.446189i
$295$	6.04064e7			2.35297
$296$		−	4.34966e7i		−	1.67718i
$297$	7.26491e6	−	7.93710e6i	0.277307	−	0.302965i
$298$	−2.43466e7			−0.920005
$299$			8.23931e6i			0.308232i
$300$	−2.05033e7	−	5.17141e6i	−0.759382	−	0.191534i
$301$	4.15328e7			1.52297
$302$		−	1.44328e7i		−	0.523999i
$303$	−8.94122e6	+	3.54497e7i	−0.321417	+	1.27434i
$304$	−719710.			−0.0256175
$305$		−	2.47178e7i		−	0.871185i
$306$	−3.68192e6	−	1.98351e6i	−0.128502	−	0.0692263i
$307$	−2.14579e7			−0.741605			−0.370803	−	0.928712i	$-0.620917\pi$
$307$	−2.14579e7			−0.741605			−0.370803	+	0.928712i	$0.620917\pi$
$308$			1.01675e7i			0.347985i
$309$	−4.75074e7	−	1.19825e7i	−1.61022	−	0.406135i
$310$	−4.06314e7			−1.36388
$311$			4.04195e7i			1.34372i	0.740677	+	0.671861i	$0.234505\pi$
$311$			4.04195e7i			1.34372i	−0.740677	+	0.671861i	$0.765495\pi$
$312$	4.24496e6	−	1.68302e7i	0.139769	−	0.554147i
$313$	−4.26968e7			−1.39239			−0.696197	−	0.717850i	$-0.745126\pi$
$313$	−4.26968e7			−1.39239			−0.696197	+	0.717850i	$0.745126\pi$
$314$			4.84785e6i			0.156589i
$315$	−2.93923e7	+	5.45598e7i	−0.940377	+	1.74558i
$316$	1.19860e7			0.379850
$317$		−	2.01769e7i		−	0.633397i	−0.948526	−	0.316698i	$-0.897426\pi$
$317$		−	2.01769e7i		−	0.633397i	0.948526	−	0.316698i	$-0.102574\pi$
$318$	1.84658e7	+	4.65750e6i	0.574231	+	0.144834i
$319$	−1.54453e7			−0.475801
$320$			2.76210e7i			0.842927i
$321$	−1.07073e6	+	4.24516e6i	−0.0323716	+	0.128345i
$322$	−1.41889e7			−0.424993
$323$			4.69226e6i			0.139243i
$324$	1.19342e7	+	1.81158e7i	0.350879	+	0.532626i
$325$	−2.44397e7			−0.711944
$326$		−	8.60717e6i		−	0.248432i
$327$	−3.32117e6	−	837676.i	−0.0949835	−	0.0239570i
$328$	4.74354e7			1.34425
$329$		−	5.76597e6i		−	0.161914i
$330$	−3.24256e6	+	1.28559e7i	−0.0902289	+	0.357734i
$331$	−1.73310e6			−0.0477902			−0.0238951	−	0.999714i	$-0.507607\pi$
$331$	−1.73310e6			−0.0477902			−0.0238951	+	0.999714i	$0.507607\pi$
$332$		−	3.88607e7i		−	1.06193i
$333$	5.53160e7	+	2.97997e7i	1.49802	+	0.807010i
$334$	2.06842e7			0.555135
$335$			1.00486e7i			0.267284i
$336$	−2.18016e6	−	549887.i	−0.0574739	−	0.0144963i
$337$	1.18743e6			0.0310253			0.0155127	−	0.999880i	$-0.495062\pi$
$337$	1.18743e6			0.0310253			0.0155127	+	0.999880i	$0.495062\pi$
$338$			1.54264e7i			0.399498i
$339$	−1.08404e7	+	4.29795e7i	−0.278258	+	1.10322i
$340$	−9.07513e6			−0.230896
$341$		−	2.47268e7i		−	0.623598i
$342$	−6.55500e6	+	1.21678e7i	−0.163868	+	0.304182i
$343$	1.26177e7			0.312677
$344$			4.60015e7i			1.13005i
$345$	3.15938e7	+	7.96868e6i	0.769386	+	0.194057i
$346$	−2.40243e7			−0.579993
$347$			5.18585e7i			1.24117i	0.784139	+	0.620585i	$0.213105\pi$
$347$			5.18585e7i			1.24117i	−0.784139	+	0.620585i	$0.786895\pi$
$348$	7.61567e6	−	3.01942e7i	0.180705	−	0.716450i
$349$	3.89353e7			0.915941			0.457970	−	0.888967i	$-0.348577\pi$
$349$	3.89353e7			0.915941			0.457970	+	0.888967i	$0.348577\pi$
$350$		−	4.20876e7i		−	0.981635i
$351$	1.84952e7	+	1.69289e7i	0.427700	+	0.391478i
$352$	−1.81373e7			−0.415858
$353$			9.88439e6i			0.224712i	0.993668	+	0.112356i	$0.0358397\pi$
$353$			9.88439e6i			0.224712i	−0.993668	+	0.112356i	$0.964160\pi$
$354$	4.08087e7	+	1.02929e7i	0.919906	+	0.232021i
$355$	8.38031e7			1.87316
$356$		−	3.44176e7i		−	0.762835i
$357$	−3.58507e6	+	1.42139e7i	−0.0787940	+	0.312398i
$358$	−3.34521e7			−0.729078
$359$			6.05198e7i			1.30802i	0.756486	+	0.654010i	$0.226915\pi$
$359$			6.05198e7i			1.30802i	−0.756486	+	0.654010i	$0.773085\pi$
$360$	−6.04301e7	−	3.25547e7i	−1.29523	−	0.697761i
$361$	−3.15392e7			−0.670392
$362$		−	3.07766e7i		−	0.648776i
$363$	3.85560e7	+	9.72471e6i	0.806069	+	0.203309i
$364$	−2.36925e7			−0.491255
$365$		−	7.77843e7i		−	1.59961i
$366$	4.21177e6	−	1.66986e7i	0.0859057	−	0.340594i
$367$	6.38169e7			1.29103			0.645517	−	0.763746i	$-0.276642\pi$
$367$	6.38169e7			1.29103			0.645517	+	0.763746i	$0.276642\pi$
$368$			1.18215e6i			0.0237207i
$369$	−3.24982e7	+	6.03252e7i	−0.646816	+	1.20066i
$370$	−7.74225e7			−1.52849
$371$		−	6.67514e7i		−	1.30719i
$372$	4.83386e7	+	1.21921e7i	0.939000	+	0.236837i
$373$	−2.92176e7			−0.563012			−0.281506	−	0.959560i	$-0.590834\pi$
$373$	−2.92176e7			−0.563012			−0.281506	+	0.959560i	$0.590834\pi$
$374$			3.13615e6i			0.0599490i
$375$	−4.38686e6	+	1.73928e7i	−0.0831879	+	0.329819i
$376$	6.38636e6			0.120141
$377$		−	3.59911e7i		−	0.671693i
$378$	−2.91532e7	+	3.18507e7i	−0.539773	+	0.589716i
$379$	−2.53543e7			−0.465731			−0.232865	−	0.972509i	$-0.574810\pi$
$379$	−2.53543e7			−0.465731			−0.232865	+	0.972509i	$0.574810\pi$
$380$			2.99910e7i			0.546562i
$381$	−6.70595e7	−	1.69140e7i	−1.21251	−	0.305823i
$382$	5.12639e7			0.919648
$383$		−	5.56098e7i		−	0.989818i	−0.868945	−	0.494909i	$-0.835201\pi$
$383$		−	5.56098e7i		−	0.989818i	0.868945	−	0.494909i	$-0.164799\pi$
$384$	9.31489e6	−	3.69311e7i	0.164507	−	0.652228i
$385$	4.64724e7			0.814354
$386$		−	2.20115e7i		−	0.382726i
$387$	−5.85016e7	−	3.15158e7i	−1.00933	−	0.543746i
$388$	−1.42338e7			−0.243683
$389$			6.46420e7i			1.09816i	0.835769	+	0.549080i	$0.185022\pi$
$389$			6.46420e7i			1.09816i	−0.835769	+	0.549080i	$0.814978\pi$
$390$	−2.99572e7	−	7.55589e6i	−0.505018	−	0.127377i
$391$	7.70719e6			0.128933
$392$		−	4.53977e7i		−	0.753660i
$393$	8.61882e6	−	3.41714e7i	0.141994	−	0.562970i
$394$	1.35262e7			0.221150
$395$		−	5.47843e7i		−	0.888925i
$396$	7.71525e6	−	1.43215e7i	0.124241	−	0.230624i
$397$	1.30380e7			0.208372			0.104186	−	0.994558i	$-0.466776\pi$
$397$	1.30380e7			0.208372			0.104186	+	0.994558i	$0.466776\pi$
$398$			5.15674e7i			0.817948i
$399$	4.69732e7	+	1.18477e7i	0.739489	+	0.186516i
$400$	−3.50652e6			−0.0547894
$401$			5.50337e7i			0.853484i	0.904373	+	0.426742i	$0.140339\pi$
$401$			5.50337e7i			0.853484i	−0.904373	+	0.426742i	$0.859661\pi$
$402$	−1.71223e6	+	6.78857e6i	−0.0263563	+	0.104496i
$403$	5.76191e7			0.880341
$404$			5.52732e7i			0.838245i
$405$	8.28019e7	−	5.45475e7i	1.24645	−	0.821126i
$406$	6.19803e7			0.926137
$407$		−	4.71165e7i		−	0.698860i
$408$	−1.57432e7	−	3.97081e6i	−0.231800	−	0.0584653i
$409$	−6.18847e7			−0.904509			−0.452255	−	0.891889i	$-0.649380\pi$
$409$	−6.18847e7			−0.904509			−0.452255	+	0.891889i	$0.649380\pi$
$410$		−	8.44335e7i		−	1.22508i
$411$	1.05080e7	−	4.16614e7i	0.151354	−	0.600079i
$412$	−7.40738e7			−1.05919
$413$		−	1.47518e8i		−	2.09409i
$414$	1.99860e7	+	1.07668e7i	0.281660	+	0.151735i
$415$	−1.77621e8			−2.48513
$416$		−	4.22640e7i		−	0.587072i
$417$	−5.77019e7	−	1.45537e7i	−0.795759	−	0.200709i
$418$	1.03642e7			0.141908
$419$			3.97307e7i			0.540113i	0.962845	+	0.270056i	$0.0870423\pi$
$419$			3.97307e7i			0.540113i	−0.962845	+	0.270056i	$0.912958\pi$
$420$	−2.29143e7	+	9.08493e7i	−0.309285	+	1.22624i
$421$	−2.64517e7			−0.354492			−0.177246	−	0.984167i	$-0.556719\pi$
$421$	−2.64517e7			−0.354492			−0.177246	+	0.984167i	$0.556719\pi$
$422$		−	187296.i		−	0.00249225i
$423$	−4.37532e6	+	8.12174e6i	−0.0578081	+	0.107307i
$424$	7.39335e7			0.969937
$425$			2.28613e7i			0.297807i
$426$	5.66148e7	+	1.42796e7i	0.732321	+	0.184708i
$427$	−6.03633e7			−0.775335
$428$			6.61908e6i			0.0844240i
$429$	4.59824e6	−	1.82309e7i	0.0582398	−	0.230906i
$430$	8.18811e7			1.02986
$431$			2.17717e7i			0.271932i	0.990714	+	0.135966i	$0.0434137\pi$
$431$			2.17717e7i			0.271932i	−0.990714	+	0.135966i	$0.956586\pi$
$432$	2.65363e6	+	2.42890e6i	0.0329147	+	0.0301271i
$433$	−1.24511e8			−1.53371			−0.766855	−	0.641820i	$-0.778180\pi$
$433$	−1.24511e8			−1.53371			−0.766855	+	0.641820i	$0.778180\pi$
$434$			9.92258e7i			1.21382i
$435$	−1.38009e8	−	3.48089e7i	−1.67663	−	0.422886i
$436$	−5.17838e6			−0.0624791
$437$		−	2.54703e7i		−	0.305203i
$438$	1.32540e7	−	5.25488e7i	0.157734	−	0.625374i
$439$	1.16388e8			1.37567			0.687834	−	0.725868i	$-0.258562\pi$
$439$	1.16388e8			1.37567			0.687834	+	0.725868i	$0.258562\pi$
$440$			5.14726e7i			0.604252i
$441$	5.77337e7	+	3.11022e7i	0.673153	+	0.362639i
$442$	−7.30794e6			−0.0846308
$443$			3.48643e7i			0.401024i	0.979691	+	0.200512i	$0.0642605\pi$
$443$			3.48643e7i			0.401024i	−0.979691	+	0.200512i	$0.935739\pi$
$444$	9.21085e7	+	2.32319e7i	1.05233	+	0.265421i
$445$	−1.57312e8			−1.78518
$446$			7.28141e7i			0.820750i
$447$	3.33918e7	−	1.32390e8i	0.373867	−	1.48229i
$448$	6.74533e7			0.750186
$449$			8.56415e7i			0.946118i	0.881031	+	0.473059i	$0.156850\pi$
$449$			8.56415e7i			0.946118i	−0.881031	+	0.473059i	$0.843150\pi$
$450$	−3.19368e7	+	5.92831e7i	−0.350473	+	0.650569i
$451$	5.13832e7			0.560134
$452$			6.70138e7i			0.725687i
$453$	7.84814e7	+	1.97948e7i	0.844252	+	0.212940i
$454$	−8.59407e7			−0.918399
$455$			1.08291e8i			1.14963i
$456$	−1.31225e7	+	5.20273e7i	−0.138395	+	0.548702i
$457$	2.62710e7			0.275251			0.137625	−	0.990484i	$-0.456053\pi$
$457$	2.62710e7			0.275251			0.137625	+	0.990484i	$0.456053\pi$
$458$		−	8.77603e7i		−	0.913486i
$459$	1.58356e7	−	1.73008e7i	0.163755	−	0.178907i
$460$	4.92611e7			0.506094
$461$			1.16159e8i			1.18563i	0.805338	+	0.592816i	$0.201984\pi$
$461$			1.16159e8i			1.18563i	−0.805338	+	0.592816i	$0.798016\pi$
$462$	3.13954e7	+	7.91863e6i	0.318375	+	0.0803016i
$463$	−8.82682e7			−0.889327			−0.444663	−	0.895698i	$-0.646677\pi$
$463$	−8.82682e7			−0.889327			−0.444663	+	0.895698i	$0.646677\pi$
$464$		−	5.16388e6i		−	0.0516919i
$465$	5.57265e7	−	2.20941e8i	0.554247	−	2.19745i
$466$	−5.61754e7			−0.555122
$467$		−	5.93568e7i		−	0.582800i	−0.956601	−	0.291400i	$-0.905879\pi$
$467$		−	5.93568e7i		−	0.582800i	0.956601	−	0.291400i	$-0.0941211\pi$
$468$	3.33724e7	+	1.79783e7i	0.325574	+	0.175392i
$469$	2.45398e7			0.237877
$470$		−	1.13675e7i		−	0.109489i
$471$	−2.63612e7	−	6.64890e6i	−0.252291	−	0.0636337i
$472$	1.63390e8			1.55382
$473$			4.98299e7i			0.470876i
$474$	9.33494e6	−	3.70107e7i	0.0876550	−	0.347530i
$475$	7.55507e7			0.704949
$476$			2.21624e7i			0.205492i
$477$	−5.06522e7	+	9.40236e7i	−0.466706	+	0.866327i
$478$	3.70490e7			0.339229
$479$			1.40261e8i			1.27624i	0.769938	+	0.638119i	$0.220287\pi$
$479$			1.40261e8i			1.27624i	−0.769938	+	0.638119i	$0.779713\pi$
$480$	−1.62062e8	−	4.08758e7i	−1.46541	−	0.369609i
$481$	1.09792e8			0.986589
$482$			7.49762e7i			0.669549i
$483$	1.94603e7	−	7.71551e7i	0.172706	−	0.684736i
$484$	6.01166e7			0.530223
$485$			6.50583e7i			0.570266i
$486$	6.52330e7	−	2.27417e7i	0.568275	−	0.198114i
$487$	−3.05361e7			−0.264379			−0.132189	−	0.991224i	$-0.542201\pi$
$487$	−3.05361e7			−0.264379			−0.132189	+	0.991224i	$0.542201\pi$
$488$		−	6.68580e7i		−	0.575300i
$489$	4.68032e7	+	1.18049e7i	0.400267	+	0.100956i
$490$	−8.08064e7			−0.686843
$491$		−	1.19081e8i		−	1.00600i	−0.864286	−	0.503001i	$-0.832229\pi$
$491$		−	1.19081e8i		−	1.00600i	0.864286	−	0.503001i	$-0.167771\pi$
$492$	−2.53357e7	+	1.00450e8i	−0.212734	+	0.843437i
$493$	−3.36667e7			−0.280970
$494$			2.41508e7i			0.200333i
$495$	−6.54594e7	−	3.52641e7i	−0.539705	−	0.290748i
$496$	8.26699e6			0.0677489
$497$		−	2.04655e8i		−	1.66707i
$498$	−1.19995e8	−	3.02656e7i	−0.971574	−	0.245053i
$499$	1.70501e8			1.37222			0.686112	−	0.727496i	$-0.259316\pi$
$499$	1.70501e8			1.37222			0.686112	+	0.727496i	$0.259316\pi$
$500$			2.71189e7i			0.216951i
$501$	−2.83686e7	+	1.12474e8i	−0.225593	+	0.894417i
$502$	2.80017e7			0.221347
$503$			1.26740e8i			0.995884i	0.867210	+	0.497942i	$0.165911\pi$
$503$			1.26740e8i			0.995884i	−0.867210	+	0.497942i	$0.834089\pi$
$504$	−7.95018e7	+	1.47576e8i	−0.620991	+	1.15272i
$505$	2.52637e8			1.96166
$506$		−	1.70235e7i		−	0.131400i
$507$	−8.38843e7	−	2.11575e7i	−0.643660	−	0.162346i
$508$	−1.04559e8			−0.797576
$509$		−	1.05768e8i		−	0.802051i	−0.916067	−	0.401025i	$-0.868654\pi$
$509$		−	1.05768e8i		−	0.802051i	0.916067	−	0.401025i	$-0.131346\pi$
$510$	−7.06791e6	+	2.80224e7i	−0.0532820	+	0.211249i
$511$	−1.89957e8			−1.42361
$512$		−	1.19670e7i		−	0.0891609i
$513$	−5.71745e7	−	5.23324e7i	−0.423497	−	0.387631i
$514$	8.57178e7			0.631222
$515$			3.38569e8i			2.47871i
$516$	−9.74130e7	−	2.45698e7i	−0.709035	−	0.178835i
$517$	6.91786e6			0.0500611
$518$			1.89073e8i			1.36032i
$519$	3.29497e7	−	1.30637e8i	0.235695	−	0.934469i
$520$	−1.19943e8			−0.853030
$521$		−	2.34980e8i		−	1.66157i	−0.556596	−	0.830783i	$-0.687893\pi$
$521$		−	2.34980e8i		−	1.66157i	0.556596	−	0.830783i	$-0.312107\pi$
$522$	−8.73032e7	−	4.70317e7i	−0.613788	−	0.330658i
$523$	−1.43654e7			−0.100418			−0.0502090	−	0.998739i	$-0.515989\pi$
$523$	−1.43654e7			−0.100418			−0.0502090	+	0.998739i	$0.515989\pi$
$524$		−	5.32802e7i		−	0.370315i
$525$	2.28860e8	+	5.77237e7i	1.58158	+	0.398911i
$526$	5.66755e7			0.389438
$527$		−	5.38979e7i		−	0.368247i
$528$	659740.	−	2.61570e6i	0.00448199	−	0.0177700i
$529$	1.06200e8			0.717395
$530$		−	1.31599e8i		−	0.883946i
$531$	−1.11939e8	+	2.07789e8i	−0.747652	+	1.38784i
$532$	7.32408e7			0.486428
$533$			1.19735e8i			0.790748i
$534$	−1.06276e8	−	2.68051e7i	−0.697926	−	0.176033i
$535$	3.02538e7			0.197569
$536$			2.71801e7i			0.176505i
$537$	4.58800e7	−	1.81902e8i	0.296279	−	1.17467i
$538$	−1.27506e8			−0.818812
$539$		−	4.91759e7i		−	0.314041i
$540$	1.01214e8	−	1.10579e8i	0.642778	−	0.702251i
$541$	−1.60232e8			−1.01195			−0.505973	−	0.862550i	$-0.668866\pi$
$541$	−1.60232e8			−1.01195			−0.505973	+	0.862550i	$0.668866\pi$
$542$		−	1.10249e8i		−	0.692434i
$543$	1.67354e8	+	4.22105e7i	1.04529	+	0.263646i
$544$	−3.95345e7			−0.245572
$545$			2.36688e7i			0.146213i
$546$	−1.84522e7	+	7.31583e7i	−0.113363	+	0.449455i
$547$	−1.89807e8			−1.15971			−0.579856	−	0.814719i	$-0.696891\pi$
$547$	−1.89807e8			−1.15971			−0.579856	+	0.814719i	$0.696891\pi$
$548$		−	6.49586e7i		−	0.394725i
$549$	8.50256e7	+	4.58047e7i	0.513845	+	0.276817i
$550$	5.04956e7			0.303505
$551$			1.11260e8i			0.665094i
$552$	8.54565e7	+	2.15541e7i	0.508075	+	0.128148i
$553$	−1.33789e8			−0.791123
$554$			1.37627e8i			0.809419i
$555$	1.06186e8	−	4.21000e8i	0.621138	−	2.46265i
$556$	−8.99690e7			−0.523442
$557$		−	1.48324e8i		−	0.858313i	−0.903230	−	0.429157i	$-0.858811\pi$
$557$		−	1.48324e8i		−	0.858313i	0.903230	−	0.429157i	$-0.141189\pi$
$558$	7.52943e7	−	1.39766e8i	0.433371	−	0.804449i
$559$	−1.16115e8			−0.664742
$560$			1.55373e7i			0.0884729i
$561$	−1.70535e7	−	4.30128e6i	−0.0965882	−	0.0243618i
$562$	8.96026e7			0.504791
$563$			8.63238e7i			0.483733i	0.970310	+	0.241866i	$0.0777596\pi$
$563$			8.63238e7i			0.483733i	−0.970310	+	0.241866i	$0.922240\pi$
$564$	−3.41101e6	+	1.35238e7i	−0.0190128	+	0.0753808i
$565$	3.06300e8			1.69825
$566$		−	5.98879e7i		−	0.330286i
$567$	−1.33210e8	−	2.02210e8i	−0.730784	−	1.10931i
$568$	2.26675e8			1.23697
$569$			1.57880e7i			0.0857018i	0.999081	+	0.0428509i	$0.0136440\pi$
$569$			1.57880e7i			0.0857018i	−0.999081	+	0.0428509i	$0.986356\pi$
$570$	9.26069e7	+	2.33576e7i	0.500056	+	0.126126i
$571$	−7.29151e7			−0.391660			−0.195830	−	0.980638i	$-0.562740\pi$
$571$	−7.29151e7			−0.391660			−0.195830	+	0.980638i	$0.562740\pi$
$572$		−	2.84256e7i		−	0.151887i
$573$	−7.03092e7	+	2.78758e8i	−0.373721	+	1.48171i
$574$	−2.06195e8			−1.09029
$575$		−	1.24094e8i		−	0.652753i
$576$	−9.50123e7	−	5.11848e7i	−0.497178	−	0.267839i
$577$	−2.95505e8			−1.53829			−0.769143	−	0.639077i	$-0.779317\pi$
$577$	−2.95505e8			−1.53829			−0.769143	+	0.639077i	$0.779317\pi$
$578$			6.83597e6i			0.0354011i
$579$	1.19692e8	+	3.01891e7i	0.616638	+	0.155530i
$580$	−2.15183e8			−1.10287
$581$			4.33767e8i			2.21171i
$582$	−1.10856e7	+	4.39514e7i	−0.0562327	+	0.222948i
$583$	8.00866e7			0.404161
$584$		−	2.10395e8i		−	1.05632i
$585$	8.21734e7	−	1.52535e8i	0.410453	−	0.761908i
$586$	1.54362e7			0.0767091
$587$		−	2.81473e8i		−	1.39163i	−0.718223	−	0.695813i	$-0.755044\pi$
$587$		−	2.81473e8i		−	1.39163i	0.718223	−	0.695813i	$-0.244956\pi$
$588$	9.61344e7	+	2.42473e7i	0.472875	+	0.119270i
$589$	−1.78118e8			−0.871692
$590$		−	2.90830e8i		−	1.41606i
$591$	−1.85513e7	+	7.35513e7i	−0.0898695	+	0.356310i
$592$	1.57526e7			0.0759255
$593$		−	1.54424e8i		−	0.740542i	−0.928924	−	0.370271i	$-0.879265\pi$
$593$		−	1.54424e8i		−	0.740542i	0.928924	−	0.370271i	$-0.120735\pi$
$594$	−3.82136e7	−	3.49773e7i	−0.182330	−	0.166889i
$595$	1.01297e8			0.480892
$596$		−	2.06423e8i		−	0.975032i
$597$	−2.80408e8	−	7.07254e7i	−1.31785	−	0.332393i
$598$	3.96686e7			0.185500
$599$			4.40428e7i			0.204925i	0.994737	+	0.102462i	$0.0326722\pi$
$599$			4.40428e7i			0.204925i	−0.994737	+	0.102462i	$0.967328\pi$
$600$	−6.39345e7	+	2.53484e8i	−0.295993	+	1.17354i
$601$	6.29814e6			0.0290127			0.0145064	−	0.999895i	$-0.495382\pi$
$601$	6.29814e6			0.0290127			0.0145064	+	0.999895i	$0.495382\pi$
$602$		−	1.99962e8i		−	0.916553i
$603$	−3.45659e7	−	1.86212e7i	−0.157650	−	0.0849291i
$604$	1.22368e8			0.555340
$605$		−	2.74775e8i		−	1.24083i
$606$	1.70674e8	+	4.30480e7i	0.766920	+	0.193435i
$607$	1.85596e8			0.829857			0.414929	−	0.909854i	$-0.363807\pi$
$607$	1.85596e8			0.829857			0.414929	+	0.909854i	$0.363807\pi$
$608$			1.30651e8i			0.581303i
$609$	−8.50068e7	+	3.37030e8i	−0.376359	+	1.49217i
$610$	−1.19005e8			−0.524296
$611$			1.61202e7i			0.0706718i
$612$	1.68172e7	−	3.12171e7i	0.0733668	−	0.136188i
$613$	1.98985e8			0.863849			0.431925	−	0.901910i	$-0.357835\pi$
$613$	1.98985e8			0.863849			0.431925	+	0.901910i	$0.357835\pi$
$614$			1.03310e8i			0.446312i
$615$	4.59124e8	+	1.15802e8i	1.97381	+	0.497840i
$616$	1.25701e8			0.537770
$617$			2.78856e7i			0.118720i	0.998237	+	0.0593602i	$0.0189060\pi$
$617$			2.78856e7i			0.118720i	−0.998237	+	0.0593602i	$0.981094\pi$
$618$	−5.76902e7	+	2.28727e8i	−0.244420	+	0.969063i
$619$	−5.67493e7			−0.239270			−0.119635	−	0.992818i	$-0.538172\pi$
$619$	−5.67493e7			−0.239270			−0.119635	+	0.992818i	$0.538172\pi$
$620$		−	3.44493e8i		−	1.44546i
$621$	−8.59577e7	+	9.39111e7i	−0.358930	+	0.392141i
$622$	1.94602e8			0.808677
$623$			3.84172e8i			1.58877i
$624$	6.09518e6	+	1.53734e6i	0.0250861	+	0.00632728i
$625$	−1.75825e8			−0.720179
$626$			2.05566e8i			0.837970i
$627$	−1.42146e7	+	5.63572e7i	−0.0576676	+	0.228637i
$628$	−4.11025e7			−0.165954
$629$		−	1.02702e8i		−	0.412691i
$630$	2.62681e8	+	1.41511e8i	1.05053	+	0.565936i
$631$	2.56661e8			1.02158			0.510789	−	0.859706i	$-0.329353\pi$
$631$	2.56661e8			1.02158			0.510789	+	0.859706i	$0.329353\pi$
$632$		−	1.48184e8i		−	0.587015i
$633$	1.01846e6	+	256879.i	0.00401544	+	0.00101279i
$634$	−9.71425e7			−0.381190
$635$			4.77910e8i			1.86649i
$636$	−3.94885e7	+	1.56562e8i	−0.153497	+	0.608576i
$637$	1.14591e8			0.443335
$638$			7.43623e7i			0.286346i
$639$	−1.55296e8	+	2.88270e8i	−0.595193	+	1.10483i
$640$	−2.63195e8			−1.00401
$641$			2.88737e8i			1.09630i	0.836382	+	0.548148i	$0.184667\pi$
$641$			2.88737e8i			1.09630i	−0.836382	+	0.548148i	$0.815333\pi$
$642$	2.04386e7	+	5.15508e6i	0.0772405	+	0.0194818i
$643$	2.59604e8			0.976515			0.488257	−	0.872700i	$-0.337633\pi$
$643$	2.59604e8			0.976515			0.488257	+	0.872700i	$0.337633\pi$
$644$		−	1.20300e8i		−	0.450412i
$645$	−1.12301e8	+	4.45245e8i	−0.418509	+	1.65928i
$646$	2.25911e7			0.0837992
$647$		−	7.59613e7i		−	0.280465i	−0.990119	−	0.140233i	$-0.955215\pi$
$647$		−	7.59613e7i		−	0.280465i	0.990119	−	0.140233i	$-0.0447851\pi$
$648$	2.23967e8	−	1.47543e8i	0.823112	−	0.542243i
$649$	1.76988e8			0.647457
$650$			1.17666e8i			0.428461i
$651$	−5.39560e8	−	1.36090e8i	−1.95568	−	0.493267i
$652$	7.29758e7			0.263291
$653$		−	2.19060e8i		−	0.786726i	−0.919383	−	0.393363i	$-0.871312\pi$
$653$		−	2.19060e8i		−	0.786726i	0.919383	−	0.393363i	$-0.128688\pi$
$654$	−4.03303e6	+	1.59900e7i	−0.0144178	+	0.0571629i
$655$	−2.43528e8			−0.866611
$656$			1.71791e7i			0.0608540i
$657$	2.67566e8	+	1.44143e8i	0.943486	+	0.508272i
$658$	−2.77606e7			−0.0974430
$659$			2.11942e8i			0.740562i	0.928920	+	0.370281i	$0.120739\pi$
$659$			2.11942e8i			0.740562i	−0.928920	+	0.370281i	$0.879261\pi$
$660$	−1.08999e8	−	2.74920e7i	−0.379131	−	0.0956255i
$661$	−2.69285e8			−0.932412			−0.466206	−	0.884676i	$-0.654379\pi$
$661$	−2.69285e8			−0.932412			−0.466206	+	0.884676i	$0.654379\pi$
$662$			8.34408e6i			0.0287610i
$663$	1.00229e7	−	3.97384e7i	0.0343918	−	0.136355i
$664$	−4.80438e8			−1.64109
$665$		−	3.34762e8i		−	1.13834i
$666$	1.43472e8	−	2.66322e8i	0.485674	−	0.901538i
$667$	1.82748e8			0.615849
$668$			1.75370e8i			0.588338i
$669$	−3.95941e8	−	9.98656e7i	−1.32237	−	0.333532i
$670$	4.83797e7			0.160857
$671$		−	7.24223e7i		−	0.239720i
$672$	−9.98228e7	+	3.95772e8i	−0.328944	+	1.30418i
$673$	5.63865e7			0.184982			0.0924912	−	0.995714i	$-0.470517\pi$
$673$	5.63865e7			0.184982			0.0924912	+	0.995714i	$0.470517\pi$
$674$		−	5.71692e6i		−	0.0186716i
$675$	−2.78562e8	−	2.54971e8i	−0.905754	−	0.829046i
$676$	−1.30793e8			−0.423393
$677$		−	3.19306e8i		−	1.02906i	−0.857472	−	0.514530i	$-0.827966\pi$
$677$		−	3.19306e8i		−	1.02906i	0.857472	−	0.514530i	$-0.172034\pi$
$678$	2.06927e8	+	5.21918e7i	0.663939	+	0.167461i
$679$	1.58879e8			0.507524
$680$			1.12197e8i			0.356823i
$681$	1.17869e8	−	4.67320e8i	0.373214	−	1.47970i
$682$	−1.19048e8			−0.375293
$683$		−	2.74590e7i		−	0.0861833i	−0.999071	−	0.0430916i	$-0.986279\pi$
$683$		−	2.74590e7i		−	0.0861833i	0.999071	−	0.0430916i	$-0.0137207\pi$
$684$	−1.03164e8	−	5.55765e7i	−0.322375	−	0.173669i
$685$	−2.96906e8			−0.923735
$686$		−	6.07484e7i		−	0.188175i
$687$	4.77215e8	+	1.20365e8i	1.47178	+	0.371217i
$688$	−1.66598e7			−0.0511569
$689$			1.86620e8i			0.570559i
$690$	3.83656e7	−	1.52110e8i	0.116787	−	0.463031i
$691$	2.47550e8			0.750289			0.375145	−	0.926966i	$-0.377593\pi$
$691$	2.47550e8			0.750289			0.375145	+	0.926966i	$0.377593\pi$
$692$		−	2.03690e8i		−	0.614683i
$693$	−8.61184e7	+	1.59858e8i	−0.258759	+	0.480325i
$694$	2.49675e8			0.746959
$695$			4.11221e8i			1.22496i
$696$	−3.73293e8	−	9.41531e7i	−1.10719	−	0.279259i
$697$	1.12002e8			0.330770
$698$		−	1.87456e8i		−	0.551230i
$699$	7.70453e7	−	3.05465e8i	0.225587	−	0.894396i
$700$	3.56839e8			1.04035
$701$			3.39193e8i			0.984674i	0.870405	+	0.492337i	$0.163857\pi$
$701$			3.39193e8i			0.984674i	−0.870405	+	0.492337i	$0.836143\pi$
$702$	8.15050e7	−	8.90463e7i	0.235599	−	0.257398i
$703$	−3.39402e8			−0.976896
$704$			8.09287e7i			0.231945i
$705$	6.18131e7	+	1.55907e7i	0.176406	+	0.0444937i
$706$	4.75889e7			0.135236
$707$		−	6.16965e8i		−	1.74583i
$708$	−8.72682e7	+	3.45996e8i	−0.245899	+	0.974926i
$709$	2.51182e8			0.704774			0.352387	−	0.935854i	$-0.385370\pi$
$709$	2.51182e8			0.704774			0.352387	+	0.935854i	$0.385370\pi$
$710$		−	4.03474e8i		−	1.12730i
$711$	1.88450e8	+	1.01521e8i	0.524309	+	0.282454i
$712$	−4.25507e8			−1.17887
$713$			2.92565e8i			0.807150i
$714$	6.84335e7	+	1.72605e7i	0.188007	+	0.0474197i
$715$	−1.29925e8			−0.355447
$716$		−	2.83623e8i		−	0.772685i
$717$	−5.08133e7	+	2.01462e8i	−0.137854	+	0.546556i
$718$	2.91376e8			0.787191
$719$		−	2.17110e8i		−	0.584108i	−0.956402	−	0.292054i	$-0.905661\pi$
$719$		−	2.17110e8i		−	0.584108i	0.956402	−	0.292054i	$-0.0943388\pi$
$720$	1.17899e7	−	2.18852e7i	0.0315874	−	0.0586345i
$721$	8.26818e8			2.20599
$722$			1.51847e8i			0.403455i
$723$	−4.07698e8	−	1.02831e8i	−1.07876	−	0.272088i
$724$	2.60939e8			0.687579
$725$			5.42072e8i			1.42247i
$726$	4.68201e7	−	1.85630e8i	0.122355	−	0.485107i
$727$	2.39617e8			0.623611			0.311805	−	0.950146i	$-0.399066\pi$
$727$	2.39617e8			0.623611			0.311805	+	0.950146i	$0.399066\pi$
$728$			2.92912e8i			0.759177i
$729$	3.41946e7	+	3.85908e8i	0.0882622	+	0.996097i
$730$	−3.74497e8			−0.962674
$731$			1.08616e8i			0.278062i
$732$	1.41579e8	+	3.57095e7i	0.360965	+	0.0910437i
$733$	−3.31931e8			−0.842821			−0.421410	−	0.906870i	$-0.638465\pi$
$733$	−3.31931e8			−0.842821			−0.421410	+	0.906870i	$0.638465\pi$
$734$		−	3.07250e8i		−	0.776968i
$735$	1.10827e8	−	4.39401e8i	0.279116	−	1.10662i
$736$	2.14599e8			0.538263
$737$			2.94422e7i			0.0735474i
$738$	2.90439e8	+	1.56464e8i	0.722579	+	0.389266i
$739$	−4.51759e8			−1.11937			−0.559685	−	0.828705i	$-0.689078\pi$
$739$	−4.51759e8			−1.11937			−0.559685	+	0.828705i	$0.689078\pi$
$740$		−	6.56425e8i		−	1.61991i
$741$	−1.31325e8	−	3.31232e7i	−0.322770	−	0.0814101i
$742$	−3.21378e8			−0.786692
$743$		−	6.09560e7i		−	0.148611i	−0.997236	−	0.0743053i	$-0.976326\pi$
$743$		−	6.09560e7i		−	0.148611i	0.997236	−	0.0743053i	$-0.0236739\pi$
$744$	1.50732e8	−	5.97614e8i	0.366005	−	1.45112i
$745$	−9.43496e8			−2.28177
$746$			1.40669e8i			0.338831i
$747$	3.29150e8	−	6.10989e8i	0.789646	−	1.46579i
$748$	−2.65898e7			−0.0635346
$749$		−	7.38827e7i		−	0.175832i
$750$	8.37385e7	+	2.11208e7i	0.198491	+	0.0500640i
$751$	2.99949e8			0.708155			0.354077	−	0.935216i	$-0.384795\pi$
$751$	2.99949e8			0.708155			0.354077	+	0.935216i	$0.384795\pi$
$752$			2.31287e6i			0.00543873i
$753$	−3.84048e7	+	1.52265e8i	−0.0899499	+	0.356628i
$754$	−1.73281e8			−0.404238
$755$		−	5.59309e8i		−	1.29960i
$756$	−2.70045e8	−	2.47175e8i	−0.624988	−	0.572057i
$757$	5.14554e8			1.18616			0.593080	−	0.805143i	$-0.297912\pi$
$757$	5.14554e8			1.18616			0.593080	+	0.805143i	$0.297912\pi$
$758$			1.22070e8i			0.280285i
$759$	9.25686e7	+	2.33479e7i	0.211709	+	0.0533978i
$760$	3.70780e8			0.844648
$761$			1.01850e8i			0.231105i	0.993301	+	0.115552i	$0.0368638\pi$
$761$			1.01850e8i			0.231105i	−0.993301	+	0.115552i	$0.963136\pi$
$762$	−8.14331e7	+	3.22862e8i	−0.184050	+	0.729712i
$763$	5.78016e7			0.130127
$764$			4.34640e8i			0.974653i
$765$	−1.42684e8	−	7.68663e7i	−0.318706	−	0.171693i
$766$	−2.67736e8			−0.595691
$767$			4.12423e8i			0.914023i
$768$	−4.25854e8	−	1.07410e8i	−0.940107	−	0.237117i
$769$	8.67940e8			1.90858			0.954290	−	0.298881i	$-0.0966135\pi$
$769$	8.67940e8			1.90858			0.954290	+	0.298881i	$0.0966135\pi$
$770$		−	2.23744e8i		−	0.490093i
$771$	−1.17563e8	+	4.66108e8i	−0.256512	+	1.01701i
$772$	1.86624e8			0.405617
$773$			3.29837e8i			0.714104i	0.934085	+	0.357052i	$0.116218\pi$
$773$			3.29837e8i			0.714104i	−0.934085	+	0.357052i	$0.883782\pi$
$774$	−1.51735e8	+	2.81659e8i	−0.327236	+	0.607436i
$775$	−8.67817e8			−1.86433
$776$			1.75973e8i			0.376584i
$777$	−1.02812e9	−	2.59317e8i	−2.19171	−	0.552799i
$778$	3.11222e8			0.660894
$779$		−	3.70137e8i		−	0.782978i
$780$	6.40625e7	−	2.53991e8i	0.134996	−	0.535224i
$781$	2.45540e8			0.515429
$782$		−	3.71067e7i		−	0.0775946i
$783$	3.75482e8	−	4.10224e8i	0.782175	−	0.854547i
$784$	1.64411e7			0.0341180
$785$			1.87867e8i			0.388366i
$786$	−1.64520e8	−	4.14957e7i	−0.338806	−	0.0854547i
$787$	−9.09009e8			−1.86485			−0.932425	−	0.361363i	$-0.882312\pi$
$787$	−9.09009e8			−1.86485			−0.932425	+	0.361363i	$0.882312\pi$
$788$			1.14681e8i			0.234377i
$789$	−7.77312e7	+	3.08184e8i	−0.158257	+	0.627451i
$790$	−2.63762e8			−0.534972
$791$		−	7.48014e8i		−	1.51140i
$792$	−1.77058e8	−	9.53842e7i	−0.356402	−	0.192000i
$793$	1.68760e8			0.338416
$794$		−	6.27721e7i		−	0.125402i
$795$	7.15597e8	+	1.80490e8i	1.42419	+	0.359213i
$796$	−4.37213e8			−0.866870
$797$			1.96968e8i			0.389064i	0.980896	+	0.194532i	$0.0623189\pi$
$797$			1.96968e8i			0.389064i	−0.980896	+	0.194532i	$0.937681\pi$
$798$	5.70415e7	−	2.26155e8i	0.112249	−	0.445038i
$799$	1.50791e7			0.0295621
$800$			6.36550e8i			1.24326i
$801$	2.91517e8	−	5.41131e8i	0.567239	−	1.05294i
$802$	2.64963e8			0.513643
$803$		−	2.27905e8i		−	0.440157i
$804$	−5.75568e7	−	1.45171e7i	−0.110746	−	0.0279327i
$805$	−5.49857e8			−1.05405
$806$		−	2.77410e8i		−	0.529806i
$807$	1.74876e8	−	6.93340e8i	0.332744	−	1.31925i
$808$	6.83347e8			1.29541
$809$		−	5.75611e7i		−	0.108714i	−0.998522	−	0.0543568i	$-0.982689\pi$
$809$		−	5.75611e7i		−	0.108714i	0.998522	−	0.0543568i	$-0.0173108\pi$
$810$	−2.62622e8	−	3.98654e8i	−0.494169	−	0.750138i
$811$	4.98931e8			0.935358			0.467679	−	0.883898i	$-0.345090\pi$
$811$	4.98931e8			0.935358			0.467679	+	0.883898i	$0.345090\pi$
$812$			5.25499e8i			0.981530i
$813$	5.99504e8	+	1.51209e8i	1.11563	+	0.281388i
$814$	−2.26845e8			−0.420587
$815$		−	3.33550e8i		−	0.616153i
$816$	1.43806e6	−	5.70153e6i	0.00264671	−	0.0104935i
$817$	3.58948e8			0.658211
$818$			2.97947e8i			0.544351i
$819$	−3.72506e8	−	2.00675e8i	−0.678080	−	0.365294i
$820$	7.15868e8			1.29835
$821$			1.87818e8i			0.339397i	0.985496	+	0.169698i	$0.0542793\pi$
$821$			1.87818e8i			0.339397i	−0.985496	+	0.169698i	$0.945721\pi$
$822$	−2.00581e8	−	5.05911e7i	−0.361139	−	0.0910875i
$823$	−8.77742e8			−1.57459			−0.787294	−	0.616577i	$-0.788519\pi$
$823$	−8.77742e8			−1.57459			−0.787294	+	0.616577i	$0.788519\pi$
$824$			9.15779e8i			1.63685i
$825$	−6.92554e7	+	2.74580e8i	−0.123337	+	0.488998i
$826$	−7.10234e8			−1.26026
$827$			3.45001e8i			0.609964i	0.952358	+	0.304982i	$0.0986505\pi$
$827$			3.45001e8i			0.609964i	−0.952358	+	0.304982i	$0.901349\pi$
$828$	−9.12862e7	+	1.69451e8i	−0.160810	+	0.298506i
$829$	7.34590e8			1.28938			0.644691	−	0.764443i	$-0.276986\pi$
$829$	7.34590e8			1.28938			0.644691	+	0.764443i	$0.276986\pi$
$830$			8.55164e8i			1.49560i
$831$	−7.48373e8	−	1.88757e8i	−1.30411	−	0.328927i
$832$	−1.88582e8			−0.327439
$833$		−	1.07190e8i		−	0.185447i
$834$	−7.00697e7	+	2.77809e8i	−0.120790	+	0.478903i
$835$	8.01565e8			1.37683
$836$			8.78724e7i			0.150395i
$837$	6.56738e8	+	6.01119e8i	1.11999	+	1.02514i
$838$	1.91286e8			0.325050
$839$		−	3.10505e8i		−	0.525754i	−0.964829	−	0.262877i	$-0.915329\pi$
$839$		−	3.10505e8i		−	0.525754i	0.964829	−	0.262877i	$-0.0846713\pi$
$840$	1.12318e9	+	2.83291e8i	1.89500	+	0.477964i
$841$	−2.03458e8			−0.342048
$842$			1.27353e8i			0.213340i
$843$	−1.22891e8	+	4.87233e8i	−0.205134	+	0.813305i
$844$	1.58799e6			0.00264131
$845$			5.97814e8i			0.990823i
$846$	3.91025e7	+	2.10652e7i	0.0645794	+	0.0347900i
$847$	−6.71027e8			−1.10431
$848$			2.67756e7i			0.0439088i
$849$	3.25653e8	+	8.21371e7i	0.532147	+	0.134220i
$850$	1.10067e8			0.179226
$851$			5.57479e8i			0.904565i
$852$	−1.21069e8	+	4.80008e8i	−0.195756	+	0.776122i
$853$	−2.94179e8			−0.473984			−0.236992	−	0.971512i	$-0.576162\pi$
$853$	−2.94179e8			−0.473984			−0.236992	+	0.971512i	$0.576162\pi$
$854$			2.90622e8i			0.466611i
$855$	−2.54023e8	+	4.71533e8i	−0.406420	+	0.754422i
$856$	8.18321e7			0.130467
$857$			3.63429e8i			0.577401i	0.957420	+	0.288700i	$0.0932231\pi$
$857$			3.63429e8i			0.577401i	−0.957420	+	0.288700i	$0.906777\pi$
$858$	−8.77734e7	−	2.21385e7i	−0.138964	−	0.0350498i
$859$	2.86351e8			0.451772			0.225886	−	0.974154i	$-0.427472\pi$
$859$	2.86351e8			0.451772			0.225886	+	0.974154i	$0.427472\pi$
$860$			6.94228e8i			1.09146i
$861$	2.82799e8	−	1.12123e9i	0.443066	−	1.75664i
$862$	1.04821e8			0.163654
$863$		−	9.98994e8i		−	1.55428i	−0.629325	−	0.777142i	$-0.716669\pi$
$863$		−	9.98994e8i		−	1.55428i	0.629325	−	0.777142i	$-0.283331\pi$
$864$	4.40926e8	−	4.81723e8i	0.683635	−	0.746889i
$865$	−9.31005e8			−1.43848
$866$			5.99464e8i			0.923016i
$867$	−3.71720e7	−	9.37563e6i	−0.0570372	−	0.0143861i
$868$	−8.41284e8			−1.28642
$869$		−	1.60516e8i		−	0.244602i
$870$	−1.67589e8	+	6.64449e8i	−0.254501	+	1.00903i
$871$	−6.86070e7			−0.103828
$872$			6.40207e7i			0.0965542i
$873$	−2.23791e8	−	1.20560e8i	−0.336356	−	0.181201i
$874$	−1.22628e8			−0.183677
$875$		−	3.02704e8i		−	0.451849i
$876$	4.45534e8	+	1.12374e8i	0.662779	+	0.167168i
$877$	1.12433e9			1.66684			0.833421	−	0.552638i	$-0.186379\pi$
$877$	1.12433e9			1.66684			0.833421	+	0.552638i	$0.186379\pi$
$878$		−	5.60354e8i		−	0.827902i
$879$	−2.11709e7	+	8.39374e7i	−0.0311726	+	0.123592i
$880$	−1.86412e7			−0.0273543
$881$		−	4.77630e8i		−	0.698495i	−0.937030	−	0.349248i	$-0.886437\pi$
$881$		−	4.77630e8i		−	0.698495i	0.937030	−	0.349248i	$-0.113563\pi$
$882$	1.49743e8	−	2.77962e8i	0.218243	−	0.405116i
$883$	−1.71911e8			−0.249702			−0.124851	−	0.992175i	$-0.539845\pi$
$883$	−1.71911e8			−0.249702			−0.124851	+	0.992175i	$0.539845\pi$
$884$		−	6.19603e7i		−	0.0896926i
$885$	1.58144e9	+	3.98877e8i	2.28152	+	0.575452i
$886$	1.67856e8			0.241344
$887$		−	4.28226e8i		−	0.613624i	−0.951770	−	0.306812i	$-0.900738\pi$
$887$		−	4.28226e8i		−	0.613624i	0.951770	−	0.306812i	$-0.0992623\pi$
$888$	2.87218e8	−	1.13874e9i	0.410178	−	1.62625i
$889$	1.16710e9			1.66113
$890$			7.57389e8i			1.07436i
$891$	2.42606e8	−	1.59822e8i	0.342980	−	0.225946i
$892$	−6.17353e8			−0.869840
$893$		−	4.98325e7i		−	0.0699774i
$894$	−6.37398e8	−	1.60766e8i	−0.892068	−	0.225000i
$895$	−1.29636e9			−1.80824
$896$			6.42749e8i			0.893546i
$897$	−5.44060e7	+	2.15706e8i	−0.0753823	+	0.298872i
$898$	4.12325e8			0.569391
$899$		−	1.27799e9i		−	1.75893i
$900$	−5.02631e8	−	2.70776e8i	−0.689480	−	0.371435i
$901$	1.74567e8			0.238665
$902$		−	2.47387e8i		−	0.337099i
$903$	1.08733e9	+	2.74250e8i	1.47672	+	0.372464i
$904$	8.28497e8			1.12146
$905$		−	1.19267e9i		−	1.60907i
$906$	9.53031e7	−	3.77853e8i	0.128151	−	0.508087i
$907$	−2.22702e8			−0.298471			−0.149236	−	0.988802i	$-0.547681\pi$
$907$	−2.22702e8			−0.298471			−0.149236	+	0.988802i	$0.547681\pi$
$908$		−	7.28647e8i		−	0.973329i
$909$	−4.68164e8	+	8.69035e8i	−0.623313	+	1.15703i
$910$	5.21374e8			0.691871
$911$			1.16229e9i			1.53731i	0.639664	+	0.768655i	$0.279073\pi$
$911$			1.16229e9i			1.53731i	−0.639664	+	0.768655i	$0.720927\pi$
$912$	−1.88421e7	−	4.75241e6i	−0.0248396	−	0.00626512i
$913$	−5.20422e8			−0.683823
$914$		−	1.26483e8i		−	0.165651i
$915$	1.63217e8	−	6.47115e8i	0.213060	−	0.844730i
$916$	7.44075e8			0.968122
$917$			5.94718e8i			0.771264i
$918$	−8.32954e7	−	7.62411e7i	−0.107670	−	0.0985511i
$919$	−4.30449e8			−0.554594			−0.277297	−	0.960784i	$-0.589439\pi$
$919$	−4.30449e8			−0.554594			−0.277297	+	0.960784i	$0.589439\pi$
$920$		−	6.09019e8i		−	0.782109i
$921$	−5.61771e8	−	1.41692e8i	−0.719085	−	0.181370i
$922$	5.59253e8			0.713535
$923$			5.72164e8i			0.727638i
$924$	−6.71380e7	+	2.66185e8i	−0.0851045	+	0.337418i
$925$	−1.65361e9			−2.08934
$926$			4.24972e8i			0.535214i
$927$	−1.16463e9	−	6.27404e8i	−1.46200	−	0.787605i
$928$	−9.37415e8			−1.17297
$929$		−	8.38176e8i		−	1.04541i	−0.852513	−	0.522707i	$-0.824922\pi$
$929$		−	8.38176e8i		−	1.04541i	0.852513	−	0.522707i	$-0.175078\pi$
$930$	−1.06373e9	−	2.68298e8i	−1.32246	−	0.333556i
$931$	−3.54236e8			−0.438979
$932$		−	4.76282e8i		−	0.588324i
$933$	−2.66899e8	+	1.05819e9i	−0.328626	+	1.30292i
$934$	−2.85776e8			−0.350740
$935$			1.21534e8i			0.148684i
$936$	2.22267e8	−	4.12585e8i	0.271049	−	0.503137i
$937$	−1.21450e8			−0.147631			−0.0738156	−	0.997272i	$-0.523518\pi$
$937$	−1.21450e8			−0.147631			−0.0738156	+	0.997272i	$0.523518\pi$
$938$		−	1.18148e8i		−	0.143159i
$939$	−1.11781e9	−	2.81937e8i	−1.35011	−	0.340529i
$940$	9.63793e7			0.116038
$941$		−	1.37005e9i		−	1.64425i	−0.569305	−	0.822127i	$-0.692788\pi$
$941$		−	1.37005e9i		−	1.64425i	0.569305	−	0.822127i	$-0.307212\pi$
$942$	−3.20115e7	+	1.26917e8i	−0.0382959	+	0.151834i
$943$	−6.07962e8			−0.725005
$944$			5.91731e7i			0.0703409i
$945$	−1.12976e9	+	1.23430e9i	−1.33873	+	1.46259i
$946$	2.39909e8			0.283382
$947$			7.43740e8i			0.875732i	0.899040	+	0.437866i	$0.144266\pi$
$947$			7.43740e8i			0.875732i	−0.899040	+	0.437866i	$0.855734\pi$
$948$	3.13794e8	+	7.91462e7i	0.368316	+	0.0928977i
$949$	5.31071e8			0.621375
$950$		−	3.63743e8i		−	0.424251i
$951$	1.33232e8	−	5.28232e8i	0.154906	−	0.614163i
$952$	2.73995e8			0.317564
$953$		−	2.38153e8i		−	0.275155i	−0.990491	−	0.137578i	$-0.956068\pi$
$953$		−	2.38153e8i		−	0.275155i	0.990491	−	0.137578i	$-0.0439316\pi$
$954$	4.52682e8	+	2.43867e8i	0.521372	+	0.280872i
$955$	1.98661e9			2.28088
$956$			3.14120e8i			0.359519i
$957$	−4.04360e8	−	1.01989e8i	−0.461352	−	0.116364i
$958$	6.75296e8			0.768064
$959$			7.25073e8i			0.822103i
$960$	−1.82388e8	+	7.23122e8i	−0.206150	+	0.817330i
$961$	1.15846e9			1.30530
$962$		−	5.28600e8i		−	0.593748i
$963$	−5.60635e7	+	1.04069e8i	−0.0627772	+	0.116531i
$964$	−6.35685e8			−0.709595
$965$		−	8.53004e8i		−	0.949225i
$966$	−3.71467e8	−	9.36926e7i	−0.412087	−	0.103938i
$967$	−2.62642e8			−0.290459			−0.145230	−	0.989398i	$-0.546392\pi$
$967$	−2.62642e8			−0.290459			−0.145230	+	0.989398i	$0.546392\pi$
$968$		−	7.43226e8i		−	0.819398i
$969$	−3.09840e7	+	1.22844e8i	−0.0340539	+	0.135015i
$970$	3.13227e8			0.343197
$971$			9.76282e8i			1.06639i	0.845991	+	0.533197i	$0.179010\pi$
$971$			9.76282e8i			1.06639i	−0.845991	+	0.533197i	$0.820990\pi$
$972$	1.92815e8	+	5.53077e8i	0.209963	+	0.602264i
$973$	1.00424e9			1.09018
$974$			1.47018e8i			0.159108i
$975$	−6.39834e8	−	1.61381e8i	−0.690324	−	0.174116i
$976$	2.42131e7			0.0260436
$977$		−	5.35664e8i		−	0.574392i	−0.957872	−	0.287196i	$-0.907277\pi$
$977$		−	5.35664e8i		−	0.574392i	0.957872	−	0.287196i	$-0.0927231\pi$
$978$	5.68351e7	−	2.25337e8i	0.0607575	−	0.240888i
$979$	−4.60920e8			−0.491221
$980$		−	6.85116e8i		−	0.727924i
$981$	−8.14172e7	−	4.38609e7i	−0.0862401	−	0.0464591i
$982$	−5.73323e8			−0.605431
$983$		−	1.45757e9i		−	1.53451i	−0.641343	−	0.767254i	$-0.721622\pi$
$983$		−	1.45757e9i		−	1.53451i	0.641343	−	0.767254i	$-0.278378\pi$
$984$	1.24186e9	+	3.13227e8i	1.30343	+	0.328756i
$985$	5.24174e8			0.548488
$986$			1.62090e8i			0.169093i
$987$	3.80740e7	−	1.50954e8i	0.0395983	−	0.156997i
$988$	−2.04763e8			−0.212315
$989$		−	5.89583e8i		−	0.609476i
$990$	−1.69781e8	+	3.15157e8i	−0.174978	+	0.324804i
$991$	7.72082e8			0.793309			0.396655	−	0.917968i	$-0.370171\pi$
$991$	7.72082e8			0.793309			0.396655	+	0.917968i	$0.370171\pi$
$992$		−	1.50073e9i		−	1.53733i
$993$	−4.53726e7	−	1.14440e7i	−0.0463390	−	0.0116878i
$994$	−9.85323e8			−1.00327
$995$			1.99837e9i			2.02865i
$996$	2.56606e8	−	1.01738e9i	0.259710	−	1.02968i
$997$	−4.09058e8			−0.412762			−0.206381	−	0.978472i	$-0.566169\pi$
$997$	−4.09058e8			−0.412762			−0.206381	+	0.978472i	$0.566169\pi$
$998$		−	8.20886e8i		−	0.825831i
$999$	1.25140e9	+	1.14542e9i	1.25517	+	1.14887i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	51.7.b.a.35.13	✓	32
3.2	odd	2	inner	51.7.b.a.35.20	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.7.b.a.35.13	✓	32	1.1	even	1	trivial
51.7.b.a.35.20	yes	32	3.2	odd	2	inner

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-4.81455i q^{2} +(26.1801 + 6.60322i) q^{3} +40.8201 q^{4} -186.576i q^{5} +(31.7916 - 126.045i) q^{6} -455.638 q^{7} -504.662i q^{8} +(641.795 + 345.746i) q^{9} -898.281 q^{10} -546.662i q^{11} +(1068.67 + 269.544i) q^{12} +1273.85 q^{13} +2193.69i q^{14} +(1232.01 - 4884.59i) q^{15} +182.767 q^{16} -1191.58i q^{17} +(1664.61 - 3089.95i) q^{18} -3937.85 q^{19} -7616.07i q^{20} +(-11928.6 - 3008.68i) q^{21} -2631.93 q^{22} +6468.05i q^{23} +(3332.39 - 13212.1i) q^{24} -19185.7 q^{25} -6133.00i q^{26} +(14519.2 + 13289.6i) q^{27} -18599.2 q^{28} -28253.9i q^{29} +(-23517.1 - 5931.55i) q^{30} +45232.4 q^{31} -33178.3i q^{32} +(3609.73 - 14311.7i) q^{33} -5736.91 q^{34} +85011.2i q^{35} +(26198.1 + 14113.4i) q^{36} +86189.5 q^{37} +18959.0i q^{38} +(33349.4 + 8411.49i) q^{39} -94158.0 q^{40} +93994.5i q^{41} +(-14485.4 + 57431.0i) q^{42} -91153.1 q^{43} -22314.8i q^{44} +(64508.1 - 119744. i) q^{45} +31140.8 q^{46} +12654.7i q^{47} +(4784.86 + 1206.85i) q^{48} +89956.7 q^{49} +92370.8i q^{50} +(7868.25 - 31195.6i) q^{51} +51998.5 q^{52} +146501. i q^{53} +(63983.4 - 69903.5i) q^{54} -101994. q^{55} +229943. i q^{56} +(-103093. - 26002.5i) q^{57} -136030. q^{58} +323762. i q^{59} +(50290.6 - 199389. i) q^{60} +132481. q^{61} -217773. i q^{62} +(-292426. - 157535. i) q^{63} -148041. q^{64} -237670. i q^{65} +(-68904.2 - 17379.2i) q^{66} -53858.1 q^{67} -48640.3i q^{68} +(-42710.0 + 169334. i) q^{69} +409291. q^{70} +449162. i q^{71} +(174485. - 323889. i) q^{72} +416903. q^{73} -414964. i q^{74} +(-502285. - 126688. i) q^{75} -160744. q^{76} +249080. i q^{77} +(40497.6 - 160562. i) q^{78} +293630. q^{79} -34100.0i q^{80} +(292360. + 443796. i) q^{81} +452541. q^{82} -952000. i q^{83} +(-486928. - 122815. i) q^{84} -222320. q^{85} +438861. i q^{86} +(186567. - 739689. i) q^{87} -275879. q^{88} -843153. i q^{89} +(-576512. - 310577. i) q^{90} -580412. q^{91} +264027. i q^{92} +(1.18419e6 + 298679. i) q^{93} +60926.8 q^{94} +734710. i q^{95} +(219084. - 868611. i) q^{96} -348695. q^{97} -433101. i q^{98} +(189006. - 350845. 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

Varieties

Fields

Representations

Groups

Database

Properties

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