LMFDB - Embedded newform 51.7.b.a.35.22

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			35.22
Character	$\chi$	$=$	51.35
Dual form			51.7.b.a.35.11

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+6.35177i q^{2} +(-6.87539 - 26.1099i) q^{3} +23.6551 q^{4} -214.273i q^{5} +(165.844 - 43.6709i) q^{6} -487.138 q^{7} +556.764i q^{8} +(-634.458 + 359.032i) q^{9} +1361.01 q^{10} +1788.18i q^{11} +(-162.638 - 617.633i) q^{12} -1815.64 q^{13} -3094.18i q^{14} +(-5594.65 + 1473.21i) q^{15} -2022.51 q^{16} +1191.58i q^{17} +(-2280.49 - 4029.93i) q^{18} +152.498 q^{19} -5068.64i q^{20} +(3349.26 + 12719.1i) q^{21} -11358.1 q^{22} -12165.6i q^{23} +(14537.1 - 3827.97i) q^{24} -30287.8 q^{25} -11532.5i q^{26} +(13736.5 + 14097.2i) q^{27} -11523.3 q^{28} -38981.9i q^{29} +(-9357.47 - 35535.9i) q^{30} +4346.21 q^{31} +22786.4i q^{32} +(46689.4 - 12294.5i) q^{33} -7568.62 q^{34} +104380. i q^{35} +(-15008.2 + 8492.93i) q^{36} -36390.1 q^{37} +968.629i q^{38} +(12483.2 + 47406.2i) q^{39} +119299. q^{40} -46210.1i q^{41} +(-80789.0 + 21273.7i) q^{42} -97432.9 q^{43} +42299.6i q^{44} +(76930.8 + 135947. i) q^{45} +77272.7 q^{46} -37496.3i q^{47} +(13905.6 + 52807.7i) q^{48} +119654. q^{49} -192381. i q^{50} +(31112.0 - 8192.56i) q^{51} -42949.1 q^{52} -137705. i q^{53} +(-89541.9 + 87250.7i) q^{54} +383159. q^{55} -271221. i q^{56} +(-1048.48 - 3981.70i) q^{57} +247604. q^{58} -99800.3i q^{59} +(-132342. + 34848.9i) q^{60} -113868. q^{61} +27606.1i q^{62} +(309068. - 174898. i) q^{63} -274175. q^{64} +389042. i q^{65} +(78091.5 + 296560. i) q^{66} -374493. q^{67} +28186.9i q^{68} +(-317642. + 83642.9i) q^{69} -662999. q^{70} +178897. i q^{71} +(-199896. - 353244. i) q^{72} -189276. q^{73} -231141. i q^{74} +(208240. + 790812. i) q^{75} +3607.34 q^{76} -871092. i q^{77} +(-301113. + 79290.6i) q^{78} +769979. q^{79} +433369. i q^{80} +(273633. - 455582. i) q^{81} +293516. q^{82} -155977. i q^{83} +(79227.1 + 300872. i) q^{84} +255322. q^{85} -618871. i q^{86} +(-1.01782e6 + 268016. i) q^{87} -995597. q^{88} +18567.4i q^{89} +(-863503. + 488646. i) q^{90} +884467. q^{91} -287777. i q^{92} +(-29881.9 - 113479. i) q^{93} +238167. q^{94} -32676.1i q^{95} +(594952. - 156665. i) q^{96} -1.12203e6 q^{97} +760015. i q^{98} +(-642015. - 1.13453e6i) 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 $

$2$			6.35177i			0.793971i	0.917825	+	0.396985i	$0.129944\pi$
$2$			6.35177i			0.793971i	−0.917825	+	0.396985i	$0.870056\pi$
$3$	−6.87539	−	26.1099i	−0.254644	−	0.967035i
$3$	−6.87539	−	26.1099i	−0.254644	−	0.967035i
$4$	23.6551			0.369611
$5$		−	214.273i		−	1.71418i	−0.515166	−	0.857091i	$-0.672270\pi$
$5$		−	214.273i		−	1.71418i	0.515166	−	0.857091i	$-0.327730\pi$
$6$	165.844	−	43.6709i	0.767797	−	0.202180i
$7$	−487.138			−1.42023			−0.710113	−	0.704087i	$-0.751356\pi$
$7$	−487.138			−1.42023			−0.710113	+	0.704087i	$0.751356\pi$
$8$			556.764i			1.08743i
$9$	−634.458	+	359.032i	−0.870313	+	0.492500i
$10$	1361.01			1.36101
$11$			1788.18i			1.34349i	0.740783	+	0.671744i	$0.234455\pi$
$11$			1788.18i			1.34349i	−0.740783	+	0.671744i	$0.765545\pi$
$12$	−162.638	−	617.633i	−0.0941192	−	0.357426i
$13$	−1815.64			−0.826418			−0.413209	−	0.910636i	$-0.635592\pi$
$13$	−1815.64			−0.826418			−0.413209	+	0.910636i	$0.635592\pi$
$14$		−	3094.18i		−	1.12762i
$15$	−5594.65	+	1473.21i	−1.65767	+	0.436506i
$16$	−2022.51			−0.493778
$17$			1191.58i			0.242536i
$17$			1191.58i			0.242536i
$18$	−2280.49	−	4029.93i	−0.391030	−	0.691003i
$19$	152.498			0.0222332			0.0111166	−	0.999938i	$-0.496461\pi$
$19$	152.498			0.0222332			0.0111166	+	0.999938i	$0.496461\pi$
$20$		−	5068.64i		−	0.633579i
$21$	3349.26	+	12719.1i	0.361652	+	1.37341i
$22$	−11358.1			−1.06669
$23$		−	12165.6i		−	0.999881i	−0.866060	−	0.499940i	$-0.833355\pi$
$23$		−	12165.6i		−	0.999881i	0.866060	−	0.499940i	$-0.166645\pi$
$24$	14537.1	−	3827.97i	1.05158	−	0.276908i
$25$	−30287.8			−1.93842
$26$		−	11532.5i		−	0.656151i
$27$	13736.5	+	14097.2i	0.697884	+	0.716211i
$28$	−11523.3			−0.524931
$29$		−	38981.9i		−	1.59834i	−0.601105	−	0.799170i	$-0.705273\pi$
$29$		−	38981.9i		−	1.59834i	0.601105	−	0.799170i	$-0.294727\pi$
$30$	−9357.47	−	35535.9i	−0.346573	−	1.31614i
$31$	4346.21			0.145890			0.0729450	−	0.997336i	$-0.476760\pi$
$31$	4346.21			0.145890			0.0729450	+	0.997336i	$0.476760\pi$
$32$			22786.4i			0.695386i
$33$	46689.4	−	12294.5i	1.29920	−	0.342112i
$34$	−7568.62			−0.192566
$35$			104380.i			2.43453i
$36$	−15008.2	+	8492.93i	−0.321677	+	0.182033i
$37$	−36390.1			−0.718419			−0.359210	−	0.933257i	$-0.616954\pi$
$37$	−36390.1			−0.718419			−0.359210	+	0.933257i	$0.616954\pi$
$38$			968.629i			0.0176525i
$39$	12483.2	+	47406.2i	0.210442	+	0.799175i
$40$	119299.			1.86405
$41$		−	46210.1i		−	0.670479i	−0.942133	−	0.335239i	$-0.891183\pi$
$41$		−	46210.1i		−	0.670479i	0.942133	−	0.335239i	$-0.108817\pi$
$42$	−80789.0	+	21273.7i	−1.09045	+	0.287141i
$43$	−97432.9			−1.22546			−0.612731	−	0.790291i	$-0.709929\pi$
$43$	−97432.9			−1.22546			−0.612731	+	0.790291i	$0.709929\pi$
$44$			42299.6i			0.496568i
$45$	76930.8	+	135947.i	0.844233	+	1.49187i
$46$	77272.7			0.793876
$47$		−	37496.3i		−	0.361156i	−0.983561	−	0.180578i	$-0.942203\pi$
$47$		−	37496.3i		−	0.361156i	0.983561	−	0.180578i	$-0.0577968\pi$
$48$	13905.6	+	52807.7i	0.125738	+	0.477500i
$49$	119654.			1.01704
$50$		−	192381.i		−	1.53905i
$51$	31112.0	−	8192.56i	0.234540	−	0.0617603i
$52$	−42949.1			−0.305453
$53$		−	137705.i		−	0.924961i	−0.886629	−	0.462481i	$-0.846959\pi$
$53$		−	137705.i		−	0.924961i	0.886629	−	0.462481i	$-0.153041\pi$
$54$	−89541.9	+	87250.7i	−0.568650	+	0.554100i
$55$	383159.			2.30298
$56$		−	271221.i		−	1.54440i
$57$	−1048.48	−	3981.70i	−0.00566156	−	0.0215003i
$58$	247604.			1.26903
$59$		−	99800.3i		−	0.485932i	−0.970035	−	0.242966i	$-0.921880\pi$
$59$		−	99800.3i		−	0.485932i	0.970035	−	0.242966i	$-0.0781204\pi$
$60$	−132342.	+	34848.9i	−0.612693	+	0.161337i
$61$	−113868.			−0.501662			−0.250831	−	0.968031i	$-0.580704\pi$
$61$	−113868.			−0.501662			−0.250831	+	0.968031i	$0.580704\pi$
$62$			27606.1i			0.115832i
$63$	309068.	−	174898.i	1.23604	−	0.699461i
$64$	−274175.			−1.04589
$65$			389042.i			1.41663i
$66$	78091.5	+	296560.i	0.271627	+	1.03153i
$67$	−374493.			−1.24514			−0.622572	−	0.782562i	$-0.713912\pi$
$67$	−374493.			−1.24514			−0.622572	+	0.782562i	$0.713912\pi$
$68$			28186.9i			0.0896437i
$69$	−317642.	+	83642.9i	−0.966920	+	0.254614i
$70$	−662999.			−1.93294
$71$			178897.i			0.499837i	0.968267	+	0.249918i	$0.0804038\pi$
$71$			178897.i			0.499837i	−0.968267	+	0.249918i	$0.919596\pi$
$72$	−199896.	−	353244.i	−0.535559	−	0.946405i
$73$	−189276.			−0.486549			−0.243274	−	0.969957i	$-0.578222\pi$
$73$	−189276.			−0.486549			−0.243274	+	0.969957i	$0.578222\pi$
$74$		−	231141.i		−	0.570404i
$75$	208240.	+	790812.i	0.493607	+	1.87452i
$76$	3607.34			0.00821763
$77$		−	871092.i		−	1.90806i
$78$	−301113.	+	79290.6i	−0.634521	+	0.167085i
$79$	769979.			1.56170			0.780850	−	0.624718i	$-0.214786\pi$
$79$	769979.			1.56170			0.780850	+	0.624718i	$0.214786\pi$
$80$			433369.i			0.846424i
$81$	273633.	−	455582.i	0.514888	−	0.857257i
$82$	293516.			0.532341
$83$		−	155977.i		−	0.272789i	−0.990655	−	0.136394i	$-0.956449\pi$
$83$		−	155977.i		−	0.272789i	0.990655	−	0.136394i	$-0.0435514\pi$
$84$	79227.1	+	300872.i	0.133671	+	0.507626i
$85$	255322.			0.415750
$86$		−	618871.i		−	0.972982i
$87$	−1.01782e6	+	268016.i	−1.54565	+	0.407008i
$88$	−995597.			−1.46095
$89$			18567.4i			0.0263378i	0.999913	+	0.0131689i	$0.00419192\pi$
$89$			18567.4i			0.0263378i	−0.999913	+	0.0131689i	$0.995808\pi$
$90$	−863503.	+	488646.i	−1.18450	+	0.670297i
$91$	884467.			1.17370
$92$		−	287777.i		−	0.369566i
$93$	−29881.9	−	113479.i	−0.0371500	−	0.141081i
$94$	238167.			0.286747
$95$		−	32676.1i		−	0.0381117i
$96$	594952.	−	156665.i	0.672462	−	0.177076i
$97$	−1.12203e6			−1.22939			−0.614693	−	0.788766i	$-0.710720\pi$
$97$	−1.12203e6			−1.22939			−0.614693	+	0.788766i	$0.710720\pi$
$98$			760015.i			0.807503i
$99$	−642015.	−	1.13453e6i	−0.661668	−	1.16926i
$100$	−716459.			−0.716459
$101$			1.12866e6i			1.09546i	0.836654	+	0.547732i	$0.184509\pi$
$101$			1.12866e6i			1.09546i	−0.836654	+	0.547732i	$0.815491\pi$
$102$	52037.2	+	197616.i	0.0490359	+	0.186218i
$103$	962305.			0.880645			0.440323	−	0.897840i	$-0.354864\pi$
$103$	962305.			0.880645			0.440323	+	0.897840i	$0.354864\pi$
$104$		−	1.01088e6i		−	0.898672i
$105$	2.72536e6	−	717655.i	2.35427	−	0.619938i
$106$	874673.			0.734392
$107$		−	1.68946e6i		−	1.37910i	−0.724236	−	0.689552i	$-0.757807\pi$
$107$		−	1.68946e6i		−	1.37910i	0.724236	−	0.689552i	$-0.242193\pi$
$108$	324937.	+	333470.i	0.257945	+	0.264719i
$109$	1.01017e6			0.780038			0.390019	−	0.920807i	$-0.372469\pi$
$109$	1.01017e6			0.780038			0.390019	+	0.920807i	$0.372469\pi$
$110$			2.43373e6i			1.82850i
$111$	250196.	+	950143.i	0.182941	+	0.694737i
$112$	985242.			0.701276
$113$			1.07976e6i			0.748325i	0.927363	+	0.374163i	$0.122070\pi$
$113$			1.07976e6i			0.748325i	−0.927363	+	0.374163i	$0.877930\pi$
$114$	25290.8	−	6659.70i	0.0170706	−	0.00449511i
$115$	−2.60674e6			−1.71398
$116$		−	922120.i		−	0.590763i
$117$	1.15195e6	−	651873.i	0.719242	−	0.407010i
$118$	633908.			0.385816
$119$		−	580462.i		−	0.344456i
$120$	−820230.	−	3.11490e6i	−0.474670	−	1.80260i
$121$	−1.42604e6			−0.804962
$122$		−	723261.i		−	0.398305i
$123$	−1.20654e6	+	317712.i	−0.648376	+	0.170734i
$124$	102810.			0.0539224
$125$			3.14183e6i			1.60862i
$126$	1.11091e6	+	1.96313e6i	0.555352	+	0.981381i
$127$	3.41306e6			1.66622			0.833110	−	0.553107i	$-0.186558\pi$
$127$	3.41306e6			1.66622			0.833110	+	0.553107i	$0.186558\pi$
$128$		−	283164.i		−	0.135023i
$129$	669889.	+	2.54397e6i	0.312057	+	1.18507i
$130$	−2.47110e6			−1.12476
$131$			3.23463e6i			1.43883i	0.694579	+	0.719416i	$0.255591\pi$
$131$			3.23463e6i			1.43883i	−0.694579	+	0.719416i	$0.744409\pi$
$132$	1.10444e6	−	290826.i	0.480198	−	0.126448i
$133$	−74287.3			−0.0315762
$134$		−	2.37869e6i		−	0.988608i
$135$	3.02064e6	−	2.94335e6i	1.22771	−	1.19630i
$136$	−663428.			−0.263741
$137$			4.11616e6i			1.60078i	0.599481	+	0.800389i	$0.295374\pi$
$137$			4.11616e6i			1.60078i	−0.599481	+	0.800389i	$0.704626\pi$
$138$	−531280.	−	2.01759e6i	−0.202156	−	0.767706i
$139$	5.17866e6			1.92829			0.964146	−	0.265371i	$-0.0854944\pi$
$139$	5.17866e6			1.92829			0.964146	+	0.265371i	$0.0854944\pi$
$140$			2.46912e6i			0.899826i
$141$	−979025.	+	257801.i	−0.349250	+	0.0919662i
$142$	−1.13631e6			−0.396856
$143$		−	3.24670e6i		−	1.11028i
$144$	1.28320e6	−	726147.i	0.429741	−	0.243185i
$145$	−8.35276e6			−2.73984
$146$		−	1.20224e6i		−	0.386306i
$147$	−822669.	−	3.12416e6i	−0.258984	−	0.983517i
$148$	−860810.			−0.265535
$149$		−	3.37945e6i		−	1.02161i	−0.859695	−	0.510807i	$-0.829347\pi$
$149$		−	3.37945e6i		−	1.02161i	0.859695	−	0.510807i	$-0.170653\pi$
$150$	−5.02305e6	+	1.32269e6i	−1.48831	+	0.391909i
$151$	−5.07358e6			−1.47361			−0.736807	−	0.676103i	$-0.763667\pi$
$151$	−5.07358e6			−1.47361			−0.736807	+	0.676103i	$0.763667\pi$
$152$			84905.2i			0.0241771i
$153$	−427815.	−	756006.i	−0.119449	−	0.211082i
$154$	5.53297e6			1.51494
$155$		−	931273.i		−	0.250082i
$156$	295292.	+	1.12140e6i	0.0777817	+	0.295383i
$157$	2.01317e6			0.520213			0.260106	−	0.965580i	$-0.416242\pi$
$157$	2.01317e6			0.520213			0.260106	+	0.965580i	$0.416242\pi$
$158$			4.89073e6i			1.23994i
$159$	−3.59548e6	+	946779.i	−0.894470	+	0.235536i
$160$	4.88250e6			1.19202
$161$			5.92630e6i			1.42006i
$162$	2.89375e6	+	1.73805e6i	0.680637	+	0.408806i
$163$	−5.53182e6			−1.27734			−0.638668	−	0.769482i	$-0.720514\pi$
$163$	−5.53182e6			−1.27734			−0.638668	+	0.769482i	$0.720514\pi$
$164$		−	1.09310e6i		−	0.247816i
$165$	−2.63437e6	−	1.00043e7i	−0.586441	−	2.22706i
$166$	990730.			0.216586
$167$		−	2.31058e6i		−	0.496102i	−0.968747	−	0.248051i	$-0.920210\pi$
$167$		−	2.31058e6i		−	0.496102i	0.968747	−	0.248051i	$-0.0797901\pi$
$168$	−7.08156e6	+	1.86475e6i	−1.49349	+	0.393272i
$169$	−1.53026e6			−0.317034
$170$			1.62175e6i			0.330093i
$171$	−96753.3	+	54751.5i	−0.0193498	+	0.0109498i
$172$	−2.30478e6			−0.452944
$173$			1.97759e6i			0.381943i	0.981596	+	0.190972i	$0.0611638\pi$
$173$			1.97759e6i			0.381943i	−0.981596	+	0.190972i	$0.938836\pi$
$174$	−1.70237e6	−	6.46492e6i	−0.323152	−	1.22720i
$175$	1.47543e7			2.75299
$176$		−	3.61662e6i		−	0.663384i
$177$	−2.60578e6	+	686166.i	−0.469913	+	0.123740i
$178$	−117936.			−0.0209115
$179$		−	5.11096e6i		−	0.891135i	−0.895248	−	0.445568i	$-0.853002\pi$
$179$		−	5.11096e6i		−	0.891135i	0.895248	−	0.445568i	$-0.146998\pi$
$180$	1.81980e6	+	3.21584e6i	0.312038	+	0.551412i
$181$	−8.27932e6			−1.39624			−0.698118	−	0.715983i	$-0.745979\pi$
$181$	−8.27932e6			−1.39624			−0.698118	+	0.715983i	$0.745979\pi$
$182$			5.61792e6i			0.931884i
$183$	782885.	+	2.97308e6i	0.127745	+	0.485125i
$184$	6.77335e6			1.08730
$185$			7.79740e6i			1.23150i
$186$	720793.	−	189803.i	0.112014	−	0.0294960i
$187$	−2.13076e6			−0.325844
$188$		−	886977.i		−	0.133487i
$189$	−6.69155e6	−	6.86726e6i	−0.991154	−	1.01718i
$190$	207551.			0.0302596
$191$		−	9.44505e6i		−	1.35551i	−0.735286	−	0.677757i	$-0.762952\pi$
$191$		−	9.44505e6i		−	1.35551i	0.735286	−	0.677757i	$-0.237048\pi$
$192$	1.88506e6	+	7.15868e6i	0.266331	+	1.01142i
$193$	−839115.			−0.116721			−0.0583606	−	0.998296i	$-0.518587\pi$
$193$	−839115.			−0.116721			−0.0583606	+	0.998296i	$0.518587\pi$
$194$		−	7.12686e6i		−	0.976097i
$195$	1.01579e7	−	2.67482e6i	1.36993	−	0.360736i
$196$	2.83043e6			0.375910
$197$		−	3.87609e6i		−	0.506985i	−0.967337	−	0.253492i	$-0.918421\pi$
$197$		−	3.87609e6i		−	0.506985i	0.967337	−	0.253492i	$-0.0815793\pi$
$198$	7.20625e6	−	4.07793e6i	0.928354	−	0.525345i
$199$	1.19508e6			0.151648			0.0758242	−	0.997121i	$-0.475841\pi$
$199$	1.19508e6			0.151648			0.0758242	+	0.997121i	$0.475841\pi$
$200$		−	1.68632e7i		−	2.10789i
$201$	2.57479e6	+	9.77800e6i	0.317069	+	1.20410i
$202$	−7.16897e6			−0.869766
$203$			1.89896e7i			2.27000i
$204$	735957.	−	193796.i	0.0866886	−	0.0228272i
$205$	−9.90155e6			−1.14932
$206$			6.11234e6i			0.699207i
$207$	4.36782e6	+	7.71853e6i	0.492441	+	0.870209i
$208$	3.67215e6			0.408066
$209$			272694.i			0.0298701i
$210$	4.55838e6	+	1.73109e7i	0.492212	+	1.86922i
$211$	179511.			0.0191092			0.00955460	−	0.999954i	$-0.496959\pi$
$211$	179511.			0.0191092			0.00955460	+	0.999954i	$0.496959\pi$
$212$		−	3.25743e6i		−	0.341875i
$213$	4.67099e6	−	1.22999e6i	0.483360	−	0.127281i
$214$	1.07311e7			1.09497
$215$			2.08772e7i			2.10067i
$216$	−7.84881e6	+	7.64797e6i	−0.778829	+	0.758901i
$217$	−2.11720e6			−0.207197
$218$			6.41637e6i			0.619327i
$219$	1.30135e6	+	4.94198e6i	0.123897	+	0.470510i
$220$	9.06365e6			0.851207
$221$		−	2.16348e6i		−	0.200436i
$222$	−6.03509e6	+	1.58919e6i	−0.551600	+	0.145250i
$223$	3.63621e6			0.327894			0.163947	−	0.986469i	$-0.447577\pi$
$223$	3.63621e6			0.327894			0.163947	+	0.986469i	$0.447577\pi$
$224$		−	1.11001e7i		−	0.987605i
$225$	1.92163e7	−	1.08743e7i	1.68703	−	0.954669i
$226$	−6.85836e6			−0.594148
$227$		−	1.20834e7i		−	1.03303i	−0.856279	−	0.516514i	$-0.827229\pi$
$227$		−	1.20834e7i		−	1.03303i	0.856279	−	0.516514i	$-0.172771\pi$
$228$	−24801.9	−	94187.5i	−0.00209257	−	0.00794673i
$229$	−1.42322e7			−1.18513			−0.592566	−	0.805522i	$-0.701885\pi$
$229$	−1.42322e7			−1.18513			−0.592566	+	0.805522i	$0.701885\pi$
$230$		−	1.65574e7i		−	1.36085i
$231$	−2.27442e7	+	5.98910e6i	−1.84516	+	0.485876i
$232$	2.17037e7			1.73808
$233$		−	1.33774e7i		−	1.05756i	−0.848759	−	0.528779i	$-0.822650\pi$
$233$		−	1.33774e7i		−	1.05756i	0.848759	−	0.528779i	$-0.177350\pi$
$234$	4.14054e6	+	7.31690e6i	0.323154	+	0.571057i
$235$	−8.03442e6			−0.619086
$236$		−	2.36078e6i		−	0.179606i
$237$	−5.29391e6	−	2.01041e7i	−0.397678	−	1.51022i
$238$	3.68696e6			0.273488
$239$			8.99537e6i			0.658909i	0.944171	+	0.329455i	$0.106865\pi$
$239$			8.99537e6i			0.658909i	−0.944171	+	0.329455i	$0.893135\pi$
$240$	1.13152e7	−	2.97958e6i	0.818522	−	0.215537i
$241$	1.72296e7			1.23090			0.615452	−	0.788174i	$-0.288973\pi$
$241$	1.72296e7			1.23090			0.615452	+	0.788174i	$0.288973\pi$
$242$		−	9.05786e6i		−	0.639116i
$243$	−1.37765e7	−	4.01223e6i	−0.960111	−	0.279620i
$244$	−2.69355e6			−0.185420
$245$		−	2.56386e7i		−	1.74340i
$246$	−2.01803e6	−	7.66367e6i	−0.135557	−	0.514792i
$247$	−276881.			−0.0183739
$248$			2.41981e6i			0.158645i
$249$	−4.07255e6	+	1.07240e6i	−0.263796	+	0.0694641i
$250$	−1.99562e7			−1.27719
$251$			1.44074e7i			0.911095i	0.890212	+	0.455547i	$0.150556\pi$
$251$			1.44074e7i			0.911095i	−0.890212	+	0.455547i	$0.849444\pi$
$252$	7.31104e6	−	4.13723e6i	0.456854	−	0.258528i
$253$	2.17542e7			1.34333
$254$			2.16789e7i			1.32293i
$255$	−1.75544e6	−	6.66645e6i	−0.105868	−	0.402045i
$256$	−1.57486e7			−0.938689
$257$		−	2.46410e7i		−	1.45164i	−0.687884	−	0.725821i	$-0.741460\pi$
$257$		−	2.46410e7i		−	1.45164i	0.687884	−	0.725821i	$-0.258540\pi$
$258$	−1.61587e7	+	4.25498e6i	−0.940907	+	0.247764i
$259$	1.77270e7			1.02032
$260$			9.20281e6i			0.523601i
$261$	1.39958e7	+	2.47324e7i	0.787182	+	1.39106i
$262$	−2.05456e7			−1.14239
$263$			2.40534e7i			1.32223i	0.750283	+	0.661117i	$0.229917\pi$
$263$			2.40534e7i			1.32223i	−0.750283	+	0.661117i	$0.770083\pi$
$264$	6.84512e6	+	2.59950e7i	0.372023	+	1.41279i
$265$	−2.95065e7			−1.58555
$266$		−	471856.i		−	0.0250706i
$267$	484793.	−	127658.i	0.0254696	−	0.00670678i
$268$	−8.85867e6			−0.460219
$269$		−	1.52024e7i		−	0.781007i	−0.920601	−	0.390504i	$-0.872301\pi$
$269$		−	1.52024e7i		−	0.781007i	0.920601	−	0.390504i	$-0.127699\pi$
$270$	1.86954e7	+	1.91864e7i	0.949827	+	0.974769i
$271$	1.28811e7			0.647210			0.323605	−	0.946192i	$-0.395105\pi$
$271$	1.28811e7			0.647210			0.323605	+	0.946192i	$0.395105\pi$
$272$		−	2.40998e6i		−	0.119759i
$273$	−6.08105e6	−	2.30934e7i	−0.298876	−	1.13501i
$274$	−2.61449e7			−1.27097
$275$		−	5.41601e7i		−	2.60424i
$276$	−7.51384e6	+	1.97858e6i	−0.357384	+	0.0941079i
$277$	−2.14785e7			−1.01056			−0.505282	−	0.862954i	$-0.668612\pi$
$277$	−2.14785e7			−1.01056			−0.505282	+	0.862954i	$0.668612\pi$
$278$			3.28936e7i			1.53101i
$279$	−2.75749e6	+	1.56043e6i	−0.126970	+	0.0718507i
$280$	−5.81152e7			−2.64738
$281$			2.87201e7i			1.29439i	0.762323	+	0.647197i	$0.224059\pi$
$281$			2.87201e7i			1.29439i	−0.762323	+	0.647197i	$0.775941\pi$
$282$	−1.63749e6	−	6.21854e6i	−0.0730184	−	0.277294i
$283$	2.42413e7			1.06954			0.534770	−	0.844998i	$-0.320398\pi$
$283$	2.42413e7			1.06954			0.534770	+	0.844998i	$0.320398\pi$
$284$			4.23183e6i			0.184745i
$285$	−853170.	+	224661.i	−0.0368554	+	0.00970493i
$286$	2.06223e7			0.881532
$287$			2.25107e7i			0.952232i
$288$	−8.18105e6	−	1.44570e7i	−0.342477	−	0.605203i
$289$	−1.41986e6			−0.0588235
$290$		−	5.30548e7i		−	2.17536i
$291$	7.71438e6	+	2.92961e7i	0.313056	+	1.18886i
$292$	−4.47733e6			−0.179834
$293$		−	3.54040e6i		−	0.140750i	−0.997521	−	0.0703752i	$-0.977580\pi$
$293$		−	3.54040e6i		−	0.140750i	0.997521	−	0.0703752i	$-0.0224197\pi$
$294$	1.98440e7	−	5.22540e6i	0.780883	−	0.205626i
$295$	−2.13845e7			−0.832976
$296$		−	2.02607e7i		−	0.781231i
$297$	−2.52083e7	+	2.45633e7i	−0.962221	+	0.937600i
$298$	2.14655e7			0.811132
$299$			2.20883e7i			0.826319i
$300$	4.92594e6	+	1.87067e7i	0.182442	+	0.692841i
$301$	4.74632e7			1.74043
$302$		−	3.22262e7i		−	1.17001i
$303$	2.94692e7	−	7.75996e6i	1.05935	−	0.278953i
$304$	−308428.			−0.0109783
$305$			2.43987e7i			0.859939i
$306$	4.80197e6	−	2.71738e6i	0.167593	−	0.0948388i
$307$	1.07146e7			0.370307			0.185154	−	0.982710i	$-0.440722\pi$
$307$	1.07146e7			0.370307			0.185154	+	0.982710i	$0.440722\pi$
$308$		−	2.06057e7i		−	0.705238i
$309$	−6.61622e6	−	2.51257e7i	−0.224251	−	0.851615i
$310$	5.91523e6			0.198558
$311$			4.15654e7i			1.38182i	0.722942	+	0.690908i	$0.242789\pi$
$311$			4.15654e7i			1.38182i	−0.722942	+	0.690908i	$0.757211\pi$
$312$	−2.63941e7	+	6.95022e6i	−0.869047	+	0.228842i
$313$	−1.33555e7			−0.435539			−0.217770	−	0.976000i	$-0.569878\pi$
$313$	−1.33555e7			−0.435539			−0.217770	+	0.976000i	$0.569878\pi$
$314$			1.27872e7i			0.413034i
$315$	−3.74759e7	−	6.62249e7i	−1.19900	−	2.11880i
$316$	1.82139e7			0.577221
$317$		−	2.91393e6i		−	0.0914748i	−0.998953	−	0.0457374i	$-0.985436\pi$
$317$		−	2.91393e6i		−	0.0914748i	0.998953	−	0.0457374i	$-0.0145637\pi$
$318$	−6.01372e6	−	2.28377e7i	−0.187009	−	0.710183i
$319$	6.97068e7			2.14735
$320$			5.87481e7i			1.79285i
$321$	−4.41118e7	+	1.16157e7i	−1.33364	+	0.351181i
$322$	−3.76425e7			−1.12748
$323$			181713.i			0.00539234i
$324$	6.47280e6	−	1.07768e7i	0.190308	−	0.316851i
$325$	5.49917e7			1.60194
$326$		−	3.51368e7i		−	1.01417i
$327$	−6.94532e6	−	2.63755e7i	−0.198632	−	0.754324i
$328$	2.57281e7			0.729099
$329$			1.82658e7i			0.512923i
$330$	6.35447e7	−	1.67329e7i	1.76822	−	0.465617i
$331$	4.94216e7			1.36280			0.681401	−	0.731910i	$-0.261371\pi$
$331$	4.94216e7			1.36280			0.681401	+	0.731910i	$0.261371\pi$
$332$		−	3.68965e6i		−	0.100826i
$333$	2.30880e7	−	1.30652e7i	0.625250	−	0.353821i
$334$	1.46762e7			0.393890
$335$			8.02437e7i			2.13440i
$336$	−6.77393e6	−	2.57246e7i	−0.178576	−	0.678158i
$337$	−3.48728e7			−0.911166			−0.455583	−	0.890193i	$-0.650569\pi$
$337$	−3.48728e7			−0.911166			−0.455583	+	0.890193i	$0.650569\pi$
$338$		−	9.71987e6i		−	0.251716i
$339$	2.81924e7	−	7.42375e6i	0.723657	−	0.190557i
$340$	6.03967e6			0.153666
$341$			7.77181e6i			0.196001i
$342$	−347769.	−	614554.i	−0.00869385	−	0.0153632i
$343$	−976799.			−0.0242060
$344$		−	5.42472e7i		−	1.33261i
$345$	1.79224e7	+	6.80619e7i	0.436454	+	1.65748i
$346$	−1.25612e7			−0.303252
$347$		−	4.10781e7i		−	0.983156i	−0.870834	−	0.491578i	$-0.836420\pi$
$347$		−	4.10781e7i		−	0.983156i	0.870834	−	0.491578i	$-0.163580\pi$
$348$	−2.40765e7	+	6.33994e6i	−0.571289	+	0.150434i
$349$	2.35139e7			0.553156			0.276578	−	0.960991i	$-0.410800\pi$
$349$	2.35139e7			0.553156			0.276578	+	0.960991i	$0.410800\pi$
$350$			9.37159e7i			2.18579i
$351$	−2.49405e7	−	2.55954e7i	−0.576744	−	0.591889i
$352$	−4.07463e7			−0.934243
$353$		−	4.94492e7i		−	1.12418i	−0.827076	−	0.562089i	$-0.809998\pi$
$353$		−	4.94492e7i		−	1.12418i	0.827076	−	0.562089i	$-0.190002\pi$
$354$	−4.35837e6	−	1.65513e7i	−0.0982458	−	0.373097i
$355$	3.83328e7			0.856811
$356$			439212.i			0.00973475i
$357$	−1.51558e7	+	3.99091e6i	−0.333101	+	0.0877136i
$358$	3.24636e7			0.707535
$359$			1.01375e7i			0.219103i	0.993981	+	0.109552i	$0.0349415\pi$
$359$			1.01375e7i			0.219103i	−0.993981	+	0.109552i	$0.965059\pi$
$360$	−7.56905e7	+	4.28323e7i	−1.62231	+	0.918045i
$361$	−4.70226e7			−0.999506
$362$		−	5.25883e7i		−	1.10857i
$363$	9.80458e6	+	3.72338e7i	0.204979	+	0.778426i
$364$	2.09221e7			0.433812
$365$			4.05566e7i			0.834033i
$366$	−1.88843e7	+	4.97270e6i	−0.385175	+	0.101426i
$367$	−5.45232e6			−0.110302			−0.0551510	−	0.998478i	$-0.517564\pi$
$367$	−5.45232e6			−0.110302			−0.0551510	+	0.998478i	$0.517564\pi$
$368$			2.46050e7i			0.493719i
$369$	1.65909e7	+	2.93183e7i	0.330211	+	0.583526i
$370$	−4.95273e7			−0.977776
$371$			6.70815e7i			1.31365i
$372$	−706858.	−	2.68436e6i	−0.0137310	−	0.0521449i
$373$	−8.27614e7			−1.59478			−0.797391	−	0.603463i	$-0.793787\pi$
$373$	−8.27614e7			−1.59478			−0.797391	+	0.603463i	$0.793787\pi$
$374$		−	1.35341e7i		−	0.258710i
$375$	8.20330e7	−	2.16013e7i	1.55559	−	0.409625i
$376$	2.08766e7			0.392732
$377$			7.07771e7i			1.32090i
$378$	4.36193e7	−	4.25031e7i	0.807612	−	0.786947i
$379$	−3.11189e7			−0.571618			−0.285809	−	0.958287i	$-0.592262\pi$
$379$	−3.11189e7			−0.571618			−0.285809	+	0.958287i	$0.592262\pi$
$380$		−	772954.i		−	0.0140865i
$381$	−2.34661e7	−	8.91147e7i	−0.424293	−	1.61129i
$382$	5.99927e7			1.07624
$383$			3.24145e7i			0.576957i	0.957486	+	0.288479i	$0.0931494\pi$
$383$			3.24145e7i			0.576957i	−0.957486	+	0.288479i	$0.906851\pi$
$384$	−7.39339e6	+	1.94686e6i	−0.130572	+	0.0343828i
$385$	−1.86651e8			−3.27076
$386$		−	5.32986e6i		−	0.0926732i
$387$	6.18171e7	−	3.49815e7i	1.06654	−	0.603540i
$388$	−2.65416e7			−0.454394
$389$			3.38064e7i			0.574314i	0.957883	+	0.287157i	$0.0927102\pi$
$389$			3.38064e7i			0.574314i	−0.957883	+	0.287157i	$0.907290\pi$
$390$	1.69898e7	+	6.45203e7i	0.286414	+	1.08768i
$391$	1.44962e7			0.242507
$392$			6.66192e7i			1.10596i
$393$	8.44559e7	−	2.22393e7i	1.39140	−	0.366390i
$394$	2.46200e7			0.402531
$395$		−	1.64986e8i		−	2.67704i
$396$	−1.51869e7	−	2.68373e7i	−0.244559	−	0.432169i
$397$	−3.42933e7			−0.548072			−0.274036	−	0.961719i	$-0.588359\pi$
$397$	−3.42933e7			−0.548072			−0.274036	+	0.961719i	$0.588359\pi$
$398$			7.59087e6i			0.120404i
$399$	510754.	+	1.93964e6i	0.00804069	+	0.0305353i
$400$	6.12574e7			0.957147
$401$		−	1.30045e7i		−	0.201679i	−0.994903	−	0.100840i	$-0.967847\pi$
$401$		−	1.30045e7i		−	0.201679i	0.994903	−	0.100840i	$-0.0321529\pi$
$402$	−6.21076e7	+	1.63545e7i	−0.956019	+	0.251743i
$403$	−7.89114e6			−0.120566
$404$			2.66985e7i			0.404895i
$405$	−9.76187e7	−	5.86320e7i	−1.46949	−	0.882612i
$406$	−1.20617e8			−1.80232
$407$		−	6.50722e7i		−	0.965188i
$408$	4.56133e6	+	1.73221e7i	0.0671600	+	0.255046i
$409$	1.53722e7			0.224681			0.112341	−	0.993670i	$-0.464165\pi$
$409$	1.53722e7			0.224681			0.112341	+	0.993670i	$0.464165\pi$
$410$		−	6.28924e7i		−	0.912528i
$411$	1.07473e8	−	2.83002e7i	1.54801	−	0.407629i
$412$	2.27634e7			0.325496
$413$			4.86165e7i			0.690134i
$414$	−4.90263e7	+	2.77434e7i	−0.690920	+	0.390984i
$415$	−3.34216e7			−0.467609
$416$		−	4.13719e7i		−	0.574679i
$417$	−3.56053e7	−	1.35215e8i	−0.491028	−	1.86473i
$418$	−1.73209e6			−0.0237160
$419$		−	8.18825e7i		−	1.11314i	−0.830802	−	0.556569i	$-0.812118\pi$
$419$		−	8.18825e7i		−	1.11314i	0.830802	−	0.556569i	$-0.187882\pi$
$420$	6.44687e7	−	1.69762e7i	0.870163	−	0.229136i
$421$	−4.36687e7			−0.585226			−0.292613	−	0.956231i	$-0.594525\pi$
$421$	−4.36687e7			−0.585226			−0.292613	+	0.956231i	$0.594525\pi$
$422$			1.14021e6i			0.0151722i
$423$	1.34624e7	+	2.37898e7i	0.177869	+	0.314318i
$424$	7.66695e7			1.00583
$425$		−	3.60902e7i		−	0.470135i
$426$	7.81260e6	+	2.96691e7i	0.101057	+	0.383773i
$427$	5.54693e7			0.712474
$428$		−	3.99644e7i		−	0.509732i
$429$	−8.47710e7	+	2.23223e7i	−1.07368	+	0.282727i
$430$	−1.32607e8			−1.66787
$431$			2.39809e7i			0.299526i	0.988722	+	0.149763i	$0.0478511\pi$
$431$			2.39809e7i			0.299526i	−0.988722	+	0.149763i	$0.952149\pi$
$432$	−2.77822e7	−	2.85117e7i	−0.344600	−	0.353649i
$433$	7.82361e7			0.963704			0.481852	−	0.876253i	$-0.339964\pi$
$433$	7.82361e7			0.963704			0.481852	+	0.876253i	$0.339964\pi$
$434$		−	1.34480e7i		−	0.164508i
$435$	5.74285e7	+	2.18090e8i	0.697685	+	2.64952i
$436$	2.38957e7			0.288310
$437$		−	1.85522e6i		−	0.0222306i
$438$	−3.13903e7	+	8.26584e6i	−0.373571	+	0.0983705i
$439$	−7.65437e7			−0.904723			−0.452362	−	0.891835i	$-0.649418\pi$
$439$	−7.65437e7			−0.904723			−0.452362	+	0.891835i	$0.649418\pi$
$440$			2.13329e8i			2.50433i
$441$	−7.59155e7	+	4.29597e7i	−0.885146	+	0.500894i
$442$	1.37419e7			0.159140
$443$		−	4.83048e7i		−	0.555622i	−0.960636	−	0.277811i	$-0.910391\pi$
$443$		−	4.83048e7i		−	0.555622i	0.960636	−	0.277811i	$-0.0896089\pi$
$444$	5.91841e6	+	2.24757e7i	0.0676170	+	0.256782i
$445$	3.97848e6			0.0451478
$446$			2.30963e7i			0.260339i
$447$	−8.82372e7	+	2.32350e7i	−0.987936	+	0.260148i
$448$	1.33561e8			1.48541
$449$			6.18093e7i			0.682833i	0.939912	+	0.341417i	$0.110907\pi$
$449$			6.18093e7i			0.682833i	−0.939912	+	0.341417i	$0.889093\pi$
$450$	6.90709e7	+	1.22058e8i	0.757980	+	1.33945i
$451$	8.26321e7			0.900781
$452$			2.55417e7i			0.276589i
$453$	3.48828e7	+	1.32471e8i	0.375247	+	1.42504i
$454$	7.67510e7			0.820194
$455$		−	1.89517e8i		−	2.01193i
$456$	2.21687e6	−	583757.i	0.0233801	−	0.00615655i
$457$	−5.88422e7			−0.616510			−0.308255	−	0.951304i	$-0.599745\pi$
$457$	−5.88422e7			−0.616510			−0.308255	+	0.951304i	$0.599745\pi$
$458$		−	9.03999e7i		−	0.940961i
$459$	−1.67979e7	+	1.63681e7i	−0.173707	+	0.169262i
$460$	−6.16627e7			−0.633504
$461$		−	1.42554e8i		−	1.45504i	−0.686085	−	0.727521i	$-0.740672\pi$
$461$		−	1.42554e8i		−	1.45504i	0.686085	−	0.727521i	$-0.259328\pi$
$462$	−3.80413e7	−	1.44466e8i	−0.385771	−	1.46500i
$463$	−7.66224e7			−0.771992			−0.385996	−	0.922501i	$-0.626142\pi$
$463$	−7.66224e7			−0.771992			−0.385996	+	0.922501i	$0.626142\pi$
$464$			7.88414e7i			0.789224i
$465$	−2.43155e7	+	6.40287e6i	−0.241838	+	0.0636818i
$466$	8.49702e7			0.839670
$467$			1.18640e8i			1.16488i	0.812873	+	0.582440i	$0.197902\pi$
$467$			1.18640e8i			1.16488i	−0.812873	+	0.582440i	$0.802098\pi$
$468$	2.72494e7	−	1.54201e7i	0.265839	−	0.150435i
$469$	1.82430e8			1.76839
$470$		−	5.10328e7i		−	0.491536i
$471$	−1.38413e7	−	5.25637e7i	−0.132469	−	0.503064i
$472$	5.55652e7			0.528418
$473$		−	1.74228e8i		−	1.64640i
$474$	1.27697e8	−	3.36257e7i	1.19907	−	0.315745i
$475$	−4.61881e6			−0.0430972
$476$		−	1.37309e7i		−	0.127314i
$477$	4.94407e7	+	8.73683e7i	0.455543	+	0.805006i
$478$	−5.71365e7			−0.523155
$479$		−	4.04655e7i		−	0.368196i	−0.982908	−	0.184098i	$-0.941064\pi$
$479$		−	4.04655e7i		−	0.368196i	0.982908	−	0.184098i	$-0.0589363\pi$
$480$	−3.35691e7	−	1.27482e8i	−0.303540	−	1.15272i
$481$	6.60713e7			0.593714
$482$			1.09438e8i			0.977302i
$483$	1.54735e8	−	4.07456e7i	1.37324	−	0.361609i
$484$	−3.37331e7			−0.297522
$485$			2.40420e8i			2.10739i
$486$	2.54848e7	−	8.75054e7i	0.222010	−	0.762300i
$487$	−2.71676e7			−0.235215			−0.117607	−	0.993060i	$-0.537522\pi$
$487$	−2.71676e7			−0.235215			−0.117607	+	0.993060i	$0.537522\pi$
$488$		−	6.33975e7i		−	0.545523i
$489$	3.80334e7	+	1.44435e8i	0.325266	+	1.23523i
$490$	1.62850e8			1.38421
$491$			4.47113e7i			0.377722i	0.982004	+	0.188861i	$0.0604796\pi$
$491$			4.47113e7i			0.377722i	−0.982004	+	0.188861i	$0.939520\pi$
$492$	−2.85408e7	+	7.51551e6i	−0.239647	+	0.0631049i
$493$	4.64500e7			0.387654
$494$		−	1.75868e6i		−	0.0145883i
$495$	−2.43098e8	+	1.37566e8i	−2.00432	+	1.13422i
$496$	−8.79026e6			−0.0720372
$497$		−	8.71476e7i		−	0.709882i
$498$	−6.81166e6	−	2.58679e7i	−0.0551524	−	0.209446i
$499$	−2.31136e8			−1.86023			−0.930113	−	0.367272i	$-0.880292\pi$
$499$	−2.31136e8			−1.86023			−0.930113	+	0.367272i	$0.880292\pi$
$500$			7.43202e7i			0.594562i
$501$	−6.03290e7	+	1.58861e7i	−0.479748	+	0.126329i
$502$	−9.15122e7			−0.723382
$503$		−	1.69165e8i		−	1.32925i	−0.747177	−	0.664625i	$-0.768591\pi$
$503$		−	1.69165e8i		−	1.32925i	0.747177	−	0.664625i	$-0.231409\pi$
$504$	9.73771e7	+	1.72078e8i	0.760615	+	1.34411i
$505$	2.41840e8			1.87782
$506$			1.38178e8i			1.06656i
$507$	1.05212e7	+	3.99551e7i	0.0807308	+	0.306583i
$508$	8.07361e7			0.615852
$509$			1.68108e8i			1.27478i	0.770540	+	0.637391i	$0.219987\pi$
$509$			1.68108e8i			1.27478i	−0.770540	+	0.637391i	$0.780013\pi$
$510$	4.23438e7	−	1.11502e7i	0.319212	−	0.0840563i
$511$	9.22034e7			0.691010
$512$		−	1.18154e8i		−	0.880315i
$513$	2.09478e6	+	2.14978e6i	0.0155162	+	0.0159237i
$514$	1.56514e8			1.15256
$515$		−	2.06196e8i		−	1.50959i
$516$	1.58463e7	+	6.01777e7i	0.115340	+	0.438013i
$517$	6.70502e7			0.485208
$518$			1.12598e8i			0.810103i
$519$	5.16349e7	−	1.35967e7i	0.369352	−	0.0972596i
$520$	−2.16605e8			−1.54049
$521$		−	1.71182e8i		−	1.21044i	−0.796057	−	0.605221i	$-0.793085\pi$
$521$		−	1.71182e8i		−	1.21044i	0.796057	−	0.605221i	$-0.206915\pi$
$522$	−1.57094e8	+	8.88978e7i	−1.10446	+	0.624999i
$523$	−9.56859e7			−0.668872			−0.334436	−	0.942419i	$-0.608546\pi$
$523$	−9.56859e7			−0.668872			−0.334436	+	0.942419i	$0.608546\pi$
$524$			7.65153e7i			0.531808i
$525$	−1.01442e8	−	3.85234e8i	−0.701033	−	2.66224i
$526$	−1.52781e8			−1.04982
$527$			5.17884e6i			0.0353835i
$528$	−9.44298e7	+	2.48657e7i	−0.641516	+	0.168927i
$529$	35281.4			0.000238330
$530$		−	1.87418e8i		−	1.25888i
$531$	3.58315e7	+	6.33191e7i	0.239321	+	0.422913i
$532$	−1.75727e6			−0.0116709
$533$			8.39008e7i			0.554096i
$534$	810853.	+	3.07929e6i	0.00532499	+	0.0202221i
$535$	−3.62006e8			−2.36404
$536$		−	2.08505e8i		−	1.35401i
$537$	−1.33447e8	+	3.51399e7i	−0.861759	+	0.226922i
$538$	9.65620e7			0.620097
$539$			2.13964e8i			1.36639i
$540$	7.14534e7	−	6.96251e7i	0.453776	−	0.442165i
$541$	−1.41289e8			−0.892312			−0.446156	−	0.894955i	$-0.647207\pi$
$541$	−1.41289e8			−0.892312			−0.446156	+	0.894955i	$0.647207\pi$
$542$			8.18178e7i			0.513866i
$543$	5.69236e7	+	2.16173e8i	0.355543	+	1.35021i
$544$	−2.71518e7			−0.168656
$545$		−	2.16452e8i		−	1.33713i
$546$	1.46684e8	−	3.86254e7i	0.901164	−	0.237299i
$547$	−1.26863e8			−0.775127			−0.387564	−	0.921843i	$-0.626683\pi$
$547$	−1.26863e8			−0.775127			−0.387564	+	0.921843i	$0.626683\pi$
$548$			9.73682e7i			0.591664i
$549$	7.22443e7	−	4.08822e7i	0.436603	−	0.247068i
$550$	3.44012e8			2.06769
$551$		−	5.94465e6i		−	0.0355362i
$552$	−4.65694e7	−	1.76852e8i	−0.276875	−	1.05146i
$553$	−3.75086e8			−2.21797
$554$		−	1.36426e8i		−	0.802359i
$555$	2.03590e8	−	5.36102e7i	1.19090	−	0.313595i
$556$	1.22502e8			0.712717
$557$		−	6.48692e7i		−	0.375382i	−0.982228	−	0.187691i	$-0.939900\pi$
$557$		−	6.48692e7i		−	0.375382i	0.982228	−	0.187691i	$-0.0601003\pi$
$558$	−9.91147e6	−	1.75149e7i	−0.0570474	−	0.100810i
$559$	1.76903e8			1.01274
$560$		−	2.11110e8i		−	1.20211i
$561$	1.46498e7	+	5.56340e7i	0.0829742	+	0.315102i
$562$	−1.82423e8			−1.02771
$563$			7.56366e7i			0.423845i	0.977286	+	0.211922i	$0.0679724\pi$
$563$			7.56366e7i			0.423845i	−0.977286	+	0.211922i	$0.932028\pi$
$564$	−2.31589e7	+	6.09831e6i	−0.129086	+	0.0339917i
$565$	2.31362e8			1.28277
$566$			1.53975e8i			0.849184i
$567$	−1.33297e8	+	2.21931e8i	−0.731258	+	1.21750i
$568$	−9.96036e7			−0.543538
$569$		−	1.80647e8i		−	0.980603i	−0.871553	−	0.490302i	$-0.836887\pi$
$569$		−	1.80647e8i		−	0.980603i	0.871553	−	0.490302i	$-0.163113\pi$
$570$	−1.42699e6	−	5.41913e6i	−0.00770543	−	0.0292621i
$571$	1.29391e8			0.695015			0.347507	−	0.937677i	$-0.387028\pi$
$571$	1.29391e8			0.695015			0.347507	+	0.937677i	$0.387028\pi$
$572$		−	7.68008e7i		−	0.410372i
$573$	−2.46610e8	+	6.49384e7i	−1.31083	+	0.345174i
$574$	−1.42983e8			−0.756044
$575$			3.68467e8i			1.93819i
$576$	1.73952e8	−	9.84375e7i	0.910254	−	0.515102i
$577$	7.99462e7			0.416169			0.208085	−	0.978111i	$-0.433277\pi$
$577$	7.99462e7			0.416169			0.208085	+	0.978111i	$0.433277\pi$
$578$		−	9.01860e6i		−	0.0467042i
$579$	5.76925e6	+	2.19093e7i	0.0297224	+	0.112873i
$580$	−1.97585e8			−1.01268
$581$			7.59823e7i			0.387422i
$582$	−1.86082e8	+	4.89999e7i	−0.943919	+	0.248557i
$583$	2.46243e8			1.24268
$584$		−	1.05382e8i		−	0.529088i
$585$	−1.39679e8	−	2.46831e8i	−0.697689	−	1.23291i
$586$	2.24878e7			0.111752
$587$		−	3.37859e7i		−	0.167040i	−0.996506	−	0.0835201i	$-0.973384\pi$
$587$		−	3.37859e7i		−	0.167040i	0.996506	−	0.0835201i	$-0.0266163\pi$
$588$	−1.94603e7	−	7.39023e7i	−0.0957233	−	0.363518i
$589$	662786.			0.00324360
$590$		−	1.35829e8i		−	0.661358i
$591$	−1.01204e8	+	2.66496e7i	−0.490272	+	0.129101i
$592$	7.35994e7			0.354739
$593$			3.22407e8i			1.54611i	0.634338	+	0.773056i	$0.281273\pi$
$593$			3.22407e8i			1.54611i	−0.634338	+	0.773056i	$0.718727\pi$
$594$	−1.56020e8	−	1.60117e8i	−0.744427	−	0.763975i
$595$	−1.24377e8			−0.590459
$596$		−	7.99411e7i		−	0.377599i
$597$	−8.21665e6	−	3.12035e7i	−0.0386164	−	0.146649i
$598$	−1.40299e8			−0.656073
$599$			4.90496e7i			0.228221i	0.993468	+	0.114110i	$0.0364018\pi$
$599$			4.90496e7i			0.228221i	−0.993468	+	0.114110i	$0.963598\pi$
$600$	−4.40296e8	+	1.15941e8i	−2.03841	+	0.536763i
$601$	1.26390e8			0.582221			0.291111	−	0.956689i	$-0.405975\pi$
$601$	1.26390e8			0.582221			0.291111	+	0.956689i	$0.405975\pi$
$602$			3.01475e8i			1.38185i
$603$	2.37600e8	−	1.34455e8i	1.08367	−	0.613233i
$604$	−1.20016e8			−0.544663
$605$			3.05561e8i			1.37985i
$606$	4.92895e7	+	1.87181e8i	0.221481	+	0.841094i
$607$	−2.19519e8			−0.981536			−0.490768	−	0.871290i	$-0.663284\pi$
$607$	−2.19519e8			−0.981536			−0.490768	+	0.871290i	$0.663284\pi$
$608$			3.47487e6i			0.0154607i
$609$	4.95816e8	−	1.30561e8i	2.19517	−	0.578043i
$610$	−1.54975e8			−0.682767
$611$			6.80797e7i			0.298465i
$612$	−1.01200e7	−	1.78834e7i	−0.0441495	−	0.0780181i
$613$	−2.37427e8			−1.03074			−0.515370	−	0.856968i	$-0.672345\pi$
$613$	−2.37427e8			−1.03074			−0.515370	+	0.856968i	$0.672345\pi$
$614$			6.80568e7i			0.294013i
$615$	6.80771e7	+	2.58529e8i	0.292668	+	1.11143i
$616$	4.84993e8			2.07488
$617$			4.47246e7i			0.190411i	0.995458	+	0.0952053i	$0.0303507\pi$
$617$			4.47246e7i			0.190411i	−0.995458	+	0.0952053i	$0.969649\pi$
$618$	1.59593e8	−	4.20247e7i	0.676157	−	0.178049i
$619$	2.11726e8			0.892695			0.446347	−	0.894860i	$-0.352725\pi$
$619$	2.11726e8			0.892695			0.446347	+	0.894860i	$0.352725\pi$
$620$		−	2.20293e7i		−	0.0924328i
$621$	1.71500e8	−	1.67112e8i	0.716125	−	0.697801i
$622$	−2.64013e8			−1.09712
$623$		−	9.04486e6i		−	0.0374057i
$624$	−2.52475e7	−	9.58797e7i	−0.103912	−	0.394614i
$625$	1.99962e8			0.819043
$626$		−	8.48310e7i		−	0.345805i
$627$	7.12001e6	−	1.87488e6i	0.0288854	−	0.00760624i
$628$	4.76216e7			0.192276
$629$		−	4.33616e7i		−	0.174242i
$630$	4.20645e8	−	2.38038e8i	1.68226	−	0.951973i
$631$	3.16146e7			0.125835			0.0629173	−	0.998019i	$-0.479960\pi$
$631$	3.16146e7			0.125835			0.0629173	+	0.998019i	$0.479960\pi$
$632$			4.28697e8i			1.69824i
$633$	−1.23421e6	−	4.68701e6i	−0.00486605	−	0.0184793i
$634$	1.85086e7			0.0726283
$635$		−	7.31325e8i		−	2.85620i
$636$	−8.50514e7	+	2.23961e7i	−0.330605	+	0.0870566i
$637$	−2.17249e8			−0.840503
$638$			4.42761e8i			1.70493i
$639$	−6.42298e7	−	1.13503e8i	−0.246169	−	0.435014i
$640$	−6.06742e7			−0.231454
$641$			8.03800e6i			0.0305192i	0.999884	+	0.0152596i	$0.00485748\pi$
$641$			8.03800e6i			0.0305192i	−0.999884	+	0.0152596i	$0.995143\pi$
$642$	−7.37803e7	−	2.80188e8i	−0.278827	−	1.05887i
$643$	−1.79294e7			−0.0674425			−0.0337213	−	0.999431i	$-0.510736\pi$
$643$	−1.79294e7			−0.0674425			−0.0337213	+	0.999431i	$0.510736\pi$
$644$			1.40187e8i			0.524868i
$645$	5.45102e8	−	1.43539e8i	2.03142	−	0.534922i
$646$	−1.15420e6			−0.00428136
$647$			2.33792e8i			0.863211i	0.902063	+	0.431605i	$0.142053\pi$
$647$			2.33792e8i			0.863211i	−0.902063	+	0.431605i	$0.857947\pi$
$648$	2.53652e8	+	1.52349e8i	0.932208	+	0.559905i
$649$	1.78461e8			0.652844
$650$			3.49294e8i			1.27189i
$651$	1.45566e7	+	5.52800e7i	0.0527614	+	0.200366i
$652$	−1.30856e8			−0.472117
$653$		−	3.00491e8i		−	1.07917i	−0.841930	−	0.539587i	$-0.818581\pi$
$653$		−	3.00491e8i		−	1.07917i	0.841930	−	0.539587i	$-0.181419\pi$
$654$	1.67531e8	−	4.41151e7i	0.598911	−	0.157708i
$655$	6.93092e8			2.46642
$656$			9.34605e7i			0.331067i
$657$	1.20088e8	−	6.79561e7i	0.423450	−	0.239625i
$658$	−1.16020e8			−0.407246
$659$		−	3.62789e7i		−	0.126764i	−0.997989	−	0.0633822i	$-0.979811\pi$
$659$		−	3.62789e7i		−	0.126764i	0.997989	−	0.0633822i	$-0.0201887\pi$
$660$	−6.23161e7	−	2.36651e8i	−0.216755	−	0.823147i
$661$	−5.07516e8			−1.75730			−0.878650	−	0.477467i	$-0.841555\pi$
$661$	−5.07516e8			−1.75730			−0.878650	+	0.477467i	$0.841555\pi$
$662$			3.13914e8i			1.08202i
$663$	−5.64882e7	+	1.48747e7i	−0.193828	+	0.0510398i
$664$	8.68425e7			0.296639
$665$			1.59177e7i			0.0541273i
$666$	8.29872e7	+	1.46649e8i	0.280924	+	0.496430i
$667$	−4.74236e8			−1.59815
$668$		−	5.46569e7i		−	0.183365i
$669$	−2.50004e7	−	9.49412e7i	−0.0834964	−	0.317085i
$670$	−5.09689e8			−1.69465
$671$		−	2.03616e8i		−	0.673977i
$672$	−2.89823e8	+	7.63176e7i	−0.955049	+	0.251488i
$673$	3.85589e8			1.26497			0.632485	−	0.774573i	$-0.282035\pi$
$673$	3.85589e8			1.26497			0.632485	+	0.774573i	$0.282035\pi$
$674$		−	2.21504e8i		−	0.723439i
$675$	−4.16047e8	−	4.26972e8i	−1.35279	−	1.38831i
$676$	−3.61985e7			−0.117179
$677$		−	2.10320e8i		−	0.677819i	−0.940819	−	0.338910i	$-0.889942\pi$
$677$		−	2.10320e8i		−	0.677819i	0.940819	−	0.338910i	$-0.110058\pi$
$678$	4.71539e7	+	1.79071e8i	0.151296	+	0.574562i
$679$	5.46582e8			1.74601
$680$			1.42154e8i			0.452099i
$681$	−3.15497e8	+	8.30782e7i	−0.998974	+	0.263055i
$682$	−4.93647e7			−0.155619
$683$		−	1.67096e7i		−	0.0524450i	−0.999656	−	0.0262225i	$-0.991652\pi$
$683$		−	1.67096e7i		−	0.0524450i	0.999656	−	0.0262225i	$-0.00834783\pi$
$684$	−2.28871e6	+	1.29515e6i	−0.00715190	+	0.00404718i
$685$	8.81981e8			2.74402
$686$		−	6.20440e6i		−	0.0192189i
$687$	9.78523e7	+	3.71603e8i	0.301787	+	1.14606i
$688$	1.97059e8			0.605106
$689$			2.50023e8i			0.764404i
$690$	−4.32314e8	+	1.13839e8i	−1.31599	+	0.346532i
$691$	−1.27470e8			−0.386343			−0.193172	−	0.981165i	$-0.561877\pi$
$691$	−1.27470e8			−0.386343			−0.193172	+	0.981165i	$0.561877\pi$
$692$			4.67801e7i			0.141170i
$693$	3.12750e8	+	5.52671e8i	0.939718	+	1.66061i
$694$	2.60919e8			0.780597
$695$		−	1.10965e9i		−	3.30544i
$696$	−1.49222e8	−	5.66683e8i	−0.442593	−	1.68079i
$697$	5.50629e7			0.162615
$698$			1.49355e8i			0.439190i
$699$	−3.49283e8	+	9.19749e7i	−1.02270	+	0.269301i
$700$	3.49014e8			1.01753
$701$			5.24585e8i			1.52287i	0.648243	+	0.761433i	$0.275504\pi$
$701$			5.24585e8i			1.52287i	−0.648243	+	0.761433i	$0.724496\pi$
$702$	1.62576e8	−	1.58416e8i	0.469943	−	0.457918i
$703$	−5.54940e6			−0.0159728
$704$		−	4.90275e8i		−	1.40515i
$705$	5.52398e7	+	2.09778e8i	0.157647	+	0.598678i
$706$	3.14090e8			0.892565
$707$		−	5.49812e8i		−	1.55581i
$708$	−6.16399e7	+	1.62313e7i	−0.173685	+	0.0457355i
$709$	2.79644e8			0.784633			0.392316	−	0.919830i	$-0.371674\pi$
$709$	2.79644e8			0.784633			0.392316	+	0.919830i	$0.371674\pi$
$710$			2.43481e8i			0.680283i
$711$	−4.88520e8	+	2.76447e8i	−1.35917	+	0.769137i
$712$	−1.03376e7			−0.0286406
$713$		−	5.28740e7i		−	0.145873i
$714$	−2.53493e7	−	9.62663e7i	−0.0696420	−	0.264472i
$715$	−6.95678e8			−1.90323
$716$		−	1.20900e8i		−	0.329373i
$717$	2.34869e8	−	6.18467e7i	0.637188	−	0.167787i
$718$	−6.43912e7			−0.173962
$719$			2.52270e8i			0.678702i	0.940660	+	0.339351i	$0.110207\pi$
$719$			2.52270e8i			0.678702i	−0.940660	+	0.339351i	$0.889793\pi$
$720$	−1.55593e8	−	2.74955e8i	−0.416863	−	0.736654i
$721$	−4.68775e8			−1.25072
$722$		−	2.98677e8i		−	0.793578i
$723$	−1.18460e8	−	4.49864e8i	−0.313443	−	1.19033i
$724$	−1.95848e8			−0.516064
$725$			1.18067e9i			3.09825i
$726$	−2.36500e8	+	6.22764e7i	−0.618047	+	0.162747i
$727$	9.96358e7			0.259306			0.129653	−	0.991559i	$-0.458614\pi$
$727$	9.96358e7			0.259306			0.129653	+	0.991559i	$0.458614\pi$
$728$			4.92440e8i			1.27632i
$729$	−1.00401e7	+	3.87290e8i	−0.0259152	+	0.999664i
$730$	−2.57606e8			−0.662198
$731$		−	1.16099e8i		−	0.297218i
$732$	1.85192e7	+	7.03284e7i	0.0472160	+	0.179307i
$733$	6.85192e7			0.173981			0.0869903	−	0.996209i	$-0.472275\pi$
$733$	6.85192e7			0.173981			0.0869903	+	0.996209i	$0.472275\pi$
$734$		−	3.46319e7i		−	0.0875765i
$735$	−6.69423e8	+	1.76276e8i	−1.68593	+	0.443946i
$736$	2.77209e8			0.695303
$737$		−	6.69663e8i		−	1.67284i
$738$	−1.86223e8	+	1.05382e8i	−0.463303	+	0.262177i
$739$	3.80330e8			0.942382			0.471191	−	0.882031i	$-0.343824\pi$
$739$	3.80330e8			0.942382			0.471191	+	0.882031i	$0.343824\pi$
$740$			1.84448e8i			0.455176i
$741$	1.90366e6	+	7.22934e6i	0.00467881	+	0.0177682i
$742$	−4.26086e8			−1.04300
$743$		−	5.14155e7i		−	0.125351i	−0.998034	−	0.0626754i	$-0.980037\pi$
$743$		−	5.14155e7i		−	0.125351i	0.998034	−	0.0626754i	$-0.0199633\pi$
$744$	6.31812e7	−	1.66372e7i	0.153415	−	0.0403981i
$745$	−7.24123e8			−1.75123
$746$		−	5.25681e8i		−	1.26621i
$747$	5.60008e7	+	9.89609e7i	0.134348	+	0.237412i
$748$	−5.04033e7			−0.120435
$749$			8.23001e8i			1.95864i
$750$	1.37206e8	+	5.21054e8i	0.325230	+	1.23509i
$751$	2.18343e8			0.515489			0.257744	−	0.966213i	$-0.417021\pi$
$751$	2.18343e8			0.515489			0.257744	+	0.966213i	$0.417021\pi$
$752$			7.58367e7i			0.178330i
$753$	3.76175e8	−	9.90563e7i	0.881060	−	0.232005i
$754$	−4.49560e8			−1.04875
$755$			1.08713e9i			2.52604i
$756$	−1.58289e8	−	1.62446e8i	−0.366341	−	0.375961i
$757$	1.26532e7			0.0291684			0.0145842	−	0.999894i	$-0.495358\pi$
$757$	1.26532e7			0.0291684			0.0145842	+	0.999894i	$0.495358\pi$
$758$		−	1.97660e8i		−	0.453848i
$759$	−1.49569e8	−	5.68002e8i	−0.342071	−	1.29905i
$760$	1.81929e7			0.0414439
$761$		−	8.36458e7i		−	0.189797i	−0.995487	−	0.0948987i	$-0.969747\pi$
$761$		−	8.36458e7i		−	0.189797i	0.995487	−	0.0948987i	$-0.0302527\pi$
$762$	5.66036e8	−	1.49051e8i	1.27932	−	0.336876i
$763$	−4.92093e8			−1.10783
$764$		−	2.23423e8i		−	0.501012i
$765$	−1.61991e8	+	9.16690e7i	−0.361833	+	0.204757i
$766$	−2.05890e8			−0.458087
$767$			1.81201e8i			0.401583i
$768$	1.08278e8	+	4.11195e8i	0.239032	+	0.907745i
$769$	3.90789e8			0.859337			0.429669	−	0.902987i	$-0.358630\pi$
$769$	3.90789e8			0.859337			0.429669	+	0.902987i	$0.358630\pi$
$770$		−	1.18556e9i		−	2.59689i
$771$	−6.43376e8	+	1.69417e8i	−1.40379	+	0.369652i
$772$	−1.98493e7			−0.0431414
$773$			2.91270e8i			0.630605i	0.948991	+	0.315303i	$0.102106\pi$
$773$			2.91270e8i			0.630605i	−0.948991	+	0.315303i	$0.897894\pi$
$774$	2.22195e8	+	3.92647e8i	0.479193	+	0.846798i
$775$	−1.31637e8			−0.282795
$776$		−	6.24705e8i		−	1.33687i
$777$	−1.21880e8	−	4.62851e8i	−0.259818	−	0.986683i
$778$	−2.14730e8			−0.455989
$779$		−	7.04692e6i		−	0.0149069i
$780$	2.40285e8	−	6.32730e7i	0.506341	−	0.133332i
$781$	−3.19901e8			−0.671525
$782$			9.20764e7i			0.192543i
$783$	5.49535e8	−	5.35473e8i	1.14475	−	1.11546i
$784$	−2.42002e8			−0.502193
$785$		−	4.31367e8i		−	0.891739i
$786$	1.41259e8	+	5.36444e8i	0.290903	+	1.10473i
$787$	−6.97895e8			−1.43175			−0.715873	−	0.698230i	$-0.753971\pi$
$787$	−6.97895e8			−1.43175			−0.715873	+	0.698230i	$0.753971\pi$
$788$		−	9.16892e7i		−	0.187387i
$789$	6.28032e8	−	1.65376e8i	1.27865	−	0.336699i
$790$	1.04795e9			2.12549
$791$		−	5.25990e8i		−	1.06279i
$792$	6.31664e8	−	3.57451e8i	1.27148	−	0.719518i
$793$	2.06743e8			0.414582
$794$		−	2.17823e8i		−	0.435153i
$795$	2.02869e8	+	7.70413e8i	0.403751	+	1.53328i
$796$	2.82697e7			0.0560508
$797$		−	8.61715e8i		−	1.70211i	−0.525073	−	0.851057i	$-0.675962\pi$
$797$		−	8.61715e8i		−	1.70211i	0.525073	−	0.851057i	$-0.324038\pi$
$798$	−1.23201e7	+	3.24419e6i	−0.0242441	+	0.00638407i
$799$	4.46797e7			0.0875931
$800$		−	6.90149e8i		−	1.34795i
$801$	−6.66628e6	−	1.17802e7i	−0.0129714	−	0.0229222i
$802$	8.26017e7			0.160128
$803$		−	3.38460e8i		−	0.653673i
$804$	6.09068e7	+	2.31299e8i	0.117192	+	0.445047i
$805$	1.26984e9			2.43424
$806$		−	5.01227e7i		−	0.0957259i
$807$	−3.96933e8	+	1.04522e8i	−0.755261	+	0.198879i
$808$	−6.28396e8			−1.19124
$809$		−	7.23236e8i		−	1.36595i	−0.730442	−	0.682975i	$-0.760686\pi$
$809$		−	7.23236e8i		−	1.36595i	0.730442	−	0.682975i	$-0.239314\pi$
$810$	3.72417e8	−	6.20051e8i	0.700768	−	1.16674i
$811$	8.90755e8			1.66992			0.834961	−	0.550310i	$-0.185490\pi$
$811$	8.90755e8			1.66992			0.834961	+	0.550310i	$0.185490\pi$
$812$			4.49199e8i			0.839018i
$813$	−8.85626e7	−	3.36325e8i	−0.164808	−	0.625875i
$814$	4.13323e8			0.766331
$815$			1.18532e9i			2.18959i
$816$	−6.29245e7	+	1.65696e7i	−0.115811	+	0.0304958i
$817$	−1.48583e7			−0.0272460
$818$			9.76407e7i			0.178390i
$819$	−5.61157e8	+	3.17552e8i	−1.02149	+	0.578047i
$820$	−2.34222e8			−0.424802
$821$		−	4.81595e8i		−	0.870266i	−0.900366	−	0.435133i	$-0.856701\pi$
$821$		−	4.81595e8i		−	0.870266i	0.900366	−	0.435133i	$-0.143299\pi$
$822$	1.79756e8	+	6.82642e8i	0.323645	+	1.22907i
$823$	−8.12461e8			−1.45748			−0.728741	−	0.684789i	$-0.759894\pi$
$823$	−8.12461e8			−1.45748			−0.728741	+	0.684789i	$0.759894\pi$
$824$			5.35777e8i			0.957641i
$825$	−1.41412e9	+	3.72372e8i	−2.51839	+	0.663155i
$826$	−3.08800e8			−0.547946
$827$			2.15815e8i			0.381562i	0.981633	+	0.190781i	$0.0611020\pi$
$827$			2.15815e8i			0.381562i	−0.981633	+	0.190781i	$0.938898\pi$
$828$	1.03321e8	+	1.82582e8i	0.182011	+	0.321638i
$829$	−1.00349e9			−1.76137			−0.880683	−	0.473707i	$-0.842916\pi$
$829$	−1.00349e9			−1.76137			−0.880683	+	0.473707i	$0.842916\pi$
$830$		−	2.12286e8i		−	0.371268i
$831$	1.47673e8	+	5.60802e8i	0.257334	+	0.977252i
$832$	4.97802e8			0.864345
$833$			1.42577e8i			0.246669i
$834$	8.58851e8	−	2.26157e8i	1.48054	−	0.389862i
$835$	−4.95093e8			−0.850409
$836$			6.45059e6i			0.0110403i
$837$	5.97015e7	+	6.12692e7i	0.101814	+	0.104488i
$838$	5.20098e8			0.883799
$839$		−	5.53392e8i		−	0.937017i	−0.883459	−	0.468508i	$-0.844792\pi$
$839$		−	5.53392e8i		−	0.937017i	0.883459	−	0.468508i	$-0.155208\pi$
$840$	3.99565e8	+	1.51739e9i	0.674139	+	2.56011i
$841$	−9.24766e8			−1.55469
$842$		−	2.77373e8i		−	0.464652i
$843$	7.49879e8	−	1.97462e8i	1.25172	−	0.329610i
$844$	4.24634e6			0.00706297
$845$			3.27893e8i			0.543454i
$846$	−1.51107e8	+	8.55098e7i	−0.249560	+	0.141223i
$847$	6.94677e8			1.14323
$848$			2.78511e8i			0.456725i
$849$	−1.66669e8	−	6.32940e8i	−0.272352	−	1.03428i
$850$	2.29237e8			0.373274
$851$			4.42706e8i			0.718334i
$852$	1.10493e8	−	2.90955e7i	0.178655	−	0.0470442i
$853$	6.56767e8			1.05819			0.529095	−	0.848562i	$-0.322531\pi$
$853$	6.56767e8			1.05819			0.529095	+	0.848562i	$0.322531\pi$
$854$			3.52328e8i			0.565683i
$855$	1.17318e7	+	2.07316e7i	0.0187700	+	0.0331691i
$856$	9.40633e8			1.49968
$857$			8.40672e8i			1.33562i	0.744330	+	0.667812i	$0.232769\pi$
$857$			8.40672e8i			1.33562i	−0.744330	+	0.667812i	$0.767231\pi$
$858$	−1.41786e8	−	5.38446e8i	−0.224477	−	0.852472i
$859$	8.18475e8			1.29130			0.645648	−	0.763635i	$-0.276587\pi$
$859$	8.18475e8			1.29130			0.645648	+	0.763635i	$0.276587\pi$
$860$			4.93852e8i			0.776428i
$861$	5.87752e8	−	1.54770e8i	0.920841	−	0.242480i
$862$	−1.52321e8			−0.237815
$863$		−	1.06508e8i		−	0.165710i	−0.996562	−	0.0828548i	$-0.973596\pi$
$863$		−	1.06508e8i		−	0.165710i	0.996562	−	0.0828548i	$-0.0264038\pi$
$864$	−3.21224e8	+	3.13004e8i	−0.498043	+	0.485299i
$865$	4.23744e8			0.654720
$866$			4.96938e8i			0.765153i
$867$	9.76207e6	+	3.70724e7i	0.0149791	+	0.0568844i
$868$	−5.00825e7			−0.0765821
$869$			1.37686e9i			2.09813i
$870$	−1.38526e9	+	3.64772e8i	−2.10364	+	0.553942i
$871$	6.79945e8			1.02901
$872$			5.62428e8i			0.848237i
$873$	7.11879e8	−	4.02844e8i	1.06995	−	0.605472i
$874$	1.17839e7			0.0176504
$875$		−	1.53050e9i		−	2.28460i
$876$	3.07834e7	+	1.16903e8i	0.0457936	+	0.173905i
$877$	7.24761e8			1.07447			0.537237	−	0.843431i	$-0.319468\pi$
$877$	7.24761e8			1.07447			0.537237	+	0.843431i	$0.319468\pi$
$878$		−	4.86188e8i		−	0.718324i
$879$	−9.24397e7	+	2.43417e7i	−0.136111	+	0.0358413i
$880$	−7.74944e8			−1.13716
$881$		−	1.05391e9i		−	1.54126i	−0.637283	−	0.770630i	$-0.719942\pi$
$881$		−	1.05391e9i		−	1.54126i	0.637283	−	0.770630i	$-0.280058\pi$
$882$	−2.72870e8	−	4.82198e8i	−0.397695	−	0.702780i
$883$	−5.57224e8			−0.809371			−0.404686	−	0.914456i	$-0.632619\pi$
$883$	−5.57224e8			−0.809371			−0.404686	+	0.914456i	$0.632619\pi$
$884$		−	5.11772e7i		−	0.0740831i
$885$	1.47027e8	+	5.58347e8i	0.212112	+	0.805517i
$886$	3.06821e8			0.441147
$887$		−	3.75397e8i		−	0.537922i	−0.963151	−	0.268961i	$-0.913320\pi$
$887$		−	3.75397e8i		−	0.537922i	0.963151	−	0.268961i	$-0.0866803\pi$
$888$	−5.29006e8	+	1.39300e8i	−0.755478	+	0.198936i
$889$	−1.66263e9			−2.36641
$890$			2.52704e7i			0.0358461i
$891$	8.14663e8	+	4.89306e8i	1.15172	+	0.691747i
$892$	8.60148e7			0.121193
$893$		−	5.71809e6i		−	0.00802965i
$894$	−1.47583e8	−	5.60462e8i	−0.206550	−	0.784393i
$895$	−1.09514e9			−1.52757
$896$			1.37940e8i			0.191763i
$897$	5.76723e8	−	1.51865e8i	0.799079	−	0.210417i
$898$	−3.92598e8			−0.542150
$899$		−	1.69423e8i		−	0.233182i
$900$	4.54563e8	−	2.57232e8i	0.623544	−	0.352856i
$901$	1.64087e8			0.224336
$902$			5.24860e8i			0.715193i
$903$	−3.26328e8	−	1.23926e9i	−0.443192	−	1.68306i
$904$	−6.01170e8			−0.813752
$905$			1.77403e9i			2.39340i
$906$	−8.41424e8	+	2.21568e8i	−1.13144	+	0.297935i
$907$	−6.65060e8			−0.891331			−0.445666	−	0.895199i	$-0.647033\pi$
$907$	−6.65060e8			−0.891331			−0.445666	+	0.895199i	$0.647033\pi$
$908$		−	2.85834e8i		−	0.381818i
$909$	−4.05224e8	−	7.16086e8i	−0.539515	−	0.953396i
$910$	1.20377e9			1.59742
$911$		−	1.76438e8i		−	0.233366i	−0.993169	−	0.116683i	$-0.962774\pi$
$911$		−	1.76438e8i		−	0.233366i	0.993169	−	0.116683i	$-0.0372262\pi$
$912$	2.12057e6	+	8.05304e6i	0.00279555	+	0.0106164i
$913$	2.78916e8			0.366489
$914$		−	3.73752e8i		−	0.489491i
$915$	6.37050e8	−	1.67751e8i	0.831591	−	0.218979i
$916$	−3.36665e8			−0.438037
$917$		−	1.57571e9i		−	2.04347i
$918$	−1.03966e8	−	1.06696e8i	−0.134389	−	0.137918i
$919$	−5.36704e8			−0.691494			−0.345747	−	0.938328i	$-0.612374\pi$
$919$	−5.36704e8			−0.691494			−0.345747	+	0.938328i	$0.612374\pi$
$920$		−	1.45134e9i		−	1.86383i
$921$	−7.36673e7	−	2.79758e8i	−0.0942966	−	0.358100i
$922$	9.05467e8			1.15526
$923$		−	3.24813e8i		−	0.413074i
$924$	−5.38015e8	+	1.41673e8i	−0.681990	+	0.179585i
$925$	1.10217e9			1.39260
$926$		−	4.86687e8i		−	0.612939i
$927$	−6.10542e8	+	3.45498e8i	−0.766437	+	0.433717i
$928$	8.88257e8			1.11146
$929$			7.69413e8i			0.959649i	0.877364	+	0.479824i	$0.159300\pi$
$929$			7.69413e8i			0.959649i	−0.877364	+	0.479824i	$0.840700\pi$
$930$	−4.06695e7	−	1.54446e8i	−0.0505615	−	0.192012i
$931$	1.82470e7			0.0226121
$932$		−	3.16444e8i		−	0.390885i
$933$	1.08527e9	−	2.85778e8i	1.33626	−	0.351872i
$934$	−7.53575e8			−0.924881
$935$			4.56563e8i			0.558555i
$936$	3.62940e8	+	6.41363e8i	0.442595	+	0.782126i
$937$	−2.73895e8			−0.332939			−0.166470	−	0.986047i	$-0.553237\pi$
$937$	−2.73895e8			−0.332939			−0.166470	+	0.986047i	$0.553237\pi$
$938$			1.15875e9i			1.40405i
$939$	9.18243e7	+	3.48711e8i	0.110908	+	0.421182i
$940$	−1.90055e8			−0.228821
$941$		−	5.17644e8i		−	0.621245i	−0.950533	−	0.310622i	$-0.899463\pi$
$941$		−	5.17644e8i		−	0.621245i	0.950533	−	0.310622i	$-0.100537\pi$
$942$	3.33872e8	−	8.79168e7i	0.399418	−	0.105177i
$943$	−5.62171e8			−0.670399
$944$			2.01847e8i			0.239942i
$945$	−1.47147e9	+	1.43382e9i	−1.74363	+	1.69902i
$946$	1.10665e9			1.30719
$947$			1.53275e9i			1.80477i	0.430933	+	0.902384i	$0.358185\pi$
$947$			1.53275e9i			1.80477i	−0.430933	+	0.902384i	$0.641815\pi$
$948$	−1.25228e8	−	4.75564e8i	−0.146986	−	0.558193i
$949$	3.43657e8			0.402093
$950$		−	2.93376e7i		−	0.0342179i
$951$	−7.60825e7	+	2.00344e7i	−0.0884593	+	0.0232935i
$952$	3.23181e8			0.374572
$953$			1.93142e8i			0.223151i	0.993756	+	0.111575i	$0.0355896\pi$
$953$			1.93142e8i			0.223151i	−0.993756	+	0.111575i	$0.964410\pi$
$954$	−5.54943e8	+	3.14036e8i	−0.639151	+	0.361688i
$955$	−2.02382e9			−2.32360
$956$			2.12786e8i			0.243540i
$957$	−4.79262e8	−	1.82004e9i	−0.546810	−	2.07656i
$958$	2.57028e8			0.292337
$959$		−	2.00514e9i		−	2.27347i
$960$	1.53391e9	−	4.03916e8i	1.73375	−	0.456539i
$961$	−8.68614e8			−0.978716
$962$			4.19669e8i			0.471392i
$963$	6.06571e8	+	1.07189e9i	0.679208	+	1.20025i
$964$	4.07568e8			0.454955
$965$			1.79799e8i			0.200081i
$966$	2.58807e8	+	9.82842e8i	0.287107	+	1.09032i
$967$	1.37798e9			1.52393			0.761965	−	0.647618i	$-0.224235\pi$
$967$	1.37798e9			1.52393			0.761965	+	0.647618i	$0.224235\pi$
$968$		−	7.93968e8i		−	0.875340i
$969$	4.74451e6	−	1.24935e6i	0.00521458	−	0.00137313i
$970$	−1.52709e9			−1.67321
$971$			5.16726e7i			0.0564421i	0.999602	+	0.0282210i	$0.00898423\pi$
$971$			5.16726e7i			0.0564421i	−0.999602	+	0.0282210i	$0.991016\pi$
$972$	−3.25885e8	−	9.49097e7i	−0.354867	−	0.103350i
$973$	−2.52272e9			−2.73861
$974$		−	1.72562e8i		−	0.186753i
$975$	−3.78089e8	−	1.43583e9i	−0.407925	−	1.54913i
$976$	2.30299e8			0.247709
$977$			1.56821e9i			1.68160i	0.541349	+	0.840798i	$0.317914\pi$
$977$			1.56821e9i			1.68160i	−0.541349	+	0.840798i	$0.682086\pi$
$978$	−9.17420e8	+	2.41579e8i	−0.980735	+	0.258252i
$979$	−3.32019e7			−0.0353846
$980$		−	6.06483e8i		−	0.644378i
$981$	−6.40911e8	+	3.62684e8i	−0.678877	+	0.384168i
$982$	−2.83995e8			−0.299900
$983$		−	2.83489e8i		−	0.298453i	−0.988803	−	0.149227i	$-0.952322\pi$
$983$		−	2.83489e8i		−	0.298453i	0.988803	−	0.149227i	$-0.0476784\pi$
$984$	−1.76891e8	−	6.71760e8i	−0.185661	−	0.705064i
$985$	−8.30540e8			−0.869064
$986$			2.95039e8i			0.307786i
$987$	4.76920e8	−	1.25585e8i	0.496014	−	0.130613i
$988$	−6.54963e6			−0.00679119
$989$			1.18532e9i			1.22532i
$990$	−8.73789e8	−	1.54410e9i	−0.900536	−	1.59137i
$991$	4.06516e8			0.417693			0.208846	−	0.977948i	$-0.433029\pi$
$991$	4.06516e8			0.417693			0.208846	+	0.977948i	$0.433029\pi$
$992$			9.90344e7i			0.101450i
$993$	−3.39793e8	−	1.29039e9i	−0.347030	−	1.31788i
$994$	5.53541e8			0.563625
$995$		−	2.56073e8i		−	0.259953i
$996$	−9.63365e7	+	2.53678e7i	−0.0975019	+	0.0256746i
$997$	4.29286e8			0.433173			0.216587	−	0.976263i	$-0.430508\pi$
$997$	4.29286e8			0.433173			0.216587	+	0.976263i	$0.430508\pi$
$998$		−	1.46812e9i		−	1.47697i
$999$	−4.99871e8	−	5.12997e8i	−0.501374	−	0.514540i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	51.7.b.a.35.22	yes	32
3.2	odd	2	inner	51.7.b.a.35.11	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.7.b.a.35.11	✓	32	3.2	odd	2	inner
51.7.b.a.35.22	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+6.35177i q^{2} +(-6.87539 - 26.1099i) q^{3} +23.6551 q^{4} -214.273i q^{5} +(165.844 - 43.6709i) q^{6} -487.138 q^{7} +556.764i q^{8} +(-634.458 + 359.032i) q^{9} +1361.01 q^{10} +1788.18i q^{11} +(-162.638 - 617.633i) q^{12} -1815.64 q^{13} -3094.18i q^{14} +(-5594.65 + 1473.21i) q^{15} -2022.51 q^{16} +1191.58i q^{17} +(-2280.49 - 4029.93i) q^{18} +152.498 q^{19} -5068.64i q^{20} +(3349.26 + 12719.1i) q^{21} -11358.1 q^{22} -12165.6i q^{23} +(14537.1 - 3827.97i) q^{24} -30287.8 q^{25} -11532.5i q^{26} +(13736.5 + 14097.2i) q^{27} -11523.3 q^{28} -38981.9i q^{29} +(-9357.47 - 35535.9i) q^{30} +4346.21 q^{31} +22786.4i q^{32} +(46689.4 - 12294.5i) q^{33} -7568.62 q^{34} +104380. i q^{35} +(-15008.2 + 8492.93i) q^{36} -36390.1 q^{37} +968.629i q^{38} +(12483.2 + 47406.2i) q^{39} +119299. q^{40} -46210.1i q^{41} +(-80789.0 + 21273.7i) q^{42} -97432.9 q^{43} +42299.6i q^{44} +(76930.8 + 135947. i) q^{45} +77272.7 q^{46} -37496.3i q^{47} +(13905.6 + 52807.7i) q^{48} +119654. q^{49} -192381. i q^{50} +(31112.0 - 8192.56i) q^{51} -42949.1 q^{52} -137705. i q^{53} +(-89541.9 + 87250.7i) q^{54} +383159. q^{55} -271221. i q^{56} +(-1048.48 - 3981.70i) q^{57} +247604. q^{58} -99800.3i q^{59} +(-132342. + 34848.9i) q^{60} -113868. q^{61} +27606.1i q^{62} +(309068. - 174898. i) q^{63} -274175. q^{64} +389042. i q^{65} +(78091.5 + 296560. i) q^{66} -374493. q^{67} +28186.9i q^{68} +(-317642. + 83642.9i) q^{69} -662999. q^{70} +178897. i q^{71} +(-199896. - 353244. i) q^{72} -189276. q^{73} -231141. i q^{74} +(208240. + 790812. i) q^{75} +3607.34 q^{76} -871092. i q^{77} +(-301113. + 79290.6i) q^{78} +769979. q^{79} +433369. i q^{80} +(273633. - 455582. i) q^{81} +293516. q^{82} -155977. i q^{83} +(79227.1 + 300872. i) q^{84} +255322. q^{85} -618871. i q^{86} +(-1.01782e6 + 268016. i) q^{87} -995597. q^{88} +18567.4i q^{89} +(-863503. + 488646. i) q^{90} +884467. q^{91} -287777. i q^{92} +(-29881.9 - 113479. i) q^{93} +238167. q^{94} -32676.1i q^{95} +(594952. - 156665. i) q^{96} -1.12203e6 q^{97} +760015. i q^{98} +(-642015. - 1.13453e6i) 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

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Representations

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Database

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