LMFDB - Embedded newform 320.9.b.d.191.13

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [320,9,Mod(191,320)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(320, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) N = Newforms(chi, 9, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("320.191"); S:= CuspForms(chi, 9); N := Newforms(S);

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	320.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,-38800] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$130.361155220$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 5 x^{15} + 26 x^{14} - 834 x^{13} + 4390 x^{12} - 61783 x^{11} + 466168 x^{10} + \cdots + 206161212459445 $ x^16 - 5x^15 + 26x^14 - 834x^13 + 4390x^12 - 61783x^11 + 466168x^10 - 1105435x^9 + 46850799x^8 - 116275535x^7 + 626274432x^6 - 8558999923x^5 + 34408048994x^4 - 448299413930x^3 + 1712647133330x^2 + 15986651928135*x + 206161212459445
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{120}\cdot 5^{16} $
Twist minimal:	no (minimal twist has level 20)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.13
Root			$9.29581 + 2.24761i$ of defining polynomial
Character	$\chi$	$=$	320.191
Dual form			320.9.b.d.191.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+98.1237i q^{3} -279.508 q^{5} -820.952i q^{7} -3067.27 q^{9} +21147.7i q^{11} +36649.1 q^{13} -27426.4i q^{15} -110563. q^{17} -184840. i q^{19} +80554.9 q^{21} -181906. i q^{23} +78125.0 q^{25} +342818. i q^{27} +229647. q^{29} -388118. i q^{31} -2.07510e6 q^{33} +229463. i q^{35} -1.32611e6 q^{37} +3.59615e6i q^{39} -303036. q^{41} +962361. i q^{43} +857327. q^{45} -8.76124e6i q^{47} +5.09084e6 q^{49} -1.08489e7i q^{51} +577555. q^{53} -5.91097e6i q^{55} +1.81371e7 q^{57} -4.11725e6i q^{59} -2.10168e7 q^{61} +2.51808e6i q^{63} -1.02437e7 q^{65} -3.85926e7i q^{67} +1.78493e7 q^{69} -2.07744e7i q^{71} -2.70559e7 q^{73} +7.66592e6i q^{75} +1.73613e7 q^{77} -5.62358e7i q^{79} -5.37629e7 q^{81} +6.22189e7i q^{83} +3.09033e7 q^{85} +2.25338e7i q^{87} +6.04409e7 q^{89} -3.00871e7i q^{91} +3.80836e7 q^{93} +5.16642e7i q^{95} +1.31334e8 q^{97} -6.48658e7i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 38800 q^{9} - 51392 q^{13} + 27552 q^{17} - 414496 q^{21} + 1250000 q^{25} - 2764896 q^{29} - 5521600 q^{33} - 9009472 q^{37} - 8576448 q^{41} - 1580000 q^{45} - 32803600 q^{49} - 2452032 q^{53}+ \cdots + 171851232 q^{97}+O(q^{100}) $ 16 * q - 38800 * q^9 - 51392 * q^13 + 27552 * q^17 - 414496 * q^21 + 1250000 * q^25 - 2764896 * q^29 - 5521600 * q^33 - 9009472 * q^37 - 8576448 * q^41 - 1580000 * q^45 - 32803600 * q^49 - 2452032 * q^53 + 11957760 * q^57 - 8371712 * q^61 + 9060000 * q^65 - 7527264 * q^69 + 61907232 * q^73 + 156997440 * q^77 + 140586672 * q^81 + 106960000 * q^85 + 106647456 * q^89 - 105563840 * q^93 + 171851232 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/320\mathbb{Z}\right)^\times$.

$n$	$191$	$257$	$261$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	98.1237i	1.21140i	0.795692	+	0.605702i	$0.207108\pi$
$3$	98.1237i	1.21140i	−0.795692	+	0.605702i	$0.792892\pi$
$4$	0	0
$5$	−279.508	−0.447214
$5$	−279.508	−0.447214
$6$	0	0
$7$	− 820.952i	− 0.341921i	−0.985278	−	0.170960i	$-0.945313\pi$
$7$	− 820.952i	− 0.341921i	0.985278	−	0.170960i	$-0.0546870\pi$
$8$	0	0
$9$	−3067.27	−0.467500
$10$	0	0
$11$	21147.7i	1.44442i	0.691674	+	0.722210i	$0.256873\pi$
$11$	21147.7i	1.44442i	−0.691674	+	0.722210i	$0.743127\pi$
$12$	0	0
$13$	36649.1	1.28319	0.641593	−	0.767045i	$-0.278274\pi$
$13$	36649.1	1.28319	0.641593	+	0.767045i	$0.278274\pi$
$14$	0	0
$15$	− 27426.4i	− 0.541756i
$16$	0	0
$17$	−110563.	−1.32378	−0.661888	−	0.749603i	$-0.730245\pi$
$17$	−110563.	−1.32378	−0.661888	+	0.749603i	$0.730245\pi$
$18$	0	0
$19$	− 184840.i	− 1.41834i	−0.705037	−	0.709170i	$-0.749070\pi$
$19$	− 184840.i	− 1.41834i	0.705037	−	0.709170i	$-0.250930\pi$
$20$	0	0
$21$	80554.9	0.414204
$22$	0	0
$23$	− 181906.i	− 0.650035i	−0.945708	−	0.325017i	$-0.894630\pi$
$23$	− 181906.i	− 0.650035i	0.945708	−	0.325017i	$-0.105370\pi$
$24$	0	0
$25$	78125.0	0.200000
$26$	0	0
$27$	342818.i	0.645073i
$28$	0	0
$29$	229647.	0.324690	0.162345	−	0.986734i	$-0.448094\pi$
$29$	229647.	0.324690	0.162345	+	0.986734i	$0.448094\pi$
$30$	0	0
$31$	− 388118.i	− 0.420259i	−0.977674	−	0.210129i	$-0.932611\pi$
$31$	− 388118.i	− 0.420259i	0.977674	−	0.210129i	$-0.0673885\pi$
$32$	0	0
$33$	−2.07510e6	−1.74978
$34$	0	0
$35$	229463.i	0.152912i
$36$	0	0
$37$	−1.32611e6	−0.707576	−0.353788	−	0.935326i	$-0.615107\pi$
$37$	−1.32611e6	−0.707576	−0.353788	+	0.935326i	$0.615107\pi$
$38$	0	0
$39$	3.59615e6i	1.55446i
$40$	0	0
$41$	−303036.	−0.107240	−0.0536202	−	0.998561i	$-0.517076\pi$
$41$	−303036.	−0.107240	−0.0536202	+	0.998561i	$0.517076\pi$
$42$	0	0
$43$	962361.i	0.281491i	0.990046	+	0.140745i	$0.0449499\pi$
$43$	962361.i	0.281491i	−0.990046	+	0.140745i	$0.955050\pi$
$44$	0	0
$45$	857327.	0.209072
$46$	0	0
$47$	− 8.76124e6i	− 1.79545i	−0.440552	−	0.897727i	$-0.645217\pi$
$47$	− 8.76124e6i	− 1.79545i	0.440552	−	0.897727i	$-0.354783\pi$
$48$	0	0
$49$	5.09084e6	0.883090
$50$	0	0
$51$	− 1.08489e7i	− 1.60363i
$52$	0	0
$53$	577555.	0.0731965	0.0365982	−	0.999330i	$-0.488348\pi$
$53$	577555.	0.0731965	0.0365982	+	0.999330i	$0.488348\pi$
$54$	0	0
$55$	− 5.91097e6i	− 0.645964i
$56$	0	0
$57$	1.81371e7	1.71818
$58$	0	0
$59$	− 4.11725e6i	− 0.339781i	−0.985463	−	0.169891i	$-0.945659\pi$
$59$	− 4.11725e6i	− 0.339781i	0.985463	−	0.169891i	$-0.0543414\pi$
$60$	0	0
$61$	−2.10168e7	−1.51791	−0.758956	−	0.651142i	$-0.774290\pi$
$61$	−2.10168e7	−1.51791	−0.758956	+	0.651142i	$0.774290\pi$
$62$	0	0
$63$	2.51808e6i	0.159848i
$64$	0	0
$65$	−1.02437e7	−0.573858
$66$	0	0
$67$	− 3.85926e7i	− 1.91516i	−0.288168	−	0.957580i	$-0.593046\pi$
$67$	− 3.85926e7i	− 1.91516i	0.288168	−	0.957580i	$-0.406954\pi$
$68$	0	0
$69$	1.78493e7	0.787455
$70$	0	0
$71$	− 2.07744e7i	− 0.817515i	−0.912643	−	0.408758i	$-0.865962\pi$
$71$	− 2.07744e7i	− 0.817515i	0.912643	−	0.408758i	$-0.134038\pi$
$72$	0	0
$73$	−2.70559e7	−0.952732	−0.476366	−	0.879247i	$-0.658046\pi$
$73$	−2.70559e7	−0.952732	−0.476366	+	0.879247i	$0.658046\pi$
$74$	0	0
$75$	7.66592e6i	0.242281i
$76$	0	0
$77$	1.73613e7	0.493877
$78$	0	0
$79$	− 5.62358e7i	− 1.44379i	−0.692002	−	0.721896i	$-0.743271\pi$
$79$	− 5.62358e7i	− 1.44379i	0.692002	−	0.721896i	$-0.256729\pi$
$80$	0	0
$81$	−5.37629e7	−1.24894
$82$	0	0
$83$	6.22189e7i	1.31102i	0.755186	+	0.655511i	$0.227547\pi$
$83$	6.22189e7i	1.31102i	−0.755186	+	0.655511i	$0.772453\pi$
$84$	0	0
$85$	3.09033e7	0.592011
$86$	0	0
$87$	2.25338e7i	0.393331i
$88$	0	0
$89$	6.04409e7	0.963321	0.481661	−	0.876358i	$-0.340034\pi$
$89$	6.04409e7	0.963321	0.481661	+	0.876358i	$0.340034\pi$
$90$	0	0
$91$	− 3.00871e7i	− 0.438748i
$92$	0	0
$93$	3.80836e7	0.509103
$94$	0	0
$95$	5.16642e7i	0.634301i
$96$	0	0
$97$	1.31334e8	1.48351	0.741755	−	0.670671i	$-0.233994\pi$
$97$	1.31334e8	1.48351	0.741755	+	0.670671i	$0.233994\pi$
$98$	0	0
$99$	− 6.48658e7i	− 0.675266i
$100$	0	0
$101$	1.55397e8	1.49333	0.746665	−	0.665200i	$-0.231654\pi$
$101$	1.55397e8	1.49333	0.746665	+	0.665200i	$0.231654\pi$
$102$	0	0
$103$	3.83213e7i	0.340480i	0.985403	+	0.170240i	$0.0544543\pi$
$103$	3.83213e7i	0.340480i	−0.985403	+	0.170240i	$0.945546\pi$
$104$	0	0
$105$	−2.25158e7	−0.185238
$106$	0	0
$107$	− 3.36904e7i	− 0.257022i	−0.991708	−	0.128511i	$-0.958980\pi$
$107$	− 3.36904e7i	− 0.257022i	0.991708	−	0.128511i	$-0.0410198\pi$
$108$	0	0
$109$	−1.87899e8	−1.33112	−0.665562	−	0.746343i	$-0.731808\pi$
$109$	−1.87899e8	−1.33112	−0.665562	+	0.746343i	$0.731808\pi$
$110$	0	0
$111$	− 1.30123e8i	− 0.857161i
$112$	0	0
$113$	2.44191e8	1.49767	0.748835	−	0.662757i	$-0.230614\pi$
$113$	2.44191e8	1.49767	0.748835	+	0.662757i	$0.230614\pi$
$114$	0	0
$115$	5.08444e7i	0.290704i
$116$	0	0
$117$	−1.12413e8	−0.599889
$118$	0	0
$119$	9.07670e7i	0.452627i
$120$	0	0
$121$	−2.32868e8	−1.08635
$122$	0	0
$123$	− 2.97350e7i	− 0.129911i
$124$	0	0
$125$	−2.18366e7	−0.0894427
$126$	0	0
$127$	4.32402e7i	0.166216i	0.996541	+	0.0831080i	$0.0264846\pi$
$127$	4.32402e7i	0.166216i	−0.996541	+	0.0831080i	$0.973515\pi$
$128$	0	0
$129$	−9.44304e7	−0.340999
$130$	0	0
$131$	3.46986e8i	1.17822i	0.808052	+	0.589111i	$0.200522\pi$
$131$	3.46986e8i	1.17822i	−0.808052	+	0.589111i	$0.799478\pi$
$132$	0	0
$133$	−1.51744e8	−0.484960
$134$	0	0
$135$	− 9.58206e7i	− 0.288485i
$136$	0	0
$137$	4.54595e8	1.29045	0.645227	−	0.763991i	$-0.276763\pi$
$137$	4.54595e8	1.29045	0.645227	+	0.763991i	$0.276763\pi$
$138$	0	0
$139$	− 1.24500e8i	− 0.333511i	−0.985998	−	0.166755i	$-0.946671\pi$
$139$	− 1.24500e8i	− 0.333511i	0.985998	−	0.166755i	$-0.0533290\pi$
$140$	0	0
$141$	8.59686e8	2.17502
$142$	0	0
$143$	7.75046e8i	1.85346i
$144$	0	0
$145$	−6.41883e7	−0.145206
$146$	0	0
$147$	4.99532e8i	1.06978i
$148$	0	0
$149$	2.17967e8	0.442228	0.221114	−	0.975248i	$-0.429031\pi$
$149$	2.17967e8	0.442228	0.221114	+	0.975248i	$0.429031\pi$
$150$	0	0
$151$	2.00315e8i	0.385306i	0.981267	+	0.192653i	$0.0617091\pi$
$151$	2.00315e8i	0.385306i	−0.981267	+	0.192653i	$0.938291\pi$
$152$	0	0
$153$	3.39126e8	0.618865
$154$	0	0
$155$	1.08482e8i	0.187945i
$156$	0	0
$157$	2.01644e7	0.0331885	0.0165942	−	0.999862i	$-0.494718\pi$
$157$	2.01644e7	0.0331885	0.0165942	+	0.999862i	$0.494718\pi$
$158$	0	0
$159$	5.66719e7i	0.0886705i
$160$	0	0
$161$	−1.49336e8	−0.222260
$162$	0	0
$163$	1.92932e7i	0.0273309i	0.999907	+	0.0136654i	$0.00434998\pi$
$163$	1.92932e7i	0.0273309i	−0.999907	+	0.0136654i	$0.995650\pi$
$164$	0	0
$165$	5.80007e8	0.782523
$166$	0	0
$167$	− 2.03394e8i	− 0.261500i	−0.991415	−	0.130750i	$-0.958261\pi$
$167$	− 2.03394e8i	− 0.261500i	0.991415	−	0.130750i	$-0.0417385\pi$
$168$	0	0
$169$	5.27425e8	0.646567
$170$	0	0
$171$	5.66952e8i	0.663074i
$172$	0	0
$173$	4.16190e8	0.464630	0.232315	−	0.972641i	$-0.425370\pi$
$173$	4.16190e8	0.464630	0.232315	+	0.972641i	$0.425370\pi$
$174$	0	0
$175$	− 6.41369e7i	− 0.0683842i
$176$	0	0
$177$	4.04000e8	0.411612
$178$	0	0
$179$	5.41911e8i	0.527856i	0.964542	+	0.263928i	$0.0850182\pi$
$179$	5.41911e8i	0.527856i	−0.964542	+	0.263928i	$0.914982\pi$
$180$	0	0
$181$	1.43210e9	1.33432	0.667159	−	0.744915i	$-0.267510\pi$
$181$	1.43210e9	1.33432	0.667159	+	0.744915i	$0.267510\pi$
$182$	0	0
$183$	− 2.06224e9i	− 1.83880i
$184$	0	0
$185$	3.70660e8	0.316438
$186$	0	0
$187$	− 2.33816e9i	− 1.91209i
$188$	0	0
$189$	2.81437e8	0.220564
$190$	0	0
$191$	1.75647e9i	1.31980i	0.751355	+	0.659898i	$0.229400\pi$
$191$	1.75647e9i	1.31980i	−0.751355	+	0.659898i	$0.770600\pi$
$192$	0	0
$193$	−6.79519e8	−0.489748	−0.244874	−	0.969555i	$-0.578747\pi$
$193$	−6.79519e8	−0.489748	−0.244874	+	0.969555i	$0.578747\pi$
$194$	0	0
$195$	− 1.00515e9i	− 0.695174i
$196$	0	0
$197$	−4.80726e8	−0.319178	−0.159589	−	0.987184i	$-0.551017\pi$
$197$	−4.80726e8	−0.319178	−0.159589	+	0.987184i	$0.551017\pi$
$198$	0	0
$199$	− 6.20901e8i	− 0.395922i	−0.980210	−	0.197961i	$-0.936568\pi$
$199$	− 6.20901e8i	− 0.395922i	0.980210	−	0.197961i	$-0.0634320\pi$
$200$	0	0
$201$	3.78685e9	2.32003
$202$	0	0
$203$	− 1.88529e8i	− 0.111018i
$204$	0	0
$205$	8.47011e7	0.0479594
$206$	0	0
$207$	5.57955e8i	0.303891i
$208$	0	0
$209$	3.90894e9	2.04868
$210$	0	0
$211$	2.32298e9i	1.17197i	0.810322	+	0.585984i	$0.199292\pi$
$211$	2.32298e9i	1.17197i	−0.810322	+	0.585984i	$0.800708\pi$
$212$	0	0
$213$	2.03846e9	0.990341
$214$	0	0
$215$	− 2.68988e8i	− 0.125886i
$216$	0	0
$217$	−3.18626e8	−0.143695
$218$	0	0
$219$	− 2.65483e9i	− 1.15414i
$220$	0	0
$221$	−4.05204e9	−1.69865
$222$	0	0
$223$	− 1.52445e9i	− 0.616446i	−0.951314	−	0.308223i	$-0.900266\pi$
$223$	− 1.52445e9i	− 0.616446i	0.951314	−	0.308223i	$-0.0997343\pi$
$224$	0	0
$225$	−2.39630e8	−0.0935000
$226$	0	0
$227$	− 7.14722e8i	− 0.269174i	−0.990902	−	0.134587i	$-0.957029\pi$
$227$	− 7.14722e8i	− 0.269174i	0.990902	−	0.134587i	$-0.0429708\pi$
$228$	0	0
$229$	4.05044e9	1.47286	0.736428	−	0.676516i	$-0.236511\pi$
$229$	4.05044e9	1.47286	0.736428	+	0.676516i	$0.236511\pi$
$230$	0	0
$231$	1.70355e9i	0.598285i
$232$	0	0
$233$	1.91668e9	0.650318	0.325159	−	0.945659i	$-0.394582\pi$
$233$	1.91668e9	0.650318	0.325159	+	0.945659i	$0.394582\pi$
$234$	0	0
$235$	2.44884e9i	0.802952i
$236$	0	0
$237$	5.51807e9	1.74901
$238$	0	0
$239$	− 3.09701e9i	− 0.949186i	−0.880205	−	0.474593i	$-0.842595\pi$
$239$	− 3.09701e9i	− 0.949186i	0.880205	−	0.474593i	$-0.157405\pi$
$240$	0	0
$241$	−1.93463e9	−0.573496	−0.286748	−	0.958006i	$-0.592574\pi$
$241$	−1.93463e9	−0.573496	−0.286748	+	0.958006i	$0.592574\pi$
$242$	0	0
$243$	− 3.02619e9i	− 0.867903i
$244$	0	0
$245$	−1.42293e9	−0.394930
$246$	0	0
$247$	− 6.77420e9i	− 1.82000i
$248$	0	0
$249$	−6.10515e9	−1.58818
$250$	0	0
$251$	− 5.70277e9i	− 1.43678i	−0.695639	−	0.718391i	$-0.744879\pi$
$251$	− 5.70277e9i	− 1.43678i	0.695639	−	0.718391i	$-0.255121\pi$
$252$	0	0
$253$	3.84691e9	0.938923
$254$	0	0
$255$	3.03235e9i	0.717164i
$256$	0	0
$257$	1.76094e9	0.403656	0.201828	−	0.979421i	$-0.435312\pi$
$257$	1.76094e9	0.403656	0.201828	+	0.979421i	$0.435312\pi$
$258$	0	0
$259$	1.08867e9i	0.241935i
$260$	0	0
$261$	−7.04389e8	−0.151793
$262$	0	0
$263$	2.58682e8i	0.0540683i	0.999635	+	0.0270342i	$0.00860629\pi$
$263$	2.58682e8i	0.0540683i	−0.999635	+	0.0270342i	$0.991394\pi$
$264$	0	0
$265$	−1.61432e8	−0.0327345
$266$	0	0
$267$	5.93069e9i	1.16697i
$268$	0	0
$269$	3.03788e9	0.580178	0.290089	−	0.957000i	$-0.406315\pi$
$269$	3.03788e9	0.580178	0.290089	+	0.957000i	$0.406315\pi$
$270$	0	0
$271$	1.65010e9i	0.305937i	0.988231	+	0.152968i	$0.0488833\pi$
$271$	1.65010e9i	0.305937i	−0.988231	+	0.152968i	$0.951117\pi$
$272$	0	0
$273$	2.95226e9	0.531501
$274$	0	0
$275$	1.65217e9i	0.288884i
$276$	0	0
$277$	5.17850e9	0.879599	0.439800	−	0.898096i	$-0.355049\pi$
$277$	5.17850e9	0.879599	0.439800	+	0.898096i	$0.355049\pi$
$278$	0	0
$279$	1.19046e9i	0.196471i
$280$	0	0
$281$	8.47832e9	1.35983	0.679915	−	0.733291i	$-0.262017\pi$
$281$	8.47832e9	1.35983	0.679915	+	0.733291i	$0.262017\pi$
$282$	0	0
$283$	− 3.20005e8i	− 0.0498898i	−0.999689	−	0.0249449i	$-0.992059\pi$
$283$	− 3.20005e8i	− 0.0498898i	0.999689	−	0.0249449i	$-0.00794102\pi$
$284$	0	0
$285$	−5.06949e9	−0.768395
$286$	0	0
$287$	2.48778e8i	0.0366677i
$288$	0	0
$289$	5.24844e9	0.752383
$290$	0	0
$291$	1.28870e10i	1.79713i
$292$	0	0
$293$	−7.87248e9	−1.06817	−0.534086	−	0.845430i	$-0.679344\pi$
$293$	−7.87248e9	−1.06817	−0.534086	+	0.845430i	$0.679344\pi$
$294$	0	0
$295$	1.15081e9i	0.151955i
$296$	0	0
$297$	−7.24983e9	−0.931756
$298$	0	0
$299$	− 6.66670e9i	− 0.834116i
$300$	0	0
$301$	7.90052e8	0.0962475
$302$	0	0
$303$	1.52481e10i	1.80903i
$304$	0	0
$305$	5.87436e9	0.678831
$306$	0	0
$307$	− 1.47434e10i	− 1.65976i	−0.557941	−	0.829881i	$-0.688408\pi$
$307$	− 1.47434e10i	− 1.65976i	0.557941	−	0.829881i	$-0.311592\pi$
$308$	0	0
$309$	−3.76023e9	−0.412459
$310$	0	0
$311$	6.36308e9i	0.680184i	0.940392	+	0.340092i	$0.110458\pi$
$311$	6.36308e9i	0.680184i	−0.940392	+	0.340092i	$0.889542\pi$
$312$	0	0
$313$	4.35717e9	0.453970	0.226985	−	0.973898i	$-0.427113\pi$
$313$	4.35717e9	0.453970	0.226985	+	0.973898i	$0.427113\pi$
$314$	0	0
$315$	− 7.03824e8i	− 0.0714862i
$316$	0	0
$317$	1.02074e9	0.101083	0.0505413	−	0.998722i	$-0.483905\pi$
$317$	1.02074e9	0.101083	0.0505413	+	0.998722i	$0.483905\pi$
$318$	0	0
$319$	4.85652e9i	0.468989i
$320$	0	0
$321$	3.30583e9	0.311358
$322$	0	0
$323$	2.04364e10i	1.87757i
$324$	0	0
$325$	2.86321e9	0.256637
$326$	0	0
$327$	− 1.84373e10i	− 1.61253i
$328$	0	0
$329$	−7.19256e9	−0.613903
$330$	0	0
$331$	1.01219e10i	0.843235i	0.906774	+	0.421617i	$0.138537\pi$
$331$	1.01219e10i	0.843235i	−0.906774	+	0.421617i	$0.861463\pi$
$332$	0	0
$333$	4.06754e9	0.330792
$334$	0	0
$335$	1.07870e10i	0.856485i
$336$	0	0
$337$	−1.94388e9	−0.150713	−0.0753563	−	0.997157i	$-0.524009\pi$
$337$	−1.94388e9	−0.150713	−0.0753563	+	0.997157i	$0.524009\pi$
$338$	0	0
$339$	2.39609e10i	1.81428i
$340$	0	0
$341$	8.20782e9	0.607030
$342$	0	0
$343$	− 8.91196e9i	− 0.643868i
$344$	0	0
$345$	−4.98904e9	−0.352160
$346$	0	0
$347$	− 2.15696e10i	− 1.48773i	−0.668330	−	0.743865i	$-0.732991\pi$
$347$	− 2.15696e10i	− 1.48773i	0.668330	−	0.743865i	$-0.267009\pi$
$348$	0	0
$349$	−1.37934e10	−0.929760	−0.464880	−	0.885374i	$-0.653903\pi$
$349$	−1.37934e10	−0.929760	−0.464880	+	0.885374i	$0.653903\pi$
$350$	0	0
$351$	1.25640e10i	0.827749i
$352$	0	0
$353$	−1.69884e10	−1.09409	−0.547047	−	0.837102i	$-0.684248\pi$
$353$	−1.69884e10	−1.09409	−0.547047	+	0.837102i	$0.684248\pi$
$354$	0	0
$355$	5.80663e9i	0.365604i
$356$	0	0
$357$	−8.90639e9	−0.548314
$358$	0	0
$359$	− 1.35600e10i	− 0.816358i	−0.912902	−	0.408179i	$-0.866164\pi$
$359$	− 1.35600e10i	− 0.816358i	0.912902	−	0.408179i	$-0.133836\pi$
$360$	0	0
$361$	−1.71821e10	−1.01169
$362$	0	0
$363$	− 2.28499e10i	− 1.31601i
$364$	0	0
$365$	7.56236e9	0.426075
$366$	0	0
$367$	9.93203e9i	0.547487i	0.961803	+	0.273743i	$0.0882619\pi$
$367$	9.93203e9i	0.547487i	−0.961803	+	0.273743i	$0.911738\pi$
$368$	0	0
$369$	9.29491e8	0.0501349
$370$	0	0
$371$	− 4.74145e8i	− 0.0250274i
$372$	0	0
$373$	−2.08933e10	−1.07937	−0.539686	−	0.841867i	$-0.681457\pi$
$373$	−2.08933e10	−1.07937	−0.539686	+	0.841867i	$0.681457\pi$
$374$	0	0
$375$	− 2.14269e9i	− 0.108351i
$376$	0	0
$377$	8.41636e9	0.416638
$378$	0	0
$379$	− 2.78733e10i	− 1.35093i	−0.737394	−	0.675463i	$-0.763944\pi$
$379$	− 2.78733e10i	− 1.35093i	0.737394	−	0.675463i	$-0.236056\pi$
$380$	0	0
$381$	−4.24289e9	−0.201355
$382$	0	0
$383$	− 3.14260e10i	− 1.46048i	−0.683193	−	0.730238i	$-0.739409\pi$
$383$	− 3.14260e10i	− 1.46048i	0.683193	−	0.730238i	$-0.260591\pi$
$384$	0	0
$385$	−4.85262e9	−0.220869
$386$	0	0
$387$	− 2.95182e9i	− 0.131597i
$388$	0	0
$389$	2.45574e10	1.07247	0.536233	−	0.844070i	$-0.319847\pi$
$389$	2.45574e10	1.07247	0.536233	+	0.844070i	$0.319847\pi$
$390$	0	0
$391$	2.01121e10i	0.860500i
$392$	0	0
$393$	−3.40476e10	−1.42730
$394$	0	0
$395$	1.57184e10i	0.645683i
$396$	0	0
$397$	−3.64363e10	−1.46681	−0.733403	−	0.679794i	$-0.762069\pi$
$397$	−3.64363e10	−1.46681	−0.733403	+	0.679794i	$0.762069\pi$
$398$	0	0
$399$	− 1.48897e10i	− 0.587483i
$400$	0	0
$401$	−1.07267e10	−0.414847	−0.207423	−	0.978251i	$-0.566508\pi$
$401$	−1.07267e10	−0.414847	−0.207423	+	0.978251i	$0.566508\pi$
$402$	0	0
$403$	− 1.42242e10i	− 0.539270i
$404$	0	0
$405$	1.50272e10	0.558545
$406$	0	0
$407$	− 2.80443e10i	− 1.02204i
$408$	0	0
$409$	−3.91731e7	−0.00139989	−0.000699947	−	1.00000i	$-0.500223\pi$
$409$	−3.91731e7	−0.00139989	−0.000699947		1.00000i	$0.500223\pi$
$410$	0	0
$411$	4.46066e10i	1.56326i
$412$	0	0
$413$	−3.38006e9	−0.116178
$414$	0	0
$415$	− 1.73907e10i	− 0.586307i
$416$	0	0
$417$	1.22164e10	0.404016
$418$	0	0
$419$	− 3.53613e10i	− 1.14729i	−0.819105	−	0.573644i	$-0.805529\pi$
$419$	− 3.53613e10i	− 1.14729i	0.819105	−	0.573644i	$-0.194471\pi$
$420$	0	0
$421$	−9.81579e9	−0.312462	−0.156231	−	0.987721i	$-0.549934\pi$
$421$	−9.81579e9	−0.312462	−0.156231	+	0.987721i	$0.549934\pi$
$422$	0	0
$423$	2.68731e10i	0.839375i
$424$	0	0
$425$	−8.63774e9	−0.264755
$426$	0	0
$427$	1.72537e10i	0.519005i
$428$	0	0
$429$	−7.60504e10	−2.24529
$430$	0	0
$431$	9.11726e9i	0.264214i	0.991235	+	0.132107i	$0.0421742\pi$
$431$	9.11726e9i	0.264214i	−0.991235	+	0.132107i	$0.957826\pi$
$432$	0	0
$433$	4.67005e9	0.132852	0.0664262	−	0.997791i	$-0.478840\pi$
$433$	4.67005e9	0.132852	0.0664262	+	0.997791i	$0.478840\pi$
$434$	0	0
$435$	− 6.29840e9i	− 0.175903i
$436$	0	0
$437$	−3.36235e10	−0.921970
$438$	0	0
$439$	3.39406e10i	0.913822i	0.889512	+	0.456911i	$0.151044\pi$
$439$	3.39406e10i	0.913822i	−0.889512	+	0.456911i	$0.848956\pi$
$440$	0	0
$441$	−1.56150e10	−0.412844
$442$	0	0
$443$	− 7.20845e9i	− 0.187166i	−0.995611	−	0.0935830i	$-0.970168\pi$
$443$	− 7.20845e9i	− 0.187166i	0.995611	−	0.0935830i	$-0.0298320\pi$
$444$	0	0
$445$	−1.68938e10	−0.430810
$446$	0	0
$447$	2.13878e10i	0.535717i
$448$	0	0
$449$	−2.32977e10	−0.573229	−0.286614	−	0.958046i	$-0.592530\pi$
$449$	−2.32977e10	−0.573229	−0.286614	+	0.958046i	$0.592530\pi$
$450$	0	0
$451$	− 6.40852e9i	− 0.154900i
$452$	0	0
$453$	−1.96556e10	−0.466761
$454$	0	0
$455$	8.40961e9i	0.196214i
$456$	0	0
$457$	1.72652e10	0.395829	0.197915	−	0.980219i	$-0.436583\pi$
$457$	1.72652e10	0.395829	0.197915	+	0.980219i	$0.436583\pi$
$458$	0	0
$459$	− 3.79030e10i	− 0.853932i
$460$	0	0
$461$	5.65725e10	1.25257	0.626285	−	0.779594i	$-0.284575\pi$
$461$	5.65725e10	1.25257	0.626285	+	0.779594i	$0.284575\pi$
$462$	0	0
$463$	− 7.57511e10i	− 1.64841i	−0.566292	−	0.824205i	$-0.691623\pi$
$463$	− 7.57511e10i	− 1.64841i	0.566292	−	0.824205i	$-0.308377\pi$
$464$	0	0
$465$	−1.06447e10	−0.227678
$466$	0	0
$467$	− 5.63076e10i	− 1.18386i	−0.805990	−	0.591929i	$-0.798367\pi$
$467$	− 5.63076e10i	− 1.18386i	0.805990	−	0.591929i	$-0.201633\pi$
$468$	0	0
$469$	−3.16827e10	−0.654833
$470$	0	0
$471$	1.97861e9i	0.0402046i
$472$	0	0
$473$	−2.03518e10	−0.406591
$474$	0	0
$475$	− 1.44406e10i	− 0.283668i
$476$	0	0
$477$	−1.77152e9	−0.0342193
$478$	0	0
$479$	− 8.37112e10i	− 1.59016i	−0.606502	−	0.795082i	$-0.707428\pi$
$479$	− 8.37112e10i	− 1.59016i	0.606502	−	0.795082i	$-0.292572\pi$
$480$	0	0
$481$	−4.86008e10	−0.907952
$482$	0	0
$483$	− 1.46534e10i	− 0.269247i
$484$	0	0
$485$	−3.67090e10	−0.663446
$486$	0	0
$487$	− 3.58034e10i	− 0.636514i	−0.948004	−	0.318257i	$-0.896902\pi$
$487$	− 3.58034e10i	− 0.636514i	0.948004	−	0.318257i	$-0.103098\pi$
$488$	0	0
$489$	−1.89312e9	−0.0331088
$490$	0	0
$491$	5.45993e10i	0.939423i	0.882820	+	0.469712i	$0.155642\pi$
$491$	5.45993e10i	0.939423i	−0.882820	+	0.469712i	$0.844358\pi$
$492$	0	0
$493$	−2.53905e10	−0.429817
$494$	0	0
$495$	1.81305e10i	0.301988i
$496$	0	0
$497$	−1.70548e10	−0.279525
$498$	0	0
$499$	6.97805e10i	1.12547i	0.826639	+	0.562733i	$0.190250\pi$
$499$	6.97805e10i	1.12547i	−0.826639	+	0.562733i	$0.809750\pi$
$500$	0	0
$501$	1.99578e10	0.316782
$502$	0	0
$503$	6.40878e10i	1.00116i	0.865691	+	0.500580i	$0.166880\pi$
$503$	6.40878e10i	1.00116i	−0.865691	+	0.500580i	$0.833120\pi$
$504$	0	0
$505$	−4.34347e10	−0.667838
$506$	0	0
$507$	5.17529e10i	0.783254i
$508$	0	0
$509$	1.03640e11	1.54402	0.772012	−	0.635608i	$-0.219250\pi$
$509$	1.03640e11	1.54402	0.772012	+	0.635608i	$0.219250\pi$
$510$	0	0
$511$	2.22116e10i	0.325759i
$512$	0	0
$513$	6.33664e10	0.914933
$514$	0	0
$515$	− 1.07111e10i	− 0.152267i
$516$	0	0
$517$	1.85281e11	2.59339
$518$	0	0
$519$	4.08382e10i	0.562855i
$520$	0	0
$521$	−3.35410e10	−0.455225	−0.227612	−	0.973752i	$-0.573092\pi$
$521$	−3.35410e10	−0.455225	−0.227612	+	0.973752i	$0.573092\pi$
$522$	0	0
$523$	3.32143e10i	0.443934i	0.975054	+	0.221967i	$0.0712477\pi$
$523$	3.32143e10i	0.443934i	−0.975054	+	0.221967i	$0.928752\pi$
$524$	0	0
$525$	6.29335e9	0.0828408
$526$	0	0
$527$	4.29115e10i	0.556329i
$528$	0	0
$529$	4.52211e10	0.577455
$530$	0	0
$531$	1.26287e10i	0.158848i
$532$	0	0
$533$	−1.11060e10	−0.137609
$534$	0	0
$535$	9.41675e9i	0.114944i
$536$	0	0
$537$	−5.31743e10	−0.639447
$538$	0	0
$539$	1.07660e11i	1.27555i
$540$	0	0
$541$	−1.39040e11	−1.62312	−0.811562	−	0.584266i	$-0.801382\pi$
$541$	−1.39040e11	−1.62312	−0.811562	+	0.584266i	$0.801382\pi$
$542$	0	0
$543$	1.40523e11i	1.61640i
$544$	0	0
$545$	5.25193e10	0.595296
$546$	0	0
$547$	− 7.05870e10i	− 0.788452i	−0.919013	−	0.394226i	$-0.871013\pi$
$547$	− 7.05870e10i	− 0.788452i	0.919013	−	0.394226i	$-0.128987\pi$
$548$	0	0
$549$	6.44640e10	0.709623
$550$	0	0
$551$	− 4.24479e10i	− 0.460521i
$552$	0	0
$553$	−4.61669e10	−0.493662
$554$	0	0
$555$	3.63705e10i	0.383334i
$556$	0	0
$557$	−1.36274e11	−1.41577	−0.707884	−	0.706328i	$-0.750350\pi$
$557$	−1.36274e11	−1.41577	−0.707884	+	0.706328i	$0.750350\pi$
$558$	0	0
$559$	3.52696e10i	0.361205i
$560$	0	0
$561$	2.29429e11	2.31631
$562$	0	0
$563$	9.28198e10i	0.923862i	0.886916	+	0.461931i	$0.152843\pi$
$563$	9.28198e10i	0.923862i	−0.886916	+	0.461931i	$0.847157\pi$
$564$	0	0
$565$	−6.82535e10	−0.669778
$566$	0	0
$567$	4.41368e10i	0.427040i
$568$	0	0
$569$	−3.45537e10	−0.329644	−0.164822	−	0.986323i	$-0.552705\pi$
$569$	−3.45537e10	−0.329644	−0.164822	+	0.986323i	$0.552705\pi$
$570$	0	0
$571$	− 9.27891e10i	− 0.872876i	−0.899734	−	0.436438i	$-0.856240\pi$
$571$	− 9.27891e10i	− 0.872876i	0.899734	−	0.436438i	$-0.143760\pi$
$572$	0	0
$573$	−1.72351e11	−1.59881
$574$	0	0
$575$	− 1.42114e10i	− 0.130007i
$576$	0	0
$577$	−3.75444e10	−0.338721	−0.169361	−	0.985554i	$-0.554170\pi$
$577$	−3.75444e10	−0.338721	−0.169361	+	0.985554i	$0.554170\pi$
$578$	0	0
$579$	− 6.66770e10i	− 0.593282i
$580$	0	0
$581$	5.10787e10	0.448266
$582$	0	0
$583$	1.22140e10i	0.105726i
$584$	0	0
$585$	3.14203e10	0.268279
$586$	0	0
$587$	2.66542e10i	0.224498i	0.993680	+	0.112249i	$0.0358054\pi$
$587$	2.66542e10i	0.224498i	−0.993680	+	0.112249i	$0.964195\pi$
$588$	0	0
$589$	−7.17395e10	−0.596070
$590$	0	0
$591$	− 4.71706e10i	− 0.386653i
$592$	0	0
$593$	−1.07542e11	−0.869681	−0.434840	−	0.900508i	$-0.643195\pi$
$593$	−1.07542e11	−0.869681	−0.434840	+	0.900508i	$0.643195\pi$
$594$	0	0
$595$	− 2.53701e10i	− 0.202421i
$596$	0	0
$597$	6.09251e10	0.479622
$598$	0	0
$599$	5.10104e10i	0.396234i	0.980178	+	0.198117i	$0.0634825\pi$
$599$	5.10104e10i	0.396234i	−0.980178	+	0.198117i	$0.936517\pi$
$600$	0	0
$601$	1.64304e11	1.25936	0.629681	−	0.776854i	$-0.283186\pi$
$601$	1.64304e11	1.25936	0.629681	+	0.776854i	$0.283186\pi$
$602$	0	0
$603$	1.18374e11i	0.895337i
$604$	0	0
$605$	6.50886e10	0.485829
$606$	0	0
$607$	− 2.64985e11i	− 1.95194i	−0.217909	−	0.975969i	$-0.569923\pi$
$607$	− 2.64985e11i	− 1.95194i	0.217909	−	0.975969i	$-0.430077\pi$
$608$	0	0
$609$	1.84992e10	0.134488
$610$	0	0
$611$	− 3.21092e11i	− 2.30390i
$612$	0	0
$613$	2.58975e11	1.83407	0.917036	−	0.398804i	$-0.130575\pi$
$613$	2.58975e11	1.83407	0.917036	+	0.398804i	$0.130575\pi$
$614$	0	0
$615$	8.31118e9i	0.0580982i
$616$	0	0
$617$	2.43563e10	0.168063	0.0840313	−	0.996463i	$-0.473220\pi$
$617$	2.43563e10	0.168063	0.0840313	+	0.996463i	$0.473220\pi$
$618$	0	0
$619$	− 6.16901e10i	− 0.420197i	−0.977680	−	0.210098i	$-0.932622\pi$
$619$	− 6.16901e10i	− 0.420197i	0.977680	−	0.210098i	$-0.0673784\pi$
$620$	0	0
$621$	6.23608e10	0.419320
$622$	0	0
$623$	− 4.96191e10i	− 0.329380i
$624$	0	0
$625$	6.10352e9	0.0400000
$626$	0	0
$627$	3.83560e11i	2.48178i
$628$	0	0
$629$	1.46619e11	0.936673
$630$	0	0
$631$	2.91259e10i	0.183722i	0.995772	+	0.0918611i	$0.0292816\pi$
$631$	2.91259e10i	0.183722i	−0.995772	+	0.0918611i	$0.970718\pi$
$632$	0	0
$633$	−2.27940e11	−1.41973
$634$	0	0
$635$	− 1.20860e10i	− 0.0743340i
$636$	0	0
$637$	1.86575e11	1.13317
$638$	0	0
$639$	6.37207e10i	0.382188i
$640$	0	0
$641$	2.97316e11	1.76111	0.880555	−	0.473943i	$-0.157170\pi$
$641$	2.97316e11	1.76111	0.880555	+	0.473943i	$0.157170\pi$
$642$	0	0
$643$	− 2.02204e11i	− 1.18289i	−0.806344	−	0.591447i	$-0.798557\pi$
$643$	− 2.02204e11i	− 1.18289i	0.806344	−	0.591447i	$-0.201443\pi$
$644$	0	0
$645$	2.63941e10	0.152499
$646$	0	0
$647$	2.98562e11i	1.70380i	0.523708	+	0.851898i	$0.324548\pi$
$647$	2.98562e11i	1.70380i	−0.523708	+	0.851898i	$0.675452\pi$
$648$	0	0
$649$	8.70705e10	0.490786
$650$	0	0
$651$	− 3.12648e10i	− 0.174073i
$652$	0	0
$653$	−3.34280e10	−0.183847	−0.0919237	−	0.995766i	$-0.529302\pi$
$653$	−3.34280e10	−0.183847	−0.0919237	+	0.995766i	$0.529302\pi$
$654$	0	0
$655$	− 9.69855e10i	− 0.526916i
$656$	0	0
$657$	8.29877e10	0.445402
$658$	0	0
$659$	− 7.70116e10i	− 0.408333i	−0.978936	−	0.204166i	$-0.934552\pi$
$659$	− 7.70116e10i	− 0.408333i	0.978936	−	0.204166i	$-0.0654484\pi$
$660$	0	0
$661$	2.47987e11	1.29904	0.649520	−	0.760344i	$-0.274970\pi$
$661$	2.47987e11	1.29904	0.649520	+	0.760344i	$0.274970\pi$
$662$	0	0
$663$	− 3.97601e11i	− 2.05775i
$664$	0	0
$665$	4.24138e10	0.216881
$666$	0	0
$667$	− 4.17743e10i	− 0.211060i
$668$	0	0
$669$	1.49585e11	0.746765
$670$	0	0
$671$	− 4.44457e11i	− 2.19250i
$672$	0	0
$673$	1.03219e11	0.503151	0.251576	−	0.967838i	$-0.419051\pi$
$673$	1.03219e11	0.503151	0.251576	+	0.967838i	$0.419051\pi$
$674$	0	0
$675$	2.67827e10i	0.129015i
$676$	0	0
$677$	3.83549e10	0.182585	0.0912926	−	0.995824i	$-0.470900\pi$
$677$	3.83549e10	0.182585	0.0912926	+	0.995824i	$0.470900\pi$
$678$	0	0
$679$	− 1.07819e11i	− 0.507243i
$680$	0	0
$681$	7.01312e10	0.326079
$682$	0	0
$683$	− 2.47284e10i	− 0.113635i	−0.998385	−	0.0568177i	$-0.981905\pi$
$683$	− 2.47284e10i	− 0.113635i	0.998385	−	0.0568177i	$-0.0180954\pi$
$684$	0	0
$685$	−1.27063e11	−0.577108
$686$	0	0
$687$	3.97444e11i	1.78422i
$688$	0	0
$689$	2.11669e10	0.0939247
$690$	0	0
$691$	− 3.63864e10i	− 0.159598i	−0.996811	−	0.0797988i	$-0.974572\pi$
$691$	− 3.63864e10i	− 0.159598i	0.996811	−	0.0797988i	$-0.0254278\pi$
$692$	0	0
$693$	−5.32517e10	−0.230887
$694$	0	0
$695$	3.47988e10i	0.149151i
$696$	0	0
$697$	3.35046e10	0.141962
$698$	0	0
$699$	1.88072e11i	0.787797i
$700$	0	0
$701$	−1.36852e11	−0.566732	−0.283366	−	0.959012i	$-0.591451\pi$
$701$	−1.36852e11	−0.566732	−0.283366	+	0.959012i	$0.591451\pi$
$702$	0	0
$703$	2.45118e11i	1.00358i
$704$	0	0
$705$	−2.40290e11	−0.972699
$706$	0	0
$707$	− 1.27573e11i	− 0.510601i
$708$	0	0
$709$	−1.84677e11	−0.730848	−0.365424	−	0.930841i	$-0.619076\pi$
$709$	−1.84677e11	−0.730848	−0.365424	+	0.930841i	$0.619076\pi$
$710$	0	0
$711$	1.72490e11i	0.674972i
$712$	0	0
$713$	−7.06011e10	−0.273183
$714$	0	0
$715$	− 2.16632e11i	− 0.828892i
$716$	0	0
$717$	3.03890e11	1.14985
$718$	0	0
$719$	− 2.26289e11i	− 0.846735i	−0.905958	−	0.423368i	$-0.860848\pi$
$719$	− 2.26289e11i	− 0.846735i	0.905958	−	0.423368i	$-0.139152\pi$
$720$	0	0
$721$	3.14600e10	0.116417
$722$	0	0
$723$	− 1.89833e11i	− 0.694735i
$724$	0	0
$725$	1.79412e10	0.0649380
$726$	0	0
$727$	− 6.89485e10i	− 0.246824i	−0.992356	−	0.123412i	$-0.960616\pi$
$727$	− 6.89485e10i	− 0.246824i	0.992356	−	0.123412i	$-0.0393837\pi$
$728$	0	0
$729$	−5.57976e10	−0.197563
$730$	0	0
$731$	− 1.06402e11i	− 0.372631i
$732$	0	0
$733$	−3.05441e11	−1.05806	−0.529031	−	0.848603i	$-0.677444\pi$
$733$	−3.05441e11	−1.05806	−0.529031	+	0.848603i	$0.677444\pi$
$734$	0	0
$735$	− 1.39623e11i	− 0.478420i
$736$	0	0
$737$	8.16147e11	2.76629
$738$	0	0
$739$	3.94495e11i	1.32271i	0.750074	+	0.661354i	$0.230018\pi$
$739$	3.94495e11i	1.32271i	−0.750074	+	0.661354i	$0.769982\pi$
$740$	0	0
$741$	6.64710e11	2.20475
$742$	0	0
$743$	− 5.69341e10i	− 0.186817i	−0.995628	−	0.0934086i	$-0.970224\pi$
$743$	− 5.69341e10i	− 0.186817i	0.995628	−	0.0934086i	$-0.0297763\pi$
$744$	0	0
$745$	−6.09237e10	−0.197770
$746$	0	0
$747$	− 1.90842e11i	− 0.612902i
$748$	0	0
$749$	−2.76582e10	−0.0878813
$750$	0	0
$751$	− 2.12038e11i	− 0.666583i	−0.942824	−	0.333292i	$-0.891841\pi$
$751$	− 2.12038e11i	− 0.666583i	0.942824	−	0.333292i	$-0.108159\pi$
$752$	0	0
$753$	5.59577e11	1.74052
$754$	0	0
$755$	− 5.59897e10i	− 0.172314i
$756$	0	0
$757$	6.46697e10	0.196933	0.0984663	−	0.995140i	$-0.468606\pi$
$757$	6.46697e10	0.196933	0.0984663	+	0.995140i	$0.468606\pi$
$758$	0	0
$759$	3.77473e11i	1.13741i
$760$	0	0
$761$	−2.88384e11	−0.859869	−0.429935	−	0.902860i	$-0.641463\pi$
$761$	−2.88384e11	−0.859869	−0.429935	+	0.902860i	$0.641463\pi$
$762$	0	0
$763$	1.54256e11i	0.455139i
$764$	0	0
$765$	−9.47887e10	−0.276765
$766$	0	0
$767$	− 1.50893e11i	− 0.436002i
$768$	0	0
$769$	3.09585e11	0.885267	0.442633	−	0.896703i	$-0.354044\pi$
$769$	3.09585e11	0.885267	0.442633	+	0.896703i	$0.354044\pi$
$770$	0	0
$771$	1.72790e11i	0.488991i
$772$	0	0
$773$	−5.21097e11	−1.45949	−0.729744	−	0.683721i	$-0.760361\pi$
$773$	−5.21097e11	−1.45949	−0.729744	+	0.683721i	$0.760361\pi$
$774$	0	0
$775$	− 3.03217e10i	− 0.0840518i
$776$	0	0
$777$	−1.06825e11	−0.293081
$778$	0	0
$779$	5.60130e10i	0.152103i
$780$	0	0
$781$	4.39332e11	1.18083
$782$	0	0
$783$	7.87272e10i	0.209449i
$784$	0	0
$785$	−5.63613e9	−0.0148423
$786$	0	0
$787$	5.00687e11i	1.30517i	0.757715	+	0.652585i	$0.226316\pi$
$787$	5.00687e11i	1.30517i	−0.757715	+	0.652585i	$0.773684\pi$
$788$	0	0
$789$	−2.53828e10	−0.0654986
$790$	0	0
$791$	− 2.00469e11i	− 0.512084i
$792$	0	0
$793$	−7.70245e11	−1.94776
$794$	0	0
$795$	− 1.58403e10i	− 0.0396546i
$796$	0	0
$797$	−4.22774e11	−1.04779	−0.523896	−	0.851783i	$-0.675522\pi$
$797$	−4.22774e11	−1.04779	−0.523896	+	0.851783i	$0.675522\pi$
$798$	0	0
$799$	9.68670e11i	2.37678i
$800$	0	0
$801$	−1.85388e11	−0.450352
$802$	0	0
$803$	− 5.72172e11i	− 1.37615i
$804$	0	0
$805$	4.17408e10	0.0993978
$806$	0	0
$807$	2.98088e11i	0.702830i
$808$	0	0
$809$	1.06326e9	0.00248226	0.00124113	−	0.999999i	$-0.499605\pi$
$809$	1.06326e9	0.00248226	0.00124113	+	0.999999i	$0.499605\pi$
$810$	0	0
$811$	2.59999e11i	0.601020i	0.953779	+	0.300510i	$0.0971569\pi$
$811$	2.59999e11i	0.601020i	−0.953779	+	0.300510i	$0.902843\pi$
$812$	0	0
$813$	−1.61913e11	−0.370613
$814$	0	0
$815$	− 5.39261e9i	− 0.0122227i
$816$	0	0
$817$	1.77882e11	0.399250
$818$	0	0
$819$	9.22853e10i	0.205115i
$820$	0	0
$821$	2.68005e11	0.589889	0.294944	−	0.955514i	$-0.404699\pi$
$821$	2.68005e11	0.589889	0.294944	+	0.955514i	$0.404699\pi$
$822$	0	0
$823$	8.13596e10i	0.177341i	0.996061	+	0.0886705i	$0.0282618\pi$
$823$	8.13596e10i	0.177341i	−0.996061	+	0.0886705i	$0.971738\pi$
$824$	0	0
$825$	−1.62117e11	−0.349955
$826$	0	0
$827$	3.60895e11i	0.771540i	0.922595	+	0.385770i	$0.126064\pi$
$827$	3.60895e11i	0.771540i	−0.922595	+	0.385770i	$0.873936\pi$
$828$	0	0
$829$	−3.75987e11	−0.796077	−0.398038	−	0.917369i	$-0.630309\pi$
$829$	−3.75987e11	−0.796077	−0.398038	+	0.917369i	$0.630309\pi$
$830$	0	0
$831$	5.08134e11i	1.06555i
$832$	0	0
$833$	−5.62859e11	−1.16901
$834$	0	0
$835$	5.68503e10i	0.116946i
$836$	0	0
$837$	1.33054e11	0.271098
$838$	0	0
$839$	− 3.78533e11i	− 0.763935i	−0.924176	−	0.381967i	$-0.875247\pi$
$839$	− 3.78533e11i	− 0.763935i	0.924176	−	0.381967i	$-0.124753\pi$
$840$	0	0
$841$	−4.47509e11	−0.894576
$842$	0	0
$843$	8.31925e11i	1.64730i
$844$	0	0
$845$	−1.47420e11	−0.289154
$846$	0	0
$847$	1.91174e11i	0.371445i
$848$	0	0
$849$	3.14001e10	0.0604366
$850$	0	0
$851$	2.41228e11i	0.459949i
$852$	0	0
$853$	3.42638e11	0.647202	0.323601	−	0.946194i	$-0.395106\pi$
$853$	3.42638e11	0.647202	0.323601	+	0.946194i	$0.395106\pi$
$854$	0	0
$855$	− 1.58468e11i	− 0.296536i
$856$	0	0
$857$	−3.33508e11	−0.618277	−0.309138	−	0.951017i	$-0.600041\pi$
$857$	−3.33508e11	−0.618277	−0.309138	+	0.951017i	$0.600041\pi$
$858$	0	0
$859$	− 1.00496e12i	− 1.84576i	−0.385084	−	0.922881i	$-0.625828\pi$
$859$	− 1.00496e12i	− 1.84576i	0.385084	−	0.922881i	$-0.374172\pi$
$860$	0	0
$861$	−2.44110e10	−0.0444194
$862$	0	0
$863$	4.79204e11i	0.863928i	0.901891	+	0.431964i	$0.142179\pi$
$863$	4.79204e11i	0.863928i	−0.901891	+	0.431964i	$0.857821\pi$
$864$	0	0
$865$	−1.16329e11	−0.207789
$866$	0	0
$867$	5.14997e11i	0.911440i
$868$	0	0
$869$	1.18926e12	2.08544
$870$	0	0
$871$	− 1.41438e12i	− 2.45751i
$872$	0	0
$873$	−4.02837e11	−0.693541
$874$	0	0
$875$	1.79268e10i	0.0305823i
$876$	0	0
$877$	−4.59890e11	−0.777420	−0.388710	−	0.921360i	$-0.627079\pi$
$877$	−4.59890e11	−0.777420	−0.388710	+	0.921360i	$0.627079\pi$
$878$	0	0
$879$	− 7.72477e11i	− 1.29399i
$880$	0	0
$881$	7.20232e11	1.19555	0.597777	−	0.801663i	$-0.296051\pi$
$881$	7.20232e11	1.19555	0.597777	+	0.801663i	$0.296051\pi$
$882$	0	0
$883$	8.35891e11i	1.37501i	0.726178	+	0.687507i	$0.241295\pi$
$883$	8.35891e11i	1.37501i	−0.726178	+	0.687507i	$0.758705\pi$
$884$	0	0
$885$	−1.12921e11	−0.184079
$886$	0	0
$887$	2.07019e11i	0.334439i	0.985920	+	0.167219i	$0.0534788\pi$
$887$	2.07019e11i	0.334439i	−0.985920	+	0.167219i	$0.946521\pi$
$888$	0	0
$889$	3.54981e10	0.0568327
$890$	0	0
$891$	− 1.13696e12i	− 1.80400i
$892$	0	0
$893$	−1.61942e12	−2.54657
$894$	0	0
$895$	− 1.51469e11i	− 0.236064i
$896$	0	0
$897$	6.54162e11	1.01045
$898$	0	0
$899$	− 8.91301e10i	− 0.136454i
$900$	0	0
$901$	−6.38563e10	−0.0968957
$902$	0	0
$903$	7.75228e10i	0.116595i
$904$	0	0
$905$	−4.00284e11	−0.596725
$906$	0	0
$907$	− 1.16689e12i	− 1.72426i	−0.506690	−	0.862128i	$-0.669131\pi$
$907$	− 1.16689e12i	− 1.72426i	0.506690	−	0.862128i	$-0.330869\pi$
$908$	0	0
$909$	−4.76643e11	−0.698132
$910$	0	0
$911$	1.15580e12i	1.67806i	0.544084	+	0.839031i	$0.316877\pi$
$911$	1.15580e12i	1.67806i	−0.544084	+	0.839031i	$0.683123\pi$
$912$	0	0
$913$	−1.31579e12	−1.89366
$914$	0	0
$915$	5.76414e11i	0.822338i
$916$	0	0
$917$	2.84859e11	0.402858
$918$	0	0
$919$	− 1.20572e11i	− 0.169038i	−0.996422	−	0.0845192i	$-0.973065\pi$
$919$	− 1.20572e11i	− 0.169038i	0.996422	−	0.0845192i	$-0.0269354\pi$
$920$	0	0
$921$	1.44668e12	2.01064
$922$	0	0
$923$	− 7.61364e11i	− 1.04902i
$924$	0	0
$925$	−1.03602e11	−0.141515
$926$	0	0
$927$	− 1.17542e11i	− 0.159174i
$928$	0	0
$929$	7.85328e11	1.05436	0.527179	−	0.849754i	$-0.323250\pi$
$929$	7.85328e11	1.05436	0.527179	+	0.849754i	$0.323250\pi$
$930$	0	0
$931$	− 9.40988e11i	− 1.25252i
$932$	0	0
$933$	−6.24370e11	−0.823977
$934$	0	0
$935$	6.53536e11i	0.855112i
$936$	0	0
$937$	−4.90137e11	−0.635856	−0.317928	−	0.948115i	$-0.602987\pi$
$937$	−4.90137e11	−0.635856	−0.317928	+	0.948115i	$0.602987\pi$
$938$	0	0
$939$	4.27542e11i	0.549941i
$940$	0	0
$941$	−5.66225e11	−0.722155	−0.361077	−	0.932536i	$-0.617591\pi$
$941$	−5.66225e11	−0.722155	−0.361077	+	0.932536i	$0.617591\pi$
$942$	0	0
$943$	5.51241e10i	0.0697100i
$944$	0	0
$945$	−7.86641e10	−0.0986391
$946$	0	0
$947$	− 3.38620e10i	− 0.0421029i	−0.999778	−	0.0210515i	$-0.993299\pi$
$947$	− 3.38620e10i	− 0.0421029i	0.999778	−	0.0210515i	$-0.00670139\pi$
$948$	0	0
$949$	−9.91575e11	−1.22253
$950$	0	0
$951$	1.00159e11i	0.122452i
$952$	0	0
$953$	−2.54903e11	−0.309032	−0.154516	−	0.987990i	$-0.549382\pi$
$953$	−2.54903e11	−0.309032	−0.154516	+	0.987990i	$0.549382\pi$
$954$	0	0
$955$	− 4.90948e11i	− 0.590231i
$956$	0	0
$957$	−4.76540e11	−0.568135
$958$	0	0
$959$	− 3.73201e11i	− 0.441233i
$960$	0	0
$961$	7.02256e11	0.823383
$962$	0	0
$963$	1.03337e11i	0.120158i
$964$	0	0
$965$	1.89931e11	0.219022
$966$	0	0
$967$	− 9.19624e11i	− 1.05173i	−0.850568	−	0.525865i	$-0.823742\pi$
$967$	− 9.19624e11i	− 1.05173i	0.850568	−	0.525865i	$-0.176258\pi$
$968$	0	0
$969$	−2.00530e12	−2.27449
$970$	0	0
$971$	1.35436e12i	1.52355i	0.647840	+	0.761776i	$0.275672\pi$
$971$	1.35436e12i	1.52355i	−0.647840	+	0.761776i	$0.724328\pi$
$972$	0	0
$973$	−1.02208e11	−0.114034
$974$	0	0
$975$	2.80949e11i	0.310891i
$976$	0	0
$977$	−1.38499e12	−1.52008	−0.760042	−	0.649874i	$-0.774822\pi$
$977$	−1.38499e12	−1.52008	−0.760042	+	0.649874i	$0.774822\pi$
$978$	0	0
$979$	1.27819e12i	1.39144i
$980$	0	0
$981$	5.76336e11	0.622300
$982$	0	0
$983$	− 1.57780e12i	− 1.68981i	−0.534916	−	0.844905i	$-0.679657\pi$
$983$	− 1.57780e12i	− 1.68981i	0.534916	−	0.844905i	$-0.320343\pi$
$984$	0	0
$985$	1.34367e11	0.142741
$986$	0	0
$987$	− 7.05761e11i	− 0.743685i
$988$	0	0
$989$	1.75060e11	0.182979
$990$	0	0
$991$	1.83814e11i	0.190583i	0.995449	+	0.0952917i	$0.0303784\pi$
$991$	1.83814e11i	0.190583i	−0.995449	+	0.0952917i	$0.969622\pi$
$992$	0	0
$993$	−9.93195e11	−1.02150
$994$	0	0
$995$	1.73547e11i	0.177062i
$996$	0	0
$997$	2.34014e11	0.236844	0.118422	−	0.992963i	$-0.462217\pi$
$997$	2.34014e11	0.236844	0.118422	+	0.992963i	$0.462217\pi$
$998$	0	0
$999$	− 4.54615e11i	− 0.456438i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.9.b.d.191.13		16
4.3	odd	2	inner	320.9.b.d.191.4		16
8.3	odd	2		20.9.b.a.11.1	✓	16
8.5	even	2		20.9.b.a.11.2	yes	16
24.5	odd	2		180.9.c.a.91.15		16
24.11	even	2		180.9.c.a.91.16		16
40.3	even	4		100.9.d.c.99.16		32
40.13	odd	4		100.9.d.c.99.18		32
40.19	odd	2		100.9.b.d.51.16		16
40.27	even	4		100.9.d.c.99.17		32
40.29	even	2		100.9.b.d.51.15		16
40.37	odd	4		100.9.d.c.99.15		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.9.b.a.11.1	✓	16	8.3	odd	2
20.9.b.a.11.2	yes	16	8.5	even	2
100.9.b.d.51.15		16	40.29	even	2
100.9.b.d.51.16		16	40.19	odd	2
100.9.d.c.99.15		32	40.37	odd	4
100.9.d.c.99.16		32	40.3	even	4
100.9.d.c.99.17		32	40.27	even	4
100.9.d.c.99.18		32	40.13	odd	4
180.9.c.a.91.15		16	24.5	odd	2
180.9.c.a.91.16		16	24.11	even	2
320.9.b.d.191.4		16	4.3	odd	2	inner
320.9.b.d.191.13		16	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:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	320.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+98.1237i q^{3} -279.508 q^{5} -820.952i q^{7} -3067.27 q^{9} +21147.7i q^{11} +36649.1 q^{13} -27426.4i q^{15} -110563. q^{17} -184840. i q^{19} +80554.9 q^{21} -181906. i q^{23} +78125.0 q^{25} +342818. i q^{27} +229647. q^{29} -388118. i q^{31} -2.07510e6 q^{33} +229463. i q^{35} -1.32611e6 q^{37} +3.59615e6i q^{39} -303036. q^{41} +962361. i q^{43} +857327. q^{45} -8.76124e6i q^{47} +5.09084e6 q^{49} -1.08489e7i q^{51} +577555. q^{53} -5.91097e6i q^{55} +1.81371e7 q^{57} -4.11725e6i q^{59} -2.10168e7 q^{61} +2.51808e6i q^{63} -1.02437e7 q^{65} -3.85926e7i q^{67} +1.78493e7 q^{69} -2.07744e7i q^{71} -2.70559e7 q^{73} +7.66592e6i q^{75} +1.73613e7 q^{77} -5.62358e7i q^{79} -5.37629e7 q^{81} +6.22189e7i q^{83} +3.09033e7 q^{85} +2.25338e7i q^{87} +6.04409e7 q^{89} -3.00871e7i q^{91} +3.80836e7 q^{93} +5.16642e7i q^{95} +1.31334e8 q^{97} -6.48658e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 38800 q^{9} - 51392 q^{13} + 27552 q^{17} - 414496 q^{21} + 1250000 q^{25} - 2764896 q^{29} - 5521600 q^{33} - 9009472 q^{37} - 8576448 q^{41} - 1580000 q^{45} - 32803600 q^{49} - 2452032 q^{53}+ \cdots + 171851232 q^{97}+O(q^{100}) \) 16 * q - 38800 * q^9 - 51392 * q^13 + 27552 * q^17 - 414496 * q^21 + 1250000 * q^25 - 2764896 * q^29 - 5521600 * q^33 - 9009472 * q^37 - 8576448 * q^41 - 1580000 * q^45 - 32803600 * q^49 - 2452032 * q^53 + 11957760 * q^57 - 8371712 * q^61 + 9060000 * q^65 - 7527264 * q^69 + 61907232 * q^73 + 156997440 * q^77 + 140586672 * q^81 + 106960000 * q^85 + 106647456 * q^89 - 105563840 * q^93 + 171851232 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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