LMFDB - Embedded newform 324.9.c.a.161.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,9,Mod(161,324)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(324, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 9, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("324.161");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	324.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$131.990669660$
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} - 3 x^{15} + 1777 x^{14} - 2036 x^{13} - 1362401 x^{12} - 229443563 x^{11} + \cdots + 41\!\cdots\!00 $ x^16 - 3x^15 + 1777x^14 - 2036x^13 - 1362401x^12 - 229443563x^11 + 12565251279x^10 - 924267732266x^9 + 53903003620983x^8 - 669407211102857x^7 + 37488371175442929x^6 - 415198738773859682x^5 + 80523268287397086568x^4 - 10099200795405062595224x^3 + 445357305762676329633376x^2 - 15926706368268795333431520*x + 417889304743672757045502400
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{28}\cdot 3^{88} $
Twist minimal:	no (minimal twist has level 36)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.12
Root			$-45.2983 + 25.4240i$ of defining polynomial
Character	$\chi$	$=$	324.161
Dual form			324.9.c.a.161.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+466.761i q^{5} -1577.33 q^{7} -10955.1i q^{11} -21081.8 q^{13} +106309. i q^{17} +128508. q^{19} +230034. i q^{23} +172759. q^{25} -135593. i q^{29} -1.32481e6 q^{31} -736235. i q^{35} -916812. q^{37} +1.28599e6i q^{41} -3.51031e6 q^{43} +4.86808e6i q^{47} -3.27684e6 q^{49} -1.24495e7i q^{53} +5.11339e6 q^{55} +1.04175e7i q^{59} +1.85403e7 q^{61} -9.84016e6i q^{65} -3.42607e7 q^{67} -3.79956e7i q^{71} +4.43751e7 q^{73} +1.72797e7i q^{77} -3.76224e6 q^{79} -1.70100e7i q^{83} -4.96207e7 q^{85} +1.82163e7i q^{89} +3.32529e7 q^{91} +5.99826e7i q^{95} +6.93557e7 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 1846 q^{7} - 3370 q^{13} + 84518 q^{19} - 911606 q^{25} + 756614 q^{31} - 1671664 q^{37} - 679024 q^{43} + 6052890 q^{49} - 17497638 q^{55} - 34901410 q^{61} + 70053452 q^{67} + 28071218 q^{73} - 61378918 q^{79}+ \cdots + 66842156 q^{97}+O(q^{100}) $ 16 * q - 1846 * q^7 - 3370 * q^13 + 84518 * q^19 - 911606 * q^25 + 756614 * q^31 - 1671664 * q^37 - 679024 * q^43 + 6052890 * q^49 - 17497638 * q^55 - 34901410 * q^61 + 70053452 * q^67 + 28071218 * q^73 - 61378918 * q^79 + 79786044 * q^85 - 55534214 * q^91 + 66842156 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$.

$n$	$163$	$245$
$\chi(n)$	$1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	466.761i	0.746817i	0.927667	+	0.373409i	$0.121811\pi$
$5$	466.761i	0.746817i	−0.927667	+	0.373409i	$0.878189\pi$
$6$	0	0
$7$	−1577.33	−0.656946	−0.328473	−	0.944513i	$-0.606534\pi$
$7$	−1577.33	−0.656946	−0.328473	+	0.944513i	$0.606534\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 10955.1i	− 0.748245i	−0.927379	−	0.374122i	$-0.877944\pi$
$11$	− 10955.1i	− 0.748245i	0.927379	−	0.374122i	$-0.122056\pi$
$12$	0	0
$13$	−21081.8	−0.738133	−0.369066	−	0.929403i	$-0.620322\pi$
$13$	−21081.8	−0.738133	−0.369066	+	0.929403i	$0.620322\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	106309.i	1.27284i	0.771344	+	0.636419i	$0.219585\pi$
$17$	106309.i	1.27284i	−0.771344	+	0.636419i	$0.780415\pi$
$18$	0	0
$19$	128508.	0.986090	0.493045	−	0.870004i	$-0.335884\pi$
$19$	128508.	0.986090	0.493045	+	0.870004i	$0.335884\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	230034.i	0.822016i	0.911632	+	0.411008i	$0.134823\pi$
$23$	230034.i	0.822016i	−0.911632	+	0.411008i	$0.865177\pi$
$24$	0	0
$25$	172759.	0.442264
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 135593.i	− 0.191710i	−0.995395	−	0.0958549i	$-0.969442\pi$
$29$	− 135593.i	− 0.191710i	0.995395	−	0.0958549i	$-0.0305585\pi$
$30$	0	0
$31$	−1.32481e6	−1.43452	−0.717262	−	0.696804i	$-0.754605\pi$
$31$	−1.32481e6	−1.43452	−0.717262	+	0.696804i	$0.754605\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 736235.i	− 0.490619i
$36$	0	0
$37$	−916812.	−0.489185	−0.244593	−	0.969626i	$-0.578654\pi$
$37$	−916812.	−0.489185	−0.244593	+	0.969626i	$0.578654\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.28599e6i	0.455094i	0.973767	+	0.227547i	$0.0730706\pi$
$41$	1.28599e6i	0.455094i	−0.973767	+	0.227547i	$0.926929\pi$
$42$	0	0
$43$	−3.51031e6	−1.02677	−0.513383	−	0.858160i	$-0.671608\pi$
$43$	−3.51031e6	−1.02677	−0.513383	+	0.858160i	$0.671608\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.86808e6i	0.997623i	0.866711	+	0.498811i	$0.166230\pi$
$47$	4.86808e6i	0.997623i	−0.866711	+	0.498811i	$0.833770\pi$
$48$	0	0
$49$	−3.27684e6	−0.568421
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 1.24495e7i	− 1.57779i	−0.614527	−	0.788895i	$-0.710653\pi$
$53$	− 1.24495e7i	− 1.57779i	0.614527	−	0.788895i	$-0.289347\pi$
$54$	0	0
$55$	5.11339e6	0.558802
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.04175e7i	0.859718i	0.902896	+	0.429859i	$0.141437\pi$
$59$	1.04175e7i	0.859718i	−0.902896	+	0.429859i	$0.858563\pi$
$60$	0	0
$61$	1.85403e7	1.33905	0.669526	−	0.742789i	$-0.266497\pi$
$61$	1.85403e7	1.33905	0.669526	+	0.742789i	$0.266497\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 9.84016e6i	− 0.551250i
$66$	0	0
$67$	−3.42607e7	−1.70019	−0.850095	−	0.526630i	$-0.823455\pi$
$67$	−3.42607e7	−1.70019	−0.850095	+	0.526630i	$0.823455\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 3.79956e7i	− 1.49520i	−0.664148	−	0.747601i	$-0.731206\pi$
$71$	− 3.79956e7i	− 1.49520i	0.664148	−	0.747601i	$-0.268794\pi$
$72$	0	0
$73$	4.43751e7	1.56260	0.781301	−	0.624155i	$-0.214556\pi$
$73$	4.43751e7	1.56260	0.781301	+	0.624155i	$0.214556\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.72797e7i	0.491557i
$78$	0	0
$79$	−3.76224e6	−0.0965913	−0.0482957	−	0.998833i	$-0.515379\pi$
$79$	−3.76224e6	−0.0965913	−0.0482957	+	0.998833i	$0.515379\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 1.70100e7i	− 0.358421i	−0.983811	−	0.179210i	$-0.942646\pi$
$83$	− 1.70100e7i	− 0.358421i	0.983811	−	0.179210i	$-0.0573542\pi$
$84$	0	0
$85$	−4.96207e7	−0.950577
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.82163e7i	0.290335i	0.989407	+	0.145168i	$0.0463722\pi$
$89$	1.82163e7i	0.290335i	−0.989407	+	0.145168i	$0.953628\pi$
$90$	0	0
$91$	3.32529e7	0.484914
$92$	0	0
$93$	0	0
$94$	0	0
$95$	5.99826e7i	0.736429i
$96$	0	0
$97$	6.93557e7	0.783421	0.391711	−	0.920088i	$-0.371883\pi$
$97$	6.93557e7	0.783421	0.391711	+	0.920088i	$0.371883\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.05630e8i	1.01509i	0.861626	+	0.507544i	$0.169446\pi$
$101$	1.05630e8i	1.01509i	−0.861626	+	0.507544i	$0.830554\pi$
$102$	0	0
$103$	1.21627e8	1.08064	0.540319	−	0.841460i	$-0.318304\pi$
$103$	1.21627e8	1.08064	0.540319	+	0.841460i	$0.318304\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1.98700e8i	− 1.51587i	−0.652330	−	0.757935i	$-0.726208\pi$
$107$	− 1.98700e8i	− 1.51587i	0.652330	−	0.757935i	$-0.273792\pi$
$108$	0	0
$109$	−8.94122e7	−0.633418	−0.316709	−	0.948523i	$-0.602578\pi$
$109$	−8.94122e7	−0.633418	−0.316709	+	0.948523i	$0.602578\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 2.04528e8i	− 1.25441i	−0.778856	−	0.627203i	$-0.784199\pi$
$113$	− 2.04528e8i	− 1.25441i	0.778856	−	0.627203i	$-0.215801\pi$
$114$	0	0
$115$	−1.07371e8	−0.613895
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 1.67684e8i	− 0.836186i
$120$	0	0
$121$	9.43457e7	0.440130
$122$	0	0
$123$	0	0
$124$	0	0
$125$	2.62966e8i	1.07711i
$126$	0	0
$127$	−2.19577e8	−0.844058	−0.422029	−	0.906582i	$-0.638682\pi$
$127$	−2.19577e8	−0.844058	−0.422029	+	0.906582i	$0.638682\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 3.51599e8i	− 1.19388i	−0.802285	−	0.596942i	$-0.796382\pi$
$131$	− 3.51599e8i	− 1.19388i	0.802285	−	0.596942i	$-0.203618\pi$
$132$	0	0
$133$	−2.02700e8	−0.647808
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 5.49676e8i	− 1.56036i	−0.625555	−	0.780180i	$-0.715128\pi$
$137$	− 5.49676e8i	− 1.56036i	0.625555	−	0.780180i	$-0.284872\pi$
$138$	0	0
$139$	−1.15593e8	−0.309650	−0.154825	−	0.987942i	$-0.549481\pi$
$139$	−1.15593e8	−0.309650	−0.154825	+	0.987942i	$0.549481\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2.30952e8i	0.552304i
$144$	0	0
$145$	6.32894e7	0.143172
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 5.24849e8i	− 1.06485i	−0.846477	−	0.532426i	$-0.821280\pi$
$149$	− 5.24849e8i	− 1.06485i	0.846477	−	0.532426i	$-0.178720\pi$
$150$	0	0
$151$	1.60088e7	0.0307929	0.0153965	−	0.999881i	$-0.495099\pi$
$151$	1.60088e7	0.0307929	0.0153965	+	0.999881i	$0.495099\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 6.18370e8i	− 1.07133i
$156$	0	0
$157$	7.75730e8	1.27677	0.638384	−	0.769718i	$-0.279603\pi$
$157$	7.75730e8	1.27677	0.638384	+	0.769718i	$0.279603\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 3.62839e8i	− 0.540020i
$162$	0	0
$163$	−8.44040e8	−1.19567	−0.597837	−	0.801618i	$-0.703973\pi$
$163$	−8.44040e8	−1.19567	−0.597837	+	0.801618i	$0.703973\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 7.04198e8i	− 0.905375i	−0.891669	−	0.452688i	$-0.850465\pi$
$167$	− 7.04198e8i	− 0.905375i	0.891669	−	0.452688i	$-0.149535\pi$
$168$	0	0
$169$	−3.71288e8	−0.455160
$170$	0	0
$171$	0	0
$172$	0	0
$173$	9.00284e8i	1.00507i	0.864558	+	0.502533i	$0.167599\pi$
$173$	9.00284e8i	1.00507i	−0.864558	+	0.502533i	$0.832401\pi$
$174$	0	0
$175$	−2.72498e8	−0.290544
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 4.93044e8i	− 0.480257i	−0.970741	−	0.240128i	$-0.922810\pi$
$179$	− 4.93044e8i	− 0.480257i	0.970741	−	0.240128i	$-0.0771895\pi$
$180$	0	0
$181$	2.98103e8	0.277749	0.138874	−	0.990310i	$-0.455652\pi$
$181$	2.98103e8	0.277749	0.138874	+	0.990310i	$0.455652\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 4.27932e8i	− 0.365332i
$186$	0	0
$187$	1.16462e9	0.952394
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 1.57061e9i	− 1.18014i	−0.807350	−	0.590072i	$-0.799099\pi$
$191$	− 1.57061e9i	− 1.18014i	0.807350	−	0.590072i	$-0.200901\pi$
$192$	0	0
$193$	1.90271e9	1.37133	0.685666	−	0.727916i	$-0.259511\pi$
$193$	1.90271e9	1.37133	0.685666	+	0.727916i	$0.259511\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 2.70298e9i	− 1.79464i	−0.441378	−	0.897321i	$-0.645510\pi$
$197$	− 2.70298e9i	− 1.79464i	0.441378	−	0.897321i	$-0.354490\pi$
$198$	0	0
$199$	1.73691e9	1.10755	0.553777	−	0.832665i	$-0.313186\pi$
$199$	1.73691e9	1.10755	0.553777	+	0.832665i	$0.313186\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.13874e8i	0.125943i
$204$	0	0
$205$	−6.00249e8	−0.339872
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 1.40781e9i	− 0.737837i
$210$	0	0
$211$	−3.53379e9	−1.78283	−0.891416	−	0.453186i	$-0.850288\pi$
$211$	−3.53379e9	−1.78283	−0.891416	+	0.453186i	$0.850288\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 1.63847e9i	− 0.766806i
$216$	0	0
$217$	2.08966e9	0.942405
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 2.24118e9i	− 0.939523i
$222$	0	0
$223$	2.80290e9	1.13341	0.566706	−	0.823920i	$-0.308218\pi$
$223$	2.80290e9	1.13341	0.566706	+	0.823920i	$0.308218\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 1.87240e9i	− 0.705174i	−0.935779	−	0.352587i	$-0.885302\pi$
$227$	− 1.87240e9i	− 0.705174i	0.935779	−	0.352587i	$-0.114698\pi$
$228$	0	0
$229$	−4.56368e9	−1.65948	−0.829742	−	0.558148i	$-0.811512\pi$
$229$	−4.56368e9	−1.65948	−0.829742	+	0.558148i	$0.811512\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 3.37063e9i	− 1.14364i	−0.820381	−	0.571818i	$-0.806238\pi$
$233$	− 3.37063e9i	− 1.14364i	0.820381	−	0.571818i	$-0.193762\pi$
$234$	0	0
$235$	−2.27223e9	−0.745042
$236$	0	0
$237$	0	0
$238$	0	0
$239$	3.97567e9i	1.21848i	0.792986	+	0.609240i	$0.208526\pi$
$239$	3.97567e9i	1.21848i	−0.792986	+	0.609240i	$0.791474\pi$
$240$	0	0
$241$	−3.27327e9	−0.970316	−0.485158	−	0.874426i	$-0.661238\pi$
$241$	−3.27327e9	−0.970316	−0.485158	+	0.874426i	$0.661238\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 1.52950e9i	− 0.424507i
$246$	0	0
$247$	−2.70919e9	−0.727865
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 6.04583e9i	− 1.52321i	−0.648039	−	0.761607i	$-0.724410\pi$
$251$	− 6.04583e9i	− 1.52321i	0.648039	−	0.761607i	$-0.275590\pi$
$252$	0	0
$253$	2.52003e9	0.615069
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 5.54259e9i	− 1.27052i	−0.772300	−	0.635258i	$-0.780894\pi$
$257$	− 5.54259e9i	− 1.27052i	0.772300	−	0.635258i	$-0.219106\pi$
$258$	0	0
$259$	1.44611e9	0.321369
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 2.88296e9i	− 0.602581i	−0.953532	−	0.301291i	$-0.902583\pi$
$263$	− 2.88296e9i	− 0.602581i	0.953532	−	0.301291i	$-0.0974175\pi$
$264$	0	0
$265$	5.81095e9	1.17832
$266$	0	0
$267$	0	0
$268$	0	0
$269$	2.93406e9i	0.560350i	0.959949	+	0.280175i	$0.0903925\pi$
$269$	2.93406e9i	0.560350i	−0.959949	+	0.280175i	$0.909608\pi$
$270$	0	0
$271$	−1.52216e9	−0.282217	−0.141109	−	0.989994i	$-0.545067\pi$
$271$	−1.52216e9	−0.282217	−0.141109	+	0.989994i	$0.545067\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 1.89259e9i	− 0.330922i
$276$	0	0
$277$	−5.35390e8	−0.0909393	−0.0454696	−	0.998966i	$-0.514478\pi$
$277$	−5.35390e8	−0.0909393	−0.0454696	+	0.998966i	$0.514478\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	4.79146e9i	0.768497i	0.923230	+	0.384248i	$0.125539\pi$
$281$	4.79146e9i	0.768497i	−0.923230	+	0.384248i	$0.874461\pi$
$282$	0	0
$283$	4.63044e9	0.721899	0.360949	−	0.932585i	$-0.382453\pi$
$283$	4.63044e9	0.721899	0.360949	+	0.932585i	$0.382453\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 2.02842e9i	− 0.298973i
$288$	0	0
$289$	−4.32577e9	−0.620115
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 1.08308e10i	− 1.46957i	−0.678299	−	0.734786i	$-0.737283\pi$
$293$	− 1.08308e10i	− 1.46957i	0.678299	−	0.734786i	$-0.262717\pi$
$294$	0	0
$295$	−4.86249e9	−0.642052
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 4.84953e9i	− 0.606757i
$300$	0	0
$301$	5.53691e9	0.674530
$302$	0	0
$303$	0	0
$304$	0	0
$305$	8.65388e9i	1.00003i
$306$	0	0
$307$	−5.45595e9	−0.614210	−0.307105	−	0.951676i	$-0.599360\pi$
$307$	−5.45595e9	−0.614210	−0.307105	+	0.951676i	$0.599360\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1.45863e10i	1.55921i	0.626272	+	0.779605i	$0.284580\pi$
$311$	1.45863e10i	1.55921i	−0.626272	+	0.779605i	$0.715420\pi$
$312$	0	0
$313$	−1.96679e9	−0.204918	−0.102459	−	0.994737i	$-0.532671\pi$
$313$	−1.96679e9	−0.204918	−0.102459	+	0.994737i	$0.532671\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6.92492e9i	0.685769i	0.939378	+	0.342885i	$0.111404\pi$
$317$	6.92492e9i	0.685769i	−0.939378	+	0.342885i	$0.888596\pi$
$318$	0	0
$319$	−1.48543e9	−0.143446
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1.36615e10i	1.25513i
$324$	0	0
$325$	−3.64208e9	−0.326450
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 7.67856e9i	− 0.655385i
$330$	0	0
$331$	1.75109e10	1.45880	0.729401	−	0.684086i	$-0.239799\pi$
$331$	1.75109e10	1.45880	0.729401	+	0.684086i	$0.239799\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 1.59916e10i	− 1.26973i
$336$	0	0
$337$	3.25944e9	0.252711	0.126355	−	0.991985i	$-0.459672\pi$
$337$	3.25944e9	0.252711	0.126355	+	0.991985i	$0.459672\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	1.45134e10i	1.07337i
$342$	0	0
$343$	1.42616e10	1.03037
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 2.34912e10i	− 1.62027i	−0.586243	−	0.810135i	$-0.699394\pi$
$347$	− 2.34912e10i	− 1.62027i	0.586243	−	0.810135i	$-0.300606\pi$
$348$	0	0
$349$	−1.84032e10	−1.24048	−0.620242	−	0.784410i	$-0.712966\pi$
$349$	−1.84032e10	−1.24048	−0.620242	+	0.784410i	$0.712966\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 5.46001e9i	− 0.351637i	−0.984423	−	0.175818i	$-0.943743\pi$
$353$	− 5.46001e9i	− 0.351637i	0.984423	−	0.175818i	$-0.0562572\pi$
$354$	0	0
$355$	1.77349e10	1.11664
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 1.20718e10i	− 0.726763i	−0.931640	−	0.363382i	$-0.881622\pi$
$359$	− 1.20718e10i	− 0.726763i	0.931640	−	0.363382i	$-0.118378\pi$
$360$	0	0
$361$	−4.69191e8	−0.0276262
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2.07126e10i	1.16698i
$366$	0	0
$367$	4.61047e9	0.254145	0.127072	−	0.991893i	$-0.459442\pi$
$367$	4.61047e9	0.254145	0.127072	+	0.991893i	$0.459442\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1.96370e10i	1.03652i
$372$	0	0
$373$	3.24556e9	0.167670	0.0838348	−	0.996480i	$-0.473283\pi$
$373$	3.24556e9	0.167670	0.0838348	+	0.996480i	$0.473283\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.85854e9i	0.141507i
$378$	0	0
$379$	−7.63733e9	−0.370156	−0.185078	−	0.982724i	$-0.559254\pi$
$379$	−7.63733e9	−0.370156	−0.185078	+	0.982724i	$0.559254\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	2.08424e10i	0.968619i	0.874897	+	0.484309i	$0.160929\pi$
$383$	2.08424e10i	0.968619i	−0.874897	+	0.484309i	$0.839071\pi$
$384$	0	0
$385$	−8.06549e9	−0.367103
$386$	0	0
$387$	0	0
$388$	0	0
$389$	4.33164e10i	1.89171i	0.324596	+	0.945853i	$0.394772\pi$
$389$	4.33164e10i	1.89171i	−0.324596	+	0.945853i	$0.605228\pi$
$390$	0	0
$391$	−2.44546e10	−1.04629
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 1.75607e9i	− 0.0721361i
$396$	0	0
$397$	−1.75923e10	−0.708206	−0.354103	−	0.935206i	$-0.615214\pi$
$397$	−1.75923e10	−0.708206	−0.354103	+	0.935206i	$0.615214\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.51012e10i	0.970770i	0.874301	+	0.485385i	$0.161321\pi$
$401$	2.51012e10i	0.970770i	−0.874301	+	0.485385i	$0.838679\pi$
$402$	0	0
$403$	2.79294e10	1.05887
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.00437e10i	0.366030i
$408$	0	0
$409$	−2.91003e10	−1.03993	−0.519964	−	0.854188i	$-0.674055\pi$
$409$	−2.91003e10	−1.03993	−0.519964	+	0.854188i	$0.674055\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 1.64318e10i	− 0.564789i
$414$	0	0
$415$	7.93962e9	0.267675
$416$	0	0
$417$	0	0
$418$	0	0
$419$	4.02874e10i	1.30711i	0.756877	+	0.653557i	$0.226724\pi$
$419$	4.02874e10i	1.30711i	−0.756877	+	0.653557i	$0.773276\pi$
$420$	0	0
$421$	−1.82772e10	−0.581812	−0.290906	−	0.956752i	$-0.593957\pi$
$421$	−1.82772e10	−0.581812	−0.290906	+	0.956752i	$0.593957\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1.83658e10i	0.562930i
$426$	0	0
$427$	−2.92441e10	−0.879685
$428$	0	0
$429$	0	0
$430$	0	0
$431$	3.30045e10i	0.956454i	0.878236	+	0.478227i	$0.158720\pi$
$431$	3.30045e10i	0.956454i	−0.878236	+	0.478227i	$0.841280\pi$
$432$	0	0
$433$	4.27716e10	1.21676	0.608379	−	0.793647i	$-0.291820\pi$
$433$	4.27716e10	1.21676	0.608379	+	0.793647i	$0.291820\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.95612e10i	0.810582i
$438$	0	0
$439$	−3.71875e10	−1.00124	−0.500620	−	0.865667i	$-0.666895\pi$
$439$	−3.71875e10	−1.00124	−0.500620	+	0.865667i	$0.666895\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	6.12526e9i	0.159041i	0.996833	+	0.0795205i	$0.0253389\pi$
$443$	6.12526e9i	0.159041i	−0.996833	+	0.0795205i	$0.974661\pi$
$444$	0	0
$445$	−8.50265e9	−0.216827
$446$	0	0
$447$	0	0
$448$	0	0
$449$	1.21732e10i	0.299516i	0.988723	+	0.149758i	$0.0478494\pi$
$449$	1.21732e10i	0.299516i	−0.988723	+	0.149758i	$0.952151\pi$
$450$	0	0
$451$	1.40881e10	0.340522
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.55212e10i	0.362142i
$456$	0	0
$457$	6.39080e10	1.46518	0.732589	−	0.680671i	$-0.238312\pi$
$457$	6.39080e10	1.46518	0.732589	+	0.680671i	$0.238312\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 1.19193e10i	− 0.263905i	−0.991256	−	0.131952i	$-0.957875\pi$
$461$	− 1.19193e10i	− 0.263905i	0.991256	−	0.131952i	$-0.0421246\pi$
$462$	0	0
$463$	−3.87770e10	−0.843820	−0.421910	−	0.906638i	$-0.638640\pi$
$463$	−3.87770e10	−0.843820	−0.421910	+	0.906638i	$0.638640\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 8.53122e8i	− 0.0179367i	−0.999960	−	0.00896837i	$-0.997145\pi$
$467$	− 8.53122e8i	− 0.0179367i	0.999960	−	0.00896837i	$-0.00285476\pi$
$468$	0	0
$469$	5.40404e10	1.11693
$470$	0	0
$471$	0	0
$472$	0	0
$473$	3.84556e10i	0.768272i
$474$	0	0
$475$	2.22010e10	0.436112
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 1.01215e10i	− 0.192266i	−0.995369	−	0.0961329i	$-0.969353\pi$
$479$	− 1.01215e10i	− 0.192266i	0.995369	−	0.0961329i	$-0.0306474\pi$
$480$	0	0
$481$	1.93281e10	0.361084
$482$	0	0
$483$	0	0
$484$	0	0
$485$	3.23725e10i	0.585072i
$486$	0	0
$487$	−5.42141e10	−0.963821	−0.481911	−	0.876220i	$-0.660057\pi$
$487$	−5.42141e10	−0.963821	−0.481911	+	0.876220i	$0.660057\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 6.61539e10i	− 1.13823i	−0.822258	−	0.569115i	$-0.807286\pi$
$491$	− 6.61539e10i	− 1.13823i	0.822258	−	0.569115i	$-0.192714\pi$
$492$	0	0
$493$	1.44147e10	0.244015
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5.99315e10i	0.982268i
$498$	0	0
$499$	9.22923e10	1.48855	0.744275	−	0.667873i	$-0.232795\pi$
$499$	9.22923e10	1.48855	0.744275	+	0.667873i	$0.232795\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 2.93013e10i	− 0.457736i	−0.973457	−	0.228868i	$-0.926498\pi$
$503$	− 2.93013e10i	− 0.457736i	0.973457	−	0.228868i	$-0.0735024\pi$
$504$	0	0
$505$	−4.93041e10	−0.758085
$506$	0	0
$507$	0	0
$508$	0	0
$509$	7.86993e10i	1.17246i	0.810143	+	0.586232i	$0.199389\pi$
$509$	7.86993e10i	1.17246i	−0.810143	+	0.586232i	$0.800611\pi$
$510$	0	0
$511$	−6.99942e10	−1.02655
$512$	0	0
$513$	0	0
$514$	0	0
$515$	5.67706e10i	0.807039i
$516$	0	0
$517$	5.33301e10	0.746466
$518$	0	0
$519$	0	0
$520$	0	0
$521$	3.59283e10i	0.487625i	0.969822	+	0.243812i	$0.0783981\pi$
$521$	3.59283e10i	0.487625i	−0.969822	+	0.243812i	$0.921602\pi$
$522$	0	0
$523$	−1.73999e10	−0.232562	−0.116281	−	0.993216i	$-0.537097\pi$
$523$	−1.73999e10	−0.232562	−0.116281	+	0.993216i	$0.537097\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 1.40839e11i	− 1.82592i
$528$	0	0
$529$	2.53955e10	0.324290
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 2.71109e10i	− 0.335920i
$534$	0	0
$535$	9.27452e10	1.13208
$536$	0	0
$537$	0	0
$538$	0	0
$539$	3.58979e10i	0.425318i
$540$	0	0
$541$	−4.58718e10	−0.535497	−0.267748	−	0.963489i	$-0.586280\pi$
$541$	−4.58718e10	−0.535497	−0.267748	+	0.963489i	$0.586280\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 4.17341e10i	− 0.473048i
$546$	0	0
$547$	−1.03482e11	−1.15589	−0.577946	−	0.816075i	$-0.696146\pi$
$547$	−1.03482e11	−1.15589	−0.577946	+	0.816075i	$0.696146\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 1.74248e10i	− 0.189043i
$552$	0	0
$553$	5.93429e9	0.0634553
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.33768e11i	1.38973i	0.719139	+	0.694867i	$0.244537\pi$
$557$	1.33768e11i	1.38973i	−0.719139	+	0.694867i	$0.755463\pi$
$558$	0	0
$559$	7.40036e10	0.757889
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 2.95704e10i	− 0.294323i	−0.989112	−	0.147162i	$-0.952986\pi$
$563$	− 2.95704e10i	− 0.294323i	0.989112	−	0.147162i	$-0.0470137\pi$
$564$	0	0
$565$	9.54655e10	0.936812
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 1.92259e11i	− 1.83416i	−0.398699	−	0.917082i	$-0.630538\pi$
$569$	− 1.92259e11i	− 1.83416i	0.398699	−	0.917082i	$-0.369462\pi$
$570$	0	0
$571$	−1.11124e10	−0.104535	−0.0522677	−	0.998633i	$-0.516645\pi$
$571$	−1.11124e10	−0.104535	−0.0522677	+	0.998633i	$0.516645\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3.97405e10i	0.363548i
$576$	0	0
$577$	−2.11777e11	−1.91063	−0.955314	−	0.295594i	$-0.904482\pi$
$577$	−2.11777e11	−1.91063	−0.955314	+	0.295594i	$0.904482\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	2.68304e10i	0.235463i
$582$	0	0
$583$	−1.36385e11	−1.18057
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.83784e10i	0.154794i	0.997000	+	0.0773970i	$0.0246609\pi$
$587$	1.83784e10i	0.154794i	−0.997000	+	0.0773970i	$0.975339\pi$
$588$	0	0
$589$	−1.70249e11	−1.41457
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 5.18489e10i	− 0.419296i	−0.977777	−	0.209648i	$-0.932768\pi$
$593$	− 5.18489e10i	− 0.419296i	0.977777	−	0.209648i	$-0.0672318\pi$
$594$	0	0
$595$	7.82681e10	0.624478
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 2.40425e10i	− 0.186755i	−0.995631	−	0.0933775i	$-0.970234\pi$
$599$	− 2.40425e10i	− 0.186755i	0.995631	−	0.0933775i	$-0.0297664\pi$
$600$	0	0
$601$	7.21267e10	0.552838	0.276419	−	0.961037i	$-0.410852\pi$
$601$	7.21267e10	0.552838	0.276419	+	0.961037i	$0.410852\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	4.40369e10i	0.328696i
$606$	0	0
$607$	6.59221e10	0.485597	0.242799	−	0.970077i	$-0.421935\pi$
$607$	6.59221e10	0.485597	0.242799	+	0.970077i	$0.421935\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 1.02628e11i	− 0.736378i
$612$	0	0
$613$	1.10453e11	0.782234	0.391117	−	0.920341i	$-0.372089\pi$
$613$	1.10453e11	0.782234	0.391117	+	0.920341i	$0.372089\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 6.08425e10i	− 0.419823i	−0.977720	−	0.209912i	$-0.932682\pi$
$617$	− 6.08425e10i	− 0.419823i	0.977720	−	0.209912i	$-0.0673177\pi$
$618$	0	0
$619$	−3.52013e10	−0.239771	−0.119885	−	0.992788i	$-0.538253\pi$
$619$	−3.52013e10	−0.239771	−0.119885	+	0.992788i	$0.538253\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 2.87331e10i	− 0.190735i
$624$	0	0
$625$	−5.52579e10	−0.362138
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 9.74651e10i	− 0.622653i
$630$	0	0
$631$	−9.36211e10	−0.590549	−0.295275	−	0.955412i	$-0.595411\pi$
$631$	−9.36211e10	−0.590549	−0.295275	+	0.955412i	$0.595411\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 1.02490e11i	− 0.630357i
$636$	0	0
$637$	6.90816e10	0.419570
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 1.33852e11i	− 0.792850i	−0.918067	−	0.396425i	$-0.870251\pi$
$641$	− 1.33852e11i	− 0.792850i	0.918067	−	0.396425i	$-0.129749\pi$
$642$	0	0
$643$	2.74043e10	0.160316	0.0801578	−	0.996782i	$-0.474458\pi$
$643$	2.74043e10	0.160316	0.0801578	+	0.996782i	$0.474458\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 3.10935e11i	− 1.77440i	−0.461381	−	0.887202i	$-0.652646\pi$
$647$	− 3.10935e11i	− 1.77440i	0.461381	−	0.887202i	$-0.347354\pi$
$648$	0	0
$649$	1.14124e11	0.643280
$650$	0	0
$651$	0	0
$652$	0	0
$653$	9.52563e10i	0.523891i	0.965083	+	0.261945i	$0.0843641\pi$
$653$	9.52563e10i	0.523891i	−0.965083	+	0.261945i	$0.915636\pi$
$654$	0	0
$655$	1.64112e11	0.891612
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 2.16953e11i	− 1.15034i	−0.818036	−	0.575168i	$-0.804937\pi$
$659$	− 2.16953e11i	− 1.15034i	0.818036	−	0.575168i	$-0.195063\pi$
$660$	0	0
$661$	−1.28088e10	−0.0670968	−0.0335484	−	0.999437i	$-0.510681\pi$
$661$	−1.28088e10	−0.0670968	−0.0335484	+	0.999437i	$0.510681\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 9.46123e10i	− 0.483794i
$666$	0	0
$667$	3.11909e10	0.157589
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 2.03110e11i	− 1.00194i
$672$	0	0
$673$	−1.15334e11	−0.562208	−0.281104	−	0.959677i	$-0.590701\pi$
$673$	−1.15334e11	−0.562208	−0.281104	+	0.959677i	$0.590701\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	3.31800e11i	1.57951i	0.613424	+	0.789754i	$0.289792\pi$
$677$	3.31800e11i	1.57951i	−0.613424	+	0.789754i	$0.710208\pi$
$678$	0	0
$679$	−1.09397e11	−0.514666
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 2.28809e11i	− 1.05145i	−0.850653	−	0.525727i	$-0.823793\pi$
$683$	− 2.28809e11i	− 1.05145i	0.850653	−	0.525727i	$-0.176207\pi$
$684$	0	0
$685$	2.56567e11	1.16530
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.62459e11i	1.16462i
$690$	0	0
$691$	−6.81289e10	−0.298827	−0.149413	−	0.988775i	$-0.547738\pi$
$691$	−6.81289e10	−0.298827	−0.149413	+	0.988775i	$0.547738\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 5.39541e10i	− 0.231252i
$696$	0	0
$697$	−1.36712e11	−0.579261
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 3.51559e11i	− 1.45588i	−0.685640	−	0.727941i	$-0.740477\pi$
$701$	− 3.51559e11i	− 1.45588i	0.685640	−	0.727941i	$-0.259523\pi$
$702$	0	0
$703$	−1.17818e11	−0.482381
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 1.66614e11i	− 0.666858i
$708$	0	0
$709$	−5.60206e10	−0.221698	−0.110849	−	0.993837i	$-0.535357\pi$
$709$	−5.60206e10	−0.221698	−0.110849	+	0.993837i	$0.535357\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 3.04752e11i	− 1.17920i
$714$	0	0
$715$	−1.07799e11	−0.412470
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 1.91675e11i	− 0.717216i	−0.933488	−	0.358608i	$-0.883252\pi$
$719$	− 1.91675e11i	− 0.717216i	0.933488	−	0.358608i	$-0.116748\pi$
$720$	0	0
$721$	−1.91845e11	−0.709921
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 2.34249e10i	− 0.0847864i
$726$	0	0
$727$	3.32166e11	1.18910	0.594549	−	0.804059i	$-0.297330\pi$
$727$	3.32166e11	1.18910	0.594549	+	0.804059i	$0.297330\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 3.73176e11i	− 1.30691i
$732$	0	0
$733$	3.56926e10	0.123641	0.0618205	−	0.998087i	$-0.480309\pi$
$733$	3.56926e10	0.123641	0.0618205	+	0.998087i	$0.480309\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.75328e11i	1.27216i
$738$	0	0
$739$	5.37492e11	1.80216	0.901081	−	0.433650i	$-0.142775\pi$
$739$	5.37492e11	1.80216	0.901081	+	0.433650i	$0.142775\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 4.07427e11i	− 1.33689i	−0.743764	−	0.668443i	$-0.766961\pi$
$743$	− 4.07427e11i	− 1.33689i	0.743764	−	0.668443i	$-0.233039\pi$
$744$	0	0
$745$	2.44979e11	0.795250
$746$	0	0
$747$	0	0
$748$	0	0
$749$	3.13415e11i	0.995846i
$750$	0	0
$751$	−2.17032e11	−0.682281	−0.341141	−	0.940012i	$-0.610813\pi$
$751$	−2.17032e11	−0.682281	−0.341141	+	0.940012i	$0.610813\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	7.47228e9i	0.0229967i
$756$	0	0
$757$	1.61808e11	0.492739	0.246370	−	0.969176i	$-0.420762\pi$
$757$	1.61808e11	0.492739	0.246370	+	0.969176i	$0.420762\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	5.27348e11i	1.57238i	0.617983	+	0.786192i	$0.287950\pi$
$761$	5.27348e11i	1.57238i	−0.617983	+	0.786192i	$0.712050\pi$
$762$	0	0
$763$	1.41032e11	0.416122
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 2.19620e11i	− 0.634586i
$768$	0	0
$769$	3.01064e11	0.860902	0.430451	−	0.902614i	$-0.358355\pi$
$769$	3.01064e11	0.860902	0.430451	+	0.902614i	$0.358355\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	7.50947e10i	0.210325i	0.994455	+	0.105163i	$0.0335363\pi$
$773$	7.50947e10i	0.210325i	−0.994455	+	0.105163i	$0.966464\pi$
$774$	0	0
$775$	−2.28874e11	−0.634438
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.65260e11i	0.448764i
$780$	0	0
$781$	−4.16244e11	−1.11878
$782$	0	0
$783$	0	0
$784$	0	0
$785$	3.62080e11i	0.953512i
$786$	0	0
$787$	2.27263e11	0.592419	0.296210	−	0.955123i	$-0.404277\pi$
$787$	2.27263e11	0.592419	0.296210	+	0.955123i	$0.404277\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	3.22607e11i	0.824078i
$792$	0	0
$793$	−3.90863e11	−0.988398
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 4.79744e11i	− 1.18899i	−0.804101	−	0.594493i	$-0.797353\pi$
$797$	− 4.79744e11i	− 1.18899i	0.804101	−	0.594493i	$-0.202647\pi$
$798$	0	0
$799$	−5.17519e11	−1.26981
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 4.86132e11i	− 1.16921i
$804$	0	0
$805$	1.69359e11	0.403296
$806$	0	0
$807$	0	0
$808$	0	0
$809$	7.81491e11i	1.82444i	0.409700	+	0.912220i	$0.365633\pi$
$809$	7.81491e11i	1.82444i	−0.409700	+	0.912220i	$0.634367\pi$
$810$	0	0
$811$	−4.47395e11	−1.03421	−0.517103	−	0.855923i	$-0.672990\pi$
$811$	−4.47395e11	−1.03421	−0.517103	+	0.855923i	$0.672990\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 3.93965e11i	− 0.892950i
$816$	0	0
$817$	−4.51104e11	−1.01248
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.85764e11i	0.408872i	0.978880	+	0.204436i	$0.0655361\pi$
$821$	1.85764e11i	0.408872i	−0.978880	+	0.204436i	$0.934464\pi$
$822$	0	0
$823$	−4.54477e10	−0.0990632	−0.0495316	−	0.998773i	$-0.515773\pi$
$823$	−4.54477e10	−0.0990632	−0.0495316	+	0.998773i	$0.515773\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 4.80346e11i	− 1.02691i	−0.858117	−	0.513454i	$-0.828366\pi$
$827$	− 4.80346e11i	− 1.02691i	0.858117	−	0.513454i	$-0.171634\pi$
$828$	0	0
$829$	−4.62670e11	−0.979610	−0.489805	−	0.871832i	$-0.662932\pi$
$829$	−4.62670e11	−0.979610	−0.489805	+	0.871832i	$0.662932\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 3.48356e11i	− 0.723508i
$834$	0	0
$835$	3.28692e11	0.676150
$836$	0	0
$837$	0	0
$838$	0	0
$839$	5.08581e11i	1.02639i	0.858272	+	0.513195i	$0.171538\pi$
$839$	5.08581e11i	1.02639i	−0.858272	+	0.513195i	$0.828462\pi$
$840$	0	0
$841$	4.81861e11	0.963247
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 1.73303e11i	− 0.339922i
$846$	0	0
$847$	−1.48814e11	−0.289142
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 2.10898e11i	− 0.402118i
$852$	0	0
$853$	−5.24179e11	−0.990110	−0.495055	−	0.868862i	$-0.664852\pi$
$853$	−5.24179e11	−0.990110	−0.495055	+	0.868862i	$0.664852\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	7.30185e10i	0.135366i	0.997707	+	0.0676830i	$0.0215607\pi$
$857$	7.30185e10i	0.135366i	−0.997707	+	0.0676830i	$0.978439\pi$
$858$	0	0
$859$	4.44067e11	0.815598	0.407799	−	0.913072i	$-0.366296\pi$
$859$	4.44067e11	0.815598	0.407799	+	0.913072i	$0.366296\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 4.99140e10i	− 0.0899868i	−0.998987	−	0.0449934i	$-0.985673\pi$
$863$	− 4.99140e10i	− 0.0899868i	0.998987	−	0.0449934i	$-0.0143267\pi$
$864$	0	0
$865$	−4.20217e11	−0.750601
$866$	0	0
$867$	0	0
$868$	0	0
$869$	4.12155e10i	0.0722740i
$870$	0	0
$871$	7.22278e11	1.25497
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 4.14783e11i	− 0.707602i
$876$	0	0
$877$	−2.27533e11	−0.384632	−0.192316	−	0.981333i	$-0.561600\pi$
$877$	−2.27533e11	−0.384632	−0.192316	+	0.981333i	$0.561600\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	8.85209e11i	1.46941i	0.678388	+	0.734704i	$0.262679\pi$
$881$	8.85209e11i	1.46941i	−0.678388	+	0.734704i	$0.737321\pi$
$882$	0	0
$883$	3.77081e11	0.620286	0.310143	−	0.950690i	$-0.399623\pi$
$883$	3.77081e11	0.620286	0.310143	+	0.950690i	$0.399623\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.08596e12i	1.75436i	0.480160	+	0.877181i	$0.340579\pi$
$887$	1.08596e12i	1.75436i	−0.480160	+	0.877181i	$0.659421\pi$
$888$	0	0
$889$	3.46345e11	0.554501
$890$	0	0
$891$	0	0
$892$	0	0
$893$	6.25589e11i	0.983746i
$894$	0	0
$895$	2.30133e11	0.358664
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.79635e11i	0.275012i
$900$	0	0
$901$	1.32349e12	2.00827
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.39143e11i	0.207427i
$906$	0	0
$907$	1.07960e11	0.159527	0.0797634	−	0.996814i	$-0.474584\pi$
$907$	1.07960e11	0.159527	0.0797634	+	0.996814i	$0.474584\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 5.51004e11i	− 0.799984i	−0.916519	−	0.399992i	$-0.869013\pi$
$911$	− 5.51004e11i	− 0.799984i	0.916519	−	0.399992i	$-0.130987\pi$
$912$	0	0
$913$	−1.86346e11	−0.268186
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5.54586e11i	0.784317i
$918$	0	0
$919$	−4.32195e11	−0.605923	−0.302961	−	0.953003i	$-0.597975\pi$
$919$	−4.32195e11	−0.605923	−0.302961	+	0.953003i	$0.597975\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	8.01016e11i	1.10366i
$924$	0	0
$925$	−1.58388e11	−0.216349
$926$	0	0
$927$	0	0
$928$	0	0
$929$	8.56563e11i	1.15000i	0.818155	+	0.574998i	$0.194997\pi$
$929$	8.56563e11i	1.15000i	−0.818155	+	0.574998i	$0.805003\pi$
$930$	0	0
$931$	−4.21101e11	−0.560515
$932$	0	0
$933$	0	0
$934$	0	0
$935$	5.43597e11i	0.711264i
$936$	0	0
$937$	6.13380e11	0.795740	0.397870	−	0.917442i	$-0.369750\pi$
$937$	6.13380e11	0.795740	0.397870	+	0.917442i	$0.369750\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 1.16251e11i	− 0.148265i	−0.997248	−	0.0741327i	$-0.976381\pi$
$941$	− 1.16251e11i	− 0.148265i	0.997248	−	0.0741327i	$-0.0236189\pi$
$942$	0	0
$943$	−2.95821e11	−0.374095
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.24509e11i	0.154811i	0.997000	+	0.0774054i	$0.0246636\pi$
$947$	1.24509e11i	0.154811i	−0.997000	+	0.0774054i	$0.975336\pi$
$948$	0	0
$949$	−9.35508e11	−1.15341
$950$	0	0
$951$	0	0
$952$	0	0
$953$	7.28266e11i	0.882914i	0.897283	+	0.441457i	$0.145538\pi$
$953$	7.28266e11i	0.882914i	−0.897283	+	0.441457i	$0.854462\pi$
$954$	0	0
$955$	7.33100e11	0.881352
$956$	0	0
$957$	0	0
$958$	0	0
$959$	8.67020e11i	1.02507i
$960$	0	0
$961$	9.02238e11	1.05786
$962$	0	0
$963$	0	0
$964$	0	0
$965$	8.88109e11i	1.02413i
$966$	0	0
$967$	−7.77539e11	−0.889235	−0.444617	−	0.895721i	$-0.646660\pi$
$967$	−7.77539e11	−0.889235	−0.444617	+	0.895721i	$0.646660\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 1.23280e12i	− 1.38681i	−0.720549	−	0.693404i	$-0.756110\pi$
$971$	− 1.23280e12i	− 1.38681i	0.720549	−	0.693404i	$-0.243890\pi$
$972$	0	0
$973$	1.82327e11	0.203423
$974$	0	0
$975$	0	0
$976$	0	0
$977$	7.10332e11i	0.779620i	0.920895	+	0.389810i	$0.127459\pi$
$977$	7.10332e11i	0.779620i	−0.920895	+	0.389810i	$0.872541\pi$
$978$	0	0
$979$	1.99560e11	0.217242
$980$	0	0
$981$	0	0
$982$	0	0
$983$	5.03756e11i	0.539519i	0.962928	+	0.269759i	$0.0869442\pi$
$983$	5.03756e11i	0.539519i	−0.962928	+	0.269759i	$0.913056\pi$
$984$	0	0
$985$	1.26164e12	1.34027
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 8.07489e11i	− 0.844018i
$990$	0	0
$991$	−7.14718e11	−0.741038	−0.370519	−	0.928825i	$-0.620820\pi$
$991$	−7.14718e11	−0.741038	−0.370519	+	0.928825i	$0.620820\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	8.10721e11i	0.827140i
$996$	0	0
$997$	−8.23742e11	−0.833702	−0.416851	−	0.908975i	$-0.636866\pi$
$997$	−8.23742e11	−0.833702	−0.416851	+	0.908975i	$0.636866\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.9.c.a.161.12		16
3.2	odd	2	inner	324.9.c.a.161.5		16
9.2	odd	6		36.9.g.a.5.3	✓	16
9.4	even	3		36.9.g.a.29.3	yes	16
9.5	odd	6		108.9.g.a.89.3		16
9.7	even	3		108.9.g.a.17.3		16
36.7	odd	6		432.9.q.b.17.3		16
36.11	even	6		144.9.q.c.113.6		16
36.23	even	6		432.9.q.b.305.3		16
36.31	odd	6		144.9.q.c.65.6		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.9.g.a.5.3	✓	16	9.2	odd	6
36.9.g.a.29.3	yes	16	9.4	even	3
108.9.g.a.17.3		16	9.7	even	3
108.9.g.a.89.3		16	9.5	odd	6
144.9.q.c.65.6		16	36.31	odd	6
144.9.q.c.113.6		16	36.11	even	6
324.9.c.a.161.5		16	3.2	odd	2	inner
324.9.c.a.161.12		16	1.1	even	1	trivial
432.9.q.b.17.3		16	36.7	odd	6
432.9.q.b.305.3		16	36.23	even	6

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	324.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+466.761i q^{5} -1577.33 q^{7} -10955.1i q^{11} -21081.8 q^{13} +106309. i q^{17} +128508. q^{19} +230034. i q^{23} +172759. q^{25} -135593. i q^{29} -1.32481e6 q^{31} -736235. i q^{35} -916812. q^{37} +1.28599e6i q^{41} -3.51031e6 q^{43} +4.86808e6i q^{47} -3.27684e6 q^{49} -1.24495e7i q^{53} +5.11339e6 q^{55} +1.04175e7i q^{59} +1.85403e7 q^{61} -9.84016e6i q^{65} -3.42607e7 q^{67} -3.79956e7i q^{71} +4.43751e7 q^{73} +1.72797e7i q^{77} -3.76224e6 q^{79} -1.70100e7i q^{83} -4.96207e7 q^{85} +1.82163e7i q^{89} +3.32529e7 q^{91} +5.99826e7i q^{95} +6.93557e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 1846 q^{7} - 3370 q^{13} + 84518 q^{19} - 911606 q^{25} + 756614 q^{31} - 1671664 q^{37} - 679024 q^{43} + 6052890 q^{49} - 17497638 q^{55} - 34901410 q^{61} + 70053452 q^{67} + 28071218 q^{73} - 61378918 q^{79}+ \cdots + 66842156 q^{97}+O(q^{100}) \) 16 * q - 1846 * q^7 - 3370 * q^13 + 84518 * q^19 - 911606 * q^25 + 756614 * q^31 - 1671664 * q^37 - 679024 * q^43 + 6052890 * q^49 - 17497638 * q^55 - 34901410 * q^61 + 70053452 * q^67 + 28071218 * q^73 - 61378918 * q^79 + 79786044 * q^85 - 55534214 * q^91 + 66842156 * q^97

Introduction

L-functions

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Varieties

Fields

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Database

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