LMFDB - Embedded newform 324.7.c.b.161.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,7,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, 7, names="a")

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

chi := DirichletCharacter("324.161");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 7 $
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:	$74.5375230928$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2 x^{11} + 59 x^{10} + 602 x^{9} + 26655 x^{8} + 692184 x^{7} - 4209870 x^{6} + \cdots + 166668145981081 $ x^12 - 2x^11 + 59x^10 + 602x^9 + 26655x^8 + 692184x^7 - 4209870x^6 - 34154736x^5 + 2261179815x^4 - 21421205158x^3 + 1095551670899x^2 - 17316244703642*x + 166668145981081
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{18}\cdot 3^{48} $
Twist minimal:	no (minimal twist has level 36)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.8
Root			$-16.7265 + 5.11172i$ of defining polynomial
Character	$\chi$	$=$	324.161
Dual form			324.7.c.b.161.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+59.2773i q^{5} -153.748 q^{7} +O(q^{10})$	$q+59.2773i q^{5} -153.748 q^{7} +678.075i q^{11} +1652.14 q^{13} -6420.23i q^{17} +2857.79 q^{19} +7015.90i q^{23} +12111.2 q^{25} -9269.79i q^{29} -52237.9 q^{31} -9113.74i q^{35} +58462.5 q^{37} +118669. i q^{41} +89654.8 q^{43} +67861.6i q^{47} -94010.7 q^{49} -221631. i q^{53} -40194.5 q^{55} +168337. i q^{59} -212380. q^{61} +97934.4i q^{65} -108742. q^{67} +408102. i q^{71} +39555.6 q^{73} -104252. i q^{77} +418332. q^{79} +25444.6i q^{83} +380574. q^{85} +1.26291e6i q^{89} -254013. q^{91} +169402. i q^{95} -870065. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 240 q^{7}+O(q^{10}) $ 12 * q + 240 * q^7	$ 12 q + 240 q^{7} + 1680 q^{13} - 12900 q^{19} - 48156 q^{25} - 6240 q^{31} + 12768 q^{37} - 142860 q^{43} + 477828 q^{49} + 5400 q^{55} + 187152 q^{61} - 208668 q^{67} + 221820 q^{73} - 246936 q^{79} - 755136 q^{85} + 91488 q^{91} + 2066964 q^{97}+O(q^{100}) $ 12 * q + 240 * q^7 + 1680 * q^13 - 12900 * q^19 - 48156 * q^25 - 6240 * q^31 + 12768 * q^37 - 142860 * q^43 + 477828 * q^49 + 5400 * q^55 + 187152 * q^61 - 208668 * q^67 + 221820 * q^73 - 246936 * q^79 - 755136 * q^85 + 91488 * q^91 + 2066964 * q^97

Display 100 coefficients 10 coefficients

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
$5$	59.2773i	0.474218i	0.971483	+	0.237109i	$0.0761999\pi$
$5$	59.2773i	0.474218i	−0.971483	+	0.237109i	$0.923800\pi$
$6$	0	0
$7$	−153.748	−0.448244	−0.224122	−	0.974561i	$-0.571951\pi$
$7$	−153.748	−0.448244	−0.224122	+	0.974561i	$0.571951\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	678.075i	0.509448i	0.967014	+	0.254724i	$0.0819846\pi$
$11$	678.075i	0.509448i	−0.967014	+	0.254724i	$0.918015\pi$
$12$	0	0
$13$	1652.14	0.751998	0.375999	−	0.926620i	$-0.377300\pi$
$13$	1652.14	0.751998	0.375999	+	0.926620i	$0.377300\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 6420.23i	− 1.30678i	−0.757020	−	0.653392i	$-0.773345\pi$
$17$	− 6420.23i	− 1.30678i	0.757020	−	0.653392i	$-0.226655\pi$
$18$	0	0
$19$	2857.79	0.416648	0.208324	−	0.978060i	$-0.433199\pi$
$19$	2857.79	0.416648	0.208324	+	0.978060i	$0.433199\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7015.90i	0.576633i	0.957535	+	0.288317i	$0.0930956\pi$
$23$	7015.90i	0.576633i	−0.957535	+	0.288317i	$0.906904\pi$
$24$	0	0
$25$	12111.2	0.775117
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 9269.79i	− 0.380081i	−0.981776	−	0.190040i	$-0.939138\pi$
$29$	− 9269.79i	− 0.380081i	0.981776	−	0.190040i	$-0.0608619\pi$
$30$	0	0
$31$	−52237.9	−1.75348	−0.876740	−	0.480964i	$-0.840287\pi$
$31$	−52237.9	−1.75348	−0.876740	+	0.480964i	$0.840287\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 9113.74i	− 0.212565i
$36$	0	0
$37$	58462.5	1.15418	0.577088	−	0.816682i	$-0.304189\pi$
$37$	58462.5	1.15418	0.577088	+	0.816682i	$0.304189\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	118669.i	1.72181i	0.508767	+	0.860904i	$0.330101\pi$
$41$	118669.i	1.72181i	−0.508767	+	0.860904i	$0.669899\pi$
$42$	0	0
$43$	89654.8	1.12763	0.563817	−	0.825900i	$-0.309332\pi$
$43$	89654.8	1.12763	0.563817	+	0.825900i	$0.309332\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	67861.6i	0.653628i	0.945089	+	0.326814i	$0.105975\pi$
$47$	67861.6i	0.653628i	−0.945089	+	0.326814i	$0.894025\pi$
$48$	0	0
$49$	−94010.7	−0.799078
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 221631.i	− 1.48869i	−0.667797	−	0.744343i	$-0.732763\pi$
$53$	− 221631.i	− 1.48869i	0.667797	−	0.744343i	$-0.267237\pi$
$54$	0	0
$55$	−40194.5	−0.241590
$56$	0	0
$57$	0	0
$58$	0	0
$59$	168337.i	0.819643i	0.912166	+	0.409821i	$0.134409\pi$
$59$	168337.i	0.819643i	−0.912166	+	0.409821i	$0.865591\pi$
$60$	0	0
$61$	−212380.	−0.935672	−0.467836	−	0.883815i	$-0.654966\pi$
$61$	−212380.	−0.935672	−0.467836	+	0.883815i	$0.654966\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	97934.4i	0.356611i
$66$	0	0
$67$	−108742.	−0.361554	−0.180777	−	0.983524i	$-0.557861\pi$
$67$	−108742.	−0.361554	−0.180777	+	0.983524i	$0.557861\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	408102.i	1.14023i	0.821564	+	0.570117i	$0.193102\pi$
$71$	408102.i	1.14023i	−0.821564	+	0.570117i	$0.806898\pi$
$72$	0	0
$73$	39555.6	0.101681	0.0508404	−	0.998707i	$-0.483810\pi$
$73$	39555.6	0.101681	0.0508404	+	0.998707i	$0.483810\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 104252.i	− 0.228357i
$78$	0	0
$79$	418332.	0.848477	0.424238	−	0.905551i	$-0.360542\pi$
$79$	418332.	0.848477	0.424238	+	0.905551i	$0.360542\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	25444.6i	0.0445002i	0.999752	+	0.0222501i	$0.00708301\pi$
$83$	25444.6i	0.0445002i	−0.999752	+	0.0222501i	$0.992917\pi$
$84$	0	0
$85$	380574.	0.619701
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.26291e6i	1.79145i	0.444611	+	0.895724i	$0.353342\pi$
$89$	1.26291e6i	1.79145i	−0.444611	+	0.895724i	$0.646658\pi$
$90$	0	0
$91$	−254013.	−0.337079
$92$	0	0
$93$	0	0
$94$	0	0
$95$	169402.i	0.197582i
$96$	0	0
$97$	−870065.	−0.953315	−0.476658	−	0.879089i	$-0.658152\pi$
$97$	−870065.	−0.953315	−0.476658	+	0.879089i	$0.658152\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.42394e6i	1.38206i	0.722826	+	0.691030i	$0.242843\pi$
$101$	1.42394e6i	1.38206i	−0.722826	+	0.691030i	$0.757157\pi$
$102$	0	0
$103$	1.62513e6	1.48722	0.743612	−	0.668611i	$-0.233111\pi$
$103$	1.62513e6	1.48722	0.743612	+	0.668611i	$0.233111\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.10780e6i	0.904292i	0.891944	+	0.452146i	$0.149341\pi$
$107$	1.10780e6i	0.904292i	−0.891944	+	0.452146i	$0.850659\pi$
$108$	0	0
$109$	−513511.	−0.396525	−0.198262	−	0.980149i	$-0.563530\pi$
$109$	−513511.	−0.396525	−0.198262	+	0.980149i	$0.563530\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 490277.i	− 0.339786i	−0.985462	−	0.169893i	$-0.945658\pi$
$113$	− 490277.i	− 0.339786i	0.985462	−	0.169893i	$-0.0543423\pi$
$114$	0	0
$115$	−415883.	−0.273450
$116$	0	0
$117$	0	0
$118$	0	0
$119$	987095.i	0.585758i
$120$	0	0
$121$	1.31177e6	0.740463
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.64413e6i	0.841793i
$126$	0	0
$127$	−114175.	−0.0557389	−0.0278694	−	0.999612i	$-0.508872\pi$
$127$	−114175.	−0.0557389	−0.0278694	+	0.999612i	$0.508872\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	4.43929e6i	1.97470i	0.158571	+	0.987348i	$0.449311\pi$
$131$	4.43929e6i	1.97470i	−0.158571	+	0.987348i	$0.550689\pi$
$132$	0	0
$133$	−439378.	−0.186760
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.84469e6i	1.10630i	0.833081	+	0.553151i	$0.186575\pi$
$137$	2.84469e6i	1.10630i	−0.833081	+	0.553151i	$0.813425\pi$
$138$	0	0
$139$	−4.05879e6	−1.51131	−0.755654	−	0.654972i	$-0.772681\pi$
$139$	−4.05879e6	−1.51131	−0.755654	+	0.654972i	$0.772681\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.12028e6i	0.383104i
$144$	0	0
$145$	549488.	0.180241
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 584375.i	− 0.176658i	−0.996091	−	0.0883289i	$-0.971847\pi$
$149$	− 584375.i	− 0.176658i	0.996091	−	0.0883289i	$-0.0281526\pi$
$150$	0	0
$151$	4.26200e6	1.23789	0.618946	−	0.785434i	$-0.287560\pi$
$151$	4.26200e6	1.23789	0.618946	+	0.785434i	$0.287560\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 3.09652e6i	− 0.831533i
$156$	0	0
$157$	−4.28178e6	−1.10643	−0.553217	−	0.833037i	$-0.686600\pi$
$157$	−4.28178e6	−1.10643	−0.553217	+	0.833037i	$0.686600\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 1.07868e6i	− 0.258472i
$162$	0	0
$163$	1.85229e6	0.427707	0.213853	−	0.976866i	$-0.431399\pi$
$163$	1.85229e6	0.427707	0.213853	+	0.976866i	$0.431399\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.98350e6i	1.92884i	0.264377	+	0.964420i	$0.414834\pi$
$167$	8.98350e6i	1.92884i	−0.264377	+	0.964420i	$0.585166\pi$
$168$	0	0
$169$	−2.09724e6	−0.434499
$170$	0	0
$171$	0	0
$172$	0	0
$173$	5.49299e6i	1.06089i	0.847719	+	0.530445i	$0.177975\pi$
$173$	5.49299e6i	1.06089i	−0.847719	+	0.530445i	$0.822025\pi$
$174$	0	0
$175$	−1.86207e6	−0.347441
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 2.48807e6i	− 0.433814i	−0.976192	−	0.216907i	$-0.930403\pi$
$179$	− 2.48807e6i	− 0.433814i	0.976192	−	0.216907i	$-0.0695969\pi$
$180$	0	0
$181$	5.06731e6	0.854558	0.427279	−	0.904120i	$-0.359472\pi$
$181$	5.06731e6	0.854558	0.427279	+	0.904120i	$0.359472\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3.46550e6i	0.547331i
$186$	0	0
$187$	4.35340e6	0.665738
$188$	0	0
$189$	0	0
$190$	0	0
$191$	5.42580e6i	0.778688i	0.921092	+	0.389344i	$0.127298\pi$
$191$	5.42580e6i	0.778688i	−0.921092	+	0.389344i	$0.872702\pi$
$192$	0	0
$193$	5.63550e6	0.783900	0.391950	−	0.919987i	$-0.371801\pi$
$193$	5.63550e6	0.783900	0.391950	+	0.919987i	$0.371801\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 1.52261e6i	− 0.199154i	−0.995030	−	0.0995771i	$-0.968251\pi$
$197$	− 1.52261e6i	− 0.199154i	0.995030	−	0.0995771i	$-0.0317490\pi$
$198$	0	0
$199$	−9.65115e6	−1.22467	−0.612336	−	0.790597i	$-0.709770\pi$
$199$	−9.65115e6	−1.22467	−0.612336	+	0.790597i	$0.709770\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.42521e6i	0.170369i
$204$	0	0
$205$	−7.03436e6	−0.816513
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.93779e6i	0.212260i
$210$	0	0
$211$	−3.47882e6	−0.370326	−0.185163	−	0.982708i	$-0.559281\pi$
$211$	−3.47882e6	−0.370326	−0.185163	+	0.982708i	$0.559281\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	5.31450e6i	0.534745i
$216$	0	0
$217$	8.03146e6	0.785987
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 1.06071e7i	− 0.982699i
$222$	0	0
$223$	8.00414e6	0.721772	0.360886	−	0.932610i	$-0.382474\pi$
$223$	8.00414e6	0.721772	0.360886	+	0.932610i	$0.382474\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 1.96422e7i	− 1.67924i	−0.543177	−	0.839618i	$-0.682779\pi$
$227$	− 1.96422e7i	− 1.67924i	0.543177	−	0.839618i	$-0.317221\pi$
$228$	0	0
$229$	1.05478e7	0.878324	0.439162	−	0.898408i	$-0.355275\pi$
$229$	1.05478e7	0.878324	0.439162	+	0.898408i	$0.355275\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	5.28004e6i	0.417416i	0.977978	+	0.208708i	$0.0669259\pi$
$233$	5.28004e6i	0.417416i	−0.977978	+	0.208708i	$0.933074\pi$
$234$	0	0
$235$	−4.02265e6	−0.309962
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 2.01877e7i	− 1.47874i	−0.673298	−	0.739371i	$-0.735123\pi$
$239$	− 2.01877e7i	− 1.47874i	0.673298	−	0.739371i	$-0.264877\pi$
$240$	0	0
$241$	−6.40797e6	−0.457793	−0.228897	−	0.973451i	$-0.573512\pi$
$241$	−6.40797e6	−0.457793	−0.228897	+	0.973451i	$0.573512\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 5.57270e6i	− 0.378937i
$246$	0	0
$247$	4.72147e6	0.313318
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.55033e7i	0.980400i	0.871610	+	0.490200i	$0.163076\pi$
$251$	1.55033e7i	0.980400i	−0.871610	+	0.490200i	$0.836924\pi$
$252$	0	0
$253$	−4.75731e6	−0.293765
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 2.17571e7i	− 1.28175i	−0.767647	−	0.640873i	$-0.778573\pi$
$257$	− 2.17571e7i	− 1.28175i	0.767647	−	0.640873i	$-0.221427\pi$
$258$	0	0
$259$	−8.98846e6	−0.517352
$260$	0	0
$261$	0	0
$262$	0	0
$263$	7.25779e6i	0.398967i	0.979901	+	0.199484i	$0.0639265\pi$
$263$	7.25779e6i	0.398967i	−0.979901	+	0.199484i	$0.936074\pi$
$264$	0	0
$265$	1.31377e7	0.705962
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 2.87490e7i	− 1.47695i	−0.674280	−	0.738476i	$-0.735546\pi$
$269$	− 2.87490e7i	− 1.47695i	0.674280	−	0.738476i	$-0.264454\pi$
$270$	0	0
$271$	−1.61888e7	−0.813405	−0.406703	−	0.913561i	$-0.633321\pi$
$271$	−1.61888e7	−0.813405	−0.406703	+	0.913561i	$0.633321\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	8.21231e6i	0.394882i
$276$	0	0
$277$	−1.53506e7	−0.722248	−0.361124	−	0.932518i	$-0.617607\pi$
$277$	−1.53506e7	−0.722248	−0.361124	+	0.932518i	$0.617607\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 1.88829e7i	− 0.851041i	−0.904949	−	0.425520i	$-0.860091\pi$
$281$	− 1.88829e7i	− 0.851041i	0.904949	−	0.425520i	$-0.139909\pi$
$282$	0	0
$283$	211924.	0.00935022	0.00467511	−	0.999989i	$-0.498512\pi$
$283$	211924.	0.00935022	0.00467511	+	0.999989i	$0.498512\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 1.82450e7i	− 0.771790i
$288$	0	0
$289$	−1.70818e7	−0.707683
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 1.49466e7i	− 0.594211i	−0.954845	−	0.297106i	$-0.903979\pi$
$293$	− 1.49466e7i	− 0.594211i	0.954845	−	0.297106i	$-0.0960214\pi$
$294$	0	0
$295$	−9.97858e6	−0.388689
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.15912e7i	0.433627i
$300$	0	0
$301$	−1.37842e7	−0.505455
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 1.25893e7i	− 0.443713i
$306$	0	0
$307$	4.02227e7	1.39013	0.695066	−	0.718946i	$-0.255375\pi$
$307$	4.02227e7	1.39013	0.695066	+	0.718946i	$0.255375\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	556257.i	0.0184924i	0.999957	+	0.00924622i	$0.00294320\pi$
$311$	556257.i	0.0184924i	−0.999957	+	0.00924622i	$0.997057\pi$
$312$	0	0
$313$	5.01770e7	1.63633	0.818166	−	0.574981i	$-0.194991\pi$
$313$	5.01770e7	1.63633	0.818166	+	0.574981i	$0.194991\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 4.67637e7i	− 1.46802i	−0.679141	−	0.734008i	$-0.737647\pi$
$317$	− 4.67637e7i	− 1.46802i	0.679141	−	0.734008i	$-0.262353\pi$
$318$	0	0
$319$	6.28561e6	0.193631
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 1.83476e7i	− 0.544469i
$324$	0	0
$325$	2.00094e7	0.582887
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 1.04336e7i	− 0.292985i
$330$	0	0
$331$	3.44759e6	0.0950673	0.0475337	−	0.998870i	$-0.484864\pi$
$331$	3.44759e6	0.0950673	0.0475337	+	0.998870i	$0.484864\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 6.44594e6i	− 0.171456i
$336$	0	0
$337$	−7.07135e7	−1.84762	−0.923810	−	0.382850i	$-0.874942\pi$
$337$	−7.07135e7	−1.84762	−0.923810	+	0.382850i	$0.874942\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 3.54213e7i	− 0.893307i
$342$	0	0
$343$	3.25422e7	0.806425
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3.69276e7i	0.883817i	0.897060	+	0.441908i	$0.145698\pi$
$347$	3.69276e7i	0.883817i	−0.897060	+	0.441908i	$0.854302\pi$
$348$	0	0
$349$	4.04904e7	0.952523	0.476261	−	0.879304i	$-0.341992\pi$
$349$	4.04904e7	0.952523	0.476261	+	0.879304i	$0.341992\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 4.32188e7i	− 0.982536i	−0.871008	−	0.491268i	$-0.836534\pi$
$353$	− 4.32188e7i	− 0.982536i	0.871008	−	0.491268i	$-0.163466\pi$
$354$	0	0
$355$	−2.41912e7	−0.540720
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 3.23781e7i	− 0.699791i	−0.936789	−	0.349896i	$-0.886217\pi$
$359$	− 3.23781e7i	− 0.699791i	0.936789	−	0.349896i	$-0.113783\pi$
$360$	0	0
$361$	−3.88789e7	−0.826405
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2.34475e6i	0.0482189i
$366$	0	0
$367$	−1.96028e7	−0.396570	−0.198285	−	0.980144i	$-0.563537\pi$
$367$	−1.96028e7	−0.396570	−0.198285	+	0.980144i	$0.563537\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3.40753e7i	0.667294i
$372$	0	0
$373$	1.81112e6	0.0348996	0.0174498	−	0.999848i	$-0.494445\pi$
$373$	1.81112e6	0.0348996	0.0174498	+	0.999848i	$0.494445\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 1.53150e7i	− 0.285820i
$378$	0	0
$379$	−1.03932e7	−0.190912	−0.0954559	−	0.995434i	$-0.530431\pi$
$379$	−1.03932e7	−0.190912	−0.0954559	+	0.995434i	$0.530431\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 2.05292e7i	− 0.365406i	−0.983168	−	0.182703i	$-0.941515\pi$
$383$	− 2.05292e7i	− 0.365406i	0.983168	−	0.182703i	$-0.0584846\pi$
$384$	0	0
$385$	6.17980e6	0.108291
$386$	0	0
$387$	0	0
$388$	0	0
$389$	2.34222e7i	0.397904i	0.980009	+	0.198952i	$0.0637539\pi$
$389$	2.34222e7i	0.397904i	−0.980009	+	0.198952i	$0.936246\pi$
$390$	0	0
$391$	4.50437e7	0.753535
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2.47976e7i	0.402363i
$396$	0	0
$397$	2.83340e7	0.452831	0.226415	−	0.974031i	$-0.427299\pi$
$397$	2.83340e7	0.452831	0.226415	+	0.974031i	$0.427299\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.71851e7i	0.421597i	0.977530	+	0.210798i	$0.0676063\pi$
$401$	2.71851e7i	0.421597i	−0.977530	+	0.210798i	$0.932394\pi$
$402$	0	0
$403$	−8.63044e7	−1.31861
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.96419e7i	0.587992i
$408$	0	0
$409$	6.99717e7	1.02271	0.511355	−	0.859369i	$-0.329144\pi$
$409$	6.99717e7	1.02271	0.511355	+	0.859369i	$0.329144\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 2.58815e7i	− 0.367400i
$414$	0	0
$415$	−1.50829e6	−0.0211028
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 6.62723e7i	− 0.900928i	−0.892795	−	0.450464i	$-0.851259\pi$
$419$	− 6.62723e7i	− 0.900928i	0.892795	−	0.450464i	$-0.148741\pi$
$420$	0	0
$421$	−2.54183e6	−0.0340644	−0.0170322	−	0.999855i	$-0.505422\pi$
$421$	−2.54183e6	−0.0340644	−0.0170322	+	0.999855i	$0.505422\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 7.77567e7i	− 1.01291i
$426$	0	0
$427$	3.26529e7	0.419409
$428$	0	0
$429$	0	0
$430$	0	0
$431$	1.56169e8i	1.95057i	0.220946	+	0.975286i	$0.429086\pi$
$431$	1.56169e8i	1.95057i	−0.220946	+	0.975286i	$0.570914\pi$
$432$	0	0
$433$	−8.12049e7	−1.00027	−0.500136	−	0.865947i	$-0.666717\pi$
$433$	−8.12049e7	−1.00027	−0.500136	+	0.865947i	$0.666717\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.00499e7i	0.240253i
$438$	0	0
$439$	2.02858e7	0.239772	0.119886	−	0.992788i	$-0.461747\pi$
$439$	2.02858e7	0.239772	0.119886	+	0.992788i	$0.461747\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.39555e8i	1.60522i	0.596503	+	0.802611i	$0.296556\pi$
$443$	1.39555e8i	1.60522i	−0.596503	+	0.802611i	$0.703444\pi$
$444$	0	0
$445$	−7.48622e7	−0.849537
$446$	0	0
$447$	0	0
$448$	0	0
$449$	1.19418e8i	1.31926i	0.751588	+	0.659632i	$0.229288\pi$
$449$	1.19418e8i	1.31926i	−0.751588	+	0.659632i	$0.770712\pi$
$450$	0	0
$451$	−8.04663e7	−0.877172
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 1.50572e7i	− 0.159849i
$456$	0	0
$457$	−3.24323e7	−0.339805	−0.169902	−	0.985461i	$-0.554345\pi$
$457$	−3.24323e7	−0.339805	−0.169902	+	0.985461i	$0.554345\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 3.63260e7i	− 0.370779i	−0.982665	−	0.185389i	$-0.940645\pi$
$461$	− 3.63260e7i	− 0.370779i	0.982665	−	0.185389i	$-0.0593546\pi$
$462$	0	0
$463$	4.44544e7	0.447890	0.223945	−	0.974602i	$-0.428106\pi$
$463$	4.44544e7	0.447890	0.223945	+	0.974602i	$0.428106\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.52522e8i	1.49755i	0.662822	+	0.748777i	$0.269359\pi$
$467$	1.52522e8i	1.49755i	−0.662822	+	0.748777i	$0.730641\pi$
$468$	0	0
$469$	1.67189e7	0.162065
$470$	0	0
$471$	0	0
$472$	0	0
$473$	6.07927e7i	0.574471i
$474$	0	0
$475$	3.46112e7	0.322951
$476$	0	0
$477$	0	0
$478$	0	0
$479$	1.10522e8i	1.00564i	0.864391	+	0.502821i	$0.167705\pi$
$479$	1.10522e8i	1.00564i	−0.864391	+	0.502821i	$0.832295\pi$
$480$	0	0
$481$	9.65882e7	0.867938
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 5.15751e7i	− 0.452079i
$486$	0	0
$487$	1.83933e8	1.59247	0.796237	−	0.604985i	$-0.206821\pi$
$487$	1.83933e8	1.59247	0.796237	+	0.604985i	$0.206821\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	7.99636e6i	0.0675535i	0.999429	+	0.0337768i	$0.0107535\pi$
$491$	7.99636e6i	0.0675535i	−0.999429	+	0.0337768i	$0.989246\pi$
$492$	0	0
$493$	−5.95142e7	−0.496683
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 6.27447e7i	− 0.511103i
$498$	0	0
$499$	4.12336e7	0.331856	0.165928	−	0.986138i	$-0.446938\pi$
$499$	4.12336e7	0.331856	0.165928	+	0.986138i	$0.446938\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	4.69138e7i	0.368635i	0.982867	+	0.184317i	$0.0590075\pi$
$503$	4.69138e7i	0.368635i	−0.982867	+	0.184317i	$0.940993\pi$
$504$	0	0
$505$	−8.44072e7	−0.655398
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 5.52136e7i	− 0.418690i	−0.977842	−	0.209345i	$-0.932867\pi$
$509$	− 5.52136e7i	− 0.418690i	0.977842	−	0.209345i	$-0.0671332\pi$
$510$	0	0
$511$	−6.08158e6	−0.0455778
$512$	0	0
$513$	0	0
$514$	0	0
$515$	9.63333e7i	0.705269i
$516$	0	0
$517$	−4.60153e7	−0.332989
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 5.59371e7i	− 0.395537i	−0.980249	−	0.197768i	$-0.936631\pi$
$521$	− 5.59371e7i	− 0.395537i	0.980249	−	0.197768i	$-0.0633694\pi$
$522$	0	0
$523$	−1.43384e8	−1.00230	−0.501149	−	0.865361i	$-0.667089\pi$
$523$	−1.43384e8	−1.00230	−0.501149	+	0.865361i	$0.667089\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.35380e8i	2.29142i
$528$	0	0
$529$	9.88131e7	0.667494
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.96057e8i	1.29480i
$534$	0	0
$535$	−6.56672e7	−0.428832
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 6.37463e7i	− 0.407088i
$540$	0	0
$541$	1.75550e8	1.10868	0.554342	−	0.832289i	$-0.312970\pi$
$541$	1.75550e8	1.10868	0.554342	+	0.832289i	$0.312970\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 3.04395e7i	− 0.188039i
$546$	0	0
$547$	−6.74524e6	−0.0412131	−0.0206066	−	0.999788i	$-0.506560\pi$
$547$	−6.74524e6	−0.0412131	−0.0206066	+	0.999788i	$0.506560\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 2.64911e7i	− 0.158360i
$552$	0	0
$553$	−6.43176e7	−0.380324
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 2.14625e8i	− 1.24198i	−0.783819	−	0.620990i	$-0.786731\pi$
$557$	− 2.14625e8i	− 1.24198i	0.783819	−	0.620990i	$-0.213269\pi$
$558$	0	0
$559$	1.48122e8	0.847979
$560$	0	0
$561$	0	0
$562$	0	0
$563$	2.53692e8i	1.42161i	0.703387	+	0.710807i	$0.251670\pi$
$563$	2.53692e8i	1.42161i	−0.703387	+	0.710807i	$0.748330\pi$
$564$	0	0
$565$	2.90623e7	0.161133
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 1.09905e8i	− 0.596598i	−0.954472	−	0.298299i	$-0.903581\pi$
$569$	− 1.09905e8i	− 0.596598i	0.954472	−	0.298299i	$-0.0964193\pi$
$570$	0	0
$571$	−1.27816e8	−0.686558	−0.343279	−	0.939234i	$-0.611538\pi$
$571$	−1.27816e8	−0.686558	−0.343279	+	0.939234i	$0.611538\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	8.49710e7i	0.446958i
$576$	0	0
$577$	−5.97876e7	−0.311232	−0.155616	−	0.987818i	$-0.549736\pi$
$577$	−5.97876e7	−0.311232	−0.155616	+	0.987818i	$0.549736\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 3.91205e6i	− 0.0199469i
$582$	0	0
$583$	1.50283e8	0.758408
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.34025e8i	0.662633i	0.943520	+	0.331316i	$0.107493\pi$
$587$	1.34025e8i	0.662633i	−0.943520	+	0.331316i	$0.892507\pi$
$588$	0	0
$589$	−1.49285e8	−0.730584
$590$	0	0
$591$	0	0
$592$	0	0
$593$	2.75560e8i	1.32146i	0.750626	+	0.660728i	$0.229752\pi$
$593$	2.75560e8i	1.32146i	−0.750626	+	0.660728i	$0.770248\pi$
$594$	0	0
$595$	−5.85123e7	−0.277777
$596$	0	0
$597$	0	0
$598$	0	0
$599$	2.94870e8i	1.37199i	0.727608	+	0.685993i	$0.240632\pi$
$599$	2.94870e8i	1.37199i	−0.727608	+	0.685993i	$0.759368\pi$
$600$	0	0
$601$	1.35591e8	0.624610	0.312305	−	0.949982i	$-0.398899\pi$
$601$	1.35591e8	0.624610	0.312305	+	0.949982i	$0.398899\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	7.77585e7i	0.351141i
$606$	0	0
$607$	−3.39905e8	−1.51982	−0.759908	−	0.650031i	$-0.774756\pi$
$607$	−3.39905e8	−1.51982	−0.759908	+	0.650031i	$0.774756\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.12117e8i	0.491527i
$612$	0	0
$613$	3.34467e8	1.45202	0.726008	−	0.687686i	$-0.241373\pi$
$613$	3.34467e8	1.45202	0.726008	+	0.687686i	$0.241373\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 3.45178e8i	− 1.46956i	−0.678305	−	0.734780i	$-0.737285\pi$
$617$	− 3.45178e8i	− 1.46956i	0.678305	−	0.734780i	$-0.262715\pi$
$618$	0	0
$619$	−1.14620e8	−0.483269	−0.241635	−	0.970367i	$-0.577684\pi$
$619$	−1.14620e8	−0.483269	−0.241635	+	0.970367i	$0.577684\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 1.94170e8i	− 0.803005i
$624$	0	0
$625$	9.17782e7	0.375923
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 3.75342e8i	− 1.50826i
$630$	0	0
$631$	−3.72217e8	−1.48152	−0.740760	−	0.671769i	$-0.765535\pi$
$631$	−3.72217e8	−1.48152	−0.740760	+	0.671769i	$0.765535\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 6.76796e6i	− 0.0264324i
$636$	0	0
$637$	−1.55319e8	−0.600905
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 3.85099e8i	− 1.46217i	−0.682285	−	0.731087i	$-0.739013\pi$
$641$	− 3.85099e8i	− 1.46217i	0.682285	−	0.731087i	$-0.260987\pi$
$642$	0	0
$643$	2.24477e8	0.844381	0.422191	−	0.906507i	$-0.361261\pi$
$643$	2.24477e8	0.844381	0.422191	+	0.906507i	$0.361261\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 1.62221e8i	− 0.598955i	−0.954103	−	0.299477i	$-0.903188\pi$
$647$	− 1.62221e8i	− 0.598955i	0.954103	−	0.299477i	$-0.0968123\pi$
$648$	0	0
$649$	−1.14145e8	−0.417565
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 2.77189e8i	− 0.995490i	−0.867324	−	0.497745i	$-0.834162\pi$
$653$	− 2.77189e8i	− 0.995490i	0.867324	−	0.497745i	$-0.165838\pi$
$654$	0	0
$655$	−2.63149e8	−0.936436
$656$	0	0
$657$	0	0
$658$	0	0
$659$	3.93222e7i	0.137398i	0.997637	+	0.0686992i	$0.0218849\pi$
$659$	3.93222e7i	0.137398i	−0.997637	+	0.0686992i	$0.978115\pi$
$660$	0	0
$661$	2.79219e8	0.966807	0.483404	−	0.875398i	$-0.339400\pi$
$661$	2.79219e8	0.966807	0.483404	+	0.875398i	$0.339400\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 2.60451e7i	− 0.0885649i
$666$	0	0
$667$	6.50359e7	0.219167
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 1.44009e8i	− 0.476676i
$672$	0	0
$673$	−1.26948e8	−0.416467	−0.208234	−	0.978079i	$-0.566771\pi$
$673$	−1.26948e8	−0.416467	−0.208234	+	0.978079i	$0.566771\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	5.51813e8i	1.77839i	0.457532	+	0.889193i	$0.348734\pi$
$677$	5.51813e8i	1.77839i	−0.457532	+	0.889193i	$0.651266\pi$
$678$	0	0
$679$	1.33770e8	0.427318
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1.16196e7i	0.0364695i	0.999834	+	0.0182347i	$0.00580462\pi$
$683$	1.16196e7i	0.0364695i	−0.999834	+	0.0182347i	$0.994195\pi$
$684$	0	0
$685$	−1.68626e8	−0.524628
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 3.66166e8i	− 1.11949i
$690$	0	0
$691$	6.54400e8	1.98340	0.991698	−	0.128591i	$-0.0410454\pi$
$691$	6.54400e8	1.98340	0.991698	+	0.128591i	$0.0410454\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 2.40594e8i	− 0.716689i
$696$	0	0
$697$	7.61880e8	2.25003
$698$	0	0
$699$	0	0
$700$	0	0
$701$	8.65379e7i	0.251219i	0.992080	+	0.125609i	$0.0400886\pi$
$701$	8.65379e7i	0.251219i	−0.992080	+	0.125609i	$0.959911\pi$
$702$	0	0
$703$	1.67073e8	0.480885
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 2.18927e8i	− 0.619500i
$708$	0	0
$709$	−6.79982e8	−1.90791	−0.953957	−	0.299945i	$-0.903032\pi$
$709$	−6.79982e8	−1.90791	−0.953957	+	0.299945i	$0.903032\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 3.66496e8i	− 1.01112i
$714$	0	0
$715$	−6.64069e7	−0.181675
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1.62934e8i	0.438354i	0.975685	+	0.219177i	$0.0703372\pi$
$719$	1.62934e8i	0.438354i	−0.975685	+	0.219177i	$0.929663\pi$
$720$	0	0
$721$	−2.49860e8	−0.666639
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 1.12268e8i	− 0.294607i
$726$	0	0
$727$	1.53459e8	0.399382	0.199691	−	0.979859i	$-0.436006\pi$
$727$	1.53459e8	0.399382	0.199691	+	0.979859i	$0.436006\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 5.75604e8i	− 1.47357i
$732$	0	0
$733$	−3.49512e8	−0.887463	−0.443732	−	0.896160i	$-0.646346\pi$
$733$	−3.49512e8	−0.887463	−0.443732	+	0.896160i	$0.646346\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 7.37354e7i	− 0.184193i
$738$	0	0
$739$	5.10104e7	0.126394	0.0631968	−	0.998001i	$-0.479870\pi$
$739$	5.10104e7	0.126394	0.0631968	+	0.998001i	$0.479870\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 4.67479e8i	− 1.13971i	−0.821744	−	0.569857i	$-0.806999\pi$
$743$	− 4.67479e8i	− 1.13971i	0.821744	−	0.569857i	$-0.193001\pi$
$744$	0	0
$745$	3.46402e7	0.0837743
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 1.70321e8i	− 0.405343i
$750$	0	0
$751$	2.77214e8	0.654478	0.327239	−	0.944942i	$-0.393882\pi$
$751$	2.77214e8	0.654478	0.327239	+	0.944942i	$0.393882\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2.52640e8i	0.587031i
$756$	0	0
$757$	6.09490e8	1.40501	0.702504	−	0.711680i	$-0.252065\pi$
$757$	6.09490e8	1.40501	0.702504	+	0.711680i	$0.252065\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 2.92189e8i	− 0.662995i	−0.943456	−	0.331498i	$-0.892446\pi$
$761$	− 2.92189e8i	− 0.662995i	0.943456	−	0.331498i	$-0.107554\pi$
$762$	0	0
$763$	7.89511e7	0.177740
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.78117e8i	0.616370i
$768$	0	0
$769$	−6.56484e8	−1.44359	−0.721797	−	0.692105i	$-0.756683\pi$
$769$	−6.56484e8	−1.44359	−0.721797	+	0.692105i	$0.756683\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 4.93327e8i	− 1.06806i	−0.845465	−	0.534031i	$-0.820677\pi$
$773$	− 4.93327e8i	− 1.06806i	0.845465	−	0.534031i	$-0.179323\pi$
$774$	0	0
$775$	−6.32664e8	−1.35915
$776$	0	0
$777$	0	0
$778$	0	0
$779$	3.39130e8i	0.717388i
$780$	0	0
$781$	−2.76724e8	−0.580890
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 2.53813e8i	− 0.524692i
$786$	0	0
$787$	−1.42429e8	−0.292196	−0.146098	−	0.989270i	$-0.546671\pi$
$787$	−1.42429e8	−0.292196	−0.146098	+	0.989270i	$0.546671\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	7.53789e7i	0.152307i
$792$	0	0
$793$	−3.50881e8	−0.703624
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 6.22911e7i	− 0.123041i	−0.998106	−	0.0615207i	$-0.980405\pi$
$797$	− 6.22911e7i	− 0.123041i	0.998106	−	0.0615207i	$-0.0195950\pi$
$798$	0	0
$799$	4.35687e8	0.854150
$800$	0	0
$801$	0	0
$802$	0	0
$803$	2.68217e7i	0.0518011i
$804$	0	0
$805$	6.39411e7	0.122572
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 6.87496e8i	− 1.29845i	−0.760597	−	0.649224i	$-0.775094\pi$
$809$	− 6.87496e8i	− 1.29845i	0.760597	−	0.649224i	$-0.224906\pi$
$810$	0	0
$811$	1.75961e8	0.329879	0.164939	−	0.986304i	$-0.447257\pi$
$811$	1.75961e8	0.329879	0.164939	+	0.986304i	$0.447257\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.09799e8i	0.202826i
$816$	0	0
$817$	2.56214e8	0.469826
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 2.37506e8i	− 0.429186i	−0.976704	−	0.214593i	$-0.931158\pi$
$821$	− 2.37506e8i	− 0.429186i	0.976704	−	0.214593i	$-0.0688424\pi$
$822$	0	0
$823$	−3.57816e8	−0.641889	−0.320945	−	0.947098i	$-0.604000\pi$
$823$	−3.57816e8	−0.641889	−0.320945	+	0.947098i	$0.604000\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 8.43721e8i	− 1.49170i	−0.666113	−	0.745851i	$-0.732043\pi$
$827$	− 8.43721e8i	− 1.49170i	0.666113	−	0.745851i	$-0.267957\pi$
$828$	0	0
$829$	1.00864e9	1.77040	0.885199	−	0.465212i	$-0.154022\pi$
$829$	1.00864e9	1.77040	0.885199	+	0.465212i	$0.154022\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	6.03570e8i	1.04422i
$834$	0	0
$835$	−5.32517e8	−0.914691
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1.45477e8i	0.246325i	0.992387	+	0.123162i	$0.0393036\pi$
$839$	1.45477e8i	0.246325i	−0.992387	+	0.123162i	$0.960696\pi$
$840$	0	0
$841$	5.08894e8	0.855539
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 1.24319e8i	− 0.206047i
$846$	0	0
$847$	−2.01682e8	−0.331908
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4.10167e8i	0.665536i
$852$	0	0
$853$	5.15395e8	0.830411	0.415206	−	0.909728i	$-0.363710\pi$
$853$	5.15395e8	0.830411	0.415206	+	0.909728i	$0.363710\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	4.35869e8i	0.692490i	0.938144	+	0.346245i	$0.112543\pi$
$857$	4.35869e8i	0.692490i	−0.938144	+	0.346245i	$0.887457\pi$
$858$	0	0
$859$	−2.91458e8	−0.459829	−0.229915	−	0.973211i	$-0.573845\pi$
$859$	−2.91458e8	−0.459829	−0.229915	+	0.973211i	$0.573845\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 9.92214e8i	− 1.54374i	−0.635783	−	0.771868i	$-0.719323\pi$
$863$	− 9.92214e8i	− 1.54374i	0.635783	−	0.771868i	$-0.280677\pi$
$864$	0	0
$865$	−3.25609e8	−0.503093
$866$	0	0
$867$	0	0
$868$	0	0
$869$	2.83661e8i	0.432255i
$870$	0	0
$871$	−1.79657e8	−0.271888
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 2.52781e8i	− 0.377328i
$876$	0	0
$877$	−1.18701e9	−1.75976	−0.879881	−	0.475194i	$-0.842378\pi$
$877$	−1.18701e9	−1.75976	−0.879881	+	0.475194i	$0.842378\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1.23545e9i	1.80675i	0.428847	+	0.903377i	$0.358920\pi$
$881$	1.23545e9i	1.80675i	−0.428847	+	0.903377i	$0.641080\pi$
$882$	0	0
$883$	8.66631e8	1.25879	0.629394	−	0.777087i	$-0.283303\pi$
$883$	8.66631e8	1.25879	0.629394	+	0.777087i	$0.283303\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 8.13039e8i	− 1.16504i	−0.812817	−	0.582519i	$-0.802067\pi$
$887$	− 8.13039e8i	− 1.16504i	0.812817	−	0.582519i	$-0.197933\pi$
$888$	0	0
$889$	1.75541e7	0.0249846
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.93934e8i	0.272333i
$894$	0	0
$895$	1.47486e8	0.205723
$896$	0	0
$897$	0	0
$898$	0	0
$899$	4.84235e8i	0.666464i
$900$	0	0
$901$	−1.42292e9	−1.94539
$902$	0	0
$903$	0	0
$904$	0	0
$905$	3.00376e8i	0.405247i
$906$	0	0
$907$	2.13387e7	0.0285987	0.0142994	−	0.999898i	$-0.495448\pi$
$907$	2.13387e7	0.0285987	0.0142994	+	0.999898i	$0.495448\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 2.26935e8i	− 0.300155i	−0.988674	−	0.150078i	$-0.952048\pi$
$911$	− 2.26935e8i	− 0.300155i	0.988674	−	0.150078i	$-0.0479524\pi$
$912$	0	0
$913$	−1.72534e7	−0.0226705
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 6.82531e8i	− 0.885145i
$918$	0	0
$919$	1.26885e9	1.63479	0.817396	−	0.576077i	$-0.195417\pi$
$919$	1.26885e9	1.63479	0.817396	+	0.576077i	$0.195417\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	6.74242e8i	0.857454i
$924$	0	0
$925$	7.08051e8	0.894621
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 5.57807e8i	− 0.695723i	−0.937546	−	0.347862i	$-0.886908\pi$
$929$	− 5.57807e8i	− 0.695723i	0.937546	−	0.347862i	$-0.113092\pi$
$930$	0	0
$931$	−2.68663e8	−0.332934
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2.58058e8i	0.315705i
$936$	0	0
$937$	−1.39223e8	−0.169236	−0.0846182	−	0.996413i	$-0.526967\pi$
$937$	−1.39223e8	−0.169236	−0.0846182	+	0.996413i	$0.526967\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	4.06544e8i	0.487908i	0.969787	+	0.243954i	$0.0784447\pi$
$941$	4.06544e8i	0.487908i	−0.969787	+	0.243954i	$0.921555\pi$
$942$	0	0
$943$	−8.32568e8	−0.992852
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.00135e9i	1.17906i	0.807748	+	0.589529i	$0.200686\pi$
$947$	1.00135e9i	1.17906i	−0.807748	+	0.589529i	$0.799314\pi$
$948$	0	0
$949$	6.53514e7	0.0764638
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1.33438e9i	1.54170i	0.637017	+	0.770849i	$0.280168\pi$
$953$	1.33438e9i	1.54170i	−0.637017	+	0.770849i	$0.719832\pi$
$954$	0	0
$955$	−3.21627e8	−0.369268
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 4.37365e8i	− 0.495893i
$960$	0	0
$961$	1.84130e9	2.07469
$962$	0	0
$963$	0	0
$964$	0	0
$965$	3.34057e8i	0.371740i
$966$	0	0
$967$	1.60966e9	1.78014	0.890072	−	0.455820i	$-0.150654\pi$
$967$	1.60966e9	1.78014	0.890072	+	0.455820i	$0.150654\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1.52289e9i	1.66345i	0.555185	+	0.831727i	$0.312648\pi$
$971$	1.52289e9i	1.66345i	−0.555185	+	0.831727i	$0.687352\pi$
$972$	0	0
$973$	6.24030e8	0.677434
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.27041e8i	0.136226i	0.997678	+	0.0681129i	$0.0216978\pi$
$977$	1.27041e8i	0.136226i	−0.997678	+	0.0681129i	$0.978302\pi$
$978$	0	0
$979$	−8.56351e8	−0.912649
$980$	0	0
$981$	0	0
$982$	0	0
$983$	8.84822e8i	0.931527i	0.884909	+	0.465763i	$0.154220\pi$
$983$	8.84822e8i	0.931527i	−0.884909	+	0.465763i	$0.845780\pi$
$984$	0	0
$985$	9.02561e7	0.0944426
$986$	0	0
$987$	0	0
$988$	0	0
$989$	6.29009e8i	0.650232i
$990$	0	0
$991$	−1.07134e9	−1.10079	−0.550396	−	0.834904i	$-0.685523\pi$
$991$	−1.07134e9	−1.10079	−0.550396	+	0.834904i	$0.685523\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 5.72094e8i	− 0.580762i
$996$	0	0
$997$	−1.58853e9	−1.60291	−0.801455	−	0.598056i	$-0.795940\pi$
$997$	−1.58853e9	−1.60291	−0.801455	+	0.598056i	$0.795940\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.7.c.b.161.8		12
3.2	odd	2	inner	324.7.c.b.161.5		12
9.2	odd	6		36.7.g.a.5.4	✓	12
9.4	even	3		36.7.g.a.29.4	yes	12
9.5	odd	6		108.7.g.a.89.3		12
9.7	even	3		108.7.g.a.17.3		12
36.7	odd	6		432.7.q.c.17.3		12
36.11	even	6		144.7.q.b.113.3		12
36.23	even	6		432.7.q.c.305.3		12
36.31	odd	6		144.7.q.b.65.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.7.g.a.5.4	✓	12	9.2	odd	6
36.7.g.a.29.4	yes	12	9.4	even	3
108.7.g.a.17.3		12	9.7	even	3
108.7.g.a.89.3		12	9.5	odd	6
144.7.q.b.65.3		12	36.31	odd	6
144.7.q.b.113.3		12	36.11	even	6
324.7.c.b.161.5		12	3.2	odd	2	inner
324.7.c.b.161.8		12	1.1	even	1	trivial
432.7.q.c.17.3		12	36.7	odd	6
432.7.q.c.305.3		12	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:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	324.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+59.2773i q^{5} -153.748 q^{7} +O(q^{10})\)	\(q+59.2773i q^{5} -153.748 q^{7} +678.075i q^{11} +1652.14 q^{13} -6420.23i q^{17} +2857.79 q^{19} +7015.90i q^{23} +12111.2 q^{25} -9269.79i q^{29} -52237.9 q^{31} -9113.74i q^{35} +58462.5 q^{37} +118669. i q^{41} +89654.8 q^{43} +67861.6i q^{47} -94010.7 q^{49} -221631. i q^{53} -40194.5 q^{55} +168337. i q^{59} -212380. q^{61} +97934.4i q^{65} -108742. q^{67} +408102. i q^{71} +39555.6 q^{73} -104252. i q^{77} +418332. q^{79} +25444.6i q^{83} +380574. q^{85} +1.26291e6i q^{89} -254013. q^{91} +169402. i q^{95} -870065. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 240 q^{7}+O(q^{10}) \) 12 * q + 240 * q^7	\( 12 q + 240 q^{7} + 1680 q^{13} - 12900 q^{19} - 48156 q^{25} - 6240 q^{31} + 12768 q^{37} - 142860 q^{43} + 477828 q^{49} + 5400 q^{55} + 187152 q^{61} - 208668 q^{67} + 221820 q^{73} - 246936 q^{79} - 755136 q^{85} + 91488 q^{91} + 2066964 q^{97}+O(q^{100}) \) 12 * q + 240 * q^7 + 1680 * q^13 - 12900 * q^19 - 48156 * q^25 - 6240 * q^31 + 12768 * q^37 - 142860 * q^43 + 477828 * q^49 + 5400 * q^55 + 187152 * q^61 - 208668 * q^67 + 221820 * q^73 - 246936 * q^79 - 755136 * q^85 + 91488 * q^91 + 2066964 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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