LMFDB - Embedded newform 324.8.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,8,Mod(1,324)]

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, 0]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("324.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	324.a (trivial)

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:	yes
Analytic conductor:	$101.212748257$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 1289x^{5} + 4994x^{4} + 496633x^{3} - 2291461x^{2} - 56851263x + 373225328 $ x^7 - 3x^6 - 1289x^5 + 4994x^4 + 496633x^3 - 2291461x^2 - 56851263x + 373225328
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{15} $
Twist minimal:	no (minimal twist has level 36)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$21.5027$ of defining polynomial
Character	$\chi$	$=$	324.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-510.715 q^{5} -753.430 q^{7} +O(q^{10})$	$q-510.715 q^{5} -753.430 q^{7} +7361.21 q^{11} -1123.87 q^{13} -3525.44 q^{17} -22018.0 q^{19} +22397.1 q^{23} +182705. q^{25} +27841.9 q^{29} +261308. q^{31} +384788. q^{35} +491408. q^{37} -760983. q^{41} +478927. q^{43} -689600. q^{47} -255887. q^{49} +723038. q^{53} -3.75948e6 q^{55} -1.15900e6 q^{59} +2.43260e6 q^{61} +573979. q^{65} -2.37742e6 q^{67} +216067. q^{71} -1.03883e6 q^{73} -5.54615e6 q^{77} -4.13148e6 q^{79} +444882. q^{83} +1.80049e6 q^{85} +3.13504e6 q^{89} +846760. q^{91} +1.12449e7 q^{95} -522831. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 321 q^{5} + 83 q^{7}+O(q^{10}) $ 7 * q - 321 * q^5 + 83 * q^7	$ 7 q - 321 q^{5} + 83 q^{7} + 111 q^{11} + 1847 q^{13} - 48 q^{17} + 10124 q^{19} - 19119 q^{23} + 73378 q^{25} - 6045 q^{29} + 153089 q^{31} + 13713 q^{35} + 69674 q^{37} - 446631 q^{41} + 384347 q^{43} - 298413 q^{47} + 351876 q^{49} - 454038 q^{53} - 1263483 q^{55} - 2619543 q^{59} + 146231 q^{61} - 2535735 q^{65} - 1637419 q^{67} - 4353492 q^{71} - 2132260 q^{73} - 9785451 q^{77} - 2402185 q^{79} - 12936357 q^{83} + 1015002 q^{85} - 19684830 q^{89} - 492203 q^{91} - 22685196 q^{95} + 2853257 q^{97}+O(q^{100}) $ 7 * q - 321 * q^5 + 83 * q^7 + 111 * q^11 + 1847 * q^13 - 48 * q^17 + 10124 * q^19 - 19119 * q^23 + 73378 * q^25 - 6045 * q^29 + 153089 * q^31 + 13713 * q^35 + 69674 * q^37 - 446631 * q^41 + 384347 * q^43 - 298413 * q^47 + 351876 * q^49 - 454038 * q^53 - 1263483 * q^55 - 2619543 * q^59 + 146231 * q^61 - 2535735 * q^65 - 1637419 * q^67 - 4353492 * q^71 - 2132260 * q^73 - 9785451 * q^77 - 2402185 * q^79 - 12936357 * q^83 + 1015002 * q^85 - 19684830 * q^89 - 492203 * q^91 - 22685196 * q^95 + 2853257 * q^97

Display 100 coefficients 10 coefficients

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
$5$	−510.715	−1.82719	−0.913595	−	0.406626i	$-0.866705\pi$
$5$	−510.715	−1.82719	−0.913595	+	0.406626i	$0.866705\pi$
$6$	0	0
$7$	−753.430	−0.830232	−0.415116	−	0.909768i	$-0.636259\pi$
$7$	−753.430	−0.830232	−0.415116	+	0.909768i	$0.636259\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	7361.21	1.66753	0.833767	−	0.552116i	$-0.186180\pi$
$11$	7361.21	1.66753	0.833767	+	0.552116i	$0.186180\pi$
$12$	0	0
$13$	−1123.87	−0.141878	−0.0709391	−	0.997481i	$-0.522600\pi$
$13$	−1123.87	−0.141878	−0.0709391	+	0.997481i	$0.522600\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3525.44	−0.174037	−0.0870185	−	0.996207i	$-0.527734\pi$
$17$	−3525.44	−0.174037	−0.0870185	+	0.996207i	$0.527734\pi$
$18$	0	0
$19$	−22018.0	−0.736445	−0.368223	−	0.929738i	$-0.620034\pi$
$19$	−22018.0	−0.736445	−0.368223	+	0.929738i	$0.620034\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	22397.1	0.383835	0.191917	−	0.981411i	$-0.438529\pi$
$23$	22397.1	0.383835	0.191917	+	0.981411i	$0.438529\pi$
$24$	0	0
$25$	182705.	2.33862
$26$	0	0
$27$	0	0
$28$	0	0
$29$	27841.9	0.211985	0.105993	−	0.994367i	$-0.466198\pi$
$29$	27841.9	0.211985	0.105993	+	0.994367i	$0.466198\pi$
$30$	0	0
$31$	261308.	1.57539	0.787693	−	0.616069i	$-0.211276\pi$
$31$	261308.	1.57539	0.787693	+	0.616069i	$0.211276\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	384788.	1.51699
$36$	0	0
$37$	491408.	1.59491	0.797455	−	0.603379i	$-0.206179\pi$
$37$	491408.	1.59491	0.797455	+	0.603379i	$0.206179\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−760983.	−1.72437	−0.862187	−	0.506590i	$-0.830906\pi$
$41$	−760983.	−1.72437	−0.862187	+	0.506590i	$0.830906\pi$
$42$	0	0
$43$	478927.	0.918607	0.459304	−	0.888279i	$-0.348099\pi$
$43$	478927.	0.918607	0.459304	+	0.888279i	$0.348099\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−689600.	−0.968847	−0.484423	−	0.874834i	$-0.660971\pi$
$47$	−689600.	−0.968847	−0.484423	+	0.874834i	$0.660971\pi$
$48$	0	0
$49$	−255887.	−0.310714
$50$	0	0
$51$	0	0
$52$	0	0
$53$	723038.	0.667107	0.333553	−	0.942731i	$-0.391752\pi$
$53$	723038.	0.667107	0.333553	+	0.942731i	$0.391752\pi$
$54$	0	0
$55$	−3.75948e6	−3.04690
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1.15900e6	−0.734685	−0.367342	−	0.930086i	$-0.619732\pi$
$59$	−1.15900e6	−0.734685	−0.367342	+	0.930086i	$0.619732\pi$
$60$	0	0
$61$	2.43260e6	1.37220	0.686098	−	0.727509i	$-0.259322\pi$
$61$	2.43260e6	1.37220	0.686098	+	0.727509i	$0.259322\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	573979.	0.259238
$66$	0	0
$67$	−2.37742e6	−0.965705	−0.482852	−	0.875702i	$-0.660399\pi$
$67$	−2.37742e6	−0.965705	−0.482852	+	0.875702i	$0.660399\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	216067.	0.0716447	0.0358223	−	0.999358i	$-0.488595\pi$
$71$	216067.	0.0716447	0.0358223	+	0.999358i	$0.488595\pi$
$72$	0	0
$73$	−1.03883e6	−0.312547	−0.156274	−	0.987714i	$-0.549948\pi$
$73$	−1.03883e6	−0.312547	−0.156274	+	0.987714i	$0.549948\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−5.54615e6	−1.38444
$78$	0	0
$79$	−4.13148e6	−0.942782	−0.471391	−	0.881924i	$-0.656248\pi$
$79$	−4.13148e6	−0.942782	−0.471391	+	0.881924i	$0.656248\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	444882.	0.0854028	0.0427014	−	0.999088i	$-0.486404\pi$
$83$	444882.	0.0854028	0.0427014	+	0.999088i	$0.486404\pi$
$84$	0	0
$85$	1.80049e6	0.317999
$86$	0	0
$87$	0	0
$88$	0	0
$89$	3.13504e6	0.471388	0.235694	−	0.971827i	$-0.424264\pi$
$89$	3.13504e6	0.471388	0.235694	+	0.971827i	$0.424264\pi$
$90$	0	0
$91$	846760.	0.117792
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.12449e7	1.34562
$96$	0	0
$97$	−522831.	−0.0581648	−0.0290824	−	0.999577i	$-0.509259\pi$
$97$	−522831.	−0.0581648	−0.0290824	+	0.999577i	$0.509259\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1.71690e7	−1.65814	−0.829070	−	0.559144i	$-0.811130\pi$
$101$	−1.71690e7	−1.65814	−0.829070	+	0.559144i	$0.811130\pi$
$102$	0	0
$103$	−5.53482e6	−0.499083	−0.249542	−	0.968364i	$-0.580280\pi$
$103$	−5.53482e6	−0.499083	−0.249542	+	0.968364i	$0.580280\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.22969e7	−1.75955	−0.879774	−	0.475393i	$-0.842306\pi$
$107$	−2.22969e7	−1.75955	−0.879774	+	0.475393i	$0.842306\pi$
$108$	0	0
$109$	1.70033e7	1.25759	0.628796	−	0.777571i	$-0.283548\pi$
$109$	1.70033e7	1.25759	0.628796	+	0.777571i	$0.283548\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1.26746e7	−0.826339	−0.413170	−	0.910654i	$-0.635578\pi$
$113$	−1.26746e7	−0.826339	−0.413170	+	0.910654i	$0.635578\pi$
$114$	0	0
$115$	−1.14385e7	−0.701338
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.65617e6	0.144491
$120$	0	0
$121$	3.47002e7	1.78067
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−5.34104e7	−2.44591
$126$	0	0
$127$	3.42261e7	1.48267	0.741335	−	0.671135i	$-0.234193\pi$
$127$	3.42261e7	1.48267	0.741335	+	0.671135i	$0.234193\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1.01803e7	−0.395651	−0.197825	−	0.980237i	$-0.563388\pi$
$131$	−1.01803e7	−0.395651	−0.197825	+	0.980237i	$0.563388\pi$
$132$	0	0
$133$	1.65890e7	0.611420
$134$	0	0
$135$	0	0
$136$	0	0
$137$	9.97772e6	0.331520	0.165760	−	0.986166i	$-0.446992\pi$
$137$	9.97772e6	0.331520	0.165760	+	0.986166i	$0.446992\pi$
$138$	0	0
$139$	−1.34787e7	−0.425694	−0.212847	−	0.977086i	$-0.568274\pi$
$139$	−1.34787e7	−0.425694	−0.212847	+	0.977086i	$0.568274\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−8.27307e6	−0.236587
$144$	0	0
$145$	−1.42193e7	−0.387337
$146$	0	0
$147$	0	0
$148$	0	0
$149$	8.61459e6	0.213345	0.106673	−	0.994294i	$-0.465980\pi$
$149$	8.61459e6	0.213345	0.106673	+	0.994294i	$0.465980\pi$
$150$	0	0
$151$	−2.10933e7	−0.498569	−0.249284	−	0.968430i	$-0.580195\pi$
$151$	−2.10933e7	−0.498569	−0.249284	+	0.968430i	$0.580195\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1.33454e8	−2.87853
$156$	0	0
$157$	−1.27875e7	−0.263717	−0.131858	−	0.991269i	$-0.542094\pi$
$157$	−1.27875e7	−0.263717	−0.131858	+	0.991269i	$0.542094\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1.68746e7	−0.318672
$162$	0	0
$163$	−5.85105e7	−1.05822	−0.529111	−	0.848553i	$-0.677475\pi$
$163$	−5.85105e7	−1.05822	−0.529111	+	0.848553i	$0.677475\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−4.03685e7	−0.670710	−0.335355	−	0.942092i	$-0.608856\pi$
$167$	−4.03685e7	−0.670710	−0.335355	+	0.942092i	$0.608856\pi$
$168$	0	0
$169$	−6.14854e7	−0.979871
$170$	0	0
$171$	0	0
$172$	0	0
$173$	5.73075e7	0.841492	0.420746	−	0.907179i	$-0.361768\pi$
$173$	5.73075e7	0.841492	0.420746	+	0.907179i	$0.361768\pi$
$174$	0	0
$175$	−1.37655e8	−1.94160
$176$	0	0
$177$	0	0
$178$	0	0
$179$	7.30450e7	0.951929	0.475965	−	0.879464i	$-0.342099\pi$
$179$	7.30450e7	0.951929	0.475965	+	0.879464i	$0.342099\pi$
$180$	0	0
$181$	−6.99247e7	−0.876507	−0.438254	−	0.898851i	$-0.644403\pi$
$181$	−6.99247e7	−0.876507	−0.438254	+	0.898851i	$0.644403\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−2.50969e8	−2.91420
$186$	0	0
$187$	−2.59515e7	−0.290213
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4.96742e7	−0.515838	−0.257919	−	0.966166i	$-0.583037\pi$
$191$	−4.96742e7	−0.515838	−0.257919	+	0.966166i	$0.583037\pi$
$192$	0	0
$193$	1.58856e8	1.59057	0.795285	−	0.606236i	$-0.207321\pi$
$193$	1.58856e8	1.59057	0.795285	+	0.606236i	$0.207321\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−5.00143e7	−0.466082	−0.233041	−	0.972467i	$-0.574868\pi$
$197$	−5.00143e7	−0.466082	−0.233041	+	0.972467i	$0.574868\pi$
$198$	0	0
$199$	1.12730e7	0.101404	0.0507021	−	0.998714i	$-0.483854\pi$
$199$	1.12730e7	0.101404	0.0507021	+	0.998714i	$0.483854\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−2.09769e7	−0.175997
$204$	0	0
$205$	3.88645e8	3.15076
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1.62079e8	−1.22805
$210$	0	0
$211$	−6.24163e7	−0.457414	−0.228707	−	0.973495i	$-0.573450\pi$
$211$	−6.24163e7	−0.457414	−0.228707	+	0.973495i	$0.573450\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2.44595e8	−1.67847
$216$	0	0
$217$	−1.96877e8	−1.30794
$218$	0	0
$219$	0	0
$220$	0	0
$221$	3.96214e6	0.0246921
$222$	0	0
$223$	1.96521e8	1.18670	0.593351	−	0.804944i	$-0.297804\pi$
$223$	1.96521e8	1.18670	0.593351	+	0.804944i	$0.297804\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−3.99961e7	−0.226948	−0.113474	−	0.993541i	$-0.536198\pi$
$227$	−3.99961e7	−0.226948	−0.113474	+	0.993541i	$0.536198\pi$
$228$	0	0
$229$	2.15384e8	1.18519	0.592597	−	0.805499i	$-0.298103\pi$
$229$	2.15384e8	1.18519	0.592597	+	0.805499i	$0.298103\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3.23461e8	1.67524	0.837619	−	0.546255i	$-0.183947\pi$
$233$	3.23461e8	1.67524	0.837619	+	0.546255i	$0.183947\pi$
$234$	0	0
$235$	3.52189e8	1.77027
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1.22439e8	−0.580132	−0.290066	−	0.957007i	$-0.593677\pi$
$239$	−1.22439e8	−0.580132	−0.290066	+	0.957007i	$0.593677\pi$
$240$	0	0
$241$	−3.51350e8	−1.61689	−0.808446	−	0.588571i	$-0.799691\pi$
$241$	−3.51350e8	−1.61689	−0.808446	+	0.588571i	$0.799691\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.30685e8	0.567734
$246$	0	0
$247$	2.47455e7	0.104486
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−2.55899e8	−1.02143	−0.510716	−	0.859749i	$-0.670620\pi$
$251$	−2.55899e8	−1.02143	−0.510716	+	0.859749i	$0.670620\pi$
$252$	0	0
$253$	1.64870e8	0.640057
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2.07793e8	−0.763599	−0.381799	−	0.924245i	$-0.624695\pi$
$257$	−2.07793e8	−0.763599	−0.381799	+	0.924245i	$0.624695\pi$
$258$	0	0
$259$	−3.70241e8	−1.32415
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−2.26100e8	−0.766400	−0.383200	−	0.923665i	$-0.625178\pi$
$263$	−2.26100e8	−0.766400	−0.383200	+	0.923665i	$0.625178\pi$
$264$	0	0
$265$	−3.69266e8	−1.21893
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−1.89571e8	−0.593799	−0.296899	−	0.954909i	$-0.595953\pi$
$269$	−1.89571e8	−0.593799	−0.296899	+	0.954909i	$0.595953\pi$
$270$	0	0
$271$	2.44703e8	0.746873	0.373436	−	0.927656i	$-0.378179\pi$
$271$	2.44703e8	0.746873	0.373436	+	0.927656i	$0.378179\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.34493e9	3.89973
$276$	0	0
$277$	−4.23523e8	−1.19729	−0.598643	−	0.801016i	$-0.704293\pi$
$277$	−4.23523e8	−1.19729	−0.598643	+	0.801016i	$0.704293\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5.04318e8	1.35591	0.677957	−	0.735101i	$-0.262865\pi$
$281$	5.04318e8	1.35591	0.677957	+	0.735101i	$0.262865\pi$
$282$	0	0
$283$	2.72662e8	0.715110	0.357555	−	0.933892i	$-0.383611\pi$
$283$	2.72662e8	0.715110	0.357555	+	0.933892i	$0.383611\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.73347e8	1.43163
$288$	0	0
$289$	−3.97910e8	−0.969711
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−2.48917e8	−0.578119	−0.289060	−	0.957311i	$-0.593343\pi$
$293$	−2.48917e8	−0.578119	−0.289060	+	0.957311i	$0.593343\pi$
$294$	0	0
$295$	5.91918e8	1.34241
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2.51715e7	−0.0544578
$300$	0	0
$301$	−3.60838e8	−0.762657
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−1.24237e9	−2.50726
$306$	0	0
$307$	−1.30629e8	−0.257664	−0.128832	−	0.991666i	$-0.541123\pi$
$307$	−1.30629e8	−0.257664	−0.128832	+	0.991666i	$0.541123\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	8.25571e7	0.155630	0.0778150	−	0.996968i	$-0.475206\pi$
$311$	8.25571e7	0.155630	0.0778150	+	0.996968i	$0.475206\pi$
$312$	0	0
$313$	−4.60076e8	−0.848056	−0.424028	−	0.905649i	$-0.639384\pi$
$313$	−4.60076e8	−0.848056	−0.424028	+	0.905649i	$0.639384\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6.48756e7	0.114386	0.0571931	−	0.998363i	$-0.481785\pi$
$317$	6.48756e7	0.114386	0.0571931	+	0.998363i	$0.481785\pi$
$318$	0	0
$319$	2.04950e8	0.353492
$320$	0	0
$321$	0	0
$322$	0	0
$323$	7.76231e7	0.128169
$324$	0	0
$325$	−2.05337e8	−0.331799
$326$	0	0
$327$	0	0
$328$	0	0
$329$	5.19565e8	0.804368
$330$	0	0
$331$	−5.36688e8	−0.813437	−0.406718	−	0.913554i	$-0.633327\pi$
$331$	−5.36688e8	−0.813437	−0.406718	+	0.913554i	$0.633327\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1.21418e9	1.76453
$336$	0	0
$337$	−4.44816e8	−0.633104	−0.316552	−	0.948575i	$-0.602525\pi$
$337$	−4.44816e8	−0.633104	−0.316552	+	0.948575i	$0.602525\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	1.92354e9	2.62701
$342$	0	0
$343$	8.13274e8	1.08820
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−8.09313e8	−1.03983	−0.519916	−	0.854217i	$-0.674037\pi$
$347$	−8.09313e8	−1.03983	−0.519916	+	0.854217i	$0.674037\pi$
$348$	0	0
$349$	−7.41251e7	−0.0933419	−0.0466709	−	0.998910i	$-0.514861\pi$
$349$	−7.41251e7	−0.0933419	−0.0466709	+	0.998910i	$0.514861\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−3.81326e8	−0.461407	−0.230704	−	0.973024i	$-0.574103\pi$
$353$	−3.81326e8	−0.461407	−0.230704	+	0.973024i	$0.574103\pi$
$354$	0	0
$355$	−1.10349e8	−0.130908
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.14988e9	1.31166	0.655830	−	0.754909i	$-0.272319\pi$
$359$	1.14988e9	1.31166	0.655830	+	0.754909i	$0.272319\pi$
$360$	0	0
$361$	−4.09079e8	−0.457649
$362$	0	0
$363$	0	0
$364$	0	0
$365$	5.30547e8	0.571083
$366$	0	0
$367$	1.59460e9	1.68391	0.841957	−	0.539545i	$-0.181404\pi$
$367$	1.59460e9	1.68391	0.841957	+	0.539545i	$0.181404\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−5.44758e8	−0.553854
$372$	0	0
$373$	1.59560e9	1.59200	0.796001	−	0.605296i	$-0.206945\pi$
$373$	1.59560e9	1.59200	0.796001	+	0.605296i	$0.206945\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3.12907e7	−0.0300761
$378$	0	0
$379$	−4.66799e8	−0.440446	−0.220223	−	0.975450i	$-0.570679\pi$
$379$	−4.66799e8	−0.440446	−0.220223	+	0.975450i	$0.570679\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	8.52458e8	0.775314	0.387657	−	0.921804i	$-0.373285\pi$
$383$	8.52458e8	0.775314	0.387657	+	0.921804i	$0.373285\pi$
$384$	0	0
$385$	2.83250e9	2.52963
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1.75950e7	−0.0151554	−0.00757769	−	0.999971i	$-0.502412\pi$
$389$	−1.75950e7	−0.0151554	−0.00757769	+	0.999971i	$0.502412\pi$
$390$	0	0
$391$	−7.89595e7	−0.0668014
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2.11001e9	1.72264
$396$	0	0
$397$	−1.41133e8	−0.113204	−0.0566020	−	0.998397i	$-0.518027\pi$
$397$	−1.41133e8	−0.113204	−0.0566020	+	0.998397i	$0.518027\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−3.47458e8	−0.269090	−0.134545	−	0.990908i	$-0.542957\pi$
$401$	−3.47458e8	−0.269090	−0.134545	+	0.990908i	$0.542957\pi$
$402$	0	0
$403$	−2.93677e8	−0.223513
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.61736e9	2.65957
$408$	0	0
$409$	8.86514e8	0.640699	0.320349	−	0.947299i	$-0.396200\pi$
$409$	8.86514e8	0.640699	0.320349	+	0.947299i	$0.396200\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	8.73224e8	0.609959
$414$	0	0
$415$	−2.27208e8	−0.156047
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−2.17951e7	−0.0144747	−0.00723735	−	0.999974i	$-0.502304\pi$
$419$	−2.17951e7	−0.0144747	−0.00723735	+	0.999974i	$0.502304\pi$
$420$	0	0
$421$	−4.58963e8	−0.299772	−0.149886	−	0.988703i	$-0.547891\pi$
$421$	−4.58963e8	−0.299772	−0.149886	+	0.988703i	$0.547891\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−6.44114e8	−0.407007
$426$	0	0
$427$	−1.83279e9	−1.13924
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−5.21929e8	−0.314008	−0.157004	−	0.987598i	$-0.550184\pi$
$431$	−5.21929e8	−0.314008	−0.157004	+	0.987598i	$0.550184\pi$
$432$	0	0
$433$	1.52520e9	0.902855	0.451428	−	0.892308i	$-0.350915\pi$
$433$	1.52520e9	0.902855	0.451428	+	0.892308i	$0.350915\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.93139e8	−0.282673
$438$	0	0
$439$	−1.72040e9	−0.970519	−0.485259	−	0.874370i	$-0.661275\pi$
$439$	−1.72040e9	−0.970519	−0.485259	+	0.874370i	$0.661275\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.47150e9	0.804170	0.402085	−	0.915602i	$-0.368286\pi$
$443$	1.47150e9	0.804170	0.402085	+	0.915602i	$0.368286\pi$
$444$	0	0
$445$	−1.60111e9	−0.861315
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−3.44117e9	−1.79409	−0.897044	−	0.441942i	$-0.854290\pi$
$449$	−3.44117e9	−1.79409	−0.897044	+	0.441942i	$0.854290\pi$
$450$	0	0
$451$	−5.60176e9	−2.87545
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−4.32453e8	−0.215228
$456$	0	0
$457$	−1.19848e9	−0.587386	−0.293693	−	0.955900i	$-0.594884\pi$
$457$	−1.19848e9	−0.587386	−0.293693	+	0.955900i	$0.594884\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−2.60563e9	−1.23868	−0.619341	−	0.785122i	$-0.712600\pi$
$461$	−2.60563e9	−1.23868	−0.619341	+	0.785122i	$0.712600\pi$
$462$	0	0
$463$	−1.43755e8	−0.0673114	−0.0336557	−	0.999433i	$-0.510715\pi$
$463$	−1.43755e8	−0.0673114	−0.0336557	+	0.999433i	$0.510715\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	2.52732e8	0.114829	0.0574145	−	0.998350i	$-0.481714\pi$
$467$	2.52732e8	0.114829	0.0574145	+	0.998350i	$0.481714\pi$
$468$	0	0
$469$	1.79122e9	0.801759
$470$	0	0
$471$	0	0
$472$	0	0
$473$	3.52548e9	1.53181
$474$	0	0
$475$	−4.02279e9	−1.72227
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−3.80400e9	−1.58149	−0.790745	−	0.612146i	$-0.790307\pi$
$479$	−3.80400e9	−1.58149	−0.790745	+	0.612146i	$0.790307\pi$
$480$	0	0
$481$	−5.52280e8	−0.226283
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.67018e8	0.106278
$486$	0	0
$487$	7.16333e8	0.281037	0.140519	−	0.990078i	$-0.455123\pi$
$487$	7.16333e8	0.281037	0.140519	+	0.990078i	$0.455123\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	8.52981e7	0.0325203	0.0162601	−	0.999868i	$-0.494824\pi$
$491$	8.52981e7	0.0325203	0.0162601	+	0.999868i	$0.494824\pi$
$492$	0	0
$493$	−9.81548e7	−0.0368933
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.62791e8	−0.0594817
$498$	0	0
$499$	2.18843e9	0.788463	0.394231	−	0.919011i	$-0.371011\pi$
$499$	2.18843e9	0.788463	0.394231	+	0.919011i	$0.371011\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−3.15563e9	−1.10560	−0.552800	−	0.833314i	$-0.686441\pi$
$503$	−3.15563e9	−1.10560	−0.552800	+	0.833314i	$0.686441\pi$
$504$	0	0
$505$	8.76849e9	3.02974
$506$	0	0
$507$	0	0
$508$	0	0
$509$	1.25553e9	0.422004	0.211002	−	0.977486i	$-0.432327\pi$
$509$	1.25553e9	0.422004	0.211002	+	0.977486i	$0.432327\pi$
$510$	0	0
$511$	7.82687e8	0.259487
$512$	0	0
$513$	0	0
$514$	0	0
$515$	2.82671e9	0.911920
$516$	0	0
$517$	−5.07629e9	−1.61559
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1.09928e9	−0.340546	−0.170273	−	0.985397i	$-0.554465\pi$
$521$	−1.09928e9	−0.340546	−0.170273	+	0.985397i	$0.554465\pi$
$522$	0	0
$523$	−2.73274e8	−0.0835299	−0.0417650	−	0.999127i	$-0.513298\pi$
$523$	−2.73274e8	−0.0835299	−0.0417650	+	0.999127i	$0.513298\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−9.21225e8	−0.274175
$528$	0	0
$529$	−2.90320e9	−0.852671
$530$	0	0
$531$	0	0
$532$	0	0
$533$	8.55249e8	0.244651
$534$	0	0
$535$	1.13874e10	3.21503
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.88364e9	−0.518127
$540$	0	0
$541$	9.44221e8	0.256380	0.128190	−	0.991750i	$-0.459083\pi$
$541$	9.44221e8	0.256380	0.128190	+	0.991750i	$0.459083\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−8.68382e9	−2.29786
$546$	0	0
$547$	3.68427e9	0.962488	0.481244	−	0.876587i	$-0.340185\pi$
$547$	3.68427e9	0.962488	0.481244	+	0.876587i	$0.340185\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−6.13022e8	−0.156115
$552$	0	0
$553$	3.11278e9	0.782728
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−6.88688e9	−1.68861	−0.844305	−	0.535862i	$-0.819987\pi$
$557$	−6.88688e9	−1.68861	−0.844305	+	0.535862i	$0.819987\pi$
$558$	0	0
$559$	−5.38253e8	−0.130330
$560$	0	0
$561$	0	0
$562$	0	0
$563$	4.26690e9	1.00770	0.503852	−	0.863790i	$-0.331916\pi$
$563$	4.26690e9	1.00770	0.503852	+	0.863790i	$0.331916\pi$
$564$	0	0
$565$	6.47309e9	1.50988
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−6.98525e9	−1.58960	−0.794802	−	0.606869i	$-0.792425\pi$
$569$	−6.98525e9	−1.58960	−0.794802	+	0.606869i	$0.792425\pi$
$570$	0	0
$571$	−7.42403e9	−1.66884	−0.834418	−	0.551133i	$-0.814196\pi$
$571$	−7.42403e9	−1.66884	−0.834418	+	0.551133i	$0.814196\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	4.09205e9	0.897643
$576$	0	0
$577$	−7.64018e9	−1.65572	−0.827862	−	0.560932i	$-0.810443\pi$
$577$	−7.64018e9	−1.65572	−0.827862	+	0.560932i	$0.810443\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−3.35188e8	−0.0709041
$582$	0	0
$583$	5.32243e9	1.11242
$584$	0	0
$585$	0	0
$586$	0	0
$587$	7.41997e9	1.51415	0.757075	−	0.653328i	$-0.226628\pi$
$587$	7.41997e9	1.51415	0.757075	+	0.653328i	$0.226628\pi$
$588$	0	0
$589$	−5.75348e9	−1.16018
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1.66653e9	0.328188	0.164094	−	0.986445i	$-0.447530\pi$
$593$	1.66653e9	0.328188	0.164094	+	0.986445i	$0.447530\pi$
$594$	0	0
$595$	−1.35654e9	−0.264013
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−3.29631e9	−0.626664	−0.313332	−	0.949644i	$-0.601445\pi$
$599$	−3.29631e9	−0.626664	−0.313332	+	0.949644i	$0.601445\pi$
$600$	0	0
$601$	−1.96066e9	−0.368418	−0.184209	−	0.982887i	$-0.558972\pi$
$601$	−1.96066e9	−0.368418	−0.184209	+	0.982887i	$0.558972\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1.77219e10	−3.25362
$606$	0	0
$607$	1.49028e8	0.0270463	0.0135231	−	0.999909i	$-0.495695\pi$
$607$	1.49028e8	0.0270463	0.0135231	+	0.999909i	$0.495695\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	7.75024e8	0.137458
$612$	0	0
$613$	−5.35015e9	−0.938111	−0.469056	−	0.883169i	$-0.655406\pi$
$613$	−5.35015e9	−0.938111	−0.469056	+	0.883169i	$0.655406\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−6.56919e9	−1.12594	−0.562968	−	0.826479i	$-0.690341\pi$
$617$	−6.56919e9	−1.12594	−0.562968	+	0.826479i	$0.690341\pi$
$618$	0	0
$619$	−6.77471e9	−1.14808	−0.574042	−	0.818826i	$-0.694625\pi$
$619$	−6.77471e9	−1.14808	−0.574042	+	0.818826i	$0.694625\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−2.36203e9	−0.391361
$624$	0	0
$625$	1.30037e10	2.13052
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1.73243e9	−0.277573
$630$	0	0
$631$	9.33253e9	1.47876	0.739378	−	0.673290i	$-0.235120\pi$
$631$	9.33253e9	1.47876	0.739378	+	0.673290i	$0.235120\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−1.74798e10	−2.70912
$636$	0	0
$637$	2.87584e8	0.0440836
$638$	0	0
$639$	0	0
$640$	0	0
$641$	9.36670e9	1.40470	0.702350	−	0.711832i	$-0.252134\pi$
$641$	9.36670e9	1.40470	0.702350	+	0.711832i	$0.252134\pi$
$642$	0	0
$643$	−1.06799e10	−1.58427	−0.792135	−	0.610346i	$-0.791030\pi$
$643$	−1.06799e10	−1.58427	−0.792135	+	0.610346i	$0.791030\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	9.47097e9	1.37477	0.687384	−	0.726294i	$-0.258759\pi$
$647$	9.47097e9	1.37477	0.687384	+	0.726294i	$0.258759\pi$
$648$	0	0
$649$	−8.53164e9	−1.22511
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−9.16674e9	−1.28830	−0.644152	−	0.764897i	$-0.722790\pi$
$653$	−9.16674e9	−1.28830	−0.644152	+	0.764897i	$0.722790\pi$
$654$	0	0
$655$	5.19924e9	0.722928
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−4.53624e9	−0.617443	−0.308721	−	0.951153i	$-0.599901\pi$
$659$	−4.53624e9	−0.617443	−0.308721	+	0.951153i	$0.599901\pi$
$660$	0	0
$661$	−7.46697e9	−1.00563	−0.502816	−	0.864393i	$-0.667703\pi$
$661$	−7.46697e9	−1.00563	−0.502816	+	0.864393i	$0.667703\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−8.47226e9	−1.11718
$666$	0	0
$667$	6.23577e8	0.0813672
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1.79069e10	2.28818
$672$	0	0
$673$	−1.03666e10	−1.31094	−0.655468	−	0.755223i	$-0.727529\pi$
$673$	−1.03666e10	−1.31094	−0.655468	+	0.755223i	$0.727529\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−7.38282e9	−0.914454	−0.457227	−	0.889350i	$-0.651157\pi$
$677$	−7.38282e9	−0.914454	−0.457227	+	0.889350i	$0.651157\pi$
$678$	0	0
$679$	3.93916e8	0.0482903
$680$	0	0
$681$	0	0
$682$	0	0
$683$	2.23161e9	0.268007	0.134004	−	0.990981i	$-0.457217\pi$
$683$	2.23161e9	0.268007	0.134004	+	0.990981i	$0.457217\pi$
$684$	0	0
$685$	−5.09577e9	−0.605749
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−8.12603e8	−0.0946479
$690$	0	0
$691$	9.86474e9	1.13740	0.568699	−	0.822546i	$-0.307447\pi$
$691$	9.86474e9	1.13740	0.568699	+	0.822546i	$0.307447\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	6.88379e9	0.777823
$696$	0	0
$697$	2.68280e9	0.300105
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−2.92787e8	−0.0321025	−0.0160512	−	0.999871i	$-0.505109\pi$
$701$	−2.92787e8	−0.0321025	−0.0160512	+	0.999871i	$0.505109\pi$
$702$	0	0
$703$	−1.08198e10	−1.17456
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.29357e10	1.37664
$708$	0	0
$709$	8.51705e9	0.897485	0.448742	−	0.893661i	$-0.351872\pi$
$709$	8.51705e9	0.897485	0.448742	+	0.893661i	$0.351872\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	5.85254e9	0.604687
$714$	0	0
$715$	4.22518e9	0.432289
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−4.76002e9	−0.477593	−0.238797	−	0.971070i	$-0.576753\pi$
$719$	−4.76002e9	−0.477593	−0.238797	+	0.971070i	$0.576753\pi$
$720$	0	0
$721$	4.17010e9	0.414355
$722$	0	0
$723$	0	0
$724$	0	0
$725$	5.08684e9	0.495753
$726$	0	0
$727$	−1.02646e10	−0.990763	−0.495382	−	0.868675i	$-0.664972\pi$
$727$	−1.02646e10	−0.990763	−0.495382	+	0.868675i	$0.664972\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1.68843e9	−0.159872
$732$	0	0
$733$	−8.21959e9	−0.770879	−0.385440	−	0.922733i	$-0.625950\pi$
$733$	−8.21959e9	−0.770879	−0.385440	+	0.922733i	$0.625950\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.75007e10	−1.61035
$738$	0	0
$739$	7.78199e9	0.709309	0.354654	−	0.934997i	$-0.384599\pi$
$739$	7.78199e9	0.709309	0.354654	+	0.934997i	$0.384599\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	7.17479e9	0.641724	0.320862	−	0.947126i	$-0.396027\pi$
$743$	7.17479e9	0.641724	0.320862	+	0.947126i	$0.396027\pi$
$744$	0	0
$745$	−4.39960e9	−0.389822
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1.67991e10	1.46083
$750$	0	0
$751$	1.09555e10	0.943830	0.471915	−	0.881644i	$-0.343563\pi$
$751$	1.09555e10	0.943830	0.471915	+	0.881644i	$0.343563\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1.07727e10	0.910979
$756$	0	0
$757$	1.54904e10	1.29786	0.648928	−	0.760850i	$-0.275218\pi$
$757$	1.54904e10	1.29786	0.648928	+	0.760850i	$0.275218\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.14863e10	−0.944784	−0.472392	−	0.881388i	$-0.656609\pi$
$761$	−1.14863e10	−0.944784	−0.472392	+	0.881388i	$0.656609\pi$
$762$	0	0
$763$	−1.28108e10	−1.04409
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.30257e9	0.104236
$768$	0	0
$769$	1.33596e10	1.05938	0.529690	−	0.848191i	$-0.322308\pi$
$769$	1.33596e10	1.05938	0.529690	+	0.848191i	$0.322308\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	9.13322e9	0.711207	0.355603	−	0.934637i	$-0.384275\pi$
$773$	9.13322e9	0.711207	0.355603	+	0.934637i	$0.384275\pi$
$774$	0	0
$775$	4.77422e10	3.68423
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.67553e10	1.26991
$780$	0	0
$781$	1.59051e9	0.119470
$782$	0	0
$783$	0	0
$784$	0	0
$785$	6.53078e9	0.481860
$786$	0	0
$787$	3.59969e9	0.263241	0.131620	−	0.991300i	$-0.457982\pi$
$787$	3.59969e9	0.263241	0.131620	+	0.991300i	$0.457982\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	9.54939e9	0.686053
$792$	0	0
$793$	−2.73393e9	−0.194685
$794$	0	0
$795$	0	0
$796$	0	0
$797$	3.64195e8	0.0254818	0.0127409	−	0.999919i	$-0.495944\pi$
$797$	3.64195e8	0.0254818	0.0127409	+	0.999919i	$0.495944\pi$
$798$	0	0
$799$	2.43114e9	0.168615
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−7.64706e9	−0.521183
$804$	0	0
$805$	8.61812e9	0.582274
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1.30554e10	−0.866904	−0.433452	−	0.901177i	$-0.642705\pi$
$809$	−1.30554e10	−0.866904	−0.433452	+	0.901177i	$0.642705\pi$
$810$	0	0
$811$	−8.16417e9	−0.537451	−0.268726	−	0.963217i	$-0.586602\pi$
$811$	−8.16417e9	−0.537451	−0.268726	+	0.963217i	$0.586602\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	2.98822e10	1.93357
$816$	0	0
$817$	−1.05450e10	−0.676504
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−2.91296e10	−1.83710	−0.918551	−	0.395304i	$-0.870639\pi$
$821$	−2.91296e10	−1.83710	−0.918551	+	0.395304i	$0.870639\pi$
$822$	0	0
$823$	−1.02052e10	−0.638146	−0.319073	−	0.947730i	$-0.603372\pi$
$823$	−1.02052e10	−0.638146	−0.319073	+	0.947730i	$0.603372\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−1.27246e10	−0.782306	−0.391153	−	0.920326i	$-0.627924\pi$
$827$	−1.27246e10	−0.782306	−0.391153	+	0.920326i	$0.627924\pi$
$828$	0	0
$829$	−1.02153e10	−0.622746	−0.311373	−	0.950288i	$-0.600789\pi$
$829$	−1.02153e10	−0.622746	−0.311373	+	0.950288i	$0.600789\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	9.02112e8	0.0540758
$834$	0	0
$835$	2.06168e10	1.22551
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1.60196e10	0.936452	0.468226	−	0.883609i	$-0.344893\pi$
$839$	1.60196e10	0.936452	0.468226	+	0.883609i	$0.344893\pi$
$840$	0	0
$841$	−1.64747e10	−0.955062
$842$	0	0
$843$	0	0
$844$	0	0
$845$	3.14015e10	1.79041
$846$	0	0
$847$	−2.61442e10	−1.47837
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1.10061e10	0.612181
$852$	0	0
$853$	−1.62378e9	−0.0895788	−0.0447894	−	0.998996i	$-0.514262\pi$
$853$	−1.62378e9	−0.0895788	−0.0447894	+	0.998996i	$0.514262\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.44248e10	1.32556	0.662779	−	0.748815i	$-0.269377\pi$
$857$	2.44248e10	1.32556	0.662779	+	0.748815i	$0.269377\pi$
$858$	0	0
$859$	−2.10374e10	−1.13244	−0.566221	−	0.824253i	$-0.691595\pi$
$859$	−2.10374e10	−1.13244	−0.566221	+	0.824253i	$0.691595\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	4.62099e9	0.244735	0.122368	−	0.992485i	$-0.460951\pi$
$863$	4.62099e9	0.244735	0.122368	+	0.992485i	$0.460951\pi$
$864$	0	0
$865$	−2.92678e10	−1.53756
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−3.04127e10	−1.57212
$870$	0	0
$871$	2.67192e9	0.137012
$872$	0	0
$873$	0	0
$874$	0	0
$875$	4.02410e10	2.03068
$876$	0	0
$877$	2.26492e10	1.13385	0.566923	−	0.823771i	$-0.308134\pi$
$877$	2.26492e10	1.13385	0.566923	+	0.823771i	$0.308134\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	2.79479e10	1.37700	0.688499	−	0.725237i	$-0.258270\pi$
$881$	2.79479e10	1.37700	0.688499	+	0.725237i	$0.258270\pi$
$882$	0	0
$883$	−1.13730e10	−0.555920	−0.277960	−	0.960593i	$-0.589658\pi$
$883$	−1.13730e10	−0.555920	−0.277960	+	0.960593i	$0.589658\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1.77467e10	−0.853856	−0.426928	−	0.904286i	$-0.640404\pi$
$887$	−1.77467e10	−0.853856	−0.426928	+	0.904286i	$0.640404\pi$
$888$	0	0
$889$	−2.57870e10	−1.23096
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.51836e10	0.713502
$894$	0	0
$895$	−3.73052e10	−1.73936
$896$	0	0
$897$	0	0
$898$	0	0
$899$	7.27530e9	0.333958
$900$	0	0
$901$	−2.54902e9	−0.116101
$902$	0	0
$903$	0	0
$904$	0	0
$905$	3.57116e10	1.60154
$906$	0	0
$907$	−3.26910e10	−1.45480	−0.727399	−	0.686215i	$-0.759271\pi$
$907$	−3.26910e10	−1.45480	−0.727399	+	0.686215i	$0.759271\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−3.30504e10	−1.44831	−0.724156	−	0.689636i	$-0.757771\pi$
$911$	−3.30504e10	−1.44831	−0.724156	+	0.689636i	$0.757771\pi$
$912$	0	0
$913$	3.27487e9	0.142412
$914$	0	0
$915$	0	0
$916$	0	0
$917$	7.67015e9	0.328482
$918$	0	0
$919$	−2.39009e10	−1.01580	−0.507901	−	0.861415i	$-0.669578\pi$
$919$	−2.39009e10	−1.01580	−0.507901	+	0.861415i	$0.669578\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−2.42832e8	−0.0101648
$924$	0	0
$925$	8.97825e10	3.72989
$926$	0	0
$927$	0	0
$928$	0	0
$929$	3.56513e10	1.45888	0.729441	−	0.684044i	$-0.239780\pi$
$929$	3.56513e10	1.45888	0.729441	+	0.684044i	$0.239780\pi$
$930$	0	0
$931$	5.63412e9	0.228824
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1.32538e10	0.530274
$936$	0	0
$937$	1.89148e10	0.751127	0.375563	−	0.926797i	$-0.377449\pi$
$937$	1.89148e10	0.751127	0.375563	+	0.926797i	$0.377449\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−3.41865e10	−1.33749	−0.668746	−	0.743491i	$-0.733168\pi$
$941$	−3.41865e10	−1.33749	−0.668746	+	0.743491i	$0.733168\pi$
$942$	0	0
$943$	−1.70438e10	−0.661874
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−7.95457e9	−0.304363	−0.152182	−	0.988353i	$-0.548630\pi$
$947$	−7.95457e9	−0.304363	−0.152182	+	0.988353i	$0.548630\pi$
$948$	0	0
$949$	1.16752e9	0.0443436
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1.91341e9	0.0716114	0.0358057	−	0.999359i	$-0.488600\pi$
$953$	1.91341e9	0.0716114	0.0358057	+	0.999359i	$0.488600\pi$
$954$	0	0
$955$	2.53693e10	0.942534
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−7.51751e9	−0.275238
$960$	0	0
$961$	4.07692e10	1.48184
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−8.11301e10	−2.90627
$966$	0	0
$967$	3.98306e10	1.41652	0.708262	−	0.705950i	$-0.249480\pi$
$967$	3.98306e10	1.41652	0.708262	+	0.705950i	$0.249480\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	5.28356e10	1.85208	0.926040	−	0.377426i	$-0.123191\pi$
$971$	5.28356e10	1.85208	0.926040	+	0.377426i	$0.123191\pi$
$972$	0	0
$973$	1.01553e10	0.353425
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.77304e10	1.29437	0.647187	−	0.762331i	$-0.275945\pi$
$977$	3.77304e10	1.29437	0.647187	+	0.762331i	$0.275945\pi$
$978$	0	0
$979$	2.30777e10	0.786055
$980$	0	0
$981$	0	0
$982$	0	0
$983$	4.32946e10	1.45377	0.726886	−	0.686758i	$-0.240967\pi$
$983$	4.32946e10	1.45377	0.726886	+	0.686758i	$0.240967\pi$
$984$	0	0
$985$	2.55430e10	0.851620
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.07266e10	0.352593
$990$	0	0
$991$	−2.23295e10	−0.728822	−0.364411	−	0.931238i	$-0.618730\pi$
$991$	−2.23295e10	−0.728822	−0.364411	+	0.931238i	$0.618730\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−5.75731e9	−0.185284
$996$	0	0
$997$	−3.93744e10	−1.25829	−0.629145	−	0.777288i	$-0.716595\pi$
$997$	−3.93744e10	−1.25829	−0.629145	+	0.777288i	$0.716595\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.8.a.c.1.1		7
3.2	odd	2		324.8.a.d.1.7		7
9.2	odd	6		108.8.e.a.37.1		14
9.4	even	3		36.8.e.a.25.3	yes	14
9.5	odd	6		108.8.e.a.73.1		14
9.7	even	3		36.8.e.a.13.3	✓	14
36.7	odd	6		144.8.i.d.49.5		14
36.11	even	6		432.8.i.d.145.1		14
36.23	even	6		432.8.i.d.289.1		14
36.31	odd	6		144.8.i.d.97.5		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.8.e.a.13.3	✓	14	9.7	even	3
36.8.e.a.25.3	yes	14	9.4	even	3
108.8.e.a.37.1		14	9.2	odd	6
108.8.e.a.73.1		14	9.5	odd	6
144.8.i.d.49.5		14	36.7	odd	6
144.8.i.d.97.5		14	36.31	odd	6
324.8.a.c.1.1		7	1.1	even	1	trivial
324.8.a.d.1.7		7	3.2	odd	2
432.8.i.d.145.1		14	36.11	even	6
432.8.i.d.289.1		14	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:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	324.a (trivial)

\(f(q)\)	\(=\)	\(q-510.715 q^{5} -753.430 q^{7} +O(q^{10})\)	\(q-510.715 q^{5} -753.430 q^{7} +7361.21 q^{11} -1123.87 q^{13} -3525.44 q^{17} -22018.0 q^{19} +22397.1 q^{23} +182705. q^{25} +27841.9 q^{29} +261308. q^{31} +384788. q^{35} +491408. q^{37} -760983. q^{41} +478927. q^{43} -689600. q^{47} -255887. q^{49} +723038. q^{53} -3.75948e6 q^{55} -1.15900e6 q^{59} +2.43260e6 q^{61} +573979. q^{65} -2.37742e6 q^{67} +216067. q^{71} -1.03883e6 q^{73} -5.54615e6 q^{77} -4.13148e6 q^{79} +444882. q^{83} +1.80049e6 q^{85} +3.13504e6 q^{89} +846760. q^{91} +1.12449e7 q^{95} -522831. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 321 q^{5} + 83 q^{7}+O(q^{10}) \) 7 * q - 321 * q^5 + 83 * q^7	\( 7 q - 321 q^{5} + 83 q^{7} + 111 q^{11} + 1847 q^{13} - 48 q^{17} + 10124 q^{19} - 19119 q^{23} + 73378 q^{25} - 6045 q^{29} + 153089 q^{31} + 13713 q^{35} + 69674 q^{37} - 446631 q^{41} + 384347 q^{43} - 298413 q^{47} + 351876 q^{49} - 454038 q^{53} - 1263483 q^{55} - 2619543 q^{59} + 146231 q^{61} - 2535735 q^{65} - 1637419 q^{67} - 4353492 q^{71} - 2132260 q^{73} - 9785451 q^{77} - 2402185 q^{79} - 12936357 q^{83} + 1015002 q^{85} - 19684830 q^{89} - 492203 q^{91} - 22685196 q^{95} + 2853257 q^{97}+O(q^{100}) \) 7 * q - 321 * q^5 + 83 * q^7 + 111 * q^11 + 1847 * q^13 - 48 * q^17 + 10124 * q^19 - 19119 * q^23 + 73378 * q^25 - 6045 * q^29 + 153089 * q^31 + 13713 * q^35 + 69674 * q^37 - 446631 * q^41 + 384347 * q^43 - 298413 * q^47 + 351876 * q^49 - 454038 * q^53 - 1263483 * q^55 - 2619543 * q^59 + 146231 * q^61 - 2535735 * q^65 - 1637419 * q^67 - 4353492 * q^71 - 2132260 * q^73 - 9785451 * q^77 - 2402185 * q^79 - 12936357 * q^83 + 1015002 * q^85 - 19684830 * q^89 - 492203 * q^91 - 22685196 * q^95 + 2853257 * q^97

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