LMFDB - Embedded newform 324.8.a.c.1.5

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			1.5
Root			$-17.5402$ 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)

$f(q)$	$=$	$q+200.155 q^{5} -288.547 q^{7} +O(q^{10})$	$q+200.155 q^{5} -288.547 q^{7} +3481.77 q^{11} -54.9514 q^{13} -19611.2 q^{17} -34448.6 q^{19} +60004.1 q^{23} -38063.1 q^{25} -3861.33 q^{29} -77711.8 q^{31} -57754.1 q^{35} +522297. q^{37} +201092. q^{41} -943156. q^{43} -545509. q^{47} -740283. q^{49} -1.50659e6 q^{53} +696893. q^{55} +2.63485e6 q^{59} -195017. q^{61} -10998.8 q^{65} +2.70256e6 q^{67} -4.32952e6 q^{71} +3.99999e6 q^{73} -1.00466e6 q^{77} -541795. q^{79} -1.58864e6 q^{83} -3.92527e6 q^{85} +5.32762e6 q^{89} +15856.1 q^{91} -6.89505e6 q^{95} +421195. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	200.155	0.716095	0.358048	−	0.933703i	$-0.383443\pi$
$5$	200.155	0.716095	0.358048	+	0.933703i	$0.383443\pi$
$6$	0	0
$7$	−288.547	−0.317961	−0.158981	−	0.987282i	$-0.550821\pi$
$7$	−288.547	−0.317961	−0.158981	+	0.987282i	$0.550821\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3481.77	0.788726	0.394363	−	0.918955i	$-0.370965\pi$
$11$	3481.77	0.788726	0.394363	+	0.918955i	$0.370965\pi$
$12$	0	0
$13$	−54.9514	−0.00693708	−0.00346854	−	0.999994i	$-0.501104\pi$
$13$	−54.9514	−0.00693708	−0.00346854	+	0.999994i	$0.501104\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−19611.2	−0.968127	−0.484063	−	0.875033i	$-0.660840\pi$
$17$	−19611.2	−0.968127	−0.484063	+	0.875033i	$0.660840\pi$
$18$	0	0
$19$	−34448.6	−1.15222	−0.576108	−	0.817374i	$-0.695429\pi$
$19$	−34448.6	−1.15222	−0.576108	+	0.817374i	$0.695429\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	60004.1	1.02833	0.514166	−	0.857691i	$-0.328102\pi$
$23$	60004.1	1.02833	0.514166	+	0.857691i	$0.328102\pi$
$24$	0	0
$25$	−38063.1	−0.487207
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−3861.33	−0.0293998	−0.0146999	−	0.999892i	$-0.504679\pi$
$29$	−3861.33	−0.0293998	−0.0146999	+	0.999892i	$0.504679\pi$
$30$	0	0
$31$	−77711.8	−0.468512	−0.234256	−	0.972175i	$-0.575265\pi$
$31$	−77711.8	−0.468512	−0.234256	+	0.972175i	$0.575265\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−57754.1	−0.227691
$36$	0	0
$37$	522297.	1.69516	0.847581	−	0.530666i	$-0.178058\pi$
$37$	522297.	1.69516	0.847581	+	0.530666i	$0.178058\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	201092.	0.455671	0.227836	−	0.973700i	$-0.426835\pi$
$41$	201092.	0.455671	0.227836	+	0.973700i	$0.426835\pi$
$42$	0	0
$43$	−943156.	−1.80902	−0.904511	−	0.426450i	$-0.859764\pi$
$43$	−943156.	−1.80902	−0.904511	+	0.426450i	$0.859764\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−545509.	−0.766407	−0.383204	−	0.923664i	$-0.625179\pi$
$47$	−545509.	−0.766407	−0.383204	+	0.923664i	$0.625179\pi$
$48$	0	0
$49$	−740283.	−0.898901
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−1.50659e6	−1.39005	−0.695024	−	0.718987i	$-0.744606\pi$
$53$	−1.50659e6	−1.39005	−0.695024	+	0.718987i	$0.744606\pi$
$54$	0	0
$55$	696893.	0.564803
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2.63485e6	1.67022	0.835110	−	0.550083i	$-0.185404\pi$
$59$	2.63485e6	1.67022	0.835110	+	0.550083i	$0.185404\pi$
$60$	0	0
$61$	−195017.	−0.110006	−0.0550031	−	0.998486i	$-0.517517\pi$
$61$	−195017.	−0.110006	−0.0550031	+	0.998486i	$0.517517\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−10998.8	−0.00496761
$66$	0	0
$67$	2.70256e6	1.09778	0.548888	−	0.835896i	$-0.315051\pi$
$67$	2.70256e6	1.09778	0.548888	+	0.835896i	$0.315051\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−4.32952e6	−1.43561	−0.717804	−	0.696246i	$-0.754852\pi$
$71$	−4.32952e6	−1.43561	−0.717804	+	0.696246i	$0.754852\pi$
$72$	0	0
$73$	3.99999e6	1.20345	0.601726	−	0.798702i	$-0.294480\pi$
$73$	3.99999e6	1.20345	0.601726	+	0.798702i	$0.294480\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.00466e6	−0.250784
$78$	0	0
$79$	−541795.	−0.123635	−0.0618173	−	0.998087i	$-0.519690\pi$
$79$	−541795.	−0.123635	−0.0618173	+	0.998087i	$0.519690\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−1.58864e6	−0.304967	−0.152484	−	0.988306i	$-0.548727\pi$
$83$	−1.58864e6	−0.304967	−0.152484	+	0.988306i	$0.548727\pi$
$84$	0	0
$85$	−3.92527e6	−0.693271
$86$	0	0
$87$	0	0
$88$	0	0
$89$	5.32762e6	0.801066	0.400533	−	0.916282i	$-0.368825\pi$
$89$	5.32762e6	0.801066	0.400533	+	0.916282i	$0.368825\pi$
$90$	0	0
$91$	15856.1	0.00220572
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−6.89505e6	−0.825096
$96$	0	0
$97$	421195.	0.0468578	0.0234289	−	0.999726i	$-0.492542\pi$
$97$	421195.	0.0468578	0.0234289	+	0.999726i	$0.492542\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−9.97738e6	−0.963589	−0.481794	−	0.876284i	$-0.660015\pi$
$101$	−9.97738e6	−0.963589	−0.481794	+	0.876284i	$0.660015\pi$
$102$	0	0
$103$	2.88235e6	0.259906	0.129953	−	0.991520i	$-0.458517\pi$
$103$	2.88235e6	0.259906	0.129953	+	0.991520i	$0.458517\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.78835e6	0.141127	0.0705634	−	0.997507i	$-0.477520\pi$
$107$	1.78835e6	0.141127	0.0705634	+	0.997507i	$0.477520\pi$
$108$	0	0
$109$	−1.96681e7	−1.45468	−0.727342	−	0.686275i	$-0.759245\pi$
$109$	−1.96681e7	−1.45468	−0.727342	+	0.686275i	$0.759245\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2.52196e7	−1.64423	−0.822116	−	0.569320i	$-0.807206\pi$
$113$	−2.52196e7	−1.64423	−0.822116	+	0.569320i	$0.807206\pi$
$114$	0	0
$115$	1.20101e7	0.736384
$116$	0	0
$117$	0	0
$118$	0	0
$119$	5.65875e6	0.307827
$120$	0	0
$121$	−7.36443e6	−0.377912
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−2.32556e7	−1.06498
$126$	0	0
$127$	−2.87962e7	−1.24745	−0.623723	−	0.781646i	$-0.714380\pi$
$127$	−2.87962e7	−1.24745	−0.623723	+	0.781646i	$0.714380\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−4.05304e7	−1.57518	−0.787592	−	0.616197i	$-0.788672\pi$
$131$	−4.05304e7	−1.57518	−0.787592	+	0.616197i	$0.788672\pi$
$132$	0	0
$133$	9.94005e6	0.366360
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−3.04867e7	−1.01295	−0.506476	−	0.862254i	$-0.669052\pi$
$137$	−3.04867e7	−1.01295	−0.506476	+	0.862254i	$0.669052\pi$
$138$	0	0
$139$	3.22000e7	1.01696	0.508481	−	0.861073i	$-0.330207\pi$
$139$	3.22000e7	1.01696	0.508481	+	0.861073i	$0.330207\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−191328.	−0.00547145
$144$	0	0
$145$	−772864.	−0.0210531
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1.11956e7	0.277265	0.138632	−	0.990344i	$-0.455729\pi$
$149$	1.11956e7	0.277265	0.138632	+	0.990344i	$0.455729\pi$
$150$	0	0
$151$	2.69922e7	0.637997	0.318999	−	0.947755i	$-0.396654\pi$
$151$	2.69922e7	0.637997	0.318999	+	0.947755i	$0.396654\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1.55544e7	−0.335499
$156$	0	0
$157$	−3.43860e7	−0.709142	−0.354571	−	0.935029i	$-0.615373\pi$
$157$	−3.43860e7	−0.709142	−0.354571	+	0.935029i	$0.615373\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1.73140e7	−0.326969
$162$	0	0
$163$	5.17875e7	0.936631	0.468316	−	0.883561i	$-0.344861\pi$
$163$	5.17875e7	0.936631	0.468316	+	0.883561i	$0.344861\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.00946e8	−1.67718	−0.838590	−	0.544762i	$-0.816620\pi$
$167$	−1.00946e8	−1.67718	−0.838590	+	0.544762i	$0.816620\pi$
$168$	0	0
$169$	−6.27455e7	−0.999952
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1.18594e8	−1.74141	−0.870703	−	0.491810i	$-0.836336\pi$
$173$	−1.18594e8	−1.74141	−0.870703	+	0.491810i	$0.836336\pi$
$174$	0	0
$175$	1.09830e7	0.154913
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−1.11247e8	−1.44978	−0.724890	−	0.688865i	$-0.758109\pi$
$179$	−1.11247e8	−1.44978	−0.724890	+	0.688865i	$0.758109\pi$
$180$	0	0
$181$	1.25752e8	1.57630	0.788150	−	0.615483i	$-0.211039\pi$
$181$	1.25752e8	1.57630	0.788150	+	0.615483i	$0.211039\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.04540e8	1.21390
$186$	0	0
$187$	−6.82816e7	−0.763586
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3.89537e7	0.404512	0.202256	−	0.979333i	$-0.435173\pi$
$191$	3.89537e7	0.404512	0.202256	+	0.979333i	$0.435173\pi$
$192$	0	0
$193$	−3.05887e7	−0.306275	−0.153137	−	0.988205i	$-0.548938\pi$
$193$	−3.05887e7	−0.306275	−0.153137	+	0.988205i	$0.548938\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1.41658e8	−1.32011	−0.660055	−	0.751217i	$-0.729467\pi$
$197$	−1.41658e8	−1.32011	−0.660055	+	0.751217i	$0.729467\pi$
$198$	0	0
$199$	−4.62200e7	−0.415761	−0.207881	−	0.978154i	$-0.566657\pi$
$199$	−4.62200e7	−0.415761	−0.207881	+	0.978154i	$0.566657\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.11418e6	0.00934799
$204$	0	0
$205$	4.02495e7	0.326304
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1.19942e8	−0.908782
$210$	0	0
$211$	8.96761e7	0.657186	0.328593	−	0.944472i	$-0.393426\pi$
$211$	8.96761e7	0.657186	0.328593	+	0.944472i	$0.393426\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1.88777e8	−1.29543
$216$	0	0
$217$	2.24235e7	0.148969
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.07766e6	0.00671597
$222$	0	0
$223$	−7.49255e7	−0.452442	−0.226221	−	0.974076i	$-0.572637\pi$
$223$	−7.49255e7	−0.452442	−0.226221	+	0.974076i	$0.572637\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−6.56019e7	−0.372243	−0.186121	−	0.982527i	$-0.559592\pi$
$227$	−6.56019e7	−0.372243	−0.186121	+	0.982527i	$0.559592\pi$
$228$	0	0
$229$	−1.33819e8	−0.736367	−0.368184	−	0.929753i	$-0.620020\pi$
$229$	−1.33819e8	−0.736367	−0.368184	+	0.929753i	$0.620020\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2.18183e6	0.0112999	0.00564996	−	0.999984i	$-0.498202\pi$
$233$	2.18183e6	0.0112999	0.00564996	+	0.999984i	$0.498202\pi$
$234$	0	0
$235$	−1.09186e8	−0.548821
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−8.04166e7	−0.381025	−0.190512	−	0.981685i	$-0.561015\pi$
$239$	−8.04166e7	−0.381025	−0.190512	+	0.981685i	$0.561015\pi$
$240$	0	0
$241$	−1.81087e7	−0.0833349	−0.0416674	−	0.999132i	$-0.513267\pi$
$241$	−1.81087e7	−0.0833349	−0.0416674	+	0.999132i	$0.513267\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.48171e8	−0.643699
$246$	0	0
$247$	1.89300e6	0.00799301
$248$	0	0
$249$	0	0
$250$	0	0
$251$	2.72149e8	1.08630	0.543148	−	0.839637i	$-0.317232\pi$
$251$	2.72149e8	1.08630	0.543148	+	0.839637i	$0.317232\pi$
$252$	0	0
$253$	2.08920e8	0.811072
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3.29406e8	−1.21050	−0.605251	−	0.796035i	$-0.706927\pi$
$257$	−3.29406e8	−1.21050	−0.605251	+	0.796035i	$0.706927\pi$
$258$	0	0
$259$	−1.50707e8	−0.538996
$260$	0	0
$261$	0	0
$262$	0	0
$263$	5.04397e8	1.70973	0.854865	−	0.518851i	$-0.173640\pi$
$263$	5.04397e8	1.70973	0.854865	+	0.518851i	$0.173640\pi$
$264$	0	0
$265$	−3.01551e8	−0.995406
$266$	0	0
$267$	0	0
$268$	0	0
$269$	4.80450e8	1.50493	0.752463	−	0.658635i	$-0.228866\pi$
$269$	4.80450e8	1.50493	0.752463	+	0.658635i	$0.228866\pi$
$270$	0	0
$271$	−5.32155e8	−1.62422	−0.812112	−	0.583502i	$-0.801682\pi$
$271$	−5.32155e8	−1.62422	−0.812112	+	0.583502i	$0.801682\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.32527e8	−0.384273
$276$	0	0
$277$	5.31051e8	1.50126	0.750632	−	0.660720i	$-0.229749\pi$
$277$	5.31051e8	1.50126	0.750632	+	0.660720i	$0.229749\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−2.00139e8	−0.538095	−0.269048	−	0.963127i	$-0.586709\pi$
$281$	−2.00139e8	−0.538095	−0.269048	+	0.963127i	$0.586709\pi$
$282$	0	0
$283$	6.42124e8	1.68410	0.842048	−	0.539403i	$-0.181350\pi$
$283$	6.42124e8	1.68410	0.842048	+	0.539403i	$0.181350\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−5.80246e7	−0.144886
$288$	0	0
$289$	−2.57409e7	−0.0627310
$290$	0	0
$291$	0	0
$292$	0	0
$293$	2.91844e8	0.677821	0.338910	−	0.940819i	$-0.389942\pi$
$293$	2.91844e8	0.677821	0.338910	+	0.940819i	$0.389942\pi$
$294$	0	0
$295$	5.27377e8	1.19604
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−3.29730e6	−0.00713362
$300$	0	0
$301$	2.72145e8	0.575199
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−3.90335e7	−0.0787750
$306$	0	0
$307$	−2.98628e8	−0.589041	−0.294521	−	0.955645i	$-0.595160\pi$
$307$	−2.98628e8	−0.589041	−0.294521	+	0.955645i	$0.595160\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−5.12935e7	−0.0966942	−0.0483471	−	0.998831i	$-0.515395\pi$
$311$	−5.12935e7	−0.0966942	−0.0483471	+	0.998831i	$0.515395\pi$
$312$	0	0
$313$	3.31540e8	0.611126	0.305563	−	0.952172i	$-0.401155\pi$
$313$	3.31540e8	0.611126	0.305563	+	0.952172i	$0.401155\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8.27243e8	1.45857	0.729283	−	0.684212i	$-0.239854\pi$
$317$	8.27243e8	1.45857	0.729283	+	0.684212i	$0.239854\pi$
$318$	0	0
$319$	−1.34443e7	−0.0231884
$320$	0	0
$321$	0	0
$322$	0	0
$323$	6.75577e8	1.11549
$324$	0	0
$325$	2.09162e6	0.00337980
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.57405e8	0.243688
$330$	0	0
$331$	−5.52167e8	−0.836897	−0.418449	−	0.908240i	$-0.637426\pi$
$331$	−5.52167e8	−0.836897	−0.418449	+	0.908240i	$0.637426\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	5.40930e8	0.786112
$336$	0	0
$337$	−4.64829e8	−0.661589	−0.330795	−	0.943703i	$-0.607317\pi$
$337$	−4.64829e8	−0.661589	−0.330795	+	0.943703i	$0.607317\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−2.70575e8	−0.369528
$342$	0	0
$343$	4.51238e8	0.603777
$344$	0	0
$345$	0	0
$346$	0	0
$347$	9.48628e8	1.21883	0.609414	−	0.792852i	$-0.291405\pi$
$347$	9.48628e8	1.21883	0.609414	+	0.792852i	$0.291405\pi$
$348$	0	0
$349$	−1.28794e9	−1.62184	−0.810918	−	0.585159i	$-0.801032\pi$
$349$	−1.28794e9	−1.62184	−0.810918	+	0.585159i	$0.801032\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−6.18825e8	−0.748783	−0.374392	−	0.927271i	$-0.622148\pi$
$353$	−6.18825e8	−0.748783	−0.374392	+	0.927271i	$0.622148\pi$
$354$	0	0
$355$	−8.66574e8	−1.02803
$356$	0	0
$357$	0	0
$358$	0	0
$359$	8.53032e8	0.973049	0.486525	−	0.873667i	$-0.338264\pi$
$359$	8.53032e8	0.973049	0.486525	+	0.873667i	$0.338264\pi$
$360$	0	0
$361$	2.92832e8	0.327600
$362$	0	0
$363$	0	0
$364$	0	0
$365$	8.00617e8	0.861787
$366$	0	0
$367$	−6.70293e7	−0.0707837	−0.0353919	−	0.999374i	$-0.511268\pi$
$367$	−6.70293e7	−0.0707837	−0.0353919	+	0.999374i	$0.511268\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	4.34723e8	0.441981
$372$	0	0
$373$	8.23900e8	0.822041	0.411021	−	0.911626i	$-0.365172\pi$
$373$	8.23900e8	0.822041	0.411021	+	0.911626i	$0.365172\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	212185.	0.000203949 0
$378$	0	0
$379$	1.74077e9	1.64250	0.821249	−	0.570570i	$-0.193278\pi$
$379$	1.74077e9	1.64250	0.821249	+	0.570570i	$0.193278\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	1.13184e9	1.02942	0.514708	−	0.857366i	$-0.327900\pi$
$383$	1.13184e9	1.02942	0.514708	+	0.857366i	$0.327900\pi$
$384$	0	0
$385$	−2.01087e8	−0.179585
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1.65158e8	−0.142257	−0.0711287	−	0.997467i	$-0.522660\pi$
$389$	−1.65158e8	−0.142257	−0.0711287	+	0.997467i	$0.522660\pi$
$390$	0	0
$391$	−1.17675e9	−0.995555
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1.08443e8	−0.0885342
$396$	0	0
$397$	−4.70125e8	−0.377092	−0.188546	−	0.982064i	$-0.560377\pi$
$397$	−4.70125e8	−0.377092	−0.188546	+	0.982064i	$0.560377\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.16607e9	0.903062	0.451531	−	0.892255i	$-0.350878\pi$
$401$	1.16607e9	0.903062	0.451531	+	0.892255i	$0.350878\pi$
$402$	0	0
$403$	4.27037e6	0.00325011
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.81852e9	1.33702
$408$	0	0
$409$	8.94495e8	0.646467	0.323233	−	0.946319i	$-0.395230\pi$
$409$	8.94495e8	0.646467	0.323233	+	0.946319i	$0.395230\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−7.60279e8	−0.531065
$414$	0	0
$415$	−3.17975e8	−0.218386
$416$	0	0
$417$	0	0
$418$	0	0
$419$	1.91968e9	1.27491	0.637454	−	0.770489i	$-0.279988\pi$
$419$	1.91968e9	1.27491	0.637454	+	0.770489i	$0.279988\pi$
$420$	0	0
$421$	−1.16760e9	−0.762620	−0.381310	−	0.924447i	$-0.624527\pi$
$421$	−1.16760e9	−0.762620	−0.381310	+	0.924447i	$0.624527\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	7.46461e8	0.471678
$426$	0	0
$427$	5.62716e7	0.0349777
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−1.29658e8	−0.0780064	−0.0390032	−	0.999239i	$-0.512418\pi$
$431$	−1.29658e8	−0.0780064	−0.0390032	+	0.999239i	$0.512418\pi$
$432$	0	0
$433$	−6.61741e8	−0.391724	−0.195862	−	0.980631i	$-0.562750\pi$
$433$	−6.61741e8	−0.391724	−0.195862	+	0.980631i	$0.562750\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.06705e9	−1.18486
$438$	0	0
$439$	4.60219e8	0.259620	0.129810	−	0.991539i	$-0.458563\pi$
$439$	4.60219e8	0.259620	0.129810	+	0.991539i	$0.458563\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−1.01835e9	−0.556526	−0.278263	−	0.960505i	$-0.589759\pi$
$443$	−1.01835e9	−0.556526	−0.278263	+	0.960505i	$0.589759\pi$
$444$	0	0
$445$	1.06635e9	0.573640
$446$	0	0
$447$	0	0
$448$	0	0
$449$	3.14105e8	0.163762	0.0818809	−	0.996642i	$-0.473907\pi$
$449$	3.14105e8	0.163762	0.0818809	+	0.996642i	$0.473907\pi$
$450$	0	0
$451$	7.00157e8	0.359400
$452$	0	0
$453$	0	0
$454$	0	0
$455$	3.17367e6	0.00157951
$456$	0	0
$457$	2.77353e9	1.35934	0.679668	−	0.733520i	$-0.262124\pi$
$457$	2.77353e9	1.35934	0.679668	+	0.733520i	$0.262124\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.32039e9	0.627695	0.313848	−	0.949473i	$-0.398382\pi$
$461$	1.32039e9	0.627695	0.313848	+	0.949473i	$0.398382\pi$
$462$	0	0
$463$	−8.00467e8	−0.374809	−0.187404	−	0.982283i	$-0.560007\pi$
$463$	−8.00467e8	−0.374809	−0.187404	+	0.982283i	$0.560007\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	2.31110e9	1.05005	0.525025	−	0.851087i	$-0.324056\pi$
$467$	2.31110e9	1.05005	0.525025	+	0.851087i	$0.324056\pi$
$468$	0	0
$469$	−7.79817e8	−0.349050
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−3.28386e9	−1.42682
$474$	0	0
$475$	1.31122e9	0.561368
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−4.28873e9	−1.78301	−0.891506	−	0.453008i	$-0.850351\pi$
$479$	−4.28873e9	−1.78301	−0.891506	+	0.453008i	$0.850351\pi$
$480$	0	0
$481$	−2.87009e7	−0.0117595
$482$	0	0
$483$	0	0
$484$	0	0
$485$	8.43041e7	0.0335547
$486$	0	0
$487$	−9.67183e8	−0.379452	−0.189726	−	0.981837i	$-0.560760\pi$
$487$	−9.67183e8	−0.379452	−0.189726	+	0.981837i	$0.560760\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−5.09040e8	−0.194074	−0.0970368	−	0.995281i	$-0.530936\pi$
$491$	−5.09040e8	−0.194074	−0.0970368	+	0.995281i	$0.530936\pi$
$492$	0	0
$493$	7.57252e7	0.0284627
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.24927e9	0.456467
$498$	0	0
$499$	1.90024e9	0.684631	0.342316	−	0.939585i	$-0.388789\pi$
$499$	1.90024e9	0.684631	0.342316	+	0.939585i	$0.388789\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	2.89903e9	1.01570	0.507849	−	0.861446i	$-0.330441\pi$
$503$	2.89903e9	1.01570	0.507849	+	0.861446i	$0.330441\pi$
$504$	0	0
$505$	−1.99702e9	−0.690021
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−5.39791e9	−1.81432	−0.907158	−	0.420790i	$-0.861753\pi$
$509$	−5.39791e9	−1.81432	−0.907158	+	0.420790i	$0.861753\pi$
$510$	0	0
$511$	−1.15419e9	−0.382651
$512$	0	0
$513$	0	0
$514$	0	0
$515$	5.76917e8	0.186118
$516$	0	0
$517$	−1.89934e9	−0.604485
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−7.06492e8	−0.218864	−0.109432	−	0.993994i	$-0.534903\pi$
$521$	−7.06492e8	−0.218864	−0.109432	+	0.993994i	$0.534903\pi$
$522$	0	0
$523$	1.01604e9	0.310568	0.155284	−	0.987870i	$-0.450371\pi$
$523$	1.01604e9	0.310568	0.155284	+	0.987870i	$0.450371\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.52402e9	0.453579
$528$	0	0
$529$	1.95661e8	0.0574657
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−1.10503e7	−0.00316103
$534$	0	0
$535$	3.57947e8	0.101060
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−2.57750e9	−0.708986
$540$	0	0
$541$	−3.43848e9	−0.933633	−0.466816	−	0.884354i	$-0.654599\pi$
$541$	−3.43848e9	−0.933633	−0.466816	+	0.884354i	$0.654599\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−3.93666e9	−1.04169
$546$	0	0
$547$	6.01526e9	1.57144	0.785721	−	0.618580i	$-0.212292\pi$
$547$	6.01526e9	1.57144	0.785721	+	0.618580i	$0.212292\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.33017e8	0.0338749
$552$	0	0
$553$	1.56334e8	0.0393110
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−7.35443e9	−1.80325	−0.901625	−	0.432518i	$-0.857625\pi$
$557$	−7.35443e9	−1.80325	−0.901625	+	0.432518i	$0.857625\pi$
$558$	0	0
$559$	5.18277e7	0.0125493
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1.51427e9	0.357622	0.178811	−	0.983883i	$-0.442775\pi$
$563$	1.51427e9	0.357622	0.178811	+	0.983883i	$0.442775\pi$
$564$	0	0
$565$	−5.04782e9	−1.17743
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−8.02359e9	−1.82589	−0.912947	−	0.408078i	$-0.866199\pi$
$569$	−8.02359e9	−1.82589	−0.912947	+	0.408078i	$0.866199\pi$
$570$	0	0
$571$	−8.05656e8	−0.181102	−0.0905510	−	0.995892i	$-0.528863\pi$
$571$	−8.05656e8	−0.181102	−0.0905510	+	0.995892i	$0.528863\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−2.28394e9	−0.501011
$576$	0	0
$577$	−2.54010e9	−0.550473	−0.275237	−	0.961376i	$-0.588756\pi$
$577$	−2.54010e9	−0.550473	−0.275237	+	0.961376i	$0.588756\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	4.58399e8	0.0969678
$582$	0	0
$583$	−5.24560e9	−1.09637
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2.56088e8	0.0522584	0.0261292	−	0.999659i	$-0.491682\pi$
$587$	2.56088e8	0.0522584	0.0261292	+	0.999659i	$0.491682\pi$
$588$	0	0
$589$	2.67706e9	0.539827
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−1.32316e9	−0.260567	−0.130283	−	0.991477i	$-0.541589\pi$
$593$	−1.32316e9	−0.260567	−0.130283	+	0.991477i	$0.541589\pi$
$594$	0	0
$595$	1.13263e9	0.220433
$596$	0	0
$597$	0	0
$598$	0	0
$599$	6.18012e9	1.17490	0.587452	−	0.809259i	$-0.300131\pi$
$599$	6.18012e9	1.17490	0.587452	+	0.809259i	$0.300131\pi$
$600$	0	0
$601$	−1.63231e8	−0.0306720	−0.0153360	−	0.999882i	$-0.504882\pi$
$601$	−1.63231e8	−0.0306720	−0.0153360	+	0.999882i	$0.504882\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1.47403e9	−0.270621
$606$	0	0
$607$	−5.66003e9	−1.02721	−0.513603	−	0.858028i	$-0.671690\pi$
$607$	−5.66003e9	−1.02721	−0.513603	+	0.858028i	$0.671690\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.99765e7	0.00531663
$612$	0	0
$613$	9.43896e9	1.65505	0.827527	−	0.561425i	$-0.189747\pi$
$613$	9.43896e9	1.65505	0.827527	+	0.561425i	$0.189747\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−6.10059e9	−1.04562	−0.522810	−	0.852449i	$-0.675116\pi$
$617$	−6.10059e9	−1.04562	−0.522810	+	0.852449i	$0.675116\pi$
$618$	0	0
$619$	−1.90109e9	−0.322170	−0.161085	−	0.986941i	$-0.551499\pi$
$619$	−1.90109e9	−0.322170	−0.161085	+	0.986941i	$0.551499\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−1.53727e9	−0.254708
$624$	0	0
$625$	−1.68104e9	−0.275422
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1.02428e10	−1.64113
$630$	0	0
$631$	8.99285e9	1.42493	0.712466	−	0.701706i	$-0.247578\pi$
$631$	8.99285e9	1.42493	0.712466	+	0.701706i	$0.247578\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−5.76369e9	−0.893290
$636$	0	0
$637$	4.06796e7	0.00623575
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−1.08363e10	−1.62509	−0.812546	−	0.582897i	$-0.801919\pi$
$641$	−1.08363e10	−1.62509	−0.812546	+	0.582897i	$0.801919\pi$
$642$	0	0
$643$	−1.88473e9	−0.279583	−0.139791	−	0.990181i	$-0.544643\pi$
$643$	−1.88473e9	−0.279583	−0.139791	+	0.990181i	$0.544643\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	2.33910e8	0.0339535	0.0169767	−	0.999856i	$-0.494596\pi$
$647$	2.33910e8	0.0339535	0.0169767	+	0.999856i	$0.494596\pi$
$648$	0	0
$649$	9.17394e9	1.31735
$650$	0	0
$651$	0	0
$652$	0	0
$653$	7.37340e9	1.03627	0.518133	−	0.855300i	$-0.326627\pi$
$653$	7.37340e9	1.03627	0.518133	+	0.855300i	$0.326627\pi$
$654$	0	0
$655$	−8.11235e9	−1.12798
$656$	0	0
$657$	0	0
$658$	0	0
$659$	5.14251e9	0.699965	0.349983	−	0.936756i	$-0.386187\pi$
$659$	5.14251e9	0.699965	0.349983	+	0.936756i	$0.386187\pi$
$660$	0	0
$661$	1.19277e10	1.60639	0.803197	−	0.595713i	$-0.203131\pi$
$661$	1.19277e10	1.60639	0.803197	+	0.595713i	$0.203131\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.98955e9	0.262348
$666$	0	0
$667$	−2.31696e8	−0.0302327
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−6.79004e8	−0.0867648
$672$	0	0
$673$	6.34080e9	0.801847	0.400923	−	0.916112i	$-0.368689\pi$
$673$	6.34080e9	0.801847	0.400923	+	0.916112i	$0.368689\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−3.17062e9	−0.392721	−0.196360	−	0.980532i	$-0.562912\pi$
$677$	−3.17062e9	−0.392721	−0.196360	+	0.980532i	$0.562912\pi$
$678$	0	0
$679$	−1.21535e8	−0.0148990
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1.80345e9	−0.216586	−0.108293	−	0.994119i	$-0.534539\pi$
$683$	−1.80345e9	−0.216586	−0.108293	+	0.994119i	$0.534539\pi$
$684$	0	0
$685$	−6.10206e9	−0.725370
$686$	0	0
$687$	0	0
$688$	0	0
$689$	8.27892e7	0.00964287
$690$	0	0
$691$	7.52078e9	0.867141	0.433571	−	0.901120i	$-0.357253\pi$
$691$	7.52078e9	0.867141	0.433571	+	0.901120i	$0.357253\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	6.44499e9	0.728242
$696$	0	0
$697$	−3.94365e9	−0.441147
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−1.33896e10	−1.46810	−0.734049	−	0.679097i	$-0.762372\pi$
$701$	−1.33896e10	−1.46810	−0.734049	+	0.679097i	$0.762372\pi$
$702$	0	0
$703$	−1.79924e10	−1.95319
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.87895e9	0.306384
$708$	0	0
$709$	2.25864e9	0.238005	0.119002	−	0.992894i	$-0.462030\pi$
$709$	2.25864e9	0.238005	0.119002	+	0.992894i	$0.462030\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−4.66302e9	−0.481786
$714$	0	0
$715$	−3.82952e7	−0.00391808
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1.82486e10	1.83096	0.915480	−	0.402364i	$-0.131811\pi$
$719$	1.82486e10	1.83096	0.915480	+	0.402364i	$0.131811\pi$
$720$	0	0
$721$	−8.31696e8	−0.0826402
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1.46974e8	0.0143238
$726$	0	0
$727$	−2.29774e9	−0.221784	−0.110892	−	0.993832i	$-0.535371\pi$
$727$	−2.29774e9	−0.221784	−0.110892	+	0.993832i	$0.535371\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1.84964e10	1.75136
$732$	0	0
$733$	−5.28126e9	−0.495306	−0.247653	−	0.968849i	$-0.579659\pi$
$733$	−5.28126e9	−0.495306	−0.247653	+	0.968849i	$0.579659\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	9.40970e9	0.865844
$738$	0	0
$739$	−1.40530e10	−1.28090	−0.640448	−	0.768002i	$-0.721251\pi$
$739$	−1.40530e10	−1.28090	−0.640448	+	0.768002i	$0.721251\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−3.77227e9	−0.337398	−0.168699	−	0.985668i	$-0.553957\pi$
$743$	−3.77227e9	−0.337398	−0.168699	+	0.985668i	$0.553957\pi$
$744$	0	0
$745$	2.24085e9	0.198548
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−5.16024e8	−0.0448729
$750$	0	0
$751$	1.52219e10	1.31138	0.655692	−	0.755029i	$-0.272377\pi$
$751$	1.52219e10	1.31138	0.655692	+	0.755029i	$0.272377\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	5.40261e9	0.456867
$756$	0	0
$757$	2.23445e9	0.187213	0.0936064	−	0.995609i	$-0.470160\pi$
$757$	2.23445e9	0.187213	0.0936064	+	0.995609i	$0.470160\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−9.61278e9	−0.790683	−0.395342	−	0.918534i	$-0.629374\pi$
$761$	−9.61278e9	−0.790683	−0.395342	+	0.918534i	$0.629374\pi$
$762$	0	0
$763$	5.67517e9	0.462533
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.44788e8	−0.0115864
$768$	0	0
$769$	−2.02365e10	−1.60470	−0.802349	−	0.596855i	$-0.796417\pi$
$769$	−2.02365e10	−1.60470	−0.802349	+	0.596855i	$0.796417\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	6.13371e9	0.477634	0.238817	−	0.971065i	$-0.423240\pi$
$773$	6.13371e9	0.477634	0.238817	+	0.971065i	$0.423240\pi$
$774$	0	0
$775$	2.95795e9	0.228262
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−6.92734e9	−0.525031
$780$	0	0
$781$	−1.50744e10	−1.13230
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−6.88253e9	−0.507814
$786$	0	0
$787$	5.59056e9	0.408831	0.204415	−	0.978884i	$-0.434471\pi$
$787$	5.59056e9	0.408831	0.204415	+	0.978884i	$0.434471\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	7.27704e9	0.522802
$792$	0	0
$793$	1.07164e7	0.000763122 0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−6.22473e9	−0.435528	−0.217764	−	0.976001i	$-0.569876\pi$
$797$	−6.22473e9	−0.435528	−0.217764	+	0.976001i	$0.569876\pi$
$798$	0	0
$799$	1.06981e10	0.741979
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1.39271e10	0.949194
$804$	0	0
$805$	−3.46548e9	−0.234141
$806$	0	0
$807$	0	0
$808$	0	0
$809$	2.54549e10	1.69025	0.845125	−	0.534569i	$-0.179526\pi$
$809$	2.54549e10	1.69025	0.845125	+	0.534569i	$0.179526\pi$
$810$	0	0
$811$	−6.89981e9	−0.454217	−0.227109	−	0.973869i	$-0.572927\pi$
$811$	−6.89981e9	−0.454217	−0.227109	+	0.973869i	$0.572927\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.03655e10	0.670717
$816$	0	0
$817$	3.24904e10	2.08438
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.17460e10	0.740780	0.370390	−	0.928876i	$-0.379224\pi$
$821$	1.17460e10	0.740780	0.370390	+	0.928876i	$0.379224\pi$
$822$	0	0
$823$	2.05115e10	1.28262	0.641311	−	0.767281i	$-0.278391\pi$
$823$	2.05115e10	1.28262	0.641311	+	0.767281i	$0.278391\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.40362e10	0.862939	0.431470	−	0.902127i	$-0.357995\pi$
$827$	1.40362e10	0.862939	0.431470	+	0.902127i	$0.357995\pi$
$828$	0	0
$829$	1.15591e10	0.704666	0.352333	−	0.935875i	$-0.385388\pi$
$829$	1.15591e10	0.704666	0.352333	+	0.935875i	$0.385388\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.45178e10	0.870250
$834$	0	0
$835$	−2.02048e10	−1.20102
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1.50426e10	0.879336	0.439668	−	0.898160i	$-0.355096\pi$
$839$	1.50426e10	0.879336	0.439668	+	0.898160i	$0.355096\pi$
$840$	0	0
$841$	−1.72350e10	−0.999136
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.25588e10	−0.716061
$846$	0	0
$847$	2.12499e9	0.120161
$848$	0	0
$849$	0	0
$850$	0	0
$851$	3.13399e10	1.74319
$852$	0	0
$853$	−1.56572e10	−0.863757	−0.431879	−	0.901932i	$-0.642149\pi$
$853$	−1.56572e10	−0.863757	−0.431879	+	0.901932i	$0.642149\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−2.44791e10	−1.32850	−0.664252	−	0.747509i	$-0.731250\pi$
$857$	−2.44791e10	−1.32850	−0.664252	+	0.747509i	$0.731250\pi$
$858$	0	0
$859$	3.98702e9	0.214621	0.107310	−	0.994226i	$-0.465776\pi$
$859$	3.98702e9	0.214621	0.107310	+	0.994226i	$0.465776\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1.19886e10	−0.634938	−0.317469	−	0.948269i	$-0.602833\pi$
$863$	−1.19886e10	−0.634938	−0.317469	+	0.948269i	$0.602833\pi$
$864$	0	0
$865$	−2.37371e10	−1.24701
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1.88641e9	−0.0975139
$870$	0	0
$871$	−1.48509e8	−0.00761536
$872$	0	0
$873$	0	0
$874$	0	0
$875$	6.71034e9	0.338623
$876$	0	0
$877$	1.87396e10	0.938127	0.469064	−	0.883164i	$-0.344592\pi$
$877$	1.87396e10	0.938127	0.469064	+	0.883164i	$0.344592\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−2.08570e10	−1.02763	−0.513815	−	0.857901i	$-0.671768\pi$
$881$	−2.08570e10	−1.02763	−0.513815	+	0.857901i	$0.671768\pi$
$882$	0	0
$883$	1.67880e10	0.820609	0.410305	−	0.911949i	$-0.365422\pi$
$883$	1.67880e10	0.820609	0.410305	+	0.911949i	$0.365422\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.37185e10	0.660046	0.330023	−	0.943973i	$-0.392944\pi$
$887$	1.37185e10	0.660046	0.330023	+	0.943973i	$0.392944\pi$
$888$	0	0
$889$	8.30906e9	0.396639
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.87920e10	0.883066
$894$	0	0
$895$	−2.22666e10	−1.03818
$896$	0	0
$897$	0	0
$898$	0	0
$899$	3.00071e8	0.0137742
$900$	0	0
$901$	2.95460e10	1.34574
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2.51698e10	1.12878
$906$	0	0
$907$	−1.25592e10	−0.558901	−0.279451	−	0.960160i	$-0.590152\pi$
$907$	−1.25592e10	−0.558901	−0.279451	+	0.960160i	$0.590152\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	3.94315e10	1.72794	0.863970	−	0.503543i	$-0.167970\pi$
$911$	3.94315e10	1.72794	0.863970	+	0.503543i	$0.167970\pi$
$912$	0	0
$913$	−5.53130e9	−0.240536
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.16949e10	0.500847
$918$	0	0
$919$	2.96366e9	0.125958	0.0629788	−	0.998015i	$-0.479940\pi$
$919$	2.96366e9	0.125958	0.0629788	+	0.998015i	$0.479940\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	2.37913e8	0.00995892
$924$	0	0
$925$	−1.98802e10	−0.825895
$926$	0	0
$927$	0	0
$928$	0	0
$929$	6.63588e8	0.0271546	0.0135773	−	0.999908i	$-0.495678\pi$
$929$	6.63588e8	0.0271546	0.0135773	+	0.999908i	$0.495678\pi$
$930$	0	0
$931$	2.55017e10	1.03573
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−1.36669e10	−0.546801
$936$	0	0
$937$	3.26999e10	1.29855	0.649274	−	0.760555i	$-0.275073\pi$
$937$	3.26999e10	1.29855	0.649274	+	0.760555i	$0.275073\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	2.30454e10	0.901615	0.450807	−	0.892621i	$-0.351136\pi$
$941$	2.30454e10	0.901615	0.450807	+	0.892621i	$0.351136\pi$
$942$	0	0
$943$	1.20663e10	0.468581
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4.80920e10	−1.84013	−0.920064	−	0.391769i	$-0.871863\pi$
$947$	−4.80920e10	−1.84013	−0.920064	+	0.391769i	$0.871863\pi$
$948$	0	0
$949$	−2.19805e8	−0.00834845
$950$	0	0
$951$	0	0
$952$	0	0
$953$	3.57125e10	1.33658	0.668290	−	0.743901i	$-0.267026\pi$
$953$	3.57125e10	1.33658	0.668290	+	0.743901i	$0.267026\pi$
$954$	0	0
$955$	7.79676e9	0.289669
$956$	0	0
$957$	0	0
$958$	0	0
$959$	8.79687e9	0.322079
$960$	0	0
$961$	−2.14735e10	−0.780496
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−6.12248e9	−0.219322
$966$	0	0
$967$	−8.62144e9	−0.306610	−0.153305	−	0.988179i	$-0.548992\pi$
$967$	−8.62144e9	−0.306610	−0.153305	+	0.988179i	$0.548992\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	3.03801e9	0.106493	0.0532467	−	0.998581i	$-0.483043\pi$
$971$	3.03801e9	0.106493	0.0532467	+	0.998581i	$0.483043\pi$
$972$	0	0
$973$	−9.29124e9	−0.323354
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.35210e9	0.0806909	0.0403454	−	0.999186i	$-0.487154\pi$
$977$	2.35210e9	0.0806909	0.0403454	+	0.999186i	$0.487154\pi$
$978$	0	0
$979$	1.85496e10	0.631822
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−6.40896e9	−0.215204	−0.107602	−	0.994194i	$-0.534317\pi$
$983$	−6.40896e9	−0.215204	−0.107602	+	0.994194i	$0.534317\pi$
$984$	0	0
$985$	−2.83536e10	−0.945325
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−5.65932e10	−1.86027
$990$	0	0
$991$	1.48616e10	0.485073	0.242537	−	0.970142i	$-0.422021\pi$
$991$	1.48616e10	0.485073	0.242537	+	0.970142i	$0.422021\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−9.25115e9	−0.297725
$996$	0	0
$997$	−4.30152e10	−1.37464	−0.687320	−	0.726354i	$-0.741213\pi$
$997$	−4.30152e10	−1.37464	−0.687320	+	0.726354i	$0.741213\pi$
$998$	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.1.5		7
3.2	odd	2		324.8.a.d.1.3		7
9.2	odd	6		108.8.e.a.37.5		14
9.4	even	3		36.8.e.a.25.5	yes	14
9.5	odd	6		108.8.e.a.73.5		14
9.7	even	3		36.8.e.a.13.5	✓	14
36.7	odd	6		144.8.i.d.49.3		14
36.11	even	6		432.8.i.d.145.5		14
36.23	even	6		432.8.i.d.289.5		14
36.31	odd	6		144.8.i.d.97.3		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.8.e.a.13.5	✓	14	9.7	even	3
36.8.e.a.25.5	yes	14	9.4	even	3
108.8.e.a.37.5		14	9.2	odd	6
108.8.e.a.73.5		14	9.5	odd	6
144.8.i.d.49.3		14	36.7	odd	6
144.8.i.d.97.3		14	36.31	odd	6
324.8.a.c.1.5		7	1.1	even	1	trivial
324.8.a.d.1.3		7	3.2	odd	2
432.8.i.d.145.5		14	36.11	even	6
432.8.i.d.289.5		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)

\(f(q)\)	\(=\)	\(q+200.155 q^{5} -288.547 q^{7} +O(q^{10})\)	\(q+200.155 q^{5} -288.547 q^{7} +3481.77 q^{11} -54.9514 q^{13} -19611.2 q^{17} -34448.6 q^{19} +60004.1 q^{23} -38063.1 q^{25} -3861.33 q^{29} -77711.8 q^{31} -57754.1 q^{35} +522297. q^{37} +201092. q^{41} -943156. q^{43} -545509. q^{47} -740283. q^{49} -1.50659e6 q^{53} +696893. q^{55} +2.63485e6 q^{59} -195017. q^{61} -10998.8 q^{65} +2.70256e6 q^{67} -4.32952e6 q^{71} +3.99999e6 q^{73} -1.00466e6 q^{77} -541795. q^{79} -1.58864e6 q^{83} -3.92527e6 q^{85} +5.32762e6 q^{89} +15856.1 q^{91} -6.89505e6 q^{95} +421195. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 321 q^{5} + 83 q^{7}+O(q^{10}) \) 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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