LMFDB - Embedded newform 8036.2.a.p.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8036,2,Mod(1,8036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8036, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8036 = 2^{2} \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8036.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:	$64.1677830643$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 18x^{8} + 32x^{7} + 110x^{6} - 154x^{5} - 282x^{4} + 256x^{3} + 253x^{2} - 126x - 49 $ x^10 - 2x^9 - 18x^8 + 32x^7 + 110x^6 - 154x^5 - 282x^4 + 256x^3 + 253x^2 - 126*x - 49
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$1.20067$ of defining polynomial
Character	$\chi$	$=$	8036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.20067 q^{3} -0.960855 q^{5} -1.55839 q^{9} +O(q^{10})$	$q+1.20067 q^{3} -0.960855 q^{5} -1.55839 q^{9} -4.42245 q^{11} +0.696329 q^{13} -1.15367 q^{15} +1.02043 q^{17} -7.03402 q^{19} -4.50530 q^{23} -4.07676 q^{25} -5.47312 q^{27} +8.50757 q^{29} +2.30265 q^{31} -5.30991 q^{33} +6.54332 q^{37} +0.836062 q^{39} +1.00000 q^{41} +6.37372 q^{43} +1.49739 q^{45} +0.286914 q^{47} +1.22520 q^{51} +6.03283 q^{53} +4.24934 q^{55} -8.44553 q^{57} -14.2347 q^{59} +5.32167 q^{61} -0.669072 q^{65} +0.778309 q^{67} -5.40938 q^{69} -0.249785 q^{71} +6.06573 q^{73} -4.89484 q^{75} +1.79493 q^{79} -1.89624 q^{81} +0.273880 q^{83} -0.980482 q^{85} +10.2148 q^{87} +10.8123 q^{89} +2.76472 q^{93} +6.75867 q^{95} +18.9582 q^{97} +6.89192 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 2 q^{3} + 4 q^{5} + 10 q^{9}+O(q^{10}) $ 10 * q + 2 * q^3 + 4 * q^5 + 10 * q^9	$ 10 q + 2 q^{3} + 4 q^{5} + 10 q^{9} + 6 q^{13} - 4 q^{15} + 12 q^{17} + 8 q^{19} + 4 q^{23} + 16 q^{25} + 8 q^{27} + 2 q^{29} + 8 q^{31} - 6 q^{33} - 2 q^{37} - 2 q^{39} + 10 q^{41} - 2 q^{43} + 44 q^{45} - 14 q^{47} + 14 q^{51} + 8 q^{53} + 8 q^{55} - 10 q^{57} + 24 q^{59} + 14 q^{61} + 2 q^{65} - 8 q^{67} + 16 q^{69} + 10 q^{71} + 44 q^{73} - 50 q^{75} + 10 q^{79} - 14 q^{81} + 20 q^{83} + 8 q^{85} + 20 q^{87} + 6 q^{89} + 8 q^{93} + 4 q^{95} + 46 q^{97} + 2 q^{99}+O(q^{100}) $ 10 * q + 2 * q^3 + 4 * q^5 + 10 * q^9 + 6 * q^13 - 4 * q^15 + 12 * q^17 + 8 * q^19 + 4 * q^23 + 16 * q^25 + 8 * q^27 + 2 * q^29 + 8 * q^31 - 6 * q^33 - 2 * q^37 - 2 * q^39 + 10 * q^41 - 2 * q^43 + 44 * q^45 - 14 * q^47 + 14 * q^51 + 8 * q^53 + 8 * q^55 - 10 * q^57 + 24 * q^59 + 14 * q^61 + 2 * q^65 - 8 * q^67 + 16 * q^69 + 10 * q^71 + 44 * q^73 - 50 * q^75 + 10 * q^79 - 14 * q^81 + 20 * q^83 + 8 * q^85 + 20 * q^87 + 6 * q^89 + 8 * q^93 + 4 * q^95 + 46 * q^97 + 2 * q^99

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
$3$	1.20067	0.693207	0.346604	−	0.938012i	$-0.387335\pi$
$3$	1.20067	0.693207	0.346604	+	0.938012i	$0.387335\pi$
$4$	0	0
$5$	−0.960855	−0.429708	−0.214854	−	0.976646i	$-0.568927\pi$
$5$	−0.960855	−0.429708	−0.214854	+	0.976646i	$0.568927\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−1.55839	−0.519464
$10$	0	0
$11$	−4.42245	−1.33342	−0.666710	−	0.745317i	$-0.732298\pi$
$11$	−4.42245	−1.33342	−0.666710	+	0.745317i	$0.732298\pi$
$12$	0	0
$13$	0.696329	0.193127	0.0965635	−	0.995327i	$-0.469215\pi$
$13$	0.696329	0.193127	0.0965635	+	0.995327i	$0.469215\pi$
$14$	0	0
$15$	−1.15367	−0.297876
$16$	0	0
$17$	1.02043	0.247490	0.123745	−	0.992314i	$-0.460510\pi$
$17$	1.02043	0.247490	0.123745	+	0.992314i	$0.460510\pi$
$18$	0	0
$19$	−7.03402	−1.61371	−0.806857	−	0.590747i	$-0.798833\pi$
$19$	−7.03402	−1.61371	−0.806857	+	0.590747i	$0.798833\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.50530	−0.939420	−0.469710	−	0.882821i	$-0.655641\pi$
$23$	−4.50530	−0.939420	−0.469710	+	0.882821i	$0.655641\pi$
$24$	0	0
$25$	−4.07676	−0.815351
$26$	0	0
$27$	−5.47312	−1.05330
$28$	0	0
$29$	8.50757	1.57982	0.789909	−	0.613225i	$-0.210128\pi$
$29$	8.50757	1.57982	0.789909	+	0.613225i	$0.210128\pi$
$30$	0	0
$31$	2.30265	0.413568	0.206784	−	0.978387i	$-0.433700\pi$
$31$	2.30265	0.413568	0.206784	+	0.978387i	$0.433700\pi$
$32$	0	0
$33$	−5.30991	−0.924336
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.54332	1.07572	0.537858	−	0.843036i	$-0.319234\pi$
$37$	6.54332	1.07572	0.537858	+	0.843036i	$0.319234\pi$
$38$	0	0
$39$	0.836062	0.133877
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	6.37372	0.971983	0.485992	−	0.873964i	$-0.338459\pi$
$43$	6.37372	0.971983	0.485992	+	0.873964i	$0.338459\pi$
$44$	0	0
$45$	1.49739	0.223218
$46$	0	0
$47$	0.286914	0.0418507	0.0209253	−	0.999781i	$-0.493339\pi$
$47$	0.286914	0.0418507	0.0209253	+	0.999781i	$0.493339\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	1.22520	0.171562
$52$	0	0
$53$	6.03283	0.828673	0.414337	−	0.910124i	$-0.364014\pi$
$53$	6.03283	0.828673	0.414337	+	0.910124i	$0.364014\pi$
$54$	0	0
$55$	4.24934	0.572981
$56$	0	0
$57$	−8.44553	−1.11864
$58$	0	0
$59$	−14.2347	−1.85319	−0.926597	−	0.376055i	$-0.877280\pi$
$59$	−14.2347	−1.85319	−0.926597	+	0.376055i	$0.877280\pi$
$60$	0	0
$61$	5.32167	0.681370	0.340685	−	0.940177i	$-0.389341\pi$
$61$	5.32167	0.681370	0.340685	+	0.940177i	$0.389341\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−0.669072	−0.0829881
$66$	0	0
$67$	0.778309	0.0950856	0.0475428	−	0.998869i	$-0.484861\pi$
$67$	0.778309	0.0950856	0.0475428	+	0.998869i	$0.484861\pi$
$68$	0	0
$69$	−5.40938	−0.651212
$70$	0	0
$71$	−0.249785	−0.0296441	−0.0148220	−	0.999890i	$-0.504718\pi$
$71$	−0.249785	−0.0296441	−0.0148220	+	0.999890i	$0.504718\pi$
$72$	0	0
$73$	6.06573	0.709939	0.354970	−	0.934878i	$-0.384491\pi$
$73$	6.06573	0.709939	0.354970	+	0.934878i	$0.384491\pi$
$74$	0	0
$75$	−4.89484	−0.565207
$76$	0	0
$77$	0	0
$78$	0	0
$79$	1.79493	0.201945	0.100973	−	0.994889i	$-0.467805\pi$
$79$	1.79493	0.201945	0.100973	+	0.994889i	$0.467805\pi$
$80$	0	0
$81$	−1.89624	−0.210693
$82$	0	0
$83$	0.273880	0.0300622	0.0150311	−	0.999887i	$-0.495215\pi$
$83$	0.273880	0.0300622	0.0150311	+	0.999887i	$0.495215\pi$
$84$	0	0
$85$	−0.980482	−0.106348
$86$	0	0
$87$	10.2148	1.09514
$88$	0	0
$89$	10.8123	1.14610	0.573052	−	0.819519i	$-0.305759\pi$
$89$	10.8123	1.14610	0.573052	+	0.819519i	$0.305759\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	2.76472	0.286689
$94$	0	0
$95$	6.75867	0.693425
$96$	0	0
$97$	18.9582	1.92492	0.962458	−	0.271431i	$-0.0874969\pi$
$97$	18.9582	1.92492	0.962458	+	0.271431i	$0.0874969\pi$
$98$	0	0
$99$	6.89192	0.692664
$100$	0	0
$101$	18.2970	1.82062	0.910309	−	0.413929i	$-0.135844\pi$
$101$	18.2970	1.82062	0.910309	+	0.413929i	$0.135844\pi$
$102$	0	0
$103$	0.562381	0.0554130	0.0277065	−	0.999616i	$-0.491180\pi$
$103$	0.562381	0.0554130	0.0277065	+	0.999616i	$0.491180\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.07805	−0.104219	−0.0521094	−	0.998641i	$-0.516594\pi$
$107$	−1.07805	−0.104219	−0.0521094	+	0.998641i	$0.516594\pi$
$108$	0	0
$109$	−2.25003	−0.215514	−0.107757	−	0.994177i	$-0.534367\pi$
$109$	−2.25003	−0.215514	−0.107757	+	0.994177i	$0.534367\pi$
$110$	0	0
$111$	7.85637	0.745693
$112$	0	0
$113$	8.09996	0.761980	0.380990	−	0.924579i	$-0.375583\pi$
$113$	8.09996	0.761980	0.380990	+	0.924579i	$0.375583\pi$
$114$	0	0
$115$	4.32894	0.403676
$116$	0	0
$117$	−1.08515	−0.100323
$118$	0	0
$119$	0	0
$120$	0	0
$121$	8.55811	0.778010
$122$	0	0
$123$	1.20067	0.108261
$124$	0	0
$125$	8.72145	0.780070
$126$	0	0
$127$	9.24445	0.820313	0.410156	−	0.912015i	$-0.365474\pi$
$127$	9.24445	0.820313	0.410156	+	0.912015i	$0.365474\pi$
$128$	0	0
$129$	7.65273	0.673785
$130$	0	0
$131$	−5.82273	−0.508734	−0.254367	−	0.967108i	$-0.581867\pi$
$131$	−5.82273	−0.508734	−0.254367	+	0.967108i	$0.581867\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	5.25888	0.452612
$136$	0	0
$137$	−22.0115	−1.88057	−0.940285	−	0.340388i	$-0.889442\pi$
$137$	−22.0115	−1.88057	−0.940285	+	0.340388i	$0.889442\pi$
$138$	0	0
$139$	12.8321	1.08841	0.544204	−	0.838953i	$-0.316832\pi$
$139$	12.8321	1.08841	0.544204	+	0.838953i	$0.316832\pi$
$140$	0	0
$141$	0.344489	0.0290112
$142$	0	0
$143$	−3.07948	−0.257519
$144$	0	0
$145$	−8.17455	−0.678859
$146$	0	0
$147$	0	0
$148$	0	0
$149$	14.9482	1.22460	0.612301	−	0.790625i	$-0.290244\pi$
$149$	14.9482	1.22460	0.612301	+	0.790625i	$0.290244\pi$
$150$	0	0
$151$	−15.6267	−1.27169	−0.635843	−	0.771818i	$-0.719347\pi$
$151$	−15.6267	−1.27169	−0.635843	+	0.771818i	$0.719347\pi$
$152$	0	0
$153$	−1.59022	−0.128562
$154$	0	0
$155$	−2.21251	−0.177713
$156$	0	0
$157$	14.8900	1.18835	0.594175	−	0.804336i	$-0.297479\pi$
$157$	14.8900	1.18835	0.594175	+	0.804336i	$0.297479\pi$
$158$	0	0
$159$	7.24344	0.574442
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−3.18475	−0.249449	−0.124724	−	0.992191i	$-0.539805\pi$
$163$	−3.18475	−0.249449	−0.124724	+	0.992191i	$0.539805\pi$
$164$	0	0
$165$	5.10205	0.397194
$166$	0	0
$167$	14.1043	1.09142	0.545712	−	0.837973i	$-0.316259\pi$
$167$	14.1043	1.09142	0.545712	+	0.837973i	$0.316259\pi$
$168$	0	0
$169$	−12.5151	−0.962702
$170$	0	0
$171$	10.9618	0.838266
$172$	0	0
$173$	−7.31878	−0.556436	−0.278218	−	0.960518i	$-0.589744\pi$
$173$	−7.31878	−0.556436	−0.278218	+	0.960518i	$0.589744\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−17.0911	−1.28465
$178$	0	0
$179$	−5.41426	−0.404681	−0.202340	−	0.979315i	$-0.564855\pi$
$179$	−5.41426	−0.404681	−0.202340	+	0.979315i	$0.564855\pi$
$180$	0	0
$181$	−19.4477	−1.44554	−0.722768	−	0.691090i	$-0.757131\pi$
$181$	−19.4477	−1.44554	−0.722768	+	0.691090i	$0.757131\pi$
$182$	0	0
$183$	6.38957	0.472331
$184$	0	0
$185$	−6.28718	−0.462243
$186$	0	0
$187$	−4.51279	−0.330008
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−13.1177	−0.949165	−0.474583	−	0.880211i	$-0.657401\pi$
$191$	−13.1177	−0.949165	−0.474583	+	0.880211i	$0.657401\pi$
$192$	0	0
$193$	1.64643	0.118512	0.0592561	−	0.998243i	$-0.481127\pi$
$193$	1.64643	0.118512	0.0592561	+	0.998243i	$0.481127\pi$
$194$	0	0
$195$	−0.803334	−0.0575279
$196$	0	0
$197$	7.34791	0.523517	0.261758	−	0.965133i	$-0.415698\pi$
$197$	7.34791	0.523517	0.261758	+	0.965133i	$0.415698\pi$
$198$	0	0
$199$	−5.91693	−0.419440	−0.209720	−	0.977761i	$-0.567255\pi$
$199$	−5.91693	−0.419440	−0.209720	+	0.977761i	$0.567255\pi$
$200$	0	0
$201$	0.934492	0.0659140
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−0.960855	−0.0671090
$206$	0	0
$207$	7.02102	0.487995
$208$	0	0
$209$	31.1076	2.15176
$210$	0	0
$211$	−23.2386	−1.59981	−0.799906	−	0.600125i	$-0.795117\pi$
$211$	−23.2386	−1.59981	−0.799906	+	0.600125i	$0.795117\pi$
$212$	0	0
$213$	−0.299910	−0.0205495
$214$	0	0
$215$	−6.12422	−0.417668
$216$	0	0
$217$	0	0
$218$	0	0
$219$	7.28293	0.492135
$220$	0	0
$221$	0.710553	0.0477970
$222$	0	0
$223$	22.8794	1.53212	0.766059	−	0.642771i	$-0.222215\pi$
$223$	22.8794	1.53212	0.766059	+	0.642771i	$0.222215\pi$
$224$	0	0
$225$	6.35319	0.423546
$226$	0	0
$227$	−16.7210	−1.10981	−0.554905	−	0.831914i	$-0.687245\pi$
$227$	−16.7210	−1.10981	−0.554905	+	0.831914i	$0.687245\pi$
$228$	0	0
$229$	5.07185	0.335157	0.167579	−	0.985859i	$-0.446405\pi$
$229$	5.07185	0.335157	0.167579	+	0.985859i	$0.446405\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	14.6473	0.959577	0.479789	−	0.877384i	$-0.340713\pi$
$233$	14.6473	0.959577	0.479789	+	0.877384i	$0.340713\pi$
$234$	0	0
$235$	−0.275683	−0.0179835
$236$	0	0
$237$	2.15512	0.139990
$238$	0	0
$239$	12.1788	0.787779	0.393890	−	0.919158i	$-0.371129\pi$
$239$	12.1788	0.787779	0.393890	+	0.919158i	$0.371129\pi$
$240$	0	0
$241$	24.2926	1.56482	0.782411	−	0.622763i	$-0.213990\pi$
$241$	24.2926	1.56482	0.782411	+	0.622763i	$0.213990\pi$
$242$	0	0
$243$	14.1426	0.907249
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−4.89799	−0.311652
$248$	0	0
$249$	0.328839	0.0208393
$250$	0	0
$251$	−21.7394	−1.37218	−0.686089	−	0.727518i	$-0.740674\pi$
$251$	−21.7394	−1.37218	−0.686089	+	0.727518i	$0.740674\pi$
$252$	0	0
$253$	19.9245	1.25264
$254$	0	0
$255$	−1.17724	−0.0737213
$256$	0	0
$257$	−25.2802	−1.57693	−0.788466	−	0.615078i	$-0.789125\pi$
$257$	−25.2802	−1.57693	−0.788466	+	0.615078i	$0.789125\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−13.2581	−0.820658
$262$	0	0
$263$	27.7075	1.70852	0.854260	−	0.519846i	$-0.174011\pi$
$263$	27.7075	1.70852	0.854260	+	0.519846i	$0.174011\pi$
$264$	0	0
$265$	−5.79668	−0.356087
$266$	0	0
$267$	12.9820	0.794487
$268$	0	0
$269$	12.1912	0.743311	0.371655	−	0.928371i	$-0.378790\pi$
$269$	12.1912	0.743311	0.371655	+	0.928371i	$0.378790\pi$
$270$	0	0
$271$	23.2192	1.41046	0.705232	−	0.708977i	$-0.250843\pi$
$271$	23.2192	1.41046	0.705232	+	0.708977i	$0.250843\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	18.0293	1.08721
$276$	0	0
$277$	−9.58468	−0.575888	−0.287944	−	0.957647i	$-0.592972\pi$
$277$	−9.58468	−0.575888	−0.287944	+	0.957647i	$0.592972\pi$
$278$	0	0
$279$	−3.58843	−0.214834
$280$	0	0
$281$	−16.2612	−0.970062	−0.485031	−	0.874497i	$-0.661192\pi$
$281$	−16.2612	−0.970062	−0.485031	+	0.874497i	$0.661192\pi$
$282$	0	0
$283$	6.92241	0.411495	0.205747	−	0.978605i	$-0.434038\pi$
$283$	6.92241	0.411495	0.205747	+	0.978605i	$0.434038\pi$
$284$	0	0
$285$	8.11493	0.480687
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−15.9587	−0.938749
$290$	0	0
$291$	22.7626	1.33437
$292$	0	0
$293$	8.64464	0.505025	0.252513	−	0.967594i	$-0.418743\pi$
$293$	8.64464	0.505025	0.252513	+	0.967594i	$0.418743\pi$
$294$	0	0
$295$	13.6774	0.796332
$296$	0	0
$297$	24.2046	1.40450
$298$	0	0
$299$	−3.13717	−0.181427
$300$	0	0
$301$	0	0
$302$	0	0
$303$	21.9686	1.26207
$304$	0	0
$305$	−5.11336	−0.292790
$306$	0	0
$307$	14.4004	0.821875	0.410937	−	0.911664i	$-0.365201\pi$
$307$	14.4004	0.821875	0.410937	+	0.911664i	$0.365201\pi$
$308$	0	0
$309$	0.675233	0.0384127
$310$	0	0
$311$	22.3514	1.26743	0.633716	−	0.773566i	$-0.281529\pi$
$311$	22.3514	1.26743	0.633716	+	0.773566i	$0.281529\pi$
$312$	0	0
$313$	−22.8805	−1.29328	−0.646641	−	0.762795i	$-0.723827\pi$
$313$	−22.8805	−1.29328	−0.646641	+	0.762795i	$0.723827\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−23.1999	−1.30303	−0.651517	−	0.758634i	$-0.725867\pi$
$317$	−23.1999	−1.30303	−0.651517	+	0.758634i	$0.725867\pi$
$318$	0	0
$319$	−37.6244	−2.10656
$320$	0	0
$321$	−1.29438	−0.0722452
$322$	0	0
$323$	−7.17770	−0.399378
$324$	0	0
$325$	−2.83877	−0.157466
$326$	0	0
$327$	−2.70154	−0.149396
$328$	0	0
$329$	0	0
$330$	0	0
$331$	13.1383	0.722147	0.361073	−	0.932537i	$-0.382410\pi$
$331$	13.1383	0.722147	0.361073	+	0.932537i	$0.382410\pi$
$332$	0	0
$333$	−10.1971	−0.558795
$334$	0	0
$335$	−0.747842	−0.0408590
$336$	0	0
$337$	−28.4357	−1.54899	−0.774495	−	0.632580i	$-0.781996\pi$
$337$	−28.4357	−1.54899	−0.774495	+	0.632580i	$0.781996\pi$
$338$	0	0
$339$	9.72538	0.528210
$340$	0	0
$341$	−10.1834	−0.551460
$342$	0	0
$343$	0	0
$344$	0	0
$345$	5.19763	0.279831
$346$	0	0
$347$	4.99947	0.268386	0.134193	−	0.990955i	$-0.457156\pi$
$347$	4.99947	0.268386	0.134193	+	0.990955i	$0.457156\pi$
$348$	0	0
$349$	−7.09183	−0.379617	−0.189809	−	0.981821i	$-0.560787\pi$
$349$	−7.09183	−0.379617	−0.189809	+	0.981821i	$0.560787\pi$
$350$	0	0
$351$	−3.81110	−0.203421
$352$	0	0
$353$	3.71132	0.197533	0.0987667	−	0.995111i	$-0.468510\pi$
$353$	3.71132	0.197533	0.0987667	+	0.995111i	$0.468510\pi$
$354$	0	0
$355$	0.240007	0.0127383
$356$	0	0
$357$	0	0
$358$	0	0
$359$	10.5218	0.555320	0.277660	−	0.960679i	$-0.410441\pi$
$359$	10.5218	0.555320	0.277660	+	0.960679i	$0.410441\pi$
$360$	0	0
$361$	30.4774	1.60407
$362$	0	0
$363$	10.2755	0.539322
$364$	0	0
$365$	−5.82828	−0.305066
$366$	0	0
$367$	25.2458	1.31782	0.658910	−	0.752222i	$-0.271018\pi$
$367$	25.2458	1.31782	0.658910	+	0.752222i	$0.271018\pi$
$368$	0	0
$369$	−1.55839	−0.0811266
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−2.71198	−0.140421	−0.0702106	−	0.997532i	$-0.522367\pi$
$373$	−2.71198	−0.140421	−0.0702106	+	0.997532i	$0.522367\pi$
$374$	0	0
$375$	10.4716	0.540750
$376$	0	0
$377$	5.92407	0.305105
$378$	0	0
$379$	28.8541	1.48214	0.741069	−	0.671429i	$-0.234319\pi$
$379$	28.8541	1.48214	0.741069	+	0.671429i	$0.234319\pi$
$380$	0	0
$381$	11.0995	0.568647
$382$	0	0
$383$	−28.3527	−1.44876	−0.724379	−	0.689402i	$-0.757873\pi$
$383$	−28.3527	−1.44876	−0.724379	+	0.689402i	$0.757873\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−9.93275	−0.504910
$388$	0	0
$389$	19.2733	0.977195	0.488598	−	0.872509i	$-0.337509\pi$
$389$	19.2733	0.977195	0.488598	+	0.872509i	$0.337509\pi$
$390$	0	0
$391$	−4.59733	−0.232497
$392$	0	0
$393$	−6.99118	−0.352658
$394$	0	0
$395$	−1.72467	−0.0867774
$396$	0	0
$397$	32.8499	1.64869	0.824346	−	0.566087i	$-0.191543\pi$
$397$	32.8499	1.64869	0.824346	+	0.566087i	$0.191543\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	13.8111	0.689695	0.344847	−	0.938659i	$-0.387931\pi$
$401$	13.8111	0.689695	0.344847	+	0.938659i	$0.387931\pi$
$402$	0	0
$403$	1.60340	0.0798712
$404$	0	0
$405$	1.82201	0.0905364
$406$	0	0
$407$	−28.9375	−1.43438
$408$	0	0
$409$	−11.9398	−0.590385	−0.295193	−	0.955438i	$-0.595384\pi$
$409$	−11.9398	−0.590385	−0.295193	+	0.955438i	$0.595384\pi$
$410$	0	0
$411$	−26.4286	−1.30362
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−0.263159	−0.0129180
$416$	0	0
$417$	15.4072	0.754492
$418$	0	0
$419$	−7.86680	−0.384318	−0.192159	−	0.981364i	$-0.561549\pi$
$419$	−7.86680	−0.384318	−0.192159	+	0.981364i	$0.561549\pi$
$420$	0	0
$421$	−25.6643	−1.25080	−0.625400	−	0.780304i	$-0.715064\pi$
$421$	−25.6643	−1.25080	−0.625400	+	0.780304i	$0.715064\pi$
$422$	0	0
$423$	−0.447124	−0.0217399
$424$	0	0
$425$	−4.16003	−0.201791
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−3.69744	−0.178514
$430$	0	0
$431$	12.4449	0.599449	0.299725	−	0.954026i	$-0.403105\pi$
$431$	12.4449	0.599449	0.299725	+	0.954026i	$0.403105\pi$
$432$	0	0
$433$	1.50279	0.0722196	0.0361098	−	0.999348i	$-0.488503\pi$
$433$	1.50279	0.0722196	0.0361098	+	0.999348i	$0.488503\pi$
$434$	0	0
$435$	−9.81493	−0.470590
$436$	0	0
$437$	31.6903	1.51595
$438$	0	0
$439$	−30.8960	−1.47459	−0.737293	−	0.675573i	$-0.763896\pi$
$439$	−30.8960	−1.47459	−0.737293	+	0.675573i	$0.763896\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−14.8122	−0.703749	−0.351874	−	0.936047i	$-0.614456\pi$
$443$	−14.8122	−0.703749	−0.351874	+	0.936047i	$0.614456\pi$
$444$	0	0
$445$	−10.3891	−0.492489
$446$	0	0
$447$	17.9478	0.848902
$448$	0	0
$449$	−38.4709	−1.81555	−0.907776	−	0.419455i	$-0.862221\pi$
$449$	−38.4709	−1.81555	−0.907776	+	0.419455i	$0.862221\pi$
$450$	0	0
$451$	−4.42245	−0.208245
$452$	0	0
$453$	−18.7626	−0.881542
$454$	0	0
$455$	0	0
$456$	0	0
$457$	25.5711	1.19616	0.598082	−	0.801435i	$-0.295930\pi$
$457$	25.5711	1.19616	0.598082	+	0.801435i	$0.295930\pi$
$458$	0	0
$459$	−5.58492	−0.260682
$460$	0	0
$461$	7.94612	0.370088	0.185044	−	0.982730i	$-0.440757\pi$
$461$	7.94612	0.370088	0.185044	+	0.982730i	$0.440757\pi$
$462$	0	0
$463$	33.6444	1.56359	0.781795	−	0.623536i	$-0.214305\pi$
$463$	33.6444	1.56359	0.781795	+	0.623536i	$0.214305\pi$
$464$	0	0
$465$	−2.65650	−0.123192
$466$	0	0
$467$	−36.6702	−1.69690	−0.848448	−	0.529279i	$-0.822463\pi$
$467$	−36.6702	−1.69690	−0.848448	+	0.529279i	$0.822463\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	17.8780	0.823773
$472$	0	0
$473$	−28.1875	−1.29606
$474$	0	0
$475$	28.6760	1.31574
$476$	0	0
$477$	−9.40152	−0.430466
$478$	0	0
$479$	19.4725	0.889722	0.444861	−	0.895600i	$-0.353253\pi$
$479$	19.4725	0.889722	0.444861	+	0.895600i	$0.353253\pi$
$480$	0	0
$481$	4.55631	0.207750
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−18.2161	−0.827151
$486$	0	0
$487$	22.6738	1.02745	0.513723	−	0.857956i	$-0.328266\pi$
$487$	22.6738	1.02745	0.513723	+	0.857956i	$0.328266\pi$
$488$	0	0
$489$	−3.82383	−0.172920
$490$	0	0
$491$	16.6279	0.750408	0.375204	−	0.926942i	$-0.377573\pi$
$491$	16.6279	0.750408	0.375204	+	0.926942i	$0.377573\pi$
$492$	0	0
$493$	8.68136	0.390989
$494$	0	0
$495$	−6.62214	−0.297643
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−30.6666	−1.37282	−0.686412	−	0.727213i	$-0.740815\pi$
$499$	−30.6666	−1.37282	−0.686412	+	0.727213i	$0.740815\pi$
$500$	0	0
$501$	16.9346	0.756583
$502$	0	0
$503$	−5.49221	−0.244885	−0.122443	−	0.992476i	$-0.539073\pi$
$503$	−5.49221	−0.244885	−0.122443	+	0.992476i	$0.539073\pi$
$504$	0	0
$505$	−17.5808	−0.782333
$506$	0	0
$507$	−15.0265	−0.667352
$508$	0	0
$509$	20.8187	0.922774	0.461387	−	0.887199i	$-0.347352\pi$
$509$	20.8187	0.922774	0.461387	+	0.887199i	$0.347352\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	38.4980	1.69973
$514$	0	0
$515$	−0.540366	−0.0238114
$516$	0	0
$517$	−1.26886	−0.0558045
$518$	0	0
$519$	−8.78743	−0.385725
$520$	0	0
$521$	−15.7425	−0.689692	−0.344846	−	0.938659i	$-0.612069\pi$
$521$	−15.7425	−0.689692	−0.344846	+	0.938659i	$0.612069\pi$
$522$	0	0
$523$	19.2884	0.843425	0.421713	−	0.906730i	$-0.361429\pi$
$523$	19.2884	0.843425	0.421713	+	0.906730i	$0.361429\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.34969	0.102354
$528$	0	0
$529$	−2.70228	−0.117490
$530$	0	0
$531$	22.1832	0.962668
$532$	0	0
$533$	0.696329	0.0301614
$534$	0	0
$535$	1.03585	0.0447836
$536$	0	0
$537$	−6.50074	−0.280527
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−16.2247	−0.697555	−0.348777	−	0.937206i	$-0.613403\pi$
$541$	−16.2247	−0.697555	−0.348777	+	0.937206i	$0.613403\pi$
$542$	0	0
$543$	−23.3503	−1.00206
$544$	0	0
$545$	2.16195	0.0926079
$546$	0	0
$547$	5.86146	0.250618	0.125309	−	0.992118i	$-0.460008\pi$
$547$	5.86146	0.250618	0.125309	+	0.992118i	$0.460008\pi$
$548$	0	0
$549$	−8.29325	−0.353947
$550$	0	0
$551$	−59.8424	−2.54937
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−7.54883	−0.320430
$556$	0	0
$557$	−41.9290	−1.77659	−0.888295	−	0.459273i	$-0.848110\pi$
$557$	−41.9290	−1.77659	−0.888295	+	0.459273i	$0.848110\pi$
$558$	0	0
$559$	4.43821	0.187716
$560$	0	0
$561$	−5.41837	−0.228764
$562$	0	0
$563$	25.1667	1.06065	0.530325	−	0.847795i	$-0.322070\pi$
$563$	25.1667	1.06065	0.530325	+	0.847795i	$0.322070\pi$
$564$	0	0
$565$	−7.78289	−0.327429
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−20.0437	−0.840276	−0.420138	−	0.907460i	$-0.638018\pi$
$569$	−20.0437	−0.840276	−0.420138	+	0.907460i	$0.638018\pi$
$570$	0	0
$571$	30.0571	1.25785	0.628925	−	0.777466i	$-0.283495\pi$
$571$	30.0571	1.25785	0.628925	+	0.777466i	$0.283495\pi$
$572$	0	0
$573$	−15.7501	−0.657968
$574$	0	0
$575$	18.3670	0.765957
$576$	0	0
$577$	34.7832	1.44805	0.724023	−	0.689776i	$-0.242291\pi$
$577$	34.7832	1.44805	0.724023	+	0.689776i	$0.242291\pi$
$578$	0	0
$579$	1.97681	0.0821535
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−26.6799	−1.10497
$584$	0	0
$585$	1.04268	0.0431093
$586$	0	0
$587$	38.7289	1.59851	0.799256	−	0.600991i	$-0.205227\pi$
$587$	38.7289	1.59851	0.799256	+	0.600991i	$0.205227\pi$
$588$	0	0
$589$	−16.1969	−0.667381
$590$	0	0
$591$	8.82241	0.362905
$592$	0	0
$593$	31.3187	1.28611	0.643053	−	0.765821i	$-0.277667\pi$
$593$	31.3187	1.28611	0.643053	+	0.765821i	$0.277667\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−7.10428	−0.290759
$598$	0	0
$599$	−31.7113	−1.29569	−0.647845	−	0.761772i	$-0.724329\pi$
$599$	−31.7113	−1.29569	−0.647845	+	0.761772i	$0.724329\pi$
$600$	0	0
$601$	7.97320	0.325234	0.162617	−	0.986689i	$-0.448007\pi$
$601$	7.97320	0.325234	0.162617	+	0.986689i	$0.448007\pi$
$602$	0	0
$603$	−1.21291	−0.0493935
$604$	0	0
$605$	−8.22310	−0.334317
$606$	0	0
$607$	−10.7159	−0.434944	−0.217472	−	0.976067i	$-0.569781\pi$
$607$	−10.7159	−0.434944	−0.217472	+	0.976067i	$0.569781\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0.199786	0.00808249
$612$	0	0
$613$	6.31151	0.254919	0.127460	−	0.991844i	$-0.459318\pi$
$613$	6.31151	0.254919	0.127460	+	0.991844i	$0.459318\pi$
$614$	0	0
$615$	−1.15367	−0.0465205
$616$	0	0
$617$	19.7164	0.793752	0.396876	−	0.917872i	$-0.370094\pi$
$617$	19.7164	0.793752	0.396876	+	0.917872i	$0.370094\pi$
$618$	0	0
$619$	−23.0555	−0.926678	−0.463339	−	0.886181i	$-0.653349\pi$
$619$	−23.0555	−0.926678	−0.463339	+	0.886181i	$0.653349\pi$
$620$	0	0
$621$	24.6581	0.989494
$622$	0	0
$623$	0	0
$624$	0	0
$625$	12.0037	0.480149
$626$	0	0
$627$	37.3500	1.49161
$628$	0	0
$629$	6.67698	0.266229
$630$	0	0
$631$	−17.4544	−0.694847	−0.347424	−	0.937708i	$-0.612943\pi$
$631$	−17.4544	−0.694847	−0.347424	+	0.937708i	$0.612943\pi$
$632$	0	0
$633$	−27.9019	−1.10900
$634$	0	0
$635$	−8.88258	−0.352495
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0.389263	0.0153990
$640$	0	0
$641$	2.45391	0.0969235	0.0484618	−	0.998825i	$-0.484568\pi$
$641$	2.45391	0.0969235	0.0484618	+	0.998825i	$0.484568\pi$
$642$	0	0
$643$	2.18234	0.0860631	0.0430315	−	0.999074i	$-0.486298\pi$
$643$	2.18234	0.0860631	0.0430315	+	0.999074i	$0.486298\pi$
$644$	0	0
$645$	−7.35317	−0.289531
$646$	0	0
$647$	−8.23779	−0.323861	−0.161930	−	0.986802i	$-0.551772\pi$
$647$	−8.23779	−0.323861	−0.161930	+	0.986802i	$0.551772\pi$
$648$	0	0
$649$	62.9521	2.47109
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−39.5156	−1.54636	−0.773182	−	0.634184i	$-0.781336\pi$
$653$	−39.5156	−1.54636	−0.773182	+	0.634184i	$0.781336\pi$
$654$	0	0
$655$	5.59480	0.218607
$656$	0	0
$657$	−9.45278	−0.368788
$658$	0	0
$659$	−0.710018	−0.0276584	−0.0138292	−	0.999904i	$-0.504402\pi$
$659$	−0.710018	−0.0276584	−0.0138292	+	0.999904i	$0.504402\pi$
$660$	0	0
$661$	−3.85365	−0.149889	−0.0749447	−	0.997188i	$-0.523878\pi$
$661$	−3.85365	−0.149889	−0.0749447	+	0.997188i	$0.523878\pi$
$662$	0	0
$663$	0.853139	0.0331332
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−38.3292	−1.48411
$668$	0	0
$669$	27.4706	1.06207
$670$	0	0
$671$	−23.5349	−0.908553
$672$	0	0
$673$	6.80389	0.262271	0.131135	−	0.991364i	$-0.458138\pi$
$673$	6.80389	0.262271	0.131135	+	0.991364i	$0.458138\pi$
$674$	0	0
$675$	22.3126	0.858812
$676$	0	0
$677$	26.1026	1.00320	0.501602	−	0.865098i	$-0.332744\pi$
$677$	26.1026	1.00320	0.501602	+	0.865098i	$0.332744\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−20.0764	−0.769328
$682$	0	0
$683$	−11.7058	−0.447910	−0.223955	−	0.974600i	$-0.571897\pi$
$683$	−11.7058	−0.447910	−0.223955	+	0.974600i	$0.571897\pi$
$684$	0	0
$685$	21.1499	0.808095
$686$	0	0
$687$	6.08962	0.232333
$688$	0	0
$689$	4.20084	0.160039
$690$	0	0
$691$	27.9879	1.06471	0.532355	−	0.846521i	$-0.321307\pi$
$691$	27.9879	1.06471	0.532355	+	0.846521i	$0.321307\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−12.3298	−0.467697
$696$	0	0
$697$	1.02043	0.0386514
$698$	0	0
$699$	17.5866	0.665186
$700$	0	0
$701$	−0.331564	−0.0125230	−0.00626149	−	0.999980i	$-0.501993\pi$
$701$	−0.331564	−0.0125230	−0.00626149	+	0.999980i	$0.501993\pi$
$702$	0	0
$703$	−46.0258	−1.73590
$704$	0	0
$705$	−0.331004	−0.0124663
$706$	0	0
$707$	0	0
$708$	0	0
$709$	19.2339	0.722345	0.361172	−	0.932499i	$-0.382377\pi$
$709$	19.2339	0.722345	0.361172	+	0.932499i	$0.382377\pi$
$710$	0	0
$711$	−2.79720	−0.104903
$712$	0	0
$713$	−10.3741	−0.388514
$714$	0	0
$715$	2.95894	0.110658
$716$	0	0
$717$	14.6227	0.546094
$718$	0	0
$719$	−14.2799	−0.532550	−0.266275	−	0.963897i	$-0.585793\pi$
$719$	−14.2799	−0.532550	−0.266275	+	0.963897i	$0.585793\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	29.1673	1.08475
$724$	0	0
$725$	−34.6833	−1.28811
$726$	0	0
$727$	43.4351	1.61092	0.805460	−	0.592650i	$-0.201918\pi$
$727$	43.4351	1.61092	0.805460	+	0.592650i	$0.201918\pi$
$728$	0	0
$729$	22.6693	0.839605
$730$	0	0
$731$	6.50391	0.240556
$732$	0	0
$733$	51.2125	1.89158	0.945789	−	0.324782i	$-0.105291\pi$
$733$	51.2125	1.89158	0.945789	+	0.324782i	$0.105291\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−3.44204	−0.126789
$738$	0	0
$739$	7.81873	0.287617	0.143808	−	0.989606i	$-0.454065\pi$
$739$	7.81873	0.287617	0.143808	+	0.989606i	$0.454065\pi$
$740$	0	0
$741$	−5.88087	−0.216039
$742$	0	0
$743$	−46.1171	−1.69187	−0.845935	−	0.533285i	$-0.820957\pi$
$743$	−46.1171	−1.69187	−0.845935	+	0.533285i	$0.820957\pi$
$744$	0	0
$745$	−14.3630	−0.526220
$746$	0	0
$747$	−0.426812	−0.0156162
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−44.2495	−1.61469	−0.807343	−	0.590083i	$-0.799095\pi$
$751$	−44.2495	−1.61469	−0.807343	+	0.590083i	$0.799095\pi$
$752$	0	0
$753$	−26.1018	−0.951203
$754$	0	0
$755$	15.0150	0.546453
$756$	0	0
$757$	33.9225	1.23293	0.616466	−	0.787381i	$-0.288564\pi$
$757$	33.9225	1.23293	0.616466	+	0.787381i	$0.288564\pi$
$758$	0	0
$759$	23.9227	0.868340
$760$	0	0
$761$	−3.04413	−0.110350	−0.0551748	−	0.998477i	$-0.517572\pi$
$761$	−3.04413	−0.110350	−0.0551748	+	0.998477i	$0.517572\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	1.52798	0.0552441
$766$	0	0
$767$	−9.91201	−0.357902
$768$	0	0
$769$	−10.2605	−0.370003	−0.185002	−	0.982738i	$-0.559229\pi$
$769$	−10.2605	−0.370003	−0.185002	+	0.982738i	$0.559229\pi$
$770$	0	0
$771$	−30.3531	−1.09314
$772$	0	0
$773$	42.1375	1.51558	0.757791	−	0.652497i	$-0.226279\pi$
$773$	42.1375	1.51558	0.757791	+	0.652497i	$0.226279\pi$
$774$	0	0
$775$	−9.38735	−0.337204
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−7.03402	−0.252020
$780$	0	0
$781$	1.10466	0.0395280
$782$	0	0
$783$	−46.5630	−1.66403
$784$	0	0
$785$	−14.3071	−0.510643
$786$	0	0
$787$	6.73060	0.239920	0.119960	−	0.992779i	$-0.461723\pi$
$787$	6.73060	0.239920	0.119960	+	0.992779i	$0.461723\pi$
$788$	0	0
$789$	33.2676	1.18436
$790$	0	0
$791$	0	0
$792$	0	0
$793$	3.70564	0.131591
$794$	0	0
$795$	−6.95990	−0.246842
$796$	0	0
$797$	−38.8945	−1.37771	−0.688857	−	0.724897i	$-0.741887\pi$
$797$	−38.8945	−1.37771	−0.688857	+	0.724897i	$0.741887\pi$
$798$	0	0
$799$	0.292774	0.0103576
$800$	0	0
$801$	−16.8498	−0.595359
$802$	0	0
$803$	−26.8254	−0.946648
$804$	0	0
$805$	0	0
$806$	0	0
$807$	14.6376	0.515268
$808$	0	0
$809$	5.28280	0.185733	0.0928667	−	0.995679i	$-0.470397\pi$
$809$	5.28280	0.185733	0.0928667	+	0.995679i	$0.470397\pi$
$810$	0	0
$811$	55.9839	1.96586	0.982929	−	0.183983i	$-0.0588990\pi$
$811$	55.9839	1.96586	0.982929	+	0.183983i	$0.0588990\pi$
$812$	0	0
$813$	27.8786	0.977744
$814$	0	0
$815$	3.06008	0.107190
$816$	0	0
$817$	−44.8328	−1.56850
$818$	0	0
$819$	0	0
$820$	0	0
$821$	8.60213	0.300217	0.150108	−	0.988670i	$-0.452038\pi$
$821$	8.60213	0.300217	0.150108	+	0.988670i	$0.452038\pi$
$822$	0	0
$823$	4.71187	0.164246	0.0821228	−	0.996622i	$-0.473830\pi$
$823$	4.71187	0.164246	0.0821228	+	0.996622i	$0.473830\pi$
$824$	0	0
$825$	21.6472	0.753659
$826$	0	0
$827$	27.2953	0.949152	0.474576	−	0.880215i	$-0.342601\pi$
$827$	27.2953	0.949152	0.474576	+	0.880215i	$0.342601\pi$
$828$	0	0
$829$	3.32378	0.115440	0.0577199	−	0.998333i	$-0.481617\pi$
$829$	3.32378	0.115440	0.0577199	+	0.998333i	$0.481617\pi$
$830$	0	0
$831$	−11.5080	−0.399209
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−13.5522	−0.468993
$836$	0	0
$837$	−12.6027	−0.435613
$838$	0	0
$839$	1.30879	0.0451844	0.0225922	−	0.999745i	$-0.492808\pi$
$839$	1.30879	0.0451844	0.0225922	+	0.999745i	$0.492808\pi$
$840$	0	0
$841$	43.3788	1.49582
$842$	0	0
$843$	−19.5243	−0.672454
$844$	0	0
$845$	12.0252	0.413680
$846$	0	0
$847$	0	0
$848$	0	0
$849$	8.31153	0.285251
$850$	0	0
$851$	−29.4796	−1.01055
$852$	0	0
$853$	32.9494	1.12817	0.564084	−	0.825718i	$-0.309230\pi$
$853$	32.9494	1.12817	0.564084	+	0.825718i	$0.309230\pi$
$854$	0	0
$855$	−10.5327	−0.360209
$856$	0	0
$857$	12.5820	0.429792	0.214896	−	0.976637i	$-0.431059\pi$
$857$	12.5820	0.429792	0.214896	+	0.976637i	$0.431059\pi$
$858$	0	0
$859$	44.1748	1.50723	0.753613	−	0.657319i	$-0.228309\pi$
$859$	44.1748	1.50723	0.753613	+	0.657319i	$0.228309\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	7.62354	0.259508	0.129754	−	0.991546i	$-0.458581\pi$
$863$	7.62354	0.259508	0.129754	+	0.991546i	$0.458581\pi$
$864$	0	0
$865$	7.03228	0.239105
$866$	0	0
$867$	−19.1612	−0.650747
$868$	0	0
$869$	−7.93799	−0.269278
$870$	0	0
$871$	0.541959	0.0183636
$872$	0	0
$873$	−29.5443	−0.999924
$874$	0	0
$875$	0	0
$876$	0	0
$877$	40.5396	1.36892	0.684462	−	0.729049i	$-0.260037\pi$
$877$	40.5396	1.36892	0.684462	+	0.729049i	$0.260037\pi$
$878$	0	0
$879$	10.3794	0.350087
$880$	0	0
$881$	15.9206	0.536379	0.268189	−	0.963366i	$-0.413575\pi$
$881$	15.9206	0.536379	0.268189	+	0.963366i	$0.413575\pi$
$882$	0	0
$883$	17.8403	0.600374	0.300187	−	0.953880i	$-0.402951\pi$
$883$	17.8403	0.600374	0.300187	+	0.953880i	$0.402951\pi$
$884$	0	0
$885$	16.4221	0.552023
$886$	0	0
$887$	−17.0709	−0.573185	−0.286592	−	0.958053i	$-0.592523\pi$
$887$	−17.0709	−0.573185	−0.286592	+	0.958053i	$0.592523\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	8.38603	0.280943
$892$	0	0
$893$	−2.01816	−0.0675350
$894$	0	0
$895$	5.20232	0.173894
$896$	0	0
$897$	−3.76671	−0.125767
$898$	0	0
$899$	19.5900	0.653362
$900$	0	0
$901$	6.15606	0.205088
$902$	0	0
$903$	0	0
$904$	0	0
$905$	18.6864	0.621158
$906$	0	0
$907$	43.5612	1.44643	0.723213	−	0.690625i	$-0.242664\pi$
$907$	43.5612	1.44643	0.723213	+	0.690625i	$0.242664\pi$
$908$	0	0
$909$	−28.5139	−0.945746
$910$	0	0
$911$	−42.9086	−1.42162	−0.710812	−	0.703382i	$-0.751673\pi$
$911$	−42.9086	−1.42162	−0.710812	+	0.703382i	$0.751673\pi$
$912$	0	0
$913$	−1.21122	−0.0400856
$914$	0	0
$915$	−6.13945	−0.202964
$916$	0	0
$917$	0	0
$918$	0	0
$919$	31.5185	1.03970	0.519849	−	0.854258i	$-0.325988\pi$
$919$	31.5185	1.03970	0.519849	+	0.854258i	$0.325988\pi$
$920$	0	0
$921$	17.2901	0.569729
$922$	0	0
$923$	−0.173933	−0.00572507
$924$	0	0
$925$	−26.6755	−0.877086
$926$	0	0
$927$	−0.876409	−0.0287851
$928$	0	0
$929$	13.8605	0.454747	0.227374	−	0.973808i	$-0.426986\pi$
$929$	13.8605	0.454747	0.227374	+	0.973808i	$0.426986\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	26.8367	0.878593
$934$	0	0
$935$	4.33614	0.141807
$936$	0	0
$937$	−19.8346	−0.647967	−0.323983	−	0.946063i	$-0.605022\pi$
$937$	−19.8346	−0.647967	−0.323983	+	0.946063i	$0.605022\pi$
$938$	0	0
$939$	−27.4719	−0.896512
$940$	0	0
$941$	3.35579	0.109396	0.0546978	−	0.998503i	$-0.482580\pi$
$941$	3.35579	0.109396	0.0546978	+	0.998503i	$0.482580\pi$
$942$	0	0
$943$	−4.50530	−0.146713
$944$	0	0
$945$	0	0
$946$	0	0
$947$	28.4951	0.925965	0.462983	−	0.886367i	$-0.346779\pi$
$947$	28.4951	0.925965	0.462983	+	0.886367i	$0.346779\pi$
$948$	0	0
$949$	4.22374	0.137108
$950$	0	0
$951$	−27.8554	−0.903273
$952$	0	0
$953$	−22.8478	−0.740114	−0.370057	−	0.929009i	$-0.620662\pi$
$953$	−22.8478	−0.740114	−0.370057	+	0.929009i	$0.620662\pi$
$954$	0	0
$955$	12.6042	0.407863
$956$	0	0
$957$	−45.1744	−1.46028
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−25.6978	−0.828961
$962$	0	0
$963$	1.68002	0.0541379
$964$	0	0
$965$	−1.58198	−0.0509256
$966$	0	0
$967$	−43.2554	−1.39100	−0.695500	−	0.718526i	$-0.744817\pi$
$967$	−43.2554	−1.39100	−0.695500	+	0.718526i	$0.744817\pi$
$968$	0	0
$969$	−8.61804	−0.276851
$970$	0	0
$971$	7.07103	0.226920	0.113460	−	0.993543i	$-0.463807\pi$
$971$	7.07103	0.226920	0.113460	+	0.993543i	$0.463807\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−3.40842	−0.109157
$976$	0	0
$977$	2.10191	0.0672461	0.0336230	−	0.999435i	$-0.489295\pi$
$977$	2.10191	0.0672461	0.0336230	+	0.999435i	$0.489295\pi$
$978$	0	0
$979$	−47.8170	−1.52824
$980$	0	0
$981$	3.50643	0.111952
$982$	0	0
$983$	−9.55506	−0.304759	−0.152379	−	0.988322i	$-0.548694\pi$
$983$	−9.55506	−0.304759	−0.152379	+	0.988322i	$0.548694\pi$
$984$	0	0
$985$	−7.06027	−0.224959
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−28.7155	−0.913100
$990$	0	0
$991$	−22.5928	−0.717685	−0.358843	−	0.933398i	$-0.616829\pi$
$991$	−22.5928	−0.717685	−0.358843	+	0.933398i	$0.616829\pi$
$992$	0	0
$993$	15.7748	0.500597
$994$	0	0
$995$	5.68531	0.180237
$996$	0	0
$997$	−0.674933	−0.0213754	−0.0106877	−	0.999943i	$-0.503402\pi$
$997$	−0.674933	−0.0213754	−0.0106877	+	0.999943i	$0.503402\pi$
$998$	0	0
$999$	−35.8124	−1.13305

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.p.1.7	yes	10
7.6	odd	2		8036.2.a.o.1.4	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8036.2.a.o.1.4	✓	10	7.6	odd	2
8036.2.a.p.1.7	yes	10	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8036.a (trivial)

\(f(q)\)	\(=\)	\(q+1.20067 q^{3} -0.960855 q^{5} -1.55839 q^{9} +O(q^{10})\)	\(q+1.20067 q^{3} -0.960855 q^{5} -1.55839 q^{9} -4.42245 q^{11} +0.696329 q^{13} -1.15367 q^{15} +1.02043 q^{17} -7.03402 q^{19} -4.50530 q^{23} -4.07676 q^{25} -5.47312 q^{27} +8.50757 q^{29} +2.30265 q^{31} -5.30991 q^{33} +6.54332 q^{37} +0.836062 q^{39} +1.00000 q^{41} +6.37372 q^{43} +1.49739 q^{45} +0.286914 q^{47} +1.22520 q^{51} +6.03283 q^{53} +4.24934 q^{55} -8.44553 q^{57} -14.2347 q^{59} +5.32167 q^{61} -0.669072 q^{65} +0.778309 q^{67} -5.40938 q^{69} -0.249785 q^{71} +6.06573 q^{73} -4.89484 q^{75} +1.79493 q^{79} -1.89624 q^{81} +0.273880 q^{83} -0.980482 q^{85} +10.2148 q^{87} +10.8123 q^{89} +2.76472 q^{93} +6.75867 q^{95} +18.9582 q^{97} +6.89192 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{3} + 4 q^{5} + 10 q^{9}+O(q^{10}) \) 10 * q + 2 * q^3 + 4 * q^5 + 10 * q^9	\( 10 q + 2 q^{3} + 4 q^{5} + 10 q^{9} + 6 q^{13} - 4 q^{15} + 12 q^{17} + 8 q^{19} + 4 q^{23} + 16 q^{25} + 8 q^{27} + 2 q^{29} + 8 q^{31} - 6 q^{33} - 2 q^{37} - 2 q^{39} + 10 q^{41} - 2 q^{43} + 44 q^{45} - 14 q^{47} + 14 q^{51} + 8 q^{53} + 8 q^{55} - 10 q^{57} + 24 q^{59} + 14 q^{61} + 2 q^{65} - 8 q^{67} + 16 q^{69} + 10 q^{71} + 44 q^{73} - 50 q^{75} + 10 q^{79} - 14 q^{81} + 20 q^{83} + 8 q^{85} + 20 q^{87} + 6 q^{89} + 8 q^{93} + 4 q^{95} + 46 q^{97} + 2 q^{99}+O(q^{100}) \) 10 * q + 2 * q^3 + 4 * q^5 + 10 * q^9 + 6 * q^13 - 4 * q^15 + 12 * q^17 + 8 * q^19 + 4 * q^23 + 16 * q^25 + 8 * q^27 + 2 * q^29 + 8 * q^31 - 6 * q^33 - 2 * q^37 - 2 * q^39 + 10 * q^41 - 2 * q^43 + 44 * q^45 - 14 * q^47 + 14 * q^51 + 8 * q^53 + 8 * q^55 - 10 * q^57 + 24 * q^59 + 14 * q^61 + 2 * q^65 - 8 * q^67 + 16 * q^69 + 10 * q^71 + 44 * q^73 - 50 * q^75 + 10 * q^79 - 14 * q^81 + 20 * q^83 + 8 * q^85 + 20 * q^87 + 6 * q^89 + 8 * q^93 + 4 * q^95 + 46 * q^97 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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