LMFDB - Embedded newform 7605.2.a.y.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7605,2,Mod(1,7605)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7605, 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("7605.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7605 = 3^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7605.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:	$60.7262307372$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 195)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	7605.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.732051 q^{2} -1.46410 q^{4} -1.00000 q^{5} -4.46410 q^{7} -2.53590 q^{8} -0.732051 q^{10} -3.46410 q^{11} -3.26795 q^{14} +1.07180 q^{16} +6.73205 q^{17} +5.46410 q^{19} +1.46410 q^{20} -2.53590 q^{22} +0.535898 q^{23} +1.00000 q^{25} +6.53590 q^{28} +2.73205 q^{29} +3.19615 q^{31} +5.85641 q^{32} +4.92820 q^{34} +4.46410 q^{35} -4.00000 q^{37} +4.00000 q^{38} +2.53590 q^{40} -5.26795 q^{41} -0.267949 q^{43} +5.07180 q^{44} +0.392305 q^{46} -0.196152 q^{47} +12.9282 q^{49} +0.732051 q^{50} +6.92820 q^{53} +3.46410 q^{55} +11.3205 q^{56} +2.00000 q^{58} -7.26795 q^{59} +4.46410 q^{61} +2.33975 q^{62} +2.14359 q^{64} -12.4641 q^{67} -9.85641 q^{68} +3.26795 q^{70} -12.7321 q^{71} +15.3923 q^{73} -2.92820 q^{74} -8.00000 q^{76} +15.4641 q^{77} +1.92820 q^{79} -1.07180 q^{80} -3.85641 q^{82} -2.53590 q^{83} -6.73205 q^{85} -0.196152 q^{86} +8.78461 q^{88} -1.26795 q^{89} -0.784610 q^{92} -0.143594 q^{94} -5.46410 q^{95} +16.4641 q^{97} +9.46410 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 4 q^{4} - 2 q^{5} - 2 q^{7} - 12 q^{8} + 2 q^{10} - 10 q^{14} + 16 q^{16} + 10 q^{17} + 4 q^{19} - 4 q^{20} - 12 q^{22} + 8 q^{23} + 2 q^{25} + 20 q^{28} + 2 q^{29} - 4 q^{31} - 16 q^{32}+ \cdots + 12 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 4 * q^4 - 2 * q^5 - 2 * q^7 - 12 * q^8 + 2 * q^10 - 10 * q^14 + 16 * q^16 + 10 * q^17 + 4 * q^19 - 4 * q^20 - 12 * q^22 + 8 * q^23 + 2 * q^25 + 20 * q^28 + 2 * q^29 - 4 * q^31 - 16 * q^32 - 4 * q^34 + 2 * q^35 - 8 * q^37 + 8 * q^38 + 12 * q^40 - 14 * q^41 - 4 * q^43 + 24 * q^44 - 20 * q^46 + 10 * q^47 + 12 * q^49 - 2 * q^50 - 12 * q^56 + 4 * q^58 - 18 * q^59 + 2 * q^61 + 22 * q^62 + 32 * q^64 - 18 * q^67 + 8 * q^68 + 10 * q^70 - 22 * q^71 + 10 * q^73 + 8 * q^74 - 16 * q^76 + 24 * q^77 - 10 * q^79 - 16 * q^80 + 20 * q^82 - 12 * q^83 - 10 * q^85 + 10 * q^86 - 24 * q^88 - 6 * q^89 + 40 * q^92 - 28 * q^94 - 4 * q^95 + 26 * q^97 + 12 * q^98

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 $

$2$	0.732051	0.517638	0.258819	−	0.965926i	$-0.416667\pi$
$2$	0.732051	0.517638	0.258819	+	0.965926i	$0.416667\pi$
$3$	0	0
$3$	0	0
$4$	−1.46410	−0.732051
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−4.46410	−1.68727	−0.843636	−	0.536916i	$-0.819589\pi$
$7$	−4.46410	−1.68727	−0.843636	+	0.536916i	$0.819589\pi$
$8$	−2.53590	−0.896575
$9$	0	0
$10$	−0.732051	−0.231495
$11$	−3.46410	−1.04447	−0.522233	−	0.852803i	$-0.674901\pi$
$11$	−3.46410	−1.04447	−0.522233	+	0.852803i	$0.674901\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−3.26795	−0.873396
$15$	0	0
$16$	1.07180	0.267949
$17$	6.73205	1.63276	0.816381	−	0.577514i	$-0.195977\pi$
$17$	6.73205	1.63276	0.816381	+	0.577514i	$0.195977\pi$
$18$	0	0
$19$	5.46410	1.25355	0.626775	−	0.779200i	$-0.284374\pi$
$19$	5.46410	1.25355	0.626775	+	0.779200i	$0.284374\pi$
$20$	1.46410	0.327383
$21$	0	0
$22$	−2.53590	−0.540655
$23$	0.535898	0.111743	0.0558713	−	0.998438i	$-0.482206\pi$
$23$	0.535898	0.111743	0.0558713	+	0.998438i	$0.482206\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	6.53590	1.23517
$29$	2.73205	0.507329	0.253665	−	0.967292i	$-0.418364\pi$
$29$	2.73205	0.507329	0.253665	+	0.967292i	$0.418364\pi$
$30$	0	0
$31$	3.19615	0.574046	0.287023	−	0.957924i	$-0.407334\pi$
$31$	3.19615	0.574046	0.287023	+	0.957924i	$0.407334\pi$
$32$	5.85641	1.03528
$33$	0	0
$34$	4.92820	0.845180
$35$	4.46410	0.754571
$36$	0	0
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	4.00000	0.648886
$39$	0	0
$40$	2.53590	0.400961
$41$	−5.26795	−0.822715	−0.411358	−	0.911474i	$-0.634945\pi$
$41$	−5.26795	−0.822715	−0.411358	+	0.911474i	$0.634945\pi$
$42$	0	0
$43$	−0.267949	−0.0408619	−0.0204309	−	0.999791i	$-0.506504\pi$
$43$	−0.267949	−0.0408619	−0.0204309	+	0.999791i	$0.506504\pi$
$44$	5.07180	0.764602
$45$	0	0
$46$	0.392305	0.0578422
$47$	−0.196152	−0.0286118	−0.0143059	−	0.999898i	$-0.504554\pi$
$47$	−0.196152	−0.0286118	−0.0143059	+	0.999898i	$0.504554\pi$
$48$	0	0
$49$	12.9282	1.84689
$50$	0.732051	0.103528
$51$	0	0
$52$	0	0
$53$	6.92820	0.951662	0.475831	−	0.879537i	$-0.342147\pi$
$53$	6.92820	0.951662	0.475831	+	0.879537i	$0.342147\pi$
$54$	0	0
$55$	3.46410	0.467099
$56$	11.3205	1.51277
$57$	0	0
$58$	2.00000	0.262613
$59$	−7.26795	−0.946206	−0.473103	−	0.881007i	$-0.656866\pi$
$59$	−7.26795	−0.946206	−0.473103	+	0.881007i	$0.656866\pi$
$60$	0	0
$61$	4.46410	0.571570	0.285785	−	0.958294i	$-0.407746\pi$
$61$	4.46410	0.571570	0.285785	+	0.958294i	$0.407746\pi$
$62$	2.33975	0.297148
$63$	0	0
$64$	2.14359	0.267949
$65$	0	0
$66$	0	0
$67$	−12.4641	−1.52273	−0.761366	−	0.648322i	$-0.775471\pi$
$67$	−12.4641	−1.52273	−0.761366	+	0.648322i	$0.775471\pi$
$68$	−9.85641	−1.19526
$69$	0	0
$70$	3.26795	0.390595
$71$	−12.7321	−1.51102	−0.755508	−	0.655139i	$-0.772610\pi$
$71$	−12.7321	−1.51102	−0.755508	+	0.655139i	$0.772610\pi$
$72$	0	0
$73$	15.3923	1.80153	0.900767	−	0.434304i	$-0.143006\pi$
$73$	15.3923	1.80153	0.900767	+	0.434304i	$0.143006\pi$
$74$	−2.92820	−0.340397
$75$	0	0
$76$	−8.00000	−0.917663
$77$	15.4641	1.76230
$78$	0	0
$79$	1.92820	0.216940	0.108470	−	0.994100i	$-0.465405\pi$
$79$	1.92820	0.216940	0.108470	+	0.994100i	$0.465405\pi$
$80$	−1.07180	−0.119831
$81$	0	0
$82$	−3.85641	−0.425869
$83$	−2.53590	−0.278351	−0.139176	−	0.990268i	$-0.544445\pi$
$83$	−2.53590	−0.278351	−0.139176	+	0.990268i	$0.544445\pi$
$84$	0	0
$85$	−6.73205	−0.730193
$86$	−0.196152	−0.0211517
$87$	0	0
$88$	8.78461	0.936443
$89$	−1.26795	−0.134402	−0.0672012	−	0.997739i	$-0.521407\pi$
$89$	−1.26795	−0.134402	−0.0672012	+	0.997739i	$0.521407\pi$
$90$	0	0
$91$	0	0
$92$	−0.784610	−0.0818012
$93$	0	0
$94$	−0.143594	−0.0148105
$95$	−5.46410	−0.560605
$96$	0	0
$97$	16.4641	1.67168	0.835838	−	0.548976i	$-0.184982\pi$
$97$	16.4641	1.67168	0.835838	+	0.548976i	$0.184982\pi$
$98$	9.46410	0.956019
$99$	0	0
$100$	−1.46410	−0.146410
$101$	4.92820	0.490375	0.245187	−	0.969476i	$-0.421151\pi$
$101$	4.92820	0.490375	0.245187	+	0.969476i	$0.421151\pi$
$102$	0	0
$103$	−3.19615	−0.314926	−0.157463	−	0.987525i	$-0.550332\pi$
$103$	−3.19615	−0.314926	−0.157463	+	0.987525i	$0.550332\pi$
$104$	0	0
$105$	0	0
$106$	5.07180	0.492616
$107$	−17.1244	−1.65547	−0.827737	−	0.561116i	$-0.810372\pi$
$107$	−17.1244	−1.65547	−0.827737	+	0.561116i	$0.810372\pi$
$108$	0	0
$109$	8.26795	0.791926	0.395963	−	0.918266i	$-0.370411\pi$
$109$	8.26795	0.791926	0.395963	+	0.918266i	$0.370411\pi$
$110$	2.53590	0.241788
$111$	0	0
$112$	−4.78461	−0.452103
$113$	−19.4641	−1.83103	−0.915514	−	0.402285i	$-0.868216\pi$
$113$	−19.4641	−1.83103	−0.915514	+	0.402285i	$0.868216\pi$
$114$	0	0
$115$	−0.535898	−0.0499728
$116$	−4.00000	−0.371391
$117$	0	0
$118$	−5.32051	−0.489792
$119$	−30.0526	−2.75491
$120$	0	0
$121$	1.00000	0.0909091
$122$	3.26795	0.295866
$123$	0	0
$124$	−4.67949	−0.420231
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	14.6603	1.30089	0.650444	−	0.759555i	$-0.274583\pi$
$127$	14.6603	1.30089	0.650444	+	0.759555i	$0.274583\pi$
$128$	−10.1436	−0.896575
$129$	0	0
$130$	0	0
$131$	−1.26795	−0.110781	−0.0553906	−	0.998465i	$-0.517640\pi$
$131$	−1.26795	−0.110781	−0.0553906	+	0.998465i	$0.517640\pi$
$132$	0	0
$133$	−24.3923	−2.11508
$134$	−9.12436	−0.788224
$135$	0	0
$136$	−17.0718	−1.46389
$137$	9.12436	0.779546	0.389773	−	0.920911i	$-0.372553\pi$
$137$	9.12436	0.779546	0.389773	+	0.920911i	$0.372553\pi$
$138$	0	0
$139$	−17.9282	−1.52065	−0.760325	−	0.649543i	$-0.774960\pi$
$139$	−17.9282	−1.52065	−0.760325	+	0.649543i	$0.774960\pi$
$140$	−6.53590	−0.552384
$141$	0	0
$142$	−9.32051	−0.782160
$143$	0	0
$144$	0	0
$145$	−2.73205	−0.226884
$146$	11.2679	0.932542
$147$	0	0
$148$	5.85641	0.481394
$149$	−4.00000	−0.327693	−0.163846	−	0.986486i	$-0.552390\pi$
$149$	−4.00000	−0.327693	−0.163846	+	0.986486i	$0.552390\pi$
$150$	0	0
$151$	17.8564	1.45313	0.726567	−	0.687096i	$-0.241115\pi$
$151$	17.8564	1.45313	0.726567	+	0.687096i	$0.241115\pi$
$152$	−13.8564	−1.12390
$153$	0	0
$154$	11.3205	0.912233
$155$	−3.19615	−0.256721
$156$	0	0
$157$	9.73205	0.776702	0.388351	−	0.921511i	$-0.373045\pi$
$157$	9.73205	0.776702	0.388351	+	0.921511i	$0.373045\pi$
$158$	1.41154	0.112296
$159$	0	0
$160$	−5.85641	−0.462990
$161$	−2.39230	−0.188540
$162$	0	0
$163$	5.53590	0.433605	0.216803	−	0.976215i	$-0.430437\pi$
$163$	5.53590	0.433605	0.216803	+	0.976215i	$0.430437\pi$
$164$	7.71281	0.602270
$165$	0	0
$166$	−1.85641	−0.144085
$167$	−17.4641	−1.35141	−0.675706	−	0.737171i	$-0.736161\pi$
$167$	−17.4641	−1.35141	−0.675706	+	0.737171i	$0.736161\pi$
$168$	0	0
$169$	0	0
$170$	−4.92820	−0.377976
$171$	0	0
$172$	0.392305	0.0299130
$173$	−3.26795	−0.248458	−0.124229	−	0.992254i	$-0.539646\pi$
$173$	−3.26795	−0.248458	−0.124229	+	0.992254i	$0.539646\pi$
$174$	0	0
$175$	−4.46410	−0.337454
$176$	−3.71281	−0.279864
$177$	0	0
$178$	−0.928203	−0.0695718
$179$	−9.12436	−0.681986	−0.340993	−	0.940066i	$-0.610763\pi$
$179$	−9.12436	−0.681986	−0.340993	+	0.940066i	$0.610763\pi$
$180$	0	0
$181$	−14.5359	−1.08044	−0.540222	−	0.841522i	$-0.681660\pi$
$181$	−14.5359	−1.08044	−0.540222	+	0.841522i	$0.681660\pi$
$182$	0	0
$183$	0	0
$184$	−1.35898	−0.100186
$185$	4.00000	0.294086
$186$	0	0
$187$	−23.3205	−1.70536
$188$	0.287187	0.0209453
$189$	0	0
$190$	−4.00000	−0.290191
$191$	27.5167	1.99104	0.995518	−	0.0945740i	$-0.0301489\pi$
$191$	27.5167	1.99104	0.995518	+	0.0945740i	$0.0301489\pi$
$192$	0	0
$193$	−6.07180	−0.437057	−0.218529	−	0.975831i	$-0.570126\pi$
$193$	−6.07180	−0.437057	−0.218529	+	0.975831i	$0.570126\pi$
$194$	12.0526	0.865323
$195$	0	0
$196$	−18.9282	−1.35201
$197$	2.92820	0.208626	0.104313	−	0.994545i	$-0.466736\pi$
$197$	2.92820	0.208626	0.104313	+	0.994545i	$0.466736\pi$
$198$	0	0
$199$	−15.0000	−1.06332	−0.531661	−	0.846957i	$-0.678432\pi$
$199$	−15.0000	−1.06332	−0.531661	+	0.846957i	$0.678432\pi$
$200$	−2.53590	−0.179315
$201$	0	0
$202$	3.60770	0.253837
$203$	−12.1962	−0.856002
$204$	0	0
$205$	5.26795	0.367930
$206$	−2.33975	−0.163018
$207$	0	0
$208$	0	0
$209$	−18.9282	−1.30929
$210$	0	0
$211$	11.5359	0.794164	0.397082	−	0.917783i	$-0.370023\pi$
$211$	11.5359	0.794164	0.397082	+	0.917783i	$0.370023\pi$
$212$	−10.1436	−0.696665
$213$	0	0
$214$	−12.5359	−0.856936
$215$	0.267949	0.0182740
$216$	0	0
$217$	−14.2679	−0.968572
$218$	6.05256	0.409931
$219$	0	0
$220$	−5.07180	−0.341940
$221$	0	0
$222$	0	0
$223$	−11.8564	−0.793964	−0.396982	−	0.917826i	$-0.629942\pi$
$223$	−11.8564	−0.793964	−0.396982	+	0.917826i	$0.629942\pi$
$224$	−26.1436	−1.74679
$225$	0	0
$226$	−14.2487	−0.947810
$227$	11.6603	0.773918	0.386959	−	0.922097i	$-0.373525\pi$
$227$	11.6603	0.773918	0.386959	+	0.922097i	$0.373525\pi$
$228$	0	0
$229$	−18.3923	−1.21540	−0.607699	−	0.794168i	$-0.707907\pi$
$229$	−18.3923	−1.21540	−0.607699	+	0.794168i	$0.707907\pi$
$230$	−0.392305	−0.0258678
$231$	0	0
$232$	−6.92820	−0.454859
$233$	−23.8564	−1.56289	−0.781443	−	0.623977i	$-0.785516\pi$
$233$	−23.8564	−1.56289	−0.781443	+	0.623977i	$0.785516\pi$
$234$	0	0
$235$	0.196152	0.0127956
$236$	10.6410	0.692671
$237$	0	0
$238$	−22.0000	−1.42605
$239$	−5.46410	−0.353443	−0.176722	−	0.984261i	$-0.556549\pi$
$239$	−5.46410	−0.353443	−0.176722	+	0.984261i	$0.556549\pi$
$240$	0	0
$241$	1.60770	0.103561	0.0517804	−	0.998658i	$-0.483510\pi$
$241$	1.60770	0.103561	0.0517804	+	0.998658i	$0.483510\pi$
$242$	0.732051	0.0470580
$243$	0	0
$244$	−6.53590	−0.418418
$245$	−12.9282	−0.825953
$246$	0	0
$247$	0	0
$248$	−8.10512	−0.514675
$249$	0	0
$250$	−0.732051	−0.0462990
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	−1.85641	−0.116711
$254$	10.7321	0.673389
$255$	0	0
$256$	−11.7128	−0.732051
$257$	−10.7321	−0.669447	−0.334723	−	0.942316i	$-0.608643\pi$
$257$	−10.7321	−0.669447	−0.334723	+	0.942316i	$0.608643\pi$
$258$	0	0
$259$	17.8564	1.10954
$260$	0	0
$261$	0	0
$262$	−0.928203	−0.0573446
$263$	6.33975	0.390925	0.195463	−	0.980711i	$-0.437379\pi$
$263$	6.33975	0.390925	0.195463	+	0.980711i	$0.437379\pi$
$264$	0	0
$265$	−6.92820	−0.425596
$266$	−17.8564	−1.09485
$267$	0	0
$268$	18.2487	1.11472
$269$	−15.6603	−0.954823	−0.477411	−	0.878680i	$-0.658425\pi$
$269$	−15.6603	−0.954823	−0.477411	+	0.878680i	$0.658425\pi$
$270$	0	0
$271$	3.73205	0.226706	0.113353	−	0.993555i	$-0.463841\pi$
$271$	3.73205	0.226706	0.113353	+	0.993555i	$0.463841\pi$
$272$	7.21539	0.437497
$273$	0	0
$274$	6.67949	0.403523
$275$	−3.46410	−0.208893
$276$	0	0
$277$	20.2487	1.21663	0.608314	−	0.793697i	$-0.291846\pi$
$277$	20.2487	1.21663	0.608314	+	0.793697i	$0.291846\pi$
$278$	−13.1244	−0.787147
$279$	0	0
$280$	−11.3205	−0.676530
$281$	−21.1244	−1.26017	−0.630087	−	0.776525i	$-0.716981\pi$
$281$	−21.1244	−1.26017	−0.630087	+	0.776525i	$0.716981\pi$
$282$	0	0
$283$	−0.267949	−0.0159279	−0.00796396	−	0.999968i	$-0.502535\pi$
$283$	−0.267949	−0.0159279	−0.00796396	+	0.999968i	$0.502535\pi$
$284$	18.6410	1.10614
$285$	0	0
$286$	0	0
$287$	23.5167	1.38814
$288$	0	0
$289$	28.3205	1.66591
$290$	−2.00000	−0.117444
$291$	0	0
$292$	−22.5359	−1.31881
$293$	−24.5885	−1.43647	−0.718237	−	0.695799i	$-0.755050\pi$
$293$	−24.5885	−1.43647	−0.718237	+	0.695799i	$0.755050\pi$
$294$	0	0
$295$	7.26795	0.423156
$296$	10.1436	0.589584
$297$	0	0
$298$	−2.92820	−0.169626
$299$	0	0
$300$	0	0
$301$	1.19615	0.0689451
$302$	13.0718	0.752197
$303$	0	0
$304$	5.85641	0.335888
$305$	−4.46410	−0.255614
$306$	0	0
$307$	12.0718	0.688974	0.344487	−	0.938791i	$-0.388053\pi$
$307$	12.0718	0.688974	0.344487	+	0.938791i	$0.388053\pi$
$308$	−22.6410	−1.29009
$309$	0	0
$310$	−2.33975	−0.132889
$311$	−10.1962	−0.578171	−0.289085	−	0.957303i	$-0.593351\pi$
$311$	−10.1962	−0.578171	−0.289085	+	0.957303i	$0.593351\pi$
$312$	0	0
$313$	−10.8038	−0.610670	−0.305335	−	0.952245i	$-0.598768\pi$
$313$	−10.8038	−0.610670	−0.305335	+	0.952245i	$0.598768\pi$
$314$	7.12436	0.402051
$315$	0	0
$316$	−2.82309	−0.158811
$317$	17.4641	0.980882	0.490441	−	0.871474i	$-0.336836\pi$
$317$	17.4641	0.980882	0.490441	+	0.871474i	$0.336836\pi$
$318$	0	0
$319$	−9.46410	−0.529888
$320$	−2.14359	−0.119831
$321$	0	0
$322$	−1.75129	−0.0975955
$323$	36.7846	2.04675
$324$	0	0
$325$	0	0
$326$	4.05256	0.224450
$327$	0	0
$328$	13.3590	0.737626
$329$	0.875644	0.0482758
$330$	0	0
$331$	−8.12436	−0.446555	−0.223277	−	0.974755i	$-0.571676\pi$
$331$	−8.12436	−0.446555	−0.223277	+	0.974755i	$0.571676\pi$
$332$	3.71281	0.203767
$333$	0	0
$334$	−12.7846	−0.699543
$335$	12.4641	0.680987
$336$	0	0
$337$	31.0526	1.69154	0.845770	−	0.533547i	$-0.179141\pi$
$337$	31.0526	1.69154	0.845770	+	0.533547i	$0.179141\pi$
$338$	0	0
$339$	0	0
$340$	9.85641	0.534539
$341$	−11.0718	−0.599571
$342$	0	0
$343$	−26.4641	−1.42893
$344$	0.679492	0.0366357
$345$	0	0
$346$	−2.39230	−0.128611
$347$	−6.53590	−0.350865	−0.175433	−	0.984491i	$-0.556132\pi$
$347$	−6.53590	−0.350865	−0.175433	+	0.984491i	$0.556132\pi$
$348$	0	0
$349$	−29.9808	−1.60483	−0.802417	−	0.596764i	$-0.796453\pi$
$349$	−29.9808	−1.60483	−0.802417	+	0.596764i	$0.796453\pi$
$350$	−3.26795	−0.174679
$351$	0	0
$352$	−20.2872	−1.08131
$353$	36.9282	1.96549	0.982745	−	0.184966i	$-0.0592174\pi$
$353$	36.9282	1.96549	0.982745	+	0.184966i	$0.0592174\pi$
$354$	0	0
$355$	12.7321	0.675747
$356$	1.85641	0.0983893
$357$	0	0
$358$	−6.67949	−0.353022
$359$	33.6603	1.77652	0.888260	−	0.459341i	$-0.151914\pi$
$359$	33.6603	1.77652	0.888260	+	0.459341i	$0.151914\pi$
$360$	0	0
$361$	10.8564	0.571390
$362$	−10.6410	−0.559279
$363$	0	0
$364$	0	0
$365$	−15.3923	−0.805670
$366$	0	0
$367$	−18.1244	−0.946084	−0.473042	−	0.881040i	$-0.656844\pi$
$367$	−18.1244	−0.946084	−0.473042	+	0.881040i	$0.656844\pi$
$368$	0.574374	0.0299413
$369$	0	0
$370$	2.92820	0.152230
$371$	−30.9282	−1.60571
$372$	0	0
$373$	−1.87564	−0.0971172	−0.0485586	−	0.998820i	$-0.515463\pi$
$373$	−1.87564	−0.0971172	−0.0485586	+	0.998820i	$0.515463\pi$
$374$	−17.0718	−0.882762
$375$	0	0
$376$	0.497423	0.0256526
$377$	0	0
$378$	0	0
$379$	−19.7321	−1.01357	−0.506784	−	0.862073i	$-0.669166\pi$
$379$	−19.7321	−1.01357	−0.506784	+	0.862073i	$0.669166\pi$
$380$	8.00000	0.410391
$381$	0	0
$382$	20.1436	1.03064
$383$	8.33975	0.426141	0.213071	−	0.977037i	$-0.431654\pi$
$383$	8.33975	0.426141	0.213071	+	0.977037i	$0.431654\pi$
$384$	0	0
$385$	−15.4641	−0.788124
$386$	−4.44486	−0.226238
$387$	0	0
$388$	−24.1051	−1.22375
$389$	−17.4641	−0.885465	−0.442733	−	0.896654i	$-0.645991\pi$
$389$	−17.4641	−0.885465	−0.442733	+	0.896654i	$0.645991\pi$
$390$	0	0
$391$	3.60770	0.182449
$392$	−32.7846	−1.65587
$393$	0	0
$394$	2.14359	0.107993
$395$	−1.92820	−0.0970184
$396$	0	0
$397$	24.3205	1.22061	0.610306	−	0.792166i	$-0.291047\pi$
$397$	24.3205	1.22061	0.610306	+	0.792166i	$0.291047\pi$
$398$	−10.9808	−0.550416
$399$	0	0
$400$	1.07180	0.0535898
$401$	−15.3205	−0.765070	−0.382535	−	0.923941i	$-0.624949\pi$
$401$	−15.3205	−0.765070	−0.382535	+	0.923941i	$0.624949\pi$
$402$	0	0
$403$	0	0
$404$	−7.21539	−0.358979
$405$	0	0
$406$	−8.92820	−0.443099
$407$	13.8564	0.686837
$408$	0	0
$409$	−2.66025	−0.131541	−0.0657705	−	0.997835i	$-0.520951\pi$
$409$	−2.66025	−0.131541	−0.0657705	+	0.997835i	$0.520951\pi$
$410$	3.85641	0.190454
$411$	0	0
$412$	4.67949	0.230542
$413$	32.4449	1.59651
$414$	0	0
$415$	2.53590	0.124482
$416$	0	0
$417$	0	0
$418$	−13.8564	−0.677739
$419$	−20.4449	−0.998797	−0.499398	−	0.866372i	$-0.666446\pi$
$419$	−20.4449	−0.998797	−0.499398	+	0.866372i	$0.666446\pi$
$420$	0	0
$421$	23.5885	1.14963	0.574816	−	0.818283i	$-0.305074\pi$
$421$	23.5885	1.14963	0.574816	+	0.818283i	$0.305074\pi$
$422$	8.44486	0.411090
$423$	0	0
$424$	−17.5692	−0.853237
$425$	6.73205	0.326552
$426$	0	0
$427$	−19.9282	−0.964393
$428$	25.0718	1.21189
$429$	0	0
$430$	0.196152	0.00945931
$431$	23.3205	1.12331	0.561655	−	0.827372i	$-0.310165\pi$
$431$	23.3205	1.12331	0.561655	+	0.827372i	$0.310165\pi$
$432$	0	0
$433$	4.80385	0.230858	0.115429	−	0.993316i	$-0.463176\pi$
$433$	4.80385	0.230858	0.115429	+	0.993316i	$0.463176\pi$
$434$	−10.4449	−0.501370
$435$	0	0
$436$	−12.1051	−0.579730
$437$	2.92820	0.140075
$438$	0	0
$439$	15.3923	0.734635	0.367317	−	0.930096i	$-0.380276\pi$
$439$	15.3923	0.734635	0.367317	+	0.930096i	$0.380276\pi$
$440$	−8.78461	−0.418790
$441$	0	0
$442$	0	0
$443$	−3.12436	−0.148443	−0.0742213	−	0.997242i	$-0.523647\pi$
$443$	−3.12436	−0.148443	−0.0742213	+	0.997242i	$0.523647\pi$
$444$	0	0
$445$	1.26795	0.0601066
$446$	−8.67949	−0.410986
$447$	0	0
$448$	−9.56922	−0.452103
$449$	−11.0718	−0.522510	−0.261255	−	0.965270i	$-0.584136\pi$
$449$	−11.0718	−0.522510	−0.261255	+	0.965270i	$0.584136\pi$
$450$	0	0
$451$	18.2487	0.859298
$452$	28.4974	1.34041
$453$	0	0
$454$	8.53590	0.400610
$455$	0	0
$456$	0	0
$457$	−5.39230	−0.252241	−0.126121	−	0.992015i	$-0.540253\pi$
$457$	−5.39230	−0.252241	−0.126121	+	0.992015i	$0.540253\pi$
$458$	−13.4641	−0.629136
$459$	0	0
$460$	0.784610	0.0365826
$461$	11.5167	0.536384	0.268192	−	0.963365i	$-0.413574\pi$
$461$	11.5167	0.536384	0.268192	+	0.963365i	$0.413574\pi$
$462$	0	0
$463$	−1.78461	−0.0829378	−0.0414689	−	0.999140i	$-0.513204\pi$
$463$	−1.78461	−0.0829378	−0.0414689	+	0.999140i	$0.513204\pi$
$464$	2.92820	0.135938
$465$	0	0
$466$	−17.4641	−0.809009
$467$	23.6603	1.09487	0.547433	−	0.836850i	$-0.315605\pi$
$467$	23.6603	1.09487	0.547433	+	0.836850i	$0.315605\pi$
$468$	0	0
$469$	55.6410	2.56926
$470$	0.143594	0.00662348
$471$	0	0
$472$	18.4308	0.848345
$473$	0.928203	0.0426788
$474$	0	0
$475$	5.46410	0.250710
$476$	44.0000	2.01674
$477$	0	0
$478$	−4.00000	−0.182956
$479$	−35.9090	−1.64072	−0.820361	−	0.571846i	$-0.806228\pi$
$479$	−35.9090	−1.64072	−0.820361	+	0.571846i	$0.806228\pi$
$480$	0	0
$481$	0	0
$482$	1.17691	0.0536070
$483$	0	0
$484$	−1.46410	−0.0665501
$485$	−16.4641	−0.747596
$486$	0	0
$487$	−20.7846	−0.941841	−0.470920	−	0.882176i	$-0.656078\pi$
$487$	−20.7846	−0.941841	−0.470920	+	0.882176i	$0.656078\pi$
$488$	−11.3205	−0.512455
$489$	0	0
$490$	−9.46410	−0.427545
$491$	−7.12436	−0.321518	−0.160759	−	0.986994i	$-0.551394\pi$
$491$	−7.12436	−0.321518	−0.160759	+	0.986994i	$0.551394\pi$
$492$	0	0
$493$	18.3923	0.828348
$494$	0	0
$495$	0	0
$496$	3.42563	0.153815
$497$	56.8372	2.54950
$498$	0	0
$499$	0.392305	0.0175620	0.00878099	−	0.999961i	$-0.497205\pi$
$499$	0.392305	0.0175620	0.00878099	+	0.999961i	$0.497205\pi$
$500$	1.46410	0.0654766
$501$	0	0
$502$	0	0
$503$	30.7846	1.37262	0.686309	−	0.727310i	$-0.259230\pi$
$503$	30.7846	1.37262	0.686309	+	0.727310i	$0.259230\pi$
$504$	0	0
$505$	−4.92820	−0.219302
$506$	−1.35898	−0.0604142
$507$	0	0
$508$	−21.4641	−0.952316
$509$	−12.9282	−0.573033	−0.286516	−	0.958075i	$-0.592497\pi$
$509$	−12.9282	−0.573033	−0.286516	+	0.958075i	$0.592497\pi$
$510$	0	0
$511$	−68.7128	−3.03968
$512$	11.7128	0.517638
$513$	0	0
$514$	−7.85641	−0.346531
$515$	3.19615	0.140839
$516$	0	0
$517$	0.679492	0.0298840
$518$	13.0718	0.574342
$519$	0	0
$520$	0	0
$521$	12.7321	0.557801	0.278901	−	0.960320i	$-0.410030\pi$
$521$	12.7321	0.557801	0.278901	+	0.960320i	$0.410030\pi$
$522$	0	0
$523$	−37.4641	−1.63819	−0.819095	−	0.573657i	$-0.805524\pi$
$523$	−37.4641	−1.63819	−0.819095	+	0.573657i	$0.805524\pi$
$524$	1.85641	0.0810975
$525$	0	0
$526$	4.64102	0.202358
$527$	21.5167	0.937280
$528$	0	0
$529$	−22.7128	−0.987514
$530$	−5.07180	−0.220305
$531$	0	0
$532$	35.7128	1.54835
$533$	0	0
$534$	0	0
$535$	17.1244	0.740350
$536$	31.6077	1.36524
$537$	0	0
$538$	−11.4641	−0.494253
$539$	−44.7846	−1.92901
$540$	0	0
$541$	17.5885	0.756187	0.378093	−	0.925767i	$-0.376580\pi$
$541$	17.5885	0.756187	0.378093	+	0.925767i	$0.376580\pi$
$542$	2.73205	0.117352
$543$	0	0
$544$	39.4256	1.69036
$545$	−8.26795	−0.354160
$546$	0	0
$547$	−24.6603	−1.05440	−0.527198	−	0.849742i	$-0.676757\pi$
$547$	−24.6603	−1.05440	−0.527198	+	0.849742i	$0.676757\pi$
$548$	−13.3590	−0.570668
$549$	0	0
$550$	−2.53590	−0.108131
$551$	14.9282	0.635963
$552$	0	0
$553$	−8.60770	−0.366036
$554$	14.8231	0.629773
$555$	0	0
$556$	26.2487	1.11319
$557$	−21.4641	−0.909463	−0.454732	−	0.890629i	$-0.650265\pi$
$557$	−21.4641	−0.909463	−0.454732	+	0.890629i	$0.650265\pi$
$558$	0	0
$559$	0	0
$560$	4.78461	0.202187
$561$	0	0
$562$	−15.4641	−0.652314
$563$	3.46410	0.145994	0.0729972	−	0.997332i	$-0.476744\pi$
$563$	3.46410	0.145994	0.0729972	+	0.997332i	$0.476744\pi$
$564$	0	0
$565$	19.4641	0.818861
$566$	−0.196152	−0.00824490
$567$	0	0
$568$	32.2872	1.35474
$569$	44.1962	1.85280	0.926400	−	0.376542i	$-0.122887\pi$
$569$	44.1962	1.85280	0.926400	+	0.376542i	$0.122887\pi$
$570$	0	0
$571$	−3.85641	−0.161386	−0.0806928	−	0.996739i	$-0.525713\pi$
$571$	−3.85641	−0.161386	−0.0806928	+	0.996739i	$0.525713\pi$
$572$	0	0
$573$	0	0
$574$	17.2154	0.718557
$575$	0.535898	0.0223485
$576$	0	0
$577$	−11.7128	−0.487611	−0.243805	−	0.969824i	$-0.578396\pi$
$577$	−11.7128	−0.487611	−0.243805	+	0.969824i	$0.578396\pi$
$578$	20.7321	0.862340
$579$	0	0
$580$	4.00000	0.166091
$581$	11.3205	0.469654
$582$	0	0
$583$	−24.0000	−0.993978
$584$	−39.0333	−1.61521
$585$	0	0
$586$	−18.0000	−0.743573
$587$	25.6603	1.05911	0.529556	−	0.848275i	$-0.322359\pi$
$587$	25.6603	1.05911	0.529556	+	0.848275i	$0.322359\pi$
$588$	0	0
$589$	17.4641	0.719596
$590$	5.32051	0.219042
$591$	0	0
$592$	−4.28719	−0.176202
$593$	−21.3205	−0.875528	−0.437764	−	0.899090i	$-0.644230\pi$
$593$	−21.3205	−0.875528	−0.437764	+	0.899090i	$0.644230\pi$
$594$	0	0
$595$	30.0526	1.23203
$596$	5.85641	0.239888
$597$	0	0
$598$	0	0
$599$	23.1244	0.944836	0.472418	−	0.881375i	$-0.343381\pi$
$599$	23.1244	0.944836	0.472418	+	0.881375i	$0.343381\pi$
$600$	0	0
$601$	−22.2487	−0.907544	−0.453772	−	0.891118i	$-0.649922\pi$
$601$	−22.2487	−0.907544	−0.453772	+	0.891118i	$0.649922\pi$
$602$	0.875644	0.0356886
$603$	0	0
$604$	−26.1436	−1.06377
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	17.6077	0.714674	0.357337	−	0.933975i	$-0.383685\pi$
$607$	17.6077	0.714674	0.357337	+	0.933975i	$0.383685\pi$
$608$	32.0000	1.29777
$609$	0	0
$610$	−3.26795	−0.132315
$611$	0	0
$612$	0	0
$613$	−2.21539	−0.0894788	−0.0447394	−	0.998999i	$-0.514246\pi$
$613$	−2.21539	−0.0894788	−0.0447394	+	0.998999i	$0.514246\pi$
$614$	8.83717	0.356639
$615$	0	0
$616$	−39.2154	−1.58003
$617$	−24.6410	−0.992010	−0.496005	−	0.868320i	$-0.665200\pi$
$617$	−24.6410	−0.992010	−0.496005	+	0.868320i	$0.665200\pi$
$618$	0	0
$619$	−11.5885	−0.465779	−0.232890	−	0.972503i	$-0.574818\pi$
$619$	−11.5885	−0.465779	−0.232890	+	0.972503i	$0.574818\pi$
$620$	4.67949	0.187933
$621$	0	0
$622$	−7.46410	−0.299283
$623$	5.66025	0.226773
$624$	0	0
$625$	1.00000	0.0400000
$626$	−7.90897	−0.316106
$627$	0	0
$628$	−14.2487	−0.568585
$629$	−26.9282	−1.07370
$630$	0	0
$631$	−10.4115	−0.414477	−0.207238	−	0.978290i	$-0.566448\pi$
$631$	−10.4115	−0.414477	−0.207238	+	0.978290i	$0.566448\pi$
$632$	−4.88973	−0.194503
$633$	0	0
$634$	12.7846	0.507742
$635$	−14.6603	−0.581774
$636$	0	0
$637$	0	0
$638$	−6.92820	−0.274290
$639$	0	0
$640$	10.1436	0.400961
$641$	−38.7846	−1.53190	−0.765950	−	0.642900i	$-0.777731\pi$
$641$	−38.7846	−1.53190	−0.765950	+	0.642900i	$0.777731\pi$
$642$	0	0
$643$	3.78461	0.149250	0.0746252	−	0.997212i	$-0.476224\pi$
$643$	3.78461	0.149250	0.0746252	+	0.997212i	$0.476224\pi$
$644$	3.50258	0.138021
$645$	0	0
$646$	26.9282	1.05948
$647$	−2.00000	−0.0786281	−0.0393141	−	0.999227i	$-0.512517\pi$
$647$	−2.00000	−0.0786281	−0.0393141	+	0.999227i	$0.512517\pi$
$648$	0	0
$649$	25.1769	0.988280
$650$	0	0
$651$	0	0
$652$	−8.10512	−0.317421
$653$	0.483340	0.0189145	0.00945727	−	0.999955i	$-0.496990\pi$
$653$	0.483340	0.0189145	0.00945727	+	0.999955i	$0.496990\pi$
$654$	0	0
$655$	1.26795	0.0495429
$656$	−5.64617	−0.220446
$657$	0	0
$658$	0.641016	0.0249894
$659$	37.1769	1.44821	0.724103	−	0.689691i	$-0.242254\pi$
$659$	37.1769	1.44821	0.724103	+	0.689691i	$0.242254\pi$
$660$	0	0
$661$	−21.0526	−0.818850	−0.409425	−	0.912344i	$-0.634271\pi$
$661$	−21.0526	−0.818850	−0.409425	+	0.912344i	$0.634271\pi$
$662$	−5.94744	−0.231154
$663$	0	0
$664$	6.43078	0.249563
$665$	24.3923	0.945893
$666$	0	0
$667$	1.46410	0.0566902
$668$	25.5692	0.989303
$669$	0	0
$670$	9.12436	0.352505
$671$	−15.4641	−0.596985
$672$	0	0
$673$	−18.6603	−0.719300	−0.359650	−	0.933087i	$-0.617104\pi$
$673$	−18.6603	−0.719300	−0.359650	+	0.933087i	$0.617104\pi$
$674$	22.7321	0.875606
$675$	0	0
$676$	0	0
$677$	−32.7846	−1.26001	−0.630007	−	0.776589i	$-0.716948\pi$
$677$	−32.7846	−1.26001	−0.630007	+	0.776589i	$0.716948\pi$
$678$	0	0
$679$	−73.4974	−2.82057
$680$	17.0718	0.654674
$681$	0	0
$682$	−8.10512	−0.310361
$683$	−41.2679	−1.57907	−0.789537	−	0.613703i	$-0.789679\pi$
$683$	−41.2679	−1.57907	−0.789537	+	0.613703i	$0.789679\pi$
$684$	0	0
$685$	−9.12436	−0.348624
$686$	−19.3731	−0.739667
$687$	0	0
$688$	−0.287187	−0.0109489
$689$	0	0
$690$	0	0
$691$	−43.5885	−1.65818	−0.829092	−	0.559113i	$-0.811142\pi$
$691$	−43.5885	−1.65818	−0.829092	+	0.559113i	$0.811142\pi$
$692$	4.78461	0.181884
$693$	0	0
$694$	−4.78461	−0.181621
$695$	17.9282	0.680056
$696$	0	0
$697$	−35.4641	−1.34330
$698$	−21.9474	−0.830723
$699$	0	0
$700$	6.53590	0.247034
$701$	8.78461	0.331790	0.165895	−	0.986143i	$-0.446949\pi$
$701$	8.78461	0.331790	0.165895	+	0.986143i	$0.446949\pi$
$702$	0	0
$703$	−21.8564	−0.824330
$704$	−7.42563	−0.279864
$705$	0	0
$706$	27.0333	1.01741
$707$	−22.0000	−0.827395
$708$	0	0
$709$	−30.6603	−1.15147	−0.575735	−	0.817636i	$-0.695284\pi$
$709$	−30.6603	−1.15147	−0.575735	+	0.817636i	$0.695284\pi$
$710$	9.32051	0.349792
$711$	0	0
$712$	3.21539	0.120502
$713$	1.71281	0.0641453
$714$	0	0
$715$	0	0
$716$	13.3590	0.499249
$717$	0	0
$718$	24.6410	0.919595
$719$	−34.7321	−1.29529	−0.647643	−	0.761944i	$-0.724245\pi$
$719$	−34.7321	−1.29529	−0.647643	+	0.761944i	$0.724245\pi$
$720$	0	0
$721$	14.2679	0.531366
$722$	7.94744	0.295773
$723$	0	0
$724$	21.2820	0.790941
$725$	2.73205	0.101466
$726$	0	0
$727$	−6.26795	−0.232465	−0.116233	−	0.993222i	$-0.537082\pi$
$727$	−6.26795	−0.232465	−0.116233	+	0.993222i	$0.537082\pi$
$728$	0	0
$729$	0	0
$730$	−11.2679	−0.417046
$731$	−1.80385	−0.0667177
$732$	0	0
$733$	−2.32051	−0.0857099	−0.0428550	−	0.999081i	$-0.513645\pi$
$733$	−2.32051	−0.0857099	−0.0428550	+	0.999081i	$0.513645\pi$
$734$	−13.2679	−0.489729
$735$	0	0
$736$	3.13844	0.115684
$737$	43.1769	1.59044
$738$	0	0
$739$	−10.3923	−0.382287	−0.191144	−	0.981562i	$-0.561220\pi$
$739$	−10.3923	−0.382287	−0.191144	+	0.981562i	$0.561220\pi$
$740$	−5.85641	−0.215286
$741$	0	0
$742$	−22.6410	−0.831178
$743$	−39.9090	−1.46412	−0.732059	−	0.681241i	$-0.761440\pi$
$743$	−39.9090	−1.46412	−0.732059	+	0.681241i	$0.761440\pi$
$744$	0	0
$745$	4.00000	0.146549
$746$	−1.37307	−0.0502716
$747$	0	0
$748$	34.1436	1.24841
$749$	76.4449	2.79323
$750$	0	0
$751$	−21.3205	−0.777996	−0.388998	−	0.921239i	$-0.627179\pi$
$751$	−21.3205	−0.777996	−0.388998	+	0.921239i	$0.627179\pi$
$752$	−0.210236	−0.00766650
$753$	0	0
$754$	0	0
$755$	−17.8564	−0.649861
$756$	0	0
$757$	26.9282	0.978722	0.489361	−	0.872081i	$-0.337230\pi$
$757$	26.9282	0.978722	0.489361	+	0.872081i	$0.337230\pi$
$758$	−14.4449	−0.524661
$759$	0	0
$760$	13.8564	0.502625
$761$	5.85641	0.212295	0.106147	−	0.994350i	$-0.466148\pi$
$761$	5.85641	0.212295	0.106147	+	0.994350i	$0.466148\pi$
$762$	0	0
$763$	−36.9090	−1.33619
$764$	−40.2872	−1.45754
$765$	0	0
$766$	6.10512	0.220587
$767$	0	0
$768$	0	0
$769$	50.3923	1.81719	0.908596	−	0.417675i	$-0.137155\pi$
$769$	50.3923	1.81719	0.908596	+	0.417675i	$0.137155\pi$
$770$	−11.3205	−0.407963
$771$	0	0
$772$	8.88973	0.319948
$773$	7.85641	0.282575	0.141288	−	0.989969i	$-0.454876\pi$
$773$	7.85641	0.282575	0.141288	+	0.989969i	$0.454876\pi$
$774$	0	0
$775$	3.19615	0.114809
$776$	−41.7513	−1.49878
$777$	0	0
$778$	−12.7846	−0.458350
$779$	−28.7846	−1.03132
$780$	0	0
$781$	44.1051	1.57821
$782$	2.64102	0.0944425
$783$	0	0
$784$	13.8564	0.494872
$785$	−9.73205	−0.347352
$786$	0	0
$787$	−21.5359	−0.767672	−0.383836	−	0.923401i	$-0.625397\pi$
$787$	−21.5359	−0.767672	−0.383836	+	0.923401i	$0.625397\pi$
$788$	−4.28719	−0.152725
$789$	0	0
$790$	−1.41154	−0.0502204
$791$	86.8897	3.08944
$792$	0	0
$793$	0	0
$794$	17.8038	0.631835
$795$	0	0
$796$	21.9615	0.778406
$797$	30.4449	1.07841	0.539206	−	0.842174i	$-0.318724\pi$
$797$	30.4449	1.07841	0.539206	+	0.842174i	$0.318724\pi$
$798$	0	0
$799$	−1.32051	−0.0467162
$800$	5.85641	0.207055
$801$	0	0
$802$	−11.2154	−0.396029
$803$	−53.3205	−1.88164
$804$	0	0
$805$	2.39230	0.0843177
$806$	0	0
$807$	0	0
$808$	−12.4974	−0.439658
$809$	−36.0000	−1.26569	−0.632846	−	0.774277i	$-0.718114\pi$
$809$	−36.0000	−1.26569	−0.632846	+	0.774277i	$0.718114\pi$
$810$	0	0
$811$	−0.947441	−0.0332692	−0.0166346	−	0.999862i	$-0.505295\pi$
$811$	−0.947441	−0.0332692	−0.0166346	+	0.999862i	$0.505295\pi$
$812$	17.8564	0.626637
$813$	0	0
$814$	10.1436	0.355533
$815$	−5.53590	−0.193914
$816$	0	0
$817$	−1.46410	−0.0512224
$818$	−1.94744	−0.0680907
$819$	0	0
$820$	−7.71281	−0.269343
$821$	0.928203	0.0323945	0.0161973	−	0.999869i	$-0.494844\pi$
$821$	0.928203	0.0323945	0.0161973	+	0.999869i	$0.494844\pi$
$822$	0	0
$823$	−13.1769	−0.459318	−0.229659	−	0.973271i	$-0.573761\pi$
$823$	−13.1769	−0.459318	−0.229659	+	0.973271i	$0.573761\pi$
$824$	8.10512	0.282355
$825$	0	0
$826$	23.7513	0.826413
$827$	2.58846	0.0900095	0.0450047	−	0.998987i	$-0.485670\pi$
$827$	2.58846	0.0900095	0.0450047	+	0.998987i	$0.485670\pi$
$828$	0	0
$829$	−38.1769	−1.32594	−0.662970	−	0.748646i	$-0.730704\pi$
$829$	−38.1769	−1.32594	−0.662970	+	0.748646i	$0.730704\pi$
$830$	1.85641	0.0644368
$831$	0	0
$832$	0	0
$833$	87.0333	3.01553
$834$	0	0
$835$	17.4641	0.604370
$836$	27.7128	0.958468
$837$	0	0
$838$	−14.9667	−0.517015
$839$	20.5359	0.708978	0.354489	−	0.935060i	$-0.384655\pi$
$839$	20.5359	0.708978	0.354489	+	0.935060i	$0.384655\pi$
$840$	0	0
$841$	−21.5359	−0.742617
$842$	17.2679	0.595093
$843$	0	0
$844$	−16.8897	−0.581368
$845$	0	0
$846$	0	0
$847$	−4.46410	−0.153388
$848$	7.42563	0.254997
$849$	0	0
$850$	4.92820	0.169036
$851$	−2.14359	−0.0734814
$852$	0	0
$853$	−16.6077	−0.568637	−0.284318	−	0.958730i	$-0.591767\pi$
$853$	−16.6077	−0.568637	−0.284318	+	0.958730i	$0.591767\pi$
$854$	−14.5885	−0.499207
$855$	0	0
$856$	43.4256	1.48426
$857$	17.1244	0.584957	0.292478	−	0.956272i	$-0.405520\pi$
$857$	17.1244	0.584957	0.292478	+	0.956272i	$0.405520\pi$
$858$	0	0
$859$	7.78461	0.265607	0.132804	−	0.991142i	$-0.457602\pi$
$859$	7.78461	0.265607	0.132804	+	0.991142i	$0.457602\pi$
$860$	−0.392305	−0.0133775
$861$	0	0
$862$	17.0718	0.581468
$863$	−34.3923	−1.17073	−0.585364	−	0.810771i	$-0.699048\pi$
$863$	−34.3923	−1.17073	−0.585364	+	0.810771i	$0.699048\pi$
$864$	0	0
$865$	3.26795	0.111114
$866$	3.51666	0.119501
$867$	0	0
$868$	20.8897	0.709044
$869$	−6.67949	−0.226586
$870$	0	0
$871$	0	0
$872$	−20.9667	−0.710021
$873$	0	0
$874$	2.14359	0.0725081
$875$	4.46410	0.150914
$876$	0	0
$877$	18.6410	0.629462	0.314731	−	0.949181i	$-0.398086\pi$
$877$	18.6410	0.629462	0.314731	+	0.949181i	$0.398086\pi$
$878$	11.2679	0.380275
$879$	0	0
$880$	3.71281	0.125159
$881$	25.1769	0.848232	0.424116	−	0.905608i	$-0.360585\pi$
$881$	25.1769	0.848232	0.424116	+	0.905608i	$0.360585\pi$
$882$	0	0
$883$	−0.660254	−0.0222193	−0.0111097	−	0.999938i	$-0.503536\pi$
$883$	−0.660254	−0.0222193	−0.0111097	+	0.999938i	$0.503536\pi$
$884$	0	0
$885$	0	0
$886$	−2.28719	−0.0768396
$887$	2.98076	0.100084	0.0500421	−	0.998747i	$-0.484064\pi$
$887$	2.98076	0.100084	0.0500421	+	0.998747i	$0.484064\pi$
$888$	0	0
$889$	−65.4449	−2.19495
$890$	0.928203	0.0311134
$891$	0	0
$892$	17.3590	0.581222
$893$	−1.07180	−0.0358663
$894$	0	0
$895$	9.12436	0.304994
$896$	45.2820	1.51277
$897$	0	0
$898$	−8.10512	−0.270471
$899$	8.73205	0.291230
$900$	0	0
$901$	46.6410	1.55384
$902$	13.3590	0.444806
$903$	0	0
$904$	49.3590	1.64166
$905$	14.5359	0.483190
$906$	0	0
$907$	−24.5359	−0.814701	−0.407351	−	0.913272i	$-0.633547\pi$
$907$	−24.5359	−0.814701	−0.407351	+	0.913272i	$0.633547\pi$
$908$	−17.0718	−0.566547
$909$	0	0
$910$	0	0
$911$	7.71281	0.255537	0.127768	−	0.991804i	$-0.459219\pi$
$911$	7.71281	0.255537	0.127768	+	0.991804i	$0.459219\pi$
$912$	0	0
$913$	8.78461	0.290728
$914$	−3.94744	−0.130570
$915$	0	0
$916$	26.9282	0.889733
$917$	5.66025	0.186918
$918$	0	0
$919$	−19.0718	−0.629121	−0.314560	−	0.949238i	$-0.601857\pi$
$919$	−19.0718	−0.629121	−0.314560	+	0.949238i	$0.601857\pi$
$920$	1.35898	0.0448044
$921$	0	0
$922$	8.43078	0.277653
$923$	0	0
$924$	0	0
$925$	−4.00000	−0.131519
$926$	−1.30642	−0.0429318
$927$	0	0
$928$	16.0000	0.525226
$929$	−15.3205	−0.502650	−0.251325	−	0.967903i	$-0.580866\pi$
$929$	−15.3205	−0.502650	−0.251325	+	0.967903i	$0.580866\pi$
$930$	0	0
$931$	70.6410	2.31517
$932$	34.9282	1.14411
$933$	0	0
$934$	17.3205	0.566744
$935$	23.3205	0.762662
$936$	0	0
$937$	32.2487	1.05352	0.526760	−	0.850014i	$-0.323407\pi$
$937$	32.2487	1.05352	0.526760	+	0.850014i	$0.323407\pi$
$938$	40.7321	1.32995
$939$	0	0
$940$	−0.287187	−0.00936701
$941$	9.41154	0.306808	0.153404	−	0.988164i	$-0.450976\pi$
$941$	9.41154	0.306808	0.153404	+	0.988164i	$0.450976\pi$
$942$	0	0
$943$	−2.82309	−0.0919323
$944$	−7.78976	−0.253535
$945$	0	0
$946$	0.679492	0.0220922
$947$	−5.41154	−0.175852	−0.0879258	−	0.996127i	$-0.528024\pi$
$947$	−5.41154	−0.175852	−0.0879258	+	0.996127i	$0.528024\pi$
$948$	0	0
$949$	0	0
$950$	4.00000	0.129777
$951$	0	0
$952$	76.2102	2.46999
$953$	−24.2487	−0.785493	−0.392746	−	0.919647i	$-0.628475\pi$
$953$	−24.2487	−0.785493	−0.392746	+	0.919647i	$0.628475\pi$
$954$	0	0
$955$	−27.5167	−0.890418
$956$	8.00000	0.258738
$957$	0	0
$958$	−26.2872	−0.849300
$959$	−40.7321	−1.31531
$960$	0	0
$961$	−20.7846	−0.670471
$962$	0	0
$963$	0	0
$964$	−2.35383	−0.0758117
$965$	6.07180	0.195458
$966$	0	0
$967$	16.2487	0.522523	0.261262	−	0.965268i	$-0.415861\pi$
$967$	16.2487	0.522523	0.261262	+	0.965268i	$0.415861\pi$
$968$	−2.53590	−0.0815069
$969$	0	0
$970$	−12.0526	−0.386984
$971$	−61.1769	−1.96326	−0.981630	−	0.190793i	$-0.938894\pi$
$971$	−61.1769	−1.96326	−0.981630	+	0.190793i	$0.938894\pi$
$972$	0	0
$973$	80.0333	2.56575
$974$	−15.2154	−0.487533
$975$	0	0
$976$	4.78461	0.153152
$977$	14.9282	0.477596	0.238798	−	0.971069i	$-0.423247\pi$
$977$	14.9282	0.477596	0.238798	+	0.971069i	$0.423247\pi$
$978$	0	0
$979$	4.39230	0.140379
$980$	18.9282	0.604639
$981$	0	0
$982$	−5.21539	−0.166430
$983$	3.85641	0.123000	0.0615001	−	0.998107i	$-0.480412\pi$
$983$	3.85641	0.123000	0.0615001	+	0.998107i	$0.480412\pi$
$984$	0	0
$985$	−2.92820	−0.0933003
$986$	13.4641	0.428784
$987$	0	0
$988$	0	0
$989$	−0.143594	−0.00456601
$990$	0	0
$991$	15.0718	0.478771	0.239386	−	0.970925i	$-0.423054\pi$
$991$	15.0718	0.478771	0.239386	+	0.970925i	$0.423054\pi$
$992$	18.7180	0.594296
$993$	0	0
$994$	41.6077	1.31972
$995$	15.0000	0.475532
$996$	0	0
$997$	−26.4115	−0.836462	−0.418231	−	0.908341i	$-0.637350\pi$
$997$	−26.4115	−0.836462	−0.418231	+	0.908341i	$0.637350\pi$
$998$	0.287187	0.00909075
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7605.2.a.y.1.2		2
3.2	odd	2		2535.2.a.s.1.1		2
13.2	odd	12		585.2.bu.a.316.1		4
13.7	odd	12		585.2.bu.a.361.1		4
13.12	even	2		7605.2.a.bk.1.1		2
39.2	even	12		195.2.bb.a.121.2	✓	4
39.20	even	12		195.2.bb.a.166.2	yes	4
39.38	odd	2		2535.2.a.n.1.2		2
195.2	odd	12		975.2.w.a.199.2		4
195.59	even	12		975.2.bc.h.751.1		4
195.98	odd	12		975.2.w.a.49.2		4
195.119	even	12		975.2.bc.h.901.1		4
195.137	odd	12		975.2.w.f.49.1		4
195.158	odd	12		975.2.w.f.199.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
195.2.bb.a.121.2	✓	4	39.2	even	12
195.2.bb.a.166.2	yes	4	39.20	even	12
585.2.bu.a.316.1		4	13.2	odd	12
585.2.bu.a.361.1		4	13.7	odd	12
975.2.w.a.49.2		4	195.98	odd	12
975.2.w.a.199.2		4	195.2	odd	12
975.2.w.f.49.1		4	195.137	odd	12
975.2.w.f.199.1		4	195.158	odd	12
975.2.bc.h.751.1		4	195.59	even	12
975.2.bc.h.901.1		4	195.119	even	12
2535.2.a.n.1.2		2	39.38	odd	2
2535.2.a.s.1.1		2	3.2	odd	2
7605.2.a.y.1.2		2	1.1	even	1	trivial
7605.2.a.bk.1.1		2	13.12	even	2

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:	\( N \)	\(=\)	\( 7605 = 3^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7605.a (trivial)

\(f(q)\)	\(=\)	\(q+0.732051 q^{2} -1.46410 q^{4} -1.00000 q^{5} -4.46410 q^{7} -2.53590 q^{8} -0.732051 q^{10} -3.46410 q^{11} -3.26795 q^{14} +1.07180 q^{16} +6.73205 q^{17} +5.46410 q^{19} +1.46410 q^{20} -2.53590 q^{22} +0.535898 q^{23} +1.00000 q^{25} +6.53590 q^{28} +2.73205 q^{29} +3.19615 q^{31} +5.85641 q^{32} +4.92820 q^{34} +4.46410 q^{35} -4.00000 q^{37} +4.00000 q^{38} +2.53590 q^{40} -5.26795 q^{41} -0.267949 q^{43} +5.07180 q^{44} +0.392305 q^{46} -0.196152 q^{47} +12.9282 q^{49} +0.732051 q^{50} +6.92820 q^{53} +3.46410 q^{55} +11.3205 q^{56} +2.00000 q^{58} -7.26795 q^{59} +4.46410 q^{61} +2.33975 q^{62} +2.14359 q^{64} -12.4641 q^{67} -9.85641 q^{68} +3.26795 q^{70} -12.7321 q^{71} +15.3923 q^{73} -2.92820 q^{74} -8.00000 q^{76} +15.4641 q^{77} +1.92820 q^{79} -1.07180 q^{80} -3.85641 q^{82} -2.53590 q^{83} -6.73205 q^{85} -0.196152 q^{86} +8.78461 q^{88} -1.26795 q^{89} -0.784610 q^{92} -0.143594 q^{94} -5.46410 q^{95} +16.4641 q^{97} +9.46410 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 4 q^{4} - 2 q^{5} - 2 q^{7} - 12 q^{8} + 2 q^{10} - 10 q^{14} + 16 q^{16} + 10 q^{17} + 4 q^{19} - 4 q^{20} - 12 q^{22} + 8 q^{23} + 2 q^{25} + 20 q^{28} + 2 q^{29} - 4 q^{31} - 16 q^{32}+ \cdots + 12 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 4 * q^4 - 2 * q^5 - 2 * q^7 - 12 * q^8 + 2 * q^10 - 10 * q^14 + 16 * q^16 + 10 * q^17 + 4 * q^19 - 4 * q^20 - 12 * q^22 + 8 * q^23 + 2 * q^25 + 20 * q^28 + 2 * q^29 - 4 * q^31 - 16 * q^32 - 4 * q^34 + 2 * q^35 - 8 * q^37 + 8 * q^38 + 12 * q^40 - 14 * q^41 - 4 * q^43 + 24 * q^44 - 20 * q^46 + 10 * q^47 + 12 * q^49 - 2 * q^50 - 12 * q^56 + 4 * q^58 - 18 * q^59 + 2 * q^61 + 22 * q^62 + 32 * q^64 - 18 * q^67 + 8 * q^68 + 10 * q^70 - 22 * q^71 + 10 * q^73 + 8 * q^74 - 16 * q^76 + 24 * q^77 - 10 * q^79 - 16 * q^80 + 20 * q^82 - 12 * q^83 - 10 * q^85 + 10 * q^86 - 24 * q^88 - 6 * q^89 + 40 * q^92 - 28 * q^94 - 4 * q^95 + 26 * q^97 + 12 * q^98

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