LMFDB - Embedded newform 9464.2.a.bs.1.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9464,2,Mod(1,9464)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9464, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9464.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 9464 = 2^{3} \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9464.a (trivial)

Newform invariants

comment:select newform

sage:traces = [15,0,-3,0,4,0,15,0,16,0,15,0,0,0,8,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$75.5704204729$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 3 x^{14} - 26 x^{13} + 78 x^{12} + 253 x^{11} - 782 x^{10} - 1087 x^{9} + 3776 x^{8} + \cdots - 344 $ x^15 - 3x^14 - 26x^13 + 78x^12 + 253x^11 - 782x^10 - 1087x^9 + 3776x^8 + 1576x^7 - 8849x^6 + 1791x^5 + 8496x^4 - 5353x^3 - 1252x^2 + 1724x - 344
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$0.739098$ of defining polynomial
Character	$\chi$	$=$	9464.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.739098 q^{3} +1.43777 q^{5} +1.00000 q^{7} -2.45373 q^{9} +4.73322 q^{11} -1.06266 q^{15} +2.85457 q^{17} -0.0123052 q^{19} -0.739098 q^{21} +8.75886 q^{23} -2.93281 q^{25} +4.03085 q^{27} -6.55507 q^{29} -6.65606 q^{31} -3.49831 q^{33} +1.43777 q^{35} -2.21341 q^{37} +1.13704 q^{41} +8.89935 q^{43} -3.52791 q^{45} +13.0192 q^{47} +1.00000 q^{49} -2.10981 q^{51} +6.20713 q^{53} +6.80529 q^{55} +0.00909476 q^{57} -8.21482 q^{59} +8.05714 q^{61} -2.45373 q^{63} +9.19433 q^{67} -6.47366 q^{69} -0.560824 q^{71} -11.2553 q^{73} +2.16763 q^{75} +4.73322 q^{77} -9.20505 q^{79} +4.38201 q^{81} -10.9039 q^{83} +4.10423 q^{85} +4.84484 q^{87} +0.0602602 q^{89} +4.91948 q^{93} -0.0176921 q^{95} +16.0537 q^{97} -11.6141 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - 3 q^{3} + 4 q^{5} + 15 q^{7} + 16 q^{9} + 15 q^{11} + 8 q^{15} + 2 q^{17} + 13 q^{19} - 3 q^{21} - 10 q^{23} + 23 q^{25} - 9 q^{27} + 25 q^{29} + 19 q^{31} - 24 q^{33} + 4 q^{35} - 2 q^{37} + 30 q^{41}+ \cdots + 70 q^{99}+O(q^{100}) $ 15 * q - 3 * q^3 + 4 * q^5 + 15 * q^7 + 16 * q^9 + 15 * q^11 + 8 * q^15 + 2 * q^17 + 13 * q^19 - 3 * q^21 - 10 * q^23 + 23 * q^25 - 9 * q^27 + 25 * q^29 + 19 * q^31 - 24 * q^33 + 4 * q^35 - 2 * q^37 + 30 * q^41 + 15 * q^43 - 2 * q^45 + 13 * q^47 + 15 * q^49 + q^51 + 62 * q^53 - 3 * q^55 - 2 * q^57 + 22 * q^59 - q^61 + 16 * q^63 + 29 * q^67 + 32 * q^69 + 20 * q^71 + 21 * q^73 - 46 * q^75 + 15 * q^77 + 29 * q^79 + 7 * q^81 + 23 * q^83 + 8 * q^85 - 22 * q^87 + 9 * q^89 + 10 * q^93 + 28 * q^95 + 11 * q^97 + 70 * q^99

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
$3$	−0.739098	−0.426719	−0.213359	−	0.976974i	$-0.568440\pi$
$3$	−0.739098	−0.426719	−0.213359	+	0.976974i	$0.568440\pi$
$4$	0	0
$5$	1.43777	0.642992	0.321496	−	0.946911i	$-0.395814\pi$
$5$	1.43777	0.642992	0.321496	+	0.946911i	$0.395814\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.45373	−0.817911
$10$	0	0
$11$	4.73322	1.42712	0.713559	−	0.700595i	$-0.247082\pi$
$11$	4.73322	1.42712	0.713559	+	0.700595i	$0.247082\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−1.06266	−0.274377
$16$	0	0
$17$	2.85457	0.692336	0.346168	−	0.938173i	$-0.387483\pi$
$17$	2.85457	0.692336	0.346168	+	0.938173i	$0.387483\pi$
$18$	0	0
$19$	−0.0123052	−0.00282301	−0.00141150	−	0.999999i	$-0.500449\pi$
$19$	−0.0123052	−0.00282301	−0.00141150	+	0.999999i	$0.500449\pi$
$20$	0	0
$21$	−0.739098	−0.161284
$22$	0	0
$23$	8.75886	1.82635	0.913175	−	0.407569i	$-0.133623\pi$
$23$	8.75886	1.82635	0.913175	+	0.407569i	$0.133623\pi$
$24$	0	0
$25$	−2.93281	−0.586561
$26$	0	0
$27$	4.03085	0.775737
$28$	0	0
$29$	−6.55507	−1.21725	−0.608623	−	0.793459i	$-0.708278\pi$
$29$	−6.55507	−1.21725	−0.608623	+	0.793459i	$0.708278\pi$
$30$	0	0
$31$	−6.65606	−1.19546	−0.597732	−	0.801696i	$-0.703931\pi$
$31$	−6.65606	−1.19546	−0.597732	+	0.801696i	$0.703931\pi$
$32$	0	0
$33$	−3.49831	−0.608978
$34$	0	0
$35$	1.43777	0.243028
$36$	0	0
$37$	−2.21341	−0.363882	−0.181941	−	0.983309i	$-0.558238\pi$
$37$	−2.21341	−0.363882	−0.181941	+	0.983309i	$0.558238\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.13704	0.177577	0.0887883	−	0.996051i	$-0.471701\pi$
$41$	1.13704	0.177577	0.0887883	+	0.996051i	$0.471701\pi$
$42$	0	0
$43$	8.89935	1.35714	0.678569	−	0.734537i	$-0.262600\pi$
$43$	8.89935	1.35714	0.678569	+	0.734537i	$0.262600\pi$
$44$	0	0
$45$	−3.52791	−0.525910
$46$	0	0
$47$	13.0192	1.89905	0.949525	−	0.313692i	$-0.101566\pi$
$47$	13.0192	1.89905	0.949525	+	0.313692i	$0.101566\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.10981	−0.295433
$52$	0	0
$53$	6.20713	0.852615	0.426308	−	0.904578i	$-0.359814\pi$
$53$	6.20713	0.852615	0.426308	+	0.904578i	$0.359814\pi$
$54$	0	0
$55$	6.80529	0.917626
$56$	0	0
$57$	0.00909476	0.00120463
$58$	0	0
$59$	−8.21482	−1.06948	−0.534739	−	0.845017i	$-0.679590\pi$
$59$	−8.21482	−1.06948	−0.534739	+	0.845017i	$0.679590\pi$
$60$	0	0
$61$	8.05714	1.03161	0.515805	−	0.856706i	$-0.327493\pi$
$61$	8.05714	1.03161	0.515805	+	0.856706i	$0.327493\pi$
$62$	0	0
$63$	−2.45373	−0.309141
$64$	0	0
$65$	0	0
$66$	0	0
$67$	9.19433	1.12327	0.561633	−	0.827387i	$-0.310173\pi$
$67$	9.19433	1.12327	0.561633	+	0.827387i	$0.310173\pi$
$68$	0	0
$69$	−6.47366	−0.779337
$70$	0	0
$71$	−0.560824	−0.0665576	−0.0332788	−	0.999446i	$-0.510595\pi$
$71$	−0.560824	−0.0665576	−0.0332788	+	0.999446i	$0.510595\pi$
$72$	0	0
$73$	−11.2553	−1.31733	−0.658665	−	0.752436i	$-0.728879\pi$
$73$	−11.2553	−1.31733	−0.658665	+	0.752436i	$0.728879\pi$
$74$	0	0
$75$	2.16763	0.250297
$76$	0	0
$77$	4.73322	0.539400
$78$	0	0
$79$	−9.20505	−1.03565	−0.517824	−	0.855487i	$-0.673258\pi$
$79$	−9.20505	−1.03565	−0.517824	+	0.855487i	$0.673258\pi$
$80$	0	0
$81$	4.38201	0.486890
$82$	0	0
$83$	−10.9039	−1.19686	−0.598431	−	0.801175i	$-0.704209\pi$
$83$	−10.9039	−1.19686	−0.598431	+	0.801175i	$0.704209\pi$
$84$	0	0
$85$	4.10423	0.445166
$86$	0	0
$87$	4.84484	0.519422
$88$	0	0
$89$	0.0602602	0.00638757	0.00319379	−	0.999995i	$-0.498983\pi$
$89$	0.0602602	0.00638757	0.00319379	+	0.999995i	$0.498983\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	4.91948	0.510127
$94$	0	0
$95$	−0.0176921	−0.00181517
$96$	0	0
$97$	16.0537	1.63001	0.815006	−	0.579453i	$-0.196734\pi$
$97$	16.0537	1.63001	0.815006	+	0.579453i	$0.196734\pi$
$98$	0	0
$99$	−11.6141	−1.16726
$100$	0	0
$101$	14.8575	1.47838	0.739190	−	0.673497i	$-0.235209\pi$
$101$	14.8575	1.47838	0.739190	+	0.673497i	$0.235209\pi$
$102$	0	0
$103$	−11.8433	−1.16695	−0.583477	−	0.812130i	$-0.698308\pi$
$103$	−11.8433	−1.16695	−0.583477	+	0.812130i	$0.698308\pi$
$104$	0	0
$105$	−1.06266	−0.103705
$106$	0	0
$107$	−8.59602	−0.831009	−0.415504	−	0.909591i	$-0.636395\pi$
$107$	−8.59602	−0.831009	−0.415504	+	0.909591i	$0.636395\pi$
$108$	0	0
$109$	4.79814	0.459578	0.229789	−	0.973240i	$-0.426196\pi$
$109$	4.79814	0.459578	0.229789	+	0.973240i	$0.426196\pi$
$110$	0	0
$111$	1.63593	0.155275
$112$	0	0
$113$	15.2321	1.43291	0.716456	−	0.697632i	$-0.245763\pi$
$113$	15.2321	1.43291	0.716456	+	0.697632i	$0.245763\pi$
$114$	0	0
$115$	12.5933	1.17433
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.85457	0.261678
$120$	0	0
$121$	11.4033	1.03667
$122$	0	0
$123$	−0.840388	−0.0757752
$124$	0	0
$125$	−11.4056	−1.02015
$126$	0	0
$127$	−10.5724	−0.938145	−0.469072	−	0.883160i	$-0.655412\pi$
$127$	−10.5724	−0.938145	−0.469072	+	0.883160i	$0.655412\pi$
$128$	0	0
$129$	−6.57749	−0.579116
$130$	0	0
$131$	11.4595	1.00122	0.500609	−	0.865674i	$-0.333110\pi$
$131$	11.4595	1.00122	0.500609	+	0.865674i	$0.333110\pi$
$132$	0	0
$133$	−0.0123052	−0.00106700
$134$	0	0
$135$	5.79544	0.498792
$136$	0	0
$137$	17.2627	1.47486	0.737428	−	0.675426i	$-0.236040\pi$
$137$	17.2627	1.47486	0.737428	+	0.675426i	$0.236040\pi$
$138$	0	0
$139$	−9.89549	−0.839324	−0.419662	−	0.907680i	$-0.637851\pi$
$139$	−9.89549	−0.839324	−0.419662	+	0.907680i	$0.637851\pi$
$140$	0	0
$141$	−9.62249	−0.810360
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−9.42471	−0.782680
$146$	0	0
$147$	−0.739098	−0.0609598
$148$	0	0
$149$	−12.8608	−1.05360	−0.526799	−	0.849990i	$-0.676608\pi$
$149$	−12.8608	−1.05360	−0.526799	+	0.849990i	$0.676608\pi$
$150$	0	0
$151$	−5.75509	−0.468343	−0.234171	−	0.972195i	$-0.575238\pi$
$151$	−5.75509	−0.468343	−0.234171	+	0.972195i	$0.575238\pi$
$152$	0	0
$153$	−7.00437	−0.566269
$154$	0	0
$155$	−9.56991	−0.768673
$156$	0	0
$157$	−0.00669191	−0.000534072 0	−0.000267036	−	1.00000i	$-0.500085\pi$
$157$	−0.00669191	−0.000534072 0	−0.000267036		1.00000i	$0.500085\pi$
$158$	0	0
$159$	−4.58768	−0.363827
$160$	0	0
$161$	8.75886	0.690295
$162$	0	0
$163$	−4.06876	−0.318690	−0.159345	−	0.987223i	$-0.550938\pi$
$163$	−4.06876	−0.318690	−0.159345	+	0.987223i	$0.550938\pi$
$164$	0	0
$165$	−5.02978	−0.391568
$166$	0	0
$167$	14.2107	1.09966	0.549828	−	0.835278i	$-0.314693\pi$
$167$	14.2107	1.09966	0.549828	+	0.835278i	$0.314693\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	0.0301937	0.00230897
$172$	0	0
$173$	18.5209	1.40812	0.704059	−	0.710141i	$-0.251369\pi$
$173$	18.5209	1.40812	0.704059	+	0.710141i	$0.251369\pi$
$174$	0	0
$175$	−2.93281	−0.221699
$176$	0	0
$177$	6.07156	0.456366
$178$	0	0
$179$	−15.3133	−1.14457	−0.572285	−	0.820055i	$-0.693943\pi$
$179$	−15.3133	−1.14457	−0.572285	+	0.820055i	$0.693943\pi$
$180$	0	0
$181$	−24.3037	−1.80648	−0.903239	−	0.429138i	$-0.858817\pi$
$181$	−24.3037	−1.80648	−0.903239	+	0.429138i	$0.858817\pi$
$182$	0	0
$183$	−5.95502	−0.440207
$184$	0	0
$185$	−3.18238	−0.233973
$186$	0	0
$187$	13.5113	0.988046
$188$	0	0
$189$	4.03085	0.293201
$190$	0	0
$191$	13.0831	0.946656	0.473328	−	0.880886i	$-0.343052\pi$
$191$	13.0831	0.946656	0.473328	+	0.880886i	$0.343052\pi$
$192$	0	0
$193$	−7.27999	−0.524025	−0.262013	−	0.965064i	$-0.584386\pi$
$193$	−7.27999	−0.524025	−0.262013	+	0.965064i	$0.584386\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	17.4616	1.24409	0.622044	−	0.782982i	$-0.286302\pi$
$197$	17.4616	1.24409	0.622044	+	0.782982i	$0.286302\pi$
$198$	0	0
$199$	−18.9308	−1.34197	−0.670986	−	0.741470i	$-0.734129\pi$
$199$	−18.9308	−1.34197	−0.670986	+	0.741470i	$0.734129\pi$
$200$	0	0
$201$	−6.79551	−0.479318
$202$	0	0
$203$	−6.55507	−0.460076
$204$	0	0
$205$	1.63481	0.114180
$206$	0	0
$207$	−21.4919	−1.49379
$208$	0	0
$209$	−0.0582432	−0.00402877
$210$	0	0
$211$	−5.39681	−0.371532	−0.185766	−	0.982594i	$-0.559477\pi$
$211$	−5.39681	−0.371532	−0.185766	+	0.982594i	$0.559477\pi$
$212$	0	0
$213$	0.414504	0.0284014
$214$	0	0
$215$	12.7952	0.872629
$216$	0	0
$217$	−6.65606	−0.451843
$218$	0	0
$219$	8.31876	0.562130
$220$	0	0
$221$	0	0
$222$	0	0
$223$	14.0955	0.943903	0.471952	−	0.881624i	$-0.343550\pi$
$223$	14.0955	0.943903	0.471952	+	0.881624i	$0.343550\pi$
$224$	0	0
$225$	7.19633	0.479755
$226$	0	0
$227$	−4.54506	−0.301666	−0.150833	−	0.988559i	$-0.548196\pi$
$227$	−4.54506	−0.301666	−0.150833	+	0.988559i	$0.548196\pi$
$228$	0	0
$229$	−9.69784	−0.640851	−0.320426	−	0.947274i	$-0.603826\pi$
$229$	−9.69784	−0.640851	−0.320426	+	0.947274i	$0.603826\pi$
$230$	0	0
$231$	−3.49831	−0.230172
$232$	0	0
$233$	−7.26297	−0.475813	−0.237906	−	0.971288i	$-0.576461\pi$
$233$	−7.26297	−0.475813	−0.237906	+	0.971288i	$0.576461\pi$
$234$	0	0
$235$	18.7187	1.22107
$236$	0	0
$237$	6.80344	0.441931
$238$	0	0
$239$	23.6215	1.52794	0.763972	−	0.645249i	$-0.223246\pi$
$239$	23.6215	1.52794	0.763972	+	0.645249i	$0.223246\pi$
$240$	0	0
$241$	−0.721163	−0.0464542	−0.0232271	−	0.999730i	$-0.507394\pi$
$241$	−0.721163	−0.0464542	−0.0232271	+	0.999730i	$0.507394\pi$
$242$	0	0
$243$	−15.3313	−0.983502
$244$	0	0
$245$	1.43777	0.0918560
$246$	0	0
$247$	0	0
$248$	0	0
$249$	8.05907	0.510723
$250$	0	0
$251$	9.70022	0.612272	0.306136	−	0.951988i	$-0.400964\pi$
$251$	9.70022	0.612272	0.306136	+	0.951988i	$0.400964\pi$
$252$	0	0
$253$	41.4576	2.60642
$254$	0	0
$255$	−3.03343	−0.189961
$256$	0	0
$257$	4.84021	0.301924	0.150962	−	0.988540i	$-0.451763\pi$
$257$	4.84021	0.301924	0.150962	+	0.988540i	$0.451763\pi$
$258$	0	0
$259$	−2.21341	−0.137535
$260$	0	0
$261$	16.0844	0.995600
$262$	0	0
$263$	3.54533	0.218615	0.109307	−	0.994008i	$-0.465137\pi$
$263$	3.54533	0.218615	0.109307	+	0.994008i	$0.465137\pi$
$264$	0	0
$265$	8.92445	0.548225
$266$	0	0
$267$	−0.0445382	−0.00272570
$268$	0	0
$269$	21.5909	1.31642	0.658210	−	0.752834i	$-0.271314\pi$
$269$	21.5909	1.31642	0.658210	+	0.752834i	$0.271314\pi$
$270$	0	0
$271$	18.3590	1.11523	0.557614	−	0.830101i	$-0.311717\pi$
$271$	18.3590	1.11523	0.557614	+	0.830101i	$0.311717\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−13.8816	−0.837093
$276$	0	0
$277$	−1.20391	−0.0723358	−0.0361679	−	0.999346i	$-0.511515\pi$
$277$	−1.20391	−0.0723358	−0.0361679	+	0.999346i	$0.511515\pi$
$278$	0	0
$279$	16.3322	0.977783
$280$	0	0
$281$	−1.91102	−0.114002	−0.0570011	−	0.998374i	$-0.518154\pi$
$281$	−1.91102	−0.114002	−0.0570011	+	0.998374i	$0.518154\pi$
$282$	0	0
$283$	18.8183	1.11863	0.559315	−	0.828955i	$-0.311064\pi$
$283$	18.8183	1.11863	0.559315	+	0.828955i	$0.311064\pi$
$284$	0	0
$285$	0.0130762	0.000774568 0
$286$	0	0
$287$	1.13704	0.0671176
$288$	0	0
$289$	−8.85140	−0.520671
$290$	0	0
$291$	−11.8653	−0.695556
$292$	0	0
$293$	25.3667	1.48194	0.740970	−	0.671538i	$-0.234366\pi$
$293$	25.3667	1.48194	0.740970	+	0.671538i	$0.234366\pi$
$294$	0	0
$295$	−11.8110	−0.687666
$296$	0	0
$297$	19.0789	1.10707
$298$	0	0
$299$	0	0
$300$	0	0
$301$	8.89935	0.512950
$302$	0	0
$303$	−10.9812	−0.630852
$304$	0	0
$305$	11.5843	0.663317
$306$	0	0
$307$	0.385991	0.0220297	0.0110148	−	0.999939i	$-0.496494\pi$
$307$	0.385991	0.0220297	0.0110148	+	0.999939i	$0.496494\pi$
$308$	0	0
$309$	8.75335	0.497961
$310$	0	0
$311$	−7.06003	−0.400338	−0.200169	−	0.979761i	$-0.564149\pi$
$311$	−7.06003	−0.400338	−0.200169	+	0.979761i	$0.564149\pi$
$312$	0	0
$313$	−30.7682	−1.73912	−0.869561	−	0.493826i	$-0.835598\pi$
$313$	−30.7682	−1.73912	−0.869561	+	0.493826i	$0.835598\pi$
$314$	0	0
$315$	−3.52791	−0.198775
$316$	0	0
$317$	18.2431	1.02464	0.512318	−	0.858796i	$-0.328787\pi$
$317$	18.2431	1.02464	0.512318	+	0.858796i	$0.328787\pi$
$318$	0	0
$319$	−31.0266	−1.73716
$320$	0	0
$321$	6.35331	0.354607
$322$	0	0
$323$	−0.0351261	−0.00195447
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−3.54629	−0.196111
$328$	0	0
$329$	13.0192	0.717773
$330$	0	0
$331$	−11.9871	−0.658871	−0.329436	−	0.944178i	$-0.606858\pi$
$331$	−11.9871	−0.658871	−0.329436	+	0.944178i	$0.606858\pi$
$332$	0	0
$333$	5.43112	0.297623
$334$	0	0
$335$	13.2194	0.722251
$336$	0	0
$337$	28.3914	1.54658	0.773289	−	0.634054i	$-0.218610\pi$
$337$	28.3914	1.54658	0.773289	+	0.634054i	$0.218610\pi$
$338$	0	0
$339$	−11.2580	−0.611450
$340$	0	0
$341$	−31.5046	−1.70607
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−9.30766	−0.501107
$346$	0	0
$347$	28.4813	1.52896	0.764479	−	0.644649i	$-0.222996\pi$
$347$	28.4813	1.52896	0.764479	+	0.644649i	$0.222996\pi$
$348$	0	0
$349$	−10.4237	−0.557970	−0.278985	−	0.960295i	$-0.589998\pi$
$349$	−10.4237	−0.557970	−0.278985	+	0.960295i	$0.589998\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	17.7421	0.944315	0.472158	−	0.881514i	$-0.343475\pi$
$353$	17.7421	0.944315	0.472158	+	0.881514i	$0.343475\pi$
$354$	0	0
$355$	−0.806338	−0.0427960
$356$	0	0
$357$	−2.10981	−0.111663
$358$	0	0
$359$	−15.7513	−0.831320	−0.415660	−	0.909520i	$-0.636449\pi$
$359$	−15.7513	−0.831320	−0.415660	+	0.909520i	$0.636449\pi$
$360$	0	0
$361$	−18.9998	−0.999992
$362$	0	0
$363$	−8.42819	−0.442365
$364$	0	0
$365$	−16.1825	−0.847033
$366$	0	0
$367$	−1.17630	−0.0614025	−0.0307013	−	0.999529i	$-0.509774\pi$
$367$	−1.17630	−0.0614025	−0.0307013	+	0.999529i	$0.509774\pi$
$368$	0	0
$369$	−2.79001	−0.145242
$370$	0	0
$371$	6.20713	0.322258
$372$	0	0
$373$	0.296248	0.0153391	0.00766957	−	0.999971i	$-0.497559\pi$
$373$	0.296248	0.0153391	0.00766957	+	0.999971i	$0.497559\pi$
$374$	0	0
$375$	8.42985	0.435315
$376$	0	0
$377$	0	0
$378$	0	0
$379$	14.6639	0.753235	0.376617	−	0.926369i	$-0.377087\pi$
$379$	14.6639	0.753235	0.376617	+	0.926369i	$0.377087\pi$
$380$	0	0
$381$	7.81401	0.400324
$382$	0	0
$383$	30.4945	1.55820	0.779099	−	0.626901i	$-0.215677\pi$
$383$	30.4945	1.55820	0.779099	+	0.626901i	$0.215677\pi$
$384$	0	0
$385$	6.80529	0.346830
$386$	0	0
$387$	−21.8366	−1.11002
$388$	0	0
$389$	15.3289	0.777206	0.388603	−	0.921405i	$-0.372958\pi$
$389$	15.3289	0.777206	0.388603	+	0.921405i	$0.372958\pi$
$390$	0	0
$391$	25.0028	1.26445
$392$	0	0
$393$	−8.46967	−0.427238
$394$	0	0
$395$	−13.2348	−0.665914
$396$	0	0
$397$	−26.6483	−1.33744	−0.668721	−	0.743514i	$-0.733158\pi$
$397$	−26.6483	−1.33744	−0.668721	+	0.743514i	$0.733158\pi$
$398$	0	0
$399$	0.00909476	0.000455307 0
$400$	0	0
$401$	34.0084	1.69830	0.849149	−	0.528153i	$-0.177115\pi$
$401$	34.0084	1.69830	0.849149	+	0.528153i	$0.177115\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	6.30034	0.313066
$406$	0	0
$407$	−10.4765	−0.519303
$408$	0	0
$409$	−18.7181	−0.925549	−0.462774	−	0.886476i	$-0.653146\pi$
$409$	−18.7181	−0.925549	−0.462774	+	0.886476i	$0.653146\pi$
$410$	0	0
$411$	−12.7589	−0.629348
$412$	0	0
$413$	−8.21482	−0.404225
$414$	0	0
$415$	−15.6774	−0.769572
$416$	0	0
$417$	7.31374	0.358155
$418$	0	0
$419$	−10.7245	−0.523926	−0.261963	−	0.965078i	$-0.584370\pi$
$419$	−10.7245	−0.523926	−0.261963	+	0.965078i	$0.584370\pi$
$420$	0	0
$421$	−7.94362	−0.387148	−0.193574	−	0.981086i	$-0.562008\pi$
$421$	−7.94362	−0.387148	−0.193574	+	0.981086i	$0.562008\pi$
$422$	0	0
$423$	−31.9457	−1.55325
$424$	0	0
$425$	−8.37192	−0.406098
$426$	0	0
$427$	8.05714	0.389912
$428$	0	0
$429$	0	0
$430$	0	0
$431$	7.94545	0.382719	0.191359	−	0.981520i	$-0.438710\pi$
$431$	7.94545	0.382719	0.191359	+	0.981520i	$0.438710\pi$
$432$	0	0
$433$	21.3383	1.02545	0.512727	−	0.858551i	$-0.328635\pi$
$433$	21.3383	1.02545	0.512727	+	0.858551i	$0.328635\pi$
$434$	0	0
$435$	6.96579	0.333984
$436$	0	0
$437$	−0.107780	−0.00515580
$438$	0	0
$439$	8.61980	0.411401	0.205700	−	0.978615i	$-0.434053\pi$
$439$	8.61980	0.411401	0.205700	+	0.978615i	$0.434053\pi$
$440$	0	0
$441$	−2.45373	−0.116844
$442$	0	0
$443$	6.37389	0.302833	0.151416	−	0.988470i	$-0.451617\pi$
$443$	6.37389	0.302833	0.151416	+	0.988470i	$0.451617\pi$
$444$	0	0
$445$	0.0866406	0.00410716
$446$	0	0
$447$	9.50540	0.449590
$448$	0	0
$449$	−28.3857	−1.33960	−0.669802	−	0.742539i	$-0.733621\pi$
$449$	−28.3857	−1.33960	−0.669802	+	0.742539i	$0.733621\pi$
$450$	0	0
$451$	5.38188	0.253423
$452$	0	0
$453$	4.25358	0.199851
$454$	0	0
$455$	0	0
$456$	0	0
$457$	15.2569	0.713687	0.356844	−	0.934164i	$-0.383853\pi$
$457$	15.2569	0.713687	0.356844	+	0.934164i	$0.383853\pi$
$458$	0	0
$459$	11.5063	0.537070
$460$	0	0
$461$	−39.0810	−1.82018	−0.910092	−	0.414406i	$-0.863989\pi$
$461$	−39.0810	−1.82018	−0.910092	+	0.414406i	$0.863989\pi$
$462$	0	0
$463$	−17.0493	−0.792347	−0.396174	−	0.918176i	$-0.629662\pi$
$463$	−17.0493	−0.792347	−0.396174	+	0.918176i	$0.629662\pi$
$464$	0	0
$465$	7.07310	0.328007
$466$	0	0
$467$	−37.8487	−1.75143	−0.875713	−	0.482831i	$-0.839608\pi$
$467$	−37.8487	−1.75143	−0.875713	+	0.482831i	$0.839608\pi$
$468$	0	0
$469$	9.19433	0.424555
$470$	0	0
$471$	0.00494598	0.000227899 0
$472$	0	0
$473$	42.1225	1.93680
$474$	0	0
$475$	0.0360888	0.00165587
$476$	0	0
$477$	−15.2306	−0.697364
$478$	0	0
$479$	−9.07594	−0.414690	−0.207345	−	0.978268i	$-0.566482\pi$
$479$	−9.07594	−0.414690	−0.207345	+	0.978268i	$0.566482\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−6.47366	−0.294562
$484$	0	0
$485$	23.0817	1.04808
$486$	0	0
$487$	20.4498	0.926669	0.463334	−	0.886183i	$-0.346653\pi$
$487$	20.4498	0.926669	0.463334	+	0.886183i	$0.346653\pi$
$488$	0	0
$489$	3.00721	0.135991
$490$	0	0
$491$	−35.7051	−1.61135	−0.805674	−	0.592359i	$-0.798197\pi$
$491$	−35.7051	−1.61135	−0.805674	+	0.592359i	$0.798197\pi$
$492$	0	0
$493$	−18.7119	−0.842744
$494$	0	0
$495$	−16.6984	−0.750536
$496$	0	0
$497$	−0.560824	−0.0251564
$498$	0	0
$499$	27.3366	1.22375	0.611877	−	0.790953i	$-0.290415\pi$
$499$	27.3366	1.22375	0.611877	+	0.790953i	$0.290415\pi$
$500$	0	0
$501$	−10.5031	−0.469243
$502$	0	0
$503$	−20.1944	−0.900424	−0.450212	−	0.892922i	$-0.648652\pi$
$503$	−20.1944	−0.900424	−0.450212	+	0.892922i	$0.648652\pi$
$504$	0	0
$505$	21.3618	0.950586
$506$	0	0
$507$	0	0
$508$	0	0
$509$	18.4517	0.817858	0.408929	−	0.912566i	$-0.365902\pi$
$509$	18.4517	0.817858	0.408929	+	0.912566i	$0.365902\pi$
$510$	0	0
$511$	−11.2553	−0.497904
$512$	0	0
$513$	−0.0496004	−0.00218991
$514$	0	0
$515$	−17.0280	−0.750342
$516$	0	0
$517$	61.6228	2.71017
$518$	0	0
$519$	−13.6888	−0.600871
$520$	0	0
$521$	−3.89874	−0.170807	−0.0854035	−	0.996346i	$-0.527218\pi$
$521$	−3.89874	−0.170807	−0.0854035	+	0.996346i	$0.527218\pi$
$522$	0	0
$523$	18.3776	0.803596	0.401798	−	0.915728i	$-0.368385\pi$
$523$	18.3776	0.803596	0.401798	+	0.915728i	$0.368385\pi$
$524$	0	0
$525$	2.16763	0.0946033
$526$	0	0
$527$	−19.0002	−0.827663
$528$	0	0
$529$	53.7177	2.33555
$530$	0	0
$531$	20.1570	0.874738
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−12.3591	−0.534332
$536$	0	0
$537$	11.3180	0.488409
$538$	0	0
$539$	4.73322	0.203874
$540$	0	0
$541$	−5.76656	−0.247924	−0.123962	−	0.992287i	$-0.539560\pi$
$541$	−5.76656	−0.247924	−0.123962	+	0.992287i	$0.539560\pi$
$542$	0	0
$543$	17.9628	0.770858
$544$	0	0
$545$	6.89863	0.295505
$546$	0	0
$547$	0.511830	0.0218843	0.0109421	−	0.999940i	$-0.496517\pi$
$547$	0.511830	0.0218843	0.0109421	+	0.999940i	$0.496517\pi$
$548$	0	0
$549$	−19.7701	−0.843766
$550$	0	0
$551$	0.0806616	0.00343630
$552$	0	0
$553$	−9.20505	−0.391438
$554$	0	0
$555$	2.35209	0.0998407
$556$	0	0
$557$	−11.7391	−0.497400	−0.248700	−	0.968581i	$-0.580003\pi$
$557$	−11.7391	−0.497400	−0.248700	+	0.968581i	$0.580003\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−9.98619	−0.421617
$562$	0	0
$563$	−0.773900	−0.0326160	−0.0163080	−	0.999867i	$-0.505191\pi$
$563$	−0.773900	−0.0326160	−0.0163080	+	0.999867i	$0.505191\pi$
$564$	0	0
$565$	21.9003	0.921351
$566$	0	0
$567$	4.38201	0.184027
$568$	0	0
$569$	38.9667	1.63357	0.816785	−	0.576942i	$-0.195754\pi$
$569$	38.9667	1.63357	0.816785	+	0.576942i	$0.195754\pi$
$570$	0	0
$571$	16.8130	0.703603	0.351801	−	0.936075i	$-0.385569\pi$
$571$	16.8130	0.703603	0.351801	+	0.936075i	$0.385569\pi$
$572$	0	0
$573$	−9.66966	−0.403956
$574$	0	0
$575$	−25.6881	−1.07127
$576$	0	0
$577$	43.1012	1.79433	0.897163	−	0.441699i	$-0.145624\pi$
$577$	43.1012	1.79433	0.897163	+	0.441699i	$0.145624\pi$
$578$	0	0
$579$	5.38063	0.223611
$580$	0	0
$581$	−10.9039	−0.452371
$582$	0	0
$583$	29.3797	1.21678
$584$	0	0
$585$	0	0
$586$	0	0
$587$	39.4085	1.62656	0.813282	−	0.581870i	$-0.197679\pi$
$587$	39.4085	1.62656	0.813282	+	0.581870i	$0.197679\pi$
$588$	0	0
$589$	0.0819042	0.00337480
$590$	0	0
$591$	−12.9058	−0.530876
$592$	0	0
$593$	28.0980	1.15385	0.576923	−	0.816799i	$-0.304253\pi$
$593$	28.0980	1.15385	0.576923	+	0.816799i	$0.304253\pi$
$594$	0	0
$595$	4.10423	0.168257
$596$	0	0
$597$	13.9917	0.572644
$598$	0	0
$599$	9.70891	0.396695	0.198348	−	0.980132i	$-0.436442\pi$
$599$	9.70891	0.396695	0.198348	+	0.980132i	$0.436442\pi$
$600$	0	0
$601$	−5.64199	−0.230141	−0.115071	−	0.993357i	$-0.536709\pi$
$601$	−5.64199	−0.230141	−0.115071	+	0.993357i	$0.536709\pi$
$602$	0	0
$603$	−22.5604	−0.918732
$604$	0	0
$605$	16.3954	0.666569
$606$	0	0
$607$	−29.6547	−1.20365	−0.601824	−	0.798629i	$-0.705559\pi$
$607$	−29.6547	−1.20365	−0.601824	+	0.798629i	$0.705559\pi$
$608$	0	0
$609$	4.84484	0.196323
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−3.33755	−0.134802	−0.0674012	−	0.997726i	$-0.521471\pi$
$613$	−3.33755	−0.134802	−0.0674012	+	0.997726i	$0.521471\pi$
$614$	0	0
$615$	−1.20829	−0.0487229
$616$	0	0
$617$	−27.0763	−1.09005	−0.545025	−	0.838420i	$-0.683480\pi$
$617$	−27.0763	−1.09005	−0.545025	+	0.838420i	$0.683480\pi$
$618$	0	0
$619$	−2.80966	−0.112930	−0.0564648	−	0.998405i	$-0.517983\pi$
$619$	−2.80966	−0.112930	−0.0564648	+	0.998405i	$0.517983\pi$
$620$	0	0
$621$	35.3056	1.41677
$622$	0	0
$623$	0.0602602	0.00241428
$624$	0	0
$625$	−1.73461	−0.0693843
$626$	0	0
$627$	0.0430475	0.00171915
$628$	0	0
$629$	−6.31834	−0.251929
$630$	0	0
$631$	7.59709	0.302435	0.151218	−	0.988500i	$-0.451681\pi$
$631$	7.59709	0.302435	0.151218	+	0.988500i	$0.451681\pi$
$632$	0	0
$633$	3.98877	0.158539
$634$	0	0
$635$	−15.2007	−0.603219
$636$	0	0
$637$	0	0
$638$	0	0
$639$	1.37611	0.0544382
$640$	0	0
$641$	28.1025	1.10998	0.554991	−	0.831856i	$-0.312722\pi$
$641$	28.1025	1.10998	0.554991	+	0.831856i	$0.312722\pi$
$642$	0	0
$643$	15.9447	0.628797	0.314398	−	0.949291i	$-0.398197\pi$
$643$	15.9447	0.628797	0.314398	+	0.949291i	$0.398197\pi$
$644$	0	0
$645$	−9.45695	−0.372367
$646$	0	0
$647$	−6.54759	−0.257412	−0.128706	−	0.991683i	$-0.541082\pi$
$647$	−6.54759	−0.257412	−0.128706	+	0.991683i	$0.541082\pi$
$648$	0	0
$649$	−38.8825	−1.52627
$650$	0	0
$651$	4.91948	0.192810
$652$	0	0
$653$	43.4384	1.69988	0.849939	−	0.526881i	$-0.176639\pi$
$653$	43.4384	1.69988	0.849939	+	0.526881i	$0.176639\pi$
$654$	0	0
$655$	16.4761	0.643775
$656$	0	0
$657$	27.6175	1.07746
$658$	0	0
$659$	4.61021	0.179588	0.0897941	−	0.995960i	$-0.471379\pi$
$659$	4.61021	0.179588	0.0897941	+	0.995960i	$0.471379\pi$
$660$	0	0
$661$	−15.7249	−0.611626	−0.305813	−	0.952092i	$-0.598928\pi$
$661$	−15.7249	−0.611626	−0.305813	+	0.952092i	$0.598928\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−0.0176921	−0.000686070 0
$666$	0	0
$667$	−57.4150	−2.22312
$668$	0	0
$669$	−10.4179	−0.402781
$670$	0	0
$671$	38.1362	1.47223
$672$	0	0
$673$	6.17462	0.238014	0.119007	−	0.992893i	$-0.462029\pi$
$673$	6.17462	0.238014	0.119007	+	0.992893i	$0.462029\pi$
$674$	0	0
$675$	−11.8217	−0.455017
$676$	0	0
$677$	14.9856	0.575945	0.287972	−	0.957639i	$-0.407019\pi$
$677$	14.9856	0.575945	0.287972	+	0.957639i	$0.407019\pi$
$678$	0	0
$679$	16.0537	0.616086
$680$	0	0
$681$	3.35925	0.128727
$682$	0	0
$683$	11.2056	0.428770	0.214385	−	0.976749i	$-0.431225\pi$
$683$	11.2056	0.428770	0.214385	+	0.976749i	$0.431225\pi$
$684$	0	0
$685$	24.8199	0.948320
$686$	0	0
$687$	7.16766	0.273463
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−3.29602	−0.125387	−0.0626933	−	0.998033i	$-0.519969\pi$
$691$	−3.29602	−0.125387	−0.0626933	+	0.998033i	$0.519969\pi$
$692$	0	0
$693$	−11.6141	−0.441181
$694$	0	0
$695$	−14.2275	−0.539679
$696$	0	0
$697$	3.24578	0.122943
$698$	0	0
$699$	5.36805	0.203038
$700$	0	0
$701$	−25.2549	−0.953864	−0.476932	−	0.878940i	$-0.658251\pi$
$701$	−25.2549	−0.953864	−0.476932	+	0.878940i	$0.658251\pi$
$702$	0	0
$703$	0.0272365	0.00102724
$704$	0	0
$705$	−13.8350	−0.521055
$706$	0	0
$707$	14.8575	0.558775
$708$	0	0
$709$	23.2928	0.874778	0.437389	−	0.899272i	$-0.355903\pi$
$709$	23.2928	0.874778	0.437389	+	0.899272i	$0.355903\pi$
$710$	0	0
$711$	22.5867	0.847069
$712$	0	0
$713$	−58.2995	−2.18333
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−17.4586	−0.652002
$718$	0	0
$719$	−3.96844	−0.147998	−0.0739989	−	0.997258i	$-0.523576\pi$
$719$	−3.96844	−0.147998	−0.0739989	+	0.997258i	$0.523576\pi$
$720$	0	0
$721$	−11.8433	−0.441067
$722$	0	0
$723$	0.533011	0.0198229
$724$	0	0
$725$	19.2248	0.713990
$726$	0	0
$727$	1.38311	0.0512966	0.0256483	−	0.999671i	$-0.491835\pi$
$727$	1.38311	0.0512966	0.0256483	+	0.999671i	$0.491835\pi$
$728$	0	0
$729$	−1.81471	−0.0672116
$730$	0	0
$731$	25.4039	0.939595
$732$	0	0
$733$	22.4616	0.829637	0.414819	−	0.909904i	$-0.363845\pi$
$733$	22.4616	0.829637	0.414819	+	0.909904i	$0.363845\pi$
$734$	0	0
$735$	−1.06266	−0.0391967
$736$	0	0
$737$	43.5187	1.60303
$738$	0	0
$739$	40.8199	1.50158	0.750792	−	0.660539i	$-0.229672\pi$
$739$	40.8199	1.50158	0.750792	+	0.660539i	$0.229672\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	14.0414	0.515130	0.257565	−	0.966261i	$-0.417080\pi$
$743$	14.0414	0.515130	0.257565	+	0.966261i	$0.417080\pi$
$744$	0	0
$745$	−18.4909	−0.677455
$746$	0	0
$747$	26.7553	0.978926
$748$	0	0
$749$	−8.59602	−0.314092
$750$	0	0
$751$	−43.8260	−1.59924	−0.799618	−	0.600510i	$-0.794964\pi$
$751$	−43.8260	−1.59924	−0.799618	+	0.600510i	$0.794964\pi$
$752$	0	0
$753$	−7.16941	−0.261268
$754$	0	0
$755$	−8.27452	−0.301141
$756$	0	0
$757$	−26.8248	−0.974963	−0.487482	−	0.873133i	$-0.662084\pi$
$757$	−26.8248	−0.974963	−0.487482	+	0.873133i	$0.662084\pi$
$758$	0	0
$759$	−30.6412	−1.11221
$760$	0	0
$761$	−11.3939	−0.413027	−0.206513	−	0.978444i	$-0.566212\pi$
$761$	−11.3939	−0.413027	−0.206513	+	0.978444i	$0.566212\pi$
$762$	0	0
$763$	4.79814	0.173704
$764$	0	0
$765$	−10.0707	−0.364107
$766$	0	0
$767$	0	0
$768$	0	0
$769$	15.6620	0.564785	0.282392	−	0.959299i	$-0.408872\pi$
$769$	15.6620	0.564785	0.282392	+	0.959299i	$0.408872\pi$
$770$	0	0
$771$	−3.57739	−0.128837
$772$	0	0
$773$	36.6682	1.31886	0.659432	−	0.751764i	$-0.270797\pi$
$773$	36.6682	1.31886	0.659432	+	0.751764i	$0.270797\pi$
$774$	0	0
$775$	19.5209	0.701213
$776$	0	0
$777$	1.63593	0.0586885
$778$	0	0
$779$	−0.0139916	−0.000501300 0
$780$	0	0
$781$	−2.65450	−0.0949856
$782$	0	0
$783$	−26.4225	−0.944263
$784$	0	0
$785$	−0.00962144	−0.000343404 0
$786$	0	0
$787$	33.9486	1.21014	0.605069	−	0.796173i	$-0.293146\pi$
$787$	33.9486	1.21014	0.605069	+	0.796173i	$0.293146\pi$
$788$	0	0
$789$	−2.62035	−0.0932869
$790$	0	0
$791$	15.2321	0.541590
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−6.59605	−0.233938
$796$	0	0
$797$	−26.6818	−0.945118	−0.472559	−	0.881299i	$-0.656670\pi$
$797$	−26.6818	−0.945118	−0.472559	+	0.881299i	$0.656670\pi$
$798$	0	0
$799$	37.1644	1.31478
$800$	0	0
$801$	−0.147863	−0.00522447
$802$	0	0
$803$	−53.2737	−1.87999
$804$	0	0
$805$	12.5933	0.443854
$806$	0	0
$807$	−15.9578	−0.561741
$808$	0	0
$809$	26.7497	0.940469	0.470235	−	0.882541i	$-0.344169\pi$
$809$	26.7497	0.940469	0.470235	+	0.882541i	$0.344169\pi$
$810$	0	0
$811$	37.4354	1.31453	0.657267	−	0.753658i	$-0.271712\pi$
$811$	37.4354	1.31453	0.657267	+	0.753658i	$0.271712\pi$
$812$	0	0
$813$	−13.5691	−0.475888
$814$	0	0
$815$	−5.84995	−0.204915
$816$	0	0
$817$	−0.109508	−0.00383121
$818$	0	0
$819$	0	0
$820$	0	0
$821$	23.8525	0.832458	0.416229	−	0.909260i	$-0.363352\pi$
$821$	23.8525	0.832458	0.416229	+	0.909260i	$0.363352\pi$
$822$	0	0
$823$	12.9901	0.452805	0.226402	−	0.974034i	$-0.427304\pi$
$823$	12.9901	0.452805	0.226402	+	0.974034i	$0.427304\pi$
$824$	0	0
$825$	10.2599	0.357203
$826$	0	0
$827$	−36.2537	−1.26067	−0.630333	−	0.776325i	$-0.717082\pi$
$827$	−36.2537	−1.26067	−0.630333	+	0.776325i	$0.717082\pi$
$828$	0	0
$829$	−13.3569	−0.463904	−0.231952	−	0.972727i	$-0.574511\pi$
$829$	−13.3569	−0.463904	−0.231952	+	0.972727i	$0.574511\pi$
$830$	0	0
$831$	0.889806	0.0308670
$832$	0	0
$833$	2.85457	0.0989052
$834$	0	0
$835$	20.4317	0.707069
$836$	0	0
$837$	−26.8295	−0.927365
$838$	0	0
$839$	−43.0223	−1.48529	−0.742647	−	0.669683i	$-0.766430\pi$
$839$	−43.0223	−1.48529	−0.742647	+	0.669683i	$0.766430\pi$
$840$	0	0
$841$	13.9690	0.481689
$842$	0	0
$843$	1.41243	0.0486468
$844$	0	0
$845$	0	0
$846$	0	0
$847$	11.4033	0.391823
$848$	0	0
$849$	−13.9086	−0.477340
$850$	0	0
$851$	−19.3869	−0.664576
$852$	0	0
$853$	19.7144	0.675008	0.337504	−	0.941324i	$-0.390417\pi$
$853$	19.7144	0.675008	0.337504	+	0.941324i	$0.390417\pi$
$854$	0	0
$855$	0.0434117	0.00148465
$856$	0	0
$857$	33.8090	1.15489	0.577447	−	0.816428i	$-0.304049\pi$
$857$	33.8090	1.15489	0.577447	+	0.816428i	$0.304049\pi$
$858$	0	0
$859$	−45.0185	−1.53601	−0.768006	−	0.640443i	$-0.778751\pi$
$859$	−45.0185	−1.53601	−0.768006	+	0.640443i	$0.778751\pi$
$860$	0	0
$861$	−0.840388	−0.0286403
$862$	0	0
$863$	−2.68668	−0.0914557	−0.0457279	−	0.998954i	$-0.514561\pi$
$863$	−2.68668	−0.0914557	−0.0457279	+	0.998954i	$0.514561\pi$
$864$	0	0
$865$	26.6289	0.905409
$866$	0	0
$867$	6.54206	0.222180
$868$	0	0
$869$	−43.5695	−1.47799
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−39.3916	−1.33320
$874$	0	0
$875$	−11.4056	−0.385579
$876$	0	0
$877$	30.5122	1.03032	0.515161	−	0.857093i	$-0.327732\pi$
$877$	30.5122	1.03032	0.515161	+	0.857093i	$0.327732\pi$
$878$	0	0
$879$	−18.7485	−0.632371
$880$	0	0
$881$	−29.6039	−0.997381	−0.498690	−	0.866780i	$-0.666186\pi$
$881$	−29.6039	−0.997381	−0.498690	+	0.866780i	$0.666186\pi$
$882$	0	0
$883$	−31.4145	−1.05718	−0.528592	−	0.848876i	$-0.677280\pi$
$883$	−31.4145	−1.05718	−0.528592	+	0.848876i	$0.677280\pi$
$884$	0	0
$885$	8.72952	0.293440
$886$	0	0
$887$	−2.91909	−0.0980134	−0.0490067	−	0.998798i	$-0.515606\pi$
$887$	−2.91909	−0.0980134	−0.0490067	+	0.998798i	$0.515606\pi$
$888$	0	0
$889$	−10.5724	−0.354585
$890$	0	0
$891$	20.7410	0.694850
$892$	0	0
$893$	−0.160204	−0.00536103
$894$	0	0
$895$	−22.0171	−0.735949
$896$	0	0
$897$	0	0
$898$	0	0
$899$	43.6310	1.45517
$900$	0	0
$901$	17.7187	0.590296
$902$	0	0
$903$	−6.57749	−0.218885
$904$	0	0
$905$	−34.9432	−1.16155
$906$	0	0
$907$	24.2232	0.804319	0.402160	−	0.915570i	$-0.368260\pi$
$907$	24.2232	0.804319	0.402160	+	0.915570i	$0.368260\pi$
$908$	0	0
$909$	−36.4564	−1.20918
$910$	0	0
$911$	−25.3807	−0.840901	−0.420450	−	0.907316i	$-0.638128\pi$
$911$	−25.3807	−0.840901	−0.420450	+	0.907316i	$0.638128\pi$
$912$	0	0
$913$	−51.6106	−1.70806
$914$	0	0
$915$	−8.56197	−0.283050
$916$	0	0
$917$	11.4595	0.378425
$918$	0	0
$919$	−33.2005	−1.09518	−0.547591	−	0.836746i	$-0.684455\pi$
$919$	−33.2005	−1.09518	−0.547591	+	0.836746i	$0.684455\pi$
$920$	0	0
$921$	−0.285286	−0.00940048
$922$	0	0
$923$	0	0
$924$	0	0
$925$	6.49150	0.213439
$926$	0	0
$927$	29.0603	0.954464
$928$	0	0
$929$	3.18867	0.104617	0.0523084	−	0.998631i	$-0.483342\pi$
$929$	3.18867	0.104617	0.0523084	+	0.998631i	$0.483342\pi$
$930$	0	0
$931$	−0.0123052	−0.000403287 0
$932$	0	0
$933$	5.21806	0.170831
$934$	0	0
$935$	19.4262	0.635305
$936$	0	0
$937$	−24.6622	−0.805679	−0.402840	−	0.915271i	$-0.631977\pi$
$937$	−24.6622	−0.805679	−0.402840	+	0.915271i	$0.631977\pi$
$938$	0	0
$939$	22.7407	0.742116
$940$	0	0
$941$	22.7134	0.740435	0.370218	−	0.928945i	$-0.379283\pi$
$941$	22.7134	0.740435	0.370218	+	0.928945i	$0.379283\pi$
$942$	0	0
$943$	9.95922	0.324317
$944$	0	0
$945$	5.79544	0.188526
$946$	0	0
$947$	29.0536	0.944114	0.472057	−	0.881568i	$-0.343512\pi$
$947$	29.0536	0.944114	0.472057	+	0.881568i	$0.343512\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−13.4835	−0.437232
$952$	0	0
$953$	8.40319	0.272206	0.136103	−	0.990695i	$-0.456542\pi$
$953$	8.40319	0.272206	0.136103	+	0.990695i	$0.456542\pi$
$954$	0	0
$955$	18.8105	0.608692
$956$	0	0
$957$	22.9317	0.741276
$958$	0	0
$959$	17.2627	0.557443
$960$	0	0
$961$	13.3031	0.429133
$962$	0	0
$963$	21.0924	0.679691
$964$	0	0
$965$	−10.4670	−0.336944
$966$	0	0
$967$	−40.1233	−1.29028	−0.645140	−	0.764065i	$-0.723201\pi$
$967$	−40.1233	−1.29028	−0.645140	+	0.764065i	$0.723201\pi$
$968$	0	0
$969$	0.0259617	0.000834009 0
$970$	0	0
$971$	−3.03848	−0.0975093	−0.0487547	−	0.998811i	$-0.515525\pi$
$971$	−3.03848	−0.0975093	−0.0487547	+	0.998811i	$0.515525\pi$
$972$	0	0
$973$	−9.89549	−0.317235
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−34.7205	−1.11081	−0.555403	−	0.831581i	$-0.687436\pi$
$977$	−34.7205	−1.11081	−0.555403	+	0.831581i	$0.687436\pi$
$978$	0	0
$979$	0.285225	0.00911582
$980$	0	0
$981$	−11.7733	−0.375894
$982$	0	0
$983$	−48.2951	−1.54037	−0.770187	−	0.637818i	$-0.779837\pi$
$983$	−48.2951	−1.54037	−0.770187	+	0.637818i	$0.779837\pi$
$984$	0	0
$985$	25.1058	0.799939
$986$	0	0
$987$	−9.62249	−0.306287
$988$	0	0
$989$	77.9482	2.47861
$990$	0	0
$991$	46.7064	1.48368	0.741838	−	0.670579i	$-0.233954\pi$
$991$	46.7064	1.48368	0.741838	+	0.670579i	$0.233954\pi$
$992$	0	0
$993$	8.85966	0.281153
$994$	0	0
$995$	−27.2183	−0.862877
$996$	0	0
$997$	57.8202	1.83118	0.915591	−	0.402110i	$-0.131723\pi$
$997$	57.8202	1.83118	0.915591	+	0.402110i	$0.131723\pi$
$998$	0	0
$999$	−8.92191	−0.282277

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9464.2.a.bs.1.8	yes	15
13.12	even	2		9464.2.a.br.1.8	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9464.2.a.br.1.8	✓	15	13.12	even	2
9464.2.a.bs.1.8	yes	15	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 9464 = 2^{3} \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9464.a (trivial)

\(f(q)\)	\(=\)	\(q-0.739098 q^{3} +1.43777 q^{5} +1.00000 q^{7} -2.45373 q^{9} +4.73322 q^{11} -1.06266 q^{15} +2.85457 q^{17} -0.0123052 q^{19} -0.739098 q^{21} +8.75886 q^{23} -2.93281 q^{25} +4.03085 q^{27} -6.55507 q^{29} -6.65606 q^{31} -3.49831 q^{33} +1.43777 q^{35} -2.21341 q^{37} +1.13704 q^{41} +8.89935 q^{43} -3.52791 q^{45} +13.0192 q^{47} +1.00000 q^{49} -2.10981 q^{51} +6.20713 q^{53} +6.80529 q^{55} +0.00909476 q^{57} -8.21482 q^{59} +8.05714 q^{61} -2.45373 q^{63} +9.19433 q^{67} -6.47366 q^{69} -0.560824 q^{71} -11.2553 q^{73} +2.16763 q^{75} +4.73322 q^{77} -9.20505 q^{79} +4.38201 q^{81} -10.9039 q^{83} +4.10423 q^{85} +4.84484 q^{87} +0.0602602 q^{89} +4.91948 q^{93} -0.0176921 q^{95} +16.0537 q^{97} -11.6141 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 3 q^{3} + 4 q^{5} + 15 q^{7} + 16 q^{9} + 15 q^{11} + 8 q^{15} + 2 q^{17} + 13 q^{19} - 3 q^{21} - 10 q^{23} + 23 q^{25} - 9 q^{27} + 25 q^{29} + 19 q^{31} - 24 q^{33} + 4 q^{35} - 2 q^{37} + 30 q^{41}+ \cdots + 70 q^{99}+O(q^{100}) \) 15 * q - 3 * q^3 + 4 * q^5 + 15 * q^7 + 16 * q^9 + 15 * q^11 + 8 * q^15 + 2 * q^17 + 13 * q^19 - 3 * q^21 - 10 * q^23 + 23 * q^25 - 9 * q^27 + 25 * q^29 + 19 * q^31 - 24 * q^33 + 4 * q^35 - 2 * q^37 + 30 * q^41 + 15 * q^43 - 2 * q^45 + 13 * q^47 + 15 * q^49 + q^51 + 62 * q^53 - 3 * q^55 - 2 * q^57 + 22 * q^59 - q^61 + 16 * q^63 + 29 * q^67 + 32 * q^69 + 20 * q^71 + 21 * q^73 - 46 * q^75 + 15 * q^77 + 29 * q^79 + 7 * q^81 + 23 * q^83 + 8 * q^85 - 22 * q^87 + 9 * q^89 + 10 * q^93 + 28 * q^95 + 11 * q^97 + 70 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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