LMFDB - Embedded newform 9464.2.a.bs.1.4

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			1.4
Root			$2.23382$ 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)

$f(q)$	$=$	$q-2.23382 q^{3} -3.65424 q^{5} +1.00000 q^{7} +1.98994 q^{9} +3.59386 q^{11} +8.16292 q^{15} +0.523266 q^{17} -4.88563 q^{19} -2.23382 q^{21} -5.85374 q^{23} +8.35350 q^{25} +2.25628 q^{27} -0.335543 q^{29} +8.80205 q^{31} -8.02802 q^{33} -3.65424 q^{35} +5.06510 q^{37} -3.59478 q^{41} +9.42197 q^{43} -7.27175 q^{45} -11.0569 q^{47} +1.00000 q^{49} -1.16888 q^{51} -2.73964 q^{53} -13.1328 q^{55} +10.9136 q^{57} +10.7238 q^{59} -0.0368175 q^{61} +1.98994 q^{63} +0.533398 q^{67} +13.0762 q^{69} +2.19349 q^{71} -4.19000 q^{73} -18.6602 q^{75} +3.59386 q^{77} +4.08884 q^{79} -11.0100 q^{81} +5.52286 q^{83} -1.91214 q^{85} +0.749542 q^{87} -16.0882 q^{89} -19.6622 q^{93} +17.8533 q^{95} -12.3593 q^{97} +7.15157 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.23382	−1.28970	−0.644848	−	0.764311i	$-0.723079\pi$
$3$	−2.23382	−1.28970	−0.644848	+	0.764311i	$0.723079\pi$
$4$	0	0
$5$	−3.65424	−1.63423	−0.817114	−	0.576476i	$-0.804427\pi$
$5$	−3.65424	−1.63423	−0.817114	+	0.576476i	$0.804427\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.98994	0.663315
$10$	0	0
$11$	3.59386	1.08359	0.541794	−	0.840511i	$-0.317745\pi$
$11$	3.59386	1.08359	0.541794	+	0.840511i	$0.317745\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	8.16292	2.10766
$16$	0	0
$17$	0.523266	0.126911	0.0634553	−	0.997985i	$-0.479788\pi$
$17$	0.523266	0.126911	0.0634553	+	0.997985i	$0.479788\pi$
$18$	0	0
$19$	−4.88563	−1.12084	−0.560421	−	0.828208i	$-0.689361\pi$
$19$	−4.88563	−1.12084	−0.560421	+	0.828208i	$0.689361\pi$
$20$	0	0
$21$	−2.23382	−0.487459
$22$	0	0
$23$	−5.85374	−1.22059	−0.610294	−	0.792175i	$-0.708949\pi$
$23$	−5.85374	−1.22059	−0.610294	+	0.792175i	$0.708949\pi$
$24$	0	0
$25$	8.35350	1.67070
$26$	0	0
$27$	2.25628	0.434221
$28$	0	0
$29$	−0.335543	−0.0623087	−0.0311544	−	0.999515i	$-0.509918\pi$
$29$	−0.335543	−0.0623087	−0.0311544	+	0.999515i	$0.509918\pi$
$30$	0	0
$31$	8.80205	1.58089	0.790447	−	0.612530i	$-0.209848\pi$
$31$	8.80205	1.58089	0.790447	+	0.612530i	$0.209848\pi$
$32$	0	0
$33$	−8.02802	−1.39750
$34$	0	0
$35$	−3.65424	−0.617680
$36$	0	0
$37$	5.06510	0.832697	0.416348	−	0.909205i	$-0.363310\pi$
$37$	5.06510	0.832697	0.416348	+	0.909205i	$0.363310\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3.59478	−0.561410	−0.280705	−	0.959794i	$-0.590568\pi$
$41$	−3.59478	−0.561410	−0.280705	+	0.959794i	$0.590568\pi$
$42$	0	0
$43$	9.42197	1.43684	0.718419	−	0.695611i	$-0.244866\pi$
$43$	9.42197	1.43684	0.718419	+	0.695611i	$0.244866\pi$
$44$	0	0
$45$	−7.27175	−1.08401
$46$	0	0
$47$	−11.0569	−1.61282	−0.806410	−	0.591357i	$-0.798592\pi$
$47$	−11.0569	−1.61282	−0.806410	+	0.591357i	$0.798592\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.16888	−0.163676
$52$	0	0
$53$	−2.73964	−0.376318	−0.188159	−	0.982139i	$-0.560252\pi$
$53$	−2.73964	−0.376318	−0.188159	+	0.982139i	$0.560252\pi$
$54$	0	0
$55$	−13.1328	−1.77083
$56$	0	0
$57$	10.9136	1.44554
$58$	0	0
$59$	10.7238	1.39613	0.698063	−	0.716036i	$-0.254046\pi$
$59$	10.7238	1.39613	0.698063	+	0.716036i	$0.254046\pi$
$60$	0	0
$61$	−0.0368175	−0.00471399	−0.00235700	−	0.999997i	$-0.500750\pi$
$61$	−0.0368175	−0.00471399	−0.00235700	+	0.999997i	$0.500750\pi$
$62$	0	0
$63$	1.98994	0.250709
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0.533398	0.0651650	0.0325825	−	0.999469i	$-0.489627\pi$
$67$	0.533398	0.0651650	0.0325825	+	0.999469i	$0.489627\pi$
$68$	0	0
$69$	13.0762	1.57419
$70$	0	0
$71$	2.19349	0.260320	0.130160	−	0.991493i	$-0.458451\pi$
$71$	2.19349	0.260320	0.130160	+	0.991493i	$0.458451\pi$
$72$	0	0
$73$	−4.19000	−0.490402	−0.245201	−	0.969472i	$-0.578854\pi$
$73$	−4.19000	−0.490402	−0.245201	+	0.969472i	$0.578854\pi$
$74$	0	0
$75$	−18.6602	−2.15470
$76$	0	0
$77$	3.59386	0.409558
$78$	0	0
$79$	4.08884	0.460031	0.230015	−	0.973187i	$-0.426122\pi$
$79$	4.08884	0.460031	0.230015	+	0.973187i	$0.426122\pi$
$80$	0	0
$81$	−11.0100	−1.22333
$82$	0	0
$83$	5.52286	0.606213	0.303106	−	0.952957i	$-0.401976\pi$
$83$	5.52286	0.606213	0.303106	+	0.952957i	$0.401976\pi$
$84$	0	0
$85$	−1.91214	−0.207401
$86$	0	0
$87$	0.749542	0.0803593
$88$	0	0
$89$	−16.0882	−1.70534	−0.852672	−	0.522446i	$-0.825019\pi$
$89$	−16.0882	−1.70534	−0.852672	+	0.522446i	$0.825019\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−19.6622	−2.03887
$94$	0	0
$95$	17.8533	1.83171
$96$	0	0
$97$	−12.3593	−1.25490	−0.627449	−	0.778657i	$-0.715901\pi$
$97$	−12.3593	−1.25490	−0.627449	+	0.778657i	$0.715901\pi$
$98$	0	0
$99$	7.15157	0.718760
$100$	0	0
$101$	15.5254	1.54484	0.772418	−	0.635114i	$-0.219047\pi$
$101$	15.5254	1.54484	0.772418	+	0.635114i	$0.219047\pi$
$102$	0	0
$103$	−11.3488	−1.11823	−0.559114	−	0.829091i	$-0.688859\pi$
$103$	−11.3488	−1.11823	−0.559114	+	0.829091i	$0.688859\pi$
$104$	0	0
$105$	8.16292	0.796619
$106$	0	0
$107$	−12.2867	−1.18780	−0.593900	−	0.804539i	$-0.702413\pi$
$107$	−12.2867	−1.18780	−0.593900	+	0.804539i	$0.702413\pi$
$108$	0	0
$109$	3.68889	0.353332	0.176666	−	0.984271i	$-0.443469\pi$
$109$	3.68889	0.353332	0.176666	+	0.984271i	$0.443469\pi$
$110$	0	0
$111$	−11.3145	−1.07393
$112$	0	0
$113$	−20.2282	−1.90291	−0.951456	−	0.307784i	$-0.900413\pi$
$113$	−20.2282	−1.90291	−0.951456	+	0.307784i	$0.900413\pi$
$114$	0	0
$115$	21.3910	1.99472
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.523266	0.0479677
$120$	0	0
$121$	1.91580	0.174163
$122$	0	0
$123$	8.03008	0.724048
$124$	0	0
$125$	−12.2545	−1.09608
$126$	0	0
$127$	−5.07362	−0.450211	−0.225105	−	0.974334i	$-0.572273\pi$
$127$	−5.07362	−0.450211	−0.225105	+	0.974334i	$0.572273\pi$
$128$	0	0
$129$	−21.0470	−1.85308
$130$	0	0
$131$	−10.7310	−0.937571	−0.468785	−	0.883312i	$-0.655308\pi$
$131$	−10.7310	−0.937571	−0.468785	+	0.883312i	$0.655308\pi$
$132$	0	0
$133$	−4.88563	−0.423638
$134$	0	0
$135$	−8.24500	−0.709616
$136$	0	0
$137$	3.18036	0.271717	0.135858	−	0.990728i	$-0.456621\pi$
$137$	3.18036	0.271717	0.135858	+	0.990728i	$0.456621\pi$
$138$	0	0
$139$	−13.1951	−1.11919	−0.559596	−	0.828765i	$-0.689044\pi$
$139$	−13.1951	−1.11919	−0.559596	+	0.828765i	$0.689044\pi$
$140$	0	0
$141$	24.6992	2.08005
$142$	0	0
$143$	0	0
$144$	0	0
$145$	1.22616	0.101827
$146$	0	0
$147$	−2.23382	−0.184242
$148$	0	0
$149$	13.9497	1.14281	0.571404	−	0.820669i	$-0.306399\pi$
$149$	13.9497	1.14281	0.571404	+	0.820669i	$0.306399\pi$
$150$	0	0
$151$	0.797580	0.0649061	0.0324531	−	0.999473i	$-0.489668\pi$
$151$	0.797580	0.0649061	0.0324531	+	0.999473i	$0.489668\pi$
$152$	0	0
$153$	1.04127	0.0841817
$154$	0	0
$155$	−32.1648	−2.58354
$156$	0	0
$157$	−23.2501	−1.85556	−0.927778	−	0.373131i	$-0.878284\pi$
$157$	−23.2501	−1.85556	−0.927778	+	0.373131i	$0.878284\pi$
$158$	0	0
$159$	6.11985	0.485336
$160$	0	0
$161$	−5.85374	−0.461339
$162$	0	0
$163$	17.4334	1.36549	0.682744	−	0.730657i	$-0.260786\pi$
$163$	17.4334	1.36549	0.682744	+	0.730657i	$0.260786\pi$
$164$	0	0
$165$	29.3363	2.28383
$166$	0	0
$167$	−20.9467	−1.62090	−0.810451	−	0.585807i	$-0.800778\pi$
$167$	−20.9467	−1.62090	−0.810451	+	0.585807i	$0.800778\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−9.72214	−0.743471
$172$	0	0
$173$	22.5761	1.71643	0.858214	−	0.513293i	$-0.171574\pi$
$173$	22.5761	1.71643	0.858214	+	0.513293i	$0.171574\pi$
$174$	0	0
$175$	8.35350	0.631466
$176$	0	0
$177$	−23.9551	−1.80058
$178$	0	0
$179$	−19.0378	−1.42296	−0.711478	−	0.702709i	$-0.751974\pi$
$179$	−19.0378	−1.42296	−0.711478	+	0.702709i	$0.751974\pi$
$180$	0	0
$181$	6.34003	0.471251	0.235626	−	0.971844i	$-0.424286\pi$
$181$	6.34003	0.471251	0.235626	+	0.971844i	$0.424286\pi$
$182$	0	0
$183$	0.0822435	0.00607962
$184$	0	0
$185$	−18.5091	−1.36082
$186$	0	0
$187$	1.88054	0.137519
$188$	0	0
$189$	2.25628	0.164120
$190$	0	0
$191$	16.3591	1.18371	0.591853	−	0.806046i	$-0.298397\pi$
$191$	16.3591	1.18371	0.591853	+	0.806046i	$0.298397\pi$
$192$	0	0
$193$	18.7785	1.35171	0.675853	−	0.737036i	$-0.263775\pi$
$193$	18.7785	1.35171	0.675853	+	0.737036i	$0.263775\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	16.7832	1.19575	0.597877	−	0.801588i	$-0.296011\pi$
$197$	16.7832	1.19575	0.597877	+	0.801588i	$0.296011\pi$
$198$	0	0
$199$	−13.1248	−0.930394	−0.465197	−	0.885207i	$-0.654016\pi$
$199$	−13.1248	−0.930394	−0.465197	+	0.885207i	$0.654016\pi$
$200$	0	0
$201$	−1.19151	−0.0840430
$202$	0	0
$203$	−0.335543	−0.0235505
$204$	0	0
$205$	13.1362	0.917471
$206$	0	0
$207$	−11.6486	−0.809635
$208$	0	0
$209$	−17.5583	−1.21453
$210$	0	0
$211$	20.6865	1.42412	0.712058	−	0.702120i	$-0.247763\pi$
$211$	20.6865	1.42412	0.712058	+	0.702120i	$0.247763\pi$
$212$	0	0
$213$	−4.89986	−0.335733
$214$	0	0
$215$	−34.4302	−2.34812
$216$	0	0
$217$	8.80205	0.597522
$218$	0	0
$219$	9.35969	0.632469
$220$	0	0
$221$	0	0
$222$	0	0
$223$	22.0881	1.47913	0.739564	−	0.673086i	$-0.235032\pi$
$223$	22.0881	1.47913	0.739564	+	0.673086i	$0.235032\pi$
$224$	0	0
$225$	16.6230	1.10820
$226$	0	0
$227$	−28.8399	−1.91417	−0.957085	−	0.289807i	$-0.906409\pi$
$227$	−28.8399	−1.91417	−0.957085	+	0.289807i	$0.906409\pi$
$228$	0	0
$229$	−17.3604	−1.14721	−0.573603	−	0.819134i	$-0.694455\pi$
$229$	−17.3604	−1.14721	−0.573603	+	0.819134i	$0.694455\pi$
$230$	0	0
$231$	−8.02802	−0.528205
$232$	0	0
$233$	−10.1724	−0.666415	−0.333208	−	0.942854i	$-0.608131\pi$
$233$	−10.1724	−0.666415	−0.333208	+	0.942854i	$0.608131\pi$
$234$	0	0
$235$	40.4047	2.63571
$236$	0	0
$237$	−9.13373	−0.593300
$238$	0	0
$239$	−18.0660	−1.16859	−0.584295	−	0.811541i	$-0.698629\pi$
$239$	−18.0660	−1.16859	−0.584295	+	0.811541i	$0.698629\pi$
$240$	0	0
$241$	16.2933	1.04954	0.524772	−	0.851243i	$-0.324151\pi$
$241$	16.2933	1.04954	0.524772	+	0.851243i	$0.324151\pi$
$242$	0	0
$243$	17.8254	1.14350
$244$	0	0
$245$	−3.65424	−0.233461
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−12.3371	−0.781830
$250$	0	0
$251$	−0.911182	−0.0575133	−0.0287566	−	0.999586i	$-0.509155\pi$
$251$	−0.911182	−0.0575133	−0.0287566	+	0.999586i	$0.509155\pi$
$252$	0	0
$253$	−21.0375	−1.32262
$254$	0	0
$255$	4.27138	0.267484
$256$	0	0
$257$	27.2763	1.70145	0.850726	−	0.525610i	$-0.176163\pi$
$257$	27.2763	1.70145	0.850726	+	0.525610i	$0.176163\pi$
$258$	0	0
$259$	5.06510	0.314730
$260$	0	0
$261$	−0.667712	−0.0413303
$262$	0	0
$263$	−12.4532	−0.767897	−0.383948	−	0.923355i	$-0.625436\pi$
$263$	−12.4532	−0.767897	−0.383948	+	0.923355i	$0.625436\pi$
$264$	0	0
$265$	10.0113	0.614989
$266$	0	0
$267$	35.9381	2.19938
$268$	0	0
$269$	5.82364	0.355074	0.177537	−	0.984114i	$-0.443187\pi$
$269$	5.82364	0.355074	0.177537	+	0.984114i	$0.443187\pi$
$270$	0	0
$271$	−3.08086	−0.187149	−0.0935744	−	0.995612i	$-0.529829\pi$
$271$	−3.08086	−0.187149	−0.0935744	+	0.995612i	$0.529829\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	30.0213	1.81035
$276$	0	0
$277$	7.29344	0.438220	0.219110	−	0.975700i	$-0.429685\pi$
$277$	7.29344	0.438220	0.219110	+	0.975700i	$0.429685\pi$
$278$	0	0
$279$	17.5156	1.04863
$280$	0	0
$281$	−0.556271	−0.0331844	−0.0165922	−	0.999862i	$-0.505282\pi$
$281$	−0.556271	−0.0331844	−0.0165922	+	0.999862i	$0.505282\pi$
$282$	0	0
$283$	−9.61970	−0.571832	−0.285916	−	0.958255i	$-0.592298\pi$
$283$	−9.61970	−0.571832	−0.285916	+	0.958255i	$0.592298\pi$
$284$	0	0
$285$	−39.8810	−2.36235
$286$	0	0
$287$	−3.59478	−0.212193
$288$	0	0
$289$	−16.7262	−0.983894
$290$	0	0
$291$	27.6085	1.61844
$292$	0	0
$293$	15.8707	0.927174	0.463587	−	0.886051i	$-0.346562\pi$
$293$	15.8707	0.927174	0.463587	+	0.886051i	$0.346562\pi$
$294$	0	0
$295$	−39.1876	−2.28159
$296$	0	0
$297$	8.10874	0.470517
$298$	0	0
$299$	0	0
$300$	0	0
$301$	9.42197	0.543073
$302$	0	0
$303$	−34.6810	−1.99237
$304$	0	0
$305$	0.134540	0.00770374
$306$	0	0
$307$	−27.0574	−1.54425	−0.772124	−	0.635472i	$-0.780805\pi$
$307$	−27.0574	−1.54425	−0.772124	+	0.635472i	$0.780805\pi$
$308$	0	0
$309$	25.3511	1.44217
$310$	0	0
$311$	3.61878	0.205202	0.102601	−	0.994723i	$-0.467283\pi$
$311$	3.61878	0.205202	0.102601	+	0.994723i	$0.467283\pi$
$312$	0	0
$313$	10.1368	0.572968	0.286484	−	0.958085i	$-0.407513\pi$
$313$	10.1368	0.572968	0.286484	+	0.958085i	$0.407513\pi$
$314$	0	0
$315$	−7.27175	−0.409716
$316$	0	0
$317$	−6.45061	−0.362302	−0.181151	−	0.983455i	$-0.557982\pi$
$317$	−6.45061	−0.362302	−0.181151	+	0.983455i	$0.557982\pi$
$318$	0	0
$319$	−1.20589	−0.0675170
$320$	0	0
$321$	27.4463	1.53190
$322$	0	0
$323$	−2.55649	−0.142247
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−8.24031	−0.455690
$328$	0	0
$329$	−11.0569	−0.609588
$330$	0	0
$331$	−8.25809	−0.453906	−0.226953	−	0.973906i	$-0.572876\pi$
$331$	−8.25809	−0.453906	−0.226953	+	0.973906i	$0.572876\pi$
$332$	0	0
$333$	10.0793	0.552340
$334$	0	0
$335$	−1.94917	−0.106494
$336$	0	0
$337$	14.2570	0.776629	0.388315	−	0.921527i	$-0.373057\pi$
$337$	14.2570	0.776629	0.388315	+	0.921527i	$0.373057\pi$
$338$	0	0
$339$	45.1862	2.45418
$340$	0	0
$341$	31.6333	1.71304
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−47.7836	−2.57258
$346$	0	0
$347$	−10.0018	−0.536927	−0.268463	−	0.963290i	$-0.586516\pi$
$347$	−10.0018	−0.536927	−0.268463	+	0.963290i	$0.586516\pi$
$348$	0	0
$349$	15.4589	0.827498	0.413749	−	0.910391i	$-0.364219\pi$
$349$	15.4589	0.827498	0.413749	+	0.910391i	$0.364219\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	11.5355	0.613974	0.306987	−	0.951714i	$-0.400679\pi$
$353$	11.5355	0.613974	0.306987	+	0.951714i	$0.400679\pi$
$354$	0	0
$355$	−8.01556	−0.425422
$356$	0	0
$357$	−1.16888	−0.0618637
$358$	0	0
$359$	7.97359	0.420830	0.210415	−	0.977612i	$-0.432518\pi$
$359$	7.97359	0.420830	0.210415	+	0.977612i	$0.432518\pi$
$360$	0	0
$361$	4.86942	0.256285
$362$	0	0
$363$	−4.27954	−0.224618
$364$	0	0
$365$	15.3113	0.801428
$366$	0	0
$367$	−2.12105	−0.110718	−0.0553590	−	0.998467i	$-0.517630\pi$
$367$	−2.12105	−0.110718	−0.0553590	+	0.998467i	$0.517630\pi$
$368$	0	0
$369$	−7.15341	−0.372391
$370$	0	0
$371$	−2.73964	−0.142235
$372$	0	0
$373$	35.2329	1.82429	0.912144	−	0.409869i	$-0.134426\pi$
$373$	35.2329	1.82429	0.912144	+	0.409869i	$0.134426\pi$
$374$	0	0
$375$	27.3744	1.41361
$376$	0	0
$377$	0	0
$378$	0	0
$379$	15.3183	0.786846	0.393423	−	0.919358i	$-0.371291\pi$
$379$	15.3183	0.786846	0.393423	+	0.919358i	$0.371291\pi$
$380$	0	0
$381$	11.3335	0.580635
$382$	0	0
$383$	38.8011	1.98264	0.991321	−	0.131465i	$-0.0419680\pi$
$383$	38.8011	1.98264	0.991321	+	0.131465i	$0.0419680\pi$
$384$	0	0
$385$	−13.1328	−0.669311
$386$	0	0
$387$	18.7492	0.953076
$388$	0	0
$389$	−7.56180	−0.383398	−0.191699	−	0.981454i	$-0.561400\pi$
$389$	−7.56180	−0.383398	−0.191699	+	0.981454i	$0.561400\pi$
$390$	0	0
$391$	−3.06306	−0.154906
$392$	0	0
$393$	23.9711	1.20918
$394$	0	0
$395$	−14.9416	−0.751795
$396$	0	0
$397$	−0.959334	−0.0481476	−0.0240738	−	0.999710i	$-0.507664\pi$
$397$	−0.959334	−0.0481476	−0.0240738	+	0.999710i	$0.507664\pi$
$398$	0	0
$399$	10.9136	0.546364
$400$	0	0
$401$	12.7776	0.638085	0.319043	−	0.947740i	$-0.396639\pi$
$401$	12.7776	0.638085	0.319043	+	0.947740i	$0.396639\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	40.2331	1.99920
$406$	0	0
$407$	18.2032	0.902300
$408$	0	0
$409$	26.2802	1.29947	0.649736	−	0.760160i	$-0.274880\pi$
$409$	26.2802	1.29947	0.649736	+	0.760160i	$0.274880\pi$
$410$	0	0
$411$	−7.10436	−0.350432
$412$	0	0
$413$	10.7238	0.527686
$414$	0	0
$415$	−20.1819	−0.990690
$416$	0	0
$417$	29.4754	1.44342
$418$	0	0
$419$	−18.4420	−0.900950	−0.450475	−	0.892789i	$-0.648745\pi$
$419$	−18.4420	−0.900950	−0.450475	+	0.892789i	$0.648745\pi$
$420$	0	0
$421$	37.3294	1.81932	0.909661	−	0.415351i	$-0.136341\pi$
$421$	37.3294	1.81932	0.909661	+	0.415351i	$0.136341\pi$
$422$	0	0
$423$	−22.0027	−1.06981
$424$	0	0
$425$	4.37110	0.212030
$426$	0	0
$427$	−0.0368175	−0.00178172
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−24.0791	−1.15985	−0.579925	−	0.814670i	$-0.696918\pi$
$431$	−24.0791	−1.15985	−0.579925	+	0.814670i	$0.696918\pi$
$432$	0	0
$433$	−21.5670	−1.03645	−0.518223	−	0.855246i	$-0.673406\pi$
$433$	−21.5670	−1.03645	−0.518223	+	0.855246i	$0.673406\pi$
$434$	0	0
$435$	−2.73901	−0.131325
$436$	0	0
$437$	28.5992	1.36809
$438$	0	0
$439$	6.43596	0.307172	0.153586	−	0.988135i	$-0.450918\pi$
$439$	6.43596	0.307172	0.153586	+	0.988135i	$0.450918\pi$
$440$	0	0
$441$	1.98994	0.0947593
$442$	0	0
$443$	39.8586	1.89374	0.946870	−	0.321618i	$-0.104227\pi$
$443$	39.8586	1.89374	0.946870	+	0.321618i	$0.104227\pi$
$444$	0	0
$445$	58.7902	2.78692
$446$	0	0
$447$	−31.1612	−1.47387
$448$	0	0
$449$	35.9047	1.69445	0.847224	−	0.531237i	$-0.178272\pi$
$449$	35.9047	1.69445	0.847224	+	0.531237i	$0.178272\pi$
$450$	0	0
$451$	−12.9191	−0.608337
$452$	0	0
$453$	−1.78165	−0.0837092
$454$	0	0
$455$	0	0
$456$	0	0
$457$	3.89831	0.182355	0.0911777	−	0.995835i	$-0.470937\pi$
$457$	3.89831	0.182355	0.0911777	+	0.995835i	$0.470937\pi$
$458$	0	0
$459$	1.18063	0.0551073
$460$	0	0
$461$	28.9131	1.34662	0.673310	−	0.739361i	$-0.264872\pi$
$461$	28.9131	1.34662	0.673310	+	0.739361i	$0.264872\pi$
$462$	0	0
$463$	6.52035	0.303026	0.151513	−	0.988455i	$-0.451585\pi$
$463$	6.52035	0.303026	0.151513	+	0.988455i	$0.451585\pi$
$464$	0	0
$465$	71.8504	3.33198
$466$	0	0
$467$	−7.61110	−0.352200	−0.176100	−	0.984372i	$-0.556348\pi$
$467$	−7.61110	−0.352200	−0.176100	+	0.984372i	$0.556348\pi$
$468$	0	0
$469$	0.533398	0.0246300
$470$	0	0
$471$	51.9364	2.39310
$472$	0	0
$473$	33.8612	1.55694
$474$	0	0
$475$	−40.8122	−1.87259
$476$	0	0
$477$	−5.45173	−0.249617
$478$	0	0
$479$	21.1050	0.964312	0.482156	−	0.876086i	$-0.339854\pi$
$479$	21.1050	0.964312	0.482156	+	0.876086i	$0.339854\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	13.0762	0.594987
$484$	0	0
$485$	45.1640	2.05079
$486$	0	0
$487$	35.0845	1.58983	0.794916	−	0.606719i	$-0.207515\pi$
$487$	35.0845	1.58983	0.794916	+	0.606719i	$0.207515\pi$
$488$	0	0
$489$	−38.9430	−1.76106
$490$	0	0
$491$	21.5183	0.971107	0.485554	−	0.874207i	$-0.338618\pi$
$491$	21.5183	0.971107	0.485554	+	0.874207i	$0.338618\pi$
$492$	0	0
$493$	−0.175578	−0.00790764
$494$	0	0
$495$	−26.1336	−1.17462
$496$	0	0
$497$	2.19349	0.0983916
$498$	0	0
$499$	−8.79408	−0.393677	−0.196838	−	0.980436i	$-0.563067\pi$
$499$	−8.79408	−0.393677	−0.196838	+	0.980436i	$0.563067\pi$
$500$	0	0
$501$	46.7910	2.09047
$502$	0	0
$503$	−3.41762	−0.152384	−0.0761922	−	0.997093i	$-0.524276\pi$
$503$	−3.41762	−0.152384	−0.0761922	+	0.997093i	$0.524276\pi$
$504$	0	0
$505$	−56.7337	−2.52462
$506$	0	0
$507$	0	0
$508$	0	0
$509$	16.2480	0.720180	0.360090	−	0.932918i	$-0.382746\pi$
$509$	16.2480	0.720180	0.360090	+	0.932918i	$0.382746\pi$
$510$	0	0
$511$	−4.19000	−0.185354
$512$	0	0
$513$	−11.0234	−0.486693
$514$	0	0
$515$	41.4712	1.82744
$516$	0	0
$517$	−39.7370	−1.74763
$518$	0	0
$519$	−50.4309	−2.21367
$520$	0	0
$521$	−15.2329	−0.667365	−0.333683	−	0.942686i	$-0.608291\pi$
$521$	−15.2329	−0.667365	−0.333683	+	0.942686i	$0.608291\pi$
$522$	0	0
$523$	−17.4048	−0.761058	−0.380529	−	0.924769i	$-0.624258\pi$
$523$	−17.4048	−0.761058	−0.380529	+	0.924769i	$0.624258\pi$
$524$	0	0
$525$	−18.6602	−0.814398
$526$	0	0
$527$	4.60581	0.200632
$528$	0	0
$529$	11.2663	0.489837
$530$	0	0
$531$	21.3399	0.926071
$532$	0	0
$533$	0	0
$534$	0	0
$535$	44.8986	1.94114
$536$	0	0
$537$	42.5271	1.83518
$538$	0	0
$539$	3.59386	0.154798
$540$	0	0
$541$	29.3275	1.26089	0.630444	−	0.776235i	$-0.282873\pi$
$541$	29.3275	1.26089	0.630444	+	0.776235i	$0.282873\pi$
$542$	0	0
$543$	−14.1625	−0.607770
$544$	0	0
$545$	−13.4801	−0.577425
$546$	0	0
$547$	−3.07670	−0.131550	−0.0657750	−	0.997834i	$-0.520952\pi$
$547$	−3.07670	−0.131550	−0.0657750	+	0.997834i	$0.520952\pi$
$548$	0	0
$549$	−0.0732647	−0.00312686
$550$	0	0
$551$	1.63934	0.0698382
$552$	0	0
$553$	4.08884	0.173875
$554$	0	0
$555$	41.3460	1.75504
$556$	0	0
$557$	−33.5265	−1.42057	−0.710283	−	0.703917i	$-0.751433\pi$
$557$	−33.5265	−1.42057	−0.710283	+	0.703917i	$0.751433\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−4.20079	−0.177357
$562$	0	0
$563$	−4.44274	−0.187239	−0.0936195	−	0.995608i	$-0.529844\pi$
$563$	−4.44274	−0.187239	−0.0936195	+	0.995608i	$0.529844\pi$
$564$	0	0
$565$	73.9189	3.10979
$566$	0	0
$567$	−11.0100	−0.462375
$568$	0	0
$569$	−25.6346	−1.07466	−0.537330	−	0.843372i	$-0.680567\pi$
$569$	−25.6346	−1.07466	−0.537330	+	0.843372i	$0.680567\pi$
$570$	0	0
$571$	−21.1422	−0.884773	−0.442386	−	0.896825i	$-0.645868\pi$
$571$	−21.1422	−0.884773	−0.442386	+	0.896825i	$0.645868\pi$
$572$	0	0
$573$	−36.5433	−1.52662
$574$	0	0
$575$	−48.8992	−2.03924
$576$	0	0
$577$	18.6402	0.776002	0.388001	−	0.921659i	$-0.373166\pi$
$577$	18.6402	0.776002	0.388001	+	0.921659i	$0.373166\pi$
$578$	0	0
$579$	−41.9478	−1.74329
$580$	0	0
$581$	5.52286	0.229127
$582$	0	0
$583$	−9.84586	−0.407774
$584$	0	0
$585$	0	0
$586$	0	0
$587$	31.2610	1.29028	0.645139	−	0.764065i	$-0.276800\pi$
$587$	31.2610	1.29028	0.645139	+	0.764065i	$0.276800\pi$
$588$	0	0
$589$	−43.0036	−1.77193
$590$	0	0
$591$	−37.4906	−1.54216
$592$	0	0
$593$	9.97063	0.409445	0.204722	−	0.978820i	$-0.434371\pi$
$593$	9.97063	0.409445	0.204722	+	0.978820i	$0.434371\pi$
$594$	0	0
$595$	−1.91214	−0.0783901
$596$	0	0
$597$	29.3185	1.19992
$598$	0	0
$599$	−15.9628	−0.652222	−0.326111	−	0.945331i	$-0.605738\pi$
$599$	−15.9628	−0.652222	−0.326111	+	0.945331i	$0.605738\pi$
$600$	0	0
$601$	−23.9242	−0.975889	−0.487944	−	0.872875i	$-0.662253\pi$
$601$	−23.9242	−0.975889	−0.487944	+	0.872875i	$0.662253\pi$
$602$	0	0
$603$	1.06143	0.0432249
$604$	0	0
$605$	−7.00079	−0.284622
$606$	0	0
$607$	47.1499	1.91376	0.956878	−	0.290489i	$-0.0938181\pi$
$607$	47.1499	1.91376	0.956878	+	0.290489i	$0.0938181\pi$
$608$	0	0
$609$	0.749542	0.0303730
$610$	0	0
$611$	0	0
$612$	0	0
$613$	3.75121	0.151510	0.0757550	−	0.997126i	$-0.475863\pi$
$613$	3.75121	0.151510	0.0757550	+	0.997126i	$0.475863\pi$
$614$	0	0
$615$	−29.3439	−1.18326
$616$	0	0
$617$	37.0517	1.49165	0.745823	−	0.666145i	$-0.232057\pi$
$617$	37.0517	1.49165	0.745823	+	0.666145i	$0.232057\pi$
$618$	0	0
$619$	−38.0711	−1.53021	−0.765104	−	0.643906i	$-0.777313\pi$
$619$	−38.0711	−1.53021	−0.765104	+	0.643906i	$0.777313\pi$
$620$	0	0
$621$	−13.2077	−0.530006
$622$	0	0
$623$	−16.0882	−0.644560
$624$	0	0
$625$	3.01351	0.120540
$626$	0	0
$627$	39.2220	1.56637
$628$	0	0
$629$	2.65039	0.105678
$630$	0	0
$631$	−4.94117	−0.196705	−0.0983525	−	0.995152i	$-0.531357\pi$
$631$	−4.94117	−0.196705	−0.0983525	+	0.995152i	$0.531357\pi$
$632$	0	0
$633$	−46.2099	−1.83668
$634$	0	0
$635$	18.5402	0.735747
$636$	0	0
$637$	0	0
$638$	0	0
$639$	4.36493	0.172674
$640$	0	0
$641$	6.19200	0.244569	0.122285	−	0.992495i	$-0.460978\pi$
$641$	6.19200	0.244569	0.122285	+	0.992495i	$0.460978\pi$
$642$	0	0
$643$	44.6852	1.76221	0.881107	−	0.472918i	$-0.156799\pi$
$643$	44.6852	1.76221	0.881107	+	0.472918i	$0.156799\pi$
$644$	0	0
$645$	76.9108	3.02836
$646$	0	0
$647$	3.96747	0.155977	0.0779886	−	0.996954i	$-0.475150\pi$
$647$	3.96747	0.155977	0.0779886	+	0.996954i	$0.475150\pi$
$648$	0	0
$649$	38.5400	1.51283
$650$	0	0
$651$	−19.6622	−0.770622
$652$	0	0
$653$	−31.2980	−1.22478	−0.612392	−	0.790554i	$-0.709792\pi$
$653$	−31.2980	−1.22478	−0.612392	+	0.790554i	$0.709792\pi$
$654$	0	0
$655$	39.2137	1.53220
$656$	0	0
$657$	−8.33786	−0.325291
$658$	0	0
$659$	−17.9143	−0.697842	−0.348921	−	0.937152i	$-0.613452\pi$
$659$	−17.9143	−0.697842	−0.348921	+	0.937152i	$0.613452\pi$
$660$	0	0
$661$	−5.53764	−0.215389	−0.107695	−	0.994184i	$-0.534347\pi$
$661$	−5.53764	−0.215389	−0.107695	+	0.994184i	$0.534347\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	17.8533	0.692321
$666$	0	0
$667$	1.96418	0.0760534
$668$	0	0
$669$	−49.3408	−1.90763
$670$	0	0
$671$	−0.132317	−0.00510803
$672$	0	0
$673$	35.1270	1.35405	0.677023	−	0.735962i	$-0.263270\pi$
$673$	35.1270	1.35405	0.677023	+	0.735962i	$0.263270\pi$
$674$	0	0
$675$	18.8478	0.725454
$676$	0	0
$677$	−27.3688	−1.05187	−0.525934	−	0.850525i	$-0.676284\pi$
$677$	−27.3688	−1.05187	−0.525934	+	0.850525i	$0.676284\pi$
$678$	0	0
$679$	−12.3593	−0.474307
$680$	0	0
$681$	64.4231	2.46870
$682$	0	0
$683$	−33.8210	−1.29412	−0.647062	−	0.762437i	$-0.724002\pi$
$683$	−33.8210	−1.29412	−0.647062	+	0.762437i	$0.724002\pi$
$684$	0	0
$685$	−11.6218	−0.444047
$686$	0	0
$687$	38.7799	1.47955
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−7.95802	−0.302737	−0.151369	−	0.988477i	$-0.548368\pi$
$691$	−7.95802	−0.302737	−0.151369	+	0.988477i	$0.548368\pi$
$692$	0	0
$693$	7.15157	0.271666
$694$	0	0
$695$	48.2181	1.82902
$696$	0	0
$697$	−1.88102	−0.0712488
$698$	0	0
$699$	22.7233	0.859473
$700$	0	0
$701$	−6.13153	−0.231585	−0.115792	−	0.993273i	$-0.536941\pi$
$701$	−6.13153	−0.231585	−0.115792	+	0.993273i	$0.536941\pi$
$702$	0	0
$703$	−24.7462	−0.933321
$704$	0	0
$705$	−90.2568	−3.39927
$706$	0	0
$707$	15.5254	0.583893
$708$	0	0
$709$	10.4070	0.390845	0.195422	−	0.980719i	$-0.437392\pi$
$709$	10.4070	0.390845	0.195422	+	0.980719i	$0.437392\pi$
$710$	0	0
$711$	8.13657	0.305145
$712$	0	0
$713$	−51.5249	−1.92962
$714$	0	0
$715$	0	0
$716$	0	0
$717$	40.3561	1.50713
$718$	0	0
$719$	0.420339	0.0156760	0.00783799	−	0.999969i	$-0.497505\pi$
$719$	0.420339	0.0156760	0.00783799	+	0.999969i	$0.497505\pi$
$720$	0	0
$721$	−11.3488	−0.422651
$722$	0	0
$723$	−36.3963	−1.35359
$724$	0	0
$725$	−2.80296	−0.104099
$726$	0	0
$727$	−23.4398	−0.869336	−0.434668	−	0.900591i	$-0.643134\pi$
$727$	−23.4398	−0.869336	−0.434668	+	0.900591i	$0.643134\pi$
$728$	0	0
$729$	−6.78884	−0.251439
$730$	0	0
$731$	4.93020	0.182350
$732$	0	0
$733$	−49.4516	−1.82654	−0.913268	−	0.407359i	$-0.866450\pi$
$733$	−49.4516	−1.82654	−0.913268	+	0.407359i	$0.866450\pi$
$734$	0	0
$735$	8.16292	0.301094
$736$	0	0
$737$	1.91696	0.0706120
$738$	0	0
$739$	31.5841	1.16184	0.580920	−	0.813961i	$-0.302693\pi$
$739$	31.5841	1.16184	0.580920	+	0.813961i	$0.302693\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−44.6986	−1.63983	−0.819916	−	0.572484i	$-0.805980\pi$
$743$	−44.6986	−1.63983	−0.819916	+	0.572484i	$0.805980\pi$
$744$	0	0
$745$	−50.9758	−1.86761
$746$	0	0
$747$	10.9902	0.402110
$748$	0	0
$749$	−12.2867	−0.448946
$750$	0	0
$751$	18.4336	0.672651	0.336326	−	0.941746i	$-0.390816\pi$
$751$	18.4336	0.672651	0.336326	+	0.941746i	$0.390816\pi$
$752$	0	0
$753$	2.03541	0.0741746
$754$	0	0
$755$	−2.91455	−0.106071
$756$	0	0
$757$	4.04446	0.146998	0.0734991	−	0.997295i	$-0.476583\pi$
$757$	4.04446	0.146998	0.0734991	+	0.997295i	$0.476583\pi$
$758$	0	0
$759$	46.9939	1.70577
$760$	0	0
$761$	35.0672	1.27118	0.635592	−	0.772025i	$-0.280756\pi$
$761$	35.0672	1.27118	0.635592	+	0.772025i	$0.280756\pi$
$762$	0	0
$763$	3.68889	0.133547
$764$	0	0
$765$	−3.80506	−0.137572
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−20.4729	−0.738271	−0.369136	−	0.929376i	$-0.620346\pi$
$769$	−20.4729	−0.738271	−0.369136	+	0.929376i	$0.620346\pi$
$770$	0	0
$771$	−60.9304	−2.19435
$772$	0	0
$773$	26.8165	0.964522	0.482261	−	0.876028i	$-0.339816\pi$
$773$	26.8165	0.964522	0.482261	+	0.876028i	$0.339816\pi$
$774$	0	0
$775$	73.5279	2.64120
$776$	0	0
$777$	−11.3145	−0.405906
$778$	0	0
$779$	17.5628	0.629251
$780$	0	0
$781$	7.88310	0.282079
$782$	0	0
$783$	−0.757078	−0.0270558
$784$	0	0
$785$	84.9614	3.03240
$786$	0	0
$787$	1.22600	0.0437022	0.0218511	−	0.999761i	$-0.493044\pi$
$787$	1.22600	0.0437022	0.0218511	+	0.999761i	$0.493044\pi$
$788$	0	0
$789$	27.8182	0.990353
$790$	0	0
$791$	−20.2282	−0.719233
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−22.3634	−0.793149
$796$	0	0
$797$	2.03881	0.0722184	0.0361092	−	0.999348i	$-0.488504\pi$
$797$	2.03881	0.0722184	0.0361092	+	0.999348i	$0.488504\pi$
$798$	0	0
$799$	−5.78571	−0.204684
$800$	0	0
$801$	−32.0146	−1.13118
$802$	0	0
$803$	−15.0582	−0.531394
$804$	0	0
$805$	21.3910	0.753933
$806$	0	0
$807$	−13.0090	−0.457937
$808$	0	0
$809$	49.0474	1.72442	0.862208	−	0.506554i	$-0.169081\pi$
$809$	49.0474	1.72442	0.862208	+	0.506554i	$0.169081\pi$
$810$	0	0
$811$	−13.9488	−0.489810	−0.244905	−	0.969547i	$-0.578757\pi$
$811$	−13.9488	−0.489810	−0.244905	+	0.969547i	$0.578757\pi$
$812$	0	0
$813$	6.88208	0.241365
$814$	0	0
$815$	−63.7059	−2.23152
$816$	0	0
$817$	−46.0323	−1.61047
$818$	0	0
$819$	0	0
$820$	0	0
$821$	15.0578	0.525522	0.262761	−	0.964861i	$-0.415367\pi$
$821$	15.0578	0.525522	0.262761	+	0.964861i	$0.415367\pi$
$822$	0	0
$823$	20.1510	0.702419	0.351209	−	0.936297i	$-0.385771\pi$
$823$	20.1510	0.702419	0.351209	+	0.936297i	$0.385771\pi$
$824$	0	0
$825$	−67.0621	−2.33480
$826$	0	0
$827$	−37.1684	−1.29247	−0.646235	−	0.763138i	$-0.723657\pi$
$827$	−37.1684	−1.29247	−0.646235	+	0.763138i	$0.723657\pi$
$828$	0	0
$829$	8.06684	0.280173	0.140086	−	0.990139i	$-0.455262\pi$
$829$	8.06684	0.280173	0.140086	+	0.990139i	$0.455262\pi$
$830$	0	0
$831$	−16.2922	−0.565171
$832$	0	0
$833$	0.523266	0.0181301
$834$	0	0
$835$	76.5442	2.64892
$836$	0	0
$837$	19.8599	0.686458
$838$	0	0
$839$	21.1736	0.730994	0.365497	−	0.930812i	$-0.380899\pi$
$839$	21.1736	0.730994	0.365497	+	0.930812i	$0.380899\pi$
$840$	0	0
$841$	−28.8874	−0.996118
$842$	0	0
$843$	1.24261	0.0427977
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1.91580	0.0658275
$848$	0	0
$849$	21.4887	0.737489
$850$	0	0
$851$	−29.6497	−1.01638
$852$	0	0
$853$	−9.01944	−0.308820	−0.154410	−	0.988007i	$-0.549348\pi$
$853$	−9.01944	−0.308820	−0.154410	+	0.988007i	$0.549348\pi$
$854$	0	0
$855$	35.5271	1.21500
$856$	0	0
$857$	−25.4950	−0.870892	−0.435446	−	0.900215i	$-0.643409\pi$
$857$	−25.4950	−0.870892	−0.435446	+	0.900215i	$0.643409\pi$
$858$	0	0
$859$	−17.5592	−0.599112	−0.299556	−	0.954079i	$-0.596839\pi$
$859$	−17.5592	−0.599112	−0.299556	+	0.954079i	$0.596839\pi$
$860$	0	0
$861$	8.03008	0.273664
$862$	0	0
$863$	−26.2220	−0.892606	−0.446303	−	0.894882i	$-0.647260\pi$
$863$	−26.2220	−0.892606	−0.446303	+	0.894882i	$0.647260\pi$
$864$	0	0
$865$	−82.4985	−2.80503
$866$	0	0
$867$	37.3633	1.26892
$868$	0	0
$869$	14.6947	0.498484
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−24.5944	−0.832393
$874$	0	0
$875$	−12.2545	−0.414279
$876$	0	0
$877$	−58.4076	−1.97229	−0.986143	−	0.165898i	$-0.946948\pi$
$877$	−58.4076	−1.97229	−0.986143	+	0.165898i	$0.946948\pi$
$878$	0	0
$879$	−35.4522	−1.19577
$880$	0	0
$881$	−3.26593	−0.110032	−0.0550160	−	0.998485i	$-0.517521\pi$
$881$	−3.26593	−0.110032	−0.0550160	+	0.998485i	$0.517521\pi$
$882$	0	0
$883$	−30.8801	−1.03920	−0.519598	−	0.854411i	$-0.673918\pi$
$883$	−30.8801	−1.03920	−0.519598	+	0.854411i	$0.673918\pi$
$884$	0	0
$885$	87.5379	2.94255
$886$	0	0
$887$	−10.8847	−0.365473	−0.182736	−	0.983162i	$-0.558495\pi$
$887$	−10.8847	−0.365473	−0.182736	+	0.983162i	$0.558495\pi$
$888$	0	0
$889$	−5.07362	−0.170164
$890$	0	0
$891$	−39.5682	−1.32558
$892$	0	0
$893$	54.0201	1.80771
$894$	0	0
$895$	69.5689	2.32543
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−2.95346	−0.0985035
$900$	0	0
$901$	−1.43356	−0.0477587
$902$	0	0
$903$	−21.0470	−0.700399
$904$	0	0
$905$	−23.1680	−0.770132
$906$	0	0
$907$	−23.1014	−0.767069	−0.383534	−	0.923527i	$-0.625293\pi$
$907$	−23.1014	−0.767069	−0.383534	+	0.923527i	$0.625293\pi$
$908$	0	0
$909$	30.8947	1.02471
$910$	0	0
$911$	23.3012	0.772002	0.386001	−	0.922498i	$-0.373856\pi$
$911$	23.3012	0.772002	0.386001	+	0.922498i	$0.373856\pi$
$912$	0	0
$913$	19.8484	0.656885
$914$	0	0
$915$	−0.300538	−0.00993548
$916$	0	0
$917$	−10.7310	−0.354368
$918$	0	0
$919$	15.8450	0.522680	0.261340	−	0.965247i	$-0.415836\pi$
$919$	15.8450	0.522680	0.261340	+	0.965247i	$0.415836\pi$
$920$	0	0
$921$	60.4413	1.99161
$922$	0	0
$923$	0	0
$924$	0	0
$925$	42.3113	1.39119
$926$	0	0
$927$	−22.5834	−0.741738
$928$	0	0
$929$	35.3986	1.16139	0.580695	−	0.814121i	$-0.302781\pi$
$929$	35.3986	1.16139	0.580695	+	0.814121i	$0.302781\pi$
$930$	0	0
$931$	−4.88563	−0.160120
$932$	0	0
$933$	−8.08371	−0.264649
$934$	0	0
$935$	−6.87196	−0.224737
$936$	0	0
$937$	−11.8676	−0.387698	−0.193849	−	0.981031i	$-0.562097\pi$
$937$	−11.8676	−0.387698	−0.193849	+	0.981031i	$0.562097\pi$
$938$	0	0
$939$	−22.6439	−0.738955
$940$	0	0
$941$	20.3178	0.662342	0.331171	−	0.943571i	$-0.392556\pi$
$941$	20.3178	0.662342	0.331171	+	0.943571i	$0.392556\pi$
$942$	0	0
$943$	21.0429	0.685251
$944$	0	0
$945$	−8.24500	−0.268210
$946$	0	0
$947$	8.27959	0.269051	0.134525	−	0.990910i	$-0.457049\pi$
$947$	8.27959	0.269051	0.134525	+	0.990910i	$0.457049\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	14.4095	0.467260
$952$	0	0
$953$	1.78597	0.0578533	0.0289267	−	0.999582i	$-0.490791\pi$
$953$	1.78597	0.0578533	0.0289267	+	0.999582i	$0.490791\pi$
$954$	0	0
$955$	−59.7803	−1.93444
$956$	0	0
$957$	2.69374	0.0870764
$958$	0	0
$959$	3.18036	0.102699
$960$	0	0
$961$	46.4761	1.49923
$962$	0	0
$963$	−24.4499	−0.787886
$964$	0	0
$965$	−68.6212	−2.20900
$966$	0	0
$967$	41.8505	1.34582	0.672911	−	0.739723i	$-0.265044\pi$
$967$	41.8505	1.34582	0.672911	+	0.739723i	$0.265044\pi$
$968$	0	0
$969$	5.71072	0.183455
$970$	0	0
$971$	−19.0102	−0.610067	−0.305034	−	0.952342i	$-0.598668\pi$
$971$	−19.0102	−0.610067	−0.305034	+	0.952342i	$0.598668\pi$
$972$	0	0
$973$	−13.1951	−0.423015
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−52.8495	−1.69081	−0.845403	−	0.534129i	$-0.820640\pi$
$977$	−52.8495	−1.69081	−0.845403	+	0.534129i	$0.820640\pi$
$978$	0	0
$979$	−57.8186	−1.84789
$980$	0	0
$981$	7.34069	0.234370
$982$	0	0
$983$	28.6422	0.913545	0.456772	−	0.889584i	$-0.349005\pi$
$983$	28.6422	0.913545	0.456772	+	0.889584i	$0.349005\pi$
$984$	0	0
$985$	−61.3299	−1.95413
$986$	0	0
$987$	24.6992	0.786184
$988$	0	0
$989$	−55.1538	−1.75379
$990$	0	0
$991$	27.0860	0.860415	0.430208	−	0.902730i	$-0.358440\pi$
$991$	27.0860	0.860415	0.430208	+	0.902730i	$0.358440\pi$
$992$	0	0
$993$	18.4471	0.585400
$994$	0	0
$995$	47.9613	1.52048
$996$	0	0
$997$	45.5672	1.44313	0.721564	−	0.692348i	$-0.243424\pi$
$997$	45.5672	1.44313	0.721564	+	0.692348i	$0.243424\pi$
$998$	0	0
$999$	11.4283	0.361575

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9464.2.a.bs.1.4	yes	15
13.12	even	2		9464.2.a.br.1.4	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9464.2.a.br.1.4	✓	15	13.12	even	2
9464.2.a.bs.1.4	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)

\(f(q)\)	\(=\)	\(q-2.23382 q^{3} -3.65424 q^{5} +1.00000 q^{7} +1.98994 q^{9} +3.59386 q^{11} +8.16292 q^{15} +0.523266 q^{17} -4.88563 q^{19} -2.23382 q^{21} -5.85374 q^{23} +8.35350 q^{25} +2.25628 q^{27} -0.335543 q^{29} +8.80205 q^{31} -8.02802 q^{33} -3.65424 q^{35} +5.06510 q^{37} -3.59478 q^{41} +9.42197 q^{43} -7.27175 q^{45} -11.0569 q^{47} +1.00000 q^{49} -1.16888 q^{51} -2.73964 q^{53} -13.1328 q^{55} +10.9136 q^{57} +10.7238 q^{59} -0.0368175 q^{61} +1.98994 q^{63} +0.533398 q^{67} +13.0762 q^{69} +2.19349 q^{71} -4.19000 q^{73} -18.6602 q^{75} +3.59386 q^{77} +4.08884 q^{79} -11.0100 q^{81} +5.52286 q^{83} -1.91214 q^{85} +0.749542 q^{87} -16.0882 q^{89} -19.6622 q^{93} +17.8533 q^{95} -12.3593 q^{97} +7.15157 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 3 q^{3} + 4 q^{5} + 15 q^{7} + 16 q^{9} + 15 q^{11} + 8 q^{15} + 2 q^{17} + 13 q^{19} - 3 q^{21} - 10 q^{23} + 23 q^{25} - 9 q^{27} + 25 q^{29} + 19 q^{31} - 24 q^{33} + 4 q^{35} - 2 q^{37} + 30 q^{41}+ \cdots + 70 q^{99}+O(q^{100}) \) 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

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Database

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