LMFDB - Embedded newform 5625.2.a.bf.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5625,2,Mod(1,5625)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5625, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5625.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5625 = 3^{2} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5625.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:	$44.9158511370$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 225)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	5625.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.71285 q^{2} +0.933844 q^{4} -3.13880 q^{7} +1.82616 q^{8} +O(q^{10})$	$q-1.71285 q^{2} +0.933844 q^{4} -3.13880 q^{7} +1.82616 q^{8} +2.19293 q^{11} -4.96314 q^{13} +5.37629 q^{14} -4.99562 q^{16} +1.99373 q^{17} +5.62373 q^{19} -3.75616 q^{22} -6.50395 q^{23} +8.50110 q^{26} -2.93115 q^{28} -2.30955 q^{29} +2.04089 q^{31} +4.90442 q^{32} -3.41496 q^{34} +6.99745 q^{37} -9.63258 q^{38} -10.4685 q^{41} -5.28634 q^{43} +2.04786 q^{44} +11.1403 q^{46} +9.28514 q^{47} +2.85207 q^{49} -4.63480 q^{52} -5.22981 q^{53} -5.73196 q^{56} +3.95591 q^{58} +0.139376 q^{59} +10.6293 q^{61} -3.49573 q^{62} +1.59073 q^{64} +0.617360 q^{67} +1.86184 q^{68} -8.92453 q^{71} -3.25434 q^{73} -11.9856 q^{74} +5.25169 q^{76} -6.88319 q^{77} -3.00772 q^{79} +17.9310 q^{82} -15.2888 q^{83} +9.05468 q^{86} +4.00465 q^{88} +0.342131 q^{89} +15.5783 q^{91} -6.07368 q^{92} -15.9040 q^{94} -12.1264 q^{97} -4.88516 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 32 q^{4}+O(q^{10}) $ 24 * q + 32 * q^4	$ 24 q + 32 q^{4} + 56 q^{16} + 36 q^{19} + 52 q^{31} + 60 q^{34} + 60 q^{46} + 72 q^{49} + 68 q^{61} + 108 q^{64} + 88 q^{76} + 84 q^{79} + 80 q^{91} + 100 q^{94}+O(q^{100}) $ 24 * q + 32 * q^4 + 56 * q^16 + 36 * q^19 + 52 * q^31 + 60 * q^34 + 60 * q^46 + 72 * q^49 + 68 * q^61 + 108 * q^64 + 88 * q^76 + 84 * q^79 + 80 * q^91 + 100 * q^94

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.71285	−1.21117	−0.605583	−	0.795782i	$-0.707060\pi$
$2$	−1.71285	−1.21117	−0.605583	+	0.795782i	$0.707060\pi$
$3$	0	0
$3$	0	0
$4$	0.933844	0.466922
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.13880	−1.18636	−0.593178	−	0.805072i	$-0.702127\pi$
$7$	−3.13880	−1.18636	−0.593178	+	0.805072i	$0.702127\pi$
$8$	1.82616	0.645646
$9$	0	0
$10$	0	0
$11$	2.19293	0.661195	0.330597	−	0.943772i	$-0.392750\pi$
$11$	2.19293	0.661195	0.330597	+	0.943772i	$0.392750\pi$
$12$	0	0
$13$	−4.96314	−1.37653	−0.688264	−	0.725460i	$-0.741627\pi$
$13$	−4.96314	−1.37653	−0.688264	+	0.725460i	$0.741627\pi$
$14$	5.37629	1.43687
$15$	0	0
$16$	−4.99562	−1.24891
$17$	1.99373	0.483551	0.241776	−	0.970332i	$-0.422270\pi$
$17$	1.99373	0.483551	0.241776	+	0.970332i	$0.422270\pi$
$18$	0	0
$19$	5.62373	1.29017	0.645086	−	0.764110i	$-0.276822\pi$
$19$	5.62373	1.29017	0.645086	+	0.764110i	$0.276822\pi$
$20$	0	0
$21$	0	0
$22$	−3.75616	−0.800816
$23$	−6.50395	−1.35617	−0.678083	−	0.734985i	$-0.737189\pi$
$23$	−6.50395	−1.35617	−0.678083	+	0.734985i	$0.737189\pi$
$24$	0	0
$25$	0	0
$26$	8.50110	1.66720
$27$	0	0
$28$	−2.93115	−0.553936
$29$	−2.30955	−0.428873	−0.214436	−	0.976738i	$-0.568791\pi$
$29$	−2.30955	−0.428873	−0.214436	+	0.976738i	$0.568791\pi$
$30$	0	0
$31$	2.04089	0.366554	0.183277	−	0.983061i	$-0.441329\pi$
$31$	2.04089	0.366554	0.183277	+	0.983061i	$0.441329\pi$
$32$	4.90442	0.866986
$33$	0	0
$34$	−3.41496	−0.585660
$35$	0	0
$36$	0	0
$37$	6.99745	1.15037	0.575187	−	0.818022i	$-0.304929\pi$
$37$	6.99745	1.15037	0.575187	+	0.818022i	$0.304929\pi$
$38$	−9.63258	−1.56261
$39$	0	0
$40$	0	0
$41$	−10.4685	−1.63491	−0.817454	−	0.575993i	$-0.804615\pi$
$41$	−10.4685	−1.63491	−0.817454	+	0.575993i	$0.804615\pi$
$42$	0	0
$43$	−5.28634	−0.806159	−0.403079	−	0.915165i	$-0.632060\pi$
$43$	−5.28634	−0.806159	−0.403079	+	0.915165i	$0.632060\pi$
$44$	2.04786	0.308726
$45$	0	0
$46$	11.1403	1.64254
$47$	9.28514	1.35438	0.677189	−	0.735809i	$-0.263198\pi$
$47$	9.28514	1.35438	0.677189	+	0.735809i	$0.263198\pi$
$48$	0	0
$49$	2.85207	0.407439
$50$	0	0
$51$	0	0
$52$	−4.63480	−0.642732
$53$	−5.22981	−0.718369	−0.359185	−	0.933267i	$-0.616945\pi$
$53$	−5.22981	−0.718369	−0.359185	+	0.933267i	$0.616945\pi$
$54$	0	0
$55$	0	0
$56$	−5.73196	−0.765965
$57$	0	0
$58$	3.95591	0.519436
$59$	0.139376	0.0181453	0.00907263	−	0.999959i	$-0.497112\pi$
$59$	0.139376	0.0181453	0.00907263	+	0.999959i	$0.497112\pi$
$60$	0	0
$61$	10.6293	1.36094	0.680470	−	0.732776i	$-0.261776\pi$
$61$	10.6293	1.36094	0.680470	+	0.732776i	$0.261776\pi$
$62$	−3.49573	−0.443958
$63$	0	0
$64$	1.59073	0.198842
$65$	0	0
$66$	0	0
$67$	0.617360	0.0754226	0.0377113	−	0.999289i	$-0.487993\pi$
$67$	0.617360	0.0754226	0.0377113	+	0.999289i	$0.487993\pi$
$68$	1.86184	0.225781
$69$	0	0
$70$	0	0
$71$	−8.92453	−1.05915	−0.529573	−	0.848264i	$-0.677648\pi$
$71$	−8.92453	−1.05915	−0.529573	+	0.848264i	$0.677648\pi$
$72$	0	0
$73$	−3.25434	−0.380892	−0.190446	−	0.981698i	$-0.560993\pi$
$73$	−3.25434	−0.380892	−0.190446	+	0.981698i	$0.560993\pi$
$74$	−11.9856	−1.39329
$75$	0	0
$76$	5.25169	0.602410
$77$	−6.88319	−0.784412
$78$	0	0
$79$	−3.00772	−0.338395	−0.169198	−	0.985582i	$-0.554118\pi$
$79$	−3.00772	−0.338395	−0.169198	+	0.985582i	$0.554118\pi$
$80$	0	0
$81$	0	0
$82$	17.9310	1.98015
$83$	−15.2888	−1.67816	−0.839081	−	0.544006i	$-0.816907\pi$
$83$	−15.2888	−1.67816	−0.839081	+	0.544006i	$0.816907\pi$
$84$	0	0
$85$	0	0
$86$	9.05468	0.976392
$87$	0	0
$88$	4.00465	0.426897
$89$	0.342131	0.0362658	0.0181329	−	0.999836i	$-0.494228\pi$
$89$	0.342131	0.0362658	0.0181329	+	0.999836i	$0.494228\pi$
$90$	0	0
$91$	15.5783	1.63305
$92$	−6.07368	−0.633224
$93$	0	0
$94$	−15.9040	−1.64038
$95$	0	0
$96$	0	0
$97$	−12.1264	−1.23125	−0.615626	−	0.788038i	$-0.711097\pi$
$97$	−12.1264	−1.23125	−0.615626	+	0.788038i	$0.711097\pi$
$98$	−4.88516	−0.493476
$99$	0	0
$100$	0	0
$101$	17.4920	1.74052	0.870260	−	0.492593i	$-0.163951\pi$
$101$	17.4920	1.74052	0.870260	+	0.492593i	$0.163951\pi$
$102$	0	0
$103$	14.7338	1.45177	0.725883	−	0.687818i	$-0.241431\pi$
$103$	14.7338	1.45177	0.725883	+	0.687818i	$0.241431\pi$
$104$	−9.06350	−0.888749
$105$	0	0
$106$	8.95786	0.870064
$107$	6.21837	0.601153	0.300576	−	0.953758i	$-0.402821\pi$
$107$	6.21837	0.601153	0.300576	+	0.953758i	$0.402821\pi$
$108$	0	0
$109$	18.6659	1.78787	0.893936	−	0.448195i	$-0.147933\pi$
$109$	18.6659	1.78787	0.893936	+	0.448195i	$0.147933\pi$
$110$	0	0
$111$	0	0
$112$	15.6803	1.48165
$113$	−11.3727	−1.06985	−0.534926	−	0.844899i	$-0.679661\pi$
$113$	−11.3727	−1.06985	−0.534926	+	0.844899i	$0.679661\pi$
$114$	0	0
$115$	0	0
$116$	−2.15676	−0.200250
$117$	0	0
$118$	−0.238730	−0.0219769
$119$	−6.25793	−0.573663
$120$	0	0
$121$	−6.19104	−0.562822
$122$	−18.2063	−1.64832
$123$	0	0
$124$	1.90587	0.171152
$125$	0	0
$126$	0	0
$127$	−15.4961	−1.37505	−0.687526	−	0.726159i	$-0.741303\pi$
$127$	−15.4961	−1.37505	−0.687526	+	0.726159i	$0.741303\pi$
$128$	−12.5335	−1.10782
$129$	0	0
$130$	0	0
$131$	−21.3916	−1.86899	−0.934496	−	0.355973i	$-0.884150\pi$
$131$	−21.3916	−1.86899	−0.934496	+	0.355973i	$0.884150\pi$
$132$	0	0
$133$	−17.6518	−1.53060
$134$	−1.05744	−0.0913492
$135$	0	0
$136$	3.64088	0.312203
$137$	0.965182	0.0824610	0.0412305	−	0.999150i	$-0.486872\pi$
$137$	0.965182	0.0824610	0.0412305	+	0.999150i	$0.486872\pi$
$138$	0	0
$139$	5.91823	0.501977	0.250989	−	0.967990i	$-0.419244\pi$
$139$	5.91823	0.501977	0.250989	+	0.967990i	$0.419244\pi$
$140$	0	0
$141$	0	0
$142$	15.2864	1.28280
$143$	−10.8838	−0.910153
$144$	0	0
$145$	0	0
$146$	5.57419	0.461324
$147$	0	0
$148$	6.53453	0.537135
$149$	21.7197	1.77934	0.889672	−	0.456600i	$-0.150933\pi$
$149$	21.7197	1.77934	0.889672	+	0.456600i	$0.150933\pi$
$150$	0	0
$151$	2.46090	0.200265	0.100132	−	0.994974i	$-0.468073\pi$
$151$	2.46090	0.200265	0.100132	+	0.994974i	$0.468073\pi$
$152$	10.2698	0.832993
$153$	0	0
$154$	11.7898	0.950053
$155$	0	0
$156$	0	0
$157$	−0.459855	−0.0367004	−0.0183502	−	0.999832i	$-0.505841\pi$
$157$	−0.459855	−0.0367004	−0.0183502	+	0.999832i	$0.505841\pi$
$158$	5.15177	0.409853
$159$	0	0
$160$	0	0
$161$	20.4146	1.60890
$162$	0	0
$163$	4.01627	0.314579	0.157289	−	0.987553i	$-0.449724\pi$
$163$	4.01627	0.314579	0.157289	+	0.987553i	$0.449724\pi$
$164$	−9.77597	−0.763375
$165$	0	0
$166$	26.1873	2.03253
$167$	−0.941160	−0.0728292	−0.0364146	−	0.999337i	$-0.511594\pi$
$167$	−0.941160	−0.0728292	−0.0364146	+	0.999337i	$0.511594\pi$
$168$	0	0
$169$	11.6328	0.894830
$170$	0	0
$171$	0	0
$172$	−4.93662	−0.376413
$173$	−6.87222	−0.522485	−0.261243	−	0.965273i	$-0.584132\pi$
$173$	−6.87222	−0.522485	−0.261243	+	0.965273i	$0.584132\pi$
$174$	0	0
$175$	0	0
$176$	−10.9551	−0.825770
$177$	0	0
$178$	−0.586018	−0.0439239
$179$	13.9799	1.04491	0.522455	−	0.852667i	$-0.325016\pi$
$179$	13.9799	1.04491	0.522455	+	0.852667i	$0.325016\pi$
$180$	0	0
$181$	1.69822	0.126228	0.0631138	−	0.998006i	$-0.479897\pi$
$181$	1.69822	0.126228	0.0631138	+	0.998006i	$0.479897\pi$
$182$	−26.6833	−1.97790
$183$	0	0
$184$	−11.8773	−0.875603
$185$	0	0
$186$	0	0
$187$	4.37212	0.319721
$188$	8.67088	0.632389
$189$	0	0
$190$	0	0
$191$	−3.46210	−0.250509	−0.125255	−	0.992125i	$-0.539975\pi$
$191$	−3.46210	−0.250509	−0.125255	+	0.992125i	$0.539975\pi$
$192$	0	0
$193$	12.4916	0.899165	0.449583	−	0.893239i	$-0.351573\pi$
$193$	12.4916	0.899165	0.449583	+	0.893239i	$0.351573\pi$
$194$	20.7707	1.49125
$195$	0	0
$196$	2.66339	0.190242
$197$	11.0698	0.788687	0.394344	−	0.918963i	$-0.370972\pi$
$197$	11.0698	0.788687	0.394344	+	0.918963i	$0.370972\pi$
$198$	0	0
$199$	−13.5576	−0.961075	−0.480538	−	0.876974i	$-0.659558\pi$
$199$	−13.5576	−0.961075	−0.480538	+	0.876974i	$0.659558\pi$
$200$	0	0
$201$	0	0
$202$	−29.9611	−2.10806
$203$	7.24922	0.508795
$204$	0	0
$205$	0	0
$206$	−25.2368	−1.75833
$207$	0	0
$208$	24.7940	1.71915
$209$	12.3325	0.853054
$210$	0	0
$211$	−7.76765	−0.534747	−0.267374	−	0.963593i	$-0.586156\pi$
$211$	−7.76765	−0.534747	−0.267374	+	0.963593i	$0.586156\pi$
$212$	−4.88383	−0.335423
$213$	0	0
$214$	−10.6511	−0.728096
$215$	0	0
$216$	0	0
$217$	−6.40594	−0.434863
$218$	−31.9719	−2.16541
$219$	0	0
$220$	0	0
$221$	−9.89518	−0.665622
$222$	0	0
$223$	16.3163	1.09262	0.546312	−	0.837582i	$-0.316031\pi$
$223$	16.3163	1.09262	0.546312	+	0.837582i	$0.316031\pi$
$224$	−15.3940	−1.02855
$225$	0	0
$226$	19.4797	1.29577
$227$	9.68468	0.642795	0.321397	−	0.946944i	$-0.395848\pi$
$227$	9.68468	0.642795	0.321397	+	0.946944i	$0.395848\pi$
$228$	0	0
$229$	−3.56664	−0.235690	−0.117845	−	0.993032i	$-0.537599\pi$
$229$	−3.56664	−0.235690	−0.117845	+	0.993032i	$0.537599\pi$
$230$	0	0
$231$	0	0
$232$	−4.21761	−0.276900
$233$	−11.7770	−0.771539	−0.385770	−	0.922595i	$-0.626064\pi$
$233$	−11.7770	−0.771539	−0.385770	+	0.922595i	$0.626064\pi$
$234$	0	0
$235$	0	0
$236$	0.130156	0.00847242
$237$	0	0
$238$	10.7189	0.694801
$239$	−14.6782	−0.949453	−0.474727	−	0.880133i	$-0.657453\pi$
$239$	−14.6782	−0.949453	−0.474727	+	0.880133i	$0.657453\pi$
$240$	0	0
$241$	−6.46925	−0.416721	−0.208360	−	0.978052i	$-0.566813\pi$
$241$	−6.46925	−0.416721	−0.208360	+	0.978052i	$0.566813\pi$
$242$	10.6043	0.681670
$243$	0	0
$244$	9.92609	0.635453
$245$	0	0
$246$	0	0
$247$	−27.9114	−1.77596
$248$	3.72699	0.236664
$249$	0	0
$250$	0	0
$251$	5.57460	0.351865	0.175933	−	0.984402i	$-0.443706\pi$
$251$	5.57460	0.351865	0.175933	+	0.984402i	$0.443706\pi$
$252$	0	0
$253$	−14.2627	−0.896690
$254$	26.5424	1.66542
$255$	0	0
$256$	18.2865	1.14291
$257$	6.36817	0.397236	0.198618	−	0.980077i	$-0.436355\pi$
$257$	6.36817	0.397236	0.198618	+	0.980077i	$0.436355\pi$
$258$	0	0
$259$	−21.9636	−1.36475
$260$	0	0
$261$	0	0
$262$	36.6405	2.26366
$263$	25.5040	1.57264	0.786322	−	0.617817i	$-0.211983\pi$
$263$	25.5040	1.57264	0.786322	+	0.617817i	$0.211983\pi$
$264$	0	0
$265$	0	0
$266$	30.2348	1.85381
$267$	0	0
$268$	0.576518	0.0352165
$269$	−24.0654	−1.46729	−0.733646	−	0.679532i	$-0.762183\pi$
$269$	−24.0654	−1.46729	−0.733646	+	0.679532i	$0.762183\pi$
$270$	0	0
$271$	20.7735	1.26190	0.630950	−	0.775823i	$-0.282665\pi$
$271$	20.7735	1.26190	0.630950	+	0.775823i	$0.282665\pi$
$272$	−9.95994	−0.603910
$273$	0	0
$274$	−1.65321	−0.0998740
$275$	0	0
$276$	0	0
$277$	10.1358	0.609004	0.304502	−	0.952512i	$-0.401510\pi$
$277$	10.1358	0.609004	0.304502	+	0.952512i	$0.401510\pi$
$278$	−10.1370	−0.607978
$279$	0	0
$280$	0	0
$281$	21.5055	1.28291	0.641457	−	0.767159i	$-0.278330\pi$
$281$	21.5055	1.28291	0.641457	+	0.767159i	$0.278330\pi$
$282$	0	0
$283$	9.46070	0.562380	0.281190	−	0.959652i	$-0.409271\pi$
$283$	9.46070	0.562380	0.281190	+	0.959652i	$0.409271\pi$
$284$	−8.33412	−0.494539
$285$	0	0
$286$	18.6424	1.10235
$287$	32.8586	1.93958
$288$	0	0
$289$	−13.0250	−0.766178
$290$	0	0
$291$	0	0
$292$	−3.03905	−0.177847
$293$	1.85448	0.108340	0.0541700	−	0.998532i	$-0.482749\pi$
$293$	1.85448	0.108340	0.0541700	+	0.998532i	$0.482749\pi$
$294$	0	0
$295$	0	0
$296$	12.7785	0.742734
$297$	0	0
$298$	−37.2025	−2.15508
$299$	32.2800	1.86680
$300$	0	0
$301$	16.5928	0.956391
$302$	−4.21514	−0.242554
$303$	0	0
$304$	−28.0940	−1.61130
$305$	0	0
$306$	0	0
$307$	3.43812	0.196224	0.0981118	−	0.995175i	$-0.468720\pi$
$307$	3.43812	0.196224	0.0981118	+	0.995175i	$0.468720\pi$
$308$	−6.42782	−0.366259
$309$	0	0
$310$	0	0
$311$	7.11249	0.403312	0.201656	−	0.979456i	$-0.435368\pi$
$311$	7.11249	0.403312	0.201656	+	0.979456i	$0.435368\pi$
$312$	0	0
$313$	−12.7359	−0.719875	−0.359937	−	0.932976i	$-0.617202\pi$
$313$	−12.7359	−0.719875	−0.359937	+	0.932976i	$0.617202\pi$
$314$	0.787662	0.0444503
$315$	0	0
$316$	−2.80874	−0.158004
$317$	−0.555733	−0.0312131	−0.0156065	−	0.999878i	$-0.504968\pi$
$317$	−0.555733	−0.0312131	−0.0156065	+	0.999878i	$0.504968\pi$
$318$	0	0
$319$	−5.06469	−0.283568
$320$	0	0
$321$	0	0
$322$	−34.9671	−1.94864
$323$	11.2122	0.623864
$324$	0	0
$325$	0	0
$326$	−6.87926	−0.381007
$327$	0	0
$328$	−19.1172	−1.05557
$329$	−29.1442	−1.60677
$330$	0	0
$331$	27.2175	1.49601	0.748004	−	0.663694i	$-0.231012\pi$
$331$	27.2175	1.49601	0.748004	+	0.663694i	$0.231012\pi$
$332$	−14.2773	−0.783571
$333$	0	0
$334$	1.61206	0.0882082
$335$	0	0
$336$	0	0
$337$	18.4639	1.00579	0.502896	−	0.864347i	$-0.332268\pi$
$337$	18.4639	1.00579	0.502896	+	0.864347i	$0.332268\pi$
$338$	−19.9252	−1.08379
$339$	0	0
$340$	0	0
$341$	4.47553	0.242364
$342$	0	0
$343$	13.0195	0.702988
$344$	−9.65370	−0.520493
$345$	0	0
$346$	11.7711	0.632816
$347$	−11.1988	−0.601186	−0.300593	−	0.953753i	$-0.597184\pi$
$347$	−11.1988	−0.601186	−0.300593	+	0.953753i	$0.597184\pi$
$348$	0	0
$349$	20.1790	1.08015	0.540077	−	0.841615i	$-0.318395\pi$
$349$	20.1790	1.08015	0.540077	+	0.841615i	$0.318395\pi$
$350$	0	0
$351$	0	0
$352$	10.7551	0.573247
$353$	6.16774	0.328275	0.164138	−	0.986437i	$-0.447516\pi$
$353$	6.16774	0.328275	0.164138	+	0.986437i	$0.447516\pi$
$354$	0	0
$355$	0	0
$356$	0.319497	0.0169333
$357$	0	0
$358$	−23.9455	−1.26556
$359$	−31.6854	−1.67229	−0.836146	−	0.548506i	$-0.815197\pi$
$359$	−31.6854	−1.67229	−0.836146	+	0.548506i	$0.815197\pi$
$360$	0	0
$361$	12.6263	0.664542
$362$	−2.90879	−0.152882
$363$	0	0
$364$	14.5477	0.762508
$365$	0	0
$366$	0	0
$367$	3.66523	0.191323	0.0956616	−	0.995414i	$-0.469503\pi$
$367$	3.66523	0.191323	0.0956616	+	0.995414i	$0.469503\pi$
$368$	32.4913	1.69372
$369$	0	0
$370$	0	0
$371$	16.4153	0.852241
$372$	0	0
$373$	12.9435	0.670191	0.335095	−	0.942184i	$-0.391231\pi$
$373$	12.9435	0.670191	0.335095	+	0.942184i	$0.391231\pi$
$374$	−7.48878	−0.387236
$375$	0	0
$376$	16.9562	0.874448
$377$	11.4626	0.590355
$378$	0	0
$379$	19.7828	1.01617	0.508086	−	0.861306i	$-0.330353\pi$
$379$	19.7828	1.01617	0.508086	+	0.861306i	$0.330353\pi$
$380$	0	0
$381$	0	0
$382$	5.93005	0.303408
$383$	3.33495	0.170408	0.0852040	−	0.996364i	$-0.472846\pi$
$383$	3.33495	0.170408	0.0852040	+	0.996364i	$0.472846\pi$
$384$	0	0
$385$	0	0
$386$	−21.3962	−1.08904
$387$	0	0
$388$	−11.3242	−0.574899
$389$	15.2410	0.772749	0.386374	−	0.922342i	$-0.373727\pi$
$389$	15.2410	0.772749	0.386374	+	0.922342i	$0.373727\pi$
$390$	0	0
$391$	−12.9671	−0.655776
$392$	5.20834	0.263061
$393$	0	0
$394$	−18.9608	−0.955231
$395$	0	0
$396$	0	0
$397$	20.6687	1.03733	0.518665	−	0.854977i	$-0.326429\pi$
$397$	20.6687	1.03733	0.518665	+	0.854977i	$0.326429\pi$
$398$	23.2221	1.16402
$399$	0	0
$400$	0	0
$401$	−23.4868	−1.17287	−0.586436	−	0.809995i	$-0.699470\pi$
$401$	−23.4868	−1.17287	−0.586436	+	0.809995i	$0.699470\pi$
$402$	0	0
$403$	−10.1292	−0.504572
$404$	16.3348	0.812687
$405$	0	0
$406$	−12.4168	−0.616235
$407$	15.3450	0.760621
$408$	0	0
$409$	17.3544	0.858122	0.429061	−	0.903276i	$-0.358845\pi$
$409$	17.3544	0.858122	0.429061	+	0.903276i	$0.358845\pi$
$410$	0	0
$411$	0	0
$412$	13.7591	0.677862
$413$	−0.437475	−0.0215267
$414$	0	0
$415$	0	0
$416$	−24.3413	−1.19343
$417$	0	0
$418$	−21.1236	−1.03319
$419$	−13.9947	−0.683684	−0.341842	−	0.939757i	$-0.611051\pi$
$419$	−13.9947	−0.683684	−0.341842	+	0.939757i	$0.611051\pi$
$420$	0	0
$421$	20.0565	0.977495	0.488748	−	0.872425i	$-0.337454\pi$
$421$	20.0565	0.977495	0.488748	+	0.872425i	$0.337454\pi$
$422$	13.3048	0.647668
$423$	0	0
$424$	−9.55047	−0.463812
$425$	0	0
$426$	0	0
$427$	−33.3632	−1.61456
$428$	5.80699	0.280692
$429$	0	0
$430$	0	0
$431$	18.2657	0.879827	0.439913	−	0.898040i	$-0.355009\pi$
$431$	18.2657	0.879827	0.439913	+	0.898040i	$0.355009\pi$
$432$	0	0
$433$	−10.2094	−0.490633	−0.245317	−	0.969443i	$-0.578892\pi$
$433$	−10.2094	−0.490633	−0.245317	+	0.969443i	$0.578892\pi$
$434$	10.9724	0.526691
$435$	0	0
$436$	17.4311	0.834797
$437$	−36.5764	−1.74969
$438$	0	0
$439$	9.06031	0.432425	0.216212	−	0.976346i	$-0.430630\pi$
$439$	9.06031	0.432425	0.216212	+	0.976346i	$0.430630\pi$
$440$	0	0
$441$	0	0
$442$	16.9489	0.806178
$443$	1.36623	0.0649117	0.0324558	−	0.999473i	$-0.489667\pi$
$443$	1.36623	0.0649117	0.0324558	+	0.999473i	$0.489667\pi$
$444$	0	0
$445$	0	0
$446$	−27.9474	−1.32335
$447$	0	0
$448$	−4.99300	−0.235897
$449$	−28.7680	−1.35764	−0.678822	−	0.734303i	$-0.737509\pi$
$449$	−28.7680	−1.35764	−0.678822	+	0.734303i	$0.737509\pi$
$450$	0	0
$451$	−22.9568	−1.08099
$452$	−10.6203	−0.499538
$453$	0	0
$454$	−16.5884	−0.778531
$455$	0	0
$456$	0	0
$457$	39.0545	1.82689	0.913446	−	0.406959i	$-0.133411\pi$
$457$	39.0545	1.82689	0.913446	+	0.406959i	$0.133411\pi$
$458$	6.10910	0.285460
$459$	0	0
$460$	0	0
$461$	−37.4464	−1.74405	−0.872025	−	0.489461i	$-0.837194\pi$
$461$	−37.4464	−1.74405	−0.872025	+	0.489461i	$0.837194\pi$
$462$	0	0
$463$	−7.10759	−0.330318	−0.165159	−	0.986267i	$-0.552814\pi$
$463$	−7.10759	−0.330318	−0.165159	+	0.986267i	$0.552814\pi$
$464$	11.5376	0.535622
$465$	0	0
$466$	20.1723	0.934462
$467$	−4.40519	−0.203848	−0.101924	−	0.994792i	$-0.532500\pi$
$467$	−4.40519	−0.203848	−0.101924	+	0.994792i	$0.532500\pi$
$468$	0	0
$469$	−1.93777	−0.0894779
$470$	0	0
$471$	0	0
$472$	0.254524	0.0117154
$473$	−11.5926	−0.533028
$474$	0	0
$475$	0	0
$476$	−5.84393	−0.267856
$477$	0	0
$478$	25.1415	1.14994
$479$	−20.9201	−0.955864	−0.477932	−	0.878397i	$-0.658614\pi$
$479$	−20.9201	−0.955864	−0.477932	+	0.878397i	$0.658614\pi$
$480$	0	0
$481$	−34.7294	−1.58352
$482$	11.0808	0.504718
$483$	0	0
$484$	−5.78147	−0.262794
$485$	0	0
$486$	0	0
$487$	−5.05656	−0.229135	−0.114567	−	0.993415i	$-0.536548\pi$
$487$	−5.05656	−0.229135	−0.114567	+	0.993415i	$0.536548\pi$
$488$	19.4108	0.878684
$489$	0	0
$490$	0	0
$491$	12.9532	0.584571	0.292285	−	0.956331i	$-0.405584\pi$
$491$	12.9532	0.584571	0.292285	+	0.956331i	$0.405584\pi$
$492$	0	0
$493$	−4.60462	−0.207382
$494$	47.8079	2.15098
$495$	0	0
$496$	−10.1955	−0.457791
$497$	28.0123	1.25652
$498$	0	0
$499$	27.2629	1.22045	0.610227	−	0.792226i	$-0.291078\pi$
$499$	27.2629	1.22045	0.610227	+	0.792226i	$0.291078\pi$
$500$	0	0
$501$	0	0
$502$	−9.54843	−0.426167
$503$	23.5107	1.04829	0.524145	−	0.851629i	$-0.324385\pi$
$503$	23.5107	1.04829	0.524145	+	0.851629i	$0.324385\pi$
$504$	0	0
$505$	0	0
$506$	24.4299	1.08604
$507$	0	0
$508$	−14.4709	−0.642043
$509$	27.2586	1.20822	0.604108	−	0.796902i	$-0.293529\pi$
$509$	27.2586	1.20822	0.604108	+	0.796902i	$0.293529\pi$
$510$	0	0
$511$	10.2147	0.451873
$512$	−6.25498	−0.276434
$513$	0	0
$514$	−10.9077	−0.481118
$515$	0	0
$516$	0	0
$517$	20.3617	0.895507
$518$	37.6203	1.65294
$519$	0	0
$520$	0	0
$521$	1.21541	0.0532479	0.0266239	−	0.999646i	$-0.491524\pi$
$521$	1.21541	0.0532479	0.0266239	+	0.999646i	$0.491524\pi$
$522$	0	0
$523$	27.0226	1.18162	0.590808	−	0.806812i	$-0.298809\pi$
$523$	27.0226	1.18162	0.590808	+	0.806812i	$0.298809\pi$
$524$	−19.9764	−0.872674
$525$	0	0
$526$	−43.6844	−1.90473
$527$	4.06898	0.177248
$528$	0	0
$529$	19.3013	0.839189
$530$	0	0
$531$	0	0
$532$	−16.4840	−0.714672
$533$	51.9568	2.25050
$534$	0	0
$535$	0	0
$536$	1.12740	0.0486962
$537$	0	0
$538$	41.2203	1.77713
$539$	6.25441	0.269396
$540$	0	0
$541$	−0.222481	−0.00956521	−0.00478261	−	0.999989i	$-0.501522\pi$
$541$	−0.222481	−0.00956521	−0.00478261	+	0.999989i	$0.501522\pi$
$542$	−35.5818	−1.52837
$543$	0	0
$544$	9.77809	0.419232
$545$	0	0
$546$	0	0
$547$	22.5774	0.965340	0.482670	−	0.875802i	$-0.339667\pi$
$547$	22.5774	0.965340	0.482670	+	0.875802i	$0.339667\pi$
$548$	0.901330	0.0385029
$549$	0	0
$550$	0	0
$551$	−12.9883	−0.553319
$552$	0	0
$553$	9.44064	0.401457
$554$	−17.3612	−0.737605
$555$	0	0
$556$	5.52670	0.234384
$557$	42.6859	1.80866	0.904330	−	0.426834i	$-0.140371\pi$
$557$	42.6859	1.80866	0.904330	+	0.426834i	$0.140371\pi$
$558$	0	0
$559$	26.2368	1.10970
$560$	0	0
$561$	0	0
$562$	−36.8357	−1.55382
$563$	−3.77858	−0.159248	−0.0796241	−	0.996825i	$-0.525372\pi$
$563$	−3.77858	−0.159248	−0.0796241	+	0.996825i	$0.525372\pi$
$564$	0	0
$565$	0	0
$566$	−16.2047	−0.681135
$567$	0	0
$568$	−16.2976	−0.683833
$569$	−10.0826	−0.422684	−0.211342	−	0.977412i	$-0.567783\pi$
$569$	−10.0826	−0.422684	−0.211342	+	0.977412i	$0.567783\pi$
$570$	0	0
$571$	12.0914	0.506010	0.253005	−	0.967465i	$-0.418581\pi$
$571$	12.0914	0.506010	0.253005	+	0.967465i	$0.418581\pi$
$572$	−10.1638	−0.424971
$573$	0	0
$574$	−56.2818	−2.34916
$575$	0	0
$576$	0	0
$577$	−12.0222	−0.500491	−0.250245	−	0.968182i	$-0.580511\pi$
$577$	−12.0222	−0.500491	−0.250245	+	0.968182i	$0.580511\pi$
$578$	22.3099	0.927969
$579$	0	0
$580$	0	0
$581$	47.9885	1.99090
$582$	0	0
$583$	−11.4686	−0.474982
$584$	−5.94296	−0.245921
$585$	0	0
$586$	−3.17644	−0.131218
$587$	41.1247	1.69740	0.848699	−	0.528876i	$-0.177386\pi$
$587$	41.1247	1.69740	0.848699	+	0.528876i	$0.177386\pi$
$588$	0	0
$589$	11.4774	0.472917
$590$	0	0
$591$	0	0
$592$	−34.9566	−1.43671
$593$	12.7151	0.522147	0.261073	−	0.965319i	$-0.415924\pi$
$593$	12.7151	0.522147	0.261073	+	0.965319i	$0.415924\pi$
$594$	0	0
$595$	0	0
$596$	20.2828	0.830815
$597$	0	0
$598$	−55.2907	−2.26101
$599$	43.1612	1.76352	0.881760	−	0.471698i	$-0.156359\pi$
$599$	43.1612	1.76352	0.881760	+	0.471698i	$0.156359\pi$
$600$	0	0
$601$	10.5930	0.432096	0.216048	−	0.976383i	$-0.430683\pi$
$601$	10.5930	0.432096	0.216048	+	0.976383i	$0.430683\pi$
$602$	−28.4209	−1.15835
$603$	0	0
$604$	2.29809	0.0935081
$605$	0	0
$606$	0	0
$607$	−20.7969	−0.844121	−0.422061	−	0.906568i	$-0.638693\pi$
$607$	−20.7969	−0.844121	−0.422061	+	0.906568i	$0.638693\pi$
$608$	27.5811	1.11856
$609$	0	0
$610$	0	0
$611$	−46.0835	−1.86434
$612$	0	0
$613$	−17.6191	−0.711631	−0.355815	−	0.934556i	$-0.615797\pi$
$613$	−17.6191	−0.711631	−0.355815	+	0.934556i	$0.615797\pi$
$614$	−5.88897	−0.237659
$615$	0	0
$616$	−12.5698	−0.506452
$617$	34.8582	1.40334	0.701670	−	0.712503i	$-0.252438\pi$
$617$	34.8582	1.40334	0.701670	+	0.712503i	$0.252438\pi$
$618$	0	0
$619$	−18.5999	−0.747593	−0.373797	−	0.927511i	$-0.621944\pi$
$619$	−18.5999	−0.747593	−0.373797	+	0.927511i	$0.621944\pi$
$620$	0	0
$621$	0	0
$622$	−12.1826	−0.488478
$623$	−1.07388	−0.0430241
$624$	0	0
$625$	0	0
$626$	21.8146	0.871888
$627$	0	0
$628$	−0.429433	−0.0171363
$629$	13.9510	0.556265
$630$	0	0
$631$	28.8895	1.15007	0.575036	−	0.818128i	$-0.304988\pi$
$631$	28.8895	1.15007	0.575036	+	0.818128i	$0.304988\pi$
$632$	−5.49259	−0.218483
$633$	0	0
$634$	0.951885	0.0378042
$635$	0	0
$636$	0	0
$637$	−14.1552	−0.560851
$638$	8.67504	0.343448
$639$	0	0
$640$	0	0
$641$	−11.6616	−0.460606	−0.230303	−	0.973119i	$-0.573972\pi$
$641$	−11.6616	−0.460606	−0.230303	+	0.973119i	$0.573972\pi$
$642$	0	0
$643$	1.09694	0.0432590	0.0216295	−	0.999766i	$-0.493115\pi$
$643$	1.09694	0.0432590	0.0216295	+	0.999766i	$0.493115\pi$
$644$	19.0641	0.751229
$645$	0	0
$646$	−19.2048	−0.755602
$647$	4.54285	0.178598	0.0892989	−	0.996005i	$-0.471537\pi$
$647$	4.54285	0.178598	0.0892989	+	0.996005i	$0.471537\pi$
$648$	0	0
$649$	0.305643	0.0119975
$650$	0	0
$651$	0	0
$652$	3.75057	0.146884
$653$	−11.2430	−0.439971	−0.219986	−	0.975503i	$-0.570601\pi$
$653$	−11.2430	−0.439971	−0.219986	+	0.975503i	$0.570601\pi$
$654$	0	0
$655$	0	0
$656$	52.2968	2.04185
$657$	0	0
$658$	49.9196	1.94607
$659$	28.5072	1.11048	0.555241	−	0.831689i	$-0.312626\pi$
$659$	28.5072	1.11048	0.555241	+	0.831689i	$0.312626\pi$
$660$	0	0
$661$	−21.2841	−0.827855	−0.413928	−	0.910310i	$-0.635843\pi$
$661$	−21.2841	−0.827855	−0.413928	+	0.910310i	$0.635843\pi$
$662$	−46.6194	−1.81191
$663$	0	0
$664$	−27.9198	−1.08350
$665$	0	0
$666$	0	0
$667$	15.0212	0.581623
$668$	−0.878897	−0.0340056
$669$	0	0
$670$	0	0
$671$	23.3093	0.899846
$672$	0	0
$673$	32.3997	1.24892	0.624458	−	0.781058i	$-0.285320\pi$
$673$	32.3997	1.24892	0.624458	+	0.781058i	$0.285320\pi$
$674$	−31.6258	−1.21818
$675$	0	0
$676$	10.8632	0.417816
$677$	−23.7772	−0.913833	−0.456916	−	0.889510i	$-0.651046\pi$
$677$	−23.7772	−0.913833	−0.456916	+	0.889510i	$0.651046\pi$
$678$	0	0
$679$	38.0624	1.46070
$680$	0	0
$681$	0	0
$682$	−7.66590	−0.293542
$683$	11.1998	0.428548	0.214274	−	0.976774i	$-0.431261\pi$
$683$	11.1998	0.428548	0.214274	+	0.976774i	$0.431261\pi$
$684$	0	0
$685$	0	0
$686$	−22.3004	−0.851435
$687$	0	0
$688$	26.4085	1.00682
$689$	25.9563	0.988856
$690$	0	0
$691$	36.5801	1.39157	0.695785	−	0.718250i	$-0.255056\pi$
$691$	36.5801	1.39157	0.695785	+	0.718250i	$0.255056\pi$
$692$	−6.41759	−0.243960
$693$	0	0
$694$	19.1819	0.728135
$695$	0	0
$696$	0	0
$697$	−20.8714	−0.790562
$698$	−34.5635	−1.30825
$699$	0	0
$700$	0	0
$701$	−9.66732	−0.365130	−0.182565	−	0.983194i	$-0.558440\pi$
$701$	−9.66732	−0.365130	−0.182565	+	0.983194i	$0.558440\pi$
$702$	0	0
$703$	39.3518	1.48418
$704$	3.48838	0.131473
$705$	0	0
$706$	−10.5644	−0.397596
$707$	−54.9039	−2.06487
$708$	0	0
$709$	−15.4111	−0.578776	−0.289388	−	0.957212i	$-0.593452\pi$
$709$	−15.4111	−0.578776	−0.289388	+	0.957212i	$0.593452\pi$
$710$	0	0
$711$	0	0
$712$	0.624786	0.0234148
$713$	−13.2738	−0.497108
$714$	0	0
$715$	0	0
$716$	13.0551	0.487892
$717$	0	0
$718$	54.2723	2.02542
$719$	12.5352	0.467485	0.233743	−	0.972299i	$-0.424903\pi$
$719$	12.5352	0.467485	0.233743	+	0.972299i	$0.424903\pi$
$720$	0	0
$721$	−46.2465	−1.72231
$722$	−21.6269	−0.804870
$723$	0	0
$724$	1.58587	0.0589384
$725$	0	0
$726$	0	0
$727$	28.8874	1.07138	0.535688	−	0.844416i	$-0.320052\pi$
$727$	28.8874	1.07138	0.535688	+	0.844416i	$0.320052\pi$
$728$	28.4485	1.05437
$729$	0	0
$730$	0	0
$731$	−10.5395	−0.389819
$732$	0	0
$733$	1.45611	0.0537825	0.0268913	−	0.999638i	$-0.491439\pi$
$733$	1.45611	0.0537825	0.0268913	+	0.999638i	$0.491439\pi$
$734$	−6.27797	−0.231724
$735$	0	0
$736$	−31.8981	−1.17578
$737$	1.35383	0.0498690
$738$	0	0
$739$	30.1602	1.10946	0.554730	−	0.832030i	$-0.312822\pi$
$739$	30.1602	1.10946	0.554730	+	0.832030i	$0.312822\pi$
$740$	0	0
$741$	0	0
$742$	−28.1169	−1.03221
$743$	24.3151	0.892035	0.446017	−	0.895024i	$-0.352842\pi$
$743$	24.3151	0.892035	0.446017	+	0.895024i	$0.352842\pi$
$744$	0	0
$745$	0	0
$746$	−22.1703	−0.811712
$747$	0	0
$748$	4.08288	0.149285
$749$	−19.5182	−0.713181
$750$	0	0
$751$	42.1647	1.53861	0.769307	−	0.638880i	$-0.220602\pi$
$751$	42.1647	1.53861	0.769307	+	0.638880i	$0.220602\pi$
$752$	−46.3851	−1.69149
$753$	0	0
$754$	−19.6337	−0.715018
$755$	0	0
$756$	0	0
$757$	11.8803	0.431796	0.215898	−	0.976416i	$-0.430732\pi$
$757$	11.8803	0.431796	0.215898	+	0.976416i	$0.430732\pi$
$758$	−33.8848	−1.23075
$759$	0	0
$760$	0	0
$761$	10.6044	0.384408	0.192204	−	0.981355i	$-0.438436\pi$
$761$	10.6044	0.384408	0.192204	+	0.981355i	$0.438436\pi$
$762$	0	0
$763$	−58.5886	−2.12105
$764$	−3.23307	−0.116968
$765$	0	0
$766$	−5.71226	−0.206392
$767$	−0.691745	−0.0249775
$768$	0	0
$769$	35.5740	1.28283	0.641415	−	0.767194i	$-0.278348\pi$
$769$	35.5740	1.28283	0.641415	+	0.767194i	$0.278348\pi$
$770$	0	0
$771$	0	0
$772$	11.6652	0.419840
$773$	50.1932	1.80532	0.902661	−	0.430352i	$-0.141611\pi$
$773$	50.1932	1.80532	0.902661	+	0.430352i	$0.141611\pi$
$774$	0	0
$775$	0	0
$776$	−22.1448	−0.794953
$777$	0	0
$778$	−26.1055	−0.935927
$779$	−58.8721	−2.10931
$780$	0	0
$781$	−19.5709	−0.700302
$782$	22.2107	0.794253
$783$	0	0
$784$	−14.2479	−0.508853
$785$	0	0
$786$	0	0
$787$	28.5286	1.01693	0.508467	−	0.861081i	$-0.330212\pi$
$787$	28.5286	1.01693	0.508467	+	0.861081i	$0.330212\pi$
$788$	10.3374	0.368256
$789$	0	0
$790$	0	0
$791$	35.6966	1.26923
$792$	0	0
$793$	−52.7546	−1.87337
$794$	−35.4023	−1.25638
$795$	0	0
$796$	−12.6607	−0.448747
$797$	5.60113	0.198402	0.0992012	−	0.995067i	$-0.468371\pi$
$797$	5.60113	0.198402	0.0992012	+	0.995067i	$0.468371\pi$
$798$	0	0
$799$	18.5121	0.654911
$800$	0	0
$801$	0	0
$802$	40.2292	1.42054
$803$	−7.13656	−0.251844
$804$	0	0
$805$	0	0
$806$	17.3498	0.611120
$807$	0	0
$808$	31.9432	1.12376
$809$	−18.0958	−0.636215	−0.318108	−	0.948055i	$-0.603047\pi$
$809$	−18.0958	−0.636215	−0.318108	+	0.948055i	$0.603047\pi$
$810$	0	0
$811$	−27.2433	−0.956643	−0.478321	−	0.878185i	$-0.658755\pi$
$811$	−27.2433	−0.956643	−0.478321	+	0.878185i	$0.658755\pi$
$812$	6.76964	0.237568
$813$	0	0
$814$	−26.2836	−0.921238
$815$	0	0
$816$	0	0
$817$	−29.7289	−1.04008
$818$	−29.7255	−1.03933
$819$	0	0
$820$	0	0
$821$	−17.6235	−0.615066	−0.307533	−	0.951537i	$-0.599503\pi$
$821$	−17.6235	−0.615066	−0.307533	+	0.951537i	$0.599503\pi$
$822$	0	0
$823$	18.7935	0.655100	0.327550	−	0.944834i	$-0.393777\pi$
$823$	18.7935	0.655100	0.327550	+	0.944834i	$0.393777\pi$
$824$	26.9063	0.937326
$825$	0	0
$826$	0.749327	0.0260724
$827$	57.3362	1.99378	0.996888	−	0.0788358i	$-0.0251203\pi$
$827$	57.3362	1.99378	0.996888	+	0.0788358i	$0.0251203\pi$
$828$	0	0
$829$	−19.0424	−0.661370	−0.330685	−	0.943741i	$-0.607280\pi$
$829$	−19.0424	−0.661370	−0.330685	+	0.943741i	$0.607280\pi$
$830$	0	0
$831$	0	0
$832$	−7.89504	−0.273711
$833$	5.68627	0.197017
$834$	0	0
$835$	0	0
$836$	11.5166	0.398310
$837$	0	0
$838$	23.9707	0.828054
$839$	−8.31661	−0.287121	−0.143561	−	0.989642i	$-0.545855\pi$
$839$	−8.31661	−0.287121	−0.143561	+	0.989642i	$0.545855\pi$
$840$	0	0
$841$	−23.6660	−0.816068
$842$	−34.3538	−1.18391
$843$	0	0
$844$	−7.25378	−0.249685
$845$	0	0
$846$	0	0
$847$	19.4324	0.667706
$848$	26.1261	0.897176
$849$	0	0
$850$	0	0
$851$	−45.5111	−1.56010
$852$	0	0
$853$	8.45800	0.289596	0.144798	−	0.989461i	$-0.453747\pi$
$853$	8.45800	0.289596	0.144798	+	0.989461i	$0.453747\pi$
$854$	57.1460	1.95550
$855$	0	0
$856$	11.3558	0.388132
$857$	7.19182	0.245668	0.122834	−	0.992427i	$-0.460802\pi$
$857$	7.19182	0.245668	0.122834	+	0.992427i	$0.460802\pi$
$858$	0	0
$859$	−5.66691	−0.193353	−0.0966763	−	0.995316i	$-0.530821\pi$
$859$	−5.66691	−0.193353	−0.0966763	+	0.995316i	$0.530821\pi$
$860$	0	0
$861$	0	0
$862$	−31.2863	−1.06562
$863$	16.6027	0.565164	0.282582	−	0.959243i	$-0.408809\pi$
$863$	16.6027	0.565164	0.282582	+	0.959243i	$0.408809\pi$
$864$	0	0
$865$	0	0
$866$	17.4872	0.594238
$867$	0	0
$868$	−5.98215	−0.203047
$869$	−6.59574	−0.223745
$870$	0	0
$871$	−3.06405	−0.103821
$872$	34.0870	1.15433
$873$	0	0
$874$	62.6498	2.11916
$875$	0	0
$876$	0	0
$877$	52.8626	1.78504	0.892521	−	0.451005i	$-0.148934\pi$
$877$	52.8626	1.78504	0.892521	+	0.451005i	$0.148934\pi$
$878$	−15.5189	−0.523738
$879$	0	0
$880$	0	0
$881$	−24.4945	−0.825241	−0.412621	−	0.910903i	$-0.635386\pi$
$881$	−24.4945	−0.825241	−0.412621	+	0.910903i	$0.635386\pi$
$882$	0	0
$883$	−19.2310	−0.647175	−0.323588	−	0.946198i	$-0.604889\pi$
$883$	−19.2310	−0.647175	−0.323588	+	0.946198i	$0.604889\pi$
$884$	−9.24056	−0.310794
$885$	0	0
$886$	−2.34015	−0.0786188
$887$	−22.9423	−0.770328	−0.385164	−	0.922848i	$-0.625855\pi$
$887$	−22.9423	−0.770328	−0.385164	+	0.922848i	$0.625855\pi$
$888$	0	0
$889$	48.6390	1.63130
$890$	0	0
$891$	0	0
$892$	15.2369	0.510170
$893$	52.2171	1.74738
$894$	0	0
$895$	0	0
$896$	39.3402	1.31426
$897$	0	0
$898$	49.2751	1.64433
$899$	−4.71353	−0.157205
$900$	0	0
$901$	−10.4268	−0.347368
$902$	39.3215	1.30926
$903$	0	0
$904$	−20.7684	−0.690746
$905$	0	0
$906$	0	0
$907$	−29.6759	−0.985374	−0.492687	−	0.870207i	$-0.663985\pi$
$907$	−29.6759	−0.985374	−0.492687	+	0.870207i	$0.663985\pi$
$908$	9.04398	0.300135
$909$	0	0
$910$	0	0
$911$	−13.9333	−0.461632	−0.230816	−	0.972997i	$-0.574140\pi$
$911$	−13.9333	−0.461632	−0.230816	+	0.972997i	$0.574140\pi$
$912$	0	0
$913$	−33.5273	−1.10959
$914$	−66.8944	−2.21267
$915$	0	0
$916$	−3.33068	−0.110049
$917$	67.1440	2.21729
$918$	0	0
$919$	43.4563	1.43349	0.716745	−	0.697335i	$-0.245631\pi$
$919$	43.4563	1.43349	0.716745	+	0.697335i	$0.245631\pi$
$920$	0	0
$921$	0	0
$922$	64.1399	2.11233
$923$	44.2937	1.45795
$924$	0	0
$925$	0	0
$926$	12.1742	0.400069
$927$	0	0
$928$	−11.3270	−0.371827
$929$	−15.4714	−0.507599	−0.253799	−	0.967257i	$-0.581680\pi$
$929$	−15.4714	−0.507599	−0.253799	+	0.967257i	$0.581680\pi$
$930$	0	0
$931$	16.0393	0.525666
$932$	−10.9979	−0.360249
$933$	0	0
$934$	7.54542	0.246894
$935$	0	0
$936$	0	0
$937$	−47.3825	−1.54792	−0.773960	−	0.633235i	$-0.781727\pi$
$937$	−47.3825	−1.54792	−0.773960	+	0.633235i	$0.781727\pi$
$938$	3.31910	0.108373
$939$	0	0
$940$	0	0
$941$	−5.99458	−0.195418	−0.0977088	−	0.995215i	$-0.531151\pi$
$941$	−5.99458	−0.195418	−0.0977088	+	0.995215i	$0.531151\pi$
$942$	0	0
$943$	68.0867	2.21721
$944$	−0.696272	−0.0226617
$945$	0	0
$946$	19.8563	0.645585
$947$	−26.1426	−0.849519	−0.424759	−	0.905306i	$-0.639641\pi$
$947$	−26.1426	−0.849519	−0.424759	+	0.905306i	$0.639641\pi$
$948$	0	0
$949$	16.1518	0.524309
$950$	0	0
$951$	0	0
$952$	−11.4280	−0.370383
$953$	−2.02015	−0.0654390	−0.0327195	−	0.999465i	$-0.510417\pi$
$953$	−2.02015	−0.0654390	−0.0327195	+	0.999465i	$0.510417\pi$
$954$	0	0
$955$	0	0
$956$	−13.7071	−0.443321
$957$	0	0
$958$	35.8329	1.15771
$959$	−3.02951	−0.0978281
$960$	0	0
$961$	−26.8348	−0.865638
$962$	59.4861	1.91791
$963$	0	0
$964$	−6.04127	−0.194576
$965$	0	0
$966$	0	0
$967$	33.3078	1.07111	0.535554	−	0.844501i	$-0.320103\pi$
$967$	33.3078	1.07111	0.535554	+	0.844501i	$0.320103\pi$
$968$	−11.3058	−0.363383
$969$	0	0
$970$	0	0
$971$	18.2869	0.586854	0.293427	−	0.955982i	$-0.405204\pi$
$971$	18.2869	0.586854	0.293427	+	0.955982i	$0.405204\pi$
$972$	0	0
$973$	−18.5761	−0.595524
$974$	8.66111	0.277520
$975$	0	0
$976$	−53.0999	−1.69968
$977$	19.7212	0.630938	0.315469	−	0.948936i	$-0.397838\pi$
$977$	19.7212	0.630938	0.315469	+	0.948936i	$0.397838\pi$
$978$	0	0
$979$	0.750271	0.0239788
$980$	0	0
$981$	0	0
$982$	−22.1869	−0.708012
$983$	−47.2559	−1.50723	−0.753614	−	0.657317i	$-0.771691\pi$
$983$	−47.2559	−1.50723	−0.753614	+	0.657317i	$0.771691\pi$
$984$	0	0
$985$	0	0
$986$	7.88702	0.251174
$987$	0	0
$988$	−26.0649	−0.829234
$989$	34.3821	1.09329
$990$	0	0
$991$	−20.8290	−0.661655	−0.330827	−	0.943691i	$-0.607328\pi$
$991$	−20.8290	−0.661655	−0.330827	+	0.943691i	$0.607328\pi$
$992$	10.0094	0.317797
$993$	0	0
$994$	−47.9808	−1.52186
$995$	0	0
$996$	0	0
$997$	34.6495	1.09736	0.548680	−	0.836032i	$-0.315130\pi$
$997$	34.6495	1.09736	0.548680	+	0.836032i	$0.315130\pi$
$998$	−46.6972	−1.47817
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.bf.1.5		24
3.2	odd	2	inner	5625.2.a.bf.1.19		24
5.4	even	2	inner	5625.2.a.bf.1.20		24
15.14	odd	2	inner	5625.2.a.bf.1.6		24
25.8	odd	20		225.2.m.c.64.2	✓	24
25.22	odd	20		225.2.m.c.109.2	yes	24
75.8	even	20		225.2.m.c.64.5	yes	24
75.47	even	20		225.2.m.c.109.5	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.2.m.c.64.2	✓	24	25.8	odd	20
225.2.m.c.64.5	yes	24	75.8	even	20
225.2.m.c.109.2	yes	24	25.22	odd	20
225.2.m.c.109.5	yes	24	75.47	even	20
5625.2.a.bf.1.5		24	1.1	even	1	trivial
5625.2.a.bf.1.6		24	15.14	odd	2	inner
5625.2.a.bf.1.19		24	3.2	odd	2	inner
5625.2.a.bf.1.20		24	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 5625 = 3^{2} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5625.a (trivial)

\(f(q)\)	\(=\)	\(q-1.71285 q^{2} +0.933844 q^{4} -3.13880 q^{7} +1.82616 q^{8} +O(q^{10})\)	\(q-1.71285 q^{2} +0.933844 q^{4} -3.13880 q^{7} +1.82616 q^{8} +2.19293 q^{11} -4.96314 q^{13} +5.37629 q^{14} -4.99562 q^{16} +1.99373 q^{17} +5.62373 q^{19} -3.75616 q^{22} -6.50395 q^{23} +8.50110 q^{26} -2.93115 q^{28} -2.30955 q^{29} +2.04089 q^{31} +4.90442 q^{32} -3.41496 q^{34} +6.99745 q^{37} -9.63258 q^{38} -10.4685 q^{41} -5.28634 q^{43} +2.04786 q^{44} +11.1403 q^{46} +9.28514 q^{47} +2.85207 q^{49} -4.63480 q^{52} -5.22981 q^{53} -5.73196 q^{56} +3.95591 q^{58} +0.139376 q^{59} +10.6293 q^{61} -3.49573 q^{62} +1.59073 q^{64} +0.617360 q^{67} +1.86184 q^{68} -8.92453 q^{71} -3.25434 q^{73} -11.9856 q^{74} +5.25169 q^{76} -6.88319 q^{77} -3.00772 q^{79} +17.9310 q^{82} -15.2888 q^{83} +9.05468 q^{86} +4.00465 q^{88} +0.342131 q^{89} +15.5783 q^{91} -6.07368 q^{92} -15.9040 q^{94} -12.1264 q^{97} -4.88516 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 32 q^{4}+O(q^{10}) \) 24 * q + 32 * q^4	\( 24 q + 32 q^{4} + 56 q^{16} + 36 q^{19} + 52 q^{31} + 60 q^{34} + 60 q^{46} + 72 q^{49} + 68 q^{61} + 108 q^{64} + 88 q^{76} + 84 q^{79} + 80 q^{91} + 100 q^{94}+O(q^{100}) \) 24 * q + 32 * q^4 + 56 * q^16 + 36 * q^19 + 52 * q^31 + 60 * q^34 + 60 * q^46 + 72 * q^49 + 68 * q^61 + 108 * q^64 + 88 * q^76 + 84 * q^79 + 80 * q^91 + 100 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more