LMFDB - Embedded newform 5625.2.a.bf.1.15

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			1.15
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)

$f(q)$	$=$	$q+0.899356 q^{2} -1.19116 q^{4} -3.86000 q^{7} -2.86999 q^{8} +O(q^{10})$	$q+0.899356 q^{2} -1.19116 q^{4} -3.86000 q^{7} -2.86999 q^{8} -4.88410 q^{11} +1.48340 q^{13} -3.47151 q^{14} -0.198827 q^{16} -3.74702 q^{17} +7.15099 q^{19} -4.39255 q^{22} -6.63244 q^{23} +1.33411 q^{26} +4.59787 q^{28} -8.70929 q^{29} -1.54537 q^{31} +5.56116 q^{32} -3.36991 q^{34} -2.52680 q^{37} +6.43129 q^{38} +0.951509 q^{41} -6.07225 q^{43} +5.81774 q^{44} -5.96492 q^{46} +3.79857 q^{47} +7.89958 q^{49} -1.76697 q^{52} -7.19680 q^{53} +11.0781 q^{56} -7.83276 q^{58} +4.20444 q^{59} -5.70110 q^{61} -1.38984 q^{62} +5.39912 q^{64} +0.532552 q^{67} +4.46329 q^{68} +0.452972 q^{71} -2.95779 q^{73} -2.27250 q^{74} -8.51796 q^{76} +18.8526 q^{77} +4.98465 q^{79} +0.855746 q^{82} +7.29026 q^{83} -5.46112 q^{86} +14.0173 q^{88} +10.9949 q^{89} -5.72593 q^{91} +7.90028 q^{92} +3.41627 q^{94} -2.65180 q^{97} +7.10454 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.899356	0.635941	0.317971	−	0.948101i	$-0.396999\pi$
$2$	0.899356	0.635941	0.317971	+	0.948101i	$0.396999\pi$
$3$	0	0
$3$	0	0
$4$	−1.19116	−0.595579
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.86000	−1.45894	−0.729471	−	0.684012i	$-0.760234\pi$
$7$	−3.86000	−1.45894	−0.729471	+	0.684012i	$0.760234\pi$
$8$	−2.86999	−1.01469
$9$	0	0
$10$	0	0
$11$	−4.88410	−1.47261	−0.736306	−	0.676648i	$-0.763432\pi$
$11$	−4.88410	−1.47261	−0.736306	+	0.676648i	$0.763432\pi$
$12$	0	0
$13$	1.48340	0.411422	0.205711	−	0.978613i	$-0.434049\pi$
$13$	1.48340	0.411422	0.205711	+	0.978613i	$0.434049\pi$
$14$	−3.47151	−0.927801
$15$	0	0
$16$	−0.198827	−0.0497068
$17$	−3.74702	−0.908786	−0.454393	−	0.890801i	$-0.650144\pi$
$17$	−3.74702	−0.908786	−0.454393	+	0.890801i	$0.650144\pi$
$18$	0	0
$19$	7.15099	1.64055	0.820275	−	0.571969i	$-0.193820\pi$
$19$	7.15099	1.64055	0.820275	+	0.571969i	$0.193820\pi$
$20$	0	0
$21$	0	0
$22$	−4.39255	−0.936495
$23$	−6.63244	−1.38296	−0.691479	−	0.722396i	$-0.743041\pi$
$23$	−6.63244	−1.38296	−0.691479	+	0.722396i	$0.743041\pi$
$24$	0	0
$25$	0	0
$26$	1.33411	0.261640
$27$	0	0
$28$	4.59787	0.868915
$29$	−8.70929	−1.61728	−0.808638	−	0.588307i	$-0.799795\pi$
$29$	−8.70929	−1.61728	−0.808638	+	0.588307i	$0.799795\pi$
$30$	0	0
$31$	−1.54537	−0.277556	−0.138778	−	0.990323i	$-0.544317\pi$
$31$	−1.54537	−0.277556	−0.138778	+	0.990323i	$0.544317\pi$
$32$	5.56116	0.983084
$33$	0	0
$34$	−3.36991	−0.577934
$35$	0	0
$36$	0	0
$37$	−2.52680	−0.415404	−0.207702	−	0.978192i	$-0.566598\pi$
$37$	−2.52680	−0.415404	−0.207702	+	0.978192i	$0.566598\pi$
$38$	6.43129	1.04329
$39$	0	0
$40$	0	0
$41$	0.951509	0.148601	0.0743004	−	0.997236i	$-0.476328\pi$
$41$	0.951509	0.148601	0.0743004	+	0.997236i	$0.476328\pi$
$42$	0	0
$43$	−6.07225	−0.926010	−0.463005	−	0.886356i	$-0.653229\pi$
$43$	−6.07225	−0.926010	−0.463005	+	0.886356i	$0.653229\pi$
$44$	5.81774	0.877057
$45$	0	0
$46$	−5.96492	−0.879480
$47$	3.79857	0.554078	0.277039	−	0.960859i	$-0.410647\pi$
$47$	3.79857	0.554078	0.277039	+	0.960859i	$0.410647\pi$
$48$	0	0
$49$	7.89958	1.12851
$50$	0	0
$51$	0	0
$52$	−1.76697	−0.245034
$53$	−7.19680	−0.988556	−0.494278	−	0.869304i	$-0.664568\pi$
$53$	−7.19680	−0.988556	−0.494278	+	0.869304i	$0.664568\pi$
$54$	0	0
$55$	0	0
$56$	11.0781	1.48038
$57$	0	0
$58$	−7.83276	−1.02849
$59$	4.20444	0.547371	0.273686	−	0.961819i	$-0.411757\pi$
$59$	4.20444	0.547371	0.273686	+	0.961819i	$0.411757\pi$
$60$	0	0
$61$	−5.70110	−0.729951	−0.364976	−	0.931017i	$-0.618923\pi$
$61$	−5.70110	−0.729951	−0.364976	+	0.931017i	$0.618923\pi$
$62$	−1.38984	−0.176509
$63$	0	0
$64$	5.39912	0.674890
$65$	0	0
$66$	0	0
$67$	0.532552	0.0650616	0.0325308	−	0.999471i	$-0.489643\pi$
$67$	0.532552	0.0650616	0.0325308	+	0.999471i	$0.489643\pi$
$68$	4.46329	0.541254
$69$	0	0
$70$	0	0
$71$	0.452972	0.0537579	0.0268789	−	0.999639i	$-0.491443\pi$
$71$	0.452972	0.0537579	0.0268789	+	0.999639i	$0.491443\pi$
$72$	0	0
$73$	−2.95779	−0.346183	−0.173092	−	0.984906i	$-0.555376\pi$
$73$	−2.95779	−0.346183	−0.173092	+	0.984906i	$0.555376\pi$
$74$	−2.27250	−0.264173
$75$	0	0
$76$	−8.51796	−0.977077
$77$	18.8526	2.14846
$78$	0	0
$79$	4.98465	0.560817	0.280409	−	0.959881i	$-0.409530\pi$
$79$	4.98465	0.560817	0.280409	+	0.959881i	$0.409530\pi$
$80$	0	0
$81$	0	0
$82$	0.855746	0.0945013
$83$	7.29026	0.800210	0.400105	−	0.916469i	$-0.368974\pi$
$83$	7.29026	0.800210	0.400105	+	0.916469i	$0.368974\pi$
$84$	0	0
$85$	0	0
$86$	−5.46112	−0.588888
$87$	0	0
$88$	14.0173	1.49425
$89$	10.9949	1.16546	0.582729	−	0.812666i	$-0.301985\pi$
$89$	10.9949	1.16546	0.582729	+	0.812666i	$0.301985\pi$
$90$	0	0
$91$	−5.72593	−0.600241
$92$	7.90028	0.823661
$93$	0	0
$94$	3.41627	0.352361
$95$	0	0
$96$	0	0
$97$	−2.65180	−0.269249	−0.134625	−	0.990897i	$-0.542983\pi$
$97$	−2.65180	−0.269249	−0.134625	+	0.990897i	$0.542983\pi$
$98$	7.10454	0.717667
$99$	0	0
$100$	0	0
$101$	−6.20716	−0.617636	−0.308818	−	0.951121i	$-0.599933\pi$
$101$	−6.20716	−0.617636	−0.308818	+	0.951121i	$0.599933\pi$
$102$	0	0
$103$	5.41309	0.533368	0.266684	−	0.963784i	$-0.414072\pi$
$103$	5.41309	0.533368	0.266684	+	0.963784i	$0.414072\pi$
$104$	−4.25735	−0.417468
$105$	0	0
$106$	−6.47249	−0.628663
$107$	5.46344	0.528170	0.264085	−	0.964499i	$-0.414930\pi$
$107$	5.46344	0.528170	0.264085	+	0.964499i	$0.414930\pi$
$108$	0	0
$109$	15.5735	1.49167	0.745834	−	0.666131i	$-0.232051\pi$
$109$	15.5735	1.49167	0.745834	+	0.666131i	$0.232051\pi$
$110$	0	0
$111$	0	0
$112$	0.767472	0.0725193
$113$	−4.86032	−0.457220	−0.228610	−	0.973518i	$-0.573418\pi$
$113$	−4.86032	−0.457220	−0.228610	+	0.973518i	$0.573418\pi$
$114$	0	0
$115$	0	0
$116$	10.3741	0.963215
$117$	0	0
$118$	3.78129	0.348096
$119$	14.4635	1.32587
$120$	0	0
$121$	12.8545	1.16859
$122$	−5.12732	−0.464206
$123$	0	0
$124$	1.84078	0.165307
$125$	0	0
$126$	0	0
$127$	17.9527	1.59304	0.796521	−	0.604611i	$-0.206671\pi$
$127$	17.9527	1.59304	0.796521	+	0.604611i	$0.206671\pi$
$128$	−6.26659	−0.553893
$129$	0	0
$130$	0	0
$131$	−17.0189	−1.48695	−0.743473	−	0.668766i	$-0.766823\pi$
$131$	−17.0189	−1.48695	−0.743473	+	0.668766i	$0.766823\pi$
$132$	0	0
$133$	−27.6028	−2.39347
$134$	0.478954	0.0413754
$135$	0	0
$136$	10.7539	0.922140
$137$	8.66415	0.740229	0.370114	−	0.928986i	$-0.379319\pi$
$137$	8.66415	0.740229	0.370114	+	0.928986i	$0.379319\pi$
$138$	0	0
$139$	13.0907	1.11034	0.555171	−	0.831736i	$-0.312653\pi$
$139$	13.0907	1.11034	0.555171	+	0.831736i	$0.312653\pi$
$140$	0	0
$141$	0	0
$142$	0.407383	0.0341868
$143$	−7.24509	−0.605865
$144$	0	0
$145$	0	0
$146$	−2.66011	−0.220152
$147$	0	0
$148$	3.00982	0.247406
$149$	12.8221	1.05043	0.525215	−	0.850970i	$-0.323985\pi$
$149$	12.8221	1.05043	0.525215	+	0.850970i	$0.323985\pi$
$150$	0	0
$151$	7.61093	0.619369	0.309684	−	0.950839i	$-0.399777\pi$
$151$	7.61093	0.619369	0.309684	+	0.950839i	$0.399777\pi$
$152$	−20.5233	−1.66466
$153$	0	0
$154$	16.9552	1.36629
$155$	0	0
$156$	0	0
$157$	−0.549040	−0.0438181	−0.0219091	−	0.999760i	$-0.506974\pi$
$157$	−0.549040	−0.0438181	−0.0219091	+	0.999760i	$0.506974\pi$
$158$	4.48298	0.356647
$159$	0	0
$160$	0	0
$161$	25.6012	2.01766
$162$	0	0
$163$	17.1669	1.34462	0.672308	−	0.740272i	$-0.265303\pi$
$163$	17.1669	1.34462	0.672308	+	0.740272i	$0.265303\pi$
$164$	−1.13340	−0.0885035
$165$	0	0
$166$	6.55654	0.508886
$167$	−0.282439	−0.0218558	−0.0109279	−	0.999940i	$-0.503479\pi$
$167$	−0.282439	−0.0218558	−0.0109279	+	0.999940i	$0.503479\pi$
$168$	0	0
$169$	−10.7995	−0.830732
$170$	0	0
$171$	0	0
$172$	7.23301	0.551512
$173$	8.90884	0.677327	0.338663	−	0.940908i	$-0.390025\pi$
$173$	8.90884	0.677327	0.338663	+	0.940908i	$0.390025\pi$
$174$	0	0
$175$	0	0
$176$	0.971092	0.0731988
$177$	0	0
$178$	9.88835	0.741163
$179$	−3.73566	−0.279217	−0.139608	−	0.990207i	$-0.544584\pi$
$179$	−3.73566	−0.279217	−0.139608	+	0.990207i	$0.544584\pi$
$180$	0	0
$181$	21.3927	1.59011	0.795054	−	0.606539i	$-0.207442\pi$
$181$	21.3927	1.59011	0.795054	+	0.606539i	$0.207442\pi$
$182$	−5.14965	−0.381718
$183$	0	0
$184$	19.0350	1.40328
$185$	0	0
$186$	0	0
$187$	18.3008	1.33829
$188$	−4.52470	−0.329997
$189$	0	0
$190$	0	0
$191$	−9.82146	−0.710656	−0.355328	−	0.934742i	$-0.615631\pi$
$191$	−9.82146	−0.710656	−0.355328	+	0.934742i	$0.615631\pi$
$192$	0	0
$193$	−24.6614	−1.77517	−0.887584	−	0.460646i	$-0.847618\pi$
$193$	−24.6614	−1.77517	−0.887584	+	0.460646i	$0.847618\pi$
$194$	−2.38491	−0.171227
$195$	0	0
$196$	−9.40964	−0.672117
$197$	19.5735	1.39455	0.697276	−	0.716802i	$-0.254395\pi$
$197$	19.5735	1.39455	0.697276	+	0.716802i	$0.254395\pi$
$198$	0	0
$199$	7.09016	0.502608	0.251304	−	0.967908i	$-0.419141\pi$
$199$	7.09016	0.502608	0.251304	+	0.967908i	$0.419141\pi$
$200$	0	0
$201$	0	0
$202$	−5.58245	−0.392780
$203$	33.6179	2.35951
$204$	0	0
$205$	0	0
$206$	4.86830	0.339191
$207$	0	0
$208$	−0.294941	−0.0204505
$209$	−34.9262	−2.41589
$210$	0	0
$211$	−7.70878	−0.530694	−0.265347	−	0.964153i	$-0.585487\pi$
$211$	−7.70878	−0.530694	−0.265347	+	0.964153i	$0.585487\pi$
$212$	8.57252	0.588763
$213$	0	0
$214$	4.91358	0.335885
$215$	0	0
$216$	0	0
$217$	5.96512	0.404938
$218$	14.0061	0.948613
$219$	0	0
$220$	0	0
$221$	−5.55834	−0.373895
$222$	0	0
$223$	−12.1609	−0.814354	−0.407177	−	0.913349i	$-0.633487\pi$
$223$	−12.1609	−0.814354	−0.407177	+	0.913349i	$0.633487\pi$
$224$	−21.4661	−1.43426
$225$	0	0
$226$	−4.37116	−0.290765
$227$	−4.43043	−0.294058	−0.147029	−	0.989132i	$-0.546971\pi$
$227$	−4.43043	−0.294058	−0.147029	+	0.989132i	$0.546971\pi$
$228$	0	0
$229$	20.9907	1.38710	0.693551	−	0.720407i	$-0.256045\pi$
$229$	20.9907	1.38710	0.693551	+	0.720407i	$0.256045\pi$
$230$	0	0
$231$	0	0
$232$	24.9956	1.64104
$233$	−8.67005	−0.567994	−0.283997	−	0.958825i	$-0.591661\pi$
$233$	−8.67005	−0.567994	−0.283997	+	0.958825i	$0.591661\pi$
$234$	0	0
$235$	0	0
$236$	−5.00815	−0.326003
$237$	0	0
$238$	13.0078	0.843172
$239$	20.7268	1.34071	0.670354	−	0.742041i	$-0.266142\pi$
$239$	20.7268	1.34071	0.670354	+	0.742041i	$0.266142\pi$
$240$	0	0
$241$	1.61804	0.104227	0.0521136	−	0.998641i	$-0.483404\pi$
$241$	1.61804	0.104227	0.0521136	+	0.998641i	$0.483404\pi$
$242$	11.5607	0.743153
$243$	0	0
$244$	6.79091	0.434744
$245$	0	0
$246$	0	0
$247$	10.6078	0.674959
$248$	4.43519	0.281635
$249$	0	0
$250$	0	0
$251$	−3.16599	−0.199836	−0.0999179	−	0.994996i	$-0.531858\pi$
$251$	−3.16599	−0.199836	−0.0999179	+	0.994996i	$0.531858\pi$
$252$	0	0
$253$	32.3935	2.03656
$254$	16.1458	1.01308
$255$	0	0
$256$	−16.4341	−1.02713
$257$	−7.74573	−0.483165	−0.241583	−	0.970380i	$-0.577666\pi$
$257$	−7.74573	−0.483165	−0.241583	+	0.970380i	$0.577666\pi$
$258$	0	0
$259$	9.75346	0.606050
$260$	0	0
$261$	0	0
$262$	−15.3060	−0.945610
$263$	−1.28753	−0.0793928	−0.0396964	−	0.999212i	$-0.512639\pi$
$263$	−1.28753	−0.0793928	−0.0396964	+	0.999212i	$0.512639\pi$
$264$	0	0
$265$	0	0
$266$	−24.8248	−1.52210
$267$	0	0
$268$	−0.634354	−0.0387493
$269$	−27.0215	−1.64753	−0.823764	−	0.566933i	$-0.808130\pi$
$269$	−27.0215	−1.64753	−0.823764	+	0.566933i	$0.808130\pi$
$270$	0	0
$271$	−1.41663	−0.0860543	−0.0430272	−	0.999074i	$-0.513700\pi$
$271$	−1.41663	−0.0860543	−0.0430272	+	0.999074i	$0.513700\pi$
$272$	0.745009	0.0451728
$273$	0	0
$274$	7.79216	0.470742
$275$	0	0
$276$	0	0
$277$	−20.4167	−1.22672	−0.613361	−	0.789803i	$-0.710183\pi$
$277$	−20.4167	−1.22672	−0.613361	+	0.789803i	$0.710183\pi$
$278$	11.7732	0.706112
$279$	0	0
$280$	0	0
$281$	−14.3919	−0.858551	−0.429276	−	0.903174i	$-0.641231\pi$
$281$	−14.3919	−0.858551	−0.429276	+	0.903174i	$0.641231\pi$
$282$	0	0
$283$	−27.4467	−1.63154	−0.815770	−	0.578377i	$-0.803686\pi$
$283$	−27.4467	−1.63154	−0.815770	+	0.578377i	$0.803686\pi$
$284$	−0.539561	−0.0320171
$285$	0	0
$286$	−6.51592	−0.385295
$287$	−3.67282	−0.216800
$288$	0	0
$289$	−2.95984	−0.174108
$290$	0	0
$291$	0	0
$292$	3.52320	0.206179
$293$	−28.6959	−1.67643	−0.838217	−	0.545337i	$-0.816402\pi$
$293$	−28.6959	−1.67643	−0.838217	+	0.545337i	$0.816402\pi$
$294$	0	0
$295$	0	0
$296$	7.25190	0.421508
$297$	0	0
$298$	11.5317	0.668012
$299$	−9.83858	−0.568980
$300$	0	0
$301$	23.4389	1.35099
$302$	6.84494	0.393882
$303$	0	0
$304$	−1.42181	−0.0815464
$305$	0	0
$306$	0	0
$307$	−13.9691	−0.797261	−0.398630	−	0.917112i	$-0.630514\pi$
$307$	−13.9691	−0.797261	−0.398630	+	0.917112i	$0.630514\pi$
$308$	−22.4565	−1.27958
$309$	0	0
$310$	0	0
$311$	11.7476	0.666147	0.333074	−	0.942901i	$-0.391914\pi$
$311$	11.7476	0.666147	0.333074	+	0.942901i	$0.391914\pi$
$312$	0	0
$313$	17.4988	0.989092	0.494546	−	0.869151i	$-0.335334\pi$
$313$	17.4988	0.989092	0.494546	+	0.869151i	$0.335334\pi$
$314$	−0.493782	−0.0278658
$315$	0	0
$316$	−5.93751	−0.334011
$317$	27.3765	1.53762	0.768810	−	0.639477i	$-0.220849\pi$
$317$	27.3765	1.53762	0.768810	+	0.639477i	$0.220849\pi$
$318$	0	0
$319$	42.5371	2.38162
$320$	0	0
$321$	0	0
$322$	23.0246	1.28311
$323$	−26.7949	−1.49091
$324$	0	0
$325$	0	0
$326$	15.4392	0.855096
$327$	0	0
$328$	−2.73082	−0.150784
$329$	−14.6625	−0.808368
$330$	0	0
$331$	−8.54425	−0.469634	−0.234817	−	0.972040i	$-0.575449\pi$
$331$	−8.54425	−0.469634	−0.234817	+	0.972040i	$0.575449\pi$
$332$	−8.68385	−0.476588
$333$	0	0
$334$	−0.254013	−0.0138990
$335$	0	0
$336$	0	0
$337$	−9.94865	−0.541937	−0.270969	−	0.962588i	$-0.587344\pi$
$337$	−9.94865	−0.541937	−0.270969	+	0.962588i	$0.587344\pi$
$338$	−9.71261	−0.528297
$339$	0	0
$340$	0	0
$341$	7.54774	0.408733
$342$	0	0
$343$	−3.47237	−0.187490
$344$	17.4273	0.939617
$345$	0	0
$346$	8.01222	0.430740
$347$	−19.6978	−1.05743	−0.528717	−	0.848798i	$-0.677327\pi$
$347$	−19.6978	−1.05743	−0.528717	+	0.848798i	$0.677327\pi$
$348$	0	0
$349$	4.55598	0.243876	0.121938	−	0.992538i	$-0.461089\pi$
$349$	4.55598	0.243876	0.121938	+	0.992538i	$0.461089\pi$
$350$	0	0
$351$	0	0
$352$	−27.1613	−1.44770
$353$	−22.5366	−1.19950	−0.599751	−	0.800186i	$-0.704734\pi$
$353$	−22.5366	−1.19950	−0.599751	+	0.800186i	$0.704734\pi$
$354$	0	0
$355$	0	0
$356$	−13.0967	−0.694122
$357$	0	0
$358$	−3.35969	−0.177565
$359$	14.5125	0.765938	0.382969	−	0.923761i	$-0.374902\pi$
$359$	14.5125	0.765938	0.382969	+	0.923761i	$0.374902\pi$
$360$	0	0
$361$	32.1367	1.69141
$362$	19.2397	1.01122
$363$	0	0
$364$	6.82049	0.357491
$365$	0	0
$366$	0	0
$367$	−20.9000	−1.09097	−0.545486	−	0.838120i	$-0.683655\pi$
$367$	−20.9000	−1.09097	−0.545486	+	0.838120i	$0.683655\pi$
$368$	1.31871	0.0687424
$369$	0	0
$370$	0	0
$371$	27.7796	1.44225
$372$	0	0
$373$	−9.04096	−0.468123	−0.234062	−	0.972222i	$-0.575202\pi$
$373$	−9.04096	−0.468123	−0.234062	+	0.972222i	$0.575202\pi$
$374$	16.4590	0.851073
$375$	0	0
$376$	−10.9019	−0.562220
$377$	−12.9194	−0.665383
$378$	0	0
$379$	−19.4528	−0.999222	−0.499611	−	0.866250i	$-0.666524\pi$
$379$	−19.4528	−0.999222	−0.499611	+	0.866250i	$0.666524\pi$
$380$	0	0
$381$	0	0
$382$	−8.83300	−0.451935
$383$	8.34622	0.426472	0.213236	−	0.977001i	$-0.431600\pi$
$383$	8.34622	0.426472	0.213236	+	0.977001i	$0.431600\pi$
$384$	0	0
$385$	0	0
$386$	−22.1794	−1.12890
$387$	0	0
$388$	3.15871	0.160359
$389$	−18.3095	−0.928328	−0.464164	−	0.885749i	$-0.653645\pi$
$389$	−18.3095	−0.928328	−0.464164	+	0.885749i	$0.653645\pi$
$390$	0	0
$391$	24.8519	1.25681
$392$	−22.6717	−1.14509
$393$	0	0
$394$	17.6035	0.886853
$395$	0	0
$396$	0	0
$397$	−38.3586	−1.92516	−0.962582	−	0.270991i	$-0.912649\pi$
$397$	−38.3586	−1.92516	−0.962582	+	0.270991i	$0.912649\pi$
$398$	6.37659	0.319629
$399$	0	0
$400$	0	0
$401$	34.4606	1.72088	0.860439	−	0.509553i	$-0.170189\pi$
$401$	34.4606	1.72088	0.860439	+	0.509553i	$0.170189\pi$
$402$	0	0
$403$	−2.29240	−0.114193
$404$	7.39371	0.367851
$405$	0	0
$406$	30.2344	1.50051
$407$	12.3412	0.611729
$408$	0	0
$409$	−13.7208	−0.678448	−0.339224	−	0.940706i	$-0.610164\pi$
$409$	−13.7208	−0.678448	−0.339224	+	0.940706i	$0.610164\pi$
$410$	0	0
$411$	0	0
$412$	−6.44785	−0.317663
$413$	−16.2291	−0.798583
$414$	0	0
$415$	0	0
$416$	8.24944	0.404462
$417$	0	0
$418$	−31.4111	−1.53637
$419$	−11.7686	−0.574936	−0.287468	−	0.957790i	$-0.592813\pi$
$419$	−11.7686	−0.574936	−0.287468	+	0.957790i	$0.592813\pi$
$420$	0	0
$421$	1.82949	0.0891640	0.0445820	−	0.999006i	$-0.485804\pi$
$421$	1.82949	0.0891640	0.0445820	+	0.999006i	$0.485804\pi$
$422$	−6.93294	−0.337490
$423$	0	0
$424$	20.6547	1.00308
$425$	0	0
$426$	0	0
$427$	22.0062	1.06496
$428$	−6.50782	−0.314567
$429$	0	0
$430$	0	0
$431$	32.2801	1.55488	0.777438	−	0.628960i	$-0.216519\pi$
$431$	32.2801	1.55488	0.777438	+	0.628960i	$0.216519\pi$
$432$	0	0
$433$	−13.5499	−0.651169	−0.325584	−	0.945513i	$-0.605561\pi$
$433$	−13.5499	−0.651169	−0.325584	+	0.945513i	$0.605561\pi$
$434$	5.36477	0.257517
$435$	0	0
$436$	−18.5505	−0.888407
$437$	−47.4285	−2.26881
$438$	0	0
$439$	33.3166	1.59011	0.795056	−	0.606535i	$-0.207441\pi$
$439$	33.3166	1.59011	0.795056	+	0.606535i	$0.207441\pi$
$440$	0	0
$441$	0	0
$442$	−4.99893	−0.237775
$443$	6.95526	0.330454	0.165227	−	0.986256i	$-0.447164\pi$
$443$	6.95526	0.330454	0.165227	+	0.986256i	$0.447164\pi$
$444$	0	0
$445$	0	0
$446$	−10.9370	−0.517881
$447$	0	0
$448$	−20.8406	−0.984625
$449$	7.36708	0.347674	0.173837	−	0.984774i	$-0.444383\pi$
$449$	7.36708	0.347674	0.173837	+	0.984774i	$0.444383\pi$
$450$	0	0
$451$	−4.64727	−0.218831
$452$	5.78940	0.272311
$453$	0	0
$454$	−3.98453	−0.187003
$455$	0	0
$456$	0	0
$457$	−27.0579	−1.26572	−0.632859	−	0.774267i	$-0.718119\pi$
$457$	−27.0579	−1.26572	−0.632859	+	0.774267i	$0.718119\pi$
$458$	18.8781	0.882116
$459$	0	0
$460$	0	0
$461$	−15.3784	−0.716244	−0.358122	−	0.933675i	$-0.616583\pi$
$461$	−15.3784	−0.716244	−0.358122	+	0.933675i	$0.616583\pi$
$462$	0	0
$463$	15.2830	0.710263	0.355131	−	0.934816i	$-0.384436\pi$
$463$	15.2830	0.710263	0.355131	+	0.934816i	$0.384436\pi$
$464$	1.73164	0.0803895
$465$	0	0
$466$	−7.79747	−0.361211
$467$	−34.9596	−1.61774	−0.808868	−	0.587990i	$-0.799919\pi$
$467$	−34.9596	−1.61774	−0.808868	+	0.587990i	$0.799919\pi$
$468$	0	0
$469$	−2.05565	−0.0949211
$470$	0	0
$471$	0	0
$472$	−12.0667	−0.555414
$473$	29.6575	1.36365
$474$	0	0
$475$	0	0
$476$	−17.2283	−0.789658
$477$	0	0
$478$	18.6408	0.852612
$479$	40.9009	1.86881	0.934405	−	0.356214i	$-0.115932\pi$
$479$	40.9009	1.86881	0.934405	+	0.356214i	$0.115932\pi$
$480$	0	0
$481$	−3.74827	−0.170906
$482$	1.45520	0.0662824
$483$	0	0
$484$	−15.3117	−0.695986
$485$	0	0
$486$	0	0
$487$	17.0134	0.770953	0.385476	−	0.922718i	$-0.374037\pi$
$487$	17.0134	0.770953	0.385476	+	0.922718i	$0.374037\pi$
$488$	16.3621	0.740677
$489$	0	0
$490$	0	0
$491$	17.9582	0.810440	0.405220	−	0.914219i	$-0.367195\pi$
$491$	17.9582	0.810440	0.405220	+	0.914219i	$0.367195\pi$
$492$	0	0
$493$	32.6339	1.46976
$494$	9.54020	0.429234
$495$	0	0
$496$	0.307261	0.0137964
$497$	−1.74847	−0.0784296
$498$	0	0
$499$	14.3327	0.641619	0.320809	−	0.947144i	$-0.396045\pi$
$499$	14.3327	0.641619	0.320809	+	0.947144i	$0.396045\pi$
$500$	0	0
$501$	0	0
$502$	−2.84736	−0.127084
$503$	−38.1401	−1.70058	−0.850291	−	0.526312i	$-0.823574\pi$
$503$	−38.1401	−1.70058	−0.850291	+	0.526312i	$0.823574\pi$
$504$	0	0
$505$	0	0
$506$	29.1333	1.29513
$507$	0	0
$508$	−21.3845	−0.948782
$509$	37.9593	1.68252	0.841258	−	0.540634i	$-0.181815\pi$
$509$	37.9593	1.68252	0.841258	+	0.540634i	$0.181815\pi$
$510$	0	0
$511$	11.4171	0.505061
$512$	−2.24697	−0.0993031
$513$	0	0
$514$	−6.96617	−0.307265
$515$	0	0
$516$	0	0
$517$	−18.5526	−0.815943
$518$	8.77184	0.385412
$519$	0	0
$520$	0	0
$521$	−30.0950	−1.31849	−0.659243	−	0.751930i	$-0.729123\pi$
$521$	−30.0950	−1.31849	−0.659243	+	0.751930i	$0.729123\pi$
$522$	0	0
$523$	34.8172	1.52245	0.761226	−	0.648487i	$-0.224598\pi$
$523$	34.8172	1.52245	0.761226	+	0.648487i	$0.224598\pi$
$524$	20.2722	0.885594
$525$	0	0
$526$	−1.15795	−0.0504891
$527$	5.79052	0.252239
$528$	0	0
$529$	20.9892	0.912574
$530$	0	0
$531$	0	0
$532$	32.8793	1.42550
$533$	1.41147	0.0611376
$534$	0	0
$535$	0	0
$536$	−1.52842	−0.0660177
$537$	0	0
$538$	−24.3019	−1.04773
$539$	−38.5824	−1.66186
$540$	0	0
$541$	6.70785	0.288393	0.144197	−	0.989549i	$-0.453940\pi$
$541$	6.70785	0.288393	0.144197	+	0.989549i	$0.453940\pi$
$542$	−1.27406	−0.0547255
$543$	0	0
$544$	−20.8378	−0.893412
$545$	0	0
$546$	0	0
$547$	5.77229	0.246805	0.123403	−	0.992357i	$-0.460619\pi$
$547$	5.77229	0.246805	0.123403	+	0.992357i	$0.460619\pi$
$548$	−10.3204	−0.440865
$549$	0	0
$550$	0	0
$551$	−62.2801	−2.65322
$552$	0	0
$553$	−19.2407	−0.818200
$554$	−18.3619	−0.780123
$555$	0	0
$556$	−15.5931	−0.661296
$557$	−6.81225	−0.288644	−0.144322	−	0.989531i	$-0.546100\pi$
$557$	−6.81225	−0.288644	−0.144322	+	0.989531i	$0.546100\pi$
$558$	0	0
$559$	−9.00760	−0.380981
$560$	0	0
$561$	0	0
$562$	−12.9435	−0.545988
$563$	23.6138	0.995202	0.497601	−	0.867406i	$-0.334214\pi$
$563$	23.6138	0.995202	0.497601	+	0.867406i	$0.334214\pi$
$564$	0	0
$565$	0	0
$566$	−24.6844	−1.03756
$567$	0	0
$568$	−1.30002	−0.0545478
$569$	−6.95092	−0.291398	−0.145699	−	0.989329i	$-0.546543\pi$
$569$	−6.95092	−0.291398	−0.145699	+	0.989329i	$0.546543\pi$
$570$	0	0
$571$	19.9277	0.833948	0.416974	−	0.908919i	$-0.363091\pi$
$571$	19.9277	0.833948	0.416974	+	0.908919i	$0.363091\pi$
$572$	8.63005	0.360841
$573$	0	0
$574$	−3.30318	−0.137872
$575$	0	0
$576$	0	0
$577$	33.0804	1.37715	0.688577	−	0.725163i	$-0.258235\pi$
$577$	33.0804	1.37715	0.688577	+	0.725163i	$0.258235\pi$
$578$	−2.66195	−0.110723
$579$	0	0
$580$	0	0
$581$	−28.1404	−1.16746
$582$	0	0
$583$	35.1499	1.45576
$584$	8.48883	0.351270
$585$	0	0
$586$	−25.8079	−1.06611
$587$	2.38572	0.0984694	0.0492347	−	0.998787i	$-0.484322\pi$
$587$	2.38572	0.0984694	0.0492347	+	0.998787i	$0.484322\pi$
$588$	0	0
$589$	−11.0509	−0.455345
$590$	0	0
$591$	0	0
$592$	0.502397	0.0206484
$593$	20.4095	0.838118	0.419059	−	0.907959i	$-0.362360\pi$
$593$	20.4095	0.838118	0.419059	+	0.907959i	$0.362360\pi$
$594$	0	0
$595$	0	0
$596$	−15.2732	−0.625614
$597$	0	0
$598$	−8.84839	−0.361838
$599$	−1.21706	−0.0497278	−0.0248639	−	0.999691i	$-0.507915\pi$
$599$	−1.21706	−0.0497278	−0.0248639	+	0.999691i	$0.507915\pi$
$600$	0	0
$601$	−19.9467	−0.813642	−0.406821	−	0.913508i	$-0.633363\pi$
$601$	−19.9467	−0.813642	−0.406821	+	0.913508i	$0.633363\pi$
$602$	21.0799	0.859153
$603$	0	0
$604$	−9.06582	−0.368883
$605$	0	0
$606$	0	0
$607$	13.9064	0.564442	0.282221	−	0.959349i	$-0.408929\pi$
$607$	13.9064	0.564442	0.282221	+	0.959349i	$0.408929\pi$
$608$	39.7678	1.61280
$609$	0	0
$610$	0	0
$611$	5.63481	0.227960
$612$	0	0
$613$	26.2104	1.05863	0.529314	−	0.848426i	$-0.322449\pi$
$613$	26.2104	1.05863	0.529314	+	0.848426i	$0.322449\pi$
$614$	−12.5632	−0.507011
$615$	0	0
$616$	−54.1068	−2.18003
$617$	−33.3980	−1.34455	−0.672277	−	0.740300i	$-0.734684\pi$
$617$	−33.3980	−1.34455	−0.672277	+	0.740300i	$0.734684\pi$
$618$	0	0
$619$	−2.42159	−0.0973321	−0.0486661	−	0.998815i	$-0.515497\pi$
$619$	−2.42159	−0.0973321	−0.0486661	+	0.998815i	$0.515497\pi$
$620$	0	0
$621$	0	0
$622$	10.5653	0.423630
$623$	−42.4403	−1.70034
$624$	0	0
$625$	0	0
$626$	15.7377	0.629004
$627$	0	0
$628$	0.653993	0.0260972
$629$	9.46799	0.377513
$630$	0	0
$631$	7.17308	0.285556	0.142778	−	0.989755i	$-0.454397\pi$
$631$	7.17308	0.285556	0.142778	+	0.989755i	$0.454397\pi$
$632$	−14.3059	−0.569058
$633$	0	0
$634$	24.6213	0.977836
$635$	0	0
$636$	0	0
$637$	11.7183	0.464294
$638$	38.2560	1.51457
$639$	0	0
$640$	0	0
$641$	42.0370	1.66036	0.830181	−	0.557494i	$-0.188237\pi$
$641$	42.0370	1.66036	0.830181	+	0.557494i	$0.188237\pi$
$642$	0	0
$643$	8.35291	0.329407	0.164703	−	0.986343i	$-0.447333\pi$
$643$	8.35291	0.329407	0.164703	+	0.986343i	$0.447333\pi$
$644$	−30.4951	−1.20167
$645$	0	0
$646$	−24.0982	−0.948130
$647$	−33.4576	−1.31535	−0.657677	−	0.753300i	$-0.728461\pi$
$647$	−33.4576	−1.31535	−0.657677	+	0.753300i	$0.728461\pi$
$648$	0	0
$649$	−20.5349	−0.806066
$650$	0	0
$651$	0	0
$652$	−20.4485	−0.800825
$653$	14.0432	0.549554	0.274777	−	0.961508i	$-0.411396\pi$
$653$	14.0432	0.549554	0.274777	+	0.961508i	$0.411396\pi$
$654$	0	0
$655$	0	0
$656$	−0.189186	−0.00738646
$657$	0	0
$658$	−13.1868	−0.514075
$659$	23.0287	0.897070	0.448535	−	0.893765i	$-0.351946\pi$
$659$	23.0287	0.897070	0.448535	+	0.893765i	$0.351946\pi$
$660$	0	0
$661$	9.79872	0.381126	0.190563	−	0.981675i	$-0.438969\pi$
$661$	9.79872	0.381126	0.190563	+	0.981675i	$0.438969\pi$
$662$	−7.68432	−0.298660
$663$	0	0
$664$	−20.9230	−0.811968
$665$	0	0
$666$	0	0
$667$	57.7638	2.23662
$668$	0.336429	0.0130168
$669$	0	0
$670$	0	0
$671$	27.8448	1.07494
$672$	0	0
$673$	3.39038	0.130690	0.0653448	−	0.997863i	$-0.479185\pi$
$673$	3.39038	0.130690	0.0653448	+	0.997863i	$0.479185\pi$
$674$	−8.94738	−0.344640
$675$	0	0
$676$	12.8639	0.494766
$677$	−29.0347	−1.11589	−0.557947	−	0.829877i	$-0.688411\pi$
$677$	−29.0347	−1.11589	−0.557947	+	0.829877i	$0.688411\pi$
$678$	0	0
$679$	10.2359	0.392819
$680$	0	0
$681$	0	0
$682$	6.78811	0.259930
$683$	−36.3512	−1.39094	−0.695470	−	0.718555i	$-0.744804\pi$
$683$	−36.3512	−1.39094	−0.695470	+	0.718555i	$0.744804\pi$
$684$	0	0
$685$	0	0
$686$	−3.12290	−0.119233
$687$	0	0
$688$	1.20733	0.0460290
$689$	−10.6758	−0.406714
$690$	0	0
$691$	2.59072	0.0985557	0.0492778	−	0.998785i	$-0.484308\pi$
$691$	2.59072	0.0985557	0.0492778	+	0.998785i	$0.484308\pi$
$692$	−10.6118	−0.403401
$693$	0	0
$694$	−17.7153	−0.672465
$695$	0	0
$696$	0	0
$697$	−3.56532	−0.135046
$698$	4.09745	0.155091
$699$	0	0
$700$	0	0
$701$	−18.1438	−0.685283	−0.342641	−	0.939466i	$-0.611322\pi$
$701$	−18.1438	−0.685283	−0.342641	+	0.939466i	$0.611322\pi$
$702$	0	0
$703$	−18.0692	−0.681491
$704$	−26.3699	−0.993851
$705$	0	0
$706$	−20.2684	−0.762813
$707$	23.9596	0.901095
$708$	0	0
$709$	31.0035	1.16436	0.582180	−	0.813060i	$-0.302200\pi$
$709$	31.0035	1.16436	0.582180	+	0.813060i	$0.302200\pi$
$710$	0	0
$711$	0	0
$712$	−31.5553	−1.18258
$713$	10.2496	0.383849
$714$	0	0
$715$	0	0
$716$	4.44977	0.166296
$717$	0	0
$718$	13.0519	0.487092
$719$	22.7855	0.849756	0.424878	−	0.905251i	$-0.360317\pi$
$719$	22.7855	0.849756	0.424878	+	0.905251i	$0.360317\pi$
$720$	0	0
$721$	−20.8945	−0.778153
$722$	28.9024	1.07563
$723$	0	0
$724$	−25.4821	−0.947035
$725$	0	0
$726$	0	0
$727$	6.50807	0.241371	0.120685	−	0.992691i	$-0.461491\pi$
$727$	6.50807	0.241371	0.120685	+	0.992691i	$0.461491\pi$
$728$	16.4334	0.609061
$729$	0	0
$730$	0	0
$731$	22.7529	0.841545
$732$	0	0
$733$	28.0555	1.03625	0.518126	−	0.855304i	$-0.326630\pi$
$733$	28.0555	1.03625	0.518126	+	0.855304i	$0.326630\pi$
$734$	−18.7966	−0.693793
$735$	0	0
$736$	−36.8840	−1.35956
$737$	−2.60104	−0.0958106
$738$	0	0
$739$	5.61297	0.206476	0.103238	−	0.994657i	$-0.467080\pi$
$739$	5.61297	0.206476	0.103238	+	0.994657i	$0.467080\pi$
$740$	0	0
$741$	0	0
$742$	24.9838	0.917183
$743$	3.68704	0.135265	0.0676323	−	0.997710i	$-0.478456\pi$
$743$	3.68704	0.135265	0.0676323	+	0.997710i	$0.478456\pi$
$744$	0	0
$745$	0	0
$746$	−8.13105	−0.297699
$747$	0	0
$748$	−21.7992	−0.797057
$749$	−21.0888	−0.770570
$750$	0	0
$751$	−0.177985	−0.00649477	−0.00324739	−	0.999995i	$-0.501034\pi$
$751$	−0.177985	−0.00649477	−0.00324739	+	0.999995i	$0.501034\pi$
$752$	−0.755259	−0.0275414
$753$	0	0
$754$	−11.6191	−0.423144
$755$	0	0
$756$	0	0
$757$	−46.8854	−1.70408	−0.852040	−	0.523477i	$-0.824635\pi$
$757$	−46.8854	−1.70408	−0.852040	+	0.523477i	$0.824635\pi$
$758$	−17.4950	−0.635446
$759$	0	0
$760$	0	0
$761$	−20.8659	−0.756390	−0.378195	−	0.925726i	$-0.623455\pi$
$761$	−20.8659	−0.756390	−0.378195	+	0.925726i	$0.623455\pi$
$762$	0	0
$763$	−60.1136	−2.17626
$764$	11.6989	0.423252
$765$	0	0
$766$	7.50623	0.271211
$767$	6.23688	0.225201
$768$	0	0
$769$	−43.0187	−1.55129	−0.775647	−	0.631167i	$-0.782576\pi$
$769$	−43.0187	−1.55129	−0.775647	+	0.631167i	$0.782576\pi$
$770$	0	0
$771$	0	0
$772$	29.3756	1.05725
$773$	11.7880	0.423984	0.211992	−	0.977271i	$-0.432005\pi$
$773$	11.7880	0.423984	0.211992	+	0.977271i	$0.432005\pi$
$774$	0	0
$775$	0	0
$776$	7.61063	0.273206
$777$	0	0
$778$	−16.4668	−0.590362
$779$	6.80424	0.243787
$780$	0	0
$781$	−2.21236	−0.0791645
$782$	22.3507	0.799259
$783$	0	0
$784$	−1.57065	−0.0560946
$785$	0	0
$786$	0	0
$787$	48.6870	1.73551	0.867753	−	0.496996i	$-0.165564\pi$
$787$	48.6870	1.73551	0.867753	+	0.496996i	$0.165564\pi$
$788$	−23.3151	−0.830566
$789$	0	0
$790$	0	0
$791$	18.7608	0.667057
$792$	0	0
$793$	−8.45703	−0.300318
$794$	−34.4981	−1.22429
$795$	0	0
$796$	−8.44551	−0.299343
$797$	16.6192	0.588684	0.294342	−	0.955700i	$-0.404900\pi$
$797$	16.6192	0.588684	0.294342	+	0.955700i	$0.404900\pi$
$798$	0	0
$799$	−14.2333	−0.503539
$800$	0	0
$801$	0	0
$802$	30.9923	1.09438
$803$	14.4462	0.509794
$804$	0	0
$805$	0	0
$806$	−2.06169	−0.0726199
$807$	0	0
$808$	17.8145	0.626712
$809$	−7.67930	−0.269990	−0.134995	−	0.990846i	$-0.543102\pi$
$809$	−7.67930	−0.269990	−0.134995	+	0.990846i	$0.543102\pi$
$810$	0	0
$811$	40.3534	1.41700	0.708499	−	0.705712i	$-0.249373\pi$
$811$	40.3534	1.41700	0.708499	+	0.705712i	$0.249373\pi$
$812$	−40.0442	−1.40527
$813$	0	0
$814$	11.0991	0.389024
$815$	0	0
$816$	0	0
$817$	−43.4226	−1.51917
$818$	−12.3399	−0.431453
$819$	0	0
$820$	0	0
$821$	−1.69254	−0.0590701	−0.0295351	−	0.999564i	$-0.509403\pi$
$821$	−1.69254	−0.0590701	−0.0295351	+	0.999564i	$0.509403\pi$
$822$	0	0
$823$	30.1933	1.05247	0.526236	−	0.850339i	$-0.323603\pi$
$823$	30.1933	1.05247	0.526236	+	0.850339i	$0.323603\pi$
$824$	−15.5355	−0.541205
$825$	0	0
$826$	−14.5958	−0.507851
$827$	16.5536	0.575624	0.287812	−	0.957687i	$-0.407072\pi$
$827$	16.5536	0.575624	0.287812	+	0.957687i	$0.407072\pi$
$828$	0	0
$829$	8.16964	0.283743	0.141872	−	0.989885i	$-0.454688\pi$
$829$	8.16964	0.283743	0.141872	+	0.989885i	$0.454688\pi$
$830$	0	0
$831$	0	0
$832$	8.00907	0.277665
$833$	−29.5999	−1.02557
$834$	0	0
$835$	0	0
$836$	41.6026	1.43886
$837$	0	0
$838$	−10.5842	−0.365625
$839$	−15.4017	−0.531726	−0.265863	−	0.964011i	$-0.585657\pi$
$839$	−15.4017	−0.531726	−0.265863	+	0.964011i	$0.585657\pi$
$840$	0	0
$841$	46.8518	1.61558
$842$	1.64537	0.0567030
$843$	0	0
$844$	9.18237	0.316070
$845$	0	0
$846$	0	0
$847$	−49.6182	−1.70490
$848$	1.43092	0.0491379
$849$	0	0
$850$	0	0
$851$	16.7589	0.574487
$852$	0	0
$853$	38.5951	1.32147	0.660735	−	0.750619i	$-0.270245\pi$
$853$	38.5951	1.32147	0.660735	+	0.750619i	$0.270245\pi$
$854$	19.7914	0.677249
$855$	0	0
$856$	−15.6800	−0.535931
$857$	−48.3995	−1.65330	−0.826648	−	0.562719i	$-0.809755\pi$
$857$	−48.3995	−1.65330	−0.826648	+	0.562719i	$0.809755\pi$
$858$	0	0
$859$	−2.51234	−0.0857199	−0.0428600	−	0.999081i	$-0.513647\pi$
$859$	−2.51234	−0.0857199	−0.0428600	+	0.999081i	$0.513647\pi$
$860$	0	0
$861$	0	0
$862$	29.0313	0.988809
$863$	−23.5845	−0.802827	−0.401414	−	0.915897i	$-0.631481\pi$
$863$	−23.5845	−0.802827	−0.401414	+	0.915897i	$0.631481\pi$
$864$	0	0
$865$	0	0
$866$	−12.1862	−0.414105
$867$	0	0
$868$	−7.10539	−0.241173
$869$	−24.3456	−0.825866
$870$	0	0
$871$	0.789990	0.0267678
$872$	−44.6957	−1.51359
$873$	0	0
$874$	−42.6551	−1.44283
$875$	0	0
$876$	0	0
$877$	46.7589	1.57894	0.789468	−	0.613791i	$-0.210356\pi$
$877$	46.7589	1.57894	0.789468	+	0.613791i	$0.210356\pi$
$878$	29.9635	1.01122
$879$	0	0
$880$	0	0
$881$	−10.3281	−0.347963	−0.173981	−	0.984749i	$-0.555663\pi$
$881$	−10.3281	−0.347963	−0.173981	+	0.984749i	$0.555663\pi$
$882$	0	0
$883$	44.6829	1.50370	0.751850	−	0.659334i	$-0.229162\pi$
$883$	44.6829	1.50370	0.751850	+	0.659334i	$0.229162\pi$
$884$	6.62086	0.222684
$885$	0	0
$886$	6.25526	0.210149
$887$	54.9993	1.84669	0.923347	−	0.383966i	$-0.125442\pi$
$887$	54.9993	1.84669	0.923347	+	0.383966i	$0.125442\pi$
$888$	0	0
$889$	−69.2972	−2.32415
$890$	0	0
$891$	0	0
$892$	14.4856	0.485012
$893$	27.1635	0.908994
$894$	0	0
$895$	0	0
$896$	24.1890	0.808098
$897$	0	0
$898$	6.62563	0.221100
$899$	13.4591	0.448885
$900$	0	0
$901$	26.9665	0.898386
$902$	−4.17955	−0.139164
$903$	0	0
$904$	13.9491	0.463939
$905$	0	0
$906$	0	0
$907$	−5.84187	−0.193976	−0.0969880	−	0.995286i	$-0.530921\pi$
$907$	−5.84187	−0.193976	−0.0969880	+	0.995286i	$0.530921\pi$
$908$	5.27734	0.175135
$909$	0	0
$910$	0	0
$911$	−3.63802	−0.120533	−0.0602664	−	0.998182i	$-0.519195\pi$
$911$	−3.63802	−0.120533	−0.0602664	+	0.998182i	$0.519195\pi$
$912$	0	0
$913$	−35.6064	−1.17840
$914$	−24.3347	−0.804922
$915$	0	0
$916$	−25.0032	−0.826129
$917$	65.6928	2.16937
$918$	0	0
$919$	−19.1153	−0.630554	−0.315277	−	0.949000i	$-0.602097\pi$
$919$	−19.1153	−0.630554	−0.315277	+	0.949000i	$0.602097\pi$
$920$	0	0
$921$	0	0
$922$	−13.8307	−0.455489
$923$	0.671940	0.0221172
$924$	0	0
$925$	0	0
$926$	13.7449	0.451685
$927$	0	0
$928$	−48.4338	−1.58992
$929$	14.9289	0.489802	0.244901	−	0.969548i	$-0.421245\pi$
$929$	14.9289	0.489802	0.244901	+	0.969548i	$0.421245\pi$
$930$	0	0
$931$	56.4898	1.85138
$932$	10.3274	0.338285
$933$	0	0
$934$	−31.4411	−1.02878
$935$	0	0
$936$	0	0
$937$	16.5899	0.541967	0.270984	−	0.962584i	$-0.412651\pi$
$937$	16.5899	0.541967	0.270984	+	0.962584i	$0.412651\pi$
$938$	−1.84876	−0.0603643
$939$	0	0
$940$	0	0
$941$	−25.1007	−0.818260	−0.409130	−	0.912476i	$-0.634168\pi$
$941$	−25.1007	−0.818260	−0.409130	+	0.912476i	$0.634168\pi$
$942$	0	0
$943$	−6.31082	−0.205509
$944$	−0.835956	−0.0272080
$945$	0	0
$946$	26.6727	0.867203
$947$	−38.7058	−1.25777	−0.628884	−	0.777499i	$-0.716488\pi$
$947$	−38.7058	−1.25777	−0.628884	+	0.777499i	$0.716488\pi$
$948$	0	0
$949$	−4.38760	−0.142427
$950$	0	0
$951$	0	0
$952$	−41.5100	−1.34535
$953$	−20.8292	−0.674725	−0.337362	−	0.941375i	$-0.609535\pi$
$953$	−20.8292	−0.674725	−0.337362	+	0.941375i	$0.609535\pi$
$954$	0	0
$955$	0	0
$956$	−24.6889	−0.798498
$957$	0	0
$958$	36.7845	1.18845
$959$	−33.4436	−1.07995
$960$	0	0
$961$	−28.6118	−0.922963
$962$	−3.37103	−0.108686
$963$	0	0
$964$	−1.92734	−0.0620756
$965$	0	0
$966$	0	0
$967$	40.5604	1.30433	0.652167	−	0.758076i	$-0.273860\pi$
$967$	40.5604	1.30433	0.652167	+	0.758076i	$0.273860\pi$
$968$	−36.8922	−1.18576
$969$	0	0
$970$	0	0
$971$	−42.1799	−1.35362	−0.676809	−	0.736159i	$-0.736638\pi$
$971$	−42.1799	−1.35362	−0.676809	+	0.736159i	$0.736638\pi$
$972$	0	0
$973$	−50.5302	−1.61992
$974$	15.3012	0.490281
$975$	0	0
$976$	1.13353	0.0362835
$977$	−29.0797	−0.930343	−0.465172	−	0.885220i	$-0.654007\pi$
$977$	−29.0797	−0.930343	−0.465172	+	0.885220i	$0.654007\pi$
$978$	0	0
$979$	−53.7003	−1.71627
$980$	0	0
$981$	0	0
$982$	16.1508	0.515392
$983$	43.3451	1.38249	0.691247	−	0.722618i	$-0.257061\pi$
$983$	43.3451	1.38249	0.691247	+	0.722618i	$0.257061\pi$
$984$	0	0
$985$	0	0
$986$	29.3495	0.934679
$987$	0	0
$988$	−12.6356	−0.401991
$989$	40.2738	1.28063
$990$	0	0
$991$	22.6880	0.720709	0.360355	−	0.932815i	$-0.382656\pi$
$991$	22.6880	0.720709	0.360355	+	0.932815i	$0.382656\pi$
$992$	−8.59404	−0.272861
$993$	0	0
$994$	−1.57250	−0.0498766
$995$	0	0
$996$	0	0
$997$	42.0712	1.33241	0.666204	−	0.745770i	$-0.267918\pi$
$997$	42.0712	1.33241	0.666204	+	0.745770i	$0.267918\pi$
$998$	12.8902	0.408032
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.bf.1.15		24
3.2	odd	2	inner	5625.2.a.bf.1.9		24
5.4	even	2	inner	5625.2.a.bf.1.10		24
15.14	odd	2	inner	5625.2.a.bf.1.16		24
25.2	odd	20		225.2.m.c.154.4	yes	24
25.13	odd	20		225.2.m.c.19.4	yes	24
75.2	even	20		225.2.m.c.154.3	yes	24
75.38	even	20		225.2.m.c.19.3	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.2.m.c.19.3	✓	24	75.38	even	20
225.2.m.c.19.4	yes	24	25.13	odd	20
225.2.m.c.154.3	yes	24	75.2	even	20
225.2.m.c.154.4	yes	24	25.2	odd	20
5625.2.a.bf.1.9		24	3.2	odd	2	inner
5625.2.a.bf.1.10		24	5.4	even	2	inner
5625.2.a.bf.1.15		24	1.1	even	1	trivial
5625.2.a.bf.1.16		24	15.14	odd	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)

\(f(q)\)	\(=\)	\(q+0.899356 q^{2} -1.19116 q^{4} -3.86000 q^{7} -2.86999 q^{8} +O(q^{10})\)	\(q+0.899356 q^{2} -1.19116 q^{4} -3.86000 q^{7} -2.86999 q^{8} -4.88410 q^{11} +1.48340 q^{13} -3.47151 q^{14} -0.198827 q^{16} -3.74702 q^{17} +7.15099 q^{19} -4.39255 q^{22} -6.63244 q^{23} +1.33411 q^{26} +4.59787 q^{28} -8.70929 q^{29} -1.54537 q^{31} +5.56116 q^{32} -3.36991 q^{34} -2.52680 q^{37} +6.43129 q^{38} +0.951509 q^{41} -6.07225 q^{43} +5.81774 q^{44} -5.96492 q^{46} +3.79857 q^{47} +7.89958 q^{49} -1.76697 q^{52} -7.19680 q^{53} +11.0781 q^{56} -7.83276 q^{58} +4.20444 q^{59} -5.70110 q^{61} -1.38984 q^{62} +5.39912 q^{64} +0.532552 q^{67} +4.46329 q^{68} +0.452972 q^{71} -2.95779 q^{73} -2.27250 q^{74} -8.51796 q^{76} +18.8526 q^{77} +4.98465 q^{79} +0.855746 q^{82} +7.29026 q^{83} -5.46112 q^{86} +14.0173 q^{88} +10.9949 q^{89} -5.72593 q^{91} +7.90028 q^{92} +3.41627 q^{94} -2.65180 q^{97} +7.10454 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 32 q^{4}+O(q^{10}) \) 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