LMFDB - Embedded newform 5625.2.a.v.1.7

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:	$1$
Dimension:	$8$
Coefficient field:	8.8.33620000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 10x^{6} + 30x^{4} - 25x^{2} + 5 $ x^8 - 10x^6 + 30x^4 - 25*x^2 + 5
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 225)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$2.03035$ of defining polynomial
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)

$f(q)$	$=$	$q+2.03035 q^{2} +2.12233 q^{4} +0.504300 q^{7} +0.248380 q^{8} +O(q^{10})$	$q+2.03035 q^{2} +2.12233 q^{4} +0.504300 q^{7} +0.248380 q^{8} +3.43869 q^{11} -4.93831 q^{13} +1.02391 q^{14} -3.74037 q^{16} -1.17741 q^{17} -5.93831 q^{19} +6.98175 q^{22} -2.97817 q^{23} -10.0265 q^{26} +1.07029 q^{28} -5.01799 q^{29} +3.56166 q^{31} -8.09103 q^{32} -2.39057 q^{34} +1.50101 q^{37} -12.0569 q^{38} -8.78461 q^{41} +10.2500 q^{43} +7.29804 q^{44} -6.04673 q^{46} +2.85291 q^{47} -6.74568 q^{49} -10.4807 q^{52} -13.2638 q^{53} +0.125258 q^{56} -10.1883 q^{58} -9.50068 q^{59} -8.48073 q^{61} +7.23142 q^{62} -8.94691 q^{64} +9.70289 q^{67} -2.49886 q^{68} +13.7157 q^{71} +12.4389 q^{73} +3.04759 q^{74} -12.6031 q^{76} +1.73413 q^{77} -14.6926 q^{79} -17.8359 q^{82} -0.356179 q^{83} +20.8111 q^{86} +0.854102 q^{88} +6.51041 q^{89} -2.49039 q^{91} -6.32066 q^{92} +5.79241 q^{94} -10.9807 q^{97} -13.6961 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{4}+O(q^{10}) $ 8 * q + 4 * q^4	$ 8 q + 4 q^{4} - 10 q^{13} - 8 q^{16} - 18 q^{19} + 30 q^{28} - 26 q^{31} - 20 q^{34} + 40 q^{43} - 30 q^{46} - 16 q^{49} - 40 q^{52} - 10 q^{58} - 24 q^{61} - 34 q^{64} + 40 q^{67} - 40 q^{73} - 44 q^{76} - 42 q^{79} - 60 q^{82} - 20 q^{88} - 40 q^{91} - 10 q^{94} + 40 q^{97}+O(q^{100}) $ 8 * q + 4 * q^4 - 10 * q^13 - 8 * q^16 - 18 * q^19 + 30 * q^28 - 26 * q^31 - 20 * q^34 + 40 * q^43 - 30 * q^46 - 16 * q^49 - 40 * q^52 - 10 * q^58 - 24 * q^61 - 34 * q^64 + 40 * q^67 - 40 * q^73 - 44 * q^76 - 42 * q^79 - 60 * q^82 - 20 * q^88 - 40 * q^91 - 10 * q^94 + 40 * q^97

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 $

$2$	2.03035	1.43568	0.717838	−	0.696210i	$-0.245132\pi$
$2$	2.03035	1.43568	0.717838	+	0.696210i	$0.245132\pi$
$3$	0	0
$3$	0	0
$4$	2.12233	1.06117
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.504300	0.190607	0.0953037	−	0.995448i	$-0.469618\pi$
$7$	0.504300	0.190607	0.0953037	+	0.995448i	$0.469618\pi$
$8$	0.248380	0.0878157
$9$	0	0
$10$	0	0
$11$	3.43869	1.03680	0.518402	−	0.855137i	$-0.326527\pi$
$11$	3.43869	1.03680	0.518402	+	0.855137i	$0.326527\pi$
$12$	0	0
$13$	−4.93831	−1.36964	−0.684820	−	0.728712i	$-0.740119\pi$
$13$	−4.93831	−1.36964	−0.684820	+	0.728712i	$0.740119\pi$
$14$	1.02391	0.273650
$15$	0	0
$16$	−3.74037	−0.935092
$17$	−1.17741	−0.285565	−0.142782	−	0.989754i	$-0.545605\pi$
$17$	−1.17741	−0.285565	−0.142782	+	0.989754i	$0.545605\pi$
$18$	0	0
$19$	−5.93831	−1.36234	−0.681171	−	0.732125i	$-0.738529\pi$
$19$	−5.93831	−1.36234	−0.681171	+	0.732125i	$0.738529\pi$
$20$	0	0
$21$	0	0
$22$	6.98175	1.48851
$23$	−2.97817	−0.620990	−0.310495	−	0.950575i	$-0.600495\pi$
$23$	−2.97817	−0.620990	−0.310495	+	0.950575i	$0.600495\pi$
$24$	0	0
$25$	0	0
$26$	−10.0265	−1.96636
$27$	0	0
$28$	1.07029	0.202266
$29$	−5.01799	−0.931817	−0.465909	−	0.884833i	$-0.654272\pi$
$29$	−5.01799	−0.931817	−0.465909	+	0.884833i	$0.654272\pi$
$30$	0	0
$31$	3.56166	0.639692	0.319846	−	0.947470i	$-0.396369\pi$
$31$	3.56166	0.639692	0.319846	+	0.947470i	$0.396369\pi$
$32$	−8.09103	−1.43031
$33$	0	0
$34$	−2.39057	−0.409979
$35$	0	0
$36$	0	0
$37$	1.50101	0.246765	0.123383	−	0.992359i	$-0.460626\pi$
$37$	1.50101	0.246765	0.123383	+	0.992359i	$0.460626\pi$
$38$	−12.0569	−1.95588
$39$	0	0
$40$	0	0
$41$	−8.78461	−1.37192	−0.685962	−	0.727637i	$-0.740619\pi$
$41$	−8.78461	−1.37192	−0.685962	+	0.727637i	$0.740619\pi$
$42$	0	0
$43$	10.2500	1.56311	0.781554	−	0.623838i	$-0.214427\pi$
$43$	10.2500	1.56311	0.781554	+	0.623838i	$0.214427\pi$
$44$	7.29804	1.10022
$45$	0	0
$46$	−6.04673	−0.891541
$47$	2.85291	0.416139	0.208070	−	0.978114i	$-0.433282\pi$
$47$	2.85291	0.416139	0.208070	+	0.978114i	$0.433282\pi$
$48$	0	0
$49$	−6.74568	−0.963669
$50$	0	0
$51$	0	0
$52$	−10.4807	−1.45342
$53$	−13.2638	−1.82193	−0.910964	−	0.412485i	$-0.864661\pi$
$53$	−13.2638	−1.82193	−0.910964	+	0.412485i	$0.864661\pi$
$54$	0	0
$55$	0	0
$56$	0.125258	0.0167383
$57$	0	0
$58$	−10.1883	−1.33779
$59$	−9.50068	−1.23688	−0.618442	−	0.785831i	$-0.712236\pi$
$59$	−9.50068	−1.23688	−0.618442	+	0.785831i	$0.712236\pi$
$60$	0	0
$61$	−8.48073	−1.08585	−0.542923	−	0.839782i	$-0.682683\pi$
$61$	−8.48073	−1.08585	−0.542923	+	0.839782i	$0.682683\pi$
$62$	7.23142	0.918391
$63$	0	0
$64$	−8.94691	−1.11836
$65$	0	0
$66$	0	0
$67$	9.70289	1.18540	0.592698	−	0.805425i	$-0.298063\pi$
$67$	9.70289	1.18540	0.592698	+	0.805425i	$0.298063\pi$
$68$	−2.49886	−0.303032
$69$	0	0
$70$	0	0
$71$	13.7157	1.62776	0.813878	−	0.581036i	$-0.197353\pi$
$71$	13.7157	1.62776	0.813878	+	0.581036i	$0.197353\pi$
$72$	0	0
$73$	12.4389	1.45587	0.727933	−	0.685649i	$-0.240481\pi$
$73$	12.4389	1.45587	0.727933	+	0.685649i	$0.240481\pi$
$74$	3.04759	0.354275
$75$	0	0
$76$	−12.6031	−1.44567
$77$	1.73413	0.197622
$78$	0	0
$79$	−14.6926	−1.65305	−0.826523	−	0.562903i	$-0.809684\pi$
$79$	−14.6926	−1.65305	−0.826523	+	0.562903i	$0.809684\pi$
$80$	0	0
$81$	0	0
$82$	−17.8359	−1.96964
$83$	−0.356179	−0.0390957	−0.0195479	−	0.999809i	$-0.506223\pi$
$83$	−0.356179	−0.0390957	−0.0195479	+	0.999809i	$0.506223\pi$
$84$	0	0
$85$	0	0
$86$	20.8111	2.24412
$87$	0	0
$88$	0.854102	0.0910476
$89$	6.51041	0.690102	0.345051	−	0.938584i	$-0.387862\pi$
$89$	6.51041	0.690102	0.345051	+	0.938584i	$0.387862\pi$
$90$	0	0
$91$	−2.49039	−0.261063
$92$	−6.32066	−0.658974
$93$	0	0
$94$	5.79241	0.597441
$95$	0	0
$96$	0	0
$97$	−10.9807	−1.11492	−0.557460	−	0.830203i	$-0.688224\pi$
$97$	−10.9807	−1.11492	−0.557460	+	0.830203i	$0.688224\pi$
$98$	−13.6961	−1.38352
$99$	0	0
$100$	0	0
$101$	−8.10395	−0.806373	−0.403187	−	0.915118i	$-0.632097\pi$
$101$	−8.10395	−0.806373	−0.403187	+	0.915118i	$0.632097\pi$
$102$	0	0
$103$	−5.39981	−0.532060	−0.266030	−	0.963965i	$-0.585712\pi$
$103$	−5.39981	−0.532060	−0.266030	+	0.963965i	$0.585712\pi$
$104$	−1.22658	−0.120276
$105$	0	0
$106$	−26.9303	−2.61570
$107$	10.6007	1.02481	0.512403	−	0.858745i	$-0.328755\pi$
$107$	10.6007	1.02481	0.512403	+	0.858745i	$0.328755\pi$
$108$	0	0
$109$	−2.79878	−0.268074	−0.134037	−	0.990976i	$-0.542794\pi$
$109$	−2.79878	−0.268074	−0.134037	+	0.990976i	$0.542794\pi$
$110$	0	0
$111$	0	0
$112$	−1.88627	−0.178235
$113$	−7.13868	−0.671551	−0.335775	−	0.941942i	$-0.608998\pi$
$113$	−7.13868	−0.671551	−0.335775	+	0.941942i	$0.608998\pi$
$114$	0	0
$115$	0	0
$116$	−10.6498	−0.988813
$117$	0	0
$118$	−19.2897	−1.77576
$119$	−0.593769	−0.0544307
$120$	0	0
$121$	0.824573	0.0749612
$122$	−17.2189	−1.55892
$123$	0	0
$124$	7.55902	0.678820
$125$	0	0
$126$	0	0
$127$	4.52681	0.401690	0.200845	−	0.979623i	$-0.435631\pi$
$127$	4.52681	0.401690	0.200845	+	0.979623i	$0.435631\pi$
$128$	−1.98332	−0.175303
$129$	0	0
$130$	0	0
$131$	9.68911	0.846541	0.423271	−	0.906003i	$-0.360882\pi$
$131$	9.68911	0.846541	0.423271	+	0.906003i	$0.360882\pi$
$132$	0	0
$133$	−2.99469	−0.259672
$134$	19.7003	1.70185
$135$	0	0
$136$	−0.292446	−0.0250771
$137$	3.98461	0.340428	0.170214	−	0.985407i	$-0.445554\pi$
$137$	3.98461	0.340428	0.170214	+	0.985407i	$0.445554\pi$
$138$	0	0
$139$	4.96147	0.420826	0.210413	−	0.977613i	$-0.432519\pi$
$139$	4.96147	0.420826	0.210413	+	0.977613i	$0.432519\pi$
$140$	0	0
$141$	0	0
$142$	27.8477	2.33693
$143$	−16.9813	−1.42005
$144$	0	0
$145$	0	0
$146$	25.2554	2.09015
$147$	0	0
$148$	3.18565	0.261859
$149$	−5.62123	−0.460509	−0.230254	−	0.973130i	$-0.573956\pi$
$149$	−5.62123	−0.460509	−0.230254	+	0.973130i	$0.573956\pi$
$150$	0	0
$151$	−2.35414	−0.191577	−0.0957886	−	0.995402i	$-0.530537\pi$
$151$	−2.35414	−0.191577	−0.0957886	+	0.995402i	$0.530537\pi$
$152$	−1.47496	−0.119635
$153$	0	0
$154$	3.52089	0.283722
$155$	0	0
$156$	0	0
$157$	−20.7688	−1.65753	−0.828767	−	0.559594i	$-0.810957\pi$
$157$	−20.7688	−1.65753	−0.828767	+	0.559594i	$0.810957\pi$
$158$	−29.8311	−2.37324
$159$	0	0
$160$	0	0
$161$	−1.50189	−0.118365
$162$	0	0
$163$	6.99363	0.547784	0.273892	−	0.961761i	$-0.411689\pi$
$163$	6.99363	0.547784	0.273892	+	0.961761i	$0.411689\pi$
$164$	−18.6439	−1.45584
$165$	0	0
$166$	−0.723169	−0.0561288
$167$	14.8830	1.15168	0.575841	−	0.817562i	$-0.304675\pi$
$167$	14.8830	1.15168	0.575841	+	0.817562i	$0.304675\pi$
$168$	0	0
$169$	11.3869	0.875914
$170$	0	0
$171$	0	0
$172$	21.7539	1.65872
$173$	−1.56398	−0.118907	−0.0594535	−	0.998231i	$-0.518936\pi$
$173$	−1.56398	−0.118907	−0.0594535	+	0.998231i	$0.518936\pi$
$174$	0	0
$175$	0	0
$176$	−12.8620	−0.969506
$177$	0	0
$178$	13.2184	0.990763
$179$	11.9728	0.894887	0.447444	−	0.894312i	$-0.352334\pi$
$179$	11.9728	0.894887	0.447444	+	0.894312i	$0.352334\pi$
$180$	0	0
$181$	−0.834224	−0.0620074	−0.0310037	−	0.999519i	$-0.509870\pi$
$181$	−0.834224	−0.0620074	−0.0310037	+	0.999519i	$0.509870\pi$
$182$	−5.05636	−0.374803
$183$	0	0
$184$	−0.739717	−0.0545327
$185$	0	0
$186$	0	0
$187$	−4.04876	−0.296074
$188$	6.05482	0.441593
$189$	0	0
$190$	0	0
$191$	−26.4677	−1.91514	−0.957569	−	0.288204i	$-0.906942\pi$
$191$	−26.4677	−1.91514	−0.957569	+	0.288204i	$0.906942\pi$
$192$	0	0
$193$	3.97684	0.286259	0.143130	−	0.989704i	$-0.454283\pi$
$193$	3.97684	0.286259	0.143130	+	0.989704i	$0.454283\pi$
$194$	−22.2947	−1.60067
$195$	0	0
$196$	−14.3166	−1.02261
$197$	23.3619	1.66447	0.832234	−	0.554425i	$-0.187062\pi$
$197$	23.3619	1.66447	0.832234	+	0.554425i	$0.187062\pi$
$198$	0	0
$199$	−4.43506	−0.314393	−0.157197	−	0.987567i	$-0.550246\pi$
$199$	−4.43506	−0.314393	−0.157197	+	0.987567i	$0.550246\pi$
$200$	0	0
$201$	0	0
$202$	−16.4539	−1.15769
$203$	−2.53057	−0.177611
$204$	0	0
$205$	0	0
$206$	−10.9635	−0.763865
$207$	0	0
$208$	18.4711	1.28074
$209$	−20.4200	−1.41248
$210$	0	0
$211$	−6.29448	−0.433330	−0.216665	−	0.976246i	$-0.569518\pi$
$211$	−6.29448	−0.433330	−0.216665	+	0.976246i	$0.569518\pi$
$212$	−28.1503	−1.93337
$213$	0	0
$214$	21.5231	1.47129
$215$	0	0
$216$	0	0
$217$	1.79614	0.121930
$218$	−5.68250	−0.384868
$219$	0	0
$220$	0	0
$221$	5.81443	0.391121
$222$	0	0
$223$	−24.4297	−1.63593	−0.817966	−	0.575267i	$-0.804898\pi$
$223$	−24.4297	−1.63593	−0.817966	+	0.575267i	$0.804898\pi$
$224$	−4.08030	−0.272627
$225$	0	0
$226$	−14.4940	−0.964129
$227$	−13.4464	−0.892470	−0.446235	−	0.894916i	$-0.647235\pi$
$227$	−13.4464	−0.892470	−0.446235	+	0.894916i	$0.647235\pi$
$228$	0	0
$229$	15.1498	1.00112	0.500562	−	0.865701i	$-0.333127\pi$
$229$	15.1498	1.00112	0.500562	+	0.865701i	$0.333127\pi$
$230$	0	0
$231$	0	0
$232$	−1.24637	−0.0818281
$233$	3.19112	0.209057	0.104529	−	0.994522i	$-0.466667\pi$
$233$	3.19112	0.209057	0.104529	+	0.994522i	$0.466667\pi$
$234$	0	0
$235$	0	0
$236$	−20.1636	−1.31254
$237$	0	0
$238$	−1.20556	−0.0781449
$239$	22.6132	1.46272	0.731362	−	0.681990i	$-0.238885\pi$
$239$	22.6132	1.46272	0.731362	+	0.681990i	$0.238885\pi$
$240$	0	0
$241$	−19.1614	−1.23430	−0.617148	−	0.786847i	$-0.711712\pi$
$241$	−19.1614	−1.23430	−0.617148	+	0.786847i	$0.711712\pi$
$242$	1.67417	0.107620
$243$	0	0
$244$	−17.9989	−1.15226
$245$	0	0
$246$	0	0
$247$	29.3252	1.86592
$248$	0.884645	0.0561750
$249$	0	0
$250$	0	0
$251$	21.1160	1.33283	0.666414	−	0.745582i	$-0.267828\pi$
$251$	21.1160	1.33283	0.666414	+	0.745582i	$0.267828\pi$
$252$	0	0
$253$	−10.2410	−0.643845
$254$	9.19103	0.576696
$255$	0	0
$256$	13.8670	0.866685
$257$	18.4143	1.14865	0.574325	−	0.818627i	$-0.305265\pi$
$257$	18.4143	1.14865	0.574325	+	0.818627i	$0.305265\pi$
$258$	0	0
$259$	0.756961	0.0470353
$260$	0	0
$261$	0	0
$262$	19.6723	1.21536
$263$	0.595251	0.0367047	0.0183524	−	0.999832i	$-0.494158\pi$
$263$	0.595251	0.0367047	0.0183524	+	0.999832i	$0.494158\pi$
$264$	0	0
$265$	0	0
$266$	−6.08027	−0.372805
$267$	0	0
$268$	20.5928	1.25790
$269$	7.81440	0.476452	0.238226	−	0.971210i	$-0.423434\pi$
$269$	7.81440	0.476452	0.238226	+	0.971210i	$0.423434\pi$
$270$	0	0
$271$	−21.0713	−1.27999	−0.639994	−	0.768380i	$-0.721063\pi$
$271$	−21.0713	−1.27999	−0.639994	+	0.768380i	$0.721063\pi$
$272$	4.40396	0.267029
$273$	0	0
$274$	8.09017	0.488745
$275$	0	0
$276$	0	0
$277$	−19.9383	−1.19798	−0.598988	−	0.800758i	$-0.704430\pi$
$277$	−19.9383	−1.19798	−0.598988	+	0.800758i	$0.704430\pi$
$278$	10.0735	0.604171
$279$	0	0
$280$	0	0
$281$	−10.2829	−0.613425	−0.306712	−	0.951802i	$-0.599229\pi$
$281$	−10.2829	−0.613425	−0.306712	+	0.951802i	$0.599229\pi$
$282$	0	0
$283$	9.67231	0.574959	0.287480	−	0.957787i	$-0.407183\pi$
$283$	9.67231	0.574959	0.287480	+	0.957787i	$0.407183\pi$
$284$	29.1093	1.72732
$285$	0	0
$286$	−34.4780	−2.03873
$287$	−4.43007	−0.261499
$288$	0	0
$289$	−15.6137	−0.918453
$290$	0	0
$291$	0	0
$292$	26.3995	1.54492
$293$	−11.0903	−0.647900	−0.323950	−	0.946074i	$-0.605011\pi$
$293$	−11.0903	−0.647900	−0.323950	+	0.946074i	$0.605011\pi$
$294$	0	0
$295$	0	0
$296$	0.372822	0.0216699
$297$	0	0
$298$	−11.4131	−0.661142
$299$	14.7071	0.850533
$300$	0	0
$301$	5.16906	0.297940
$302$	−4.77973	−0.275043
$303$	0	0
$304$	22.2115	1.27391
$305$	0	0
$306$	0	0
$307$	−2.14760	−0.122570	−0.0612850	−	0.998120i	$-0.519520\pi$
$307$	−2.14760	−0.122570	−0.0612850	+	0.998120i	$0.519520\pi$
$308$	3.68040	0.209710
$309$	0	0
$310$	0	0
$311$	−11.0409	−0.626074	−0.313037	−	0.949741i	$-0.601346\pi$
$311$	−11.0409	−0.626074	−0.313037	+	0.949741i	$0.601346\pi$
$312$	0	0
$313$	11.6300	0.657364	0.328682	−	0.944441i	$-0.393396\pi$
$313$	11.6300	0.657364	0.328682	+	0.944441i	$0.393396\pi$
$314$	−42.1681	−2.37968
$315$	0	0
$316$	−31.1826	−1.75416
$317$	−10.5341	−0.591652	−0.295826	−	0.955242i	$-0.595595\pi$
$317$	−10.5341	−0.591652	−0.295826	+	0.955242i	$0.595595\pi$
$318$	0	0
$319$	−17.2553	−0.966111
$320$	0	0
$321$	0	0
$322$	−3.04936	−0.169934
$323$	6.99184	0.389037
$324$	0	0
$325$	0	0
$326$	14.1995	0.786440
$327$	0	0
$328$	−2.18192	−0.120477
$329$	1.43872	0.0793192
$330$	0	0
$331$	−23.3836	−1.28528	−0.642639	−	0.766169i	$-0.722161\pi$
$331$	−23.3836	−1.28528	−0.642639	+	0.766169i	$0.722161\pi$
$332$	−0.755931	−0.0414871
$333$	0	0
$334$	30.2177	1.65344
$335$	0	0
$336$	0	0
$337$	−4.17535	−0.227446	−0.113723	−	0.993512i	$-0.536278\pi$
$337$	−4.17535	−0.227446	−0.113723	+	0.993512i	$0.536278\pi$
$338$	23.1194	1.25753
$339$	0	0
$340$	0	0
$341$	12.2474	0.663235
$342$	0	0
$343$	−6.93194	−0.374290
$344$	2.54589	0.137265
$345$	0	0
$346$	−3.17543	−0.170712
$347$	−8.35473	−0.448505	−0.224253	−	0.974531i	$-0.571994\pi$
$347$	−8.35473	−0.448505	−0.224253	+	0.974531i	$0.571994\pi$
$348$	0	0
$349$	8.84117	0.473257	0.236628	−	0.971600i	$-0.423958\pi$
$349$	8.84117	0.473257	0.236628	+	0.971600i	$0.423958\pi$
$350$	0	0
$351$	0	0
$352$	−27.8225	−1.48295
$353$	−9.31415	−0.495742	−0.247871	−	0.968793i	$-0.579731\pi$
$353$	−9.31415	−0.495742	−0.247871	+	0.968793i	$0.579731\pi$
$354$	0	0
$355$	0	0
$356$	13.8173	0.732313
$357$	0	0
$358$	24.3090	1.28477
$359$	−7.05633	−0.372419	−0.186209	−	0.982510i	$-0.559620\pi$
$359$	−7.05633	−0.372419	−0.186209	+	0.982510i	$0.559620\pi$
$360$	0	0
$361$	16.2635	0.855973
$362$	−1.69377	−0.0890225
$363$	0	0
$364$	−5.28543	−0.277032
$365$	0	0
$366$	0	0
$367$	−3.33747	−0.174215	−0.0871073	−	0.996199i	$-0.527762\pi$
$367$	−3.33747	−0.174215	−0.0871073	+	0.996199i	$0.527762\pi$
$368$	11.1394	0.580683
$369$	0	0
$370$	0	0
$371$	−6.68895	−0.347273
$372$	0	0
$373$	−16.8630	−0.873135	−0.436567	−	0.899672i	$-0.643806\pi$
$373$	−16.8630	−0.873135	−0.436567	+	0.899672i	$0.643806\pi$
$374$	−8.22041	−0.425067
$375$	0	0
$376$	0.708606	0.0365436
$377$	24.7804	1.27625
$378$	0	0
$379$	−1.95859	−0.100606	−0.0503029	−	0.998734i	$-0.516019\pi$
$379$	−1.95859	−0.100606	−0.0503029	+	0.998734i	$0.516019\pi$
$380$	0	0
$381$	0	0
$382$	−53.7388	−2.74952
$383$	−14.8352	−0.758041	−0.379021	−	0.925388i	$-0.623739\pi$
$383$	−14.8352	−0.758041	−0.379021	+	0.925388i	$0.623739\pi$
$384$	0	0
$385$	0	0
$386$	8.07438	0.410975
$387$	0	0
$388$	−23.3047	−1.18312
$389$	29.9006	1.51602	0.758009	−	0.652244i	$-0.226172\pi$
$389$	29.9006	1.51602	0.758009	+	0.652244i	$0.226172\pi$
$390$	0	0
$391$	3.50653	0.177333
$392$	−1.67549	−0.0846252
$393$	0	0
$394$	47.4329	2.38964
$395$	0	0
$396$	0	0
$397$	−2.77850	−0.139449	−0.0697244	−	0.997566i	$-0.522212\pi$
$397$	−2.77850	−0.139449	−0.0697244	+	0.997566i	$0.522212\pi$
$398$	−9.00474	−0.451367
$399$	0	0
$400$	0	0
$401$	−8.54941	−0.426937	−0.213469	−	0.976950i	$-0.568476\pi$
$401$	−8.54941	−0.426937	−0.213469	+	0.976950i	$0.568476\pi$
$402$	0	0
$403$	−17.5886	−0.876148
$404$	−17.1993	−0.855697
$405$	0	0
$406$	−5.13795	−0.254992
$407$	5.16152	0.255847
$408$	0	0
$409$	6.22439	0.307776	0.153888	−	0.988088i	$-0.450820\pi$
$409$	6.22439	0.307776	0.153888	+	0.988088i	$0.450820\pi$
$410$	0	0
$411$	0	0
$412$	−11.4602	−0.564604
$413$	−4.79119	−0.235759
$414$	0	0
$415$	0	0
$416$	39.9560	1.95900
$417$	0	0
$418$	−41.4598	−2.02786
$419$	−37.4992	−1.83196	−0.915978	−	0.401228i	$-0.868583\pi$
$419$	−37.4992	−1.83196	−0.915978	+	0.401228i	$0.868583\pi$
$420$	0	0
$421$	9.47043	0.461561	0.230780	−	0.973006i	$-0.425872\pi$
$421$	9.47043	0.461561	0.230780	+	0.973006i	$0.425872\pi$
$422$	−12.7800	−0.622121
$423$	0	0
$424$	−3.29448	−0.159994
$425$	0	0
$426$	0	0
$427$	−4.27683	−0.206970
$428$	22.4982	1.08749
$429$	0	0
$430$	0	0
$431$	9.88297	0.476046	0.238023	−	0.971260i	$-0.423501\pi$
$431$	9.88297	0.476046	0.238023	+	0.971260i	$0.423501\pi$
$432$	0	0
$433$	−24.2827	−1.16695	−0.583477	−	0.812130i	$-0.698308\pi$
$433$	−24.2827	−1.16695	−0.583477	+	0.812130i	$0.698308\pi$
$434$	3.64680	0.175052
$435$	0	0
$436$	−5.93994	−0.284471
$437$	17.6853	0.846001
$438$	0	0
$439$	−1.75659	−0.0838374	−0.0419187	−	0.999121i	$-0.513347\pi$
$439$	−1.75659	−0.0838374	−0.0419187	+	0.999121i	$0.513347\pi$
$440$	0	0
$441$	0	0
$442$	11.8053	0.561523
$443$	37.3529	1.77469	0.887346	−	0.461105i	$-0.152547\pi$
$443$	37.3529	1.77469	0.887346	+	0.461105i	$0.152547\pi$
$444$	0	0
$445$	0	0
$446$	−49.6009	−2.34867
$447$	0	0
$448$	−4.51192	−0.213168
$449$	−12.9452	−0.610923	−0.305461	−	0.952204i	$-0.598811\pi$
$449$	−12.9452	−0.610923	−0.305461	+	0.952204i	$0.598811\pi$
$450$	0	0
$451$	−30.2075	−1.42242
$452$	−15.1507	−0.712627
$453$	0	0
$454$	−27.3010	−1.28130
$455$	0	0
$456$	0	0
$457$	22.2891	1.04264	0.521320	−	0.853361i	$-0.325440\pi$
$457$	22.2891	1.04264	0.521320	+	0.853361i	$0.325440\pi$
$458$	30.7594	1.43729
$459$	0	0
$460$	0	0
$461$	0.537120	0.0250162	0.0125081	−	0.999922i	$-0.496018\pi$
$461$	0.537120	0.0250162	0.0125081	+	0.999922i	$0.496018\pi$
$462$	0	0
$463$	22.3508	1.03873	0.519364	−	0.854553i	$-0.326169\pi$
$463$	22.3508	1.03873	0.519364	+	0.854553i	$0.326169\pi$
$464$	18.7691	0.871335
$465$	0	0
$466$	6.47911	0.300139
$467$	37.4759	1.73418	0.867089	−	0.498154i	$-0.165988\pi$
$467$	37.4759	1.73418	0.867089	+	0.498154i	$0.165988\pi$
$468$	0	0
$469$	4.89316	0.225945
$470$	0	0
$471$	0	0
$472$	−2.35978	−0.108618
$473$	35.2465	1.62064
$474$	0	0
$475$	0	0
$476$	−1.26018	−0.0577601
$477$	0	0
$478$	45.9127	2.10000
$479$	18.6191	0.850727	0.425364	−	0.905023i	$-0.360146\pi$
$479$	18.6191	0.850727	0.425364	+	0.905023i	$0.360146\pi$
$480$	0	0
$481$	−7.41247	−0.337980
$482$	−38.9045	−1.77205
$483$	0	0
$484$	1.75002	0.0795463
$485$	0	0
$486$	0	0
$487$	1.79111	0.0811629	0.0405814	−	0.999176i	$-0.487079\pi$
$487$	1.79111	0.0811629	0.0405814	+	0.999176i	$0.487079\pi$
$488$	−2.10645	−0.0953544
$489$	0	0
$490$	0	0
$491$	27.2755	1.23092	0.615462	−	0.788166i	$-0.288969\pi$
$491$	27.2755	1.23092	0.615462	+	0.788166i	$0.288969\pi$
$492$	0	0
$493$	5.90825	0.266094
$494$	59.5405	2.67885
$495$	0	0
$496$	−13.3219	−0.598171
$497$	6.91683	0.310262
$498$	0	0
$499$	10.5722	0.473275	0.236637	−	0.971598i	$-0.423955\pi$
$499$	10.5722	0.473275	0.236637	+	0.971598i	$0.423955\pi$
$500$	0	0
$501$	0	0
$502$	42.8729	1.91351
$503$	27.0940	1.20806	0.604031	−	0.796961i	$-0.293560\pi$
$503$	27.0940	1.20806	0.604031	+	0.796961i	$0.293560\pi$
$504$	0	0
$505$	0	0
$506$	−20.7928	−0.924353
$507$	0	0
$508$	9.60740	0.426260
$509$	−28.1026	−1.24563	−0.622813	−	0.782371i	$-0.714010\pi$
$509$	−28.1026	−1.24563	−0.622813	+	0.782371i	$0.714010\pi$
$510$	0	0
$511$	6.27294	0.277499
$512$	32.1215	1.41958
$513$	0	0
$514$	37.3875	1.64909
$515$	0	0
$516$	0	0
$517$	9.81026	0.431455
$518$	1.53690	0.0675274
$519$	0	0
$520$	0	0
$521$	−33.0509	−1.44798	−0.723992	−	0.689808i	$-0.757695\pi$
$521$	−33.0509	−1.44798	−0.723992	+	0.689808i	$0.757695\pi$
$522$	0	0
$523$	40.7409	1.78148	0.890739	−	0.454516i	$-0.150188\pi$
$523$	40.7409	1.78148	0.890739	+	0.454516i	$0.150188\pi$
$524$	20.5635	0.898321
$525$	0	0
$526$	1.20857	0.0526961
$527$	−4.19354	−0.182674
$528$	0	0
$529$	−14.1305	−0.614371
$530$	0	0
$531$	0	0
$532$	−6.35572	−0.275555
$533$	43.3811	1.87904
$534$	0	0
$535$	0	0
$536$	2.41001	0.104096
$537$	0	0
$538$	15.8660	0.684031
$539$	−23.1963	−0.999135
$540$	0	0
$541$	45.2061	1.94356	0.971781	−	0.235884i	$-0.0757986\pi$
$541$	45.2061	1.94356	0.971781	+	0.235884i	$0.0757986\pi$
$542$	−42.7821	−1.83765
$543$	0	0
$544$	9.52648	0.408445
$545$	0	0
$546$	0	0
$547$	−2.43892	−0.104281	−0.0521404	−	0.998640i	$-0.516604\pi$
$547$	−2.43892	−0.104281	−0.0521404	+	0.998640i	$0.516604\pi$
$548$	8.45668	0.361251
$549$	0	0
$550$	0	0
$551$	29.7984	1.26945
$552$	0	0
$553$	−7.40947	−0.315083
$554$	−40.4818	−1.71991
$555$	0	0
$556$	10.5299	0.446567
$557$	7.91914	0.335545	0.167772	−	0.985826i	$-0.446343\pi$
$557$	7.91914	0.335545	0.167772	+	0.985826i	$0.446343\pi$
$558$	0	0
$559$	−50.6176	−2.14089
$560$	0	0
$561$	0	0
$562$	−20.8779	−0.880680
$563$	24.0511	1.01363	0.506816	−	0.862054i	$-0.330822\pi$
$563$	24.0511	1.01363	0.506816	+	0.862054i	$0.330822\pi$
$564$	0	0
$565$	0	0
$566$	19.6382	0.825455
$567$	0	0
$568$	3.40671	0.142942
$569$	14.6355	0.613554	0.306777	−	0.951781i	$-0.400749\pi$
$569$	14.6355	0.613554	0.306777	+	0.951781i	$0.400749\pi$
$570$	0	0
$571$	21.1399	0.884677	0.442339	−	0.896848i	$-0.354149\pi$
$571$	21.1399	0.884677	0.442339	+	0.896848i	$0.354149\pi$
$572$	−36.0400	−1.50691
$573$	0	0
$574$	−8.99461	−0.375428
$575$	0	0
$576$	0	0
$577$	3.94894	0.164396	0.0821982	−	0.996616i	$-0.473806\pi$
$577$	3.94894	0.164396	0.0821982	+	0.996616i	$0.473806\pi$
$578$	−31.7013	−1.31860
$579$	0	0
$580$	0	0
$581$	−0.179621	−0.00745193
$582$	0	0
$583$	−45.6102	−1.88898
$584$	3.08958	0.127848
$585$	0	0
$586$	−22.5172	−0.930175
$587$	20.2733	0.836769	0.418384	−	0.908270i	$-0.362596\pi$
$587$	20.2733	0.836769	0.418384	+	0.908270i	$0.362596\pi$
$588$	0	0
$589$	−21.1502	−0.871479
$590$	0	0
$591$	0	0
$592$	−5.61435	−0.230748
$593$	30.2561	1.24247	0.621234	−	0.783625i	$-0.286632\pi$
$593$	30.2561	1.24247	0.621234	+	0.783625i	$0.286632\pi$
$594$	0	0
$595$	0	0
$596$	−11.9301	−0.488677
$597$	0	0
$598$	29.8606	1.22109
$599$	1.02686	0.0419563	0.0209781	−	0.999780i	$-0.493322\pi$
$599$	1.02686	0.0419563	0.0209781	+	0.999780i	$0.493322\pi$
$600$	0	0
$601$	−16.9840	−0.692791	−0.346396	−	0.938089i	$-0.612594\pi$
$601$	−16.9840	−0.692791	−0.346396	+	0.938089i	$0.612594\pi$
$602$	10.4950	0.427745
$603$	0	0
$604$	−4.99627	−0.203295
$605$	0	0
$606$	0	0
$607$	−10.7660	−0.436977	−0.218488	−	0.975840i	$-0.570113\pi$
$607$	−10.7660	−0.436977	−0.218488	+	0.975840i	$0.570113\pi$
$608$	48.0470	1.94856
$609$	0	0
$610$	0	0
$611$	−14.0885	−0.569961
$612$	0	0
$613$	5.04568	0.203793	0.101896	−	0.994795i	$-0.467509\pi$
$613$	5.04568	0.203793	0.101896	+	0.994795i	$0.467509\pi$
$614$	−4.36039	−0.175971
$615$	0	0
$616$	0.430723	0.0173543
$617$	−31.0901	−1.25164	−0.625820	−	0.779968i	$-0.715236\pi$
$617$	−31.0901	−1.25164	−0.625820	+	0.779968i	$0.715236\pi$
$618$	0	0
$619$	−34.7983	−1.39866	−0.699331	−	0.714798i	$-0.746519\pi$
$619$	−34.7983	−1.39866	−0.699331	+	0.714798i	$0.746519\pi$
$620$	0	0
$621$	0	0
$622$	−22.4170	−0.898840
$623$	3.28319	0.131538
$624$	0	0
$625$	0	0
$626$	23.6129	0.943763
$627$	0	0
$628$	−44.0784	−1.75892
$629$	−1.76732	−0.0704675
$630$	0	0
$631$	−9.26324	−0.368764	−0.184382	−	0.982855i	$-0.559028\pi$
$631$	−9.26324	−0.368764	−0.184382	+	0.982855i	$0.559028\pi$
$632$	−3.64935	−0.145163
$633$	0	0
$634$	−21.3879	−0.849420
$635$	0	0
$636$	0	0
$637$	33.3123	1.31988
$638$	−35.0343	−1.38702
$639$	0	0
$640$	0	0
$641$	43.6553	1.72428	0.862141	−	0.506669i	$-0.169123\pi$
$641$	43.6553	1.72428	0.862141	+	0.506669i	$0.169123\pi$
$642$	0	0
$643$	35.4133	1.39656	0.698281	−	0.715824i	$-0.253949\pi$
$643$	35.4133	1.39656	0.698281	+	0.715824i	$0.253949\pi$
$644$	−3.18751	−0.125605
$645$	0	0
$646$	14.1959	0.558531
$647$	33.3262	1.31019	0.655093	−	0.755548i	$-0.272629\pi$
$647$	33.3262	1.31019	0.655093	+	0.755548i	$0.272629\pi$
$648$	0	0
$649$	−32.6699	−1.28240
$650$	0	0
$651$	0	0
$652$	14.8428	0.581290
$653$	18.7055	0.732001	0.366000	−	0.930615i	$-0.380727\pi$
$653$	18.7055	0.732001	0.366000	+	0.930615i	$0.380727\pi$
$654$	0	0
$655$	0	0
$656$	32.8577	1.28288
$657$	0	0
$658$	2.92111	0.113877
$659$	−25.9218	−1.00977	−0.504885	−	0.863187i	$-0.668465\pi$
$659$	−25.9218	−1.00977	−0.504885	+	0.863187i	$0.668465\pi$
$660$	0	0
$661$	23.0397	0.896140	0.448070	−	0.893999i	$-0.352112\pi$
$661$	23.0397	0.896140	0.448070	+	0.893999i	$0.352112\pi$
$662$	−47.4770	−1.84524
$663$	0	0
$664$	−0.0884679	−0.00343322
$665$	0	0
$666$	0	0
$667$	14.9444	0.578650
$668$	31.5867	1.22213
$669$	0	0
$670$	0	0
$671$	−29.1626	−1.12581
$672$	0	0
$673$	6.14688	0.236945	0.118472	−	0.992957i	$-0.462200\pi$
$673$	6.14688	0.236945	0.118472	+	0.992957i	$0.462200\pi$
$674$	−8.47744	−0.326539
$675$	0	0
$676$	24.1668	0.929491
$677$	35.5427	1.36602	0.683009	−	0.730410i	$-0.260671\pi$
$677$	35.5427	1.36602	0.683009	+	0.730410i	$0.260671\pi$
$678$	0	0
$679$	−5.53756	−0.212512
$680$	0	0
$681$	0	0
$682$	24.8666	0.952191
$683$	−16.9532	−0.648696	−0.324348	−	0.945938i	$-0.605145\pi$
$683$	−16.9532	−0.648696	−0.324348	+	0.945938i	$0.605145\pi$
$684$	0	0
$685$	0	0
$686$	−14.0743	−0.537359
$687$	0	0
$688$	−38.3387	−1.46165
$689$	65.5009	2.49539
$690$	0	0
$691$	−15.9751	−0.607720	−0.303860	−	0.952717i	$-0.598275\pi$
$691$	−15.9751	−0.607720	−0.303860	+	0.952717i	$0.598275\pi$
$692$	−3.31928	−0.126180
$693$	0	0
$694$	−16.9631	−0.643909
$695$	0	0
$696$	0	0
$697$	10.3431	0.391773
$698$	17.9507	0.679444
$699$	0	0
$700$	0	0
$701$	−19.5403	−0.738026	−0.369013	−	0.929424i	$-0.620304\pi$
$701$	−19.5403	−0.738026	−0.369013	+	0.929424i	$0.620304\pi$
$702$	0	0
$703$	−8.91349	−0.336179
$704$	−30.7656	−1.15952
$705$	0	0
$706$	−18.9110	−0.711725
$707$	−4.08682	−0.153701
$708$	0	0
$709$	−13.1729	−0.494719	−0.247360	−	0.968924i	$-0.579563\pi$
$709$	−13.1729	−0.494719	−0.247360	+	0.968924i	$0.579563\pi$
$710$	0	0
$711$	0	0
$712$	1.61706	0.0606017
$713$	−10.6072	−0.397243
$714$	0	0
$715$	0	0
$716$	25.4102	0.949625
$717$	0	0
$718$	−14.3268	−0.534673
$719$	8.36032	0.311787	0.155894	−	0.987774i	$-0.450174\pi$
$719$	8.36032	0.311787	0.155894	+	0.987774i	$0.450174\pi$
$720$	0	0
$721$	−2.72312	−0.101414
$722$	33.0206	1.22890
$723$	0	0
$724$	−1.77050	−0.0658002
$725$	0	0
$726$	0	0
$727$	0.264217	0.00979928	0.00489964	−	0.999988i	$-0.498440\pi$
$727$	0.264217	0.00979928	0.00489964	+	0.999988i	$0.498440\pi$
$728$	−0.618563	−0.0229255
$729$	0	0
$730$	0	0
$731$	−12.0685	−0.446368
$732$	0	0
$733$	−17.9811	−0.664149	−0.332074	−	0.943253i	$-0.607749\pi$
$733$	−17.9811	−0.664149	−0.332074	+	0.943253i	$0.607749\pi$
$734$	−6.77625	−0.250116
$735$	0	0
$736$	24.0964	0.888206
$737$	33.3652	1.22902
$738$	0	0
$739$	−37.5141	−1.37998	−0.689990	−	0.723819i	$-0.742385\pi$
$739$	−37.5141	−1.37998	−0.689990	+	0.723819i	$0.742385\pi$
$740$	0	0
$741$	0	0
$742$	−13.5809	−0.498572
$743$	−18.6523	−0.684285	−0.342142	−	0.939648i	$-0.611153\pi$
$743$	−18.6523	−0.684285	−0.342142	+	0.939648i	$0.611153\pi$
$744$	0	0
$745$	0	0
$746$	−34.2379	−1.25354
$747$	0	0
$748$	−8.59281	−0.314184
$749$	5.34592	0.195336
$750$	0	0
$751$	51.3728	1.87462	0.937310	−	0.348496i	$-0.113307\pi$
$751$	51.3728	1.87462	0.937310	+	0.348496i	$0.113307\pi$
$752$	−10.6709	−0.389128
$753$	0	0
$754$	50.3129	1.83229
$755$	0	0
$756$	0	0
$757$	33.8427	1.23003	0.615016	−	0.788514i	$-0.289149\pi$
$757$	33.8427	1.23003	0.615016	+	0.788514i	$0.289149\pi$
$758$	−3.97662	−0.144437
$759$	0	0
$760$	0	0
$761$	10.8295	0.392568	0.196284	−	0.980547i	$-0.437113\pi$
$761$	10.8295	0.392568	0.196284	+	0.980547i	$0.437113\pi$
$762$	0	0
$763$	−1.41142	−0.0510969
$764$	−56.1734	−2.03228
$765$	0	0
$766$	−30.1206	−1.08830
$767$	46.9173	1.69408
$768$	0	0
$769$	−33.1017	−1.19368	−0.596838	−	0.802362i	$-0.703577\pi$
$769$	−33.1017	−1.19368	−0.596838	+	0.802362i	$0.703577\pi$
$770$	0	0
$771$	0	0
$772$	8.44018	0.303769
$773$	−10.0757	−0.362397	−0.181198	−	0.983447i	$-0.557998\pi$
$773$	−10.0757	−0.362397	−0.181198	+	0.983447i	$0.557998\pi$
$774$	0	0
$775$	0	0
$776$	−2.72739	−0.0979075
$777$	0	0
$778$	60.7087	2.17651
$779$	52.1657	1.86903
$780$	0	0
$781$	47.1640	1.68766
$782$	7.11950	0.254593
$783$	0	0
$784$	25.2313	0.901119
$785$	0	0
$786$	0	0
$787$	−6.50658	−0.231934	−0.115967	−	0.993253i	$-0.536997\pi$
$787$	−6.50658	−0.231934	−0.115967	+	0.993253i	$0.536997\pi$
$788$	49.5818	1.76628
$789$	0	0
$790$	0	0
$791$	−3.60003	−0.128002
$792$	0	0
$793$	41.8805	1.48722
$794$	−5.64133	−0.200203
$795$	0	0
$796$	−9.41268	−0.333623
$797$	−16.8384	−0.596447	−0.298224	−	0.954496i	$-0.596394\pi$
$797$	−16.8384	−0.596447	−0.298224	+	0.954496i	$0.596394\pi$
$798$	0	0
$799$	−3.35905	−0.118835
$800$	0	0
$801$	0	0
$802$	−17.3583	−0.612944
$803$	42.7736	1.50945
$804$	0	0
$805$	0	0
$806$	−35.7110	−1.25787
$807$	0	0
$808$	−2.01286	−0.0708122
$809$	41.6930	1.46585	0.732924	−	0.680310i	$-0.238155\pi$
$809$	41.6930	1.46585	0.732924	+	0.680310i	$0.238155\pi$
$810$	0	0
$811$	−34.6657	−1.21728	−0.608638	−	0.793448i	$-0.708284\pi$
$811$	−34.6657	−1.21728	−0.608638	+	0.793448i	$0.708284\pi$
$812$	−5.37071	−0.188475
$813$	0	0
$814$	10.4797	0.367314
$815$	0	0
$816$	0	0
$817$	−60.8675	−2.12949
$818$	12.6377	0.441867
$819$	0	0
$820$	0	0
$821$	−26.0643	−0.909649	−0.454825	−	0.890581i	$-0.650298\pi$
$821$	−26.0643	−0.909649	−0.454825	+	0.890581i	$0.650298\pi$
$822$	0	0
$823$	12.4237	0.433062	0.216531	−	0.976276i	$-0.430526\pi$
$823$	12.4237	0.433062	0.216531	+	0.976276i	$0.430526\pi$
$824$	−1.34121	−0.0467232
$825$	0	0
$826$	−9.72781	−0.338474
$827$	31.6260	1.09975	0.549873	−	0.835249i	$-0.314676\pi$
$827$	31.6260	1.09975	0.549873	+	0.835249i	$0.314676\pi$
$828$	0	0
$829$	−39.4793	−1.37117	−0.685587	−	0.727991i	$-0.740454\pi$
$829$	−39.4793	−1.37117	−0.685587	+	0.727991i	$0.740454\pi$
$830$	0	0
$831$	0	0
$832$	44.1826	1.53176
$833$	7.94246	0.275190
$834$	0	0
$835$	0	0
$836$	−43.3380	−1.49888
$837$	0	0
$838$	−76.1366	−2.63010
$839$	−19.0298	−0.656981	−0.328490	−	0.944507i	$-0.606540\pi$
$839$	−19.0298	−0.656981	−0.328490	+	0.944507i	$0.606540\pi$
$840$	0	0
$841$	−3.81979	−0.131717
$842$	19.2283	0.662652
$843$	0	0
$844$	−13.3590	−0.459835
$845$	0	0
$846$	0	0
$847$	0.415832	0.0142881
$848$	49.6116	1.70367
$849$	0	0
$850$	0	0
$851$	−4.47027	−0.153239
$852$	0	0
$853$	−28.8943	−0.989321	−0.494660	−	0.869086i	$-0.664708\pi$
$853$	−28.8943	−0.989321	−0.494660	+	0.869086i	$0.664708\pi$
$854$	−8.68348	−0.297142
$855$	0	0
$856$	2.63300	0.0899941
$857$	−54.7084	−1.86880	−0.934401	−	0.356222i	$-0.884065\pi$
$857$	−54.7084	−1.86880	−0.934401	+	0.356222i	$0.884065\pi$
$858$	0	0
$859$	17.3470	0.591874	0.295937	−	0.955208i	$-0.404368\pi$
$859$	17.3470	0.591874	0.295937	+	0.955208i	$0.404368\pi$
$860$	0	0
$861$	0	0
$862$	20.0659	0.683448
$863$	31.7103	1.07943	0.539715	−	0.841848i	$-0.318532\pi$
$863$	31.7103	1.07943	0.539715	+	0.841848i	$0.318532\pi$
$864$	0	0
$865$	0	0
$866$	−49.3025	−1.67537
$867$	0	0
$868$	3.81201	0.129388
$869$	−50.5232	−1.71388
$870$	0	0
$871$	−47.9158	−1.62357
$872$	−0.695160	−0.0235411
$873$	0	0
$874$	35.9073	1.21458
$875$	0	0
$876$	0	0
$877$	−32.2346	−1.08849	−0.544243	−	0.838928i	$-0.683183\pi$
$877$	−32.2346	−1.08849	−0.544243	+	0.838928i	$0.683183\pi$
$878$	−3.56649	−0.120363
$879$	0	0
$880$	0	0
$881$	−37.2286	−1.25426	−0.627132	−	0.778913i	$-0.715771\pi$
$881$	−37.2286	−1.25426	−0.627132	+	0.778913i	$0.715771\pi$
$882$	0	0
$883$	−5.44208	−0.183140	−0.0915702	−	0.995799i	$-0.529189\pi$
$883$	−5.44208	−0.183140	−0.0915702	+	0.995799i	$0.529189\pi$
$884$	12.3402	0.415044
$885$	0	0
$886$	75.8396	2.54788
$887$	−47.5809	−1.59761	−0.798805	−	0.601590i	$-0.794534\pi$
$887$	−47.5809	−1.59761	−0.798805	+	0.601590i	$0.794534\pi$
$888$	0	0
$889$	2.28287	0.0765650
$890$	0	0
$891$	0	0
$892$	−51.8479	−1.73600
$893$	−16.9414	−0.566924
$894$	0	0
$895$	0	0
$896$	−1.00019	−0.0334140
$897$	0	0
$898$	−26.2834	−0.877087
$899$	−17.8723	−0.596076
$900$	0	0
$901$	15.6170	0.520279
$902$	−61.3319	−2.04213
$903$	0	0
$904$	−1.77311	−0.0589727
$905$	0	0
$906$	0	0
$907$	−21.0436	−0.698742	−0.349371	−	0.936985i	$-0.613605\pi$
$907$	−21.0436	−0.698742	−0.349371	+	0.936985i	$0.613605\pi$
$908$	−28.5378	−0.947059
$909$	0	0
$910$	0	0
$911$	−4.41837	−0.146387	−0.0731935	−	0.997318i	$-0.523319\pi$
$911$	−4.41837	−0.146387	−0.0731935	+	0.997318i	$0.523319\pi$
$912$	0	0
$913$	−1.22479	−0.0405346
$914$	45.2547	1.49689
$915$	0	0
$916$	32.1528	1.06236
$917$	4.88621	0.161357
$918$	0	0
$919$	43.7856	1.44435	0.722176	−	0.691709i	$-0.243142\pi$
$919$	43.7856	1.44435	0.722176	+	0.691709i	$0.243142\pi$
$920$	0	0
$921$	0	0
$922$	1.09054	0.0359151
$923$	−67.7324	−2.22944
$924$	0	0
$925$	0	0
$926$	45.3800	1.49128
$927$	0	0
$928$	40.6007	1.33278
$929$	2.98260	0.0978559	0.0489279	−	0.998802i	$-0.484420\pi$
$929$	2.98260	0.0978559	0.0489279	+	0.998802i	$0.484420\pi$
$930$	0	0
$931$	40.0579	1.31285
$932$	6.77263	0.221845
$933$	0	0
$934$	76.0893	2.48972
$935$	0	0
$936$	0	0
$937$	−25.4557	−0.831602	−0.415801	−	0.909456i	$-0.636499\pi$
$937$	−25.4557	−0.831602	−0.415801	+	0.909456i	$0.636499\pi$
$938$	9.93485	0.324384
$939$	0	0
$940$	0	0
$941$	17.5071	0.570716	0.285358	−	0.958421i	$-0.407888\pi$
$941$	17.5071	0.570716	0.285358	+	0.958421i	$0.407888\pi$
$942$	0	0
$943$	26.1620	0.851952
$944$	35.5360	1.15660
$945$	0	0
$946$	71.5628	2.32671
$947$	−27.4729	−0.892748	−0.446374	−	0.894847i	$-0.647285\pi$
$947$	−27.4729	−0.892748	−0.446374	+	0.894847i	$0.647285\pi$
$948$	0	0
$949$	−61.4272	−1.99401
$950$	0	0
$951$	0	0
$952$	−0.147480	−0.00477987
$953$	−26.2625	−0.850726	−0.425363	−	0.905023i	$-0.639854\pi$
$953$	−26.2625	−0.850726	−0.425363	+	0.905023i	$0.639854\pi$
$954$	0	0
$955$	0	0
$956$	47.9927	1.55219
$957$	0	0
$958$	37.8033	1.22137
$959$	2.00944	0.0648881
$960$	0	0
$961$	−18.3146	−0.590794
$962$	−15.0499	−0.485229
$963$	0	0
$964$	−40.6670	−1.30979
$965$	0	0
$966$	0	0
$967$	−16.1701	−0.519996	−0.259998	−	0.965609i	$-0.583722\pi$
$967$	−16.1701	−0.519996	−0.259998	+	0.965609i	$0.583722\pi$
$968$	0.204808	0.00658276
$969$	0	0
$970$	0	0
$971$	44.6961	1.43437	0.717183	−	0.696885i	$-0.245431\pi$
$971$	44.6961	1.43437	0.717183	+	0.696885i	$0.245431\pi$
$972$	0	0
$973$	2.50207	0.0802126
$974$	3.63658	0.116524
$975$	0	0
$976$	31.7211	1.01537
$977$	−22.3665	−0.715568	−0.357784	−	0.933804i	$-0.616468\pi$
$977$	−22.3665	−0.715568	−0.357784	+	0.933804i	$0.616468\pi$
$978$	0	0
$979$	22.3873	0.715500
$980$	0	0
$981$	0	0
$982$	55.3788	1.76721
$983$	−4.89446	−0.156109	−0.0780545	−	0.996949i	$-0.524871\pi$
$983$	−4.89446	−0.156109	−0.0780545	+	0.996949i	$0.524871\pi$
$984$	0	0
$985$	0	0
$986$	11.9958	0.382025
$987$	0	0
$988$	62.2378	1.98005
$989$	−30.5261	−0.970675
$990$	0	0
$991$	−13.3635	−0.424505	−0.212252	−	0.977215i	$-0.568080\pi$
$991$	−13.3635	−0.424505	−0.212252	+	0.977215i	$0.568080\pi$
$992$	−28.8175	−0.914955
$993$	0	0
$994$	14.0436	0.445436
$995$	0	0
$996$	0	0
$997$	−43.6763	−1.38324	−0.691621	−	0.722261i	$-0.743103\pi$
$997$	−43.6763	−1.38324	−0.691621	+	0.722261i	$0.743103\pi$
$998$	21.4652	0.679470
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.v.1.7		8
3.2	odd	2	inner	5625.2.a.v.1.2		8
5.4	even	2		5625.2.a.w.1.2		8
15.14	odd	2		5625.2.a.w.1.7		8
25.4	even	10		225.2.h.e.91.4	yes	16
25.19	even	10		225.2.h.e.136.4	yes	16
75.29	odd	10		225.2.h.e.91.1	✓	16
75.44	odd	10		225.2.h.e.136.1	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.2.h.e.91.1	✓	16	75.29	odd	10
225.2.h.e.91.4	yes	16	25.4	even	10
225.2.h.e.136.1	yes	16	75.44	odd	10
225.2.h.e.136.4	yes	16	25.19	even	10
5625.2.a.v.1.2		8	3.2	odd	2	inner
5625.2.a.v.1.7		8	1.1	even	1	trivial
5625.2.a.w.1.2		8	5.4	even	2
5625.2.a.w.1.7		8	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+2.03035 q^{2} +2.12233 q^{4} +0.504300 q^{7} +0.248380 q^{8} +O(q^{10})\)	\(q+2.03035 q^{2} +2.12233 q^{4} +0.504300 q^{7} +0.248380 q^{8} +3.43869 q^{11} -4.93831 q^{13} +1.02391 q^{14} -3.74037 q^{16} -1.17741 q^{17} -5.93831 q^{19} +6.98175 q^{22} -2.97817 q^{23} -10.0265 q^{26} +1.07029 q^{28} -5.01799 q^{29} +3.56166 q^{31} -8.09103 q^{32} -2.39057 q^{34} +1.50101 q^{37} -12.0569 q^{38} -8.78461 q^{41} +10.2500 q^{43} +7.29804 q^{44} -6.04673 q^{46} +2.85291 q^{47} -6.74568 q^{49} -10.4807 q^{52} -13.2638 q^{53} +0.125258 q^{56} -10.1883 q^{58} -9.50068 q^{59} -8.48073 q^{61} +7.23142 q^{62} -8.94691 q^{64} +9.70289 q^{67} -2.49886 q^{68} +13.7157 q^{71} +12.4389 q^{73} +3.04759 q^{74} -12.6031 q^{76} +1.73413 q^{77} -14.6926 q^{79} -17.8359 q^{82} -0.356179 q^{83} +20.8111 q^{86} +0.854102 q^{88} +6.51041 q^{89} -2.49039 q^{91} -6.32066 q^{92} +5.79241 q^{94} -10.9807 q^{97} -13.6961 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{4}+O(q^{10}) \) 8 * q + 4 * q^4	\( 8 q + 4 q^{4} - 10 q^{13} - 8 q^{16} - 18 q^{19} + 30 q^{28} - 26 q^{31} - 20 q^{34} + 40 q^{43} - 30 q^{46} - 16 q^{49} - 40 q^{52} - 10 q^{58} - 24 q^{61} - 34 q^{64} + 40 q^{67} - 40 q^{73} - 44 q^{76} - 42 q^{79} - 60 q^{82} - 20 q^{88} - 40 q^{91} - 10 q^{94} + 40 q^{97}+O(q^{100}) \) 8 * q + 4 * q^4 - 10 * q^13 - 8 * q^16 - 18 * q^19 + 30 * q^28 - 26 * q^31 - 20 * q^34 + 40 * q^43 - 30 * q^46 - 16 * q^49 - 40 * q^52 - 10 * q^58 - 24 * q^61 - 34 * q^64 + 40 * q^67 - 40 * q^73 - 44 * q^76 - 42 * q^79 - 60 * q^82 - 20 * q^88 - 40 * q^91 - 10 * q^94 + 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more