LMFDB - Embedded newform 225.10.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,10,Mod(1,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 10, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("225.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	225.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:	$115.883063137$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 5)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-8.00000 q^{2} -448.000 q^{4} -4242.00 q^{7} +7680.00 q^{8} +O(q^{10})$

$q-8.00000 q^{2} -448.000 q^{4} -4242.00 q^{7} +7680.00 q^{8} +46208.0 q^{11} +115934. q^{13} +33936.0 q^{14} +167936. q^{16} +494842. q^{17} -1.00874e6 q^{19} -369664. q^{22} -532554. q^{23} -927472. q^{26} +1.90042e6 q^{28} -4.19639e6 q^{29} -3.36503e6 q^{31} -5.27565e6 q^{32} -3.95874e6 q^{34} +1.49314e7 q^{37} +8.06992e6 q^{38} -1.10563e7 q^{41} +6.39679e6 q^{43} -2.07012e7 q^{44} +4.26043e6 q^{46} -3.55592e7 q^{47} -2.23590e7 q^{49} -5.19384e7 q^{52} +3.97386e7 q^{53} -3.25786e7 q^{56} +3.35711e7 q^{58} +8.51856e7 q^{59} +4.57486e7 q^{61} +2.69202e7 q^{62} -4.37780e7 q^{64} +4.52862e7 q^{67} -2.21689e8 q^{68} +1.89967e8 q^{71} -4.12171e8 q^{73} -1.19451e8 q^{74} +4.51916e8 q^{76} -1.96014e8 q^{77} +9.50408e7 q^{79} +8.84501e7 q^{82} +2.61706e8 q^{83} -5.11744e7 q^{86} +3.54877e8 q^{88} +1.99386e7 q^{89} -4.91792e8 q^{91} +2.38584e8 q^{92} +2.84473e8 q^{94} +1.95034e7 q^{97} +1.78872e8 q^{98} +O(q^{100})$

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 $

$2$	−8.00000	−0.353553	−0.176777	−	0.984251i	$-0.556567\pi$
$2$	−8.00000	−0.353553	−0.176777	+	0.984251i	$0.556567\pi$
$3$	0	0
$3$	0	0
$4$	−448.000	−0.875000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4242.00	−0.667774	−0.333887	−	0.942613i	$-0.608360\pi$
$7$	−4242.00	−0.667774	−0.333887	+	0.942613i	$0.608360\pi$
$8$	7680.00	0.662913
$9$	0	0
$10$	0	0
$11$	46208.0	0.951590	0.475795	−	0.879556i	$-0.342160\pi$
$11$	46208.0	0.951590	0.475795	+	0.879556i	$0.342160\pi$
$12$	0	0
$13$	115934.	1.12581	0.562906	−	0.826521i	$-0.309683\pi$
$13$	115934.	1.12581	0.562906	+	0.826521i	$0.309683\pi$
$14$	33936.0	0.236094
$15$	0	0
$16$	167936.	0.640625
$17$	494842.	1.43697	0.718483	−	0.695545i	$-0.244837\pi$
$17$	494842.	1.43697	0.718483	+	0.695545i	$0.244837\pi$
$18$	0	0
$19$	−1.00874e6	−1.77578	−0.887888	−	0.460060i	$-0.847828\pi$
$19$	−1.00874e6	−1.77578	−0.887888	+	0.460060i	$0.847828\pi$
$20$	0	0
$21$	0	0
$22$	−369664.	−0.336438
$23$	−532554.	−0.396815	−0.198408	−	0.980120i	$-0.563577\pi$
$23$	−532554.	−0.396815	−0.198408	+	0.980120i	$0.563577\pi$
$24$	0	0
$25$	0	0
$26$	−927472.	−0.398035
$27$	0	0
$28$	1.90042e6	0.584302
$29$	−4.19639e6	−1.10175	−0.550877	−	0.834586i	$-0.685707\pi$
$29$	−4.19639e6	−1.10175	−0.550877	+	0.834586i	$0.685707\pi$
$30$	0	0
$31$	−3.36503e6	−0.654427	−0.327213	−	0.944950i	$-0.606110\pi$
$31$	−3.36503e6	−0.654427	−0.327213	+	0.944950i	$0.606110\pi$
$32$	−5.27565e6	−0.889408
$33$	0	0
$34$	−3.95874e6	−0.508044
$35$	0	0
$36$	0	0
$37$	1.49314e7	1.30976	0.654880	−	0.755733i	$-0.272719\pi$
$37$	1.49314e7	1.30976	0.654880	+	0.755733i	$0.272719\pi$
$38$	8.06992e6	0.627831
$39$	0	0
$40$	0	0
$41$	−1.10563e7	−0.611056	−0.305528	−	0.952183i	$-0.598833\pi$
$41$	−1.10563e7	−0.611056	−0.305528	+	0.952183i	$0.598833\pi$
$42$	0	0
$43$	6.39679e6	0.285335	0.142667	−	0.989771i	$-0.454432\pi$
$43$	6.39679e6	0.285335	0.142667	+	0.989771i	$0.454432\pi$
$44$	−2.07012e7	−0.832642
$45$	0	0
$46$	4.26043e6	0.140295
$47$	−3.55592e7	−1.06295	−0.531473	−	0.847075i	$-0.678361\pi$
$47$	−3.55592e7	−1.06295	−0.531473	+	0.847075i	$0.678361\pi$
$48$	0	0
$49$	−2.23590e7	−0.554078
$50$	0	0
$51$	0	0
$52$	−5.19384e7	−0.985085
$53$	3.97386e7	0.691785	0.345892	−	0.938274i	$-0.387576\pi$
$53$	3.97386e7	0.691785	0.345892	+	0.938274i	$0.387576\pi$
$54$	0	0
$55$	0	0
$56$	−3.25786e7	−0.442676
$57$	0	0
$58$	3.35711e7	0.389529
$59$	8.51856e7	0.915234	0.457617	−	0.889149i	$-0.348703\pi$
$59$	8.51856e7	0.915234	0.457617	+	0.889149i	$0.348703\pi$
$60$	0	0
$61$	4.57486e7	0.423052	0.211526	−	0.977372i	$-0.432157\pi$
$61$	4.57486e7	0.423052	0.211526	+	0.977372i	$0.432157\pi$
$62$	2.69202e7	0.231375
$63$	0	0
$64$	−4.37780e7	−0.326172
$65$	0	0
$66$	0	0
$67$	4.52862e7	0.274555	0.137277	−	0.990533i	$-0.456165\pi$
$67$	4.52862e7	0.274555	0.137277	+	0.990533i	$0.456165\pi$
$68$	−2.21689e8	−1.25735
$69$	0	0
$70$	0	0
$71$	1.89967e8	0.887190	0.443595	−	0.896227i	$-0.353703\pi$
$71$	1.89967e8	0.887190	0.443595	+	0.896227i	$0.353703\pi$
$72$	0	0
$73$	−4.12171e8	−1.69873	−0.849365	−	0.527805i	$-0.823015\pi$
$73$	−4.12171e8	−1.69873	−0.849365	+	0.527805i	$0.823015\pi$
$74$	−1.19451e8	−0.463070
$75$	0	0
$76$	4.51916e8	1.55380
$77$	−1.96014e8	−0.635447
$78$	0	0
$79$	9.50408e7	0.274529	0.137265	−	0.990534i	$-0.456169\pi$
$79$	9.50408e7	0.274529	0.137265	+	0.990534i	$0.456169\pi$
$80$	0	0
$81$	0	0
$82$	8.84501e7	0.216041
$83$	2.61706e8	0.605289	0.302645	−	0.953104i	$-0.402131\pi$
$83$	2.61706e8	0.605289	0.302645	+	0.953104i	$0.402131\pi$
$84$	0	0
$85$	0	0
$86$	−5.11744e7	−0.100881
$87$	0	0
$88$	3.54877e8	0.630821
$89$	1.99386e7	0.0336853	0.0168426	−	0.999858i	$-0.494639\pi$
$89$	1.99386e7	0.0336853	0.0168426	+	0.999858i	$0.494639\pi$
$90$	0	0
$91$	−4.91792e8	−0.751788
$92$	2.38584e8	0.347213
$93$	0	0
$94$	2.84473e8	0.375808
$95$	0	0
$96$	0	0
$97$	1.95034e7	0.0223685	0.0111842	−	0.999937i	$-0.496440\pi$
$97$	1.95034e7	0.0223685	0.0111842	+	0.999937i	$0.496440\pi$
$98$	1.78872e8	0.195896
$99$	0	0
$100$	0	0
$101$	2.16672e8	0.207184	0.103592	−	0.994620i	$-0.466966\pi$
$101$	2.16672e8	0.207184	0.103592	+	0.994620i	$0.466966\pi$
$102$	0	0
$103$	7.48234e8	0.655043	0.327522	−	0.944844i	$-0.393787\pi$
$103$	7.48234e8	0.655043	0.327522	+	0.944844i	$0.393787\pi$
$104$	8.90373e8	0.746315
$105$	0	0
$106$	−3.17909e8	−0.244583
$107$	1.05711e9	0.779637	0.389819	−	0.920892i	$-0.372538\pi$
$107$	1.05711e9	0.779637	0.389819	+	0.920892i	$0.372538\pi$
$108$	0	0
$109$	−2.49659e9	−1.69406	−0.847029	−	0.531547i	$-0.821611\pi$
$109$	−2.49659e9	−1.69406	−0.847029	+	0.531547i	$0.821611\pi$
$110$	0	0
$111$	0	0
$112$	−7.12385e8	−0.427793
$113$	1.11566e9	0.643695	0.321848	−	0.946791i	$-0.395696\pi$
$113$	1.11566e9	0.643695	0.321848	+	0.946791i	$0.395696\pi$
$114$	0	0
$115$	0	0
$116$	1.87998e9	0.964035
$117$	0	0
$118$	−6.81485e8	−0.323584
$119$	−2.09912e9	−0.959568
$120$	0	0
$121$	−2.22768e8	−0.0944756
$122$	−3.65989e8	−0.149572
$123$	0	0
$124$	1.50753e9	0.572623
$125$	0	0
$126$	0	0
$127$	−3.12054e8	−0.106442	−0.0532210	−	0.998583i	$-0.516949\pi$
$127$	−3.12054e8	−0.106442	−0.0532210	+	0.998583i	$0.516949\pi$
$128$	3.05136e9	1.00473
$129$	0	0
$130$	0	0
$131$	2.81701e9	0.835733	0.417867	−	0.908508i	$-0.362778\pi$
$131$	2.81701e9	0.835733	0.417867	+	0.908508i	$0.362778\pi$
$132$	0	0
$133$	4.27908e9	1.18582
$134$	−3.62289e8	−0.0970697
$135$	0	0
$136$	3.80039e9	0.952583
$137$	−7.15311e9	−1.73481	−0.867406	−	0.497600i	$-0.834215\pi$
$137$	−7.15311e9	−1.73481	−0.867406	+	0.497600i	$0.834215\pi$
$138$	0	0
$139$	2.52323e9	0.573310	0.286655	−	0.958034i	$-0.407457\pi$
$139$	2.52323e9	0.573310	0.286655	+	0.958034i	$0.407457\pi$
$140$	0	0
$141$	0	0
$142$	−1.51974e9	−0.313669
$143$	5.35708e9	1.07131
$144$	0	0
$145$	0	0
$146$	3.29737e9	0.600592
$147$	0	0
$148$	−6.68925e9	−1.14604
$149$	−1.46531e9	−0.243551	−0.121775	−	0.992558i	$-0.538859\pi$
$149$	−1.46531e9	−0.243551	−0.121775	+	0.992558i	$0.538859\pi$
$150$	0	0
$151$	3.65515e9	0.572149	0.286075	−	0.958207i	$-0.407650\pi$
$151$	3.65515e9	0.572149	0.286075	+	0.958207i	$0.407650\pi$
$152$	−7.74712e9	−1.17718
$153$	0	0
$154$	1.56811e9	0.224665
$155$	0	0
$156$	0	0
$157$	−5.55191e9	−0.729279	−0.364639	−	0.931149i	$-0.618808\pi$
$157$	−5.55191e9	−0.729279	−0.364639	+	0.931149i	$0.618808\pi$
$158$	−7.60327e8	−0.0970607
$159$	0	0
$160$	0	0
$161$	2.25909e9	0.264983
$162$	0	0
$163$	1.67637e10	1.86006	0.930029	−	0.367485i	$-0.119781\pi$
$163$	1.67637e10	1.86006	0.930029	+	0.367485i	$0.119781\pi$
$164$	4.95321e9	0.534674
$165$	0	0
$166$	−2.09365e9	−0.214002
$167$	1.39549e10	1.38836	0.694182	−	0.719799i	$-0.255766\pi$
$167$	1.39549e10	1.38836	0.694182	+	0.719799i	$0.255766\pi$
$168$	0	0
$169$	2.83619e9	0.267452
$170$	0	0
$171$	0	0
$172$	−2.86576e9	−0.249668
$173$	1.57000e10	1.33258	0.666289	−	0.745694i	$-0.267882\pi$
$173$	1.57000e10	1.33258	0.666289	+	0.745694i	$0.267882\pi$
$174$	0	0
$175$	0	0
$176$	7.75999e9	0.609613
$177$	0	0
$178$	−1.59509e8	−0.0119095
$179$	2.51902e10	1.83397	0.916985	−	0.398921i	$-0.130615\pi$
$179$	2.51902e10	1.83397	0.916985	+	0.398921i	$0.130615\pi$
$180$	0	0
$181$	1.04482e10	0.723579	0.361789	−	0.932260i	$-0.382166\pi$
$181$	1.04482e10	0.723579	0.361789	+	0.932260i	$0.382166\pi$
$182$	3.93434e9	0.265797
$183$	0	0
$184$	−4.09001e9	−0.263054
$185$	0	0
$186$	0	0
$187$	2.28657e10	1.36740
$188$	1.59305e10	0.930078
$189$	0	0
$190$	0	0
$191$	9.68625e9	0.526630	0.263315	−	0.964710i	$-0.415184\pi$
$191$	9.68625e9	0.526630	0.263315	+	0.964710i	$0.415184\pi$
$192$	0	0
$193$	2.04431e10	1.06057	0.530285	−	0.847820i	$-0.322085\pi$
$193$	2.04431e10	1.06057	0.530285	+	0.847820i	$0.322085\pi$
$194$	−1.56027e8	−0.00790845
$195$	0	0
$196$	1.00169e10	0.484818
$197$	−2.52431e10	−1.19411	−0.597056	−	0.802200i	$-0.703663\pi$
$197$	−2.52431e10	−1.19411	−0.597056	+	0.802200i	$0.703663\pi$
$198$	0	0
$199$	8.06736e9	0.364664	0.182332	−	0.983237i	$-0.441635\pi$
$199$	8.06736e9	0.364664	0.182332	+	0.983237i	$0.441635\pi$
$200$	0	0
$201$	0	0
$202$	−1.73338e9	−0.0732506
$203$	1.78011e10	0.735723
$204$	0	0
$205$	0	0
$206$	−5.98587e9	−0.231593
$207$	0	0
$208$	1.94695e10	0.721223
$209$	−4.66119e10	−1.68981
$210$	0	0
$211$	−8.04589e9	−0.279449	−0.139725	−	0.990190i	$-0.544622\pi$
$211$	−8.04589e9	−0.279449	−0.139725	+	0.990190i	$0.544622\pi$
$212$	−1.78029e10	−0.605312
$213$	0	0
$214$	−8.45687e9	−0.275643
$215$	0	0
$216$	0	0
$217$	1.42744e10	0.437009
$218$	1.99727e10	0.598940
$219$	0	0
$220$	0	0
$221$	5.73690e10	1.61775
$222$	0	0
$223$	−3.18618e10	−0.862777	−0.431388	−	0.902166i	$-0.641976\pi$
$223$	−3.18618e10	−0.862777	−0.431388	+	0.902166i	$0.641976\pi$
$224$	2.23793e10	0.593923
$225$	0	0
$226$	−8.92531e9	−0.227581
$227$	3.02621e10	0.756454	0.378227	−	0.925713i	$-0.376534\pi$
$227$	3.02621e10	0.756454	0.378227	+	0.925713i	$0.376534\pi$
$228$	0	0
$229$	2.06101e10	0.495245	0.247623	−	0.968857i	$-0.420351\pi$
$229$	2.06101e10	0.495245	0.247623	+	0.968857i	$0.420351\pi$
$230$	0	0
$231$	0	0
$232$	−3.22283e10	−0.730367
$233$	3.61735e9	0.0804061	0.0402031	−	0.999192i	$-0.487200\pi$
$233$	3.61735e9	0.0804061	0.0402031	+	0.999192i	$0.487200\pi$
$234$	0	0
$235$	0	0
$236$	−3.81632e10	−0.800830
$237$	0	0
$238$	1.67930e10	0.339259
$239$	−2.21916e9	−0.0439945	−0.0219973	−	0.999758i	$-0.507003\pi$
$239$	−2.21916e9	−0.0439945	−0.0219973	+	0.999758i	$0.507003\pi$
$240$	0	0
$241$	2.57977e10	0.492611	0.246306	−	0.969192i	$-0.420783\pi$
$241$	2.57977e10	0.492611	0.246306	+	0.969192i	$0.420783\pi$
$242$	1.78215e9	0.0334022
$243$	0	0
$244$	−2.04954e10	−0.370171
$245$	0	0
$246$	0	0
$247$	−1.16947e11	−1.99919
$248$	−2.58434e10	−0.433828
$249$	0	0
$250$	0	0
$251$	8.91848e10	1.41827	0.709135	−	0.705072i	$-0.249086\pi$
$251$	8.91848e10	1.41827	0.709135	+	0.705072i	$0.249086\pi$
$252$	0	0
$253$	−2.46083e10	−0.377606
$254$	2.49643e9	0.0376329
$255$	0	0
$256$	−1.99649e9	−0.0290527
$257$	−1.10058e11	−1.57371	−0.786853	−	0.617141i	$-0.788291\pi$
$257$	−1.10058e11	−1.57371	−0.786853	+	0.617141i	$0.788291\pi$
$258$	0	0
$259$	−6.33388e10	−0.874623
$260$	0	0
$261$	0	0
$262$	−2.25361e10	−0.295476
$263$	2.06374e10	0.265983	0.132992	−	0.991117i	$-0.457542\pi$
$263$	2.06374e10	0.265983	0.132992	+	0.991117i	$0.457542\pi$
$264$	0	0
$265$	0	0
$266$	−3.42326e10	−0.419249
$267$	0	0
$268$	−2.02882e10	−0.240235
$269$	2.64890e10	0.308446	0.154223	−	0.988036i	$-0.450713\pi$
$269$	2.64890e10	0.308446	0.154223	+	0.988036i	$0.450713\pi$
$270$	0	0
$271$	3.44004e10	0.387438	0.193719	−	0.981057i	$-0.437945\pi$
$271$	3.44004e10	0.387438	0.193719	+	0.981057i	$0.437945\pi$
$272$	8.31018e10	0.920556
$273$	0	0
$274$	5.72249e10	0.613349
$275$	0	0
$276$	0	0
$277$	1.06023e11	1.08203	0.541017	−	0.841012i	$-0.318040\pi$
$277$	1.06023e11	1.08203	0.541017	+	0.841012i	$0.318040\pi$
$278$	−2.01858e10	−0.202696
$279$	0	0
$280$	0	0
$281$	−1.00299e11	−0.959662	−0.479831	−	0.877361i	$-0.659302\pi$
$281$	−1.00299e11	−0.959662	−0.479831	+	0.877361i	$0.659302\pi$
$282$	0	0
$283$	2.49296e9	0.0231035	0.0115517	−	0.999933i	$-0.496323\pi$
$283$	2.49296e9	0.0231035	0.0115517	+	0.999933i	$0.496323\pi$
$284$	−8.51054e10	−0.776291
$285$	0	0
$286$	−4.28566e10	−0.378766
$287$	4.69007e10	0.408047
$288$	0	0
$289$	1.26281e11	1.06487
$290$	0	0
$291$	0	0
$292$	1.84653e11	1.48639
$293$	−8.47068e10	−0.671451	−0.335725	−	0.941960i	$-0.608981\pi$
$293$	−8.47068e10	−0.671451	−0.335725	+	0.941960i	$0.608981\pi$
$294$	0	0
$295$	0	0
$296$	1.14673e11	0.868256
$297$	0	0
$298$	1.17224e10	0.0861083
$299$	−6.17411e10	−0.446739
$300$	0	0
$301$	−2.71352e10	−0.190539
$302$	−2.92412e10	−0.202285
$303$	0	0
$304$	−1.69404e11	−1.13761
$305$	0	0
$306$	0	0
$307$	−6.35507e9	−0.0408317	−0.0204159	−	0.999792i	$-0.506499\pi$
$307$	−6.35507e9	−0.0408317	−0.0204159	+	0.999792i	$0.506499\pi$
$308$	8.78144e10	0.556016
$309$	0	0
$310$	0	0
$311$	−2.71253e11	−1.64419	−0.822096	−	0.569350i	$-0.807195\pi$
$311$	−2.71253e11	−1.64419	−0.822096	+	0.569350i	$0.807195\pi$
$312$	0	0
$313$	−1.63499e11	−0.962865	−0.481432	−	0.876483i	$-0.659883\pi$
$313$	−1.63499e11	−0.962865	−0.481432	+	0.876483i	$0.659883\pi$
$314$	4.44153e10	0.257839
$315$	0	0
$316$	−4.25783e10	−0.240213
$317$	2.53606e11	1.41056	0.705282	−	0.708927i	$-0.250820\pi$
$317$	2.53606e11	1.41056	0.705282	+	0.708927i	$0.250820\pi$
$318$	0	0
$319$	−1.93907e11	−1.04842
$320$	0	0
$321$	0	0
$322$	−1.80728e10	−0.0936856
$323$	−4.99167e11	−2.55173
$324$	0	0
$325$	0	0
$326$	−1.34110e11	−0.657630
$327$	0	0
$328$	−8.49121e10	−0.405077
$329$	1.50842e11	0.709808
$330$	0	0
$331$	−2.40064e11	−1.09926	−0.549631	−	0.835408i	$-0.685232\pi$
$331$	−2.40064e11	−1.09926	−0.549631	+	0.835408i	$0.685232\pi$
$332$	−1.17244e11	−0.529628
$333$	0	0
$334$	−1.11639e11	−0.490861
$335$	0	0
$336$	0	0
$337$	5.13812e10	0.217005	0.108502	−	0.994096i	$-0.465394\pi$
$337$	5.13812e10	0.217005	0.108502	+	0.994096i	$0.465394\pi$
$338$	−2.26895e10	−0.0945585
$339$	0	0
$340$	0	0
$341$	−1.55491e11	−0.622746
$342$	0	0
$343$	2.66027e11	1.03777
$344$	4.91274e10	0.189152
$345$	0	0
$346$	−1.25600e11	−0.471137
$347$	2.76560e11	1.02401	0.512007	−	0.858981i	$-0.328902\pi$
$347$	2.76560e11	1.02401	0.512007	+	0.858981i	$0.328902\pi$
$348$	0	0
$349$	4.66592e11	1.68354	0.841770	−	0.539837i	$-0.181514\pi$
$349$	4.66592e11	1.68354	0.841770	+	0.539837i	$0.181514\pi$
$350$	0	0
$351$	0	0
$352$	−2.43777e11	−0.846352
$353$	−1.00998e11	−0.346198	−0.173099	−	0.984904i	$-0.555378\pi$
$353$	−1.00998e11	−0.346198	−0.173099	+	0.984904i	$0.555378\pi$
$354$	0	0
$355$	0	0
$356$	−8.93251e9	−0.0294746
$357$	0	0
$358$	−2.01521e11	−0.648406
$359$	1.66447e11	0.528874	0.264437	−	0.964403i	$-0.414814\pi$
$359$	1.66447e11	0.528874	0.264437	+	0.964403i	$0.414814\pi$
$360$	0	0
$361$	6.94869e11	2.15338
$362$	−8.35852e10	−0.255824
$363$	0	0
$364$	2.20323e11	0.657814
$365$	0	0
$366$	0	0
$367$	2.97381e11	0.855688	0.427844	−	0.903853i	$-0.359273\pi$
$367$	2.97381e11	0.855688	0.427844	+	0.903853i	$0.359273\pi$
$368$	−8.94350e10	−0.254210
$369$	0	0
$370$	0	0
$371$	−1.68571e11	−0.461956
$372$	0	0
$373$	−1.95714e11	−0.523517	−0.261759	−	0.965133i	$-0.584302\pi$
$373$	−1.95714e11	−0.523517	−0.261759	+	0.965133i	$0.584302\pi$
$374$	−1.82925e11	−0.483450
$375$	0	0
$376$	−2.73094e11	−0.704640
$377$	−4.86504e11	−1.24037
$378$	0	0
$379$	−1.67009e11	−0.415781	−0.207890	−	0.978152i	$-0.566660\pi$
$379$	−1.67009e11	−0.415781	−0.207890	+	0.978152i	$0.566660\pi$
$380$	0	0
$381$	0	0
$382$	−7.74900e10	−0.186192
$383$	7.20782e10	0.171163	0.0855814	−	0.996331i	$-0.472725\pi$
$383$	7.20782e10	0.171163	0.0855814	+	0.996331i	$0.472725\pi$
$384$	0	0
$385$	0	0
$386$	−1.63545e11	−0.374968
$387$	0	0
$388$	−8.73750e9	−0.0195724
$389$	−7.92249e11	−1.75424	−0.877119	−	0.480272i	$-0.840538\pi$
$389$	−7.92249e11	−1.75424	−0.877119	+	0.480272i	$0.840538\pi$
$390$	0	0
$391$	−2.63530e11	−0.570210
$392$	−1.71717e11	−0.367305
$393$	0	0
$394$	2.01945e11	0.422182
$395$	0	0
$396$	0	0
$397$	1.47288e11	0.297584	0.148792	−	0.988869i	$-0.452462\pi$
$397$	1.47288e11	0.297584	0.148792	+	0.988869i	$0.452462\pi$
$398$	−6.45389e10	−0.128928
$399$	0	0
$400$	0	0
$401$	3.22101e11	0.622075	0.311037	−	0.950398i	$-0.399324\pi$
$401$	3.22101e11	0.622075	0.311037	+	0.950398i	$0.399324\pi$
$402$	0	0
$403$	−3.90121e11	−0.736761
$404$	−9.70690e10	−0.181286
$405$	0	0
$406$	−1.42409e11	−0.260117
$407$	6.89948e11	1.24635
$408$	0	0
$409$	−3.96423e11	−0.700493	−0.350247	−	0.936658i	$-0.613902\pi$
$409$	−3.96423e11	−0.700493	−0.350247	+	0.936658i	$0.613902\pi$
$410$	0	0
$411$	0	0
$412$	−3.35209e11	−0.573163
$413$	−3.61357e11	−0.611170
$414$	0	0
$415$	0	0
$416$	−6.11627e11	−1.00131
$417$	0	0
$418$	3.72895e11	0.597438
$419$	1.23162e12	1.95215	0.976074	−	0.217441i	$-0.0697708\pi$
$419$	1.23162e12	1.95215	0.976074	+	0.217441i	$0.0697708\pi$
$420$	0	0
$421$	−1.17500e12	−1.82293	−0.911465	−	0.411379i	$-0.865047\pi$
$421$	−1.17500e12	−1.82293	−0.911465	+	0.411379i	$0.865047\pi$
$422$	6.43671e10	0.0988002
$423$	0	0
$424$	3.05192e11	0.458593
$425$	0	0
$426$	0	0
$427$	−1.94066e11	−0.282503
$428$	−4.73585e11	−0.682183
$429$	0	0
$430$	0	0
$431$	2.32327e11	0.324304	0.162152	−	0.986766i	$-0.448157\pi$
$431$	2.32327e11	0.324304	0.162152	+	0.986766i	$0.448157\pi$
$432$	0	0
$433$	1.51554e11	0.207191	0.103596	−	0.994620i	$-0.466965\pi$
$433$	1.51554e11	0.207191	0.103596	+	0.994620i	$0.466965\pi$
$434$	−1.14196e11	−0.154506
$435$	0	0
$436$	1.11847e12	1.48230
$437$	5.37209e11	0.704655
$438$	0	0
$439$	−5.21385e11	−0.669990	−0.334995	−	0.942220i	$-0.608735\pi$
$439$	−5.21385e11	−0.669990	−0.334995	+	0.942220i	$0.608735\pi$
$440$	0	0
$441$	0	0
$442$	−4.58952e11	−0.571962
$443$	1.10169e12	1.35908	0.679539	−	0.733639i	$-0.262180\pi$
$443$	1.10169e12	1.35908	0.679539	+	0.733639i	$0.262180\pi$
$444$	0	0
$445$	0	0
$446$	2.54894e11	0.305038
$447$	0	0
$448$	1.85706e11	0.217809
$449$	−5.85672e11	−0.680058	−0.340029	−	0.940415i	$-0.610437\pi$
$449$	−5.85672e11	−0.680058	−0.340029	+	0.940415i	$0.610437\pi$
$450$	0	0
$451$	−5.10888e11	−0.581475
$452$	−4.99817e11	−0.563233
$453$	0	0
$454$	−2.42097e11	−0.267447
$455$	0	0
$456$	0	0
$457$	5.70804e11	0.612159	0.306079	−	0.952006i	$-0.400983\pi$
$457$	5.70804e11	0.612159	0.306079	+	0.952006i	$0.400983\pi$
$458$	−1.64881e11	−0.175096
$459$	0	0
$460$	0	0
$461$	7.60279e11	0.784005	0.392002	−	0.919964i	$-0.371782\pi$
$461$	7.60279e11	0.784005	0.392002	+	0.919964i	$0.371782\pi$
$462$	0	0
$463$	7.14289e11	0.722370	0.361185	−	0.932494i	$-0.382372\pi$
$463$	7.14289e11	0.722370	0.361185	+	0.932494i	$0.382372\pi$
$464$	−7.04725e11	−0.705812
$465$	0	0
$466$	−2.89388e10	−0.0284279
$467$	7.82521e11	0.761325	0.380662	−	0.924714i	$-0.375696\pi$
$467$	7.82521e11	0.761325	0.380662	+	0.924714i	$0.375696\pi$
$468$	0	0
$469$	−1.92104e11	−0.183340
$470$	0	0
$471$	0	0
$472$	6.54226e11	0.606720
$473$	2.95583e11	0.271522
$474$	0	0
$475$	0	0
$476$	9.40406e11	0.839622
$477$	0	0
$478$	1.77533e10	0.0155544
$479$	6.95738e11	0.603860	0.301930	−	0.953330i	$-0.402369\pi$
$479$	6.95738e11	0.603860	0.301930	+	0.953330i	$0.402369\pi$
$480$	0	0
$481$	1.73105e12	1.47454
$482$	−2.06382e11	−0.174164
$483$	0	0
$484$	9.98003e10	0.0826661
$485$	0	0
$486$	0	0
$487$	1.65196e12	1.33082	0.665408	−	0.746480i	$-0.268258\pi$
$487$	1.65196e12	1.33082	0.665408	+	0.746480i	$0.268258\pi$
$488$	3.51350e11	0.280447
$489$	0	0
$490$	0	0
$491$	−1.19989e12	−0.931694	−0.465847	−	0.884865i	$-0.654250\pi$
$491$	−1.19989e12	−0.931694	−0.465847	+	0.884865i	$0.654250\pi$
$492$	0	0
$493$	−2.07655e12	−1.58318
$494$	9.35578e11	0.706820
$495$	0	0
$496$	−5.65109e11	−0.419242
$497$	−8.05842e11	−0.592442
$498$	0	0
$499$	1.43146e12	1.03354	0.516768	−	0.856125i	$-0.327135\pi$
$499$	1.43146e12	1.03354	0.516768	+	0.856125i	$0.327135\pi$
$500$	0	0
$501$	0	0
$502$	−7.13478e11	−0.501434
$503$	1.41833e12	0.987919	0.493959	−	0.869485i	$-0.335549\pi$
$503$	1.41833e12	0.987919	0.493959	+	0.869485i	$0.335549\pi$
$504$	0	0
$505$	0	0
$506$	1.96866e11	0.133504
$507$	0	0
$508$	1.39800e11	0.0931367
$509$	1.16463e12	0.769059	0.384529	−	0.923113i	$-0.374364\pi$
$509$	1.16463e12	0.769059	0.384529	+	0.923113i	$0.374364\pi$
$510$	0	0
$511$	1.74843e12	1.13437
$512$	−1.54632e12	−0.994455
$513$	0	0
$514$	8.80466e11	0.556389
$515$	0	0
$516$	0	0
$517$	−1.64312e12	−1.01149
$518$	5.06711e11	0.309226
$519$	0	0
$520$	0	0
$521$	8.36946e11	0.497654	0.248827	−	0.968548i	$-0.419955\pi$
$521$	8.36946e11	0.497654	0.248827	+	0.968548i	$0.419955\pi$
$522$	0	0
$523$	2.60288e12	1.52124	0.760618	−	0.649199i	$-0.224896\pi$
$523$	2.60288e12	1.52124	0.760618	+	0.649199i	$0.224896\pi$
$524$	−1.26202e12	−0.731267
$525$	0	0
$526$	−1.65099e11	−0.0940394
$527$	−1.66516e12	−0.940389
$528$	0	0
$529$	−1.51754e12	−0.842538
$530$	0	0
$531$	0	0
$532$	−1.91703e12	−1.03759
$533$	−1.28180e12	−0.687934
$534$	0	0
$535$	0	0
$536$	3.47798e11	0.182006
$537$	0	0
$538$	−2.11912e11	−0.109052
$539$	−1.03317e12	−0.527255
$540$	0	0
$541$	3.07406e12	1.54286	0.771428	−	0.636317i	$-0.219543\pi$
$541$	3.07406e12	1.54286	0.771428	+	0.636317i	$0.219543\pi$
$542$	−2.75203e11	−0.136980
$543$	0	0
$544$	−2.61061e12	−1.27805
$545$	0	0
$546$	0	0
$547$	−1.32972e12	−0.635062	−0.317531	−	0.948248i	$-0.602854\pi$
$547$	−1.32972e12	−0.635062	−0.317531	+	0.948248i	$0.602854\pi$
$548$	3.20460e12	1.51796
$549$	0	0
$550$	0	0
$551$	4.23307e12	1.95647
$552$	0	0
$553$	−4.03163e11	−0.183323
$554$	−8.48183e11	−0.382557
$555$	0	0
$556$	−1.13041e12	−0.501647
$557$	2.49418e12	1.09794	0.548971	−	0.835841i	$-0.315020\pi$
$557$	2.49418e12	1.09794	0.548971	+	0.835841i	$0.315020\pi$
$558$	0	0
$559$	7.41606e11	0.321233
$560$	0	0
$561$	0	0
$562$	8.02392e11	0.339292
$563$	1.06689e12	0.447541	0.223770	−	0.974642i	$-0.428163\pi$
$563$	1.06689e12	0.447541	0.223770	+	0.974642i	$0.428163\pi$
$564$	0	0
$565$	0	0
$566$	−1.99437e10	−0.00816831
$567$	0	0
$568$	1.45895e12	0.588129
$569$	4.57278e12	1.82884	0.914420	−	0.404768i	$-0.132648\pi$
$569$	4.57278e12	1.82884	0.914420	+	0.404768i	$0.132648\pi$
$570$	0	0
$571$	−2.32305e12	−0.914525	−0.457263	−	0.889332i	$-0.651170\pi$
$571$	−2.32305e12	−0.914525	−0.457263	+	0.889332i	$0.651170\pi$
$572$	−2.39997e12	−0.937398
$573$	0	0
$574$	−3.75205e11	−0.144266
$575$	0	0
$576$	0	0
$577$	−1.01144e12	−0.379881	−0.189940	−	0.981796i	$-0.560829\pi$
$577$	−1.01144e12	−0.379881	−0.189940	+	0.981796i	$0.560829\pi$
$578$	−1.01025e12	−0.376489
$579$	0	0
$580$	0	0
$581$	−1.11016e12	−0.404196
$582$	0	0
$583$	1.83624e12	0.658296
$584$	−3.16547e12	−1.12611
$585$	0	0
$586$	6.77654e11	0.237394
$587$	−1.10143e12	−0.382899	−0.191449	−	0.981502i	$-0.561319\pi$
$587$	−1.10143e12	−0.382899	−0.191449	+	0.981502i	$0.561319\pi$
$588$	0	0
$589$	3.39444e12	1.16211
$590$	0	0
$591$	0	0
$592$	2.50751e12	0.839065
$593$	9.84366e11	0.326897	0.163448	−	0.986552i	$-0.447738\pi$
$593$	9.84366e11	0.326897	0.163448	+	0.986552i	$0.447738\pi$
$594$	0	0
$595$	0	0
$596$	6.56457e11	0.213107
$597$	0	0
$598$	4.93929e11	0.157946
$599$	−4.51397e12	−1.43264	−0.716321	−	0.697771i	$-0.754175\pi$
$599$	−4.51397e12	−1.43264	−0.716321	+	0.697771i	$0.754175\pi$
$600$	0	0
$601$	−1.44212e12	−0.450885	−0.225443	−	0.974256i	$-0.572383\pi$
$601$	−1.44212e12	−0.450885	−0.225443	+	0.974256i	$0.572383\pi$
$602$	2.17082e11	0.0673657
$603$	0	0
$604$	−1.63751e12	−0.500630
$605$	0	0
$606$	0	0
$607$	4.12105e12	1.23214	0.616069	−	0.787692i	$-0.288724\pi$
$607$	4.12105e12	1.23214	0.616069	+	0.787692i	$0.288724\pi$
$608$	5.32176e12	1.57939
$609$	0	0
$610$	0	0
$611$	−4.12252e12	−1.19668
$612$	0	0
$613$	−1.38470e12	−0.396080	−0.198040	−	0.980194i	$-0.563458\pi$
$613$	−1.38470e12	−0.396080	−0.198040	+	0.980194i	$0.563458\pi$
$614$	5.08406e10	0.0144362
$615$	0	0
$616$	−1.50539e12	−0.421246
$617$	6.22714e11	0.172984	0.0864918	−	0.996253i	$-0.472434\pi$
$617$	6.22714e11	0.172984	0.0864918	+	0.996253i	$0.472434\pi$
$618$	0	0
$619$	1.89438e12	0.518631	0.259315	−	0.965793i	$-0.416503\pi$
$619$	1.89438e12	0.518631	0.259315	+	0.965793i	$0.416503\pi$
$620$	0	0
$621$	0	0
$622$	2.17002e12	0.581309
$623$	−8.45797e10	−0.0224942
$624$	0	0
$625$	0	0
$626$	1.30799e12	0.340424
$627$	0	0
$628$	2.48725e12	0.638119
$629$	7.38866e12	1.88208
$630$	0	0
$631$	−1.10160e12	−0.276624	−0.138312	−	0.990389i	$-0.544168\pi$
$631$	−1.10160e12	−0.276624	−0.138312	+	0.990389i	$0.544168\pi$
$632$	7.29914e11	0.181989
$633$	0	0
$634$	−2.02885e12	−0.498710
$635$	0	0
$636$	0	0
$637$	−2.59217e12	−0.623787
$638$	1.55125e12	0.370672
$639$	0	0
$640$	0	0
$641$	4.85252e12	1.13529	0.567644	−	0.823274i	$-0.307855\pi$
$641$	4.85252e12	1.13529	0.567644	+	0.823274i	$0.307855\pi$
$642$	0	0
$643$	−1.57487e12	−0.363325	−0.181662	−	0.983361i	$-0.558148\pi$
$643$	−1.57487e12	−0.363325	−0.181662	+	0.983361i	$0.558148\pi$
$644$	−1.01207e12	−0.231860
$645$	0	0
$646$	3.99334e12	0.902172
$647$	6.93025e12	1.55482	0.777410	−	0.628995i	$-0.216533\pi$
$647$	6.93025e12	1.55482	0.777410	+	0.628995i	$0.216533\pi$
$648$	0	0
$649$	3.93626e12	0.870928
$650$	0	0
$651$	0	0
$652$	−7.51015e12	−1.62755
$653$	−8.15499e12	−1.75515	−0.877575	−	0.479440i	$-0.840840\pi$
$653$	−8.15499e12	−1.75515	−0.877575	+	0.479440i	$0.840840\pi$
$654$	0	0
$655$	0	0
$656$	−1.85674e12	−0.391458
$657$	0	0
$658$	−1.20674e12	−0.250955
$659$	2.92553e12	0.604255	0.302127	−	0.953268i	$-0.402303\pi$
$659$	2.92553e12	0.604255	0.302127	+	0.953268i	$0.402303\pi$
$660$	0	0
$661$	−1.62921e12	−0.331947	−0.165974	−	0.986130i	$-0.553077\pi$
$661$	−1.62921e12	−0.331947	−0.165974	+	0.986130i	$0.553077\pi$
$662$	1.92051e12	0.388648
$663$	0	0
$664$	2.00990e12	0.401254
$665$	0	0
$666$	0	0
$667$	2.23480e12	0.437193
$668$	−6.25181e12	−1.21482
$669$	0	0
$670$	0	0
$671$	2.11395e12	0.402572
$672$	0	0
$673$	2.87521e12	0.540258	0.270129	−	0.962824i	$-0.412934\pi$
$673$	2.87521e12	0.540258	0.270129	+	0.962824i	$0.412934\pi$
$674$	−4.11049e11	−0.0767228
$675$	0	0
$676$	−1.27061e12	−0.234020
$677$	3.19620e12	0.584769	0.292385	−	0.956301i	$-0.405551\pi$
$677$	3.19620e12	0.584769	0.292385	+	0.956301i	$0.405551\pi$
$678$	0	0
$679$	−8.27332e10	−0.0149371
$680$	0	0
$681$	0	0
$682$	1.24393e12	0.220174
$683$	−5.47901e11	−0.0963406	−0.0481703	−	0.998839i	$-0.515339\pi$
$683$	−5.47901e11	−0.0963406	−0.0481703	+	0.998839i	$0.515339\pi$
$684$	0	0
$685$	0	0
$686$	−2.12822e12	−0.366908
$687$	0	0
$688$	1.07425e12	0.182792
$689$	4.60705e12	0.778819
$690$	0	0
$691$	5.90996e11	0.0986128	0.0493064	−	0.998784i	$-0.484299\pi$
$691$	5.90996e11	0.0986128	0.0493064	+	0.998784i	$0.484299\pi$
$692$	−7.03361e12	−1.16601
$693$	0	0
$694$	−2.21248e12	−0.362044
$695$	0	0
$696$	0	0
$697$	−5.47110e12	−0.878066
$698$	−3.73274e12	−0.595221
$699$	0	0
$700$	0	0
$701$	−7.12738e12	−1.11481	−0.557403	−	0.830242i	$-0.688202\pi$
$701$	−7.12738e12	−1.11481	−0.557403	+	0.830242i	$0.688202\pi$
$702$	0	0
$703$	−1.50619e13	−2.32584
$704$	−2.02290e12	−0.310382
$705$	0	0
$706$	8.07981e11	0.122400
$707$	−9.19122e11	−0.138352
$708$	0	0
$709$	9.09269e11	0.135140	0.0675701	−	0.997715i	$-0.478475\pi$
$709$	9.09269e11	0.135140	0.0675701	+	0.997715i	$0.478475\pi$
$710$	0	0
$711$	0	0
$712$	1.53129e11	0.0223304
$713$	1.79206e12	0.259687
$714$	0	0
$715$	0	0
$716$	−1.12852e13	−1.60472
$717$	0	0
$718$	−1.33158e12	−0.186985
$719$	−1.08017e13	−1.50734	−0.753669	−	0.657254i	$-0.771718\pi$
$719$	−1.08017e13	−1.50734	−0.753669	+	0.657254i	$0.771718\pi$
$720$	0	0
$721$	−3.17401e12	−0.437421
$722$	−5.55895e12	−0.761334
$723$	0	0
$724$	−4.68077e12	−0.633132
$725$	0	0
$726$	0	0
$727$	9.77691e12	1.29807	0.649033	−	0.760760i	$-0.275174\pi$
$727$	9.77691e12	1.29807	0.649033	+	0.760760i	$0.275174\pi$
$728$	−3.77696e12	−0.498370
$729$	0	0
$730$	0	0
$731$	3.16540e12	0.410016
$732$	0	0
$733$	−3.52295e11	−0.0450753	−0.0225377	−	0.999746i	$-0.507175\pi$
$733$	−3.52295e11	−0.0450753	−0.0225377	+	0.999746i	$0.507175\pi$
$734$	−2.37905e12	−0.302531
$735$	0	0
$736$	2.80957e12	0.352931
$737$	2.09258e12	0.261264
$738$	0	0
$739$	−2.72124e12	−0.335635	−0.167818	−	0.985818i	$-0.553672\pi$
$739$	−2.72124e12	−0.335635	−0.167818	+	0.985818i	$0.553672\pi$
$740$	0	0
$741$	0	0
$742$	1.34857e12	0.163326
$743$	−5.85311e12	−0.704590	−0.352295	−	0.935889i	$-0.614599\pi$
$743$	−5.85311e12	−0.704590	−0.352295	+	0.935889i	$0.614599\pi$
$744$	0	0
$745$	0	0
$746$	1.56571e12	0.185091
$747$	0	0
$748$	−1.02438e13	−1.19648
$749$	−4.48426e12	−0.520622
$750$	0	0
$751$	−5.13723e12	−0.589317	−0.294659	−	0.955603i	$-0.595206\pi$
$751$	−5.13723e12	−0.589317	−0.294659	+	0.955603i	$0.595206\pi$
$752$	−5.97166e12	−0.680950
$753$	0	0
$754$	3.89203e12	0.438536
$755$	0	0
$756$	0	0
$757$	1.45713e13	1.61275	0.806373	−	0.591407i	$-0.201427\pi$
$757$	1.45713e13	1.61275	0.806373	+	0.591407i	$0.201427\pi$
$758$	1.33607e12	0.147001
$759$	0	0
$760$	0	0
$761$	−1.03936e13	−1.12340	−0.561700	−	0.827341i	$-0.689852\pi$
$761$	−1.03936e13	−1.12340	−0.561700	+	0.827341i	$0.689852\pi$
$762$	0	0
$763$	1.05905e13	1.13125
$764$	−4.33944e12	−0.460801
$765$	0	0
$766$	−5.76626e11	−0.0605152
$767$	9.87591e12	1.03038
$768$	0	0
$769$	−3.91664e12	−0.403873	−0.201936	−	0.979399i	$-0.564723\pi$
$769$	−3.91664e12	−0.403873	−0.201936	+	0.979399i	$0.564723\pi$
$770$	0	0
$771$	0	0
$772$	−9.15851e12	−0.927998
$773$	−6.36894e12	−0.641592	−0.320796	−	0.947148i	$-0.603950\pi$
$773$	−6.36894e12	−0.641592	−0.320796	+	0.947148i	$0.603950\pi$
$774$	0	0
$775$	0	0
$776$	1.49786e11	0.0148284
$777$	0	0
$778$	6.33800e12	0.620217
$779$	1.11529e13	1.08510
$780$	0	0
$781$	8.77802e12	0.844242
$782$	2.10824e12	0.201600
$783$	0	0
$784$	−3.75489e12	−0.354956
$785$	0	0
$786$	0	0
$787$	−6.55343e11	−0.0608951	−0.0304475	−	0.999536i	$-0.509693\pi$
$787$	−6.55343e11	−0.0608951	−0.0304475	+	0.999536i	$0.509693\pi$
$788$	1.13089e13	1.04485
$789$	0	0
$790$	0	0
$791$	−4.73265e12	−0.429843
$792$	0	0
$793$	5.30382e12	0.476277
$794$	−1.17830e12	−0.105212
$795$	0	0
$796$	−3.61418e12	−0.319081
$797$	3.83799e12	0.336931	0.168466	−	0.985708i	$-0.446119\pi$
$797$	3.83799e12	0.336931	0.168466	+	0.985708i	$0.446119\pi$
$798$	0	0
$799$	−1.75962e13	−1.52742
$800$	0	0
$801$	0	0
$802$	−2.57681e12	−0.219937
$803$	−1.90456e13	−1.61650
$804$	0	0
$805$	0	0
$806$	3.12097e12	0.260484
$807$	0	0
$808$	1.66404e12	0.137345
$809$	−2.67581e12	−0.219627	−0.109814	−	0.993952i	$-0.535025\pi$
$809$	−2.67581e12	−0.219627	−0.109814	+	0.993952i	$0.535025\pi$
$810$	0	0
$811$	−2.32586e13	−1.88795	−0.943973	−	0.330023i	$-0.892943\pi$
$811$	−2.32586e13	−1.88795	−0.943973	+	0.330023i	$0.892943\pi$
$812$	−7.97489e12	−0.643758
$813$	0	0
$814$	−5.51959e12	−0.440653
$815$	0	0
$816$	0	0
$817$	−6.45270e12	−0.506690
$818$	3.17138e12	0.247662
$819$	0	0
$820$	0	0
$821$	−1.65740e13	−1.27316	−0.636580	−	0.771211i	$-0.719651\pi$
$821$	−1.65740e13	−1.27316	−0.636580	+	0.771211i	$0.719651\pi$
$822$	0	0
$823$	−1.37332e13	−1.04345	−0.521726	−	0.853113i	$-0.674712\pi$
$823$	−1.37332e13	−1.04345	−0.521726	+	0.853113i	$0.674712\pi$
$824$	5.74644e12	0.434236
$825$	0	0
$826$	2.89086e12	0.216081
$827$	1.80036e13	1.33840	0.669198	−	0.743085i	$-0.266638\pi$
$827$	1.80036e13	1.33840	0.669198	+	0.743085i	$0.266638\pi$
$828$	0	0
$829$	2.11116e13	1.55248	0.776240	−	0.630438i	$-0.217125\pi$
$829$	2.11116e13	1.55248	0.776240	+	0.630438i	$0.217125\pi$
$830$	0	0
$831$	0	0
$832$	−5.07536e12	−0.367208
$833$	−1.10642e13	−0.796191
$834$	0	0
$835$	0	0
$836$	2.08821e13	1.47858
$837$	0	0
$838$	−9.85294e12	−0.690188
$839$	3.47988e12	0.242458	0.121229	−	0.992625i	$-0.461317\pi$
$839$	3.47988e12	0.242458	0.121229	+	0.992625i	$0.461317\pi$
$840$	0	0
$841$	3.10254e12	0.213863
$842$	9.40003e12	0.644503
$843$	0	0
$844$	3.60456e12	0.244518
$845$	0	0
$846$	0	0
$847$	9.44984e11	0.0630883
$848$	6.67354e12	0.443175
$849$	0	0
$850$	0	0
$851$	−7.95175e12	−0.519733
$852$	0	0
$853$	−1.12686e13	−0.728784	−0.364392	−	0.931246i	$-0.618723\pi$
$853$	−1.12686e13	−0.728784	−0.364392	+	0.931246i	$0.618723\pi$
$854$	1.55253e12	0.0998800
$855$	0	0
$856$	8.11860e12	0.516832
$857$	3.75227e12	0.237618	0.118809	−	0.992917i	$-0.462092\pi$
$857$	3.75227e12	0.237618	0.118809	+	0.992917i	$0.462092\pi$
$858$	0	0
$859$	1.57014e13	0.983944	0.491972	−	0.870611i	$-0.336276\pi$
$859$	1.57014e13	0.983944	0.491972	+	0.870611i	$0.336276\pi$
$860$	0	0
$861$	0	0
$862$	−1.85862e12	−0.114659
$863$	−1.39316e13	−0.854975	−0.427487	−	0.904021i	$-0.640601\pi$
$863$	−1.39316e13	−0.854975	−0.427487	+	0.904021i	$0.640601\pi$
$864$	0	0
$865$	0	0
$866$	−1.21243e12	−0.0732531
$867$	0	0
$868$	−6.39495e12	−0.382383
$869$	4.39165e12	0.261239
$870$	0	0
$871$	5.25021e12	0.309097
$872$	−1.91738e13	−1.12301
$873$	0	0
$874$	−4.29767e12	−0.249133
$875$	0	0
$876$	0	0
$877$	1.59832e13	0.912357	0.456179	−	0.889888i	$-0.349218\pi$
$877$	1.59832e13	0.912357	0.456179	+	0.889888i	$0.349218\pi$
$878$	4.17108e12	0.236877
$879$	0	0
$880$	0	0
$881$	3.16227e13	1.76851	0.884255	−	0.467005i	$-0.154667\pi$
$881$	3.16227e13	1.76851	0.884255	+	0.467005i	$0.154667\pi$
$882$	0	0
$883$	1.48526e13	0.822202	0.411101	−	0.911590i	$-0.365144\pi$
$883$	1.48526e13	0.822202	0.411101	+	0.911590i	$0.365144\pi$
$884$	−2.57013e13	−1.41553
$885$	0	0
$886$	−8.81356e12	−0.480507
$887$	−9.39325e12	−0.509518	−0.254759	−	0.967005i	$-0.581996\pi$
$887$	−9.39325e12	−0.509518	−0.254759	+	0.967005i	$0.581996\pi$
$888$	0	0
$889$	1.32373e12	0.0710792
$890$	0	0
$891$	0	0
$892$	1.42741e13	0.754930
$893$	3.58699e13	1.88755
$894$	0	0
$895$	0	0
$896$	−1.29439e13	−0.670930
$897$	0	0
$898$	4.68538e12	0.240437
$899$	1.41210e13	0.721018
$900$	0	0
$901$	1.96643e13	0.994071
$902$	4.08710e12	0.205582
$903$	0	0
$904$	8.56830e12	0.426714
$905$	0	0
$906$	0	0
$907$	−2.64429e13	−1.29741	−0.648704	−	0.761041i	$-0.724689\pi$
$907$	−2.64429e13	−1.29741	−0.648704	+	0.761041i	$0.724689\pi$
$908$	−1.35574e13	−0.661898
$909$	0	0
$910$	0	0
$911$	3.41645e13	1.64339	0.821697	−	0.569924i	$-0.193028\pi$
$911$	3.41645e13	1.64339	0.821697	+	0.569924i	$0.193028\pi$
$912$	0	0
$913$	1.20929e13	0.575987
$914$	−4.56643e12	−0.216431
$915$	0	0
$916$	−9.23333e12	−0.433340
$917$	−1.19498e13	−0.558081
$918$	0	0
$919$	3.14366e13	1.45384	0.726920	−	0.686723i	$-0.240951\pi$
$919$	3.14366e13	1.45384	0.726920	+	0.686723i	$0.240951\pi$
$920$	0	0
$921$	0	0
$922$	−6.08223e12	−0.277188
$923$	2.20237e13	0.998809
$924$	0	0
$925$	0	0
$926$	−5.71431e12	−0.255396
$927$	0	0
$928$	2.21387e13	0.979909
$929$	8.77007e12	0.386307	0.193153	−	0.981169i	$-0.438128\pi$
$929$	8.77007e12	0.386307	0.193153	+	0.981169i	$0.438128\pi$
$930$	0	0
$931$	2.25545e13	0.983918
$932$	−1.62057e12	−0.0703553
$933$	0	0
$934$	−6.26017e12	−0.269169
$935$	0	0
$936$	0	0
$937$	1.66294e13	0.704770	0.352385	−	0.935855i	$-0.385371\pi$
$937$	1.66294e13	0.704770	0.352385	+	0.935855i	$0.385371\pi$
$938$	1.53683e12	0.0648206
$939$	0	0
$940$	0	0
$941$	7.56052e12	0.314339	0.157169	−	0.987572i	$-0.449763\pi$
$941$	7.56052e12	0.314339	0.157169	+	0.987572i	$0.449763\pi$
$942$	0	0
$943$	5.88806e12	0.242476
$944$	1.43057e13	0.586322
$945$	0	0
$946$	−2.36466e12	−0.0959974
$947$	3.40925e13	1.37747	0.688737	−	0.725011i	$-0.258165\pi$
$947$	3.40925e13	1.37747	0.688737	+	0.725011i	$0.258165\pi$
$948$	0	0
$949$	−4.77846e13	−1.91245
$950$	0	0
$951$	0	0
$952$	−1.61212e13	−0.636110
$953$	3.45885e13	1.35836	0.679178	−	0.733973i	$-0.262336\pi$
$953$	3.45885e13	1.35836	0.679178	+	0.733973i	$0.262336\pi$
$954$	0	0
$955$	0	0
$956$	9.94185e11	0.0384952
$957$	0	0
$958$	−5.56591e12	−0.213497
$959$	3.03435e13	1.15846
$960$	0	0
$961$	−1.51162e13	−0.571726
$962$	−1.38484e13	−0.521329
$963$	0	0
$964$	−1.15574e13	−0.431035
$965$	0	0
$966$	0	0
$967$	−4.59891e13	−1.69136	−0.845680	−	0.533691i	$-0.820805\pi$
$967$	−4.59891e13	−1.69136	−0.845680	+	0.533691i	$0.820805\pi$
$968$	−1.71086e12	−0.0626290
$969$	0	0
$970$	0	0
$971$	−2.72663e13	−0.984327	−0.492164	−	0.870503i	$-0.663794\pi$
$971$	−2.72663e13	−0.984327	−0.492164	+	0.870503i	$0.663794\pi$
$972$	0	0
$973$	−1.07035e13	−0.382842
$974$	−1.32156e13	−0.470515
$975$	0	0
$976$	7.68284e12	0.271018
$977$	−4.92228e13	−1.72839	−0.864193	−	0.503160i	$-0.832171\pi$
$977$	−4.92228e13	−1.72839	−0.864193	+	0.503160i	$0.832171\pi$
$978$	0	0
$979$	9.21324e11	0.0320546
$980$	0	0
$981$	0	0
$982$	9.59909e12	0.329404
$983$	3.98337e13	1.36069	0.680346	−	0.732891i	$-0.261830\pi$
$983$	3.98337e13	1.36069	0.680346	+	0.732891i	$0.261830\pi$
$984$	0	0
$985$	0	0
$986$	1.66124e13	0.559740
$987$	0	0
$988$	5.23924e13	1.74929
$989$	−3.40664e12	−0.113225
$990$	0	0
$991$	−2.56388e13	−0.844434	−0.422217	−	0.906495i	$-0.638748\pi$
$991$	−2.56388e13	−0.844434	−0.422217	+	0.906495i	$0.638748\pi$
$992$	1.77527e13	0.582052
$993$	0	0
$994$	6.44674e12	0.209460
$995$	0	0
$996$	0	0
$997$	4.97241e13	1.59382	0.796909	−	0.604099i	$-0.206467\pi$
$997$	4.97241e13	1.59382	0.796909	+	0.604099i	$0.206467\pi$
$998$	−1.14517e13	−0.365410
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.10.a.b.1.1		1
3.2	odd	2		25.10.a.a.1.1		1
5.2	odd	4		225.10.b.d.199.1		2
5.3	odd	4		225.10.b.d.199.2		2
5.4	even	2		45.10.a.c.1.1		1
12.11	even	2		400.10.a.c.1.1		1
15.2	even	4		25.10.b.a.24.2		2
15.8	even	4		25.10.b.a.24.1		2
15.14	odd	2		5.10.a.a.1.1	✓	1
60.23	odd	4		400.10.c.e.49.1		2
60.47	odd	4		400.10.c.e.49.2		2
60.59	even	2		80.10.a.d.1.1		1
105.104	even	2		245.10.a.a.1.1		1
120.29	odd	2		320.10.a.h.1.1		1
120.59	even	2		320.10.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.10.a.a.1.1	✓	1	15.14	odd	2
25.10.a.a.1.1		1	3.2	odd	2
25.10.b.a.24.1		2	15.8	even	4
25.10.b.a.24.2		2	15.2	even	4
45.10.a.c.1.1		1	5.4	even	2
80.10.a.d.1.1		1	60.59	even	2
225.10.a.b.1.1		1	1.1	even	1	trivial
225.10.b.d.199.1		2	5.2	odd	4
225.10.b.d.199.2		2	5.3	odd	4
245.10.a.a.1.1		1	105.104	even	2
320.10.a.c.1.1		1	120.59	even	2
320.10.a.h.1.1		1	120.29	odd	2
400.10.a.c.1.1		1	12.11	even	2
400.10.c.e.49.1		2	60.23	odd	4
400.10.c.e.49.2		2	60.47	odd	4

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:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	225.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more