LMFDB - Embedded newform 2574.2.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2574,2,Mod(1,2574)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2574, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2574.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2574 = 2 \cdot 3^{2} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2574.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:	$20.5534934803$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 858)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2574.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{4} +3.00000 q^{5} +5.00000 q^{7} -1.00000 q^{8} +O(q^{10})$

$q-1.00000 q^{2} +1.00000 q^{4} +3.00000 q^{5} +5.00000 q^{7} -1.00000 q^{8} -3.00000 q^{10} +1.00000 q^{11} +1.00000 q^{13} -5.00000 q^{14} +1.00000 q^{16} +2.00000 q^{19} +3.00000 q^{20} -1.00000 q^{22} -3.00000 q^{23} +4.00000 q^{25} -1.00000 q^{26} +5.00000 q^{28} +3.00000 q^{29} +8.00000 q^{31} -1.00000 q^{32} +15.0000 q^{35} -10.0000 q^{37} -2.00000 q^{38} -3.00000 q^{40} +3.00000 q^{41} +5.00000 q^{43} +1.00000 q^{44} +3.00000 q^{46} -12.0000 q^{47} +18.0000 q^{49} -4.00000 q^{50} +1.00000 q^{52} +6.00000 q^{53} +3.00000 q^{55} -5.00000 q^{56} -3.00000 q^{58} +9.00000 q^{59} -13.0000 q^{61} -8.00000 q^{62} +1.00000 q^{64} +3.00000 q^{65} +5.00000 q^{67} -15.0000 q^{70} -6.00000 q^{71} -7.00000 q^{73} +10.0000 q^{74} +2.00000 q^{76} +5.00000 q^{77} -10.0000 q^{79} +3.00000 q^{80} -3.00000 q^{82} -6.00000 q^{83} -5.00000 q^{86} -1.00000 q^{88} +5.00000 q^{91} -3.00000 q^{92} +12.0000 q^{94} +6.00000 q^{95} -10.0000 q^{97} -18.0000 q^{98} +O(q^{100})$

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	0	0
$7$	5.00000	1.88982	0.944911	−	0.327327i	$-0.106148\pi$
$7$	5.00000	1.88982	0.944911	+	0.327327i	$0.106148\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−3.00000	−0.948683
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−5.00000	−1.33631
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	3.00000	0.670820
$21$	0	0
$22$	−1.00000	−0.213201
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	−1.00000	−0.196116
$27$	0	0
$28$	5.00000	0.944911
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	0	0
$35$	15.0000	2.53546
$36$	0	0
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	−2.00000	−0.324443
$39$	0	0
$40$	−3.00000	−0.474342
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	0	0
$43$	5.00000	0.762493	0.381246	−	0.924473i	$-0.375495\pi$
$43$	5.00000	0.762493	0.381246	+	0.924473i	$0.375495\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	3.00000	0.442326
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	0	0
$49$	18.0000	2.57143
$50$	−4.00000	−0.565685
$51$	0	0
$52$	1.00000	0.138675
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	3.00000	0.404520
$56$	−5.00000	−0.668153
$57$	0	0
$58$	−3.00000	−0.393919
$59$	9.00000	1.17170	0.585850	−	0.810419i	$-0.300761\pi$
$59$	9.00000	1.17170	0.585850	+	0.810419i	$0.300761\pi$
$60$	0	0
$61$	−13.0000	−1.66448	−0.832240	−	0.554416i	$-0.812942\pi$
$61$	−13.0000	−1.66448	−0.832240	+	0.554416i	$0.812942\pi$
$62$	−8.00000	−1.01600
$63$	0	0
$64$	1.00000	0.125000
$65$	3.00000	0.372104
$66$	0	0
$67$	5.00000	0.610847	0.305424	−	0.952217i	$-0.401202\pi$
$67$	5.00000	0.610847	0.305424	+	0.952217i	$0.401202\pi$
$68$	0	0
$69$	0	0
$70$	−15.0000	−1.79284
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	0	0
$73$	−7.00000	−0.819288	−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000	−0.819288	−0.409644	+	0.912245i	$0.634347\pi$
$74$	10.0000	1.16248
$75$	0	0
$76$	2.00000	0.229416
$77$	5.00000	0.569803
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	3.00000	0.335410
$81$	0	0
$82$	−3.00000	−0.331295
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	0	0
$85$	0	0
$86$	−5.00000	−0.539164
$87$	0	0
$88$	−1.00000	−0.106600
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	5.00000	0.524142
$92$	−3.00000	−0.312772
$93$	0	0
$94$	12.0000	1.23771
$95$	6.00000	0.615587
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	−18.0000	−1.81827
$99$	0	0
$100$	4.00000	0.400000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	−19.0000	−1.87213	−0.936063	−	0.351833i	$-0.885559\pi$
$103$	−19.0000	−1.87213	−0.936063	+	0.351833i	$0.885559\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	−6.00000	−0.582772
$107$	3.00000	0.290021	0.145010	−	0.989430i	$-0.453678\pi$
$107$	3.00000	0.290021	0.145010	+	0.989430i	$0.453678\pi$
$108$	0	0
$109$	−16.0000	−1.53252	−0.766261	−	0.642529i	$-0.777885\pi$
$109$	−16.0000	−1.53252	−0.766261	+	0.642529i	$0.777885\pi$
$110$	−3.00000	−0.286039
$111$	0	0
$112$	5.00000	0.472456
$113$	15.0000	1.41108	0.705541	−	0.708669i	$-0.250704\pi$
$113$	15.0000	1.41108	0.705541	+	0.708669i	$0.250704\pi$
$114$	0	0
$115$	−9.00000	−0.839254
$116$	3.00000	0.278543
$117$	0	0
$118$	−9.00000	−0.828517
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	13.0000	1.17696
$123$	0	0
$124$	8.00000	0.718421
$125$	−3.00000	−0.268328
$126$	0	0
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−3.00000	−0.263117
$131$	−15.0000	−1.31056	−0.655278	−	0.755388i	$-0.727449\pi$
$131$	−15.0000	−1.31056	−0.655278	+	0.755388i	$0.727449\pi$
$132$	0	0
$133$	10.0000	0.867110
$134$	−5.00000	−0.431934
$135$	0	0
$136$	0	0
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	0	0
$139$	−16.0000	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	15.0000	1.26773
$141$	0	0
$142$	6.00000	0.503509
$143$	1.00000	0.0836242
$144$	0	0
$145$	9.00000	0.747409
$146$	7.00000	0.579324
$147$	0	0
$148$	−10.0000	−0.821995
$149$	−12.0000	−0.983078	−0.491539	−	0.870855i	$-0.663566\pi$
$149$	−12.0000	−0.983078	−0.491539	+	0.870855i	$0.663566\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	−2.00000	−0.162221
$153$	0	0
$154$	−5.00000	−0.402911
$155$	24.0000	1.92773
$156$	0	0
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	10.0000	0.795557
$159$	0	0
$160$	−3.00000	−0.237171
$161$	−15.0000	−1.18217
$162$	0	0
$163$	−1.00000	−0.0783260	−0.0391630	−	0.999233i	$-0.512469\pi$
$163$	−1.00000	−0.0783260	−0.0391630	+	0.999233i	$0.512469\pi$
$164$	3.00000	0.234261
$165$	0	0
$166$	6.00000	0.465690
$167$	−9.00000	−0.696441	−0.348220	−	0.937413i	$-0.613214\pi$
$167$	−9.00000	−0.696441	−0.348220	+	0.937413i	$0.613214\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	5.00000	0.381246
$173$	9.00000	0.684257	0.342129	−	0.939653i	$-0.388852\pi$
$173$	9.00000	0.684257	0.342129	+	0.939653i	$0.388852\pi$
$174$	0	0
$175$	20.0000	1.51186
$176$	1.00000	0.0753778
$177$	0	0
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	−5.00000	−0.370625
$183$	0	0
$184$	3.00000	0.221163
$185$	−30.0000	−2.20564
$186$	0	0
$187$	0	0
$188$	−12.0000	−0.875190
$189$	0	0
$190$	−6.00000	−0.435286
$191$	3.00000	0.217072	0.108536	−	0.994092i	$-0.465384\pi$
$191$	3.00000	0.217072	0.108536	+	0.994092i	$0.465384\pi$
$192$	0	0
$193$	−22.0000	−1.58359	−0.791797	−	0.610784i	$-0.790854\pi$
$193$	−22.0000	−1.58359	−0.791797	+	0.610784i	$0.790854\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	18.0000	1.28571
$197$	−12.0000	−0.854965	−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000	−0.854965	−0.427482	+	0.904024i	$0.640599\pi$
$198$	0	0
$199$	11.0000	0.779769	0.389885	−	0.920864i	$-0.372515\pi$
$199$	11.0000	0.779769	0.389885	+	0.920864i	$0.372515\pi$
$200$	−4.00000	−0.282843
$201$	0	0
$202$	−6.00000	−0.422159
$203$	15.0000	1.05279
$204$	0	0
$205$	9.00000	0.628587
$206$	19.0000	1.32379
$207$	0	0
$208$	1.00000	0.0693375
$209$	2.00000	0.138343
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	−3.00000	−0.205076
$215$	15.0000	1.02299
$216$	0	0
$217$	40.0000	2.71538
$218$	16.0000	1.08366
$219$	0	0
$220$	3.00000	0.202260
$221$	0	0
$222$	0	0
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	−5.00000	−0.334077
$225$	0	0
$226$	−15.0000	−0.997785
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	0	0
$229$	23.0000	1.51988	0.759941	−	0.649992i	$-0.225228\pi$
$229$	23.0000	1.51988	0.759941	+	0.649992i	$0.225228\pi$
$230$	9.00000	0.593442
$231$	0	0
$232$	−3.00000	−0.196960
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	−36.0000	−2.34838
$236$	9.00000	0.585850
$237$	0	0
$238$	0	0
$239$	−15.0000	−0.970269	−0.485135	−	0.874439i	$-0.661229\pi$
$239$	−15.0000	−0.970269	−0.485135	+	0.874439i	$0.661229\pi$
$240$	0	0
$241$	26.0000	1.67481	0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000	1.67481	0.837404	+	0.546585i	$0.184072\pi$
$242$	−1.00000	−0.0642824
$243$	0	0
$244$	−13.0000	−0.832240
$245$	54.0000	3.44993
$246$	0	0
$247$	2.00000	0.127257
$248$	−8.00000	−0.508001
$249$	0	0
$250$	3.00000	0.189737
$251$	18.0000	1.13615	0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000	1.13615	0.568075	+	0.822977i	$0.307688\pi$
$252$	0	0
$253$	−3.00000	−0.188608
$254$	−2.00000	−0.125491
$255$	0	0
$256$	1.00000	0.0625000
$257$	21.0000	1.30994	0.654972	−	0.755653i	$-0.272680\pi$
$257$	21.0000	1.30994	0.654972	+	0.755653i	$0.272680\pi$
$258$	0	0
$259$	−50.0000	−3.10685
$260$	3.00000	0.186052
$261$	0	0
$262$	15.0000	0.926703
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	0	0
$265$	18.0000	1.10573
$266$	−10.0000	−0.613139
$267$	0	0
$268$	5.00000	0.305424
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	0	0
$273$	0	0
$274$	−18.0000	−1.08742
$275$	4.00000	0.241209
$276$	0	0
$277$	−13.0000	−0.781094	−0.390547	−	0.920583i	$-0.627714\pi$
$277$	−13.0000	−0.781094	−0.390547	+	0.920583i	$0.627714\pi$
$278$	16.0000	0.959616
$279$	0	0
$280$	−15.0000	−0.896421
$281$	9.00000	0.536895	0.268447	−	0.963294i	$-0.413489\pi$
$281$	9.00000	0.536895	0.268447	+	0.963294i	$0.413489\pi$
$282$	0	0
$283$	17.0000	1.01055	0.505273	−	0.862960i	$-0.331392\pi$
$283$	17.0000	1.01055	0.505273	+	0.862960i	$0.331392\pi$
$284$	−6.00000	−0.356034
$285$	0	0
$286$	−1.00000	−0.0591312
$287$	15.0000	0.885422
$288$	0	0
$289$	−17.0000	−1.00000
$290$	−9.00000	−0.528498
$291$	0	0
$292$	−7.00000	−0.409644
$293$	−30.0000	−1.75262	−0.876309	−	0.481749i	$-0.840002\pi$
$293$	−30.0000	−1.75262	−0.876309	+	0.481749i	$0.840002\pi$
$294$	0	0
$295$	27.0000	1.57200
$296$	10.0000	0.581238
$297$	0	0
$298$	12.0000	0.695141
$299$	−3.00000	−0.173494
$300$	0	0
$301$	25.0000	1.44098
$302$	−8.00000	−0.460348
$303$	0	0
$304$	2.00000	0.114708
$305$	−39.0000	−2.23313
$306$	0	0
$307$	2.00000	0.114146	0.0570730	−	0.998370i	$-0.481823\pi$
$307$	2.00000	0.114146	0.0570730	+	0.998370i	$0.481823\pi$
$308$	5.00000	0.284901
$309$	0	0
$310$	−24.0000	−1.36311
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	−13.0000	−0.734803	−0.367402	−	0.930062i	$-0.619753\pi$
$313$	−13.0000	−0.734803	−0.367402	+	0.930062i	$0.619753\pi$
$314$	−14.0000	−0.790066
$315$	0	0
$316$	−10.0000	−0.562544
$317$	9.00000	0.505490	0.252745	−	0.967533i	$-0.418667\pi$
$317$	9.00000	0.505490	0.252745	+	0.967533i	$0.418667\pi$
$318$	0	0
$319$	3.00000	0.167968
$320$	3.00000	0.167705
$321$	0	0
$322$	15.0000	0.835917
$323$	0	0
$324$	0	0
$325$	4.00000	0.221880
$326$	1.00000	0.0553849
$327$	0	0
$328$	−3.00000	−0.165647
$329$	−60.0000	−3.30791
$330$	0	0
$331$	−19.0000	−1.04433	−0.522167	−	0.852843i	$-0.674876\pi$
$331$	−19.0000	−1.04433	−0.522167	+	0.852843i	$0.674876\pi$
$332$	−6.00000	−0.329293
$333$	0	0
$334$	9.00000	0.492458
$335$	15.0000	0.819538
$336$	0	0
$337$	26.0000	1.41631	0.708155	−	0.706057i	$-0.249528\pi$
$337$	26.0000	1.41631	0.708155	+	0.706057i	$0.249528\pi$
$338$	−1.00000	−0.0543928
$339$	0	0
$340$	0	0
$341$	8.00000	0.433224
$342$	0	0
$343$	55.0000	2.96972
$344$	−5.00000	−0.269582
$345$	0	0
$346$	−9.00000	−0.483843
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	0	0
$349$	20.0000	1.07058	0.535288	−	0.844670i	$-0.320203\pi$
$349$	20.0000	1.07058	0.535288	+	0.844670i	$0.320203\pi$
$350$	−20.0000	−1.06904
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	−30.0000	−1.59674	−0.798369	−	0.602168i	$-0.794304\pi$
$353$	−30.0000	−1.59674	−0.798369	+	0.602168i	$0.794304\pi$
$354$	0	0
$355$	−18.0000	−0.955341
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−33.0000	−1.74167	−0.870837	−	0.491572i	$-0.836422\pi$
$359$	−33.0000	−1.74167	−0.870837	+	0.491572i	$0.836422\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	−14.0000	−0.735824
$363$	0	0
$364$	5.00000	0.262071
$365$	−21.0000	−1.09919
$366$	0	0
$367$	−16.0000	−0.835193	−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000	−0.835193	−0.417597	+	0.908633i	$0.637127\pi$
$368$	−3.00000	−0.156386
$369$	0	0
$370$	30.0000	1.55963
$371$	30.0000	1.55752
$372$	0	0
$373$	−13.0000	−0.673114	−0.336557	−	0.941663i	$-0.609263\pi$
$373$	−13.0000	−0.673114	−0.336557	+	0.941663i	$0.609263\pi$
$374$	0	0
$375$	0	0
$376$	12.0000	0.618853
$377$	3.00000	0.154508
$378$	0	0
$379$	8.00000	0.410932	0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000	0.410932	0.205466	+	0.978664i	$0.434129\pi$
$380$	6.00000	0.307794
$381$	0	0
$382$	−3.00000	−0.153493
$383$	18.0000	0.919757	0.459879	−	0.887982i	$-0.347893\pi$
$383$	18.0000	0.919757	0.459879	+	0.887982i	$0.347893\pi$
$384$	0	0
$385$	15.0000	0.764471
$386$	22.0000	1.11977
$387$	0	0
$388$	−10.0000	−0.507673
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	0	0
$391$	0	0
$392$	−18.0000	−0.909137
$393$	0	0
$394$	12.0000	0.604551
$395$	−30.0000	−1.50946
$396$	0	0
$397$	35.0000	1.75660	0.878300	−	0.478110i	$-0.158678\pi$
$397$	35.0000	1.75660	0.878300	+	0.478110i	$0.158678\pi$
$398$	−11.0000	−0.551380
$399$	0	0
$400$	4.00000	0.200000
$401$	36.0000	1.79775	0.898877	−	0.438201i	$-0.144384\pi$
$401$	36.0000	1.79775	0.898877	+	0.438201i	$0.144384\pi$
$402$	0	0
$403$	8.00000	0.398508
$404$	6.00000	0.298511
$405$	0	0
$406$	−15.0000	−0.744438
$407$	−10.0000	−0.495682
$408$	0	0
$409$	23.0000	1.13728	0.568638	−	0.822588i	$-0.307470\pi$
$409$	23.0000	1.13728	0.568638	+	0.822588i	$0.307470\pi$
$410$	−9.00000	−0.444478
$411$	0	0
$412$	−19.0000	−0.936063
$413$	45.0000	2.21431
$414$	0	0
$415$	−18.0000	−0.883585
$416$	−1.00000	−0.0490290
$417$	0	0
$418$	−2.00000	−0.0978232
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	11.0000	0.536107	0.268054	−	0.963404i	$-0.413620\pi$
$421$	11.0000	0.536107	0.268054	+	0.963404i	$0.413620\pi$
$422$	4.00000	0.194717
$423$	0	0
$424$	−6.00000	−0.291386
$425$	0	0
$426$	0	0
$427$	−65.0000	−3.14557
$428$	3.00000	0.145010
$429$	0	0
$430$	−15.0000	−0.723364
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	−19.0000	−0.913082	−0.456541	−	0.889702i	$-0.650912\pi$
$433$	−19.0000	−0.913082	−0.456541	+	0.889702i	$0.650912\pi$
$434$	−40.0000	−1.92006
$435$	0	0
$436$	−16.0000	−0.766261
$437$	−6.00000	−0.287019
$438$	0	0
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	−3.00000	−0.143019
$441$	0	0
$442$	0	0
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	0	0
$445$	0	0
$446$	4.00000	0.189405
$447$	0	0
$448$	5.00000	0.236228
$449$	12.0000	0.566315	0.283158	−	0.959073i	$-0.408618\pi$
$449$	12.0000	0.566315	0.283158	+	0.959073i	$0.408618\pi$
$450$	0	0
$451$	3.00000	0.141264
$452$	15.0000	0.705541
$453$	0	0
$454$	−12.0000	−0.563188
$455$	15.0000	0.703211
$456$	0	0
$457$	11.0000	0.514558	0.257279	−	0.966337i	$-0.417174\pi$
$457$	11.0000	0.514558	0.257279	+	0.966337i	$0.417174\pi$
$458$	−23.0000	−1.07472
$459$	0	0
$460$	−9.00000	−0.419627
$461$	−12.0000	−0.558896	−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000	−0.558896	−0.279448	+	0.960161i	$0.590151\pi$
$462$	0	0
$463$	−22.0000	−1.02243	−0.511213	−	0.859454i	$-0.670804\pi$
$463$	−22.0000	−1.02243	−0.511213	+	0.859454i	$0.670804\pi$
$464$	3.00000	0.139272
$465$	0	0
$466$	−6.00000	−0.277945
$467$	−36.0000	−1.66588	−0.832941	−	0.553362i	$-0.813345\pi$
$467$	−36.0000	−1.66588	−0.832941	+	0.553362i	$0.813345\pi$
$468$	0	0
$469$	25.0000	1.15439
$470$	36.0000	1.66056
$471$	0	0
$472$	−9.00000	−0.414259
$473$	5.00000	0.229900
$474$	0	0
$475$	8.00000	0.367065
$476$	0	0
$477$	0	0
$478$	15.0000	0.686084
$479$	3.00000	0.137073	0.0685367	−	0.997649i	$-0.478167\pi$
$479$	3.00000	0.137073	0.0685367	+	0.997649i	$0.478167\pi$
$480$	0	0
$481$	−10.0000	−0.455961
$482$	−26.0000	−1.18427
$483$	0	0
$484$	1.00000	0.0454545
$485$	−30.0000	−1.36223
$486$	0	0
$487$	38.0000	1.72194	0.860972	−	0.508652i	$-0.169856\pi$
$487$	38.0000	1.72194	0.860972	+	0.508652i	$0.169856\pi$
$488$	13.0000	0.588482
$489$	0	0
$490$	−54.0000	−2.43947
$491$	9.00000	0.406164	0.203082	−	0.979162i	$-0.434904\pi$
$491$	9.00000	0.406164	0.203082	+	0.979162i	$0.434904\pi$
$492$	0	0
$493$	0	0
$494$	−2.00000	−0.0899843
$495$	0	0
$496$	8.00000	0.359211
$497$	−30.0000	−1.34568
$498$	0	0
$499$	23.0000	1.02962	0.514811	−	0.857304i	$-0.327862\pi$
$499$	23.0000	1.02962	0.514811	+	0.857304i	$0.327862\pi$
$500$	−3.00000	−0.134164
$501$	0	0
$502$	−18.0000	−0.803379
$503$	−36.0000	−1.60516	−0.802580	−	0.596544i	$-0.796540\pi$
$503$	−36.0000	−1.60516	−0.802580	+	0.596544i	$0.796540\pi$
$504$	0	0
$505$	18.0000	0.800989
$506$	3.00000	0.133366
$507$	0	0
$508$	2.00000	0.0887357
$509$	42.0000	1.86162	0.930809	−	0.365507i	$-0.119104\pi$
$509$	42.0000	1.86162	0.930809	+	0.365507i	$0.119104\pi$
$510$	0	0
$511$	−35.0000	−1.54831
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−21.0000	−0.926270
$515$	−57.0000	−2.51172
$516$	0	0
$517$	−12.0000	−0.527759
$518$	50.0000	2.19687
$519$	0	0
$520$	−3.00000	−0.131559
$521$	33.0000	1.44576	0.722878	−	0.690976i	$-0.242819\pi$
$521$	33.0000	1.44576	0.722878	+	0.690976i	$0.242819\pi$
$522$	0	0
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	−15.0000	−0.655278
$525$	0	0
$526$	−24.0000	−1.04645
$527$	0	0
$528$	0	0
$529$	−14.0000	−0.608696
$530$	−18.0000	−0.781870
$531$	0	0
$532$	10.0000	0.433555
$533$	3.00000	0.129944
$534$	0	0
$535$	9.00000	0.389104
$536$	−5.00000	−0.215967
$537$	0	0
$538$	0	0
$539$	18.0000	0.775315
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	−8.00000	−0.343629
$543$	0	0
$544$	0	0
$545$	−48.0000	−2.05609
$546$	0	0
$547$	−43.0000	−1.83855	−0.919274	−	0.393619i	$-0.871223\pi$
$547$	−43.0000	−1.83855	−0.919274	+	0.393619i	$0.871223\pi$
$548$	18.0000	0.768922
$549$	0	0
$550$	−4.00000	−0.170561
$551$	6.00000	0.255609
$552$	0	0
$553$	−50.0000	−2.12622
$554$	13.0000	0.552317
$555$	0	0
$556$	−16.0000	−0.678551
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	0	0
$559$	5.00000	0.211477
$560$	15.0000	0.633866
$561$	0	0
$562$	−9.00000	−0.379642
$563$	−24.0000	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$564$	0	0
$565$	45.0000	1.89316
$566$	−17.0000	−0.714563
$567$	0	0
$568$	6.00000	0.251754
$569$	18.0000	0.754599	0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000	0.754599	0.377300	+	0.926091i	$0.376853\pi$
$570$	0	0
$571$	−19.0000	−0.795125	−0.397563	−	0.917575i	$-0.630144\pi$
$571$	−19.0000	−0.795125	−0.397563	+	0.917575i	$0.630144\pi$
$572$	1.00000	0.0418121
$573$	0	0
$574$	−15.0000	−0.626088
$575$	−12.0000	−0.500435
$576$	0	0
$577$	32.0000	1.33218	0.666089	−	0.745873i	$-0.267967\pi$
$577$	32.0000	1.33218	0.666089	+	0.745873i	$0.267967\pi$
$578$	17.0000	0.707107
$579$	0	0
$580$	9.00000	0.373705
$581$	−30.0000	−1.24461
$582$	0	0
$583$	6.00000	0.248495
$584$	7.00000	0.289662
$585$	0	0
$586$	30.0000	1.23929
$587$	15.0000	0.619116	0.309558	−	0.950881i	$-0.399819\pi$
$587$	15.0000	0.619116	0.309558	+	0.950881i	$0.399819\pi$
$588$	0	0
$589$	16.0000	0.659269
$590$	−27.0000	−1.11157
$591$	0	0
$592$	−10.0000	−0.410997
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	0	0
$595$	0	0
$596$	−12.0000	−0.491539
$597$	0	0
$598$	3.00000	0.122679
$599$	−15.0000	−0.612883	−0.306442	−	0.951889i	$-0.599138\pi$
$599$	−15.0000	−0.612883	−0.306442	+	0.951889i	$0.599138\pi$
$600$	0	0
$601$	20.0000	0.815817	0.407909	−	0.913023i	$-0.366258\pi$
$601$	20.0000	0.815817	0.407909	+	0.913023i	$0.366258\pi$
$602$	−25.0000	−1.01892
$603$	0	0
$604$	8.00000	0.325515
$605$	3.00000	0.121967
$606$	0	0
$607$	−22.0000	−0.892952	−0.446476	−	0.894795i	$-0.647321\pi$
$607$	−22.0000	−0.892952	−0.446476	+	0.894795i	$0.647321\pi$
$608$	−2.00000	−0.0811107
$609$	0	0
$610$	39.0000	1.57906
$611$	−12.0000	−0.485468
$612$	0	0
$613$	−16.0000	−0.646234	−0.323117	−	0.946359i	$-0.604731\pi$
$613$	−16.0000	−0.646234	−0.323117	+	0.946359i	$0.604731\pi$
$614$	−2.00000	−0.0807134
$615$	0	0
$616$	−5.00000	−0.201456
$617$	−30.0000	−1.20775	−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−30.0000	−1.20775	−0.603877	+	0.797077i	$0.706378\pi$
$618$	0	0
$619$	5.00000	0.200967	0.100483	−	0.994939i	$-0.467961\pi$
$619$	5.00000	0.200967	0.100483	+	0.994939i	$0.467961\pi$
$620$	24.0000	0.963863
$621$	0	0
$622$	24.0000	0.962312
$623$	0	0
$624$	0	0
$625$	−29.0000	−1.16000
$626$	13.0000	0.519584
$627$	0	0
$628$	14.0000	0.558661
$629$	0	0
$630$	0	0
$631$	−10.0000	−0.398094	−0.199047	−	0.979990i	$-0.563785\pi$
$631$	−10.0000	−0.398094	−0.199047	+	0.979990i	$0.563785\pi$
$632$	10.0000	0.397779
$633$	0	0
$634$	−9.00000	−0.357436
$635$	6.00000	0.238103
$636$	0	0
$637$	18.0000	0.713186
$638$	−3.00000	−0.118771
$639$	0	0
$640$	−3.00000	−0.118585
$641$	33.0000	1.30342	0.651711	−	0.758468i	$-0.274052\pi$
$641$	33.0000	1.30342	0.651711	+	0.758468i	$0.274052\pi$
$642$	0	0
$643$	−4.00000	−0.157745	−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000	−0.157745	−0.0788723	+	0.996885i	$0.525132\pi$
$644$	−15.0000	−0.591083
$645$	0	0
$646$	0	0
$647$	−24.0000	−0.943537	−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000	−0.943537	−0.471769	+	0.881722i	$0.656384\pi$
$648$	0	0
$649$	9.00000	0.353281
$650$	−4.00000	−0.156893
$651$	0	0
$652$	−1.00000	−0.0391630
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	0	0
$655$	−45.0000	−1.75830
$656$	3.00000	0.117130
$657$	0	0
$658$	60.0000	2.33904
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	19.0000	0.738456
$663$	0	0
$664$	6.00000	0.232845
$665$	30.0000	1.16335
$666$	0	0
$667$	−9.00000	−0.348481
$668$	−9.00000	−0.348220
$669$	0	0
$670$	−15.0000	−0.579501
$671$	−13.0000	−0.501859
$672$	0	0
$673$	44.0000	1.69608	0.848038	−	0.529936i	$-0.177784\pi$
$673$	44.0000	1.69608	0.848038	+	0.529936i	$0.177784\pi$
$674$	−26.0000	−1.00148
$675$	0	0
$676$	1.00000	0.0384615
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	0	0
$679$	−50.0000	−1.91882
$680$	0	0
$681$	0	0
$682$	−8.00000	−0.306336
$683$	−39.0000	−1.49229	−0.746147	−	0.665782i	$-0.768098\pi$
$683$	−39.0000	−1.49229	−0.746147	+	0.665782i	$0.768098\pi$
$684$	0	0
$685$	54.0000	2.06323
$686$	−55.0000	−2.09991
$687$	0	0
$688$	5.00000	0.190623
$689$	6.00000	0.228582
$690$	0	0
$691$	−16.0000	−0.608669	−0.304334	−	0.952565i	$-0.598434\pi$
$691$	−16.0000	−0.608669	−0.304334	+	0.952565i	$0.598434\pi$
$692$	9.00000	0.342129
$693$	0	0
$694$	12.0000	0.455514
$695$	−48.0000	−1.82074
$696$	0	0
$697$	0	0
$698$	−20.0000	−0.757011
$699$	0	0
$700$	20.0000	0.755929
$701$	−9.00000	−0.339925	−0.169963	−	0.985451i	$-0.554365\pi$
$701$	−9.00000	−0.339925	−0.169963	+	0.985451i	$0.554365\pi$
$702$	0	0
$703$	−20.0000	−0.754314
$704$	1.00000	0.0376889
$705$	0	0
$706$	30.0000	1.12906
$707$	30.0000	1.12827
$708$	0	0
$709$	−1.00000	−0.0375558	−0.0187779	−	0.999824i	$-0.505978\pi$
$709$	−1.00000	−0.0375558	−0.0187779	+	0.999824i	$0.505978\pi$
$710$	18.0000	0.675528
$711$	0	0
$712$	0	0
$713$	−24.0000	−0.898807
$714$	0	0
$715$	3.00000	0.112194
$716$	0	0
$717$	0	0
$718$	33.0000	1.23155
$719$	51.0000	1.90198	0.950990	−	0.309223i	$-0.100069\pi$
$719$	51.0000	1.90198	0.950990	+	0.309223i	$0.100069\pi$
$720$	0	0
$721$	−95.0000	−3.53798
$722$	15.0000	0.558242
$723$	0	0
$724$	14.0000	0.520306
$725$	12.0000	0.445669
$726$	0	0
$727$	44.0000	1.63187	0.815935	−	0.578144i	$-0.196223\pi$
$727$	44.0000	1.63187	0.815935	+	0.578144i	$0.196223\pi$
$728$	−5.00000	−0.185312
$729$	0	0
$730$	21.0000	0.777245
$731$	0	0
$732$	0	0
$733$	−34.0000	−1.25582	−0.627909	−	0.778287i	$-0.716089\pi$
$733$	−34.0000	−1.25582	−0.627909	+	0.778287i	$0.716089\pi$
$734$	16.0000	0.590571
$735$	0	0
$736$	3.00000	0.110581
$737$	5.00000	0.184177
$738$	0	0
$739$	−4.00000	−0.147142	−0.0735712	−	0.997290i	$-0.523440\pi$
$739$	−4.00000	−0.147142	−0.0735712	+	0.997290i	$0.523440\pi$
$740$	−30.0000	−1.10282
$741$	0	0
$742$	−30.0000	−1.10133
$743$	−33.0000	−1.21065	−0.605326	−	0.795977i	$-0.706957\pi$
$743$	−33.0000	−1.21065	−0.605326	+	0.795977i	$0.706957\pi$
$744$	0	0
$745$	−36.0000	−1.31894
$746$	13.0000	0.475964
$747$	0	0
$748$	0	0
$749$	15.0000	0.548088
$750$	0	0
$751$	35.0000	1.27717	0.638584	−	0.769552i	$-0.279520\pi$
$751$	35.0000	1.27717	0.638584	+	0.769552i	$0.279520\pi$
$752$	−12.0000	−0.437595
$753$	0	0
$754$	−3.00000	−0.109254
$755$	24.0000	0.873449
$756$	0	0
$757$	26.0000	0.944986	0.472493	−	0.881334i	$-0.343354\pi$
$757$	26.0000	0.944986	0.472493	+	0.881334i	$0.343354\pi$
$758$	−8.00000	−0.290573
$759$	0	0
$760$	−6.00000	−0.217643
$761$	−15.0000	−0.543750	−0.271875	−	0.962333i	$-0.587644\pi$
$761$	−15.0000	−0.543750	−0.271875	+	0.962333i	$0.587644\pi$
$762$	0	0
$763$	−80.0000	−2.89619
$764$	3.00000	0.108536
$765$	0	0
$766$	−18.0000	−0.650366
$767$	9.00000	0.324971
$768$	0	0
$769$	−31.0000	−1.11789	−0.558944	−	0.829205i	$-0.688793\pi$
$769$	−31.0000	−1.11789	−0.558944	+	0.829205i	$0.688793\pi$
$770$	−15.0000	−0.540562
$771$	0	0
$772$	−22.0000	−0.791797
$773$	6.00000	0.215805	0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000	0.215805	0.107903	+	0.994161i	$0.465587\pi$
$774$	0	0
$775$	32.0000	1.14947
$776$	10.0000	0.358979
$777$	0	0
$778$	18.0000	0.645331
$779$	6.00000	0.214972
$780$	0	0
$781$	−6.00000	−0.214697
$782$	0	0
$783$	0	0
$784$	18.0000	0.642857
$785$	42.0000	1.49904
$786$	0	0
$787$	8.00000	0.285169	0.142585	−	0.989783i	$-0.454459\pi$
$787$	8.00000	0.285169	0.142585	+	0.989783i	$0.454459\pi$
$788$	−12.0000	−0.427482
$789$	0	0
$790$	30.0000	1.06735
$791$	75.0000	2.66669
$792$	0	0
$793$	−13.0000	−0.461644
$794$	−35.0000	−1.24210
$795$	0	0
$796$	11.0000	0.389885
$797$	−36.0000	−1.27519	−0.637593	−	0.770374i	$-0.720070\pi$
$797$	−36.0000	−1.27519	−0.637593	+	0.770374i	$0.720070\pi$
$798$	0	0
$799$	0	0
$800$	−4.00000	−0.141421
$801$	0	0
$802$	−36.0000	−1.27120
$803$	−7.00000	−0.247025
$804$	0	0
$805$	−45.0000	−1.58604
$806$	−8.00000	−0.281788
$807$	0	0
$808$	−6.00000	−0.211079
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	0	0
$811$	−28.0000	−0.983213	−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000	−0.983213	−0.491606	+	0.870817i	$0.663590\pi$
$812$	15.0000	0.526397
$813$	0	0
$814$	10.0000	0.350500
$815$	−3.00000	−0.105085
$816$	0	0
$817$	10.0000	0.349856
$818$	−23.0000	−0.804176
$819$	0	0
$820$	9.00000	0.314294
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	0	0
$823$	−49.0000	−1.70803	−0.854016	−	0.520246i	$-0.825840\pi$
$823$	−49.0000	−1.70803	−0.854016	+	0.520246i	$0.825840\pi$
$824$	19.0000	0.661896
$825$	0	0
$826$	−45.0000	−1.56575
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	0	0
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	18.0000	0.624789
$831$	0	0
$832$	1.00000	0.0346688
$833$	0	0
$834$	0	0
$835$	−27.0000	−0.934374
$836$	2.00000	0.0691714
$837$	0	0
$838$	12.0000	0.414533
$839$	−42.0000	−1.45000	−0.725001	−	0.688748i	$-0.758161\pi$
$839$	−42.0000	−1.45000	−0.725001	+	0.688748i	$0.758161\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	−11.0000	−0.379085
$843$	0	0
$844$	−4.00000	−0.137686
$845$	3.00000	0.103203
$846$	0	0
$847$	5.00000	0.171802
$848$	6.00000	0.206041
$849$	0	0
$850$	0	0
$851$	30.0000	1.02839
$852$	0	0
$853$	−16.0000	−0.547830	−0.273915	−	0.961754i	$-0.588319\pi$
$853$	−16.0000	−0.547830	−0.273915	+	0.961754i	$0.588319\pi$
$854$	65.0000	2.22425
$855$	0	0
$856$	−3.00000	−0.102538
$857$	−12.0000	−0.409912	−0.204956	−	0.978771i	$-0.565705\pi$
$857$	−12.0000	−0.409912	−0.204956	+	0.978771i	$0.565705\pi$
$858$	0	0
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	15.0000	0.511496
$861$	0	0
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	27.0000	0.918028
$866$	19.0000	0.645646
$867$	0	0
$868$	40.0000	1.35769
$869$	−10.0000	−0.339227
$870$	0	0
$871$	5.00000	0.169419
$872$	16.0000	0.541828
$873$	0	0
$874$	6.00000	0.202953
$875$	−15.0000	−0.507093
$876$	0	0
$877$	−16.0000	−0.540282	−0.270141	−	0.962821i	$-0.587070\pi$
$877$	−16.0000	−0.540282	−0.270141	+	0.962821i	$0.587070\pi$
$878$	−8.00000	−0.269987
$879$	0	0
$880$	3.00000	0.101130
$881$	27.0000	0.909653	0.454827	−	0.890580i	$-0.349701\pi$
$881$	27.0000	0.909653	0.454827	+	0.890580i	$0.349701\pi$
$882$	0	0
$883$	2.00000	0.0673054	0.0336527	−	0.999434i	$-0.489286\pi$
$883$	2.00000	0.0673054	0.0336527	+	0.999434i	$0.489286\pi$
$884$	0	0
$885$	0	0
$886$	12.0000	0.403148
$887$	−36.0000	−1.20876	−0.604381	−	0.796696i	$-0.706579\pi$
$887$	−36.0000	−1.20876	−0.604381	+	0.796696i	$0.706579\pi$
$888$	0	0
$889$	10.0000	0.335389
$890$	0	0
$891$	0	0
$892$	−4.00000	−0.133930
$893$	−24.0000	−0.803129
$894$	0	0
$895$	0	0
$896$	−5.00000	−0.167038
$897$	0	0
$898$	−12.0000	−0.400445
$899$	24.0000	0.800445
$900$	0	0
$901$	0	0
$902$	−3.00000	−0.0998891
$903$	0	0
$904$	−15.0000	−0.498893
$905$	42.0000	1.39613
$906$	0	0
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	12.0000	0.398234
$909$	0	0
$910$	−15.0000	−0.497245
$911$	24.0000	0.795155	0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000	0.795155	0.397578	+	0.917568i	$0.369851\pi$
$912$	0	0
$913$	−6.00000	−0.198571
$914$	−11.0000	−0.363848
$915$	0	0
$916$	23.0000	0.759941
$917$	−75.0000	−2.47672
$918$	0	0
$919$	20.0000	0.659739	0.329870	−	0.944027i	$-0.392995\pi$
$919$	20.0000	0.659739	0.329870	+	0.944027i	$0.392995\pi$
$920$	9.00000	0.296721
$921$	0	0
$922$	12.0000	0.395199
$923$	−6.00000	−0.197492
$924$	0	0
$925$	−40.0000	−1.31519
$926$	22.0000	0.722965
$927$	0	0
$928$	−3.00000	−0.0984798
$929$	36.0000	1.18112	0.590561	−	0.806993i	$-0.298907\pi$
$929$	36.0000	1.18112	0.590561	+	0.806993i	$0.298907\pi$
$930$	0	0
$931$	36.0000	1.17985
$932$	6.00000	0.196537
$933$	0	0
$934$	36.0000	1.17796
$935$	0	0
$936$	0	0
$937$	−4.00000	−0.130674	−0.0653372	−	0.997863i	$-0.520812\pi$
$937$	−4.00000	−0.130674	−0.0653372	+	0.997863i	$0.520812\pi$
$938$	−25.0000	−0.816279
$939$	0	0
$940$	−36.0000	−1.17419
$941$	−48.0000	−1.56476	−0.782378	−	0.622804i	$-0.785993\pi$
$941$	−48.0000	−1.56476	−0.782378	+	0.622804i	$0.785993\pi$
$942$	0	0
$943$	−9.00000	−0.293080
$944$	9.00000	0.292925
$945$	0	0
$946$	−5.00000	−0.162564
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	0	0
$949$	−7.00000	−0.227230
$950$	−8.00000	−0.259554
$951$	0	0
$952$	0	0
$953$	48.0000	1.55487	0.777436	−	0.628962i	$-0.216520\pi$
$953$	48.0000	1.55487	0.777436	+	0.628962i	$0.216520\pi$
$954$	0	0
$955$	9.00000	0.291233
$956$	−15.0000	−0.485135
$957$	0	0
$958$	−3.00000	−0.0969256
$959$	90.0000	2.90625
$960$	0	0
$961$	33.0000	1.06452
$962$	10.0000	0.322413
$963$	0	0
$964$	26.0000	0.837404
$965$	−66.0000	−2.12462
$966$	0	0
$967$	−37.0000	−1.18984	−0.594920	−	0.803785i	$-0.702816\pi$
$967$	−37.0000	−1.18984	−0.594920	+	0.803785i	$0.702816\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	30.0000	0.963242
$971$	30.0000	0.962746	0.481373	−	0.876516i	$-0.340138\pi$
$971$	30.0000	0.962746	0.481373	+	0.876516i	$0.340138\pi$
$972$	0	0
$973$	−80.0000	−2.56468
$974$	−38.0000	−1.21760
$975$	0	0
$976$	−13.0000	−0.416120
$977$	−12.0000	−0.383914	−0.191957	−	0.981403i	$-0.561483\pi$
$977$	−12.0000	−0.383914	−0.191957	+	0.981403i	$0.561483\pi$
$978$	0	0
$979$	0	0
$980$	54.0000	1.72497
$981$	0	0
$982$	−9.00000	−0.287202
$983$	54.0000	1.72233	0.861166	−	0.508323i	$-0.169735\pi$
$983$	54.0000	1.72233	0.861166	+	0.508323i	$0.169735\pi$
$984$	0	0
$985$	−36.0000	−1.14706
$986$	0	0
$987$	0	0
$988$	2.00000	0.0636285
$989$	−15.0000	−0.476972
$990$	0	0
$991$	−7.00000	−0.222362	−0.111181	−	0.993800i	$-0.535463\pi$
$991$	−7.00000	−0.222362	−0.111181	+	0.993800i	$0.535463\pi$
$992$	−8.00000	−0.254000
$993$	0	0
$994$	30.0000	0.951542
$995$	33.0000	1.04617
$996$	0	0
$997$	53.0000	1.67853	0.839263	−	0.543725i	$-0.182987\pi$
$997$	53.0000	1.67853	0.839263	+	0.543725i	$0.182987\pi$
$998$	−23.0000	−0.728052
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2574.2.a.p.1.1		1
3.2	odd	2		858.2.a.j.1.1	✓	1
12.11	even	2		6864.2.a.a.1.1		1
33.32	even	2		9438.2.a.j.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
858.2.a.j.1.1	✓	1	3.2	odd	2
2574.2.a.p.1.1		1	1.1	even	1	trivial
6864.2.a.a.1.1		1	12.11	even	2
9438.2.a.j.1.1		1	33.32	even	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:	\( N \)	\(=\)	\( 2574 = 2 \cdot 3^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2574.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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