LMFDB - Embedded newform 2574.2.a.i.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)

$f(q)$

$=$

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

$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -3.00000 q^{7} -1.00000 q^{8} -1.00000 q^{10} -1.00000 q^{11} -1.00000 q^{13} +3.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} -2.00000 q^{19} +1.00000 q^{20} +1.00000 q^{22} +1.00000 q^{23} -4.00000 q^{25} +1.00000 q^{26} -3.00000 q^{28} +9.00000 q^{29} -4.00000 q^{31} -1.00000 q^{32} -4.00000 q^{34} -3.00000 q^{35} -6.00000 q^{37} +2.00000 q^{38} -1.00000 q^{40} -1.00000 q^{41} +11.0000 q^{43} -1.00000 q^{44} -1.00000 q^{46} +2.00000 q^{49} +4.00000 q^{50} -1.00000 q^{52} +10.0000 q^{53} -1.00000 q^{55} +3.00000 q^{56} -9.00000 q^{58} +3.00000 q^{59} +5.00000 q^{61} +4.00000 q^{62} +1.00000 q^{64} -1.00000 q^{65} +3.00000 q^{67} +4.00000 q^{68} +3.00000 q^{70} -10.0000 q^{71} +9.00000 q^{73} +6.00000 q^{74} -2.00000 q^{76} +3.00000 q^{77} +10.0000 q^{79} +1.00000 q^{80} +1.00000 q^{82} +6.00000 q^{83} +4.00000 q^{85} -11.0000 q^{86} +1.00000 q^{88} +8.00000 q^{89} +3.00000 q^{91} +1.00000 q^{92} -2.00000 q^{95} +2.00000 q^{97} -2.00000 q^{98} +O(q^{100})$

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
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	0	0
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	3.00000	0.801784
$15$	0	0
$16$	1.00000	0.250000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	1.00000	0.213201
$23$	1.00000	0.208514	0.104257	−	0.994550i	$-0.466753\pi$
$23$	1.00000	0.208514	0.104257	+	0.994550i	$0.466753\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	1.00000	0.196116
$27$	0	0
$28$	−3.00000	−0.566947
$29$	9.00000	1.67126	0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000	1.67126	0.835629	+	0.549294i	$0.185103\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−4.00000	−0.685994
$35$	−3.00000	−0.507093
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	2.00000	0.324443
$39$	0	0
$40$	−1.00000	−0.158114
$41$	−1.00000	−0.156174	−0.0780869	−	0.996947i	$-0.524881\pi$
$41$	−1.00000	−0.156174	−0.0780869	+	0.996947i	$0.524881\pi$
$42$	0	0
$43$	11.0000	1.67748	0.838742	−	0.544529i	$-0.183292\pi$
$43$	11.0000	1.67748	0.838742	+	0.544529i	$0.183292\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−1.00000	−0.147442
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	4.00000	0.565685
$51$	0	0
$52$	−1.00000	−0.138675
$53$	10.0000	1.37361	0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000	1.37361	0.686803	+	0.726844i	$0.259014\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	3.00000	0.400892
$57$	0	0
$58$	−9.00000	−1.18176
$59$	3.00000	0.390567	0.195283	−	0.980747i	$-0.437437\pi$
$59$	3.00000	0.390567	0.195283	+	0.980747i	$0.437437\pi$
$60$	0	0
$61$	5.00000	0.640184	0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000	0.640184	0.320092	+	0.947386i	$0.396286\pi$
$62$	4.00000	0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	−1.00000	−0.124035
$66$	0	0
$67$	3.00000	0.366508	0.183254	−	0.983066i	$-0.441337\pi$
$67$	3.00000	0.366508	0.183254	+	0.983066i	$0.441337\pi$
$68$	4.00000	0.485071
$69$	0	0
$70$	3.00000	0.358569
$71$	−10.0000	−1.18678	−0.593391	−	0.804914i	$-0.702211\pi$
$71$	−10.0000	−1.18678	−0.593391	+	0.804914i	$0.702211\pi$
$72$	0	0
$73$	9.00000	1.05337	0.526685	−	0.850060i	$-0.323435\pi$
$73$	9.00000	1.05337	0.526685	+	0.850060i	$0.323435\pi$
$74$	6.00000	0.697486
$75$	0	0
$76$	−2.00000	−0.229416
$77$	3.00000	0.341882
$78$	0	0
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	1.00000	0.110432
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	4.00000	0.433861
$86$	−11.0000	−1.18616
$87$	0	0
$88$	1.00000	0.106600
$89$	8.00000	0.847998	0.423999	−	0.905663i	$-0.360626\pi$
$89$	8.00000	0.847998	0.423999	+	0.905663i	$0.360626\pi$
$90$	0	0
$91$	3.00000	0.314485
$92$	1.00000	0.104257
$93$	0	0
$94$	0	0
$95$	−2.00000	−0.205196
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	−2.00000	−0.202031
$99$	0	0
$100$	−4.00000	−0.400000
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	0	0
$103$	−11.0000	−1.08386	−0.541931	−	0.840423i	$-0.682307\pi$
$103$	−11.0000	−1.08386	−0.541931	+	0.840423i	$0.682307\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	−10.0000	−0.971286
$107$	−7.00000	−0.676716	−0.338358	−	0.941018i	$-0.609871\pi$
$107$	−7.00000	−0.676716	−0.338358	+	0.941018i	$0.609871\pi$
$108$	0	0
$109$	16.0000	1.53252	0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000	1.53252	0.766261	+	0.642529i	$0.222115\pi$
$110$	1.00000	0.0953463
$111$	0	0
$112$	−3.00000	−0.283473
$113$	−5.00000	−0.470360	−0.235180	−	0.971952i	$-0.575568\pi$
$113$	−5.00000	−0.470360	−0.235180	+	0.971952i	$0.575568\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	9.00000	0.835629
$117$	0	0
$118$	−3.00000	−0.276172
$119$	−12.0000	−1.10004
$120$	0	0
$121$	1.00000	0.0909091
$122$	−5.00000	−0.452679
$123$	0	0
$124$	−4.00000	−0.359211
$125$	−9.00000	−0.804984
$126$	0	0
$127$	−14.0000	−1.24230	−0.621150	−	0.783692i	$-0.713334\pi$
$127$	−14.0000	−1.24230	−0.621150	+	0.783692i	$0.713334\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	1.00000	0.0877058
$131$	3.00000	0.262111	0.131056	−	0.991375i	$-0.458163\pi$
$131$	3.00000	0.262111	0.131056	+	0.991375i	$0.458163\pi$
$132$	0	0
$133$	6.00000	0.520266
$134$	−3.00000	−0.259161
$135$	0	0
$136$	−4.00000	−0.342997
$137$	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
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	−3.00000	−0.253546
$141$	0	0
$142$	10.0000	0.839181
$143$	1.00000	0.0836242
$144$	0	0
$145$	9.00000	0.747409
$146$	−9.00000	−0.744845
$147$	0	0
$148$	−6.00000	−0.493197
$149$	−4.00000	−0.327693	−0.163846	−	0.986486i	$-0.552390\pi$
$149$	−4.00000	−0.327693	−0.163846	+	0.986486i	$0.552390\pi$
$150$	0	0
$151$	24.0000	1.95309	0.976546	−	0.215308i	$-0.0690756\pi$
$151$	24.0000	1.95309	0.976546	+	0.215308i	$0.0690756\pi$
$152$	2.00000	0.162221
$153$	0	0
$154$	−3.00000	−0.241747
$155$	−4.00000	−0.321288
$156$	0	0
$157$	−22.0000	−1.75579	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−22.0000	−1.75579	−0.877896	+	0.478852i	$0.841053\pi$
$158$	−10.0000	−0.795557
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	−3.00000	−0.236433
$162$	0	0
$163$	9.00000	0.704934	0.352467	−	0.935824i	$-0.385343\pi$
$163$	9.00000	0.704934	0.352467	+	0.935824i	$0.385343\pi$
$164$	−1.00000	−0.0780869
$165$	0	0
$166$	−6.00000	−0.465690
$167$	3.00000	0.232147	0.116073	−	0.993241i	$-0.462969\pi$
$167$	3.00000	0.232147	0.116073	+	0.993241i	$0.462969\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−4.00000	−0.306786
$171$	0	0
$172$	11.0000	0.838742
$173$	19.0000	1.44454	0.722272	−	0.691609i	$-0.243098\pi$
$173$	19.0000	1.44454	0.722272	+	0.691609i	$0.243098\pi$
$174$	0	0
$175$	12.0000	0.907115
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	−8.00000	−0.599625
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	−3.00000	−0.222375
$183$	0	0
$184$	−1.00000	−0.0737210
$185$	−6.00000	−0.441129
$186$	0	0
$187$	−4.00000	−0.292509
$188$	0	0
$189$	0	0
$190$	2.00000	0.145095
$191$	15.0000	1.08536	0.542681	−	0.839939i	$-0.317409\pi$
$191$	15.0000	1.08536	0.542681	+	0.839939i	$0.317409\pi$
$192$	0	0
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	2.00000	0.142857
$197$	−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
$199$	3.00000	0.212664	0.106332	−	0.994331i	$-0.466089\pi$
$199$	3.00000	0.212664	0.106332	+	0.994331i	$0.466089\pi$
$200$	4.00000	0.282843
$201$	0	0
$202$	−10.0000	−0.703598
$203$	−27.0000	−1.89503
$204$	0	0
$205$	−1.00000	−0.0698430
$206$	11.0000	0.766406
$207$	0	0
$208$	−1.00000	−0.0693375
$209$	2.00000	0.138343
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	10.0000	0.686803
$213$	0	0
$214$	7.00000	0.478510
$215$	11.0000	0.750194
$216$	0	0
$217$	12.0000	0.814613
$218$	−16.0000	−1.08366
$219$	0	0
$220$	−1.00000	−0.0674200
$221$	−4.00000	−0.269069
$222$	0	0
$223$	−20.0000	−1.33930	−0.669650	−	0.742677i	$-0.733556\pi$
$223$	−20.0000	−1.33930	−0.669650	+	0.742677i	$0.733556\pi$
$224$	3.00000	0.200446
$225$	0	0
$226$	5.00000	0.332595
$227$	−8.00000	−0.530979	−0.265489	−	0.964114i	$-0.585534\pi$
$227$	−8.00000	−0.530979	−0.265489	+	0.964114i	$0.585534\pi$
$228$	0	0
$229$	25.0000	1.65205	0.826023	−	0.563636i	$-0.190598\pi$
$229$	25.0000	1.65205	0.826023	+	0.563636i	$0.190598\pi$
$230$	−1.00000	−0.0659380
$231$	0	0
$232$	−9.00000	−0.590879
$233$	−14.0000	−0.917170	−0.458585	−	0.888650i	$-0.651644\pi$
$233$	−14.0000	−0.917170	−0.458585	+	0.888650i	$0.651644\pi$
$234$	0	0
$235$	0	0
$236$	3.00000	0.195283
$237$	0	0
$238$	12.0000	0.777844
$239$	5.00000	0.323423	0.161712	−	0.986838i	$-0.448299\pi$
$239$	5.00000	0.323423	0.161712	+	0.986838i	$0.448299\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	−1.00000	−0.0642824
$243$	0	0
$244$	5.00000	0.320092
$245$	2.00000	0.127775
$246$	0	0
$247$	2.00000	0.127257
$248$	4.00000	0.254000
$249$	0	0
$250$	9.00000	0.569210
$251$	6.00000	0.378717	0.189358	−	0.981908i	$-0.439359\pi$
$251$	6.00000	0.378717	0.189358	+	0.981908i	$0.439359\pi$
$252$	0	0
$253$	−1.00000	−0.0628695
$254$	14.0000	0.878438
$255$	0	0
$256$	1.00000	0.0625000
$257$	25.0000	1.55946	0.779729	−	0.626118i	$-0.215357\pi$
$257$	25.0000	1.55946	0.779729	+	0.626118i	$0.215357\pi$
$258$	0	0
$259$	18.0000	1.11847
$260$	−1.00000	−0.0620174
$261$	0	0
$262$	−3.00000	−0.185341
$263$	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
$265$	10.0000	0.614295
$266$	−6.00000	−0.367884
$267$	0	0
$268$	3.00000	0.183254
$269$	4.00000	0.243884	0.121942	−	0.992537i	$-0.461088\pi$
$269$	4.00000	0.243884	0.121942	+	0.992537i	$0.461088\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	4.00000	0.242536
$273$	0	0
$274$	−18.0000	−1.08742
$275$	4.00000	0.241209
$276$	0	0
$277$	5.00000	0.300421	0.150210	−	0.988654i	$-0.452005\pi$
$277$	5.00000	0.300421	0.150210	+	0.988654i	$0.452005\pi$
$278$	0	0
$279$	0	0
$280$	3.00000	0.179284
$281$	29.0000	1.72999	0.864997	−	0.501776i	$-0.167320\pi$
$281$	29.0000	1.72999	0.864997	+	0.501776i	$0.167320\pi$
$282$	0	0
$283$	−9.00000	−0.534994	−0.267497	−	0.963559i	$-0.586197\pi$
$283$	−9.00000	−0.534994	−0.267497	+	0.963559i	$0.586197\pi$
$284$	−10.0000	−0.593391
$285$	0	0
$286$	−1.00000	−0.0591312
$287$	3.00000	0.177084
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	−9.00000	−0.528498
$291$	0	0
$292$	9.00000	0.526685
$293$	18.0000	1.05157	0.525786	−	0.850617i	$-0.323771\pi$
$293$	18.0000	1.05157	0.525786	+	0.850617i	$0.323771\pi$
$294$	0	0
$295$	3.00000	0.174667
$296$	6.00000	0.348743
$297$	0	0
$298$	4.00000	0.231714
$299$	−1.00000	−0.0578315
$300$	0	0
$301$	−33.0000	−1.90209
$302$	−24.0000	−1.38104
$303$	0	0
$304$	−2.00000	−0.114708
$305$	5.00000	0.286299
$306$	0	0
$307$	−10.0000	−0.570730	−0.285365	−	0.958419i	$-0.592115\pi$
$307$	−10.0000	−0.570730	−0.285365	+	0.958419i	$0.592115\pi$
$308$	3.00000	0.170941
$309$	0	0
$310$	4.00000	0.227185
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	0	0
$313$	19.0000	1.07394	0.536972	−	0.843600i	$-0.319568\pi$
$313$	19.0000	1.07394	0.536972	+	0.843600i	$0.319568\pi$
$314$	22.0000	1.24153
$315$	0	0
$316$	10.0000	0.562544
$317$	3.00000	0.168497	0.0842484	−	0.996445i	$-0.473151\pi$
$317$	3.00000	0.168497	0.0842484	+	0.996445i	$0.473151\pi$
$318$	0	0
$319$	−9.00000	−0.503903
$320$	1.00000	0.0559017
$321$	0	0
$322$	3.00000	0.167183
$323$	−8.00000	−0.445132
$324$	0	0
$325$	4.00000	0.221880
$326$	−9.00000	−0.498464
$327$	0	0
$328$	1.00000	0.0552158
$329$	0	0
$330$	0	0
$331$	19.0000	1.04433	0.522167	−	0.852843i	$-0.325124\pi$
$331$	19.0000	1.04433	0.522167	+	0.852843i	$0.325124\pi$
$332$	6.00000	0.329293
$333$	0	0
$334$	−3.00000	−0.164153
$335$	3.00000	0.163908
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	−1.00000	−0.0543928
$339$	0	0
$340$	4.00000	0.216930
$341$	4.00000	0.216612
$342$	0	0
$343$	15.0000	0.809924
$344$	−11.0000	−0.593080
$345$	0	0
$346$	−19.0000	−1.02145
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	0	0
$349$	−20.0000	−1.07058	−0.535288	−	0.844670i	$-0.679797\pi$
$349$	−20.0000	−1.07058	−0.535288	+	0.844670i	$0.679797\pi$
$350$	−12.0000	−0.641427
$351$	0	0
$352$	1.00000	0.0533002
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	0	0
$355$	−10.0000	−0.530745
$356$	8.00000	0.423999
$357$	0	0
$358$	12.0000	0.634220
$359$	−13.0000	−0.686114	−0.343057	−	0.939315i	$-0.611462\pi$
$359$	−13.0000	−0.686114	−0.343057	+	0.939315i	$0.611462\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	−14.0000	−0.735824
$363$	0	0
$364$	3.00000	0.157243
$365$	9.00000	0.471082
$366$	0	0
$367$	−32.0000	−1.67039	−0.835193	−	0.549957i	$-0.814644\pi$
$367$	−32.0000	−1.67039	−0.835193	+	0.549957i	$0.814644\pi$
$368$	1.00000	0.0521286
$369$	0	0
$370$	6.00000	0.311925
$371$	−30.0000	−1.55752
$372$	0	0
$373$	−11.0000	−0.569558	−0.284779	−	0.958593i	$-0.591920\pi$
$373$	−11.0000	−0.569558	−0.284779	+	0.958593i	$0.591920\pi$
$374$	4.00000	0.206835
$375$	0	0
$376$	0	0
$377$	−9.00000	−0.463524
$378$	0	0
$379$	24.0000	1.23280	0.616399	−	0.787434i	$-0.288591\pi$
$379$	24.0000	1.23280	0.616399	+	0.787434i	$0.288591\pi$
$380$	−2.00000	−0.102598
$381$	0	0
$382$	−15.0000	−0.767467
$383$	−14.0000	−0.715367	−0.357683	−	0.933843i	$-0.616433\pi$
$383$	−14.0000	−0.715367	−0.357683	+	0.933843i	$0.616433\pi$
$384$	0	0
$385$	3.00000	0.152894
$386$	6.00000	0.305392
$387$	0	0
$388$	2.00000	0.101535
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	4.00000	0.202289
$392$	−2.00000	−0.101015
$393$	0	0
$394$	12.0000	0.604551
$395$	10.0000	0.503155
$396$	0	0
$397$	−19.0000	−0.953583	−0.476791	−	0.879017i	$-0.658200\pi$
$397$	−19.0000	−0.953583	−0.476791	+	0.879017i	$0.658200\pi$
$398$	−3.00000	−0.150376
$399$	0	0
$400$	−4.00000	−0.200000
$401$	24.0000	1.19850	0.599251	−	0.800561i	$-0.295465\pi$
$401$	24.0000	1.19850	0.599251	+	0.800561i	$0.295465\pi$
$402$	0	0
$403$	4.00000	0.199254
$404$	10.0000	0.497519
$405$	0	0
$406$	27.0000	1.33999
$407$	6.00000	0.297409
$408$	0	0
$409$	7.00000	0.346128	0.173064	−	0.984911i	$-0.444633\pi$
$409$	7.00000	0.346128	0.173064	+	0.984911i	$0.444633\pi$
$410$	1.00000	0.0493865
$411$	0	0
$412$	−11.0000	−0.541931
$413$	−9.00000	−0.442861
$414$	0	0
$415$	6.00000	0.294528
$416$	1.00000	0.0490290
$417$	0	0
$418$	−2.00000	−0.0978232
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	5.00000	0.243685	0.121843	−	0.992549i	$-0.461120\pi$
$421$	5.00000	0.243685	0.121843	+	0.992549i	$0.461120\pi$
$422$	−20.0000	−0.973585
$423$	0	0
$424$	−10.0000	−0.485643
$425$	−16.0000	−0.776114
$426$	0	0
$427$	−15.0000	−0.725901
$428$	−7.00000	−0.338358
$429$	0	0
$430$	−11.0000	−0.530467
$431$	−16.0000	−0.770693	−0.385346	−	0.922772i	$-0.625918\pi$
$431$	−16.0000	−0.770693	−0.385346	+	0.922772i	$0.625918\pi$
$432$	0	0
$433$	5.00000	0.240285	0.120142	−	0.992757i	$-0.461665\pi$
$433$	5.00000	0.240285	0.120142	+	0.992757i	$0.461665\pi$
$434$	−12.0000	−0.576018
$435$	0	0
$436$	16.0000	0.766261
$437$	−2.00000	−0.0956730
$438$	0	0
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	1.00000	0.0476731
$441$	0	0
$442$	4.00000	0.190261
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	8.00000	0.379236
$446$	20.0000	0.947027
$447$	0	0
$448$	−3.00000	−0.141737
$449$	36.0000	1.69895	0.849473	−	0.527633i	$-0.176920\pi$
$449$	36.0000	1.69895	0.849473	+	0.527633i	$0.176920\pi$
$450$	0	0
$451$	1.00000	0.0470882
$452$	−5.00000	−0.235180
$453$	0	0
$454$	8.00000	0.375459
$455$	3.00000	0.140642
$456$	0	0
$457$	−37.0000	−1.73079	−0.865393	−	0.501093i	$-0.832931\pi$
$457$	−37.0000	−1.73079	−0.865393	+	0.501093i	$0.832931\pi$
$458$	−25.0000	−1.16817
$459$	0	0
$460$	1.00000	0.0466252
$461$	−24.0000	−1.11779	−0.558896	−	0.829238i	$-0.688775\pi$
$461$	−24.0000	−1.11779	−0.558896	+	0.829238i	$0.688775\pi$
$462$	0	0
$463$	−34.0000	−1.58011	−0.790057	−	0.613033i	$-0.789949\pi$
$463$	−34.0000	−1.58011	−0.790057	+	0.613033i	$0.789949\pi$
$464$	9.00000	0.417815
$465$	0	0
$466$	14.0000	0.648537
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	−9.00000	−0.415581
$470$	0	0
$471$	0	0
$472$	−3.00000	−0.138086
$473$	−11.0000	−0.505781
$474$	0	0
$475$	8.00000	0.367065
$476$	−12.0000	−0.550019
$477$	0	0
$478$	−5.00000	−0.228695
$479$	31.0000	1.41643	0.708213	−	0.705999i	$-0.249502\pi$
$479$	31.0000	1.41643	0.708213	+	0.705999i	$0.249502\pi$
$480$	0	0
$481$	6.00000	0.273576
$482$	−2.00000	−0.0910975
$483$	0	0
$484$	1.00000	0.0454545
$485$	2.00000	0.0908153
$486$	0	0
$487$	−2.00000	−0.0906287	−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	−2.00000	−0.0906287	−0.0453143	+	0.998973i	$0.514429\pi$
$488$	−5.00000	−0.226339
$489$	0	0
$490$	−2.00000	−0.0903508
$491$	−29.0000	−1.30875	−0.654376	−	0.756169i	$-0.727069\pi$
$491$	−29.0000	−1.30875	−0.654376	+	0.756169i	$0.727069\pi$
$492$	0	0
$493$	36.0000	1.62136
$494$	−2.00000	−0.0899843
$495$	0	0
$496$	−4.00000	−0.179605
$497$	30.0000	1.34568
$498$	0	0
$499$	17.0000	0.761025	0.380512	−	0.924776i	$-0.375748\pi$
$499$	17.0000	0.761025	0.380512	+	0.924776i	$0.375748\pi$
$500$	−9.00000	−0.402492
$501$	0	0
$502$	−6.00000	−0.267793
$503$	−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
$505$	10.0000	0.444994
$506$	1.00000	0.0444554
$507$	0	0
$508$	−14.0000	−0.621150
$509$	−34.0000	−1.50702	−0.753512	−	0.657434i	$-0.771642\pi$
$509$	−34.0000	−1.50702	−0.753512	+	0.657434i	$0.771642\pi$
$510$	0	0
$511$	−27.0000	−1.19441
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−25.0000	−1.10270
$515$	−11.0000	−0.484718
$516$	0	0
$517$	0	0
$518$	−18.0000	−0.790875
$519$	0	0
$520$	1.00000	0.0438529
$521$	37.0000	1.62100	0.810500	−	0.585739i	$-0.199196\pi$
$521$	37.0000	1.62100	0.810500	+	0.585739i	$0.199196\pi$
$522$	0	0
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	3.00000	0.131056
$525$	0	0
$526$	−24.0000	−1.04645
$527$	−16.0000	−0.696971
$528$	0	0
$529$	−22.0000	−0.956522
$530$	−10.0000	−0.434372
$531$	0	0
$532$	6.00000	0.260133
$533$	1.00000	0.0433148
$534$	0	0
$535$	−7.00000	−0.302636
$536$	−3.00000	−0.129580
$537$	0	0
$538$	−4.00000	−0.172452
$539$	−2.00000	−0.0861461
$540$	0	0
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	16.0000	0.687259
$543$	0	0
$544$	−4.00000	−0.171499
$545$	16.0000	0.685365
$546$	0	0
$547$	−5.00000	−0.213785	−0.106892	−	0.994271i	$-0.534090\pi$
$547$	−5.00000	−0.213785	−0.106892	+	0.994271i	$0.534090\pi$
$548$	18.0000	0.768922
$549$	0	0
$550$	−4.00000	−0.170561
$551$	−18.0000	−0.766826
$552$	0	0
$553$	−30.0000	−1.27573
$554$	−5.00000	−0.212430
$555$	0	0
$556$	0	0
$557$	14.0000	0.593199	0.296600	−	0.955002i	$-0.404147\pi$
$557$	14.0000	0.593199	0.296600	+	0.955002i	$0.404147\pi$
$558$	0	0
$559$	−11.0000	−0.465250
$560$	−3.00000	−0.126773
$561$	0	0
$562$	−29.0000	−1.22329
$563$	−32.0000	−1.34864	−0.674320	−	0.738440i	$-0.735563\pi$
$563$	−32.0000	−1.34864	−0.674320	+	0.738440i	$0.735563\pi$
$564$	0	0
$565$	−5.00000	−0.210352
$566$	9.00000	0.378298
$567$	0	0
$568$	10.0000	0.419591
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	0	0
$571$	−29.0000	−1.21361	−0.606806	−	0.794850i	$-0.707550\pi$
$571$	−29.0000	−1.21361	−0.606806	+	0.794850i	$0.707550\pi$
$572$	1.00000	0.0418121
$573$	0	0
$574$	−3.00000	−0.125218
$575$	−4.00000	−0.166812
$576$	0	0
$577$	4.00000	0.166522	0.0832611	−	0.996528i	$-0.473466\pi$
$577$	4.00000	0.166522	0.0832611	+	0.996528i	$0.473466\pi$
$578$	1.00000	0.0415945
$579$	0	0
$580$	9.00000	0.373705
$581$	−18.0000	−0.746766
$582$	0	0
$583$	−10.0000	−0.414158
$584$	−9.00000	−0.372423
$585$	0	0
$586$	−18.0000	−0.743573
$587$	37.0000	1.52715	0.763577	−	0.645717i	$-0.223441\pi$
$587$	37.0000	1.52715	0.763577	+	0.645717i	$0.223441\pi$
$588$	0	0
$589$	8.00000	0.329634
$590$	−3.00000	−0.123508
$591$	0	0
$592$	−6.00000	−0.246598
$593$	−26.0000	−1.06769	−0.533846	−	0.845582i	$-0.679254\pi$
$593$	−26.0000	−1.06769	−0.533846	+	0.845582i	$0.679254\pi$
$594$	0	0
$595$	−12.0000	−0.491952
$596$	−4.00000	−0.163846
$597$	0	0
$598$	1.00000	0.0408930
$599$	13.0000	0.531166	0.265583	−	0.964088i	$-0.414436\pi$
$599$	13.0000	0.531166	0.265583	+	0.964088i	$0.414436\pi$
$600$	0	0
$601$	−24.0000	−0.978980	−0.489490	−	0.872009i	$-0.662817\pi$
$601$	−24.0000	−0.978980	−0.489490	+	0.872009i	$0.662817\pi$
$602$	33.0000	1.34498
$603$	0	0
$604$	24.0000	0.976546
$605$	1.00000	0.0406558
$606$	0	0
$607$	10.0000	0.405887	0.202944	−	0.979190i	$-0.434949\pi$
$607$	10.0000	0.405887	0.202944	+	0.979190i	$0.434949\pi$
$608$	2.00000	0.0811107
$609$	0	0
$610$	−5.00000	−0.202444
$611$	0	0
$612$	0	0
$613$	24.0000	0.969351	0.484675	−	0.874694i	$-0.338938\pi$
$613$	24.0000	0.969351	0.484675	+	0.874694i	$0.338938\pi$
$614$	10.0000	0.403567
$615$	0	0
$616$	−3.00000	−0.120873
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	0	0
$619$	11.0000	0.442127	0.221064	−	0.975259i	$-0.429047\pi$
$619$	11.0000	0.442127	0.221064	+	0.975259i	$0.429047\pi$
$620$	−4.00000	−0.160644
$621$	0	0
$622$	−8.00000	−0.320771
$623$	−24.0000	−0.961540
$624$	0	0
$625$	11.0000	0.440000
$626$	−19.0000	−0.759393
$627$	0	0
$628$	−22.0000	−0.877896
$629$	−24.0000	−0.956943
$630$	0	0
$631$	30.0000	1.19428	0.597141	−	0.802137i	$-0.296303\pi$
$631$	30.0000	1.19428	0.597141	+	0.802137i	$0.296303\pi$
$632$	−10.0000	−0.397779
$633$	0	0
$634$	−3.00000	−0.119145
$635$	−14.0000	−0.555573
$636$	0	0
$637$	−2.00000	−0.0792429
$638$	9.00000	0.356313
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	−35.0000	−1.38242	−0.691208	−	0.722655i	$-0.742921\pi$
$641$	−35.0000	−1.38242	−0.691208	+	0.722655i	$0.742921\pi$
$642$	0	0
$643$	−36.0000	−1.41970	−0.709851	−	0.704352i	$-0.751238\pi$
$643$	−36.0000	−1.41970	−0.709851	+	0.704352i	$0.751238\pi$
$644$	−3.00000	−0.118217
$645$	0	0
$646$	8.00000	0.314756
$647$	24.0000	0.943537	0.471769	−	0.881722i	$-0.343616\pi$
$647$	24.0000	0.943537	0.471769	+	0.881722i	$0.343616\pi$
$648$	0	0
$649$	−3.00000	−0.117760
$650$	−4.00000	−0.156893
$651$	0	0
$652$	9.00000	0.352467
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	0	0
$655$	3.00000	0.117220
$656$	−1.00000	−0.0390434
$657$	0	0
$658$	0	0
$659$	40.0000	1.55818	0.779089	−	0.626913i	$-0.215682\pi$
$659$	40.0000	1.55818	0.779089	+	0.626913i	$0.215682\pi$
$660$	0	0
$661$	−14.0000	−0.544537	−0.272268	−	0.962221i	$-0.587774\pi$
$661$	−14.0000	−0.544537	−0.272268	+	0.962221i	$0.587774\pi$
$662$	−19.0000	−0.738456
$663$	0	0
$664$	−6.00000	−0.232845
$665$	6.00000	0.232670
$666$	0	0
$667$	9.00000	0.348481
$668$	3.00000	0.116073
$669$	0	0
$670$	−3.00000	−0.115900
$671$	−5.00000	−0.193023
$672$	0	0
$673$	−16.0000	−0.616755	−0.308377	−	0.951264i	$-0.599786\pi$
$673$	−16.0000	−0.616755	−0.308377	+	0.951264i	$0.599786\pi$
$674$	−2.00000	−0.0770371
$675$	0	0
$676$	1.00000	0.0384615
$677$	22.0000	0.845529	0.422764	−	0.906240i	$-0.361060\pi$
$677$	22.0000	0.845529	0.422764	+	0.906240i	$0.361060\pi$
$678$	0	0
$679$	−6.00000	−0.230259
$680$	−4.00000	−0.153393
$681$	0	0
$682$	−4.00000	−0.153168
$683$	−29.0000	−1.10965	−0.554827	−	0.831966i	$-0.687216\pi$
$683$	−29.0000	−1.10965	−0.554827	+	0.831966i	$0.687216\pi$
$684$	0	0
$685$	18.0000	0.687745
$686$	−15.0000	−0.572703
$687$	0	0
$688$	11.0000	0.419371
$689$	−10.0000	−0.380970
$690$	0	0
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	19.0000	0.722272
$693$	0	0
$694$	−12.0000	−0.455514
$695$	0	0
$696$	0	0
$697$	−4.00000	−0.151511
$698$	20.0000	0.757011
$699$	0	0
$700$	12.0000	0.453557
$701$	−35.0000	−1.32193	−0.660966	−	0.750416i	$-0.729853\pi$
$701$	−35.0000	−1.32193	−0.660966	+	0.750416i	$0.729853\pi$
$702$	0	0
$703$	12.0000	0.452589
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	18.0000	0.677439
$707$	−30.0000	−1.12827
$708$	0	0
$709$	−39.0000	−1.46468	−0.732338	−	0.680941i	$-0.761571\pi$
$709$	−39.0000	−1.46468	−0.732338	+	0.680941i	$0.761571\pi$
$710$	10.0000	0.375293
$711$	0	0
$712$	−8.00000	−0.299813
$713$	−4.00000	−0.149801
$714$	0	0
$715$	1.00000	0.0373979
$716$	−12.0000	−0.448461
$717$	0	0
$718$	13.0000	0.485156
$719$	15.0000	0.559406	0.279703	−	0.960087i	$-0.409764\pi$
$719$	15.0000	0.559406	0.279703	+	0.960087i	$0.409764\pi$
$720$	0	0
$721$	33.0000	1.22898
$722$	15.0000	0.558242
$723$	0	0
$724$	14.0000	0.520306
$725$	−36.0000	−1.33701
$726$	0	0
$727$	−4.00000	−0.148352	−0.0741759	−	0.997245i	$-0.523633\pi$
$727$	−4.00000	−0.148352	−0.0741759	+	0.997245i	$0.523633\pi$
$728$	−3.00000	−0.111187
$729$	0	0
$730$	−9.00000	−0.333105
$731$	44.0000	1.62740
$732$	0	0
$733$	−2.00000	−0.0738717	−0.0369358	−	0.999318i	$-0.511760\pi$
$733$	−2.00000	−0.0738717	−0.0369358	+	0.999318i	$0.511760\pi$
$734$	32.0000	1.18114
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	−3.00000	−0.110506
$738$	0	0
$739$	−24.0000	−0.882854	−0.441427	−	0.897297i	$-0.645528\pi$
$739$	−24.0000	−0.882854	−0.441427	+	0.897297i	$0.645528\pi$
$740$	−6.00000	−0.220564
$741$	0	0
$742$	30.0000	1.10133
$743$	−29.0000	−1.06391	−0.531953	−	0.846774i	$-0.678542\pi$
$743$	−29.0000	−1.06391	−0.531953	+	0.846774i	$0.678542\pi$
$744$	0	0
$745$	−4.00000	−0.146549
$746$	11.0000	0.402739
$747$	0	0
$748$	−4.00000	−0.146254
$749$	21.0000	0.767323
$750$	0	0
$751$	−53.0000	−1.93400	−0.966999	−	0.254781i	$-0.917997\pi$
$751$	−53.0000	−1.93400	−0.966999	+	0.254781i	$0.917997\pi$
$752$	0	0
$753$	0	0
$754$	9.00000	0.327761
$755$	24.0000	0.873449
$756$	0	0
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	−24.0000	−0.871719
$759$	0	0
$760$	2.00000	0.0725476
$761$	−35.0000	−1.26875	−0.634375	−	0.773026i	$-0.718742\pi$
$761$	−35.0000	−1.26875	−0.634375	+	0.773026i	$0.718742\pi$
$762$	0	0
$763$	−48.0000	−1.73772
$764$	15.0000	0.542681
$765$	0	0
$766$	14.0000	0.505841
$767$	−3.00000	−0.108324
$768$	0	0
$769$	49.0000	1.76699	0.883493	−	0.468445i	$-0.155186\pi$
$769$	49.0000	1.76699	0.883493	+	0.468445i	$0.155186\pi$
$770$	−3.00000	−0.108112
$771$	0	0
$772$	−6.00000	−0.215945
$773$	2.00000	0.0719350	0.0359675	−	0.999353i	$-0.488549\pi$
$773$	2.00000	0.0719350	0.0359675	+	0.999353i	$0.488549\pi$
$774$	0	0
$775$	16.0000	0.574737
$776$	−2.00000	−0.0717958
$777$	0	0
$778$	−18.0000	−0.645331
$779$	2.00000	0.0716574
$780$	0	0
$781$	10.0000	0.357828
$782$	−4.00000	−0.143040
$783$	0	0
$784$	2.00000	0.0714286
$785$	−22.0000	−0.785214
$786$	0	0
$787$	44.0000	1.56843	0.784215	−	0.620489i	$-0.213066\pi$
$787$	44.0000	1.56843	0.784215	+	0.620489i	$0.213066\pi$
$788$	−12.0000	−0.427482
$789$	0	0
$790$	−10.0000	−0.355784
$791$	15.0000	0.533339
$792$	0	0
$793$	−5.00000	−0.177555
$794$	19.0000	0.674285
$795$	0	0
$796$	3.00000	0.106332
$797$	−28.0000	−0.991811	−0.495905	−	0.868377i	$-0.665164\pi$
$797$	−28.0000	−0.991811	−0.495905	+	0.868377i	$0.665164\pi$
$798$	0	0
$799$	0	0
$800$	4.00000	0.141421
$801$	0	0
$802$	−24.0000	−0.847469
$803$	−9.00000	−0.317603
$804$	0	0
$805$	−3.00000	−0.105736
$806$	−4.00000	−0.140894
$807$	0	0
$808$	−10.0000	−0.351799
$809$	50.0000	1.75791	0.878953	−	0.476908i	$-0.158243\pi$
$809$	50.0000	1.75791	0.878953	+	0.476908i	$0.158243\pi$
$810$	0	0
$811$	56.0000	1.96643	0.983213	−	0.182462i	$-0.0584065\pi$
$811$	56.0000	1.96643	0.983213	+	0.182462i	$0.0584065\pi$
$812$	−27.0000	−0.947514
$813$	0	0
$814$	−6.00000	−0.210300
$815$	9.00000	0.315256
$816$	0	0
$817$	−22.0000	−0.769683
$818$	−7.00000	−0.244749
$819$	0	0
$820$	−1.00000	−0.0349215
$821$	−34.0000	−1.18661	−0.593304	−	0.804978i	$-0.702177\pi$
$821$	−34.0000	−1.18661	−0.593304	+	0.804978i	$0.702177\pi$
$822$	0	0
$823$	7.00000	0.244005	0.122002	−	0.992530i	$-0.461068\pi$
$823$	7.00000	0.244005	0.122002	+	0.992530i	$0.461068\pi$
$824$	11.0000	0.383203
$825$	0	0
$826$	9.00000	0.313150
$827$	−28.0000	−0.973655	−0.486828	−	0.873498i	$-0.661846\pi$
$827$	−28.0000	−0.973655	−0.486828	+	0.873498i	$0.661846\pi$
$828$	0	0
$829$	−6.00000	−0.208389	−0.104194	−	0.994557i	$-0.533226\pi$
$829$	−6.00000	−0.208389	−0.104194	+	0.994557i	$0.533226\pi$
$830$	−6.00000	−0.208263
$831$	0	0
$832$	−1.00000	−0.0346688
$833$	8.00000	0.277184
$834$	0	0
$835$	3.00000	0.103819
$836$	2.00000	0.0691714
$837$	0	0
$838$	0	0
$839$	−14.0000	−0.483334	−0.241667	−	0.970359i	$-0.577694\pi$
$839$	−14.0000	−0.483334	−0.241667	+	0.970359i	$0.577694\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	−5.00000	−0.172311
$843$	0	0
$844$	20.0000	0.688428
$845$	1.00000	0.0344010
$846$	0	0
$847$	−3.00000	−0.103081
$848$	10.0000	0.343401
$849$	0	0
$850$	16.0000	0.548795
$851$	−6.00000	−0.205677
$852$	0	0
$853$	16.0000	0.547830	0.273915	−	0.961754i	$-0.411681\pi$
$853$	16.0000	0.547830	0.273915	+	0.961754i	$0.411681\pi$
$854$	15.0000	0.513289
$855$	0	0
$856$	7.00000	0.239255
$857$	12.0000	0.409912	0.204956	−	0.978771i	$-0.434295\pi$
$857$	12.0000	0.409912	0.204956	+	0.978771i	$0.434295\pi$
$858$	0	0
$859$	32.0000	1.09183	0.545913	−	0.837842i	$-0.316183\pi$
$859$	32.0000	1.09183	0.545913	+	0.837842i	$0.316183\pi$
$860$	11.0000	0.375097
$861$	0	0
$862$	16.0000	0.544962
$863$	−32.0000	−1.08929	−0.544646	−	0.838666i	$-0.683336\pi$
$863$	−32.0000	−1.08929	−0.544646	+	0.838666i	$0.683336\pi$
$864$	0	0
$865$	19.0000	0.646019
$866$	−5.00000	−0.169907
$867$	0	0
$868$	12.0000	0.407307
$869$	−10.0000	−0.339227
$870$	0	0
$871$	−3.00000	−0.101651
$872$	−16.0000	−0.541828
$873$	0	0
$874$	2.00000	0.0676510
$875$	27.0000	0.912767
$876$	0	0
$877$	36.0000	1.21563	0.607817	−	0.794077i	$-0.292045\pi$
$877$	36.0000	1.21563	0.607817	+	0.794077i	$0.292045\pi$
$878$	8.00000	0.269987
$879$	0	0
$880$	−1.00000	−0.0337100
$881$	−9.00000	−0.303218	−0.151609	−	0.988441i	$-0.548445\pi$
$881$	−9.00000	−0.303218	−0.151609	+	0.988441i	$0.548445\pi$
$882$	0	0
$883$	26.0000	0.874970	0.437485	−	0.899226i	$-0.355869\pi$
$883$	26.0000	0.874970	0.437485	+	0.899226i	$0.355869\pi$
$884$	−4.00000	−0.134535
$885$	0	0
$886$	0	0
$887$	32.0000	1.07445	0.537227	−	0.843437i	$-0.319472\pi$
$887$	32.0000	1.07445	0.537227	+	0.843437i	$0.319472\pi$
$888$	0	0
$889$	42.0000	1.40863
$890$	−8.00000	−0.268161
$891$	0	0
$892$	−20.0000	−0.669650
$893$	0	0
$894$	0	0
$895$	−12.0000	−0.401116
$896$	3.00000	0.100223
$897$	0	0
$898$	−36.0000	−1.20134
$899$	−36.0000	−1.20067
$900$	0	0
$901$	40.0000	1.33259
$902$	−1.00000	−0.0332964
$903$	0	0
$904$	5.00000	0.166298
$905$	14.0000	0.465376
$906$	0	0
$907$	−20.0000	−0.664089	−0.332045	−	0.943264i	$-0.607738\pi$
$907$	−20.0000	−0.664089	−0.332045	+	0.943264i	$0.607738\pi$
$908$	−8.00000	−0.265489
$909$	0	0
$910$	−3.00000	−0.0994490
$911$	16.0000	0.530104	0.265052	−	0.964234i	$-0.414611\pi$
$911$	16.0000	0.530104	0.265052	+	0.964234i	$0.414611\pi$
$912$	0	0
$913$	−6.00000	−0.198571
$914$	37.0000	1.22385
$915$	0	0
$916$	25.0000	0.826023
$917$	−9.00000	−0.297206
$918$	0	0
$919$	20.0000	0.659739	0.329870	−	0.944027i	$-0.392995\pi$
$919$	20.0000	0.659739	0.329870	+	0.944027i	$0.392995\pi$
$920$	−1.00000	−0.0329690
$921$	0	0
$922$	24.0000	0.790398
$923$	10.0000	0.329154
$924$	0	0
$925$	24.0000	0.789115
$926$	34.0000	1.11731
$927$	0	0
$928$	−9.00000	−0.295439
$929$	12.0000	0.393707	0.196854	−	0.980433i	$-0.436928\pi$
$929$	12.0000	0.393707	0.196854	+	0.980433i	$0.436928\pi$
$930$	0	0
$931$	−4.00000	−0.131095
$932$	−14.0000	−0.458585
$933$	0	0
$934$	0	0
$935$	−4.00000	−0.130814
$936$	0	0
$937$	20.0000	0.653372	0.326686	−	0.945133i	$-0.394068\pi$
$937$	20.0000	0.653372	0.326686	+	0.945133i	$0.394068\pi$
$938$	9.00000	0.293860
$939$	0	0
$940$	0	0
$941$	−24.0000	−0.782378	−0.391189	−	0.920310i	$-0.627936\pi$
$941$	−24.0000	−0.782378	−0.391189	+	0.920310i	$0.627936\pi$
$942$	0	0
$943$	−1.00000	−0.0325645
$944$	3.00000	0.0976417
$945$	0	0
$946$	11.0000	0.357641
$947$	−4.00000	−0.129983	−0.0649913	−	0.997886i	$-0.520702\pi$
$947$	−4.00000	−0.129983	−0.0649913	+	0.997886i	$0.520702\pi$
$948$	0	0
$949$	−9.00000	−0.292152
$950$	−8.00000	−0.259554
$951$	0	0
$952$	12.0000	0.388922
$953$	−44.0000	−1.42530	−0.712650	−	0.701520i	$-0.752505\pi$
$953$	−44.0000	−1.42530	−0.712650	+	0.701520i	$0.752505\pi$
$954$	0	0
$955$	15.0000	0.485389
$956$	5.00000	0.161712
$957$	0	0
$958$	−31.0000	−1.00156
$959$	−54.0000	−1.74375
$960$	0	0
$961$	−15.0000	−0.483871
$962$	−6.00000	−0.193448
$963$	0	0
$964$	2.00000	0.0644157
$965$	−6.00000	−0.193147
$966$	0	0
$967$	−13.0000	−0.418052	−0.209026	−	0.977910i	$-0.567029\pi$
$967$	−13.0000	−0.418052	−0.209026	+	0.977910i	$0.567029\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	−2.00000	−0.0642161
$971$	−34.0000	−1.09111	−0.545556	−	0.838074i	$-0.683681\pi$
$971$	−34.0000	−1.09111	−0.545556	+	0.838074i	$0.683681\pi$
$972$	0	0
$973$	0	0
$974$	2.00000	0.0640841
$975$	0	0
$976$	5.00000	0.160046
$977$	48.0000	1.53566	0.767828	−	0.640656i	$-0.221338\pi$
$977$	48.0000	1.53566	0.767828	+	0.640656i	$0.221338\pi$
$978$	0	0
$979$	−8.00000	−0.255681
$980$	2.00000	0.0638877
$981$	0	0
$982$	29.0000	0.925427
$983$	−38.0000	−1.21201	−0.606006	−	0.795460i	$-0.707229\pi$
$983$	−38.0000	−1.21201	−0.606006	+	0.795460i	$0.707229\pi$
$984$	0	0
$985$	−12.0000	−0.382352
$986$	−36.0000	−1.14647
$987$	0	0
$988$	2.00000	0.0636285
$989$	11.0000	0.349780
$990$	0	0
$991$	−47.0000	−1.49300	−0.746502	−	0.665383i	$-0.768268\pi$
$991$	−47.0000	−1.49300	−0.746502	+	0.665383i	$0.768268\pi$
$992$	4.00000	0.127000
$993$	0	0
$994$	−30.0000	−0.951542
$995$	3.00000	0.0951064
$996$	0	0
$997$	−37.0000	−1.17180	−0.585901	−	0.810383i	$-0.699259\pi$
$997$	−37.0000	−1.17180	−0.585901	+	0.810383i	$0.699259\pi$
$998$	−17.0000	−0.538126
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
858.2.a.g.1.1	✓	1	3.2	odd	2
2574.2.a.i.1.1		1	1.1	even	1	trivial
6864.2.a.u.1.1		1	12.11	even	2
9438.2.a.d.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

Related objects

Downloads

Learn more