LMFDB - Embedded newform 2842.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2842,2,Mod(1,2842)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2842, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2842.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2842 = 2 \cdot 7^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2842.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:	$22.6934842544$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 406)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2842.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} -2.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} -2.00000 q^{9} +3.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} +1.00000 q^{13} +3.00000 q^{15} +1.00000 q^{16} +4.00000 q^{17} -2.00000 q^{18} +4.00000 q^{19} +3.00000 q^{20} -1.00000 q^{22} -2.00000 q^{23} +1.00000 q^{24} +4.00000 q^{25} +1.00000 q^{26} -5.00000 q^{27} +1.00000 q^{29} +3.00000 q^{30} +1.00000 q^{31} +1.00000 q^{32} -1.00000 q^{33} +4.00000 q^{34} -2.00000 q^{36} +6.00000 q^{37} +4.00000 q^{38} +1.00000 q^{39} +3.00000 q^{40} +3.00000 q^{43} -1.00000 q^{44} -6.00000 q^{45} -2.00000 q^{46} +9.00000 q^{47} +1.00000 q^{48} +4.00000 q^{50} +4.00000 q^{51} +1.00000 q^{52} +3.00000 q^{53} -5.00000 q^{54} -3.00000 q^{55} +4.00000 q^{57} +1.00000 q^{58} +3.00000 q^{60} -6.00000 q^{61} +1.00000 q^{62} +1.00000 q^{64} +3.00000 q^{65} -1.00000 q^{66} +2.00000 q^{67} +4.00000 q^{68} -2.00000 q^{69} -8.00000 q^{71} -2.00000 q^{72} +6.00000 q^{74} +4.00000 q^{75} +4.00000 q^{76} +1.00000 q^{78} -13.0000 q^{79} +3.00000 q^{80} +1.00000 q^{81} +12.0000 q^{85} +3.00000 q^{86} +1.00000 q^{87} -1.00000 q^{88} +14.0000 q^{89} -6.00000 q^{90} -2.00000 q^{92} +1.00000 q^{93} +9.00000 q^{94} +12.0000 q^{95} +1.00000 q^{96} -16.0000 q^{97} +2.00000 q^{99} +O(q^{100})\)

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
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	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	$0.265942\pi$
$6$	1.00000	0.408248
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	−2.00000	−0.666667
$10$	3.00000	0.948683
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	1.00000	0.288675
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	0	0
$15$	3.00000	0.774597
$16$	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	$0.338794\pi$
$18$	−2.00000	−0.471405
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	3.00000	0.670820
$21$	0	0
$22$	−1.00000	−0.213201
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	1.00000	0.204124
$25$	4.00000	0.800000
$26$	1.00000	0.196116
$27$	−5.00000	−0.962250
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	3.00000	0.547723
$31$	1.00000	0.179605	0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000	0.179605	0.0898027	+	0.995960i	$0.471376\pi$
$32$	1.00000	0.176777
$33$	−1.00000	−0.174078
$34$	4.00000	0.685994
$35$	0	0
$36$	−2.00000	−0.333333
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	4.00000	0.648886
$39$	1.00000	0.160128
$40$	3.00000	0.474342
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	3.00000	0.457496	0.228748	−	0.973486i	$-0.426537\pi$
$43$	3.00000	0.457496	0.228748	+	0.973486i	$0.426537\pi$
$44$	−1.00000	−0.150756
$45$	−6.00000	−0.894427
$46$	−2.00000	−0.294884
$47$	9.00000	1.31278	0.656392	−	0.754420i	$-0.272082\pi$
$47$	9.00000	1.31278	0.656392	+	0.754420i	$0.272082\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	4.00000	0.565685
$51$	4.00000	0.560112
$52$	1.00000	0.138675
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	−5.00000	−0.680414
$55$	−3.00000	−0.404520
$56$	0	0
$57$	4.00000	0.529813
$58$	1.00000	0.131306
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	3.00000	0.387298
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	1.00000	0.127000
$63$	0	0
$64$	1.00000	0.125000
$65$	3.00000	0.372104
$66$	−1.00000	−0.123091
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	4.00000	0.485071
$69$	−2.00000	−0.240772
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	−2.00000	−0.235702
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	6.00000	0.697486
$75$	4.00000	0.461880
$76$	4.00000	0.458831
$77$	0	0
$78$	1.00000	0.113228
$79$	−13.0000	−1.46261	−0.731307	−	0.682048i	$-0.761089\pi$
$79$	−13.0000	−1.46261	−0.731307	+	0.682048i	$0.761089\pi$
$80$	3.00000	0.335410
$81$	1.00000	0.111111
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	12.0000	1.30158
$86$	3.00000	0.323498
$87$	1.00000	0.107211
$88$	−1.00000	−0.106600
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	−6.00000	−0.632456
$91$	0	0
$92$	−2.00000	−0.208514
$93$	1.00000	0.103695
$94$	9.00000	0.928279
$95$	12.0000	1.23117
$96$	1.00000	0.102062
$97$	−16.0000	−1.62455	−0.812277	−	0.583272i	$-0.801772\pi$
$97$	−16.0000	−1.62455	−0.812277	+	0.583272i	$0.801772\pi$
$98$	0	0
$99$	2.00000	0.201008
$100$	4.00000	0.400000
$101$	4.00000	0.398015	0.199007	−	0.979998i	$-0.436228\pi$
$101$	4.00000	0.398015	0.199007	+	0.979998i	$0.436228\pi$
$102$	4.00000	0.396059
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	3.00000	0.291386
$107$	−14.0000	−1.35343	−0.676716	−	0.736245i	$-0.736597\pi$
$107$	−14.0000	−1.35343	−0.676716	+	0.736245i	$0.736597\pi$
$108$	−5.00000	−0.481125
$109$	15.0000	1.43674	0.718370	−	0.695662i	$-0.244889\pi$
$109$	15.0000	1.43674	0.718370	+	0.695662i	$0.244889\pi$
$110$	−3.00000	−0.286039
$111$	6.00000	0.569495
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	4.00000	0.374634
$115$	−6.00000	−0.559503
$116$	1.00000	0.0928477
$117$	−2.00000	−0.184900
$118$	0	0
$119$	0	0
$120$	3.00000	0.273861
$121$	−10.0000	−0.909091
$122$	−6.00000	−0.543214
$123$	0	0
$124$	1.00000	0.0898027
$125$	−3.00000	−0.268328
$126$	0	0
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	1.00000	0.0883883
$129$	3.00000	0.264135
$130$	3.00000	0.263117
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	−1.00000	−0.0870388
$133$	0	0
$134$	2.00000	0.172774
$135$	−15.0000	−1.29099
$136$	4.00000	0.342997
$137$	−10.0000	−0.854358	−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000	−0.854358	−0.427179	+	0.904167i	$0.640493\pi$
$138$	−2.00000	−0.170251
$139$	14.0000	1.18746	0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000	1.18746	0.593732	+	0.804663i	$0.297654\pi$
$140$	0	0
$141$	9.00000	0.757937
$142$	−8.00000	−0.671345
$143$	−1.00000	−0.0836242
$144$	−2.00000	−0.166667
$145$	3.00000	0.249136
$146$	0	0
$147$	0	0
$148$	6.00000	0.493197
$149$	−11.0000	−0.901155	−0.450578	−	0.892737i	$-0.648782\pi$
$149$	−11.0000	−0.901155	−0.450578	+	0.892737i	$0.648782\pi$
$150$	4.00000	0.326599
$151$	14.0000	1.13930	0.569652	−	0.821886i	$-0.307078\pi$
$151$	14.0000	1.13930	0.569652	+	0.821886i	$0.307078\pi$
$152$	4.00000	0.324443
$153$	−8.00000	−0.646762
$154$	0	0
$155$	3.00000	0.240966
$156$	1.00000	0.0800641
$157$	12.0000	0.957704	0.478852	−	0.877896i	$-0.341053\pi$
$157$	12.0000	0.957704	0.478852	+	0.877896i	$0.341053\pi$
$158$	−13.0000	−1.03422
$159$	3.00000	0.237915
$160$	3.00000	0.237171
$161$	0	0
$162$	1.00000	0.0785674
$163$	−21.0000	−1.64485	−0.822423	−	0.568876i	$-0.807379\pi$
$163$	−21.0000	−1.64485	−0.822423	+	0.568876i	$0.807379\pi$
$164$	0	0
$165$	−3.00000	−0.233550
$166$	0	0
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	12.0000	0.920358
$171$	−8.00000	−0.611775
$172$	3.00000	0.228748
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	1.00000	0.0758098
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	14.0000	1.04934
$179$	−24.0000	−1.79384	−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000	−1.79384	−0.896922	+	0.442189i	$0.854202\pi$
$180$	−6.00000	−0.447214
$181$	−23.0000	−1.70958	−0.854788	−	0.518977i	$-0.826313\pi$
$181$	−23.0000	−1.70958	−0.854788	+	0.518977i	$0.826313\pi$
$182$	0	0
$183$	−6.00000	−0.443533
$184$	−2.00000	−0.147442
$185$	18.0000	1.32339
$186$	1.00000	0.0733236
$187$	−4.00000	−0.292509
$188$	9.00000	0.656392
$189$	0	0
$190$	12.0000	0.870572
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	1.00000	0.0721688
$193$	12.0000	0.863779	0.431889	−	0.901927i	$-0.357847\pi$
$193$	12.0000	0.863779	0.431889	+	0.901927i	$0.357847\pi$
$194$	−16.0000	−1.14873
$195$	3.00000	0.214834
$196$	0	0
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	2.00000	0.142134
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	4.00000	0.282843
$201$	2.00000	0.141069
$202$	4.00000	0.281439
$203$	0	0
$204$	4.00000	0.280056
$205$	0	0
$206$	−4.00000	−0.278693
$207$	4.00000	0.278019
$208$	1.00000	0.0693375
$209$	−4.00000	−0.276686
$210$	0	0
$211$	−9.00000	−0.619586	−0.309793	−	0.950804i	$-0.600260\pi$
$211$	−9.00000	−0.619586	−0.309793	+	0.950804i	$0.600260\pi$
$212$	3.00000	0.206041
$213$	−8.00000	−0.548151
$214$	−14.0000	−0.957020
$215$	9.00000	0.613795
$216$	−5.00000	−0.340207
$217$	0	0
$218$	15.0000	1.01593
$219$	0	0
$220$	−3.00000	−0.202260
$221$	4.00000	0.269069
$222$	6.00000	0.402694
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	0	0
$225$	−8.00000	−0.533333
$226$	0	0
$227$	−14.0000	−0.929213	−0.464606	−	0.885517i	$-0.653804\pi$
$227$	−14.0000	−0.929213	−0.464606	+	0.885517i	$0.653804\pi$
$228$	4.00000	0.264906
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	−6.00000	−0.395628
$231$	0	0
$232$	1.00000	0.0656532
$233$	11.0000	0.720634	0.360317	−	0.932830i	$-0.382669\pi$
$233$	11.0000	0.720634	0.360317	+	0.932830i	$0.382669\pi$
$234$	−2.00000	−0.130744
$235$	27.0000	1.76129
$236$	0	0
$237$	−13.0000	−0.844441
$238$	0	0
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	3.00000	0.193649
$241$	29.0000	1.86805	0.934027	−	0.357202i	$-0.116269\pi$
$241$	29.0000	1.86805	0.934027	+	0.357202i	$0.116269\pi$
$242$	−10.0000	−0.642824
$243$	16.0000	1.02640
$244$	−6.00000	−0.384111
$245$	0	0
$246$	0	0
$247$	4.00000	0.254514
$248$	1.00000	0.0635001
$249$	0	0
$250$	−3.00000	−0.189737
$251$	17.0000	1.07303	0.536515	−	0.843891i	$-0.319740\pi$
$251$	17.0000	1.07303	0.536515	+	0.843891i	$0.319740\pi$
$252$	0	0
$253$	2.00000	0.125739
$254$	−16.0000	−1.00393
$255$	12.0000	0.751469
$256$	1.00000	0.0625000
$257$	9.00000	0.561405	0.280702	−	0.959795i	$-0.409433\pi$
$257$	9.00000	0.561405	0.280702	+	0.959795i	$0.409433\pi$
$258$	3.00000	0.186772
$259$	0	0
$260$	3.00000	0.186052
$261$	−2.00000	−0.123797
$262$	0	0
$263$	5.00000	0.308313	0.154157	−	0.988046i	$-0.450734\pi$
$263$	5.00000	0.308313	0.154157	+	0.988046i	$0.450734\pi$
$264$	−1.00000	−0.0615457
$265$	9.00000	0.552866
$266$	0	0
$267$	14.0000	0.856786
$268$	2.00000	0.122169
$269$	28.0000	1.70719	0.853595	−	0.520937i	$-0.174417\pi$
$269$	28.0000	1.70719	0.853595	+	0.520937i	$0.174417\pi$
$270$	−15.0000	−0.912871
$271$	−25.0000	−1.51864	−0.759321	−	0.650716i	$-0.774469\pi$
$271$	−25.0000	−1.51864	−0.759321	+	0.650716i	$0.774469\pi$
$272$	4.00000	0.242536
$273$	0	0
$274$	−10.0000	−0.604122
$275$	−4.00000	−0.241209
$276$	−2.00000	−0.120386
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	14.0000	0.839664
$279$	−2.00000	−0.119737
$280$	0	0
$281$	−21.0000	−1.25275	−0.626377	−	0.779520i	$-0.715463\pi$
$281$	−21.0000	−1.25275	−0.626377	+	0.779520i	$0.715463\pi$
$282$	9.00000	0.535942
$283$	2.00000	0.118888	0.0594438	−	0.998232i	$-0.481067\pi$
$283$	2.00000	0.118888	0.0594438	+	0.998232i	$0.481067\pi$
$284$	−8.00000	−0.474713
$285$	12.0000	0.710819
$286$	−1.00000	−0.0591312
$287$	0	0
$288$	−2.00000	−0.117851
$289$	−1.00000	−0.0588235
$290$	3.00000	0.176166
$291$	−16.0000	−0.937937
$292$	0	0
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	0	0
$295$	0	0
$296$	6.00000	0.348743
$297$	5.00000	0.290129
$298$	−11.0000	−0.637213
$299$	−2.00000	−0.115663
$300$	4.00000	0.230940
$301$	0	0
$302$	14.0000	0.805609
$303$	4.00000	0.229794
$304$	4.00000	0.229416
$305$	−18.0000	−1.03068
$306$	−8.00000	−0.457330
$307$	−9.00000	−0.513657	−0.256829	−	0.966457i	$-0.582678\pi$
$307$	−9.00000	−0.513657	−0.256829	+	0.966457i	$0.582678\pi$
$308$	0	0
$309$	−4.00000	−0.227552
$310$	3.00000	0.170389
$311$	−20.0000	−1.13410	−0.567048	−	0.823685i	$-0.691915\pi$
$311$	−20.0000	−1.13410	−0.567048	+	0.823685i	$0.691915\pi$
$312$	1.00000	0.0566139
$313$	5.00000	0.282617	0.141308	−	0.989966i	$-0.454869\pi$
$313$	5.00000	0.282617	0.141308	+	0.989966i	$0.454869\pi$
$314$	12.0000	0.677199
$315$	0	0
$316$	−13.0000	−0.731307
$317$	22.0000	1.23564	0.617822	−	0.786318i	$-0.288015\pi$
$317$	22.0000	1.23564	0.617822	+	0.786318i	$0.288015\pi$
$318$	3.00000	0.168232
$319$	−1.00000	−0.0559893
$320$	3.00000	0.167705
$321$	−14.0000	−0.781404
$322$	0	0
$323$	16.0000	0.890264
$324$	1.00000	0.0555556
$325$	4.00000	0.221880
$326$	−21.0000	−1.16308
$327$	15.0000	0.829502
$328$	0	0
$329$	0	0
$330$	−3.00000	−0.165145
$331$	27.0000	1.48405	0.742027	−	0.670370i	$-0.233865\pi$
$331$	27.0000	1.48405	0.742027	+	0.670370i	$0.233865\pi$
$332$	0	0
$333$	−12.0000	−0.657596
$334$	−12.0000	−0.656611
$335$	6.00000	0.327815
$336$	0	0
$337$	26.0000	1.41631	0.708155	−	0.706057i	$-0.249528\pi$
$337$	26.0000	1.41631	0.708155	+	0.706057i	$0.249528\pi$
$338$	−12.0000	−0.652714
$339$	0	0
$340$	12.0000	0.650791
$341$	−1.00000	−0.0541530
$342$	−8.00000	−0.432590
$343$	0	0
$344$	3.00000	0.161749
$345$	−6.00000	−0.323029
$346$	−14.0000	−0.752645
$347$	10.0000	0.536828	0.268414	−	0.963304i	$-0.413500\pi$
$347$	10.0000	0.536828	0.268414	+	0.963304i	$0.413500\pi$
$348$	1.00000	0.0536056
$349$	3.00000	0.160586	0.0802932	−	0.996771i	$-0.474414\pi$
$349$	3.00000	0.160586	0.0802932	+	0.996771i	$0.474414\pi$
$350$	0	0
$351$	−5.00000	−0.266880
$352$	−1.00000	−0.0533002
$353$	34.0000	1.80964	0.904819	−	0.425797i	$-0.140006\pi$
$353$	34.0000	1.80964	0.904819	+	0.425797i	$0.140006\pi$
$354$	0	0
$355$	−24.0000	−1.27379
$356$	14.0000	0.741999
$357$	0	0
$358$	−24.0000	−1.26844
$359$	−21.0000	−1.10834	−0.554169	−	0.832404i	$-0.686964\pi$
$359$	−21.0000	−1.10834	−0.554169	+	0.832404i	$0.686964\pi$
$360$	−6.00000	−0.316228
$361$	−3.00000	−0.157895
$362$	−23.0000	−1.20885
$363$	−10.0000	−0.524864
$364$	0	0
$365$	0	0
$366$	−6.00000	−0.313625
$367$	−16.0000	−0.835193	−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000	−0.835193	−0.417597	+	0.908633i	$0.637127\pi$
$368$	−2.00000	−0.104257
$369$	0	0
$370$	18.0000	0.935775
$371$	0	0
$372$	1.00000	0.0518476
$373$	17.0000	0.880227	0.440113	−	0.897942i	$-0.354938\pi$
$373$	17.0000	0.880227	0.440113	+	0.897942i	$0.354938\pi$
$374$	−4.00000	−0.206835
$375$	−3.00000	−0.154919
$376$	9.00000	0.464140
$377$	1.00000	0.0515026
$378$	0	0
$379$	36.0000	1.84920	0.924598	−	0.380945i	$-0.124401\pi$
$379$	36.0000	1.84920	0.924598	+	0.380945i	$0.124401\pi$
$380$	12.0000	0.615587
$381$	−16.0000	−0.819705
$382$	8.00000	0.409316
$383$	−30.0000	−1.53293	−0.766464	−	0.642287i	$-0.777986\pi$
$383$	−30.0000	−1.53293	−0.766464	+	0.642287i	$0.777986\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	12.0000	0.610784
$387$	−6.00000	−0.304997
$388$	−16.0000	−0.812277
$389$	−4.00000	−0.202808	−0.101404	−	0.994845i	$-0.532333\pi$
$389$	−4.00000	−0.202808	−0.101404	+	0.994845i	$0.532333\pi$
$390$	3.00000	0.151911
$391$	−8.00000	−0.404577
$392$	0	0
$393$	0	0
$394$	−2.00000	−0.100759
$395$	−39.0000	−1.96230
$396$	2.00000	0.100504
$397$	−11.0000	−0.552074	−0.276037	−	0.961147i	$-0.589021\pi$
$397$	−11.0000	−0.552074	−0.276037	+	0.961147i	$0.589021\pi$
$398$	20.0000	1.00251
$399$	0	0
$400$	4.00000	0.200000
$401$	39.0000	1.94757	0.973784	−	0.227477i	$-0.0730475\pi$
$401$	39.0000	1.94757	0.973784	+	0.227477i	$0.0730475\pi$
$402$	2.00000	0.0997509
$403$	1.00000	0.0498135
$404$	4.00000	0.199007
$405$	3.00000	0.149071
$406$	0	0
$407$	−6.00000	−0.297409
$408$	4.00000	0.198030
$409$	−28.0000	−1.38451	−0.692255	−	0.721653i	$-0.743383\pi$
$409$	−28.0000	−1.38451	−0.692255	+	0.721653i	$0.743383\pi$
$410$	0	0
$411$	−10.0000	−0.493264
$412$	−4.00000	−0.197066
$413$	0	0
$414$	4.00000	0.196589
$415$	0	0
$416$	1.00000	0.0490290
$417$	14.0000	0.685583
$418$	−4.00000	−0.195646
$419$	16.0000	0.781651	0.390826	−	0.920465i	$-0.372190\pi$
$419$	16.0000	0.781651	0.390826	+	0.920465i	$0.372190\pi$
$420$	0	0
$421$	26.0000	1.26716	0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000	1.26716	0.633581	+	0.773676i	$0.281584\pi$
$422$	−9.00000	−0.438113
$423$	−18.0000	−0.875190
$424$	3.00000	0.145693
$425$	16.0000	0.776114
$426$	−8.00000	−0.387601
$427$	0	0
$428$	−14.0000	−0.676716
$429$	−1.00000	−0.0482805
$430$	9.00000	0.434019
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	−5.00000	−0.240563
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	0	0
$435$	3.00000	0.143839
$436$	15.0000	0.718370
$437$	−8.00000	−0.382692
$438$	0	0
$439$	−2.00000	−0.0954548	−0.0477274	−	0.998860i	$-0.515198\pi$
$439$	−2.00000	−0.0954548	−0.0477274	+	0.998860i	$0.515198\pi$
$440$	−3.00000	−0.143019
$441$	0	0
$442$	4.00000	0.190261
$443$	20.0000	0.950229	0.475114	−	0.879924i	$-0.342407\pi$
$443$	20.0000	0.950229	0.475114	+	0.879924i	$0.342407\pi$
$444$	6.00000	0.284747
$445$	42.0000	1.99099
$446$	4.00000	0.189405
$447$	−11.0000	−0.520282
$448$	0	0
$449$	−16.0000	−0.755087	−0.377543	−	0.925992i	$-0.623231\pi$
$449$	−16.0000	−0.755087	−0.377543	+	0.925992i	$0.623231\pi$
$450$	−8.00000	−0.377124
$451$	0	0
$452$	0	0
$453$	14.0000	0.657777
$454$	−14.0000	−0.657053
$455$	0	0
$456$	4.00000	0.187317
$457$	−26.0000	−1.21623	−0.608114	−	0.793849i	$-0.708074\pi$
$457$	−26.0000	−1.21623	−0.608114	+	0.793849i	$0.708074\pi$
$458$	−14.0000	−0.654177
$459$	−20.0000	−0.933520
$460$	−6.00000	−0.279751
$461$	28.0000	1.30409	0.652045	−	0.758180i	$-0.273911\pi$
$461$	28.0000	1.30409	0.652045	+	0.758180i	$0.273911\pi$
$462$	0	0
$463$	−20.0000	−0.929479	−0.464739	−	0.885448i	$-0.653852\pi$
$463$	−20.0000	−0.929479	−0.464739	+	0.885448i	$0.653852\pi$
$464$	1.00000	0.0464238
$465$	3.00000	0.139122
$466$	11.0000	0.509565
$467$	−13.0000	−0.601568	−0.300784	−	0.953692i	$-0.597248\pi$
$467$	−13.0000	−0.601568	−0.300784	+	0.953692i	$0.597248\pi$
$468$	−2.00000	−0.0924500
$469$	0	0
$470$	27.0000	1.24542
$471$	12.0000	0.552931
$472$	0	0
$473$	−3.00000	−0.137940
$474$	−13.0000	−0.597110
$475$	16.0000	0.734130
$476$	0	0
$477$	−6.00000	−0.274721
$478$	−12.0000	−0.548867
$479$	−21.0000	−0.959514	−0.479757	−	0.877401i	$-0.659275\pi$
$479$	−21.0000	−0.959514	−0.479757	+	0.877401i	$0.659275\pi$
$480$	3.00000	0.136931
$481$	6.00000	0.273576
$482$	29.0000	1.32091
$483$	0	0
$484$	−10.0000	−0.454545
$485$	−48.0000	−2.17957
$486$	16.0000	0.725775
$487$	−20.0000	−0.906287	−0.453143	−	0.891438i	$-0.649697\pi$
$487$	−20.0000	−0.906287	−0.453143	+	0.891438i	$0.649697\pi$
$488$	−6.00000	−0.271607
$489$	−21.0000	−0.949653
$490$	0	0
$491$	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
$493$	4.00000	0.180151
$494$	4.00000	0.179969
$495$	6.00000	0.269680
$496$	1.00000	0.0449013
$497$	0	0
$498$	0	0
$499$	−38.0000	−1.70111	−0.850557	−	0.525883i	$-0.823735\pi$
$499$	−38.0000	−1.70111	−0.850557	+	0.525883i	$0.823735\pi$
$500$	−3.00000	−0.134164
$501$	−12.0000	−0.536120
$502$	17.0000	0.758747
$503$	11.0000	0.490466	0.245233	−	0.969464i	$-0.421136\pi$
$503$	11.0000	0.490466	0.245233	+	0.969464i	$0.421136\pi$
$504$	0	0
$505$	12.0000	0.533993
$506$	2.00000	0.0889108
$507$	−12.0000	−0.532939
$508$	−16.0000	−0.709885
$509$	7.00000	0.310270	0.155135	−	0.987893i	$-0.450419\pi$
$509$	7.00000	0.310270	0.155135	+	0.987893i	$0.450419\pi$
$510$	12.0000	0.531369
$511$	0	0
$512$	1.00000	0.0441942
$513$	−20.0000	−0.883022
$514$	9.00000	0.396973
$515$	−12.0000	−0.528783
$516$	3.00000	0.132068
$517$	−9.00000	−0.395820
$518$	0	0
$519$	−14.0000	−0.614532
$520$	3.00000	0.131559
$521$	−25.0000	−1.09527	−0.547635	−	0.836717i	$-0.684472\pi$
$521$	−25.0000	−1.09527	−0.547635	+	0.836717i	$0.684472\pi$
$522$	−2.00000	−0.0875376
$523$	34.0000	1.48672	0.743358	−	0.668894i	$-0.233232\pi$
$523$	34.0000	1.48672	0.743358	+	0.668894i	$0.233232\pi$
$524$	0	0
$525$	0	0
$526$	5.00000	0.218010
$527$	4.00000	0.174243
$528$	−1.00000	−0.0435194
$529$	−19.0000	−0.826087
$530$	9.00000	0.390935
$531$	0	0
$532$	0	0
$533$	0	0
$534$	14.0000	0.605839
$535$	−42.0000	−1.81582
$536$	2.00000	0.0863868
$537$	−24.0000	−1.03568
$538$	28.0000	1.20717
$539$	0	0
$540$	−15.0000	−0.645497
$541$	−20.0000	−0.859867	−0.429934	−	0.902861i	$-0.641463\pi$
$541$	−20.0000	−0.859867	−0.429934	+	0.902861i	$0.641463\pi$
$542$	−25.0000	−1.07384
$543$	−23.0000	−0.987024
$544$	4.00000	0.171499
$545$	45.0000	1.92759
$546$	0	0
$547$	−14.0000	−0.598597	−0.299298	−	0.954160i	$-0.596753\pi$
$547$	−14.0000	−0.598597	−0.299298	+	0.954160i	$0.596753\pi$
$548$	−10.0000	−0.427179
$549$	12.0000	0.512148
$550$	−4.00000	−0.170561
$551$	4.00000	0.170406
$552$	−2.00000	−0.0851257
$553$	0	0
$554$	10.0000	0.424859
$555$	18.0000	0.764057
$556$	14.0000	0.593732
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	−2.00000	−0.0846668
$559$	3.00000	0.126886
$560$	0	0
$561$	−4.00000	−0.168880
$562$	−21.0000	−0.885832
$563$	−9.00000	−0.379305	−0.189652	−	0.981851i	$-0.560736\pi$
$563$	−9.00000	−0.379305	−0.189652	+	0.981851i	$0.560736\pi$
$564$	9.00000	0.378968
$565$	0	0
$566$	2.00000	0.0840663
$567$	0	0
$568$	−8.00000	−0.335673
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	12.0000	0.502625
$571$	18.0000	0.753277	0.376638	−	0.926360i	$-0.377080\pi$
$571$	18.0000	0.753277	0.376638	+	0.926360i	$0.377080\pi$
$572$	−1.00000	−0.0418121
$573$	8.00000	0.334205
$574$	0	0
$575$	−8.00000	−0.333623
$576$	−2.00000	−0.0833333
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	−1.00000	−0.0415945
$579$	12.0000	0.498703
$580$	3.00000	0.124568
$581$	0	0
$582$	−16.0000	−0.663221
$583$	−3.00000	−0.124247
$584$	0	0
$585$	−6.00000	−0.248069
$586$	−18.0000	−0.743573
$587$	−30.0000	−1.23823	−0.619116	−	0.785299i	$-0.712509\pi$
$587$	−30.0000	−1.23823	−0.619116	+	0.785299i	$0.712509\pi$
$588$	0	0
$589$	4.00000	0.164817
$590$	0	0
$591$	−2.00000	−0.0822690
$592$	6.00000	0.246598
$593$	3.00000	0.123195	0.0615976	−	0.998101i	$-0.480380\pi$
$593$	3.00000	0.123195	0.0615976	+	0.998101i	$0.480380\pi$
$594$	5.00000	0.205152
$595$	0	0
$596$	−11.0000	−0.450578
$597$	20.0000	0.818546
$598$	−2.00000	−0.0817861
$599$	15.0000	0.612883	0.306442	−	0.951889i	$-0.400862\pi$
$599$	15.0000	0.612883	0.306442	+	0.951889i	$0.400862\pi$
$600$	4.00000	0.163299
$601$	28.0000	1.14214	0.571072	−	0.820900i	$-0.306528\pi$
$601$	28.0000	1.14214	0.571072	+	0.820900i	$0.306528\pi$
$602$	0	0
$603$	−4.00000	−0.162893
$604$	14.0000	0.569652
$605$	−30.0000	−1.21967
$606$	4.00000	0.162489
$607$	7.00000	0.284121	0.142061	−	0.989858i	$-0.454627\pi$
$607$	7.00000	0.284121	0.142061	+	0.989858i	$0.454627\pi$
$608$	4.00000	0.162221
$609$	0	0
$610$	−18.0000	−0.728799
$611$	9.00000	0.364101
$612$	−8.00000	−0.323381
$613$	−29.0000	−1.17130	−0.585649	−	0.810564i	$-0.699160\pi$
$613$	−29.0000	−1.17130	−0.585649	+	0.810564i	$0.699160\pi$
$614$	−9.00000	−0.363210
$615$	0	0
$616$	0	0
$617$	−36.0000	−1.44931	−0.724653	−	0.689114i	$-0.758000\pi$
$617$	−36.0000	−1.44931	−0.724653	+	0.689114i	$0.758000\pi$
$618$	−4.00000	−0.160904
$619$	3.00000	0.120580	0.0602901	−	0.998181i	$-0.480797\pi$
$619$	3.00000	0.120580	0.0602901	+	0.998181i	$0.480797\pi$
$620$	3.00000	0.120483
$621$	10.0000	0.401286
$622$	−20.0000	−0.801927
$623$	0	0
$624$	1.00000	0.0400320
$625$	−29.0000	−1.16000
$626$	5.00000	0.199840
$627$	−4.00000	−0.159745
$628$	12.0000	0.478852
$629$	24.0000	0.956943
$630$	0	0
$631$	−12.0000	−0.477712	−0.238856	−	0.971055i	$-0.576772\pi$
$631$	−12.0000	−0.477712	−0.238856	+	0.971055i	$0.576772\pi$
$632$	−13.0000	−0.517112
$633$	−9.00000	−0.357718
$634$	22.0000	0.873732
$635$	−48.0000	−1.90482
$636$	3.00000	0.118958
$637$	0	0
$638$	−1.00000	−0.0395904
$639$	16.0000	0.632950
$640$	3.00000	0.118585
$641$	12.0000	0.473972	0.236986	−	0.971513i	$-0.423841\pi$
$641$	12.0000	0.473972	0.236986	+	0.971513i	$0.423841\pi$
$642$	−14.0000	−0.552536
$643$	−12.0000	−0.473234	−0.236617	−	0.971603i	$-0.576039\pi$
$643$	−12.0000	−0.473234	−0.236617	+	0.971603i	$0.576039\pi$
$644$	0	0
$645$	9.00000	0.354375
$646$	16.0000	0.629512
$647$	20.0000	0.786281	0.393141	−	0.919478i	$-0.371389\pi$
$647$	20.0000	0.786281	0.393141	+	0.919478i	$0.371389\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	4.00000	0.156893
$651$	0	0
$652$	−21.0000	−0.822423
$653$	20.0000	0.782660	0.391330	−	0.920250i	$-0.372015\pi$
$653$	20.0000	0.782660	0.391330	+	0.920250i	$0.372015\pi$
$654$	15.0000	0.586546
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−43.0000	−1.67504	−0.837521	−	0.546405i	$-0.815996\pi$
$659$	−43.0000	−1.67504	−0.837521	+	0.546405i	$0.815996\pi$
$660$	−3.00000	−0.116775
$661$	2.00000	0.0777910	0.0388955	−	0.999243i	$-0.487616\pi$
$661$	2.00000	0.0777910	0.0388955	+	0.999243i	$0.487616\pi$
$662$	27.0000	1.04938
$663$	4.00000	0.155347
$664$	0	0
$665$	0	0
$666$	−12.0000	−0.464991
$667$	−2.00000	−0.0774403
$668$	−12.0000	−0.464294
$669$	4.00000	0.154649
$670$	6.00000	0.231800
$671$	6.00000	0.231627
$672$	0	0
$673$	−19.0000	−0.732396	−0.366198	−	0.930537i	$-0.619341\pi$
$673$	−19.0000	−0.732396	−0.366198	+	0.930537i	$0.619341\pi$
$674$	26.0000	1.00148
$675$	−20.0000	−0.769800
$676$	−12.0000	−0.461538
$677$	−40.0000	−1.53732	−0.768662	−	0.639655i	$-0.779077\pi$
$677$	−40.0000	−1.53732	−0.768662	+	0.639655i	$0.779077\pi$
$678$	0	0
$679$	0	0
$680$	12.0000	0.460179
$681$	−14.0000	−0.536481
$682$	−1.00000	−0.0382920
$683$	−24.0000	−0.918334	−0.459167	−	0.888350i	$-0.651852\pi$
$683$	−24.0000	−0.918334	−0.459167	+	0.888350i	$0.651852\pi$
$684$	−8.00000	−0.305888
$685$	−30.0000	−1.14624
$686$	0	0
$687$	−14.0000	−0.534133
$688$	3.00000	0.114374
$689$	3.00000	0.114291
$690$	−6.00000	−0.228416
$691$	48.0000	1.82601	0.913003	−	0.407953i	$-0.133757\pi$
$691$	48.0000	1.82601	0.913003	+	0.407953i	$0.133757\pi$
$692$	−14.0000	−0.532200
$693$	0	0
$694$	10.0000	0.379595
$695$	42.0000	1.59315
$696$	1.00000	0.0379049
$697$	0	0
$698$	3.00000	0.113552
$699$	11.0000	0.416058
$700$	0	0
$701$	−35.0000	−1.32193	−0.660966	−	0.750416i	$-0.729853\pi$
$701$	−35.0000	−1.32193	−0.660966	+	0.750416i	$0.729853\pi$
$702$	−5.00000	−0.188713
$703$	24.0000	0.905177
$704$	−1.00000	−0.0376889
$705$	27.0000	1.01688
$706$	34.0000	1.27961
$707$	0	0
$708$	0	0
$709$	9.00000	0.338002	0.169001	−	0.985616i	$-0.445946\pi$
$709$	9.00000	0.338002	0.169001	+	0.985616i	$0.445946\pi$
$710$	−24.0000	−0.900704
$711$	26.0000	0.975076
$712$	14.0000	0.524672
$713$	−2.00000	−0.0749006
$714$	0	0
$715$	−3.00000	−0.112194
$716$	−24.0000	−0.896922
$717$	−12.0000	−0.448148
$718$	−21.0000	−0.783713
$719$	2.00000	0.0745874	0.0372937	−	0.999304i	$-0.488126\pi$
$719$	2.00000	0.0745874	0.0372937	+	0.999304i	$0.488126\pi$
$720$	−6.00000	−0.223607
$721$	0	0
$722$	−3.00000	−0.111648
$723$	29.0000	1.07852
$724$	−23.0000	−0.854788
$725$	4.00000	0.148556
$726$	−10.0000	−0.371135
$727$	−12.0000	−0.445055	−0.222528	−	0.974926i	$-0.571431\pi$
$727$	−12.0000	−0.445055	−0.222528	+	0.974926i	$0.571431\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	12.0000	0.443836
$732$	−6.00000	−0.221766
$733$	38.0000	1.40356	0.701781	−	0.712393i	$-0.252388\pi$
$733$	38.0000	1.40356	0.701781	+	0.712393i	$0.252388\pi$
$734$	−16.0000	−0.590571
$735$	0	0
$736$	−2.00000	−0.0737210
$737$	−2.00000	−0.0736709
$738$	0	0
$739$	−47.0000	−1.72892	−0.864461	−	0.502699i	$-0.832340\pi$
$739$	−47.0000	−1.72892	−0.864461	+	0.502699i	$0.832340\pi$
$740$	18.0000	0.661693
$741$	4.00000	0.146944
$742$	0	0
$743$	−20.0000	−0.733729	−0.366864	−	0.930274i	$-0.619569\pi$
$743$	−20.0000	−0.733729	−0.366864	+	0.930274i	$0.619569\pi$
$744$	1.00000	0.0366618
$745$	−33.0000	−1.20903
$746$	17.0000	0.622414
$747$	0	0
$748$	−4.00000	−0.146254
$749$	0	0
$750$	−3.00000	−0.109545
$751$	40.0000	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$752$	9.00000	0.328196
$753$	17.0000	0.619514
$754$	1.00000	0.0364179
$755$	42.0000	1.52854
$756$	0	0
$757$	−28.0000	−1.01768	−0.508839	−	0.860862i	$-0.669925\pi$
$757$	−28.0000	−1.01768	−0.508839	+	0.860862i	$0.669925\pi$
$758$	36.0000	1.30758
$759$	2.00000	0.0725954
$760$	12.0000	0.435286
$761$	18.0000	0.652499	0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000	0.652499	0.326250	+	0.945284i	$0.394215\pi$
$762$	−16.0000	−0.579619
$763$	0	0
$764$	8.00000	0.289430
$765$	−24.0000	−0.867722
$766$	−30.0000	−1.08394
$767$	0	0
$768$	1.00000	0.0360844
$769$	−4.00000	−0.144244	−0.0721218	−	0.997396i	$-0.522977\pi$
$769$	−4.00000	−0.144244	−0.0721218	+	0.997396i	$0.522977\pi$
$770$	0	0
$771$	9.00000	0.324127
$772$	12.0000	0.431889
$773$	20.0000	0.719350	0.359675	−	0.933078i	$-0.382888\pi$
$773$	20.0000	0.719350	0.359675	+	0.933078i	$0.382888\pi$
$774$	−6.00000	−0.215666
$775$	4.00000	0.143684
$776$	−16.0000	−0.574367
$777$	0	0
$778$	−4.00000	−0.143407
$779$	0	0
$780$	3.00000	0.107417
$781$	8.00000	0.286263
$782$	−8.00000	−0.286079
$783$	−5.00000	−0.178685
$784$	0	0
$785$	36.0000	1.28490
$786$	0	0
$787$	12.0000	0.427754	0.213877	−	0.976861i	$-0.431391\pi$
$787$	12.0000	0.427754	0.213877	+	0.976861i	$0.431391\pi$
$788$	−2.00000	−0.0712470
$789$	5.00000	0.178005
$790$	−39.0000	−1.38756
$791$	0	0
$792$	2.00000	0.0710669
$793$	−6.00000	−0.213066
$794$	−11.0000	−0.390375
$795$	9.00000	0.319197
$796$	20.0000	0.708881
$797$	−12.0000	−0.425062	−0.212531	−	0.977154i	$-0.568171\pi$
$797$	−12.0000	−0.425062	−0.212531	+	0.977154i	$0.568171\pi$
$798$	0	0
$799$	36.0000	1.27359
$800$	4.00000	0.141421
$801$	−28.0000	−0.989331
$802$	39.0000	1.37714
$803$	0	0
$804$	2.00000	0.0705346
$805$	0	0
$806$	1.00000	0.0352235
$807$	28.0000	0.985647
$808$	4.00000	0.140720
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	3.00000	0.105409
$811$	30.0000	1.05344	0.526721	−	0.850038i	$-0.323421\pi$
$811$	30.0000	1.05344	0.526721	+	0.850038i	$0.323421\pi$
$812$	0	0
$813$	−25.0000	−0.876788
$814$	−6.00000	−0.210300
$815$	−63.0000	−2.20679
$816$	4.00000	0.140028
$817$	12.0000	0.419827
$818$	−28.0000	−0.978997
$819$	0	0
$820$	0	0
$821$	53.0000	1.84971	0.924856	−	0.380317i	$-0.124185\pi$
$821$	53.0000	1.84971	0.924856	+	0.380317i	$0.124185\pi$
$822$	−10.0000	−0.348790
$823$	−8.00000	−0.278862	−0.139431	−	0.990232i	$-0.544527\pi$
$823$	−8.00000	−0.278862	−0.139431	+	0.990232i	$0.544527\pi$
$824$	−4.00000	−0.139347
$825$	−4.00000	−0.139262
$826$	0	0
$827$	47.0000	1.63435	0.817175	−	0.576390i	$-0.195539\pi$
$827$	47.0000	1.63435	0.817175	+	0.576390i	$0.195539\pi$
$828$	4.00000	0.139010
$829$	20.0000	0.694629	0.347314	−	0.937749i	$-0.387094\pi$
$829$	20.0000	0.694629	0.347314	+	0.937749i	$0.387094\pi$
$830$	0	0
$831$	10.0000	0.346896
$832$	1.00000	0.0346688
$833$	0	0
$834$	14.0000	0.484780
$835$	−36.0000	−1.24583
$836$	−4.00000	−0.138343
$837$	−5.00000	−0.172825
$838$	16.0000	0.552711
$839$	21.0000	0.725001	0.362500	−	0.931984i	$-0.381923\pi$
$839$	21.0000	0.725001	0.362500	+	0.931984i	$0.381923\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	26.0000	0.896019
$843$	−21.0000	−0.723278
$844$	−9.00000	−0.309793
$845$	−36.0000	−1.23844
$846$	−18.0000	−0.618853
$847$	0	0
$848$	3.00000	0.103020
$849$	2.00000	0.0686398
$850$	16.0000	0.548795
$851$	−12.0000	−0.411355
$852$	−8.00000	−0.274075
$853$	26.0000	0.890223	0.445112	−	0.895475i	$-0.353164\pi$
$853$	26.0000	0.890223	0.445112	+	0.895475i	$0.353164\pi$
$854$	0	0
$855$	−24.0000	−0.820783
$856$	−14.0000	−0.478510
$857$	41.0000	1.40053	0.700267	−	0.713881i	$-0.253064\pi$
$857$	41.0000	1.40053	0.700267	+	0.713881i	$0.253064\pi$
$858$	−1.00000	−0.0341394
$859$	−31.0000	−1.05771	−0.528853	−	0.848713i	$-0.677378\pi$
$859$	−31.0000	−1.05771	−0.528853	+	0.848713i	$0.677378\pi$
$860$	9.00000	0.306897
$861$	0	0
$862$	−24.0000	−0.817443
$863$	58.0000	1.97434	0.987171	−	0.159664i	$-0.0510410\pi$
$863$	58.0000	1.97434	0.987171	+	0.159664i	$0.0510410\pi$
$864$	−5.00000	−0.170103
$865$	−42.0000	−1.42804
$866$	−34.0000	−1.15537
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	13.0000	0.440995
$870$	3.00000	0.101710
$871$	2.00000	0.0677674
$872$	15.0000	0.507964
$873$	32.0000	1.08304
$874$	−8.00000	−0.270604
$875$	0	0
$876$	0	0
$877$	31.0000	1.04680	0.523398	−	0.852088i	$-0.324664\pi$
$877$	31.0000	1.04680	0.523398	+	0.852088i	$0.324664\pi$
$878$	−2.00000	−0.0674967
$879$	−18.0000	−0.607125
$880$	−3.00000	−0.101130
$881$	26.0000	0.875962	0.437981	−	0.898984i	$-0.355694\pi$
$881$	26.0000	0.875962	0.437981	+	0.898984i	$0.355694\pi$
$882$	0	0
$883$	−6.00000	−0.201916	−0.100958	−	0.994891i	$-0.532191\pi$
$883$	−6.00000	−0.201916	−0.100958	+	0.994891i	$0.532191\pi$
$884$	4.00000	0.134535
$885$	0	0
$886$	20.0000	0.671913
$887$	−3.00000	−0.100730	−0.0503651	−	0.998731i	$-0.516038\pi$
$887$	−3.00000	−0.100730	−0.0503651	+	0.998731i	$0.516038\pi$
$888$	6.00000	0.201347
$889$	0	0
$890$	42.0000	1.40784
$891$	−1.00000	−0.0335013
$892$	4.00000	0.133930
$893$	36.0000	1.20469
$894$	−11.0000	−0.367895
$895$	−72.0000	−2.40669
$896$	0	0
$897$	−2.00000	−0.0667781
$898$	−16.0000	−0.533927
$899$	1.00000	0.0333519
$900$	−8.00000	−0.266667
$901$	12.0000	0.399778
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−69.0000	−2.29364
$906$	14.0000	0.465119
$907$	28.0000	0.929725	0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000	0.929725	0.464862	+	0.885383i	$0.346104\pi$
$908$	−14.0000	−0.464606
$909$	−8.00000	−0.265343
$910$	0	0
$911$	−9.00000	−0.298183	−0.149092	−	0.988823i	$-0.547635\pi$
$911$	−9.00000	−0.298183	−0.149092	+	0.988823i	$0.547635\pi$
$912$	4.00000	0.132453
$913$	0	0
$914$	−26.0000	−0.860004
$915$	−18.0000	−0.595062
$916$	−14.0000	−0.462573
$917$	0	0
$918$	−20.0000	−0.660098
$919$	10.0000	0.329870	0.164935	−	0.986304i	$-0.447259\pi$
$919$	10.0000	0.329870	0.164935	+	0.986304i	$0.447259\pi$
$920$	−6.00000	−0.197814
$921$	−9.00000	−0.296560
$922$	28.0000	0.922131
$923$	−8.00000	−0.263323
$924$	0	0
$925$	24.0000	0.789115
$926$	−20.0000	−0.657241
$927$	8.00000	0.262754
$928$	1.00000	0.0328266
$929$	−54.0000	−1.77168	−0.885841	−	0.463988i	$-0.846418\pi$
$929$	−54.0000	−1.77168	−0.885841	+	0.463988i	$0.846418\pi$
$930$	3.00000	0.0983739
$931$	0	0
$932$	11.0000	0.360317
$933$	−20.0000	−0.654771
$934$	−13.0000	−0.425373
$935$	−12.0000	−0.392442
$936$	−2.00000	−0.0653720
$937$	−2.00000	−0.0653372	−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	0	0
$939$	5.00000	0.163169
$940$	27.0000	0.880643
$941$	27.0000	0.880175	0.440087	−	0.897955i	$-0.354947\pi$
$941$	27.0000	0.880175	0.440087	+	0.897955i	$0.354947\pi$
$942$	12.0000	0.390981
$943$	0	0
$944$	0	0
$945$	0	0
$946$	−3.00000	−0.0975384
$947$	19.0000	0.617417	0.308709	−	0.951157i	$-0.400103\pi$
$947$	19.0000	0.617417	0.308709	+	0.951157i	$0.400103\pi$
$948$	−13.0000	−0.422220
$949$	0	0
$950$	16.0000	0.519109
$951$	22.0000	0.713399
$952$	0	0
$953$	47.0000	1.52248	0.761240	−	0.648471i	$-0.224591\pi$
$953$	47.0000	1.52248	0.761240	+	0.648471i	$0.224591\pi$
$954$	−6.00000	−0.194257
$955$	24.0000	0.776622
$956$	−12.0000	−0.388108
$957$	−1.00000	−0.0323254
$958$	−21.0000	−0.678479
$959$	0	0
$960$	3.00000	0.0968246
$961$	−30.0000	−0.967742
$962$	6.00000	0.193448
$963$	28.0000	0.902287
$964$	29.0000	0.934027
$965$	36.0000	1.15888
$966$	0	0
$967$	−31.0000	−0.996893	−0.498446	−	0.866921i	$-0.666096\pi$
$967$	−31.0000	−0.996893	−0.498446	+	0.866921i	$0.666096\pi$
$968$	−10.0000	−0.321412
$969$	16.0000	0.513994
$970$	−48.0000	−1.54119
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	16.0000	0.513200
$973$	0	0
$974$	−20.0000	−0.640841
$975$	4.00000	0.128103
$976$	−6.00000	−0.192055
$977$	53.0000	1.69562	0.847810	−	0.530300i	$-0.177921\pi$
$977$	53.0000	1.69562	0.847810	+	0.530300i	$0.177921\pi$
$978$	−21.0000	−0.671506
$979$	−14.0000	−0.447442
$980$	0	0
$981$	−30.0000	−0.957826
$982$	9.00000	0.287202
$983$	−39.0000	−1.24391	−0.621953	−	0.783054i	$-0.713661\pi$
$983$	−39.0000	−1.24391	−0.621953	+	0.783054i	$0.713661\pi$
$984$	0	0
$985$	−6.00000	−0.191176
$986$	4.00000	0.127386
$987$	0	0
$988$	4.00000	0.127257
$989$	−6.00000	−0.190789
$990$	6.00000	0.190693
$991$	−2.00000	−0.0635321	−0.0317660	−	0.999495i	$-0.510113\pi$
$991$	−2.00000	−0.0635321	−0.0317660	+	0.999495i	$0.510113\pi$
$992$	1.00000	0.0317500
$993$	27.0000	0.856819
$994$	0	0
$995$	60.0000	1.90213
$996$	0	0
$997$	14.0000	0.443384	0.221692	−	0.975117i	$-0.428842\pi$
$997$	14.0000	0.443384	0.221692	+	0.975117i	$0.428842\pi$
$998$	−38.0000	−1.20287
$999$	−30.0000	−0.949158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2842.2.a.f.1.1		1
7.6	odd	2		406.2.a.d.1.1	✓	1
21.20	even	2		3654.2.a.l.1.1		1
28.27	even	2		3248.2.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
406.2.a.d.1.1	✓	1	7.6	odd	2
2842.2.a.f.1.1		1	1.1	even	1	trivial
3248.2.a.k.1.1		1	28.27	even	2
3654.2.a.l.1.1		1	21.20	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:	\( N \)	\(=\)	\( 2842 = 2 \cdot 7^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2842.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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