LMFDB - Embedded newform 2358.2.a.w.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2358,2,Mod(1,2358)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2358, 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("2358.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2358 = 2 \cdot 3^{2} \cdot 131 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2358.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:	$18.8287247966$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 786)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2358.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} +2.00000 q^{7} +1.00000 q^{8} +O(q^{10})$

\(q+1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} +2.00000 q^{7} +1.00000 q^{8} +2.00000 q^{10} +3.00000 q^{11} +3.00000 q^{13} +2.00000 q^{14} +1.00000 q^{16} +5.00000 q^{17} +1.00000 q^{19} +2.00000 q^{20} +3.00000 q^{22} -4.00000 q^{23} -1.00000 q^{25} +3.00000 q^{26} +2.00000 q^{28} -9.00000 q^{29} -5.00000 q^{31} +1.00000 q^{32} +5.00000 q^{34} +4.00000 q^{35} -8.00000 q^{37} +1.00000 q^{38} +2.00000 q^{40} +12.0000 q^{41} -6.00000 q^{43} +3.00000 q^{44} -4.00000 q^{46} +8.00000 q^{47} -3.00000 q^{49} -1.00000 q^{50} +3.00000 q^{52} -12.0000 q^{53} +6.00000 q^{55} +2.00000 q^{56} -9.00000 q^{58} +5.00000 q^{59} -3.00000 q^{61} -5.00000 q^{62} +1.00000 q^{64} +6.00000 q^{65} +5.00000 q^{68} +4.00000 q^{70} +8.00000 q^{71} -2.00000 q^{73} -8.00000 q^{74} +1.00000 q^{76} +6.00000 q^{77} -8.00000 q^{79} +2.00000 q^{80} +12.0000 q^{82} +14.0000 q^{83} +10.0000 q^{85} -6.00000 q^{86} +3.00000 q^{88} +14.0000 q^{89} +6.00000 q^{91} -4.00000 q^{92} +8.00000 q^{94} +2.00000 q^{95} +12.0000 q^{97} -3.00000 q^{98} +O(q^{100})\)

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
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	2.00000	0.632456
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	0	0
$13$	3.00000	0.832050	0.416025	−	0.909353i	$-0.363423\pi$
$13$	3.00000	0.832050	0.416025	+	0.909353i	$0.363423\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	5.00000	1.21268	0.606339	−	0.795206i	$-0.292637\pi$
$17$	5.00000	1.21268	0.606339	+	0.795206i	$0.292637\pi$
$18$	0	0
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	2.00000	0.447214
$21$	0	0
$22$	3.00000	0.639602
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	3.00000	0.588348
$27$	0	0
$28$	2.00000	0.377964
$29$	−9.00000	−1.67126	−0.835629	−	0.549294i	$-0.814897\pi$
$29$	−9.00000	−1.67126	−0.835629	+	0.549294i	$0.814897\pi$
$30$	0	0
$31$	−5.00000	−0.898027	−0.449013	−	0.893525i	$-0.648224\pi$
$31$	−5.00000	−0.898027	−0.449013	+	0.893525i	$0.648224\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	5.00000	0.857493
$35$	4.00000	0.676123
$36$	0	0
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	2.00000	0.316228
$41$	12.0000	1.87409	0.937043	−	0.349215i	$-0.113552\pi$
$41$	12.0000	1.87409	0.937043	+	0.349215i	$0.113552\pi$
$42$	0	0
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	3.00000	0.452267
$45$	0	0
$46$	−4.00000	−0.589768
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	−1.00000	−0.141421
$51$	0	0
$52$	3.00000	0.416025
$53$	−12.0000	−1.64833	−0.824163	−	0.566352i	$-0.808354\pi$
$53$	−12.0000	−1.64833	−0.824163	+	0.566352i	$0.808354\pi$
$54$	0	0
$55$	6.00000	0.809040
$56$	2.00000	0.267261
$57$	0	0
$58$	−9.00000	−1.18176
$59$	5.00000	0.650945	0.325472	−	0.945552i	$-0.394477\pi$
$59$	5.00000	0.650945	0.325472	+	0.945552i	$0.394477\pi$
$60$	0	0
$61$	−3.00000	−0.384111	−0.192055	−	0.981384i	$-0.561515\pi$
$61$	−3.00000	−0.384111	−0.192055	+	0.981384i	$0.561515\pi$
$62$	−5.00000	−0.635001
$63$	0	0
$64$	1.00000	0.125000
$65$	6.00000	0.744208
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	5.00000	0.606339
$69$	0	0
$70$	4.00000	0.478091
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	−8.00000	−0.929981
$75$	0	0
$76$	1.00000	0.114708
$77$	6.00000	0.683763
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	2.00000	0.223607
$81$	0	0
$82$	12.0000	1.32518
$83$	14.0000	1.53670	0.768350	−	0.640030i	$-0.221078\pi$
$83$	14.0000	1.53670	0.768350	+	0.640030i	$0.221078\pi$
$84$	0	0
$85$	10.0000	1.08465
$86$	−6.00000	−0.646997
$87$	0	0
$88$	3.00000	0.319801
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	0	0
$91$	6.00000	0.628971
$92$	−4.00000	−0.417029
$93$	0	0
$94$	8.00000	0.825137
$95$	2.00000	0.205196
$96$	0	0
$97$	12.0000	1.21842	0.609208	−	0.793011i	$-0.291488\pi$
$97$	12.0000	1.21842	0.609208	+	0.793011i	$0.291488\pi$
$98$	−3.00000	−0.303046
$99$	0	0
$100$	−1.00000	−0.100000
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	3.00000	0.294174
$105$	0	0
$106$	−12.0000	−1.16554
$107$	−3.00000	−0.290021	−0.145010	−	0.989430i	$-0.546322\pi$
$107$	−3.00000	−0.290021	−0.145010	+	0.989430i	$0.546322\pi$
$108$	0	0
$109$	−13.0000	−1.24517	−0.622587	−	0.782551i	$-0.713918\pi$
$109$	−13.0000	−1.24517	−0.622587	+	0.782551i	$0.713918\pi$
$110$	6.00000	0.572078
$111$	0	0
$112$	2.00000	0.188982
$113$	−4.00000	−0.376288	−0.188144	−	0.982141i	$-0.560247\pi$
$113$	−4.00000	−0.376288	−0.188144	+	0.982141i	$0.560247\pi$
$114$	0	0
$115$	−8.00000	−0.746004
$116$	−9.00000	−0.835629
$117$	0	0
$118$	5.00000	0.460287
$119$	10.0000	0.916698
$120$	0	0
$121$	−2.00000	−0.181818
$122$	−3.00000	−0.271607
$123$	0	0
$124$	−5.00000	−0.449013
$125$	−12.0000	−1.07331
$126$	0	0
$127$	15.0000	1.33103	0.665517	−	0.746382i	$-0.268211\pi$
$127$	15.0000	1.33103	0.665517	+	0.746382i	$0.268211\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	6.00000	0.526235
$131$	1.00000	0.0873704
$131$	1.00000	0.0873704
$132$	0	0
$133$	2.00000	0.173422
$134$	0	0
$135$	0	0
$136$	5.00000	0.428746
$137$	7.00000	0.598050	0.299025	−	0.954245i	$-0.403339\pi$
$137$	7.00000	0.598050	0.299025	+	0.954245i	$0.403339\pi$
$138$	0	0
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	4.00000	0.338062
$141$	0	0
$142$	8.00000	0.671345
$143$	9.00000	0.752618
$144$	0	0
$145$	−18.0000	−1.49482
$146$	−2.00000	−0.165521
$147$	0	0
$148$	−8.00000	−0.657596
$149$	−9.00000	−0.737309	−0.368654	−	0.929567i	$-0.620181\pi$
$149$	−9.00000	−0.737309	−0.368654	+	0.929567i	$0.620181\pi$
$150$	0	0
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	1.00000	0.0811107
$153$	0	0
$154$	6.00000	0.483494
$155$	−10.0000	−0.803219
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	−8.00000	−0.636446
$159$	0	0
$160$	2.00000	0.158114
$161$	−8.00000	−0.630488
$162$	0	0
$163$	−1.00000	−0.0783260	−0.0391630	−	0.999233i	$-0.512469\pi$
$163$	−1.00000	−0.0783260	−0.0391630	+	0.999233i	$0.512469\pi$
$164$	12.0000	0.937043
$165$	0	0
$166$	14.0000	1.08661
$167$	−21.0000	−1.62503	−0.812514	−	0.582941i	$-0.801902\pi$
$167$	−21.0000	−1.62503	−0.812514	+	0.582941i	$0.801902\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	10.0000	0.766965
$171$	0	0
$172$	−6.00000	−0.457496
$173$	11.0000	0.836315	0.418157	−	0.908375i	$-0.362676\pi$
$173$	11.0000	0.836315	0.418157	+	0.908375i	$0.362676\pi$
$174$	0	0
$175$	−2.00000	−0.151186
$176$	3.00000	0.226134
$177$	0	0
$178$	14.0000	1.04934
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	0	0
$181$	20.0000	1.48659	0.743294	−	0.668965i	$-0.233262\pi$
$181$	20.0000	1.48659	0.743294	+	0.668965i	$0.233262\pi$
$182$	6.00000	0.444750
$183$	0	0
$184$	−4.00000	−0.294884
$185$	−16.0000	−1.17634
$186$	0	0
$187$	15.0000	1.09691
$188$	8.00000	0.583460
$189$	0	0
$190$	2.00000	0.145095
$191$	5.00000	0.361787	0.180894	−	0.983503i	$-0.442101\pi$
$191$	5.00000	0.361787	0.180894	+	0.983503i	$0.442101\pi$
$192$	0	0
$193$	19.0000	1.36765	0.683825	−	0.729646i	$-0.260315\pi$
$193$	19.0000	1.36765	0.683825	+	0.729646i	$0.260315\pi$
$194$	12.0000	0.861550
$195$	0	0
$196$	−3.00000	−0.214286
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	0	0
$199$	−17.0000	−1.20510	−0.602549	−	0.798082i	$-0.705848\pi$
$199$	−17.0000	−1.20510	−0.602549	+	0.798082i	$0.705848\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	−14.0000	−0.985037
$203$	−18.0000	−1.26335
$204$	0	0
$205$	24.0000	1.67623
$206$	−8.00000	−0.557386
$207$	0	0
$208$	3.00000	0.208013
$209$	3.00000	0.207514
$210$	0	0
$211$	2.00000	0.137686	0.0688428	−	0.997628i	$-0.478069\pi$
$211$	2.00000	0.137686	0.0688428	+	0.997628i	$0.478069\pi$
$212$	−12.0000	−0.824163
$213$	0	0
$214$	−3.00000	−0.205076
$215$	−12.0000	−0.818393
$216$	0	0
$217$	−10.0000	−0.678844
$218$	−13.0000	−0.880471
$219$	0	0
$220$	6.00000	0.404520
$221$	15.0000	1.00901
$222$	0	0
$223$	−19.0000	−1.27233	−0.636167	−	0.771551i	$-0.719481\pi$
$223$	−19.0000	−1.27233	−0.636167	+	0.771551i	$0.719481\pi$
$224$	2.00000	0.133631
$225$	0	0
$226$	−4.00000	−0.266076
$227$	6.00000	0.398234	0.199117	−	0.979976i	$-0.436193\pi$
$227$	6.00000	0.398234	0.199117	+	0.979976i	$0.436193\pi$
$228$	0	0
$229$	20.0000	1.32164	0.660819	−	0.750546i	$-0.270209\pi$
$229$	20.0000	1.32164	0.660819	+	0.750546i	$0.270209\pi$
$230$	−8.00000	−0.527504
$231$	0	0
$232$	−9.00000	−0.590879
$233$	−20.0000	−1.31024	−0.655122	−	0.755523i	$-0.727383\pi$
$233$	−20.0000	−1.31024	−0.655122	+	0.755523i	$0.727383\pi$
$234$	0	0
$235$	16.0000	1.04372
$236$	5.00000	0.325472
$237$	0	0
$238$	10.0000	0.648204
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	0	0
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	−2.00000	−0.128565
$243$	0	0
$244$	−3.00000	−0.192055
$245$	−6.00000	−0.383326
$246$	0	0
$247$	3.00000	0.190885
$248$	−5.00000	−0.317500
$249$	0	0
$250$	−12.0000	−0.758947
$251$	22.0000	1.38863	0.694314	−	0.719672i	$-0.255708\pi$
$251$	22.0000	1.38863	0.694314	+	0.719672i	$0.255708\pi$
$252$	0	0
$253$	−12.0000	−0.754434
$254$	15.0000	0.941184
$255$	0	0
$256$	1.00000	0.0625000
$257$	−26.0000	−1.62184	−0.810918	−	0.585160i	$-0.801032\pi$
$257$	−26.0000	−1.62184	−0.810918	+	0.585160i	$0.801032\pi$
$258$	0	0
$259$	−16.0000	−0.994192
$260$	6.00000	0.372104
$261$	0	0
$262$	1.00000	0.0617802
$263$	21.0000	1.29492	0.647458	−	0.762101i	$-0.275832\pi$
$263$	21.0000	1.29492	0.647458	+	0.762101i	$0.275832\pi$
$264$	0	0
$265$	−24.0000	−1.47431
$266$	2.00000	0.122628
$267$	0	0
$268$	0	0
$269$	8.00000	0.487769	0.243884	−	0.969804i	$-0.421578\pi$
$269$	8.00000	0.487769	0.243884	+	0.969804i	$0.421578\pi$
$270$	0	0
$271$	4.00000	0.242983	0.121491	−	0.992592i	$-0.461232\pi$
$271$	4.00000	0.242983	0.121491	+	0.992592i	$0.461232\pi$
$272$	5.00000	0.303170
$273$	0	0
$274$	7.00000	0.422885
$275$	−3.00000	−0.180907
$276$	0	0
$277$	3.00000	0.180253	0.0901263	−	0.995930i	$-0.471273\pi$
$277$	3.00000	0.180253	0.0901263	+	0.995930i	$0.471273\pi$
$278$	12.0000	0.719712
$279$	0	0
$280$	4.00000	0.239046
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	0	0
$283$	−14.0000	−0.832214	−0.416107	−	0.909316i	$-0.636606\pi$
$283$	−14.0000	−0.832214	−0.416107	+	0.909316i	$0.636606\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	9.00000	0.532181
$287$	24.0000	1.41668
$288$	0	0
$289$	8.00000	0.470588
$290$	−18.0000	−1.05700
$291$	0	0
$292$	−2.00000	−0.117041
$293$	−29.0000	−1.69420	−0.847099	−	0.531435i	$-0.821653\pi$
$293$	−29.0000	−1.69420	−0.847099	+	0.531435i	$0.821653\pi$
$294$	0	0
$295$	10.0000	0.582223
$296$	−8.00000	−0.464991
$297$	0	0
$298$	−9.00000	−0.521356
$299$	−12.0000	−0.693978
$300$	0	0
$301$	−12.0000	−0.691669
$302$	−12.0000	−0.690522
$303$	0	0
$304$	1.00000	0.0573539
$305$	−6.00000	−0.343559
$306$	0	0
$307$	10.0000	0.570730	0.285365	−	0.958419i	$-0.407885\pi$
$307$	10.0000	0.570730	0.285365	+	0.958419i	$0.407885\pi$
$308$	6.00000	0.341882
$309$	0	0
$310$	−10.0000	−0.567962
$311$	33.0000	1.87126	0.935629	−	0.352985i	$-0.114833\pi$
$311$	33.0000	1.87126	0.935629	+	0.352985i	$0.114833\pi$
$312$	0	0
$313$	8.00000	0.452187	0.226093	−	0.974106i	$-0.427405\pi$
$313$	8.00000	0.452187	0.226093	+	0.974106i	$0.427405\pi$
$314$	2.00000	0.112867
$315$	0	0
$316$	−8.00000	−0.450035
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	0	0
$319$	−27.0000	−1.51171
$320$	2.00000	0.111803
$321$	0	0
$322$	−8.00000	−0.445823
$323$	5.00000	0.278207
$324$	0	0
$325$	−3.00000	−0.166410
$326$	−1.00000	−0.0553849
$327$	0	0
$328$	12.0000	0.662589
$329$	16.0000	0.882109
$330$	0	0
$331$	−17.0000	−0.934405	−0.467202	−	0.884150i	$-0.654738\pi$
$331$	−17.0000	−0.934405	−0.467202	+	0.884150i	$0.654738\pi$
$332$	14.0000	0.768350
$333$	0	0
$334$	−21.0000	−1.14907
$335$	0	0
$336$	0	0
$337$	1.00000	0.0544735	0.0272367	−	0.999629i	$-0.491329\pi$
$337$	1.00000	0.0544735	0.0272367	+	0.999629i	$0.491329\pi$
$338$	−4.00000	−0.217571
$339$	0	0
$340$	10.0000	0.542326
$341$	−15.0000	−0.812296
$342$	0	0
$343$	−20.0000	−1.07990
$344$	−6.00000	−0.323498
$345$	0	0
$346$	11.0000	0.591364
$347$	30.0000	1.61048	0.805242	−	0.592946i	$-0.202035\pi$
$347$	30.0000	1.61048	0.805242	+	0.592946i	$0.202035\pi$
$348$	0	0
$349$	−30.0000	−1.60586	−0.802932	−	0.596071i	$-0.796728\pi$
$349$	−30.0000	−1.60586	−0.802932	+	0.596071i	$0.796728\pi$
$350$	−2.00000	−0.106904
$351$	0	0
$352$	3.00000	0.159901
$353$	−28.0000	−1.49029	−0.745145	−	0.666903i	$-0.767620\pi$
$353$	−28.0000	−1.49029	−0.745145	+	0.666903i	$0.767620\pi$
$354$	0	0
$355$	16.0000	0.849192
$356$	14.0000	0.741999
$357$	0	0
$358$	4.00000	0.211407
$359$	14.0000	0.738892	0.369446	−	0.929252i	$-0.379548\pi$
$359$	14.0000	0.738892	0.369446	+	0.929252i	$0.379548\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	20.0000	1.05118
$363$	0	0
$364$	6.00000	0.314485
$365$	−4.00000	−0.209370
$366$	0	0
$367$	−20.0000	−1.04399	−0.521996	−	0.852948i	$-0.674812\pi$
$367$	−20.0000	−1.04399	−0.521996	+	0.852948i	$0.674812\pi$
$368$	−4.00000	−0.208514
$369$	0	0
$370$	−16.0000	−0.831800
$371$	−24.0000	−1.24602
$372$	0	0
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	15.0000	0.775632
$375$	0	0
$376$	8.00000	0.412568
$377$	−27.0000	−1.39057
$378$	0	0
$379$	10.0000	0.513665	0.256833	−	0.966456i	$-0.417321\pi$
$379$	10.0000	0.513665	0.256833	+	0.966456i	$0.417321\pi$
$380$	2.00000	0.102598
$381$	0	0
$382$	5.00000	0.255822
$383$	9.00000	0.459879	0.229939	−	0.973205i	$-0.426147\pi$
$383$	9.00000	0.459879	0.229939	+	0.973205i	$0.426147\pi$
$384$	0	0
$385$	12.0000	0.611577
$386$	19.0000	0.967075
$387$	0	0
$388$	12.0000	0.609208
$389$	21.0000	1.06474	0.532371	−	0.846511i	$-0.321301\pi$
$389$	21.0000	1.06474	0.532371	+	0.846511i	$0.321301\pi$
$390$	0	0
$391$	−20.0000	−1.01144
$392$	−3.00000	−0.151523
$393$	0	0
$394$	−6.00000	−0.302276
$395$	−16.0000	−0.805047
$396$	0	0
$397$	−14.0000	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	−17.0000	−0.852133
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	17.0000	0.848939	0.424470	−	0.905442i	$-0.360461\pi$
$401$	17.0000	0.848939	0.424470	+	0.905442i	$0.360461\pi$
$402$	0	0
$403$	−15.0000	−0.747203
$404$	−14.0000	−0.696526
$405$	0	0
$406$	−18.0000	−0.893325
$407$	−24.0000	−1.18964
$408$	0	0
$409$	−22.0000	−1.08783	−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000	−1.08783	−0.543915	+	0.839140i	$0.683059\pi$
$410$	24.0000	1.18528
$411$	0	0
$412$	−8.00000	−0.394132
$413$	10.0000	0.492068
$414$	0	0
$415$	28.0000	1.37447
$416$	3.00000	0.147087
$417$	0	0
$418$	3.00000	0.146735
$419$	−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
$421$	−27.0000	−1.31590	−0.657950	−	0.753062i	$-0.728576\pi$
$421$	−27.0000	−1.31590	−0.657950	+	0.753062i	$0.728576\pi$
$422$	2.00000	0.0973585
$423$	0	0
$424$	−12.0000	−0.582772
$425$	−5.00000	−0.242536
$426$	0	0
$427$	−6.00000	−0.290360
$428$	−3.00000	−0.145010
$429$	0	0
$430$	−12.0000	−0.578691
$431$	−28.0000	−1.34871	−0.674356	−	0.738406i	$-0.735579\pi$
$431$	−28.0000	−1.34871	−0.674356	+	0.738406i	$0.735579\pi$
$432$	0	0
$433$	6.00000	0.288342	0.144171	−	0.989553i	$-0.453949\pi$
$433$	6.00000	0.288342	0.144171	+	0.989553i	$0.453949\pi$
$434$	−10.0000	−0.480015
$435$	0	0
$436$	−13.0000	−0.622587
$437$	−4.00000	−0.191346
$438$	0	0
$439$	10.0000	0.477274	0.238637	−	0.971109i	$-0.423299\pi$
$439$	10.0000	0.477274	0.238637	+	0.971109i	$0.423299\pi$
$440$	6.00000	0.286039
$441$	0	0
$442$	15.0000	0.713477
$443$	−6.00000	−0.285069	−0.142534	−	0.989790i	$-0.545525\pi$
$443$	−6.00000	−0.285069	−0.142534	+	0.989790i	$0.545525\pi$
$444$	0	0
$445$	28.0000	1.32733
$446$	−19.0000	−0.899676
$447$	0	0
$448$	2.00000	0.0944911
$449$	−23.0000	−1.08544	−0.542719	−	0.839915i	$-0.682605\pi$
$449$	−23.0000	−1.08544	−0.542719	+	0.839915i	$0.682605\pi$
$450$	0	0
$451$	36.0000	1.69517
$452$	−4.00000	−0.188144
$453$	0	0
$454$	6.00000	0.281594
$455$	12.0000	0.562569
$456$	0	0
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	20.0000	0.934539
$459$	0	0
$460$	−8.00000	−0.373002
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	0	0
$463$	13.0000	0.604161	0.302081	−	0.953282i	$-0.402319\pi$
$463$	13.0000	0.604161	0.302081	+	0.953282i	$0.402319\pi$
$464$	−9.00000	−0.417815
$465$	0	0
$466$	−20.0000	−0.926482
$467$	−25.0000	−1.15686	−0.578431	−	0.815731i	$-0.696335\pi$
$467$	−25.0000	−1.15686	−0.578431	+	0.815731i	$0.696335\pi$
$468$	0	0
$469$	0	0
$470$	16.0000	0.738025
$471$	0	0
$472$	5.00000	0.230144
$473$	−18.0000	−0.827641
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	10.0000	0.458349
$477$	0	0
$478$	−8.00000	−0.365911
$479$	12.0000	0.548294	0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000	0.548294	0.274147	+	0.961688i	$0.411605\pi$
$480$	0	0
$481$	−24.0000	−1.09431
$482$	8.00000	0.364390
$483$	0	0
$484$	−2.00000	−0.0909091
$485$	24.0000	1.08978
$486$	0	0
$487$	28.0000	1.26880	0.634401	−	0.773004i	$-0.281247\pi$
$487$	28.0000	1.26880	0.634401	+	0.773004i	$0.281247\pi$
$488$	−3.00000	−0.135804
$489$	0	0
$490$	−6.00000	−0.271052
$491$	−18.0000	−0.812329	−0.406164	−	0.913800i	$-0.633134\pi$
$491$	−18.0000	−0.812329	−0.406164	+	0.913800i	$0.633134\pi$
$492$	0	0
$493$	−45.0000	−2.02670
$494$	3.00000	0.134976
$495$	0	0
$496$	−5.00000	−0.224507
$497$	16.0000	0.717698
$498$	0	0
$499$	40.0000	1.79065	0.895323	−	0.445418i	$-0.146945\pi$
$499$	40.0000	1.79065	0.895323	+	0.445418i	$0.146945\pi$
$500$	−12.0000	−0.536656
$501$	0	0
$502$	22.0000	0.981908
$503$	2.00000	0.0891756	0.0445878	−	0.999005i	$-0.485803\pi$
$503$	2.00000	0.0891756	0.0445878	+	0.999005i	$0.485803\pi$
$504$	0	0
$505$	−28.0000	−1.24598
$506$	−12.0000	−0.533465
$507$	0	0
$508$	15.0000	0.665517
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	0	0
$511$	−4.00000	−0.176950
$512$	1.00000	0.0441942
$513$	0	0
$514$	−26.0000	−1.14681
$515$	−16.0000	−0.705044
$516$	0	0
$517$	24.0000	1.05552
$518$	−16.0000	−0.703000
$519$	0	0
$520$	6.00000	0.263117
$521$	−27.0000	−1.18289	−0.591446	−	0.806345i	$-0.701443\pi$
$521$	−27.0000	−1.18289	−0.591446	+	0.806345i	$0.701443\pi$
$522$	0	0
$523$	−23.0000	−1.00572	−0.502860	−	0.864368i	$-0.667719\pi$
$523$	−23.0000	−1.00572	−0.502860	+	0.864368i	$0.667719\pi$
$524$	1.00000	0.0436852
$525$	0	0
$526$	21.0000	0.915644
$527$	−25.0000	−1.08902
$528$	0	0
$529$	−7.00000	−0.304348
$530$	−24.0000	−1.04249
$531$	0	0
$532$	2.00000	0.0867110
$533$	36.0000	1.55933
$534$	0	0
$535$	−6.00000	−0.259403
$536$	0	0
$537$	0	0
$538$	8.00000	0.344904
$539$	−9.00000	−0.387657
$540$	0	0
$541$	10.0000	0.429934	0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000	0.429934	0.214967	+	0.976621i	$0.431036\pi$
$542$	4.00000	0.171815
$543$	0	0
$544$	5.00000	0.214373
$545$	−26.0000	−1.11372
$546$	0	0
$547$	−1.00000	−0.0427569	−0.0213785	−	0.999771i	$-0.506805\pi$
$547$	−1.00000	−0.0427569	−0.0213785	+	0.999771i	$0.506805\pi$
$548$	7.00000	0.299025
$549$	0	0
$550$	−3.00000	−0.127920
$551$	−9.00000	−0.383413
$552$	0	0
$553$	−16.0000	−0.680389
$554$	3.00000	0.127458
$555$	0	0
$556$	12.0000	0.508913
$557$	−24.0000	−1.01691	−0.508456	−	0.861088i	$-0.669784\pi$
$557$	−24.0000	−1.01691	−0.508456	+	0.861088i	$0.669784\pi$
$558$	0	0
$559$	−18.0000	−0.761319
$560$	4.00000	0.169031
$561$	0	0
$562$	10.0000	0.421825
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	0	0
$565$	−8.00000	−0.336563
$566$	−14.0000	−0.588464
$567$	0	0
$568$	8.00000	0.335673
$569$	22.0000	0.922288	0.461144	−	0.887325i	$-0.347439\pi$
$569$	22.0000	0.922288	0.461144	+	0.887325i	$0.347439\pi$
$570$	0	0
$571$	23.0000	0.962520	0.481260	−	0.876578i	$-0.340179\pi$
$571$	23.0000	0.962520	0.481260	+	0.876578i	$0.340179\pi$
$572$	9.00000	0.376309
$573$	0	0
$574$	24.0000	1.00174
$575$	4.00000	0.166812
$576$	0	0
$577$	5.00000	0.208153	0.104076	−	0.994569i	$-0.466811\pi$
$577$	5.00000	0.208153	0.104076	+	0.994569i	$0.466811\pi$
$578$	8.00000	0.332756
$579$	0	0
$580$	−18.0000	−0.747409
$581$	28.0000	1.16164
$582$	0	0
$583$	−36.0000	−1.49097
$584$	−2.00000	−0.0827606
$585$	0	0
$586$	−29.0000	−1.19798
$587$	33.0000	1.36206	0.681028	−	0.732257i	$-0.261533\pi$
$587$	33.0000	1.36206	0.681028	+	0.732257i	$0.261533\pi$
$588$	0	0
$589$	−5.00000	−0.206021
$590$	10.0000	0.411693
$591$	0	0
$592$	−8.00000	−0.328798
$593$	−19.0000	−0.780236	−0.390118	−	0.920765i	$-0.627566\pi$
$593$	−19.0000	−0.780236	−0.390118	+	0.920765i	$0.627566\pi$
$594$	0	0
$595$	20.0000	0.819920
$596$	−9.00000	−0.368654
$597$	0	0
$598$	−12.0000	−0.490716
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	0	0
$601$	−19.0000	−0.775026	−0.387513	−	0.921864i	$-0.626666\pi$
$601$	−19.0000	−0.775026	−0.387513	+	0.921864i	$0.626666\pi$
$602$	−12.0000	−0.489083
$603$	0	0
$604$	−12.0000	−0.488273
$605$	−4.00000	−0.162623
$606$	0	0
$607$	44.0000	1.78590	0.892952	−	0.450151i	$-0.148630\pi$
$607$	44.0000	1.78590	0.892952	+	0.450151i	$0.148630\pi$
$608$	1.00000	0.0405554
$609$	0	0
$610$	−6.00000	−0.242933
$611$	24.0000	0.970936
$612$	0	0
$613$	−3.00000	−0.121169	−0.0605844	−	0.998163i	$-0.519296\pi$
$613$	−3.00000	−0.121169	−0.0605844	+	0.998163i	$0.519296\pi$
$614$	10.0000	0.403567
$615$	0	0
$616$	6.00000	0.241747
$617$	−38.0000	−1.52982	−0.764911	−	0.644136i	$-0.777217\pi$
$617$	−38.0000	−1.52982	−0.764911	+	0.644136i	$0.777217\pi$
$618$	0	0
$619$	−31.0000	−1.24600	−0.622998	−	0.782224i	$-0.714085\pi$
$619$	−31.0000	−1.24600	−0.622998	+	0.782224i	$0.714085\pi$
$620$	−10.0000	−0.401610
$621$	0	0
$622$	33.0000	1.32318
$623$	28.0000	1.12180
$624$	0	0
$625$	−19.0000	−0.760000
$626$	8.00000	0.319744
$627$	0	0
$628$	2.00000	0.0798087
$629$	−40.0000	−1.59490
$630$	0	0
$631$	26.0000	1.03504	0.517522	−	0.855670i	$-0.326855\pi$
$631$	26.0000	1.03504	0.517522	+	0.855670i	$0.326855\pi$
$632$	−8.00000	−0.318223
$633$	0	0
$634$	−18.0000	−0.714871
$635$	30.0000	1.19051
$636$	0	0
$637$	−9.00000	−0.356593
$638$	−27.0000	−1.06894
$639$	0	0
$640$	2.00000	0.0790569
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	0	0
$643$	21.0000	0.828159	0.414080	−	0.910241i	$-0.364104\pi$
$643$	21.0000	0.828159	0.414080	+	0.910241i	$0.364104\pi$
$644$	−8.00000	−0.315244
$645$	0	0
$646$	5.00000	0.196722
$647$	31.0000	1.21874	0.609368	−	0.792888i	$-0.291423\pi$
$647$	31.0000	1.21874	0.609368	+	0.792888i	$0.291423\pi$
$648$	0	0
$649$	15.0000	0.588802
$650$	−3.00000	−0.117670
$651$	0	0
$652$	−1.00000	−0.0391630
$653$	38.0000	1.48705	0.743527	−	0.668705i	$-0.233151\pi$
$653$	38.0000	1.48705	0.743527	+	0.668705i	$0.233151\pi$
$654$	0	0
$655$	2.00000	0.0781465
$656$	12.0000	0.468521
$657$	0	0
$658$	16.0000	0.623745
$659$	−13.0000	−0.506408	−0.253204	−	0.967413i	$-0.581484\pi$
$659$	−13.0000	−0.506408	−0.253204	+	0.967413i	$0.581484\pi$
$660$	0	0
$661$	−20.0000	−0.777910	−0.388955	−	0.921257i	$-0.627164\pi$
$661$	−20.0000	−0.777910	−0.388955	+	0.921257i	$0.627164\pi$
$662$	−17.0000	−0.660724
$663$	0	0
$664$	14.0000	0.543305
$665$	4.00000	0.155113
$666$	0	0
$667$	36.0000	1.39393
$668$	−21.0000	−0.812514
$669$	0	0
$670$	0	0
$671$	−9.00000	−0.347441
$672$	0	0
$673$	16.0000	0.616755	0.308377	−	0.951264i	$-0.400214\pi$
$673$	16.0000	0.616755	0.308377	+	0.951264i	$0.400214\pi$
$674$	1.00000	0.0385186
$675$	0	0
$676$	−4.00000	−0.153846
$677$	−30.0000	−1.15299	−0.576497	−	0.817099i	$-0.695581\pi$
$677$	−30.0000	−1.15299	−0.576497	+	0.817099i	$0.695581\pi$
$678$	0	0
$679$	24.0000	0.921035
$680$	10.0000	0.383482
$681$	0	0
$682$	−15.0000	−0.574380
$683$	−41.0000	−1.56882	−0.784411	−	0.620242i	$-0.787034\pi$
$683$	−41.0000	−1.56882	−0.784411	+	0.620242i	$0.787034\pi$
$684$	0	0
$685$	14.0000	0.534913
$686$	−20.0000	−0.763604
$687$	0	0
$688$	−6.00000	−0.228748
$689$	−36.0000	−1.37149
$690$	0	0
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	11.0000	0.418157
$693$	0	0
$694$	30.0000	1.13878
$695$	24.0000	0.910372
$696$	0	0
$697$	60.0000	2.27266
$698$	−30.0000	−1.13552
$699$	0	0
$700$	−2.00000	−0.0755929
$701$	−50.0000	−1.88847	−0.944237	−	0.329267i	$-0.893198\pi$
$701$	−50.0000	−1.88847	−0.944237	+	0.329267i	$0.893198\pi$
$702$	0	0
$703$	−8.00000	−0.301726
$704$	3.00000	0.113067
$705$	0	0
$706$	−28.0000	−1.05379
$707$	−28.0000	−1.05305
$708$	0	0
$709$	18.0000	0.676004	0.338002	−	0.941145i	$-0.390249\pi$
$709$	18.0000	0.676004	0.338002	+	0.941145i	$0.390249\pi$
$710$	16.0000	0.600469
$711$	0	0
$712$	14.0000	0.524672
$713$	20.0000	0.749006
$714$	0	0
$715$	18.0000	0.673162
$716$	4.00000	0.149487
$717$	0	0
$718$	14.0000	0.522475
$719$	−8.00000	−0.298350	−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000	−0.298350	−0.149175	+	0.988811i	$0.547662\pi$
$720$	0	0
$721$	−16.0000	−0.595871
$722$	−18.0000	−0.669891
$723$	0	0
$724$	20.0000	0.743294
$725$	9.00000	0.334252
$726$	0	0
$727$	40.0000	1.48352	0.741759	−	0.670667i	$-0.233992\pi$
$727$	40.0000	1.48352	0.741759	+	0.670667i	$0.233992\pi$
$728$	6.00000	0.222375
$729$	0	0
$730$	−4.00000	−0.148047
$731$	−30.0000	−1.10959
$732$	0	0
$733$	6.00000	0.221615	0.110808	−	0.993842i	$-0.464656\pi$
$733$	6.00000	0.221615	0.110808	+	0.993842i	$0.464656\pi$
$734$	−20.0000	−0.738213
$735$	0	0
$736$	−4.00000	−0.147442
$737$	0	0
$738$	0	0
$739$	−10.0000	−0.367856	−0.183928	−	0.982940i	$-0.558881\pi$
$739$	−10.0000	−0.367856	−0.183928	+	0.982940i	$0.558881\pi$
$740$	−16.0000	−0.588172
$741$	0	0
$742$	−24.0000	−0.881068
$743$	12.0000	0.440237	0.220119	−	0.975473i	$-0.429356\pi$
$743$	12.0000	0.440237	0.220119	+	0.975473i	$0.429356\pi$
$744$	0	0
$745$	−18.0000	−0.659469
$746$	10.0000	0.366126
$747$	0	0
$748$	15.0000	0.548454
$749$	−6.00000	−0.219235
$750$	0	0
$751$	37.0000	1.35015	0.675075	−	0.737749i	$-0.264111\pi$
$751$	37.0000	1.35015	0.675075	+	0.737749i	$0.264111\pi$
$752$	8.00000	0.291730
$753$	0	0
$754$	−27.0000	−0.983282
$755$	−24.0000	−0.873449
$756$	0	0
$757$	19.0000	0.690567	0.345283	−	0.938498i	$-0.387783\pi$
$757$	19.0000	0.690567	0.345283	+	0.938498i	$0.387783\pi$
$758$	10.0000	0.363216
$759$	0	0
$760$	2.00000	0.0725476
$761$	13.0000	0.471250	0.235625	−	0.971844i	$-0.424286\pi$
$761$	13.0000	0.471250	0.235625	+	0.971844i	$0.424286\pi$
$762$	0	0
$763$	−26.0000	−0.941263
$764$	5.00000	0.180894
$765$	0	0
$766$	9.00000	0.325183
$767$	15.0000	0.541619
$768$	0	0
$769$	−17.0000	−0.613036	−0.306518	−	0.951865i	$-0.599164\pi$
$769$	−17.0000	−0.613036	−0.306518	+	0.951865i	$0.599164\pi$
$770$	12.0000	0.432450
$771$	0	0
$772$	19.0000	0.683825
$773$	53.0000	1.90628	0.953139	−	0.302534i	$-0.0978324\pi$
$773$	53.0000	1.90628	0.953139	+	0.302534i	$0.0978324\pi$
$774$	0	0
$775$	5.00000	0.179605
$776$	12.0000	0.430775
$777$	0	0
$778$	21.0000	0.752886
$779$	12.0000	0.429945
$780$	0	0
$781$	24.0000	0.858788
$782$	−20.0000	−0.715199
$783$	0	0
$784$	−3.00000	−0.107143
$785$	4.00000	0.142766
$786$	0	0
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	−6.00000	−0.213741
$789$	0	0
$790$	−16.0000	−0.569254
$791$	−8.00000	−0.284447
$792$	0	0
$793$	−9.00000	−0.319599
$794$	−14.0000	−0.496841
$795$	0	0
$796$	−17.0000	−0.602549
$797$	30.0000	1.06265	0.531327	−	0.847167i	$-0.321693\pi$
$797$	30.0000	1.06265	0.531327	+	0.847167i	$0.321693\pi$
$798$	0	0
$799$	40.0000	1.41510
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	17.0000	0.600291
$803$	−6.00000	−0.211735
$804$	0	0
$805$	−16.0000	−0.563926
$806$	−15.0000	−0.528352
$807$	0	0
$808$	−14.0000	−0.492518
$809$	−26.0000	−0.914111	−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000	−0.914111	−0.457056	+	0.889438i	$0.651096\pi$
$810$	0	0
$811$	−18.0000	−0.632065	−0.316033	−	0.948748i	$-0.602351\pi$
$811$	−18.0000	−0.632065	−0.316033	+	0.948748i	$0.602351\pi$
$812$	−18.0000	−0.631676
$813$	0	0
$814$	−24.0000	−0.841200
$815$	−2.00000	−0.0700569
$816$	0	0
$817$	−6.00000	−0.209913
$818$	−22.0000	−0.769212
$819$	0	0
$820$	24.0000	0.838116
$821$	−20.0000	−0.698005	−0.349002	−	0.937122i	$-0.613479\pi$
$821$	−20.0000	−0.698005	−0.349002	+	0.937122i	$0.613479\pi$
$822$	0	0
$823$	47.0000	1.63832	0.819159	−	0.573567i	$-0.194441\pi$
$823$	47.0000	1.63832	0.819159	+	0.573567i	$0.194441\pi$
$824$	−8.00000	−0.278693
$825$	0	0
$826$	10.0000	0.347945
$827$	−33.0000	−1.14752	−0.573761	−	0.819023i	$-0.694516\pi$
$827$	−33.0000	−1.14752	−0.573761	+	0.819023i	$0.694516\pi$
$828$	0	0
$829$	−15.0000	−0.520972	−0.260486	−	0.965478i	$-0.583883\pi$
$829$	−15.0000	−0.520972	−0.260486	+	0.965478i	$0.583883\pi$
$830$	28.0000	0.971894
$831$	0	0
$832$	3.00000	0.104006
$833$	−15.0000	−0.519719
$834$	0	0
$835$	−42.0000	−1.45347
$836$	3.00000	0.103757
$837$	0	0
$838$	−12.0000	−0.414533
$839$	29.0000	1.00119	0.500596	−	0.865681i	$-0.333114\pi$
$839$	29.0000	1.00119	0.500596	+	0.865681i	$0.333114\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	−27.0000	−0.930481
$843$	0	0
$844$	2.00000	0.0688428
$845$	−8.00000	−0.275208
$846$	0	0
$847$	−4.00000	−0.137442
$848$	−12.0000	−0.412082
$849$	0	0
$850$	−5.00000	−0.171499
$851$	32.0000	1.09695
$852$	0	0
$853$	−44.0000	−1.50653	−0.753266	−	0.657716i	$-0.771523\pi$
$853$	−44.0000	−1.50653	−0.753266	+	0.657716i	$0.771523\pi$
$854$	−6.00000	−0.205316
$855$	0	0
$856$	−3.00000	−0.102538
$857$	45.0000	1.53717	0.768585	−	0.639747i	$-0.220961\pi$
$857$	45.0000	1.53717	0.768585	+	0.639747i	$0.220961\pi$
$858$	0	0
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	−12.0000	−0.409197
$861$	0	0
$862$	−28.0000	−0.953684
$863$	−5.00000	−0.170202	−0.0851010	−	0.996372i	$-0.527121\pi$
$863$	−5.00000	−0.170202	−0.0851010	+	0.996372i	$0.527121\pi$
$864$	0	0
$865$	22.0000	0.748022
$866$	6.00000	0.203888
$867$	0	0
$868$	−10.0000	−0.339422
$869$	−24.0000	−0.814144
$870$	0	0
$871$	0	0
$872$	−13.0000	−0.440236
$873$	0	0
$874$	−4.00000	−0.135302
$875$	−24.0000	−0.811348
$876$	0	0
$877$	−42.0000	−1.41824	−0.709120	−	0.705088i	$-0.750907\pi$
$877$	−42.0000	−1.41824	−0.709120	+	0.705088i	$0.750907\pi$
$878$	10.0000	0.337484
$879$	0	0
$880$	6.00000	0.202260
$881$	50.0000	1.68454	0.842271	−	0.539054i	$-0.181218\pi$
$881$	50.0000	1.68454	0.842271	+	0.539054i	$0.181218\pi$
$882$	0	0
$883$	−17.0000	−0.572096	−0.286048	−	0.958215i	$-0.592342\pi$
$883$	−17.0000	−0.572096	−0.286048	+	0.958215i	$0.592342\pi$
$884$	15.0000	0.504505
$885$	0	0
$886$	−6.00000	−0.201574
$887$	24.0000	0.805841	0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000	0.805841	0.402921	+	0.915235i	$0.367995\pi$
$888$	0	0
$889$	30.0000	1.00617
$890$	28.0000	0.938562
$891$	0	0
$892$	−19.0000	−0.636167
$893$	8.00000	0.267710
$894$	0	0
$895$	8.00000	0.267411
$896$	2.00000	0.0668153
$897$	0	0
$898$	−23.0000	−0.767520
$899$	45.0000	1.50083
$900$	0	0
$901$	−60.0000	−1.99889
$902$	36.0000	1.19867
$903$	0	0
$904$	−4.00000	−0.133038
$905$	40.0000	1.32964
$906$	0	0
$907$	−34.0000	−1.12895	−0.564476	−	0.825450i	$-0.690922\pi$
$907$	−34.0000	−1.12895	−0.564476	+	0.825450i	$0.690922\pi$
$908$	6.00000	0.199117
$909$	0	0
$910$	12.0000	0.397796
$911$	−31.0000	−1.02708	−0.513538	−	0.858067i	$-0.671665\pi$
$911$	−31.0000	−1.02708	−0.513538	+	0.858067i	$0.671665\pi$
$912$	0	0
$913$	42.0000	1.39000
$914$	−10.0000	−0.330771
$915$	0	0
$916$	20.0000	0.660819
$917$	2.00000	0.0660458
$918$	0	0
$919$	25.0000	0.824674	0.412337	−	0.911031i	$-0.364713\pi$
$919$	25.0000	0.824674	0.412337	+	0.911031i	$0.364713\pi$
$920$	−8.00000	−0.263752
$921$	0	0
$922$	6.00000	0.197599
$923$	24.0000	0.789970
$924$	0	0
$925$	8.00000	0.263038
$926$	13.0000	0.427207
$927$	0	0
$928$	−9.00000	−0.295439
$929$	−14.0000	−0.459325	−0.229663	−	0.973270i	$-0.573762\pi$
$929$	−14.0000	−0.459325	−0.229663	+	0.973270i	$0.573762\pi$
$930$	0	0
$931$	−3.00000	−0.0983210
$932$	−20.0000	−0.655122
$933$	0	0
$934$	−25.0000	−0.818025
$935$	30.0000	0.981105
$936$	0	0
$937$	17.0000	0.555366	0.277683	−	0.960673i	$-0.410434\pi$
$937$	17.0000	0.555366	0.277683	+	0.960673i	$0.410434\pi$
$938$	0	0
$939$	0	0
$940$	16.0000	0.521862
$941$	49.0000	1.59735	0.798677	−	0.601760i	$-0.205534\pi$
$941$	49.0000	1.59735	0.798677	+	0.601760i	$0.205534\pi$
$942$	0	0
$943$	−48.0000	−1.56310
$944$	5.00000	0.162736
$945$	0	0
$946$	−18.0000	−0.585230
$947$	40.0000	1.29983	0.649913	−	0.760009i	$-0.274805\pi$
$947$	40.0000	1.29983	0.649913	+	0.760009i	$0.274805\pi$
$948$	0	0
$949$	−6.00000	−0.194768
$950$	−1.00000	−0.0324443
$951$	0	0
$952$	10.0000	0.324102
$953$	−8.00000	−0.259145	−0.129573	−	0.991570i	$-0.541361\pi$
$953$	−8.00000	−0.259145	−0.129573	+	0.991570i	$0.541361\pi$
$954$	0	0
$955$	10.0000	0.323592
$956$	−8.00000	−0.258738
$957$	0	0
$958$	12.0000	0.387702
$959$	14.0000	0.452084
$960$	0	0
$961$	−6.00000	−0.193548
$962$	−24.0000	−0.773791
$963$	0	0
$964$	8.00000	0.257663
$965$	38.0000	1.22326
$966$	0	0
$967$	33.0000	1.06121	0.530604	−	0.847620i	$-0.321965\pi$
$967$	33.0000	1.06121	0.530604	+	0.847620i	$0.321965\pi$
$968$	−2.00000	−0.0642824
$969$	0	0
$970$	24.0000	0.770594
$971$	−20.0000	−0.641831	−0.320915	−	0.947108i	$-0.603990\pi$
$971$	−20.0000	−0.641831	−0.320915	+	0.947108i	$0.603990\pi$
$972$	0	0
$973$	24.0000	0.769405
$974$	28.0000	0.897178
$975$	0	0
$976$	−3.00000	−0.0960277
$977$	−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
$979$	42.0000	1.34233
$980$	−6.00000	−0.191663
$981$	0	0
$982$	−18.0000	−0.574403
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	−12.0000	−0.382352
$986$	−45.0000	−1.43309
$987$	0	0
$988$	3.00000	0.0954427
$989$	24.0000	0.763156
$990$	0	0
$991$	−46.0000	−1.46124	−0.730619	−	0.682785i	$-0.760768\pi$
$991$	−46.0000	−1.46124	−0.730619	+	0.682785i	$0.760768\pi$
$992$	−5.00000	−0.158750
$993$	0	0
$994$	16.0000	0.507489
$995$	−34.0000	−1.07787
$996$	0	0
$997$	35.0000	1.10846	0.554231	−	0.832363i	$-0.313013\pi$
$997$	35.0000	1.10846	0.554231	+	0.832363i	$0.313013\pi$
$998$	40.0000	1.26618
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2358.2.a.w.1.1		1
3.2	odd	2		786.2.a.a.1.1	✓	1
12.11	even	2		6288.2.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
786.2.a.a.1.1	✓	1	3.2	odd	2
2358.2.a.w.1.1		1	1.1	even	1	trivial
6288.2.a.i.1.1		1	12.11	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:	\( N \)	\(=\)	\( 2358 = 2 \cdot 3^{2} \cdot 131 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2358.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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