LMFDB - Embedded newform 1950.2.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1950,2,Mod(1,1950)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1950, 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("1950.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1950.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:	$15.5708283941$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 390)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1950.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -6.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} +1.00000 q^{16} +1.00000 q^{18} +6.00000 q^{19} -6.00000 q^{22} +6.00000 q^{23} -1.00000 q^{24} +1.00000 q^{26} -1.00000 q^{27} +2.00000 q^{29} +4.00000 q^{31} +1.00000 q^{32} +6.00000 q^{33} +1.00000 q^{36} +10.0000 q^{37} +6.00000 q^{38} -1.00000 q^{39} -6.00000 q^{41} +8.00000 q^{43} -6.00000 q^{44} +6.00000 q^{46} +8.00000 q^{47} -1.00000 q^{48} -7.00000 q^{49} +1.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} -6.00000 q^{57} +2.00000 q^{58} +10.0000 q^{59} -6.00000 q^{61} +4.00000 q^{62} +1.00000 q^{64} +6.00000 q^{66} +4.00000 q^{67} -6.00000 q^{69} -8.00000 q^{71} +1.00000 q^{72} +6.00000 q^{73} +10.0000 q^{74} +6.00000 q^{76} -1.00000 q^{78} +16.0000 q^{79} +1.00000 q^{81} -6.00000 q^{82} +4.00000 q^{83} +8.00000 q^{86} -2.00000 q^{87} -6.00000 q^{88} -10.0000 q^{89} +6.00000 q^{92} -4.00000 q^{93} +8.00000 q^{94} -1.00000 q^{96} +2.00000 q^{97} -7.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

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
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−6.00000	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	−1.00000	−0.288675
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	1.00000	0.235702
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	0	0
$21$	0	0
$22$	−6.00000	−1.27920
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	1.00000	0.196116
$27$	−1.00000	−0.192450
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	1.00000	0.176777
$33$	6.00000	1.04447
$34$	0	0
$35$	0	0
$36$	1.00000	0.166667
$37$	10.0000	1.64399	0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000	1.64399	0.821995	+	0.569495i	$0.192861\pi$
$38$	6.00000	0.973329
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	−6.00000	−0.904534
$45$	0	0
$46$	6.00000	0.884652
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	−1.00000	−0.144338
$49$	−7.00000	−1.00000
$50$	0	0
$51$	0	0
$52$	1.00000	0.138675
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	0	0
$57$	−6.00000	−0.794719
$58$	2.00000	0.262613
$59$	10.0000	1.30189	0.650945	−	0.759125i	$-0.274373\pi$
$59$	10.0000	1.30189	0.650945	+	0.759125i	$0.274373\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	4.00000	0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	6.00000	0.738549
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	−6.00000	−0.722315
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	1.00000	0.117851
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	10.0000	1.16248
$75$	0	0
$76$	6.00000	0.688247
$77$	0	0
$78$	−1.00000	−0.113228
$79$	16.0000	1.80014	0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000	1.80014	0.900070	+	0.435745i	$0.143515\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	0	0
$86$	8.00000	0.862662
$87$	−2.00000	−0.214423
$88$	−6.00000	−0.639602
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	0	0
$92$	6.00000	0.625543
$93$	−4.00000	−0.414781
$94$	8.00000	0.825137
$95$	0	0
$96$	−1.00000	−0.102062
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	−7.00000	−0.707107
$99$	−6.00000	−0.603023
$100$	0	0
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	0	0
$103$	−10.0000	−0.985329	−0.492665	−	0.870219i	$-0.663977\pi$
$103$	−10.0000	−0.985329	−0.492665	+	0.870219i	$0.663977\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	−6.00000	−0.582772
$107$	16.0000	1.54678	0.773389	−	0.633932i	$-0.218560\pi$
$107$	16.0000	1.54678	0.773389	+	0.633932i	$0.218560\pi$
$108$	−1.00000	−0.0962250
$109$	−16.0000	−1.53252	−0.766261	−	0.642529i	$-0.777885\pi$
$109$	−16.0000	−1.53252	−0.766261	+	0.642529i	$0.777885\pi$
$110$	0	0
$111$	−10.0000	−0.949158
$112$	0	0
$113$	8.00000	0.752577	0.376288	−	0.926503i	$-0.377200\pi$
$113$	8.00000	0.752577	0.376288	+	0.926503i	$0.377200\pi$
$114$	−6.00000	−0.561951
$115$	0	0
$116$	2.00000	0.185695
$117$	1.00000	0.0924500
$118$	10.0000	0.920575
$119$	0	0
$120$	0	0
$121$	25.0000	2.27273
$122$	−6.00000	−0.543214
$123$	6.00000	0.541002
$124$	4.00000	0.359211
$125$	0	0
$126$	0	0
$127$	−18.0000	−1.59724	−0.798621	−	0.601834i	$-0.794437\pi$
$127$	−18.0000	−1.59724	−0.798621	+	0.601834i	$0.794437\pi$
$128$	1.00000	0.0883883
$129$	−8.00000	−0.704361
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	6.00000	0.522233
$133$	0	0
$134$	4.00000	0.345547
$135$	0	0
$136$	0	0
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	−6.00000	−0.510754
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	−8.00000	−0.671345
$143$	−6.00000	−0.501745
$144$	1.00000	0.0833333
$145$	0	0
$146$	6.00000	0.496564
$147$	7.00000	0.577350
$148$	10.0000	0.821995
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	6.00000	0.486664
$153$	0	0
$154$	0	0
$155$	0	0
$156$	−1.00000	−0.0800641
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	16.0000	1.27289
$159$	6.00000	0.475831
$160$	0	0
$161$	0	0
$162$	1.00000	0.0785674
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	4.00000	0.310460
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	6.00000	0.458831
$172$	8.00000	0.609994
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	−2.00000	−0.151620
$175$	0	0
$176$	−6.00000	−0.452267
$177$	−10.0000	−0.751646
$178$	−10.0000	−0.749532
$179$	−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
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	6.00000	0.443533
$184$	6.00000	0.442326
$185$	0	0
$186$	−4.00000	−0.293294
$187$	0	0
$188$	8.00000	0.583460
$189$	0	0
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	−1.00000	−0.0721688
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	−7.00000	−0.500000
$197$	−22.0000	−1.56744	−0.783718	−	0.621117i	$-0.786679\pi$
$197$	−22.0000	−1.56744	−0.783718	+	0.621117i	$0.786679\pi$
$198$	−6.00000	−0.426401
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	−2.00000	−0.140720
$203$	0	0
$204$	0	0
$205$	0	0
$206$	−10.0000	−0.696733
$207$	6.00000	0.417029
$208$	1.00000	0.0693375
$209$	−36.0000	−2.49017
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	−6.00000	−0.412082
$213$	8.00000	0.548151
$214$	16.0000	1.09374
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	−16.0000	−1.08366
$219$	−6.00000	−0.405442
$220$	0	0
$221$	0	0
$222$	−10.0000	−0.671156
$223$	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
$225$	0	0
$226$	8.00000	0.532152
$227$	−20.0000	−1.32745	−0.663723	−	0.747978i	$-0.731025\pi$
$227$	−20.0000	−1.32745	−0.663723	+	0.747978i	$0.731025\pi$
$228$	−6.00000	−0.397360
$229$	20.0000	1.32164	0.660819	−	0.750546i	$-0.270209\pi$
$229$	20.0000	1.32164	0.660819	+	0.750546i	$0.270209\pi$
$230$	0	0
$231$	0	0
$232$	2.00000	0.131306
$233$	−8.00000	−0.524097	−0.262049	−	0.965055i	$-0.584398\pi$
$233$	−8.00000	−0.524097	−0.262049	+	0.965055i	$0.584398\pi$
$234$	1.00000	0.0653720
$235$	0	0
$236$	10.0000	0.650945
$237$	−16.0000	−1.03931
$238$	0	0
$239$	−28.0000	−1.81117	−0.905585	−	0.424165i	$-0.860568\pi$
$239$	−28.0000	−1.81117	−0.905585	+	0.424165i	$0.860568\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	25.0000	1.60706
$243$	−1.00000	−0.0641500
$244$	−6.00000	−0.384111
$245$	0	0
$246$	6.00000	0.382546
$247$	6.00000	0.381771
$248$	4.00000	0.254000
$249$	−4.00000	−0.253490
$250$	0	0
$251$	8.00000	0.504956	0.252478	−	0.967603i	$-0.418755\pi$
$251$	8.00000	0.504956	0.252478	+	0.967603i	$0.418755\pi$
$252$	0	0
$253$	−36.0000	−2.26330
$254$	−18.0000	−1.12942
$255$	0	0
$256$	1.00000	0.0625000
$257$	−24.0000	−1.49708	−0.748539	−	0.663090i	$-0.769245\pi$
$257$	−24.0000	−1.49708	−0.748539	+	0.663090i	$0.769245\pi$
$258$	−8.00000	−0.498058
$259$	0	0
$260$	0	0
$261$	2.00000	0.123797
$262$	12.0000	0.741362
$263$	−6.00000	−0.369976	−0.184988	−	0.982741i	$-0.559225\pi$
$263$	−6.00000	−0.369976	−0.184988	+	0.982741i	$0.559225\pi$
$264$	6.00000	0.369274
$265$	0	0
$266$	0	0
$267$	10.0000	0.611990
$268$	4.00000	0.244339
$269$	14.0000	0.853595	0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000	0.853595	0.426798	+	0.904347i	$0.359642\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	0	0
$273$	0	0
$274$	2.00000	0.120824
$275$	0	0
$276$	−6.00000	−0.361158
$277$	26.0000	1.56219	0.781094	−	0.624413i	$-0.214662\pi$
$277$	26.0000	1.56219	0.781094	+	0.624413i	$0.214662\pi$
$278$	8.00000	0.479808
$279$	4.00000	0.239474
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	−8.00000	−0.476393
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	−8.00000	−0.474713
$285$	0	0
$286$	−6.00000	−0.354787
$287$	0	0
$288$	1.00000	0.0589256
$289$	−17.0000	−1.00000
$290$	0	0
$291$	−2.00000	−0.117242
$292$	6.00000	0.351123
$293$	−14.0000	−0.817889	−0.408944	−	0.912559i	$-0.634103\pi$
$293$	−14.0000	−0.817889	−0.408944	+	0.912559i	$0.634103\pi$
$294$	7.00000	0.408248
$295$	0	0
$296$	10.0000	0.581238
$297$	6.00000	0.348155
$298$	0	0
$299$	6.00000	0.346989
$300$	0	0
$301$	0	0
$302$	−8.00000	−0.460348
$303$	2.00000	0.114897
$304$	6.00000	0.344124
$305$	0	0
$306$	0	0
$307$	−28.0000	−1.59804	−0.799022	−	0.601302i	$-0.794649\pi$
$307$	−28.0000	−1.59804	−0.799022	+	0.601302i	$0.794649\pi$
$308$	0	0
$309$	10.0000	0.568880
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	−1.00000	−0.0566139
$313$	−16.0000	−0.904373	−0.452187	−	0.891923i	$-0.649356\pi$
$313$	−16.0000	−0.904373	−0.452187	+	0.891923i	$0.649356\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	16.0000	0.900070
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	6.00000	0.336463
$319$	−12.0000	−0.671871
$320$	0	0
$321$	−16.0000	−0.893033
$322$	0	0
$323$	0	0
$324$	1.00000	0.0555556
$325$	0	0
$326$	12.0000	0.664619
$327$	16.0000	0.884802
$328$	−6.00000	−0.331295
$329$	0	0
$330$	0	0
$331$	−26.0000	−1.42909	−0.714545	−	0.699590i	$-0.753366\pi$
$331$	−26.0000	−1.42909	−0.714545	+	0.699590i	$0.753366\pi$
$332$	4.00000	0.219529
$333$	10.0000	0.547997
$334$	8.00000	0.437741
$335$	0	0
$336$	0	0
$337$	8.00000	0.435788	0.217894	−	0.975972i	$-0.430081\pi$
$337$	8.00000	0.435788	0.217894	+	0.975972i	$0.430081\pi$
$338$	1.00000	0.0543928
$339$	−8.00000	−0.434500
$340$	0	0
$341$	−24.0000	−1.29967
$342$	6.00000	0.324443
$343$	0	0
$344$	8.00000	0.431331
$345$	0	0
$346$	18.0000	0.967686
$347$	32.0000	1.71785	0.858925	−	0.512101i	$-0.171133\pi$
$347$	32.0000	1.71785	0.858925	+	0.512101i	$0.171133\pi$
$348$	−2.00000	−0.107211
$349$	8.00000	0.428230	0.214115	−	0.976808i	$-0.431313\pi$
$349$	8.00000	0.428230	0.214115	+	0.976808i	$0.431313\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	−6.00000	−0.319801
$353$	−10.0000	−0.532246	−0.266123	−	0.963939i	$-0.585743\pi$
$353$	−10.0000	−0.532246	−0.266123	+	0.963939i	$0.585743\pi$
$354$	−10.0000	−0.531494
$355$	0	0
$356$	−10.0000	−0.529999
$357$	0	0
$358$	−12.0000	−0.634220
$359$	−12.0000	−0.633336	−0.316668	−	0.948536i	$-0.602564\pi$
$359$	−12.0000	−0.633336	−0.316668	+	0.948536i	$0.602564\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	−2.00000	−0.105118
$363$	−25.0000	−1.31216
$364$	0	0
$365$	0	0
$366$	6.00000	0.313625
$367$	6.00000	0.313197	0.156599	−	0.987662i	$-0.449947\pi$
$367$	6.00000	0.313197	0.156599	+	0.987662i	$0.449947\pi$
$368$	6.00000	0.312772
$369$	−6.00000	−0.312348
$370$	0	0
$371$	0	0
$372$	−4.00000	−0.207390
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	0	0
$375$	0	0
$376$	8.00000	0.412568
$377$	2.00000	0.103005
$378$	0	0
$379$	22.0000	1.13006	0.565032	−	0.825069i	$-0.308864\pi$
$379$	22.0000	1.13006	0.565032	+	0.825069i	$0.308864\pi$
$380$	0	0
$381$	18.0000	0.922168
$382$	−8.00000	−0.409316
$383$	−20.0000	−1.02195	−0.510976	−	0.859595i	$-0.670716\pi$
$383$	−20.0000	−1.02195	−0.510976	+	0.859595i	$0.670716\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	10.0000	0.508987
$387$	8.00000	0.406663
$388$	2.00000	0.101535
$389$	−14.0000	−0.709828	−0.354914	−	0.934899i	$-0.615490\pi$
$389$	−14.0000	−0.709828	−0.354914	+	0.934899i	$0.615490\pi$
$390$	0	0
$391$	0	0
$392$	−7.00000	−0.353553
$393$	−12.0000	−0.605320
$394$	−22.0000	−1.10834
$395$	0	0
$396$	−6.00000	−0.301511
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	16.0000	0.802008
$399$	0	0
$400$	0	0
$401$	14.0000	0.699127	0.349563	−	0.936913i	$-0.386330\pi$
$401$	14.0000	0.699127	0.349563	+	0.936913i	$0.386330\pi$
$402$	−4.00000	−0.199502
$403$	4.00000	0.199254
$404$	−2.00000	−0.0995037
$405$	0	0
$406$	0	0
$407$	−60.0000	−2.97409
$408$	0	0
$409$	30.0000	1.48340	0.741702	−	0.670729i	$-0.234019\pi$
$409$	30.0000	1.48340	0.741702	+	0.670729i	$0.234019\pi$
$410$	0	0
$411$	−2.00000	−0.0986527
$412$	−10.0000	−0.492665
$413$	0	0
$414$	6.00000	0.294884
$415$	0	0
$416$	1.00000	0.0490290
$417$	−8.00000	−0.391762
$418$	−36.0000	−1.76082
$419$	28.0000	1.36789	0.683945	−	0.729534i	$-0.260263\pi$
$419$	28.0000	1.36789	0.683945	+	0.729534i	$0.260263\pi$
$420$	0	0
$421$	−8.00000	−0.389896	−0.194948	−	0.980814i	$-0.562454\pi$
$421$	−8.00000	−0.389896	−0.194948	+	0.980814i	$0.562454\pi$
$422$	4.00000	0.194717
$423$	8.00000	0.388973
$424$	−6.00000	−0.291386
$425$	0	0
$426$	8.00000	0.387601
$427$	0	0
$428$	16.0000	0.773389
$429$	6.00000	0.289683
$430$	0	0
$431$	−8.00000	−0.385346	−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000	−0.385346	−0.192673	+	0.981263i	$0.561716\pi$
$432$	−1.00000	−0.0481125
$433$	24.0000	1.15337	0.576683	−	0.816968i	$-0.304347\pi$
$433$	24.0000	1.15337	0.576683	+	0.816968i	$0.304347\pi$
$434$	0	0
$435$	0	0
$436$	−16.0000	−0.766261
$437$	36.0000	1.72211
$438$	−6.00000	−0.286691
$439$	16.0000	0.763638	0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000	0.763638	0.381819	+	0.924237i	$0.375298\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	0	0
$443$	−32.0000	−1.52037	−0.760183	−	0.649709i	$-0.774891\pi$
$443$	−32.0000	−1.52037	−0.760183	+	0.649709i	$0.774891\pi$
$444$	−10.0000	−0.474579
$445$	0	0
$446$	4.00000	0.189405
$447$	0	0
$448$	0	0
$449$	−34.0000	−1.60456	−0.802280	−	0.596948i	$-0.796380\pi$
$449$	−34.0000	−1.60456	−0.802280	+	0.596948i	$0.796380\pi$
$450$	0	0
$451$	36.0000	1.69517
$452$	8.00000	0.376288
$453$	8.00000	0.375873
$454$	−20.0000	−0.938647
$455$	0	0
$456$	−6.00000	−0.280976
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	20.0000	0.934539
$459$	0	0
$460$	0	0
$461$	36.0000	1.67669	0.838344	−	0.545142i	$-0.183524\pi$
$461$	36.0000	1.67669	0.838344	+	0.545142i	$0.183524\pi$
$462$	0	0
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	−8.00000	−0.370593
$467$	−40.0000	−1.85098	−0.925490	−	0.378773i	$-0.876346\pi$
$467$	−40.0000	−1.85098	−0.925490	+	0.378773i	$0.876346\pi$
$468$	1.00000	0.0462250
$469$	0	0
$470$	0	0
$471$	2.00000	0.0921551
$472$	10.0000	0.460287
$473$	−48.0000	−2.20704
$474$	−16.0000	−0.734904
$475$	0	0
$476$	0	0
$477$	−6.00000	−0.274721
$478$	−28.0000	−1.28069
$479$	20.0000	0.913823	0.456912	−	0.889512i	$-0.348956\pi$
$479$	20.0000	0.913823	0.456912	+	0.889512i	$0.348956\pi$
$480$	0	0
$481$	10.0000	0.455961
$482$	−22.0000	−1.00207
$483$	0	0
$484$	25.0000	1.13636
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	−12.0000	−0.543772	−0.271886	−	0.962329i	$-0.587647\pi$
$487$	−12.0000	−0.543772	−0.271886	+	0.962329i	$0.587647\pi$
$488$	−6.00000	−0.271607
$489$	−12.0000	−0.542659
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	6.00000	0.270501
$493$	0	0
$494$	6.00000	0.269953
$495$	0	0
$496$	4.00000	0.179605
$497$	0	0
$498$	−4.00000	−0.179244
$499$	−6.00000	−0.268597	−0.134298	−	0.990941i	$-0.542878\pi$
$499$	−6.00000	−0.268597	−0.134298	+	0.990941i	$0.542878\pi$
$500$	0	0
$501$	−8.00000	−0.357414
$502$	8.00000	0.357057
$503$	−26.0000	−1.15928	−0.579641	−	0.814872i	$-0.696807\pi$
$503$	−26.0000	−1.15928	−0.579641	+	0.814872i	$0.696807\pi$
$504$	0	0
$505$	0	0
$506$	−36.0000	−1.60040
$507$	−1.00000	−0.0444116
$508$	−18.0000	−0.798621
$509$	24.0000	1.06378	0.531891	−	0.846813i	$-0.321482\pi$
$509$	24.0000	1.06378	0.531891	+	0.846813i	$0.321482\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	−6.00000	−0.264906
$514$	−24.0000	−1.05859
$515$	0	0
$516$	−8.00000	−0.352180
$517$	−48.0000	−2.11104
$518$	0	0
$519$	−18.0000	−0.790112
$520$	0	0
$521$	−10.0000	−0.438108	−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000	−0.438108	−0.219054	+	0.975713i	$0.570297\pi$
$522$	2.00000	0.0875376
$523$	40.0000	1.74908	0.874539	−	0.484955i	$-0.161164\pi$
$523$	40.0000	1.74908	0.874539	+	0.484955i	$0.161164\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	−6.00000	−0.261612
$527$	0	0
$528$	6.00000	0.261116
$529$	13.0000	0.565217
$530$	0	0
$531$	10.0000	0.433963
$532$	0	0
$533$	−6.00000	−0.259889
$534$	10.0000	0.432742
$535$	0	0
$536$	4.00000	0.172774
$537$	12.0000	0.517838
$538$	14.0000	0.603583
$539$	42.0000	1.80907
$540$	0	0
$541$	4.00000	0.171973	0.0859867	−	0.996296i	$-0.472596\pi$
$541$	4.00000	0.171973	0.0859867	+	0.996296i	$0.472596\pi$
$542$	−16.0000	−0.687259
$543$	2.00000	0.0858282
$544$	0	0
$545$	0	0
$546$	0	0
$547$	12.0000	0.513083	0.256541	−	0.966533i	$-0.417417\pi$
$547$	12.0000	0.513083	0.256541	+	0.966533i	$0.417417\pi$
$548$	2.00000	0.0854358
$549$	−6.00000	−0.256074
$550$	0	0
$551$	12.0000	0.511217
$552$	−6.00000	−0.255377
$553$	0	0
$554$	26.0000	1.10463
$555$	0	0
$556$	8.00000	0.339276
$557$	−2.00000	−0.0847427	−0.0423714	−	0.999102i	$-0.513491\pi$
$557$	−2.00000	−0.0847427	−0.0423714	+	0.999102i	$0.513491\pi$
$558$	4.00000	0.169334
$559$	8.00000	0.338364
$560$	0	0
$561$	0	0
$562$	−10.0000	−0.421825
$563$	−24.0000	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$564$	−8.00000	−0.336861
$565$	0	0
$566$	−4.00000	−0.168133
$567$	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
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	−6.00000	−0.250873
$573$	8.00000	0.334205
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	−17.0000	−0.707107
$579$	−10.0000	−0.415586
$580$	0	0
$581$	0	0
$582$	−2.00000	−0.0829027
$583$	36.0000	1.49097
$584$	6.00000	0.248282
$585$	0	0
$586$	−14.0000	−0.578335
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	7.00000	0.288675
$589$	24.0000	0.988903
$590$	0	0
$591$	22.0000	0.904959
$592$	10.0000	0.410997
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	6.00000	0.246183
$595$	0	0
$596$	0	0
$597$	−16.0000	−0.654836
$598$	6.00000	0.245358
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	−8.00000	−0.325515
$605$	0	0
$606$	2.00000	0.0812444
$607$	2.00000	0.0811775	0.0405887	−	0.999176i	$-0.487077\pi$
$607$	2.00000	0.0811775	0.0405887	+	0.999176i	$0.487077\pi$
$608$	6.00000	0.243332
$609$	0	0
$610$	0	0
$611$	8.00000	0.323645
$612$	0	0
$613$	−38.0000	−1.53481	−0.767403	−	0.641165i	$-0.778451\pi$
$613$	−38.0000	−1.53481	−0.767403	+	0.641165i	$0.778451\pi$
$614$	−28.0000	−1.12999
$615$	0	0
$616$	0	0
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	10.0000	0.402259
$619$	38.0000	1.52735	0.763674	−	0.645601i	$-0.223393\pi$
$619$	38.0000	1.52735	0.763674	+	0.645601i	$0.223393\pi$
$620$	0	0
$621$	−6.00000	−0.240772
$622$	0	0
$623$	0	0
$624$	−1.00000	−0.0400320
$625$	0	0
$626$	−16.0000	−0.639489
$627$	36.0000	1.43770
$628$	−2.00000	−0.0798087
$629$	0	0
$630$	0	0
$631$	32.0000	1.27390	0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000	1.27390	0.636950	+	0.770905i	$0.280196\pi$
$632$	16.0000	0.636446
$633$	−4.00000	−0.158986
$634$	18.0000	0.714871
$635$	0	0
$636$	6.00000	0.237915
$637$	−7.00000	−0.277350
$638$	−12.0000	−0.475085
$639$	−8.00000	−0.316475
$640$	0	0
$641$	−42.0000	−1.65890	−0.829450	−	0.558581i	$-0.811346\pi$
$641$	−42.0000	−1.65890	−0.829450	+	0.558581i	$0.811346\pi$
$642$	−16.0000	−0.631470
$643$	44.0000	1.73519	0.867595	−	0.497271i	$-0.165665\pi$
$643$	44.0000	1.73519	0.867595	+	0.497271i	$0.165665\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	14.0000	0.550397	0.275198	−	0.961387i	$-0.411256\pi$
$647$	14.0000	0.550397	0.275198	+	0.961387i	$0.411256\pi$
$648$	1.00000	0.0392837
$649$	−60.0000	−2.35521
$650$	0	0
$651$	0	0
$652$	12.0000	0.469956
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	16.0000	0.625650
$655$	0	0
$656$	−6.00000	−0.234261
$657$	6.00000	0.234082
$658$	0	0
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	0	0
$661$	−28.0000	−1.08907	−0.544537	−	0.838737i	$-0.683295\pi$
$661$	−28.0000	−1.08907	−0.544537	+	0.838737i	$0.683295\pi$
$662$	−26.0000	−1.01052
$663$	0	0
$664$	4.00000	0.155230
$665$	0	0
$666$	10.0000	0.387492
$667$	12.0000	0.464642
$668$	8.00000	0.309529
$669$	−4.00000	−0.154649
$670$	0	0
$671$	36.0000	1.38976
$672$	0	0
$673$	44.0000	1.69608	0.848038	−	0.529936i	$-0.177784\pi$
$673$	44.0000	1.69608	0.848038	+	0.529936i	$0.177784\pi$
$674$	8.00000	0.308148
$675$	0	0
$676$	1.00000	0.0384615
$677$	38.0000	1.46046	0.730229	−	0.683202i	$-0.239413\pi$
$677$	38.0000	1.46046	0.730229	+	0.683202i	$0.239413\pi$
$678$	−8.00000	−0.307238
$679$	0	0
$680$	0	0
$681$	20.0000	0.766402
$682$	−24.0000	−0.919007
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	6.00000	0.229416
$685$	0	0
$686$	0	0
$687$	−20.0000	−0.763048
$688$	8.00000	0.304997
$689$	−6.00000	−0.228582
$690$	0	0
$691$	22.0000	0.836919	0.418460	−	0.908235i	$-0.362570\pi$
$691$	22.0000	0.836919	0.418460	+	0.908235i	$0.362570\pi$
$692$	18.0000	0.684257
$693$	0	0
$694$	32.0000	1.21470
$695$	0	0
$696$	−2.00000	−0.0758098
$697$	0	0
$698$	8.00000	0.302804
$699$	8.00000	0.302588
$700$	0	0
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	−1.00000	−0.0377426
$703$	60.0000	2.26294
$704$	−6.00000	−0.226134
$705$	0	0
$706$	−10.0000	−0.376355
$707$	0	0
$708$	−10.0000	−0.375823
$709$	−44.0000	−1.65245	−0.826227	−	0.563337i	$-0.809517\pi$
$709$	−44.0000	−1.65245	−0.826227	+	0.563337i	$0.809517\pi$
$710$	0	0
$711$	16.0000	0.600047
$712$	−10.0000	−0.374766
$713$	24.0000	0.898807
$714$	0	0
$715$	0	0
$716$	−12.0000	−0.448461
$717$	28.0000	1.04568
$718$	−12.0000	−0.447836
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	0	0
$721$	0	0
$722$	17.0000	0.632674
$723$	22.0000	0.818189
$724$	−2.00000	−0.0743294
$725$	0	0
$726$	−25.0000	−0.927837
$727$	38.0000	1.40934	0.704671	−	0.709534i	$-0.251095\pi$
$727$	38.0000	1.40934	0.704671	+	0.709534i	$0.251095\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	6.00000	0.221766
$733$	−26.0000	−0.960332	−0.480166	−	0.877178i	$-0.659424\pi$
$733$	−26.0000	−0.960332	−0.480166	+	0.877178i	$0.659424\pi$
$734$	6.00000	0.221464
$735$	0	0
$736$	6.00000	0.221163
$737$	−24.0000	−0.884051
$738$	−6.00000	−0.220863
$739$	−22.0000	−0.809283	−0.404642	−	0.914475i	$-0.632604\pi$
$739$	−22.0000	−0.809283	−0.404642	+	0.914475i	$0.632604\pi$
$740$	0	0
$741$	−6.00000	−0.220416
$742$	0	0
$743$	52.0000	1.90769	0.953847	−	0.300291i	$-0.0970839\pi$
$743$	52.0000	1.90769	0.953847	+	0.300291i	$0.0970839\pi$
$744$	−4.00000	−0.146647
$745$	0	0
$746$	−10.0000	−0.366126
$747$	4.00000	0.146352
$748$	0	0
$749$	0	0
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	8.00000	0.291730
$753$	−8.00000	−0.291536
$754$	2.00000	0.0728357
$755$	0	0
$756$	0	0
$757$	10.0000	0.363456	0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000	0.363456	0.181728	+	0.983349i	$0.441831\pi$
$758$	22.0000	0.799076
$759$	36.0000	1.30672
$760$	0	0
$761$	30.0000	1.08750	0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000	1.08750	0.543750	+	0.839248i	$0.317004\pi$
$762$	18.0000	0.652071
$763$	0	0
$764$	−8.00000	−0.289430
$765$	0	0
$766$	−20.0000	−0.722629
$767$	10.0000	0.361079
$768$	−1.00000	−0.0360844
$769$	−54.0000	−1.94729	−0.973645	−	0.228069i	$-0.926759\pi$
$769$	−54.0000	−1.94729	−0.973645	+	0.228069i	$0.926759\pi$
$770$	0	0
$771$	24.0000	0.864339
$772$	10.0000	0.359908
$773$	−18.0000	−0.647415	−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000	−0.647415	−0.323708	+	0.946157i	$0.604929\pi$
$774$	8.00000	0.287554
$775$	0	0
$776$	2.00000	0.0717958
$777$	0	0
$778$	−14.0000	−0.501924
$779$	−36.0000	−1.28983
$780$	0	0
$781$	48.0000	1.71758
$782$	0	0
$783$	−2.00000	−0.0714742
$784$	−7.00000	−0.250000
$785$	0	0
$786$	−12.0000	−0.428026
$787$	−12.0000	−0.427754	−0.213877	−	0.976861i	$-0.568609\pi$
$787$	−12.0000	−0.427754	−0.213877	+	0.976861i	$0.568609\pi$
$788$	−22.0000	−0.783718
$789$	6.00000	0.213606
$790$	0	0
$791$	0	0
$792$	−6.00000	−0.213201
$793$	−6.00000	−0.213066
$794$	−18.0000	−0.638796
$795$	0	0
$796$	16.0000	0.567105
$797$	42.0000	1.48772	0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000	1.48772	0.743858	+	0.668338i	$0.232994\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−10.0000	−0.353333
$802$	14.0000	0.494357
$803$	−36.0000	−1.27041
$804$	−4.00000	−0.141069
$805$	0	0
$806$	4.00000	0.140894
$807$	−14.0000	−0.492823
$808$	−2.00000	−0.0703598
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	2.00000	0.0702295	0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000	0.0702295	0.0351147	+	0.999383i	$0.488820\pi$
$812$	0	0
$813$	16.0000	0.561144
$814$	−60.0000	−2.10300
$815$	0	0
$816$	0	0
$817$	48.0000	1.67931
$818$	30.0000	1.04893
$819$	0	0
$820$	0	0
$821$	4.00000	0.139601	0.0698005	−	0.997561i	$-0.477764\pi$
$821$	4.00000	0.139601	0.0698005	+	0.997561i	$0.477764\pi$
$822$	−2.00000	−0.0697580
$823$	−54.0000	−1.88232	−0.941161	−	0.337959i	$-0.890263\pi$
$823$	−54.0000	−1.88232	−0.941161	+	0.337959i	$0.890263\pi$
$824$	−10.0000	−0.348367
$825$	0	0
$826$	0	0
$827$	−20.0000	−0.695468	−0.347734	−	0.937593i	$-0.613049\pi$
$827$	−20.0000	−0.695468	−0.347734	+	0.937593i	$0.613049\pi$
$828$	6.00000	0.208514
$829$	26.0000	0.903017	0.451509	−	0.892267i	$-0.350886\pi$
$829$	26.0000	0.903017	0.451509	+	0.892267i	$0.350886\pi$
$830$	0	0
$831$	−26.0000	−0.901930
$832$	1.00000	0.0346688
$833$	0	0
$834$	−8.00000	−0.277017
$835$	0	0
$836$	−36.0000	−1.24509
$837$	−4.00000	−0.138260
$838$	28.0000	0.967244
$839$	−16.0000	−0.552381	−0.276191	−	0.961103i	$-0.589072\pi$
$839$	−16.0000	−0.552381	−0.276191	+	0.961103i	$0.589072\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−8.00000	−0.275698
$843$	10.0000	0.344418
$844$	4.00000	0.137686
$845$	0	0
$846$	8.00000	0.275046
$847$	0	0
$848$	−6.00000	−0.206041
$849$	4.00000	0.137280
$850$	0	0
$851$	60.0000	2.05677
$852$	8.00000	0.274075
$853$	14.0000	0.479351	0.239675	−	0.970853i	$-0.422959\pi$
$853$	14.0000	0.479351	0.239675	+	0.970853i	$0.422959\pi$
$854$	0	0
$855$	0	0
$856$	16.0000	0.546869
$857$	−52.0000	−1.77629	−0.888143	−	0.459567i	$-0.848005\pi$
$857$	−52.0000	−1.77629	−0.888143	+	0.459567i	$0.848005\pi$
$858$	6.00000	0.204837
$859$	12.0000	0.409435	0.204717	−	0.978821i	$-0.434372\pi$
$859$	12.0000	0.409435	0.204717	+	0.978821i	$0.434372\pi$
$860$	0	0
$861$	0	0
$862$	−8.00000	−0.272481
$863$	−32.0000	−1.08929	−0.544646	−	0.838666i	$-0.683336\pi$
$863$	−32.0000	−1.08929	−0.544646	+	0.838666i	$0.683336\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	24.0000	0.815553
$867$	17.0000	0.577350
$868$	0	0
$869$	−96.0000	−3.25658
$870$	0	0
$871$	4.00000	0.135535
$872$	−16.0000	−0.541828
$873$	2.00000	0.0676897
$874$	36.0000	1.21772
$875$	0	0
$876$	−6.00000	−0.202721
$877$	−22.0000	−0.742887	−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000	−0.742887	−0.371444	+	0.928456i	$0.621137\pi$
$878$	16.0000	0.539974
$879$	14.0000	0.472208
$880$	0	0
$881$	−54.0000	−1.81931	−0.909653	−	0.415369i	$-0.863653\pi$
$881$	−54.0000	−1.81931	−0.909653	+	0.415369i	$0.863653\pi$
$882$	−7.00000	−0.235702
$883$	−16.0000	−0.538443	−0.269221	−	0.963078i	$-0.586766\pi$
$883$	−16.0000	−0.538443	−0.269221	+	0.963078i	$0.586766\pi$
$884$	0	0
$885$	0	0
$886$	−32.0000	−1.07506
$887$	−26.0000	−0.872995	−0.436497	−	0.899706i	$-0.643781\pi$
$887$	−26.0000	−0.872995	−0.436497	+	0.899706i	$0.643781\pi$
$888$	−10.0000	−0.335578
$889$	0	0
$890$	0	0
$891$	−6.00000	−0.201008
$892$	4.00000	0.133930
$893$	48.0000	1.60626
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−6.00000	−0.200334
$898$	−34.0000	−1.13459
$899$	8.00000	0.266815
$900$	0	0
$901$	0	0
$902$	36.0000	1.19867
$903$	0	0
$904$	8.00000	0.266076
$905$	0	0
$906$	8.00000	0.265782
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	−20.0000	−0.663723
$909$	−2.00000	−0.0663358
$910$	0	0
$911$	−16.0000	−0.530104	−0.265052	−	0.964234i	$-0.585389\pi$
$911$	−16.0000	−0.530104	−0.265052	+	0.964234i	$0.585389\pi$
$912$	−6.00000	−0.198680
$913$	−24.0000	−0.794284
$914$	18.0000	0.595387
$915$	0	0
$916$	20.0000	0.660819
$917$	0	0
$918$	0	0
$919$	−24.0000	−0.791687	−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000	−0.791687	−0.395843	+	0.918318i	$0.629548\pi$
$920$	0	0
$921$	28.0000	0.922631
$922$	36.0000	1.18560
$923$	−8.00000	−0.263323
$924$	0	0
$925$	0	0
$926$	−16.0000	−0.525793
$927$	−10.0000	−0.328443
$928$	2.00000	0.0656532
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	0	0
$931$	−42.0000	−1.37649
$932$	−8.00000	−0.262049
$933$	0	0
$934$	−40.0000	−1.30884
$935$	0	0
$936$	1.00000	0.0326860
$937$	16.0000	0.522697	0.261349	−	0.965244i	$-0.415833\pi$
$937$	16.0000	0.522697	0.261349	+	0.965244i	$0.415833\pi$
$938$	0	0
$939$	16.0000	0.522140
$940$	0	0
$941$	−24.0000	−0.782378	−0.391189	−	0.920310i	$-0.627936\pi$
$941$	−24.0000	−0.782378	−0.391189	+	0.920310i	$0.627936\pi$
$942$	2.00000	0.0651635
$943$	−36.0000	−1.17232
$944$	10.0000	0.325472
$945$	0	0
$946$	−48.0000	−1.56061
$947$	4.00000	0.129983	0.0649913	−	0.997886i	$-0.479298\pi$
$947$	4.00000	0.129983	0.0649913	+	0.997886i	$0.479298\pi$
$948$	−16.0000	−0.519656
$949$	6.00000	0.194768
$950$	0	0
$951$	−18.0000	−0.583690
$952$	0	0
$953$	44.0000	1.42530	0.712650	−	0.701520i	$-0.247495\pi$
$953$	44.0000	1.42530	0.712650	+	0.701520i	$0.247495\pi$
$954$	−6.00000	−0.194257
$955$	0	0
$956$	−28.0000	−0.905585
$957$	12.0000	0.387905
$958$	20.0000	0.646171
$959$	0	0
$960$	0	0
$961$	−15.0000	−0.483871
$962$	10.0000	0.322413
$963$	16.0000	0.515593
$964$	−22.0000	−0.708572
$965$	0	0
$966$	0	0
$967$	32.0000	1.02905	0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000	1.02905	0.514525	+	0.857475i	$0.327968\pi$
$968$	25.0000	0.803530
$969$	0	0
$970$	0	0
$971$	40.0000	1.28366	0.641831	−	0.766846i	$-0.278175\pi$
$971$	40.0000	1.28366	0.641831	+	0.766846i	$0.278175\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	−12.0000	−0.384505
$975$	0	0
$976$	−6.00000	−0.192055
$977$	50.0000	1.59964	0.799821	−	0.600239i	$-0.204928\pi$
$977$	50.0000	1.59964	0.799821	+	0.600239i	$0.204928\pi$
$978$	−12.0000	−0.383718
$979$	60.0000	1.91761
$980$	0	0
$981$	−16.0000	−0.510841
$982$	−12.0000	−0.382935
$983$	12.0000	0.382741	0.191370	−	0.981518i	$-0.438707\pi$
$983$	12.0000	0.382741	0.191370	+	0.981518i	$0.438707\pi$
$984$	6.00000	0.191273
$985$	0	0
$986$	0	0
$987$	0	0
$988$	6.00000	0.190885
$989$	48.0000	1.52631
$990$	0	0
$991$	8.00000	0.254128	0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000	0.254128	0.127064	+	0.991894i	$0.459445\pi$
$992$	4.00000	0.127000
$993$	26.0000	0.825085
$994$	0	0
$995$	0	0
$996$	−4.00000	−0.126745
$997$	14.0000	0.443384	0.221692	−	0.975117i	$-0.428842\pi$
$997$	14.0000	0.443384	0.221692	+	0.975117i	$0.428842\pi$
$998$	−6.00000	−0.189927
$999$	−10.0000	−0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1950.2.a.r.1.1		1
3.2	odd	2		5850.2.a.o.1.1		1
5.2	odd	4		390.2.e.a.79.2	yes	2
5.3	odd	4		390.2.e.a.79.1	✓	2
5.4	even	2		1950.2.a.i.1.1		1
15.2	even	4		1170.2.e.d.469.1		2
15.8	even	4		1170.2.e.d.469.2		2
15.14	odd	2		5850.2.a.bs.1.1		1
20.3	even	4		3120.2.l.a.1249.2		2
20.7	even	4		3120.2.l.a.1249.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.e.a.79.1	✓	2	5.3	odd	4
390.2.e.a.79.2	yes	2	5.2	odd	4
1170.2.e.d.469.1		2	15.2	even	4
1170.2.e.d.469.2		2	15.8	even	4
1950.2.a.i.1.1		1	5.4	even	2
1950.2.a.r.1.1		1	1.1	even	1	trivial
3120.2.l.a.1249.1		2	20.7	even	4
3120.2.l.a.1249.2		2	20.3	even	4
5850.2.a.o.1.1		1	3.2	odd	2
5850.2.a.bs.1.1		1	15.14	odd	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:	\( N \)	\(=\)	\( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1950.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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