LMFDB - Embedded newform 1950.2.a.s.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:	yes
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)

$f(q)$

$=$

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

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +4.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} +1.00000 q^{16} +1.00000 q^{18} +1.00000 q^{19} +4.00000 q^{22} -4.00000 q^{23} -1.00000 q^{24} +1.00000 q^{26} -1.00000 q^{27} -3.00000 q^{29} +4.00000 q^{31} +1.00000 q^{32} -4.00000 q^{33} +1.00000 q^{36} +5.00000 q^{37} +1.00000 q^{38} -1.00000 q^{39} +9.00000 q^{41} -2.00000 q^{43} +4.00000 q^{44} -4.00000 q^{46} +3.00000 q^{47} -1.00000 q^{48} -7.00000 q^{49} +1.00000 q^{52} -1.00000 q^{53} -1.00000 q^{54} -1.00000 q^{57} -3.00000 q^{58} +10.0000 q^{59} +4.00000 q^{61} +4.00000 q^{62} +1.00000 q^{64} -4.00000 q^{66} +9.00000 q^{67} +4.00000 q^{69} +7.00000 q^{71} +1.00000 q^{72} -4.00000 q^{73} +5.00000 q^{74} +1.00000 q^{76} -1.00000 q^{78} +11.0000 q^{79} +1.00000 q^{81} +9.00000 q^{82} -6.00000 q^{83} -2.00000 q^{86} +3.00000 q^{87} +4.00000 q^{88} +10.0000 q^{89} -4.00000 q^{92} -4.00000 q^{93} +3.00000 q^{94} -1.00000 q^{96} +12.0000 q^{97} -7.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	−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
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	0	0
$21$	0	0
$22$	4.00000	0.852803
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	1.00000	0.196116
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	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
$33$	−4.00000	−0.696311
$34$	0	0
$35$	0	0
$36$	1.00000	0.166667
$37$	5.00000	0.821995	0.410997	−	0.911636i	$-0.365181\pi$
$37$	5.00000	0.821995	0.410997	+	0.911636i	$0.365181\pi$
$38$	1.00000	0.162221
$39$	−1.00000	−0.160128
$40$	0	0
$41$	9.00000	1.40556	0.702782	−	0.711405i	$-0.251941\pi$
$41$	9.00000	1.40556	0.702782	+	0.711405i	$0.251941\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	−4.00000	−0.589768
$47$	3.00000	0.437595	0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000	0.437595	0.218797	+	0.975770i	$0.429787\pi$
$48$	−1.00000	−0.144338
$49$	−7.00000	−1.00000
$50$	0	0
$51$	0	0
$52$	1.00000	0.138675
$53$	−1.00000	−0.137361	−0.0686803	−	0.997639i	$-0.521879\pi$
$53$	−1.00000	−0.137361	−0.0686803	+	0.997639i	$0.521879\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	0	0
$57$	−1.00000	−0.132453
$58$	−3.00000	−0.393919
$59$	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
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	4.00000	0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	−4.00000	−0.492366
$67$	9.00000	1.09952	0.549762	−	0.835321i	$-0.314718\pi$
$67$	9.00000	1.09952	0.549762	+	0.835321i	$0.314718\pi$
$68$	0	0
$69$	4.00000	0.481543
$70$	0	0
$71$	7.00000	0.830747	0.415374	−	0.909651i	$-0.363651\pi$
$71$	7.00000	0.830747	0.415374	+	0.909651i	$0.363651\pi$
$72$	1.00000	0.117851
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	5.00000	0.581238
$75$	0	0
$76$	1.00000	0.114708
$77$	0	0
$78$	−1.00000	−0.113228
$79$	11.0000	1.23760	0.618798	−	0.785550i	$-0.287620\pi$
$79$	11.0000	1.23760	0.618798	+	0.785550i	$0.287620\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	9.00000	0.993884
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	0	0
$85$	0	0
$86$	−2.00000	−0.215666
$87$	3.00000	0.321634
$88$	4.00000	0.426401
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	0	0
$92$	−4.00000	−0.417029
$93$	−4.00000	−0.414781
$94$	3.00000	0.309426
$95$	0	0
$96$	−1.00000	−0.102062
$97$	12.0000	1.21842	0.609208	−	0.793011i	$-0.291488\pi$
$97$	12.0000	1.21842	0.609208	+	0.793011i	$0.291488\pi$
$98$	−7.00000	−0.707107
$99$	4.00000	0.402015
$100$	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
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	−1.00000	−0.0971286
$107$	−9.00000	−0.870063	−0.435031	−	0.900415i	$-0.643263\pi$
$107$	−9.00000	−0.870063	−0.435031	+	0.900415i	$0.643263\pi$
$108$	−1.00000	−0.0962250
$109$	−1.00000	−0.0957826	−0.0478913	−	0.998853i	$-0.515250\pi$
$109$	−1.00000	−0.0957826	−0.0478913	+	0.998853i	$0.515250\pi$
$110$	0	0
$111$	−5.00000	−0.474579
$112$	0	0
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	−1.00000	−0.0936586
$115$	0	0
$116$	−3.00000	−0.278543
$117$	1.00000	0.0924500
$118$	10.0000	0.920575
$119$	0	0
$120$	0	0
$121$	5.00000	0.454545
$122$	4.00000	0.362143
$123$	−9.00000	−0.811503
$124$	4.00000	0.359211
$125$	0	0
$126$	0	0
$127$	17.0000	1.50851	0.754253	−	0.656584i	$-0.227999\pi$
$127$	17.0000	1.50851	0.754253	+	0.656584i	$0.227999\pi$
$128$	1.00000	0.0883883
$129$	2.00000	0.176090
$130$	0	0
$131$	−3.00000	−0.262111	−0.131056	−	0.991375i	$-0.541837\pi$
$131$	−3.00000	−0.262111	−0.131056	+	0.991375i	$0.541837\pi$
$132$	−4.00000	−0.348155
$133$	0	0
$134$	9.00000	0.777482
$135$	0	0
$136$	0	0
$137$	−23.0000	−1.96502	−0.982511	−	0.186203i	$-0.940382\pi$
$137$	−23.0000	−1.96502	−0.982511	+	0.186203i	$0.940382\pi$
$138$	4.00000	0.340503
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	0	0
$141$	−3.00000	−0.252646
$142$	7.00000	0.587427
$143$	4.00000	0.334497
$144$	1.00000	0.0833333
$145$	0	0
$146$	−4.00000	−0.331042
$147$	7.00000	0.577350
$148$	5.00000	0.410997
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	1.00000	0.0811107
$153$	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	$0.525431\pi$
$158$	11.0000	0.875113
$159$	1.00000	0.0793052
$160$	0	0
$161$	0	0
$162$	1.00000	0.0785674
$163$	−8.00000	−0.626608	−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000	−0.626608	−0.313304	+	0.949653i	$0.601436\pi$
$164$	9.00000	0.702782
$165$	0	0
$166$	−6.00000	−0.465690
$167$	−7.00000	−0.541676	−0.270838	−	0.962625i	$-0.587301\pi$
$167$	−7.00000	−0.541676	−0.270838	+	0.962625i	$0.587301\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	1.00000	0.0764719
$172$	−2.00000	−0.152499
$173$	3.00000	0.228086	0.114043	−	0.993476i	$-0.463620\pi$
$173$	3.00000	0.228086	0.114043	+	0.993476i	$0.463620\pi$
$174$	3.00000	0.227429
$175$	0	0
$176$	4.00000	0.301511
$177$	−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
$181$	−12.0000	−0.891953	−0.445976	−	0.895045i	$-0.647144\pi$
$181$	−12.0000	−0.891953	−0.445976	+	0.895045i	$0.647144\pi$
$182$	0	0
$183$	−4.00000	−0.295689
$184$	−4.00000	−0.294884
$185$	0	0
$186$	−4.00000	−0.293294
$187$	0	0
$188$	3.00000	0.218797
$189$	0	0
$190$	0	0
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	−1.00000	−0.0721688
$193$	−20.0000	−1.43963	−0.719816	−	0.694165i	$-0.755774\pi$
$193$	−20.0000	−1.43963	−0.719816	+	0.694165i	$0.755774\pi$
$194$	12.0000	0.861550
$195$	0	0
$196$	−7.00000	−0.500000
$197$	−12.0000	−0.854965	−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000	−0.854965	−0.427482	+	0.904024i	$0.640599\pi$
$198$	4.00000	0.284268
$199$	1.00000	0.0708881	0.0354441	−	0.999372i	$-0.488715\pi$
$199$	1.00000	0.0708881	0.0354441	+	0.999372i	$0.488715\pi$
$200$	0	0
$201$	−9.00000	−0.634811
$202$	−2.00000	−0.140720
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−4.00000	−0.278019
$208$	1.00000	0.0693375
$209$	4.00000	0.276686
$210$	0	0
$211$	24.0000	1.65223	0.826114	−	0.563503i	$-0.190547\pi$
$211$	24.0000	1.65223	0.826114	+	0.563503i	$0.190547\pi$
$212$	−1.00000	−0.0686803
$213$	−7.00000	−0.479632
$214$	−9.00000	−0.615227
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	−1.00000	−0.0677285
$219$	4.00000	0.270295
$220$	0	0
$221$	0	0
$222$	−5.00000	−0.335578
$223$	14.0000	0.937509	0.468755	−	0.883328i	$-0.344703\pi$
$223$	14.0000	0.937509	0.468755	+	0.883328i	$0.344703\pi$
$224$	0	0
$225$	0	0
$226$	−2.00000	−0.133038
$227$	10.0000	0.663723	0.331862	−	0.943328i	$-0.392323\pi$
$227$	10.0000	0.663723	0.331862	+	0.943328i	$0.392323\pi$
$228$	−1.00000	−0.0662266
$229$	5.00000	0.330409	0.165205	−	0.986259i	$-0.447172\pi$
$229$	5.00000	0.330409	0.165205	+	0.986259i	$0.447172\pi$
$230$	0	0
$231$	0	0
$232$	−3.00000	−0.196960
$233$	−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
$237$	−11.0000	−0.714527
$238$	0	0
$239$	−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
$241$	−12.0000	−0.772988	−0.386494	−	0.922292i	$-0.626314\pi$
$241$	−12.0000	−0.772988	−0.386494	+	0.922292i	$0.626314\pi$
$242$	5.00000	0.321412
$243$	−1.00000	−0.0641500
$244$	4.00000	0.256074
$245$	0	0
$246$	−9.00000	−0.573819
$247$	1.00000	0.0636285
$248$	4.00000	0.254000
$249$	6.00000	0.380235
$250$	0	0
$251$	23.0000	1.45175	0.725874	−	0.687828i	$-0.241436\pi$
$251$	23.0000	1.45175	0.725874	+	0.687828i	$0.241436\pi$
$252$	0	0
$253$	−16.0000	−1.00591
$254$	17.0000	1.06667
$255$	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	$0.769245\pi$
$258$	2.00000	0.124515
$259$	0	0
$260$	0	0
$261$	−3.00000	−0.185695
$262$	−3.00000	−0.185341
$263$	−6.00000	−0.369976	−0.184988	−	0.982741i	$-0.559225\pi$
$263$	−6.00000	−0.369976	−0.184988	+	0.982741i	$0.559225\pi$
$264$	−4.00000	−0.246183
$265$	0	0
$266$	0	0
$267$	−10.0000	−0.611990
$268$	9.00000	0.549762
$269$	−11.0000	−0.670682	−0.335341	−	0.942097i	$-0.608852\pi$
$269$	−11.0000	−0.670682	−0.335341	+	0.942097i	$0.608852\pi$
$270$	0	0
$271$	24.0000	1.45790	0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000	1.45790	0.728948	+	0.684569i	$0.240010\pi$
$272$	0	0
$273$	0	0
$274$	−23.0000	−1.38948
$275$	0	0
$276$	4.00000	0.240772
$277$	26.0000	1.56219	0.781094	−	0.624413i	$-0.214662\pi$
$277$	26.0000	1.56219	0.781094	+	0.624413i	$0.214662\pi$
$278$	−12.0000	−0.719712
$279$	4.00000	0.239474
$280$	0	0
$281$	−5.00000	−0.298275	−0.149137	−	0.988816i	$-0.547650\pi$
$281$	−5.00000	−0.298275	−0.149137	+	0.988816i	$0.547650\pi$
$282$	−3.00000	−0.178647
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	7.00000	0.415374
$285$	0	0
$286$	4.00000	0.236525
$287$	0	0
$288$	1.00000	0.0589256
$289$	−17.0000	−1.00000
$290$	0	0
$291$	−12.0000	−0.703452
$292$	−4.00000	−0.234082
$293$	16.0000	0.934730	0.467365	−	0.884064i	$-0.345203\pi$
$293$	16.0000	0.934730	0.467365	+	0.884064i	$0.345203\pi$
$294$	7.00000	0.408248
$295$	0	0
$296$	5.00000	0.290619
$297$	−4.00000	−0.232104
$298$	0	0
$299$	−4.00000	−0.231326
$300$	0	0
$301$	0	0
$302$	12.0000	0.690522
$303$	2.00000	0.114897
$304$	1.00000	0.0573539
$305$	0	0
$306$	0	0
$307$	−3.00000	−0.171219	−0.0856095	−	0.996329i	$-0.527284\pi$
$307$	−3.00000	−0.171219	−0.0856095	+	0.996329i	$0.527284\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−30.0000	−1.70114	−0.850572	−	0.525859i	$-0.823744\pi$
$311$	−30.0000	−1.70114	−0.850572	+	0.525859i	$0.823744\pi$
$312$	−1.00000	−0.0566139
$313$	−1.00000	−0.0565233	−0.0282617	−	0.999601i	$-0.508997\pi$
$313$	−1.00000	−0.0565233	−0.0282617	+	0.999601i	$0.508997\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	11.0000	0.618798
$317$	−32.0000	−1.79730	−0.898650	−	0.438667i	$-0.855451\pi$
$317$	−32.0000	−1.79730	−0.898650	+	0.438667i	$0.855451\pi$
$318$	1.00000	0.0560772
$319$	−12.0000	−0.671871
$320$	0	0
$321$	9.00000	0.502331
$322$	0	0
$323$	0	0
$324$	1.00000	0.0555556
$325$	0	0
$326$	−8.00000	−0.443079
$327$	1.00000	0.0553001
$328$	9.00000	0.496942
$329$	0	0
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	−6.00000	−0.329293
$333$	5.00000	0.273998
$334$	−7.00000	−0.383023
$335$	0	0
$336$	0	0
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	1.00000	0.0543928
$339$	2.00000	0.108625
$340$	0	0
$341$	16.0000	0.866449
$342$	1.00000	0.0540738
$343$	0	0
$344$	−2.00000	−0.107833
$345$	0	0
$346$	3.00000	0.161281
$347$	17.0000	0.912608	0.456304	−	0.889824i	$-0.349173\pi$
$347$	17.0000	0.912608	0.456304	+	0.889824i	$0.349173\pi$
$348$	3.00000	0.160817
$349$	18.0000	0.963518	0.481759	−	0.876304i	$-0.339998\pi$
$349$	18.0000	0.963518	0.481759	+	0.876304i	$0.339998\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	4.00000	0.213201
$353$	15.0000	0.798369	0.399185	−	0.916871i	$-0.369293\pi$
$353$	15.0000	0.798369	0.399185	+	0.916871i	$0.369293\pi$
$354$	−10.0000	−0.531494
$355$	0	0
$356$	10.0000	0.529999
$357$	0	0
$358$	−12.0000	−0.634220
$359$	−17.0000	−0.897226	−0.448613	−	0.893726i	$-0.648082\pi$
$359$	−17.0000	−0.897226	−0.448613	+	0.893726i	$0.648082\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	−12.0000	−0.630706
$363$	−5.00000	−0.262432
$364$	0	0
$365$	0	0
$366$	−4.00000	−0.209083
$367$	21.0000	1.09619	0.548096	−	0.836416i	$-0.315353\pi$
$367$	21.0000	1.09619	0.548096	+	0.836416i	$0.315353\pi$
$368$	−4.00000	−0.208514
$369$	9.00000	0.468521
$370$	0	0
$371$	0	0
$372$	−4.00000	−0.207390
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	0	0
$375$	0	0
$376$	3.00000	0.154713
$377$	−3.00000	−0.154508
$378$	0	0
$379$	−28.0000	−1.43826	−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000	−1.43826	−0.719132	+	0.694874i	$0.755460\pi$
$380$	0	0
$381$	−17.0000	−0.870936
$382$	−18.0000	−0.920960
$383$	5.00000	0.255488	0.127744	−	0.991807i	$-0.459226\pi$
$383$	5.00000	0.255488	0.127744	+	0.991807i	$0.459226\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−20.0000	−1.01797
$387$	−2.00000	−0.101666
$388$	12.0000	0.609208
$389$	−19.0000	−0.963338	−0.481669	−	0.876353i	$-0.659969\pi$
$389$	−19.0000	−0.963338	−0.481669	+	0.876353i	$0.659969\pi$
$390$	0	0
$391$	0	0
$392$	−7.00000	−0.353553
$393$	3.00000	0.151330
$394$	−12.0000	−0.604551
$395$	0	0
$396$	4.00000	0.201008
$397$	−3.00000	−0.150566	−0.0752828	−	0.997162i	$-0.523986\pi$
$397$	−3.00000	−0.150566	−0.0752828	+	0.997162i	$0.523986\pi$
$398$	1.00000	0.0501255
$399$	0	0
$400$	0	0
$401$	−26.0000	−1.29838	−0.649189	−	0.760627i	$-0.724892\pi$
$401$	−26.0000	−1.29838	−0.649189	+	0.760627i	$0.724892\pi$
$402$	−9.00000	−0.448879
$403$	4.00000	0.199254
$404$	−2.00000	−0.0995037
$405$	0	0
$406$	0	0
$407$	20.0000	0.991363
$408$	0	0
$409$	−40.0000	−1.97787	−0.988936	−	0.148340i	$-0.952607\pi$
$409$	−40.0000	−1.97787	−0.988936	+	0.148340i	$0.952607\pi$
$410$	0	0
$411$	23.0000	1.13451
$412$	0	0
$413$	0	0
$414$	−4.00000	−0.196589
$415$	0	0
$416$	1.00000	0.0490290
$417$	12.0000	0.587643
$418$	4.00000	0.195646
$419$	−37.0000	−1.80757	−0.903784	−	0.427989i	$-0.859222\pi$
$419$	−37.0000	−1.80757	−0.903784	+	0.427989i	$0.859222\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	24.0000	1.16830
$423$	3.00000	0.145865
$424$	−1.00000	−0.0485643
$425$	0	0
$426$	−7.00000	−0.339151
$427$	0	0
$428$	−9.00000	−0.435031
$429$	−4.00000	−0.193122
$430$	0	0
$431$	−3.00000	−0.144505	−0.0722525	−	0.997386i	$-0.523019\pi$
$431$	−3.00000	−0.144505	−0.0722525	+	0.997386i	$0.523019\pi$
$432$	−1.00000	−0.0481125
$433$	19.0000	0.913082	0.456541	−	0.889702i	$-0.349088\pi$
$433$	19.0000	0.913082	0.456541	+	0.889702i	$0.349088\pi$
$434$	0	0
$435$	0	0
$436$	−1.00000	−0.0478913
$437$	−4.00000	−0.191346
$438$	4.00000	0.191127
$439$	−19.0000	−0.906821	−0.453410	−	0.891302i	$-0.649793\pi$
$439$	−19.0000	−0.906821	−0.453410	+	0.891302i	$0.649793\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	0	0
$443$	3.00000	0.142534	0.0712672	−	0.997457i	$-0.477296\pi$
$443$	3.00000	0.142534	0.0712672	+	0.997457i	$0.477296\pi$
$444$	−5.00000	−0.237289
$445$	0	0
$446$	14.0000	0.662919
$447$	0	0
$448$	0	0
$449$	−9.00000	−0.424736	−0.212368	−	0.977190i	$-0.568118\pi$
$449$	−9.00000	−0.424736	−0.212368	+	0.977190i	$0.568118\pi$
$450$	0	0
$451$	36.0000	1.69517
$452$	−2.00000	−0.0940721
$453$	−12.0000	−0.563809
$454$	10.0000	0.469323
$455$	0	0
$456$	−1.00000	−0.0468293
$457$	−32.0000	−1.49690	−0.748448	−	0.663193i	$-0.769201\pi$
$457$	−32.0000	−1.49690	−0.748448	+	0.663193i	$0.769201\pi$
$458$	5.00000	0.233635
$459$	0	0
$460$	0	0
$461$	−24.0000	−1.11779	−0.558896	−	0.829238i	$-0.688775\pi$
$461$	−24.0000	−1.11779	−0.558896	+	0.829238i	$0.688775\pi$
$462$	0	0
$463$	14.0000	0.650635	0.325318	−	0.945605i	$-0.394529\pi$
$463$	14.0000	0.650635	0.325318	+	0.945605i	$0.394529\pi$
$464$	−3.00000	−0.139272
$465$	0	0
$466$	−8.00000	−0.370593
$467$	−15.0000	−0.694117	−0.347059	−	0.937843i	$-0.612820\pi$
$467$	−15.0000	−0.694117	−0.347059	+	0.937843i	$0.612820\pi$
$468$	1.00000	0.0462250
$469$	0	0
$470$	0	0
$471$	2.00000	0.0921551
$472$	10.0000	0.460287
$473$	−8.00000	−0.367840
$474$	−11.0000	−0.505247
$475$	0	0
$476$	0	0
$477$	−1.00000	−0.0457869
$478$	−8.00000	−0.365911
$479$	25.0000	1.14228	0.571140	−	0.820853i	$-0.306501\pi$
$479$	25.0000	1.14228	0.571140	+	0.820853i	$0.306501\pi$
$480$	0	0
$481$	5.00000	0.227980
$482$	−12.0000	−0.546585
$483$	0	0
$484$	5.00000	0.227273
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	38.0000	1.72194	0.860972	−	0.508652i	$-0.169856\pi$
$487$	38.0000	1.72194	0.860972	+	0.508652i	$0.169856\pi$
$488$	4.00000	0.181071
$489$	8.00000	0.361773
$490$	0	0
$491$	−32.0000	−1.44414	−0.722070	−	0.691820i	$-0.756809\pi$
$491$	−32.0000	−1.44414	−0.722070	+	0.691820i	$0.756809\pi$
$492$	−9.00000	−0.405751
$493$	0	0
$494$	1.00000	0.0449921
$495$	0	0
$496$	4.00000	0.179605
$497$	0	0
$498$	6.00000	0.268866
$499$	−1.00000	−0.0447661	−0.0223831	−	0.999749i	$-0.507125\pi$
$499$	−1.00000	−0.0447661	−0.0223831	+	0.999749i	$0.507125\pi$
$500$	0	0
$501$	7.00000	0.312737
$502$	23.0000	1.02654
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	0	0
$505$	0	0
$506$	−16.0000	−0.711287
$507$	−1.00000	−0.0444116
$508$	17.0000	0.754253
$509$	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
$513$	−1.00000	−0.0441511
$514$	−24.0000	−1.05859
$515$	0	0
$516$	2.00000	0.0880451
$517$	12.0000	0.527759
$518$	0	0
$519$	−3.00000	−0.131685
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	−3.00000	−0.131306
$523$	30.0000	1.31181	0.655904	−	0.754844i	$-0.272288\pi$
$523$	30.0000	1.31181	0.655904	+	0.754844i	$0.272288\pi$
$524$	−3.00000	−0.131056
$525$	0	0
$526$	−6.00000	−0.261612
$527$	0	0
$528$	−4.00000	−0.174078
$529$	−7.00000	−0.304348
$530$	0	0
$531$	10.0000	0.433963
$532$	0	0
$533$	9.00000	0.389833
$534$	−10.0000	−0.432742
$535$	0	0
$536$	9.00000	0.388741
$537$	12.0000	0.517838
$538$	−11.0000	−0.474244
$539$	−28.0000	−1.20605
$540$	0	0
$541$	−26.0000	−1.11783	−0.558914	−	0.829226i	$-0.688782\pi$
$541$	−26.0000	−1.11783	−0.558914	+	0.829226i	$0.688782\pi$
$542$	24.0000	1.03089
$543$	12.0000	0.514969
$544$	0	0
$545$	0	0
$546$	0	0
$547$	22.0000	0.940652	0.470326	−	0.882493i	$-0.344136\pi$
$547$	22.0000	0.940652	0.470326	+	0.882493i	$0.344136\pi$
$548$	−23.0000	−0.982511
$549$	4.00000	0.170716
$550$	0	0
$551$	−3.00000	−0.127804
$552$	4.00000	0.170251
$553$	0	0
$554$	26.0000	1.10463
$555$	0	0
$556$	−12.0000	−0.508913
$557$	−42.0000	−1.77960	−0.889799	−	0.456354i	$-0.849155\pi$
$557$	−42.0000	−1.77960	−0.889799	+	0.456354i	$0.849155\pi$
$558$	4.00000	0.169334
$559$	−2.00000	−0.0845910
$560$	0	0
$561$	0	0
$562$	−5.00000	−0.210912
$563$	21.0000	0.885044	0.442522	−	0.896758i	$-0.354084\pi$
$563$	21.0000	0.885044	0.442522	+	0.896758i	$0.354084\pi$
$564$	−3.00000	−0.126323
$565$	0	0
$566$	−4.00000	−0.168133
$567$	0	0
$568$	7.00000	0.293713
$569$	−8.00000	−0.335377	−0.167689	−	0.985840i	$-0.553630\pi$
$569$	−8.00000	−0.335377	−0.167689	+	0.985840i	$0.553630\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	4.00000	0.167248
$573$	18.0000	0.751961
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	−12.0000	−0.499567	−0.249783	−	0.968302i	$-0.580359\pi$
$577$	−12.0000	−0.499567	−0.249783	+	0.968302i	$0.580359\pi$
$578$	−17.0000	−0.707107
$579$	20.0000	0.831172
$580$	0	0
$581$	0	0
$582$	−12.0000	−0.497416
$583$	−4.00000	−0.165663
$584$	−4.00000	−0.165521
$585$	0	0
$586$	16.0000	0.660954
$587$	−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
$589$	4.00000	0.164817
$590$	0	0
$591$	12.0000	0.493614
$592$	5.00000	0.205499
$593$	1.00000	0.0410651	0.0205325	−	0.999789i	$-0.493464\pi$
$593$	1.00000	0.0410651	0.0205325	+	0.999789i	$0.493464\pi$
$594$	−4.00000	−0.164122
$595$	0	0
$596$	0	0
$597$	−1.00000	−0.0409273
$598$	−4.00000	−0.163572
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	45.0000	1.83559	0.917794	−	0.397057i	$-0.129968\pi$
$601$	45.0000	1.83559	0.917794	+	0.397057i	$0.129968\pi$
$602$	0	0
$603$	9.00000	0.366508
$604$	12.0000	0.488273
$605$	0	0
$606$	2.00000	0.0812444
$607$	7.00000	0.284121	0.142061	−	0.989858i	$-0.454627\pi$
$607$	7.00000	0.284121	0.142061	+	0.989858i	$0.454627\pi$
$608$	1.00000	0.0405554
$609$	0	0
$610$	0	0
$611$	3.00000	0.121367
$612$	0	0
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	−3.00000	−0.121070
$615$	0	0
$616$	0	0
$617$	27.0000	1.08698	0.543490	−	0.839416i	$-0.317103\pi$
$617$	27.0000	1.08698	0.543490	+	0.839416i	$0.317103\pi$
$618$	0	0
$619$	8.00000	0.321547	0.160774	−	0.986991i	$-0.448601\pi$
$619$	8.00000	0.321547	0.160774	+	0.986991i	$0.448601\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	−30.0000	−1.20289
$623$	0	0
$624$	−1.00000	−0.0400320
$625$	0	0
$626$	−1.00000	−0.0399680
$627$	−4.00000	−0.159745
$628$	−2.00000	−0.0798087
$629$	0	0
$630$	0	0
$631$	−28.0000	−1.11466	−0.557331	−	0.830290i	$-0.688175\pi$
$631$	−28.0000	−1.11466	−0.557331	+	0.830290i	$0.688175\pi$
$632$	11.0000	0.437557
$633$	−24.0000	−0.953914
$634$	−32.0000	−1.27088
$635$	0	0
$636$	1.00000	0.0396526
$637$	−7.00000	−0.277350
$638$	−12.0000	−0.475085
$639$	7.00000	0.276916
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	9.00000	0.355202
$643$	19.0000	0.749287	0.374643	−	0.927169i	$-0.377765\pi$
$643$	19.0000	0.749287	0.374643	+	0.927169i	$0.377765\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	24.0000	0.943537	0.471769	−	0.881722i	$-0.343616\pi$
$647$	24.0000	0.943537	0.471769	+	0.881722i	$0.343616\pi$
$648$	1.00000	0.0392837
$649$	40.0000	1.57014
$650$	0	0
$651$	0	0
$652$	−8.00000	−0.313304
$653$	26.0000	1.01746	0.508729	−	0.860927i	$-0.330115\pi$
$653$	26.0000	1.01746	0.508729	+	0.860927i	$0.330115\pi$
$654$	1.00000	0.0391031
$655$	0	0
$656$	9.00000	0.351391
$657$	−4.00000	−0.156055
$658$	0	0
$659$	13.0000	0.506408	0.253204	−	0.967413i	$-0.418516\pi$
$659$	13.0000	0.506408	0.253204	+	0.967413i	$0.418516\pi$
$660$	0	0
$661$	−13.0000	−0.505641	−0.252821	−	0.967513i	$-0.581358\pi$
$661$	−13.0000	−0.505641	−0.252821	+	0.967513i	$0.581358\pi$
$662$	4.00000	0.155464
$663$	0	0
$664$	−6.00000	−0.232845
$665$	0	0
$666$	5.00000	0.193746
$667$	12.0000	0.464642
$668$	−7.00000	−0.270838
$669$	−14.0000	−0.541271
$670$	0	0
$671$	16.0000	0.617673
$672$	0	0
$673$	−11.0000	−0.424019	−0.212009	−	0.977268i	$-0.568001\pi$
$673$	−11.0000	−0.424019	−0.212009	+	0.977268i	$0.568001\pi$
$674$	−22.0000	−0.847408
$675$	0	0
$676$	1.00000	0.0384615
$677$	−2.00000	−0.0768662	−0.0384331	−	0.999261i	$-0.512237\pi$
$677$	−2.00000	−0.0768662	−0.0384331	+	0.999261i	$0.512237\pi$
$678$	2.00000	0.0768095
$679$	0	0
$680$	0	0
$681$	−10.0000	−0.383201
$682$	16.0000	0.612672
$683$	28.0000	1.07139	0.535695	−	0.844411i	$-0.320050\pi$
$683$	28.0000	1.07139	0.535695	+	0.844411i	$0.320050\pi$
$684$	1.00000	0.0382360
$685$	0	0
$686$	0	0
$687$	−5.00000	−0.190762
$688$	−2.00000	−0.0762493
$689$	−1.00000	−0.0380970
$690$	0	0
$691$	−3.00000	−0.114125	−0.0570627	−	0.998371i	$-0.518173\pi$
$691$	−3.00000	−0.114125	−0.0570627	+	0.998371i	$0.518173\pi$
$692$	3.00000	0.114043
$693$	0	0
$694$	17.0000	0.645311
$695$	0	0
$696$	3.00000	0.113715
$697$	0	0
$698$	18.0000	0.681310
$699$	8.00000	0.302588
$700$	0	0
$701$	−38.0000	−1.43524	−0.717620	−	0.696435i	$-0.754769\pi$
$701$	−38.0000	−1.43524	−0.717620	+	0.696435i	$0.754769\pi$
$702$	−1.00000	−0.0377426
$703$	5.00000	0.188579
$704$	4.00000	0.150756
$705$	0	0
$706$	15.0000	0.564532
$707$	0	0
$708$	−10.0000	−0.375823
$709$	46.0000	1.72757	0.863783	−	0.503864i	$-0.168089\pi$
$709$	46.0000	1.72757	0.863783	+	0.503864i	$0.168089\pi$
$710$	0	0
$711$	11.0000	0.412532
$712$	10.0000	0.374766
$713$	−16.0000	−0.599205
$714$	0	0
$715$	0	0
$716$	−12.0000	−0.448461
$717$	8.00000	0.298765
$718$	−17.0000	−0.634434
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	0	0
$721$	0	0
$722$	−18.0000	−0.669891
$723$	12.0000	0.446285
$724$	−12.0000	−0.445976
$725$	0	0
$726$	−5.00000	−0.185567
$727$	−52.0000	−1.92857	−0.964287	−	0.264861i	$-0.914674\pi$
$727$	−52.0000	−1.92857	−0.964287	+	0.264861i	$0.914674\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	−4.00000	−0.147844
$733$	49.0000	1.80986	0.904928	−	0.425564i	$-0.139924\pi$
$733$	49.0000	1.80986	0.904928	+	0.425564i	$0.139924\pi$
$734$	21.0000	0.775124
$735$	0	0
$736$	−4.00000	−0.147442
$737$	36.0000	1.32608
$738$	9.00000	0.331295
$739$	53.0000	1.94964	0.974818	−	0.223001i	$-0.0715853\pi$
$739$	53.0000	1.94964	0.974818	+	0.223001i	$0.0715853\pi$
$740$	0	0
$741$	−1.00000	−0.0367359
$742$	0	0
$743$	27.0000	0.990534	0.495267	−	0.868741i	$-0.335070\pi$
$743$	27.0000	0.990534	0.495267	+	0.868741i	$0.335070\pi$
$744$	−4.00000	−0.146647
$745$	0	0
$746$	10.0000	0.366126
$747$	−6.00000	−0.219529
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−43.0000	−1.56909	−0.784546	−	0.620070i	$-0.787104\pi$
$751$	−43.0000	−1.56909	−0.784546	+	0.620070i	$0.787104\pi$
$752$	3.00000	0.109399
$753$	−23.0000	−0.838167
$754$	−3.00000	−0.109254
$755$	0	0
$756$	0	0
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	−28.0000	−1.01701
$759$	16.0000	0.580763
$760$	0	0
$761$	−15.0000	−0.543750	−0.271875	−	0.962333i	$-0.587644\pi$
$761$	−15.0000	−0.543750	−0.271875	+	0.962333i	$0.587644\pi$
$762$	−17.0000	−0.615845
$763$	0	0
$764$	−18.0000	−0.651217
$765$	0	0
$766$	5.00000	0.180657
$767$	10.0000	0.361079
$768$	−1.00000	−0.0360844
$769$	16.0000	0.576975	0.288487	−	0.957484i	$-0.406848\pi$
$769$	16.0000	0.576975	0.288487	+	0.957484i	$0.406848\pi$
$770$	0	0
$771$	24.0000	0.864339
$772$	−20.0000	−0.719816
$773$	32.0000	1.15096	0.575480	−	0.817816i	$-0.304815\pi$
$773$	32.0000	1.15096	0.575480	+	0.817816i	$0.304815\pi$
$774$	−2.00000	−0.0718885
$775$	0	0
$776$	12.0000	0.430775
$777$	0	0
$778$	−19.0000	−0.681183
$779$	9.00000	0.322458
$780$	0	0
$781$	28.0000	1.00192
$782$	0	0
$783$	3.00000	0.107211
$784$	−7.00000	−0.250000
$785$	0	0
$786$	3.00000	0.107006
$787$	−32.0000	−1.14068	−0.570338	−	0.821410i	$-0.693188\pi$
$787$	−32.0000	−1.14068	−0.570338	+	0.821410i	$0.693188\pi$
$788$	−12.0000	−0.427482
$789$	6.00000	0.213606
$790$	0	0
$791$	0	0
$792$	4.00000	0.142134
$793$	4.00000	0.142044
$794$	−3.00000	−0.106466
$795$	0	0
$796$	1.00000	0.0354441
$797$	22.0000	0.779280	0.389640	−	0.920967i	$-0.372599\pi$
$797$	22.0000	0.779280	0.389640	+	0.920967i	$0.372599\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	10.0000	0.353333
$802$	−26.0000	−0.918092
$803$	−16.0000	−0.564628
$804$	−9.00000	−0.317406
$805$	0	0
$806$	4.00000	0.140894
$807$	11.0000	0.387218
$808$	−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
$811$	12.0000	0.421377	0.210688	−	0.977553i	$-0.432429\pi$
$811$	12.0000	0.421377	0.210688	+	0.977553i	$0.432429\pi$
$812$	0	0
$813$	−24.0000	−0.841717
$814$	20.0000	0.701000
$815$	0	0
$816$	0	0
$817$	−2.00000	−0.0699711
$818$	−40.0000	−1.39857
$819$	0	0
$820$	0	0
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	23.0000	0.802217
$823$	31.0000	1.08059	0.540296	−	0.841475i	$-0.318312\pi$
$823$	31.0000	1.08059	0.540296	+	0.841475i	$0.318312\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	−4.00000	−0.139010
$829$	46.0000	1.59765	0.798823	−	0.601566i	$-0.205456\pi$
$829$	46.0000	1.59765	0.798823	+	0.601566i	$0.205456\pi$
$830$	0	0
$831$	−26.0000	−0.901930
$832$	1.00000	0.0346688
$833$	0	0
$834$	12.0000	0.415526
$835$	0	0
$836$	4.00000	0.138343
$837$	−4.00000	−0.138260
$838$	−37.0000	−1.27814
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	2.00000	0.0689246
$843$	5.00000	0.172209
$844$	24.0000	0.826114
$845$	0	0
$846$	3.00000	0.103142
$847$	0	0
$848$	−1.00000	−0.0343401
$849$	4.00000	0.137280
$850$	0	0
$851$	−20.0000	−0.685591
$852$	−7.00000	−0.239816
$853$	−41.0000	−1.40381	−0.701907	−	0.712269i	$-0.747668\pi$
$853$	−41.0000	−1.40381	−0.701907	+	0.712269i	$0.747668\pi$
$854$	0	0
$855$	0	0
$856$	−9.00000	−0.307614
$857$	−42.0000	−1.43469	−0.717346	−	0.696717i	$-0.754643\pi$
$857$	−42.0000	−1.43469	−0.717346	+	0.696717i	$0.754643\pi$
$858$	−4.00000	−0.136558
$859$	22.0000	0.750630	0.375315	−	0.926897i	$-0.377534\pi$
$859$	22.0000	0.750630	0.375315	+	0.926897i	$0.377534\pi$
$860$	0	0
$861$	0	0
$862$	−3.00000	−0.102180
$863$	−57.0000	−1.94030	−0.970151	−	0.242500i	$-0.922032\pi$
$863$	−57.0000	−1.94030	−0.970151	+	0.242500i	$0.922032\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	19.0000	0.645646
$867$	17.0000	0.577350
$868$	0	0
$869$	44.0000	1.49260
$870$	0	0
$871$	9.00000	0.304953
$872$	−1.00000	−0.0338643
$873$	12.0000	0.406138
$874$	−4.00000	−0.135302
$875$	0	0
$876$	4.00000	0.135147
$877$	23.0000	0.776655	0.388327	−	0.921521i	$-0.373053\pi$
$877$	23.0000	0.776655	0.388327	+	0.921521i	$0.373053\pi$
$878$	−19.0000	−0.641219
$879$	−16.0000	−0.539667
$880$	0	0
$881$	56.0000	1.88669	0.943344	−	0.331816i	$-0.107661\pi$
$881$	56.0000	1.88669	0.943344	+	0.331816i	$0.107661\pi$
$882$	−7.00000	−0.235702
$883$	−36.0000	−1.21150	−0.605748	−	0.795656i	$-0.707126\pi$
$883$	−36.0000	−1.21150	−0.605748	+	0.795656i	$0.707126\pi$
$884$	0	0
$885$	0	0
$886$	3.00000	0.100787
$887$	44.0000	1.47738	0.738688	−	0.674048i	$-0.235446\pi$
$887$	44.0000	1.47738	0.738688	+	0.674048i	$0.235446\pi$
$888$	−5.00000	−0.167789
$889$	0	0
$890$	0	0
$891$	4.00000	0.134005
$892$	14.0000	0.468755
$893$	3.00000	0.100391
$894$	0	0
$895$	0	0
$896$	0	0
$897$	4.00000	0.133556
$898$	−9.00000	−0.300334
$899$	−12.0000	−0.400222
$900$	0	0
$901$	0	0
$902$	36.0000	1.19867
$903$	0	0
$904$	−2.00000	−0.0665190
$905$	0	0
$906$	−12.0000	−0.398673
$907$	30.0000	0.996134	0.498067	−	0.867139i	$-0.334043\pi$
$907$	30.0000	0.996134	0.498067	+	0.867139i	$0.334043\pi$
$908$	10.0000	0.331862
$909$	−2.00000	−0.0663358
$910$	0	0
$911$	−6.00000	−0.198789	−0.0993944	−	0.995048i	$-0.531691\pi$
$911$	−6.00000	−0.198789	−0.0993944	+	0.995048i	$0.531691\pi$
$912$	−1.00000	−0.0331133
$913$	−24.0000	−0.794284
$914$	−32.0000	−1.05847
$915$	0	0
$916$	5.00000	0.165205
$917$	0	0
$918$	0	0
$919$	1.00000	0.0329870	0.0164935	−	0.999864i	$-0.494750\pi$
$919$	1.00000	0.0329870	0.0164935	+	0.999864i	$0.494750\pi$
$920$	0	0
$921$	3.00000	0.0988534
$922$	−24.0000	−0.790398
$923$	7.00000	0.230408
$924$	0	0
$925$	0	0
$926$	14.0000	0.460069
$927$	0	0
$928$	−3.00000	−0.0984798
$929$	1.00000	0.0328089	0.0164045	−	0.999865i	$-0.494778\pi$
$929$	1.00000	0.0328089	0.0164045	+	0.999865i	$0.494778\pi$
$930$	0	0
$931$	−7.00000	−0.229416
$932$	−8.00000	−0.262049
$933$	30.0000	0.982156
$934$	−15.0000	−0.490815
$935$	0	0
$936$	1.00000	0.0326860
$937$	−34.0000	−1.11073	−0.555366	−	0.831606i	$-0.687422\pi$
$937$	−34.0000	−1.11073	−0.555366	+	0.831606i	$0.687422\pi$
$938$	0	0
$939$	1.00000	0.0326338
$940$	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
$946$	−8.00000	−0.260102
$947$	4.00000	0.129983	0.0649913	−	0.997886i	$-0.479298\pi$
$947$	4.00000	0.129983	0.0649913	+	0.997886i	$0.479298\pi$
$948$	−11.0000	−0.357263
$949$	−4.00000	−0.129845
$950$	0	0
$951$	32.0000	1.03767
$952$	0	0
$953$	−56.0000	−1.81402	−0.907009	−	0.421111i	$-0.861640\pi$
$953$	−56.0000	−1.81402	−0.907009	+	0.421111i	$0.861640\pi$
$954$	−1.00000	−0.0323762
$955$	0	0
$956$	−8.00000	−0.258738
$957$	12.0000	0.387905
$958$	25.0000	0.807713
$959$	0	0
$960$	0	0
$961$	−15.0000	−0.483871
$962$	5.00000	0.161206
$963$	−9.00000	−0.290021
$964$	−12.0000	−0.386494
$965$	0	0
$966$	0	0
$967$	−8.00000	−0.257263	−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000	−0.257263	−0.128631	+	0.991692i	$0.541058\pi$
$968$	5.00000	0.160706
$969$	0	0
$970$	0	0
$971$	15.0000	0.481373	0.240686	−	0.970603i	$-0.422627\pi$
$971$	15.0000	0.481373	0.240686	+	0.970603i	$0.422627\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	38.0000	1.21760
$975$	0	0
$976$	4.00000	0.128037
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	8.00000	0.255812
$979$	40.0000	1.27841
$980$	0	0
$981$	−1.00000	−0.0319275
$982$	−32.0000	−1.02116
$983$	32.0000	1.02064	0.510321	−	0.859984i	$-0.329527\pi$
$983$	32.0000	1.02064	0.510321	+	0.859984i	$0.329527\pi$
$984$	−9.00000	−0.286910
$985$	0	0
$986$	0	0
$987$	0	0
$988$	1.00000	0.0318142
$989$	8.00000	0.254385
$990$	0	0
$991$	−7.00000	−0.222362	−0.111181	−	0.993800i	$-0.535463\pi$
$991$	−7.00000	−0.222362	−0.111181	+	0.993800i	$0.535463\pi$
$992$	4.00000	0.127000
$993$	−4.00000	−0.126936
$994$	0	0
$995$	0	0
$996$	6.00000	0.190117
$997$	14.0000	0.443384	0.221692	−	0.975117i	$-0.428842\pi$
$997$	14.0000	0.443384	0.221692	+	0.975117i	$0.428842\pi$
$998$	−1.00000	−0.0316544
$999$	−5.00000	−0.158193

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1950.2.a.s.1.1	yes	1
3.2	odd	2		5850.2.a.l.1.1		1
5.2	odd	4		1950.2.e.h.1249.2		2
5.3	odd	4		1950.2.e.h.1249.1		2
5.4	even	2		1950.2.a.j.1.1	✓	1
15.2	even	4		5850.2.e.f.5149.1		2
15.8	even	4		5850.2.e.f.5149.2		2
15.14	odd	2		5850.2.a.bp.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1950.2.a.j.1.1	✓	1	5.4	even	2
1950.2.a.s.1.1	yes	1	1.1	even	1	trivial
1950.2.e.h.1249.1		2	5.3	odd	4
1950.2.e.h.1249.2		2	5.2	odd	4
5850.2.a.l.1.1		1	3.2	odd	2
5850.2.a.bp.1.1		1	15.14	odd	2
5850.2.e.f.5149.1		2	15.2	even	4
5850.2.e.f.5149.2		2	15.8	even	4

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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