LMFDB - Embedded newform 1950.2.a.x.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)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +4.00000 q^{11} +1.00000 q^{12} +1.00000 q^{13} -4.00000 q^{14} +1.00000 q^{16} -4.00000 q^{17} +1.00000 q^{18} +7.00000 q^{19} -4.00000 q^{21} +4.00000 q^{22} +4.00000 q^{23} +1.00000 q^{24} +1.00000 q^{26} +1.00000 q^{27} -4.00000 q^{28} +5.00000 q^{29} +4.00000 q^{31} +1.00000 q^{32} +4.00000 q^{33} -4.00000 q^{34} +1.00000 q^{36} +9.00000 q^{37} +7.00000 q^{38} +1.00000 q^{39} -5.00000 q^{41} -4.00000 q^{42} -10.0000 q^{43} +4.00000 q^{44} +4.00000 q^{46} +3.00000 q^{47} +1.00000 q^{48} +9.00000 q^{49} -4.00000 q^{51} +1.00000 q^{52} +9.00000 q^{53} +1.00000 q^{54} -4.00000 q^{56} +7.00000 q^{57} +5.00000 q^{58} -6.00000 q^{59} +4.00000 q^{61} +4.00000 q^{62} -4.00000 q^{63} +1.00000 q^{64} +4.00000 q^{66} -7.00000 q^{67} -4.00000 q^{68} +4.00000 q^{69} -15.0000 q^{71} +1.00000 q^{72} +12.0000 q^{73} +9.00000 q^{74} +7.00000 q^{76} -16.0000 q^{77} +1.00000 q^{78} +7.00000 q^{79} +1.00000 q^{81} -5.00000 q^{82} +6.00000 q^{83} -4.00000 q^{84} -10.0000 q^{86} +5.00000 q^{87} +4.00000 q^{88} +14.0000 q^{89} -4.00000 q^{91} +4.00000 q^{92} +4.00000 q^{93} +3.00000 q^{94} +1.00000 q^{96} -16.0000 q^{97} +9.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	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
$14$	−4.00000	−1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	1.00000	0.235702
$19$	7.00000	1.60591	0.802955	−	0.596040i	$-0.203260\pi$
$19$	7.00000	1.60591	0.802955	+	0.596040i	$0.203260\pi$
$20$	0	0
$21$	−4.00000	−0.872872
$22$	4.00000	0.852803
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	1.00000	0.196116
$27$	1.00000	0.192450
$28$	−4.00000	−0.755929
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	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
$34$	−4.00000	−0.685994
$35$	0	0
$36$	1.00000	0.166667
$37$	9.00000	1.47959	0.739795	−	0.672832i	$-0.234922\pi$
$37$	9.00000	1.47959	0.739795	+	0.672832i	$0.234922\pi$
$38$	7.00000	1.13555
$39$	1.00000	0.160128
$40$	0	0
$41$	−5.00000	−0.780869	−0.390434	−	0.920631i	$-0.627675\pi$
$41$	−5.00000	−0.780869	−0.390434	+	0.920631i	$0.627675\pi$
$42$	−4.00000	−0.617213
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	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
$49$	9.00000	1.28571
$50$	0	0
$51$	−4.00000	−0.560112
$52$	1.00000	0.138675
$53$	9.00000	1.23625	0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000	1.23625	0.618123	+	0.786082i	$0.287894\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	−4.00000	−0.534522
$57$	7.00000	0.927173
$58$	5.00000	0.656532
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	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
$63$	−4.00000	−0.503953
$64$	1.00000	0.125000
$65$	0	0
$66$	4.00000	0.492366
$67$	−7.00000	−0.855186	−0.427593	−	0.903971i	$-0.640638\pi$
$67$	−7.00000	−0.855186	−0.427593	+	0.903971i	$0.640638\pi$
$68$	−4.00000	−0.485071
$69$	4.00000	0.481543
$70$	0	0
$71$	−15.0000	−1.78017	−0.890086	−	0.455792i	$-0.849356\pi$
$71$	−15.0000	−1.78017	−0.890086	+	0.455792i	$0.849356\pi$
$72$	1.00000	0.117851
$73$	12.0000	1.40449	0.702247	−	0.711934i	$-0.252180\pi$
$73$	12.0000	1.40449	0.702247	+	0.711934i	$0.252180\pi$
$74$	9.00000	1.04623
$75$	0	0
$76$	7.00000	0.802955
$77$	−16.0000	−1.82337
$78$	1.00000	0.113228
$79$	7.00000	0.787562	0.393781	−	0.919204i	$-0.371167\pi$
$79$	7.00000	0.787562	0.393781	+	0.919204i	$0.371167\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−5.00000	−0.552158
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	−4.00000	−0.436436
$85$	0	0
$86$	−10.0000	−1.07833
$87$	5.00000	0.536056
$88$	4.00000	0.426401
$89$	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
$91$	−4.00000	−0.419314
$92$	4.00000	0.417029
$93$	4.00000	0.414781
$94$	3.00000	0.309426
$95$	0	0
$96$	1.00000	0.102062
$97$	−16.0000	−1.62455	−0.812277	−	0.583272i	$-0.801772\pi$
$97$	−16.0000	−1.62455	−0.812277	+	0.583272i	$0.801772\pi$
$98$	9.00000	0.909137
$99$	4.00000	0.402015
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	−4.00000	−0.396059
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	9.00000	0.874157
$107$	5.00000	0.483368	0.241684	−	0.970355i	$-0.422300\pi$
$107$	5.00000	0.483368	0.241684	+	0.970355i	$0.422300\pi$
$108$	1.00000	0.0962250
$109$	−11.0000	−1.05361	−0.526804	−	0.849987i	$-0.676610\pi$
$109$	−11.0000	−1.05361	−0.526804	+	0.849987i	$0.676610\pi$
$110$	0	0
$111$	9.00000	0.854242
$112$	−4.00000	−0.377964
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	7.00000	0.655610
$115$	0	0
$116$	5.00000	0.464238
$117$	1.00000	0.0924500
$118$	−6.00000	−0.552345
$119$	16.0000	1.46672
$120$	0	0
$121$	5.00000	0.454545
$122$	4.00000	0.362143
$123$	−5.00000	−0.450835
$124$	4.00000	0.359211
$125$	0	0
$126$	−4.00000	−0.356348
$127$	−5.00000	−0.443678	−0.221839	−	0.975083i	$-0.571206\pi$
$127$	−5.00000	−0.443678	−0.221839	+	0.975083i	$0.571206\pi$
$128$	1.00000	0.0883883
$129$	−10.0000	−0.880451
$130$	0	0
$131$	17.0000	1.48530	0.742648	−	0.669681i	$-0.233569\pi$
$131$	17.0000	1.48530	0.742648	+	0.669681i	$0.233569\pi$
$132$	4.00000	0.348155
$133$	−28.0000	−2.42791
$134$	−7.00000	−0.604708
$135$	0	0
$136$	−4.00000	−0.342997
$137$	−19.0000	−1.62328	−0.811640	−	0.584158i	$-0.801425\pi$
$137$	−19.0000	−1.62328	−0.811640	+	0.584158i	$0.801425\pi$
$138$	4.00000	0.340503
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	3.00000	0.252646
$142$	−15.0000	−1.25877
$143$	4.00000	0.334497
$144$	1.00000	0.0833333
$145$	0	0
$146$	12.0000	0.993127
$147$	9.00000	0.742307
$148$	9.00000	0.739795
$149$	−4.00000	−0.327693	−0.163846	−	0.986486i	$-0.552390\pi$
$149$	−4.00000	−0.327693	−0.163846	+	0.986486i	$0.552390\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	7.00000	0.567775
$153$	−4.00000	−0.323381
$154$	−16.0000	−1.28932
$155$	0	0
$156$	1.00000	0.0800641
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	7.00000	0.556890
$159$	9.00000	0.713746
$160$	0	0
$161$	−16.0000	−1.26098
$162$	1.00000	0.0785674
$163$	−24.0000	−1.87983	−0.939913	−	0.341415i	$-0.889094\pi$
$163$	−24.0000	−1.87983	−0.939913	+	0.341415i	$0.889094\pi$
$164$	−5.00000	−0.390434
$165$	0	0
$166$	6.00000	0.465690
$167$	9.00000	0.696441	0.348220	−	0.937413i	$-0.386786\pi$
$167$	9.00000	0.696441	0.348220	+	0.937413i	$0.386786\pi$
$168$	−4.00000	−0.308607
$169$	1.00000	0.0769231
$170$	0	0
$171$	7.00000	0.535303
$172$	−10.0000	−0.762493
$173$	13.0000	0.988372	0.494186	−	0.869356i	$-0.335466\pi$
$173$	13.0000	0.988372	0.494186	+	0.869356i	$0.335466\pi$
$174$	5.00000	0.379049
$175$	0	0
$176$	4.00000	0.301511
$177$	−6.00000	−0.450988
$178$	14.0000	1.04934
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	−4.00000	−0.297318	−0.148659	−	0.988889i	$-0.547496\pi$
$181$	−4.00000	−0.297318	−0.148659	+	0.988889i	$0.547496\pi$
$182$	−4.00000	−0.296500
$183$	4.00000	0.295689
$184$	4.00000	0.294884
$185$	0	0
$186$	4.00000	0.293294
$187$	−16.0000	−1.17004
$188$	3.00000	0.218797
$189$	−4.00000	−0.290957
$190$	0	0
$191$	10.0000	0.723575	0.361787	−	0.932261i	$-0.382167\pi$
$191$	10.0000	0.723575	0.361787	+	0.932261i	$0.382167\pi$
$192$	1.00000	0.0721688
$193$	−12.0000	−0.863779	−0.431889	−	0.901927i	$-0.642153\pi$
$193$	−12.0000	−0.863779	−0.431889	+	0.901927i	$0.642153\pi$
$194$	−16.0000	−1.14873
$195$	0	0
$196$	9.00000	0.642857
$197$	−16.0000	−1.13995	−0.569976	−	0.821661i	$-0.693048\pi$
$197$	−16.0000	−1.13995	−0.569976	+	0.821661i	$0.693048\pi$
$198$	4.00000	0.284268
$199$	−3.00000	−0.212664	−0.106332	−	0.994331i	$-0.533911\pi$
$199$	−3.00000	−0.212664	−0.106332	+	0.994331i	$0.533911\pi$
$200$	0	0
$201$	−7.00000	−0.493742
$202$	−10.0000	−0.703598
$203$	−20.0000	−1.40372
$204$	−4.00000	−0.280056
$205$	0	0
$206$	8.00000	0.557386
$207$	4.00000	0.278019
$208$	1.00000	0.0693375
$209$	28.0000	1.93680
$210$	0	0
$211$	−16.0000	−1.10149	−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000	−1.10149	−0.550743	+	0.834675i	$0.685655\pi$
$212$	9.00000	0.618123
$213$	−15.0000	−1.02778
$214$	5.00000	0.341793
$215$	0	0
$216$	1.00000	0.0680414
$217$	−16.0000	−1.08615
$218$	−11.0000	−0.745014
$219$	12.0000	0.810885
$220$	0	0
$221$	−4.00000	−0.269069
$222$	9.00000	0.604040
$223$	−2.00000	−0.133930	−0.0669650	−	0.997755i	$-0.521332\pi$
$223$	−2.00000	−0.133930	−0.0669650	+	0.997755i	$0.521332\pi$
$224$	−4.00000	−0.267261
$225$	0	0
$226$	−14.0000	−0.931266
$227$	6.00000	0.398234	0.199117	−	0.979976i	$-0.436193\pi$
$227$	6.00000	0.398234	0.199117	+	0.979976i	$0.436193\pi$
$228$	7.00000	0.463586
$229$	−17.0000	−1.12339	−0.561696	−	0.827344i	$-0.689851\pi$
$229$	−17.0000	−1.12339	−0.561696	+	0.827344i	$0.689851\pi$
$230$	0	0
$231$	−16.0000	−1.05272
$232$	5.00000	0.328266
$233$	8.00000	0.524097	0.262049	−	0.965055i	$-0.415602\pi$
$233$	8.00000	0.524097	0.262049	+	0.965055i	$0.415602\pi$
$234$	1.00000	0.0653720
$235$	0	0
$236$	−6.00000	−0.390567
$237$	7.00000	0.454699
$238$	16.0000	1.03713
$239$	16.0000	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$240$	0	0
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	5.00000	0.321412
$243$	1.00000	0.0641500
$244$	4.00000	0.256074
$245$	0	0
$246$	−5.00000	−0.318788
$247$	7.00000	0.445399
$248$	4.00000	0.254000
$249$	6.00000	0.380235
$250$	0	0
$251$	−5.00000	−0.315597	−0.157799	−	0.987471i	$-0.550440\pi$
$251$	−5.00000	−0.315597	−0.157799	+	0.987471i	$0.550440\pi$
$252$	−4.00000	−0.251976
$253$	16.0000	1.00591
$254$	−5.00000	−0.313728
$255$	0	0
$256$	1.00000	0.0625000
$257$	8.00000	0.499026	0.249513	−	0.968371i	$-0.419729\pi$
$257$	8.00000	0.499026	0.249513	+	0.968371i	$0.419729\pi$
$258$	−10.0000	−0.622573
$259$	−36.0000	−2.23693
$260$	0	0
$261$	5.00000	0.309492
$262$	17.0000	1.05026
$263$	30.0000	1.84988	0.924940	−	0.380114i	$-0.124115\pi$
$263$	30.0000	1.84988	0.924940	+	0.380114i	$0.124115\pi$
$264$	4.00000	0.246183
$265$	0	0
$266$	−28.0000	−1.71679
$267$	14.0000	0.856786
$268$	−7.00000	−0.427593
$269$	5.00000	0.304855	0.152428	−	0.988315i	$-0.451291\pi$
$269$	5.00000	0.304855	0.152428	+	0.988315i	$0.451291\pi$
$270$	0	0
$271$	32.0000	1.94386	0.971931	−	0.235267i	$-0.0755965\pi$
$271$	32.0000	1.94386	0.971931	+	0.235267i	$0.0755965\pi$
$272$	−4.00000	−0.242536
$273$	−4.00000	−0.242091
$274$	−19.0000	−1.14783
$275$	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	$0.214662\pi$
$278$	−4.00000	−0.239904
$279$	4.00000	0.239474
$280$	0	0
$281$	17.0000	1.01413	0.507067	−	0.861906i	$-0.330729\pi$
$281$	17.0000	1.01413	0.507067	+	0.861906i	$0.330729\pi$
$282$	3.00000	0.178647
$283$	8.00000	0.475551	0.237775	−	0.971320i	$-0.423582\pi$
$283$	8.00000	0.475551	0.237775	+	0.971320i	$0.423582\pi$
$284$	−15.0000	−0.890086
$285$	0	0
$286$	4.00000	0.236525
$287$	20.0000	1.18056
$288$	1.00000	0.0589256
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−16.0000	−0.937937
$292$	12.0000	0.702247
$293$	−12.0000	−0.701047	−0.350524	−	0.936554i	$-0.613996\pi$
$293$	−12.0000	−0.701047	−0.350524	+	0.936554i	$0.613996\pi$
$294$	9.00000	0.524891
$295$	0	0
$296$	9.00000	0.523114
$297$	4.00000	0.232104
$298$	−4.00000	−0.231714
$299$	4.00000	0.231326
$300$	0	0
$301$	40.0000	2.30556
$302$	−4.00000	−0.230174
$303$	−10.0000	−0.574485
$304$	7.00000	0.401478
$305$	0	0
$306$	−4.00000	−0.228665
$307$	21.0000	1.19853	0.599267	−	0.800549i	$-0.295459\pi$
$307$	21.0000	1.19853	0.599267	+	0.800549i	$0.295459\pi$
$308$	−16.0000	−0.911685
$309$	8.00000	0.455104
$310$	0	0
$311$	2.00000	0.113410	0.0567048	−	0.998391i	$-0.481941\pi$
$311$	2.00000	0.113410	0.0567048	+	0.998391i	$0.481941\pi$
$312$	1.00000	0.0566139
$313$	−23.0000	−1.30004	−0.650018	−	0.759918i	$-0.725239\pi$
$313$	−23.0000	−1.30004	−0.650018	+	0.759918i	$0.725239\pi$
$314$	−14.0000	−0.790066
$315$	0	0
$316$	7.00000	0.393781
$317$	−28.0000	−1.57264	−0.786318	−	0.617822i	$-0.788015\pi$
$317$	−28.0000	−1.57264	−0.786318	+	0.617822i	$0.788015\pi$
$318$	9.00000	0.504695
$319$	20.0000	1.11979
$320$	0	0
$321$	5.00000	0.279073
$322$	−16.0000	−0.891645
$323$	−28.0000	−1.55796
$324$	1.00000	0.0555556
$325$	0	0
$326$	−24.0000	−1.32924
$327$	−11.0000	−0.608301
$328$	−5.00000	−0.276079
$329$	−12.0000	−0.661581
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	6.00000	0.329293
$333$	9.00000	0.493197
$334$	9.00000	0.492458
$335$	0	0
$336$	−4.00000	−0.218218
$337$	−34.0000	−1.85210	−0.926049	−	0.377403i	$-0.876817\pi$
$337$	−34.0000	−1.85210	−0.926049	+	0.377403i	$0.876817\pi$
$338$	1.00000	0.0543928
$339$	−14.0000	−0.760376
$340$	0	0
$341$	16.0000	0.866449
$342$	7.00000	0.378517
$343$	−8.00000	−0.431959
$344$	−10.0000	−0.539164
$345$	0	0
$346$	13.0000	0.698884
$347$	11.0000	0.590511	0.295255	−	0.955418i	$-0.404595\pi$
$347$	11.0000	0.590511	0.295255	+	0.955418i	$0.404595\pi$
$348$	5.00000	0.268028
$349$	−34.0000	−1.81998	−0.909989	−	0.414632i	$-0.863910\pi$
$349$	−34.0000	−1.81998	−0.909989	+	0.414632i	$0.863910\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	4.00000	0.213201
$353$	−21.0000	−1.11772	−0.558859	−	0.829263i	$-0.688761\pi$
$353$	−21.0000	−1.11772	−0.558859	+	0.829263i	$0.688761\pi$
$354$	−6.00000	−0.318896
$355$	0	0
$356$	14.0000	0.741999
$357$	16.0000	0.846810
$358$	−4.00000	−0.211407
$359$	−15.0000	−0.791670	−0.395835	−	0.918322i	$-0.629545\pi$
$359$	−15.0000	−0.791670	−0.395835	+	0.918322i	$0.629545\pi$
$360$	0	0
$361$	30.0000	1.57895
$362$	−4.00000	−0.210235
$363$	5.00000	0.262432
$364$	−4.00000	−0.209657
$365$	0	0
$366$	4.00000	0.209083
$367$	7.00000	0.365397	0.182699	−	0.983169i	$-0.441517\pi$
$367$	7.00000	0.365397	0.182699	+	0.983169i	$0.441517\pi$
$368$	4.00000	0.208514
$369$	−5.00000	−0.260290
$370$	0	0
$371$	−36.0000	−1.86903
$372$	4.00000	0.207390
$373$	22.0000	1.13912	0.569558	−	0.821951i	$-0.307114\pi$
$373$	22.0000	1.13912	0.569558	+	0.821951i	$0.307114\pi$
$374$	−16.0000	−0.827340
$375$	0	0
$376$	3.00000	0.154713
$377$	5.00000	0.257513
$378$	−4.00000	−0.205738
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	−5.00000	−0.256158
$382$	10.0000	0.511645
$383$	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
$386$	−12.0000	−0.610784
$387$	−10.0000	−0.508329
$388$	−16.0000	−0.812277
$389$	5.00000	0.253510	0.126755	−	0.991934i	$-0.459544\pi$
$389$	5.00000	0.253510	0.126755	+	0.991934i	$0.459544\pi$
$390$	0	0
$391$	−16.0000	−0.809155
$392$	9.00000	0.454569
$393$	17.0000	0.857537
$394$	−16.0000	−0.806068
$395$	0	0
$396$	4.00000	0.201008
$397$	−15.0000	−0.752828	−0.376414	−	0.926451i	$-0.622843\pi$
$397$	−15.0000	−0.752828	−0.376414	+	0.926451i	$0.622843\pi$
$398$	−3.00000	−0.150376
$399$	−28.0000	−1.40175
$400$	0	0
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	−7.00000	−0.349128
$403$	4.00000	0.199254
$404$	−10.0000	−0.497519
$405$	0	0
$406$	−20.0000	−0.992583
$407$	36.0000	1.78445
$408$	−4.00000	−0.198030
$409$	−24.0000	−1.18672	−0.593362	−	0.804936i	$-0.702200\pi$
$409$	−24.0000	−1.18672	−0.593362	+	0.804936i	$0.702200\pi$
$410$	0	0
$411$	−19.0000	−0.937201
$412$	8.00000	0.394132
$413$	24.0000	1.18096
$414$	4.00000	0.196589
$415$	0	0
$416$	1.00000	0.0490290
$417$	−4.00000	−0.195881
$418$	28.0000	1.36952
$419$	23.0000	1.12362	0.561812	−	0.827265i	$-0.310105\pi$
$419$	23.0000	1.12362	0.561812	+	0.827265i	$0.310105\pi$
$420$	0	0
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	−16.0000	−0.778868
$423$	3.00000	0.145865
$424$	9.00000	0.437079
$425$	0	0
$426$	−15.0000	−0.726752
$427$	−16.0000	−0.774294
$428$	5.00000	0.241684
$429$	4.00000	0.193122
$430$	0	0
$431$	19.0000	0.915198	0.457599	−	0.889159i	$-0.348710\pi$
$431$	19.0000	0.915198	0.457599	+	0.889159i	$0.348710\pi$
$432$	1.00000	0.0481125
$433$	−19.0000	−0.913082	−0.456541	−	0.889702i	$-0.650912\pi$
$433$	−19.0000	−0.913082	−0.456541	+	0.889702i	$0.650912\pi$
$434$	−16.0000	−0.768025
$435$	0	0
$436$	−11.0000	−0.526804
$437$	28.0000	1.33942
$438$	12.0000	0.573382
$439$	−7.00000	−0.334092	−0.167046	−	0.985949i	$-0.553423\pi$
$439$	−7.00000	−0.334092	−0.167046	+	0.985949i	$0.553423\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	−4.00000	−0.190261
$443$	25.0000	1.18779	0.593893	−	0.804544i	$-0.297590\pi$
$443$	25.0000	1.18779	0.593893	+	0.804544i	$0.297590\pi$
$444$	9.00000	0.427121
$445$	0	0
$446$	−2.00000	−0.0947027
$447$	−4.00000	−0.189194
$448$	−4.00000	−0.188982
$449$	−27.0000	−1.27421	−0.637104	−	0.770778i	$-0.719868\pi$
$449$	−27.0000	−1.27421	−0.637104	+	0.770778i	$0.719868\pi$
$450$	0	0
$451$	−20.0000	−0.941763
$452$	−14.0000	−0.658505
$453$	−4.00000	−0.187936
$454$	6.00000	0.281594
$455$	0	0
$456$	7.00000	0.327805
$457$	4.00000	0.187112	0.0935561	−	0.995614i	$-0.470177\pi$
$457$	4.00000	0.187112	0.0935561	+	0.995614i	$0.470177\pi$
$458$	−17.0000	−0.794358
$459$	−4.00000	−0.186704
$460$	0	0
$461$	−24.0000	−1.11779	−0.558896	−	0.829238i	$-0.688775\pi$
$461$	−24.0000	−1.11779	−0.558896	+	0.829238i	$0.688775\pi$
$462$	−16.0000	−0.744387
$463$	−26.0000	−1.20832	−0.604161	−	0.796862i	$-0.706492\pi$
$463$	−26.0000	−1.20832	−0.604161	+	0.796862i	$0.706492\pi$
$464$	5.00000	0.232119
$465$	0	0
$466$	8.00000	0.370593
$467$	3.00000	0.138823	0.0694117	−	0.997588i	$-0.477888\pi$
$467$	3.00000	0.138823	0.0694117	+	0.997588i	$0.477888\pi$
$468$	1.00000	0.0462250
$469$	28.0000	1.29292
$470$	0	0
$471$	−14.0000	−0.645086
$472$	−6.00000	−0.276172
$473$	−40.0000	−1.83920
$474$	7.00000	0.321521
$475$	0	0
$476$	16.0000	0.733359
$477$	9.00000	0.412082
$478$	16.0000	0.731823
$479$	15.0000	0.685367	0.342684	−	0.939451i	$-0.388664\pi$
$479$	15.0000	0.685367	0.342684	+	0.939451i	$0.388664\pi$
$480$	0	0
$481$	9.00000	0.410365
$482$	8.00000	0.364390
$483$	−16.0000	−0.728025
$484$	5.00000	0.227273
$485$	0	0
$486$	1.00000	0.0453609
$487$	22.0000	0.996915	0.498458	−	0.866914i	$-0.333900\pi$
$487$	22.0000	0.996915	0.498458	+	0.866914i	$0.333900\pi$
$488$	4.00000	0.181071
$489$	−24.0000	−1.08532
$490$	0	0
$491$	−40.0000	−1.80517	−0.902587	−	0.430507i	$-0.858335\pi$
$491$	−40.0000	−1.80517	−0.902587	+	0.430507i	$0.858335\pi$
$492$	−5.00000	−0.225417
$493$	−20.0000	−0.900755
$494$	7.00000	0.314945
$495$	0	0
$496$	4.00000	0.179605
$497$	60.0000	2.69137
$498$	6.00000	0.268866
$499$	−23.0000	−1.02962	−0.514811	−	0.857304i	$-0.672138\pi$
$499$	−23.0000	−1.02962	−0.514811	+	0.857304i	$0.672138\pi$
$500$	0	0
$501$	9.00000	0.402090
$502$	−5.00000	−0.223161
$503$	36.0000	1.60516	0.802580	−	0.596544i	$-0.203460\pi$
$503$	36.0000	1.60516	0.802580	+	0.596544i	$0.203460\pi$
$504$	−4.00000	−0.178174
$505$	0	0
$506$	16.0000	0.711287
$507$	1.00000	0.0444116
$508$	−5.00000	−0.221839
$509$	−4.00000	−0.177297	−0.0886484	−	0.996063i	$-0.528255\pi$
$509$	−4.00000	−0.177297	−0.0886484	+	0.996063i	$0.528255\pi$
$510$	0	0
$511$	−48.0000	−2.12339
$512$	1.00000	0.0441942
$513$	7.00000	0.309058
$514$	8.00000	0.352865
$515$	0	0
$516$	−10.0000	−0.440225
$517$	12.0000	0.527759
$518$	−36.0000	−1.58175
$519$	13.0000	0.570637
$520$	0	0
$521$	36.0000	1.57719	0.788594	−	0.614914i	$-0.210809\pi$
$521$	36.0000	1.57719	0.788594	+	0.614914i	$0.210809\pi$
$522$	5.00000	0.218844
$523$	−14.0000	−0.612177	−0.306089	−	0.952003i	$-0.599020\pi$
$523$	−14.0000	−0.612177	−0.306089	+	0.952003i	$0.599020\pi$
$524$	17.0000	0.742648
$525$	0	0
$526$	30.0000	1.30806
$527$	−16.0000	−0.696971
$528$	4.00000	0.174078
$529$	−7.00000	−0.304348
$530$	0	0
$531$	−6.00000	−0.260378
$532$	−28.0000	−1.21395
$533$	−5.00000	−0.216574
$534$	14.0000	0.605839
$535$	0	0
$536$	−7.00000	−0.302354
$537$	−4.00000	−0.172613
$538$	5.00000	0.215565
$539$	36.0000	1.55063
$540$	0	0
$541$	18.0000	0.773880	0.386940	−	0.922105i	$-0.373532\pi$
$541$	18.0000	0.773880	0.386940	+	0.922105i	$0.373532\pi$
$542$	32.0000	1.37452
$543$	−4.00000	−0.171656
$544$	−4.00000	−0.171499
$545$	0	0
$546$	−4.00000	−0.171184
$547$	6.00000	0.256541	0.128271	−	0.991739i	$-0.459057\pi$
$547$	6.00000	0.256541	0.128271	+	0.991739i	$0.459057\pi$
$548$	−19.0000	−0.811640
$549$	4.00000	0.170716
$550$	0	0
$551$	35.0000	1.49105
$552$	4.00000	0.170251
$553$	−28.0000	−1.19068
$554$	26.0000	1.10463
$555$	0	0
$556$	−4.00000	−0.169638
$557$	−38.0000	−1.61011	−0.805056	−	0.593199i	$-0.797865\pi$
$557$	−38.0000	−1.61011	−0.805056	+	0.593199i	$0.797865\pi$
$558$	4.00000	0.169334
$559$	−10.0000	−0.422955
$560$	0	0
$561$	−16.0000	−0.675521
$562$	17.0000	0.717102
$563$	39.0000	1.64365	0.821827	−	0.569737i	$-0.192955\pi$
$563$	39.0000	1.64365	0.821827	+	0.569737i	$0.192955\pi$
$564$	3.00000	0.126323
$565$	0	0
$566$	8.00000	0.336265
$567$	−4.00000	−0.167984
$568$	−15.0000	−0.629386
$569$	8.00000	0.335377	0.167689	−	0.985840i	$-0.446370\pi$
$569$	8.00000	0.335377	0.167689	+	0.985840i	$0.446370\pi$
$570$	0	0
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	4.00000	0.167248
$573$	10.0000	0.417756
$574$	20.0000	0.834784
$575$	0	0
$576$	1.00000	0.0416667
$577$	−8.00000	−0.333044	−0.166522	−	0.986038i	$-0.553254\pi$
$577$	−8.00000	−0.333044	−0.166522	+	0.986038i	$0.553254\pi$
$578$	−1.00000	−0.0415945
$579$	−12.0000	−0.498703
$580$	0	0
$581$	−24.0000	−0.995688
$582$	−16.0000	−0.663221
$583$	36.0000	1.49097
$584$	12.0000	0.496564
$585$	0	0
$586$	−12.0000	−0.495715
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	9.00000	0.371154
$589$	28.0000	1.15372
$590$	0	0
$591$	−16.0000	−0.658152
$592$	9.00000	0.369898
$593$	29.0000	1.19089	0.595444	−	0.803397i	$-0.296976\pi$
$593$	29.0000	1.19089	0.595444	+	0.803397i	$0.296976\pi$
$594$	4.00000	0.164122
$595$	0	0
$596$	−4.00000	−0.163846
$597$	−3.00000	−0.122782
$598$	4.00000	0.163572
$599$	−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
$601$	−3.00000	−0.122373	−0.0611863	−	0.998126i	$-0.519488\pi$
$601$	−3.00000	−0.122373	−0.0611863	+	0.998126i	$0.519488\pi$
$602$	40.0000	1.63028
$603$	−7.00000	−0.285062
$604$	−4.00000	−0.162758
$605$	0	0
$606$	−10.0000	−0.406222
$607$	−3.00000	−0.121766	−0.0608831	−	0.998145i	$-0.519392\pi$
$607$	−3.00000	−0.121766	−0.0608831	+	0.998145i	$0.519392\pi$
$608$	7.00000	0.283887
$609$	−20.0000	−0.810441
$610$	0	0
$611$	3.00000	0.121367
$612$	−4.00000	−0.161690
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	21.0000	0.847491
$615$	0	0
$616$	−16.0000	−0.644658
$617$	7.00000	0.281809	0.140905	−	0.990023i	$-0.454999\pi$
$617$	7.00000	0.281809	0.140905	+	0.990023i	$0.454999\pi$
$618$	8.00000	0.321807
$619$	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
$622$	2.00000	0.0801927
$623$	−56.0000	−2.24359
$624$	1.00000	0.0400320
$625$	0	0
$626$	−23.0000	−0.919265
$627$	28.0000	1.11821
$628$	−14.0000	−0.558661
$629$	−36.0000	−1.43541
$630$	0	0
$631$	40.0000	1.59237	0.796187	−	0.605050i	$-0.206847\pi$
$631$	40.0000	1.59237	0.796187	+	0.605050i	$0.206847\pi$
$632$	7.00000	0.278445
$633$	−16.0000	−0.635943
$634$	−28.0000	−1.11202
$635$	0	0
$636$	9.00000	0.356873
$637$	9.00000	0.356593
$638$	20.0000	0.791808
$639$	−15.0000	−0.593391
$640$	0	0
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	5.00000	0.197334
$643$	−5.00000	−0.197181	−0.0985904	−	0.995128i	$-0.531433\pi$
$643$	−5.00000	−0.197181	−0.0985904	+	0.995128i	$0.531433\pi$
$644$	−16.0000	−0.630488
$645$	0	0
$646$	−28.0000	−1.10165
$647$	−36.0000	−1.41531	−0.707653	−	0.706560i	$-0.750246\pi$
$647$	−36.0000	−1.41531	−0.707653	+	0.706560i	$0.750246\pi$
$648$	1.00000	0.0392837
$649$	−24.0000	−0.942082
$650$	0	0
$651$	−16.0000	−0.627089
$652$	−24.0000	−0.939913
$653$	−10.0000	−0.391330	−0.195665	−	0.980671i	$-0.562687\pi$
$653$	−10.0000	−0.391330	−0.195665	+	0.980671i	$0.562687\pi$
$654$	−11.0000	−0.430134
$655$	0	0
$656$	−5.00000	−0.195217
$657$	12.0000	0.468165
$658$	−12.0000	−0.467809
$659$	33.0000	1.28550	0.642749	−	0.766077i	$-0.277794\pi$
$659$	33.0000	1.28550	0.642749	+	0.766077i	$0.277794\pi$
$660$	0	0
$661$	−7.00000	−0.272268	−0.136134	−	0.990690i	$-0.543468\pi$
$661$	−7.00000	−0.272268	−0.136134	+	0.990690i	$0.543468\pi$
$662$	−28.0000	−1.08825
$663$	−4.00000	−0.155347
$664$	6.00000	0.232845
$665$	0	0
$666$	9.00000	0.348743
$667$	20.0000	0.774403
$668$	9.00000	0.348220
$669$	−2.00000	−0.0773245
$670$	0	0
$671$	16.0000	0.617673
$672$	−4.00000	−0.154303
$673$	19.0000	0.732396	0.366198	−	0.930537i	$-0.380659\pi$
$673$	19.0000	0.732396	0.366198	+	0.930537i	$0.380659\pi$
$674$	−34.0000	−1.30963
$675$	0	0
$676$	1.00000	0.0384615
$677$	−14.0000	−0.538064	−0.269032	−	0.963131i	$-0.586704\pi$
$677$	−14.0000	−0.538064	−0.269032	+	0.963131i	$0.586704\pi$
$678$	−14.0000	−0.537667
$679$	64.0000	2.45609
$680$	0	0
$681$	6.00000	0.229920
$682$	16.0000	0.612672
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	7.00000	0.267652
$685$	0	0
$686$	−8.00000	−0.305441
$687$	−17.0000	−0.648590
$688$	−10.0000	−0.381246
$689$	9.00000	0.342873
$690$	0	0
$691$	3.00000	0.114125	0.0570627	−	0.998371i	$-0.481827\pi$
$691$	3.00000	0.114125	0.0570627	+	0.998371i	$0.481827\pi$
$692$	13.0000	0.494186
$693$	−16.0000	−0.607790
$694$	11.0000	0.417554
$695$	0	0
$696$	5.00000	0.189525
$697$	20.0000	0.757554
$698$	−34.0000	−1.28692
$699$	8.00000	0.302588
$700$	0	0
$701$	34.0000	1.28416	0.642081	−	0.766637i	$-0.278071\pi$
$701$	34.0000	1.28416	0.642081	+	0.766637i	$0.278071\pi$
$702$	1.00000	0.0377426
$703$	63.0000	2.37609
$704$	4.00000	0.150756
$705$	0	0
$706$	−21.0000	−0.790345
$707$	40.0000	1.50435
$708$	−6.00000	−0.225494
$709$	−38.0000	−1.42712	−0.713560	−	0.700594i	$-0.752918\pi$
$709$	−38.0000	−1.42712	−0.713560	+	0.700594i	$0.752918\pi$
$710$	0	0
$711$	7.00000	0.262521
$712$	14.0000	0.524672
$713$	16.0000	0.599205
$714$	16.0000	0.598785
$715$	0	0
$716$	−4.00000	−0.149487
$717$	16.0000	0.597531
$718$	−15.0000	−0.559795
$719$	−12.0000	−0.447524	−0.223762	−	0.974644i	$-0.571834\pi$
$719$	−12.0000	−0.447524	−0.223762	+	0.974644i	$0.571834\pi$
$720$	0	0
$721$	−32.0000	−1.19174
$722$	30.0000	1.11648
$723$	8.00000	0.297523
$724$	−4.00000	−0.148659
$725$	0	0
$726$	5.00000	0.185567
$727$	28.0000	1.03846	0.519231	−	0.854634i	$-0.326218\pi$
$727$	28.0000	1.03846	0.519231	+	0.854634i	$0.326218\pi$
$728$	−4.00000	−0.148250
$729$	1.00000	0.0370370
$730$	0	0
$731$	40.0000	1.47945
$732$	4.00000	0.147844
$733$	−3.00000	−0.110808	−0.0554038	−	0.998464i	$-0.517645\pi$
$733$	−3.00000	−0.110808	−0.0554038	+	0.998464i	$0.517645\pi$
$734$	7.00000	0.258375
$735$	0	0
$736$	4.00000	0.147442
$737$	−28.0000	−1.03139
$738$	−5.00000	−0.184053
$739$	−37.0000	−1.36107	−0.680534	−	0.732717i	$-0.738252\pi$
$739$	−37.0000	−1.36107	−0.680534	+	0.732717i	$0.738252\pi$
$740$	0	0
$741$	7.00000	0.257151
$742$	−36.0000	−1.32160
$743$	−21.0000	−0.770415	−0.385208	−	0.922830i	$-0.625870\pi$
$743$	−21.0000	−0.770415	−0.385208	+	0.922830i	$0.625870\pi$
$744$	4.00000	0.146647
$745$	0	0
$746$	22.0000	0.805477
$747$	6.00000	0.219529
$748$	−16.0000	−0.585018
$749$	−20.0000	−0.730784
$750$	0	0
$751$	−23.0000	−0.839282	−0.419641	−	0.907690i	$-0.637844\pi$
$751$	−23.0000	−0.839282	−0.419641	+	0.907690i	$0.637844\pi$
$752$	3.00000	0.109399
$753$	−5.00000	−0.182210
$754$	5.00000	0.182089
$755$	0	0
$756$	−4.00000	−0.145479
$757$	8.00000	0.290765	0.145382	−	0.989376i	$-0.453559\pi$
$757$	8.00000	0.290765	0.145382	+	0.989376i	$0.453559\pi$
$758$	−20.0000	−0.726433
$759$	16.0000	0.580763
$760$	0	0
$761$	35.0000	1.26875	0.634375	−	0.773026i	$-0.281258\pi$
$761$	35.0000	1.26875	0.634375	+	0.773026i	$0.281258\pi$
$762$	−5.00000	−0.181131
$763$	44.0000	1.59291
$764$	10.0000	0.361787
$765$	0	0
$766$	5.00000	0.180657
$767$	−6.00000	−0.216647
$768$	1.00000	0.0360844
$769$	40.0000	1.44244	0.721218	−	0.692708i	$-0.243582\pi$
$769$	40.0000	1.44244	0.721218	+	0.692708i	$0.243582\pi$
$770$	0	0
$771$	8.00000	0.288113
$772$	−12.0000	−0.431889
$773$	−8.00000	−0.287740	−0.143870	−	0.989597i	$-0.545955\pi$
$773$	−8.00000	−0.287740	−0.143870	+	0.989597i	$0.545955\pi$
$774$	−10.0000	−0.359443
$775$	0	0
$776$	−16.0000	−0.574367
$777$	−36.0000	−1.29149
$778$	5.00000	0.179259
$779$	−35.0000	−1.25401
$780$	0	0
$781$	−60.0000	−2.14697
$782$	−16.0000	−0.572159
$783$	5.00000	0.178685
$784$	9.00000	0.321429
$785$	0	0
$786$	17.0000	0.606370
$787$	−32.0000	−1.14068	−0.570338	−	0.821410i	$-0.693188\pi$
$787$	−32.0000	−1.14068	−0.570338	+	0.821410i	$0.693188\pi$
$788$	−16.0000	−0.569976
$789$	30.0000	1.06803
$790$	0	0
$791$	56.0000	1.99113
$792$	4.00000	0.142134
$793$	4.00000	0.142044
$794$	−15.0000	−0.532330
$795$	0	0
$796$	−3.00000	−0.106332
$797$	−30.0000	−1.06265	−0.531327	−	0.847167i	$-0.678307\pi$
$797$	−30.0000	−1.06265	−0.531327	+	0.847167i	$0.678307\pi$
$798$	−28.0000	−0.991189
$799$	−12.0000	−0.424529
$800$	0	0
$801$	14.0000	0.494666
$802$	−30.0000	−1.05934
$803$	48.0000	1.69388
$804$	−7.00000	−0.246871
$805$	0	0
$806$	4.00000	0.140894
$807$	5.00000	0.176008
$808$	−10.0000	−0.351799
$809$	18.0000	0.632846	0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000	0.632846	0.316423	+	0.948618i	$0.397518\pi$
$810$	0	0
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	−20.0000	−0.701862
$813$	32.0000	1.12229
$814$	36.0000	1.26180
$815$	0	0
$816$	−4.00000	−0.140028
$817$	−70.0000	−2.44899
$818$	−24.0000	−0.839140
$819$	−4.00000	−0.139771
$820$	0	0
$821$	42.0000	1.46581	0.732905	−	0.680331i	$-0.238164\pi$
$821$	42.0000	1.46581	0.732905	+	0.680331i	$0.238164\pi$
$822$	−19.0000	−0.662701
$823$	−19.0000	−0.662298	−0.331149	−	0.943578i	$-0.607436\pi$
$823$	−19.0000	−0.662298	−0.331149	+	0.943578i	$0.607436\pi$
$824$	8.00000	0.278693
$825$	0	0
$826$	24.0000	0.835067
$827$	−4.00000	−0.139094	−0.0695468	−	0.997579i	$-0.522155\pi$
$827$	−4.00000	−0.139094	−0.0695468	+	0.997579i	$0.522155\pi$
$828$	4.00000	0.139010
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	0	0
$831$	26.0000	0.901930
$832$	1.00000	0.0346688
$833$	−36.0000	−1.24733
$834$	−4.00000	−0.138509
$835$	0	0
$836$	28.0000	0.968400
$837$	4.00000	0.138260
$838$	23.0000	0.794522
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	14.0000	0.482472
$843$	17.0000	0.585511
$844$	−16.0000	−0.550743
$845$	0	0
$846$	3.00000	0.103142
$847$	−20.0000	−0.687208
$848$	9.00000	0.309061
$849$	8.00000	0.274559
$850$	0	0
$851$	36.0000	1.23406
$852$	−15.0000	−0.513892
$853$	19.0000	0.650548	0.325274	−	0.945620i	$-0.394544\pi$
$853$	19.0000	0.650548	0.325274	+	0.945620i	$0.394544\pi$
$854$	−16.0000	−0.547509
$855$	0	0
$856$	5.00000	0.170896
$857$	−50.0000	−1.70797	−0.853984	−	0.520300i	$-0.825820\pi$
$857$	−50.0000	−1.70797	−0.853984	+	0.520300i	$0.825820\pi$
$858$	4.00000	0.136558
$859$	−2.00000	−0.0682391	−0.0341196	−	0.999418i	$-0.510863\pi$
$859$	−2.00000	−0.0682391	−0.0341196	+	0.999418i	$0.510863\pi$
$860$	0	0
$861$	20.0000	0.681598
$862$	19.0000	0.647143
$863$	39.0000	1.32758	0.663788	−	0.747921i	$-0.268948\pi$
$863$	39.0000	1.32758	0.663788	+	0.747921i	$0.268948\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−19.0000	−0.645646
$867$	−1.00000	−0.0339618
$868$	−16.0000	−0.543075
$869$	28.0000	0.949835
$870$	0	0
$871$	−7.00000	−0.237186
$872$	−11.0000	−0.372507
$873$	−16.0000	−0.541518
$874$	28.0000	0.947114
$875$	0	0
$876$	12.0000	0.405442
$877$	43.0000	1.45201	0.726003	−	0.687691i	$-0.241376\pi$
$877$	43.0000	1.45201	0.726003	+	0.687691i	$0.241376\pi$
$878$	−7.00000	−0.236239
$879$	−12.0000	−0.404750
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	9.00000	0.303046
$883$	40.0000	1.34611	0.673054	−	0.739594i	$-0.264982\pi$
$883$	40.0000	1.34611	0.673054	+	0.739594i	$0.264982\pi$
$884$	−4.00000	−0.134535
$885$	0	0
$886$	25.0000	0.839891
$887$	−36.0000	−1.20876	−0.604381	−	0.796696i	$-0.706579\pi$
$887$	−36.0000	−1.20876	−0.604381	+	0.796696i	$0.706579\pi$
$888$	9.00000	0.302020
$889$	20.0000	0.670778
$890$	0	0
$891$	4.00000	0.134005
$892$	−2.00000	−0.0669650
$893$	21.0000	0.702738
$894$	−4.00000	−0.133780
$895$	0	0
$896$	−4.00000	−0.133631
$897$	4.00000	0.133556
$898$	−27.0000	−0.901002
$899$	20.0000	0.667037
$900$	0	0
$901$	−36.0000	−1.19933
$902$	−20.0000	−0.665927
$903$	40.0000	1.33112
$904$	−14.0000	−0.465633
$905$	0	0
$906$	−4.00000	−0.132891
$907$	50.0000	1.66022	0.830111	−	0.557598i	$-0.188277\pi$
$907$	50.0000	1.66022	0.830111	+	0.557598i	$0.188277\pi$
$908$	6.00000	0.199117
$909$	−10.0000	−0.331679
$910$	0	0
$911$	−18.0000	−0.596367	−0.298183	−	0.954509i	$-0.596381\pi$
$911$	−18.0000	−0.596367	−0.298183	+	0.954509i	$0.596381\pi$
$912$	7.00000	0.231793
$913$	24.0000	0.794284
$914$	4.00000	0.132308
$915$	0	0
$916$	−17.0000	−0.561696
$917$	−68.0000	−2.24556
$918$	−4.00000	−0.132020
$919$	29.0000	0.956622	0.478311	−	0.878191i	$-0.341249\pi$
$919$	29.0000	0.956622	0.478311	+	0.878191i	$0.341249\pi$
$920$	0	0
$921$	21.0000	0.691974
$922$	−24.0000	−0.790398
$923$	−15.0000	−0.493731
$924$	−16.0000	−0.526361
$925$	0	0
$926$	−26.0000	−0.854413
$927$	8.00000	0.262754
$928$	5.00000	0.164133
$929$	11.0000	0.360898	0.180449	−	0.983584i	$-0.442245\pi$
$929$	11.0000	0.360898	0.180449	+	0.983584i	$0.442245\pi$
$930$	0	0
$931$	63.0000	2.06474
$932$	8.00000	0.262049
$933$	2.00000	0.0654771
$934$	3.00000	0.0981630
$935$	0	0
$936$	1.00000	0.0326860
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	28.0000	0.914232
$939$	−23.0000	−0.750577
$940$	0	0
$941$	52.0000	1.69515	0.847576	−	0.530674i	$-0.178061\pi$
$941$	52.0000	1.69515	0.847576	+	0.530674i	$0.178061\pi$
$942$	−14.0000	−0.456145
$943$	−20.0000	−0.651290
$944$	−6.00000	−0.195283
$945$	0	0
$946$	−40.0000	−1.30051
$947$	−8.00000	−0.259965	−0.129983	−	0.991516i	$-0.541492\pi$
$947$	−8.00000	−0.259965	−0.129983	+	0.991516i	$0.541492\pi$
$948$	7.00000	0.227349
$949$	12.0000	0.389536
$950$	0	0
$951$	−28.0000	−0.907962
$952$	16.0000	0.518563
$953$	36.0000	1.16615	0.583077	−	0.812417i	$-0.301849\pi$
$953$	36.0000	1.16615	0.583077	+	0.812417i	$0.301849\pi$
$954$	9.00000	0.291386
$955$	0	0
$956$	16.0000	0.517477
$957$	20.0000	0.646508
$958$	15.0000	0.484628
$959$	76.0000	2.45417
$960$	0	0
$961$	−15.0000	−0.483871
$962$	9.00000	0.290172
$963$	5.00000	0.161123
$964$	8.00000	0.257663
$965$	0	0
$966$	−16.0000	−0.514792
$967$	−44.0000	−1.41494	−0.707472	−	0.706741i	$-0.750165\pi$
$967$	−44.0000	−1.41494	−0.707472	+	0.706741i	$0.750165\pi$
$968$	5.00000	0.160706
$969$	−28.0000	−0.899490
$970$	0	0
$971$	−21.0000	−0.673922	−0.336961	−	0.941519i	$-0.609399\pi$
$971$	−21.0000	−0.673922	−0.336961	+	0.941519i	$0.609399\pi$
$972$	1.00000	0.0320750
$973$	16.0000	0.512936
$974$	22.0000	0.704925
$975$	0	0
$976$	4.00000	0.128037
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	−24.0000	−0.767435
$979$	56.0000	1.78977
$980$	0	0
$981$	−11.0000	−0.351203
$982$	−40.0000	−1.27645
$983$	−48.0000	−1.53096	−0.765481	−	0.643458i	$-0.777499\pi$
$983$	−48.0000	−1.53096	−0.765481	+	0.643458i	$0.777499\pi$
$984$	−5.00000	−0.159394
$985$	0	0
$986$	−20.0000	−0.636930
$987$	−12.0000	−0.381964
$988$	7.00000	0.222700
$989$	−40.0000	−1.27193
$990$	0	0
$991$	13.0000	0.412959	0.206479	−	0.978451i	$-0.433799\pi$
$991$	13.0000	0.412959	0.206479	+	0.978451i	$0.433799\pi$
$992$	4.00000	0.127000
$993$	−28.0000	−0.888553
$994$	60.0000	1.90308
$995$	0	0
$996$	6.00000	0.190117
$997$	−10.0000	−0.316703	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	−10.0000	−0.316703	−0.158352	+	0.987383i	$0.550618\pi$
$998$	−23.0000	−0.728052
$999$	9.00000	0.284747

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1950.2.a.x.1.1	yes	1
3.2	odd	2		5850.2.a.a.1.1		1
5.2	odd	4		1950.2.e.n.1249.2		2
5.3	odd	4		1950.2.e.n.1249.1		2
5.4	even	2		1950.2.a.e.1.1	✓	1
15.2	even	4		5850.2.e.c.5149.1		2
15.8	even	4		5850.2.e.c.5149.2		2
15.14	odd	2		5850.2.a.by.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1950.2.a.e.1.1	✓	1	5.4	even	2
1950.2.a.x.1.1	yes	1	1.1	even	1	trivial
1950.2.e.n.1249.1		2	5.3	odd	4
1950.2.e.n.1249.2		2	5.2	odd	4
5850.2.a.a.1.1		1	3.2	odd	2
5850.2.a.by.1.1		1	15.14	odd	2
5850.2.e.c.5149.1		2	15.2	even	4
5850.2.e.c.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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