LMFDB - Embedded newform 1950.2.a.q.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:	$1$
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)

$f(q)$

$=$

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

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{18} -8.00000 q^{19} +1.00000 q^{21} -3.00000 q^{22} +4.00000 q^{23} -1.00000 q^{24} +1.00000 q^{26} -1.00000 q^{27} -1.00000 q^{28} -7.00000 q^{29} +1.00000 q^{31} +1.00000 q^{32} +3.00000 q^{33} -1.00000 q^{34} +1.00000 q^{36} +4.00000 q^{37} -8.00000 q^{38} -1.00000 q^{39} -6.00000 q^{41} +1.00000 q^{42} -12.0000 q^{43} -3.00000 q^{44} +4.00000 q^{46} +3.00000 q^{47} -1.00000 q^{48} -6.00000 q^{49} +1.00000 q^{51} +1.00000 q^{52} +5.00000 q^{53} -1.00000 q^{54} -1.00000 q^{56} +8.00000 q^{57} -7.00000 q^{58} -9.00000 q^{59} +5.00000 q^{61} +1.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +3.00000 q^{66} -11.0000 q^{67} -1.00000 q^{68} -4.00000 q^{69} +8.00000 q^{71} +1.00000 q^{72} +4.00000 q^{74} -8.00000 q^{76} +3.00000 q^{77} -1.00000 q^{78} -8.00000 q^{79} +1.00000 q^{81} -6.00000 q^{82} +7.00000 q^{83} +1.00000 q^{84} -12.0000 q^{86} +7.00000 q^{87} -3.00000 q^{88} -8.00000 q^{89} -1.00000 q^{91} +4.00000 q^{92} -1.00000 q^{93} +3.00000 q^{94} -1.00000 q^{96} +6.00000 q^{97} -6.00000 q^{98} -3.00000 q^{99} +O(q^{100})\)

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
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	−1.00000	−0.288675
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536	−0.121268	−	0.992620i	$-0.538696\pi$
$17$	−1.00000	−0.242536	−0.121268	+	0.992620i	$0.538696\pi$
$18$	1.00000	0.235702
$19$	−8.00000	−1.83533	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	−8.00000	−1.83533	−0.917663	+	0.397360i	$0.869927\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	−3.00000	−0.639602
$23$	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
$28$	−1.00000	−0.188982
$29$	−7.00000	−1.29987	−0.649934	−	0.759991i	$-0.725203\pi$
$29$	−7.00000	−1.29987	−0.649934	+	0.759991i	$0.725203\pi$
$30$	0	0
$31$	1.00000	0.179605	0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000	0.179605	0.0898027	+	0.995960i	$0.471376\pi$
$32$	1.00000	0.176777
$33$	3.00000	0.522233
$34$	−1.00000	−0.171499
$35$	0	0
$36$	1.00000	0.166667
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	−8.00000	−1.29777
$39$	−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	$0.655213\pi$
$42$	1.00000	0.154303
$43$	−12.0000	−1.82998	−0.914991	−	0.403473i	$-0.867803\pi$
$43$	−12.0000	−1.82998	−0.914991	+	0.403473i	$0.867803\pi$
$44$	−3.00000	−0.452267
$45$	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
$49$	−6.00000	−0.857143
$50$	0	0
$51$	1.00000	0.140028
$52$	1.00000	0.138675
$53$	5.00000	0.686803	0.343401	−	0.939189i	$-0.388421\pi$
$53$	5.00000	0.686803	0.343401	+	0.939189i	$0.388421\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	−1.00000	−0.133631
$57$	8.00000	1.05963
$58$	−7.00000	−0.919145
$59$	−9.00000	−1.17170	−0.585850	−	0.810419i	$-0.699239\pi$
$59$	−9.00000	−1.17170	−0.585850	+	0.810419i	$0.699239\pi$
$60$	0	0
$61$	5.00000	0.640184	0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000	0.640184	0.320092	+	0.947386i	$0.396286\pi$
$62$	1.00000	0.127000
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	3.00000	0.369274
$67$	−11.0000	−1.34386	−0.671932	−	0.740613i	$-0.734535\pi$
$67$	−11.0000	−1.34386	−0.671932	+	0.740613i	$0.734535\pi$
$68$	−1.00000	−0.121268
$69$	−4.00000	−0.481543
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	1.00000	0.117851
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	4.00000	0.464991
$75$	0	0
$76$	−8.00000	−0.917663
$77$	3.00000	0.341882
$78$	−1.00000	−0.113228
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	7.00000	0.768350	0.384175	−	0.923260i	$-0.374486\pi$
$83$	7.00000	0.768350	0.384175	+	0.923260i	$0.374486\pi$
$84$	1.00000	0.109109
$85$	0	0
$86$	−12.0000	−1.29399
$87$	7.00000	0.750479
$88$	−3.00000	−0.319801
$89$	−8.00000	−0.847998	−0.423999	−	0.905663i	$-0.639374\pi$
$89$	−8.00000	−0.847998	−0.423999	+	0.905663i	$0.639374\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	4.00000	0.417029
$93$	−1.00000	−0.103695
$94$	3.00000	0.309426
$95$	0	0
$96$	−1.00000	−0.102062
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	−6.00000	−0.606092
$99$	−3.00000	−0.301511
$100$	0	0
$101$	1.00000	0.0995037	0.0497519	−	0.998762i	$-0.484157\pi$
$101$	1.00000	0.0995037	0.0497519	+	0.998762i	$0.484157\pi$
$102$	1.00000	0.0990148
$103$	−14.0000	−1.37946	−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000	−1.37946	−0.689730	+	0.724066i	$0.742271\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	5.00000	0.485643
$107$	−18.0000	−1.74013	−0.870063	−	0.492941i	$-0.835922\pi$
$107$	−18.0000	−1.74013	−0.870063	+	0.492941i	$0.835922\pi$
$108$	−1.00000	−0.0962250
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	−1.00000	−0.0944911
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	8.00000	0.749269
$115$	0	0
$116$	−7.00000	−0.649934
$117$	1.00000	0.0924500
$118$	−9.00000	−0.828517
$119$	1.00000	0.0916698
$120$	0	0
$121$	−2.00000	−0.181818
$122$	5.00000	0.452679
$123$	6.00000	0.541002
$124$	1.00000	0.0898027
$125$	0	0
$126$	−1.00000	−0.0890871
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	1.00000	0.0883883
$129$	12.0000	1.05654
$130$	0	0
$131$	−10.0000	−0.873704	−0.436852	−	0.899533i	$-0.643907\pi$
$131$	−10.0000	−0.873704	−0.436852	+	0.899533i	$0.643907\pi$
$132$	3.00000	0.261116
$133$	8.00000	0.693688
$134$	−11.0000	−0.950255
$135$	0	0
$136$	−1.00000	−0.0857493
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	−4.00000	−0.340503
$139$	−14.0000	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$140$	0	0
$141$	−3.00000	−0.252646
$142$	8.00000	0.671345
$143$	−3.00000	−0.250873
$144$	1.00000	0.0833333
$145$	0	0
$146$	0	0
$147$	6.00000	0.494872
$148$	4.00000	0.328798
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	13.0000	1.05792	0.528962	−	0.848645i	$-0.322581\pi$
$151$	13.0000	1.05792	0.528962	+	0.848645i	$0.322581\pi$
$152$	−8.00000	−0.648886
$153$	−1.00000	−0.0808452
$154$	3.00000	0.241747
$155$	0	0
$156$	−1.00000	−0.0800641
$157$	−15.0000	−1.19713	−0.598565	−	0.801074i	$-0.704262\pi$
$157$	−15.0000	−1.19713	−0.598565	+	0.801074i	$0.704262\pi$
$158$	−8.00000	−0.636446
$159$	−5.00000	−0.396526
$160$	0	0
$161$	−4.00000	−0.315244
$162$	1.00000	0.0785674
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	7.00000	0.543305
$167$	20.0000	1.54765	0.773823	−	0.633402i	$-0.218342\pi$
$167$	20.0000	1.54765	0.773823	+	0.633402i	$0.218342\pi$
$168$	1.00000	0.0771517
$169$	1.00000	0.0769231
$170$	0	0
$171$	−8.00000	−0.611775
$172$	−12.0000	−0.914991
$173$	−5.00000	−0.380143	−0.190071	−	0.981770i	$-0.560872\pi$
$173$	−5.00000	−0.380143	−0.190071	+	0.981770i	$0.560872\pi$
$174$	7.00000	0.530669
$175$	0	0
$176$	−3.00000	−0.226134
$177$	9.00000	0.676481
$178$	−8.00000	−0.599625
$179$	10.0000	0.747435	0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000	0.747435	0.373718	+	0.927543i	$0.378083\pi$
$180$	0	0
$181$	1.00000	0.0743294	0.0371647	−	0.999309i	$-0.488167\pi$
$181$	1.00000	0.0743294	0.0371647	+	0.999309i	$0.488167\pi$
$182$	−1.00000	−0.0741249
$183$	−5.00000	−0.369611
$184$	4.00000	0.294884
$185$	0	0
$186$	−1.00000	−0.0733236
$187$	3.00000	0.219382
$188$	3.00000	0.218797
$189$	1.00000	0.0727393
$190$	0	0
$191$	−10.0000	−0.723575	−0.361787	−	0.932261i	$-0.617833\pi$
$191$	−10.0000	−0.723575	−0.361787	+	0.932261i	$0.617833\pi$
$192$	−1.00000	−0.0721688
$193$	6.00000	0.431889	0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000	0.431889	0.215945	+	0.976406i	$0.430717\pi$
$194$	6.00000	0.430775
$195$	0	0
$196$	−6.00000	−0.428571
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	−3.00000	−0.213201
$199$	−26.0000	−1.84309	−0.921546	−	0.388270i	$-0.873073\pi$
$199$	−26.0000	−1.84309	−0.921546	+	0.388270i	$0.873073\pi$
$200$	0	0
$201$	11.0000	0.775880
$202$	1.00000	0.0703598
$203$	7.00000	0.491304
$204$	1.00000	0.0700140
$205$	0	0
$206$	−14.0000	−0.975426
$207$	4.00000	0.278019
$208$	1.00000	0.0693375
$209$	24.0000	1.66011
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	5.00000	0.343401
$213$	−8.00000	−0.548151
$214$	−18.0000	−1.23045
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	−1.00000	−0.0678844
$218$	2.00000	0.135457
$219$	0	0
$220$	0	0
$221$	−1.00000	−0.0672673
$222$	−4.00000	−0.268462
$223$	24.0000	1.60716	0.803579	−	0.595198i	$-0.202926\pi$
$223$	24.0000	1.60716	0.803579	+	0.595198i	$0.202926\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−2.00000	−0.133038
$227$	−3.00000	−0.199117	−0.0995585	−	0.995032i	$-0.531743\pi$
$227$	−3.00000	−0.199117	−0.0995585	+	0.995032i	$0.531743\pi$
$228$	8.00000	0.529813
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	0	0
$231$	−3.00000	−0.197386
$232$	−7.00000	−0.459573
$233$	26.0000	1.70332	0.851658	−	0.524097i	$-0.175597\pi$
$233$	26.0000	1.70332	0.851658	+	0.524097i	$0.175597\pi$
$234$	1.00000	0.0653720
$235$	0	0
$236$	−9.00000	−0.585850
$237$	8.00000	0.519656
$238$	1.00000	0.0648204
$239$	3.00000	0.194054	0.0970269	−	0.995282i	$-0.469067\pi$
$239$	3.00000	0.194054	0.0970269	+	0.995282i	$0.469067\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	−2.00000	−0.128565
$243$	−1.00000	−0.0641500
$244$	5.00000	0.320092
$245$	0	0
$246$	6.00000	0.382546
$247$	−8.00000	−0.509028
$248$	1.00000	0.0635001
$249$	−7.00000	−0.443607
$250$	0	0
$251$	18.0000	1.13615	0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000	1.13615	0.568075	+	0.822977i	$0.307688\pi$
$252$	−1.00000	−0.0629941
$253$	−12.0000	−0.754434
$254$	12.0000	0.752947
$255$	0	0
$256$	1.00000	0.0625000
$257$	−7.00000	−0.436648	−0.218324	−	0.975876i	$-0.570059\pi$
$257$	−7.00000	−0.436648	−0.218324	+	0.975876i	$0.570059\pi$
$258$	12.0000	0.747087
$259$	−4.00000	−0.248548
$260$	0	0
$261$	−7.00000	−0.433289
$262$	−10.0000	−0.617802
$263$	30.0000	1.84988	0.924940	−	0.380114i	$-0.124115\pi$
$263$	30.0000	1.84988	0.924940	+	0.380114i	$0.124115\pi$
$264$	3.00000	0.184637
$265$	0	0
$266$	8.00000	0.490511
$267$	8.00000	0.489592
$268$	−11.0000	−0.671932
$269$	1.00000	0.0609711	0.0304855	−	0.999535i	$-0.490295\pi$
$269$	1.00000	0.0609711	0.0304855	+	0.999535i	$0.490295\pi$
$270$	0	0
$271$	1.00000	0.0607457	0.0303728	−	0.999539i	$-0.490331\pi$
$271$	1.00000	0.0607457	0.0303728	+	0.999539i	$0.490331\pi$
$272$	−1.00000	−0.0606339
$273$	1.00000	0.0605228
$274$	12.0000	0.724947
$275$	0	0
$276$	−4.00000	−0.240772
$277$	−18.0000	−1.08152	−0.540758	−	0.841178i	$-0.681862\pi$
$277$	−18.0000	−1.08152	−0.540758	+	0.841178i	$0.681862\pi$
$278$	−14.0000	−0.839664
$279$	1.00000	0.0598684
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	−3.00000	−0.178647
$283$	−14.0000	−0.832214	−0.416107	−	0.909316i	$-0.636606\pi$
$283$	−14.0000	−0.832214	−0.416107	+	0.909316i	$0.636606\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	−3.00000	−0.177394
$287$	6.00000	0.354169
$288$	1.00000	0.0589256
$289$	−16.0000	−0.941176
$290$	0	0
$291$	−6.00000	−0.351726
$292$	0	0
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	6.00000	0.349927
$295$	0	0
$296$	4.00000	0.232495
$297$	3.00000	0.174078
$298$	10.0000	0.579284
$299$	4.00000	0.231326
$300$	0	0
$301$	12.0000	0.691669
$302$	13.0000	0.748066
$303$	−1.00000	−0.0574485
$304$	−8.00000	−0.458831
$305$	0	0
$306$	−1.00000	−0.0571662
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	3.00000	0.170941
$309$	14.0000	0.796432
$310$	0	0
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	−1.00000	−0.0566139
$313$	3.00000	0.169570	0.0847850	−	0.996399i	$-0.472980\pi$
$313$	3.00000	0.169570	0.0847850	+	0.996399i	$0.472980\pi$
$314$	−15.0000	−0.846499
$315$	0	0
$316$	−8.00000	−0.450035
$317$	20.0000	1.12331	0.561656	−	0.827371i	$-0.310164\pi$
$317$	20.0000	1.12331	0.561656	+	0.827371i	$0.310164\pi$
$318$	−5.00000	−0.280386
$319$	21.0000	1.17577
$320$	0	0
$321$	18.0000	1.00466
$322$	−4.00000	−0.222911
$323$	8.00000	0.445132
$324$	1.00000	0.0555556
$325$	0	0
$326$	−4.00000	−0.221540
$327$	−2.00000	−0.110600
$328$	−6.00000	−0.331295
$329$	−3.00000	−0.165395
$330$	0	0
$331$	28.0000	1.53902	0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000	1.53902	0.769510	+	0.638635i	$0.220501\pi$
$332$	7.00000	0.384175
$333$	4.00000	0.219199
$334$	20.0000	1.09435
$335$	0	0
$336$	1.00000	0.0545545
$337$	13.0000	0.708155	0.354078	−	0.935216i	$-0.384795\pi$
$337$	13.0000	0.708155	0.354078	+	0.935216i	$0.384795\pi$
$338$	1.00000	0.0543928
$339$	2.00000	0.108625
$340$	0	0
$341$	−3.00000	−0.162459
$342$	−8.00000	−0.432590
$343$	13.0000	0.701934
$344$	−12.0000	−0.646997
$345$	0	0
$346$	−5.00000	−0.268802
$347$	−14.0000	−0.751559	−0.375780	−	0.926709i	$-0.622625\pi$
$347$	−14.0000	−0.751559	−0.375780	+	0.926709i	$0.622625\pi$
$348$	7.00000	0.375239
$349$	−28.0000	−1.49881	−0.749403	−	0.662114i	$-0.769659\pi$
$349$	−28.0000	−1.49881	−0.749403	+	0.662114i	$0.769659\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	−3.00000	−0.159901
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	9.00000	0.478345
$355$	0	0
$356$	−8.00000	−0.423999
$357$	−1.00000	−0.0529256
$358$	10.0000	0.528516
$359$	−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
$361$	45.0000	2.36842
$362$	1.00000	0.0525588
$363$	2.00000	0.104973
$364$	−1.00000	−0.0524142
$365$	0	0
$366$	−5.00000	−0.261354
$367$	−28.0000	−1.46159	−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000	−1.46159	−0.730794	+	0.682598i	$0.760850\pi$
$368$	4.00000	0.208514
$369$	−6.00000	−0.312348
$370$	0	0
$371$	−5.00000	−0.259587
$372$	−1.00000	−0.0518476
$373$	13.0000	0.673114	0.336557	−	0.941663i	$-0.390737\pi$
$373$	13.0000	0.673114	0.336557	+	0.941663i	$0.390737\pi$
$374$	3.00000	0.155126
$375$	0	0
$376$	3.00000	0.154713
$377$	−7.00000	−0.360518
$378$	1.00000	0.0514344
$379$	11.0000	0.565032	0.282516	−	0.959263i	$-0.408831\pi$
$379$	11.0000	0.565032	0.282516	+	0.959263i	$0.408831\pi$
$380$	0	0
$381$	−12.0000	−0.614779
$382$	−10.0000	−0.511645
$383$	−32.0000	−1.63512	−0.817562	−	0.575841i	$-0.804675\pi$
$383$	−32.0000	−1.63512	−0.817562	+	0.575841i	$0.804675\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	6.00000	0.305392
$387$	−12.0000	−0.609994
$388$	6.00000	0.304604
$389$	34.0000	1.72387	0.861934	−	0.507020i	$-0.169253\pi$
$389$	34.0000	1.72387	0.861934	+	0.507020i	$0.169253\pi$
$390$	0	0
$391$	−4.00000	−0.202289
$392$	−6.00000	−0.303046
$393$	10.0000	0.504433
$394$	6.00000	0.302276
$395$	0	0
$396$	−3.00000	−0.150756
$397$	26.0000	1.30490	0.652451	−	0.757831i	$-0.273741\pi$
$397$	26.0000	1.30490	0.652451	+	0.757831i	$0.273741\pi$
$398$	−26.0000	−1.30326
$399$	−8.00000	−0.400501
$400$	0	0
$401$	−28.0000	−1.39825	−0.699127	−	0.714998i	$-0.746428\pi$
$401$	−28.0000	−1.39825	−0.699127	+	0.714998i	$0.746428\pi$
$402$	11.0000	0.548630
$403$	1.00000	0.0498135
$404$	1.00000	0.0497519
$405$	0	0
$406$	7.00000	0.347404
$407$	−12.0000	−0.594818
$408$	1.00000	0.0495074
$409$	−2.00000	−0.0988936	−0.0494468	−	0.998777i	$-0.515746\pi$
$409$	−2.00000	−0.0988936	−0.0494468	+	0.998777i	$0.515746\pi$
$410$	0	0
$411$	−12.0000	−0.591916
$412$	−14.0000	−0.689730
$413$	9.00000	0.442861
$414$	4.00000	0.196589
$415$	0	0
$416$	1.00000	0.0490290
$417$	14.0000	0.685583
$418$	24.0000	1.17388
$419$	6.00000	0.293119	0.146560	−	0.989202i	$-0.453180\pi$
$419$	6.00000	0.293119	0.146560	+	0.989202i	$0.453180\pi$
$420$	0	0
$421$	−12.0000	−0.584844	−0.292422	−	0.956289i	$-0.594461\pi$
$421$	−12.0000	−0.584844	−0.292422	+	0.956289i	$0.594461\pi$
$422$	12.0000	0.584151
$423$	3.00000	0.145865
$424$	5.00000	0.242821
$425$	0	0
$426$	−8.00000	−0.387601
$427$	−5.00000	−0.241967
$428$	−18.0000	−0.870063
$429$	3.00000	0.144841
$430$	0	0
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	−1.00000	−0.0481125
$433$	−10.0000	−0.480569	−0.240285	−	0.970702i	$-0.577241\pi$
$433$	−10.0000	−0.480569	−0.240285	+	0.970702i	$0.577241\pi$
$434$	−1.00000	−0.0480015
$435$	0	0
$436$	2.00000	0.0957826
$437$	−32.0000	−1.53077
$438$	0	0
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	−1.00000	−0.0475651
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	−4.00000	−0.189832
$445$	0	0
$446$	24.0000	1.13643
$447$	−10.0000	−0.472984
$448$	−1.00000	−0.0472456
$449$	36.0000	1.69895	0.849473	−	0.527633i	$-0.176920\pi$
$449$	36.0000	1.69895	0.849473	+	0.527633i	$0.176920\pi$
$450$	0	0
$451$	18.0000	0.847587
$452$	−2.00000	−0.0940721
$453$	−13.0000	−0.610793
$454$	−3.00000	−0.140797
$455$	0	0
$456$	8.00000	0.374634
$457$	−6.00000	−0.280668	−0.140334	−	0.990104i	$-0.544818\pi$
$457$	−6.00000	−0.280668	−0.140334	+	0.990104i	$0.544818\pi$
$458$	14.0000	0.654177
$459$	1.00000	0.0466760
$460$	0	0
$461$	26.0000	1.21094	0.605470	−	0.795868i	$-0.292985\pi$
$461$	26.0000	1.21094	0.605470	+	0.795868i	$0.292985\pi$
$462$	−3.00000	−0.139573
$463$	−11.0000	−0.511213	−0.255607	−	0.966781i	$-0.582275\pi$
$463$	−11.0000	−0.511213	−0.255607	+	0.966781i	$0.582275\pi$
$464$	−7.00000	−0.324967
$465$	0	0
$466$	26.0000	1.20443
$467$	24.0000	1.11059	0.555294	−	0.831654i	$-0.312606\pi$
$467$	24.0000	1.11059	0.555294	+	0.831654i	$0.312606\pi$
$468$	1.00000	0.0462250
$469$	11.0000	0.507933
$470$	0	0
$471$	15.0000	0.691164
$472$	−9.00000	−0.414259
$473$	36.0000	1.65528
$474$	8.00000	0.367452
$475$	0	0
$476$	1.00000	0.0458349
$477$	5.00000	0.228934
$478$	3.00000	0.137217
$479$	23.0000	1.05090	0.525448	−	0.850825i	$-0.323898\pi$
$479$	23.0000	1.05090	0.525448	+	0.850825i	$0.323898\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	−10.0000	−0.455488
$483$	4.00000	0.182006
$484$	−2.00000	−0.0909091
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	−25.0000	−1.13286	−0.566429	−	0.824110i	$-0.691675\pi$
$487$	−25.0000	−1.13286	−0.566429	+	0.824110i	$0.691675\pi$
$488$	5.00000	0.226339
$489$	4.00000	0.180886
$490$	0	0
$491$	18.0000	0.812329	0.406164	−	0.913800i	$-0.366866\pi$
$491$	18.0000	0.812329	0.406164	+	0.913800i	$0.366866\pi$
$492$	6.00000	0.270501
$493$	7.00000	0.315264
$494$	−8.00000	−0.359937
$495$	0	0
$496$	1.00000	0.0449013
$497$	−8.00000	−0.358849
$498$	−7.00000	−0.313678
$499$	25.0000	1.11915	0.559577	−	0.828778i	$-0.310964\pi$
$499$	25.0000	1.11915	0.559577	+	0.828778i	$0.310964\pi$
$500$	0	0
$501$	−20.0000	−0.893534
$502$	18.0000	0.803379
$503$	−22.0000	−0.980932	−0.490466	−	0.871460i	$-0.663173\pi$
$503$	−22.0000	−0.980932	−0.490466	+	0.871460i	$0.663173\pi$
$504$	−1.00000	−0.0445435
$505$	0	0
$506$	−12.0000	−0.533465
$507$	−1.00000	−0.0444116
$508$	12.0000	0.532414
$509$	34.0000	1.50702	0.753512	−	0.657434i	$-0.228358\pi$
$509$	34.0000	1.50702	0.753512	+	0.657434i	$0.228358\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	8.00000	0.353209
$514$	−7.00000	−0.308757
$515$	0	0
$516$	12.0000	0.528271
$517$	−9.00000	−0.395820
$518$	−4.00000	−0.175750
$519$	5.00000	0.219476
$520$	0	0
$521$	38.0000	1.66481	0.832405	−	0.554168i	$-0.186963\pi$
$521$	38.0000	1.66481	0.832405	+	0.554168i	$0.186963\pi$
$522$	−7.00000	−0.306382
$523$	−2.00000	−0.0874539	−0.0437269	−	0.999044i	$-0.513923\pi$
$523$	−2.00000	−0.0874539	−0.0437269	+	0.999044i	$0.513923\pi$
$524$	−10.0000	−0.436852
$525$	0	0
$526$	30.0000	1.30806
$527$	−1.00000	−0.0435607
$528$	3.00000	0.130558
$529$	−7.00000	−0.304348
$530$	0	0
$531$	−9.00000	−0.390567
$532$	8.00000	0.346844
$533$	−6.00000	−0.259889
$534$	8.00000	0.346194
$535$	0	0
$536$	−11.0000	−0.475128
$537$	−10.0000	−0.431532
$538$	1.00000	0.0431131
$539$	18.0000	0.775315
$540$	0	0
$541$	−8.00000	−0.343947	−0.171973	−	0.985102i	$-0.555014\pi$
$541$	−8.00000	−0.343947	−0.171973	+	0.985102i	$0.555014\pi$
$542$	1.00000	0.0429537
$543$	−1.00000	−0.0429141
$544$	−1.00000	−0.0428746
$545$	0	0
$546$	1.00000	0.0427960
$547$	−6.00000	−0.256541	−0.128271	−	0.991739i	$-0.540943\pi$
$547$	−6.00000	−0.256541	−0.128271	+	0.991739i	$0.540943\pi$
$548$	12.0000	0.512615
$549$	5.00000	0.213395
$550$	0	0
$551$	56.0000	2.38568
$552$	−4.00000	−0.170251
$553$	8.00000	0.340195
$554$	−18.0000	−0.764747
$555$	0	0
$556$	−14.0000	−0.593732
$557$	−6.00000	−0.254228	−0.127114	−	0.991888i	$-0.540571\pi$
$557$	−6.00000	−0.254228	−0.127114	+	0.991888i	$0.540571\pi$
$558$	1.00000	0.0423334
$559$	−12.0000	−0.507546
$560$	0	0
$561$	−3.00000	−0.126660
$562$	−18.0000	−0.759284
$563$	−36.0000	−1.51722	−0.758610	−	0.651546i	$-0.774121\pi$
$563$	−36.0000	−1.51722	−0.758610	+	0.651546i	$0.774121\pi$
$564$	−3.00000	−0.126323
$565$	0	0
$566$	−14.0000	−0.588464
$567$	−1.00000	−0.0419961
$568$	8.00000	0.335673
$569$	3.00000	0.125767	0.0628833	−	0.998021i	$-0.479970\pi$
$569$	3.00000	0.125767	0.0628833	+	0.998021i	$0.479970\pi$
$570$	0	0
$571$	−36.0000	−1.50655	−0.753277	−	0.657704i	$-0.771528\pi$
$571$	−36.0000	−1.50655	−0.753277	+	0.657704i	$0.771528\pi$
$572$	−3.00000	−0.125436
$573$	10.0000	0.417756
$574$	6.00000	0.250435
$575$	0	0
$576$	1.00000	0.0416667
$577$	−14.0000	−0.582828	−0.291414	−	0.956597i	$-0.594126\pi$
$577$	−14.0000	−0.582828	−0.291414	+	0.956597i	$0.594126\pi$
$578$	−16.0000	−0.665512
$579$	−6.00000	−0.249351
$580$	0	0
$581$	−7.00000	−0.290409
$582$	−6.00000	−0.248708
$583$	−15.0000	−0.621237
$584$	0	0
$585$	0	0
$586$	6.00000	0.247858
$587$	−39.0000	−1.60970	−0.804851	−	0.593477i	$-0.797755\pi$
$587$	−39.0000	−1.60970	−0.804851	+	0.593477i	$0.797755\pi$
$588$	6.00000	0.247436
$589$	−8.00000	−0.329634
$590$	0	0
$591$	−6.00000	−0.246807
$592$	4.00000	0.164399
$593$	34.0000	1.39621	0.698106	−	0.715994i	$-0.254026\pi$
$593$	34.0000	1.39621	0.698106	+	0.715994i	$0.254026\pi$
$594$	3.00000	0.123091
$595$	0	0
$596$	10.0000	0.409616
$597$	26.0000	1.06411
$598$	4.00000	0.163572
$599$	−26.0000	−1.06233	−0.531166	−	0.847268i	$-0.678246\pi$
$599$	−26.0000	−1.06233	−0.531166	+	0.847268i	$0.678246\pi$
$600$	0	0
$601$	−35.0000	−1.42768	−0.713840	−	0.700309i	$-0.753046\pi$
$601$	−35.0000	−1.42768	−0.713840	+	0.700309i	$0.753046\pi$
$602$	12.0000	0.489083
$603$	−11.0000	−0.447955
$604$	13.0000	0.528962
$605$	0	0
$606$	−1.00000	−0.0406222
$607$	−22.0000	−0.892952	−0.446476	−	0.894795i	$-0.647321\pi$
$607$	−22.0000	−0.892952	−0.446476	+	0.894795i	$0.647321\pi$
$608$	−8.00000	−0.324443
$609$	−7.00000	−0.283654
$610$	0	0
$611$	3.00000	0.121367
$612$	−1.00000	−0.0404226
$613$	−34.0000	−1.37325	−0.686624	−	0.727013i	$-0.740908\pi$
$613$	−34.0000	−1.37325	−0.686624	+	0.727013i	$0.740908\pi$
$614$	−12.0000	−0.484281
$615$	0	0
$616$	3.00000	0.120873
$617$	−34.0000	−1.36879	−0.684394	−	0.729112i	$-0.739933\pi$
$617$	−34.0000	−1.36879	−0.684394	+	0.729112i	$0.739933\pi$
$618$	14.0000	0.563163
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	18.0000	0.721734
$623$	8.00000	0.320513
$624$	−1.00000	−0.0400320
$625$	0	0
$626$	3.00000	0.119904
$627$	−24.0000	−0.958468
$628$	−15.0000	−0.598565
$629$	−4.00000	−0.159490
$630$	0	0
$631$	−12.0000	−0.477712	−0.238856	−	0.971055i	$-0.576772\pi$
$631$	−12.0000	−0.477712	−0.238856	+	0.971055i	$0.576772\pi$
$632$	−8.00000	−0.318223
$633$	−12.0000	−0.476957
$634$	20.0000	0.794301
$635$	0	0
$636$	−5.00000	−0.198263
$637$	−6.00000	−0.237729
$638$	21.0000	0.831398
$639$	8.00000	0.316475
$640$	0	0
$641$	−27.0000	−1.06644	−0.533218	−	0.845978i	$-0.679017\pi$
$641$	−27.0000	−1.06644	−0.533218	+	0.845978i	$0.679017\pi$
$642$	18.0000	0.710403
$643$	20.0000	0.788723	0.394362	−	0.918955i	$-0.370966\pi$
$643$	20.0000	0.788723	0.394362	+	0.918955i	$0.370966\pi$
$644$	−4.00000	−0.157622
$645$	0	0
$646$	8.00000	0.314756
$647$	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
$649$	27.0000	1.05984
$650$	0	0
$651$	1.00000	0.0391931
$652$	−4.00000	−0.156652
$653$	−21.0000	−0.821794	−0.410897	−	0.911682i	$-0.634784\pi$
$653$	−21.0000	−0.821794	−0.410897	+	0.911682i	$0.634784\pi$
$654$	−2.00000	−0.0782062
$655$	0	0
$656$	−6.00000	−0.234261
$657$	0	0
$658$	−3.00000	−0.116952
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	−46.0000	−1.78919	−0.894596	−	0.446875i	$-0.852537\pi$
$661$	−46.0000	−1.78919	−0.894596	+	0.446875i	$0.852537\pi$
$662$	28.0000	1.08825
$663$	1.00000	0.0388368
$664$	7.00000	0.271653
$665$	0	0
$666$	4.00000	0.154997
$667$	−28.0000	−1.08416
$668$	20.0000	0.773823
$669$	−24.0000	−0.927894
$670$	0	0
$671$	−15.0000	−0.579069
$672$	1.00000	0.0385758
$673$	−43.0000	−1.65753	−0.828764	−	0.559598i	$-0.810955\pi$
$673$	−43.0000	−1.65753	−0.828764	+	0.559598i	$0.810955\pi$
$674$	13.0000	0.500741
$675$	0	0
$676$	1.00000	0.0384615
$677$	22.0000	0.845529	0.422764	−	0.906240i	$-0.361060\pi$
$677$	22.0000	0.845529	0.422764	+	0.906240i	$0.361060\pi$
$678$	2.00000	0.0768095
$679$	−6.00000	−0.230259
$680$	0	0
$681$	3.00000	0.114960
$682$	−3.00000	−0.114876
$683$	27.0000	1.03313	0.516563	−	0.856249i	$-0.327211\pi$
$683$	27.0000	1.03313	0.516563	+	0.856249i	$0.327211\pi$
$684$	−8.00000	−0.305888
$685$	0	0
$686$	13.0000	0.496342
$687$	−14.0000	−0.534133
$688$	−12.0000	−0.457496
$689$	5.00000	0.190485
$690$	0	0
$691$	−43.0000	−1.63580	−0.817899	−	0.575362i	$-0.804861\pi$
$691$	−43.0000	−1.63580	−0.817899	+	0.575362i	$0.804861\pi$
$692$	−5.00000	−0.190071
$693$	3.00000	0.113961
$694$	−14.0000	−0.531433
$695$	0	0
$696$	7.00000	0.265334
$697$	6.00000	0.227266
$698$	−28.0000	−1.05982
$699$	−26.0000	−0.983410
$700$	0	0
$701$	15.0000	0.566542	0.283271	−	0.959040i	$-0.408580\pi$
$701$	15.0000	0.566542	0.283271	+	0.959040i	$0.408580\pi$
$702$	−1.00000	−0.0377426
$703$	−32.0000	−1.20690
$704$	−3.00000	−0.113067
$705$	0	0
$706$	14.0000	0.526897
$707$	−1.00000	−0.0376089
$708$	9.00000	0.338241
$709$	38.0000	1.42712	0.713560	−	0.700594i	$-0.247082\pi$
$709$	38.0000	1.42712	0.713560	+	0.700594i	$0.247082\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	−8.00000	−0.299813
$713$	4.00000	0.149801
$714$	−1.00000	−0.0374241
$715$	0	0
$716$	10.0000	0.373718
$717$	−3.00000	−0.112037
$718$	−15.0000	−0.559795
$719$	−26.0000	−0.969636	−0.484818	−	0.874615i	$-0.661114\pi$
$719$	−26.0000	−0.969636	−0.484818	+	0.874615i	$0.661114\pi$
$720$	0	0
$721$	14.0000	0.521387
$722$	45.0000	1.67473
$723$	10.0000	0.371904
$724$	1.00000	0.0371647
$725$	0	0
$726$	2.00000	0.0742270
$727$	14.0000	0.519231	0.259616	−	0.965712i	$-0.416404\pi$
$727$	14.0000	0.519231	0.259616	+	0.965712i	$0.416404\pi$
$728$	−1.00000	−0.0370625
$729$	1.00000	0.0370370
$730$	0	0
$731$	12.0000	0.443836
$732$	−5.00000	−0.184805
$733$	28.0000	1.03420	0.517102	−	0.855924i	$-0.327011\pi$
$733$	28.0000	1.03420	0.517102	+	0.855924i	$0.327011\pi$
$734$	−28.0000	−1.03350
$735$	0	0
$736$	4.00000	0.147442
$737$	33.0000	1.21557
$738$	−6.00000	−0.220863
$739$	−47.0000	−1.72892	−0.864461	−	0.502699i	$-0.832340\pi$
$739$	−47.0000	−1.72892	−0.864461	+	0.502699i	$0.832340\pi$
$740$	0	0
$741$	8.00000	0.293887
$742$	−5.00000	−0.183556
$743$	−29.0000	−1.06391	−0.531953	−	0.846774i	$-0.678542\pi$
$743$	−29.0000	−1.06391	−0.531953	+	0.846774i	$0.678542\pi$
$744$	−1.00000	−0.0366618
$745$	0	0
$746$	13.0000	0.475964
$747$	7.00000	0.256117
$748$	3.00000	0.109691
$749$	18.0000	0.657706
$750$	0	0
$751$	28.0000	1.02173	0.510867	−	0.859660i	$-0.329324\pi$
$751$	28.0000	1.02173	0.510867	+	0.859660i	$0.329324\pi$
$752$	3.00000	0.109399
$753$	−18.0000	−0.655956
$754$	−7.00000	−0.254925
$755$	0	0
$756$	1.00000	0.0363696
$757$	43.0000	1.56286	0.781431	−	0.623992i	$-0.214490\pi$
$757$	43.0000	1.56286	0.781431	+	0.623992i	$0.214490\pi$
$758$	11.0000	0.399538
$759$	12.0000	0.435572
$760$	0	0
$761$	−36.0000	−1.30500	−0.652499	−	0.757789i	$-0.726280\pi$
$761$	−36.0000	−1.30500	−0.652499	+	0.757789i	$0.726280\pi$
$762$	−12.0000	−0.434714
$763$	−2.00000	−0.0724049
$764$	−10.0000	−0.361787
$765$	0	0
$766$	−32.0000	−1.15621
$767$	−9.00000	−0.324971
$768$	−1.00000	−0.0360844
$769$	8.00000	0.288487	0.144244	−	0.989542i	$-0.453925\pi$
$769$	8.00000	0.288487	0.144244	+	0.989542i	$0.453925\pi$
$770$	0	0
$771$	7.00000	0.252099
$772$	6.00000	0.215945
$773$	−46.0000	−1.65451	−0.827253	−	0.561830i	$-0.810097\pi$
$773$	−46.0000	−1.65451	−0.827253	+	0.561830i	$0.810097\pi$
$774$	−12.0000	−0.431331
$775$	0	0
$776$	6.00000	0.215387
$777$	4.00000	0.143499
$778$	34.0000	1.21896
$779$	48.0000	1.71978
$780$	0	0
$781$	−24.0000	−0.858788
$782$	−4.00000	−0.143040
$783$	7.00000	0.250160
$784$	−6.00000	−0.214286
$785$	0	0
$786$	10.0000	0.356688
$787$	7.00000	0.249523	0.124762	−	0.992187i	$-0.460183\pi$
$787$	7.00000	0.249523	0.124762	+	0.992187i	$0.460183\pi$
$788$	6.00000	0.213741
$789$	−30.0000	−1.06803
$790$	0	0
$791$	2.00000	0.0711118
$792$	−3.00000	−0.106600
$793$	5.00000	0.177555
$794$	26.0000	0.922705
$795$	0	0
$796$	−26.0000	−0.921546
$797$	−11.0000	−0.389640	−0.194820	−	0.980839i	$-0.562412\pi$
$797$	−11.0000	−0.389640	−0.194820	+	0.980839i	$0.562412\pi$
$798$	−8.00000	−0.283197
$799$	−3.00000	−0.106132
$800$	0	0
$801$	−8.00000	−0.282666
$802$	−28.0000	−0.988714
$803$	0	0
$804$	11.0000	0.387940
$805$	0	0
$806$	1.00000	0.0352235
$807$	−1.00000	−0.0352017
$808$	1.00000	0.0351799
$809$	−26.0000	−0.914111	−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000	−0.914111	−0.457056	+	0.889438i	$0.651096\pi$
$810$	0	0
$811$	−47.0000	−1.65039	−0.825197	−	0.564846i	$-0.808936\pi$
$811$	−47.0000	−1.65039	−0.825197	+	0.564846i	$0.808936\pi$
$812$	7.00000	0.245652
$813$	−1.00000	−0.0350715
$814$	−12.0000	−0.420600
$815$	0	0
$816$	1.00000	0.0350070
$817$	96.0000	3.35861
$818$	−2.00000	−0.0699284
$819$	−1.00000	−0.0349428
$820$	0	0
$821$	−52.0000	−1.81481	−0.907406	−	0.420255i	$-0.861941\pi$
$821$	−52.0000	−1.81481	−0.907406	+	0.420255i	$0.861941\pi$
$822$	−12.0000	−0.418548
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	−14.0000	−0.487713
$825$	0	0
$826$	9.00000	0.313150
$827$	3.00000	0.104320	0.0521601	−	0.998639i	$-0.483389\pi$
$827$	3.00000	0.104320	0.0521601	+	0.998639i	$0.483389\pi$
$828$	4.00000	0.139010
$829$	47.0000	1.63238	0.816189	−	0.577785i	$-0.196083\pi$
$829$	47.0000	1.63238	0.816189	+	0.577785i	$0.196083\pi$
$830$	0	0
$831$	18.0000	0.624413
$832$	1.00000	0.0346688
$833$	6.00000	0.207888
$834$	14.0000	0.484780
$835$	0	0
$836$	24.0000	0.830057
$837$	−1.00000	−0.0345651
$838$	6.00000	0.207267
$839$	16.0000	0.552381	0.276191	−	0.961103i	$-0.410928\pi$
$839$	16.0000	0.552381	0.276191	+	0.961103i	$0.410928\pi$
$840$	0	0
$841$	20.0000	0.689655
$842$	−12.0000	−0.413547
$843$	18.0000	0.619953
$844$	12.0000	0.413057
$845$	0	0
$846$	3.00000	0.103142
$847$	2.00000	0.0687208
$848$	5.00000	0.171701
$849$	14.0000	0.480479
$850$	0	0
$851$	16.0000	0.548473
$852$	−8.00000	−0.274075
$853$	8.00000	0.273915	0.136957	−	0.990577i	$-0.456268\pi$
$853$	8.00000	0.273915	0.136957	+	0.990577i	$0.456268\pi$
$854$	−5.00000	−0.171096
$855$	0	0
$856$	−18.0000	−0.615227
$857$	10.0000	0.341593	0.170797	−	0.985306i	$-0.445366\pi$
$857$	10.0000	0.341593	0.170797	+	0.985306i	$0.445366\pi$
$858$	3.00000	0.102418
$859$	−24.0000	−0.818869	−0.409435	−	0.912339i	$-0.634274\pi$
$859$	−24.0000	−0.818869	−0.409435	+	0.912339i	$0.634274\pi$
$860$	0	0
$861$	−6.00000	−0.204479
$862$	12.0000	0.408722
$863$	−9.00000	−0.306364	−0.153182	−	0.988198i	$-0.548952\pi$
$863$	−9.00000	−0.306364	−0.153182	+	0.988198i	$0.548952\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−10.0000	−0.339814
$867$	16.0000	0.543388
$868$	−1.00000	−0.0339422
$869$	24.0000	0.814144
$870$	0	0
$871$	−11.0000	−0.372721
$872$	2.00000	0.0677285
$873$	6.00000	0.203069
$874$	−32.0000	−1.08242
$875$	0	0
$876$	0	0
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	−8.00000	−0.269987
$879$	−6.00000	−0.202375
$880$	0	0
$881$	−21.0000	−0.707508	−0.353754	−	0.935339i	$-0.615095\pi$
$881$	−21.0000	−0.707508	−0.353754	+	0.935339i	$0.615095\pi$
$882$	−6.00000	−0.202031
$883$	48.0000	1.61533	0.807664	−	0.589643i	$-0.200731\pi$
$883$	48.0000	1.61533	0.807664	+	0.589643i	$0.200731\pi$
$884$	−1.00000	−0.0336336
$885$	0	0
$886$	−12.0000	−0.403148
$887$	−56.0000	−1.88030	−0.940148	−	0.340766i	$-0.889313\pi$
$887$	−56.0000	−1.88030	−0.940148	+	0.340766i	$0.889313\pi$
$888$	−4.00000	−0.134231
$889$	−12.0000	−0.402467
$890$	0	0
$891$	−3.00000	−0.100504
$892$	24.0000	0.803579
$893$	−24.0000	−0.803129
$894$	−10.0000	−0.334450
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	−4.00000	−0.133556
$898$	36.0000	1.20134
$899$	−7.00000	−0.233463
$900$	0	0
$901$	−5.00000	−0.166574
$902$	18.0000	0.599334
$903$	−12.0000	−0.399335
$904$	−2.00000	−0.0665190
$905$	0	0
$906$	−13.0000	−0.431896
$907$	8.00000	0.265636	0.132818	−	0.991140i	$-0.457597\pi$
$907$	8.00000	0.265636	0.132818	+	0.991140i	$0.457597\pi$
$908$	−3.00000	−0.0995585
$909$	1.00000	0.0331679
$910$	0	0
$911$	28.0000	0.927681	0.463841	−	0.885919i	$-0.346471\pi$
$911$	28.0000	0.927681	0.463841	+	0.885919i	$0.346471\pi$
$912$	8.00000	0.264906
$913$	−21.0000	−0.694999
$914$	−6.00000	−0.198462
$915$	0	0
$916$	14.0000	0.462573
$917$	10.0000	0.330229
$918$	1.00000	0.0330049
$919$	−14.0000	−0.461817	−0.230909	−	0.972975i	$-0.574170\pi$
$919$	−14.0000	−0.461817	−0.230909	+	0.972975i	$0.574170\pi$
$920$	0	0
$921$	12.0000	0.395413
$922$	26.0000	0.856264
$923$	8.00000	0.263323
$924$	−3.00000	−0.0986928
$925$	0	0
$926$	−11.0000	−0.361482
$927$	−14.0000	−0.459820
$928$	−7.00000	−0.229786
$929$	−34.0000	−1.11550	−0.557752	−	0.830008i	$-0.688336\pi$
$929$	−34.0000	−1.11550	−0.557752	+	0.830008i	$0.688336\pi$
$930$	0	0
$931$	48.0000	1.57314
$932$	26.0000	0.851658
$933$	−18.0000	−0.589294
$934$	24.0000	0.785304
$935$	0	0
$936$	1.00000	0.0326860
$937$	21.0000	0.686040	0.343020	−	0.939328i	$-0.388550\pi$
$937$	21.0000	0.686040	0.343020	+	0.939328i	$0.388550\pi$
$938$	11.0000	0.359163
$939$	−3.00000	−0.0979013
$940$	0	0
$941$	−60.0000	−1.95594	−0.977972	−	0.208736i	$-0.933065\pi$
$941$	−60.0000	−1.95594	−0.977972	+	0.208736i	$0.933065\pi$
$942$	15.0000	0.488726
$943$	−24.0000	−0.781548
$944$	−9.00000	−0.292925
$945$	0	0
$946$	36.0000	1.17046
$947$	47.0000	1.52729	0.763647	−	0.645634i	$-0.223407\pi$
$947$	47.0000	1.52729	0.763647	+	0.645634i	$0.223407\pi$
$948$	8.00000	0.259828
$949$	0	0
$950$	0	0
$951$	−20.0000	−0.648544
$952$	1.00000	0.0324102
$953$	57.0000	1.84641	0.923206	−	0.384307i	$-0.125559\pi$
$953$	57.0000	1.84641	0.923206	+	0.384307i	$0.125559\pi$
$954$	5.00000	0.161881
$955$	0	0
$956$	3.00000	0.0970269
$957$	−21.0000	−0.678834
$958$	23.0000	0.743096
$959$	−12.0000	−0.387500
$960$	0	0
$961$	−30.0000	−0.967742
$962$	4.00000	0.128965
$963$	−18.0000	−0.580042
$964$	−10.0000	−0.322078
$965$	0	0
$966$	4.00000	0.128698
$967$	7.00000	0.225105	0.112552	−	0.993646i	$-0.464097\pi$
$967$	7.00000	0.225105	0.112552	+	0.993646i	$0.464097\pi$
$968$	−2.00000	−0.0642824
$969$	−8.00000	−0.256997
$970$	0	0
$971$	42.0000	1.34784	0.673922	−	0.738802i	$-0.264608\pi$
$971$	42.0000	1.34784	0.673922	+	0.738802i	$0.264608\pi$
$972$	−1.00000	−0.0320750
$973$	14.0000	0.448819
$974$	−25.0000	−0.801052
$975$	0	0
$976$	5.00000	0.160046
$977$	52.0000	1.66363	0.831814	−	0.555055i	$-0.187303\pi$
$977$	52.0000	1.66363	0.831814	+	0.555055i	$0.187303\pi$
$978$	4.00000	0.127906
$979$	24.0000	0.767043
$980$	0	0
$981$	2.00000	0.0638551
$982$	18.0000	0.574403
$983$	−11.0000	−0.350846	−0.175423	−	0.984493i	$-0.556129\pi$
$983$	−11.0000	−0.350846	−0.175423	+	0.984493i	$0.556129\pi$
$984$	6.00000	0.191273
$985$	0	0
$986$	7.00000	0.222925
$987$	3.00000	0.0954911
$988$	−8.00000	−0.254514
$989$	−48.0000	−1.52631
$990$	0	0
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	1.00000	0.0317500
$993$	−28.0000	−0.888553
$994$	−8.00000	−0.253745
$995$	0	0
$996$	−7.00000	−0.221803
$997$	−17.0000	−0.538395	−0.269198	−	0.963085i	$-0.586759\pi$
$997$	−17.0000	−0.538395	−0.269198	+	0.963085i	$0.586759\pi$
$998$	25.0000	0.791361
$999$	−4.00000	−0.126554

Twists

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

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