LMFDB - Embedded newform 1650.2.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1650,2,Mod(1,1650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1650, 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("1650.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1650 = 2 \cdot 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1650.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:	$13.1753163335$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 66)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1650.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{12} +6.00000 q^{13} -4.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} +4.00000 q^{21} +1.00000 q^{22} -4.00000 q^{23} -1.00000 q^{24} -6.00000 q^{26} +1.00000 q^{27} +4.00000 q^{28} +6.00000 q^{29} -1.00000 q^{32} -1.00000 q^{33} +2.00000 q^{34} +1.00000 q^{36} -6.00000 q^{37} -4.00000 q^{38} +6.00000 q^{39} -6.00000 q^{41} -4.00000 q^{42} -4.00000 q^{43} -1.00000 q^{44} +4.00000 q^{46} +12.0000 q^{47} +1.00000 q^{48} +9.00000 q^{49} -2.00000 q^{51} +6.00000 q^{52} -2.00000 q^{53} -1.00000 q^{54} -4.00000 q^{56} +4.00000 q^{57} -6.00000 q^{58} +12.0000 q^{59} -14.0000 q^{61} +4.00000 q^{63} +1.00000 q^{64} +1.00000 q^{66} -4.00000 q^{67} -2.00000 q^{68} -4.00000 q^{69} -12.0000 q^{71} -1.00000 q^{72} +6.00000 q^{73} +6.00000 q^{74} +4.00000 q^{76} -4.00000 q^{77} -6.00000 q^{78} -4.00000 q^{79} +1.00000 q^{81} +6.00000 q^{82} -4.00000 q^{83} +4.00000 q^{84} +4.00000 q^{86} +6.00000 q^{87} +1.00000 q^{88} +10.0000 q^{89} +24.0000 q^{91} -4.00000 q^{92} -12.0000 q^{94} -1.00000 q^{96} +14.0000 q^{97} -9.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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	$-0.227186\pi$
$7$	4.00000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	1.00000	0.288675
$13$	6.00000	1.66410	0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000	1.66410	0.832050	+	0.554700i	$0.187167\pi$
$14$	−4.00000	−1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	−1.00000	−0.235702
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	4.00000	0.872872
$22$	1.00000	0.213201
$23$	−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
$26$	−6.00000	−1.17670
$27$	1.00000	0.192450
$28$	4.00000	0.755929
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−1.00000	−0.176777
$33$	−1.00000	−0.174078
$34$	2.00000	0.342997
$35$	0	0
$36$	1.00000	0.166667
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	−4.00000	−0.648886
$39$	6.00000	0.960769
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	−4.00000	−0.617213
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	4.00000	0.589768
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	1.00000	0.144338
$49$	9.00000	1.28571
$50$	0	0
$51$	−2.00000	−0.280056
$52$	6.00000	0.832050
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	−4.00000	−0.534522
$57$	4.00000	0.529813
$58$	−6.00000	−0.787839
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	0	0
$63$	4.00000	0.503953
$64$	1.00000	0.125000
$65$	0	0
$66$	1.00000	0.123091
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	−2.00000	−0.242536
$69$	−4.00000	−0.481543
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	−1.00000	−0.117851
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	6.00000	0.697486
$75$	0	0
$76$	4.00000	0.458831
$77$	−4.00000	−0.455842
$78$	−6.00000	−0.679366
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	4.00000	0.436436
$85$	0	0
$86$	4.00000	0.431331
$87$	6.00000	0.643268
$88$	1.00000	0.106600
$89$	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
$91$	24.0000	2.51588
$92$	−4.00000	−0.417029
$93$	0	0
$94$	−12.0000	−1.23771
$95$	0	0
$96$	−1.00000	−0.102062
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	−9.00000	−0.909137
$99$	−1.00000	−0.100504
$100$	0	0
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	2.00000	0.198030
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	2.00000	0.194257
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	1.00000	0.0962250
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	0	0
$111$	−6.00000	−0.569495
$112$	4.00000	0.377964
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	−4.00000	−0.374634
$115$	0	0
$116$	6.00000	0.557086
$117$	6.00000	0.554700
$118$	−12.0000	−1.10469
$119$	−8.00000	−0.733359
$120$	0	0
$121$	1.00000	0.0909091
$122$	14.0000	1.26750
$123$	−6.00000	−0.541002
$124$	0	0
$125$	0	0
$126$	−4.00000	−0.356348
$127$	−12.0000	−1.06483	−0.532414	−	0.846484i	$-0.678715\pi$
$127$	−12.0000	−1.06483	−0.532414	+	0.846484i	$0.678715\pi$
$128$	−1.00000	−0.0883883
$129$	−4.00000	−0.352180
$130$	0	0
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	−1.00000	−0.0870388
$133$	16.0000	1.38738
$134$	4.00000	0.345547
$135$	0	0
$136$	2.00000	0.171499
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	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
$141$	12.0000	1.01058
$142$	12.0000	1.00702
$143$	−6.00000	−0.501745
$144$	1.00000	0.0833333
$145$	0	0
$146$	−6.00000	−0.496564
$147$	9.00000	0.742307
$148$	−6.00000	−0.493197
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	−4.00000	−0.324443
$153$	−2.00000	−0.161690
$154$	4.00000	0.322329
$155$	0	0
$156$	6.00000	0.480384
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	4.00000	0.318223
$159$	−2.00000	−0.158610
$160$	0	0
$161$	−16.0000	−1.26098
$162$	−1.00000	−0.0785674
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	4.00000	0.310460
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	−4.00000	−0.308607
$169$	23.0000	1.76923
$170$	0	0
$171$	4.00000	0.305888
$172$	−4.00000	−0.304997
$173$	10.0000	0.760286	0.380143	−	0.924928i	$-0.375875\pi$
$173$	10.0000	0.760286	0.380143	+	0.924928i	$0.375875\pi$
$174$	−6.00000	−0.454859
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	12.0000	0.901975
$178$	−10.0000	−0.749532
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	−24.0000	−1.77900
$183$	−14.0000	−1.03491
$184$	4.00000	0.294884
$185$	0	0
$186$	0	0
$187$	2.00000	0.146254
$188$	12.0000	0.875190
$189$	4.00000	0.290957
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	1.00000	0.0721688
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	−14.0000	−1.00514
$195$	0	0
$196$	9.00000	0.642857
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	1.00000	0.0710669
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	−14.0000	−0.985037
$203$	24.0000	1.68447
$204$	−2.00000	−0.140028
$205$	0	0
$206$	0	0
$207$	−4.00000	−0.278019
$208$	6.00000	0.416025
$209$	−4.00000	−0.276686
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	−2.00000	−0.137361
$213$	−12.0000	−0.822226
$214$	4.00000	0.273434
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	6.00000	0.406371
$219$	6.00000	0.405442
$220$	0	0
$221$	−12.0000	−0.807207
$222$	6.00000	0.402694
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	−4.00000	−0.267261
$225$	0	0
$226$	2.00000	0.133038
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	4.00000	0.264906
$229$	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
$231$	−4.00000	−0.263181
$232$	−6.00000	−0.393919
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	−6.00000	−0.392232
$235$	0	0
$236$	12.0000	0.781133
$237$	−4.00000	−0.259828
$238$	8.00000	0.518563
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	−1.00000	−0.0642824
$243$	1.00000	0.0641500
$244$	−14.0000	−0.896258
$245$	0	0
$246$	6.00000	0.382546
$247$	24.0000	1.52708
$248$	0	0
$249$	−4.00000	−0.253490
$250$	0	0
$251$	−4.00000	−0.252478	−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000	−0.252478	−0.126239	+	0.992000i	$0.540291\pi$
$252$	4.00000	0.251976
$253$	4.00000	0.251478
$254$	12.0000	0.752947
$255$	0	0
$256$	1.00000	0.0625000
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	4.00000	0.249029
$259$	−24.0000	−1.49129
$260$	0	0
$261$	6.00000	0.371391
$262$	−4.00000	−0.247121
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	1.00000	0.0615457
$265$	0	0
$266$	−16.0000	−0.981023
$267$	10.0000	0.611990
$268$	−4.00000	−0.244339
$269$	26.0000	1.58525	0.792624	−	0.609711i	$-0.208714\pi$
$269$	26.0000	1.58525	0.792624	+	0.609711i	$0.208714\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	−2.00000	−0.121268
$273$	24.0000	1.45255
$274$	2.00000	0.120824
$275$	0	0
$276$	−4.00000	−0.240772
$277$	−26.0000	−1.56219	−0.781094	−	0.624413i	$-0.785338\pi$
$277$	−26.0000	−1.56219	−0.781094	+	0.624413i	$0.785338\pi$
$278$	4.00000	0.239904
$279$	0	0
$280$	0	0
$281$	−22.0000	−1.31241	−0.656205	−	0.754583i	$-0.727839\pi$
$281$	−22.0000	−1.31241	−0.656205	+	0.754583i	$0.727839\pi$
$282$	−12.0000	−0.714590
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	−12.0000	−0.712069
$285$	0	0
$286$	6.00000	0.354787
$287$	−24.0000	−1.41668
$288$	−1.00000	−0.0589256
$289$	−13.0000	−0.764706
$290$	0	0
$291$	14.0000	0.820695
$292$	6.00000	0.351123
$293$	−22.0000	−1.28525	−0.642627	−	0.766179i	$-0.722155\pi$
$293$	−22.0000	−1.28525	−0.642627	+	0.766179i	$0.722155\pi$
$294$	−9.00000	−0.524891
$295$	0	0
$296$	6.00000	0.348743
$297$	−1.00000	−0.0580259
$298$	10.0000	0.579284
$299$	−24.0000	−1.38796
$300$	0	0
$301$	−16.0000	−0.922225
$302$	−4.00000	−0.230174
$303$	14.0000	0.804279
$304$	4.00000	0.229416
$305$	0	0
$306$	2.00000	0.114332
$307$	−4.00000	−0.228292	−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000	−0.228292	−0.114146	+	0.993464i	$0.536413\pi$
$308$	−4.00000	−0.227921
$309$	0	0
$310$	0	0
$311$	4.00000	0.226819	0.113410	−	0.993548i	$-0.463823\pi$
$311$	4.00000	0.226819	0.113410	+	0.993548i	$0.463823\pi$
$312$	−6.00000	−0.339683
$313$	−26.0000	−1.46961	−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−26.0000	−1.46961	−0.734803	+	0.678280i	$0.762726\pi$
$314$	−10.0000	−0.564333
$315$	0	0
$316$	−4.00000	−0.225018
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	2.00000	0.112154
$319$	−6.00000	−0.335936
$320$	0	0
$321$	−4.00000	−0.223258
$322$	16.0000	0.891645
$323$	−8.00000	−0.445132
$324$	1.00000	0.0555556
$325$	0	0
$326$	−20.0000	−1.10770
$327$	−6.00000	−0.331801
$328$	6.00000	0.331295
$329$	48.0000	2.64633
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	−4.00000	−0.219529
$333$	−6.00000	−0.328798
$334$	0	0
$335$	0	0
$336$	4.00000	0.218218
$337$	−18.0000	−0.980522	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000	−0.980522	−0.490261	+	0.871576i	$0.663099\pi$
$338$	−23.0000	−1.25104
$339$	−2.00000	−0.108625
$340$	0	0
$341$	0	0
$342$	−4.00000	−0.216295
$343$	8.00000	0.431959
$344$	4.00000	0.215666
$345$	0	0
$346$	−10.0000	−0.537603
$347$	4.00000	0.214731	0.107366	−	0.994220i	$-0.465758\pi$
$347$	4.00000	0.214731	0.107366	+	0.994220i	$0.465758\pi$
$348$	6.00000	0.321634
$349$	−6.00000	−0.321173	−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000	−0.321173	−0.160586	+	0.987022i	$0.551338\pi$
$350$	0	0
$351$	6.00000	0.320256
$352$	1.00000	0.0533002
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	−12.0000	−0.637793
$355$	0	0
$356$	10.0000	0.529999
$357$	−8.00000	−0.423405
$358$	−20.0000	−1.05703
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	2.00000	0.105118
$363$	1.00000	0.0524864
$364$	24.0000	1.25794
$365$	0	0
$366$	14.0000	0.731792
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	−4.00000	−0.208514
$369$	−6.00000	−0.312348
$370$	0	0
$371$	−8.00000	−0.415339
$372$	0	0
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	−2.00000	−0.103418
$375$	0	0
$376$	−12.0000	−0.618853
$377$	36.0000	1.85409
$378$	−4.00000	−0.205738
$379$	−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
$381$	−12.0000	−0.614779
$382$	12.0000	0.613973
$383$	4.00000	0.204390	0.102195	−	0.994764i	$-0.467413\pi$
$383$	4.00000	0.204390	0.102195	+	0.994764i	$0.467413\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	10.0000	0.508987
$387$	−4.00000	−0.203331
$388$	14.0000	0.710742
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	−9.00000	−0.454569
$393$	4.00000	0.201773
$394$	−2.00000	−0.100759
$395$	0	0
$396$	−1.00000	−0.0502519
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	16.0000	0.802008
$399$	16.0000	0.801002
$400$	0	0
$401$	−38.0000	−1.89763	−0.948815	−	0.315833i	$-0.897716\pi$
$401$	−38.0000	−1.89763	−0.948815	+	0.315833i	$0.897716\pi$
$402$	4.00000	0.199502
$403$	0	0
$404$	14.0000	0.696526
$405$	0	0
$406$	−24.0000	−1.19110
$407$	6.00000	0.297409
$408$	2.00000	0.0990148
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	0	0
$411$	−2.00000	−0.0986527
$412$	0	0
$413$	48.0000	2.36193
$414$	4.00000	0.196589
$415$	0	0
$416$	−6.00000	−0.294174
$417$	−4.00000	−0.195881
$418$	4.00000	0.195646
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	4.00000	0.194717
$423$	12.0000	0.583460
$424$	2.00000	0.0971286
$425$	0	0
$426$	12.0000	0.581402
$427$	−56.0000	−2.71003
$428$	−4.00000	−0.193347
$429$	−6.00000	−0.289683
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	1.00000	0.0481125
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	0	0
$435$	0	0
$436$	−6.00000	−0.287348
$437$	−16.0000	−0.765384
$438$	−6.00000	−0.286691
$439$	4.00000	0.190910	0.0954548	−	0.995434i	$-0.469569\pi$
$439$	4.00000	0.190910	0.0954548	+	0.995434i	$0.469569\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	12.0000	0.570782
$443$	−28.0000	−1.33032	−0.665160	−	0.746701i	$-0.731637\pi$
$443$	−28.0000	−1.33032	−0.665160	+	0.746701i	$0.731637\pi$
$444$	−6.00000	−0.284747
$445$	0	0
$446$	0	0
$447$	−10.0000	−0.472984
$448$	4.00000	0.188982
$449$	−22.0000	−1.03824	−0.519122	−	0.854700i	$-0.673741\pi$
$449$	−22.0000	−1.03824	−0.519122	+	0.854700i	$0.673741\pi$
$450$	0	0
$451$	6.00000	0.282529
$452$	−2.00000	−0.0940721
$453$	4.00000	0.187936
$454$	12.0000	0.563188
$455$	0	0
$456$	−4.00000	−0.187317
$457$	−34.0000	−1.59045	−0.795226	−	0.606313i	$-0.792648\pi$
$457$	−34.0000	−1.59045	−0.795226	+	0.606313i	$0.792648\pi$
$458$	−14.0000	−0.654177
$459$	−2.00000	−0.0933520
$460$	0	0
$461$	14.0000	0.652045	0.326023	−	0.945362i	$-0.394291\pi$
$461$	14.0000	0.652045	0.326023	+	0.945362i	$0.394291\pi$
$462$	4.00000	0.186097
$463$	24.0000	1.11537	0.557687	−	0.830051i	$-0.311689\pi$
$463$	24.0000	1.11537	0.557687	+	0.830051i	$0.311689\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	10.0000	0.463241
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	6.00000	0.277350
$469$	−16.0000	−0.738811
$470$	0	0
$471$	10.0000	0.460776
$472$	−12.0000	−0.552345
$473$	4.00000	0.183920
$474$	4.00000	0.183726
$475$	0	0
$476$	−8.00000	−0.366679
$477$	−2.00000	−0.0915737
$478$	−8.00000	−0.365911
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−36.0000	−1.64146
$482$	−10.0000	−0.455488
$483$	−16.0000	−0.728025
$484$	1.00000	0.0454545
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	16.0000	0.725029	0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000	0.725029	0.362515	+	0.931978i	$0.381918\pi$
$488$	14.0000	0.633750
$489$	20.0000	0.904431
$490$	0	0
$491$	−28.0000	−1.26362	−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000	−1.26362	−0.631811	+	0.775122i	$0.717688\pi$
$492$	−6.00000	−0.270501
$493$	−12.0000	−0.540453
$494$	−24.0000	−1.07981
$495$	0	0
$496$	0	0
$497$	−48.0000	−2.15309
$498$	4.00000	0.179244
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	0	0
$501$	0	0
$502$	4.00000	0.178529
$503$	−32.0000	−1.42681	−0.713405	−	0.700752i	$-0.752848\pi$
$503$	−32.0000	−1.42681	−0.713405	+	0.700752i	$0.752848\pi$
$504$	−4.00000	−0.178174
$505$	0	0
$506$	−4.00000	−0.177822
$507$	23.0000	1.02147
$508$	−12.0000	−0.532414
$509$	−22.0000	−0.975133	−0.487566	−	0.873086i	$-0.662115\pi$
$509$	−22.0000	−0.975133	−0.487566	+	0.873086i	$0.662115\pi$
$510$	0	0
$511$	24.0000	1.06170
$512$	−1.00000	−0.0441942
$513$	4.00000	0.176604
$514$	2.00000	0.0882162
$515$	0	0
$516$	−4.00000	−0.176090
$517$	−12.0000	−0.527759
$518$	24.0000	1.05450
$519$	10.0000	0.438951
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	−6.00000	−0.262613
$523$	−20.0000	−0.874539	−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$524$	4.00000	0.174741
$525$	0	0
$526$	−24.0000	−1.04645
$527$	0	0
$528$	−1.00000	−0.0435194
$529$	−7.00000	−0.304348
$530$	0	0
$531$	12.0000	0.520756
$532$	16.0000	0.693688
$533$	−36.0000	−1.55933
$534$	−10.0000	−0.432742
$535$	0	0
$536$	4.00000	0.172774
$537$	20.0000	0.863064
$538$	−26.0000	−1.12094
$539$	−9.00000	−0.387657
$540$	0	0
$541$	−38.0000	−1.63375	−0.816874	−	0.576816i	$-0.804295\pi$
$541$	−38.0000	−1.63375	−0.816874	+	0.576816i	$0.804295\pi$
$542$	−20.0000	−0.859074
$543$	−2.00000	−0.0858282
$544$	2.00000	0.0857493
$545$	0	0
$546$	−24.0000	−1.02711
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	−2.00000	−0.0854358
$549$	−14.0000	−0.597505
$550$	0	0
$551$	24.0000	1.02243
$552$	4.00000	0.170251
$553$	−16.0000	−0.680389
$554$	26.0000	1.10463
$555$	0	0
$556$	−4.00000	−0.169638
$557$	−30.0000	−1.27114	−0.635570	−	0.772043i	$-0.719235\pi$
$557$	−30.0000	−1.27114	−0.635570	+	0.772043i	$0.719235\pi$
$558$	0	0
$559$	−24.0000	−1.01509
$560$	0	0
$561$	2.00000	0.0844401
$562$	22.0000	0.928014
$563$	20.0000	0.842900	0.421450	−	0.906852i	$-0.361521\pi$
$563$	20.0000	0.842900	0.421450	+	0.906852i	$0.361521\pi$
$564$	12.0000	0.505291
$565$	0	0
$566$	4.00000	0.168133
$567$	4.00000	0.167984
$568$	12.0000	0.503509
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	0	0
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	−6.00000	−0.250873
$573$	−12.0000	−0.501307
$574$	24.0000	1.00174
$575$	0	0
$576$	1.00000	0.0416667
$577$	46.0000	1.91501	0.957503	−	0.288425i	$-0.0931316\pi$
$577$	46.0000	1.91501	0.957503	+	0.288425i	$0.0931316\pi$
$578$	13.0000	0.540729
$579$	−10.0000	−0.415586
$580$	0	0
$581$	−16.0000	−0.663792
$582$	−14.0000	−0.580319
$583$	2.00000	0.0828315
$584$	−6.00000	−0.248282
$585$	0	0
$586$	22.0000	0.908812
$587$	36.0000	1.48588	0.742940	−	0.669359i	$-0.233431\pi$
$587$	36.0000	1.48588	0.742940	+	0.669359i	$0.233431\pi$
$588$	9.00000	0.371154
$589$	0	0
$590$	0	0
$591$	2.00000	0.0822690
$592$	−6.00000	−0.246598
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	1.00000	0.0410305
$595$	0	0
$596$	−10.0000	−0.409616
$597$	−16.0000	−0.654836
$598$	24.0000	0.981433
$599$	36.0000	1.47092	0.735460	−	0.677568i	$-0.236966\pi$
$599$	36.0000	1.47092	0.735460	+	0.677568i	$0.236966\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	16.0000	0.652111
$603$	−4.00000	−0.162893
$604$	4.00000	0.162758
$605$	0	0
$606$	−14.0000	−0.568711
$607$	−28.0000	−1.13648	−0.568242	−	0.822861i	$-0.692376\pi$
$607$	−28.0000	−1.13648	−0.568242	+	0.822861i	$0.692376\pi$
$608$	−4.00000	−0.162221
$609$	24.0000	0.972529
$610$	0	0
$611$	72.0000	2.91281
$612$	−2.00000	−0.0808452
$613$	6.00000	0.242338	0.121169	−	0.992632i	$-0.461336\pi$
$613$	6.00000	0.242338	0.121169	+	0.992632i	$0.461336\pi$
$614$	4.00000	0.161427
$615$	0	0
$616$	4.00000	0.161165
$617$	22.0000	0.885687	0.442843	−	0.896599i	$-0.353970\pi$
$617$	22.0000	0.885687	0.442843	+	0.896599i	$0.353970\pi$
$618$	0	0
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	−4.00000	−0.160385
$623$	40.0000	1.60257
$624$	6.00000	0.240192
$625$	0	0
$626$	26.0000	1.03917
$627$	−4.00000	−0.159745
$628$	10.0000	0.399043
$629$	12.0000	0.478471
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	4.00000	0.159111
$633$	−4.00000	−0.158986
$634$	18.0000	0.714871
$635$	0	0
$636$	−2.00000	−0.0793052
$637$	54.0000	2.13956
$638$	6.00000	0.237542
$639$	−12.0000	−0.474713
$640$	0	0
$641$	42.0000	1.65890	0.829450	−	0.558581i	$-0.188654\pi$
$641$	42.0000	1.65890	0.829450	+	0.558581i	$0.188654\pi$
$642$	4.00000	0.157867
$643$	28.0000	1.10421	0.552106	−	0.833774i	$-0.313824\pi$
$643$	28.0000	1.10421	0.552106	+	0.833774i	$0.313824\pi$
$644$	−16.0000	−0.630488
$645$	0	0
$646$	8.00000	0.314756
$647$	28.0000	1.10079	0.550397	−	0.834903i	$-0.314476\pi$
$647$	28.0000	1.10079	0.550397	+	0.834903i	$0.314476\pi$
$648$	−1.00000	−0.0392837
$649$	−12.0000	−0.471041
$650$	0	0
$651$	0	0
$652$	20.0000	0.783260
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	6.00000	0.234619
$655$	0	0
$656$	−6.00000	−0.234261
$657$	6.00000	0.234082
$658$	−48.0000	−1.87123
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	−18.0000	−0.700119	−0.350059	−	0.936727i	$-0.613839\pi$
$661$	−18.0000	−0.700119	−0.350059	+	0.936727i	$0.613839\pi$
$662$	−20.0000	−0.777322
$663$	−12.0000	−0.466041
$664$	4.00000	0.155230
$665$	0	0
$666$	6.00000	0.232495
$667$	−24.0000	−0.929284
$668$	0	0
$669$	0	0
$670$	0	0
$671$	14.0000	0.540464
$672$	−4.00000	−0.154303
$673$	−26.0000	−1.00223	−0.501113	−	0.865382i	$-0.667076\pi$
$673$	−26.0000	−1.00223	−0.501113	+	0.865382i	$0.667076\pi$
$674$	18.0000	0.693334
$675$	0	0
$676$	23.0000	0.884615
$677$	−46.0000	−1.76792	−0.883962	−	0.467559i	$-0.845134\pi$
$677$	−46.0000	−1.76792	−0.883962	+	0.467559i	$0.845134\pi$
$678$	2.00000	0.0768095
$679$	56.0000	2.14908
$680$	0	0
$681$	−12.0000	−0.459841
$682$	0	0
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	4.00000	0.152944
$685$	0	0
$686$	−8.00000	−0.305441
$687$	14.0000	0.534133
$688$	−4.00000	−0.152499
$689$	−12.0000	−0.457164
$690$	0	0
$691$	−12.0000	−0.456502	−0.228251	−	0.973602i	$-0.573301\pi$
$691$	−12.0000	−0.456502	−0.228251	+	0.973602i	$0.573301\pi$
$692$	10.0000	0.380143
$693$	−4.00000	−0.151947
$694$	−4.00000	−0.151838
$695$	0	0
$696$	−6.00000	−0.227429
$697$	12.0000	0.454532
$698$	6.00000	0.227103
$699$	−10.0000	−0.378235
$700$	0	0
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	−6.00000	−0.226455
$703$	−24.0000	−0.905177
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	18.0000	0.677439
$707$	56.0000	2.10610
$708$	12.0000	0.450988
$709$	−18.0000	−0.676004	−0.338002	−	0.941145i	$-0.609751\pi$
$709$	−18.0000	−0.676004	−0.338002	+	0.941145i	$0.609751\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	−10.0000	−0.374766
$713$	0	0
$714$	8.00000	0.299392
$715$	0	0
$716$	20.0000	0.747435
$717$	8.00000	0.298765
$718$	0	0
$719$	12.0000	0.447524	0.223762	−	0.974644i	$-0.428166\pi$
$719$	12.0000	0.447524	0.223762	+	0.974644i	$0.428166\pi$
$720$	0	0
$721$	0	0
$722$	3.00000	0.111648
$723$	10.0000	0.371904
$724$	−2.00000	−0.0743294
$725$	0	0
$726$	−1.00000	−0.0371135
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	−24.0000	−0.889499
$729$	1.00000	0.0370370
$730$	0	0
$731$	8.00000	0.295891
$732$	−14.0000	−0.517455
$733$	22.0000	0.812589	0.406294	−	0.913742i	$-0.366821\pi$
$733$	22.0000	0.812589	0.406294	+	0.913742i	$0.366821\pi$
$734$	−16.0000	−0.590571
$735$	0	0
$736$	4.00000	0.147442
$737$	4.00000	0.147342
$738$	6.00000	0.220863
$739$	44.0000	1.61857	0.809283	−	0.587419i	$-0.199856\pi$
$739$	44.0000	1.61857	0.809283	+	0.587419i	$0.199856\pi$
$740$	0	0
$741$	24.0000	0.881662
$742$	8.00000	0.293689
$743$	32.0000	1.17397	0.586983	−	0.809599i	$-0.300316\pi$
$743$	32.0000	1.17397	0.586983	+	0.809599i	$0.300316\pi$
$744$	0	0
$745$	0	0
$746$	−14.0000	−0.512576
$747$	−4.00000	−0.146352
$748$	2.00000	0.0731272
$749$	−16.0000	−0.584627
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	12.0000	0.437595
$753$	−4.00000	−0.145768
$754$	−36.0000	−1.31104
$755$	0	0
$756$	4.00000	0.145479
$757$	−38.0000	−1.38113	−0.690567	−	0.723269i	$-0.742639\pi$
$757$	−38.0000	−1.38113	−0.690567	+	0.723269i	$0.742639\pi$
$758$	28.0000	1.01701
$759$	4.00000	0.145191
$760$	0	0
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	12.0000	0.434714
$763$	−24.0000	−0.868858
$764$	−12.0000	−0.434145
$765$	0	0
$766$	−4.00000	−0.144526
$767$	72.0000	2.59977
$768$	1.00000	0.0360844
$769$	10.0000	0.360609	0.180305	−	0.983611i	$-0.442292\pi$
$769$	10.0000	0.360609	0.180305	+	0.983611i	$0.442292\pi$
$770$	0	0
$771$	−2.00000	−0.0720282
$772$	−10.0000	−0.359908
$773$	−18.0000	−0.647415	−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000	−0.647415	−0.323708	+	0.946157i	$0.604929\pi$
$774$	4.00000	0.143777
$775$	0	0
$776$	−14.0000	−0.502571
$777$	−24.0000	−0.860995
$778$	30.0000	1.07555
$779$	−24.0000	−0.859889
$780$	0	0
$781$	12.0000	0.429394
$782$	−8.00000	−0.286079
$783$	6.00000	0.214423
$784$	9.00000	0.321429
$785$	0	0
$786$	−4.00000	−0.142675
$787$	20.0000	0.712923	0.356462	−	0.934310i	$-0.383983\pi$
$787$	20.0000	0.712923	0.356462	+	0.934310i	$0.383983\pi$
$788$	2.00000	0.0712470
$789$	24.0000	0.854423
$790$	0	0
$791$	−8.00000	−0.284447
$792$	1.00000	0.0355335
$793$	−84.0000	−2.98293
$794$	22.0000	0.780751
$795$	0	0
$796$	−16.0000	−0.567105
$797$	54.0000	1.91278	0.956389	−	0.292096i	$-0.0943526\pi$
$797$	54.0000	1.91278	0.956389	+	0.292096i	$0.0943526\pi$
$798$	−16.0000	−0.566394
$799$	−24.0000	−0.849059
$800$	0	0
$801$	10.0000	0.353333
$802$	38.0000	1.34183
$803$	−6.00000	−0.211735
$804$	−4.00000	−0.141069
$805$	0	0
$806$	0	0
$807$	26.0000	0.915243
$808$	−14.0000	−0.492518
$809$	10.0000	0.351581	0.175791	−	0.984428i	$-0.443752\pi$
$809$	10.0000	0.351581	0.175791	+	0.984428i	$0.443752\pi$
$810$	0	0
$811$	−28.0000	−0.983213	−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000	−0.983213	−0.491606	+	0.870817i	$0.663590\pi$
$812$	24.0000	0.842235
$813$	20.0000	0.701431
$814$	−6.00000	−0.210300
$815$	0	0
$816$	−2.00000	−0.0700140
$817$	−16.0000	−0.559769
$818$	14.0000	0.489499
$819$	24.0000	0.838628
$820$	0	0
$821$	−2.00000	−0.0698005	−0.0349002	−	0.999391i	$-0.511111\pi$
$821$	−2.00000	−0.0698005	−0.0349002	+	0.999391i	$0.511111\pi$
$822$	2.00000	0.0697580
$823$	40.0000	1.39431	0.697156	−	0.716919i	$-0.254448\pi$
$823$	40.0000	1.39431	0.697156	+	0.716919i	$0.254448\pi$
$824$	0	0
$825$	0	0
$826$	−48.0000	−1.67013
$827$	52.0000	1.80822	0.904109	−	0.427303i	$-0.140536\pi$
$827$	52.0000	1.80822	0.904109	+	0.427303i	$0.140536\pi$
$828$	−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
$832$	6.00000	0.208013
$833$	−18.0000	−0.623663
$834$	4.00000	0.138509
$835$	0	0
$836$	−4.00000	−0.138343
$837$	0	0
$838$	12.0000	0.414533
$839$	12.0000	0.414286	0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000	0.414286	0.207143	+	0.978311i	$0.433583\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	10.0000	0.344623
$843$	−22.0000	−0.757720
$844$	−4.00000	−0.137686
$845$	0	0
$846$	−12.0000	−0.412568
$847$	4.00000	0.137442
$848$	−2.00000	−0.0686803
$849$	−4.00000	−0.137280
$850$	0	0
$851$	24.0000	0.822709
$852$	−12.0000	−0.411113
$853$	6.00000	0.205436	0.102718	−	0.994711i	$-0.467246\pi$
$853$	6.00000	0.205436	0.102718	+	0.994711i	$0.467246\pi$
$854$	56.0000	1.91628
$855$	0	0
$856$	4.00000	0.136717
$857$	−26.0000	−0.888143	−0.444072	−	0.895991i	$-0.646466\pi$
$857$	−26.0000	−0.888143	−0.444072	+	0.895991i	$0.646466\pi$
$858$	6.00000	0.204837
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	0	0
$861$	−24.0000	−0.817918
$862$	0	0
$863$	−20.0000	−0.680808	−0.340404	−	0.940279i	$-0.610564\pi$
$863$	−20.0000	−0.680808	−0.340404	+	0.940279i	$0.610564\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	2.00000	0.0679628
$867$	−13.0000	−0.441503
$868$	0	0
$869$	4.00000	0.135691
$870$	0	0
$871$	−24.0000	−0.813209
$872$	6.00000	0.203186
$873$	14.0000	0.473828
$874$	16.0000	0.541208
$875$	0	0
$876$	6.00000	0.202721
$877$	−42.0000	−1.41824	−0.709120	−	0.705088i	$-0.750907\pi$
$877$	−42.0000	−1.41824	−0.709120	+	0.705088i	$0.750907\pi$
$878$	−4.00000	−0.134993
$879$	−22.0000	−0.742042
$880$	0	0
$881$	−14.0000	−0.471672	−0.235836	−	0.971793i	$-0.575783\pi$
$881$	−14.0000	−0.471672	−0.235836	+	0.971793i	$0.575783\pi$
$882$	−9.00000	−0.303046
$883$	−4.00000	−0.134611	−0.0673054	−	0.997732i	$-0.521440\pi$
$883$	−4.00000	−0.134611	−0.0673054	+	0.997732i	$0.521440\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	28.0000	0.940678
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	6.00000	0.201347
$889$	−48.0000	−1.60987
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	48.0000	1.60626
$894$	10.0000	0.334450
$895$	0	0
$896$	−4.00000	−0.133631
$897$	−24.0000	−0.801337
$898$	22.0000	0.734150
$899$	0	0
$900$	0	0
$901$	4.00000	0.133259
$902$	−6.00000	−0.199778
$903$	−16.0000	−0.532447
$904$	2.00000	0.0665190
$905$	0	0
$906$	−4.00000	−0.132891
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	−12.0000	−0.398234
$909$	14.0000	0.464351
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	4.00000	0.132453
$913$	4.00000	0.132381
$914$	34.0000	1.12462
$915$	0	0
$916$	14.0000	0.462573
$917$	16.0000	0.528367
$918$	2.00000	0.0660098
$919$	−4.00000	−0.131948	−0.0659739	−	0.997821i	$-0.521015\pi$
$919$	−4.00000	−0.131948	−0.0659739	+	0.997821i	$0.521015\pi$
$920$	0	0
$921$	−4.00000	−0.131804
$922$	−14.0000	−0.461065
$923$	−72.0000	−2.36991
$924$	−4.00000	−0.131590
$925$	0	0
$926$	−24.0000	−0.788689
$927$	0	0
$928$	−6.00000	−0.196960
$929$	42.0000	1.37798	0.688988	−	0.724773i	$-0.258055\pi$
$929$	42.0000	1.37798	0.688988	+	0.724773i	$0.258055\pi$
$930$	0	0
$931$	36.0000	1.17985
$932$	−10.0000	−0.327561
$933$	4.00000	0.130954
$934$	12.0000	0.392652
$935$	0	0
$936$	−6.00000	−0.196116
$937$	38.0000	1.24141	0.620703	−	0.784046i	$-0.286847\pi$
$937$	38.0000	1.24141	0.620703	+	0.784046i	$0.286847\pi$
$938$	16.0000	0.522419
$939$	−26.0000	−0.848478
$940$	0	0
$941$	−42.0000	−1.36916	−0.684580	−	0.728937i	$-0.740015\pi$
$941$	−42.0000	−1.36916	−0.684580	+	0.728937i	$0.740015\pi$
$942$	−10.0000	−0.325818
$943$	24.0000	0.781548
$944$	12.0000	0.390567
$945$	0	0
$946$	−4.00000	−0.130051
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	−4.00000	−0.129914
$949$	36.0000	1.16861
$950$	0	0
$951$	−18.0000	−0.583690
$952$	8.00000	0.259281
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	2.00000	0.0647524
$955$	0	0
$956$	8.00000	0.258738
$957$	−6.00000	−0.193952
$958$	0	0
$959$	−8.00000	−0.258333
$960$	0	0
$961$	−31.0000	−1.00000
$962$	36.0000	1.16069
$963$	−4.00000	−0.128898
$964$	10.0000	0.322078
$965$	0	0
$966$	16.0000	0.514792
$967$	44.0000	1.41494	0.707472	−	0.706741i	$-0.249835\pi$
$967$	44.0000	1.41494	0.707472	+	0.706741i	$0.249835\pi$
$968$	−1.00000	−0.0321412
$969$	−8.00000	−0.256997
$970$	0	0
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	1.00000	0.0320750
$973$	−16.0000	−0.512936
$974$	−16.0000	−0.512673
$975$	0	0
$976$	−14.0000	−0.448129
$977$	−26.0000	−0.831814	−0.415907	−	0.909407i	$-0.636536\pi$
$977$	−26.0000	−0.831814	−0.415907	+	0.909407i	$0.636536\pi$
$978$	−20.0000	−0.639529
$979$	−10.0000	−0.319601
$980$	0	0
$981$	−6.00000	−0.191565
$982$	28.0000	0.893516
$983$	−36.0000	−1.14822	−0.574111	−	0.818778i	$-0.694652\pi$
$983$	−36.0000	−1.14822	−0.574111	+	0.818778i	$0.694652\pi$
$984$	6.00000	0.191273
$985$	0	0
$986$	12.0000	0.382158
$987$	48.0000	1.52786
$988$	24.0000	0.763542
$989$	16.0000	0.508770
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	0	0
$993$	20.0000	0.634681
$994$	48.0000	1.52247
$995$	0	0
$996$	−4.00000	−0.126745
$997$	14.0000	0.443384	0.221692	−	0.975117i	$-0.428842\pi$
$997$	14.0000	0.443384	0.221692	+	0.975117i	$0.428842\pi$
$998$	4.00000	0.126618
$999$	−6.00000	−0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1650.2.a.k.1.1		1
3.2	odd	2		4950.2.a.bu.1.1		1
5.2	odd	4		1650.2.c.e.199.1		2
5.3	odd	4		1650.2.c.e.199.2		2
5.4	even	2		66.2.a.b.1.1	✓	1
15.2	even	4		4950.2.c.p.199.2		2
15.8	even	4		4950.2.c.p.199.1		2
15.14	odd	2		198.2.a.a.1.1		1
20.19	odd	2		528.2.a.j.1.1		1
35.34	odd	2		3234.2.a.t.1.1		1
40.19	odd	2		2112.2.a.e.1.1		1
40.29	even	2		2112.2.a.r.1.1		1
45.4	even	6		1782.2.e.e.1189.1		2
45.14	odd	6		1782.2.e.v.1189.1		2
45.29	odd	6		1782.2.e.v.595.1		2
45.34	even	6		1782.2.e.e.595.1		2
55.4	even	10		726.2.e.g.511.1		4
55.9	even	10		726.2.e.g.565.1		4
55.14	even	10		726.2.e.g.493.1		4
55.19	odd	10		726.2.e.o.493.1		4
55.24	odd	10		726.2.e.o.565.1		4
55.29	odd	10		726.2.e.o.511.1		4
55.39	odd	10		726.2.e.o.487.1		4
55.49	even	10		726.2.e.g.487.1		4
55.54	odd	2		726.2.a.c.1.1		1
60.59	even	2		1584.2.a.f.1.1		1
105.104	even	2		9702.2.a.x.1.1		1
120.29	odd	2		6336.2.a.bw.1.1		1
120.59	even	2		6336.2.a.cj.1.1		1
165.164	even	2		2178.2.a.g.1.1		1
220.219	even	2		5808.2.a.bc.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
66.2.a.b.1.1	✓	1	5.4	even	2
198.2.a.a.1.1		1	15.14	odd	2
528.2.a.j.1.1		1	20.19	odd	2
726.2.a.c.1.1		1	55.54	odd	2
726.2.e.g.487.1		4	55.49	even	10
726.2.e.g.493.1		4	55.14	even	10
726.2.e.g.511.1		4	55.4	even	10
726.2.e.g.565.1		4	55.9	even	10
726.2.e.o.487.1		4	55.39	odd	10
726.2.e.o.493.1		4	55.19	odd	10
726.2.e.o.511.1		4	55.29	odd	10
726.2.e.o.565.1		4	55.24	odd	10
1584.2.a.f.1.1		1	60.59	even	2
1650.2.a.k.1.1		1	1.1	even	1	trivial
1650.2.c.e.199.1		2	5.2	odd	4
1650.2.c.e.199.2		2	5.3	odd	4
1782.2.e.e.595.1		2	45.34	even	6
1782.2.e.e.1189.1		2	45.4	even	6
1782.2.e.v.595.1		2	45.29	odd	6
1782.2.e.v.1189.1		2	45.14	odd	6
2112.2.a.e.1.1		1	40.19	odd	2
2112.2.a.r.1.1		1	40.29	even	2
2178.2.a.g.1.1		1	165.164	even	2
3234.2.a.t.1.1		1	35.34	odd	2
4950.2.a.bu.1.1		1	3.2	odd	2
4950.2.c.p.199.1		2	15.8	even	4
4950.2.c.p.199.2		2	15.2	even	4
5808.2.a.bc.1.1		1	220.219	even	2
6336.2.a.bw.1.1		1	120.29	odd	2
6336.2.a.cj.1.1		1	120.59	even	2
9702.2.a.x.1.1		1	105.104	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1650 = 2 \cdot 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1650.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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