LMFDB - Embedded newform 4650.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4650,2,Mod(1,4650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4650, 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("4650.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4650.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:	$37.1304369399$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 186)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4650.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -3.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} -2.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} -1.00000 q^{18} -5.00000 q^{19} -2.00000 q^{21} +3.00000 q^{22} -4.00000 q^{23} +1.00000 q^{24} -1.00000 q^{26} -1.00000 q^{27} +2.00000 q^{28} +1.00000 q^{31} -1.00000 q^{32} +3.00000 q^{33} +3.00000 q^{34} +1.00000 q^{36} +2.00000 q^{37} +5.00000 q^{38} -1.00000 q^{39} +2.00000 q^{41} +2.00000 q^{42} +6.00000 q^{43} -3.00000 q^{44} +4.00000 q^{46} +7.00000 q^{47} -1.00000 q^{48} -3.00000 q^{49} +3.00000 q^{51} +1.00000 q^{52} -14.0000 q^{53} +1.00000 q^{54} -2.00000 q^{56} +5.00000 q^{57} +10.0000 q^{59} +7.00000 q^{61} -1.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} -3.00000 q^{66} +7.00000 q^{67} -3.00000 q^{68} +4.00000 q^{69} -3.00000 q^{71} -1.00000 q^{72} +6.00000 q^{73} -2.00000 q^{74} -5.00000 q^{76} -6.00000 q^{77} +1.00000 q^{78} +15.0000 q^{79} +1.00000 q^{81} -2.00000 q^{82} +1.00000 q^{83} -2.00000 q^{84} -6.00000 q^{86} +3.00000 q^{88} +10.0000 q^{89} +2.00000 q^{91} -4.00000 q^{92} -1.00000 q^{93} -7.00000 q^{94} +1.00000 q^{96} -13.0000 q^{97} +3.00000 q^{98} -3.00000 q^{99} +O(q^{100})\)

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
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	−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	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	−1.00000	−0.235702
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	3.00000	0.639602
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	−1.00000	−0.196116
$27$	−1.00000	−0.192450
$28$	2.00000	0.377964
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	−1.00000	−0.176777
$33$	3.00000	0.522233
$34$	3.00000	0.514496
$35$	0	0
$36$	1.00000	0.166667
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	5.00000	0.811107
$39$	−1.00000	−0.160128
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	2.00000	0.308607
$43$	6.00000	0.914991	0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000	0.914991	0.457496	+	0.889212i	$0.348747\pi$
$44$	−3.00000	−0.452267
$45$	0	0
$46$	4.00000	0.589768
$47$	7.00000	1.02105	0.510527	−	0.859861i	$-0.329450\pi$
$47$	7.00000	1.02105	0.510527	+	0.859861i	$0.329450\pi$
$48$	−1.00000	−0.144338
$49$	−3.00000	−0.428571
$50$	0	0
$51$	3.00000	0.420084
$52$	1.00000	0.138675
$53$	−14.0000	−1.92305	−0.961524	−	0.274721i	$-0.911414\pi$
$53$	−14.0000	−1.92305	−0.961524	+	0.274721i	$0.911414\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	−2.00000	−0.267261
$57$	5.00000	0.662266
$58$	0	0
$59$	10.0000	1.30189	0.650945	−	0.759125i	$-0.274373\pi$
$59$	10.0000	1.30189	0.650945	+	0.759125i	$0.274373\pi$
$60$	0	0
$61$	7.00000	0.896258	0.448129	−	0.893969i	$-0.352090\pi$
$61$	7.00000	0.896258	0.448129	+	0.893969i	$0.352090\pi$
$62$	−1.00000	−0.127000
$63$	2.00000	0.251976
$64$	1.00000	0.125000
$65$	0	0
$66$	−3.00000	−0.369274
$67$	7.00000	0.855186	0.427593	−	0.903971i	$-0.359362\pi$
$67$	7.00000	0.855186	0.427593	+	0.903971i	$0.359362\pi$
$68$	−3.00000	−0.363803
$69$	4.00000	0.481543
$70$	0	0
$71$	−3.00000	−0.356034	−0.178017	−	0.984027i	$-0.556968\pi$
$71$	−3.00000	−0.356034	−0.178017	+	0.984027i	$0.556968\pi$
$72$	−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	$0.385800\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	−5.00000	−0.573539
$77$	−6.00000	−0.683763
$78$	1.00000	0.113228
$79$	15.0000	1.68763	0.843816	−	0.536633i	$-0.180304\pi$
$79$	15.0000	1.68763	0.843816	+	0.536633i	$0.180304\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−2.00000	−0.220863
$83$	1.00000	0.109764	0.0548821	−	0.998493i	$-0.482522\pi$
$83$	1.00000	0.109764	0.0548821	+	0.998493i	$0.482522\pi$
$84$	−2.00000	−0.218218
$85$	0	0
$86$	−6.00000	−0.646997
$87$	0	0
$88$	3.00000	0.319801
$89$	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
$91$	2.00000	0.209657
$92$	−4.00000	−0.417029
$93$	−1.00000	−0.103695
$94$	−7.00000	−0.721995
$95$	0	0
$96$	1.00000	0.102062
$97$	−13.0000	−1.31995	−0.659975	−	0.751288i	$-0.729433\pi$
$97$	−13.0000	−1.31995	−0.659975	+	0.751288i	$0.729433\pi$
$98$	3.00000	0.303046
$99$	−3.00000	−0.301511
$100$	0	0
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	−3.00000	−0.297044
$103$	−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
$106$	14.0000	1.35980
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	−1.00000	−0.0962250
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	2.00000	0.188982
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	−5.00000	−0.468293
$115$	0	0
$116$	0	0
$117$	1.00000	0.0924500
$118$	−10.0000	−0.920575
$119$	−6.00000	−0.550019
$120$	0	0
$121$	−2.00000	−0.181818
$122$	−7.00000	−0.633750
$123$	−2.00000	−0.180334
$124$	1.00000	0.0898027
$125$	0	0
$126$	−2.00000	−0.178174
$127$	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
$129$	−6.00000	−0.528271
$130$	0	0
$131$	−18.0000	−1.57267	−0.786334	−	0.617802i	$-0.788023\pi$
$131$	−18.0000	−1.57267	−0.786334	+	0.617802i	$0.788023\pi$
$132$	3.00000	0.261116
$133$	−10.0000	−0.867110
$134$	−7.00000	−0.604708
$135$	0	0
$136$	3.00000	0.257248
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	−4.00000	−0.340503
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	0	0
$141$	−7.00000	−0.589506
$142$	3.00000	0.251754
$143$	−3.00000	−0.250873
$144$	1.00000	0.0833333
$145$	0	0
$146$	−6.00000	−0.496564
$147$	3.00000	0.247436
$148$	2.00000	0.164399
$149$	5.00000	0.409616	0.204808	−	0.978802i	$-0.434343\pi$
$149$	5.00000	0.409616	0.204808	+	0.978802i	$0.434343\pi$
$150$	0	0
$151$	−3.00000	−0.244137	−0.122068	−	0.992522i	$-0.538953\pi$
$151$	−3.00000	−0.244137	−0.122068	+	0.992522i	$0.538953\pi$
$152$	5.00000	0.405554
$153$	−3.00000	−0.242536
$154$	6.00000	0.483494
$155$	0	0
$156$	−1.00000	−0.0800641
$157$	−8.00000	−0.638470	−0.319235	−	0.947676i	$-0.603426\pi$
$157$	−8.00000	−0.638470	−0.319235	+	0.947676i	$0.603426\pi$
$158$	−15.0000	−1.19334
$159$	14.0000	1.11027
$160$	0	0
$161$	−8.00000	−0.630488
$162$	−1.00000	−0.0785674
$163$	21.0000	1.64485	0.822423	−	0.568876i	$-0.192621\pi$
$163$	21.0000	1.64485	0.822423	+	0.568876i	$0.192621\pi$
$164$	2.00000	0.156174
$165$	0	0
$166$	−1.00000	−0.0776151
$167$	2.00000	0.154765	0.0773823	−	0.997001i	$-0.475344\pi$
$167$	2.00000	0.154765	0.0773823	+	0.997001i	$0.475344\pi$
$168$	2.00000	0.154303
$169$	−12.0000	−0.923077
$170$	0	0
$171$	−5.00000	−0.382360
$172$	6.00000	0.457496
$173$	21.0000	1.59660	0.798300	−	0.602260i	$-0.205733\pi$
$173$	21.0000	1.59660	0.798300	+	0.602260i	$0.205733\pi$
$174$	0	0
$175$	0	0
$176$	−3.00000	−0.226134
$177$	−10.0000	−0.751646
$178$	−10.0000	−0.749532
$179$	−5.00000	−0.373718	−0.186859	−	0.982387i	$-0.559831\pi$
$179$	−5.00000	−0.373718	−0.186859	+	0.982387i	$0.559831\pi$
$180$	0	0
$181$	−18.0000	−1.33793	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000	−1.33793	−0.668965	+	0.743294i	$0.733262\pi$
$182$	−2.00000	−0.148250
$183$	−7.00000	−0.517455
$184$	4.00000	0.294884
$185$	0	0
$186$	1.00000	0.0733236
$187$	9.00000	0.658145
$188$	7.00000	0.510527
$189$	−2.00000	−0.145479
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	−1.00000	−0.0721688
$193$	21.0000	1.51161	0.755807	−	0.654795i	$-0.227245\pi$
$193$	21.0000	1.51161	0.755807	+	0.654795i	$0.227245\pi$
$194$	13.0000	0.933346
$195$	0	0
$196$	−3.00000	−0.214286
$197$	22.0000	1.56744	0.783718	−	0.621117i	$-0.213321\pi$
$197$	22.0000	1.56744	0.783718	+	0.621117i	$0.213321\pi$
$198$	3.00000	0.213201
$199$	15.0000	1.06332	0.531661	−	0.846957i	$-0.321568\pi$
$199$	15.0000	1.06332	0.531661	+	0.846957i	$0.321568\pi$
$200$	0	0
$201$	−7.00000	−0.493742
$202$	−2.00000	−0.140720
$203$	0	0
$204$	3.00000	0.210042
$205$	0	0
$206$	14.0000	0.975426
$207$	−4.00000	−0.278019
$208$	1.00000	0.0693375
$209$	15.0000	1.03757
$210$	0	0
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	−14.0000	−0.961524
$213$	3.00000	0.205557
$214$	−12.0000	−0.820303
$215$	0	0
$216$	1.00000	0.0680414
$217$	2.00000	0.135769
$218$	−10.0000	−0.677285
$219$	−6.00000	−0.405442
$220$	0	0
$221$	−3.00000	−0.201802
$222$	2.00000	0.134231
$223$	−9.00000	−0.602685	−0.301342	−	0.953516i	$-0.597435\pi$
$223$	−9.00000	−0.602685	−0.301342	+	0.953516i	$0.597435\pi$
$224$	−2.00000	−0.133631
$225$	0	0
$226$	14.0000	0.931266
$227$	22.0000	1.46019	0.730096	−	0.683345i	$-0.239475\pi$
$227$	22.0000	1.46019	0.730096	+	0.683345i	$0.239475\pi$
$228$	5.00000	0.331133
$229$	−5.00000	−0.330409	−0.165205	−	0.986259i	$-0.552828\pi$
$229$	−5.00000	−0.330409	−0.165205	+	0.986259i	$0.552828\pi$
$230$	0	0
$231$	6.00000	0.394771
$232$	0	0
$233$	−4.00000	−0.262049	−0.131024	−	0.991379i	$-0.541827\pi$
$233$	−4.00000	−0.262049	−0.131024	+	0.991379i	$0.541827\pi$
$234$	−1.00000	−0.0653720
$235$	0	0
$236$	10.0000	0.650945
$237$	−15.0000	−0.974355
$238$	6.00000	0.388922
$239$	20.0000	1.29369	0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000	1.29369	0.646846	+	0.762620i	$0.276088\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	2.00000	0.128565
$243$	−1.00000	−0.0641500
$244$	7.00000	0.448129
$245$	0	0
$246$	2.00000	0.127515
$247$	−5.00000	−0.318142
$248$	−1.00000	−0.0635001
$249$	−1.00000	−0.0633724
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	2.00000	0.125988
$253$	12.0000	0.754434
$254$	−12.0000	−0.752947
$255$	0	0
$256$	1.00000	0.0625000
$257$	−28.0000	−1.74659	−0.873296	−	0.487190i	$-0.838022\pi$
$257$	−28.0000	−1.74659	−0.873296	+	0.487190i	$0.838022\pi$
$258$	6.00000	0.373544
$259$	4.00000	0.248548
$260$	0	0
$261$	0	0
$262$	18.0000	1.11204
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	−3.00000	−0.184637
$265$	0	0
$266$	10.0000	0.613139
$267$	−10.0000	−0.611990
$268$	7.00000	0.427593
$269$	10.0000	0.609711	0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000	0.609711	0.304855	+	0.952399i	$0.401392\pi$
$270$	0	0
$271$	7.00000	0.425220	0.212610	−	0.977137i	$-0.431804\pi$
$271$	7.00000	0.425220	0.212610	+	0.977137i	$0.431804\pi$
$272$	−3.00000	−0.181902
$273$	−2.00000	−0.121046
$274$	−2.00000	−0.120824
$275$	0	0
$276$	4.00000	0.240772
$277$	17.0000	1.02143	0.510716	−	0.859750i	$-0.329381\pi$
$277$	17.0000	1.02143	0.510716	+	0.859750i	$0.329381\pi$
$278$	−20.0000	−1.19952
$279$	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	$0.680403\pi$
$282$	7.00000	0.416844
$283$	−9.00000	−0.534994	−0.267497	−	0.963559i	$-0.586197\pi$
$283$	−9.00000	−0.534994	−0.267497	+	0.963559i	$0.586197\pi$
$284$	−3.00000	−0.178017
$285$	0	0
$286$	3.00000	0.177394
$287$	4.00000	0.236113
$288$	−1.00000	−0.0589256
$289$	−8.00000	−0.470588
$290$	0	0
$291$	13.0000	0.762073
$292$	6.00000	0.351123
$293$	−14.0000	−0.817889	−0.408944	−	0.912559i	$-0.634103\pi$
$293$	−14.0000	−0.817889	−0.408944	+	0.912559i	$0.634103\pi$
$294$	−3.00000	−0.174964
$295$	0	0
$296$	−2.00000	−0.116248
$297$	3.00000	0.174078
$298$	−5.00000	−0.289642
$299$	−4.00000	−0.231326
$300$	0	0
$301$	12.0000	0.691669
$302$	3.00000	0.172631
$303$	−2.00000	−0.114897
$304$	−5.00000	−0.286770
$305$	0	0
$306$	3.00000	0.171499
$307$	−8.00000	−0.456584	−0.228292	−	0.973593i	$-0.573314\pi$
$307$	−8.00000	−0.456584	−0.228292	+	0.973593i	$0.573314\pi$
$308$	−6.00000	−0.341882
$309$	14.0000	0.796432
$310$	0	0
$311$	27.0000	1.53103	0.765515	−	0.643418i	$-0.222484\pi$
$311$	27.0000	1.53103	0.765515	+	0.643418i	$0.222484\pi$
$312$	1.00000	0.0566139
$313$	26.0000	1.46961	0.734803	−	0.678280i	$-0.237274\pi$
$313$	26.0000	1.46961	0.734803	+	0.678280i	$0.237274\pi$
$314$	8.00000	0.451466
$315$	0	0
$316$	15.0000	0.843816
$317$	17.0000	0.954815	0.477408	−	0.878682i	$-0.341577\pi$
$317$	17.0000	0.954815	0.477408	+	0.878682i	$0.341577\pi$
$318$	−14.0000	−0.785081
$319$	0	0
$320$	0	0
$321$	−12.0000	−0.669775
$322$	8.00000	0.445823
$323$	15.0000	0.834622
$324$	1.00000	0.0555556
$325$	0	0
$326$	−21.0000	−1.16308
$327$	−10.0000	−0.553001
$328$	−2.00000	−0.110432
$329$	14.0000	0.771845
$330$	0	0
$331$	32.0000	1.75888	0.879440	−	0.476011i	$-0.157918\pi$
$331$	32.0000	1.75888	0.879440	+	0.476011i	$0.157918\pi$
$332$	1.00000	0.0548821
$333$	2.00000	0.109599
$334$	−2.00000	−0.109435
$335$	0	0
$336$	−2.00000	−0.109109
$337$	32.0000	1.74315	0.871576	−	0.490261i	$-0.163099\pi$
$337$	32.0000	1.74315	0.871576	+	0.490261i	$0.163099\pi$
$338$	12.0000	0.652714
$339$	14.0000	0.760376
$340$	0	0
$341$	−3.00000	−0.162459
$342$	5.00000	0.270369
$343$	−20.0000	−1.07990
$344$	−6.00000	−0.323498
$345$	0	0
$346$	−21.0000	−1.12897
$347$	7.00000	0.375780	0.187890	−	0.982190i	$-0.439835\pi$
$347$	7.00000	0.375780	0.187890	+	0.982190i	$0.439835\pi$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	3.00000	0.159901
$353$	−9.00000	−0.479022	−0.239511	−	0.970894i	$-0.576987\pi$
$353$	−9.00000	−0.479022	−0.239511	+	0.970894i	$0.576987\pi$
$354$	10.0000	0.531494
$355$	0	0
$356$	10.0000	0.529999
$357$	6.00000	0.317554
$358$	5.00000	0.264258
$359$	−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
$361$	6.00000	0.315789
$362$	18.0000	0.946059
$363$	2.00000	0.104973
$364$	2.00000	0.104828
$365$	0	0
$366$	7.00000	0.365896
$367$	17.0000	0.887393	0.443696	−	0.896177i	$-0.353667\pi$
$367$	17.0000	0.887393	0.443696	+	0.896177i	$0.353667\pi$
$368$	−4.00000	−0.208514
$369$	2.00000	0.104116
$370$	0	0
$371$	−28.0000	−1.45369
$372$	−1.00000	−0.0518476
$373$	16.0000	0.828449	0.414224	−	0.910175i	$-0.364053\pi$
$373$	16.0000	0.828449	0.414224	+	0.910175i	$0.364053\pi$
$374$	−9.00000	−0.465379
$375$	0	0
$376$	−7.00000	−0.360997
$377$	0	0
$378$	2.00000	0.102869
$379$	−35.0000	−1.79783	−0.898915	−	0.438124i	$-0.855643\pi$
$379$	−35.0000	−1.79783	−0.898915	+	0.438124i	$0.855643\pi$
$380$	0	0
$381$	−12.0000	−0.614779
$382$	8.00000	0.409316
$383$	26.0000	1.32854	0.664269	−	0.747494i	$-0.268743\pi$
$383$	26.0000	1.32854	0.664269	+	0.747494i	$0.268743\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−21.0000	−1.06887
$387$	6.00000	0.304997
$388$	−13.0000	−0.659975
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	3.00000	0.151523
$393$	18.0000	0.907980
$394$	−22.0000	−1.10834
$395$	0	0
$396$	−3.00000	−0.150756
$397$	22.0000	1.10415	0.552074	−	0.833795i	$-0.313837\pi$
$397$	22.0000	1.10415	0.552074	+	0.833795i	$0.313837\pi$
$398$	−15.0000	−0.751882
$399$	10.0000	0.500626
$400$	0	0
$401$	−3.00000	−0.149813	−0.0749064	−	0.997191i	$-0.523866\pi$
$401$	−3.00000	−0.149813	−0.0749064	+	0.997191i	$0.523866\pi$
$402$	7.00000	0.349128
$403$	1.00000	0.0498135
$404$	2.00000	0.0995037
$405$	0	0
$406$	0	0
$407$	−6.00000	−0.297409
$408$	−3.00000	−0.148522
$409$	30.0000	1.48340	0.741702	−	0.670729i	$-0.234019\pi$
$409$	30.0000	1.48340	0.741702	+	0.670729i	$0.234019\pi$
$410$	0	0
$411$	−2.00000	−0.0986527
$412$	−14.0000	−0.689730
$413$	20.0000	0.984136
$414$	4.00000	0.196589
$415$	0	0
$416$	−1.00000	−0.0490290
$417$	−20.0000	−0.979404
$418$	−15.0000	−0.733674
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	8.00000	0.389434
$423$	7.00000	0.340352
$424$	14.0000	0.679900
$425$	0	0
$426$	−3.00000	−0.145350
$427$	14.0000	0.677507
$428$	12.0000	0.580042
$429$	3.00000	0.144841
$430$	0	0
$431$	32.0000	1.54139	0.770693	−	0.637207i	$-0.219910\pi$
$431$	32.0000	1.54139	0.770693	+	0.637207i	$0.219910\pi$
$432$	−1.00000	−0.0481125
$433$	−24.0000	−1.15337	−0.576683	−	0.816968i	$-0.695653\pi$
$433$	−24.0000	−1.15337	−0.576683	+	0.816968i	$0.695653\pi$
$434$	−2.00000	−0.0960031
$435$	0	0
$436$	10.0000	0.478913
$437$	20.0000	0.956730
$438$	6.00000	0.286691
$439$	−20.0000	−0.954548	−0.477274	−	0.878755i	$-0.658375\pi$
$439$	−20.0000	−0.954548	−0.477274	+	0.878755i	$0.658375\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	3.00000	0.142695
$443$	6.00000	0.285069	0.142534	−	0.989790i	$-0.454475\pi$
$443$	6.00000	0.285069	0.142534	+	0.989790i	$0.454475\pi$
$444$	−2.00000	−0.0949158
$445$	0	0
$446$	9.00000	0.426162
$447$	−5.00000	−0.236492
$448$	2.00000	0.0944911
$449$	−15.0000	−0.707894	−0.353947	−	0.935266i	$-0.615161\pi$
$449$	−15.0000	−0.707894	−0.353947	+	0.935266i	$0.615161\pi$
$450$	0	0
$451$	−6.00000	−0.282529
$452$	−14.0000	−0.658505
$453$	3.00000	0.140952
$454$	−22.0000	−1.03251
$455$	0	0
$456$	−5.00000	−0.234146
$457$	22.0000	1.02912	0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000	1.02912	0.514558	+	0.857455i	$0.327956\pi$
$458$	5.00000	0.233635
$459$	3.00000	0.140028
$460$	0	0
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	−6.00000	−0.279145
$463$	−19.0000	−0.883005	−0.441502	−	0.897260i	$-0.645554\pi$
$463$	−19.0000	−0.883005	−0.441502	+	0.897260i	$0.645554\pi$
$464$	0	0
$465$	0	0
$466$	4.00000	0.185296
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	1.00000	0.0462250
$469$	14.0000	0.646460
$470$	0	0
$471$	8.00000	0.368621
$472$	−10.0000	−0.460287
$473$	−18.0000	−0.827641
$474$	15.0000	0.688973
$475$	0	0
$476$	−6.00000	−0.275010
$477$	−14.0000	−0.641016
$478$	−20.0000	−0.914779
$479$	35.0000	1.59919	0.799595	−	0.600539i	$-0.205047\pi$
$479$	35.0000	1.59919	0.799595	+	0.600539i	$0.205047\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	−2.00000	−0.0910975
$483$	8.00000	0.364013
$484$	−2.00000	−0.0909091
$485$	0	0
$486$	1.00000	0.0453609
$487$	17.0000	0.770344	0.385172	−	0.922845i	$-0.374142\pi$
$487$	17.0000	0.770344	0.385172	+	0.922845i	$0.374142\pi$
$488$	−7.00000	−0.316875
$489$	−21.0000	−0.949653
$490$	0	0
$491$	−28.0000	−1.26362	−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000	−1.26362	−0.631811	+	0.775122i	$0.717688\pi$
$492$	−2.00000	−0.0901670
$493$	0	0
$494$	5.00000	0.224961
$495$	0	0
$496$	1.00000	0.0449013
$497$	−6.00000	−0.269137
$498$	1.00000	0.0448111
$499$	40.0000	1.79065	0.895323	−	0.445418i	$-0.146945\pi$
$499$	40.0000	1.79065	0.895323	+	0.445418i	$0.146945\pi$
$500$	0	0
$501$	−2.00000	−0.0893534
$502$	−12.0000	−0.535586
$503$	−9.00000	−0.401290	−0.200645	−	0.979664i	$-0.564304\pi$
$503$	−9.00000	−0.401290	−0.200645	+	0.979664i	$0.564304\pi$
$504$	−2.00000	−0.0890871
$505$	0	0
$506$	−12.0000	−0.533465
$507$	12.0000	0.532939
$508$	12.0000	0.532414
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	12.0000	0.530849
$512$	−1.00000	−0.0441942
$513$	5.00000	0.220755
$514$	28.0000	1.23503
$515$	0	0
$516$	−6.00000	−0.264135
$517$	−21.0000	−0.923579
$518$	−4.00000	−0.175750
$519$	−21.0000	−0.921798
$520$	0	0
$521$	12.0000	0.525730	0.262865	−	0.964833i	$-0.415333\pi$
$521$	12.0000	0.525730	0.262865	+	0.964833i	$0.415333\pi$
$522$	0	0
$523$	26.0000	1.13690	0.568450	−	0.822718i	$-0.307543\pi$
$523$	26.0000	1.13690	0.568450	+	0.822718i	$0.307543\pi$
$524$	−18.0000	−0.786334
$525$	0	0
$526$	24.0000	1.04645
$527$	−3.00000	−0.130682
$528$	3.00000	0.130558
$529$	−7.00000	−0.304348
$530$	0	0
$531$	10.0000	0.433963
$532$	−10.0000	−0.433555
$533$	2.00000	0.0866296
$534$	10.0000	0.432742
$535$	0	0
$536$	−7.00000	−0.302354
$537$	5.00000	0.215766
$538$	−10.0000	−0.431131
$539$	9.00000	0.387657
$540$	0	0
$541$	−8.00000	−0.343947	−0.171973	−	0.985102i	$-0.555014\pi$
$541$	−8.00000	−0.343947	−0.171973	+	0.985102i	$0.555014\pi$
$542$	−7.00000	−0.300676
$543$	18.0000	0.772454
$544$	3.00000	0.128624
$545$	0	0
$546$	2.00000	0.0855921
$547$	−8.00000	−0.342055	−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000	−0.342055	−0.171028	+	0.985266i	$0.554709\pi$
$548$	2.00000	0.0854358
$549$	7.00000	0.298753
$550$	0	0
$551$	0	0
$552$	−4.00000	−0.170251
$553$	30.0000	1.27573
$554$	−17.0000	−0.722261
$555$	0	0
$556$	20.0000	0.848189
$557$	32.0000	1.35588	0.677942	−	0.735116i	$-0.262872\pi$
$557$	32.0000	1.35588	0.677942	+	0.735116i	$0.262872\pi$
$558$	−1.00000	−0.0423334
$559$	6.00000	0.253773
$560$	0	0
$561$	−9.00000	−0.379980
$562$	18.0000	0.759284
$563$	−24.0000	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$564$	−7.00000	−0.294753
$565$	0	0
$566$	9.00000	0.378298
$567$	2.00000	0.0839921
$568$	3.00000	0.125877
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	0	0
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	−3.00000	−0.125436
$573$	8.00000	0.334205
$574$	−4.00000	−0.166957
$575$	0	0
$576$	1.00000	0.0416667
$577$	−43.0000	−1.79011	−0.895057	−	0.445952i	$-0.852865\pi$
$577$	−43.0000	−1.79011	−0.895057	+	0.445952i	$0.852865\pi$
$578$	8.00000	0.332756
$579$	−21.0000	−0.872730
$580$	0	0
$581$	2.00000	0.0829740
$582$	−13.0000	−0.538867
$583$	42.0000	1.73946
$584$	−6.00000	−0.248282
$585$	0	0
$586$	14.0000	0.578335
$587$	17.0000	0.701665	0.350833	−	0.936438i	$-0.385899\pi$
$587$	17.0000	0.701665	0.350833	+	0.936438i	$0.385899\pi$
$588$	3.00000	0.123718
$589$	−5.00000	−0.206021
$590$	0	0
$591$	−22.0000	−0.904959
$592$	2.00000	0.0821995
$593$	−14.0000	−0.574911	−0.287456	−	0.957794i	$-0.592809\pi$
$593$	−14.0000	−0.574911	−0.287456	+	0.957794i	$0.592809\pi$
$594$	−3.00000	−0.123091
$595$	0	0
$596$	5.00000	0.204808
$597$	−15.0000	−0.613909
$598$	4.00000	0.163572
$599$	15.0000	0.612883	0.306442	−	0.951889i	$-0.400862\pi$
$599$	15.0000	0.612883	0.306442	+	0.951889i	$0.400862\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	−12.0000	−0.489083
$603$	7.00000	0.285062
$604$	−3.00000	−0.122068
$605$	0	0
$606$	2.00000	0.0812444
$607$	22.0000	0.892952	0.446476	−	0.894795i	$-0.352679\pi$
$607$	22.0000	0.892952	0.446476	+	0.894795i	$0.352679\pi$
$608$	5.00000	0.202777
$609$	0	0
$610$	0	0
$611$	7.00000	0.283190
$612$	−3.00000	−0.121268
$613$	−19.0000	−0.767403	−0.383701	−	0.923457i	$-0.625351\pi$
$613$	−19.0000	−0.767403	−0.383701	+	0.923457i	$0.625351\pi$
$614$	8.00000	0.322854
$615$	0	0
$616$	6.00000	0.241747
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	−14.0000	−0.563163
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	−27.0000	−1.08260
$623$	20.0000	0.801283
$624$	−1.00000	−0.0400320
$625$	0	0
$626$	−26.0000	−1.03917
$627$	−15.0000	−0.599042
$628$	−8.00000	−0.319235
$629$	−6.00000	−0.239236
$630$	0	0
$631$	32.0000	1.27390	0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000	1.27390	0.636950	+	0.770905i	$0.280196\pi$
$632$	−15.0000	−0.596668
$633$	8.00000	0.317971
$634$	−17.0000	−0.675156
$635$	0	0
$636$	14.0000	0.555136
$637$	−3.00000	−0.118864
$638$	0	0
$639$	−3.00000	−0.118678
$640$	0	0
$641$	27.0000	1.06644	0.533218	−	0.845978i	$-0.320983\pi$
$641$	27.0000	1.06644	0.533218	+	0.845978i	$0.320983\pi$
$642$	12.0000	0.473602
$643$	−4.00000	−0.157745	−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000	−0.157745	−0.0788723	+	0.996885i	$0.525132\pi$
$644$	−8.00000	−0.315244
$645$	0	0
$646$	−15.0000	−0.590167
$647$	−38.0000	−1.49393	−0.746967	−	0.664861i	$-0.768491\pi$
$647$	−38.0000	−1.49393	−0.746967	+	0.664861i	$0.768491\pi$
$648$	−1.00000	−0.0392837
$649$	−30.0000	−1.17760
$650$	0	0
$651$	−2.00000	−0.0783862
$652$	21.0000	0.822423
$653$	31.0000	1.21312	0.606562	−	0.795036i	$-0.292548\pi$
$653$	31.0000	1.21312	0.606562	+	0.795036i	$0.292548\pi$
$654$	10.0000	0.391031
$655$	0	0
$656$	2.00000	0.0780869
$657$	6.00000	0.234082
$658$	−14.0000	−0.545777
$659$	30.0000	1.16863	0.584317	−	0.811525i	$-0.301362\pi$
$659$	30.0000	1.16863	0.584317	+	0.811525i	$0.301362\pi$
$660$	0	0
$661$	32.0000	1.24466	0.622328	−	0.782757i	$-0.286187\pi$
$661$	32.0000	1.24466	0.622328	+	0.782757i	$0.286187\pi$
$662$	−32.0000	−1.24372
$663$	3.00000	0.116510
$664$	−1.00000	−0.0388075
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	0	0
$668$	2.00000	0.0773823
$669$	9.00000	0.347960
$670$	0	0
$671$	−21.0000	−0.810696
$672$	2.00000	0.0771517
$673$	−24.0000	−0.925132	−0.462566	−	0.886585i	$-0.653071\pi$
$673$	−24.0000	−0.925132	−0.462566	+	0.886585i	$0.653071\pi$
$674$	−32.0000	−1.23259
$675$	0	0
$676$	−12.0000	−0.461538
$677$	−48.0000	−1.84479	−0.922395	−	0.386248i	$-0.873771\pi$
$677$	−48.0000	−1.84479	−0.922395	+	0.386248i	$0.873771\pi$
$678$	−14.0000	−0.537667
$679$	−26.0000	−0.997788
$680$	0	0
$681$	−22.0000	−0.843042
$682$	3.00000	0.114876
$683$	−24.0000	−0.918334	−0.459167	−	0.888350i	$-0.651852\pi$
$683$	−24.0000	−0.918334	−0.459167	+	0.888350i	$0.651852\pi$
$684$	−5.00000	−0.191180
$685$	0	0
$686$	20.0000	0.763604
$687$	5.00000	0.190762
$688$	6.00000	0.228748
$689$	−14.0000	−0.533358
$690$	0	0
$691$	−23.0000	−0.874961	−0.437481	−	0.899228i	$-0.644129\pi$
$691$	−23.0000	−0.874961	−0.437481	+	0.899228i	$0.644129\pi$
$692$	21.0000	0.798300
$693$	−6.00000	−0.227921
$694$	−7.00000	−0.265716
$695$	0	0
$696$	0	0
$697$	−6.00000	−0.227266
$698$	0	0
$699$	4.00000	0.151294
$700$	0	0
$701$	−23.0000	−0.868698	−0.434349	−	0.900745i	$-0.643022\pi$
$701$	−23.0000	−0.868698	−0.434349	+	0.900745i	$0.643022\pi$
$702$	1.00000	0.0377426
$703$	−10.0000	−0.377157
$704$	−3.00000	−0.113067
$705$	0	0
$706$	9.00000	0.338719
$707$	4.00000	0.150435
$708$	−10.0000	−0.375823
$709$	−35.0000	−1.31445	−0.657226	−	0.753693i	$-0.728270\pi$
$709$	−35.0000	−1.31445	−0.657226	+	0.753693i	$0.728270\pi$
$710$	0	0
$711$	15.0000	0.562544
$712$	−10.0000	−0.374766
$713$	−4.00000	−0.149801
$714$	−6.00000	−0.224544
$715$	0	0
$716$	−5.00000	−0.186859
$717$	−20.0000	−0.746914
$718$	15.0000	0.559795
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	−28.0000	−1.04277
$722$	−6.00000	−0.223297
$723$	−2.00000	−0.0743808
$724$	−18.0000	−0.668965
$725$	0	0
$726$	−2.00000	−0.0742270
$727$	32.0000	1.18681	0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000	1.18681	0.593407	+	0.804902i	$0.297782\pi$
$728$	−2.00000	−0.0741249
$729$	1.00000	0.0370370
$730$	0	0
$731$	−18.0000	−0.665754
$732$	−7.00000	−0.258727
$733$	46.0000	1.69905	0.849524	−	0.527549i	$-0.176889\pi$
$733$	46.0000	1.69905	0.849524	+	0.527549i	$0.176889\pi$
$734$	−17.0000	−0.627481
$735$	0	0
$736$	4.00000	0.147442
$737$	−21.0000	−0.773545
$738$	−2.00000	−0.0736210
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	5.00000	0.183680
$742$	28.0000	1.02791
$743$	16.0000	0.586983	0.293492	−	0.955962i	$-0.405183\pi$
$743$	16.0000	0.586983	0.293492	+	0.955962i	$0.405183\pi$
$744$	1.00000	0.0366618
$745$	0	0
$746$	−16.0000	−0.585802
$747$	1.00000	0.0365881
$748$	9.00000	0.329073
$749$	24.0000	0.876941
$750$	0	0
$751$	22.0000	0.802791	0.401396	−	0.915905i	$-0.368525\pi$
$751$	22.0000	0.802791	0.401396	+	0.915905i	$0.368525\pi$
$752$	7.00000	0.255264
$753$	−12.0000	−0.437304
$754$	0	0
$755$	0	0
$756$	−2.00000	−0.0727393
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	35.0000	1.27126
$759$	−12.0000	−0.435572
$760$	0	0
$761$	−3.00000	−0.108750	−0.0543750	−	0.998521i	$-0.517317\pi$
$761$	−3.00000	−0.108750	−0.0543750	+	0.998521i	$0.517317\pi$
$762$	12.0000	0.434714
$763$	20.0000	0.724049
$764$	−8.00000	−0.289430
$765$	0	0
$766$	−26.0000	−0.939418
$767$	10.0000	0.361079
$768$	−1.00000	−0.0360844
$769$	30.0000	1.08183	0.540914	−	0.841078i	$-0.318079\pi$
$769$	30.0000	1.08183	0.540914	+	0.841078i	$0.318079\pi$
$770$	0	0
$771$	28.0000	1.00840
$772$	21.0000	0.755807
$773$	36.0000	1.29483	0.647415	−	0.762138i	$-0.275850\pi$
$773$	36.0000	1.29483	0.647415	+	0.762138i	$0.275850\pi$
$774$	−6.00000	−0.215666
$775$	0	0
$776$	13.0000	0.466673
$777$	−4.00000	−0.143499
$778$	−30.0000	−1.07555
$779$	−10.0000	−0.358287
$780$	0	0
$781$	9.00000	0.322045
$782$	−12.0000	−0.429119
$783$	0	0
$784$	−3.00000	−0.107143
$785$	0	0
$786$	−18.0000	−0.642039
$787$	−8.00000	−0.285169	−0.142585	−	0.989783i	$-0.545541\pi$
$787$	−8.00000	−0.285169	−0.142585	+	0.989783i	$0.545541\pi$
$788$	22.0000	0.783718
$789$	24.0000	0.854423
$790$	0	0
$791$	−28.0000	−0.995565
$792$	3.00000	0.106600
$793$	7.00000	0.248577
$794$	−22.0000	−0.780751
$795$	0	0
$796$	15.0000	0.531661
$797$	12.0000	0.425062	0.212531	−	0.977154i	$-0.431829\pi$
$797$	12.0000	0.425062	0.212531	+	0.977154i	$0.431829\pi$
$798$	−10.0000	−0.353996
$799$	−21.0000	−0.742927
$800$	0	0
$801$	10.0000	0.353333
$802$	3.00000	0.105934
$803$	−18.0000	−0.635206
$804$	−7.00000	−0.246871
$805$	0	0
$806$	−1.00000	−0.0352235
$807$	−10.0000	−0.352017
$808$	−2.00000	−0.0703598
$809$	15.0000	0.527372	0.263686	−	0.964609i	$-0.415062\pi$
$809$	15.0000	0.527372	0.263686	+	0.964609i	$0.415062\pi$
$810$	0	0
$811$	52.0000	1.82597	0.912983	−	0.407997i	$-0.133772\pi$
$811$	52.0000	1.82597	0.912983	+	0.407997i	$0.133772\pi$
$812$	0	0
$813$	−7.00000	−0.245501
$814$	6.00000	0.210300
$815$	0	0
$816$	3.00000	0.105021
$817$	−30.0000	−1.04957
$818$	−30.0000	−1.04893
$819$	2.00000	0.0698857
$820$	0	0
$821$	−38.0000	−1.32621	−0.663105	−	0.748527i	$-0.730762\pi$
$821$	−38.0000	−1.32621	−0.663105	+	0.748527i	$0.730762\pi$
$822$	2.00000	0.0697580
$823$	−9.00000	−0.313720	−0.156860	−	0.987621i	$-0.550137\pi$
$823$	−9.00000	−0.313720	−0.156860	+	0.987621i	$0.550137\pi$
$824$	14.0000	0.487713
$825$	0	0
$826$	−20.0000	−0.695889
$827$	−43.0000	−1.49526	−0.747628	−	0.664117i	$-0.768807\pi$
$827$	−43.0000	−1.49526	−0.747628	+	0.664117i	$0.768807\pi$
$828$	−4.00000	−0.139010
$829$	−10.0000	−0.347314	−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000	−0.347314	−0.173657	+	0.984806i	$0.555558\pi$
$830$	0	0
$831$	−17.0000	−0.589723
$832$	1.00000	0.0346688
$833$	9.00000	0.311832
$834$	20.0000	0.692543
$835$	0	0
$836$	15.0000	0.518786
$837$	−1.00000	−0.0345651
$838$	0	0
$839$	20.0000	0.690477	0.345238	−	0.938515i	$-0.387798\pi$
$839$	20.0000	0.690477	0.345238	+	0.938515i	$0.387798\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	−22.0000	−0.758170
$843$	18.0000	0.619953
$844$	−8.00000	−0.275371
$845$	0	0
$846$	−7.00000	−0.240665
$847$	−4.00000	−0.137442
$848$	−14.0000	−0.480762
$849$	9.00000	0.308879
$850$	0	0
$851$	−8.00000	−0.274236
$852$	3.00000	0.102778
$853$	−54.0000	−1.84892	−0.924462	−	0.381273i	$-0.875486\pi$
$853$	−54.0000	−1.84892	−0.924462	+	0.381273i	$0.875486\pi$
$854$	−14.0000	−0.479070
$855$	0	0
$856$	−12.0000	−0.410152
$857$	−28.0000	−0.956462	−0.478231	−	0.878234i	$-0.658722\pi$
$857$	−28.0000	−0.956462	−0.478231	+	0.878234i	$0.658722\pi$
$858$	−3.00000	−0.102418
$859$	10.0000	0.341196	0.170598	−	0.985341i	$-0.445430\pi$
$859$	10.0000	0.341196	0.170598	+	0.985341i	$0.445430\pi$
$860$	0	0
$861$	−4.00000	−0.136320
$862$	−32.0000	−1.08992
$863$	6.00000	0.204242	0.102121	−	0.994772i	$-0.467437\pi$
$863$	6.00000	0.204242	0.102121	+	0.994772i	$0.467437\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	24.0000	0.815553
$867$	8.00000	0.271694
$868$	2.00000	0.0678844
$869$	−45.0000	−1.52652
$870$	0	0
$871$	7.00000	0.237186
$872$	−10.0000	−0.338643
$873$	−13.0000	−0.439983
$874$	−20.0000	−0.676510
$875$	0	0
$876$	−6.00000	−0.202721
$877$	−8.00000	−0.270141	−0.135070	−	0.990836i	$-0.543126\pi$
$877$	−8.00000	−0.270141	−0.135070	+	0.990836i	$0.543126\pi$
$878$	20.0000	0.674967
$879$	14.0000	0.472208
$880$	0	0
$881$	7.00000	0.235836	0.117918	−	0.993023i	$-0.462378\pi$
$881$	7.00000	0.235836	0.117918	+	0.993023i	$0.462378\pi$
$882$	3.00000	0.101015
$883$	26.0000	0.874970	0.437485	−	0.899226i	$-0.355869\pi$
$883$	26.0000	0.874970	0.437485	+	0.899226i	$0.355869\pi$
$884$	−3.00000	−0.100901
$885$	0	0
$886$	−6.00000	−0.201574
$887$	−8.00000	−0.268614	−0.134307	−	0.990940i	$-0.542881\pi$
$887$	−8.00000	−0.268614	−0.134307	+	0.990940i	$0.542881\pi$
$888$	2.00000	0.0671156
$889$	24.0000	0.804934
$890$	0	0
$891$	−3.00000	−0.100504
$892$	−9.00000	−0.301342
$893$	−35.0000	−1.17123
$894$	5.00000	0.167225
$895$	0	0
$896$	−2.00000	−0.0668153
$897$	4.00000	0.133556
$898$	15.0000	0.500556
$899$	0	0
$900$	0	0
$901$	42.0000	1.39922
$902$	6.00000	0.199778
$903$	−12.0000	−0.399335
$904$	14.0000	0.465633
$905$	0	0
$906$	−3.00000	−0.0996683
$907$	37.0000	1.22856	0.614282	−	0.789086i	$-0.289446\pi$
$907$	37.0000	1.22856	0.614282	+	0.789086i	$0.289446\pi$
$908$	22.0000	0.730096
$909$	2.00000	0.0663358
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	5.00000	0.165567
$913$	−3.00000	−0.0992855
$914$	−22.0000	−0.727695
$915$	0	0
$916$	−5.00000	−0.165205
$917$	−36.0000	−1.18882
$918$	−3.00000	−0.0990148
$919$	−20.0000	−0.659739	−0.329870	−	0.944027i	$-0.607005\pi$
$919$	−20.0000	−0.659739	−0.329870	+	0.944027i	$0.607005\pi$
$920$	0	0
$921$	8.00000	0.263609
$922$	18.0000	0.592798
$923$	−3.00000	−0.0987462
$924$	6.00000	0.197386
$925$	0	0
$926$	19.0000	0.624379
$927$	−14.0000	−0.459820
$928$	0	0
$929$	−10.0000	−0.328089	−0.164045	−	0.986453i	$-0.552454\pi$
$929$	−10.0000	−0.328089	−0.164045	+	0.986453i	$0.552454\pi$
$930$	0	0
$931$	15.0000	0.491605
$932$	−4.00000	−0.131024
$933$	−27.0000	−0.883940
$934$	−12.0000	−0.392652
$935$	0	0
$936$	−1.00000	−0.0326860
$937$	7.00000	0.228680	0.114340	−	0.993442i	$-0.463525\pi$
$937$	7.00000	0.228680	0.114340	+	0.993442i	$0.463525\pi$
$938$	−14.0000	−0.457116
$939$	−26.0000	−0.848478
$940$	0	0
$941$	12.0000	0.391189	0.195594	−	0.980685i	$-0.437336\pi$
$941$	12.0000	0.391189	0.195594	+	0.980685i	$0.437336\pi$
$942$	−8.00000	−0.260654
$943$	−8.00000	−0.260516
$944$	10.0000	0.325472
$945$	0	0
$946$	18.0000	0.585230
$947$	−3.00000	−0.0974869	−0.0487435	−	0.998811i	$-0.515522\pi$
$947$	−3.00000	−0.0974869	−0.0487435	+	0.998811i	$0.515522\pi$
$948$	−15.0000	−0.487177
$949$	6.00000	0.194768
$950$	0	0
$951$	−17.0000	−0.551263
$952$	6.00000	0.194461
$953$	−39.0000	−1.26333	−0.631667	−	0.775240i	$-0.717629\pi$
$953$	−39.0000	−1.26333	−0.631667	+	0.775240i	$0.717629\pi$
$954$	14.0000	0.453267
$955$	0	0
$956$	20.0000	0.646846
$957$	0	0
$958$	−35.0000	−1.13080
$959$	4.00000	0.129167
$960$	0	0
$961$	1.00000	0.0322581
$962$	−2.00000	−0.0644826
$963$	12.0000	0.386695
$964$	2.00000	0.0644157
$965$	0	0
$966$	−8.00000	−0.257396
$967$	−43.0000	−1.38279	−0.691393	−	0.722478i	$-0.743003\pi$
$967$	−43.0000	−1.38279	−0.691393	+	0.722478i	$0.743003\pi$
$968$	2.00000	0.0642824
$969$	−15.0000	−0.481869
$970$	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
$973$	40.0000	1.28234
$974$	−17.0000	−0.544715
$975$	0	0
$976$	7.00000	0.224065
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	21.0000	0.671506
$979$	−30.0000	−0.958804
$980$	0	0
$981$	10.0000	0.319275
$982$	28.0000	0.893516
$983$	36.0000	1.14822	0.574111	−	0.818778i	$-0.305348\pi$
$983$	36.0000	1.14822	0.574111	+	0.818778i	$0.305348\pi$
$984$	2.00000	0.0637577
$985$	0	0
$986$	0	0
$987$	−14.0000	−0.445625
$988$	−5.00000	−0.159071
$989$	−24.0000	−0.763156
$990$	0	0
$991$	12.0000	0.381193	0.190596	−	0.981669i	$-0.438958\pi$
$991$	12.0000	0.381193	0.190596	+	0.981669i	$0.438958\pi$
$992$	−1.00000	−0.0317500
$993$	−32.0000	−1.01549
$994$	6.00000	0.190308
$995$	0	0
$996$	−1.00000	−0.0316862
$997$	−18.0000	−0.570066	−0.285033	−	0.958518i	$-0.592005\pi$
$997$	−18.0000	−0.570066	−0.285033	+	0.958518i	$0.592005\pi$
$998$	−40.0000	−1.26618
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4650.2.a.i.1.1		1
5.2	odd	4		4650.2.d.r.3349.1		2
5.3	odd	4		4650.2.d.r.3349.2		2
5.4	even	2		186.2.a.c.1.1	✓	1
15.14	odd	2		558.2.a.a.1.1		1
20.19	odd	2		1488.2.a.f.1.1		1
35.34	odd	2		9114.2.a.t.1.1		1
40.19	odd	2		5952.2.a.v.1.1		1
40.29	even	2		5952.2.a.d.1.1		1
60.59	even	2		4464.2.a.h.1.1		1
155.154	odd	2		5766.2.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
186.2.a.c.1.1	✓	1	5.4	even	2
558.2.a.a.1.1		1	15.14	odd	2
1488.2.a.f.1.1		1	20.19	odd	2
4464.2.a.h.1.1		1	60.59	even	2
4650.2.a.i.1.1		1	1.1	even	1	trivial
4650.2.d.r.3349.1		2	5.2	odd	4
4650.2.d.r.3349.2		2	5.3	odd	4
5766.2.a.h.1.1		1	155.154	odd	2
5952.2.a.d.1.1		1	40.29	even	2
5952.2.a.v.1.1		1	40.19	odd	2
9114.2.a.t.1.1		1	35.34	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4650.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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