LMFDB - Embedded newform 1650.2.a.m.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)

$f(q)$

$=$

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

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{12} +4.00000 q^{13} -2.00000 q^{14} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} +2.00000 q^{21} -1.00000 q^{22} -6.00000 q^{23} -1.00000 q^{24} +4.00000 q^{26} -1.00000 q^{27} -2.00000 q^{28} +6.00000 q^{29} +8.00000 q^{31} +1.00000 q^{32} +1.00000 q^{33} +6.00000 q^{34} +1.00000 q^{36} +10.0000 q^{37} -4.00000 q^{38} -4.00000 q^{39} +6.00000 q^{41} +2.00000 q^{42} -8.00000 q^{43} -1.00000 q^{44} -6.00000 q^{46} +6.00000 q^{47} -1.00000 q^{48} -3.00000 q^{49} -6.00000 q^{51} +4.00000 q^{52} -1.00000 q^{54} -2.00000 q^{56} +4.00000 q^{57} +6.00000 q^{58} +8.00000 q^{61} +8.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} +1.00000 q^{66} +4.00000 q^{67} +6.00000 q^{68} +6.00000 q^{69} +6.00000 q^{71} +1.00000 q^{72} -2.00000 q^{73} +10.0000 q^{74} -4.00000 q^{76} +2.00000 q^{77} -4.00000 q^{78} +14.0000 q^{79} +1.00000 q^{81} +6.00000 q^{82} +12.0000 q^{83} +2.00000 q^{84} -8.00000 q^{86} -6.00000 q^{87} -1.00000 q^{88} -6.00000 q^{89} -8.00000 q^{91} -6.00000 q^{92} -8.00000 q^{93} +6.00000 q^{94} -1.00000 q^{96} -14.0000 q^{97} -3.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	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
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	1.00000	0.235702
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	−1.00000	−0.213201
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	4.00000	0.784465
$27$	−1.00000	−0.192450
$28$	−2.00000	−0.377964
$29$	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
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	1.00000	0.176777
$33$	1.00000	0.174078
$34$	6.00000	1.02899
$35$	0	0
$36$	1.00000	0.166667
$37$	10.0000	1.64399	0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000	1.64399	0.821995	+	0.569495i	$0.192861\pi$
$38$	−4.00000	−0.648886
$39$	−4.00000	−0.640513
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	2.00000	0.308607
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−6.00000	−0.884652
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	−1.00000	−0.144338
$49$	−3.00000	−0.428571
$50$	0	0
$51$	−6.00000	−0.840168
$52$	4.00000	0.554700
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	−2.00000	−0.267261
$57$	4.00000	0.529813
$58$	6.00000	0.787839
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	8.00000	1.01600
$63$	−2.00000	−0.251976
$64$	1.00000	0.125000
$65$	0	0
$66$	1.00000	0.123091
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	6.00000	0.727607
$69$	6.00000	0.722315
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	1.00000	0.117851
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	10.0000	1.16248
$75$	0	0
$76$	−4.00000	−0.458831
$77$	2.00000	0.227921
$78$	−4.00000	−0.452911
$79$	14.0000	1.57512	0.787562	−	0.616236i	$-0.211343\pi$
$79$	14.0000	1.57512	0.787562	+	0.616236i	$0.211343\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	2.00000	0.218218
$85$	0	0
$86$	−8.00000	−0.862662
$87$	−6.00000	−0.643268
$88$	−1.00000	−0.106600
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	−8.00000	−0.838628
$92$	−6.00000	−0.625543
$93$	−8.00000	−0.829561
$94$	6.00000	0.618853
$95$	0	0
$96$	−1.00000	−0.102062
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	−3.00000	−0.303046
$99$	−1.00000	−0.100504
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	−6.00000	−0.594089
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	4.00000	0.392232
$105$	0	0
$106$	0	0
$107$	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
$109$	−4.00000	−0.383131	−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000	−0.383131	−0.191565	+	0.981480i	$0.561356\pi$
$110$	0	0
$111$	−10.0000	−0.949158
$112$	−2.00000	−0.188982
$113$	−18.0000	−1.69330	−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000	−1.69330	−0.846649	+	0.532152i	$0.821383\pi$
$114$	4.00000	0.374634
$115$	0	0
$116$	6.00000	0.557086
$117$	4.00000	0.369800
$118$	0	0
$119$	−12.0000	−1.10004
$120$	0	0
$121$	1.00000	0.0909091
$122$	8.00000	0.724286
$123$	−6.00000	−0.541002
$124$	8.00000	0.718421
$125$	0	0
$126$	−2.00000	−0.178174
$127$	−14.0000	−1.24230	−0.621150	−	0.783692i	$-0.713334\pi$
$127$	−14.0000	−1.24230	−0.621150	+	0.783692i	$0.713334\pi$
$128$	1.00000	0.0883883
$129$	8.00000	0.704361
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	1.00000	0.0870388
$133$	8.00000	0.693688
$134$	4.00000	0.345547
$135$	0	0
$136$	6.00000	0.514496
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	6.00000	0.510754
$139$	−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
$141$	−6.00000	−0.505291
$142$	6.00000	0.503509
$143$	−4.00000	−0.334497
$144$	1.00000	0.0833333
$145$	0	0
$146$	−2.00000	−0.165521
$147$	3.00000	0.247436
$148$	10.0000	0.821995
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	−4.00000	−0.324443
$153$	6.00000	0.485071
$154$	2.00000	0.161165
$155$	0	0
$156$	−4.00000	−0.320256
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	14.0000	1.11378
$159$	0	0
$160$	0	0
$161$	12.0000	0.945732
$162$	1.00000	0.0785674
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	12.0000	0.931381
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	2.00000	0.154303
$169$	3.00000	0.230769
$170$	0	0
$171$	−4.00000	−0.305888
$172$	−8.00000	−0.609994
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	−6.00000	−0.454859
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	−6.00000	−0.449719
$179$	24.0000	1.79384	0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000	1.79384	0.896922	+	0.442189i	$0.145798\pi$
$180$	0	0
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	−8.00000	−0.592999
$183$	−8.00000	−0.591377
$184$	−6.00000	−0.442326
$185$	0	0
$186$	−8.00000	−0.586588
$187$	−6.00000	−0.438763
$188$	6.00000	0.437595
$189$	2.00000	0.145479
$190$	0	0
$191$	18.0000	1.30243	0.651217	−	0.758891i	$-0.274259\pi$
$191$	18.0000	1.30243	0.651217	+	0.758891i	$0.274259\pi$
$192$	−1.00000	−0.0721688
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	−14.0000	−1.00514
$195$	0	0
$196$	−3.00000	−0.214286
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	−1.00000	−0.0710669
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	6.00000	0.422159
$203$	−12.0000	−0.842235
$204$	−6.00000	−0.420084
$205$	0	0
$206$	4.00000	0.278693
$207$	−6.00000	−0.417029
$208$	4.00000	0.277350
$209$	4.00000	0.276686
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	0	0
$213$	−6.00000	−0.411113
$214$	12.0000	0.820303
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	−16.0000	−1.08615
$218$	−4.00000	−0.270914
$219$	2.00000	0.135147
$220$	0	0
$221$	24.0000	1.61441
$222$	−10.0000	−0.671156
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	−2.00000	−0.133631
$225$	0	0
$226$	−18.0000	−1.19734
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	4.00000	0.264906
$229$	−22.0000	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	0	0
$231$	−2.00000	−0.131590
$232$	6.00000	0.393919
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	4.00000	0.261488
$235$	0	0
$236$	0	0
$237$	−14.0000	−0.909398
$238$	−12.0000	−0.777844
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	1.00000	0.0642824
$243$	−1.00000	−0.0641500
$244$	8.00000	0.512148
$245$	0	0
$246$	−6.00000	−0.382546
$247$	−16.0000	−1.01806
$248$	8.00000	0.508001
$249$	−12.0000	−0.760469
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	−2.00000	−0.125988
$253$	6.00000	0.377217
$254$	−14.0000	−0.878438
$255$	0	0
$256$	1.00000	0.0625000
$257$	30.0000	1.87135	0.935674	−	0.352865i	$-0.114792\pi$
$257$	30.0000	1.87135	0.935674	+	0.352865i	$0.114792\pi$
$258$	8.00000	0.498058
$259$	−20.0000	−1.24274
$260$	0	0
$261$	6.00000	0.371391
$262$	−12.0000	−0.741362
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	1.00000	0.0615457
$265$	0	0
$266$	8.00000	0.490511
$267$	6.00000	0.367194
$268$	4.00000	0.244339
$269$	−24.0000	−1.46331	−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000	−1.46331	−0.731653	+	0.681677i	$0.761251\pi$
$270$	0	0
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	6.00000	0.363803
$273$	8.00000	0.484182
$274$	18.0000	1.08742
$275$	0	0
$276$	6.00000	0.361158
$277$	16.0000	0.961347	0.480673	−	0.876900i	$-0.340392\pi$
$277$	16.0000	0.961347	0.480673	+	0.876900i	$0.340392\pi$
$278$	−4.00000	−0.239904
$279$	8.00000	0.478947
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	−6.00000	−0.357295
$283$	−8.00000	−0.475551	−0.237775	−	0.971320i	$-0.576418\pi$
$283$	−8.00000	−0.475551	−0.237775	+	0.971320i	$0.576418\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	−4.00000	−0.236525
$287$	−12.0000	−0.708338
$288$	1.00000	0.0589256
$289$	19.0000	1.11765
$290$	0	0
$291$	14.0000	0.820695
$292$	−2.00000	−0.117041
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	3.00000	0.174964
$295$	0	0
$296$	10.0000	0.581238
$297$	1.00000	0.0580259
$298$	−6.00000	−0.347571
$299$	−24.0000	−1.38796
$300$	0	0
$301$	16.0000	0.922225
$302$	−10.0000	−0.575435
$303$	−6.00000	−0.344691
$304$	−4.00000	−0.229416
$305$	0	0
$306$	6.00000	0.342997
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	2.00000	0.113961
$309$	−4.00000	−0.227552
$310$	0	0
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	−4.00000	−0.226455
$313$	−26.0000	−1.46961	−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−26.0000	−1.46961	−0.734803	+	0.678280i	$0.762726\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	14.0000	0.787562
$317$	−12.0000	−0.673987	−0.336994	−	0.941507i	$-0.609410\pi$
$317$	−12.0000	−0.673987	−0.336994	+	0.941507i	$0.609410\pi$
$318$	0	0
$319$	−6.00000	−0.335936
$320$	0	0
$321$	−12.0000	−0.669775
$322$	12.0000	0.668734
$323$	−24.0000	−1.33540
$324$	1.00000	0.0555556
$325$	0	0
$326$	4.00000	0.221540
$327$	4.00000	0.221201
$328$	6.00000	0.331295
$329$	−12.0000	−0.661581
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	12.0000	0.658586
$333$	10.0000	0.547997
$334$	−12.0000	−0.656611
$335$	0	0
$336$	2.00000	0.109109
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	3.00000	0.163178
$339$	18.0000	0.977626
$340$	0	0
$341$	−8.00000	−0.433224
$342$	−4.00000	−0.216295
$343$	20.0000	1.07990
$344$	−8.00000	−0.431331
$345$	0	0
$346$	−6.00000	−0.322562
$347$	−36.0000	−1.93258	−0.966291	−	0.257454i	$-0.917117\pi$
$347$	−36.0000	−1.93258	−0.966291	+	0.257454i	$0.917117\pi$
$348$	−6.00000	−0.321634
$349$	−4.00000	−0.214115	−0.107058	−	0.994253i	$-0.534143\pi$
$349$	−4.00000	−0.214115	−0.107058	+	0.994253i	$0.534143\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	−1.00000	−0.0533002
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	0	0
$355$	0	0
$356$	−6.00000	−0.317999
$357$	12.0000	0.635107
$358$	24.0000	1.26844
$359$	−12.0000	−0.633336	−0.316668	−	0.948536i	$-0.602564\pi$
$359$	−12.0000	−0.633336	−0.316668	+	0.948536i	$0.602564\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	−22.0000	−1.15629
$363$	−1.00000	−0.0524864
$364$	−8.00000	−0.419314
$365$	0	0
$366$	−8.00000	−0.418167
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	−6.00000	−0.312772
$369$	6.00000	0.312348
$370$	0	0
$371$	0	0
$372$	−8.00000	−0.414781
$373$	−20.0000	−1.03556	−0.517780	−	0.855514i	$-0.673242\pi$
$373$	−20.0000	−1.03556	−0.517780	+	0.855514i	$0.673242\pi$
$374$	−6.00000	−0.310253
$375$	0	0
$376$	6.00000	0.309426
$377$	24.0000	1.23606
$378$	2.00000	0.102869
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	0	0
$381$	14.0000	0.717242
$382$	18.0000	0.920960
$383$	−6.00000	−0.306586	−0.153293	−	0.988181i	$-0.548988\pi$
$383$	−6.00000	−0.306586	−0.153293	+	0.988181i	$0.548988\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−14.0000	−0.712581
$387$	−8.00000	−0.406663
$388$	−14.0000	−0.710742
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	−36.0000	−1.82060
$392$	−3.00000	−0.151523
$393$	12.0000	0.605320
$394$	−6.00000	−0.302276
$395$	0	0
$396$	−1.00000	−0.0502519
$397$	−26.0000	−1.30490	−0.652451	−	0.757831i	$-0.726259\pi$
$397$	−26.0000	−1.30490	−0.652451	+	0.757831i	$0.726259\pi$
$398$	−4.00000	−0.200502
$399$	−8.00000	−0.400501
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	−4.00000	−0.199502
$403$	32.0000	1.59403
$404$	6.00000	0.298511
$405$	0	0
$406$	−12.0000	−0.595550
$407$	−10.0000	−0.495682
$408$	−6.00000	−0.297044
$409$	−34.0000	−1.68119	−0.840596	−	0.541663i	$-0.817795\pi$
$409$	−34.0000	−1.68119	−0.840596	+	0.541663i	$0.817795\pi$
$410$	0	0
$411$	−18.0000	−0.887875
$412$	4.00000	0.197066
$413$	0	0
$414$	−6.00000	−0.294884
$415$	0	0
$416$	4.00000	0.196116
$417$	4.00000	0.195881
$418$	4.00000	0.195646
$419$	24.0000	1.17248	0.586238	−	0.810139i	$-0.300608\pi$
$419$	24.0000	1.17248	0.586238	+	0.810139i	$0.300608\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	8.00000	0.389434
$423$	6.00000	0.291730
$424$	0	0
$425$	0	0
$426$	−6.00000	−0.290701
$427$	−16.0000	−0.774294
$428$	12.0000	0.580042
$429$	4.00000	0.193122
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	−1.00000	−0.0481125
$433$	−26.0000	−1.24948	−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000	−1.24948	−0.624740	+	0.780833i	$0.714795\pi$
$434$	−16.0000	−0.768025
$435$	0	0
$436$	−4.00000	−0.191565
$437$	24.0000	1.14808
$438$	2.00000	0.0955637
$439$	−10.0000	−0.477274	−0.238637	−	0.971109i	$-0.576701\pi$
$439$	−10.0000	−0.477274	−0.238637	+	0.971109i	$0.576701\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	24.0000	1.14156
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	−10.0000	−0.474579
$445$	0	0
$446$	16.0000	0.757622
$447$	6.00000	0.283790
$448$	−2.00000	−0.0944911
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	−6.00000	−0.282529
$452$	−18.0000	−0.846649
$453$	10.0000	0.469841
$454$	12.0000	0.563188
$455$	0	0
$456$	4.00000	0.187317
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	−22.0000	−1.02799
$459$	−6.00000	−0.280056
$460$	0	0
$461$	42.0000	1.95614	0.978068	−	0.208288i	$-0.0667892\pi$
$461$	42.0000	1.95614	0.978068	+	0.208288i	$0.0667892\pi$
$462$	−2.00000	−0.0930484
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	18.0000	0.833834
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	4.00000	0.184900
$469$	−8.00000	−0.369406
$470$	0	0
$471$	2.00000	0.0921551
$472$	0	0
$473$	8.00000	0.367840
$474$	−14.0000	−0.643041
$475$	0	0
$476$	−12.0000	−0.550019
$477$	0	0
$478$	−12.0000	−0.548867
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	40.0000	1.82384
$482$	−10.0000	−0.455488
$483$	−12.0000	−0.546019
$484$	1.00000	0.0454545
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	−20.0000	−0.906287	−0.453143	−	0.891438i	$-0.649697\pi$
$487$	−20.0000	−0.906287	−0.453143	+	0.891438i	$0.649697\pi$
$488$	8.00000	0.362143
$489$	−4.00000	−0.180886
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	−6.00000	−0.270501
$493$	36.0000	1.62136
$494$	−16.0000	−0.719874
$495$	0	0
$496$	8.00000	0.359211
$497$	−12.0000	−0.538274
$498$	−12.0000	−0.537733
$499$	−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
$501$	12.0000	0.536120
$502$	0	0
$503$	−12.0000	−0.535054	−0.267527	−	0.963550i	$-0.586206\pi$
$503$	−12.0000	−0.535054	−0.267527	+	0.963550i	$0.586206\pi$
$504$	−2.00000	−0.0890871
$505$	0	0
$506$	6.00000	0.266733
$507$	−3.00000	−0.133235
$508$	−14.0000	−0.621150
$509$	24.0000	1.06378	0.531891	−	0.846813i	$-0.321482\pi$
$509$	24.0000	1.06378	0.531891	+	0.846813i	$0.321482\pi$
$510$	0	0
$511$	4.00000	0.176950
$512$	1.00000	0.0441942
$513$	4.00000	0.176604
$514$	30.0000	1.32324
$515$	0	0
$516$	8.00000	0.352180
$517$	−6.00000	−0.263880
$518$	−20.0000	−0.878750
$519$	6.00000	0.263371
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	6.00000	0.262613
$523$	16.0000	0.699631	0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000	0.699631	0.349816	+	0.936819i	$0.386244\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	0	0
$527$	48.0000	2.09091
$528$	1.00000	0.0435194
$529$	13.0000	0.565217
$530$	0	0
$531$	0	0
$532$	8.00000	0.346844
$533$	24.0000	1.03956
$534$	6.00000	0.259645
$535$	0	0
$536$	4.00000	0.172774
$537$	−24.0000	−1.03568
$538$	−24.0000	−1.03471
$539$	3.00000	0.129219
$540$	0	0
$541$	20.0000	0.859867	0.429934	−	0.902861i	$-0.358537\pi$
$541$	20.0000	0.859867	0.429934	+	0.902861i	$0.358537\pi$
$542$	2.00000	0.0859074
$543$	22.0000	0.944110
$544$	6.00000	0.257248
$545$	0	0
$546$	8.00000	0.342368
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	18.0000	0.768922
$549$	8.00000	0.341432
$550$	0	0
$551$	−24.0000	−1.02243
$552$	6.00000	0.255377
$553$	−28.0000	−1.19068
$554$	16.0000	0.679775
$555$	0	0
$556$	−4.00000	−0.169638
$557$	−18.0000	−0.762684	−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000	−0.762684	−0.381342	+	0.924434i	$0.624538\pi$
$558$	8.00000	0.338667
$559$	−32.0000	−1.35346
$560$	0	0
$561$	6.00000	0.253320
$562$	−6.00000	−0.253095
$563$	12.0000	0.505740	0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000	0.505740	0.252870	+	0.967500i	$0.418626\pi$
$564$	−6.00000	−0.252646
$565$	0	0
$566$	−8.00000	−0.336265
$567$	−2.00000	−0.0839921
$568$	6.00000	0.251754
$569$	18.0000	0.754599	0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000	0.754599	0.377300	+	0.926091i	$0.376853\pi$
$570$	0	0
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	−4.00000	−0.167248
$573$	−18.0000	−0.751961
$574$	−12.0000	−0.500870
$575$	0	0
$576$	1.00000	0.0416667
$577$	34.0000	1.41544	0.707719	−	0.706494i	$-0.249724\pi$
$577$	34.0000	1.41544	0.707719	+	0.706494i	$0.249724\pi$
$578$	19.0000	0.790296
$579$	14.0000	0.581820
$580$	0	0
$581$	−24.0000	−0.995688
$582$	14.0000	0.580319
$583$	0	0
$584$	−2.00000	−0.0827606
$585$	0	0
$586$	6.00000	0.247858
$587$	−24.0000	−0.990586	−0.495293	−	0.868726i	$-0.664939\pi$
$587$	−24.0000	−0.990586	−0.495293	+	0.868726i	$0.664939\pi$
$588$	3.00000	0.123718
$589$	−32.0000	−1.31854
$590$	0	0
$591$	6.00000	0.246807
$592$	10.0000	0.410997
$593$	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
$596$	−6.00000	−0.245770
$597$	4.00000	0.163709
$598$	−24.0000	−0.981433
$599$	30.0000	1.22577	0.612883	−	0.790173i	$-0.290010\pi$
$599$	30.0000	1.22577	0.612883	+	0.790173i	$0.290010\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	16.0000	0.652111
$603$	4.00000	0.162893
$604$	−10.0000	−0.406894
$605$	0	0
$606$	−6.00000	−0.243733
$607$	−14.0000	−0.568242	−0.284121	−	0.958788i	$-0.591702\pi$
$607$	−14.0000	−0.568242	−0.284121	+	0.958788i	$0.591702\pi$
$608$	−4.00000	−0.162221
$609$	12.0000	0.486265
$610$	0	0
$611$	24.0000	0.970936
$612$	6.00000	0.242536
$613$	16.0000	0.646234	0.323117	−	0.946359i	$-0.395269\pi$
$613$	16.0000	0.646234	0.323117	+	0.946359i	$0.395269\pi$
$614$	−20.0000	−0.807134
$615$	0	0
$616$	2.00000	0.0805823
$617$	30.0000	1.20775	0.603877	−	0.797077i	$-0.293622\pi$
$617$	30.0000	1.20775	0.603877	+	0.797077i	$0.293622\pi$
$618$	−4.00000	−0.160904
$619$	44.0000	1.76851	0.884255	−	0.467005i	$-0.154667\pi$
$619$	44.0000	1.76851	0.884255	+	0.467005i	$0.154667\pi$
$620$	0	0
$621$	6.00000	0.240772
$622$	−18.0000	−0.721734
$623$	12.0000	0.480770
$624$	−4.00000	−0.160128
$625$	0	0
$626$	−26.0000	−1.03917
$627$	−4.00000	−0.159745
$628$	−2.00000	−0.0798087
$629$	60.0000	2.39236
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	14.0000	0.556890
$633$	−8.00000	−0.317971
$634$	−12.0000	−0.476581
$635$	0	0
$636$	0	0
$637$	−12.0000	−0.475457
$638$	−6.00000	−0.237542
$639$	6.00000	0.237356
$640$	0	0
$641$	6.00000	0.236986	0.118493	−	0.992955i	$-0.462194\pi$
$641$	6.00000	0.236986	0.118493	+	0.992955i	$0.462194\pi$
$642$	−12.0000	−0.473602
$643$	4.00000	0.157745	0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000	0.157745	0.0788723	+	0.996885i	$0.474868\pi$
$644$	12.0000	0.472866
$645$	0	0
$646$	−24.0000	−0.944267
$647$	−6.00000	−0.235884	−0.117942	−	0.993020i	$-0.537630\pi$
$647$	−6.00000	−0.235884	−0.117942	+	0.993020i	$0.537630\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	0	0
$651$	16.0000	0.627089
$652$	4.00000	0.156652
$653$	36.0000	1.40879	0.704394	−	0.709809i	$-0.251219\pi$
$653$	36.0000	1.40879	0.704394	+	0.709809i	$0.251219\pi$
$654$	4.00000	0.156412
$655$	0	0
$656$	6.00000	0.234261
$657$	−2.00000	−0.0780274
$658$	−12.0000	−0.467809
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	0	0
$661$	−22.0000	−0.855701	−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000	−0.855701	−0.427850	+	0.903850i	$0.640729\pi$
$662$	−4.00000	−0.155464
$663$	−24.0000	−0.932083
$664$	12.0000	0.465690
$665$	0	0
$666$	10.0000	0.387492
$667$	−36.0000	−1.39393
$668$	−12.0000	−0.464294
$669$	−16.0000	−0.618596
$670$	0	0
$671$	−8.00000	−0.308837
$672$	2.00000	0.0771517
$673$	−14.0000	−0.539660	−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000	−0.539660	−0.269830	+	0.962908i	$0.586968\pi$
$674$	−2.00000	−0.0770371
$675$	0	0
$676$	3.00000	0.115385
$677$	−30.0000	−1.15299	−0.576497	−	0.817099i	$-0.695581\pi$
$677$	−30.0000	−1.15299	−0.576497	+	0.817099i	$0.695581\pi$
$678$	18.0000	0.691286
$679$	28.0000	1.07454
$680$	0	0
$681$	−12.0000	−0.459841
$682$	−8.00000	−0.306336
$683$	24.0000	0.918334	0.459167	−	0.888350i	$-0.348148\pi$
$683$	24.0000	0.918334	0.459167	+	0.888350i	$0.348148\pi$
$684$	−4.00000	−0.152944
$685$	0	0
$686$	20.0000	0.763604
$687$	22.0000	0.839352
$688$	−8.00000	−0.304997
$689$	0	0
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	−6.00000	−0.228086
$693$	2.00000	0.0759737
$694$	−36.0000	−1.36654
$695$	0	0
$696$	−6.00000	−0.227429
$697$	36.0000	1.36360
$698$	−4.00000	−0.151402
$699$	−18.0000	−0.680823
$700$	0	0
$701$	−6.00000	−0.226617	−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000	−0.226617	−0.113308	+	0.993560i	$0.536145\pi$
$702$	−4.00000	−0.150970
$703$	−40.0000	−1.50863
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	6.00000	0.225813
$707$	−12.0000	−0.451306
$708$	0	0
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	0	0
$711$	14.0000	0.525041
$712$	−6.00000	−0.224860
$713$	−48.0000	−1.79761
$714$	12.0000	0.449089
$715$	0	0
$716$	24.0000	0.896922
$717$	12.0000	0.448148
$718$	−12.0000	−0.447836
$719$	30.0000	1.11881	0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000	1.11881	0.559406	+	0.828894i	$0.311029\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	−3.00000	−0.111648
$723$	10.0000	0.371904
$724$	−22.0000	−0.817624
$725$	0	0
$726$	−1.00000	−0.0371135
$727$	28.0000	1.03846	0.519231	−	0.854634i	$-0.326218\pi$
$727$	28.0000	1.03846	0.519231	+	0.854634i	$0.326218\pi$
$728$	−8.00000	−0.296500
$729$	1.00000	0.0370370
$730$	0	0
$731$	−48.0000	−1.77534
$732$	−8.00000	−0.295689
$733$	4.00000	0.147743	0.0738717	−	0.997268i	$-0.476464\pi$
$733$	4.00000	0.147743	0.0738717	+	0.997268i	$0.476464\pi$
$734$	−8.00000	−0.295285
$735$	0	0
$736$	−6.00000	−0.221163
$737$	−4.00000	−0.147342
$738$	6.00000	0.220863
$739$	8.00000	0.294285	0.147142	−	0.989115i	$-0.452992\pi$
$739$	8.00000	0.294285	0.147142	+	0.989115i	$0.452992\pi$
$740$	0	0
$741$	16.0000	0.587775
$742$	0	0
$743$	−36.0000	−1.32071	−0.660356	−	0.750953i	$-0.729595\pi$
$743$	−36.0000	−1.32071	−0.660356	+	0.750953i	$0.729595\pi$
$744$	−8.00000	−0.293294
$745$	0	0
$746$	−20.0000	−0.732252
$747$	12.0000	0.439057
$748$	−6.00000	−0.219382
$749$	−24.0000	−0.876941
$750$	0	0
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	6.00000	0.218797
$753$	0	0
$754$	24.0000	0.874028
$755$	0	0
$756$	2.00000	0.0727393
$757$	34.0000	1.23575	0.617876	−	0.786276i	$-0.287994\pi$
$757$	34.0000	1.23575	0.617876	+	0.786276i	$0.287994\pi$
$758$	20.0000	0.726433
$759$	−6.00000	−0.217786
$760$	0	0
$761$	18.0000	0.652499	0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000	0.652499	0.326250	+	0.945284i	$0.394215\pi$
$762$	14.0000	0.507166
$763$	8.00000	0.289619
$764$	18.0000	0.651217
$765$	0	0
$766$	−6.00000	−0.216789
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	−34.0000	−1.22607	−0.613036	−	0.790055i	$-0.710052\pi$
$769$	−34.0000	−1.22607	−0.613036	+	0.790055i	$0.710052\pi$
$770$	0	0
$771$	−30.0000	−1.08042
$772$	−14.0000	−0.503871
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	−8.00000	−0.287554
$775$	0	0
$776$	−14.0000	−0.502571
$777$	20.0000	0.717496
$778$	0	0
$779$	−24.0000	−0.859889
$780$	0	0
$781$	−6.00000	−0.214697
$782$	−36.0000	−1.28736
$783$	−6.00000	−0.214423
$784$	−3.00000	−0.107143
$785$	0	0
$786$	12.0000	0.428026
$787$	−32.0000	−1.14068	−0.570338	−	0.821410i	$-0.693188\pi$
$787$	−32.0000	−1.14068	−0.570338	+	0.821410i	$0.693188\pi$
$788$	−6.00000	−0.213741
$789$	0	0
$790$	0	0
$791$	36.0000	1.28001
$792$	−1.00000	−0.0355335
$793$	32.0000	1.13635
$794$	−26.0000	−0.922705
$795$	0	0
$796$	−4.00000	−0.141776
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	−8.00000	−0.283197
$799$	36.0000	1.27359
$800$	0	0
$801$	−6.00000	−0.212000
$802$	30.0000	1.05934
$803$	2.00000	0.0705785
$804$	−4.00000	−0.141069
$805$	0	0
$806$	32.0000	1.12715
$807$	24.0000	0.844840
$808$	6.00000	0.211079
$809$	−42.0000	−1.47664	−0.738321	−	0.674450i	$-0.764381\pi$
$809$	−42.0000	−1.47664	−0.738321	+	0.674450i	$0.764381\pi$
$810$	0	0
$811$	8.00000	0.280918	0.140459	−	0.990086i	$-0.455142\pi$
$811$	8.00000	0.280918	0.140459	+	0.990086i	$0.455142\pi$
$812$	−12.0000	−0.421117
$813$	−2.00000	−0.0701431
$814$	−10.0000	−0.350500
$815$	0	0
$816$	−6.00000	−0.210042
$817$	32.0000	1.11954
$818$	−34.0000	−1.18878
$819$	−8.00000	−0.279543
$820$	0	0
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	−18.0000	−0.627822
$823$	−8.00000	−0.278862	−0.139431	−	0.990232i	$-0.544527\pi$
$823$	−8.00000	−0.278862	−0.139431	+	0.990232i	$0.544527\pi$
$824$	4.00000	0.139347
$825$	0	0
$826$	0	0
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	−6.00000	−0.208514
$829$	14.0000	0.486240	0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000	0.486240	0.243120	+	0.969996i	$0.421829\pi$
$830$	0	0
$831$	−16.0000	−0.555034
$832$	4.00000	0.138675
$833$	−18.0000	−0.623663
$834$	4.00000	0.138509
$835$	0	0
$836$	4.00000	0.138343
$837$	−8.00000	−0.276520
$838$	24.0000	0.829066
$839$	18.0000	0.621429	0.310715	−	0.950503i	$-0.399432\pi$
$839$	18.0000	0.621429	0.310715	+	0.950503i	$0.399432\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−10.0000	−0.344623
$843$	6.00000	0.206651
$844$	8.00000	0.275371
$845$	0	0
$846$	6.00000	0.206284
$847$	−2.00000	−0.0687208
$848$	0	0
$849$	8.00000	0.274559
$850$	0	0
$851$	−60.0000	−2.05677
$852$	−6.00000	−0.205557
$853$	−8.00000	−0.273915	−0.136957	−	0.990577i	$-0.543732\pi$
$853$	−8.00000	−0.273915	−0.136957	+	0.990577i	$0.543732\pi$
$854$	−16.0000	−0.547509
$855$	0	0
$856$	12.0000	0.410152
$857$	−54.0000	−1.84460	−0.922302	−	0.386469i	$-0.873695\pi$
$857$	−54.0000	−1.84460	−0.922302	+	0.386469i	$0.873695\pi$
$858$	4.00000	0.136558
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	0	0
$861$	12.0000	0.408959
$862$	0	0
$863$	−42.0000	−1.42970	−0.714848	−	0.699280i	$-0.753504\pi$
$863$	−42.0000	−1.42970	−0.714848	+	0.699280i	$0.753504\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−26.0000	−0.883516
$867$	−19.0000	−0.645274
$868$	−16.0000	−0.543075
$869$	−14.0000	−0.474917
$870$	0	0
$871$	16.0000	0.542139
$872$	−4.00000	−0.135457
$873$	−14.0000	−0.473828
$874$	24.0000	0.811812
$875$	0	0
$876$	2.00000	0.0675737
$877$	52.0000	1.75592	0.877958	−	0.478738i	$-0.158906\pi$
$877$	52.0000	1.75592	0.877958	+	0.478738i	$0.158906\pi$
$878$	−10.0000	−0.337484
$879$	−6.00000	−0.202375
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	−3.00000	−0.101015
$883$	28.0000	0.942275	0.471138	−	0.882060i	$-0.343844\pi$
$883$	28.0000	0.942275	0.471138	+	0.882060i	$0.343844\pi$
$884$	24.0000	0.807207
$885$	0	0
$886$	24.0000	0.806296
$887$	36.0000	1.20876	0.604381	−	0.796696i	$-0.293421\pi$
$887$	36.0000	1.20876	0.604381	+	0.796696i	$0.293421\pi$
$888$	−10.0000	−0.335578
$889$	28.0000	0.939090
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	16.0000	0.535720
$893$	−24.0000	−0.803129
$894$	6.00000	0.200670
$895$	0	0
$896$	−2.00000	−0.0668153
$897$	24.0000	0.801337
$898$	6.00000	0.200223
$899$	48.0000	1.60089
$900$	0	0
$901$	0	0
$902$	−6.00000	−0.199778
$903$	−16.0000	−0.532447
$904$	−18.0000	−0.598671
$905$	0	0
$906$	10.0000	0.332228
$907$	−20.0000	−0.664089	−0.332045	−	0.943264i	$-0.607738\pi$
$907$	−20.0000	−0.664089	−0.332045	+	0.943264i	$0.607738\pi$
$908$	12.0000	0.398234
$909$	6.00000	0.199007
$910$	0	0
$911$	−42.0000	−1.39152	−0.695761	−	0.718273i	$-0.744933\pi$
$911$	−42.0000	−1.39152	−0.695761	+	0.718273i	$0.744933\pi$
$912$	4.00000	0.132453
$913$	−12.0000	−0.397142
$914$	10.0000	0.330771
$915$	0	0
$916$	−22.0000	−0.726900
$917$	24.0000	0.792550
$918$	−6.00000	−0.198030
$919$	2.00000	0.0659739	0.0329870	−	0.999456i	$-0.489498\pi$
$919$	2.00000	0.0659739	0.0329870	+	0.999456i	$0.489498\pi$
$920$	0	0
$921$	20.0000	0.659022
$922$	42.0000	1.38320
$923$	24.0000	0.789970
$924$	−2.00000	−0.0657952
$925$	0	0
$926$	4.00000	0.131448
$927$	4.00000	0.131377
$928$	6.00000	0.196960
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	12.0000	0.393284
$932$	18.0000	0.589610
$933$	18.0000	0.589294
$934$	12.0000	0.392652
$935$	0	0
$936$	4.00000	0.130744
$937$	22.0000	0.718709	0.359354	−	0.933201i	$-0.382997\pi$
$937$	22.0000	0.718709	0.359354	+	0.933201i	$0.382997\pi$
$938$	−8.00000	−0.261209
$939$	26.0000	0.848478
$940$	0	0
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	2.00000	0.0651635
$943$	−36.0000	−1.17232
$944$	0	0
$945$	0	0
$946$	8.00000	0.260102
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	−14.0000	−0.454699
$949$	−8.00000	−0.259691
$950$	0	0
$951$	12.0000	0.389127
$952$	−12.0000	−0.388922
$953$	42.0000	1.36051	0.680257	−	0.732974i	$-0.261868\pi$
$953$	42.0000	1.36051	0.680257	+	0.732974i	$0.261868\pi$
$954$	0	0
$955$	0	0
$956$	−12.0000	−0.388108
$957$	6.00000	0.193952
$958$	−24.0000	−0.775405
$959$	−36.0000	−1.16250
$960$	0	0
$961$	33.0000	1.06452
$962$	40.0000	1.28965
$963$	12.0000	0.386695
$964$	−10.0000	−0.322078
$965$	0	0
$966$	−12.0000	−0.386094
$967$	−14.0000	−0.450210	−0.225105	−	0.974335i	$-0.572272\pi$
$967$	−14.0000	−0.450210	−0.225105	+	0.974335i	$0.572272\pi$
$968$	1.00000	0.0321412
$969$	24.0000	0.770991
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	−1.00000	−0.0320750
$973$	8.00000	0.256468
$974$	−20.0000	−0.640841
$975$	0	0
$976$	8.00000	0.256074
$977$	−54.0000	−1.72761	−0.863807	−	0.503824i	$-0.831926\pi$
$977$	−54.0000	−1.72761	−0.863807	+	0.503824i	$0.831926\pi$
$978$	−4.00000	−0.127906
$979$	6.00000	0.191761
$980$	0	0
$981$	−4.00000	−0.127710
$982$	12.0000	0.382935
$983$	−30.0000	−0.956851	−0.478426	−	0.878128i	$-0.658792\pi$
$983$	−30.0000	−0.956851	−0.478426	+	0.878128i	$0.658792\pi$
$984$	−6.00000	−0.191273
$985$	0	0
$986$	36.0000	1.14647
$987$	12.0000	0.381964
$988$	−16.0000	−0.509028
$989$	48.0000	1.52631
$990$	0	0
$991$	56.0000	1.77890	0.889449	−	0.457034i	$-0.151088\pi$
$991$	56.0000	1.77890	0.889449	+	0.457034i	$0.151088\pi$
$992$	8.00000	0.254000
$993$	4.00000	0.126936
$994$	−12.0000	−0.380617
$995$	0	0
$996$	−12.0000	−0.380235
$997$	−44.0000	−1.39349	−0.696747	−	0.717317i	$-0.745370\pi$
$997$	−44.0000	−1.39349	−0.696747	+	0.717317i	$0.745370\pi$
$998$	−4.00000	−0.126618
$999$	−10.0000	−0.316386

Twists

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

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