LMFDB - Embedded newform 3330.2.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3330,2,Mod(1,3330)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3330, 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("3330.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3330.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:	$26.5901838731$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	3330.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{8} +O(q^{10})$

$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{8} -1.00000 q^{10} +2.00000 q^{11} -2.00000 q^{13} +1.00000 q^{16} -2.00000 q^{17} -8.00000 q^{19} -1.00000 q^{20} +2.00000 q^{22} -4.00000 q^{23} +1.00000 q^{25} -2.00000 q^{26} -10.0000 q^{29} +2.00000 q^{31} +1.00000 q^{32} -2.00000 q^{34} +1.00000 q^{37} -8.00000 q^{38} -1.00000 q^{40} +8.00000 q^{41} -6.00000 q^{43} +2.00000 q^{44} -4.00000 q^{46} +6.00000 q^{47} -7.00000 q^{49} +1.00000 q^{50} -2.00000 q^{52} -2.00000 q^{55} -10.0000 q^{58} -4.00000 q^{59} -4.00000 q^{61} +2.00000 q^{62} +1.00000 q^{64} +2.00000 q^{65} -16.0000 q^{67} -2.00000 q^{68} +4.00000 q^{71} +6.00000 q^{73} +1.00000 q^{74} -8.00000 q^{76} +10.0000 q^{79} -1.00000 q^{80} +8.00000 q^{82} +16.0000 q^{83} +2.00000 q^{85} -6.00000 q^{86} +2.00000 q^{88} +2.00000 q^{89} -4.00000 q^{92} +6.00000 q^{94} +8.00000 q^{95} -8.00000 q^{97} -7.00000 q^{98} +O(q^{100})$

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
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	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	$0.577979\pi$
$18$	0	0
$19$	−8.00000	−1.83533	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	−8.00000	−1.83533	−0.917663	+	0.397360i	$0.869927\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	2.00000	0.426401
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−2.00000	−0.392232
$27$	0	0
$28$	0	0
$29$	−10.0000	−1.85695	−0.928477	−	0.371391i	$-0.878881\pi$
$29$	−10.0000	−1.85695	−0.928477	+	0.371391i	$0.878881\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−2.00000	−0.342997
$35$	0	0
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	−8.00000	−1.29777
$39$	0	0
$40$	−1.00000	−0.158114
$41$	8.00000	1.24939	0.624695	−	0.780869i	$-0.285223\pi$
$41$	8.00000	1.24939	0.624695	+	0.780869i	$0.285223\pi$
$42$	0	0
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	2.00000	0.301511
$45$	0	0
$46$	−4.00000	−0.589768
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	1.00000	0.141421
$51$	0	0
$52$	−2.00000	−0.277350
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	−2.00000	−0.269680
$56$	0	0
$57$	0	0
$58$	−10.0000	−1.31306
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	2.00000	0.254000
$63$	0	0
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	0	0
$67$	−16.0000	−1.95471	−0.977356	−	0.211604i	$-0.932131\pi$
$67$	−16.0000	−1.95471	−0.977356	+	0.211604i	$0.932131\pi$
$68$	−2.00000	−0.242536
$69$	0	0
$70$	0	0
$71$	4.00000	0.474713	0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000	0.474713	0.237356	+	0.971423i	$0.423719\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	−8.00000	−0.917663
$77$	0	0
$78$	0	0
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	8.00000	0.883452
$83$	16.0000	1.75623	0.878114	−	0.478451i	$-0.158802\pi$
$83$	16.0000	1.75623	0.878114	+	0.478451i	$0.158802\pi$
$84$	0	0
$85$	2.00000	0.216930
$86$	−6.00000	−0.646997
$87$	0	0
$88$	2.00000	0.213201
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	0	0
$92$	−4.00000	−0.417029
$93$	0	0
$94$	6.00000	0.618853
$95$	8.00000	0.820783
$96$	0	0
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	−7.00000	−0.707107
$99$	0	0
$100$	1.00000	0.100000
$101$	−18.0000	−1.79107	−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000	−1.79107	−0.895533	+	0.444994i	$0.853206\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	0	0
$107$	−8.00000	−0.773389	−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000	−0.773389	−0.386695	+	0.922208i	$0.626383\pi$
$108$	0	0
$109$	−4.00000	−0.383131	−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000	−0.383131	−0.191565	+	0.981480i	$0.561356\pi$
$110$	−2.00000	−0.190693
$111$	0	0
$112$	0	0
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	−10.0000	−0.928477
$117$	0	0
$118$	−4.00000	−0.368230
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−4.00000	−0.362143
$123$	0	0
$124$	2.00000	0.179605
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	2.00000	0.175412
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	0	0
$134$	−16.0000	−1.38219
$135$	0	0
$136$	−2.00000	−0.171499
$137$	22.0000	1.87959	0.939793	−	0.341743i	$-0.111017\pi$
$137$	22.0000	1.87959	0.939793	+	0.341743i	$0.111017\pi$
$138$	0	0
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	0	0
$141$	0	0
$142$	4.00000	0.335673
$143$	−4.00000	−0.334497
$144$	0	0
$145$	10.0000	0.830455
$146$	6.00000	0.496564
$147$	0	0
$148$	1.00000	0.0821995
$149$	−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
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	−8.00000	−0.648886
$153$	0	0
$154$	0	0
$155$	−2.00000	−0.160644
$156$	0	0
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	10.0000	0.795557
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	0	0
$162$	0	0
$163$	−18.0000	−1.40987	−0.704934	−	0.709273i	$-0.749024\pi$
$163$	−18.0000	−1.40987	−0.704934	+	0.709273i	$0.749024\pi$
$164$	8.00000	0.624695
$165$	0	0
$166$	16.0000	1.24184
$167$	4.00000	0.309529	0.154765	−	0.987951i	$-0.450538\pi$
$167$	4.00000	0.309529	0.154765	+	0.987951i	$0.450538\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	2.00000	0.153393
$171$	0	0
$172$	−6.00000	−0.457496
$173$	8.00000	0.608229	0.304114	−	0.952636i	$-0.401639\pi$
$173$	8.00000	0.608229	0.304114	+	0.952636i	$0.401639\pi$
$174$	0	0
$175$	0	0
$176$	2.00000	0.150756
$177$	0	0
$178$	2.00000	0.149906
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	0	0
$183$	0	0
$184$	−4.00000	−0.294884
$185$	−1.00000	−0.0735215
$186$	0	0
$187$	−4.00000	−0.292509
$188$	6.00000	0.437595
$189$	0	0
$190$	8.00000	0.580381
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	0	0
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	−8.00000	−0.574367
$195$	0	0
$196$	−7.00000	−0.500000
$197$	−16.0000	−1.13995	−0.569976	−	0.821661i	$-0.693048\pi$
$197$	−16.0000	−1.13995	−0.569976	+	0.821661i	$0.693048\pi$
$198$	0	0
$199$	−26.0000	−1.84309	−0.921546	−	0.388270i	$-0.873073\pi$
$199$	−26.0000	−1.84309	−0.921546	+	0.388270i	$0.873073\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	−18.0000	−1.26648
$203$	0	0
$204$	0	0
$205$	−8.00000	−0.558744
$206$	−4.00000	−0.278693
$207$	0	0
$208$	−2.00000	−0.138675
$209$	−16.0000	−1.10674
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	0	0
$213$	0	0
$214$	−8.00000	−0.546869
$215$	6.00000	0.409197
$216$	0	0
$217$	0	0
$218$	−4.00000	−0.270914
$219$	0	0
$220$	−2.00000	−0.134840
$221$	4.00000	0.269069
$222$	0	0
$223$	24.0000	1.60716	0.803579	−	0.595198i	$-0.202926\pi$
$223$	24.0000	1.60716	0.803579	+	0.595198i	$0.202926\pi$
$224$	0	0
$225$	0	0
$226$	2.00000	0.133038
$227$	28.0000	1.85843	0.929213	−	0.369546i	$-0.120487\pi$
$227$	28.0000	1.85843	0.929213	+	0.369546i	$0.120487\pi$
$228$	0	0
$229$	26.0000	1.71813	0.859064	−	0.511868i	$-0.171046\pi$
$229$	26.0000	1.71813	0.859064	+	0.511868i	$0.171046\pi$
$230$	4.00000	0.263752
$231$	0	0
$232$	−10.0000	−0.656532
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	0	0
$235$	−6.00000	−0.391397
$236$	−4.00000	−0.260378
$237$	0	0
$238$	0	0
$239$	16.0000	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$240$	0	0
$241$	−2.00000	−0.128831	−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000	−0.128831	−0.0644157	+	0.997923i	$0.520518\pi$
$242$	−7.00000	−0.449977
$243$	0	0
$244$	−4.00000	−0.256074
$245$	7.00000	0.447214
$246$	0	0
$247$	16.0000	1.01806
$248$	2.00000	0.127000
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	−8.00000	−0.502956
$254$	−8.00000	−0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	22.0000	1.37232	0.686161	−	0.727450i	$-0.259294\pi$
$257$	22.0000	1.37232	0.686161	+	0.727450i	$0.259294\pi$
$258$	0	0
$259$	0	0
$260$	2.00000	0.124035
$261$	0	0
$262$	−12.0000	−0.741362
$263$	10.0000	0.616626	0.308313	−	0.951285i	$-0.400236\pi$
$263$	10.0000	0.616626	0.308313	+	0.951285i	$0.400236\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	−16.0000	−0.977356
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	−2.00000	−0.121268
$273$	0	0
$274$	22.0000	1.32907
$275$	2.00000	0.120605
$276$	0	0
$277$	−14.0000	−0.841178	−0.420589	−	0.907251i	$-0.638177\pi$
$277$	−14.0000	−0.841178	−0.420589	+	0.907251i	$0.638177\pi$
$278$	−12.0000	−0.719712
$279$	0	0
$280$	0	0
$281$	26.0000	1.55103	0.775515	−	0.631329i	$-0.217490\pi$
$281$	26.0000	1.55103	0.775515	+	0.631329i	$0.217490\pi$
$282$	0	0
$283$	18.0000	1.06999	0.534994	−	0.844856i	$-0.320314\pi$
$283$	18.0000	1.06999	0.534994	+	0.844856i	$0.320314\pi$
$284$	4.00000	0.237356
$285$	0	0
$286$	−4.00000	−0.236525
$287$	0	0
$288$	0	0
$289$	−13.0000	−0.764706
$290$	10.0000	0.587220
$291$	0	0
$292$	6.00000	0.351123
$293$	12.0000	0.701047	0.350524	−	0.936554i	$-0.386004\pi$
$293$	12.0000	0.701047	0.350524	+	0.936554i	$0.386004\pi$
$294$	0	0
$295$	4.00000	0.232889
$296$	1.00000	0.0581238
$297$	0	0
$298$	−6.00000	−0.347571
$299$	8.00000	0.462652
$300$	0	0
$301$	0	0
$302$	−8.00000	−0.460348
$303$	0	0
$304$	−8.00000	−0.458831
$305$	4.00000	0.229039
$306$	0	0
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	0	0
$309$	0	0
$310$	−2.00000	−0.113592
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−28.0000	−1.58265	−0.791327	−	0.611393i	$-0.790609\pi$
$313$	−28.0000	−1.58265	−0.791327	+	0.611393i	$0.790609\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	10.0000	0.562544
$317$	24.0000	1.34797	0.673987	−	0.738743i	$-0.264580\pi$
$317$	24.0000	1.34797	0.673987	+	0.738743i	$0.264580\pi$
$318$	0	0
$319$	−20.0000	−1.11979
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	0	0
$323$	16.0000	0.890264
$324$	0	0
$325$	−2.00000	−0.110940
$326$	−18.0000	−0.996928
$327$	0	0
$328$	8.00000	0.441726
$329$	0	0
$330$	0	0
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	16.0000	0.878114
$333$	0	0
$334$	4.00000	0.218870
$335$	16.0000	0.874173
$336$	0	0
$337$	−34.0000	−1.85210	−0.926049	−	0.377403i	$-0.876817\pi$
$337$	−34.0000	−1.85210	−0.926049	+	0.377403i	$0.876817\pi$
$338$	−9.00000	−0.489535
$339$	0	0
$340$	2.00000	0.108465
$341$	4.00000	0.216612
$342$	0	0
$343$	0	0
$344$	−6.00000	−0.323498
$345$	0	0
$346$	8.00000	0.430083
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	0	0
$349$	26.0000	1.39175	0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000	1.39175	0.695874	+	0.718164i	$0.255017\pi$
$350$	0	0
$351$	0	0
$352$	2.00000	0.106600
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	0	0
$355$	−4.00000	−0.212298
$356$	2.00000	0.106000
$357$	0	0
$358$	0	0
$359$	32.0000	1.68890	0.844448	−	0.535638i	$-0.179929\pi$
$359$	32.0000	1.68890	0.844448	+	0.535638i	$0.179929\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	14.0000	0.735824
$363$	0	0
$364$	0	0
$365$	−6.00000	−0.314054
$366$	0	0
$367$	−16.0000	−0.835193	−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000	−0.835193	−0.417597	+	0.908633i	$0.637127\pi$
$368$	−4.00000	−0.208514
$369$	0	0
$370$	−1.00000	−0.0519875
$371$	0	0
$372$	0	0
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	−4.00000	−0.206835
$375$	0	0
$376$	6.00000	0.309426
$377$	20.0000	1.03005
$378$	0	0
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	8.00000	0.410391
$381$	0	0
$382$	−8.00000	−0.409316
$383$	8.00000	0.408781	0.204390	−	0.978889i	$-0.434479\pi$
$383$	8.00000	0.408781	0.204390	+	0.978889i	$0.434479\pi$
$384$	0	0
$385$	0	0
$386$	−4.00000	−0.203595
$387$	0	0
$388$	−8.00000	−0.406138
$389$	10.0000	0.507020	0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000	0.507020	0.253510	+	0.967333i	$0.418415\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	−7.00000	−0.353553
$393$	0	0
$394$	−16.0000	−0.806068
$395$	−10.0000	−0.503155
$396$	0	0
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	−26.0000	−1.30326
$399$	0	0
$400$	1.00000	0.0500000
$401$	14.0000	0.699127	0.349563	−	0.936913i	$-0.386330\pi$
$401$	14.0000	0.699127	0.349563	+	0.936913i	$0.386330\pi$
$402$	0	0
$403$	−4.00000	−0.199254
$404$	−18.0000	−0.895533
$405$	0	0
$406$	0	0
$407$	2.00000	0.0991363
$408$	0	0
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	−8.00000	−0.395092
$411$	0	0
$412$	−4.00000	−0.197066
$413$	0	0
$414$	0	0
$415$	−16.0000	−0.785409
$416$	−2.00000	−0.0980581
$417$	0	0
$418$	−16.0000	−0.782586
$419$	−2.00000	−0.0977064	−0.0488532	−	0.998806i	$-0.515557\pi$
$419$	−2.00000	−0.0977064	−0.0488532	+	0.998806i	$0.515557\pi$
$420$	0	0
$421$	−32.0000	−1.55958	−0.779792	−	0.626038i	$-0.784675\pi$
$421$	−32.0000	−1.55958	−0.779792	+	0.626038i	$0.784675\pi$
$422$	20.0000	0.973585
$423$	0	0
$424$	0	0
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	0	0
$428$	−8.00000	−0.386695
$429$	0	0
$430$	6.00000	0.289346
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	0	0
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	0	0
$435$	0	0
$436$	−4.00000	−0.191565
$437$	32.0000	1.53077
$438$	0	0
$439$	6.00000	0.286364	0.143182	−	0.989696i	$-0.454267\pi$
$439$	6.00000	0.286364	0.143182	+	0.989696i	$0.454267\pi$
$440$	−2.00000	−0.0953463
$441$	0	0
$442$	4.00000	0.190261
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	0	0
$445$	−2.00000	−0.0948091
$446$	24.0000	1.13643
$447$	0	0
$448$	0	0
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	0	0
$451$	16.0000	0.753411
$452$	2.00000	0.0940721
$453$	0	0
$454$	28.0000	1.31411
$455$	0	0
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	26.0000	1.21490
$459$	0	0
$460$	4.00000	0.186501
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	0	0
$463$	36.0000	1.67306	0.836531	−	0.547920i	$-0.184580\pi$
$463$	36.0000	1.67306	0.836531	+	0.547920i	$0.184580\pi$
$464$	−10.0000	−0.464238
$465$	0	0
$466$	−10.0000	−0.463241
$467$	36.0000	1.66588	0.832941	−	0.553362i	$-0.186655\pi$
$467$	36.0000	1.66588	0.832941	+	0.553362i	$0.186655\pi$
$468$	0	0
$469$	0	0
$470$	−6.00000	−0.276759
$471$	0	0
$472$	−4.00000	−0.184115
$473$	−12.0000	−0.551761
$474$	0	0
$475$	−8.00000	−0.367065
$476$	0	0
$477$	0	0
$478$	16.0000	0.731823
$479$	−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
$481$	−2.00000	−0.0911922
$482$	−2.00000	−0.0910975
$483$	0	0
$484$	−7.00000	−0.318182
$485$	8.00000	0.363261
$486$	0	0
$487$	−12.0000	−0.543772	−0.271886	−	0.962329i	$-0.587647\pi$
$487$	−12.0000	−0.543772	−0.271886	+	0.962329i	$0.587647\pi$
$488$	−4.00000	−0.181071
$489$	0	0
$490$	7.00000	0.316228
$491$	26.0000	1.17336	0.586682	−	0.809818i	$-0.300434\pi$
$491$	26.0000	1.17336	0.586682	+	0.809818i	$0.300434\pi$
$492$	0	0
$493$	20.0000	0.900755
$494$	16.0000	0.719874
$495$	0	0
$496$	2.00000	0.0898027
$497$	0	0
$498$	0	0
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−12.0000	−0.535586
$503$	−12.0000	−0.535054	−0.267527	−	0.963550i	$-0.586206\pi$
$503$	−12.0000	−0.535054	−0.267527	+	0.963550i	$0.586206\pi$
$504$	0	0
$505$	18.0000	0.800989
$506$	−8.00000	−0.355643
$507$	0	0
$508$	−8.00000	−0.354943
$509$	−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
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	22.0000	0.970378
$515$	4.00000	0.176261
$516$	0	0
$517$	12.0000	0.527759
$518$	0	0
$519$	0	0
$520$	2.00000	0.0877058
$521$	−36.0000	−1.57719	−0.788594	−	0.614914i	$-0.789191\pi$
$521$	−36.0000	−1.57719	−0.788594	+	0.614914i	$0.789191\pi$
$522$	0	0
$523$	34.0000	1.48672	0.743358	−	0.668894i	$-0.233232\pi$
$523$	34.0000	1.48672	0.743358	+	0.668894i	$0.233232\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	10.0000	0.436021
$527$	−4.00000	−0.174243
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−16.0000	−0.693037
$534$	0	0
$535$	8.00000	0.345870
$536$	−16.0000	−0.691095
$537$	0	0
$538$	−10.0000	−0.431131
$539$	−14.0000	−0.603023
$540$	0	0
$541$	24.0000	1.03184	0.515920	−	0.856637i	$-0.327450\pi$
$541$	24.0000	1.03184	0.515920	+	0.856637i	$0.327450\pi$
$542$	8.00000	0.343629
$543$	0	0
$544$	−2.00000	−0.0857493
$545$	4.00000	0.171341
$546$	0	0
$547$	26.0000	1.11168	0.555840	−	0.831289i	$-0.312397\pi$
$547$	26.0000	1.11168	0.555840	+	0.831289i	$0.312397\pi$
$548$	22.0000	0.939793
$549$	0	0
$550$	2.00000	0.0852803
$551$	80.0000	3.40811
$552$	0	0
$553$	0	0
$554$	−14.0000	−0.594803
$555$	0	0
$556$	−12.0000	−0.508913
$557$	−42.0000	−1.77960	−0.889799	−	0.456354i	$-0.849155\pi$
$557$	−42.0000	−1.77960	−0.889799	+	0.456354i	$0.849155\pi$
$558$	0	0
$559$	12.0000	0.507546
$560$	0	0
$561$	0	0
$562$	26.0000	1.09674
$563$	−12.0000	−0.505740	−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000	−0.505740	−0.252870	+	0.967500i	$0.581374\pi$
$564$	0	0
$565$	−2.00000	−0.0841406
$566$	18.0000	0.756596
$567$	0	0
$568$	4.00000	0.167836
$569$	−14.0000	−0.586911	−0.293455	−	0.955973i	$-0.594805\pi$
$569$	−14.0000	−0.586911	−0.293455	+	0.955973i	$0.594805\pi$
$570$	0	0
$571$	−12.0000	−0.502184	−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000	−0.502184	−0.251092	+	0.967963i	$0.580790\pi$
$572$	−4.00000	−0.167248
$573$	0	0
$574$	0	0
$575$	−4.00000	−0.166812
$576$	0	0
$577$	−32.0000	−1.33218	−0.666089	−	0.745873i	$-0.732033\pi$
$577$	−32.0000	−1.33218	−0.666089	+	0.745873i	$0.732033\pi$
$578$	−13.0000	−0.540729
$579$	0	0
$580$	10.0000	0.415227
$581$	0	0
$582$	0	0
$583$	0	0
$584$	6.00000	0.248282
$585$	0	0
$586$	12.0000	0.495715
$587$	36.0000	1.48588	0.742940	−	0.669359i	$-0.233431\pi$
$587$	36.0000	1.48588	0.742940	+	0.669359i	$0.233431\pi$
$588$	0	0
$589$	−16.0000	−0.659269
$590$	4.00000	0.164677
$591$	0	0
$592$	1.00000	0.0410997
$593$	−22.0000	−0.903432	−0.451716	−	0.892162i	$-0.649188\pi$
$593$	−22.0000	−0.903432	−0.451716	+	0.892162i	$0.649188\pi$
$594$	0	0
$595$	0	0
$596$	−6.00000	−0.245770
$597$	0	0
$598$	8.00000	0.327144
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	0	0
$603$	0	0
$604$	−8.00000	−0.325515
$605$	7.00000	0.284590
$606$	0	0
$607$	28.0000	1.13648	0.568242	−	0.822861i	$-0.307624\pi$
$607$	28.0000	1.13648	0.568242	+	0.822861i	$0.307624\pi$
$608$	−8.00000	−0.324443
$609$	0	0
$610$	4.00000	0.161955
$611$	−12.0000	−0.485468
$612$	0	0
$613$	−14.0000	−0.565455	−0.282727	−	0.959200i	$-0.591239\pi$
$613$	−14.0000	−0.565455	−0.282727	+	0.959200i	$0.591239\pi$
$614$	12.0000	0.484281
$615$	0	0
$616$	0	0
$617$	30.0000	1.20775	0.603877	−	0.797077i	$-0.293622\pi$
$617$	30.0000	1.20775	0.603877	+	0.797077i	$0.293622\pi$
$618$	0	0
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	−2.00000	−0.0803219
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	−28.0000	−1.11911
$627$	0	0
$628$	−2.00000	−0.0798087
$629$	−2.00000	−0.0797452
$630$	0	0
$631$	−46.0000	−1.83123	−0.915616	−	0.402055i	$-0.868296\pi$
$631$	−46.0000	−1.83123	−0.915616	+	0.402055i	$0.868296\pi$
$632$	10.0000	0.397779
$633$	0	0
$634$	24.0000	0.953162
$635$	8.00000	0.317470
$636$	0	0
$637$	14.0000	0.554700
$638$	−20.0000	−0.791808
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	−8.00000	−0.315981	−0.157991	−	0.987441i	$-0.550502\pi$
$641$	−8.00000	−0.315981	−0.157991	+	0.987441i	$0.550502\pi$
$642$	0	0
$643$	14.0000	0.552106	0.276053	−	0.961142i	$-0.410973\pi$
$643$	14.0000	0.552106	0.276053	+	0.961142i	$0.410973\pi$
$644$	0	0
$645$	0	0
$646$	16.0000	0.629512
$647$	8.00000	0.314512	0.157256	−	0.987558i	$-0.449735\pi$
$647$	8.00000	0.314512	0.157256	+	0.987558i	$0.449735\pi$
$648$	0	0
$649$	−8.00000	−0.314027
$650$	−2.00000	−0.0784465
$651$	0	0
$652$	−18.0000	−0.704934
$653$	30.0000	1.17399	0.586995	−	0.809590i	$-0.300311\pi$
$653$	30.0000	1.17399	0.586995	+	0.809590i	$0.300311\pi$
$654$	0	0
$655$	12.0000	0.468879
$656$	8.00000	0.312348
$657$	0	0
$658$	0	0
$659$	−30.0000	−1.16863	−0.584317	−	0.811525i	$-0.698638\pi$
$659$	−30.0000	−1.16863	−0.584317	+	0.811525i	$0.698638\pi$
$660$	0	0
$661$	16.0000	0.622328	0.311164	−	0.950356i	$-0.399281\pi$
$661$	16.0000	0.622328	0.311164	+	0.950356i	$0.399281\pi$
$662$	−12.0000	−0.466393
$663$	0	0
$664$	16.0000	0.620920
$665$	0	0
$666$	0	0
$667$	40.0000	1.54881
$668$	4.00000	0.154765
$669$	0	0
$670$	16.0000	0.618134
$671$	−8.00000	−0.308837
$672$	0	0
$673$	−38.0000	−1.46479	−0.732396	−	0.680879i	$-0.761598\pi$
$673$	−38.0000	−1.46479	−0.732396	+	0.680879i	$0.761598\pi$
$674$	−34.0000	−1.30963
$675$	0	0
$676$	−9.00000	−0.346154
$677$	−12.0000	−0.461197	−0.230599	−	0.973049i	$-0.574068\pi$
$677$	−12.0000	−0.461197	−0.230599	+	0.973049i	$0.574068\pi$
$678$	0	0
$679$	0	0
$680$	2.00000	0.0766965
$681$	0	0
$682$	4.00000	0.153168
$683$	−4.00000	−0.153056	−0.0765279	−	0.997067i	$-0.524383\pi$
$683$	−4.00000	−0.153056	−0.0765279	+	0.997067i	$0.524383\pi$
$684$	0	0
$685$	−22.0000	−0.840577
$686$	0	0
$687$	0	0
$688$	−6.00000	−0.228748
$689$	0	0
$690$	0	0
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	8.00000	0.304114
$693$	0	0
$694$	−12.0000	−0.455514
$695$	12.0000	0.455186
$696$	0	0
$697$	−16.0000	−0.606043
$698$	26.0000	0.984115
$699$	0	0
$700$	0	0
$701$	−50.0000	−1.88847	−0.944237	−	0.329267i	$-0.893198\pi$
$701$	−50.0000	−1.88847	−0.944237	+	0.329267i	$0.893198\pi$
$702$	0	0
$703$	−8.00000	−0.301726
$704$	2.00000	0.0753778
$705$	0	0
$706$	−14.0000	−0.526897
$707$	0	0
$708$	0	0
$709$	−12.0000	−0.450669	−0.225335	−	0.974281i	$-0.572348\pi$
$709$	−12.0000	−0.450669	−0.225335	+	0.974281i	$0.572348\pi$
$710$	−4.00000	−0.150117
$711$	0	0
$712$	2.00000	0.0749532
$713$	−8.00000	−0.299602
$714$	0	0
$715$	4.00000	0.149592
$716$	0	0
$717$	0	0
$718$	32.0000	1.19423
$719$	−4.00000	−0.149175	−0.0745874	−	0.997214i	$-0.523764\pi$
$719$	−4.00000	−0.149175	−0.0745874	+	0.997214i	$0.523764\pi$
$720$	0	0
$721$	0	0
$722$	45.0000	1.67473
$723$	0	0
$724$	14.0000	0.520306
$725$	−10.0000	−0.371391
$726$	0	0
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	0	0
$730$	−6.00000	−0.222070
$731$	12.0000	0.443836
$732$	0	0
$733$	38.0000	1.40356	0.701781	−	0.712393i	$-0.252388\pi$
$733$	38.0000	1.40356	0.701781	+	0.712393i	$0.252388\pi$
$734$	−16.0000	−0.590571
$735$	0	0
$736$	−4.00000	−0.147442
$737$	−32.0000	−1.17874
$738$	0	0
$739$	12.0000	0.441427	0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000	0.441427	0.220714	+	0.975339i	$0.429161\pi$
$740$	−1.00000	−0.0367607
$741$	0	0
$742$	0	0
$743$	10.0000	0.366864	0.183432	−	0.983032i	$-0.441279\pi$
$743$	10.0000	0.366864	0.183432	+	0.983032i	$0.441279\pi$
$744$	0	0
$745$	6.00000	0.219823
$746$	10.0000	0.366126
$747$	0	0
$748$	−4.00000	−0.146254
$749$	0	0
$750$	0	0
$751$	20.0000	0.729810	0.364905	−	0.931045i	$-0.381101\pi$
$751$	20.0000	0.729810	0.364905	+	0.931045i	$0.381101\pi$
$752$	6.00000	0.218797
$753$	0	0
$754$	20.0000	0.728357
$755$	8.00000	0.291150
$756$	0	0
$757$	10.0000	0.363456	0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000	0.363456	0.181728	+	0.983349i	$0.441831\pi$
$758$	−20.0000	−0.726433
$759$	0	0
$760$	8.00000	0.290191
$761$	−40.0000	−1.45000	−0.724999	−	0.688749i	$-0.758160\pi$
$761$	−40.0000	−1.45000	−0.724999	+	0.688749i	$0.758160\pi$
$762$	0	0
$763$	0	0
$764$	−8.00000	−0.289430
$765$	0	0
$766$	8.00000	0.289052
$767$	8.00000	0.288863
$768$	0	0
$769$	2.00000	0.0721218	0.0360609	−	0.999350i	$-0.488519\pi$
$769$	2.00000	0.0721218	0.0360609	+	0.999350i	$0.488519\pi$
$770$	0	0
$771$	0	0
$772$	−4.00000	−0.143963
$773$	−36.0000	−1.29483	−0.647415	−	0.762138i	$-0.724150\pi$
$773$	−36.0000	−1.29483	−0.647415	+	0.762138i	$0.724150\pi$
$774$	0	0
$775$	2.00000	0.0718421
$776$	−8.00000	−0.287183
$777$	0	0
$778$	10.0000	0.358517
$779$	−64.0000	−2.29304
$780$	0	0
$781$	8.00000	0.286263
$782$	8.00000	0.286079
$783$	0	0
$784$	−7.00000	−0.250000
$785$	2.00000	0.0713831
$786$	0	0
$787$	32.0000	1.14068	0.570338	−	0.821410i	$-0.306812\pi$
$787$	32.0000	1.14068	0.570338	+	0.821410i	$0.306812\pi$
$788$	−16.0000	−0.569976
$789$	0	0
$790$	−10.0000	−0.355784
$791$	0	0
$792$	0	0
$793$	8.00000	0.284088
$794$	14.0000	0.496841
$795$	0	0
$796$	−26.0000	−0.921546
$797$	34.0000	1.20434	0.602171	−	0.798367i	$-0.294303\pi$
$797$	34.0000	1.20434	0.602171	+	0.798367i	$0.294303\pi$
$798$	0	0
$799$	−12.0000	−0.424529
$800$	1.00000	0.0353553
$801$	0	0
$802$	14.0000	0.494357
$803$	12.0000	0.423471
$804$	0	0
$805$	0	0
$806$	−4.00000	−0.140894
$807$	0	0
$808$	−18.0000	−0.633238
$809$	−2.00000	−0.0703163	−0.0351581	−	0.999382i	$-0.511193\pi$
$809$	−2.00000	−0.0703163	−0.0351581	+	0.999382i	$0.511193\pi$
$810$	0	0
$811$	44.0000	1.54505	0.772524	−	0.634985i	$-0.218994\pi$
$811$	44.0000	1.54505	0.772524	+	0.634985i	$0.218994\pi$
$812$	0	0
$813$	0	0
$814$	2.00000	0.0701000
$815$	18.0000	0.630512
$816$	0	0
$817$	48.0000	1.67931
$818$	10.0000	0.349642
$819$	0	0
$820$	−8.00000	−0.279372
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	0	0
$823$	−32.0000	−1.11545	−0.557725	−	0.830026i	$-0.688326\pi$
$823$	−32.0000	−1.11545	−0.557725	+	0.830026i	$0.688326\pi$
$824$	−4.00000	−0.139347
$825$	0	0
$826$	0	0
$827$	36.0000	1.25184	0.625921	−	0.779886i	$-0.284723\pi$
$827$	36.0000	1.25184	0.625921	+	0.779886i	$0.284723\pi$
$828$	0	0
$829$	20.0000	0.694629	0.347314	−	0.937749i	$-0.387094\pi$
$829$	20.0000	0.694629	0.347314	+	0.937749i	$0.387094\pi$
$830$	−16.0000	−0.555368
$831$	0	0
$832$	−2.00000	−0.0693375
$833$	14.0000	0.485071
$834$	0	0
$835$	−4.00000	−0.138426
$836$	−16.0000	−0.553372
$837$	0	0
$838$	−2.00000	−0.0690889
$839$	−12.0000	−0.414286	−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	−32.0000	−1.10279
$843$	0	0
$844$	20.0000	0.688428
$845$	9.00000	0.309609
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	−2.00000	−0.0685994
$851$	−4.00000	−0.137118
$852$	0	0
$853$	6.00000	0.205436	0.102718	−	0.994711i	$-0.467246\pi$
$853$	6.00000	0.205436	0.102718	+	0.994711i	$0.467246\pi$
$854$	0	0
$855$	0	0
$856$	−8.00000	−0.273434
$857$	42.0000	1.43469	0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000	1.43469	0.717346	+	0.696717i	$0.245357\pi$
$858$	0	0
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	6.00000	0.204598
$861$	0	0
$862$	24.0000	0.817443
$863$	18.0000	0.612727	0.306364	−	0.951915i	$-0.400888\pi$
$863$	18.0000	0.612727	0.306364	+	0.951915i	$0.400888\pi$
$864$	0	0
$865$	−8.00000	−0.272008
$866$	14.0000	0.475739
$867$	0	0
$868$	0	0
$869$	20.0000	0.678454
$870$	0	0
$871$	32.0000	1.08428
$872$	−4.00000	−0.135457
$873$	0	0
$874$	32.0000	1.08242
$875$	0	0
$876$	0	0
$877$	−30.0000	−1.01303	−0.506514	−	0.862232i	$-0.669066\pi$
$877$	−30.0000	−1.01303	−0.506514	+	0.862232i	$0.669066\pi$
$878$	6.00000	0.202490
$879$	0	0
$880$	−2.00000	−0.0674200
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	14.0000	0.471138	0.235569	−	0.971858i	$-0.424305\pi$
$883$	14.0000	0.471138	0.235569	+	0.971858i	$0.424305\pi$
$884$	4.00000	0.134535
$885$	0	0
$886$	−24.0000	−0.806296
$887$	−30.0000	−1.00730	−0.503651	−	0.863907i	$-0.668010\pi$
$887$	−30.0000	−1.00730	−0.503651	+	0.863907i	$0.668010\pi$
$888$	0	0
$889$	0	0
$890$	−2.00000	−0.0670402
$891$	0	0
$892$	24.0000	0.803579
$893$	−48.0000	−1.60626
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−30.0000	−1.00111
$899$	−20.0000	−0.667037
$900$	0	0
$901$	0	0
$902$	16.0000	0.532742
$903$	0	0
$904$	2.00000	0.0665190
$905$	−14.0000	−0.465376
$906$	0	0
$907$	−54.0000	−1.79304	−0.896520	−	0.443003i	$-0.853913\pi$
$907$	−54.0000	−1.79304	−0.896520	+	0.443003i	$0.853913\pi$
$908$	28.0000	0.929213
$909$	0	0
$910$	0	0
$911$	16.0000	0.530104	0.265052	−	0.964234i	$-0.414611\pi$
$911$	16.0000	0.530104	0.265052	+	0.964234i	$0.414611\pi$
$912$	0	0
$913$	32.0000	1.05905
$914$	0	0
$915$	0	0
$916$	26.0000	0.859064
$917$	0	0
$918$	0	0
$919$	42.0000	1.38545	0.692726	−	0.721201i	$-0.256409\pi$
$919$	42.0000	1.38545	0.692726	+	0.721201i	$0.256409\pi$
$920$	4.00000	0.131876
$921$	0	0
$922$	−18.0000	−0.592798
$923$	−8.00000	−0.263323
$924$	0	0
$925$	1.00000	0.0328798
$926$	36.0000	1.18303
$927$	0	0
$928$	−10.0000	−0.328266
$929$	−28.0000	−0.918650	−0.459325	−	0.888268i	$-0.651909\pi$
$929$	−28.0000	−0.918650	−0.459325	+	0.888268i	$0.651909\pi$
$930$	0	0
$931$	56.0000	1.83533
$932$	−10.0000	−0.327561
$933$	0	0
$934$	36.0000	1.17796
$935$	4.00000	0.130814
$936$	0	0
$937$	−34.0000	−1.11073	−0.555366	−	0.831606i	$-0.687422\pi$
$937$	−34.0000	−1.11073	−0.555366	+	0.831606i	$0.687422\pi$
$938$	0	0
$939$	0	0
$940$	−6.00000	−0.195698
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	0	0
$943$	−32.0000	−1.04206
$944$	−4.00000	−0.130189
$945$	0	0
$946$	−12.0000	−0.390154
$947$	44.0000	1.42981	0.714904	−	0.699223i	$-0.246470\pi$
$947$	44.0000	1.42981	0.714904	+	0.699223i	$0.246470\pi$
$948$	0	0
$949$	−12.0000	−0.389536
$950$	−8.00000	−0.259554
$951$	0	0
$952$	0	0
$953$	−26.0000	−0.842223	−0.421111	−	0.907009i	$-0.638360\pi$
$953$	−26.0000	−0.842223	−0.421111	+	0.907009i	$0.638360\pi$
$954$	0	0
$955$	8.00000	0.258874
$956$	16.0000	0.517477
$957$	0	0
$958$	−24.0000	−0.775405
$959$	0	0
$960$	0	0
$961$	−27.0000	−0.870968
$962$	−2.00000	−0.0644826
$963$	0	0
$964$	−2.00000	−0.0644157
$965$	4.00000	0.128765
$966$	0	0
$967$	32.0000	1.02905	0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000	1.02905	0.514525	+	0.857475i	$0.327968\pi$
$968$	−7.00000	−0.224989
$969$	0	0
$970$	8.00000	0.256865
$971$	−38.0000	−1.21948	−0.609739	−	0.792602i	$-0.708726\pi$
$971$	−38.0000	−1.21948	−0.609739	+	0.792602i	$0.708726\pi$
$972$	0	0
$973$	0	0
$974$	−12.0000	−0.384505
$975$	0	0
$976$	−4.00000	−0.128037
$977$	−62.0000	−1.98356	−0.991778	−	0.127971i	$-0.959153\pi$
$977$	−62.0000	−1.98356	−0.991778	+	0.127971i	$0.959153\pi$
$978$	0	0
$979$	4.00000	0.127841
$980$	7.00000	0.223607
$981$	0	0
$982$	26.0000	0.829693
$983$	18.0000	0.574111	0.287055	−	0.957914i	$-0.407324\pi$
$983$	18.0000	0.574111	0.287055	+	0.957914i	$0.407324\pi$
$984$	0	0
$985$	16.0000	0.509802
$986$	20.0000	0.636930
$987$	0	0
$988$	16.0000	0.509028
$989$	24.0000	0.763156
$990$	0	0
$991$	−38.0000	−1.20711	−0.603555	−	0.797321i	$-0.706250\pi$
$991$	−38.0000	−1.20711	−0.603555	+	0.797321i	$0.706250\pi$
$992$	2.00000	0.0635001
$993$	0	0
$994$	0	0
$995$	26.0000	0.824255
$996$	0	0
$997$	−6.00000	−0.190022	−0.0950110	−	0.995476i	$-0.530289\pi$
$997$	−6.00000	−0.190022	−0.0950110	+	0.995476i	$0.530289\pi$
$998$	−20.0000	−0.633089
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3330.2.a.o.1.1	yes	1
3.2	odd	2		3330.2.a.i.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3330.2.a.i.1.1	✓	1	3.2	odd	2
3330.2.a.o.1.1	yes	1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3330.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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