LMFDB - Embedded newform 3330.2.a.p.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:	no (minimal twist has level 370)
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)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$

$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} -3.00000 q^{11} +1.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} -6.00000 q^{19} -1.00000 q^{20} -3.00000 q^{22} -2.00000 q^{23} +1.00000 q^{25} +1.00000 q^{28} +3.00000 q^{29} +3.00000 q^{31} +1.00000 q^{32} -3.00000 q^{34} -1.00000 q^{35} -1.00000 q^{37} -6.00000 q^{38} -1.00000 q^{40} -3.00000 q^{41} -1.00000 q^{43} -3.00000 q^{44} -2.00000 q^{46} -4.00000 q^{47} -6.00000 q^{49} +1.00000 q^{50} -13.0000 q^{53} +3.00000 q^{55} +1.00000 q^{56} +3.00000 q^{58} -15.0000 q^{61} +3.00000 q^{62} +1.00000 q^{64} -3.00000 q^{68} -1.00000 q^{70} +2.00000 q^{71} -1.00000 q^{74} -6.00000 q^{76} -3.00000 q^{77} -8.00000 q^{79} -1.00000 q^{80} -3.00000 q^{82} +4.00000 q^{83} +3.00000 q^{85} -1.00000 q^{86} -3.00000 q^{88} +18.0000 q^{89} -2.00000 q^{92} -4.00000 q^{94} +6.00000 q^{95} -7.00000 q^{97} -6.00000 q^{98} +O(q^{100})$

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
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	1.00000	0.267261
$15$	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	$0.618522\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	−3.00000	−0.639602
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	1.00000	0.188982
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−3.00000	−0.514496
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	−6.00000	−0.973329
$39$	0	0
$40$	−1.00000	−0.158114
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	0	0
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	−3.00000	−0.452267
$45$	0	0
$46$	−2.00000	−0.294884
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	1.00000	0.141421
$51$	0	0
$52$	0	0
$53$	−13.0000	−1.78569	−0.892844	−	0.450367i	$-0.851293\pi$
$53$	−13.0000	−1.78569	−0.892844	+	0.450367i	$0.851293\pi$
$54$	0	0
$55$	3.00000	0.404520
$56$	1.00000	0.133631
$57$	0	0
$58$	3.00000	0.393919
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−15.0000	−1.92055	−0.960277	−	0.279050i	$-0.909981\pi$
$61$	−15.0000	−1.92055	−0.960277	+	0.279050i	$0.909981\pi$
$62$	3.00000	0.381000
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	−3.00000	−0.363803
$69$	0	0
$70$	−1.00000	−0.119523
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	0	0
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	−1.00000	−0.116248
$75$	0	0
$76$	−6.00000	−0.688247
$77$	−3.00000	−0.341882
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	−3.00000	−0.331295
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	3.00000	0.325396
$86$	−1.00000	−0.107833
$87$	0	0
$88$	−3.00000	−0.319801
$89$	18.0000	1.90800	0.953998	−	0.299813i	$-0.0969242\pi$
$89$	18.0000	1.90800	0.953998	+	0.299813i	$0.0969242\pi$
$90$	0	0
$91$	0	0
$92$	−2.00000	−0.208514
$93$	0	0
$94$	−4.00000	−0.412568
$95$	6.00000	0.615587
$96$	0	0
$97$	−7.00000	−0.710742	−0.355371	−	0.934725i	$-0.615646\pi$
$97$	−7.00000	−0.710742	−0.355371	+	0.934725i	$0.615646\pi$
$98$	−6.00000	−0.606092
$99$	0	0
$100$	1.00000	0.100000
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	0	0
$106$	−13.0000	−1.26267
$107$	−2.00000	−0.193347	−0.0966736	−	0.995316i	$-0.530820\pi$
$107$	−2.00000	−0.193347	−0.0966736	+	0.995316i	$0.530820\pi$
$108$	0	0
$109$	−3.00000	−0.287348	−0.143674	−	0.989625i	$-0.545892\pi$
$109$	−3.00000	−0.287348	−0.143674	+	0.989625i	$0.545892\pi$
$110$	3.00000	0.286039
$111$	0	0
$112$	1.00000	0.0944911
$113$	7.00000	0.658505	0.329252	−	0.944242i	$-0.393203\pi$
$113$	7.00000	0.658505	0.329252	+	0.944242i	$0.393203\pi$
$114$	0	0
$115$	2.00000	0.186501
$116$	3.00000	0.278543
$117$	0	0
$118$	0	0
$119$	−3.00000	−0.275010
$120$	0	0
$121$	−2.00000	−0.181818
$122$	−15.0000	−1.35804
$123$	0	0
$124$	3.00000	0.269408
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	−6.00000	−0.520266
$134$	0	0
$135$	0	0
$136$	−3.00000	−0.257248
$137$	−8.00000	−0.683486	−0.341743	−	0.939793i	$-0.611017\pi$
$137$	−8.00000	−0.683486	−0.341743	+	0.939793i	$0.611017\pi$
$138$	0	0
$139$	3.00000	0.254457	0.127228	−	0.991873i	$-0.459392\pi$
$139$	3.00000	0.254457	0.127228	+	0.991873i	$0.459392\pi$
$140$	−1.00000	−0.0845154
$141$	0	0
$142$	2.00000	0.167836
$143$	0	0
$144$	0	0
$145$	−3.00000	−0.249136
$146$	0	0
$147$	0	0
$148$	−1.00000	−0.0821995
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	0	0
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	−6.00000	−0.486664
$153$	0	0
$154$	−3.00000	−0.241747
$155$	−3.00000	−0.240966
$156$	0	0
$157$	−3.00000	−0.239426	−0.119713	−	0.992809i	$-0.538197\pi$
$157$	−3.00000	−0.239426	−0.119713	+	0.992809i	$0.538197\pi$
$158$	−8.00000	−0.636446
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	−2.00000	−0.157622
$162$	0	0
$163$	−5.00000	−0.391630	−0.195815	−	0.980641i	$-0.562735\pi$
$163$	−5.00000	−0.391630	−0.195815	+	0.980641i	$0.562735\pi$
$164$	−3.00000	−0.234261
$165$	0	0
$166$	4.00000	0.310460
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	3.00000	0.230089
$171$	0	0
$172$	−1.00000	−0.0762493
$173$	−9.00000	−0.684257	−0.342129	−	0.939653i	$-0.611148\pi$
$173$	−9.00000	−0.684257	−0.342129	+	0.939653i	$0.611148\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	−3.00000	−0.226134
$177$	0	0
$178$	18.0000	1.34916
$179$	−24.0000	−1.79384	−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000	−1.79384	−0.896922	+	0.442189i	$0.854202\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	0	0
$184$	−2.00000	−0.147442
$185$	1.00000	0.0735215
$186$	0	0
$187$	9.00000	0.658145
$188$	−4.00000	−0.291730
$189$	0	0
$190$	6.00000	0.435286
$191$	−21.0000	−1.51951	−0.759753	−	0.650211i	$-0.774680\pi$
$191$	−21.0000	−1.51951	−0.759753	+	0.650211i	$0.774680\pi$
$192$	0	0
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	−7.00000	−0.502571
$195$	0	0
$196$	−6.00000	−0.428571
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	10.0000	0.703598
$203$	3.00000	0.210559
$204$	0	0
$205$	3.00000	0.209529
$206$	8.00000	0.557386
$207$	0	0
$208$	0	0
$209$	18.0000	1.24509
$210$	0	0
$211$	−3.00000	−0.206529	−0.103264	−	0.994654i	$-0.532929\pi$
$211$	−3.00000	−0.206529	−0.103264	+	0.994654i	$0.532929\pi$
$212$	−13.0000	−0.892844
$213$	0	0
$214$	−2.00000	−0.136717
$215$	1.00000	0.0681994
$216$	0	0
$217$	3.00000	0.203653
$218$	−3.00000	−0.203186
$219$	0	0
$220$	3.00000	0.202260
$221$	0	0
$222$	0	0
$223$	23.0000	1.54019	0.770097	−	0.637927i	$-0.220208\pi$
$223$	23.0000	1.54019	0.770097	+	0.637927i	$0.220208\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	7.00000	0.465633
$227$	−13.0000	−0.862840	−0.431420	−	0.902151i	$-0.641987\pi$
$227$	−13.0000	−0.862840	−0.431420	+	0.902151i	$0.641987\pi$
$228$	0	0
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	2.00000	0.131876
$231$	0	0
$232$	3.00000	0.196960
$233$	−4.00000	−0.262049	−0.131024	−	0.991379i	$-0.541827\pi$
$233$	−4.00000	−0.262049	−0.131024	+	0.991379i	$0.541827\pi$
$234$	0	0
$235$	4.00000	0.260931
$236$	0	0
$237$	0	0
$238$	−3.00000	−0.194461
$239$	−9.00000	−0.582162	−0.291081	−	0.956698i	$-0.594015\pi$
$239$	−9.00000	−0.582162	−0.291081	+	0.956698i	$0.594015\pi$
$240$	0	0
$241$	24.0000	1.54598	0.772988	−	0.634421i	$-0.218761\pi$
$241$	24.0000	1.54598	0.772988	+	0.634421i	$0.218761\pi$
$242$	−2.00000	−0.128565
$243$	0	0
$244$	−15.0000	−0.960277
$245$	6.00000	0.383326
$246$	0	0
$247$	0	0
$248$	3.00000	0.190500
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	−6.00000	−0.378717	−0.189358	−	0.981908i	$-0.560641\pi$
$251$	−6.00000	−0.378717	−0.189358	+	0.981908i	$0.560641\pi$
$252$	0	0
$253$	6.00000	0.377217
$254$	−4.00000	−0.250982
$255$	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
$259$	−1.00000	−0.0621370
$260$	0	0
$261$	0	0
$262$	12.0000	0.741362
$263$	1.00000	0.0616626	0.0308313	−	0.999525i	$-0.490185\pi$
$263$	1.00000	0.0616626	0.0308313	+	0.999525i	$0.490185\pi$
$264$	0	0
$265$	13.0000	0.798584
$266$	−6.00000	−0.367884
$267$	0	0
$268$	0	0
$269$	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
$271$	−10.0000	−0.607457	−0.303728	−	0.952759i	$-0.598232\pi$
$271$	−10.0000	−0.607457	−0.303728	+	0.952759i	$0.598232\pi$
$272$	−3.00000	−0.181902
$273$	0	0
$274$	−8.00000	−0.483298
$275$	−3.00000	−0.180907
$276$	0	0
$277$	−20.0000	−1.20168	−0.600842	−	0.799368i	$-0.705168\pi$
$277$	−20.0000	−1.20168	−0.600842	+	0.799368i	$0.705168\pi$
$278$	3.00000	0.179928
$279$	0	0
$280$	−1.00000	−0.0597614
$281$	24.0000	1.43172	0.715860	−	0.698244i	$-0.246035\pi$
$281$	24.0000	1.43172	0.715860	+	0.698244i	$0.246035\pi$
$282$	0	0
$283$	−20.0000	−1.18888	−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000	−1.18888	−0.594438	+	0.804141i	$0.702626\pi$
$284$	2.00000	0.118678
$285$	0	0
$286$	0	0
$287$	−3.00000	−0.177084
$288$	0	0
$289$	−8.00000	−0.470588
$290$	−3.00000	−0.176166
$291$	0	0
$292$	0	0
$293$	−3.00000	−0.175262	−0.0876309	−	0.996153i	$-0.527930\pi$
$293$	−3.00000	−0.175262	−0.0876309	+	0.996153i	$0.527930\pi$
$294$	0	0
$295$	0	0
$296$	−1.00000	−0.0581238
$297$	0	0
$298$	−18.0000	−1.04271
$299$	0	0
$300$	0	0
$301$	−1.00000	−0.0576390
$302$	16.0000	0.920697
$303$	0	0
$304$	−6.00000	−0.344124
$305$	15.0000	0.858898
$306$	0	0
$307$	34.0000	1.94048	0.970241	−	0.242140i	$-0.0778494\pi$
$307$	34.0000	1.94048	0.970241	+	0.242140i	$0.0778494\pi$
$308$	−3.00000	−0.170941
$309$	0	0
$310$	−3.00000	−0.170389
$311$	−25.0000	−1.41762	−0.708810	−	0.705399i	$-0.750768\pi$
$311$	−25.0000	−1.41762	−0.708810	+	0.705399i	$0.750768\pi$
$312$	0	0
$313$	26.0000	1.46961	0.734803	−	0.678280i	$-0.237274\pi$
$313$	26.0000	1.46961	0.734803	+	0.678280i	$0.237274\pi$
$314$	−3.00000	−0.169300
$315$	0	0
$316$	−8.00000	−0.450035
$317$	−21.0000	−1.17948	−0.589739	−	0.807594i	$-0.700769\pi$
$317$	−21.0000	−1.17948	−0.589739	+	0.807594i	$0.700769\pi$
$318$	0	0
$319$	−9.00000	−0.503903
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	−2.00000	−0.111456
$323$	18.0000	1.00155
$324$	0	0
$325$	0	0
$326$	−5.00000	−0.276924
$327$	0	0
$328$	−3.00000	−0.165647
$329$	−4.00000	−0.220527
$330$	0	0
$331$	30.0000	1.64895	0.824475	−	0.565899i	$-0.191471\pi$
$331$	30.0000	1.64895	0.824475	+	0.565899i	$0.191471\pi$
$332$	4.00000	0.219529
$333$	0	0
$334$	−12.0000	−0.656611
$335$	0	0
$336$	0	0
$337$	12.0000	0.653682	0.326841	−	0.945079i	$-0.394016\pi$
$337$	12.0000	0.653682	0.326841	+	0.945079i	$0.394016\pi$
$338$	−13.0000	−0.707107
$339$	0	0
$340$	3.00000	0.162698
$341$	−9.00000	−0.487377
$342$	0	0
$343$	−13.0000	−0.701934
$344$	−1.00000	−0.0539164
$345$	0	0
$346$	−9.00000	−0.483843
$347$	28.0000	1.50312	0.751559	−	0.659665i	$-0.229302\pi$
$347$	28.0000	1.50312	0.751559	+	0.659665i	$0.229302\pi$
$348$	0	0
$349$	−22.0000	−1.17763	−0.588817	−	0.808267i	$-0.700406\pi$
$349$	−22.0000	−1.17763	−0.588817	+	0.808267i	$0.700406\pi$
$350$	1.00000	0.0534522
$351$	0	0
$352$	−3.00000	−0.159901
$353$	−19.0000	−1.01127	−0.505634	−	0.862748i	$-0.668741\pi$
$353$	−19.0000	−1.01127	−0.505634	+	0.862748i	$0.668741\pi$
$354$	0	0
$355$	−2.00000	−0.106149
$356$	18.0000	0.953998
$357$	0	0
$358$	−24.0000	−1.26844
$359$	−16.0000	−0.844448	−0.422224	−	0.906492i	$-0.638750\pi$
$359$	−16.0000	−0.844448	−0.422224	+	0.906492i	$0.638750\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	2.00000	0.105118
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	13.0000	0.678594	0.339297	−	0.940679i	$-0.389811\pi$
$367$	13.0000	0.678594	0.339297	+	0.940679i	$0.389811\pi$
$368$	−2.00000	−0.104257
$369$	0	0
$370$	1.00000	0.0519875
$371$	−13.0000	−0.674926
$372$	0	0
$373$	22.0000	1.13912	0.569558	−	0.821951i	$-0.307114\pi$
$373$	22.0000	1.13912	0.569558	+	0.821951i	$0.307114\pi$
$374$	9.00000	0.465379
$375$	0	0
$376$	−4.00000	−0.206284
$377$	0	0
$378$	0	0
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	6.00000	0.307794
$381$	0	0
$382$	−21.0000	−1.07445
$383$	12.0000	0.613171	0.306586	−	0.951843i	$-0.400813\pi$
$383$	12.0000	0.613171	0.306586	+	0.951843i	$0.400813\pi$
$384$	0	0
$385$	3.00000	0.152894
$386$	−10.0000	−0.508987
$387$	0	0
$388$	−7.00000	−0.355371
$389$	−15.0000	−0.760530	−0.380265	−	0.924878i	$-0.624167\pi$
$389$	−15.0000	−0.760530	−0.380265	+	0.924878i	$0.624167\pi$
$390$	0	0
$391$	6.00000	0.303433
$392$	−6.00000	−0.303046
$393$	0	0
$394$	18.0000	0.906827
$395$	8.00000	0.402524
$396$	0	0
$397$	18.0000	0.903394	0.451697	−	0.892171i	$-0.350819\pi$
$397$	18.0000	0.903394	0.451697	+	0.892171i	$0.350819\pi$
$398$	8.00000	0.401004
$399$	0	0
$400$	1.00000	0.0500000
$401$	−34.0000	−1.69788	−0.848939	−	0.528490i	$-0.822758\pi$
$401$	−34.0000	−1.69788	−0.848939	+	0.528490i	$0.822758\pi$
$402$	0	0
$403$	0	0
$404$	10.0000	0.497519
$405$	0	0
$406$	3.00000	0.148888
$407$	3.00000	0.148704
$408$	0	0
$409$	0	0		−	1.00000i	$-0.5\pi$
$409$	0	0			1.00000i	$0.5\pi$
$410$	3.00000	0.148159
$411$	0	0
$412$	8.00000	0.394132
$413$	0	0
$414$	0	0
$415$	−4.00000	−0.196352
$416$	0	0
$417$	0	0
$418$	18.0000	0.880409
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	34.0000	1.65706	0.828529	−	0.559946i	$-0.189178\pi$
$421$	34.0000	1.65706	0.828529	+	0.559946i	$0.189178\pi$
$422$	−3.00000	−0.146038
$423$	0	0
$424$	−13.0000	−0.631336
$425$	−3.00000	−0.145521
$426$	0	0
$427$	−15.0000	−0.725901
$428$	−2.00000	−0.0966736
$429$	0	0
$430$	1.00000	0.0482243
$431$	31.0000	1.49322	0.746609	−	0.665263i	$-0.231681\pi$
$431$	31.0000	1.49322	0.746609	+	0.665263i	$0.231681\pi$
$432$	0	0
$433$	−22.0000	−1.05725	−0.528626	−	0.848855i	$-0.677293\pi$
$433$	−22.0000	−1.05725	−0.528626	+	0.848855i	$0.677293\pi$
$434$	3.00000	0.144005
$435$	0	0
$436$	−3.00000	−0.143674
$437$	12.0000	0.574038
$438$	0	0
$439$	37.0000	1.76591	0.882957	−	0.469454i	$-0.155549\pi$
$439$	37.0000	1.76591	0.882957	+	0.469454i	$0.155549\pi$
$440$	3.00000	0.143019
$441$	0	0
$442$	0	0
$443$	34.0000	1.61539	0.807694	−	0.589601i	$-0.200715\pi$
$443$	34.0000	1.61539	0.807694	+	0.589601i	$0.200715\pi$
$444$	0	0
$445$	−18.0000	−0.853282
$446$	23.0000	1.08908
$447$	0	0
$448$	1.00000	0.0472456
$449$	10.0000	0.471929	0.235965	−	0.971762i	$-0.424175\pi$
$449$	10.0000	0.471929	0.235965	+	0.971762i	$0.424175\pi$
$450$	0	0
$451$	9.00000	0.423793
$452$	7.00000	0.329252
$453$	0	0
$454$	−13.0000	−0.610120
$455$	0	0
$456$	0	0
$457$	9.00000	0.421002	0.210501	−	0.977594i	$-0.432490\pi$
$457$	9.00000	0.421002	0.210501	+	0.977594i	$0.432490\pi$
$458$	−6.00000	−0.280362
$459$	0	0
$460$	2.00000	0.0932505
$461$	1.00000	0.0465746	0.0232873	−	0.999729i	$-0.492587\pi$
$461$	1.00000	0.0465746	0.0232873	+	0.999729i	$0.492587\pi$
$462$	0	0
$463$	12.0000	0.557687	0.278844	−	0.960337i	$-0.410049\pi$
$463$	12.0000	0.557687	0.278844	+	0.960337i	$0.410049\pi$
$464$	3.00000	0.139272
$465$	0	0
$466$	−4.00000	−0.185296
$467$	37.0000	1.71216	0.856078	−	0.516847i	$-0.172894\pi$
$467$	37.0000	1.71216	0.856078	+	0.516847i	$0.172894\pi$
$468$	0	0
$469$	0	0
$470$	4.00000	0.184506
$471$	0	0
$472$	0	0
$473$	3.00000	0.137940
$474$	0	0
$475$	−6.00000	−0.275299
$476$	−3.00000	−0.137505
$477$	0	0
$478$	−9.00000	−0.411650
$479$	8.00000	0.365529	0.182765	−	0.983157i	$-0.441495\pi$
$479$	8.00000	0.365529	0.182765	+	0.983157i	$0.441495\pi$
$480$	0	0
$481$	0	0
$482$	24.0000	1.09317
$483$	0	0
$484$	−2.00000	−0.0909091
$485$	7.00000	0.317854
$486$	0	0
$487$	34.0000	1.54069	0.770344	−	0.637629i	$-0.220085\pi$
$487$	34.0000	1.54069	0.770344	+	0.637629i	$0.220085\pi$
$488$	−15.0000	−0.679018
$489$	0	0
$490$	6.00000	0.271052
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	0	0
$493$	−9.00000	−0.405340
$494$	0	0
$495$	0	0
$496$	3.00000	0.134704
$497$	2.00000	0.0897123
$498$	0	0
$499$	−22.0000	−0.984855	−0.492428	−	0.870353i	$-0.663890\pi$
$499$	−22.0000	−0.984855	−0.492428	+	0.870353i	$0.663890\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−6.00000	−0.267793
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	0	0
$505$	−10.0000	−0.444994
$506$	6.00000	0.266733
$507$	0	0
$508$	−4.00000	−0.177471
$509$	−36.0000	−1.59567	−0.797836	−	0.602875i	$-0.794022\pi$
$509$	−36.0000	−1.59567	−0.797836	+	0.602875i	$0.794022\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	22.0000	0.970378
$515$	−8.00000	−0.352522
$516$	0	0
$517$	12.0000	0.527759
$518$	−1.00000	−0.0439375
$519$	0	0
$520$	0	0
$521$	23.0000	1.00765	0.503824	−	0.863806i	$-0.331926\pi$
$521$	23.0000	1.00765	0.503824	+	0.863806i	$0.331926\pi$
$522$	0	0
$523$	−44.0000	−1.92399	−0.961993	−	0.273075i	$-0.911959\pi$
$523$	−44.0000	−1.92399	−0.961993	+	0.273075i	$0.911959\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	1.00000	0.0436021
$527$	−9.00000	−0.392046
$528$	0	0
$529$	−19.0000	−0.826087
$530$	13.0000	0.564684
$531$	0	0
$532$	−6.00000	−0.260133
$533$	0	0
$534$	0	0
$535$	2.00000	0.0864675
$536$	0	0
$537$	0	0
$538$	10.0000	0.431131
$539$	18.0000	0.775315
$540$	0	0
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	−10.0000	−0.429537
$543$	0	0
$544$	−3.00000	−0.128624
$545$	3.00000	0.128506
$546$	0	0
$547$	23.0000	0.983409	0.491704	−	0.870762i	$-0.336374\pi$
$547$	23.0000	0.983409	0.491704	+	0.870762i	$0.336374\pi$
$548$	−8.00000	−0.341743
$549$	0	0
$550$	−3.00000	−0.127920
$551$	−18.0000	−0.766826
$552$	0	0
$553$	−8.00000	−0.340195
$554$	−20.0000	−0.849719
$555$	0	0
$556$	3.00000	0.127228
$557$	40.0000	1.69485	0.847427	−	0.530912i	$-0.178150\pi$
$557$	40.0000	1.69485	0.847427	+	0.530912i	$0.178150\pi$
$558$	0	0
$559$	0	0
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	24.0000	1.01238
$563$	11.0000	0.463595	0.231797	−	0.972764i	$-0.425539\pi$
$563$	11.0000	0.463595	0.231797	+	0.972764i	$0.425539\pi$
$564$	0	0
$565$	−7.00000	−0.294492
$566$	−20.0000	−0.840663
$567$	0	0
$568$	2.00000	0.0839181
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	0	0
$571$	−31.0000	−1.29731	−0.648655	−	0.761083i	$-0.724668\pi$
$571$	−31.0000	−1.29731	−0.648655	+	0.761083i	$0.724668\pi$
$572$	0	0
$573$	0	0
$574$	−3.00000	−0.125218
$575$	−2.00000	−0.0834058
$576$	0	0
$577$	30.0000	1.24892	0.624458	−	0.781058i	$-0.285320\pi$
$577$	30.0000	1.24892	0.624458	+	0.781058i	$0.285320\pi$
$578$	−8.00000	−0.332756
$579$	0	0
$580$	−3.00000	−0.124568
$581$	4.00000	0.165948
$582$	0	0
$583$	39.0000	1.61521
$584$	0	0
$585$	0	0
$586$	−3.00000	−0.123929
$587$	−35.0000	−1.44460	−0.722302	−	0.691577i	$-0.756916\pi$
$587$	−35.0000	−1.44460	−0.722302	+	0.691577i	$0.756916\pi$
$588$	0	0
$589$	−18.0000	−0.741677
$590$	0	0
$591$	0	0
$592$	−1.00000	−0.0410997
$593$	−12.0000	−0.492781	−0.246390	−	0.969171i	$-0.579245\pi$
$593$	−12.0000	−0.492781	−0.246390	+	0.969171i	$0.579245\pi$
$594$	0	0
$595$	3.00000	0.122988
$596$	−18.0000	−0.737309
$597$	0	0
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	21.0000	0.856608	0.428304	−	0.903635i	$-0.359111\pi$
$601$	21.0000	0.856608	0.428304	+	0.903635i	$0.359111\pi$
$602$	−1.00000	−0.0407570
$603$	0	0
$604$	16.0000	0.651031
$605$	2.00000	0.0813116
$606$	0	0
$607$	22.0000	0.892952	0.446476	−	0.894795i	$-0.352679\pi$
$607$	22.0000	0.892952	0.446476	+	0.894795i	$0.352679\pi$
$608$	−6.00000	−0.243332
$609$	0	0
$610$	15.0000	0.607332
$611$	0	0
$612$	0	0
$613$	11.0000	0.444286	0.222143	−	0.975014i	$-0.428695\pi$
$613$	11.0000	0.444286	0.222143	+	0.975014i	$0.428695\pi$
$614$	34.0000	1.37213
$615$	0	0
$616$	−3.00000	−0.120873
$617$	−36.0000	−1.44931	−0.724653	−	0.689114i	$-0.758000\pi$
$617$	−36.0000	−1.44931	−0.724653	+	0.689114i	$0.758000\pi$
$618$	0	0
$619$	−37.0000	−1.48716	−0.743578	−	0.668649i	$-0.766873\pi$
$619$	−37.0000	−1.48716	−0.743578	+	0.668649i	$0.766873\pi$
$620$	−3.00000	−0.120483
$621$	0	0
$622$	−25.0000	−1.00241
$623$	18.0000	0.721155
$624$	0	0
$625$	1.00000	0.0400000
$626$	26.0000	1.03917
$627$	0	0
$628$	−3.00000	−0.119713
$629$	3.00000	0.119618
$630$	0	0
$631$	−41.0000	−1.63218	−0.816092	−	0.577922i	$-0.803864\pi$
$631$	−41.0000	−1.63218	−0.816092	+	0.577922i	$0.803864\pi$
$632$	−8.00000	−0.318223
$633$	0	0
$634$	−21.0000	−0.834017
$635$	4.00000	0.158735
$636$	0	0
$637$	0	0
$638$	−9.00000	−0.356313
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	33.0000	1.30342	0.651711	−	0.758468i	$-0.274052\pi$
$641$	33.0000	1.30342	0.651711	+	0.758468i	$0.274052\pi$
$642$	0	0
$643$	11.0000	0.433798	0.216899	−	0.976194i	$-0.430406\pi$
$643$	11.0000	0.433798	0.216899	+	0.976194i	$0.430406\pi$
$644$	−2.00000	−0.0788110
$645$	0	0
$646$	18.0000	0.708201
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	−5.00000	−0.195815
$653$	−30.0000	−1.17399	−0.586995	−	0.809590i	$-0.699689\pi$
$653$	−30.0000	−1.17399	−0.586995	+	0.809590i	$0.699689\pi$
$654$	0	0
$655$	−12.0000	−0.468879
$656$	−3.00000	−0.117130
$657$	0	0
$658$	−4.00000	−0.155936
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	−13.0000	−0.505641	−0.252821	−	0.967513i	$-0.581358\pi$
$661$	−13.0000	−0.505641	−0.252821	+	0.967513i	$0.581358\pi$
$662$	30.0000	1.16598
$663$	0	0
$664$	4.00000	0.155230
$665$	6.00000	0.232670
$666$	0	0
$667$	−6.00000	−0.232321
$668$	−12.0000	−0.464294
$669$	0	0
$670$	0	0
$671$	45.0000	1.73721
$672$	0	0
$673$	−34.0000	−1.31060	−0.655302	−	0.755367i	$-0.727459\pi$
$673$	−34.0000	−1.31060	−0.655302	+	0.755367i	$0.727459\pi$
$674$	12.0000	0.462223
$675$	0	0
$676$	−13.0000	−0.500000
$677$	−34.0000	−1.30673	−0.653363	−	0.757045i	$-0.726642\pi$
$677$	−34.0000	−1.30673	−0.653363	+	0.757045i	$0.726642\pi$
$678$	0	0
$679$	−7.00000	−0.268635
$680$	3.00000	0.115045
$681$	0	0
$682$	−9.00000	−0.344628
$683$	31.0000	1.18618	0.593091	−	0.805135i	$-0.297907\pi$
$683$	31.0000	1.18618	0.593091	+	0.805135i	$0.297907\pi$
$684$	0	0
$685$	8.00000	0.305664
$686$	−13.0000	−0.496342
$687$	0	0
$688$	−1.00000	−0.0381246
$689$	0	0
$690$	0	0
$691$	5.00000	0.190209	0.0951045	−	0.995467i	$-0.469681\pi$
$691$	5.00000	0.190209	0.0951045	+	0.995467i	$0.469681\pi$
$692$	−9.00000	−0.342129
$693$	0	0
$694$	28.0000	1.06287
$695$	−3.00000	−0.113796
$696$	0	0
$697$	9.00000	0.340899
$698$	−22.0000	−0.832712
$699$	0	0
$700$	1.00000	0.0377964
$701$	22.0000	0.830929	0.415464	−	0.909610i	$-0.363619\pi$
$701$	22.0000	0.830929	0.415464	+	0.909610i	$0.363619\pi$
$702$	0	0
$703$	6.00000	0.226294
$704$	−3.00000	−0.113067
$705$	0	0
$706$	−19.0000	−0.715074
$707$	10.0000	0.376089
$708$	0	0
$709$	−19.0000	−0.713560	−0.356780	−	0.934188i	$-0.616125\pi$
$709$	−19.0000	−0.713560	−0.356780	+	0.934188i	$0.616125\pi$
$710$	−2.00000	−0.0750587
$711$	0	0
$712$	18.0000	0.674579
$713$	−6.00000	−0.224702
$714$	0	0
$715$	0	0
$716$	−24.0000	−0.896922
$717$	0	0
$718$	−16.0000	−0.597115
$719$	18.0000	0.671287	0.335643	−	0.941989i	$-0.391046\pi$
$719$	18.0000	0.671287	0.335643	+	0.941989i	$0.391046\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	17.0000	0.632674
$723$	0	0
$724$	2.00000	0.0743294
$725$	3.00000	0.111417
$726$	0	0
$727$	52.0000	1.92857	0.964287	−	0.264861i	$-0.0853260\pi$
$727$	52.0000	1.92857	0.964287	+	0.264861i	$0.0853260\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	3.00000	0.110959
$732$	0	0
$733$	−41.0000	−1.51437	−0.757185	−	0.653201i	$-0.773426\pi$
$733$	−41.0000	−1.51437	−0.757185	+	0.653201i	$0.773426\pi$
$734$	13.0000	0.479839
$735$	0	0
$736$	−2.00000	−0.0737210
$737$	0	0
$738$	0	0
$739$	27.0000	0.993211	0.496606	−	0.867976i	$-0.334580\pi$
$739$	27.0000	0.993211	0.496606	+	0.867976i	$0.334580\pi$
$740$	1.00000	0.0367607
$741$	0	0
$742$	−13.0000	−0.477245
$743$	−3.00000	−0.110059	−0.0550297	−	0.998485i	$-0.517525\pi$
$743$	−3.00000	−0.110059	−0.0550297	+	0.998485i	$0.517525\pi$
$744$	0	0
$745$	18.0000	0.659469
$746$	22.0000	0.805477
$747$	0	0
$748$	9.00000	0.329073
$749$	−2.00000	−0.0730784
$750$	0	0
$751$	−22.0000	−0.802791	−0.401396	−	0.915905i	$-0.631475\pi$
$751$	−22.0000	−0.802791	−0.401396	+	0.915905i	$0.631475\pi$
$752$	−4.00000	−0.145865
$753$	0	0
$754$	0	0
$755$	−16.0000	−0.582300
$756$	0	0
$757$	−50.0000	−1.81728	−0.908640	−	0.417579i	$-0.862879\pi$
$757$	−50.0000	−1.81728	−0.908640	+	0.417579i	$0.862879\pi$
$758$	16.0000	0.581146
$759$	0	0
$760$	6.00000	0.217643
$761$	21.0000	0.761249	0.380625	−	0.924730i	$-0.375709\pi$
$761$	21.0000	0.761249	0.380625	+	0.924730i	$0.375709\pi$
$762$	0	0
$763$	−3.00000	−0.108607
$764$	−21.0000	−0.759753
$765$	0	0
$766$	12.0000	0.433578
$767$	0	0
$768$	0	0
$769$	8.00000	0.288487	0.144244	−	0.989542i	$-0.453925\pi$
$769$	8.00000	0.288487	0.144244	+	0.989542i	$0.453925\pi$
$770$	3.00000	0.108112
$771$	0	0
$772$	−10.0000	−0.359908
$773$	41.0000	1.47467	0.737334	−	0.675529i	$-0.236085\pi$
$773$	41.0000	1.47467	0.737334	+	0.675529i	$0.236085\pi$
$774$	0	0
$775$	3.00000	0.107763
$776$	−7.00000	−0.251285
$777$	0	0
$778$	−15.0000	−0.537776
$779$	18.0000	0.644917
$780$	0	0
$781$	−6.00000	−0.214697
$782$	6.00000	0.214560
$783$	0	0
$784$	−6.00000	−0.214286
$785$	3.00000	0.107075
$786$	0	0
$787$	−10.0000	−0.356462	−0.178231	−	0.983989i	$-0.557037\pi$
$787$	−10.0000	−0.356462	−0.178231	+	0.983989i	$0.557037\pi$
$788$	18.0000	0.641223
$789$	0	0
$790$	8.00000	0.284627
$791$	7.00000	0.248891
$792$	0	0
$793$	0	0
$794$	18.0000	0.638796
$795$	0	0
$796$	8.00000	0.283552
$797$	26.0000	0.920967	0.460484	−	0.887668i	$-0.347676\pi$
$797$	26.0000	0.920967	0.460484	+	0.887668i	$0.347676\pi$
$798$	0	0
$799$	12.0000	0.424529
$800$	1.00000	0.0353553
$801$	0	0
$802$	−34.0000	−1.20058
$803$	0	0
$804$	0	0
$805$	2.00000	0.0704907
$806$	0	0
$807$	0	0
$808$	10.0000	0.351799
$809$	−34.0000	−1.19538	−0.597688	−	0.801729i	$-0.703914\pi$
$809$	−34.0000	−1.19538	−0.597688	+	0.801729i	$0.703914\pi$
$810$	0	0
$811$	−16.0000	−0.561836	−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000	−0.561836	−0.280918	+	0.959732i	$0.590639\pi$
$812$	3.00000	0.105279
$813$	0	0
$814$	3.00000	0.105150
$815$	5.00000	0.175142
$816$	0	0
$817$	6.00000	0.209913
$818$	0	0
$819$	0	0
$820$	3.00000	0.104765
$821$	−8.00000	−0.279202	−0.139601	−	0.990208i	$-0.544582\pi$
$821$	−8.00000	−0.279202	−0.139601	+	0.990208i	$0.544582\pi$
$822$	0	0
$823$	24.0000	0.836587	0.418294	−	0.908312i	$-0.362628\pi$
$823$	24.0000	0.836587	0.418294	+	0.908312i	$0.362628\pi$
$824$	8.00000	0.278693
$825$	0	0
$826$	0	0
$827$	11.0000	0.382507	0.191254	−	0.981541i	$-0.438745\pi$
$827$	11.0000	0.382507	0.191254	+	0.981541i	$0.438745\pi$
$828$	0	0
$829$	−39.0000	−1.35453	−0.677263	−	0.735741i	$-0.736834\pi$
$829$	−39.0000	−1.35453	−0.677263	+	0.735741i	$0.736834\pi$
$830$	−4.00000	−0.138842
$831$	0	0
$832$	0	0
$833$	18.0000	0.623663
$834$	0	0
$835$	12.0000	0.415277
$836$	18.0000	0.622543
$837$	0	0
$838$	−12.0000	−0.414533
$839$	−36.0000	−1.24286	−0.621429	−	0.783470i	$-0.713448\pi$
$839$	−36.0000	−1.24286	−0.621429	+	0.783470i	$0.713448\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	34.0000	1.17172
$843$	0	0
$844$	−3.00000	−0.103264
$845$	13.0000	0.447214
$846$	0	0
$847$	−2.00000	−0.0687208
$848$	−13.0000	−0.446422
$849$	0	0
$850$	−3.00000	−0.102899
$851$	2.00000	0.0685591
$852$	0	0
$853$	22.0000	0.753266	0.376633	−	0.926363i	$-0.377082\pi$
$853$	22.0000	0.753266	0.376633	+	0.926363i	$0.377082\pi$
$854$	−15.0000	−0.513289
$855$	0	0
$856$	−2.00000	−0.0683586
$857$	−33.0000	−1.12726	−0.563629	−	0.826028i	$-0.690595\pi$
$857$	−33.0000	−1.12726	−0.563629	+	0.826028i	$0.690595\pi$
$858$	0	0
$859$	−36.0000	−1.22830	−0.614152	−	0.789188i	$-0.710502\pi$
$859$	−36.0000	−1.22830	−0.614152	+	0.789188i	$0.710502\pi$
$860$	1.00000	0.0340997
$861$	0	0
$862$	31.0000	1.05586
$863$	−33.0000	−1.12333	−0.561667	−	0.827364i	$-0.689840\pi$
$863$	−33.0000	−1.12333	−0.561667	+	0.827364i	$0.689840\pi$
$864$	0	0
$865$	9.00000	0.306009
$866$	−22.0000	−0.747590
$867$	0	0
$868$	3.00000	0.101827
$869$	24.0000	0.814144
$870$	0	0
$871$	0	0
$872$	−3.00000	−0.101593
$873$	0	0
$874$	12.0000	0.405906
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−43.0000	−1.45201	−0.726003	−	0.687691i	$-0.758624\pi$
$877$	−43.0000	−1.45201	−0.726003	+	0.687691i	$0.758624\pi$
$878$	37.0000	1.24869
$879$	0	0
$880$	3.00000	0.101130
$881$	9.00000	0.303218	0.151609	−	0.988441i	$-0.451555\pi$
$881$	9.00000	0.303218	0.151609	+	0.988441i	$0.451555\pi$
$882$	0	0
$883$	29.0000	0.975928	0.487964	−	0.872864i	$-0.337740\pi$
$883$	29.0000	0.975928	0.487964	+	0.872864i	$0.337740\pi$
$884$	0	0
$885$	0	0
$886$	34.0000	1.14225
$887$	33.0000	1.10803	0.554016	−	0.832506i	$-0.313095\pi$
$887$	33.0000	1.10803	0.554016	+	0.832506i	$0.313095\pi$
$888$	0	0
$889$	−4.00000	−0.134156
$890$	−18.0000	−0.603361
$891$	0	0
$892$	23.0000	0.770097
$893$	24.0000	0.803129
$894$	0	0
$895$	24.0000	0.802232
$896$	1.00000	0.0334077
$897$	0	0
$898$	10.0000	0.333704
$899$	9.00000	0.300167
$900$	0	0
$901$	39.0000	1.29928
$902$	9.00000	0.299667
$903$	0	0
$904$	7.00000	0.232817
$905$	−2.00000	−0.0664822
$906$	0	0
$907$	−12.0000	−0.398453	−0.199227	−	0.979953i	$-0.563843\pi$
$907$	−12.0000	−0.398453	−0.199227	+	0.979953i	$0.563843\pi$
$908$	−13.0000	−0.431420
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	−12.0000	−0.397142
$914$	9.00000	0.297694
$915$	0	0
$916$	−6.00000	−0.198246
$917$	12.0000	0.396275
$918$	0	0
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	2.00000	0.0659380
$921$	0	0
$922$	1.00000	0.0329332
$923$	0	0
$924$	0	0
$925$	−1.00000	−0.0328798
$926$	12.0000	0.394344
$927$	0	0
$928$	3.00000	0.0984798
$929$	3.00000	0.0984268	0.0492134	−	0.998788i	$-0.484329\pi$
$929$	3.00000	0.0984268	0.0492134	+	0.998788i	$0.484329\pi$
$930$	0	0
$931$	36.0000	1.17985
$932$	−4.00000	−0.131024
$933$	0	0
$934$	37.0000	1.21068
$935$	−9.00000	−0.294331
$936$	0	0
$937$	−14.0000	−0.457360	−0.228680	−	0.973502i	$-0.573441\pi$
$937$	−14.0000	−0.457360	−0.228680	+	0.973502i	$0.573441\pi$
$938$	0	0
$939$	0	0
$940$	4.00000	0.130466
$941$	6.00000	0.195594	0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000	0.195594	0.0977972	+	0.995206i	$0.468820\pi$
$942$	0	0
$943$	6.00000	0.195387
$944$	0	0
$945$	0	0
$946$	3.00000	0.0975384
$947$	−9.00000	−0.292461	−0.146230	−	0.989251i	$-0.546714\pi$
$947$	−9.00000	−0.292461	−0.146230	+	0.989251i	$0.546714\pi$
$948$	0	0
$949$	0	0
$950$	−6.00000	−0.194666
$951$	0	0
$952$	−3.00000	−0.0972306
$953$	−36.0000	−1.16615	−0.583077	−	0.812417i	$-0.698151\pi$
$953$	−36.0000	−1.16615	−0.583077	+	0.812417i	$0.698151\pi$
$954$	0	0
$955$	21.0000	0.679544
$956$	−9.00000	−0.291081
$957$	0	0
$958$	8.00000	0.258468
$959$	−8.00000	−0.258333
$960$	0	0
$961$	−22.0000	−0.709677
$962$	0	0
$963$	0	0
$964$	24.0000	0.772988
$965$	10.0000	0.321911
$966$	0	0
$967$	48.0000	1.54358	0.771788	−	0.635880i	$-0.219363\pi$
$967$	48.0000	1.54358	0.771788	+	0.635880i	$0.219363\pi$
$968$	−2.00000	−0.0642824
$969$	0	0
$970$	7.00000	0.224756
$971$	−9.00000	−0.288824	−0.144412	−	0.989518i	$-0.546129\pi$
$971$	−9.00000	−0.288824	−0.144412	+	0.989518i	$0.546129\pi$
$972$	0	0
$973$	3.00000	0.0961756
$974$	34.0000	1.08943
$975$	0	0
$976$	−15.0000	−0.480138
$977$	−35.0000	−1.11975	−0.559875	−	0.828577i	$-0.689151\pi$
$977$	−35.0000	−1.11975	−0.559875	+	0.828577i	$0.689151\pi$
$978$	0	0
$979$	−54.0000	−1.72585
$980$	6.00000	0.191663
$981$	0	0
$982$	20.0000	0.638226
$983$	27.0000	0.861166	0.430583	−	0.902551i	$-0.358308\pi$
$983$	27.0000	0.861166	0.430583	+	0.902551i	$0.358308\pi$
$984$	0	0
$985$	−18.0000	−0.573528
$986$	−9.00000	−0.286618
$987$	0	0
$988$	0	0
$989$	2.00000	0.0635963
$990$	0	0
$991$	33.0000	1.04828	0.524140	−	0.851632i	$-0.324387\pi$
$991$	33.0000	1.04828	0.524140	+	0.851632i	$0.324387\pi$
$992$	3.00000	0.0952501
$993$	0	0
$994$	2.00000	0.0634361
$995$	−8.00000	−0.253617
$996$	0	0
$997$	−12.0000	−0.380044	−0.190022	−	0.981780i	$-0.560856\pi$
$997$	−12.0000	−0.380044	−0.190022	+	0.981780i	$0.560856\pi$
$998$	−22.0000	−0.696398
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3330.2.a.p.1.1		1
3.2	odd	2		370.2.a.c.1.1	✓	1
12.11	even	2		2960.2.a.c.1.1		1
15.2	even	4		1850.2.b.c.149.1		2
15.8	even	4		1850.2.b.c.149.2		2
15.14	odd	2		1850.2.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.a.c.1.1	✓	1	3.2	odd	2
1850.2.a.i.1.1		1	15.14	odd	2
1850.2.b.c.149.1		2	15.2	even	4
1850.2.b.c.149.2		2	15.8	even	4
2960.2.a.c.1.1		1	12.11	even	2
3330.2.a.p.1.1		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)

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