LMFDB - Embedded newform 153.2.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [153,2,Mod(1,153)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(153, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("153.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 153 = 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	153.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:	$1.22171115093$
Analytic rank:	$0$
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	$\chi$	$=$	153.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} +2.00000 q^{4} +1.00000 q^{5} -2.00000 q^{7} +O(q^{10})$

$q+2.00000 q^{2} +2.00000 q^{4} +1.00000 q^{5} -2.00000 q^{7} +2.00000 q^{10} +3.00000 q^{11} -5.00000 q^{13} -4.00000 q^{14} -4.00000 q^{16} +1.00000 q^{17} -1.00000 q^{19} +2.00000 q^{20} +6.00000 q^{22} +7.00000 q^{23} -4.00000 q^{25} -10.0000 q^{26} -4.00000 q^{28} -6.00000 q^{29} +4.00000 q^{31} -8.00000 q^{32} +2.00000 q^{34} -2.00000 q^{35} +10.0000 q^{37} -2.00000 q^{38} -9.00000 q^{41} +1.00000 q^{43} +6.00000 q^{44} +14.0000 q^{46} +12.0000 q^{47} -3.00000 q^{49} -8.00000 q^{50} -10.0000 q^{52} +12.0000 q^{53} +3.00000 q^{55} -12.0000 q^{58} -6.00000 q^{59} +2.00000 q^{61} +8.00000 q^{62} -8.00000 q^{64} -5.00000 q^{65} +4.00000 q^{67} +2.00000 q^{68} -4.00000 q^{70} +8.00000 q^{71} +20.0000 q^{74} -2.00000 q^{76} -6.00000 q^{77} -6.00000 q^{79} -4.00000 q^{80} -18.0000 q^{82} -4.00000 q^{83} +1.00000 q^{85} +2.00000 q^{86} -2.00000 q^{89} +10.0000 q^{91} +14.0000 q^{92} +24.0000 q^{94} -1.00000 q^{95} +8.00000 q^{97} -6.00000 q^{98} +O(q^{100})$

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 $

$2$	2.00000	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$2$	2.00000	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$3$	0	0
$3$	0	0
$4$	2.00000	1.00000
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	0	0
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	0	0
$9$	0	0
$10$	2.00000	0.632456
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	0	0
$13$	−5.00000	−1.38675	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000	−1.38675	−0.693375	+	0.720577i	$0.743877\pi$
$14$	−4.00000	−1.06904
$15$	0	0
$16$	−4.00000	−1.00000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	2.00000	0.447214
$21$	0	0
$22$	6.00000	1.27920
$23$	7.00000	1.45960	0.729800	−	0.683660i	$-0.239613\pi$
$23$	7.00000	1.45960	0.729800	+	0.683660i	$0.239613\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	−10.0000	−1.96116
$27$	0	0
$28$	−4.00000	−0.755929
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	−8.00000	−1.41421
$33$	0	0
$34$	2.00000	0.342997
$35$	−2.00000	−0.338062
$36$	0	0
$37$	10.0000	1.64399	0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000	1.64399	0.821995	+	0.569495i	$0.192861\pi$
$38$	−2.00000	−0.324443
$39$	0	0
$40$	0	0
$41$	−9.00000	−1.40556	−0.702782	−	0.711405i	$-0.748059\pi$
$41$	−9.00000	−1.40556	−0.702782	+	0.711405i	$0.748059\pi$
$42$	0	0
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	6.00000	0.904534
$45$	0	0
$46$	14.0000	2.06419
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	−8.00000	−1.13137
$51$	0	0
$52$	−10.0000	−1.38675
$53$	12.0000	1.64833	0.824163	−	0.566352i	$-0.191646\pi$
$53$	12.0000	1.64833	0.824163	+	0.566352i	$0.191646\pi$
$54$	0	0
$55$	3.00000	0.404520
$56$	0	0
$57$	0	0
$58$	−12.0000	−1.57568
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	8.00000	1.01600
$63$	0	0
$64$	−8.00000	−1.00000
$65$	−5.00000	−0.620174
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	2.00000	0.242536
$69$	0	0
$70$	−4.00000	−0.478091
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	20.0000	2.32495
$75$	0	0
$76$	−2.00000	−0.229416
$77$	−6.00000	−0.683763
$78$	0	0
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	−4.00000	−0.447214
$81$	0	0
$82$	−18.0000	−1.98777
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	1.00000	0.108465
$86$	2.00000	0.215666
$87$	0	0
$88$	0	0
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	10.0000	1.04828
$92$	14.0000	1.45960
$93$	0	0
$94$	24.0000	2.47541
$95$	−1.00000	−0.102598
$96$	0	0
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	−6.00000	−0.606092
$99$	0	0
$100$	−8.00000	−0.800000
$101$	−4.00000	−0.398015	−0.199007	−	0.979998i	$-0.563772\pi$
$101$	−4.00000	−0.398015	−0.199007	+	0.979998i	$0.563772\pi$
$102$	0	0
$103$	−19.0000	−1.87213	−0.936063	−	0.351833i	$-0.885559\pi$
$103$	−19.0000	−1.87213	−0.936063	+	0.351833i	$0.885559\pi$
$104$	0	0
$105$	0	0
$106$	24.0000	2.33109
$107$	11.0000	1.06341	0.531705	−	0.846930i	$-0.321551\pi$
$107$	11.0000	1.06341	0.531705	+	0.846930i	$0.321551\pi$
$108$	0	0
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	6.00000	0.572078
$111$	0	0
$112$	8.00000	0.755929
$113$	1.00000	0.0940721	0.0470360	−	0.998893i	$-0.485022\pi$
$113$	1.00000	0.0940721	0.0470360	+	0.998893i	$0.485022\pi$
$114$	0	0
$115$	7.00000	0.652753
$116$	−12.0000	−1.11417
$117$	0	0
$118$	−12.0000	−1.10469
$119$	−2.00000	−0.183340
$120$	0	0
$121$	−2.00000	−0.181818
$122$	4.00000	0.362143
$123$	0	0
$124$	8.00000	0.718421
$125$	−9.00000	−0.804984
$126$	0	0
$127$	11.0000	0.976092	0.488046	−	0.872818i	$-0.337710\pi$
$127$	11.0000	0.976092	0.488046	+	0.872818i	$0.337710\pi$
$128$	0	0
$129$	0	0
$130$	−10.0000	−0.877058
$131$	−11.0000	−0.961074	−0.480537	−	0.876974i	$-0.659558\pi$
$131$	−11.0000	−0.961074	−0.480537	+	0.876974i	$0.659558\pi$
$132$	0	0
$133$	2.00000	0.173422
$134$	8.00000	0.691095
$135$	0	0
$136$	0	0
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	0	0
$139$	−14.0000	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$140$	−4.00000	−0.338062
$141$	0	0
$142$	16.0000	1.34269
$143$	−15.0000	−1.25436
$144$	0	0
$145$	−6.00000	−0.498273
$146$	0	0
$147$	0	0
$148$	20.0000	1.64399
$149$	2.00000	0.163846	0.0819232	−	0.996639i	$-0.473894\pi$
$149$	2.00000	0.163846	0.0819232	+	0.996639i	$0.473894\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	0	0
$153$	0	0
$154$	−12.0000	−0.966988
$155$	4.00000	0.321288
$156$	0	0
$157$	−5.00000	−0.399043	−0.199522	−	0.979893i	$-0.563939\pi$
$157$	−5.00000	−0.399043	−0.199522	+	0.979893i	$0.563939\pi$
$158$	−12.0000	−0.954669
$159$	0	0
$160$	−8.00000	−0.632456
$161$	−14.0000	−1.10335
$162$	0	0
$163$	−24.0000	−1.87983	−0.939913	−	0.341415i	$-0.889094\pi$
$163$	−24.0000	−1.87983	−0.939913	+	0.341415i	$0.889094\pi$
$164$	−18.0000	−1.40556
$165$	0	0
$166$	−8.00000	−0.620920
$167$	23.0000	1.77979	0.889897	−	0.456162i	$-0.150776\pi$
$167$	23.0000	1.77979	0.889897	+	0.456162i	$0.150776\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	2.00000	0.153393
$171$	0	0
$172$	2.00000	0.152499
$173$	−23.0000	−1.74866	−0.874329	−	0.485334i	$-0.838698\pi$
$173$	−23.0000	−1.74866	−0.874329	+	0.485334i	$0.838698\pi$
$174$	0	0
$175$	8.00000	0.604743
$176$	−12.0000	−0.904534
$177$	0	0
$178$	−4.00000	−0.299813
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	20.0000	1.48250
$183$	0	0
$184$	0	0
$185$	10.0000	0.735215
$186$	0	0
$187$	3.00000	0.219382
$188$	24.0000	1.75038
$189$	0	0
$190$	−2.00000	−0.145095
$191$	14.0000	1.01300	0.506502	−	0.862239i	$-0.330938\pi$
$191$	14.0000	1.01300	0.506502	+	0.862239i	$0.330938\pi$
$192$	0	0
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	16.0000	1.14873
$195$	0	0
$196$	−6.00000	−0.428571
$197$	−3.00000	−0.213741	−0.106871	−	0.994273i	$-0.534083\pi$
$197$	−3.00000	−0.213741	−0.106871	+	0.994273i	$0.534083\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	0	0
$202$	−8.00000	−0.562878
$203$	12.0000	0.842235
$204$	0	0
$205$	−9.00000	−0.628587
$206$	−38.0000	−2.64759
$207$	0	0
$208$	20.0000	1.38675
$209$	−3.00000	−0.207514
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	24.0000	1.64833
$213$	0	0
$214$	22.0000	1.50389
$215$	1.00000	0.0681994
$216$	0	0
$217$	−8.00000	−0.543075
$218$	0	0
$219$	0	0
$220$	6.00000	0.404520
$221$	−5.00000	−0.336336
$222$	0	0
$223$	15.0000	1.00447	0.502237	−	0.864730i	$-0.332510\pi$
$223$	15.0000	1.00447	0.502237	+	0.864730i	$0.332510\pi$
$224$	16.0000	1.06904
$225$	0	0
$226$	2.00000	0.133038
$227$	−15.0000	−0.995585	−0.497792	−	0.867296i	$-0.665856\pi$
$227$	−15.0000	−0.995585	−0.497792	+	0.867296i	$0.665856\pi$
$228$	0	0
$229$	−18.0000	−1.18947	−0.594737	−	0.803921i	$-0.702744\pi$
$229$	−18.0000	−1.18947	−0.594737	+	0.803921i	$0.702744\pi$
$230$	14.0000	0.923133
$231$	0	0
$232$	0	0
$233$	15.0000	0.982683	0.491341	−	0.870967i	$-0.336507\pi$
$233$	15.0000	0.982683	0.491341	+	0.870967i	$0.336507\pi$
$234$	0	0
$235$	12.0000	0.782794
$236$	−12.0000	−0.781133
$237$	0	0
$238$	−4.00000	−0.259281
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	−4.00000	−0.257130
$243$	0	0
$244$	4.00000	0.256074
$245$	−3.00000	−0.191663
$246$	0	0
$247$	5.00000	0.318142
$248$	0	0
$249$	0	0
$250$	−18.0000	−1.13842
$251$	−18.0000	−1.13615	−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000	−1.13615	−0.568075	+	0.822977i	$0.692312\pi$
$252$	0	0
$253$	21.0000	1.32026
$254$	22.0000	1.38040
$255$	0	0
$256$	16.0000	1.00000
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	−20.0000	−1.24274
$260$	−10.0000	−0.620174
$261$	0	0
$262$	−22.0000	−1.35916
$263$	2.00000	0.123325	0.0616626	−	0.998097i	$-0.480360\pi$
$263$	2.00000	0.123325	0.0616626	+	0.998097i	$0.480360\pi$
$264$	0	0
$265$	12.0000	0.737154
$266$	4.00000	0.245256
$267$	0	0
$268$	8.00000	0.488678
$269$	31.0000	1.89010	0.945052	−	0.326921i	$-0.106011\pi$
$269$	31.0000	1.89010	0.945052	+	0.326921i	$0.106011\pi$
$270$	0	0
$271$	−13.0000	−0.789694	−0.394847	−	0.918747i	$-0.629202\pi$
$271$	−13.0000	−0.789694	−0.394847	+	0.918747i	$0.629202\pi$
$272$	−4.00000	−0.242536
$273$	0	0
$274$	−36.0000	−2.17484
$275$	−12.0000	−0.723627
$276$	0	0
$277$	−4.00000	−0.240337	−0.120168	−	0.992754i	$-0.538343\pi$
$277$	−4.00000	−0.240337	−0.120168	+	0.992754i	$0.538343\pi$
$278$	−28.0000	−1.67933
$279$	0	0
$280$	0	0
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	0	0
$283$	2.00000	0.118888	0.0594438	−	0.998232i	$-0.481067\pi$
$283$	2.00000	0.118888	0.0594438	+	0.998232i	$0.481067\pi$
$284$	16.0000	0.949425
$285$	0	0
$286$	−30.0000	−1.77394
$287$	18.0000	1.06251
$288$	0	0
$289$	1.00000	0.0588235
$290$	−12.0000	−0.704664
$291$	0	0
$292$	0	0
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	−6.00000	−0.349334
$296$	0	0
$297$	0	0
$298$	4.00000	0.231714
$299$	−35.0000	−2.02410
$300$	0	0
$301$	−2.00000	−0.115278
$302$	−32.0000	−1.84139
$303$	0	0
$304$	4.00000	0.229416
$305$	2.00000	0.114520
$306$	0	0
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	−12.0000	−0.683763
$309$	0	0
$310$	8.00000	0.454369
$311$	16.0000	0.907277	0.453638	−	0.891186i	$-0.350126\pi$
$311$	16.0000	0.907277	0.453638	+	0.891186i	$0.350126\pi$
$312$	0	0
$313$	−16.0000	−0.904373	−0.452187	−	0.891923i	$-0.649356\pi$
$313$	−16.0000	−0.904373	−0.452187	+	0.891923i	$0.649356\pi$
$314$	−10.0000	−0.564333
$315$	0	0
$316$	−12.0000	−0.675053
$317$	−10.0000	−0.561656	−0.280828	−	0.959758i	$-0.590609\pi$
$317$	−10.0000	−0.561656	−0.280828	+	0.959758i	$0.590609\pi$
$318$	0	0
$319$	−18.0000	−1.00781
$320$	−8.00000	−0.447214
$321$	0	0
$322$	−28.0000	−1.56038
$323$	−1.00000	−0.0556415
$324$	0	0
$325$	20.0000	1.10940
$326$	−48.0000	−2.65847
$327$	0	0
$328$	0	0
$329$	−24.0000	−1.32316
$330$	0	0
$331$	7.00000	0.384755	0.192377	−	0.981321i	$-0.438380\pi$
$331$	7.00000	0.384755	0.192377	+	0.981321i	$0.438380\pi$
$332$	−8.00000	−0.439057
$333$	0	0
$334$	46.0000	2.51701
$335$	4.00000	0.218543
$336$	0	0
$337$	22.0000	1.19842	0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000	1.19842	0.599208	+	0.800593i	$0.295482\pi$
$338$	24.0000	1.30543
$339$	0	0
$340$	2.00000	0.108465
$341$	12.0000	0.649836
$342$	0	0
$343$	20.0000	1.07990
$344$	0	0
$345$	0	0
$346$	−46.0000	−2.47298
$347$	−4.00000	−0.214731	−0.107366	−	0.994220i	$-0.534242\pi$
$347$	−4.00000	−0.214731	−0.107366	+	0.994220i	$0.534242\pi$
$348$	0	0
$349$	−15.0000	−0.802932	−0.401466	−	0.915874i	$-0.631499\pi$
$349$	−15.0000	−0.802932	−0.401466	+	0.915874i	$0.631499\pi$
$350$	16.0000	0.855236
$351$	0	0
$352$	−24.0000	−1.27920
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	0	0
$355$	8.00000	0.424596
$356$	−4.00000	−0.212000
$357$	0	0
$358$	24.0000	1.26844
$359$	−6.00000	−0.316668	−0.158334	−	0.987386i	$-0.550612\pi$
$359$	−6.00000	−0.316668	−0.158334	+	0.987386i	$0.550612\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	−28.0000	−1.47165
$363$	0	0
$364$	20.0000	1.04828
$365$	0	0
$366$	0	0
$367$	−2.00000	−0.104399	−0.0521996	−	0.998637i	$-0.516623\pi$
$367$	−2.00000	−0.104399	−0.0521996	+	0.998637i	$0.516623\pi$
$368$	−28.0000	−1.45960
$369$	0	0
$370$	20.0000	1.03975
$371$	−24.0000	−1.24602
$372$	0	0
$373$	30.0000	1.55334	0.776671	−	0.629907i	$-0.216907\pi$
$373$	30.0000	1.55334	0.776671	+	0.629907i	$0.216907\pi$
$374$	6.00000	0.310253
$375$	0	0
$376$	0	0
$377$	30.0000	1.54508
$378$	0	0
$379$	−14.0000	−0.719132	−0.359566	−	0.933120i	$-0.617075\pi$
$379$	−14.0000	−0.719132	−0.359566	+	0.933120i	$0.617075\pi$
$380$	−2.00000	−0.102598
$381$	0	0
$382$	28.0000	1.43260
$383$	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
$385$	−6.00000	−0.305788
$386$	−8.00000	−0.407189
$387$	0	0
$388$	16.0000	0.812277
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	7.00000	0.354005
$392$	0	0
$393$	0	0
$394$	−6.00000	−0.302276
$395$	−6.00000	−0.301893
$396$	0	0
$397$	12.0000	0.602263	0.301131	−	0.953583i	$-0.402636\pi$
$397$	12.0000	0.602263	0.301131	+	0.953583i	$0.402636\pi$
$398$	32.0000	1.60402
$399$	0	0
$400$	16.0000	0.800000
$401$	7.00000	0.349563	0.174782	−	0.984607i	$-0.444078\pi$
$401$	7.00000	0.349563	0.174782	+	0.984607i	$0.444078\pi$
$402$	0	0
$403$	−20.0000	−0.996271
$404$	−8.00000	−0.398015
$405$	0	0
$406$	24.0000	1.19110
$407$	30.0000	1.48704
$408$	0	0
$409$	5.00000	0.247234	0.123617	−	0.992330i	$-0.460551\pi$
$409$	5.00000	0.247234	0.123617	+	0.992330i	$0.460551\pi$
$410$	−18.0000	−0.888957
$411$	0	0
$412$	−38.0000	−1.87213
$413$	12.0000	0.590481
$414$	0	0
$415$	−4.00000	−0.196352
$416$	40.0000	1.96116
$417$	0	0
$418$	−6.00000	−0.293470
$419$	−4.00000	−0.195413	−0.0977064	−	0.995215i	$-0.531151\pi$
$419$	−4.00000	−0.195413	−0.0977064	+	0.995215i	$0.531151\pi$
$420$	0	0
$421$	−5.00000	−0.243685	−0.121843	−	0.992549i	$-0.538880\pi$
$421$	−5.00000	−0.243685	−0.121843	+	0.992549i	$0.538880\pi$
$422$	40.0000	1.94717
$423$	0	0
$424$	0	0
$425$	−4.00000	−0.194029
$426$	0	0
$427$	−4.00000	−0.193574
$428$	22.0000	1.06341
$429$	0	0
$430$	2.00000	0.0964486
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	−25.0000	−1.20142	−0.600712	−	0.799466i	$-0.705116\pi$
$433$	−25.0000	−1.20142	−0.600712	+	0.799466i	$0.705116\pi$
$434$	−16.0000	−0.768025
$435$	0	0
$436$	0	0
$437$	−7.00000	−0.334855
$438$	0	0
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	0	0
$441$	0	0
$442$	−10.0000	−0.475651
$443$	−20.0000	−0.950229	−0.475114	−	0.879924i	$-0.657593\pi$
$443$	−20.0000	−0.950229	−0.475114	+	0.879924i	$0.657593\pi$
$444$	0	0
$445$	−2.00000	−0.0948091
$446$	30.0000	1.42054
$447$	0	0
$448$	16.0000	0.755929
$449$	−14.0000	−0.660701	−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000	−0.660701	−0.330350	+	0.943858i	$0.607167\pi$
$450$	0	0
$451$	−27.0000	−1.27138
$452$	2.00000	0.0940721
$453$	0	0
$454$	−30.0000	−1.40797
$455$	10.0000	0.468807
$456$	0	0
$457$	−15.0000	−0.701670	−0.350835	−	0.936437i	$-0.614102\pi$
$457$	−15.0000	−0.701670	−0.350835	+	0.936437i	$0.614102\pi$
$458$	−36.0000	−1.68217
$459$	0	0
$460$	14.0000	0.652753
$461$	10.0000	0.465746	0.232873	−	0.972507i	$-0.425187\pi$
$461$	10.0000	0.465746	0.232873	+	0.972507i	$0.425187\pi$
$462$	0	0
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	24.0000	1.11417
$465$	0	0
$466$	30.0000	1.38972
$467$	−30.0000	−1.38823	−0.694117	−	0.719862i	$-0.744205\pi$
$467$	−30.0000	−1.38823	−0.694117	+	0.719862i	$0.744205\pi$
$468$	0	0
$469$	−8.00000	−0.369406
$470$	24.0000	1.10704
$471$	0	0
$472$	0	0
$473$	3.00000	0.137940
$474$	0	0
$475$	4.00000	0.183533
$476$	−4.00000	−0.183340
$477$	0	0
$478$	16.0000	0.731823
$479$	−21.0000	−0.959514	−0.479757	−	0.877401i	$-0.659275\pi$
$479$	−21.0000	−0.959514	−0.479757	+	0.877401i	$0.659275\pi$
$480$	0	0
$481$	−50.0000	−2.27980
$482$	36.0000	1.63976
$483$	0	0
$484$	−4.00000	−0.181818
$485$	8.00000	0.363261
$486$	0	0
$487$	−22.0000	−0.996915	−0.498458	−	0.866914i	$-0.666100\pi$
$487$	−22.0000	−0.996915	−0.498458	+	0.866914i	$0.666100\pi$
$488$	0	0
$489$	0	0
$490$	−6.00000	−0.271052
$491$	−4.00000	−0.180517	−0.0902587	−	0.995918i	$-0.528769\pi$
$491$	−4.00000	−0.180517	−0.0902587	+	0.995918i	$0.528769\pi$
$492$	0	0
$493$	−6.00000	−0.270226
$494$	10.0000	0.449921
$495$	0	0
$496$	−16.0000	−0.718421
$497$	−16.0000	−0.717698
$498$	0	0
$499$	−16.0000	−0.716258	−0.358129	−	0.933672i	$-0.616585\pi$
$499$	−16.0000	−0.716258	−0.358129	+	0.933672i	$0.616585\pi$
$500$	−18.0000	−0.804984
$501$	0	0
$502$	−36.0000	−1.60676
$503$	9.00000	0.401290	0.200645	−	0.979664i	$-0.435696\pi$
$503$	9.00000	0.401290	0.200645	+	0.979664i	$0.435696\pi$
$504$	0	0
$505$	−4.00000	−0.177998
$506$	42.0000	1.86713
$507$	0	0
$508$	22.0000	0.976092
$509$	−20.0000	−0.886484	−0.443242	−	0.896402i	$-0.646172\pi$
$509$	−20.0000	−0.886484	−0.443242	+	0.896402i	$0.646172\pi$
$510$	0	0
$511$	0	0
$512$	32.0000	1.41421
$513$	0	0
$514$	12.0000	0.529297
$515$	−19.0000	−0.837240
$516$	0	0
$517$	36.0000	1.58328
$518$	−40.0000	−1.75750
$519$	0	0
$520$	0	0
$521$	35.0000	1.53338	0.766689	−	0.642019i	$-0.221903\pi$
$521$	35.0000	1.53338	0.766689	+	0.642019i	$0.221903\pi$
$522$	0	0
$523$	36.0000	1.57417	0.787085	−	0.616844i	$-0.211589\pi$
$523$	36.0000	1.57417	0.787085	+	0.616844i	$0.211589\pi$
$524$	−22.0000	−0.961074
$525$	0	0
$526$	4.00000	0.174408
$527$	4.00000	0.174243
$528$	0	0
$529$	26.0000	1.13043
$530$	24.0000	1.04249
$531$	0	0
$532$	4.00000	0.173422
$533$	45.0000	1.94917
$534$	0	0
$535$	11.0000	0.475571
$536$	0	0
$537$	0	0
$538$	62.0000	2.67301
$539$	−9.00000	−0.387657
$540$	0	0
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	−26.0000	−1.11680
$543$	0	0
$544$	−8.00000	−0.342997
$545$	0	0
$546$	0	0
$547$	−14.0000	−0.598597	−0.299298	−	0.954160i	$-0.596753\pi$
$547$	−14.0000	−0.598597	−0.299298	+	0.954160i	$0.596753\pi$
$548$	−36.0000	−1.53784
$549$	0	0
$550$	−24.0000	−1.02336
$551$	6.00000	0.255609
$552$	0	0
$553$	12.0000	0.510292
$554$	−8.00000	−0.339887
$555$	0	0
$556$	−28.0000	−1.18746
$557$	−30.0000	−1.27114	−0.635570	−	0.772043i	$-0.719235\pi$
$557$	−30.0000	−1.27114	−0.635570	+	0.772043i	$0.719235\pi$
$558$	0	0
$559$	−5.00000	−0.211477
$560$	8.00000	0.338062
$561$	0	0
$562$	60.0000	2.53095
$563$	8.00000	0.337160	0.168580	−	0.985688i	$-0.446082\pi$
$563$	8.00000	0.337160	0.168580	+	0.985688i	$0.446082\pi$
$564$	0	0
$565$	1.00000	0.0420703
$566$	4.00000	0.168133
$567$	0	0
$568$	0	0
$569$	16.0000	0.670755	0.335377	−	0.942084i	$-0.391136\pi$
$569$	16.0000	0.670755	0.335377	+	0.942084i	$0.391136\pi$
$570$	0	0
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	−30.0000	−1.25436
$573$	0	0
$574$	36.0000	1.50261
$575$	−28.0000	−1.16768
$576$	0	0
$577$	37.0000	1.54033	0.770165	−	0.637845i	$-0.220174\pi$
$577$	37.0000	1.54033	0.770165	+	0.637845i	$0.220174\pi$
$578$	2.00000	0.0831890
$579$	0	0
$580$	−12.0000	−0.498273
$581$	8.00000	0.331896
$582$	0	0
$583$	36.0000	1.49097
$584$	0	0
$585$	0	0
$586$	12.0000	0.495715
$587$	−32.0000	−1.32078	−0.660391	−	0.750922i	$-0.729609\pi$
$587$	−32.0000	−1.32078	−0.660391	+	0.750922i	$0.729609\pi$
$588$	0	0
$589$	−4.00000	−0.164817
$590$	−12.0000	−0.494032
$591$	0	0
$592$	−40.0000	−1.64399
$593$	−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
$595$	−2.00000	−0.0819920
$596$	4.00000	0.163846
$597$	0	0
$598$	−70.0000	−2.86251
$599$	−30.0000	−1.22577	−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000	−1.22577	−0.612883	+	0.790173i	$0.709990\pi$
$600$	0	0
$601$	40.0000	1.63163	0.815817	−	0.578310i	$-0.196288\pi$
$601$	40.0000	1.63163	0.815817	+	0.578310i	$0.196288\pi$
$602$	−4.00000	−0.163028
$603$	0	0
$604$	−32.0000	−1.30206
$605$	−2.00000	−0.0813116
$606$	0	0
$607$	26.0000	1.05531	0.527654	−	0.849460i	$-0.323072\pi$
$607$	26.0000	1.05531	0.527654	+	0.849460i	$0.323072\pi$
$608$	8.00000	0.324443
$609$	0	0
$610$	4.00000	0.161955
$611$	−60.0000	−2.42734
$612$	0	0
$613$	−17.0000	−0.686624	−0.343312	−	0.939222i	$-0.611549\pi$
$613$	−17.0000	−0.686624	−0.343312	+	0.939222i	$0.611549\pi$
$614$	24.0000	0.968561
$615$	0	0
$616$	0	0
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	0	0
$619$	18.0000	0.723481	0.361741	−	0.932279i	$-0.382183\pi$
$619$	18.0000	0.723481	0.361741	+	0.932279i	$0.382183\pi$
$620$	8.00000	0.321288
$621$	0	0
$622$	32.0000	1.28308
$623$	4.00000	0.160257
$624$	0	0
$625$	11.0000	0.440000
$626$	−32.0000	−1.27898
$627$	0	0
$628$	−10.0000	−0.399043
$629$	10.0000	0.398726
$630$	0	0
$631$	−29.0000	−1.15447	−0.577236	−	0.816577i	$-0.695869\pi$
$631$	−29.0000	−1.15447	−0.577236	+	0.816577i	$0.695869\pi$
$632$	0	0
$633$	0	0
$634$	−20.0000	−0.794301
$635$	11.0000	0.436522
$636$	0	0
$637$	15.0000	0.594322
$638$	−36.0000	−1.42525
$639$	0	0
$640$	0	0
$641$	−15.0000	−0.592464	−0.296232	−	0.955116i	$-0.595730\pi$
$641$	−15.0000	−0.592464	−0.296232	+	0.955116i	$0.595730\pi$
$642$	0	0
$643$	44.0000	1.73519	0.867595	−	0.497271i	$-0.165665\pi$
$643$	44.0000	1.73519	0.867595	+	0.497271i	$0.165665\pi$
$644$	−28.0000	−1.10335
$645$	0	0
$646$	−2.00000	−0.0786889
$647$	−10.0000	−0.393141	−0.196570	−	0.980490i	$-0.562980\pi$
$647$	−10.0000	−0.393141	−0.196570	+	0.980490i	$0.562980\pi$
$648$	0	0
$649$	−18.0000	−0.706562
$650$	40.0000	1.56893
$651$	0	0
$652$	−48.0000	−1.87983
$653$	9.00000	0.352197	0.176099	−	0.984373i	$-0.443652\pi$
$653$	9.00000	0.352197	0.176099	+	0.984373i	$0.443652\pi$
$654$	0	0
$655$	−11.0000	−0.429806
$656$	36.0000	1.40556
$657$	0	0
$658$	−48.0000	−1.87123
$659$	22.0000	0.856998	0.428499	−	0.903542i	$-0.359042\pi$
$659$	22.0000	0.856998	0.428499	+	0.903542i	$0.359042\pi$
$660$	0	0
$661$	−7.00000	−0.272268	−0.136134	−	0.990690i	$-0.543468\pi$
$661$	−7.00000	−0.272268	−0.136134	+	0.990690i	$0.543468\pi$
$662$	14.0000	0.544125
$663$	0	0
$664$	0	0
$665$	2.00000	0.0775567
$666$	0	0
$667$	−42.0000	−1.62625
$668$	46.0000	1.77979
$669$	0	0
$670$	8.00000	0.309067
$671$	6.00000	0.231627
$672$	0	0
$673$	−28.0000	−1.07932	−0.539660	−	0.841883i	$-0.681447\pi$
$673$	−28.0000	−1.07932	−0.539660	+	0.841883i	$0.681447\pi$
$674$	44.0000	1.69482
$675$	0	0
$676$	24.0000	0.923077
$677$	−9.00000	−0.345898	−0.172949	−	0.984931i	$-0.555330\pi$
$677$	−9.00000	−0.345898	−0.172949	+	0.984931i	$0.555330\pi$
$678$	0	0
$679$	−16.0000	−0.614024
$680$	0	0
$681$	0	0
$682$	24.0000	0.919007
$683$	29.0000	1.10965	0.554827	−	0.831966i	$-0.312784\pi$
$683$	29.0000	1.10965	0.554827	+	0.831966i	$0.312784\pi$
$684$	0	0
$685$	−18.0000	−0.687745
$686$	40.0000	1.52721
$687$	0	0
$688$	−4.00000	−0.152499
$689$	−60.0000	−2.28582
$690$	0	0
$691$	40.0000	1.52167	0.760836	−	0.648944i	$-0.224789\pi$
$691$	40.0000	1.52167	0.760836	+	0.648944i	$0.224789\pi$
$692$	−46.0000	−1.74866
$693$	0	0
$694$	−8.00000	−0.303676
$695$	−14.0000	−0.531050
$696$	0	0
$697$	−9.00000	−0.340899
$698$	−30.0000	−1.13552
$699$	0	0
$700$	16.0000	0.604743
$701$	48.0000	1.81293	0.906467	−	0.422276i	$-0.138769\pi$
$701$	48.0000	1.81293	0.906467	+	0.422276i	$0.138769\pi$
$702$	0	0
$703$	−10.0000	−0.377157
$704$	−24.0000	−0.904534
$705$	0	0
$706$	36.0000	1.35488
$707$	8.00000	0.300871
$708$	0	0
$709$	−16.0000	−0.600893	−0.300446	−	0.953799i	$-0.597136\pi$
$709$	−16.0000	−0.600893	−0.300446	+	0.953799i	$0.597136\pi$
$710$	16.0000	0.600469
$711$	0	0
$712$	0	0
$713$	28.0000	1.04861
$714$	0	0
$715$	−15.0000	−0.560968
$716$	24.0000	0.896922
$717$	0	0
$718$	−12.0000	−0.447836
$719$	13.0000	0.484818	0.242409	−	0.970174i	$-0.422062\pi$
$719$	13.0000	0.484818	0.242409	+	0.970174i	$0.422062\pi$
$720$	0	0
$721$	38.0000	1.41519
$722$	−36.0000	−1.33978
$723$	0	0
$724$	−28.0000	−1.04061
$725$	24.0000	0.891338
$726$	0	0
$727$	4.00000	0.148352	0.0741759	−	0.997245i	$-0.476367\pi$
$727$	4.00000	0.148352	0.0741759	+	0.997245i	$0.476367\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1.00000	0.0369863
$732$	0	0
$733$	10.0000	0.369358	0.184679	−	0.982799i	$-0.440875\pi$
$733$	10.0000	0.369358	0.184679	+	0.982799i	$0.440875\pi$
$734$	−4.00000	−0.147643
$735$	0	0
$736$	−56.0000	−2.06419
$737$	12.0000	0.442026
$738$	0	0
$739$	19.0000	0.698926	0.349463	−	0.936950i	$-0.386364\pi$
$739$	19.0000	0.698926	0.349463	+	0.936950i	$0.386364\pi$
$740$	20.0000	0.735215
$741$	0	0
$742$	−48.0000	−1.76214
$743$	−12.0000	−0.440237	−0.220119	−	0.975473i	$-0.570644\pi$
$743$	−12.0000	−0.440237	−0.220119	+	0.975473i	$0.570644\pi$
$744$	0	0
$745$	2.00000	0.0732743
$746$	60.0000	2.19676
$747$	0	0
$748$	6.00000	0.219382
$749$	−22.0000	−0.803863
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	−48.0000	−1.75038
$753$	0	0
$754$	60.0000	2.18507
$755$	−16.0000	−0.582300
$756$	0	0
$757$	1.00000	0.0363456	0.0181728	−	0.999835i	$-0.494215\pi$
$757$	1.00000	0.0363456	0.0181728	+	0.999835i	$0.494215\pi$
$758$	−28.0000	−1.01701
$759$	0	0
$760$	0	0
$761$	20.0000	0.724999	0.362500	−	0.931984i	$-0.381923\pi$
$761$	20.0000	0.724999	0.362500	+	0.931984i	$0.381923\pi$
$762$	0	0
$763$	0	0
$764$	28.0000	1.01300
$765$	0	0
$766$	24.0000	0.867155
$767$	30.0000	1.08324
$768$	0	0
$769$	13.0000	0.468792	0.234396	−	0.972141i	$-0.424689\pi$
$769$	13.0000	0.468792	0.234396	+	0.972141i	$0.424689\pi$
$770$	−12.0000	−0.432450
$771$	0	0
$772$	−8.00000	−0.287926
$773$	28.0000	1.00709	0.503545	−	0.863969i	$-0.332029\pi$
$773$	28.0000	1.00709	0.503545	+	0.863969i	$0.332029\pi$
$774$	0	0
$775$	−16.0000	−0.574737
$776$	0	0
$777$	0	0
$778$	−12.0000	−0.430221
$779$	9.00000	0.322458
$780$	0	0
$781$	24.0000	0.858788
$782$	14.0000	0.500639
$783$	0	0
$784$	12.0000	0.428571
$785$	−5.00000	−0.178458
$786$	0	0
$787$	28.0000	0.998092	0.499046	−	0.866575i	$-0.333684\pi$
$787$	28.0000	0.998092	0.499046	+	0.866575i	$0.333684\pi$
$788$	−6.00000	−0.213741
$789$	0	0
$790$	−12.0000	−0.426941
$791$	−2.00000	−0.0711118
$792$	0	0
$793$	−10.0000	−0.355110
$794$	24.0000	0.851728
$795$	0	0
$796$	32.0000	1.13421
$797$	−20.0000	−0.708436	−0.354218	−	0.935163i	$-0.615253\pi$
$797$	−20.0000	−0.708436	−0.354218	+	0.935163i	$0.615253\pi$
$798$	0	0
$799$	12.0000	0.424529
$800$	32.0000	1.13137
$801$	0	0
$802$	14.0000	0.494357
$803$	0	0
$804$	0	0
$805$	−14.0000	−0.493435
$806$	−40.0000	−1.40894
$807$	0	0
$808$	0	0
$809$	23.0000	0.808637	0.404318	−	0.914618i	$-0.367509\pi$
$809$	23.0000	0.808637	0.404318	+	0.914618i	$0.367509\pi$
$810$	0	0
$811$	−8.00000	−0.280918	−0.140459	−	0.990086i	$-0.544858\pi$
$811$	−8.00000	−0.280918	−0.140459	+	0.990086i	$0.544858\pi$
$812$	24.0000	0.842235
$813$	0	0
$814$	60.0000	2.10300
$815$	−24.0000	−0.840683
$816$	0	0
$817$	−1.00000	−0.0349856
$818$	10.0000	0.349642
$819$	0	0
$820$	−18.0000	−0.628587
$821$	9.00000	0.314102	0.157051	−	0.987590i	$-0.449801\pi$
$821$	9.00000	0.314102	0.157051	+	0.987590i	$0.449801\pi$
$822$	0	0
$823$	−34.0000	−1.18517	−0.592583	−	0.805510i	$-0.701892\pi$
$823$	−34.0000	−1.18517	−0.592583	+	0.805510i	$0.701892\pi$
$824$	0	0
$825$	0	0
$826$	24.0000	0.835067
$827$	−39.0000	−1.35616	−0.678081	−	0.734987i	$-0.737188\pi$
$827$	−39.0000	−1.35616	−0.678081	+	0.734987i	$0.737188\pi$
$828$	0	0
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	−8.00000	−0.277684
$831$	0	0
$832$	40.0000	1.38675
$833$	−3.00000	−0.103944
$834$	0	0
$835$	23.0000	0.795948
$836$	−6.00000	−0.207514
$837$	0	0
$838$	−8.00000	−0.276355
$839$	29.0000	1.00119	0.500596	−	0.865681i	$-0.333114\pi$
$839$	29.0000	1.00119	0.500596	+	0.865681i	$0.333114\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−10.0000	−0.344623
$843$	0	0
$844$	40.0000	1.37686
$845$	12.0000	0.412813
$846$	0	0
$847$	4.00000	0.137442
$848$	−48.0000	−1.64833
$849$	0	0
$850$	−8.00000	−0.274398
$851$	70.0000	2.39957
$852$	0	0
$853$	14.0000	0.479351	0.239675	−	0.970853i	$-0.422959\pi$
$853$	14.0000	0.479351	0.239675	+	0.970853i	$0.422959\pi$
$854$	−8.00000	−0.273754
$855$	0	0
$856$	0	0
$857$	58.0000	1.98124	0.990621	−	0.136637i	$-0.0436295\pi$
$857$	58.0000	1.98124	0.990621	+	0.136637i	$0.0436295\pi$
$858$	0	0
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	2.00000	0.0681994
$861$	0	0
$862$	0	0
$863$	−38.0000	−1.29354	−0.646768	−	0.762687i	$-0.723880\pi$
$863$	−38.0000	−1.29354	−0.646768	+	0.762687i	$0.723880\pi$
$864$	0	0
$865$	−23.0000	−0.782023
$866$	−50.0000	−1.69907
$867$	0	0
$868$	−16.0000	−0.543075
$869$	−18.0000	−0.610608
$870$	0	0
$871$	−20.0000	−0.677674
$872$	0	0
$873$	0	0
$874$	−14.0000	−0.473557
$875$	18.0000	0.608511
$876$	0	0
$877$	6.00000	0.202606	0.101303	−	0.994856i	$-0.467699\pi$
$877$	6.00000	0.202606	0.101303	+	0.994856i	$0.467699\pi$
$878$	56.0000	1.88991
$879$	0	0
$880$	−12.0000	−0.404520
$881$	−46.0000	−1.54978	−0.774890	−	0.632096i	$-0.782195\pi$
$881$	−46.0000	−1.54978	−0.774890	+	0.632096i	$0.782195\pi$
$882$	0	0
$883$	−21.0000	−0.706706	−0.353353	−	0.935490i	$-0.614959\pi$
$883$	−21.0000	−0.706706	−0.353353	+	0.935490i	$0.614959\pi$
$884$	−10.0000	−0.336336
$885$	0	0
$886$	−40.0000	−1.34383
$887$	9.00000	0.302190	0.151095	−	0.988519i	$-0.451720\pi$
$887$	9.00000	0.302190	0.151095	+	0.988519i	$0.451720\pi$
$888$	0	0
$889$	−22.0000	−0.737856
$890$	−4.00000	−0.134080
$891$	0	0
$892$	30.0000	1.00447
$893$	−12.0000	−0.401565
$894$	0	0
$895$	12.0000	0.401116
$896$	0	0
$897$	0	0
$898$	−28.0000	−0.934372
$899$	−24.0000	−0.800445
$900$	0	0
$901$	12.0000	0.399778
$902$	−54.0000	−1.79800
$903$	0	0
$904$	0	0
$905$	−14.0000	−0.465376
$906$	0	0
$907$	−46.0000	−1.52740	−0.763702	−	0.645568i	$-0.776621\pi$
$907$	−46.0000	−1.52740	−0.763702	+	0.645568i	$0.776621\pi$
$908$	−30.0000	−0.995585
$909$	0	0
$910$	20.0000	0.662994
$911$	47.0000	1.55718	0.778590	−	0.627533i	$-0.215935\pi$
$911$	47.0000	1.55718	0.778590	+	0.627533i	$0.215935\pi$
$912$	0	0
$913$	−12.0000	−0.397142
$914$	−30.0000	−0.992312
$915$	0	0
$916$	−36.0000	−1.18947
$917$	22.0000	0.726504
$918$	0	0
$919$	15.0000	0.494804	0.247402	−	0.968913i	$-0.420423\pi$
$919$	15.0000	0.494804	0.247402	+	0.968913i	$0.420423\pi$
$920$	0	0
$921$	0	0
$922$	20.0000	0.658665
$923$	−40.0000	−1.31662
$924$	0	0
$925$	−40.0000	−1.31519
$926$	−32.0000	−1.05159
$927$	0	0
$928$	48.0000	1.57568
$929$	−27.0000	−0.885841	−0.442921	−	0.896561i	$-0.646058\pi$
$929$	−27.0000	−0.885841	−0.442921	+	0.896561i	$0.646058\pi$
$930$	0	0
$931$	3.00000	0.0983210
$932$	30.0000	0.982683
$933$	0	0
$934$	−60.0000	−1.96326
$935$	3.00000	0.0981105
$936$	0	0
$937$	−26.0000	−0.849383	−0.424691	−	0.905338i	$-0.639617\pi$
$937$	−26.0000	−0.849383	−0.424691	+	0.905338i	$0.639617\pi$
$938$	−16.0000	−0.522419
$939$	0	0
$940$	24.0000	0.782794
$941$	−30.0000	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$942$	0	0
$943$	−63.0000	−2.05156
$944$	24.0000	0.781133
$945$	0	0
$946$	6.00000	0.195077
$947$	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
$949$	0	0
$950$	8.00000	0.259554
$951$	0	0
$952$	0	0
$953$	8.00000	0.259145	0.129573	−	0.991570i	$-0.458639\pi$
$953$	8.00000	0.259145	0.129573	+	0.991570i	$0.458639\pi$
$954$	0	0
$955$	14.0000	0.453029
$956$	16.0000	0.517477
$957$	0	0
$958$	−42.0000	−1.35696
$959$	36.0000	1.16250
$960$	0	0
$961$	−15.0000	−0.483871
$962$	−100.000	−3.22413
$963$	0	0
$964$	36.0000	1.15948
$965$	−4.00000	−0.128765
$966$	0	0
$967$	33.0000	1.06121	0.530604	−	0.847620i	$-0.321965\pi$
$967$	33.0000	1.06121	0.530604	+	0.847620i	$0.321965\pi$
$968$	0	0
$969$	0	0
$970$	16.0000	0.513729
$971$	−48.0000	−1.54039	−0.770197	−	0.637806i	$-0.779842\pi$
$971$	−48.0000	−1.54039	−0.770197	+	0.637806i	$0.779842\pi$
$972$	0	0
$973$	28.0000	0.897639
$974$	−44.0000	−1.40985
$975$	0	0
$976$	−8.00000	−0.256074
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	0	0
$979$	−6.00000	−0.191761
$980$	−6.00000	−0.191663
$981$	0	0
$982$	−8.00000	−0.255290
$983$	21.0000	0.669796	0.334898	−	0.942254i	$-0.391298\pi$
$983$	21.0000	0.669796	0.334898	+	0.942254i	$0.391298\pi$
$984$	0	0
$985$	−3.00000	−0.0955879
$986$	−12.0000	−0.382158
$987$	0	0
$988$	10.0000	0.318142
$989$	7.00000	0.222587
$990$	0	0
$991$	60.0000	1.90596	0.952981	−	0.303029i	$-0.0979978\pi$
$991$	60.0000	1.90596	0.952981	+	0.303029i	$0.0979978\pi$
$992$	−32.0000	−1.01600
$993$	0	0
$994$	−32.0000	−1.01498
$995$	16.0000	0.507234
$996$	0	0
$997$	−38.0000	−1.20347	−0.601736	−	0.798695i	$-0.705524\pi$
$997$	−38.0000	−1.20347	−0.601736	+	0.798695i	$0.705524\pi$
$998$	−32.0000	−1.01294
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	153.2.a.d.1.1	yes	1
3.2	odd	2		153.2.a.a.1.1	✓	1
4.3	odd	2		2448.2.a.m.1.1		1
5.4	even	2		3825.2.a.b.1.1		1
7.6	odd	2		7497.2.a.p.1.1		1
8.3	odd	2		9792.2.a.w.1.1		1
8.5	even	2		9792.2.a.p.1.1		1
12.11	even	2		2448.2.a.g.1.1		1
15.14	odd	2		3825.2.a.o.1.1		1
17.16	even	2		2601.2.a.l.1.1		1
21.20	even	2		7497.2.a.a.1.1		1
24.5	odd	2		9792.2.a.bm.1.1		1
24.11	even	2		9792.2.a.bp.1.1		1
51.50	odd	2		2601.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
153.2.a.a.1.1	✓	1	3.2	odd	2
153.2.a.d.1.1	yes	1	1.1	even	1	trivial
2448.2.a.g.1.1		1	12.11	even	2
2448.2.a.m.1.1		1	4.3	odd	2
2601.2.a.b.1.1		1	51.50	odd	2
2601.2.a.l.1.1		1	17.16	even	2
3825.2.a.b.1.1		1	5.4	even	2
3825.2.a.o.1.1		1	15.14	odd	2
7497.2.a.a.1.1		1	21.20	even	2
7497.2.a.p.1.1		1	7.6	odd	2
9792.2.a.p.1.1		1	8.5	even	2
9792.2.a.w.1.1		1	8.3	odd	2
9792.2.a.bm.1.1		1	24.5	odd	2
9792.2.a.bp.1.1		1	24.11	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:	\( N \)	\(=\)	\( 153 = 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	153.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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