LMFDB - Embedded newform 333.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [333,2,Mod(1,333)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(333, 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("333.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 333 = 3^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	333.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:	$2.65901838731$
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	$\chi$	$=$	333.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{4} +2.00000 q^{5} -4.00000 q^{7} +3.00000 q^{8} +O(q^{10})$

$q-1.00000 q^{2} -1.00000 q^{4} +2.00000 q^{5} -4.00000 q^{7} +3.00000 q^{8} -2.00000 q^{10} -4.00000 q^{11} -2.00000 q^{13} +4.00000 q^{14} -1.00000 q^{16} +6.00000 q^{17} -6.00000 q^{19} -2.00000 q^{20} +4.00000 q^{22} -8.00000 q^{23} -1.00000 q^{25} +2.00000 q^{26} +4.00000 q^{28} -6.00000 q^{29} +2.00000 q^{31} -5.00000 q^{32} -6.00000 q^{34} -8.00000 q^{35} -1.00000 q^{37} +6.00000 q^{38} +6.00000 q^{40} -10.0000 q^{43} +4.00000 q^{44} +8.00000 q^{46} +12.0000 q^{47} +9.00000 q^{49} +1.00000 q^{50} +2.00000 q^{52} -4.00000 q^{53} -8.00000 q^{55} -12.0000 q^{56} +6.00000 q^{58} +4.00000 q^{59} +10.0000 q^{61} -2.00000 q^{62} +7.00000 q^{64} -4.00000 q^{65} -4.00000 q^{67} -6.00000 q^{68} +8.00000 q^{70} +12.0000 q^{71} -10.0000 q^{73} +1.00000 q^{74} +6.00000 q^{76} +16.0000 q^{77} +10.0000 q^{79} -2.00000 q^{80} +12.0000 q^{85} +10.0000 q^{86} -12.0000 q^{88} +2.00000 q^{89} +8.00000 q^{91} +8.00000 q^{92} -12.0000 q^{94} -12.0000 q^{95} -2.00000 q^{97} -9.00000 q^{98} +O(q^{100})$

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	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	3.00000	1.06066
$9$	0	0
$10$	−2.00000	−0.632456
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	4.00000	1.06904
$15$	0	0
$16$	−1.00000	−0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	−2.00000	−0.447214
$21$	0	0
$22$	4.00000	0.852803
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	2.00000	0.392232
$27$	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
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	−5.00000	−0.883883
$33$	0	0
$34$	−6.00000	−1.02899
$35$	−8.00000	−1.35225
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	6.00000	0.973329
$39$	0	0
$40$	6.00000	0.948683
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	8.00000	1.17954
$47$	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
$49$	9.00000	1.28571
$50$	1.00000	0.141421
$51$	0	0
$52$	2.00000	0.277350
$53$	−4.00000	−0.549442	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000	−0.549442	−0.274721	+	0.961524i	$0.588586\pi$
$54$	0	0
$55$	−8.00000	−1.07872
$56$	−12.0000	−1.60357
$57$	0	0
$58$	6.00000	0.787839
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	−2.00000	−0.254000
$63$	0	0
$64$	7.00000	0.875000
$65$	−4.00000	−0.496139
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	−6.00000	−0.727607
$69$	0	0
$70$	8.00000	0.956183
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	6.00000	0.688247
$77$	16.0000	1.82337
$78$	0	0
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	−2.00000	−0.223607
$81$	0	0
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	12.0000	1.30158
$86$	10.0000	1.07833
$87$	0	0
$88$	−12.0000	−1.27920
$89$	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
$91$	8.00000	0.838628
$92$	8.00000	0.834058
$93$	0	0
$94$	−12.0000	−1.23771
$95$	−12.0000	−1.23117
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	−9.00000	−0.909137
$99$	0	0
$100$	1.00000	0.100000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	6.00000	0.591198	0.295599	−	0.955312i	$-0.404481\pi$
$103$	6.00000	0.591198	0.295599	+	0.955312i	$0.404481\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	4.00000	0.388514
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	8.00000	0.762770
$111$	0	0
$112$	4.00000	0.377964
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	−16.0000	−1.49201
$116$	6.00000	0.557086
$117$	0	0
$118$	−4.00000	−0.368230
$119$	−24.0000	−2.20008
$120$	0	0
$121$	5.00000	0.454545
$122$	−10.0000	−0.905357
$123$	0	0
$124$	−2.00000	−0.179605
$125$	−12.0000	−1.07331
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	3.00000	0.265165
$129$	0	0
$130$	4.00000	0.350823
$131$	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
$133$	24.0000	2.08106
$134$	4.00000	0.345547
$135$	0	0
$136$	18.0000	1.54349
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	8.00000	0.676123
$141$	0	0
$142$	−12.0000	−1.00702
$143$	8.00000	0.668994
$144$	0	0
$145$	−12.0000	−0.996546
$146$	10.0000	0.827606
$147$	0	0
$148$	1.00000	0.0821995
$149$	8.00000	0.655386	0.327693	−	0.944784i	$-0.393729\pi$
$149$	8.00000	0.655386	0.327693	+	0.944784i	$0.393729\pi$
$150$	0	0
$151$	−20.0000	−1.62758	−0.813788	−	0.581161i	$-0.802599\pi$
$151$	−20.0000	−1.62758	−0.813788	+	0.581161i	$0.802599\pi$
$152$	−18.0000	−1.45999
$153$	0	0
$154$	−16.0000	−1.28932
$155$	4.00000	0.321288
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	−10.0000	−0.795557
$159$	0	0
$160$	−10.0000	−0.790569
$161$	32.0000	2.52195
$162$	0	0
$163$	−6.00000	−0.469956	−0.234978	−	0.972001i	$-0.575502\pi$
$163$	−6.00000	−0.469956	−0.234978	+	0.972001i	$0.575502\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−12.0000	−0.920358
$171$	0	0
$172$	10.0000	0.762493
$173$	−12.0000	−0.912343	−0.456172	−	0.889892i	$-0.650780\pi$
$173$	−12.0000	−0.912343	−0.456172	+	0.889892i	$0.650780\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	4.00000	0.301511
$177$	0	0
$178$	−2.00000	−0.149906
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	−8.00000	−0.592999
$183$	0	0
$184$	−24.0000	−1.76930
$185$	−2.00000	−0.147043
$186$	0	0
$187$	−24.0000	−1.75505
$188$	−12.0000	−0.875190
$189$	0	0
$190$	12.0000	0.870572
$191$	−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
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	−9.00000	−0.642857
$197$	−24.0000	−1.70993	−0.854965	−	0.518686i	$-0.826421\pi$
$197$	−24.0000	−1.70993	−0.854965	+	0.518686i	$0.826421\pi$
$198$	0	0
$199$	2.00000	0.141776	0.0708881	−	0.997484i	$-0.477417\pi$
$199$	2.00000	0.141776	0.0708881	+	0.997484i	$0.477417\pi$
$200$	−3.00000	−0.212132
$201$	0	0
$202$	0	0
$203$	24.0000	1.68447
$204$	0	0
$205$	0	0
$206$	−6.00000	−0.418040
$207$	0	0
$208$	2.00000	0.138675
$209$	24.0000	1.66011
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	4.00000	0.274721
$213$	0	0
$214$	−12.0000	−0.820303
$215$	−20.0000	−1.36399
$216$	0	0
$217$	−8.00000	−0.543075
$218$	10.0000	0.677285
$219$	0	0
$220$	8.00000	0.539360
$221$	−12.0000	−0.807207
$222$	0	0
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	20.0000	1.33631
$225$	0	0
$226$	6.00000	0.399114
$227$	4.00000	0.265489	0.132745	−	0.991150i	$-0.457621\pi$
$227$	4.00000	0.265489	0.132745	+	0.991150i	$0.457621\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	16.0000	1.05501
$231$	0	0
$232$	−18.0000	−1.18176
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	24.0000	1.56559
$236$	−4.00000	−0.260378
$237$	0	0
$238$	24.0000	1.55569
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	−5.00000	−0.321412
$243$	0	0
$244$	−10.0000	−0.640184
$245$	18.0000	1.14998
$246$	0	0
$247$	12.0000	0.763542
$248$	6.00000	0.381000
$249$	0	0
$250$	12.0000	0.758947
$251$	20.0000	1.26239	0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000	1.26239	0.631194	+	0.775625i	$0.282565\pi$
$252$	0	0
$253$	32.0000	2.01182
$254$	8.00000	0.501965
$255$	0	0
$256$	−17.0000	−1.06250
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	0	0
$259$	4.00000	0.248548
$260$	4.00000	0.248069
$261$	0	0
$262$	−12.0000	−0.741362
$263$	20.0000	1.23325	0.616626	−	0.787256i	$-0.288499\pi$
$263$	20.0000	1.23325	0.616626	+	0.787256i	$0.288499\pi$
$264$	0	0
$265$	−8.00000	−0.491436
$266$	−24.0000	−1.47153
$267$	0	0
$268$	4.00000	0.244339
$269$	24.0000	1.46331	0.731653	−	0.681677i	$-0.238749\pi$
$269$	24.0000	1.46331	0.731653	+	0.681677i	$0.238749\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	−6.00000	−0.363803
$273$	0	0
$274$	12.0000	0.724947
$275$	4.00000	0.241209
$276$	0	0
$277$	−18.0000	−1.08152	−0.540758	−	0.841178i	$-0.681862\pi$
$277$	−18.0000	−1.08152	−0.540758	+	0.841178i	$0.681862\pi$
$278$	−4.00000	−0.239904
$279$	0	0
$280$	−24.0000	−1.43427
$281$	−30.0000	−1.78965	−0.894825	−	0.446417i	$-0.852700\pi$
$281$	−30.0000	−1.78965	−0.894825	+	0.446417i	$0.852700\pi$
$282$	0	0
$283$	−26.0000	−1.54554	−0.772770	−	0.634686i	$-0.781129\pi$
$283$	−26.0000	−1.54554	−0.772770	+	0.634686i	$0.781129\pi$
$284$	−12.0000	−0.712069
$285$	0	0
$286$	−8.00000	−0.473050
$287$	0	0
$288$	0	0
$289$	19.0000	1.11765
$290$	12.0000	0.704664
$291$	0	0
$292$	10.0000	0.585206
$293$	−28.0000	−1.63578	−0.817889	−	0.575376i	$-0.804856\pi$
$293$	−28.0000	−1.63578	−0.817889	+	0.575376i	$0.804856\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	−3.00000	−0.174371
$297$	0	0
$298$	−8.00000	−0.463428
$299$	16.0000	0.925304
$300$	0	0
$301$	40.0000	2.30556
$302$	20.0000	1.15087
$303$	0	0
$304$	6.00000	0.344124
$305$	20.0000	1.14520
$306$	0	0
$307$	−32.0000	−1.82634	−0.913168	−	0.407583i	$-0.866372\pi$
$307$	−32.0000	−1.82634	−0.913168	+	0.407583i	$0.866372\pi$
$308$	−16.0000	−0.911685
$309$	0	0
$310$	−4.00000	−0.227185
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	34.0000	1.92179	0.960897	−	0.276907i	$-0.0893093\pi$
$313$	34.0000	1.92179	0.960897	+	0.276907i	$0.0893093\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	−10.0000	−0.562544
$317$	20.0000	1.12331	0.561656	−	0.827371i	$-0.310164\pi$
$317$	20.0000	1.12331	0.561656	+	0.827371i	$0.310164\pi$
$318$	0	0
$319$	24.0000	1.34374
$320$	14.0000	0.782624
$321$	0	0
$322$	−32.0000	−1.78329
$323$	−36.0000	−2.00309
$324$	0	0
$325$	2.00000	0.110940
$326$	6.00000	0.332309
$327$	0	0
$328$	0	0
$329$	−48.0000	−2.64633
$330$	0	0
$331$	10.0000	0.549650	0.274825	−	0.961494i	$-0.411380\pi$
$331$	10.0000	0.549650	0.274825	+	0.961494i	$0.411380\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−8.00000	−0.437087
$336$	0	0
$337$	26.0000	1.41631	0.708155	−	0.706057i	$-0.249528\pi$
$337$	26.0000	1.41631	0.708155	+	0.706057i	$0.249528\pi$
$338$	9.00000	0.489535
$339$	0	0
$340$	−12.0000	−0.650791
$341$	−8.00000	−0.433224
$342$	0	0
$343$	−8.00000	−0.431959
$344$	−30.0000	−1.61749
$345$	0	0
$346$	12.0000	0.645124
$347$	−20.0000	−1.07366	−0.536828	−	0.843692i	$-0.680378\pi$
$347$	−20.0000	−1.07366	−0.536828	+	0.843692i	$0.680378\pi$
$348$	0	0
$349$	6.00000	0.321173	0.160586	−	0.987022i	$-0.448662\pi$
$349$	6.00000	0.321173	0.160586	+	0.987022i	$0.448662\pi$
$350$	−4.00000	−0.213809
$351$	0	0
$352$	20.0000	1.06600
$353$	10.0000	0.532246	0.266123	−	0.963939i	$-0.414257\pi$
$353$	10.0000	0.532246	0.266123	+	0.963939i	$0.414257\pi$
$354$	0	0
$355$	24.0000	1.27379
$356$	−2.00000	−0.106000
$357$	0	0
$358$	12.0000	0.634220
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	−2.00000	−0.105118
$363$	0	0
$364$	−8.00000	−0.419314
$365$	−20.0000	−1.04685
$366$	0	0
$367$	−16.0000	−0.835193	−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000	−0.835193	−0.417597	+	0.908633i	$0.637127\pi$
$368$	8.00000	0.417029
$369$	0	0
$370$	2.00000	0.103975
$371$	16.0000	0.830679
$372$	0	0
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	24.0000	1.24101
$375$	0	0
$376$	36.0000	1.85656
$377$	12.0000	0.618031
$378$	0	0
$379$	−24.0000	−1.23280	−0.616399	−	0.787434i	$-0.711409\pi$
$379$	−24.0000	−1.23280	−0.616399	+	0.787434i	$0.711409\pi$
$380$	12.0000	0.615587
$381$	0	0
$382$	8.00000	0.409316
$383$	−32.0000	−1.63512	−0.817562	−	0.575841i	$-0.804675\pi$
$383$	−32.0000	−1.63512	−0.817562	+	0.575841i	$0.804675\pi$
$384$	0	0
$385$	32.0000	1.63087
$386$	14.0000	0.712581
$387$	0	0
$388$	2.00000	0.101535
$389$	14.0000	0.709828	0.354914	−	0.934899i	$-0.384510\pi$
$389$	14.0000	0.709828	0.354914	+	0.934899i	$0.384510\pi$
$390$	0	0
$391$	−48.0000	−2.42746
$392$	27.0000	1.36371
$393$	0	0
$394$	24.0000	1.20910
$395$	20.0000	1.00631
$396$	0	0
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	−2.00000	−0.100251
$399$	0	0
$400$	1.00000	0.0500000
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	0	0
$403$	−4.00000	−0.199254
$404$	0	0
$405$	0	0
$406$	−24.0000	−1.19110
$407$	4.00000	0.198273
$408$	0	0
$409$	−34.0000	−1.68119	−0.840596	−	0.541663i	$-0.817795\pi$
$409$	−34.0000	−1.68119	−0.840596	+	0.541663i	$0.817795\pi$
$410$	0	0
$411$	0	0
$412$	−6.00000	−0.295599
$413$	−16.0000	−0.787309
$414$	0	0
$415$	0	0
$416$	10.0000	0.490290
$417$	0	0
$418$	−24.0000	−1.17388
$419$	−16.0000	−0.781651	−0.390826	−	0.920465i	$-0.627810\pi$
$419$	−16.0000	−0.781651	−0.390826	+	0.920465i	$0.627810\pi$
$420$	0	0
$421$	30.0000	1.46211	0.731055	−	0.682318i	$-0.239028\pi$
$421$	30.0000	1.46211	0.731055	+	0.682318i	$0.239028\pi$
$422$	4.00000	0.194717
$423$	0	0
$424$	−12.0000	−0.582772
$425$	−6.00000	−0.291043
$426$	0	0
$427$	−40.0000	−1.93574
$428$	−12.0000	−0.580042
$429$	0	0
$430$	20.0000	0.964486
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	0	0
$433$	−30.0000	−1.44171	−0.720854	−	0.693087i	$-0.756250\pi$
$433$	−30.0000	−1.44171	−0.720854	+	0.693087i	$0.756250\pi$
$434$	8.00000	0.384012
$435$	0	0
$436$	10.0000	0.478913
$437$	48.0000	2.29615
$438$	0	0
$439$	−14.0000	−0.668184	−0.334092	−	0.942541i	$-0.608430\pi$
$439$	−14.0000	−0.668184	−0.334092	+	0.942541i	$0.608430\pi$
$440$	−24.0000	−1.14416
$441$	0	0
$442$	12.0000	0.570782
$443$	−16.0000	−0.760183	−0.380091	−	0.924949i	$-0.624107\pi$
$443$	−16.0000	−0.760183	−0.380091	+	0.924949i	$0.624107\pi$
$444$	0	0
$445$	4.00000	0.189618
$446$	−16.0000	−0.757622
$447$	0	0
$448$	−28.0000	−1.32288
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	0	0
$452$	6.00000	0.282216
$453$	0	0
$454$	−4.00000	−0.187729
$455$	16.0000	0.750092
$456$	0	0
$457$	30.0000	1.40334	0.701670	−	0.712502i	$-0.252438\pi$
$457$	30.0000	1.40334	0.701670	+	0.712502i	$0.252438\pi$
$458$	−10.0000	−0.467269
$459$	0	0
$460$	16.0000	0.746004
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	0	0
$463$	−10.0000	−0.464739	−0.232370	−	0.972628i	$-0.574648\pi$
$463$	−10.0000	−0.464739	−0.232370	+	0.972628i	$0.574648\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	0	0
$467$	−4.00000	−0.185098	−0.0925490	−	0.995708i	$-0.529501\pi$
$467$	−4.00000	−0.185098	−0.0925490	+	0.995708i	$0.529501\pi$
$468$	0	0
$469$	16.0000	0.738811
$470$	−24.0000	−1.10704
$471$	0	0
$472$	12.0000	0.552345
$473$	40.0000	1.83920
$474$	0	0
$475$	6.00000	0.275299
$476$	24.0000	1.10004
$477$	0	0
$478$	0	0
$479$	−8.00000	−0.365529	−0.182765	−	0.983157i	$-0.558505\pi$
$479$	−8.00000	−0.365529	−0.182765	+	0.983157i	$0.558505\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	−14.0000	−0.637683
$483$	0	0
$484$	−5.00000	−0.227273
$485$	−4.00000	−0.181631
$486$	0	0
$487$	−30.0000	−1.35943	−0.679715	−	0.733476i	$-0.737896\pi$
$487$	−30.0000	−1.35943	−0.679715	+	0.733476i	$0.737896\pi$
$488$	30.0000	1.35804
$489$	0	0
$490$	−18.0000	−0.813157
$491$	16.0000	0.722070	0.361035	−	0.932552i	$-0.382424\pi$
$491$	16.0000	0.722070	0.361035	+	0.932552i	$0.382424\pi$
$492$	0	0
$493$	−36.0000	−1.62136
$494$	−12.0000	−0.539906
$495$	0	0
$496$	−2.00000	−0.0898027
$497$	−48.0000	−2.15309
$498$	0	0
$499$	−6.00000	−0.268597	−0.134298	−	0.990941i	$-0.542878\pi$
$499$	−6.00000	−0.268597	−0.134298	+	0.990941i	$0.542878\pi$
$500$	12.0000	0.536656
$501$	0	0
$502$	−20.0000	−0.892644
$503$	32.0000	1.42681	0.713405	−	0.700752i	$-0.247152\pi$
$503$	32.0000	1.42681	0.713405	+	0.700752i	$0.247152\pi$
$504$	0	0
$505$	0	0
$506$	−32.0000	−1.42257
$507$	0	0
$508$	8.00000	0.354943
$509$	28.0000	1.24108	0.620539	−	0.784176i	$-0.286914\pi$
$509$	28.0000	1.24108	0.620539	+	0.784176i	$0.286914\pi$
$510$	0	0
$511$	40.0000	1.76950
$512$	11.0000	0.486136
$513$	0	0
$514$	6.00000	0.264649
$515$	12.0000	0.528783
$516$	0	0
$517$	−48.0000	−2.11104
$518$	−4.00000	−0.175750
$519$	0	0
$520$	−12.0000	−0.526235
$521$	−24.0000	−1.05146	−0.525730	−	0.850652i	$-0.676208\pi$
$521$	−24.0000	−1.05146	−0.525730	+	0.850652i	$0.676208\pi$
$522$	0	0
$523$	14.0000	0.612177	0.306089	−	0.952003i	$-0.400980\pi$
$523$	14.0000	0.612177	0.306089	+	0.952003i	$0.400980\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	−20.0000	−0.872041
$527$	12.0000	0.522728
$528$	0	0
$529$	41.0000	1.78261
$530$	8.00000	0.347498
$531$	0	0
$532$	−24.0000	−1.04053
$533$	0	0
$534$	0	0
$535$	24.0000	1.03761
$536$	−12.0000	−0.518321
$537$	0	0
$538$	−24.0000	−1.03471
$539$	−36.0000	−1.55063
$540$	0	0
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	−20.0000	−0.859074
$543$	0	0
$544$	−30.0000	−1.28624
$545$	−20.0000	−0.856706
$546$	0	0
$547$	−22.0000	−0.940652	−0.470326	−	0.882493i	$-0.655864\pi$
$547$	−22.0000	−0.940652	−0.470326	+	0.882493i	$0.655864\pi$
$548$	12.0000	0.512615
$549$	0	0
$550$	−4.00000	−0.170561
$551$	36.0000	1.53365
$552$	0	0
$553$	−40.0000	−1.70097
$554$	18.0000	0.764747
$555$	0	0
$556$	−4.00000	−0.169638
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	0	0
$559$	20.0000	0.845910
$560$	8.00000	0.338062
$561$	0	0
$562$	30.0000	1.26547
$563$	36.0000	1.51722	0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000	1.51722	0.758610	+	0.651546i	$0.225879\pi$
$564$	0	0
$565$	−12.0000	−0.504844
$566$	26.0000	1.09286
$567$	0	0
$568$	36.0000	1.51053
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	−8.00000	−0.334790	−0.167395	−	0.985890i	$-0.553535\pi$
$571$	−8.00000	−0.334790	−0.167395	+	0.985890i	$0.553535\pi$
$572$	−8.00000	−0.334497
$573$	0	0
$574$	0	0
$575$	8.00000	0.333623
$576$	0	0
$577$	18.0000	0.749350	0.374675	−	0.927156i	$-0.377754\pi$
$577$	18.0000	0.749350	0.374675	+	0.927156i	$0.377754\pi$
$578$	−19.0000	−0.790296
$579$	0	0
$580$	12.0000	0.498273
$581$	0	0
$582$	0	0
$583$	16.0000	0.662652
$584$	−30.0000	−1.24141
$585$	0	0
$586$	28.0000	1.15667
$587$	−28.0000	−1.15568	−0.577842	−	0.816149i	$-0.696105\pi$
$587$	−28.0000	−1.15568	−0.577842	+	0.816149i	$0.696105\pi$
$588$	0	0
$589$	−12.0000	−0.494451
$590$	−8.00000	−0.329355
$591$	0	0
$592$	1.00000	0.0410997
$593$	−16.0000	−0.657041	−0.328521	−	0.944497i	$-0.606550\pi$
$593$	−16.0000	−0.657041	−0.328521	+	0.944497i	$0.606550\pi$
$594$	0	0
$595$	−48.0000	−1.96781
$596$	−8.00000	−0.327693
$597$	0	0
$598$	−16.0000	−0.654289
$599$	20.0000	0.817178	0.408589	−	0.912719i	$-0.366021\pi$
$599$	20.0000	0.817178	0.408589	+	0.912719i	$0.366021\pi$
$600$	0	0
$601$	14.0000	0.571072	0.285536	−	0.958368i	$-0.407828\pi$
$601$	14.0000	0.571072	0.285536	+	0.958368i	$0.407828\pi$
$602$	−40.0000	−1.63028
$603$	0	0
$604$	20.0000	0.813788
$605$	10.0000	0.406558
$606$	0	0
$607$	22.0000	0.892952	0.446476	−	0.894795i	$-0.352679\pi$
$607$	22.0000	0.892952	0.446476	+	0.894795i	$0.352679\pi$
$608$	30.0000	1.21666
$609$	0	0
$610$	−20.0000	−0.809776
$611$	−24.0000	−0.970936
$612$	0	0
$613$	−6.00000	−0.242338	−0.121169	−	0.992632i	$-0.538664\pi$
$613$	−6.00000	−0.242338	−0.121169	+	0.992632i	$0.538664\pi$
$614$	32.0000	1.29141
$615$	0	0
$616$	48.0000	1.93398
$617$	4.00000	0.161034	0.0805170	−	0.996753i	$-0.474343\pi$
$617$	4.00000	0.161034	0.0805170	+	0.996753i	$0.474343\pi$
$618$	0	0
$619$	−16.0000	−0.643094	−0.321547	−	0.946894i	$-0.604203\pi$
$619$	−16.0000	−0.643094	−0.321547	+	0.946894i	$0.604203\pi$
$620$	−4.00000	−0.160644
$621$	0	0
$622$	0	0
$623$	−8.00000	−0.320513
$624$	0	0
$625$	−19.0000	−0.760000
$626$	−34.0000	−1.35891
$627$	0	0
$628$	−2.00000	−0.0798087
$629$	−6.00000	−0.239236
$630$	0	0
$631$	38.0000	1.51276	0.756378	−	0.654135i	$-0.226967\pi$
$631$	38.0000	1.51276	0.756378	+	0.654135i	$0.226967\pi$
$632$	30.0000	1.19334
$633$	0	0
$634$	−20.0000	−0.794301
$635$	−16.0000	−0.634941
$636$	0	0
$637$	−18.0000	−0.713186
$638$	−24.0000	−0.950169
$639$	0	0
$640$	6.00000	0.237171
$641$	40.0000	1.57991	0.789953	−	0.613168i	$-0.210105\pi$
$641$	40.0000	1.57991	0.789953	+	0.613168i	$0.210105\pi$
$642$	0	0
$643$	−10.0000	−0.394362	−0.197181	−	0.980367i	$-0.563179\pi$
$643$	−10.0000	−0.394362	−0.197181	+	0.980367i	$0.563179\pi$
$644$	−32.0000	−1.26098
$645$	0	0
$646$	36.0000	1.41640
$647$	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
$649$	−16.0000	−0.628055
$650$	−2.00000	−0.0784465
$651$	0	0
$652$	6.00000	0.234978
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	0	0
$655$	24.0000	0.937758
$656$	0	0
$657$	0	0
$658$	48.0000	1.87123
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	0	0
$661$	26.0000	1.01128	0.505641	−	0.862744i	$-0.331256\pi$
$661$	26.0000	1.01128	0.505641	+	0.862744i	$0.331256\pi$
$662$	−10.0000	−0.388661
$663$	0	0
$664$	0	0
$665$	48.0000	1.86136
$666$	0	0
$667$	48.0000	1.85857
$668$	0	0
$669$	0	0
$670$	8.00000	0.309067
$671$	−40.0000	−1.54418
$672$	0	0
$673$	6.00000	0.231283	0.115642	−	0.993291i	$-0.463108\pi$
$673$	6.00000	0.231283	0.115642	+	0.993291i	$0.463108\pi$
$674$	−26.0000	−1.00148
$675$	0	0
$676$	9.00000	0.346154
$677$	−4.00000	−0.153732	−0.0768662	−	0.997041i	$-0.524491\pi$
$677$	−4.00000	−0.153732	−0.0768662	+	0.997041i	$0.524491\pi$
$678$	0	0
$679$	8.00000	0.307012
$680$	36.0000	1.38054
$681$	0	0
$682$	8.00000	0.306336
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	0	0
$685$	−24.0000	−0.916993
$686$	8.00000	0.305441
$687$	0	0
$688$	10.0000	0.381246
$689$	8.00000	0.304776
$690$	0	0
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	12.0000	0.456172
$693$	0	0
$694$	20.0000	0.759190
$695$	8.00000	0.303457
$696$	0	0
$697$	0	0
$698$	−6.00000	−0.227103
$699$	0	0
$700$	−4.00000	−0.151186
$701$	6.00000	0.226617	0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000	0.226617	0.113308	+	0.993560i	$0.463855\pi$
$702$	0	0
$703$	6.00000	0.226294
$704$	−28.0000	−1.05529
$705$	0	0
$706$	−10.0000	−0.376355
$707$	0	0
$708$	0	0
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	−24.0000	−0.900704
$711$	0	0
$712$	6.00000	0.224860
$713$	−16.0000	−0.599205
$714$	0	0
$715$	16.0000	0.598366
$716$	12.0000	0.448461
$717$	0	0
$718$	0	0
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	0	0
$721$	−24.0000	−0.893807
$722$	−17.0000	−0.632674
$723$	0	0
$724$	−2.00000	−0.0743294
$725$	6.00000	0.222834
$726$	0	0
$727$	34.0000	1.26099	0.630495	−	0.776193i	$-0.282852\pi$
$727$	34.0000	1.26099	0.630495	+	0.776193i	$0.282852\pi$
$728$	24.0000	0.889499
$729$	0	0
$730$	20.0000	0.740233
$731$	−60.0000	−2.21918
$732$	0	0
$733$	−50.0000	−1.84679	−0.923396	−	0.383849i	$-0.874598\pi$
$733$	−50.0000	−1.84679	−0.923396	+	0.383849i	$0.874598\pi$
$734$	16.0000	0.590571
$735$	0	0
$736$	40.0000	1.47442
$737$	16.0000	0.589368
$738$	0	0
$739$	−24.0000	−0.882854	−0.441427	−	0.897297i	$-0.645528\pi$
$739$	−24.0000	−0.882854	−0.441427	+	0.897297i	$0.645528\pi$
$740$	2.00000	0.0735215
$741$	0	0
$742$	−16.0000	−0.587378
$743$	−36.0000	−1.32071	−0.660356	−	0.750953i	$-0.729595\pi$
$743$	−36.0000	−1.32071	−0.660356	+	0.750953i	$0.729595\pi$
$744$	0	0
$745$	16.0000	0.586195
$746$	22.0000	0.805477
$747$	0	0
$748$	24.0000	0.877527
$749$	−48.0000	−1.75388
$750$	0	0
$751$	−8.00000	−0.291924	−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000	−0.291924	−0.145962	+	0.989290i	$0.546628\pi$
$752$	−12.0000	−0.437595
$753$	0	0
$754$	−12.0000	−0.437014
$755$	−40.0000	−1.45575
$756$	0	0
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	24.0000	0.871719
$759$	0	0
$760$	−36.0000	−1.30586
$761$	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
$763$	40.0000	1.44810
$764$	8.00000	0.289430
$765$	0	0
$766$	32.0000	1.15621
$767$	−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	$0.488519\pi$
$770$	−32.0000	−1.15320
$771$	0	0
$772$	14.0000	0.503871
$773$	24.0000	0.863220	0.431610	−	0.902060i	$-0.357946\pi$
$773$	24.0000	0.863220	0.431610	+	0.902060i	$0.357946\pi$
$774$	0	0
$775$	−2.00000	−0.0718421
$776$	−6.00000	−0.215387
$777$	0	0
$778$	−14.0000	−0.501924
$779$	0	0
$780$	0	0
$781$	−48.0000	−1.71758
$782$	48.0000	1.71648
$783$	0	0
$784$	−9.00000	−0.321429
$785$	4.00000	0.142766
$786$	0	0
$787$	−44.0000	−1.56843	−0.784215	−	0.620489i	$-0.786934\pi$
$787$	−44.0000	−1.56843	−0.784215	+	0.620489i	$0.786934\pi$
$788$	24.0000	0.854965
$789$	0	0
$790$	−20.0000	−0.711568
$791$	24.0000	0.853342
$792$	0	0
$793$	−20.0000	−0.710221
$794$	2.00000	0.0709773
$795$	0	0
$796$	−2.00000	−0.0708881
$797$	2.00000	0.0708436	0.0354218	−	0.999372i	$-0.488723\pi$
$797$	2.00000	0.0708436	0.0354218	+	0.999372i	$0.488723\pi$
$798$	0	0
$799$	72.0000	2.54718
$800$	5.00000	0.176777
$801$	0	0
$802$	18.0000	0.635602
$803$	40.0000	1.41157
$804$	0	0
$805$	64.0000	2.25570
$806$	4.00000	0.140894
$807$	0	0
$808$	0	0
$809$	22.0000	0.773479	0.386739	−	0.922189i	$-0.373601\pi$
$809$	22.0000	0.773479	0.386739	+	0.922189i	$0.373601\pi$
$810$	0	0
$811$	−16.0000	−0.561836	−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000	−0.561836	−0.280918	+	0.959732i	$0.590639\pi$
$812$	−24.0000	−0.842235
$813$	0	0
$814$	−4.00000	−0.140200
$815$	−12.0000	−0.420342
$816$	0	0
$817$	60.0000	2.09913
$818$	34.0000	1.18878
$819$	0	0
$820$	0	0
$821$	−28.0000	−0.977207	−0.488603	−	0.872506i	$-0.662493\pi$
$821$	−28.0000	−0.977207	−0.488603	+	0.872506i	$0.662493\pi$
$822$	0	0
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	18.0000	0.627060
$825$	0	0
$826$	16.0000	0.556711
$827$	44.0000	1.53003	0.765015	−	0.644013i	$-0.222732\pi$
$827$	44.0000	1.53003	0.765015	+	0.644013i	$0.222732\pi$
$828$	0	0
$829$	−10.0000	−0.347314	−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000	−0.347314	−0.173657	+	0.984806i	$0.555558\pi$
$830$	0	0
$831$	0	0
$832$	−14.0000	−0.485363
$833$	54.0000	1.87099
$834$	0	0
$835$	0	0
$836$	−24.0000	−0.830057
$837$	0	0
$838$	16.0000	0.552711
$839$	52.0000	1.79524	0.897620	−	0.440771i	$-0.145295\pi$
$839$	52.0000	1.79524	0.897620	+	0.440771i	$0.145295\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−30.0000	−1.03387
$843$	0	0
$844$	4.00000	0.137686
$845$	−18.0000	−0.619219
$846$	0	0
$847$	−20.0000	−0.687208
$848$	4.00000	0.137361
$849$	0	0
$850$	6.00000	0.205798
$851$	8.00000	0.274236
$852$	0	0
$853$	−46.0000	−1.57501	−0.787505	−	0.616308i	$-0.788628\pi$
$853$	−46.0000	−1.57501	−0.787505	+	0.616308i	$0.788628\pi$
$854$	40.0000	1.36877
$855$	0	0
$856$	36.0000	1.23045
$857$	−6.00000	−0.204956	−0.102478	−	0.994735i	$-0.532677\pi$
$857$	−6.00000	−0.204956	−0.102478	+	0.994735i	$0.532677\pi$
$858$	0	0
$859$	22.0000	0.750630	0.375315	−	0.926897i	$-0.377534\pi$
$859$	22.0000	0.750630	0.375315	+	0.926897i	$0.377534\pi$
$860$	20.0000	0.681994
$861$	0	0
$862$	24.0000	0.817443
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	0	0
$865$	−24.0000	−0.816024
$866$	30.0000	1.01944
$867$	0	0
$868$	8.00000	0.271538
$869$	−40.0000	−1.35691
$870$	0	0
$871$	8.00000	0.271070
$872$	−30.0000	−1.01593
$873$	0	0
$874$	−48.0000	−1.62362
$875$	48.0000	1.62270
$876$	0	0
$877$	2.00000	0.0675352	0.0337676	−	0.999430i	$-0.489249\pi$
$877$	2.00000	0.0675352	0.0337676	+	0.999430i	$0.489249\pi$
$878$	14.0000	0.472477
$879$	0	0
$880$	8.00000	0.269680
$881$	−28.0000	−0.943344	−0.471672	−	0.881774i	$-0.656349\pi$
$881$	−28.0000	−0.943344	−0.471672	+	0.881774i	$0.656349\pi$
$882$	0	0
$883$	6.00000	0.201916	0.100958	−	0.994891i	$-0.467809\pi$
$883$	6.00000	0.201916	0.100958	+	0.994891i	$0.467809\pi$
$884$	12.0000	0.403604
$885$	0	0
$886$	16.0000	0.537531
$887$	−52.0000	−1.74599	−0.872995	−	0.487730i	$-0.837825\pi$
$887$	−52.0000	−1.74599	−0.872995	+	0.487730i	$0.837825\pi$
$888$	0	0
$889$	32.0000	1.07325
$890$	−4.00000	−0.134080
$891$	0	0
$892$	−16.0000	−0.535720
$893$	−72.0000	−2.40939
$894$	0	0
$895$	−24.0000	−0.802232
$896$	−12.0000	−0.400892
$897$	0	0
$898$	6.00000	0.200223
$899$	−12.0000	−0.400222
$900$	0	0
$901$	−24.0000	−0.799556
$902$	0	0
$903$	0	0
$904$	−18.0000	−0.598671
$905$	4.00000	0.132964
$906$	0	0
$907$	−26.0000	−0.863316	−0.431658	−	0.902037i	$-0.642071\pi$
$907$	−26.0000	−0.863316	−0.431658	+	0.902037i	$0.642071\pi$
$908$	−4.00000	−0.132745
$909$	0	0
$910$	−16.0000	−0.530395
$911$	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
$913$	0	0
$914$	−30.0000	−0.992312
$915$	0	0
$916$	−10.0000	−0.330409
$917$	−48.0000	−1.58510
$918$	0	0
$919$	38.0000	1.25350	0.626752	−	0.779219i	$-0.284384\pi$
$919$	38.0000	1.25350	0.626752	+	0.779219i	$0.284384\pi$
$920$	−48.0000	−1.58251
$921$	0	0
$922$	−30.0000	−0.987997
$923$	−24.0000	−0.789970
$924$	0	0
$925$	1.00000	0.0328798
$926$	10.0000	0.328620
$927$	0	0
$928$	30.0000	0.984798
$929$	−24.0000	−0.787414	−0.393707	−	0.919236i	$-0.628808\pi$
$929$	−24.0000	−0.787414	−0.393707	+	0.919236i	$0.628808\pi$
$930$	0	0
$931$	−54.0000	−1.76978
$932$	0	0
$933$	0	0
$934$	4.00000	0.130884
$935$	−48.0000	−1.56977
$936$	0	0
$937$	22.0000	0.718709	0.359354	−	0.933201i	$-0.382997\pi$
$937$	22.0000	0.718709	0.359354	+	0.933201i	$0.382997\pi$
$938$	−16.0000	−0.522419
$939$	0	0
$940$	−24.0000	−0.782794
$941$	28.0000	0.912774	0.456387	−	0.889781i	$-0.349143\pi$
$941$	28.0000	0.912774	0.456387	+	0.889781i	$0.349143\pi$
$942$	0	0
$943$	0	0
$944$	−4.00000	−0.130189
$945$	0	0
$946$	−40.0000	−1.30051
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	0	0
$949$	20.0000	0.649227
$950$	−6.00000	−0.194666
$951$	0	0
$952$	−72.0000	−2.33353
$953$	20.0000	0.647864	0.323932	−	0.946080i	$-0.394995\pi$
$953$	20.0000	0.647864	0.323932	+	0.946080i	$0.394995\pi$
$954$	0	0
$955$	−16.0000	−0.517748
$956$	0	0
$957$	0	0
$958$	8.00000	0.258468
$959$	48.0000	1.55000
$960$	0	0
$961$	−27.0000	−0.870968
$962$	−2.00000	−0.0644826
$963$	0	0
$964$	−14.0000	−0.450910
$965$	−28.0000	−0.901352
$966$	0	0
$967$	−2.00000	−0.0643157	−0.0321578	−	0.999483i	$-0.510238\pi$
$967$	−2.00000	−0.0643157	−0.0321578	+	0.999483i	$0.510238\pi$
$968$	15.0000	0.482118
$969$	0	0
$970$	4.00000	0.128432
$971$	−16.0000	−0.513464	−0.256732	−	0.966483i	$-0.582646\pi$
$971$	−16.0000	−0.513464	−0.256732	+	0.966483i	$0.582646\pi$
$972$	0	0
$973$	−16.0000	−0.512936
$974$	30.0000	0.961262
$975$	0	0
$976$	−10.0000	−0.320092
$977$	38.0000	1.21573	0.607864	−	0.794041i	$-0.292027\pi$
$977$	38.0000	1.21573	0.607864	+	0.794041i	$0.292027\pi$
$978$	0	0
$979$	−8.00000	−0.255681
$980$	−18.0000	−0.574989
$981$	0	0
$982$	−16.0000	−0.510581
$983$	36.0000	1.14822	0.574111	−	0.818778i	$-0.305348\pi$
$983$	36.0000	1.14822	0.574111	+	0.818778i	$0.305348\pi$
$984$	0	0
$985$	−48.0000	−1.52941
$986$	36.0000	1.14647
$987$	0	0
$988$	−12.0000	−0.381771
$989$	80.0000	2.54385
$990$	0	0
$991$	42.0000	1.33417	0.667087	−	0.744980i	$-0.267541\pi$
$991$	42.0000	1.33417	0.667087	+	0.744980i	$0.267541\pi$
$992$	−10.0000	−0.317500
$993$	0	0
$994$	48.0000	1.52247
$995$	4.00000	0.126809
$996$	0	0
$997$	−6.00000	−0.190022	−0.0950110	−	0.995476i	$-0.530289\pi$
$997$	−6.00000	−0.190022	−0.0950110	+	0.995476i	$0.530289\pi$
$998$	6.00000	0.189927
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	333.2.a.a.1.1	✓	1
3.2	odd	2		333.2.a.c.1.1	yes	1
4.3	odd	2		5328.2.a.u.1.1		1
5.4	even	2		8325.2.a.z.1.1		1
12.11	even	2		5328.2.a.g.1.1		1
15.14	odd	2		8325.2.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
333.2.a.a.1.1	✓	1	1.1	even	1	trivial
333.2.a.c.1.1	yes	1	3.2	odd	2
5328.2.a.g.1.1		1	12.11	even	2
5328.2.a.u.1.1		1	4.3	odd	2
8325.2.a.h.1.1		1	15.14	odd	2
8325.2.a.z.1.1		1	5.4	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:	\( N \)	\(=\)	\( 333 = 3^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	333.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more