LMFDB - Embedded newform 234.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [234,2,Mod(1,234)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(234, 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("234.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 234 = 2 \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	234.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:	$1.86849940730$
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	$\chi$	$=$	234.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

$q-1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} -2.00000 q^{7} -1.00000 q^{8} +2.00000 q^{10} -4.00000 q^{11} -1.00000 q^{13} +2.00000 q^{14} +1.00000 q^{16} -6.00000 q^{19} -2.00000 q^{20} +4.00000 q^{22} +4.00000 q^{23} -1.00000 q^{25} +1.00000 q^{26} -2.00000 q^{28} -8.00000 q^{29} -2.00000 q^{31} -1.00000 q^{32} +4.00000 q^{35} +6.00000 q^{37} +6.00000 q^{38} +2.00000 q^{40} +6.00000 q^{41} -8.00000 q^{43} -4.00000 q^{44} -4.00000 q^{46} +8.00000 q^{47} -3.00000 q^{49} +1.00000 q^{50} -1.00000 q^{52} +12.0000 q^{53} +8.00000 q^{55} +2.00000 q^{56} +8.00000 q^{58} +4.00000 q^{59} +10.0000 q^{61} +2.00000 q^{62} +1.00000 q^{64} +2.00000 q^{65} -2.00000 q^{67} -4.00000 q^{70} -16.0000 q^{71} +14.0000 q^{73} -6.00000 q^{74} -6.00000 q^{76} +8.00000 q^{77} -4.00000 q^{79} -2.00000 q^{80} -6.00000 q^{82} -12.0000 q^{83} +8.00000 q^{86} +4.00000 q^{88} -6.00000 q^{89} +2.00000 q^{91} +4.00000 q^{92} -8.00000 q^{94} +12.0000 q^{95} -10.0000 q^{97} +3.00000 q^{98} +O(q^{100})$

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
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	0	0
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	−1.00000	−0.353553
$9$	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
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	2.00000	0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	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
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	1.00000	0.196116
$27$	0	0
$28$	−2.00000	−0.377964
$29$	−8.00000	−1.48556	−0.742781	−	0.669534i	$-0.766494\pi$
$29$	−8.00000	−1.48556	−0.742781	+	0.669534i	$0.766494\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	0	0
$35$	4.00000	0.676123
$36$	0	0
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	6.00000	0.973329
$39$	0	0
$40$	2.00000	0.316228
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	−4.00000	−0.589768
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	1.00000	0.141421
$51$	0	0
$52$	−1.00000	−0.138675
$53$	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
$55$	8.00000	1.07872
$56$	2.00000	0.267261
$57$	0	0
$58$	8.00000	1.05045
$59$	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
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	0	0
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	0	0
$69$	0	0
$70$	−4.00000	−0.478091
$71$	−16.0000	−1.89885	−0.949425	−	0.313993i	$-0.898333\pi$
$71$	−16.0000	−1.89885	−0.949425	+	0.313993i	$0.898333\pi$
$72$	0	0
$73$	14.0000	1.63858	0.819288	−	0.573382i	$-0.194369\pi$
$73$	14.0000	1.63858	0.819288	+	0.573382i	$0.194369\pi$
$74$	−6.00000	−0.697486
$75$	0	0
$76$	−6.00000	−0.688247
$77$	8.00000	0.911685
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	−2.00000	−0.223607
$81$	0	0
$82$	−6.00000	−0.662589
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	0	0
$85$	0	0
$86$	8.00000	0.862662
$87$	0	0
$88$	4.00000	0.426401
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	4.00000	0.417029
$93$	0	0
$94$	−8.00000	−0.825137
$95$	12.0000	1.23117
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	3.00000	0.303046
$99$	0	0
$100$	−1.00000	−0.100000
$101$	−16.0000	−1.59206	−0.796030	−	0.605257i	$-0.793070\pi$
$101$	−16.0000	−1.59206	−0.796030	+	0.605257i	$0.793070\pi$
$102$	0	0
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	−12.0000	−1.16554
$107$	−20.0000	−1.93347	−0.966736	−	0.255774i	$-0.917670\pi$
$107$	−20.0000	−1.93347	−0.966736	+	0.255774i	$0.917670\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	−8.00000	−0.762770
$111$	0	0
$112$	−2.00000	−0.188982
$113$	4.00000	0.376288	0.188144	−	0.982141i	$-0.439753\pi$
$113$	4.00000	0.376288	0.188144	+	0.982141i	$0.439753\pi$
$114$	0	0
$115$	−8.00000	−0.746004
$116$	−8.00000	−0.742781
$117$	0	0
$118$	−4.00000	−0.368230
$119$	0	0
$120$	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	$0.615500\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−2.00000	−0.175412
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	12.0000	1.04053
$134$	2.00000	0.172774
$135$	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
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	4.00000	0.338062
$141$	0	0
$142$	16.0000	1.34269
$143$	4.00000	0.334497
$144$	0	0
$145$	16.0000	1.32873
$146$	−14.0000	−1.15865
$147$	0	0
$148$	6.00000	0.493197
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	18.0000	1.46482	0.732410	−	0.680864i	$-0.238396\pi$
$151$	18.0000	1.46482	0.732410	+	0.680864i	$0.238396\pi$
$152$	6.00000	0.486664
$153$	0	0
$154$	−8.00000	−0.644658
$155$	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	$0.474569\pi$
$158$	4.00000	0.318223
$159$	0	0
$160$	2.00000	0.158114
$161$	−8.00000	−0.630488
$162$	0	0
$163$	−10.0000	−0.783260	−0.391630	−	0.920123i	$-0.628089\pi$
$163$	−10.0000	−0.783260	−0.391630	+	0.920123i	$0.628089\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	12.0000	0.931381
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	−8.00000	−0.609994
$173$	16.0000	1.21646	0.608229	−	0.793762i	$-0.291880\pi$
$173$	16.0000	1.21646	0.608229	+	0.793762i	$0.291880\pi$
$174$	0	0
$175$	2.00000	0.151186
$176$	−4.00000	−0.301511
$177$	0	0
$178$	6.00000	0.449719
$179$	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
$181$	−18.0000	−1.33793	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000	−1.33793	−0.668965	+	0.743294i	$0.733262\pi$
$182$	−2.00000	−0.148250
$183$	0	0
$184$	−4.00000	−0.294884
$185$	−12.0000	−0.882258
$186$	0	0
$187$	0	0
$188$	8.00000	0.583460
$189$	0	0
$190$	−12.0000	−0.870572
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	−3.00000	−0.214286
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	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
$202$	16.0000	1.12576
$203$	16.0000	1.12298
$204$	0	0
$205$	−12.0000	−0.838116
$206$	−4.00000	−0.278693
$207$	0	0
$208$	−1.00000	−0.0693375
$209$	24.0000	1.66011
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	12.0000	0.824163
$213$	0	0
$214$	20.0000	1.36717
$215$	16.0000	1.09119
$216$	0	0
$217$	4.00000	0.271538
$218$	−10.0000	−0.677285
$219$	0	0
$220$	8.00000	0.539360
$221$	0	0
$222$	0	0
$223$	6.00000	0.401790	0.200895	−	0.979613i	$-0.435615\pi$
$223$	6.00000	0.401790	0.200895	+	0.979613i	$0.435615\pi$
$224$	2.00000	0.133631
$225$	0	0
$226$	−4.00000	−0.266076
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	0	0
$229$	−30.0000	−1.98246	−0.991228	−	0.132164i	$-0.957808\pi$
$229$	−30.0000	−1.98246	−0.991228	+	0.132164i	$0.957808\pi$
$230$	8.00000	0.527504
$231$	0	0
$232$	8.00000	0.525226
$233$	20.0000	1.31024	0.655122	−	0.755523i	$-0.272617\pi$
$233$	20.0000	1.31024	0.655122	+	0.755523i	$0.272617\pi$
$234$	0	0
$235$	−16.0000	−1.04372
$236$	4.00000	0.260378
$237$	0	0
$238$	0	0
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	6.00000	0.386494	0.193247	−	0.981150i	$-0.438098\pi$
$241$	6.00000	0.386494	0.193247	+	0.981150i	$0.438098\pi$
$242$	−5.00000	−0.321412
$243$	0	0
$244$	10.0000	0.640184
$245$	6.00000	0.383326
$246$	0	0
$247$	6.00000	0.381771
$248$	2.00000	0.127000
$249$	0	0
$250$	−12.0000	−0.758947
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	−16.0000	−1.00591
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	12.0000	0.748539	0.374270	−	0.927320i	$-0.377893\pi$
$257$	12.0000	0.748539	0.374270	+	0.927320i	$0.377893\pi$
$258$	0	0
$259$	−12.0000	−0.745644
$260$	2.00000	0.124035
$261$	0	0
$262$	−4.00000	−0.247121
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	−24.0000	−1.47431
$266$	−12.0000	−0.735767
$267$	0	0
$268$	−2.00000	−0.122169
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	22.0000	1.33640	0.668202	−	0.743980i	$-0.267064\pi$
$271$	22.0000	1.33640	0.668202	+	0.743980i	$0.267064\pi$
$272$	0	0
$273$	0	0
$274$	18.0000	1.08742
$275$	4.00000	0.241209
$276$	0	0
$277$	−6.00000	−0.360505	−0.180253	−	0.983620i	$-0.557691\pi$
$277$	−6.00000	−0.360505	−0.180253	+	0.983620i	$0.557691\pi$
$278$	20.0000	1.19952
$279$	0	0
$280$	−4.00000	−0.239046
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	−20.0000	−1.18888	−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000	−1.18888	−0.594438	+	0.804141i	$0.702626\pi$
$284$	−16.0000	−0.949425
$285$	0	0
$286$	−4.00000	−0.236525
$287$	−12.0000	−0.708338
$288$	0	0
$289$	−17.0000	−1.00000
$290$	−16.0000	−0.939552
$291$	0	0
$292$	14.0000	0.819288
$293$	2.00000	0.116841	0.0584206	−	0.998292i	$-0.481394\pi$
$293$	2.00000	0.116841	0.0584206	+	0.998292i	$0.481394\pi$
$294$	0	0
$295$	−8.00000	−0.465778
$296$	−6.00000	−0.348743
$297$	0	0
$298$	6.00000	0.347571
$299$	−4.00000	−0.231326
$300$	0	0
$301$	16.0000	0.922225
$302$	−18.0000	−1.03578
$303$	0	0
$304$	−6.00000	−0.344124
$305$	−20.0000	−1.14520
$306$	0	0
$307$	2.00000	0.114146	0.0570730	−	0.998370i	$-0.481823\pi$
$307$	2.00000	0.114146	0.0570730	+	0.998370i	$0.481823\pi$
$308$	8.00000	0.455842
$309$	0	0
$310$	−4.00000	−0.227185
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	26.0000	1.46961	0.734803	−	0.678280i	$-0.237274\pi$
$313$	26.0000	1.46961	0.734803	+	0.678280i	$0.237274\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	−4.00000	−0.225018
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	0	0
$319$	32.0000	1.79166
$320$	−2.00000	−0.111803
$321$	0	0
$322$	8.00000	0.445823
$323$	0	0
$324$	0	0
$325$	1.00000	0.0554700
$326$	10.0000	0.553849
$327$	0	0
$328$	−6.00000	−0.331295
$329$	−16.0000	−0.882109
$330$	0	0
$331$	−10.0000	−0.549650	−0.274825	−	0.961494i	$-0.588620\pi$
$331$	−10.0000	−0.549650	−0.274825	+	0.961494i	$0.588620\pi$
$332$	−12.0000	−0.658586
$333$	0	0
$334$	0	0
$335$	4.00000	0.218543
$336$	0	0
$337$	−34.0000	−1.85210	−0.926049	−	0.377403i	$-0.876817\pi$
$337$	−34.0000	−1.85210	−0.926049	+	0.377403i	$0.876817\pi$
$338$	−1.00000	−0.0543928
$339$	0	0
$340$	0	0
$341$	8.00000	0.433224
$342$	0	0
$343$	20.0000	1.07990
$344$	8.00000	0.431331
$345$	0	0
$346$	−16.0000	−0.860165
$347$	−24.0000	−1.28839	−0.644194	−	0.764862i	$-0.722807\pi$
$347$	−24.0000	−1.28839	−0.644194	+	0.764862i	$0.722807\pi$
$348$	0	0
$349$	22.0000	1.17763	0.588817	−	0.808267i	$-0.299594\pi$
$349$	22.0000	1.17763	0.588817	+	0.808267i	$0.299594\pi$
$350$	−2.00000	−0.106904
$351$	0	0
$352$	4.00000	0.213201
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	0	0
$355$	32.0000	1.69838
$356$	−6.00000	−0.317999
$357$	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
$362$	18.0000	0.946059
$363$	0	0
$364$	2.00000	0.104828
$365$	−28.0000	−1.46559
$366$	0	0
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	4.00000	0.208514
$369$	0	0
$370$	12.0000	0.623850
$371$	−24.0000	−1.24602
$372$	0	0
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	0	0
$375$	0	0
$376$	−8.00000	−0.412568
$377$	8.00000	0.412021
$378$	0	0
$379$	10.0000	0.513665	0.256833	−	0.966456i	$-0.417321\pi$
$379$	10.0000	0.513665	0.256833	+	0.966456i	$0.417321\pi$
$380$	12.0000	0.615587
$381$	0	0
$382$	4.00000	0.204658
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	−16.0000	−0.815436
$386$	−2.00000	−0.101797
$387$	0	0
$388$	−10.0000	−0.507673
$389$	−12.0000	−0.608424	−0.304212	−	0.952604i	$-0.598393\pi$
$389$	−12.0000	−0.608424	−0.304212	+	0.952604i	$0.598393\pi$
$390$	0	0
$391$	0	0
$392$	3.00000	0.151523
$393$	0	0
$394$	18.0000	0.906827
$395$	8.00000	0.402524
$396$	0	0
$397$	−10.0000	−0.501886	−0.250943	−	0.968002i	$-0.580741\pi$
$397$	−10.0000	−0.501886	−0.250943	+	0.968002i	$0.580741\pi$
$398$	−8.00000	−0.401004
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	0	0
$403$	2.00000	0.0996271
$404$	−16.0000	−0.796030
$405$	0	0
$406$	−16.0000	−0.794067
$407$	−24.0000	−1.18964
$408$	0	0
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	12.0000	0.592638
$411$	0	0
$412$	4.00000	0.197066
$413$	−8.00000	−0.393654
$414$	0	0
$415$	24.0000	1.17811
$416$	1.00000	0.0490290
$417$	0	0
$418$	−24.0000	−1.17388
$419$	36.0000	1.75872	0.879358	−	0.476162i	$-0.157972\pi$
$419$	36.0000	1.75872	0.879358	+	0.476162i	$0.157972\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	−8.00000	−0.389434
$423$	0	0
$424$	−12.0000	−0.582772
$425$	0	0
$426$	0	0
$427$	−20.0000	−0.967868
$428$	−20.0000	−0.966736
$429$	0	0
$430$	−16.0000	−0.771589
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	−4.00000	−0.192006
$435$	0	0
$436$	10.0000	0.478913
$437$	−24.0000	−1.14808
$438$	0	0
$439$	−4.00000	−0.190910	−0.0954548	−	0.995434i	$-0.530431\pi$
$439$	−4.00000	−0.190910	−0.0954548	+	0.995434i	$0.530431\pi$
$440$	−8.00000	−0.381385
$441$	0	0
$442$	0	0
$443$	36.0000	1.71041	0.855206	−	0.518289i	$-0.173431\pi$
$443$	36.0000	1.71041	0.855206	+	0.518289i	$0.173431\pi$
$444$	0	0
$445$	12.0000	0.568855
$446$	−6.00000	−0.284108
$447$	0	0
$448$	−2.00000	−0.0944911
$449$	26.0000	1.22702	0.613508	−	0.789689i	$-0.289758\pi$
$449$	26.0000	1.22702	0.613508	+	0.789689i	$0.289758\pi$
$450$	0	0
$451$	−24.0000	−1.13012
$452$	4.00000	0.188144
$453$	0	0
$454$	−12.0000	−0.563188
$455$	−4.00000	−0.187523
$456$	0	0
$457$	−2.00000	−0.0935561	−0.0467780	−	0.998905i	$-0.514895\pi$
$457$	−2.00000	−0.0935561	−0.0467780	+	0.998905i	$0.514895\pi$
$458$	30.0000	1.40181
$459$	0	0
$460$	−8.00000	−0.373002
$461$	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
$463$	22.0000	1.02243	0.511213	−	0.859454i	$-0.329196\pi$
$463$	22.0000	1.02243	0.511213	+	0.859454i	$0.329196\pi$
$464$	−8.00000	−0.371391
$465$	0	0
$466$	−20.0000	−0.926482
$467$	−8.00000	−0.370196	−0.185098	−	0.982720i	$-0.559260\pi$
$467$	−8.00000	−0.370196	−0.185098	+	0.982720i	$0.559260\pi$
$468$	0	0
$469$	4.00000	0.184703
$470$	16.0000	0.738025
$471$	0	0
$472$	−4.00000	−0.184115
$473$	32.0000	1.47136
$474$	0	0
$475$	6.00000	0.275299
$476$	0	0
$477$	0	0
$478$	24.0000	1.09773
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−6.00000	−0.273576
$482$	−6.00000	−0.273293
$483$	0	0
$484$	5.00000	0.227273
$485$	20.0000	0.908153
$486$	0	0
$487$	38.0000	1.72194	0.860972	−	0.508652i	$-0.169856\pi$
$487$	38.0000	1.72194	0.860972	+	0.508652i	$0.169856\pi$
$488$	−10.0000	−0.452679
$489$	0	0
$490$	−6.00000	−0.271052
$491$	−16.0000	−0.722070	−0.361035	−	0.932552i	$-0.617576\pi$
$491$	−16.0000	−0.722070	−0.361035	+	0.932552i	$0.617576\pi$
$492$	0	0
$493$	0	0
$494$	−6.00000	−0.269953
$495$	0	0
$496$	−2.00000	−0.0898027
$497$	32.0000	1.43540
$498$	0	0
$499$	22.0000	0.984855	0.492428	−	0.870353i	$-0.336110\pi$
$499$	22.0000	0.984855	0.492428	+	0.870353i	$0.336110\pi$
$500$	12.0000	0.536656
$501$	0	0
$502$	0	0
$503$	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
$505$	32.0000	1.42398
$506$	16.0000	0.711287
$507$	0	0
$508$	−8.00000	−0.354943
$509$	−10.0000	−0.443242	−0.221621	−	0.975133i	$-0.571135\pi$
$509$	−10.0000	−0.443242	−0.221621	+	0.975133i	$0.571135\pi$
$510$	0	0
$511$	−28.0000	−1.23865
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−12.0000	−0.529297
$515$	−8.00000	−0.352522
$516$	0	0
$517$	−32.0000	−1.40736
$518$	12.0000	0.527250
$519$	0	0
$520$	−2.00000	−0.0877058
$521$	−36.0000	−1.57719	−0.788594	−	0.614914i	$-0.789191\pi$
$521$	−36.0000	−1.57719	−0.788594	+	0.614914i	$0.789191\pi$
$522$	0	0
$523$	−12.0000	−0.524723	−0.262362	−	0.964970i	$-0.584501\pi$
$523$	−12.0000	−0.524723	−0.262362	+	0.964970i	$0.584501\pi$
$524$	4.00000	0.174741
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−7.00000	−0.304348
$530$	24.0000	1.04249
$531$	0	0
$532$	12.0000	0.520266
$533$	−6.00000	−0.259889
$534$	0	0
$535$	40.0000	1.72935
$536$	2.00000	0.0863868
$537$	0	0
$538$	0	0
$539$	12.0000	0.516877
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	−22.0000	−0.944981
$543$	0	0
$544$	0	0
$545$	−20.0000	−0.856706
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	−18.0000	−0.768922
$549$	0	0
$550$	−4.00000	−0.170561
$551$	48.0000	2.04487
$552$	0	0
$553$	8.00000	0.340195
$554$	6.00000	0.254916
$555$	0	0
$556$	−20.0000	−0.848189
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	4.00000	0.169031
$561$	0	0
$562$	10.0000	0.421825
$563$	−20.0000	−0.842900	−0.421450	−	0.906852i	$-0.638479\pi$
$563$	−20.0000	−0.842900	−0.421450	+	0.906852i	$0.638479\pi$
$564$	0	0
$565$	−8.00000	−0.336563
$566$	20.0000	0.840663
$567$	0	0
$568$	16.0000	0.671345
$569$	−4.00000	−0.167689	−0.0838444	−	0.996479i	$-0.526720\pi$
$569$	−4.00000	−0.167689	−0.0838444	+	0.996479i	$0.526720\pi$
$570$	0	0
$571$	36.0000	1.50655	0.753277	−	0.657704i	$-0.228472\pi$
$571$	36.0000	1.50655	0.753277	+	0.657704i	$0.228472\pi$
$572$	4.00000	0.167248
$573$	0	0
$574$	12.0000	0.500870
$575$	−4.00000	−0.166812
$576$	0	0
$577$	−30.0000	−1.24892	−0.624458	−	0.781058i	$-0.714680\pi$
$577$	−30.0000	−1.24892	−0.624458	+	0.781058i	$0.714680\pi$
$578$	17.0000	0.707107
$579$	0	0
$580$	16.0000	0.664364
$581$	24.0000	0.995688
$582$	0	0
$583$	−48.0000	−1.98796
$584$	−14.0000	−0.579324
$585$	0	0
$586$	−2.00000	−0.0826192
$587$	28.0000	1.15568	0.577842	−	0.816149i	$-0.303895\pi$
$587$	28.0000	1.15568	0.577842	+	0.816149i	$0.303895\pi$
$588$	0	0
$589$	12.0000	0.494451
$590$	8.00000	0.329355
$591$	0	0
$592$	6.00000	0.246598
$593$	−38.0000	−1.56047	−0.780236	−	0.625485i	$-0.784901\pi$
$593$	−38.0000	−1.56047	−0.780236	+	0.625485i	$0.784901\pi$
$594$	0	0
$595$	0	0
$596$	−6.00000	−0.245770
$597$	0	0
$598$	4.00000	0.163572
$599$	−40.0000	−1.63436	−0.817178	−	0.576386i	$-0.804463\pi$
$599$	−40.0000	−1.63436	−0.817178	+	0.576386i	$0.804463\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	−16.0000	−0.652111
$603$	0	0
$604$	18.0000	0.732410
$605$	−10.0000	−0.406558
$606$	0	0
$607$	24.0000	0.974130	0.487065	−	0.873366i	$-0.338067\pi$
$607$	24.0000	0.974130	0.487065	+	0.873366i	$0.338067\pi$
$608$	6.00000	0.243332
$609$	0	0
$610$	20.0000	0.809776
$611$	−8.00000	−0.323645
$612$	0	0
$613$	−34.0000	−1.37325	−0.686624	−	0.727013i	$-0.740908\pi$
$613$	−34.0000	−1.37325	−0.686624	+	0.727013i	$0.740908\pi$
$614$	−2.00000	−0.0807134
$615$	0	0
$616$	−8.00000	−0.322329
$617$	14.0000	0.563619	0.281809	−	0.959470i	$-0.409065\pi$
$617$	14.0000	0.563619	0.281809	+	0.959470i	$0.409065\pi$
$618$	0	0
$619$	34.0000	1.36658	0.683288	−	0.730149i	$-0.260549\pi$
$619$	34.0000	1.36658	0.683288	+	0.730149i	$0.260549\pi$
$620$	4.00000	0.160644
$621$	0	0
$622$	12.0000	0.481156
$623$	12.0000	0.480770
$624$	0	0
$625$	−19.0000	−0.760000
$626$	−26.0000	−1.03917
$627$	0	0
$628$	2.00000	0.0798087
$629$	0	0
$630$	0	0
$631$	−14.0000	−0.557331	−0.278666	−	0.960388i	$-0.589892\pi$
$631$	−14.0000	−0.557331	−0.278666	+	0.960388i	$0.589892\pi$
$632$	4.00000	0.159111
$633$	0	0
$634$	−6.00000	−0.238290
$635$	16.0000	0.634941
$636$	0	0
$637$	3.00000	0.118864
$638$	−32.0000	−1.26689
$639$	0	0
$640$	2.00000	0.0790569
$641$	−40.0000	−1.57991	−0.789953	−	0.613168i	$-0.789895\pi$
$641$	−40.0000	−1.57991	−0.789953	+	0.613168i	$0.789895\pi$
$642$	0	0
$643$	−10.0000	−0.394362	−0.197181	−	0.980367i	$-0.563179\pi$
$643$	−10.0000	−0.394362	−0.197181	+	0.980367i	$0.563179\pi$
$644$	−8.00000	−0.315244
$645$	0	0
$646$	0	0
$647$	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
$650$	−1.00000	−0.0392232
$651$	0	0
$652$	−10.0000	−0.391630
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	−8.00000	−0.312586
$656$	6.00000	0.234261
$657$	0	0
$658$	16.0000	0.623745
$659$	−48.0000	−1.86981	−0.934907	−	0.354892i	$-0.884518\pi$
$659$	−48.0000	−1.86981	−0.934907	+	0.354892i	$0.884518\pi$
$660$	0	0
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	10.0000	0.388661
$663$	0	0
$664$	12.0000	0.465690
$665$	−24.0000	−0.930680
$666$	0	0
$667$	−32.0000	−1.23904
$668$	0	0
$669$	0	0
$670$	−4.00000	−0.154533
$671$	−40.0000	−1.54418
$672$	0	0
$673$	22.0000	0.848038	0.424019	−	0.905653i	$-0.360619\pi$
$673$	22.0000	0.848038	0.424019	+	0.905653i	$0.360619\pi$
$674$	34.0000	1.30963
$675$	0	0
$676$	1.00000	0.0384615
$677$	−12.0000	−0.461197	−0.230599	−	0.973049i	$-0.574068\pi$
$677$	−12.0000	−0.461197	−0.230599	+	0.973049i	$0.574068\pi$
$678$	0	0
$679$	20.0000	0.767530
$680$	0	0
$681$	0	0
$682$	−8.00000	−0.306336
$683$	−4.00000	−0.153056	−0.0765279	−	0.997067i	$-0.524383\pi$
$683$	−4.00000	−0.153056	−0.0765279	+	0.997067i	$0.524383\pi$
$684$	0	0
$685$	36.0000	1.37549
$686$	−20.0000	−0.763604
$687$	0	0
$688$	−8.00000	−0.304997
$689$	−12.0000	−0.457164
$690$	0	0
$691$	−26.0000	−0.989087	−0.494543	−	0.869153i	$-0.664665\pi$
$691$	−26.0000	−0.989087	−0.494543	+	0.869153i	$0.664665\pi$
$692$	16.0000	0.608229
$693$	0	0
$694$	24.0000	0.911028
$695$	40.0000	1.51729
$696$	0	0
$697$	0	0
$698$	−22.0000	−0.832712
$699$	0	0
$700$	2.00000	0.0755929
$701$	−12.0000	−0.453234	−0.226617	−	0.973984i	$-0.572767\pi$
$701$	−12.0000	−0.453234	−0.226617	+	0.973984i	$0.572767\pi$
$702$	0	0
$703$	−36.0000	−1.35777
$704$	−4.00000	−0.150756
$705$	0	0
$706$	−14.0000	−0.526897
$707$	32.0000	1.20348
$708$	0	0
$709$	−22.0000	−0.826227	−0.413114	−	0.910679i	$-0.635559\pi$
$709$	−22.0000	−0.826227	−0.413114	+	0.910679i	$0.635559\pi$
$710$	−32.0000	−1.20094
$711$	0	0
$712$	6.00000	0.224860
$713$	−8.00000	−0.299602
$714$	0	0
$715$	−8.00000	−0.299183
$716$	12.0000	0.448461
$717$	0	0
$718$	0	0
$719$	28.0000	1.04422	0.522112	−	0.852877i	$-0.325144\pi$
$719$	28.0000	1.04422	0.522112	+	0.852877i	$0.325144\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	−17.0000	−0.632674
$723$	0	0
$724$	−18.0000	−0.668965
$725$	8.00000	0.297113
$726$	0	0
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	−2.00000	−0.0741249
$729$	0	0
$730$	28.0000	1.03633
$731$	0	0
$732$	0	0
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	−8.00000	−0.295285
$735$	0	0
$736$	−4.00000	−0.147442
$737$	8.00000	0.294684
$738$	0	0
$739$	−54.0000	−1.98642	−0.993211	−	0.116326i	$-0.962888\pi$
$739$	−54.0000	−1.98642	−0.993211	+	0.116326i	$0.962888\pi$
$740$	−12.0000	−0.441129
$741$	0	0
$742$	24.0000	0.881068
$743$	−40.0000	−1.46746	−0.733729	−	0.679442i	$-0.762222\pi$
$743$	−40.0000	−1.46746	−0.733729	+	0.679442i	$0.762222\pi$
$744$	0	0
$745$	12.0000	0.439646
$746$	10.0000	0.366126
$747$	0	0
$748$	0	0
$749$	40.0000	1.46157
$750$	0	0
$751$	−12.0000	−0.437886	−0.218943	−	0.975738i	$-0.570261\pi$
$751$	−12.0000	−0.437886	−0.218943	+	0.975738i	$0.570261\pi$
$752$	8.00000	0.291730
$753$	0	0
$754$	−8.00000	−0.291343
$755$	−36.0000	−1.31017
$756$	0	0
$757$	−18.0000	−0.654221	−0.327111	−	0.944986i	$-0.606075\pi$
$757$	−18.0000	−0.654221	−0.327111	+	0.944986i	$0.606075\pi$
$758$	−10.0000	−0.363216
$759$	0	0
$760$	−12.0000	−0.435286
$761$	30.0000	1.08750	0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000	1.08750	0.543750	+	0.839248i	$0.317004\pi$
$762$	0	0
$763$	−20.0000	−0.724049
$764$	−4.00000	−0.144715
$765$	0	0
$766$	0	0
$767$	−4.00000	−0.144432
$768$	0	0
$769$	−30.0000	−1.08183	−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000	−1.08183	−0.540914	+	0.841078i	$0.681921\pi$
$770$	16.0000	0.576600
$771$	0	0
$772$	2.00000	0.0719816
$773$	−14.0000	−0.503545	−0.251773	−	0.967786i	$-0.581013\pi$
$773$	−14.0000	−0.503545	−0.251773	+	0.967786i	$0.581013\pi$
$774$	0	0
$775$	2.00000	0.0718421
$776$	10.0000	0.358979
$777$	0	0
$778$	12.0000	0.430221
$779$	−36.0000	−1.28983
$780$	0	0
$781$	64.0000	2.29010
$782$	0	0
$783$	0	0
$784$	−3.00000	−0.107143
$785$	−4.00000	−0.142766
$786$	0	0
$787$	−22.0000	−0.784215	−0.392108	−	0.919919i	$-0.628254\pi$
$787$	−22.0000	−0.784215	−0.392108	+	0.919919i	$0.628254\pi$
$788$	−18.0000	−0.641223
$789$	0	0
$790$	−8.00000	−0.284627
$791$	−8.00000	−0.284447
$792$	0	0
$793$	−10.0000	−0.355110
$794$	10.0000	0.354887
$795$	0	0
$796$	8.00000	0.283552
$797$	−12.0000	−0.425062	−0.212531	−	0.977154i	$-0.568171\pi$
$797$	−12.0000	−0.425062	−0.212531	+	0.977154i	$0.568171\pi$
$798$	0	0
$799$	0	0
$800$	1.00000	0.0353553
$801$	0	0
$802$	6.00000	0.211867
$803$	−56.0000	−1.97620
$804$	0	0
$805$	16.0000	0.563926
$806$	−2.00000	−0.0704470
$807$	0	0
$808$	16.0000	0.562878
$809$	36.0000	1.26569	0.632846	−	0.774277i	$-0.281886\pi$
$809$	36.0000	1.26569	0.632846	+	0.774277i	$0.281886\pi$
$810$	0	0
$811$	50.0000	1.75574	0.877869	−	0.478901i	$-0.158965\pi$
$811$	50.0000	1.75574	0.877869	+	0.478901i	$0.158965\pi$
$812$	16.0000	0.561490
$813$	0	0
$814$	24.0000	0.841200
$815$	20.0000	0.700569
$816$	0	0
$817$	48.0000	1.67931
$818$	26.0000	0.909069
$819$	0	0
$820$	−12.0000	−0.419058
$821$	34.0000	1.18661	0.593304	−	0.804978i	$-0.297823\pi$
$821$	34.0000	1.18661	0.593304	+	0.804978i	$0.297823\pi$
$822$	0	0
$823$	−12.0000	−0.418294	−0.209147	−	0.977884i	$-0.567069\pi$
$823$	−12.0000	−0.418294	−0.209147	+	0.977884i	$0.567069\pi$
$824$	−4.00000	−0.139347
$825$	0	0
$826$	8.00000	0.278356
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	0	0
$829$	46.0000	1.59765	0.798823	−	0.601566i	$-0.205456\pi$
$829$	46.0000	1.59765	0.798823	+	0.601566i	$0.205456\pi$
$830$	−24.0000	−0.833052
$831$	0	0
$832$	−1.00000	−0.0346688
$833$	0	0
$834$	0	0
$835$	0	0
$836$	24.0000	0.830057
$837$	0	0
$838$	−36.0000	−1.24360
$839$	16.0000	0.552381	0.276191	−	0.961103i	$-0.410928\pi$
$839$	16.0000	0.552381	0.276191	+	0.961103i	$0.410928\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	−10.0000	−0.344623
$843$	0	0
$844$	8.00000	0.275371
$845$	−2.00000	−0.0688021
$846$	0	0
$847$	−10.0000	−0.343604
$848$	12.0000	0.412082
$849$	0	0
$850$	0	0
$851$	24.0000	0.822709
$852$	0	0
$853$	−26.0000	−0.890223	−0.445112	−	0.895475i	$-0.646836\pi$
$853$	−26.0000	−0.890223	−0.445112	+	0.895475i	$0.646836\pi$
$854$	20.0000	0.684386
$855$	0	0
$856$	20.0000	0.683586
$857$	−28.0000	−0.956462	−0.478231	−	0.878234i	$-0.658722\pi$
$857$	−28.0000	−0.956462	−0.478231	+	0.878234i	$0.658722\pi$
$858$	0	0
$859$	8.00000	0.272956	0.136478	−	0.990643i	$-0.456422\pi$
$859$	8.00000	0.272956	0.136478	+	0.990643i	$0.456422\pi$
$860$	16.0000	0.545595
$861$	0	0
$862$	0	0
$863$	8.00000	0.272323	0.136162	−	0.990687i	$-0.456523\pi$
$863$	8.00000	0.272323	0.136162	+	0.990687i	$0.456523\pi$
$864$	0	0
$865$	−32.0000	−1.08803
$866$	2.00000	0.0679628
$867$	0	0
$868$	4.00000	0.135769
$869$	16.0000	0.542763
$870$	0	0
$871$	2.00000	0.0677674
$872$	−10.0000	−0.338643
$873$	0	0
$874$	24.0000	0.811812
$875$	−24.0000	−0.811348
$876$	0	0
$877$	6.00000	0.202606	0.101303	−	0.994856i	$-0.467699\pi$
$877$	6.00000	0.202606	0.101303	+	0.994856i	$0.467699\pi$
$878$	4.00000	0.134993
$879$	0	0
$880$	8.00000	0.269680
$881$	36.0000	1.21287	0.606435	−	0.795133i	$-0.292599\pi$
$881$	36.0000	1.21287	0.606435	+	0.795133i	$0.292599\pi$
$882$	0	0
$883$	−20.0000	−0.673054	−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000	−0.673054	−0.336527	+	0.941674i	$0.609252\pi$
$884$	0	0
$885$	0	0
$886$	−36.0000	−1.20944
$887$	44.0000	1.47738	0.738688	−	0.674048i	$-0.235446\pi$
$887$	44.0000	1.47738	0.738688	+	0.674048i	$0.235446\pi$
$888$	0	0
$889$	16.0000	0.536623
$890$	−12.0000	−0.402241
$891$	0	0
$892$	6.00000	0.200895
$893$	−48.0000	−1.60626
$894$	0	0
$895$	−24.0000	−0.802232
$896$	2.00000	0.0668153
$897$	0	0
$898$	−26.0000	−0.867631
$899$	16.0000	0.533630
$900$	0	0
$901$	0	0
$902$	24.0000	0.799113
$903$	0	0
$904$	−4.00000	−0.133038
$905$	36.0000	1.19668
$906$	0	0
$907$	12.0000	0.398453	0.199227	−	0.979953i	$-0.436157\pi$
$907$	12.0000	0.398453	0.199227	+	0.979953i	$0.436157\pi$
$908$	12.0000	0.398234
$909$	0	0
$910$	4.00000	0.132599
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	0	0
$913$	48.0000	1.58857
$914$	2.00000	0.0661541
$915$	0	0
$916$	−30.0000	−0.991228
$917$	−8.00000	−0.264183
$918$	0	0
$919$	12.0000	0.395843	0.197922	−	0.980218i	$-0.436581\pi$
$919$	12.0000	0.395843	0.197922	+	0.980218i	$0.436581\pi$
$920$	8.00000	0.263752
$921$	0	0
$922$	−30.0000	−0.987997
$923$	16.0000	0.526646
$924$	0	0
$925$	−6.00000	−0.197279
$926$	−22.0000	−0.722965
$927$	0	0
$928$	8.00000	0.262613
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	18.0000	0.589926
$932$	20.0000	0.655122
$933$	0	0
$934$	8.00000	0.261768
$935$	0	0
$936$	0	0
$937$	30.0000	0.980057	0.490029	−	0.871706i	$-0.336986\pi$
$937$	30.0000	0.980057	0.490029	+	0.871706i	$0.336986\pi$
$938$	−4.00000	−0.130605
$939$	0	0
$940$	−16.0000	−0.521862
$941$	26.0000	0.847576	0.423788	−	0.905761i	$-0.360700\pi$
$941$	26.0000	0.847576	0.423788	+	0.905761i	$0.360700\pi$
$942$	0	0
$943$	24.0000	0.781548
$944$	4.00000	0.130189
$945$	0	0
$946$	−32.0000	−1.04041
$947$	36.0000	1.16984	0.584921	−	0.811090i	$-0.301125\pi$
$947$	36.0000	1.16984	0.584921	+	0.811090i	$0.301125\pi$
$948$	0	0
$949$	−14.0000	−0.454459
$950$	−6.00000	−0.194666
$951$	0	0
$952$	0	0
$953$	28.0000	0.907009	0.453504	−	0.891254i	$-0.350174\pi$
$953$	28.0000	0.907009	0.453504	+	0.891254i	$0.350174\pi$
$954$	0	0
$955$	8.00000	0.258874
$956$	−24.0000	−0.776215
$957$	0	0
$958$	0	0
$959$	36.0000	1.16250
$960$	0	0
$961$	−27.0000	−0.870968
$962$	6.00000	0.193448
$963$	0	0
$964$	6.00000	0.193247
$965$	−4.00000	−0.128765
$966$	0	0
$967$	−58.0000	−1.86515	−0.932577	−	0.360971i	$-0.882445\pi$
$967$	−58.0000	−1.86515	−0.932577	+	0.360971i	$0.882445\pi$
$968$	−5.00000	−0.160706
$969$	0	0
$970$	−20.0000	−0.642161
$971$	48.0000	1.54039	0.770197	−	0.637806i	$-0.220158\pi$
$971$	48.0000	1.54039	0.770197	+	0.637806i	$0.220158\pi$
$972$	0	0
$973$	40.0000	1.28234
$974$	−38.0000	−1.21760
$975$	0	0
$976$	10.0000	0.320092
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	0	0
$979$	24.0000	0.767043
$980$	6.00000	0.191663
$981$	0	0
$982$	16.0000	0.510581
$983$	−32.0000	−1.02064	−0.510321	−	0.859984i	$-0.670473\pi$
$983$	−32.0000	−1.02064	−0.510321	+	0.859984i	$0.670473\pi$
$984$	0	0
$985$	36.0000	1.14706
$986$	0	0
$987$	0	0
$988$	6.00000	0.190885
$989$	−32.0000	−1.01754
$990$	0	0
$991$	−48.0000	−1.52477	−0.762385	−	0.647124i	$-0.775972\pi$
$991$	−48.0000	−1.52477	−0.762385	+	0.647124i	$0.775972\pi$
$992$	2.00000	0.0635001
$993$	0	0
$994$	−32.0000	−1.01498
$995$	−16.0000	−0.507234
$996$	0	0
$997$	38.0000	1.20347	0.601736	−	0.798695i	$-0.294476\pi$
$997$	38.0000	1.20347	0.601736	+	0.798695i	$0.294476\pi$
$998$	−22.0000	−0.696398
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	234.2.a.a.1.1	✓	1
3.2	odd	2		234.2.a.d.1.1	yes	1
4.3	odd	2		1872.2.a.g.1.1		1
5.2	odd	4		5850.2.e.d.5149.1		2
5.3	odd	4		5850.2.e.d.5149.2		2
5.4	even	2		5850.2.a.bv.1.1		1
8.3	odd	2		7488.2.a.bu.1.1		1
8.5	even	2		7488.2.a.bp.1.1		1
9.2	odd	6		2106.2.e.e.1405.1		2
9.4	even	3		2106.2.e.z.703.1		2
9.5	odd	6		2106.2.e.e.703.1		2
9.7	even	3		2106.2.e.z.1405.1		2
12.11	even	2		1872.2.a.p.1.1		1
13.5	odd	4		3042.2.b.c.1351.2		2
13.8	odd	4		3042.2.b.c.1351.1		2
13.12	even	2		3042.2.a.o.1.1		1
15.2	even	4		5850.2.e.bd.5149.2		2
15.8	even	4		5850.2.e.bd.5149.1		2
15.14	odd	2		5850.2.a.v.1.1		1
24.5	odd	2		7488.2.a.j.1.1		1
24.11	even	2		7488.2.a.s.1.1		1
39.5	even	4		3042.2.b.b.1351.1		2
39.8	even	4		3042.2.b.b.1351.2		2
39.38	odd	2		3042.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
234.2.a.a.1.1	✓	1	1.1	even	1	trivial
234.2.a.d.1.1	yes	1	3.2	odd	2
1872.2.a.g.1.1		1	4.3	odd	2
1872.2.a.p.1.1		1	12.11	even	2
2106.2.e.e.703.1		2	9.5	odd	6
2106.2.e.e.1405.1		2	9.2	odd	6
2106.2.e.z.703.1		2	9.4	even	3
2106.2.e.z.1405.1		2	9.7	even	3
3042.2.a.b.1.1		1	39.38	odd	2
3042.2.a.o.1.1		1	13.12	even	2
3042.2.b.b.1351.1		2	39.5	even	4
3042.2.b.b.1351.2		2	39.8	even	4
3042.2.b.c.1351.1		2	13.8	odd	4
3042.2.b.c.1351.2		2	13.5	odd	4
5850.2.a.v.1.1		1	15.14	odd	2
5850.2.a.bv.1.1		1	5.4	even	2
5850.2.e.d.5149.1		2	5.2	odd	4
5850.2.e.d.5149.2		2	5.3	odd	4
5850.2.e.bd.5149.1		2	15.8	even	4
5850.2.e.bd.5149.2		2	15.2	even	4
7488.2.a.j.1.1		1	24.5	odd	2
7488.2.a.s.1.1		1	24.11	even	2
7488.2.a.bp.1.1		1	8.5	even	2
7488.2.a.bu.1.1		1	8.3	odd	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:	\( N \)	\(=\)	\( 234 = 2 \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	234.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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