LMFDB - Embedded newform 2790.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2790,2,Mod(1,2790)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2790, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2790.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2790.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:	$22.2782621639$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 930)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2790.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

f(q)

=

q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{7} -1.00000 q^{8} -1.00000 q^{10} +4.00000 q^{13} +2.00000 q^{14} +1.00000 q^{16} -6.00000 q^{17} +1.00000 q^{20} +1.00000 q^{25} -4.00000 q^{26} -2.00000 q^{28} -8.00000 q^{29} -1.00000 q^{31} -1.00000 q^{32} +6.00000 q^{34} -2.00000 q^{35} +4.00000 q^{37} -1.00000 q^{40} -10.0000 q^{41} +8.00000 q^{43} +4.00000 q^{47} -3.00000 q^{49} -1.00000 q^{50} +4.00000 q^{52} +14.0000 q^{53} +2.00000 q^{56} +8.00000 q^{58} -14.0000 q^{59} -6.00000 q^{61} +1.00000 q^{62} +1.00000 q^{64} +4.00000 q^{65} -10.0000 q^{67} -6.00000 q^{68} +2.00000 q^{70} -6.00000 q^{71} -8.00000 q^{73} -4.00000 q^{74} +8.00000 q^{79} +1.00000 q^{80} +10.0000 q^{82} +12.0000 q^{83} -6.00000 q^{85} -8.00000 q^{86} -16.0000 q^{89} -8.00000 q^{91} -4.00000 q^{94} -10.0000 q^{97} +3.00000 q^{98} +O(q^{100})

Coefficient data

For each $n$ we display the coefficients of the $q$ -expansion $a_n$ , the Satake parameters $\alpha_p$ , and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$ .

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−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
$10$	−1.00000	−0.316228
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−4.00000	−0.784465
$27$	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
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−1.00000	−0.176777
$33$	0	0
$34$	6.00000	1.02899
$35$	−2.00000	−0.338062
$36$	0	0
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	0	0
$39$	0	0
$40$	−1.00000	−0.158114
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	−1.00000	−0.141421
$51$	0	0
$52$	4.00000	0.554700
$53$	14.0000	1.92305	0.961524	−	0.274721i	$-0.0885855\pi$
$53$	14.0000	1.92305	0.961524	+	0.274721i	$0.0885855\pi$
$54$	0	0
$55$	0	0
$56$	2.00000	0.267261
$57$	0	0
$58$	8.00000	1.05045
$59$	−14.0000	−1.82264	−0.911322	−	0.411693i	$-0.864937\pi$
$59$	−14.0000	−1.82264	−0.911322	+	0.411693i	$0.864937\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	1.00000	0.127000
$63$	0	0
$64$	1.00000	0.125000
$65$	4.00000	0.496139
$66$	0	0
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	−6.00000	−0.727607
$69$	0	0
$70$	2.00000	0.239046
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	0	0
$73$	−8.00000	−0.936329	−0.468165	−	0.883641i	$-0.655085\pi$
$73$	−8.00000	−0.936329	−0.468165	+	0.883641i	$0.655085\pi$
$74$	−4.00000	−0.464991
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	10.0000	1.10432
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	0	0
$85$	−6.00000	−0.650791
$86$	−8.00000	−0.862662
$87$	0	0
$88$	0	0
$89$	−16.0000	−1.69600	−0.847998	−	0.529999i	$-0.822192\pi$
$89$	−16.0000	−1.69600	−0.847998	+	0.529999i	$0.822192\pi$
$90$	0	0
$91$	−8.00000	−0.838628
$92$	0	0
$93$	0	0
$94$	−4.00000	−0.412568
$95$	0	0
$96$	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
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	−6.00000	−0.591198	−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000	−0.591198	−0.295599	+	0.955312i	$0.595519\pi$
$104$	−4.00000	−0.392232
$105$	0	0
$106$	−14.0000	−1.35980
$107$	−8.00000	−0.773389	−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000	−0.773389	−0.386695	+	0.922208i	$0.626383\pi$
$108$	0	0
$109$	18.0000	1.72409	0.862044	−	0.506834i	$-0.169184\pi$
$109$	18.0000	1.72409	0.862044	+	0.506834i	$0.169184\pi$
$110$	0	0
$111$	0	0
$112$	−2.00000	−0.188982
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	0	0
$115$	0	0
$116$	−8.00000	−0.742781
$117$	0	0
$118$	14.0000	1.28880
$119$	12.0000	1.10004
$120$	0	0
$121$	−11.0000	−1.00000
$122$	6.00000	0.543214
$123$	0	0
$124$	−1.00000	−0.0898027
$125$	1.00000	0.0894427
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−4.00000	−0.350823
$131$	18.0000	1.57267	0.786334	−	0.617802i	$-0.211977\pi$
$131$	18.0000	1.57267	0.786334	+	0.617802i	$0.211977\pi$
$132$	0	0
$133$	0	0
$134$	10.0000	0.863868
$135$	0	0
$136$	6.00000	0.514496
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	−2.00000	−0.169031
$141$	0	0
$142$	6.00000	0.503509
$143$	0	0
$144$	0	0
$145$	−8.00000	−0.664364
$146$	8.00000	0.662085
$147$	0	0
$148$	4.00000	0.328798
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1.00000	−0.0803219
$156$	0	0
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	−8.00000	−0.636446
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	0	0
$162$	0	0
$163$	6.00000	0.469956	0.234978	−	0.972001i	$-0.424498\pi$
$163$	6.00000	0.469956	0.234978	+	0.972001i	$0.424498\pi$
$164$	−10.0000	−0.780869
$165$	0	0
$166$	−12.0000	−0.931381
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	6.00000	0.460179
$171$	0	0
$172$	8.00000	0.609994
$173$	−10.0000	−0.760286	−0.380143	−	0.924928i	$-0.624125\pi$
$173$	−10.0000	−0.760286	−0.380143	+	0.924928i	$0.624125\pi$
$174$	0	0
$175$	−2.00000	−0.151186
$176$	0	0
$177$	0	0
$178$	16.0000	1.19925
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	8.00000	0.592999
$183$	0	0
$184$	0	0
$185$	4.00000	0.294086
$186$	0	0
$187$	0	0
$188$	4.00000	0.291730
$189$	0	0
$190$	0	0
$191$	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
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	−3.00000	−0.214286
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	0	0
$199$	−24.0000	−1.70131	−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000	−1.70131	−0.850657	+	0.525720i	$0.823796\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	−6.00000	−0.422159
$203$	16.0000	1.12298
$204$	0	0
$205$	−10.0000	−0.698430
$206$	6.00000	0.418040
$207$	0	0
$208$	4.00000	0.277350
$209$	0	0
$210$	0	0
$211$	16.0000	1.10149	0.550743	−	0.834675i	$-0.314345\pi$
$211$	16.0000	1.10149	0.550743	+	0.834675i	$0.314345\pi$
$212$	14.0000	0.961524
$213$	0	0
$214$	8.00000	0.546869
$215$	8.00000	0.545595
$216$	0	0
$217$	2.00000	0.135769
$218$	−18.0000	−1.21911
$219$	0	0
$220$	0	0
$221$	−24.0000	−1.61441
$222$	0	0
$223$	−20.0000	−1.33930	−0.669650	−	0.742677i	$-0.733556\pi$
$223$	−20.0000	−1.33930	−0.669650	+	0.742677i	$0.733556\pi$
$224$	2.00000	0.133631
$225$	0	0
$226$	14.0000	0.931266
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	0	0
$232$	8.00000	0.525226
$233$	14.0000	0.917170	0.458585	−	0.888650i	$-0.348356\pi$
$233$	14.0000	0.917170	0.458585	+	0.888650i	$0.348356\pi$
$234$	0	0
$235$	4.00000	0.260931
$236$	−14.0000	−0.911322
$237$	0	0
$238$	−12.0000	−0.777844
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	0	0
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	11.0000	0.707107
$243$	0	0
$244$	−6.00000	−0.384111
$245$	−3.00000	−0.191663
$246$	0	0
$247$	0	0
$248$	1.00000	0.0635001
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	1.00000	0.0625000
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	0	0
$259$	−8.00000	−0.497096
$260$	4.00000	0.248069
$261$	0	0
$262$	−18.0000	−1.11204
$263$	−16.0000	−0.986602	−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	0	0
$265$	14.0000	0.860013
$266$	0	0
$267$	0	0
$268$	−10.0000	−0.610847
$269$	−24.0000	−1.46331	−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000	−1.46331	−0.731653	+	0.681677i	$0.761251\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	−6.00000	−0.363803
$273$	0	0
$274$	−6.00000	−0.362473
$275$	0	0
$276$	0	0
$277$	16.0000	0.961347	0.480673	−	0.876900i	$-0.340392\pi$
$277$	16.0000	0.961347	0.480673	+	0.876900i	$0.340392\pi$
$278$	−4.00000	−0.239904
$279$	0	0
$280$	2.00000	0.119523
$281$	−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
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	−6.00000	−0.356034
$285$	0	0
$286$	0	0
$287$	20.0000	1.18056
$288$	0	0
$289$	19.0000	1.11765
$290$	8.00000	0.469776
$291$	0	0
$292$	−8.00000	−0.468165
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	0	0
$295$	−14.0000	−0.815112
$296$	−4.00000	−0.232495
$297$	0	0
$298$	18.0000	1.04271
$299$	0	0
$300$	0	0
$301$	−16.0000	−0.922225
$302$	8.00000	0.460348
$303$	0	0
$304$	0	0
$305$	−6.00000	−0.343559
$306$	0	0
$307$	10.0000	0.570730	0.285365	−	0.958419i	$-0.407885\pi$
$307$	10.0000	0.570730	0.285365	+	0.958419i	$0.407885\pi$
$308$	0	0
$309$	0	0
$310$	1.00000	0.0567962
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	0	0
$313$	−16.0000	−0.904373	−0.452187	−	0.891923i	$-0.649356\pi$
$313$	−16.0000	−0.904373	−0.452187	+	0.891923i	$0.649356\pi$
$314$	18.0000	1.01580
$315$	0	0
$316$	8.00000	0.450035
$317$	10.0000	0.561656	0.280828	−	0.959758i	$-0.409391\pi$
$317$	10.0000	0.561656	0.280828	+	0.959758i	$0.409391\pi$
$318$	0	0
$319$	0	0
$320$	1.00000	0.0559017
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	4.00000	0.221880
$326$	−6.00000	−0.332309
$327$	0	0
$328$	10.0000	0.552158
$329$	−8.00000	−0.441054
$330$	0	0
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	12.0000	0.658586
$333$	0	0
$334$	−8.00000	−0.437741
$335$	−10.0000	−0.546358
$336$	0	0
$337$	4.00000	0.217894	0.108947	−	0.994048i	$-0.465252\pi$
$337$	4.00000	0.217894	0.108947	+	0.994048i	$0.465252\pi$
$338$	−3.00000	−0.163178
$339$	0	0
$340$	−6.00000	−0.325396
$341$	0	0
$342$	0	0
$343$	20.0000	1.07990
$344$	−8.00000	−0.431331
$345$	0	0
$346$	10.0000	0.537603
$347$	−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
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	2.00000	0.106904
$351$	0	0
$352$	0	0
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	0	0
$355$	−6.00000	−0.318447
$356$	−16.0000	−0.847998
$357$	0	0
$358$	0	0
$359$	−18.0000	−0.950004	−0.475002	−	0.879985i	$-0.657553\pi$
$359$	−18.0000	−0.950004	−0.475002	+	0.879985i	$0.657553\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	6.00000	0.315353
$363$	0	0
$364$	−8.00000	−0.419314
$365$	−8.00000	−0.418739
$366$	0	0
$367$	−36.0000	−1.87918	−0.939592	−	0.342296i	$-0.888796\pi$
$367$	−36.0000	−1.87918	−0.939592	+	0.342296i	$0.888796\pi$
$368$	0	0
$369$	0	0
$370$	−4.00000	−0.207950
$371$	−28.0000	−1.45369
$372$	0	0
$373$	6.00000	0.310668	0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000	0.310668	0.155334	+	0.987862i	$0.450355\pi$
$374$	0	0
$375$	0	0
$376$	−4.00000	−0.206284
$377$	−32.0000	−1.64808
$378$	0	0
$379$	8.00000	0.410932	0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000	0.410932	0.205466	+	0.978664i	$0.434129\pi$
$380$	0	0
$381$	0	0
$382$	−14.0000	−0.716302
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	2.00000	0.101797
$387$	0	0
$388$	−10.0000	−0.507673
$389$	32.0000	1.62246	0.811232	−	0.584724i	$-0.198797\pi$
$389$	32.0000	1.62246	0.811232	+	0.584724i	$0.198797\pi$
$390$	0	0
$391$	0	0
$392$	3.00000	0.151523
$393$	0	0
$394$	−2.00000	−0.100759
$395$	8.00000	0.402524
$396$	0	0
$397$	18.0000	0.903394	0.451697	−	0.892171i	$-0.350819\pi$
$397$	18.0000	0.903394	0.451697	+	0.892171i	$0.350819\pi$
$398$	24.0000	1.20301
$399$	0	0
$400$	1.00000	0.0500000
$401$	−4.00000	−0.199750	−0.0998752	−	0.995000i	$-0.531844\pi$
$401$	−4.00000	−0.199750	−0.0998752	+	0.995000i	$0.531844\pi$
$402$	0	0
$403$	−4.00000	−0.199254
$404$	6.00000	0.298511
$405$	0	0
$406$	−16.0000	−0.794067
$407$	0	0
$408$	0	0
$409$	30.0000	1.48340	0.741702	−	0.670729i	$-0.234019\pi$
$409$	30.0000	1.48340	0.741702	+	0.670729i	$0.234019\pi$
$410$	10.0000	0.493865
$411$	0	0
$412$	−6.00000	−0.295599
$413$	28.0000	1.37779
$414$	0	0
$415$	12.0000	0.589057
$416$	−4.00000	−0.196116
$417$	0	0
$418$	0	0
$419$	6.00000	0.293119	0.146560	−	0.989202i	$-0.453180\pi$
$419$	6.00000	0.293119	0.146560	+	0.989202i	$0.453180\pi$
$420$	0	0
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	−16.0000	−0.778868
$423$	0	0
$424$	−14.0000	−0.679900
$425$	−6.00000	−0.291043
$426$	0	0
$427$	12.0000	0.580721
$428$	−8.00000	−0.386695
$429$	0	0
$430$	−8.00000	−0.385794
$431$	6.00000	0.289010	0.144505	−	0.989504i	$-0.453841\pi$
$431$	6.00000	0.289010	0.144505	+	0.989504i	$0.453841\pi$
$432$	0	0
$433$	−16.0000	−0.768911	−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000	−0.768911	−0.384455	+	0.923144i	$0.625611\pi$
$434$	−2.00000	−0.0960031
$435$	0	0
$436$	18.0000	0.862044
$437$	0	0
$438$	0	0
$439$	24.0000	1.14546	0.572729	−	0.819745i	$-0.305885\pi$
$439$	24.0000	1.14546	0.572729	+	0.819745i	$0.305885\pi$
$440$	0	0
$441$	0	0
$442$	24.0000	1.14156
$443$	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
$445$	−16.0000	−0.758473
$446$	20.0000	0.947027
$447$	0	0
$448$	−2.00000	−0.0944911
$449$	20.0000	0.943858	0.471929	−	0.881636i	$-0.343558\pi$
$449$	20.0000	0.943858	0.471929	+	0.881636i	$0.343558\pi$
$450$	0	0
$451$	0	0
$452$	−14.0000	−0.658505
$453$	0	0
$454$	12.0000	0.563188
$455$	−8.00000	−0.375046
$456$	0	0
$457$	8.00000	0.374224	0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000	0.374224	0.187112	+	0.982339i	$0.440087\pi$
$458$	10.0000	0.467269
$459$	0	0
$460$	0	0
$461$	12.0000	0.558896	0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000	0.558896	0.279448	+	0.960161i	$0.409849\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	−8.00000	−0.371391
$465$	0	0
$466$	−14.0000	−0.648537
$467$	32.0000	1.48078	0.740392	−	0.672176i	$-0.234640\pi$
$467$	32.0000	1.48078	0.740392	+	0.672176i	$0.234640\pi$
$468$	0	0
$469$	20.0000	0.923514
$470$	−4.00000	−0.184506
$471$	0	0
$472$	14.0000	0.644402
$473$	0	0
$474$	0	0
$475$	0	0
$476$	12.0000	0.550019
$477$	0	0
$478$	−12.0000	−0.548867
$479$	42.0000	1.91903	0.959514	−	0.281659i	$-0.0908848\pi$
$479$	42.0000	1.91903	0.959514	+	0.281659i	$0.0908848\pi$
$480$	0	0
$481$	16.0000	0.729537
$482$	14.0000	0.637683
$483$	0	0
$484$	−11.0000	−0.500000
$485$	−10.0000	−0.454077
$486$	0	0
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	6.00000	0.271607
$489$	0	0
$490$	3.00000	0.135526
$491$	−36.0000	−1.62466	−0.812329	−	0.583200i	$-0.801800\pi$
$491$	−36.0000	−1.62466	−0.812329	+	0.583200i	$0.801800\pi$
$492$	0	0
$493$	48.0000	2.16181
$494$	0	0
$495$	0	0
$496$	−1.00000	−0.0449013
$497$	12.0000	0.538274
$498$	0	0
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	0	0
$503$	20.0000	0.891756	0.445878	−	0.895094i	$-0.352892\pi$
$503$	20.0000	0.891756	0.445878	+	0.895094i	$0.352892\pi$
$504$	0	0
$505$	6.00000	0.266996
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−4.00000	−0.177297	−0.0886484	−	0.996063i	$-0.528255\pi$
$509$	−4.00000	−0.177297	−0.0886484	+	0.996063i	$0.528255\pi$
$510$	0	0
$511$	16.0000	0.707798
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	2.00000	0.0882162
$515$	−6.00000	−0.264392
$516$	0	0
$517$	0	0
$518$	8.00000	0.351500
$519$	0	0
$520$	−4.00000	−0.175412
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	0	0
$523$	24.0000	1.04945	0.524723	−	0.851273i	$-0.324169\pi$
$523$	24.0000	1.04945	0.524723	+	0.851273i	$0.324169\pi$
$524$	18.0000	0.786334
$525$	0	0
$526$	16.0000	0.697633
$527$	6.00000	0.261364
$528$	0	0
$529$	−23.0000	−1.00000
$530$	−14.0000	−0.608121
$531$	0	0
$532$	0	0
$533$	−40.0000	−1.73259
$534$	0	0
$535$	−8.00000	−0.345870
$536$	10.0000	0.431934
$537$	0	0
$538$	24.0000	1.03471
$539$	0	0
$540$	0	0
$541$	18.0000	0.773880	0.386940	−	0.922105i	$-0.373532\pi$
$541$	18.0000	0.773880	0.386940	+	0.922105i	$0.373532\pi$
$542$	16.0000	0.687259
$543$	0	0
$544$	6.00000	0.257248
$545$	18.0000	0.771035
$546$	0	0
$547$	6.00000	0.256541	0.128271	−	0.991739i	$-0.459057\pi$
$547$	6.00000	0.256541	0.128271	+	0.991739i	$0.459057\pi$
$548$	6.00000	0.256307
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−16.0000	−0.680389
$554$	−16.0000	−0.679775
$555$	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
$559$	32.0000	1.35346
$560$	−2.00000	−0.0845154
$561$	0	0
$562$	30.0000	1.26547
$563$	32.0000	1.34864	0.674320	−	0.738440i	$-0.264437\pi$
$563$	32.0000	1.34864	0.674320	+	0.738440i	$0.264437\pi$
$564$	0	0
$565$	−14.0000	−0.588984
$566$	−14.0000	−0.588464
$567$	0	0
$568$	6.00000	0.251754
$569$	−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
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	0	0
$573$	0	0
$574$	−20.0000	−0.834784
$575$	0	0
$576$	0	0
$577$	−6.00000	−0.249783	−0.124892	−	0.992170i	$-0.539858\pi$
$577$	−6.00000	−0.249783	−0.124892	+	0.992170i	$0.539858\pi$
$578$	−19.0000	−0.790296
$579$	0	0
$580$	−8.00000	−0.332182
$581$	−24.0000	−0.995688
$582$	0	0
$583$	0	0
$584$	8.00000	0.331042
$585$	0	0
$586$	6.00000	0.247858
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	0	0
$589$	0	0
$590$	14.0000	0.576371
$591$	0	0
$592$	4.00000	0.164399
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	0	0
$595$	12.0000	0.491952
$596$	−18.0000	−0.737309
$597$	0	0
$598$	0	0
$599$	−42.0000	−1.71607	−0.858037	−	0.513588i	$-0.828316\pi$
$599$	−42.0000	−1.71607	−0.858037	+	0.513588i	$0.828316\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	16.0000	0.652111
$603$	0	0
$604$	−8.00000	−0.325515
$605$	−11.0000	−0.447214
$606$	0	0
$607$	2.00000	0.0811775	0.0405887	−	0.999176i	$-0.487077\pi$
$607$	2.00000	0.0811775	0.0405887	+	0.999176i	$0.487077\pi$
$608$	0	0
$609$	0	0
$610$	6.00000	0.242933
$611$	16.0000	0.647291
$612$	0	0
$613$	16.0000	0.646234	0.323117	−	0.946359i	$-0.395269\pi$
$613$	16.0000	0.646234	0.323117	+	0.946359i	$0.395269\pi$
$614$	−10.0000	−0.403567
$615$	0	0
$616$	0	0
$617$	−38.0000	−1.52982	−0.764911	−	0.644136i	$-0.777217\pi$
$617$	−38.0000	−1.52982	−0.764911	+	0.644136i	$0.777217\pi$
$618$	0	0
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	−1.00000	−0.0401610
$621$	0	0
$622$	18.0000	0.721734
$623$	32.0000	1.28205
$624$	0	0
$625$	1.00000	0.0400000
$626$	16.0000	0.639489
$627$	0	0
$628$	−18.0000	−0.718278
$629$	−24.0000	−0.956943
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	−8.00000	−0.318223
$633$	0	0
$634$	−10.0000	−0.397151
$635$	0	0
$636$	0	0
$637$	−12.0000	−0.475457
$638$	0	0
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	44.0000	1.73790	0.868948	−	0.494904i	$-0.164797\pi$
$641$	44.0000	1.73790	0.868948	+	0.494904i	$0.164797\pi$
$642$	0	0
$643$	−16.0000	−0.630978	−0.315489	−	0.948929i	$-0.602169\pi$
$643$	−16.0000	−0.630978	−0.315489	+	0.948929i	$0.602169\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	16.0000	0.629025	0.314512	−	0.949253i	$-0.398159\pi$
$647$	16.0000	0.629025	0.314512	+	0.949253i	$0.398159\pi$
$648$	0	0
$649$	0	0
$650$	−4.00000	−0.156893
$651$	0	0
$652$	6.00000	0.234978
$653$	10.0000	0.391330	0.195665	−	0.980671i	$-0.437313\pi$
$653$	10.0000	0.391330	0.195665	+	0.980671i	$0.437313\pi$
$654$	0	0
$655$	18.0000	0.703318
$656$	−10.0000	−0.390434
$657$	0	0
$658$	8.00000	0.311872
$659$	30.0000	1.16863	0.584317	−	0.811525i	$-0.301362\pi$
$659$	30.0000	1.16863	0.584317	+	0.811525i	$0.301362\pi$
$660$	0	0
$661$	−42.0000	−1.63361	−0.816805	−	0.576913i	$-0.804257\pi$
$661$	−42.0000	−1.63361	−0.816805	+	0.576913i	$0.804257\pi$
$662$	12.0000	0.466393
$663$	0	0
$664$	−12.0000	−0.465690
$665$	0	0
$666$	0	0
$667$	0	0
$668$	8.00000	0.309529
$669$	0	0
$670$	10.0000	0.386334
$671$	0	0
$672$	0	0
$673$	−20.0000	−0.770943	−0.385472	−	0.922720i	$-0.625961\pi$
$673$	−20.0000	−0.770943	−0.385472	+	0.922720i	$0.625961\pi$
$674$	−4.00000	−0.154074
$675$	0	0
$676$	3.00000	0.115385
$677$	22.0000	0.845529	0.422764	−	0.906240i	$-0.361060\pi$
$677$	22.0000	0.845529	0.422764	+	0.906240i	$0.361060\pi$
$678$	0	0
$679$	20.0000	0.767530
$680$	6.00000	0.230089
$681$	0	0
$682$	0	0
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	0	0
$685$	6.00000	0.229248
$686$	−20.0000	−0.763604
$687$	0	0
$688$	8.00000	0.304997
$689$	56.0000	2.13343
$690$	0	0
$691$	−4.00000	−0.152167	−0.0760836	−	0.997101i	$-0.524242\pi$
$691$	−4.00000	−0.152167	−0.0760836	+	0.997101i	$0.524242\pi$
$692$	−10.0000	−0.380143
$693$	0	0
$694$	20.0000	0.759190
$695$	4.00000	0.151729
$696$	0	0
$697$	60.0000	2.27266
$698$	−2.00000	−0.0757011
$699$	0	0
$700$	−2.00000	−0.0755929
$701$	−30.0000	−1.13308	−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000	−1.13308	−0.566542	+	0.824033i	$0.691719\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	14.0000	0.526897
$707$	−12.0000	−0.451306
$708$	0	0
$709$	46.0000	1.72757	0.863783	−	0.503864i	$-0.168089\pi$
$709$	46.0000	1.72757	0.863783	+	0.503864i	$0.168089\pi$
$710$	6.00000	0.225176
$711$	0	0
$712$	16.0000	0.599625
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	18.0000	0.671754
$719$	40.0000	1.49175	0.745874	−	0.666087i	$-0.232032\pi$
$719$	40.0000	1.49175	0.745874	+	0.666087i	$0.232032\pi$
$720$	0	0
$721$	12.0000	0.446903
$722$	19.0000	0.707107
$723$	0	0
$724$	−6.00000	−0.222988
$725$	−8.00000	−0.297113
$726$	0	0
$727$	−10.0000	−0.370879	−0.185440	−	0.982656i	$-0.559371\pi$
$727$	−10.0000	−0.370879	−0.185440	+	0.982656i	$0.559371\pi$
$728$	8.00000	0.296500
$729$	0	0
$730$	8.00000	0.296093
$731$	−48.0000	−1.77534
$732$	0	0
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	36.0000	1.32878
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	4.00000	0.147043
$741$	0	0
$742$	28.0000	1.02791
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	0	0
$745$	−18.0000	−0.659469
$746$	−6.00000	−0.219676
$747$	0	0
$748$	0	0
$749$	16.0000	0.584627
$750$	0	0
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	4.00000	0.145865
$753$	0	0
$754$	32.0000	1.16537
$755$	−8.00000	−0.291150
$756$	0	0
$757$	−12.0000	−0.436147	−0.218074	−	0.975932i	$-0.569977\pi$
$757$	−12.0000	−0.436147	−0.218074	+	0.975932i	$0.569977\pi$
$758$	−8.00000	−0.290573
$759$	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
$763$	−36.0000	−1.30329
$764$	14.0000	0.506502
$765$	0	0
$766$	0	0
$767$	−56.0000	−2.02204
$768$	0	0
$769$	30.0000	1.08183	0.540914	−	0.841078i	$-0.318079\pi$
$769$	30.0000	1.08183	0.540914	+	0.841078i	$0.318079\pi$
$770$	0	0
$771$	0	0
$772$	−2.00000	−0.0719816
$773$	46.0000	1.65451	0.827253	−	0.561830i	$-0.189903\pi$
$773$	46.0000	1.65451	0.827253	+	0.561830i	$0.189903\pi$
$774$	0	0
$775$	−1.00000	−0.0359211
$776$	10.0000	0.358979
$777$	0	0
$778$	−32.0000	−1.14726
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−3.00000	−0.107143
$785$	−18.0000	−0.642448
$786$	0	0
$787$	44.0000	1.56843	0.784215	−	0.620489i	$-0.213066\pi$
$787$	44.0000	1.56843	0.784215	+	0.620489i	$0.213066\pi$
$788$	2.00000	0.0712470
$789$	0	0
$790$	−8.00000	−0.284627
$791$	28.0000	0.995565
$792$	0	0
$793$	−24.0000	−0.852265
$794$	−18.0000	−0.638796
$795$	0	0
$796$	−24.0000	−0.850657
$797$	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
$799$	−24.0000	−0.849059
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	4.00000	0.141245
$803$	0	0
$804$	0	0
$805$	0	0
$806$	4.00000	0.140894
$807$	0	0
$808$	−6.00000	−0.211079
$809$	−4.00000	−0.140633	−0.0703163	−	0.997525i	$-0.522401\pi$
$809$	−4.00000	−0.140633	−0.0703163	+	0.997525i	$0.522401\pi$
$810$	0	0
$811$	56.0000	1.96643	0.983213	−	0.182462i	$-0.0584065\pi$
$811$	56.0000	1.96643	0.983213	+	0.182462i	$0.0584065\pi$
$812$	16.0000	0.561490
$813$	0	0
$814$	0	0
$815$	6.00000	0.210171
$816$	0	0
$817$	0	0
$818$	−30.0000	−1.04893
$819$	0	0
$820$	−10.0000	−0.349215
$821$	48.0000	1.67521	0.837606	−	0.546275i	$-0.183955\pi$
$821$	48.0000	1.67521	0.837606	+	0.546275i	$0.183955\pi$
$822$	0	0
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	6.00000	0.209020
$825$	0	0
$826$	−28.0000	−0.974245
$827$	−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
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	−12.0000	−0.416526
$831$	0	0
$832$	4.00000	0.138675
$833$	18.0000	0.623663
$834$	0	0
$835$	8.00000	0.276851
$836$	0	0
$837$	0	0
$838$	−6.00000	−0.207267
$839$	30.0000	1.03572	0.517858	−	0.855467i	$-0.326730\pi$
$839$	30.0000	1.03572	0.517858	+	0.855467i	$0.326730\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	6.00000	0.206774
$843$	0	0
$844$	16.0000	0.550743
$845$	3.00000	0.103203
$846$	0	0
$847$	22.0000	0.755929
$848$	14.0000	0.480762
$849$	0	0
$850$	6.00000	0.205798
$851$	0	0
$852$	0	0
$853$	18.0000	0.616308	0.308154	−	0.951336i	$-0.400289\pi$
$853$	18.0000	0.616308	0.308154	+	0.951336i	$0.400289\pi$
$854$	−12.0000	−0.410632
$855$	0	0
$856$	8.00000	0.273434
$857$	10.0000	0.341593	0.170797	−	0.985306i	$-0.445366\pi$
$857$	10.0000	0.341593	0.170797	+	0.985306i	$0.445366\pi$
$858$	0	0
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	8.00000	0.272798
$861$	0	0
$862$	−6.00000	−0.204361
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	−10.0000	−0.340010
$866$	16.0000	0.543702
$867$	0	0
$868$	2.00000	0.0678844
$869$	0	0
$870$	0	0
$871$	−40.0000	−1.35535
$872$	−18.0000	−0.609557
$873$	0	0
$874$	0	0
$875$	−2.00000	−0.0676123
$876$	0	0
$877$	−6.00000	−0.202606	−0.101303	−	0.994856i	$-0.532301\pi$
$877$	−6.00000	−0.202606	−0.101303	+	0.994856i	$0.532301\pi$
$878$	−24.0000	−0.809961
$879$	0	0
$880$	0	0
$881$	48.0000	1.61716	0.808581	−	0.588386i	$-0.200236\pi$
$881$	48.0000	1.61716	0.808581	+	0.588386i	$0.200236\pi$
$882$	0	0
$883$	−12.0000	−0.403832	−0.201916	−	0.979403i	$-0.564717\pi$
$883$	−12.0000	−0.403832	−0.201916	+	0.979403i	$0.564717\pi$
$884$	−24.0000	−0.807207
$885$	0	0
$886$	−36.0000	−1.20944
$887$	56.0000	1.88030	0.940148	−	0.340766i	$-0.110687\pi$
$887$	56.0000	1.88030	0.940148	+	0.340766i	$0.110687\pi$
$888$	0	0
$889$	0	0
$890$	16.0000	0.536321
$891$	0	0
$892$	−20.0000	−0.669650
$893$	0	0
$894$	0	0
$895$	0	0
$896$	2.00000	0.0668153
$897$	0	0
$898$	−20.0000	−0.667409
$899$	8.00000	0.266815
$900$	0	0
$901$	−84.0000	−2.79845
$902$	0	0
$903$	0	0
$904$	14.0000	0.465633
$905$	−6.00000	−0.199447
$906$	0	0
$907$	2.00000	0.0664089	0.0332045	−	0.999449i	$-0.489429\pi$
$907$	2.00000	0.0664089	0.0332045	+	0.999449i	$0.489429\pi$
$908$	−12.0000	−0.398234
$909$	0	0
$910$	8.00000	0.265197
$911$	24.0000	0.795155	0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000	0.795155	0.397578	+	0.917568i	$0.369851\pi$
$912$	0	0
$913$	0	0
$914$	−8.00000	−0.264616
$915$	0	0
$916$	−10.0000	−0.330409
$917$	−36.0000	−1.18882
$918$	0	0
$919$	−60.0000	−1.97922	−0.989609	−	0.143787i	$-0.954072\pi$
$919$	−60.0000	−1.97922	−0.989609	+	0.143787i	$0.954072\pi$
$920$	0	0
$921$	0	0
$922$	−12.0000	−0.395199
$923$	−24.0000	−0.789970
$924$	0	0
$925$	4.00000	0.131519
$926$	0	0
$927$	0	0
$928$	8.00000	0.262613
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	0	0
$931$	0	0
$932$	14.0000	0.458585
$933$	0	0
$934$	−32.0000	−1.04707
$935$	0	0
$936$	0	0
$937$	−42.0000	−1.37208	−0.686040	−	0.727564i	$-0.740653\pi$
$937$	−42.0000	−1.37208	−0.686040	+	0.727564i	$0.740653\pi$
$938$	−20.0000	−0.653023
$939$	0	0
$940$	4.00000	0.130466
$941$	8.00000	0.260793	0.130396	−	0.991462i	$-0.458375\pi$
$941$	8.00000	0.260793	0.130396	+	0.991462i	$0.458375\pi$
$942$	0	0
$943$	0	0
$944$	−14.0000	−0.455661
$945$	0	0
$946$	0	0
$947$	28.0000	0.909878	0.454939	−	0.890523i	$-0.349661\pi$
$947$	28.0000	0.909878	0.454939	+	0.890523i	$0.349661\pi$
$948$	0	0
$949$	−32.0000	−1.03876
$950$	0	0
$951$	0	0
$952$	−12.0000	−0.388922
$953$	46.0000	1.49009	0.745043	−	0.667016i	$-0.232429\pi$
$953$	46.0000	1.49009	0.745043	+	0.667016i	$0.232429\pi$
$954$	0	0
$955$	14.0000	0.453029
$956$	12.0000	0.388108
$957$	0	0
$958$	−42.0000	−1.35696
$959$	−12.0000	−0.387500
$960$	0	0
$961$	1.00000	0.0322581
$962$	−16.0000	−0.515861
$963$	0	0
$964$	−14.0000	−0.450910
$965$	−2.00000	−0.0643823
$966$	0	0
$967$	52.0000	1.67221	0.836104	−	0.548572i	$-0.184828\pi$
$967$	52.0000	1.67221	0.836104	+	0.548572i	$0.184828\pi$
$968$	11.0000	0.353553
$969$	0	0
$970$	10.0000	0.321081
$971$	−18.0000	−0.577647	−0.288824	−	0.957382i	$-0.593264\pi$
$971$	−18.0000	−0.577647	−0.288824	+	0.957382i	$0.593264\pi$
$972$	0	0
$973$	−8.00000	−0.256468
$974$	8.00000	0.256337
$975$	0	0
$976$	−6.00000	−0.192055
$977$	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
$979$	0	0
$980$	−3.00000	−0.0958315
$981$	0	0
$982$	36.0000	1.14881
$983$	40.0000	1.27580	0.637901	−	0.770118i	$-0.279803\pi$
$983$	40.0000	1.27580	0.637901	+	0.770118i	$0.279803\pi$
$984$	0	0
$985$	2.00000	0.0637253
$986$	−48.0000	−1.52863
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	1.00000	0.0317500
$993$	0	0
$994$	−12.0000	−0.380617
$995$	−24.0000	−0.760851
$996$	0	0
$997$	6.00000	0.190022	0.0950110	−	0.995476i	$-0.469711\pi$
$997$	6.00000	0.190022	0.0950110	+	0.995476i	$0.469711\pi$
$998$	−20.0000	−0.633089
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2790.2.a.f.1.1		1
3.2	odd	2		930.2.a.k.1.1	✓	1
12.11	even	2		7440.2.a.u.1.1		1
15.2	even	4		4650.2.d.g.3349.2		2
15.8	even	4		4650.2.d.g.3349.1		2
15.14	odd	2		4650.2.a.t.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.a.k.1.1	✓	1	3.2	odd	2
2790.2.a.f.1.1		1	1.1	even	1	trivial
4650.2.a.t.1.1		1	15.14	odd	2
4650.2.d.g.3349.1		2	15.8	even	4
4650.2.d.g.3349.2		2	15.2	even	4
7440.2.a.u.1.1		1	12.11	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 $\F_{q}$

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2790.2.a.f.1.1

Level	$2790$
Weight	$2$
Character	2790.1
Self dual	yes
Analytic conductor	$22.278$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$