LMFDB - Embedded newform 1350.2.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1350,2,Mod(1,1350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1350, 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("1350.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1350 = 2 \cdot 3^{3} \cdot 5^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1350.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:	$10.7798042729$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 270)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1350.1

$q$ -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} +4.00000 q^{7} -1.00000 q^{8} -5.00000 q^{11} -3.00000 q^{13} -4.00000 q^{14} +1.00000 q^{16} -1.00000 q^{17} -6.00000 q^{19} +5.00000 q^{22} -1.00000 q^{23} +3.00000 q^{26} +4.00000 q^{28} -9.00000 q^{29} -5.00000 q^{31} -1.00000 q^{32} +1.00000 q^{34} -2.00000 q^{37} +6.00000 q^{38} -2.00000 q^{41} +1.00000 q^{43} -5.00000 q^{44} +1.00000 q^{46} +13.0000 q^{47} +9.00000 q^{49} -3.00000 q^{52} -4.00000 q^{56} +9.00000 q^{58} -4.00000 q^{59} +8.00000 q^{61} +5.00000 q^{62} +1.00000 q^{64} -4.00000 q^{67} -1.00000 q^{68} -6.00000 q^{71} -2.00000 q^{73} +2.00000 q^{74} -6.00000 q^{76} -20.0000 q^{77} +9.00000 q^{79} +2.00000 q^{82} -4.00000 q^{83} -1.00000 q^{86} +5.00000 q^{88} -14.0000 q^{89} -12.0000 q^{91} -1.00000 q^{92} -13.0000 q^{94} -10.0000 q^{97} -9.00000 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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.00000	1.51186	0.755929	−	0.654654i	$-0.227186\pi$
$7$	4.00000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	−5.00000	−1.50756	−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000	−1.50756	−0.753778	+	0.657129i	$0.771771\pi$
$12$	0	0
$13$	−3.00000	−0.832050	−0.416025	−	0.909353i	$-0.636577\pi$
$13$	−3.00000	−0.832050	−0.416025	+	0.909353i	$0.636577\pi$
$14$	−4.00000	−1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536	−0.121268	−	0.992620i	$-0.538696\pi$
$17$	−1.00000	−0.242536	−0.121268	+	0.992620i	$0.538696\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	0	0
$22$	5.00000	1.06600
$23$	−1.00000	−0.208514	−0.104257	−	0.994550i	$-0.533247\pi$
$23$	−1.00000	−0.208514	−0.104257	+	0.994550i	$0.533247\pi$
$24$	0	0
$25$	0	0
$26$	3.00000	0.588348
$27$	0	0
$28$	4.00000	0.755929
$29$	−9.00000	−1.67126	−0.835629	−	0.549294i	$-0.814897\pi$
$29$	−9.00000	−1.67126	−0.835629	+	0.549294i	$0.814897\pi$
$30$	0	0
$31$	−5.00000	−0.898027	−0.449013	−	0.893525i	$-0.648224\pi$
$31$	−5.00000	−0.898027	−0.449013	+	0.893525i	$0.648224\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	1.00000	0.171499
$35$	0	0
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	6.00000	0.973329
$39$	0	0
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	−5.00000	−0.753778
$45$	0	0
$46$	1.00000	0.147442
$47$	13.0000	1.89624	0.948122	−	0.317905i	$-0.102979\pi$
$47$	13.0000	1.89624	0.948122	+	0.317905i	$0.102979\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	0	0
$51$	0	0
$52$	−3.00000	−0.416025
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0	0
$56$	−4.00000	−0.534522
$57$	0	0
$58$	9.00000	1.18176
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	5.00000	0.635001
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	−1.00000	−0.121268
$69$	0	0
$70$	0	0
$71$	−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
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	−6.00000	−0.688247
$77$	−20.0000	−2.27921
$78$	0	0
$79$	9.00000	1.01258	0.506290	−	0.862364i	$-0.331017\pi$
$79$	9.00000	1.01258	0.506290	+	0.862364i	$0.331017\pi$
$80$	0	0
$81$	0	0
$82$	2.00000	0.220863
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	0	0
$86$	−1.00000	−0.107833
$87$	0	0
$88$	5.00000	0.533002
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	0	0
$91$	−12.0000	−1.25794
$92$	−1.00000	−0.104257
$93$	0	0
$94$	−13.0000	−1.34085
$95$	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	$0.669494\pi$
$98$	−9.00000	−0.909137
$99$	0	0
$100$	0	0
$101$	17.0000	1.69156	0.845782	−	0.533529i	$-0.179135\pi$
$101$	17.0000	1.69156	0.845782	+	0.533529i	$0.179135\pi$
$102$	0	0
$103$	−10.0000	−0.985329	−0.492665	−	0.870219i	$-0.663977\pi$
$103$	−10.0000	−0.985329	−0.492665	+	0.870219i	$0.663977\pi$
$104$	3.00000	0.294174
$105$	0	0
$106$	0	0
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	0	0
$109$	−8.00000	−0.766261	−0.383131	−	0.923694i	$-0.625154\pi$
$109$	−8.00000	−0.766261	−0.383131	+	0.923694i	$0.625154\pi$
$110$	0	0
$111$	0	0
$112$	4.00000	0.377964
$113$	3.00000	0.282216	0.141108	−	0.989994i	$-0.454933\pi$
$113$	3.00000	0.282216	0.141108	+	0.989994i	$0.454933\pi$
$114$	0	0
$115$	0	0
$116$	−9.00000	−0.835629
$117$	0	0
$118$	4.00000	0.368230
$119$	−4.00000	−0.366679
$120$	0	0
$121$	14.0000	1.27273
$122$	−8.00000	−0.724286
$123$	0	0
$124$	−5.00000	−0.449013
$125$	0	0
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	−3.00000	−0.262111	−0.131056	−	0.991375i	$-0.541837\pi$
$131$	−3.00000	−0.262111	−0.131056	+	0.991375i	$0.541837\pi$
$132$	0	0
$133$	−24.0000	−2.08106
$134$	4.00000	0.345547
$135$	0	0
$136$	1.00000	0.0857493
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	0	0
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	0	0
$142$	6.00000	0.503509
$143$	15.0000	1.25436
$144$	0	0
$145$	0	0
$146$	2.00000	0.165521
$147$	0	0
$148$	−2.00000	−0.164399
$149$	−13.0000	−1.06500	−0.532501	−	0.846430i	$-0.678748\pi$
$149$	−13.0000	−1.06500	−0.532501	+	0.846430i	$0.678748\pi$
$150$	0	0
$151$	13.0000	1.05792	0.528962	−	0.848645i	$-0.322581\pi$
$151$	13.0000	1.05792	0.528962	+	0.848645i	$0.322581\pi$
$152$	6.00000	0.486664
$153$	0	0
$154$	20.0000	1.61165
$155$	0	0
$156$	0	0
$157$	−13.0000	−1.03751	−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000	−1.03751	−0.518756	+	0.854922i	$0.673605\pi$
$158$	−9.00000	−0.716002
$159$	0	0
$160$	0	0
$161$	−4.00000	−0.315244
$162$	0	0
$163$	−19.0000	−1.48819	−0.744097	−	0.668071i	$-0.767120\pi$
$163$	−19.0000	−1.48819	−0.744097	+	0.668071i	$0.767120\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	4.00000	0.310460
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	0	0
$171$	0	0
$172$	1.00000	0.0762493
$173$	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
$175$	0	0
$176$	−5.00000	−0.376889
$177$	0	0
$178$	14.0000	1.04934
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	0	0
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	12.0000	0.889499
$183$	0	0
$184$	1.00000	0.0737210
$185$	0	0
$186$	0	0
$187$	5.00000	0.365636
$188$	13.0000	0.948122
$189$	0	0
$190$	0	0
$191$	−6.00000	−0.434145	−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000	−0.434145	−0.217072	+	0.976156i	$0.569651\pi$
$192$	0	0
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	9.00000	0.642857
$197$	22.0000	1.56744	0.783718	−	0.621117i	$-0.213321\pi$
$197$	22.0000	1.56744	0.783718	+	0.621117i	$0.213321\pi$
$198$	0	0
$199$	3.00000	0.212664	0.106332	−	0.994331i	$-0.466089\pi$
$199$	3.00000	0.212664	0.106332	+	0.994331i	$0.466089\pi$
$200$	0	0
$201$	0	0
$202$	−17.0000	−1.19612
$203$	−36.0000	−2.52670
$204$	0	0
$205$	0	0
$206$	10.0000	0.696733
$207$	0	0
$208$	−3.00000	−0.208013
$209$	30.0000	2.07514
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	0	0
$213$	0	0
$214$	6.00000	0.410152
$215$	0	0
$216$	0	0
$217$	−20.0000	−1.35769
$218$	8.00000	0.541828
$219$	0	0
$220$	0	0
$221$	3.00000	0.201802
$222$	0	0
$223$	22.0000	1.47323	0.736614	−	0.676313i	$-0.236423\pi$
$223$	22.0000	1.47323	0.736614	+	0.676313i	$0.236423\pi$
$224$	−4.00000	−0.267261
$225$	0	0
$226$	−3.00000	−0.199557
$227$	2.00000	0.132745	0.0663723	−	0.997795i	$-0.478857\pi$
$227$	2.00000	0.132745	0.0663723	+	0.997795i	$0.478857\pi$
$228$	0	0
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	0	0
$231$	0	0
$232$	9.00000	0.590879
$233$	−14.0000	−0.917170	−0.458585	−	0.888650i	$-0.651644\pi$
$233$	−14.0000	−0.917170	−0.458585	+	0.888650i	$0.651644\pi$
$234$	0	0
$235$	0	0
$236$	−4.00000	−0.260378
$237$	0	0
$238$	4.00000	0.259281
$239$	16.0000	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$240$	0	0
$241$	−23.0000	−1.48156	−0.740780	−	0.671748i	$-0.765544\pi$
$241$	−23.0000	−1.48156	−0.740780	+	0.671748i	$0.765544\pi$
$242$	−14.0000	−0.899954
$243$	0	0
$244$	8.00000	0.512148
$245$	0	0
$246$	0	0
$247$	18.0000	1.14531
$248$	5.00000	0.317500
$249$	0	0
$250$	0	0
$251$	15.0000	0.946792	0.473396	−	0.880850i	$-0.343028\pi$
$251$	15.0000	0.946792	0.473396	+	0.880850i	$0.343028\pi$
$252$	0	0
$253$	5.00000	0.314347
$254$	−4.00000	−0.250982
$255$	0	0
$256$	1.00000	0.0625000
$257$	−21.0000	−1.30994	−0.654972	−	0.755653i	$-0.727320\pi$
$257$	−21.0000	−1.30994	−0.654972	+	0.755653i	$0.727320\pi$
$258$	0	0
$259$	−8.00000	−0.497096
$260$	0	0
$261$	0	0
$262$	3.00000	0.185341
$263$	8.00000	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$264$	0	0
$265$	0	0
$266$	24.0000	1.47153
$267$	0	0
$268$	−4.00000	−0.244339
$269$	9.00000	0.548740	0.274370	−	0.961624i	$-0.411531\pi$
$269$	9.00000	0.548740	0.274370	+	0.961624i	$0.411531\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	−1.00000	−0.0606339
$273$	0	0
$274$	−18.0000	−1.08742
$275$	0	0
$276$	0	0
$277$	−26.0000	−1.56219	−0.781094	−	0.624413i	$-0.785338\pi$
$277$	−26.0000	−1.56219	−0.781094	+	0.624413i	$0.785338\pi$
$278$	8.00000	0.479808
$279$	0	0
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	0	0
$283$	20.0000	1.18888	0.594438	−	0.804141i	$-0.297374\pi$
$283$	20.0000	1.18888	0.594438	+	0.804141i	$0.297374\pi$
$284$	−6.00000	−0.356034
$285$	0	0
$286$	−15.0000	−0.886969
$287$	−8.00000	−0.472225
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	0	0
$292$	−2.00000	−0.117041
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	0	0
$296$	2.00000	0.116248
$297$	0	0
$298$	13.0000	0.753070
$299$	3.00000	0.173494
$300$	0	0
$301$	4.00000	0.230556
$302$	−13.0000	−0.748066
$303$	0	0
$304$	−6.00000	−0.344124
$305$	0	0
$306$	0	0
$307$	21.0000	1.19853	0.599267	−	0.800549i	$-0.295459\pi$
$307$	21.0000	1.19853	0.599267	+	0.800549i	$0.295459\pi$
$308$	−20.0000	−1.13961
$309$	0	0
$310$	0	0
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	0	0
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	13.0000	0.733632
$315$	0	0
$316$	9.00000	0.506290
$317$	−4.00000	−0.224662	−0.112331	−	0.993671i	$-0.535832\pi$
$317$	−4.00000	−0.224662	−0.112331	+	0.993671i	$0.535832\pi$
$318$	0	0
$319$	45.0000	2.51952
$320$	0	0
$321$	0	0
$322$	4.00000	0.222911
$323$	6.00000	0.333849
$324$	0	0
$325$	0	0
$326$	19.0000	1.05231
$327$	0	0
$328$	2.00000	0.110432
$329$	52.0000	2.86685
$330$	0	0
$331$	10.0000	0.549650	0.274825	−	0.961494i	$-0.411380\pi$
$331$	10.0000	0.549650	0.274825	+	0.961494i	$0.411380\pi$
$332$	−4.00000	−0.219529
$333$	0	0
$334$	12.0000	0.656611
$335$	0	0
$336$	0	0
$337$	16.0000	0.871576	0.435788	−	0.900049i	$-0.356470\pi$
$337$	16.0000	0.871576	0.435788	+	0.900049i	$0.356470\pi$
$338$	4.00000	0.217571
$339$	0	0
$340$	0	0
$341$	25.0000	1.35383
$342$	0	0
$343$	8.00000	0.431959
$344$	−1.00000	−0.0539164
$345$	0	0
$346$	−16.0000	−0.860165
$347$	6.00000	0.322097	0.161048	−	0.986947i	$-0.448512\pi$
$347$	6.00000	0.322097	0.161048	+	0.986947i	$0.448512\pi$
$348$	0	0
$349$	16.0000	0.856460	0.428230	−	0.903670i	$-0.359137\pi$
$349$	16.0000	0.856460	0.428230	+	0.903670i	$0.359137\pi$
$350$	0	0
$351$	0	0
$352$	5.00000	0.266501
$353$	−11.0000	−0.585471	−0.292735	−	0.956193i	$-0.594566\pi$
$353$	−11.0000	−0.585471	−0.292735	+	0.956193i	$0.594566\pi$
$354$	0	0
$355$	0	0
$356$	−14.0000	−0.741999
$357$	0	0
$358$	−20.0000	−1.05703
$359$	6.00000	0.316668	0.158334	−	0.987386i	$-0.449388\pi$
$359$	6.00000	0.316668	0.158334	+	0.987386i	$0.449388\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	22.0000	1.15629
$363$	0	0
$364$	−12.0000	−0.628971
$365$	0	0
$366$	0	0
$367$	22.0000	1.14839	0.574195	−	0.818718i	$-0.305315\pi$
$367$	22.0000	1.14839	0.574195	+	0.818718i	$0.305315\pi$
$368$	−1.00000	−0.0521286
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−21.0000	−1.08734	−0.543669	−	0.839299i	$-0.682965\pi$
$373$	−21.0000	−1.08734	−0.543669	+	0.839299i	$0.682965\pi$
$374$	−5.00000	−0.258544
$375$	0	0
$376$	−13.0000	−0.670424
$377$	27.0000	1.39057
$378$	0	0
$379$	−28.0000	−1.43826	−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000	−1.43826	−0.719132	+	0.694874i	$0.755460\pi$
$380$	0	0
$381$	0	0
$382$	6.00000	0.306987
$383$	19.0000	0.970855	0.485427	−	0.874277i	$-0.338664\pi$
$383$	19.0000	0.970855	0.485427	+	0.874277i	$0.338664\pi$
$384$	0	0
$385$	0	0
$386$	−4.00000	−0.203595
$387$	0	0
$388$	−10.0000	−0.507673
$389$	−5.00000	−0.253510	−0.126755	−	0.991934i	$-0.540456\pi$
$389$	−5.00000	−0.253510	−0.126755	+	0.991934i	$0.540456\pi$
$390$	0	0
$391$	1.00000	0.0505722
$392$	−9.00000	−0.454569
$393$	0	0
$394$	−22.0000	−1.10834
$395$	0	0
$396$	0	0
$397$	13.0000	0.652451	0.326226	−	0.945292i	$-0.394223\pi$
$397$	13.0000	0.652451	0.326226	+	0.945292i	$0.394223\pi$
$398$	−3.00000	−0.150376
$399$	0	0
$400$	0	0
$401$	−16.0000	−0.799002	−0.399501	−	0.916733i	$-0.630817\pi$
$401$	−16.0000	−0.799002	−0.399501	+	0.916733i	$0.630817\pi$
$402$	0	0
$403$	15.0000	0.747203
$404$	17.0000	0.845782
$405$	0	0
$406$	36.0000	1.78665
$407$	10.0000	0.495682
$408$	0	0
$409$	11.0000	0.543915	0.271957	−	0.962309i	$-0.412329\pi$
$409$	11.0000	0.543915	0.271957	+	0.962309i	$0.412329\pi$
$410$	0	0
$411$	0	0
$412$	−10.0000	−0.492665
$413$	−16.0000	−0.787309
$414$	0	0
$415$	0	0
$416$	3.00000	0.147087
$417$	0	0
$418$	−30.0000	−1.46735
$419$	−23.0000	−1.12362	−0.561812	−	0.827265i	$-0.689895\pi$
$419$	−23.0000	−1.12362	−0.561812	+	0.827265i	$0.689895\pi$
$420$	0	0
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	32.0000	1.54859
$428$	−6.00000	−0.290021
$429$	0	0
$430$	0	0
$431$	−8.00000	−0.385346	−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000	−0.385346	−0.192673	+	0.981263i	$0.561716\pi$
$432$	0	0
$433$	22.0000	1.05725	0.528626	−	0.848855i	$-0.322707\pi$
$433$	22.0000	1.05725	0.528626	+	0.848855i	$0.322707\pi$
$434$	20.0000	0.960031
$435$	0	0
$436$	−8.00000	−0.383131
$437$	6.00000	0.287019
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	0	0
$442$	−3.00000	−0.142695
$443$	30.0000	1.42534	0.712672	−	0.701498i	$-0.247485\pi$
$443$	30.0000	1.42534	0.712672	+	0.701498i	$0.247485\pi$
$444$	0	0
$445$	0	0
$446$	−22.0000	−1.04173
$447$	0	0
$448$	4.00000	0.188982
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	10.0000	0.470882
$452$	3.00000	0.141108
$453$	0	0
$454$	−2.00000	−0.0938647
$455$	0	0
$456$	0	0
$457$	16.0000	0.748448	0.374224	−	0.927338i	$-0.377909\pi$
$457$	16.0000	0.748448	0.374224	+	0.927338i	$0.377909\pi$
$458$	−2.00000	−0.0934539
$459$	0	0
$460$	0	0
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	0	0
$463$	−6.00000	−0.278844	−0.139422	−	0.990233i	$-0.544524\pi$
$463$	−6.00000	−0.278844	−0.139422	+	0.990233i	$0.544524\pi$
$464$	−9.00000	−0.417815
$465$	0	0
$466$	14.0000	0.648537
$467$	−6.00000	−0.277647	−0.138823	−	0.990317i	$-0.544332\pi$
$467$	−6.00000	−0.277647	−0.138823	+	0.990317i	$0.544332\pi$
$468$	0	0
$469$	−16.0000	−0.738811
$470$	0	0
$471$	0	0
$472$	4.00000	0.184115
$473$	−5.00000	−0.229900
$474$	0	0
$475$	0	0
$476$	−4.00000	−0.183340
$477$	0	0
$478$	−16.0000	−0.731823
$479$	−2.00000	−0.0913823	−0.0456912	−	0.998956i	$-0.514549\pi$
$479$	−2.00000	−0.0913823	−0.0456912	+	0.998956i	$0.514549\pi$
$480$	0	0
$481$	6.00000	0.273576
$482$	23.0000	1.04762
$483$	0	0
$484$	14.0000	0.636364
$485$	0	0
$486$	0	0
$487$	−12.0000	−0.543772	−0.271886	−	0.962329i	$-0.587647\pi$
$487$	−12.0000	−0.543772	−0.271886	+	0.962329i	$0.587647\pi$
$488$	−8.00000	−0.362143
$489$	0	0
$490$	0	0
$491$	24.0000	1.08310	0.541552	−	0.840667i	$-0.317837\pi$
$491$	24.0000	1.08310	0.541552	+	0.840667i	$0.317837\pi$
$492$	0	0
$493$	9.00000	0.405340
$494$	−18.0000	−0.809858
$495$	0	0
$496$	−5.00000	−0.224507
$497$	−24.0000	−1.07655
$498$	0	0
$499$	−12.0000	−0.537194	−0.268597	−	0.963253i	$-0.586560\pi$
$499$	−12.0000	−0.537194	−0.268597	+	0.963253i	$0.586560\pi$
$500$	0	0
$501$	0	0
$502$	−15.0000	−0.669483
$503$	−9.00000	−0.401290	−0.200645	−	0.979664i	$-0.564304\pi$
$503$	−9.00000	−0.401290	−0.200645	+	0.979664i	$0.564304\pi$
$504$	0	0
$505$	0	0
$506$	−5.00000	−0.222277
$507$	0	0
$508$	4.00000	0.177471
$509$	−21.0000	−0.930809	−0.465404	−	0.885098i	$-0.654091\pi$
$509$	−21.0000	−0.930809	−0.465404	+	0.885098i	$0.654091\pi$
$510$	0	0
$511$	−8.00000	−0.353899
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	21.0000	0.926270
$515$	0	0
$516$	0	0
$517$	−65.0000	−2.85870
$518$	8.00000	0.351500
$519$	0	0
$520$	0	0
$521$	−22.0000	−0.963837	−0.481919	−	0.876216i	$-0.660060\pi$
$521$	−22.0000	−0.963837	−0.481919	+	0.876216i	$0.660060\pi$
$522$	0	0
$523$	23.0000	1.00572	0.502860	−	0.864368i	$-0.332281\pi$
$523$	23.0000	1.00572	0.502860	+	0.864368i	$0.332281\pi$
$524$	−3.00000	−0.131056
$525$	0	0
$526$	−8.00000	−0.348817
$527$	5.00000	0.217803
$528$	0	0
$529$	−22.0000	−0.956522
$530$	0	0
$531$	0	0
$532$	−24.0000	−1.04053
$533$	6.00000	0.259889
$534$	0	0
$535$	0	0
$536$	4.00000	0.172774
$537$	0	0
$538$	−9.00000	−0.388018
$539$	−45.0000	−1.93829
$540$	0	0
$541$	10.0000	0.429934	0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000	0.429934	0.214967	+	0.976621i	$0.431036\pi$
$542$	−16.0000	−0.687259
$543$	0	0
$544$	1.00000	0.0428746
$545$	0	0
$546$	0	0
$547$	25.0000	1.06892	0.534461	−	0.845193i	$-0.320514\pi$
$547$	25.0000	1.06892	0.534461	+	0.845193i	$0.320514\pi$
$548$	18.0000	0.768922
$549$	0	0
$550$	0	0
$551$	54.0000	2.30048
$552$	0	0
$553$	36.0000	1.53088
$554$	26.0000	1.10463
$555$	0	0
$556$	−8.00000	−0.339276
$557$	−42.0000	−1.77960	−0.889799	−	0.456354i	$-0.849155\pi$
$557$	−42.0000	−1.77960	−0.889799	+	0.456354i	$0.849155\pi$
$558$	0	0
$559$	−3.00000	−0.126886
$560$	0	0
$561$	0	0
$562$	18.0000	0.759284
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	0	0
$565$	0	0
$566$	−20.0000	−0.840663
$567$	0	0
$568$	6.00000	0.251754
$569$	−20.0000	−0.838444	−0.419222	−	0.907884i	$-0.637697\pi$
$569$	−20.0000	−0.838444	−0.419222	+	0.907884i	$0.637697\pi$
$570$	0	0
$571$	−32.0000	−1.33916	−0.669579	−	0.742741i	$-0.733526\pi$
$571$	−32.0000	−1.33916	−0.669579	+	0.742741i	$0.733526\pi$
$572$	15.0000	0.627182
$573$	0	0
$574$	8.00000	0.333914
$575$	0	0
$576$	0	0
$577$	−32.0000	−1.33218	−0.666089	−	0.745873i	$-0.732033\pi$
$577$	−32.0000	−1.33218	−0.666089	+	0.745873i	$0.732033\pi$
$578$	16.0000	0.665512
$579$	0	0
$580$	0	0
$581$	−16.0000	−0.663792
$582$	0	0
$583$	0	0
$584$	2.00000	0.0827606
$585$	0	0
$586$	−6.00000	−0.247858
$587$	−36.0000	−1.48588	−0.742940	−	0.669359i	$-0.766569\pi$
$587$	−36.0000	−1.48588	−0.742940	+	0.669359i	$0.766569\pi$
$588$	0	0
$589$	30.0000	1.23613
$590$	0	0
$591$	0	0
$592$	−2.00000	−0.0821995
$593$	−21.0000	−0.862367	−0.431183	−	0.902264i	$-0.641904\pi$
$593$	−21.0000	−0.862367	−0.431183	+	0.902264i	$0.641904\pi$
$594$	0	0
$595$	0	0
$596$	−13.0000	−0.532501
$597$	0	0
$598$	−3.00000	−0.122679
$599$	−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
$601$	−19.0000	−0.775026	−0.387513	−	0.921864i	$-0.626666\pi$
$601$	−19.0000	−0.775026	−0.387513	+	0.921864i	$0.626666\pi$
$602$	−4.00000	−0.163028
$603$	0	0
$604$	13.0000	0.528962
$605$	0	0
$606$	0	0
$607$	−10.0000	−0.405887	−0.202944	−	0.979190i	$-0.565051\pi$
$607$	−10.0000	−0.405887	−0.202944	+	0.979190i	$0.565051\pi$
$608$	6.00000	0.243332
$609$	0	0
$610$	0	0
$611$	−39.0000	−1.57777
$612$	0	0
$613$	−23.0000	−0.928961	−0.464481	−	0.885583i	$-0.653759\pi$
$613$	−23.0000	−0.928961	−0.464481	+	0.885583i	$0.653759\pi$
$614$	−21.0000	−0.847491
$615$	0	0
$616$	20.0000	0.805823
$617$	29.0000	1.16750	0.583748	−	0.811935i	$-0.301586\pi$
$617$	29.0000	1.16750	0.583748	+	0.811935i	$0.301586\pi$
$618$	0	0
$619$	26.0000	1.04503	0.522514	−	0.852631i	$-0.324994\pi$
$619$	26.0000	1.04503	0.522514	+	0.852631i	$0.324994\pi$
$620$	0	0
$621$	0	0
$622$	−18.0000	−0.721734
$623$	−56.0000	−2.24359
$624$	0	0
$625$	0	0
$626$	14.0000	0.559553
$627$	0	0
$628$	−13.0000	−0.518756
$629$	2.00000	0.0797452
$630$	0	0
$631$	−4.00000	−0.159237	−0.0796187	−	0.996825i	$-0.525370\pi$
$631$	−4.00000	−0.159237	−0.0796187	+	0.996825i	$0.525370\pi$
$632$	−9.00000	−0.358001
$633$	0	0
$634$	4.00000	0.158860
$635$	0	0
$636$	0	0
$637$	−27.0000	−1.06978
$638$	−45.0000	−1.78157
$639$	0	0
$640$	0	0
$641$	−20.0000	−0.789953	−0.394976	−	0.918691i	$-0.629247\pi$
$641$	−20.0000	−0.789953	−0.394976	+	0.918691i	$0.629247\pi$
$642$	0	0
$643$	21.0000	0.828159	0.414080	−	0.910241i	$-0.364104\pi$
$643$	21.0000	0.828159	0.414080	+	0.910241i	$0.364104\pi$
$644$	−4.00000	−0.157622
$645$	0	0
$646$	−6.00000	−0.236067
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	20.0000	0.785069
$650$	0	0
$651$	0	0
$652$	−19.0000	−0.744097
$653$	26.0000	1.01746	0.508729	−	0.860927i	$-0.330115\pi$
$653$	26.0000	1.01746	0.508729	+	0.860927i	$0.330115\pi$
$654$	0	0
$655$	0	0
$656$	−2.00000	−0.0780869
$657$	0	0
$658$	−52.0000	−2.02717
$659$	20.0000	0.779089	0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000	0.779089	0.389545	+	0.921008i	$0.372632\pi$
$660$	0	0
$661$	44.0000	1.71140	0.855701	−	0.517471i	$-0.173126\pi$
$661$	44.0000	1.71140	0.855701	+	0.517471i	$0.173126\pi$
$662$	−10.0000	−0.388661
$663$	0	0
$664$	4.00000	0.155230
$665$	0	0
$666$	0	0
$667$	9.00000	0.348481
$668$	−12.0000	−0.464294
$669$	0	0
$670$	0	0
$671$	−40.0000	−1.54418
$672$	0	0
$673$	48.0000	1.85026	0.925132	−	0.379646i	$-0.123954\pi$
$673$	48.0000	1.85026	0.925132	+	0.379646i	$0.123954\pi$
$674$	−16.0000	−0.616297
$675$	0	0
$676$	−4.00000	−0.153846
$677$	−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
$679$	−40.0000	−1.53506
$680$	0	0
$681$	0	0
$682$	−25.0000	−0.957299
$683$	44.0000	1.68361	0.841807	−	0.539779i	$-0.181492\pi$
$683$	44.0000	1.68361	0.841807	+	0.539779i	$0.181492\pi$
$684$	0	0
$685$	0	0
$686$	−8.00000	−0.305441
$687$	0	0
$688$	1.00000	0.0381246
$689$	0	0
$690$	0	0
$691$	22.0000	0.836919	0.418460	−	0.908235i	$-0.362570\pi$
$691$	22.0000	0.836919	0.418460	+	0.908235i	$0.362570\pi$
$692$	16.0000	0.608229
$693$	0	0
$694$	−6.00000	−0.227757
$695$	0	0
$696$	0	0
$697$	2.00000	0.0757554
$698$	−16.0000	−0.605609
$699$	0	0
$700$	0	0
$701$	31.0000	1.17085	0.585427	−	0.810725i	$-0.300927\pi$
$701$	31.0000	1.17085	0.585427	+	0.810725i	$0.300927\pi$
$702$	0	0
$703$	12.0000	0.452589
$704$	−5.00000	−0.188445
$705$	0	0
$706$	11.0000	0.413990
$707$	68.0000	2.55740
$708$	0	0
$709$	−24.0000	−0.901339	−0.450669	−	0.892691i	$-0.648815\pi$
$709$	−24.0000	−0.901339	−0.450669	+	0.892691i	$0.648815\pi$
$710$	0	0
$711$	0	0
$712$	14.0000	0.524672
$713$	5.00000	0.187251
$714$	0	0
$715$	0	0
$716$	20.0000	0.747435
$717$	0	0
$718$	−6.00000	−0.223918
$719$	−20.0000	−0.745874	−0.372937	−	0.927857i	$-0.621649\pi$
$719$	−20.0000	−0.745874	−0.372937	+	0.927857i	$0.621649\pi$
$720$	0	0
$721$	−40.0000	−1.48968
$722$	−17.0000	−0.632674
$723$	0	0
$724$	−22.0000	−0.817624
$725$	0	0
$726$	0	0
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	12.0000	0.444750
$729$	0	0
$730$	0	0
$731$	−1.00000	−0.0369863
$732$	0	0
$733$	−46.0000	−1.69905	−0.849524	−	0.527549i	$-0.823111\pi$
$733$	−46.0000	−1.69905	−0.849524	+	0.527549i	$0.823111\pi$
$734$	−22.0000	−0.812035
$735$	0	0
$736$	1.00000	0.0368605
$737$	20.0000	0.736709
$738$	0	0
$739$	10.0000	0.367856	0.183928	−	0.982940i	$-0.441119\pi$
$739$	10.0000	0.367856	0.183928	+	0.982940i	$0.441119\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	9.00000	0.330178	0.165089	−	0.986279i	$-0.447209\pi$
$743$	9.00000	0.330178	0.165089	+	0.986279i	$0.447209\pi$
$744$	0	0
$745$	0	0
$746$	21.0000	0.768865
$747$	0	0
$748$	5.00000	0.182818
$749$	−24.0000	−0.876941
$750$	0	0
$751$	−7.00000	−0.255434	−0.127717	−	0.991811i	$-0.540765\pi$
$751$	−7.00000	−0.255434	−0.127717	+	0.991811i	$0.540765\pi$
$752$	13.0000	0.474061
$753$	0	0
$754$	−27.0000	−0.983282
$755$	0	0
$756$	0	0
$757$	43.0000	1.56286	0.781431	−	0.623992i	$-0.214490\pi$
$757$	43.0000	1.56286	0.781431	+	0.623992i	$0.214490\pi$
$758$	28.0000	1.01701
$759$	0	0
$760$	0	0
$761$	12.0000	0.435000	0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000	0.435000	0.217500	+	0.976060i	$0.430210\pi$
$762$	0	0
$763$	−32.0000	−1.15848
$764$	−6.00000	−0.217072
$765$	0	0
$766$	−19.0000	−0.686498
$767$	12.0000	0.433295
$768$	0	0
$769$	3.00000	0.108183	0.0540914	−	0.998536i	$-0.482774\pi$
$769$	3.00000	0.108183	0.0540914	+	0.998536i	$0.482774\pi$
$770$	0	0
$771$	0	0
$772$	4.00000	0.143963
$773$	18.0000	0.647415	0.323708	−	0.946157i	$-0.395071\pi$
$773$	18.0000	0.647415	0.323708	+	0.946157i	$0.395071\pi$
$774$	0	0
$775$	0	0
$776$	10.0000	0.358979
$777$	0	0
$778$	5.00000	0.179259
$779$	12.0000	0.429945
$780$	0	0
$781$	30.0000	1.07348
$782$	−1.00000	−0.0357599
$783$	0	0
$784$	9.00000	0.321429
$785$	0	0
$786$	0	0
$787$	13.0000	0.463400	0.231700	−	0.972787i	$-0.425571\pi$
$787$	13.0000	0.463400	0.231700	+	0.972787i	$0.425571\pi$
$788$	22.0000	0.783718
$789$	0	0
$790$	0	0
$791$	12.0000	0.426671
$792$	0	0
$793$	−24.0000	−0.852265
$794$	−13.0000	−0.461353
$795$	0	0
$796$	3.00000	0.106332
$797$	14.0000	0.495905	0.247953	−	0.968772i	$-0.420242\pi$
$797$	14.0000	0.495905	0.247953	+	0.968772i	$0.420242\pi$
$798$	0	0
$799$	−13.0000	−0.459907
$800$	0	0
$801$	0	0
$802$	16.0000	0.564980
$803$	10.0000	0.352892
$804$	0	0
$805$	0	0
$806$	−15.0000	−0.528352
$807$	0	0
$808$	−17.0000	−0.598058
$809$	12.0000	0.421898	0.210949	−	0.977497i	$-0.432345\pi$
$809$	12.0000	0.421898	0.210949	+	0.977497i	$0.432345\pi$
$810$	0	0
$811$	−26.0000	−0.912983	−0.456492	−	0.889728i	$-0.650894\pi$
$811$	−26.0000	−0.912983	−0.456492	+	0.889728i	$0.650894\pi$
$812$	−36.0000	−1.26335
$813$	0	0
$814$	−10.0000	−0.350500
$815$	0	0
$816$	0	0
$817$	−6.00000	−0.209913
$818$	−11.0000	−0.384606
$819$	0	0
$820$	0	0
$821$	−46.0000	−1.60541	−0.802706	−	0.596376i	$-0.796607\pi$
$821$	−46.0000	−1.60541	−0.802706	+	0.596376i	$0.796607\pi$
$822$	0	0
$823$	24.0000	0.836587	0.418294	−	0.908312i	$-0.362628\pi$
$823$	24.0000	0.836587	0.418294	+	0.908312i	$0.362628\pi$
$824$	10.0000	0.348367
$825$	0	0
$826$	16.0000	0.556711
$827$	−22.0000	−0.765015	−0.382507	−	0.923952i	$-0.624939\pi$
$827$	−22.0000	−0.765015	−0.382507	+	0.923952i	$0.624939\pi$
$828$	0	0
$829$	0	0		−	1.00000i	$-0.5\pi$
$829$	0	0			1.00000i	$0.5\pi$
$830$	0	0
$831$	0	0
$832$	−3.00000	−0.104006
$833$	−9.00000	−0.311832
$834$	0	0
$835$	0	0
$836$	30.0000	1.03757
$837$	0	0
$838$	23.0000	0.794522
$839$	−34.0000	−1.17381	−0.586905	−	0.809656i	$-0.699654\pi$
$839$	−34.0000	−1.17381	−0.586905	+	0.809656i	$0.699654\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	26.0000	0.896019
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	56.0000	1.92418
$848$	0	0
$849$	0	0
$850$	0	0
$851$	2.00000	0.0685591
$852$	0	0
$853$	−19.0000	−0.650548	−0.325274	−	0.945620i	$-0.605456\pi$
$853$	−19.0000	−0.650548	−0.325274	+	0.945620i	$0.605456\pi$
$854$	−32.0000	−1.09502
$855$	0	0
$856$	6.00000	0.205076
$857$	6.00000	0.204956	0.102478	−	0.994735i	$-0.467323\pi$
$857$	6.00000	0.204956	0.102478	+	0.994735i	$0.467323\pi$
$858$	0	0
$859$	2.00000	0.0682391	0.0341196	−	0.999418i	$-0.489137\pi$
$859$	2.00000	0.0682391	0.0341196	+	0.999418i	$0.489137\pi$
$860$	0	0
$861$	0	0
$862$	8.00000	0.272481
$863$	−39.0000	−1.32758	−0.663788	−	0.747921i	$-0.731052\pi$
$863$	−39.0000	−1.32758	−0.663788	+	0.747921i	$0.731052\pi$
$864$	0	0
$865$	0	0
$866$	−22.0000	−0.747590
$867$	0	0
$868$	−20.0000	−0.678844
$869$	−45.0000	−1.52652
$870$	0	0
$871$	12.0000	0.406604
$872$	8.00000	0.270914
$873$	0	0
$874$	−6.00000	−0.202953
$875$	0	0
$876$	0	0
$877$	−1.00000	−0.0337676	−0.0168838	−	0.999857i	$-0.505375\pi$
$877$	−1.00000	−0.0337676	−0.0168838	+	0.999857i	$0.505375\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	0	0
$883$	36.0000	1.21150	0.605748	−	0.795656i	$-0.292874\pi$
$883$	36.0000	1.21150	0.605748	+	0.795656i	$0.292874\pi$
$884$	3.00000	0.100901
$885$	0	0
$886$	−30.0000	−1.00787
$887$	−51.0000	−1.71241	−0.856206	−	0.516634i	$-0.827185\pi$
$887$	−51.0000	−1.71241	−0.856206	+	0.516634i	$0.827185\pi$
$888$	0	0
$889$	16.0000	0.536623
$890$	0	0
$891$	0	0
$892$	22.0000	0.736614
$893$	−78.0000	−2.61017
$894$	0	0
$895$	0	0
$896$	−4.00000	−0.133631
$897$	0	0
$898$	0	0
$899$	45.0000	1.50083
$900$	0	0
$901$	0	0
$902$	−10.0000	−0.332964
$903$	0	0
$904$	−3.00000	−0.0997785
$905$	0	0
$906$	0	0
$907$	−3.00000	−0.0996134	−0.0498067	−	0.998759i	$-0.515861\pi$
$907$	−3.00000	−0.0996134	−0.0498067	+	0.998759i	$0.515861\pi$
$908$	2.00000	0.0663723
$909$	0	0
$910$	0	0
$911$	−36.0000	−1.19273	−0.596367	−	0.802712i	$-0.703390\pi$
$911$	−36.0000	−1.19273	−0.596367	+	0.802712i	$0.703390\pi$
$912$	0	0
$913$	20.0000	0.661903
$914$	−16.0000	−0.529233
$915$	0	0
$916$	2.00000	0.0660819
$917$	−12.0000	−0.396275
$918$	0	0
$919$	33.0000	1.08857	0.544285	−	0.838901i	$-0.316801\pi$
$919$	33.0000	1.08857	0.544285	+	0.838901i	$0.316801\pi$
$920$	0	0
$921$	0	0
$922$	6.00000	0.197599
$923$	18.0000	0.592477
$924$	0	0
$925$	0	0
$926$	6.00000	0.197172
$927$	0	0
$928$	9.00000	0.295439
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	0	0
$931$	−54.0000	−1.76978
$932$	−14.0000	−0.458585
$933$	0	0
$934$	6.00000	0.196326
$935$	0	0
$936$	0	0
$937$	−40.0000	−1.30674	−0.653372	−	0.757037i	$-0.726646\pi$
$937$	−40.0000	−1.30674	−0.653372	+	0.757037i	$0.726646\pi$
$938$	16.0000	0.522419
$939$	0	0
$940$	0	0
$941$	31.0000	1.01057	0.505286	−	0.862952i	$-0.331387\pi$
$941$	31.0000	1.01057	0.505286	+	0.862952i	$0.331387\pi$
$942$	0	0
$943$	2.00000	0.0651290
$944$	−4.00000	−0.130189
$945$	0	0
$946$	5.00000	0.162564
$947$	−48.0000	−1.55979	−0.779895	−	0.625910i	$-0.784728\pi$
$947$	−48.0000	−1.55979	−0.779895	+	0.625910i	$0.784728\pi$
$948$	0	0
$949$	6.00000	0.194768
$950$	0	0
$951$	0	0
$952$	4.00000	0.129641
$953$	61.0000	1.97598	0.987992	−	0.154506i	$-0.0493785\pi$
$953$	61.0000	1.97598	0.987992	+	0.154506i	$0.0493785\pi$
$954$	0	0
$955$	0	0
$956$	16.0000	0.517477
$957$	0	0
$958$	2.00000	0.0646171
$959$	72.0000	2.32500
$960$	0	0
$961$	−6.00000	−0.193548
$962$	−6.00000	−0.193448
$963$	0	0
$964$	−23.0000	−0.740780
$965$	0	0
$966$	0	0
$967$	2.00000	0.0643157	0.0321578	−	0.999483i	$-0.489762\pi$
$967$	2.00000	0.0643157	0.0321578	+	0.999483i	$0.489762\pi$
$968$	−14.0000	−0.449977
$969$	0	0
$970$	0	0
$971$	−9.00000	−0.288824	−0.144412	−	0.989518i	$-0.546129\pi$
$971$	−9.00000	−0.288824	−0.144412	+	0.989518i	$0.546129\pi$
$972$	0	0
$973$	−32.0000	−1.02587
$974$	12.0000	0.384505
$975$	0	0
$976$	8.00000	0.256074
$977$	39.0000	1.24772	0.623860	−	0.781536i	$-0.285563\pi$
$977$	39.0000	1.24772	0.623860	+	0.781536i	$0.285563\pi$
$978$	0	0
$979$	70.0000	2.23721
$980$	0	0
$981$	0	0
$982$	−24.0000	−0.765871
$983$	−29.0000	−0.924956	−0.462478	−	0.886631i	$-0.653040\pi$
$983$	−29.0000	−0.924956	−0.462478	+	0.886631i	$0.653040\pi$
$984$	0	0
$985$	0	0
$986$	−9.00000	−0.286618
$987$	0	0
$988$	18.0000	0.572656
$989$	−1.00000	−0.0317982
$990$	0	0
$991$	−17.0000	−0.540023	−0.270011	−	0.962857i	$-0.587027\pi$
$991$	−17.0000	−0.540023	−0.270011	+	0.962857i	$0.587027\pi$
$992$	5.00000	0.158750
$993$	0	0
$994$	24.0000	0.761234
$995$	0	0
$996$	0	0
$997$	15.0000	0.475055	0.237527	−	0.971381i	$-0.423663\pi$
$997$	15.0000	0.475055	0.237527	+	0.971381i	$0.423663\pi$
$998$	12.0000	0.379853
$999$	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	1350.2.a.j.1.1		1
3.2	odd	2		1350.2.a.v.1.1		1
5.2	odd	4		270.2.c.a.109.1	✓	2
5.3	odd	4		270.2.c.a.109.2	yes	2
5.4	even	2		1350.2.a.l.1.1		1
15.2	even	4		270.2.c.b.109.2	yes	2
15.8	even	4		270.2.c.b.109.1	yes	2
15.14	odd	2		1350.2.a.b.1.1		1
20.3	even	4		2160.2.f.d.1729.1		2
20.7	even	4		2160.2.f.d.1729.2		2
45.2	even	12		810.2.i.c.109.2		4
45.7	odd	12		810.2.i.d.109.1		4
45.13	odd	12		810.2.i.d.379.1		4
45.22	odd	12		810.2.i.d.379.2		4
45.23	even	12		810.2.i.c.379.2		4
45.32	even	12		810.2.i.c.379.1		4
45.38	even	12		810.2.i.c.109.1		4
45.43	odd	12		810.2.i.d.109.2		4
60.23	odd	4		2160.2.f.e.1729.2		2
60.47	odd	4		2160.2.f.e.1729.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.2.c.a.109.1	✓	2	5.2	odd	4
270.2.c.a.109.2	yes	2	5.3	odd	4
270.2.c.b.109.1	yes	2	15.8	even	4
270.2.c.b.109.2	yes	2	15.2	even	4
810.2.i.c.109.1		4	45.38	even	12
810.2.i.c.109.2		4	45.2	even	12
810.2.i.c.379.1		4	45.32	even	12
810.2.i.c.379.2		4	45.23	even	12
810.2.i.d.109.1		4	45.7	odd	12
810.2.i.d.109.2		4	45.43	odd	12
810.2.i.d.379.1		4	45.13	odd	12
810.2.i.d.379.2		4	45.22	odd	12
1350.2.a.b.1.1		1	15.14	odd	2
1350.2.a.j.1.1		1	1.1	even	1	trivial
1350.2.a.l.1.1		1	5.4	even	2
1350.2.a.v.1.1		1	3.2	odd	2
2160.2.f.d.1729.1		2	20.3	even	4
2160.2.f.d.1729.2		2	20.7	even	4
2160.2.f.e.1729.1		2	60.47	odd	4
2160.2.f.e.1729.2		2	60.23	odd	4

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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

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	1350.2.a.j.1.1

Level	$1350$
Weight	$2$
Character	1350.1
Self dual	yes
Analytic conductor	$10.780$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$