LMFDB - Embedded newform 1350.2.a.d.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:	$0$
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)

$f(q)$

$=$

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

$q-1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{7} -1.00000 q^{8} +3.00000 q^{11} -5.00000 q^{13} +2.00000 q^{14} +1.00000 q^{16} +3.00000 q^{17} -4.00000 q^{19} -3.00000 q^{22} +9.00000 q^{23} +5.00000 q^{26} -2.00000 q^{28} -3.00000 q^{29} +5.00000 q^{31} -1.00000 q^{32} -3.00000 q^{34} +10.0000 q^{37} +4.00000 q^{38} +1.00000 q^{43} +3.00000 q^{44} -9.00000 q^{46} -9.00000 q^{47} -3.00000 q^{49} -5.00000 q^{52} +12.0000 q^{53} +2.00000 q^{56} +3.00000 q^{58} +12.0000 q^{59} +2.00000 q^{61} -5.00000 q^{62} +1.00000 q^{64} +4.00000 q^{67} +3.00000 q^{68} +12.0000 q^{71} +10.0000 q^{73} -10.0000 q^{74} -4.00000 q^{76} -6.00000 q^{77} -13.0000 q^{79} -6.00000 q^{83} -1.00000 q^{86} -3.00000 q^{88} -12.0000 q^{89} +10.0000 q^{91} +9.00000 q^{92} +9.00000 q^{94} -2.00000 q^{97} +3.00000 q^{98} +O(q^{100})$

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
$5$	0	0
$5$	0	0
$6$	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
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	0	0
$13$	−5.00000	−1.38675	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000	−1.38675	−0.693375	+	0.720577i	$0.743877\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	0	0
$22$	−3.00000	−0.639602
$23$	9.00000	1.87663	0.938315	−	0.345782i	$-0.112386\pi$
$23$	9.00000	1.87663	0.938315	+	0.345782i	$0.112386\pi$
$24$	0	0
$25$	0	0
$26$	5.00000	0.980581
$27$	0	0
$28$	−2.00000	−0.377964
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	0	0
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−3.00000	−0.514496
$35$	0	0
$36$	0	0
$37$	10.0000	1.64399	0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000	1.64399	0.821995	+	0.569495i	$0.192861\pi$
$38$	4.00000	0.648886
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	3.00000	0.452267
$45$	0	0
$46$	−9.00000	−1.32698
$47$	−9.00000	−1.31278	−0.656392	−	0.754420i	$-0.727918\pi$
$47$	−9.00000	−1.31278	−0.656392	+	0.754420i	$0.727918\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	0	0
$52$	−5.00000	−0.693375
$53$	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
$55$	0	0
$56$	2.00000	0.267261
$57$	0	0
$58$	3.00000	0.393919
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	−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	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	3.00000	0.363803
$69$	0	0
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	−10.0000	−1.16248
$75$	0	0
$76$	−4.00000	−0.458831
$77$	−6.00000	−0.683763
$78$	0	0
$79$	−13.0000	−1.46261	−0.731307	−	0.682048i	$-0.761089\pi$
$79$	−13.0000	−1.46261	−0.731307	+	0.682048i	$0.761089\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	0	0
$85$	0	0
$86$	−1.00000	−0.107833
$87$	0	0
$88$	−3.00000	−0.319801
$89$	−12.0000	−1.27200	−0.635999	−	0.771690i	$-0.719412\pi$
$89$	−12.0000	−1.27200	−0.635999	+	0.771690i	$0.719412\pi$
$90$	0	0
$91$	10.0000	1.04828
$92$	9.00000	0.938315
$93$	0	0
$94$	9.00000	0.928279
$95$	0	0
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	3.00000	0.303046
$99$	0	0
$100$	0	0
$101$	15.0000	1.49256	0.746278	−	0.665635i	$-0.231839\pi$
$101$	15.0000	1.49256	0.746278	+	0.665635i	$0.231839\pi$
$102$	0	0
$103$	−2.00000	−0.197066	−0.0985329	−	0.995134i	$-0.531415\pi$
$103$	−2.00000	−0.197066	−0.0985329	+	0.995134i	$0.531415\pi$
$104$	5.00000	0.490290
$105$	0	0
$106$	−12.0000	−1.16554
$107$	−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
$109$	20.0000	1.91565	0.957826	−	0.287348i	$-0.0927736\pi$
$109$	20.0000	1.91565	0.957826	+	0.287348i	$0.0927736\pi$
$110$	0	0
$111$	0	0
$112$	−2.00000	−0.188982
$113$	15.0000	1.41108	0.705541	−	0.708669i	$-0.250704\pi$
$113$	15.0000	1.41108	0.705541	+	0.708669i	$0.250704\pi$
$114$	0	0
$115$	0	0
$116$	−3.00000	−0.278543
$117$	0	0
$118$	−12.0000	−1.10469
$119$	−6.00000	−0.550019
$120$	0	0
$121$	−2.00000	−0.181818
$122$	−2.00000	−0.181071
$123$	0	0
$124$	5.00000	0.449013
$125$	0	0
$126$	0	0
$127$	−2.00000	−0.177471	−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000	−0.177471	−0.0887357	+	0.996055i	$0.528283\pi$
$128$	−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
$133$	8.00000	0.693688
$134$	−4.00000	−0.345547
$135$	0	0
$136$	−3.00000	−0.257248
$137$	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
$139$	2.00000	0.169638	0.0848189	−	0.996396i	$-0.472969\pi$
$139$	2.00000	0.169638	0.0848189	+	0.996396i	$0.472969\pi$
$140$	0	0
$141$	0	0
$142$	−12.0000	−1.00702
$143$	−15.0000	−1.25436
$144$	0	0
$145$	0	0
$146$	−10.0000	−0.827606
$147$	0	0
$148$	10.0000	0.821995
$149$	21.0000	1.72039	0.860194	−	0.509968i	$-0.170343\pi$
$149$	21.0000	1.72039	0.860194	+	0.509968i	$0.170343\pi$
$150$	0	0
$151$	11.0000	0.895167	0.447584	−	0.894242i	$-0.352285\pi$
$151$	11.0000	0.895167	0.447584	+	0.894242i	$0.352285\pi$
$152$	4.00000	0.324443
$153$	0	0
$154$	6.00000	0.483494
$155$	0	0
$156$	0	0
$157$	13.0000	1.03751	0.518756	−	0.854922i	$-0.326395\pi$
$157$	13.0000	1.03751	0.518756	+	0.854922i	$0.326395\pi$
$158$	13.0000	1.03422
$159$	0	0
$160$	0	0
$161$	−18.0000	−1.41860
$162$	0	0
$163$	−11.0000	−0.861586	−0.430793	−	0.902451i	$-0.641766\pi$
$163$	−11.0000	−0.861586	−0.430793	+	0.902451i	$0.641766\pi$
$164$	0	0
$165$	0	0
$166$	6.00000	0.465690
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	0	0
$172$	1.00000	0.0762493
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	0	0
$175$	0	0
$176$	3.00000	0.226134
$177$	0	0
$178$	12.0000	0.899438
$179$	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
$181$	20.0000	1.48659	0.743294	−	0.668965i	$-0.233262\pi$
$181$	20.0000	1.48659	0.743294	+	0.668965i	$0.233262\pi$
$182$	−10.0000	−0.741249
$183$	0	0
$184$	−9.00000	−0.663489
$185$	0	0
$186$	0	0
$187$	9.00000	0.658145
$188$	−9.00000	−0.656392
$189$	0	0
$190$	0	0
$191$	6.00000	0.434145	0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000	0.434145	0.217072	+	0.976156i	$0.430349\pi$
$192$	0	0
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	−3.00000	−0.214286
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	−7.00000	−0.496217	−0.248108	−	0.968732i	$-0.579809\pi$
$199$	−7.00000	−0.496217	−0.248108	+	0.968732i	$0.579809\pi$
$200$	0	0
$201$	0	0
$202$	−15.0000	−1.05540
$203$	6.00000	0.421117
$204$	0	0
$205$	0	0
$206$	2.00000	0.139347
$207$	0	0
$208$	−5.00000	−0.346688
$209$	−12.0000	−0.830057
$210$	0	0
$211$	−22.0000	−1.51454	−0.757271	−	0.653101i	$-0.773468\pi$
$211$	−22.0000	−1.51454	−0.757271	+	0.653101i	$0.773468\pi$
$212$	12.0000	0.824163
$213$	0	0
$214$	6.00000	0.410152
$215$	0	0
$216$	0	0
$217$	−10.0000	−0.678844
$218$	−20.0000	−1.35457
$219$	0	0
$220$	0	0
$221$	−15.0000	−1.00901
$222$	0	0
$223$	−8.00000	−0.535720	−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000	−0.535720	−0.267860	+	0.963458i	$0.586316\pi$
$224$	2.00000	0.133631
$225$	0	0
$226$	−15.0000	−0.997785
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	−4.00000	−0.264327	−0.132164	−	0.991228i	$-0.542192\pi$
$229$	−4.00000	−0.264327	−0.132164	+	0.991228i	$0.542192\pi$
$230$	0	0
$231$	0	0
$232$	3.00000	0.196960
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	0	0
$236$	12.0000	0.781133
$237$	0	0
$238$	6.00000	0.388922
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	−19.0000	−1.22390	−0.611949	−	0.790897i	$-0.709614\pi$
$241$	−19.0000	−1.22390	−0.611949	+	0.790897i	$0.709614\pi$
$242$	2.00000	0.128565
$243$	0	0
$244$	2.00000	0.128037
$245$	0	0
$246$	0	0
$247$	20.0000	1.27257
$248$	−5.00000	−0.317500
$249$	0	0
$250$	0	0
$251$	−9.00000	−0.568075	−0.284037	−	0.958813i	$-0.591674\pi$
$251$	−9.00000	−0.568075	−0.284037	+	0.958813i	$0.591674\pi$
$252$	0	0
$253$	27.0000	1.69748
$254$	2.00000	0.125491
$255$	0	0
$256$	1.00000	0.0625000
$257$	15.0000	0.935674	0.467837	−	0.883815i	$-0.345033\pi$
$257$	15.0000	0.935674	0.467837	+	0.883815i	$0.345033\pi$
$258$	0	0
$259$	−20.0000	−1.24274
$260$	0	0
$261$	0	0
$262$	3.00000	0.185341
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	0	0
$265$	0	0
$266$	−8.00000	−0.490511
$267$	0	0
$268$	4.00000	0.244339
$269$	−9.00000	−0.548740	−0.274370	−	0.961624i	$-0.588469\pi$
$269$	−9.00000	−0.548740	−0.274370	+	0.961624i	$0.588469\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	3.00000	0.181902
$273$	0	0
$274$	−6.00000	−0.362473
$275$	0	0
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	−2.00000	−0.119952
$279$	0	0
$280$	0	0
$281$	−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
$283$	−32.0000	−1.90220	−0.951101	−	0.308879i	$-0.900046\pi$
$283$	−32.0000	−1.90220	−0.951101	+	0.308879i	$0.900046\pi$
$284$	12.0000	0.712069
$285$	0	0
$286$	15.0000	0.886969
$287$	0	0
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	0	0
$292$	10.0000	0.585206
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	0	0
$295$	0	0
$296$	−10.0000	−0.581238
$297$	0	0
$298$	−21.0000	−1.21650
$299$	−45.0000	−2.60242
$300$	0	0
$301$	−2.00000	−0.115278
$302$	−11.0000	−0.632979
$303$	0	0
$304$	−4.00000	−0.229416
$305$	0	0
$306$	0	0
$307$	−11.0000	−0.627803	−0.313902	−	0.949456i	$-0.601636\pi$
$307$	−11.0000	−0.627803	−0.313902	+	0.949456i	$0.601636\pi$
$308$	−6.00000	−0.341882
$309$	0	0
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	−8.00000	−0.452187	−0.226093	−	0.974106i	$-0.572595\pi$
$313$	−8.00000	−0.452187	−0.226093	+	0.974106i	$0.572595\pi$
$314$	−13.0000	−0.733632
$315$	0	0
$316$	−13.0000	−0.731307
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	−9.00000	−0.503903
$320$	0	0
$321$	0	0
$322$	18.0000	1.00310
$323$	−12.0000	−0.667698
$324$	0	0
$325$	0	0
$326$	11.0000	0.609234
$327$	0	0
$328$	0	0
$329$	18.0000	0.992372
$330$	0	0
$331$	14.0000	0.769510	0.384755	−	0.923019i	$-0.374286\pi$
$331$	14.0000	0.769510	0.384755	+	0.923019i	$0.374286\pi$
$332$	−6.00000	−0.329293
$333$	0	0
$334$	−12.0000	−0.656611
$335$	0	0
$336$	0	0
$337$	−32.0000	−1.74315	−0.871576	−	0.490261i	$-0.836901\pi$
$337$	−32.0000	−1.74315	−0.871576	+	0.490261i	$0.836901\pi$
$338$	−12.0000	−0.652714
$339$	0	0
$340$	0	0
$341$	15.0000	0.812296
$342$	0	0
$343$	20.0000	1.07990
$344$	−1.00000	−0.0539164
$345$	0	0
$346$	18.0000	0.967686
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	0	0
$351$	0	0
$352$	−3.00000	−0.159901
$353$	−3.00000	−0.159674	−0.0798369	−	0.996808i	$-0.525440\pi$
$353$	−3.00000	−0.159674	−0.0798369	+	0.996808i	$0.525440\pi$
$354$	0	0
$355$	0	0
$356$	−12.0000	−0.635999
$357$	0	0
$358$	−12.0000	−0.634220
$359$	12.0000	0.633336	0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000	0.633336	0.316668	+	0.948536i	$0.397436\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	−20.0000	−1.05118
$363$	0	0
$364$	10.0000	0.524142
$365$	0	0
$366$	0	0
$367$	−26.0000	−1.35719	−0.678594	−	0.734513i	$-0.737411\pi$
$367$	−26.0000	−1.35719	−0.678594	+	0.734513i	$0.737411\pi$
$368$	9.00000	0.469157
$369$	0	0
$370$	0	0
$371$	−24.0000	−1.24602
$372$	0	0
$373$	1.00000	0.0517780	0.0258890	−	0.999665i	$-0.491758\pi$
$373$	1.00000	0.0517780	0.0258890	+	0.999665i	$0.491758\pi$
$374$	−9.00000	−0.465379
$375$	0	0
$376$	9.00000	0.464140
$377$	15.0000	0.772539
$378$	0	0
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	0	0
$381$	0	0
$382$	−6.00000	−0.306987
$383$	9.00000	0.459879	0.229939	−	0.973205i	$-0.426147\pi$
$383$	9.00000	0.459879	0.229939	+	0.973205i	$0.426147\pi$
$384$	0	0
$385$	0	0
$386$	−10.0000	−0.508987
$387$	0	0
$388$	−2.00000	−0.101535
$389$	9.00000	0.456318	0.228159	−	0.973624i	$-0.426729\pi$
$389$	9.00000	0.456318	0.228159	+	0.973624i	$0.426729\pi$
$390$	0	0
$391$	27.0000	1.36545
$392$	3.00000	0.151523
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	7.00000	0.351320	0.175660	−	0.984451i	$-0.443794\pi$
$397$	7.00000	0.351320	0.175660	+	0.984451i	$0.443794\pi$
$398$	7.00000	0.350878
$399$	0	0
$400$	0	0
$401$	36.0000	1.79775	0.898877	−	0.438201i	$-0.144384\pi$
$401$	36.0000	1.79775	0.898877	+	0.438201i	$0.144384\pi$
$402$	0	0
$403$	−25.0000	−1.24534
$404$	15.0000	0.746278
$405$	0	0
$406$	−6.00000	−0.297775
$407$	30.0000	1.48704
$408$	0	0
$409$	23.0000	1.13728	0.568638	−	0.822588i	$-0.307470\pi$
$409$	23.0000	1.13728	0.568638	+	0.822588i	$0.307470\pi$
$410$	0	0
$411$	0	0
$412$	−2.00000	−0.0985329
$413$	−24.0000	−1.18096
$414$	0	0
$415$	0	0
$416$	5.00000	0.245145
$417$	0	0
$418$	12.0000	0.586939
$419$	−3.00000	−0.146560	−0.0732798	−	0.997311i	$-0.523347\pi$
$419$	−3.00000	−0.146560	−0.0732798	+	0.997311i	$0.523347\pi$
$420$	0	0
$421$	−4.00000	−0.194948	−0.0974740	−	0.995238i	$-0.531076\pi$
$421$	−4.00000	−0.194948	−0.0974740	+	0.995238i	$0.531076\pi$
$422$	22.0000	1.07094
$423$	0	0
$424$	−12.0000	−0.582772
$425$	0	0
$426$	0	0
$427$	−4.00000	−0.193574
$428$	−6.00000	−0.290021
$429$	0	0
$430$	0	0
$431$	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	$0.515303\pi$
$434$	10.0000	0.480015
$435$	0	0
$436$	20.0000	0.957826
$437$	−36.0000	−1.72211
$438$	0	0
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	0	0
$441$	0	0
$442$	15.0000	0.713477
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	0	0
$445$	0	0
$446$	8.00000	0.378811
$447$	0	0
$448$	−2.00000	−0.0944911
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	0	0
$452$	15.0000	0.705541
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	28.0000	1.30978	0.654892	−	0.755722i	$-0.272714\pi$
$457$	28.0000	1.30978	0.654892	+	0.755722i	$0.272714\pi$
$458$	4.00000	0.186908
$459$	0	0
$460$	0	0
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	0	0
$463$	−38.0000	−1.76601	−0.883005	−	0.469364i	$-0.844483\pi$
$463$	−38.0000	−1.76601	−0.883005	+	0.469364i	$0.844483\pi$
$464$	−3.00000	−0.139272
$465$	0	0
$466$	6.00000	0.277945
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	0	0
$469$	−8.00000	−0.369406
$470$	0	0
$471$	0	0
$472$	−12.0000	−0.552345
$473$	3.00000	0.137940
$474$	0	0
$475$	0	0
$476$	−6.00000	−0.275010
$477$	0	0
$478$	6.00000	0.274434
$479$	36.0000	1.64488	0.822441	−	0.568850i	$-0.192612\pi$
$479$	36.0000	1.64488	0.822441	+	0.568850i	$0.192612\pi$
$480$	0	0
$481$	−50.0000	−2.27980
$482$	19.0000	0.865426
$483$	0	0
$484$	−2.00000	−0.0909091
$485$	0	0
$486$	0	0
$487$	−20.0000	−0.906287	−0.453143	−	0.891438i	$-0.649697\pi$
$487$	−20.0000	−0.906287	−0.453143	+	0.891438i	$0.649697\pi$
$488$	−2.00000	−0.0905357
$489$	0	0
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	0	0
$493$	−9.00000	−0.405340
$494$	−20.0000	−0.899843
$495$	0	0
$496$	5.00000	0.224507
$497$	−24.0000	−1.07655
$498$	0	0
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	0	0
$501$	0	0
$502$	9.00000	0.401690
$503$	9.00000	0.401290	0.200645	−	0.979664i	$-0.435696\pi$
$503$	9.00000	0.401290	0.200645	+	0.979664i	$0.435696\pi$
$504$	0	0
$505$	0	0
$506$	−27.0000	−1.20030
$507$	0	0
$508$	−2.00000	−0.0887357
$509$	9.00000	0.398918	0.199459	−	0.979906i	$-0.436082\pi$
$509$	9.00000	0.398918	0.199459	+	0.979906i	$0.436082\pi$
$510$	0	0
$511$	−20.0000	−0.884748
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−15.0000	−0.661622
$515$	0	0
$516$	0	0
$517$	−27.0000	−1.18746
$518$	20.0000	0.878750
$519$	0	0
$520$	0	0
$521$	−12.0000	−0.525730	−0.262865	−	0.964833i	$-0.584667\pi$
$521$	−12.0000	−0.525730	−0.262865	+	0.964833i	$0.584667\pi$
$522$	0	0
$523$	31.0000	1.35554	0.677768	−	0.735276i	$-0.262948\pi$
$523$	31.0000	1.35554	0.677768	+	0.735276i	$0.262948\pi$
$524$	−3.00000	−0.131056
$525$	0	0
$526$	12.0000	0.523225
$527$	15.0000	0.653410
$528$	0	0
$529$	58.0000	2.52174
$530$	0	0
$531$	0	0
$532$	8.00000	0.346844
$533$	0	0
$534$	0	0
$535$	0	0
$536$	−4.00000	−0.172774
$537$	0	0
$538$	9.00000	0.388018
$539$	−9.00000	−0.387657
$540$	0	0
$541$	−22.0000	−0.945854	−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000	−0.945854	−0.472927	+	0.881102i	$0.656803\pi$
$542$	−8.00000	−0.343629
$543$	0	0
$544$	−3.00000	−0.128624
$545$	0	0
$546$	0	0
$547$	13.0000	0.555840	0.277920	−	0.960604i	$-0.410355\pi$
$547$	13.0000	0.555840	0.277920	+	0.960604i	$0.410355\pi$
$548$	6.00000	0.256307
$549$	0	0
$550$	0	0
$551$	12.0000	0.511217
$552$	0	0
$553$	26.0000	1.10563
$554$	−10.0000	−0.424859
$555$	0	0
$556$	2.00000	0.0848189
$557$	−12.0000	−0.508456	−0.254228	−	0.967144i	$-0.581821\pi$
$557$	−12.0000	−0.508456	−0.254228	+	0.967144i	$0.581821\pi$
$558$	0	0
$559$	−5.00000	−0.211477
$560$	0	0
$561$	0	0
$562$	30.0000	1.26547
$563$	−6.00000	−0.252870	−0.126435	−	0.991975i	$-0.540353\pi$
$563$	−6.00000	−0.252870	−0.126435	+	0.991975i	$0.540353\pi$
$564$	0	0
$565$	0	0
$566$	32.0000	1.34506
$567$	0	0
$568$	−12.0000	−0.503509
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	−10.0000	−0.418487	−0.209243	−	0.977864i	$-0.567100\pi$
$571$	−10.0000	−0.418487	−0.209243	+	0.977864i	$0.567100\pi$
$572$	−15.0000	−0.627182
$573$	0	0
$574$	0	0
$575$	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	$0.732033\pi$
$578$	8.00000	0.332756
$579$	0	0
$580$	0	0
$581$	12.0000	0.497844
$582$	0	0
$583$	36.0000	1.49097
$584$	−10.0000	−0.413803
$585$	0	0
$586$	18.0000	0.743573
$587$	18.0000	0.742940	0.371470	−	0.928445i	$-0.378854\pi$
$587$	18.0000	0.742940	0.371470	+	0.928445i	$0.378854\pi$
$588$	0	0
$589$	−20.0000	−0.824086
$590$	0	0
$591$	0	0
$592$	10.0000	0.410997
$593$	27.0000	1.10876	0.554379	−	0.832265i	$-0.312956\pi$
$593$	27.0000	1.10876	0.554379	+	0.832265i	$0.312956\pi$
$594$	0	0
$595$	0	0
$596$	21.0000	0.860194
$597$	0	0
$598$	45.0000	1.84019
$599$	−30.0000	−1.22577	−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000	−1.22577	−0.612883	+	0.790173i	$0.709990\pi$
$600$	0	0
$601$	−19.0000	−0.775026	−0.387513	−	0.921864i	$-0.626666\pi$
$601$	−19.0000	−0.775026	−0.387513	+	0.921864i	$0.626666\pi$
$602$	2.00000	0.0815139
$603$	0	0
$604$	11.0000	0.447584
$605$	0	0
$606$	0	0
$607$	34.0000	1.38002	0.690009	−	0.723801i	$-0.257607\pi$
$607$	34.0000	1.38002	0.690009	+	0.723801i	$0.257607\pi$
$608$	4.00000	0.162221
$609$	0	0
$610$	0	0
$611$	45.0000	1.82051
$612$	0	0
$613$	19.0000	0.767403	0.383701	−	0.923457i	$-0.374649\pi$
$613$	19.0000	0.767403	0.383701	+	0.923457i	$0.374649\pi$
$614$	11.0000	0.443924
$615$	0	0
$616$	6.00000	0.241747
$617$	−3.00000	−0.120775	−0.0603877	−	0.998175i	$-0.519234\pi$
$617$	−3.00000	−0.120775	−0.0603877	+	0.998175i	$0.519234\pi$
$618$	0	0
$619$	−34.0000	−1.36658	−0.683288	−	0.730149i	$-0.739451\pi$
$619$	−34.0000	−1.36658	−0.683288	+	0.730149i	$0.739451\pi$
$620$	0	0
$621$	0	0
$622$	24.0000	0.962312
$623$	24.0000	0.961540
$624$	0	0
$625$	0	0
$626$	8.00000	0.319744
$627$	0	0
$628$	13.0000	0.518756
$629$	30.0000	1.19618
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	13.0000	0.517112
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	15.0000	0.594322
$638$	9.00000	0.356313
$639$	0	0
$640$	0	0
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	0	0
$643$	−23.0000	−0.907031	−0.453516	−	0.891248i	$-0.649830\pi$
$643$	−23.0000	−0.907031	−0.453516	+	0.891248i	$0.649830\pi$
$644$	−18.0000	−0.709299
$645$	0	0
$646$	12.0000	0.472134
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	0	0
$649$	36.0000	1.41312
$650$	0	0
$651$	0	0
$652$	−11.0000	−0.430793
$653$	−36.0000	−1.40879	−0.704394	−	0.709809i	$-0.748781\pi$
$653$	−36.0000	−1.40879	−0.704394	+	0.709809i	$0.748781\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	−18.0000	−0.701713
$659$	48.0000	1.86981	0.934907	−	0.354892i	$-0.115482\pi$
$659$	48.0000	1.86981	0.934907	+	0.354892i	$0.115482\pi$
$660$	0	0
$661$	2.00000	0.0777910	0.0388955	−	0.999243i	$-0.487616\pi$
$661$	2.00000	0.0777910	0.0388955	+	0.999243i	$0.487616\pi$
$662$	−14.0000	−0.544125
$663$	0	0
$664$	6.00000	0.232845
$665$	0	0
$666$	0	0
$667$	−27.0000	−1.04544
$668$	12.0000	0.464294
$669$	0	0
$670$	0	0
$671$	6.00000	0.231627
$672$	0	0
$673$	22.0000	0.848038	0.424019	−	0.905653i	$-0.360619\pi$
$673$	22.0000	0.848038	0.424019	+	0.905653i	$0.360619\pi$
$674$	32.0000	1.23259
$675$	0	0
$676$	12.0000	0.461538
$677$	−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
$679$	4.00000	0.153506
$680$	0	0
$681$	0	0
$682$	−15.0000	−0.574380
$683$	48.0000	1.83667	0.918334	−	0.395805i	$-0.129534\pi$
$683$	48.0000	1.83667	0.918334	+	0.395805i	$0.129534\pi$
$684$	0	0
$685$	0	0
$686$	−20.0000	−0.763604
$687$	0	0
$688$	1.00000	0.0381246
$689$	−60.0000	−2.28582
$690$	0	0
$691$	−10.0000	−0.380418	−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000	−0.380418	−0.190209	+	0.981744i	$0.560917\pi$
$692$	−18.0000	−0.684257
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	−14.0000	−0.529908
$699$	0	0
$700$	0	0
$701$	21.0000	0.793159	0.396580	−	0.918000i	$-0.370197\pi$
$701$	21.0000	0.793159	0.396580	+	0.918000i	$0.370197\pi$
$702$	0	0
$703$	−40.0000	−1.50863
$704$	3.00000	0.113067
$705$	0	0
$706$	3.00000	0.112906
$707$	−30.0000	−1.12827
$708$	0	0
$709$	20.0000	0.751116	0.375558	−	0.926799i	$-0.377451\pi$
$709$	20.0000	0.751116	0.375558	+	0.926799i	$0.377451\pi$
$710$	0	0
$711$	0	0
$712$	12.0000	0.449719
$713$	45.0000	1.68526
$714$	0	0
$715$	0	0
$716$	12.0000	0.448461
$717$	0	0
$718$	−12.0000	−0.447836
$719$	−30.0000	−1.11881	−0.559406	−	0.828894i	$-0.688971\pi$
$719$	−30.0000	−1.11881	−0.559406	+	0.828894i	$0.688971\pi$
$720$	0	0
$721$	4.00000	0.148968
$722$	3.00000	0.111648
$723$	0	0
$724$	20.0000	0.743294
$725$	0	0
$726$	0	0
$727$	−44.0000	−1.63187	−0.815935	−	0.578144i	$-0.803777\pi$
$727$	−44.0000	−1.63187	−0.815935	+	0.578144i	$0.803777\pi$
$728$	−10.0000	−0.370625
$729$	0	0
$730$	0	0
$731$	3.00000	0.110959
$732$	0	0
$733$	22.0000	0.812589	0.406294	−	0.913742i	$-0.366821\pi$
$733$	22.0000	0.812589	0.406294	+	0.913742i	$0.366821\pi$
$734$	26.0000	0.959678
$735$	0	0
$736$	−9.00000	−0.331744
$737$	12.0000	0.442026
$738$	0	0
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	0	0
$742$	24.0000	0.881068
$743$	27.0000	0.990534	0.495267	−	0.868741i	$-0.335070\pi$
$743$	27.0000	0.990534	0.495267	+	0.868741i	$0.335070\pi$
$744$	0	0
$745$	0	0
$746$	−1.00000	−0.0366126
$747$	0	0
$748$	9.00000	0.329073
$749$	12.0000	0.438470
$750$	0	0
$751$	−25.0000	−0.912263	−0.456131	−	0.889912i	$-0.650765\pi$
$751$	−25.0000	−0.912263	−0.456131	+	0.889912i	$0.650765\pi$
$752$	−9.00000	−0.328196
$753$	0	0
$754$	−15.0000	−0.546268
$755$	0	0
$756$	0	0
$757$	13.0000	0.472493	0.236247	−	0.971693i	$-0.424083\pi$
$757$	13.0000	0.472493	0.236247	+	0.971693i	$0.424083\pi$
$758$	16.0000	0.581146
$759$	0	0
$760$	0	0
$761$	−18.0000	−0.652499	−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000	−0.652499	−0.326250	+	0.945284i	$0.605785\pi$
$762$	0	0
$763$	−40.0000	−1.44810
$764$	6.00000	0.217072
$765$	0	0
$766$	−9.00000	−0.325183
$767$	−60.0000	−2.16647
$768$	0	0
$769$	23.0000	0.829401	0.414701	−	0.909958i	$-0.363886\pi$
$769$	23.0000	0.829401	0.414701	+	0.909958i	$0.363886\pi$
$770$	0	0
$771$	0	0
$772$	10.0000	0.359908
$773$	42.0000	1.51064	0.755318	−	0.655359i	$-0.227483\pi$
$773$	42.0000	1.51064	0.755318	+	0.655359i	$0.227483\pi$
$774$	0	0
$775$	0	0
$776$	2.00000	0.0717958
$777$	0	0
$778$	−9.00000	−0.322666
$779$	0	0
$780$	0	0
$781$	36.0000	1.28818
$782$	−27.0000	−0.965518
$783$	0	0
$784$	−3.00000	−0.107143
$785$	0	0
$786$	0	0
$787$	1.00000	0.0356462	0.0178231	−	0.999841i	$-0.494326\pi$
$787$	1.00000	0.0356462	0.0178231	+	0.999841i	$0.494326\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−30.0000	−1.06668
$792$	0	0
$793$	−10.0000	−0.355110
$794$	−7.00000	−0.248421
$795$	0	0
$796$	−7.00000	−0.248108
$797$	36.0000	1.27519	0.637593	−	0.770374i	$-0.279930\pi$
$797$	36.0000	1.27519	0.637593	+	0.770374i	$0.279930\pi$
$798$	0	0
$799$	−27.0000	−0.955191
$800$	0	0
$801$	0	0
$802$	−36.0000	−1.27120
$803$	30.0000	1.05868
$804$	0	0
$805$	0	0
$806$	25.0000	0.880587
$807$	0	0
$808$	−15.0000	−0.527698
$809$	24.0000	0.843795	0.421898	−	0.906644i	$-0.361364\pi$
$809$	24.0000	0.843795	0.421898	+	0.906644i	$0.361364\pi$
$810$	0	0
$811$	38.0000	1.33436	0.667180	−	0.744896i	$-0.267501\pi$
$811$	38.0000	1.33436	0.667180	+	0.744896i	$0.267501\pi$
$812$	6.00000	0.210559
$813$	0	0
$814$	−30.0000	−1.05150
$815$	0	0
$816$	0	0
$817$	−4.00000	−0.139942
$818$	−23.0000	−0.804176
$819$	0	0
$820$	0	0
$821$	−42.0000	−1.46581	−0.732905	−	0.680331i	$-0.761836\pi$
$821$	−42.0000	−1.46581	−0.732905	+	0.680331i	$0.761836\pi$
$822$	0	0
$823$	4.00000	0.139431	0.0697156	−	0.997567i	$-0.477791\pi$
$823$	4.00000	0.139431	0.0697156	+	0.997567i	$0.477791\pi$
$824$	2.00000	0.0696733
$825$	0	0
$826$	24.0000	0.835067
$827$	42.0000	1.46048	0.730242	−	0.683189i	$-0.239408\pi$
$827$	42.0000	1.46048	0.730242	+	0.683189i	$0.239408\pi$
$828$	0	0
$829$	44.0000	1.52818	0.764092	−	0.645108i	$-0.223188\pi$
$829$	44.0000	1.52818	0.764092	+	0.645108i	$0.223188\pi$
$830$	0	0
$831$	0	0
$832$	−5.00000	−0.173344
$833$	−9.00000	−0.311832
$834$	0	0
$835$	0	0
$836$	−12.0000	−0.415029
$837$	0	0
$838$	3.00000	0.103633
$839$	6.00000	0.207143	0.103572	−	0.994622i	$-0.466973\pi$
$839$	6.00000	0.207143	0.103572	+	0.994622i	$0.466973\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	4.00000	0.137849
$843$	0	0
$844$	−22.0000	−0.757271
$845$	0	0
$846$	0	0
$847$	4.00000	0.137442
$848$	12.0000	0.412082
$849$	0	0
$850$	0	0
$851$	90.0000	3.08516
$852$	0	0
$853$	19.0000	0.650548	0.325274	−	0.945620i	$-0.394544\pi$
$853$	19.0000	0.650548	0.325274	+	0.945620i	$0.394544\pi$
$854$	4.00000	0.136877
$855$	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
$862$	0	0
$863$	−21.0000	−0.714848	−0.357424	−	0.933942i	$-0.616345\pi$
$863$	−21.0000	−0.714848	−0.357424	+	0.933942i	$0.616345\pi$
$864$	0	0
$865$	0	0
$866$	2.00000	0.0679628
$867$	0	0
$868$	−10.0000	−0.339422
$869$	−39.0000	−1.32298
$870$	0	0
$871$	−20.0000	−0.677674
$872$	−20.0000	−0.677285
$873$	0	0
$874$	36.0000	1.21772
$875$	0	0
$876$	0	0
$877$	1.00000	0.0337676	0.0168838	−	0.999857i	$-0.494625\pi$
$877$	1.00000	0.0337676	0.0168838	+	0.999857i	$0.494625\pi$
$878$	−8.00000	−0.269987
$879$	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
$883$	4.00000	0.134611	0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000	0.134611	0.0673054	+	0.997732i	$0.478560\pi$
$884$	−15.0000	−0.504505
$885$	0	0
$886$	−12.0000	−0.403148
$887$	3.00000	0.100730	0.0503651	−	0.998731i	$-0.483962\pi$
$887$	3.00000	0.100730	0.0503651	+	0.998731i	$0.483962\pi$
$888$	0	0
$889$	4.00000	0.134156
$890$	0	0
$891$	0	0
$892$	−8.00000	−0.267860
$893$	36.0000	1.20469
$894$	0	0
$895$	0	0
$896$	2.00000	0.0668153
$897$	0	0
$898$	6.00000	0.200223
$899$	−15.0000	−0.500278
$900$	0	0
$901$	36.0000	1.19933
$902$	0	0
$903$	0	0
$904$	−15.0000	−0.498893
$905$	0	0
$906$	0	0
$907$	−59.0000	−1.95906	−0.979531	−	0.201291i	$-0.935486\pi$
$907$	−59.0000	−1.95906	−0.979531	+	0.201291i	$0.935486\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	−18.0000	−0.595713
$914$	−28.0000	−0.926158
$915$	0	0
$916$	−4.00000	−0.132164
$917$	6.00000	0.198137
$918$	0	0
$919$	−25.0000	−0.824674	−0.412337	−	0.911031i	$-0.635287\pi$
$919$	−25.0000	−0.824674	−0.412337	+	0.911031i	$0.635287\pi$
$920$	0	0
$921$	0	0
$922$	−18.0000	−0.592798
$923$	−60.0000	−1.97492
$924$	0	0
$925$	0	0
$926$	38.0000	1.24876
$927$	0	0
$928$	3.00000	0.0984798
$929$	36.0000	1.18112	0.590561	−	0.806993i	$-0.298907\pi$
$929$	36.0000	1.18112	0.590561	+	0.806993i	$0.298907\pi$
$930$	0	0
$931$	12.0000	0.393284
$932$	−6.00000	−0.196537
$933$	0	0
$934$	12.0000	0.392652
$935$	0	0
$936$	0	0
$937$	22.0000	0.718709	0.359354	−	0.933201i	$-0.382997\pi$
$937$	22.0000	0.718709	0.359354	+	0.933201i	$0.382997\pi$
$938$	8.00000	0.261209
$939$	0	0
$940$	0	0
$941$	33.0000	1.07577	0.537885	−	0.843018i	$-0.319224\pi$
$941$	33.0000	1.07577	0.537885	+	0.843018i	$0.319224\pi$
$942$	0	0
$943$	0	0
$944$	12.0000	0.390567
$945$	0	0
$946$	−3.00000	−0.0975384
$947$	42.0000	1.36482	0.682408	−	0.730971i	$-0.260933\pi$
$947$	42.0000	1.36482	0.682408	+	0.730971i	$0.260933\pi$
$948$	0	0
$949$	−50.0000	−1.62307
$950$	0	0
$951$	0	0
$952$	6.00000	0.194461
$953$	−15.0000	−0.485898	−0.242949	−	0.970039i	$-0.578115\pi$
$953$	−15.0000	−0.485898	−0.242949	+	0.970039i	$0.578115\pi$
$954$	0	0
$955$	0	0
$956$	−6.00000	−0.194054
$957$	0	0
$958$	−36.0000	−1.16311
$959$	−12.0000	−0.387500
$960$	0	0
$961$	−6.00000	−0.193548
$962$	50.0000	1.61206
$963$	0	0
$964$	−19.0000	−0.611949
$965$	0	0
$966$	0	0
$967$	−2.00000	−0.0643157	−0.0321578	−	0.999483i	$-0.510238\pi$
$967$	−2.00000	−0.0643157	−0.0321578	+	0.999483i	$0.510238\pi$
$968$	2.00000	0.0642824
$969$	0	0
$970$	0	0
$971$	−57.0000	−1.82922	−0.914609	−	0.404341i	$-0.867501\pi$
$971$	−57.0000	−1.82922	−0.914609	+	0.404341i	$0.867501\pi$
$972$	0	0
$973$	−4.00000	−0.128234
$974$	20.0000	0.640841
$975$	0	0
$976$	2.00000	0.0640184
$977$	−33.0000	−1.05576	−0.527882	−	0.849318i	$-0.677014\pi$
$977$	−33.0000	−1.05576	−0.527882	+	0.849318i	$0.677014\pi$
$978$	0	0
$979$	−36.0000	−1.15056
$980$	0	0
$981$	0	0
$982$	12.0000	0.382935
$983$	−39.0000	−1.24391	−0.621953	−	0.783054i	$-0.713661\pi$
$983$	−39.0000	−1.24391	−0.621953	+	0.783054i	$0.713661\pi$
$984$	0	0
$985$	0	0
$986$	9.00000	0.286618
$987$	0	0
$988$	20.0000	0.636285
$989$	9.00000	0.286183
$990$	0	0
$991$	5.00000	0.158830	0.0794151	−	0.996842i	$-0.474695\pi$
$991$	5.00000	0.158830	0.0794151	+	0.996842i	$0.474695\pi$
$992$	−5.00000	−0.158750
$993$	0	0
$994$	24.0000	0.761234
$995$	0	0
$996$	0	0
$997$	37.0000	1.17180	0.585901	−	0.810383i	$-0.300741\pi$
$997$	37.0000	1.17180	0.585901	+	0.810383i	$0.300741\pi$
$998$	4.00000	0.126618
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1350.2.a.d.1.1		1
3.2	odd	2		1350.2.a.o.1.1		1
5.2	odd	4		1350.2.c.j.649.1		2
5.3	odd	4		1350.2.c.j.649.2		2
5.4	even	2		270.2.a.c.1.1	yes	1
15.2	even	4		1350.2.c.c.649.2		2
15.8	even	4		1350.2.c.c.649.1		2
15.14	odd	2		270.2.a.b.1.1	✓	1
20.19	odd	2		2160.2.a.b.1.1		1
40.19	odd	2		8640.2.a.bn.1.1		1
40.29	even	2		8640.2.a.bx.1.1		1
45.4	even	6		810.2.e.d.541.1		2
45.14	odd	6		810.2.e.i.541.1		2
45.29	odd	6		810.2.e.i.271.1		2
45.34	even	6		810.2.e.d.271.1		2
60.59	even	2		2160.2.a.q.1.1		1
120.29	odd	2		8640.2.a.y.1.1		1
120.59	even	2		8640.2.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.2.a.b.1.1	✓	1	15.14	odd	2
270.2.a.c.1.1	yes	1	5.4	even	2
810.2.e.d.271.1		2	45.34	even	6
810.2.e.d.541.1		2	45.4	even	6
810.2.e.i.271.1		2	45.29	odd	6
810.2.e.i.541.1		2	45.14	odd	6
1350.2.a.d.1.1		1	1.1	even	1	trivial
1350.2.a.o.1.1		1	3.2	odd	2
1350.2.c.c.649.1		2	15.8	even	4
1350.2.c.c.649.2		2	15.2	even	4
1350.2.c.j.649.1		2	5.2	odd	4
1350.2.c.j.649.2		2	5.3	odd	4
2160.2.a.b.1.1		1	20.19	odd	2
2160.2.a.q.1.1		1	60.59	even	2
8640.2.a.e.1.1		1	120.59	even	2
8640.2.a.y.1.1		1	120.29	odd	2
8640.2.a.bn.1.1		1	40.19	odd	2
8640.2.a.bx.1.1		1	40.29	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1350.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more