LMFDB - Embedded newform 450.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,2,Mod(1,450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(450, 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("450.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	450.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:	$3.59326809096$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 30)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

$q-1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{7} -1.00000 q^{8} -2.00000 q^{11} -6.00000 q^{13} +2.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} +2.00000 q^{22} -4.00000 q^{23} +6.00000 q^{26} -2.00000 q^{28} -8.00000 q^{31} -1.00000 q^{32} -2.00000 q^{34} -2.00000 q^{37} -2.00000 q^{41} +4.00000 q^{43} -2.00000 q^{44} +4.00000 q^{46} -8.00000 q^{47} -3.00000 q^{49} -6.00000 q^{52} +6.00000 q^{53} +2.00000 q^{56} -10.0000 q^{59} +2.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} +8.00000 q^{67} +2.00000 q^{68} -12.0000 q^{71} +4.00000 q^{73} +2.00000 q^{74} +4.00000 q^{77} +2.00000 q^{82} -4.00000 q^{83} -4.00000 q^{86} +2.00000 q^{88} +10.0000 q^{89} +12.0000 q^{91} -4.00000 q^{92} +8.00000 q^{94} +8.00000 q^{97} +3.00000 q^{98} +O(q^{100})$

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
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	2.00000	0.426401
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	0	0
$26$	6.00000	1.17670
$27$	0	0
$28$	−2.00000	−0.377964
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−2.00000	−0.342997
$35$	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	$0.552568\pi$
$38$	0	0
$39$	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
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	4.00000	0.589768
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	0	0
$52$	−6.00000	−0.832050
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	0	0
$56$	2.00000	0.267261
$57$	0	0
$58$	0	0
$59$	−10.0000	−1.30189	−0.650945	−	0.759125i	$-0.725627\pi$
$59$	−10.0000	−1.30189	−0.650945	+	0.759125i	$0.725627\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	8.00000	1.01600
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	2.00000	0.242536
$69$	0	0
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	0	0
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	0	0
$77$	4.00000	0.455842
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	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
$86$	−4.00000	−0.431331
$87$	0	0
$88$	2.00000	0.213201
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	12.0000	1.25794
$92$	−4.00000	−0.417029
$93$	0	0
$94$	8.00000	0.825137
$95$	0	0
$96$	0	0
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	3.00000	0.303046
$99$	0	0
$100$	0	0
$101$	8.00000	0.796030	0.398015	−	0.917379i	$-0.369699\pi$
$101$	8.00000	0.796030	0.398015	+	0.917379i	$0.369699\pi$
$102$	0	0
$103$	14.0000	1.37946	0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000	1.37946	0.689730	+	0.724066i	$0.257729\pi$
$104$	6.00000	0.588348
$105$	0	0
$106$	−6.00000	−0.582772
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	0	0
$112$	−2.00000	−0.188982
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	10.0000	0.920575
$119$	−4.00000	−0.366679
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−2.00000	−0.181071
$123$	0	0
$124$	−8.00000	−0.718421
$125$	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
$131$	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
$134$	−8.00000	−0.691095
$135$	0	0
$136$	−2.00000	−0.171499
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	0	0
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	0	0
$141$	0	0
$142$	12.0000	1.00702
$143$	12.0000	1.00349
$144$	0	0
$145$	0	0
$146$	−4.00000	−0.331042
$147$	0	0
$148$	−2.00000	−0.164399
$149$	20.0000	1.63846	0.819232	−	0.573462i	$-0.194400\pi$
$149$	20.0000	1.63846	0.819232	+	0.573462i	$0.194400\pi$
$150$	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
$154$	−4.00000	−0.322329
$155$	0	0
$156$	0	0
$157$	−22.0000	−1.75579	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−22.0000	−1.75579	−0.877896	+	0.478852i	$0.841053\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	8.00000	0.630488
$162$	0	0
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	4.00000	0.310460
$167$	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
$169$	23.0000	1.76923
$170$	0	0
$171$	0	0
$172$	4.00000	0.304997
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	0	0
$175$	0	0
$176$	−2.00000	−0.150756
$177$	0	0
$178$	−10.0000	−0.749532
$179$	−10.0000	−0.747435	−0.373718	−	0.927543i	$-0.621917\pi$
$179$	−10.0000	−0.747435	−0.373718	+	0.927543i	$0.621917\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	−12.0000	−0.889499
$183$	0	0
$184$	4.00000	0.294884
$185$	0	0
$186$	0	0
$187$	−4.00000	−0.292509
$188$	−8.00000	−0.583460
$189$	0	0
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	0	0
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	−8.00000	−0.574367
$195$	0	0
$196$	−3.00000	−0.214286
$197$	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
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	0	0
$202$	−8.00000	−0.562878
$203$	0	0
$204$	0	0
$205$	0	0
$206$	−14.0000	−0.975426
$207$	0	0
$208$	−6.00000	−0.416025
$209$	0	0
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	−12.0000	−0.820303
$215$	0	0
$216$	0	0
$217$	16.0000	1.08615
$218$	−10.0000	−0.677285
$219$	0	0
$220$	0	0
$221$	−12.0000	−0.807207
$222$	0	0
$223$	−26.0000	−1.74109	−0.870544	−	0.492090i	$-0.836233\pi$
$223$	−26.0000	−1.74109	−0.870544	+	0.492090i	$0.836233\pi$
$224$	2.00000	0.133631
$225$	0	0
$226$	−6.00000	−0.399114
$227$	−28.0000	−1.85843	−0.929213	−	0.369546i	$-0.879513\pi$
$227$	−28.0000	−1.85843	−0.929213	+	0.369546i	$0.879513\pi$
$228$	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
$232$	0	0
$233$	−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
$236$	−10.0000	−0.650945
$237$	0	0
$238$	4.00000	0.259281
$239$	−20.0000	−1.29369	−0.646846	−	0.762620i	$-0.723912\pi$
$239$	−20.0000	−1.29369	−0.646846	+	0.762620i	$0.723912\pi$
$240$	0	0
$241$	22.0000	1.41714	0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000	1.41714	0.708572	+	0.705638i	$0.249340\pi$
$242$	7.00000	0.449977
$243$	0	0
$244$	2.00000	0.128037
$245$	0	0
$246$	0	0
$247$	0	0
$248$	8.00000	0.508001
$249$	0	0
$250$	0	0
$251$	18.0000	1.13615	0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000	1.13615	0.568075	+	0.822977i	$0.307688\pi$
$252$	0	0
$253$	8.00000	0.502956
$254$	2.00000	0.125491
$255$	0	0
$256$	1.00000	0.0625000
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	4.00000	0.248548
$260$	0	0
$261$	0	0
$262$	−18.0000	−1.11204
$263$	−4.00000	−0.246651	−0.123325	−	0.992366i	$-0.539356\pi$
$263$	−4.00000	−0.246651	−0.123325	+	0.992366i	$0.539356\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	8.00000	0.488678
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	2.00000	0.121268
$273$	0	0
$274$	18.0000	1.08742
$275$	0	0
$276$	0	0
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	20.0000	1.19952
$279$	0	0
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	−16.0000	−0.951101	−0.475551	−	0.879688i	$-0.657751\pi$
$283$	−16.0000	−0.951101	−0.475551	+	0.879688i	$0.657751\pi$
$284$	−12.0000	−0.712069
$285$	0	0
$286$	−12.0000	−0.709575
$287$	4.00000	0.236113
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	0	0
$292$	4.00000	0.234082
$293$	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
$298$	−20.0000	−1.15857
$299$	24.0000	1.38796
$300$	0	0
$301$	−8.00000	−0.461112
$302$	8.00000	0.460348
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	4.00000	0.227921
$309$	0	0
$310$	0	0
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	4.00000	0.226093	0.113047	−	0.993590i	$-0.463939\pi$
$313$	4.00000	0.226093	0.113047	+	0.993590i	$0.463939\pi$
$314$	22.0000	1.24153
$315$	0	0
$316$	0	0
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	−8.00000	−0.445823
$323$	0	0
$324$	0	0
$325$	0	0
$326$	16.0000	0.886158
$327$	0	0
$328$	2.00000	0.110432
$329$	16.0000	0.882109
$330$	0	0
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	−4.00000	−0.219529
$333$	0	0
$334$	−12.0000	−0.656611
$335$	0	0
$336$	0	0
$337$	28.0000	1.52526	0.762629	−	0.646837i	$-0.223908\pi$
$337$	28.0000	1.52526	0.762629	+	0.646837i	$0.223908\pi$
$338$	−23.0000	−1.25104
$339$	0	0
$340$	0	0
$341$	16.0000	0.866449
$342$	0	0
$343$	20.0000	1.07990
$344$	−4.00000	−0.215666
$345$	0	0
$346$	14.0000	0.752645
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	0	0
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	0	0
$352$	2.00000	0.106600
$353$	−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
$355$	0	0
$356$	10.0000	0.529999
$357$	0	0
$358$	10.0000	0.528516
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	−2.00000	−0.105118
$363$	0	0
$364$	12.0000	0.628971
$365$	0	0
$366$	0	0
$367$	−2.00000	−0.104399	−0.0521996	−	0.998637i	$-0.516623\pi$
$367$	−2.00000	−0.104399	−0.0521996	+	0.998637i	$0.516623\pi$
$368$	−4.00000	−0.208514
$369$	0	0
$370$	0	0
$371$	−12.0000	−0.623009
$372$	0	0
$373$	−6.00000	−0.310668	−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000	−0.310668	−0.155334	+	0.987862i	$0.549645\pi$
$374$	4.00000	0.206835
$375$	0	0
$376$	8.00000	0.412568
$377$	0	0
$378$	0	0
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	0	0
$381$	0	0
$382$	12.0000	0.613973
$383$	16.0000	0.817562	0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000	0.817562	0.408781	+	0.912633i	$0.365954\pi$
$384$	0	0
$385$	0	0
$386$	−4.00000	−0.203595
$387$	0	0
$388$	8.00000	0.406138
$389$	−20.0000	−1.01404	−0.507020	−	0.861934i	$-0.669253\pi$
$389$	−20.0000	−1.01404	−0.507020	+	0.861934i	$0.669253\pi$
$390$	0	0
$391$	−8.00000	−0.404577
$392$	3.00000	0.151523
$393$	0	0
$394$	−22.0000	−1.10834
$395$	0	0
$396$	0	0
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−22.0000	−1.09863	−0.549314	−	0.835616i	$-0.685111\pi$
$401$	−22.0000	−1.09863	−0.549314	+	0.835616i	$0.685111\pi$
$402$	0	0
$403$	48.0000	2.39105
$404$	8.00000	0.398015
$405$	0	0
$406$	0	0
$407$	4.00000	0.198273
$408$	0	0
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	0	0
$411$	0	0
$412$	14.0000	0.689730
$413$	20.0000	0.984136
$414$	0	0
$415$	0	0
$416$	6.00000	0.294174
$417$	0	0
$418$	0	0
$419$	10.0000	0.488532	0.244266	−	0.969708i	$-0.421453\pi$
$419$	10.0000	0.488532	0.244266	+	0.969708i	$0.421453\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	−12.0000	−0.584151
$423$	0	0
$424$	−6.00000	−0.291386
$425$	0	0
$426$	0	0
$427$	−4.00000	−0.193574
$428$	12.0000	0.580042
$429$	0	0
$430$	0	0
$431$	−32.0000	−1.54139	−0.770693	−	0.637207i	$-0.780090\pi$
$431$	−32.0000	−1.54139	−0.770693	+	0.637207i	$0.780090\pi$
$432$	0	0
$433$	4.00000	0.192228	0.0961139	−	0.995370i	$-0.469359\pi$
$433$	4.00000	0.192228	0.0961139	+	0.995370i	$0.469359\pi$
$434$	−16.0000	−0.768025
$435$	0	0
$436$	10.0000	0.478913
$437$	0	0
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	0	0
$442$	12.0000	0.570782
$443$	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
$445$	0	0
$446$	26.0000	1.23114
$447$	0	0
$448$	−2.00000	−0.0944911
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	0	0
$451$	4.00000	0.188353
$452$	6.00000	0.282216
$453$	0	0
$454$	28.0000	1.31411
$455$	0	0
$456$	0	0
$457$	−32.0000	−1.49690	−0.748448	−	0.663193i	$-0.769201\pi$
$457$	−32.0000	−1.49690	−0.748448	+	0.663193i	$0.769201\pi$
$458$	10.0000	0.467269
$459$	0	0
$460$	0	0
$461$	−12.0000	−0.558896	−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000	−0.558896	−0.279448	+	0.960161i	$0.590151\pi$
$462$	0	0
$463$	−6.00000	−0.278844	−0.139422	−	0.990233i	$-0.544524\pi$
$463$	−6.00000	−0.278844	−0.139422	+	0.990233i	$0.544524\pi$
$464$	0	0
$465$	0	0
$466$	14.0000	0.648537
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	0	0
$469$	−16.0000	−0.738811
$470$	0	0
$471$	0	0
$472$	10.0000	0.460287
$473$	−8.00000	−0.367840
$474$	0	0
$475$	0	0
$476$	−4.00000	−0.183340
$477$	0	0
$478$	20.0000	0.914779
$479$	−20.0000	−0.913823	−0.456912	−	0.889512i	$-0.651044\pi$
$479$	−20.0000	−0.913823	−0.456912	+	0.889512i	$0.651044\pi$
$480$	0	0
$481$	12.0000	0.547153
$482$	−22.0000	−1.00207
$483$	0	0
$484$	−7.00000	−0.318182
$485$	0	0
$486$	0	0
$487$	18.0000	0.815658	0.407829	−	0.913058i	$-0.366286\pi$
$487$	18.0000	0.815658	0.407829	+	0.913058i	$0.366286\pi$
$488$	−2.00000	−0.0905357
$489$	0	0
$490$	0	0
$491$	18.0000	0.812329	0.406164	−	0.913800i	$-0.366866\pi$
$491$	18.0000	0.812329	0.406164	+	0.913800i	$0.366866\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	−8.00000	−0.359211
$497$	24.0000	1.07655
$498$	0	0
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	0	0
$501$	0	0
$502$	−18.0000	−0.803379
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	0	0
$505$	0	0
$506$	−8.00000	−0.355643
$507$	0	0
$508$	−2.00000	−0.0887357
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	−8.00000	−0.353899
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	18.0000	0.793946
$515$	0	0
$516$	0	0
$517$	16.0000	0.703679
$518$	−4.00000	−0.175750
$519$	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
$523$	−16.0000	−0.699631	−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000	−0.699631	−0.349816	+	0.936819i	$0.613756\pi$
$524$	18.0000	0.786334
$525$	0	0
$526$	4.00000	0.174408
$527$	−16.0000	−0.696971
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	0	0
$533$	12.0000	0.519778
$534$	0	0
$535$	0	0
$536$	−8.00000	−0.345547
$537$	0	0
$538$	0	0
$539$	6.00000	0.258438
$540$	0	0
$541$	−38.0000	−1.63375	−0.816874	−	0.576816i	$-0.804295\pi$
$541$	−38.0000	−1.63375	−0.816874	+	0.576816i	$0.804295\pi$
$542$	8.00000	0.343629
$543$	0	0
$544$	−2.00000	−0.0857493
$545$	0	0
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	−18.0000	−0.768922
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	2.00000	0.0849719
$555$	0	0
$556$	−20.0000	−0.848189
$557$	−18.0000	−0.762684	−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000	−0.762684	−0.381342	+	0.924434i	$0.624538\pi$
$558$	0	0
$559$	−24.0000	−1.01509
$560$	0	0
$561$	0	0
$562$	−18.0000	−0.759284
$563$	−44.0000	−1.85438	−0.927189	−	0.374593i	$-0.877783\pi$
$563$	−44.0000	−1.85438	−0.927189	+	0.374593i	$0.877783\pi$
$564$	0	0
$565$	0	0
$566$	16.0000	0.672530
$567$	0	0
$568$	12.0000	0.503509
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	0	0
$571$	−8.00000	−0.334790	−0.167395	−	0.985890i	$-0.553535\pi$
$571$	−8.00000	−0.334790	−0.167395	+	0.985890i	$0.553535\pi$
$572$	12.0000	0.501745
$573$	0	0
$574$	−4.00000	−0.166957
$575$	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	$0.732033\pi$
$578$	13.0000	0.540729
$579$	0	0
$580$	0	0
$581$	8.00000	0.331896
$582$	0	0
$583$	−12.0000	−0.496989
$584$	−4.00000	−0.165521
$585$	0	0
$586$	−6.00000	−0.247858
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	−2.00000	−0.0821995
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	0	0
$595$	0	0
$596$	20.0000	0.819232
$597$	0	0
$598$	−24.0000	−0.981433
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	8.00000	0.326056
$603$	0	0
$604$	−8.00000	−0.325515
$605$	0	0
$606$	0	0
$607$	−22.0000	−0.892952	−0.446476	−	0.894795i	$-0.647321\pi$
$607$	−22.0000	−0.892952	−0.446476	+	0.894795i	$0.647321\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	48.0000	1.94187
$612$	0	0
$613$	−26.0000	−1.05013	−0.525065	−	0.851062i	$-0.675959\pi$
$613$	−26.0000	−1.05013	−0.525065	+	0.851062i	$0.675959\pi$
$614$	12.0000	0.484281
$615$	0	0
$616$	−4.00000	−0.161165
$617$	2.00000	0.0805170	0.0402585	−	0.999189i	$-0.487182\pi$
$617$	2.00000	0.0805170	0.0402585	+	0.999189i	$0.487182\pi$
$618$	0	0
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	0	0
$621$	0	0
$622$	12.0000	0.481156
$623$	−20.0000	−0.801283
$624$	0	0
$625$	0	0
$626$	−4.00000	−0.159872
$627$	0	0
$628$	−22.0000	−0.877896
$629$	−4.00000	−0.159490
$630$	0	0
$631$	32.0000	1.27390	0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000	1.27390	0.636950	+	0.770905i	$0.280196\pi$
$632$	0	0
$633$	0	0
$634$	−2.00000	−0.0794301
$635$	0	0
$636$	0	0
$637$	18.0000	0.713186
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	0	0
$643$	24.0000	0.946468	0.473234	−	0.880937i	$-0.343087\pi$
$643$	24.0000	0.946468	0.473234	+	0.880937i	$0.343087\pi$
$644$	8.00000	0.315244
$645$	0	0
$646$	0	0
$647$	−48.0000	−1.88707	−0.943537	−	0.331266i	$-0.892524\pi$
$647$	−48.0000	−1.88707	−0.943537	+	0.331266i	$0.892524\pi$
$648$	0	0
$649$	20.0000	0.785069
$650$	0	0
$651$	0	0
$652$	−16.0000	−0.626608
$653$	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
$658$	−16.0000	−0.623745
$659$	50.0000	1.94772	0.973862	−	0.227142i	$-0.0729380\pi$
$659$	50.0000	1.94772	0.973862	+	0.227142i	$0.0729380\pi$
$660$	0	0
$661$	2.00000	0.0777910	0.0388955	−	0.999243i	$-0.487616\pi$
$661$	2.00000	0.0777910	0.0388955	+	0.999243i	$0.487616\pi$
$662$	8.00000	0.310929
$663$	0	0
$664$	4.00000	0.155230
$665$	0	0
$666$	0	0
$667$	0	0
$668$	12.0000	0.464294
$669$	0	0
$670$	0	0
$671$	−4.00000	−0.154418
$672$	0	0
$673$	−36.0000	−1.38770	−0.693849	−	0.720121i	$-0.744086\pi$
$673$	−36.0000	−1.38770	−0.693849	+	0.720121i	$0.744086\pi$
$674$	−28.0000	−1.07852
$675$	0	0
$676$	23.0000	0.884615
$677$	2.00000	0.0768662	0.0384331	−	0.999261i	$-0.487763\pi$
$677$	2.00000	0.0768662	0.0384331	+	0.999261i	$0.487763\pi$
$678$	0	0
$679$	−16.0000	−0.614024
$680$	0	0
$681$	0	0
$682$	−16.0000	−0.612672
$683$	−4.00000	−0.153056	−0.0765279	−	0.997067i	$-0.524383\pi$
$683$	−4.00000	−0.153056	−0.0765279	+	0.997067i	$0.524383\pi$
$684$	0	0
$685$	0	0
$686$	−20.0000	−0.763604
$687$	0	0
$688$	4.00000	0.152499
$689$	−36.0000	−1.37149
$690$	0	0
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	−14.0000	−0.532200
$693$	0	0
$694$	−12.0000	−0.455514
$695$	0	0
$696$	0	0
$697$	−4.00000	−0.151511
$698$	−10.0000	−0.378506
$699$	0	0
$700$	0	0
$701$	−32.0000	−1.20862	−0.604312	−	0.796748i	$-0.706552\pi$
$701$	−32.0000	−1.20862	−0.604312	+	0.796748i	$0.706552\pi$
$702$	0	0
$703$	0	0
$704$	−2.00000	−0.0753778
$705$	0	0
$706$	14.0000	0.526897
$707$	−16.0000	−0.601742
$708$	0	0
$709$	30.0000	1.12667	0.563337	−	0.826227i	$-0.309517\pi$
$709$	30.0000	1.12667	0.563337	+	0.826227i	$0.309517\pi$
$710$	0	0
$711$	0	0
$712$	−10.0000	−0.374766
$713$	32.0000	1.19841
$714$	0	0
$715$	0	0
$716$	−10.0000	−0.373718
$717$	0	0
$718$	0	0
$719$	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
$721$	−28.0000	−1.04277
$722$	19.0000	0.707107
$723$	0	0
$724$	2.00000	0.0743294
$725$	0	0
$726$	0	0
$727$	18.0000	0.667583	0.333792	−	0.942647i	$-0.391672\pi$
$727$	18.0000	0.667583	0.333792	+	0.942647i	$0.391672\pi$
$728$	−12.0000	−0.444750
$729$	0	0
$730$	0	0
$731$	8.00000	0.295891
$732$	0	0
$733$	14.0000	0.517102	0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000	0.517102	0.258551	+	0.965998i	$0.416755\pi$
$734$	2.00000	0.0738213
$735$	0	0
$736$	4.00000	0.147442
$737$	−16.0000	−0.589368
$738$	0	0
$739$	40.0000	1.47142	0.735712	−	0.677295i	$-0.236848\pi$
$739$	40.0000	1.47142	0.735712	+	0.677295i	$0.236848\pi$
$740$	0	0
$741$	0	0
$742$	12.0000	0.440534
$743$	−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
$745$	0	0
$746$	6.00000	0.219676
$747$	0	0
$748$	−4.00000	−0.146254
$749$	−24.0000	−0.876941
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	−8.00000	−0.291730
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	−20.0000	−0.726433
$759$	0	0
$760$	0	0
$761$	18.0000	0.652499	0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000	0.652499	0.326250	+	0.945284i	$0.394215\pi$
$762$	0	0
$763$	−20.0000	−0.724049
$764$	−12.0000	−0.434145
$765$	0	0
$766$	−16.0000	−0.578103
$767$	60.0000	2.16647
$768$	0	0
$769$	−30.0000	−1.08183	−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000	−1.08183	−0.540914	+	0.841078i	$0.681921\pi$
$770$	0	0
$771$	0	0
$772$	4.00000	0.143963
$773$	−54.0000	−1.94225	−0.971123	−	0.238581i	$-0.923318\pi$
$773$	−54.0000	−1.94225	−0.971123	+	0.238581i	$0.923318\pi$
$774$	0	0
$775$	0	0
$776$	−8.00000	−0.287183
$777$	0	0
$778$	20.0000	0.717035
$779$	0	0
$780$	0	0
$781$	24.0000	0.858788
$782$	8.00000	0.286079
$783$	0	0
$784$	−3.00000	−0.107143
$785$	0	0
$786$	0	0
$787$	−32.0000	−1.14068	−0.570338	−	0.821410i	$-0.693188\pi$
$787$	−32.0000	−1.14068	−0.570338	+	0.821410i	$0.693188\pi$
$788$	22.0000	0.783718
$789$	0	0
$790$	0	0
$791$	−12.0000	−0.426671
$792$	0	0
$793$	−12.0000	−0.426132
$794$	2.00000	0.0709773
$795$	0	0
$796$	0	0
$797$	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
$799$	−16.0000	−0.566039
$800$	0	0
$801$	0	0
$802$	22.0000	0.776847
$803$	−8.00000	−0.282314
$804$	0	0
$805$	0	0
$806$	−48.0000	−1.69073
$807$	0	0
$808$	−8.00000	−0.281439
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	0	0
$811$	52.0000	1.82597	0.912983	−	0.407997i	$-0.133772\pi$
$811$	52.0000	1.82597	0.912983	+	0.407997i	$0.133772\pi$
$812$	0	0
$813$	0	0
$814$	−4.00000	−0.140200
$815$	0	0
$816$	0	0
$817$	0	0
$818$	10.0000	0.349642
$819$	0	0
$820$	0	0
$821$	8.00000	0.279202	0.139601	−	0.990208i	$-0.455418\pi$
$821$	8.00000	0.279202	0.139601	+	0.990208i	$0.455418\pi$
$822$	0	0
$823$	−6.00000	−0.209147	−0.104573	−	0.994517i	$-0.533348\pi$
$823$	−6.00000	−0.209147	−0.104573	+	0.994517i	$0.533348\pi$
$824$	−14.0000	−0.487713
$825$	0	0
$826$	−20.0000	−0.695889
$827$	−28.0000	−0.973655	−0.486828	−	0.873498i	$-0.661846\pi$
$827$	−28.0000	−0.973655	−0.486828	+	0.873498i	$0.661846\pi$
$828$	0	0
$829$	−30.0000	−1.04194	−0.520972	−	0.853574i	$-0.674430\pi$
$829$	−30.0000	−1.04194	−0.520972	+	0.853574i	$0.674430\pi$
$830$	0	0
$831$	0	0
$832$	−6.00000	−0.208013
$833$	−6.00000	−0.207888
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	−10.0000	−0.345444
$839$	−40.0000	−1.38095	−0.690477	−	0.723355i	$-0.742599\pi$
$839$	−40.0000	−1.38095	−0.690477	+	0.723355i	$0.742599\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	−22.0000	−0.758170
$843$	0	0
$844$	12.0000	0.413057
$845$	0	0
$846$	0	0
$847$	14.0000	0.481046
$848$	6.00000	0.206041
$849$	0	0
$850$	0	0
$851$	8.00000	0.274236
$852$	0	0
$853$	14.0000	0.479351	0.239675	−	0.970853i	$-0.422959\pi$
$853$	14.0000	0.479351	0.239675	+	0.970853i	$0.422959\pi$
$854$	4.00000	0.136877
$855$	0	0
$856$	−12.0000	−0.410152
$857$	42.0000	1.43469	0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000	1.43469	0.717346	+	0.696717i	$0.245357\pi$
$858$	0	0
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	0	0
$861$	0	0
$862$	32.0000	1.08992
$863$	36.0000	1.22545	0.612727	−	0.790295i	$-0.290072\pi$
$863$	36.0000	1.22545	0.612727	+	0.790295i	$0.290072\pi$
$864$	0	0
$865$	0	0
$866$	−4.00000	−0.135926
$867$	0	0
$868$	16.0000	0.543075
$869$	0	0
$870$	0	0
$871$	−48.0000	−1.62642
$872$	−10.0000	−0.338643
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−22.0000	−0.742887	−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000	−0.742887	−0.371444	+	0.928456i	$0.621137\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	0	0
$883$	24.0000	0.807664	0.403832	−	0.914833i	$-0.367678\pi$
$883$	24.0000	0.807664	0.403832	+	0.914833i	$0.367678\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	−36.0000	−1.20944
$887$	12.0000	0.402921	0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000	0.402921	0.201460	+	0.979497i	$0.435431\pi$
$888$	0	0
$889$	4.00000	0.134156
$890$	0	0
$891$	0	0
$892$	−26.0000	−0.870544
$893$	0	0
$894$	0	0
$895$	0	0
$896$	2.00000	0.0668153
$897$	0	0
$898$	30.0000	1.00111
$899$	0	0
$900$	0	0
$901$	12.0000	0.399778
$902$	−4.00000	−0.133185
$903$	0	0
$904$	−6.00000	−0.199557
$905$	0	0
$906$	0	0
$907$	−12.0000	−0.398453	−0.199227	−	0.979953i	$-0.563843\pi$
$907$	−12.0000	−0.398453	−0.199227	+	0.979953i	$0.563843\pi$
$908$	−28.0000	−0.929213
$909$	0	0
$910$	0	0
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	0	0
$913$	8.00000	0.264761
$914$	32.0000	1.05847
$915$	0	0
$916$	−10.0000	−0.330409
$917$	−36.0000	−1.18882
$918$	0	0
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	0	0
$921$	0	0
$922$	12.0000	0.395199
$923$	72.0000	2.36991
$924$	0	0
$925$	0	0
$926$	6.00000	0.197172
$927$	0	0
$928$	0	0
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	0	0
$931$	0	0
$932$	−14.0000	−0.458585
$933$	0	0
$934$	−12.0000	−0.392652
$935$	0	0
$936$	0	0
$937$	8.00000	0.261349	0.130674	−	0.991425i	$-0.458286\pi$
$937$	8.00000	0.261349	0.130674	+	0.991425i	$0.458286\pi$
$938$	16.0000	0.522419
$939$	0	0
$940$	0	0
$941$	28.0000	0.912774	0.456387	−	0.889781i	$-0.349143\pi$
$941$	28.0000	0.912774	0.456387	+	0.889781i	$0.349143\pi$
$942$	0	0
$943$	8.00000	0.260516
$944$	−10.0000	−0.325472
$945$	0	0
$946$	8.00000	0.260102
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	0	0
$949$	−24.0000	−0.779073
$950$	0	0
$951$	0	0
$952$	4.00000	0.129641
$953$	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
$955$	0	0
$956$	−20.0000	−0.646846
$957$	0	0
$958$	20.0000	0.646171
$959$	36.0000	1.16250
$960$	0	0
$961$	33.0000	1.06452
$962$	−12.0000	−0.386896
$963$	0	0
$964$	22.0000	0.708572
$965$	0	0
$966$	0	0
$967$	38.0000	1.22200	0.610999	−	0.791632i	$-0.290768\pi$
$967$	38.0000	1.22200	0.610999	+	0.791632i	$0.290768\pi$
$968$	7.00000	0.224989
$969$	0	0
$970$	0	0
$971$	18.0000	0.577647	0.288824	−	0.957382i	$-0.406736\pi$
$971$	18.0000	0.577647	0.288824	+	0.957382i	$0.406736\pi$
$972$	0	0
$973$	40.0000	1.28234
$974$	−18.0000	−0.576757
$975$	0	0
$976$	2.00000	0.0640184
$977$	42.0000	1.34370	0.671850	−	0.740688i	$-0.265500\pi$
$977$	42.0000	1.34370	0.671850	+	0.740688i	$0.265500\pi$
$978$	0	0
$979$	−20.0000	−0.639203
$980$	0	0
$981$	0	0
$982$	−18.0000	−0.574403
$983$	16.0000	0.510321	0.255160	−	0.966899i	$-0.417872\pi$
$983$	16.0000	0.510321	0.255160	+	0.966899i	$0.417872\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−16.0000	−0.508770
$990$	0	0
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	8.00000	0.254000
$993$	0	0
$994$	−24.0000	−0.761234
$995$	0	0
$996$	0	0
$997$	18.0000	0.570066	0.285033	−	0.958518i	$-0.407995\pi$
$997$	18.0000	0.570066	0.285033	+	0.958518i	$0.407995\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.2.a.b.1.1		1
3.2	odd	2		150.2.a.c.1.1		1
4.3	odd	2		3600.2.a.bg.1.1		1
5.2	odd	4		90.2.c.a.19.1		2
5.3	odd	4		90.2.c.a.19.2		2
5.4	even	2		450.2.a.f.1.1		1
12.11	even	2		1200.2.a.g.1.1		1
15.2	even	4		30.2.c.a.19.2	yes	2
15.8	even	4		30.2.c.a.19.1	✓	2
15.14	odd	2		150.2.a.a.1.1		1
20.3	even	4		720.2.f.f.289.2		2
20.7	even	4		720.2.f.f.289.1		2
20.19	odd	2		3600.2.a.o.1.1		1
21.20	even	2		7350.2.a.cc.1.1		1
24.5	odd	2		4800.2.a.l.1.1		1
24.11	even	2		4800.2.a.cj.1.1		1
40.3	even	4		2880.2.f.c.1729.1		2
40.13	odd	4		2880.2.f.e.1729.1		2
40.27	even	4		2880.2.f.c.1729.2		2
40.37	odd	4		2880.2.f.e.1729.2		2
45.2	even	12		810.2.i.e.109.2		4
45.7	odd	12		810.2.i.b.109.1		4
45.13	odd	12		810.2.i.b.379.1		4
45.22	odd	12		810.2.i.b.379.2		4
45.23	even	12		810.2.i.e.379.2		4
45.32	even	12		810.2.i.e.379.1		4
45.38	even	12		810.2.i.e.109.1		4
45.43	odd	12		810.2.i.b.109.2		4
60.23	odd	4		240.2.f.a.49.1		2
60.47	odd	4		240.2.f.a.49.2		2
60.59	even	2		1200.2.a.m.1.1		1
105.2	even	12		1470.2.n.h.949.2		4
105.17	odd	12		1470.2.n.a.79.1		4
105.23	even	12		1470.2.n.h.949.1		4
105.32	even	12		1470.2.n.h.79.1		4
105.38	odd	12		1470.2.n.a.79.2		4
105.47	odd	12		1470.2.n.a.949.2		4
105.53	even	12		1470.2.n.h.79.2		4
105.62	odd	4		1470.2.g.g.589.2		2
105.68	odd	12		1470.2.n.a.949.1		4
105.83	odd	4		1470.2.g.g.589.1		2
105.104	even	2		7350.2.a.bg.1.1		1
120.29	odd	2		4800.2.a.cg.1.1		1
120.53	even	4		960.2.f.h.769.1		2
120.59	even	2		4800.2.a.m.1.1		1
120.77	even	4		960.2.f.h.769.2		2
120.83	odd	4		960.2.f.i.769.2		2
120.107	odd	4		960.2.f.i.769.1		2
240.53	even	4		3840.2.d.y.2689.1		2
240.77	even	4		3840.2.d.y.2689.2		2
240.83	odd	4		3840.2.d.x.2689.2		2
240.107	odd	4		3840.2.d.x.2689.1		2
240.173	even	4		3840.2.d.g.2689.2		2
240.197	even	4		3840.2.d.g.2689.1		2
240.203	odd	4		3840.2.d.j.2689.1		2
240.227	odd	4		3840.2.d.j.2689.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.2.c.a.19.1	✓	2	15.8	even	4
30.2.c.a.19.2	yes	2	15.2	even	4
90.2.c.a.19.1		2	5.2	odd	4
90.2.c.a.19.2		2	5.3	odd	4
150.2.a.a.1.1		1	15.14	odd	2
150.2.a.c.1.1		1	3.2	odd	2
240.2.f.a.49.1		2	60.23	odd	4
240.2.f.a.49.2		2	60.47	odd	4
450.2.a.b.1.1		1	1.1	even	1	trivial
450.2.a.f.1.1		1	5.4	even	2
720.2.f.f.289.1		2	20.7	even	4
720.2.f.f.289.2		2	20.3	even	4
810.2.i.b.109.1		4	45.7	odd	12
810.2.i.b.109.2		4	45.43	odd	12
810.2.i.b.379.1		4	45.13	odd	12
810.2.i.b.379.2		4	45.22	odd	12
810.2.i.e.109.1		4	45.38	even	12
810.2.i.e.109.2		4	45.2	even	12
810.2.i.e.379.1		4	45.32	even	12
810.2.i.e.379.2		4	45.23	even	12
960.2.f.h.769.1		2	120.53	even	4
960.2.f.h.769.2		2	120.77	even	4
960.2.f.i.769.1		2	120.107	odd	4
960.2.f.i.769.2		2	120.83	odd	4
1200.2.a.g.1.1		1	12.11	even	2
1200.2.a.m.1.1		1	60.59	even	2
1470.2.g.g.589.1		2	105.83	odd	4
1470.2.g.g.589.2		2	105.62	odd	4
1470.2.n.a.79.1		4	105.17	odd	12
1470.2.n.a.79.2		4	105.38	odd	12
1470.2.n.a.949.1		4	105.68	odd	12
1470.2.n.a.949.2		4	105.47	odd	12
1470.2.n.h.79.1		4	105.32	even	12
1470.2.n.h.79.2		4	105.53	even	12
1470.2.n.h.949.1		4	105.23	even	12
1470.2.n.h.949.2		4	105.2	even	12
2880.2.f.c.1729.1		2	40.3	even	4
2880.2.f.c.1729.2		2	40.27	even	4
2880.2.f.e.1729.1		2	40.13	odd	4
2880.2.f.e.1729.2		2	40.37	odd	4
3600.2.a.o.1.1		1	20.19	odd	2
3600.2.a.bg.1.1		1	4.3	odd	2
3840.2.d.g.2689.1		2	240.197	even	4
3840.2.d.g.2689.2		2	240.173	even	4
3840.2.d.j.2689.1		2	240.203	odd	4
3840.2.d.j.2689.2		2	240.227	odd	4
3840.2.d.x.2689.1		2	240.107	odd	4
3840.2.d.x.2689.2		2	240.83	odd	4
3840.2.d.y.2689.1		2	240.53	even	4
3840.2.d.y.2689.2		2	240.77	even	4
4800.2.a.l.1.1		1	24.5	odd	2
4800.2.a.m.1.1		1	120.59	even	2
4800.2.a.cg.1.1		1	120.29	odd	2
4800.2.a.cj.1.1		1	24.11	even	2
7350.2.a.bg.1.1		1	105.104	even	2
7350.2.a.cc.1.1		1	21.20	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:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	450.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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