LMFDB - Embedded newform 5440.2.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5440,2,Mod(1,5440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5440, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5440.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5440 = 2^{6} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5440.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:	$43.4386186996$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 170)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	5440.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{3} +1.00000 q^{5} -2.00000 q^{7} -2.00000 q^{9} +O(q^{10})$

$q+1.00000 q^{3} +1.00000 q^{5} -2.00000 q^{7} -2.00000 q^{9} +1.00000 q^{13} +1.00000 q^{15} -1.00000 q^{17} -1.00000 q^{19} -2.00000 q^{21} +6.00000 q^{23} +1.00000 q^{25} -5.00000 q^{27} +3.00000 q^{29} -5.00000 q^{31} -2.00000 q^{35} -8.00000 q^{37} +1.00000 q^{39} +6.00000 q^{41} -10.0000 q^{43} -2.00000 q^{45} +3.00000 q^{47} -3.00000 q^{49} -1.00000 q^{51} +3.00000 q^{53} -1.00000 q^{57} +3.00000 q^{59} -11.0000 q^{61} +4.00000 q^{63} +1.00000 q^{65} +2.00000 q^{67} +6.00000 q^{69} -9.00000 q^{71} +11.0000 q^{73} +1.00000 q^{75} -8.00000 q^{79} +1.00000 q^{81} -12.0000 q^{83} -1.00000 q^{85} +3.00000 q^{87} +15.0000 q^{89} -2.00000 q^{91} -5.00000 q^{93} -1.00000 q^{95} -7.00000 q^{97} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	0	0
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.00000	−0.962250
$28$	0	0
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	−5.00000	−0.898027	−0.449013	−	0.893525i	$-0.648224\pi$
$31$	−5.00000	−0.898027	−0.449013	+	0.893525i	$0.648224\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.00000	−0.338062
$36$	0	0
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	0	0
$45$	−2.00000	−0.298142
$46$	0	0
$47$	3.00000	0.437595	0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000	0.437595	0.218797	+	0.975770i	$0.429787\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	−1.00000	−0.140028
$52$	0	0
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	3.00000	0.390567	0.195283	−	0.980747i	$-0.437437\pi$
$59$	3.00000	0.390567	0.195283	+	0.980747i	$0.437437\pi$
$60$	0	0
$61$	−11.0000	−1.40841	−0.704203	−	0.709999i	$-0.748695\pi$
$61$	−11.0000	−1.40841	−0.704203	+	0.709999i	$0.748695\pi$
$62$	0	0
$63$	4.00000	0.503953
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	0	0
$69$	6.00000	0.722315
$70$	0	0
$71$	−9.00000	−1.06810	−0.534052	−	0.845452i	$-0.679331\pi$
$71$	−9.00000	−1.06810	−0.534052	+	0.845452i	$0.679331\pi$
$72$	0	0
$73$	11.0000	1.28745	0.643726	−	0.765256i	$-0.277388\pi$
$73$	11.0000	1.28745	0.643726	+	0.765256i	$0.277388\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	0	0
$85$	−1.00000	−0.108465
$86$	0	0
$87$	3.00000	0.321634
$88$	0	0
$89$	15.0000	1.59000	0.794998	−	0.606612i	$-0.207472\pi$
$89$	15.0000	1.59000	0.794998	+	0.606612i	$0.207472\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	0	0
$93$	−5.00000	−0.518476
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	−7.00000	−0.710742	−0.355371	−	0.934725i	$-0.615646\pi$
$97$	−7.00000	−0.710742	−0.355371	+	0.934725i	$0.615646\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	0	0
$105$	−2.00000	−0.195180
$106$	0	0
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	−11.0000	−1.05361	−0.526804	−	0.849987i	$-0.676610\pi$
$109$	−11.0000	−1.05361	−0.526804	+	0.849987i	$0.676610\pi$
$110$	0	0
$111$	−8.00000	−0.759326
$112$	0	0
$113$	9.00000	0.846649	0.423324	−	0.905978i	$-0.360863\pi$
$113$	9.00000	0.846649	0.423324	+	0.905978i	$0.360863\pi$
$114$	0	0
$115$	6.00000	0.559503
$116$	0	0
$117$	−2.00000	−0.184900
$118$	0	0
$119$	2.00000	0.183340
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	6.00000	0.541002
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−11.0000	−0.976092	−0.488046	−	0.872818i	$-0.662290\pi$
$127$	−11.0000	−0.976092	−0.488046	+	0.872818i	$0.662290\pi$
$128$	0	0
$129$	−10.0000	−0.880451
$130$	0	0
$131$	−18.0000	−1.57267	−0.786334	−	0.617802i	$-0.788023\pi$
$131$	−18.0000	−1.57267	−0.786334	+	0.617802i	$0.788023\pi$
$132$	0	0
$133$	2.00000	0.173422
$134$	0	0
$135$	−5.00000	−0.430331
$136$	0	0
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	0	0
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	0	0
$141$	3.00000	0.252646
$142$	0	0
$143$	0	0
$144$	0	0
$145$	3.00000	0.249136
$146$	0	0
$147$	−3.00000	−0.247436
$148$	0	0
$149$	12.0000	0.983078	0.491539	−	0.870855i	$-0.336434\pi$
$149$	12.0000	0.983078	0.491539	+	0.870855i	$0.336434\pi$
$150$	0	0
$151$	22.0000	1.79033	0.895167	−	0.445730i	$-0.147056\pi$
$151$	22.0000	1.79033	0.895167	+	0.445730i	$0.147056\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	0	0
$155$	−5.00000	−0.401610
$156$	0	0
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	0	0
$159$	3.00000	0.237915
$160$	0	0
$161$	−12.0000	−0.945732
$162$	0	0
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	18.0000	1.39288	0.696441	−	0.717614i	$-0.254766\pi$
$167$	18.0000	1.39288	0.696441	+	0.717614i	$0.254766\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	2.00000	0.152944
$172$	0	0
$173$	−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
$175$	−2.00000	−0.151186
$176$	0	0
$177$	3.00000	0.225494
$178$	0	0
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	0	0
$183$	−11.0000	−0.813143
$184$	0	0
$185$	−8.00000	−0.588172
$186$	0	0
$187$	0	0
$188$	0	0
$189$	10.0000	0.727393
$190$	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
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	0	0
$195$	1.00000	0.0716115
$196$	0	0
$197$	24.0000	1.70993	0.854965	−	0.518686i	$-0.173579\pi$
$197$	24.0000	1.70993	0.854965	+	0.518686i	$0.173579\pi$
$198$	0	0
$199$	−11.0000	−0.779769	−0.389885	−	0.920864i	$-0.627485\pi$
$199$	−11.0000	−0.779769	−0.389885	+	0.920864i	$0.627485\pi$
$200$	0	0
$201$	2.00000	0.141069
$202$	0	0
$203$	−6.00000	−0.421117
$204$	0	0
$205$	6.00000	0.419058
$206$	0	0
$207$	−12.0000	−0.834058
$208$	0	0
$209$	0	0
$210$	0	0
$211$	2.00000	0.137686	0.0688428	−	0.997628i	$-0.478069\pi$
$211$	2.00000	0.137686	0.0688428	+	0.997628i	$0.478069\pi$
$212$	0	0
$213$	−9.00000	−0.616670
$214$	0	0
$215$	−10.0000	−0.681994
$216$	0	0
$217$	10.0000	0.678844
$218$	0	0
$219$	11.0000	0.743311
$220$	0	0
$221$	−1.00000	−0.0672673
$222$	0	0
$223$	1.00000	0.0669650	0.0334825	−	0.999439i	$-0.489340\pi$
$223$	1.00000	0.0669650	0.0334825	+	0.999439i	$0.489340\pi$
$224$	0	0
$225$	−2.00000	−0.133333
$226$	0	0
$227$	3.00000	0.199117	0.0995585	−	0.995032i	$-0.468257\pi$
$227$	3.00000	0.199117	0.0995585	+	0.995032i	$0.468257\pi$
$228$	0	0
$229$	22.0000	1.45380	0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000	1.45380	0.726900	+	0.686743i	$0.240960\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	9.00000	0.589610	0.294805	−	0.955557i	$-0.404745\pi$
$233$	9.00000	0.589610	0.294805	+	0.955557i	$0.404745\pi$
$234$	0	0
$235$	3.00000	0.195698
$236$	0	0
$237$	−8.00000	−0.519656
$238$	0	0
$239$	−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
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	0	0
$243$	16.0000	1.02640
$244$	0	0
$245$	−3.00000	−0.191663
$246$	0	0
$247$	−1.00000	−0.0636285
$248$	0	0
$249$	−12.0000	−0.760469
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−1.00000	−0.0626224
$256$	0	0
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	16.0000	0.994192
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	−9.00000	−0.554964	−0.277482	−	0.960731i	$-0.589500\pi$
$263$	−9.00000	−0.554964	−0.277482	+	0.960731i	$0.589500\pi$
$264$	0	0
$265$	3.00000	0.184289
$266$	0	0
$267$	15.0000	0.917985
$268$	0	0
$269$	−15.0000	−0.914566	−0.457283	−	0.889321i	$-0.651177\pi$
$269$	−15.0000	−0.914566	−0.457283	+	0.889321i	$0.651177\pi$
$270$	0	0
$271$	−32.0000	−1.94386	−0.971931	−	0.235267i	$-0.924404\pi$
$271$	−32.0000	−1.94386	−0.971931	+	0.235267i	$0.924404\pi$
$272$	0	0
$273$	−2.00000	−0.121046
$274$	0	0
$275$	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	$0.402873\pi$
$278$	0	0
$279$	10.0000	0.598684
$280$	0	0
$281$	27.0000	1.61068	0.805342	−	0.592810i	$-0.201981\pi$
$281$	27.0000	1.61068	0.805342	+	0.592810i	$0.201981\pi$
$282$	0	0
$283$	−13.0000	−0.772770	−0.386385	−	0.922338i	$-0.626276\pi$
$283$	−13.0000	−0.772770	−0.386385	+	0.922338i	$0.626276\pi$
$284$	0	0
$285$	−1.00000	−0.0592349
$286$	0	0
$287$	−12.0000	−0.708338
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−7.00000	−0.410347
$292$	0	0
$293$	3.00000	0.175262	0.0876309	−	0.996153i	$-0.472070\pi$
$293$	3.00000	0.175262	0.0876309	+	0.996153i	$0.472070\pi$
$294$	0	0
$295$	3.00000	0.174667
$296$	0	0
$297$	0	0
$298$	0	0
$299$	6.00000	0.346989
$300$	0	0
$301$	20.0000	1.15278
$302$	0	0
$303$	−6.00000	−0.344691
$304$	0	0
$305$	−11.0000	−0.629858
$306$	0	0
$307$	20.0000	1.14146	0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000	1.14146	0.570730	+	0.821138i	$0.306660\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	0	0
$315$	4.00000	0.225374
$316$	0	0
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−12.0000	−0.669775
$322$	0	0
$323$	1.00000	0.0556415
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	−11.0000	−0.608301
$328$	0	0
$329$	−6.00000	−0.330791
$330$	0	0
$331$	−31.0000	−1.70391	−0.851957	−	0.523612i	$-0.824584\pi$
$331$	−31.0000	−1.70391	−0.851957	+	0.523612i	$0.824584\pi$
$332$	0	0
$333$	16.0000	0.876795
$334$	0	0
$335$	2.00000	0.109272
$336$	0	0
$337$	−13.0000	−0.708155	−0.354078	−	0.935216i	$-0.615205\pi$
$337$	−13.0000	−0.708155	−0.354078	+	0.935216i	$0.615205\pi$
$338$	0	0
$339$	9.00000	0.488813
$340$	0	0
$341$	0	0
$342$	0	0
$343$	20.0000	1.07990
$344$	0	0
$345$	6.00000	0.323029
$346$	0	0
$347$	−15.0000	−0.805242	−0.402621	−	0.915367i	$-0.631901\pi$
$347$	−15.0000	−0.805242	−0.402621	+	0.915367i	$0.631901\pi$
$348$	0	0
$349$	28.0000	1.49881	0.749403	−	0.662114i	$-0.230341\pi$
$349$	28.0000	1.49881	0.749403	+	0.662114i	$0.230341\pi$
$350$	0	0
$351$	−5.00000	−0.266880
$352$	0	0
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	0	0
$355$	−9.00000	−0.477670
$356$	0	0
$357$	2.00000	0.105851
$358$	0	0
$359$	−30.0000	−1.58334	−0.791670	−	0.610949i	$-0.790788\pi$
$359$	−30.0000	−1.58334	−0.791670	+	0.610949i	$0.790788\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	−11.0000	−0.577350
$364$	0	0
$365$	11.0000	0.575766
$366$	0	0
$367$	22.0000	1.14839	0.574195	−	0.818718i	$-0.305315\pi$
$367$	22.0000	1.14839	0.574195	+	0.818718i	$0.305315\pi$
$368$	0	0
$369$	−12.0000	−0.624695
$370$	0	0
$371$	−6.00000	−0.311504
$372$	0	0
$373$	22.0000	1.13912	0.569558	−	0.821951i	$-0.307114\pi$
$373$	22.0000	1.13912	0.569558	+	0.821951i	$0.307114\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	3.00000	0.154508
$378$	0	0
$379$	8.00000	0.410932	0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000	0.410932	0.205466	+	0.978664i	$0.434129\pi$
$380$	0	0
$381$	−11.0000	−0.563547
$382$	0	0
$383$	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
$386$	0	0
$387$	20.0000	1.01666
$388$	0	0
$389$	−12.0000	−0.608424	−0.304212	−	0.952604i	$-0.598393\pi$
$389$	−12.0000	−0.608424	−0.304212	+	0.952604i	$0.598393\pi$
$390$	0	0
$391$	−6.00000	−0.303433
$392$	0	0
$393$	−18.0000	−0.907980
$394$	0	0
$395$	−8.00000	−0.402524
$396$	0	0
$397$	−20.0000	−1.00377	−0.501886	−	0.864934i	$-0.667360\pi$
$397$	−20.0000	−1.00377	−0.501886	+	0.864934i	$0.667360\pi$
$398$	0	0
$399$	2.00000	0.100125
$400$	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
$403$	−5.00000	−0.249068
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	0	0
$408$	0	0
$409$	29.0000	1.43396	0.716979	−	0.697095i	$-0.245524\pi$
$409$	29.0000	1.43396	0.716979	+	0.697095i	$0.245524\pi$
$410$	0	0
$411$	−18.0000	−0.887875
$412$	0	0
$413$	−6.00000	−0.295241
$414$	0	0
$415$	−12.0000	−0.589057
$416$	0	0
$417$	8.00000	0.391762
$418$	0	0
$419$	6.00000	0.293119	0.146560	−	0.989202i	$-0.453180\pi$
$419$	6.00000	0.293119	0.146560	+	0.989202i	$0.453180\pi$
$420$	0	0
$421$	4.00000	0.194948	0.0974740	−	0.995238i	$-0.468924\pi$
$421$	4.00000	0.194948	0.0974740	+	0.995238i	$0.468924\pi$
$422$	0	0
$423$	−6.00000	−0.291730
$424$	0	0
$425$	−1.00000	−0.0485071
$426$	0	0
$427$	22.0000	1.06465
$428$	0	0
$429$	0	0
$430$	0	0
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	0	0
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	0	0
$435$	3.00000	0.143839
$436$	0	0
$437$	−6.00000	−0.287019
$438$	0	0
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	6.00000	0.285714
$442$	0	0
$443$	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
$445$	15.0000	0.711068
$446$	0	0
$447$	12.0000	0.567581
$448$	0	0
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	22.0000	1.03365
$454$	0	0
$455$	−2.00000	−0.0937614
$456$	0	0
$457$	26.0000	1.21623	0.608114	−	0.793849i	$-0.291926\pi$
$457$	26.0000	1.21623	0.608114	+	0.793849i	$0.291926\pi$
$458$	0	0
$459$	5.00000	0.233380
$460$	0	0
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	0	0
$463$	19.0000	0.883005	0.441502	−	0.897260i	$-0.354446\pi$
$463$	19.0000	0.883005	0.441502	+	0.897260i	$0.354446\pi$
$464$	0	0
$465$	−5.00000	−0.231869
$466$	0	0
$467$	24.0000	1.11059	0.555294	−	0.831654i	$-0.312606\pi$
$467$	24.0000	1.11059	0.555294	+	0.831654i	$0.312606\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	0	0
$471$	−14.0000	−0.645086
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	−6.00000	−0.274721
$478$	0	0
$479$	−27.0000	−1.23366	−0.616831	−	0.787096i	$-0.711584\pi$
$479$	−27.0000	−1.23366	−0.616831	+	0.787096i	$0.711584\pi$
$480$	0	0
$481$	−8.00000	−0.364769
$482$	0	0
$483$	−12.0000	−0.546019
$484$	0	0
$485$	−7.00000	−0.317854
$486$	0	0
$487$	22.0000	0.996915	0.498458	−	0.866914i	$-0.333900\pi$
$487$	22.0000	0.996915	0.498458	+	0.866914i	$0.333900\pi$
$488$	0	0
$489$	−16.0000	−0.723545
$490$	0	0
$491$	−39.0000	−1.76005	−0.880023	−	0.474932i	$-0.842473\pi$
$491$	−39.0000	−1.76005	−0.880023	+	0.474932i	$0.842473\pi$
$492$	0	0
$493$	−3.00000	−0.135113
$494$	0	0
$495$	0	0
$496$	0	0
$497$	18.0000	0.807410
$498$	0	0
$499$	−22.0000	−0.984855	−0.492428	−	0.870353i	$-0.663890\pi$
$499$	−22.0000	−0.984855	−0.492428	+	0.870353i	$0.663890\pi$
$500$	0	0
$501$	18.0000	0.804181
$502$	0	0
$503$	12.0000	0.535054	0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000	0.535054	0.267527	+	0.963550i	$0.413794\pi$
$504$	0	0
$505$	−6.00000	−0.266996
$506$	0	0
$507$	−12.0000	−0.532939
$508$	0	0
$509$	36.0000	1.59567	0.797836	−	0.602875i	$-0.205978\pi$
$509$	36.0000	1.59567	0.797836	+	0.602875i	$0.205978\pi$
$510$	0	0
$511$	−22.0000	−0.973223
$512$	0	0
$513$	5.00000	0.220755
$514$	0	0
$515$	−8.00000	−0.352522
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−18.0000	−0.790112
$520$	0	0
$521$	36.0000	1.57719	0.788594	−	0.614914i	$-0.210809\pi$
$521$	36.0000	1.57719	0.788594	+	0.614914i	$0.210809\pi$
$522$	0	0
$523$	2.00000	0.0874539	0.0437269	−	0.999044i	$-0.486077\pi$
$523$	2.00000	0.0874539	0.0437269	+	0.999044i	$0.486077\pi$
$524$	0	0
$525$	−2.00000	−0.0872872
$526$	0	0
$527$	5.00000	0.217803
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	−6.00000	−0.260378
$532$	0	0
$533$	6.00000	0.259889
$534$	0	0
$535$	−12.0000	−0.518805
$536$	0	0
$537$	−12.0000	−0.517838
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	0	0
$543$	10.0000	0.429141
$544$	0	0
$545$	−11.0000	−0.471188
$546$	0	0
$547$	−31.0000	−1.32546	−0.662732	−	0.748857i	$-0.730603\pi$
$547$	−31.0000	−1.32546	−0.662732	+	0.748857i	$0.730603\pi$
$548$	0	0
$549$	22.0000	0.938937
$550$	0	0
$551$	−3.00000	−0.127804
$552$	0	0
$553$	16.0000	0.680389
$554$	0	0
$555$	−8.00000	−0.339581
$556$	0	0
$557$	−39.0000	−1.65248	−0.826242	−	0.563316i	$-0.809525\pi$
$557$	−39.0000	−1.65248	−0.826242	+	0.563316i	$0.809525\pi$
$558$	0	0
$559$	−10.0000	−0.422955
$560$	0	0
$561$	0	0
$562$	0	0
$563$	18.0000	0.758610	0.379305	−	0.925272i	$-0.376163\pi$
$563$	18.0000	0.758610	0.379305	+	0.925272i	$0.376163\pi$
$564$	0	0
$565$	9.00000	0.378633
$566$	0	0
$567$	−2.00000	−0.0839921
$568$	0	0
$569$	27.0000	1.13190	0.565949	−	0.824440i	$-0.308510\pi$
$569$	27.0000	1.13190	0.565949	+	0.824440i	$0.308510\pi$
$570$	0	0
$571$	38.0000	1.59025	0.795125	−	0.606445i	$-0.207405\pi$
$571$	38.0000	1.59025	0.795125	+	0.606445i	$0.207405\pi$
$572$	0	0
$573$	−12.0000	−0.501307
$574$	0	0
$575$	6.00000	0.250217
$576$	0	0
$577$	−28.0000	−1.16566	−0.582828	−	0.812596i	$-0.698054\pi$
$577$	−28.0000	−1.16566	−0.582828	+	0.812596i	$0.698054\pi$
$578$	0	0
$579$	2.00000	0.0831172
$580$	0	0
$581$	24.0000	0.995688
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−2.00000	−0.0826898
$586$	0	0
$587$	−42.0000	−1.73353	−0.866763	−	0.498721i	$-0.833803\pi$
$587$	−42.0000	−1.73353	−0.866763	+	0.498721i	$0.833803\pi$
$588$	0	0
$589$	5.00000	0.206021
$590$	0	0
$591$	24.0000	0.987228
$592$	0	0
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	0	0
$595$	2.00000	0.0819920
$596$	0	0
$597$	−11.0000	−0.450200
$598$	0	0
$599$	18.0000	0.735460	0.367730	−	0.929933i	$-0.380135\pi$
$599$	18.0000	0.735460	0.367730	+	0.929933i	$0.380135\pi$
$600$	0	0
$601$	8.00000	0.326327	0.163163	−	0.986599i	$-0.447830\pi$
$601$	8.00000	0.326327	0.163163	+	0.986599i	$0.447830\pi$
$602$	0	0
$603$	−4.00000	−0.162893
$604$	0	0
$605$	−11.0000	−0.447214
$606$	0	0
$607$	−26.0000	−1.05531	−0.527654	−	0.849460i	$-0.676928\pi$
$607$	−26.0000	−1.05531	−0.527654	+	0.849460i	$0.676928\pi$
$608$	0	0
$609$	−6.00000	−0.243132
$610$	0	0
$611$	3.00000	0.121367
$612$	0	0
$613$	1.00000	0.0403896	0.0201948	−	0.999796i	$-0.493571\pi$
$613$	1.00000	0.0403896	0.0201948	+	0.999796i	$0.493571\pi$
$614$	0	0
$615$	6.00000	0.241943
$616$	0	0
$617$	−39.0000	−1.57008	−0.785040	−	0.619445i	$-0.787358\pi$
$617$	−39.0000	−1.57008	−0.785040	+	0.619445i	$0.787358\pi$
$618$	0	0
$619$	−46.0000	−1.84890	−0.924448	−	0.381308i	$-0.875474\pi$
$619$	−46.0000	−1.84890	−0.924448	+	0.381308i	$0.875474\pi$
$620$	0	0
$621$	−30.0000	−1.20386
$622$	0	0
$623$	−30.0000	−1.20192
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	8.00000	0.318981
$630$	0	0
$631$	−32.0000	−1.27390	−0.636950	−	0.770905i	$-0.719804\pi$
$631$	−32.0000	−1.27390	−0.636950	+	0.770905i	$0.719804\pi$
$632$	0	0
$633$	2.00000	0.0794929
$634$	0	0
$635$	−11.0000	−0.436522
$636$	0	0
$637$	−3.00000	−0.118864
$638$	0	0
$639$	18.0000	0.712069
$640$	0	0
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	0	0
$643$	−4.00000	−0.157745	−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000	−0.157745	−0.0788723	+	0.996885i	$0.525132\pi$
$644$	0	0
$645$	−10.0000	−0.393750
$646$	0	0
$647$	33.0000	1.29736	0.648682	−	0.761060i	$-0.275321\pi$
$647$	33.0000	1.29736	0.648682	+	0.761060i	$0.275321\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	10.0000	0.391931
$652$	0	0
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	0	0
$655$	−18.0000	−0.703318
$656$	0	0
$657$	−22.0000	−0.858302
$658$	0	0
$659$	9.00000	0.350590	0.175295	−	0.984516i	$-0.443912\pi$
$659$	9.00000	0.350590	0.175295	+	0.984516i	$0.443912\pi$
$660$	0	0
$661$	−20.0000	−0.777910	−0.388955	−	0.921257i	$-0.627164\pi$
$661$	−20.0000	−0.777910	−0.388955	+	0.921257i	$0.627164\pi$
$662$	0	0
$663$	−1.00000	−0.0388368
$664$	0	0
$665$	2.00000	0.0775567
$666$	0	0
$667$	18.0000	0.696963
$668$	0	0
$669$	1.00000	0.0386622
$670$	0	0
$671$	0	0
$672$	0	0
$673$	23.0000	0.886585	0.443292	−	0.896377i	$-0.353810\pi$
$673$	23.0000	0.886585	0.443292	+	0.896377i	$0.353810\pi$
$674$	0	0
$675$	−5.00000	−0.192450
$676$	0	0
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	0	0
$679$	14.0000	0.537271
$680$	0	0
$681$	3.00000	0.114960
$682$	0	0
$683$	−3.00000	−0.114792	−0.0573959	−	0.998351i	$-0.518280\pi$
$683$	−3.00000	−0.114792	−0.0573959	+	0.998351i	$0.518280\pi$
$684$	0	0
$685$	−18.0000	−0.687745
$686$	0	0
$687$	22.0000	0.839352
$688$	0	0
$689$	3.00000	0.114291
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	8.00000	0.303457
$696$	0	0
$697$	−6.00000	−0.227266
$698$	0	0
$699$	9.00000	0.340411
$700$	0	0
$701$	36.0000	1.35970	0.679851	−	0.733351i	$-0.262045\pi$
$701$	36.0000	1.35970	0.679851	+	0.733351i	$0.262045\pi$
$702$	0	0
$703$	8.00000	0.301726
$704$	0	0
$705$	3.00000	0.112987
$706$	0	0
$707$	12.0000	0.451306
$708$	0	0
$709$	−41.0000	−1.53979	−0.769894	−	0.638172i	$-0.779691\pi$
$709$	−41.0000	−1.53979	−0.769894	+	0.638172i	$0.779691\pi$
$710$	0	0
$711$	16.0000	0.600047
$712$	0	0
$713$	−30.0000	−1.12351
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−6.00000	−0.224074
$718$	0	0
$719$	39.0000	1.45445	0.727227	−	0.686397i	$-0.240809\pi$
$719$	39.0000	1.45445	0.727227	+	0.686397i	$0.240809\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	0	0
$723$	−10.0000	−0.371904
$724$	0	0
$725$	3.00000	0.111417
$726$	0	0
$727$	19.0000	0.704671	0.352335	−	0.935874i	$-0.385388\pi$
$727$	19.0000	0.704671	0.352335	+	0.935874i	$0.385388\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	10.0000	0.369863
$732$	0	0
$733$	−14.0000	−0.517102	−0.258551	−	0.965998i	$-0.583245\pi$
$733$	−14.0000	−0.517102	−0.258551	+	0.965998i	$0.583245\pi$
$734$	0	0
$735$	−3.00000	−0.110657
$736$	0	0
$737$	0	0
$738$	0	0
$739$	29.0000	1.06678	0.533391	−	0.845869i	$-0.320917\pi$
$739$	29.0000	1.06678	0.533391	+	0.845869i	$0.320917\pi$
$740$	0	0
$741$	−1.00000	−0.0367359
$742$	0	0
$743$	−30.0000	−1.10059	−0.550297	−	0.834969i	$-0.685485\pi$
$743$	−30.0000	−1.10059	−0.550297	+	0.834969i	$0.685485\pi$
$744$	0	0
$745$	12.0000	0.439646
$746$	0	0
$747$	24.0000	0.878114
$748$	0	0
$749$	24.0000	0.876941
$750$	0	0
$751$	25.0000	0.912263	0.456131	−	0.889912i	$-0.349235\pi$
$751$	25.0000	0.912263	0.456131	+	0.889912i	$0.349235\pi$
$752$	0	0
$753$	−12.0000	−0.437304
$754$	0	0
$755$	22.0000	0.800662
$756$	0	0
$757$	13.0000	0.472493	0.236247	−	0.971693i	$-0.424083\pi$
$757$	13.0000	0.472493	0.236247	+	0.971693i	$0.424083\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	0	0
$763$	22.0000	0.796453
$764$	0	0
$765$	2.00000	0.0723102
$766$	0	0
$767$	3.00000	0.108324
$768$	0	0
$769$	−31.0000	−1.11789	−0.558944	−	0.829205i	$-0.688793\pi$
$769$	−31.0000	−1.11789	−0.558944	+	0.829205i	$0.688793\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	0	0
$773$	−42.0000	−1.51064	−0.755318	−	0.655359i	$-0.772517\pi$
$773$	−42.0000	−1.51064	−0.755318	+	0.655359i	$0.772517\pi$
$774$	0	0
$775$	−5.00000	−0.179605
$776$	0	0
$777$	16.0000	0.573997
$778$	0	0
$779$	−6.00000	−0.214972
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−15.0000	−0.536056
$784$	0	0
$785$	−14.0000	−0.499681
$786$	0	0
$787$	−7.00000	−0.249523	−0.124762	−	0.992187i	$-0.539817\pi$
$787$	−7.00000	−0.249523	−0.124762	+	0.992187i	$0.539817\pi$
$788$	0	0
$789$	−9.00000	−0.320408
$790$	0	0
$791$	−18.0000	−0.640006
$792$	0	0
$793$	−11.0000	−0.390621
$794$	0	0
$795$	3.00000	0.106399
$796$	0	0
$797$	−6.00000	−0.212531	−0.106265	−	0.994338i	$-0.533889\pi$
$797$	−6.00000	−0.212531	−0.106265	+	0.994338i	$0.533889\pi$
$798$	0	0
$799$	−3.00000	−0.106132
$800$	0	0
$801$	−30.0000	−1.06000
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−12.0000	−0.422944
$806$	0	0
$807$	−15.0000	−0.528025
$808$	0	0
$809$	42.0000	1.47664	0.738321	−	0.674450i	$-0.235619\pi$
$809$	42.0000	1.47664	0.738321	+	0.674450i	$0.235619\pi$
$810$	0	0
$811$	2.00000	0.0702295	0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000	0.0702295	0.0351147	+	0.999383i	$0.488820\pi$
$812$	0	0
$813$	−32.0000	−1.12229
$814$	0	0
$815$	−16.0000	−0.560456
$816$	0	0
$817$	10.0000	0.349856
$818$	0	0
$819$	4.00000	0.139771
$820$	0	0
$821$	−15.0000	−0.523504	−0.261752	−	0.965135i	$-0.584300\pi$
$821$	−15.0000	−0.523504	−0.261752	+	0.965135i	$0.584300\pi$
$822$	0	0
$823$	10.0000	0.348578	0.174289	−	0.984695i	$-0.444237\pi$
$823$	10.0000	0.348578	0.174289	+	0.984695i	$0.444237\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	0	0
$829$	−32.0000	−1.11141	−0.555703	−	0.831381i	$-0.687551\pi$
$829$	−32.0000	−1.11141	−0.555703	+	0.831381i	$0.687551\pi$
$830$	0	0
$831$	10.0000	0.346896
$832$	0	0
$833$	3.00000	0.103944
$834$	0	0
$835$	18.0000	0.622916
$836$	0	0
$837$	25.0000	0.864126
$838$	0	0
$839$	21.0000	0.725001	0.362500	−	0.931984i	$-0.381923\pi$
$839$	21.0000	0.725001	0.362500	+	0.931984i	$0.381923\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	27.0000	0.929929
$844$	0	0
$845$	−12.0000	−0.412813
$846$	0	0
$847$	22.0000	0.755929
$848$	0	0
$849$	−13.0000	−0.446159
$850$	0	0
$851$	−48.0000	−1.64542
$852$	0	0
$853$	−2.00000	−0.0684787	−0.0342393	−	0.999414i	$-0.510901\pi$
$853$	−2.00000	−0.0684787	−0.0342393	+	0.999414i	$0.510901\pi$
$854$	0	0
$855$	2.00000	0.0683986
$856$	0	0
$857$	33.0000	1.12726	0.563629	−	0.826028i	$-0.309405\pi$
$857$	33.0000	1.12726	0.563629	+	0.826028i	$0.309405\pi$
$858$	0	0
$859$	−1.00000	−0.0341196	−0.0170598	−	0.999854i	$-0.505431\pi$
$859$	−1.00000	−0.0341196	−0.0170598	+	0.999854i	$0.505431\pi$
$860$	0	0
$861$	−12.0000	−0.408959
$862$	0	0
$863$	48.0000	1.63394	0.816970	−	0.576681i	$-0.195652\pi$
$863$	48.0000	1.63394	0.816970	+	0.576681i	$0.195652\pi$
$864$	0	0
$865$	−18.0000	−0.612018
$866$	0	0
$867$	1.00000	0.0339618
$868$	0	0
$869$	0	0
$870$	0	0
$871$	2.00000	0.0677674
$872$	0	0
$873$	14.0000	0.473828
$874$	0	0
$875$	−2.00000	−0.0676123
$876$	0	0
$877$	40.0000	1.35070	0.675352	−	0.737496i	$-0.263992\pi$
$877$	40.0000	1.35070	0.675352	+	0.737496i	$0.263992\pi$
$878$	0	0
$879$	3.00000	0.101187
$880$	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
$883$	38.0000	1.27880	0.639401	−	0.768874i	$-0.279182\pi$
$883$	38.0000	1.27880	0.639401	+	0.768874i	$0.279182\pi$
$884$	0	0
$885$	3.00000	0.100844
$886$	0	0
$887$	30.0000	1.00730	0.503651	−	0.863907i	$-0.331990\pi$
$887$	30.0000	1.00730	0.503651	+	0.863907i	$0.331990\pi$
$888$	0	0
$889$	22.0000	0.737856
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−3.00000	−0.100391
$894$	0	0
$895$	−12.0000	−0.401116
$896$	0	0
$897$	6.00000	0.200334
$898$	0	0
$899$	−15.0000	−0.500278
$900$	0	0
$901$	−3.00000	−0.0999445
$902$	0	0
$903$	20.0000	0.665558
$904$	0	0
$905$	10.0000	0.332411
$906$	0	0
$907$	47.0000	1.56061	0.780305	−	0.625400i	$-0.215064\pi$
$907$	47.0000	1.56061	0.780305	+	0.625400i	$0.215064\pi$
$908$	0	0
$909$	12.0000	0.398015
$910$	0	0
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−11.0000	−0.363649
$916$	0	0
$917$	36.0000	1.18882
$918$	0	0
$919$	−2.00000	−0.0659739	−0.0329870	−	0.999456i	$-0.510502\pi$
$919$	−2.00000	−0.0659739	−0.0329870	+	0.999456i	$0.510502\pi$
$920$	0	0
$921$	20.0000	0.659022
$922$	0	0
$923$	−9.00000	−0.296239
$924$	0	0
$925$	−8.00000	−0.263038
$926$	0	0
$927$	16.0000	0.525509
$928$	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
$931$	3.00000	0.0983210
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	44.0000	1.43742	0.718709	−	0.695311i	$-0.244734\pi$
$937$	44.0000	1.43742	0.718709	+	0.695311i	$0.244734\pi$
$938$	0	0
$939$	14.0000	0.456873
$940$	0	0
$941$	15.0000	0.488986	0.244493	−	0.969651i	$-0.421378\pi$
$941$	15.0000	0.488986	0.244493	+	0.969651i	$0.421378\pi$
$942$	0	0
$943$	36.0000	1.17232
$944$	0	0
$945$	10.0000	0.325300
$946$	0	0
$947$	51.0000	1.65728	0.828639	−	0.559784i	$-0.189116\pi$
$947$	51.0000	1.65728	0.828639	+	0.559784i	$0.189116\pi$
$948$	0	0
$949$	11.0000	0.357075
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−24.0000	−0.777436	−0.388718	−	0.921357i	$-0.627082\pi$
$953$	−24.0000	−0.777436	−0.388718	+	0.921357i	$0.627082\pi$
$954$	0	0
$955$	−12.0000	−0.388311
$956$	0	0
$957$	0	0
$958$	0	0
$959$	36.0000	1.16250
$960$	0	0
$961$	−6.00000	−0.193548
$962$	0	0
$963$	24.0000	0.773389
$964$	0	0
$965$	2.00000	0.0643823
$966$	0	0
$967$	40.0000	1.28631	0.643157	−	0.765735i	$-0.277624\pi$
$967$	40.0000	1.28631	0.643157	+	0.765735i	$0.277624\pi$
$968$	0	0
$969$	1.00000	0.0321246
$970$	0	0
$971$	−21.0000	−0.673922	−0.336961	−	0.941519i	$-0.609399\pi$
$971$	−21.0000	−0.673922	−0.336961	+	0.941519i	$0.609399\pi$
$972$	0	0
$973$	−16.0000	−0.512936
$974$	0	0
$975$	1.00000	0.0320256
$976$	0	0
$977$	−12.0000	−0.383914	−0.191957	−	0.981403i	$-0.561483\pi$
$977$	−12.0000	−0.383914	−0.191957	+	0.981403i	$0.561483\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	22.0000	0.702406
$982$	0	0
$983$	−6.00000	−0.191370	−0.0956851	−	0.995412i	$-0.530504\pi$
$983$	−6.00000	−0.191370	−0.0956851	+	0.995412i	$0.530504\pi$
$984$	0	0
$985$	24.0000	0.764704
$986$	0	0
$987$	−6.00000	−0.190982
$988$	0	0
$989$	−60.0000	−1.90789
$990$	0	0
$991$	−29.0000	−0.921215	−0.460608	−	0.887604i	$-0.652368\pi$
$991$	−29.0000	−0.921215	−0.460608	+	0.887604i	$0.652368\pi$
$992$	0	0
$993$	−31.0000	−0.983755
$994$	0	0
$995$	−11.0000	−0.348723
$996$	0	0
$997$	46.0000	1.45683	0.728417	−	0.685134i	$-0.240256\pi$
$997$	46.0000	1.45683	0.728417	+	0.685134i	$0.240256\pi$
$998$	0	0
$999$	40.0000	1.26554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5440.2.a.r.1.1		1
4.3	odd	2		5440.2.a.k.1.1		1
8.3	odd	2		170.2.a.e.1.1	✓	1
8.5	even	2		1360.2.a.d.1.1		1
24.11	even	2		1530.2.a.g.1.1		1
40.3	even	4		850.2.c.e.749.1		2
40.19	odd	2		850.2.a.b.1.1		1
40.27	even	4		850.2.c.e.749.2		2
40.29	even	2		6800.2.a.t.1.1		1
56.27	even	2		8330.2.a.q.1.1		1
120.59	even	2		7650.2.a.bo.1.1		1
136.67	odd	2		2890.2.a.n.1.1		1
136.115	odd	4		2890.2.b.b.2311.2		2
136.123	odd	4		2890.2.b.b.2311.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
170.2.a.e.1.1	✓	1	8.3	odd	2
850.2.a.b.1.1		1	40.19	odd	2
850.2.c.e.749.1		2	40.3	even	4
850.2.c.e.749.2		2	40.27	even	4
1360.2.a.d.1.1		1	8.5	even	2
1530.2.a.g.1.1		1	24.11	even	2
2890.2.a.n.1.1		1	136.67	odd	2
2890.2.b.b.2311.1		2	136.123	odd	4
2890.2.b.b.2311.2		2	136.115	odd	4
5440.2.a.k.1.1		1	4.3	odd	2
5440.2.a.r.1.1		1	1.1	even	1	trivial
6800.2.a.t.1.1		1	40.29	even	2
7650.2.a.bo.1.1		1	120.59	even	2
8330.2.a.q.1.1		1	56.27	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:	\( N \)	\(=\)	\( 5440 = 2^{6} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5440.a (trivial)

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