LMFDB - Embedded newform 135.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [135,2,Mod(1,135)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(135, 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("135.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 135 = 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	135.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:	$1.07798042729$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	135.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} +2.00000 q^{4} -1.00000 q^{5} -3.00000 q^{7} +O(q^{10})$

$q-2.00000 q^{2} +2.00000 q^{4} -1.00000 q^{5} -3.00000 q^{7} +2.00000 q^{10} -2.00000 q^{11} -5.00000 q^{13} +6.00000 q^{14} -4.00000 q^{16} -8.00000 q^{17} +1.00000 q^{19} -2.00000 q^{20} +4.00000 q^{22} +6.00000 q^{23} +1.00000 q^{25} +10.0000 q^{26} -6.00000 q^{28} +2.00000 q^{29} +8.00000 q^{32} +16.0000 q^{34} +3.00000 q^{35} +5.00000 q^{37} -2.00000 q^{38} -10.0000 q^{41} +4.00000 q^{43} -4.00000 q^{44} -12.0000 q^{46} +4.00000 q^{47} +2.00000 q^{49} -2.00000 q^{50} -10.0000 q^{52} -2.00000 q^{53} +2.00000 q^{55} -4.00000 q^{58} -8.00000 q^{59} +7.00000 q^{61} -8.00000 q^{64} +5.00000 q^{65} -9.00000 q^{67} -16.0000 q^{68} -6.00000 q^{70} +2.00000 q^{71} -5.00000 q^{73} -10.0000 q^{74} +2.00000 q^{76} +6.00000 q^{77} -3.00000 q^{79} +4.00000 q^{80} +20.0000 q^{82} +6.00000 q^{83} +8.00000 q^{85} -8.00000 q^{86} -12.0000 q^{89} +15.0000 q^{91} +12.0000 q^{92} -8.00000 q^{94} -1.00000 q^{95} -13.0000 q^{97} -4.00000 q^{98} +O(q^{100})$

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 $

$2$	−2.00000	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$2$	−2.00000	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$3$	0	0
$3$	0	0
$4$	2.00000	1.00000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	0	0
$9$	0	0
$10$	2.00000	0.632456
$11$	−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
$13$	−5.00000	−1.38675	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000	−1.38675	−0.693375	+	0.720577i	$0.743877\pi$
$14$	6.00000	1.60357
$15$	0	0
$16$	−4.00000	−1.00000
$17$	−8.00000	−1.94029	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	−8.00000	−1.94029	−0.970143	+	0.242536i	$0.922021\pi$
$18$	0	0
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	−2.00000	−0.447214
$21$	0	0
$22$	4.00000	0.852803
$23$	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
$26$	10.0000	1.96116
$27$	0	0
$28$	−6.00000	−1.13389
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	8.00000	1.41421
$33$	0	0
$34$	16.0000	2.74398
$35$	3.00000	0.507093
$36$	0	0
$37$	5.00000	0.821995	0.410997	−	0.911636i	$-0.365181\pi$
$37$	5.00000	0.821995	0.410997	+	0.911636i	$0.365181\pi$
$38$	−2.00000	−0.324443
$39$	0	0
$40$	0	0
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	−12.0000	−1.76930
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	−2.00000	−0.282843
$51$	0	0
$52$	−10.0000	−1.38675
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	0	0
$57$	0	0
$58$	−4.00000	−0.525226
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	7.00000	0.896258	0.448129	−	0.893969i	$-0.352090\pi$
$61$	7.00000	0.896258	0.448129	+	0.893969i	$0.352090\pi$
$62$	0	0
$63$	0	0
$64$	−8.00000	−1.00000
$65$	5.00000	0.620174
$66$	0	0
$67$	−9.00000	−1.09952	−0.549762	−	0.835321i	$-0.685282\pi$
$67$	−9.00000	−1.09952	−0.549762	+	0.835321i	$0.685282\pi$
$68$	−16.0000	−1.94029
$69$	0	0
$70$	−6.00000	−0.717137
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	0	0
$73$	−5.00000	−0.585206	−0.292603	−	0.956234i	$-0.594521\pi$
$73$	−5.00000	−0.585206	−0.292603	+	0.956234i	$0.594521\pi$
$74$	−10.0000	−1.16248
$75$	0	0
$76$	2.00000	0.229416
$77$	6.00000	0.683763
$78$	0	0
$79$	−3.00000	−0.337526	−0.168763	−	0.985657i	$-0.553977\pi$
$79$	−3.00000	−0.337526	−0.168763	+	0.985657i	$0.553977\pi$
$80$	4.00000	0.447214
$81$	0	0
$82$	20.0000	2.20863
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	8.00000	0.867722
$86$	−8.00000	−0.862662
$87$	0	0
$88$	0	0
$89$	−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
$91$	15.0000	1.57243
$92$	12.0000	1.25109
$93$	0	0
$94$	−8.00000	−0.825137
$95$	−1.00000	−0.102598
$96$	0	0
$97$	−13.0000	−1.31995	−0.659975	−	0.751288i	$-0.729433\pi$
$97$	−13.0000	−1.31995	−0.659975	+	0.751288i	$0.729433\pi$
$98$	−4.00000	−0.404061
$99$	0	0
$100$	2.00000	0.200000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	17.0000	1.67506	0.837530	−	0.546392i	$-0.183999\pi$
$103$	17.0000	1.67506	0.837530	+	0.546392i	$0.183999\pi$
$104$	0	0
$105$	0	0
$106$	4.00000	0.388514
$107$	6.00000	0.580042	0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000	0.580042	0.290021	+	0.957020i	$0.406338\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	−4.00000	−0.381385
$111$	0	0
$112$	12.0000	1.13389
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	0	0
$115$	−6.00000	−0.559503
$116$	4.00000	0.371391
$117$	0	0
$118$	16.0000	1.47292
$119$	24.0000	2.20008
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−14.0000	−1.26750
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	0	0
$129$	0	0
$130$	−10.0000	−0.877058
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	−3.00000	−0.260133
$134$	18.0000	1.55496
$135$	0	0
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	−13.0000	−1.10265	−0.551323	−	0.834292i	$-0.685877\pi$
$139$	−13.0000	−1.10265	−0.551323	+	0.834292i	$0.685877\pi$
$140$	6.00000	0.507093
$141$	0	0
$142$	−4.00000	−0.335673
$143$	10.0000	0.836242
$144$	0	0
$145$	−2.00000	−0.166091
$146$	10.0000	0.827606
$147$	0	0
$148$	10.0000	0.821995
$149$	−4.00000	−0.327693	−0.163846	−	0.986486i	$-0.552390\pi$
$149$	−4.00000	−0.327693	−0.163846	+	0.986486i	$0.552390\pi$
$150$	0	0
$151$	1.00000	0.0813788	0.0406894	−	0.999172i	$-0.487045\pi$
$151$	1.00000	0.0813788	0.0406894	+	0.999172i	$0.487045\pi$
$152$	0	0
$153$	0	0
$154$	−12.0000	−0.966988
$155$	0	0
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	6.00000	0.477334
$159$	0	0
$160$	−8.00000	−0.632456
$161$	−18.0000	−1.41860
$162$	0	0
$163$	−19.0000	−1.48819	−0.744097	−	0.668071i	$-0.767120\pi$
$163$	−19.0000	−1.48819	−0.744097	+	0.668071i	$0.767120\pi$
$164$	−20.0000	−1.56174
$165$	0	0
$166$	−12.0000	−0.931381
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	−16.0000	−1.22714
$171$	0	0
$172$	8.00000	0.609994
$173$	−12.0000	−0.912343	−0.456172	−	0.889892i	$-0.650780\pi$
$173$	−12.0000	−0.912343	−0.456172	+	0.889892i	$0.650780\pi$
$174$	0	0
$175$	−3.00000	−0.226779
$176$	8.00000	0.603023
$177$	0	0
$178$	24.0000	1.79888
$179$	22.0000	1.64436	0.822179	−	0.569230i	$-0.192758\pi$
$179$	22.0000	1.64436	0.822179	+	0.569230i	$0.192758\pi$
$180$	0	0
$181$	5.00000	0.371647	0.185824	−	0.982583i	$-0.440505\pi$
$181$	5.00000	0.371647	0.185824	+	0.982583i	$0.440505\pi$
$182$	−30.0000	−2.22375
$183$	0	0
$184$	0	0
$185$	−5.00000	−0.367607
$186$	0	0
$187$	16.0000	1.17004
$188$	8.00000	0.583460
$189$	0	0
$190$	2.00000	0.145095
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	0	0
$193$	5.00000	0.359908	0.179954	−	0.983675i	$-0.442405\pi$
$193$	5.00000	0.359908	0.179954	+	0.983675i	$0.442405\pi$
$194$	26.0000	1.86669
$195$	0	0
$196$	4.00000	0.285714
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	−17.0000	−1.20510	−0.602549	−	0.798082i	$-0.705848\pi$
$199$	−17.0000	−1.20510	−0.602549	+	0.798082i	$0.705848\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−6.00000	−0.421117
$204$	0	0
$205$	10.0000	0.698430
$206$	−34.0000	−2.36889
$207$	0	0
$208$	20.0000	1.38675
$209$	−2.00000	−0.138343
$210$	0	0
$211$	23.0000	1.58339	0.791693	−	0.610920i	$-0.209200\pi$
$211$	23.0000	1.58339	0.791693	+	0.610920i	$0.209200\pi$
$212$	−4.00000	−0.274721
$213$	0	0
$214$	−12.0000	−0.820303
$215$	−4.00000	−0.272798
$216$	0	0
$217$	0	0
$218$	20.0000	1.35457
$219$	0	0
$220$	4.00000	0.269680
$221$	40.0000	2.69069
$222$	0	0
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	−24.0000	−1.60357
$225$	0	0
$226$	−20.0000	−1.33038
$227$	−10.0000	−0.663723	−0.331862	−	0.943328i	$-0.607677\pi$
$227$	−10.0000	−0.663723	−0.331862	+	0.943328i	$0.607677\pi$
$228$	0	0
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	12.0000	0.791257
$231$	0	0
$232$	0	0
$233$	−24.0000	−1.57229	−0.786146	−	0.618041i	$-0.787927\pi$
$233$	−24.0000	−1.57229	−0.786146	+	0.618041i	$0.787927\pi$
$234$	0	0
$235$	−4.00000	−0.260931
$236$	−16.0000	−1.04151
$237$	0	0
$238$	−48.0000	−3.11138
$239$	−26.0000	−1.68180	−0.840900	−	0.541190i	$-0.817974\pi$
$239$	−26.0000	−1.68180	−0.840900	+	0.541190i	$0.817974\pi$
$240$	0	0
$241$	1.00000	0.0644157	0.0322078	−	0.999481i	$-0.489746\pi$
$241$	1.00000	0.0644157	0.0322078	+	0.999481i	$0.489746\pi$
$242$	14.0000	0.899954
$243$	0	0
$244$	14.0000	0.896258
$245$	−2.00000	−0.127775
$246$	0	0
$247$	−5.00000	−0.318142
$248$	0	0
$249$	0	0
$250$	2.00000	0.126491
$251$	6.00000	0.378717	0.189358	−	0.981908i	$-0.439359\pi$
$251$	6.00000	0.378717	0.189358	+	0.981908i	$0.439359\pi$
$252$	0	0
$253$	−12.0000	−0.754434
$254$	16.0000	1.00393
$255$	0	0
$256$	16.0000	1.00000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	−15.0000	−0.932055
$260$	10.0000	0.620174
$261$	0	0
$262$	−24.0000	−1.48272
$263$	−28.0000	−1.72655	−0.863277	−	0.504730i	$-0.831592\pi$
$263$	−28.0000	−1.72655	−0.863277	+	0.504730i	$0.831592\pi$
$264$	0	0
$265$	2.00000	0.122859
$266$	6.00000	0.367884
$267$	0	0
$268$	−18.0000	−1.09952
$269$	16.0000	0.975537	0.487769	−	0.872973i	$-0.337811\pi$
$269$	16.0000	0.975537	0.487769	+	0.872973i	$0.337811\pi$
$270$	0	0
$271$	13.0000	0.789694	0.394847	−	0.918747i	$-0.370798\pi$
$271$	13.0000	0.789694	0.394847	+	0.918747i	$0.370798\pi$
$272$	32.0000	1.94029
$273$	0	0
$274$	12.0000	0.724947
$275$	−2.00000	−0.120605
$276$	0	0
$277$	−30.0000	−1.80253	−0.901263	−	0.433273i	$-0.857359\pi$
$277$	−30.0000	−1.80253	−0.901263	+	0.433273i	$0.857359\pi$
$278$	26.0000	1.55938
$279$	0	0
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	12.0000	0.713326	0.356663	−	0.934233i	$-0.383914\pi$
$283$	12.0000	0.713326	0.356663	+	0.934233i	$0.383914\pi$
$284$	4.00000	0.237356
$285$	0	0
$286$	−20.0000	−1.18262
$287$	30.0000	1.77084
$288$	0	0
$289$	47.0000	2.76471
$290$	4.00000	0.234888
$291$	0	0
$292$	−10.0000	−0.585206
$293$	18.0000	1.05157	0.525786	−	0.850617i	$-0.323771\pi$
$293$	18.0000	1.05157	0.525786	+	0.850617i	$0.323771\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	0	0
$297$	0	0
$298$	8.00000	0.463428
$299$	−30.0000	−1.73494
$300$	0	0
$301$	−12.0000	−0.691669
$302$	−2.00000	−0.115087
$303$	0	0
$304$	−4.00000	−0.229416
$305$	−7.00000	−0.400819
$306$	0	0
$307$	16.0000	0.913168	0.456584	−	0.889680i	$-0.349073\pi$
$307$	16.0000	0.913168	0.456584	+	0.889680i	$0.349073\pi$
$308$	12.0000	0.683763
$309$	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
$313$	−7.00000	−0.395663	−0.197832	−	0.980236i	$-0.563390\pi$
$313$	−7.00000	−0.395663	−0.197832	+	0.980236i	$0.563390\pi$
$314$	−4.00000	−0.225733
$315$	0	0
$316$	−6.00000	−0.337526
$317$	20.0000	1.12331	0.561656	−	0.827371i	$-0.310164\pi$
$317$	20.0000	1.12331	0.561656	+	0.827371i	$0.310164\pi$
$318$	0	0
$319$	−4.00000	−0.223957
$320$	8.00000	0.447214
$321$	0	0
$322$	36.0000	2.00620
$323$	−8.00000	−0.445132
$324$	0	0
$325$	−5.00000	−0.277350
$326$	38.0000	2.10463
$327$	0	0
$328$	0	0
$329$	−12.0000	−0.661581
$330$	0	0
$331$	−21.0000	−1.15426	−0.577132	−	0.816651i	$-0.695828\pi$
$331$	−21.0000	−1.15426	−0.577132	+	0.816651i	$0.695828\pi$
$332$	12.0000	0.658586
$333$	0	0
$334$	24.0000	1.31322
$335$	9.00000	0.491723
$336$	0	0
$337$	7.00000	0.381314	0.190657	−	0.981657i	$-0.438938\pi$
$337$	7.00000	0.381314	0.190657	+	0.981657i	$0.438938\pi$
$338$	−24.0000	−1.30543
$339$	0	0
$340$	16.0000	0.867722
$341$	0	0
$342$	0	0
$343$	15.0000	0.809924
$344$	0	0
$345$	0	0
$346$	24.0000	1.29025
$347$	10.0000	0.536828	0.268414	−	0.963304i	$-0.413500\pi$
$347$	10.0000	0.536828	0.268414	+	0.963304i	$0.413500\pi$
$348$	0	0
$349$	19.0000	1.01705	0.508523	−	0.861048i	$-0.330192\pi$
$349$	19.0000	1.01705	0.508523	+	0.861048i	$0.330192\pi$
$350$	6.00000	0.320713
$351$	0	0
$352$	−16.0000	−0.852803
$353$	−12.0000	−0.638696	−0.319348	−	0.947638i	$-0.603464\pi$
$353$	−12.0000	−0.638696	−0.319348	+	0.947638i	$0.603464\pi$
$354$	0	0
$355$	−2.00000	−0.106149
$356$	−24.0000	−1.27200
$357$	0	0
$358$	−44.0000	−2.32547
$359$	−18.0000	−0.950004	−0.475002	−	0.879985i	$-0.657553\pi$
$359$	−18.0000	−0.950004	−0.475002	+	0.879985i	$0.657553\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	−10.0000	−0.525588
$363$	0	0
$364$	30.0000	1.57243
$365$	5.00000	0.261712
$366$	0	0
$367$	21.0000	1.09619	0.548096	−	0.836416i	$-0.315353\pi$
$367$	21.0000	1.09619	0.548096	+	0.836416i	$0.315353\pi$
$368$	−24.0000	−1.25109
$369$	0	0
$370$	10.0000	0.519875
$371$	6.00000	0.311504
$372$	0	0
$373$	−11.0000	−0.569558	−0.284779	−	0.958593i	$-0.591920\pi$
$373$	−11.0000	−0.569558	−0.284779	+	0.958593i	$0.591920\pi$
$374$	−32.0000	−1.65468
$375$	0	0
$376$	0	0
$377$	−10.0000	−0.515026
$378$	0	0
$379$	−11.0000	−0.565032	−0.282516	−	0.959263i	$-0.591169\pi$
$379$	−11.0000	−0.565032	−0.282516	+	0.959263i	$0.591169\pi$
$380$	−2.00000	−0.102598
$381$	0	0
$382$	8.00000	0.409316
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	0	0
$385$	−6.00000	−0.305788
$386$	−10.0000	−0.508987
$387$	0	0
$388$	−26.0000	−1.31995
$389$	24.0000	1.21685	0.608424	−	0.793612i	$-0.291802\pi$
$389$	24.0000	1.21685	0.608424	+	0.793612i	$0.291802\pi$
$390$	0	0
$391$	−48.0000	−2.42746
$392$	0	0
$393$	0	0
$394$	0	0
$395$	3.00000	0.150946
$396$	0	0
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	34.0000	1.70427
$399$	0	0
$400$	−4.00000	−0.200000
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	12.0000	0.595550
$407$	−10.0000	−0.495682
$408$	0	0
$409$	−7.00000	−0.346128	−0.173064	−	0.984911i	$-0.555367\pi$
$409$	−7.00000	−0.346128	−0.173064	+	0.984911i	$0.555367\pi$
$410$	−20.0000	−0.987730
$411$	0	0
$412$	34.0000	1.67506
$413$	24.0000	1.18096
$414$	0	0
$415$	−6.00000	−0.294528
$416$	−40.0000	−1.96116
$417$	0	0
$418$	4.00000	0.195646
$419$	32.0000	1.56330	0.781651	−	0.623716i	$-0.214378\pi$
$419$	32.0000	1.56330	0.781651	+	0.623716i	$0.214378\pi$
$420$	0	0
$421$	1.00000	0.0487370	0.0243685	−	0.999703i	$-0.492242\pi$
$421$	1.00000	0.0487370	0.0243685	+	0.999703i	$0.492242\pi$
$422$	−46.0000	−2.23924
$423$	0	0
$424$	0	0
$425$	−8.00000	−0.388057
$426$	0	0
$427$	−21.0000	−1.01626
$428$	12.0000	0.580042
$429$	0	0
$430$	8.00000	0.385794
$431$	30.0000	1.44505	0.722525	−	0.691345i	$-0.242982\pi$
$431$	30.0000	1.44505	0.722525	+	0.691345i	$0.242982\pi$
$432$	0	0
$433$	−38.0000	−1.82616	−0.913082	−	0.407777i	$-0.866304\pi$
$433$	−38.0000	−1.82616	−0.913082	+	0.407777i	$0.866304\pi$
$434$	0	0
$435$	0	0
$436$	−20.0000	−0.957826
$437$	6.00000	0.287019
$438$	0	0
$439$	−32.0000	−1.52728	−0.763638	−	0.645644i	$-0.776589\pi$
$439$	−32.0000	−1.52728	−0.763638	+	0.645644i	$0.776589\pi$
$440$	0	0
$441$	0	0
$442$	−80.0000	−3.80521
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	0	0
$445$	12.0000	0.568855
$446$	−16.0000	−0.757622
$447$	0	0
$448$	24.0000	1.13389
$449$	4.00000	0.188772	0.0943858	−	0.995536i	$-0.469911\pi$
$449$	4.00000	0.188772	0.0943858	+	0.995536i	$0.469911\pi$
$450$	0	0
$451$	20.0000	0.941763
$452$	20.0000	0.940721
$453$	0	0
$454$	20.0000	0.938647
$455$	−15.0000	−0.703211
$456$	0	0
$457$	22.0000	1.02912	0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000	1.02912	0.514558	+	0.857455i	$0.327956\pi$
$458$	−12.0000	−0.560723
$459$	0	0
$460$	−12.0000	−0.559503
$461$	−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
$463$	33.0000	1.53364	0.766820	−	0.641862i	$-0.221838\pi$
$463$	33.0000	1.53364	0.766820	+	0.641862i	$0.221838\pi$
$464$	−8.00000	−0.371391
$465$	0	0
$466$	48.0000	2.22356
$467$	−28.0000	−1.29569	−0.647843	−	0.761774i	$-0.724329\pi$
$467$	−28.0000	−1.29569	−0.647843	+	0.761774i	$0.724329\pi$
$468$	0	0
$469$	27.0000	1.24674
$470$	8.00000	0.369012
$471$	0	0
$472$	0	0
$473$	−8.00000	−0.367840
$474$	0	0
$475$	1.00000	0.0458831
$476$	48.0000	2.20008
$477$	0	0
$478$	52.0000	2.37842
$479$	6.00000	0.274147	0.137073	−	0.990561i	$-0.456230\pi$
$479$	6.00000	0.274147	0.137073	+	0.990561i	$0.456230\pi$
$480$	0	0
$481$	−25.0000	−1.13990
$482$	−2.00000	−0.0910975
$483$	0	0
$484$	−14.0000	−0.636364
$485$	13.0000	0.590300
$486$	0	0
$487$	5.00000	0.226572	0.113286	−	0.993562i	$-0.463862\pi$
$487$	5.00000	0.226572	0.113286	+	0.993562i	$0.463862\pi$
$488$	0	0
$489$	0	0
$490$	4.00000	0.180702
$491$	−22.0000	−0.992846	−0.496423	−	0.868081i	$-0.665354\pi$
$491$	−22.0000	−0.992846	−0.496423	+	0.868081i	$0.665354\pi$
$492$	0	0
$493$	−16.0000	−0.720604
$494$	10.0000	0.449921
$495$	0	0
$496$	0	0
$497$	−6.00000	−0.269137
$498$	0	0
$499$	16.0000	0.716258	0.358129	−	0.933672i	$-0.383415\pi$
$499$	16.0000	0.716258	0.358129	+	0.933672i	$0.383415\pi$
$500$	−2.00000	−0.0894427
$501$	0	0
$502$	−12.0000	−0.535586
$503$	26.0000	1.15928	0.579641	−	0.814872i	$-0.303193\pi$
$503$	26.0000	1.15928	0.579641	+	0.814872i	$0.303193\pi$
$504$	0	0
$505$	0	0
$506$	24.0000	1.06693
$507$	0	0
$508$	−16.0000	−0.709885
$509$	34.0000	1.50702	0.753512	−	0.657434i	$-0.228358\pi$
$509$	34.0000	1.50702	0.753512	+	0.657434i	$0.228358\pi$
$510$	0	0
$511$	15.0000	0.663561
$512$	−32.0000	−1.41421
$513$	0	0
$514$	0	0
$515$	−17.0000	−0.749110
$516$	0	0
$517$	−8.00000	−0.351840
$518$	30.0000	1.31812
$519$	0	0
$520$	0	0
$521$	38.0000	1.66481	0.832405	−	0.554168i	$-0.186963\pi$
$521$	38.0000	1.66481	0.832405	+	0.554168i	$0.186963\pi$
$522$	0	0
$523$	19.0000	0.830812	0.415406	−	0.909636i	$-0.363640\pi$
$523$	19.0000	0.830812	0.415406	+	0.909636i	$0.363640\pi$
$524$	24.0000	1.04844
$525$	0	0
$526$	56.0000	2.44172
$527$	0	0
$528$	0	0
$529$	13.0000	0.565217
$530$	−4.00000	−0.173749
$531$	0	0
$532$	−6.00000	−0.260133
$533$	50.0000	2.16574
$534$	0	0
$535$	−6.00000	−0.259403
$536$	0	0
$537$	0	0
$538$	−32.0000	−1.37962
$539$	−4.00000	−0.172292
$540$	0	0
$541$	33.0000	1.41878	0.709390	−	0.704816i	$-0.248970\pi$
$541$	33.0000	1.41878	0.709390	+	0.704816i	$0.248970\pi$
$542$	−26.0000	−1.11680
$543$	0	0
$544$	−64.0000	−2.74398
$545$	10.0000	0.428353
$546$	0	0
$547$	37.0000	1.58201	0.791003	−	0.611812i	$-0.209559\pi$
$547$	37.0000	1.58201	0.791003	+	0.611812i	$0.209559\pi$
$548$	−12.0000	−0.512615
$549$	0	0
$550$	4.00000	0.170561
$551$	2.00000	0.0852029
$552$	0	0
$553$	9.00000	0.382719
$554$	60.0000	2.54916
$555$	0	0
$556$	−26.0000	−1.10265
$557$	12.0000	0.508456	0.254228	−	0.967144i	$-0.418179\pi$
$557$	12.0000	0.508456	0.254228	+	0.967144i	$0.418179\pi$
$558$	0	0
$559$	−20.0000	−0.845910
$560$	−12.0000	−0.507093
$561$	0	0
$562$	0	0
$563$	36.0000	1.51722	0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000	1.51722	0.758610	+	0.651546i	$0.225879\pi$
$564$	0	0
$565$	−10.0000	−0.420703
$566$	−24.0000	−1.00880
$567$	0	0
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	5.00000	0.209243	0.104622	−	0.994512i	$-0.466637\pi$
$571$	5.00000	0.209243	0.104622	+	0.994512i	$0.466637\pi$
$572$	20.0000	0.836242
$573$	0	0
$574$	−60.0000	−2.50435
$575$	6.00000	0.250217
$576$	0	0
$577$	17.0000	0.707719	0.353860	−	0.935299i	$-0.384869\pi$
$577$	17.0000	0.707719	0.353860	+	0.935299i	$0.384869\pi$
$578$	−94.0000	−3.90988
$579$	0	0
$580$	−4.00000	−0.166091
$581$	−18.0000	−0.746766
$582$	0	0
$583$	4.00000	0.165663
$584$	0	0
$585$	0	0
$586$	−36.0000	−1.48715
$587$	−18.0000	−0.742940	−0.371470	−	0.928445i	$-0.621146\pi$
$587$	−18.0000	−0.742940	−0.371470	+	0.928445i	$0.621146\pi$
$588$	0	0
$589$	0	0
$590$	−16.0000	−0.658710
$591$	0	0
$592$	−20.0000	−0.821995
$593$	−22.0000	−0.903432	−0.451716	−	0.892162i	$-0.649188\pi$
$593$	−22.0000	−0.903432	−0.451716	+	0.892162i	$0.649188\pi$
$594$	0	0
$595$	−24.0000	−0.983904
$596$	−8.00000	−0.327693
$597$	0	0
$598$	60.0000	2.45358
$599$	−10.0000	−0.408589	−0.204294	−	0.978909i	$-0.565490\pi$
$599$	−10.0000	−0.408589	−0.204294	+	0.978909i	$0.565490\pi$
$600$	0	0
$601$	−34.0000	−1.38689	−0.693444	−	0.720510i	$-0.743908\pi$
$601$	−34.0000	−1.38689	−0.693444	+	0.720510i	$0.743908\pi$
$602$	24.0000	0.978167
$603$	0	0
$604$	2.00000	0.0813788
$605$	7.00000	0.284590
$606$	0	0
$607$	1.00000	0.0405887	0.0202944	−	0.999794i	$-0.493540\pi$
$607$	1.00000	0.0405887	0.0202944	+	0.999794i	$0.493540\pi$
$608$	8.00000	0.324443
$609$	0	0
$610$	14.0000	0.566843
$611$	−20.0000	−0.809113
$612$	0	0
$613$	1.00000	0.0403896	0.0201948	−	0.999796i	$-0.493571\pi$
$613$	1.00000	0.0403896	0.0201948	+	0.999796i	$0.493571\pi$
$614$	−32.0000	−1.29141
$615$	0	0
$616$	0	0
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	0	0
$619$	−19.0000	−0.763674	−0.381837	−	0.924230i	$-0.624709\pi$
$619$	−19.0000	−0.763674	−0.381837	+	0.924230i	$0.624709\pi$
$620$	0	0
$621$	0	0
$622$	48.0000	1.92462
$623$	36.0000	1.44231
$624$	0	0
$625$	1.00000	0.0400000
$626$	14.0000	0.559553
$627$	0	0
$628$	4.00000	0.159617
$629$	−40.0000	−1.59490
$630$	0	0
$631$	−11.0000	−0.437903	−0.218952	−	0.975736i	$-0.570264\pi$
$631$	−11.0000	−0.437903	−0.218952	+	0.975736i	$0.570264\pi$
$632$	0	0
$633$	0	0
$634$	−40.0000	−1.58860
$635$	8.00000	0.317470
$636$	0	0
$637$	−10.0000	−0.396214
$638$	8.00000	0.316723
$639$	0	0
$640$	0	0
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	−12.0000	−0.473234	−0.236617	−	0.971603i	$-0.576039\pi$
$643$	−12.0000	−0.473234	−0.236617	+	0.971603i	$0.576039\pi$
$644$	−36.0000	−1.41860
$645$	0	0
$646$	16.0000	0.629512
$647$	−38.0000	−1.49393	−0.746967	−	0.664861i	$-0.768491\pi$
$647$	−38.0000	−1.49393	−0.746967	+	0.664861i	$0.768491\pi$
$648$	0	0
$649$	16.0000	0.628055
$650$	10.0000	0.392232
$651$	0	0
$652$	−38.0000	−1.48819
$653$	−4.00000	−0.156532	−0.0782660	−	0.996933i	$-0.524938\pi$
$653$	−4.00000	−0.156532	−0.0782660	+	0.996933i	$0.524938\pi$
$654$	0	0
$655$	−12.0000	−0.468879
$656$	40.0000	1.56174
$657$	0	0
$658$	24.0000	0.935617
$659$	−32.0000	−1.24654	−0.623272	−	0.782006i	$-0.714197\pi$
$659$	−32.0000	−1.24654	−0.623272	+	0.782006i	$0.714197\pi$
$660$	0	0
$661$	7.00000	0.272268	0.136134	−	0.990690i	$-0.456532\pi$
$661$	7.00000	0.272268	0.136134	+	0.990690i	$0.456532\pi$
$662$	42.0000	1.63238
$663$	0	0
$664$	0	0
$665$	3.00000	0.116335
$666$	0	0
$667$	12.0000	0.464642
$668$	−24.0000	−0.928588
$669$	0	0
$670$	−18.0000	−0.695401
$671$	−14.0000	−0.540464
$672$	0	0
$673$	33.0000	1.27206	0.636028	−	0.771666i	$-0.280576\pi$
$673$	33.0000	1.27206	0.636028	+	0.771666i	$0.280576\pi$
$674$	−14.0000	−0.539260
$675$	0	0
$676$	24.0000	0.923077
$677$	12.0000	0.461197	0.230599	−	0.973049i	$-0.425932\pi$
$677$	12.0000	0.461197	0.230599	+	0.973049i	$0.425932\pi$
$678$	0	0
$679$	39.0000	1.49668
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−18.0000	−0.688751	−0.344375	−	0.938832i	$-0.611909\pi$
$683$	−18.0000	−0.688751	−0.344375	+	0.938832i	$0.611909\pi$
$684$	0	0
$685$	6.00000	0.229248
$686$	−30.0000	−1.14541
$687$	0	0
$688$	−16.0000	−0.609994
$689$	10.0000	0.380970
$690$	0	0
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	−24.0000	−0.912343
$693$	0	0
$694$	−20.0000	−0.759190
$695$	13.0000	0.493118
$696$	0	0
$697$	80.0000	3.03022
$698$	−38.0000	−1.43832
$699$	0	0
$700$	−6.00000	−0.226779
$701$	−34.0000	−1.28416	−0.642081	−	0.766637i	$-0.721929\pi$
$701$	−34.0000	−1.28416	−0.642081	+	0.766637i	$0.721929\pi$
$702$	0	0
$703$	5.00000	0.188579
$704$	16.0000	0.603023
$705$	0	0
$706$	24.0000	0.903252
$707$	0	0
$708$	0	0
$709$	25.0000	0.938895	0.469447	−	0.882960i	$-0.344453\pi$
$709$	25.0000	0.938895	0.469447	+	0.882960i	$0.344453\pi$
$710$	4.00000	0.150117
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−10.0000	−0.373979
$716$	44.0000	1.64436
$717$	0	0
$718$	36.0000	1.34351
$719$	30.0000	1.11881	0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000	1.11881	0.559406	+	0.828894i	$0.311029\pi$
$720$	0	0
$721$	−51.0000	−1.89934
$722$	36.0000	1.33978
$723$	0	0
$724$	10.0000	0.371647
$725$	2.00000	0.0742781
$726$	0	0
$727$	−16.0000	−0.593407	−0.296704	−	0.954970i	$-0.595887\pi$
$727$	−16.0000	−0.593407	−0.296704	+	0.954970i	$0.595887\pi$
$728$	0	0
$729$	0	0
$730$	−10.0000	−0.370117
$731$	−32.0000	−1.18356
$732$	0	0
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	−42.0000	−1.55025
$735$	0	0
$736$	48.0000	1.76930
$737$	18.0000	0.663039
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	−10.0000	−0.367607
$741$	0	0
$742$	−12.0000	−0.440534
$743$	−32.0000	−1.17397	−0.586983	−	0.809599i	$-0.699684\pi$
$743$	−32.0000	−1.17397	−0.586983	+	0.809599i	$0.699684\pi$
$744$	0	0
$745$	4.00000	0.146549
$746$	22.0000	0.805477
$747$	0	0
$748$	32.0000	1.17004
$749$	−18.0000	−0.657706
$750$	0	0
$751$	25.0000	0.912263	0.456131	−	0.889912i	$-0.349235\pi$
$751$	25.0000	0.912263	0.456131	+	0.889912i	$0.349235\pi$
$752$	−16.0000	−0.583460
$753$	0	0
$754$	20.0000	0.728357
$755$	−1.00000	−0.0363937
$756$	0	0
$757$	−53.0000	−1.92632	−0.963159	−	0.268933i	$-0.913329\pi$
$757$	−53.0000	−1.92632	−0.963159	+	0.268933i	$0.913329\pi$
$758$	22.0000	0.799076
$759$	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
$763$	30.0000	1.08607
$764$	−8.00000	−0.289430
$765$	0	0
$766$	48.0000	1.73431
$767$	40.0000	1.44432
$768$	0	0
$769$	−37.0000	−1.33425	−0.667127	−	0.744944i	$-0.732476\pi$
$769$	−37.0000	−1.33425	−0.667127	+	0.744944i	$0.732476\pi$
$770$	12.0000	0.432450
$771$	0	0
$772$	10.0000	0.359908
$773$	18.0000	0.647415	0.323708	−	0.946157i	$-0.395071\pi$
$773$	18.0000	0.647415	0.323708	+	0.946157i	$0.395071\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	−48.0000	−1.72088
$779$	−10.0000	−0.358287
$780$	0	0
$781$	−4.00000	−0.143131
$782$	96.0000	3.43295
$783$	0	0
$784$	−8.00000	−0.285714
$785$	−2.00000	−0.0713831
$786$	0	0
$787$	−41.0000	−1.46149	−0.730746	−	0.682649i	$-0.760828\pi$
$787$	−41.0000	−1.46149	−0.730746	+	0.682649i	$0.760828\pi$
$788$	0	0
$789$	0	0
$790$	−6.00000	−0.213470
$791$	−30.0000	−1.06668
$792$	0	0
$793$	−35.0000	−1.24289
$794$	4.00000	0.141955
$795$	0	0
$796$	−34.0000	−1.20510
$797$	−16.0000	−0.566749	−0.283375	−	0.959009i	$-0.591454\pi$
$797$	−16.0000	−0.566749	−0.283375	+	0.959009i	$0.591454\pi$
$798$	0	0
$799$	−32.0000	−1.13208
$800$	8.00000	0.282843
$801$	0	0
$802$	−12.0000	−0.423735
$803$	10.0000	0.352892
$804$	0	0
$805$	18.0000	0.634417
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−16.0000	−0.562530	−0.281265	−	0.959630i	$-0.590754\pi$
$809$	−16.0000	−0.562530	−0.281265	+	0.959630i	$0.590754\pi$
$810$	0	0
$811$	−12.0000	−0.421377	−0.210688	−	0.977553i	$-0.567571\pi$
$811$	−12.0000	−0.421377	−0.210688	+	0.977553i	$0.567571\pi$
$812$	−12.0000	−0.421117
$813$	0	0
$814$	20.0000	0.701000
$815$	19.0000	0.665541
$816$	0	0
$817$	4.00000	0.139942
$818$	14.0000	0.489499
$819$	0	0
$820$	20.0000	0.698430
$821$	48.0000	1.67521	0.837606	−	0.546275i	$-0.183955\pi$
$821$	48.0000	1.67521	0.837606	+	0.546275i	$0.183955\pi$
$822$	0	0
$823$	11.0000	0.383436	0.191718	−	0.981450i	$-0.438594\pi$
$823$	11.0000	0.383436	0.191718	+	0.981450i	$0.438594\pi$
$824$	0	0
$825$	0	0
$826$	−48.0000	−1.67013
$827$	28.0000	0.973655	0.486828	−	0.873498i	$-0.338154\pi$
$827$	28.0000	0.973655	0.486828	+	0.873498i	$0.338154\pi$
$828$	0	0
$829$	−51.0000	−1.77130	−0.885652	−	0.464350i	$-0.846288\pi$
$829$	−51.0000	−1.77130	−0.885652	+	0.464350i	$0.846288\pi$
$830$	12.0000	0.416526
$831$	0	0
$832$	40.0000	1.38675
$833$	−16.0000	−0.554367
$834$	0	0
$835$	12.0000	0.415277
$836$	−4.00000	−0.138343
$837$	0	0
$838$	−64.0000	−2.21084
$839$	−4.00000	−0.138095	−0.0690477	−	0.997613i	$-0.521996\pi$
$839$	−4.00000	−0.138095	−0.0690477	+	0.997613i	$0.521996\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−2.00000	−0.0689246
$843$	0	0
$844$	46.0000	1.58339
$845$	−12.0000	−0.412813
$846$	0	0
$847$	21.0000	0.721569
$848$	8.00000	0.274721
$849$	0	0
$850$	16.0000	0.548795
$851$	30.0000	1.02839
$852$	0	0
$853$	21.0000	0.719026	0.359513	−	0.933140i	$-0.382943\pi$
$853$	21.0000	0.719026	0.359513	+	0.933140i	$0.382943\pi$
$854$	42.0000	1.43721
$855$	0	0
$856$	0	0
$857$	−26.0000	−0.888143	−0.444072	−	0.895991i	$-0.646466\pi$
$857$	−26.0000	−0.888143	−0.444072	+	0.895991i	$0.646466\pi$
$858$	0	0
$859$	7.00000	0.238837	0.119418	−	0.992844i	$-0.461897\pi$
$859$	7.00000	0.238837	0.119418	+	0.992844i	$0.461897\pi$
$860$	−8.00000	−0.272798
$861$	0	0
$862$	−60.0000	−2.04361
$863$	−4.00000	−0.136162	−0.0680808	−	0.997680i	$-0.521688\pi$
$863$	−4.00000	−0.136162	−0.0680808	+	0.997680i	$0.521688\pi$
$864$	0	0
$865$	12.0000	0.408012
$866$	76.0000	2.58259
$867$	0	0
$868$	0	0
$869$	6.00000	0.203536
$870$	0	0
$871$	45.0000	1.52477
$872$	0	0
$873$	0	0
$874$	−12.0000	−0.405906
$875$	3.00000	0.101419
$876$	0	0
$877$	9.00000	0.303908	0.151954	−	0.988388i	$-0.451443\pi$
$877$	9.00000	0.303908	0.151954	+	0.988388i	$0.451443\pi$
$878$	64.0000	2.15990
$879$	0	0
$880$	−8.00000	−0.269680
$881$	−38.0000	−1.28025	−0.640126	−	0.768270i	$-0.721118\pi$
$881$	−38.0000	−1.28025	−0.640126	+	0.768270i	$0.721118\pi$
$882$	0	0
$883$	−19.0000	−0.639401	−0.319700	−	0.947519i	$-0.603582\pi$
$883$	−19.0000	−0.639401	−0.319700	+	0.947519i	$0.603582\pi$
$884$	80.0000	2.69069
$885$	0	0
$886$	24.0000	0.806296
$887$	−48.0000	−1.61168	−0.805841	−	0.592132i	$-0.798286\pi$
$887$	−48.0000	−1.61168	−0.805841	+	0.592132i	$0.798286\pi$
$888$	0	0
$889$	24.0000	0.804934
$890$	−24.0000	−0.804482
$891$	0	0
$892$	16.0000	0.535720
$893$	4.00000	0.133855
$894$	0	0
$895$	−22.0000	−0.735379
$896$	0	0
$897$	0	0
$898$	−8.00000	−0.266963
$899$	0	0
$900$	0	0
$901$	16.0000	0.533037
$902$	−40.0000	−1.33185
$903$	0	0
$904$	0	0
$905$	−5.00000	−0.166206
$906$	0	0
$907$	−21.0000	−0.697294	−0.348647	−	0.937254i	$-0.613359\pi$
$907$	−21.0000	−0.697294	−0.348647	+	0.937254i	$0.613359\pi$
$908$	−20.0000	−0.663723
$909$	0	0
$910$	30.0000	0.994490
$911$	40.0000	1.32526	0.662630	−	0.748947i	$-0.269440\pi$
$911$	40.0000	1.32526	0.662630	+	0.748947i	$0.269440\pi$
$912$	0	0
$913$	−12.0000	−0.397142
$914$	−44.0000	−1.45539
$915$	0	0
$916$	12.0000	0.396491
$917$	−36.0000	−1.18882
$918$	0	0
$919$	40.0000	1.31948	0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000	1.31948	0.659739	+	0.751495i	$0.270667\pi$
$920$	0	0
$921$	0	0
$922$	24.0000	0.790398
$923$	−10.0000	−0.329154
$924$	0	0
$925$	5.00000	0.164399
$926$	−66.0000	−2.16889
$927$	0	0
$928$	16.0000	0.525226
$929$	−34.0000	−1.11550	−0.557752	−	0.830008i	$-0.688336\pi$
$929$	−34.0000	−1.11550	−0.557752	+	0.830008i	$0.688336\pi$
$930$	0	0
$931$	2.00000	0.0655474
$932$	−48.0000	−1.57229
$933$	0	0
$934$	56.0000	1.83238
$935$	−16.0000	−0.523256
$936$	0	0
$937$	33.0000	1.07806	0.539032	−	0.842286i	$-0.318790\pi$
$937$	33.0000	1.07806	0.539032	+	0.842286i	$0.318790\pi$
$938$	−54.0000	−1.76316
$939$	0	0
$940$	−8.00000	−0.260931
$941$	−22.0000	−0.717180	−0.358590	−	0.933495i	$-0.616742\pi$
$941$	−22.0000	−0.717180	−0.358590	+	0.933495i	$0.616742\pi$
$942$	0	0
$943$	−60.0000	−1.95387
$944$	32.0000	1.04151
$945$	0	0
$946$	16.0000	0.520205
$947$	48.0000	1.55979	0.779895	−	0.625910i	$-0.215272\pi$
$947$	48.0000	1.55979	0.779895	+	0.625910i	$0.215272\pi$
$948$	0	0
$949$	25.0000	0.811534
$950$	−2.00000	−0.0648886
$951$	0	0
$952$	0	0
$953$	40.0000	1.29573	0.647864	−	0.761756i	$-0.275663\pi$
$953$	40.0000	1.29573	0.647864	+	0.761756i	$0.275663\pi$
$954$	0	0
$955$	4.00000	0.129437
$956$	−52.0000	−1.68180
$957$	0	0
$958$	−12.0000	−0.387702
$959$	18.0000	0.581250
$960$	0	0
$961$	−31.0000	−1.00000
$962$	50.0000	1.61206
$963$	0	0
$964$	2.00000	0.0644157
$965$	−5.00000	−0.160956
$966$	0	0
$967$	−13.0000	−0.418052	−0.209026	−	0.977910i	$-0.567029\pi$
$967$	−13.0000	−0.418052	−0.209026	+	0.977910i	$0.567029\pi$
$968$	0	0
$969$	0	0
$970$	−26.0000	−0.834810
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	0	0
$973$	39.0000	1.25028
$974$	−10.0000	−0.320421
$975$	0	0
$976$	−28.0000	−0.896258
$977$	−62.0000	−1.98356	−0.991778	−	0.127971i	$-0.959153\pi$
$977$	−62.0000	−1.98356	−0.991778	+	0.127971i	$0.959153\pi$
$978$	0	0
$979$	24.0000	0.767043
$980$	−4.00000	−0.127775
$981$	0	0
$982$	44.0000	1.40410
$983$	54.0000	1.72233	0.861166	−	0.508323i	$-0.169735\pi$
$983$	54.0000	1.72233	0.861166	+	0.508323i	$0.169735\pi$
$984$	0	0
$985$	0	0
$986$	32.0000	1.01909
$987$	0	0
$988$	−10.0000	−0.318142
$989$	24.0000	0.763156
$990$	0	0
$991$	−25.0000	−0.794151	−0.397076	−	0.917786i	$-0.629975\pi$
$991$	−25.0000	−0.794151	−0.397076	+	0.917786i	$0.629975\pi$
$992$	0	0
$993$	0	0
$994$	12.0000	0.380617
$995$	17.0000	0.538936
$996$	0	0
$997$	−42.0000	−1.33015	−0.665077	−	0.746775i	$-0.731601\pi$
$997$	−42.0000	−1.33015	−0.665077	+	0.746775i	$0.731601\pi$
$998$	−32.0000	−1.01294
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	135.2.a.a.1.1	✓	1
3.2	odd	2		135.2.a.b.1.1	yes	1
4.3	odd	2		2160.2.a.j.1.1		1
5.2	odd	4		675.2.b.a.649.1		2
5.3	odd	4		675.2.b.a.649.2		2
5.4	even	2		675.2.a.i.1.1		1
7.6	odd	2		6615.2.a.a.1.1		1
8.3	odd	2		8640.2.a.ce.1.1		1
8.5	even	2		8640.2.a.bh.1.1		1
9.2	odd	6		405.2.e.b.271.1		2
9.4	even	3		405.2.e.h.136.1		2
9.5	odd	6		405.2.e.b.136.1		2
9.7	even	3		405.2.e.h.271.1		2
12.11	even	2		2160.2.a.v.1.1		1
15.2	even	4		675.2.b.b.649.2		2
15.8	even	4		675.2.b.b.649.1		2
15.14	odd	2		675.2.a.a.1.1		1
21.20	even	2		6615.2.a.j.1.1		1
24.5	odd	2		8640.2.a.c.1.1		1
24.11	even	2		8640.2.a.bb.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.2.a.a.1.1	✓	1	1.1	even	1	trivial
135.2.a.b.1.1	yes	1	3.2	odd	2
405.2.e.b.136.1		2	9.5	odd	6
405.2.e.b.271.1		2	9.2	odd	6
405.2.e.h.136.1		2	9.4	even	3
405.2.e.h.271.1		2	9.7	even	3
675.2.a.a.1.1		1	15.14	odd	2
675.2.a.i.1.1		1	5.4	even	2
675.2.b.a.649.1		2	5.2	odd	4
675.2.b.a.649.2		2	5.3	odd	4
675.2.b.b.649.1		2	15.8	even	4
675.2.b.b.649.2		2	15.2	even	4
2160.2.a.j.1.1		1	4.3	odd	2
2160.2.a.v.1.1		1	12.11	even	2
6615.2.a.a.1.1		1	7.6	odd	2
6615.2.a.j.1.1		1	21.20	even	2
8640.2.a.c.1.1		1	24.5	odd	2
8640.2.a.bb.1.1		1	24.11	even	2
8640.2.a.bh.1.1		1	8.5	even	2
8640.2.a.ce.1.1		1	8.3	odd	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:	\( N \)	\(=\)	\( 135 = 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	135.a (trivial)

Introduction

L-functions

Modular forms

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Representations

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Database

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