LMFDB - Embedded newform 2550.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2550,2,Mod(1,2550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2550, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2550.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:	$20.3618525154$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 102)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2550.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -4.00000 q^{11} -1.00000 q^{12} +2.00000 q^{13} +1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} +4.00000 q^{22} +1.00000 q^{24} -2.00000 q^{26} -1.00000 q^{27} -10.0000 q^{29} +8.00000 q^{31} -1.00000 q^{32} +4.00000 q^{33} +1.00000 q^{34} +1.00000 q^{36} +2.00000 q^{37} -4.00000 q^{38} -2.00000 q^{39} +10.0000 q^{41} -12.0000 q^{43} -4.00000 q^{44} -1.00000 q^{48} -7.00000 q^{49} +1.00000 q^{51} +2.00000 q^{52} -6.00000 q^{53} +1.00000 q^{54} -4.00000 q^{57} +10.0000 q^{58} +12.0000 q^{59} -10.0000 q^{61} -8.00000 q^{62} +1.00000 q^{64} -4.00000 q^{66} +12.0000 q^{67} -1.00000 q^{68} -1.00000 q^{72} -10.0000 q^{73} -2.00000 q^{74} +4.00000 q^{76} +2.00000 q^{78} -8.00000 q^{79} +1.00000 q^{81} -10.0000 q^{82} -4.00000 q^{83} +12.0000 q^{86} +10.0000 q^{87} +4.00000 q^{88} -6.00000 q^{89} -8.00000 q^{93} +1.00000 q^{96} +14.0000 q^{97} +7.00000 q^{98} -4.00000 q^{99} +O(q^{100})$

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	−1.00000	−0.288675
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−1.00000	−0.235702
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	0	0
$22$	4.00000	0.852803
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	−2.00000	−0.392232
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−10.0000	−1.85695	−0.928477	−	0.371391i	$-0.878881\pi$
$29$	−10.0000	−1.85695	−0.928477	+	0.371391i	$0.878881\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	−1.00000	−0.176777
$33$	4.00000	0.696311
$34$	1.00000	0.171499
$35$	0	0
$36$	1.00000	0.166667
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	−4.00000	−0.648886
$39$	−2.00000	−0.320256
$40$	0	0
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	−12.0000	−1.82998	−0.914991	−	0.403473i	$-0.867803\pi$
$43$	−12.0000	−1.82998	−0.914991	+	0.403473i	$0.867803\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	−1.00000	−0.144338
$49$	−7.00000	−1.00000
$50$	0	0
$51$	1.00000	0.140028
$52$	2.00000	0.277350
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	0	0
$57$	−4.00000	−0.529813
$58$	10.0000	1.31306
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	−8.00000	−1.01600
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	−4.00000	−0.492366
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	−1.00000	−0.121268
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	−1.00000	−0.117851
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	4.00000	0.458831
$77$	0	0
$78$	2.00000	0.226455
$79$	−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
$82$	−10.0000	−1.10432
$83$	−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
$86$	12.0000	1.29399
$87$	10.0000	1.07211
$88$	4.00000	0.426401
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−8.00000	−0.829561
$94$	0	0
$95$	0	0
$96$	1.00000	0.102062
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	7.00000	0.707107
$99$	−4.00000	−0.402015
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	−1.00000	−0.0990148
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	6.00000	0.582772
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	−1.00000	−0.0962250
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	0	0
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	4.00000	0.374634
$115$	0	0
$116$	−10.0000	−0.928477
$117$	2.00000	0.184900
$118$	−12.0000	−1.10469
$119$	0	0
$120$	0	0
$121$	5.00000	0.454545
$122$	10.0000	0.905357
$123$	−10.0000	−0.901670
$124$	8.00000	0.718421
$125$	0	0
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	−1.00000	−0.0883883
$129$	12.0000	1.05654
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	4.00000	0.348155
$133$	0	0
$134$	−12.0000	−1.03664
$135$	0	0
$136$	1.00000	0.0857493
$137$	−10.0000	−0.854358	−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000	−0.854358	−0.427179	+	0.904167i	$0.640493\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−8.00000	−0.668994
$144$	1.00000	0.0833333
$145$	0	0
$146$	10.0000	0.827606
$147$	7.00000	0.577350
$148$	2.00000	0.164399
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	24.0000	1.95309	0.976546	−	0.215308i	$-0.0690756\pi$
$151$	24.0000	1.95309	0.976546	+	0.215308i	$0.0690756\pi$
$152$	−4.00000	−0.324443
$153$	−1.00000	−0.0808452
$154$	0	0
$155$	0	0
$156$	−2.00000	−0.160128
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	8.00000	0.636446
$159$	6.00000	0.475831
$160$	0	0
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	10.0000	0.780869
$165$	0	0
$166$	4.00000	0.310460
$167$	−16.0000	−1.23812	−0.619059	−	0.785345i	$-0.712486\pi$
$167$	−16.0000	−1.23812	−0.619059	+	0.785345i	$0.712486\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	4.00000	0.305888
$172$	−12.0000	−0.914991
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	−10.0000	−0.758098
$175$	0	0
$176$	−4.00000	−0.301511
$177$	−12.0000	−0.901975
$178$	6.00000	0.449719
$179$	−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
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	0	0
$183$	10.0000	0.739221
$184$	0	0
$185$	0	0
$186$	8.00000	0.586588
$187$	4.00000	0.292509
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	−1.00000	−0.0721688
$193$	−18.0000	−1.29567	−0.647834	−	0.761781i	$-0.724325\pi$
$193$	−18.0000	−1.29567	−0.647834	+	0.761781i	$0.724325\pi$
$194$	−14.0000	−1.00514
$195$	0	0
$196$	−7.00000	−0.500000
$197$	−14.0000	−0.997459	−0.498729	−	0.866758i	$-0.666200\pi$
$197$	−14.0000	−0.997459	−0.498729	+	0.866758i	$0.666200\pi$
$198$	4.00000	0.284268
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	−12.0000	−0.846415
$202$	10.0000	0.703598
$203$	0	0
$204$	1.00000	0.0700140
$205$	0	0
$206$	−8.00000	−0.557386
$207$	0	0
$208$	2.00000	0.138675
$209$	−16.0000	−1.10674
$210$	0	0
$211$	−28.0000	−1.92760	−0.963800	−	0.266627i	$-0.914091\pi$
$211$	−28.0000	−1.92760	−0.963800	+	0.266627i	$0.914091\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	−4.00000	−0.273434
$215$	0	0
$216$	1.00000	0.0680414
$217$	0	0
$218$	10.0000	0.677285
$219$	10.0000	0.675737
$220$	0	0
$221$	−2.00000	−0.134535
$222$	2.00000	0.134231
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	0	0
$225$	0	0
$226$	2.00000	0.133038
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	−4.00000	−0.264906
$229$	−26.0000	−1.71813	−0.859064	−	0.511868i	$-0.828954\pi$
$229$	−26.0000	−1.71813	−0.859064	+	0.511868i	$0.828954\pi$
$230$	0	0
$231$	0	0
$232$	10.0000	0.656532
$233$	−26.0000	−1.70332	−0.851658	−	0.524097i	$-0.824403\pi$
$233$	−26.0000	−1.70332	−0.851658	+	0.524097i	$0.824403\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	12.0000	0.781133
$237$	8.00000	0.519656
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	−5.00000	−0.321412
$243$	−1.00000	−0.0641500
$244$	−10.0000	−0.640184
$245$	0	0
$246$	10.0000	0.637577
$247$	8.00000	0.509028
$248$	−8.00000	−0.508001
$249$	4.00000	0.253490
$250$	0	0
$251$	28.0000	1.76734	0.883672	−	0.468106i	$-0.155064\pi$
$251$	28.0000	1.76734	0.883672	+	0.468106i	$0.155064\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	1.00000	0.0625000
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	−12.0000	−0.747087
$259$	0	0
$260$	0	0
$261$	−10.0000	−0.618984
$262$	12.0000	0.741362
$263$	8.00000	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$264$	−4.00000	−0.246183
$265$	0	0
$266$	0	0
$267$	6.00000	0.367194
$268$	12.0000	0.733017
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	−1.00000	−0.0606339
$273$	0	0
$274$	10.0000	0.604122
$275$	0	0
$276$	0	0
$277$	−30.0000	−1.80253	−0.901263	−	0.433273i	$-0.857359\pi$
$277$	−30.0000	−1.80253	−0.901263	+	0.433273i	$0.857359\pi$
$278$	4.00000	0.239904
$279$	8.00000	0.478947
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	−12.0000	−0.713326	−0.356663	−	0.934233i	$-0.616086\pi$
$283$	−12.0000	−0.713326	−0.356663	+	0.934233i	$0.616086\pi$
$284$	0	0
$285$	0	0
$286$	8.00000	0.473050
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	1.00000	0.0588235
$290$	0	0
$291$	−14.0000	−0.820695
$292$	−10.0000	−0.585206
$293$	26.0000	1.51894	0.759468	−	0.650545i	$-0.225459\pi$
$293$	26.0000	1.51894	0.759468	+	0.650545i	$0.225459\pi$
$294$	−7.00000	−0.408248
$295$	0	0
$296$	−2.00000	−0.116248
$297$	4.00000	0.232104
$298$	10.0000	0.579284
$299$	0	0
$300$	0	0
$301$	0	0
$302$	−24.0000	−1.38104
$303$	10.0000	0.574485
$304$	4.00000	0.229416
$305$	0	0
$306$	1.00000	0.0571662
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	2.00000	0.113228
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	−8.00000	−0.450035
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	−6.00000	−0.336463
$319$	40.0000	2.23957
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	−4.00000	−0.222566
$324$	1.00000	0.0555556
$325$	0	0
$326$	4.00000	0.221540
$327$	10.0000	0.553001
$328$	−10.0000	−0.552158
$329$	0	0
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	−4.00000	−0.219529
$333$	2.00000	0.109599
$334$	16.0000	0.875481
$335$	0	0
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	9.00000	0.489535
$339$	2.00000	0.108625
$340$	0	0
$341$	−32.0000	−1.73290
$342$	−4.00000	−0.216295
$343$	0	0
$344$	12.0000	0.646997
$345$	0	0
$346$	6.00000	0.322562
$347$	−28.0000	−1.50312	−0.751559	−	0.659665i	$-0.770698\pi$
$347$	−28.0000	−1.50312	−0.751559	+	0.659665i	$0.770698\pi$
$348$	10.0000	0.536056
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	4.00000	0.213201
$353$	30.0000	1.59674	0.798369	−	0.602168i	$-0.205696\pi$
$353$	30.0000	1.59674	0.798369	+	0.602168i	$0.205696\pi$
$354$	12.0000	0.637793
$355$	0	0
$356$	−6.00000	−0.317999
$357$	0	0
$358$	12.0000	0.634220
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	−14.0000	−0.735824
$363$	−5.00000	−0.262432
$364$	0	0
$365$	0	0
$366$	−10.0000	−0.522708
$367$	−24.0000	−1.25279	−0.626395	−	0.779506i	$-0.715470\pi$
$367$	−24.0000	−1.25279	−0.626395	+	0.779506i	$0.715470\pi$
$368$	0	0
$369$	10.0000	0.520579
$370$	0	0
$371$	0	0
$372$	−8.00000	−0.414781
$373$	−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
$376$	0	0
$377$	−20.0000	−1.03005
$378$	0	0
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	0	0
$381$	0	0
$382$	16.0000	0.818631
$383$	16.0000	0.817562	0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000	0.817562	0.408781	+	0.912633i	$0.365954\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	18.0000	0.916176
$387$	−12.0000	−0.609994
$388$	14.0000	0.710742
$389$	−26.0000	−1.31825	−0.659126	−	0.752032i	$-0.729074\pi$
$389$	−26.0000	−1.31825	−0.659126	+	0.752032i	$0.729074\pi$
$390$	0	0
$391$	0	0
$392$	7.00000	0.353553
$393$	12.0000	0.605320
$394$	14.0000	0.705310
$395$	0	0
$396$	−4.00000	−0.201008
$397$	26.0000	1.30490	0.652451	−	0.757831i	$-0.273741\pi$
$397$	26.0000	1.30490	0.652451	+	0.757831i	$0.273741\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−14.0000	−0.699127	−0.349563	−	0.936913i	$-0.613670\pi$
$401$	−14.0000	−0.699127	−0.349563	+	0.936913i	$0.613670\pi$
$402$	12.0000	0.598506
$403$	16.0000	0.797017
$404$	−10.0000	−0.497519
$405$	0	0
$406$	0	0
$407$	−8.00000	−0.396545
$408$	−1.00000	−0.0495074
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	0	0
$411$	10.0000	0.493264
$412$	8.00000	0.394132
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	4.00000	0.195881
$418$	16.0000	0.782586
$419$	4.00000	0.195413	0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000	0.195413	0.0977064	+	0.995215i	$0.468849\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	28.0000	1.36302
$423$	0	0
$424$	6.00000	0.291386
$425$	0	0
$426$	0	0
$427$	0	0
$428$	4.00000	0.193347
$429$	8.00000	0.386244
$430$	0	0
$431$	−8.00000	−0.385346	−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000	−0.385346	−0.192673	+	0.981263i	$0.561716\pi$
$432$	−1.00000	−0.0481125
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	0	0
$435$	0	0
$436$	−10.0000	−0.478913
$437$	0	0
$438$	−10.0000	−0.477818
$439$	16.0000	0.763638	0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000	0.763638	0.381819	+	0.924237i	$0.375298\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	2.00000	0.0951303
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	−2.00000	−0.0949158
$445$	0	0
$446$	16.0000	0.757622
$447$	10.0000	0.472984
$448$	0	0
$449$	2.00000	0.0943858	0.0471929	−	0.998886i	$-0.484972\pi$
$449$	2.00000	0.0943858	0.0471929	+	0.998886i	$0.484972\pi$
$450$	0	0
$451$	−40.0000	−1.88353
$452$	−2.00000	−0.0940721
$453$	−24.0000	−1.12762
$454$	4.00000	0.187729
$455$	0	0
$456$	4.00000	0.187317
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	26.0000	1.21490
$459$	1.00000	0.0466760
$460$	0	0
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	0	0
$463$	−32.0000	−1.48717	−0.743583	−	0.668644i	$-0.766875\pi$
$463$	−32.0000	−1.48717	−0.743583	+	0.668644i	$0.766875\pi$
$464$	−10.0000	−0.464238
$465$	0	0
$466$	26.0000	1.20443
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	2.00000	0.0924500
$469$	0	0
$470$	0	0
$471$	−2.00000	−0.0921551
$472$	−12.0000	−0.552345
$473$	48.0000	2.20704
$474$	−8.00000	−0.367452
$475$	0	0
$476$	0	0
$477$	−6.00000	−0.274721
$478$	0	0
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	−2.00000	−0.0910975
$483$	0	0
$484$	5.00000	0.227273
$485$	0	0
$486$	1.00000	0.0453609
$487$	16.0000	0.725029	0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000	0.725029	0.362515	+	0.931978i	$0.381918\pi$
$488$	10.0000	0.452679
$489$	4.00000	0.180886
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	−10.0000	−0.450835
$493$	10.0000	0.450377
$494$	−8.00000	−0.359937
$495$	0	0
$496$	8.00000	0.359211
$497$	0	0
$498$	−4.00000	−0.179244
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	16.0000	0.714827
$502$	−28.0000	−1.24970
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	9.00000	0.399704
$508$	0	0
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	−4.00000	−0.176604
$514$	2.00000	0.0882162
$515$	0	0
$516$	12.0000	0.528271
$517$	0	0
$518$	0	0
$519$	6.00000	0.263371
$520$	0	0
$521$	−22.0000	−0.963837	−0.481919	−	0.876216i	$-0.660060\pi$
$521$	−22.0000	−0.963837	−0.481919	+	0.876216i	$0.660060\pi$
$522$	10.0000	0.437688
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	−8.00000	−0.348817
$527$	−8.00000	−0.348485
$528$	4.00000	0.174078
$529$	−23.0000	−1.00000
$530$	0	0
$531$	12.0000	0.520756
$532$	0	0
$533$	20.0000	0.866296
$534$	−6.00000	−0.259645
$535$	0	0
$536$	−12.0000	−0.518321
$537$	12.0000	0.517838
$538$	−6.00000	−0.258678
$539$	28.0000	1.20605
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	−16.0000	−0.687259
$543$	−14.0000	−0.600798
$544$	1.00000	0.0428746
$545$	0	0
$546$	0	0
$547$	12.0000	0.513083	0.256541	−	0.966533i	$-0.417417\pi$
$547$	12.0000	0.513083	0.256541	+	0.966533i	$0.417417\pi$
$548$	−10.0000	−0.427179
$549$	−10.0000	−0.426790
$550$	0	0
$551$	−40.0000	−1.70406
$552$	0	0
$553$	0	0
$554$	30.0000	1.27458
$555$	0	0
$556$	−4.00000	−0.169638
$557$	34.0000	1.44063	0.720313	−	0.693649i	$-0.243998\pi$
$557$	34.0000	1.44063	0.720313	+	0.693649i	$0.243998\pi$
$558$	−8.00000	−0.338667
$559$	−24.0000	−1.01509
$560$	0	0
$561$	−4.00000	−0.168880
$562$	6.00000	0.253095
$563$	−36.0000	−1.51722	−0.758610	−	0.651546i	$-0.774121\pi$
$563$	−36.0000	−1.51722	−0.758610	+	0.651546i	$0.774121\pi$
$564$	0	0
$565$	0	0
$566$	12.0000	0.504398
$567$	0	0
$568$	0	0
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	28.0000	1.17176	0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000	1.17176	0.585882	+	0.810397i	$0.300748\pi$
$572$	−8.00000	−0.334497
$573$	16.0000	0.668410
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	−1.00000	−0.0415945
$579$	18.0000	0.748054
$580$	0	0
$581$	0	0
$582$	14.0000	0.580319
$583$	24.0000	0.993978
$584$	10.0000	0.413803
$585$	0	0
$586$	−26.0000	−1.07405
$587$	20.0000	0.825488	0.412744	−	0.910847i	$-0.364570\pi$
$587$	20.0000	0.825488	0.412744	+	0.910847i	$0.364570\pi$
$588$	7.00000	0.288675
$589$	32.0000	1.31854
$590$	0	0
$591$	14.0000	0.575883
$592$	2.00000	0.0821995
$593$	14.0000	0.574911	0.287456	−	0.957794i	$-0.407191\pi$
$593$	14.0000	0.574911	0.287456	+	0.957794i	$0.407191\pi$
$594$	−4.00000	−0.164122
$595$	0	0
$596$	−10.0000	−0.409616
$597$	0	0
$598$	0	0
$599$	−8.00000	−0.326871	−0.163436	−	0.986554i	$-0.552258\pi$
$599$	−8.00000	−0.326871	−0.163436	+	0.986554i	$0.552258\pi$
$600$	0	0
$601$	−6.00000	−0.244745	−0.122373	−	0.992484i	$-0.539050\pi$
$601$	−6.00000	−0.244745	−0.122373	+	0.992484i	$0.539050\pi$
$602$	0	0
$603$	12.0000	0.488678
$604$	24.0000	0.976546
$605$	0	0
$606$	−10.0000	−0.406222
$607$	24.0000	0.974130	0.487065	−	0.873366i	$-0.338067\pi$
$607$	24.0000	0.974130	0.487065	+	0.873366i	$0.338067\pi$
$608$	−4.00000	−0.162221
$609$	0	0
$610$	0	0
$611$	0	0
$612$	−1.00000	−0.0404226
$613$	−38.0000	−1.53481	−0.767403	−	0.641165i	$-0.778451\pi$
$613$	−38.0000	−1.53481	−0.767403	+	0.641165i	$0.778451\pi$
$614$	−12.0000	−0.484281
$615$	0	0
$616$	0	0
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	8.00000	0.321807
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	−2.00000	−0.0800641
$625$	0	0
$626$	10.0000	0.399680
$627$	16.0000	0.638978
$628$	2.00000	0.0798087
$629$	−2.00000	−0.0797452
$630$	0	0
$631$	−24.0000	−0.955425	−0.477712	−	0.878516i	$-0.658534\pi$
$631$	−24.0000	−0.955425	−0.477712	+	0.878516i	$0.658534\pi$
$632$	8.00000	0.318223
$633$	28.0000	1.11290
$634$	6.00000	0.238290
$635$	0	0
$636$	6.00000	0.237915
$637$	−14.0000	−0.554700
$638$	−40.0000	−1.58362
$639$	0	0
$640$	0	0
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	4.00000	0.157867
$643$	28.0000	1.10421	0.552106	−	0.833774i	$-0.313824\pi$
$643$	28.0000	1.10421	0.552106	+	0.833774i	$0.313824\pi$
$644$	0	0
$645$	0	0
$646$	4.00000	0.157378
$647$	40.0000	1.57256	0.786281	−	0.617869i	$-0.212004\pi$
$647$	40.0000	1.57256	0.786281	+	0.617869i	$0.212004\pi$
$648$	−1.00000	−0.0392837
$649$	−48.0000	−1.88416
$650$	0	0
$651$	0	0
$652$	−4.00000	−0.156652
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	−10.0000	−0.391031
$655$	0	0
$656$	10.0000	0.390434
$657$	−10.0000	−0.390137
$658$	0	0
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	0	0
$661$	6.00000	0.233373	0.116686	−	0.993169i	$-0.462773\pi$
$661$	6.00000	0.233373	0.116686	+	0.993169i	$0.462773\pi$
$662$	20.0000	0.777322
$663$	2.00000	0.0776736
$664$	4.00000	0.155230
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	0	0
$668$	−16.0000	−0.619059
$669$	16.0000	0.618596
$670$	0	0
$671$	40.0000	1.54418
$672$	0	0
$673$	46.0000	1.77317	0.886585	−	0.462566i	$-0.153071\pi$
$673$	46.0000	1.77317	0.886585	+	0.462566i	$0.153071\pi$
$674$	−14.0000	−0.539260
$675$	0	0
$676$	−9.00000	−0.346154
$677$	−46.0000	−1.76792	−0.883962	−	0.467559i	$-0.845134\pi$
$677$	−46.0000	−1.76792	−0.883962	+	0.467559i	$0.845134\pi$
$678$	−2.00000	−0.0768095
$679$	0	0
$680$	0	0
$681$	4.00000	0.153280
$682$	32.0000	1.22534
$683$	20.0000	0.765279	0.382639	−	0.923898i	$-0.375015\pi$
$683$	20.0000	0.765279	0.382639	+	0.923898i	$0.375015\pi$
$684$	4.00000	0.152944
$685$	0	0
$686$	0	0
$687$	26.0000	0.991962
$688$	−12.0000	−0.457496
$689$	−12.0000	−0.457164
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	−6.00000	−0.228086
$693$	0	0
$694$	28.0000	1.06287
$695$	0	0
$696$	−10.0000	−0.379049
$697$	−10.0000	−0.378777
$698$	−14.0000	−0.529908
$699$	26.0000	0.983410
$700$	0	0
$701$	46.0000	1.73740	0.868698	−	0.495342i	$-0.164957\pi$
$701$	46.0000	1.73740	0.868698	+	0.495342i	$0.164957\pi$
$702$	2.00000	0.0754851
$703$	8.00000	0.301726
$704$	−4.00000	−0.150756
$705$	0	0
$706$	−30.0000	−1.12906
$707$	0	0
$708$	−12.0000	−0.450988
$709$	46.0000	1.72757	0.863783	−	0.503864i	$-0.168089\pi$
$709$	46.0000	1.72757	0.863783	+	0.503864i	$0.168089\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	6.00000	0.224860
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−12.0000	−0.448461
$717$	0	0
$718$	24.0000	0.895672
$719$	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
$721$	0	0
$722$	3.00000	0.111648
$723$	−2.00000	−0.0743808
$724$	14.0000	0.520306
$725$	0	0
$726$	5.00000	0.185567
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	12.0000	0.443836
$732$	10.0000	0.369611
$733$	−46.0000	−1.69905	−0.849524	−	0.527549i	$-0.823111\pi$
$733$	−46.0000	−1.69905	−0.849524	+	0.527549i	$0.823111\pi$
$734$	24.0000	0.885856
$735$	0	0
$736$	0	0
$737$	−48.0000	−1.76810
$738$	−10.0000	−0.368105
$739$	52.0000	1.91285	0.956425	−	0.291977i	$-0.0943129\pi$
$739$	52.0000	1.91285	0.956425	+	0.291977i	$0.0943129\pi$
$740$	0	0
$741$	−8.00000	−0.293887
$742$	0	0
$743$	16.0000	0.586983	0.293492	−	0.955962i	$-0.405183\pi$
$743$	16.0000	0.586983	0.293492	+	0.955962i	$0.405183\pi$
$744$	8.00000	0.293294
$745$	0	0
$746$	6.00000	0.219676
$747$	−4.00000	−0.146352
$748$	4.00000	0.146254
$749$	0	0
$750$	0	0
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	0	0
$753$	−28.0000	−1.02038
$754$	20.0000	0.728357
$755$	0	0
$756$	0	0
$757$	10.0000	0.363456	0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000	0.363456	0.181728	+	0.983349i	$0.441831\pi$
$758$	4.00000	0.145287
$759$	0	0
$760$	0	0
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	0	0
$763$	0	0
$764$	−16.0000	−0.578860
$765$	0	0
$766$	−16.0000	−0.578103
$767$	24.0000	0.866590
$768$	−1.00000	−0.0360844
$769$	−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
$771$	2.00000	0.0720282
$772$	−18.0000	−0.647834
$773$	−38.0000	−1.36677	−0.683383	−	0.730061i	$-0.739492\pi$
$773$	−38.0000	−1.36677	−0.683383	+	0.730061i	$0.739492\pi$
$774$	12.0000	0.431331
$775$	0	0
$776$	−14.0000	−0.502571
$777$	0	0
$778$	26.0000	0.932145
$779$	40.0000	1.43315
$780$	0	0
$781$	0	0
$782$	0	0
$783$	10.0000	0.357371
$784$	−7.00000	−0.250000
$785$	0	0
$786$	−12.0000	−0.428026
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	−14.0000	−0.498729
$789$	−8.00000	−0.284808
$790$	0	0
$791$	0	0
$792$	4.00000	0.142134
$793$	−20.0000	−0.710221
$794$	−26.0000	−0.922705
$795$	0	0
$796$	0	0
$797$	−14.0000	−0.495905	−0.247953	−	0.968772i	$-0.579758\pi$
$797$	−14.0000	−0.495905	−0.247953	+	0.968772i	$0.579758\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−6.00000	−0.212000
$802$	14.0000	0.494357
$803$	40.0000	1.41157
$804$	−12.0000	−0.423207
$805$	0	0
$806$	−16.0000	−0.563576
$807$	−6.00000	−0.211210
$808$	10.0000	0.351799
$809$	−22.0000	−0.773479	−0.386739	−	0.922189i	$-0.626399\pi$
$809$	−22.0000	−0.773479	−0.386739	+	0.922189i	$0.626399\pi$
$810$	0	0
$811$	−20.0000	−0.702295	−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000	−0.702295	−0.351147	+	0.936320i	$0.614208\pi$
$812$	0	0
$813$	−16.0000	−0.561144
$814$	8.00000	0.280400
$815$	0	0
$816$	1.00000	0.0350070
$817$	−48.0000	−1.67931
$818$	−26.0000	−0.909069
$819$	0	0
$820$	0	0
$821$	−34.0000	−1.18661	−0.593304	−	0.804978i	$-0.702177\pi$
$821$	−34.0000	−1.18661	−0.593304	+	0.804978i	$0.702177\pi$
$822$	−10.0000	−0.348790
$823$	−32.0000	−1.11545	−0.557725	−	0.830026i	$-0.688326\pi$
$823$	−32.0000	−1.11545	−0.557725	+	0.830026i	$0.688326\pi$
$824$	−8.00000	−0.278693
$825$	0	0
$826$	0	0
$827$	20.0000	0.695468	0.347734	−	0.937593i	$-0.386951\pi$
$827$	20.0000	0.695468	0.347734	+	0.937593i	$0.386951\pi$
$828$	0	0
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	0	0
$831$	30.0000	1.04069
$832$	2.00000	0.0693375
$833$	7.00000	0.242536
$834$	−4.00000	−0.138509
$835$	0	0
$836$	−16.0000	−0.553372
$837$	−8.00000	−0.276520
$838$	−4.00000	−0.138178
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	−22.0000	−0.758170
$843$	6.00000	0.206651
$844$	−28.0000	−0.963800
$845$	0	0
$846$	0	0
$847$	0	0
$848$	−6.00000	−0.206041
$849$	12.0000	0.411839
$850$	0	0
$851$	0	0
$852$	0	0
$853$	2.00000	0.0684787	0.0342393	−	0.999414i	$-0.489099\pi$
$853$	2.00000	0.0684787	0.0342393	+	0.999414i	$0.489099\pi$
$854$	0	0
$855$	0	0
$856$	−4.00000	−0.136717
$857$	6.00000	0.204956	0.102478	−	0.994735i	$-0.467323\pi$
$857$	6.00000	0.204956	0.102478	+	0.994735i	$0.467323\pi$
$858$	−8.00000	−0.273115
$859$	−36.0000	−1.22830	−0.614152	−	0.789188i	$-0.710502\pi$
$859$	−36.0000	−1.22830	−0.614152	+	0.789188i	$0.710502\pi$
$860$	0	0
$861$	0	0
$862$	8.00000	0.272481
$863$	32.0000	1.08929	0.544646	−	0.838666i	$-0.316664\pi$
$863$	32.0000	1.08929	0.544646	+	0.838666i	$0.316664\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−14.0000	−0.475739
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	32.0000	1.08553
$870$	0	0
$871$	24.0000	0.813209
$872$	10.0000	0.338643
$873$	14.0000	0.473828
$874$	0	0
$875$	0	0
$876$	10.0000	0.337869
$877$	−22.0000	−0.742887	−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000	−0.742887	−0.371444	+	0.928456i	$0.621137\pi$
$878$	−16.0000	−0.539974
$879$	−26.0000	−0.876958
$880$	0	0
$881$	2.00000	0.0673817	0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000	0.0673817	0.0336909	+	0.999432i	$0.489274\pi$
$882$	7.00000	0.235702
$883$	−36.0000	−1.21150	−0.605748	−	0.795656i	$-0.707126\pi$
$883$	−36.0000	−1.21150	−0.605748	+	0.795656i	$0.707126\pi$
$884$	−2.00000	−0.0672673
$885$	0	0
$886$	−4.00000	−0.134383
$887$	48.0000	1.61168	0.805841	−	0.592132i	$-0.201714\pi$
$887$	48.0000	1.61168	0.805841	+	0.592132i	$0.201714\pi$
$888$	2.00000	0.0671156
$889$	0	0
$890$	0	0
$891$	−4.00000	−0.134005
$892$	−16.0000	−0.535720
$893$	0	0
$894$	−10.0000	−0.334450
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−2.00000	−0.0667409
$899$	−80.0000	−2.66815
$900$	0	0
$901$	6.00000	0.199889
$902$	40.0000	1.33185
$903$	0	0
$904$	2.00000	0.0665190
$905$	0	0
$906$	24.0000	0.797347
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	−4.00000	−0.132745
$909$	−10.0000	−0.331679
$910$	0	0
$911$	8.00000	0.265052	0.132526	−	0.991180i	$-0.457691\pi$
$911$	8.00000	0.265052	0.132526	+	0.991180i	$0.457691\pi$
$912$	−4.00000	−0.132453
$913$	16.0000	0.529523
$914$	10.0000	0.330771
$915$	0	0
$916$	−26.0000	−0.859064
$917$	0	0
$918$	−1.00000	−0.0330049
$919$	24.0000	0.791687	0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000	0.791687	0.395843	+	0.918318i	$0.370452\pi$
$920$	0	0
$921$	−12.0000	−0.395413
$922$	−30.0000	−0.987997
$923$	0	0
$924$	0	0
$925$	0	0
$926$	32.0000	1.05159
$927$	8.00000	0.262754
$928$	10.0000	0.328266
$929$	2.00000	0.0656179	0.0328089	−	0.999462i	$-0.489555\pi$
$929$	2.00000	0.0656179	0.0328089	+	0.999462i	$0.489555\pi$
$930$	0	0
$931$	−28.0000	−0.917663
$932$	−26.0000	−0.851658
$933$	0	0
$934$	−12.0000	−0.392652
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	22.0000	0.718709	0.359354	−	0.933201i	$-0.382997\pi$
$937$	22.0000	0.718709	0.359354	+	0.933201i	$0.382997\pi$
$938$	0	0
$939$	10.0000	0.326338
$940$	0	0
$941$	−10.0000	−0.325991	−0.162995	−	0.986627i	$-0.552116\pi$
$941$	−10.0000	−0.325991	−0.162995	+	0.986627i	$0.552116\pi$
$942$	2.00000	0.0651635
$943$	0	0
$944$	12.0000	0.390567
$945$	0	0
$946$	−48.0000	−1.56061
$947$	28.0000	0.909878	0.454939	−	0.890523i	$-0.349661\pi$
$947$	28.0000	0.909878	0.454939	+	0.890523i	$0.349661\pi$
$948$	8.00000	0.259828
$949$	−20.0000	−0.649227
$950$	0	0
$951$	6.00000	0.194563
$952$	0	0
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	0	0
$957$	−40.0000	−1.29302
$958$	−24.0000	−0.775405
$959$	0	0
$960$	0	0
$961$	33.0000	1.06452
$962$	−4.00000	−0.128965
$963$	4.00000	0.128898
$964$	2.00000	0.0644157
$965$	0	0
$966$	0	0
$967$	8.00000	0.257263	0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000	0.257263	0.128631	+	0.991692i	$0.458942\pi$
$968$	−5.00000	−0.160706
$969$	4.00000	0.128499
$970$	0	0
$971$	−4.00000	−0.128366	−0.0641831	−	0.997938i	$-0.520444\pi$
$971$	−4.00000	−0.128366	−0.0641831	+	0.997938i	$0.520444\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	−16.0000	−0.512673
$975$	0	0
$976$	−10.0000	−0.320092
$977$	46.0000	1.47167	0.735835	−	0.677161i	$-0.236790\pi$
$977$	46.0000	1.47167	0.735835	+	0.677161i	$0.236790\pi$
$978$	−4.00000	−0.127906
$979$	24.0000	0.767043
$980$	0	0
$981$	−10.0000	−0.319275
$982$	−12.0000	−0.382935
$983$	−48.0000	−1.53096	−0.765481	−	0.643458i	$-0.777499\pi$
$983$	−48.0000	−1.53096	−0.765481	+	0.643458i	$0.777499\pi$
$984$	10.0000	0.318788
$985$	0	0
$986$	−10.0000	−0.318465
$987$	0	0
$988$	8.00000	0.254514
$989$	0	0
$990$	0	0
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	−8.00000	−0.254000
$993$	20.0000	0.634681
$994$	0	0
$995$	0	0
$996$	4.00000	0.126745
$997$	2.00000	0.0633406	0.0316703	−	0.999498i	$-0.489917\pi$
$997$	2.00000	0.0633406	0.0316703	+	0.999498i	$0.489917\pi$
$998$	−4.00000	−0.126618
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2550.2.a.c.1.1		1
3.2	odd	2		7650.2.a.ca.1.1		1
5.2	odd	4		2550.2.d.m.2449.1		2
5.3	odd	4		2550.2.d.m.2449.2		2
5.4	even	2		102.2.a.c.1.1	✓	1
15.14	odd	2		306.2.a.b.1.1		1
20.19	odd	2		816.2.a.b.1.1		1
35.34	odd	2		4998.2.a.be.1.1		1
40.19	odd	2		3264.2.a.bc.1.1		1
40.29	even	2		3264.2.a.m.1.1		1
60.59	even	2		2448.2.a.p.1.1		1
85.4	even	4		1734.2.b.b.577.1		2
85.9	even	8		1734.2.f.e.829.2		4
85.19	even	8		1734.2.f.e.1483.2		4
85.49	even	8		1734.2.f.e.1483.1		4
85.59	even	8		1734.2.f.e.829.1		4
85.64	even	4		1734.2.b.b.577.2		2
85.84	even	2		1734.2.a.j.1.1		1
120.29	odd	2		9792.2.a.k.1.1		1
120.59	even	2		9792.2.a.l.1.1		1
255.254	odd	2		5202.2.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
102.2.a.c.1.1	✓	1	5.4	even	2
306.2.a.b.1.1		1	15.14	odd	2
816.2.a.b.1.1		1	20.19	odd	2
1734.2.a.j.1.1		1	85.84	even	2
1734.2.b.b.577.1		2	85.4	even	4
1734.2.b.b.577.2		2	85.64	even	4
1734.2.f.e.829.1		4	85.59	even	8
1734.2.f.e.829.2		4	85.9	even	8
1734.2.f.e.1483.1		4	85.49	even	8
1734.2.f.e.1483.2		4	85.19	even	8
2448.2.a.p.1.1		1	60.59	even	2
2550.2.a.c.1.1		1	1.1	even	1	trivial
2550.2.d.m.2449.1		2	5.2	odd	4
2550.2.d.m.2449.2		2	5.3	odd	4
3264.2.a.m.1.1		1	40.29	even	2
3264.2.a.bc.1.1		1	40.19	odd	2
4998.2.a.be.1.1		1	35.34	odd	2
5202.2.a.c.1.1		1	255.254	odd	2
7650.2.a.ca.1.1		1	3.2	odd	2
9792.2.a.k.1.1		1	120.29	odd	2
9792.2.a.l.1.1		1	120.59	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:	\( N \)	\(=\)	\( 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2550.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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