LMFDB - Embedded newform 2550.2.a.e.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2550,2,Mod(1,2550)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [1,-1,-1,1,0,1,0,-1,1,0,6,-1,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$20.3618525154$
Analytic rank:	$0$
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	$\chi$	$=$	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} +6.00000 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} -6.00000 q^{22} +5.00000 q^{23} +1.00000 q^{24} -2.00000 q^{26} -1.00000 q^{27} -2.00000 q^{31} -1.00000 q^{32} -6.00000 q^{33} +1.00000 q^{34} +1.00000 q^{36} -3.00000 q^{37} -4.00000 q^{38} -2.00000 q^{39} +5.00000 q^{41} -2.00000 q^{43} +6.00000 q^{44} -5.00000 q^{46} -1.00000 q^{48} -7.00000 q^{49} +1.00000 q^{51} +2.00000 q^{52} -1.00000 q^{53} +1.00000 q^{54} -4.00000 q^{57} -3.00000 q^{59} +5.00000 q^{61} +2.00000 q^{62} +1.00000 q^{64} +6.00000 q^{66} +2.00000 q^{67} -1.00000 q^{68} -5.00000 q^{69} +5.00000 q^{71} -1.00000 q^{72} +3.00000 q^{74} +4.00000 q^{76} +2.00000 q^{78} -8.00000 q^{79} +1.00000 q^{81} -5.00000 q^{82} +1.00000 q^{83} +2.00000 q^{86} -6.00000 q^{88} +14.0000 q^{89} +5.00000 q^{92} +2.00000 q^{93} +1.00000 q^{96} -16.0000 q^{97} +7.00000 q^{98} +6.00000 q^{99} +O(q^{100})

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
$11$	6.00000	1.80907	0.904534	−	0.426401i	$-0.140219\pi$
$11$	6.00000	1.80907	0.904534	+	0.426401i	$0.140219\pi$
$12$	−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
$22$	−6.00000	−1.27920
$23$	5.00000	1.04257	0.521286	−	0.853382i	$-0.325452\pi$
$23$	5.00000	1.04257	0.521286	+	0.853382i	$0.325452\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	−2.00000	−0.392232
$27$	−1.00000	−0.192450
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	−1.00000	−0.176777
$33$	−6.00000	−1.04447
$34$	1.00000	0.171499
$35$	0	0
$36$	1.00000	0.166667
$37$	−3.00000	−0.493197	−0.246598	−	0.969118i	$-0.579313\pi$
$37$	−3.00000	−0.493197	−0.246598	+	0.969118i	$0.579313\pi$
$38$	−4.00000	−0.648886
$39$	−2.00000	−0.320256
$40$	0	0
$41$	5.00000	0.780869	0.390434	−	0.920631i	$-0.372325\pi$
$41$	5.00000	0.780869	0.390434	+	0.920631i	$0.372325\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	6.00000	0.904534
$45$	0	0
$46$	−5.00000	−0.737210
$47$	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
$53$	−1.00000	−0.137361	−0.0686803	−	0.997639i	$-0.521879\pi$
$53$	−1.00000	−0.137361	−0.0686803	+	0.997639i	$0.521879\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	0	0
$57$	−4.00000	−0.529813
$58$	0	0
$59$	−3.00000	−0.390567	−0.195283	−	0.980747i	$-0.562563\pi$
$59$	−3.00000	−0.390567	−0.195283	+	0.980747i	$0.562563\pi$
$60$	0	0
$61$	5.00000	0.640184	0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000	0.640184	0.320092	+	0.947386i	$0.396286\pi$
$62$	2.00000	0.254000
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	6.00000	0.738549
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	−1.00000	−0.121268
$69$	−5.00000	−0.601929
$70$	0	0
$71$	5.00000	0.593391	0.296695	−	0.954972i	$-0.404115\pi$
$71$	5.00000	0.593391	0.296695	+	0.954972i	$0.404115\pi$
$72$	−1.00000	−0.117851
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	3.00000	0.348743
$75$	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
$82$	−5.00000	−0.552158
$83$	1.00000	0.109764	0.0548821	−	0.998493i	$-0.482522\pi$
$83$	1.00000	0.109764	0.0548821	+	0.998493i	$0.482522\pi$
$84$	0	0
$85$	0	0
$86$	2.00000	0.215666
$87$	0	0
$88$	−6.00000	−0.639602
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	0	0
$91$	0	0
$92$	5.00000	0.521286
$93$	2.00000	0.207390
$94$	0	0
$95$	0	0
$96$	1.00000	0.102062
$97$	−16.0000	−1.62455	−0.812277	−	0.583272i	$-0.801772\pi$
$97$	−16.0000	−1.62455	−0.812277	+	0.583272i	$0.801772\pi$
$98$	7.00000	0.707107
$99$	6.00000	0.603023
$100$	0	0
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	−1.00000	−0.0990148
$103$	13.0000	1.28093	0.640464	−	0.767988i	$-0.278742\pi$
$103$	13.0000	1.28093	0.640464	+	0.767988i	$0.278742\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	1.00000	0.0971286
$107$	−16.0000	−1.54678	−0.773389	−	0.633932i	$-0.781440\pi$
$107$	−16.0000	−1.54678	−0.773389	+	0.633932i	$0.781440\pi$
$108$	−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
$111$	3.00000	0.284747
$112$	0	0
$113$	13.0000	1.22294	0.611469	−	0.791269i	$-0.290579\pi$
$113$	13.0000	1.22294	0.611469	+	0.791269i	$0.290579\pi$
$114$	4.00000	0.374634
$115$	0	0
$116$	0	0
$117$	2.00000	0.184900
$118$	3.00000	0.276172
$119$	0	0
$120$	0	0
$121$	25.0000	2.27273
$122$	−5.00000	−0.452679
$123$	−5.00000	−0.450835
$124$	−2.00000	−0.179605
$125$	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
$129$	2.00000	0.176090
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	−6.00000	−0.522233
$133$	0	0
$134$	−2.00000	−0.172774
$135$	0	0
$136$	1.00000	0.0857493
$137$	20.0000	1.70872	0.854358	−	0.519685i	$-0.173951\pi$
$137$	20.0000	1.70872	0.854358	+	0.519685i	$0.173951\pi$
$138$	5.00000	0.425628
$139$	1.00000	0.0848189	0.0424094	−	0.999100i	$-0.486497\pi$
$139$	1.00000	0.0848189	0.0424094	+	0.999100i	$0.486497\pi$
$140$	0	0
$141$	0	0
$142$	−5.00000	−0.419591
$143$	12.0000	1.00349
$144$	1.00000	0.0833333
$145$	0	0
$146$	0	0
$147$	7.00000	0.577350
$148$	−3.00000	−0.246598
$149$	−5.00000	−0.409616	−0.204808	−	0.978802i	$-0.565657\pi$
$149$	−5.00000	−0.409616	−0.204808	+	0.978802i	$0.565657\pi$
$150$	0	0
$151$	9.00000	0.732410	0.366205	−	0.930534i	$-0.380657\pi$
$151$	9.00000	0.732410	0.366205	+	0.930534i	$0.380657\pi$
$152$	−4.00000	−0.324443
$153$	−1.00000	−0.0808452
$154$	0	0
$155$	0	0
$156$	−2.00000	−0.160128
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	8.00000	0.636446
$159$	1.00000	0.0793052
$160$	0	0
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−9.00000	−0.704934	−0.352467	−	0.935824i	$-0.614657\pi$
$163$	−9.00000	−0.704934	−0.352467	+	0.935824i	$0.614657\pi$
$164$	5.00000	0.390434
$165$	0	0
$166$	−1.00000	−0.0776151
$167$	24.0000	1.85718	0.928588	−	0.371113i	$-0.121024\pi$
$167$	24.0000	1.85718	0.928588	+	0.371113i	$0.121024\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	4.00000	0.305888
$172$	−2.00000	−0.152499
$173$	24.0000	1.82469	0.912343	−	0.409426i	$-0.134271\pi$
$173$	24.0000	1.82469	0.912343	+	0.409426i	$0.134271\pi$
$174$	0	0
$175$	0	0
$176$	6.00000	0.452267
$177$	3.00000	0.225494
$178$	−14.0000	−1.04934
$179$	3.00000	0.224231	0.112115	−	0.993695i	$-0.464237\pi$
$179$	3.00000	0.224231	0.112115	+	0.993695i	$0.464237\pi$
$180$	0	0
$181$	19.0000	1.41226	0.706129	−	0.708083i	$-0.250440\pi$
$181$	19.0000	1.41226	0.706129	+	0.708083i	$0.250440\pi$
$182$	0	0
$183$	−5.00000	−0.369611
$184$	−5.00000	−0.368605
$185$	0	0
$186$	−2.00000	−0.146647
$187$	−6.00000	−0.438763
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−6.00000	−0.434145	−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000	−0.434145	−0.217072	+	0.976156i	$0.569651\pi$
$192$	−1.00000	−0.0721688
$193$	12.0000	0.863779	0.431889	−	0.901927i	$-0.357847\pi$
$193$	12.0000	0.863779	0.431889	+	0.901927i	$0.357847\pi$
$194$	16.0000	1.14873
$195$	0	0
$196$	−7.00000	−0.500000
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	−6.00000	−0.426401
$199$	−10.0000	−0.708881	−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000	−0.708881	−0.354441	+	0.935079i	$0.615329\pi$
$200$	0	0
$201$	−2.00000	−0.141069
$202$	−10.0000	−0.703598
$203$	0	0
$204$	1.00000	0.0700140
$205$	0	0
$206$	−13.0000	−0.905753
$207$	5.00000	0.347524
$208$	2.00000	0.138675
$209$	24.0000	1.66011
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	−1.00000	−0.0686803
$213$	−5.00000	−0.342594
$214$	16.0000	1.09374
$215$	0	0
$216$	1.00000	0.0680414
$217$	0	0
$218$	10.0000	0.677285
$219$	0	0
$220$	0	0
$221$	−2.00000	−0.134535
$222$	−3.00000	−0.201347
$223$	−1.00000	−0.0669650	−0.0334825	−	0.999439i	$-0.510660\pi$
$223$	−1.00000	−0.0669650	−0.0334825	+	0.999439i	$0.510660\pi$
$224$	0	0
$225$	0	0
$226$	−13.0000	−0.864747
$227$	6.00000	0.398234	0.199117	−	0.979976i	$-0.436193\pi$
$227$	6.00000	0.398234	0.199117	+	0.979976i	$0.436193\pi$
$228$	−4.00000	−0.264906
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.00000	−0.0655122	−0.0327561	−	0.999463i	$-0.510428\pi$
$233$	−1.00000	−0.0655122	−0.0327561	+	0.999463i	$0.510428\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	−3.00000	−0.195283
$237$	8.00000	0.519656
$238$	0	0
$239$	−30.0000	−1.94054	−0.970269	−	0.242028i	$-0.922188\pi$
$239$	−30.0000	−1.94054	−0.970269	+	0.242028i	$0.922188\pi$
$240$	0	0
$241$	−18.0000	−1.15948	−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000	−1.15948	−0.579741	+	0.814801i	$0.696846\pi$
$242$	−25.0000	−1.60706
$243$	−1.00000	−0.0641500
$244$	5.00000	0.320092
$245$	0	0
$246$	5.00000	0.318788
$247$	8.00000	0.509028
$248$	2.00000	0.127000
$249$	−1.00000	−0.0633724
$250$	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
$253$	30.0000	1.88608
$254$	0	0
$255$	0	0
$256$	1.00000	0.0625000
$257$	28.0000	1.74659	0.873296	−	0.487190i	$-0.161978\pi$
$257$	28.0000	1.74659	0.873296	+	0.487190i	$0.161978\pi$
$258$	−2.00000	−0.124515
$259$	0	0
$260$	0	0
$261$	0	0
$262$	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	$0.420670\pi$
$264$	6.00000	0.369274
$265$	0	0
$266$	0	0
$267$	−14.0000	−0.856786
$268$	2.00000	0.122169
$269$	26.0000	1.58525	0.792624	−	0.609711i	$-0.208714\pi$
$269$	26.0000	1.58525	0.792624	+	0.609711i	$0.208714\pi$
$270$	0	0
$271$	11.0000	0.668202	0.334101	−	0.942537i	$-0.391567\pi$
$271$	11.0000	0.668202	0.334101	+	0.942537i	$0.391567\pi$
$272$	−1.00000	−0.0606339
$273$	0	0
$274$	−20.0000	−1.20824
$275$	0	0
$276$	−5.00000	−0.300965
$277$	25.0000	1.50210	0.751052	−	0.660243i	$-0.229547\pi$
$277$	25.0000	1.50210	0.751052	+	0.660243i	$0.229547\pi$
$278$	−1.00000	−0.0599760
$279$	−2.00000	−0.119737
$280$	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
$283$	−17.0000	−1.01055	−0.505273	−	0.862960i	$-0.668608\pi$
$283$	−17.0000	−1.01055	−0.505273	+	0.862960i	$0.668608\pi$
$284$	5.00000	0.296695
$285$	0	0
$286$	−12.0000	−0.709575
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	1.00000	0.0588235
$290$	0	0
$291$	16.0000	0.937937
$292$	0	0
$293$	21.0000	1.22683	0.613417	−	0.789760i	$-0.289795\pi$
$293$	21.0000	1.22683	0.613417	+	0.789760i	$0.289795\pi$
$294$	−7.00000	−0.408248
$295$	0	0
$296$	3.00000	0.174371
$297$	−6.00000	−0.348155
$298$	5.00000	0.289642
$299$	10.0000	0.578315
$300$	0	0
$301$	0	0
$302$	−9.00000	−0.517892
$303$	−10.0000	−0.574485
$304$	4.00000	0.229416
$305$	0	0
$306$	1.00000	0.0571662
$307$	2.00000	0.114146	0.0570730	−	0.998370i	$-0.481823\pi$
$307$	2.00000	0.114146	0.0570730	+	0.998370i	$0.481823\pi$
$308$	0	0
$309$	−13.0000	−0.739544
$310$	0	0
$311$	15.0000	0.850572	0.425286	−	0.905059i	$-0.360174\pi$
$311$	15.0000	0.850572	0.425286	+	0.905059i	$0.360174\pi$
$312$	2.00000	0.113228
$313$	20.0000	1.13047	0.565233	−	0.824931i	$-0.308786\pi$
$313$	20.0000	1.13047	0.565233	+	0.824931i	$0.308786\pi$
$314$	18.0000	1.01580
$315$	0	0
$316$	−8.00000	−0.450035
$317$	14.0000	0.786318	0.393159	−	0.919470i	$-0.371382\pi$
$317$	14.0000	0.786318	0.393159	+	0.919470i	$0.371382\pi$
$318$	−1.00000	−0.0560772
$319$	0	0
$320$	0	0
$321$	16.0000	0.893033
$322$	0	0
$323$	−4.00000	−0.222566
$324$	1.00000	0.0555556
$325$	0	0
$326$	9.00000	0.498464
$327$	10.0000	0.553001
$328$	−5.00000	−0.276079
$329$	0	0
$330$	0	0
$331$	10.0000	0.549650	0.274825	−	0.961494i	$-0.411380\pi$
$331$	10.0000	0.549650	0.274825	+	0.961494i	$0.411380\pi$
$332$	1.00000	0.0548821
$333$	−3.00000	−0.164399
$334$	−24.0000	−1.31322
$335$	0	0
$336$	0	0
$337$	−6.00000	−0.326841	−0.163420	−	0.986557i	$-0.552253\pi$
$337$	−6.00000	−0.326841	−0.163420	+	0.986557i	$0.552253\pi$
$338$	9.00000	0.489535
$339$	−13.0000	−0.706063
$340$	0	0
$341$	−12.0000	−0.649836
$342$	−4.00000	−0.216295
$343$	0	0
$344$	2.00000	0.107833
$345$	0	0
$346$	−24.0000	−1.29025
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	0	0
$349$	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
$352$	−6.00000	−0.319801
$353$	−20.0000	−1.06449	−0.532246	−	0.846590i	$-0.678652\pi$
$353$	−20.0000	−1.06449	−0.532246	+	0.846590i	$0.678652\pi$
$354$	−3.00000	−0.159448
$355$	0	0
$356$	14.0000	0.741999
$357$	0	0
$358$	−3.00000	−0.158555
$359$	−4.00000	−0.211112	−0.105556	−	0.994413i	$-0.533662\pi$
$359$	−4.00000	−0.211112	−0.105556	+	0.994413i	$0.533662\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	−19.0000	−0.998618
$363$	−25.0000	−1.31216
$364$	0	0
$365$	0	0
$366$	5.00000	0.261354
$367$	−14.0000	−0.730794	−0.365397	−	0.930852i	$-0.619067\pi$
$367$	−14.0000	−0.730794	−0.365397	+	0.930852i	$0.619067\pi$
$368$	5.00000	0.260643
$369$	5.00000	0.260290
$370$	0	0
$371$	0	0
$372$	2.00000	0.103695
$373$	4.00000	0.207112	0.103556	−	0.994624i	$-0.466978\pi$
$373$	4.00000	0.207112	0.103556	+	0.994624i	$0.466978\pi$
$374$	6.00000	0.310253
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	31.0000	1.59236	0.796182	−	0.605058i	$-0.206850\pi$
$379$	31.0000	1.59236	0.796182	+	0.605058i	$0.206850\pi$
$380$	0	0
$381$	0	0
$382$	6.00000	0.306987
$383$	−4.00000	−0.204390	−0.102195	−	0.994764i	$-0.532587\pi$
$383$	−4.00000	−0.204390	−0.102195	+	0.994764i	$0.532587\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−12.0000	−0.610784
$387$	−2.00000	−0.101666
$388$	−16.0000	−0.812277
$389$	19.0000	0.963338	0.481669	−	0.876353i	$-0.340031\pi$
$389$	19.0000	0.963338	0.481669	+	0.876353i	$0.340031\pi$
$390$	0	0
$391$	−5.00000	−0.252861
$392$	7.00000	0.353553
$393$	12.0000	0.605320
$394$	−6.00000	−0.302276
$395$	0	0
$396$	6.00000	0.301511
$397$	21.0000	1.05396	0.526980	−	0.849878i	$-0.323324\pi$
$397$	21.0000	1.05396	0.526980	+	0.849878i	$0.323324\pi$
$398$	10.0000	0.501255
$399$	0	0
$400$	0	0
$401$	−39.0000	−1.94757	−0.973784	−	0.227477i	$-0.926952\pi$
$401$	−39.0000	−1.94757	−0.973784	+	0.227477i	$0.926952\pi$
$402$	2.00000	0.0997509
$403$	−4.00000	−0.199254
$404$	10.0000	0.497519
$405$	0	0
$406$	0	0
$407$	−18.0000	−0.892227
$408$	−1.00000	−0.0495074
$409$	11.0000	0.543915	0.271957	−	0.962309i	$-0.412329\pi$
$409$	11.0000	0.543915	0.271957	+	0.962309i	$0.412329\pi$
$410$	0	0
$411$	−20.0000	−0.986527
$412$	13.0000	0.640464
$413$	0	0
$414$	−5.00000	−0.245737
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	−1.00000	−0.0489702
$418$	−24.0000	−1.17388
$419$	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
$421$	−28.0000	−1.36464	−0.682318	−	0.731055i	$-0.739028\pi$
$421$	−28.0000	−1.36464	−0.682318	+	0.731055i	$0.739028\pi$
$422$	−12.0000	−0.584151
$423$	0	0
$424$	1.00000	0.0485643
$425$	0	0
$426$	5.00000	0.242251
$427$	0	0
$428$	−16.0000	−0.773389
$429$	−12.0000	−0.579365
$430$	0	0
$431$	−28.0000	−1.34871	−0.674356	−	0.738406i	$-0.735579\pi$
$431$	−28.0000	−1.34871	−0.674356	+	0.738406i	$0.735579\pi$
$432$	−1.00000	−0.0481125
$433$	−6.00000	−0.288342	−0.144171	−	0.989553i	$-0.546051\pi$
$433$	−6.00000	−0.288342	−0.144171	+	0.989553i	$0.546051\pi$
$434$	0	0
$435$	0	0
$436$	−10.0000	−0.478913
$437$	20.0000	0.956730
$438$	0	0
$439$	−4.00000	−0.190910	−0.0954548	−	0.995434i	$-0.530431\pi$
$439$	−4.00000	−0.190910	−0.0954548	+	0.995434i	$0.530431\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	2.00000	0.0951303
$443$	−11.0000	−0.522626	−0.261313	−	0.965254i	$-0.584155\pi$
$443$	−11.0000	−0.522626	−0.261313	+	0.965254i	$0.584155\pi$
$444$	3.00000	0.142374
$445$	0	0
$446$	1.00000	0.0473514
$447$	5.00000	0.236492
$448$	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
$451$	30.0000	1.41264
$452$	13.0000	0.611469
$453$	−9.00000	−0.422857
$454$	−6.00000	−0.281594
$455$	0	0
$456$	4.00000	0.187317
$457$	−25.0000	−1.16945	−0.584725	−	0.811231i	$-0.698798\pi$
$457$	−25.0000	−1.16945	−0.584725	+	0.811231i	$0.698798\pi$
$458$	6.00000	0.280362
$459$	1.00000	0.0466760
$460$	0	0
$461$	−35.0000	−1.63011	−0.815056	−	0.579382i	$-0.803294\pi$
$461$	−35.0000	−1.63011	−0.815056	+	0.579382i	$0.803294\pi$
$462$	0	0
$463$	−27.0000	−1.25480	−0.627398	−	0.778699i	$-0.715880\pi$
$463$	−27.0000	−1.25480	−0.627398	+	0.778699i	$0.715880\pi$
$464$	0	0
$465$	0	0
$466$	1.00000	0.0463241
$467$	−33.0000	−1.52706	−0.763529	−	0.645774i	$-0.776535\pi$
$467$	−33.0000	−1.52706	−0.763529	+	0.645774i	$0.776535\pi$
$468$	2.00000	0.0924500
$469$	0	0
$470$	0	0
$471$	18.0000	0.829396
$472$	3.00000	0.138086
$473$	−12.0000	−0.551761
$474$	−8.00000	−0.367452
$475$	0	0
$476$	0	0
$477$	−1.00000	−0.0457869
$478$	30.0000	1.37217
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	0	0
$481$	−6.00000	−0.273576
$482$	18.0000	0.819878
$483$	0	0
$484$	25.0000	1.13636
$485$	0	0
$486$	1.00000	0.0453609
$487$	26.0000	1.17817	0.589086	−	0.808070i	$-0.299488\pi$
$487$	26.0000	1.17817	0.589086	+	0.808070i	$0.299488\pi$
$488$	−5.00000	−0.226339
$489$	9.00000	0.406994
$490$	0	0
$491$	−43.0000	−1.94056	−0.970281	−	0.241979i	$-0.922203\pi$
$491$	−43.0000	−1.94056	−0.970281	+	0.241979i	$0.922203\pi$
$492$	−5.00000	−0.225417
$493$	0	0
$494$	−8.00000	−0.359937
$495$	0	0
$496$	−2.00000	−0.0898027
$497$	0	0
$498$	1.00000	0.0448111
$499$	−41.0000	−1.83541	−0.917706	−	0.397260i	$-0.869961\pi$
$499$	−41.0000	−1.83541	−0.917706	+	0.397260i	$0.869961\pi$
$500$	0	0
$501$	−24.0000	−1.07224
$502$	12.0000	0.535586
$503$	−5.00000	−0.222939	−0.111469	−	0.993768i	$-0.535556\pi$
$503$	−5.00000	−0.222939	−0.111469	+	0.993768i	$0.535556\pi$
$504$	0	0
$505$	0	0
$506$	−30.0000	−1.33366
$507$	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
$514$	−28.0000	−1.23503
$515$	0	0
$516$	2.00000	0.0880451
$517$	0	0
$518$	0	0
$519$	−24.0000	−1.05348
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	0	0
$523$	−26.0000	−1.13690	−0.568450	−	0.822718i	$-0.692457\pi$
$523$	−26.0000	−1.13690	−0.568450	+	0.822718i	$0.692457\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	−8.00000	−0.348817
$527$	2.00000	0.0871214
$528$	−6.00000	−0.261116
$529$	2.00000	0.0869565
$530$	0	0
$531$	−3.00000	−0.130189
$532$	0	0
$533$	10.0000	0.433148
$534$	14.0000	0.605839
$535$	0	0
$536$	−2.00000	−0.0863868
$537$	−3.00000	−0.129460
$538$	−26.0000	−1.12094
$539$	−42.0000	−1.80907
$540$	0	0
$541$	−25.0000	−1.07483	−0.537417	−	0.843317i	$-0.680600\pi$
$541$	−25.0000	−1.07483	−0.537417	+	0.843317i	$0.680600\pi$
$542$	−11.0000	−0.472490
$543$	−19.0000	−0.815368
$544$	1.00000	0.0428746
$545$	0	0
$546$	0	0
$547$	−13.0000	−0.555840	−0.277920	−	0.960604i	$-0.589645\pi$
$547$	−13.0000	−0.555840	−0.277920	+	0.960604i	$0.589645\pi$
$548$	20.0000	0.854358
$549$	5.00000	0.213395
$550$	0	0
$551$	0	0
$552$	5.00000	0.212814
$553$	0	0
$554$	−25.0000	−1.06215
$555$	0	0
$556$	1.00000	0.0424094
$557$	−11.0000	−0.466085	−0.233042	−	0.972467i	$-0.574868\pi$
$557$	−11.0000	−0.466085	−0.233042	+	0.972467i	$0.574868\pi$
$558$	2.00000	0.0846668
$559$	−4.00000	−0.169182
$560$	0	0
$561$	6.00000	0.253320
$562$	6.00000	0.253095
$563$	−21.0000	−0.885044	−0.442522	−	0.896758i	$-0.645916\pi$
$563$	−21.0000	−0.885044	−0.442522	+	0.896758i	$0.645916\pi$
$564$	0	0
$565$	0	0
$566$	17.0000	0.714563
$567$	0	0
$568$	−5.00000	−0.209795
$569$	−26.0000	−1.08998	−0.544988	−	0.838444i	$-0.683466\pi$
$569$	−26.0000	−1.08998	−0.544988	+	0.838444i	$0.683466\pi$
$570$	0	0
$571$	23.0000	0.962520	0.481260	−	0.876578i	$-0.340179\pi$
$571$	23.0000	0.962520	0.481260	+	0.876578i	$0.340179\pi$
$572$	12.0000	0.501745
$573$	6.00000	0.250654
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	−7.00000	−0.291414	−0.145707	−	0.989328i	$-0.546546\pi$
$577$	−7.00000	−0.291414	−0.145707	+	0.989328i	$0.546546\pi$
$578$	−1.00000	−0.0415945
$579$	−12.0000	−0.498703
$580$	0	0
$581$	0	0
$582$	−16.0000	−0.663221
$583$	−6.00000	−0.248495
$584$	0	0
$585$	0	0
$586$	−21.0000	−0.867502
$587$	45.0000	1.85735	0.928674	−	0.370896i	$-0.120949\pi$
$587$	45.0000	1.85735	0.928674	+	0.370896i	$0.120949\pi$
$588$	7.00000	0.288675
$589$	−8.00000	−0.329634
$590$	0	0
$591$	−6.00000	−0.246807
$592$	−3.00000	−0.123299
$593$	24.0000	0.985562	0.492781	−	0.870153i	$-0.335980\pi$
$593$	24.0000	0.985562	0.492781	+	0.870153i	$0.335980\pi$
$594$	6.00000	0.246183
$595$	0	0
$596$	−5.00000	−0.204808
$597$	10.0000	0.409273
$598$	−10.0000	−0.408930
$599$	−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
$601$	14.0000	0.571072	0.285536	−	0.958368i	$-0.407828\pi$
$601$	14.0000	0.571072	0.285536	+	0.958368i	$0.407828\pi$
$602$	0	0
$603$	2.00000	0.0814463
$604$	9.00000	0.366205
$605$	0	0
$606$	10.0000	0.406222
$607$	34.0000	1.38002	0.690009	−	0.723801i	$-0.257607\pi$
$607$	34.0000	1.38002	0.690009	+	0.723801i	$0.257607\pi$
$608$	−4.00000	−0.162221
$609$	0	0
$610$	0	0
$611$	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	$0.778451\pi$
$614$	−2.00000	−0.0807134
$615$	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	$0.461462\pi$
$618$	13.0000	0.522937
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	0	0
$621$	−5.00000	−0.200643
$622$	−15.0000	−0.601445
$623$	0	0
$624$	−2.00000	−0.0800641
$625$	0	0
$626$	−20.0000	−0.799361
$627$	−24.0000	−0.958468
$628$	−18.0000	−0.718278
$629$	3.00000	0.119618
$630$	0	0
$631$	41.0000	1.63218	0.816092	−	0.577922i	$-0.196136\pi$
$631$	41.0000	1.63218	0.816092	+	0.577922i	$0.196136\pi$
$632$	8.00000	0.318223
$633$	−12.0000	−0.476957
$634$	−14.0000	−0.556011
$635$	0	0
$636$	1.00000	0.0396526
$637$	−14.0000	−0.554700
$638$	0	0
$639$	5.00000	0.197797
$640$	0	0
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	−16.0000	−0.631470
$643$	−37.0000	−1.45914	−0.729569	−	0.683907i	$-0.760279\pi$
$643$	−37.0000	−1.45914	−0.729569	+	0.683907i	$0.760279\pi$
$644$	0	0
$645$	0	0
$646$	4.00000	0.157378
$647$	10.0000	0.393141	0.196570	−	0.980490i	$-0.437020\pi$
$647$	10.0000	0.393141	0.196570	+	0.980490i	$0.437020\pi$
$648$	−1.00000	−0.0392837
$649$	−18.0000	−0.706562
$650$	0	0
$651$	0	0
$652$	−9.00000	−0.352467
$653$	24.0000	0.939193	0.469596	−	0.882881i	$-0.344399\pi$
$653$	24.0000	0.939193	0.469596	+	0.882881i	$0.344399\pi$
$654$	−10.0000	−0.391031
$655$	0	0
$656$	5.00000	0.195217
$657$	0	0
$658$	0	0
$659$	16.0000	0.623272	0.311636	−	0.950202i	$-0.399123\pi$
$659$	16.0000	0.623272	0.311636	+	0.950202i	$0.399123\pi$
$660$	0	0
$661$	26.0000	1.01128	0.505641	−	0.862744i	$-0.331256\pi$
$661$	26.0000	1.01128	0.505641	+	0.862744i	$0.331256\pi$
$662$	−10.0000	−0.388661
$663$	2.00000	0.0776736
$664$	−1.00000	−0.0388075
$665$	0	0
$666$	3.00000	0.116248
$667$	0	0
$668$	24.0000	0.928588
$669$	1.00000	0.0386622
$670$	0	0
$671$	30.0000	1.15814
$672$	0	0
$673$	16.0000	0.616755	0.308377	−	0.951264i	$-0.400214\pi$
$673$	16.0000	0.616755	0.308377	+	0.951264i	$0.400214\pi$
$674$	6.00000	0.231111
$675$	0	0
$676$	−9.00000	−0.346154
$677$	−36.0000	−1.38359	−0.691796	−	0.722093i	$-0.743180\pi$
$677$	−36.0000	−1.38359	−0.691796	+	0.722093i	$0.743180\pi$
$678$	13.0000	0.499262
$679$	0	0
$680$	0	0
$681$	−6.00000	−0.229920
$682$	12.0000	0.459504
$683$	30.0000	1.14792	0.573959	−	0.818884i	$-0.305407\pi$
$683$	30.0000	1.14792	0.573959	+	0.818884i	$0.305407\pi$
$684$	4.00000	0.152944
$685$	0	0
$686$	0	0
$687$	6.00000	0.228914
$688$	−2.00000	−0.0762493
$689$	−2.00000	−0.0761939
$690$	0	0
$691$	−33.0000	−1.25538	−0.627690	−	0.778464i	$-0.715999\pi$
$691$	−33.0000	−1.25538	−0.627690	+	0.778464i	$0.715999\pi$
$692$	24.0000	0.912343
$693$	0	0
$694$	−12.0000	−0.455514
$695$	0	0
$696$	0	0
$697$	−5.00000	−0.189389
$698$	−14.0000	−0.529908
$699$	1.00000	0.0378235
$700$	0	0
$701$	31.0000	1.17085	0.585427	−	0.810725i	$-0.300927\pi$
$701$	31.0000	1.17085	0.585427	+	0.810725i	$0.300927\pi$
$702$	2.00000	0.0754851
$703$	−12.0000	−0.452589
$704$	6.00000	0.226134
$705$	0	0
$706$	20.0000	0.752710
$707$	0	0
$708$	3.00000	0.112747
$709$	−34.0000	−1.27690	−0.638448	−	0.769665i	$-0.720423\pi$
$709$	−34.0000	−1.27690	−0.638448	+	0.769665i	$0.720423\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	−14.0000	−0.524672
$713$	−10.0000	−0.374503
$714$	0	0
$715$	0	0
$716$	3.00000	0.112115
$717$	30.0000	1.12037
$718$	4.00000	0.149279
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	3.00000	0.111648
$723$	18.0000	0.669427
$724$	19.0000	0.706129
$725$	0	0
$726$	25.0000	0.927837
$727$	−52.0000	−1.92857	−0.964287	−	0.264861i	$-0.914674\pi$
$727$	−52.0000	−1.92857	−0.964287	+	0.264861i	$0.914674\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	2.00000	0.0739727
$732$	−5.00000	−0.184805
$733$	14.0000	0.517102	0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000	0.517102	0.258551	+	0.965998i	$0.416755\pi$
$734$	14.0000	0.516749
$735$	0	0
$736$	−5.00000	−0.184302
$737$	12.0000	0.442026
$738$	−5.00000	−0.184053
$739$	−28.0000	−1.03000	−0.514998	−	0.857191i	$-0.672207\pi$
$739$	−28.0000	−1.03000	−0.514998	+	0.857191i	$0.672207\pi$
$740$	0	0
$741$	−8.00000	−0.293887
$742$	0	0
$743$	−19.0000	−0.697042	−0.348521	−	0.937301i	$-0.613316\pi$
$743$	−19.0000	−0.697042	−0.348521	+	0.937301i	$0.613316\pi$
$744$	−2.00000	−0.0733236
$745$	0	0
$746$	−4.00000	−0.146450
$747$	1.00000	0.0365881
$748$	−6.00000	−0.219382
$749$	0	0
$750$	0	0
$751$	−12.0000	−0.437886	−0.218943	−	0.975738i	$-0.570261\pi$
$751$	−12.0000	−0.437886	−0.218943	+	0.975738i	$0.570261\pi$
$752$	0	0
$753$	12.0000	0.437304
$754$	0	0
$755$	0	0
$756$	0	0
$757$	40.0000	1.45382	0.726912	−	0.686730i	$-0.240955\pi$
$757$	40.0000	1.45382	0.726912	+	0.686730i	$0.240955\pi$
$758$	−31.0000	−1.12597
$759$	−30.0000	−1.08893
$760$	0	0
$761$	−36.0000	−1.30500	−0.652499	−	0.757789i	$-0.726280\pi$
$761$	−36.0000	−1.30500	−0.652499	+	0.757789i	$0.726280\pi$
$762$	0	0
$763$	0	0
$764$	−6.00000	−0.217072
$765$	0	0
$766$	4.00000	0.144526
$767$	−6.00000	−0.216647
$768$	−1.00000	−0.0360844
$769$	35.0000	1.26213	0.631066	−	0.775729i	$-0.282618\pi$
$769$	35.0000	1.26213	0.631066	+	0.775729i	$0.282618\pi$
$770$	0	0
$771$	−28.0000	−1.00840
$772$	12.0000	0.431889
$773$	−3.00000	−0.107903	−0.0539513	−	0.998544i	$-0.517182\pi$
$773$	−3.00000	−0.107903	−0.0539513	+	0.998544i	$0.517182\pi$
$774$	2.00000	0.0718885
$775$	0	0
$776$	16.0000	0.574367
$777$	0	0
$778$	−19.0000	−0.681183
$779$	20.0000	0.716574
$780$	0	0
$781$	30.0000	1.07348
$782$	5.00000	0.178800
$783$	0	0
$784$	−7.00000	−0.250000
$785$	0	0
$786$	−12.0000	−0.428026
$787$	21.0000	0.748569	0.374285	−	0.927314i	$-0.377888\pi$
$787$	21.0000	0.748569	0.374285	+	0.927314i	$0.377888\pi$
$788$	6.00000	0.213741
$789$	−8.00000	−0.284808
$790$	0	0
$791$	0	0
$792$	−6.00000	−0.213201
$793$	10.0000	0.355110
$794$	−21.0000	−0.745262
$795$	0	0
$796$	−10.0000	−0.354441
$797$	−19.0000	−0.673015	−0.336507	−	0.941681i	$-0.609246\pi$
$797$	−19.0000	−0.673015	−0.336507	+	0.941681i	$0.609246\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	14.0000	0.494666
$802$	39.0000	1.37714
$803$	0	0
$804$	−2.00000	−0.0705346
$805$	0	0
$806$	4.00000	0.140894
$807$	−26.0000	−0.915243
$808$	−10.0000	−0.351799
$809$	18.0000	0.632846	0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000	0.632846	0.316423	+	0.948618i	$0.397518\pi$
$810$	0	0
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	0	0
$813$	−11.0000	−0.385787
$814$	18.0000	0.630900
$815$	0	0
$816$	1.00000	0.0350070
$817$	−8.00000	−0.279885
$818$	−11.0000	−0.384606
$819$	0	0
$820$	0	0
$821$	36.0000	1.25641	0.628204	−	0.778048i	$-0.283790\pi$
$821$	36.0000	1.25641	0.628204	+	0.778048i	$0.283790\pi$
$822$	20.0000	0.697580
$823$	−32.0000	−1.11545	−0.557725	−	0.830026i	$-0.688326\pi$
$823$	−32.0000	−1.11545	−0.557725	+	0.830026i	$0.688326\pi$
$824$	−13.0000	−0.452876
$825$	0	0
$826$	0	0
$827$	−10.0000	−0.347734	−0.173867	−	0.984769i	$-0.555626\pi$
$827$	−10.0000	−0.347734	−0.173867	+	0.984769i	$0.555626\pi$
$828$	5.00000	0.173762
$829$	16.0000	0.555703	0.277851	−	0.960624i	$-0.410378\pi$
$829$	16.0000	0.555703	0.277851	+	0.960624i	$0.410378\pi$
$830$	0	0
$831$	−25.0000	−0.867240
$832$	2.00000	0.0693375
$833$	7.00000	0.242536
$834$	1.00000	0.0346272
$835$	0	0
$836$	24.0000	0.830057
$837$	2.00000	0.0691301
$838$	−4.00000	−0.138178
$839$	−15.0000	−0.517858	−0.258929	−	0.965896i	$-0.583369\pi$
$839$	−15.0000	−0.517858	−0.258929	+	0.965896i	$0.583369\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	28.0000	0.964944
$843$	6.00000	0.206651
$844$	12.0000	0.413057
$845$	0	0
$846$	0	0
$847$	0	0
$848$	−1.00000	−0.0343401
$849$	17.0000	0.583438
$850$	0	0
$851$	−15.0000	−0.514193
$852$	−5.00000	−0.171297
$853$	42.0000	1.43805	0.719026	−	0.694983i	$-0.244588\pi$
$853$	42.0000	1.43805	0.719026	+	0.694983i	$0.244588\pi$
$854$	0	0
$855$	0	0
$856$	16.0000	0.546869
$857$	11.0000	0.375753	0.187876	−	0.982193i	$-0.439840\pi$
$857$	11.0000	0.375753	0.187876	+	0.982193i	$0.439840\pi$
$858$	12.0000	0.409673
$859$	−16.0000	−0.545913	−0.272956	−	0.962026i	$-0.588002\pi$
$859$	−16.0000	−0.545913	−0.272956	+	0.962026i	$0.588002\pi$
$860$	0	0
$861$	0	0
$862$	28.0000	0.953684
$863$	−28.0000	−0.953131	−0.476566	−	0.879139i	$-0.658119\pi$
$863$	−28.0000	−0.953131	−0.476566	+	0.879139i	$0.658119\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	6.00000	0.203888
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	−48.0000	−1.62829
$870$	0	0
$871$	4.00000	0.135535
$872$	10.0000	0.338643
$873$	−16.0000	−0.541518
$874$	−20.0000	−0.676510
$875$	0	0
$876$	0	0
$877$	−2.00000	−0.0675352	−0.0337676	−	0.999430i	$-0.510751\pi$
$877$	−2.00000	−0.0675352	−0.0337676	+	0.999430i	$0.510751\pi$
$878$	4.00000	0.134993
$879$	−21.0000	−0.708312
$880$	0	0
$881$	−53.0000	−1.78562	−0.892808	−	0.450438i	$-0.851268\pi$
$881$	−53.0000	−1.78562	−0.892808	+	0.450438i	$0.851268\pi$
$882$	7.00000	0.235702
$883$	−6.00000	−0.201916	−0.100958	−	0.994891i	$-0.532191\pi$
$883$	−6.00000	−0.201916	−0.100958	+	0.994891i	$0.532191\pi$
$884$	−2.00000	−0.0672673
$885$	0	0
$886$	11.0000	0.369552
$887$	3.00000	0.100730	0.0503651	−	0.998731i	$-0.483962\pi$
$887$	3.00000	0.100730	0.0503651	+	0.998731i	$0.483962\pi$
$888$	−3.00000	−0.100673
$889$	0	0
$890$	0	0
$891$	6.00000	0.201008
$892$	−1.00000	−0.0334825
$893$	0	0
$894$	−5.00000	−0.167225
$895$	0	0
$896$	0	0
$897$	−10.0000	−0.333890
$898$	−2.00000	−0.0667409
$899$	0	0
$900$	0	0
$901$	1.00000	0.0333148
$902$	−30.0000	−0.998891
$903$	0	0
$904$	−13.0000	−0.432374
$905$	0	0
$906$	9.00000	0.299005
$907$	17.0000	0.564476	0.282238	−	0.959344i	$-0.408923\pi$
$907$	17.0000	0.564476	0.282238	+	0.959344i	$0.408923\pi$
$908$	6.00000	0.199117
$909$	10.0000	0.331679
$910$	0	0
$911$	−32.0000	−1.06021	−0.530104	−	0.847933i	$-0.677847\pi$
$911$	−32.0000	−1.06021	−0.530104	+	0.847933i	$0.677847\pi$
$912$	−4.00000	−0.132453
$913$	6.00000	0.198571
$914$	25.0000	0.826927
$915$	0	0
$916$	−6.00000	−0.198246
$917$	0	0
$918$	−1.00000	−0.0330049
$919$	49.0000	1.61636	0.808180	−	0.588935i	$-0.200453\pi$
$919$	49.0000	1.61636	0.808180	+	0.588935i	$0.200453\pi$
$920$	0	0
$921$	−2.00000	−0.0659022
$922$	35.0000	1.15266
$923$	10.0000	0.329154
$924$	0	0
$925$	0	0
$926$	27.0000	0.887275
$927$	13.0000	0.426976
$928$	0	0
$929$	−13.0000	−0.426516	−0.213258	−	0.976996i	$-0.568408\pi$
$929$	−13.0000	−0.426516	−0.213258	+	0.976996i	$0.568408\pi$
$930$	0	0
$931$	−28.0000	−0.917663
$932$	−1.00000	−0.0327561
$933$	−15.0000	−0.491078
$934$	33.0000	1.07979
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	−23.0000	−0.751377	−0.375689	−	0.926746i	$-0.622594\pi$
$937$	−23.0000	−0.751377	−0.375689	+	0.926746i	$0.622594\pi$
$938$	0	0
$939$	−20.0000	−0.652675
$940$	0	0
$941$	10.0000	0.325991	0.162995	−	0.986627i	$-0.447884\pi$
$941$	10.0000	0.325991	0.162995	+	0.986627i	$0.447884\pi$
$942$	−18.0000	−0.586472
$943$	25.0000	0.814112
$944$	−3.00000	−0.0976417
$945$	0	0
$946$	12.0000	0.390154
$947$	48.0000	1.55979	0.779895	−	0.625910i	$-0.215272\pi$
$947$	48.0000	1.55979	0.779895	+	0.625910i	$0.215272\pi$
$948$	8.00000	0.259828
$949$	0	0
$950$	0	0
$951$	−14.0000	−0.453981
$952$	0	0
$953$	−14.0000	−0.453504	−0.226752	−	0.973952i	$-0.572811\pi$
$953$	−14.0000	−0.453504	−0.226752	+	0.973952i	$0.572811\pi$
$954$	1.00000	0.0323762
$955$	0	0
$956$	−30.0000	−0.970269
$957$	0	0
$958$	16.0000	0.516937
$959$	0	0
$960$	0	0
$961$	−27.0000	−0.870968
$962$	6.00000	0.193448
$963$	−16.0000	−0.515593
$964$	−18.0000	−0.579741
$965$	0	0
$966$	0	0
$967$	−57.0000	−1.83300	−0.916498	−	0.400039i	$-0.868997\pi$
$967$	−57.0000	−1.83300	−0.916498	+	0.400039i	$0.868997\pi$
$968$	−25.0000	−0.803530
$969$	4.00000	0.128499
$970$	0	0
$971$	51.0000	1.63667	0.818334	−	0.574743i	$-0.194898\pi$
$971$	51.0000	1.63667	0.818334	+	0.574743i	$0.194898\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	−26.0000	−0.833094
$975$	0	0
$976$	5.00000	0.160046
$977$	−34.0000	−1.08776	−0.543878	−	0.839164i	$-0.683045\pi$
$977$	−34.0000	−1.08776	−0.543878	+	0.839164i	$0.683045\pi$
$978$	−9.00000	−0.287788
$979$	84.0000	2.68465
$980$	0	0
$981$	−10.0000	−0.319275
$982$	43.0000	1.37219
$983$	−8.00000	−0.255160	−0.127580	−	0.991828i	$-0.540721\pi$
$983$	−8.00000	−0.255160	−0.127580	+	0.991828i	$0.540721\pi$
$984$	5.00000	0.159394
$985$	0	0
$986$	0	0
$987$	0	0
$988$	8.00000	0.254514
$989$	−10.0000	−0.317982
$990$	0	0
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	2.00000	0.0635001
$993$	−10.0000	−0.317340
$994$	0	0
$995$	0	0
$996$	−1.00000	−0.0316862
$997$	22.0000	0.696747	0.348373	−	0.937356i	$-0.386734\pi$
$997$	22.0000	0.696747	0.348373	+	0.937356i	$0.386734\pi$
$998$	41.0000	1.29783
$999$	3.00000	0.0949158

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2550.2.a.e.1.1	✓	1
3.2	odd	2		7650.2.a.bv.1.1		1
5.2	odd	4		2550.2.d.t.2449.1		2
5.3	odd	4		2550.2.d.t.2449.2		2
5.4	even	2		2550.2.a.bd.1.1	yes	1
15.14	odd	2		7650.2.a.p.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2550.2.a.e.1.1	✓	1	1.1	even	1	trivial
2550.2.a.bd.1.1	yes	1	5.4	even	2
2550.2.d.t.2449.1		2	5.2	odd	4
2550.2.d.t.2449.2		2	5.3	odd	4
7650.2.a.p.1.1		1	15.14	odd	2
7650.2.a.bv.1.1		1	3.2	odd	2

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

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$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2550.2.a.e.1.1

Level	$2550$
Weight	$2$
Character	2550.1
Self dual	yes
Analytic conductor	$20.362$
Analytic rank	$0$
Dimension	$1$
CM	no
Inner twists	$1$