LMFDB - Embedded newform 550.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [550,2,Mod(1,550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(550, 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("550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 550 = 2 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	550.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:	$4.39177211117$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 110)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	550.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -5.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -5.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +1.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} -5.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} -2.00000 q^{18} -7.00000 q^{19} +5.00000 q^{21} +1.00000 q^{22} +6.00000 q^{23} -1.00000 q^{24} -2.00000 q^{26} +5.00000 q^{27} -5.00000 q^{28} -3.00000 q^{29} -7.00000 q^{31} +1.00000 q^{32} -1.00000 q^{33} -3.00000 q^{34} -2.00000 q^{36} +7.00000 q^{37} -7.00000 q^{38} +2.00000 q^{39} +6.00000 q^{41} +5.00000 q^{42} -8.00000 q^{43} +1.00000 q^{44} +6.00000 q^{46} -6.00000 q^{47} -1.00000 q^{48} +18.0000 q^{49} +3.00000 q^{51} -2.00000 q^{52} +3.00000 q^{53} +5.00000 q^{54} -5.00000 q^{56} +7.00000 q^{57} -3.00000 q^{58} -6.00000 q^{59} -1.00000 q^{61} -7.00000 q^{62} +10.0000 q^{63} +1.00000 q^{64} -1.00000 q^{66} -8.00000 q^{67} -3.00000 q^{68} -6.00000 q^{69} +3.00000 q^{71} -2.00000 q^{72} -2.00000 q^{73} +7.00000 q^{74} -7.00000 q^{76} -5.00000 q^{77} +2.00000 q^{78} -10.0000 q^{79} +1.00000 q^{81} +6.00000 q^{82} +6.00000 q^{83} +5.00000 q^{84} -8.00000 q^{86} +3.00000 q^{87} +1.00000 q^{88} +9.00000 q^{89} +10.0000 q^{91} +6.00000 q^{92} +7.00000 q^{93} -6.00000 q^{94} -1.00000 q^{96} +4.00000 q^{97} +18.0000 q^{98} -2.00000 q^{99} +O(q^{100})\)

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	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	−5.00000	−1.88982	−0.944911	−	0.327327i	$-0.893852\pi$
$7$	−5.00000	−1.88982	−0.944911	+	0.327327i	$0.893852\pi$
$8$	1.00000	0.353553
$9$	−2.00000	−0.666667
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−1.00000	−0.288675
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	−5.00000	−1.33631
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	−2.00000	−0.471405
$19$	−7.00000	−1.60591	−0.802955	−	0.596040i	$-0.796740\pi$
$19$	−7.00000	−1.60591	−0.802955	+	0.596040i	$0.796740\pi$
$20$	0	0
$21$	5.00000	1.09109
$22$	1.00000	0.213201
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−2.00000	−0.392232
$27$	5.00000	0.962250
$28$	−5.00000	−0.944911
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	0	0
$31$	−7.00000	−1.25724	−0.628619	−	0.777714i	$-0.716379\pi$
$31$	−7.00000	−1.25724	−0.628619	+	0.777714i	$0.716379\pi$
$32$	1.00000	0.176777
$33$	−1.00000	−0.174078
$34$	−3.00000	−0.514496
$35$	0	0
$36$	−2.00000	−0.333333
$37$	7.00000	1.15079	0.575396	−	0.817875i	$-0.304848\pi$
$37$	7.00000	1.15079	0.575396	+	0.817875i	$0.304848\pi$
$38$	−7.00000	−1.13555
$39$	2.00000	0.320256
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	5.00000	0.771517
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	6.00000	0.884652
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	−1.00000	−0.144338
$49$	18.0000	2.57143
$50$	0	0
$51$	3.00000	0.420084
$52$	−2.00000	−0.277350
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	5.00000	0.680414
$55$	0	0
$56$	−5.00000	−0.668153
$57$	7.00000	0.927173
$58$	−3.00000	−0.393919
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	−7.00000	−0.889001
$63$	10.0000	1.25988
$64$	1.00000	0.125000
$65$	0	0
$66$	−1.00000	−0.123091
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	−3.00000	−0.363803
$69$	−6.00000	−0.722315
$70$	0	0
$71$	3.00000	0.356034	0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000	0.356034	0.178017	+	0.984027i	$0.443032\pi$
$72$	−2.00000	−0.235702
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	7.00000	0.813733
$75$	0	0
$76$	−7.00000	−0.802955
$77$	−5.00000	−0.569803
$78$	2.00000	0.226455
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	5.00000	0.545545
$85$	0	0
$86$	−8.00000	−0.862662
$87$	3.00000	0.321634
$88$	1.00000	0.106600
$89$	9.00000	0.953998	0.476999	−	0.878904i	$-0.341725\pi$
$89$	9.00000	0.953998	0.476999	+	0.878904i	$0.341725\pi$
$90$	0	0
$91$	10.0000	1.04828
$92$	6.00000	0.625543
$93$	7.00000	0.725866
$94$	−6.00000	−0.618853
$95$	0	0
$96$	−1.00000	−0.102062
$97$	4.00000	0.406138	0.203069	−	0.979164i	$-0.434908\pi$
$97$	4.00000	0.406138	0.203069	+	0.979164i	$0.434908\pi$
$98$	18.0000	1.81827
$99$	−2.00000	−0.201008
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	3.00000	0.297044
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	3.00000	0.291386
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	5.00000	0.481125
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	0	0
$111$	−7.00000	−0.664411
$112$	−5.00000	−0.472456
$113$	−12.0000	−1.12887	−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000	−1.12887	−0.564433	+	0.825479i	$0.690905\pi$
$114$	7.00000	0.655610
$115$	0	0
$116$	−3.00000	−0.278543
$117$	4.00000	0.369800
$118$	−6.00000	−0.552345
$119$	15.0000	1.37505
$120$	0	0
$121$	1.00000	0.0909091
$122$	−1.00000	−0.0905357
$123$	−6.00000	−0.541002
$124$	−7.00000	−0.628619
$125$	0	0
$126$	10.0000	0.890871
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	1.00000	0.0883883
$129$	8.00000	0.704361
$130$	0	0
$131$	−3.00000	−0.262111	−0.131056	−	0.991375i	$-0.541837\pi$
$131$	−3.00000	−0.262111	−0.131056	+	0.991375i	$0.541837\pi$
$132$	−1.00000	−0.0870388
$133$	35.0000	3.03488
$134$	−8.00000	−0.691095
$135$	0	0
$136$	−3.00000	−0.257248
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	−6.00000	−0.510754
$139$	−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
$141$	6.00000	0.505291
$142$	3.00000	0.251754
$143$	−2.00000	−0.167248
$144$	−2.00000	−0.166667
$145$	0	0
$146$	−2.00000	−0.165521
$147$	−18.0000	−1.48461
$148$	7.00000	0.575396
$149$	15.0000	1.22885	0.614424	−	0.788976i	$-0.289388\pi$
$149$	15.0000	1.22885	0.614424	+	0.788976i	$0.289388\pi$
$150$	0	0
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	−7.00000	−0.567775
$153$	6.00000	0.485071
$154$	−5.00000	−0.402911
$155$	0	0
$156$	2.00000	0.160128
$157$	7.00000	0.558661	0.279330	−	0.960195i	$-0.409888\pi$
$157$	7.00000	0.558661	0.279330	+	0.960195i	$0.409888\pi$
$158$	−10.0000	−0.795557
$159$	−3.00000	−0.237915
$160$	0	0
$161$	−30.0000	−2.36433
$162$	1.00000	0.0785674
$163$	−5.00000	−0.391630	−0.195815	−	0.980641i	$-0.562735\pi$
$163$	−5.00000	−0.391630	−0.195815	+	0.980641i	$0.562735\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	6.00000	0.465690
$167$	−21.0000	−1.62503	−0.812514	−	0.582941i	$-0.801902\pi$
$167$	−21.0000	−1.62503	−0.812514	+	0.582941i	$0.801902\pi$
$168$	5.00000	0.385758
$169$	−9.00000	−0.692308
$170$	0	0
$171$	14.0000	1.07061
$172$	−8.00000	−0.609994
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	3.00000	0.227429
$175$	0	0
$176$	1.00000	0.0753778
$177$	6.00000	0.450988
$178$	9.00000	0.674579
$179$	−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
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	10.0000	0.741249
$183$	1.00000	0.0739221
$184$	6.00000	0.442326
$185$	0	0
$186$	7.00000	0.513265
$187$	−3.00000	−0.219382
$188$	−6.00000	−0.437595
$189$	−25.0000	−1.81848
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	−1.00000	−0.0721688
$193$	−23.0000	−1.65558	−0.827788	−	0.561041i	$-0.810401\pi$
$193$	−23.0000	−1.65558	−0.827788	+	0.561041i	$0.810401\pi$
$194$	4.00000	0.287183
$195$	0	0
$196$	18.0000	1.28571
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	−2.00000	−0.142134
$199$	11.0000	0.779769	0.389885	−	0.920864i	$-0.372515\pi$
$199$	11.0000	0.779769	0.389885	+	0.920864i	$0.372515\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	−6.00000	−0.422159
$203$	15.0000	1.05279
$204$	3.00000	0.210042
$205$	0	0
$206$	4.00000	0.278693
$207$	−12.0000	−0.834058
$208$	−2.00000	−0.138675
$209$	−7.00000	−0.484200
$210$	0	0
$211$	5.00000	0.344214	0.172107	−	0.985078i	$-0.444942\pi$
$211$	5.00000	0.344214	0.172107	+	0.985078i	$0.444942\pi$
$212$	3.00000	0.206041
$213$	−3.00000	−0.205557
$214$	0	0
$215$	0	0
$216$	5.00000	0.340207
$217$	35.0000	2.37595
$218$	14.0000	0.948200
$219$	2.00000	0.135147
$220$	0	0
$221$	6.00000	0.403604
$222$	−7.00000	−0.469809
$223$	−14.0000	−0.937509	−0.468755	−	0.883328i	$-0.655297\pi$
$223$	−14.0000	−0.937509	−0.468755	+	0.883328i	$0.655297\pi$
$224$	−5.00000	−0.334077
$225$	0	0
$226$	−12.0000	−0.798228
$227$	18.0000	1.19470	0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000	1.19470	0.597351	+	0.801980i	$0.296220\pi$
$228$	7.00000	0.463586
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	0	0
$231$	5.00000	0.328976
$232$	−3.00000	−0.196960
$233$	−27.0000	−1.76883	−0.884414	−	0.466702i	$-0.845442\pi$
$233$	−27.0000	−1.76883	−0.884414	+	0.466702i	$0.845442\pi$
$234$	4.00000	0.261488
$235$	0	0
$236$	−6.00000	−0.390567
$237$	10.0000	0.649570
$238$	15.0000	0.972306
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	1.00000	0.0642824
$243$	−16.0000	−1.02640
$244$	−1.00000	−0.0640184
$245$	0	0
$246$	−6.00000	−0.382546
$247$	14.0000	0.890799
$248$	−7.00000	−0.444500
$249$	−6.00000	−0.380235
$250$	0	0
$251$	−30.0000	−1.89358	−0.946792	−	0.321847i	$-0.895696\pi$
$251$	−30.0000	−1.89358	−0.946792	+	0.321847i	$0.895696\pi$
$252$	10.0000	0.629941
$253$	6.00000	0.377217
$254$	16.0000	1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	8.00000	0.498058
$259$	−35.0000	−2.17479
$260$	0	0
$261$	6.00000	0.371391
$262$	−3.00000	−0.185341
$263$	9.00000	0.554964	0.277482	−	0.960731i	$-0.410500\pi$
$263$	9.00000	0.554964	0.277482	+	0.960731i	$0.410500\pi$
$264$	−1.00000	−0.0615457
$265$	0	0
$266$	35.0000	2.14599
$267$	−9.00000	−0.550791
$268$	−8.00000	−0.488678
$269$	−12.0000	−0.731653	−0.365826	−	0.930683i	$-0.619214\pi$
$269$	−12.0000	−0.731653	−0.365826	+	0.930683i	$0.619214\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	−3.00000	−0.181902
$273$	−10.0000	−0.605228
$274$	−12.0000	−0.724947
$275$	0	0
$276$	−6.00000	−0.361158
$277$	4.00000	0.240337	0.120168	−	0.992754i	$-0.461657\pi$
$277$	4.00000	0.240337	0.120168	+	0.992754i	$0.461657\pi$
$278$	−4.00000	−0.239904
$279$	14.0000	0.838158
$280$	0	0
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	6.00000	0.357295
$283$	−2.00000	−0.118888	−0.0594438	−	0.998232i	$-0.518933\pi$
$283$	−2.00000	−0.118888	−0.0594438	+	0.998232i	$0.518933\pi$
$284$	3.00000	0.178017
$285$	0	0
$286$	−2.00000	−0.118262
$287$	−30.0000	−1.77084
$288$	−2.00000	−0.117851
$289$	−8.00000	−0.470588
$290$	0	0
$291$	−4.00000	−0.234484
$292$	−2.00000	−0.117041
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	−18.0000	−1.04978
$295$	0	0
$296$	7.00000	0.406867
$297$	5.00000	0.290129
$298$	15.0000	0.868927
$299$	−12.0000	−0.693978
$300$	0	0
$301$	40.0000	2.30556
$302$	2.00000	0.115087
$303$	6.00000	0.344691
$304$	−7.00000	−0.401478
$305$	0	0
$306$	6.00000	0.342997
$307$	10.0000	0.570730	0.285365	−	0.958419i	$-0.407885\pi$
$307$	10.0000	0.570730	0.285365	+	0.958419i	$0.407885\pi$
$308$	−5.00000	−0.284901
$309$	−4.00000	−0.227552
$310$	0	0
$311$	−15.0000	−0.850572	−0.425286	−	0.905059i	$-0.639826\pi$
$311$	−15.0000	−0.850572	−0.425286	+	0.905059i	$0.639826\pi$
$312$	2.00000	0.113228
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	7.00000	0.395033
$315$	0	0
$316$	−10.0000	−0.562544
$317$	21.0000	1.17948	0.589739	−	0.807594i	$-0.299231\pi$
$317$	21.0000	1.17948	0.589739	+	0.807594i	$0.299231\pi$
$318$	−3.00000	−0.168232
$319$	−3.00000	−0.167968
$320$	0	0
$321$	0	0
$322$	−30.0000	−1.67183
$323$	21.0000	1.16847
$324$	1.00000	0.0555556
$325$	0	0
$326$	−5.00000	−0.276924
$327$	−14.0000	−0.774202
$328$	6.00000	0.331295
$329$	30.0000	1.65395
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	6.00000	0.329293
$333$	−14.0000	−0.767195
$334$	−21.0000	−1.14907
$335$	0	0
$336$	5.00000	0.272772
$337$	−5.00000	−0.272367	−0.136184	−	0.990684i	$-0.543484\pi$
$337$	−5.00000	−0.272367	−0.136184	+	0.990684i	$0.543484\pi$
$338$	−9.00000	−0.489535
$339$	12.0000	0.651751
$340$	0	0
$341$	−7.00000	−0.379071
$342$	14.0000	0.757033
$343$	−55.0000	−2.96972
$344$	−8.00000	−0.431331
$345$	0	0
$346$	−18.0000	−0.967686
$347$	−30.0000	−1.61048	−0.805242	−	0.592946i	$-0.797965\pi$
$347$	−30.0000	−1.61048	−0.805242	+	0.592946i	$0.797965\pi$
$348$	3.00000	0.160817
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	0	0
$351$	−10.0000	−0.533761
$352$	1.00000	0.0533002
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	6.00000	0.318896
$355$	0	0
$356$	9.00000	0.476999
$357$	−15.0000	−0.793884
$358$	−12.0000	−0.634220
$359$	−12.0000	−0.633336	−0.316668	−	0.948536i	$-0.602564\pi$
$359$	−12.0000	−0.633336	−0.316668	+	0.948536i	$0.602564\pi$
$360$	0	0
$361$	30.0000	1.57895
$362$	−22.0000	−1.15629
$363$	−1.00000	−0.0524864
$364$	10.0000	0.524142
$365$	0	0
$366$	1.00000	0.0522708
$367$	4.00000	0.208798	0.104399	−	0.994535i	$-0.466708\pi$
$367$	4.00000	0.208798	0.104399	+	0.994535i	$0.466708\pi$
$368$	6.00000	0.312772
$369$	−12.0000	−0.624695
$370$	0	0
$371$	−15.0000	−0.778761
$372$	7.00000	0.362933
$373$	−2.00000	−0.103556	−0.0517780	−	0.998659i	$-0.516489\pi$
$373$	−2.00000	−0.103556	−0.0517780	+	0.998659i	$0.516489\pi$
$374$	−3.00000	−0.155126
$375$	0	0
$376$	−6.00000	−0.309426
$377$	6.00000	0.309016
$378$	−25.0000	−1.28586
$379$	26.0000	1.33553	0.667765	−	0.744372i	$-0.267251\pi$
$379$	26.0000	1.33553	0.667765	+	0.744372i	$0.267251\pi$
$380$	0	0
$381$	−16.0000	−0.819705
$382$	−12.0000	−0.613973
$383$	18.0000	0.919757	0.459879	−	0.887982i	$-0.347893\pi$
$383$	18.0000	0.919757	0.459879	+	0.887982i	$0.347893\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−23.0000	−1.17067
$387$	16.0000	0.813326
$388$	4.00000	0.203069
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	−18.0000	−0.910299
$392$	18.0000	0.909137
$393$	3.00000	0.151330
$394$	0	0
$395$	0	0
$396$	−2.00000	−0.100504
$397$	22.0000	1.10415	0.552074	−	0.833795i	$-0.313837\pi$
$397$	22.0000	1.10415	0.552074	+	0.833795i	$0.313837\pi$
$398$	11.0000	0.551380
$399$	−35.0000	−1.75219
$400$	0	0
$401$	3.00000	0.149813	0.0749064	−	0.997191i	$-0.476134\pi$
$401$	3.00000	0.149813	0.0749064	+	0.997191i	$0.476134\pi$
$402$	8.00000	0.399004
$403$	14.0000	0.697390
$404$	−6.00000	−0.298511
$405$	0	0
$406$	15.0000	0.744438
$407$	7.00000	0.346977
$408$	3.00000	0.148522
$409$	−4.00000	−0.197787	−0.0988936	−	0.995098i	$-0.531530\pi$
$409$	−4.00000	−0.197787	−0.0988936	+	0.995098i	$0.531530\pi$
$410$	0	0
$411$	12.0000	0.591916
$412$	4.00000	0.197066
$413$	30.0000	1.47620
$414$	−12.0000	−0.589768
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	4.00000	0.195881
$418$	−7.00000	−0.342381
$419$	24.0000	1.17248	0.586238	−	0.810139i	$-0.300608\pi$
$419$	24.0000	1.17248	0.586238	+	0.810139i	$0.300608\pi$
$420$	0	0
$421$	8.00000	0.389896	0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000	0.389896	0.194948	+	0.980814i	$0.437546\pi$
$422$	5.00000	0.243396
$423$	12.0000	0.583460
$424$	3.00000	0.145693
$425$	0	0
$426$	−3.00000	−0.145350
$427$	5.00000	0.241967
$428$	0	0
$429$	2.00000	0.0965609
$430$	0	0
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	5.00000	0.240563
$433$	28.0000	1.34559	0.672797	−	0.739827i	$-0.265093\pi$
$433$	28.0000	1.34559	0.672797	+	0.739827i	$0.265093\pi$
$434$	35.0000	1.68005
$435$	0	0
$436$	14.0000	0.670478
$437$	−42.0000	−2.00913
$438$	2.00000	0.0955637
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	0	0
$441$	−36.0000	−1.71429
$442$	6.00000	0.285391
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	−7.00000	−0.332205
$445$	0	0
$446$	−14.0000	−0.662919
$447$	−15.0000	−0.709476
$448$	−5.00000	−0.236228
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	0	0
$451$	6.00000	0.282529
$452$	−12.0000	−0.564433
$453$	−2.00000	−0.0939682
$454$	18.0000	0.844782
$455$	0	0
$456$	7.00000	0.327805
$457$	−17.0000	−0.795226	−0.397613	−	0.917553i	$-0.630161\pi$
$457$	−17.0000	−0.795226	−0.397613	+	0.917553i	$0.630161\pi$
$458$	14.0000	0.654177
$459$	−15.0000	−0.700140
$460$	0	0
$461$	3.00000	0.139724	0.0698620	−	0.997557i	$-0.477744\pi$
$461$	3.00000	0.139724	0.0698620	+	0.997557i	$0.477744\pi$
$462$	5.00000	0.232621
$463$	22.0000	1.02243	0.511213	−	0.859454i	$-0.329196\pi$
$463$	22.0000	1.02243	0.511213	+	0.859454i	$0.329196\pi$
$464$	−3.00000	−0.139272
$465$	0	0
$466$	−27.0000	−1.25075
$467$	39.0000	1.80470	0.902352	−	0.430999i	$-0.141839\pi$
$467$	39.0000	1.80470	0.902352	+	0.430999i	$0.141839\pi$
$468$	4.00000	0.184900
$469$	40.0000	1.84703
$470$	0	0
$471$	−7.00000	−0.322543
$472$	−6.00000	−0.276172
$473$	−8.00000	−0.367840
$474$	10.0000	0.459315
$475$	0	0
$476$	15.0000	0.687524
$477$	−6.00000	−0.274721
$478$	6.00000	0.274434
$479$	−36.0000	−1.64488	−0.822441	−	0.568850i	$-0.807388\pi$
$479$	−36.0000	−1.64488	−0.822441	+	0.568850i	$0.807388\pi$
$480$	0	0
$481$	−14.0000	−0.638345
$482$	−22.0000	−1.00207
$483$	30.0000	1.36505
$484$	1.00000	0.0454545
$485$	0	0
$486$	−16.0000	−0.725775
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	−1.00000	−0.0452679
$489$	5.00000	0.226108
$490$	0	0
$491$	33.0000	1.48927	0.744635	−	0.667472i	$-0.232624\pi$
$491$	33.0000	1.48927	0.744635	+	0.667472i	$0.232624\pi$
$492$	−6.00000	−0.270501
$493$	9.00000	0.405340
$494$	14.0000	0.629890
$495$	0	0
$496$	−7.00000	−0.314309
$497$	−15.0000	−0.672842
$498$	−6.00000	−0.268866
$499$	44.0000	1.96971	0.984855	−	0.173379i	$-0.0554684\pi$
$499$	44.0000	1.96971	0.984855	+	0.173379i	$0.0554684\pi$
$500$	0	0
$501$	21.0000	0.938211
$502$	−30.0000	−1.33897
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	10.0000	0.445435
$505$	0	0
$506$	6.00000	0.266733
$507$	9.00000	0.399704
$508$	16.0000	0.709885
$509$	−12.0000	−0.531891	−0.265945	−	0.963988i	$-0.585684\pi$
$509$	−12.0000	−0.531891	−0.265945	+	0.963988i	$0.585684\pi$
$510$	0	0
$511$	10.0000	0.442374
$512$	1.00000	0.0441942
$513$	−35.0000	−1.54529
$514$	−6.00000	−0.264649
$515$	0	0
$516$	8.00000	0.352180
$517$	−6.00000	−0.263880
$518$	−35.0000	−1.53781
$519$	18.0000	0.790112
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	6.00000	0.262613
$523$	16.0000	0.699631	0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000	0.699631	0.349816	+	0.936819i	$0.386244\pi$
$524$	−3.00000	−0.131056
$525$	0	0
$526$	9.00000	0.392419
$527$	21.0000	0.914774
$528$	−1.00000	−0.0435194
$529$	13.0000	0.565217
$530$	0	0
$531$	12.0000	0.520756
$532$	35.0000	1.51744
$533$	−12.0000	−0.519778
$534$	−9.00000	−0.389468
$535$	0	0
$536$	−8.00000	−0.345547
$537$	12.0000	0.517838
$538$	−12.0000	−0.517357
$539$	18.0000	0.775315
$540$	0	0
$541$	41.0000	1.76273	0.881364	−	0.472438i	$-0.156626\pi$
$541$	41.0000	1.76273	0.881364	+	0.472438i	$0.156626\pi$
$542$	20.0000	0.859074
$543$	22.0000	0.944110
$544$	−3.00000	−0.128624
$545$	0	0
$546$	−10.0000	−0.427960
$547$	−8.00000	−0.342055	−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000	−0.342055	−0.171028	+	0.985266i	$0.554709\pi$
$548$	−12.0000	−0.512615
$549$	2.00000	0.0853579
$550$	0	0
$551$	21.0000	0.894630
$552$	−6.00000	−0.255377
$553$	50.0000	2.12622
$554$	4.00000	0.169944
$555$	0	0
$556$	−4.00000	−0.169638
$557$	6.00000	0.254228	0.127114	−	0.991888i	$-0.459429\pi$
$557$	6.00000	0.254228	0.127114	+	0.991888i	$0.459429\pi$
$558$	14.0000	0.592667
$559$	16.0000	0.676728
$560$	0	0
$561$	3.00000	0.126660
$562$	30.0000	1.26547
$563$	6.00000	0.252870	0.126435	−	0.991975i	$-0.459647\pi$
$563$	6.00000	0.252870	0.126435	+	0.991975i	$0.459647\pi$
$564$	6.00000	0.252646
$565$	0	0
$566$	−2.00000	−0.0840663
$567$	−5.00000	−0.209980
$568$	3.00000	0.125877
$569$	24.0000	1.00613	0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000	1.00613	0.503066	+	0.864248i	$0.332205\pi$
$570$	0	0
$571$	−25.0000	−1.04622	−0.523109	−	0.852266i	$-0.675228\pi$
$571$	−25.0000	−1.04622	−0.523109	+	0.852266i	$0.675228\pi$
$572$	−2.00000	−0.0836242
$573$	12.0000	0.501307
$574$	−30.0000	−1.25218
$575$	0	0
$576$	−2.00000	−0.0833333
$577$	−38.0000	−1.58196	−0.790980	−	0.611842i	$-0.790429\pi$
$577$	−38.0000	−1.58196	−0.790980	+	0.611842i	$0.790429\pi$
$578$	−8.00000	−0.332756
$579$	23.0000	0.955847
$580$	0	0
$581$	−30.0000	−1.24461
$582$	−4.00000	−0.165805
$583$	3.00000	0.124247
$584$	−2.00000	−0.0827606
$585$	0	0
$586$	6.00000	0.247858
$587$	−3.00000	−0.123823	−0.0619116	−	0.998082i	$-0.519720\pi$
$587$	−3.00000	−0.123823	−0.0619116	+	0.998082i	$0.519720\pi$
$588$	−18.0000	−0.742307
$589$	49.0000	2.01901
$590$	0	0
$591$	0	0
$592$	7.00000	0.287698
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	5.00000	0.205152
$595$	0	0
$596$	15.0000	0.614424
$597$	−11.0000	−0.450200
$598$	−12.0000	−0.490716
$599$	9.00000	0.367730	0.183865	−	0.982952i	$-0.441139\pi$
$599$	9.00000	0.367730	0.183865	+	0.982952i	$0.441139\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	40.0000	1.63028
$603$	16.0000	0.651570
$604$	2.00000	0.0813788
$605$	0	0
$606$	6.00000	0.243733
$607$	49.0000	1.98885	0.994424	−	0.105453i	$-0.0336291\pi$
$607$	49.0000	1.98885	0.994424	+	0.105453i	$0.0336291\pi$
$608$	−7.00000	−0.283887
$609$	−15.0000	−0.607831
$610$	0	0
$611$	12.0000	0.485468
$612$	6.00000	0.242536
$613$	−2.00000	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	10.0000	0.403567
$615$	0	0
$616$	−5.00000	−0.201456
$617$	48.0000	1.93241	0.966204	−	0.257780i	$-0.0829910\pi$
$617$	48.0000	1.93241	0.966204	+	0.257780i	$0.0829910\pi$
$618$	−4.00000	−0.160904
$619$	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
$621$	30.0000	1.20386
$622$	−15.0000	−0.601445
$623$	−45.0000	−1.80289
$624$	2.00000	0.0800641
$625$	0	0
$626$	−14.0000	−0.559553
$627$	7.00000	0.279553
$628$	7.00000	0.279330
$629$	−21.0000	−0.837325
$630$	0	0
$631$	35.0000	1.39333	0.696664	−	0.717398i	$-0.254667\pi$
$631$	35.0000	1.39333	0.696664	+	0.717398i	$0.254667\pi$
$632$	−10.0000	−0.397779
$633$	−5.00000	−0.198732
$634$	21.0000	0.834017
$635$	0	0
$636$	−3.00000	−0.118958
$637$	−36.0000	−1.42637
$638$	−3.00000	−0.118771
$639$	−6.00000	−0.237356
$640$	0	0
$641$	−33.0000	−1.30342	−0.651711	−	0.758468i	$-0.725948\pi$
$641$	−33.0000	−1.30342	−0.651711	+	0.758468i	$0.725948\pi$
$642$	0	0
$643$	−5.00000	−0.197181	−0.0985904	−	0.995128i	$-0.531433\pi$
$643$	−5.00000	−0.197181	−0.0985904	+	0.995128i	$0.531433\pi$
$644$	−30.0000	−1.18217
$645$	0	0
$646$	21.0000	0.826234
$647$	−6.00000	−0.235884	−0.117942	−	0.993020i	$-0.537630\pi$
$647$	−6.00000	−0.235884	−0.117942	+	0.993020i	$0.537630\pi$
$648$	1.00000	0.0392837
$649$	−6.00000	−0.235521
$650$	0	0
$651$	−35.0000	−1.37176
$652$	−5.00000	−0.195815
$653$	45.0000	1.76099	0.880493	−	0.474059i	$-0.157212\pi$
$653$	45.0000	1.76099	0.880493	+	0.474059i	$0.157212\pi$
$654$	−14.0000	−0.547443
$655$	0	0
$656$	6.00000	0.234261
$657$	4.00000	0.156055
$658$	30.0000	1.16952
$659$	−21.0000	−0.818044	−0.409022	−	0.912525i	$-0.634130\pi$
$659$	−21.0000	−0.818044	−0.409022	+	0.912525i	$0.634130\pi$
$660$	0	0
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	8.00000	0.310929
$663$	−6.00000	−0.233021
$664$	6.00000	0.232845
$665$	0	0
$666$	−14.0000	−0.542489
$667$	−18.0000	−0.696963
$668$	−21.0000	−0.812514
$669$	14.0000	0.541271
$670$	0	0
$671$	−1.00000	−0.0386046
$672$	5.00000	0.192879
$673$	1.00000	0.0385472	0.0192736	−	0.999814i	$-0.493865\pi$
$673$	1.00000	0.0385472	0.0192736	+	0.999814i	$0.493865\pi$
$674$	−5.00000	−0.192593
$675$	0	0
$676$	−9.00000	−0.346154
$677$	−12.0000	−0.461197	−0.230599	−	0.973049i	$-0.574068\pi$
$677$	−12.0000	−0.461197	−0.230599	+	0.973049i	$0.574068\pi$
$678$	12.0000	0.460857
$679$	−20.0000	−0.767530
$680$	0	0
$681$	−18.0000	−0.689761
$682$	−7.00000	−0.268044
$683$	9.00000	0.344375	0.172188	−	0.985064i	$-0.444916\pi$
$683$	9.00000	0.344375	0.172188	+	0.985064i	$0.444916\pi$
$684$	14.0000	0.535303
$685$	0	0
$686$	−55.0000	−2.09991
$687$	−14.0000	−0.534133
$688$	−8.00000	−0.304997
$689$	−6.00000	−0.228582
$690$	0	0
$691$	−10.0000	−0.380418	−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000	−0.380418	−0.190209	+	0.981744i	$0.560917\pi$
$692$	−18.0000	−0.684257
$693$	10.0000	0.379869
$694$	−30.0000	−1.13878
$695$	0	0
$696$	3.00000	0.113715
$697$	−18.0000	−0.681799
$698$	−10.0000	−0.378506
$699$	27.0000	1.02123
$700$	0	0
$701$	39.0000	1.47301	0.736505	−	0.676432i	$-0.236475\pi$
$701$	39.0000	1.47301	0.736505	+	0.676432i	$0.236475\pi$
$702$	−10.0000	−0.377426
$703$	−49.0000	−1.84807
$704$	1.00000	0.0376889
$705$	0	0
$706$	−6.00000	−0.225813
$707$	30.0000	1.12827
$708$	6.00000	0.225494
$709$	−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
$711$	20.0000	0.750059
$712$	9.00000	0.337289
$713$	−42.0000	−1.57291
$714$	−15.0000	−0.561361
$715$	0	0
$716$	−12.0000	−0.448461
$717$	−6.00000	−0.224074
$718$	−12.0000	−0.447836
$719$	−51.0000	−1.90198	−0.950990	−	0.309223i	$-0.899931\pi$
$719$	−51.0000	−1.90198	−0.950990	+	0.309223i	$0.899931\pi$
$720$	0	0
$721$	−20.0000	−0.744839
$722$	30.0000	1.11648
$723$	22.0000	0.818189
$724$	−22.0000	−0.817624
$725$	0	0
$726$	−1.00000	−0.0371135
$727$	10.0000	0.370879	0.185440	−	0.982656i	$-0.440629\pi$
$727$	10.0000	0.370879	0.185440	+	0.982656i	$0.440629\pi$
$728$	10.0000	0.370625
$729$	13.0000	0.481481
$730$	0	0
$731$	24.0000	0.887672
$732$	1.00000	0.0369611
$733$	4.00000	0.147743	0.0738717	−	0.997268i	$-0.476464\pi$
$733$	4.00000	0.147743	0.0738717	+	0.997268i	$0.476464\pi$
$734$	4.00000	0.147643
$735$	0	0
$736$	6.00000	0.221163
$737$	−8.00000	−0.294684
$738$	−12.0000	−0.441726
$739$	−16.0000	−0.588570	−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000	−0.588570	−0.294285	+	0.955718i	$0.595081\pi$
$740$	0	0
$741$	−14.0000	−0.514303
$742$	−15.0000	−0.550667
$743$	3.00000	0.110059	0.0550297	−	0.998485i	$-0.482475\pi$
$743$	3.00000	0.110059	0.0550297	+	0.998485i	$0.482475\pi$
$744$	7.00000	0.256632
$745$	0	0
$746$	−2.00000	−0.0732252
$747$	−12.0000	−0.439057
$748$	−3.00000	−0.109691
$749$	0	0
$750$	0	0
$751$	5.00000	0.182453	0.0912263	−	0.995830i	$-0.470921\pi$
$751$	5.00000	0.182453	0.0912263	+	0.995830i	$0.470921\pi$
$752$	−6.00000	−0.218797
$753$	30.0000	1.09326
$754$	6.00000	0.218507
$755$	0	0
$756$	−25.0000	−0.909241
$757$	−26.0000	−0.944986	−0.472493	−	0.881334i	$-0.656646\pi$
$757$	−26.0000	−0.944986	−0.472493	+	0.881334i	$0.656646\pi$
$758$	26.0000	0.944363
$759$	−6.00000	−0.217786
$760$	0	0
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	−16.0000	−0.579619
$763$	−70.0000	−2.53417
$764$	−12.0000	−0.434145
$765$	0	0
$766$	18.0000	0.650366
$767$	12.0000	0.433295
$768$	−1.00000	−0.0360844
$769$	−10.0000	−0.360609	−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000	−0.360609	−0.180305	+	0.983611i	$0.557708\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	−23.0000	−0.827788
$773$	−9.00000	−0.323708	−0.161854	−	0.986815i	$-0.551747\pi$
$773$	−9.00000	−0.323708	−0.161854	+	0.986815i	$0.551747\pi$
$774$	16.0000	0.575108
$775$	0	0
$776$	4.00000	0.143592
$777$	35.0000	1.25562
$778$	18.0000	0.645331
$779$	−42.0000	−1.50481
$780$	0	0
$781$	3.00000	0.107348
$782$	−18.0000	−0.643679
$783$	−15.0000	−0.536056
$784$	18.0000	0.642857
$785$	0	0
$786$	3.00000	0.107006
$787$	−8.00000	−0.285169	−0.142585	−	0.989783i	$-0.545541\pi$
$787$	−8.00000	−0.285169	−0.142585	+	0.989783i	$0.545541\pi$
$788$	0	0
$789$	−9.00000	−0.320408
$790$	0	0
$791$	60.0000	2.13335
$792$	−2.00000	−0.0710669
$793$	2.00000	0.0710221
$794$	22.0000	0.780751
$795$	0	0
$796$	11.0000	0.389885
$797$	−30.0000	−1.06265	−0.531327	−	0.847167i	$-0.678307\pi$
$797$	−30.0000	−1.06265	−0.531327	+	0.847167i	$0.678307\pi$
$798$	−35.0000	−1.23899
$799$	18.0000	0.636794
$800$	0	0
$801$	−18.0000	−0.635999
$802$	3.00000	0.105934
$803$	−2.00000	−0.0705785
$804$	8.00000	0.282138
$805$	0	0
$806$	14.0000	0.493129
$807$	12.0000	0.422420
$808$	−6.00000	−0.211079
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	0	0
$811$	−13.0000	−0.456492	−0.228246	−	0.973604i	$-0.573299\pi$
$811$	−13.0000	−0.456492	−0.228246	+	0.973604i	$0.573299\pi$
$812$	15.0000	0.526397
$813$	−20.0000	−0.701431
$814$	7.00000	0.245350
$815$	0	0
$816$	3.00000	0.105021
$817$	56.0000	1.95919
$818$	−4.00000	−0.139857
$819$	−20.0000	−0.698857
$820$	0	0
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	12.0000	0.418548
$823$	4.00000	0.139431	0.0697156	−	0.997567i	$-0.477791\pi$
$823$	4.00000	0.139431	0.0697156	+	0.997567i	$0.477791\pi$
$824$	4.00000	0.139347
$825$	0	0
$826$	30.0000	1.04383
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	−12.0000	−0.417029
$829$	−52.0000	−1.80603	−0.903017	−	0.429604i	$-0.858653\pi$
$829$	−52.0000	−1.80603	−0.903017	+	0.429604i	$0.858653\pi$
$830$	0	0
$831$	−4.00000	−0.138758
$832$	−2.00000	−0.0693375
$833$	−54.0000	−1.87099
$834$	4.00000	0.138509
$835$	0	0
$836$	−7.00000	−0.242100
$837$	−35.0000	−1.20978
$838$	24.0000	0.829066
$839$	48.0000	1.65714	0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000	1.65714	0.828572	+	0.559883i	$0.189154\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	8.00000	0.275698
$843$	−30.0000	−1.03325
$844$	5.00000	0.172107
$845$	0	0
$846$	12.0000	0.412568
$847$	−5.00000	−0.171802
$848$	3.00000	0.103020
$849$	2.00000	0.0686398
$850$	0	0
$851$	42.0000	1.43974
$852$	−3.00000	−0.102778
$853$	34.0000	1.16414	0.582069	−	0.813139i	$-0.302243\pi$
$853$	34.0000	1.16414	0.582069	+	0.813139i	$0.302243\pi$
$854$	5.00000	0.171096
$855$	0	0
$856$	0	0
$857$	−3.00000	−0.102478	−0.0512390	−	0.998686i	$-0.516317\pi$
$857$	−3.00000	−0.102478	−0.0512390	+	0.998686i	$0.516317\pi$
$858$	2.00000	0.0682789
$859$	14.0000	0.477674	0.238837	−	0.971060i	$-0.423234\pi$
$859$	14.0000	0.477674	0.238837	+	0.971060i	$0.423234\pi$
$860$	0	0
$861$	30.0000	1.02240
$862$	−24.0000	−0.817443
$863$	6.00000	0.204242	0.102121	−	0.994772i	$-0.467437\pi$
$863$	6.00000	0.204242	0.102121	+	0.994772i	$0.467437\pi$
$864$	5.00000	0.170103
$865$	0	0
$866$	28.0000	0.951479
$867$	8.00000	0.271694
$868$	35.0000	1.18798
$869$	−10.0000	−0.339227
$870$	0	0
$871$	16.0000	0.542139
$872$	14.0000	0.474100
$873$	−8.00000	−0.270759
$874$	−42.0000	−1.42067
$875$	0	0
$876$	2.00000	0.0675737
$877$	22.0000	0.742887	0.371444	−	0.928456i	$-0.378863\pi$
$877$	22.0000	0.742887	0.371444	+	0.928456i	$0.378863\pi$
$878$	−16.0000	−0.539974
$879$	−6.00000	−0.202375
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	−36.0000	−1.21218
$883$	1.00000	0.0336527	0.0168263	−	0.999858i	$-0.494644\pi$
$883$	1.00000	0.0336527	0.0168263	+	0.999858i	$0.494644\pi$
$884$	6.00000	0.201802
$885$	0	0
$886$	−12.0000	−0.403148
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	−7.00000	−0.234905
$889$	−80.0000	−2.68311
$890$	0	0
$891$	1.00000	0.0335013
$892$	−14.0000	−0.468755
$893$	42.0000	1.40548
$894$	−15.0000	−0.501675
$895$	0	0
$896$	−5.00000	−0.167038
$897$	12.0000	0.400668
$898$	−30.0000	−1.00111
$899$	21.0000	0.700389
$900$	0	0
$901$	−9.00000	−0.299833
$902$	6.00000	0.199778
$903$	−40.0000	−1.33112
$904$	−12.0000	−0.399114
$905$	0	0
$906$	−2.00000	−0.0664455
$907$	−17.0000	−0.564476	−0.282238	−	0.959344i	$-0.591077\pi$
$907$	−17.0000	−0.564476	−0.282238	+	0.959344i	$0.591077\pi$
$908$	18.0000	0.597351
$909$	12.0000	0.398015
$910$	0	0
$911$	15.0000	0.496972	0.248486	−	0.968635i	$-0.420067\pi$
$911$	15.0000	0.496972	0.248486	+	0.968635i	$0.420067\pi$
$912$	7.00000	0.231793
$913$	6.00000	0.198571
$914$	−17.0000	−0.562310
$915$	0	0
$916$	14.0000	0.462573
$917$	15.0000	0.495344
$918$	−15.0000	−0.495074
$919$	−4.00000	−0.131948	−0.0659739	−	0.997821i	$-0.521015\pi$
$919$	−4.00000	−0.131948	−0.0659739	+	0.997821i	$0.521015\pi$
$920$	0	0
$921$	−10.0000	−0.329511
$922$	3.00000	0.0987997
$923$	−6.00000	−0.197492
$924$	5.00000	0.164488
$925$	0	0
$926$	22.0000	0.722965
$927$	−8.00000	−0.262754
$928$	−3.00000	−0.0984798
$929$	−45.0000	−1.47640	−0.738201	−	0.674581i	$-0.764324\pi$
$929$	−45.0000	−1.47640	−0.738201	+	0.674581i	$0.764324\pi$
$930$	0	0
$931$	−126.000	−4.12948
$932$	−27.0000	−0.884414
$933$	15.0000	0.491078
$934$	39.0000	1.27612
$935$	0	0
$936$	4.00000	0.130744
$937$	−26.0000	−0.849383	−0.424691	−	0.905338i	$-0.639617\pi$
$937$	−26.0000	−0.849383	−0.424691	+	0.905338i	$0.639617\pi$
$938$	40.0000	1.30605
$939$	14.0000	0.456873
$940$	0	0
$941$	−15.0000	−0.488986	−0.244493	−	0.969651i	$-0.578622\pi$
$941$	−15.0000	−0.488986	−0.244493	+	0.969651i	$0.578622\pi$
$942$	−7.00000	−0.228072
$943$	36.0000	1.17232
$944$	−6.00000	−0.195283
$945$	0	0
$946$	−8.00000	−0.260102
$947$	−3.00000	−0.0974869	−0.0487435	−	0.998811i	$-0.515522\pi$
$947$	−3.00000	−0.0974869	−0.0487435	+	0.998811i	$0.515522\pi$
$948$	10.0000	0.324785
$949$	4.00000	0.129845
$950$	0	0
$951$	−21.0000	−0.680972
$952$	15.0000	0.486153
$953$	27.0000	0.874616	0.437308	−	0.899312i	$-0.355932\pi$
$953$	27.0000	0.874616	0.437308	+	0.899312i	$0.355932\pi$
$954$	−6.00000	−0.194257
$955$	0	0
$956$	6.00000	0.194054
$957$	3.00000	0.0969762
$958$	−36.0000	−1.16311
$959$	60.0000	1.93750
$960$	0	0
$961$	18.0000	0.580645
$962$	−14.0000	−0.451378
$963$	0	0
$964$	−22.0000	−0.708572
$965$	0	0
$966$	30.0000	0.965234
$967$	13.0000	0.418052	0.209026	−	0.977910i	$-0.432971\pi$
$967$	13.0000	0.418052	0.209026	+	0.977910i	$0.432971\pi$
$968$	1.00000	0.0321412
$969$	−21.0000	−0.674617
$970$	0	0
$971$	−24.0000	−0.770197	−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000	−0.770197	−0.385098	+	0.922876i	$0.625832\pi$
$972$	−16.0000	−0.513200
$973$	20.0000	0.641171
$974$	−8.00000	−0.256337
$975$	0	0
$976$	−1.00000	−0.0320092
$977$	−12.0000	−0.383914	−0.191957	−	0.981403i	$-0.561483\pi$
$977$	−12.0000	−0.383914	−0.191957	+	0.981403i	$0.561483\pi$
$978$	5.00000	0.159882
$979$	9.00000	0.287641
$980$	0	0
$981$	−28.0000	−0.893971
$982$	33.0000	1.05307
$983$	6.00000	0.191370	0.0956851	−	0.995412i	$-0.469496\pi$
$983$	6.00000	0.191370	0.0956851	+	0.995412i	$0.469496\pi$
$984$	−6.00000	−0.191273
$985$	0	0
$986$	9.00000	0.286618
$987$	−30.0000	−0.954911
$988$	14.0000	0.445399
$989$	−48.0000	−1.52631
$990$	0	0
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	−7.00000	−0.222250
$993$	−8.00000	−0.253872
$994$	−15.0000	−0.475771
$995$	0	0
$996$	−6.00000	−0.190117
$997$	28.0000	0.886769	0.443384	−	0.896332i	$-0.353778\pi$
$997$	28.0000	0.886769	0.443384	+	0.896332i	$0.353778\pi$
$998$	44.0000	1.39280
$999$	35.0000	1.10735

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	550.2.a.i.1.1		1
3.2	odd	2		4950.2.a.a.1.1		1
4.3	odd	2		4400.2.a.w.1.1		1
5.2	odd	4		550.2.b.b.199.2		2
5.3	odd	4		550.2.b.b.199.1		2
5.4	even	2		110.2.a.a.1.1	✓	1
11.10	odd	2		6050.2.a.i.1.1		1
15.2	even	4		4950.2.c.a.199.1		2
15.8	even	4		4950.2.c.a.199.2		2
15.14	odd	2		990.2.a.l.1.1		1
20.3	even	4		4400.2.b.g.4049.2		2
20.7	even	4		4400.2.b.g.4049.1		2
20.19	odd	2		880.2.a.c.1.1		1
35.34	odd	2		5390.2.a.h.1.1		1
40.19	odd	2		3520.2.a.z.1.1		1
40.29	even	2		3520.2.a.l.1.1		1
55.54	odd	2		1210.2.a.k.1.1		1
60.59	even	2		7920.2.a.s.1.1		1
220.219	even	2		9680.2.a.j.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
110.2.a.a.1.1	✓	1	5.4	even	2
550.2.a.i.1.1		1	1.1	even	1	trivial
550.2.b.b.199.1		2	5.3	odd	4
550.2.b.b.199.2		2	5.2	odd	4
880.2.a.c.1.1		1	20.19	odd	2
990.2.a.l.1.1		1	15.14	odd	2
1210.2.a.k.1.1		1	55.54	odd	2
3520.2.a.l.1.1		1	40.29	even	2
3520.2.a.z.1.1		1	40.19	odd	2
4400.2.a.w.1.1		1	4.3	odd	2
4400.2.b.g.4049.1		2	20.7	even	4
4400.2.b.g.4049.2		2	20.3	even	4
4950.2.a.a.1.1		1	3.2	odd	2
4950.2.c.a.199.1		2	15.2	even	4
4950.2.c.a.199.2		2	15.8	even	4
5390.2.a.h.1.1		1	35.34	odd	2
6050.2.a.i.1.1		1	11.10	odd	2
7920.2.a.s.1.1		1	60.59	even	2
9680.2.a.j.1.1		1	220.219	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:	\( N \)	\(=\)	\( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	550.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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