LMFDB - Embedded newform 5550.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5550,2,Mod(1,5550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5550, 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("5550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5550 = 2 \cdot 3 \cdot 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5550.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:	$44.3169731218$
Analytic rank:	$2$
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	$\chi$	$=$	5550.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{11} -1.00000 q^{12} -4.00000 q^{13} +4.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} -7.00000 q^{19} +4.00000 q^{21} +2.00000 q^{22} -3.00000 q^{23} +1.00000 q^{24} +4.00000 q^{26} -1.00000 q^{27} -4.00000 q^{28} -8.00000 q^{31} -1.00000 q^{32} +2.00000 q^{33} +2.00000 q^{34} +1.00000 q^{36} -1.00000 q^{37} +7.00000 q^{38} +4.00000 q^{39} +9.00000 q^{41} -4.00000 q^{42} -11.0000 q^{43} -2.00000 q^{44} +3.00000 q^{46} -6.00000 q^{47} -1.00000 q^{48} +9.00000 q^{49} +2.00000 q^{51} -4.00000 q^{52} -9.00000 q^{53} +1.00000 q^{54} +4.00000 q^{56} +7.00000 q^{57} -15.0000 q^{59} +8.00000 q^{62} -4.00000 q^{63} +1.00000 q^{64} -2.00000 q^{66} -12.0000 q^{67} -2.00000 q^{68} +3.00000 q^{69} +12.0000 q^{71} -1.00000 q^{72} +15.0000 q^{73} +1.00000 q^{74} -7.00000 q^{76} +8.00000 q^{77} -4.00000 q^{78} +7.00000 q^{79} +1.00000 q^{81} -9.00000 q^{82} -8.00000 q^{83} +4.00000 q^{84} +11.0000 q^{86} +2.00000 q^{88} -8.00000 q^{89} +16.0000 q^{91} -3.00000 q^{92} +8.00000 q^{93} +6.00000 q^{94} +1.00000 q^{96} +18.0000 q^{97} -9.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

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
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	−1.00000	−0.288675
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	4.00000	1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	−1.00000	−0.235702
$19$	−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
$21$	4.00000	0.872872
$22$	2.00000	0.426401
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	4.00000	0.784465
$27$	−1.00000	−0.192450
$28$	−4.00000	−0.755929
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	−1.00000	−0.176777
$33$	2.00000	0.348155
$34$	2.00000	0.342997
$35$	0	0
$36$	1.00000	0.166667
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	7.00000	1.13555
$39$	4.00000	0.640513
$40$	0	0
$41$	9.00000	1.40556	0.702782	−	0.711405i	$-0.251941\pi$
$41$	9.00000	1.40556	0.702782	+	0.711405i	$0.251941\pi$
$42$	−4.00000	−0.617213
$43$	−11.0000	−1.67748	−0.838742	−	0.544529i	$-0.816708\pi$
$43$	−11.0000	−1.67748	−0.838742	+	0.544529i	$0.816708\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	3.00000	0.442326
$47$	−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
$49$	9.00000	1.28571
$50$	0	0
$51$	2.00000	0.280056
$52$	−4.00000	−0.554700
$53$	−9.00000	−1.23625	−0.618123	−	0.786082i	$-0.712106\pi$
$53$	−9.00000	−1.23625	−0.618123	+	0.786082i	$0.712106\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	4.00000	0.534522
$57$	7.00000	0.927173
$58$	0	0
$59$	−15.0000	−1.95283	−0.976417	−	0.215894i	$-0.930733\pi$
$59$	−15.0000	−1.95283	−0.976417	+	0.215894i	$0.930733\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	8.00000	1.01600
$63$	−4.00000	−0.503953
$64$	1.00000	0.125000
$65$	0	0
$66$	−2.00000	−0.246183
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	−2.00000	−0.242536
$69$	3.00000	0.361158
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	−1.00000	−0.117851
$73$	15.0000	1.75562	0.877809	−	0.479012i	$-0.159005\pi$
$73$	15.0000	1.75562	0.877809	+	0.479012i	$0.159005\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	−7.00000	−0.802955
$77$	8.00000	0.911685
$78$	−4.00000	−0.452911
$79$	7.00000	0.787562	0.393781	−	0.919204i	$-0.371167\pi$
$79$	7.00000	0.787562	0.393781	+	0.919204i	$0.371167\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−9.00000	−0.993884
$83$	−8.00000	−0.878114	−0.439057	−	0.898459i	$-0.644687\pi$
$83$	−8.00000	−0.878114	−0.439057	+	0.898459i	$0.644687\pi$
$84$	4.00000	0.436436
$85$	0	0
$86$	11.0000	1.18616
$87$	0	0
$88$	2.00000	0.213201
$89$	−8.00000	−0.847998	−0.423999	−	0.905663i	$-0.639374\pi$
$89$	−8.00000	−0.847998	−0.423999	+	0.905663i	$0.639374\pi$
$90$	0	0
$91$	16.0000	1.67726
$92$	−3.00000	−0.312772
$93$	8.00000	0.829561
$94$	6.00000	0.618853
$95$	0	0
$96$	1.00000	0.102062
$97$	18.0000	1.82762	0.913812	−	0.406138i	$-0.133125\pi$
$97$	18.0000	1.82762	0.913812	+	0.406138i	$0.133125\pi$
$98$	−9.00000	−0.909137
$99$	−2.00000	−0.201008
$100$	0	0
$101$	−19.0000	−1.89057	−0.945285	−	0.326245i	$-0.894217\pi$
$101$	−19.0000	−1.89057	−0.945285	+	0.326245i	$0.894217\pi$
$102$	−2.00000	−0.198030
$103$	11.0000	1.08386	0.541931	−	0.840423i	$-0.317693\pi$
$103$	11.0000	1.08386	0.541931	+	0.840423i	$0.317693\pi$
$104$	4.00000	0.392232
$105$	0	0
$106$	9.00000	0.874157
$107$	−14.0000	−1.35343	−0.676716	−	0.736245i	$-0.736597\pi$
$107$	−14.0000	−1.35343	−0.676716	+	0.736245i	$0.736597\pi$
$108$	−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
$111$	1.00000	0.0949158
$112$	−4.00000	−0.377964
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	−7.00000	−0.655610
$115$	0	0
$116$	0	0
$117$	−4.00000	−0.369800
$118$	15.0000	1.38086
$119$	8.00000	0.733359
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	−9.00000	−0.811503
$124$	−8.00000	−0.718421
$125$	0	0
$126$	4.00000	0.356348
$127$	10.0000	0.887357	0.443678	−	0.896186i	$-0.353673\pi$
$127$	10.0000	0.887357	0.443678	+	0.896186i	$0.353673\pi$
$128$	−1.00000	−0.0883883
$129$	11.0000	0.968496
$130$	0	0
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	2.00000	0.174078
$133$	28.0000	2.42791
$134$	12.0000	1.03664
$135$	0	0
$136$	2.00000	0.171499
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	−3.00000	−0.255377
$139$	−16.0000	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	0	0
$141$	6.00000	0.505291
$142$	−12.0000	−1.00702
$143$	8.00000	0.668994
$144$	1.00000	0.0833333
$145$	0	0
$146$	−15.0000	−1.24141
$147$	−9.00000	−0.742307
$148$	−1.00000	−0.0821995
$149$	−3.00000	−0.245770	−0.122885	−	0.992421i	$-0.539215\pi$
$149$	−3.00000	−0.245770	−0.122885	+	0.992421i	$0.539215\pi$
$150$	0	0
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	7.00000	0.567775
$153$	−2.00000	−0.161690
$154$	−8.00000	−0.644658
$155$	0	0
$156$	4.00000	0.320256
$157$	−13.0000	−1.03751	−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000	−1.03751	−0.518756	+	0.854922i	$0.673605\pi$
$158$	−7.00000	−0.556890
$159$	9.00000	0.713746
$160$	0	0
$161$	12.0000	0.945732
$162$	−1.00000	−0.0785674
$163$	9.00000	0.704934	0.352467	−	0.935824i	$-0.385343\pi$
$163$	9.00000	0.704934	0.352467	+	0.935824i	$0.385343\pi$
$164$	9.00000	0.702782
$165$	0	0
$166$	8.00000	0.620920
$167$	−5.00000	−0.386912	−0.193456	−	0.981109i	$-0.561970\pi$
$167$	−5.00000	−0.386912	−0.193456	+	0.981109i	$0.561970\pi$
$168$	−4.00000	−0.308607
$169$	3.00000	0.230769
$170$	0	0
$171$	−7.00000	−0.535303
$172$	−11.0000	−0.838742
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	0	0
$175$	0	0
$176$	−2.00000	−0.150756
$177$	15.0000	1.12747
$178$	8.00000	0.599625
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	5.00000	0.371647	0.185824	−	0.982583i	$-0.440505\pi$
$181$	5.00000	0.371647	0.185824	+	0.982583i	$0.440505\pi$
$182$	−16.0000	−1.18600
$183$	0	0
$184$	3.00000	0.221163
$185$	0	0
$186$	−8.00000	−0.586588
$187$	4.00000	0.292509
$188$	−6.00000	−0.437595
$189$	4.00000	0.290957
$190$	0	0
$191$	21.0000	1.51951	0.759753	−	0.650211i	$-0.225320\pi$
$191$	21.0000	1.51951	0.759753	+	0.650211i	$0.225320\pi$
$192$	−1.00000	−0.0721688
$193$	−16.0000	−1.15171	−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000	−1.15171	−0.575853	+	0.817554i	$0.695330\pi$
$194$	−18.0000	−1.29232
$195$	0	0
$196$	9.00000	0.642857
$197$	−19.0000	−1.35369	−0.676847	−	0.736124i	$-0.736654\pi$
$197$	−19.0000	−1.35369	−0.676847	+	0.736124i	$0.736654\pi$
$198$	2.00000	0.142134
$199$	7.00000	0.496217	0.248108	−	0.968732i	$-0.420191\pi$
$199$	7.00000	0.496217	0.248108	+	0.968732i	$0.420191\pi$
$200$	0	0
$201$	12.0000	0.846415
$202$	19.0000	1.33684
$203$	0	0
$204$	2.00000	0.140028
$205$	0	0
$206$	−11.0000	−0.766406
$207$	−3.00000	−0.208514
$208$	−4.00000	−0.277350
$209$	14.0000	0.968400
$210$	0	0
$211$	22.0000	1.51454	0.757271	−	0.653101i	$-0.226532\pi$
$211$	22.0000	1.51454	0.757271	+	0.653101i	$0.226532\pi$
$212$	−9.00000	−0.618123
$213$	−12.0000	−0.822226
$214$	14.0000	0.957020
$215$	0	0
$216$	1.00000	0.0680414
$217$	32.0000	2.17230
$218$	10.0000	0.677285
$219$	−15.0000	−1.01361
$220$	0	0
$221$	8.00000	0.538138
$222$	−1.00000	−0.0671156
$223$	6.00000	0.401790	0.200895	−	0.979613i	$-0.435615\pi$
$223$	6.00000	0.401790	0.200895	+	0.979613i	$0.435615\pi$
$224$	4.00000	0.267261
$225$	0	0
$226$	−6.00000	−0.399114
$227$	9.00000	0.597351	0.298675	−	0.954355i	$-0.403455\pi$
$227$	9.00000	0.597351	0.298675	+	0.954355i	$0.403455\pi$
$228$	7.00000	0.463586
$229$	−1.00000	−0.0660819	−0.0330409	−	0.999454i	$-0.510519\pi$
$229$	−1.00000	−0.0660819	−0.0330409	+	0.999454i	$0.510519\pi$
$230$	0	0
$231$	−8.00000	−0.526361
$232$	0	0
$233$	−15.0000	−0.982683	−0.491341	−	0.870967i	$-0.663493\pi$
$233$	−15.0000	−0.982683	−0.491341	+	0.870967i	$0.663493\pi$
$234$	4.00000	0.261488
$235$	0	0
$236$	−15.0000	−0.976417
$237$	−7.00000	−0.454699
$238$	−8.00000	−0.518563
$239$	−29.0000	−1.87585	−0.937927	−	0.346833i	$-0.887257\pi$
$239$	−29.0000	−1.87585	−0.937927	+	0.346833i	$0.887257\pi$
$240$	0	0
$241$	−8.00000	−0.515325	−0.257663	−	0.966235i	$-0.582952\pi$
$241$	−8.00000	−0.515325	−0.257663	+	0.966235i	$0.582952\pi$
$242$	7.00000	0.449977
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	9.00000	0.573819
$247$	28.0000	1.78160
$248$	8.00000	0.508001
$249$	8.00000	0.506979
$250$	0	0
$251$	23.0000	1.45175	0.725874	−	0.687828i	$-0.241436\pi$
$251$	23.0000	1.45175	0.725874	+	0.687828i	$0.241436\pi$
$252$	−4.00000	−0.251976
$253$	6.00000	0.377217
$254$	−10.0000	−0.627456
$255$	0	0
$256$	1.00000	0.0625000
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	−11.0000	−0.684830
$259$	4.00000	0.248548
$260$	0	0
$261$	0	0
$262$	−8.00000	−0.494242
$263$	14.0000	0.863277	0.431638	−	0.902047i	$-0.357936\pi$
$263$	14.0000	0.863277	0.431638	+	0.902047i	$0.357936\pi$
$264$	−2.00000	−0.123091
$265$	0	0
$266$	−28.0000	−1.71679
$267$	8.00000	0.489592
$268$	−12.0000	−0.733017
$269$	−15.0000	−0.914566	−0.457283	−	0.889321i	$-0.651177\pi$
$269$	−15.0000	−0.914566	−0.457283	+	0.889321i	$0.651177\pi$
$270$	0	0
$271$	−22.0000	−1.33640	−0.668202	−	0.743980i	$-0.732936\pi$
$271$	−22.0000	−1.33640	−0.668202	+	0.743980i	$0.732936\pi$
$272$	−2.00000	−0.121268
$273$	−16.0000	−0.968364
$274$	18.0000	1.08742
$275$	0	0
$276$	3.00000	0.180579
$277$	8.00000	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$278$	16.0000	0.959616
$279$	−8.00000	−0.478947
$280$	0	0
$281$	4.00000	0.238620	0.119310	−	0.992857i	$-0.461932\pi$
$281$	4.00000	0.238620	0.119310	+	0.992857i	$0.461932\pi$
$282$	−6.00000	−0.357295
$283$	21.0000	1.24832	0.624160	−	0.781296i	$-0.285441\pi$
$283$	21.0000	1.24832	0.624160	+	0.781296i	$0.285441\pi$
$284$	12.0000	0.712069
$285$	0	0
$286$	−8.00000	−0.473050
$287$	−36.0000	−2.12501
$288$	−1.00000	−0.0589256
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−18.0000	−1.05518
$292$	15.0000	0.877809
$293$	−29.0000	−1.69420	−0.847099	−	0.531435i	$-0.821653\pi$
$293$	−29.0000	−1.69420	−0.847099	+	0.531435i	$0.821653\pi$
$294$	9.00000	0.524891
$295$	0	0
$296$	1.00000	0.0581238
$297$	2.00000	0.116052
$298$	3.00000	0.173785
$299$	12.0000	0.693978
$300$	0	0
$301$	44.0000	2.53612
$302$	10.0000	0.575435
$303$	19.0000	1.09152
$304$	−7.00000	−0.401478
$305$	0	0
$306$	2.00000	0.114332
$307$	−34.0000	−1.94048	−0.970241	−	0.242140i	$-0.922151\pi$
$307$	−34.0000	−1.94048	−0.970241	+	0.242140i	$0.922151\pi$
$308$	8.00000	0.455842
$309$	−11.0000	−0.625768
$310$	0	0
$311$	−15.0000	−0.850572	−0.425286	−	0.905059i	$-0.639826\pi$
$311$	−15.0000	−0.850572	−0.425286	+	0.905059i	$0.639826\pi$
$312$	−4.00000	−0.226455
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	13.0000	0.733632
$315$	0	0
$316$	7.00000	0.393781
$317$	15.0000	0.842484	0.421242	−	0.906948i	$-0.361594\pi$
$317$	15.0000	0.842484	0.421242	+	0.906948i	$0.361594\pi$
$318$	−9.00000	−0.504695
$319$	0	0
$320$	0	0
$321$	14.0000	0.781404
$322$	−12.0000	−0.668734
$323$	14.0000	0.778981
$324$	1.00000	0.0555556
$325$	0	0
$326$	−9.00000	−0.498464
$327$	10.0000	0.553001
$328$	−9.00000	−0.496942
$329$	24.0000	1.32316
$330$	0	0
$331$	16.0000	0.879440	0.439720	−	0.898135i	$-0.355078\pi$
$331$	16.0000	0.879440	0.439720	+	0.898135i	$0.355078\pi$
$332$	−8.00000	−0.439057
$333$	−1.00000	−0.0547997
$334$	5.00000	0.273588
$335$	0	0
$336$	4.00000	0.218218
$337$	−25.0000	−1.36184	−0.680918	−	0.732359i	$-0.738419\pi$
$337$	−25.0000	−1.36184	−0.680918	+	0.732359i	$0.738419\pi$
$338$	−3.00000	−0.163178
$339$	−6.00000	−0.325875
$340$	0	0
$341$	16.0000	0.866449
$342$	7.00000	0.378517
$343$	−8.00000	−0.431959
$344$	11.0000	0.593080
$345$	0	0
$346$	−14.0000	−0.752645
$347$	9.00000	0.483145	0.241573	−	0.970383i	$-0.422337\pi$
$347$	9.00000	0.483145	0.241573	+	0.970383i	$0.422337\pi$
$348$	0	0
$349$	1.00000	0.0535288	0.0267644	−	0.999642i	$-0.491480\pi$
$349$	1.00000	0.0535288	0.0267644	+	0.999642i	$0.491480\pi$
$350$	0	0
$351$	4.00000	0.213504
$352$	2.00000	0.106600
$353$	16.0000	0.851594	0.425797	−	0.904819i	$-0.359994\pi$
$353$	16.0000	0.851594	0.425797	+	0.904819i	$0.359994\pi$
$354$	−15.0000	−0.797241
$355$	0	0
$356$	−8.00000	−0.423999
$357$	−8.00000	−0.423405
$358$	−12.0000	−0.634220
$359$	−30.0000	−1.58334	−0.791670	−	0.610949i	$-0.790788\pi$
$359$	−30.0000	−1.58334	−0.791670	+	0.610949i	$0.790788\pi$
$360$	0	0
$361$	30.0000	1.57895
$362$	−5.00000	−0.262794
$363$	7.00000	0.367405
$364$	16.0000	0.838628
$365$	0	0
$366$	0	0
$367$	−10.0000	−0.521996	−0.260998	−	0.965339i	$-0.584052\pi$
$367$	−10.0000	−0.521996	−0.260998	+	0.965339i	$0.584052\pi$
$368$	−3.00000	−0.156386
$369$	9.00000	0.468521
$370$	0	0
$371$	36.0000	1.86903
$372$	8.00000	0.414781
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	−4.00000	−0.206835
$375$	0	0
$376$	6.00000	0.309426
$377$	0	0
$378$	−4.00000	−0.205738
$379$	−6.00000	−0.308199	−0.154100	−	0.988055i	$-0.549248\pi$
$379$	−6.00000	−0.308199	−0.154100	+	0.988055i	$0.549248\pi$
$380$	0	0
$381$	−10.0000	−0.512316
$382$	−21.0000	−1.07445
$383$	−21.0000	−1.07305	−0.536525	−	0.843884i	$-0.680263\pi$
$383$	−21.0000	−1.07305	−0.536525	+	0.843884i	$0.680263\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	16.0000	0.814379
$387$	−11.0000	−0.559161
$388$	18.0000	0.913812
$389$	−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
$391$	6.00000	0.303433
$392$	−9.00000	−0.454569
$393$	−8.00000	−0.403547
$394$	19.0000	0.957206
$395$	0	0
$396$	−2.00000	−0.100504
$397$	7.00000	0.351320	0.175660	−	0.984451i	$-0.443794\pi$
$397$	7.00000	0.351320	0.175660	+	0.984451i	$0.443794\pi$
$398$	−7.00000	−0.350878
$399$	−28.0000	−1.40175
$400$	0	0
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	−12.0000	−0.598506
$403$	32.0000	1.59403
$404$	−19.0000	−0.945285
$405$	0	0
$406$	0	0
$407$	2.00000	0.0991363
$408$	−2.00000	−0.0990148
$409$	8.00000	0.395575	0.197787	−	0.980245i	$-0.436624\pi$
$409$	8.00000	0.395575	0.197787	+	0.980245i	$0.436624\pi$
$410$	0	0
$411$	18.0000	0.887875
$412$	11.0000	0.541931
$413$	60.0000	2.95241
$414$	3.00000	0.147442
$415$	0	0
$416$	4.00000	0.196116
$417$	16.0000	0.783523
$418$	−14.0000	−0.684762
$419$	−30.0000	−1.46560	−0.732798	−	0.680446i	$-0.761786\pi$
$419$	−30.0000	−1.46560	−0.732798	+	0.680446i	$0.761786\pi$
$420$	0	0
$421$	20.0000	0.974740	0.487370	−	0.873195i	$-0.337956\pi$
$421$	20.0000	0.974740	0.487370	+	0.873195i	$0.337956\pi$
$422$	−22.0000	−1.07094
$423$	−6.00000	−0.291730
$424$	9.00000	0.437079
$425$	0	0
$426$	12.0000	0.581402
$427$	0	0
$428$	−14.0000	−0.676716
$429$	−8.00000	−0.386244
$430$	0	0
$431$	21.0000	1.01153	0.505767	−	0.862670i	$-0.331209\pi$
$431$	21.0000	1.01153	0.505767	+	0.862670i	$0.331209\pi$
$432$	−1.00000	−0.0481125
$433$	30.0000	1.44171	0.720854	−	0.693087i	$-0.243750\pi$
$433$	30.0000	1.44171	0.720854	+	0.693087i	$0.243750\pi$
$434$	−32.0000	−1.53605
$435$	0	0
$436$	−10.0000	−0.478913
$437$	21.0000	1.00457
$438$	15.0000	0.716728
$439$	25.0000	1.19318	0.596592	−	0.802544i	$-0.296521\pi$
$439$	25.0000	1.19318	0.596592	+	0.802544i	$0.296521\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	−8.00000	−0.380521
$443$	14.0000	0.665160	0.332580	−	0.943075i	$-0.392081\pi$
$443$	14.0000	0.665160	0.332580	+	0.943075i	$0.392081\pi$
$444$	1.00000	0.0474579
$445$	0	0
$446$	−6.00000	−0.284108
$447$	3.00000	0.141895
$448$	−4.00000	−0.188982
$449$	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
$451$	−18.0000	−0.847587
$452$	6.00000	0.282216
$453$	10.0000	0.469841
$454$	−9.00000	−0.422391
$455$	0	0
$456$	−7.00000	−0.327805
$457$	32.0000	1.49690	0.748448	−	0.663193i	$-0.230799\pi$
$457$	32.0000	1.49690	0.748448	+	0.663193i	$0.230799\pi$
$458$	1.00000	0.0467269
$459$	2.00000	0.0933520
$460$	0	0
$461$	−26.0000	−1.21094	−0.605470	−	0.795868i	$-0.707015\pi$
$461$	−26.0000	−1.21094	−0.605470	+	0.795868i	$0.707015\pi$
$462$	8.00000	0.372194
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	0	0
$465$	0	0
$466$	15.0000	0.694862
$467$	−36.0000	−1.66588	−0.832941	−	0.553362i	$-0.813345\pi$
$467$	−36.0000	−1.66588	−0.832941	+	0.553362i	$0.813345\pi$
$468$	−4.00000	−0.184900
$469$	48.0000	2.21643
$470$	0	0
$471$	13.0000	0.599008
$472$	15.0000	0.690431
$473$	22.0000	1.01156
$474$	7.00000	0.321521
$475$	0	0
$476$	8.00000	0.366679
$477$	−9.00000	−0.412082
$478$	29.0000	1.32643
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	8.00000	0.364390
$483$	−12.0000	−0.546019
$484$	−7.00000	−0.318182
$485$	0	0
$486$	1.00000	0.0453609
$487$	32.0000	1.45006	0.725029	−	0.688718i	$-0.241826\pi$
$487$	32.0000	1.45006	0.725029	+	0.688718i	$0.241826\pi$
$488$	0	0
$489$	−9.00000	−0.406994
$490$	0	0
$491$	30.0000	1.35388	0.676941	−	0.736038i	$-0.263305\pi$
$491$	30.0000	1.35388	0.676941	+	0.736038i	$0.263305\pi$
$492$	−9.00000	−0.405751
$493$	0	0
$494$	−28.0000	−1.25978
$495$	0	0
$496$	−8.00000	−0.359211
$497$	−48.0000	−2.15309
$498$	−8.00000	−0.358489
$499$	17.0000	0.761025	0.380512	−	0.924776i	$-0.375748\pi$
$499$	17.0000	0.761025	0.380512	+	0.924776i	$0.375748\pi$
$500$	0	0
$501$	5.00000	0.223384
$502$	−23.0000	−1.02654
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	4.00000	0.178174
$505$	0	0
$506$	−6.00000	−0.266733
$507$	−3.00000	−0.133235
$508$	10.0000	0.443678
$509$	11.0000	0.487566	0.243783	−	0.969830i	$-0.421611\pi$
$509$	11.0000	0.487566	0.243783	+	0.969830i	$0.421611\pi$
$510$	0	0
$511$	−60.0000	−2.65424
$512$	−1.00000	−0.0441942
$513$	7.00000	0.309058
$514$	−2.00000	−0.0882162
$515$	0	0
$516$	11.0000	0.484248
$517$	12.0000	0.527759
$518$	−4.00000	−0.175750
$519$	−14.0000	−0.614532
$520$	0	0
$521$	13.0000	0.569540	0.284770	−	0.958596i	$-0.408083\pi$
$521$	13.0000	0.569540	0.284770	+	0.958596i	$0.408083\pi$
$522$	0	0
$523$	−28.0000	−1.22435	−0.612177	−	0.790721i	$-0.709706\pi$
$523$	−28.0000	−1.22435	−0.612177	+	0.790721i	$0.709706\pi$
$524$	8.00000	0.349482
$525$	0	0
$526$	−14.0000	−0.610429
$527$	16.0000	0.696971
$528$	2.00000	0.0870388
$529$	−14.0000	−0.608696
$530$	0	0
$531$	−15.0000	−0.650945
$532$	28.0000	1.21395
$533$	−36.0000	−1.55933
$534$	−8.00000	−0.346194
$535$	0	0
$536$	12.0000	0.518321
$537$	−12.0000	−0.517838
$538$	15.0000	0.646696
$539$	−18.0000	−0.775315
$540$	0	0
$541$	−14.0000	−0.601907	−0.300954	−	0.953639i	$-0.597305\pi$
$541$	−14.0000	−0.601907	−0.300954	+	0.953639i	$0.597305\pi$
$542$	22.0000	0.944981
$543$	−5.00000	−0.214571
$544$	2.00000	0.0857493
$545$	0	0
$546$	16.0000	0.684737
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	−18.0000	−0.768922
$549$	0	0
$550$	0	0
$551$	0	0
$552$	−3.00000	−0.127688
$553$	−28.0000	−1.19068
$554$	−8.00000	−0.339887
$555$	0	0
$556$	−16.0000	−0.678551
$557$	20.0000	0.847427	0.423714	−	0.905796i	$-0.360726\pi$
$557$	20.0000	0.847427	0.423714	+	0.905796i	$0.360726\pi$
$558$	8.00000	0.338667
$559$	44.0000	1.86100
$560$	0	0
$561$	−4.00000	−0.168880
$562$	−4.00000	−0.168730
$563$	−21.0000	−0.885044	−0.442522	−	0.896758i	$-0.645916\pi$
$563$	−21.0000	−0.885044	−0.442522	+	0.896758i	$0.645916\pi$
$564$	6.00000	0.252646
$565$	0	0
$566$	−21.0000	−0.882696
$567$	−4.00000	−0.167984
$568$	−12.0000	−0.503509
$569$	−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
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	8.00000	0.334497
$573$	−21.0000	−0.877288
$574$	36.0000	1.50261
$575$	0	0
$576$	1.00000	0.0416667
$577$	12.0000	0.499567	0.249783	−	0.968302i	$-0.419641\pi$
$577$	12.0000	0.499567	0.249783	+	0.968302i	$0.419641\pi$
$578$	13.0000	0.540729
$579$	16.0000	0.664937
$580$	0	0
$581$	32.0000	1.32758
$582$	18.0000	0.746124
$583$	18.0000	0.745484
$584$	−15.0000	−0.620704
$585$	0	0
$586$	29.0000	1.19798
$587$	15.0000	0.619116	0.309558	−	0.950881i	$-0.399819\pi$
$587$	15.0000	0.619116	0.309558	+	0.950881i	$0.399819\pi$
$588$	−9.00000	−0.371154
$589$	56.0000	2.30744
$590$	0	0
$591$	19.0000	0.781556
$592$	−1.00000	−0.0410997
$593$	41.0000	1.68367	0.841834	−	0.539736i	$-0.181476\pi$
$593$	41.0000	1.68367	0.841834	+	0.539736i	$0.181476\pi$
$594$	−2.00000	−0.0820610
$595$	0	0
$596$	−3.00000	−0.122885
$597$	−7.00000	−0.286491
$598$	−12.0000	−0.490716
$599$	14.0000	0.572024	0.286012	−	0.958226i	$-0.407670\pi$
$599$	14.0000	0.572024	0.286012	+	0.958226i	$0.407670\pi$
$600$	0	0
$601$	−5.00000	−0.203954	−0.101977	−	0.994787i	$-0.532517\pi$
$601$	−5.00000	−0.203954	−0.101977	+	0.994787i	$0.532517\pi$
$602$	−44.0000	−1.79331
$603$	−12.0000	−0.488678
$604$	−10.0000	−0.406894
$605$	0	0
$606$	−19.0000	−0.771822
$607$	16.0000	0.649420	0.324710	−	0.945814i	$-0.394733\pi$
$607$	16.0000	0.649420	0.324710	+	0.945814i	$0.394733\pi$
$608$	7.00000	0.283887
$609$	0	0
$610$	0	0
$611$	24.0000	0.970936
$612$	−2.00000	−0.0808452
$613$	23.0000	0.928961	0.464481	−	0.885583i	$-0.346241\pi$
$613$	23.0000	0.928961	0.464481	+	0.885583i	$0.346241\pi$
$614$	34.0000	1.37213
$615$	0	0
$616$	−8.00000	−0.322329
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	11.0000	0.442485
$619$	28.0000	1.12542	0.562708	−	0.826656i	$-0.309760\pi$
$619$	28.0000	1.12542	0.562708	+	0.826656i	$0.309760\pi$
$620$	0	0
$621$	3.00000	0.120386
$622$	15.0000	0.601445
$623$	32.0000	1.28205
$624$	4.00000	0.160128
$625$	0	0
$626$	−10.0000	−0.399680
$627$	−14.0000	−0.559106
$628$	−13.0000	−0.518756
$629$	2.00000	0.0797452
$630$	0	0
$631$	−20.0000	−0.796187	−0.398094	−	0.917345i	$-0.630328\pi$
$631$	−20.0000	−0.796187	−0.398094	+	0.917345i	$0.630328\pi$
$632$	−7.00000	−0.278445
$633$	−22.0000	−0.874421
$634$	−15.0000	−0.595726
$635$	0	0
$636$	9.00000	0.356873
$637$	−36.0000	−1.42637
$638$	0	0
$639$	12.0000	0.474713
$640$	0	0
$641$	−27.0000	−1.06644	−0.533218	−	0.845978i	$-0.679017\pi$
$641$	−27.0000	−1.06644	−0.533218	+	0.845978i	$0.679017\pi$
$642$	−14.0000	−0.552536
$643$	−20.0000	−0.788723	−0.394362	−	0.918955i	$-0.629034\pi$
$643$	−20.0000	−0.788723	−0.394362	+	0.918955i	$0.629034\pi$
$644$	12.0000	0.472866
$645$	0	0
$646$	−14.0000	−0.550823
$647$	7.00000	0.275198	0.137599	−	0.990488i	$-0.456061\pi$
$647$	7.00000	0.275198	0.137599	+	0.990488i	$0.456061\pi$
$648$	−1.00000	−0.0392837
$649$	30.0000	1.17760
$650$	0	0
$651$	−32.0000	−1.25418
$652$	9.00000	0.352467
$653$	16.0000	0.626128	0.313064	−	0.949732i	$-0.398644\pi$
$653$	16.0000	0.626128	0.313064	+	0.949732i	$0.398644\pi$
$654$	−10.0000	−0.391031
$655$	0	0
$656$	9.00000	0.351391
$657$	15.0000	0.585206
$658$	−24.0000	−0.935617
$659$	−6.00000	−0.233727	−0.116863	−	0.993148i	$-0.537284\pi$
$659$	−6.00000	−0.233727	−0.116863	+	0.993148i	$0.537284\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	−16.0000	−0.621858
$663$	−8.00000	−0.310694
$664$	8.00000	0.310460
$665$	0	0
$666$	1.00000	0.0387492
$667$	0	0
$668$	−5.00000	−0.193456
$669$	−6.00000	−0.231973
$670$	0	0
$671$	0	0
$672$	−4.00000	−0.154303
$673$	39.0000	1.50334	0.751670	−	0.659540i	$-0.229249\pi$
$673$	39.0000	1.50334	0.751670	+	0.659540i	$0.229249\pi$
$674$	25.0000	0.962964
$675$	0	0
$676$	3.00000	0.115385
$677$	−17.0000	−0.653363	−0.326682	−	0.945134i	$-0.605930\pi$
$677$	−17.0000	−0.653363	−0.326682	+	0.945134i	$0.605930\pi$
$678$	6.00000	0.230429
$679$	−72.0000	−2.76311
$680$	0	0
$681$	−9.00000	−0.344881
$682$	−16.0000	−0.612672
$683$	32.0000	1.22445	0.612223	−	0.790685i	$-0.290275\pi$
$683$	32.0000	1.22445	0.612223	+	0.790685i	$0.290275\pi$
$684$	−7.00000	−0.267652
$685$	0	0
$686$	8.00000	0.305441
$687$	1.00000	0.0381524
$688$	−11.0000	−0.419371
$689$	36.0000	1.37149
$690$	0	0
$691$	−10.0000	−0.380418	−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000	−0.380418	−0.190209	+	0.981744i	$0.560917\pi$
$692$	14.0000	0.532200
$693$	8.00000	0.303895
$694$	−9.00000	−0.341635
$695$	0	0
$696$	0	0
$697$	−18.0000	−0.681799
$698$	−1.00000	−0.0378506
$699$	15.0000	0.567352
$700$	0	0
$701$	16.0000	0.604312	0.302156	−	0.953259i	$-0.402294\pi$
$701$	16.0000	0.604312	0.302156	+	0.953259i	$0.402294\pi$
$702$	−4.00000	−0.150970
$703$	7.00000	0.264010
$704$	−2.00000	−0.0753778
$705$	0	0
$706$	−16.0000	−0.602168
$707$	76.0000	2.85827
$708$	15.0000	0.563735
$709$	30.0000	1.12667	0.563337	−	0.826227i	$-0.309517\pi$
$709$	30.0000	1.12667	0.563337	+	0.826227i	$0.309517\pi$
$710$	0	0
$711$	7.00000	0.262521
$712$	8.00000	0.299813
$713$	24.0000	0.898807
$714$	8.00000	0.299392
$715$	0	0
$716$	12.0000	0.448461
$717$	29.0000	1.08302
$718$	30.0000	1.11959
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	0	0
$721$	−44.0000	−1.63865
$722$	−30.0000	−1.11648
$723$	8.00000	0.297523
$724$	5.00000	0.185824
$725$	0	0
$726$	−7.00000	−0.259794
$727$	32.0000	1.18681	0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000	1.18681	0.593407	+	0.804902i	$0.297782\pi$
$728$	−16.0000	−0.592999
$729$	1.00000	0.0370370
$730$	0	0
$731$	22.0000	0.813699
$732$	0	0
$733$	−6.00000	−0.221615	−0.110808	−	0.993842i	$-0.535344\pi$
$733$	−6.00000	−0.221615	−0.110808	+	0.993842i	$0.535344\pi$
$734$	10.0000	0.369107
$735$	0	0
$736$	3.00000	0.110581
$737$	24.0000	0.884051
$738$	−9.00000	−0.331295
$739$	−2.00000	−0.0735712	−0.0367856	−	0.999323i	$-0.511712\pi$
$739$	−2.00000	−0.0735712	−0.0367856	+	0.999323i	$0.511712\pi$
$740$	0	0
$741$	−28.0000	−1.02861
$742$	−36.0000	−1.32160
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	−8.00000	−0.293294
$745$	0	0
$746$	−10.0000	−0.366126
$747$	−8.00000	−0.292705
$748$	4.00000	0.146254
$749$	56.0000	2.04620
$750$	0	0
$751$	−18.0000	−0.656829	−0.328415	−	0.944534i	$-0.606514\pi$
$751$	−18.0000	−0.656829	−0.328415	+	0.944534i	$0.606514\pi$
$752$	−6.00000	−0.218797
$753$	−23.0000	−0.838167
$754$	0	0
$755$	0	0
$756$	4.00000	0.145479
$757$	40.0000	1.45382	0.726912	−	0.686730i	$-0.240955\pi$
$757$	40.0000	1.45382	0.726912	+	0.686730i	$0.240955\pi$
$758$	6.00000	0.217930
$759$	−6.00000	−0.217786
$760$	0	0
$761$	−21.0000	−0.761249	−0.380625	−	0.924730i	$-0.624291\pi$
$761$	−21.0000	−0.761249	−0.380625	+	0.924730i	$0.624291\pi$
$762$	10.0000	0.362262
$763$	40.0000	1.44810
$764$	21.0000	0.759753
$765$	0	0
$766$	21.0000	0.758761
$767$	60.0000	2.16647
$768$	−1.00000	−0.0360844
$769$	24.0000	0.865462	0.432731	−	0.901523i	$-0.357550\pi$
$769$	24.0000	0.865462	0.432731	+	0.901523i	$0.357550\pi$
$770$	0	0
$771$	−2.00000	−0.0720282
$772$	−16.0000	−0.575853
$773$	6.00000	0.215805	0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000	0.215805	0.107903	+	0.994161i	$0.465587\pi$
$774$	11.0000	0.395387
$775$	0	0
$776$	−18.0000	−0.646162
$777$	−4.00000	−0.143499
$778$	26.0000	0.932145
$779$	−63.0000	−2.25721
$780$	0	0
$781$	−24.0000	−0.858788
$782$	−6.00000	−0.214560
$783$	0	0
$784$	9.00000	0.321429
$785$	0	0
$786$	8.00000	0.285351
$787$	14.0000	0.499046	0.249523	−	0.968369i	$-0.419726\pi$
$787$	14.0000	0.499046	0.249523	+	0.968369i	$0.419726\pi$
$788$	−19.0000	−0.676847
$789$	−14.0000	−0.498413
$790$	0	0
$791$	−24.0000	−0.853342
$792$	2.00000	0.0710669
$793$	0	0
$794$	−7.00000	−0.248421
$795$	0	0
$796$	7.00000	0.248108
$797$	−22.0000	−0.779280	−0.389640	−	0.920967i	$-0.627401\pi$
$797$	−22.0000	−0.779280	−0.389640	+	0.920967i	$0.627401\pi$
$798$	28.0000	0.991189
$799$	12.0000	0.424529
$800$	0	0
$801$	−8.00000	−0.282666
$802$	18.0000	0.635602
$803$	−30.0000	−1.05868
$804$	12.0000	0.423207
$805$	0	0
$806$	−32.0000	−1.12715
$807$	15.0000	0.528025
$808$	19.0000	0.668418
$809$	8.00000	0.281265	0.140633	−	0.990062i	$-0.455086\pi$
$809$	8.00000	0.281265	0.140633	+	0.990062i	$0.455086\pi$
$810$	0	0
$811$	−22.0000	−0.772524	−0.386262	−	0.922389i	$-0.626234\pi$
$811$	−22.0000	−0.772524	−0.386262	+	0.922389i	$0.626234\pi$
$812$	0	0
$813$	22.0000	0.771574
$814$	−2.00000	−0.0701000
$815$	0	0
$816$	2.00000	0.0700140
$817$	77.0000	2.69389
$818$	−8.00000	−0.279713
$819$	16.0000	0.559085
$820$	0	0
$821$	15.0000	0.523504	0.261752	−	0.965135i	$-0.415700\pi$
$821$	15.0000	0.523504	0.261752	+	0.965135i	$0.415700\pi$
$822$	−18.0000	−0.627822
$823$	−32.0000	−1.11545	−0.557725	−	0.830026i	$-0.688326\pi$
$823$	−32.0000	−1.11545	−0.557725	+	0.830026i	$0.688326\pi$
$824$	−11.0000	−0.383203
$825$	0	0
$826$	−60.0000	−2.08767
$827$	28.0000	0.973655	0.486828	−	0.873498i	$-0.338154\pi$
$827$	28.0000	0.973655	0.486828	+	0.873498i	$0.338154\pi$
$828$	−3.00000	−0.104257
$829$	−28.0000	−0.972480	−0.486240	−	0.873825i	$-0.661632\pi$
$829$	−28.0000	−0.972480	−0.486240	+	0.873825i	$0.661632\pi$
$830$	0	0
$831$	−8.00000	−0.277517
$832$	−4.00000	−0.138675
$833$	−18.0000	−0.623663
$834$	−16.0000	−0.554035
$835$	0	0
$836$	14.0000	0.484200
$837$	8.00000	0.276520
$838$	30.0000	1.03633
$839$	−32.0000	−1.10476	−0.552381	−	0.833592i	$-0.686281\pi$
$839$	−32.0000	−1.10476	−0.552381	+	0.833592i	$0.686281\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	−20.0000	−0.689246
$843$	−4.00000	−0.137767
$844$	22.0000	0.757271
$845$	0	0
$846$	6.00000	0.206284
$847$	28.0000	0.962091
$848$	−9.00000	−0.309061
$849$	−21.0000	−0.720718
$850$	0	0
$851$	3.00000	0.102839
$852$	−12.0000	−0.411113
$853$	−38.0000	−1.30110	−0.650548	−	0.759465i	$-0.725461\pi$
$853$	−38.0000	−1.30110	−0.650548	+	0.759465i	$0.725461\pi$
$854$	0	0
$855$	0	0
$856$	14.0000	0.478510
$857$	−44.0000	−1.50301	−0.751506	−	0.659727i	$-0.770672\pi$
$857$	−44.0000	−1.50301	−0.751506	+	0.659727i	$0.770672\pi$
$858$	8.00000	0.273115
$859$	−23.0000	−0.784750	−0.392375	−	0.919805i	$-0.628346\pi$
$859$	−23.0000	−0.784750	−0.392375	+	0.919805i	$0.628346\pi$
$860$	0	0
$861$	36.0000	1.22688
$862$	−21.0000	−0.715263
$863$	−42.0000	−1.42970	−0.714848	−	0.699280i	$-0.753504\pi$
$863$	−42.0000	−1.42970	−0.714848	+	0.699280i	$0.753504\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−30.0000	−1.01944
$867$	13.0000	0.441503
$868$	32.0000	1.08615
$869$	−14.0000	−0.474917
$870$	0	0
$871$	48.0000	1.62642
$872$	10.0000	0.338643
$873$	18.0000	0.609208
$874$	−21.0000	−0.710336
$875$	0	0
$876$	−15.0000	−0.506803
$877$	−17.0000	−0.574049	−0.287025	−	0.957923i	$-0.592666\pi$
$877$	−17.0000	−0.574049	−0.287025	+	0.957923i	$0.592666\pi$
$878$	−25.0000	−0.843709
$879$	29.0000	0.978146
$880$	0	0
$881$	−5.00000	−0.168454	−0.0842271	−	0.996447i	$-0.526842\pi$
$881$	−5.00000	−0.168454	−0.0842271	+	0.996447i	$0.526842\pi$
$882$	−9.00000	−0.303046
$883$	−40.0000	−1.34611	−0.673054	−	0.739594i	$-0.735018\pi$
$883$	−40.0000	−1.34611	−0.673054	+	0.739594i	$0.735018\pi$
$884$	8.00000	0.269069
$885$	0	0
$886$	−14.0000	−0.470339
$887$	−28.0000	−0.940148	−0.470074	−	0.882627i	$-0.655773\pi$
$887$	−28.0000	−0.940148	−0.470074	+	0.882627i	$0.655773\pi$
$888$	−1.00000	−0.0335578
$889$	−40.0000	−1.34156
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	6.00000	0.200895
$893$	42.0000	1.40548
$894$	−3.00000	−0.100335
$895$	0	0
$896$	4.00000	0.133631
$897$	−12.0000	−0.400668
$898$	−2.00000	−0.0667409
$899$	0	0
$900$	0	0
$901$	18.0000	0.599667
$902$	18.0000	0.599334
$903$	−44.0000	−1.46423
$904$	−6.00000	−0.199557
$905$	0	0
$906$	−10.0000	−0.332228
$907$	−13.0000	−0.431658	−0.215829	−	0.976431i	$-0.569245\pi$
$907$	−13.0000	−0.431658	−0.215829	+	0.976431i	$0.569245\pi$
$908$	9.00000	0.298675
$909$	−19.0000	−0.630190
$910$	0	0
$911$	−31.0000	−1.02708	−0.513538	−	0.858067i	$-0.671665\pi$
$911$	−31.0000	−1.02708	−0.513538	+	0.858067i	$0.671665\pi$
$912$	7.00000	0.231793
$913$	16.0000	0.529523
$914$	−32.0000	−1.05847
$915$	0	0
$916$	−1.00000	−0.0330409
$917$	−32.0000	−1.05673
$918$	−2.00000	−0.0660098
$919$	8.00000	0.263896	0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000	0.263896	0.131948	+	0.991257i	$0.457877\pi$
$920$	0	0
$921$	34.0000	1.12034
$922$	26.0000	0.856264
$923$	−48.0000	−1.57994
$924$	−8.00000	−0.263181
$925$	0	0
$926$	−4.00000	−0.131448
$927$	11.0000	0.361287
$928$	0	0
$929$	−22.0000	−0.721797	−0.360898	−	0.932605i	$-0.617530\pi$
$929$	−22.0000	−0.721797	−0.360898	+	0.932605i	$0.617530\pi$
$930$	0	0
$931$	−63.0000	−2.06474
$932$	−15.0000	−0.491341
$933$	15.0000	0.491078
$934$	36.0000	1.17796
$935$	0	0
$936$	4.00000	0.130744
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	−48.0000	−1.56726
$939$	−10.0000	−0.326338
$940$	0	0
$941$	−46.0000	−1.49956	−0.749779	−	0.661689i	$-0.769840\pi$
$941$	−46.0000	−1.49956	−0.749779	+	0.661689i	$0.769840\pi$
$942$	−13.0000	−0.423563
$943$	−27.0000	−0.879241
$944$	−15.0000	−0.488208
$945$	0	0
$946$	−22.0000	−0.715282
$947$	−21.0000	−0.682408	−0.341204	−	0.939989i	$-0.610835\pi$
$947$	−21.0000	−0.682408	−0.341204	+	0.939989i	$0.610835\pi$
$948$	−7.00000	−0.227349
$949$	−60.0000	−1.94768
$950$	0	0
$951$	−15.0000	−0.486408
$952$	−8.00000	−0.259281
$953$	−1.00000	−0.0323932	−0.0161966	−	0.999869i	$-0.505156\pi$
$953$	−1.00000	−0.0323932	−0.0161966	+	0.999869i	$0.505156\pi$
$954$	9.00000	0.291386
$955$	0	0
$956$	−29.0000	−0.937927
$957$	0	0
$958$	24.0000	0.775405
$959$	72.0000	2.32500
$960$	0	0
$961$	33.0000	1.06452
$962$	−4.00000	−0.128965
$963$	−14.0000	−0.451144
$964$	−8.00000	−0.257663
$965$	0	0
$966$	12.0000	0.386094
$967$	−39.0000	−1.25416	−0.627078	−	0.778957i	$-0.715749\pi$
$967$	−39.0000	−1.25416	−0.627078	+	0.778957i	$0.715749\pi$
$968$	7.00000	0.224989
$969$	−14.0000	−0.449745
$970$	0	0
$971$	−32.0000	−1.02693	−0.513464	−	0.858111i	$-0.671638\pi$
$971$	−32.0000	−1.02693	−0.513464	+	0.858111i	$0.671638\pi$
$972$	−1.00000	−0.0320750
$973$	64.0000	2.05175
$974$	−32.0000	−1.02535
$975$	0	0
$976$	0	0
$977$	28.0000	0.895799	0.447900	−	0.894084i	$-0.352172\pi$
$977$	28.0000	0.895799	0.447900	+	0.894084i	$0.352172\pi$
$978$	9.00000	0.287788
$979$	16.0000	0.511362
$980$	0	0
$981$	−10.0000	−0.319275
$982$	−30.0000	−0.957338
$983$	−12.0000	−0.382741	−0.191370	−	0.981518i	$-0.561293\pi$
$983$	−12.0000	−0.382741	−0.191370	+	0.981518i	$0.561293\pi$
$984$	9.00000	0.286910
$985$	0	0
$986$	0	0
$987$	−24.0000	−0.763928
$988$	28.0000	0.890799
$989$	33.0000	1.04934
$990$	0	0
$991$	−21.0000	−0.667087	−0.333543	−	0.942735i	$-0.608244\pi$
$991$	−21.0000	−0.667087	−0.333543	+	0.942735i	$0.608244\pi$
$992$	8.00000	0.254000
$993$	−16.0000	−0.507745
$994$	48.0000	1.52247
$995$	0	0
$996$	8.00000	0.253490
$997$	12.0000	0.380044	0.190022	−	0.981780i	$-0.439144\pi$
$997$	12.0000	0.380044	0.190022	+	0.981780i	$0.439144\pi$
$998$	−17.0000	−0.538126
$999$	1.00000	0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5550.2.a.a.1.1	✓	1
5.4	even	2		5550.2.a.bq.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5550.2.a.a.1.1	✓	1	1.1	even	1	trivial
5550.2.a.bq.1.1	yes	1	5.4	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:	\( N \)	\(=\)	\( 5550 = 2 \cdot 3 \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5550.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more