LMFDB - Embedded newform 550.2.a.e.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [550,2,Mod(1,550)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

Level:	$ N $	$=$	$ 550 = 2 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	550.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-1,1,1,0,-1,-3,-1,-2,0,1,1,-4] 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:	$4.39177211117$
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)

$f(q)$

$=$

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +1.00000 q^{11} +1.00000 q^{12} -4.00000 q^{13} +3.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} +2.00000 q^{18} -5.00000 q^{19} -3.00000 q^{21} -1.00000 q^{22} -4.00000 q^{23} -1.00000 q^{24} +4.00000 q^{26} -5.00000 q^{27} -3.00000 q^{28} +5.00000 q^{29} +7.00000 q^{31} -1.00000 q^{32} +1.00000 q^{33} +3.00000 q^{34} -2.00000 q^{36} +7.00000 q^{37} +5.00000 q^{38} -4.00000 q^{39} -8.00000 q^{41} +3.00000 q^{42} +6.00000 q^{43} +1.00000 q^{44} +4.00000 q^{46} -8.00000 q^{47} +1.00000 q^{48} +2.00000 q^{49} -3.00000 q^{51} -4.00000 q^{52} -9.00000 q^{53} +5.00000 q^{54} +3.00000 q^{56} -5.00000 q^{57} -5.00000 q^{58} -13.0000 q^{61} -7.00000 q^{62} +6.00000 q^{63} +1.00000 q^{64} -1.00000 q^{66} +12.0000 q^{67} -3.00000 q^{68} -4.00000 q^{69} -3.00000 q^{71} +2.00000 q^{72} +6.00000 q^{73} -7.00000 q^{74} -5.00000 q^{76} -3.00000 q^{77} +4.00000 q^{78} +1.00000 q^{81} +8.00000 q^{82} -4.00000 q^{83} -3.00000 q^{84} -6.00000 q^{86} +5.00000 q^{87} -1.00000 q^{88} -15.0000 q^{89} +12.0000 q^{91} -4.00000 q^{92} +7.00000 q^{93} +8.00000 q^{94} -1.00000 q^{96} +12.0000 q^{97} -2.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

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	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	−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
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	3.00000	0.801784
$15$	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
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	0	0
$21$	−3.00000	−0.654654
$22$	−1.00000	−0.213201
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	4.00000	0.784465
$27$	−5.00000	−0.962250
$28$	−3.00000	−0.566947
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	0	0
$31$	7.00000	1.25724	0.628619	−	0.777714i	$-0.283621\pi$
$31$	7.00000	1.25724	0.628619	+	0.777714i	$0.283621\pi$
$32$	−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	$0.304848\pi$
$38$	5.00000	0.811107
$39$	−4.00000	−0.640513
$40$	0	0
$41$	−8.00000	−1.24939	−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000	−1.24939	−0.624695	+	0.780869i	$0.714777\pi$
$42$	3.00000	0.462910
$43$	6.00000	0.914991	0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000	0.914991	0.457496	+	0.889212i	$0.348747\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	4.00000	0.589768
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	1.00000	0.144338
$49$	2.00000	0.285714
$50$	0	0
$51$	−3.00000	−0.420084
$52$	−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	$0.712106\pi$
$54$	5.00000	0.680414
$55$	0	0
$56$	3.00000	0.400892
$57$	−5.00000	−0.662266
$58$	−5.00000	−0.656532
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−13.0000	−1.66448	−0.832240	−	0.554416i	$-0.812942\pi$
$61$	−13.0000	−1.66448	−0.832240	+	0.554416i	$0.812942\pi$
$62$	−7.00000	−0.889001
$63$	6.00000	0.755929
$64$	1.00000	0.125000
$65$	0	0
$66$	−1.00000	−0.123091
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	−3.00000	−0.363803
$69$	−4.00000	−0.481543
$70$	0	0
$71$	−3.00000	−0.356034	−0.178017	−	0.984027i	$-0.556968\pi$
$71$	−3.00000	−0.356034	−0.178017	+	0.984027i	$0.556968\pi$
$72$	2.00000	0.235702
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	−7.00000	−0.813733
$75$	0	0
$76$	−5.00000	−0.573539
$77$	−3.00000	−0.341882
$78$	4.00000	0.452911
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	8.00000	0.883452
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	−3.00000	−0.327327
$85$	0	0
$86$	−6.00000	−0.646997
$87$	5.00000	0.536056
$88$	−1.00000	−0.106600
$89$	−15.0000	−1.59000	−0.794998	−	0.606612i	$-0.792528\pi$
$89$	−15.0000	−1.59000	−0.794998	+	0.606612i	$0.792528\pi$
$90$	0	0
$91$	12.0000	1.25794
$92$	−4.00000	−0.417029
$93$	7.00000	0.725866
$94$	8.00000	0.825137
$95$	0	0
$96$	−1.00000	−0.102062
$97$	12.0000	1.21842	0.609208	−	0.793011i	$-0.291488\pi$
$97$	12.0000	1.21842	0.609208	+	0.793011i	$0.291488\pi$
$98$	−2.00000	−0.202031
$99$	−2.00000	−0.201008
$100$	0	0
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	3.00000	0.297044
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	4.00000	0.392232
$105$	0	0
$106$	9.00000	0.874157
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	−5.00000	−0.481125
$109$	−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
$111$	7.00000	0.664411
$112$	−3.00000	−0.283473
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	5.00000	0.468293
$115$	0	0
$116$	5.00000	0.464238
$117$	8.00000	0.739600
$118$	0	0
$119$	9.00000	0.825029
$120$	0	0
$121$	1.00000	0.0909091
$122$	13.0000	1.17696
$123$	−8.00000	−0.721336
$124$	7.00000	0.628619
$125$	0	0
$126$	−6.00000	−0.534522
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	−1.00000	−0.0883883
$129$	6.00000	0.528271
$130$	0	0
$131$	7.00000	0.611593	0.305796	−	0.952097i	$-0.401077\pi$
$131$	7.00000	0.611593	0.305796	+	0.952097i	$0.401077\pi$
$132$	1.00000	0.0870388
$133$	15.0000	1.30066
$134$	−12.0000	−1.03664
$135$	0	0
$136$	3.00000	0.257248
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	4.00000	0.340503
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	3.00000	0.251754
$143$	−4.00000	−0.334497
$144$	−2.00000	−0.166667
$145$	0	0
$146$	−6.00000	−0.496564
$147$	2.00000	0.164957
$148$	7.00000	0.575396
$149$	−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
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	5.00000	0.405554
$153$	6.00000	0.485071
$154$	3.00000	0.241747
$155$	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	$0.673605\pi$
$158$	0	0
$159$	−9.00000	−0.713746
$160$	0	0
$161$	12.0000	0.945732
$162$	−1.00000	−0.0785674
$163$	21.0000	1.64485	0.822423	−	0.568876i	$-0.192621\pi$
$163$	21.0000	1.64485	0.822423	+	0.568876i	$0.192621\pi$
$164$	−8.00000	−0.624695
$165$	0	0
$166$	4.00000	0.310460
$167$	−23.0000	−1.77979	−0.889897	−	0.456162i	$-0.849224\pi$
$167$	−23.0000	−1.77979	−0.889897	+	0.456162i	$0.849224\pi$
$168$	3.00000	0.231455
$169$	3.00000	0.230769
$170$	0	0
$171$	10.0000	0.764719
$172$	6.00000	0.457496
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	−5.00000	−0.379049
$175$	0	0
$176$	1.00000	0.0753778
$177$	0	0
$178$	15.0000	1.12430
$179$	10.0000	0.747435	0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000	0.747435	0.373718	+	0.927543i	$0.378083\pi$
$180$	0	0
$181$	−18.0000	−1.33793	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000	−1.33793	−0.668965	+	0.743294i	$0.733262\pi$
$182$	−12.0000	−0.889499
$183$	−13.0000	−0.960988
$184$	4.00000	0.294884
$185$	0	0
$186$	−7.00000	−0.513265
$187$	−3.00000	−0.219382
$188$	−8.00000	−0.583460
$189$	15.0000	1.09109
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	1.00000	0.0721688
$193$	1.00000	0.0719816	0.0359908	−	0.999352i	$-0.488541\pi$
$193$	1.00000	0.0719816	0.0359908	+	0.999352i	$0.488541\pi$
$194$	−12.0000	−0.861550
$195$	0	0
$196$	2.00000	0.142857
$197$	22.0000	1.56744	0.783718	−	0.621117i	$-0.213321\pi$
$197$	22.0000	1.56744	0.783718	+	0.621117i	$0.213321\pi$
$198$	2.00000	0.142134
$199$	25.0000	1.77220	0.886102	−	0.463491i	$-0.153403\pi$
$199$	25.0000	1.77220	0.886102	+	0.463491i	$0.153403\pi$
$200$	0	0
$201$	12.0000	0.846415
$202$	−2.00000	−0.140720
$203$	−15.0000	−1.05279
$204$	−3.00000	−0.210042
$205$	0	0
$206$	4.00000	0.278693
$207$	8.00000	0.556038
$208$	−4.00000	−0.277350
$209$	−5.00000	−0.345857
$210$	0	0
$211$	−13.0000	−0.894957	−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000	−0.894957	−0.447478	+	0.894295i	$0.647678\pi$
$212$	−9.00000	−0.618123
$213$	−3.00000	−0.205557
$214$	−12.0000	−0.820303
$215$	0	0
$216$	5.00000	0.340207
$217$	−21.0000	−1.42557
$218$	10.0000	0.677285
$219$	6.00000	0.405442
$220$	0	0
$221$	12.0000	0.807207
$222$	−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	$0.655297\pi$
$224$	3.00000	0.200446
$225$	0	0
$226$	−6.00000	−0.399114
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	−5.00000	−0.331133
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	0	0
$231$	−3.00000	−0.197386
$232$	−5.00000	−0.328266
$233$	−19.0000	−1.24473	−0.622366	−	0.782727i	$-0.713828\pi$
$233$	−19.0000	−1.24473	−0.622366	+	0.782727i	$0.713828\pi$
$234$	−8.00000	−0.522976
$235$	0	0
$236$	0	0
$237$	0	0
$238$	−9.00000	−0.583383
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−8.00000	−0.515325	−0.257663	−	0.966235i	$-0.582952\pi$
$241$	−8.00000	−0.515325	−0.257663	+	0.966235i	$0.582952\pi$
$242$	−1.00000	−0.0642824
$243$	16.0000	1.02640
$244$	−13.0000	−0.832240
$245$	0	0
$246$	8.00000	0.510061
$247$	20.0000	1.27257
$248$	−7.00000	−0.444500
$249$	−4.00000	−0.253490
$250$	0	0
$251$	2.00000	0.126239	0.0631194	−	0.998006i	$-0.479895\pi$
$251$	2.00000	0.126239	0.0631194	+	0.998006i	$0.479895\pi$
$252$	6.00000	0.377964
$253$	−4.00000	−0.251478
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	−28.0000	−1.74659	−0.873296	−	0.487190i	$-0.838022\pi$
$257$	−28.0000	−1.74659	−0.873296	+	0.487190i	$0.838022\pi$
$258$	−6.00000	−0.373544
$259$	−21.0000	−1.30488
$260$	0	0
$261$	−10.0000	−0.618984
$262$	−7.00000	−0.432461
$263$	−9.00000	−0.554964	−0.277482	−	0.960731i	$-0.589500\pi$
$263$	−9.00000	−0.554964	−0.277482	+	0.960731i	$0.589500\pi$
$264$	−1.00000	−0.0615457
$265$	0	0
$266$	−15.0000	−0.919709
$267$	−15.0000	−0.917985
$268$	12.0000	0.733017
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	22.0000	1.33640	0.668202	−	0.743980i	$-0.267064\pi$
$271$	22.0000	1.33640	0.668202	+	0.743980i	$0.267064\pi$
$272$	−3.00000	−0.181902
$273$	12.0000	0.726273
$274$	−12.0000	−0.724947
$275$	0	0
$276$	−4.00000	−0.240772
$277$	−28.0000	−1.68236	−0.841178	−	0.540758i	$-0.818138\pi$
$277$	−28.0000	−1.68236	−0.841178	+	0.540758i	$0.818138\pi$
$278$	−20.0000	−1.19952
$279$	−14.0000	−0.838158
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	8.00000	0.476393
$283$	16.0000	0.951101	0.475551	−	0.879688i	$-0.342249\pi$
$283$	16.0000	0.951101	0.475551	+	0.879688i	$0.342249\pi$
$284$	−3.00000	−0.178017
$285$	0	0
$286$	4.00000	0.236525
$287$	24.0000	1.41668
$288$	2.00000	0.117851
$289$	−8.00000	−0.470588
$290$	0	0
$291$	12.0000	0.703452
$292$	6.00000	0.351123
$293$	16.0000	0.934730	0.467365	−	0.884064i	$-0.345203\pi$
$293$	16.0000	0.934730	0.467365	+	0.884064i	$0.345203\pi$
$294$	−2.00000	−0.116642
$295$	0	0
$296$	−7.00000	−0.406867
$297$	−5.00000	−0.290129
$298$	5.00000	0.289642
$299$	16.0000	0.925304
$300$	0	0
$301$	−18.0000	−1.03750
$302$	−2.00000	−0.115087
$303$	2.00000	0.114897
$304$	−5.00000	−0.286770
$305$	0	0
$306$	−6.00000	−0.342997
$307$	−8.00000	−0.456584	−0.228292	−	0.973593i	$-0.573314\pi$
$307$	−8.00000	−0.456584	−0.228292	+	0.973593i	$0.573314\pi$
$308$	−3.00000	−0.170941
$309$	−4.00000	−0.227552
$310$	0	0
$311$	−13.0000	−0.737162	−0.368581	−	0.929596i	$-0.620156\pi$
$311$	−13.0000	−0.737162	−0.368581	+	0.929596i	$0.620156\pi$
$312$	4.00000	0.226455
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	13.0000	0.733632
$315$	0	0
$316$	0	0
$317$	−23.0000	−1.29181	−0.645904	−	0.763418i	$-0.723520\pi$
$317$	−23.0000	−1.29181	−0.645904	+	0.763418i	$0.723520\pi$
$318$	9.00000	0.504695
$319$	5.00000	0.279946
$320$	0	0
$321$	12.0000	0.669775
$322$	−12.0000	−0.668734
$323$	15.0000	0.834622
$324$	1.00000	0.0555556
$325$	0	0
$326$	−21.0000	−1.16308
$327$	−10.0000	−0.553001
$328$	8.00000	0.441726
$329$	24.0000	1.32316
$330$	0	0
$331$	−18.0000	−0.989369	−0.494685	−	0.869072i	$-0.664716\pi$
$331$	−18.0000	−0.989369	−0.494685	+	0.869072i	$0.664716\pi$
$332$	−4.00000	−0.219529
$333$	−14.0000	−0.767195
$334$	23.0000	1.25850
$335$	0	0
$336$	−3.00000	−0.163663
$337$	7.00000	0.381314	0.190657	−	0.981657i	$-0.438938\pi$
$337$	7.00000	0.381314	0.190657	+	0.981657i	$0.438938\pi$
$338$	−3.00000	−0.163178
$339$	6.00000	0.325875
$340$	0	0
$341$	7.00000	0.379071
$342$	−10.0000	−0.540738
$343$	15.0000	0.809924
$344$	−6.00000	−0.323498
$345$	0	0
$346$	14.0000	0.752645
$347$	−18.0000	−0.966291	−0.483145	−	0.875540i	$-0.660506\pi$
$347$	−18.0000	−0.966291	−0.483145	+	0.875540i	$0.660506\pi$
$348$	5.00000	0.268028
$349$	−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
$351$	20.0000	1.06752
$352$	−1.00000	−0.0533002
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	0	0
$355$	0	0
$356$	−15.0000	−0.794998
$357$	9.00000	0.476331
$358$	−10.0000	−0.528516
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	18.0000	0.946059
$363$	1.00000	0.0524864
$364$	12.0000	0.628971
$365$	0	0
$366$	13.0000	0.679521
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	−4.00000	−0.208514
$369$	16.0000	0.832927
$370$	0	0
$371$	27.0000	1.40177
$372$	7.00000	0.362933
$373$	−4.00000	−0.207112	−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000	−0.207112	−0.103556	+	0.994624i	$0.533022\pi$
$374$	3.00000	0.155126
$375$	0	0
$376$	8.00000	0.412568
$377$	−20.0000	−1.03005
$378$	−15.0000	−0.771517
$379$	−10.0000	−0.513665	−0.256833	−	0.966456i	$-0.582679\pi$
$379$	−10.0000	−0.513665	−0.256833	+	0.966456i	$0.582679\pi$
$380$	0	0
$381$	−8.00000	−0.409852
$382$	8.00000	0.409316
$383$	6.00000	0.306586	0.153293	−	0.988181i	$-0.451012\pi$
$383$	6.00000	0.306586	0.153293	+	0.988181i	$0.451012\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−1.00000	−0.0508987
$387$	−12.0000	−0.609994
$388$	12.0000	0.609208
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	−2.00000	−0.101015
$393$	7.00000	0.353103
$394$	−22.0000	−1.10834
$395$	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	$0.313837\pi$
$398$	−25.0000	−1.25314
$399$	15.0000	0.750939
$400$	0	0
$401$	27.0000	1.34832	0.674158	−	0.738587i	$-0.264507\pi$
$401$	27.0000	1.34832	0.674158	+	0.738587i	$0.264507\pi$
$402$	−12.0000	−0.598506
$403$	−28.0000	−1.39478
$404$	2.00000	0.0995037
$405$	0	0
$406$	15.0000	0.744438
$407$	7.00000	0.346977
$408$	3.00000	0.148522
$409$	20.0000	0.988936	0.494468	−	0.869196i	$-0.335363\pi$
$409$	20.0000	0.988936	0.494468	+	0.869196i	$0.335363\pi$
$410$	0	0
$411$	12.0000	0.591916
$412$	−4.00000	−0.197066
$413$	0	0
$414$	−8.00000	−0.393179
$415$	0	0
$416$	4.00000	0.196116
$417$	20.0000	0.979404
$418$	5.00000	0.244558
$419$	−20.0000	−0.977064	−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000	−0.977064	−0.488532	+	0.872546i	$0.662467\pi$
$420$	0	0
$421$	−28.0000	−1.36464	−0.682318	−	0.731055i	$-0.739028\pi$
$421$	−28.0000	−1.36464	−0.682318	+	0.731055i	$0.739028\pi$
$422$	13.0000	0.632830
$423$	16.0000	0.777947
$424$	9.00000	0.437079
$425$	0	0
$426$	3.00000	0.145350
$427$	39.0000	1.88734
$428$	12.0000	0.580042
$429$	−4.00000	−0.193122
$430$	0	0
$431$	32.0000	1.54139	0.770693	−	0.637207i	$-0.219910\pi$
$431$	32.0000	1.54139	0.770693	+	0.637207i	$0.219910\pi$
$432$	−5.00000	−0.240563
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	21.0000	1.00803
$435$	0	0
$436$	−10.0000	−0.478913
$437$	20.0000	0.956730
$438$	−6.00000	−0.286691
$439$	10.0000	0.477274	0.238637	−	0.971109i	$-0.423299\pi$
$439$	10.0000	0.477274	0.238637	+	0.971109i	$0.423299\pi$
$440$	0	0
$441$	−4.00000	−0.190476
$442$	−12.0000	−0.570782
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	7.00000	0.332205
$445$	0	0
$446$	14.0000	0.662919
$447$	−5.00000	−0.236492
$448$	−3.00000	−0.141737
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	0	0
$451$	−8.00000	−0.376705
$452$	6.00000	0.282216
$453$	2.00000	0.0939682
$454$	−12.0000	−0.563188
$455$	0	0
$456$	5.00000	0.234146
$457$	27.0000	1.26301	0.631503	−	0.775373i	$-0.282438\pi$
$457$	27.0000	1.26301	0.631503	+	0.775373i	$0.282438\pi$
$458$	0	0
$459$	15.0000	0.700140
$460$	0	0
$461$	−13.0000	−0.605470	−0.302735	−	0.953075i	$-0.597900\pi$
$461$	−13.0000	−0.605470	−0.302735	+	0.953075i	$0.597900\pi$
$462$	3.00000	0.139573
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	5.00000	0.232119
$465$	0	0
$466$	19.0000	0.880158
$467$	−3.00000	−0.138823	−0.0694117	−	0.997588i	$-0.522112\pi$
$467$	−3.00000	−0.138823	−0.0694117	+	0.997588i	$0.522112\pi$
$468$	8.00000	0.369800
$469$	−36.0000	−1.66233
$470$	0	0
$471$	−13.0000	−0.599008
$472$	0	0
$473$	6.00000	0.275880
$474$	0	0
$475$	0	0
$476$	9.00000	0.412514
$477$	18.0000	0.824163
$478$	0	0
$479$	−10.0000	−0.456912	−0.228456	−	0.973554i	$-0.573368\pi$
$479$	−10.0000	−0.456912	−0.228456	+	0.973554i	$0.573368\pi$
$480$	0	0
$481$	−28.0000	−1.27669
$482$	8.00000	0.364390
$483$	12.0000	0.546019
$484$	1.00000	0.0454545
$485$	0	0
$486$	−16.0000	−0.725775
$487$	22.0000	0.996915	0.498458	−	0.866914i	$-0.333900\pi$
$487$	22.0000	0.996915	0.498458	+	0.866914i	$0.333900\pi$
$488$	13.0000	0.588482
$489$	21.0000	0.949653
$490$	0	0
$491$	−33.0000	−1.48927	−0.744635	−	0.667472i	$-0.767376\pi$
$491$	−33.0000	−1.48927	−0.744635	+	0.667472i	$0.767376\pi$
$492$	−8.00000	−0.360668
$493$	−15.0000	−0.675566
$494$	−20.0000	−0.899843
$495$	0	0
$496$	7.00000	0.314309
$497$	9.00000	0.403705
$498$	4.00000	0.179244
$499$	−40.0000	−1.79065	−0.895323	−	0.445418i	$-0.853055\pi$
$499$	−40.0000	−1.79065	−0.895323	+	0.445418i	$0.853055\pi$
$500$	0	0
$501$	−23.0000	−1.02756
$502$	−2.00000	−0.0892644
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	−6.00000	−0.267261
$505$	0	0
$506$	4.00000	0.177822
$507$	3.00000	0.133235
$508$	−8.00000	−0.354943
$509$	−20.0000	−0.886484	−0.443242	−	0.896402i	$-0.646172\pi$
$509$	−20.0000	−0.886484	−0.443242	+	0.896402i	$0.646172\pi$
$510$	0	0
$511$	−18.0000	−0.796273
$512$	−1.00000	−0.0441942
$513$	25.0000	1.10378
$514$	28.0000	1.23503
$515$	0	0
$516$	6.00000	0.264135
$517$	−8.00000	−0.351840
$518$	21.0000	0.922687
$519$	−14.0000	−0.614532
$520$	0	0
$521$	22.0000	0.963837	0.481919	−	0.876216i	$-0.339940\pi$
$521$	22.0000	0.963837	0.481919	+	0.876216i	$0.339940\pi$
$522$	10.0000	0.437688
$523$	16.0000	0.699631	0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000	0.699631	0.349816	+	0.936819i	$0.386244\pi$
$524$	7.00000	0.305796
$525$	0	0
$526$	9.00000	0.392419
$527$	−21.0000	−0.914774
$528$	1.00000	0.0435194
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	15.0000	0.650332
$533$	32.0000	1.38607
$534$	15.0000	0.649113
$535$	0	0
$536$	−12.0000	−0.518321
$537$	10.0000	0.431532
$538$	0	0
$539$	2.00000	0.0861461
$540$	0	0
$541$	17.0000	0.730887	0.365444	−	0.930834i	$-0.380917\pi$
$541$	17.0000	0.730887	0.365444	+	0.930834i	$0.380917\pi$
$542$	−22.0000	−0.944981
$543$	−18.0000	−0.772454
$544$	3.00000	0.128624
$545$	0	0
$546$	−12.0000	−0.513553
$547$	−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
$549$	26.0000	1.10965
$550$	0	0
$551$	−25.0000	−1.06504
$552$	4.00000	0.170251
$553$	0	0
$554$	28.0000	1.18961
$555$	0	0
$556$	20.0000	0.848189
$557$	−28.0000	−1.18640	−0.593199	−	0.805056i	$-0.702135\pi$
$557$	−28.0000	−1.18640	−0.593199	+	0.805056i	$0.702135\pi$
$558$	14.0000	0.592667
$559$	−24.0000	−1.01509
$560$	0	0
$561$	−3.00000	−0.126660
$562$	18.0000	0.759284
$563$	−4.00000	−0.168580	−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000	−0.168580	−0.0842900	+	0.996441i	$0.526862\pi$
$564$	−8.00000	−0.336861
$565$	0	0
$566$	−16.0000	−0.672530
$567$	−3.00000	−0.125988
$568$	3.00000	0.125877
$569$	20.0000	0.838444	0.419222	−	0.907884i	$-0.362303\pi$
$569$	20.0000	0.838444	0.419222	+	0.907884i	$0.362303\pi$
$570$	0	0
$571$	−3.00000	−0.125546	−0.0627730	−	0.998028i	$-0.519994\pi$
$571$	−3.00000	−0.125546	−0.0627730	+	0.998028i	$0.519994\pi$
$572$	−4.00000	−0.167248
$573$	−8.00000	−0.334205
$574$	−24.0000	−1.00174
$575$	0	0
$576$	−2.00000	−0.0833333
$577$	−8.00000	−0.333044	−0.166522	−	0.986038i	$-0.553254\pi$
$577$	−8.00000	−0.333044	−0.166522	+	0.986038i	$0.553254\pi$
$578$	8.00000	0.332756
$579$	1.00000	0.0415586
$580$	0	0
$581$	12.0000	0.497844
$582$	−12.0000	−0.497416
$583$	−9.00000	−0.372742
$584$	−6.00000	−0.248282
$585$	0	0
$586$	−16.0000	−0.660954
$587$	7.00000	0.288921	0.144460	−	0.989511i	$-0.453855\pi$
$587$	7.00000	0.288921	0.144460	+	0.989511i	$0.453855\pi$
$588$	2.00000	0.0824786
$589$	−35.0000	−1.44215
$590$	0	0
$591$	22.0000	0.904959
$592$	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
$596$	−5.00000	−0.204808
$597$	25.0000	1.02318
$598$	−16.0000	−0.654289
$599$	−5.00000	−0.204294	−0.102147	−	0.994769i	$-0.532571\pi$
$599$	−5.00000	−0.204294	−0.102147	+	0.994769i	$0.532571\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	18.0000	0.733625
$603$	−24.0000	−0.977356
$604$	2.00000	0.0813788
$605$	0	0
$606$	−2.00000	−0.0812444
$607$	7.00000	0.284121	0.142061	−	0.989858i	$-0.454627\pi$
$607$	7.00000	0.284121	0.142061	+	0.989858i	$0.454627\pi$
$608$	5.00000	0.202777
$609$	−15.0000	−0.607831
$610$	0	0
$611$	32.0000	1.29458
$612$	6.00000	0.242536
$613$	6.00000	0.242338	0.121169	−	0.992632i	$-0.461336\pi$
$613$	6.00000	0.242338	0.121169	+	0.992632i	$0.461336\pi$
$614$	8.00000	0.322854
$615$	0	0
$616$	3.00000	0.120873
$617$	12.0000	0.483102	0.241551	−	0.970388i	$-0.422344\pi$
$617$	12.0000	0.483102	0.241551	+	0.970388i	$0.422344\pi$
$618$	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
$621$	20.0000	0.802572
$622$	13.0000	0.521253
$623$	45.0000	1.80289
$624$	−4.00000	−0.160128
$625$	0	0
$626$	14.0000	0.559553
$627$	−5.00000	−0.199681
$628$	−13.0000	−0.518756
$629$	−21.0000	−0.837325
$630$	0	0
$631$	−43.0000	−1.71180	−0.855901	−	0.517139i	$-0.826997\pi$
$631$	−43.0000	−1.71180	−0.855901	+	0.517139i	$0.826997\pi$
$632$	0	0
$633$	−13.0000	−0.516704
$634$	23.0000	0.913447
$635$	0	0
$636$	−9.00000	−0.356873
$637$	−8.00000	−0.316972
$638$	−5.00000	−0.197952
$639$	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	$0.725948\pi$
$642$	−12.0000	−0.473602
$643$	1.00000	0.0394362	0.0197181	−	0.999806i	$-0.493723\pi$
$643$	1.00000	0.0394362	0.0197181	+	0.999806i	$0.493723\pi$
$644$	12.0000	0.472866
$645$	0	0
$646$	−15.0000	−0.590167
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	−1.00000	−0.0392837
$649$	0	0
$650$	0	0
$651$	−21.0000	−0.823055
$652$	21.0000	0.822423
$653$	21.0000	0.821794	0.410897	−	0.911682i	$-0.365216\pi$
$653$	21.0000	0.821794	0.410897	+	0.911682i	$0.365216\pi$
$654$	10.0000	0.391031
$655$	0	0
$656$	−8.00000	−0.312348
$657$	−12.0000	−0.468165
$658$	−24.0000	−0.935617
$659$	−35.0000	−1.36341	−0.681703	−	0.731629i	$-0.738760\pi$
$659$	−35.0000	−1.36341	−0.681703	+	0.731629i	$0.738760\pi$
$660$	0	0
$661$	12.0000	0.466746	0.233373	−	0.972387i	$-0.425024\pi$
$661$	12.0000	0.466746	0.233373	+	0.972387i	$0.425024\pi$
$662$	18.0000	0.699590
$663$	12.0000	0.466041
$664$	4.00000	0.155230
$665$	0	0
$666$	14.0000	0.542489
$667$	−20.0000	−0.774403
$668$	−23.0000	−0.889897
$669$	−14.0000	−0.541271
$670$	0	0
$671$	−13.0000	−0.501859
$672$	3.00000	0.115728
$673$	1.00000	0.0385472	0.0192736	−	0.999814i	$-0.493865\pi$
$673$	1.00000	0.0385472	0.0192736	+	0.999814i	$0.493865\pi$
$674$	−7.00000	−0.269630
$675$	0	0
$676$	3.00000	0.115385
$677$	−8.00000	−0.307465	−0.153732	−	0.988113i	$-0.549129\pi$
$677$	−8.00000	−0.307465	−0.153732	+	0.988113i	$0.549129\pi$
$678$	−6.00000	−0.230429
$679$	−36.0000	−1.38155
$680$	0	0
$681$	12.0000	0.459841
$682$	−7.00000	−0.268044
$683$	−49.0000	−1.87493	−0.937466	−	0.348076i	$-0.886835\pi$
$683$	−49.0000	−1.87493	−0.937466	+	0.348076i	$0.886835\pi$
$684$	10.0000	0.382360
$685$	0	0
$686$	−15.0000	−0.572703
$687$	0	0
$688$	6.00000	0.228748
$689$	36.0000	1.37149
$690$	0	0
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	−14.0000	−0.532200
$693$	6.00000	0.227921
$694$	18.0000	0.683271
$695$	0	0
$696$	−5.00000	−0.189525
$697$	24.0000	0.909065
$698$	10.0000	0.378506
$699$	−19.0000	−0.718646
$700$	0	0
$701$	27.0000	1.01978	0.509888	−	0.860241i	$-0.329687\pi$
$701$	27.0000	1.01978	0.509888	+	0.860241i	$0.329687\pi$
$702$	−20.0000	−0.754851
$703$	−35.0000	−1.32005
$704$	1.00000	0.0376889
$705$	0	0
$706$	14.0000	0.526897
$707$	−6.00000	−0.225653
$708$	0	0
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	0	0
$711$	0	0
$712$	15.0000	0.562149
$713$	−28.0000	−1.04861
$714$	−9.00000	−0.336817
$715$	0	0
$716$	10.0000	0.373718
$717$	0	0
$718$	0	0
$719$	15.0000	0.559406	0.279703	−	0.960087i	$-0.409764\pi$
$719$	15.0000	0.559406	0.279703	+	0.960087i	$0.409764\pi$
$720$	0	0
$721$	12.0000	0.446903
$722$	−6.00000	−0.223297
$723$	−8.00000	−0.297523
$724$	−18.0000	−0.668965
$725$	0	0
$726$	−1.00000	−0.0371135
$727$	−18.0000	−0.667583	−0.333792	−	0.942647i	$-0.608328\pi$
$727$	−18.0000	−0.667583	−0.333792	+	0.942647i	$0.608328\pi$
$728$	−12.0000	−0.444750
$729$	13.0000	0.481481
$730$	0	0
$731$	−18.0000	−0.665754
$732$	−13.0000	−0.480494
$733$	−14.0000	−0.517102	−0.258551	−	0.965998i	$-0.583245\pi$
$733$	−14.0000	−0.517102	−0.258551	+	0.965998i	$0.583245\pi$
$734$	8.00000	0.295285
$735$	0	0
$736$	4.00000	0.147442
$737$	12.0000	0.442026
$738$	−16.0000	−0.588968
$739$	40.0000	1.47142	0.735712	−	0.677295i	$-0.236848\pi$
$739$	40.0000	1.47142	0.735712	+	0.677295i	$0.236848\pi$
$740$	0	0
$741$	20.0000	0.734718
$742$	−27.0000	−0.991201
$743$	1.00000	0.0366864	0.0183432	−	0.999832i	$-0.494161\pi$
$743$	1.00000	0.0366864	0.0183432	+	0.999832i	$0.494161\pi$
$744$	−7.00000	−0.256632
$745$	0	0
$746$	4.00000	0.146450
$747$	8.00000	0.292705
$748$	−3.00000	−0.109691
$749$	−36.0000	−1.31541
$750$	0	0
$751$	27.0000	0.985244	0.492622	−	0.870243i	$-0.336039\pi$
$751$	27.0000	0.985244	0.492622	+	0.870243i	$0.336039\pi$
$752$	−8.00000	−0.291730
$753$	2.00000	0.0728841
$754$	20.0000	0.728357
$755$	0	0
$756$	15.0000	0.545545
$757$	−18.0000	−0.654221	−0.327111	−	0.944986i	$-0.606075\pi$
$757$	−18.0000	−0.654221	−0.327111	+	0.944986i	$0.606075\pi$
$758$	10.0000	0.363216
$759$	−4.00000	−0.145191
$760$	0	0
$761$	12.0000	0.435000	0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000	0.435000	0.217500	+	0.976060i	$0.430210\pi$
$762$	8.00000	0.289809
$763$	30.0000	1.08607
$764$	−8.00000	−0.289430
$765$	0	0
$766$	−6.00000	−0.216789
$767$	0	0
$768$	1.00000	0.0360844
$769$	30.0000	1.08183	0.540914	−	0.841078i	$-0.318079\pi$
$769$	30.0000	1.08183	0.540914	+	0.841078i	$0.318079\pi$
$770$	0	0
$771$	−28.0000	−1.00840
$772$	1.00000	0.0359908
$773$	11.0000	0.395643	0.197821	−	0.980238i	$-0.436613\pi$
$773$	11.0000	0.395643	0.197821	+	0.980238i	$0.436613\pi$
$774$	12.0000	0.431331
$775$	0	0
$776$	−12.0000	−0.430775
$777$	−21.0000	−0.753371
$778$	30.0000	1.07555
$779$	40.0000	1.43315
$780$	0	0
$781$	−3.00000	−0.107348
$782$	−12.0000	−0.429119
$783$	−25.0000	−0.893427
$784$	2.00000	0.0714286
$785$	0	0
$786$	−7.00000	−0.249682
$787$	−38.0000	−1.35455	−0.677277	−	0.735728i	$-0.736840\pi$
$787$	−38.0000	−1.35455	−0.677277	+	0.735728i	$0.736840\pi$
$788$	22.0000	0.783718
$789$	−9.00000	−0.320408
$790$	0	0
$791$	−18.0000	−0.640006
$792$	2.00000	0.0710669
$793$	52.0000	1.84657
$794$	−22.0000	−0.780751
$795$	0	0
$796$	25.0000	0.886102
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	−15.0000	−0.530994
$799$	24.0000	0.849059
$800$	0	0
$801$	30.0000	1.06000
$802$	−27.0000	−0.953403
$803$	6.00000	0.211735
$804$	12.0000	0.423207
$805$	0	0
$806$	28.0000	0.986258
$807$	0	0
$808$	−2.00000	−0.0703598
$809$	10.0000	0.351581	0.175791	−	0.984428i	$-0.443752\pi$
$809$	10.0000	0.351581	0.175791	+	0.984428i	$0.443752\pi$
$810$	0	0
$811$	37.0000	1.29925	0.649623	−	0.760257i	$-0.274927\pi$
$811$	37.0000	1.29925	0.649623	+	0.760257i	$0.274927\pi$
$812$	−15.0000	−0.526397
$813$	22.0000	0.771574
$814$	−7.00000	−0.245350
$815$	0	0
$816$	−3.00000	−0.105021
$817$	−30.0000	−1.04957
$818$	−20.0000	−0.699284
$819$	−24.0000	−0.838628
$820$	0	0
$821$	22.0000	0.767805	0.383903	−	0.923374i	$-0.374580\pi$
$821$	22.0000	0.767805	0.383903	+	0.923374i	$0.374580\pi$
$822$	−12.0000	−0.418548
$823$	26.0000	0.906303	0.453152	−	0.891434i	$-0.350300\pi$
$823$	26.0000	0.906303	0.453152	+	0.891434i	$0.350300\pi$
$824$	4.00000	0.139347
$825$	0	0
$826$	0	0
$827$	22.0000	0.765015	0.382507	−	0.923952i	$-0.375061\pi$
$827$	22.0000	0.765015	0.382507	+	0.923952i	$0.375061\pi$
$828$	8.00000	0.278019
$829$	−20.0000	−0.694629	−0.347314	−	0.937749i	$-0.612906\pi$
$829$	−20.0000	−0.694629	−0.347314	+	0.937749i	$0.612906\pi$
$830$	0	0
$831$	−28.0000	−0.971309
$832$	−4.00000	−0.138675
$833$	−6.00000	−0.207888
$834$	−20.0000	−0.692543
$835$	0	0
$836$	−5.00000	−0.172929
$837$	−35.0000	−1.20978
$838$	20.0000	0.690889
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	28.0000	0.964944
$843$	−18.0000	−0.619953
$844$	−13.0000	−0.447478
$845$	0	0
$846$	−16.0000	−0.550091
$847$	−3.00000	−0.103081
$848$	−9.00000	−0.309061
$849$	16.0000	0.549119
$850$	0	0
$851$	−28.0000	−0.959828
$852$	−3.00000	−0.102778
$853$	−44.0000	−1.50653	−0.753266	−	0.657716i	$-0.771523\pi$
$853$	−44.0000	−1.50653	−0.753266	+	0.657716i	$0.771523\pi$
$854$	−39.0000	−1.33455
$855$	0	0
$856$	−12.0000	−0.410152
$857$	17.0000	0.580709	0.290354	−	0.956919i	$-0.406227\pi$
$857$	17.0000	0.580709	0.290354	+	0.956919i	$0.406227\pi$
$858$	4.00000	0.136558
$859$	−10.0000	−0.341196	−0.170598	−	0.985341i	$-0.554570\pi$
$859$	−10.0000	−0.341196	−0.170598	+	0.985341i	$0.554570\pi$
$860$	0	0
$861$	24.0000	0.817918
$862$	−32.0000	−1.08992
$863$	36.0000	1.22545	0.612727	−	0.790295i	$-0.290072\pi$
$863$	36.0000	1.22545	0.612727	+	0.790295i	$0.290072\pi$
$864$	5.00000	0.170103
$865$	0	0
$866$	14.0000	0.475739
$867$	−8.00000	−0.271694
$868$	−21.0000	−0.712786
$869$	0	0
$870$	0	0
$871$	−48.0000	−1.62642
$872$	10.0000	0.338643
$873$	−24.0000	−0.812277
$874$	−20.0000	−0.676510
$875$	0	0
$876$	6.00000	0.202721
$877$	2.00000	0.0675352	0.0337676	−	0.999430i	$-0.489249\pi$
$877$	2.00000	0.0675352	0.0337676	+	0.999430i	$0.489249\pi$
$878$	−10.0000	−0.337484
$879$	16.0000	0.539667
$880$	0	0
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	4.00000	0.134687
$883$	−9.00000	−0.302874	−0.151437	−	0.988467i	$-0.548390\pi$
$883$	−9.00000	−0.302874	−0.151437	+	0.988467i	$0.548390\pi$
$884$	12.0000	0.403604
$885$	0	0
$886$	4.00000	0.134383
$887$	32.0000	1.07445	0.537227	−	0.843437i	$-0.319472\pi$
$887$	32.0000	1.07445	0.537227	+	0.843437i	$0.319472\pi$
$888$	−7.00000	−0.234905
$889$	24.0000	0.804934
$890$	0	0
$891$	1.00000	0.0335013
$892$	−14.0000	−0.468755
$893$	40.0000	1.33855
$894$	5.00000	0.167225
$895$	0	0
$896$	3.00000	0.100223
$897$	16.0000	0.534224
$898$	−30.0000	−1.00111
$899$	35.0000	1.16732
$900$	0	0
$901$	27.0000	0.899500
$902$	8.00000	0.266371
$903$	−18.0000	−0.599002
$904$	−6.00000	−0.199557
$905$	0	0
$906$	−2.00000	−0.0664455
$907$	−23.0000	−0.763702	−0.381851	−	0.924224i	$-0.624713\pi$
$907$	−23.0000	−0.763702	−0.381851	+	0.924224i	$0.624713\pi$
$908$	12.0000	0.398234
$909$	−4.00000	−0.132672
$910$	0	0
$911$	37.0000	1.22586	0.612932	−	0.790135i	$-0.289990\pi$
$911$	37.0000	1.22586	0.612932	+	0.790135i	$0.289990\pi$
$912$	−5.00000	−0.165567
$913$	−4.00000	−0.132381
$914$	−27.0000	−0.893081
$915$	0	0
$916$	0	0
$917$	−21.0000	−0.693481
$918$	−15.0000	−0.495074
$919$	50.0000	1.64935	0.824674	−	0.565608i	$-0.191359\pi$
$919$	50.0000	1.64935	0.824674	+	0.565608i	$0.191359\pi$
$920$	0	0
$921$	−8.00000	−0.263609
$922$	13.0000	0.428132
$923$	12.0000	0.394985
$924$	−3.00000	−0.0986928
$925$	0	0
$926$	−16.0000	−0.525793
$927$	8.00000	0.262754
$928$	−5.00000	−0.164133
$929$	15.0000	0.492134	0.246067	−	0.969253i	$-0.420862\pi$
$929$	15.0000	0.492134	0.246067	+	0.969253i	$0.420862\pi$
$930$	0	0
$931$	−10.0000	−0.327737
$932$	−19.0000	−0.622366
$933$	−13.0000	−0.425601
$934$	3.00000	0.0981630
$935$	0	0
$936$	−8.00000	−0.261488
$937$	42.0000	1.37208	0.686040	−	0.727564i	$-0.259347\pi$
$937$	42.0000	1.37208	0.686040	+	0.727564i	$0.259347\pi$
$938$	36.0000	1.17544
$939$	−14.0000	−0.456873
$940$	0	0
$941$	17.0000	0.554184	0.277092	−	0.960843i	$-0.410629\pi$
$941$	17.0000	0.554184	0.277092	+	0.960843i	$0.410629\pi$
$942$	13.0000	0.423563
$943$	32.0000	1.04206
$944$	0	0
$945$	0	0
$946$	−6.00000	−0.195077
$947$	−53.0000	−1.72227	−0.861134	−	0.508378i	$-0.830245\pi$
$947$	−53.0000	−1.72227	−0.861134	+	0.508378i	$0.830245\pi$
$948$	0	0
$949$	−24.0000	−0.779073
$950$	0	0
$951$	−23.0000	−0.745826
$952$	−9.00000	−0.291692
$953$	−29.0000	−0.939402	−0.469701	−	0.882826i	$-0.655638\pi$
$953$	−29.0000	−0.939402	−0.469701	+	0.882826i	$0.655638\pi$
$954$	−18.0000	−0.582772
$955$	0	0
$956$	0	0
$957$	5.00000	0.161627
$958$	10.0000	0.323085
$959$	−36.0000	−1.16250
$960$	0	0
$961$	18.0000	0.580645
$962$	28.0000	0.902756
$963$	−24.0000	−0.773389
$964$	−8.00000	−0.257663
$965$	0	0
$966$	−12.0000	−0.386094
$967$	−13.0000	−0.418052	−0.209026	−	0.977910i	$-0.567029\pi$
$967$	−13.0000	−0.418052	−0.209026	+	0.977910i	$0.567029\pi$
$968$	−1.00000	−0.0321412
$969$	15.0000	0.481869
$970$	0	0
$971$	−28.0000	−0.898563	−0.449281	−	0.893390i	$-0.648320\pi$
$971$	−28.0000	−0.898563	−0.449281	+	0.893390i	$0.648320\pi$
$972$	16.0000	0.513200
$973$	−60.0000	−1.92351
$974$	−22.0000	−0.704925
$975$	0	0
$976$	−13.0000	−0.416120
$977$	−28.0000	−0.895799	−0.447900	−	0.894084i	$-0.647828\pi$
$977$	−28.0000	−0.895799	−0.447900	+	0.894084i	$0.647828\pi$
$978$	−21.0000	−0.671506
$979$	−15.0000	−0.479402
$980$	0	0
$981$	20.0000	0.638551
$982$	33.0000	1.05307
$983$	−14.0000	−0.446531	−0.223265	−	0.974758i	$-0.571672\pi$
$983$	−14.0000	−0.446531	−0.223265	+	0.974758i	$0.571672\pi$
$984$	8.00000	0.255031
$985$	0	0
$986$	15.0000	0.477697
$987$	24.0000	0.763928
$988$	20.0000	0.636285
$989$	−24.0000	−0.763156
$990$	0	0
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	−7.00000	−0.222250
$993$	−18.0000	−0.571213
$994$	−9.00000	−0.285463
$995$	0	0
$996$	−4.00000	−0.126745
$997$	2.00000	0.0633406	0.0316703	−	0.999498i	$-0.489917\pi$
$997$	2.00000	0.0633406	0.0316703	+	0.999498i	$0.489917\pi$
$998$	40.0000	1.26618
$999$	−35.0000	−1.10735

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	550.2.a.e.1.1		1
3.2	odd	2		4950.2.a.ba.1.1		1
4.3	odd	2		4400.2.a.k.1.1		1
5.2	odd	4		110.2.b.a.89.1	✓	2
5.3	odd	4		110.2.b.a.89.2	yes	2
5.4	even	2		550.2.a.j.1.1		1
11.10	odd	2		6050.2.a.bk.1.1		1
15.2	even	4		990.2.c.d.199.2		2
15.8	even	4		990.2.c.d.199.1		2
15.14	odd	2		4950.2.a.q.1.1		1
20.3	even	4		880.2.b.a.529.1		2
20.7	even	4		880.2.b.a.529.2		2
20.19	odd	2		4400.2.a.s.1.1		1
55.32	even	4		1210.2.b.a.969.2		2
55.43	even	4		1210.2.b.a.969.1		2
55.54	odd	2		6050.2.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
110.2.b.a.89.1	✓	2	5.2	odd	4
110.2.b.a.89.2	yes	2	5.3	odd	4
550.2.a.e.1.1		1	1.1	even	1	trivial
550.2.a.j.1.1		1	5.4	even	2
880.2.b.a.529.1		2	20.3	even	4
880.2.b.a.529.2		2	20.7	even	4
990.2.c.d.199.1		2	15.8	even	4
990.2.c.d.199.2		2	15.2	even	4
1210.2.b.a.969.1		2	55.43	even	4
1210.2.b.a.969.2		2	55.32	even	4
4400.2.a.k.1.1		1	4.3	odd	2
4400.2.a.s.1.1		1	20.19	odd	2
4950.2.a.q.1.1		1	15.14	odd	2
4950.2.a.ba.1.1		1	3.2	odd	2
6050.2.a.f.1.1		1	55.54	odd	2
6050.2.a.bk.1.1		1	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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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