LMFDB - Embedded newform 58.2.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [58,2,Mod(1,58)] 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("58.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(58, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 58 = 2 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	58.a (trivial)

Newform invariants

comment:select newform

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

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

Self dual:	yes
Analytic conductor:	$0.463132331723$
Analytic rank:	$1$
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	$\chi$	$=$	58.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q-1.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} +3.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +6.00000 q^{9} +3.00000 q^{10} -1.00000 q^{11} -3.00000 q^{12} +3.00000 q^{13} +2.00000 q^{14} +9.00000 q^{15} +1.00000 q^{16} -4.00000 q^{17} -6.00000 q^{18} -8.00000 q^{19} -3.00000 q^{20} +6.00000 q^{21} +1.00000 q^{22} +3.00000 q^{24} +4.00000 q^{25} -3.00000 q^{26} -9.00000 q^{27} -2.00000 q^{28} -1.00000 q^{29} -9.00000 q^{30} +3.00000 q^{31} -1.00000 q^{32} +3.00000 q^{33} +4.00000 q^{34} +6.00000 q^{35} +6.00000 q^{36} -8.00000 q^{37} +8.00000 q^{38} -9.00000 q^{39} +3.00000 q^{40} -2.00000 q^{41} -6.00000 q^{42} +7.00000 q^{43} -1.00000 q^{44} -18.0000 q^{45} +11.0000 q^{47} -3.00000 q^{48} -3.00000 q^{49} -4.00000 q^{50} +12.0000 q^{51} +3.00000 q^{52} +1.00000 q^{53} +9.00000 q^{54} +3.00000 q^{55} +2.00000 q^{56} +24.0000 q^{57} +1.00000 q^{58} -4.00000 q^{59} +9.00000 q^{60} +4.00000 q^{61} -3.00000 q^{62} -12.0000 q^{63} +1.00000 q^{64} -9.00000 q^{65} -3.00000 q^{66} -4.00000 q^{67} -4.00000 q^{68} -6.00000 q^{70} -2.00000 q^{71} -6.00000 q^{72} -12.0000 q^{73} +8.00000 q^{74} -12.0000 q^{75} -8.00000 q^{76} +2.00000 q^{77} +9.00000 q^{78} -7.00000 q^{79} -3.00000 q^{80} +9.00000 q^{81} +2.00000 q^{82} +6.00000 q^{84} +12.0000 q^{85} -7.00000 q^{86} +3.00000 q^{87} +1.00000 q^{88} -6.00000 q^{89} +18.0000 q^{90} -6.00000 q^{91} -9.00000 q^{93} -11.0000 q^{94} +24.0000 q^{95} +3.00000 q^{96} -6.00000 q^{97} +3.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−3.00000	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$3$	−3.00000	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$4$	1.00000	0.500000
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	3.00000	1.22474
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	−1.00000	−0.353553
$9$	6.00000	2.00000
$10$	3.00000	0.948683
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	−3.00000	−0.866025
$13$	3.00000	0.832050	0.416025	−	0.909353i	$-0.363423\pi$
$13$	3.00000	0.832050	0.416025	+	0.909353i	$0.363423\pi$
$14$	2.00000	0.534522
$15$	9.00000	2.32379
$16$	1.00000	0.250000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	−6.00000	−1.41421
$19$	−8.00000	−1.83533	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	−8.00000	−1.83533	−0.917663	+	0.397360i	$0.869927\pi$
$20$	−3.00000	−0.670820
$21$	6.00000	1.30931
$22$	1.00000	0.213201
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	3.00000	0.612372
$25$	4.00000	0.800000
$26$	−3.00000	−0.588348
$27$	−9.00000	−1.73205
$28$	−2.00000	−0.377964
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	−9.00000	−1.64317
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	−1.00000	−0.176777
$33$	3.00000	0.522233
$34$	4.00000	0.685994
$35$	6.00000	1.01419
$36$	6.00000	1.00000
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	8.00000	1.29777
$39$	−9.00000	−1.44115
$40$	3.00000	0.474342
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	−6.00000	−0.925820
$43$	7.00000	1.06749	0.533745	−	0.845645i	$-0.320784\pi$
$43$	7.00000	1.06749	0.533745	+	0.845645i	$0.320784\pi$
$44$	−1.00000	−0.150756
$45$	−18.0000	−2.68328
$46$	0	0
$47$	11.0000	1.60451	0.802257	−	0.596978i	$-0.203632\pi$
$47$	11.0000	1.60451	0.802257	+	0.596978i	$0.203632\pi$
$48$	−3.00000	−0.433013
$49$	−3.00000	−0.428571
$50$	−4.00000	−0.565685
$51$	12.0000	1.68034
$52$	3.00000	0.416025
$53$	1.00000	0.137361	0.0686803	−	0.997639i	$-0.478121\pi$
$53$	1.00000	0.137361	0.0686803	+	0.997639i	$0.478121\pi$
$54$	9.00000	1.22474
$55$	3.00000	0.404520
$56$	2.00000	0.267261
$57$	24.0000	3.17888
$58$	1.00000	0.131306
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	9.00000	1.16190
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	−3.00000	−0.381000
$63$	−12.0000	−1.51186
$64$	1.00000	0.125000
$65$	−9.00000	−1.11631
$66$	−3.00000	−0.369274
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	−4.00000	−0.485071
$69$	0	0
$70$	−6.00000	−0.717137
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	−6.00000	−0.707107
$73$	−12.0000	−1.40449	−0.702247	−	0.711934i	$-0.747820\pi$
$73$	−12.0000	−1.40449	−0.702247	+	0.711934i	$0.747820\pi$
$74$	8.00000	0.929981
$75$	−12.0000	−1.38564
$76$	−8.00000	−0.917663
$77$	2.00000	0.227921
$78$	9.00000	1.01905
$79$	−7.00000	−0.787562	−0.393781	−	0.919204i	$-0.628833\pi$
$79$	−7.00000	−0.787562	−0.393781	+	0.919204i	$0.628833\pi$
$80$	−3.00000	−0.335410
$81$	9.00000	1.00000
$82$	2.00000	0.220863
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	6.00000	0.654654
$85$	12.0000	1.30158
$86$	−7.00000	−0.754829
$87$	3.00000	0.321634
$88$	1.00000	0.106600
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	18.0000	1.89737
$91$	−6.00000	−0.628971
$92$	0	0
$93$	−9.00000	−0.933257
$94$	−11.0000	−1.13456
$95$	24.0000	2.46235
$96$	3.00000	0.306186
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	3.00000	0.303046
$99$	−6.00000	−0.603023
$100$	4.00000	0.400000
$101$	8.00000	0.796030	0.398015	−	0.917379i	$-0.369699\pi$
$101$	8.00000	0.796030	0.398015	+	0.917379i	$0.369699\pi$
$102$	−12.0000	−1.18818
$103$	−6.00000	−0.591198	−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000	−0.591198	−0.295599	+	0.955312i	$0.595519\pi$
$104$	−3.00000	−0.294174
$105$	−18.0000	−1.75662
$106$	−1.00000	−0.0971286
$107$	−2.00000	−0.193347	−0.0966736	−	0.995316i	$-0.530820\pi$
$107$	−2.00000	−0.193347	−0.0966736	+	0.995316i	$0.530820\pi$
$108$	−9.00000	−0.866025
$109$	1.00000	0.0957826	0.0478913	−	0.998853i	$-0.484750\pi$
$109$	1.00000	0.0957826	0.0478913	+	0.998853i	$0.484750\pi$
$110$	−3.00000	−0.286039
$111$	24.0000	2.27798
$112$	−2.00000	−0.188982
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	−24.0000	−2.24781
$115$	0	0
$116$	−1.00000	−0.0928477
$117$	18.0000	1.66410
$118$	4.00000	0.368230
$119$	8.00000	0.733359
$120$	−9.00000	−0.821584
$121$	−10.0000	−0.909091
$122$	−4.00000	−0.362143
$123$	6.00000	0.541002
$124$	3.00000	0.269408
$125$	3.00000	0.268328
$126$	12.0000	1.06904
$127$	−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
$129$	−21.0000	−1.84895
$130$	9.00000	0.789352
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	3.00000	0.261116
$133$	16.0000	1.38738
$134$	4.00000	0.345547
$135$	27.0000	2.32379
$136$	4.00000	0.342997
$137$	−20.0000	−1.70872	−0.854358	−	0.519685i	$-0.826049\pi$
$137$	−20.0000	−1.70872	−0.854358	+	0.519685i	$0.826049\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	6.00000	0.507093
$141$	−33.0000	−2.77910
$142$	2.00000	0.167836
$143$	−3.00000	−0.250873
$144$	6.00000	0.500000
$145$	3.00000	0.249136
$146$	12.0000	0.993127
$147$	9.00000	0.742307
$148$	−8.00000	−0.657596
$149$	3.00000	0.245770	0.122885	−	0.992421i	$-0.460785\pi$
$149$	3.00000	0.245770	0.122885	+	0.992421i	$0.460785\pi$
$150$	12.0000	0.979796
$151$	10.0000	0.813788	0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000	0.813788	0.406894	+	0.913475i	$0.366612\pi$
$152$	8.00000	0.648886
$153$	−24.0000	−1.94029
$154$	−2.00000	−0.161165
$155$	−9.00000	−0.722897
$156$	−9.00000	−0.720577
$157$	22.0000	1.75579	0.877896	−	0.478852i	$-0.158947\pi$
$157$	22.0000	1.75579	0.877896	+	0.478852i	$0.158947\pi$
$158$	7.00000	0.556890
$159$	−3.00000	−0.237915
$160$	3.00000	0.237171
$161$	0	0
$162$	−9.00000	−0.707107
$163$	19.0000	1.48819	0.744097	−	0.668071i	$-0.232880\pi$
$163$	19.0000	1.48819	0.744097	+	0.668071i	$0.232880\pi$
$164$	−2.00000	−0.156174
$165$	−9.00000	−0.700649
$166$	0	0
$167$	−22.0000	−1.70241	−0.851206	−	0.524832i	$-0.824128\pi$
$167$	−22.0000	−1.70241	−0.851206	+	0.524832i	$0.824128\pi$
$168$	−6.00000	−0.462910
$169$	−4.00000	−0.307692
$170$	−12.0000	−0.920358
$171$	−48.0000	−3.67065
$172$	7.00000	0.533745
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	−3.00000	−0.227429
$175$	−8.00000	−0.604743
$176$	−1.00000	−0.0753778
$177$	12.0000	0.901975
$178$	6.00000	0.449719
$179$	−14.0000	−1.04641	−0.523205	−	0.852207i	$-0.675264\pi$
$179$	−14.0000	−1.04641	−0.523205	+	0.852207i	$0.675264\pi$
$180$	−18.0000	−1.34164
$181$	−13.0000	−0.966282	−0.483141	−	0.875542i	$-0.660504\pi$
$181$	−13.0000	−0.966282	−0.483141	+	0.875542i	$0.660504\pi$
$182$	6.00000	0.444750
$183$	−12.0000	−0.887066
$184$	0	0
$185$	24.0000	1.76452
$186$	9.00000	0.659912
$187$	4.00000	0.292509
$188$	11.0000	0.802257
$189$	18.0000	1.30931
$190$	−24.0000	−1.74114
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	−3.00000	−0.216506
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	6.00000	0.430775
$195$	27.0000	1.93351
$196$	−3.00000	−0.214286
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	6.00000	0.426401
$199$	−2.00000	−0.141776	−0.0708881	−	0.997484i	$-0.522583\pi$
$199$	−2.00000	−0.141776	−0.0708881	+	0.997484i	$0.522583\pi$
$200$	−4.00000	−0.282843
$201$	12.0000	0.846415
$202$	−8.00000	−0.562878
$203$	2.00000	0.140372
$204$	12.0000	0.840168
$205$	6.00000	0.419058
$206$	6.00000	0.418040
$207$	0	0
$208$	3.00000	0.208013
$209$	8.00000	0.553372
$210$	18.0000	1.24212
$211$	−25.0000	−1.72107	−0.860535	−	0.509390i	$-0.829871\pi$
$211$	−25.0000	−1.72107	−0.860535	+	0.509390i	$0.829871\pi$
$212$	1.00000	0.0686803
$213$	6.00000	0.411113
$214$	2.00000	0.136717
$215$	−21.0000	−1.43219
$216$	9.00000	0.612372
$217$	−6.00000	−0.407307
$218$	−1.00000	−0.0677285
$219$	36.0000	2.43265
$220$	3.00000	0.202260
$221$	−12.0000	−0.807207
$222$	−24.0000	−1.61077
$223$	26.0000	1.74109	0.870544	−	0.492090i	$-0.163767\pi$
$223$	26.0000	1.74109	0.870544	+	0.492090i	$0.163767\pi$
$224$	2.00000	0.133631
$225$	24.0000	1.60000
$226$	−18.0000	−1.19734
$227$	−18.0000	−1.19470	−0.597351	−	0.801980i	$-0.703780\pi$
$227$	−18.0000	−1.19470	−0.597351	+	0.801980i	$0.703780\pi$
$228$	24.0000	1.58944
$229$	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
$231$	−6.00000	−0.394771
$232$	1.00000	0.0656532
$233$	−25.0000	−1.63780	−0.818902	−	0.573933i	$-0.805417\pi$
$233$	−25.0000	−1.63780	−0.818902	+	0.573933i	$0.805417\pi$
$234$	−18.0000	−1.17670
$235$	−33.0000	−2.15268
$236$	−4.00000	−0.260378
$237$	21.0000	1.36410
$238$	−8.00000	−0.518563
$239$	20.0000	1.29369	0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000	1.29369	0.646846	+	0.762620i	$0.276088\pi$
$240$	9.00000	0.580948
$241$	17.0000	1.09507	0.547533	−	0.836784i	$-0.315567\pi$
$241$	17.0000	1.09507	0.547533	+	0.836784i	$0.315567\pi$
$242$	10.0000	0.642824
$243$	0	0
$244$	4.00000	0.256074
$245$	9.00000	0.574989
$246$	−6.00000	−0.382546
$247$	−24.0000	−1.52708
$248$	−3.00000	−0.190500
$249$	0	0
$250$	−3.00000	−0.189737
$251$	−7.00000	−0.441836	−0.220918	−	0.975292i	$-0.570905\pi$
$251$	−7.00000	−0.441836	−0.220918	+	0.975292i	$0.570905\pi$
$252$	−12.0000	−0.755929
$253$	0	0
$254$	8.00000	0.501965
$255$	−36.0000	−2.25441
$256$	1.00000	0.0625000
$257$	21.0000	1.30994	0.654972	−	0.755653i	$-0.272680\pi$
$257$	21.0000	1.30994	0.654972	+	0.755653i	$0.272680\pi$
$258$	21.0000	1.30740
$259$	16.0000	0.994192
$260$	−9.00000	−0.558156
$261$	−6.00000	−0.371391
$262$	−12.0000	−0.741362
$263$	−17.0000	−1.04826	−0.524132	−	0.851637i	$-0.675610\pi$
$263$	−17.0000	−1.04826	−0.524132	+	0.851637i	$0.675610\pi$
$264$	−3.00000	−0.184637
$265$	−3.00000	−0.184289
$266$	−16.0000	−0.981023
$267$	18.0000	1.10158
$268$	−4.00000	−0.244339
$269$	20.0000	1.21942	0.609711	−	0.792624i	$-0.291286\pi$
$269$	20.0000	1.21942	0.609711	+	0.792624i	$0.291286\pi$
$270$	−27.0000	−1.64317
$271$	13.0000	0.789694	0.394847	−	0.918747i	$-0.370798\pi$
$271$	13.0000	0.789694	0.394847	+	0.918747i	$0.370798\pi$
$272$	−4.00000	−0.242536
$273$	18.0000	1.08941
$274$	20.0000	1.20824
$275$	−4.00000	−0.241209
$276$	0	0
$277$	14.0000	0.841178	0.420589	−	0.907251i	$-0.361823\pi$
$277$	14.0000	0.841178	0.420589	+	0.907251i	$0.361823\pi$
$278$	0	0
$279$	18.0000	1.07763
$280$	−6.00000	−0.358569
$281$	−13.0000	−0.775515	−0.387757	−	0.921761i	$-0.626750\pi$
$281$	−13.0000	−0.775515	−0.387757	+	0.921761i	$0.626750\pi$
$282$	33.0000	1.96512
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	−2.00000	−0.118678
$285$	−72.0000	−4.26491
$286$	3.00000	0.177394
$287$	4.00000	0.236113
$288$	−6.00000	−0.353553
$289$	−1.00000	−0.0588235
$290$	−3.00000	−0.176166
$291$	18.0000	1.05518
$292$	−12.0000	−0.702247
$293$	−34.0000	−1.98630	−0.993151	−	0.116841i	$-0.962723\pi$
$293$	−34.0000	−1.98630	−0.993151	+	0.116841i	$0.962723\pi$
$294$	−9.00000	−0.524891
$295$	12.0000	0.698667
$296$	8.00000	0.464991
$297$	9.00000	0.522233
$298$	−3.00000	−0.173785
$299$	0	0
$300$	−12.0000	−0.692820
$301$	−14.0000	−0.806947
$302$	−10.0000	−0.575435
$303$	−24.0000	−1.37876
$304$	−8.00000	−0.458831
$305$	−12.0000	−0.687118
$306$	24.0000	1.37199
$307$	−29.0000	−1.65512	−0.827559	−	0.561379i	$-0.810271\pi$
$307$	−29.0000	−1.65512	−0.827559	+	0.561379i	$0.810271\pi$
$308$	2.00000	0.113961
$309$	18.0000	1.02398
$310$	9.00000	0.511166
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	9.00000	0.509525
$313$	25.0000	1.41308	0.706542	−	0.707671i	$-0.250254\pi$
$313$	25.0000	1.41308	0.706542	+	0.707671i	$0.250254\pi$
$314$	−22.0000	−1.24153
$315$	36.0000	2.02837
$316$	−7.00000	−0.393781
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	3.00000	0.168232
$319$	1.00000	0.0559893
$320$	−3.00000	−0.167705
$321$	6.00000	0.334887
$322$	0	0
$323$	32.0000	1.78053
$324$	9.00000	0.500000
$325$	12.0000	0.665640
$326$	−19.0000	−1.05231
$327$	−3.00000	−0.165900
$328$	2.00000	0.110432
$329$	−22.0000	−1.21290
$330$	9.00000	0.495434
$331$	3.00000	0.164895	0.0824475	−	0.996595i	$-0.473726\pi$
$331$	3.00000	0.164895	0.0824475	+	0.996595i	$0.473726\pi$
$332$	0	0
$333$	−48.0000	−2.63038
$334$	22.0000	1.20379
$335$	12.0000	0.655630
$336$	6.00000	0.327327
$337$	4.00000	0.217894	0.108947	−	0.994048i	$-0.465252\pi$
$337$	4.00000	0.217894	0.108947	+	0.994048i	$0.465252\pi$
$338$	4.00000	0.217571
$339$	−54.0000	−2.93288
$340$	12.0000	0.650791
$341$	−3.00000	−0.162459
$342$	48.0000	2.59554
$343$	20.0000	1.07990
$344$	−7.00000	−0.377415
$345$	0	0
$346$	14.0000	0.752645
$347$	18.0000	0.966291	0.483145	−	0.875540i	$-0.339494\pi$
$347$	18.0000	0.966291	0.483145	+	0.875540i	$0.339494\pi$
$348$	3.00000	0.160817
$349$	−19.0000	−1.01705	−0.508523	−	0.861048i	$-0.669808\pi$
$349$	−19.0000	−1.01705	−0.508523	+	0.861048i	$0.669808\pi$
$350$	8.00000	0.427618
$351$	−27.0000	−1.44115
$352$	1.00000	0.0533002
$353$	−2.00000	−0.106449	−0.0532246	−	0.998583i	$-0.516950\pi$
$353$	−2.00000	−0.106449	−0.0532246	+	0.998583i	$0.516950\pi$
$354$	−12.0000	−0.637793
$355$	6.00000	0.318447
$356$	−6.00000	−0.317999
$357$	−24.0000	−1.27021
$358$	14.0000	0.739923
$359$	9.00000	0.475002	0.237501	−	0.971387i	$-0.423672\pi$
$359$	9.00000	0.475002	0.237501	+	0.971387i	$0.423672\pi$
$360$	18.0000	0.948683
$361$	45.0000	2.36842
$362$	13.0000	0.683265
$363$	30.0000	1.57459
$364$	−6.00000	−0.314485
$365$	36.0000	1.88433
$366$	12.0000	0.627250
$367$	−16.0000	−0.835193	−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000	−0.835193	−0.417597	+	0.908633i	$0.637127\pi$
$368$	0	0
$369$	−12.0000	−0.624695
$370$	−24.0000	−1.24770
$371$	−2.00000	−0.103835
$372$	−9.00000	−0.466628
$373$	−1.00000	−0.0517780	−0.0258890	−	0.999665i	$-0.508242\pi$
$373$	−1.00000	−0.0517780	−0.0258890	+	0.999665i	$0.508242\pi$
$374$	−4.00000	−0.206835
$375$	−9.00000	−0.464758
$376$	−11.0000	−0.567282
$377$	−3.00000	−0.154508
$378$	−18.0000	−0.925820
$379$	−28.0000	−1.43826	−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000	−1.43826	−0.719132	+	0.694874i	$0.755460\pi$
$380$	24.0000	1.23117
$381$	24.0000	1.22956
$382$	0	0
$383$	−34.0000	−1.73732	−0.868659	−	0.495410i	$-0.835018\pi$
$383$	−34.0000	−1.73732	−0.868659	+	0.495410i	$0.835018\pi$
$384$	3.00000	0.153093
$385$	−6.00000	−0.305788
$386$	−10.0000	−0.508987
$387$	42.0000	2.13498
$388$	−6.00000	−0.304604
$389$	16.0000	0.811232	0.405616	−	0.914044i	$-0.367057\pi$
$389$	16.0000	0.811232	0.405616	+	0.914044i	$0.367057\pi$
$390$	−27.0000	−1.36720
$391$	0	0
$392$	3.00000	0.151523
$393$	−36.0000	−1.81596
$394$	−2.00000	−0.100759
$395$	21.0000	1.05662
$396$	−6.00000	−0.301511
$397$	19.0000	0.953583	0.476791	−	0.879017i	$-0.341800\pi$
$397$	19.0000	0.953583	0.476791	+	0.879017i	$0.341800\pi$
$398$	2.00000	0.100251
$399$	−48.0000	−2.40301
$400$	4.00000	0.200000
$401$	−5.00000	−0.249688	−0.124844	−	0.992176i	$-0.539843\pi$
$401$	−5.00000	−0.249688	−0.124844	+	0.992176i	$0.539843\pi$
$402$	−12.0000	−0.598506
$403$	9.00000	0.448322
$404$	8.00000	0.398015
$405$	−27.0000	−1.34164
$406$	−2.00000	−0.0992583
$407$	8.00000	0.396545
$408$	−12.0000	−0.594089
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	−6.00000	−0.296319
$411$	60.0000	2.95958
$412$	−6.00000	−0.295599
$413$	8.00000	0.393654
$414$	0	0
$415$	0	0
$416$	−3.00000	−0.147087
$417$	0	0
$418$	−8.00000	−0.391293
$419$	−14.0000	−0.683945	−0.341972	−	0.939710i	$-0.611095\pi$
$419$	−14.0000	−0.683945	−0.341972	+	0.939710i	$0.611095\pi$
$420$	−18.0000	−0.878310
$421$	−28.0000	−1.36464	−0.682318	−	0.731055i	$-0.739028\pi$
$421$	−28.0000	−1.36464	−0.682318	+	0.731055i	$0.739028\pi$
$422$	25.0000	1.21698
$423$	66.0000	3.20903
$424$	−1.00000	−0.0485643
$425$	−16.0000	−0.776114
$426$	−6.00000	−0.290701
$427$	−8.00000	−0.387147
$428$	−2.00000	−0.0966736
$429$	9.00000	0.434524
$430$	21.0000	1.01271
$431$	−20.0000	−0.963366	−0.481683	−	0.876346i	$-0.659974\pi$
$431$	−20.0000	−0.963366	−0.481683	+	0.876346i	$0.659974\pi$
$432$	−9.00000	−0.433013
$433$	−16.0000	−0.768911	−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000	−0.768911	−0.384455	+	0.923144i	$0.625611\pi$
$434$	6.00000	0.288009
$435$	−9.00000	−0.431517
$436$	1.00000	0.0478913
$437$	0	0
$438$	−36.0000	−1.72015
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	−3.00000	−0.143019
$441$	−18.0000	−0.857143
$442$	12.0000	0.570782
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	24.0000	1.13899
$445$	18.0000	0.853282
$446$	−26.0000	−1.23114
$447$	−9.00000	−0.425685
$448$	−2.00000	−0.0944911
$449$	22.0000	1.03824	0.519122	−	0.854700i	$-0.326259\pi$
$449$	22.0000	1.03824	0.519122	+	0.854700i	$0.326259\pi$
$450$	−24.0000	−1.13137
$451$	2.00000	0.0941763
$452$	18.0000	0.846649
$453$	−30.0000	−1.40952
$454$	18.0000	0.844782
$455$	18.0000	0.843853
$456$	−24.0000	−1.12390
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	−14.0000	−0.654177
$459$	36.0000	1.68034
$460$	0	0
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	6.00000	0.279145
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	−1.00000	−0.0464238
$465$	27.0000	1.25210
$466$	25.0000	1.15810
$467$	23.0000	1.06431	0.532157	−	0.846646i	$-0.321382\pi$
$467$	23.0000	1.06431	0.532157	+	0.846646i	$0.321382\pi$
$468$	18.0000	0.832050
$469$	8.00000	0.369406
$470$	33.0000	1.52218
$471$	−66.0000	−3.04112
$472$	4.00000	0.184115
$473$	−7.00000	−0.321860
$474$	−21.0000	−0.964562
$475$	−32.0000	−1.46826
$476$	8.00000	0.366679
$477$	6.00000	0.274721
$478$	−20.0000	−0.914779
$479$	−11.0000	−0.502603	−0.251301	−	0.967909i	$-0.580859\pi$
$479$	−11.0000	−0.502603	−0.251301	+	0.967909i	$0.580859\pi$
$480$	−9.00000	−0.410792
$481$	−24.0000	−1.09431
$482$	−17.0000	−0.774329
$483$	0	0
$484$	−10.0000	−0.454545
$485$	18.0000	0.817338
$486$	0	0
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	−4.00000	−0.181071
$489$	−57.0000	−2.57763
$490$	−9.00000	−0.406579
$491$	5.00000	0.225647	0.112823	−	0.993615i	$-0.464011\pi$
$491$	5.00000	0.225647	0.112823	+	0.993615i	$0.464011\pi$
$492$	6.00000	0.270501
$493$	4.00000	0.180151
$494$	24.0000	1.07981
$495$	18.0000	0.809040
$496$	3.00000	0.134704
$497$	4.00000	0.179425
$498$	0	0
$499$	−8.00000	−0.358129	−0.179065	−	0.983837i	$-0.557307\pi$
$499$	−8.00000	−0.358129	−0.179065	+	0.983837i	$0.557307\pi$
$500$	3.00000	0.134164
$501$	66.0000	2.94866
$502$	7.00000	0.312425
$503$	−19.0000	−0.847168	−0.423584	−	0.905857i	$-0.639228\pi$
$503$	−19.0000	−0.847168	−0.423584	+	0.905857i	$0.639228\pi$
$504$	12.0000	0.534522
$505$	−24.0000	−1.06799
$506$	0	0
$507$	12.0000	0.532939
$508$	−8.00000	−0.354943
$509$	21.0000	0.930809	0.465404	−	0.885098i	$-0.345909\pi$
$509$	21.0000	0.930809	0.465404	+	0.885098i	$0.345909\pi$
$510$	36.0000	1.59411
$511$	24.0000	1.06170
$512$	−1.00000	−0.0441942
$513$	72.0000	3.17888
$514$	−21.0000	−0.926270
$515$	18.0000	0.793175
$516$	−21.0000	−0.924473
$517$	−11.0000	−0.483779
$518$	−16.0000	−0.703000
$519$	42.0000	1.84360
$520$	9.00000	0.394676
$521$	27.0000	1.18289	0.591446	−	0.806345i	$-0.298557\pi$
$521$	27.0000	1.18289	0.591446	+	0.806345i	$0.298557\pi$
$522$	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	$0.386244\pi$
$524$	12.0000	0.524222
$525$	24.0000	1.04745
$526$	17.0000	0.741235
$527$	−12.0000	−0.522728
$528$	3.00000	0.130558
$529$	−23.0000	−1.00000
$530$	3.00000	0.130312
$531$	−24.0000	−1.04151
$532$	16.0000	0.693688
$533$	−6.00000	−0.259889
$534$	−18.0000	−0.778936
$535$	6.00000	0.259403
$536$	4.00000	0.172774
$537$	42.0000	1.81243
$538$	−20.0000	−0.862261
$539$	3.00000	0.129219
$540$	27.0000	1.16190
$541$	−8.00000	−0.343947	−0.171973	−	0.985102i	$-0.555014\pi$
$541$	−8.00000	−0.343947	−0.171973	+	0.985102i	$0.555014\pi$
$542$	−13.0000	−0.558398
$543$	39.0000	1.67365
$544$	4.00000	0.171499
$545$	−3.00000	−0.128506
$546$	−18.0000	−0.770329
$547$	−2.00000	−0.0855138	−0.0427569	−	0.999086i	$-0.513614\pi$
$547$	−2.00000	−0.0855138	−0.0427569	+	0.999086i	$0.513614\pi$
$548$	−20.0000	−0.854358
$549$	24.0000	1.02430
$550$	4.00000	0.170561
$551$	8.00000	0.340811
$552$	0	0
$553$	14.0000	0.595341
$554$	−14.0000	−0.594803
$555$	−72.0000	−3.05623
$556$	0	0
$557$	−18.0000	−0.762684	−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000	−0.762684	−0.381342	+	0.924434i	$0.624538\pi$
$558$	−18.0000	−0.762001
$559$	21.0000	0.888205
$560$	6.00000	0.253546
$561$	−12.0000	−0.506640
$562$	13.0000	0.548372
$563$	−9.00000	−0.379305	−0.189652	−	0.981851i	$-0.560736\pi$
$563$	−9.00000	−0.379305	−0.189652	+	0.981851i	$0.560736\pi$
$564$	−33.0000	−1.38955
$565$	−54.0000	−2.27180
$566$	0	0
$567$	−18.0000	−0.755929
$568$	2.00000	0.0839181
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	72.0000	3.01575
$571$	−36.0000	−1.50655	−0.753277	−	0.657704i	$-0.771528\pi$
$571$	−36.0000	−1.50655	−0.753277	+	0.657704i	$0.771528\pi$
$572$	−3.00000	−0.125436
$573$	0	0
$574$	−4.00000	−0.166957
$575$	0	0
$576$	6.00000	0.250000
$577$	−44.0000	−1.83174	−0.915872	−	0.401470i	$-0.868499\pi$
$577$	−44.0000	−1.83174	−0.915872	+	0.401470i	$0.868499\pi$
$578$	1.00000	0.0415945
$579$	−30.0000	−1.24676
$580$	3.00000	0.124568
$581$	0	0
$582$	−18.0000	−0.746124
$583$	−1.00000	−0.0414158
$584$	12.0000	0.496564
$585$	−54.0000	−2.23263
$586$	34.0000	1.40453
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	9.00000	0.371154
$589$	−24.0000	−0.988903
$590$	−12.0000	−0.494032
$591$	−6.00000	−0.246807
$592$	−8.00000	−0.328798
$593$	−17.0000	−0.698106	−0.349053	−	0.937103i	$-0.613497\pi$
$593$	−17.0000	−0.698106	−0.349053	+	0.937103i	$0.613497\pi$
$594$	−9.00000	−0.369274
$595$	−24.0000	−0.983904
$596$	3.00000	0.122885
$597$	6.00000	0.245564
$598$	0	0
$599$	13.0000	0.531166	0.265583	−	0.964088i	$-0.414436\pi$
$599$	13.0000	0.531166	0.265583	+	0.964088i	$0.414436\pi$
$600$	12.0000	0.489898
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	14.0000	0.570597
$603$	−24.0000	−0.977356
$604$	10.0000	0.406894
$605$	30.0000	1.21967
$606$	24.0000	0.974933
$607$	29.0000	1.17707	0.588537	−	0.808470i	$-0.299704\pi$
$607$	29.0000	1.17707	0.588537	+	0.808470i	$0.299704\pi$
$608$	8.00000	0.324443
$609$	−6.00000	−0.243132
$610$	12.0000	0.485866
$611$	33.0000	1.33504
$612$	−24.0000	−0.970143
$613$	−27.0000	−1.09052	−0.545260	−	0.838267i	$-0.683569\pi$
$613$	−27.0000	−1.09052	−0.545260	+	0.838267i	$0.683569\pi$
$614$	29.0000	1.17034
$615$	−18.0000	−0.725830
$616$	−2.00000	−0.0805823
$617$	12.0000	0.483102	0.241551	−	0.970388i	$-0.422344\pi$
$617$	12.0000	0.483102	0.241551	+	0.970388i	$0.422344\pi$
$618$	−18.0000	−0.724066
$619$	31.0000	1.24600	0.622998	−	0.782224i	$-0.285915\pi$
$619$	31.0000	1.24600	0.622998	+	0.782224i	$0.285915\pi$
$620$	−9.00000	−0.361449
$621$	0	0
$622$	0	0
$623$	12.0000	0.480770
$624$	−9.00000	−0.360288
$625$	−29.0000	−1.16000
$626$	−25.0000	−0.999201
$627$	−24.0000	−0.958468
$628$	22.0000	0.877896
$629$	32.0000	1.27592
$630$	−36.0000	−1.43427
$631$	22.0000	0.875806	0.437903	−	0.899022i	$-0.355721\pi$
$631$	22.0000	0.875806	0.437903	+	0.899022i	$0.355721\pi$
$632$	7.00000	0.278445
$633$	75.0000	2.98098
$634$	0	0
$635$	24.0000	0.952411
$636$	−3.00000	−0.118958
$637$	−9.00000	−0.356593
$638$	−1.00000	−0.0395904
$639$	−12.0000	−0.474713
$640$	3.00000	0.118585
$641$	36.0000	1.42191	0.710957	−	0.703235i	$-0.248262\pi$
$641$	36.0000	1.42191	0.710957	+	0.703235i	$0.248262\pi$
$642$	−6.00000	−0.236801
$643$	2.00000	0.0788723	0.0394362	−	0.999222i	$-0.487444\pi$
$643$	2.00000	0.0788723	0.0394362	+	0.999222i	$0.487444\pi$
$644$	0	0
$645$	63.0000	2.48062
$646$	−32.0000	−1.25902
$647$	−20.0000	−0.786281	−0.393141	−	0.919478i	$-0.628611\pi$
$647$	−20.0000	−0.786281	−0.393141	+	0.919478i	$0.628611\pi$
$648$	−9.00000	−0.353553
$649$	4.00000	0.157014
$650$	−12.0000	−0.470679
$651$	18.0000	0.705476
$652$	19.0000	0.744097
$653$	−14.0000	−0.547862	−0.273931	−	0.961749i	$-0.588324\pi$
$653$	−14.0000	−0.547862	−0.273931	+	0.961749i	$0.588324\pi$
$654$	3.00000	0.117309
$655$	−36.0000	−1.40664
$656$	−2.00000	−0.0780869
$657$	−72.0000	−2.80899
$658$	22.0000	0.857649
$659$	21.0000	0.818044	0.409022	−	0.912525i	$-0.365870\pi$
$659$	21.0000	0.818044	0.409022	+	0.912525i	$0.365870\pi$
$660$	−9.00000	−0.350325
$661$	46.0000	1.78919	0.894596	−	0.446875i	$-0.147463\pi$
$661$	46.0000	1.78919	0.894596	+	0.446875i	$0.147463\pi$
$662$	−3.00000	−0.116598
$663$	36.0000	1.39812
$664$	0	0
$665$	−48.0000	−1.86136
$666$	48.0000	1.85996
$667$	0	0
$668$	−22.0000	−0.851206
$669$	−78.0000	−3.01565
$670$	−12.0000	−0.463600
$671$	−4.00000	−0.154418
$672$	−6.00000	−0.231455
$673$	1.00000	0.0385472	0.0192736	−	0.999814i	$-0.493865\pi$
$673$	1.00000	0.0385472	0.0192736	+	0.999814i	$0.493865\pi$
$674$	−4.00000	−0.154074
$675$	−36.0000	−1.38564
$676$	−4.00000	−0.153846
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	54.0000	2.07386
$679$	12.0000	0.460518
$680$	−12.0000	−0.460179
$681$	54.0000	2.06928
$682$	3.00000	0.114876
$683$	−16.0000	−0.612223	−0.306111	−	0.951996i	$-0.599028\pi$
$683$	−16.0000	−0.612223	−0.306111	+	0.951996i	$0.599028\pi$
$684$	−48.0000	−1.83533
$685$	60.0000	2.29248
$686$	−20.0000	−0.763604
$687$	−42.0000	−1.60240
$688$	7.00000	0.266872
$689$	3.00000	0.114291
$690$	0	0
$691$	4.00000	0.152167	0.0760836	−	0.997101i	$-0.475758\pi$
$691$	4.00000	0.152167	0.0760836	+	0.997101i	$0.475758\pi$
$692$	−14.0000	−0.532200
$693$	12.0000	0.455842
$694$	−18.0000	−0.683271
$695$	0	0
$696$	−3.00000	−0.113715
$697$	8.00000	0.303022
$698$	19.0000	0.719161
$699$	75.0000	2.83676
$700$	−8.00000	−0.302372
$701$	31.0000	1.17085	0.585427	−	0.810725i	$-0.300927\pi$
$701$	31.0000	1.17085	0.585427	+	0.810725i	$0.300927\pi$
$702$	27.0000	1.01905
$703$	64.0000	2.41381
$704$	−1.00000	−0.0376889
$705$	99.0000	3.72856
$706$	2.00000	0.0752710
$707$	−16.0000	−0.601742
$708$	12.0000	0.450988
$709$	35.0000	1.31445	0.657226	−	0.753693i	$-0.271730\pi$
$709$	35.0000	1.31445	0.657226	+	0.753693i	$0.271730\pi$
$710$	−6.00000	−0.225176
$711$	−42.0000	−1.57512
$712$	6.00000	0.224860
$713$	0	0
$714$	24.0000	0.898177
$715$	9.00000	0.336581
$716$	−14.0000	−0.523205
$717$	−60.0000	−2.24074
$718$	−9.00000	−0.335877
$719$	−6.00000	−0.223762	−0.111881	−	0.993722i	$-0.535688\pi$
$719$	−6.00000	−0.223762	−0.111881	+	0.993722i	$0.535688\pi$
$720$	−18.0000	−0.670820
$721$	12.0000	0.446903
$722$	−45.0000	−1.67473
$723$	−51.0000	−1.89671
$724$	−13.0000	−0.483141
$725$	−4.00000	−0.148556
$726$	−30.0000	−1.11340
$727$	32.0000	1.18681	0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000	1.18681	0.593407	+	0.804902i	$0.297782\pi$
$728$	6.00000	0.222375
$729$	−27.0000	−1.00000
$730$	−36.0000	−1.33242
$731$	−28.0000	−1.03562
$732$	−12.0000	−0.443533
$733$	−48.0000	−1.77292	−0.886460	−	0.462805i	$-0.846843\pi$
$733$	−48.0000	−1.77292	−0.886460	+	0.462805i	$0.846843\pi$
$734$	16.0000	0.590571
$735$	−27.0000	−0.995910
$736$	0	0
$737$	4.00000	0.147342
$738$	12.0000	0.441726
$739$	1.00000	0.0367856	0.0183928	−	0.999831i	$-0.494145\pi$
$739$	1.00000	0.0367856	0.0183928	+	0.999831i	$0.494145\pi$
$740$	24.0000	0.882258
$741$	72.0000	2.64499
$742$	2.00000	0.0734223
$743$	−28.0000	−1.02722	−0.513610	−	0.858024i	$-0.671692\pi$
$743$	−28.0000	−1.02722	−0.513610	+	0.858024i	$0.671692\pi$
$744$	9.00000	0.329956
$745$	−9.00000	−0.329734
$746$	1.00000	0.0366126
$747$	0	0
$748$	4.00000	0.146254
$749$	4.00000	0.146157
$750$	9.00000	0.328634
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	11.0000	0.401129
$753$	21.0000	0.765283
$754$	3.00000	0.109254
$755$	−30.0000	−1.09181
$756$	18.0000	0.654654
$757$	−28.0000	−1.01768	−0.508839	−	0.860862i	$-0.669925\pi$
$757$	−28.0000	−1.01768	−0.508839	+	0.860862i	$0.669925\pi$
$758$	28.0000	1.01701
$759$	0	0
$760$	−24.0000	−0.870572
$761$	34.0000	1.23250	0.616250	−	0.787551i	$-0.288651\pi$
$761$	34.0000	1.23250	0.616250	+	0.787551i	$0.288651\pi$
$762$	−24.0000	−0.869428
$763$	−2.00000	−0.0724049
$764$	0	0
$765$	72.0000	2.60317
$766$	34.0000	1.22847
$767$	−12.0000	−0.433295
$768$	−3.00000	−0.108253
$769$	48.0000	1.73092	0.865462	−	0.500974i	$-0.167025\pi$
$769$	48.0000	1.73092	0.865462	+	0.500974i	$0.167025\pi$
$770$	6.00000	0.216225
$771$	−63.0000	−2.26889
$772$	10.0000	0.359908
$773$	−6.00000	−0.215805	−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000	−0.215805	−0.107903	+	0.994161i	$0.534413\pi$
$774$	−42.0000	−1.50966
$775$	12.0000	0.431053
$776$	6.00000	0.215387
$777$	−48.0000	−1.72199
$778$	−16.0000	−0.573628
$779$	16.0000	0.573259
$780$	27.0000	0.966755
$781$	2.00000	0.0715656
$782$	0	0
$783$	9.00000	0.321634
$784$	−3.00000	−0.107143
$785$	−66.0000	−2.35564
$786$	36.0000	1.28408
$787$	46.0000	1.63972	0.819861	−	0.572562i	$-0.194050\pi$
$787$	46.0000	1.63972	0.819861	+	0.572562i	$0.194050\pi$
$788$	2.00000	0.0712470
$789$	51.0000	1.81565
$790$	−21.0000	−0.747146
$791$	−36.0000	−1.28001
$792$	6.00000	0.213201
$793$	12.0000	0.426132
$794$	−19.0000	−0.674285
$795$	9.00000	0.319197
$796$	−2.00000	−0.0708881
$797$	12.0000	0.425062	0.212531	−	0.977154i	$-0.431829\pi$
$797$	12.0000	0.425062	0.212531	+	0.977154i	$0.431829\pi$
$798$	48.0000	1.69918
$799$	−44.0000	−1.55661
$800$	−4.00000	−0.141421
$801$	−36.0000	−1.27200
$802$	5.00000	0.176556
$803$	12.0000	0.423471
$804$	12.0000	0.423207
$805$	0	0
$806$	−9.00000	−0.317011
$807$	−60.0000	−2.11210
$808$	−8.00000	−0.281439
$809$	24.0000	0.843795	0.421898	−	0.906644i	$-0.361364\pi$
$809$	24.0000	0.843795	0.421898	+	0.906644i	$0.361364\pi$
$810$	27.0000	0.948683
$811$	6.00000	0.210688	0.105344	−	0.994436i	$-0.466406\pi$
$811$	6.00000	0.210688	0.105344	+	0.994436i	$0.466406\pi$
$812$	2.00000	0.0701862
$813$	−39.0000	−1.36779
$814$	−8.00000	−0.280400
$815$	−57.0000	−1.99662
$816$	12.0000	0.420084
$817$	−56.0000	−1.95919
$818$	−22.0000	−0.769212
$819$	−36.0000	−1.25794
$820$	6.00000	0.209529
$821$	−37.0000	−1.29131	−0.645654	−	0.763630i	$-0.723415\pi$
$821$	−37.0000	−1.29131	−0.645654	+	0.763630i	$0.723415\pi$
$822$	−60.0000	−2.09274
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	6.00000	0.209020
$825$	12.0000	0.417786
$826$	−8.00000	−0.278356
$827$	23.0000	0.799788	0.399894	−	0.916561i	$-0.369047\pi$
$827$	23.0000	0.799788	0.399894	+	0.916561i	$0.369047\pi$
$828$	0	0
$829$	−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
$831$	−42.0000	−1.45696
$832$	3.00000	0.104006
$833$	12.0000	0.415775
$834$	0	0
$835$	66.0000	2.28402
$836$	8.00000	0.276686
$837$	−27.0000	−0.933257
$838$	14.0000	0.483622
$839$	−45.0000	−1.55357	−0.776786	−	0.629764i	$-0.783151\pi$
$839$	−45.0000	−1.55357	−0.776786	+	0.629764i	$0.783151\pi$
$840$	18.0000	0.621059
$841$	1.00000	0.0344828
$842$	28.0000	0.964944
$843$	39.0000	1.34323
$844$	−25.0000	−0.860535
$845$	12.0000	0.412813
$846$	−66.0000	−2.26913
$847$	20.0000	0.687208
$848$	1.00000	0.0343401
$849$	0	0
$850$	16.0000	0.548795
$851$	0	0
$852$	6.00000	0.205557
$853$	30.0000	1.02718	0.513590	−	0.858036i	$-0.328315\pi$
$853$	30.0000	1.02718	0.513590	+	0.858036i	$0.328315\pi$
$854$	8.00000	0.273754
$855$	144.000	4.92470
$856$	2.00000	0.0683586
$857$	−11.0000	−0.375753	−0.187876	−	0.982193i	$-0.560160\pi$
$857$	−11.0000	−0.375753	−0.187876	+	0.982193i	$0.560160\pi$
$858$	−9.00000	−0.307255
$859$	−51.0000	−1.74010	−0.870049	−	0.492966i	$-0.835913\pi$
$859$	−51.0000	−1.74010	−0.870049	+	0.492966i	$0.835913\pi$
$860$	−21.0000	−0.716094
$861$	−12.0000	−0.408959
$862$	20.0000	0.681203
$863$	6.00000	0.204242	0.102121	−	0.994772i	$-0.467437\pi$
$863$	6.00000	0.204242	0.102121	+	0.994772i	$0.467437\pi$
$864$	9.00000	0.306186
$865$	42.0000	1.42804
$866$	16.0000	0.543702
$867$	3.00000	0.101885
$868$	−6.00000	−0.203653
$869$	7.00000	0.237459
$870$	9.00000	0.305129
$871$	−12.0000	−0.406604
$872$	−1.00000	−0.0338643
$873$	−36.0000	−1.21842
$874$	0	0
$875$	−6.00000	−0.202837
$876$	36.0000	1.21633
$877$	−23.0000	−0.776655	−0.388327	−	0.921521i	$-0.626947\pi$
$877$	−23.0000	−0.776655	−0.388327	+	0.921521i	$0.626947\pi$
$878$	8.00000	0.269987
$879$	102.000	3.44037
$880$	3.00000	0.101130
$881$	42.0000	1.41502	0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000	1.41502	0.707508	+	0.706705i	$0.249819\pi$
$882$	18.0000	0.606092
$883$	−42.0000	−1.41341	−0.706706	−	0.707507i	$-0.749820\pi$
$883$	−42.0000	−1.41341	−0.706706	+	0.707507i	$0.749820\pi$
$884$	−12.0000	−0.403604
$885$	−36.0000	−1.21013
$886$	−12.0000	−0.403148
$887$	−1.00000	−0.0335767	−0.0167884	−	0.999859i	$-0.505344\pi$
$887$	−1.00000	−0.0335767	−0.0167884	+	0.999859i	$0.505344\pi$
$888$	−24.0000	−0.805387
$889$	16.0000	0.536623
$890$	−18.0000	−0.603361
$891$	−9.00000	−0.301511
$892$	26.0000	0.870544
$893$	−88.0000	−2.94481
$894$	9.00000	0.301005
$895$	42.0000	1.40391
$896$	2.00000	0.0668153
$897$	0	0
$898$	−22.0000	−0.734150
$899$	−3.00000	−0.100056
$900$	24.0000	0.800000
$901$	−4.00000	−0.133259
$902$	−2.00000	−0.0665927
$903$	42.0000	1.39767
$904$	−18.0000	−0.598671
$905$	39.0000	1.29640
$906$	30.0000	0.996683
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	−18.0000	−0.597351
$909$	48.0000	1.59206
$910$	−18.0000	−0.596694
$911$	−3.00000	−0.0993944	−0.0496972	−	0.998764i	$-0.515826\pi$
$911$	−3.00000	−0.0993944	−0.0496972	+	0.998764i	$0.515826\pi$
$912$	24.0000	0.794719
$913$	0	0
$914$	10.0000	0.330771
$915$	36.0000	1.19012
$916$	14.0000	0.462573
$917$	−24.0000	−0.792550
$918$	−36.0000	−1.18818
$919$	−6.00000	−0.197922	−0.0989609	−	0.995091i	$-0.531552\pi$
$919$	−6.00000	−0.197922	−0.0989609	+	0.995091i	$0.531552\pi$
$920$	0	0
$921$	87.0000	2.86675
$922$	30.0000	0.987997
$923$	−6.00000	−0.197492
$924$	−6.00000	−0.197386
$925$	−32.0000	−1.05215
$926$	16.0000	0.525793
$927$	−36.0000	−1.18240
$928$	1.00000	0.0328266
$929$	18.0000	0.590561	0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000	0.590561	0.295280	+	0.955411i	$0.404587\pi$
$930$	−27.0000	−0.885365
$931$	24.0000	0.786568
$932$	−25.0000	−0.818902
$933$	0	0
$934$	−23.0000	−0.752583
$935$	−12.0000	−0.392442
$936$	−18.0000	−0.588348
$937$	14.0000	0.457360	0.228680	−	0.973502i	$-0.426559\pi$
$937$	14.0000	0.457360	0.228680	+	0.973502i	$0.426559\pi$
$938$	−8.00000	−0.261209
$939$	−75.0000	−2.44753
$940$	−33.0000	−1.07634
$941$	−15.0000	−0.488986	−0.244493	−	0.969651i	$-0.578622\pi$
$941$	−15.0000	−0.488986	−0.244493	+	0.969651i	$0.578622\pi$
$942$	66.0000	2.15040
$943$	0	0
$944$	−4.00000	−0.130189
$945$	−54.0000	−1.75662
$946$	7.00000	0.227590
$947$	51.0000	1.65728	0.828639	−	0.559784i	$-0.189116\pi$
$947$	51.0000	1.65728	0.828639	+	0.559784i	$0.189116\pi$
$948$	21.0000	0.682048
$949$	−36.0000	−1.16861
$950$	32.0000	1.03822
$951$	0	0
$952$	−8.00000	−0.259281
$953$	15.0000	0.485898	0.242949	−	0.970039i	$-0.421885\pi$
$953$	15.0000	0.485898	0.242949	+	0.970039i	$0.421885\pi$
$954$	−6.00000	−0.194257
$955$	0	0
$956$	20.0000	0.646846
$957$	−3.00000	−0.0969762
$958$	11.0000	0.355394
$959$	40.0000	1.29167
$960$	9.00000	0.290474
$961$	−22.0000	−0.709677
$962$	24.0000	0.773791
$963$	−12.0000	−0.386695
$964$	17.0000	0.547533
$965$	−30.0000	−0.965734
$966$	0	0
$967$	59.0000	1.89731	0.948656	−	0.316310i	$-0.102444\pi$
$967$	59.0000	1.89731	0.948656	+	0.316310i	$0.102444\pi$
$968$	10.0000	0.321412
$969$	−96.0000	−3.08396
$970$	−18.0000	−0.577945
$971$	−52.0000	−1.66876	−0.834380	−	0.551190i	$-0.814174\pi$
$971$	−52.0000	−1.66876	−0.834380	+	0.551190i	$0.814174\pi$
$972$	0	0
$973$	0	0
$974$	−2.00000	−0.0640841
$975$	−36.0000	−1.15292
$976$	4.00000	0.128037
$977$	37.0000	1.18373	0.591867	−	0.806035i	$-0.298391\pi$
$977$	37.0000	1.18373	0.591867	+	0.806035i	$0.298391\pi$
$978$	57.0000	1.82266
$979$	6.00000	0.191761
$980$	9.00000	0.287494
$981$	6.00000	0.191565
$982$	−5.00000	−0.159556
$983$	39.0000	1.24391	0.621953	−	0.783054i	$-0.286339\pi$
$983$	39.0000	1.24391	0.621953	+	0.783054i	$0.286339\pi$
$984$	−6.00000	−0.191273
$985$	−6.00000	−0.191176
$986$	−4.00000	−0.127386
$987$	66.0000	2.10080
$988$	−24.0000	−0.763542
$989$	0	0
$990$	−18.0000	−0.572078
$991$	26.0000	0.825917	0.412959	−	0.910750i	$-0.364495\pi$
$991$	26.0000	0.825917	0.412959	+	0.910750i	$0.364495\pi$
$992$	−3.00000	−0.0952501
$993$	−9.00000	−0.285606
$994$	−4.00000	−0.126872
$995$	6.00000	0.190213
$996$	0	0
$997$	16.0000	0.506725	0.253363	−	0.967371i	$-0.418463\pi$
$997$	16.0000	0.506725	0.253363	+	0.967371i	$0.418463\pi$
$998$	8.00000	0.253236
$999$	72.0000	2.27798

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	58.2.a.a.1.1	✓	1
3.2	odd	2		522.2.a.k.1.1		1
4.3	odd	2		464.2.a.f.1.1		1
5.2	odd	4		1450.2.b.f.349.1		2
5.3	odd	4		1450.2.b.f.349.2		2
5.4	even	2		1450.2.a.i.1.1		1
7.6	odd	2		2842.2.a.d.1.1		1
8.3	odd	2		1856.2.a.b.1.1		1
8.5	even	2		1856.2.a.p.1.1		1
11.10	odd	2		7018.2.a.c.1.1		1
12.11	even	2		4176.2.a.bh.1.1		1
13.12	even	2		9802.2.a.d.1.1		1
29.12	odd	4		1682.2.b.e.1681.1		2
29.17	odd	4		1682.2.b.e.1681.2		2
29.28	even	2		1682.2.a.j.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
58.2.a.a.1.1	✓	1	1.1	even	1	trivial
464.2.a.f.1.1		1	4.3	odd	2
522.2.a.k.1.1		1	3.2	odd	2
1450.2.a.i.1.1		1	5.4	even	2
1450.2.b.f.349.1		2	5.2	odd	4
1450.2.b.f.349.2		2	5.3	odd	4
1682.2.a.j.1.1		1	29.28	even	2
1682.2.b.e.1681.1		2	29.12	odd	4
1682.2.b.e.1681.2		2	29.17	odd	4
1856.2.a.b.1.1		1	8.3	odd	2
1856.2.a.p.1.1		1	8.5	even	2
2842.2.a.d.1.1		1	7.6	odd	2
4176.2.a.bh.1.1		1	12.11	even	2
7018.2.a.c.1.1		1	11.10	odd	2
9802.2.a.d.1.1		1	13.12	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 58 = 2 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	58.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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