LMFDB - Embedded newform 1058.2.a.g.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1058,2,Mod(1,1058)] 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("1058.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1058, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1058 = 2 \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1058.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,2,-4,2,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$8.44817253385$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	1058.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -1.73205 q^{5} -2.00000 q^{6} +3.46410 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.73205 q^{10} -3.46410 q^{11} -2.00000 q^{12} -5.00000 q^{13} +3.46410 q^{14} +3.46410 q^{15} +1.00000 q^{16} +6.92820 q^{17} +1.00000 q^{18} -3.46410 q^{19} -1.73205 q^{20} -6.92820 q^{21} -3.46410 q^{22} -2.00000 q^{24} -2.00000 q^{25} -5.00000 q^{26} +4.00000 q^{27} +3.46410 q^{28} -3.00000 q^{29} +3.46410 q^{30} -8.00000 q^{31} +1.00000 q^{32} +6.92820 q^{33} +6.92820 q^{34} -6.00000 q^{35} +1.00000 q^{36} -3.46410 q^{38} +10.0000 q^{39} -1.73205 q^{40} -9.00000 q^{41} -6.92820 q^{42} -6.92820 q^{43} -3.46410 q^{44} -1.73205 q^{45} -6.00000 q^{47} -2.00000 q^{48} +5.00000 q^{49} -2.00000 q^{50} -13.8564 q^{51} -5.00000 q^{52} +1.73205 q^{53} +4.00000 q^{54} +6.00000 q^{55} +3.46410 q^{56} +6.92820 q^{57} -3.00000 q^{58} -6.00000 q^{59} +3.46410 q^{60} -5.19615 q^{61} -8.00000 q^{62} +3.46410 q^{63} +1.00000 q^{64} +8.66025 q^{65} +6.92820 q^{66} +6.92820 q^{67} +6.92820 q^{68} -6.00000 q^{70} +6.00000 q^{71} +1.00000 q^{72} -11.0000 q^{73} +4.00000 q^{75} -3.46410 q^{76} -12.0000 q^{77} +10.0000 q^{78} +6.92820 q^{79} -1.73205 q^{80} -11.0000 q^{81} -9.00000 q^{82} -6.92820 q^{84} -12.0000 q^{85} -6.92820 q^{86} +6.00000 q^{87} -3.46410 q^{88} +1.73205 q^{89} -1.73205 q^{90} -17.3205 q^{91} +16.0000 q^{93} -6.00000 q^{94} +6.00000 q^{95} -2.00000 q^{96} -12.1244 q^{97} +5.00000 q^{98} -3.46410 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 4 q^{3} + 2 q^{4} - 4 q^{6} + 2 q^{8} + 2 q^{9} - 4 q^{12} - 10 q^{13} + 2 q^{16} + 2 q^{18} - 4 q^{24} - 4 q^{25} - 10 q^{26} + 8 q^{27} - 6 q^{29} - 16 q^{31} + 2 q^{32} - 12 q^{35} + 2 q^{36}+ \cdots + 10 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 - 4 * q^3 + 2 * q^4 - 4 * q^6 + 2 * q^8 + 2 * q^9 - 4 * q^12 - 10 * q^13 + 2 * q^16 + 2 * q^18 - 4 * q^24 - 4 * q^25 - 10 * q^26 + 8 * q^27 - 6 * q^29 - 16 * q^31 + 2 * q^32 - 12 * q^35 + 2 * q^36 + 20 * q^39 - 18 * q^41 - 12 * q^47 - 4 * q^48 + 10 * q^49 - 4 * q^50 - 10 * q^52 + 8 * q^54 + 12 * q^55 - 6 * q^58 - 12 * q^59 - 16 * q^62 + 2 * q^64 - 12 * q^70 + 12 * q^71 + 2 * q^72 - 22 * q^73 + 8 * q^75 - 24 * q^77 + 20 * q^78 - 22 * q^81 - 18 * q^82 - 24 * q^85 + 12 * q^87 + 32 * q^93 - 12 * q^94 + 12 * q^95 - 4 * q^96 + 10 * q^98

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
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	1.00000	0.500000
$5$	−1.73205	−0.774597	−0.387298	−	0.921954i	$-0.626592\pi$
$5$	−1.73205	−0.774597	−0.387298	+	0.921954i	$0.626592\pi$
$6$	−2.00000	−0.816497
$7$	3.46410	1.30931	0.654654	−	0.755929i	$-0.272814\pi$
$7$	3.46410	1.30931	0.654654	+	0.755929i	$0.272814\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−1.73205	−0.547723
$11$	−3.46410	−1.04447	−0.522233	−	0.852803i	$-0.674901\pi$
$11$	−3.46410	−1.04447	−0.522233	+	0.852803i	$0.674901\pi$
$12$	−2.00000	−0.577350
$13$	−5.00000	−1.38675	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000	−1.38675	−0.693375	+	0.720577i	$0.743877\pi$
$14$	3.46410	0.925820
$15$	3.46410	0.894427
$16$	1.00000	0.250000
$17$	6.92820	1.68034	0.840168	−	0.542326i	$-0.182456\pi$
$17$	6.92820	1.68034	0.840168	+	0.542326i	$0.182456\pi$
$18$	1.00000	0.235702
$19$	−3.46410	−0.794719	−0.397360	−	0.917663i	$-0.630073\pi$
$19$	−3.46410	−0.794719	−0.397360	+	0.917663i	$0.630073\pi$
$20$	−1.73205	−0.387298
$21$	−6.92820	−1.51186
$22$	−3.46410	−0.738549
$23$	0	0
$23$	0	0
$24$	−2.00000	−0.408248
$25$	−2.00000	−0.400000
$26$	−5.00000	−0.980581
$27$	4.00000	0.769800
$28$	3.46410	0.654654
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	3.46410	0.632456
$31$	−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
$33$	6.92820	1.20605
$34$	6.92820	1.18818
$35$	−6.00000	−1.01419
$36$	1.00000	0.166667
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	−3.46410	−0.561951
$39$	10.0000	1.60128
$40$	−1.73205	−0.273861
$41$	−9.00000	−1.40556	−0.702782	−	0.711405i	$-0.748059\pi$
$41$	−9.00000	−1.40556	−0.702782	+	0.711405i	$0.748059\pi$
$42$	−6.92820	−1.06904
$43$	−6.92820	−1.05654	−0.528271	−	0.849076i	$-0.677159\pi$
$43$	−6.92820	−1.05654	−0.528271	+	0.849076i	$0.677159\pi$
$44$	−3.46410	−0.522233
$45$	−1.73205	−0.258199
$46$	0	0
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	−2.00000	−0.288675
$49$	5.00000	0.714286
$50$	−2.00000	−0.282843
$51$	−13.8564	−1.94029
$52$	−5.00000	−0.693375
$53$	1.73205	0.237915	0.118958	−	0.992899i	$-0.462045\pi$
$53$	1.73205	0.237915	0.118958	+	0.992899i	$0.462045\pi$
$54$	4.00000	0.544331
$55$	6.00000	0.809040
$56$	3.46410	0.462910
$57$	6.92820	0.917663
$58$	−3.00000	−0.393919
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	3.46410	0.447214
$61$	−5.19615	−0.665299	−0.332650	−	0.943051i	$-0.607943\pi$
$61$	−5.19615	−0.665299	−0.332650	+	0.943051i	$0.607943\pi$
$62$	−8.00000	−1.01600
$63$	3.46410	0.436436
$64$	1.00000	0.125000
$65$	8.66025	1.07417
$66$	6.92820	0.852803
$67$	6.92820	0.846415	0.423207	−	0.906033i	$-0.360904\pi$
$67$	6.92820	0.846415	0.423207	+	0.906033i	$0.360904\pi$
$68$	6.92820	0.840168
$69$	0	0
$70$	−6.00000	−0.717137
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	1.00000	0.117851
$73$	−11.0000	−1.28745	−0.643726	−	0.765256i	$-0.722612\pi$
$73$	−11.0000	−1.28745	−0.643726	+	0.765256i	$0.722612\pi$
$74$	0	0
$75$	4.00000	0.461880
$76$	−3.46410	−0.397360
$77$	−12.0000	−1.36753
$78$	10.0000	1.13228
$79$	6.92820	0.779484	0.389742	−	0.920924i	$-0.372564\pi$
$79$	6.92820	0.779484	0.389742	+	0.920924i	$0.372564\pi$
$80$	−1.73205	−0.193649
$81$	−11.0000	−1.22222
$82$	−9.00000	−0.993884
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	−6.92820	−0.755929
$85$	−12.0000	−1.30158
$86$	−6.92820	−0.747087
$87$	6.00000	0.643268
$88$	−3.46410	−0.369274
$89$	1.73205	0.183597	0.0917985	−	0.995778i	$-0.470738\pi$
$89$	1.73205	0.183597	0.0917985	+	0.995778i	$0.470738\pi$
$90$	−1.73205	−0.182574
$91$	−17.3205	−1.81568
$92$	0	0
$93$	16.0000	1.65912
$94$	−6.00000	−0.618853
$95$	6.00000	0.615587
$96$	−2.00000	−0.204124
$97$	−12.1244	−1.23104	−0.615521	−	0.788121i	$-0.711054\pi$
$97$	−12.1244	−1.23104	−0.615521	+	0.788121i	$0.711054\pi$
$98$	5.00000	0.505076
$99$	−3.46410	−0.348155
$100$	−2.00000	−0.200000
$101$	3.00000	0.298511	0.149256	−	0.988799i	$-0.452312\pi$
$101$	3.00000	0.298511	0.149256	+	0.988799i	$0.452312\pi$
$102$	−13.8564	−1.37199
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	−5.00000	−0.490290
$105$	12.0000	1.17108
$106$	1.73205	0.168232
$107$	3.46410	0.334887	0.167444	−	0.985882i	$-0.446449\pi$
$107$	3.46410	0.334887	0.167444	+	0.985882i	$0.446449\pi$
$108$	4.00000	0.384900
$109$	−15.5885	−1.49310	−0.746552	−	0.665327i	$-0.768292\pi$
$109$	−15.5885	−1.49310	−0.746552	+	0.665327i	$0.768292\pi$
$110$	6.00000	0.572078
$111$	0	0
$112$	3.46410	0.327327
$113$	5.19615	0.488813	0.244406	−	0.969673i	$-0.421407\pi$
$113$	5.19615	0.488813	0.244406	+	0.969673i	$0.421407\pi$
$114$	6.92820	0.648886
$115$	0	0
$116$	−3.00000	−0.278543
$117$	−5.00000	−0.462250
$118$	−6.00000	−0.552345
$119$	24.0000	2.20008
$120$	3.46410	0.316228
$121$	1.00000	0.0909091
$122$	−5.19615	−0.470438
$123$	18.0000	1.62301
$124$	−8.00000	−0.718421
$125$	12.1244	1.08444
$126$	3.46410	0.308607
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	1.00000	0.0883883
$129$	13.8564	1.21999
$130$	8.66025	0.759555
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	6.92820	0.603023
$133$	−12.0000	−1.04053
$134$	6.92820	0.598506
$135$	−6.92820	−0.596285
$136$	6.92820	0.594089
$137$	15.5885	1.33181	0.665906	−	0.746036i	$-0.268045\pi$
$137$	15.5885	1.33181	0.665906	+	0.746036i	$0.268045\pi$
$138$	0	0
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	−6.00000	−0.507093
$141$	12.0000	1.01058
$142$	6.00000	0.503509
$143$	17.3205	1.44841
$144$	1.00000	0.0833333
$145$	5.19615	0.431517
$146$	−11.0000	−0.910366
$147$	−10.0000	−0.824786
$148$	0	0
$149$	−5.19615	−0.425685	−0.212843	−	0.977086i	$-0.568272\pi$
$149$	−5.19615	−0.425685	−0.212843	+	0.977086i	$0.568272\pi$
$150$	4.00000	0.326599
$151$	14.0000	1.13930	0.569652	−	0.821886i	$-0.307078\pi$
$151$	14.0000	1.13930	0.569652	+	0.821886i	$0.307078\pi$
$152$	−3.46410	−0.280976
$153$	6.92820	0.560112
$154$	−12.0000	−0.966988
$155$	13.8564	1.11297
$156$	10.0000	0.800641
$157$	5.19615	0.414698	0.207349	−	0.978267i	$-0.433516\pi$
$157$	5.19615	0.414698	0.207349	+	0.978267i	$0.433516\pi$
$158$	6.92820	0.551178
$159$	−3.46410	−0.274721
$160$	−1.73205	−0.136931
$161$	0	0
$162$	−11.0000	−0.864242
$163$	−2.00000	−0.156652	−0.0783260	−	0.996928i	$-0.524958\pi$
$163$	−2.00000	−0.156652	−0.0783260	+	0.996928i	$0.524958\pi$
$164$	−9.00000	−0.702782
$165$	−12.0000	−0.934199
$166$	0	0
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	−6.92820	−0.534522
$169$	12.0000	0.923077
$170$	−12.0000	−0.920358
$171$	−3.46410	−0.264906
$172$	−6.92820	−0.528271
$173$	21.0000	1.59660	0.798300	−	0.602260i	$-0.205733\pi$
$173$	21.0000	1.59660	0.798300	+	0.602260i	$0.205733\pi$
$174$	6.00000	0.454859
$175$	−6.92820	−0.523723
$176$	−3.46410	−0.261116
$177$	12.0000	0.901975
$178$	1.73205	0.129823
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	−1.73205	−0.129099
$181$	−13.8564	−1.02994	−0.514969	−	0.857209i	$-0.672197\pi$
$181$	−13.8564	−1.02994	−0.514969	+	0.857209i	$0.672197\pi$
$182$	−17.3205	−1.28388
$183$	10.3923	0.768221
$184$	0	0
$185$	0	0
$186$	16.0000	1.17318
$187$	−24.0000	−1.75505
$188$	−6.00000	−0.437595
$189$	13.8564	1.00791
$190$	6.00000	0.435286
$191$	−6.92820	−0.501307	−0.250654	−	0.968077i	$-0.580646\pi$
$191$	−6.92820	−0.501307	−0.250654	+	0.968077i	$0.580646\pi$
$192$	−2.00000	−0.144338
$193$	−19.0000	−1.36765	−0.683825	−	0.729646i	$-0.739685\pi$
$193$	−19.0000	−1.36765	−0.683825	+	0.729646i	$0.739685\pi$
$194$	−12.1244	−0.870478
$195$	−17.3205	−1.24035
$196$	5.00000	0.357143
$197$	−15.0000	−1.06871	−0.534353	−	0.845262i	$-0.679445\pi$
$197$	−15.0000	−1.06871	−0.534353	+	0.845262i	$0.679445\pi$
$198$	−3.46410	−0.246183
$199$	−3.46410	−0.245564	−0.122782	−	0.992434i	$-0.539182\pi$
$199$	−3.46410	−0.245564	−0.122782	+	0.992434i	$0.539182\pi$
$200$	−2.00000	−0.141421
$201$	−13.8564	−0.977356
$202$	3.00000	0.211079
$203$	−10.3923	−0.729397
$204$	−13.8564	−0.970143
$205$	15.5885	1.08875
$206$	0	0
$207$	0	0
$208$	−5.00000	−0.346688
$209$	12.0000	0.830057
$210$	12.0000	0.828079
$211$	10.0000	0.688428	0.344214	−	0.938891i	$-0.388145\pi$
$211$	10.0000	0.688428	0.344214	+	0.938891i	$0.388145\pi$
$212$	1.73205	0.118958
$213$	−12.0000	−0.822226
$214$	3.46410	0.236801
$215$	12.0000	0.818393
$216$	4.00000	0.272166
$217$	−27.7128	−1.88127
$218$	−15.5885	−1.05578
$219$	22.0000	1.48662
$220$	6.00000	0.404520
$221$	−34.6410	−2.33021
$222$	0	0
$223$	−10.0000	−0.669650	−0.334825	−	0.942280i	$-0.608677\pi$
$223$	−10.0000	−0.669650	−0.334825	+	0.942280i	$0.608677\pi$
$224$	3.46410	0.231455
$225$	−2.00000	−0.133333
$226$	5.19615	0.345643
$227$	6.92820	0.459841	0.229920	−	0.973209i	$-0.426153\pi$
$227$	6.92820	0.459841	0.229920	+	0.973209i	$0.426153\pi$
$228$	6.92820	0.458831
$229$	−13.8564	−0.915657	−0.457829	−	0.889041i	$-0.651373\pi$
$229$	−13.8564	−0.915657	−0.457829	+	0.889041i	$0.651373\pi$
$230$	0	0
$231$	24.0000	1.57908
$232$	−3.00000	−0.196960
$233$	15.0000	0.982683	0.491341	−	0.870967i	$-0.336507\pi$
$233$	15.0000	0.982683	0.491341	+	0.870967i	$0.336507\pi$
$234$	−5.00000	−0.326860
$235$	10.3923	0.677919
$236$	−6.00000	−0.390567
$237$	−13.8564	−0.900070
$238$	24.0000	1.55569
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	3.46410	0.223607
$241$	25.9808	1.67357	0.836784	−	0.547533i	$-0.184433\pi$
$241$	25.9808	1.67357	0.836784	+	0.547533i	$0.184433\pi$
$242$	1.00000	0.0642824
$243$	10.0000	0.641500
$244$	−5.19615	−0.332650
$245$	−8.66025	−0.553283
$246$	18.0000	1.14764
$247$	17.3205	1.10208
$248$	−8.00000	−0.508001
$249$	0	0
$250$	12.1244	0.766812
$251$	3.46410	0.218652	0.109326	−	0.994006i	$-0.465131\pi$
$251$	3.46410	0.218652	0.109326	+	0.994006i	$0.465131\pi$
$252$	3.46410	0.218218
$253$	0	0
$254$	16.0000	1.00393
$255$	24.0000	1.50294
$256$	1.00000	0.0625000
$257$	−21.0000	−1.30994	−0.654972	−	0.755653i	$-0.727320\pi$
$257$	−21.0000	−1.30994	−0.654972	+	0.755653i	$0.727320\pi$
$258$	13.8564	0.862662
$259$	0	0
$260$	8.66025	0.537086
$261$	−3.00000	−0.185695
$262$	6.00000	0.370681
$263$	−17.3205	−1.06803	−0.534014	−	0.845476i	$-0.679317\pi$
$263$	−17.3205	−1.06803	−0.534014	+	0.845476i	$0.679317\pi$
$264$	6.92820	0.426401
$265$	−3.00000	−0.184289
$266$	−12.0000	−0.735767
$267$	−3.46410	−0.212000
$268$	6.92820	0.423207
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	−6.92820	−0.421637
$271$	−14.0000	−0.850439	−0.425220	−	0.905090i	$-0.639803\pi$
$271$	−14.0000	−0.850439	−0.425220	+	0.905090i	$0.639803\pi$
$272$	6.92820	0.420084
$273$	34.6410	2.09657
$274$	15.5885	0.941733
$275$	6.92820	0.417786
$276$	0	0
$277$	−26.0000	−1.56219	−0.781094	−	0.624413i	$-0.785338\pi$
$277$	−26.0000	−1.56219	−0.781094	+	0.624413i	$0.785338\pi$
$278$	8.00000	0.479808
$279$	−8.00000	−0.478947
$280$	−6.00000	−0.358569
$281$	−6.92820	−0.413302	−0.206651	−	0.978415i	$-0.566256\pi$
$281$	−6.92820	−0.413302	−0.206651	+	0.978415i	$0.566256\pi$
$282$	12.0000	0.714590
$283$	27.7128	1.64736	0.823678	−	0.567058i	$-0.191918\pi$
$283$	27.7128	1.64736	0.823678	+	0.567058i	$0.191918\pi$
$284$	6.00000	0.356034
$285$	−12.0000	−0.710819
$286$	17.3205	1.02418
$287$	−31.1769	−1.84032
$288$	1.00000	0.0589256
$289$	31.0000	1.82353
$290$	5.19615	0.305129
$291$	24.2487	1.42148
$292$	−11.0000	−0.643726
$293$	−32.9090	−1.92256	−0.961281	−	0.275570i	$-0.911134\pi$
$293$	−32.9090	−1.92256	−0.961281	+	0.275570i	$0.911134\pi$
$294$	−10.0000	−0.583212
$295$	10.3923	0.605063
$296$	0	0
$297$	−13.8564	−0.804030
$298$	−5.19615	−0.301005
$299$	0	0
$300$	4.00000	0.230940
$301$	−24.0000	−1.38334
$302$	14.0000	0.805609
$303$	−6.00000	−0.344691
$304$	−3.46410	−0.198680
$305$	9.00000	0.515339
$306$	6.92820	0.396059
$307$	−26.0000	−1.48390	−0.741949	−	0.670456i	$-0.766098\pi$
$307$	−26.0000	−1.48390	−0.741949	+	0.670456i	$0.766098\pi$
$308$	−12.0000	−0.683763
$309$	0	0
$310$	13.8564	0.786991
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	10.0000	0.566139
$313$	−25.9808	−1.46852	−0.734260	−	0.678869i	$-0.762471\pi$
$313$	−25.9808	−1.46852	−0.734260	+	0.678869i	$0.762471\pi$
$314$	5.19615	0.293236
$315$	−6.00000	−0.338062
$316$	6.92820	0.389742
$317$	21.0000	1.17948	0.589739	−	0.807594i	$-0.299231\pi$
$317$	21.0000	1.17948	0.589739	+	0.807594i	$0.299231\pi$
$318$	−3.46410	−0.194257
$319$	10.3923	0.581857
$320$	−1.73205	−0.0968246
$321$	−6.92820	−0.386695
$322$	0	0
$323$	−24.0000	−1.33540
$324$	−11.0000	−0.611111
$325$	10.0000	0.554700
$326$	−2.00000	−0.110770
$327$	31.1769	1.72409
$328$	−9.00000	−0.496942
$329$	−20.7846	−1.14589
$330$	−12.0000	−0.660578
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	0	0
$333$	0	0
$334$	12.0000	0.656611
$335$	−12.0000	−0.655630
$336$	−6.92820	−0.377964
$337$	−1.73205	−0.0943508	−0.0471754	−	0.998887i	$-0.515022\pi$
$337$	−1.73205	−0.0943508	−0.0471754	+	0.998887i	$0.515022\pi$
$338$	12.0000	0.652714
$339$	−10.3923	−0.564433
$340$	−12.0000	−0.650791
$341$	27.7128	1.50073
$342$	−3.46410	−0.187317
$343$	−6.92820	−0.374088
$344$	−6.92820	−0.373544
$345$	0	0
$346$	21.0000	1.12897
$347$	−18.0000	−0.966291	−0.483145	−	0.875540i	$-0.660506\pi$
$347$	−18.0000	−0.966291	−0.483145	+	0.875540i	$0.660506\pi$
$348$	6.00000	0.321634
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	−6.92820	−0.370328
$351$	−20.0000	−1.06752
$352$	−3.46410	−0.184637
$353$	−3.00000	−0.159674	−0.0798369	−	0.996808i	$-0.525440\pi$
$353$	−3.00000	−0.159674	−0.0798369	+	0.996808i	$0.525440\pi$
$354$	12.0000	0.637793
$355$	−10.3923	−0.551566
$356$	1.73205	0.0917985
$357$	−48.0000	−2.54043
$358$	12.0000	0.634220
$359$	24.2487	1.27980	0.639899	−	0.768459i	$-0.278976\pi$
$359$	24.2487	1.27980	0.639899	+	0.768459i	$0.278976\pi$
$360$	−1.73205	−0.0912871
$361$	−7.00000	−0.368421
$362$	−13.8564	−0.728277
$363$	−2.00000	−0.104973
$364$	−17.3205	−0.907841
$365$	19.0526	0.997257
$366$	10.3923	0.543214
$367$	24.2487	1.26577	0.632886	−	0.774245i	$-0.281870\pi$
$367$	24.2487	1.26577	0.632886	+	0.774245i	$0.281870\pi$
$368$	0	0
$369$	−9.00000	−0.468521
$370$	0	0
$371$	6.00000	0.311504
$372$	16.0000	0.829561
$373$	0	0		−	1.00000i	$-0.5\pi$
$373$	0	0			1.00000i	$0.5\pi$
$374$	−24.0000	−1.24101
$375$	−24.2487	−1.25220
$376$	−6.00000	−0.309426
$377$	15.0000	0.772539
$378$	13.8564	0.712697
$379$	31.1769	1.60145	0.800725	−	0.599032i	$-0.204448\pi$
$379$	31.1769	1.60145	0.800725	+	0.599032i	$0.204448\pi$
$380$	6.00000	0.307794
$381$	−32.0000	−1.63941
$382$	−6.92820	−0.354478
$383$	−6.92820	−0.354015	−0.177007	−	0.984210i	$-0.556642\pi$
$383$	−6.92820	−0.354015	−0.177007	+	0.984210i	$0.556642\pi$
$384$	−2.00000	−0.102062
$385$	20.7846	1.05928
$386$	−19.0000	−0.967075
$387$	−6.92820	−0.352180
$388$	−12.1244	−0.615521
$389$	13.8564	0.702548	0.351274	−	0.936273i	$-0.385749\pi$
$389$	13.8564	0.702548	0.351274	+	0.936273i	$0.385749\pi$
$390$	−17.3205	−0.877058
$391$	0	0
$392$	5.00000	0.252538
$393$	−12.0000	−0.605320
$394$	−15.0000	−0.755689
$395$	−12.0000	−0.603786
$396$	−3.46410	−0.174078
$397$	−1.00000	−0.0501886	−0.0250943	−	0.999685i	$-0.507989\pi$
$397$	−1.00000	−0.0501886	−0.0250943	+	0.999685i	$0.507989\pi$
$398$	−3.46410	−0.173640
$399$	24.0000	1.20150
$400$	−2.00000	−0.100000
$401$	29.4449	1.47041	0.735203	−	0.677847i	$-0.237087\pi$
$401$	29.4449	1.47041	0.735203	+	0.677847i	$0.237087\pi$
$402$	−13.8564	−0.691095
$403$	40.0000	1.99254
$404$	3.00000	0.149256
$405$	19.0526	0.946729
$406$	−10.3923	−0.515761
$407$	0	0
$408$	−13.8564	−0.685994
$409$	2.00000	0.0988936	0.0494468	−	0.998777i	$-0.484254\pi$
$409$	2.00000	0.0988936	0.0494468	+	0.998777i	$0.484254\pi$
$410$	15.5885	0.769859
$411$	−31.1769	−1.53784
$412$	0	0
$413$	−20.7846	−1.02274
$414$	0	0
$415$	0	0
$416$	−5.00000	−0.245145
$417$	−16.0000	−0.783523
$418$	12.0000	0.586939
$419$	−13.8564	−0.676930	−0.338465	−	0.940979i	$-0.609908\pi$
$419$	−13.8564	−0.676930	−0.338465	+	0.940979i	$0.609908\pi$
$420$	12.0000	0.585540
$421$	6.92820	0.337660	0.168830	−	0.985645i	$-0.446001\pi$
$421$	6.92820	0.337660	0.168830	+	0.985645i	$0.446001\pi$
$422$	10.0000	0.486792
$423$	−6.00000	−0.291730
$424$	1.73205	0.0841158
$425$	−13.8564	−0.672134
$426$	−12.0000	−0.581402
$427$	−18.0000	−0.871081
$428$	3.46410	0.167444
$429$	−34.6410	−1.67248
$430$	12.0000	0.578691
$431$	24.2487	1.16802	0.584010	−	0.811747i	$-0.301483\pi$
$431$	24.2487	1.16802	0.584010	+	0.811747i	$0.301483\pi$
$432$	4.00000	0.192450
$433$	−29.4449	−1.41503	−0.707515	−	0.706698i	$-0.750184\pi$
$433$	−29.4449	−1.41503	−0.707515	+	0.706698i	$0.750184\pi$
$434$	−27.7128	−1.33026
$435$	−10.3923	−0.498273
$436$	−15.5885	−0.746552
$437$	0	0
$438$	22.0000	1.05120
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	6.00000	0.286039
$441$	5.00000	0.238095
$442$	−34.6410	−1.64771
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	0	0
$445$	−3.00000	−0.142214
$446$	−10.0000	−0.473514
$447$	10.3923	0.491539
$448$	3.46410	0.163663
$449$	−33.0000	−1.55737	−0.778683	−	0.627417i	$-0.784112\pi$
$449$	−33.0000	−1.55737	−0.778683	+	0.627417i	$0.784112\pi$
$450$	−2.00000	−0.0942809
$451$	31.1769	1.46806
$452$	5.19615	0.244406
$453$	−28.0000	−1.31555
$454$	6.92820	0.325157
$455$	30.0000	1.40642
$456$	6.92820	0.324443
$457$	12.1244	0.567153	0.283577	−	0.958950i	$-0.408479\pi$
$457$	12.1244	0.567153	0.283577	+	0.958950i	$0.408479\pi$
$458$	−13.8564	−0.647467
$459$	27.7128	1.29352
$460$	0	0
$461$	−27.0000	−1.25752	−0.628758	−	0.777601i	$-0.716436\pi$
$461$	−27.0000	−1.25752	−0.628758	+	0.777601i	$0.716436\pi$
$462$	24.0000	1.11658
$463$	−28.0000	−1.30127	−0.650635	−	0.759390i	$-0.725497\pi$
$463$	−28.0000	−1.30127	−0.650635	+	0.759390i	$0.725497\pi$
$464$	−3.00000	−0.139272
$465$	−27.7128	−1.28515
$466$	15.0000	0.694862
$467$	20.7846	0.961797	0.480899	−	0.876776i	$-0.340311\pi$
$467$	20.7846	0.961797	0.480899	+	0.876776i	$0.340311\pi$
$468$	−5.00000	−0.231125
$469$	24.0000	1.10822
$470$	10.3923	0.479361
$471$	−10.3923	−0.478852
$472$	−6.00000	−0.276172
$473$	24.0000	1.10352
$474$	−13.8564	−0.636446
$475$	6.92820	0.317888
$476$	24.0000	1.10004
$477$	1.73205	0.0793052
$478$	12.0000	0.548867
$479$	−34.6410	−1.58279	−0.791394	−	0.611306i	$-0.790644\pi$
$479$	−34.6410	−1.58279	−0.791394	+	0.611306i	$0.790644\pi$
$480$	3.46410	0.158114
$481$	0	0
$482$	25.9808	1.18339
$483$	0	0
$484$	1.00000	0.0454545
$485$	21.0000	0.953561
$486$	10.0000	0.453609
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	−5.19615	−0.235219
$489$	4.00000	0.180886
$490$	−8.66025	−0.391230
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	18.0000	0.811503
$493$	−20.7846	−0.936092
$494$	17.3205	0.779287
$495$	6.00000	0.269680
$496$	−8.00000	−0.359211
$497$	20.7846	0.932317
$498$	0	0
$499$	−32.0000	−1.43252	−0.716258	−	0.697835i	$-0.754147\pi$
$499$	−32.0000	−1.43252	−0.716258	+	0.697835i	$0.754147\pi$
$500$	12.1244	0.542218
$501$	−24.0000	−1.07224
$502$	3.46410	0.154610
$503$	−17.3205	−0.772283	−0.386142	−	0.922440i	$-0.626192\pi$
$503$	−17.3205	−0.772283	−0.386142	+	0.922440i	$0.626192\pi$
$504$	3.46410	0.154303
$505$	−5.19615	−0.231226
$506$	0	0
$507$	−24.0000	−1.06588
$508$	16.0000	0.709885
$509$	−15.0000	−0.664863	−0.332432	−	0.943127i	$-0.607869\pi$
$509$	−15.0000	−0.664863	−0.332432	+	0.943127i	$0.607869\pi$
$510$	24.0000	1.06274
$511$	−38.1051	−1.68567
$512$	1.00000	0.0441942
$513$	−13.8564	−0.611775
$514$	−21.0000	−0.926270
$515$	0	0
$516$	13.8564	0.609994
$517$	20.7846	0.914106
$518$	0	0
$519$	−42.0000	−1.84360
$520$	8.66025	0.379777
$521$	20.7846	0.910590	0.455295	−	0.890341i	$-0.349534\pi$
$521$	20.7846	0.910590	0.455295	+	0.890341i	$0.349534\pi$
$522$	−3.00000	−0.131306
$523$	13.8564	0.605898	0.302949	−	0.953007i	$-0.402029\pi$
$523$	13.8564	0.605898	0.302949	+	0.953007i	$0.402029\pi$
$524$	6.00000	0.262111
$525$	13.8564	0.604743
$526$	−17.3205	−0.755210
$527$	−55.4256	−2.41438
$528$	6.92820	0.301511
$529$	0	0
$530$	−3.00000	−0.130312
$531$	−6.00000	−0.260378
$532$	−12.0000	−0.520266
$533$	45.0000	1.94917
$534$	−3.46410	−0.149906
$535$	−6.00000	−0.259403
$536$	6.92820	0.299253
$537$	−24.0000	−1.03568
$538$	6.00000	0.258678
$539$	−17.3205	−0.746047
$540$	−6.92820	−0.298142
$541$	17.0000	0.730887	0.365444	−	0.930834i	$-0.380917\pi$
$541$	17.0000	0.730887	0.365444	+	0.930834i	$0.380917\pi$
$542$	−14.0000	−0.601351
$543$	27.7128	1.18927
$544$	6.92820	0.297044
$545$	27.0000	1.15655
$546$	34.6410	1.48250
$547$	40.0000	1.71028	0.855138	−	0.518400i	$-0.173472\pi$
$547$	40.0000	1.71028	0.855138	+	0.518400i	$0.173472\pi$
$548$	15.5885	0.665906
$549$	−5.19615	−0.221766
$550$	6.92820	0.295420
$551$	10.3923	0.442727
$552$	0	0
$553$	24.0000	1.02058
$554$	−26.0000	−1.10463
$555$	0	0
$556$	8.00000	0.339276
$557$	36.3731	1.54118	0.770588	−	0.637333i	$-0.219963\pi$
$557$	36.3731	1.54118	0.770588	+	0.637333i	$0.219963\pi$
$558$	−8.00000	−0.338667
$559$	34.6410	1.46516
$560$	−6.00000	−0.253546
$561$	48.0000	2.02656
$562$	−6.92820	−0.292249
$563$	34.6410	1.45994	0.729972	−	0.683477i	$-0.239533\pi$
$563$	34.6410	1.45994	0.729972	+	0.683477i	$0.239533\pi$
$564$	12.0000	0.505291
$565$	−9.00000	−0.378633
$566$	27.7128	1.16486
$567$	−38.1051	−1.60026
$568$	6.00000	0.251754
$569$	−29.4449	−1.23439	−0.617196	−	0.786809i	$-0.711732\pi$
$569$	−29.4449	−1.23439	−0.617196	+	0.786809i	$0.711732\pi$
$570$	−12.0000	−0.502625
$571$	−31.1769	−1.30471	−0.652357	−	0.757912i	$-0.726220\pi$
$571$	−31.1769	−1.30471	−0.652357	+	0.757912i	$0.726220\pi$
$572$	17.3205	0.724207
$573$	13.8564	0.578860
$574$	−31.1769	−1.30130
$575$	0	0
$576$	1.00000	0.0416667
$577$	−34.0000	−1.41544	−0.707719	−	0.706494i	$-0.750276\pi$
$577$	−34.0000	−1.41544	−0.707719	+	0.706494i	$0.750276\pi$
$578$	31.0000	1.28943
$579$	38.0000	1.57923
$580$	5.19615	0.215758
$581$	0	0
$582$	24.2487	1.00514
$583$	−6.00000	−0.248495
$584$	−11.0000	−0.455183
$585$	8.66025	0.358057
$586$	−32.9090	−1.35946
$587$	−36.0000	−1.48588	−0.742940	−	0.669359i	$-0.766569\pi$
$587$	−36.0000	−1.48588	−0.742940	+	0.669359i	$0.766569\pi$
$588$	−10.0000	−0.412393
$589$	27.7128	1.14189
$590$	10.3923	0.427844
$591$	30.0000	1.23404
$592$	0	0
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	−13.8564	−0.568535
$595$	−41.5692	−1.70417
$596$	−5.19615	−0.212843
$597$	6.92820	0.283552
$598$	0	0
$599$	30.0000	1.22577	0.612883	−	0.790173i	$-0.290010\pi$
$599$	30.0000	1.22577	0.612883	+	0.790173i	$0.290010\pi$
$600$	4.00000	0.163299
$601$	5.00000	0.203954	0.101977	−	0.994787i	$-0.467483\pi$
$601$	5.00000	0.203954	0.101977	+	0.994787i	$0.467483\pi$
$602$	−24.0000	−0.978167
$603$	6.92820	0.282138
$604$	14.0000	0.569652
$605$	−1.73205	−0.0704179
$606$	−6.00000	−0.243733
$607$	−22.0000	−0.892952	−0.446476	−	0.894795i	$-0.647321\pi$
$607$	−22.0000	−0.892952	−0.446476	+	0.894795i	$0.647321\pi$
$608$	−3.46410	−0.140488
$609$	20.7846	0.842235
$610$	9.00000	0.364399
$611$	30.0000	1.21367
$612$	6.92820	0.280056
$613$	15.5885	0.629612	0.314806	−	0.949156i	$-0.398061\pi$
$613$	15.5885	0.629612	0.314806	+	0.949156i	$0.398061\pi$
$614$	−26.0000	−1.04927
$615$	−31.1769	−1.25717
$616$	−12.0000	−0.483494
$617$	−41.5692	−1.67351	−0.836757	−	0.547575i	$-0.815551\pi$
$617$	−41.5692	−1.67351	−0.836757	+	0.547575i	$0.815551\pi$
$618$	0	0
$619$	−13.8564	−0.556936	−0.278468	−	0.960446i	$-0.589827\pi$
$619$	−13.8564	−0.556936	−0.278468	+	0.960446i	$0.589827\pi$
$620$	13.8564	0.556487
$621$	0	0
$622$	−18.0000	−0.721734
$623$	6.00000	0.240385
$624$	10.0000	0.400320
$625$	−11.0000	−0.440000
$626$	−25.9808	−1.03840
$627$	−24.0000	−0.958468
$628$	5.19615	0.207349
$629$	0	0
$630$	−6.00000	−0.239046
$631$	13.8564	0.551615	0.275807	−	0.961213i	$-0.411055\pi$
$631$	13.8564	0.551615	0.275807	+	0.961213i	$0.411055\pi$
$632$	6.92820	0.275589
$633$	−20.0000	−0.794929
$634$	21.0000	0.834017
$635$	−27.7128	−1.09975
$636$	−3.46410	−0.137361
$637$	−25.0000	−0.990536
$638$	10.3923	0.411435
$639$	6.00000	0.237356
$640$	−1.73205	−0.0684653
$641$	5.19615	0.205236	0.102618	−	0.994721i	$-0.467278\pi$
$641$	5.19615	0.205236	0.102618	+	0.994721i	$0.467278\pi$
$642$	−6.92820	−0.273434
$643$	−38.1051	−1.50272	−0.751360	−	0.659893i	$-0.770602\pi$
$643$	−38.1051	−1.50272	−0.751360	+	0.659893i	$0.770602\pi$
$644$	0	0
$645$	−24.0000	−0.944999
$646$	−24.0000	−0.944267
$647$	−6.00000	−0.235884	−0.117942	−	0.993020i	$-0.537630\pi$
$647$	−6.00000	−0.235884	−0.117942	+	0.993020i	$0.537630\pi$
$648$	−11.0000	−0.432121
$649$	20.7846	0.815867
$650$	10.0000	0.392232
$651$	55.4256	2.17230
$652$	−2.00000	−0.0783260
$653$	27.0000	1.05659	0.528296	−	0.849060i	$-0.322831\pi$
$653$	27.0000	1.05659	0.528296	+	0.849060i	$0.322831\pi$
$654$	31.1769	1.21911
$655$	−10.3923	−0.406061
$656$	−9.00000	−0.351391
$657$	−11.0000	−0.429151
$658$	−20.7846	−0.810268
$659$	−27.7128	−1.07954	−0.539769	−	0.841813i	$-0.681488\pi$
$659$	−27.7128	−1.07954	−0.539769	+	0.841813i	$0.681488\pi$
$660$	−12.0000	−0.467099
$661$	29.4449	1.14527	0.572636	−	0.819810i	$-0.305921\pi$
$661$	29.4449	1.14527	0.572636	+	0.819810i	$0.305921\pi$
$662$	−8.00000	−0.310929
$663$	69.2820	2.69069
$664$	0	0
$665$	20.7846	0.805993
$666$	0	0
$667$	0	0
$668$	12.0000	0.464294
$669$	20.0000	0.773245
$670$	−12.0000	−0.463600
$671$	18.0000	0.694882
$672$	−6.92820	−0.267261
$673$	−26.0000	−1.00223	−0.501113	−	0.865382i	$-0.667076\pi$
$673$	−26.0000	−1.00223	−0.501113	+	0.865382i	$0.667076\pi$
$674$	−1.73205	−0.0667161
$675$	−8.00000	−0.307920
$676$	12.0000	0.461538
$677$	43.3013	1.66420	0.832102	−	0.554623i	$-0.187138\pi$
$677$	43.3013	1.66420	0.832102	+	0.554623i	$0.187138\pi$
$678$	−10.3923	−0.399114
$679$	−42.0000	−1.61181
$680$	−12.0000	−0.460179
$681$	−13.8564	−0.530979
$682$	27.7128	1.06118
$683$	24.0000	0.918334	0.459167	−	0.888350i	$-0.348148\pi$
$683$	24.0000	0.918334	0.459167	+	0.888350i	$0.348148\pi$
$684$	−3.46410	−0.132453
$685$	−27.0000	−1.03162
$686$	−6.92820	−0.264520
$687$	27.7128	1.05731
$688$	−6.92820	−0.264135
$689$	−8.66025	−0.329929
$690$	0	0
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	21.0000	0.798300
$693$	−12.0000	−0.455842
$694$	−18.0000	−0.683271
$695$	−13.8564	−0.525603
$696$	6.00000	0.227429
$697$	−62.3538	−2.36182
$698$	−10.0000	−0.378506
$699$	−30.0000	−1.13470
$700$	−6.92820	−0.261861
$701$	−13.8564	−0.523349	−0.261675	−	0.965156i	$-0.584275\pi$
$701$	−13.8564	−0.523349	−0.261675	+	0.965156i	$0.584275\pi$
$702$	−20.0000	−0.754851
$703$	0	0
$704$	−3.46410	−0.130558
$705$	−20.7846	−0.782794
$706$	−3.00000	−0.112906
$707$	10.3923	0.390843
$708$	12.0000	0.450988
$709$	1.73205	0.0650485	0.0325243	−	0.999471i	$-0.489645\pi$
$709$	1.73205	0.0650485	0.0325243	+	0.999471i	$0.489645\pi$
$710$	−10.3923	−0.390016
$711$	6.92820	0.259828
$712$	1.73205	0.0649113
$713$	0	0
$714$	−48.0000	−1.79635
$715$	−30.0000	−1.12194
$716$	12.0000	0.448461
$717$	−24.0000	−0.896296
$718$	24.2487	0.904954
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	−1.73205	−0.0645497
$721$	0	0
$722$	−7.00000	−0.260513
$723$	−51.9615	−1.93247
$724$	−13.8564	−0.514969
$725$	6.00000	0.222834
$726$	−2.00000	−0.0742270
$727$	10.3923	0.385429	0.192715	−	0.981255i	$-0.438271\pi$
$727$	10.3923	0.385429	0.192715	+	0.981255i	$0.438271\pi$
$728$	−17.3205	−0.641941
$729$	13.0000	0.481481
$730$	19.0526	0.705167
$731$	−48.0000	−1.77534
$732$	10.3923	0.384111
$733$	19.0526	0.703722	0.351861	−	0.936052i	$-0.385549\pi$
$733$	19.0526	0.703722	0.351861	+	0.936052i	$0.385549\pi$
$734$	24.2487	0.895036
$735$	17.3205	0.638877
$736$	0	0
$737$	−24.0000	−0.884051
$738$	−9.00000	−0.331295
$739$	2.00000	0.0735712	0.0367856	−	0.999323i	$-0.488288\pi$
$739$	2.00000	0.0735712	0.0367856	+	0.999323i	$0.488288\pi$
$740$	0	0
$741$	−34.6410	−1.27257
$742$	6.00000	0.220267
$743$	−38.1051	−1.39794	−0.698971	−	0.715150i	$-0.746358\pi$
$743$	−38.1051	−1.39794	−0.698971	+	0.715150i	$0.746358\pi$
$744$	16.0000	0.586588
$745$	9.00000	0.329734
$746$	0	0
$747$	0	0
$748$	−24.0000	−0.877527
$749$	12.0000	0.438470
$750$	−24.2487	−0.885438
$751$	24.2487	0.884848	0.442424	−	0.896806i	$-0.354119\pi$
$751$	24.2487	0.884848	0.442424	+	0.896806i	$0.354119\pi$
$752$	−6.00000	−0.218797
$753$	−6.92820	−0.252478
$754$	15.0000	0.546268
$755$	−24.2487	−0.882501
$756$	13.8564	0.503953
$757$	22.5167	0.818382	0.409191	−	0.912449i	$-0.365811\pi$
$757$	22.5167	0.818382	0.409191	+	0.912449i	$0.365811\pi$
$758$	31.1769	1.13240
$759$	0	0
$760$	6.00000	0.217643
$761$	21.0000	0.761249	0.380625	−	0.924730i	$-0.375709\pi$
$761$	21.0000	0.761249	0.380625	+	0.924730i	$0.375709\pi$
$762$	−32.0000	−1.15924
$763$	−54.0000	−1.95493
$764$	−6.92820	−0.250654
$765$	−12.0000	−0.433861
$766$	−6.92820	−0.250326
$767$	30.0000	1.08324
$768$	−2.00000	−0.0721688
$769$	−6.92820	−0.249837	−0.124919	−	0.992167i	$-0.539867\pi$
$769$	−6.92820	−0.249837	−0.124919	+	0.992167i	$0.539867\pi$
$770$	20.7846	0.749025
$771$	42.0000	1.51259
$772$	−19.0000	−0.683825
$773$	−34.6410	−1.24595	−0.622975	−	0.782241i	$-0.714076\pi$
$773$	−34.6410	−1.24595	−0.622975	+	0.782241i	$0.714076\pi$
$774$	−6.92820	−0.249029
$775$	16.0000	0.574737
$776$	−12.1244	−0.435239
$777$	0	0
$778$	13.8564	0.496776
$779$	31.1769	1.11703
$780$	−17.3205	−0.620174
$781$	−20.7846	−0.743732
$782$	0	0
$783$	−12.0000	−0.428845
$784$	5.00000	0.178571
$785$	−9.00000	−0.321224
$786$	−12.0000	−0.428026
$787$	−24.2487	−0.864373	−0.432187	−	0.901784i	$-0.642258\pi$
$787$	−24.2487	−0.864373	−0.432187	+	0.901784i	$0.642258\pi$
$788$	−15.0000	−0.534353
$789$	34.6410	1.23325
$790$	−12.0000	−0.426941
$791$	18.0000	0.640006
$792$	−3.46410	−0.123091
$793$	25.9808	0.922604
$794$	−1.00000	−0.0354887
$795$	6.00000	0.212798
$796$	−3.46410	−0.122782
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	24.0000	0.849591
$799$	−41.5692	−1.47061
$800$	−2.00000	−0.0707107
$801$	1.73205	0.0611990
$802$	29.4449	1.03973
$803$	38.1051	1.34470
$804$	−13.8564	−0.488678
$805$	0	0
$806$	40.0000	1.40894
$807$	−12.0000	−0.422420
$808$	3.00000	0.105540
$809$	−42.0000	−1.47664	−0.738321	−	0.674450i	$-0.764381\pi$
$809$	−42.0000	−1.47664	−0.738321	+	0.674450i	$0.764381\pi$
$810$	19.0526	0.669439
$811$	10.0000	0.351147	0.175574	−	0.984466i	$-0.443822\pi$
$811$	10.0000	0.351147	0.175574	+	0.984466i	$0.443822\pi$
$812$	−10.3923	−0.364698
$813$	28.0000	0.982003
$814$	0	0
$815$	3.46410	0.121342
$816$	−13.8564	−0.485071
$817$	24.0000	0.839654
$818$	2.00000	0.0699284
$819$	−17.3205	−0.605228
$820$	15.5885	0.544373
$821$	3.00000	0.104701	0.0523504	−	0.998629i	$-0.483329\pi$
$821$	3.00000	0.104701	0.0523504	+	0.998629i	$0.483329\pi$
$822$	−31.1769	−1.08742
$823$	−2.00000	−0.0697156	−0.0348578	−	0.999392i	$-0.511098\pi$
$823$	−2.00000	−0.0697156	−0.0348578	+	0.999392i	$0.511098\pi$
$824$	0	0
$825$	−13.8564	−0.482418
$826$	−20.7846	−0.723189
$827$	−45.0333	−1.56596	−0.782981	−	0.622046i	$-0.786302\pi$
$827$	−45.0333	−1.56596	−0.782981	+	0.622046i	$0.786302\pi$
$828$	0	0
$829$	−7.00000	−0.243120	−0.121560	−	0.992584i	$-0.538790\pi$
$829$	−7.00000	−0.243120	−0.121560	+	0.992584i	$0.538790\pi$
$830$	0	0
$831$	52.0000	1.80386
$832$	−5.00000	−0.173344
$833$	34.6410	1.20024
$834$	−16.0000	−0.554035
$835$	−20.7846	−0.719281
$836$	12.0000	0.415029
$837$	−32.0000	−1.10608
$838$	−13.8564	−0.478662
$839$	−3.46410	−0.119594	−0.0597970	−	0.998211i	$-0.519045\pi$
$839$	−3.46410	−0.119594	−0.0597970	+	0.998211i	$0.519045\pi$
$840$	12.0000	0.414039
$841$	−20.0000	−0.689655
$842$	6.92820	0.238762
$843$	13.8564	0.477240
$844$	10.0000	0.344214
$845$	−20.7846	−0.715012
$846$	−6.00000	−0.206284
$847$	3.46410	0.119028
$848$	1.73205	0.0594789
$849$	−55.4256	−1.90220
$850$	−13.8564	−0.475271
$851$	0	0
$852$	−12.0000	−0.411113
$853$	26.0000	0.890223	0.445112	−	0.895475i	$-0.353164\pi$
$853$	26.0000	0.890223	0.445112	+	0.895475i	$0.353164\pi$
$854$	−18.0000	−0.615947
$855$	6.00000	0.205196
$856$	3.46410	0.118401
$857$	15.0000	0.512390	0.256195	−	0.966625i	$-0.417531\pi$
$857$	15.0000	0.512390	0.256195	+	0.966625i	$0.417531\pi$
$858$	−34.6410	−1.18262
$859$	2.00000	0.0682391	0.0341196	−	0.999418i	$-0.489137\pi$
$859$	2.00000	0.0682391	0.0341196	+	0.999418i	$0.489137\pi$
$860$	12.0000	0.409197
$861$	62.3538	2.12501
$862$	24.2487	0.825914
$863$	6.00000	0.204242	0.102121	−	0.994772i	$-0.467437\pi$
$863$	6.00000	0.204242	0.102121	+	0.994772i	$0.467437\pi$
$864$	4.00000	0.136083
$865$	−36.3731	−1.23672
$866$	−29.4449	−1.00058
$867$	−62.0000	−2.10563
$868$	−27.7128	−0.940634
$869$	−24.0000	−0.814144
$870$	−10.3923	−0.352332
$871$	−34.6410	−1.17377
$872$	−15.5885	−0.527892
$873$	−12.1244	−0.410347
$874$	0	0
$875$	42.0000	1.41986
$876$	22.0000	0.743311
$877$	22.0000	0.742887	0.371444	−	0.928456i	$-0.378863\pi$
$877$	22.0000	0.742887	0.371444	+	0.928456i	$0.378863\pi$
$878$	8.00000	0.269987
$879$	65.8179	2.21998
$880$	6.00000	0.202260
$881$	29.4449	0.992023	0.496011	−	0.868316i	$-0.334797\pi$
$881$	29.4449	0.992023	0.496011	+	0.868316i	$0.334797\pi$
$882$	5.00000	0.168359
$883$	−38.0000	−1.27880	−0.639401	−	0.768874i	$-0.720818\pi$
$883$	−38.0000	−1.27880	−0.639401	+	0.768874i	$0.720818\pi$
$884$	−34.6410	−1.16510
$885$	−20.7846	−0.698667
$886$	24.0000	0.806296
$887$	12.0000	0.402921	0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000	0.402921	0.201460	+	0.979497i	$0.435431\pi$
$888$	0	0
$889$	55.4256	1.85892
$890$	−3.00000	−0.100560
$891$	38.1051	1.27657
$892$	−10.0000	−0.334825
$893$	20.7846	0.695530
$894$	10.3923	0.347571
$895$	−20.7846	−0.694753
$896$	3.46410	0.115728
$897$	0	0
$898$	−33.0000	−1.10122
$899$	24.0000	0.800445
$900$	−2.00000	−0.0666667
$901$	12.0000	0.399778
$902$	31.1769	1.03808
$903$	48.0000	1.59734
$904$	5.19615	0.172821
$905$	24.0000	0.797787
$906$	−28.0000	−0.930238
$907$	−31.1769	−1.03521	−0.517606	−	0.855619i	$-0.673177\pi$
$907$	−31.1769	−1.03521	−0.517606	+	0.855619i	$0.673177\pi$
$908$	6.92820	0.229920
$909$	3.00000	0.0995037
$910$	30.0000	0.994490
$911$	20.7846	0.688625	0.344312	−	0.938855i	$-0.388112\pi$
$911$	20.7846	0.688625	0.344312	+	0.938855i	$0.388112\pi$
$912$	6.92820	0.229416
$913$	0	0
$914$	12.1244	0.401038
$915$	−18.0000	−0.595062
$916$	−13.8564	−0.457829
$917$	20.7846	0.686368
$918$	27.7128	0.914659
$919$	45.0333	1.48551	0.742756	−	0.669562i	$-0.233518\pi$
$919$	45.0333	1.48551	0.742756	+	0.669562i	$0.233518\pi$
$920$	0	0
$921$	52.0000	1.71346
$922$	−27.0000	−0.889198
$923$	−30.0000	−0.987462
$924$	24.0000	0.789542
$925$	0	0
$926$	−28.0000	−0.920137
$927$	0	0
$928$	−3.00000	−0.0984798
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	−27.7128	−0.908739
$931$	−17.3205	−0.567657
$932$	15.0000	0.491341
$933$	36.0000	1.17859
$934$	20.7846	0.680093
$935$	41.5692	1.35946
$936$	−5.00000	−0.163430
$937$	20.7846	0.679004	0.339502	−	0.940605i	$-0.389742\pi$
$937$	20.7846	0.679004	0.339502	+	0.940605i	$0.389742\pi$
$938$	24.0000	0.783628
$939$	51.9615	1.69570
$940$	10.3923	0.338960
$941$	6.92820	0.225853	0.112926	−	0.993603i	$-0.463978\pi$
$941$	6.92820	0.225853	0.112926	+	0.993603i	$0.463978\pi$
$942$	−10.3923	−0.338600
$943$	0	0
$944$	−6.00000	−0.195283
$945$	−24.0000	−0.780720
$946$	24.0000	0.780307
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	−13.8564	−0.450035
$949$	55.0000	1.78538
$950$	6.92820	0.224781
$951$	−42.0000	−1.36194
$952$	24.0000	0.777844
$953$	−22.5167	−0.729386	−0.364693	−	0.931128i	$-0.618826\pi$
$953$	−22.5167	−0.729386	−0.364693	+	0.931128i	$0.618826\pi$
$954$	1.73205	0.0560772
$955$	12.0000	0.388311
$956$	12.0000	0.388108
$957$	−20.7846	−0.671871
$958$	−34.6410	−1.11920
$959$	54.0000	1.74375
$960$	3.46410	0.111803
$961$	33.0000	1.06452
$962$	0	0
$963$	3.46410	0.111629
$964$	25.9808	0.836784
$965$	32.9090	1.05938
$966$	0	0
$967$	−50.0000	−1.60789	−0.803946	−	0.594703i	$-0.797270\pi$
$967$	−50.0000	−1.60789	−0.803946	+	0.594703i	$0.797270\pi$
$968$	1.00000	0.0321412
$969$	48.0000	1.54198
$970$	21.0000	0.674269
$971$	3.46410	0.111168	0.0555842	−	0.998454i	$-0.482298\pi$
$971$	3.46410	0.111168	0.0555842	+	0.998454i	$0.482298\pi$
$972$	10.0000	0.320750
$973$	27.7128	0.888432
$974$	2.00000	0.0640841
$975$	−20.0000	−0.640513
$976$	−5.19615	−0.166325
$977$	−36.3731	−1.16368	−0.581839	−	0.813304i	$-0.697667\pi$
$977$	−36.3731	−1.16368	−0.581839	+	0.813304i	$0.697667\pi$
$978$	4.00000	0.127906
$979$	−6.00000	−0.191761
$980$	−8.66025	−0.276642
$981$	−15.5885	−0.497701
$982$	0	0
$983$	6.92820	0.220975	0.110488	−	0.993877i	$-0.464759\pi$
$983$	6.92820	0.220975	0.110488	+	0.993877i	$0.464759\pi$
$984$	18.0000	0.573819
$985$	25.9808	0.827816
$986$	−20.7846	−0.661917
$987$	41.5692	1.32316
$988$	17.3205	0.551039
$989$	0	0
$990$	6.00000	0.190693
$991$	52.0000	1.65183	0.825917	−	0.563791i	$-0.190658\pi$
$991$	52.0000	1.65183	0.825917	+	0.563791i	$0.190658\pi$
$992$	−8.00000	−0.254000
$993$	16.0000	0.507745
$994$	20.7846	0.659248
$995$	6.00000	0.190213
$996$	0	0
$997$	5.00000	0.158352	0.0791758	−	0.996861i	$-0.474771\pi$
$997$	5.00000	0.158352	0.0791758	+	0.996861i	$0.474771\pi$
$998$	−32.0000	−1.01294
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1058.2.a.g.1.1	✓	2
3.2	odd	2		9522.2.a.t.1.2		2
4.3	odd	2		8464.2.a.bh.1.1		2
23.22	odd	2	inner	1058.2.a.g.1.2	yes	2
69.68	even	2		9522.2.a.t.1.1		2
92.91	even	2		8464.2.a.bh.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1058.2.a.g.1.1	✓	2	1.1	even	1	trivial
1058.2.a.g.1.2	yes	2	23.22	odd	2	inner
8464.2.a.bh.1.1		2	4.3	odd	2
8464.2.a.bh.1.2		2	92.91	even	2
9522.2.a.t.1.1		2	69.68	even	2
9522.2.a.t.1.2		2	3.2	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:	\( N \)	\(=\)	\( 1058 = 2 \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1058.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -1.73205 q^{5} -2.00000 q^{6} +3.46410 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.73205 q^{10} -3.46410 q^{11} -2.00000 q^{12} -5.00000 q^{13} +3.46410 q^{14} +3.46410 q^{15} +1.00000 q^{16} +6.92820 q^{17} +1.00000 q^{18} -3.46410 q^{19} -1.73205 q^{20} -6.92820 q^{21} -3.46410 q^{22} -2.00000 q^{24} -2.00000 q^{25} -5.00000 q^{26} +4.00000 q^{27} +3.46410 q^{28} -3.00000 q^{29} +3.46410 q^{30} -8.00000 q^{31} +1.00000 q^{32} +6.92820 q^{33} +6.92820 q^{34} -6.00000 q^{35} +1.00000 q^{36} -3.46410 q^{38} +10.0000 q^{39} -1.73205 q^{40} -9.00000 q^{41} -6.92820 q^{42} -6.92820 q^{43} -3.46410 q^{44} -1.73205 q^{45} -6.00000 q^{47} -2.00000 q^{48} +5.00000 q^{49} -2.00000 q^{50} -13.8564 q^{51} -5.00000 q^{52} +1.73205 q^{53} +4.00000 q^{54} +6.00000 q^{55} +3.46410 q^{56} +6.92820 q^{57} -3.00000 q^{58} -6.00000 q^{59} +3.46410 q^{60} -5.19615 q^{61} -8.00000 q^{62} +3.46410 q^{63} +1.00000 q^{64} +8.66025 q^{65} +6.92820 q^{66} +6.92820 q^{67} +6.92820 q^{68} -6.00000 q^{70} +6.00000 q^{71} +1.00000 q^{72} -11.0000 q^{73} +4.00000 q^{75} -3.46410 q^{76} -12.0000 q^{77} +10.0000 q^{78} +6.92820 q^{79} -1.73205 q^{80} -11.0000 q^{81} -9.00000 q^{82} -6.92820 q^{84} -12.0000 q^{85} -6.92820 q^{86} +6.00000 q^{87} -3.46410 q^{88} +1.73205 q^{89} -1.73205 q^{90} -17.3205 q^{91} +16.0000 q^{93} -6.00000 q^{94} +6.00000 q^{95} -2.00000 q^{96} -12.1244 q^{97} +5.00000 q^{98} -3.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 4 q^{3} + 2 q^{4} - 4 q^{6} + 2 q^{8} + 2 q^{9} - 4 q^{12} - 10 q^{13} + 2 q^{16} + 2 q^{18} - 4 q^{24} - 4 q^{25} - 10 q^{26} + 8 q^{27} - 6 q^{29} - 16 q^{31} + 2 q^{32} - 12 q^{35} + 2 q^{36}+ \cdots + 10 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 - 4 * q^3 + 2 * q^4 - 4 * q^6 + 2 * q^8 + 2 * q^9 - 4 * q^12 - 10 * q^13 + 2 * q^16 + 2 * q^18 - 4 * q^24 - 4 * q^25 - 10 * q^26 + 8 * q^27 - 6 * q^29 - 16 * q^31 + 2 * q^32 - 12 * q^35 + 2 * q^36 + 20 * q^39 - 18 * q^41 - 12 * q^47 - 4 * q^48 + 10 * q^49 - 4 * q^50 - 10 * q^52 + 8 * q^54 + 12 * q^55 - 6 * q^58 - 12 * q^59 - 16 * q^62 + 2 * q^64 - 12 * q^70 + 12 * q^71 + 2 * q^72 - 22 * q^73 + 8 * q^75 - 24 * q^77 + 20 * q^78 - 22 * q^81 - 18 * q^82 - 24 * q^85 + 12 * q^87 + 32 * q^93 - 12 * q^94 + 12 * q^95 - 4 * q^96 + 10 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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