LMFDB - Embedded newform 5041.2.a.c.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5041,2,Mod(1,5041)] 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("5041.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5041, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5041 = 71^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5041.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,-4,4,0,12,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$40.2525876589$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{24})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 1 $ x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.93185$ of defining polynomial
Character	$\chi$	$=$	5041.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.73205 q^{2} +0.732051 q^{3} +1.00000 q^{4} +1.73205 q^{5} +1.26795 q^{6} -1.41421 q^{7} -1.73205 q^{8} -2.46410 q^{9} +3.00000 q^{10} +2.44949 q^{11} +0.732051 q^{12} -0.378937 q^{13} -2.44949 q^{14} +1.26795 q^{15} -5.00000 q^{16} -0.896575 q^{17} -4.26795 q^{18} -6.92820 q^{19} +1.73205 q^{20} -1.03528 q^{21} +4.24264 q^{22} +1.79315 q^{23} -1.26795 q^{24} -2.00000 q^{25} -0.656339 q^{26} -4.00000 q^{27} -1.41421 q^{28} -2.53590 q^{29} +2.19615 q^{30} +6.31319 q^{31} -5.19615 q^{32} +1.79315 q^{33} -1.55291 q^{34} -2.44949 q^{35} -2.46410 q^{36} -9.19615 q^{37} -12.0000 q^{38} -0.277401 q^{39} -3.00000 q^{40} +4.00240 q^{41} -1.79315 q^{42} +6.19615 q^{43} +2.44949 q^{44} -4.26795 q^{45} +3.10583 q^{46} -2.44949 q^{47} -3.66025 q^{48} -5.00000 q^{49} -3.46410 q^{50} -0.656339 q^{51} -0.378937 q^{52} -13.1440 q^{53} -6.92820 q^{54} +4.24264 q^{55} +2.44949 q^{56} -5.07180 q^{57} -4.39230 q^{58} -11.5911 q^{59} +1.26795 q^{60} +13.7632 q^{61} +10.9348 q^{62} +3.48477 q^{63} +1.00000 q^{64} -0.656339 q^{65} +3.10583 q^{66} -14.7985 q^{67} -0.896575 q^{68} +1.31268 q^{69} -4.24264 q^{70} +4.26795 q^{72} +5.53590 q^{73} -15.9282 q^{74} -1.46410 q^{75} -6.92820 q^{76} -3.46410 q^{77} -0.480473 q^{78} +2.19615 q^{79} -8.66025 q^{80} +4.46410 q^{81} +6.93237 q^{82} -14.1962 q^{83} -1.03528 q^{84} -1.55291 q^{85} +10.7321 q^{86} -1.85641 q^{87} -4.24264 q^{88} +12.9282 q^{89} -7.39230 q^{90} +0.535898 q^{91} +1.79315 q^{92} +4.62158 q^{93} -4.24264 q^{94} -12.0000 q^{95} -3.80385 q^{96} +6.17449 q^{97} -8.66025 q^{98} -6.03579 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 4 q^{4} + 12 q^{6} + 4 q^{9} + 12 q^{10} - 4 q^{12} + 12 q^{15} - 20 q^{16} - 24 q^{18} - 12 q^{24} - 8 q^{25} - 16 q^{27} - 24 q^{29} - 12 q^{30} + 4 q^{36} - 16 q^{37} - 48 q^{38} - 12 q^{40}+ \cdots - 36 q^{96}+O(q^{100}) $ 4 * q - 4 * q^3 + 4 * q^4 + 12 * q^6 + 4 * q^9 + 12 * q^10 - 4 * q^12 + 12 * q^15 - 20 * q^16 - 24 * q^18 - 12 * q^24 - 8 * q^25 - 16 * q^27 - 24 * q^29 - 12 * q^30 + 4 * q^36 - 16 * q^37 - 48 * q^38 - 12 * q^40 + 4 * q^43 - 24 * q^45 + 20 * q^48 - 20 * q^49 - 48 * q^57 + 24 * q^58 + 12 * q^60 + 4 * q^64 + 24 * q^72 + 36 * q^73 - 36 * q^74 + 8 * q^75 - 12 * q^79 + 4 * q^81 - 36 * q^83 + 36 * q^86 + 48 * q^87 + 24 * q^89 + 12 * q^90 + 16 * q^91 - 48 * q^95 - 36 * q^96

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 $

$2$	1.73205	1.22474	0.612372	−	0.790569i	$-0.290215\pi$
$2$	1.73205	1.22474	0.612372	+	0.790569i	$0.290215\pi$
$3$	0.732051	0.422650	0.211325	−	0.977416i	$-0.432222\pi$
$3$	0.732051	0.422650	0.211325	+	0.977416i	$0.432222\pi$
$4$	1.00000	0.500000
$5$	1.73205	0.774597	0.387298	−	0.921954i	$-0.373408\pi$
$5$	1.73205	0.774597	0.387298	+	0.921954i	$0.373408\pi$
$6$	1.26795	0.517638
$7$	−1.41421	−0.534522	−0.267261	−	0.963624i	$-0.586119\pi$
$7$	−1.41421	−0.534522	−0.267261	+	0.963624i	$0.586119\pi$
$8$	−1.73205	−0.612372
$9$	−2.46410	−0.821367
$10$	3.00000	0.948683
$11$	2.44949	0.738549	0.369274	−	0.929320i	$-0.379606\pi$
$11$	2.44949	0.738549	0.369274	+	0.929320i	$0.379606\pi$
$12$	0.732051	0.211325
$13$	−0.378937	−0.105098	−0.0525492	−	0.998618i	$-0.516735\pi$
$13$	−0.378937	−0.105098	−0.0525492	+	0.998618i	$0.516735\pi$
$14$	−2.44949	−0.654654
$15$	1.26795	0.327383
$16$	−5.00000	−1.25000
$17$	−0.896575	−0.217451	−0.108726	−	0.994072i	$-0.534677\pi$
$17$	−0.896575	−0.217451	−0.108726	+	0.994072i	$0.534677\pi$
$18$	−4.26795	−1.00597
$19$	−6.92820	−1.58944	−0.794719	−	0.606977i	$-0.792382\pi$
$19$	−6.92820	−1.58944	−0.794719	+	0.606977i	$0.792382\pi$
$20$	1.73205	0.387298
$21$	−1.03528	−0.225916
$22$	4.24264	0.904534
$23$	1.79315	0.373898	0.186949	−	0.982370i	$-0.440140\pi$
$23$	1.79315	0.373898	0.186949	+	0.982370i	$0.440140\pi$
$24$	−1.26795	−0.258819
$25$	−2.00000	−0.400000
$26$	−0.656339	−0.128719
$27$	−4.00000	−0.769800
$28$	−1.41421	−0.267261
$29$	−2.53590	−0.470905	−0.235452	−	0.971886i	$-0.575657\pi$
$29$	−2.53590	−0.470905	−0.235452	+	0.971886i	$0.575657\pi$
$30$	2.19615	0.400961
$31$	6.31319	1.13388	0.566941	−	0.823758i	$-0.308127\pi$
$31$	6.31319	1.13388	0.566941	+	0.823758i	$0.308127\pi$
$32$	−5.19615	−0.918559
$33$	1.79315	0.312148
$34$	−1.55291	−0.266323
$35$	−2.44949	−0.414039
$36$	−2.46410	−0.410684
$37$	−9.19615	−1.51184	−0.755919	−	0.654665i	$-0.772810\pi$
$37$	−9.19615	−1.51184	−0.755919	+	0.654665i	$0.772810\pi$
$38$	−12.0000	−1.94666
$39$	−0.277401	−0.0444198
$40$	−3.00000	−0.474342
$41$	4.00240	0.625070	0.312535	−	0.949906i	$-0.398822\pi$
$41$	4.00240	0.625070	0.312535	+	0.949906i	$0.398822\pi$
$42$	−1.79315	−0.276689
$43$	6.19615	0.944904	0.472452	−	0.881356i	$-0.343369\pi$
$43$	6.19615	0.944904	0.472452	+	0.881356i	$0.343369\pi$
$44$	2.44949	0.369274
$45$	−4.26795	−0.636228
$46$	3.10583	0.457929
$47$	−2.44949	−0.357295	−0.178647	−	0.983913i	$-0.557172\pi$
$47$	−2.44949	−0.357295	−0.178647	+	0.983913i	$0.557172\pi$
$48$	−3.66025	−0.528312
$49$	−5.00000	−0.714286
$50$	−3.46410	−0.489898
$51$	−0.656339	−0.0919058
$52$	−0.378937	−0.0525492
$53$	−13.1440	−1.80547	−0.902735	−	0.430196i	$-0.858444\pi$
$53$	−13.1440	−1.80547	−0.902735	+	0.430196i	$0.858444\pi$
$54$	−6.92820	−0.942809
$55$	4.24264	0.572078
$56$	2.44949	0.327327
$57$	−5.07180	−0.671776
$58$	−4.39230	−0.576738
$59$	−11.5911	−1.50903	−0.754517	−	0.656281i	$-0.772129\pi$
$59$	−11.5911	−1.50903	−0.754517	+	0.656281i	$0.772129\pi$
$60$	1.26795	0.163692
$61$	13.7632	1.76220	0.881098	−	0.472933i	$-0.156805\pi$
$61$	13.7632	1.76220	0.881098	+	0.472933i	$0.156805\pi$
$62$	10.9348	1.38872
$63$	3.48477	0.439039
$64$	1.00000	0.125000
$65$	−0.656339	−0.0814088
$66$	3.10583	0.382301
$67$	−14.7985	−1.80792	−0.903961	−	0.427616i	$-0.859354\pi$
$67$	−14.7985	−1.80792	−0.903961	+	0.427616i	$0.859354\pi$
$68$	−0.896575	−0.108726
$69$	1.31268	0.158028
$70$	−4.24264	−0.507093
$71$	0	0
$71$	0	0
$72$	4.26795	0.502983
$73$	5.53590	0.647928	0.323964	−	0.946069i	$-0.394984\pi$
$73$	5.53590	0.647928	0.323964	+	0.946069i	$0.394984\pi$
$74$	−15.9282	−1.85162
$75$	−1.46410	−0.169060
$76$	−6.92820	−0.794719
$77$	−3.46410	−0.394771
$78$	−0.480473	−0.0544029
$79$	2.19615	0.247086	0.123543	−	0.992339i	$-0.460574\pi$
$79$	2.19615	0.247086	0.123543	+	0.992339i	$0.460574\pi$
$80$	−8.66025	−0.968246
$81$	4.46410	0.496011
$82$	6.93237	0.765552
$83$	−14.1962	−1.55823	−0.779115	−	0.626881i	$-0.784331\pi$
$83$	−14.1962	−1.55823	−0.779115	+	0.626881i	$0.784331\pi$
$84$	−1.03528	−0.112958
$85$	−1.55291	−0.168437
$86$	10.7321	1.15727
$87$	−1.85641	−0.199028
$88$	−4.24264	−0.452267
$89$	12.9282	1.37039	0.685193	−	0.728361i	$-0.259718\pi$
$89$	12.9282	1.37039	0.685193	+	0.728361i	$0.259718\pi$
$90$	−7.39230	−0.779217
$91$	0.535898	0.0561774
$92$	1.79315	0.186949
$93$	4.62158	0.479235
$94$	−4.24264	−0.437595
$95$	−12.0000	−1.23117
$96$	−3.80385	−0.388229
$97$	6.17449	0.626925	0.313462	−	0.949601i	$-0.398511\pi$
$97$	6.17449	0.626925	0.313462	+	0.949601i	$0.398511\pi$
$98$	−8.66025	−0.874818
$99$	−6.03579	−0.606620
$100$	−2.00000	−0.200000
$101$	5.19615	0.517036	0.258518	−	0.966006i	$-0.416766\pi$
$101$	5.19615	0.517036	0.258518	+	0.966006i	$0.416766\pi$
$102$	−1.13681	−0.112561
$103$	−6.19615	−0.610525	−0.305263	−	0.952268i	$-0.598744\pi$
$103$	−6.19615	−0.610525	−0.305263	+	0.952268i	$0.598744\pi$
$104$	0.656339	0.0643593
$105$	−1.79315	−0.174994
$106$	−22.7661	−2.21124
$107$	14.1962	1.37239	0.686197	−	0.727416i	$-0.259279\pi$
$107$	14.1962	1.37239	0.686197	+	0.727416i	$0.259279\pi$
$108$	−4.00000	−0.384900
$109$	−13.3923	−1.28275	−0.641375	−	0.767227i	$-0.721636\pi$
$109$	−13.3923	−1.28275	−0.641375	+	0.767227i	$0.721636\pi$
$110$	7.34847	0.700649
$111$	−6.73205	−0.638978
$112$	7.07107	0.668153
$113$	−14.9372	−1.40517	−0.702586	−	0.711599i	$-0.747971\pi$
$113$	−14.9372	−1.40517	−0.702586	+	0.711599i	$0.747971\pi$
$114$	−8.78461	−0.822754
$115$	3.10583	0.289620
$116$	−2.53590	−0.235452
$117$	0.933740	0.0863243
$118$	−20.0764	−1.84818
$119$	1.26795	0.116233
$120$	−2.19615	−0.200480
$121$	−5.00000	−0.454545
$122$	23.8386	2.15824
$123$	2.92996	0.264186
$124$	6.31319	0.566941
$125$	−12.1244	−1.08444
$126$	6.03579	0.537711
$127$	7.45001	0.661081	0.330541	−	0.943792i	$-0.392769\pi$
$127$	7.45001	0.661081	0.330541	+	0.943792i	$0.392769\pi$
$128$	12.1244	1.07165
$129$	4.53590	0.399364
$130$	−1.13681	−0.0997050
$131$	2.53590	0.221562	0.110781	−	0.993845i	$-0.464665\pi$
$131$	2.53590	0.221562	0.110781	+	0.993845i	$0.464665\pi$
$132$	1.79315	0.156074
$133$	9.79796	0.849591
$134$	−25.6317	−2.21424
$135$	−6.92820	−0.596285
$136$	1.55291	0.133161
$137$	2.68973	0.229799	0.114899	−	0.993377i	$-0.463345\pi$
$137$	2.68973	0.229799	0.114899	+	0.993377i	$0.463345\pi$
$138$	2.27362	0.193544
$139$	1.51575	0.128564	0.0642821	−	0.997932i	$-0.479524\pi$
$139$	1.51575	0.128564	0.0642821	+	0.997932i	$0.479524\pi$
$140$	−2.44949	−0.207020
$141$	−1.79315	−0.151011
$142$	0	0
$143$	−0.928203	−0.0776203
$144$	12.3205	1.02671
$145$	−4.39230	−0.364761
$146$	9.58846	0.793546
$147$	−3.66025	−0.301893
$148$	−9.19615	−0.755919
$149$	−6.45189	−0.528560	−0.264280	−	0.964446i	$-0.585134\pi$
$149$	−6.45189	−0.528560	−0.264280	+	0.964446i	$0.585134\pi$
$150$	−2.53590	−0.207055
$151$	−14.3923	−1.17123	−0.585615	−	0.810590i	$-0.699147\pi$
$151$	−14.3923	−1.17123	−0.585615	+	0.810590i	$0.699147\pi$
$152$	12.0000	0.973329
$153$	2.20925	0.178608
$154$	−6.00000	−0.483494
$155$	10.9348	0.878302
$156$	−0.277401	−0.0222099
$157$	4.80385	0.383389	0.191694	−	0.981455i	$-0.438602\pi$
$157$	4.80385	0.383389	0.191694	+	0.981455i	$0.438602\pi$
$158$	3.80385	0.302618
$159$	−9.62209	−0.763082
$160$	−9.00000	−0.711512
$161$	−2.53590	−0.199857
$162$	7.73205	0.607487
$163$	9.52056	0.745708	0.372854	−	0.927890i	$-0.378379\pi$
$163$	9.52056	0.745708	0.372854	+	0.927890i	$0.378379\pi$
$164$	4.00240	0.312535
$165$	3.10583	0.241788
$166$	−24.5885	−1.90843
$167$	13.8564	1.07224	0.536120	−	0.844141i	$-0.319889\pi$
$167$	13.8564	1.07224	0.536120	+	0.844141i	$0.319889\pi$
$168$	1.79315	0.138345
$169$	−12.8564	−0.988954
$170$	−2.68973	−0.206293
$171$	17.0718	1.30551
$172$	6.19615	0.472452
$173$	13.6245	1.03585	0.517926	−	0.855426i	$-0.326704\pi$
$173$	13.6245	1.03585	0.517926	+	0.855426i	$0.326704\pi$
$174$	−3.21539	−0.243758
$175$	2.82843	0.213809
$176$	−12.2474	−0.923186
$177$	−8.48528	−0.637793
$178$	22.3923	1.67837
$179$	−6.00000	−0.448461	−0.224231	−	0.974536i	$-0.571987\pi$
$179$	−6.00000	−0.448461	−0.224231	+	0.974536i	$0.571987\pi$
$180$	−4.26795	−0.318114
$181$	−12.8666	−0.956369	−0.478184	−	0.878260i	$-0.658705\pi$
$181$	−12.8666	−0.956369	−0.478184	+	0.878260i	$0.658705\pi$
$182$	0.928203	0.0688030
$183$	10.0754	0.744792
$184$	−3.10583	−0.228965
$185$	−15.9282	−1.17106
$186$	8.00481	0.586941
$187$	−2.19615	−0.160599
$188$	−2.44949	−0.178647
$189$	5.65685	0.411476
$190$	−20.7846	−1.50787
$191$	20.7846	1.50392	0.751961	−	0.659208i	$-0.229108\pi$
$191$	20.7846	1.50392	0.751961	+	0.659208i	$0.229108\pi$
$192$	0.732051	0.0528312
$193$	−19.0783	−1.37328	−0.686642	−	0.726995i	$-0.740916\pi$
$193$	−19.0783	−1.37328	−0.686642	+	0.726995i	$0.740916\pi$
$194$	10.6945	0.767823
$195$	−0.480473	−0.0344074
$196$	−5.00000	−0.357143
$197$	15.8338	1.12811	0.564054	−	0.825738i	$-0.309241\pi$
$197$	15.8338	1.12811	0.564054	+	0.825738i	$0.309241\pi$
$198$	−10.4543	−0.742955
$199$	−6.92820	−0.491127	−0.245564	−	0.969380i	$-0.578973\pi$
$199$	−6.92820	−0.491127	−0.245564	+	0.969380i	$0.578973\pi$
$200$	3.46410	0.244949
$201$	−10.8332	−0.764117
$202$	9.00000	0.633238
$203$	3.58630	0.251709
$204$	−0.656339	−0.0459529
$205$	6.93237	0.484178
$206$	−10.7321	−0.747737
$207$	−4.41851	−0.307107
$208$	1.89469	0.131373
$209$	−16.9706	−1.17388
$210$	−3.10583	−0.214323
$211$	14.1421	0.973585	0.486792	−	0.873518i	$-0.338167\pi$
$211$	14.1421	0.973585	0.486792	+	0.873518i	$0.338167\pi$
$212$	−13.1440	−0.902735
$213$	0	0
$214$	24.5885	1.68083
$215$	10.7321	0.731920
$216$	6.92820	0.471405
$217$	−8.92820	−0.606086
$218$	−23.1962	−1.57104
$219$	4.05256	0.273847
$220$	4.24264	0.286039
$221$	0.339746	0.0228538
$222$	−11.6603	−0.782585
$223$	−3.46410	−0.231973	−0.115987	−	0.993251i	$-0.537003\pi$
$223$	−3.46410	−0.231973	−0.115987	+	0.993251i	$0.537003\pi$
$224$	7.34847	0.490990
$225$	4.92820	0.328547
$226$	−25.8719	−1.72098
$227$	28.0812	1.86381	0.931907	−	0.362696i	$-0.118144\pi$
$227$	28.0812	1.86381	0.931907	+	0.362696i	$0.118144\pi$
$228$	−5.07180	−0.335888
$229$	8.00000	0.528655	0.264327	−	0.964433i	$-0.414850\pi$
$229$	8.00000	0.528655	0.264327	+	0.964433i	$0.414850\pi$
$230$	5.37945	0.354711
$231$	−2.53590	−0.166850
$232$	4.39230	0.288369
$233$	−1.39230	−0.0912129	−0.0456065	−	0.998959i	$-0.514522\pi$
$233$	−1.39230	−0.0912129	−0.0456065	+	0.998959i	$0.514522\pi$
$234$	1.61729	0.105725
$235$	−4.24264	−0.276759
$236$	−11.5911	−0.754517
$237$	1.60770	0.104431
$238$	2.19615	0.142355
$239$	5.37945	0.347968	0.173984	−	0.984748i	$-0.444336\pi$
$239$	5.37945	0.347968	0.173984	+	0.984748i	$0.444336\pi$
$240$	−6.33975	−0.409229
$241$	−20.1779	−1.29977	−0.649887	−	0.760031i	$-0.725184\pi$
$241$	−20.1779	−1.29977	−0.649887	+	0.760031i	$0.725184\pi$
$242$	−8.66025	−0.556702
$243$	15.2679	0.979439
$244$	13.7632	0.881098
$245$	−8.66025	−0.553283
$246$	5.07484	0.323560
$247$	2.62536	0.167047
$248$	−10.9348	−0.694359
$249$	−10.3923	−0.658586
$250$	−21.0000	−1.32816
$251$	−11.6603	−0.735989	−0.367994	−	0.929828i	$-0.619955\pi$
$251$	−11.6603	−0.735989	−0.367994	+	0.929828i	$0.619955\pi$
$252$	3.48477	0.219520
$253$	4.39230	0.276142
$254$	12.9038	0.809656
$255$	−1.13681	−0.0711899
$256$	19.0000	1.18750
$257$	14.2808	0.890814	0.445407	−	0.895328i	$-0.353059\pi$
$257$	14.2808	0.890814	0.445407	+	0.895328i	$0.353059\pi$
$258$	7.85641	0.489119
$259$	13.0053	0.808111
$260$	−0.656339	−0.0407044
$261$	6.24871	0.386786
$262$	4.39230	0.271357
$263$	−19.8564	−1.22440	−0.612199	−	0.790704i	$-0.709715\pi$
$263$	−19.8564	−1.22440	−0.612199	+	0.790704i	$0.709715\pi$
$264$	−3.10583	−0.191151
$265$	−22.7661	−1.39851
$266$	16.9706	1.04053
$267$	9.46410	0.579194
$268$	−14.7985	−0.903961
$269$	−20.3166	−1.23873	−0.619363	−	0.785104i	$-0.712609\pi$
$269$	−20.3166	−1.23873	−0.619363	+	0.785104i	$0.712609\pi$
$270$	−12.0000	−0.730297
$271$	−21.4641	−1.30385	−0.651926	−	0.758283i	$-0.726039\pi$
$271$	−21.4641	−1.30385	−0.651926	+	0.758283i	$0.726039\pi$
$272$	4.48288	0.271814
$273$	0.392305	0.0237434
$274$	4.65874	0.281445
$275$	−4.89898	−0.295420
$276$	1.31268	0.0790139
$277$	15.4641	0.929148	0.464574	−	0.885534i	$-0.346208\pi$
$277$	15.4641	0.929148	0.464574	+	0.885534i	$0.346208\pi$
$278$	2.62536	0.157458
$279$	−15.5563	−0.931334
$280$	4.24264	0.253546
$281$	−4.72311	−0.281757	−0.140879	−	0.990027i	$-0.544993\pi$
$281$	−4.72311	−0.281757	−0.140879	+	0.990027i	$0.544993\pi$
$282$	−3.10583	−0.184949
$283$	14.1421	0.840663	0.420331	−	0.907371i	$-0.361914\pi$
$283$	14.1421	0.840663	0.420331	+	0.907371i	$0.361914\pi$
$284$	0	0
$285$	−8.78461	−0.520355
$286$	−1.60770	−0.0950650
$287$	−5.66025	−0.334114
$288$	12.8038	0.754474
$289$	−16.1962	−0.952715
$290$	−7.60770	−0.446739
$291$	4.52004	0.264970
$292$	5.53590	0.323964
$293$	9.92820	0.580012	0.290006	−	0.957025i	$-0.406343\pi$
$293$	9.92820	0.580012	0.290006	+	0.957025i	$0.406343\pi$
$294$	−6.33975	−0.369741
$295$	−20.0764	−1.16889
$296$	15.9282	0.925808
$297$	−9.79796	−0.568535
$298$	−11.1750	−0.647351
$299$	−0.679492	−0.0392960
$300$	−1.46410	−0.0845299
$301$	−8.76268	−0.505073
$302$	−24.9282	−1.43446
$303$	3.80385	0.218525
$304$	34.6410	1.98680
$305$	23.8386	1.36499
$306$	3.82654	0.218749
$307$	−28.9406	−1.65173	−0.825864	−	0.563869i	$-0.809312\pi$
$307$	−28.9406	−1.65173	−0.825864	+	0.563869i	$0.809312\pi$
$308$	−3.46410	−0.197386
$309$	−4.53590	−0.258038
$310$	18.9396	1.07570
$311$	−6.33975	−0.359494	−0.179747	−	0.983713i	$-0.557528\pi$
$311$	−6.33975	−0.359494	−0.179747	+	0.983713i	$0.557528\pi$
$312$	0.480473	0.0272014
$313$	−12.8038	−0.723716	−0.361858	−	0.932233i	$-0.617858\pi$
$313$	−12.8038	−0.723716	−0.361858	+	0.932233i	$0.617858\pi$
$314$	8.32051	0.469553
$315$	6.03579	0.340078
$316$	2.19615	0.123543
$317$	19.1798	1.07725	0.538623	−	0.842547i	$-0.318945\pi$
$317$	19.1798	1.07725	0.538623	+	0.842547i	$0.318945\pi$
$318$	−16.6660	−0.934580
$319$	−6.21166	−0.347786
$320$	1.73205	0.0968246
$321$	10.3923	0.580042
$322$	−4.39230	−0.244774
$323$	6.21166	0.345626
$324$	4.46410	0.248006
$325$	0.757875	0.0420393
$326$	16.4901	0.913302
$327$	−9.80385	−0.542154
$328$	−6.93237	−0.382776
$329$	3.46410	0.190982
$330$	5.37945	0.296129
$331$	8.58682	0.471974	0.235987	−	0.971756i	$-0.424168\pi$
$331$	8.58682	0.471974	0.235987	+	0.971756i	$0.424168\pi$
$332$	−14.1962	−0.779115
$333$	22.6603	1.24177
$334$	24.0000	1.31322
$335$	−25.6317	−1.40041
$336$	5.17638	0.282395
$337$	2.72689	0.148543	0.0742716	−	0.997238i	$-0.476337\pi$
$337$	2.72689	0.148543	0.0742716	+	0.997238i	$0.476337\pi$
$338$	−22.2679	−1.21122
$339$	−10.9348	−0.593895
$340$	−1.55291	−0.0842186
$341$	15.4641	0.837428
$342$	29.5692	1.59892
$343$	16.9706	0.916324
$344$	−10.7321	−0.578633
$345$	2.27362	0.122408
$346$	23.5983	1.26865
$347$	−3.58630	−0.192523	−0.0962614	−	0.995356i	$-0.530688\pi$
$347$	−3.58630	−0.192523	−0.0962614	+	0.995356i	$0.530688\pi$
$348$	−1.85641	−0.0995138
$349$	19.2814	1.03211	0.516054	−	0.856556i	$-0.327401\pi$
$349$	19.2814	1.03211	0.516054	+	0.856556i	$0.327401\pi$
$350$	4.89898	0.261861
$351$	1.51575	0.0809047
$352$	−12.7279	−0.678401
$353$	−2.44949	−0.130373	−0.0651866	−	0.997873i	$-0.520764\pi$
$353$	−2.44949	−0.130373	−0.0651866	+	0.997873i	$0.520764\pi$
$354$	−14.6969	−0.781133
$355$	0	0
$356$	12.9282	0.685193
$357$	0.928203	0.0491257
$358$	−10.3923	−0.549250
$359$	−8.87564	−0.468439	−0.234219	−	0.972184i	$-0.575253\pi$
$359$	−8.87564	−0.468439	−0.234219	+	0.972184i	$0.575253\pi$
$360$	7.39230	0.389609
$361$	29.0000	1.52632
$362$	−22.2856	−1.17131
$363$	−3.66025	−0.192114
$364$	0.535898	0.0280887
$365$	9.58846	0.501883
$366$	17.4510	0.912180
$367$	−4.73205	−0.247011	−0.123506	−	0.992344i	$-0.539414\pi$
$367$	−4.73205	−0.247011	−0.123506	+	0.992344i	$0.539414\pi$
$368$	−8.96575	−0.467372
$369$	−9.86233	−0.513412
$370$	−27.5885	−1.43426
$371$	18.5885	0.965065
$372$	4.62158	0.239618
$373$	5.07180	0.262608	0.131304	−	0.991342i	$-0.458084\pi$
$373$	5.07180	0.262608	0.131304	+	0.991342i	$0.458084\pi$
$374$	−3.80385	−0.196692
$375$	−8.87564	−0.458336
$376$	4.24264	0.218797
$377$	0.960947	0.0494913
$378$	9.79796	0.503953
$379$	3.12436	0.160487	0.0802437	−	0.996775i	$-0.474430\pi$
$379$	3.12436	0.160487	0.0802437	+	0.996775i	$0.474430\pi$
$380$	−12.0000	−0.615587
$381$	5.45378	0.279406
$382$	36.0000	1.84192
$383$	25.6317	1.30972	0.654860	−	0.755751i	$-0.272728\pi$
$383$	25.6317	1.30972	0.654860	+	0.755751i	$0.272728\pi$
$384$	8.87564	0.452933
$385$	−6.00000	−0.305788
$386$	−33.0446	−1.68192
$387$	−15.2679	−0.776113
$388$	6.17449	0.313462
$389$	20.7327	1.05119	0.525596	−	0.850735i	$-0.323843\pi$
$389$	20.7327	1.05119	0.525596	+	0.850735i	$0.323843\pi$
$390$	−0.832204	−0.0421403
$391$	−1.60770	−0.0813046
$392$	8.66025	0.437409
$393$	1.85641	0.0936433
$394$	27.4249	1.38164
$395$	3.80385	0.191392
$396$	−6.03579	−0.303310
$397$	6.07296	0.304793	0.152396	−	0.988319i	$-0.451301\pi$
$397$	6.07296	0.304793	0.152396	+	0.988319i	$0.451301\pi$
$398$	−12.0000	−0.601506
$399$	7.17260	0.359079
$400$	10.0000	0.500000
$401$	−38.1194	−1.90359	−0.951796	−	0.306732i	$-0.900764\pi$
$401$	−38.1194	−1.90359	−0.951796	+	0.306732i	$0.900764\pi$
$402$	−18.7637	−0.935849
$403$	−2.39230	−0.119169
$404$	5.19615	0.258518
$405$	7.73205	0.384209
$406$	6.21166	0.308279
$407$	−22.5259	−1.11657
$408$	1.13681	0.0562806
$409$	21.7846	1.07718	0.538590	−	0.842568i	$-0.318957\pi$
$409$	21.7846	1.07718	0.538590	+	0.842568i	$0.318957\pi$
$410$	12.0072	0.592994
$411$	1.96902	0.0971244
$412$	−6.19615	−0.305263
$413$	16.3923	0.806613
$414$	−7.65308	−0.376128
$415$	−24.5885	−1.20700
$416$	1.96902	0.0965390
$417$	1.10961	0.0543376
$418$	−29.3939	−1.43770
$419$	0.588457	0.0287480	0.0143740	−	0.999897i	$-0.495424\pi$
$419$	0.588457	0.0287480	0.0143740	+	0.999897i	$0.495424\pi$
$420$	−1.79315	−0.0874968
$421$	−27.4620	−1.33842	−0.669209	−	0.743075i	$-0.733367\pi$
$421$	−27.4620	−1.33842	−0.669209	+	0.743075i	$0.733367\pi$
$422$	24.4949	1.19239
$423$	6.03579	0.293470
$424$	22.7661	1.10562
$425$	1.79315	0.0869806
$426$	0	0
$427$	−19.4641	−0.941934
$428$	14.1962	0.686197
$429$	−0.679492	−0.0328062
$430$	18.5885	0.896415
$431$	19.5167	0.940084	0.470042	−	0.882644i	$-0.344239\pi$
$431$	19.5167	0.940084	0.470042	+	0.882644i	$0.344239\pi$
$432$	20.0000	0.962250
$433$	−6.55343	−0.314938	−0.157469	−	0.987524i	$-0.550333\pi$
$433$	−6.55343	−0.314938	−0.157469	+	0.987524i	$0.550333\pi$
$434$	−15.4641	−0.742301
$435$	−3.21539	−0.154166
$436$	−13.3923	−0.641375
$437$	−12.4233	−0.594288
$438$	7.01924	0.335392
$439$	−33.5622	−1.60184	−0.800918	−	0.598774i	$-0.795655\pi$
$439$	−33.5622	−1.60184	−0.800918	+	0.598774i	$0.795655\pi$
$440$	−7.34847	−0.350325
$441$	12.3205	0.586691
$442$	0.588457	0.0279901
$443$	1.96902	0.0935508	0.0467754	−	0.998905i	$-0.485105\pi$
$443$	1.96902	0.0935508	0.0467754	+	0.998905i	$0.485105\pi$
$444$	−6.73205	−0.319489
$445$	22.3923	1.06150
$446$	−6.00000	−0.284108
$447$	−4.72311	−0.223396
$448$	−1.41421	−0.0668153
$449$	27.9053	1.31693	0.658467	−	0.752610i	$-0.271205\pi$
$449$	27.9053	1.31693	0.658467	+	0.752610i	$0.271205\pi$
$450$	8.53590	0.402386
$451$	9.80385	0.461645
$452$	−14.9372	−0.702586
$453$	−10.5359	−0.495020
$454$	48.6381	2.28270
$455$	0.928203	0.0435148
$456$	8.78461	0.411377
$457$	−16.5916	−0.776123	−0.388062	−	0.921633i	$-0.626855\pi$
$457$	−16.5916	−0.776123	−0.388062	+	0.921633i	$0.626855\pi$
$458$	13.8564	0.647467
$459$	3.58630	0.167394
$460$	3.10583	0.144810
$461$	33.2204	1.54723	0.773615	−	0.633657i	$-0.218447\pi$
$461$	33.2204	1.54723	0.773615	+	0.633657i	$0.218447\pi$
$462$	−4.39230	−0.204349
$463$	18.0000	0.836531	0.418265	−	0.908325i	$-0.362638\pi$
$463$	18.0000	0.836531	0.418265	+	0.908325i	$0.362638\pi$
$464$	12.6795	0.588631
$465$	8.00481	0.371214
$466$	−2.41154	−0.111713
$467$	12.0716	0.558606	0.279303	−	0.960203i	$-0.409897\pi$
$467$	12.0716	0.558606	0.279303	+	0.960203i	$0.409897\pi$
$468$	0.933740	0.0431622
$469$	20.9282	0.966375
$470$	−7.34847	−0.338960
$471$	3.51666	0.162039
$472$	20.0764	0.924091
$473$	15.1774	0.697858
$474$	2.78461	0.127901
$475$	13.8564	0.635776
$476$	1.26795	0.0581164
$477$	32.3882	1.48295
$478$	9.31749	0.426172
$479$	24.9754	1.14115	0.570577	−	0.821244i	$-0.306720\pi$
$479$	24.9754	1.14115	0.570577	+	0.821244i	$0.306720\pi$
$480$	−6.58846	−0.300721
$481$	3.48477	0.158892
$482$	−34.9492	−1.59189
$483$	−1.85641	−0.0844694
$484$	−5.00000	−0.227273
$485$	10.6945	0.485614
$486$	26.4449	1.19956
$487$	19.6975	0.892577	0.446288	−	0.894889i	$-0.352746\pi$
$487$	19.6975	0.892577	0.446288	+	0.894889i	$0.352746\pi$
$488$	−23.8386	−1.07912
$489$	6.96953	0.315173
$490$	−15.0000	−0.677631
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	2.92996	0.132093
$493$	2.27362	0.102399
$494$	4.54725	0.204590
$495$	−10.4543	−0.469886
$496$	−31.5660	−1.41735
$497$	0	0
$498$	−18.0000	−0.806599
$499$	−38.1051	−1.70582	−0.852910	−	0.522059i	$-0.825164\pi$
$499$	−38.1051	−1.70582	−0.852910	+	0.522059i	$0.825164\pi$
$500$	−12.1244	−0.542218
$501$	10.1436	0.453182
$502$	−20.1962	−0.901398
$503$	−18.5885	−0.828818	−0.414409	−	0.910091i	$-0.636012\pi$
$503$	−18.5885	−0.828818	−0.414409	+	0.910091i	$0.636012\pi$
$504$	−6.03579	−0.268856
$505$	9.00000	0.400495
$506$	7.60770	0.338203
$507$	−9.41154	−0.417981
$508$	7.45001	0.330541
$509$	−6.46410	−0.286516	−0.143258	−	0.989685i	$-0.545758\pi$
$509$	−6.46410	−0.286516	−0.143258	+	0.989685i	$0.545758\pi$
$510$	−1.96902	−0.0871895
$511$	−7.82894	−0.346332
$512$	8.66025	0.382733
$513$	27.7128	1.22355
$514$	24.7351	1.09102
$515$	−10.7321	−0.472911
$516$	4.53590	0.199682
$517$	−6.00000	−0.263880
$518$	22.5259	0.989730
$519$	9.97382	0.437802
$520$	1.13681	0.0498525
$521$	20.0718	0.879361	0.439681	−	0.898154i	$-0.355092\pi$
$521$	20.0718	0.879361	0.439681	+	0.898154i	$0.355092\pi$
$522$	10.8231	0.473714
$523$	35.7071	1.56136	0.780681	−	0.624930i	$-0.214873\pi$
$523$	35.7071	1.56136	0.780681	+	0.624930i	$0.214873\pi$
$524$	2.53590	0.110781
$525$	2.07055	0.0903663
$526$	−34.3923	−1.49958
$527$	−5.66025	−0.246565
$528$	−8.96575	−0.390184
$529$	−19.7846	−0.860200
$530$	−39.4321	−1.71282
$531$	28.5617	1.23947
$532$	9.79796	0.424795
$533$	−1.51666	−0.0656939
$534$	16.3923	0.709364
$535$	24.5885	1.06305
$536$	25.6317	1.10712
$537$	−4.39230	−0.189542
$538$	−35.1894	−1.51712
$539$	−12.2474	−0.527535
$540$	−6.92820	−0.298142
$541$	20.9358	0.900100	0.450050	−	0.893003i	$-0.351406\pi$
$541$	20.9358	0.900100	0.450050	+	0.893003i	$0.351406\pi$
$542$	−37.1769	−1.59689
$543$	−9.41902	−0.404209
$544$	4.65874	0.199742
$545$	−23.1962	−0.993614
$546$	0.679492	0.0290796
$547$	−24.9808	−1.06810	−0.534050	−	0.845453i	$-0.679331\pi$
$547$	−24.9808	−1.06810	−0.534050	+	0.845453i	$0.679331\pi$
$548$	2.68973	0.114899
$549$	−33.9139	−1.44741
$550$	−8.48528	−0.361814
$551$	17.5692	0.748474
$552$	−2.27362	−0.0967719
$553$	−3.10583	−0.132073
$554$	26.7846	1.13797
$555$	−11.6603	−0.494950
$556$	1.51575	0.0642821
$557$	−40.8564	−1.73114	−0.865571	−	0.500787i	$-0.833044\pi$
$557$	−40.8564	−1.73114	−0.865571	+	0.500787i	$0.833044\pi$
$558$	−26.9444	−1.14065
$559$	−2.34795	−0.0993079
$560$	12.2474	0.517549
$561$	−1.60770	−0.0678769
$562$	−8.18067	−0.345081
$563$	−31.6675	−1.33463	−0.667313	−	0.744777i	$-0.732556\pi$
$563$	−31.6675	−1.33463	−0.667313	+	0.744777i	$0.732556\pi$
$564$	−1.79315	−0.0755053
$565$	−25.8719	−1.08844
$566$	24.4949	1.02960
$567$	−6.31319	−0.265129
$568$	0	0
$569$	−43.7321	−1.83334	−0.916671	−	0.399642i	$-0.869135\pi$
$569$	−43.7321	−1.83334	−0.916671	+	0.399642i	$0.869135\pi$
$570$	−15.2154	−0.637303
$571$	−7.26795	−0.304154	−0.152077	−	0.988369i	$-0.548596\pi$
$571$	−7.26795	−0.304154	−0.152077	+	0.988369i	$0.548596\pi$
$572$	−0.928203	−0.0388101
$573$	15.2154	0.635632
$574$	−9.80385	−0.409205
$575$	−3.58630	−0.149559
$576$	−2.46410	−0.102671
$577$	37.3923	1.55666	0.778331	−	0.627854i	$-0.216067\pi$
$577$	37.3923	1.55666	0.778331	+	0.627854i	$0.216067\pi$
$578$	−28.0526	−1.16683
$579$	−13.9663	−0.580418
$580$	−4.39230	−0.182381
$581$	20.0764	0.832909
$582$	7.82894	0.324520
$583$	−32.1962	−1.33343
$584$	−9.58846	−0.396773
$585$	1.61729	0.0668665
$586$	17.1962	0.710367
$587$	0.928203	0.0383110	0.0191555	−	0.999817i	$-0.493902\pi$
$587$	0.928203	0.0383110	0.0191555	+	0.999817i	$0.493902\pi$
$588$	−3.66025	−0.150946
$589$	−43.7391	−1.80224
$590$	−34.7733	−1.43160
$591$	11.5911	0.476795
$592$	45.9808	1.88980
$593$	−13.7321	−0.563908	−0.281954	−	0.959428i	$-0.590983\pi$
$593$	−13.7321	−0.563908	−0.281954	+	0.959428i	$0.590983\pi$
$594$	−16.9706	−0.696311
$595$	2.19615	0.0900335
$596$	−6.45189	−0.264280
$597$	−5.07180	−0.207575
$598$	−1.17691	−0.0481276
$599$	17.6269	0.720216	0.360108	−	0.932911i	$-0.382740\pi$
$599$	17.6269	0.720216	0.360108	+	0.932911i	$0.382740\pi$
$600$	2.53590	0.103528
$601$	35.1151	1.43237	0.716187	−	0.697908i	$-0.245886\pi$
$601$	35.1151	1.43237	0.716187	+	0.697908i	$0.245886\pi$
$602$	−15.1774	−0.618585
$603$	36.4649	1.48497
$604$	−14.3923	−0.585615
$605$	−8.66025	−0.352089
$606$	6.58846	0.267638
$607$	−30.6322	−1.24332	−0.621662	−	0.783286i	$-0.713542\pi$
$607$	−30.6322	−1.24332	−0.621662	+	0.783286i	$0.713542\pi$
$608$	36.0000	1.45999
$609$	2.62536	0.106385
$610$	41.2896	1.67177
$611$	0.928203	0.0375511
$612$	2.20925	0.0893038
$613$	7.00000	0.282727	0.141364	−	0.989958i	$-0.454851\pi$
$613$	7.00000	0.282727	0.141364	+	0.989958i	$0.454851\pi$
$614$	−50.1266	−2.02295
$615$	5.07484	0.204637
$616$	6.00000	0.241747
$617$	−15.0000	−0.603877	−0.301939	−	0.953327i	$-0.597634\pi$
$617$	−15.0000	−0.603877	−0.301939	+	0.953327i	$0.597634\pi$
$618$	−7.85641	−0.316031
$619$	−34.2185	−1.37536	−0.687679	−	0.726015i	$-0.741370\pi$
$619$	−34.2185	−1.37536	−0.687679	+	0.726015i	$0.741370\pi$
$620$	10.9348	0.439151
$621$	−7.17260	−0.287827
$622$	−10.9808	−0.440288
$623$	−18.2832	−0.732503
$624$	1.38701	0.0555247
$625$	−11.0000	−0.440000
$626$	−22.1769	−0.886368
$627$	−12.4233	−0.496139
$628$	4.80385	0.191694
$629$	8.24504	0.328751
$630$	10.4543	0.416509
$631$	−9.69642	−0.386009	−0.193004	−	0.981198i	$-0.561823\pi$
$631$	−9.69642	−0.386009	−0.193004	+	0.981198i	$0.561823\pi$
$632$	−3.80385	−0.151309
$633$	10.3528	0.411485
$634$	33.2204	1.31935
$635$	12.9038	0.512071
$636$	−9.62209	−0.381541
$637$	1.89469	0.0750702
$638$	−10.7589	−0.425949
$639$	0	0
$640$	21.0000	0.830098
$641$	20.0718	0.792788	0.396394	−	0.918080i	$-0.370261\pi$
$641$	20.0718	0.792788	0.396394	+	0.918080i	$0.370261\pi$
$642$	18.0000	0.710403
$643$	−8.19615	−0.323225	−0.161612	−	0.986854i	$-0.551669\pi$
$643$	−8.19615	−0.323225	−0.161612	+	0.986854i	$0.551669\pi$
$644$	−2.53590	−0.0999284
$645$	7.85641	0.309346
$646$	10.7589	0.423303
$647$	−15.8038	−0.621313	−0.310657	−	0.950522i	$-0.600549\pi$
$647$	−15.8038	−0.621313	−0.310657	+	0.950522i	$0.600549\pi$
$648$	−7.73205	−0.303744
$649$	−28.3923	−1.11450
$650$	1.31268	0.0514875
$651$	−6.53590	−0.256162
$652$	9.52056	0.372854
$653$	16.2499	0.635906	0.317953	−	0.948106i	$-0.397005\pi$
$653$	16.2499	0.635906	0.317953	+	0.948106i	$0.397005\pi$
$654$	−16.9808	−0.664000
$655$	4.39230	0.171622
$656$	−20.0120	−0.781338
$657$	−13.6410	−0.532187
$658$	6.00000	0.233904
$659$	−34.0526	−1.32650	−0.663250	−	0.748398i	$-0.730823\pi$
$659$	−34.0526	−1.32650	−0.663250	+	0.748398i	$0.730823\pi$
$660$	3.10583	0.120894
$661$	−39.4593	−1.53479	−0.767394	−	0.641176i	$-0.778447\pi$
$661$	−39.4593	−1.53479	−0.767394	+	0.641176i	$0.778447\pi$
$662$	14.8728	0.578048
$663$	0.248711	0.00965915
$664$	24.5885	0.954217
$665$	16.9706	0.658090
$666$	39.2487	1.52086
$667$	−4.54725	−0.176070
$668$	13.8564	0.536120
$669$	−2.53590	−0.0980435
$670$	−44.3954	−1.71514
$671$	33.7128	1.30147
$672$	5.37945	0.207517
$673$	−0.693504	−0.0267326	−0.0133663	−	0.999911i	$-0.504255\pi$
$673$	−0.693504	−0.0267326	−0.0133663	+	0.999911i	$0.504255\pi$
$674$	4.72311	0.181928
$675$	8.00000	0.307920
$676$	−12.8564	−0.494477
$677$	28.6410	1.10076	0.550382	−	0.834913i	$-0.314482\pi$
$677$	28.6410	1.10076	0.550382	+	0.834913i	$0.314482\pi$
$678$	−18.9396	−0.727370
$679$	−8.73205	−0.335105
$680$	2.68973	0.103146
$681$	20.5569	0.787741
$682$	26.7846	1.02564
$683$	−39.0160	−1.49290	−0.746452	−	0.665439i	$-0.768244\pi$
$683$	−39.0160	−1.49290	−0.746452	+	0.665439i	$0.768244\pi$
$684$	17.0718	0.652756
$685$	4.65874	0.178001
$686$	29.3939	1.12226
$687$	5.85641	0.223436
$688$	−30.9808	−1.18113
$689$	4.98076	0.189752
$690$	3.93803	0.149918
$691$	−14.1421	−0.537992	−0.268996	−	0.963141i	$-0.586692\pi$
$691$	−14.1421	−0.537992	−0.268996	+	0.963141i	$0.586692\pi$
$692$	13.6245	0.517926
$693$	8.53590	0.324252
$694$	−6.21166	−0.235791
$695$	2.62536	0.0995854
$696$	3.21539	0.121879
$697$	−3.58846	−0.135923
$698$	33.3963	1.26407
$699$	−1.01924	−0.0385511
$700$	2.82843	0.106904
$701$	−29.6341	−1.11927	−0.559633	−	0.828741i	$-0.689058\pi$
$701$	−29.6341	−1.11927	−0.559633	+	0.828741i	$0.689058\pi$
$702$	2.62536	0.0990876
$703$	63.7128	2.40297
$704$	2.44949	0.0923186
$705$	−3.10583	−0.116972
$706$	−4.24264	−0.159674
$707$	−7.34847	−0.276368
$708$	−8.48528	−0.318896
$709$	40.9478	1.53783	0.768914	−	0.639352i	$-0.220797\pi$
$709$	40.9478	1.53783	0.768914	+	0.639352i	$0.220797\pi$
$710$	0	0
$711$	−5.41154	−0.202949
$712$	−22.3923	−0.839187
$713$	11.3205	0.423956
$714$	1.60770	0.0601665
$715$	−1.60770	−0.0601244
$716$	−6.00000	−0.224231
$717$	3.93803	0.147069
$718$	−15.3731	−0.573718
$719$	18.5885	0.693232	0.346616	−	0.938007i	$-0.387331\pi$
$719$	18.5885	0.693232	0.346616	+	0.938007i	$0.387331\pi$
$720$	21.3397	0.795285
$721$	8.76268	0.326339
$722$	50.2295	1.86935
$723$	−14.7713	−0.549349
$724$	−12.8666	−0.478184
$725$	5.07180	0.188362
$726$	−6.33975	−0.235290
$727$	−10.1769	−0.377440	−0.188720	−	0.982031i	$-0.560434\pi$
$727$	−10.1769	−0.377440	−0.188720	+	0.982031i	$0.560434\pi$
$728$	−0.928203	−0.0344015
$729$	−2.21539	−0.0820515
$730$	16.6077	0.614678
$731$	−5.55532	−0.205471
$732$	10.0754	0.372396
$733$	−31.7047	−1.17104	−0.585519	−	0.810658i	$-0.699109\pi$
$733$	−31.7047	−1.17104	−0.585519	+	0.810658i	$0.699109\pi$
$734$	−8.19615	−0.302526
$735$	−6.33975	−0.233845
$736$	−9.31749	−0.343447
$737$	−36.2487	−1.33524
$738$	−17.0821	−0.628799
$739$	3.60770	0.132711	0.0663556	−	0.997796i	$-0.478863\pi$
$739$	3.60770	0.132711	0.0663556	+	0.997796i	$0.478863\pi$
$740$	−15.9282	−0.585532
$741$	1.92189	0.0706025
$742$	32.1962	1.18196
$743$	20.2523	0.742983	0.371492	−	0.928436i	$-0.378846\pi$
$743$	20.2523	0.742983	0.371492	+	0.928436i	$0.378846\pi$
$744$	−8.00481	−0.293471
$745$	−11.1750	−0.409421
$746$	8.78461	0.321627
$747$	34.9808	1.27988
$748$	−2.19615	−0.0802993
$749$	−20.0764	−0.733575
$750$	−15.3731	−0.561345
$751$	−50.0251	−1.82544	−0.912720	−	0.408585i	$-0.866022\pi$
$751$	−50.0251	−1.82544	−0.912720	+	0.408585i	$0.866022\pi$
$752$	12.2474	0.446619
$753$	−8.53590	−0.311065
$754$	1.66441	0.0606142
$755$	−24.9282	−0.907230
$756$	5.65685	0.205738
$757$	−13.4858	−0.490150	−0.245075	−	0.969504i	$-0.578813\pi$
$757$	−13.4858	−0.490150	−0.245075	+	0.969504i	$0.578813\pi$
$758$	5.41154	0.196556
$759$	3.21539	0.116711
$760$	20.7846	0.753937
$761$	4.83461	0.175254	0.0876272	−	0.996153i	$-0.472072\pi$
$761$	4.83461	0.175254	0.0876272	+	0.996153i	$0.472072\pi$
$762$	9.44623	0.342201
$763$	18.9396	0.685659
$764$	20.7846	0.751961
$765$	3.82654	0.138349
$766$	44.3954	1.60407
$767$	4.39230	0.158597
$768$	13.9090	0.501897
$769$	25.7704	0.929305	0.464652	−	0.885493i	$-0.346179\pi$
$769$	25.7704	0.929305	0.464652	+	0.885493i	$0.346179\pi$
$770$	−10.3923	−0.374513
$771$	10.4543	0.376502
$772$	−19.0783	−0.686642
$773$	25.8719	0.930549	0.465275	−	0.885166i	$-0.345956\pi$
$773$	25.8719	0.930549	0.465275	+	0.885166i	$0.345956\pi$
$774$	−26.4449	−0.950541
$775$	−12.6264	−0.453553
$776$	−10.6945	−0.383911
$777$	9.52056	0.341548
$778$	35.9101	1.28744
$779$	−27.7295	−0.993511
$780$	−0.480473	−0.0172037
$781$	0	0
$782$	−2.78461	−0.0995774
$783$	10.1436	0.362502
$784$	25.0000	0.892857
$785$	8.32051	0.296972
$786$	3.21539	0.114689
$787$	−18.0000	−0.641631	−0.320815	−	0.947142i	$-0.603957\pi$
$787$	−18.0000	−0.641631	−0.320815	+	0.947142i	$0.603957\pi$
$788$	15.8338	0.564054
$789$	−14.5359	−0.517492
$790$	6.58846	0.234407
$791$	21.1244	0.751096
$792$	10.4543	0.371477
$793$	−5.21539	−0.185204
$794$	10.5187	0.373294
$795$	−16.6660	−0.591081
$796$	−6.92820	−0.245564
$797$	−33.9282	−1.20180	−0.600899	−	0.799325i	$-0.705191\pi$
$797$	−33.9282	−1.20180	−0.600899	+	0.799325i	$0.705191\pi$
$798$	12.4233	0.439781
$799$	2.19615	0.0776943
$800$	10.3923	0.367423
$801$	−31.8564	−1.12559
$802$	−66.0247	−2.33141
$803$	13.5601	0.478526
$804$	−10.8332	−0.382059
$805$	−4.39230	−0.154808
$806$	−4.14359	−0.145952
$807$	−14.8728	−0.523547
$808$	−9.00000	−0.316619
$809$	−0.175865	−0.00618310	−0.00309155	−	0.999995i	$-0.500984\pi$
$809$	−0.175865	−0.00618310	−0.00309155	+	0.999995i	$0.500984\pi$
$810$	13.3923	0.470558
$811$	−9.41154	−0.330484	−0.165242	−	0.986253i	$-0.552841\pi$
$811$	−9.41154	−0.330484	−0.165242	+	0.986253i	$0.552841\pi$
$812$	3.58630	0.125855
$813$	−15.7128	−0.551072
$814$	−39.0160	−1.36751
$815$	16.4901	0.577623
$816$	3.28169	0.114882
$817$	−42.9282	−1.50187
$818$	37.7321	1.31927
$819$	−1.32051	−0.0461423
$820$	6.93237	0.242089
$821$	45.2487	1.57919	0.789595	−	0.613628i	$-0.210290\pi$
$821$	45.2487	1.57919	0.789595	+	0.613628i	$0.210290\pi$
$822$	3.41044	0.118953
$823$	−1.41421	−0.0492964	−0.0246482	−	0.999696i	$-0.507847\pi$
$823$	−1.41421	−0.0492964	−0.0246482	+	0.999696i	$0.507847\pi$
$824$	10.7321	0.373869
$825$	−3.58630	−0.124859
$826$	28.3923	0.987895
$827$	1.96902	0.0684694	0.0342347	−	0.999414i	$-0.489101\pi$
$827$	1.96902	0.0684694	0.0342347	+	0.999414i	$0.489101\pi$
$828$	−4.41851	−0.153554
$829$	38.3205	1.33093	0.665463	−	0.746431i	$-0.268234\pi$
$829$	38.3205	1.33093	0.665463	+	0.746431i	$0.268234\pi$
$830$	−42.5885	−1.47827
$831$	11.3205	0.392704
$832$	−0.378937	−0.0131373
$833$	4.48288	0.155322
$834$	1.92189	0.0665497
$835$	24.0000	0.830554
$836$	−16.9706	−0.586939
$837$	−25.2528	−0.872863
$838$	1.01924	0.0352090
$839$	−5.07180	−0.175098	−0.0875489	−	0.996160i	$-0.527903\pi$
$839$	−5.07180	−0.175098	−0.0875489	+	0.996160i	$0.527903\pi$
$840$	3.10583	0.107161
$841$	−22.5692	−0.778249
$842$	−47.5656	−1.63922
$843$	−3.45756	−0.119085
$844$	14.1421	0.486792
$845$	−22.2679	−0.766041
$846$	10.4543	0.359426
$847$	7.07107	0.242965
$848$	65.7201	2.25684
$849$	10.3528	0.355306
$850$	3.10583	0.106529
$851$	−16.4901	−0.565273
$852$	0	0
$853$	29.3923	1.00637	0.503187	−	0.864178i	$-0.332161\pi$
$853$	29.3923	1.00637	0.503187	+	0.864178i	$0.332161\pi$
$854$	−33.7128	−1.15363
$855$	29.5692	1.01125
$856$	−24.5885	−0.840416
$857$	14.0718	0.480683	0.240342	−	0.970688i	$-0.422741\pi$
$857$	14.0718	0.480683	0.240342	+	0.970688i	$0.422741\pi$
$858$	−1.17691	−0.0401792
$859$	19.5216	0.666068	0.333034	−	0.942915i	$-0.391928\pi$
$859$	19.5216	0.666068	0.333034	+	0.942915i	$0.391928\pi$
$860$	10.7321	0.365960
$861$	−4.14359	−0.141213
$862$	33.8038	1.15136
$863$	−12.9038	−0.439250	−0.219625	−	0.975584i	$-0.570483\pi$
$863$	−12.9038	−0.439250	−0.219625	+	0.975584i	$0.570483\pi$
$864$	20.7846	0.707107
$865$	23.5983	0.802367
$866$	−11.3509	−0.385718
$867$	−11.8564	−0.402665
$868$	−8.92820	−0.303043
$869$	5.37945	0.182485
$870$	−5.56922	−0.188814
$871$	5.60770	0.190010
$872$	23.1962	0.785521
$873$	−15.2146	−0.514935
$874$	−21.5178	−0.727851
$875$	17.1464	0.579655
$876$	4.05256	0.136923
$877$	−8.90897	−0.300834	−0.150417	−	0.988623i	$-0.548062\pi$
$877$	−8.90897	−0.300834	−0.150417	+	0.988623i	$0.548062\pi$
$878$	−58.1314	−1.96184
$879$	7.26795	0.245142
$880$	−21.2132	−0.715097
$881$	58.5167	1.97148	0.985738	−	0.168286i	$-0.0538233\pi$
$881$	58.5167	1.97148	0.985738	+	0.168286i	$0.0538233\pi$
$882$	21.3397	0.718547
$883$	7.62587	0.256631	0.128315	−	0.991733i	$-0.459043\pi$
$883$	7.62587	0.256631	0.128315	+	0.991733i	$0.459043\pi$
$884$	0.339746	0.0114269
$885$	−14.6969	−0.494032
$886$	3.41044	0.114576
$887$	−15.3533	−0.515513	−0.257756	−	0.966210i	$-0.582983\pi$
$887$	−15.3533	−0.515513	−0.257756	+	0.966210i	$0.582983\pi$
$888$	11.6603	0.391293
$889$	−10.5359	−0.353363
$890$	38.7846	1.30006
$891$	10.9348	0.366329
$892$	−3.46410	−0.115987
$893$	16.9706	0.567898
$894$	−8.18067	−0.273603
$895$	−10.3923	−0.347376
$896$	−17.1464	−0.572822
$897$	−0.497423	−0.0166085
$898$	48.3335	1.61291
$899$	−16.0096	−0.533951
$900$	4.92820	0.164273
$901$	11.7846	0.392602
$902$	16.9808	0.565398
$903$	−6.41473	−0.213469
$904$	25.8719	0.860488
$905$	−22.2856	−0.740800
$906$	−18.2487	−0.606273
$907$	−24.3934	−0.809968	−0.404984	−	0.914324i	$-0.632723\pi$
$907$	−24.3934	−0.809968	−0.404984	+	0.914324i	$0.632723\pi$
$908$	28.0812	0.931907
$909$	−12.8038	−0.424677
$910$	1.60770	0.0532946
$911$	−56.1624	−1.86074	−0.930372	−	0.366618i	$-0.880516\pi$
$911$	−56.1624	−1.86074	−0.930372	+	0.366618i	$0.880516\pi$
$912$	25.3590	0.839720
$913$	−34.7733	−1.15083
$914$	−28.7375	−0.950553
$915$	17.4510	0.576913
$916$	8.00000	0.264327
$917$	−3.58630	−0.118430
$918$	6.21166	0.205015
$919$	−16.1112	−0.531458	−0.265729	−	0.964048i	$-0.585613\pi$
$919$	−16.1112	−0.531458	−0.265729	+	0.964048i	$0.585613\pi$
$920$	−5.37945	−0.177355
$921$	−21.1860	−0.698102
$922$	57.5394	1.89496
$923$	0	0
$924$	−2.53590	−0.0834249
$925$	18.3923	0.604735
$926$	31.1769	1.02454
$927$	15.2679	0.501465
$928$	13.1769	0.432553
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	13.8647	0.454643
$931$	34.6410	1.13531
$932$	−1.39230	−0.0456065
$933$	−4.64102	−0.151940
$934$	20.9086	0.684150
$935$	−3.80385	−0.124399
$936$	−1.61729	−0.0528626
$937$	58.0571	1.89664	0.948321	−	0.317312i	$-0.102780\pi$
$937$	58.0571	1.89664	0.948321	+	0.317312i	$0.102780\pi$
$938$	36.2487	1.18356
$939$	−9.37307	−0.305878
$940$	−4.24264	−0.138380
$941$	31.9808	1.04254	0.521272	−	0.853391i	$-0.325458\pi$
$941$	31.9808	1.04254	0.521272	+	0.853391i	$0.325458\pi$
$942$	6.09103	0.198457
$943$	7.17691	0.233712
$944$	57.9555	1.88629
$945$	9.79796	0.318728
$946$	26.2880	0.854698
$947$	17.6603	0.573881	0.286941	−	0.957948i	$-0.407362\pi$
$947$	17.6603	0.573881	0.286941	+	0.957948i	$0.407362\pi$
$948$	1.60770	0.0522155
$949$	−2.09776	−0.0680961
$950$	24.0000	0.778663
$951$	14.0406	0.455298
$952$	−2.19615	−0.0711777
$953$	−23.7846	−0.770459	−0.385230	−	0.922821i	$-0.625878\pi$
$953$	−23.7846	−0.770459	−0.385230	+	0.922821i	$0.625878\pi$
$954$	56.0980	1.81624
$955$	36.0000	1.16493
$956$	5.37945	0.173984
$957$	−4.54725	−0.146992
$958$	43.2586	1.39762
$959$	−3.80385	−0.122833
$960$	1.26795	0.0409229
$961$	8.85641	0.285691
$962$	6.03579	0.194602
$963$	−34.9808	−1.12724
$964$	−20.1779	−0.649887
$965$	−33.0446	−1.06374
$966$	−3.21539	−0.103453
$967$	3.38323	0.108797	0.0543987	−	0.998519i	$-0.482676\pi$
$967$	3.38323	0.108797	0.0543987	+	0.998519i	$0.482676\pi$
$968$	8.66025	0.278351
$969$	4.54725	0.146079
$970$	18.5235	0.594753
$971$	19.9474	0.640144	0.320072	−	0.947393i	$-0.396293\pi$
$971$	19.9474	0.640144	0.320072	+	0.947393i	$0.396293\pi$
$972$	15.2679	0.489720
$973$	−2.14359	−0.0687205
$974$	34.1170	1.09318
$975$	0.554803	0.0177679
$976$	−68.8160	−2.20275
$977$	20.6603	0.660980	0.330490	−	0.943809i	$-0.392786\pi$
$977$	20.6603	0.660980	0.330490	+	0.943809i	$0.392786\pi$
$978$	12.0716	0.386007
$979$	31.6675	1.01210
$980$	−8.66025	−0.276642
$981$	33.0000	1.05361
$982$	0	0
$983$	41.2295	1.31502	0.657508	−	0.753448i	$-0.271611\pi$
$983$	41.2295	1.31502	0.657508	+	0.753448i	$0.271611\pi$
$984$	−5.07484	−0.161780
$985$	27.4249	0.873829
$986$	3.93803	0.125413
$987$	2.53590	0.0807185
$988$	2.62536	0.0835237
$989$	11.1106	0.353298
$990$	−18.1074	−0.575490
$991$	42.7038	1.35653	0.678266	−	0.734817i	$-0.262732\pi$
$991$	42.7038	1.35653	0.678266	+	0.734817i	$0.262732\pi$
$992$	−32.8043	−1.04154
$993$	6.28599	0.199480
$994$	0	0
$995$	−12.0000	−0.380426
$996$	−10.3923	−0.329293
$997$	−48.5692	−1.53820	−0.769101	−	0.639127i	$-0.779296\pi$
$997$	−48.5692	−1.53820	−0.769101	+	0.639127i	$0.779296\pi$
$998$	−66.0000	−2.08919
$999$	36.7846	1.16381

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5041.2.a.c.1.3	✓	4
71.70	odd	2	inner	5041.2.a.c.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5041.2.a.c.1.3	✓	4	1.1	even	1	trivial
5041.2.a.c.1.4	yes	4	71.70	odd	2	inner

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:	\( N \)	\(=\)	\( 5041 = 71^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5041.a (trivial)

\(f(q)\)	\(=\)	\(q+1.73205 q^{2} +0.732051 q^{3} +1.00000 q^{4} +1.73205 q^{5} +1.26795 q^{6} -1.41421 q^{7} -1.73205 q^{8} -2.46410 q^{9} +3.00000 q^{10} +2.44949 q^{11} +0.732051 q^{12} -0.378937 q^{13} -2.44949 q^{14} +1.26795 q^{15} -5.00000 q^{16} -0.896575 q^{17} -4.26795 q^{18} -6.92820 q^{19} +1.73205 q^{20} -1.03528 q^{21} +4.24264 q^{22} +1.79315 q^{23} -1.26795 q^{24} -2.00000 q^{25} -0.656339 q^{26} -4.00000 q^{27} -1.41421 q^{28} -2.53590 q^{29} +2.19615 q^{30} +6.31319 q^{31} -5.19615 q^{32} +1.79315 q^{33} -1.55291 q^{34} -2.44949 q^{35} -2.46410 q^{36} -9.19615 q^{37} -12.0000 q^{38} -0.277401 q^{39} -3.00000 q^{40} +4.00240 q^{41} -1.79315 q^{42} +6.19615 q^{43} +2.44949 q^{44} -4.26795 q^{45} +3.10583 q^{46} -2.44949 q^{47} -3.66025 q^{48} -5.00000 q^{49} -3.46410 q^{50} -0.656339 q^{51} -0.378937 q^{52} -13.1440 q^{53} -6.92820 q^{54} +4.24264 q^{55} +2.44949 q^{56} -5.07180 q^{57} -4.39230 q^{58} -11.5911 q^{59} +1.26795 q^{60} +13.7632 q^{61} +10.9348 q^{62} +3.48477 q^{63} +1.00000 q^{64} -0.656339 q^{65} +3.10583 q^{66} -14.7985 q^{67} -0.896575 q^{68} +1.31268 q^{69} -4.24264 q^{70} +4.26795 q^{72} +5.53590 q^{73} -15.9282 q^{74} -1.46410 q^{75} -6.92820 q^{76} -3.46410 q^{77} -0.480473 q^{78} +2.19615 q^{79} -8.66025 q^{80} +4.46410 q^{81} +6.93237 q^{82} -14.1962 q^{83} -1.03528 q^{84} -1.55291 q^{85} +10.7321 q^{86} -1.85641 q^{87} -4.24264 q^{88} +12.9282 q^{89} -7.39230 q^{90} +0.535898 q^{91} +1.79315 q^{92} +4.62158 q^{93} -4.24264 q^{94} -12.0000 q^{95} -3.80385 q^{96} +6.17449 q^{97} -8.66025 q^{98} -6.03579 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 4 q^{4} + 12 q^{6} + 4 q^{9} + 12 q^{10} - 4 q^{12} + 12 q^{15} - 20 q^{16} - 24 q^{18} - 12 q^{24} - 8 q^{25} - 16 q^{27} - 24 q^{29} - 12 q^{30} + 4 q^{36} - 16 q^{37} - 48 q^{38} - 12 q^{40}+ \cdots - 36 q^{96}+O(q^{100}) \) 4 * q - 4 * q^3 + 4 * q^4 + 12 * q^6 + 4 * q^9 + 12 * q^10 - 4 * q^12 + 12 * q^15 - 20 * q^16 - 24 * q^18 - 12 * q^24 - 8 * q^25 - 16 * q^27 - 24 * q^29 - 12 * q^30 + 4 * q^36 - 16 * q^37 - 48 * q^38 - 12 * q^40 + 4 * q^43 - 24 * q^45 + 20 * q^48 - 20 * q^49 - 48 * q^57 + 24 * q^58 + 12 * q^60 + 4 * q^64 + 24 * q^72 + 36 * q^73 - 36 * q^74 + 8 * q^75 - 12 * q^79 + 4 * q^81 - 36 * q^83 + 36 * q^86 + 48 * q^87 + 24 * q^89 + 12 * q^90 + 16 * q^91 - 48 * q^95 - 36 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more