LMFDB - Embedded newform 507.2.a.l.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,2,Mod(1,507)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("507.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 507 = 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	507.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$4.04841538248$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.24698$ of defining polynomial
Character	$\chi$	$=$	507.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.69202 q^{2} -1.00000 q^{3} +5.24698 q^{4} -1.04892 q^{5} -2.69202 q^{6} +0.554958 q^{7} +8.74094 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.69202 q^{2} -1.00000 q^{3} +5.24698 q^{4} -1.04892 q^{5} -2.69202 q^{6} +0.554958 q^{7} +8.74094 q^{8} +1.00000 q^{9} -2.82371 q^{10} +2.91185 q^{11} -5.24698 q^{12} +1.49396 q^{14} +1.04892 q^{15} +13.0368 q^{16} -4.85086 q^{17} +2.69202 q^{18} -0.753020 q^{19} -5.50365 q^{20} -0.554958 q^{21} +7.83877 q^{22} +5.76271 q^{23} -8.74094 q^{24} -3.89977 q^{25} -1.00000 q^{27} +2.91185 q^{28} -1.91185 q^{29} +2.82371 q^{30} -9.51573 q^{31} +17.6136 q^{32} -2.91185 q^{33} -13.0586 q^{34} -0.582105 q^{35} +5.24698 q^{36} +5.75302 q^{37} -2.02715 q^{38} -9.16852 q^{40} +4.91185 q^{41} -1.49396 q^{42} -11.0978 q^{43} +15.2784 q^{44} -1.04892 q^{45} +15.5133 q^{46} +0.753020 q^{47} -13.0368 q^{48} -6.69202 q^{49} -10.4983 q^{50} +4.85086 q^{51} -7.58211 q^{53} -2.69202 q^{54} -3.05429 q^{55} +4.85086 q^{56} +0.753020 q^{57} -5.14675 q^{58} +4.09783 q^{59} +5.50365 q^{60} -3.42327 q^{61} -25.6165 q^{62} +0.554958 q^{63} +21.3424 q^{64} -7.83877 q^{66} -1.87263 q^{67} -25.4523 q^{68} -5.76271 q^{69} -1.56704 q^{70} -10.5036 q^{71} +8.74094 q^{72} +10.4765 q^{73} +15.4873 q^{74} +3.89977 q^{75} -3.95108 q^{76} +1.61596 q^{77} +1.33513 q^{79} -13.6746 q^{80} +1.00000 q^{81} +13.2228 q^{82} +2.64310 q^{83} -2.91185 q^{84} +5.08815 q^{85} -29.8756 q^{86} +1.91185 q^{87} +25.4523 q^{88} +9.92692 q^{89} -2.82371 q^{90} +30.2368 q^{92} +9.51573 q^{93} +2.02715 q^{94} +0.789856 q^{95} -17.6136 q^{96} +17.0737 q^{97} -18.0151 q^{98} +2.91185 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} - 3 q^{3} + 11 q^{4} + 6 q^{5} - 3 q^{6} + 2 q^{7} + 12 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 - 3 * q^3 + 11 * q^4 + 6 * q^5 - 3 * q^6 + 2 * q^7 + 12 * q^8 + 3 * q^9	$ 3 q + 3 q^{2} - 3 q^{3} + 11 q^{4} + 6 q^{5} - 3 q^{6} + 2 q^{7} + 12 q^{8} + 3 q^{9} - q^{10} + 5 q^{11} - 11 q^{12} - 5 q^{14} - 6 q^{15} + 11 q^{16} - q^{17} + 3 q^{18} - 7 q^{19} + 15 q^{20} - 2 q^{21} - 9 q^{22} - 12 q^{24} + 11 q^{25} - 3 q^{27} + 5 q^{28} - 2 q^{29} + q^{30} - 16 q^{31} + 22 q^{32} - 5 q^{33} - 8 q^{34} + 4 q^{35} + 11 q^{36} + 22 q^{37} + 3 q^{40} + 11 q^{41} + 5 q^{42} - 15 q^{43} + 16 q^{44} + 6 q^{45} - 7 q^{46} + 7 q^{47} - 11 q^{48} - 15 q^{49} - 3 q^{50} + q^{51} - 17 q^{53} - 3 q^{54} + 3 q^{55} + q^{56} + 7 q^{57} + 12 q^{58} - 6 q^{59} - 15 q^{60} - 13 q^{61} - 2 q^{62} + 2 q^{63} + 9 q^{66} + 11 q^{67} - 13 q^{68} - 24 q^{70} + 12 q^{72} + 6 q^{73} + 15 q^{74} - 11 q^{75} - 21 q^{76} + 15 q^{77} + 3 q^{79} - 20 q^{80} + 3 q^{81} - 3 q^{82} + 12 q^{83} - 5 q^{84} + 19 q^{85} - 29 q^{86} + 2 q^{87} + 13 q^{88} + q^{89} - q^{90} + 7 q^{92} + 16 q^{93} - 21 q^{95} - 22 q^{96} - 5 q^{97} - 29 q^{98} + 5 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 - 3 * q^3 + 11 * q^4 + 6 * q^5 - 3 * q^6 + 2 * q^7 + 12 * q^8 + 3 * q^9 - q^10 + 5 * q^11 - 11 * q^12 - 5 * q^14 - 6 * q^15 + 11 * q^16 - q^17 + 3 * q^18 - 7 * q^19 + 15 * q^20 - 2 * q^21 - 9 * q^22 - 12 * q^24 + 11 * q^25 - 3 * q^27 + 5 * q^28 - 2 * q^29 + q^30 - 16 * q^31 + 22 * q^32 - 5 * q^33 - 8 * q^34 + 4 * q^35 + 11 * q^36 + 22 * q^37 + 3 * q^40 + 11 * q^41 + 5 * q^42 - 15 * q^43 + 16 * q^44 + 6 * q^45 - 7 * q^46 + 7 * q^47 - 11 * q^48 - 15 * q^49 - 3 * q^50 + q^51 - 17 * q^53 - 3 * q^54 + 3 * q^55 + q^56 + 7 * q^57 + 12 * q^58 - 6 * q^59 - 15 * q^60 - 13 * q^61 - 2 * q^62 + 2 * q^63 + 9 * q^66 + 11 * q^67 - 13 * q^68 - 24 * q^70 + 12 * q^72 + 6 * q^73 + 15 * q^74 - 11 * q^75 - 21 * q^76 + 15 * q^77 + 3 * q^79 - 20 * q^80 + 3 * q^81 - 3 * q^82 + 12 * q^83 - 5 * q^84 + 19 * q^85 - 29 * q^86 + 2 * q^87 + 13 * q^88 + q^89 - q^90 + 7 * q^92 + 16 * q^93 - 21 * q^95 - 22 * q^96 - 5 * q^97 - 29 * q^98 + 5 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	2.69202	1.90355	0.951773	−	0.306802i	$-0.0992590\pi$
$2$	2.69202	1.90355	0.951773	+	0.306802i	$0.0992590\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	5.24698	2.62349
$5$	−1.04892	−0.469090	−0.234545	−	0.972105i	$-0.575360\pi$
$5$	−1.04892	−0.469090	−0.234545	+	0.972105i	$0.575360\pi$
$6$	−2.69202	−1.09901
$7$	0.554958	0.209754	0.104877	−	0.994485i	$-0.466555\pi$
$7$	0.554958	0.209754	0.104877	+	0.994485i	$0.466555\pi$
$8$	8.74094	3.09039
$9$	1.00000	0.333333
$10$	−2.82371	−0.892935
$11$	2.91185	0.877957	0.438979	−	0.898498i	$-0.355340\pi$
$11$	2.91185	0.877957	0.438979	+	0.898498i	$0.355340\pi$
$12$	−5.24698	−1.51467
$13$	0	0
$13$	0	0
$14$	1.49396	0.399277
$15$	1.04892	0.270829
$16$	13.0368	3.25921
$17$	−4.85086	−1.17651	−0.588253	−	0.808677i	$-0.700184\pi$
$17$	−4.85086	−1.17651	−0.588253	+	0.808677i	$0.700184\pi$
$18$	2.69202	0.634516
$19$	−0.753020	−0.172755	−0.0863774	−	0.996262i	$-0.527529\pi$
$19$	−0.753020	−0.172755	−0.0863774	+	0.996262i	$0.527529\pi$
$20$	−5.50365	−1.23065
$21$	−0.554958	−0.121102
$22$	7.83877	1.67123
$23$	5.76271	1.20161	0.600804	−	0.799396i	$-0.294847\pi$
$23$	5.76271	1.20161	0.600804	+	0.799396i	$0.294847\pi$
$24$	−8.74094	−1.78424
$25$	−3.89977	−0.779954
$26$	0	0
$27$	−1.00000	−0.192450
$28$	2.91185	0.550289
$29$	−1.91185	−0.355022	−0.177511	−	0.984119i	$-0.556805\pi$
$29$	−1.91185	−0.355022	−0.177511	+	0.984119i	$0.556805\pi$
$30$	2.82371	0.515536
$31$	−9.51573	−1.70908	−0.854538	−	0.519389i	$-0.826159\pi$
$31$	−9.51573	−1.70908	−0.854538	+	0.519389i	$0.826159\pi$
$32$	17.6136	3.11367
$33$	−2.91185	−0.506889
$34$	−13.0586	−2.23953
$35$	−0.582105	−0.0983937
$36$	5.24698	0.874497
$37$	5.75302	0.945791	0.472895	−	0.881119i	$-0.343209\pi$
$37$	5.75302	0.945791	0.472895	+	0.881119i	$0.343209\pi$
$38$	−2.02715	−0.328847
$39$	0	0
$40$	−9.16852	−1.44967
$41$	4.91185	0.767103	0.383551	−	0.923520i	$-0.374701\pi$
$41$	4.91185	0.767103	0.383551	+	0.923520i	$0.374701\pi$
$42$	−1.49396	−0.230523
$43$	−11.0978	−1.69240	−0.846202	−	0.532862i	$-0.821116\pi$
$43$	−11.0978	−1.69240	−0.846202	+	0.532862i	$0.821116\pi$
$44$	15.2784	2.30331
$45$	−1.04892	−0.156363
$46$	15.5133	2.28732
$47$	0.753020	0.109839	0.0549197	−	0.998491i	$-0.482510\pi$
$47$	0.753020	0.109839	0.0549197	+	0.998491i	$0.482510\pi$
$48$	−13.0368	−1.88171
$49$	−6.69202	−0.956003
$50$	−10.4983	−1.48468
$51$	4.85086	0.679256
$52$	0	0
$53$	−7.58211	−1.04148	−0.520741	−	0.853715i	$-0.674344\pi$
$53$	−7.58211	−1.04148	−0.520741	+	0.853715i	$0.674344\pi$
$54$	−2.69202	−0.366338
$55$	−3.05429	−0.411841
$56$	4.85086	0.648223
$57$	0.753020	0.0997400
$58$	−5.14675	−0.675802
$59$	4.09783	0.533493	0.266746	−	0.963767i	$-0.414051\pi$
$59$	4.09783	0.533493	0.266746	+	0.963767i	$0.414051\pi$
$60$	5.50365	0.710518
$61$	−3.42327	−0.438305	−0.219153	−	0.975691i	$-0.570329\pi$
$61$	−3.42327	−0.438305	−0.219153	+	0.975691i	$0.570329\pi$
$62$	−25.6165	−3.25330
$63$	0.554958	0.0699182
$64$	21.3424	2.66780
$65$	0	0
$66$	−7.83877	−0.964886
$67$	−1.87263	−0.228778	−0.114389	−	0.993436i	$-0.536491\pi$
$67$	−1.87263	−0.228778	−0.114389	+	0.993436i	$0.536491\pi$
$68$	−25.4523	−3.08655
$69$	−5.76271	−0.693749
$70$	−1.56704	−0.187297
$71$	−10.5036	−1.24655	−0.623277	−	0.782001i	$-0.714199\pi$
$71$	−10.5036	−1.24655	−0.623277	+	0.782001i	$0.714199\pi$
$72$	8.74094	1.03013
$73$	10.4765	1.22618	0.613091	−	0.790012i	$-0.289926\pi$
$73$	10.4765	1.22618	0.613091	+	0.790012i	$0.289926\pi$
$74$	15.4873	1.80036
$75$	3.89977	0.450307
$76$	−3.95108	−0.453220
$77$	1.61596	0.184155
$78$	0	0
$79$	1.33513	0.150213	0.0751067	−	0.997176i	$-0.476070\pi$
$79$	1.33513	0.150213	0.0751067	+	0.997176i	$0.476070\pi$
$80$	−13.6746	−1.52886
$81$	1.00000	0.111111
$82$	13.2228	1.46022
$83$	2.64310	0.290118	0.145059	−	0.989423i	$-0.453663\pi$
$83$	2.64310	0.290118	0.145059	+	0.989423i	$0.453663\pi$
$84$	−2.91185	−0.317709
$85$	5.08815	0.551887
$86$	−29.8756	−3.22157
$87$	1.91185	0.204972
$88$	25.4523	2.71323
$89$	9.92692	1.05225	0.526126	−	0.850407i	$-0.323644\pi$
$89$	9.92692	1.05225	0.526126	+	0.850407i	$0.323644\pi$
$90$	−2.82371	−0.297645
$91$	0	0
$92$	30.2368	3.15241
$93$	9.51573	0.986735
$94$	2.02715	0.209084
$95$	0.789856	0.0810375
$96$	−17.6136	−1.79768
$97$	17.0737	1.73357	0.866784	−	0.498683i	$-0.166183\pi$
$97$	17.0737	1.73357	0.866784	+	0.498683i	$0.166183\pi$
$98$	−18.0151	−1.81980
$99$	2.91185	0.292652
$100$	−20.4620	−2.04620
$101$	7.32304	0.728670	0.364335	−	0.931268i	$-0.381296\pi$
$101$	7.32304	0.728670	0.364335	+	0.931268i	$0.381296\pi$
$102$	13.0586	1.29299
$103$	−4.21983	−0.415792	−0.207896	−	0.978151i	$-0.566662\pi$
$103$	−4.21983	−0.415792	−0.207896	+	0.978151i	$0.566662\pi$
$104$	0	0
$105$	0.582105	0.0568077
$106$	−20.4112	−1.98251
$107$	6.39373	0.618105	0.309053	−	0.951045i	$-0.399988\pi$
$107$	6.39373	0.618105	0.309053	+	0.951045i	$0.399988\pi$
$108$	−5.24698	−0.504891
$109$	−3.46011	−0.331418	−0.165709	−	0.986175i	$-0.552991\pi$
$109$	−3.46011	−0.331418	−0.165709	+	0.986175i	$0.552991\pi$
$110$	−8.22223	−0.783958
$111$	−5.75302	−0.546053
$112$	7.23490	0.683634
$113$	9.35690	0.880223	0.440111	−	0.897943i	$-0.354939\pi$
$113$	9.35690	0.880223	0.440111	+	0.897943i	$0.354939\pi$
$114$	2.02715	0.189860
$115$	−6.04461	−0.563662
$116$	−10.0315	−0.931398
$117$	0	0
$118$	11.0315	1.01553
$119$	−2.69202	−0.246777
$120$	9.16852	0.836968
$121$	−2.52111	−0.229191
$122$	−9.21552	−0.834334
$123$	−4.91185	−0.442887
$124$	−49.9288	−4.48374
$125$	9.33513	0.834959
$126$	1.49396	0.133092
$127$	−4.48188	−0.397702	−0.198851	−	0.980030i	$-0.563721\pi$
$127$	−4.48188	−0.397702	−0.198851	+	0.980030i	$0.563721\pi$
$128$	22.2271	1.96462
$129$	11.0978	0.977110
$130$	0	0
$131$	−9.21744	−0.805331	−0.402666	−	0.915347i	$-0.631916\pi$
$131$	−9.21744	−0.805331	−0.402666	+	0.915347i	$0.631916\pi$
$132$	−15.2784	−1.32982
$133$	−0.417895	−0.0362361
$134$	−5.04115	−0.435489
$135$	1.04892	0.0902764
$136$	−42.4010	−3.63586
$137$	−7.46980	−0.638188	−0.319094	−	0.947723i	$-0.603379\pi$
$137$	−7.46980	−0.638188	−0.319094	+	0.947723i	$0.603379\pi$
$138$	−15.5133	−1.32058
$139$	−17.9976	−1.52654	−0.763269	−	0.646081i	$-0.776407\pi$
$139$	−17.9976	−1.52654	−0.763269	+	0.646081i	$0.776407\pi$
$140$	−3.05429	−0.258135
$141$	−0.753020	−0.0634158
$142$	−28.2760	−2.37287
$143$	0	0
$144$	13.0368	1.08640
$145$	2.00538	0.166537
$146$	28.2030	2.33409
$147$	6.69202	0.551949
$148$	30.1860	2.48127
$149$	15.3351	1.25630	0.628151	−	0.778091i	$-0.283812\pi$
$149$	15.3351	1.25630	0.628151	+	0.778091i	$0.283812\pi$
$150$	10.4983	0.857180
$151$	2.53079	0.205953	0.102977	−	0.994684i	$-0.467163\pi$
$151$	2.53079	0.205953	0.102977	+	0.994684i	$0.467163\pi$
$152$	−6.58211	−0.533879
$153$	−4.85086	−0.392168
$154$	4.35019	0.350548
$155$	9.98121	0.801710
$156$	0	0
$157$	17.2392	1.37584	0.687919	−	0.725787i	$-0.258524\pi$
$157$	17.2392	1.37584	0.687919	+	0.725787i	$0.258524\pi$
$158$	3.59419	0.285938
$159$	7.58211	0.601300
$160$	−18.4752	−1.46059
$161$	3.19806	0.252043
$162$	2.69202	0.211505
$163$	15.7071	1.23027	0.615137	−	0.788420i	$-0.289101\pi$
$163$	15.7071	1.23027	0.615137	+	0.788420i	$0.289101\pi$
$164$	25.7724	2.01249
$165$	3.05429	0.237776
$166$	7.11529	0.552254
$167$	5.39612	0.417565	0.208782	−	0.977962i	$-0.433050\pi$
$167$	5.39612	0.417565	0.208782	+	0.977962i	$0.433050\pi$
$168$	−4.85086	−0.374252
$169$	0	0
$170$	13.6974	1.05054
$171$	−0.753020	−0.0575849
$172$	−58.2301	−4.44000
$173$	23.9420	1.82028	0.910138	−	0.414306i	$-0.135976\pi$
$173$	23.9420	1.82028	0.910138	+	0.414306i	$0.135976\pi$
$174$	5.14675	0.390174
$175$	−2.16421	−0.163599
$176$	37.9614	2.86145
$177$	−4.09783	−0.308012
$178$	26.7235	2.00301
$179$	18.4088	1.37594	0.687969	−	0.725740i	$-0.258502\pi$
$179$	18.4088	1.37594	0.687969	+	0.725740i	$0.258502\pi$
$180$	−5.50365	−0.410218
$181$	−3.63342	−0.270070	−0.135035	−	0.990841i	$-0.543115\pi$
$181$	−3.63342	−0.270070	−0.135035	+	0.990841i	$0.543115\pi$
$182$	0	0
$183$	3.42327	0.253056
$184$	50.3715	3.71344
$185$	−6.03444	−0.443661
$186$	25.6165	1.87830
$187$	−14.1250	−1.03292
$188$	3.95108	0.288162
$189$	−0.554958	−0.0403673
$190$	2.12631	0.154259
$191$	−21.1782	−1.53240	−0.766201	−	0.642601i	$-0.777855\pi$
$191$	−21.1782	−1.53240	−0.766201	+	0.642601i	$0.777855\pi$
$192$	−21.3424	−1.54026
$193$	−17.6112	−1.26768	−0.633840	−	0.773464i	$-0.718522\pi$
$193$	−17.6112	−1.26768	−0.633840	+	0.773464i	$0.718522\pi$
$194$	45.9627	3.29993
$195$	0	0
$196$	−35.1129	−2.50806
$197$	−4.66248	−0.332188	−0.166094	−	0.986110i	$-0.553116\pi$
$197$	−4.66248	−0.332188	−0.166094	+	0.986110i	$0.553116\pi$
$198$	7.83877	0.557077
$199$	15.0368	1.06593	0.532967	−	0.846136i	$-0.321077\pi$
$199$	15.0368	1.06593	0.532967	+	0.846136i	$0.321077\pi$
$200$	−34.0877	−2.41036
$201$	1.87263	0.132085
$202$	19.7138	1.38706
$203$	−1.06100	−0.0744675
$204$	25.4523	1.78202
$205$	−5.15213	−0.359840
$206$	−11.3599	−0.791480
$207$	5.76271	0.400536
$208$	0	0
$209$	−2.19269	−0.151671
$210$	1.56704	0.108136
$211$	−0.460107	−0.0316751	−0.0158375	−	0.999875i	$-0.505041\pi$
$211$	−0.460107	−0.0316751	−0.0158375	+	0.999875i	$0.505041\pi$
$212$	−39.7832	−2.73232
$213$	10.5036	0.719698
$214$	17.2121	1.17659
$215$	11.6407	0.793890
$216$	−8.74094	−0.594746
$217$	−5.28083	−0.358486
$218$	−9.31468	−0.630870
$219$	−10.4765	−0.707936
$220$	−16.0258	−1.08046
$221$	0	0
$222$	−15.4873	−1.03944
$223$	−16.3502	−1.09489	−0.547445	−	0.836842i	$-0.684399\pi$
$223$	−16.3502	−1.09489	−0.547445	+	0.836842i	$0.684399\pi$
$224$	9.77479	0.653106
$225$	−3.89977	−0.259985
$226$	25.1890	1.67555
$227$	6.56033	0.435425	0.217712	−	0.976013i	$-0.430141\pi$
$227$	6.56033	0.435425	0.217712	+	0.976013i	$0.430141\pi$
$228$	3.95108	0.261667
$229$	−3.95539	−0.261380	−0.130690	−	0.991423i	$-0.541719\pi$
$229$	−3.95539	−0.261380	−0.130690	+	0.991423i	$0.541719\pi$
$230$	−16.2722	−1.07296
$231$	−1.61596	−0.106322
$232$	−16.7114	−1.09716
$233$	8.35690	0.547478	0.273739	−	0.961804i	$-0.411739\pi$
$233$	8.35690	0.547478	0.273739	+	0.961804i	$0.411739\pi$
$234$	0	0
$235$	−0.789856	−0.0515245
$236$	21.5013	1.39961
$237$	−1.33513	−0.0867257
$238$	−7.24698	−0.469752
$239$	20.1008	1.30021	0.650107	−	0.759843i	$-0.274724\pi$
$239$	20.1008	1.30021	0.650107	+	0.759843i	$0.274724\pi$
$240$	13.6746	0.882689
$241$	−19.0127	−1.22471	−0.612357	−	0.790581i	$-0.709778\pi$
$241$	−19.0127	−1.22471	−0.612357	+	0.790581i	$0.709778\pi$
$242$	−6.78687	−0.436277
$243$	−1.00000	−0.0641500
$244$	−17.9618	−1.14989
$245$	7.01938	0.448452
$246$	−13.2228	−0.843056
$247$	0	0
$248$	−83.1764	−5.28171
$249$	−2.64310	−0.167500
$250$	25.1304	1.58938
$251$	−0.763774	−0.0482090	−0.0241045	−	0.999709i	$-0.507673\pi$
$251$	−0.763774	−0.0482090	−0.0241045	+	0.999709i	$0.507673\pi$
$252$	2.91185	0.183430
$253$	16.7802	1.05496
$254$	−12.0653	−0.757045
$255$	−5.08815	−0.318632
$256$	17.1511	1.07194
$257$	−13.0911	−0.816602	−0.408301	−	0.912847i	$-0.633879\pi$
$257$	−13.0911	−0.816602	−0.408301	+	0.912847i	$0.633879\pi$
$258$	29.8756	1.85997
$259$	3.19269	0.198384
$260$	0	0
$261$	−1.91185	−0.118341
$262$	−24.8135	−1.53299
$263$	18.3773	1.13320	0.566598	−	0.823995i	$-0.308259\pi$
$263$	18.3773	1.13320	0.566598	+	0.823995i	$0.308259\pi$
$264$	−25.4523	−1.56648
$265$	7.95300	0.488549
$266$	−1.12498	−0.0689771
$267$	−9.92692	−0.607518
$268$	−9.82563	−0.600196
$269$	23.6625	1.44273	0.721363	−	0.692557i	$-0.243516\pi$
$269$	23.6625	1.44273	0.721363	+	0.692557i	$0.243516\pi$
$270$	2.82371	0.171845
$271$	−19.7530	−1.19991	−0.599955	−	0.800034i	$-0.704815\pi$
$271$	−19.7530	−1.19991	−0.599955	+	0.800034i	$0.704815\pi$
$272$	−63.2398	−3.83448
$273$	0	0
$274$	−20.1089	−1.21482
$275$	−11.3556	−0.684767
$276$	−30.2368	−1.82004
$277$	1.77777	0.106816	0.0534081	−	0.998573i	$-0.482992\pi$
$277$	1.77777	0.106816	0.0534081	+	0.998573i	$0.482992\pi$
$278$	−48.4499	−2.90583
$279$	−9.51573	−0.569692
$280$	−5.08815	−0.304075
$281$	1.62133	0.0967207	0.0483603	−	0.998830i	$-0.484600\pi$
$281$	1.62133	0.0967207	0.0483603	+	0.998830i	$0.484600\pi$
$282$	−2.02715	−0.120715
$283$	4.98361	0.296245	0.148122	−	0.988969i	$-0.452677\pi$
$283$	4.98361	0.296245	0.148122	+	0.988969i	$0.452677\pi$
$284$	−55.1124	−3.27032
$285$	−0.789856	−0.0467870
$286$	0	0
$287$	2.72587	0.160903
$288$	17.6136	1.03789
$289$	6.53079	0.384164
$290$	5.39852	0.317012
$291$	−17.0737	−1.00088
$292$	54.9700	3.21688
$293$	−0.0717525	−0.00419183	−0.00209591	−	0.999998i	$-0.500667\pi$
$293$	−0.0717525	−0.00419183	−0.00209591	+	0.999998i	$0.500667\pi$
$294$	18.0151	1.05066
$295$	−4.29829	−0.250256
$296$	50.2868	2.92286
$297$	−2.91185	−0.168963
$298$	41.2825	2.39143
$299$	0	0
$300$	20.4620	1.18138
$301$	−6.15883	−0.354989
$302$	6.81295	0.392041
$303$	−7.32304	−0.420698
$304$	−9.81700	−0.563044
$305$	3.59073	0.205605
$306$	−13.0586	−0.746511
$307$	−5.19806	−0.296669	−0.148335	−	0.988937i	$-0.547391\pi$
$307$	−5.19806	−0.296669	−0.148335	+	0.988937i	$0.547391\pi$
$308$	8.47889	0.483130
$309$	4.21983	0.240058
$310$	26.8696	1.52609
$311$	−22.5429	−1.27829	−0.639145	−	0.769087i	$-0.720711\pi$
$311$	−22.5429	−1.27829	−0.639145	+	0.769087i	$0.720711\pi$
$312$	0	0
$313$	22.6612	1.28088	0.640442	−	0.768006i	$-0.278751\pi$
$313$	22.6612	1.28088	0.640442	+	0.768006i	$0.278751\pi$
$314$	46.4083	2.61897
$315$	−0.582105	−0.0327979
$316$	7.00538	0.394083
$317$	26.3424	1.47954	0.739769	−	0.672861i	$-0.234935\pi$
$317$	26.3424	1.47954	0.739769	+	0.672861i	$0.234935\pi$
$318$	20.4112	1.14460
$319$	−5.56704	−0.311694
$320$	−22.3864	−1.25144
$321$	−6.39373	−0.356863
$322$	8.60925	0.479775
$323$	3.65279	0.203247
$324$	5.24698	0.291499
$325$	0	0
$326$	42.2838	2.34188
$327$	3.46011	0.191344
$328$	42.9342	2.37065
$329$	0.417895	0.0230393
$330$	8.22223	0.452619
$331$	−11.2295	−0.617230	−0.308615	−	0.951187i	$-0.599866\pi$
$331$	−11.2295	−0.617230	−0.308615	+	0.951187i	$0.599866\pi$
$332$	13.8683	0.761123
$333$	5.75302	0.315264
$334$	14.5265	0.794854
$335$	1.96423	0.107317
$336$	−7.23490	−0.394696
$337$	2.30798	0.125724	0.0628618	−	0.998022i	$-0.479977\pi$
$337$	2.30798	0.125724	0.0628618	+	0.998022i	$0.479977\pi$
$338$	0	0
$339$	−9.35690	−0.508197
$340$	26.6974	1.44787
$341$	−27.7084	−1.50049
$342$	−2.02715	−0.109616
$343$	−7.59850	−0.410280
$344$	−97.0055	−5.23019
$345$	6.04461	0.325431
$346$	64.4523	3.46498
$347$	−9.13706	−0.490503	−0.245252	−	0.969459i	$-0.578871\pi$
$347$	−9.13706	−0.490503	−0.245252	+	0.969459i	$0.578871\pi$
$348$	10.0315	0.537743
$349$	23.9758	1.28340	0.641699	−	0.766957i	$-0.278230\pi$
$349$	23.9758	1.28340	0.641699	+	0.766957i	$0.278230\pi$
$350$	−5.82610	−0.311418
$351$	0	0
$352$	51.2881	2.73367
$353$	−27.1239	−1.44366	−0.721830	−	0.692070i	$-0.756699\pi$
$353$	−27.1239	−1.44366	−0.721830	+	0.692070i	$0.756699\pi$
$354$	−11.0315	−0.586315
$355$	11.0175	0.584746
$356$	52.0863	2.76057
$357$	2.69202	0.142477
$358$	49.5569	2.61916
$359$	−26.0790	−1.37640	−0.688200	−	0.725521i	$-0.741599\pi$
$359$	−26.0790	−1.37640	−0.688200	+	0.725521i	$0.741599\pi$
$360$	−9.16852	−0.483224
$361$	−18.4330	−0.970156
$362$	−9.78123	−0.514090
$363$	2.52111	0.132324
$364$	0	0
$365$	−10.9890	−0.575190
$366$	9.21552	0.481703
$367$	9.57434	0.499776	0.249888	−	0.968275i	$-0.419606\pi$
$367$	9.57434	0.499776	0.249888	+	0.968275i	$0.419606\pi$
$368$	75.1275	3.91629
$369$	4.91185	0.255701
$370$	−16.2448	−0.844530
$371$	−4.20775	−0.218456
$372$	49.9288	2.58869
$373$	28.1497	1.45754	0.728769	−	0.684760i	$-0.240093\pi$
$373$	28.1497	1.45754	0.728769	+	0.684760i	$0.240093\pi$
$374$	−38.0248	−1.96621
$375$	−9.33513	−0.482064
$376$	6.58211	0.339446
$377$	0	0
$378$	−1.49396	−0.0768410
$379$	16.0465	0.824255	0.412127	−	0.911126i	$-0.364786\pi$
$379$	16.0465	0.824255	0.412127	+	0.911126i	$0.364786\pi$
$380$	4.14436	0.212601
$381$	4.48188	0.229614
$382$	−57.0122	−2.91700
$383$	−24.6165	−1.25785	−0.628923	−	0.777467i	$-0.716504\pi$
$383$	−24.6165	−1.25785	−0.628923	+	0.777467i	$0.716504\pi$
$384$	−22.2271	−1.13427
$385$	−1.69501	−0.0863855
$386$	−47.4097	−2.41309
$387$	−11.0978	−0.564135
$388$	89.5852	4.54800
$389$	−17.2198	−0.873080	−0.436540	−	0.899685i	$-0.643796\pi$
$389$	−17.2198	−0.873080	−0.436540	+	0.899685i	$0.643796\pi$
$390$	0	0
$391$	−27.9541	−1.41370
$392$	−58.4946	−2.95442
$393$	9.21744	0.464958
$394$	−12.5515	−0.632335
$395$	−1.40044	−0.0704636
$396$	15.2784	0.767770
$397$	−2.03923	−0.102346	−0.0511730	−	0.998690i	$-0.516296\pi$
$397$	−2.03923	−0.102346	−0.0511730	+	0.998690i	$0.516296\pi$
$398$	40.4795	2.02905
$399$	0.417895	0.0209209
$400$	−50.8407	−2.54203
$401$	−1.46144	−0.0729806	−0.0364903	−	0.999334i	$-0.511618\pi$
$401$	−1.46144	−0.0729806	−0.0364903	+	0.999334i	$0.511618\pi$
$402$	5.04115	0.251430
$403$	0	0
$404$	38.4239	1.91166
$405$	−1.04892	−0.0521211
$406$	−2.85623	−0.141752
$407$	16.7520	0.830364
$408$	42.4010	2.09916
$409$	−29.9390	−1.48039	−0.740194	−	0.672393i	$-0.765266\pi$
$409$	−29.9390	−1.48039	−0.740194	+	0.672393i	$0.765266\pi$
$410$	−13.8696	−0.684973
$411$	7.46980	0.368458
$412$	−22.1414	−1.09083
$413$	2.27413	0.111902
$414$	15.5133	0.762439
$415$	−2.77240	−0.136092
$416$	0	0
$417$	17.9976	0.881347
$418$	−5.90276	−0.288713
$419$	6.64742	0.324748	0.162374	−	0.986729i	$-0.448085\pi$
$419$	6.64742	0.324748	0.162374	+	0.986729i	$0.448085\pi$
$420$	3.05429	0.149034
$421$	13.5646	0.661100	0.330550	−	0.943788i	$-0.392766\pi$
$421$	13.5646	0.661100	0.330550	+	0.943788i	$0.392766\pi$
$422$	−1.23862	−0.0602950
$423$	0.753020	0.0366131
$424$	−66.2747	−3.21858
$425$	18.9172	0.917620
$426$	28.2760	1.36998
$427$	−1.89977	−0.0919364
$428$	33.5478	1.62159
$429$	0	0
$430$	31.3370	1.51121
$431$	35.9463	1.73147	0.865736	−	0.500501i	$-0.166851\pi$
$431$	35.9463	1.73147	0.865736	+	0.500501i	$0.166851\pi$
$432$	−13.0368	−0.627235
$433$	−32.4741	−1.56061	−0.780303	−	0.625402i	$-0.784935\pi$
$433$	−32.4741	−1.56061	−0.780303	+	0.625402i	$0.784935\pi$
$434$	−14.2161	−0.682395
$435$	−2.00538	−0.0961505
$436$	−18.1551	−0.869472
$437$	−4.33944	−0.207583
$438$	−28.2030	−1.34759
$439$	12.8321	0.612441	0.306221	−	0.951961i	$-0.400935\pi$
$439$	12.8321	0.612441	0.306221	+	0.951961i	$0.400935\pi$
$440$	−26.6974	−1.27275
$441$	−6.69202	−0.318668
$442$	0	0
$443$	11.9608	0.568273	0.284137	−	0.958784i	$-0.408293\pi$
$443$	11.9608	0.568273	0.284137	+	0.958784i	$0.408293\pi$
$444$	−30.1860	−1.43256
$445$	−10.4125	−0.493601
$446$	−44.0151	−2.08417
$447$	−15.3351	−0.725327
$448$	11.8442	0.559584
$449$	−12.4789	−0.588915	−0.294458	−	0.955665i	$-0.595139\pi$
$449$	−12.4789	−0.588915	−0.294458	+	0.955665i	$0.595139\pi$
$450$	−10.4983	−0.494893
$451$	14.3026	0.673483
$452$	49.0954	2.30926
$453$	−2.53079	−0.118907
$454$	17.6606	0.828851
$455$	0	0
$456$	6.58211	0.308235
$457$	32.4523	1.51806	0.759028	−	0.651058i	$-0.225674\pi$
$457$	32.4523	1.51806	0.759028	+	0.651058i	$0.225674\pi$
$458$	−10.6480	−0.497549
$459$	4.85086	0.226419
$460$	−31.7159	−1.47876
$461$	24.4034	1.13658	0.568290	−	0.822828i	$-0.307605\pi$
$461$	24.4034	1.13658	0.568290	+	0.822828i	$0.307605\pi$
$462$	−4.35019	−0.202389
$463$	−33.1836	−1.54217	−0.771086	−	0.636731i	$-0.780286\pi$
$463$	−33.1836	−1.54217	−0.771086	+	0.636731i	$0.780286\pi$
$464$	−24.9245	−1.15709
$465$	−9.98121	−0.462868
$466$	22.4969	1.04215
$467$	−38.5206	−1.78252	−0.891261	−	0.453490i	$-0.850179\pi$
$467$	−38.5206	−1.78252	−0.891261	+	0.453490i	$0.850179\pi$
$468$	0	0
$469$	−1.03923	−0.0479871
$470$	−2.12631	−0.0980794
$471$	−17.2392	−0.794341
$472$	35.8189	1.64870
$473$	−32.3153	−1.48586
$474$	−3.59419	−0.165086
$475$	2.93661	0.134741
$476$	−14.1250	−0.647417
$477$	−7.58211	−0.347161
$478$	54.1118	2.47502
$479$	−8.34481	−0.381284	−0.190642	−	0.981660i	$-0.561057\pi$
$479$	−8.34481	−0.381284	−0.190642	+	0.981660i	$0.561057\pi$
$480$	18.4752	0.843272
$481$	0	0
$482$	−51.1825	−2.33130
$483$	−3.19806	−0.145517
$484$	−13.2282	−0.601282
$485$	−17.9089	−0.813200
$486$	−2.69202	−0.122113
$487$	−14.8586	−0.673309	−0.336654	−	0.941628i	$-0.609295\pi$
$487$	−14.8586	−0.673309	−0.336654	+	0.941628i	$0.609295\pi$
$488$	−29.9226	−1.35453
$489$	−15.7071	−0.710299
$490$	18.8963	0.853648
$491$	4.99894	0.225599	0.112799	−	0.993618i	$-0.464018\pi$
$491$	4.99894	0.225599	0.112799	+	0.993618i	$0.464018\pi$
$492$	−25.7724	−1.16191
$493$	9.27413	0.417686
$494$	0	0
$495$	−3.05429	−0.137280
$496$	−124.055	−5.57023
$497$	−5.82908	−0.261470
$498$	−7.11529	−0.318844
$499$	−0.385371	−0.0172516	−0.00862579	−	0.999963i	$-0.502746\pi$
$499$	−0.385371	−0.0172516	−0.00862579	+	0.999963i	$0.502746\pi$
$500$	48.9812	2.19051
$501$	−5.39612	−0.241081
$502$	−2.05610	−0.0917681
$503$	22.6179	1.00848	0.504241	−	0.863563i	$-0.331772\pi$
$503$	22.6179	1.00848	0.504241	+	0.863563i	$0.331772\pi$
$504$	4.85086	0.216074
$505$	−7.68127	−0.341812
$506$	45.1726	2.00817
$507$	0	0
$508$	−23.5163	−1.04337
$509$	−11.6039	−0.514333	−0.257166	−	0.966367i	$-0.582789\pi$
$509$	−11.6039	−0.514333	−0.257166	+	0.966367i	$0.582789\pi$
$510$	−13.6974	−0.606531
$511$	5.81402	0.257197
$512$	1.71678	0.0758715
$513$	0.753020	0.0332467
$514$	−35.2416	−1.55444
$515$	4.42626	0.195044
$516$	58.2301	2.56344
$517$	2.19269	0.0964342
$518$	8.59478	0.377633
$519$	−23.9420	−1.05094
$520$	0	0
$521$	−1.62671	−0.0712675	−0.0356337	−	0.999365i	$-0.511345\pi$
$521$	−1.62671	−0.0712675	−0.0356337	+	0.999365i	$0.511345\pi$
$522$	−5.14675	−0.225267
$523$	10.0718	0.440407	0.220203	−	0.975454i	$-0.429328\pi$
$523$	10.0718	0.440407	0.220203	+	0.975454i	$0.429328\pi$
$524$	−48.3637	−2.11278
$525$	2.16421	0.0944539
$526$	49.4722	2.15709
$527$	46.1594	2.01074
$528$	−37.9614	−1.65206
$529$	10.2088	0.443862
$530$	21.4097	0.929976
$531$	4.09783	0.177831
$532$	−2.19269	−0.0950650
$533$	0	0
$534$	−26.7235	−1.15644
$535$	−6.70650	−0.289947
$536$	−16.3685	−0.707012
$537$	−18.4088	−0.794398
$538$	63.6999	2.74630
$539$	−19.4862	−0.839330
$540$	5.50365	0.236839
$541$	−20.4674	−0.879962	−0.439981	−	0.898007i	$-0.645015\pi$
$541$	−20.4674	−0.879962	−0.439981	+	0.898007i	$0.645015\pi$
$542$	−53.1756	−2.28409
$543$	3.63342	0.155925
$544$	−85.4408	−3.66325
$545$	3.62937	0.155465
$546$	0	0
$547$	−27.5478	−1.17786	−0.588929	−	0.808185i	$-0.700450\pi$
$547$	−27.5478	−1.17786	−0.588929	+	0.808185i	$0.700450\pi$
$548$	−39.1939	−1.67428
$549$	−3.42327	−0.146102
$550$	−30.5694	−1.30348
$551$	1.43967	0.0613318
$552$	−50.3715	−2.14395
$553$	0.740939	0.0315079
$554$	4.78581	0.203329
$555$	6.03444	0.256148
$556$	−94.4331	−4.00485
$557$	37.9855	1.60950	0.804749	−	0.593615i	$-0.202300\pi$
$557$	37.9855	1.60950	0.804749	+	0.593615i	$0.202300\pi$
$558$	−25.6165	−1.08443
$559$	0	0
$560$	−7.58881	−0.320686
$561$	14.1250	0.596357
$562$	4.36467	0.184112
$563$	−29.7724	−1.25476	−0.627378	−	0.778714i	$-0.715872\pi$
$563$	−29.7724	−1.25476	−0.627378	+	0.778714i	$0.715872\pi$
$564$	−3.95108	−0.166371
$565$	−9.81461	−0.412904
$566$	13.4160	0.563916
$567$	0.554958	0.0233061
$568$	−91.8117	−3.85234
$569$	−21.9541	−0.920362	−0.460181	−	0.887825i	$-0.652216\pi$
$569$	−21.9541	−0.920362	−0.460181	+	0.887825i	$0.652216\pi$
$570$	−2.12631	−0.0890613
$571$	2.46575	0.103188	0.0515942	−	0.998668i	$-0.483570\pi$
$571$	2.46575	0.103188	0.0515942	+	0.998668i	$0.483570\pi$
$572$	0	0
$573$	21.1782	0.884732
$574$	7.33811	0.306287
$575$	−22.4733	−0.937199
$576$	21.3424	0.889268
$577$	17.4547	0.726650	0.363325	−	0.931662i	$-0.381641\pi$
$577$	17.4547	0.726650	0.363325	+	0.931662i	$0.381641\pi$
$578$	17.5810	0.731275
$579$	17.6112	0.731895
$580$	10.5222	0.436909
$581$	1.46681	0.0608536
$582$	−45.9627	−1.90521
$583$	−22.0780	−0.914377
$584$	91.5745	3.78938
$585$	0	0
$586$	−0.193159	−0.00797934
$587$	6.26337	0.258517	0.129259	−	0.991611i	$-0.458740\pi$
$587$	6.26337	0.258517	0.129259	+	0.991611i	$0.458740\pi$
$588$	35.1129	1.44803
$589$	7.16554	0.295251
$590$	−11.5711	−0.476374
$591$	4.66248	0.191789
$592$	75.0012	3.08253
$593$	22.8745	0.939345	0.469672	−	0.882841i	$-0.344372\pi$
$593$	22.8745	0.939345	0.469672	+	0.882841i	$0.344372\pi$
$594$	−7.83877	−0.321629
$595$	2.82371	0.115761
$596$	80.4631	3.29590
$597$	−15.0368	−0.615417
$598$	0	0
$599$	−1.05621	−0.0431557	−0.0215778	−	0.999767i	$-0.506869\pi$
$599$	−1.05621	−0.0431557	−0.0215778	+	0.999767i	$0.506869\pi$
$600$	34.0877	1.39162
$601$	−33.3236	−1.35930	−0.679650	−	0.733537i	$-0.737868\pi$
$601$	−33.3236	−1.35930	−0.679650	+	0.733537i	$0.737868\pi$
$602$	−16.5797	−0.675739
$603$	−1.87263	−0.0762592
$604$	13.2790	0.540316
$605$	2.64443	0.107511
$606$	−19.7138	−0.800818
$607$	16.2403	0.659172	0.329586	−	0.944125i	$-0.393091\pi$
$607$	16.2403	0.659172	0.329586	+	0.944125i	$0.393091\pi$
$608$	−13.2634	−0.537901
$609$	1.06100	0.0429938
$610$	9.66632	0.391378
$611$	0	0
$612$	−25.4523	−1.02885
$613$	11.0479	0.446219	0.223109	−	0.974793i	$-0.428379\pi$
$613$	11.0479	0.446219	0.223109	+	0.974793i	$0.428379\pi$
$614$	−13.9933	−0.564723
$615$	5.15213	0.207754
$616$	14.1250	0.569112
$617$	−4.65950	−0.187584	−0.0937922	−	0.995592i	$-0.529899\pi$
$617$	−4.65950	−0.187584	−0.0937922	+	0.995592i	$0.529899\pi$
$618$	11.3599	0.456961
$619$	31.9259	1.28321	0.641604	−	0.767036i	$-0.278269\pi$
$619$	31.9259	1.28321	0.641604	+	0.767036i	$0.278269\pi$
$620$	52.3712	2.10328
$621$	−5.76271	−0.231250
$622$	−60.6859	−2.43328
$623$	5.50902	0.220714
$624$	0	0
$625$	9.70709	0.388283
$626$	61.0043	2.43822
$627$	2.19269	0.0875674
$628$	90.4538	3.60950
$629$	−27.9071	−1.11273
$630$	−1.56704	−0.0624324
$631$	−39.4413	−1.57013	−0.785067	−	0.619411i	$-0.787372\pi$
$631$	−39.4413	−1.57013	−0.785067	+	0.619411i	$0.787372\pi$
$632$	11.6703	0.464218
$633$	0.460107	0.0182876
$634$	70.9144	2.81637
$635$	4.70112	0.186558
$636$	39.7832	1.57750
$637$	0	0
$638$	−14.9866	−0.593325
$639$	−10.5036	−0.415518
$640$	−23.3144	−0.921583
$641$	45.4510	1.79521	0.897603	−	0.440804i	$-0.145307\pi$
$641$	45.4510	1.79521	0.897603	+	0.440804i	$0.145307\pi$
$642$	−17.2121	−0.679306
$643$	−29.7469	−1.17310	−0.586552	−	0.809912i	$-0.699515\pi$
$643$	−29.7469	−1.17310	−0.586552	+	0.809912i	$0.699515\pi$
$644$	16.7802	0.661231
$645$	−11.6407	−0.458353
$646$	9.83340	0.386890
$647$	−16.9312	−0.665635	−0.332818	−	0.942991i	$-0.607999\pi$
$647$	−16.9312	−0.665635	−0.332818	+	0.942991i	$0.607999\pi$
$648$	8.74094	0.343377
$649$	11.9323	0.468384
$650$	0	0
$651$	5.28083	0.206972
$652$	82.4148	3.22761
$653$	−33.1976	−1.29912	−0.649561	−	0.760309i	$-0.725047\pi$
$653$	−33.1976	−1.29912	−0.649561	+	0.760309i	$0.725047\pi$
$654$	9.31468	0.364233
$655$	9.66833	0.377773
$656$	64.0350	2.50015
$657$	10.4765	0.408727
$658$	1.12498	0.0438564
$659$	−42.6571	−1.66168	−0.830842	−	0.556508i	$-0.812141\pi$
$659$	−42.6571	−1.66168	−0.830842	+	0.556508i	$0.812141\pi$
$660$	16.0258	0.623804
$661$	38.6902	1.50488	0.752438	−	0.658664i	$-0.228878\pi$
$661$	38.6902	1.50488	0.752438	+	0.658664i	$0.228878\pi$
$662$	−30.2301	−1.17493
$663$	0	0
$664$	23.1032	0.896578
$665$	0.438337	0.0169980
$666$	15.4873	0.600119
$667$	−11.0175	−0.426598
$668$	28.3134	1.09548
$669$	16.3502	0.632135
$670$	5.28775	0.204283
$671$	−9.96807	−0.384813
$672$	−9.77479	−0.377071
$673$	−2.59419	−0.0999986	−0.0499993	−	0.998749i	$-0.515922\pi$
$673$	−2.59419	−0.0999986	−0.0499993	+	0.998749i	$0.515922\pi$
$674$	6.21313	0.239321
$675$	3.89977	0.150102
$676$	0	0
$677$	1.75302	0.0673740	0.0336870	−	0.999432i	$-0.489275\pi$
$677$	1.75302	0.0673740	0.0336870	+	0.999432i	$0.489275\pi$
$678$	−25.1890	−0.967376
$679$	9.47517	0.363624
$680$	44.4752	1.70555
$681$	−6.56033	−0.251393
$682$	−74.5916	−2.85626
$683$	16.3351	0.625046	0.312523	−	0.949910i	$-0.398826\pi$
$683$	16.3351	0.625046	0.312523	+	0.949910i	$0.398826\pi$
$684$	−3.95108	−0.151073
$685$	7.83520	0.299368
$686$	−20.4553	−0.780988
$687$	3.95539	0.150908
$688$	−144.681	−5.51590
$689$	0	0
$690$	16.2722	0.619472
$691$	15.9105	0.605265	0.302632	−	0.953107i	$-0.402135\pi$
$691$	15.9105	0.605265	0.302632	+	0.953107i	$0.402135\pi$
$692$	125.623	4.77547
$693$	1.61596	0.0613851
$694$	−24.5972	−0.933696
$695$	18.8780	0.716083
$696$	16.7114	0.633444
$697$	−23.8267	−0.902500
$698$	64.5435	2.44301
$699$	−8.35690	−0.316087
$700$	−11.3556	−0.429200
$701$	−20.8635	−0.788005	−0.394002	−	0.919109i	$-0.628910\pi$
$701$	−20.8635	−0.788005	−0.394002	+	0.919109i	$0.628910\pi$
$702$	0	0
$703$	−4.33214	−0.163390
$704$	62.1460	2.34222
$705$	0.789856	0.0297477
$706$	−73.0182	−2.74807
$707$	4.06398	0.152842
$708$	−21.5013	−0.808067
$709$	15.6485	0.587691	0.293846	−	0.955853i	$-0.405065\pi$
$709$	15.6485	0.587691	0.293846	+	0.955853i	$0.405065\pi$
$710$	29.6592	1.11309
$711$	1.33513	0.0500711
$712$	86.7706	3.25187
$713$	−54.8364	−2.05364
$714$	7.24698	0.271211
$715$	0	0
$716$	96.5906	3.60976
$717$	−20.1008	−0.750679
$718$	−70.2054	−2.62004
$719$	−27.1594	−1.01288	−0.506438	−	0.862276i	$-0.669038\pi$
$719$	−27.1594	−1.01288	−0.506438	+	0.862276i	$0.669038\pi$
$720$	−13.6746	−0.509621
$721$	−2.34183	−0.0872143
$722$	−49.6219	−1.84674
$723$	19.0127	0.707089
$724$	−19.0645	−0.708525
$725$	7.45580	0.276901
$726$	6.78687	0.251884
$727$	31.7784	1.17859	0.589297	−	0.807916i	$-0.299405\pi$
$727$	31.7784	1.17859	0.589297	+	0.807916i	$0.299405\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−29.5826	−1.09490
$731$	53.8340	1.99112
$732$	17.9618	0.663889
$733$	46.8907	1.73195	0.865973	−	0.500090i	$-0.166700\pi$
$733$	46.8907	1.73195	0.865973	+	0.500090i	$0.166700\pi$
$734$	25.7743	0.951347
$735$	−7.01938	−0.258914
$736$	101.502	3.74141
$737$	−5.45281	−0.200857
$738$	13.2228	0.486739
$739$	17.0479	0.627115	0.313558	−	0.949569i	$-0.398479\pi$
$739$	17.0479	0.627115	0.313558	+	0.949569i	$0.398479\pi$
$740$	−31.6626	−1.16394
$741$	0	0
$742$	−11.3274	−0.415840
$743$	−11.6324	−0.426750	−0.213375	−	0.976970i	$-0.568446\pi$
$743$	−11.6324	−0.426750	−0.213375	+	0.976970i	$0.568446\pi$
$744$	83.1764	3.04940
$745$	−16.0853	−0.589319
$746$	75.7797	2.77449
$747$	2.64310	0.0967061
$748$	−74.1135	−2.70986
$749$	3.54825	0.129650
$750$	−25.1304	−0.917631
$751$	−13.0295	−0.475455	−0.237727	−	0.971332i	$-0.576402\pi$
$751$	−13.0295	−0.475455	−0.237727	+	0.971332i	$0.576402\pi$
$752$	9.81700	0.357989
$753$	0.763774	0.0278335
$754$	0	0
$755$	−2.65459	−0.0966106
$756$	−2.91185	−0.105903
$757$	22.7899	0.828311	0.414156	−	0.910206i	$-0.364077\pi$
$757$	22.7899	0.828311	0.414156	+	0.910206i	$0.364077\pi$
$758$	43.1976	1.56901
$759$	−16.7802	−0.609081
$760$	6.90408	0.250437
$761$	−38.3424	−1.38991	−0.694956	−	0.719052i	$-0.744576\pi$
$761$	−38.3424	−1.38991	−0.694956	+	0.719052i	$0.744576\pi$
$762$	12.0653	0.437080
$763$	−1.92021	−0.0695164
$764$	−111.122	−4.02024
$765$	5.08815	0.183962
$766$	−66.2683	−2.39437
$767$	0	0
$768$	−17.1511	−0.618886
$769$	−3.63879	−0.131218	−0.0656091	−	0.997845i	$-0.520899\pi$
$769$	−3.63879	−0.131218	−0.0656091	+	0.997845i	$0.520899\pi$
$770$	−4.56299	−0.164439
$771$	13.0911	0.471466
$772$	−92.4055	−3.32575
$773$	−39.3424	−1.41505	−0.707524	−	0.706689i	$-0.750188\pi$
$773$	−39.3424	−1.41505	−0.707524	+	0.706689i	$0.750188\pi$
$774$	−29.8756	−1.07386
$775$	37.1092	1.33300
$776$	149.240	5.35740
$777$	−3.19269	−0.114537
$778$	−46.3562	−1.66195
$779$	−3.69873	−0.132521
$780$	0	0
$781$	−30.5851	−1.09442
$782$	−75.2529	−2.69104
$783$	1.91185	0.0683241
$784$	−87.2428	−3.11581
$785$	−18.0825	−0.645392
$786$	24.8135	0.885070
$787$	−25.4252	−0.906310	−0.453155	−	0.891432i	$-0.649702\pi$
$787$	−25.4252	−0.906310	−0.453155	+	0.891432i	$0.649702\pi$
$788$	−24.4639	−0.871492
$789$	−18.3773	−0.654251
$790$	−3.77000	−0.134131
$791$	5.19269	0.184631
$792$	25.4523	0.904409
$793$	0	0
$794$	−5.48965	−0.194820
$795$	−7.95300	−0.282064
$796$	78.8980	2.79646
$797$	20.7138	0.733720	0.366860	−	0.930276i	$-0.380433\pi$
$797$	20.7138	0.733720	0.366860	+	0.930276i	$0.380433\pi$
$798$	1.12498	0.0398239
$799$	−3.65279	−0.129227
$800$	−68.6889	−2.42852
$801$	9.92692	0.350750
$802$	−3.93422	−0.138922
$803$	30.5060	1.07653
$804$	9.82563	0.346523
$805$	−3.35450	−0.118231
$806$	0	0
$807$	−23.6625	−0.832959
$808$	64.0103	2.25187
$809$	11.0978	0.390179	0.195090	−	0.980785i	$-0.437500\pi$
$809$	11.0978	0.390179	0.195090	+	0.980785i	$0.437500\pi$
$810$	−2.82371	−0.0992150
$811$	−4.84223	−0.170034	−0.0850169	−	0.996380i	$-0.527094\pi$
$811$	−4.84223	−0.170034	−0.0850169	+	0.996380i	$0.527094\pi$
$812$	−5.56704	−0.195365
$813$	19.7530	0.692769
$814$	45.0966	1.58064
$815$	−16.4754	−0.577109
$816$	63.2398	2.21384
$817$	8.35690	0.292371
$818$	−80.5964	−2.81799
$819$	0	0
$820$	−27.0331	−0.944037
$821$	−43.7381	−1.52647	−0.763235	−	0.646121i	$-0.776390\pi$
$821$	−43.7381	−1.52647	−0.763235	+	0.646121i	$0.776390\pi$
$822$	20.1089	0.701377
$823$	12.0998	0.421771	0.210885	−	0.977511i	$-0.432365\pi$
$823$	12.0998	0.421771	0.210885	+	0.977511i	$0.432365\pi$
$824$	−36.8853	−1.28496
$825$	11.3556	0.395350
$826$	6.12200	0.213012
$827$	−35.3212	−1.22824	−0.614120	−	0.789213i	$-0.710489\pi$
$827$	−35.3212	−1.22824	−0.614120	+	0.789213i	$0.710489\pi$
$828$	30.2368	1.05080
$829$	53.4946	1.85794	0.928971	−	0.370152i	$-0.120694\pi$
$829$	53.4946	1.85794	0.928971	+	0.370152i	$0.120694\pi$
$830$	−7.46335	−0.259057
$831$	−1.77777	−0.0616703
$832$	0	0
$833$	32.4620	1.12474
$834$	48.4499	1.67768
$835$	−5.66009	−0.195875
$836$	−11.5050	−0.397908
$837$	9.51573	0.328912
$838$	17.8950	0.618172
$839$	23.1521	0.799300	0.399650	−	0.916668i	$-0.369132\pi$
$839$	23.1521	0.799300	0.399650	+	0.916668i	$0.369132\pi$
$840$	5.08815	0.175558
$841$	−25.3448	−0.873959
$842$	36.5163	1.25844
$843$	−1.62133	−0.0558417
$844$	−2.41417	−0.0830993
$845$	0	0
$846$	2.02715	0.0696948
$847$	−1.39911	−0.0480739
$848$	−98.8467	−3.39441
$849$	−4.98361	−0.171037
$850$	50.9256	1.74673
$851$	33.1530	1.13647
$852$	55.1124	1.88812
$853$	26.7265	0.915097	0.457548	−	0.889185i	$-0.348728\pi$
$853$	26.7265	0.915097	0.457548	+	0.889185i	$0.348728\pi$
$854$	−5.11423	−0.175005
$855$	0.789856	0.0270125
$856$	55.8872	1.91019
$857$	42.6064	1.45541	0.727703	−	0.685892i	$-0.240588\pi$
$857$	42.6064	1.45541	0.727703	+	0.685892i	$0.240588\pi$
$858$	0	0
$859$	33.6079	1.14669	0.573344	−	0.819315i	$-0.305646\pi$
$859$	33.6079	1.14669	0.573344	+	0.819315i	$0.305646\pi$
$860$	61.0786	2.08276
$861$	−2.72587	−0.0928975
$862$	96.7682	3.29594
$863$	18.7047	0.636715	0.318358	−	0.947971i	$-0.396869\pi$
$863$	18.7047	0.636715	0.318358	+	0.947971i	$0.396869\pi$
$864$	−17.6136	−0.599226
$865$	−25.1132	−0.853873
$866$	−87.4210	−2.97069
$867$	−6.53079	−0.221797
$868$	−27.7084	−0.940485
$869$	3.88769	0.131881
$870$	−5.39852	−0.183027
$871$	0	0
$872$	−30.2446	−1.02421
$873$	17.0737	0.577856
$874$	−11.6819	−0.395145
$875$	5.18060	0.175136
$876$	−54.9700	−1.85726
$877$	36.8237	1.24345	0.621724	−	0.783236i	$-0.286433\pi$
$877$	36.8237	1.24345	0.621724	+	0.783236i	$0.286433\pi$
$878$	34.5442	1.16581
$879$	0.0717525	0.00242015
$880$	−39.8183	−1.34228
$881$	−41.1250	−1.38554	−0.692768	−	0.721161i	$-0.743609\pi$
$881$	−41.1250	−1.38554	−0.692768	+	0.721161i	$0.743609\pi$
$882$	−18.0151	−0.606599
$883$	30.7482	1.03476	0.517380	−	0.855756i	$-0.326907\pi$
$883$	30.7482	1.03476	0.517380	+	0.855756i	$0.326907\pi$
$884$	0	0
$885$	4.29829	0.144485
$886$	32.1987	1.08173
$887$	7.58940	0.254827	0.127414	−	0.991850i	$-0.459332\pi$
$887$	7.58940	0.254827	0.127414	+	0.991850i	$0.459332\pi$
$888$	−50.2868	−1.68751
$889$	−2.48725	−0.0834198
$890$	−28.0307	−0.939592
$891$	2.91185	0.0975508
$892$	−85.7891	−2.87243
$893$	−0.567040	−0.0189753
$894$	−41.2825	−1.38069
$895$	−19.3093	−0.645439
$896$	12.3351	0.412088
$897$	0	0
$898$	−33.5934	−1.12103
$899$	18.1927	0.606760
$900$	−20.4620	−0.682068
$901$	36.7797	1.22531
$902$	38.5029	1.28201
$903$	6.15883	0.204953
$904$	81.7881	2.72023
$905$	3.81115	0.126687
$906$	−6.81295	−0.226345
$907$	19.3333	0.641952	0.320976	−	0.947087i	$-0.395989\pi$
$907$	19.3333	0.641952	0.320976	+	0.947087i	$0.395989\pi$
$908$	34.4219	1.14233
$909$	7.32304	0.242890
$910$	0	0
$911$	22.7149	0.752577	0.376288	−	0.926503i	$-0.377200\pi$
$911$	22.7149	0.752577	0.376288	+	0.926503i	$0.377200\pi$
$912$	9.81700	0.325073
$913$	7.69633	0.254711
$914$	87.3624	2.88969
$915$	−3.59073	−0.118706
$916$	−20.7539	−0.685727
$917$	−5.11529	−0.168922
$918$	13.0586	0.430998
$919$	14.2911	0.471420	0.235710	−	0.971823i	$-0.424258\pi$
$919$	14.2911	0.471420	0.235710	+	0.971823i	$0.424258\pi$
$920$	−52.8355	−1.74194
$921$	5.19806	0.171282
$922$	65.6945	2.16353
$923$	0	0
$924$	−8.47889	−0.278935
$925$	−22.4355	−0.737674
$926$	−89.3309	−2.93560
$927$	−4.21983	−0.138597
$928$	−33.6746	−1.10542
$929$	32.7211	1.07354	0.536772	−	0.843727i	$-0.319644\pi$
$929$	32.7211	1.07354	0.536772	+	0.843727i	$0.319644\pi$
$930$	−26.8696	−0.881090
$931$	5.03923	0.165154
$932$	43.8485	1.43630
$933$	22.5429	0.738021
$934$	−103.698	−3.39311
$935$	14.8159	0.484533
$936$	0	0
$937$	−4.01400	−0.131132	−0.0655658	−	0.997848i	$-0.520885\pi$
$937$	−4.01400	−0.131132	−0.0655658	+	0.997848i	$0.520885\pi$
$938$	−2.79763	−0.0913457
$939$	−22.6612	−0.739519
$940$	−4.14436	−0.135174
$941$	−28.2669	−0.921476	−0.460738	−	0.887536i	$-0.652415\pi$
$941$	−28.2669	−0.921476	−0.460738	+	0.887536i	$0.652415\pi$
$942$	−46.4083	−1.51206
$943$	28.3056	0.921757
$944$	53.4228	1.73876
$945$	0.582105	0.0189359
$946$	−86.9934	−2.82840
$947$	−37.8455	−1.22981	−0.614906	−	0.788600i	$-0.710806\pi$
$947$	−37.8455	−1.22981	−0.614906	+	0.788600i	$0.710806\pi$
$948$	−7.00538	−0.227524
$949$	0	0
$950$	7.90541	0.256485
$951$	−26.3424	−0.854212
$952$	−23.5308	−0.762637
$953$	−40.8256	−1.32247	−0.661236	−	0.750178i	$-0.729968\pi$
$953$	−40.8256	−1.32247	−0.661236	+	0.750178i	$0.729968\pi$
$954$	−20.4112	−0.660837
$955$	22.2142	0.718834
$956$	105.469	3.41110
$957$	5.56704	0.179957
$958$	−22.4644	−0.725792
$959$	−4.14542	−0.133863
$960$	22.3864	0.722519
$961$	59.5491	1.92094
$962$	0	0
$963$	6.39373	0.206035
$964$	−99.7591	−3.21302
$965$	18.4727	0.594656
$966$	−8.60925	−0.276998
$967$	10.2798	0.330575	0.165288	−	0.986245i	$-0.447145\pi$
$967$	10.2798	0.330575	0.165288	+	0.986245i	$0.447145\pi$
$968$	−22.0368	−0.708291
$969$	−3.65279	−0.117345
$970$	−48.2111	−1.54796
$971$	−19.4805	−0.625161	−0.312580	−	0.949891i	$-0.601193\pi$
$971$	−19.4805	−0.625161	−0.312580	+	0.949891i	$0.601193\pi$
$972$	−5.24698	−0.168297
$973$	−9.98792	−0.320198
$974$	−39.9997	−1.28167
$975$	0	0
$976$	−44.6286	−1.42853
$977$	47.0538	1.50539	0.752693	−	0.658372i	$-0.228755\pi$
$977$	47.0538	1.50539	0.752693	+	0.658372i	$0.228755\pi$
$978$	−42.2838	−1.35209
$979$	28.9057	0.923831
$980$	36.8305	1.17651
$981$	−3.46011	−0.110473
$982$	13.4572	0.429438
$983$	34.4295	1.09813	0.549065	−	0.835779i	$-0.314984\pi$
$983$	34.4295	1.09813	0.549065	+	0.835779i	$0.314984\pi$
$984$	−42.9342	−1.36869
$985$	4.89056	0.155826
$986$	24.9661	0.795084
$987$	−0.417895	−0.0133017
$988$	0	0
$989$	−63.9536	−2.03361
$990$	−8.22223	−0.261319
$991$	7.30798	0.232146	0.116073	−	0.993241i	$-0.462969\pi$
$991$	7.30798	0.232146	0.116073	+	0.993241i	$0.462969\pi$
$992$	−167.606	−5.32149
$993$	11.2295	0.356358
$994$	−15.6920	−0.497721
$995$	−15.7724	−0.500019
$996$	−13.8683	−0.439434
$997$	−35.6915	−1.13036	−0.565181	−	0.824967i	$-0.691194\pi$
$997$	−35.6915	−1.13036	−0.565181	+	0.824967i	$0.691194\pi$
$998$	−1.03743	−0.0328392
$999$	−5.75302	−0.182018

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.a.l.1.3	yes	3
3.2	odd	2		1521.2.a.n.1.1		3
4.3	odd	2		8112.2.a.cp.1.1		3
13.2	odd	12		507.2.j.i.316.6		12
13.3	even	3		507.2.e.i.22.1		6
13.4	even	6		507.2.e.l.484.3		6
13.5	odd	4		507.2.b.f.337.1		6
13.6	odd	12		507.2.j.i.361.1		12
13.7	odd	12		507.2.j.i.361.6		12
13.8	odd	4		507.2.b.f.337.6		6
13.9	even	3		507.2.e.i.484.1		6
13.10	even	6		507.2.e.l.22.3		6
13.11	odd	12		507.2.j.i.316.1		12
13.12	even	2		507.2.a.i.1.1	✓	3
39.5	even	4		1521.2.b.k.1351.6		6
39.8	even	4		1521.2.b.k.1351.1		6
39.38	odd	2		1521.2.a.s.1.3		3
52.51	odd	2		8112.2.a.cg.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.a.i.1.1	✓	3	13.12	even	2
507.2.a.l.1.3	yes	3	1.1	even	1	trivial
507.2.b.f.337.1		6	13.5	odd	4
507.2.b.f.337.6		6	13.8	odd	4
507.2.e.i.22.1		6	13.3	even	3
507.2.e.i.484.1		6	13.9	even	3
507.2.e.l.22.3		6	13.10	even	6
507.2.e.l.484.3		6	13.4	even	6
507.2.j.i.316.1		12	13.11	odd	12
507.2.j.i.316.6		12	13.2	odd	12
507.2.j.i.361.1		12	13.6	odd	12
507.2.j.i.361.6		12	13.7	odd	12
1521.2.a.n.1.1		3	3.2	odd	2
1521.2.a.s.1.3		3	39.38	odd	2
1521.2.b.k.1351.1		6	39.8	even	4
1521.2.b.k.1351.6		6	39.5	even	4
8112.2.a.cg.1.3		3	52.51	odd	2
8112.2.a.cp.1.1		3	4.3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	507.a (trivial)

\(f(q)\)	\(=\)	\(q+2.69202 q^{2} -1.00000 q^{3} +5.24698 q^{4} -1.04892 q^{5} -2.69202 q^{6} +0.554958 q^{7} +8.74094 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.69202 q^{2} -1.00000 q^{3} +5.24698 q^{4} -1.04892 q^{5} -2.69202 q^{6} +0.554958 q^{7} +8.74094 q^{8} +1.00000 q^{9} -2.82371 q^{10} +2.91185 q^{11} -5.24698 q^{12} +1.49396 q^{14} +1.04892 q^{15} +13.0368 q^{16} -4.85086 q^{17} +2.69202 q^{18} -0.753020 q^{19} -5.50365 q^{20} -0.554958 q^{21} +7.83877 q^{22} +5.76271 q^{23} -8.74094 q^{24} -3.89977 q^{25} -1.00000 q^{27} +2.91185 q^{28} -1.91185 q^{29} +2.82371 q^{30} -9.51573 q^{31} +17.6136 q^{32} -2.91185 q^{33} -13.0586 q^{34} -0.582105 q^{35} +5.24698 q^{36} +5.75302 q^{37} -2.02715 q^{38} -9.16852 q^{40} +4.91185 q^{41} -1.49396 q^{42} -11.0978 q^{43} +15.2784 q^{44} -1.04892 q^{45} +15.5133 q^{46} +0.753020 q^{47} -13.0368 q^{48} -6.69202 q^{49} -10.4983 q^{50} +4.85086 q^{51} -7.58211 q^{53} -2.69202 q^{54} -3.05429 q^{55} +4.85086 q^{56} +0.753020 q^{57} -5.14675 q^{58} +4.09783 q^{59} +5.50365 q^{60} -3.42327 q^{61} -25.6165 q^{62} +0.554958 q^{63} +21.3424 q^{64} -7.83877 q^{66} -1.87263 q^{67} -25.4523 q^{68} -5.76271 q^{69} -1.56704 q^{70} -10.5036 q^{71} +8.74094 q^{72} +10.4765 q^{73} +15.4873 q^{74} +3.89977 q^{75} -3.95108 q^{76} +1.61596 q^{77} +1.33513 q^{79} -13.6746 q^{80} +1.00000 q^{81} +13.2228 q^{82} +2.64310 q^{83} -2.91185 q^{84} +5.08815 q^{85} -29.8756 q^{86} +1.91185 q^{87} +25.4523 q^{88} +9.92692 q^{89} -2.82371 q^{90} +30.2368 q^{92} +9.51573 q^{93} +2.02715 q^{94} +0.789856 q^{95} -17.6136 q^{96} +17.0737 q^{97} -18.0151 q^{98} +2.91185 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} - 3 q^{3} + 11 q^{4} + 6 q^{5} - 3 q^{6} + 2 q^{7} + 12 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 - 3 * q^3 + 11 * q^4 + 6 * q^5 - 3 * q^6 + 2 * q^7 + 12 * q^8 + 3 * q^9	\( 3 q + 3 q^{2} - 3 q^{3} + 11 q^{4} + 6 q^{5} - 3 q^{6} + 2 q^{7} + 12 q^{8} + 3 q^{9} - q^{10} + 5 q^{11} - 11 q^{12} - 5 q^{14} - 6 q^{15} + 11 q^{16} - q^{17} + 3 q^{18} - 7 q^{19} + 15 q^{20} - 2 q^{21} - 9 q^{22} - 12 q^{24} + 11 q^{25} - 3 q^{27} + 5 q^{28} - 2 q^{29} + q^{30} - 16 q^{31} + 22 q^{32} - 5 q^{33} - 8 q^{34} + 4 q^{35} + 11 q^{36} + 22 q^{37} + 3 q^{40} + 11 q^{41} + 5 q^{42} - 15 q^{43} + 16 q^{44} + 6 q^{45} - 7 q^{46} + 7 q^{47} - 11 q^{48} - 15 q^{49} - 3 q^{50} + q^{51} - 17 q^{53} - 3 q^{54} + 3 q^{55} + q^{56} + 7 q^{57} + 12 q^{58} - 6 q^{59} - 15 q^{60} - 13 q^{61} - 2 q^{62} + 2 q^{63} + 9 q^{66} + 11 q^{67} - 13 q^{68} - 24 q^{70} + 12 q^{72} + 6 q^{73} + 15 q^{74} - 11 q^{75} - 21 q^{76} + 15 q^{77} + 3 q^{79} - 20 q^{80} + 3 q^{81} - 3 q^{82} + 12 q^{83} - 5 q^{84} + 19 q^{85} - 29 q^{86} + 2 q^{87} + 13 q^{88} + q^{89} - q^{90} + 7 q^{92} + 16 q^{93} - 21 q^{95} - 22 q^{96} - 5 q^{97} - 29 q^{98} + 5 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 - 3 * q^3 + 11 * q^4 + 6 * q^5 - 3 * q^6 + 2 * q^7 + 12 * q^8 + 3 * q^9 - q^10 + 5 * q^11 - 11 * q^12 - 5 * q^14 - 6 * q^15 + 11 * q^16 - q^17 + 3 * q^18 - 7 * q^19 + 15 * q^20 - 2 * q^21 - 9 * q^22 - 12 * q^24 + 11 * q^25 - 3 * q^27 + 5 * q^28 - 2 * q^29 + q^30 - 16 * q^31 + 22 * q^32 - 5 * q^33 - 8 * q^34 + 4 * q^35 + 11 * q^36 + 22 * q^37 + 3 * q^40 + 11 * q^41 + 5 * q^42 - 15 * q^43 + 16 * q^44 + 6 * q^45 - 7 * q^46 + 7 * q^47 - 11 * q^48 - 15 * q^49 - 3 * q^50 + q^51 - 17 * q^53 - 3 * q^54 + 3 * q^55 + q^56 + 7 * q^57 + 12 * q^58 - 6 * q^59 - 15 * q^60 - 13 * q^61 - 2 * q^62 + 2 * q^63 + 9 * q^66 + 11 * q^67 - 13 * q^68 - 24 * q^70 + 12 * q^72 + 6 * q^73 + 15 * q^74 - 11 * q^75 - 21 * q^76 + 15 * q^77 + 3 * q^79 - 20 * q^80 + 3 * q^81 - 3 * q^82 + 12 * q^83 - 5 * q^84 + 19 * q^85 - 29 * q^86 + 2 * q^87 + 13 * q^88 + q^89 - q^90 + 7 * q^92 + 16 * q^93 - 21 * q^95 - 22 * q^96 - 5 * q^97 - 29 * q^98 + 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more