LMFDB - Embedded newform 6035.2.a.g.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6035,2,Mod(1,6035)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6035, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6035.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6035 = 5 \cdot 17 \cdot 71 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6035.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:	$48.1897176198$
Analytic rank:	$0$
Dimension:	$58$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	6035.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.49259 q^{2} -0.124755 q^{3} +4.21303 q^{4} +1.00000 q^{5} +0.310963 q^{6} +1.75866 q^{7} -5.51618 q^{8} -2.98444 q^{9} +O(q^{10})$	\(q-2.49259 q^{2} -0.124755 q^{3} +4.21303 q^{4} +1.00000 q^{5} +0.310963 q^{6} +1.75866 q^{7} -5.51618 q^{8} -2.98444 q^{9} -2.49259 q^{10} -5.37225 q^{11} -0.525595 q^{12} +0.699150 q^{13} -4.38362 q^{14} -0.124755 q^{15} +5.32354 q^{16} +1.00000 q^{17} +7.43899 q^{18} -1.88050 q^{19} +4.21303 q^{20} -0.219401 q^{21} +13.3908 q^{22} -1.95182 q^{23} +0.688169 q^{24} +1.00000 q^{25} -1.74270 q^{26} +0.746586 q^{27} +7.40927 q^{28} +9.84829 q^{29} +0.310963 q^{30} +3.28429 q^{31} -2.23707 q^{32} +0.670213 q^{33} -2.49259 q^{34} +1.75866 q^{35} -12.5735 q^{36} +6.35932 q^{37} +4.68732 q^{38} -0.0872222 q^{39} -5.51618 q^{40} -5.30939 q^{41} +0.546877 q^{42} -8.23831 q^{43} -22.6334 q^{44} -2.98444 q^{45} +4.86509 q^{46} -1.86886 q^{47} -0.664137 q^{48} -3.90712 q^{49} -2.49259 q^{50} -0.124755 q^{51} +2.94554 q^{52} -5.31994 q^{53} -1.86094 q^{54} -5.37225 q^{55} -9.70107 q^{56} +0.234601 q^{57} -24.5478 q^{58} +1.78230 q^{59} -0.525595 q^{60} -3.83232 q^{61} -8.18639 q^{62} -5.24860 q^{63} -5.07097 q^{64} +0.699150 q^{65} -1.67057 q^{66} -4.50249 q^{67} +4.21303 q^{68} +0.243498 q^{69} -4.38362 q^{70} +1.00000 q^{71} +16.4627 q^{72} +16.0079 q^{73} -15.8512 q^{74} -0.124755 q^{75} -7.92258 q^{76} -9.44794 q^{77} +0.217410 q^{78} +13.0423 q^{79} +5.32354 q^{80} +8.86017 q^{81} +13.2341 q^{82} -6.55844 q^{83} -0.924341 q^{84} +1.00000 q^{85} +20.5348 q^{86} -1.22862 q^{87} +29.6343 q^{88} -4.67841 q^{89} +7.43899 q^{90} +1.22956 q^{91} -8.22306 q^{92} -0.409730 q^{93} +4.65831 q^{94} -1.88050 q^{95} +0.279085 q^{96} -8.89758 q^{97} +9.73887 q^{98} +16.0331 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 58 q + q^{2} + 6 q^{3} + 69 q^{4} + 58 q^{5} + 10 q^{6} + 13 q^{7} - 3 q^{8} + 84 q^{9}+O(q^{10}) $ 58 * q + q^2 + 6 * q^3 + 69 * q^4 + 58 * q^5 + 10 * q^6 + 13 * q^7 - 3 * q^8 + 84 * q^9	\( 58 q + q^{2} + 6 q^{3} + 69 q^{4} + 58 q^{5} + 10 q^{6} + 13 q^{7} - 3 q^{8} + 84 q^{9} + q^{10} + 28 q^{11} + 18 q^{12} + 37 q^{13} + 28 q^{14} + 6 q^{15} + 83 q^{16} + 58 q^{17} - 12 q^{18} + 19 q^{19} + 69 q^{20} + 31 q^{21} + 13 q^{22} + 14 q^{23} + 13 q^{24} + 58 q^{25} + 18 q^{26} + 9 q^{27} + 8 q^{28} + 60 q^{29} + 10 q^{30} + 39 q^{31} - 30 q^{32} + 13 q^{33} + q^{34} + 13 q^{35} + 113 q^{36} + 60 q^{37} - q^{38} + 41 q^{39} - 3 q^{40} + 65 q^{41} - 30 q^{42} + 17 q^{43} + 69 q^{44} + 84 q^{45} + 24 q^{46} + 16 q^{47} + 14 q^{48} + 117 q^{49} + q^{50} + 6 q^{51} + 61 q^{52} + 5 q^{53} + 24 q^{54} + 28 q^{55} + 105 q^{56} + 8 q^{57} - 34 q^{58} + 22 q^{59} + 18 q^{60} + 113 q^{61} - 19 q^{62} + 8 q^{63} + 89 q^{64} + 37 q^{65} - 37 q^{66} + 19 q^{67} + 69 q^{68} + 75 q^{69} + 28 q^{70} + 58 q^{71} - 17 q^{72} + 49 q^{73} + 29 q^{74} + 6 q^{75} - 6 q^{76} + 17 q^{77} - 12 q^{78} + 7 q^{79} + 83 q^{80} + 134 q^{81} + 7 q^{82} - 12 q^{83} - 18 q^{84} + 58 q^{85} + 23 q^{86} - 36 q^{87} - 33 q^{88} + 52 q^{89} - 12 q^{90} + 31 q^{91} + 80 q^{92} - 37 q^{93} + 4 q^{94} + 19 q^{95} - 35 q^{96} + 26 q^{97} - 33 q^{98} + 57 q^{99}+O(q^{100}) \) 58 * q + q^2 + 6 * q^3 + 69 * q^4 + 58 * q^5 + 10 * q^6 + 13 * q^7 - 3 * q^8 + 84 * q^9 + q^10 + 28 * q^11 + 18 * q^12 + 37 * q^13 + 28 * q^14 + 6 * q^15 + 83 * q^16 + 58 * q^17 - 12 * q^18 + 19 * q^19 + 69 * q^20 + 31 * q^21 + 13 * q^22 + 14 * q^23 + 13 * q^24 + 58 * q^25 + 18 * q^26 + 9 * q^27 + 8 * q^28 + 60 * q^29 + 10 * q^30 + 39 * q^31 - 30 * q^32 + 13 * q^33 + q^34 + 13 * q^35 + 113 * q^36 + 60 * q^37 - q^38 + 41 * q^39 - 3 * q^40 + 65 * q^41 - 30 * q^42 + 17 * q^43 + 69 * q^44 + 84 * q^45 + 24 * q^46 + 16 * q^47 + 14 * q^48 + 117 * q^49 + q^50 + 6 * q^51 + 61 * q^52 + 5 * q^53 + 24 * q^54 + 28 * q^55 + 105 * q^56 + 8 * q^57 - 34 * q^58 + 22 * q^59 + 18 * q^60 + 113 * q^61 - 19 * q^62 + 8 * q^63 + 89 * q^64 + 37 * q^65 - 37 * q^66 + 19 * q^67 + 69 * q^68 + 75 * q^69 + 28 * q^70 + 58 * q^71 - 17 * q^72 + 49 * q^73 + 29 * q^74 + 6 * q^75 - 6 * q^76 + 17 * q^77 - 12 * q^78 + 7 * q^79 + 83 * q^80 + 134 * q^81 + 7 * q^82 - 12 * q^83 - 18 * q^84 + 58 * q^85 + 23 * q^86 - 36 * q^87 - 33 * q^88 + 52 * q^89 - 12 * q^90 + 31 * q^91 + 80 * q^92 - 37 * q^93 + 4 * q^94 + 19 * q^95 - 35 * q^96 + 26 * q^97 - 33 * q^98 + 57 * 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 $

$2$	−2.49259	−1.76253	−0.881265	−	0.472622i	$-0.843308\pi$
$2$	−2.49259	−1.76253	−0.881265	+	0.472622i	$0.843308\pi$
$3$	−0.124755	−0.0720271	−0.0360136	−	0.999351i	$-0.511466\pi$
$3$	−0.124755	−0.0720271	−0.0360136	+	0.999351i	$0.511466\pi$
$4$	4.21303	2.10651
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0.310963	0.126950
$7$	1.75866	0.664710	0.332355	−	0.943154i	$-0.392157\pi$
$7$	1.75866	0.664710	0.332355	+	0.943154i	$0.392157\pi$
$8$	−5.51618	−1.95026
$9$	−2.98444	−0.994812
$10$	−2.49259	−0.788228
$11$	−5.37225	−1.61979	−0.809897	−	0.586573i	$-0.800477\pi$
$11$	−5.37225	−1.61979	−0.809897	+	0.586573i	$0.800477\pi$
$12$	−0.525595	−0.151726
$13$	0.699150	0.193909	0.0969546	−	0.995289i	$-0.469090\pi$
$13$	0.699150	0.193909	0.0969546	+	0.995289i	$0.469090\pi$
$14$	−4.38362	−1.17157
$15$	−0.124755	−0.0322115
$16$	5.32354	1.33089
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	7.43899	1.75339
$19$	−1.88050	−0.431415	−0.215708	−	0.976458i	$-0.569206\pi$
$19$	−1.88050	−0.431415	−0.215708	+	0.976458i	$0.569206\pi$
$20$	4.21303	0.942061
$21$	−0.219401	−0.0478772
$22$	13.3908	2.85494
$23$	−1.95182	−0.406982	−0.203491	−	0.979077i	$-0.565229\pi$
$23$	−1.95182	−0.406982	−0.203491	+	0.979077i	$0.565229\pi$
$24$	0.688169	0.140472
$25$	1.00000	0.200000
$26$	−1.74270	−0.341771
$27$	0.746586	0.143681
$28$	7.40927	1.40022
$29$	9.84829	1.82878	0.914391	−	0.404833i	$-0.132670\pi$
$29$	9.84829	1.82878	0.914391	+	0.404833i	$0.132670\pi$
$30$	0.310963	0.0567738
$31$	3.28429	0.589875	0.294938	−	0.955517i	$-0.404701\pi$
$31$	3.28429	0.589875	0.294938	+	0.955517i	$0.404701\pi$
$32$	−2.23707	−0.395462
$33$	0.670213	0.116669
$34$	−2.49259	−0.427476
$35$	1.75866	0.297267
$36$	−12.5735	−2.09559
$37$	6.35932	1.04547	0.522733	−	0.852496i	$-0.324912\pi$
$37$	6.35932	1.04547	0.522733	+	0.852496i	$0.324912\pi$
$38$	4.68732	0.760383
$39$	−0.0872222	−0.0139667
$40$	−5.51618	−0.872184
$41$	−5.30939	−0.829187	−0.414593	−	0.910007i	$-0.636076\pi$
$41$	−5.30939	−0.829187	−0.414593	+	0.910007i	$0.636076\pi$
$42$	0.546877	0.0843849
$43$	−8.23831	−1.25633	−0.628165	−	0.778080i	$-0.716194\pi$
$43$	−8.23831	−1.25633	−0.628165	+	0.778080i	$0.716194\pi$
$44$	−22.6334	−3.41212
$45$	−2.98444	−0.444893
$46$	4.86509	0.717319
$47$	−1.86886	−0.272601	−0.136301	−	0.990668i	$-0.543521\pi$
$47$	−1.86886	−0.272601	−0.136301	+	0.990668i	$0.543521\pi$
$48$	−0.664137	−0.0958599
$49$	−3.90712	−0.558161
$50$	−2.49259	−0.352506
$51$	−0.124755	−0.0174691
$52$	2.94554	0.408472
$53$	−5.31994	−0.730750	−0.365375	−	0.930860i	$-0.619059\pi$
$53$	−5.31994	−0.730750	−0.365375	+	0.930860i	$0.619059\pi$
$54$	−1.86094	−0.253241
$55$	−5.37225	−0.724394
$56$	−9.70107	−1.29636
$57$	0.234601	0.0310736
$58$	−24.5478	−3.22328
$59$	1.78230	0.232036	0.116018	−	0.993247i	$-0.462987\pi$
$59$	1.78230	0.232036	0.116018	+	0.993247i	$0.462987\pi$
$60$	−0.525595	−0.0678540
$61$	−3.83232	−0.490679	−0.245339	−	0.969437i	$-0.578899\pi$
$61$	−3.83232	−0.490679	−0.245339	+	0.969437i	$0.578899\pi$
$62$	−8.18639	−1.03967
$63$	−5.24860	−0.661262
$64$	−5.07097	−0.633871
$65$	0.699150	0.0867189
$66$	−1.67057	−0.205633
$67$	−4.50249	−0.550067	−0.275034	−	0.961435i	$-0.588689\pi$
$67$	−4.50249	−0.550067	−0.275034	+	0.961435i	$0.588689\pi$
$68$	4.21303	0.510905
$69$	0.243498	0.0293138
$70$	−4.38362	−0.523943
$71$	1.00000	0.118678
$71$	1.00000	0.118678
$72$	16.4627	1.94015
$73$	16.0079	1.87358	0.936792	−	0.349886i	$-0.113780\pi$
$73$	16.0079	1.87358	0.936792	+	0.349886i	$0.113780\pi$
$74$	−15.8512	−1.84267
$75$	−0.124755	−0.0144054
$76$	−7.92258	−0.908783
$77$	−9.44794	−1.07669
$78$	0.217410	0.0246168
$79$	13.0423	1.46737	0.733687	−	0.679487i	$-0.237798\pi$
$79$	13.0423	1.46737	0.733687	+	0.679487i	$0.237798\pi$
$80$	5.32354	0.595190
$81$	8.86017	0.984463
$82$	13.2341	1.46147
$83$	−6.55844	−0.719883	−0.359941	−	0.932975i	$-0.617203\pi$
$83$	−6.55844	−0.719883	−0.359941	+	0.932975i	$0.617203\pi$
$84$	−0.924341	−0.100854
$85$	1.00000	0.108465
$86$	20.5348	2.21432
$87$	−1.22862	−0.131722
$88$	29.6343	3.15902
$89$	−4.67841	−0.495911	−0.247955	−	0.968771i	$-0.579759\pi$
$89$	−4.67841	−0.495911	−0.247955	+	0.968771i	$0.579759\pi$
$90$	7.43899	0.784138
$91$	1.22956	0.128893
$92$	−8.22306	−0.857313
$93$	−0.409730	−0.0424870
$94$	4.65831	0.480468
$95$	−1.88050	−0.192935
$96$	0.279085	0.0284840
$97$	−8.89758	−0.903413	−0.451706	−	0.892167i	$-0.649185\pi$
$97$	−8.89758	−0.903413	−0.451706	+	0.892167i	$0.649185\pi$
$98$	9.73887	0.983775
$99$	16.0331	1.61139
$100$	4.21303	0.421303
$101$	−1.88869	−0.187932	−0.0939659	−	0.995575i	$-0.529954\pi$
$101$	−1.88869	−0.187932	−0.0939659	+	0.995575i	$0.529954\pi$
$102$	0.310963	0.0307899
$103$	−4.46923	−0.440366	−0.220183	−	0.975459i	$-0.570665\pi$
$103$	−4.46923	−0.440366	−0.220183	+	0.975459i	$0.570665\pi$
$104$	−3.85663	−0.378174
$105$	−0.219401	−0.0214113
$106$	13.2605	1.28797
$107$	−14.3716	−1.38935	−0.694677	−	0.719322i	$-0.744453\pi$
$107$	−14.3716	−1.38935	−0.694677	+	0.719322i	$0.744453\pi$
$108$	3.14539	0.302665
$109$	12.2011	1.16865	0.584326	−	0.811519i	$-0.301359\pi$
$109$	12.2011	1.16865	0.584326	+	0.811519i	$0.301359\pi$
$110$	13.3908	1.27677
$111$	−0.793355	−0.0753019
$112$	9.36229	0.884653
$113$	−13.7393	−1.29248	−0.646240	−	0.763134i	$-0.723660\pi$
$113$	−13.7393	−1.29248	−0.646240	+	0.763134i	$0.723660\pi$
$114$	−0.584764	−0.0547682
$115$	−1.95182	−0.182008
$116$	41.4911	3.85235
$117$	−2.08657	−0.192903
$118$	−4.44255	−0.408970
$119$	1.75866	0.161216
$120$	0.688169	0.0628209
$121$	17.8610	1.62373
$122$	9.55243	0.864836
$123$	0.662371	0.0597239
$124$	13.8368	1.24258
$125$	1.00000	0.0894427
$126$	13.0826	1.16549
$127$	12.0418	1.06854	0.534269	−	0.845315i	$-0.320587\pi$
$127$	12.0418	1.06854	0.534269	+	0.845315i	$0.320587\pi$
$128$	17.1140	1.51268
$129$	1.02777	0.0904899
$130$	−1.74270	−0.152845
$131$	−13.9264	−1.21675	−0.608376	−	0.793649i	$-0.708179\pi$
$131$	−13.9264	−1.21675	−0.608376	+	0.793649i	$0.708179\pi$
$132$	2.82362	0.245765
$133$	−3.30715	−0.286766
$134$	11.2229	0.969510
$135$	0.746586	0.0642559
$136$	−5.51618	−0.473008
$137$	8.46444	0.723166	0.361583	−	0.932340i	$-0.382236\pi$
$137$	8.46444	0.723166	0.361583	+	0.932340i	$0.382236\pi$
$138$	−0.606943	−0.0516664
$139$	4.35996	0.369807	0.184904	−	0.982757i	$-0.440803\pi$
$139$	4.35996	0.369807	0.184904	+	0.982757i	$0.440803\pi$
$140$	7.40927	0.626198
$141$	0.233149	0.0196347
$142$	−2.49259	−0.209174
$143$	−3.75600	−0.314093
$144$	−15.8878	−1.32398
$145$	9.84829	0.817856
$146$	−39.9012	−3.30225
$147$	0.487432	0.0402027
$148$	26.7920	2.20229
$149$	20.3268	1.66524	0.832618	−	0.553848i	$-0.186841\pi$
$149$	20.3268	1.66524	0.832618	+	0.553848i	$0.186841\pi$
$150$	0.310963	0.0253900
$151$	15.3269	1.24729	0.623644	−	0.781708i	$-0.285651\pi$
$151$	15.3269	1.24729	0.623644	+	0.781708i	$0.285651\pi$
$152$	10.3732	0.841374
$153$	−2.98444	−0.241277
$154$	23.5499	1.89770
$155$	3.28429	0.263800
$156$	−0.367469	−0.0294211
$157$	18.8867	1.50732	0.753662	−	0.657262i	$-0.228286\pi$
$157$	18.8867	1.50732	0.753662	+	0.657262i	$0.228286\pi$
$158$	−32.5092	−2.58629
$159$	0.663687	0.0526338
$160$	−2.23707	−0.176856
$161$	−3.43258	−0.270525
$162$	−22.0848	−1.73515
$163$	−10.4966	−0.822159	−0.411080	−	0.911599i	$-0.634848\pi$
$163$	−10.4966	−0.822159	−0.411080	+	0.911599i	$0.634848\pi$
$164$	−22.3686	−1.74669
$165$	0.670213	0.0521760
$166$	16.3475	1.26882
$167$	−9.87600	−0.764228	−0.382114	−	0.924115i	$-0.624804\pi$
$167$	−9.87600	−0.764228	−0.382114	+	0.924115i	$0.624804\pi$
$168$	1.21025	0.0933731
$169$	−12.5112	−0.962399
$170$	−2.49259	−0.191173
$171$	5.61222	0.429177
$172$	−34.7082	−2.64648
$173$	2.75006	0.209083	0.104541	−	0.994521i	$-0.466663\pi$
$173$	2.75006	0.209083	0.104541	+	0.994521i	$0.466663\pi$
$174$	3.06245	0.232164
$175$	1.75866	0.132942
$176$	−28.5994	−2.15576
$177$	−0.222350	−0.0167129
$178$	11.6614	0.874058
$179$	25.9940	1.94288	0.971441	−	0.237281i	$-0.0762563\pi$
$179$	25.9940	1.94288	0.971441	+	0.237281i	$0.0762563\pi$
$180$	−12.5735	−0.937174
$181$	15.2327	1.13224	0.566118	−	0.824324i	$-0.308445\pi$
$181$	15.2327	1.13224	0.566118	+	0.824324i	$0.308445\pi$
$182$	−3.06481	−0.227179
$183$	0.478100	0.0353422
$184$	10.7666	0.793723
$185$	6.35932	0.467547
$186$	1.02129	0.0748846
$187$	−5.37225	−0.392858
$188$	−7.87356	−0.574238
$189$	1.31299	0.0955059
$190$	4.68732	0.340054
$191$	−18.2595	−1.32121	−0.660607	−	0.750732i	$-0.729701\pi$
$191$	−18.2595	−1.32121	−0.660607	+	0.750732i	$0.729701\pi$
$192$	0.632627	0.0456559
$193$	−2.53264	−0.182304	−0.0911518	−	0.995837i	$-0.529055\pi$
$193$	−2.53264	−0.182304	−0.0911518	+	0.995837i	$0.529055\pi$
$194$	22.1781	1.59229
$195$	−0.0872222	−0.00624611
$196$	−16.4608	−1.17577
$197$	−12.0612	−0.859323	−0.429661	−	0.902990i	$-0.641367\pi$
$197$	−12.0612	−0.859323	−0.429661	+	0.902990i	$0.641367\pi$
$198$	−39.9641	−2.84012
$199$	−22.8034	−1.61649	−0.808247	−	0.588844i	$-0.799583\pi$
$199$	−22.8034	−1.61649	−0.808247	+	0.588844i	$0.799583\pi$
$200$	−5.51618	−0.390053
$201$	0.561707	0.0396198
$202$	4.70774	0.331236
$203$	17.3198	1.21561
$204$	−0.525595	−0.0367990
$205$	−5.30939	−0.370824
$206$	11.1400	0.776159
$207$	5.82508	0.404871
$208$	3.72195	0.258071
$209$	10.1025	0.698804
$210$	0.546877	0.0377381
$211$	20.5703	1.41612	0.708058	−	0.706154i	$-0.249571\pi$
$211$	20.5703	1.41612	0.708058	+	0.706154i	$0.249571\pi$
$212$	−22.4130	−1.53933
$213$	−0.124755	−0.00854805
$214$	35.8225	2.44878
$215$	−8.23831	−0.561848
$216$	−4.11830	−0.280215
$217$	5.77593	0.392096
$218$	−30.4123	−2.05978
$219$	−1.99706	−0.134949
$220$	−22.6334	−1.52594
$221$	0.699150	0.0470299
$222$	1.97751	0.132722
$223$	18.4506	1.23554	0.617772	−	0.786357i	$-0.288035\pi$
$223$	18.4506	1.23554	0.617772	+	0.786357i	$0.288035\pi$
$224$	−3.93425	−0.262868
$225$	−2.98444	−0.198962
$226$	34.2464	2.27804
$227$	6.70576	0.445077	0.222538	−	0.974924i	$-0.428566\pi$
$227$	6.70576	0.445077	0.222538	+	0.974924i	$0.428566\pi$
$228$	0.988379	0.0654570
$229$	15.3655	1.01538	0.507689	−	0.861540i	$-0.330500\pi$
$229$	15.3655	1.01538	0.507689	+	0.861540i	$0.330500\pi$
$230$	4.86509	0.320795
$231$	1.17867	0.0775511
$232$	−54.3249	−3.56661
$233$	8.87130	0.581178	0.290589	−	0.956848i	$-0.406149\pi$
$233$	8.87130	0.581178	0.290589	+	0.956848i	$0.406149\pi$
$234$	5.20097	0.339998
$235$	−1.86886	−0.121911
$236$	7.50888	0.488787
$237$	−1.62709	−0.105691
$238$	−4.38362	−0.284148
$239$	−26.7730	−1.73180	−0.865899	−	0.500218i	$-0.833253\pi$
$239$	−26.7730	−1.73180	−0.865899	+	0.500218i	$0.833253\pi$
$240$	−0.664137	−0.0428698
$241$	22.5340	1.45154	0.725772	−	0.687935i	$-0.241483\pi$
$241$	22.5340	1.45154	0.725772	+	0.687935i	$0.241483\pi$
$242$	−44.5203	−2.86187
$243$	−3.34511	−0.214589
$244$	−16.1457	−1.03362
$245$	−3.90712	−0.249617
$246$	−1.65102	−0.105265
$247$	−1.31475	−0.0836555
$248$	−18.1167	−1.15041
$249$	0.818196	0.0518511
$250$	−2.49259	−0.157646
$251$	26.1843	1.65274	0.826368	−	0.563131i	$-0.190403\pi$
$251$	26.1843	1.65274	0.826368	+	0.563131i	$0.190403\pi$
$252$	−22.1125	−1.39296
$253$	10.4856	0.659227
$254$	−30.0153	−1.88333
$255$	−0.124755	−0.00781244
$256$	−32.5164	−2.03227
$257$	21.8292	1.36167	0.680833	−	0.732439i	$-0.261618\pi$
$257$	21.8292	1.36167	0.680833	+	0.732439i	$0.261618\pi$
$258$	−2.56181	−0.159491
$259$	11.1839	0.694932
$260$	2.94554	0.182674
$261$	−29.3916	−1.81929
$262$	34.7128	2.14456
$263$	0.116704	0.00719626	0.00359813	−	0.999994i	$-0.498855\pi$
$263$	0.116704	0.00719626	0.00359813	+	0.999994i	$0.498855\pi$
$264$	−3.69701	−0.227535
$265$	−5.31994	−0.326801
$266$	8.24338	0.505434
$267$	0.583654	0.0357190
$268$	−18.9691	−1.15872
$269$	−1.44421	−0.0880551	−0.0440275	−	0.999030i	$-0.514019\pi$
$269$	−1.44421	−0.0880551	−0.0440275	+	0.999030i	$0.514019\pi$
$270$	−1.86094	−0.113253
$271$	5.26470	0.319808	0.159904	−	0.987133i	$-0.448882\pi$
$271$	5.26470	0.319808	0.159904	+	0.987133i	$0.448882\pi$
$272$	5.32354	0.322787
$273$	−0.153394	−0.00928382
$274$	−21.0984	−1.27460
$275$	−5.37225	−0.323959
$276$	1.02587	0.0617498
$277$	−21.7967	−1.30964	−0.654818	−	0.755787i	$-0.727255\pi$
$277$	−21.7967	−1.30964	−0.654818	+	0.755787i	$0.727255\pi$
$278$	−10.8676	−0.651796
$279$	−9.80174	−0.586815
$280$	−9.70107	−0.579750
$281$	−5.83981	−0.348374	−0.174187	−	0.984713i	$-0.555730\pi$
$281$	−5.83981	−0.348374	−0.174187	+	0.984713i	$0.555730\pi$
$282$	−0.581146	−0.0346067
$283$	−7.84244	−0.466185	−0.233092	−	0.972455i	$-0.574884\pi$
$283$	−7.84244	−0.466185	−0.233092	+	0.972455i	$0.574884\pi$
$284$	4.21303	0.249997
$285$	0.234601	0.0138965
$286$	9.36220	0.553598
$287$	−9.33739	−0.551169
$288$	6.67640	0.393411
$289$	1.00000	0.0588235
$290$	−24.5478	−1.44150
$291$	1.11001	0.0650702
$292$	67.4418	3.94673
$293$	−6.82566	−0.398760	−0.199380	−	0.979922i	$-0.563893\pi$
$293$	−6.82566	−0.398760	−0.199380	+	0.979922i	$0.563893\pi$
$294$	−1.21497	−0.0708585
$295$	1.78230	0.103770
$296$	−35.0792	−2.03893
$297$	−4.01085	−0.232733
$298$	−50.6664	−2.93503
$299$	−1.36461	−0.0789176
$300$	−0.525595	−0.0303452
$301$	−14.4884	−0.835095
$302$	−38.2039	−2.19838
$303$	0.235623	0.0135362
$304$	−10.0109	−0.574165
$305$	−3.83232	−0.219438
$306$	7.43899	0.425259
$307$	9.09454	0.519053	0.259527	−	0.965736i	$-0.416433\pi$
$307$	9.09454	0.519053	0.259527	+	0.965736i	$0.416433\pi$
$308$	−39.8044	−2.26807
$309$	0.557557	0.0317183
$310$	−8.18639	−0.464956
$311$	18.0386	1.02287	0.511437	−	0.859321i	$-0.329114\pi$
$311$	18.0386	1.02287	0.511437	+	0.859321i	$0.329114\pi$
$312$	0.481133	0.0272388
$313$	25.3252	1.43146	0.715731	−	0.698376i	$-0.246094\pi$
$313$	25.3252	1.43146	0.715731	+	0.698376i	$0.246094\pi$
$314$	−47.0769	−2.65670
$315$	−5.24860	−0.295725
$316$	54.9476	3.09104
$317$	−10.1654	−0.570946	−0.285473	−	0.958387i	$-0.592151\pi$
$317$	−10.1654	−0.570946	−0.285473	+	0.958387i	$0.592151\pi$
$318$	−1.65430	−0.0927687
$319$	−52.9075	−2.96225
$320$	−5.07097	−0.283476
$321$	1.79292	0.100071
$322$	8.55603	0.476809
$323$	−1.88050	−0.104634
$324$	37.3281	2.07378
$325$	0.699150	0.0387818
$326$	26.1638	1.44908
$327$	−1.52214	−0.0841746
$328$	29.2875	1.61713
$329$	−3.28668	−0.181201
$330$	−1.67057	−0.0919618
$331$	16.6573	0.915568	0.457784	−	0.889063i	$-0.348643\pi$
$331$	16.6573	0.915568	0.457784	+	0.889063i	$0.348643\pi$
$332$	−27.6309	−1.51644
$333$	−18.9790	−1.04004
$334$	24.6169	1.34697
$335$	−4.50249	−0.245998
$336$	−1.16799	−0.0637190
$337$	10.1074	0.550584	0.275292	−	0.961361i	$-0.411225\pi$
$337$	10.1074	0.550584	0.275292	+	0.961361i	$0.411225\pi$
$338$	31.1853	1.69626
$339$	1.71404	0.0930937
$340$	4.21303	0.228483
$341$	−17.6440	−0.955476
$342$	−13.9890	−0.756438
$343$	−19.1819	−1.03572
$344$	45.4440	2.45018
$345$	0.243498	0.0131095
$346$	−6.85477	−0.368515
$347$	−21.8175	−1.17123	−0.585613	−	0.810591i	$-0.699146\pi$
$347$	−21.8175	−1.17123	−0.585613	+	0.810591i	$0.699146\pi$
$348$	−5.17621	−0.277474
$349$	30.2574	1.61964	0.809819	−	0.586679i	$-0.199565\pi$
$349$	30.2574	1.61964	0.809819	+	0.586679i	$0.199565\pi$
$350$	−4.38362	−0.234314
$351$	0.521976	0.0278610
$352$	12.0181	0.640567
$353$	23.4253	1.24681	0.623403	−	0.781901i	$-0.285750\pi$
$353$	23.4253	1.24681	0.623403	+	0.781901i	$0.285750\pi$
$354$	0.554229	0.0294570
$355$	1.00000	0.0530745
$356$	−19.7103	−1.04464
$357$	−0.219401	−0.0116119
$358$	−64.7925	−3.42439
$359$	9.78746	0.516563	0.258281	−	0.966070i	$-0.416844\pi$
$359$	9.78746	0.516563	0.258281	+	0.966070i	$0.416844\pi$
$360$	16.4627	0.867660
$361$	−15.4637	−0.813881
$362$	−37.9689	−1.99560
$363$	−2.22825	−0.116953
$364$	5.18019	0.271516
$365$	16.0079	0.837892
$366$	−1.19171	−0.0622917
$367$	−25.5311	−1.33271	−0.666357	−	0.745633i	$-0.732147\pi$
$367$	−25.5311	−1.33271	−0.666357	+	0.745633i	$0.732147\pi$
$368$	−10.3906	−0.541647
$369$	15.8455	0.824885
$370$	−15.8512	−0.824065
$371$	−9.35595	−0.485737
$372$	−1.72620	−0.0894995
$373$	10.3465	0.535723	0.267861	−	0.963457i	$-0.413683\pi$
$373$	10.3465	0.535723	0.267861	+	0.963457i	$0.413683\pi$
$374$	13.3908	0.692423
$375$	−0.124755	−0.00644230
$376$	10.3090	0.531644
$377$	6.88543	0.354618
$378$	−3.27275	−0.168332
$379$	−12.3010	−0.631859	−0.315930	−	0.948783i	$-0.602316\pi$
$379$	−12.3010	−0.631859	−0.315930	+	0.948783i	$0.602316\pi$
$380$	−7.92258	−0.406420
$381$	−1.50227	−0.0769637
$382$	45.5136	2.32868
$383$	23.5048	1.20104	0.600519	−	0.799610i	$-0.294961\pi$
$383$	23.5048	1.20104	0.600519	+	0.799610i	$0.294961\pi$
$384$	−2.13505	−0.108954
$385$	−9.44794	−0.481512
$386$	6.31285	0.321316
$387$	24.5867	1.24981
$388$	−37.4858	−1.90305
$389$	1.55689	0.0789375	0.0394687	−	0.999221i	$-0.487433\pi$
$389$	1.55689	0.0789375	0.0394687	+	0.999221i	$0.487433\pi$
$390$	0.217410	0.0110090
$391$	−1.95182	−0.0987077
$392$	21.5524	1.08856
$393$	1.73738	0.0876392
$394$	30.0636	1.51458
$395$	13.0423	0.656230
$396$	67.5480	3.39441
$397$	−25.2762	−1.26858	−0.634288	−	0.773097i	$-0.718707\pi$
$397$	−25.2762	−1.26858	−0.634288	+	0.773097i	$0.718707\pi$
$398$	56.8397	2.84912
$399$	0.412582	0.0206549
$400$	5.32354	0.266177
$401$	15.0593	0.752028	0.376014	−	0.926614i	$-0.377294\pi$
$401$	15.0593	0.752028	0.376014	+	0.926614i	$0.377294\pi$
$402$	−1.40011	−0.0698310
$403$	2.29621	0.114382
$404$	−7.95711	−0.395881
$405$	8.86017	0.440265
$406$	−43.1712	−2.14255
$407$	−34.1639	−1.69344
$408$	0.688169	0.0340694
$409$	0.567349	0.0280536	0.0140268	−	0.999902i	$-0.495535\pi$
$409$	0.567349	0.0280536	0.0140268	+	0.999902i	$0.495535\pi$
$410$	13.2341	0.653588
$411$	−1.05598	−0.0520875
$412$	−18.8290	−0.927637
$413$	3.13446	0.154237
$414$	−14.5196	−0.713597
$415$	−6.55844	−0.321941
$416$	−1.56405	−0.0766838
$417$	−0.543926	−0.0266362
$418$	−25.1814	−1.23166
$419$	−26.2781	−1.28377	−0.641885	−	0.766801i	$-0.721847\pi$
$419$	−26.2781	−1.28377	−0.641885	+	0.766801i	$0.721847\pi$
$420$	−0.924341	−0.0451032
$421$	38.7275	1.88746	0.943732	−	0.330712i	$-0.107289\pi$
$421$	38.7275	1.88746	0.943732	+	0.330712i	$0.107289\pi$
$422$	−51.2734	−2.49595
$423$	5.57749	0.271187
$424$	29.3457	1.42515
$425$	1.00000	0.0485071
$426$	0.310963	0.0150662
$427$	−6.73974	−0.326159
$428$	−60.5479	−2.92669
$429$	0.468579	0.0226232
$430$	20.5348	0.990274
$431$	9.94945	0.479248	0.239624	−	0.970866i	$-0.422976\pi$
$431$	9.94945	0.479248	0.239624	+	0.970866i	$0.422976\pi$
$432$	3.97448	0.191222
$433$	24.8235	1.19294	0.596471	−	0.802635i	$-0.296569\pi$
$433$	24.8235	1.19294	0.596471	+	0.802635i	$0.296569\pi$
$434$	−14.3971	−0.691081
$435$	−1.22862	−0.0589078
$436$	51.4035	2.46178
$437$	3.67039	0.175578
$438$	4.97786	0.237852
$439$	29.0194	1.38502	0.692509	−	0.721409i	$-0.256505\pi$
$439$	29.0194	1.38502	0.692509	+	0.721409i	$0.256505\pi$
$440$	29.6343	1.41276
$441$	11.6606	0.555265
$442$	−1.74270	−0.0828916
$443$	26.5521	1.26153	0.630764	−	0.775975i	$-0.282742\pi$
$443$	26.5521	1.26153	0.630764	+	0.775975i	$0.282742\pi$
$444$	−3.34243	−0.158625
$445$	−4.67841	−0.221778
$446$	−45.9899	−2.17768
$447$	−2.53586	−0.119942
$448$	−8.91809	−0.421340
$449$	−20.7474	−0.979128	−0.489564	−	0.871967i	$-0.662844\pi$
$449$	−20.7474	−0.979128	−0.489564	+	0.871967i	$0.662844\pi$
$450$	7.43899	0.350677
$451$	28.5233	1.34311
$452$	−57.8839	−2.72263
$453$	−1.91211	−0.0898386
$454$	−16.7147	−0.784462
$455$	1.22956	0.0576429
$456$	−1.29410	−0.0606017
$457$	−23.2058	−1.08552	−0.542762	−	0.839887i	$-0.682621\pi$
$457$	−23.2058	−1.08552	−0.542762	+	0.839887i	$0.682621\pi$
$458$	−38.2999	−1.78964
$459$	0.746586	0.0348477
$460$	−8.22306	−0.383402
$461$	−10.1036	−0.470573	−0.235286	−	0.971926i	$-0.575603\pi$
$461$	−10.1036	−0.470573	−0.235286	+	0.971926i	$0.575603\pi$
$462$	−2.93796	−0.136686
$463$	−22.7930	−1.05928	−0.529640	−	0.848223i	$-0.677673\pi$
$463$	−22.7930	−1.05928	−0.529640	+	0.848223i	$0.677673\pi$
$464$	52.4278	2.43390
$465$	−0.409730	−0.0190008
$466$	−22.1125	−1.02434
$467$	−33.3833	−1.54480	−0.772398	−	0.635139i	$-0.780943\pi$
$467$	−33.3833	−1.54480	−0.772398	+	0.635139i	$0.780943\pi$
$468$	−8.79077	−0.406353
$469$	−7.91834	−0.365635
$470$	4.65831	0.214872
$471$	−2.35621	−0.108568
$472$	−9.83149	−0.452531
$473$	44.2582	2.03500
$474$	4.05567	0.186283
$475$	−1.88050	−0.0862831
$476$	7.40927	0.339603
$477$	15.8770	0.726959
$478$	66.7341	3.05235
$479$	35.5256	1.62320	0.811602	−	0.584211i	$-0.198596\pi$
$479$	35.5256	1.62320	0.811602	+	0.584211i	$0.198596\pi$
$480$	0.279085	0.0127384
$481$	4.44612	0.202726
$482$	−56.1682	−2.55839
$483$	0.428230	0.0194852
$484$	75.2490	3.42041
$485$	−8.89758	−0.404018
$486$	8.33799	0.378219
$487$	1.50765	0.0683183	0.0341591	−	0.999416i	$-0.489125\pi$
$487$	1.50765	0.0683183	0.0341591	+	0.999416i	$0.489125\pi$
$488$	21.1398	0.956953
$489$	1.30950	0.0592178
$490$	9.73887	0.439958
$491$	33.5711	1.51504	0.757521	−	0.652811i	$-0.226410\pi$
$491$	33.5711	1.51504	0.757521	+	0.652811i	$0.226410\pi$
$492$	2.79059	0.125809
$493$	9.84829	0.443545
$494$	3.27713	0.147445
$495$	16.0331	0.720636
$496$	17.4840	0.785056
$497$	1.75866	0.0788866
$498$	−2.03943	−0.0913891
$499$	36.0293	1.61289	0.806447	−	0.591307i	$-0.201388\pi$
$499$	36.0293	1.61289	0.806447	+	0.591307i	$0.201388\pi$
$500$	4.21303	0.188412
$501$	1.23208	0.0550451
$502$	−65.2667	−2.91300
$503$	31.4548	1.40250	0.701250	−	0.712915i	$-0.252626\pi$
$503$	31.4548	1.40250	0.701250	+	0.712915i	$0.252626\pi$
$504$	28.9522	1.28963
$505$	−1.88869	−0.0840457
$506$	−26.1365	−1.16191
$507$	1.56083	0.0693189
$508$	50.7324	2.25089
$509$	27.2134	1.20621	0.603106	−	0.797661i	$-0.293930\pi$
$509$	27.2134	1.20621	0.603106	+	0.797661i	$0.293930\pi$
$510$	0.310963	0.0137697
$511$	28.1524	1.24539
$512$	46.8221	2.06926
$513$	−1.40395	−0.0619860
$514$	−54.4113	−2.39998
$515$	−4.46923	−0.196938
$516$	4.33001	0.190618
$517$	10.0400	0.441558
$518$	−27.8769	−1.22484
$519$	−0.343082	−0.0150596
$520$	−3.85663	−0.169125
$521$	−1.70041	−0.0744965	−0.0372482	−	0.999306i	$-0.511859\pi$
$521$	−1.70041	−0.0744965	−0.0372482	+	0.999306i	$0.511859\pi$
$522$	73.2613	3.20656
$523$	19.7094	0.861832	0.430916	−	0.902392i	$-0.358191\pi$
$523$	19.7094	0.861832	0.430916	+	0.902392i	$0.358191\pi$
$524$	−58.6722	−2.56311
$525$	−0.219401	−0.00957543
$526$	−0.290895	−0.0126836
$527$	3.28429	0.143066
$528$	3.56791	0.155273
$529$	−19.1904	−0.834365
$530$	13.2605	0.575997
$531$	−5.31916	−0.230832
$532$	−13.9331	−0.604077
$533$	−3.71206	−0.160787
$534$	−1.45481	−0.0629559
$535$	−14.3716	−0.621338
$536$	24.8366	1.07278
$537$	−3.24287	−0.139940
$538$	3.59983	0.155200
$539$	20.9900	0.904105
$540$	3.14539	0.135356
$541$	−24.3880	−1.04852	−0.524260	−	0.851558i	$-0.675658\pi$
$541$	−24.3880	−1.04852	−0.524260	+	0.851558i	$0.675658\pi$
$542$	−13.1228	−0.563671
$543$	−1.90035	−0.0815517
$544$	−2.23707	−0.0959137
$545$	12.2011	0.522637
$546$	0.382349	0.0163630
$547$	26.2292	1.12148	0.560741	−	0.827991i	$-0.310516\pi$
$547$	26.2292	1.12148	0.560741	+	0.827991i	$0.310516\pi$
$548$	35.6609	1.52336
$549$	11.4373	0.488133
$550$	13.3908	0.570987
$551$	−18.5197	−0.788965
$552$	−1.34318	−0.0571696
$553$	22.9370	0.975379
$554$	54.3303	2.30827
$555$	−0.793355	−0.0336760
$556$	18.3686	0.779004
$557$	34.5306	1.46311	0.731553	−	0.681784i	$-0.238796\pi$
$557$	34.5306	1.46311	0.731553	+	0.681784i	$0.238796\pi$
$558$	24.4318	1.03428
$559$	−5.75981	−0.243614
$560$	9.36229	0.395629
$561$	0.670213	0.0282964
$562$	14.5563	0.614019
$563$	3.05960	0.128947	0.0644734	−	0.997919i	$-0.479463\pi$
$563$	3.05960	0.128947	0.0644734	+	0.997919i	$0.479463\pi$
$564$	0.982263	0.0413607
$565$	−13.7393	−0.578015
$566$	19.5480	0.821665
$567$	15.5820	0.654383
$568$	−5.51618	−0.231454
$569$	−2.34921	−0.0984839	−0.0492420	−	0.998787i	$-0.515681\pi$
$569$	−2.34921	−0.0984839	−0.0492420	+	0.998787i	$0.515681\pi$
$570$	−0.584764	−0.0244931
$571$	−26.6239	−1.11417	−0.557087	−	0.830454i	$-0.688081\pi$
$571$	−26.6239	−1.11417	−0.557087	+	0.830454i	$0.688081\pi$
$572$	−15.8241	−0.661641
$573$	2.27796	0.0951632
$574$	23.2743	0.971452
$575$	−1.95182	−0.0813964
$576$	15.1340	0.630582
$577$	−14.1976	−0.591054	−0.295527	−	0.955334i	$-0.595495\pi$
$577$	−14.1976	−0.591054	−0.295527	+	0.955334i	$0.595495\pi$
$578$	−2.49259	−0.103678
$579$	0.315959	0.0131308
$580$	41.4911	1.72282
$581$	−11.5341	−0.478513
$582$	−2.76682	−0.114688
$583$	28.5800	1.18366
$584$	−88.3025	−3.65398
$585$	−2.08657	−0.0862690
$586$	17.0136	0.702826
$587$	19.7437	0.814909	0.407455	−	0.913225i	$-0.366416\pi$
$587$	19.7437	0.814909	0.407455	+	0.913225i	$0.366416\pi$
$588$	2.05356	0.0846875
$589$	−6.17609	−0.254481
$590$	−4.44255	−0.182897
$591$	1.50469	0.0618945
$592$	33.8541	1.39140
$593$	−24.1804	−0.992968	−0.496484	−	0.868046i	$-0.665376\pi$
$593$	−24.1804	−0.992968	−0.496484	+	0.868046i	$0.665376\pi$
$594$	9.99741	0.410199
$595$	1.75866	0.0720979
$596$	85.6373	3.50784
$597$	2.84484	0.116431
$598$	3.40143	0.139095
$599$	−2.17864	−0.0890169	−0.0445085	−	0.999009i	$-0.514172\pi$
$599$	−2.17864	−0.0890169	−0.0445085	+	0.999009i	$0.514172\pi$
$600$	0.688169	0.0280944
$601$	1.08993	0.0444593	0.0222297	−	0.999753i	$-0.492923\pi$
$601$	1.08993	0.0444593	0.0222297	+	0.999753i	$0.492923\pi$
$602$	36.1136	1.47188
$603$	13.4374	0.547213
$604$	64.5728	2.62743
$605$	17.8610	0.726154
$606$	−0.587313	−0.0238580
$607$	3.02116	0.122625	0.0613125	−	0.998119i	$-0.480471\pi$
$607$	3.02116	0.122625	0.0613125	+	0.998119i	$0.480471\pi$
$608$	4.20681	0.170609
$609$	−2.16072	−0.0875569
$610$	9.55243	0.386766
$611$	−1.30661	−0.0528599
$612$	−12.5735	−0.508254
$613$	−13.2503	−0.535173	−0.267586	−	0.963534i	$-0.586226\pi$
$613$	−13.2503	−0.535173	−0.267586	+	0.963534i	$0.586226\pi$
$614$	−22.6690	−0.914847
$615$	0.662371	0.0267094
$616$	52.1165	2.09984
$617$	−17.4194	−0.701279	−0.350639	−	0.936511i	$-0.614036\pi$
$617$	−17.4194	−0.701279	−0.350639	+	0.936511i	$0.614036\pi$
$618$	−1.38976	−0.0559045
$619$	24.2545	0.974869	0.487435	−	0.873160i	$-0.337933\pi$
$619$	24.2545	0.974869	0.487435	+	0.873160i	$0.337933\pi$
$620$	13.8368	0.555699
$621$	−1.45720	−0.0584754
$622$	−44.9628	−1.80284
$623$	−8.22772	−0.329637
$624$	−0.464331	−0.0185881
$625$	1.00000	0.0400000
$626$	−63.1253	−2.52300
$627$	−1.26033	−0.0503328
$628$	79.5702	3.17520
$629$	6.35932	0.253563
$630$	13.0826	0.521225
$631$	−12.1672	−0.484370	−0.242185	−	0.970230i	$-0.577864\pi$
$631$	−12.1672	−0.484370	−0.242185	+	0.970230i	$0.577864\pi$
$632$	−71.9437	−2.86177
$633$	−2.56624	−0.101999
$634$	25.3382	1.00631
$635$	12.0418	0.477864
$636$	2.79613	0.110874
$637$	−2.73166	−0.108232
$638$	131.877	5.22105
$639$	−2.98444	−0.118062
$640$	17.1140	0.676491
$641$	2.05507	0.0811704	0.0405852	−	0.999176i	$-0.487078\pi$
$641$	2.05507	0.0811704	0.0405852	+	0.999176i	$0.487078\pi$
$642$	−4.46903	−0.176378
$643$	−38.0631	−1.50106	−0.750531	−	0.660835i	$-0.770202\pi$
$643$	−38.0631	−1.50106	−0.750531	+	0.660835i	$0.770202\pi$
$644$	−14.4616	−0.569865
$645$	1.02777	0.0404683
$646$	4.68732	0.184420
$647$	−7.55077	−0.296851	−0.148426	−	0.988924i	$-0.547421\pi$
$647$	−7.55077	−0.296851	−0.148426	+	0.988924i	$0.547421\pi$
$648$	−48.8743	−1.91996
$649$	−9.57496	−0.375850
$650$	−1.74270	−0.0683542
$651$	−0.720575	−0.0282415
$652$	−44.2226	−1.73189
$653$	40.9725	1.60338	0.801689	−	0.597741i	$-0.203935\pi$
$653$	40.9725	1.60338	0.801689	+	0.597741i	$0.203935\pi$
$654$	3.79408	0.148360
$655$	−13.9264	−0.544148
$656$	−28.2647	−1.10355
$657$	−47.7746	−1.86386
$658$	8.19237	0.319372
$659$	13.7798	0.536783	0.268392	−	0.963310i	$-0.413508\pi$
$659$	13.7798	0.536783	0.268392	+	0.963310i	$0.413508\pi$
$660$	2.82362	0.109909
$661$	−20.4895	−0.796950	−0.398475	−	0.917179i	$-0.630460\pi$
$661$	−20.4895	−0.796950	−0.398475	+	0.917179i	$0.630460\pi$
$662$	−41.5199	−1.61372
$663$	−0.0872222	−0.00338743
$664$	36.1775	1.40396
$665$	−3.30715	−0.128246
$666$	47.3069	1.83311
$667$	−19.2221	−0.744282
$668$	−41.6078	−1.60986
$669$	−2.30180	−0.0889927
$670$	11.2229	0.433578
$671$	20.5882	0.794798
$672$	0.490815	0.0189336
$673$	49.3011	1.90042	0.950210	−	0.311611i	$-0.100869\pi$
$673$	49.3011	1.90042	0.950210	+	0.311611i	$0.100869\pi$
$674$	−25.1936	−0.970421
$675$	0.746586	0.0287361
$676$	−52.7100	−2.02731
$677$	24.7425	0.950931	0.475465	−	0.879734i	$-0.342280\pi$
$677$	24.7425	0.950931	0.475465	+	0.879734i	$0.342280\pi$
$678$	−4.27240	−0.164080
$679$	−15.6478	−0.600508
$680$	−5.51618	−0.211536
$681$	−0.836575	−0.0320576
$682$	43.9793	1.68406
$683$	−7.93114	−0.303476	−0.151738	−	0.988421i	$-0.548487\pi$
$683$	−7.93114	−0.303476	−0.151738	+	0.988421i	$0.548487\pi$
$684$	23.6444	0.904068
$685$	8.46444	0.323410
$686$	47.8127	1.82550
$687$	−1.91691	−0.0731348
$688$	−43.8570	−1.67203
$689$	−3.71943	−0.141699
$690$	−0.606943	−0.0231059
$691$	36.5211	1.38933	0.694664	−	0.719334i	$-0.255553\pi$
$691$	36.5211	1.38933	0.694664	+	0.719334i	$0.255553\pi$
$692$	11.5861	0.440436
$693$	28.1968	1.07111
$694$	54.3822	2.06432
$695$	4.35996	0.165383
$696$	6.77729	0.256892
$697$	−5.30939	−0.201107
$698$	−75.4193	−2.85466
$699$	−1.10674	−0.0418606
$700$	7.40927	0.280044
$701$	28.6632	1.08259	0.541296	−	0.840832i	$-0.317934\pi$
$701$	28.6632	1.08259	0.541296	+	0.840832i	$0.317934\pi$
$702$	−1.30107	−0.0491058
$703$	−11.9587	−0.451030
$704$	27.2425	1.02674
$705$	0.233149	0.00878090
$706$	−58.3899	−2.19753
$707$	−3.32156	−0.124920
$708$	−0.936768	−0.0352059
$709$	13.9727	0.524756	0.262378	−	0.964965i	$-0.415493\pi$
$709$	13.9727	0.524756	0.262378	+	0.964965i	$0.415493\pi$
$710$	−2.49259	−0.0935454
$711$	−38.9239	−1.45976
$712$	25.8070	0.967156
$713$	−6.41033	−0.240069
$714$	0.546877	0.0204664
$715$	−3.75600	−0.140467
$716$	109.513	4.09271
$717$	3.34005	0.124736
$718$	−24.3962	−0.910457
$719$	45.3481	1.69120	0.845600	−	0.533817i	$-0.179243\pi$
$719$	45.3481	1.69120	0.845600	+	0.533817i	$0.179243\pi$
$720$	−15.8878	−0.592102
$721$	−7.85984	−0.292716
$722$	38.5448	1.43449
$723$	−2.81123	−0.104551
$724$	64.1757	2.38507
$725$	9.84829	0.365756
$726$	5.55412	0.206133
$727$	1.44388	0.0535505	0.0267753	−	0.999641i	$-0.491476\pi$
$727$	1.44388	0.0535505	0.0267753	+	0.999641i	$0.491476\pi$
$728$	−6.78250	−0.251376
$729$	−26.1632	−0.969007
$730$	−39.9012	−1.47681
$731$	−8.23831	−0.304705
$732$	2.01425	0.0744488
$733$	−23.7608	−0.877626	−0.438813	−	0.898578i	$-0.644601\pi$
$733$	−23.7608	−0.877626	−0.438813	+	0.898578i	$0.644601\pi$
$734$	63.6387	2.34895
$735$	0.487432	0.0179792
$736$	4.36636	0.160946
$737$	24.1885	0.890995
$738$	−39.4965	−1.45389
$739$	−32.0805	−1.18010	−0.590050	−	0.807367i	$-0.700892\pi$
$739$	−32.0805	−1.18010	−0.590050	+	0.807367i	$0.700892\pi$
$740$	26.7920	0.984893
$741$	0.164021	0.00602546
$742$	23.3206	0.856126
$743$	7.69480	0.282295	0.141147	−	0.989989i	$-0.454921\pi$
$743$	7.69480	0.282295	0.141147	+	0.989989i	$0.454921\pi$
$744$	2.26014	0.0828609
$745$	20.3268	0.744716
$746$	−25.7897	−0.944227
$747$	19.5733	0.716148
$748$	−22.6334	−0.827560
$749$	−25.2747	−0.923517
$750$	0.310963	0.0113548
$751$	17.0439	0.621942	0.310971	−	0.950419i	$-0.399346\pi$
$751$	17.0439	0.621942	0.310971	+	0.950419i	$0.399346\pi$
$752$	−9.94895	−0.362801
$753$	−3.26661	−0.119042
$754$	−17.1626	−0.625024
$755$	15.3269	0.557805
$756$	5.53166	0.201185
$757$	−5.65675	−0.205598	−0.102799	−	0.994702i	$-0.532780\pi$
$757$	−5.65675	−0.205598	−0.102799	+	0.994702i	$0.532780\pi$
$758$	30.6614	1.11367
$759$	−1.30813	−0.0474822
$760$	10.3732	0.376274
$761$	−29.9216	−1.08466	−0.542328	−	0.840167i	$-0.682457\pi$
$761$	−29.9216	−1.08466	−0.542328	+	0.840167i	$0.682457\pi$
$762$	3.74455	0.135651
$763$	21.4575	0.776814
$764$	−76.9279	−2.78315
$765$	−2.98444	−0.107903
$766$	−58.5879	−2.11687
$767$	1.24610	0.0449939
$768$	4.05657	0.146379
$769$	24.7899	0.893945	0.446973	−	0.894548i	$-0.352502\pi$
$769$	24.7899	0.893945	0.446973	+	0.894548i	$0.352502\pi$
$770$	23.5499	0.848679
$771$	−2.72329	−0.0980769
$772$	−10.6701	−0.384025
$773$	−46.3703	−1.66782	−0.833912	−	0.551897i	$-0.813904\pi$
$773$	−46.3703	−1.66782	−0.833912	+	0.551897i	$0.813904\pi$
$774$	−61.2847	−2.20283
$775$	3.28429	0.117975
$776$	49.0807	1.76189
$777$	−1.39524	−0.0500540
$778$	−3.88070	−0.139130
$779$	9.98428	0.357724
$780$	−0.367469	−0.0131575
$781$	−5.37225	−0.192234
$782$	4.86509	0.173975
$783$	7.35260	0.262760
$784$	−20.7997	−0.742848
$785$	18.8867	0.674096
$786$	−4.33058	−0.154467
$787$	40.5683	1.44610	0.723051	−	0.690794i	$-0.242739\pi$
$787$	40.5683	1.44610	0.723051	+	0.690794i	$0.242739\pi$
$788$	−50.8140	−1.81017
$789$	−0.0145593	−0.000518326 0
$790$	−32.5092	−1.15662
$791$	−24.1627	−0.859125
$792$	−88.4416	−3.14264
$793$	−2.67937	−0.0951471
$794$	63.0033	2.23590
$795$	0.663687	0.0235386
$796$	−96.0715	−3.40517
$797$	−5.24534	−0.185799	−0.0928997	−	0.995675i	$-0.529614\pi$
$797$	−5.24534	−0.185799	−0.0928997	+	0.995675i	$0.529614\pi$
$798$	−1.02840	−0.0364050
$799$	−1.86886	−0.0661155
$800$	−2.23707	−0.0790925
$801$	13.9624	0.493338
$802$	−37.5368	−1.32547
$803$	−85.9985	−3.03482
$804$	2.36649	0.0834595
$805$	−3.43258	−0.120983
$806$	−5.72351	−0.201602
$807$	0.180172	0.00634235
$808$	10.4184	0.366517
$809$	−4.64465	−0.163297	−0.0816486	−	0.996661i	$-0.526019\pi$
$809$	−4.64465	−0.163297	−0.0816486	+	0.996661i	$0.526019\pi$
$810$	−22.0848	−0.775981
$811$	−32.6383	−1.14608	−0.573042	−	0.819526i	$-0.694237\pi$
$811$	−32.6383	−1.14608	−0.573042	+	0.819526i	$0.694237\pi$
$812$	72.9687	2.56070
$813$	−0.656796	−0.0230348
$814$	85.1566	2.98474
$815$	−10.4966	−0.367681
$816$	−0.664137	−0.0232494
$817$	15.4921	0.542000
$818$	−1.41417	−0.0494453
$819$	−3.66956	−0.128225
$820$	−22.3686	−0.781145
$821$	36.3923	1.27010	0.635050	−	0.772471i	$-0.280979\pi$
$821$	36.3923	1.27010	0.635050	+	0.772471i	$0.280979\pi$
$822$	2.63212	0.0918059
$823$	10.2321	0.356669	0.178335	−	0.983970i	$-0.442929\pi$
$823$	10.2321	0.356669	0.178335	+	0.983970i	$0.442929\pi$
$824$	24.6531	0.858830
$825$	0.670213	0.0233338
$826$	−7.81293	−0.271847
$827$	2.83152	0.0984617	0.0492308	−	0.998787i	$-0.484323\pi$
$827$	2.83152	0.0984617	0.0492308	+	0.998787i	$0.484323\pi$
$828$	24.5412	0.852866
$829$	24.1314	0.838117	0.419058	−	0.907959i	$-0.362360\pi$
$829$	24.1314	0.838117	0.419058	+	0.907959i	$0.362360\pi$
$830$	16.3475	0.567431
$831$	2.71924	0.0943293
$832$	−3.54537	−0.122913
$833$	−3.90712	−0.135374
$834$	1.35579	0.0469470
$835$	−9.87600	−0.341773
$836$	42.5621	1.47204
$837$	2.45200	0.0847536
$838$	65.5007	2.26268
$839$	−53.5161	−1.84758	−0.923790	−	0.382899i	$-0.874926\pi$
$839$	−53.5161	−1.84758	−0.923790	+	0.382899i	$0.874926\pi$
$840$	1.21025	0.0417577
$841$	67.9888	2.34444
$842$	−96.5320	−3.32671
$843$	0.728543	0.0250924
$844$	86.6632	2.98307
$845$	−12.5112	−0.430398
$846$	−13.9024	−0.477975
$847$	31.4114	1.07931
$848$	−28.3209	−0.972544
$849$	0.978381	0.0335779
$850$	−2.49259	−0.0854953
$851$	−12.4122	−0.425486
$852$	−0.525595	−0.0180066
$853$	14.8216	0.507482	0.253741	−	0.967272i	$-0.418339\pi$
$853$	14.8216	0.507482	0.253741	+	0.967272i	$0.418339\pi$
$854$	16.7994	0.574865
$855$	5.61222	0.191934
$856$	79.2762	2.70961
$857$	−38.4458	−1.31328	−0.656641	−	0.754203i	$-0.728023\pi$
$857$	−38.4458	−1.31328	−0.656641	+	0.754203i	$0.728023\pi$
$858$	−1.16798	−0.0398741
$859$	12.9733	0.442644	0.221322	−	0.975201i	$-0.428963\pi$
$859$	12.9733	0.442644	0.221322	+	0.975201i	$0.428963\pi$
$860$	−34.7082	−1.18354
$861$	1.16488	0.0396991
$862$	−24.8000	−0.844690
$863$	36.7040	1.24942	0.624710	−	0.780857i	$-0.285217\pi$
$863$	36.7040	1.24942	0.624710	+	0.780857i	$0.285217\pi$
$864$	−1.67017	−0.0568203
$865$	2.75006	0.0935047
$866$	−61.8749	−2.10260
$867$	−0.124755	−0.00423689
$868$	24.3342	0.825955
$869$	−70.0665	−2.37684
$870$	3.06245	0.103827
$871$	−3.14792	−0.106663
$872$	−67.3033	−2.27918
$873$	26.5543	0.898726
$874$	−9.14879	−0.309462
$875$	1.75866	0.0594535
$876$	−8.41367	−0.284272
$877$	−19.6167	−0.662408	−0.331204	−	0.943559i	$-0.607455\pi$
$877$	−19.6167	−0.662408	−0.331204	+	0.943559i	$0.607455\pi$
$878$	−72.3335	−2.44114
$879$	0.851533	0.0287215
$880$	−28.5994	−0.964085
$881$	10.6754	0.359663	0.179832	−	0.983697i	$-0.442445\pi$
$881$	10.6754	0.359663	0.179832	+	0.983697i	$0.442445\pi$
$882$	−29.0651	−0.978671
$883$	−4.24705	−0.142925	−0.0714624	−	0.997443i	$-0.522767\pi$
$883$	−4.24705	−0.142925	−0.0714624	+	0.997443i	$0.522767\pi$
$884$	2.94554	0.0990691
$885$	−0.222350	−0.00747423
$886$	−66.1836	−2.22348
$887$	44.7796	1.50355	0.751776	−	0.659418i	$-0.229197\pi$
$887$	44.7796	1.50355	0.751776	+	0.659418i	$0.229197\pi$
$888$	4.37629	0.146859
$889$	21.1774	0.710267
$890$	11.6614	0.390890
$891$	−47.5990	−1.59463
$892$	77.7329	2.60269
$893$	3.51438	0.117604
$894$	6.32087	0.211402
$895$	25.9940	0.868883
$896$	30.0977	1.00549
$897$	0.170242	0.00568421
$898$	51.7147	1.72574
$899$	32.3446	1.07875
$900$	−12.5735	−0.419117
$901$	−5.31994	−0.177233
$902$	−71.0971	−2.36727
$903$	1.80749	0.0601495
$904$	75.7882	2.52068
$905$	15.2327	0.506351
$906$	4.76611	0.158343
$907$	46.5234	1.54479	0.772393	−	0.635145i	$-0.219060\pi$
$907$	46.5234	1.54479	0.772393	+	0.635145i	$0.219060\pi$
$908$	28.2515	0.937561
$909$	5.63668	0.186957
$910$	−3.06481	−0.101597
$911$	−1.26440	−0.0418913	−0.0209457	−	0.999781i	$-0.506668\pi$
$911$	−1.26440	−0.0418913	−0.0209457	+	0.999781i	$0.506668\pi$
$912$	1.24891	0.0413554
$913$	35.2336	1.16606
$914$	57.8427	1.91327
$915$	0.478100	0.0158055
$916$	64.7351	2.13891
$917$	−24.4917	−0.808788
$918$	−1.86094	−0.0614201
$919$	7.58994	0.250369	0.125185	−	0.992133i	$-0.460048\pi$
$919$	7.58994	0.250369	0.125185	+	0.992133i	$0.460048\pi$
$920$	10.7666	0.354964
$921$	−1.13459	−0.0373859
$922$	25.1842	0.829399
$923$	0.699150	0.0230128
$924$	4.96579	0.163362
$925$	6.35932	0.209093
$926$	56.8137	1.86701
$927$	13.3381	0.438082
$928$	−22.0313	−0.723214
$929$	−20.3885	−0.668924	−0.334462	−	0.942409i	$-0.608555\pi$
$929$	−20.3885	−0.668924	−0.334462	+	0.942409i	$0.608555\pi$
$930$	1.02129	0.0334894
$931$	7.34733	0.240799
$932$	37.3750	1.22426
$933$	−2.25039	−0.0736746
$934$	83.2111	2.72275
$935$	−5.37225	−0.175691
$936$	11.5099	0.376212
$937$	−5.13771	−0.167842	−0.0839208	−	0.996472i	$-0.526744\pi$
$937$	−5.13771	−0.167842	−0.0839208	+	0.996472i	$0.526744\pi$
$938$	19.7372	0.644443
$939$	−3.15943	−0.103104
$940$	−7.87356	−0.256807
$941$	−11.4148	−0.372113	−0.186057	−	0.982539i	$-0.559571\pi$
$941$	−11.4148	−0.372113	−0.186057	+	0.982539i	$0.559571\pi$
$942$	5.87306	0.191355
$943$	10.3630	0.337464
$944$	9.48815	0.308813
$945$	1.31299	0.0427116
$946$	−110.318	−3.58674
$947$	−0.607440	−0.0197392	−0.00986958	−	0.999951i	$-0.503142\pi$
$947$	−0.607440	−0.0197392	−0.00986958	+	0.999951i	$0.503142\pi$
$948$	−6.85497	−0.222639
$949$	11.1919	0.363305
$950$	4.68732	0.152077
$951$	1.26818	0.0411236
$952$	−9.70107	−0.314413
$953$	35.2659	1.14238	0.571188	−	0.820819i	$-0.306483\pi$
$953$	35.2659	1.14238	0.571188	+	0.820819i	$0.306483\pi$
$954$	−39.5750	−1.28129
$955$	−18.2595	−0.590865
$956$	−112.795	−3.64806
$957$	6.60045	0.213362
$958$	−88.5508	−2.86095
$959$	14.8860	0.480696
$960$	0.632627	0.0204179
$961$	−20.2135	−0.652047
$962$	−11.0824	−0.357310
$963$	42.8911	1.38215
$964$	94.9365	3.05770
$965$	−2.53264	−0.0815287
$966$	−1.06740	−0.0343432
$967$	−54.8455	−1.76371	−0.881856	−	0.471519i	$-0.843706\pi$
$967$	−54.8455	−1.76371	−0.881856	+	0.471519i	$0.843706\pi$
$968$	−98.5247	−3.16670
$969$	0.234601	0.00753646
$970$	22.1781	0.712095
$971$	−45.2354	−1.45167	−0.725837	−	0.687867i	$-0.758547\pi$
$971$	−45.2354	−1.45167	−0.725837	+	0.687867i	$0.758547\pi$
$972$	−14.0930	−0.452034
$973$	7.66768	0.245815
$974$	−3.75797	−0.120413
$975$	−0.0872222	−0.00279335
$976$	−20.4015	−0.653037
$977$	52.9453	1.69387	0.846936	−	0.531695i	$-0.178445\pi$
$977$	52.9453	1.69387	0.846936	+	0.531695i	$0.178445\pi$
$978$	−3.26406	−0.104373
$979$	25.1336	0.803273
$980$	−16.4608	−0.525822
$981$	−36.4133	−1.16259
$982$	−83.6791	−2.67031
$983$	55.2739	1.76296	0.881481	−	0.472219i	$-0.156547\pi$
$983$	55.2739	1.76296	0.881481	+	0.472219i	$0.156547\pi$
$984$	−3.65375	−0.116477
$985$	−12.0612	−0.384301
$986$	−24.5478	−0.781761
$987$	0.410029	0.0130514
$988$	−5.53907	−0.176221
$989$	16.0797	0.511304
$990$	−39.9641	−1.27014
$991$	10.1898	0.323690	0.161845	−	0.986816i	$-0.448256\pi$
$991$	10.1898	0.323690	0.161845	+	0.986816i	$0.448256\pi$
$992$	−7.34719	−0.233273
$993$	−2.07808	−0.0659458
$994$	−4.38362	−0.139040
$995$	−22.8034	−0.722918
$996$	3.44708	0.109225
$997$	−22.7852	−0.721614	−0.360807	−	0.932641i	$-0.617499\pi$
$997$	−22.7852	−0.721614	−0.360807	+	0.932641i	$0.617499\pi$
$998$	−89.8065	−2.84277
$999$	4.74778	0.150213

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6035.2.a.g.1.5	✓	58

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6035.2.a.g.1.5	✓	58	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6035 = 5 \cdot 17 \cdot 71 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6035.a (trivial)

\(f(q)\)	\(=\)	\(q-2.49259 q^{2} -0.124755 q^{3} +4.21303 q^{4} +1.00000 q^{5} +0.310963 q^{6} +1.75866 q^{7} -5.51618 q^{8} -2.98444 q^{9} +O(q^{10})\)	\(q-2.49259 q^{2} -0.124755 q^{3} +4.21303 q^{4} +1.00000 q^{5} +0.310963 q^{6} +1.75866 q^{7} -5.51618 q^{8} -2.98444 q^{9} -2.49259 q^{10} -5.37225 q^{11} -0.525595 q^{12} +0.699150 q^{13} -4.38362 q^{14} -0.124755 q^{15} +5.32354 q^{16} +1.00000 q^{17} +7.43899 q^{18} -1.88050 q^{19} +4.21303 q^{20} -0.219401 q^{21} +13.3908 q^{22} -1.95182 q^{23} +0.688169 q^{24} +1.00000 q^{25} -1.74270 q^{26} +0.746586 q^{27} +7.40927 q^{28} +9.84829 q^{29} +0.310963 q^{30} +3.28429 q^{31} -2.23707 q^{32} +0.670213 q^{33} -2.49259 q^{34} +1.75866 q^{35} -12.5735 q^{36} +6.35932 q^{37} +4.68732 q^{38} -0.0872222 q^{39} -5.51618 q^{40} -5.30939 q^{41} +0.546877 q^{42} -8.23831 q^{43} -22.6334 q^{44} -2.98444 q^{45} +4.86509 q^{46} -1.86886 q^{47} -0.664137 q^{48} -3.90712 q^{49} -2.49259 q^{50} -0.124755 q^{51} +2.94554 q^{52} -5.31994 q^{53} -1.86094 q^{54} -5.37225 q^{55} -9.70107 q^{56} +0.234601 q^{57} -24.5478 q^{58} +1.78230 q^{59} -0.525595 q^{60} -3.83232 q^{61} -8.18639 q^{62} -5.24860 q^{63} -5.07097 q^{64} +0.699150 q^{65} -1.67057 q^{66} -4.50249 q^{67} +4.21303 q^{68} +0.243498 q^{69} -4.38362 q^{70} +1.00000 q^{71} +16.4627 q^{72} +16.0079 q^{73} -15.8512 q^{74} -0.124755 q^{75} -7.92258 q^{76} -9.44794 q^{77} +0.217410 q^{78} +13.0423 q^{79} +5.32354 q^{80} +8.86017 q^{81} +13.2341 q^{82} -6.55844 q^{83} -0.924341 q^{84} +1.00000 q^{85} +20.5348 q^{86} -1.22862 q^{87} +29.6343 q^{88} -4.67841 q^{89} +7.43899 q^{90} +1.22956 q^{91} -8.22306 q^{92} -0.409730 q^{93} +4.65831 q^{94} -1.88050 q^{95} +0.279085 q^{96} -8.89758 q^{97} +9.73887 q^{98} +16.0331 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 58 q + q^{2} + 6 q^{3} + 69 q^{4} + 58 q^{5} + 10 q^{6} + 13 q^{7} - 3 q^{8} + 84 q^{9}+O(q^{10}) \) 58 * q + q^2 + 6 * q^3 + 69 * q^4 + 58 * q^5 + 10 * q^6 + 13 * q^7 - 3 * q^8 + 84 * q^9	\( 58 q + q^{2} + 6 q^{3} + 69 q^{4} + 58 q^{5} + 10 q^{6} + 13 q^{7} - 3 q^{8} + 84 q^{9} + q^{10} + 28 q^{11} + 18 q^{12} + 37 q^{13} + 28 q^{14} + 6 q^{15} + 83 q^{16} + 58 q^{17} - 12 q^{18} + 19 q^{19} + 69 q^{20} + 31 q^{21} + 13 q^{22} + 14 q^{23} + 13 q^{24} + 58 q^{25} + 18 q^{26} + 9 q^{27} + 8 q^{28} + 60 q^{29} + 10 q^{30} + 39 q^{31} - 30 q^{32} + 13 q^{33} + q^{34} + 13 q^{35} + 113 q^{36} + 60 q^{37} - q^{38} + 41 q^{39} - 3 q^{40} + 65 q^{41} - 30 q^{42} + 17 q^{43} + 69 q^{44} + 84 q^{45} + 24 q^{46} + 16 q^{47} + 14 q^{48} + 117 q^{49} + q^{50} + 6 q^{51} + 61 q^{52} + 5 q^{53} + 24 q^{54} + 28 q^{55} + 105 q^{56} + 8 q^{57} - 34 q^{58} + 22 q^{59} + 18 q^{60} + 113 q^{61} - 19 q^{62} + 8 q^{63} + 89 q^{64} + 37 q^{65} - 37 q^{66} + 19 q^{67} + 69 q^{68} + 75 q^{69} + 28 q^{70} + 58 q^{71} - 17 q^{72} + 49 q^{73} + 29 q^{74} + 6 q^{75} - 6 q^{76} + 17 q^{77} - 12 q^{78} + 7 q^{79} + 83 q^{80} + 134 q^{81} + 7 q^{82} - 12 q^{83} - 18 q^{84} + 58 q^{85} + 23 q^{86} - 36 q^{87} - 33 q^{88} + 52 q^{89} - 12 q^{90} + 31 q^{91} + 80 q^{92} - 37 q^{93} + 4 q^{94} + 19 q^{95} - 35 q^{96} + 26 q^{97} - 33 q^{98} + 57 q^{99}+O(q^{100}) \) 58 * q + q^2 + 6 * q^3 + 69 * q^4 + 58 * q^5 + 10 * q^6 + 13 * q^7 - 3 * q^8 + 84 * q^9 + q^10 + 28 * q^11 + 18 * q^12 + 37 * q^13 + 28 * q^14 + 6 * q^15 + 83 * q^16 + 58 * q^17 - 12 * q^18 + 19 * q^19 + 69 * q^20 + 31 * q^21 + 13 * q^22 + 14 * q^23 + 13 * q^24 + 58 * q^25 + 18 * q^26 + 9 * q^27 + 8 * q^28 + 60 * q^29 + 10 * q^30 + 39 * q^31 - 30 * q^32 + 13 * q^33 + q^34 + 13 * q^35 + 113 * q^36 + 60 * q^37 - q^38 + 41 * q^39 - 3 * q^40 + 65 * q^41 - 30 * q^42 + 17 * q^43 + 69 * q^44 + 84 * q^45 + 24 * q^46 + 16 * q^47 + 14 * q^48 + 117 * q^49 + q^50 + 6 * q^51 + 61 * q^52 + 5 * q^53 + 24 * q^54 + 28 * q^55 + 105 * q^56 + 8 * q^57 - 34 * q^58 + 22 * q^59 + 18 * q^60 + 113 * q^61 - 19 * q^62 + 8 * q^63 + 89 * q^64 + 37 * q^65 - 37 * q^66 + 19 * q^67 + 69 * q^68 + 75 * q^69 + 28 * q^70 + 58 * q^71 - 17 * q^72 + 49 * q^73 + 29 * q^74 + 6 * q^75 - 6 * q^76 + 17 * q^77 - 12 * q^78 + 7 * q^79 + 83 * q^80 + 134 * q^81 + 7 * q^82 - 12 * q^83 - 18 * q^84 + 58 * q^85 + 23 * q^86 - 36 * q^87 - 33 * q^88 + 52 * q^89 - 12 * q^90 + 31 * q^91 + 80 * q^92 - 37 * q^93 + 4 * q^94 + 19 * q^95 - 35 * q^96 + 26 * q^97 - 33 * q^98 + 57 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more