LMFDB - Embedded newform 6041.2.a.f.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6041,2,Mod(1,6041)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6041, 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("6041.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6041 = 7 \cdot 863 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6041.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:	$48.2376278611$
Analytic rank:	$0$
Dimension:	$132$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	6041.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.49223 q^{2} +0.173328 q^{3} +4.21120 q^{4} +2.29751 q^{5} -0.431973 q^{6} +1.00000 q^{7} -5.51083 q^{8} -2.96996 q^{9} +O(q^{10})$	\(q-2.49223 q^{2} +0.173328 q^{3} +4.21120 q^{4} +2.29751 q^{5} -0.431973 q^{6} +1.00000 q^{7} -5.51083 q^{8} -2.96996 q^{9} -5.72592 q^{10} +4.74692 q^{11} +0.729919 q^{12} +4.44530 q^{13} -2.49223 q^{14} +0.398222 q^{15} +5.31183 q^{16} +7.62429 q^{17} +7.40181 q^{18} -5.56550 q^{19} +9.67528 q^{20} +0.173328 q^{21} -11.8304 q^{22} +5.50036 q^{23} -0.955179 q^{24} +0.278548 q^{25} -11.0787 q^{26} -1.03476 q^{27} +4.21120 q^{28} -0.702971 q^{29} -0.992461 q^{30} +8.98471 q^{31} -2.21665 q^{32} +0.822773 q^{33} -19.0015 q^{34} +2.29751 q^{35} -12.5071 q^{36} +9.07414 q^{37} +13.8705 q^{38} +0.770494 q^{39} -12.6612 q^{40} -3.05811 q^{41} -0.431973 q^{42} +4.67944 q^{43} +19.9902 q^{44} -6.82350 q^{45} -13.7082 q^{46} +5.17442 q^{47} +0.920688 q^{48} +1.00000 q^{49} -0.694205 q^{50} +1.32150 q^{51} +18.7201 q^{52} +7.04975 q^{53} +2.57886 q^{54} +10.9061 q^{55} -5.51083 q^{56} -0.964655 q^{57} +1.75197 q^{58} -11.8939 q^{59} +1.67700 q^{60} +1.14576 q^{61} -22.3920 q^{62} -2.96996 q^{63} -5.09927 q^{64} +10.2131 q^{65} -2.05054 q^{66} -8.35343 q^{67} +32.1074 q^{68} +0.953366 q^{69} -5.72592 q^{70} -5.29454 q^{71} +16.3669 q^{72} -12.0571 q^{73} -22.6148 q^{74} +0.0482801 q^{75} -23.4374 q^{76} +4.74692 q^{77} -1.92025 q^{78} +16.6769 q^{79} +12.2040 q^{80} +8.73052 q^{81} +7.62150 q^{82} -5.73960 q^{83} +0.729919 q^{84} +17.5169 q^{85} -11.6622 q^{86} -0.121844 q^{87} -26.1594 q^{88} -4.50184 q^{89} +17.0057 q^{90} +4.44530 q^{91} +23.1632 q^{92} +1.55730 q^{93} -12.8958 q^{94} -12.7868 q^{95} -0.384207 q^{96} -5.85081 q^{97} -2.49223 q^{98} -14.0981 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 132 q + 8 q^{2} + 10 q^{3} + 174 q^{4} + 11 q^{5} + 16 q^{6} + 132 q^{7} + 30 q^{8} + 178 q^{9}+O(q^{10}) $ 132 * q + 8 * q^2 + 10 * q^3 + 174 * q^4 + 11 * q^5 + 16 * q^6 + 132 * q^7 + 30 * q^8 + 178 * q^9	\( 132 q + 8 q^{2} + 10 q^{3} + 174 q^{4} + 11 q^{5} + 16 q^{6} + 132 q^{7} + 30 q^{8} + 178 q^{9} + 22 q^{10} + 32 q^{11} + 30 q^{12} + 16 q^{13} + 8 q^{14} + 59 q^{15} + 254 q^{16} + 11 q^{17} + 33 q^{18} + 40 q^{19} + 4 q^{20} + 10 q^{21} + 66 q^{22} + 62 q^{23} + 36 q^{24} + 235 q^{25} + 25 q^{26} + 37 q^{27} + 174 q^{28} + 28 q^{29} + 45 q^{30} + 121 q^{31} + 53 q^{32} - 13 q^{33} + 44 q^{34} + 11 q^{35} + 274 q^{36} + 61 q^{37} - 28 q^{38} + 114 q^{39} + 32 q^{40} - q^{41} + 16 q^{42} + 105 q^{43} + 54 q^{44} + 29 q^{45} + 104 q^{46} + 33 q^{47} + 16 q^{48} + 132 q^{49} - 14 q^{50} + 53 q^{51} - 11 q^{52} + 48 q^{53} + 11 q^{54} + 118 q^{55} + 30 q^{56} + 93 q^{57} + 87 q^{58} + 12 q^{59} + 41 q^{60} + 54 q^{61} - 28 q^{62} + 178 q^{63} + 376 q^{64} + 22 q^{65} + 6 q^{66} + 123 q^{67} - 47 q^{68} + 58 q^{69} + 22 q^{70} + 108 q^{71} + 97 q^{72} + q^{73} - 10 q^{74} + 23 q^{75} + 71 q^{76} + 32 q^{77} + 5 q^{78} + 204 q^{79} - 10 q^{80} + 296 q^{81} + 80 q^{82} - 10 q^{83} + 30 q^{84} + 94 q^{85} + 48 q^{86} + 4 q^{87} + 155 q^{88} + q^{89} - 66 q^{90} + 16 q^{91} + 49 q^{92} + 90 q^{93} + 79 q^{94} + 100 q^{95} + q^{96} + 18 q^{97} + 8 q^{98} + 96 q^{99}+O(q^{100}) \) 132 * q + 8 * q^2 + 10 * q^3 + 174 * q^4 + 11 * q^5 + 16 * q^6 + 132 * q^7 + 30 * q^8 + 178 * q^9 + 22 * q^10 + 32 * q^11 + 30 * q^12 + 16 * q^13 + 8 * q^14 + 59 * q^15 + 254 * q^16 + 11 * q^17 + 33 * q^18 + 40 * q^19 + 4 * q^20 + 10 * q^21 + 66 * q^22 + 62 * q^23 + 36 * q^24 + 235 * q^25 + 25 * q^26 + 37 * q^27 + 174 * q^28 + 28 * q^29 + 45 * q^30 + 121 * q^31 + 53 * q^32 - 13 * q^33 + 44 * q^34 + 11 * q^35 + 274 * q^36 + 61 * q^37 - 28 * q^38 + 114 * q^39 + 32 * q^40 - q^41 + 16 * q^42 + 105 * q^43 + 54 * q^44 + 29 * q^45 + 104 * q^46 + 33 * q^47 + 16 * q^48 + 132 * q^49 - 14 * q^50 + 53 * q^51 - 11 * q^52 + 48 * q^53 + 11 * q^54 + 118 * q^55 + 30 * q^56 + 93 * q^57 + 87 * q^58 + 12 * q^59 + 41 * q^60 + 54 * q^61 - 28 * q^62 + 178 * q^63 + 376 * q^64 + 22 * q^65 + 6 * q^66 + 123 * q^67 - 47 * q^68 + 58 * q^69 + 22 * q^70 + 108 * q^71 + 97 * q^72 + q^73 - 10 * q^74 + 23 * q^75 + 71 * q^76 + 32 * q^77 + 5 * q^78 + 204 * q^79 - 10 * q^80 + 296 * q^81 + 80 * q^82 - 10 * q^83 + 30 * q^84 + 94 * q^85 + 48 * q^86 + 4 * q^87 + 155 * q^88 + q^89 - 66 * q^90 + 16 * q^91 + 49 * q^92 + 90 * q^93 + 79 * q^94 + 100 * q^95 + q^96 + 18 * q^97 + 8 * q^98 + 96 * 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 $

$2$	−2.49223	−1.76227	−0.881136	−	0.472863i	$-0.843221\pi$
$2$	−2.49223	−1.76227	−0.881136	+	0.472863i	$0.843221\pi$
$3$	0.173328	0.100071	0.0500354	−	0.998747i	$-0.484067\pi$
$3$	0.173328	0.100071	0.0500354	+	0.998747i	$0.484067\pi$
$4$	4.21120	2.10560
$5$	2.29751	1.02748	0.513739	−	0.857947i	$-0.328260\pi$
$5$	2.29751	1.02748	0.513739	+	0.857947i	$0.328260\pi$
$6$	−0.431973	−0.176352
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−5.51083	−1.94837
$9$	−2.96996	−0.989986
$10$	−5.72592	−1.81069
$11$	4.74692	1.43125	0.715625	−	0.698485i	$-0.246142\pi$
$11$	4.74692	1.43125	0.715625	+	0.698485i	$0.246142\pi$
$12$	0.729919	0.210709
$13$	4.44530	1.23290	0.616452	−	0.787392i	$-0.288569\pi$
$13$	4.44530	1.23290	0.616452	+	0.787392i	$0.288569\pi$
$14$	−2.49223	−0.666076
$15$	0.398222	0.102821
$16$	5.31183	1.32796
$17$	7.62429	1.84916	0.924581	−	0.380985i	$-0.124415\pi$
$17$	7.62429	1.84916	0.924581	+	0.380985i	$0.124415\pi$
$18$	7.40181	1.74462
$19$	−5.56550	−1.27681	−0.638406	−	0.769700i	$-0.720406\pi$
$19$	−5.56550	−1.27681	−0.638406	+	0.769700i	$0.720406\pi$
$20$	9.67528	2.16346
$21$	0.173328	0.0378232
$22$	−11.8304	−2.52225
$23$	5.50036	1.14691	0.573453	−	0.819239i	$-0.305604\pi$
$23$	5.50036	1.14691	0.573453	+	0.819239i	$0.305604\pi$
$24$	−0.955179	−0.194975
$25$	0.278548	0.0557096
$26$	−11.0787	−2.17271
$27$	−1.03476	−0.199140
$28$	4.21120	0.795843
$29$	−0.702971	−0.130539	−0.0652693	−	0.997868i	$-0.520791\pi$
$29$	−0.702971	−0.130539	−0.0652693	+	0.997868i	$0.520791\pi$
$30$	−0.992461	−0.181198
$31$	8.98471	1.61370	0.806851	−	0.590755i	$-0.201170\pi$
$31$	8.98471	1.61370	0.806851	+	0.590755i	$0.201170\pi$
$32$	−2.21665	−0.391852
$33$	0.822773	0.143226
$34$	−19.0015	−3.25873
$35$	2.29751	0.388350
$36$	−12.5071	−2.08452
$37$	9.07414	1.49178	0.745890	−	0.666070i	$-0.232025\pi$
$37$	9.07414	1.49178	0.745890	+	0.666070i	$0.232025\pi$
$38$	13.8705	2.25009
$39$	0.770494	0.123378
$40$	−12.6612	−2.00191
$41$	−3.05811	−0.477596	−0.238798	−	0.971069i	$-0.576753\pi$
$41$	−3.05811	−0.477596	−0.238798	+	0.971069i	$0.576753\pi$
$42$	−0.431973	−0.0666548
$43$	4.67944	0.713608	0.356804	−	0.934179i	$-0.383866\pi$
$43$	4.67944	0.713608	0.356804	+	0.934179i	$0.383866\pi$
$44$	19.9902	3.01364
$45$	−6.82350	−1.01719
$46$	−13.7082	−2.02116
$47$	5.17442	0.754767	0.377383	−	0.926057i	$-0.376824\pi$
$47$	5.17442	0.754767	0.377383	+	0.926057i	$0.376824\pi$
$48$	0.920688	0.132890
$49$	1.00000	0.142857
$50$	−0.694205	−0.0981755
$51$	1.32150	0.185047
$52$	18.7201	2.59601
$53$	7.04975	0.968357	0.484179	−	0.874969i	$-0.339119\pi$
$53$	7.04975	0.968357	0.484179	+	0.874969i	$0.339119\pi$
$54$	2.57886	0.350938
$55$	10.9061	1.47058
$56$	−5.51083	−0.736415
$57$	−0.964655	−0.127772
$58$	1.75197	0.230044
$59$	−11.8939	−1.54845	−0.774224	−	0.632912i	$-0.781860\pi$
$59$	−11.8939	−1.54845	−0.774224	+	0.632912i	$0.781860\pi$
$60$	1.67700	0.216499
$61$	1.14576	0.146699	0.0733495	−	0.997306i	$-0.476631\pi$
$61$	1.14576	0.146699	0.0733495	+	0.997306i	$0.476631\pi$
$62$	−22.3920	−2.84378
$63$	−2.96996	−0.374179
$64$	−5.09927	−0.637409
$65$	10.2131	1.26678
$66$	−2.05054	−0.252404
$67$	−8.35343	−1.02053	−0.510267	−	0.860016i	$-0.670453\pi$
$67$	−8.35343	−1.02053	−0.510267	+	0.860016i	$0.670453\pi$
$68$	32.1074	3.89360
$69$	0.953366	0.114772
$70$	−5.72592	−0.684378
$71$	−5.29454	−0.628346	−0.314173	−	0.949366i	$-0.601727\pi$
$71$	−5.29454	−0.628346	−0.314173	+	0.949366i	$0.601727\pi$
$72$	16.3669	1.92886
$73$	−12.0571	−1.41117	−0.705586	−	0.708625i	$-0.749316\pi$
$73$	−12.0571	−1.41117	−0.705586	+	0.708625i	$0.749316\pi$
$74$	−22.6148	−2.62892
$75$	0.0482801	0.00557491
$76$	−23.4374	−2.68846
$77$	4.74692	0.540961
$78$	−1.92025	−0.217425
$79$	16.6769	1.87630	0.938149	−	0.346232i	$-0.112539\pi$
$79$	16.6769	1.87630	0.938149	+	0.346232i	$0.112539\pi$
$80$	12.2040	1.36445
$81$	8.73052	0.970058
$82$	7.62150	0.841654
$83$	−5.73960	−0.630003	−0.315002	−	0.949091i	$-0.602005\pi$
$83$	−5.73960	−0.630003	−0.315002	+	0.949091i	$0.602005\pi$
$84$	0.729919	0.0796407
$85$	17.5169	1.89997
$86$	−11.6622	−1.25757
$87$	−0.121844	−0.0130631
$88$	−26.1594	−2.78861
$89$	−4.50184	−0.477194	−0.238597	−	0.971119i	$-0.576688\pi$
$89$	−4.50184	−0.477194	−0.238597	+	0.971119i	$0.576688\pi$
$90$	17.0057	1.79256
$91$	4.44530	0.465994
$92$	23.1632	2.41493
$93$	1.55730	0.161485
$94$	−12.8958	−1.33010
$95$	−12.7868	−1.31190
$96$	−0.384207	−0.0392129
$97$	−5.85081	−0.594060	−0.297030	−	0.954868i	$-0.595996\pi$
$97$	−5.85081	−0.594060	−0.297030	+	0.954868i	$0.595996\pi$
$98$	−2.49223	−0.251753
$99$	−14.0981	−1.41692
$100$	1.17302	0.117302
$101$	5.64318	0.561517	0.280759	−	0.959778i	$-0.409414\pi$
$101$	5.64318	0.561517	0.280759	+	0.959778i	$0.409414\pi$
$102$	−3.29348	−0.326104
$103$	−4.24701	−0.418470	−0.209235	−	0.977865i	$-0.567097\pi$
$103$	−4.24701	−0.418470	−0.209235	+	0.977865i	$0.567097\pi$
$104$	−24.4973	−2.40216
$105$	0.398222	0.0388625
$106$	−17.5696	−1.70651
$107$	−1.53811	−0.148694	−0.0743471	−	0.997232i	$-0.523687\pi$
$107$	−1.53811	−0.148694	−0.0743471	+	0.997232i	$0.523687\pi$
$108$	−4.35758	−0.419309
$109$	−18.0590	−1.72974	−0.864870	−	0.501996i	$-0.832599\pi$
$109$	−18.0590	−1.72974	−0.864870	+	0.501996i	$0.832599\pi$
$110$	−27.1805	−2.59156
$111$	1.57280	0.149284
$112$	5.31183	0.501921
$113$	1.58533	0.149135	0.0745677	−	0.997216i	$-0.476242\pi$
$113$	1.58533	0.149135	0.0745677	+	0.997216i	$0.476242\pi$
$114$	2.40414	0.225168
$115$	12.6371	1.17842
$116$	−2.96036	−0.274862
$117$	−13.2024	−1.22056
$118$	29.6422	2.72879
$119$	7.62429	0.698918
$120$	−2.19453	−0.200333
$121$	11.5332	1.04847
$122$	−2.85549	−0.258524
$123$	−0.530055	−0.0477935
$124$	37.8365	3.39781
$125$	−10.8476	−0.970237
$126$	7.40181	0.659406
$127$	13.4958	1.19756	0.598780	−	0.800914i	$-0.295652\pi$
$127$	13.4958	1.19756	0.598780	+	0.800914i	$0.295652\pi$
$128$	17.1418	1.51514
$129$	0.811077	0.0714114
$130$	−25.4534	−2.23241
$131$	22.2050	1.94006	0.970031	−	0.242979i	$-0.0781247\pi$
$131$	22.2050	1.94006	0.970031	+	0.242979i	$0.0781247\pi$
$132$	3.46486	0.301578
$133$	−5.56550	−0.482590
$134$	20.8187	1.79846
$135$	−2.37737	−0.204611
$136$	−42.0161	−3.60285
$137$	15.8926	1.35779	0.678897	−	0.734233i	$-0.262458\pi$
$137$	15.8926	1.35779	0.678897	+	0.734233i	$0.262458\pi$
$138$	−2.37601	−0.202259
$139$	4.50633	0.382222	0.191111	−	0.981568i	$-0.438791\pi$
$139$	4.50633	0.382222	0.191111	+	0.981568i	$0.438791\pi$
$140$	9.67528	0.817710
$141$	0.896871	0.0755302
$142$	13.1952	1.10732
$143$	21.1015	1.76459
$144$	−15.7759	−1.31466
$145$	−1.61508	−0.134125
$146$	30.0489	2.48687
$147$	0.173328	0.0142958
$148$	38.2130	3.14109
$149$	0.463022	0.0379322	0.0189661	−	0.999820i	$-0.493963\pi$
$149$	0.463022	0.0379322	0.0189661	+	0.999820i	$0.493963\pi$
$150$	−0.120325	−0.00982450
$151$	−2.44742	−0.199168	−0.0995841	−	0.995029i	$-0.531751\pi$
$151$	−2.44742	−0.199168	−0.0995841	+	0.995029i	$0.531751\pi$
$152$	30.6705	2.48770
$153$	−22.6438	−1.83064
$154$	−11.8304	−0.953321
$155$	20.6425	1.65804
$156$	3.24471	0.259785
$157$	13.7617	1.09830	0.549151	−	0.835723i	$-0.314951\pi$
$157$	13.7617	1.09830	0.549151	+	0.835723i	$0.314951\pi$
$158$	−41.5626	−3.30655
$159$	1.22192	0.0969043
$160$	−5.09277	−0.402619
$161$	5.50036	0.433489
$162$	−21.7585	−1.70951
$163$	−9.48515	−0.742934	−0.371467	−	0.928446i	$-0.621145\pi$
$163$	−9.48515	−0.742934	−0.371467	+	0.928446i	$0.621145\pi$
$164$	−12.8783	−1.00563
$165$	1.89033	0.147162
$166$	14.3044	1.11024
$167$	−14.5005	−1.12208	−0.561041	−	0.827788i	$-0.689599\pi$
$167$	−14.5005	−1.12208	−0.561041	+	0.827788i	$0.689599\pi$
$168$	−0.955179	−0.0736937
$169$	6.76069	0.520053
$170$	−43.6561	−3.34827
$171$	16.5293	1.26403
$172$	19.7061	1.50257
$173$	−18.9478	−1.44057	−0.720286	−	0.693678i	$-0.755989\pi$
$173$	−18.9478	−1.44057	−0.720286	+	0.693678i	$0.755989\pi$
$174$	0.303664	0.0230207
$175$	0.278548	0.0210563
$176$	25.2148	1.90064
$177$	−2.06154	−0.154955
$178$	11.2196	0.840946
$179$	−8.94525	−0.668600	−0.334300	−	0.942467i	$-0.608500\pi$
$179$	−8.94525	−0.668600	−0.334300	+	0.942467i	$0.608500\pi$
$180$	−28.7352	−2.14179
$181$	−11.0723	−0.822994	−0.411497	−	0.911411i	$-0.634994\pi$
$181$	−11.0723	−0.822994	−0.411497	+	0.911411i	$0.634994\pi$
$182$	−11.0787	−0.821208
$183$	0.198591	0.0146803
$184$	−30.3116	−2.23460
$185$	20.8479	1.53277
$186$	−3.88115	−0.284580
$187$	36.1919	2.64661
$188$	21.7905	1.58924
$189$	−1.03476	−0.0752677
$190$	31.8676	2.31192
$191$	−9.74477	−0.705107	−0.352553	−	0.935792i	$-0.614686\pi$
$191$	−9.74477	−0.705107	−0.352553	+	0.935792i	$0.614686\pi$
$192$	−0.883846	−0.0637861
$193$	17.6514	1.27058	0.635288	−	0.772275i	$-0.280881\pi$
$193$	17.6514	1.27058	0.635288	+	0.772275i	$0.280881\pi$
$194$	14.5816	1.04690
$195$	1.77022	0.126768
$196$	4.21120	0.300800
$197$	−18.0755	−1.28782	−0.643912	−	0.765099i	$-0.722690\pi$
$197$	−18.0755	−1.28782	−0.643912	+	0.765099i	$0.722690\pi$
$198$	35.1358	2.49699
$199$	−3.02589	−0.214500	−0.107250	−	0.994232i	$-0.534205\pi$
$199$	−3.02589	−0.214500	−0.107250	+	0.994232i	$0.534205\pi$
$200$	−1.53503	−0.108543
$201$	−1.44788	−0.102126
$202$	−14.0641	−0.989546
$203$	−0.702971	−0.0493389
$204$	5.56511	0.389636
$205$	−7.02603	−0.490719
$206$	10.5845	0.737458
$207$	−16.3358	−1.13542
$208$	23.6127	1.63725
$209$	−26.4189	−1.82744
$210$	−0.992461	−0.0684863
$211$	4.69620	0.323300	0.161650	−	0.986848i	$-0.448318\pi$
$211$	4.69620	0.323300	0.161650	+	0.986848i	$0.448318\pi$
$212$	29.6879	2.03897
$213$	−0.917691	−0.0628791
$214$	3.83331	0.262040
$215$	10.7511	0.733216
$216$	5.70238	0.387998
$217$	8.98471	0.609922
$218$	45.0072	3.04827
$219$	−2.08982	−0.141217
$220$	45.9278	3.09645
$221$	33.8923	2.27984
$222$	−3.91978	−0.263078
$223$	22.1866	1.48572	0.742861	−	0.669445i	$-0.233468\pi$
$223$	22.1866	1.48572	0.742861	+	0.669445i	$0.233468\pi$
$224$	−2.21665	−0.148106
$225$	−0.827276	−0.0551517
$226$	−3.95101	−0.262817
$227$	5.94761	0.394756	0.197378	−	0.980327i	$-0.436757\pi$
$227$	5.94761	0.394756	0.197378	+	0.980327i	$0.436757\pi$
$228$	−4.06236	−0.269036
$229$	−15.7376	−1.03997	−0.519986	−	0.854175i	$-0.674063\pi$
$229$	−15.7376	−1.03997	−0.519986	+	0.854175i	$0.674063\pi$
$230$	−31.4946	−2.07669
$231$	0.822773	0.0541345
$232$	3.87395	0.254338
$233$	−2.23207	−0.146228	−0.0731140	−	0.997324i	$-0.523294\pi$
$233$	−2.23207	−0.146228	−0.0731140	+	0.997324i	$0.523294\pi$
$234$	32.9033	2.15095
$235$	11.8883	0.775506
$236$	−50.0874	−3.26041
$237$	2.89057	0.187763
$238$	−19.0015	−1.23168
$239$	−20.4047	−1.31987	−0.659936	−	0.751322i	$-0.729416\pi$
$239$	−20.4047	−1.31987	−0.659936	+	0.751322i	$0.729416\pi$
$240$	2.11529	0.136541
$241$	−3.11638	−0.200743	−0.100372	−	0.994950i	$-0.532003\pi$
$241$	−3.11638	−0.200743	−0.100372	+	0.994950i	$0.532003\pi$
$242$	−28.7434	−1.84770
$243$	4.61752	0.296214
$244$	4.82501	0.308890
$245$	2.29751	0.146782
$246$	1.32102	0.0842251
$247$	−24.7403	−1.57419
$248$	−49.5132	−3.14409
$249$	−0.994833	−0.0630450
$250$	27.0346	1.70982
$251$	−19.5061	−1.23121	−0.615606	−	0.788054i	$-0.711089\pi$
$251$	−19.5061	−1.23121	−0.615606	+	0.788054i	$0.711089\pi$
$252$	−12.5071	−0.787873
$253$	26.1098	1.64151
$254$	−33.6347	−2.11043
$255$	3.03616	0.190132
$256$	−32.5229	−2.03268
$257$	4.57248	0.285223	0.142612	−	0.989779i	$-0.454450\pi$
$257$	4.57248	0.285223	0.142612	+	0.989779i	$0.454450\pi$
$258$	−2.02139	−0.125846
$259$	9.07414	0.563839
$260$	43.0095	2.66734
$261$	2.08780	0.129231
$262$	−55.3400	−3.41892
$263$	−13.0532	−0.804896	−0.402448	−	0.915443i	$-0.631841\pi$
$263$	−13.0532	−0.804896	−0.402448	+	0.915443i	$0.631841\pi$
$264$	−4.53416	−0.279058
$265$	16.1969	0.994965
$266$	13.8705	0.850454
$267$	−0.780294	−0.0477532
$268$	−35.1780	−2.14884
$269$	−3.32800	−0.202912	−0.101456	−	0.994840i	$-0.532350\pi$
$269$	−3.32800	−0.202912	−0.101456	+	0.994840i	$0.532350\pi$
$270$	5.92495	0.360581
$271$	0.195976	0.0119047	0.00595233	−	0.999982i	$-0.498105\pi$
$271$	0.195976	0.0119047	0.00595233	+	0.999982i	$0.498105\pi$
$272$	40.4990	2.45561
$273$	0.770494	0.0466324
$274$	−39.6079	−2.39280
$275$	1.32224	0.0797343
$276$	4.01482	0.241664
$277$	−1.11536	−0.0670153	−0.0335076	−	0.999438i	$-0.510668\pi$
$277$	−1.11536	−0.0670153	−0.0335076	+	0.999438i	$0.510668\pi$
$278$	−11.2308	−0.673580
$279$	−26.6842	−1.59754
$280$	−12.6612	−0.756650
$281$	−10.2548	−0.611751	−0.305876	−	0.952071i	$-0.598949\pi$
$281$	−10.2548	−0.611751	−0.305876	+	0.952071i	$0.598949\pi$
$282$	−2.23521	−0.133105
$283$	−14.3410	−0.852485	−0.426243	−	0.904609i	$-0.640163\pi$
$283$	−14.3410	−0.852485	−0.426243	+	0.904609i	$0.640163\pi$
$284$	−22.2964	−1.32305
$285$	−2.21630	−0.131283
$286$	−52.5897	−3.10969
$287$	−3.05811	−0.180514
$288$	6.58335	0.387928
$289$	41.1298	2.41940
$290$	4.02516	0.236365
$291$	−1.01411	−0.0594481
$292$	−50.7747	−2.97137
$293$	−26.8965	−1.57131	−0.785655	−	0.618665i	$-0.787674\pi$
$293$	−26.8965	−1.57131	−0.785655	+	0.618665i	$0.787674\pi$
$294$	−0.431973	−0.0251932
$295$	−27.3262	−1.59099
$296$	−50.0060	−2.90654
$297$	−4.91192	−0.285018
$298$	−1.15396	−0.0668469
$299$	24.4508	1.41402
$300$	0.203317	0.0117385
$301$	4.67944	0.269719
$302$	6.09953	0.350988
$303$	0.978119	0.0561915
$304$	−29.5630	−1.69555
$305$	2.63238	0.150730
$306$	56.4336	3.22609
$307$	−6.02151	−0.343666	−0.171833	−	0.985126i	$-0.554969\pi$
$307$	−6.02151	−0.343666	−0.171833	+	0.985126i	$0.554969\pi$
$308$	19.9902	1.13905
$309$	−0.736124	−0.0418766
$310$	−51.4457	−2.92192
$311$	29.9192	1.69656	0.848281	−	0.529546i	$-0.177638\pi$
$311$	29.9192	1.69656	0.848281	+	0.529546i	$0.177638\pi$
$312$	−4.24606	−0.240386
$313$	26.0398	1.47185	0.735927	−	0.677061i	$-0.236747\pi$
$313$	26.0398	1.47185	0.735927	+	0.677061i	$0.236747\pi$
$314$	−34.2973	−1.93551
$315$	−6.82350	−0.384461
$316$	70.2298	3.95074
$317$	−32.8485	−1.84495	−0.922477	−	0.386052i	$-0.873838\pi$
$317$	−32.8485	−1.84495	−0.922477	+	0.386052i	$0.873838\pi$
$318$	−3.04530	−0.170772
$319$	−3.33695	−0.186833
$320$	−11.7156	−0.654923
$321$	−0.266596	−0.0148800
$322$	−13.7082	−0.763926
$323$	−42.4330	−2.36103
$324$	36.7660	2.04256
$325$	1.23823	0.0686846
$326$	23.6392	1.30925
$327$	−3.13013	−0.173097
$328$	16.8527	0.930535
$329$	5.17442	0.285275
$330$	−4.71113	−0.259339
$331$	−17.0855	−0.939106	−0.469553	−	0.882904i	$-0.655585\pi$
$331$	−17.0855	−0.939106	−0.469553	+	0.882904i	$0.655585\pi$
$332$	−24.1706	−1.32654
$333$	−26.9498	−1.47684
$334$	36.1385	1.97741
$335$	−19.1921	−1.04858
$336$	0.920688	0.0502277
$337$	4.87510	0.265564	0.132782	−	0.991145i	$-0.457609\pi$
$337$	4.87510	0.265564	0.132782	+	0.991145i	$0.457609\pi$
$338$	−16.8492	−0.916474
$339$	0.274782	0.0149241
$340$	73.7672	4.00059
$341$	42.6497	2.30961
$342$	−41.1948	−2.22756
$343$	1.00000	0.0539949
$344$	−25.7876	−1.39037
$345$	2.19037	0.117925
$346$	47.2221	2.53868
$347$	−18.7327	−1.00563	−0.502813	−	0.864396i	$-0.667701\pi$
$347$	−18.7327	−1.00563	−0.502813	+	0.864396i	$0.667701\pi$
$348$	−0.513112	−0.0275057
$349$	−12.6052	−0.674743	−0.337371	−	0.941372i	$-0.609538\pi$
$349$	−12.6052	−0.674743	−0.337371	+	0.941372i	$0.609538\pi$
$350$	−0.694205	−0.0371068
$351$	−4.59982	−0.245520
$352$	−10.5222	−0.560837
$353$	6.57308	0.349850	0.174925	−	0.984582i	$-0.444032\pi$
$353$	6.57308	0.349850	0.174925	+	0.984582i	$0.444032\pi$
$354$	5.13782	0.273072
$355$	−12.1642	−0.645611
$356$	−18.9582	−1.00478
$357$	1.32150	0.0699413
$358$	22.2936	1.17825
$359$	−22.5557	−1.19045	−0.595223	−	0.803561i	$-0.702936\pi$
$359$	−22.5557	−1.19045	−0.595223	+	0.803561i	$0.702936\pi$
$360$	37.6031	1.98186
$361$	11.9747	0.630250
$362$	27.5946	1.45034
$363$	1.99903	0.104922
$364$	18.7201	0.981198
$365$	−27.7012	−1.44995
$366$	−0.494935	−0.0258707
$367$	−31.6153	−1.65031	−0.825153	−	0.564909i	$-0.808911\pi$
$367$	−31.6153	−1.65031	−0.825153	+	0.564909i	$0.808911\pi$
$368$	29.2170	1.52304
$369$	9.08245	0.472813
$370$	−51.9578	−2.70116
$371$	7.04975	0.366005
$372$	6.55811	0.340022
$373$	−19.9713	−1.03408	−0.517038	−	0.855963i	$-0.672965\pi$
$373$	−19.9713	−1.03408	−0.517038	+	0.855963i	$0.672965\pi$
$374$	−90.1984	−4.66405
$375$	−1.88019	−0.0970924
$376$	−28.5153	−1.47057
$377$	−3.12492	−0.160941
$378$	2.57886	0.132642
$379$	33.6280	1.72736	0.863678	−	0.504045i	$-0.168155\pi$
$379$	33.6280	1.72736	0.863678	+	0.504045i	$0.168155\pi$
$380$	−53.8477	−2.76233
$381$	2.33920	0.119841
$382$	24.2862	1.24259
$383$	−1.36415	−0.0697049	−0.0348524	−	0.999392i	$-0.511096\pi$
$383$	−1.36415	−0.0697049	−0.0348524	+	0.999392i	$0.511096\pi$
$384$	2.97116	0.151621
$385$	10.9061	0.555826
$386$	−43.9913	−2.23910
$387$	−13.8977	−0.706462
$388$	−24.6390	−1.25085
$389$	−36.3694	−1.84400	−0.922000	−	0.387189i	$-0.873446\pi$
$389$	−36.3694	−1.84400	−0.922000	+	0.387189i	$0.873446\pi$
$390$	−4.41179	−0.223399
$391$	41.9364	2.12081
$392$	−5.51083	−0.278339
$393$	3.84875	0.194144
$394$	45.0482	2.26950
$395$	38.3153	1.92785
$396$	−59.3702	−2.98346
$397$	0.950346	0.0476965	0.0238483	−	0.999716i	$-0.492408\pi$
$397$	0.950346	0.0476965	0.0238483	+	0.999716i	$0.492408\pi$
$398$	7.54122	0.378007
$399$	−0.964655	−0.0482932
$400$	1.47960	0.0739800
$401$	−11.4641	−0.572489	−0.286244	−	0.958157i	$-0.592407\pi$
$401$	−11.4641	−0.572489	−0.286244	+	0.958157i	$0.592407\pi$
$402$	3.60845	0.179973
$403$	39.9397	1.98954
$404$	23.7646	1.18233
$405$	20.0584	0.996712
$406$	1.75197	0.0869486
$407$	43.0742	2.13511
$408$	−7.28257	−0.360541
$409$	27.2286	1.34637	0.673184	−	0.739475i	$-0.264926\pi$
$409$	27.2286	1.34637	0.673184	+	0.739475i	$0.264926\pi$
$410$	17.5105	0.864781
$411$	2.75463	0.135876
$412$	−17.8850	−0.881131
$413$	−11.8939	−0.585258
$414$	40.7127	2.00092
$415$	−13.1868	−0.647314
$416$	−9.85366	−0.483115
$417$	0.781073	0.0382493
$418$	65.8421	3.22044
$419$	21.5419	1.05239	0.526196	−	0.850363i	$-0.323618\pi$
$419$	21.5419	1.05239	0.526196	+	0.850363i	$0.323618\pi$
$420$	1.67700	0.0818290
$421$	39.6151	1.93072	0.965362	−	0.260916i	$-0.0840245\pi$
$421$	39.6151	1.93072	0.965362	+	0.260916i	$0.0840245\pi$
$422$	−11.7040	−0.569742
$423$	−15.3678	−0.747208
$424$	−38.8499	−1.88672
$425$	2.12373	0.103016
$426$	2.28710	0.110810
$427$	1.14576	0.0554470
$428$	−6.47728	−0.313091
$429$	3.65747	0.176584
$430$	−26.7941	−1.29213
$431$	−11.3523	−0.546820	−0.273410	−	0.961898i	$-0.588152\pi$
$431$	−11.3523	−0.546820	−0.273410	+	0.961898i	$0.588152\pi$
$432$	−5.49647	−0.264449
$433$	−13.8755	−0.666812	−0.333406	−	0.942783i	$-0.608198\pi$
$433$	−13.8755	−0.666812	−0.333406	+	0.942783i	$0.608198\pi$
$434$	−22.3920	−1.07485
$435$	−0.279939	−0.0134220
$436$	−76.0502	−3.64214
$437$	−30.6123	−1.46438
$438$	5.20832	0.248863
$439$	−21.6239	−1.03205	−0.516027	−	0.856572i	$-0.672590\pi$
$439$	−21.6239	−1.03205	−0.516027	+	0.856572i	$0.672590\pi$
$440$	−60.1015	−2.86523
$441$	−2.96996	−0.141427
$442$	−84.4673	−4.01770
$443$	8.92232	0.423912	0.211956	−	0.977279i	$-0.432017\pi$
$443$	8.92232	0.423912	0.211956	+	0.977279i	$0.432017\pi$
$444$	6.62338	0.314332
$445$	−10.3430	−0.490306
$446$	−55.2940	−2.61825
$447$	0.0802545	0.00379591
$448$	−5.09927	−0.240918
$449$	−13.4689	−0.635639	−0.317819	−	0.948151i	$-0.602951\pi$
$449$	−13.4689	−0.635639	−0.317819	+	0.948151i	$0.602951\pi$
$450$	2.06176	0.0971923
$451$	−14.5166	−0.683559
$452$	6.67615	0.314020
$453$	−0.424206	−0.0199309
$454$	−14.8228	−0.695668
$455$	10.2131	0.478798
$456$	5.31605	0.248947
$457$	31.7931	1.48722	0.743608	−	0.668615i	$-0.233113\pi$
$457$	31.7931	1.48722	0.743608	+	0.668615i	$0.233113\pi$
$458$	39.2217	1.83271
$459$	−7.88931	−0.368241
$460$	53.2176	2.48128
$461$	14.3153	0.666731	0.333366	−	0.942798i	$-0.391816\pi$
$461$	14.3153	0.666731	0.333366	+	0.942798i	$0.391816\pi$
$462$	−2.05054	−0.0953997
$463$	35.4821	1.64899	0.824496	−	0.565867i	$-0.191459\pi$
$463$	35.4821	1.64899	0.824496	+	0.565867i	$0.191459\pi$
$464$	−3.73407	−0.173350
$465$	3.57791	0.165922
$466$	5.56284	0.257693
$467$	3.44580	0.159452	0.0797262	−	0.996817i	$-0.474595\pi$
$467$	3.44580	0.159452	0.0797262	+	0.996817i	$0.474595\pi$
$468$	−55.5978	−2.57001
$469$	−8.35343	−0.385726
$470$	−29.6283	−1.36665
$471$	2.38528	0.109908
$472$	65.5450	3.01695
$473$	22.2129	1.02135
$474$	−7.20396	−0.330889
$475$	−1.55026	−0.0711307
$476$	32.1074	1.47164
$477$	−20.9374	−0.958660
$478$	50.8532	2.32597
$479$	27.2486	1.24502	0.622510	−	0.782612i	$-0.286113\pi$
$479$	27.2486	1.24502	0.622510	+	0.782612i	$0.286113\pi$
$480$	−0.882718	−0.0402904
$481$	40.3373	1.83922
$482$	7.76672	0.353765
$483$	0.953366	0.0433797
$484$	48.5687	2.20767
$485$	−13.4423	−0.610383
$486$	−11.5079	−0.522010
$487$	23.9708	1.08622	0.543110	−	0.839662i	$-0.317247\pi$
$487$	23.9708	1.08622	0.543110	+	0.839662i	$0.317247\pi$
$488$	−6.31406	−0.285824
$489$	−1.64404	−0.0743461
$490$	−5.72592	−0.258671
$491$	29.4978	1.33122	0.665609	−	0.746300i	$-0.268172\pi$
$491$	29.4978	1.33122	0.665609	+	0.746300i	$0.268172\pi$
$492$	−2.23217	−0.100634
$493$	−5.35966	−0.241387
$494$	61.6585	2.77415
$495$	−32.3906	−1.45585
$496$	47.7253	2.14293
$497$	−5.29454	−0.237492
$498$	2.47935	0.111102
$499$	11.3620	0.508634	0.254317	−	0.967121i	$-0.418149\pi$
$499$	11.3620	0.508634	0.254317	+	0.967121i	$0.418149\pi$
$500$	−45.6814	−2.04293
$501$	−2.51334	−0.112288
$502$	48.6136	2.16973
$503$	38.8200	1.73090	0.865449	−	0.500997i	$-0.167033\pi$
$503$	38.8200	1.73090	0.865449	+	0.500997i	$0.167033\pi$
$504$	16.3669	0.729041
$505$	12.9652	0.576946
$506$	−65.0715	−2.89278
$507$	1.17182	0.0520421
$508$	56.8336	2.52158
$509$	−40.3600	−1.78893	−0.894464	−	0.447141i	$-0.852442\pi$
$509$	−40.3600	−1.78893	−0.894464	+	0.447141i	$0.852442\pi$
$510$	−7.56681	−0.335064
$511$	−12.0571	−0.533373
$512$	46.7707	2.06699
$513$	5.75895	0.254264
$514$	−11.3957	−0.502641
$515$	−9.75753	−0.429968
$516$	3.41561	0.150364
$517$	24.5625	1.08026
$518$	−22.6148	−0.993638
$519$	−3.28417	−0.144159
$520$	−56.2827	−2.46816
$521$	−18.6212	−0.815811	−0.407906	−	0.913024i	$-0.633741\pi$
$521$	−18.6212	−0.815811	−0.407906	+	0.913024i	$0.633741\pi$
$522$	−5.20326	−0.227741
$523$	−14.9812	−0.655081	−0.327541	−	0.944837i	$-0.606220\pi$
$523$	−14.9812	−0.655081	−0.327541	+	0.944837i	$0.606220\pi$
$524$	93.5099	4.08500
$525$	0.0482801	0.00210712
$526$	32.5316	1.41845
$527$	68.5021	2.98400
$528$	4.37043	0.190199
$529$	7.25401	0.315392
$530$	−40.3663	−1.75340
$531$	35.3242	1.53294
$532$	−23.4374	−1.01614
$533$	−13.5942	−0.588830
$534$	1.94467	0.0841542
$535$	−3.53381	−0.152780
$536$	46.0343	1.98838
$537$	−1.55046	−0.0669073
$538$	8.29414	0.357586
$539$	4.74692	0.204464
$540$	−10.0116	−0.430830
$541$	26.6191	1.14444	0.572222	−	0.820099i	$-0.306082\pi$
$541$	26.6191	1.14444	0.572222	+	0.820099i	$0.306082\pi$
$542$	−0.488416	−0.0209793
$543$	−1.91913	−0.0823577
$544$	−16.9004	−0.724597
$545$	−41.4908	−1.77727
$546$	−1.92025	−0.0821790
$547$	−33.7569	−1.44334	−0.721671	−	0.692236i	$-0.756626\pi$
$547$	−33.7569	−1.44334	−0.721671	+	0.692236i	$0.756626\pi$
$548$	66.9269	2.85897
$549$	−3.40285	−0.145230
$550$	−3.29534	−0.140514
$551$	3.91238	0.166673
$552$	−5.25384	−0.223618
$553$	16.6769	0.709174
$554$	2.77972	0.118099
$555$	3.61352	0.153386
$556$	18.9771	0.804808
$557$	−42.8417	−1.81526	−0.907631	−	0.419769i	$-0.862111\pi$
$557$	−42.8417	−1.81526	−0.907631	+	0.419769i	$0.862111\pi$
$558$	66.5032	2.81530
$559$	20.8015	0.879811
$560$	12.2040	0.515712
$561$	6.27306	0.264849
$562$	25.5574	1.07807
$563$	−28.9599	−1.22051	−0.610257	−	0.792204i	$-0.708934\pi$
$563$	−28.9599	−1.22051	−0.610257	+	0.792204i	$0.708934\pi$
$564$	3.77691	0.159036
$565$	3.64231	0.153233
$566$	35.7411	1.50231
$567$	8.73052	0.366647
$568$	29.1773	1.22425
$569$	29.9084	1.25382	0.626912	−	0.779090i	$-0.284318\pi$
$569$	29.9084	1.25382	0.626912	+	0.779090i	$0.284318\pi$
$570$	5.52354	0.231356
$571$	−10.3344	−0.432479	−0.216240	−	0.976340i	$-0.569379\pi$
$571$	−10.3344	−0.432479	−0.216240	+	0.976340i	$0.569379\pi$
$572$	88.8626	3.71553
$573$	−1.68904	−0.0705606
$574$	7.62150	0.318115
$575$	1.53212	0.0638936
$576$	15.1446	0.631026
$577$	18.6240	0.775325	0.387663	−	0.921801i	$-0.373283\pi$
$577$	18.6240	0.775325	0.387663	+	0.921801i	$0.373283\pi$
$578$	−102.505	−4.26364
$579$	3.05948	0.127148
$580$	−6.80144	−0.282415
$581$	−5.73960	−0.238119
$582$	2.52739	0.104764
$583$	33.4646	1.38596
$584$	66.4443	2.74949
$585$	−30.3325	−1.25410
$586$	67.0322	2.76907
$587$	−12.2949	−0.507464	−0.253732	−	0.967275i	$-0.581658\pi$
$587$	−12.2949	−0.507464	−0.253732	+	0.967275i	$0.581658\pi$
$588$	0.729919	0.0301013
$589$	−50.0044	−2.06039
$590$	68.1032	2.80377
$591$	−3.13298	−0.128874
$592$	48.2003	1.98102
$593$	−15.9500	−0.654990	−0.327495	−	0.944853i	$-0.606204\pi$
$593$	−15.9500	−0.654990	−0.327495	+	0.944853i	$0.606204\pi$
$594$	12.2416	0.502280
$595$	17.5169	0.718122
$596$	1.94988	0.0798702
$597$	−0.524472	−0.0214652
$598$	−60.9369	−2.49190
$599$	12.3125	0.503075	0.251537	−	0.967848i	$-0.419064\pi$
$599$	12.3125	0.503075	0.251537	+	0.967848i	$0.419064\pi$
$600$	−0.266063	−0.0108620
$601$	8.93720	0.364556	0.182278	−	0.983247i	$-0.441653\pi$
$601$	8.93720	0.364556	0.182278	+	0.983247i	$0.441653\pi$
$602$	−11.6622	−0.475317
$603$	24.8093	1.01031
$604$	−10.3066	−0.419369
$605$	26.4977	1.07728
$606$	−2.43770	−0.0990247
$607$	−2.79850	−0.113587	−0.0567937	−	0.998386i	$-0.518088\pi$
$607$	−2.79850	−0.113587	−0.0567937	+	0.998386i	$0.518088\pi$
$608$	12.3367	0.500321
$609$	−0.121844	−0.00493739
$610$	−6.56051	−0.265627
$611$	23.0018	0.930555
$612$	−95.3578	−3.85461
$613$	−19.1133	−0.771979	−0.385990	−	0.922503i	$-0.626140\pi$
$613$	−19.1133	−0.771979	−0.385990	+	0.922503i	$0.626140\pi$
$614$	15.0070	0.605632
$615$	−1.21781	−0.0491067
$616$	−26.1594	−1.05399
$617$	10.8463	0.436654	0.218327	−	0.975876i	$-0.429940\pi$
$617$	10.8463	0.436654	0.218327	+	0.975876i	$0.429940\pi$
$618$	1.83459	0.0737980
$619$	37.8949	1.52312	0.761562	−	0.648092i	$-0.224433\pi$
$619$	37.8949	1.52312	0.761562	+	0.648092i	$0.224433\pi$
$620$	86.9296	3.49118
$621$	−5.69156	−0.228394
$622$	−74.5655	−2.98981
$623$	−4.50184	−0.180362
$624$	4.09273	0.163841
$625$	−26.3152	−1.05261
$626$	−64.8970	−2.59381
$627$	−4.57914	−0.182873
$628$	57.9533	2.31259
$629$	69.1839	2.75854
$630$	17.0057	0.677525
$631$	−37.5224	−1.49374	−0.746871	−	0.664969i	$-0.768445\pi$
$631$	−37.5224	−1.49374	−0.746871	+	0.664969i	$0.768445\pi$
$632$	−91.9035	−3.65572
$633$	0.813982	0.0323529
$634$	81.8659	3.25131
$635$	31.0068	1.23047
$636$	5.14574	0.204042
$637$	4.44530	0.176129
$638$	8.31643	0.329251
$639$	15.7246	0.622054
$640$	39.3835	1.55677
$641$	−45.0884	−1.78089	−0.890443	−	0.455096i	$-0.849605\pi$
$641$	−45.0884	−1.78089	−0.890443	+	0.455096i	$0.849605\pi$
$642$	0.664419	0.0262225
$643$	−45.7955	−1.80600	−0.903000	−	0.429642i	$-0.858640\pi$
$643$	−45.7955	−1.80600	−0.903000	+	0.429642i	$0.858640\pi$
$644$	23.1632	0.912756
$645$	1.86346	0.0733736
$646$	105.753	4.16078
$647$	−14.4987	−0.570004	−0.285002	−	0.958527i	$-0.591994\pi$
$647$	−14.4987	−0.570004	−0.285002	+	0.958527i	$0.591994\pi$
$648$	−48.1124	−1.89003
$649$	−56.4591	−2.21621
$650$	−3.08595	−0.121041
$651$	1.55730	0.0610354
$652$	−39.9439	−1.56432
$653$	31.3208	1.22568	0.612838	−	0.790209i	$-0.290028\pi$
$653$	31.3208	1.22568	0.612838	+	0.790209i	$0.290028\pi$
$654$	7.80100	0.305043
$655$	51.0163	1.99337
$656$	−16.2442	−0.634228
$657$	35.8089	1.39704
$658$	−12.8958	−0.502732
$659$	28.5345	1.11155	0.555774	−	0.831334i	$-0.312422\pi$
$659$	28.5345	1.11155	0.555774	+	0.831334i	$0.312422\pi$
$660$	7.96056	0.309864
$661$	26.4881	1.03027	0.515134	−	0.857109i	$-0.327742\pi$
$661$	26.4881	1.03027	0.515134	+	0.857109i	$0.327742\pi$
$662$	42.5811	1.65496
$663$	5.87447	0.228146
$664$	31.6300	1.22748
$665$	−12.7868	−0.495850
$666$	67.1651	2.60259
$667$	−3.86660	−0.149715
$668$	−61.0645	−2.36266
$669$	3.84555	0.148678
$670$	47.8311	1.84787
$671$	5.43881	0.209963
$672$	−0.384207	−0.0148211
$673$	19.5956	0.755354	0.377677	−	0.925938i	$-0.376723\pi$
$673$	19.5956	0.755354	0.377677	+	0.925938i	$0.376723\pi$
$674$	−12.1499	−0.467995
$675$	−0.288230	−0.0110940
$676$	28.4706	1.09502
$677$	13.2267	0.508343	0.254172	−	0.967159i	$-0.418197\pi$
$677$	13.2267	0.508343	0.254172	+	0.967159i	$0.418197\pi$
$678$	−0.684819	−0.0263003
$679$	−5.85081	−0.224534
$680$	−96.5325	−3.70185
$681$	1.03089	0.0395036
$682$	−106.293	−4.07016
$683$	16.4729	0.630317	0.315158	−	0.949039i	$-0.397942\pi$
$683$	16.4729	0.630317	0.315158	+	0.949039i	$0.397942\pi$
$684$	69.6082	2.66154
$685$	36.5133	1.39510
$686$	−2.49223	−0.0951537
$687$	−2.72777	−0.104071
$688$	24.8564	0.947642
$689$	31.3382	1.19389
$690$	−5.45890	−0.207817
$691$	35.4706	1.34937	0.674684	−	0.738107i	$-0.264280\pi$
$691$	35.4706	1.34937	0.674684	+	0.738107i	$0.264280\pi$
$692$	−79.7929	−3.03327
$693$	−14.0981	−0.535544
$694$	46.6862	1.77218
$695$	10.3533	0.392725
$696$	0.671464	0.0254518
$697$	−23.3159	−0.883153
$698$	31.4151	1.18908
$699$	−0.386880	−0.0146332
$700$	1.17302	0.0443361
$701$	8.17549	0.308784	0.154392	−	0.988010i	$-0.450658\pi$
$701$	8.17549	0.308784	0.154392	+	0.988010i	$0.450658\pi$
$702$	11.4638	0.432673
$703$	−50.5021	−1.90472
$704$	−24.2058	−0.912291
$705$	2.06057	0.0776055
$706$	−16.3816	−0.616530
$707$	5.64318	0.212233
$708$	−8.68155	−0.326273
$709$	5.62194	0.211136	0.105568	−	0.994412i	$-0.466334\pi$
$709$	5.62194	0.211136	0.105568	+	0.994412i	$0.466334\pi$
$710$	30.3161	1.13774
$711$	−49.5297	−1.85751
$712$	24.8089	0.929752
$713$	49.4192	1.85076
$714$	−3.29348	−0.123256
$715$	48.4808	1.81308
$716$	−37.6703	−1.40780
$717$	−3.53670	−0.132081
$718$	56.2140	2.09789
$719$	19.8931	0.741888	0.370944	−	0.928655i	$-0.379034\pi$
$719$	19.8931	0.741888	0.370944	+	0.928655i	$0.379034\pi$
$720$	−36.2453	−1.35078
$721$	−4.24701	−0.158167
$722$	−29.8438	−1.11067
$723$	−0.540155	−0.0200886
$724$	−46.6275	−1.73290
$725$	−0.195811	−0.00727225
$726$	−4.98203	−0.184901
$727$	28.2937	1.04936	0.524678	−	0.851301i	$-0.324186\pi$
$727$	28.2937	1.04936	0.524678	+	0.851301i	$0.324186\pi$
$728$	−24.4973	−0.907929
$729$	−25.3912	−0.940415
$730$	69.0377	2.55520
$731$	35.6774	1.31958
$732$	0.836309	0.0309109
$733$	−50.8733	−1.87905	−0.939524	−	0.342483i	$-0.888732\pi$
$733$	−50.8733	−1.87905	−0.939524	+	0.342483i	$0.888732\pi$
$734$	78.7926	2.90829
$735$	0.398222	0.0146886
$736$	−12.1924	−0.449417
$737$	−39.6530	−1.46064
$738$	−22.6355	−0.833226
$739$	0.586872	0.0215884	0.0107942	−	0.999942i	$-0.496564\pi$
$739$	0.586872	0.0215884	0.0107942	+	0.999942i	$0.496564\pi$
$740$	87.7948	3.22740
$741$	−4.28818	−0.157530
$742$	−17.5696	−0.645000
$743$	−46.8964	−1.72046	−0.860230	−	0.509906i	$-0.829680\pi$
$743$	−46.8964	−1.72046	−0.860230	+	0.509906i	$0.829680\pi$
$744$	−8.58201	−0.314632
$745$	1.06380	0.0389745
$746$	49.7731	1.82232
$747$	17.0464	0.623694
$748$	152.411	5.57271
$749$	−1.53811	−0.0562011
$750$	4.68586	0.171103
$751$	−3.87881	−0.141540	−0.0707700	−	0.997493i	$-0.522546\pi$
$751$	−3.87881	−0.141540	−0.0707700	+	0.997493i	$0.522546\pi$
$752$	27.4857	1.00230
$753$	−3.38094	−0.123208
$754$	7.78801	0.283623
$755$	−5.62297	−0.204641
$756$	−4.35758	−0.158484
$757$	7.16382	0.260373	0.130187	−	0.991489i	$-0.458442\pi$
$757$	7.16382	0.260373	0.130187	+	0.991489i	$0.458442\pi$
$758$	−83.8087	−3.04407
$759$	4.52555	0.164267
$760$	70.4657	2.55606
$761$	51.2063	1.85623	0.928114	−	0.372296i	$-0.121430\pi$
$761$	51.2063	1.85623	0.928114	+	0.372296i	$0.121430\pi$
$762$	−5.82982	−0.211192
$763$	−18.0590	−0.653780
$764$	−41.0372	−1.48467
$765$	−52.0244	−1.88095
$766$	3.39978	0.122839
$767$	−52.8717	−1.90909
$768$	−5.63712	−0.203412
$769$	10.7227	0.386672	0.193336	−	0.981133i	$-0.438069\pi$
$769$	10.7227	0.386672	0.193336	+	0.981133i	$0.438069\pi$
$770$	−27.1805	−0.979516
$771$	0.792538	0.0285425
$772$	74.3337	2.67533
$773$	−38.1132	−1.37084	−0.685419	−	0.728149i	$-0.740381\pi$
$773$	−38.1132	−1.37084	−0.685419	+	0.728149i	$0.740381\pi$
$774$	34.6364	1.24498
$775$	2.50267	0.0898987
$776$	32.2428	1.15745
$777$	1.57280	0.0564239
$778$	90.6408	3.24963
$779$	17.0199	0.609801
$780$	7.45475	0.266923
$781$	−25.1327	−0.899320
$782$	−104.515	−3.73745
$783$	0.727406	0.0259954
$784$	5.31183	0.189708
$785$	31.6176	1.12848
$786$	−9.59197	−0.342134
$787$	18.9969	0.677167	0.338583	−	0.940936i	$-0.390052\pi$
$787$	18.9969	0.677167	0.338583	+	0.940936i	$0.390052\pi$
$788$	−76.1195	−2.71165
$789$	−2.26249	−0.0805466
$790$	−95.4906	−3.39740
$791$	1.58533	0.0563679
$792$	77.6924	2.76068
$793$	5.09323	0.180866
$794$	−2.36848	−0.0840542
$795$	2.80737	0.0995670
$796$	−12.7427	−0.451652
$797$	−3.70336	−0.131180	−0.0655899	−	0.997847i	$-0.520893\pi$
$797$	−3.70336	−0.131180	−0.0655899	+	0.997847i	$0.520893\pi$
$798$	2.40414	0.0851057
$799$	39.4513	1.39569
$800$	−0.617443	−0.0218299
$801$	13.3703	0.472416
$802$	28.5711	1.00888
$803$	−57.2338	−2.01974
$804$	−6.09732	−0.215036
$805$	12.6371	0.445401
$806$	−99.5390	−3.50611
$807$	−0.576835	−0.0203055
$808$	−31.0986	−1.09404
$809$	−28.9623	−1.01826	−0.509130	−	0.860690i	$-0.670033\pi$
$809$	−28.9623	−1.01826	−0.509130	+	0.860690i	$0.670033\pi$
$810$	−49.9902	−1.75648
$811$	22.9619	0.806303	0.403151	−	0.915133i	$-0.367915\pi$
$811$	22.9619	0.806303	0.403151	+	0.915133i	$0.367915\pi$
$812$	−2.96036	−0.103888
$813$	0.0339680	0.00119131
$814$	−107.351	−3.76264
$815$	−21.7922	−0.763348
$816$	7.01960	0.245735
$817$	−26.0434	−0.911144
$818$	−67.8599	−2.37267
$819$	−13.2024	−0.461327
$820$	−29.5880	−1.03326
$821$	17.4961	0.610616	0.305308	−	0.952254i	$-0.401241\pi$
$821$	17.4961	0.610616	0.305308	+	0.952254i	$0.401241\pi$
$822$	−6.86516	−0.239450
$823$	1.39813	0.0487357	0.0243678	−	0.999703i	$-0.492243\pi$
$823$	1.39813	0.0487357	0.0243678	+	0.999703i	$0.492243\pi$
$824$	23.4045	0.815335
$825$	0.229182	0.00797908
$826$	29.6422	1.03138
$827$	−14.2691	−0.496186	−0.248093	−	0.968736i	$-0.579804\pi$
$827$	−14.2691	−0.496186	−0.248093	+	0.968736i	$0.579804\pi$
$828$	−68.7936	−2.39074
$829$	44.3472	1.54024	0.770121	−	0.637898i	$-0.220196\pi$
$829$	44.3472	1.54024	0.770121	+	0.637898i	$0.220196\pi$
$830$	32.8645	1.14074
$831$	−0.193322	−0.00670628
$832$	−22.6678	−0.785864
$833$	7.62429	0.264166
$834$	−1.94661	−0.0674057
$835$	−33.3150	−1.15291
$836$	−111.256	−3.84785
$837$	−9.29702	−0.321352
$838$	−53.6874	−1.85460
$839$	−10.3940	−0.358839	−0.179420	−	0.983773i	$-0.557422\pi$
$839$	−10.3940	−0.358839	−0.179420	+	0.983773i	$0.557422\pi$
$840$	−2.19453	−0.0757186
$841$	−28.5058	−0.982960
$842$	−98.7300	−3.40246
$843$	−1.77745	−0.0612185
$844$	19.7767	0.680740
$845$	15.5327	0.534342
$846$	38.3001	1.31678
$847$	11.5332	0.396286
$848$	37.4471	1.28594
$849$	−2.48570	−0.0853089
$850$	−5.29282	−0.181542
$851$	49.9111	1.71093
$852$	−3.86458	−0.132398
$853$	40.3933	1.38304	0.691520	−	0.722357i	$-0.256941\pi$
$853$	40.3933	1.38304	0.691520	+	0.722357i	$0.256941\pi$
$854$	−2.85549	−0.0977127
$855$	37.9762	1.29876
$856$	8.47623	0.289712
$857$	33.6418	1.14918	0.574591	−	0.818441i	$-0.305161\pi$
$857$	33.6418	1.14918	0.574591	+	0.818441i	$0.305161\pi$
$858$	−9.11525	−0.311190
$859$	−29.5627	−1.00867	−0.504333	−	0.863509i	$-0.668262\pi$
$859$	−29.5627	−1.00867	−0.504333	+	0.863509i	$0.668262\pi$
$860$	45.2749	1.54386
$861$	−0.530055	−0.0180642
$862$	28.2925	0.963646
$863$	−1.00000	−0.0340404
$863$	−1.00000	−0.0340404
$864$	2.29370	0.0780332
$865$	−43.5326	−1.48015
$866$	34.5808	1.17510
$867$	7.12894	0.242112
$868$	37.8365	1.28425
$869$	79.1639	2.68545
$870$	0.697672	0.0236533
$871$	−37.1335	−1.25822
$872$	99.5201	3.37018
$873$	17.3767	0.588111
$874$	76.2927	2.58064
$875$	−10.8476	−0.366715
$876$	−8.80067	−0.297347
$877$	6.11912	0.206628	0.103314	−	0.994649i	$-0.467055\pi$
$877$	6.11912	0.206628	0.103314	+	0.994649i	$0.467055\pi$
$878$	53.8918	1.81876
$879$	−4.66191	−0.157242
$880$	57.9313	1.95286
$881$	−42.3782	−1.42776	−0.713878	−	0.700270i	$-0.753063\pi$
$881$	−42.3782	−1.42776	−0.713878	+	0.700270i	$0.753063\pi$
$882$	7.40181	0.249232
$883$	14.4968	0.487855	0.243928	−	0.969793i	$-0.421564\pi$
$883$	14.4968	0.487855	0.243928	+	0.969793i	$0.421564\pi$
$884$	142.727	4.80044
$885$	−4.73640	−0.159212
$886$	−22.2365	−0.747048
$887$	20.6720	0.694097	0.347048	−	0.937847i	$-0.387184\pi$
$887$	20.6720	0.694097	0.347048	+	0.937847i	$0.387184\pi$
$888$	−8.66743	−0.290860
$889$	13.4958	0.452635
$890$	25.7772	0.864053
$891$	41.4431	1.38839
$892$	93.4322	3.12834
$893$	−28.7982	−0.963696
$894$	−0.200013	−0.00668942
$895$	−20.5518	−0.686971
$896$	17.1418	0.572669
$897$	4.23800	0.141503
$898$	33.5677	1.12017
$899$	−6.31600	−0.210650
$900$	−3.48383	−0.116128
$901$	53.7493	1.79065
$902$	36.1786	1.20462
$903$	0.811077	0.0269910
$904$	−8.73648	−0.290571
$905$	−25.4386	−0.845608
$906$	1.05722	0.0351237
$907$	12.8102	0.425356	0.212678	−	0.977122i	$-0.431781\pi$
$907$	12.8102	0.425356	0.212678	+	0.977122i	$0.431781\pi$
$908$	25.0466	0.831200
$909$	−16.7600	−0.555894
$910$	−25.4534	−0.843773
$911$	−37.6111	−1.24611	−0.623055	−	0.782178i	$-0.714109\pi$
$911$	−37.6111	−1.24611	−0.623055	+	0.782178i	$0.714109\pi$
$912$	−5.12409	−0.169675
$913$	−27.2454	−0.901692
$914$	−79.2356	−2.62088
$915$	0.456266	0.0150837
$916$	−66.2743	−2.18977
$917$	22.2050	0.733275
$918$	19.6620	0.648941
$919$	51.8458	1.71024	0.855118	−	0.518433i	$-0.173484\pi$
$919$	51.8458	1.71024	0.855118	+	0.518433i	$0.173484\pi$
$920$	−69.6411	−2.29600
$921$	−1.04370	−0.0343909
$922$	−35.6771	−1.17496
$923$	−23.5358	−0.774690
$924$	3.46486	0.113986
$925$	2.52758	0.0831064
$926$	−88.4295	−2.90597
$927$	12.6134	0.414279
$928$	1.55824	0.0511517
$929$	15.5155	0.509048	0.254524	−	0.967066i	$-0.418081\pi$
$929$	15.5155	0.509048	0.254524	+	0.967066i	$0.418081\pi$
$930$	−8.91697	−0.292399
$931$	−5.56550	−0.182402
$932$	−9.39972	−0.307898
$933$	5.18583	0.169777
$934$	−8.58771	−0.280998
$935$	83.1512	2.71933
$936$	72.7559	2.37810
$937$	−45.3080	−1.48015	−0.740073	−	0.672526i	$-0.765209\pi$
$937$	−45.3080	−1.48015	−0.740073	+	0.672526i	$0.765209\pi$
$938$	20.8187	0.679753
$939$	4.51341	0.147290
$940$	50.0640	1.63291
$941$	−20.3497	−0.663380	−0.331690	−	0.943388i	$-0.607619\pi$
$941$	−20.3497	−0.663380	−0.331690	+	0.943388i	$0.607619\pi$
$942$	−5.94467	−0.193688
$943$	−16.8207	−0.547758
$944$	−63.1782	−2.05627
$945$	−2.37737	−0.0773358
$946$	−55.3597	−1.79990
$947$	33.8198	1.09900	0.549498	−	0.835495i	$-0.314819\pi$
$947$	33.8198	1.09900	0.549498	+	0.835495i	$0.314819\pi$
$948$	12.1728	0.395354
$949$	−53.5972	−1.73984
$950$	3.86360	0.125352
$951$	−5.69355	−0.184626
$952$	−42.0161	−1.36175
$953$	54.0496	1.75084	0.875419	−	0.483365i	$-0.160586\pi$
$953$	54.0496	1.75084	0.875419	+	0.483365i	$0.160586\pi$
$954$	52.1809	1.68942
$955$	−22.3887	−0.724481
$956$	−85.9284	−2.77912
$957$	−0.578386	−0.0186966
$958$	−67.9097	−2.19406
$959$	15.8926	0.513198
$960$	−2.03064	−0.0655387
$961$	49.7250	1.60403
$962$	−100.530	−3.24121
$963$	4.56811	0.147205
$964$	−13.1237	−0.422686
$965$	40.5543	1.30549
$966$	−2.37601	−0.0764468
$967$	−24.7111	−0.794655	−0.397327	−	0.917677i	$-0.630062\pi$
$967$	−24.7111	−0.794655	−0.397327	+	0.917677i	$0.630062\pi$
$968$	−63.5576	−2.04282
$969$	−7.35481	−0.236271
$970$	33.5013	1.07566
$971$	3.66325	0.117559	0.0587796	−	0.998271i	$-0.481279\pi$
$971$	3.66325	0.117559	0.0587796	+	0.998271i	$0.481279\pi$
$972$	19.4453	0.623709
$973$	4.50633	0.144466
$974$	−59.7406	−1.91421
$975$	0.214620	0.00687333
$976$	6.08606	0.194810
$977$	−8.76216	−0.280326	−0.140163	−	0.990128i	$-0.544763\pi$
$977$	−8.76216	−0.280326	−0.140163	+	0.990128i	$0.544763\pi$
$978$	4.09732	0.131018
$979$	−21.3699	−0.682984
$980$	9.67528	0.309065
$981$	53.6345	1.71242
$982$	−73.5153	−2.34597
$983$	55.7992	1.77972	0.889860	−	0.456234i	$-0.150802\pi$
$983$	55.7992	1.77972	0.889860	+	0.456234i	$0.150802\pi$
$984$	2.92104	0.0931194
$985$	−41.5286	−1.32321
$986$	13.3575	0.425389
$987$	0.896871	0.0285477
$988$	−104.186	−3.31461
$989$	25.7386	0.818441
$990$	80.7248	2.56560
$991$	24.6850	0.784146	0.392073	−	0.919934i	$-0.371758\pi$
$991$	24.6850	0.784146	0.392073	+	0.919934i	$0.371758\pi$
$992$	−19.9159	−0.632332
$993$	−2.96140	−0.0939771
$994$	13.1952	0.418526
$995$	−6.95202	−0.220394
$996$	−4.18945	−0.132748
$997$	−27.8325	−0.881463	−0.440731	−	0.897639i	$-0.645281\pi$
$997$	−27.8325	−0.881463	−0.440731	+	0.897639i	$0.645281\pi$
$998$	−28.3168	−0.896352
$999$	−9.38955	−0.297072

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6041.2.a.f.1.13	✓	132

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6041.2.a.f.1.13	✓	132	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:	\( N \)	\(=\)	\( 6041 = 7 \cdot 863 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6041.a (trivial)

\(f(q)\)	\(=\)	\(q-2.49223 q^{2} +0.173328 q^{3} +4.21120 q^{4} +2.29751 q^{5} -0.431973 q^{6} +1.00000 q^{7} -5.51083 q^{8} -2.96996 q^{9} +O(q^{10})\)	\(q-2.49223 q^{2} +0.173328 q^{3} +4.21120 q^{4} +2.29751 q^{5} -0.431973 q^{6} +1.00000 q^{7} -5.51083 q^{8} -2.96996 q^{9} -5.72592 q^{10} +4.74692 q^{11} +0.729919 q^{12} +4.44530 q^{13} -2.49223 q^{14} +0.398222 q^{15} +5.31183 q^{16} +7.62429 q^{17} +7.40181 q^{18} -5.56550 q^{19} +9.67528 q^{20} +0.173328 q^{21} -11.8304 q^{22} +5.50036 q^{23} -0.955179 q^{24} +0.278548 q^{25} -11.0787 q^{26} -1.03476 q^{27} +4.21120 q^{28} -0.702971 q^{29} -0.992461 q^{30} +8.98471 q^{31} -2.21665 q^{32} +0.822773 q^{33} -19.0015 q^{34} +2.29751 q^{35} -12.5071 q^{36} +9.07414 q^{37} +13.8705 q^{38} +0.770494 q^{39} -12.6612 q^{40} -3.05811 q^{41} -0.431973 q^{42} +4.67944 q^{43} +19.9902 q^{44} -6.82350 q^{45} -13.7082 q^{46} +5.17442 q^{47} +0.920688 q^{48} +1.00000 q^{49} -0.694205 q^{50} +1.32150 q^{51} +18.7201 q^{52} +7.04975 q^{53} +2.57886 q^{54} +10.9061 q^{55} -5.51083 q^{56} -0.964655 q^{57} +1.75197 q^{58} -11.8939 q^{59} +1.67700 q^{60} +1.14576 q^{61} -22.3920 q^{62} -2.96996 q^{63} -5.09927 q^{64} +10.2131 q^{65} -2.05054 q^{66} -8.35343 q^{67} +32.1074 q^{68} +0.953366 q^{69} -5.72592 q^{70} -5.29454 q^{71} +16.3669 q^{72} -12.0571 q^{73} -22.6148 q^{74} +0.0482801 q^{75} -23.4374 q^{76} +4.74692 q^{77} -1.92025 q^{78} +16.6769 q^{79} +12.2040 q^{80} +8.73052 q^{81} +7.62150 q^{82} -5.73960 q^{83} +0.729919 q^{84} +17.5169 q^{85} -11.6622 q^{86} -0.121844 q^{87} -26.1594 q^{88} -4.50184 q^{89} +17.0057 q^{90} +4.44530 q^{91} +23.1632 q^{92} +1.55730 q^{93} -12.8958 q^{94} -12.7868 q^{95} -0.384207 q^{96} -5.85081 q^{97} -2.49223 q^{98} -14.0981 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q + 8 q^{2} + 10 q^{3} + 174 q^{4} + 11 q^{5} + 16 q^{6} + 132 q^{7} + 30 q^{8} + 178 q^{9}+O(q^{10}) \) 132 * q + 8 * q^2 + 10 * q^3 + 174 * q^4 + 11 * q^5 + 16 * q^6 + 132 * q^7 + 30 * q^8 + 178 * q^9	\( 132 q + 8 q^{2} + 10 q^{3} + 174 q^{4} + 11 q^{5} + 16 q^{6} + 132 q^{7} + 30 q^{8} + 178 q^{9} + 22 q^{10} + 32 q^{11} + 30 q^{12} + 16 q^{13} + 8 q^{14} + 59 q^{15} + 254 q^{16} + 11 q^{17} + 33 q^{18} + 40 q^{19} + 4 q^{20} + 10 q^{21} + 66 q^{22} + 62 q^{23} + 36 q^{24} + 235 q^{25} + 25 q^{26} + 37 q^{27} + 174 q^{28} + 28 q^{29} + 45 q^{30} + 121 q^{31} + 53 q^{32} - 13 q^{33} + 44 q^{34} + 11 q^{35} + 274 q^{36} + 61 q^{37} - 28 q^{38} + 114 q^{39} + 32 q^{40} - q^{41} + 16 q^{42} + 105 q^{43} + 54 q^{44} + 29 q^{45} + 104 q^{46} + 33 q^{47} + 16 q^{48} + 132 q^{49} - 14 q^{50} + 53 q^{51} - 11 q^{52} + 48 q^{53} + 11 q^{54} + 118 q^{55} + 30 q^{56} + 93 q^{57} + 87 q^{58} + 12 q^{59} + 41 q^{60} + 54 q^{61} - 28 q^{62} + 178 q^{63} + 376 q^{64} + 22 q^{65} + 6 q^{66} + 123 q^{67} - 47 q^{68} + 58 q^{69} + 22 q^{70} + 108 q^{71} + 97 q^{72} + q^{73} - 10 q^{74} + 23 q^{75} + 71 q^{76} + 32 q^{77} + 5 q^{78} + 204 q^{79} - 10 q^{80} + 296 q^{81} + 80 q^{82} - 10 q^{83} + 30 q^{84} + 94 q^{85} + 48 q^{86} + 4 q^{87} + 155 q^{88} + q^{89} - 66 q^{90} + 16 q^{91} + 49 q^{92} + 90 q^{93} + 79 q^{94} + 100 q^{95} + q^{96} + 18 q^{97} + 8 q^{98} + 96 q^{99}+O(q^{100}) \) 132 * q + 8 * q^2 + 10 * q^3 + 174 * q^4 + 11 * q^5 + 16 * q^6 + 132 * q^7 + 30 * q^8 + 178 * q^9 + 22 * q^10 + 32 * q^11 + 30 * q^12 + 16 * q^13 + 8 * q^14 + 59 * q^15 + 254 * q^16 + 11 * q^17 + 33 * q^18 + 40 * q^19 + 4 * q^20 + 10 * q^21 + 66 * q^22 + 62 * q^23 + 36 * q^24 + 235 * q^25 + 25 * q^26 + 37 * q^27 + 174 * q^28 + 28 * q^29 + 45 * q^30 + 121 * q^31 + 53 * q^32 - 13 * q^33 + 44 * q^34 + 11 * q^35 + 274 * q^36 + 61 * q^37 - 28 * q^38 + 114 * q^39 + 32 * q^40 - q^41 + 16 * q^42 + 105 * q^43 + 54 * q^44 + 29 * q^45 + 104 * q^46 + 33 * q^47 + 16 * q^48 + 132 * q^49 - 14 * q^50 + 53 * q^51 - 11 * q^52 + 48 * q^53 + 11 * q^54 + 118 * q^55 + 30 * q^56 + 93 * q^57 + 87 * q^58 + 12 * q^59 + 41 * q^60 + 54 * q^61 - 28 * q^62 + 178 * q^63 + 376 * q^64 + 22 * q^65 + 6 * q^66 + 123 * q^67 - 47 * q^68 + 58 * q^69 + 22 * q^70 + 108 * q^71 + 97 * q^72 + q^73 - 10 * q^74 + 23 * q^75 + 71 * q^76 + 32 * q^77 + 5 * q^78 + 204 * q^79 - 10 * q^80 + 296 * q^81 + 80 * q^82 - 10 * q^83 + 30 * q^84 + 94 * q^85 + 48 * q^86 + 4 * q^87 + 155 * q^88 + q^89 - 66 * q^90 + 16 * q^91 + 49 * q^92 + 90 * q^93 + 79 * q^94 + 100 * q^95 + q^96 + 18 * q^97 + 8 * q^98 + 96 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more