LMFDB - Embedded newform 6041.2.a.f.1.1

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			1.1
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)

$f(q)$	$=$	$q-2.81389 q^{2} +0.225620 q^{3} +5.91799 q^{4} -1.24518 q^{5} -0.634869 q^{6} +1.00000 q^{7} -11.0248 q^{8} -2.94910 q^{9} +O(q^{10})$	\(q-2.81389 q^{2} +0.225620 q^{3} +5.91799 q^{4} -1.24518 q^{5} -0.634869 q^{6} +1.00000 q^{7} -11.0248 q^{8} -2.94910 q^{9} +3.50380 q^{10} +4.12431 q^{11} +1.33521 q^{12} +3.68405 q^{13} -2.81389 q^{14} -0.280936 q^{15} +19.1866 q^{16} -6.21890 q^{17} +8.29844 q^{18} -1.83276 q^{19} -7.36895 q^{20} +0.225620 q^{21} -11.6054 q^{22} -7.83940 q^{23} -2.48741 q^{24} -3.44953 q^{25} -10.3665 q^{26} -1.34223 q^{27} +5.91799 q^{28} -9.64129 q^{29} +0.790525 q^{30} +6.68893 q^{31} -31.9394 q^{32} +0.930525 q^{33} +17.4993 q^{34} -1.24518 q^{35} -17.4527 q^{36} +4.67182 q^{37} +5.15720 q^{38} +0.831194 q^{39} +13.7278 q^{40} -5.34630 q^{41} -0.634869 q^{42} -6.30101 q^{43} +24.4076 q^{44} +3.67215 q^{45} +22.0592 q^{46} -1.19976 q^{47} +4.32887 q^{48} +1.00000 q^{49} +9.70661 q^{50} -1.40311 q^{51} +21.8022 q^{52} +8.21793 q^{53} +3.77690 q^{54} -5.13550 q^{55} -11.0248 q^{56} -0.413507 q^{57} +27.1295 q^{58} +12.7741 q^{59} -1.66258 q^{60} -13.8054 q^{61} -18.8219 q^{62} -2.94910 q^{63} +51.5009 q^{64} -4.58730 q^{65} -2.61840 q^{66} +3.40155 q^{67} -36.8034 q^{68} -1.76872 q^{69} +3.50380 q^{70} +11.9660 q^{71} +32.5132 q^{72} -0.0538456 q^{73} -13.1460 q^{74} -0.778282 q^{75} -10.8463 q^{76} +4.12431 q^{77} -2.33889 q^{78} -1.41362 q^{79} -23.8907 q^{80} +8.54445 q^{81} +15.0439 q^{82} -13.8220 q^{83} +1.33521 q^{84} +7.74363 q^{85} +17.7304 q^{86} -2.17526 q^{87} -45.4697 q^{88} +9.62877 q^{89} -10.3330 q^{90} +3.68405 q^{91} -46.3935 q^{92} +1.50915 q^{93} +3.37598 q^{94} +2.28211 q^{95} -7.20616 q^{96} +13.6002 q^{97} -2.81389 q^{98} -12.1630 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.81389	−1.98972	−0.994861	−	0.101250i	$-0.967716\pi$
$2$	−2.81389	−1.98972	−0.994861	+	0.101250i	$0.967716\pi$
$3$	0.225620	0.130262	0.0651308	−	0.997877i	$-0.479254\pi$
$3$	0.225620	0.130262	0.0651308	+	0.997877i	$0.479254\pi$
$4$	5.91799	2.95899
$5$	−1.24518	−0.556860	−0.278430	−	0.960456i	$-0.589814\pi$
$5$	−1.24518	−0.556860	−0.278430	+	0.960456i	$0.589814\pi$
$6$	−0.634869	−0.259184
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−11.0248	−3.89785
$9$	−2.94910	−0.983032
$10$	3.50380	1.10800
$11$	4.12431	1.24353	0.621763	−	0.783206i	$-0.286417\pi$
$11$	4.12431	1.24353	0.621763	+	0.783206i	$0.286417\pi$
$12$	1.33521	0.385443
$13$	3.68405	1.02177	0.510886	−	0.859649i	$-0.329317\pi$
$13$	3.68405	1.02177	0.510886	+	0.859649i	$0.329317\pi$
$14$	−2.81389	−0.752044
$15$	−0.280936	−0.0725375
$16$	19.1866	4.79665
$17$	−6.21890	−1.50830	−0.754152	−	0.656700i	$-0.771952\pi$
$17$	−6.21890	−1.50830	−0.754152	+	0.656700i	$0.771952\pi$
$18$	8.29844	1.95596
$19$	−1.83276	−0.420465	−0.210232	−	0.977651i	$-0.567422\pi$
$19$	−1.83276	−0.420465	−0.210232	+	0.977651i	$0.567422\pi$
$20$	−7.36895	−1.64775
$21$	0.225620	0.0492342
$22$	−11.6054	−2.47427
$23$	−7.83940	−1.63463	−0.817314	−	0.576193i	$-0.804538\pi$
$23$	−7.83940	−1.63463	−0.817314	+	0.576193i	$0.804538\pi$
$24$	−2.48741	−0.507740
$25$	−3.44953	−0.689907
$26$	−10.3665	−2.03304
$27$	−1.34223	−0.258313
$28$	5.91799	1.11839
$29$	−9.64129	−1.79034	−0.895171	−	0.445723i	$-0.852947\pi$
$29$	−9.64129	−1.79034	−0.895171	+	0.445723i	$0.852947\pi$
$30$	0.790525	0.144329
$31$	6.68893	1.20137	0.600684	−	0.799487i	$-0.294895\pi$
$31$	6.68893	1.20137	0.600684	+	0.799487i	$0.294895\pi$
$32$	−31.9394	−5.64615
$33$	0.930525	0.161984
$34$	17.4993	3.00111
$35$	−1.24518	−0.210473
$36$	−17.4527	−2.90879
$37$	4.67182	0.768042	0.384021	−	0.923324i	$-0.374539\pi$
$37$	4.67182	0.768042	0.384021	+	0.923324i	$0.374539\pi$
$38$	5.15720	0.836608
$39$	0.831194	0.133098
$40$	13.7278	2.17056
$41$	−5.34630	−0.834952	−0.417476	−	0.908688i	$-0.637085\pi$
$41$	−5.34630	−0.834952	−0.417476	+	0.908688i	$0.637085\pi$
$42$	−0.634869	−0.0979624
$43$	−6.30101	−0.960895	−0.480447	−	0.877024i	$-0.659526\pi$
$43$	−6.30101	−0.960895	−0.480447	+	0.877024i	$0.659526\pi$
$44$	24.4076	3.67959
$45$	3.67215	0.547412
$46$	22.0592	3.25245
$47$	−1.19976	−0.175002	−0.0875012	−	0.996164i	$-0.527888\pi$
$47$	−1.19976	−0.175002	−0.0875012	+	0.996164i	$0.527888\pi$
$48$	4.32887	0.624819
$49$	1.00000	0.142857
$50$	9.70661	1.37272
$51$	−1.40311	−0.196474
$52$	21.8022	3.02342
$53$	8.21793	1.12882	0.564410	−	0.825495i	$-0.309104\pi$
$53$	8.21793	1.12882	0.564410	+	0.825495i	$0.309104\pi$
$54$	3.77690	0.513971
$55$	−5.13550	−0.692470
$56$	−11.0248	−1.47325
$57$	−0.413507	−0.0547703
$58$	27.1295	3.56228
$59$	12.7741	1.66304	0.831522	−	0.555492i	$-0.187470\pi$
$59$	12.7741	1.66304	0.831522	+	0.555492i	$0.187470\pi$
$60$	−1.66258	−0.214638
$61$	−13.8054	−1.76760	−0.883800	−	0.467864i	$-0.845024\pi$
$61$	−13.8054	−1.76760	−0.883800	+	0.467864i	$0.845024\pi$
$62$	−18.8219	−2.39039
$63$	−2.94910	−0.371551
$64$	51.5009	6.43762
$65$	−4.58730	−0.568984
$66$	−2.61840	−0.322302
$67$	3.40155	0.415566	0.207783	−	0.978175i	$-0.433375\pi$
$67$	3.40155	0.415566	0.207783	+	0.978175i	$0.433375\pi$
$68$	−36.8034	−4.46306
$69$	−1.76872	−0.212929
$70$	3.50380	0.418784
$71$	11.9660	1.42010	0.710049	−	0.704152i	$-0.248673\pi$
$71$	11.9660	1.42010	0.710049	+	0.704152i	$0.248673\pi$
$72$	32.5132	3.83171
$73$	−0.0538456	−0.00630215	−0.00315108	−	0.999995i	$-0.501003\pi$
$73$	−0.0538456	−0.00630215	−0.00315108	+	0.999995i	$0.501003\pi$
$74$	−13.1460	−1.52819
$75$	−0.778282	−0.0898683
$76$	−10.8463	−1.24415
$77$	4.12431	0.470009
$78$	−2.33889	−0.264827
$79$	−1.41362	−0.159045	−0.0795223	−	0.996833i	$-0.525339\pi$
$79$	−1.41362	−0.159045	−0.0795223	+	0.996833i	$0.525339\pi$
$80$	−23.8907	−2.67106
$81$	8.54445	0.949384
$82$	15.0439	1.66132
$83$	−13.8220	−1.51717	−0.758583	−	0.651576i	$-0.774108\pi$
$83$	−13.8220	−1.51717	−0.758583	+	0.651576i	$0.774108\pi$
$84$	1.33521	0.145684
$85$	7.74363	0.839915
$86$	17.7304	1.91191
$87$	−2.17526	−0.233213
$88$	−45.4697	−4.84708
$89$	9.62877	1.02065	0.510324	−	0.859982i	$-0.329525\pi$
$89$	9.62877	1.02065	0.510324	+	0.859982i	$0.329525\pi$
$90$	−10.3330	−1.08920
$91$	3.68405	0.386193
$92$	−46.3935	−4.83685
$93$	1.50915	0.156492
$94$	3.37598	0.348206
$95$	2.28211	0.234140
$96$	−7.20616	−0.735476
$97$	13.6002	1.38089	0.690444	−	0.723385i	$-0.257415\pi$
$97$	13.6002	1.38089	0.690444	+	0.723385i	$0.257415\pi$
$98$	−2.81389	−0.284246
$99$	−12.1630	−1.22243
$100$	−20.4143	−2.04143
$101$	2.67785	0.266456	0.133228	−	0.991085i	$-0.457466\pi$
$101$	2.67785	0.266456	0.133228	+	0.991085i	$0.457466\pi$
$102$	3.94819	0.390929
$103$	−15.8953	−1.56621	−0.783104	−	0.621891i	$-0.786365\pi$
$103$	−15.8953	−1.56621	−0.783104	+	0.621891i	$0.786365\pi$
$104$	−40.6159	−3.98272
$105$	−0.280936	−0.0274166
$106$	−23.1244	−2.24604
$107$	10.5862	1.02340	0.511702	−	0.859163i	$-0.329015\pi$
$107$	10.5862	1.02340	0.511702	+	0.859163i	$0.329015\pi$
$108$	−7.94332	−0.764346
$109$	−4.53186	−0.434074	−0.217037	−	0.976163i	$-0.569639\pi$
$109$	−4.53186	−0.434074	−0.217037	+	0.976163i	$0.569639\pi$
$110$	14.4507	1.37782
$111$	1.05405	0.100046
$112$	19.1866	1.81296
$113$	7.19175	0.676543	0.338271	−	0.941049i	$-0.390158\pi$
$113$	7.19175	0.676543	0.338271	+	0.941049i	$0.390158\pi$
$114$	1.16356	0.108978
$115$	9.76144	0.910259
$116$	−57.0570	−5.29761
$117$	−10.8646	−1.00443
$118$	−35.9449	−3.30899
$119$	−6.21890	−0.570086
$120$	3.09727	0.282740
$121$	6.00992	0.546356
$122$	38.8469	3.51703
$123$	−1.20623	−0.108762
$124$	39.5850	3.55484
$125$	10.5212	0.941042
$126$	8.29844	0.739284
$127$	−5.07599	−0.450421	−0.225211	−	0.974310i	$-0.572307\pi$
$127$	−5.07599	−0.450421	−0.225211	+	0.974310i	$0.572307\pi$
$128$	−81.0392	−7.16292
$129$	−1.42163	−0.125168
$130$	12.9082	1.13212
$131$	7.09883	0.620227	0.310114	−	0.950699i	$-0.399633\pi$
$131$	7.09883	0.620227	0.310114	+	0.950699i	$0.399633\pi$
$132$	5.50683	0.479308
$133$	−1.83276	−0.158921
$134$	−9.57161	−0.826861
$135$	1.67132	0.143844
$136$	68.5621	5.87915
$137$	1.13495	0.0969653	0.0484826	−	0.998824i	$-0.484561\pi$
$137$	1.13495	0.0969653	0.0484826	+	0.998824i	$0.484561\pi$
$138$	4.97699	0.423670
$139$	3.72237	0.315728	0.157864	−	0.987461i	$-0.449539\pi$
$139$	3.72237	0.315728	0.157864	+	0.987461i	$0.449539\pi$
$140$	−7.36895	−0.622790
$141$	−0.270688	−0.0227961
$142$	−33.6709	−2.82560
$143$	15.1942	1.27060
$144$	−56.5831	−4.71526
$145$	12.0051	0.996971
$146$	0.151516	0.0125395
$147$	0.225620	0.0186088
$148$	27.6478	2.27263
$149$	18.1708	1.48861	0.744306	−	0.667839i	$-0.232780\pi$
$149$	18.1708	1.48861	0.744306	+	0.667839i	$0.232780\pi$
$150$	2.19000	0.178813
$151$	15.8005	1.28582	0.642912	−	0.765940i	$-0.277726\pi$
$151$	15.8005	1.28582	0.642912	+	0.765940i	$0.277726\pi$
$152$	20.2058	1.63891
$153$	18.3401	1.48271
$154$	−11.6054	−0.935186
$155$	−8.32891	−0.668994
$156$	4.91900	0.393835
$157$	−11.3490	−0.905749	−0.452875	−	0.891574i	$-0.649601\pi$
$157$	−11.3490	−0.905749	−0.452875	+	0.891574i	$0.649601\pi$
$158$	3.97777	0.316454
$159$	1.85413	0.147042
$160$	39.7703	3.14412
$161$	−7.83940	−0.617831
$162$	−24.0432	−1.88901
$163$	4.94342	0.387198	0.193599	−	0.981081i	$-0.437984\pi$
$163$	4.94342	0.387198	0.193599	+	0.981081i	$0.437984\pi$
$164$	−31.6393	−2.47062
$165$	−1.15867	−0.0902022
$166$	38.8937	3.01874
$167$	−4.51158	−0.349117	−0.174558	−	0.984647i	$-0.555850\pi$
$167$	−4.51158	−0.349117	−0.174558	+	0.984647i	$0.555850\pi$
$168$	−2.48741	−0.191908
$169$	0.572227	0.0440175
$170$	−21.7897	−1.67120
$171$	5.40499	0.413330
$172$	−37.2893	−2.84328
$173$	18.7126	1.42269	0.711344	−	0.702844i	$-0.248087\pi$
$173$	18.7126	1.42269	0.711344	+	0.702844i	$0.248087\pi$
$174$	6.12096	0.464028
$175$	−3.44953	−0.260760
$176$	79.1315	5.96476
$177$	2.88208	0.216631
$178$	−27.0943	−2.03081
$179$	−1.65198	−0.123475	−0.0617376	−	0.998092i	$-0.519664\pi$
$179$	−1.65198	−0.123475	−0.0617376	+	0.998092i	$0.519664\pi$
$180$	21.7317	1.61979
$181$	13.8706	1.03099	0.515497	−	0.856891i	$-0.327607\pi$
$181$	13.8706	1.03099	0.515497	+	0.856891i	$0.327607\pi$
$182$	−10.3665	−0.768418
$183$	−3.11477	−0.230250
$184$	86.4277	6.37154
$185$	−5.81724	−0.427692
$186$	−4.24660	−0.311376
$187$	−25.6487	−1.87562
$188$	−7.10014	−0.517831
$189$	−1.34223	−0.0976330
$190$	−6.42162	−0.465874
$191$	−25.6765	−1.85789	−0.928945	−	0.370219i	$-0.879283\pi$
$191$	−25.6765	−1.85789	−0.928945	+	0.370219i	$0.879283\pi$
$192$	11.6196	0.838574
$193$	−14.0195	−1.00914	−0.504572	−	0.863370i	$-0.668350\pi$
$193$	−14.0195	−1.00914	−0.504572	+	0.863370i	$0.668350\pi$
$194$	−38.2694	−2.74758
$195$	−1.03498	−0.0741167
$196$	5.91799	0.422713
$197$	10.7051	0.762705	0.381353	−	0.924430i	$-0.375458\pi$
$197$	10.7051	0.762705	0.381353	+	0.924430i	$0.375458\pi$
$198$	34.2253	2.43229
$199$	19.7333	1.39885	0.699427	−	0.714704i	$-0.253438\pi$
$199$	19.7333	1.39885	0.699427	+	0.714704i	$0.253438\pi$
$200$	38.0304	2.68915
$201$	0.767457	0.0541323
$202$	−7.53517	−0.530172
$203$	−9.64129	−0.676686
$204$	−8.30356	−0.581365
$205$	6.65709	0.464952
$206$	44.7276	3.11632
$207$	23.1191	1.60689
$208$	70.6844	4.90108
$209$	−7.55888	−0.522858
$210$	0.790525	0.0545514
$211$	−15.9611	−1.09881	−0.549403	−	0.835558i	$-0.685145\pi$
$211$	−15.9611	−1.09881	−0.549403	+	0.835558i	$0.685145\pi$
$212$	48.6336	3.34017
$213$	2.69976	0.184984
$214$	−29.7884	−2.03629
$215$	7.84588	0.535084
$216$	14.7978	1.00687
$217$	6.68893	0.454074
$218$	12.7522	0.863686
$219$	−0.0121486	−0.000820928 0
$220$	−30.3918	−2.04902
$221$	−22.9107	−1.54114
$222$	−2.96599	−0.199064
$223$	−10.0023	−0.669807	−0.334903	−	0.942253i	$-0.608704\pi$
$223$	−10.0023	−0.669807	−0.334903	+	0.942253i	$0.608704\pi$
$224$	−31.9394	−2.13404
$225$	10.1730	0.678200
$226$	−20.2368	−1.34613
$227$	28.5169	1.89273	0.946367	−	0.323094i	$-0.104723\pi$
$227$	28.5169	1.89273	0.946367	+	0.323094i	$0.104723\pi$
$228$	−2.44713	−0.162065
$229$	8.51405	0.562624	0.281312	−	0.959616i	$-0.409230\pi$
$229$	8.51405	0.562624	0.281312	+	0.959616i	$0.409230\pi$
$230$	−27.4676	−1.81116
$231$	0.930525	0.0612240
$232$	106.293	6.97849
$233$	6.71699	0.440045	0.220022	−	0.975495i	$-0.429387\pi$
$233$	6.71699	0.440045	0.220022	+	0.975495i	$0.429387\pi$
$234$	30.5719	1.99854
$235$	1.49391	0.0974519
$236$	75.5968	4.92093
$237$	−0.318940	−0.0207174
$238$	17.4993	1.13431
$239$	−24.6058	−1.59162	−0.795808	−	0.605548i	$-0.792954\pi$
$239$	−24.6058	−1.59162	−0.795808	+	0.605548i	$0.792954\pi$
$240$	−5.39022	−0.347937
$241$	3.12431	0.201255	0.100627	−	0.994924i	$-0.467915\pi$
$241$	3.12431	0.201255	0.100627	+	0.994924i	$0.467915\pi$
$242$	−16.9113	−1.08710
$243$	5.95449	0.381981
$244$	−81.7002	−5.23032
$245$	−1.24518	−0.0795515
$246$	3.39420	0.216406
$247$	−6.75199	−0.429619
$248$	−73.7441	−4.68276
$249$	−3.11852	−0.197628
$250$	−29.6054	−1.87241
$251$	6.44258	0.406652	0.203326	−	0.979111i	$-0.434825\pi$
$251$	6.44258	0.406652	0.203326	+	0.979111i	$0.434825\pi$
$252$	−17.4527	−1.09942
$253$	−32.3321	−2.03270
$254$	14.2833	0.896213
$255$	1.74712	0.109409
$256$	125.034	7.81460
$257$	−2.74640	−0.171316	−0.0856578	−	0.996325i	$-0.527299\pi$
$257$	−2.74640	−0.171316	−0.0856578	+	0.996325i	$0.527299\pi$
$258$	4.00032	0.249049
$259$	4.67182	0.290293
$260$	−27.1476	−1.68362
$261$	28.4331	1.75996
$262$	−19.9753	−1.23408
$263$	5.79836	0.357542	0.178771	−	0.983891i	$-0.442788\pi$
$263$	5.79836	0.357542	0.178771	+	0.983891i	$0.442788\pi$
$264$	−10.2588	−0.631388
$265$	−10.2328	−0.628595
$266$	5.15720	0.316208
$267$	2.17244	0.132951
$268$	20.1304	1.22966
$269$	−20.1785	−1.23031	−0.615153	−	0.788408i	$-0.710906\pi$
$269$	−20.1785	−1.23031	−0.615153	+	0.788408i	$0.710906\pi$
$270$	−4.70291	−0.286210
$271$	29.9023	1.81644	0.908218	−	0.418498i	$-0.137443\pi$
$271$	29.9023	1.81644	0.908218	+	0.418498i	$0.137443\pi$
$272$	−119.320	−7.23481
$273$	0.831194	0.0503061
$274$	−3.19363	−0.192934
$275$	−14.2269	−0.857917
$276$	−10.4673	−0.630056
$277$	1.72153	0.103436	0.0517182	−	0.998662i	$-0.483530\pi$
$277$	1.72153	0.103436	0.0517182	+	0.998662i	$0.483530\pi$
$278$	−10.4744	−0.628210
$279$	−19.7263	−1.18098
$280$	13.7278	0.820395
$281$	10.2983	0.614344	0.307172	−	0.951654i	$-0.400617\pi$
$281$	10.2983	0.614344	0.307172	+	0.951654i	$0.400617\pi$
$282$	0.761688	0.0453579
$283$	−12.0010	−0.713388	−0.356694	−	0.934221i	$-0.616096\pi$
$283$	−12.0010	−0.713388	−0.356694	+	0.934221i	$0.616096\pi$
$284$	70.8144	4.20206
$285$	0.514890	0.0304994
$286$	−42.7547	−2.52814
$287$	−5.34630	−0.315582
$288$	94.1925	5.55034
$289$	21.6747	1.27498
$290$	−33.7811	−1.98369
$291$	3.06847	0.179877
$292$	−0.318658	−0.0186480
$293$	29.8631	1.74462	0.872309	−	0.488954i	$-0.162622\pi$
$293$	29.8631	1.74462	0.872309	+	0.488954i	$0.162622\pi$
$294$	−0.634869	−0.0370263
$295$	−15.9060	−0.926083
$296$	−51.5058	−2.99372
$297$	−5.53578	−0.321219
$298$	−51.1307	−2.96192
$299$	−28.8807	−1.67022
$300$	−4.60586	−0.265920
$301$	−6.30101	−0.363184
$302$	−44.4608	−2.55843
$303$	0.604174	0.0347089
$304$	−35.1645	−2.01682
$305$	17.1902	0.984307
$306$	−51.6071	−2.95018
$307$	−19.1930	−1.09540	−0.547702	−	0.836674i	$-0.684497\pi$
$307$	−19.1930	−1.09540	−0.547702	+	0.836674i	$0.684497\pi$
$308$	24.4076	1.39075
$309$	−3.58629	−0.204017
$310$	23.4367	1.33111
$311$	3.76320	0.213391	0.106696	−	0.994292i	$-0.465973\pi$
$311$	3.76320	0.213391	0.106696	+	0.994292i	$0.465973\pi$
$312$	−9.16374	−0.518795
$313$	−29.9377	−1.69218	−0.846091	−	0.533039i	$-0.821050\pi$
$313$	−29.9377	−1.69218	−0.846091	+	0.533039i	$0.821050\pi$
$314$	31.9349	1.80219
$315$	3.67215	0.206902
$316$	−8.36578	−0.470612
$317$	19.0952	1.07249	0.536246	−	0.844062i	$-0.319842\pi$
$317$	19.0952	1.07249	0.536246	+	0.844062i	$0.319842\pi$
$318$	−5.21731	−0.292572
$319$	−39.7636	−2.22634
$320$	−64.1278	−3.58485
$321$	2.38845	0.133310
$322$	22.0592	1.22931
$323$	11.3978	0.634189
$324$	50.5660	2.80922
$325$	−12.7083	−0.704927
$326$	−13.9102	−0.770417
$327$	−1.02248	−0.0565431
$328$	58.9419	3.25452
$329$	−1.19976	−0.0661447
$330$	3.26037	0.179477
$331$	4.28947	0.235771	0.117885	−	0.993027i	$-0.462388\pi$
$331$	4.28947	0.235771	0.117885	+	0.993027i	$0.462388\pi$
$332$	−81.7987	−4.48929
$333$	−13.7776	−0.755010
$334$	12.6951	0.694645
$335$	−4.23554	−0.231412
$336$	4.32887	0.236159
$337$	−6.64071	−0.361743	−0.180871	−	0.983507i	$-0.557892\pi$
$337$	−6.64071	−0.361743	−0.180871	+	0.983507i	$0.557892\pi$
$338$	−1.61019	−0.0875826
$339$	1.62260	0.0881275
$340$	45.8267	2.48530
$341$	27.5872	1.49393
$342$	−15.2091	−0.822412
$343$	1.00000	0.0539949
$344$	69.4673	3.74543
$345$	2.20237	0.118572
$346$	−52.6551	−2.83076
$347$	−18.8861	−1.01386	−0.506929	−	0.861988i	$-0.669220\pi$
$347$	−18.8861	−1.01386	−0.506929	+	0.861988i	$0.669220\pi$
$348$	−12.8732	−0.690075
$349$	15.3084	0.819439	0.409719	−	0.912212i	$-0.365627\pi$
$349$	15.3084	0.819439	0.409719	+	0.912212i	$0.365627\pi$
$350$	9.70661	0.518840
$351$	−4.94485	−0.263937
$352$	−131.728	−7.02113
$353$	−23.1957	−1.23458	−0.617292	−	0.786734i	$-0.711770\pi$
$353$	−23.1957	−1.23458	−0.617292	+	0.786734i	$0.711770\pi$
$354$	−8.10987	−0.431035
$355$	−14.8998	−0.790797
$356$	56.9830	3.02009
$357$	−1.40311	−0.0742602
$358$	4.64851	0.245681
$359$	−27.1701	−1.43398	−0.716991	−	0.697082i	$-0.754481\pi$
$359$	−27.1701	−1.43398	−0.716991	+	0.697082i	$0.754481\pi$
$360$	−40.4847	−2.13373
$361$	−15.6410	−0.823210
$362$	−39.0304	−2.05139
$363$	1.35596	0.0711692
$364$	21.8022	1.14274
$365$	0.0670474	0.00350942
$366$	8.76462	0.458134
$367$	−10.8239	−0.565005	−0.282503	−	0.959267i	$-0.591165\pi$
$367$	−10.8239	−0.565005	−0.282503	+	0.959267i	$0.591165\pi$
$368$	−150.411	−7.84074
$369$	15.7668	0.820784
$370$	16.3691	0.850989
$371$	8.21793	0.426654
$372$	8.93116	0.463059
$373$	16.2909	0.843511	0.421755	−	0.906710i	$-0.361414\pi$
$373$	16.2909	0.843511	0.421755	+	0.906710i	$0.361414\pi$
$374$	72.1726	3.73195
$375$	2.37378	0.122582
$376$	13.2271	0.682134
$377$	−35.5190	−1.82932
$378$	3.77690	0.194263
$379$	5.44932	0.279913	0.139956	−	0.990158i	$-0.455304\pi$
$379$	5.44932	0.279913	0.139956	+	0.990158i	$0.455304\pi$
$380$	13.5055	0.692819
$381$	−1.14524	−0.0586726
$382$	72.2510	3.69668
$383$	7.33864	0.374987	0.187494	−	0.982266i	$-0.439964\pi$
$383$	7.33864	0.374987	0.187494	+	0.982266i	$0.439964\pi$
$384$	−18.2840	−0.933053
$385$	−5.13550	−0.261729
$386$	39.4493	2.00792
$387$	18.5823	0.944590
$388$	80.4857	4.08604
$389$	13.4449	0.681682	0.340841	−	0.940121i	$-0.389288\pi$
$389$	13.4449	0.681682	0.340841	+	0.940121i	$0.389288\pi$
$390$	2.91233	0.147472
$391$	48.7524	2.46552
$392$	−11.0248	−0.556836
$393$	1.60163	0.0807918
$394$	−30.1229	−1.51757
$395$	1.76021	0.0885656
$396$	−71.9804	−3.61715
$397$	−10.6630	−0.535160	−0.267580	−	0.963536i	$-0.586224\pi$
$397$	−10.6630	−0.535160	−0.267580	+	0.963536i	$0.586224\pi$
$398$	−55.5273	−2.78333
$399$	−0.413507	−0.0207012
$400$	−66.1848	−3.30924
$401$	6.09173	0.304206	0.152103	−	0.988365i	$-0.451395\pi$
$401$	6.09173	0.304206	0.152103	+	0.988365i	$0.451395\pi$
$402$	−2.15954	−0.107708
$403$	24.6424	1.22752
$404$	15.8475	0.788440
$405$	−10.6394	−0.528674
$406$	27.1295	1.34642
$407$	19.2680	0.955080
$408$	15.4689	0.765827
$409$	−11.0362	−0.545705	−0.272852	−	0.962056i	$-0.587967\pi$
$409$	−11.0362	−0.545705	−0.272852	+	0.962056i	$0.587967\pi$
$410$	−18.7323	−0.925124
$411$	0.256067	0.0126308
$412$	−94.0681	−4.63440
$413$	12.7741	0.628571
$414$	−65.0547	−3.19727
$415$	17.2109	0.844850
$416$	−117.667	−5.76908
$417$	0.839840	0.0411272
$418$	21.2699	1.04034
$419$	24.8513	1.21407	0.607034	−	0.794676i	$-0.292359\pi$
$419$	24.8513	1.21407	0.607034	+	0.794676i	$0.292359\pi$
$420$	−1.66258	−0.0811255
$421$	19.0847	0.930131	0.465066	−	0.885276i	$-0.346031\pi$
$421$	19.0847	0.930131	0.465066	+	0.885276i	$0.346031\pi$
$422$	44.9127	2.18632
$423$	3.53820	0.172033
$424$	−90.6010	−4.39998
$425$	21.4523	1.04059
$426$	−7.59682	−0.368067
$427$	−13.8054	−0.668090
$428$	62.6489	3.02825
$429$	3.42810	0.165510
$430$	−22.0774	−1.06467
$431$	9.14848	0.440667	0.220333	−	0.975425i	$-0.429285\pi$
$431$	9.14848	0.440667	0.220333	+	0.975425i	$0.429285\pi$
$432$	−25.7529	−1.23904
$433$	19.0731	0.916594	0.458297	−	0.888799i	$-0.348460\pi$
$433$	19.0731	0.916594	0.458297	+	0.888799i	$0.348460\pi$
$434$	−18.8219	−0.903482
$435$	2.70859	0.129867
$436$	−26.8195	−1.28442
$437$	14.3678	0.687303
$438$	0.0341849	0.00163342
$439$	16.6094	0.792723	0.396362	−	0.918094i	$-0.370273\pi$
$439$	16.6094	0.792723	0.396362	+	0.918094i	$0.370273\pi$
$440$	56.6178	2.69915
$441$	−2.94910	−0.140433
$442$	64.4683	3.06645
$443$	16.5280	0.785270	0.392635	−	0.919694i	$-0.371564\pi$
$443$	16.5280	0.785270	0.392635	+	0.919694i	$0.371564\pi$
$444$	6.23788	0.296036
$445$	−11.9895	−0.568358
$446$	28.1455	1.33273
$447$	4.09969	0.193909
$448$	51.5009	2.43319
$449$	−0.869622	−0.0410400	−0.0205200	−	0.999789i	$-0.506532\pi$
$449$	−0.869622	−0.0410400	−0.0205200	+	0.999789i	$0.506532\pi$
$450$	−28.6257	−1.34943
$451$	−22.0498	−1.03828
$452$	42.5607	2.00189
$453$	3.56489	0.167493
$454$	−80.2435	−3.76601
$455$	−4.58730	−0.215056
$456$	4.55883	0.213487
$457$	6.35781	0.297406	0.148703	−	0.988882i	$-0.452490\pi$
$457$	6.35781	0.297406	0.148703	+	0.988882i	$0.452490\pi$
$458$	−23.9576	−1.11947
$459$	8.34721	0.389614
$460$	57.7681	2.69345
$461$	3.94964	0.183953	0.0919765	−	0.995761i	$-0.470682\pi$
$461$	3.94964	0.183953	0.0919765	+	0.995761i	$0.470682\pi$
$462$	−2.61840	−0.121819
$463$	26.8026	1.24562	0.622810	−	0.782373i	$-0.285991\pi$
$463$	26.8026	1.24562	0.622810	+	0.782373i	$0.285991\pi$
$464$	−184.984	−8.58765
$465$	−1.87917	−0.0871442
$466$	−18.9009	−0.875567
$467$	34.4026	1.59196	0.795982	−	0.605320i	$-0.206955\pi$
$467$	34.4026	1.59196	0.795982	+	0.605320i	$0.206955\pi$
$468$	−64.2967	−2.97211
$469$	3.40155	0.157069
$470$	−4.20370	−0.193902
$471$	−2.56056	−0.117984
$472$	−140.832	−6.48230
$473$	−25.9873	−1.19490
$474$	0.897463	0.0412218
$475$	6.32217	0.290081
$476$	−36.8034	−1.68688
$477$	−24.2355	−1.10967
$478$	69.2381	3.16688
$479$	−16.6804	−0.762146	−0.381073	−	0.924545i	$-0.624445\pi$
$479$	−16.6804	−0.762146	−0.381073	+	0.924545i	$0.624445\pi$
$480$	8.97295	0.409557
$481$	17.2112	0.784764
$482$	−8.79147	−0.400441
$483$	−1.76872	−0.0804796
$484$	35.5666	1.61667
$485$	−16.9346	−0.768962
$486$	−16.7553	−0.760036
$487$	−26.9676	−1.22202	−0.611009	−	0.791623i	$-0.709236\pi$
$487$	−26.9676	−1.22202	−0.611009	+	0.791623i	$0.709236\pi$
$488$	152.202	6.88985
$489$	1.11533	0.0504370
$490$	3.50380	0.158285
$491$	−26.2886	−1.18639	−0.593193	−	0.805060i	$-0.702133\pi$
$491$	−26.2886	−1.18639	−0.593193	+	0.805060i	$0.702133\pi$
$492$	−7.13845	−0.321826
$493$	59.9582	2.70038
$494$	18.9994	0.854822
$495$	15.1451	0.680720
$496$	128.338	5.76254
$497$	11.9660	0.536747
$498$	8.77519	0.393226
$499$	12.7873	0.572436	0.286218	−	0.958165i	$-0.407602\pi$
$499$	12.7873	0.572436	0.286218	+	0.958165i	$0.407602\pi$
$500$	62.2641	2.78454
$501$	−1.01790	−0.0454765
$502$	−18.1287	−0.809125
$503$	−37.0549	−1.65220	−0.826098	−	0.563526i	$-0.809444\pi$
$503$	−37.0549	−1.65220	−0.826098	+	0.563526i	$0.809444\pi$
$504$	32.5132	1.44825
$505$	−3.33439	−0.148379
$506$	90.9790	4.04451
$507$	0.129106	0.00573378
$508$	−30.0397	−1.33279
$509$	−5.13313	−0.227522	−0.113761	−	0.993508i	$-0.536290\pi$
$509$	−5.13313	−0.227522	−0.113761	+	0.993508i	$0.536290\pi$
$510$	−4.91619	−0.217693
$511$	−0.0538456	−0.00238199
$512$	−189.753	−8.38596
$513$	2.45999	0.108611
$514$	7.72807	0.340870
$515$	19.7924	0.872159
$516$	−8.41319	−0.370370
$517$	−4.94816	−0.217620
$518$	−13.1460	−0.577602
$519$	4.22192	0.185322
$520$	50.5740	2.21782
$521$	28.7925	1.26142	0.630712	−	0.776017i	$-0.282763\pi$
$521$	28.7925	1.26142	0.630712	+	0.776017i	$0.282763\pi$
$522$	−80.0076	−3.50184
$523$	17.7568	0.776450	0.388225	−	0.921565i	$-0.373088\pi$
$523$	17.7568	0.776450	0.388225	+	0.921565i	$0.373088\pi$
$524$	42.0108	1.83525
$525$	−0.778282	−0.0339670
$526$	−16.3160	−0.711410
$527$	−41.5978	−1.81203
$528$	17.8536	0.776979
$529$	38.4561	1.67201
$530$	28.7940	1.25073
$531$	−37.6720	−1.63482
$532$	−10.8463	−0.470245
$533$	−19.6960	−0.853130
$534$	−6.11301	−0.264536
$535$	−13.1817	−0.569893
$536$	−37.5014	−1.61982
$537$	−0.372720	−0.0160841
$538$	56.7802	2.44797
$539$	4.12431	0.177647
$540$	9.89084	0.425634
$541$	−18.5554	−0.797757	−0.398878	−	0.917004i	$-0.630600\pi$
$541$	−18.5554	−0.797757	−0.398878	+	0.917004i	$0.630600\pi$
$542$	−84.1419	−3.61420
$543$	3.12948	0.134299
$544$	198.628	8.51611
$545$	5.64298	0.241718
$546$	−2.33889	−0.100095
$547$	29.9525	1.28067	0.640337	−	0.768094i	$-0.278795\pi$
$547$	29.9525	1.28067	0.640337	+	0.768094i	$0.278795\pi$
$548$	6.71662	0.286920
$549$	40.7135	1.73761
$550$	40.0331	1.70702
$551$	17.6702	0.752775
$552$	19.4998	0.829966
$553$	−1.41362	−0.0601132
$554$	−4.84419	−0.205810
$555$	−1.31248	−0.0557118
$556$	22.0290	0.934236
$557$	−8.27286	−0.350532	−0.175266	−	0.984521i	$-0.556079\pi$
$557$	−8.27286	−0.350532	−0.175266	+	0.984521i	$0.556079\pi$
$558$	55.5077	2.34983
$559$	−23.2132	−0.981815
$560$	−23.8907	−1.00957
$561$	−5.78684	−0.244321
$562$	−28.9783	−1.22237
$563$	25.7075	1.08344	0.541721	−	0.840559i	$-0.317773\pi$
$563$	25.7075	1.08344	0.541721	+	0.840559i	$0.317773\pi$
$564$	−1.60193	−0.0674534
$565$	−8.95500	−0.376740
$566$	33.7696	1.41944
$567$	8.54445	0.358833
$568$	−131.922	−5.53534
$569$	−28.1916	−1.18185	−0.590927	−	0.806725i	$-0.701238\pi$
$569$	−28.1916	−1.18185	−0.590927	+	0.806725i	$0.701238\pi$
$570$	−1.44884	−0.0606854
$571$	42.4672	1.77720	0.888599	−	0.458685i	$-0.151679\pi$
$571$	42.4672	1.77720	0.888599	+	0.458685i	$0.151679\pi$
$572$	89.9189	3.75970
$573$	−5.79313	−0.242011
$574$	15.0439	0.627921
$575$	27.0423	1.12774
$576$	−151.881	−6.32838
$577$	7.28929	0.303457	0.151729	−	0.988422i	$-0.451516\pi$
$577$	7.28929	0.303457	0.151729	+	0.988422i	$0.451516\pi$
$578$	−60.9903	−2.53686
$579$	−3.16307	−0.131453
$580$	71.0461	2.95003
$581$	−13.8220	−0.573435
$582$	−8.63433	−0.357905
$583$	33.8933	1.40372
$584$	0.593637	0.0245649
$585$	13.5284	0.559330
$586$	−84.0314	−3.47131
$587$	5.98008	0.246824	0.123412	−	0.992355i	$-0.460616\pi$
$587$	5.98008	0.246824	0.123412	+	0.992355i	$0.460616\pi$
$588$	1.33521	0.0550633
$589$	−12.2592	−0.505133
$590$	44.7578	1.84265
$591$	2.41528	0.0993512
$592$	89.6363	3.68403
$593$	−6.64002	−0.272673	−0.136336	−	0.990663i	$-0.543533\pi$
$593$	−6.64002	−0.272673	−0.136336	+	0.990663i	$0.543533\pi$
$594$	15.5771	0.639136
$595$	7.74363	0.317458
$596$	107.535	4.40479
$597$	4.45221	0.182217
$598$	81.2673	3.32327
$599$	−30.7495	−1.25639	−0.628195	−	0.778056i	$-0.716206\pi$
$599$	−30.7495	−1.25639	−0.628195	+	0.778056i	$0.716206\pi$
$600$	8.58040	0.350293
$601$	9.29276	0.379060	0.189530	−	0.981875i	$-0.439304\pi$
$601$	9.29276	0.379060	0.189530	+	0.981875i	$0.439304\pi$
$602$	17.7304	0.722635
$603$	−10.0315	−0.408515
$604$	93.5069	3.80474
$605$	−7.48342	−0.304244
$606$	−1.70008	−0.0690611
$607$	−0.544087	−0.0220838	−0.0110419	−	0.999939i	$-0.503515\pi$
$607$	−0.544087	−0.0220838	−0.0110419	+	0.999939i	$0.503515\pi$
$608$	58.5374	2.37401
$609$	−2.17526	−0.0881461
$610$	−48.3713	−1.95850
$611$	−4.41996	−0.178812
$612$	108.537	4.38733
$613$	11.7191	0.473331	0.236666	−	0.971591i	$-0.423945\pi$
$613$	11.7191	0.473331	0.236666	+	0.971591i	$0.423945\pi$
$614$	54.0071	2.17955
$615$	1.50197	0.0605653
$616$	−45.4697	−1.83202
$617$	−2.03619	−0.0819741	−0.0409871	−	0.999160i	$-0.513050\pi$
$617$	−2.03619	−0.0819741	−0.0409871	+	0.999160i	$0.513050\pi$
$618$	10.0914	0.405936
$619$	39.5680	1.59037	0.795186	−	0.606365i	$-0.207373\pi$
$619$	39.5680	1.59037	0.795186	+	0.606365i	$0.207373\pi$
$620$	−49.2904	−1.97955
$621$	10.5223	0.422245
$622$	−10.5892	−0.424589
$623$	9.62877	0.385769
$624$	15.9478	0.638422
$625$	4.14694	0.165878
$626$	84.2416	3.36697
$627$	−1.70543	−0.0681083
$628$	−67.1633	−2.68011
$629$	−29.0536	−1.15844
$630$	−10.3330	−0.411678
$631$	33.4087	1.32998	0.664990	−	0.746852i	$-0.268436\pi$
$631$	33.4087	1.32998	0.664990	+	0.746852i	$0.268436\pi$
$632$	15.5849	0.619932
$633$	−3.60113	−0.143132
$634$	−53.7317	−2.13396
$635$	6.32051	0.250822
$636$	10.9727	0.435096
$637$	3.68405	0.145967
$638$	111.891	4.42979
$639$	−35.2888	−1.39600
$640$	100.908	3.98874
$641$	14.5134	0.573243	0.286621	−	0.958044i	$-0.407468\pi$
$641$	14.5134	0.573243	0.286621	+	0.958044i	$0.407468\pi$
$642$	−6.72084	−0.265250
$643$	−40.1988	−1.58529	−0.792643	−	0.609687i	$-0.791295\pi$
$643$	−40.1988	−1.58529	−0.792643	+	0.609687i	$0.791295\pi$
$644$	−46.3935	−1.82816
$645$	1.77018	0.0697009
$646$	−32.0721	−1.26186
$647$	−31.5030	−1.23851	−0.619254	−	0.785190i	$-0.712565\pi$
$647$	−31.5030	−1.23851	−0.619254	+	0.785190i	$0.712565\pi$
$648$	−94.2008	−3.70056
$649$	52.6842	2.06804
$650$	35.7596	1.40261
$651$	1.50915	0.0591484
$652$	29.2551	1.14572
$653$	35.5305	1.39042	0.695208	−	0.718809i	$-0.255312\pi$
$653$	35.5305	1.39042	0.695208	+	0.718809i	$0.255312\pi$
$654$	2.87714	0.112505
$655$	−8.83930	−0.345380
$656$	−102.577	−4.00497
$657$	0.158796	0.00619522
$658$	3.37598	0.131610
$659$	23.1778	0.902881	0.451440	−	0.892301i	$-0.350910\pi$
$659$	23.1778	0.902881	0.451440	+	0.892301i	$0.350910\pi$
$660$	−6.85699	−0.266908
$661$	−10.5632	−0.410860	−0.205430	−	0.978672i	$-0.565859\pi$
$661$	−10.5632	−0.410860	−0.205430	+	0.978672i	$0.565859\pi$
$662$	−12.0701	−0.469118
$663$	−5.16911	−0.200752
$664$	152.385	5.91369
$665$	2.28211	0.0884966
$666$	38.7688	1.50226
$667$	75.5819	2.92654
$668$	−26.6995	−1.03303
$669$	−2.25672	−0.0872500
$670$	11.9183	0.460446
$671$	−56.9377	−2.19806
$672$	−7.20616	−0.277984
$673$	−11.2628	−0.434150	−0.217075	−	0.976155i	$-0.569652\pi$
$673$	−11.2628	−0.434150	−0.217075	+	0.976155i	$0.569652\pi$
$674$	18.6862	0.719767
$675$	4.63007	0.178212
$676$	3.38643	0.130247
$677$	−0.608062	−0.0233697	−0.0116849	−	0.999932i	$-0.503719\pi$
$677$	−0.608062	−0.0233697	−0.0116849	+	0.999932i	$0.503719\pi$
$678$	−4.56582	−0.175349
$679$	13.6002	0.521927
$680$	−85.3720	−3.27387
$681$	6.43397	0.246550
$682$	−77.6275	−2.97251
$683$	−22.0405	−0.843358	−0.421679	−	0.906745i	$-0.638559\pi$
$683$	−22.0405	−0.843358	−0.421679	+	0.906745i	$0.638559\pi$
$684$	31.9867	1.22304
$685$	−1.41321	−0.0539961
$686$	−2.81389	−0.107435
$687$	1.92094	0.0732883
$688$	−120.895	−4.60908
$689$	30.2753	1.15340
$690$	−6.19724	−0.235925
$691$	−21.1030	−0.802797	−0.401398	−	0.915904i	$-0.631476\pi$
$691$	−21.1030	−0.802797	−0.401398	+	0.915904i	$0.631476\pi$
$692$	110.741	4.20973
$693$	−12.1630	−0.462033
$694$	53.1434	2.01730
$695$	−4.63502	−0.175816
$696$	23.9818	0.909029
$697$	33.2481	1.25936
$698$	−43.0761	−1.63046
$699$	1.51548	0.0573209
$700$	−20.4143	−0.771588
$701$	27.6644	1.04487	0.522435	−	0.852679i	$-0.325024\pi$
$701$	27.6644	1.04487	0.522435	+	0.852679i	$0.325024\pi$
$702$	13.9143	0.525161
$703$	−8.56233	−0.322934
$704$	212.406	8.00534
$705$	0.337055	0.0126942
$706$	65.2703	2.45648
$707$	2.67785	0.100711
$708$	17.0561	0.641008
$709$	28.3531	1.06482	0.532412	−	0.846485i	$-0.321286\pi$
$709$	28.3531	1.06482	0.532412	+	0.846485i	$0.321286\pi$
$710$	41.9263	1.57347
$711$	4.16890	0.156346
$712$	−106.155	−3.97834
$713$	−52.4372	−1.96379
$714$	3.94819	0.147757
$715$	−18.9194	−0.707546
$716$	−9.77643	−0.365362
$717$	−5.55155	−0.207326
$718$	76.4537	2.85323
$719$	−21.7253	−0.810218	−0.405109	−	0.914268i	$-0.632766\pi$
$719$	−21.7253	−0.810218	−0.405109	+	0.914268i	$0.632766\pi$
$720$	70.4560	2.62574
$721$	−15.8953	−0.591971
$722$	44.0120	1.63796
$723$	0.704906	0.0262157
$724$	82.0861	3.05071
$725$	33.2579	1.23517
$726$	−3.81551	−0.141607
$727$	−5.78059	−0.214390	−0.107195	−	0.994238i	$-0.534187\pi$
$727$	−5.78059	−0.214390	−0.107195	+	0.994238i	$0.534187\pi$
$728$	−40.6159	−1.50533
$729$	−24.2899	−0.899626
$730$	−0.188664	−0.00698277
$731$	39.1853	1.44932
$732$	−18.4332	−0.681309
$733$	−8.04894	−0.297294	−0.148647	−	0.988890i	$-0.547492\pi$
$733$	−8.04894	−0.297294	−0.148647	+	0.988890i	$0.547492\pi$
$734$	30.4574	1.12420
$735$	−0.280936	−0.0103625
$736$	250.386	9.22935
$737$	14.0291	0.516767
$738$	−44.3659	−1.63313
$739$	−6.17166	−0.227028	−0.113514	−	0.993536i	$-0.536211\pi$
$739$	−6.17166	−0.227028	−0.113514	+	0.993536i	$0.536211\pi$
$740$	−34.4264	−1.26554
$741$	−1.52338	−0.0559628
$742$	−23.1244	−0.848923
$743$	−21.1424	−0.775640	−0.387820	−	0.921735i	$-0.626772\pi$
$743$	−21.1424	−0.775640	−0.387820	+	0.921735i	$0.626772\pi$
$744$	−16.6381	−0.609983
$745$	−22.6259	−0.828949
$746$	−45.8408	−1.67835
$747$	40.7625	1.49142
$748$	−151.788	−5.54994
$749$	10.5862	0.386811
$750$	−6.67956	−0.243903
$751$	9.58269	0.349677	0.174839	−	0.984597i	$-0.444060\pi$
$751$	9.58269	0.349677	0.174839	+	0.984597i	$0.444060\pi$
$752$	−23.0192	−0.839425
$753$	1.45357	0.0529711
$754$	99.9466	3.63984
$755$	−19.6744	−0.716024
$756$	−7.94332	−0.288896
$757$	−9.53376	−0.346510	−0.173255	−	0.984877i	$-0.555429\pi$
$757$	−9.53376	−0.346510	−0.173255	+	0.984877i	$0.555429\pi$
$758$	−15.3338	−0.556948
$759$	−7.29475	−0.264783
$760$	−25.1598	−0.912644
$761$	17.3227	0.627948	0.313974	−	0.949432i	$-0.398339\pi$
$761$	17.3227	0.627948	0.313974	+	0.949432i	$0.398339\pi$
$762$	3.22259	0.116742
$763$	−4.53186	−0.164064
$764$	−151.953	−5.49748
$765$	−22.8367	−0.825663
$766$	−20.6501	−0.746120
$767$	47.0603	1.69925
$768$	28.2100	1.01794
$769$	7.16435	0.258353	0.129177	−	0.991622i	$-0.458767\pi$
$769$	7.16435	0.258353	0.129177	+	0.991622i	$0.458767\pi$
$770$	14.4507	0.520768
$771$	−0.619641	−0.0223158
$772$	−82.9671	−2.98605
$773$	37.5855	1.35185	0.675927	−	0.736968i	$-0.263743\pi$
$773$	37.5855	1.35185	0.675927	+	0.736968i	$0.263743\pi$
$774$	−52.2885	−1.87947
$775$	−23.0737	−0.828832
$776$	−149.939	−5.38250
$777$	1.05405	0.0378140
$778$	−37.8324	−1.35636
$779$	9.79850	0.351068
$780$	−6.12502	−0.219311
$781$	49.3513	1.76593
$782$	−137.184	−4.90569
$783$	12.9408	0.462468
$784$	19.1866	0.685236
$785$	14.1315	0.504376
$786$	−4.50683	−0.160753
$787$	−16.1960	−0.577326	−0.288663	−	0.957431i	$-0.593211\pi$
$787$	−16.1960	−0.577326	−0.288663	+	0.957431i	$0.593211\pi$
$788$	63.3525	2.25684
$789$	1.30822	0.0465740
$790$	−4.95303	−0.176221
$791$	7.19175	0.255709
$792$	134.094	4.76484
$793$	−50.8598	−1.80608
$794$	30.0045	1.06482
$795$	−2.30872	−0.0818818
$796$	116.781	4.13920
$797$	−1.54570	−0.0547515	−0.0273758	−	0.999625i	$-0.508715\pi$
$797$	−1.54570	−0.0547515	−0.0273758	+	0.999625i	$0.508715\pi$
$798$	1.16356	0.0411897
$799$	7.46116	0.263957
$800$	110.176	3.89532
$801$	−28.3962	−1.00333
$802$	−17.1415	−0.605286
$803$	−0.222076	−0.00783689
$804$	4.54180	0.160177
$805$	9.76144	0.344046
$806$	−69.3410	−2.44243
$807$	−4.55267	−0.160262
$808$	−29.5227	−1.03860
$809$	48.5618	1.70734	0.853670	−	0.520814i	$-0.174372\pi$
$809$	48.5618	1.70734	0.853670	+	0.520814i	$0.174372\pi$
$810$	29.9380	1.05191
$811$	11.8743	0.416964	0.208482	−	0.978026i	$-0.433148\pi$
$811$	11.8743	0.416964	0.208482	+	0.978026i	$0.433148\pi$
$812$	−57.0570	−2.00231
$813$	6.74655	0.236612
$814$	−54.2181	−1.90034
$815$	−6.15543	−0.215615
$816$	−26.9208	−0.942417
$817$	11.5483	0.404022
$818$	31.0547	1.08580
$819$	−10.8646	−0.379640
$820$	39.3966	1.37579
$821$	6.75381	0.235709	0.117855	−	0.993031i	$-0.462398\pi$
$821$	6.75381	0.235709	0.117855	+	0.993031i	$0.462398\pi$
$822$	−0.720544	−0.0251319
$823$	−37.4394	−1.30506	−0.652528	−	0.757764i	$-0.726292\pi$
$823$	−37.4394	−1.30506	−0.652528	+	0.757764i	$0.726292\pi$
$824$	175.242	6.10485
$825$	−3.20988	−0.111754
$826$	−35.9449	−1.25068
$827$	31.0660	1.08027	0.540135	−	0.841578i	$-0.318373\pi$
$827$	31.0660	1.08027	0.540135	+	0.841578i	$0.318373\pi$
$828$	136.819	4.75478
$829$	29.7453	1.03310	0.516548	−	0.856258i	$-0.327217\pi$
$829$	29.7453	1.03310	0.516548	+	0.856258i	$0.327217\pi$
$830$	−48.4296	−1.68102
$831$	0.388410	0.0134738
$832$	189.732	6.57777
$833$	−6.21890	−0.215472
$834$	−2.36322	−0.0818316
$835$	5.61772	0.194409
$836$	−44.7333	−1.54714
$837$	−8.97810	−0.310329
$838$	−69.9290	−2.41566
$839$	−23.8695	−0.824067	−0.412033	−	0.911169i	$-0.635181\pi$
$839$	−23.8695	−0.824067	−0.412033	+	0.911169i	$0.635181\pi$
$840$	3.09727	0.106866
$841$	63.9544	2.20533
$842$	−53.7023	−1.85070
$843$	2.32350	0.0800254
$844$	−94.4575	−3.25136
$845$	−0.712525	−0.0245116
$846$	−9.95610	−0.342298
$847$	6.00992	0.206503
$848$	157.674	5.41456
$849$	−2.70767	−0.0929270
$850$	−60.3644	−2.07048
$851$	−36.6242	−1.25546
$852$	15.9771	0.547367
$853$	−22.5549	−0.772266	−0.386133	−	0.922443i	$-0.626189\pi$
$853$	−22.5549	−0.772266	−0.386133	+	0.922443i	$0.626189\pi$
$854$	38.8469	1.32931
$855$	−6.73017	−0.230167
$856$	−116.710	−3.98908
$857$	23.4946	0.802562	0.401281	−	0.915955i	$-0.368565\pi$
$857$	23.4946	0.802562	0.401281	+	0.915955i	$0.368565\pi$
$858$	−9.64630	−0.329319
$859$	−20.5325	−0.700560	−0.350280	−	0.936645i	$-0.613914\pi$
$859$	−20.5325	−0.700560	−0.350280	+	0.936645i	$0.613914\pi$
$860$	46.4318	1.58331
$861$	−1.20623	−0.0411082
$862$	−25.7428	−0.876804
$863$	−1.00000	−0.0340404
$863$	−1.00000	−0.0340404
$864$	42.8702	1.45847
$865$	−23.3005	−0.792239
$866$	−53.6696	−1.82377
$867$	4.89024	0.166081
$868$	39.5850	1.34360
$869$	−5.83020	−0.197776
$870$	−7.62168	−0.258399
$871$	12.5315	0.424614
$872$	49.9629	1.69196
$873$	−40.1082	−1.35746
$874$	−40.4293	−1.36754
$875$	10.5212	0.355680
$876$	−0.0718954	−0.00242912
$877$	−34.4856	−1.16450	−0.582248	−	0.813011i	$-0.697827\pi$
$877$	−34.4856	−1.16450	−0.582248	+	0.813011i	$0.697827\pi$
$878$	−46.7371	−1.57730
$879$	6.73769	0.227257
$880$	−98.5327	−3.32154
$881$	18.0702	0.608802	0.304401	−	0.952544i	$-0.401544\pi$
$881$	18.0702	0.608802	0.304401	+	0.952544i	$0.401544\pi$
$882$	8.29844	0.279423
$883$	15.1250	0.508998	0.254499	−	0.967073i	$-0.418089\pi$
$883$	15.1250	0.508998	0.254499	+	0.967073i	$0.418089\pi$
$884$	−135.585	−4.56023
$885$	−3.58870	−0.120633
$886$	−46.5081	−1.56247
$887$	−26.2847	−0.882555	−0.441278	−	0.897371i	$-0.645475\pi$
$887$	−26.2847	−0.882555	−0.441278	+	0.897371i	$0.645475\pi$
$888$	−11.6207	−0.389966
$889$	−5.07599	−0.170243
$890$	33.7373	1.13088
$891$	35.2400	1.18058
$892$	−59.1938	−1.98195
$893$	2.19887	0.0735823
$894$	−11.5361	−0.385825
$895$	2.05701	0.0687584
$896$	−81.0392	−2.70733
$897$	−6.51606	−0.217565
$898$	2.44702	0.0816582
$899$	−64.4899	−2.15086
$900$	60.2037	2.00679
$901$	−51.1065	−1.70260
$902$	62.0457	2.06590
$903$	−1.42163	−0.0473089
$904$	−79.2875	−2.63706
$905$	−17.2714	−0.574120
$906$	−10.0312	−0.333265
$907$	50.2750	1.66935	0.834677	−	0.550739i	$-0.185654\pi$
$907$	50.2750	1.66935	0.834677	+	0.550739i	$0.185654\pi$
$908$	168.763	5.60059
$909$	−7.89722	−0.261934
$910$	12.9082	0.427901
$911$	−0.484125	−0.0160398	−0.00801989	−	0.999968i	$-0.502553\pi$
$911$	−0.484125	−0.0160398	−0.00801989	+	0.999968i	$0.502553\pi$
$912$	−7.93380	−0.262714
$913$	−57.0064	−1.88664
$914$	−17.8902	−0.591755
$915$	3.87844	0.128217
$916$	50.3860	1.66480
$917$	7.09883	0.234424
$918$	−23.4881	−0.775224
$919$	−1.50492	−0.0496426	−0.0248213	−	0.999692i	$-0.507902\pi$
$919$	−1.50492	−0.0496426	−0.0248213	+	0.999692i	$0.507902\pi$
$920$	−107.618	−3.54806
$921$	−4.33032	−0.142689
$922$	−11.1139	−0.366016
$923$	44.0832	1.45102
$924$	5.50683	0.181162
$925$	−16.1156	−0.529877
$926$	−75.4195	−2.47844
$927$	46.8767	1.53963
$928$	307.937	10.1085
$929$	4.47406	0.146789	0.0733945	−	0.997303i	$-0.476617\pi$
$929$	4.47406	0.146789	0.0733945	+	0.997303i	$0.476617\pi$
$930$	5.28777	0.173393
$931$	−1.83276	−0.0600664
$932$	39.7511	1.30209
$933$	0.849051	0.0277967
$934$	−96.8053	−3.16757
$935$	31.9371	1.04446
$936$	119.780	3.91514
$937$	13.8454	0.452310	0.226155	−	0.974091i	$-0.427384\pi$
$937$	13.8454	0.452310	0.226155	+	0.974091i	$0.427384\pi$
$938$	−9.57161	−0.312524
$939$	−6.75454	−0.220426
$940$	8.84094	0.288360
$941$	−40.9664	−1.33547	−0.667734	−	0.744400i	$-0.732735\pi$
$941$	−40.9664	−1.33547	−0.667734	+	0.744400i	$0.732735\pi$
$942$	7.20513	0.234756
$943$	41.9118	1.36483
$944$	245.091	7.97704
$945$	1.67132	0.0543680
$946$	73.1255	2.37751
$947$	−48.6265	−1.58015	−0.790075	−	0.613010i	$-0.789958\pi$
$947$	−48.6265	−1.58015	−0.790075	+	0.613010i	$0.789958\pi$
$948$	−1.88748	−0.0613026
$949$	−0.198370	−0.00643936
$950$	−17.7899	−0.577181
$951$	4.30824	0.139704
$952$	68.5621	2.22211
$953$	14.9688	0.484888	0.242444	−	0.970165i	$-0.422051\pi$
$953$	14.9688	0.484888	0.242444	+	0.970165i	$0.422051\pi$
$954$	68.1960	2.20793
$955$	31.9719	1.03458
$956$	−145.617	−4.70958
$957$	−8.97146	−0.290006
$958$	46.9368	1.51646
$959$	1.13495	0.0366494
$960$	−14.4685	−0.466968
$961$	13.7418	0.443285
$962$	−48.4305	−1.56146
$963$	−31.2197	−1.00604
$964$	18.4896	0.595511
$965$	17.4567	0.561952
$966$	4.97699	0.160132
$967$	13.1427	0.422641	0.211320	−	0.977417i	$-0.432224\pi$
$967$	13.1427	0.422641	0.211320	+	0.977417i	$0.432224\pi$
$968$	−66.2581	−2.12962
$969$	2.57156	0.0826104
$970$	47.6522	1.53002
$971$	22.4397	0.720126	0.360063	−	0.932928i	$-0.382755\pi$
$971$	22.4397	0.720126	0.360063	+	0.932928i	$0.382755\pi$
$972$	35.2386	1.13028
$973$	3.72237	0.119334
$974$	75.8839	2.43148
$975$	−2.86723	−0.0918249
$976$	−264.879	−8.47856
$977$	23.8032	0.761533	0.380766	−	0.924671i	$-0.375660\pi$
$977$	23.8032	0.761533	0.380766	+	0.924671i	$0.375660\pi$
$978$	−3.13842	−0.100356
$979$	39.7120	1.26920
$980$	−7.36895	−0.235392
$981$	13.3649	0.426708
$982$	73.9732	2.36058
$983$	−5.39300	−0.172010	−0.0860050	−	0.996295i	$-0.527410\pi$
$983$	−5.39300	−0.172010	−0.0860050	+	0.996295i	$0.527410\pi$
$984$	13.2984	0.423939
$985$	−13.3297	−0.424720
$986$	−168.716	−5.37301
$987$	−0.270688	−0.00861611
$988$	−39.9582	−1.27124
$989$	49.3961	1.57070
$990$	−42.6166	−1.35444
$991$	39.3701	1.25063	0.625316	−	0.780371i	$-0.284970\pi$
$991$	39.3701	1.25063	0.625316	+	0.780371i	$0.284970\pi$
$992$	−213.641	−6.78310
$993$	0.967788	0.0307118
$994$	−33.6709	−1.06798
$995$	−24.5714	−0.778967
$996$	−18.4554	−0.584781
$997$	−59.4927	−1.88415	−0.942076	−	0.335400i	$-0.891128\pi$
$997$	−59.4927	−1.88415	−0.942076	+	0.335400i	$0.891128\pi$
$998$	−35.9819	−1.13899
$999$	−6.27067	−0.198395

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6041.2.a.f.1.1	✓	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)

\(f(q)\)	\(=\)	\(q-2.81389 q^{2} +0.225620 q^{3} +5.91799 q^{4} -1.24518 q^{5} -0.634869 q^{6} +1.00000 q^{7} -11.0248 q^{8} -2.94910 q^{9} +O(q^{10})\)	\(q-2.81389 q^{2} +0.225620 q^{3} +5.91799 q^{4} -1.24518 q^{5} -0.634869 q^{6} +1.00000 q^{7} -11.0248 q^{8} -2.94910 q^{9} +3.50380 q^{10} +4.12431 q^{11} +1.33521 q^{12} +3.68405 q^{13} -2.81389 q^{14} -0.280936 q^{15} +19.1866 q^{16} -6.21890 q^{17} +8.29844 q^{18} -1.83276 q^{19} -7.36895 q^{20} +0.225620 q^{21} -11.6054 q^{22} -7.83940 q^{23} -2.48741 q^{24} -3.44953 q^{25} -10.3665 q^{26} -1.34223 q^{27} +5.91799 q^{28} -9.64129 q^{29} +0.790525 q^{30} +6.68893 q^{31} -31.9394 q^{32} +0.930525 q^{33} +17.4993 q^{34} -1.24518 q^{35} -17.4527 q^{36} +4.67182 q^{37} +5.15720 q^{38} +0.831194 q^{39} +13.7278 q^{40} -5.34630 q^{41} -0.634869 q^{42} -6.30101 q^{43} +24.4076 q^{44} +3.67215 q^{45} +22.0592 q^{46} -1.19976 q^{47} +4.32887 q^{48} +1.00000 q^{49} +9.70661 q^{50} -1.40311 q^{51} +21.8022 q^{52} +8.21793 q^{53} +3.77690 q^{54} -5.13550 q^{55} -11.0248 q^{56} -0.413507 q^{57} +27.1295 q^{58} +12.7741 q^{59} -1.66258 q^{60} -13.8054 q^{61} -18.8219 q^{62} -2.94910 q^{63} +51.5009 q^{64} -4.58730 q^{65} -2.61840 q^{66} +3.40155 q^{67} -36.8034 q^{68} -1.76872 q^{69} +3.50380 q^{70} +11.9660 q^{71} +32.5132 q^{72} -0.0538456 q^{73} -13.1460 q^{74} -0.778282 q^{75} -10.8463 q^{76} +4.12431 q^{77} -2.33889 q^{78} -1.41362 q^{79} -23.8907 q^{80} +8.54445 q^{81} +15.0439 q^{82} -13.8220 q^{83} +1.33521 q^{84} +7.74363 q^{85} +17.7304 q^{86} -2.17526 q^{87} -45.4697 q^{88} +9.62877 q^{89} -10.3330 q^{90} +3.68405 q^{91} -46.3935 q^{92} +1.50915 q^{93} +3.37598 q^{94} +2.28211 q^{95} -7.20616 q^{96} +13.6002 q^{97} -2.81389 q^{98} -12.1630 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q + 8 q^{2} + 10 q^{3} + 174 q^{4} + 11 q^{5} + 16 q^{6} + 132 q^{7} + 30 q^{8} + 178 q^{9}+O(q^{10}) \) 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