LMFDB - Embedded newform 8003.2.a.c.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8003,2,Mod(1,8003)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8003, 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("8003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8003 = 53 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8003.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:	$63.9042767376$
Analytic rank:	$0$
Dimension:	$172$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	8003.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.30137 q^{2} +2.98215 q^{3} +3.29631 q^{4} +1.02491 q^{5} -6.86304 q^{6} +3.76670 q^{7} -2.98329 q^{8} +5.89324 q^{9} +O(q^{10})$	\(q-2.30137 q^{2} +2.98215 q^{3} +3.29631 q^{4} +1.02491 q^{5} -6.86304 q^{6} +3.76670 q^{7} -2.98329 q^{8} +5.89324 q^{9} -2.35871 q^{10} +0.863524 q^{11} +9.83010 q^{12} +4.18568 q^{13} -8.66858 q^{14} +3.05645 q^{15} +0.273033 q^{16} -6.12712 q^{17} -13.5625 q^{18} +6.59089 q^{19} +3.37843 q^{20} +11.2329 q^{21} -1.98729 q^{22} +0.216673 q^{23} -8.89662 q^{24} -3.94955 q^{25} -9.63280 q^{26} +8.62809 q^{27} +12.4162 q^{28} +9.49709 q^{29} -7.03402 q^{30} +7.15623 q^{31} +5.33822 q^{32} +2.57516 q^{33} +14.1008 q^{34} +3.86054 q^{35} +19.4259 q^{36} +4.64001 q^{37} -15.1681 q^{38} +12.4823 q^{39} -3.05761 q^{40} -7.01617 q^{41} -25.8510 q^{42} +7.36862 q^{43} +2.84644 q^{44} +6.04006 q^{45} -0.498644 q^{46} -8.77056 q^{47} +0.814225 q^{48} +7.18804 q^{49} +9.08939 q^{50} -18.2720 q^{51} +13.7973 q^{52} -1.00000 q^{53} -19.8564 q^{54} +0.885038 q^{55} -11.2372 q^{56} +19.6550 q^{57} -21.8563 q^{58} -4.96831 q^{59} +10.0750 q^{60} +3.19614 q^{61} -16.4691 q^{62} +22.1981 q^{63} -12.8313 q^{64} +4.28996 q^{65} -5.92640 q^{66} -7.79211 q^{67} -20.1969 q^{68} +0.646151 q^{69} -8.88454 q^{70} -14.2155 q^{71} -17.5812 q^{72} -8.12270 q^{73} -10.6784 q^{74} -11.7782 q^{75} +21.7256 q^{76} +3.25264 q^{77} -28.7265 q^{78} +1.39950 q^{79} +0.279835 q^{80} +8.05056 q^{81} +16.1468 q^{82} +8.88774 q^{83} +37.0270 q^{84} -6.27977 q^{85} -16.9579 q^{86} +28.3218 q^{87} -2.57614 q^{88} +0.571882 q^{89} -13.9004 q^{90} +15.7662 q^{91} +0.714220 q^{92} +21.3410 q^{93} +20.1843 q^{94} +6.75509 q^{95} +15.9194 q^{96} +16.3787 q^{97} -16.5423 q^{98} +5.08896 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 172 q + 8 q^{2} + 25 q^{3} + 188 q^{4} + 27 q^{5} + 10 q^{6} + 31 q^{7} + 21 q^{8} + 179 q^{9}+O(q^{10}) $ 172 * q + 8 * q^2 + 25 * q^3 + 188 * q^4 + 27 * q^5 + 10 * q^6 + 31 * q^7 + 21 * q^8 + 179 * q^9	\( 172 q + 8 q^{2} + 25 q^{3} + 188 q^{4} + 27 q^{5} + 10 q^{6} + 31 q^{7} + 21 q^{8} + 179 q^{9} + 20 q^{10} - 3 q^{11} + 66 q^{12} + 121 q^{13} + 12 q^{14} + 30 q^{15} + 212 q^{16} + 8 q^{17} + 40 q^{18} + 41 q^{19} + 64 q^{20} + 56 q^{21} + 50 q^{22} + 28 q^{23} + 30 q^{24} + 231 q^{25} + 38 q^{26} + 100 q^{27} + 80 q^{28} + 26 q^{29} + 55 q^{30} + 66 q^{31} + 65 q^{32} + 99 q^{33} + 81 q^{34} + 36 q^{35} + 212 q^{36} + 153 q^{37} + q^{38} + 20 q^{39} + 59 q^{40} + 40 q^{41} + 50 q^{42} + 39 q^{43} - 51 q^{44} + 123 q^{45} + 59 q^{46} + 29 q^{47} + 128 q^{48} + 245 q^{49} + 19 q^{50} + 36 q^{51} + 215 q^{52} - 172 q^{53} + 40 q^{54} + 40 q^{55} + 15 q^{56} + 54 q^{57} + 44 q^{58} - 54 q^{60} + 100 q^{61} - 29 q^{62} + 92 q^{63} + 253 q^{64} + 77 q^{65} + 14 q^{66} + 126 q^{67} - 27 q^{68} + 47 q^{69} + 72 q^{70} + 38 q^{71} + 65 q^{72} + 185 q^{73} + 48 q^{74} + 75 q^{75} + 38 q^{76} + 120 q^{77} + 75 q^{78} + 79 q^{79} + 43 q^{80} + 232 q^{81} + 110 q^{82} + 90 q^{83} + 158 q^{84} + 115 q^{85} + 68 q^{86} + 61 q^{87} + 15 q^{88} - 36 q^{89} - 6 q^{90} + 33 q^{91} + 139 q^{92} + 103 q^{93} - 24 q^{94} - 45 q^{95} + 34 q^{96} + 159 q^{97} - 36 q^{98} + 27 q^{99}+O(q^{100}) \) 172 * q + 8 * q^2 + 25 * q^3 + 188 * q^4 + 27 * q^5 + 10 * q^6 + 31 * q^7 + 21 * q^8 + 179 * q^9 + 20 * q^10 - 3 * q^11 + 66 * q^12 + 121 * q^13 + 12 * q^14 + 30 * q^15 + 212 * q^16 + 8 * q^17 + 40 * q^18 + 41 * q^19 + 64 * q^20 + 56 * q^21 + 50 * q^22 + 28 * q^23 + 30 * q^24 + 231 * q^25 + 38 * q^26 + 100 * q^27 + 80 * q^28 + 26 * q^29 + 55 * q^30 + 66 * q^31 + 65 * q^32 + 99 * q^33 + 81 * q^34 + 36 * q^35 + 212 * q^36 + 153 * q^37 + q^38 + 20 * q^39 + 59 * q^40 + 40 * q^41 + 50 * q^42 + 39 * q^43 - 51 * q^44 + 123 * q^45 + 59 * q^46 + 29 * q^47 + 128 * q^48 + 245 * q^49 + 19 * q^50 + 36 * q^51 + 215 * q^52 - 172 * q^53 + 40 * q^54 + 40 * q^55 + 15 * q^56 + 54 * q^57 + 44 * q^58 - 54 * q^60 + 100 * q^61 - 29 * q^62 + 92 * q^63 + 253 * q^64 + 77 * q^65 + 14 * q^66 + 126 * q^67 - 27 * q^68 + 47 * q^69 + 72 * q^70 + 38 * q^71 + 65 * q^72 + 185 * q^73 + 48 * q^74 + 75 * q^75 + 38 * q^76 + 120 * q^77 + 75 * q^78 + 79 * q^79 + 43 * q^80 + 232 * q^81 + 110 * q^82 + 90 * q^83 + 158 * q^84 + 115 * q^85 + 68 * q^86 + 61 * q^87 + 15 * q^88 - 36 * q^89 - 6 * q^90 + 33 * q^91 + 139 * q^92 + 103 * q^93 - 24 * q^94 - 45 * q^95 + 34 * q^96 + 159 * q^97 - 36 * q^98 + 27 * 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 $

$2$	−2.30137	−1.62732	−0.813658	−	0.581345i	$-0.802527\pi$
$2$	−2.30137	−1.62732	−0.813658	+	0.581345i	$0.802527\pi$
$3$	2.98215	1.72175	0.860874	−	0.508819i	$-0.169918\pi$
$3$	2.98215	1.72175	0.860874	+	0.508819i	$0.169918\pi$
$4$	3.29631	1.64815
$5$	1.02491	0.458355	0.229178	−	0.973385i	$-0.426396\pi$
$5$	1.02491	0.458355	0.229178	+	0.973385i	$0.426396\pi$
$6$	−6.86304	−2.80183
$7$	3.76670	1.42368	0.711840	−	0.702342i	$-0.247862\pi$
$7$	3.76670	1.42368	0.711840	+	0.702342i	$0.247862\pi$
$8$	−2.98329	−1.05475
$9$	5.89324	1.96441
$10$	−2.35871	−0.745888
$11$	0.863524	0.260362	0.130181	−	0.991490i	$-0.458444\pi$
$11$	0.863524	0.260362	0.130181	+	0.991490i	$0.458444\pi$
$12$	9.83010	2.83770
$13$	4.18568	1.16090	0.580449	−	0.814296i	$-0.302877\pi$
$13$	4.18568	1.16090	0.580449	+	0.814296i	$0.302877\pi$
$14$	−8.66858	−2.31677
$15$	3.05645	0.789172
$16$	0.273033	0.0682582
$17$	−6.12712	−1.48605	−0.743023	−	0.669266i	$-0.766609\pi$
$17$	−6.12712	−1.48605	−0.743023	+	0.669266i	$0.766609\pi$
$18$	−13.5625	−3.19672
$19$	6.59089	1.51205	0.756026	−	0.654541i	$-0.227138\pi$
$19$	6.59089	1.51205	0.756026	+	0.654541i	$0.227138\pi$
$20$	3.37843	0.755440
$21$	11.2329	2.45122
$22$	−1.98729	−0.423692
$23$	0.216673	0.0451794	0.0225897	−	0.999745i	$-0.492809\pi$
$23$	0.216673	0.0451794	0.0225897	+	0.999745i	$0.492809\pi$
$24$	−8.89662	−1.81601
$25$	−3.94955	−0.789910
$26$	−9.63280	−1.88915
$27$	8.62809	1.66048
$28$	12.4162	2.34644
$29$	9.49709	1.76356	0.881782	−	0.471657i	$-0.156344\pi$
$29$	9.49709	1.76356	0.881782	+	0.471657i	$0.156344\pi$
$30$	−7.03402	−1.28423
$31$	7.15623	1.28530	0.642648	−	0.766161i	$-0.277836\pi$
$31$	7.15623	1.28530	0.642648	+	0.766161i	$0.277836\pi$
$32$	5.33822	0.943674
$33$	2.57516	0.448278
$34$	14.1008	2.41826
$35$	3.86054	0.652551
$36$	19.4259	3.23766
$37$	4.64001	0.762813	0.381407	−	0.924407i	$-0.375440\pi$
$37$	4.64001	0.762813	0.381407	+	0.924407i	$0.375440\pi$
$38$	−15.1681	−2.46059
$39$	12.4823	1.99877
$40$	−3.05761	−0.483451
$41$	−7.01617	−1.09574	−0.547871	−	0.836563i	$-0.684561\pi$
$41$	−7.01617	−1.09574	−0.547871	+	0.836563i	$0.684561\pi$
$42$	−25.8510	−3.98890
$43$	7.36862	1.12370	0.561852	−	0.827238i	$-0.310089\pi$
$43$	7.36862	1.12370	0.561852	+	0.827238i	$0.310089\pi$
$44$	2.84644	0.429117
$45$	6.04006	0.900399
$46$	−0.498644	−0.0735211
$47$	−8.77056	−1.27932	−0.639659	−	0.768659i	$-0.720924\pi$
$47$	−8.77056	−1.27932	−0.639659	+	0.768659i	$0.720924\pi$
$48$	0.814225	0.117523
$49$	7.18804	1.02686
$50$	9.08939	1.28543
$51$	−18.2720	−2.55859
$52$	13.7973	1.91334
$53$	−1.00000	−0.137361
$53$	−1.00000	−0.137361
$54$	−19.8564	−2.70212
$55$	0.885038	0.119338
$56$	−11.2372	−1.50163
$57$	19.6550	2.60337
$58$	−21.8563	−2.86988
$59$	−4.96831	−0.646819	−0.323409	−	0.946259i	$-0.604829\pi$
$59$	−4.96831	−0.646819	−0.323409	+	0.946259i	$0.604829\pi$
$60$	10.0750	1.30068
$61$	3.19614	0.409224	0.204612	−	0.978843i	$-0.434407\pi$
$61$	3.19614	0.409224	0.204612	+	0.978843i	$0.434407\pi$
$62$	−16.4691	−2.09158
$63$	22.1981	2.79669
$64$	−12.8313	−1.60391
$65$	4.28996	0.532104
$66$	−5.92640	−0.729490
$67$	−7.79211	−0.951958	−0.475979	−	0.879457i	$-0.657906\pi$
$67$	−7.79211	−0.951958	−0.475979	+	0.879457i	$0.657906\pi$
$68$	−20.1969	−2.44923
$69$	0.646151	0.0777875
$70$	−8.88454	−1.06191
$71$	−14.2155	−1.68707	−0.843536	−	0.537072i	$-0.819530\pi$
$71$	−14.2155	−1.68707	−0.843536	+	0.537072i	$0.819530\pi$
$72$	−17.5812	−2.07197
$73$	−8.12270	−0.950690	−0.475345	−	0.879800i	$-0.657677\pi$
$73$	−8.12270	−0.950690	−0.475345	+	0.879800i	$0.657677\pi$
$74$	−10.6784	−1.24134
$75$	−11.7782	−1.36003
$76$	21.7256	2.49210
$77$	3.25264	0.370673
$78$	−28.7265	−3.25264
$79$	1.39950	0.157456	0.0787279	−	0.996896i	$-0.474914\pi$
$79$	1.39950	0.157456	0.0787279	+	0.996896i	$0.474914\pi$
$80$	0.279835	0.0312865
$81$	8.05056	0.894506
$82$	16.1468	1.78312
$83$	8.88774	0.975556	0.487778	−	0.872968i	$-0.337808\pi$
$83$	8.88774	0.975556	0.487778	+	0.872968i	$0.337808\pi$
$84$	37.0270	4.03998
$85$	−6.27977	−0.681137
$86$	−16.9579	−1.82862
$87$	28.3218	3.03641
$88$	−2.57614	−0.274618
$89$	0.571882	0.0606194	0.0303097	−	0.999541i	$-0.490351\pi$
$89$	0.571882	0.0606194	0.0303097	+	0.999541i	$0.490351\pi$
$90$	−13.9004	−1.46523
$91$	15.7662	1.65275
$92$	0.714220	0.0744626
$93$	21.3410	2.21296
$94$	20.1843	2.08185
$95$	6.75509	0.693057
$96$	15.9194	1.62477
$97$	16.3787	1.66300	0.831502	−	0.555521i	$-0.187481\pi$
$97$	16.3787	1.66300	0.831502	+	0.555521i	$0.187481\pi$
$98$	−16.5423	−1.67103
$99$	5.08896	0.511459
$100$	−13.0189	−1.30189
$101$	12.7568	1.26935	0.634677	−	0.772778i	$-0.281133\pi$
$101$	12.7568	1.26935	0.634677	+	0.772778i	$0.281133\pi$
$102$	42.0507	4.16364
$103$	−17.8429	−1.75811	−0.879057	−	0.476716i	$-0.841827\pi$
$103$	−17.8429	−1.75811	−0.879057	+	0.476716i	$0.841827\pi$
$104$	−12.4871	−1.22446
$105$	11.5127	1.12353
$106$	2.30137	0.223529
$107$	−12.1193	−1.17162	−0.585808	−	0.810450i	$-0.699223\pi$
$107$	−12.1193	−1.17162	−0.585808	+	0.810450i	$0.699223\pi$
$108$	28.4408	2.73672
$109$	−15.2640	−1.46202	−0.731011	−	0.682366i	$-0.760951\pi$
$109$	−15.2640	−1.46202	−0.731011	+	0.682366i	$0.760951\pi$
$110$	−2.03680	−0.194201
$111$	13.8372	1.31337
$112$	1.02843	0.0971777
$113$	9.15204	0.860951	0.430476	−	0.902602i	$-0.358346\pi$
$113$	9.15204	0.860951	0.430476	+	0.902602i	$0.358346\pi$
$114$	−45.2335	−4.23651
$115$	0.222071	0.0207082
$116$	31.3053	2.90663
$117$	24.6672	2.28049
$118$	11.4339	1.05258
$119$	−23.0790	−2.11565
$120$	−9.11826	−0.832380
$121$	−10.2543	−0.932211
$122$	−7.35551	−0.665937
$123$	−20.9233	−1.88659
$124$	23.5891	2.11837
$125$	−9.17252	−0.820415
$126$	−51.0860	−4.55110
$127$	−10.8226	−0.960350	−0.480175	−	0.877173i	$-0.659427\pi$
$127$	−10.8226	−0.960350	−0.480175	+	0.877173i	$0.659427\pi$
$128$	18.8531	1.66640
$129$	21.9744	1.93473
$130$	−9.87279	−0.865901
$131$	−3.51885	−0.307443	−0.153722	−	0.988114i	$-0.549126\pi$
$131$	−3.51885	−0.307443	−0.153722	+	0.988114i	$0.549126\pi$
$132$	8.48853	0.738832
$133$	24.8259	2.15268
$134$	17.9325	1.54913
$135$	8.84304	0.761088
$136$	18.2790	1.56741
$137$	1.52226	0.130056	0.0650279	−	0.997883i	$-0.479286\pi$
$137$	1.52226	0.130056	0.0650279	+	0.997883i	$0.479286\pi$
$138$	−1.48703	−0.126585
$139$	7.96461	0.675550	0.337775	−	0.941227i	$-0.390326\pi$
$139$	7.96461	0.675550	0.337775	+	0.941227i	$0.390326\pi$
$140$	12.7255	1.07550
$141$	−26.1552	−2.20266
$142$	32.7152	2.74540
$143$	3.61444	0.302254
$144$	1.60905	0.134087
$145$	9.73369	0.808339
$146$	18.6933	1.54707
$147$	21.4358	1.76800
$148$	15.2949	1.25723
$149$	−11.6082	−0.950979	−0.475489	−	0.879721i	$-0.657729\pi$
$149$	−11.6082	−0.950979	−0.475489	+	0.879721i	$0.657729\pi$
$150$	27.1059	2.21319
$151$	1.00000	0.0813788
$151$	1.00000	0.0813788
$152$	−19.6625	−1.59484
$153$	−36.1086	−2.91921
$154$	−7.48553	−0.603201
$155$	7.33452	0.589123
$156$	41.1457	3.29429
$157$	8.42536	0.672417	0.336208	−	0.941788i	$-0.390855\pi$
$157$	8.42536	0.672417	0.336208	+	0.941788i	$0.390855\pi$
$158$	−3.22076	−0.256230
$159$	−2.98215	−0.236500
$160$	5.47122	0.432538
$161$	0.816141	0.0643210
$162$	−18.5273	−1.45564
$163$	3.67800	0.288084	0.144042	−	0.989572i	$-0.453990\pi$
$163$	3.67800	0.288084	0.144042	+	0.989572i	$0.453990\pi$
$164$	−23.1274	−1.80595
$165$	2.63932	0.205471
$166$	−20.4540	−1.58754
$167$	7.46285	0.577493	0.288747	−	0.957406i	$-0.406762\pi$
$167$	7.46285	0.577493	0.288747	+	0.957406i	$0.406762\pi$
$168$	−33.5109	−2.58542
$169$	4.51992	0.347686
$170$	14.4521	1.10842
$171$	38.8417	2.97030
$172$	24.2893	1.85204
$173$	−11.7681	−0.894710	−0.447355	−	0.894356i	$-0.647634\pi$
$173$	−11.7681	−0.894710	−0.447355	+	0.894356i	$0.647634\pi$
$174$	−65.1789	−4.94120
$175$	−14.8768	−1.12458
$176$	0.235770	0.0177719
$177$	−14.8163	−1.11366
$178$	−1.31611	−0.0986468
$179$	−8.07427	−0.603499	−0.301750	−	0.953387i	$-0.597571\pi$
$179$	−8.07427	−0.603499	−0.301750	+	0.953387i	$0.597571\pi$
$180$	19.9099	1.48400
$181$	15.6685	1.16463	0.582314	−	0.812964i	$-0.302147\pi$
$181$	15.6685	1.16463	0.582314	+	0.812964i	$0.302147\pi$
$182$	−36.2839	−2.68954
$183$	9.53139	0.704580
$184$	−0.646397	−0.0476530
$185$	4.75561	0.349639
$186$	−49.1135	−3.60118
$187$	−5.29092	−0.386910
$188$	−28.9105	−2.10851
$189$	32.4994	2.36399
$190$	−15.5460	−1.12782
$191$	9.20737	0.666222	0.333111	−	0.942888i	$-0.391902\pi$
$191$	9.20737	0.666222	0.333111	+	0.942888i	$0.391902\pi$
$192$	−38.2649	−2.76153
$193$	21.8627	1.57371	0.786856	−	0.617137i	$-0.211707\pi$
$193$	21.8627	1.57371	0.786856	+	0.617137i	$0.211707\pi$
$194$	−37.6935	−2.70623
$195$	12.7933	0.916149
$196$	23.6940	1.69243
$197$	−22.2579	−1.58581	−0.792907	−	0.609343i	$-0.791433\pi$
$197$	−22.2579	−1.58581	−0.792907	+	0.609343i	$0.791433\pi$
$198$	−11.7116	−0.832306
$199$	−19.2109	−1.36182	−0.680911	−	0.732366i	$-0.738416\pi$
$199$	−19.2109	−1.36182	−0.680911	+	0.732366i	$0.738416\pi$
$200$	11.7826	0.833159
$201$	−23.2373	−1.63903
$202$	−29.3582	−2.06564
$203$	35.7727	2.51075
$204$	−60.2302	−4.21696
$205$	−7.19096	−0.502239
$206$	41.0632	2.86101
$207$	1.27690	0.0887510
$208$	1.14283	0.0792408
$209$	5.69139	0.393682
$210$	−26.4951	−1.82833
$211$	−7.00867	−0.482496	−0.241248	−	0.970463i	$-0.577557\pi$
$211$	−7.00867	−0.482496	−0.241248	+	0.970463i	$0.577557\pi$
$212$	−3.29631	−0.226391
$213$	−42.3929	−2.90471
$214$	27.8910	1.90659
$215$	7.55220	0.515056
$216$	−25.7401	−1.75139
$217$	26.9554	1.82985
$218$	35.1280	2.37917
$219$	−24.2231	−1.63685
$220$	2.91736	0.196688
$221$	−25.6462	−1.72515
$222$	−31.8446	−2.13727
$223$	−6.07744	−0.406975	−0.203488	−	0.979078i	$-0.565228\pi$
$223$	−6.07744	−0.406975	−0.203488	+	0.979078i	$0.565228\pi$
$224$	20.1075	1.34349
$225$	−23.2757	−1.55171
$226$	−21.0622	−1.40104
$227$	20.1276	1.33591	0.667957	−	0.744200i	$-0.267169\pi$
$227$	20.1276	1.33591	0.667957	+	0.744200i	$0.267169\pi$
$228$	64.7891	4.29076
$229$	−11.4592	−0.757245	−0.378623	−	0.925551i	$-0.623602\pi$
$229$	−11.4592	−0.757245	−0.378623	+	0.925551i	$0.623602\pi$
$230$	−0.511067	−0.0336988
$231$	9.69987	0.638204
$232$	−28.3325	−1.86012
$233$	−20.8098	−1.36329	−0.681647	−	0.731681i	$-0.738736\pi$
$233$	−20.8098	−1.36329	−0.681647	+	0.731681i	$0.738736\pi$
$234$	−56.7684	−3.71107
$235$	−8.98907	−0.586382
$236$	−16.3771	−1.06606
$237$	4.17352	0.271099
$238$	53.1134	3.44283
$239$	7.05149	0.456123	0.228061	−	0.973647i	$-0.426761\pi$
$239$	7.05149	0.456123	0.228061	+	0.973647i	$0.426761\pi$
$240$	0.834510	0.0538674
$241$	−4.47295	−0.288128	−0.144064	−	0.989568i	$-0.546017\pi$
$241$	−4.47295	−0.288128	−0.144064	+	0.989568i	$0.546017\pi$
$242$	23.5990	1.51700
$243$	−1.87626	−0.120362
$244$	10.5355	0.674464
$245$	7.36712	0.470668
$246$	48.1522	3.07007
$247$	27.5873	1.75534
$248$	−21.3491	−1.35567
$249$	26.5046	1.67966
$250$	21.1094	1.33507
$251$	−3.29159	−0.207764	−0.103882	−	0.994590i	$-0.533126\pi$
$251$	−3.29159	−0.207764	−0.103882	+	0.994590i	$0.533126\pi$
$252$	73.1717	4.60938
$253$	0.187102	0.0117630
$254$	24.9068	1.56279
$255$	−18.7272	−1.17275
$256$	−17.7255	−1.10784
$257$	17.2999	1.07914	0.539570	−	0.841941i	$-0.318587\pi$
$257$	17.2999	1.07914	0.539570	+	0.841941i	$0.318587\pi$
$258$	−50.5712	−3.14842
$259$	17.4775	1.08600
$260$	14.1410	0.876990
$261$	55.9686	3.46437
$262$	8.09818	0.500307
$263$	24.6932	1.52265	0.761324	−	0.648371i	$-0.224549\pi$
$263$	24.6932	1.52265	0.761324	+	0.648371i	$0.224549\pi$
$264$	−7.68245	−0.472822
$265$	−1.02491	−0.0629599
$266$	−57.1336	−3.50309
$267$	1.70544	0.104371
$268$	−25.6852	−1.56897
$269$	1.30272	0.0794283	0.0397142	−	0.999211i	$-0.487355\pi$
$269$	1.30272	0.0794283	0.0397142	+	0.999211i	$0.487355\pi$
$270$	−20.3511	−1.23853
$271$	−18.8618	−1.14577	−0.572887	−	0.819634i	$-0.694177\pi$
$271$	−18.8618	−1.14577	−0.572887	+	0.819634i	$0.694177\pi$
$272$	−1.67290	−0.101435
$273$	47.0173	2.84561
$274$	−3.50329	−0.211642
$275$	−3.41054	−0.205663
$276$	2.12991	0.128206
$277$	−25.2993	−1.52009	−0.760044	−	0.649871i	$-0.774823\pi$
$277$	−25.2993	−1.52009	−0.760044	+	0.649871i	$0.774823\pi$
$278$	−18.3295	−1.09933
$279$	42.1734	2.52485
$280$	−11.5171	−0.688279
$281$	6.56159	0.391432	0.195716	−	0.980661i	$-0.437297\pi$
$281$	6.56159	0.391432	0.195716	+	0.980661i	$0.437297\pi$
$282$	60.1928	3.58443
$283$	4.81080	0.285972	0.142986	−	0.989725i	$-0.454330\pi$
$283$	4.81080	0.285972	0.142986	+	0.989725i	$0.454330\pi$
$284$	−46.8588	−2.78056
$285$	20.1447	1.19327
$286$	−8.31816	−0.491863
$287$	−26.4278	−1.55998
$288$	31.4594	1.85377
$289$	20.5416	1.20833
$290$	−22.4008	−1.31542
$291$	48.8438	2.86327
$292$	−26.7749	−1.56688
$293$	−19.2109	−1.12231	−0.561156	−	0.827710i	$-0.689643\pi$
$293$	−19.2109	−1.12231	−0.561156	+	0.827710i	$0.689643\pi$
$294$	−49.3318	−2.87709
$295$	−5.09209	−0.296473
$296$	−13.8425	−0.804578
$297$	7.45056	0.432326
$298$	26.7147	1.54754
$299$	0.906923	0.0524487
$300$	−38.8245	−2.24153
$301$	27.7554	1.59979
$302$	−2.30137	−0.132429
$303$	38.0429	2.18551
$304$	1.79953	0.103210
$305$	3.27577	0.187570
$306$	83.0993	4.75047
$307$	8.02995	0.458293	0.229147	−	0.973392i	$-0.426406\pi$
$307$	8.02995	0.458293	0.229147	+	0.973392i	$0.426406\pi$
$308$	10.7217	0.610926
$309$	−53.2103	−3.02703
$310$	−16.8794	−0.958688
$311$	−6.83456	−0.387553	−0.193776	−	0.981046i	$-0.562074\pi$
$311$	−6.83456	−0.387553	−0.193776	+	0.981046i	$0.562074\pi$
$312$	−37.2384	−2.10821
$313$	−24.0777	−1.36095	−0.680476	−	0.732770i	$-0.738227\pi$
$313$	−24.0777	−1.36095	−0.680476	+	0.732770i	$0.738227\pi$
$314$	−19.3899	−1.09423
$315$	22.7511	1.28188
$316$	4.61318	0.259511
$317$	12.0555	0.677105	0.338553	−	0.940947i	$-0.390063\pi$
$317$	12.0555	0.677105	0.338553	+	0.940947i	$0.390063\pi$
$318$	6.86304	0.384860
$319$	8.20097	0.459166
$320$	−13.1510	−0.735162
$321$	−36.1416	−2.01723
$322$	−1.87824	−0.104670
$323$	−40.3832	−2.24698
$324$	26.5371	1.47428
$325$	−16.5316	−0.917006
$326$	−8.46445	−0.468803
$327$	−45.5195	−2.51723
$328$	20.9312	1.15573
$329$	−33.0361	−1.82134
$330$	−6.07405	−0.334366
$331$	21.0996	1.15974	0.579869	−	0.814709i	$-0.303104\pi$
$331$	21.0996	1.15974	0.579869	+	0.814709i	$0.303104\pi$
$332$	29.2967	1.60787
$333$	27.3447	1.49848
$334$	−17.1748	−0.939763
$335$	−7.98624	−0.436335
$336$	3.06694	0.167315
$337$	22.0181	1.19940	0.599700	−	0.800225i	$-0.295287\pi$
$337$	22.0181	1.19940	0.599700	+	0.800225i	$0.295287\pi$
$338$	−10.4020	−0.565795
$339$	27.2928	1.48234
$340$	−20.7001	−1.12262
$341$	6.17958	0.334643
$342$	−89.3891	−4.83361
$343$	0.708285	0.0382438
$344$	−21.9827	−1.18523
$345$	0.662249	0.0356543
$346$	27.0827	1.45598
$347$	27.0479	1.45201	0.726003	−	0.687691i	$-0.241376\pi$
$347$	27.0479	1.45201	0.726003	+	0.687691i	$0.241376\pi$
$348$	93.3573	5.00448
$349$	−28.2299	−1.51111	−0.755555	−	0.655085i	$-0.772633\pi$
$349$	−28.2299	−1.51111	−0.755555	+	0.655085i	$0.772633\pi$
$350$	34.2370	1.83004
$351$	36.1144	1.92764
$352$	4.60969	0.245697
$353$	−3.27578	−0.174352	−0.0871761	−	0.996193i	$-0.527784\pi$
$353$	−3.27578	−0.174352	−0.0871761	+	0.996193i	$0.527784\pi$
$354$	34.0977	1.81227
$355$	−14.5697	−0.773279
$356$	1.88510	0.0999101
$357$	−68.8252	−3.64262
$358$	18.5819	0.982083
$359$	−9.51743	−0.502311	−0.251156	−	0.967947i	$-0.580811\pi$
$359$	−9.51743	−0.502311	−0.251156	+	0.967947i	$0.580811\pi$
$360$	−18.0192	−0.949697
$361$	24.4398	1.28630
$362$	−36.0590	−1.89522
$363$	−30.5800	−1.60503
$364$	51.9703	2.72398
$365$	−8.32506	−0.435754
$366$	−21.9353	−1.14657
$367$	0.0106411	0.000555461 0	0.000277730	−	1.00000i	$-0.499912\pi$
$367$	0.0106411	0.000555461 0	0.000277730		1.00000i	$0.499912\pi$
$368$	0.0591587	0.00308386
$369$	−41.3480	−2.15249
$370$	−10.9444	−0.568973
$371$	−3.76670	−0.195557
$372$	70.3464	3.64729
$373$	−1.61894	−0.0838256	−0.0419128	−	0.999121i	$-0.513345\pi$
$373$	−1.61894	−0.0838256	−0.0419128	+	0.999121i	$0.513345\pi$
$374$	12.1764	0.629625
$375$	−27.3539	−1.41255
$376$	26.1651	1.34936
$377$	39.7518	2.04732
$378$	−74.7932	−3.84695
$379$	−18.3501	−0.942582	−0.471291	−	0.881978i	$-0.656212\pi$
$379$	−18.3501	−0.942582	−0.471291	+	0.881978i	$0.656212\pi$
$380$	22.2669	1.14227
$381$	−32.2746	−1.65348
$382$	−21.1896	−1.08415
$383$	−35.0226	−1.78957	−0.894787	−	0.446494i	$-0.852672\pi$
$383$	−35.0226	−1.78957	−0.894787	+	0.446494i	$0.852672\pi$
$384$	56.2229	2.86911
$385$	3.33367	0.169900
$386$	−50.3142	−2.56093
$387$	43.4251	2.20742
$388$	53.9892	2.74089
$389$	34.5712	1.75283	0.876414	−	0.481558i	$-0.159929\pi$
$389$	34.5712	1.75283	0.876414	+	0.481558i	$0.159929\pi$
$390$	−29.4422	−1.49086
$391$	−1.32758	−0.0671386
$392$	−21.4440	−1.08308
$393$	−10.4938	−0.529340
$394$	51.2238	2.58062
$395$	1.43436	0.0721707
$396$	16.7748	0.842964
$397$	32.9100	1.65170	0.825852	−	0.563887i	$-0.190695\pi$
$397$	32.9100	1.65170	0.825852	+	0.563887i	$0.190695\pi$
$398$	44.2113	2.21611
$399$	74.0346	3.70637
$400$	−1.07836	−0.0539178
$401$	−24.4522	−1.22109	−0.610543	−	0.791983i	$-0.709049\pi$
$401$	−24.4522	−1.22109	−0.610543	+	0.791983i	$0.709049\pi$
$402$	53.4776	2.66722
$403$	29.9537	1.49210
$404$	42.0505	2.09209
$405$	8.25113	0.410002
$406$	−82.3262	−4.08578
$407$	4.00676	0.198608
$408$	54.5107	2.69868
$409$	10.3637	0.512450	0.256225	−	0.966617i	$-0.417521\pi$
$409$	10.3637	0.512450	0.256225	+	0.966617i	$0.417521\pi$
$410$	16.5491	0.817300
$411$	4.53962	0.223923
$412$	−58.8158	−2.89764
$413$	−18.7141	−0.920863
$414$	−2.93863	−0.144426
$415$	9.10916	0.447151
$416$	22.3441	1.09551
$417$	23.7517	1.16313
$418$	−13.0980	−0.640644
$419$	35.6432	1.74128	0.870642	−	0.491918i	$-0.163704\pi$
$419$	35.6432	1.74128	0.870642	+	0.491918i	$0.163704\pi$
$420$	37.9495	1.85175
$421$	−24.7546	−1.20646	−0.603232	−	0.797566i	$-0.706121\pi$
$421$	−24.7546	−1.20646	−0.603232	+	0.797566i	$0.706121\pi$
$422$	16.1295	0.785174
$423$	−51.6870	−2.51311
$424$	2.98329	0.144881
$425$	24.1994	1.17384
$426$	97.5618	4.72688
$427$	12.0389	0.582604
$428$	−39.9489	−1.93100
$429$	10.7788	0.520406
$430$	−17.3804	−0.838158
$431$	−26.0687	−1.25569	−0.627843	−	0.778340i	$-0.716062\pi$
$431$	−26.0687	−1.25569	−0.627843	+	0.778340i	$0.716062\pi$
$432$	2.35575	0.113341
$433$	−25.1886	−1.21049	−0.605243	−	0.796041i	$-0.706924\pi$
$433$	−25.1886	−1.21049	−0.605243	+	0.796041i	$0.706924\pi$
$434$	−62.0343	−2.97774
$435$	29.0274	1.39176
$436$	−50.3147	−2.40964
$437$	1.42807	0.0683136
$438$	55.7464	2.66367
$439$	3.73501	0.178262	0.0891312	−	0.996020i	$-0.471591\pi$
$439$	3.73501	0.178262	0.0891312	+	0.996020i	$0.471591\pi$
$440$	−2.64032	−0.125872
$441$	42.3608	2.01718
$442$	59.0214	2.80736
$443$	21.4642	1.01979	0.509896	−	0.860236i	$-0.329684\pi$
$443$	21.4642	1.01979	0.509896	+	0.860236i	$0.329684\pi$
$444$	45.6118	2.16464
$445$	0.586129	0.0277852
$446$	13.9864	0.662277
$447$	−34.6174	−1.63734
$448$	−48.3317	−2.28346
$449$	−25.4051	−1.19894	−0.599469	−	0.800398i	$-0.704622\pi$
$449$	−25.4051	−1.19894	−0.599469	+	0.800398i	$0.704622\pi$
$450$	53.5659	2.52512
$451$	−6.05863	−0.285290
$452$	30.1679	1.41898
$453$	2.98215	0.140114
$454$	−46.3210	−2.17395
$455$	16.1590	0.757546
$456$	−58.6366	−2.74591
$457$	−26.0744	−1.21971	−0.609855	−	0.792513i	$-0.708772\pi$
$457$	−26.0744	−1.21971	−0.609855	+	0.792513i	$0.708772\pi$
$458$	26.3719	1.23228
$459$	−52.8653	−2.46754
$460$	0.732014	0.0341303
$461$	2.95690	0.137717	0.0688583	−	0.997626i	$-0.478064\pi$
$461$	2.95690	0.137717	0.0688583	+	0.997626i	$0.478064\pi$
$462$	−22.3230	−1.03856
$463$	31.3854	1.45860	0.729302	−	0.684192i	$-0.239845\pi$
$463$	31.3854	1.45860	0.729302	+	0.684192i	$0.239845\pi$
$464$	2.59301	0.120378
$465$	21.8727	1.01432
$466$	47.8911	2.21851
$467$	−4.53566	−0.209885	−0.104943	−	0.994478i	$-0.533466\pi$
$467$	−4.53566	−0.209885	−0.104943	+	0.994478i	$0.533466\pi$
$468$	81.3108	3.75859
$469$	−29.3505	−1.35528
$470$	20.6872	0.954229
$471$	25.1257	1.15773
$472$	14.8219	0.682233
$473$	6.36299	0.292570
$474$	−9.60481	−0.441164
$475$	−26.0311	−1.19439
$476$	−76.0756	−3.48692
$477$	−5.89324	−0.269833
$478$	−16.2281	−0.742256
$479$	−17.4030	−0.795161	−0.397581	−	0.917567i	$-0.630150\pi$
$479$	−17.4030	−0.795161	−0.397581	+	0.917567i	$0.630150\pi$
$480$	16.3160	0.744721
$481$	19.4216	0.885549
$482$	10.2939	0.468875
$483$	2.43386	0.110744
$484$	−33.8014	−1.53643
$485$	16.7867	0.762247
$486$	4.31796	0.195867
$487$	30.5766	1.38556	0.692780	−	0.721149i	$-0.256386\pi$
$487$	30.5766	1.38556	0.692780	+	0.721149i	$0.256386\pi$
$488$	−9.53501	−0.431630
$489$	10.9684	0.496007
$490$	−16.9545	−0.765925
$491$	−20.5542	−0.927597	−0.463798	−	0.885941i	$-0.653514\pi$
$491$	−20.5542	−0.927597	−0.463798	+	0.885941i	$0.653514\pi$
$492$	−68.9696	−3.10939
$493$	−58.1898	−2.62074
$494$	−63.4887	−2.85649
$495$	5.21574	0.234430
$496$	1.95388	0.0877320
$497$	−53.5456	−2.40185
$498$	−60.9969	−2.73334
$499$	30.7657	1.37726	0.688630	−	0.725113i	$-0.258212\pi$
$499$	30.7657	1.37726	0.688630	+	0.725113i	$0.258212\pi$
$500$	−30.2354	−1.35217
$501$	22.2554	0.994297
$502$	7.57518	0.338097
$503$	5.88141	0.262239	0.131120	−	0.991367i	$-0.458143\pi$
$503$	5.88141	0.262239	0.131120	+	0.991367i	$0.458143\pi$
$504$	−66.2232	−2.94982
$505$	13.0747	0.581815
$506$	−0.430592	−0.0191421
$507$	13.4791	0.598628
$508$	−35.6746	−1.58280
$509$	−12.3095	−0.545608	−0.272804	−	0.962070i	$-0.587951\pi$
$509$	−12.3095	−0.545608	−0.272804	+	0.962070i	$0.587951\pi$
$510$	43.0983	1.90843
$511$	−30.5958	−1.35348
$512$	3.08658	0.136409
$513$	56.8667	2.51073
$514$	−39.8136	−1.75610
$515$	−18.2874	−0.805841
$516$	72.4343	3.18874
$517$	−7.57360	−0.333086
$518$	−40.2223	−1.76727
$519$	−35.0942	−1.54047
$520$	−12.7982	−0.561237
$521$	32.1477	1.40842	0.704208	−	0.709994i	$-0.251302\pi$
$521$	32.1477	1.40842	0.704208	+	0.709994i	$0.251302\pi$
$522$	−128.805	−5.63762
$523$	13.2888	0.581079	0.290540	−	0.956863i	$-0.406165\pi$
$523$	13.2888	0.581079	0.290540	+	0.956863i	$0.406165\pi$
$524$	−11.5992	−0.506714
$525$	−44.3649	−1.93624
$526$	−56.8282	−2.47783
$527$	−43.8471	−1.91001
$528$	0.703103	0.0305987
$529$	−22.9531	−0.997959
$530$	2.35871	0.102456
$531$	−29.2794	−1.27062
$532$	81.8338	3.54795
$533$	−29.3674	−1.27204
$534$	−3.92485	−0.169845
$535$	−12.4212	−0.537017
$536$	23.2461	1.00408
$537$	−24.0787	−1.03907
$538$	−2.99804	−0.129255
$539$	6.20705	0.267356
$540$	29.1494	1.25439
$541$	20.9537	0.900869	0.450434	−	0.892810i	$-0.351269\pi$
$541$	20.9537	0.900869	0.450434	+	0.892810i	$0.351269\pi$
$542$	43.4081	1.86454
$543$	46.7258	2.00519
$544$	−32.7080	−1.40234
$545$	−15.6442	−0.670125
$546$	−108.204	−4.63071
$547$	−6.13372	−0.262259	−0.131129	−	0.991365i	$-0.541860\pi$
$547$	−6.13372	−0.262259	−0.131129	+	0.991365i	$0.541860\pi$
$548$	5.01785	0.214352
$549$	18.8356	0.803885
$550$	7.84891	0.334679
$551$	62.5942	2.66660
$552$	−1.92765	−0.0820464
$553$	5.27149	0.224167
$554$	58.2231	2.47366
$555$	14.1820	0.601991
$556$	26.2538	1.11341
$557$	−11.6686	−0.494413	−0.247206	−	0.968963i	$-0.579513\pi$
$557$	−11.6686	−0.494413	−0.247206	+	0.968963i	$0.579513\pi$
$558$	−97.0566	−4.10873
$559$	30.8427	1.30451
$560$	1.05405	0.0445419
$561$	−15.7783	−0.666162
$562$	−15.1007	−0.636983
$563$	−22.4138	−0.944629	−0.472315	−	0.881430i	$-0.656581\pi$
$563$	−22.4138	−0.944629	−0.472315	+	0.881430i	$0.656581\pi$
$564$	−86.2155	−3.63033
$565$	9.38005	0.394621
$566$	−11.0714	−0.465367
$567$	30.3240	1.27349
$568$	42.4090	1.77944
$569$	−4.75678	−0.199415	−0.0997073	−	0.995017i	$-0.531791\pi$
$569$	−4.75678	−0.199415	−0.0997073	+	0.995017i	$0.531791\pi$
$570$	−46.3604	−1.94183
$571$	8.35497	0.349645	0.174822	−	0.984600i	$-0.444065\pi$
$571$	8.35497	0.349645	0.174822	+	0.984600i	$0.444065\pi$
$572$	11.9143	0.498162
$573$	27.4578	1.14707
$574$	60.8202	2.53859
$575$	−0.855760	−0.0356877
$576$	−75.6179	−3.15075
$577$	−5.80685	−0.241742	−0.120871	−	0.992668i	$-0.538569\pi$
$577$	−5.80685	−0.241742	−0.120871	+	0.992668i	$0.538569\pi$
$578$	−47.2739	−1.96634
$579$	65.1979	2.70953
$580$	32.0852	1.33227
$581$	33.4775	1.38888
$582$	−112.408	−4.65945
$583$	−0.863524	−0.0357635
$584$	24.2323	1.00274
$585$	25.2818	1.04527
$586$	44.2114	1.82636
$587$	34.5297	1.42519	0.712597	−	0.701573i	$-0.247519\pi$
$587$	34.5297	1.42519	0.712597	+	0.701573i	$0.247519\pi$
$588$	70.6591	2.91393
$589$	47.1659	1.94344
$590$	11.7188	0.482455
$591$	−66.3766	−2.73037
$592$	1.26687	0.0520682
$593$	−2.15064	−0.0883161	−0.0441580	−	0.999025i	$-0.514061\pi$
$593$	−2.15064	−0.0883161	−0.0441580	+	0.999025i	$0.514061\pi$
$594$	−17.1465	−0.703530
$595$	−23.6540	−0.969720
$596$	−38.2641	−1.56736
$597$	−57.2897	−2.34471
$598$	−2.08717	−0.0853506
$599$	−1.62242	−0.0662901	−0.0331451	−	0.999451i	$-0.510552\pi$
$599$	−1.62242	−0.0662901	−0.0331451	+	0.999451i	$0.510552\pi$
$600$	35.1377	1.43449
$601$	−8.76684	−0.357607	−0.178803	−	0.983885i	$-0.557223\pi$
$601$	−8.76684	−0.357607	−0.178803	+	0.983885i	$0.557223\pi$
$602$	−63.8755	−2.60337
$603$	−45.9208	−1.87004
$604$	3.29631	0.134125
$605$	−10.5098	−0.427284
$606$	−87.5508	−3.55651
$607$	−25.2819	−1.02616	−0.513081	−	0.858340i	$-0.671496\pi$
$607$	−25.2819	−1.02616	−0.513081	+	0.858340i	$0.671496\pi$
$608$	35.1836	1.42688
$609$	106.680	4.32288
$610$	−7.53876	−0.305235
$611$	−36.7108	−1.48516
$612$	−119.025	−4.81130
$613$	−8.53372	−0.344674	−0.172337	−	0.985038i	$-0.555132\pi$
$613$	−8.53372	−0.344674	−0.172337	+	0.985038i	$0.555132\pi$
$614$	−18.4799	−0.745788
$615$	−21.4446	−0.864728
$616$	−9.70355	−0.390967
$617$	25.4705	1.02540	0.512702	−	0.858566i	$-0.328644\pi$
$617$	25.4705	1.02540	0.512702	+	0.858566i	$0.328644\pi$
$618$	122.457	4.92593
$619$	8.99297	0.361458	0.180729	−	0.983533i	$-0.442154\pi$
$619$	8.99297	0.361458	0.180729	+	0.983533i	$0.442154\pi$
$620$	24.1768	0.970965
$621$	1.86947	0.0750193
$622$	15.7289	0.630670
$623$	2.15411	0.0863025
$624$	3.40809	0.136433
$625$	10.3467	0.413869
$626$	55.4118	2.21470
$627$	16.9726	0.677821
$628$	27.7726	1.10825
$629$	−28.4299	−1.13357
$630$	−52.3587	−2.08602
$631$	−19.2999	−0.768316	−0.384158	−	0.923267i	$-0.625508\pi$
$631$	−19.2999	−0.768316	−0.384158	+	0.923267i	$0.625508\pi$
$632$	−4.17510	−0.166077
$633$	−20.9009	−0.830737
$634$	−27.7442	−1.10186
$635$	−11.0922	−0.440181
$636$	−9.83010	−0.389789
$637$	30.0868	1.19208
$638$	−18.8735	−0.747208
$639$	−83.7755	−3.31411
$640$	19.3228	0.763802
$641$	−1.48377	−0.0586054	−0.0293027	−	0.999571i	$-0.509329\pi$
$641$	−1.48377	−0.0586054	−0.0293027	+	0.999571i	$0.509329\pi$
$642$	83.1752	3.28266
$643$	−32.2039	−1.27000	−0.634999	−	0.772513i	$-0.718999\pi$
$643$	−32.2039	−1.27000	−0.634999	+	0.772513i	$0.718999\pi$
$644$	2.69025	0.106011
$645$	22.5218	0.886796
$646$	92.9367	3.65654
$647$	−0.243253	−0.00956326	−0.00478163	−	0.999989i	$-0.501522\pi$
$647$	−0.243253	−0.00956326	−0.00478163	+	0.999989i	$0.501522\pi$
$648$	−24.0171	−0.943482
$649$	−4.29026	−0.168407
$650$	38.0453	1.49226
$651$	80.3851	3.15054
$652$	12.1238	0.474806
$653$	−41.8638	−1.63826	−0.819128	−	0.573611i	$-0.805542\pi$
$653$	−41.8638	−1.63826	−0.819128	+	0.573611i	$0.805542\pi$
$654$	104.757	4.09633
$655$	−3.60652	−0.140918
$656$	−1.91564	−0.0747933
$657$	−47.8690	−1.86755
$658$	76.0283	2.96389
$659$	5.98576	0.233172	0.116586	−	0.993181i	$-0.462805\pi$
$659$	5.98576	0.233172	0.116586	+	0.993181i	$0.462805\pi$
$660$	8.70001	0.338647
$661$	−23.8982	−0.929533	−0.464767	−	0.885433i	$-0.653862\pi$
$661$	−23.8982	−0.929533	−0.464767	+	0.885433i	$0.653862\pi$
$662$	−48.5580	−1.88726
$663$	−76.4809	−2.97027
$664$	−26.5147	−1.02897
$665$	25.4444	0.986691
$666$	−62.9303	−2.43850
$667$	2.05776	0.0796768
$668$	24.5999	0.951798
$669$	−18.1239	−0.700709
$670$	18.3793	0.710054
$671$	2.75995	0.106547
$672$	59.9636	2.31315
$673$	−20.0899	−0.774409	−0.387205	−	0.921994i	$-0.626559\pi$
$673$	−20.0899	−0.774409	−0.387205	+	0.921994i	$0.626559\pi$
$674$	−50.6717	−1.95180
$675$	−34.0771	−1.31163
$676$	14.8991	0.573041
$677$	−14.8548	−0.570916	−0.285458	−	0.958391i	$-0.592146\pi$
$677$	−14.8548	−0.570916	−0.285458	+	0.958391i	$0.592146\pi$
$678$	−62.8108	−2.41223
$679$	61.6937	2.36759
$680$	18.7344	0.718430
$681$	60.0236	2.30011
$682$	−14.2215	−0.544570
$683$	−43.5634	−1.66691	−0.833454	−	0.552588i	$-0.813640\pi$
$683$	−43.5634	−1.66691	−0.833454	+	0.552588i	$0.813640\pi$
$684$	128.034	4.89551
$685$	1.56019	0.0596117
$686$	−1.63003	−0.0622347
$687$	−34.1731	−1.30378
$688$	2.01187	0.0767020
$689$	−4.18568	−0.159462
$690$	−1.52408	−0.0580208
$691$	−15.7860	−0.600527	−0.300263	−	0.953856i	$-0.597075\pi$
$691$	−15.7860	−0.600527	−0.300263	+	0.953856i	$0.597075\pi$
$692$	−38.7912	−1.47462
$693$	19.1686	0.728154
$694$	−62.2472	−2.36287
$695$	8.16304	0.309642
$696$	−84.4920	−3.20266
$697$	42.9889	1.62832
$698$	64.9674	2.45905
$699$	−62.0580	−2.34725
$700$	−49.0385	−1.85348
$701$	28.1957	1.06494	0.532469	−	0.846450i	$-0.321264\pi$
$701$	28.1957	1.06494	0.532469	+	0.846450i	$0.321264\pi$
$702$	−83.1127	−3.13689
$703$	30.5818	1.15341
$704$	−11.0801	−0.417599
$705$	−26.8068	−1.00960
$706$	7.53879	0.283726
$707$	48.0512	1.80715
$708$	−48.8390	−1.83548
$709$	27.9666	1.05031	0.525153	−	0.851008i	$-0.324008\pi$
$709$	27.9666	1.05031	0.525153	+	0.851008i	$0.324008\pi$
$710$	33.5302	1.25837
$711$	8.24757	0.309308
$712$	−1.70609	−0.0639383
$713$	1.55056	0.0580689
$714$	158.392	5.92769
$715$	3.70449	0.138540
$716$	−26.6153	−0.994659
$717$	21.0286	0.785328
$718$	21.9031	0.817418
$719$	−6.31211	−0.235402	−0.117701	−	0.993049i	$-0.537552\pi$
$719$	−6.31211	−0.235402	−0.117701	+	0.993049i	$0.537552\pi$
$720$	1.64913	0.0614596
$721$	−67.2089	−2.50299
$722$	−56.2450	−2.09322
$723$	−13.3390	−0.496084
$724$	51.6481	1.91949
$725$	−37.5092	−1.39306
$726$	70.3759	2.61189
$727$	50.8684	1.88660	0.943302	−	0.331935i	$-0.107701\pi$
$727$	50.8684	1.88660	0.943302	+	0.331935i	$0.107701\pi$
$728$	−47.0351	−1.74324
$729$	−29.7470	−1.10174
$730$	19.1591	0.709108
$731$	−45.1485	−1.66988
$732$	31.4184	1.16126
$733$	20.6622	0.763177	0.381588	−	0.924332i	$-0.375377\pi$
$733$	20.6622	0.763177	0.381588	+	0.924332i	$0.375377\pi$
$734$	−0.0244891	−0.000903910 0
$735$	21.9699	0.810371
$736$	1.15665	0.0426346
$737$	−6.72868	−0.247854
$738$	95.1570	3.50278
$739$	−9.64377	−0.354752	−0.177376	−	0.984143i	$-0.556761\pi$
$739$	−9.64377	−0.354752	−0.177376	+	0.984143i	$0.556761\pi$
$740$	15.6760	0.576260
$741$	82.2697	3.02225
$742$	8.66858	0.318233
$743$	2.22098	0.0814799	0.0407399	−	0.999170i	$-0.487028\pi$
$743$	2.22098	0.0814799	0.0407399	+	0.999170i	$0.487028\pi$
$744$	−63.6663	−2.33412
$745$	−11.8974	−0.435886
$746$	3.72579	0.136411
$747$	52.3776	1.91640
$748$	−17.4405	−0.637688
$749$	−45.6498	−1.66801
$750$	62.9514	2.29866
$751$	13.9140	0.507729	0.253865	−	0.967240i	$-0.418298\pi$
$751$	13.9140	0.507729	0.253865	+	0.967240i	$0.418298\pi$
$752$	−2.39465	−0.0873239
$753$	−9.81604	−0.357716
$754$	−91.4836	−3.33164
$755$	1.02491	0.0373004
$756$	107.128	3.89621
$757$	17.9477	0.652319	0.326160	−	0.945315i	$-0.394245\pi$
$757$	17.9477	0.652319	0.326160	+	0.945315i	$0.394245\pi$
$758$	42.2304	1.53388
$759$	0.557968	0.0202529
$760$	−20.1524	−0.731003
$761$	−14.9073	−0.540390	−0.270195	−	0.962806i	$-0.587088\pi$
$761$	−14.9073	−0.540390	−0.270195	+	0.962806i	$0.587088\pi$
$762$	74.2759	2.69073
$763$	−57.4948	−2.08145
$764$	30.3503	1.09804
$765$	−37.0082	−1.33803
$766$	80.6001	2.91220
$767$	−20.7958	−0.750891
$768$	−52.8600	−1.90742
$769$	−3.55627	−0.128243	−0.0641213	−	0.997942i	$-0.520424\pi$
$769$	−3.55627	−0.128243	−0.0641213	+	0.997942i	$0.520424\pi$
$770$	−7.67202	−0.276480
$771$	51.5911	1.85801
$772$	72.0662	2.59372
$773$	25.7717	0.926943	0.463472	−	0.886112i	$-0.346604\pi$
$773$	25.7717	0.926943	0.463472	+	0.886112i	$0.346604\pi$
$774$	−99.9372	−3.59217
$775$	−28.2639	−1.01527
$776$	−48.8623	−1.75406
$777$	52.1207	1.86982
$778$	−79.5611	−2.85240
$779$	−46.2428	−1.65682
$780$	42.1707	1.50995
$781$	−12.2755	−0.439250
$782$	3.05526	0.109256
$783$	81.9417	2.92836
$784$	1.96257	0.0700918
$785$	8.63526	0.308206
$786$	24.1500	0.861403
$787$	−18.0928	−0.644938	−0.322469	−	0.946580i	$-0.604513\pi$
$787$	−18.0928	−0.644938	−0.322469	+	0.946580i	$0.604513\pi$
$788$	−73.3691	−2.61366
$789$	73.6390	2.62162
$790$	−3.30100	−0.117444
$791$	34.4730	1.22572
$792$	−15.1818	−0.539462
$793$	13.3780	0.475068
$794$	−75.7381	−2.68784
$795$	−3.05645	−0.108401
$796$	−63.3249	−2.24449
$797$	55.6917	1.97270	0.986351	−	0.164655i	$-0.0526510\pi$
$797$	55.6917	1.97270	0.986351	+	0.164655i	$0.0526510\pi$
$798$	−170.381	−6.03143
$799$	53.7383	1.90113
$800$	−21.0836	−0.745418
$801$	3.37024	0.119081
$802$	56.2737	1.98709
$803$	−7.01415	−0.247524
$804$	−76.5972	−2.70137
$805$	0.836474	0.0294818
$806$	−68.9346	−2.42812
$807$	3.88491	0.136755
$808$	−38.0573	−1.33885
$809$	−16.9104	−0.594538	−0.297269	−	0.954794i	$-0.596076\pi$
$809$	−16.9104	−0.594538	−0.297269	+	0.954794i	$0.596076\pi$
$810$	−18.9889	−0.667202
$811$	23.3920	0.821404	0.410702	−	0.911770i	$-0.365284\pi$
$811$	23.3920	0.821404	0.410702	+	0.911770i	$0.365284\pi$
$812$	117.918	4.13810
$813$	−56.2489	−1.97273
$814$	−9.22105	−0.323198
$815$	3.76964	0.132045
$816$	−4.98886	−0.174645
$817$	48.5657	1.69910
$818$	−23.8506	−0.833918
$819$	92.9141	3.24668
$820$	−23.7036	−0.827767
$821$	37.5790	1.31152	0.655758	−	0.754971i	$-0.272349\pi$
$821$	37.5790	1.31152	0.655758	+	0.754971i	$0.272349\pi$
$822$	−10.4474	−0.364393
$823$	−20.5585	−0.716625	−0.358312	−	0.933602i	$-0.616648\pi$
$823$	−20.5585	−0.716625	−0.358312	+	0.933602i	$0.616648\pi$
$824$	53.2305	1.85437
$825$	−10.1707	−0.354100
$826$	43.0682	1.49853
$827$	−1.54728	−0.0538042	−0.0269021	−	0.999638i	$-0.508564\pi$
$827$	−1.54728	−0.0538042	−0.0269021	+	0.999638i	$0.508564\pi$
$828$	4.20907	0.146275
$829$	−11.5037	−0.399540	−0.199770	−	0.979843i	$-0.564019\pi$
$829$	−11.5037	−0.399540	−0.199770	+	0.979843i	$0.564019\pi$
$830$	−20.9636	−0.727656
$831$	−75.4464	−2.61721
$832$	−53.7077	−1.86198
$833$	−44.0420	−1.52596
$834$	−54.6615	−1.89277
$835$	7.64878	0.264697
$836$	18.7606	0.648848
$837$	61.7446	2.13420
$838$	−82.0282	−2.83362
$839$	−23.6796	−0.817511	−0.408755	−	0.912644i	$-0.634037\pi$
$839$	−23.6796	−0.817511	−0.408755	+	0.912644i	$0.634037\pi$
$840$	−34.3458	−1.18504
$841$	61.1946	2.11016
$842$	56.9694	1.96330
$843$	19.5677	0.673947
$844$	−23.1027	−0.795229
$845$	4.63253	0.159364
$846$	118.951	4.08962
$847$	−38.6250	−1.32717
$848$	−0.273033	−0.00937598
$849$	14.3465	0.492372
$850$	−55.6918	−1.91021
$851$	1.00536	0.0344634
$852$	−139.740	−4.78741
$853$	−27.6271	−0.945933	−0.472966	−	0.881080i	$-0.656817\pi$
$853$	−27.6271	−0.945933	−0.472966	+	0.881080i	$0.656817\pi$
$854$	−27.7060	−0.948080
$855$	39.8094	1.36145
$856$	36.1553	1.23576
$857$	−2.20949	−0.0754747	−0.0377373	−	0.999288i	$-0.512015\pi$
$857$	−2.20949	−0.0754747	−0.0377373	+	0.999288i	$0.512015\pi$
$858$	−24.8060	−0.846864
$859$	18.8250	0.642301	0.321150	−	0.947028i	$-0.395931\pi$
$859$	18.8250	0.642301	0.321150	+	0.947028i	$0.395931\pi$
$860$	24.8944	0.848891
$861$	−78.8118	−2.68590
$862$	59.9937	2.04340
$863$	51.1166	1.74003	0.870015	−	0.493025i	$-0.164109\pi$
$863$	51.1166	1.74003	0.870015	+	0.493025i	$0.164109\pi$
$864$	46.0587	1.56695
$865$	−12.0613	−0.410095
$866$	57.9683	1.96984
$867$	61.2583	2.08044
$868$	88.8533	3.01588
$869$	1.20850	0.0409956
$870$	−66.8027	−2.26482
$871$	−32.6153	−1.10513
$872$	45.5368	1.54207
$873$	96.5236	3.26683
$874$	−3.28651	−0.111168
$875$	−34.5501	−1.16801
$876$	−79.8469	−2.69778
$877$	13.2566	0.447644	0.223822	−	0.974630i	$-0.428146\pi$
$877$	13.2566	0.447644	0.223822	+	0.974630i	$0.428146\pi$
$878$	−8.59565	−0.290089
$879$	−57.2898	−1.93234
$880$	0.241644	0.00814582
$881$	47.4575	1.59889	0.799443	−	0.600742i	$-0.205128\pi$
$881$	47.4575	1.59889	0.799443	+	0.600742i	$0.205128\pi$
$882$	−97.4880	−3.28259
$883$	−22.9651	−0.772836	−0.386418	−	0.922324i	$-0.626288\pi$
$883$	−22.9651	−0.772836	−0.386418	+	0.922324i	$0.626288\pi$
$884$	−84.5377	−2.84331
$885$	−15.1854	−0.510451
$886$	−49.3970	−1.65952
$887$	15.5718	0.522849	0.261425	−	0.965224i	$-0.415808\pi$
$887$	15.5718	0.522849	0.261425	+	0.965224i	$0.415808\pi$
$888$	−41.2804	−1.38528
$889$	−40.7655	−1.36723
$890$	−1.34890	−0.0452153
$891$	6.95185	0.232896
$892$	−20.0331	−0.670758
$893$	−57.8058	−1.93440
$894$	79.6674	2.66448
$895$	−8.27542	−0.276617
$896$	71.0141	2.37242
$897$	2.70458	0.0903034
$898$	58.4665	1.95105
$899$	67.9633	2.26670
$900$	−76.7238	−2.55746
$901$	6.12712	0.204124
$902$	13.9432	0.464256
$903$	82.7709	2.75444
$904$	−27.3032	−0.908089
$905$	16.0588	0.533813
$906$	−6.86304	−0.228009
$907$	47.0825	1.56335	0.781675	−	0.623686i	$-0.214366\pi$
$907$	47.0825	1.56335	0.781675	+	0.623686i	$0.214366\pi$
$908$	66.3467	2.20179
$909$	75.1792	2.49354
$910$	−37.1879	−1.23277
$911$	43.7170	1.44841	0.724205	−	0.689585i	$-0.242207\pi$
$911$	43.7170	1.44841	0.724205	+	0.689585i	$0.242207\pi$
$912$	5.36647	0.177701
$913$	7.67478	0.253998
$914$	60.0069	1.98485
$915$	9.76885	0.322948
$916$	−37.7731	−1.24806
$917$	−13.2545	−0.437701
$918$	121.663	4.01547
$919$	39.9549	1.31799	0.658995	−	0.752147i	$-0.270982\pi$
$919$	39.9549	1.31799	0.658995	+	0.752147i	$0.270982\pi$
$920$	−0.662501	−0.0218420
$921$	23.9465	0.789065
$922$	−6.80493	−0.224108
$923$	−59.5017	−1.95852
$924$	31.9738	1.05186
$925$	−18.3260	−0.602554
$926$	−72.2295	−2.37361
$927$	−105.153	−3.45366
$928$	50.6976	1.66423
$929$	38.6007	1.26645	0.633223	−	0.773969i	$-0.281731\pi$
$929$	38.6007	1.26645	0.633223	+	0.773969i	$0.281731\pi$
$930$	−50.3371	−1.65062
$931$	47.3755	1.55267
$932$	−68.5955	−2.24692
$933$	−20.3817	−0.667268
$934$	10.4382	0.341549
$935$	−5.42274	−0.177342
$936$	−73.5894	−2.40534
$937$	−56.8786	−1.85814	−0.929071	−	0.369901i	$-0.879391\pi$
$937$	−56.8786	−1.85814	−0.929071	+	0.369901i	$0.879391\pi$
$938$	67.5465	2.20547
$939$	−71.8034	−2.34322
$940$	−29.6307	−0.966448
$941$	−31.4540	−1.02537	−0.512685	−	0.858577i	$-0.671349\pi$
$941$	−31.4540	−1.02537	−0.512685	+	0.858577i	$0.671349\pi$
$942$	−57.8236	−1.88399
$943$	−1.52021	−0.0495049
$944$	−1.35651	−0.0441507
$945$	33.3091	1.08354
$946$	−14.6436	−0.476104
$947$	−12.2615	−0.398445	−0.199223	−	0.979954i	$-0.563842\pi$
$947$	−12.2615	−0.398445	−0.199223	+	0.979954i	$0.563842\pi$
$948$	13.7572	0.446813
$949$	−33.9990	−1.10365
$950$	59.9071	1.94364
$951$	35.9514	1.16580
$952$	68.8514	2.23149
$953$	12.0491	0.390308	0.195154	−	0.980773i	$-0.437479\pi$
$953$	12.0491	0.390308	0.195154	+	0.980773i	$0.437479\pi$
$954$	13.5625	0.439103
$955$	9.43676	0.305366
$956$	23.2439	0.751761
$957$	24.4565	0.790568
$958$	40.0507	1.29398
$959$	5.73391	0.185158
$960$	−39.2182	−1.26576
$961$	20.2116	0.651988
$962$	−44.6963	−1.44107
$963$	−71.4219	−2.30154
$964$	−14.7442	−0.474880
$965$	22.4074	0.721319
$966$	−5.60121	−0.180216
$967$	39.4582	1.26889	0.634444	−	0.772968i	$-0.281229\pi$
$967$	39.4582	1.26889	0.634444	+	0.772968i	$0.281229\pi$
$968$	30.5916	0.983251
$969$	−120.429	−3.86873
$970$	−38.6325	−1.24042
$971$	−44.0502	−1.41364	−0.706820	−	0.707394i	$-0.749871\pi$
$971$	−44.0502	−1.41364	−0.706820	+	0.707394i	$0.749871\pi$
$972$	−6.18472	−0.198375
$973$	30.0003	0.961766
$974$	−70.3682	−2.25474
$975$	−49.2997	−1.57885
$976$	0.872651	0.0279329
$977$	32.7288	1.04709	0.523544	−	0.851999i	$-0.324610\pi$
$977$	32.7288	1.04709	0.523544	+	0.851999i	$0.324610\pi$
$978$	−25.2423	−0.807160
$979$	0.493834	0.0157830
$980$	24.2843	0.775733
$981$	−89.9541	−2.87201
$982$	47.3028	1.50949
$983$	9.28298	0.296081	0.148040	−	0.988981i	$-0.452703\pi$
$983$	9.28298	0.296081	0.148040	+	0.988981i	$0.452703\pi$
$984$	62.4202	1.98988
$985$	−22.8125	−0.726866
$986$	133.916	4.26477
$987$	−98.5187	−3.13589
$988$	90.9364	2.89307
$989$	1.59658	0.0507683
$990$	−12.0034	−0.381492
$991$	34.4009	1.09278	0.546390	−	0.837531i	$-0.316002\pi$
$991$	34.4009	1.09278	0.546390	+	0.837531i	$0.316002\pi$
$992$	38.2016	1.21290
$993$	62.9222	1.99678
$994$	123.228	3.90857
$995$	−19.6895	−0.624198
$996$	87.3673	2.76834
$997$	−9.71779	−0.307766	−0.153883	−	0.988089i	$-0.549178\pi$
$997$	−9.71779	−0.307766	−0.153883	+	0.988089i	$0.549178\pi$
$998$	−70.8032	−2.24124
$999$	40.0344	1.26663

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8003.2.a.c.1.20	✓	172

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8003.2.a.c.1.20	✓	172	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:	\( N \)	\(=\)	\( 8003 = 53 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8003.a (trivial)

\(f(q)\)	\(=\)	\(q-2.30137 q^{2} +2.98215 q^{3} +3.29631 q^{4} +1.02491 q^{5} -6.86304 q^{6} +3.76670 q^{7} -2.98329 q^{8} +5.89324 q^{9} +O(q^{10})\)	\(q-2.30137 q^{2} +2.98215 q^{3} +3.29631 q^{4} +1.02491 q^{5} -6.86304 q^{6} +3.76670 q^{7} -2.98329 q^{8} +5.89324 q^{9} -2.35871 q^{10} +0.863524 q^{11} +9.83010 q^{12} +4.18568 q^{13} -8.66858 q^{14} +3.05645 q^{15} +0.273033 q^{16} -6.12712 q^{17} -13.5625 q^{18} +6.59089 q^{19} +3.37843 q^{20} +11.2329 q^{21} -1.98729 q^{22} +0.216673 q^{23} -8.89662 q^{24} -3.94955 q^{25} -9.63280 q^{26} +8.62809 q^{27} +12.4162 q^{28} +9.49709 q^{29} -7.03402 q^{30} +7.15623 q^{31} +5.33822 q^{32} +2.57516 q^{33} +14.1008 q^{34} +3.86054 q^{35} +19.4259 q^{36} +4.64001 q^{37} -15.1681 q^{38} +12.4823 q^{39} -3.05761 q^{40} -7.01617 q^{41} -25.8510 q^{42} +7.36862 q^{43} +2.84644 q^{44} +6.04006 q^{45} -0.498644 q^{46} -8.77056 q^{47} +0.814225 q^{48} +7.18804 q^{49} +9.08939 q^{50} -18.2720 q^{51} +13.7973 q^{52} -1.00000 q^{53} -19.8564 q^{54} +0.885038 q^{55} -11.2372 q^{56} +19.6550 q^{57} -21.8563 q^{58} -4.96831 q^{59} +10.0750 q^{60} +3.19614 q^{61} -16.4691 q^{62} +22.1981 q^{63} -12.8313 q^{64} +4.28996 q^{65} -5.92640 q^{66} -7.79211 q^{67} -20.1969 q^{68} +0.646151 q^{69} -8.88454 q^{70} -14.2155 q^{71} -17.5812 q^{72} -8.12270 q^{73} -10.6784 q^{74} -11.7782 q^{75} +21.7256 q^{76} +3.25264 q^{77} -28.7265 q^{78} +1.39950 q^{79} +0.279835 q^{80} +8.05056 q^{81} +16.1468 q^{82} +8.88774 q^{83} +37.0270 q^{84} -6.27977 q^{85} -16.9579 q^{86} +28.3218 q^{87} -2.57614 q^{88} +0.571882 q^{89} -13.9004 q^{90} +15.7662 q^{91} +0.714220 q^{92} +21.3410 q^{93} +20.1843 q^{94} +6.75509 q^{95} +15.9194 q^{96} +16.3787 q^{97} -16.5423 q^{98} +5.08896 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 172 q + 8 q^{2} + 25 q^{3} + 188 q^{4} + 27 q^{5} + 10 q^{6} + 31 q^{7} + 21 q^{8} + 179 q^{9}+O(q^{10}) \) 172 * q + 8 * q^2 + 25 * q^3 + 188 * q^4 + 27 * q^5 + 10 * q^6 + 31 * q^7 + 21 * q^8 + 179 * q^9	\( 172 q + 8 q^{2} + 25 q^{3} + 188 q^{4} + 27 q^{5} + 10 q^{6} + 31 q^{7} + 21 q^{8} + 179 q^{9} + 20 q^{10} - 3 q^{11} + 66 q^{12} + 121 q^{13} + 12 q^{14} + 30 q^{15} + 212 q^{16} + 8 q^{17} + 40 q^{18} + 41 q^{19} + 64 q^{20} + 56 q^{21} + 50 q^{22} + 28 q^{23} + 30 q^{24} + 231 q^{25} + 38 q^{26} + 100 q^{27} + 80 q^{28} + 26 q^{29} + 55 q^{30} + 66 q^{31} + 65 q^{32} + 99 q^{33} + 81 q^{34} + 36 q^{35} + 212 q^{36} + 153 q^{37} + q^{38} + 20 q^{39} + 59 q^{40} + 40 q^{41} + 50 q^{42} + 39 q^{43} - 51 q^{44} + 123 q^{45} + 59 q^{46} + 29 q^{47} + 128 q^{48} + 245 q^{49} + 19 q^{50} + 36 q^{51} + 215 q^{52} - 172 q^{53} + 40 q^{54} + 40 q^{55} + 15 q^{56} + 54 q^{57} + 44 q^{58} - 54 q^{60} + 100 q^{61} - 29 q^{62} + 92 q^{63} + 253 q^{64} + 77 q^{65} + 14 q^{66} + 126 q^{67} - 27 q^{68} + 47 q^{69} + 72 q^{70} + 38 q^{71} + 65 q^{72} + 185 q^{73} + 48 q^{74} + 75 q^{75} + 38 q^{76} + 120 q^{77} + 75 q^{78} + 79 q^{79} + 43 q^{80} + 232 q^{81} + 110 q^{82} + 90 q^{83} + 158 q^{84} + 115 q^{85} + 68 q^{86} + 61 q^{87} + 15 q^{88} - 36 q^{89} - 6 q^{90} + 33 q^{91} + 139 q^{92} + 103 q^{93} - 24 q^{94} - 45 q^{95} + 34 q^{96} + 159 q^{97} - 36 q^{98} + 27 q^{99}+O(q^{100}) \) 172 * q + 8 * q^2 + 25 * q^3 + 188 * q^4 + 27 * q^5 + 10 * q^6 + 31 * q^7 + 21 * q^8 + 179 * q^9 + 20 * q^10 - 3 * q^11 + 66 * q^12 + 121 * q^13 + 12 * q^14 + 30 * q^15 + 212 * q^16 + 8 * q^17 + 40 * q^18 + 41 * q^19 + 64 * q^20 + 56 * q^21 + 50 * q^22 + 28 * q^23 + 30 * q^24 + 231 * q^25 + 38 * q^26 + 100 * q^27 + 80 * q^28 + 26 * q^29 + 55 * q^30 + 66 * q^31 + 65 * q^32 + 99 * q^33 + 81 * q^34 + 36 * q^35 + 212 * q^36 + 153 * q^37 + q^38 + 20 * q^39 + 59 * q^40 + 40 * q^41 + 50 * q^42 + 39 * q^43 - 51 * q^44 + 123 * q^45 + 59 * q^46 + 29 * q^47 + 128 * q^48 + 245 * q^49 + 19 * q^50 + 36 * q^51 + 215 * q^52 - 172 * q^53 + 40 * q^54 + 40 * q^55 + 15 * q^56 + 54 * q^57 + 44 * q^58 - 54 * q^60 + 100 * q^61 - 29 * q^62 + 92 * q^63 + 253 * q^64 + 77 * q^65 + 14 * q^66 + 126 * q^67 - 27 * q^68 + 47 * q^69 + 72 * q^70 + 38 * q^71 + 65 * q^72 + 185 * q^73 + 48 * q^74 + 75 * q^75 + 38 * q^76 + 120 * q^77 + 75 * q^78 + 79 * q^79 + 43 * q^80 + 232 * q^81 + 110 * q^82 + 90 * q^83 + 158 * q^84 + 115 * q^85 + 68 * q^86 + 61 * q^87 + 15 * q^88 - 36 * q^89 - 6 * q^90 + 33 * q^91 + 139 * q^92 + 103 * q^93 - 24 * q^94 - 45 * q^95 + 34 * q^96 + 159 * q^97 - 36 * q^98 + 27 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more