LMFDB - Embedded newform 4003.2.a.b.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4003,2,Mod(1,4003)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4003, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4003 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4003.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:	$31.9641159291$
Analytic rank:	$1$
Dimension:	$152$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Character	$\chi$	$=$	4003.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.42958 q^{2} -1.44088 q^{3} +3.90288 q^{4} +0.473210 q^{5} +3.50074 q^{6} -4.22120 q^{7} -4.62321 q^{8} -0.923870 q^{9} +O(q^{10})$	\(q-2.42958 q^{2} -1.44088 q^{3} +3.90288 q^{4} +0.473210 q^{5} +3.50074 q^{6} -4.22120 q^{7} -4.62321 q^{8} -0.923870 q^{9} -1.14970 q^{10} +2.71553 q^{11} -5.62358 q^{12} -6.52831 q^{13} +10.2558 q^{14} -0.681838 q^{15} +3.42672 q^{16} -2.95637 q^{17} +2.24462 q^{18} +2.50601 q^{19} +1.84688 q^{20} +6.08223 q^{21} -6.59761 q^{22} +0.352460 q^{23} +6.66148 q^{24} -4.77607 q^{25} +15.8611 q^{26} +5.65382 q^{27} -16.4748 q^{28} +5.98074 q^{29} +1.65658 q^{30} -0.0270305 q^{31} +0.920923 q^{32} -3.91275 q^{33} +7.18276 q^{34} -1.99751 q^{35} -3.60575 q^{36} +7.65604 q^{37} -6.08857 q^{38} +9.40650 q^{39} -2.18775 q^{40} +6.98781 q^{41} -14.7773 q^{42} -9.53851 q^{43} +10.5984 q^{44} -0.437185 q^{45} -0.856332 q^{46} +6.18705 q^{47} -4.93748 q^{48} +10.8185 q^{49} +11.6039 q^{50} +4.25978 q^{51} -25.4792 q^{52} +8.86731 q^{53} -13.7364 q^{54} +1.28502 q^{55} +19.5155 q^{56} -3.61086 q^{57} -14.5307 q^{58} -7.83611 q^{59} -2.66113 q^{60} +9.95309 q^{61} +0.0656729 q^{62} +3.89984 q^{63} -9.09089 q^{64} -3.08926 q^{65} +9.50635 q^{66} +1.86101 q^{67} -11.5384 q^{68} -0.507852 q^{69} +4.85313 q^{70} -13.0373 q^{71} +4.27124 q^{72} +8.64784 q^{73} -18.6010 q^{74} +6.88174 q^{75} +9.78068 q^{76} -11.4628 q^{77} -22.8539 q^{78} +7.74098 q^{79} +1.62156 q^{80} -5.37485 q^{81} -16.9775 q^{82} +7.49076 q^{83} +23.7382 q^{84} -1.39899 q^{85} +23.1746 q^{86} -8.61751 q^{87} -12.5545 q^{88} -12.0652 q^{89} +1.06218 q^{90} +27.5573 q^{91} +1.37561 q^{92} +0.0389477 q^{93} -15.0320 q^{94} +1.18587 q^{95} -1.32694 q^{96} -6.42719 q^{97} -26.2845 q^{98} -2.50880 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 152 q - 22 q^{2} - 18 q^{3} + 138 q^{4} - 59 q^{5} - 17 q^{6} - 19 q^{7} - 66 q^{8} + 106 q^{9}+O(q^{10}) $ 152 * q - 22 * q^2 - 18 * q^3 + 138 * q^4 - 59 * q^5 - 17 * q^6 - 19 * q^7 - 66 * q^8 + 106 * q^9	\( 152 q - 22 q^{2} - 18 q^{3} + 138 q^{4} - 59 q^{5} - 17 q^{6} - 19 q^{7} - 66 q^{8} + 106 q^{9} - 15 q^{10} - 40 q^{11} - 53 q^{12} - 59 q^{13} - 36 q^{14} - 40 q^{15} + 118 q^{16} - 93 q^{17} - 59 q^{18} - 16 q^{19} - 108 q^{20} - 62 q^{21} - 37 q^{22} - 107 q^{23} - 31 q^{24} + 101 q^{25} - 64 q^{26} - 63 q^{27} - 53 q^{28} - 124 q^{29} - 68 q^{30} - 15 q^{31} - 129 q^{32} - 49 q^{33} - 76 q^{35} + 45 q^{36} - 98 q^{37} - 125 q^{38} - 47 q^{39} - 7 q^{40} - 56 q^{41} - 84 q^{42} - 62 q^{43} - 114 q^{44} - 142 q^{45} - 3 q^{46} - 111 q^{47} - 92 q^{48} + 117 q^{49} - 64 q^{50} - 21 q^{51} - 85 q^{52} - 347 q^{53} + 3 q^{54} - 16 q^{55} - 73 q^{56} - 115 q^{57} - 29 q^{58} - 50 q^{59} - 54 q^{60} - 62 q^{61} - 55 q^{62} - 70 q^{63} + 64 q^{64} - 147 q^{65} + 34 q^{66} - 86 q^{67} - 174 q^{68} - 104 q^{69} - 7 q^{70} - 86 q^{71} - 139 q^{72} - 27 q^{73} - 52 q^{74} - 49 q^{75} - 11 q^{76} - 346 q^{77} - 59 q^{78} - 17 q^{79} - 149 q^{80} - 8 q^{81} - 31 q^{82} - 106 q^{83} - 51 q^{84} - 69 q^{85} - 85 q^{86} - 32 q^{87} - 113 q^{88} - 59 q^{89} + 10 q^{90} - 9 q^{91} - 314 q^{92} - 230 q^{93} + 7 q^{94} - 74 q^{95} - 54 q^{96} - 60 q^{97} - 77 q^{98} - 96 q^{99}+O(q^{100}) \) 152 * q - 22 * q^2 - 18 * q^3 + 138 * q^4 - 59 * q^5 - 17 * q^6 - 19 * q^7 - 66 * q^8 + 106 * q^9 - 15 * q^10 - 40 * q^11 - 53 * q^12 - 59 * q^13 - 36 * q^14 - 40 * q^15 + 118 * q^16 - 93 * q^17 - 59 * q^18 - 16 * q^19 - 108 * q^20 - 62 * q^21 - 37 * q^22 - 107 * q^23 - 31 * q^24 + 101 * q^25 - 64 * q^26 - 63 * q^27 - 53 * q^28 - 124 * q^29 - 68 * q^30 - 15 * q^31 - 129 * q^32 - 49 * q^33 - 76 * q^35 + 45 * q^36 - 98 * q^37 - 125 * q^38 - 47 * q^39 - 7 * q^40 - 56 * q^41 - 84 * q^42 - 62 * q^43 - 114 * q^44 - 142 * q^45 - 3 * q^46 - 111 * q^47 - 92 * q^48 + 117 * q^49 - 64 * q^50 - 21 * q^51 - 85 * q^52 - 347 * q^53 + 3 * q^54 - 16 * q^55 - 73 * q^56 - 115 * q^57 - 29 * q^58 - 50 * q^59 - 54 * q^60 - 62 * q^61 - 55 * q^62 - 70 * q^63 + 64 * q^64 - 147 * q^65 + 34 * q^66 - 86 * q^67 - 174 * q^68 - 104 * q^69 - 7 * q^70 - 86 * q^71 - 139 * q^72 - 27 * q^73 - 52 * q^74 - 49 * q^75 - 11 * q^76 - 346 * q^77 - 59 * q^78 - 17 * q^79 - 149 * q^80 - 8 * q^81 - 31 * q^82 - 106 * q^83 - 51 * q^84 - 69 * q^85 - 85 * q^86 - 32 * q^87 - 113 * q^88 - 59 * q^89 + 10 * q^90 - 9 * q^91 - 314 * q^92 - 230 * q^93 + 7 * q^94 - 74 * q^95 - 54 * q^96 - 60 * q^97 - 77 * 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 $

$2$	−2.42958	−1.71798	−0.858988	−	0.511996i	$-0.828906\pi$
$2$	−2.42958	−1.71798	−0.858988	+	0.511996i	$0.828906\pi$
$3$	−1.44088	−0.831891	−0.415946	−	0.909389i	$-0.636549\pi$
$3$	−1.44088	−0.831891	−0.415946	+	0.909389i	$0.636549\pi$
$4$	3.90288	1.95144
$5$	0.473210	0.211626	0.105813	−	0.994386i	$-0.466255\pi$
$5$	0.473210	0.211626	0.105813	+	0.994386i	$0.466255\pi$
$6$	3.50074	1.42917
$7$	−4.22120	−1.59546	−0.797731	−	0.603013i	$-0.793967\pi$
$7$	−4.22120	−1.59546	−0.797731	+	0.603013i	$0.793967\pi$
$8$	−4.62321	−1.63455
$9$	−0.923870	−0.307957
$10$	−1.14970	−0.363568
$11$	2.71553	0.818763	0.409381	−	0.912363i	$-0.365745\pi$
$11$	2.71553	0.818763	0.409381	+	0.912363i	$0.365745\pi$
$12$	−5.62358	−1.62339
$13$	−6.52831	−1.81063	−0.905313	−	0.424744i	$-0.860364\pi$
$13$	−6.52831	−1.81063	−0.905313	+	0.424744i	$0.860364\pi$
$14$	10.2558	2.74097
$15$	−0.681838	−0.176050
$16$	3.42672	0.856679
$17$	−2.95637	−0.717026	−0.358513	−	0.933525i	$-0.616716\pi$
$17$	−2.95637	−0.717026	−0.358513	+	0.933525i	$0.616716\pi$
$18$	2.24462	0.529062
$19$	2.50601	0.574919	0.287460	−	0.957793i	$-0.407189\pi$
$19$	2.50601	0.574919	0.287460	+	0.957793i	$0.407189\pi$
$20$	1.84688	0.412976
$21$	6.08223	1.32725
$22$	−6.59761	−1.40661
$23$	0.352460	0.0734930	0.0367465	−	0.999325i	$-0.488301\pi$
$23$	0.352460	0.0734930	0.0367465	+	0.999325i	$0.488301\pi$
$24$	6.66148	1.35977
$25$	−4.77607	−0.955214
$26$	15.8611	3.11061
$27$	5.65382	1.08808
$28$	−16.4748	−3.11345
$29$	5.98074	1.11060	0.555298	−	0.831652i	$-0.312604\pi$
$29$	5.98074	1.11060	0.555298	+	0.831652i	$0.312604\pi$
$30$	1.65658	0.302449
$31$	−0.0270305	−0.00485483	−0.00242741	−	0.999997i	$-0.500773\pi$
$31$	−0.0270305	−0.00485483	−0.00242741	+	0.999997i	$0.500773\pi$
$32$	0.920923	0.162798
$33$	−3.91275	−0.681122
$34$	7.18276	1.23183
$35$	−1.99751	−0.337641
$36$	−3.60575	−0.600959
$37$	7.65604	1.25865	0.629323	−	0.777144i	$-0.283332\pi$
$37$	7.65604	1.25865	0.629323	+	0.777144i	$0.283332\pi$
$38$	−6.08857	−0.987697
$39$	9.40650	1.50625
$40$	−2.18775	−0.345914
$41$	6.98781	1.09131	0.545656	−	0.838009i	$-0.316281\pi$
$41$	6.98781	1.09131	0.545656	+	0.838009i	$0.316281\pi$
$42$	−14.7773	−2.28019
$43$	−9.53851	−1.45461	−0.727304	−	0.686315i	$-0.759227\pi$
$43$	−9.53851	−1.45461	−0.727304	+	0.686315i	$0.759227\pi$
$44$	10.5984	1.59777
$45$	−0.437185	−0.0651717
$46$	−0.856332	−0.126259
$47$	6.18705	0.902473	0.451237	−	0.892404i	$-0.350983\pi$
$47$	6.18705	0.902473	0.451237	+	0.892404i	$0.350983\pi$
$48$	−4.93748	−0.712664
$49$	10.8185	1.54550
$50$	11.6039	1.64104
$51$	4.25978	0.596488
$52$	−25.4792	−3.53333
$53$	8.86731	1.21802	0.609010	−	0.793163i	$-0.291567\pi$
$53$	8.86731	1.21802	0.609010	+	0.793163i	$0.291567\pi$
$54$	−13.7364	−1.86929
$55$	1.28502	0.173272
$56$	19.5155	2.60787
$57$	−3.61086	−0.478270
$58$	−14.5307	−1.90798
$59$	−7.83611	−1.02017	−0.510087	−	0.860123i	$-0.670387\pi$
$59$	−7.83611	−1.02017	−0.510087	+	0.860123i	$0.670387\pi$
$60$	−2.66113	−0.343551
$61$	9.95309	1.27436	0.637181	−	0.770714i	$-0.280100\pi$
$61$	9.95309	1.27436	0.637181	+	0.770714i	$0.280100\pi$
$62$	0.0656729	0.00834047
$63$	3.89984	0.491333
$64$	−9.09089	−1.13636
$65$	−3.08926	−0.383176
$66$	9.50635	1.17015
$67$	1.86101	0.227359	0.113679	−	0.993517i	$-0.463736\pi$
$67$	1.86101	0.227359	0.113679	+	0.993517i	$0.463736\pi$
$68$	−11.5384	−1.39923
$69$	−0.507852	−0.0611382
$70$	4.85313	0.580060
$71$	−13.0373	−1.54724	−0.773619	−	0.633651i	$-0.781556\pi$
$71$	−13.0373	−1.54724	−0.773619	+	0.633651i	$0.781556\pi$
$72$	4.27124	0.503371
$73$	8.64784	1.01215	0.506077	−	0.862488i	$-0.331095\pi$
$73$	8.64784	1.01215	0.506077	+	0.862488i	$0.331095\pi$
$74$	−18.6010	−2.16232
$75$	6.88174	0.794635
$76$	9.78068	1.12192
$77$	−11.4628	−1.30631
$78$	−22.8539	−2.58769
$79$	7.74098	0.870929	0.435464	−	0.900206i	$-0.356584\pi$
$79$	7.74098	0.870929	0.435464	+	0.900206i	$0.356584\pi$
$80$	1.62156	0.181296
$81$	−5.37485	−0.597206
$82$	−16.9775	−1.87485
$83$	7.49076	0.822218	0.411109	−	0.911586i	$-0.365142\pi$
$83$	7.49076	0.822218	0.411109	+	0.911586i	$0.365142\pi$
$84$	23.7382	2.59005
$85$	−1.39899	−0.151741
$86$	23.1746	2.49898
$87$	−8.61751	−0.923895
$88$	−12.5545	−1.33831
$89$	−12.0652	−1.27891	−0.639455	−	0.768829i	$-0.720840\pi$
$89$	−12.0652	−1.27891	−0.639455	+	0.768829i	$0.720840\pi$
$90$	1.06218	0.111963
$91$	27.5573	2.88879
$92$	1.37561	0.143417
$93$	0.0389477	0.00403869
$94$	−15.0320	−1.55043
$95$	1.18587	0.121668
$96$	−1.32694	−0.135430
$97$	−6.42719	−0.652582	−0.326291	−	0.945269i	$-0.605799\pi$
$97$	−6.42719	−0.652582	−0.326291	+	0.945269i	$0.605799\pi$
$98$	−26.2845	−2.65513
$99$	−2.50880	−0.252143
$100$	−18.6404	−1.86404
$101$	6.16354	0.613295	0.306648	−	0.951823i	$-0.400793\pi$
$101$	6.16354	0.613295	0.306648	+	0.951823i	$0.400793\pi$
$102$	−10.3495	−1.02475
$103$	−4.37926	−0.431501	−0.215750	−	0.976449i	$-0.569220\pi$
$103$	−4.37926	−0.431501	−0.215750	+	0.976449i	$0.569220\pi$
$104$	30.1817	2.95956
$105$	2.87817	0.280881
$106$	−21.5439	−2.09253
$107$	1.17057	0.113163	0.0565816	−	0.998398i	$-0.481980\pi$
$107$	1.17057	0.113163	0.0565816	+	0.998398i	$0.481980\pi$
$108$	22.0662	2.12332
$109$	−1.72583	−0.165304	−0.0826521	−	0.996578i	$-0.526339\pi$
$109$	−1.72583	−0.165304	−0.0826521	+	0.996578i	$0.526339\pi$
$110$	−3.12206	−0.297676
$111$	−11.0314	−1.04706
$112$	−14.4648	−1.36680
$113$	−6.87121	−0.646389	−0.323194	−	0.946333i	$-0.604757\pi$
$113$	−6.87121	−0.646389	−0.323194	+	0.946333i	$0.604757\pi$
$114$	8.77289	0.821657
$115$	0.166788	0.0155530
$116$	23.3421	2.16726
$117$	6.03131	0.557595
$118$	19.0385	1.75264
$119$	12.4794	1.14399
$120$	3.15228	0.287763
$121$	−3.62590	−0.329627
$122$	−24.1819	−2.18932
$123$	−10.0686	−0.907853
$124$	−0.105497	−0.00947390
$125$	−4.62614	−0.413774
$126$	−9.47499	−0.844099
$127$	−17.7077	−1.57131	−0.785654	−	0.618667i	$-0.787673\pi$
$127$	−17.7077	−1.57131	−0.785654	+	0.618667i	$0.787673\pi$
$128$	20.2452	1.78944
$129$	13.7438	1.21008
$130$	7.50562	0.658287
$131$	5.88021	0.513756	0.256878	−	0.966444i	$-0.417306\pi$
$131$	5.88021	0.513756	0.256878	+	0.966444i	$0.417306\pi$
$132$	−15.2710	−1.32917
$133$	−10.5784	−0.917262
$134$	−4.52149	−0.390597
$135$	2.67545	0.230266
$136$	13.6679	1.17202
$137$	−14.8661	−1.27009	−0.635047	−	0.772474i	$-0.719019\pi$
$137$	−14.8661	−1.27009	−0.635047	+	0.772474i	$0.719019\pi$
$138$	1.23387	0.105034
$139$	11.1701	0.947434	0.473717	−	0.880677i	$-0.342912\pi$
$139$	11.1701	0.947434	0.473717	+	0.880677i	$0.342912\pi$
$140$	−7.79606	−0.658887
$141$	−8.91478	−0.750760
$142$	31.6751	2.65812
$143$	−17.7278	−1.48247
$144$	−3.16584	−0.263820
$145$	2.83015	0.235031
$146$	−21.0107	−1.73886
$147$	−15.5882	−1.28569
$148$	29.8806	2.45617
$149$	−7.96345	−0.652392	−0.326196	−	0.945302i	$-0.605767\pi$
$149$	−7.96345	−0.652392	−0.326196	+	0.945302i	$0.605767\pi$
$150$	−16.7198	−1.36516
$151$	15.5499	1.26543	0.632716	−	0.774384i	$-0.281940\pi$
$151$	15.5499	1.26543	0.632716	+	0.774384i	$0.281940\pi$
$152$	−11.5858	−0.939735
$153$	2.73131	0.220813
$154$	27.8498	2.24420
$155$	−0.0127911	−0.00102741
$156$	36.7124	2.93935
$157$	19.9856	1.59502	0.797510	−	0.603305i	$-0.206150\pi$
$157$	19.9856	1.59502	0.797510	+	0.603305i	$0.206150\pi$
$158$	−18.8074	−1.49623
$159$	−12.7767	−1.01326
$160$	0.435790	0.0344522
$161$	−1.48780	−0.117255
$162$	13.0587	1.02599
$163$	14.7854	1.15808	0.579039	−	0.815300i	$-0.303428\pi$
$163$	14.7854	1.15808	0.579039	+	0.815300i	$0.303428\pi$
$164$	27.2726	2.12963
$165$	−1.85155	−0.144143
$166$	−18.1994	−1.41255
$167$	−9.67370	−0.748573	−0.374287	−	0.927313i	$-0.622112\pi$
$167$	−9.67370	−0.748573	−0.374287	+	0.927313i	$0.622112\pi$
$168$	−28.1194	−2.16946
$169$	29.6188	2.27837
$170$	3.39896	0.260688
$171$	−2.31523	−0.177050
$172$	−37.2277	−2.83858
$173$	10.1770	0.773742	0.386871	−	0.922134i	$-0.373556\pi$
$173$	10.1770	0.773742	0.386871	+	0.922134i	$0.373556\pi$
$174$	20.9370	1.58723
$175$	20.1607	1.52401
$176$	9.30535	0.701417
$177$	11.2909	0.848675
$178$	29.3134	2.19714
$179$	22.8065	1.70463	0.852317	−	0.523025i	$-0.175197\pi$
$179$	22.8065	1.70463	0.852317	+	0.523025i	$0.175197\pi$
$180$	−1.70628	−0.127179
$181$	−12.7603	−0.948468	−0.474234	−	0.880399i	$-0.657275\pi$
$181$	−12.7603	−0.948468	−0.474234	+	0.880399i	$0.657275\pi$
$182$	−66.9527	−4.96287
$183$	−14.3412	−1.06013
$184$	−1.62950	−0.120128
$185$	3.62292	0.266362
$186$	−0.0946267	−0.00693837
$187$	−8.02812	−0.587074
$188$	24.1473	1.76112
$189$	−23.8659	−1.73599
$190$	−2.88118	−0.209022
$191$	−5.83263	−0.422035	−0.211017	−	0.977482i	$-0.567678\pi$
$191$	−5.83263	−0.422035	−0.211017	+	0.977482i	$0.567678\pi$
$192$	13.0989	0.945329
$193$	−1.90949	−0.137448	−0.0687239	−	0.997636i	$-0.521893\pi$
$193$	−1.90949	−0.137448	−0.0687239	+	0.997636i	$0.521893\pi$
$194$	15.6154	1.12112
$195$	4.45125	0.318761
$196$	42.2234	3.01595
$197$	−17.6728	−1.25914	−0.629568	−	0.776945i	$-0.716768\pi$
$197$	−17.6728	−1.25914	−0.629568	+	0.776945i	$0.716768\pi$
$198$	6.09533	0.433176
$199$	11.7783	0.834944	0.417472	−	0.908690i	$-0.362916\pi$
$199$	11.7783	0.834944	0.417472	+	0.908690i	$0.362916\pi$
$200$	22.0808	1.56135
$201$	−2.68149	−0.189138
$202$	−14.9748	−1.05363
$203$	−25.2459	−1.77191
$204$	16.6254	1.16401
$205$	3.30670	0.230950
$206$	10.6398	0.741308
$207$	−0.325627	−0.0226327
$208$	−22.3707	−1.55113
$209$	6.80516	0.470722
$210$	−6.99277	−0.482547
$211$	14.2399	0.980317	0.490159	−	0.871633i	$-0.336939\pi$
$211$	14.2399	0.980317	0.490159	+	0.871633i	$0.336939\pi$
$212$	34.6081	2.37689
$213$	18.7851	1.28713
$214$	−2.84400	−0.194412
$215$	−4.51372	−0.307833
$216$	−26.1388	−1.77852
$217$	0.114101	0.00774569
$218$	4.19304	0.283989
$219$	−12.4605	−0.842002
$220$	5.01527	0.338129
$221$	19.3001	1.29827
$222$	26.8018	1.79882
$223$	6.97758	0.467253	0.233627	−	0.972326i	$-0.424941\pi$
$223$	6.97758	0.467253	0.233627	+	0.972326i	$0.424941\pi$
$224$	−3.88740	−0.259738
$225$	4.41247	0.294165
$226$	16.6942	1.11048
$227$	−2.30847	−0.153219	−0.0766093	−	0.997061i	$-0.524409\pi$
$227$	−2.30847	−0.153219	−0.0766093	+	0.997061i	$0.524409\pi$
$228$	−14.0928	−0.933316
$229$	−9.93102	−0.656260	−0.328130	−	0.944633i	$-0.606418\pi$
$229$	−9.93102	−0.656260	−0.328130	+	0.944633i	$0.606418\pi$
$230$	−0.405225	−0.0267197
$231$	16.5165	1.08670
$232$	−27.6502	−1.81532
$233$	3.82809	0.250786	0.125393	−	0.992107i	$-0.459981\pi$
$233$	3.82809	0.250786	0.125393	+	0.992107i	$0.459981\pi$
$234$	−14.6536	−0.957934
$235$	2.92777	0.190987
$236$	−30.5834	−1.99081
$237$	−11.1538	−0.724518
$238$	−30.3199	−1.96534
$239$	29.2213	1.89017	0.945083	−	0.326830i	$-0.105981\pi$
$239$	29.2213	1.89017	0.945083	+	0.326830i	$0.105981\pi$
$240$	−2.33647	−0.150818
$241$	−30.0754	−1.93733	−0.968663	−	0.248377i	$-0.920103\pi$
$241$	−30.0754	−1.93733	−0.968663	+	0.248377i	$0.920103\pi$
$242$	8.80943	0.566292
$243$	−9.21695	−0.591267
$244$	38.8457	2.48684
$245$	5.11943	0.327068
$246$	24.4625	1.55967
$247$	−16.3600	−1.04096
$248$	0.124968	0.00793546
$249$	−10.7933	−0.683996
$250$	11.2396	0.710854
$251$	−0.407155	−0.0256994	−0.0128497	−	0.999917i	$-0.504090\pi$
$251$	−0.407155	−0.0256994	−0.0128497	+	0.999917i	$0.504090\pi$
$252$	15.2206	0.958808
$253$	0.957116	0.0601734
$254$	43.0224	2.69947
$255$	2.01577	0.126232
$256$	−31.0057	−1.93786
$257$	−14.6781	−0.915594	−0.457797	−	0.889057i	$-0.651361\pi$
$257$	−14.6781	−0.915594	−0.457797	+	0.889057i	$0.651361\pi$
$258$	−33.3918	−2.07888
$259$	−32.3177	−2.00812
$260$	−12.0570	−0.747745
$261$	−5.52542	−0.342015
$262$	−14.2865	−0.882621
$263$	−3.42094	−0.210944	−0.105472	−	0.994422i	$-0.533635\pi$
$263$	−3.42094	−0.210944	−0.105472	+	0.994422i	$0.533635\pi$
$264$	18.0894	1.11333
$265$	4.19610	0.257765
$266$	25.7011	1.57583
$267$	17.3845	1.06391
$268$	7.26331	0.443677
$269$	−14.3900	−0.877372	−0.438686	−	0.898640i	$-0.644556\pi$
$269$	−14.3900	−0.877372	−0.438686	+	0.898640i	$0.644556\pi$
$270$	−6.50022	−0.395591
$271$	−4.07265	−0.247396	−0.123698	−	0.992320i	$-0.539475\pi$
$271$	−4.07265	−0.247396	−0.123698	+	0.992320i	$0.539475\pi$
$272$	−10.1307	−0.614261
$273$	−39.7067	−2.40316
$274$	36.1184	2.18199
$275$	−12.9696	−0.782094
$276$	−1.98209	−0.119308
$277$	24.0712	1.44630	0.723148	−	0.690693i	$-0.242695\pi$
$277$	24.0712	1.44630	0.723148	+	0.690693i	$0.242695\pi$
$278$	−27.1387	−1.62767
$279$	0.0249727	0.00149508
$280$	9.23493	0.551892
$281$	7.83799	0.467576	0.233788	−	0.972288i	$-0.424888\pi$
$281$	7.83799	0.467576	0.233788	+	0.972288i	$0.424888\pi$
$282$	21.6592	1.28979
$283$	−24.1881	−1.43784	−0.718918	−	0.695095i	$-0.755362\pi$
$283$	−24.1881	−1.43784	−0.718918	+	0.695095i	$0.755362\pi$
$284$	−50.8829	−3.01934
$285$	−1.70870	−0.101214
$286$	43.0712	2.54685
$287$	−29.4969	−1.74115
$288$	−0.850813	−0.0501346
$289$	−8.25985	−0.485874
$290$	−6.87608	−0.403777
$291$	9.26080	0.542877
$292$	33.7515	1.97516
$293$	14.7265	0.860329	0.430165	−	0.902751i	$-0.358456\pi$
$293$	14.7265	0.860329	0.430165	+	0.902751i	$0.358456\pi$
$294$	37.8727	2.20878
$295$	−3.70813	−0.215896
$296$	−35.3955	−2.05732
$297$	15.3531	0.890878
$298$	19.3479	1.12079
$299$	−2.30097	−0.133068
$300$	26.8586	1.55068
$301$	40.2639	2.32077
$302$	−37.7798	−2.17398
$303$	−8.88091	−0.510195
$304$	8.58740	0.492521
$305$	4.70990	0.269688
$306$	−6.63594	−0.379351
$307$	17.6980	1.01008	0.505040	−	0.863096i	$-0.331478\pi$
$307$	17.6980	1.01008	0.505040	+	0.863096i	$0.331478\pi$
$308$	−44.7379	−2.54918
$309$	6.30997	0.358962
$310$	0.0310771	0.00176506
$311$	−3.34765	−0.189828	−0.0949139	−	0.995485i	$-0.530258\pi$
$311$	−3.34765	−0.189828	−0.0949139	+	0.995485i	$0.530258\pi$
$312$	−43.4882	−2.46203
$313$	4.56242	0.257883	0.128942	−	0.991652i	$-0.458842\pi$
$313$	4.56242	0.257883	0.128942	+	0.991652i	$0.458842\pi$
$314$	−48.5566	−2.74021
$315$	1.84544	0.103979
$316$	30.2121	1.69957
$317$	−2.62279	−0.147310	−0.0736552	−	0.997284i	$-0.523466\pi$
$317$	−2.62279	−0.147310	−0.0736552	+	0.997284i	$0.523466\pi$
$318$	31.0421	1.74076
$319$	16.2409	0.909314
$320$	−4.30190	−0.240484
$321$	−1.68665	−0.0941396
$322$	3.61475	0.201442
$323$	−7.40872	−0.412232
$324$	−20.9774	−1.16541
$325$	31.1797	1.72954
$326$	−35.9223	−1.98955
$327$	2.48671	0.137515
$328$	−32.3061	−1.78381
$329$	−26.1167	−1.43986
$330$	4.49850	0.247634
$331$	7.11368	0.391003	0.195502	−	0.980703i	$-0.437367\pi$
$331$	7.11368	0.391003	0.195502	+	0.980703i	$0.437367\pi$
$332$	29.2355	1.60451
$333$	−7.07319	−0.387608
$334$	23.5031	1.28603
$335$	0.880651	0.0481151
$336$	20.8421	1.13703
$337$	−4.64050	−0.252784	−0.126392	−	0.991980i	$-0.540340\pi$
$337$	−4.64050	−0.252784	−0.126392	+	0.991980i	$0.540340\pi$
$338$	−71.9614	−3.91418
$339$	9.90057	0.537725
$340$	−5.46008	−0.296114
$341$	−0.0734022	−0.00397495
$342$	5.62505	0.304168
$343$	−16.1187	−0.870327
$344$	44.0985	2.37763
$345$	−0.240321	−0.0129384
$346$	−24.7259	−1.32927
$347$	8.86641	0.475974	0.237987	−	0.971268i	$-0.423512\pi$
$347$	8.86641	0.475974	0.237987	+	0.971268i	$0.423512\pi$
$348$	−33.6331	−1.80292
$349$	−17.0538	−0.912867	−0.456434	−	0.889757i	$-0.650873\pi$
$349$	−17.0538	−0.912867	−0.456434	+	0.889757i	$0.650873\pi$
$350$	−48.9822	−2.61821
$351$	−36.9099	−1.97010
$352$	2.50079	0.133293
$353$	−7.62697	−0.405943	−0.202971	−	0.979185i	$-0.565060\pi$
$353$	−7.62697	−0.405943	−0.202971	+	0.979185i	$0.565060\pi$
$354$	−27.4321	−1.45800
$355$	−6.16937	−0.327436
$356$	−47.0891	−2.49572
$357$	−17.9814	−0.951674
$358$	−55.4102	−2.92852
$359$	−2.68468	−0.141692	−0.0708460	−	0.997487i	$-0.522570\pi$
$359$	−2.68468	−0.141692	−0.0708460	+	0.997487i	$0.522570\pi$
$360$	2.02120	0.106526
$361$	−12.7199	−0.669468
$362$	31.0023	1.62944
$363$	5.22448	0.274214
$364$	107.553	5.63730
$365$	4.09225	0.214198
$366$	34.8431	1.82128
$367$	−5.68366	−0.296685	−0.148342	−	0.988936i	$-0.547394\pi$
$367$	−5.68366	−0.296685	−0.148342	+	0.988936i	$0.547394\pi$
$368$	1.20778	0.0629599
$369$	−6.45582	−0.336077
$370$	−8.80219	−0.457604
$371$	−37.4307	−1.94330
$372$	0.152008	0.00788126
$373$	−9.19869	−0.476290	−0.238145	−	0.971230i	$-0.576539\pi$
$373$	−9.19869	−0.476290	−0.238145	+	0.971230i	$0.576539\pi$
$374$	19.5050	1.00858
$375$	6.66570	0.344215
$376$	−28.6040	−1.47514
$377$	−39.0441	−2.01087
$378$	57.9842	2.98238
$379$	−33.5670	−1.72422	−0.862110	−	0.506722i	$-0.830857\pi$
$379$	−33.5670	−1.72422	−0.862110	+	0.506722i	$0.830857\pi$
$380$	4.62832	0.237428
$381$	25.5147	1.30716
$382$	14.1709	0.725045
$383$	−14.6642	−0.749304	−0.374652	−	0.927166i	$-0.622238\pi$
$383$	−14.6642	−0.749304	−0.374652	+	0.927166i	$0.622238\pi$
$384$	−29.1709	−1.48862
$385$	−5.42431	−0.276448
$386$	4.63926	0.236132
$387$	8.81234	0.447956
$388$	−25.0845	−1.27347
$389$	−15.3825	−0.779924	−0.389962	−	0.920831i	$-0.627512\pi$
$389$	−15.3825	−0.779924	−0.389962	+	0.920831i	$0.627512\pi$
$390$	−10.8147	−0.547623
$391$	−1.04200	−0.0526964
$392$	−50.0162	−2.52620
$393$	−8.47266	−0.427389
$394$	42.9376	2.16317
$395$	3.66311	0.184311
$396$	−9.79153	−0.492043
$397$	19.0189	0.954532	0.477266	−	0.878759i	$-0.341628\pi$
$397$	19.0189	0.954532	0.477266	+	0.878759i	$0.341628\pi$
$398$	−28.6164	−1.43441
$399$	15.2422	0.763062
$400$	−16.3662	−0.818312
$401$	−12.4188	−0.620166	−0.310083	−	0.950709i	$-0.600357\pi$
$401$	−12.4188	−0.620166	−0.310083	+	0.950709i	$0.600357\pi$
$402$	6.51492	0.324934
$403$	0.176464	0.00879028
$404$	24.0556	1.19681
$405$	−2.54344	−0.126384
$406$	61.3370	3.04410
$407$	20.7902	1.03053
$408$	−19.6938	−0.974990
$409$	−30.0625	−1.48650	−0.743248	−	0.669016i	$-0.766716\pi$
$409$	−30.0625	−1.48650	−0.743248	+	0.669016i	$0.766716\pi$
$410$	−8.03391	−0.396767
$411$	21.4202	1.05658
$412$	−17.0917	−0.842048
$413$	33.0778	1.62765
$414$	0.791139	0.0388824
$415$	3.54470	0.174003
$416$	−6.01207	−0.294766
$417$	−16.0947	−0.788162
$418$	−16.5337	−0.808690
$419$	8.09742	0.395585	0.197793	−	0.980244i	$-0.436623\pi$
$419$	8.09742	0.395585	0.197793	+	0.980244i	$0.436623\pi$
$420$	11.2332	0.548123
$421$	9.13851	0.445384	0.222692	−	0.974889i	$-0.428516\pi$
$421$	9.13851	0.445384	0.222692	+	0.974889i	$0.428516\pi$
$422$	−34.5971	−1.68416
$423$	−5.71603	−0.277923
$424$	−40.9954	−1.99091
$425$	14.1199	0.684914
$426$	−45.6400	−2.21127
$427$	−42.0139	−2.03320
$428$	4.56859	0.220831
$429$	25.5436	1.23326
$430$	10.9665	0.528850
$431$	−0.568045	−0.0273618	−0.0136809	−	0.999906i	$-0.504355\pi$
$431$	−0.568045	−0.0273618	−0.0136809	+	0.999906i	$0.504355\pi$
$432$	19.3740	0.932133
$433$	−18.8326	−0.905035	−0.452517	−	0.891756i	$-0.649474\pi$
$433$	−18.8326	−0.905035	−0.452517	+	0.891756i	$0.649474\pi$
$434$	−0.277218	−0.0133069
$435$	−4.07790	−0.195520
$436$	−6.73570	−0.322581
$437$	0.883270	0.0422525
$438$	30.2738	1.44654
$439$	0.531438	0.0253642	0.0126821	−	0.999920i	$-0.495963\pi$
$439$	0.531438	0.0253642	0.0126821	+	0.999920i	$0.495963\pi$
$440$	−5.94090	−0.283221
$441$	−9.99490	−0.475947
$442$	−46.8913	−2.23039
$443$	−27.7046	−1.31628	−0.658141	−	0.752894i	$-0.728657\pi$
$443$	−27.7046	−1.31628	−0.658141	+	0.752894i	$0.728657\pi$
$444$	−43.0543	−2.04327
$445$	−5.70938	−0.270651
$446$	−16.9526	−0.802730
$447$	11.4744	0.542719
$448$	38.3745	1.81302
$449$	−34.8489	−1.64462	−0.822310	−	0.569040i	$-0.807315\pi$
$449$	−34.8489	−1.64462	−0.822310	+	0.569040i	$0.807315\pi$
$450$	−10.7205	−0.505368
$451$	18.9756	0.893526
$452$	−26.8175	−1.26139
$453$	−22.4055	−1.05270
$454$	5.60863	0.263226
$455$	13.0404	0.611343
$456$	16.6938	0.781757
$457$	26.6961	1.24879	0.624395	−	0.781109i	$-0.285346\pi$
$457$	26.6961	1.24879	0.624395	+	0.781109i	$0.285346\pi$
$458$	24.1282	1.12744
$459$	−16.7148	−0.780180
$460$	0.650953	0.0303508
$461$	−3.83211	−0.178479	−0.0892397	−	0.996010i	$-0.528444\pi$
$461$	−3.83211	−0.178479	−0.0892397	+	0.996010i	$0.528444\pi$
$462$	−40.1282	−1.86693
$463$	−1.51711	−0.0705061	−0.0352530	−	0.999378i	$-0.511224\pi$
$463$	−1.51711	−0.0705061	−0.0352530	+	0.999378i	$0.511224\pi$
$464$	20.4943	0.951423
$465$	0.0184304	0.000854692 0
$466$	−9.30067	−0.430845
$467$	−21.6294	−1.00089	−0.500445	−	0.865768i	$-0.666830\pi$
$467$	−21.6294	−1.00089	−0.500445	+	0.865768i	$0.666830\pi$
$468$	23.5395	1.08811
$469$	−7.85571	−0.362743
$470$	−7.11327	−0.328111
$471$	−28.7967	−1.32688
$472$	36.2280	1.66753
$473$	−25.9021	−1.19098
$474$	27.0991	1.24470
$475$	−11.9689	−0.549171
$476$	48.7058	2.23243
$477$	−8.19224	−0.375097
$478$	−70.9955	−3.24726
$479$	−34.0595	−1.55622	−0.778109	−	0.628129i	$-0.783821\pi$
$479$	−34.0595	−1.55622	−0.778109	+	0.628129i	$0.783821\pi$
$480$	−0.627921	−0.0286605
$481$	−49.9810	−2.27894
$482$	73.0707	3.32828
$483$	2.14374	0.0975438
$484$	−14.1515	−0.643248
$485$	−3.04141	−0.138103
$486$	22.3933	1.01578
$487$	−27.4575	−1.24422	−0.622108	−	0.782931i	$-0.713724\pi$
$487$	−27.4575	−1.24422	−0.622108	+	0.782931i	$0.713724\pi$
$488$	−46.0152	−2.08301
$489$	−21.3039	−0.963395
$490$	−12.4381	−0.561895
$491$	27.8544	1.25705	0.628525	−	0.777789i	$-0.283659\pi$
$491$	27.8544	1.25705	0.628525	+	0.777789i	$0.283659\pi$
$492$	−39.2965	−1.77162
$493$	−17.6813	−0.796326
$494$	39.7481	1.78835
$495$	−1.18719	−0.0533601
$496$	−0.0926259	−0.00415903
$497$	55.0329	2.46856
$498$	26.2232	1.17509
$499$	0.535608	0.0239771	0.0119886	−	0.999928i	$-0.496184\pi$
$499$	0.535608	0.0239771	0.0119886	+	0.999928i	$0.496184\pi$
$500$	−18.0553	−0.807456
$501$	13.9386	0.622732
$502$	0.989216	0.0441509
$503$	8.06094	0.359419	0.179710	−	0.983720i	$-0.442484\pi$
$503$	8.06094	0.359419	0.179710	+	0.983720i	$0.442484\pi$
$504$	−18.0298	−0.803110
$505$	2.91665	0.129789
$506$	−2.32539	−0.103376
$507$	−42.6771	−1.89536
$508$	−69.1112	−3.06631
$509$	−2.22821	−0.0987636	−0.0493818	−	0.998780i	$-0.515725\pi$
$509$	−2.22821	−0.0987636	−0.0493818	+	0.998780i	$0.515725\pi$
$510$	−4.89748	−0.216864
$511$	−36.5043	−1.61485
$512$	34.8406	1.53975
$513$	14.1686	0.625557
$514$	35.6617	1.57297
$515$	−2.07231	−0.0913168
$516$	53.6405	2.36139
$517$	16.8011	0.738912
$518$	78.5185	3.44991
$519$	−14.6638	−0.643670
$520$	14.2823	0.626321
$521$	−4.16539	−0.182489	−0.0912445	−	0.995829i	$-0.529084\pi$
$521$	−4.16539	−0.182489	−0.0912445	+	0.995829i	$0.529084\pi$
$522$	13.4245	0.587574
$523$	−10.4545	−0.457145	−0.228573	−	0.973527i	$-0.573406\pi$
$523$	−10.4545	−0.457145	−0.228573	+	0.973527i	$0.573406\pi$
$524$	22.9498	1.00256
$525$	−29.0492	−1.26781
$526$	8.31147	0.362397
$527$	0.0799123	0.00348104
$528$	−13.4079	−0.583503
$529$	−22.8758	−0.994599
$530$	−10.1948	−0.442833
$531$	7.23955	0.314170
$532$	−41.2862	−1.78998
$533$	−45.6185	−1.97596
$534$	−42.2371	−1.82778
$535$	0.553926	0.0239483
$536$	−8.60385	−0.371630
$537$	−32.8613	−1.41807
$538$	34.9617	1.50730
$539$	29.3780	1.26540
$540$	10.4419	0.449350
$541$	−40.8896	−1.75798	−0.878992	−	0.476837i	$-0.841783\pi$
$541$	−40.8896	−1.75798	−0.878992	+	0.476837i	$0.841783\pi$
$542$	9.89484	0.425020
$543$	18.3861	0.789022
$544$	−2.72259	−0.116730
$545$	−0.816679	−0.0349827
$546$	96.4707	4.12857
$547$	25.5833	1.09386	0.546931	−	0.837178i	$-0.315796\pi$
$547$	25.5833	1.09386	0.546931	+	0.837178i	$0.315796\pi$
$548$	−58.0205	−2.47851
$549$	−9.19536	−0.392448
$550$	31.5107	1.34362
$551$	14.9878	0.638502
$552$	2.34791	0.0999336
$553$	−32.6762	−1.38953
$554$	−58.4829	−2.48470
$555$	−5.22018	−0.221584
$556$	43.5955	1.84886
$557$	−16.6364	−0.704905	−0.352453	−	0.935830i	$-0.614652\pi$
$557$	−16.6364	−0.704905	−0.352453	+	0.935830i	$0.614652\pi$
$558$	−0.0606733	−0.00256850
$559$	62.2703	2.63375
$560$	−6.84491	−0.289250
$561$	11.5675	0.488382
$562$	−19.0431	−0.803283
$563$	13.0087	0.548250	0.274125	−	0.961694i	$-0.411612\pi$
$563$	13.0087	0.548250	0.274125	+	0.961694i	$0.411612\pi$
$564$	−34.7933	−1.46506
$565$	−3.25152	−0.136793
$566$	58.7671	2.47017
$567$	22.6883	0.952820
$568$	60.2740	2.52904
$569$	16.8064	0.704563	0.352281	−	0.935894i	$-0.385406\pi$
$569$	16.8064	0.704563	0.352281	+	0.935894i	$0.385406\pi$
$570$	4.15142	0.173884
$571$	−0.106134	−0.00444158	−0.00222079	−	0.999998i	$-0.500707\pi$
$571$	−0.106134	−0.00444158	−0.00222079	+	0.999998i	$0.500707\pi$
$572$	−69.1895	−2.89296
$573$	8.40412	0.351087
$574$	71.6652	2.99125
$575$	−1.68338	−0.0702016
$576$	8.39880	0.349950
$577$	32.3438	1.34649	0.673244	−	0.739420i	$-0.264900\pi$
$577$	32.3438	1.34649	0.673244	+	0.739420i	$0.264900\pi$
$578$	20.0680	0.834719
$579$	2.75134	0.114342
$580$	11.0457	0.458649
$581$	−31.6200	−1.31182
$582$	−22.4999	−0.932650
$583$	24.0794	0.997269
$584$	−39.9808	−1.65442
$585$	2.85408	0.118002
$586$	−35.7792	−1.47802
$587$	35.2976	1.45689	0.728444	−	0.685106i	$-0.240244\pi$
$587$	35.2976	1.45689	0.728444	+	0.685106i	$0.240244\pi$
$588$	−60.8387	−2.50895
$589$	−0.0677389	−0.00279113
$590$	9.00921	0.370903
$591$	25.4644	1.04746
$592$	26.2351	1.07826
$593$	27.6475	1.13535	0.567674	−	0.823254i	$-0.307844\pi$
$593$	27.6475	1.13535	0.567674	+	0.823254i	$0.307844\pi$
$594$	−37.3017	−1.53051
$595$	5.90540	0.242098
$596$	−31.0804	−1.27310
$597$	−16.9711	−0.694582
$598$	5.59040	0.228608
$599$	5.45353	0.222825	0.111413	−	0.993774i	$-0.464462\pi$
$599$	5.45353	0.222825	0.111413	+	0.993774i	$0.464462\pi$
$600$	−31.8157	−1.29887
$601$	−29.1888	−1.19063	−0.595317	−	0.803491i	$-0.702974\pi$
$601$	−29.1888	−1.19063	−0.595317	+	0.803491i	$0.702974\pi$
$602$	−97.8246	−3.98703
$603$	−1.71933	−0.0700167
$604$	60.6894	2.46942
$605$	−1.71581	−0.0697577
$606$	21.5769	0.876503
$607$	24.0021	0.974216	0.487108	−	0.873342i	$-0.338052\pi$
$607$	24.0021	0.974216	0.487108	+	0.873342i	$0.338052\pi$
$608$	2.30785	0.0935955
$609$	36.3762	1.47404
$610$	−11.4431	−0.463318
$611$	−40.3909	−1.63404
$612$	10.6600	0.430903
$613$	−1.10290	−0.0445458	−0.0222729	−	0.999752i	$-0.507090\pi$
$613$	−1.10290	−0.0445458	−0.0222729	+	0.999752i	$0.507090\pi$
$614$	−42.9989	−1.73529
$615$	−4.76455	−0.192125
$616$	52.9949	2.13522
$617$	23.1467	0.931852	0.465926	−	0.884824i	$-0.345721\pi$
$617$	23.1467	0.931852	0.465926	+	0.884824i	$0.345721\pi$
$618$	−15.3306	−0.616688
$619$	8.60339	0.345799	0.172900	−	0.984939i	$-0.444686\pi$
$619$	8.60339	0.345799	0.172900	+	0.984939i	$0.444686\pi$
$620$	−0.0499222	−0.00200492
$621$	1.99275	0.0799661
$622$	8.13340	0.326120
$623$	50.9296	2.04045
$624$	32.2334	1.29037
$625$	21.6912	0.867649
$626$	−11.0848	−0.443037
$627$	−9.80540	−0.391590
$628$	78.0012	3.11259
$629$	−22.6341	−0.902482
$630$	−4.48366	−0.178633
$631$	−25.2963	−1.00703	−0.503515	−	0.863986i	$-0.667960\pi$
$631$	−25.2963	−1.00703	−0.503515	+	0.863986i	$0.667960\pi$
$632$	−35.7882	−1.42358
$633$	−20.5180	−0.815517
$634$	6.37228	0.253076
$635$	−8.37948	−0.332530
$636$	−49.8660	−1.97732
$637$	−70.6266	−2.79833
$638$	−39.4586	−1.56218
$639$	12.0447	0.476482
$640$	9.58026	0.378693
$641$	17.4396	0.688823	0.344412	−	0.938819i	$-0.388078\pi$
$641$	17.4396	0.688823	0.344412	+	0.938819i	$0.388078\pi$
$642$	4.09786	0.161729
$643$	43.5893	1.71900	0.859498	−	0.511139i	$-0.170776\pi$
$643$	43.5893	1.71900	0.859498	+	0.511139i	$0.170776\pi$
$644$	−5.80672	−0.228817
$645$	6.50372	0.256084
$646$	18.0001	0.708205
$647$	−9.02345	−0.354749	−0.177374	−	0.984143i	$-0.556760\pi$
$647$	−9.02345	−0.354749	−0.177374	+	0.984143i	$0.556760\pi$
$648$	24.8491	0.976164
$649$	−21.2792	−0.835281
$650$	−75.7536	−2.97130
$651$	−0.164406	−0.00644358
$652$	57.7055	2.25992
$653$	−21.5540	−0.843475	−0.421737	−	0.906718i	$-0.638580\pi$
$653$	−21.5540	−0.843475	−0.421737	+	0.906718i	$0.638580\pi$
$654$	−6.04166	−0.236248
$655$	2.78258	0.108724
$656$	23.9452	0.934904
$657$	−7.98948	−0.311699
$658$	63.4528	2.47365
$659$	33.5681	1.30763	0.653813	−	0.756656i	$-0.273168\pi$
$659$	33.5681	1.30763	0.653813	+	0.756656i	$0.273168\pi$
$660$	−7.22639	−0.281287
$661$	17.0629	0.663670	0.331835	−	0.943337i	$-0.392332\pi$
$661$	17.0629	0.663670	0.331835	+	0.943337i	$0.392332\pi$
$662$	−17.2833	−0.671734
$663$	−27.8091	−1.08002
$664$	−34.6313	−1.34396
$665$	−5.00580	−0.194117
$666$	17.1849	0.665902
$667$	2.10797	0.0816210
$668$	−37.7553	−1.46080
$669$	−10.0538	−0.388704
$670$	−2.13962	−0.0826605
$671$	27.0279	1.04340
$672$	5.60127	0.216074
$673$	−9.33152	−0.359704	−0.179852	−	0.983694i	$-0.557562\pi$
$673$	−9.33152	−0.359704	−0.179852	+	0.983694i	$0.557562\pi$
$674$	11.2745	0.434277
$675$	−27.0030	−1.03935
$676$	115.599	4.44610
$677$	29.7028	1.14157	0.570786	−	0.821099i	$-0.306639\pi$
$677$	29.7028	1.14157	0.570786	+	0.821099i	$0.306639\pi$
$678$	−24.0543	−0.923799
$679$	27.1304	1.04117
$680$	6.46781	0.248029
$681$	3.32623	0.127461
$682$	0.178337	0.00682887
$683$	−3.86612	−0.147933	−0.0739664	−	0.997261i	$-0.523566\pi$
$683$	−3.86612	−0.147933	−0.0739664	+	0.997261i	$0.523566\pi$
$684$	−9.03607	−0.345503
$685$	−7.03477	−0.268785
$686$	39.1617	1.49520
$687$	14.3094	0.545937
$688$	−32.6857	−1.24613
$689$	−57.8886	−2.20538
$690$	0.583880	0.0222279
$691$	27.7296	1.05488	0.527442	−	0.849591i	$-0.323151\pi$
$691$	27.7296	1.05488	0.527442	+	0.849591i	$0.323151\pi$
$692$	39.7196	1.50991
$693$	10.5901	0.402286
$694$	−21.5417	−0.817711
$695$	5.28580	0.200502
$696$	39.8406	1.51015
$697$	−20.6586	−0.782499
$698$	41.4336	1.56828
$699$	−5.51581	−0.208627
$700$	78.6850	2.97401
$701$	−16.2452	−0.613572	−0.306786	−	0.951779i	$-0.599254\pi$
$701$	−16.2452	−0.613572	−0.306786	+	0.951779i	$0.599254\pi$
$702$	89.6757	3.38459
$703$	19.1862	0.723620
$704$	−24.6866	−0.930411
$705$	−4.21857	−0.158880
$706$	18.5304	0.697399
$707$	−26.0175	−0.978490
$708$	44.0670	1.65614
$709$	−37.5708	−1.41100	−0.705500	−	0.708710i	$-0.749277\pi$
$709$	−37.5708	−1.41100	−0.705500	+	0.708710i	$0.749277\pi$
$710$	14.9890	0.562527
$711$	−7.15166	−0.268208
$712$	55.7800	2.09044
$713$	−0.00952718	−0.000356796 0
$714$	43.6872	1.63495
$715$	−8.38898	−0.313730
$716$	89.0109	3.32649
$717$	−42.1043	−1.57241
$718$	6.52266	0.243423
$719$	−27.4001	−1.02185	−0.510926	−	0.859625i	$-0.670697\pi$
$719$	−27.4001	−1.02185	−0.510926	+	0.859625i	$0.670697\pi$
$720$	−1.49811	−0.0558312
$721$	18.4857	0.688444
$722$	30.9041	1.15013
$723$	43.3350	1.61165
$724$	−49.8020	−1.85088
$725$	−28.5644	−1.06086
$726$	−12.6933	−0.471093
$727$	−9.36260	−0.347240	−0.173620	−	0.984813i	$-0.555546\pi$
$727$	−9.36260	−0.347240	−0.173620	+	0.984813i	$0.555546\pi$
$728$	−127.403	−4.72187
$729$	29.4051	1.08908
$730$	−9.94246	−0.367987
$731$	28.1994	1.04299
$732$	−55.9719	−2.06878
$733$	−26.6562	−0.984568	−0.492284	−	0.870435i	$-0.663838\pi$
$733$	−26.6562	−0.984568	−0.492284	+	0.870435i	$0.663838\pi$
$734$	13.8089	0.509697
$735$	−7.37647	−0.272085
$736$	0.324589	0.0119645
$737$	5.05364	0.186153
$738$	15.6850	0.577372
$739$	4.58031	0.168489	0.0842446	−	0.996445i	$-0.473152\pi$
$739$	4.58031	0.168489	0.0842446	+	0.996445i	$0.473152\pi$
$740$	14.1398	0.519790
$741$	23.5728	0.865969
$742$	90.9410	3.33855
$743$	19.4743	0.714443	0.357222	−	0.934020i	$-0.383724\pi$
$743$	19.4743	0.714443	0.357222	+	0.934020i	$0.383724\pi$
$744$	−0.180063	−0.00660144
$745$	−3.76839	−0.138063
$746$	22.3490	0.818255
$747$	−6.92049	−0.253207
$748$	−31.3328	−1.14564
$749$	−4.94121	−0.180548
$750$	−16.1949	−0.591354
$751$	5.94272	0.216853	0.108426	−	0.994104i	$-0.465419\pi$
$751$	5.94272	0.216853	0.108426	+	0.994104i	$0.465419\pi$
$752$	21.2012	0.773130
$753$	0.586660	0.0213791
$754$	94.8609	3.45463
$755$	7.35837	0.267799
$756$	−93.1457	−3.38768
$757$	35.6732	1.29656	0.648282	−	0.761400i	$-0.275488\pi$
$757$	35.6732	1.29656	0.648282	+	0.761400i	$0.275488\pi$
$758$	81.5538	2.96217
$759$	−1.37909	−0.0500577
$760$	−5.48253	−0.198872
$761$	−28.0282	−1.01602	−0.508011	−	0.861351i	$-0.669619\pi$
$761$	−28.0282	−1.01602	−0.508011	+	0.861351i	$0.669619\pi$
$762$	−61.9901	−2.24566
$763$	7.28506	0.263737
$764$	−22.7641	−0.823575
$765$	1.29248	0.0467298
$766$	35.6278	1.28729
$767$	51.1565	1.84716
$768$	44.6755	1.61209
$769$	44.4412	1.60259	0.801295	−	0.598269i	$-0.204145\pi$
$769$	44.4412	1.60259	0.801295	+	0.598269i	$0.204145\pi$
$770$	13.1788	0.474931
$771$	21.1493	0.761675
$772$	−7.45249	−0.268221
$773$	−46.4098	−1.66924	−0.834622	−	0.550823i	$-0.814314\pi$
$773$	−46.4098	−1.66924	−0.834622	+	0.550823i	$0.814314\pi$
$774$	−21.4103	−0.769578
$775$	0.129100	0.00463740
$776$	29.7142	1.06668
$777$	46.5658	1.67054
$778$	37.3731	1.33989
$779$	17.5115	0.627416
$780$	17.3727	0.622042
$781$	−35.4031	−1.26682
$782$	2.53164	0.0905312
$783$	33.8140	1.20841
$784$	37.0720	1.32400
$785$	9.45737	0.337548
$786$	20.5851	0.734245
$787$	17.1169	0.610152	0.305076	−	0.952328i	$-0.401318\pi$
$787$	17.1169	0.610152	0.305076	+	0.952328i	$0.401318\pi$
$788$	−68.9749	−2.45713
$789$	4.92916	0.175483
$790$	−8.89984	−0.316642
$791$	29.0047	1.03129
$792$	11.5987	0.412141
$793$	−64.9768	−2.30739
$794$	−46.2081	−1.63986
$795$	−6.04607	−0.214432
$796$	45.9694	1.62934
$797$	−8.29473	−0.293815	−0.146907	−	0.989150i	$-0.546932\pi$
$797$	−8.29473	−0.293815	−0.146907	+	0.989150i	$0.546932\pi$
$798$	−37.0321	−1.31092
$799$	−18.2912	−0.647097
$800$	−4.39839	−0.155507
$801$	11.1467	0.393849
$802$	30.1726	1.06543
$803$	23.4835	0.828714
$804$	−10.4656	−0.369092
$805$	−0.704044	−0.0248143
$806$	−0.428733	−0.0151015
$807$	20.7342	0.729879
$808$	−28.4953	−1.00246
$809$	−30.1774	−1.06098	−0.530491	−	0.847691i	$-0.677992\pi$
$809$	−30.1774	−1.06098	−0.530491	+	0.847691i	$0.677992\pi$
$810$	6.17949	0.217125
$811$	43.0654	1.51223	0.756116	−	0.654438i	$-0.227095\pi$
$811$	43.0654	1.51223	0.756116	+	0.654438i	$0.227095\pi$
$812$	−98.5316	−3.45778
$813$	5.86819	0.205806
$814$	−50.5116	−1.77043
$815$	6.99658	0.245080
$816$	14.5970	0.510999
$817$	−23.9036	−0.836282
$818$	73.0394	2.55376
$819$	−25.4593	−0.889621
$820$	12.9057	0.450685
$821$	−33.4416	−1.16712	−0.583559	−	0.812071i	$-0.698340\pi$
$821$	−33.4416	−1.16712	−0.583559	+	0.812071i	$0.698340\pi$
$822$	−52.0422	−1.81518
$823$	−52.0514	−1.81440	−0.907198	−	0.420703i	$-0.861783\pi$
$823$	−52.0514	−1.81440	−0.907198	+	0.420703i	$0.861783\pi$
$824$	20.2462	0.705310
$825$	18.6876	0.650617
$826$	−80.3652	−2.79626
$827$	−29.1481	−1.01358	−0.506790	−	0.862069i	$-0.669168\pi$
$827$	−29.1481	−1.01358	−0.506790	+	0.862069i	$0.669168\pi$
$828$	−1.27088	−0.0441663
$829$	20.1273	0.699052	0.349526	−	0.936927i	$-0.386343\pi$
$829$	20.1273	0.699052	0.349526	+	0.936927i	$0.386343\pi$
$830$	−8.61216	−0.298932
$831$	−34.6836	−1.20316
$832$	59.3481	2.05753
$833$	−31.9836	−1.10816
$834$	39.1035	1.35404
$835$	−4.57769	−0.158418
$836$	26.5597	0.918587
$837$	−0.152826	−0.00528243
$838$	−19.6734	−0.679606
$839$	−29.4120	−1.01541	−0.507707	−	0.861530i	$-0.669507\pi$
$839$	−29.4120	−1.01541	−0.507707	+	0.861530i	$0.669507\pi$
$840$	−13.3064	−0.459114
$841$	6.76922	0.233421
$842$	−22.2028	−0.765159
$843$	−11.2936	−0.388972
$844$	55.5767	1.91303
$845$	14.0159	0.482162
$846$	13.8876	0.477464
$847$	15.3056	0.525908
$848$	30.3858	1.04345
$849$	34.8522	1.19612
$850$	−34.3054	−1.17666
$851$	2.69845	0.0925017
$852$	73.3160	2.51177
$853$	24.4616	0.837548	0.418774	−	0.908091i	$-0.362460\pi$
$853$	24.4616	0.837548	0.418774	+	0.908091i	$0.362460\pi$
$854$	102.076	3.49298
$855$	−1.09559	−0.0374684
$856$	−5.41179	−0.184971
$857$	−10.2946	−0.351658	−0.175829	−	0.984421i	$-0.556261\pi$
$857$	−10.2946	−0.351658	−0.175829	+	0.984421i	$0.556261\pi$
$858$	−62.0604	−2.11871
$859$	24.6948	0.842576	0.421288	−	0.906927i	$-0.361578\pi$
$859$	24.6948	0.842576	0.421288	+	0.906927i	$0.361578\pi$
$860$	−17.6165	−0.600718
$861$	42.5015	1.44845
$862$	1.38011	0.0470068
$863$	−43.5613	−1.48284	−0.741421	−	0.671040i	$-0.765848\pi$
$863$	−43.5613	−1.48284	−0.741421	+	0.671040i	$0.765848\pi$
$864$	5.20673	0.177137
$865$	4.81586	0.163744
$866$	45.7553	1.55483
$867$	11.9014	0.404194
$868$	0.445323	0.0151153
$869$	21.0209	0.713084
$870$	9.90759	0.335899
$871$	−12.1493	−0.411662
$872$	7.97886	0.270198
$873$	5.93789	0.200967
$874$	−2.14598	−0.0725889
$875$	19.5278	0.660162
$876$	−48.6318	−1.64312
$877$	57.5742	1.94414	0.972070	−	0.234689i	$-0.0754072\pi$
$877$	57.5742	1.94414	0.972070	+	0.234689i	$0.0754072\pi$
$878$	−1.29117	−0.0435750
$879$	−21.2190	−0.715700
$880$	4.40339	0.148438
$881$	−20.0579	−0.675767	−0.337883	−	0.941188i	$-0.609711\pi$
$881$	−20.0579	−0.675767	−0.337883	+	0.941188i	$0.609711\pi$
$882$	24.2834	0.817666
$883$	12.5227	0.421423	0.210711	−	0.977548i	$-0.432422\pi$
$883$	12.5227	0.421423	0.210711	+	0.977548i	$0.432422\pi$
$884$	75.3261	2.53349
$885$	5.34296	0.179602
$886$	67.3105	2.26134
$887$	−47.6108	−1.59862	−0.799308	−	0.600922i	$-0.794800\pi$
$887$	−47.6108	−1.59862	−0.799308	+	0.600922i	$0.794800\pi$
$888$	51.0006	1.71147
$889$	74.7478	2.50696
$890$	13.8714	0.464971
$891$	−14.5956	−0.488970
$892$	27.2327	0.911817
$893$	15.5048	0.518849
$894$	−27.8779	−0.932378
$895$	10.7922	0.360745
$896$	−85.4592	−2.85499
$897$	3.31542	0.110699
$898$	84.6683	2.82542
$899$	−0.161662	−0.00539174
$900$	17.2213	0.574045
$901$	−26.2151	−0.873352
$902$	−46.1028	−1.53506
$903$	−58.0154	−1.93063
$904$	31.7670	1.05656
$905$	−6.03832	−0.200720
$906$	54.4361	1.80852
$907$	22.5698	0.749417	0.374709	−	0.927143i	$-0.377743\pi$
$907$	22.5698	0.749417	0.374709	+	0.927143i	$0.377743\pi$
$908$	−9.00969	−0.298997
$909$	−5.69431	−0.188868
$910$	−31.6827	−1.05027
$911$	36.2098	1.19968	0.599842	−	0.800118i	$-0.295230\pi$
$911$	36.2098	1.19968	0.599842	+	0.800118i	$0.295230\pi$
$912$	−12.3734	−0.409724
$913$	20.3414	0.673201
$914$	−64.8604	−2.14539
$915$	−6.78640	−0.224351
$916$	−38.7596	−1.28065
$917$	−24.8215	−0.819679
$918$	40.6100	1.34033
$919$	−46.3987	−1.53055	−0.765276	−	0.643702i	$-0.777398\pi$
$919$	−46.3987	−1.53055	−0.765276	+	0.643702i	$0.777398\pi$
$920$	−0.771095	−0.0254222
$921$	−25.5007	−0.840278
$922$	9.31044	0.306623
$923$	85.1113	2.80147
$924$	64.4618	2.12064
$925$	−36.5658	−1.20228
$926$	3.68595	0.121128
$927$	4.04586	0.132884
$928$	5.50780	0.180802
$929$	−31.8139	−1.04378	−0.521890	−	0.853013i	$-0.674773\pi$
$929$	−31.8139	−1.04378	−0.521890	+	0.853013i	$0.674773\pi$
$930$	−0.0447783	−0.00146834
$931$	27.1113	0.888538
$932$	14.9406	0.489395
$933$	4.82356	0.157916
$934$	52.5505	1.71951
$935$	−3.79899	−0.124240
$936$	−27.8840	−0.911417
$937$	−8.62905	−0.281899	−0.140949	−	0.990017i	$-0.545015\pi$
$937$	−8.62905	−0.281899	−0.140949	+	0.990017i	$0.545015\pi$
$938$	19.0861	0.623183
$939$	−6.57389	−0.214531
$940$	11.4268	0.372700
$941$	34.7944	1.13426	0.567132	−	0.823627i	$-0.308053\pi$
$941$	34.7944	1.13426	0.567132	+	0.823627i	$0.308053\pi$
$942$	69.9641	2.27955
$943$	2.46292	0.0802038
$944$	−26.8521	−0.873962
$945$	−11.2936	−0.367380
$946$	62.9313	2.04607
$947$	31.7042	1.03025	0.515124	−	0.857116i	$-0.327746\pi$
$947$	31.7042	1.03025	0.515124	+	0.857116i	$0.327746\pi$
$948$	−43.5320	−1.41385
$949$	−56.4558	−1.83263
$950$	29.0795	0.943462
$951$	3.77912	0.122546
$952$	−57.6951	−1.86991
$953$	−19.2582	−0.623834	−0.311917	−	0.950109i	$-0.600971\pi$
$953$	−19.2582	−0.623834	−0.311917	+	0.950109i	$0.600971\pi$
$954$	19.9038	0.644408
$955$	−2.76006	−0.0893135
$956$	114.047	3.68855
$957$	−23.4011	−0.756451
$958$	82.7505	2.67355
$959$	62.7526	2.02639
$960$	6.19852	0.200056
$961$	−30.9993	−0.999976
$962$	121.433	3.91516
$963$	−1.08145	−0.0348494
$964$	−117.381	−3.78058
$965$	−0.903588	−0.0290875
$966$	−5.20841	−0.167578
$967$	24.5965	0.790972	0.395486	−	0.918472i	$-0.370576\pi$
$967$	24.5965	0.790972	0.395486	+	0.918472i	$0.370576\pi$
$968$	16.7633	0.538793
$969$	10.6751	0.342932
$970$	7.38937	0.237258
$971$	52.8165	1.69496	0.847480	−	0.530827i	$-0.178118\pi$
$971$	52.8165	1.69496	0.847480	+	0.530827i	$0.178118\pi$
$972$	−35.9726	−1.15382
$973$	−47.1511	−1.51160
$974$	66.7102	2.13753
$975$	−44.9261	−1.43879
$976$	34.1064	1.09172
$977$	−14.6001	−0.467099	−0.233550	−	0.972345i	$-0.575034\pi$
$977$	−14.6001	−0.467099	−0.233550	+	0.972345i	$0.575034\pi$
$978$	51.7596	1.65509
$979$	−32.7634	−1.04712
$980$	19.9805	0.638254
$981$	1.59444	0.0509065
$982$	−67.6745	−2.15958
$983$	−45.2811	−1.44424	−0.722122	−	0.691766i	$-0.756833\pi$
$983$	−45.2811	−1.44424	−0.722122	+	0.691766i	$0.756833\pi$
$984$	46.5491	1.48393
$985$	−8.36296	−0.266466
$986$	42.9582	1.36807
$987$	37.6310	1.19781
$988$	−63.8513	−2.03138
$989$	−3.36194	−0.106904
$990$	2.88437	0.0916714
$991$	6.48047	0.205859	0.102930	−	0.994689i	$-0.467178\pi$
$991$	6.48047	0.205859	0.102930	+	0.994689i	$0.467178\pi$
$992$	−0.0248930	−0.000790355 0
$993$	−10.2499	−0.325272
$994$	−133.707	−4.24093
$995$	5.57363	0.176696
$996$	−42.1248	−1.33478
$997$	42.3298	1.34060	0.670298	−	0.742092i	$-0.266166\pi$
$997$	42.3298	1.34060	0.670298	+	0.742092i	$0.266166\pi$
$998$	−1.30130	−0.0411921
$999$	43.2859	1.36950

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4003.2.a.b.1.16	✓	152

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4003.2.a.b.1.16	✓	152	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:	\( N \)	\(=\)	\( 4003 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4003.a (trivial)

\(f(q)\)	\(=\)	\(q-2.42958 q^{2} -1.44088 q^{3} +3.90288 q^{4} +0.473210 q^{5} +3.50074 q^{6} -4.22120 q^{7} -4.62321 q^{8} -0.923870 q^{9} +O(q^{10})\)	\(q-2.42958 q^{2} -1.44088 q^{3} +3.90288 q^{4} +0.473210 q^{5} +3.50074 q^{6} -4.22120 q^{7} -4.62321 q^{8} -0.923870 q^{9} -1.14970 q^{10} +2.71553 q^{11} -5.62358 q^{12} -6.52831 q^{13} +10.2558 q^{14} -0.681838 q^{15} +3.42672 q^{16} -2.95637 q^{17} +2.24462 q^{18} +2.50601 q^{19} +1.84688 q^{20} +6.08223 q^{21} -6.59761 q^{22} +0.352460 q^{23} +6.66148 q^{24} -4.77607 q^{25} +15.8611 q^{26} +5.65382 q^{27} -16.4748 q^{28} +5.98074 q^{29} +1.65658 q^{30} -0.0270305 q^{31} +0.920923 q^{32} -3.91275 q^{33} +7.18276 q^{34} -1.99751 q^{35} -3.60575 q^{36} +7.65604 q^{37} -6.08857 q^{38} +9.40650 q^{39} -2.18775 q^{40} +6.98781 q^{41} -14.7773 q^{42} -9.53851 q^{43} +10.5984 q^{44} -0.437185 q^{45} -0.856332 q^{46} +6.18705 q^{47} -4.93748 q^{48} +10.8185 q^{49} +11.6039 q^{50} +4.25978 q^{51} -25.4792 q^{52} +8.86731 q^{53} -13.7364 q^{54} +1.28502 q^{55} +19.5155 q^{56} -3.61086 q^{57} -14.5307 q^{58} -7.83611 q^{59} -2.66113 q^{60} +9.95309 q^{61} +0.0656729 q^{62} +3.89984 q^{63} -9.09089 q^{64} -3.08926 q^{65} +9.50635 q^{66} +1.86101 q^{67} -11.5384 q^{68} -0.507852 q^{69} +4.85313 q^{70} -13.0373 q^{71} +4.27124 q^{72} +8.64784 q^{73} -18.6010 q^{74} +6.88174 q^{75} +9.78068 q^{76} -11.4628 q^{77} -22.8539 q^{78} +7.74098 q^{79} +1.62156 q^{80} -5.37485 q^{81} -16.9775 q^{82} +7.49076 q^{83} +23.7382 q^{84} -1.39899 q^{85} +23.1746 q^{86} -8.61751 q^{87} -12.5545 q^{88} -12.0652 q^{89} +1.06218 q^{90} +27.5573 q^{91} +1.37561 q^{92} +0.0389477 q^{93} -15.0320 q^{94} +1.18587 q^{95} -1.32694 q^{96} -6.42719 q^{97} -26.2845 q^{98} -2.50880 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 152 q - 22 q^{2} - 18 q^{3} + 138 q^{4} - 59 q^{5} - 17 q^{6} - 19 q^{7} - 66 q^{8} + 106 q^{9}+O(q^{10}) \) 152 * q - 22 * q^2 - 18 * q^3 + 138 * q^4 - 59 * q^5 - 17 * q^6 - 19 * q^7 - 66 * q^8 + 106 * q^9	\( 152 q - 22 q^{2} - 18 q^{3} + 138 q^{4} - 59 q^{5} - 17 q^{6} - 19 q^{7} - 66 q^{8} + 106 q^{9} - 15 q^{10} - 40 q^{11} - 53 q^{12} - 59 q^{13} - 36 q^{14} - 40 q^{15} + 118 q^{16} - 93 q^{17} - 59 q^{18} - 16 q^{19} - 108 q^{20} - 62 q^{21} - 37 q^{22} - 107 q^{23} - 31 q^{24} + 101 q^{25} - 64 q^{26} - 63 q^{27} - 53 q^{28} - 124 q^{29} - 68 q^{30} - 15 q^{31} - 129 q^{32} - 49 q^{33} - 76 q^{35} + 45 q^{36} - 98 q^{37} - 125 q^{38} - 47 q^{39} - 7 q^{40} - 56 q^{41} - 84 q^{42} - 62 q^{43} - 114 q^{44} - 142 q^{45} - 3 q^{46} - 111 q^{47} - 92 q^{48} + 117 q^{49} - 64 q^{50} - 21 q^{51} - 85 q^{52} - 347 q^{53} + 3 q^{54} - 16 q^{55} - 73 q^{56} - 115 q^{57} - 29 q^{58} - 50 q^{59} - 54 q^{60} - 62 q^{61} - 55 q^{62} - 70 q^{63} + 64 q^{64} - 147 q^{65} + 34 q^{66} - 86 q^{67} - 174 q^{68} - 104 q^{69} - 7 q^{70} - 86 q^{71} - 139 q^{72} - 27 q^{73} - 52 q^{74} - 49 q^{75} - 11 q^{76} - 346 q^{77} - 59 q^{78} - 17 q^{79} - 149 q^{80} - 8 q^{81} - 31 q^{82} - 106 q^{83} - 51 q^{84} - 69 q^{85} - 85 q^{86} - 32 q^{87} - 113 q^{88} - 59 q^{89} + 10 q^{90} - 9 q^{91} - 314 q^{92} - 230 q^{93} + 7 q^{94} - 74 q^{95} - 54 q^{96} - 60 q^{97} - 77 q^{98} - 96 q^{99}+O(q^{100}) \) 152 * q - 22 * q^2 - 18 * q^3 + 138 * q^4 - 59 * q^5 - 17 * q^6 - 19 * q^7 - 66 * q^8 + 106 * q^9 - 15 * q^10 - 40 * q^11 - 53 * q^12 - 59 * q^13 - 36 * q^14 - 40 * q^15 + 118 * q^16 - 93 * q^17 - 59 * q^18 - 16 * q^19 - 108 * q^20 - 62 * q^21 - 37 * q^22 - 107 * q^23 - 31 * q^24 + 101 * q^25 - 64 * q^26 - 63 * q^27 - 53 * q^28 - 124 * q^29 - 68 * q^30 - 15 * q^31 - 129 * q^32 - 49 * q^33 - 76 * q^35 + 45 * q^36 - 98 * q^37 - 125 * q^38 - 47 * q^39 - 7 * q^40 - 56 * q^41 - 84 * q^42 - 62 * q^43 - 114 * q^44 - 142 * q^45 - 3 * q^46 - 111 * q^47 - 92 * q^48 + 117 * q^49 - 64 * q^50 - 21 * q^51 - 85 * q^52 - 347 * q^53 + 3 * q^54 - 16 * q^55 - 73 * q^56 - 115 * q^57 - 29 * q^58 - 50 * q^59 - 54 * q^60 - 62 * q^61 - 55 * q^62 - 70 * q^63 + 64 * q^64 - 147 * q^65 + 34 * q^66 - 86 * q^67 - 174 * q^68 - 104 * q^69 - 7 * q^70 - 86 * q^71 - 139 * q^72 - 27 * q^73 - 52 * q^74 - 49 * q^75 - 11 * q^76 - 346 * q^77 - 59 * q^78 - 17 * q^79 - 149 * q^80 - 8 * q^81 - 31 * q^82 - 106 * q^83 - 51 * q^84 - 69 * q^85 - 85 * q^86 - 32 * q^87 - 113 * q^88 - 59 * q^89 + 10 * q^90 - 9 * q^91 - 314 * q^92 - 230 * q^93 + 7 * q^94 - 74 * q^95 - 54 * q^96 - 60 * q^97 - 77 * q^98 - 96 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more