LMFDB - Embedded newform 4013.2.a.c.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4013,2,Mod(1,4013)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4013, 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("4013.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4013 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4013.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:	$32.0439663311$
Analytic rank:	$0$
Dimension:	$176$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	4013.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.61853 q^{2} +0.324920 q^{3} +4.85671 q^{4} +3.20033 q^{5} -0.850813 q^{6} -1.73793 q^{7} -7.48039 q^{8} -2.89443 q^{9} +O(q^{10})$	\(q-2.61853 q^{2} +0.324920 q^{3} +4.85671 q^{4} +3.20033 q^{5} -0.850813 q^{6} -1.73793 q^{7} -7.48039 q^{8} -2.89443 q^{9} -8.38017 q^{10} -3.28048 q^{11} +1.57804 q^{12} -3.26315 q^{13} +4.55082 q^{14} +1.03985 q^{15} +9.87421 q^{16} -1.28967 q^{17} +7.57915 q^{18} +5.11905 q^{19} +15.5431 q^{20} -0.564688 q^{21} +8.59004 q^{22} +3.69547 q^{23} -2.43053 q^{24} +5.24212 q^{25} +8.54467 q^{26} -1.91522 q^{27} -8.44062 q^{28} +5.41648 q^{29} -2.72289 q^{30} -6.73858 q^{31} -10.8952 q^{32} -1.06589 q^{33} +3.37704 q^{34} -5.56195 q^{35} -14.0574 q^{36} -3.56961 q^{37} -13.4044 q^{38} -1.06026 q^{39} -23.9397 q^{40} -0.723908 q^{41} +1.47865 q^{42} +9.24023 q^{43} -15.9323 q^{44} -9.26313 q^{45} -9.67670 q^{46} +2.66395 q^{47} +3.20833 q^{48} -3.97960 q^{49} -13.7267 q^{50} -0.419039 q^{51} -15.8482 q^{52} +6.10164 q^{53} +5.01506 q^{54} -10.4986 q^{55} +13.0004 q^{56} +1.66328 q^{57} -14.1832 q^{58} -13.4637 q^{59} +5.05026 q^{60} +3.40165 q^{61} +17.6452 q^{62} +5.03031 q^{63} +8.78092 q^{64} -10.4432 q^{65} +2.79108 q^{66} +10.0383 q^{67} -6.26355 q^{68} +1.20073 q^{69} +14.5641 q^{70} -3.27612 q^{71} +21.6514 q^{72} +5.29409 q^{73} +9.34713 q^{74} +1.70327 q^{75} +24.8617 q^{76} +5.70124 q^{77} +2.77633 q^{78} +7.34468 q^{79} +31.6008 q^{80} +8.06099 q^{81} +1.89558 q^{82} -17.6655 q^{83} -2.74253 q^{84} -4.12737 q^{85} -24.1958 q^{86} +1.75992 q^{87} +24.5393 q^{88} -7.56214 q^{89} +24.2558 q^{90} +5.67113 q^{91} +17.9478 q^{92} -2.18950 q^{93} -6.97565 q^{94} +16.3827 q^{95} -3.54006 q^{96} -1.14820 q^{97} +10.4207 q^{98} +9.49511 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 176 q + 11 q^{2} + 53 q^{3} + 191 q^{4} + 5 q^{5} + 19 q^{6} + 46 q^{7} + 36 q^{8} + 193 q^{9}+O(q^{10}) $ 176 * q + 11 * q^2 + 53 * q^3 + 191 * q^4 + 5 * q^5 + 19 * q^6 + 46 * q^7 + 36 * q^8 + 193 * q^9	\( 176 q + 11 q^{2} + 53 q^{3} + 191 q^{4} + 5 q^{5} + 19 q^{6} + 46 q^{7} + 36 q^{8} + 193 q^{9} + 43 q^{10} + 18 q^{11} + 95 q^{12} + 95 q^{13} + 2 q^{14} + 36 q^{15} + 225 q^{16} + 35 q^{17} + 46 q^{18} + 127 q^{19} + 4 q^{20} + 32 q^{21} + 60 q^{22} + 35 q^{23} + 26 q^{24} + 207 q^{25} + 19 q^{26} + 191 q^{27} + 87 q^{28} + 16 q^{29} + 28 q^{30} + 93 q^{31} + 73 q^{32} + 70 q^{33} + 45 q^{34} + 73 q^{35} + 206 q^{36} + 64 q^{37} + 35 q^{38} + 72 q^{39} + 139 q^{40} + 19 q^{41} + 35 q^{42} + 261 q^{43} + 11 q^{44} + 12 q^{45} + 58 q^{46} + 40 q^{47} + 130 q^{48} + 234 q^{49} - 14 q^{50} + 76 q^{51} + 263 q^{52} + 17 q^{53} + 28 q^{54} + 170 q^{55} - 10 q^{56} + 60 q^{57} + 52 q^{58} + 69 q^{59} + 37 q^{60} + 110 q^{61} + 71 q^{62} + 101 q^{63} + 250 q^{64} - q^{65} + 43 q^{66} + 190 q^{67} + 48 q^{68} + 45 q^{69} + 14 q^{70} + 9 q^{71} + 98 q^{72} + 182 q^{73} - 23 q^{74} + 219 q^{75} + 197 q^{76} + 25 q^{77} - 26 q^{78} + 105 q^{79} + 20 q^{80} + 236 q^{81} + 107 q^{82} + 130 q^{83} + 38 q^{84} + 73 q^{85} - 24 q^{86} + 171 q^{87} + 165 q^{88} + 40 q^{89} + 45 q^{90} + 182 q^{91} - 4 q^{92} + 23 q^{93} + 98 q^{94} + 30 q^{95} - 2 q^{96} + 168 q^{97} + 82 q^{98} + 50 q^{99}+O(q^{100}) \) 176 * q + 11 * q^2 + 53 * q^3 + 191 * q^4 + 5 * q^5 + 19 * q^6 + 46 * q^7 + 36 * q^8 + 193 * q^9 + 43 * q^10 + 18 * q^11 + 95 * q^12 + 95 * q^13 + 2 * q^14 + 36 * q^15 + 225 * q^16 + 35 * q^17 + 46 * q^18 + 127 * q^19 + 4 * q^20 + 32 * q^21 + 60 * q^22 + 35 * q^23 + 26 * q^24 + 207 * q^25 + 19 * q^26 + 191 * q^27 + 87 * q^28 + 16 * q^29 + 28 * q^30 + 93 * q^31 + 73 * q^32 + 70 * q^33 + 45 * q^34 + 73 * q^35 + 206 * q^36 + 64 * q^37 + 35 * q^38 + 72 * q^39 + 139 * q^40 + 19 * q^41 + 35 * q^42 + 261 * q^43 + 11 * q^44 + 12 * q^45 + 58 * q^46 + 40 * q^47 + 130 * q^48 + 234 * q^49 - 14 * q^50 + 76 * q^51 + 263 * q^52 + 17 * q^53 + 28 * q^54 + 170 * q^55 - 10 * q^56 + 60 * q^57 + 52 * q^58 + 69 * q^59 + 37 * q^60 + 110 * q^61 + 71 * q^62 + 101 * q^63 + 250 * q^64 - q^65 + 43 * q^66 + 190 * q^67 + 48 * q^68 + 45 * q^69 + 14 * q^70 + 9 * q^71 + 98 * q^72 + 182 * q^73 - 23 * q^74 + 219 * q^75 + 197 * q^76 + 25 * q^77 - 26 * q^78 + 105 * q^79 + 20 * q^80 + 236 * q^81 + 107 * q^82 + 130 * q^83 + 38 * q^84 + 73 * q^85 - 24 * q^86 + 171 * q^87 + 165 * q^88 + 40 * q^89 + 45 * q^90 + 182 * q^91 - 4 * q^92 + 23 * q^93 + 98 * q^94 + 30 * q^95 - 2 * q^96 + 168 * q^97 + 82 * q^98 + 50 * 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 $

$2$	−2.61853	−1.85158	−0.925791	−	0.378036i	$-0.876600\pi$
$2$	−2.61853	−1.85158	−0.925791	+	0.378036i	$0.876600\pi$
$3$	0.324920	0.187593	0.0937963	−	0.995591i	$-0.470100\pi$
$3$	0.324920	0.187593	0.0937963	+	0.995591i	$0.470100\pi$
$4$	4.85671	2.42836
$5$	3.20033	1.43123	0.715616	−	0.698494i	$-0.246146\pi$
$5$	3.20033	1.43123	0.715616	+	0.698494i	$0.246146\pi$
$6$	−0.850813	−0.347343
$7$	−1.73793	−0.656876	−0.328438	−	0.944526i	$-0.606522\pi$
$7$	−1.73793	−0.656876	−0.328438	+	0.944526i	$0.606522\pi$
$8$	−7.48039	−2.64472
$9$	−2.89443	−0.964809
$10$	−8.38017	−2.65004
$11$	−3.28048	−0.989102	−0.494551	−	0.869149i	$-0.664667\pi$
$11$	−3.28048	−0.989102	−0.494551	+	0.869149i	$0.664667\pi$
$12$	1.57804	0.455542
$13$	−3.26315	−0.905035	−0.452518	−	0.891755i	$-0.649474\pi$
$13$	−3.26315	−0.905035	−0.452518	+	0.891755i	$0.649474\pi$
$14$	4.55082	1.21626
$15$	1.03985	0.268489
$16$	9.87421	2.46855
$17$	−1.28967	−0.312791	−0.156395	−	0.987695i	$-0.549987\pi$
$17$	−1.28967	−0.312791	−0.156395	+	0.987695i	$0.549987\pi$
$18$	7.57915	1.78642
$19$	5.11905	1.17439	0.587195	−	0.809445i	$-0.300232\pi$
$19$	5.11905	1.17439	0.587195	+	0.809445i	$0.300232\pi$
$20$	15.5431	3.47554
$21$	−0.564688	−0.123225
$22$	8.59004	1.83140
$23$	3.69547	0.770558	0.385279	−	0.922800i	$-0.374105\pi$
$23$	3.69547	0.770558	0.385279	+	0.922800i	$0.374105\pi$
$24$	−2.43053	−0.496129
$25$	5.24212	1.04842
$26$	8.54467	1.67575
$27$	−1.91522	−0.368584
$28$	−8.44062	−1.59513
$29$	5.41648	1.00582	0.502908	−	0.864340i	$-0.332264\pi$
$29$	5.41648	1.00582	0.502908	+	0.864340i	$0.332264\pi$
$30$	−2.72289	−0.497129
$31$	−6.73858	−1.21028	−0.605142	−	0.796117i	$-0.706884\pi$
$31$	−6.73858	−1.21028	−0.605142	+	0.796117i	$0.706884\pi$
$32$	−10.8952	−1.92601
$33$	−1.06589	−0.185548
$34$	3.37704	0.579157
$35$	−5.56195	−0.940141
$36$	−14.0574	−2.34290
$37$	−3.56961	−0.586840	−0.293420	−	0.955984i	$-0.594793\pi$
$37$	−3.56961	−0.586840	−0.293420	+	0.955984i	$0.594793\pi$
$38$	−13.4044	−2.17448
$39$	−1.06026	−0.169778
$40$	−23.9397	−3.78520
$41$	−0.723908	−0.113055	−0.0565277	−	0.998401i	$-0.518003\pi$
$41$	−0.723908	−0.113055	−0.0565277	+	0.998401i	$0.518003\pi$
$42$	1.47865	0.228161
$43$	9.24023	1.40912	0.704561	−	0.709644i	$-0.251144\pi$
$43$	9.24023	1.40912	0.704561	+	0.709644i	$0.251144\pi$
$44$	−15.9323	−2.40189
$45$	−9.26313	−1.38087
$46$	−9.67670	−1.42675
$47$	2.66395	0.388577	0.194289	−	0.980944i	$-0.437760\pi$
$47$	2.66395	0.388577	0.194289	+	0.980944i	$0.437760\pi$
$48$	3.20833	0.463082
$49$	−3.97960	−0.568514
$50$	−13.7267	−1.94124
$51$	−0.419039	−0.0586772
$52$	−15.8482	−2.19775
$53$	6.10164	0.838125	0.419062	−	0.907957i	$-0.362359\pi$
$53$	6.10164	0.838125	0.419062	+	0.907957i	$0.362359\pi$
$54$	5.01506	0.682463
$55$	−10.4986	−1.41563
$56$	13.0004	1.73725
$57$	1.66328	0.220307
$58$	−14.1832	−1.86235
$59$	−13.4637	−1.75283	−0.876415	−	0.481557i	$-0.840072\pi$
$59$	−13.4637	−1.75283	−0.876415	+	0.481557i	$0.840072\pi$
$60$	5.05026	0.651986
$61$	3.40165	0.435537	0.217768	−	0.976000i	$-0.430122\pi$
$61$	3.40165	0.435537	0.217768	+	0.976000i	$0.430122\pi$
$62$	17.6452	2.24094
$63$	5.03031	0.633759
$64$	8.78092	1.09762
$65$	−10.4432	−1.29532
$66$	2.79108	0.343558
$67$	10.0383	1.22637	0.613185	−	0.789939i	$-0.289888\pi$
$67$	10.0383	1.22637	0.613185	+	0.789939i	$0.289888\pi$
$68$	−6.26355	−0.759567
$69$	1.20073	0.144551
$70$	14.5641	1.74075
$71$	−3.27612	−0.388804	−0.194402	−	0.980922i	$-0.562277\pi$
$71$	−3.27612	−0.388804	−0.194402	+	0.980922i	$0.562277\pi$
$72$	21.6514	2.55165
$73$	5.29409	0.619627	0.309813	−	0.950797i	$-0.399733\pi$
$73$	5.29409	0.619627	0.309813	+	0.950797i	$0.399733\pi$
$74$	9.34713	1.08658
$75$	1.70327	0.196677
$76$	24.8617	2.85184
$77$	5.70124	0.649717
$78$	2.77633	0.314358
$79$	7.34468	0.826341	0.413170	−	0.910654i	$-0.364421\pi$
$79$	7.34468	0.826341	0.413170	+	0.910654i	$0.364421\pi$
$80$	31.6008	3.53307
$81$	8.06099	0.895665
$82$	1.89558	0.209331
$83$	−17.6655	−1.93904	−0.969522	−	0.245004i	$-0.921211\pi$
$83$	−17.6655	−1.93904	−0.969522	+	0.245004i	$0.921211\pi$
$84$	−2.74253	−0.299234
$85$	−4.12737	−0.447676
$86$	−24.1958	−2.60910
$87$	1.75992	0.188684
$88$	24.5393	2.61589
$89$	−7.56214	−0.801586	−0.400793	−	0.916169i	$-0.631265\pi$
$89$	−7.56214	−0.801586	−0.400793	+	0.916169i	$0.631265\pi$
$90$	24.2558	2.55678
$91$	5.67113	0.594496
$92$	17.9478	1.87119
$93$	−2.18950	−0.227041
$94$	−6.97565	−0.719483
$95$	16.3827	1.68082
$96$	−3.54006	−0.361306
$97$	−1.14820	−0.116582	−0.0582912	−	0.998300i	$-0.518565\pi$
$97$	−1.14820	−0.116582	−0.0582912	+	0.998300i	$0.518565\pi$
$98$	10.4207	1.05265
$99$	9.49511	0.954294
$100$	25.4595	2.54595
$101$	12.4508	1.23890	0.619451	−	0.785035i	$-0.287355\pi$
$101$	12.4508	1.23890	0.619451	+	0.785035i	$0.287355\pi$
$102$	1.09727	0.108646
$103$	18.2297	1.79623	0.898114	−	0.439762i	$-0.144937\pi$
$103$	18.2297	1.79623	0.898114	+	0.439762i	$0.144937\pi$
$104$	24.4096	2.39356
$105$	−1.80719	−0.176364
$106$	−15.9773	−1.55186
$107$	19.5878	1.89363	0.946813	−	0.321786i	$-0.104283\pi$
$107$	19.5878	1.89363	0.946813	+	0.321786i	$0.104283\pi$
$108$	−9.30166	−0.895052
$109$	5.46763	0.523704	0.261852	−	0.965108i	$-0.415667\pi$
$109$	5.46763	0.523704	0.261852	+	0.965108i	$0.415667\pi$
$110$	27.4910	2.62116
$111$	−1.15984	−0.110087
$112$	−17.1607	−1.62153
$113$	−7.93118	−0.746103	−0.373051	−	0.927811i	$-0.621689\pi$
$113$	−7.93118	−0.746103	−0.373051	+	0.927811i	$0.621689\pi$
$114$	−4.35536	−0.407916
$115$	11.8267	1.10285
$116$	26.3063	2.44248
$117$	9.44495	0.873186
$118$	35.2552	3.24551
$119$	2.24135	0.205465
$120$	−7.77849	−0.710076
$121$	−0.238452	−0.0216774
$122$	−8.90734	−0.806432
$123$	−0.235212	−0.0212084
$124$	−32.7273	−2.93900
$125$	0.774869	0.0693064
$126$	−13.1720	−1.17346
$127$	−4.40218	−0.390631	−0.195315	−	0.980741i	$-0.562573\pi$
$127$	−4.40218	−0.390631	−0.195315	+	0.980741i	$0.562573\pi$
$128$	−1.20279	−0.106313
$129$	3.00233	0.264341
$130$	27.3458	2.39838
$131$	−2.75634	−0.240823	−0.120411	−	0.992724i	$-0.538421\pi$
$131$	−2.75634	−0.240823	−0.120411	+	0.992724i	$0.538421\pi$
$132$	−5.17674	−0.450577
$133$	−8.89655	−0.771428
$134$	−26.2855	−2.27073
$135$	−6.12933	−0.527529
$136$	9.64722	0.827242
$137$	4.84156	0.413642	0.206821	−	0.978379i	$-0.433688\pi$
$137$	4.84156	0.413642	0.206821	+	0.978379i	$0.433688\pi$
$138$	−3.14415	−0.267648
$139$	−15.6015	−1.32330	−0.661649	−	0.749814i	$-0.730143\pi$
$139$	−15.6015	−1.32330	−0.661649	+	0.749814i	$0.730143\pi$
$140$	−27.0128	−2.28300
$141$	0.865572	0.0728943
$142$	8.57863	0.719903
$143$	10.7047	0.895172
$144$	−28.5802	−2.38168
$145$	17.3345	1.43955
$146$	−13.8628	−1.14729
$147$	−1.29305	−0.106649
$148$	−17.3366	−1.42506
$149$	8.89447	0.728664	0.364332	−	0.931269i	$-0.381297\pi$
$149$	8.89447	0.728664	0.364332	+	0.931269i	$0.381297\pi$
$150$	−4.46007	−0.364163
$151$	−1.33708	−0.108810	−0.0544051	−	0.998519i	$-0.517326\pi$
$151$	−1.33708	−0.108810	−0.0544051	+	0.998519i	$0.517326\pi$
$152$	−38.2925	−3.10593
$153$	3.73285	0.301783
$154$	−14.9289	−1.20300
$155$	−21.5657	−1.73220
$156$	−5.14939	−0.412281
$157$	4.11528	0.328435	0.164218	−	0.986424i	$-0.447490\pi$
$157$	4.11528	0.328435	0.164218	+	0.986424i	$0.447490\pi$
$158$	−19.2323	−1.53004
$159$	1.98255	0.157226
$160$	−34.8682	−2.75657
$161$	−6.42246	−0.506161
$162$	−21.1080	−1.65840
$163$	14.7894	1.15840	0.579199	−	0.815186i	$-0.303366\pi$
$163$	14.7894	1.15840	0.579199	+	0.815186i	$0.303366\pi$
$164$	−3.51581	−0.274539
$165$	−3.41121	−0.265563
$166$	46.2578	3.59030
$167$	9.34787	0.723360	0.361680	−	0.932302i	$-0.382203\pi$
$167$	9.34787	0.723360	0.361680	+	0.932302i	$0.382203\pi$
$168$	4.22409	0.325895
$169$	−2.35184	−0.180911
$170$	10.8076	0.828908
$171$	−14.8167	−1.13306
$172$	44.8771	3.42185
$173$	7.00614	0.532667	0.266334	−	0.963881i	$-0.414188\pi$
$173$	7.00614	0.532667	0.266334	+	0.963881i	$0.414188\pi$
$174$	−4.60842	−0.349363
$175$	−9.11044	−0.688684
$176$	−32.3922	−2.44165
$177$	−4.37464	−0.328818
$178$	19.8017	1.48420
$179$	0.505220	0.0377619	0.0188810	−	0.999822i	$-0.493990\pi$
$179$	0.505220	0.0377619	0.0188810	+	0.999822i	$0.493990\pi$
$180$	−44.9883	−3.35323
$181$	10.7852	0.801656	0.400828	−	0.916153i	$-0.368723\pi$
$181$	10.7852	0.801656	0.400828	+	0.916153i	$0.368723\pi$
$182$	−14.8500	−1.10076
$183$	1.10526	0.0817035
$184$	−27.6435	−2.03791
$185$	−11.4239	−0.839904
$186$	5.73327	0.420384
$187$	4.23073	0.309382
$188$	12.9380	0.943604
$189$	3.32851	0.242114
$190$	−42.8985	−3.11218
$191$	11.9747	0.866457	0.433228	−	0.901284i	$-0.357374\pi$
$191$	11.9747	0.866457	0.433228	+	0.901284i	$0.357374\pi$
$192$	2.85310	0.205905
$193$	10.7997	0.777377	0.388688	−	0.921369i	$-0.372928\pi$
$193$	10.7997	0.777377	0.388688	+	0.921369i	$0.372928\pi$
$194$	3.00661	0.215862
$195$	−3.39319	−0.242992
$196$	−19.3278	−1.38055
$197$	−13.9138	−0.991320	−0.495660	−	0.868517i	$-0.665074\pi$
$197$	−13.9138	−0.991320	−0.495660	+	0.868517i	$0.665074\pi$
$198$	−24.8632	−1.76695
$199$	14.0572	0.996488	0.498244	−	0.867037i	$-0.333978\pi$
$199$	14.0572	0.996488	0.498244	+	0.867037i	$0.333978\pi$
$200$	−39.2131	−2.77278
$201$	3.26164	0.230058
$202$	−32.6028	−2.29393
$203$	−9.41346	−0.660696
$204$	−2.03515	−0.142489
$205$	−2.31674	−0.161808
$206$	−47.7351	−3.32586
$207$	−10.6963	−0.743441
$208$	−32.2211	−2.23413
$209$	−16.7929	−1.16159
$210$	4.73218	0.326552
$211$	16.3334	1.12443	0.562217	−	0.826990i	$-0.309948\pi$
$211$	16.3334	1.12443	0.562217	+	0.826990i	$0.309948\pi$
$212$	29.6339	2.03526
$213$	−1.06448	−0.0729368
$214$	−51.2913	−3.50620
$215$	29.5718	2.01678
$216$	14.3266	0.974799
$217$	11.7112	0.795006
$218$	−14.3172	−0.969681
$219$	1.72016	0.116237
$220$	−50.9888	−3.43766
$221$	4.20838	0.283087
$222$	3.03707	0.203835
$223$	−1.87732	−0.125715	−0.0628573	−	0.998023i	$-0.520021\pi$
$223$	−1.87732	−0.125715	−0.0628573	+	0.998023i	$0.520021\pi$
$224$	18.9350	1.26515
$225$	−15.1729	−1.01153
$226$	20.7681	1.38147
$227$	20.7283	1.37579	0.687894	−	0.725811i	$-0.258535\pi$
$227$	20.7283	1.37579	0.687894	+	0.725811i	$0.258535\pi$
$228$	8.07808	0.534984
$229$	2.74562	0.181435	0.0907177	−	0.995877i	$-0.471084\pi$
$229$	2.74562	0.181435	0.0907177	+	0.995877i	$0.471084\pi$
$230$	−30.9686	−2.04201
$231$	1.85245	0.121882
$232$	−40.5174	−2.66010
$233$	8.54233	0.559627	0.279813	−	0.960054i	$-0.409727\pi$
$233$	8.54233	0.559627	0.279813	+	0.960054i	$0.409727\pi$
$234$	−24.7319	−1.61678
$235$	8.52553	0.556144
$236$	−65.3895	−4.25649
$237$	2.38643	0.155015
$238$	−5.86906	−0.380434
$239$	24.4710	1.58290	0.791448	−	0.611236i	$-0.209327\pi$
$239$	24.4710	1.58290	0.791448	+	0.611236i	$0.209327\pi$
$240$	10.2677	0.662778
$241$	22.6833	1.46116	0.730581	−	0.682826i	$-0.239249\pi$
$241$	22.6833	1.46116	0.730581	+	0.682826i	$0.239249\pi$
$242$	0.624393	0.0401375
$243$	8.36483	0.536604
$244$	16.5208	1.05764
$245$	−12.7360	−0.813676
$246$	0.615910	0.0392690
$247$	−16.7042	−1.06286
$248$	50.4072	3.20086
$249$	−5.73989	−0.363750
$250$	−2.02902	−0.128326
$251$	−13.5024	−0.852262	−0.426131	−	0.904661i	$-0.640124\pi$
$251$	−13.5024	−0.852262	−0.426131	+	0.904661i	$0.640124\pi$
$252$	24.4308	1.53899
$253$	−12.1229	−0.762160
$254$	11.5273	0.723285
$255$	−1.34106	−0.0839807
$256$	−14.4123	−0.900769
$257$	15.5487	0.969903	0.484951	−	0.874541i	$-0.338837\pi$
$257$	15.5487	0.969903	0.484951	+	0.874541i	$0.338837\pi$
$258$	−7.86171	−0.489449
$259$	6.20373	0.385481
$260$	−50.7194	−3.14549
$261$	−15.6776	−0.970420
$262$	7.21758	0.445903
$263$	−15.1991	−0.937215	−0.468608	−	0.883406i	$-0.655244\pi$
$263$	−15.1991	−0.937215	−0.468608	+	0.883406i	$0.655244\pi$
$264$	7.97330	0.490722
$265$	19.5273	1.19955
$266$	23.2959	1.42836
$267$	−2.45709	−0.150372
$268$	48.7530	2.97806
$269$	−20.0522	−1.22261	−0.611303	−	0.791397i	$-0.709354\pi$
$269$	−20.0522	−1.22261	−0.611303	+	0.791397i	$0.709354\pi$
$270$	16.0498	0.976763
$271$	29.7172	1.80519	0.902596	−	0.430489i	$-0.141659\pi$
$271$	29.7172	1.80519	0.902596	+	0.430489i	$0.141659\pi$
$272$	−12.7345	−0.772140
$273$	1.84266	0.111523
$274$	−12.6778	−0.765892
$275$	−17.1967	−1.03700
$276$	5.83160	0.351021
$277$	18.8224	1.13093	0.565464	−	0.824773i	$-0.308697\pi$
$277$	18.8224	1.13093	0.565464	+	0.824773i	$0.308697\pi$
$278$	40.8529	2.45020
$279$	19.5043	1.16769
$280$	41.6055	2.48641
$281$	−15.1757	−0.905306	−0.452653	−	0.891687i	$-0.649522\pi$
$281$	−15.1757	−0.905306	−0.452653	+	0.891687i	$0.649522\pi$
$282$	−2.26653	−0.134970
$283$	−10.8847	−0.647028	−0.323514	−	0.946223i	$-0.604864\pi$
$283$	−10.8847	−0.647028	−0.323514	+	0.946223i	$0.604864\pi$
$284$	−15.9112	−0.944155
$285$	5.32305	0.315310
$286$	−28.0306	−1.65748
$287$	1.25810	0.0742633
$288$	31.5353	1.85823
$289$	−15.3368	−0.902162
$290$	−45.3910	−2.66545
$291$	−0.373074	−0.0218700
$292$	25.7119	1.50467
$293$	−17.0909	−0.998459	−0.499230	−	0.866470i	$-0.666384\pi$
$293$	−17.0909	−0.998459	−0.499230	+	0.866470i	$0.666384\pi$
$294$	3.38590	0.197470
$295$	−43.0884	−2.50871
$296$	26.7021	1.55203
$297$	6.28283	0.364567
$298$	−23.2905	−1.34918
$299$	−12.0589	−0.697382
$300$	8.27229	0.477601
$301$	−16.0589	−0.925617
$302$	3.50119	0.201471
$303$	4.04552	0.232409
$304$	50.5466	2.89904
$305$	10.8864	0.623354
$306$	−9.77459	−0.558776
$307$	28.2870	1.61443	0.807213	−	0.590260i	$-0.200975\pi$
$307$	28.2870	1.61443	0.807213	+	0.590260i	$0.200975\pi$
$308$	27.6893	1.57774
$309$	5.92320	0.336959
$310$	56.4705	3.20731
$311$	−4.76512	−0.270205	−0.135103	−	0.990832i	$-0.543136\pi$
$311$	−4.76512	−0.270205	−0.135103	+	0.990832i	$0.543136\pi$
$312$	7.93118	0.449015
$313$	11.6889	0.660694	0.330347	−	0.943860i	$-0.392834\pi$
$313$	11.6889	0.660694	0.330347	+	0.943860i	$0.392834\pi$
$314$	−10.7760	−0.608125
$315$	16.0987	0.907057
$316$	35.6710	2.00665
$317$	−10.0510	−0.564523	−0.282261	−	0.959338i	$-0.591085\pi$
$317$	−10.0510	−0.564523	−0.282261	+	0.959338i	$0.591085\pi$
$318$	−5.19136	−0.291117
$319$	−17.7687	−0.994854
$320$	28.1019	1.57094
$321$	6.36447	0.355230
$322$	16.8174	0.937198
$323$	−6.60188	−0.367338
$324$	39.1499	2.17499
$325$	−17.1058	−0.948861
$326$	−38.7266	−2.14487
$327$	1.77654	0.0982431
$328$	5.41511	0.298999
$329$	−4.62976	−0.255247
$330$	8.93237	0.491711
$331$	−9.75415	−0.536136	−0.268068	−	0.963400i	$-0.586385\pi$
$331$	−9.75415	−0.536136	−0.268068	+	0.963400i	$0.586385\pi$
$332$	−85.7964	−4.70869
$333$	10.3320	0.566189
$334$	−24.4777	−1.33936
$335$	32.1258	1.75522
$336$	−5.57585	−0.304188
$337$	21.4057	1.16604	0.583022	−	0.812456i	$-0.301870\pi$
$337$	21.4057	1.16604	0.583022	+	0.812456i	$0.301870\pi$
$338$	6.15837	0.334971
$339$	−2.57700	−0.139963
$340$	−20.0454	−1.08712
$341$	22.1058	1.19709
$342$	38.7980	2.09796
$343$	19.0818	1.03032
$344$	−69.1205	−3.72673
$345$	3.84274	0.206886
$346$	−18.3458	−0.986277
$347$	26.5759	1.42667	0.713334	−	0.700824i	$-0.247184\pi$
$347$	26.5759	1.42667	0.713334	+	0.700824i	$0.247184\pi$
$348$	8.54744	0.458191
$349$	6.02138	0.322317	0.161158	−	0.986929i	$-0.448477\pi$
$349$	6.02138	0.322317	0.161158	+	0.986929i	$0.448477\pi$
$350$	23.8560	1.27516
$351$	6.24964	0.333581
$352$	35.7414	1.90502
$353$	−12.8008	−0.681317	−0.340658	−	0.940187i	$-0.610650\pi$
$353$	−12.8008	−0.681317	−0.340658	+	0.940187i	$0.610650\pi$
$354$	11.4551	0.608833
$355$	−10.4847	−0.556469
$356$	−36.7271	−1.94653
$357$	0.728261	0.0385436
$358$	−1.32294	−0.0699193
$359$	−35.3163	−1.86392	−0.931961	−	0.362558i	$-0.881903\pi$
$359$	−35.3163	−1.86392	−0.931961	+	0.362558i	$0.881903\pi$
$360$	69.2918	3.65200
$361$	7.20466	0.379193
$362$	−28.2413	−1.48433
$363$	−0.0774777	−0.00406653
$364$	27.5430	1.44365
$365$	16.9429	0.886829
$366$	−2.89417	−0.151281
$367$	24.1677	1.26154	0.630772	−	0.775968i	$-0.282738\pi$
$367$	24.1677	1.26154	0.630772	+	0.775968i	$0.282738\pi$
$368$	36.4898	1.90216
$369$	2.09530	0.109077
$370$	29.9139	1.55515
$371$	−10.6042	−0.550544
$372$	−10.6338	−0.551335
$373$	17.0262	0.881585	0.440792	−	0.897609i	$-0.354697\pi$
$373$	17.0262	0.881585	0.440792	+	0.897609i	$0.354697\pi$
$374$	−11.0783	−0.572846
$375$	0.251770	0.0130014
$376$	−19.9274	−1.02768
$377$	−17.6748	−0.910299
$378$	−8.71582	−0.448293
$379$	−1.88722	−0.0969402	−0.0484701	−	0.998825i	$-0.515435\pi$
$379$	−1.88722	−0.0969402	−0.0484701	+	0.998825i	$0.515435\pi$
$380$	79.5658	4.08164
$381$	−1.43036	−0.0732794
$382$	−31.3561	−1.60432
$383$	−25.4242	−1.29911	−0.649557	−	0.760313i	$-0.725045\pi$
$383$	−25.4242	−1.29911	−0.649557	+	0.760313i	$0.725045\pi$
$384$	−0.390811	−0.0199435
$385$	18.2459	0.929895
$386$	−28.2793	−1.43938
$387$	−26.7452	−1.35953
$388$	−5.57649	−0.283103
$389$	−20.4497	−1.03684	−0.518420	−	0.855126i	$-0.673480\pi$
$389$	−20.4497	−1.03684	−0.518420	+	0.855126i	$0.673480\pi$
$390$	8.88519	0.449919
$391$	−4.76593	−0.241023
$392$	29.7690	1.50356
$393$	−0.895592	−0.0451766
$394$	36.4338	1.83551
$395$	23.5054	1.18269
$396$	46.1150	2.31737
$397$	5.97801	0.300028	0.150014	−	0.988684i	$-0.452068\pi$
$397$	5.97801	0.300028	0.150014	+	0.988684i	$0.452068\pi$
$398$	−36.8092	−1.84508
$399$	−2.89067	−0.144714
$400$	51.7618	2.58809
$401$	6.94312	0.346723	0.173361	−	0.984858i	$-0.444537\pi$
$401$	6.94312	0.346723	0.173361	+	0.984858i	$0.444537\pi$
$402$	−8.54070	−0.425971
$403$	21.9890	1.09535
$404$	60.4700	3.00849
$405$	25.7978	1.28190
$406$	24.6495	1.22333
$407$	11.7100	0.580445
$408$	3.13458	0.155185
$409$	18.6251	0.920953	0.460476	−	0.887672i	$-0.347679\pi$
$409$	18.6251	0.920953	0.460476	+	0.887672i	$0.347679\pi$
$410$	6.06647	0.299602
$411$	1.57312	0.0775962
$412$	88.5365	4.36188
$413$	23.3990	1.15139
$414$	28.0085	1.37654
$415$	−56.5356	−2.77522
$416$	35.5526	1.74311
$417$	−5.06923	−0.248241
$418$	43.9728	2.15078
$419$	30.1991	1.47532	0.737661	−	0.675171i	$-0.235930\pi$
$419$	30.1991	1.47532	0.737661	+	0.675171i	$0.235930\pi$
$420$	−8.77699	−0.428273
$421$	1.53742	0.0749291	0.0374645	−	0.999298i	$-0.488072\pi$
$421$	1.53742	0.0749291	0.0374645	+	0.999298i	$0.488072\pi$
$422$	−42.7694	−2.08198
$423$	−7.71062	−0.374903
$424$	−45.6426	−2.21660
$425$	−6.76060	−0.327937
$426$	2.78737	0.135048
$427$	−5.91183	−0.286094
$428$	95.1323	4.59839
$429$	3.47817	0.167928
$430$	−77.4347	−3.73423
$431$	8.60224	0.414355	0.207178	−	0.978303i	$-0.433572\pi$
$431$	8.60224	0.414355	0.207178	+	0.978303i	$0.433572\pi$
$432$	−18.9113	−0.909869
$433$	25.7596	1.23793	0.618963	−	0.785420i	$-0.287553\pi$
$433$	25.7596	1.23793	0.618963	+	0.785420i	$0.287553\pi$
$434$	−30.6661	−1.47202
$435$	5.63234	0.270050
$436$	26.5547	1.27174
$437$	18.9173	0.904936
$438$	−4.50429	−0.215223
$439$	−8.34558	−0.398313	−0.199156	−	0.979968i	$-0.563820\pi$
$439$	−8.34558	−0.398313	−0.199156	+	0.979968i	$0.563820\pi$
$440$	78.5338	3.74395
$441$	11.5187	0.548508
$442$	−11.0198	−0.524158
$443$	−1.31124	−0.0622990	−0.0311495	−	0.999515i	$-0.509917\pi$
$443$	−1.31124	−0.0622990	−0.0311495	+	0.999515i	$0.509917\pi$
$444$	−5.63299	−0.267330
$445$	−24.2014	−1.14725
$446$	4.91582	0.232771
$447$	2.88999	0.136692
$448$	−15.2606	−0.720997
$449$	27.7650	1.31031	0.655155	−	0.755494i	$-0.272603\pi$
$449$	27.7650	1.31031	0.655155	+	0.755494i	$0.272603\pi$
$450$	39.7308	1.87293
$451$	2.37476	0.111823
$452$	−38.5195	−1.81180
$453$	−0.434445	−0.0204120
$454$	−54.2778	−2.54738
$455$	18.1495	0.850861
$456$	−12.4420	−0.582649
$457$	−40.6027	−1.89931	−0.949657	−	0.313290i	$-0.898569\pi$
$457$	−40.6027	−1.89931	−0.949657	+	0.313290i	$0.898569\pi$
$458$	−7.18948	−0.335942
$459$	2.47000	0.115290
$460$	57.4389	2.67810
$461$	−25.9215	−1.20728	−0.603642	−	0.797255i	$-0.706284\pi$
$461$	−25.9215	−1.20728	−0.603642	+	0.797255i	$0.706284\pi$
$462$	−4.85069	−0.225675
$463$	−23.3167	−1.08362	−0.541808	−	0.840502i	$-0.682260\pi$
$463$	−23.3167	−1.08362	−0.541808	+	0.840502i	$0.682260\pi$
$464$	53.4835	2.48291
$465$	−7.00712	−0.324948
$466$	−22.3684	−1.03619
$467$	−2.75941	−0.127690	−0.0638451	−	0.997960i	$-0.520336\pi$
$467$	−2.75941	−0.127690	−0.0638451	+	0.997960i	$0.520336\pi$
$468$	45.8714	2.12041
$469$	−17.4458	−0.805573
$470$	−22.3244	−1.02975
$471$	1.33714	0.0616120
$472$	100.714	4.63574
$473$	−30.3124	−1.39376
$474$	−6.24895	−0.287024
$475$	26.8347	1.23126
$476$	10.8856	0.498941
$477$	−17.6608	−0.808630
$478$	−64.0781	−2.93086
$479$	−9.04738	−0.413385	−0.206693	−	0.978406i	$-0.566270\pi$
$479$	−9.04738	−0.413385	−0.206693	+	0.978406i	$0.566270\pi$
$480$	−11.3294	−0.517112
$481$	11.6482	0.531111
$482$	−59.3970	−2.70546
$483$	−2.08679	−0.0949520
$484$	−1.15809	−0.0526405
$485$	−3.67463	−0.166856
$486$	−21.9036	−0.993566
$487$	39.2167	1.77708	0.888540	−	0.458800i	$-0.151720\pi$
$487$	39.2167	1.77708	0.888540	+	0.458800i	$0.151720\pi$
$488$	−25.4457	−1.15187
$489$	4.80538	0.217307
$490$	33.3497	1.50659
$491$	3.39730	0.153318	0.0766589	−	0.997057i	$-0.475575\pi$
$491$	3.39730	0.153318	0.0766589	+	0.997057i	$0.475575\pi$
$492$	−1.14236	−0.0515014
$493$	−6.98547	−0.314610
$494$	43.7406	1.96798
$495$	30.3875	1.36582
$496$	−66.5382	−2.98765
$497$	5.69367	0.255396
$498$	15.0301	0.673514
$499$	−6.07404	−0.271912	−0.135956	−	0.990715i	$-0.543411\pi$
$499$	−6.07404	−0.271912	−0.135956	+	0.990715i	$0.543411\pi$
$500$	3.76331	0.168301
$501$	3.03731	0.135697
$502$	35.3564	1.57803
$503$	14.6335	0.652477	0.326238	−	0.945288i	$-0.394219\pi$
$503$	14.6335	0.652477	0.326238	+	0.945288i	$0.394219\pi$
$504$	−37.6287	−1.67611
$505$	39.8467	1.77316
$506$	31.7442	1.41120
$507$	−0.764160	−0.0339376
$508$	−21.3801	−0.948590
$509$	33.1536	1.46951	0.734754	−	0.678334i	$-0.237298\pi$
$509$	33.1536	1.46951	0.734754	+	0.678334i	$0.237298\pi$
$510$	3.51162	0.155497
$511$	−9.20076	−0.407018
$512$	40.1446	1.77416
$513$	−9.80409	−0.432861
$514$	−40.7148	−1.79585
$515$	58.3412	2.57082
$516$	14.5815	0.641913
$517$	−8.73904	−0.384343
$518$	−16.2447	−0.713749
$519$	2.27644	0.0999244
$520$	78.1189	3.42574
$521$	−14.4652	−0.633734	−0.316867	−	0.948470i	$-0.602631\pi$
$521$	−14.4652	−0.633734	−0.316867	+	0.948470i	$0.602631\pi$
$522$	41.0523	1.79681
$523$	−16.4010	−0.717163	−0.358582	−	0.933498i	$-0.616740\pi$
$523$	−16.4010	−0.717163	−0.358582	+	0.933498i	$0.616740\pi$
$524$	−13.3868	−0.584804
$525$	−2.96016	−0.129192
$526$	39.7993	1.73533
$527$	8.69054	0.378566
$528$	−10.5249	−0.458036
$529$	−9.34353	−0.406240
$530$	−51.1328	−2.22107
$531$	38.9698	1.69115
$532$	−43.2079	−1.87330
$533$	2.36222	0.102319
$534$	6.43397	0.278425
$535$	62.6875	2.71022
$536$	−75.0902	−3.24340
$537$	0.164156	0.00708386
$538$	52.5074	2.26375
$539$	13.0550	0.562319
$540$	−29.7684	−1.28103
$541$	−9.95701	−0.428085	−0.214043	−	0.976824i	$-0.568663\pi$
$541$	−9.95701	−0.428085	−0.214043	+	0.976824i	$0.568663\pi$
$542$	−77.8155	−3.34246
$543$	3.50432	0.150385
$544$	14.0512	0.602439
$545$	17.4982	0.749542
$546$	−4.82507	−0.206494
$547$	3.74729	0.160222	0.0801112	−	0.996786i	$-0.474472\pi$
$547$	3.74729	0.160222	0.0801112	+	0.996786i	$0.474472\pi$
$548$	23.5140	1.00447
$549$	−9.84583	−0.420210
$550$	45.0300	1.92009
$551$	27.7272	1.18122
$552$	−8.98193	−0.382296
$553$	−12.7645	−0.542803
$554$	−49.2870	−2.09400
$555$	−3.71186	−0.157560
$556$	−75.7718	−3.21344
$557$	7.67657	0.325267	0.162633	−	0.986687i	$-0.448001\pi$
$557$	7.67657	0.325267	0.162633	+	0.986687i	$0.448001\pi$
$558$	−51.0727	−2.16208
$559$	−30.1523	−1.27530
$560$	−54.9199	−2.32079
$561$	1.37465	0.0580378
$562$	39.7381	1.67625
$563$	25.6263	1.08002	0.540011	−	0.841658i	$-0.318420\pi$
$563$	25.6263	1.08002	0.540011	+	0.841658i	$0.318420\pi$
$564$	4.20383	0.177013
$565$	−25.3824	−1.06785
$566$	28.5019	1.19802
$567$	−14.0094	−0.588341
$568$	24.5067	1.02828
$569$	−23.4931	−0.984883	−0.492442	−	0.870345i	$-0.663896\pi$
$569$	−23.4931	−0.984883	−0.492442	+	0.870345i	$0.663896\pi$
$570$	−13.9386	−0.583823
$571$	9.77704	0.409156	0.204578	−	0.978850i	$-0.434418\pi$
$571$	9.77704	0.409156	0.204578	+	0.978850i	$0.434418\pi$
$572$	51.9896	2.17380
$573$	3.89081	0.162541
$574$	−3.29438	−0.137505
$575$	19.3721	0.807872
$576$	−25.4157	−1.05899
$577$	−7.00977	−0.291821	−0.145910	−	0.989298i	$-0.546611\pi$
$577$	−7.00977	−0.291821	−0.145910	+	0.989298i	$0.546611\pi$
$578$	40.1598	1.67043
$579$	3.50903	0.145830
$580$	84.1888	3.49575
$581$	30.7014	1.27371
$582$	0.976907	0.0404941
$583$	−20.0163	−0.828991
$584$	−39.6019	−1.63874
$585$	30.2270	1.24973
$586$	44.7530	1.84873
$587$	−15.5909	−0.643505	−0.321752	−	0.946824i	$-0.604272\pi$
$587$	−15.5909	−0.643505	−0.321752	+	0.946824i	$0.604272\pi$
$588$	−6.27998	−0.258982
$589$	−34.4951	−1.42135
$590$	112.828	4.64507
$591$	−4.52088	−0.185964
$592$	−35.2471	−1.44865
$593$	−31.1264	−1.27821	−0.639105	−	0.769120i	$-0.720695\pi$
$593$	−31.1264	−1.27821	−0.639105	+	0.769120i	$0.720695\pi$
$594$	−16.4518	−0.675025
$595$	7.17307	0.294067
$596$	43.1979	1.76945
$597$	4.56746	0.186934
$598$	31.5765	1.29126
$599$	−1.45679	−0.0595230	−0.0297615	−	0.999557i	$-0.509475\pi$
$599$	−1.45679	−0.0595230	−0.0297615	+	0.999557i	$0.509475\pi$
$600$	−12.7411	−0.520154
$601$	33.7071	1.37494	0.687470	−	0.726212i	$-0.258721\pi$
$601$	33.7071	1.37494	0.687470	+	0.726212i	$0.258721\pi$
$602$	42.0506	1.71386
$603$	−29.0551	−1.18321
$604$	−6.49382	−0.264230
$605$	−0.763124	−0.0310254
$606$	−10.5933	−0.430324
$607$	16.4836	0.669048	0.334524	−	0.942387i	$-0.391424\pi$
$607$	16.4836	0.669048	0.334524	+	0.942387i	$0.391424\pi$
$608$	−55.7729	−2.26189
$609$	−3.05862	−0.123942
$610$	−28.5064	−1.15419
$611$	−8.69288	−0.351676
$612$	18.1294	0.732837
$613$	−3.14162	−0.126889	−0.0634445	−	0.997985i	$-0.520209\pi$
$613$	−3.14162	−0.126889	−0.0634445	+	0.997985i	$0.520209\pi$
$614$	−74.0705	−2.98924
$615$	−0.752757	−0.0303541
$616$	−42.6475	−1.71832
$617$	−27.1917	−1.09470	−0.547348	−	0.836905i	$-0.684363\pi$
$617$	−27.1917	−1.09470	−0.547348	+	0.836905i	$0.684363\pi$
$618$	−15.5101	−0.623908
$619$	−22.1832	−0.891618	−0.445809	−	0.895128i	$-0.647084\pi$
$619$	−22.1832	−0.891618	−0.445809	+	0.895128i	$0.647084\pi$
$620$	−104.738	−4.20639
$621$	−7.07762	−0.284015
$622$	12.4776	0.500307
$623$	13.1425	0.526542
$624$	−10.4693	−0.419106
$625$	−23.7308	−0.949231
$626$	−30.6077	−1.22333
$627$	−5.45636	−0.217906
$628$	19.9867	0.797557
$629$	4.60361	0.183558
$630$	−42.1549	−1.67949
$631$	17.4955	0.696486	0.348243	−	0.937404i	$-0.386778\pi$
$631$	17.4955	0.696486	0.348243	+	0.937404i	$0.386778\pi$
$632$	−54.9410	−2.18544
$633$	5.30703	0.210936
$634$	26.3190	1.04526
$635$	−14.0884	−0.559083
$636$	9.62865	0.381801
$637$	12.9860	0.514526
$638$	46.5278	1.84205
$639$	9.48250	0.375122
$640$	−3.84933	−0.152158
$641$	−13.7943	−0.544843	−0.272422	−	0.962178i	$-0.587825\pi$
$641$	−13.7943	−0.544843	−0.272422	+	0.962178i	$0.587825\pi$
$642$	−16.6656	−0.657738
$643$	3.49836	0.137962	0.0689810	−	0.997618i	$-0.478025\pi$
$643$	3.49836	0.137962	0.0689810	+	0.997618i	$0.478025\pi$
$644$	−31.1920	−1.22914
$645$	9.60847	0.378333
$646$	17.2872	0.680157
$647$	−0.616381	−0.0242324	−0.0121162	−	0.999927i	$-0.503857\pi$
$647$	−0.616381	−0.0242324	−0.0121162	+	0.999927i	$0.503857\pi$
$648$	−60.2993	−2.36878
$649$	44.1675	1.73373
$650$	44.7922	1.75689
$651$	3.80520	0.149137
$652$	71.8280	2.81300
$653$	−21.6364	−0.846698	−0.423349	−	0.905967i	$-0.639146\pi$
$653$	−21.6364	−0.846698	−0.423349	+	0.905967i	$0.639146\pi$
$654$	−4.65194	−0.181905
$655$	−8.82122	−0.344673
$656$	−7.14802	−0.279083
$657$	−15.3234	−0.597821
$658$	12.1232	0.472611
$659$	−7.97960	−0.310841	−0.155421	−	0.987848i	$-0.549673\pi$
$659$	−7.97960	−0.310841	−0.155421	+	0.987848i	$0.549673\pi$
$660$	−16.5673	−0.644880
$661$	46.3360	1.80226	0.901131	−	0.433547i	$-0.142738\pi$
$661$	46.3360	1.80226	0.901131	+	0.433547i	$0.142738\pi$
$662$	25.5415	0.992700
$663$	1.36739	0.0531050
$664$	132.145	5.12822
$665$	−28.4719	−1.10409
$666$	−27.0546	−1.04834
$667$	20.0164	0.775039
$668$	45.3999	1.75658
$669$	−0.609978	−0.0235831
$670$	−84.1225	−3.24993
$671$	−11.1591	−0.430790
$672$	6.15237	0.237333
$673$	−26.7605	−1.03154	−0.515772	−	0.856726i	$-0.672495\pi$
$673$	−26.7605	−1.03154	−0.515772	+	0.856726i	$0.672495\pi$
$674$	−56.0516	−2.15903
$675$	−10.0398	−0.386432
$676$	−11.4222	−0.439316
$677$	−44.2044	−1.69891	−0.849457	−	0.527659i	$-0.823070\pi$
$677$	−44.2044	−1.69891	−0.849457	+	0.527659i	$0.823070\pi$
$678$	6.74796	0.259154
$679$	1.99550	0.0765801
$680$	30.8743	1.18398
$681$	6.73505	0.258088
$682$	−57.8847	−2.21652
$683$	36.5398	1.39816	0.699078	−	0.715046i	$-0.253594\pi$
$683$	36.5398	1.39816	0.699078	+	0.715046i	$0.253594\pi$
$684$	−71.9605	−2.75148
$685$	15.4946	0.592018
$686$	−49.9662	−1.90772
$687$	0.892105	0.0340359
$688$	91.2400	3.47849
$689$	−19.9106	−0.758533
$690$	−10.0623	−0.383066
$691$	28.0437	1.06683	0.533416	−	0.845853i	$-0.320908\pi$
$691$	28.0437	1.06683	0.533416	+	0.845853i	$0.320908\pi$
$692$	34.0268	1.29350
$693$	−16.5018	−0.626853
$694$	−69.5898	−2.64159
$695$	−49.9298	−1.89395
$696$	−13.1649	−0.499015
$697$	0.933601	0.0353627
$698$	−15.7672	−0.596796
$699$	2.77557	0.104982
$700$	−44.2468	−1.67237
$701$	−19.5229	−0.737369	−0.368684	−	0.929555i	$-0.620192\pi$
$701$	−19.5229	−0.737369	−0.368684	+	0.929555i	$0.620192\pi$
$702$	−16.3649	−0.617653
$703$	−18.2730	−0.689179
$704$	−28.8056	−1.08565
$705$	2.77012	0.104329
$706$	33.5193	1.26151
$707$	−21.6386	−0.813804
$708$	−21.2464	−0.798487
$709$	−5.55378	−0.208577	−0.104288	−	0.994547i	$-0.533256\pi$
$709$	−5.55378	−0.208577	−0.104288	+	0.994547i	$0.533256\pi$
$710$	27.4545	1.03035
$711$	−21.2586	−0.797261
$712$	56.5678	2.11997
$713$	−24.9022	−0.932594
$714$	−1.90697	−0.0713667
$715$	34.2586	1.28120
$716$	2.45371	0.0916994
$717$	7.95111	0.296940
$718$	92.4768	3.45120
$719$	32.2793	1.20382	0.601908	−	0.798565i	$-0.294407\pi$
$719$	32.2793	1.20382	0.601908	+	0.798565i	$0.294407\pi$
$720$	−91.4661	−3.40874
$721$	−31.6820	−1.17990
$722$	−18.8656	−0.702106
$723$	7.37027	0.274103
$724$	52.3805	1.94671
$725$	28.3939	1.05452
$726$	0.202878	0.00752950
$727$	35.8207	1.32852	0.664258	−	0.747503i	$-0.268748\pi$
$727$	35.8207	1.32852	0.664258	+	0.747503i	$0.268748\pi$
$728$	−42.4222	−1.57227
$729$	−21.4651	−0.795002
$730$	−44.3654	−1.64204
$731$	−11.9168	−0.440760
$732$	5.36795	0.198405
$733$	−32.8720	−1.21416	−0.607078	−	0.794643i	$-0.707658\pi$
$733$	−32.8720	−1.21416	−0.607078	+	0.794643i	$0.707658\pi$
$734$	−63.2839	−2.33585
$735$	−4.13820	−0.152640
$736$	−40.2627	−1.48410
$737$	−32.9304	−1.21301
$738$	−5.48660	−0.201965
$739$	−7.14724	−0.262916	−0.131458	−	0.991322i	$-0.541966\pi$
$739$	−7.14724	−0.262916	−0.131458	+	0.991322i	$0.541966\pi$
$740$	−55.4827	−2.03959
$741$	−5.42754	−0.199386
$742$	27.7675	1.01938
$743$	40.7829	1.49618	0.748090	−	0.663598i	$-0.230971\pi$
$743$	40.7829	1.49618	0.748090	+	0.663598i	$0.230971\pi$
$744$	16.3783	0.600458
$745$	28.4653	1.04289
$746$	−44.5837	−1.63233
$747$	51.1316	1.87081
$748$	20.5474	0.751289
$749$	−34.0422	−1.24388
$750$	−0.659269	−0.0240731
$751$	−5.33905	−0.194825	−0.0974123	−	0.995244i	$-0.531057\pi$
$751$	−5.33905	−0.194825	−0.0974123	+	0.995244i	$0.531057\pi$
$752$	26.3044	0.959224
$753$	−4.38719	−0.159878
$754$	46.2820	1.68549
$755$	−4.27911	−0.155733
$756$	16.1656	0.587938
$757$	−1.48769	−0.0540711	−0.0270355	−	0.999634i	$-0.508607\pi$
$757$	−1.48769	−0.0540711	−0.0270355	+	0.999634i	$0.508607\pi$
$758$	4.94176	0.179493
$759$	−3.93897	−0.142976
$760$	−122.549	−4.44530
$761$	4.02606	0.145945	0.0729723	−	0.997334i	$-0.476752\pi$
$761$	4.02606	0.145945	0.0729723	+	0.997334i	$0.476752\pi$
$762$	3.74544	0.135683
$763$	−9.50236	−0.344009
$764$	58.1575	2.10406
$765$	11.9464	0.431922
$766$	66.5740	2.40541
$767$	43.9342	1.58637
$768$	−4.68284	−0.168978
$769$	−18.6933	−0.674098	−0.337049	−	0.941487i	$-0.609429\pi$
$769$	−18.6933	−0.674098	−0.337049	+	0.941487i	$0.609429\pi$
$770$	−47.7774	−1.72178
$771$	5.05209	0.181947
$772$	52.4508	1.88775
$773$	23.5385	0.846620	0.423310	−	0.905985i	$-0.360868\pi$
$773$	23.5385	0.846620	0.423310	+	0.905985i	$0.360868\pi$
$774$	70.0331	2.51729
$775$	−35.3245	−1.26889
$776$	8.58901	0.308327
$777$	2.01572	0.0723134
$778$	53.5481	1.91979
$779$	−3.70572	−0.132771
$780$	−16.4798	−0.590070
$781$	10.7473	0.384567
$782$	12.4797	0.446274
$783$	−10.3737	−0.370727
$784$	−39.2954	−1.40341
$785$	13.1703	0.470067
$786$	2.34514	0.0836482
$787$	−39.1278	−1.39476	−0.697378	−	0.716704i	$-0.745650\pi$
$787$	−39.1278	−1.39476	−0.697378	+	0.716704i	$0.745650\pi$
$788$	−67.5755	−2.40728
$789$	−4.93848	−0.175815
$790$	−61.5497	−2.18984
$791$	13.7838	0.490097
$792$	−71.0271	−2.52384
$793$	−11.1001	−0.394176
$794$	−15.6536	−0.555526
$795$	6.34480	0.225027
$796$	68.2717	2.41983
$797$	−6.61400	−0.234280	−0.117140	−	0.993115i	$-0.537373\pi$
$797$	−6.61400	−0.234280	−0.117140	+	0.993115i	$0.537373\pi$
$798$	7.56930	0.267950
$799$	−3.43562	−0.121543
$800$	−57.1138	−2.01928
$801$	21.8881	0.773377
$802$	−18.1808	−0.641986
$803$	−17.3672	−0.612874
$804$	15.8408	0.558663
$805$	−20.5540	−0.724433
$806$	−57.5789	−2.02813
$807$	−6.51537	−0.229352
$808$	−93.1369	−3.27654
$809$	−28.1860	−0.990967	−0.495483	−	0.868617i	$-0.665009\pi$
$809$	−28.1860	−0.990967	−0.495483	+	0.868617i	$0.665009\pi$
$810$	−67.5525	−2.37355
$811$	5.80500	0.203841	0.101921	−	0.994793i	$-0.467501\pi$
$811$	5.80500	0.203841	0.101921	+	0.994793i	$0.467501\pi$
$812$	−45.7185	−1.60440
$813$	9.65571	0.338641
$814$	−30.6631	−1.07474
$815$	47.3311	1.65794
$816$	−4.13768	−0.144848
$817$	47.3012	1.65486
$818$	−48.7705	−1.70522
$819$	−16.4147	−0.573575
$820$	−11.2518	−0.392928
$821$	44.8681	1.56591	0.782954	−	0.622079i	$-0.213712\pi$
$821$	44.8681	1.56591	0.782954	+	0.622079i	$0.213712\pi$
$822$	−4.11926	−0.143676
$823$	8.85740	0.308750	0.154375	−	0.988012i	$-0.450664\pi$
$823$	8.85740	0.308750	0.154375	+	0.988012i	$0.450664\pi$
$824$	−136.365	−4.75052
$825$	−5.58754	−0.194533
$826$	−61.2711	−2.13189
$827$	36.0864	1.25485	0.627424	−	0.778678i	$-0.284109\pi$
$827$	36.0864	1.25485	0.627424	+	0.778678i	$0.284109\pi$
$828$	−51.9486	−1.80534
$829$	−45.0080	−1.56319	−0.781596	−	0.623785i	$-0.785594\pi$
$829$	−45.0080	−1.56319	−0.781596	+	0.623785i	$0.785594\pi$
$830$	148.040	5.13855
$831$	6.11577	0.212154
$832$	−28.6535	−0.993381
$833$	5.13237	0.177826
$834$	13.2739	0.459639
$835$	29.9163	1.03530
$836$	−81.5584	−2.82076
$837$	12.9058	0.446091
$838$	−79.0773	−2.73168
$839$	2.74095	0.0946280	0.0473140	−	0.998880i	$-0.484934\pi$
$839$	2.74095	0.0946280	0.0473140	+	0.998880i	$0.484934\pi$
$840$	13.5185	0.466432
$841$	0.338274	0.0116646
$842$	−4.02577	−0.138737
$843$	−4.93089	−0.169829
$844$	79.3264	2.73053
$845$	−7.52667	−0.258925
$846$	20.1905	0.694164
$847$	0.414412	0.0142394
$848$	60.2489	2.06896
$849$	−3.53665	−0.121378
$850$	17.7028	0.607203
$851$	−13.1914	−0.452194
$852$	−5.16986	−0.177116
$853$	44.4999	1.52365	0.761823	−	0.647785i	$-0.224305\pi$
$853$	44.4999	1.52365	0.761823	+	0.647785i	$0.224305\pi$
$854$	15.4803	0.529726
$855$	−47.4184	−1.62167
$856$	−146.524	−5.00810
$857$	−43.5175	−1.48653	−0.743265	−	0.668997i	$-0.766724\pi$
$857$	−43.5175	−1.48653	−0.743265	+	0.668997i	$0.766724\pi$
$858$	−9.10771	−0.310932
$859$	31.6083	1.07846	0.539231	−	0.842158i	$-0.318715\pi$
$859$	31.6083	1.07846	0.539231	+	0.842158i	$0.318715\pi$
$860$	143.622	4.89746
$861$	0.408782	0.0139313
$862$	−22.5253	−0.767213
$863$	−28.2933	−0.963114	−0.481557	−	0.876415i	$-0.659929\pi$
$863$	−28.2933	−0.963114	−0.481557	+	0.876415i	$0.659929\pi$
$864$	20.8666	0.709897
$865$	22.4220	0.762370
$866$	−67.4523	−2.29212
$867$	−4.98322	−0.169239
$868$	56.8778	1.93056
$869$	−24.0941	−0.817335
$870$	−14.7485	−0.500020
$871$	−32.7564	−1.10991
$872$	−40.9000	−1.38505
$873$	3.32339	0.112480
$874$	−49.5355	−1.67556
$875$	−1.34667	−0.0455257
$876$	8.35430	0.282266
$877$	2.83672	0.0957893	0.0478947	−	0.998852i	$-0.484749\pi$
$877$	2.83672	0.0957893	0.0478947	+	0.998852i	$0.484749\pi$
$878$	21.8532	0.737509
$879$	−5.55317	−0.187304
$880$	−103.666	−3.49457
$881$	1.75944	0.0592769	0.0296385	−	0.999561i	$-0.490564\pi$
$881$	1.75944	0.0592769	0.0296385	+	0.999561i	$0.490564\pi$
$882$	−30.1620	−1.01561
$883$	−23.5997	−0.794193	−0.397097	−	0.917777i	$-0.629982\pi$
$883$	−23.5997	−0.794193	−0.397097	+	0.917777i	$0.629982\pi$
$884$	20.4389	0.687435
$885$	−14.0003	−0.470615
$886$	3.43353	0.115352
$887$	54.2579	1.82180	0.910901	−	0.412624i	$-0.135388\pi$
$887$	54.2579	1.82180	0.910901	+	0.412624i	$0.135388\pi$
$888$	8.67603	0.291149
$889$	7.65068	0.256596
$890$	63.3721	2.12424
$891$	−26.4439	−0.885904
$892$	−9.11759	−0.305280
$893$	13.6369	0.456342
$894$	−7.56754	−0.253096
$895$	1.61687	0.0540461
$896$	2.09037	0.0698343
$897$	−3.91817	−0.130824
$898$	−72.7035	−2.42615
$899$	−36.4994	−1.21732
$900$	−73.6906	−2.45635
$901$	−7.86910	−0.262158
$902$	−6.21840	−0.207050
$903$	−5.21785	−0.173639
$904$	59.3283	1.97323
$905$	34.5161	1.14736
$906$	1.13761	0.0377945
$907$	−30.1528	−1.00121	−0.500604	−	0.865676i	$-0.666889\pi$
$907$	−30.1528	−1.00121	−0.500604	+	0.865676i	$0.666889\pi$
$908$	100.672	3.34090
$909$	−36.0380	−1.19530
$910$	−47.5250	−1.57544
$911$	−30.1329	−0.998349	−0.499174	−	0.866502i	$-0.666363\pi$
$911$	−30.1329	−0.998349	−0.499174	+	0.866502i	$0.666363\pi$
$912$	16.4236	0.543840
$913$	57.9514	1.91791
$914$	106.320	3.51674
$915$	3.53721	0.116937
$916$	13.3347	0.440589
$917$	4.79033	0.158191
$918$	−6.46776	−0.213468
$919$	40.8232	1.34663	0.673317	−	0.739354i	$-0.264869\pi$
$919$	40.8232	1.34663	0.673317	+	0.739354i	$0.264869\pi$
$920$	−88.4684	−2.91672
$921$	9.19103	0.302855
$922$	67.8762	2.23539
$923$	10.6905	0.351882
$924$	8.99680	0.295973
$925$	−18.7123	−0.615257
$926$	61.0554	2.00640
$927$	−52.7646	−1.73302
$928$	−59.0135	−1.93721
$929$	25.2548	0.828585	0.414292	−	0.910144i	$-0.364029\pi$
$929$	25.2548	0.828585	0.414292	+	0.910144i	$0.364029\pi$
$930$	18.3484	0.601667
$931$	−20.3718	−0.667658
$932$	41.4876	1.35897
$933$	−1.54828	−0.0506885
$934$	7.22560	0.236429
$935$	13.5397	0.442797
$936$	−70.6519	−2.30933
$937$	−1.61937	−0.0529026	−0.0264513	−	0.999650i	$-0.508421\pi$
$937$	−1.61937	−0.0529026	−0.0264513	+	0.999650i	$0.508421\pi$
$938$	45.6824	1.49158
$939$	3.79795	0.123941
$940$	41.4060	1.35052
$941$	16.0564	0.523423	0.261712	−	0.965146i	$-0.415713\pi$
$941$	16.0564	0.523423	0.261712	+	0.965146i	$0.415713\pi$
$942$	−3.50134	−0.114080
$943$	−2.67518	−0.0871157
$944$	−132.944	−4.32695
$945$	10.6523	0.346521
$946$	79.3739	2.58067
$947$	5.64379	0.183399	0.0916993	−	0.995787i	$-0.470770\pi$
$947$	5.64379	0.183399	0.0916993	+	0.995787i	$0.470770\pi$
$948$	11.5902	0.376433
$949$	−17.2754	−0.560784
$950$	−70.2675	−2.27978
$951$	−3.26578	−0.105900
$952$	−16.7662	−0.543395
$953$	−24.5162	−0.794159	−0.397079	−	0.917784i	$-0.629976\pi$
$953$	−24.5162	−0.794159	−0.397079	+	0.917784i	$0.629976\pi$
$954$	46.2452	1.49725
$955$	38.3229	1.24010
$956$	118.848	3.84383
$957$	−5.77339	−0.186627
$958$	23.6909	0.765417
$959$	−8.41428	−0.271711
$960$	9.13086	0.294697
$961$	14.4085	0.464789
$962$	−30.5011	−0.983395
$963$	−56.6955	−1.82699
$964$	110.166	3.54822
$965$	34.5625	1.11261
$966$	5.46431	0.175811
$967$	60.8983	1.95836	0.979179	−	0.202998i	$-0.0650683\pi$
$967$	60.8983	1.95836	0.979179	+	0.202998i	$0.0650683\pi$
$968$	1.78371	0.0573306
$969$	−2.14508	−0.0689100
$970$	9.62214	0.308948
$971$	−33.0784	−1.06154	−0.530768	−	0.847517i	$-0.678097\pi$
$971$	−33.0784	−1.06154	−0.530768	+	0.847517i	$0.678097\pi$
$972$	40.6255	1.30307
$973$	27.1142	0.869242
$974$	−102.690	−3.29041
$975$	−5.55803	−0.177999
$976$	33.5886	1.07515
$977$	−39.6477	−1.26844	−0.634220	−	0.773152i	$-0.718679\pi$
$977$	−39.6477	−1.26844	−0.634220	+	0.773152i	$0.718679\pi$
$978$	−12.5830	−0.402361
$979$	24.8075	0.792850
$980$	−61.8553	−1.97589
$981$	−15.8257	−0.505275
$982$	−8.89593	−0.283881
$983$	−44.0086	−1.40366	−0.701829	−	0.712346i	$-0.747633\pi$
$983$	−44.0086	−1.40366	−0.701829	+	0.712346i	$0.747633\pi$
$984$	1.75948	0.0560901
$985$	−44.5289	−1.41881
$986$	18.2917	0.582525
$987$	−1.50430	−0.0478825
$988$	−81.1276	−2.58101
$989$	34.1469	1.08581
$990$	−79.5706	−2.52892
$991$	27.3138	0.867650	0.433825	−	0.900997i	$-0.357164\pi$
$991$	27.3138	0.867650	0.433825	+	0.900997i	$0.357164\pi$
$992$	73.4180	2.33102
$993$	−3.16932	−0.100575
$994$	−14.9091	−0.472887
$995$	44.9877	1.42621
$996$	−27.8770	−0.883315
$997$	−1.09526	−0.0346871	−0.0173435	−	0.999850i	$-0.505521\pi$
$997$	−1.09526	−0.0346871	−0.0173435	+	0.999850i	$0.505521\pi$
$998$	15.9051	0.503467
$999$	6.83658	0.216300

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4013.2.a.c.1.9	✓	176

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4013.2.a.c.1.9	✓	176	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:	\( N \)	\(=\)	\( 4013 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4013.a (trivial)

\(f(q)\)	\(=\)	\(q-2.61853 q^{2} +0.324920 q^{3} +4.85671 q^{4} +3.20033 q^{5} -0.850813 q^{6} -1.73793 q^{7} -7.48039 q^{8} -2.89443 q^{9} +O(q^{10})\)	\(q-2.61853 q^{2} +0.324920 q^{3} +4.85671 q^{4} +3.20033 q^{5} -0.850813 q^{6} -1.73793 q^{7} -7.48039 q^{8} -2.89443 q^{9} -8.38017 q^{10} -3.28048 q^{11} +1.57804 q^{12} -3.26315 q^{13} +4.55082 q^{14} +1.03985 q^{15} +9.87421 q^{16} -1.28967 q^{17} +7.57915 q^{18} +5.11905 q^{19} +15.5431 q^{20} -0.564688 q^{21} +8.59004 q^{22} +3.69547 q^{23} -2.43053 q^{24} +5.24212 q^{25} +8.54467 q^{26} -1.91522 q^{27} -8.44062 q^{28} +5.41648 q^{29} -2.72289 q^{30} -6.73858 q^{31} -10.8952 q^{32} -1.06589 q^{33} +3.37704 q^{34} -5.56195 q^{35} -14.0574 q^{36} -3.56961 q^{37} -13.4044 q^{38} -1.06026 q^{39} -23.9397 q^{40} -0.723908 q^{41} +1.47865 q^{42} +9.24023 q^{43} -15.9323 q^{44} -9.26313 q^{45} -9.67670 q^{46} +2.66395 q^{47} +3.20833 q^{48} -3.97960 q^{49} -13.7267 q^{50} -0.419039 q^{51} -15.8482 q^{52} +6.10164 q^{53} +5.01506 q^{54} -10.4986 q^{55} +13.0004 q^{56} +1.66328 q^{57} -14.1832 q^{58} -13.4637 q^{59} +5.05026 q^{60} +3.40165 q^{61} +17.6452 q^{62} +5.03031 q^{63} +8.78092 q^{64} -10.4432 q^{65} +2.79108 q^{66} +10.0383 q^{67} -6.26355 q^{68} +1.20073 q^{69} +14.5641 q^{70} -3.27612 q^{71} +21.6514 q^{72} +5.29409 q^{73} +9.34713 q^{74} +1.70327 q^{75} +24.8617 q^{76} +5.70124 q^{77} +2.77633 q^{78} +7.34468 q^{79} +31.6008 q^{80} +8.06099 q^{81} +1.89558 q^{82} -17.6655 q^{83} -2.74253 q^{84} -4.12737 q^{85} -24.1958 q^{86} +1.75992 q^{87} +24.5393 q^{88} -7.56214 q^{89} +24.2558 q^{90} +5.67113 q^{91} +17.9478 q^{92} -2.18950 q^{93} -6.97565 q^{94} +16.3827 q^{95} -3.54006 q^{96} -1.14820 q^{97} +10.4207 q^{98} +9.49511 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 176 q + 11 q^{2} + 53 q^{3} + 191 q^{4} + 5 q^{5} + 19 q^{6} + 46 q^{7} + 36 q^{8} + 193 q^{9}+O(q^{10}) \) 176 * q + 11 * q^2 + 53 * q^3 + 191 * q^4 + 5 * q^5 + 19 * q^6 + 46 * q^7 + 36 * q^8 + 193 * q^9	\( 176 q + 11 q^{2} + 53 q^{3} + 191 q^{4} + 5 q^{5} + 19 q^{6} + 46 q^{7} + 36 q^{8} + 193 q^{9} + 43 q^{10} + 18 q^{11} + 95 q^{12} + 95 q^{13} + 2 q^{14} + 36 q^{15} + 225 q^{16} + 35 q^{17} + 46 q^{18} + 127 q^{19} + 4 q^{20} + 32 q^{21} + 60 q^{22} + 35 q^{23} + 26 q^{24} + 207 q^{25} + 19 q^{26} + 191 q^{27} + 87 q^{28} + 16 q^{29} + 28 q^{30} + 93 q^{31} + 73 q^{32} + 70 q^{33} + 45 q^{34} + 73 q^{35} + 206 q^{36} + 64 q^{37} + 35 q^{38} + 72 q^{39} + 139 q^{40} + 19 q^{41} + 35 q^{42} + 261 q^{43} + 11 q^{44} + 12 q^{45} + 58 q^{46} + 40 q^{47} + 130 q^{48} + 234 q^{49} - 14 q^{50} + 76 q^{51} + 263 q^{52} + 17 q^{53} + 28 q^{54} + 170 q^{55} - 10 q^{56} + 60 q^{57} + 52 q^{58} + 69 q^{59} + 37 q^{60} + 110 q^{61} + 71 q^{62} + 101 q^{63} + 250 q^{64} - q^{65} + 43 q^{66} + 190 q^{67} + 48 q^{68} + 45 q^{69} + 14 q^{70} + 9 q^{71} + 98 q^{72} + 182 q^{73} - 23 q^{74} + 219 q^{75} + 197 q^{76} + 25 q^{77} - 26 q^{78} + 105 q^{79} + 20 q^{80} + 236 q^{81} + 107 q^{82} + 130 q^{83} + 38 q^{84} + 73 q^{85} - 24 q^{86} + 171 q^{87} + 165 q^{88} + 40 q^{89} + 45 q^{90} + 182 q^{91} - 4 q^{92} + 23 q^{93} + 98 q^{94} + 30 q^{95} - 2 q^{96} + 168 q^{97} + 82 q^{98} + 50 q^{99}+O(q^{100}) \) 176 * q + 11 * q^2 + 53 * q^3 + 191 * q^4 + 5 * q^5 + 19 * q^6 + 46 * q^7 + 36 * q^8 + 193 * q^9 + 43 * q^10 + 18 * q^11 + 95 * q^12 + 95 * q^13 + 2 * q^14 + 36 * q^15 + 225 * q^16 + 35 * q^17 + 46 * q^18 + 127 * q^19 + 4 * q^20 + 32 * q^21 + 60 * q^22 + 35 * q^23 + 26 * q^24 + 207 * q^25 + 19 * q^26 + 191 * q^27 + 87 * q^28 + 16 * q^29 + 28 * q^30 + 93 * q^31 + 73 * q^32 + 70 * q^33 + 45 * q^34 + 73 * q^35 + 206 * q^36 + 64 * q^37 + 35 * q^38 + 72 * q^39 + 139 * q^40 + 19 * q^41 + 35 * q^42 + 261 * q^43 + 11 * q^44 + 12 * q^45 + 58 * q^46 + 40 * q^47 + 130 * q^48 + 234 * q^49 - 14 * q^50 + 76 * q^51 + 263 * q^52 + 17 * q^53 + 28 * q^54 + 170 * q^55 - 10 * q^56 + 60 * q^57 + 52 * q^58 + 69 * q^59 + 37 * q^60 + 110 * q^61 + 71 * q^62 + 101 * q^63 + 250 * q^64 - q^65 + 43 * q^66 + 190 * q^67 + 48 * q^68 + 45 * q^69 + 14 * q^70 + 9 * q^71 + 98 * q^72 + 182 * q^73 - 23 * q^74 + 219 * q^75 + 197 * q^76 + 25 * q^77 - 26 * q^78 + 105 * q^79 + 20 * q^80 + 236 * q^81 + 107 * q^82 + 130 * q^83 + 38 * q^84 + 73 * q^85 - 24 * q^86 + 171 * q^87 + 165 * q^88 + 40 * q^89 + 45 * q^90 + 182 * q^91 - 4 * q^92 + 23 * q^93 + 98 * q^94 + 30 * q^95 - 2 * q^96 + 168 * q^97 + 82 * q^98 + 50 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more