LMFDB - Embedded newform 6009.2.a.d.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6009,2,Mod(1,6009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6009, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6009 = 3 \cdot 2003 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6009.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:	$47.9821065746$
Analytic rank:	$0$
Dimension:	$93$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	6009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.98067 q^{2} -1.00000 q^{3} +1.92305 q^{4} +0.325740 q^{5} +1.98067 q^{6} +2.23155 q^{7} +0.152420 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.98067 q^{2} -1.00000 q^{3} +1.92305 q^{4} +0.325740 q^{5} +1.98067 q^{6} +2.23155 q^{7} +0.152420 q^{8} +1.00000 q^{9} -0.645183 q^{10} +0.122948 q^{11} -1.92305 q^{12} +5.30054 q^{13} -4.41995 q^{14} -0.325740 q^{15} -4.14799 q^{16} +1.99381 q^{17} -1.98067 q^{18} -2.20664 q^{19} +0.626413 q^{20} -2.23155 q^{21} -0.243519 q^{22} -4.55089 q^{23} -0.152420 q^{24} -4.89389 q^{25} -10.4986 q^{26} -1.00000 q^{27} +4.29137 q^{28} -1.41050 q^{29} +0.645183 q^{30} -8.32826 q^{31} +7.91094 q^{32} -0.122948 q^{33} -3.94907 q^{34} +0.726904 q^{35} +1.92305 q^{36} +9.11094 q^{37} +4.37063 q^{38} -5.30054 q^{39} +0.0496494 q^{40} +1.81451 q^{41} +4.41995 q^{42} +11.9172 q^{43} +0.236435 q^{44} +0.325740 q^{45} +9.01380 q^{46} +3.72337 q^{47} +4.14799 q^{48} -2.02020 q^{49} +9.69318 q^{50} -1.99381 q^{51} +10.1932 q^{52} -13.6564 q^{53} +1.98067 q^{54} +0.0400491 q^{55} +0.340133 q^{56} +2.20664 q^{57} +2.79374 q^{58} +11.6145 q^{59} -0.626413 q^{60} +9.51637 q^{61} +16.4955 q^{62} +2.23155 q^{63} -7.37298 q^{64} +1.72660 q^{65} +0.243519 q^{66} +5.93054 q^{67} +3.83418 q^{68} +4.55089 q^{69} -1.43976 q^{70} -2.28058 q^{71} +0.152420 q^{72} +14.5567 q^{73} -18.0457 q^{74} +4.89389 q^{75} -4.24347 q^{76} +0.274364 q^{77} +10.4986 q^{78} +2.04239 q^{79} -1.35116 q^{80} +1.00000 q^{81} -3.59395 q^{82} -8.91466 q^{83} -4.29137 q^{84} +0.649462 q^{85} -23.6040 q^{86} +1.41050 q^{87} +0.0187398 q^{88} -14.6599 q^{89} -0.645183 q^{90} +11.8284 q^{91} -8.75157 q^{92} +8.32826 q^{93} -7.37477 q^{94} -0.718792 q^{95} -7.91094 q^{96} -6.13624 q^{97} +4.00135 q^{98} +0.122948 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 93 q + 2 q^{2} - 93 q^{3} + 114 q^{4} - 20 q^{5} - 2 q^{6} + 28 q^{7} + 6 q^{8} + 93 q^{9}+O(q^{10}) $ 93 * q + 2 * q^2 - 93 * q^3 + 114 * q^4 - 20 * q^5 - 2 * q^6 + 28 * q^7 + 6 * q^8 + 93 * q^9	\( 93 q + 2 q^{2} - 93 q^{3} + 114 q^{4} - 20 q^{5} - 2 q^{6} + 28 q^{7} + 6 q^{8} + 93 q^{9} + 19 q^{10} + 10 q^{11} - 114 q^{12} + 20 q^{13} + 13 q^{14} + 20 q^{15} + 148 q^{16} - 43 q^{17} + 2 q^{18} + 50 q^{19} - 31 q^{20} - 28 q^{21} + 36 q^{22} + 21 q^{23} - 6 q^{24} + 137 q^{25} + 2 q^{26} - 93 q^{27} + 62 q^{28} - q^{29} - 19 q^{30} + 58 q^{31} + 19 q^{32} - 10 q^{33} + 30 q^{34} + 30 q^{35} + 114 q^{36} + 42 q^{37} - 6 q^{38} - 20 q^{39} + 53 q^{40} - 7 q^{41} - 13 q^{42} + 60 q^{43} + 25 q^{44} - 20 q^{45} + 57 q^{46} + 9 q^{47} - 148 q^{48} + 145 q^{49} + 41 q^{50} + 43 q^{51} + 71 q^{52} - 45 q^{53} - 2 q^{54} + 78 q^{55} + 44 q^{56} - 50 q^{57} + 40 q^{58} + 42 q^{59} + 31 q^{60} + 69 q^{61} - 42 q^{62} + 28 q^{63} + 230 q^{64} - 4 q^{65} - 36 q^{66} + 76 q^{67} - 91 q^{68} - 21 q^{69} + 57 q^{70} + 92 q^{71} + 6 q^{72} + 29 q^{73} + 59 q^{74} - 137 q^{75} + 131 q^{76} - 98 q^{77} - 2 q^{78} + 215 q^{79} - 37 q^{80} + 93 q^{81} + 50 q^{82} - 27 q^{83} - 62 q^{84} + 52 q^{85} + 82 q^{86} + q^{87} + 136 q^{88} - 14 q^{89} + 19 q^{90} + 101 q^{91} - 14 q^{92} - 58 q^{93} + 112 q^{94} + 59 q^{95} - 19 q^{96} + 38 q^{97} - 16 q^{98} + 10 q^{99}+O(q^{100}) \) 93 * q + 2 * q^2 - 93 * q^3 + 114 * q^4 - 20 * q^5 - 2 * q^6 + 28 * q^7 + 6 * q^8 + 93 * q^9 + 19 * q^10 + 10 * q^11 - 114 * q^12 + 20 * q^13 + 13 * q^14 + 20 * q^15 + 148 * q^16 - 43 * q^17 + 2 * q^18 + 50 * q^19 - 31 * q^20 - 28 * q^21 + 36 * q^22 + 21 * q^23 - 6 * q^24 + 137 * q^25 + 2 * q^26 - 93 * q^27 + 62 * q^28 - q^29 - 19 * q^30 + 58 * q^31 + 19 * q^32 - 10 * q^33 + 30 * q^34 + 30 * q^35 + 114 * q^36 + 42 * q^37 - 6 * q^38 - 20 * q^39 + 53 * q^40 - 7 * q^41 - 13 * q^42 + 60 * q^43 + 25 * q^44 - 20 * q^45 + 57 * q^46 + 9 * q^47 - 148 * q^48 + 145 * q^49 + 41 * q^50 + 43 * q^51 + 71 * q^52 - 45 * q^53 - 2 * q^54 + 78 * q^55 + 44 * q^56 - 50 * q^57 + 40 * q^58 + 42 * q^59 + 31 * q^60 + 69 * q^61 - 42 * q^62 + 28 * q^63 + 230 * q^64 - 4 * q^65 - 36 * q^66 + 76 * q^67 - 91 * q^68 - 21 * q^69 + 57 * q^70 + 92 * q^71 + 6 * q^72 + 29 * q^73 + 59 * q^74 - 137 * q^75 + 131 * q^76 - 98 * q^77 - 2 * q^78 + 215 * q^79 - 37 * q^80 + 93 * q^81 + 50 * q^82 - 27 * q^83 - 62 * q^84 + 52 * q^85 + 82 * q^86 + q^87 + 136 * q^88 - 14 * q^89 + 19 * q^90 + 101 * q^91 - 14 * q^92 - 58 * q^93 + 112 * q^94 + 59 * q^95 - 19 * q^96 + 38 * q^97 - 16 * q^98 + 10 * 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 $

$2$	−1.98067	−1.40054	−0.700272	−	0.713876i	$-0.746938\pi$
$2$	−1.98067	−1.40054	−0.700272	+	0.713876i	$0.746938\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.92305	0.961523
$5$	0.325740	0.145675	0.0728377	−	0.997344i	$-0.476794\pi$
$5$	0.325740	0.145675	0.0728377	+	0.997344i	$0.476794\pi$
$6$	1.98067	0.808604
$7$	2.23155	0.843445	0.421723	−	0.906725i	$-0.361426\pi$
$7$	2.23155	0.843445	0.421723	+	0.906725i	$0.361426\pi$
$8$	0.152420	0.0538887
$9$	1.00000	0.333333
$10$	−0.645183	−0.204025
$11$	0.122948	0.0370702	0.0185351	−	0.999828i	$-0.494100\pi$
$11$	0.122948	0.0370702	0.0185351	+	0.999828i	$0.494100\pi$
$12$	−1.92305	−0.555136
$13$	5.30054	1.47011	0.735053	−	0.678010i	$-0.237157\pi$
$13$	5.30054	1.47011	0.735053	+	0.678010i	$0.237157\pi$
$14$	−4.41995	−1.18128
$15$	−0.325740	−0.0841057
$16$	−4.14799	−1.03700
$17$	1.99381	0.483569	0.241784	−	0.970330i	$-0.422267\pi$
$17$	1.99381	0.483569	0.241784	+	0.970330i	$0.422267\pi$
$18$	−1.98067	−0.466848
$19$	−2.20664	−0.506238	−0.253119	−	0.967435i	$-0.581456\pi$
$19$	−2.20664	−0.506238	−0.253119	+	0.967435i	$0.581456\pi$
$20$	0.626413	0.140070
$21$	−2.23155	−0.486963
$22$	−0.243519	−0.0519185
$23$	−4.55089	−0.948926	−0.474463	−	0.880276i	$-0.657358\pi$
$23$	−4.55089	−0.948926	−0.474463	+	0.880276i	$0.657358\pi$
$24$	−0.152420	−0.0311127
$25$	−4.89389	−0.978779
$26$	−10.4986	−2.05895
$27$	−1.00000	−0.192450
$28$	4.29137	0.810992
$29$	−1.41050	−0.261924	−0.130962	−	0.991387i	$-0.541807\pi$
$29$	−1.41050	−0.261924	−0.130962	+	0.991387i	$0.541807\pi$
$30$	0.645183	0.117794
$31$	−8.32826	−1.49580	−0.747900	−	0.663811i	$-0.768938\pi$
$31$	−8.32826	−1.49580	−0.747900	+	0.663811i	$0.768938\pi$
$32$	7.91094	1.39847
$33$	−0.122948	−0.0214025
$34$	−3.94907	−0.677259
$35$	0.726904	0.122869
$36$	1.92305	0.320508
$37$	9.11094	1.49783	0.748914	−	0.662667i	$-0.230575\pi$
$37$	9.11094	1.49783	0.748914	+	0.662667i	$0.230575\pi$
$38$	4.37063	0.709009
$39$	−5.30054	−0.848766
$40$	0.0496494	0.00785025
$41$	1.81451	0.283379	0.141690	−	0.989911i	$-0.454746\pi$
$41$	1.81451	0.283379	0.141690	+	0.989911i	$0.454746\pi$
$42$	4.41995	0.682013
$43$	11.9172	1.81735	0.908677	−	0.417500i	$-0.137094\pi$
$43$	11.9172	1.81735	0.908677	+	0.417500i	$0.137094\pi$
$44$	0.236435	0.0356439
$45$	0.325740	0.0485585
$46$	9.01380	1.32901
$47$	3.72337	0.543110	0.271555	−	0.962423i	$-0.412462\pi$
$47$	3.72337	0.543110	0.271555	+	0.962423i	$0.412462\pi$
$48$	4.14799	0.598710
$49$	−2.02020	−0.288600
$50$	9.69318	1.37082
$51$	−1.99381	−0.279189
$52$	10.1932	1.41354
$53$	−13.6564	−1.87585	−0.937925	−	0.346837i	$-0.887256\pi$
$53$	−13.6564	−1.87585	−0.937925	+	0.346837i	$0.887256\pi$
$54$	1.98067	0.269535
$55$	0.0400491	0.00540022
$56$	0.340133	0.0454522
$57$	2.20664	0.292277
$58$	2.79374	0.366836
$59$	11.6145	1.51208	0.756040	−	0.654525i	$-0.227131\pi$
$59$	11.6145	1.51208	0.756040	+	0.654525i	$0.227131\pi$
$60$	−0.626413	−0.0808696
$61$	9.51637	1.21845	0.609223	−	0.792999i	$-0.291481\pi$
$61$	9.51637	1.21845	0.609223	+	0.792999i	$0.291481\pi$
$62$	16.4955	2.09493
$63$	2.23155	0.281148
$64$	−7.37298	−0.921623
$65$	1.72660	0.214158
$66$	0.243519	0.0299752
$67$	5.93054	0.724530	0.362265	−	0.932075i	$-0.382003\pi$
$67$	5.93054	0.724530	0.362265	+	0.932075i	$0.382003\pi$
$68$	3.83418	0.464963
$69$	4.55089	0.547863
$70$	−1.43976	−0.172084
$71$	−2.28058	−0.270655	−0.135327	−	0.990801i	$-0.543209\pi$
$71$	−2.28058	−0.270655	−0.135327	+	0.990801i	$0.543209\pi$
$72$	0.152420	0.0179629
$73$	14.5567	1.70373	0.851864	−	0.523763i	$-0.175472\pi$
$73$	14.5567	1.70373	0.851864	+	0.523763i	$0.175472\pi$
$74$	−18.0457	−2.09777
$75$	4.89389	0.565098
$76$	−4.24347	−0.486760
$77$	0.274364	0.0312667
$78$	10.4986	1.18873
$79$	2.04239	0.229787	0.114893	−	0.993378i	$-0.463347\pi$
$79$	2.04239	0.229787	0.114893	+	0.993378i	$0.463347\pi$
$80$	−1.35116	−0.151065
$81$	1.00000	0.111111
$82$	−3.59395	−0.396885
$83$	−8.91466	−0.978511	−0.489256	−	0.872140i	$-0.662731\pi$
$83$	−8.91466	−0.978511	−0.489256	+	0.872140i	$0.662731\pi$
$84$	−4.29137	−0.468226
$85$	0.649462	0.0704441
$86$	−23.6040	−2.54528
$87$	1.41050	0.151222
$88$	0.0187398	0.00199767
$89$	−14.6599	−1.55394	−0.776971	−	0.629537i	$-0.783245\pi$
$89$	−14.6599	−1.55394	−0.776971	+	0.629537i	$0.783245\pi$
$90$	−0.645183	−0.0680082
$91$	11.8284	1.23995
$92$	−8.75157	−0.912414
$93$	8.32826	0.863601
$94$	−7.37477	−0.760649
$95$	−0.718792	−0.0737465
$96$	−7.91094	−0.807407
$97$	−6.13624	−0.623041	−0.311520	−	0.950239i	$-0.600838\pi$
$97$	−6.13624	−0.623041	−0.311520	+	0.950239i	$0.600838\pi$
$98$	4.00135	0.404197
$99$	0.122948	0.0123567
$100$	−9.41118	−0.941118
$101$	7.12955	0.709417	0.354709	−	0.934977i	$-0.384580\pi$
$101$	7.12955	0.709417	0.354709	+	0.934977i	$0.384580\pi$
$102$	3.94907	0.391016
$103$	13.8122	1.36095	0.680477	−	0.732769i	$-0.261773\pi$
$103$	13.8122	1.36095	0.680477	+	0.732769i	$0.261773\pi$
$104$	0.807910	0.0792221
$105$	−0.726904	−0.0709386
$106$	27.0488	2.62721
$107$	9.74292	0.941884	0.470942	−	0.882164i	$-0.343914\pi$
$107$	9.74292	0.941884	0.470942	+	0.882164i	$0.343914\pi$
$108$	−1.92305	−0.185045
$109$	−16.7924	−1.60842	−0.804211	−	0.594344i	$-0.797412\pi$
$109$	−16.7924	−1.60842	−0.804211	+	0.594344i	$0.797412\pi$
$110$	−0.0793240	−0.00756325
$111$	−9.11094	−0.864772
$112$	−9.25642	−0.874650
$113$	7.23811	0.680905	0.340452	−	0.940262i	$-0.389420\pi$
$113$	7.23811	0.680905	0.340452	+	0.940262i	$0.389420\pi$
$114$	−4.37063	−0.409347
$115$	−1.48241	−0.138235
$116$	−2.71246	−0.251846
$117$	5.30054	0.490035
$118$	−23.0045	−2.11773
$119$	4.44927	0.407864
$120$	−0.0496494	−0.00453235
$121$	−10.9849	−0.998626
$122$	−18.8488	−1.70649
$123$	−1.81451	−0.163609
$124$	−16.0156	−1.43825
$125$	−3.22284	−0.288259
$126$	−4.41995	−0.393761
$127$	14.1467	1.25531	0.627657	−	0.778490i	$-0.284014\pi$
$127$	14.1467	1.25531	0.627657	+	0.778490i	$0.284014\pi$
$128$	−1.21846	−0.107698
$129$	−11.9172	−1.04925
$130$	−3.41982	−0.299938
$131$	−3.71422	−0.324513	−0.162256	−	0.986749i	$-0.551877\pi$
$131$	−3.71422	−0.324513	−0.162256	+	0.986749i	$0.551877\pi$
$132$	−0.236435	−0.0205790
$133$	−4.92422	−0.426984
$134$	−11.7464	−1.01474
$135$	−0.325740	−0.0280352
$136$	0.303896	0.0260589
$137$	−1.72476	−0.147356	−0.0736780	−	0.997282i	$-0.523474\pi$
$137$	−1.72476	−0.147356	−0.0736780	+	0.997282i	$0.523474\pi$
$138$	−9.01380	−0.767306
$139$	3.68524	0.312578	0.156289	−	0.987711i	$-0.450047\pi$
$139$	3.68524	0.312578	0.156289	+	0.987711i	$0.450047\pi$
$140$	1.39787	0.118142
$141$	−3.72337	−0.313565
$142$	4.51707	0.379064
$143$	0.651692	0.0544972
$144$	−4.14799	−0.345665
$145$	−0.459457	−0.0381558
$146$	−28.8319	−2.38615
$147$	2.02020	0.166623
$148$	17.5208	1.44020
$149$	−9.24595	−0.757458	−0.378729	−	0.925508i	$-0.623639\pi$
$149$	−9.24595	−0.757458	−0.378729	+	0.925508i	$0.623639\pi$
$150$	−9.69318	−0.791445
$151$	2.57511	0.209559	0.104780	−	0.994495i	$-0.466586\pi$
$151$	2.57511	0.209559	0.104780	+	0.994495i	$0.466586\pi$
$152$	−0.336337	−0.0272805
$153$	1.99381	0.161190
$154$	−0.543425	−0.0437904
$155$	−2.71285	−0.217901
$156$	−10.1932	−0.816108
$157$	17.6600	1.40942	0.704709	−	0.709496i	$-0.251077\pi$
$157$	17.6600	1.40942	0.704709	+	0.709496i	$0.251077\pi$
$158$	−4.04529	−0.321826
$159$	13.6564	1.08302
$160$	2.57691	0.203723
$161$	−10.1555	−0.800367
$162$	−1.98067	−0.155616
$163$	3.22598	0.252678	0.126339	−	0.991987i	$-0.459677\pi$
$163$	3.22598	0.252678	0.126339	+	0.991987i	$0.459677\pi$
$164$	3.48939	0.272476
$165$	−0.0400491	−0.00311782
$166$	17.6570	1.37045
$167$	−1.95819	−0.151530	−0.0757648	−	0.997126i	$-0.524140\pi$
$167$	−1.95819	−0.151530	−0.0757648	+	0.997126i	$0.524140\pi$
$168$	−0.340133	−0.0262418
$169$	15.0958	1.16121
$170$	−1.28637	−0.0986600
$171$	−2.20664	−0.168746
$172$	22.9173	1.74743
$173$	−17.8396	−1.35632	−0.678161	−	0.734914i	$-0.737223\pi$
$173$	−17.8396	−1.35632	−0.678161	+	0.734914i	$0.737223\pi$
$174$	−2.79374	−0.211793
$175$	−10.9209	−0.825546
$176$	−0.509987	−0.0384417
$177$	−11.6145	−0.873000
$178$	29.0363	2.17636
$179$	23.7340	1.77396	0.886981	−	0.461806i	$-0.152798\pi$
$179$	23.7340	1.77396	0.886981	+	0.461806i	$0.152798\pi$
$180$	0.626413	0.0466901
$181$	19.2172	1.42840	0.714201	−	0.699940i	$-0.246790\pi$
$181$	19.2172	1.42840	0.714201	+	0.699940i	$0.246790\pi$
$182$	−23.4281	−1.73661
$183$	−9.51637	−0.703471
$184$	−0.693647	−0.0511364
$185$	2.96780	0.218197
$186$	−16.4955	−1.20951
$187$	0.245135	0.0179260
$188$	7.16022	0.522213
$189$	−2.23155	−0.162321
$190$	1.42369	0.103285
$191$	10.2015	0.738152	0.369076	−	0.929399i	$-0.379674\pi$
$191$	10.2015	0.738152	0.369076	+	0.929399i	$0.379674\pi$
$192$	7.37298	0.532099
$193$	6.79602	0.489188	0.244594	−	0.969626i	$-0.421345\pi$
$193$	6.79602	0.489188	0.244594	+	0.969626i	$0.421345\pi$
$194$	12.1539	0.872596
$195$	−1.72660	−0.123644
$196$	−3.88494	−0.277496
$197$	3.77557	0.268998	0.134499	−	0.990914i	$-0.457057\pi$
$197$	3.77557	0.268998	0.134499	+	0.990914i	$0.457057\pi$
$198$	−0.243519	−0.0173062
$199$	8.91674	0.632091	0.316046	−	0.948744i	$-0.397645\pi$
$199$	8.91674	0.632091	0.316046	+	0.948744i	$0.397645\pi$
$200$	−0.745928	−0.0527451
$201$	−5.93054	−0.418308
$202$	−14.1213	−0.993570
$203$	−3.14760	−0.220918
$204$	−3.83418	−0.268446
$205$	0.591059	0.0412814
$206$	−27.3573	−1.90608
$207$	−4.55089	−0.316309
$208$	−21.9866	−1.52449
$209$	−0.271302	−0.0187664
$210$	1.43976	0.0993525
$211$	−17.3721	−1.19595	−0.597973	−	0.801517i	$-0.704027\pi$
$211$	−17.3721	−1.19595	−0.597973	+	0.801517i	$0.704027\pi$
$212$	−26.2619	−1.80367
$213$	2.28058	0.156263
$214$	−19.2975	−1.31915
$215$	3.88190	0.264744
$216$	−0.152420	−0.0103709
$217$	−18.5849	−1.26163
$218$	33.2602	2.25266
$219$	−14.5567	−0.983648
$220$	0.0770163	0.00519244
$221$	10.5683	0.710898
$222$	18.0457	1.21115
$223$	4.68355	0.313634	0.156817	−	0.987628i	$-0.449877\pi$
$223$	4.68355	0.313634	0.156817	+	0.987628i	$0.449877\pi$
$224$	17.6536	1.17953
$225$	−4.89389	−0.326260
$226$	−14.3363	−0.953637
$227$	11.2788	0.748602	0.374301	−	0.927307i	$-0.377883\pi$
$227$	11.2788	0.748602	0.374301	+	0.927307i	$0.377883\pi$
$228$	4.24347	0.281031
$229$	−21.0579	−1.39155	−0.695774	−	0.718261i	$-0.744939\pi$
$229$	−21.0579	−1.39155	−0.695774	+	0.718261i	$0.744939\pi$
$230$	2.93615	0.193604
$231$	−0.274364	−0.0180518
$232$	−0.214989	−0.0141147
$233$	−9.62839	−0.630777	−0.315388	−	0.948963i	$-0.602135\pi$
$233$	−9.62839	−0.630777	−0.315388	+	0.948963i	$0.602135\pi$
$234$	−10.4986	−0.686316
$235$	1.21285	0.0791177
$236$	22.3352	1.45390
$237$	−2.04239	−0.132667
$238$	−8.81252	−0.571231
$239$	−14.2451	−0.921440	−0.460720	−	0.887546i	$-0.652409\pi$
$239$	−14.2451	−0.921440	−0.460720	+	0.887546i	$0.652409\pi$
$240$	1.35116	0.0872173
$241$	11.9702	0.771071	0.385536	−	0.922693i	$-0.374017\pi$
$241$	11.9702	0.771071	0.385536	+	0.922693i	$0.374017\pi$
$242$	21.7574	1.39862
$243$	−1.00000	−0.0641500
$244$	18.3004	1.17156
$245$	−0.658061	−0.0420420
$246$	3.59395	0.229142
$247$	−11.6964	−0.744224
$248$	−1.26940	−0.0806067
$249$	8.91466	0.564944
$250$	6.38337	0.403720
$251$	8.96654	0.565963	0.282981	−	0.959125i	$-0.408676\pi$
$251$	8.96654	0.565963	0.282981	+	0.959125i	$0.408676\pi$
$252$	4.29137	0.270331
$253$	−0.559523	−0.0351769
$254$	−28.0198	−1.75812
$255$	−0.649462	−0.0406709
$256$	17.1593	1.07246
$257$	−11.0007	−0.686206	−0.343103	−	0.939298i	$-0.611478\pi$
$257$	−11.0007	−0.686206	−0.343103	+	0.939298i	$0.611478\pi$
$258$	23.6040	1.46952
$259$	20.3315	1.26334
$260$	3.32033	0.205918
$261$	−1.41050	−0.0873079
$262$	7.35664	0.454495
$263$	21.2241	1.30873	0.654366	−	0.756178i	$-0.272936\pi$
$263$	21.2241	1.30873	0.654366	+	0.756178i	$0.272936\pi$
$264$	−0.0187398	−0.00115335
$265$	−4.44844	−0.273265
$266$	9.75325	0.598010
$267$	14.6599	0.897169
$268$	11.4047	0.696653
$269$	−14.1880	−0.865059	−0.432529	−	0.901620i	$-0.642379\pi$
$269$	−14.1880	−0.865059	−0.432529	+	0.901620i	$0.642379\pi$
$270$	0.645183	0.0392646
$271$	18.0553	1.09678	0.548390	−	0.836223i	$-0.315241\pi$
$271$	18.0553	1.09678	0.548390	+	0.836223i	$0.315241\pi$
$272$	−8.27028	−0.501459
$273$	−11.8284	−0.715888
$274$	3.41617	0.206379
$275$	−0.601695	−0.0362836
$276$	8.75157	0.526782
$277$	−18.7774	−1.12823	−0.564113	−	0.825698i	$-0.690782\pi$
$277$	−18.7774	−1.12823	−0.564113	+	0.825698i	$0.690782\pi$
$278$	−7.29924	−0.437779
$279$	−8.32826	−0.498600
$280$	0.110795	0.00662126
$281$	−17.7717	−1.06017	−0.530086	−	0.847944i	$-0.677840\pi$
$281$	−17.7717	−1.06017	−0.530086	+	0.847944i	$0.677840\pi$
$282$	7.37477	0.439161
$283$	−13.3233	−0.791989	−0.395995	−	0.918253i	$-0.629600\pi$
$283$	−13.3233	−0.791989	−0.395995	+	0.918253i	$0.629600\pi$
$284$	−4.38566	−0.260241
$285$	0.718792	0.0425775
$286$	−1.29078	−0.0763257
$287$	4.04917	0.239015
$288$	7.91094	0.466157
$289$	−13.0247	−0.766161
$290$	0.910032	0.0534389
$291$	6.13624	0.359713
$292$	27.9931	1.63817
$293$	32.5549	1.90188	0.950938	−	0.309382i	$-0.100122\pi$
$293$	32.5549	1.90188	0.950938	+	0.309382i	$0.100122\pi$
$294$	−4.00135	−0.233363
$295$	3.78331	0.220273
$296$	1.38869	0.0807160
$297$	−0.122948	−0.00713417
$298$	18.3132	1.06085
$299$	−24.1222	−1.39502
$300$	9.41118	0.543355
$301$	26.5938	1.53284
$302$	−5.10044	−0.293497
$303$	−7.12955	−0.409582
$304$	9.15312	0.524967
$305$	3.09986	0.177498
$306$	−3.94907	−0.225753
$307$	6.46325	0.368877	0.184438	−	0.982844i	$-0.440953\pi$
$307$	6.46325	0.368877	0.184438	+	0.982844i	$0.440953\pi$
$308$	0.527615	0.0300637
$309$	−13.8122	−0.785747
$310$	5.37325	0.305180
$311$	−13.6377	−0.773322	−0.386661	−	0.922222i	$-0.626372\pi$
$311$	−13.6377	−0.773322	−0.386661	+	0.922222i	$0.626372\pi$
$312$	−0.807910	−0.0457389
$313$	−4.09147	−0.231264	−0.115632	−	0.993292i	$-0.536889\pi$
$313$	−4.09147	−0.231264	−0.115632	+	0.993292i	$0.536889\pi$
$314$	−34.9785	−1.97395
$315$	0.726904	0.0409564
$316$	3.92761	0.220945
$317$	−10.7730	−0.605074	−0.302537	−	0.953138i	$-0.597834\pi$
$317$	−10.7730	−0.605074	−0.302537	+	0.953138i	$0.597834\pi$
$318$	−27.0488	−1.51682
$319$	−0.173419	−0.00970958
$320$	−2.40167	−0.134258
$321$	−9.74292	−0.543797
$322$	20.1147	1.12095
$323$	−4.39961	−0.244801
$324$	1.92305	0.106836
$325$	−25.9403	−1.43891
$326$	−6.38959	−0.353887
$327$	16.7924	0.928623
$328$	0.276569	0.0152709
$329$	8.30888	0.458083
$330$	0.0793240	0.00436664
$331$	16.6362	0.914410	0.457205	−	0.889361i	$-0.348851\pi$
$331$	16.6362	0.914410	0.457205	+	0.889361i	$0.348851\pi$
$332$	−17.1433	−0.940861
$333$	9.11094	0.499276
$334$	3.87853	0.212224
$335$	1.93181	0.105546
$336$	9.25642	0.504979
$337$	−11.1167	−0.605564	−0.302782	−	0.953060i	$-0.597915\pi$
$337$	−11.1167	−0.605564	−0.302782	+	0.953060i	$0.597915\pi$
$338$	−29.8997	−1.62633
$339$	−7.23811	−0.393120
$340$	1.24895	0.0677336
$341$	−1.02394	−0.0554497
$342$	4.37063	0.236336
$343$	−20.1290	−1.08686
$344$	1.81642	0.0979348
$345$	1.48241	0.0798101
$346$	35.3344	1.89959
$347$	−19.6240	−1.05347	−0.526737	−	0.850028i	$-0.676585\pi$
$347$	−19.6240	−1.05347	−0.526737	+	0.850028i	$0.676585\pi$
$348$	2.71246	0.145403
$349$	32.3488	1.73159	0.865795	−	0.500398i	$-0.166813\pi$
$349$	32.3488	1.73159	0.865795	+	0.500398i	$0.166813\pi$
$350$	21.6308	1.15621
$351$	−5.30054	−0.282922
$352$	0.972635	0.0518416
$353$	−15.6978	−0.835512	−0.417756	−	0.908559i	$-0.637183\pi$
$353$	−15.6978	−0.835512	−0.417756	+	0.908559i	$0.637183\pi$
$354$	23.0045	1.22267
$355$	−0.742876	−0.0394278
$356$	−28.1916	−1.49415
$357$	−4.44927	−0.235480
$358$	−47.0092	−2.48451
$359$	13.8585	0.731424	0.365712	−	0.930728i	$-0.380826\pi$
$359$	13.8585	0.731424	0.365712	+	0.930728i	$0.380826\pi$
$360$	0.0496494	0.00261675
$361$	−14.1307	−0.743723
$362$	−38.0629	−2.00054
$363$	10.9849	0.576557
$364$	22.7466	1.19224
$365$	4.74169	0.248191
$366$	18.8488	0.985241
$367$	3.15155	0.164510	0.0822549	−	0.996611i	$-0.473788\pi$
$367$	3.15155	0.164510	0.0822549	+	0.996611i	$0.473788\pi$
$368$	18.8770	0.984033
$369$	1.81451	0.0944598
$370$	−5.87822	−0.305594
$371$	−30.4749	−1.58218
$372$	16.0156	0.830372
$373$	10.3593	0.536383	0.268191	−	0.963366i	$-0.413574\pi$
$373$	10.3593	0.536383	0.268191	+	0.963366i	$0.413574\pi$
$374$	−0.485530	−0.0251062
$375$	3.22284	0.166427
$376$	0.567518	0.0292675
$377$	−7.47643	−0.385056
$378$	4.41995	0.227338
$379$	25.3722	1.30328	0.651641	−	0.758528i	$-0.274081\pi$
$379$	25.3722	1.30328	0.651641	+	0.758528i	$0.274081\pi$
$380$	−1.38227	−0.0709089
$381$	−14.1467	−0.724756
$382$	−20.2057	−1.03381
$383$	26.8886	1.37394	0.686972	−	0.726684i	$-0.258939\pi$
$383$	26.8886	1.37394	0.686972	+	0.726684i	$0.258939\pi$
$384$	1.21846	0.0621792
$385$	0.0893714	0.00455479
$386$	−13.4607	−0.685129
$387$	11.9172	0.605785
$388$	−11.8003	−0.599068
$389$	21.8669	1.10869	0.554347	−	0.832286i	$-0.312968\pi$
$389$	21.8669	1.10869	0.554347	+	0.832286i	$0.312968\pi$
$390$	3.41982	0.173169
$391$	−9.07359	−0.458871
$392$	−0.307920	−0.0155523
$393$	3.71422	0.187358
$394$	−7.47816	−0.376744
$395$	0.665288	0.0334742
$396$	0.236435	0.0118813
$397$	−14.9057	−0.748095	−0.374048	−	0.927409i	$-0.622030\pi$
$397$	−14.9057	−0.748095	−0.374048	+	0.927409i	$0.622030\pi$
$398$	−17.6611	−0.885271
$399$	4.92422	0.246519
$400$	20.2998	1.01499
$401$	−27.7597	−1.38626	−0.693128	−	0.720815i	$-0.743768\pi$
$401$	−27.7597	−1.38626	−0.693128	+	0.720815i	$0.743768\pi$
$402$	11.7464	0.585858
$403$	−44.1443	−2.19899
$404$	13.7105	0.682121
$405$	0.325740	0.0161862
$406$	6.23435	0.309406
$407$	1.12017	0.0555249
$408$	−0.303896	−0.0150451
$409$	24.8896	1.23071	0.615355	−	0.788250i	$-0.289013\pi$
$409$	24.8896	1.23071	0.615355	+	0.788250i	$0.289013\pi$
$410$	−1.17069	−0.0578164
$411$	1.72476	0.0850761
$412$	26.5615	1.30859
$413$	25.9183	1.27536
$414$	9.01380	0.443004
$415$	−2.90386	−0.142545
$416$	41.9323	2.05590
$417$	−3.68524	−0.180467
$418$	0.537360	0.0262831
$419$	31.9013	1.55848	0.779241	−	0.626725i	$-0.215605\pi$
$419$	31.9013	1.55848	0.779241	+	0.626725i	$0.215605\pi$
$420$	−1.39787	−0.0682090
$421$	15.5297	0.756869	0.378434	−	0.925628i	$-0.376463\pi$
$421$	15.5297	0.756869	0.378434	+	0.925628i	$0.376463\pi$
$422$	34.4084	1.67497
$423$	3.72337	0.181037
$424$	−2.08151	−0.101087
$425$	−9.75747	−0.473307
$426$	−4.51707	−0.218853
$427$	21.2362	1.02769
$428$	18.7361	0.905643
$429$	−0.651692	−0.0314640
$430$	−7.68877	−0.370785
$431$	−10.8296	−0.521644	−0.260822	−	0.965387i	$-0.583993\pi$
$431$	−10.8296	−0.521644	−0.260822	+	0.965387i	$0.583993\pi$
$432$	4.14799	0.199570
$433$	30.2572	1.45407	0.727034	−	0.686602i	$-0.240898\pi$
$433$	30.2572	1.45407	0.727034	+	0.686602i	$0.240898\pi$
$434$	36.8105	1.76696
$435$	0.459457	0.0220293
$436$	−32.2926	−1.54653
$437$	10.0422	0.480383
$438$	28.8319	1.37764
$439$	−8.05521	−0.384454	−0.192227	−	0.981350i	$-0.561571\pi$
$439$	−8.05521	−0.384454	−0.192227	+	0.981350i	$0.561571\pi$
$440$	0.00610430	0.000291011 0
$441$	−2.02020	−0.0962001
$442$	−20.9322	−0.995643
$443$	38.6424	1.83596	0.917978	−	0.396631i	$-0.129821\pi$
$443$	38.6424	1.83596	0.917978	+	0.396631i	$0.129821\pi$
$444$	−17.5208	−0.831498
$445$	−4.77530	−0.226371
$446$	−9.27656	−0.439258
$447$	9.24595	0.437318
$448$	−16.4531	−0.777338
$449$	11.7581	0.554898	0.277449	−	0.960740i	$-0.410511\pi$
$449$	11.7581	0.554898	0.277449	+	0.960740i	$0.410511\pi$
$450$	9.69318	0.456941
$451$	0.223091	0.0105049
$452$	13.9192	0.654705
$453$	−2.57511	−0.120989
$454$	−22.3396	−1.04845
$455$	3.85299	0.180631
$456$	0.336337	0.0157504
$457$	9.10422	0.425878	0.212939	−	0.977066i	$-0.431696\pi$
$457$	9.10422	0.425878	0.212939	+	0.977066i	$0.431696\pi$
$458$	41.7088	1.94892
$459$	−1.99381	−0.0930629
$460$	−2.85074	−0.132916
$461$	−1.73818	−0.0809553	−0.0404776	−	0.999180i	$-0.512888\pi$
$461$	−1.73818	−0.0809553	−0.0404776	+	0.999180i	$0.512888\pi$
$462$	0.543425	0.0252824
$463$	20.7526	0.964454	0.482227	−	0.876046i	$-0.339828\pi$
$463$	20.7526	0.964454	0.482227	+	0.876046i	$0.339828\pi$
$464$	5.85074	0.271614
$465$	2.71285	0.125805
$466$	19.0706	0.883431
$467$	−6.63840	−0.307188	−0.153594	−	0.988134i	$-0.549085\pi$
$467$	−6.63840	−0.307188	−0.153594	+	0.988134i	$0.549085\pi$
$468$	10.1932	0.471180
$469$	13.2343	0.611102
$470$	−2.40226	−0.110808
$471$	−17.6600	−0.813728
$472$	1.77029	0.0814840
$473$	1.46520	0.0673698
$474$	4.04529	0.185806
$475$	10.7991	0.495495
$476$	8.55615	0.392170
$477$	−13.6564	−0.625284
$478$	28.2148	1.29052
$479$	10.9573	0.500651	0.250325	−	0.968162i	$-0.419462\pi$
$479$	10.9573	0.500651	0.250325	+	0.968162i	$0.419462\pi$
$480$	−2.57691	−0.117619
$481$	48.2929	2.20197
$482$	−23.7091	−1.07992
$483$	10.1555	0.462092
$484$	−21.1244	−0.960202
$485$	−1.99882	−0.0907617
$486$	1.98067	0.0898449
$487$	7.32135	0.331762	0.165881	−	0.986146i	$-0.446953\pi$
$487$	7.32135	0.331762	0.165881	+	0.986146i	$0.446953\pi$
$488$	1.45049	0.0656605
$489$	−3.22598	−0.145884
$490$	1.30340	0.0588816
$491$	−13.7829	−0.622015	−0.311008	−	0.950407i	$-0.600666\pi$
$491$	−13.7829	−0.622015	−0.311008	+	0.950407i	$0.600666\pi$
$492$	−3.48939	−0.157314
$493$	−2.81227	−0.126658
$494$	23.1667	1.04232
$495$	0.0400491	0.00180007
$496$	34.5455	1.55114
$497$	−5.08922	−0.228283
$498$	−17.6570	−0.791228
$499$	−20.3319	−0.910179	−0.455090	−	0.890446i	$-0.650393\pi$
$499$	−20.3319	−0.910179	−0.455090	+	0.890446i	$0.650393\pi$
$500$	−6.19766	−0.277168
$501$	1.95819	0.0874857
$502$	−17.7597	−0.792656
$503$	−39.0322	−1.74036	−0.870179	−	0.492735i	$-0.835997\pi$
$503$	−39.0322	−1.74036	−0.870179	+	0.492735i	$0.835997\pi$
$504$	0.340133	0.0151507
$505$	2.32238	0.103345
$506$	1.10823	0.0492668
$507$	−15.0958	−0.670426
$508$	27.2047	1.20701
$509$	22.8658	1.01351	0.506754	−	0.862090i	$-0.330845\pi$
$509$	22.8658	1.01351	0.506754	+	0.862090i	$0.330845\pi$
$510$	1.28637	0.0569614
$511$	32.4839	1.43700
$512$	−31.5500	−1.39433
$513$	2.20664	0.0974256
$514$	21.7888	0.961061
$515$	4.49918	0.198258
$516$	−22.9173	−1.00888
$517$	0.457782	0.0201332
$518$	−40.2699	−1.76936
$519$	17.8396	0.783072
$520$	0.263169	0.0115407
$521$	−36.1873	−1.58539	−0.792697	−	0.609616i	$-0.791323\pi$
$521$	−36.1873	−1.58539	−0.792697	+	0.609616i	$0.791323\pi$
$522$	2.79374	0.122279
$523$	−16.1527	−0.706310	−0.353155	−	0.935565i	$-0.614891\pi$
$523$	−16.1527	−0.706310	−0.353155	+	0.935565i	$0.614891\pi$
$524$	−7.14262	−0.312027
$525$	10.9209	0.476629
$526$	−42.0378	−1.83294
$527$	−16.6049	−0.723323
$528$	0.509987	0.0221943
$529$	−2.28942	−0.0995399
$530$	8.81087	0.382720
$531$	11.6145	0.504027
$532$	−9.46951	−0.410555
$533$	9.61790	0.416598
$534$	−29.0363	−1.25652
$535$	3.17366	0.137209
$536$	0.903934	0.0390440
$537$	−23.7340	−1.02420
$538$	28.1017	1.21155
$539$	−0.248380	−0.0106985
$540$	−0.626413	−0.0269565
$541$	0.0687732	0.00295679	0.00147840	−	0.999999i	$-0.499529\pi$
$541$	0.0687732	0.00295679	0.00147840	+	0.999999i	$0.499529\pi$
$542$	−35.7615	−1.53609
$543$	−19.2172	−0.824688
$544$	15.7729	0.676257
$545$	−5.46996	−0.234307
$546$	23.4281	1.00263
$547$	16.1690	0.691338	0.345669	−	0.938357i	$-0.387652\pi$
$547$	16.1690	0.691338	0.345669	+	0.938357i	$0.387652\pi$
$548$	−3.31679	−0.141686
$549$	9.51637	0.406149
$550$	1.19176	0.0508167
$551$	3.11247	0.132596
$552$	0.693647	0.0295236
$553$	4.55768	0.193812
$554$	37.1918	1.58013
$555$	−2.96780	−0.125976
$556$	7.08689	0.300551
$557$	38.1840	1.61791	0.808953	−	0.587873i	$-0.200035\pi$
$557$	38.1840	1.61791	0.808953	+	0.587873i	$0.200035\pi$
$558$	16.4955	0.698311
$559$	63.1676	2.67170
$560$	−3.01519	−0.127415
$561$	−0.245135	−0.0103496
$562$	35.1999	1.48482
$563$	8.21649	0.346284	0.173142	−	0.984897i	$-0.444608\pi$
$563$	8.21649	0.346284	0.173142	+	0.984897i	$0.444608\pi$
$564$	−7.16022	−0.301500
$565$	2.35774	0.0991910
$566$	26.3891	1.10922
$567$	2.23155	0.0937161
$568$	−0.347606	−0.0145852
$569$	12.1560	0.509605	0.254802	−	0.966993i	$-0.417990\pi$
$569$	12.1560	0.509605	0.254802	+	0.966993i	$0.417990\pi$
$570$	−1.42369	−0.0596317
$571$	27.6269	1.15615	0.578074	−	0.815984i	$-0.303804\pi$
$571$	27.6269	1.15615	0.578074	+	0.815984i	$0.303804\pi$
$572$	1.25323	0.0524003
$573$	−10.2015	−0.426173
$574$	−8.02006	−0.334751
$575$	22.2716	0.928788
$576$	−7.37298	−0.307208
$577$	17.8721	0.744025	0.372013	−	0.928228i	$-0.378668\pi$
$577$	17.8721	0.744025	0.372013	+	0.928228i	$0.378668\pi$
$578$	25.7977	1.07304
$579$	−6.79602	−0.282433
$580$	−0.883557	−0.0366877
$581$	−19.8935	−0.825320
$582$	−12.1539	−0.503793
$583$	−1.67903	−0.0695382
$584$	2.21873	0.0918117
$585$	1.72660	0.0713861
$586$	−64.4804	−2.66366
$587$	38.0662	1.57116	0.785579	−	0.618761i	$-0.212365\pi$
$587$	38.0662	1.57116	0.785579	+	0.618761i	$0.212365\pi$
$588$	3.88494	0.160212
$589$	18.3775	0.757232
$590$	−7.49348	−0.308502
$591$	−3.77557	−0.155306
$592$	−37.7920	−1.55324
$593$	−13.1955	−0.541875	−0.270937	−	0.962597i	$-0.587334\pi$
$593$	−13.1955	−0.541875	−0.270937	+	0.962597i	$0.587334\pi$
$594$	0.243519	0.00999172
$595$	1.44930	0.0594157
$596$	−17.7804	−0.728313
$597$	−8.91674	−0.364938
$598$	47.7780	1.95379
$599$	−38.8141	−1.58590	−0.792950	−	0.609286i	$-0.791456\pi$
$599$	−38.8141	−1.58590	−0.792950	+	0.609286i	$0.791456\pi$
$600$	0.745928	0.0304524
$601$	25.0598	1.02221	0.511104	−	0.859519i	$-0.329237\pi$
$601$	25.0598	1.02221	0.511104	+	0.859519i	$0.329237\pi$
$602$	−52.6734	−2.14681
$603$	5.93054	0.241510
$604$	4.95205	0.201496
$605$	−3.57822	−0.145475
$606$	14.1213	0.573638
$607$	−39.6554	−1.60956	−0.804782	−	0.593570i	$-0.797718\pi$
$607$	−39.6554	−1.60956	−0.804782	+	0.593570i	$0.797718\pi$
$608$	−17.4566	−0.707959
$609$	3.14760	0.127547
$610$	−6.13980	−0.248593
$611$	19.7359	0.798429
$612$	3.83418	0.154988
$613$	6.12117	0.247232	0.123616	−	0.992330i	$-0.460551\pi$
$613$	6.12117	0.247232	0.123616	+	0.992330i	$0.460551\pi$
$614$	−12.8015	−0.516628
$615$	−0.591059	−0.0238338
$616$	0.0418187	0.00168492
$617$	−25.6400	−1.03223	−0.516113	−	0.856521i	$-0.672621\pi$
$617$	−25.6400	−1.03223	−0.516113	+	0.856521i	$0.672621\pi$
$618$	27.3573	1.10047
$619$	−12.3918	−0.498067	−0.249033	−	0.968495i	$-0.580113\pi$
$619$	−12.3918	−0.498067	−0.249033	+	0.968495i	$0.580113\pi$
$620$	−5.21693	−0.209517
$621$	4.55089	0.182621
$622$	27.0117	1.08307
$623$	−32.7141	−1.31066
$624$	21.9866	0.880168
$625$	23.4197	0.936786
$626$	8.10385	0.323895
$627$	0.271302	0.0108348
$628$	33.9609	1.35519
$629$	18.1654	0.724303
$630$	−1.43976	−0.0573612
$631$	16.3586	0.651224	0.325612	−	0.945503i	$-0.394430\pi$
$631$	16.3586	0.651224	0.325612	+	0.945503i	$0.394430\pi$
$632$	0.311301	0.0123829
$633$	17.3721	0.690479
$634$	21.3378	0.847433
$635$	4.60813	0.182868
$636$	26.2619	1.04135
$637$	−10.7082	−0.424273
$638$	0.343485	0.0135987
$639$	−2.28058	−0.0902183
$640$	−0.396901	−0.0156889
$641$	16.1102	0.636315	0.318157	−	0.948038i	$-0.396936\pi$
$641$	16.1102	0.636315	0.318157	+	0.948038i	$0.396936\pi$
$642$	19.2975	0.761611
$643$	36.7054	1.44752	0.723760	−	0.690052i	$-0.242412\pi$
$643$	36.7054	1.44752	0.723760	+	0.690052i	$0.242412\pi$
$644$	−19.5295	−0.769571
$645$	−3.88190	−0.152850
$646$	8.71418	0.342855
$647$	33.2673	1.30787	0.653937	−	0.756549i	$-0.273116\pi$
$647$	33.2673	1.30787	0.653937	+	0.756549i	$0.273116\pi$
$648$	0.152420	0.00598763
$649$	1.42798	0.0560532
$650$	51.3791	2.01525
$651$	18.5849	0.728400
$652$	6.20371	0.242956
$653$	−8.92090	−0.349102	−0.174551	−	0.984648i	$-0.555847\pi$
$653$	−8.92090	−0.349102	−0.174551	+	0.984648i	$0.555847\pi$
$654$	−33.2602	−1.30058
$655$	−1.20987	−0.0472735
$656$	−7.52657	−0.293863
$657$	14.5567	0.567909
$658$	−16.4571	−0.641566
$659$	−16.5582	−0.645016	−0.322508	−	0.946567i	$-0.604526\pi$
$659$	−16.5582	−0.645016	−0.322508	+	0.946567i	$0.604526\pi$
$660$	−0.0770163	−0.00299786
$661$	−22.8203	−0.887606	−0.443803	−	0.896124i	$-0.646371\pi$
$661$	−22.8203	−0.887606	−0.443803	+	0.896124i	$0.646371\pi$
$662$	−32.9509	−1.28067
$663$	−10.5683	−0.410437
$664$	−1.35877	−0.0527307
$665$	−1.60402	−0.0622011
$666$	−18.0457	−0.699258
$667$	6.41904	0.248546
$668$	−3.76570	−0.145699
$669$	−4.68355	−0.181077
$670$	−3.82628	−0.147822
$671$	1.17002	0.0451681
$672$	−17.6536	−0.681004
$673$	−24.7894	−0.955560	−0.477780	−	0.878479i	$-0.658558\pi$
$673$	−24.7894	−0.955560	−0.477780	+	0.878479i	$0.658558\pi$
$674$	22.0184	0.848119
$675$	4.89389	0.188366
$676$	29.0298	1.11653
$677$	−10.6752	−0.410281	−0.205141	−	0.978733i	$-0.565765\pi$
$677$	−10.6752	−0.410281	−0.205141	+	0.978733i	$0.565765\pi$
$678$	14.3363	0.550582
$679$	−13.6933	−0.525501
$680$	0.0989912	0.00379614
$681$	−11.2788	−0.432205
$682$	2.02809	0.0776597
$683$	−24.7700	−0.947796	−0.473898	−	0.880580i	$-0.657154\pi$
$683$	−24.7700	−0.947796	−0.473898	+	0.880580i	$0.657154\pi$
$684$	−4.24347	−0.162253
$685$	−0.561823	−0.0214661
$686$	39.8689	1.52220
$687$	21.0579	0.803411
$688$	−49.4323	−1.88459
$689$	−72.3863	−2.75770
$690$	−2.93615	−0.111778
$691$	1.25708	0.0478215	0.0239108	−	0.999714i	$-0.492388\pi$
$691$	1.25708	0.0478215	0.0239108	+	0.999714i	$0.492388\pi$
$692$	−34.3064	−1.30413
$693$	0.274364	0.0104222
$694$	38.8687	1.47544
$695$	1.20043	0.0455349
$696$	0.214989	0.00814914
$697$	3.61779	0.137033
$698$	−64.0722	−2.42517
$699$	9.62839	0.364179
$700$	−21.0015	−0.793782
$701$	1.11542	0.0421287	0.0210643	−	0.999778i	$-0.493295\pi$
$701$	1.11542	0.0421287	0.0210643	+	0.999778i	$0.493295\pi$
$702$	10.4986	0.396245
$703$	−20.1046	−0.758258
$704$	−0.906494	−0.0341648
$705$	−1.21285	−0.0456786
$706$	31.0922	1.17017
$707$	15.9099	0.598354
$708$	−22.3352	−0.839410
$709$	−16.3343	−0.613446	−0.306723	−	0.951799i	$-0.599233\pi$
$709$	−16.3343	−0.613446	−0.306723	+	0.951799i	$0.599233\pi$
$710$	1.47139	0.0552203
$711$	2.04239	0.0765955
$712$	−2.23446	−0.0837399
$713$	37.9010	1.41940
$714$	8.81252	0.329800
$715$	0.212282	0.00793890
$716$	45.6416	1.70571
$717$	14.2451	0.531993
$718$	−27.4491	−1.02439
$719$	19.1752	0.715112	0.357556	−	0.933892i	$-0.383610\pi$
$719$	19.1752	0.715112	0.357556	+	0.933892i	$0.383610\pi$
$720$	−1.35116	−0.0503549
$721$	30.8225	1.14789
$722$	27.9883	1.04162
$723$	−11.9702	−0.445178
$724$	36.9555	1.37344
$725$	6.90285	0.256365
$726$	−21.7574	−0.807493
$727$	19.1912	0.711761	0.355881	−	0.934531i	$-0.384181\pi$
$727$	19.1912	0.711761	0.355881	+	0.934531i	$0.384181\pi$
$728$	1.80289	0.0668195
$729$	1.00000	0.0370370
$730$	−9.39171	−0.347603
$731$	23.7606	0.878816
$732$	−18.3004	−0.676403
$733$	−26.6859	−0.985668	−0.492834	−	0.870123i	$-0.664039\pi$
$733$	−26.6859	−0.985668	−0.492834	+	0.870123i	$0.664039\pi$
$734$	−6.24218	−0.230403
$735$	0.658061	0.0242729
$736$	−36.0018	−1.32704
$737$	0.729148	0.0268585
$738$	−3.59395	−0.132295
$739$	8.11003	0.298332	0.149166	−	0.988812i	$-0.452341\pi$
$739$	8.11003	0.298332	0.149166	+	0.988812i	$0.452341\pi$
$740$	5.70721	0.209801
$741$	11.6964	0.429678
$742$	60.3606	2.21591
$743$	−29.2111	−1.07165	−0.535826	−	0.844328i	$-0.680000\pi$
$743$	−29.2111	−1.07165	−0.535826	+	0.844328i	$0.680000\pi$
$744$	1.26940	0.0465383
$745$	−3.01178	−0.110343
$746$	−20.5183	−0.751228
$747$	−8.91466	−0.326170
$748$	0.471405	0.0172363
$749$	21.7418	0.794427
$750$	−6.38337	−0.233088
$751$	−35.8113	−1.30677	−0.653387	−	0.757024i	$-0.726653\pi$
$751$	−35.8113	−1.30677	−0.653387	+	0.757024i	$0.726653\pi$
$752$	−15.4445	−0.563203
$753$	−8.96654	−0.326759
$754$	14.8083	0.539287
$755$	0.838816	0.0305276
$756$	−4.29137	−0.156075
$757$	13.3717	0.486002	0.243001	−	0.970026i	$-0.421868\pi$
$757$	13.3717	0.486002	0.243001	+	0.970026i	$0.421868\pi$
$758$	−50.2539	−1.82530
$759$	0.559523	0.0203094
$760$	−0.109558	−0.00397410
$761$	−11.8682	−0.430222	−0.215111	−	0.976590i	$-0.569011\pi$
$761$	−11.8682	−0.430222	−0.215111	+	0.976590i	$0.569011\pi$
$762$	28.0198	1.01505
$763$	−37.4730	−1.35662
$764$	19.6179	0.709751
$765$	0.649462	0.0234814
$766$	−53.2574	−1.92427
$767$	61.5632	2.22292
$768$	−17.1593	−0.619184
$769$	27.2416	0.982356	0.491178	−	0.871059i	$-0.336566\pi$
$769$	27.2416	0.982356	0.491178	+	0.871059i	$0.336566\pi$
$770$	−0.177015	−0.00637918
$771$	11.0007	0.396181
$772$	13.0691	0.470365
$773$	−9.59284	−0.345031	−0.172515	−	0.985007i	$-0.555189\pi$
$773$	−9.59284	−0.345031	−0.172515	+	0.985007i	$0.555189\pi$
$774$	−23.6040	−0.848428
$775$	40.7576	1.46406
$776$	−0.935287	−0.0335748
$777$	−20.3315	−0.729388
$778$	−43.3110	−1.55277
$779$	−4.00398	−0.143457
$780$	−3.32033	−0.118887
$781$	−0.280393	−0.0100332
$782$	17.9718	0.642669
$783$	1.41050	0.0504072
$784$	8.37977	0.299278
$785$	5.75256	0.205318
$786$	−7.35664	−0.262403
$787$	−35.1711	−1.25372	−0.626858	−	0.779134i	$-0.715659\pi$
$787$	−35.1711	−1.25372	−0.626858	+	0.779134i	$0.715659\pi$
$788$	7.26060	0.258648
$789$	−21.2241	−0.755597
$790$	−1.31771	−0.0468822
$791$	16.1522	0.574306
$792$	0.0187398	0.000665889 0
$793$	50.4419	1.79125
$794$	29.5232	1.04774
$795$	4.44844	0.157770
$796$	17.1473	0.607770
$797$	16.5459	0.586085	0.293043	−	0.956099i	$-0.405332\pi$
$797$	16.5459	0.586085	0.293043	+	0.956099i	$0.405332\pi$
$798$	−9.75325	−0.345261
$799$	7.42368	0.262631
$800$	−38.7153	−1.36879
$801$	−14.6599	−0.517981
$802$	54.9828	1.94151
$803$	1.78971	0.0631576
$804$	−11.4047	−0.402213
$805$	−3.30806	−0.116594
$806$	87.4353	3.07978
$807$	14.1880	0.499442
$808$	1.08669	0.0382296
$809$	−8.05748	−0.283286	−0.141643	−	0.989918i	$-0.545239\pi$
$809$	−8.05748	−0.283286	−0.141643	+	0.989918i	$0.545239\pi$
$810$	−0.645183	−0.0226694
$811$	26.8893	0.944212	0.472106	−	0.881542i	$-0.343494\pi$
$811$	26.8893	0.944212	0.472106	+	0.881542i	$0.343494\pi$
$812$	−6.05298	−0.212418
$813$	−18.0553	−0.633227
$814$	−2.21869	−0.0777650
$815$	1.05083	0.0368090
$816$	8.27028	0.289518
$817$	−26.2970	−0.920014
$818$	−49.2980	−1.72366
$819$	11.8284	0.413318
$820$	1.13663	0.0396930
$821$	20.6360	0.720200	0.360100	−	0.932914i	$-0.382743\pi$
$821$	20.6360	0.720200	0.360100	+	0.932914i	$0.382743\pi$
$822$	−3.41617	−0.119153
$823$	−2.50162	−0.0872008	−0.0436004	−	0.999049i	$-0.513883\pi$
$823$	−2.50162	−0.0872008	−0.0436004	+	0.999049i	$0.513883\pi$
$824$	2.10526	0.0733401
$825$	0.601695	0.0209483
$826$	−51.3356	−1.78619
$827$	7.42127	0.258063	0.129031	−	0.991641i	$-0.458813\pi$
$827$	7.42127	0.258063	0.129031	+	0.991641i	$0.458813\pi$
$828$	−8.75157	−0.304138
$829$	14.9517	0.519295	0.259647	−	0.965704i	$-0.416394\pi$
$829$	14.9517	0.519295	0.259647	+	0.965704i	$0.416394\pi$
$830$	5.75159	0.199640
$831$	18.7774	0.651382
$832$	−39.0808	−1.35488
$833$	−4.02789	−0.139558
$834$	7.29924	0.252752
$835$	−0.637862	−0.0220741
$836$	−0.521727	−0.0180443
$837$	8.32826	0.287867
$838$	−63.1859	−2.18272
$839$	−0.00148239	−5.11778e−5 0	−2.55889e−5	−	1.00000i	$-0.500008\pi$
$839$	−0.00148239	−5.11778e−5 0	−2.55889e−5		1.00000i	$0.500008\pi$
$840$	−0.110795	−0.00382279
$841$	−27.0105	−0.931396
$842$	−30.7591	−1.06003
$843$	17.7717	0.612090
$844$	−33.4074	−1.14993
$845$	4.91729	0.169160
$846$	−7.37477	−0.253550
$847$	−24.5133	−0.842286
$848$	56.6465	1.94525
$849$	13.3233	0.457255
$850$	19.3263	0.662887
$851$	−41.4629	−1.42133
$852$	4.38566	0.150250
$853$	19.3382	0.662129	0.331064	−	0.943608i	$-0.392592\pi$
$853$	19.3382	0.662129	0.331064	+	0.943608i	$0.392592\pi$
$854$	−42.0619	−1.43933
$855$	−0.718792	−0.0245822
$856$	1.48502	0.0507569
$857$	25.0741	0.856516	0.428258	−	0.903657i	$-0.359128\pi$
$857$	25.0741	0.856516	0.428258	+	0.903657i	$0.359128\pi$
$858$	1.29078	0.0440667
$859$	39.1804	1.33682	0.668409	−	0.743794i	$-0.266976\pi$
$859$	39.1804	1.33682	0.668409	+	0.743794i	$0.266976\pi$
$860$	7.46508	0.254557
$861$	−4.04917	−0.137995
$862$	21.4498	0.730585
$863$	43.8222	1.49172	0.745862	−	0.666101i	$-0.232038\pi$
$863$	43.8222	1.49172	0.745862	+	0.666101i	$0.232038\pi$
$864$	−7.91094	−0.269136
$865$	−5.81108	−0.197583
$866$	−59.9294	−2.03648
$867$	13.0247	0.442343
$868$	−35.7396	−1.21308
$869$	0.251108	0.00851825
$870$	−0.910032	−0.0308530
$871$	31.4351	1.06514
$872$	−2.55950	−0.0866757
$873$	−6.13624	−0.207680
$874$	−19.8902	−0.672797
$875$	−7.19191	−0.243131
$876$	−27.9931	−0.945800
$877$	57.2626	1.93362	0.966810	−	0.255498i	$-0.0822394\pi$
$877$	57.2626	1.93362	0.966810	+	0.255498i	$0.0822394\pi$
$878$	15.9547	0.538445
$879$	−32.5549	−1.09805
$880$	−0.166123	−0.00560001
$881$	20.4909	0.690357	0.345178	−	0.938537i	$-0.387818\pi$
$881$	20.4909	0.690357	0.345178	+	0.938537i	$0.387818\pi$
$882$	4.00135	0.134732
$883$	−14.6082	−0.491607	−0.245803	−	0.969320i	$-0.579052\pi$
$883$	−14.6082	−0.491607	−0.245803	+	0.969320i	$0.579052\pi$
$884$	20.3232	0.683544
$885$	−3.78331	−0.127175
$886$	−76.5378	−2.57134
$887$	7.88509	0.264755	0.132378	−	0.991199i	$-0.457739\pi$
$887$	7.88509	0.264755	0.132378	+	0.991199i	$0.457739\pi$
$888$	−1.38869	−0.0466014
$889$	31.5689	1.05879
$890$	9.45829	0.317043
$891$	0.122948	0.00411892
$892$	9.00669	0.301566
$893$	−8.21615	−0.274943
$894$	−18.3132	−0.612484
$895$	7.73111	0.258423
$896$	−2.71905	−0.0908370
$897$	24.1222	0.805416
$898$	−23.2888	−0.777159
$899$	11.7470	0.391786
$900$	−9.41118	−0.313706
$901$	−27.2282	−0.907103
$902$	−0.441869	−0.0147126
$903$	−26.5938	−0.884985
$904$	1.10324	0.0366931
$905$	6.25981	0.208083
$906$	5.10044	0.169451
$907$	5.27466	0.175142	0.0875711	−	0.996158i	$-0.472090\pi$
$907$	5.27466	0.175142	0.0875711	+	0.996158i	$0.472090\pi$
$908$	21.6897	0.719798
$909$	7.12955	0.236472
$910$	−7.63148	−0.252981
$911$	40.3416	1.33658	0.668289	−	0.743902i	$-0.267027\pi$
$911$	40.3416	1.33658	0.668289	+	0.743902i	$0.267027\pi$
$912$	−9.15312	−0.303090
$913$	−1.09604	−0.0362736
$914$	−18.0324	−0.596460
$915$	−3.09986	−0.102478
$916$	−40.4954	−1.33801
$917$	−8.28845	−0.273709
$918$	3.94907	0.130339
$919$	−28.4006	−0.936849	−0.468424	−	0.883504i	$-0.655178\pi$
$919$	−28.4006	−0.936849	−0.468424	+	0.883504i	$0.655178\pi$
$920$	−0.225949	−0.00744931
$921$	−6.46325	−0.212971
$922$	3.44276	0.113381
$923$	−12.0883	−0.397891
$924$	−0.527615	−0.0173573
$925$	−44.5880	−1.46604
$926$	−41.1040	−1.35076
$927$	13.8122	0.453651
$928$	−11.1584	−0.366293
$929$	19.3477	0.634777	0.317389	−	0.948296i	$-0.397194\pi$
$929$	19.3477	0.634777	0.317389	+	0.948296i	$0.397194\pi$
$930$	−5.37325	−0.176196
$931$	4.45786	0.146101
$932$	−18.5158	−0.606507
$933$	13.6377	0.446478
$934$	13.1485	0.430231
$935$	0.0798501	0.00261138
$936$	0.807910	0.0264074
$937$	−12.6530	−0.413357	−0.206678	−	0.978409i	$-0.566265\pi$
$937$	−12.6530	−0.413357	−0.206678	+	0.978409i	$0.566265\pi$
$938$	−26.2127	−0.855875
$939$	4.09147	0.133520
$940$	2.33237	0.0760735
$941$	3.94377	0.128563	0.0642815	−	0.997932i	$-0.479524\pi$
$941$	3.94377	0.128563	0.0642815	+	0.997932i	$0.479524\pi$
$942$	34.9785	1.13966
$943$	−8.25765	−0.268906
$944$	−48.1768	−1.56802
$945$	−0.726904	−0.0236462
$946$	−2.90207	−0.0943543
$947$	10.7453	0.349174	0.174587	−	0.984642i	$-0.444141\pi$
$947$	10.7453	0.349174	0.174587	+	0.984642i	$0.444141\pi$
$948$	−3.92761	−0.127563
$949$	77.1582	2.50466
$950$	−21.3894	−0.693963
$951$	10.7730	0.349340
$952$	0.678159	0.0219792
$953$	5.77051	0.186925	0.0934625	−	0.995623i	$-0.470206\pi$
$953$	5.77051	0.186925	0.0934625	+	0.995623i	$0.470206\pi$
$954$	27.0488	0.875737
$955$	3.32303	0.107531
$956$	−27.3940	−0.885985
$957$	0.173419	0.00560583
$958$	−21.7027	−0.701183
$959$	−3.84888	−0.124287
$960$	2.40167	0.0775137
$961$	38.3600	1.23742
$962$	−95.6522	−3.08395
$963$	9.74292	0.313961
$964$	23.0193	0.741403
$965$	2.21373	0.0712626
$966$	−20.1147	−0.647180
$967$	31.0509	0.998528	0.499264	−	0.866450i	$-0.333604\pi$
$967$	31.0509	0.998528	0.499264	+	0.866450i	$0.333604\pi$
$968$	−1.67432	−0.0538146
$969$	4.39961	0.141336
$970$	3.95900	0.127116
$971$	9.04002	0.290108	0.145054	−	0.989424i	$-0.453664\pi$
$971$	9.04002	0.290108	0.145054	+	0.989424i	$0.453664\pi$
$972$	−1.92305	−0.0616817
$973$	8.22378	0.263642
$974$	−14.5012	−0.464647
$975$	25.9403	0.830754
$976$	−39.4738	−1.26353
$977$	21.2375	0.679449	0.339725	−	0.940525i	$-0.389666\pi$
$977$	21.2375	0.679449	0.339725	+	0.940525i	$0.389666\pi$
$978$	6.38959	0.204317
$979$	−1.80240	−0.0576050
$980$	−1.26548	−0.0404243
$981$	−16.7924	−0.536141
$982$	27.2994	0.871160
$983$	−54.2507	−1.73033	−0.865165	−	0.501488i	$-0.832786\pi$
$983$	−54.2507	−1.73033	−0.865165	+	0.501488i	$0.832786\pi$
$984$	−0.276569	−0.00881668
$985$	1.22985	0.0391864
$986$	5.57017	0.177390
$987$	−8.30888	−0.264475
$988$	−22.4927	−0.715589
$989$	−54.2338	−1.72453
$990$	−0.0793240	−0.00252108
$991$	45.0759	1.43188	0.715941	−	0.698161i	$-0.245998\pi$
$991$	45.0759	1.43188	0.715941	+	0.698161i	$0.245998\pi$
$992$	−65.8844	−2.09183
$993$	−16.6362	−0.527935
$994$	10.0801	0.319720
$995$	2.90454	0.0920801
$996$	17.1433	0.543206
$997$	−45.5216	−1.44169	−0.720843	−	0.693099i	$-0.756245\pi$
$997$	−45.5216	−1.44169	−0.720843	+	0.693099i	$0.756245\pi$
$998$	40.2707	1.27475
$999$	−9.11094	−0.288257

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6009.2.a.d.1.18	✓	93

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6009.2.a.d.1.18	✓	93	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:	\( N \)	\(=\)	\( 6009 = 3 \cdot 2003 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6009.a (trivial)

\(f(q)\)	\(=\)	\(q-1.98067 q^{2} -1.00000 q^{3} +1.92305 q^{4} +0.325740 q^{5} +1.98067 q^{6} +2.23155 q^{7} +0.152420 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.98067 q^{2} -1.00000 q^{3} +1.92305 q^{4} +0.325740 q^{5} +1.98067 q^{6} +2.23155 q^{7} +0.152420 q^{8} +1.00000 q^{9} -0.645183 q^{10} +0.122948 q^{11} -1.92305 q^{12} +5.30054 q^{13} -4.41995 q^{14} -0.325740 q^{15} -4.14799 q^{16} +1.99381 q^{17} -1.98067 q^{18} -2.20664 q^{19} +0.626413 q^{20} -2.23155 q^{21} -0.243519 q^{22} -4.55089 q^{23} -0.152420 q^{24} -4.89389 q^{25} -10.4986 q^{26} -1.00000 q^{27} +4.29137 q^{28} -1.41050 q^{29} +0.645183 q^{30} -8.32826 q^{31} +7.91094 q^{32} -0.122948 q^{33} -3.94907 q^{34} +0.726904 q^{35} +1.92305 q^{36} +9.11094 q^{37} +4.37063 q^{38} -5.30054 q^{39} +0.0496494 q^{40} +1.81451 q^{41} +4.41995 q^{42} +11.9172 q^{43} +0.236435 q^{44} +0.325740 q^{45} +9.01380 q^{46} +3.72337 q^{47} +4.14799 q^{48} -2.02020 q^{49} +9.69318 q^{50} -1.99381 q^{51} +10.1932 q^{52} -13.6564 q^{53} +1.98067 q^{54} +0.0400491 q^{55} +0.340133 q^{56} +2.20664 q^{57} +2.79374 q^{58} +11.6145 q^{59} -0.626413 q^{60} +9.51637 q^{61} +16.4955 q^{62} +2.23155 q^{63} -7.37298 q^{64} +1.72660 q^{65} +0.243519 q^{66} +5.93054 q^{67} +3.83418 q^{68} +4.55089 q^{69} -1.43976 q^{70} -2.28058 q^{71} +0.152420 q^{72} +14.5567 q^{73} -18.0457 q^{74} +4.89389 q^{75} -4.24347 q^{76} +0.274364 q^{77} +10.4986 q^{78} +2.04239 q^{79} -1.35116 q^{80} +1.00000 q^{81} -3.59395 q^{82} -8.91466 q^{83} -4.29137 q^{84} +0.649462 q^{85} -23.6040 q^{86} +1.41050 q^{87} +0.0187398 q^{88} -14.6599 q^{89} -0.645183 q^{90} +11.8284 q^{91} -8.75157 q^{92} +8.32826 q^{93} -7.37477 q^{94} -0.718792 q^{95} -7.91094 q^{96} -6.13624 q^{97} +4.00135 q^{98} +0.122948 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 93 q + 2 q^{2} - 93 q^{3} + 114 q^{4} - 20 q^{5} - 2 q^{6} + 28 q^{7} + 6 q^{8} + 93 q^{9}+O(q^{10}) \) 93 * q + 2 * q^2 - 93 * q^3 + 114 * q^4 - 20 * q^5 - 2 * q^6 + 28 * q^7 + 6 * q^8 + 93 * q^9	\( 93 q + 2 q^{2} - 93 q^{3} + 114 q^{4} - 20 q^{5} - 2 q^{6} + 28 q^{7} + 6 q^{8} + 93 q^{9} + 19 q^{10} + 10 q^{11} - 114 q^{12} + 20 q^{13} + 13 q^{14} + 20 q^{15} + 148 q^{16} - 43 q^{17} + 2 q^{18} + 50 q^{19} - 31 q^{20} - 28 q^{21} + 36 q^{22} + 21 q^{23} - 6 q^{24} + 137 q^{25} + 2 q^{26} - 93 q^{27} + 62 q^{28} - q^{29} - 19 q^{30} + 58 q^{31} + 19 q^{32} - 10 q^{33} + 30 q^{34} + 30 q^{35} + 114 q^{36} + 42 q^{37} - 6 q^{38} - 20 q^{39} + 53 q^{40} - 7 q^{41} - 13 q^{42} + 60 q^{43} + 25 q^{44} - 20 q^{45} + 57 q^{46} + 9 q^{47} - 148 q^{48} + 145 q^{49} + 41 q^{50} + 43 q^{51} + 71 q^{52} - 45 q^{53} - 2 q^{54} + 78 q^{55} + 44 q^{56} - 50 q^{57} + 40 q^{58} + 42 q^{59} + 31 q^{60} + 69 q^{61} - 42 q^{62} + 28 q^{63} + 230 q^{64} - 4 q^{65} - 36 q^{66} + 76 q^{67} - 91 q^{68} - 21 q^{69} + 57 q^{70} + 92 q^{71} + 6 q^{72} + 29 q^{73} + 59 q^{74} - 137 q^{75} + 131 q^{76} - 98 q^{77} - 2 q^{78} + 215 q^{79} - 37 q^{80} + 93 q^{81} + 50 q^{82} - 27 q^{83} - 62 q^{84} + 52 q^{85} + 82 q^{86} + q^{87} + 136 q^{88} - 14 q^{89} + 19 q^{90} + 101 q^{91} - 14 q^{92} - 58 q^{93} + 112 q^{94} + 59 q^{95} - 19 q^{96} + 38 q^{97} - 16 q^{98} + 10 q^{99}+O(q^{100}) \) 93 * q + 2 * q^2 - 93 * q^3 + 114 * q^4 - 20 * q^5 - 2 * q^6 + 28 * q^7 + 6 * q^8 + 93 * q^9 + 19 * q^10 + 10 * q^11 - 114 * q^12 + 20 * q^13 + 13 * q^14 + 20 * q^15 + 148 * q^16 - 43 * q^17 + 2 * q^18 + 50 * q^19 - 31 * q^20 - 28 * q^21 + 36 * q^22 + 21 * q^23 - 6 * q^24 + 137 * q^25 + 2 * q^26 - 93 * q^27 + 62 * q^28 - q^29 - 19 * q^30 + 58 * q^31 + 19 * q^32 - 10 * q^33 + 30 * q^34 + 30 * q^35 + 114 * q^36 + 42 * q^37 - 6 * q^38 - 20 * q^39 + 53 * q^40 - 7 * q^41 - 13 * q^42 + 60 * q^43 + 25 * q^44 - 20 * q^45 + 57 * q^46 + 9 * q^47 - 148 * q^48 + 145 * q^49 + 41 * q^50 + 43 * q^51 + 71 * q^52 - 45 * q^53 - 2 * q^54 + 78 * q^55 + 44 * q^56 - 50 * q^57 + 40 * q^58 + 42 * q^59 + 31 * q^60 + 69 * q^61 - 42 * q^62 + 28 * q^63 + 230 * q^64 - 4 * q^65 - 36 * q^66 + 76 * q^67 - 91 * q^68 - 21 * q^69 + 57 * q^70 + 92 * q^71 + 6 * q^72 + 29 * q^73 + 59 * q^74 - 137 * q^75 + 131 * q^76 - 98 * q^77 - 2 * q^78 + 215 * q^79 - 37 * q^80 + 93 * q^81 + 50 * q^82 - 27 * q^83 - 62 * q^84 + 52 * q^85 + 82 * q^86 + q^87 + 136 * q^88 - 14 * q^89 + 19 * q^90 + 101 * q^91 - 14 * q^92 - 58 * q^93 + 112 * q^94 + 59 * q^95 - 19 * q^96 + 38 * q^97 - 16 * q^98 + 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more