LMFDB - Embedded newform 4006.2.a.g.1.33

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4006,2,Mod(1,4006)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4006, 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("4006.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4006 = 2 \cdot 2003 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4006.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$31.9880710497$
Analytic rank:	$1$
Dimension:	$40$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.33
Character	$\chi$	$=$	4006.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.84127 q^{3} +1.00000 q^{4} +1.47284 q^{5} -1.84127 q^{6} +1.09523 q^{7} -1.00000 q^{8} +0.390277 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.84127 q^{3} +1.00000 q^{4} +1.47284 q^{5} -1.84127 q^{6} +1.09523 q^{7} -1.00000 q^{8} +0.390277 q^{9} -1.47284 q^{10} -0.101160 q^{11} +1.84127 q^{12} -3.54011 q^{13} -1.09523 q^{14} +2.71190 q^{15} +1.00000 q^{16} -3.92148 q^{17} -0.390277 q^{18} -4.99251 q^{19} +1.47284 q^{20} +2.01662 q^{21} +0.101160 q^{22} -6.77759 q^{23} -1.84127 q^{24} -2.83074 q^{25} +3.54011 q^{26} -4.80521 q^{27} +1.09523 q^{28} +7.27553 q^{29} -2.71190 q^{30} +7.41051 q^{31} -1.00000 q^{32} -0.186262 q^{33} +3.92148 q^{34} +1.61310 q^{35} +0.390277 q^{36} +2.39150 q^{37} +4.99251 q^{38} -6.51830 q^{39} -1.47284 q^{40} -9.66550 q^{41} -2.01662 q^{42} -10.0378 q^{43} -0.101160 q^{44} +0.574816 q^{45} +6.77759 q^{46} +1.56239 q^{47} +1.84127 q^{48} -5.80047 q^{49} +2.83074 q^{50} -7.22050 q^{51} -3.54011 q^{52} +1.00741 q^{53} +4.80521 q^{54} -0.148992 q^{55} -1.09523 q^{56} -9.19256 q^{57} -7.27553 q^{58} -3.27334 q^{59} +2.71190 q^{60} -2.11889 q^{61} -7.41051 q^{62} +0.427443 q^{63} +1.00000 q^{64} -5.21403 q^{65} +0.186262 q^{66} +4.55049 q^{67} -3.92148 q^{68} -12.4794 q^{69} -1.61310 q^{70} -15.4298 q^{71} -0.390277 q^{72} -1.14717 q^{73} -2.39150 q^{74} -5.21215 q^{75} -4.99251 q^{76} -0.110793 q^{77} +6.51830 q^{78} +4.89848 q^{79} +1.47284 q^{80} -10.0185 q^{81} +9.66550 q^{82} +10.7533 q^{83} +2.01662 q^{84} -5.77572 q^{85} +10.0378 q^{86} +13.3962 q^{87} +0.101160 q^{88} -4.38574 q^{89} -0.574816 q^{90} -3.87724 q^{91} -6.77759 q^{92} +13.6448 q^{93} -1.56239 q^{94} -7.35318 q^{95} -1.84127 q^{96} -4.01376 q^{97} +5.80047 q^{98} -0.0394802 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 40 q^{2} - q^{3} + 40 q^{4} - 23 q^{5} + q^{6} + 12 q^{7} - 40 q^{8} + 29 q^{9}+O(q^{10}) $ 40 * q - 40 * q^2 - q^3 + 40 * q^4 - 23 * q^5 + q^6 + 12 * q^7 - 40 * q^8 + 29 * q^9	\( 40 q - 40 q^{2} - q^{3} + 40 q^{4} - 23 q^{5} + q^{6} + 12 q^{7} - 40 q^{8} + 29 q^{9} + 23 q^{10} - 28 q^{11} - q^{12} - 12 q^{13} - 12 q^{14} - 14 q^{15} + 40 q^{16} - 10 q^{17} - 29 q^{18} + 3 q^{19} - 23 q^{20} - 40 q^{21} + 28 q^{22} - 10 q^{23} + q^{24} + 43 q^{25} + 12 q^{26} - 7 q^{27} + 12 q^{28} - 37 q^{29} + 14 q^{30} - 16 q^{31} - 40 q^{32} - 11 q^{33} + 10 q^{34} - 24 q^{35} + 29 q^{36} - 17 q^{37} - 3 q^{38} - 10 q^{39} + 23 q^{40} - 58 q^{41} + 40 q^{42} + 37 q^{43} - 28 q^{44} - 66 q^{45} + 10 q^{46} - 34 q^{47} - q^{48} + 28 q^{49} - 43 q^{50} - 7 q^{51} - 12 q^{52} - 43 q^{53} + 7 q^{54} + 28 q^{55} - 12 q^{56} - 25 q^{57} + 37 q^{58} - 92 q^{59} - 14 q^{60} - 37 q^{61} + 16 q^{62} + 14 q^{63} + 40 q^{64} - 54 q^{65} + 11 q^{66} - 10 q^{67} - 10 q^{68} - 49 q^{69} + 24 q^{70} - 87 q^{71} - 29 q^{72} + 12 q^{73} + 17 q^{74} - 31 q^{75} + 3 q^{76} - 53 q^{77} + 10 q^{78} + 14 q^{79} - 23 q^{80} - 4 q^{81} + 58 q^{82} - 42 q^{83} - 40 q^{84} - 24 q^{85} - 37 q^{86} + 24 q^{87} + 28 q^{88} - 118 q^{89} + 66 q^{90} - 35 q^{91} - 10 q^{92} - 22 q^{93} + 34 q^{94} - 18 q^{95} + q^{96} + 2 q^{97} - 28 q^{98} - 48 q^{99}+O(q^{100}) \) 40 * q - 40 * q^2 - q^3 + 40 * q^4 - 23 * q^5 + q^6 + 12 * q^7 - 40 * q^8 + 29 * q^9 + 23 * q^10 - 28 * q^11 - q^12 - 12 * q^13 - 12 * q^14 - 14 * q^15 + 40 * q^16 - 10 * q^17 - 29 * q^18 + 3 * q^19 - 23 * q^20 - 40 * q^21 + 28 * q^22 - 10 * q^23 + q^24 + 43 * q^25 + 12 * q^26 - 7 * q^27 + 12 * q^28 - 37 * q^29 + 14 * q^30 - 16 * q^31 - 40 * q^32 - 11 * q^33 + 10 * q^34 - 24 * q^35 + 29 * q^36 - 17 * q^37 - 3 * q^38 - 10 * q^39 + 23 * q^40 - 58 * q^41 + 40 * q^42 + 37 * q^43 - 28 * q^44 - 66 * q^45 + 10 * q^46 - 34 * q^47 - q^48 + 28 * q^49 - 43 * q^50 - 7 * q^51 - 12 * q^52 - 43 * q^53 + 7 * q^54 + 28 * q^55 - 12 * q^56 - 25 * q^57 + 37 * q^58 - 92 * q^59 - 14 * q^60 - 37 * q^61 + 16 * q^62 + 14 * q^63 + 40 * q^64 - 54 * q^65 + 11 * q^66 - 10 * q^67 - 10 * q^68 - 49 * q^69 + 24 * q^70 - 87 * q^71 - 29 * q^72 + 12 * q^73 + 17 * q^74 - 31 * q^75 + 3 * q^76 - 53 * q^77 + 10 * q^78 + 14 * q^79 - 23 * q^80 - 4 * q^81 + 58 * q^82 - 42 * q^83 - 40 * q^84 - 24 * q^85 - 37 * q^86 + 24 * q^87 + 28 * q^88 - 118 * q^89 + 66 * q^90 - 35 * q^91 - 10 * q^92 - 22 * q^93 + 34 * q^94 - 18 * q^95 + q^96 + 2 * q^97 - 28 * q^98 - 48 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.84127	1.06306	0.531529	−	0.847040i	$-0.321618\pi$
$3$	1.84127	1.06306	0.531529	+	0.847040i	$0.321618\pi$
$4$	1.00000	0.500000
$5$	1.47284	0.658675	0.329338	−	0.944212i	$-0.393175\pi$
$5$	1.47284	0.658675	0.329338	+	0.944212i	$0.393175\pi$
$6$	−1.84127	−0.751695
$7$	1.09523	0.413958	0.206979	−	0.978345i	$-0.433637\pi$
$7$	1.09523	0.413958	0.206979	+	0.978345i	$0.433637\pi$
$8$	−1.00000	−0.353553
$9$	0.390277	0.130092
$10$	−1.47284	−0.465754
$11$	−0.101160	−0.0305008	−0.0152504	−	0.999884i	$-0.504855\pi$
$11$	−0.101160	−0.0305008	−0.0152504	+	0.999884i	$0.504855\pi$
$12$	1.84127	0.531529
$13$	−3.54011	−0.981850	−0.490925	−	0.871202i	$-0.663341\pi$
$13$	−3.54011	−0.981850	−0.490925	+	0.871202i	$0.663341\pi$
$14$	−1.09523	−0.292713
$15$	2.71190	0.700210
$16$	1.00000	0.250000
$17$	−3.92148	−0.951098	−0.475549	−	0.879689i	$-0.657751\pi$
$17$	−3.92148	−0.951098	−0.475549	+	0.879689i	$0.657751\pi$
$18$	−0.390277	−0.0919891
$19$	−4.99251	−1.14536	−0.572680	−	0.819779i	$-0.694096\pi$
$19$	−4.99251	−1.14536	−0.572680	+	0.819779i	$0.694096\pi$
$20$	1.47284	0.329338
$21$	2.01662	0.440062
$22$	0.101160	0.0215673
$23$	−6.77759	−1.41323	−0.706613	−	0.707601i	$-0.749778\pi$
$23$	−6.77759	−1.41323	−0.706613	+	0.707601i	$0.749778\pi$
$24$	−1.84127	−0.375848
$25$	−2.83074	−0.566147
$26$	3.54011	0.694273
$27$	−4.80521	−0.924762
$28$	1.09523	0.206979
$29$	7.27553	1.35103	0.675516	−	0.737346i	$-0.263921\pi$
$29$	7.27553	1.35103	0.675516	+	0.737346i	$0.263921\pi$
$30$	−2.71190	−0.495123
$31$	7.41051	1.33097	0.665483	−	0.746413i	$-0.268225\pi$
$31$	7.41051	1.33097	0.665483	+	0.746413i	$0.268225\pi$
$32$	−1.00000	−0.176777
$33$	−0.186262	−0.0324241
$34$	3.92148	0.672528
$35$	1.61310	0.272664
$36$	0.390277	0.0650461
$37$	2.39150	0.393160	0.196580	−	0.980488i	$-0.437016\pi$
$37$	2.39150	0.393160	0.196580	+	0.980488i	$0.437016\pi$
$38$	4.99251	0.809892
$39$	−6.51830	−1.04376
$40$	−1.47284	−0.232877
$41$	−9.66550	−1.50950	−0.754748	−	0.656014i	$-0.772241\pi$
$41$	−9.66550	−1.50950	−0.754748	+	0.656014i	$0.772241\pi$
$42$	−2.01662	−0.311171
$43$	−10.0378	−1.53076	−0.765379	−	0.643580i	$-0.777448\pi$
$43$	−10.0378	−1.53076	−0.765379	+	0.643580i	$0.777448\pi$
$44$	−0.101160	−0.0152504
$45$	0.574816	0.0856885
$46$	6.77759	0.999301
$47$	1.56239	0.227899	0.113949	−	0.993487i	$-0.463650\pi$
$47$	1.56239	0.227899	0.113949	+	0.993487i	$0.463650\pi$
$48$	1.84127	0.265764
$49$	−5.80047	−0.828638
$50$	2.83074	0.400326
$51$	−7.22050	−1.01107
$52$	−3.54011	−0.490925
$53$	1.00741	0.138378	0.0691892	−	0.997604i	$-0.477959\pi$
$53$	1.00741	0.138378	0.0691892	+	0.997604i	$0.477959\pi$
$54$	4.80521	0.653906
$55$	−0.148992	−0.0200901
$56$	−1.09523	−0.146356
$57$	−9.19256	−1.21758
$58$	−7.27553	−0.955323
$59$	−3.27334	−0.426153	−0.213076	−	0.977036i	$-0.568348\pi$
$59$	−3.27334	−0.426153	−0.213076	+	0.977036i	$0.568348\pi$
$60$	2.71190	0.350105
$61$	−2.11889	−0.271296	−0.135648	−	0.990757i	$-0.543312\pi$
$61$	−2.11889	−0.271296	−0.135648	+	0.990757i	$0.543312\pi$
$62$	−7.41051	−0.941136
$63$	0.427443	0.0538528
$64$	1.00000	0.125000
$65$	−5.21403	−0.646720
$66$	0.186262	0.0229273
$67$	4.55049	0.555931	0.277965	−	0.960591i	$-0.410340\pi$
$67$	4.55049	0.555931	0.277965	+	0.960591i	$0.410340\pi$
$68$	−3.92148	−0.475549
$69$	−12.4794	−1.50234
$70$	−1.61310	−0.192803
$71$	−15.4298	−1.83118	−0.915588	−	0.402117i	$-0.868274\pi$
$71$	−15.4298	−1.83118	−0.915588	+	0.402117i	$0.868274\pi$
$72$	−0.390277	−0.0459946
$73$	−1.14717	−0.134266	−0.0671329	−	0.997744i	$-0.521385\pi$
$73$	−1.14717	−0.134266	−0.0671329	+	0.997744i	$0.521385\pi$
$74$	−2.39150	−0.278006
$75$	−5.21215	−0.601847
$76$	−4.99251	−0.572680
$77$	−0.110793	−0.0126261
$78$	6.51830	0.738052
$79$	4.89848	0.551122	0.275561	−	0.961284i	$-0.411136\pi$
$79$	4.89848	0.551122	0.275561	+	0.961284i	$0.411136\pi$
$80$	1.47284	0.164669
$81$	−10.0185	−1.11317
$82$	9.66550	1.06738
$83$	10.7533	1.18033	0.590165	−	0.807283i	$-0.299063\pi$
$83$	10.7533	1.18033	0.590165	+	0.807283i	$0.299063\pi$
$84$	2.01662	0.220031
$85$	−5.77572	−0.626465
$86$	10.0378	1.08241
$87$	13.3962	1.43622
$88$	0.101160	0.0107837
$89$	−4.38574	−0.464887	−0.232444	−	0.972610i	$-0.574672\pi$
$89$	−4.38574	−0.464887	−0.232444	+	0.972610i	$0.574672\pi$
$90$	−0.574816	−0.0605909
$91$	−3.87724	−0.406445
$92$	−6.77759	−0.706613
$93$	13.6448	1.41489
$94$	−1.56239	−0.161149
$95$	−7.35318	−0.754420
$96$	−1.84127	−0.187924
$97$	−4.01376	−0.407535	−0.203768	−	0.979019i	$-0.565319\pi$
$97$	−4.01376	−0.407535	−0.203768	+	0.979019i	$0.565319\pi$
$98$	5.80047	0.585936
$99$	−0.0394802	−0.00396791
$100$	−2.83074	−0.283074
$101$	10.9802	1.09257	0.546283	−	0.837601i	$-0.316042\pi$
$101$	10.9802	1.09257	0.546283	+	0.837601i	$0.316042\pi$
$102$	7.22050	0.714936
$103$	15.5791	1.53506	0.767528	−	0.641015i	$-0.221487\pi$
$103$	15.5791	1.53506	0.767528	+	0.641015i	$0.221487\pi$
$104$	3.54011	0.347136
$105$	2.97016	0.289858
$106$	−1.00741	−0.0978483
$107$	8.57535	0.829010	0.414505	−	0.910047i	$-0.363955\pi$
$107$	8.57535	0.829010	0.414505	+	0.910047i	$0.363955\pi$
$108$	−4.80521	−0.462381
$109$	9.76920	0.935719	0.467860	−	0.883803i	$-0.345025\pi$
$109$	9.76920	0.935719	0.467860	+	0.883803i	$0.345025\pi$
$110$	0.148992	0.0142058
$111$	4.40340	0.417952
$112$	1.09523	0.103490
$113$	−18.3394	−1.72522	−0.862611	−	0.505868i	$-0.831172\pi$
$113$	−18.3394	−1.72522	−0.862611	+	0.505868i	$0.831172\pi$
$114$	9.19256	0.860962
$115$	−9.98232	−0.930856
$116$	7.27553	0.675516
$117$	−1.38162	−0.127731
$118$	3.27334	0.301336
$119$	−4.29492	−0.393715
$120$	−2.71190	−0.247562
$121$	−10.9898	−0.999070
$122$	2.11889	0.191835
$123$	−17.7968	−1.60468
$124$	7.41051	0.665483
$125$	−11.5334	−1.03158
$126$	−0.427443	−0.0380797
$127$	−19.5826	−1.73768	−0.868839	−	0.495095i	$-0.835133\pi$
$127$	−19.5826	−1.73768	−0.868839	+	0.495095i	$0.835133\pi$
$128$	−1.00000	−0.0883883
$129$	−18.4824	−1.62728
$130$	5.21403	0.457300
$131$	−3.48311	−0.304321	−0.152160	−	0.988356i	$-0.548623\pi$
$131$	−3.48311	−0.304321	−0.152160	+	0.988356i	$0.548623\pi$
$132$	−0.186262	−0.0162120
$133$	−5.46795	−0.474131
$134$	−4.55049	−0.393102
$135$	−7.07731	−0.609118
$136$	3.92148	0.336264
$137$	11.5005	0.982556	0.491278	−	0.871003i	$-0.336530\pi$
$137$	11.5005	0.982556	0.491278	+	0.871003i	$0.336530\pi$
$138$	12.4794	1.06232
$139$	−9.50298	−0.806033	−0.403016	−	0.915193i	$-0.632038\pi$
$139$	−9.50298	−0.806033	−0.403016	+	0.915193i	$0.632038\pi$
$140$	1.61310	0.136332
$141$	2.87679	0.242269
$142$	15.4298	1.29484
$143$	0.358116	0.0299472
$144$	0.390277	0.0325231
$145$	10.7157	0.889891
$146$	1.14717	0.0949403
$147$	−10.6802	−0.880891
$148$	2.39150	0.196580
$149$	9.64399	0.790067	0.395033	−	0.918667i	$-0.370733\pi$
$149$	9.64399	0.790067	0.395033	+	0.918667i	$0.370733\pi$
$150$	5.21215	0.425570
$151$	10.5837	0.861293	0.430647	−	0.902521i	$-0.358286\pi$
$151$	10.5837	0.861293	0.430647	+	0.902521i	$0.358286\pi$
$152$	4.99251	0.404946
$153$	−1.53046	−0.123730
$154$	0.110793	0.00892797
$155$	10.9145	0.876675
$156$	−6.51830	−0.521882
$157$	12.2849	0.980441	0.490220	−	0.871599i	$-0.336916\pi$
$157$	12.2849	0.980441	0.490220	+	0.871599i	$0.336916\pi$
$158$	−4.89848	−0.389702
$159$	1.85491	0.147104
$160$	−1.47284	−0.116438
$161$	−7.42303	−0.585016
$162$	10.0185	0.787129
$163$	−13.3134	−1.04279	−0.521393	−	0.853317i	$-0.674587\pi$
$163$	−13.3134	−1.04279	−0.521393	+	0.853317i	$0.674587\pi$
$164$	−9.66550	−0.754748
$165$	−0.274335	−0.0213569
$166$	−10.7533	−0.834619
$167$	19.0753	1.47609	0.738046	−	0.674751i	$-0.235749\pi$
$167$	19.0753	1.47609	0.738046	+	0.674751i	$0.235749\pi$
$168$	−2.01662	−0.155585
$169$	−0.467617	−0.0359706
$170$	5.77572	0.442977
$171$	−1.94846	−0.149002
$172$	−10.0378	−0.765379
$173$	9.06546	0.689235	0.344617	−	0.938743i	$-0.388009\pi$
$173$	9.06546	0.689235	0.344617	+	0.938743i	$0.388009\pi$
$174$	−13.3962	−1.01556
$175$	−3.10031	−0.234361
$176$	−0.101160	−0.00762519
$177$	−6.02711	−0.453025
$178$	4.38574	0.328725
$179$	−8.51324	−0.636309	−0.318155	−	0.948039i	$-0.603063\pi$
$179$	−8.51324	−0.636309	−0.318155	+	0.948039i	$0.603063\pi$
$180$	0.574816	0.0428443
$181$	2.97370	0.221033	0.110517	−	0.993874i	$-0.464749\pi$
$181$	2.97370	0.221033	0.110517	+	0.993874i	$0.464749\pi$
$182$	3.87724	0.287400
$183$	−3.90145	−0.288404
$184$	6.77759	0.499651
$185$	3.52230	0.258965
$186$	−13.6448	−1.00048
$187$	0.396695	0.0290092
$188$	1.56239	0.113949
$189$	−5.26281	−0.382813
$190$	7.35318	0.533456
$191$	−21.0940	−1.52631	−0.763155	−	0.646215i	$-0.776351\pi$
$191$	−21.0940	−1.52631	−0.763155	+	0.646215i	$0.776351\pi$
$192$	1.84127	0.132882
$193$	24.0983	1.73463	0.867315	−	0.497760i	$-0.165844\pi$
$193$	24.0983	1.73463	0.867315	+	0.497760i	$0.165844\pi$
$194$	4.01376	0.288171
$195$	−9.60043	−0.687501
$196$	−5.80047	−0.414319
$197$	4.51745	0.321855	0.160927	−	0.986966i	$-0.448551\pi$
$197$	4.51745	0.321855	0.160927	+	0.986966i	$0.448551\pi$
$198$	0.0394802	0.00280574
$199$	−5.04739	−0.357800	−0.178900	−	0.983867i	$-0.557254\pi$
$199$	−5.04739	−0.357800	−0.178900	+	0.983867i	$0.557254\pi$
$200$	2.83074	0.200163
$201$	8.37868	0.590986
$202$	−10.9802	−0.772561
$203$	7.96838	0.559271
$204$	−7.22050	−0.505536
$205$	−14.2358	−0.994268
$206$	−15.5791	−1.08545
$207$	−2.64514	−0.183850
$208$	−3.54011	−0.245463
$209$	0.505040	0.0349344
$210$	−2.97016	−0.204960
$211$	13.0372	0.897516	0.448758	−	0.893653i	$-0.351867\pi$
$211$	13.0372	0.897516	0.448758	+	0.893653i	$0.351867\pi$
$212$	1.00741	0.0691892
$213$	−28.4104	−1.94665
$214$	−8.57535	−0.586199
$215$	−14.7842	−1.00827
$216$	4.80521	0.326953
$217$	8.11622	0.550965
$218$	−9.76920	−0.661654
$219$	−2.11225	−0.142732
$220$	−0.148992	−0.0100451
$221$	13.8825	0.933835
$222$	−4.40340	−0.295537
$223$	10.5253	0.704826	0.352413	−	0.935845i	$-0.385361\pi$
$223$	10.5253	0.704826	0.352413	+	0.935845i	$0.385361\pi$
$224$	−1.09523	−0.0731782
$225$	−1.10477	−0.0736513
$226$	18.3394	1.21992
$227$	1.44809	0.0961133	0.0480566	−	0.998845i	$-0.484697\pi$
$227$	1.44809	0.0961133	0.0480566	+	0.998845i	$0.484697\pi$
$228$	−9.19256	−0.608792
$229$	26.5776	1.75630	0.878148	−	0.478389i	$-0.158779\pi$
$229$	26.5776	1.75630	0.878148	+	0.478389i	$0.158779\pi$
$230$	9.98232	0.658215
$231$	−0.204000	−0.0134222
$232$	−7.27553	−0.477662
$233$	−18.9898	−1.24407	−0.622033	−	0.782991i	$-0.713693\pi$
$233$	−18.9898	−1.24407	−0.622033	+	0.782991i	$0.713693\pi$
$234$	1.38162	0.0903195
$235$	2.30116	0.150111
$236$	−3.27334	−0.213076
$237$	9.01942	0.585874
$238$	4.29492	0.278398
$239$	−26.5503	−1.71739	−0.858697	−	0.512484i	$-0.828725\pi$
$239$	−26.5503	−1.71739	−0.858697	+	0.512484i	$0.828725\pi$
$240$	2.71190	0.175052
$241$	14.5556	0.937606	0.468803	−	0.883303i	$-0.344685\pi$
$241$	14.5556	0.937606	0.468803	+	0.883303i	$0.344685\pi$
$242$	10.9898	0.706449
$243$	−4.03118	−0.258600
$244$	−2.11889	−0.135648
$245$	−8.54318	−0.545804
$246$	17.7968	1.13468
$247$	17.6740	1.12457
$248$	−7.41051	−0.470568
$249$	19.7998	1.25476
$250$	11.5334	0.729439
$251$	−7.00160	−0.441937	−0.220969	−	0.975281i	$-0.570922\pi$
$251$	−7.00160	−0.441937	−0.220969	+	0.975281i	$0.570922\pi$
$252$	0.427443	0.0269264
$253$	0.685618	0.0431045
$254$	19.5826	1.22872
$255$	−10.6347	−0.665968
$256$	1.00000	0.0625000
$257$	5.44238	0.339486	0.169743	−	0.985488i	$-0.445706\pi$
$257$	5.44238	0.339486	0.169743	+	0.985488i	$0.445706\pi$
$258$	18.4824	1.15066
$259$	2.61925	0.162752
$260$	−5.21403	−0.323360
$261$	2.83947	0.175759
$262$	3.48311	0.215187
$263$	30.7584	1.89665	0.948323	−	0.317308i	$-0.102779\pi$
$263$	30.7584	1.89665	0.948323	+	0.317308i	$0.102779\pi$
$264$	0.186262	0.0114636
$265$	1.48376	0.0911464
$266$	5.46795	0.335262
$267$	−8.07532	−0.494202
$268$	4.55049	0.277965
$269$	−26.8949	−1.63981	−0.819905	−	0.572500i	$-0.805974\pi$
$269$	−26.8949	−1.63981	−0.819905	+	0.572500i	$0.805974\pi$
$270$	7.07731	0.430711
$271$	−16.1235	−0.979434	−0.489717	−	0.871881i	$-0.662900\pi$
$271$	−16.1235	−0.979434	−0.489717	+	0.871881i	$0.662900\pi$
$272$	−3.92148	−0.237774
$273$	−7.13904	−0.432075
$274$	−11.5005	−0.694772
$275$	0.286356	0.0172679
$276$	−12.4794	−0.751170
$277$	5.69251	0.342030	0.171015	−	0.985268i	$-0.445295\pi$
$277$	5.69251	0.342030	0.171015	+	0.985268i	$0.445295\pi$
$278$	9.50298	0.569951
$279$	2.89215	0.173148
$280$	−1.61310	−0.0964013
$281$	9.50743	0.567166	0.283583	−	0.958948i	$-0.408477\pi$
$281$	9.50743	0.567166	0.283583	+	0.958948i	$0.408477\pi$
$282$	−2.87679	−0.171310
$283$	24.2615	1.44220	0.721098	−	0.692834i	$-0.243638\pi$
$283$	24.2615	1.44220	0.721098	+	0.692834i	$0.243638\pi$
$284$	−15.4298	−0.915588
$285$	−13.5392	−0.801993
$286$	−0.358116	−0.0211759
$287$	−10.5859	−0.624869
$288$	−0.390277	−0.0229973
$289$	−1.62202	−0.0954129
$290$	−10.7157	−0.629248
$291$	−7.39041	−0.433234
$292$	−1.14717	−0.0671329
$293$	21.2296	1.24025	0.620124	−	0.784504i	$-0.287082\pi$
$293$	21.2296	1.24025	0.620124	+	0.784504i	$0.287082\pi$
$294$	10.6802	0.622884
$295$	−4.82112	−0.280696
$296$	−2.39150	−0.139003
$297$	0.486093	0.0282060
$298$	−9.64399	−0.558662
$299$	23.9934	1.38758
$300$	−5.21215	−0.300924
$301$	−10.9938	−0.633670
$302$	−10.5837	−0.609026
$303$	20.2174	1.16146
$304$	−4.99251	−0.286340
$305$	−3.12079	−0.178696
$306$	1.53046	0.0874906
$307$	18.5265	1.05737	0.528683	−	0.848819i	$-0.322686\pi$
$307$	18.5265	1.05737	0.528683	+	0.848819i	$0.322686\pi$
$308$	−0.110793	−0.00631303
$309$	28.6854	1.63185
$310$	−10.9145	−0.619903
$311$	11.6204	0.658930	0.329465	−	0.944168i	$-0.393132\pi$
$311$	11.6204	0.658930	0.329465	+	0.944168i	$0.393132\pi$
$312$	6.51830	0.369026
$313$	−1.20905	−0.0683393	−0.0341697	−	0.999416i	$-0.510879\pi$
$313$	−1.20905	−0.0683393	−0.0341697	+	0.999416i	$0.510879\pi$
$314$	−12.2849	−0.693276
$315$	0.629556	0.0354715
$316$	4.89848	0.275561
$317$	−22.5797	−1.26820	−0.634101	−	0.773250i	$-0.718630\pi$
$317$	−22.5797	−1.26820	−0.634101	+	0.773250i	$0.718630\pi$
$318$	−1.85491	−0.104018
$319$	−0.735989	−0.0412075
$320$	1.47284	0.0823344
$321$	15.7895	0.881286
$322$	7.42303	0.413669
$323$	19.5780	1.08935
$324$	−10.0185	−0.556584
$325$	10.0211	0.555871
$326$	13.3134	0.737361
$327$	17.9877	0.994724
$328$	9.66550	0.533688
$329$	1.71118	0.0943406
$330$	0.274335	0.0151016
$331$	−25.2339	−1.38698	−0.693490	−	0.720466i	$-0.743928\pi$
$331$	−25.2339	−1.38698	−0.693490	+	0.720466i	$0.743928\pi$
$332$	10.7533	0.590165
$333$	0.933347	0.0511471
$334$	−19.0753	−1.04375
$335$	6.70215	0.366178
$336$	2.01662	0.110015
$337$	32.1595	1.75184	0.875920	−	0.482456i	$-0.160255\pi$
$337$	32.1595	1.75184	0.875920	+	0.482456i	$0.160255\pi$
$338$	0.467617	0.0254350
$339$	−33.7677	−1.83401
$340$	−5.77572	−0.313232
$341$	−0.749644	−0.0405955
$342$	1.94846	0.105361
$343$	−14.0195	−0.756980
$344$	10.0378	0.541205
$345$	−18.3802	−0.989554
$346$	−9.06546	−0.487362
$347$	−4.50544	−0.241865	−0.120932	−	0.992661i	$-0.538588\pi$
$347$	−4.50544	−0.241865	−0.120932	+	0.992661i	$0.538588\pi$
$348$	13.3962	0.718112
$349$	−15.2681	−0.817285	−0.408642	−	0.912695i	$-0.633998\pi$
$349$	−15.2681	−0.817285	−0.408642	+	0.912695i	$0.633998\pi$
$350$	3.10031	0.165718
$351$	17.0110	0.907978
$352$	0.101160	0.00539183
$353$	−25.3793	−1.35080	−0.675402	−	0.737450i	$-0.736030\pi$
$353$	−25.3793	−1.35080	−0.675402	+	0.737450i	$0.736030\pi$
$354$	6.02711	0.320337
$355$	−22.7256	−1.20615
$356$	−4.38574	−0.232444
$357$	−7.90811	−0.418542
$358$	8.51324	0.449939
$359$	−23.0978	−1.21905	−0.609527	−	0.792766i	$-0.708640\pi$
$359$	−23.0978	−1.21905	−0.609527	+	0.792766i	$0.708640\pi$
$360$	−0.574816	−0.0302955
$361$	5.92515	0.311850
$362$	−2.97370	−0.156294
$363$	−20.2351	−1.06207
$364$	−3.87724	−0.203223
$365$	−1.68960	−0.0884376
$366$	3.90145	0.203932
$367$	−22.3197	−1.16508	−0.582539	−	0.812803i	$-0.697941\pi$
$367$	−22.3197	−1.16508	−0.582539	+	0.812803i	$0.697941\pi$
$368$	−6.77759	−0.353306
$369$	−3.77222	−0.196374
$370$	−3.52230	−0.183116
$371$	1.10335	0.0572829
$372$	13.6448	0.707447
$373$	−4.47324	−0.231616	−0.115808	−	0.993272i	$-0.536946\pi$
$373$	−4.47324	−0.231616	−0.115808	+	0.993272i	$0.536946\pi$
$374$	−0.396695	−0.0205126
$375$	−21.2362	−1.09663
$376$	−1.56239	−0.0805743
$377$	−25.7562	−1.32651
$378$	5.26281	0.270690
$379$	8.22925	0.422708	0.211354	−	0.977410i	$-0.432213\pi$
$379$	8.22925	0.422708	0.211354	+	0.977410i	$0.432213\pi$
$380$	−7.35318	−0.377210
$381$	−36.0569	−1.84725
$382$	21.0940	1.07926
$383$	−18.1795	−0.928927	−0.464463	−	0.885592i	$-0.653753\pi$
$383$	−18.1795	−0.928927	−0.464463	+	0.885592i	$0.653753\pi$
$384$	−1.84127	−0.0939619
$385$	−0.163181	−0.00831647
$386$	−24.0983	−1.22657
$387$	−3.91754	−0.199140
$388$	−4.01376	−0.203768
$389$	−24.7333	−1.25403	−0.627014	−	0.779008i	$-0.715723\pi$
$389$	−24.7333	−1.25403	−0.627014	+	0.779008i	$0.715723\pi$
$390$	9.60043	0.486137
$391$	26.5782	1.34412
$392$	5.80047	0.292968
$393$	−6.41335	−0.323511
$394$	−4.51745	−0.227586
$395$	7.21469	0.363010
$396$	−0.0394802	−0.00198396
$397$	−11.0486	−0.554512	−0.277256	−	0.960796i	$-0.589425\pi$
$397$	−11.0486	−0.554512	−0.277256	+	0.960796i	$0.589425\pi$
$398$	5.04739	0.253003
$399$	−10.0680	−0.504029
$400$	−2.83074	−0.141537
$401$	−11.2061	−0.559604	−0.279802	−	0.960058i	$-0.590269\pi$
$401$	−11.2061	−0.559604	−0.279802	+	0.960058i	$0.590269\pi$
$402$	−8.37868	−0.417890
$403$	−26.2340	−1.30681
$404$	10.9802	0.546283
$405$	−14.7557	−0.733216
$406$	−7.96838	−0.395464
$407$	−0.241923	−0.0119917
$408$	7.22050	0.357468
$409$	−13.5770	−0.671337	−0.335669	−	0.941980i	$-0.608962\pi$
$409$	−13.5770	−0.671337	−0.335669	+	0.941980i	$0.608962\pi$
$410$	14.2358	0.703054
$411$	21.1756	1.04451
$412$	15.5791	0.767528
$413$	−3.58507	−0.176410
$414$	2.64514	0.130001
$415$	15.8379	0.777454
$416$	3.54011	0.173568
$417$	−17.4976	−0.856859
$418$	−0.505040	−0.0247023
$419$	−8.86331	−0.433001	−0.216500	−	0.976283i	$-0.569464\pi$
$419$	−8.86331	−0.433001	−0.216500	+	0.976283i	$0.569464\pi$
$420$	2.97016	0.144929
$421$	26.3737	1.28538	0.642688	−	0.766128i	$-0.277819\pi$
$421$	26.3737	1.28538	0.642688	+	0.766128i	$0.277819\pi$
$422$	−13.0372	−0.634639
$423$	0.609766	0.0296478
$424$	−1.00741	−0.0489242
$425$	11.1007	0.538461
$426$	28.4104	1.37649
$427$	−2.32068	−0.112305
$428$	8.57535	0.414505
$429$	0.659389	0.0318356
$430$	14.7842	0.712956
$431$	4.54192	0.218777	0.109388	−	0.993999i	$-0.465111\pi$
$431$	4.54192	0.218777	0.109388	+	0.993999i	$0.465111\pi$
$432$	−4.80521	−0.231191
$433$	−20.0081	−0.961527	−0.480764	−	0.876850i	$-0.659640\pi$
$433$	−20.0081	−0.961527	−0.480764	+	0.876850i	$0.659640\pi$
$434$	−8.11622	−0.389591
$435$	19.7305	0.946005
$436$	9.76920	0.467860
$437$	33.8372	1.61865
$438$	2.11225	0.100927
$439$	14.4500	0.689659	0.344830	−	0.938665i	$-0.387937\pi$
$439$	14.4500	0.689659	0.344830	+	0.938665i	$0.387937\pi$
$440$	0.148992	0.00710292
$441$	−2.26379	−0.107799
$442$	−13.8825	−0.660321
$443$	−19.4709	−0.925093	−0.462546	−	0.886595i	$-0.653064\pi$
$443$	−19.4709	−0.925093	−0.462546	+	0.886595i	$0.653064\pi$
$444$	4.40340	0.208976
$445$	−6.45950	−0.306210
$446$	−10.5253	−0.498387
$447$	17.7572	0.839887
$448$	1.09523	0.0517448
$449$	19.9064	0.939440	0.469720	−	0.882816i	$-0.344355\pi$
$449$	19.9064	0.939440	0.469720	+	0.882816i	$0.344355\pi$
$450$	1.10477	0.0520794
$451$	0.977758	0.0460408
$452$	−18.3394	−0.862611
$453$	19.4875	0.915604
$454$	−1.44809	−0.0679624
$455$	−5.71056	−0.267715
$456$	9.19256	0.430481
$457$	−9.72662	−0.454992	−0.227496	−	0.973779i	$-0.573054\pi$
$457$	−9.72662	−0.454992	−0.227496	+	0.973779i	$0.573054\pi$
$458$	−26.5776	−1.24189
$459$	18.8435	0.879540
$460$	−9.98232	−0.465428
$461$	−37.9936	−1.76954	−0.884769	−	0.466031i	$-0.845684\pi$
$461$	−37.9936	−1.76954	−0.884769	+	0.466031i	$0.845684\pi$
$462$	0.204000	0.00949095
$463$	−1.33520	−0.0620519	−0.0310260	−	0.999519i	$-0.509877\pi$
$463$	−1.33520	−0.0620519	−0.0310260	+	0.999519i	$0.509877\pi$
$464$	7.27553	0.337758
$465$	20.0966	0.931956
$466$	18.9898	0.879687
$467$	28.0912	1.29991	0.649953	−	0.759974i	$-0.274788\pi$
$467$	28.0912	1.29991	0.649953	+	0.759974i	$0.274788\pi$
$468$	−1.38162	−0.0638655
$469$	4.98383	0.230132
$470$	−2.30116	−0.106145
$471$	22.6198	1.04227
$472$	3.27334	0.150668
$473$	1.01542	0.0466893
$474$	−9.01942	−0.414276
$475$	14.1325	0.648442
$476$	−4.29492	−0.196857
$477$	0.393169	0.0180020
$478$	26.5503	1.21438
$479$	30.1834	1.37911	0.689556	−	0.724232i	$-0.257806\pi$
$479$	30.1834	1.37911	0.689556	+	0.724232i	$0.257806\pi$
$480$	−2.71190	−0.123781
$481$	−8.46618	−0.386024
$482$	−14.5556	−0.662987
$483$	−13.6678	−0.621906
$484$	−10.9898	−0.499535
$485$	−5.91163	−0.268433
$486$	4.03118	0.182858
$487$	−24.2861	−1.10051	−0.550255	−	0.834997i	$-0.685470\pi$
$487$	−24.2861	−1.10051	−0.550255	+	0.834997i	$0.685470\pi$
$488$	2.11889	0.0959177
$489$	−24.5136	−1.10854
$490$	8.54318	0.385941
$491$	−16.3995	−0.740097	−0.370049	−	0.929012i	$-0.620659\pi$
$491$	−16.3995	−0.740097	−0.370049	+	0.929012i	$0.620659\pi$
$492$	−17.7968	−0.802341
$493$	−28.5308	−1.28496
$494$	−17.6740	−0.795192
$495$	−0.0581482	−0.00261357
$496$	7.41051	0.332742
$497$	−16.8992	−0.758031
$498$	−19.7998	−0.887248
$499$	−8.52863	−0.381794	−0.190897	−	0.981610i	$-0.561140\pi$
$499$	−8.52863	−0.381794	−0.190897	+	0.981610i	$0.561140\pi$
$500$	−11.5334	−0.515791
$501$	35.1228	1.56917
$502$	7.00160	0.312497
$503$	−26.4871	−1.18100	−0.590500	−	0.807037i	$-0.701070\pi$
$503$	−26.4871	−1.18100	−0.590500	+	0.807037i	$0.701070\pi$
$504$	−0.427443	−0.0190398
$505$	16.1720	0.719646
$506$	−0.685618	−0.0304795
$507$	−0.861010	−0.0382388
$508$	−19.5826	−0.868839
$509$	5.28072	0.234064	0.117032	−	0.993128i	$-0.462662\pi$
$509$	5.28072	0.234064	0.117032	+	0.993128i	$0.462662\pi$
$510$	10.6347	0.470911
$511$	−1.25641	−0.0555805
$512$	−1.00000	−0.0441942
$513$	23.9900	1.05919
$514$	−5.44238	−0.240053
$515$	22.9456	1.01110
$516$	−18.4824	−0.813642
$517$	−0.158051	−0.00695108
$518$	−2.61925	−0.115083
$519$	16.6920	0.732696
$520$	5.21403	0.228650
$521$	−41.1579	−1.80316	−0.901579	−	0.432614i	$-0.857591\pi$
$521$	−41.1579	−1.80316	−0.901579	+	0.432614i	$0.857591\pi$
$522$	−2.83947	−0.124280
$523$	36.5777	1.59943	0.799715	−	0.600380i	$-0.204984\pi$
$523$	36.5777	1.59943	0.799715	+	0.600380i	$0.204984\pi$
$524$	−3.48311	−0.152160
$525$	−5.70851	−0.249140
$526$	−30.7584	−1.34113
$527$	−29.0601	−1.26588
$528$	−0.186262	−0.00810602
$529$	22.9357	0.997206
$530$	−1.48376	−0.0644502
$531$	−1.27751	−0.0554392
$532$	−5.46795	−0.237066
$533$	34.2169	1.48210
$534$	8.07532	0.349453
$535$	12.6301	0.546048
$536$	−4.55049	−0.196551
$537$	−15.6752	−0.676434
$538$	26.8949	1.15952
$539$	0.586773	0.0252741
$540$	−7.07731	−0.304559
$541$	0.774000	0.0332769	0.0166384	−	0.999862i	$-0.494704\pi$
$541$	0.774000	0.0332769	0.0166384	+	0.999862i	$0.494704\pi$
$542$	16.1235	0.692564
$543$	5.47539	0.234971
$544$	3.92148	0.168132
$545$	14.3885	0.616335
$546$	7.13904	0.305523
$547$	−33.8350	−1.44668	−0.723340	−	0.690492i	$-0.757394\pi$
$547$	−33.8350	−1.44668	−0.723340	+	0.690492i	$0.757394\pi$
$548$	11.5005	0.491278
$549$	−0.826954	−0.0352935
$550$	−0.286356	−0.0122103
$551$	−36.3231	−1.54742
$552$	12.4794	0.531158
$553$	5.36496	0.228142
$554$	−5.69251	−0.241852
$555$	6.48551	0.275295
$556$	−9.50298	−0.403016
$557$	25.1106	1.06397	0.531986	−	0.846753i	$-0.321446\pi$
$557$	25.1106	1.06397	0.531986	+	0.846753i	$0.321446\pi$
$558$	−2.89215	−0.122434
$559$	35.5351	1.50297
$560$	1.61310	0.0681660
$561$	0.730423	0.0308385
$562$	−9.50743	−0.401047
$563$	−23.1626	−0.976185	−0.488093	−	0.872792i	$-0.662307\pi$
$563$	−23.1626	−0.976185	−0.488093	+	0.872792i	$0.662307\pi$
$564$	2.87679	0.121135
$565$	−27.0110	−1.13636
$566$	−24.2615	−1.01979
$567$	−10.9726	−0.460805
$568$	15.4298	0.647419
$569$	36.1374	1.51496	0.757480	−	0.652858i	$-0.226430\pi$
$569$	36.1374	1.51496	0.757480	+	0.652858i	$0.226430\pi$
$570$	13.5392	0.567094
$571$	−40.9449	−1.71349	−0.856746	−	0.515739i	$-0.827518\pi$
$571$	−40.9449	−1.71349	−0.856746	+	0.515739i	$0.827518\pi$
$572$	0.358116	0.0149736
$573$	−38.8398	−1.62256
$574$	10.5859	0.441849
$575$	19.1856	0.800093
$576$	0.390277	0.0162615
$577$	−0.250399	−0.0104243	−0.00521213	−	0.999986i	$-0.501659\pi$
$577$	−0.250399	−0.0104243	−0.00521213	+	0.999986i	$0.501659\pi$
$578$	1.62202	0.0674671
$579$	44.3714	1.84401
$580$	10.7157	0.444945
$581$	11.7774	0.488607
$582$	7.39041	0.306343
$583$	−0.101909	−0.00422065
$584$	1.14717	0.0474702
$585$	−2.03491	−0.0841333
$586$	−21.2296	−0.876987
$587$	14.0166	0.578525	0.289263	−	0.957250i	$-0.406590\pi$
$587$	14.0166	0.578525	0.289263	+	0.957250i	$0.406590\pi$
$588$	−10.6802	−0.440445
$589$	−36.9970	−1.52444
$590$	4.82112	0.198482
$591$	8.31784	0.342150
$592$	2.39150	0.0982901
$593$	−6.90834	−0.283692	−0.141846	−	0.989889i	$-0.545304\pi$
$593$	−6.90834	−0.283692	−0.141846	+	0.989889i	$0.545304\pi$
$594$	−0.486093	−0.0199446
$595$	−6.32574	−0.259330
$596$	9.64399	0.395033
$597$	−9.29361	−0.380362
$598$	−23.9934	−0.981164
$599$	18.8997	0.772222	0.386111	−	0.922452i	$-0.373818\pi$
$599$	18.8997	0.772222	0.386111	+	0.922452i	$0.373818\pi$
$600$	5.21215	0.212785
$601$	−18.6259	−0.759768	−0.379884	−	0.925034i	$-0.624036\pi$
$601$	−18.6259	−0.759768	−0.379884	+	0.925034i	$0.624036\pi$
$602$	10.9938	0.448072
$603$	1.77595	0.0723223
$604$	10.5837	0.430647
$605$	−16.1862	−0.658062
$606$	−20.2174	−0.821277
$607$	17.5136	0.710856	0.355428	−	0.934704i	$-0.384335\pi$
$607$	17.5136	0.710856	0.355428	+	0.934704i	$0.384335\pi$
$608$	4.99251	0.202473
$609$	14.6719	0.594537
$610$	3.12079	0.126357
$611$	−5.53105	−0.223762
$612$	−1.53046	−0.0618652
$613$	−3.99612	−0.161402	−0.0807008	−	0.996738i	$-0.525716\pi$
$613$	−3.99612	−0.161402	−0.0807008	+	0.996738i	$0.525716\pi$
$614$	−18.5265	−0.747671
$615$	−26.2119	−1.05696
$616$	0.110793	0.00446398
$617$	−32.4711	−1.30724	−0.653618	−	0.756824i	$-0.726750\pi$
$617$	−32.4711	−1.30724	−0.653618	+	0.756824i	$0.726750\pi$
$618$	−28.6854	−1.15389
$619$	1.20355	0.0483749	0.0241875	−	0.999707i	$-0.492300\pi$
$619$	1.20355	0.0483749	0.0241875	+	0.999707i	$0.492300\pi$
$620$	10.9145	0.438337
$621$	32.5677	1.30690
$622$	−11.6204	−0.465934
$623$	−4.80339	−0.192444
$624$	−6.51830	−0.260941
$625$	−2.83326	−0.113331
$626$	1.20905	0.0483232
$627$	0.929916	0.0371373
$628$	12.2849	0.490220
$629$	−9.37821	−0.373934
$630$	−0.629556	−0.0250821
$631$	38.9200	1.54938	0.774690	−	0.632341i	$-0.217906\pi$
$631$	38.9200	1.54938	0.774690	+	0.632341i	$0.217906\pi$
$632$	−4.89848	−0.194851
$633$	24.0050	0.954111
$634$	22.5797	0.896755
$635$	−28.8421	−1.14457
$636$	1.85491	0.0735521
$637$	20.5343	0.813599
$638$	0.735989	0.0291381
$639$	−6.02188	−0.238222
$640$	−1.47284	−0.0582192
$641$	2.61808	0.103408	0.0517040	−	0.998662i	$-0.483535\pi$
$641$	2.61808	0.103408	0.0517040	+	0.998662i	$0.483535\pi$
$642$	−15.7895	−0.623163
$643$	16.4315	0.647994	0.323997	−	0.946058i	$-0.394973\pi$
$643$	16.4315	0.647994	0.323997	+	0.946058i	$0.394973\pi$
$644$	−7.42303	−0.292508
$645$	−27.2217	−1.07185
$646$	−19.5780	−0.770286
$647$	−35.1854	−1.38328	−0.691640	−	0.722242i	$-0.743111\pi$
$647$	−35.1854	−1.38328	−0.691640	+	0.722242i	$0.743111\pi$
$648$	10.0185	0.393564
$649$	0.331130	0.0129980
$650$	−10.0211	−0.393060
$651$	14.9442	0.585708
$652$	−13.3134	−0.521393
$653$	−34.6400	−1.35557	−0.677785	−	0.735260i	$-0.737060\pi$
$653$	−34.6400	−1.35557	−0.677785	+	0.735260i	$0.737060\pi$
$654$	−17.9877	−0.703376
$655$	−5.13007	−0.200449
$656$	−9.66550	−0.377374
$657$	−0.447713	−0.0174669
$658$	−1.71118	−0.0667088
$659$	−4.62376	−0.180116	−0.0900581	−	0.995937i	$-0.528705\pi$
$659$	−4.62376	−0.180116	−0.0900581	+	0.995937i	$0.528705\pi$
$660$	−0.274335	−0.0106785
$661$	3.67954	0.143117	0.0715587	−	0.997436i	$-0.477203\pi$
$661$	3.67954	0.143117	0.0715587	+	0.997436i	$0.477203\pi$
$662$	25.2339	0.980743
$663$	25.5614	0.992721
$664$	−10.7533	−0.417310
$665$	−8.05343	−0.312299
$666$	−0.933347	−0.0361665
$667$	−49.3105	−1.90931
$668$	19.0753	0.738046
$669$	19.3799	0.749271
$670$	−6.70215	−0.258927
$671$	0.214346	0.00827475
$672$	−2.01662	−0.0777927
$673$	−35.0692	−1.35182	−0.675910	−	0.736984i	$-0.736249\pi$
$673$	−35.0692	−1.35182	−0.675910	+	0.736984i	$0.736249\pi$
$674$	−32.1595	−1.23874
$675$	13.6023	0.523551
$676$	−0.467617	−0.0179853
$677$	−12.8217	−0.492778	−0.246389	−	0.969171i	$-0.579244\pi$
$677$	−12.8217	−0.492778	−0.246389	+	0.969171i	$0.579244\pi$
$678$	33.7677	1.29684
$679$	−4.39599	−0.168703
$680$	5.77572	0.221489
$681$	2.66633	0.102174
$682$	0.749644	0.0287054
$683$	30.3154	1.15999	0.579993	−	0.814622i	$-0.303055\pi$
$683$	30.3154	1.15999	0.579993	+	0.814622i	$0.303055\pi$
$684$	−1.94846	−0.0745012
$685$	16.9385	0.647185
$686$	14.0195	0.535266
$687$	48.9365	1.86704
$688$	−10.0378	−0.382689
$689$	−3.56634	−0.135867
$690$	18.3802	0.699721
$691$	10.6387	0.404713	0.202357	−	0.979312i	$-0.435140\pi$
$691$	10.6387	0.404713	0.202357	+	0.979312i	$0.435140\pi$
$692$	9.06546	0.344617
$693$	−0.0432400	−0.00164255
$694$	4.50544	0.171024
$695$	−13.9964	−0.530914
$696$	−13.3962	−0.507782
$697$	37.9030	1.43568
$698$	15.2681	0.577907
$699$	−34.9654	−1.32251
$700$	−3.10031	−0.117181
$701$	−19.9716	−0.754318	−0.377159	−	0.926149i	$-0.623099\pi$
$701$	−19.9716	−0.754318	−0.377159	+	0.926149i	$0.623099\pi$
$702$	−17.0110	−0.642037
$703$	−11.9396	−0.450310
$704$	−0.101160	−0.00381260
$705$	4.23706	0.159577
$706$	25.3793	0.955163
$707$	12.0258	0.452277
$708$	−6.02711	−0.226513
$709$	−11.2609	−0.422913	−0.211457	−	0.977387i	$-0.567821\pi$
$709$	−11.2609	−0.422913	−0.211457	+	0.977387i	$0.567821\pi$
$710$	22.7256	0.852877
$711$	1.91176	0.0716967
$712$	4.38574	0.164362
$713$	−50.2254	−1.88096
$714$	7.90811	0.295954
$715$	0.527449	0.0197255
$716$	−8.51324	−0.318155
$717$	−48.8862	−1.82569
$718$	23.0978	0.862001
$719$	−27.2243	−1.01529	−0.507647	−	0.861565i	$-0.669485\pi$
$719$	−27.2243	−1.01529	−0.507647	+	0.861565i	$0.669485\pi$
$720$	0.574816	0.0214221
$721$	17.0627	0.635449
$722$	−5.92515	−0.220511
$723$	26.8007	0.996729
$724$	2.97370	0.110517
$725$	−20.5951	−0.764882
$726$	20.2351	0.750996
$727$	27.0621	1.00368	0.501838	−	0.864962i	$-0.332657\pi$
$727$	27.0621	1.00368	0.501838	+	0.864962i	$0.332657\pi$
$728$	3.87724	0.143700
$729$	22.6331	0.838261
$730$	1.68960	0.0625348
$731$	39.3632	1.45590
$732$	−3.90145	−0.144202
$733$	−11.7918	−0.435539	−0.217769	−	0.976000i	$-0.569878\pi$
$733$	−11.7918	−0.435539	−0.217769	+	0.976000i	$0.569878\pi$
$734$	22.3197	0.823835
$735$	−15.7303	−0.580221
$736$	6.77759	0.249825
$737$	−0.460326	−0.0169563
$738$	3.77222	0.138857
$739$	51.2825	1.88646	0.943228	−	0.332146i	$-0.107773\pi$
$739$	51.2825	1.88646	0.943228	+	0.332146i	$0.107773\pi$
$740$	3.52230	0.129482
$741$	32.5427	1.19549
$742$	−1.10335	−0.0405051
$743$	−34.4611	−1.26426	−0.632128	−	0.774864i	$-0.717818\pi$
$743$	−34.4611	−1.26426	−0.632128	+	0.774864i	$0.717818\pi$
$744$	−13.6448	−0.500241
$745$	14.2041	0.520397
$746$	4.47324	0.163777
$747$	4.19677	0.153552
$748$	0.396695	0.0145046
$749$	9.39199	0.343176
$750$	21.2362	0.775436
$751$	37.3107	1.36149	0.680743	−	0.732522i	$-0.261657\pi$
$751$	37.3107	1.36149	0.680743	+	0.732522i	$0.261657\pi$
$752$	1.56239	0.0569747
$753$	−12.8918	−0.469805
$754$	25.7562	0.937984
$755$	15.5882	0.567312
$756$	−5.26281	−0.191407
$757$	53.4389	1.94227	0.971135	−	0.238531i	$-0.0766658\pi$
$757$	53.4389	1.94227	0.971135	+	0.238531i	$0.0766658\pi$
$758$	−8.22925	−0.298900
$759$	1.26241	0.0458225
$760$	7.35318	0.266728
$761$	−35.9537	−1.30332	−0.651659	−	0.758512i	$-0.725927\pi$
$761$	−35.9537	−1.30332	−0.651659	+	0.758512i	$0.725927\pi$
$762$	36.0569	1.30620
$763$	10.6995	0.387349
$764$	−21.0940	−0.763155
$765$	−2.25413	−0.0814982
$766$	18.1795	0.656851
$767$	11.5880	0.418418
$768$	1.84127	0.0664411
$769$	9.81528	0.353948	0.176974	−	0.984216i	$-0.443369\pi$
$769$	9.81528	0.353948	0.176974	+	0.984216i	$0.443369\pi$
$770$	0.163181	0.00588063
$771$	10.0209	0.360894
$772$	24.0983	0.867315
$773$	8.33409	0.299756	0.149878	−	0.988704i	$-0.452112\pi$
$773$	8.33409	0.299756	0.149878	+	0.988704i	$0.452112\pi$
$774$	3.91754	0.140813
$775$	−20.9772	−0.753523
$776$	4.01376	0.144086
$777$	4.82274	0.173015
$778$	24.7333	0.886732
$779$	48.2551	1.72892
$780$	−9.60043	−0.343751
$781$	1.56087	0.0558523
$782$	−26.5782	−0.950433
$783$	−34.9604	−1.24938
$784$	−5.80047	−0.207160
$785$	18.0937	0.645792
$786$	6.41335	0.228757
$787$	−36.8116	−1.31219	−0.656096	−	0.754678i	$-0.727793\pi$
$787$	−36.8116	−1.31219	−0.656096	+	0.754678i	$0.727793\pi$
$788$	4.51745	0.160927
$789$	56.6346	2.01624
$790$	−7.21469	−0.256687
$791$	−20.0858	−0.714170
$792$	0.0394802	0.00140287
$793$	7.50111	0.266372
$794$	11.0486	0.392099
$795$	2.73200	0.0968939
$796$	−5.04739	−0.178900
$797$	−13.2940	−0.470899	−0.235449	−	0.971887i	$-0.575656\pi$
$797$	−13.2940	−0.470899	−0.235449	+	0.971887i	$0.575656\pi$
$798$	10.0680	0.356402
$799$	−6.12689	−0.216754
$800$	2.83074	0.100082
$801$	−1.71165	−0.0604782
$802$	11.2061	0.395700
$803$	0.116047	0.00409521
$804$	8.37868	0.295493
$805$	−10.9329	−0.385336
$806$	26.2340	0.924054
$807$	−49.5207	−1.74321
$808$	−10.9802	−0.386280
$809$	−46.6974	−1.64179	−0.820897	−	0.571076i	$-0.806526\pi$
$809$	−46.6974	−1.64179	−0.820897	+	0.571076i	$0.806526\pi$
$810$	14.7557	0.518462
$811$	−22.5931	−0.793352	−0.396676	−	0.917959i	$-0.629836\pi$
$811$	−22.5931	−0.793352	−0.396676	+	0.917959i	$0.629836\pi$
$812$	7.96838	0.279635
$813$	−29.6878	−1.04120
$814$	0.241923	0.00847941
$815$	−19.6085	−0.686857
$816$	−7.22050	−0.252768
$817$	50.1141	1.75327
$818$	13.5770	0.474707
$819$	−1.51320	−0.0528753
$820$	−14.2358	−0.497134
$821$	−39.1263	−1.36552	−0.682758	−	0.730645i	$-0.739220\pi$
$821$	−39.1263	−1.36552	−0.682758	+	0.730645i	$0.739220\pi$
$822$	−21.1756	−0.738583
$823$	−32.2886	−1.12551	−0.562756	−	0.826623i	$-0.690259\pi$
$823$	−32.2886	−1.12551	−0.562756	+	0.826623i	$0.690259\pi$
$824$	−15.5791	−0.542724
$825$	0.527259	0.0183568
$826$	3.58507	0.124740
$827$	27.3171	0.949908	0.474954	−	0.880011i	$-0.342465\pi$
$827$	27.3171	0.949908	0.474954	+	0.880011i	$0.342465\pi$
$828$	−2.64514	−0.0919248
$829$	10.8043	0.375249	0.187625	−	0.982241i	$-0.439921\pi$
$829$	10.8043	0.375249	0.187625	+	0.982241i	$0.439921\pi$
$830$	−15.8379	−0.549743
$831$	10.4815	0.363598
$832$	−3.54011	−0.122731
$833$	22.7464	0.788116
$834$	17.4976	0.605891
$835$	28.0949	0.972265
$836$	0.505040	0.0174672
$837$	−35.6090	−1.23083
$838$	8.86331	0.306178
$839$	39.3859	1.35975	0.679876	−	0.733327i	$-0.262034\pi$
$839$	39.3859	1.35975	0.679876	+	0.733327i	$0.262034\pi$
$840$	−2.97016	−0.102480
$841$	23.9333	0.825285
$842$	−26.3737	−0.908899
$843$	17.5058	0.602930
$844$	13.0372	0.448758
$845$	−0.688727	−0.0236929
$846$	−0.609766	−0.0209642
$847$	−12.0363	−0.413573
$848$	1.00741	0.0345946
$849$	44.6719	1.53314
$850$	−11.1007	−0.380750
$851$	−16.2086	−0.555624
$852$	−28.4104	−0.973323
$853$	14.2220	0.486952	0.243476	−	0.969907i	$-0.421712\pi$
$853$	14.2220	0.486952	0.243476	+	0.969907i	$0.421712\pi$
$854$	2.32068	0.0794119
$855$	−2.86977	−0.0981442
$856$	−8.57535	−0.293099
$857$	12.1058	0.413527	0.206763	−	0.978391i	$-0.433707\pi$
$857$	12.1058	0.413527	0.206763	+	0.978391i	$0.433707\pi$
$858$	−0.659389	−0.0225112
$859$	31.4063	1.07157	0.535784	−	0.844355i	$-0.320016\pi$
$859$	31.4063	1.07157	0.535784	+	0.844355i	$0.320016\pi$
$860$	−14.7842	−0.504136
$861$	−19.4916	−0.664272
$862$	−4.54192	−0.154698
$863$	50.1386	1.70674	0.853369	−	0.521307i	$-0.174555\pi$
$863$	50.1386	1.70674	0.853369	+	0.521307i	$0.174555\pi$
$864$	4.80521	0.163476
$865$	13.3520	0.453982
$866$	20.0081	0.679902
$867$	−2.98658	−0.101429
$868$	8.11622	0.275482
$869$	−0.495528	−0.0168096
$870$	−19.7305	−0.668927
$871$	−16.1092	−0.545840
$872$	−9.76920	−0.330827
$873$	−1.56648	−0.0530172
$874$	−33.8372	−1.14456
$875$	−12.6318	−0.427032
$876$	−2.11225	−0.0713662
$877$	−51.3432	−1.73374	−0.866868	−	0.498537i	$-0.833871\pi$
$877$	−51.3432	−1.73374	−0.866868	+	0.498537i	$0.833871\pi$
$878$	−14.4500	−0.487663
$879$	39.0895	1.31845
$880$	−0.148992	−0.00502253
$881$	−22.8738	−0.770637	−0.385319	−	0.922784i	$-0.625908\pi$
$881$	−22.8738	−0.770637	−0.385319	+	0.922784i	$0.625908\pi$
$882$	2.26379	0.0762257
$883$	25.0989	0.844646	0.422323	−	0.906445i	$-0.361215\pi$
$883$	25.0989	0.844646	0.422323	+	0.906445i	$0.361215\pi$
$884$	13.8825	0.466918
$885$	−8.87698	−0.298396
$886$	19.4709	0.654139
$887$	−39.0787	−1.31213	−0.656067	−	0.754702i	$-0.727781\pi$
$887$	−39.0787	−1.31213	−0.656067	+	0.754702i	$0.727781\pi$
$888$	−4.40340	−0.147768
$889$	−21.4475	−0.719326
$890$	6.45950	0.216523
$891$	1.01347	0.0339525
$892$	10.5253	0.352413
$893$	−7.80027	−0.261026
$894$	−17.7572	−0.593890
$895$	−12.5387	−0.419121
$896$	−1.09523	−0.0365891
$897$	44.1784	1.47507
$898$	−19.9064	−0.664284
$899$	53.9154	1.79818
$900$	−1.10477	−0.0368257
$901$	−3.95053	−0.131611
$902$	−0.977758	−0.0325558
$903$	−20.2425	−0.673628
$904$	18.3394	0.609958
$905$	4.37979	0.145589
$906$	−19.4875	−0.647430
$907$	−29.0338	−0.964050	−0.482025	−	0.876157i	$-0.660099\pi$
$907$	−29.0338	−0.964050	−0.482025	+	0.876157i	$0.660099\pi$
$908$	1.44809	0.0480566
$909$	4.28530	0.142134
$910$	5.71056	0.189303
$911$	−26.1041	−0.864867	−0.432434	−	0.901666i	$-0.642345\pi$
$911$	−26.1041	−0.864867	−0.432434	+	0.901666i	$0.642345\pi$
$912$	−9.19256	−0.304396
$913$	−1.08780	−0.0360010
$914$	9.72662	0.321728
$915$	−5.74623	−0.189964
$916$	26.5776	0.878148
$917$	−3.81481	−0.125976
$918$	−18.8435	−0.621928
$919$	53.4339	1.76262	0.881311	−	0.472537i	$-0.156662\pi$
$919$	53.4339	1.76262	0.881311	+	0.472537i	$0.156662\pi$
$920$	9.98232	0.329107
$921$	34.1124	1.12404
$922$	37.9936	1.25125
$923$	54.6231	1.79794
$924$	−0.204000	−0.00671111
$925$	−6.76970	−0.222586
$926$	1.33520	0.0438773
$927$	6.08017	0.199699
$928$	−7.27553	−0.238831
$929$	−11.4611	−0.376025	−0.188013	−	0.982167i	$-0.560205\pi$
$929$	−11.4611	−0.376025	−0.188013	+	0.982167i	$0.560205\pi$
$930$	−20.0966	−0.658992
$931$	28.9589	0.949089
$932$	−18.9898	−0.622033
$933$	21.3962	0.700481
$934$	−28.0912	−0.919173
$935$	0.584269	0.0191077
$936$	1.38162	0.0451598
$937$	−17.3243	−0.565959	−0.282980	−	0.959126i	$-0.591323\pi$
$937$	−17.3243	−0.565959	−0.282980	+	0.959126i	$0.591323\pi$
$938$	−4.98383	−0.162728
$939$	−2.22618	−0.0726487
$940$	2.30116	0.0750556
$941$	32.8054	1.06943	0.534713	−	0.845034i	$-0.320420\pi$
$941$	32.8054	1.06943	0.534713	+	0.845034i	$0.320420\pi$
$942$	−22.6198	−0.736993
$943$	65.5088	2.13326
$944$	−3.27334	−0.106538
$945$	−7.75129	−0.252150
$946$	−1.01542	−0.0330143
$947$	−31.4138	−1.02081	−0.510405	−	0.859934i	$-0.670504\pi$
$947$	−31.4138	−1.02081	−0.510405	+	0.859934i	$0.670504\pi$
$948$	9.01942	0.292937
$949$	4.06110	0.131829
$950$	−14.1325	−0.458518
$951$	−41.5754	−1.34817
$952$	4.29492	0.139199
$953$	0.998762	0.0323531	0.0161765	−	0.999869i	$-0.494851\pi$
$953$	0.998762	0.0323531	0.0161765	+	0.999869i	$0.494851\pi$
$954$	−0.393169	−0.0127293
$955$	−31.0682	−1.00534
$956$	−26.5503	−0.858697
$957$	−1.35516	−0.0438060
$958$	−30.1834	−0.975180
$959$	12.5957	0.406737
$960$	2.71190	0.0875262
$961$	23.9156	0.771473
$962$	8.46618	0.272960
$963$	3.34676	0.107848
$964$	14.5556	0.468803
$965$	35.4929	1.14256
$966$	13.6678	0.439754
$967$	11.9597	0.384598	0.192299	−	0.981336i	$-0.438406\pi$
$967$	11.9597	0.384598	0.192299	+	0.981336i	$0.438406\pi$
$968$	10.9898	0.353224
$969$	36.0484	1.15804
$970$	5.91163	0.189811
$971$	0.641890	0.0205992	0.0102996	−	0.999947i	$-0.496721\pi$
$971$	0.641890	0.0205992	0.0102996	+	0.999947i	$0.496721\pi$
$972$	−4.03118	−0.129300
$973$	−10.4080	−0.333664
$974$	24.2861	0.778178
$975$	18.4516	0.590924
$976$	−2.11889	−0.0678241
$977$	36.5063	1.16794	0.583969	−	0.811776i	$-0.301499\pi$
$977$	36.5063	1.16794	0.583969	+	0.811776i	$0.301499\pi$
$978$	24.5136	0.783857
$979$	0.443659	0.0141794
$980$	−8.54318	−0.272902
$981$	3.81269	0.121730
$982$	16.3995	0.523328
$983$	8.39071	0.267622	0.133811	−	0.991007i	$-0.457278\pi$
$983$	8.39071	0.267622	0.133811	+	0.991007i	$0.457278\pi$
$984$	17.7968	0.567341
$985$	6.65349	0.211998
$986$	28.5308	0.908606
$987$	3.15075	0.100289
$988$	17.6740	0.562286
$989$	68.0324	2.16331
$990$	0.0581482	0.00184807
$991$	39.5944	1.25776	0.628879	−	0.777503i	$-0.283514\pi$
$991$	39.5944	1.25776	0.628879	+	0.777503i	$0.283514\pi$
$992$	−7.41051	−0.235284
$993$	−46.4624	−1.47444
$994$	16.8992	0.536009
$995$	−7.43401	−0.235674
$996$	19.7998	0.627379
$997$	59.9258	1.89787	0.948935	−	0.315471i	$-0.102162\pi$
$997$	59.9258	1.89787	0.948935	+	0.315471i	$0.102162\pi$
$998$	8.52863	0.269969
$999$	−11.4917	−0.363580

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4006.2.a.g.1.33	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4006.2.a.g.1.33	✓	40	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:	\( N \)	\(=\)	\( 4006 = 2 \cdot 2003 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4006.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.84127 q^{3} +1.00000 q^{4} +1.47284 q^{5} -1.84127 q^{6} +1.09523 q^{7} -1.00000 q^{8} +0.390277 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.84127 q^{3} +1.00000 q^{4} +1.47284 q^{5} -1.84127 q^{6} +1.09523 q^{7} -1.00000 q^{8} +0.390277 q^{9} -1.47284 q^{10} -0.101160 q^{11} +1.84127 q^{12} -3.54011 q^{13} -1.09523 q^{14} +2.71190 q^{15} +1.00000 q^{16} -3.92148 q^{17} -0.390277 q^{18} -4.99251 q^{19} +1.47284 q^{20} +2.01662 q^{21} +0.101160 q^{22} -6.77759 q^{23} -1.84127 q^{24} -2.83074 q^{25} +3.54011 q^{26} -4.80521 q^{27} +1.09523 q^{28} +7.27553 q^{29} -2.71190 q^{30} +7.41051 q^{31} -1.00000 q^{32} -0.186262 q^{33} +3.92148 q^{34} +1.61310 q^{35} +0.390277 q^{36} +2.39150 q^{37} +4.99251 q^{38} -6.51830 q^{39} -1.47284 q^{40} -9.66550 q^{41} -2.01662 q^{42} -10.0378 q^{43} -0.101160 q^{44} +0.574816 q^{45} +6.77759 q^{46} +1.56239 q^{47} +1.84127 q^{48} -5.80047 q^{49} +2.83074 q^{50} -7.22050 q^{51} -3.54011 q^{52} +1.00741 q^{53} +4.80521 q^{54} -0.148992 q^{55} -1.09523 q^{56} -9.19256 q^{57} -7.27553 q^{58} -3.27334 q^{59} +2.71190 q^{60} -2.11889 q^{61} -7.41051 q^{62} +0.427443 q^{63} +1.00000 q^{64} -5.21403 q^{65} +0.186262 q^{66} +4.55049 q^{67} -3.92148 q^{68} -12.4794 q^{69} -1.61310 q^{70} -15.4298 q^{71} -0.390277 q^{72} -1.14717 q^{73} -2.39150 q^{74} -5.21215 q^{75} -4.99251 q^{76} -0.110793 q^{77} +6.51830 q^{78} +4.89848 q^{79} +1.47284 q^{80} -10.0185 q^{81} +9.66550 q^{82} +10.7533 q^{83} +2.01662 q^{84} -5.77572 q^{85} +10.0378 q^{86} +13.3962 q^{87} +0.101160 q^{88} -4.38574 q^{89} -0.574816 q^{90} -3.87724 q^{91} -6.77759 q^{92} +13.6448 q^{93} -1.56239 q^{94} -7.35318 q^{95} -1.84127 q^{96} -4.01376 q^{97} +5.80047 q^{98} -0.0394802 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 40 q^{2} - q^{3} + 40 q^{4} - 23 q^{5} + q^{6} + 12 q^{7} - 40 q^{8} + 29 q^{9}+O(q^{10}) \) 40 * q - 40 * q^2 - q^3 + 40 * q^4 - 23 * q^5 + q^6 + 12 * q^7 - 40 * q^8 + 29 * q^9	\( 40 q - 40 q^{2} - q^{3} + 40 q^{4} - 23 q^{5} + q^{6} + 12 q^{7} - 40 q^{8} + 29 q^{9} + 23 q^{10} - 28 q^{11} - q^{12} - 12 q^{13} - 12 q^{14} - 14 q^{15} + 40 q^{16} - 10 q^{17} - 29 q^{18} + 3 q^{19} - 23 q^{20} - 40 q^{21} + 28 q^{22} - 10 q^{23} + q^{24} + 43 q^{25} + 12 q^{26} - 7 q^{27} + 12 q^{28} - 37 q^{29} + 14 q^{30} - 16 q^{31} - 40 q^{32} - 11 q^{33} + 10 q^{34} - 24 q^{35} + 29 q^{36} - 17 q^{37} - 3 q^{38} - 10 q^{39} + 23 q^{40} - 58 q^{41} + 40 q^{42} + 37 q^{43} - 28 q^{44} - 66 q^{45} + 10 q^{46} - 34 q^{47} - q^{48} + 28 q^{49} - 43 q^{50} - 7 q^{51} - 12 q^{52} - 43 q^{53} + 7 q^{54} + 28 q^{55} - 12 q^{56} - 25 q^{57} + 37 q^{58} - 92 q^{59} - 14 q^{60} - 37 q^{61} + 16 q^{62} + 14 q^{63} + 40 q^{64} - 54 q^{65} + 11 q^{66} - 10 q^{67} - 10 q^{68} - 49 q^{69} + 24 q^{70} - 87 q^{71} - 29 q^{72} + 12 q^{73} + 17 q^{74} - 31 q^{75} + 3 q^{76} - 53 q^{77} + 10 q^{78} + 14 q^{79} - 23 q^{80} - 4 q^{81} + 58 q^{82} - 42 q^{83} - 40 q^{84} - 24 q^{85} - 37 q^{86} + 24 q^{87} + 28 q^{88} - 118 q^{89} + 66 q^{90} - 35 q^{91} - 10 q^{92} - 22 q^{93} + 34 q^{94} - 18 q^{95} + q^{96} + 2 q^{97} - 28 q^{98} - 48 q^{99}+O(q^{100}) \) 40 * q - 40 * q^2 - q^3 + 40 * q^4 - 23 * q^5 + q^6 + 12 * q^7 - 40 * q^8 + 29 * q^9 + 23 * q^10 - 28 * q^11 - q^12 - 12 * q^13 - 12 * q^14 - 14 * q^15 + 40 * q^16 - 10 * q^17 - 29 * q^18 + 3 * q^19 - 23 * q^20 - 40 * q^21 + 28 * q^22 - 10 * q^23 + q^24 + 43 * q^25 + 12 * q^26 - 7 * q^27 + 12 * q^28 - 37 * q^29 + 14 * q^30 - 16 * q^31 - 40 * q^32 - 11 * q^33 + 10 * q^34 - 24 * q^35 + 29 * q^36 - 17 * q^37 - 3 * q^38 - 10 * q^39 + 23 * q^40 - 58 * q^41 + 40 * q^42 + 37 * q^43 - 28 * q^44 - 66 * q^45 + 10 * q^46 - 34 * q^47 - q^48 + 28 * q^49 - 43 * q^50 - 7 * q^51 - 12 * q^52 - 43 * q^53 + 7 * q^54 + 28 * q^55 - 12 * q^56 - 25 * q^57 + 37 * q^58 - 92 * q^59 - 14 * q^60 - 37 * q^61 + 16 * q^62 + 14 * q^63 + 40 * q^64 - 54 * q^65 + 11 * q^66 - 10 * q^67 - 10 * q^68 - 49 * q^69 + 24 * q^70 - 87 * q^71 - 29 * q^72 + 12 * q^73 + 17 * q^74 - 31 * q^75 + 3 * q^76 - 53 * q^77 + 10 * q^78 + 14 * q^79 - 23 * q^80 - 4 * q^81 + 58 * q^82 - 42 * q^83 - 40 * q^84 - 24 * q^85 - 37 * q^86 + 24 * q^87 + 28 * q^88 - 118 * q^89 + 66 * q^90 - 35 * q^91 - 10 * q^92 - 22 * q^93 + 34 * q^94 - 18 * q^95 + q^96 + 2 * q^97 - 28 * q^98 - 48 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more