LMFDB - Embedded newform 8023.2.a.b.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8023,2,Mod(1,8023)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8023, 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("8023.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8023 = 71 \cdot 113 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8023.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:	$64.0639775417$
Analytic rank:	$1$
Dimension:	$155$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Character	$\chi$	$=$	8023.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.45088 q^{2} +1.23773 q^{3} +4.00682 q^{4} +3.46965 q^{5} -3.03354 q^{6} -0.238635 q^{7} -4.91847 q^{8} -1.46802 q^{9} +O(q^{10})$	\(q-2.45088 q^{2} +1.23773 q^{3} +4.00682 q^{4} +3.46965 q^{5} -3.03354 q^{6} -0.238635 q^{7} -4.91847 q^{8} -1.46802 q^{9} -8.50371 q^{10} -4.10843 q^{11} +4.95937 q^{12} -3.51616 q^{13} +0.584866 q^{14} +4.29450 q^{15} +4.04096 q^{16} +4.73538 q^{17} +3.59794 q^{18} +0.127784 q^{19} +13.9023 q^{20} -0.295366 q^{21} +10.0693 q^{22} +7.33266 q^{23} -6.08775 q^{24} +7.03849 q^{25} +8.61770 q^{26} -5.53021 q^{27} -0.956166 q^{28} +4.24426 q^{29} -10.5253 q^{30} -6.66738 q^{31} -0.0669577 q^{32} -5.08514 q^{33} -11.6059 q^{34} -0.827980 q^{35} -5.88208 q^{36} -11.4908 q^{37} -0.313185 q^{38} -4.35207 q^{39} -17.0654 q^{40} -8.72732 q^{41} +0.723907 q^{42} +7.71042 q^{43} -16.4617 q^{44} -5.09351 q^{45} -17.9715 q^{46} +3.23407 q^{47} +5.00162 q^{48} -6.94305 q^{49} -17.2505 q^{50} +5.86114 q^{51} -14.0886 q^{52} +3.11748 q^{53} +13.5539 q^{54} -14.2548 q^{55} +1.17372 q^{56} +0.158163 q^{57} -10.4022 q^{58} -8.43078 q^{59} +17.2073 q^{60} -1.53365 q^{61} +16.3410 q^{62} +0.350320 q^{63} -7.91781 q^{64} -12.1999 q^{65} +12.4631 q^{66} +13.9130 q^{67} +18.9738 q^{68} +9.07587 q^{69} +2.02928 q^{70} +1.00000 q^{71} +7.22041 q^{72} -4.05216 q^{73} +28.1625 q^{74} +8.71177 q^{75} +0.512009 q^{76} +0.980415 q^{77} +10.6664 q^{78} -7.16885 q^{79} +14.0207 q^{80} -2.44087 q^{81} +21.3896 q^{82} +8.98099 q^{83} -1.18348 q^{84} +16.4301 q^{85} -18.8973 q^{86} +5.25325 q^{87} +20.2072 q^{88} +4.87409 q^{89} +12.4836 q^{90} +0.839079 q^{91} +29.3806 q^{92} -8.25243 q^{93} -7.92631 q^{94} +0.443368 q^{95} -0.0828758 q^{96} +5.98370 q^{97} +17.0166 q^{98} +6.03125 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 155 q - 21 q^{2} - 16 q^{3} + 151 q^{4} - 26 q^{5} - 10 q^{6} - 40 q^{7} - 57 q^{8} + 135 q^{9}+O(q^{10}) $ 155 * q - 21 * q^2 - 16 * q^3 + 151 * q^4 - 26 * q^5 - 10 * q^6 - 40 * q^7 - 57 * q^8 + 135 * q^9	\( 155 q - 21 q^{2} - 16 q^{3} + 151 q^{4} - 26 q^{5} - 10 q^{6} - 40 q^{7} - 57 q^{8} + 135 q^{9} - 2 q^{10} - 24 q^{11} - 32 q^{12} - 62 q^{13} - 18 q^{14} - 12 q^{15} + 155 q^{16} - 129 q^{17} - 42 q^{18} - 18 q^{19} - 59 q^{20} - 45 q^{21} - 17 q^{22} - 38 q^{23} - 27 q^{24} + 129 q^{25} - 44 q^{26} - 43 q^{27} - 100 q^{28} - 52 q^{29} - 39 q^{30} - 56 q^{31} - 145 q^{32} - 126 q^{33} - q^{34} - 49 q^{35} + 131 q^{36} - 30 q^{37} - 91 q^{38} - 29 q^{39} - 5 q^{40} - 163 q^{41} - 80 q^{42} - 15 q^{43} - 118 q^{44} - 66 q^{45} + 2 q^{46} - 111 q^{47} - 89 q^{48} + 101 q^{49} - 121 q^{50} + 5 q^{51} - 111 q^{52} - 93 q^{53} - 68 q^{54} - 60 q^{55} - 27 q^{56} - 106 q^{57} + 16 q^{58} - 79 q^{59} - 103 q^{60} - 74 q^{61} - 102 q^{62} - 118 q^{63} + 175 q^{64} - 109 q^{65} + 65 q^{66} - 18 q^{67} - 346 q^{68} - 39 q^{69} + 32 q^{70} + 155 q^{71} - 203 q^{72} - 108 q^{73} - 87 q^{74} - 22 q^{75} - 16 q^{76} - 121 q^{77} - 75 q^{78} - 6 q^{79} - 136 q^{80} + 107 q^{81} - 30 q^{82} - 116 q^{83} - 5 q^{84} - 53 q^{85} + 8 q^{86} - 100 q^{87} - 43 q^{88} - 189 q^{89} - 76 q^{90} + 14 q^{91} - 99 q^{92} - 72 q^{93} + 17 q^{94} - 18 q^{95} - 50 q^{96} - 184 q^{97} - 249 q^{98} - 114 q^{99}+O(q^{100}) \) 155 * q - 21 * q^2 - 16 * q^3 + 151 * q^4 - 26 * q^5 - 10 * q^6 - 40 * q^7 - 57 * q^8 + 135 * q^9 - 2 * q^10 - 24 * q^11 - 32 * q^12 - 62 * q^13 - 18 * q^14 - 12 * q^15 + 155 * q^16 - 129 * q^17 - 42 * q^18 - 18 * q^19 - 59 * q^20 - 45 * q^21 - 17 * q^22 - 38 * q^23 - 27 * q^24 + 129 * q^25 - 44 * q^26 - 43 * q^27 - 100 * q^28 - 52 * q^29 - 39 * q^30 - 56 * q^31 - 145 * q^32 - 126 * q^33 - q^34 - 49 * q^35 + 131 * q^36 - 30 * q^37 - 91 * q^38 - 29 * q^39 - 5 * q^40 - 163 * q^41 - 80 * q^42 - 15 * q^43 - 118 * q^44 - 66 * q^45 + 2 * q^46 - 111 * q^47 - 89 * q^48 + 101 * q^49 - 121 * q^50 + 5 * q^51 - 111 * q^52 - 93 * q^53 - 68 * q^54 - 60 * q^55 - 27 * q^56 - 106 * q^57 + 16 * q^58 - 79 * q^59 - 103 * q^60 - 74 * q^61 - 102 * q^62 - 118 * q^63 + 175 * q^64 - 109 * q^65 + 65 * q^66 - 18 * q^67 - 346 * q^68 - 39 * q^69 + 32 * q^70 + 155 * q^71 - 203 * q^72 - 108 * q^73 - 87 * q^74 - 22 * q^75 - 16 * q^76 - 121 * q^77 - 75 * q^78 - 6 * q^79 - 136 * q^80 + 107 * q^81 - 30 * q^82 - 116 * q^83 - 5 * q^84 - 53 * q^85 + 8 * q^86 - 100 * q^87 - 43 * q^88 - 189 * q^89 - 76 * q^90 + 14 * q^91 - 99 * q^92 - 72 * q^93 + 17 * q^94 - 18 * q^95 - 50 * q^96 - 184 * q^97 - 249 * q^98 - 114 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.45088	−1.73303	−0.866517	−	0.499147i	$-0.833647\pi$
$2$	−2.45088	−1.73303	−0.866517	+	0.499147i	$0.833647\pi$
$3$	1.23773	0.714605	0.357303	−	0.933989i	$-0.383696\pi$
$3$	1.23773	0.714605	0.357303	+	0.933989i	$0.383696\pi$
$4$	4.00682	2.00341
$5$	3.46965	1.55168	0.775838	−	0.630932i	$-0.217327\pi$
$5$	3.46965	1.55168	0.775838	+	0.630932i	$0.217327\pi$
$6$	−3.03354	−1.23844
$7$	−0.238635	−0.0901955	−0.0450977	−	0.998983i	$-0.514360\pi$
$7$	−0.238635	−0.0901955	−0.0450977	+	0.998983i	$0.514360\pi$
$8$	−4.91847	−1.73894
$9$	−1.46802	−0.489339
$10$	−8.50371	−2.68911
$11$	−4.10843	−1.23874	−0.619369	−	0.785100i	$-0.712612\pi$
$11$	−4.10843	−1.23874	−0.619369	+	0.785100i	$0.712612\pi$
$12$	4.95937	1.43165
$13$	−3.51616	−0.975208	−0.487604	−	0.873065i	$-0.662129\pi$
$13$	−3.51616	−0.975208	−0.487604	+	0.873065i	$0.662129\pi$
$14$	0.584866	0.156312
$15$	4.29450	1.10884
$16$	4.04096	1.01024
$17$	4.73538	1.14850	0.574249	−	0.818681i	$-0.305294\pi$
$17$	4.73538	1.14850	0.574249	+	0.818681i	$0.305294\pi$
$18$	3.59794	0.848042
$19$	0.127784	0.0293158	0.0146579	−	0.999893i	$-0.495334\pi$
$19$	0.127784	0.0293158	0.0146579	+	0.999893i	$0.495334\pi$
$20$	13.9023	3.10864
$21$	−0.295366	−0.0644542
$22$	10.0693	2.14678
$23$	7.33266	1.52897	0.764483	−	0.644644i	$-0.222994\pi$
$23$	7.33266	1.52897	0.764483	+	0.644644i	$0.222994\pi$
$24$	−6.08775	−1.24266
$25$	7.03849	1.40770
$26$	8.61770	1.69007
$27$	−5.53021	−1.06429
$28$	−0.956166	−0.180698
$29$	4.24426	0.788139	0.394069	−	0.919081i	$-0.371067\pi$
$29$	4.24426	0.788139	0.394069	+	0.919081i	$0.371067\pi$
$30$	−10.5253	−1.92165
$31$	−6.66738	−1.19750	−0.598748	−	0.800937i	$-0.704335\pi$
$31$	−6.66738	−1.19750	−0.598748	+	0.800937i	$0.704335\pi$
$32$	−0.0669577	−0.0118366
$33$	−5.08514	−0.885209
$34$	−11.6059	−1.99039
$35$	−0.827980	−0.139954
$36$	−5.88208	−0.980347
$37$	−11.4908	−1.88907	−0.944535	−	0.328411i	$-0.893487\pi$
$37$	−11.4908	−1.88907	−0.944535	+	0.328411i	$0.893487\pi$
$38$	−0.313185	−0.0508052
$39$	−4.35207	−0.696889
$40$	−17.0654	−2.69828
$41$	−8.72732	−1.36298	−0.681489	−	0.731828i	$-0.738667\pi$
$41$	−8.72732	−1.36298	−0.681489	+	0.731828i	$0.738667\pi$
$42$	0.723907	0.111701
$43$	7.71042	1.17583	0.587914	−	0.808923i	$-0.299949\pi$
$43$	7.71042	1.17583	0.587914	+	0.808923i	$0.299949\pi$
$44$	−16.4617	−2.48170
$45$	−5.09351	−0.759296
$46$	−17.9715	−2.64975
$47$	3.23407	0.471737	0.235868	−	0.971785i	$-0.424207\pi$
$47$	3.23407	0.471737	0.235868	+	0.971785i	$0.424207\pi$
$48$	5.00162	0.721922
$49$	−6.94305	−0.991865
$50$	−17.2505	−2.43959
$51$	5.86114	0.820723
$52$	−14.0886	−1.95374
$53$	3.11748	0.428219	0.214110	−	0.976810i	$-0.431315\pi$
$53$	3.11748	0.428219	0.214110	+	0.976810i	$0.431315\pi$
$54$	13.5539	1.84445
$55$	−14.2548	−1.92212
$56$	1.17372	0.156845
$57$	0.158163	0.0209492
$58$	−10.4022	−1.36587
$59$	−8.43078	−1.09759	−0.548797	−	0.835956i	$-0.684914\pi$
$59$	−8.43078	−1.09759	−0.548797	+	0.835956i	$0.684914\pi$
$60$	17.2073	2.22145
$61$	−1.53365	−0.196363	−0.0981817	−	0.995169i	$-0.531303\pi$
$61$	−1.53365	−0.196363	−0.0981817	+	0.995169i	$0.531303\pi$
$62$	16.3410	2.07530
$63$	0.350320	0.0441362
$64$	−7.91781	−0.989726
$65$	−12.1999	−1.51321
$66$	12.4631	1.53410
$67$	13.9130	1.69975	0.849874	−	0.526986i	$-0.176678\pi$
$67$	13.9130	1.69975	0.849874	+	0.526986i	$0.176678\pi$
$68$	18.9738	2.30091
$69$	9.07587	1.09261
$70$	2.02928	0.242545
$71$	1.00000	0.118678
$71$	1.00000	0.118678
$72$	7.22041	0.850933
$73$	−4.05216	−0.474270	−0.237135	−	0.971477i	$-0.576208\pi$
$73$	−4.05216	−0.474270	−0.237135	+	0.971477i	$0.576208\pi$
$74$	28.1625	3.27382
$75$	8.71177	1.00595
$76$	0.512009	0.0587315
$77$	0.980415	0.111729
$78$	10.6664	1.20773
$79$	−7.16885	−0.806558	−0.403279	−	0.915077i	$-0.632130\pi$
$79$	−7.16885	−0.806558	−0.403279	+	0.915077i	$0.632130\pi$
$80$	14.0207	1.56756
$81$	−2.44087	−0.271208
$82$	21.3896	2.36209
$83$	8.98099	0.985792	0.492896	−	0.870088i	$-0.335938\pi$
$83$	8.98099	0.985792	0.492896	+	0.870088i	$0.335938\pi$
$84$	−1.18348	−0.129128
$85$	16.4301	1.78210
$86$	−18.8973	−2.03775
$87$	5.25325	0.563208
$88$	20.2072	2.15410
$89$	4.87409	0.516652	0.258326	−	0.966058i	$-0.416829\pi$
$89$	4.87409	0.516652	0.258326	+	0.966058i	$0.416829\pi$
$90$	12.4836	1.31589
$91$	0.839079	0.0879594
$92$	29.3806	3.06314
$93$	−8.25243	−0.855737
$94$	−7.92631	−0.817536
$95$	0.443368	0.0454886
$96$	−0.0828758	−0.00845847
$97$	5.98370	0.607553	0.303776	−	0.952743i	$-0.401752\pi$
$97$	5.98370	0.607553	0.303776	+	0.952743i	$0.401752\pi$
$98$	17.0166	1.71894
$99$	6.03125	0.606163
$100$	28.2020	2.82020
$101$	−16.5129	−1.64310	−0.821550	−	0.570137i	$-0.806890\pi$
$101$	−16.5129	−1.64310	−0.821550	+	0.570137i	$0.806890\pi$
$102$	−14.3649	−1.42234
$103$	−2.41903	−0.238354	−0.119177	−	0.992873i	$-0.538026\pi$
$103$	−2.41903	−0.238354	−0.119177	+	0.992873i	$0.538026\pi$
$104$	17.2942	1.69583
$105$	−1.02482	−0.100012
$106$	−7.64058	−0.742119
$107$	−7.00584	−0.677280	−0.338640	−	0.940916i	$-0.609967\pi$
$107$	−7.00584	−0.677280	−0.338640	+	0.940916i	$0.609967\pi$
$108$	−22.1586	−2.13221
$109$	−2.35060	−0.225146	−0.112573	−	0.993643i	$-0.535909\pi$
$109$	−2.35060	−0.225146	−0.112573	+	0.993643i	$0.535909\pi$
$110$	34.9369	3.33110
$111$	−14.2225	−1.34994
$112$	−0.964313	−0.0911190
$113$	1.00000	0.0940721
$113$	1.00000	0.0940721
$114$	−0.387639	−0.0363057
$115$	25.4418	2.37246
$116$	17.0060	1.57896
$117$	5.16179	0.477208
$118$	20.6628	1.90217
$119$	−1.13003	−0.103589
$120$	−21.1224	−1.92820
$121$	5.87920	0.534473
$122$	3.75879	0.340305
$123$	−10.8021	−0.973991
$124$	−26.7150	−2.39908
$125$	7.07286	0.632615
$126$	−0.858593	−0.0764896
$127$	−19.5192	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$127$	−19.5192	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$128$	19.5395	1.72707
$129$	9.54344	0.840253
$130$	29.9004	2.62244
$131$	−12.5203	−1.09390	−0.546950	−	0.837165i	$-0.684211\pi$
$131$	−12.5203	−1.09390	−0.546950	+	0.837165i	$0.684211\pi$
$132$	−20.3752	−1.77344
$133$	−0.0304938	−0.00264415
$134$	−34.0992	−2.94572
$135$	−19.1879	−1.65143
$136$	−23.2908	−1.99717
$137$	−13.0239	−1.11270	−0.556351	−	0.830947i	$-0.687799\pi$
$137$	−13.0239	−1.11270	−0.556351	+	0.830947i	$0.687799\pi$
$138$	−22.2439	−1.89353
$139$	6.48032	0.549653	0.274827	−	0.961494i	$-0.411380\pi$
$139$	6.48032	0.549653	0.274827	+	0.961494i	$0.411380\pi$
$140$	−3.31757	−0.280385
$141$	4.00291	0.337106
$142$	−2.45088	−0.205673
$143$	14.4459	1.20803
$144$	−5.93220	−0.494350
$145$	14.7261	1.22294
$146$	9.93137	0.821926
$147$	−8.59364	−0.708792
$148$	−46.0414	−3.78458
$149$	17.0095	1.39347	0.696735	−	0.717329i	$-0.254636\pi$
$149$	17.0095	1.39347	0.696735	+	0.717329i	$0.254636\pi$
$150$	−21.3515	−1.74334
$151$	−17.8720	−1.45441	−0.727203	−	0.686423i	$-0.759180\pi$
$151$	−17.8720	−1.45441	−0.727203	+	0.686423i	$0.759180\pi$
$152$	−0.628504	−0.0509784
$153$	−6.95162	−0.562005
$154$	−2.40288	−0.193630
$155$	−23.1335	−1.85813
$156$	−17.4380	−1.39615
$157$	15.2241	1.21501	0.607507	−	0.794314i	$-0.292170\pi$
$157$	15.2241	1.21501	0.607507	+	0.794314i	$0.292170\pi$
$158$	17.5700	1.39779
$159$	3.85861	0.306008
$160$	−0.232320	−0.0183665
$161$	−1.74983	−0.137906
$162$	5.98228	0.470012
$163$	−14.9513	−1.17108	−0.585538	−	0.810645i	$-0.699117\pi$
$163$	−14.9513	−1.17108	−0.585538	+	0.810645i	$0.699117\pi$
$164$	−34.9688	−2.73060
$165$	−17.6437	−1.37356
$166$	−22.0113	−1.70841
$167$	−9.32021	−0.721220	−0.360610	−	0.932717i	$-0.617431\pi$
$167$	−9.32021	−0.721220	−0.360610	+	0.932717i	$0.617431\pi$
$168$	1.45275	0.112082
$169$	−0.636593	−0.0489687
$170$	−40.2683	−3.08844
$171$	−0.187590	−0.0143454
$172$	30.8943	2.35567
$173$	15.9813	1.21503	0.607517	−	0.794307i	$-0.292166\pi$
$173$	15.9813	1.21503	0.607517	+	0.794307i	$0.292166\pi$
$174$	−12.8751	−0.976059
$175$	−1.67963	−0.126968
$176$	−16.6020	−1.25142
$177$	−10.4351	−0.784346
$178$	−11.9458	−0.895376
$179$	−23.1123	−1.72750	−0.863749	−	0.503923i	$-0.831890\pi$
$179$	−23.1123	−1.72750	−0.863749	+	0.503923i	$0.831890\pi$
$180$	−20.4088	−1.52118
$181$	19.1817	1.42576	0.712880	−	0.701286i	$-0.247390\pi$
$181$	19.1817	1.42576	0.712880	+	0.701286i	$0.247390\pi$
$182$	−2.05648	−0.152437
$183$	−1.89825	−0.140322
$184$	−36.0655	−2.65878
$185$	−39.8690	−2.93122
$186$	20.2257	1.48302
$187$	−19.4550	−1.42269
$188$	12.9583	0.945082
$189$	1.31970	0.0959941
$190$	−1.08664	−0.0788333
$191$	3.70610	0.268164	0.134082	−	0.990970i	$-0.457191\pi$
$191$	3.70610	0.268164	0.134082	+	0.990970i	$0.457191\pi$
$192$	−9.80013	−0.707263
$193$	6.72590	0.484141	0.242070	−	0.970259i	$-0.422173\pi$
$193$	6.72590	0.484141	0.242070	+	0.970259i	$0.422173\pi$
$194$	−14.6653	−1.05291
$195$	−15.1002	−1.08135
$196$	−27.8196	−1.98711
$197$	−22.4113	−1.59674	−0.798371	−	0.602166i	$-0.794305\pi$
$197$	−22.4113	−1.59674	−0.798371	+	0.602166i	$0.794305\pi$
$198$	−14.7819	−1.05050
$199$	8.72511	0.618507	0.309253	−	0.950980i	$-0.399921\pi$
$199$	8.72511	0.618507	0.309253	+	0.950980i	$0.399921\pi$
$200$	−34.6186	−2.44791
$201$	17.2206	1.21465
$202$	40.4713	2.84755
$203$	−1.01283	−0.0710865
$204$	23.4845	1.64424
$205$	−30.2808	−2.11490
$206$	5.92875	0.413076
$207$	−10.7645	−0.748183
$208$	−14.2087	−0.985194
$209$	−0.524994	−0.0363146
$210$	2.51171	0.173324
$211$	−24.5744	−1.69177	−0.845886	−	0.533364i	$-0.820928\pi$
$211$	−24.5744	−1.69177	−0.845886	+	0.533364i	$0.820928\pi$
$212$	12.4912	0.857899
$213$	1.23773	0.0848080
$214$	17.1705	1.17375
$215$	26.7525	1.82450
$216$	27.2002	1.85074
$217$	1.59107	0.108009
$218$	5.76103	0.390186
$219$	−5.01549	−0.338916
$220$	−57.1165	−3.85079
$221$	−16.6504	−1.12003
$222$	34.8576	2.33949
$223$	22.5894	1.51270	0.756348	−	0.654170i	$-0.226982\pi$
$223$	22.5894	1.51270	0.756348	+	0.654170i	$0.226982\pi$
$224$	0.0159784	0.00106760
$225$	−10.3326	−0.688842
$226$	−2.45088	−0.163030
$227$	−28.1571	−1.86885	−0.934426	−	0.356158i	$-0.884087\pi$
$227$	−28.1571	−1.86885	−0.934426	+	0.356158i	$0.884087\pi$
$228$	0.633730	0.0419698
$229$	1.49407	0.0987308	0.0493654	−	0.998781i	$-0.484280\pi$
$229$	1.49407	0.0987308	0.0493654	+	0.998781i	$0.484280\pi$
$230$	−62.3548	−4.11155
$231$	1.21349	0.0798418
$232$	−20.8753	−1.37053
$233$	1.23704	0.0810409	0.0405204	−	0.999179i	$-0.487098\pi$
$233$	1.23704	0.0810409	0.0405204	+	0.999179i	$0.487098\pi$
$234$	−12.6509	−0.827018
$235$	11.2211	0.731983
$236$	−33.7806	−2.19893
$237$	−8.87312	−0.576371
$238$	2.76956	0.179524
$239$	−21.5462	−1.39371	−0.696854	−	0.717213i	$-0.745417\pi$
$239$	−21.5462	−1.39371	−0.696854	+	0.717213i	$0.745417\pi$
$240$	17.3539	1.12019
$241$	26.2335	1.68985	0.844925	−	0.534885i	$-0.179645\pi$
$241$	26.2335	1.68985	0.844925	+	0.534885i	$0.179645\pi$
$242$	−14.4092	−0.926260
$243$	13.5695	0.870483
$244$	−6.14505	−0.393396
$245$	−24.0900	−1.53905
$246$	26.4746	1.68796
$247$	−0.449311	−0.0285890
$248$	32.7933	2.08238
$249$	11.1161	0.704452
$250$	−17.3347	−1.09634
$251$	24.2951	1.53349	0.766746	−	0.641951i	$-0.221875\pi$
$251$	24.2951	1.53349	0.766746	+	0.641951i	$0.221875\pi$
$252$	1.40367	0.0884229
$253$	−30.1257	−1.89399
$254$	47.8393	3.00170
$255$	20.3361	1.27350
$256$	−32.0534	−2.00334
$257$	−9.20612	−0.574262	−0.287131	−	0.957891i	$-0.592701\pi$
$257$	−9.20612	−0.574262	−0.287131	+	0.957891i	$0.592701\pi$
$258$	−23.3898	−1.45619
$259$	2.74210	0.170386
$260$	−48.8827	−3.03157
$261$	−6.23064	−0.385667
$262$	30.6857	1.89577
$263$	25.9002	1.59708	0.798538	−	0.601945i	$-0.205607\pi$
$263$	25.9002	1.59708	0.798538	+	0.601945i	$0.205607\pi$
$264$	25.0111	1.53933
$265$	10.8166	0.664458
$266$	0.0747367	0.00458240
$267$	6.03281	0.369202
$268$	55.7470	3.40529
$269$	8.85267	0.539757	0.269878	−	0.962894i	$-0.413016\pi$
$269$	8.85267	0.539757	0.269878	+	0.962894i	$0.413016\pi$
$270$	47.0273	2.86199
$271$	−9.33990	−0.567359	−0.283679	−	0.958919i	$-0.591555\pi$
$271$	−9.33990	−0.567359	−0.283679	+	0.958919i	$0.591555\pi$
$272$	19.1355	1.16026
$273$	1.03856	0.0628562
$274$	31.9199	1.92835
$275$	−28.9172	−1.74377
$276$	36.3654	2.18894
$277$	22.8580	1.37341	0.686703	−	0.726939i	$-0.259057\pi$
$277$	22.8580	1.37341	0.686703	+	0.726939i	$0.259057\pi$
$278$	−15.8825	−0.952569
$279$	9.78783	0.585982
$280$	4.07240	0.243372
$281$	−8.08805	−0.482493	−0.241246	−	0.970464i	$-0.577556\pi$
$281$	−8.08805	−0.482493	−0.241246	+	0.970464i	$0.577556\pi$
$282$	−9.81065	−0.584216
$283$	−5.92450	−0.352175	−0.176087	−	0.984375i	$-0.556344\pi$
$283$	−5.92450	−0.352175	−0.176087	+	0.984375i	$0.556344\pi$
$284$	4.00682	0.237761
$285$	0.548771	0.0325064
$286$	−35.4052	−2.09355
$287$	2.08264	0.122934
$288$	0.0982952	0.00579210
$289$	5.42383	0.319049
$290$	−36.0919	−2.11939
$291$	7.40622	0.434160
$292$	−16.2363	−0.950156
$293$	−28.3775	−1.65783	−0.828917	−	0.559372i	$-0.811042\pi$
$293$	−28.3775	−1.65783	−0.828917	+	0.559372i	$0.811042\pi$
$294$	21.0620	1.22836
$295$	−29.2519	−1.70311
$296$	56.5170	3.28499
$297$	22.7205	1.31838
$298$	−41.6882	−2.41493
$299$	−25.7828	−1.49106
$300$	34.9065	2.01533
$301$	−1.83997	−0.106054
$302$	43.8022	2.52053
$303$	−20.4386	−1.17417
$304$	0.516371	0.0296159
$305$	−5.32122	−0.304692
$306$	17.0376	0.973975
$307$	2.69860	0.154017	0.0770085	−	0.997030i	$-0.475463\pi$
$307$	2.69860	0.154017	0.0770085	+	0.997030i	$0.475463\pi$
$308$	3.92834	0.223838
$309$	−2.99411	−0.170329
$310$	56.6974	3.22020
$311$	11.2895	0.640170	0.320085	−	0.947389i	$-0.396288\pi$
$311$	11.2895	0.640170	0.320085	+	0.947389i	$0.396288\pi$
$312$	21.4055	1.21185
$313$	4.33240	0.244882	0.122441	−	0.992476i	$-0.460928\pi$
$313$	4.33240	0.244882	0.122441	+	0.992476i	$0.460928\pi$
$314$	−37.3124	−2.10566
$315$	1.21549	0.0684851
$316$	−28.7243	−1.61587
$317$	−23.8146	−1.33756	−0.668780	−	0.743460i	$-0.733183\pi$
$317$	−23.8146	−1.33756	−0.668780	+	0.743460i	$0.733183\pi$
$318$	−9.45700	−0.530322
$319$	−17.4372	−0.976298
$320$	−27.4720	−1.53573
$321$	−8.67135	−0.483988
$322$	4.28862	0.238995
$323$	0.605108	0.0336691
$324$	−9.78012	−0.543340
$325$	−24.7485	−1.37280
$326$	36.6438	2.02951
$327$	−2.90941	−0.160891
$328$	42.9251	2.37014
$329$	−0.771761	−0.0425485
$330$	43.2425	2.38042
$331$	−17.0414	−0.936680	−0.468340	−	0.883548i	$-0.655148\pi$
$331$	−17.0414	−0.936680	−0.468340	+	0.883548i	$0.655148\pi$
$332$	35.9852	1.97494
$333$	16.8686	0.924396
$334$	22.8427	1.24990
$335$	48.2734	2.63746
$336$	−1.19356	−0.0651141
$337$	−12.5693	−0.684694	−0.342347	−	0.939574i	$-0.611222\pi$
$337$	−12.5693	−0.684694	−0.342347	+	0.939574i	$0.611222\pi$
$338$	1.56021	0.0848645
$339$	1.23773	0.0672244
$340$	65.8325	3.57027
$341$	27.3925	1.48339
$342$	0.459761	0.0248610
$343$	3.32730	0.179657
$344$	−37.9235	−2.04470
$345$	31.4901	1.69537
$346$	−39.1682	−2.10570
$347$	−21.4821	−1.15322	−0.576610	−	0.817019i	$-0.695625\pi$
$347$	−21.4821	−1.15322	−0.576610	+	0.817019i	$0.695625\pi$
$348$	21.0488	1.12834
$349$	−3.36761	−0.180264	−0.0901321	−	0.995930i	$-0.528729\pi$
$349$	−3.36761	−0.180264	−0.0901321	+	0.995930i	$0.528729\pi$
$350$	4.11657	0.220040
$351$	19.4451	1.03790
$352$	0.275091	0.0146624
$353$	−19.7182	−1.04949	−0.524746	−	0.851259i	$-0.675840\pi$
$353$	−19.7182	−1.04949	−0.524746	+	0.851259i	$0.675840\pi$
$354$	25.5751	1.35930
$355$	3.46965	0.184150
$356$	19.5296	1.03507
$357$	−1.39867	−0.0740255
$358$	56.6456	2.99381
$359$	−10.1999	−0.538330	−0.269165	−	0.963094i	$-0.586748\pi$
$359$	−10.1999	−0.538330	−0.269165	+	0.963094i	$0.586748\pi$
$360$	25.0523	1.32037
$361$	−18.9837	−0.999141
$362$	−47.0119	−2.47089
$363$	7.27688	0.381937
$364$	3.36204	0.176219
$365$	−14.0596	−0.735913
$366$	4.65237	0.243183
$367$	−2.12737	−0.111048	−0.0555240	−	0.998457i	$-0.517683\pi$
$367$	−2.12737	−0.111048	−0.0555240	+	0.998457i	$0.517683\pi$
$368$	29.6310	1.54462
$369$	12.8119	0.666959
$370$	97.7141	5.07991
$371$	−0.743940	−0.0386235
$372$	−33.0660	−1.71439
$373$	−10.7386	−0.556023	−0.278011	−	0.960578i	$-0.589675\pi$
$373$	−10.7386	−0.556023	−0.278011	+	0.960578i	$0.589675\pi$
$374$	47.6819	2.46557
$375$	8.75430	0.452070
$376$	−15.9067	−0.820324
$377$	−14.9235	−0.768599
$378$	−3.23443	−0.166361
$379$	−15.1125	−0.776277	−0.388138	−	0.921601i	$-0.626882\pi$
$379$	−15.1125	−0.776277	−0.388138	+	0.921601i	$0.626882\pi$
$380$	1.77649	0.0911322
$381$	−24.1596	−1.23773
$382$	−9.08321	−0.464737
$383$	19.4166	0.992139	0.496070	−	0.868283i	$-0.334776\pi$
$383$	19.4166	0.992139	0.496070	+	0.868283i	$0.334776\pi$
$384$	24.1847	1.23417
$385$	3.40170	0.173367
$386$	−16.4844	−0.839033
$387$	−11.3190	−0.575379
$388$	23.9756	1.21718
$389$	−9.16438	−0.464653	−0.232326	−	0.972638i	$-0.574634\pi$
$389$	−9.16438	−0.464653	−0.232326	+	0.972638i	$0.574634\pi$
$390$	37.0087	1.87401
$391$	34.7229	1.75601
$392$	34.1492	1.72480
$393$	−15.4967	−0.781707
$394$	54.9275	2.76721
$395$	−24.8734	−1.25152
$396$	24.1661	1.21439
$397$	−32.3707	−1.62464	−0.812320	−	0.583213i	$-0.801795\pi$
$397$	−32.3707	−1.62464	−0.812320	+	0.583213i	$0.801795\pi$
$398$	−21.3842	−1.07189
$399$	−0.0377432	−0.00188952
$400$	28.4422	1.42211
$401$	−10.3983	−0.519267	−0.259633	−	0.965707i	$-0.583602\pi$
$401$	−10.3983	−0.519267	−0.259633	+	0.965707i	$0.583602\pi$
$402$	−42.2057	−2.10503
$403$	23.4436	1.16781
$404$	−66.1644	−3.29180
$405$	−8.46897	−0.420826
$406$	2.48232	0.123195
$407$	47.2090	2.34006
$408$	−28.8278	−1.42719
$409$	−7.12882	−0.352498	−0.176249	−	0.984346i	$-0.556396\pi$
$409$	−7.12882	−0.352498	−0.176249	+	0.984346i	$0.556396\pi$
$410$	74.2146	3.66520
$411$	−16.1200	−0.795143
$412$	−9.69261	−0.477521
$413$	2.01188	0.0989980
$414$	26.3825	1.29663
$415$	31.1609	1.52963
$416$	0.235434	0.0115431
$417$	8.02090	0.392785
$418$	1.28670	0.0629344
$419$	10.9445	0.534676	0.267338	−	0.963603i	$-0.413856\pi$
$419$	10.9445	0.534676	0.267338	+	0.963603i	$0.413856\pi$
$420$	−4.10626	−0.200365
$421$	−29.1062	−1.41855	−0.709274	−	0.704933i	$-0.750977\pi$
$421$	−29.1062	−1.41855	−0.709274	+	0.704933i	$0.750977\pi$
$422$	60.2290	2.93190
$423$	−4.74767	−0.230839
$424$	−15.3333	−0.744649
$425$	33.3299	1.61674
$426$	−3.03354	−0.146975
$427$	0.365982	0.0177111
$428$	−28.0711	−1.35687
$429$	17.8802	0.863263
$430$	−65.5672	−3.16193
$431$	23.3261	1.12358	0.561789	−	0.827281i	$-0.310113\pi$
$431$	23.3261	1.12358	0.561789	+	0.827281i	$0.310113\pi$
$432$	−22.3473	−1.07519
$433$	16.2628	0.781539	0.390770	−	0.920488i	$-0.372209\pi$
$433$	16.2628	0.781539	0.390770	+	0.920488i	$0.372209\pi$
$434$	−3.89952	−0.187183
$435$	18.2270	0.873916
$436$	−9.41841	−0.451060
$437$	0.937000	0.0448228
$438$	12.2924	0.587352
$439$	−27.7490	−1.32439	−0.662193	−	0.749333i	$-0.730374\pi$
$439$	−27.7490	−1.32439	−0.662193	+	0.749333i	$0.730374\pi$
$440$	70.1120	3.34246
$441$	10.1925	0.485358
$442$	40.8081	1.94104
$443$	−24.6081	−1.16917	−0.584583	−	0.811334i	$-0.698742\pi$
$443$	−24.6081	−1.16917	−0.584583	+	0.811334i	$0.698742\pi$
$444$	−56.9870	−2.70448
$445$	16.9114	0.801676
$446$	−55.3638	−2.62155
$447$	21.0532	0.995780
$448$	1.88946	0.0892688
$449$	10.7233	0.506066	0.253033	−	0.967458i	$-0.418572\pi$
$449$	10.7233	0.506066	0.253033	+	0.967458i	$0.418572\pi$
$450$	25.3241	1.19379
$451$	35.8556	1.68837
$452$	4.00682	0.188465
$453$	−22.1208	−1.03933
$454$	69.0097	3.23879
$455$	2.91131	0.136484
$456$	−0.777920	−0.0364295
$457$	−34.8821	−1.63171	−0.815857	−	0.578254i	$-0.803734\pi$
$457$	−34.8821	−1.63171	−0.815857	+	0.578254i	$0.803734\pi$
$458$	−3.66179	−0.171104
$459$	−26.1877	−1.22234
$460$	101.941	4.75301
$461$	−0.442413	−0.0206052	−0.0103026	−	0.999947i	$-0.503279\pi$
$461$	−0.442413	−0.0206052	−0.0103026	+	0.999947i	$0.503279\pi$
$462$	−2.97412	−0.138369
$463$	12.9262	0.600732	0.300366	−	0.953824i	$-0.402891\pi$
$463$	12.9262	0.600732	0.300366	+	0.953824i	$0.402891\pi$
$464$	17.1509	0.796209
$465$	−28.6331	−1.32783
$466$	−3.03183	−0.140447
$467$	29.9026	1.38373	0.691863	−	0.722029i	$-0.256790\pi$
$467$	29.9026	1.38373	0.691863	+	0.722029i	$0.256790\pi$
$468$	20.6824	0.956042
$469$	−3.32013	−0.153310
$470$	−27.5015	−1.26855
$471$	18.8433	0.868255
$472$	41.4666	1.90865
$473$	−31.6777	−1.45654
$474$	21.7470	0.998871
$475$	0.899410	0.0412677
$476$	−4.52781	−0.207532
$477$	−4.57652	−0.209545
$478$	52.8072	2.41534
$479$	−42.4527	−1.93971	−0.969856	−	0.243679i	$-0.921646\pi$
$479$	−42.4527	−1.93971	−0.969856	+	0.243679i	$0.921646\pi$
$480$	−0.287550	−0.0131248
$481$	40.4034	1.84224
$482$	−64.2952	−2.92857
$483$	−2.16582	−0.0985482
$484$	23.5569	1.07077
$485$	20.7614	0.942725
$486$	−33.2572	−1.50858
$487$	13.4115	0.607732	0.303866	−	0.952715i	$-0.401723\pi$
$487$	13.4115	0.607732	0.303866	+	0.952715i	$0.401723\pi$
$488$	7.54320	0.341465
$489$	−18.5057	−0.836857
$490$	59.0417	2.66723
$491$	4.47274	0.201852	0.100926	−	0.994894i	$-0.467819\pi$
$491$	4.47274	0.201852	0.100926	+	0.994894i	$0.467819\pi$
$492$	−43.2820	−1.95130
$493$	20.0982	0.905176
$494$	1.10121	0.0495457
$495$	20.9263	0.940569
$496$	−26.9426	−1.20976
$497$	−0.238635	−0.0107042
$498$	−27.2442	−1.22084
$499$	22.3220	0.999269	0.499635	−	0.866236i	$-0.333468\pi$
$499$	22.3220	0.999269	0.499635	+	0.866236i	$0.333468\pi$
$500$	28.3396	1.26739
$501$	−11.5359	−0.515387
$502$	−59.5444	−2.65759
$503$	−33.0423	−1.47328	−0.736641	−	0.676284i	$-0.763589\pi$
$503$	−33.0423	−1.47328	−0.736641	+	0.676284i	$0.763589\pi$
$504$	−1.72304	−0.0767503
$505$	−57.2942	−2.54956
$506$	73.8346	3.28235
$507$	−0.787932	−0.0349933
$508$	−78.2100	−3.47001
$509$	−30.4653	−1.35035	−0.675174	−	0.737658i	$-0.735932\pi$
$509$	−30.4653	−1.35035	−0.675174	+	0.737658i	$0.735932\pi$
$510$	−49.8414	−2.20701
$511$	0.966987	0.0427770
$512$	39.4801	1.74479
$513$	−0.706675	−0.0312005
$514$	22.5631	0.995216
$515$	−8.39319	−0.369848
$516$	38.2388	1.68337
$517$	−13.2869	−0.584359
$518$	−6.72055	−0.295284
$519$	19.7805	0.868269
$520$	60.0047	2.63138
$521$	34.9578	1.53153	0.765764	−	0.643122i	$-0.222361\pi$
$521$	34.9578	1.53153	0.765764	+	0.643122i	$0.222361\pi$
$522$	15.2706	0.668375
$523$	39.4757	1.72615	0.863076	−	0.505074i	$-0.168535\pi$
$523$	39.4757	1.72615	0.863076	+	0.505074i	$0.168535\pi$
$524$	−50.1664	−2.19153
$525$	−2.07893	−0.0907320
$526$	−63.4784	−2.76779
$527$	−31.5726	−1.37532
$528$	−20.5488	−0.894273
$529$	30.7679	1.33774
$530$	−26.5102	−1.15153
$531$	12.3765	0.537096
$532$	−0.122183	−0.00529731
$533$	30.6867	1.32919
$534$	−14.7857	−0.639840
$535$	−24.3078	−1.05092
$536$	−68.4309	−2.95576
$537$	−28.6069	−1.23448
$538$	−21.6968	−0.935417
$539$	28.5251	1.22866
$540$	−76.8825	−3.30850
$541$	14.5371	0.624998	0.312499	−	0.949918i	$-0.398834\pi$
$541$	14.5371	0.624998	0.312499	+	0.949918i	$0.398834\pi$
$542$	22.8910	0.983252
$543$	23.7418	1.01886
$544$	−0.317070	−0.0135943
$545$	−8.15575	−0.349354
$546$	−2.54538	−0.108932
$547$	−38.0345	−1.62624	−0.813118	−	0.582098i	$-0.802232\pi$
$547$	−38.0345	−1.62624	−0.813118	+	0.582098i	$0.802232\pi$
$548$	−52.1842	−2.22920
$549$	2.25142	0.0960883
$550$	70.8725	3.02201
$551$	0.542350	0.0231049
$552$	−44.6394	−1.89998
$553$	1.71074	0.0727479
$554$	−56.0223	−2.38016
$555$	−49.3471	−2.09467
$556$	25.9655	1.10118
$557$	−38.8058	−1.64425	−0.822127	−	0.569304i	$-0.807213\pi$
$557$	−38.8058	−1.64425	−0.822127	+	0.569304i	$0.807213\pi$
$558$	−23.9888	−1.01553
$559$	−27.1111	−1.14668
$560$	−3.34583	−0.141387
$561$	−24.0801	−1.01666
$562$	19.8228	0.836176
$563$	−29.1466	−1.22838	−0.614192	−	0.789156i	$-0.710518\pi$
$563$	−29.1466	−1.22838	−0.614192	+	0.789156i	$0.710518\pi$
$564$	16.0389	0.675361
$565$	3.46965	0.145969
$566$	14.5202	0.610331
$567$	0.582476	0.0244617
$568$	−4.91847	−0.206375
$569$	21.1179	0.885307	0.442653	−	0.896693i	$-0.354037\pi$
$569$	21.1179	0.885307	0.442653	+	0.896693i	$0.354037\pi$
$570$	−1.34497	−0.0563347
$571$	38.6447	1.61723	0.808614	−	0.588339i	$-0.200218\pi$
$571$	38.6447	1.61723	0.808614	+	0.588339i	$0.200218\pi$
$572$	57.8822	2.42017
$573$	4.58716	0.191631
$574$	−5.10431	−0.213050
$575$	51.6109	2.15232
$576$	11.6235	0.484312
$577$	−34.3544	−1.43019	−0.715095	−	0.699027i	$-0.753617\pi$
$577$	−34.3544	−1.43019	−0.715095	+	0.699027i	$0.753617\pi$
$578$	−13.2932	−0.552923
$579$	8.32487	0.345970
$580$	59.0048	2.45004
$581$	−2.14318	−0.0889140
$582$	−18.1518	−0.752415
$583$	−12.8080	−0.530452
$584$	19.9304	0.824728
$585$	17.9096	0.740472
$586$	69.5500	2.87308
$587$	46.8706	1.93456	0.967279	−	0.253715i	$-0.0816525\pi$
$587$	46.8706	1.93456	0.967279	+	0.253715i	$0.0816525\pi$
$588$	−34.4332	−1.42000
$589$	−0.851988	−0.0351055
$590$	71.6929	2.95155
$591$	−27.7392	−1.14104
$592$	−46.4337	−1.90841
$593$	25.5203	1.04799	0.523997	−	0.851720i	$-0.324440\pi$
$593$	25.5203	1.04799	0.523997	+	0.851720i	$0.324440\pi$
$594$	−55.6852	−2.28479
$595$	−3.92080	−0.160737
$596$	68.1538	2.79169
$597$	10.7994	0.441988
$598$	63.1907	2.58406
$599$	14.2441	0.581998	0.290999	−	0.956723i	$-0.406012\pi$
$599$	14.2441	0.581998	0.290999	+	0.956723i	$0.406012\pi$
$600$	−42.8486	−1.74929
$601$	−37.4921	−1.52933	−0.764667	−	0.644426i	$-0.777096\pi$
$601$	−37.4921	−1.52933	−0.764667	+	0.644426i	$0.777096\pi$
$602$	4.50956	0.183796
$603$	−20.4246	−0.831753
$604$	−71.6100	−2.91377
$605$	20.3988	0.829329
$606$	50.0926	2.03487
$607$	8.59732	0.348955	0.174477	−	0.984661i	$-0.444176\pi$
$607$	8.59732	0.348955	0.174477	+	0.984661i	$0.444176\pi$
$608$	−0.00855616	−0.000346998 0
$609$	−1.25361	−0.0507988
$610$	13.0417	0.528043
$611$	−11.3715	−0.460042
$612$	−27.8539	−1.12593
$613$	35.3134	1.42630	0.713148	−	0.701014i	$-0.247269\pi$
$613$	35.3134	1.42630	0.713148	+	0.701014i	$0.247269\pi$
$614$	−6.61394	−0.266917
$615$	−37.4795	−1.51132
$616$	−4.82214	−0.194290
$617$	−23.4327	−0.943367	−0.471683	−	0.881768i	$-0.656353\pi$
$617$	−23.4327	−0.943367	−0.471683	+	0.881768i	$0.656353\pi$
$618$	7.33821	0.295186
$619$	12.1363	0.487799	0.243900	−	0.969800i	$-0.421573\pi$
$619$	12.1363	0.487799	0.243900	+	0.969800i	$0.421573\pi$
$620$	−92.6917	−3.72259
$621$	−40.5512	−1.62726
$622$	−27.6693	−1.10944
$623$	−1.16313	−0.0465997
$624$	−17.5865	−0.704025
$625$	−10.6521	−0.426084
$626$	−10.6182	−0.424388
$627$	−0.649802	−0.0259506
$628$	61.0001	2.43417
$629$	−54.4131	−2.16959
$630$	−2.97902	−0.118687
$631$	22.7786	0.906801	0.453400	−	0.891307i	$-0.350211\pi$
$631$	22.7786	0.906801	0.453400	+	0.891307i	$0.350211\pi$
$632$	35.2598	1.40256
$633$	−30.4165	−1.20895
$634$	58.3667	2.31804
$635$	−67.7249	−2.68758
$636$	15.4608	0.613059
$637$	24.4129	0.967275
$638$	42.7366	1.69196
$639$	−1.46802	−0.0580739
$640$	67.7954	2.67985
$641$	−13.1883	−0.520908	−0.260454	−	0.965486i	$-0.583872\pi$
$641$	−13.1883	−0.520908	−0.260454	+	0.965486i	$0.583872\pi$
$642$	21.2525	0.838768
$643$	7.07743	0.279107	0.139553	−	0.990215i	$-0.455433\pi$
$643$	7.07743	0.279107	0.139553	+	0.990215i	$0.455433\pi$
$644$	−7.01124	−0.276282
$645$	33.1124	1.30380
$646$	−1.48305	−0.0583497
$647$	32.1248	1.26296	0.631478	−	0.775394i	$-0.282449\pi$
$647$	32.1248	1.26296	0.631478	+	0.775394i	$0.282449\pi$
$648$	12.0054	0.471615
$649$	34.6373	1.35963
$650$	60.6556	2.37911
$651$	1.96932	0.0771836
$652$	−59.9071	−2.34614
$653$	−18.4185	−0.720770	−0.360385	−	0.932804i	$-0.617355\pi$
$653$	−18.4185	−0.720770	−0.360385	+	0.932804i	$0.617355\pi$
$654$	7.13061	0.278829
$655$	−43.4410	−1.69738
$656$	−35.2667	−1.37693
$657$	5.94865	0.232079
$658$	1.89149	0.0737381
$659$	−26.6424	−1.03784	−0.518921	−	0.854822i	$-0.673666\pi$
$659$	−26.6424	−1.03784	−0.518921	+	0.854822i	$0.673666\pi$
$660$	−70.6950	−2.75180
$661$	−20.8135	−0.809550	−0.404775	−	0.914416i	$-0.632650\pi$
$661$	−20.8135	−0.809550	−0.404775	+	0.914416i	$0.632650\pi$
$662$	41.7665	1.62330
$663$	−20.6087	−0.800376
$664$	−44.1728	−1.71424
$665$	−0.105803	−0.00410286
$666$	−41.3431	−1.60201
$667$	31.1217	1.20504
$668$	−37.3444	−1.44490
$669$	27.9596	1.08098
$670$	−118.312	−4.57081
$671$	6.30088	0.243243
$672$	0.0197770	0.000762916 0
$673$	21.9464	0.845972	0.422986	−	0.906136i	$-0.360982\pi$
$673$	21.9464	0.845972	0.422986	+	0.906136i	$0.360982\pi$
$674$	30.8059	1.18660
$675$	−38.9243	−1.49820
$676$	−2.55071	−0.0981044
$677$	−49.6768	−1.90924	−0.954618	−	0.297834i	$-0.903736\pi$
$677$	−49.6768	−1.90924	−0.954618	+	0.297834i	$0.903736\pi$
$678$	−3.03354	−0.116502
$679$	−1.42792	−0.0547985
$680$	−80.8111	−3.09897
$681$	−34.8510	−1.33549
$682$	−67.1357	−2.57076
$683$	46.0457	1.76189	0.880944	−	0.473220i	$-0.156909\pi$
$683$	46.0457	1.76189	0.880944	+	0.473220i	$0.156909\pi$
$684$	−0.751639	−0.0287396
$685$	−45.1882	−1.72655
$686$	−8.15481	−0.311352
$687$	1.84926	0.0705536
$688$	31.1575	1.18787
$689$	−10.9616	−0.417603
$690$	−77.1786	−2.93814
$691$	33.2473	1.26479	0.632394	−	0.774647i	$-0.282072\pi$
$691$	33.2473	1.26479	0.632394	+	0.774647i	$0.282072\pi$
$692$	64.0341	2.43421
$693$	−1.43927	−0.0546732
$694$	52.6501	1.99857
$695$	22.4845	0.852884
$696$	−25.8380	−0.979387
$697$	−41.3272	−1.56538
$698$	8.25362	0.312404
$699$	1.53112	0.0579122
$700$	−6.72997	−0.254369
$701$	38.1283	1.44008	0.720042	−	0.693930i	$-0.244122\pi$
$701$	38.1283	1.44008	0.720042	+	0.693930i	$0.244122\pi$
$702$	−47.6577	−1.79872
$703$	−1.46834	−0.0553795
$704$	32.5298	1.22601
$705$	13.8887	0.523079
$706$	48.3269	1.81881
$707$	3.94056	0.148200
$708$	−41.8114	−1.57137
$709$	−8.93817	−0.335680	−0.167840	−	0.985814i	$-0.553679\pi$
$709$	−8.93817	−0.335680	−0.167840	+	0.985814i	$0.553679\pi$
$710$	−8.50371	−0.319138
$711$	10.5240	0.394681
$712$	−23.9731	−0.898428
$713$	−48.8896	−1.83093
$714$	3.42798	0.128289
$715$	50.1223	1.87447
$716$	−92.6070	−3.46088
$717$	−26.6684	−0.995951
$718$	24.9987	0.932944
$719$	−0.415904	−0.0155106	−0.00775530	−	0.999970i	$-0.502469\pi$
$719$	−0.415904	−0.0155106	−0.00775530	+	0.999970i	$0.502469\pi$
$720$	−20.5827	−0.767071
$721$	0.577265	0.0214985
$722$	46.5267	1.73155
$723$	32.4701	1.20758
$724$	76.8574	2.85638
$725$	29.8732	1.10946
$726$	−17.8348	−0.661910
$727$	−20.1313	−0.746629	−0.373315	−	0.927705i	$-0.621779\pi$
$727$	−20.1313	−0.746629	−0.373315	+	0.927705i	$0.621779\pi$
$728$	−4.12699	−0.152956
$729$	24.1180	0.893260
$730$	34.4584	1.27536
$731$	36.5118	1.35044
$732$	−7.60592	−0.281123
$733$	3.92093	0.144823	0.0724114	−	0.997375i	$-0.476931\pi$
$733$	3.92093	0.144823	0.0724114	+	0.997375i	$0.476931\pi$
$734$	5.21394	0.192450
$735$	−29.8170	−1.09982
$736$	−0.490978	−0.0180977
$737$	−57.1607	−2.10554
$738$	−31.4003	−1.15586
$739$	−0.940360	−0.0345917	−0.0172959	−	0.999850i	$-0.505506\pi$
$739$	−0.940360	−0.0345917	−0.0172959	+	0.999850i	$0.505506\pi$
$740$	−159.748	−5.87244
$741$	−0.556127	−0.0204298
$742$	1.82331	0.0669358
$743$	36.2210	1.32882	0.664410	−	0.747369i	$-0.268683\pi$
$743$	36.2210	1.32882	0.664410	+	0.747369i	$0.268683\pi$
$744$	40.5894	1.48808
$745$	59.0169	2.16221
$746$	26.3190	0.963606
$747$	−13.1843	−0.482387
$748$	−77.9526	−2.85023
$749$	1.67184	0.0610876
$750$	−21.4558	−0.783454
$751$	−2.60847	−0.0951846	−0.0475923	−	0.998867i	$-0.515155\pi$
$751$	−2.60847	−0.0951846	−0.0475923	+	0.998867i	$0.515155\pi$
$752$	13.0687	0.476567
$753$	30.0708	1.09584
$754$	36.5757	1.33201
$755$	−62.0097	−2.25677
$756$	5.28780	0.192315
$757$	26.7231	0.971269	0.485634	−	0.874162i	$-0.338589\pi$
$757$	26.7231	0.971269	0.485634	+	0.874162i	$0.338589\pi$
$758$	37.0389	1.34531
$759$	−37.2876	−1.35345
$760$	−2.18069	−0.0791020
$761$	−2.49251	−0.0903533	−0.0451767	−	0.998979i	$-0.514385\pi$
$761$	−2.49251	−0.0903533	−0.0451767	+	0.998979i	$0.514385\pi$
$762$	59.2122	2.14503
$763$	0.560934	0.0203072
$764$	14.8497	0.537242
$765$	−24.1197	−0.872050
$766$	−47.5877	−1.71941
$767$	29.6440	1.07038
$768$	−39.6736	−1.43160
$769$	22.7160	0.819162	0.409581	−	0.912274i	$-0.365675\pi$
$769$	22.7160	0.819162	0.409581	+	0.912274i	$0.365675\pi$
$770$	−8.33716	−0.300450
$771$	−11.3947	−0.410371
$772$	26.9495	0.969932
$773$	16.1030	0.579186	0.289593	−	0.957150i	$-0.406480\pi$
$773$	16.1030	0.579186	0.289593	+	0.957150i	$0.406480\pi$
$774$	27.7416	0.997152
$775$	−46.9283	−1.68571
$776$	−29.4307	−1.05650
$777$	3.39398	0.121758
$778$	22.4608	0.805259
$779$	−1.11522	−0.0399567
$780$	−60.5037	−2.16638
$781$	−4.10843	−0.147011
$782$	−85.1018	−3.04323
$783$	−23.4716	−0.838808
$784$	−28.0566	−1.00202
$785$	52.8223	1.88531
$786$	37.9807	1.35473
$787$	18.6317	0.664150	0.332075	−	0.943253i	$-0.392251\pi$
$787$	18.6317	0.664150	0.332075	+	0.943253i	$0.392251\pi$
$788$	−89.7982	−3.19893
$789$	32.0575	1.14128
$790$	60.9618	2.16892
$791$	−0.238635	−0.00848488
$792$	−29.6645	−1.05408
$793$	5.39256	0.191495
$794$	79.3368	2.81556
$795$	13.3880	0.474825
$796$	34.9599	1.23912
$797$	29.7321	1.05316	0.526582	−	0.850124i	$-0.323473\pi$
$797$	29.7321	1.05316	0.526582	+	0.850124i	$0.323473\pi$
$798$	0.0925041	0.00327461
$799$	15.3145	0.541789
$800$	−0.471281	−0.0166623
$801$	−7.15524	−0.252818
$802$	25.4850	0.899907
$803$	16.6480	0.587496
$804$	68.9999	2.43344
$805$	−6.07130	−0.213985
$806$	−57.4575	−2.02385
$807$	10.9572	0.385713
$808$	81.2185	2.85726
$809$	−40.4357	−1.42164	−0.710821	−	0.703373i	$-0.751676\pi$
$809$	−40.4357	−1.42164	−0.710821	+	0.703373i	$0.751676\pi$
$810$	20.7564	0.729307
$811$	−37.5958	−1.32017	−0.660084	−	0.751192i	$-0.729479\pi$
$811$	−37.5958	−1.32017	−0.660084	+	0.751192i	$0.729479\pi$
$812$	−4.05822	−0.142415
$813$	−11.5603	−0.405437
$814$	−115.704	−4.05541
$815$	−51.8758	−1.81713
$816$	23.6846	0.829127
$817$	0.985272	0.0344703
$818$	17.4719	0.610891
$819$	−1.23178	−0.0430420
$820$	−121.330	−4.23701
$821$	−12.0463	−0.420418	−0.210209	−	0.977656i	$-0.567414\pi$
$821$	−12.0463	−0.420418	−0.210209	+	0.977656i	$0.567414\pi$
$822$	39.5083	1.37801
$823$	−32.3062	−1.12612	−0.563062	−	0.826415i	$-0.690377\pi$
$823$	−32.3062	−1.12612	−0.563062	+	0.826415i	$0.690377\pi$
$824$	11.8979	0.414484
$825$	−35.7917	−1.24611
$826$	−4.93087	−0.171567
$827$	−0.552530	−0.0192134	−0.00960668	−	0.999954i	$-0.503058\pi$
$827$	−0.552530	−0.0192134	−0.00960668	+	0.999954i	$0.503058\pi$
$828$	−43.1313	−1.49892
$829$	−17.4317	−0.605429	−0.302714	−	0.953081i	$-0.597893\pi$
$829$	−17.4317	−0.605429	−0.302714	+	0.953081i	$0.597893\pi$
$830$	−76.3717	−2.65090
$831$	28.2921	0.981442
$832$	27.8403	0.965189
$833$	−32.8780	−1.13916
$834$	−19.6583	−0.680711
$835$	−32.3379	−1.11910
$836$	−2.10355	−0.0727529
$837$	36.8720	1.27448
$838$	−26.8238	−0.926612
$839$	19.2787	0.665576	0.332788	−	0.943002i	$-0.392011\pi$
$839$	19.2787	0.665576	0.332788	+	0.943002i	$0.392011\pi$
$840$	5.04054	0.173915
$841$	−10.9863	−0.378837
$842$	71.3358	2.45839
$843$	−10.0108	−0.344792
$844$	−98.4652	−3.38931
$845$	−2.20876	−0.0759836
$846$	11.6360	0.400053
$847$	−1.40298	−0.0482070
$848$	12.5976	0.432604
$849$	−7.33294	−0.251666
$850$	−81.6877	−2.80187
$851$	−84.2579	−2.88832
$852$	4.95937	0.169905
$853$	30.6715	1.05017	0.525085	−	0.851049i	$-0.324033\pi$
$853$	30.6715	1.05017	0.525085	+	0.851049i	$0.324033\pi$
$854$	−0.896977	−0.0306939
$855$	−0.650872	−0.0222593
$856$	34.4580	1.17775
$857$	31.2290	1.06676	0.533380	−	0.845876i	$-0.320921\pi$
$857$	31.2290	1.06676	0.533380	+	0.845876i	$0.320921\pi$
$858$	−43.8222	−1.49607
$859$	50.7585	1.73186	0.865929	−	0.500167i	$-0.166728\pi$
$859$	50.7585	1.73186	0.865929	+	0.500167i	$0.166728\pi$
$860$	107.192	3.65523
$861$	2.57775	0.0878496
$862$	−57.1694	−1.94720
$863$	−7.56896	−0.257650	−0.128825	−	0.991667i	$-0.541121\pi$
$863$	−7.56896	−0.257650	−0.128825	+	0.991667i	$0.541121\pi$
$864$	0.370290	0.0125975
$865$	55.4495	1.88534
$866$	−39.8581	−1.35443
$867$	6.71325	0.227994
$868$	6.37512	0.216386
$869$	29.4527	0.999115
$870$	−44.6721	−1.51453
$871$	−48.9205	−1.65761
$872$	11.5613	0.391516
$873$	−8.78418	−0.297299
$874$	−2.29648	−0.0776795
$875$	−1.68783	−0.0570591
$876$	−20.0962	−0.678987
$877$	0.928544	0.0313547	0.0156774	−	0.999877i	$-0.495010\pi$
$877$	0.928544	0.0313547	0.0156774	+	0.999877i	$0.495010\pi$
$878$	68.0094	2.29521
$879$	−35.1238	−1.18470
$880$	−57.6031	−1.94180
$881$	28.8892	0.973302	0.486651	−	0.873596i	$-0.338218\pi$
$881$	28.8892	0.973302	0.486651	+	0.873596i	$0.338218\pi$
$882$	−24.9807	−0.841143
$883$	33.8466	1.13903	0.569514	−	0.821982i	$-0.307131\pi$
$883$	33.8466	1.13903	0.569514	+	0.821982i	$0.307131\pi$
$884$	−66.7150	−2.24387
$885$	−36.2060	−1.21705
$886$	60.3115	2.02620
$887$	43.2843	1.45335	0.726673	−	0.686984i	$-0.241066\pi$
$887$	43.2843	1.45335	0.726673	+	0.686984i	$0.241066\pi$
$888$	69.9530	2.34747
$889$	4.65796	0.156223
$890$	−41.4478	−1.38933
$891$	10.0281	0.335955
$892$	90.5115	3.03055
$893$	0.413263	0.0138293
$894$	−51.5988	−1.72572
$895$	−80.1918	−2.68052
$896$	−4.66281	−0.155774
$897$	−31.9123	−1.06552
$898$	−26.2816	−0.877030
$899$	−28.2981	−0.943793
$900$	−41.4010	−1.38003
$901$	14.7625	0.491809
$902$	−87.8778	−2.92601
$903$	−2.27740	−0.0757870
$904$	−4.91847	−0.163586
$905$	66.5537	2.21232
$906$	54.2154	1.80119
$907$	0.555459	0.0184437	0.00922185	−	0.999957i	$-0.497065\pi$
$907$	0.555459	0.0184437	0.00922185	+	0.999957i	$0.497065\pi$
$908$	−112.820	−3.74407
$909$	24.2413	0.804033
$910$	−7.13528	−0.236532
$911$	8.53144	0.282659	0.141330	−	0.989963i	$-0.454862\pi$
$911$	8.53144	0.282659	0.141330	+	0.989963i	$0.454862\pi$
$912$	0.639130	0.0211637
$913$	−36.8978	−1.22114
$914$	85.4918	2.82782
$915$	−6.58625	−0.217735
$916$	5.98646	0.197798
$917$	2.98777	0.0986648
$918$	64.1828	2.11835
$919$	33.9436	1.11970	0.559848	−	0.828595i	$-0.310859\pi$
$919$	33.9436	1.11970	0.559848	+	0.828595i	$0.310859\pi$
$920$	−125.135	−4.12557
$921$	3.34014	0.110061
$922$	1.08430	0.0357096
$923$	−3.51616	−0.115736
$924$	4.86224	0.159956
$925$	−80.8776	−2.65924
$926$	−31.6806	−1.04109
$927$	3.55118	0.116636
$928$	−0.284186	−0.00932886
$929$	26.2887	0.862505	0.431253	−	0.902231i	$-0.358072\pi$
$929$	26.2887	0.862505	0.431253	+	0.902231i	$0.358072\pi$
$930$	70.1763	2.30117
$931$	−0.887214	−0.0290773
$932$	4.95658	0.162358
$933$	13.9734	0.457469
$934$	−73.2876	−2.39804
$935$	−67.5020	−2.20755
$936$	−25.3881	−0.829837
$937$	−57.1651	−1.86750	−0.933752	−	0.357921i	$-0.883486\pi$
$937$	−57.1651	−1.86750	−0.933752	+	0.357921i	$0.883486\pi$
$938$	8.13726	0.265691
$939$	5.36235	0.174994
$940$	44.9608	1.46646
$941$	24.8431	0.809862	0.404931	−	0.914347i	$-0.367296\pi$
$941$	24.8431	0.809862	0.404931	+	0.914347i	$0.367296\pi$
$942$	−46.1828	−1.50472
$943$	−63.9945	−2.08395
$944$	−34.0684	−1.10883
$945$	4.57890	0.148952
$946$	77.6384	2.52424
$947$	8.42094	0.273644	0.136822	−	0.990596i	$-0.456311\pi$
$947$	8.42094	0.273644	0.136822	+	0.990596i	$0.456311\pi$
$948$	−35.5530	−1.15471
$949$	14.2481	0.462512
$950$	−2.20435	−0.0715184
$951$	−29.4761	−0.955828
$952$	5.55801	0.180136
$953$	−9.70956	−0.314524	−0.157262	−	0.987557i	$-0.550267\pi$
$953$	−9.70956	−0.314524	−0.157262	+	0.987557i	$0.550267\pi$
$954$	11.2165	0.363148
$955$	12.8589	0.416103
$956$	−86.3317	−2.79217
$957$	−21.5826	−0.697667
$958$	104.046	3.36159
$959$	3.10794	0.100361
$960$	−34.0030	−1.09744
$961$	13.4540	0.433998
$962$	−99.0240	−3.19266
$963$	10.2847	0.331420
$964$	105.113	3.38546
$965$	23.3365	0.751230
$966$	5.30817	0.170787
$967$	−5.06966	−0.163029	−0.0815146	−	0.996672i	$-0.525976\pi$
$967$	−5.06966	−0.163029	−0.0815146	+	0.996672i	$0.525976\pi$
$968$	−28.9167	−0.929418
$969$	0.748962	0.0240601
$970$	−50.8836	−1.63378
$971$	−5.43774	−0.174505	−0.0872527	−	0.996186i	$-0.527809\pi$
$971$	−5.43774	−0.174505	−0.0872527	+	0.996186i	$0.527809\pi$
$972$	54.3705	1.74393
$973$	−1.54643	−0.0495763
$974$	−32.8699	−1.05322
$975$	−30.6320	−0.981009
$976$	−6.19740	−0.198374
$977$	13.8335	0.442573	0.221287	−	0.975209i	$-0.428974\pi$
$977$	13.8335	0.442573	0.221287	+	0.975209i	$0.428974\pi$
$978$	45.3553	1.45030
$979$	−20.0248	−0.639997
$980$	−96.5242	−3.08335
$981$	3.45072	0.110173
$982$	−10.9622	−0.349817
$983$	−50.4911	−1.61042	−0.805209	−	0.592992i	$-0.797947\pi$
$983$	−50.4911	−1.61042	−0.805209	+	0.592992i	$0.797947\pi$
$984$	53.1298	1.69372
$985$	−77.7596	−2.47763
$986$	−49.2582	−1.56870
$987$	−0.955233	−0.0304054
$988$	−1.80031	−0.0572754
$989$	56.5379	1.79780
$990$	−51.2880	−1.63004
$991$	−20.7864	−0.660302	−0.330151	−	0.943928i	$-0.607100\pi$
$991$	−20.7864	−0.660302	−0.330151	+	0.943928i	$0.607100\pi$
$992$	0.446433	0.0141743
$993$	−21.0927	−0.669357
$994$	0.584866	0.0185508
$995$	30.2731	0.959722
$996$	44.5401	1.41131
$997$	30.5083	0.966206	0.483103	−	0.875564i	$-0.339510\pi$
$997$	30.5083	0.966206	0.483103	+	0.875564i	$0.339510\pi$
$998$	−54.7085	−1.73177
$999$	63.5464	2.01052

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8023.2.a.b.1.17	✓	155

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8023.2.a.b.1.17	✓	155	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:	\( N \)	\(=\)	\( 8023 = 71 \cdot 113 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8023.a (trivial)

\(f(q)\)	\(=\)	\(q-2.45088 q^{2} +1.23773 q^{3} +4.00682 q^{4} +3.46965 q^{5} -3.03354 q^{6} -0.238635 q^{7} -4.91847 q^{8} -1.46802 q^{9} +O(q^{10})\)	\(q-2.45088 q^{2} +1.23773 q^{3} +4.00682 q^{4} +3.46965 q^{5} -3.03354 q^{6} -0.238635 q^{7} -4.91847 q^{8} -1.46802 q^{9} -8.50371 q^{10} -4.10843 q^{11} +4.95937 q^{12} -3.51616 q^{13} +0.584866 q^{14} +4.29450 q^{15} +4.04096 q^{16} +4.73538 q^{17} +3.59794 q^{18} +0.127784 q^{19} +13.9023 q^{20} -0.295366 q^{21} +10.0693 q^{22} +7.33266 q^{23} -6.08775 q^{24} +7.03849 q^{25} +8.61770 q^{26} -5.53021 q^{27} -0.956166 q^{28} +4.24426 q^{29} -10.5253 q^{30} -6.66738 q^{31} -0.0669577 q^{32} -5.08514 q^{33} -11.6059 q^{34} -0.827980 q^{35} -5.88208 q^{36} -11.4908 q^{37} -0.313185 q^{38} -4.35207 q^{39} -17.0654 q^{40} -8.72732 q^{41} +0.723907 q^{42} +7.71042 q^{43} -16.4617 q^{44} -5.09351 q^{45} -17.9715 q^{46} +3.23407 q^{47} +5.00162 q^{48} -6.94305 q^{49} -17.2505 q^{50} +5.86114 q^{51} -14.0886 q^{52} +3.11748 q^{53} +13.5539 q^{54} -14.2548 q^{55} +1.17372 q^{56} +0.158163 q^{57} -10.4022 q^{58} -8.43078 q^{59} +17.2073 q^{60} -1.53365 q^{61} +16.3410 q^{62} +0.350320 q^{63} -7.91781 q^{64} -12.1999 q^{65} +12.4631 q^{66} +13.9130 q^{67} +18.9738 q^{68} +9.07587 q^{69} +2.02928 q^{70} +1.00000 q^{71} +7.22041 q^{72} -4.05216 q^{73} +28.1625 q^{74} +8.71177 q^{75} +0.512009 q^{76} +0.980415 q^{77} +10.6664 q^{78} -7.16885 q^{79} +14.0207 q^{80} -2.44087 q^{81} +21.3896 q^{82} +8.98099 q^{83} -1.18348 q^{84} +16.4301 q^{85} -18.8973 q^{86} +5.25325 q^{87} +20.2072 q^{88} +4.87409 q^{89} +12.4836 q^{90} +0.839079 q^{91} +29.3806 q^{92} -8.25243 q^{93} -7.92631 q^{94} +0.443368 q^{95} -0.0828758 q^{96} +5.98370 q^{97} +17.0166 q^{98} +6.03125 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 155 q - 21 q^{2} - 16 q^{3} + 151 q^{4} - 26 q^{5} - 10 q^{6} - 40 q^{7} - 57 q^{8} + 135 q^{9}+O(q^{10}) \) 155 * q - 21 * q^2 - 16 * q^3 + 151 * q^4 - 26 * q^5 - 10 * q^6 - 40 * q^7 - 57 * q^8 + 135 * q^9	\( 155 q - 21 q^{2} - 16 q^{3} + 151 q^{4} - 26 q^{5} - 10 q^{6} - 40 q^{7} - 57 q^{8} + 135 q^{9} - 2 q^{10} - 24 q^{11} - 32 q^{12} - 62 q^{13} - 18 q^{14} - 12 q^{15} + 155 q^{16} - 129 q^{17} - 42 q^{18} - 18 q^{19} - 59 q^{20} - 45 q^{21} - 17 q^{22} - 38 q^{23} - 27 q^{24} + 129 q^{25} - 44 q^{26} - 43 q^{27} - 100 q^{28} - 52 q^{29} - 39 q^{30} - 56 q^{31} - 145 q^{32} - 126 q^{33} - q^{34} - 49 q^{35} + 131 q^{36} - 30 q^{37} - 91 q^{38} - 29 q^{39} - 5 q^{40} - 163 q^{41} - 80 q^{42} - 15 q^{43} - 118 q^{44} - 66 q^{45} + 2 q^{46} - 111 q^{47} - 89 q^{48} + 101 q^{49} - 121 q^{50} + 5 q^{51} - 111 q^{52} - 93 q^{53} - 68 q^{54} - 60 q^{55} - 27 q^{56} - 106 q^{57} + 16 q^{58} - 79 q^{59} - 103 q^{60} - 74 q^{61} - 102 q^{62} - 118 q^{63} + 175 q^{64} - 109 q^{65} + 65 q^{66} - 18 q^{67} - 346 q^{68} - 39 q^{69} + 32 q^{70} + 155 q^{71} - 203 q^{72} - 108 q^{73} - 87 q^{74} - 22 q^{75} - 16 q^{76} - 121 q^{77} - 75 q^{78} - 6 q^{79} - 136 q^{80} + 107 q^{81} - 30 q^{82} - 116 q^{83} - 5 q^{84} - 53 q^{85} + 8 q^{86} - 100 q^{87} - 43 q^{88} - 189 q^{89} - 76 q^{90} + 14 q^{91} - 99 q^{92} - 72 q^{93} + 17 q^{94} - 18 q^{95} - 50 q^{96} - 184 q^{97} - 249 q^{98} - 114 q^{99}+O(q^{100}) \) 155 * q - 21 * q^2 - 16 * q^3 + 151 * q^4 - 26 * q^5 - 10 * q^6 - 40 * q^7 - 57 * q^8 + 135 * q^9 - 2 * q^10 - 24 * q^11 - 32 * q^12 - 62 * q^13 - 18 * q^14 - 12 * q^15 + 155 * q^16 - 129 * q^17 - 42 * q^18 - 18 * q^19 - 59 * q^20 - 45 * q^21 - 17 * q^22 - 38 * q^23 - 27 * q^24 + 129 * q^25 - 44 * q^26 - 43 * q^27 - 100 * q^28 - 52 * q^29 - 39 * q^30 - 56 * q^31 - 145 * q^32 - 126 * q^33 - q^34 - 49 * q^35 + 131 * q^36 - 30 * q^37 - 91 * q^38 - 29 * q^39 - 5 * q^40 - 163 * q^41 - 80 * q^42 - 15 * q^43 - 118 * q^44 - 66 * q^45 + 2 * q^46 - 111 * q^47 - 89 * q^48 + 101 * q^49 - 121 * q^50 + 5 * q^51 - 111 * q^52 - 93 * q^53 - 68 * q^54 - 60 * q^55 - 27 * q^56 - 106 * q^57 + 16 * q^58 - 79 * q^59 - 103 * q^60 - 74 * q^61 - 102 * q^62 - 118 * q^63 + 175 * q^64 - 109 * q^65 + 65 * q^66 - 18 * q^67 - 346 * q^68 - 39 * q^69 + 32 * q^70 + 155 * q^71 - 203 * q^72 - 108 * q^73 - 87 * q^74 - 22 * q^75 - 16 * q^76 - 121 * q^77 - 75 * q^78 - 6 * q^79 - 136 * q^80 + 107 * q^81 - 30 * q^82 - 116 * q^83 - 5 * q^84 - 53 * q^85 + 8 * q^86 - 100 * q^87 - 43 * q^88 - 189 * q^89 - 76 * q^90 + 14 * q^91 - 99 * q^92 - 72 * q^93 + 17 * q^94 - 18 * q^95 - 50 * q^96 - 184 * q^97 - 249 * q^98 - 114 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more