LMFDB - Embedded newform 8042.2.a.a.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8042,2,Mod(1,8042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8042, 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("8042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8042 = 2 \cdot 4021 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8042.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:	$64.2156933055$
Analytic rank:	$1$
Dimension:	$67$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	8042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.07802 q^{3} +1.00000 q^{4} -3.66993 q^{5} -2.07802 q^{6} -3.45056 q^{7} +1.00000 q^{8} +1.31817 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.07802 q^{3} +1.00000 q^{4} -3.66993 q^{5} -2.07802 q^{6} -3.45056 q^{7} +1.00000 q^{8} +1.31817 q^{9} -3.66993 q^{10} -0.0411660 q^{11} -2.07802 q^{12} -1.28748 q^{13} -3.45056 q^{14} +7.62619 q^{15} +1.00000 q^{16} -3.99294 q^{17} +1.31817 q^{18} +4.34352 q^{19} -3.66993 q^{20} +7.17034 q^{21} -0.0411660 q^{22} -2.81157 q^{23} -2.07802 q^{24} +8.46842 q^{25} -1.28748 q^{26} +3.49488 q^{27} -3.45056 q^{28} -2.72765 q^{29} +7.62619 q^{30} +8.98284 q^{31} +1.00000 q^{32} +0.0855438 q^{33} -3.99294 q^{34} +12.6633 q^{35} +1.31817 q^{36} +8.77471 q^{37} +4.34352 q^{38} +2.67542 q^{39} -3.66993 q^{40} +0.445901 q^{41} +7.17034 q^{42} -9.80961 q^{43} -0.0411660 q^{44} -4.83758 q^{45} -2.81157 q^{46} +5.77836 q^{47} -2.07802 q^{48} +4.90639 q^{49} +8.46842 q^{50} +8.29742 q^{51} -1.28748 q^{52} +7.93141 q^{53} +3.49488 q^{54} +0.151077 q^{55} -3.45056 q^{56} -9.02591 q^{57} -2.72765 q^{58} -4.48136 q^{59} +7.62619 q^{60} +7.09875 q^{61} +8.98284 q^{62} -4.54842 q^{63} +1.00000 q^{64} +4.72498 q^{65} +0.0855438 q^{66} -10.4509 q^{67} -3.99294 q^{68} +5.84250 q^{69} +12.6633 q^{70} +7.58055 q^{71} +1.31817 q^{72} -3.83122 q^{73} +8.77471 q^{74} -17.5975 q^{75} +4.34352 q^{76} +0.142046 q^{77} +2.67542 q^{78} -1.20276 q^{79} -3.66993 q^{80} -11.2169 q^{81} +0.445901 q^{82} +4.24142 q^{83} +7.17034 q^{84} +14.6538 q^{85} -9.80961 q^{86} +5.66811 q^{87} -0.0411660 q^{88} +7.73181 q^{89} -4.83758 q^{90} +4.44255 q^{91} -2.81157 q^{92} -18.6665 q^{93} +5.77836 q^{94} -15.9404 q^{95} -2.07802 q^{96} +12.0758 q^{97} +4.90639 q^{98} -0.0542636 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) $ 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9} - 20 q^{10} - 13 q^{11} - 11 q^{12} - 51 q^{13} - 40 q^{14} - 31 q^{15} + 67 q^{16} - 34 q^{17} + 24 q^{18} - 33 q^{19} - 20 q^{20} - 39 q^{21} - 13 q^{22} - 43 q^{23} - 11 q^{24} - 9 q^{25} - 51 q^{26} - 29 q^{27} - 40 q^{28} - 63 q^{29} - 31 q^{30} - 43 q^{31} + 67 q^{32} - 49 q^{33} - 34 q^{34} - 20 q^{35} + 24 q^{36} - 77 q^{37} - 33 q^{38} - 40 q^{39} - 20 q^{40} - 50 q^{41} - 39 q^{42} - 56 q^{43} - 13 q^{44} - 48 q^{45} - 43 q^{46} - 48 q^{47} - 11 q^{48} + q^{49} - 9 q^{50} - 18 q^{51} - 51 q^{52} - 91 q^{53} - 29 q^{54} - 58 q^{55} - 40 q^{56} - 65 q^{57} - 63 q^{58} - 17 q^{59} - 31 q^{60} - 45 q^{61} - 43 q^{62} - 67 q^{63} + 67 q^{64} - 65 q^{65} - 49 q^{66} - 112 q^{67} - 34 q^{68} - 57 q^{69} - 20 q^{70} - 75 q^{71} + 24 q^{72} - 79 q^{73} - 77 q^{74} - 5 q^{75} - 33 q^{76} - 85 q^{77} - 40 q^{78} - 80 q^{79} - 20 q^{80} - 77 q^{81} - 50 q^{82} - 22 q^{83} - 39 q^{84} - 134 q^{85} - 56 q^{86} - 49 q^{87} - 13 q^{88} - 77 q^{89} - 48 q^{90} - 17 q^{91} - 43 q^{92} - 97 q^{93} - 48 q^{94} - 73 q^{95} - 11 q^{96} - 87 q^{97} + q^{98} - 44 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9 - 20 * q^10 - 13 * q^11 - 11 * q^12 - 51 * q^13 - 40 * q^14 - 31 * q^15 + 67 * q^16 - 34 * q^17 + 24 * q^18 - 33 * q^19 - 20 * q^20 - 39 * q^21 - 13 * q^22 - 43 * q^23 - 11 * q^24 - 9 * q^25 - 51 * q^26 - 29 * q^27 - 40 * q^28 - 63 * q^29 - 31 * q^30 - 43 * q^31 + 67 * q^32 - 49 * q^33 - 34 * q^34 - 20 * q^35 + 24 * q^36 - 77 * q^37 - 33 * q^38 - 40 * q^39 - 20 * q^40 - 50 * q^41 - 39 * q^42 - 56 * q^43 - 13 * q^44 - 48 * q^45 - 43 * q^46 - 48 * q^47 - 11 * q^48 + q^49 - 9 * q^50 - 18 * q^51 - 51 * q^52 - 91 * q^53 - 29 * q^54 - 58 * q^55 - 40 * q^56 - 65 * q^57 - 63 * q^58 - 17 * q^59 - 31 * q^60 - 45 * q^61 - 43 * q^62 - 67 * q^63 + 67 * q^64 - 65 * q^65 - 49 * q^66 - 112 * q^67 - 34 * q^68 - 57 * q^69 - 20 * q^70 - 75 * q^71 + 24 * q^72 - 79 * q^73 - 77 * q^74 - 5 * q^75 - 33 * q^76 - 85 * q^77 - 40 * q^78 - 80 * q^79 - 20 * q^80 - 77 * q^81 - 50 * q^82 - 22 * q^83 - 39 * q^84 - 134 * q^85 - 56 * q^86 - 49 * q^87 - 13 * q^88 - 77 * q^89 - 48 * q^90 - 17 * q^91 - 43 * q^92 - 97 * q^93 - 48 * q^94 - 73 * q^95 - 11 * q^96 - 87 * q^97 + q^98 - 44 * 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
$3$	−2.07802	−1.19975	−0.599873	−	0.800096i	$-0.704782\pi$
$3$	−2.07802	−1.19975	−0.599873	+	0.800096i	$0.704782\pi$
$4$	1.00000	0.500000
$5$	−3.66993	−1.64124	−0.820622	−	0.571471i	$-0.806373\pi$
$5$	−3.66993	−1.64124	−0.820622	+	0.571471i	$0.806373\pi$
$6$	−2.07802	−0.848348
$7$	−3.45056	−1.30419	−0.652095	−	0.758137i	$-0.726110\pi$
$7$	−3.45056	−1.30419	−0.652095	+	0.758137i	$0.726110\pi$
$8$	1.00000	0.353553
$9$	1.31817	0.439389
$10$	−3.66993	−1.16054
$11$	−0.0411660	−0.0124120	−0.00620601	−	0.999981i	$-0.501975\pi$
$11$	−0.0411660	−0.0124120	−0.00620601	+	0.999981i	$0.501975\pi$
$12$	−2.07802	−0.599873
$13$	−1.28748	−0.357084	−0.178542	−	0.983932i	$-0.557138\pi$
$13$	−1.28748	−0.357084	−0.178542	+	0.983932i	$0.557138\pi$
$14$	−3.45056	−0.922202
$15$	7.62619	1.96908
$16$	1.00000	0.250000
$17$	−3.99294	−0.968431	−0.484216	−	0.874949i	$-0.660895\pi$
$17$	−3.99294	−0.968431	−0.484216	+	0.874949i	$0.660895\pi$
$18$	1.31817	0.310695
$19$	4.34352	0.996471	0.498235	−	0.867042i	$-0.333982\pi$
$19$	4.34352	0.996471	0.498235	+	0.867042i	$0.333982\pi$
$20$	−3.66993	−0.820622
$21$	7.17034	1.56470
$22$	−0.0411660	−0.00877662
$23$	−2.81157	−0.586253	−0.293127	−	0.956074i	$-0.594696\pi$
$23$	−2.81157	−0.586253	−0.293127	+	0.956074i	$0.594696\pi$
$24$	−2.07802	−0.424174
$25$	8.46842	1.69368
$26$	−1.28748	−0.252496
$27$	3.49488	0.672591
$28$	−3.45056	−0.652095
$29$	−2.72765	−0.506512	−0.253256	−	0.967399i	$-0.581501\pi$
$29$	−2.72765	−0.506512	−0.253256	+	0.967399i	$0.581501\pi$
$30$	7.62619	1.39235
$31$	8.98284	1.61337	0.806683	−	0.590984i	$-0.201260\pi$
$31$	8.98284	1.61337	0.806683	+	0.590984i	$0.201260\pi$
$32$	1.00000	0.176777
$33$	0.0855438	0.0148913
$34$	−3.99294	−0.684784
$35$	12.6633	2.14050
$36$	1.31817	0.219694
$37$	8.77471	1.44255	0.721277	−	0.692647i	$-0.243555\pi$
$37$	8.77471	1.44255	0.721277	+	0.692647i	$0.243555\pi$
$38$	4.34352	0.704611
$39$	2.67542	0.428410
$40$	−3.66993	−0.580268
$41$	0.445901	0.0696380	0.0348190	−	0.999394i	$-0.488915\pi$
$41$	0.445901	0.0696380	0.0348190	+	0.999394i	$0.488915\pi$
$42$	7.17034	1.10641
$43$	−9.80961	−1.49595	−0.747976	−	0.663726i	$-0.768974\pi$
$43$	−9.80961	−1.49595	−0.747976	+	0.663726i	$0.768974\pi$
$44$	−0.0411660	−0.00620601
$45$	−4.83758	−0.721144
$46$	−2.81157	−0.414544
$47$	5.77836	0.842860	0.421430	−	0.906861i	$-0.361528\pi$
$47$	5.77836	0.842860	0.421430	+	0.906861i	$0.361528\pi$
$48$	−2.07802	−0.299936
$49$	4.90639	0.700914
$50$	8.46842	1.19761
$51$	8.29742	1.16187
$52$	−1.28748	−0.178542
$53$	7.93141	1.08946	0.544731	−	0.838611i	$-0.316632\pi$
$53$	7.93141	1.08946	0.544731	+	0.838611i	$0.316632\pi$
$54$	3.49488	0.475594
$55$	0.151077	0.0203712
$56$	−3.45056	−0.461101
$57$	−9.02591	−1.19551
$58$	−2.72765	−0.358158
$59$	−4.48136	−0.583424	−0.291712	−	0.956506i	$-0.594225\pi$
$59$	−4.48136	−0.583424	−0.291712	+	0.956506i	$0.594225\pi$
$60$	7.62619	0.984538
$61$	7.09875	0.908902	0.454451	−	0.890772i	$-0.349836\pi$
$61$	7.09875	0.908902	0.454451	+	0.890772i	$0.349836\pi$
$62$	8.98284	1.14082
$63$	−4.54842	−0.573047
$64$	1.00000	0.125000
$65$	4.72498	0.586062
$66$	0.0855438	0.0105297
$67$	−10.4509	−1.27678	−0.638391	−	0.769713i	$-0.720400\pi$
$67$	−10.4509	−1.27678	−0.638391	+	0.769713i	$0.720400\pi$
$68$	−3.99294	−0.484216
$69$	5.84250	0.703354
$70$	12.6633	1.51356
$71$	7.58055	0.899646	0.449823	−	0.893118i	$-0.351487\pi$
$71$	7.58055	0.899646	0.449823	+	0.893118i	$0.351487\pi$
$72$	1.31817	0.155347
$73$	−3.83122	−0.448410	−0.224205	−	0.974542i	$-0.571979\pi$
$73$	−3.83122	−0.448410	−0.224205	+	0.974542i	$0.571979\pi$
$74$	8.77471	1.02004
$75$	−17.5975	−2.03199
$76$	4.34352	0.498235
$77$	0.142046	0.0161876
$78$	2.67542	0.302931
$79$	−1.20276	−0.135321	−0.0676604	−	0.997708i	$-0.521553\pi$
$79$	−1.20276	−0.135321	−0.0676604	+	0.997708i	$0.521553\pi$
$80$	−3.66993	−0.410311
$81$	−11.2169	−1.24633
$82$	0.445901	0.0492415
$83$	4.24142	0.465556	0.232778	−	0.972530i	$-0.425218\pi$
$83$	4.24142	0.465556	0.232778	+	0.972530i	$0.425218\pi$
$84$	7.17034	0.782348
$85$	14.6538	1.58943
$86$	−9.80961	−1.05780
$87$	5.66811	0.607685
$88$	−0.0411660	−0.00438831
$89$	7.73181	0.819570	0.409785	−	0.912182i	$-0.365604\pi$
$89$	7.73181	0.819570	0.409785	+	0.912182i	$0.365604\pi$
$90$	−4.83758	−0.509926
$91$	4.44255	0.465706
$92$	−2.81157	−0.293127
$93$	−18.6665	−1.93563
$94$	5.77836	0.595992
$95$	−15.9404	−1.63545
$96$	−2.07802	−0.212087
$97$	12.0758	1.22611	0.613057	−	0.790038i	$-0.289939\pi$
$97$	12.0758	1.22611	0.613057	+	0.790038i	$0.289939\pi$
$98$	4.90639	0.495621
$99$	−0.0542636	−0.00545370
$100$	8.46842	0.846842
$101$	5.35286	0.532630	0.266315	−	0.963886i	$-0.414194\pi$
$101$	5.35286	0.532630	0.266315	+	0.963886i	$0.414194\pi$
$102$	8.29742	0.821567
$103$	−17.2376	−1.69847	−0.849236	−	0.528014i	$-0.822937\pi$
$103$	−17.2376	−1.69847	−0.849236	+	0.528014i	$0.822937\pi$
$104$	−1.28748	−0.126248
$105$	−26.3147	−2.56805
$106$	7.93141	0.770366
$107$	6.06603	0.586425	0.293212	−	0.956047i	$-0.405276\pi$
$107$	6.06603	0.586425	0.293212	+	0.956047i	$0.405276\pi$
$108$	3.49488	0.336295
$109$	−17.9980	−1.72390	−0.861949	−	0.506995i	$-0.830756\pi$
$109$	−17.9980	−1.72390	−0.861949	+	0.506995i	$0.830756\pi$
$110$	0.151077	0.0144046
$111$	−18.2340	−1.73070
$112$	−3.45056	−0.326048
$113$	7.54476	0.709751	0.354876	−	0.934914i	$-0.384523\pi$
$113$	7.54476	0.709751	0.354876	+	0.934914i	$0.384523\pi$
$114$	−9.02591	−0.845354
$115$	10.3183	0.962185
$116$	−2.72765	−0.253256
$117$	−1.69712	−0.156899
$118$	−4.48136	−0.412543
$119$	13.7779	1.26302
$120$	7.62619	0.696173
$121$	−10.9983	−0.999846
$122$	7.09875	0.642691
$123$	−0.926591	−0.0835479
$124$	8.98284	0.806683
$125$	−12.7289	−1.13850
$126$	−4.54842	−0.405205
$127$	−14.4684	−1.28386	−0.641932	−	0.766762i	$-0.721867\pi$
$127$	−14.4684	−1.28386	−0.641932	+	0.766762i	$0.721867\pi$
$128$	1.00000	0.0883883
$129$	20.3846	1.79476
$130$	4.72498	0.414408
$131$	13.9591	1.21961	0.609806	−	0.792551i	$-0.291247\pi$
$131$	13.9591	1.21961	0.609806	+	0.792551i	$0.291247\pi$
$132$	0.0855438	0.00744563
$133$	−14.9876	−1.29959
$134$	−10.4509	−0.902821
$135$	−12.8260	−1.10389
$136$	−3.99294	−0.342392
$137$	17.7070	1.51282	0.756408	−	0.654101i	$-0.226953\pi$
$137$	17.7070	1.51282	0.756408	+	0.654101i	$0.226953\pi$
$138$	5.84250	0.497347
$139$	−10.8347	−0.918990	−0.459495	−	0.888180i	$-0.651970\pi$
$139$	−10.8347	−0.918990	−0.459495	+	0.888180i	$0.651970\pi$
$140$	12.6633	1.07025
$141$	−12.0075	−1.01122
$142$	7.58055	0.636146
$143$	0.0530006	0.00443213
$144$	1.31817	0.109847
$145$	10.0103	0.831310
$146$	−3.83122	−0.317074
$147$	−10.1956	−0.840918
$148$	8.77471	0.721277
$149$	−13.2508	−1.08555	−0.542774	−	0.839879i	$-0.682626\pi$
$149$	−13.2508	−1.08555	−0.542774	+	0.839879i	$0.682626\pi$
$150$	−17.5975	−1.43683
$151$	10.1082	0.822592	0.411296	−	0.911502i	$-0.365076\pi$
$151$	10.1082	0.822592	0.411296	+	0.911502i	$0.365076\pi$
$152$	4.34352	0.352306
$153$	−5.26336	−0.425518
$154$	0.142046	0.0114464
$155$	−32.9664	−2.64793
$156$	2.67542	0.214205
$157$	3.55120	0.283416	0.141708	−	0.989908i	$-0.454741\pi$
$157$	3.55120	0.283416	0.141708	+	0.989908i	$0.454741\pi$
$158$	−1.20276	−0.0956862
$159$	−16.4816	−1.30708
$160$	−3.66993	−0.290134
$161$	9.70151	0.764586
$162$	−11.2169	−0.881286
$163$	2.35726	0.184635	0.0923175	−	0.995730i	$-0.470573\pi$
$163$	2.35726	0.184635	0.0923175	+	0.995730i	$0.470573\pi$
$164$	0.445901	0.0348190
$165$	−0.313940	−0.0244402
$166$	4.24142	0.329198
$167$	−1.22613	−0.0948810	−0.0474405	−	0.998874i	$-0.515106\pi$
$167$	−1.22613	−0.0948810	−0.0474405	+	0.998874i	$0.515106\pi$
$168$	7.17034	0.553204
$169$	−11.3424	−0.872491
$170$	14.6538	1.12390
$171$	5.72547	0.437838
$172$	−9.80961	−0.747976
$173$	−13.1214	−0.997599	−0.498800	−	0.866717i	$-0.666226\pi$
$173$	−13.1214	−0.997599	−0.498800	+	0.866717i	$0.666226\pi$
$174$	5.66811	0.429698
$175$	−29.2208	−2.20889
$176$	−0.0411660	−0.00310300
$177$	9.31236	0.699960
$178$	7.73181	0.579523
$179$	−3.60509	−0.269457	−0.134729	−	0.990883i	$-0.543016\pi$
$179$	−3.60509	−0.269457	−0.134729	+	0.990883i	$0.543016\pi$
$180$	−4.83758	−0.360572
$181$	11.5782	0.860603	0.430301	−	0.902685i	$-0.358407\pi$
$181$	11.5782	0.860603	0.430301	+	0.902685i	$0.358407\pi$
$182$	4.44255	0.329304
$183$	−14.7513	−1.09045
$184$	−2.81157	−0.207272
$185$	−32.2026	−2.36758
$186$	−18.6665	−1.36870
$187$	0.164374	0.0120202
$188$	5.77836	0.421430
$189$	−12.0593	−0.877187
$190$	−15.9404	−1.15644
$191$	−5.72854	−0.414503	−0.207251	−	0.978288i	$-0.566452\pi$
$191$	−5.72854	−0.414503	−0.207251	+	0.978288i	$0.566452\pi$
$192$	−2.07802	−0.149968
$193$	−13.8501	−0.996955	−0.498478	−	0.866903i	$-0.666107\pi$
$193$	−13.8501	−0.996955	−0.498478	+	0.866903i	$0.666107\pi$
$194$	12.0758	0.866994
$195$	−9.81861	−0.703125
$196$	4.90639	0.350457
$197$	7.70435	0.548913	0.274456	−	0.961600i	$-0.411502\pi$
$197$	7.70435	0.548913	0.274456	+	0.961600i	$0.411502\pi$
$198$	−0.0542636	−0.00385635
$199$	−0.384256	−0.0272392	−0.0136196	−	0.999907i	$-0.504335\pi$
$199$	−0.384256	−0.0272392	−0.0136196	+	0.999907i	$0.504335\pi$
$200$	8.46842	0.598807
$201$	21.7172	1.53181
$202$	5.35286	0.376626
$203$	9.41193	0.660588
$204$	8.29742	0.580935
$205$	−1.63643	−0.114293
$206$	−17.2376	−1.20100
$207$	−3.70612	−0.257593
$208$	−1.28748	−0.0892710
$209$	−0.178805	−0.0123682
$210$	−26.3147	−1.81589
$211$	21.3716	1.47129	0.735643	−	0.677370i	$-0.236880\pi$
$211$	21.3716	1.47129	0.735643	+	0.677370i	$0.236880\pi$
$212$	7.93141	0.544731
$213$	−15.7525	−1.07935
$214$	6.06603	0.414665
$215$	36.0006	2.45522
$216$	3.49488	0.237797
$217$	−30.9959	−2.10414
$218$	−17.9980	−1.21898
$219$	7.96135	0.537978
$220$	0.151077	0.0101856
$221$	5.14085	0.345811
$222$	−18.2340	−1.22379
$223$	−7.32606	−0.490589	−0.245295	−	0.969449i	$-0.578885\pi$
$223$	−7.32606	−0.490589	−0.245295	+	0.969449i	$0.578885\pi$
$224$	−3.45056	−0.230551
$225$	11.1628	0.744185
$226$	7.54476	0.501870
$227$	3.35917	0.222956	0.111478	−	0.993767i	$-0.464442\pi$
$227$	3.35917	0.222956	0.111478	+	0.993767i	$0.464442\pi$
$228$	−9.02591	−0.597756
$229$	4.02544	0.266008	0.133004	−	0.991115i	$-0.457538\pi$
$229$	4.02544	0.266008	0.133004	+	0.991115i	$0.457538\pi$
$230$	10.3183	0.680367
$231$	−0.295174	−0.0194210
$232$	−2.72765	−0.179079
$233$	2.12563	0.139255	0.0696273	−	0.997573i	$-0.477819\pi$
$233$	2.12563	0.139255	0.0696273	+	0.997573i	$0.477819\pi$
$234$	−1.69712	−0.110944
$235$	−21.2062	−1.38334
$236$	−4.48136	−0.291712
$237$	2.49935	0.162350
$238$	13.7779	0.893089
$239$	0.817349	0.0528699	0.0264350	−	0.999651i	$-0.491585\pi$
$239$	0.817349	0.0528699	0.0264350	+	0.999651i	$0.491585\pi$
$240$	7.62619	0.492269
$241$	25.8208	1.66326	0.831631	−	0.555329i	$-0.187407\pi$
$241$	25.8208	1.66326	0.831631	+	0.555329i	$0.187407\pi$
$242$	−10.9983	−0.706998
$243$	12.8244	0.822683
$244$	7.09875	0.454451
$245$	−18.0061	−1.15037
$246$	−0.926591	−0.0590773
$247$	−5.59221	−0.355824
$248$	8.98284	0.570411
$249$	−8.81375	−0.558549
$250$	−12.7289	−0.805043
$251$	19.2137	1.21276	0.606380	−	0.795175i	$-0.292621\pi$
$251$	19.2137	1.21276	0.606380	+	0.795175i	$0.292621\pi$
$252$	−4.54842	−0.286523
$253$	0.115741	0.00727658
$254$	−14.4684	−0.907828
$255$	−30.4510	−1.90691
$256$	1.00000	0.0625000
$257$	20.3353	1.26848	0.634242	−	0.773135i	$-0.281312\pi$
$257$	20.3353	1.26848	0.634242	+	0.773135i	$0.281312\pi$
$258$	20.3846	1.26909
$259$	−30.2777	−1.88137
$260$	4.72498	0.293031
$261$	−3.59550	−0.222556
$262$	13.9591	0.862396
$263$	16.2149	0.999851	0.499925	−	0.866069i	$-0.333361\pi$
$263$	16.2149	0.999851	0.499925	+	0.866069i	$0.333361\pi$
$264$	0.0855438	0.00526486
$265$	−29.1077	−1.78807
$266$	−14.9876	−0.918948
$267$	−16.0668	−0.983275
$268$	−10.4509	−0.638391
$269$	0.420135	0.0256161	0.0128080	−	0.999918i	$-0.495923\pi$
$269$	0.420135	0.0256161	0.0128080	+	0.999918i	$0.495923\pi$
$270$	−12.8260	−0.780565
$271$	−23.0699	−1.40140	−0.700698	−	0.713458i	$-0.747128\pi$
$271$	−23.0699	−1.40140	−0.700698	+	0.713458i	$0.747128\pi$
$272$	−3.99294	−0.242108
$273$	−9.23170	−0.558728
$274$	17.7070	1.06972
$275$	−0.348611	−0.0210220
$276$	5.84250	0.351677
$277$	−26.8417	−1.61276	−0.806381	−	0.591396i	$-0.798577\pi$
$277$	−26.8417	−1.61276	−0.806381	+	0.591396i	$0.798577\pi$
$278$	−10.8347	−0.649824
$279$	11.8409	0.708895
$280$	12.6633	0.756780
$281$	−17.3013	−1.03211	−0.516055	−	0.856556i	$-0.672600\pi$
$281$	−17.3013	−1.03211	−0.516055	+	0.856556i	$0.672600\pi$
$282$	−12.0075	−0.715039
$283$	−33.2822	−1.97842	−0.989212	−	0.146492i	$-0.953202\pi$
$283$	−33.2822	−1.97842	−0.989212	+	0.146492i	$0.953202\pi$
$284$	7.58055	0.449823
$285$	33.1245	1.96213
$286$	0.0530006	0.00313399
$287$	−1.53861	−0.0908213
$288$	1.31817	0.0776737
$289$	−1.05639	−0.0621407
$290$	10.0103	0.587825
$291$	−25.0938	−1.47103
$292$	−3.83122	−0.224205
$293$	6.49404	0.379386	0.189693	−	0.981843i	$-0.439251\pi$
$293$	6.49404	0.379386	0.189693	+	0.981843i	$0.439251\pi$
$294$	−10.1956	−0.594619
$295$	16.4463	0.957541
$296$	8.77471	0.510020
$297$	−0.143870	−0.00834821
$298$	−13.2508	−0.767599
$299$	3.61985	0.209342
$300$	−17.5975	−1.01599
$301$	33.8487	1.95101
$302$	10.1082	0.581661
$303$	−11.1234	−0.639020
$304$	4.34352	0.249118
$305$	−26.0519	−1.49173
$306$	−5.26336	−0.300886
$307$	−6.26311	−0.357455	−0.178727	−	0.983899i	$-0.557198\pi$
$307$	−6.26311	−0.357455	−0.178727	+	0.983899i	$0.557198\pi$
$308$	0.142046	0.00809382
$309$	35.8201	2.03773
$310$	−32.9664	−1.87237
$311$	−27.9835	−1.58680	−0.793400	−	0.608701i	$-0.791691\pi$
$311$	−27.9835	−1.58680	−0.793400	+	0.608701i	$0.791691\pi$
$312$	2.67542	0.151466
$313$	2.37282	0.134119	0.0670597	−	0.997749i	$-0.478638\pi$
$313$	2.37282	0.134119	0.0670597	+	0.997749i	$0.478638\pi$
$314$	3.55120	0.200406
$315$	16.6924	0.940509
$316$	−1.20276	−0.0676604
$317$	0.321234	0.0180423	0.00902116	−	0.999959i	$-0.497128\pi$
$317$	0.321234	0.0180423	0.00902116	+	0.999959i	$0.497128\pi$
$318$	−16.4816	−0.924243
$319$	0.112286	0.00628684
$320$	−3.66993	−0.205156
$321$	−12.6053	−0.703560
$322$	9.70151	0.540644
$323$	−17.3434	−0.965014
$324$	−11.2169	−0.623163
$325$	−10.9030	−0.604787
$326$	2.35726	0.130557
$327$	37.4002	2.06824
$328$	0.445901	0.0246208
$329$	−19.9386	−1.09925
$330$	−0.313940	−0.0172818
$331$	21.4885	1.18111	0.590557	−	0.806996i	$-0.298908\pi$
$331$	21.4885	1.18111	0.590557	+	0.806996i	$0.298908\pi$
$332$	4.24142	0.232778
$333$	11.5665	0.633842
$334$	−1.22613	−0.0670910
$335$	38.3541	2.09551
$336$	7.17034	0.391174
$337$	−25.7783	−1.40423	−0.702117	−	0.712061i	$-0.747762\pi$
$337$	−25.7783	−1.40423	−0.702117	+	0.712061i	$0.747762\pi$
$338$	−11.3424	−0.616944
$339$	−15.6782	−0.851521
$340$	14.6538	0.794716
$341$	−0.369788	−0.0200251
$342$	5.72547	0.309598
$343$	7.22412	0.390066
$344$	−9.80961	−0.528899
$345$	−21.4416	−1.15438
$346$	−13.1214	−0.705409
$347$	−7.15996	−0.384367	−0.192183	−	0.981359i	$-0.561557\pi$
$347$	−7.15996	−0.384367	−0.192183	+	0.981359i	$0.561557\pi$
$348$	5.66811	0.303843
$349$	−13.5643	−0.726082	−0.363041	−	0.931773i	$-0.618261\pi$
$349$	−13.5643	−0.726082	−0.363041	+	0.931773i	$0.618261\pi$
$350$	−29.2208	−1.56192
$351$	−4.49961	−0.240171
$352$	−0.0411660	−0.00219416
$353$	−8.32890	−0.443303	−0.221651	−	0.975126i	$-0.571145\pi$
$353$	−8.32890	−0.443303	−0.221651	+	0.975126i	$0.571145\pi$
$354$	9.31236	0.494947
$355$	−27.8201	−1.47654
$356$	7.73181	0.409785
$357$	−28.6308	−1.51530
$358$	−3.60509	−0.190535
$359$	29.6768	1.56628	0.783140	−	0.621845i	$-0.213617\pi$
$359$	29.6768	1.56628	0.783140	+	0.621845i	$0.213617\pi$
$360$	−4.83758	−0.254963
$361$	−0.133871	−0.00704582
$362$	11.5782	0.608538
$363$	22.8547	1.19956
$364$	4.44255	0.232853
$365$	14.0603	0.735951
$366$	−14.7513	−0.771065
$367$	−0.0919967	−0.00480219	−0.00240110	−	0.999997i	$-0.500764\pi$
$367$	−0.0919967	−0.00480219	−0.00240110	+	0.999997i	$0.500764\pi$
$368$	−2.81157	−0.146563
$369$	0.587771	0.0305982
$370$	−32.2026	−1.67413
$371$	−27.3678	−1.42087
$372$	−18.6665	−0.967814
$373$	−21.5505	−1.11584	−0.557921	−	0.829894i	$-0.688401\pi$
$373$	−21.5505	−1.11584	−0.557921	+	0.829894i	$0.688401\pi$
$374$	0.164374	0.00849956
$375$	26.4508	1.36591
$376$	5.77836	0.297996
$377$	3.51181	0.180867
$378$	−12.0593	−0.620265
$379$	26.9935	1.38656	0.693281	−	0.720667i	$-0.256164\pi$
$379$	26.9935	1.38656	0.693281	+	0.720667i	$0.256164\pi$
$380$	−15.9404	−0.817726
$381$	30.0656	1.54031
$382$	−5.72854	−0.293098
$383$	−7.01748	−0.358576	−0.179288	−	0.983797i	$-0.557379\pi$
$383$	−7.01748	−0.358576	−0.179288	+	0.983797i	$0.557379\pi$
$384$	−2.07802	−0.106043
$385$	−0.521299	−0.0265679
$386$	−13.8501	−0.704954
$387$	−12.9307	−0.657304
$388$	12.0758	0.613057
$389$	14.4447	0.732373	0.366187	−	0.930541i	$-0.380663\pi$
$389$	14.4447	0.732373	0.366187	+	0.930541i	$0.380663\pi$
$390$	−9.81861	−0.497184
$391$	11.2264	0.567746
$392$	4.90639	0.247810
$393$	−29.0073	−1.46322
$394$	7.70435	0.388140
$395$	4.41404	0.222094
$396$	−0.0542636	−0.00272685
$397$	−20.9947	−1.05369	−0.526847	−	0.849960i	$-0.676626\pi$
$397$	−20.9947	−1.05369	−0.526847	+	0.849960i	$0.676626\pi$
$398$	−0.384256	−0.0192610
$399$	31.1445	1.55917
$400$	8.46842	0.423421
$401$	−30.2401	−1.51012	−0.755060	−	0.655655i	$-0.772393\pi$
$401$	−30.2401	−1.51012	−0.755060	+	0.655655i	$0.772393\pi$
$402$	21.7172	1.08315
$403$	−11.5653	−0.576107
$404$	5.35286	0.266315
$405$	41.1654	2.04553
$406$	9.41193	0.467106
$407$	−0.361220	−0.0179050
$408$	8.29742	0.410783
$409$	4.69838	0.232320	0.116160	−	0.993231i	$-0.462941\pi$
$409$	4.69838	0.232320	0.116160	+	0.993231i	$0.462941\pi$
$410$	−1.63643	−0.0808174
$411$	−36.7956	−1.81499
$412$	−17.2376	−0.849236
$413$	15.4632	0.760896
$414$	−3.70612	−0.182146
$415$	−15.5657	−0.764091
$416$	−1.28748	−0.0631241
$417$	22.5148	1.10255
$418$	−0.178805	−0.00874565
$419$	27.6248	1.34956	0.674779	−	0.738020i	$-0.264239\pi$
$419$	27.6248	1.34956	0.674779	+	0.738020i	$0.264239\pi$
$420$	−26.3147	−1.28402
$421$	−4.20494	−0.204936	−0.102468	−	0.994736i	$-0.532674\pi$
$421$	−4.20494	−0.204936	−0.102468	+	0.994736i	$0.532674\pi$
$422$	21.3716	1.04036
$423$	7.61683	0.370343
$424$	7.93141	0.385183
$425$	−33.8139	−1.64022
$426$	−15.7525	−0.763213
$427$	−24.4947	−1.18538
$428$	6.06603	0.293212
$429$	−0.110136	−0.00531743
$430$	36.0006	1.73610
$431$	27.4258	1.32105	0.660527	−	0.750803i	$-0.270333\pi$
$431$	27.4258	1.32105	0.660527	+	0.750803i	$0.270333\pi$
$432$	3.49488	0.168148
$433$	−17.7193	−0.851534	−0.425767	−	0.904833i	$-0.639996\pi$
$433$	−17.7193	−0.851534	−0.425767	+	0.904833i	$0.639996\pi$
$434$	−30.9959	−1.48785
$435$	−20.8016	−0.997360
$436$	−17.9980	−0.861949
$437$	−12.2121	−0.584184
$438$	7.96135	0.380408
$439$	−11.2775	−0.538247	−0.269124	−	0.963106i	$-0.586734\pi$
$439$	−11.2775	−0.538247	−0.269124	+	0.963106i	$0.586734\pi$
$440$	0.151077	0.00720229
$441$	6.46744	0.307973
$442$	5.14085	0.244526
$443$	6.29496	0.299083	0.149541	−	0.988755i	$-0.452220\pi$
$443$	6.29496	0.299083	0.149541	+	0.988755i	$0.452220\pi$
$444$	−18.2340	−0.865349
$445$	−28.3752	−1.34511
$446$	−7.32606	−0.346899
$447$	27.5354	1.30238
$448$	−3.45056	−0.163024
$449$	30.4039	1.43485	0.717425	−	0.696636i	$-0.245321\pi$
$449$	30.4039	1.43485	0.717425	+	0.696636i	$0.245321\pi$
$450$	11.1628	0.526218
$451$	−0.0183560	−0.000864348 0
$452$	7.54476	0.354876
$453$	−21.0050	−0.986901
$454$	3.35917	0.157653
$455$	−16.3039	−0.764337
$456$	−9.02591	−0.422677
$457$	−35.3996	−1.65592	−0.827962	−	0.560785i	$-0.810500\pi$
$457$	−35.3996	−1.65592	−0.827962	+	0.560785i	$0.810500\pi$
$458$	4.02544	0.188096
$459$	−13.9549	−0.651358
$460$	10.3183	0.481092
$461$	0.985230	0.0458867	0.0229434	−	0.999737i	$-0.492696\pi$
$461$	0.985230	0.0458867	0.0229434	+	0.999737i	$0.492696\pi$
$462$	−0.295174	−0.0137328
$463$	−31.2127	−1.45058	−0.725288	−	0.688446i	$-0.758293\pi$
$463$	−31.2127	−1.45058	−0.725288	+	0.688446i	$0.758293\pi$
$464$	−2.72765	−0.126628
$465$	68.5049	3.17684
$466$	2.12563	0.0984679
$467$	−41.4673	−1.91888	−0.959440	−	0.281913i	$-0.909031\pi$
$467$	−41.4673	−1.91888	−0.959440	+	0.281913i	$0.909031\pi$
$468$	−1.69712	−0.0784493
$469$	36.0615	1.66517
$470$	−21.2062	−0.978169
$471$	−7.37946	−0.340027
$472$	−4.48136	−0.206272
$473$	0.403823	0.0185678
$474$	2.49935	0.114799
$475$	36.7827	1.68771
$476$	13.7779	0.631510
$477$	10.4549	0.478697
$478$	0.817349	0.0373847
$479$	25.1228	1.14789	0.573946	−	0.818893i	$-0.305412\pi$
$479$	25.1228	1.14789	0.573946	+	0.818893i	$0.305412\pi$
$480$	7.62619	0.348087
$481$	−11.2973	−0.515113
$482$	25.8208	1.17610
$483$	−20.1599	−0.917308
$484$	−10.9983	−0.499923
$485$	−44.3175	−2.01235
$486$	12.8244	0.581725
$487$	−1.05337	−0.0477326	−0.0238663	−	0.999715i	$-0.507598\pi$
$487$	−1.05337	−0.0477326	−0.0238663	+	0.999715i	$0.507598\pi$
$488$	7.09875	0.321345
$489$	−4.89844	−0.221515
$490$	−18.0061	−0.813435
$491$	−11.2134	−0.506054	−0.253027	−	0.967459i	$-0.581426\pi$
$491$	−11.2134	−0.506054	−0.253027	+	0.967459i	$0.581426\pi$
$492$	−0.926591	−0.0417739
$493$	10.8914	0.490522
$494$	−5.59221	−0.251605
$495$	0.199144	0.00895085
$496$	8.98284	0.403342
$497$	−26.1572	−1.17331
$498$	−8.81375	−0.394954
$499$	14.7997	0.662524	0.331262	−	0.943539i	$-0.392526\pi$
$499$	14.7997	0.662524	0.331262	+	0.943539i	$0.392526\pi$
$500$	−12.7289	−0.569252
$501$	2.54793	0.113833
$502$	19.2137	0.857551
$503$	31.2459	1.39319	0.696594	−	0.717466i	$-0.254698\pi$
$503$	31.2459	1.39319	0.696594	+	0.717466i	$0.254698\pi$
$504$	−4.54842	−0.202603
$505$	−19.6447	−0.874176
$506$	0.115741	0.00514532
$507$	23.5697	1.04677
$508$	−14.4684	−0.641932
$509$	−17.5494	−0.777861	−0.388931	−	0.921267i	$-0.627155\pi$
$509$	−17.5494	−0.777861	−0.388931	+	0.921267i	$0.627155\pi$
$510$	−30.4510	−1.34839
$511$	13.2199	0.584812
$512$	1.00000	0.0441942
$513$	15.1801	0.670217
$514$	20.3353	0.896953
$515$	63.2609	2.78761
$516$	20.3846	0.897381
$517$	−0.237872	−0.0104616
$518$	−30.2777	−1.33033
$519$	27.2665	1.19687
$520$	4.72498	0.207204
$521$	−22.8519	−1.00116	−0.500580	−	0.865690i	$-0.666880\pi$
$521$	−22.8519	−1.00116	−0.500580	+	0.865690i	$0.666880\pi$
$522$	−3.59550	−0.157371
$523$	30.8207	1.34769	0.673847	−	0.738871i	$-0.264641\pi$
$523$	30.8207	1.34769	0.673847	+	0.738871i	$0.264641\pi$
$524$	13.9591	0.609806
$525$	60.7214	2.65010
$526$	16.2149	0.707001
$527$	−35.8680	−1.56243
$528$	0.0855438	0.00372281
$529$	−15.0951	−0.656307
$530$	−29.1077	−1.26436
$531$	−5.90718	−0.256350
$532$	−14.9876	−0.649794
$533$	−0.574090	−0.0248666
$534$	−16.0668	−0.695280
$535$	−22.2619	−0.962466
$536$	−10.4509	−0.451410
$537$	7.49145	0.323280
$538$	0.420135	0.0181133
$539$	−0.201977	−0.00869975
$540$	−12.8260	−0.551943
$541$	20.3551	0.875134	0.437567	−	0.899186i	$-0.355840\pi$
$541$	20.3551	0.875134	0.437567	+	0.899186i	$0.355840\pi$
$542$	−23.0699	−0.990936
$543$	−24.0598	−1.03250
$544$	−3.99294	−0.171196
$545$	66.0516	2.82934
$546$	−9.23170	−0.395080
$547$	20.3339	0.869416	0.434708	−	0.900571i	$-0.356852\pi$
$547$	20.3339	0.869416	0.434708	+	0.900571i	$0.356852\pi$
$548$	17.7070	0.756408
$549$	9.35733	0.399361
$550$	−0.348611	−0.0148648
$551$	−11.8476	−0.504724
$552$	5.84250	0.248673
$553$	4.15019	0.176484
$554$	−26.8417	−1.14039
$555$	66.9177	2.84050
$556$	−10.8347	−0.459495
$557$	−45.6190	−1.93294	−0.966469	−	0.256782i	$-0.917338\pi$
$557$	−45.6190	−1.93294	−0.966469	+	0.256782i	$0.917338\pi$
$558$	11.8409	0.501264
$559$	12.6297	0.534180
$560$	12.6633	0.535124
$561$	−0.341572	−0.0144212
$562$	−17.3013	−0.729812
$563$	29.8794	1.25926	0.629632	−	0.776893i	$-0.283206\pi$
$563$	29.8794	1.25926	0.629632	+	0.776893i	$0.283206\pi$
$564$	−12.0075	−0.505609
$565$	−27.6888	−1.16488
$566$	−33.2822	−1.39896
$567$	38.7048	1.62545
$568$	7.58055	0.318073
$569$	−29.2356	−1.22562	−0.612811	−	0.790229i	$-0.709961\pi$
$569$	−29.2356	−1.22562	−0.612811	+	0.790229i	$0.709961\pi$
$570$	33.1245	1.38743
$571$	−20.3269	−0.850655	−0.425327	−	0.905040i	$-0.639841\pi$
$571$	−20.3269	−0.850655	−0.425327	+	0.905040i	$0.639841\pi$
$572$	0.0530006	0.00221607
$573$	11.9040	0.497298
$574$	−1.53861	−0.0642203
$575$	−23.8096	−0.992927
$576$	1.31817	0.0549236
$577$	10.7226	0.446386	0.223193	−	0.974774i	$-0.428352\pi$
$577$	10.7226	0.446386	0.223193	+	0.974774i	$0.428352\pi$
$578$	−1.05639	−0.0439401
$579$	28.7809	1.19609
$580$	10.0103	0.415655
$581$	−14.6353	−0.607174
$582$	−25.0938	−1.04017
$583$	−0.326504	−0.0135224
$584$	−3.83122	−0.158537
$585$	6.22831	0.257509
$586$	6.49404	0.268266
$587$	5.29535	0.218563	0.109281	−	0.994011i	$-0.465145\pi$
$587$	5.29535	0.218563	0.109281	+	0.994011i	$0.465145\pi$
$588$	−10.1956	−0.420459
$589$	39.0171	1.60767
$590$	16.4463	0.677084
$591$	−16.0098	−0.658555
$592$	8.77471	0.360639
$593$	19.4576	0.799027	0.399513	−	0.916727i	$-0.369179\pi$
$593$	19.4576	0.799027	0.399513	+	0.916727i	$0.369179\pi$
$594$	−0.143870	−0.00590308
$595$	−50.5640	−2.07292
$596$	−13.2508	−0.542774
$597$	0.798492	0.0326801
$598$	3.61985	0.148027
$599$	18.2888	0.747259	0.373629	−	0.927578i	$-0.378113\pi$
$599$	18.2888	0.747259	0.373629	+	0.927578i	$0.378113\pi$
$600$	−17.5975	−0.718416
$601$	−0.625965	−0.0255337	−0.0127668	−	0.999919i	$-0.504064\pi$
$601$	−0.625965	−0.0255337	−0.0127668	+	0.999919i	$0.504064\pi$
$602$	33.8487	1.37957
$603$	−13.7760	−0.561003
$604$	10.1082	0.411296
$605$	40.3631	1.64099
$606$	−11.1234	−0.451855
$607$	8.69823	0.353050	0.176525	−	0.984296i	$-0.443514\pi$
$607$	8.69823	0.353050	0.176525	+	0.984296i	$0.443514\pi$
$608$	4.34352	0.176153
$609$	−19.5582	−0.792538
$610$	−26.0519	−1.05481
$611$	−7.43955	−0.300972
$612$	−5.26336	−0.212759
$613$	28.3724	1.14595	0.572975	−	0.819573i	$-0.305789\pi$
$613$	28.3724	1.14595	0.572975	+	0.819573i	$0.305789\pi$
$614$	−6.26311	−0.252759
$615$	3.40053	0.137122
$616$	0.142046	0.00572319
$617$	16.5979	0.668208	0.334104	−	0.942536i	$-0.391566\pi$
$617$	16.5979	0.668208	0.334104	+	0.942536i	$0.391566\pi$
$618$	35.8201	1.44089
$619$	41.6792	1.67523	0.837614	−	0.546263i	$-0.183950\pi$
$619$	41.6792	1.67523	0.837614	+	0.546263i	$0.183950\pi$
$620$	−32.9664	−1.32396
$621$	−9.82612	−0.394308
$622$	−27.9835	−1.12204
$623$	−26.6791	−1.06888
$624$	2.67542	0.107102
$625$	4.37198	0.174879
$626$	2.37282	0.0948368
$627$	0.371561	0.0148387
$628$	3.55120	0.141708
$629$	−35.0370	−1.39701
$630$	16.6924	0.665041
$631$	−23.1327	−0.920897	−0.460449	−	0.887686i	$-0.652311\pi$
$631$	−23.1327	−0.920897	−0.460449	+	0.887686i	$0.652311\pi$
$632$	−1.20276	−0.0478431
$633$	−44.4107	−1.76517
$634$	0.321234	0.0127578
$635$	53.0981	2.10713
$636$	−16.4816	−0.653539
$637$	−6.31691	−0.250285
$638$	0.112286	0.00444546
$639$	9.99242	0.395294
$640$	−3.66993	−0.145067
$641$	−39.9116	−1.57641	−0.788207	−	0.615410i	$-0.788990\pi$
$641$	−39.9116	−1.57641	−0.788207	+	0.615410i	$0.788990\pi$
$642$	−12.6053	−0.497492
$643$	12.5138	0.493496	0.246748	−	0.969080i	$-0.420638\pi$
$643$	12.5138	0.493496	0.246748	+	0.969080i	$0.420638\pi$
$644$	9.70151	0.382293
$645$	−74.8100	−2.94564
$646$	−17.3434	−0.682368
$647$	40.4521	1.59034	0.795169	−	0.606388i	$-0.207382\pi$
$647$	40.4521	1.59034	0.795169	+	0.606388i	$0.207382\pi$
$648$	−11.2169	−0.440643
$649$	0.184480	0.00724147
$650$	−10.9030	−0.427649
$651$	64.4100	2.52443
$652$	2.35726	0.0923175
$653$	−42.5996	−1.66705	−0.833526	−	0.552480i	$-0.813681\pi$
$653$	−42.5996	−1.66705	−0.833526	+	0.552480i	$0.813681\pi$
$654$	37.4002	1.46247
$655$	−51.2290	−2.00168
$656$	0.445901	0.0174095
$657$	−5.05018	−0.197026
$658$	−19.9386	−0.777287
$659$	7.20073	0.280501	0.140250	−	0.990116i	$-0.455209\pi$
$659$	7.20073	0.280501	0.140250	+	0.990116i	$0.455209\pi$
$660$	−0.313940	−0.0122201
$661$	5.49090	0.213571	0.106786	−	0.994282i	$-0.465944\pi$
$661$	5.49090	0.213571	0.106786	+	0.994282i	$0.465944\pi$
$662$	21.4885	0.835173
$663$	−10.6828	−0.414885
$664$	4.24142	0.164599
$665$	55.0034	2.13294
$666$	11.5665	0.448194
$667$	7.66898	0.296944
$668$	−1.22613	−0.0474405
$669$	15.2237	0.588582
$670$	38.3541	1.48175
$671$	−0.292227	−0.0112813
$672$	7.17034	0.276602
$673$	6.34218	0.244473	0.122237	−	0.992501i	$-0.460993\pi$
$673$	6.34218	0.244473	0.122237	+	0.992501i	$0.460993\pi$
$674$	−25.7783	−0.992944
$675$	29.5961	1.13916
$676$	−11.3424	−0.436246
$677$	−29.6556	−1.13976	−0.569879	−	0.821729i	$-0.693010\pi$
$677$	−29.6556	−1.13976	−0.569879	+	0.821729i	$0.693010\pi$
$678$	−15.6782	−0.602116
$679$	−41.6684	−1.59909
$680$	14.6538	0.561949
$681$	−6.98041	−0.267490
$682$	−0.369788	−0.0141599
$683$	−12.0924	−0.462702	−0.231351	−	0.972870i	$-0.574315\pi$
$683$	−12.0924	−0.462702	−0.231351	+	0.972870i	$0.574315\pi$
$684$	5.72547	0.218919
$685$	−64.9837	−2.48290
$686$	7.22412	0.275818
$687$	−8.36494	−0.319142
$688$	−9.80961	−0.373988
$689$	−10.2116	−0.389029
$690$	−21.4416	−0.816267
$691$	43.8358	1.66759	0.833795	−	0.552074i	$-0.186163\pi$
$691$	43.8358	1.66759	0.833795	+	0.552074i	$0.186163\pi$
$692$	−13.1214	−0.498800
$693$	0.187240	0.00711266
$694$	−7.15996	−0.271788
$695$	39.7627	1.50829
$696$	5.66811	0.214849
$697$	−1.78046	−0.0674396
$698$	−13.5643	−0.513417
$699$	−4.41710	−0.167070
$700$	−29.2208	−1.10444
$701$	−25.8701	−0.977101	−0.488551	−	0.872536i	$-0.662474\pi$
$701$	−25.8701	−0.977101	−0.488551	+	0.872536i	$0.662474\pi$
$702$	−4.49961	−0.169827
$703$	38.1131	1.43746
$704$	−0.0411660	−0.00155150
$705$	44.0669	1.65965
$706$	−8.32890	−0.313462
$707$	−18.4704	−0.694651
$708$	9.31236	0.349980
$709$	16.0301	0.602024	0.301012	−	0.953620i	$-0.402676\pi$
$709$	16.0301	0.602024	0.301012	+	0.953620i	$0.402676\pi$
$710$	−27.8201	−1.04407
$711$	−1.58543	−0.0594584
$712$	7.73181	0.289762
$713$	−25.2559	−0.945841
$714$	−28.6308	−1.07148
$715$	−0.194509	−0.00727421
$716$	−3.60509	−0.134729
$717$	−1.69847	−0.0634304
$718$	29.6768	1.10753
$719$	0.101795	0.00379630	0.00189815	−	0.999998i	$-0.499396\pi$
$719$	0.101795	0.00379630	0.00189815	+	0.999998i	$0.499396\pi$
$720$	−4.83758	−0.180286
$721$	59.4795	2.21513
$722$	−0.133871	−0.00498215
$723$	−53.6561	−1.99549
$724$	11.5782	0.430301
$725$	−23.0989	−0.857871
$726$	22.8547	0.848217
$727$	−12.1973	−0.452374	−0.226187	−	0.974084i	$-0.572626\pi$
$727$	−12.1973	−0.452374	−0.226187	+	0.974084i	$0.572626\pi$
$728$	4.44255	0.164652
$729$	7.00154	0.259316
$730$	14.0603	0.520396
$731$	39.1692	1.44873
$732$	−14.7513	−0.545225
$733$	34.5973	1.27788	0.638940	−	0.769256i	$-0.279373\pi$
$733$	34.5973	1.27788	0.638940	+	0.769256i	$0.279373\pi$
$734$	−0.0919967	−0.00339566
$735$	37.4171	1.38015
$736$	−2.81157	−0.103636
$737$	0.430222	0.0158474
$738$	0.587771	0.0216362
$739$	−20.8959	−0.768668	−0.384334	−	0.923194i	$-0.625569\pi$
$739$	−20.8959	−0.768668	−0.384334	+	0.923194i	$0.625569\pi$
$740$	−32.2026	−1.18379
$741$	11.6207	0.426898
$742$	−27.3678	−1.00470
$743$	−39.9977	−1.46737	−0.733686	−	0.679488i	$-0.762202\pi$
$743$	−39.9977	−1.46737	−0.733686	+	0.679488i	$0.762202\pi$
$744$	−18.6665	−0.684348
$745$	48.6296	1.78165
$746$	−21.5505	−0.789020
$747$	5.59089	0.204560
$748$	0.164374	0.00601009
$749$	−20.9312	−0.764810
$750$	26.4508	0.965847
$751$	19.7539	0.720829	0.360414	−	0.932792i	$-0.382635\pi$
$751$	19.7539	0.720829	0.360414	+	0.932792i	$0.382635\pi$
$752$	5.77836	0.210715
$753$	−39.9265	−1.45500
$754$	3.51181	0.127892
$755$	−37.0964	−1.35007
$756$	−12.0593	−0.438593
$757$	−37.1317	−1.34958	−0.674788	−	0.738012i	$-0.735765\pi$
$757$	−37.1317	−1.34958	−0.674788	+	0.738012i	$0.735765\pi$
$758$	26.9935	0.980447
$759$	−0.240512	−0.00873005
$760$	−15.9404	−0.578220
$761$	39.1816	1.42033	0.710165	−	0.704035i	$-0.248620\pi$
$761$	39.1816	1.42033	0.710165	+	0.704035i	$0.248620\pi$
$762$	30.0656	1.08916
$763$	62.1033	2.24829
$764$	−5.72854	−0.207251
$765$	19.3162	0.698379
$766$	−7.01748	−0.253552
$767$	5.76969	0.208331
$768$	−2.07802	−0.0749841
$769$	27.1506	0.979076	0.489538	−	0.871982i	$-0.337166\pi$
$769$	27.1506	0.979076	0.489538	+	0.871982i	$0.337166\pi$
$770$	−0.521299	−0.0187863
$771$	−42.2572	−1.52186
$772$	−13.8501	−0.498478
$773$	−4.52177	−0.162637	−0.0813185	−	0.996688i	$-0.525913\pi$
$773$	−4.52177	−0.162637	−0.0813185	+	0.996688i	$0.525913\pi$
$774$	−12.9307	−0.464784
$775$	76.0704	2.73253
$776$	12.0758	0.433497
$777$	62.9177	2.25716
$778$	14.4447	0.517866
$779$	1.93678	0.0693923
$780$	−9.81861	−0.351563
$781$	−0.312061	−0.0111664
$782$	11.2264	0.401457
$783$	−9.53282	−0.340675
$784$	4.90639	0.175228
$785$	−13.0327	−0.465155
$786$	−29.0073	−1.03466
$787$	17.5622	0.626023	0.313012	−	0.949749i	$-0.398662\pi$
$787$	17.5622	0.626023	0.313012	+	0.949749i	$0.398662\pi$
$788$	7.70435	0.274456
$789$	−33.6948	−1.19957
$790$	4.41404	0.157044
$791$	−26.0337	−0.925651
$792$	−0.0542636	−0.00192817
$793$	−9.13953	−0.324554
$794$	−20.9947	−0.745074
$795$	60.4864	2.14523
$796$	−0.384256	−0.0136196
$797$	−38.5622	−1.36594	−0.682972	−	0.730445i	$-0.739313\pi$
$797$	−38.5622	−1.36594	−0.682972	+	0.730445i	$0.739313\pi$
$798$	31.1445	1.10250
$799$	−23.0727	−0.816252
$800$	8.46842	0.299404
$801$	10.1918	0.360110
$802$	−30.2401	−1.06782
$803$	0.157716	0.00556567
$804$	21.7172	0.765906
$805$	−35.6039	−1.25487
$806$	−11.5653	−0.407369
$807$	−0.873049	−0.0307328
$808$	5.35286	0.188313
$809$	22.2328	0.781664	0.390832	−	0.920462i	$-0.372187\pi$
$809$	22.2328	0.781664	0.390832	+	0.920462i	$0.372187\pi$
$810$	41.1654	1.44641
$811$	−22.5824	−0.792975	−0.396488	−	0.918040i	$-0.629771\pi$
$811$	−22.5824	−0.792975	−0.396488	+	0.918040i	$0.629771\pi$
$812$	9.41193	0.330294
$813$	47.9397	1.68132
$814$	−0.361220	−0.0126608
$815$	−8.65100	−0.303031
$816$	8.29742	0.290468
$817$	−42.6082	−1.49067
$818$	4.69838	0.164275
$819$	5.85601	0.204626
$820$	−1.63643	−0.0571465
$821$	36.4722	1.27289	0.636444	−	0.771323i	$-0.280405\pi$
$821$	36.4722	1.27289	0.636444	+	0.771323i	$0.280405\pi$
$822$	−36.7956	−1.28339
$823$	−41.4219	−1.44388	−0.721938	−	0.691958i	$-0.756748\pi$
$823$	−41.4219	−1.44388	−0.721938	+	0.691958i	$0.756748\pi$
$824$	−17.2376	−0.600500
$825$	0.724420	0.0252211
$826$	15.4632	0.538035
$827$	10.3127	0.358608	0.179304	−	0.983794i	$-0.442615\pi$
$827$	10.3127	0.358608	0.179304	+	0.983794i	$0.442615\pi$
$828$	−3.70612	−0.128796
$829$	−1.45972	−0.0506982	−0.0253491	−	0.999679i	$-0.508070\pi$
$829$	−1.45972	−0.0506982	−0.0253491	+	0.999679i	$0.508070\pi$
$830$	−15.5657	−0.540294
$831$	55.7776	1.93490
$832$	−1.28748	−0.0446355
$833$	−19.5910	−0.678787
$834$	22.5148	0.779623
$835$	4.49983	0.155723
$836$	−0.178805	−0.00618411
$837$	31.3940	1.08514
$838$	27.6248	0.954282
$839$	−41.5883	−1.43579	−0.717894	−	0.696153i	$-0.754894\pi$
$839$	−41.5883	−1.43579	−0.717894	+	0.696153i	$0.754894\pi$
$840$	−26.3147	−0.907943
$841$	−21.5599	−0.743446
$842$	−4.20494	−0.144912
$843$	35.9525	1.23827
$844$	21.3716	0.735643
$845$	41.6258	1.43197
$846$	7.61683	0.261872
$847$	37.9504	1.30399
$848$	7.93141	0.272366
$849$	69.1611	2.37360
$850$	−33.8139	−1.15981
$851$	−24.6707	−0.845702
$852$	−15.7525	−0.539673
$853$	36.7505	1.25831	0.629157	−	0.777279i	$-0.283400\pi$
$853$	36.7505	1.25831	0.629157	+	0.777279i	$0.283400\pi$
$854$	−24.4947	−0.838191
$855$	−21.0121	−0.718599
$856$	6.06603	0.207332
$857$	−40.0036	−1.36650	−0.683249	−	0.730186i	$-0.739433\pi$
$857$	−40.0036	−1.36650	−0.683249	+	0.730186i	$0.739433\pi$
$858$	−0.110136	−0.00375999
$859$	−11.6933	−0.398971	−0.199485	−	0.979901i	$-0.563927\pi$
$859$	−11.6933	−0.398971	−0.199485	+	0.979901i	$0.563927\pi$
$860$	36.0006	1.22761
$861$	3.19726	0.108962
$862$	27.4258	0.934126
$863$	−13.9631	−0.475311	−0.237655	−	0.971350i	$-0.576379\pi$
$863$	−13.9631	−0.475311	−0.237655	+	0.971350i	$0.576379\pi$
$864$	3.49488	0.118898
$865$	48.1546	1.63730
$866$	−17.7193	−0.602125
$867$	2.19520	0.0745530
$868$	−30.9959	−1.05207
$869$	0.0495127	0.00167960
$870$	−20.8016	−0.705240
$871$	13.4554	0.455918
$872$	−17.9980	−0.609490
$873$	15.9179	0.538741
$874$	−12.2121	−0.413081
$875$	43.9217	1.48483
$876$	7.96135	0.268989
$877$	46.4762	1.56939	0.784695	−	0.619882i	$-0.212820\pi$
$877$	46.4762	1.56939	0.784695	+	0.619882i	$0.212820\pi$
$878$	−11.2775	−0.380598
$879$	−13.4947	−0.455166
$880$	0.151077	0.00509279
$881$	−25.6684	−0.864792	−0.432396	−	0.901684i	$-0.642332\pi$
$881$	−25.6684	−0.864792	−0.432396	+	0.901684i	$0.642332\pi$
$882$	6.46744	0.217770
$883$	−2.14740	−0.0722658	−0.0361329	−	0.999347i	$-0.511504\pi$
$883$	−2.14740	−0.0722658	−0.0361329	+	0.999347i	$0.511504\pi$
$884$	5.14085	0.172906
$885$	−34.1758	−1.14881
$886$	6.29496	0.211483
$887$	−40.5477	−1.36146	−0.680729	−	0.732535i	$-0.738337\pi$
$887$	−40.5477	−1.36146	−0.680729	+	0.732535i	$0.738337\pi$
$888$	−18.2340	−0.611894
$889$	49.9242	1.67440
$890$	−28.3752	−0.951139
$891$	0.461756	0.0154694
$892$	−7.32606	−0.245295
$893$	25.0984	0.839886
$894$	27.5354	0.920923
$895$	13.2304	0.442245
$896$	−3.45056	−0.115275
$897$	−7.52213	−0.251157
$898$	30.4039	1.01459
$899$	−24.5021	−0.817189
$900$	11.1628	0.372092
$901$	−31.6697	−1.05507
$902$	−0.0183560	−0.000611187 0
$903$	−70.3383	−2.34071
$904$	7.54476	0.250935
$905$	−42.4913	−1.41246
$906$	−21.0050	−0.697844
$907$	27.3524	0.908223	0.454112	−	0.890945i	$-0.349957\pi$
$907$	27.3524	0.908223	0.454112	+	0.890945i	$0.349957\pi$
$908$	3.35917	0.111478
$909$	7.05596	0.234031
$910$	−16.3039	−0.540468
$911$	36.5608	1.21131	0.605656	−	0.795726i	$-0.292911\pi$
$911$	36.5608	1.21131	0.605656	+	0.795726i	$0.292911\pi$
$912$	−9.02591	−0.298878
$913$	−0.174602	−0.00577849
$914$	−35.3996	−1.17091
$915$	54.1364	1.78970
$916$	4.02544	0.133004
$917$	−48.1668	−1.59061
$918$	−13.9549	−0.460580
$919$	−42.6438	−1.40669	−0.703345	−	0.710849i	$-0.748311\pi$
$919$	−42.6438	−1.40669	−0.703345	+	0.710849i	$0.748311\pi$
$920$	10.3183	0.340184
$921$	13.0149	0.428855
$922$	0.985230	0.0324468
$923$	−9.75984	−0.321249
$924$	−0.295174	−0.00971052
$925$	74.3079	2.44323
$926$	−31.2127	−1.02571
$927$	−22.7220	−0.746289
$928$	−2.72765	−0.0895395
$929$	27.3470	0.897224	0.448612	−	0.893727i	$-0.351918\pi$
$929$	27.3470	0.897224	0.448612	+	0.893727i	$0.351918\pi$
$930$	68.5049	2.24636
$931$	21.3110	0.698440
$932$	2.12563	0.0696273
$933$	58.1503	1.90376
$934$	−41.4673	−1.35685
$935$	−0.603240	−0.0197281
$936$	−1.69712	−0.0554720
$937$	15.0547	0.491816	0.245908	−	0.969293i	$-0.420914\pi$
$937$	15.0547	0.491816	0.245908	+	0.969293i	$0.420914\pi$
$938$	36.0615	1.17745
$939$	−4.93076	−0.160909
$940$	−21.2062	−0.691670
$941$	49.2447	1.60533	0.802666	−	0.596429i	$-0.203414\pi$
$941$	49.2447	1.60533	0.802666	+	0.596429i	$0.203414\pi$
$942$	−7.37946	−0.240436
$943$	−1.25368	−0.0408255
$944$	−4.48136	−0.145856
$945$	44.2569	1.43968
$946$	0.403823	0.0131294
$947$	38.9341	1.26519	0.632594	−	0.774484i	$-0.281990\pi$
$947$	38.9341	1.26519	0.632594	+	0.774484i	$0.281990\pi$
$948$	2.49935	0.0811752
$949$	4.93263	0.160120
$950$	36.7827	1.19339
$951$	−0.667531	−0.0216462
$952$	13.7779	0.446545
$953$	−48.4828	−1.57051	−0.785256	−	0.619171i	$-0.787469\pi$
$953$	−48.4828	−1.57051	−0.785256	+	0.619171i	$0.787469\pi$
$954$	10.4549	0.338490
$955$	21.0234	0.680301
$956$	0.817349	0.0264350
$957$	−0.233334	−0.00754260
$958$	25.1228	0.811682
$959$	−61.0993	−1.97300
$960$	7.62619	0.246134
$961$	49.6915	1.60295
$962$	−11.2973	−0.364240
$963$	7.99603	0.257668
$964$	25.8208	0.831631
$965$	50.8291	1.63625
$966$	−20.1599	−0.648635
$967$	−17.5222	−0.563476	−0.281738	−	0.959491i	$-0.590911\pi$
$967$	−17.5222	−0.563476	−0.281738	+	0.959491i	$0.590911\pi$
$968$	−10.9983	−0.353499
$969$	36.0400	1.15777
$970$	−44.3175	−1.42295
$971$	41.9650	1.34672	0.673360	−	0.739315i	$-0.264850\pi$
$971$	41.9650	1.34672	0.673360	+	0.739315i	$0.264850\pi$
$972$	12.8244	0.411342
$973$	37.3859	1.19854
$974$	−1.05337	−0.0337520
$975$	22.6565	0.725590
$976$	7.09875	0.227225
$977$	13.2720	0.424607	0.212304	−	0.977204i	$-0.431903\pi$
$977$	13.2720	0.424607	0.212304	+	0.977204i	$0.431903\pi$
$978$	−4.89844	−0.156635
$979$	−0.318288	−0.0101725
$980$	−18.0061	−0.575185
$981$	−23.7244	−0.757461
$982$	−11.2134	−0.357834
$983$	−13.9796	−0.445879	−0.222939	−	0.974832i	$-0.571565\pi$
$983$	−13.9796	−0.445879	−0.222939	+	0.974832i	$0.571565\pi$
$984$	−0.926591	−0.0295386
$985$	−28.2745	−0.900900
$986$	10.8914	0.346852
$987$	41.4328	1.31882
$988$	−5.59221	−0.177912
$989$	27.5804	0.877007
$990$	0.199144	0.00632921
$991$	36.1350	1.14787	0.573933	−	0.818902i	$-0.305417\pi$
$991$	36.1350	1.14787	0.573933	+	0.818902i	$0.305417\pi$
$992$	8.98284	0.285206
$993$	−44.6535	−1.41704
$994$	−26.1572	−0.829655
$995$	1.41020	0.0447062
$996$	−8.81375	−0.279274
$997$	55.1563	1.74682	0.873409	−	0.486988i	$-0.161904\pi$
$997$	55.1563	1.74682	0.873409	+	0.486988i	$0.161904\pi$
$998$	14.7997	0.468475
$999$	30.6666	0.970249

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.a.1.14	✓	67

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.a.1.14	✓	67	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:	\( N \)	\(=\)	\( 8042 = 2 \cdot 4021 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8042.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.07802 q^{3} +1.00000 q^{4} -3.66993 q^{5} -2.07802 q^{6} -3.45056 q^{7} +1.00000 q^{8} +1.31817 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.07802 q^{3} +1.00000 q^{4} -3.66993 q^{5} -2.07802 q^{6} -3.45056 q^{7} +1.00000 q^{8} +1.31817 q^{9} -3.66993 q^{10} -0.0411660 q^{11} -2.07802 q^{12} -1.28748 q^{13} -3.45056 q^{14} +7.62619 q^{15} +1.00000 q^{16} -3.99294 q^{17} +1.31817 q^{18} +4.34352 q^{19} -3.66993 q^{20} +7.17034 q^{21} -0.0411660 q^{22} -2.81157 q^{23} -2.07802 q^{24} +8.46842 q^{25} -1.28748 q^{26} +3.49488 q^{27} -3.45056 q^{28} -2.72765 q^{29} +7.62619 q^{30} +8.98284 q^{31} +1.00000 q^{32} +0.0855438 q^{33} -3.99294 q^{34} +12.6633 q^{35} +1.31817 q^{36} +8.77471 q^{37} +4.34352 q^{38} +2.67542 q^{39} -3.66993 q^{40} +0.445901 q^{41} +7.17034 q^{42} -9.80961 q^{43} -0.0411660 q^{44} -4.83758 q^{45} -2.81157 q^{46} +5.77836 q^{47} -2.07802 q^{48} +4.90639 q^{49} +8.46842 q^{50} +8.29742 q^{51} -1.28748 q^{52} +7.93141 q^{53} +3.49488 q^{54} +0.151077 q^{55} -3.45056 q^{56} -9.02591 q^{57} -2.72765 q^{58} -4.48136 q^{59} +7.62619 q^{60} +7.09875 q^{61} +8.98284 q^{62} -4.54842 q^{63} +1.00000 q^{64} +4.72498 q^{65} +0.0855438 q^{66} -10.4509 q^{67} -3.99294 q^{68} +5.84250 q^{69} +12.6633 q^{70} +7.58055 q^{71} +1.31817 q^{72} -3.83122 q^{73} +8.77471 q^{74} -17.5975 q^{75} +4.34352 q^{76} +0.142046 q^{77} +2.67542 q^{78} -1.20276 q^{79} -3.66993 q^{80} -11.2169 q^{81} +0.445901 q^{82} +4.24142 q^{83} +7.17034 q^{84} +14.6538 q^{85} -9.80961 q^{86} +5.66811 q^{87} -0.0411660 q^{88} +7.73181 q^{89} -4.83758 q^{90} +4.44255 q^{91} -2.81157 q^{92} -18.6665 q^{93} +5.77836 q^{94} -15.9404 q^{95} -2.07802 q^{96} +12.0758 q^{97} +4.90639 q^{98} -0.0542636 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9} - 20 q^{10} - 13 q^{11} - 11 q^{12} - 51 q^{13} - 40 q^{14} - 31 q^{15} + 67 q^{16} - 34 q^{17} + 24 q^{18} - 33 q^{19} - 20 q^{20} - 39 q^{21} - 13 q^{22} - 43 q^{23} - 11 q^{24} - 9 q^{25} - 51 q^{26} - 29 q^{27} - 40 q^{28} - 63 q^{29} - 31 q^{30} - 43 q^{31} + 67 q^{32} - 49 q^{33} - 34 q^{34} - 20 q^{35} + 24 q^{36} - 77 q^{37} - 33 q^{38} - 40 q^{39} - 20 q^{40} - 50 q^{41} - 39 q^{42} - 56 q^{43} - 13 q^{44} - 48 q^{45} - 43 q^{46} - 48 q^{47} - 11 q^{48} + q^{49} - 9 q^{50} - 18 q^{51} - 51 q^{52} - 91 q^{53} - 29 q^{54} - 58 q^{55} - 40 q^{56} - 65 q^{57} - 63 q^{58} - 17 q^{59} - 31 q^{60} - 45 q^{61} - 43 q^{62} - 67 q^{63} + 67 q^{64} - 65 q^{65} - 49 q^{66} - 112 q^{67} - 34 q^{68} - 57 q^{69} - 20 q^{70} - 75 q^{71} + 24 q^{72} - 79 q^{73} - 77 q^{74} - 5 q^{75} - 33 q^{76} - 85 q^{77} - 40 q^{78} - 80 q^{79} - 20 q^{80} - 77 q^{81} - 50 q^{82} - 22 q^{83} - 39 q^{84} - 134 q^{85} - 56 q^{86} - 49 q^{87} - 13 q^{88} - 77 q^{89} - 48 q^{90} - 17 q^{91} - 43 q^{92} - 97 q^{93} - 48 q^{94} - 73 q^{95} - 11 q^{96} - 87 q^{97} + q^{98} - 44 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9 - 20 * q^10 - 13 * q^11 - 11 * q^12 - 51 * q^13 - 40 * q^14 - 31 * q^15 + 67 * q^16 - 34 * q^17 + 24 * q^18 - 33 * q^19 - 20 * q^20 - 39 * q^21 - 13 * q^22 - 43 * q^23 - 11 * q^24 - 9 * q^25 - 51 * q^26 - 29 * q^27 - 40 * q^28 - 63 * q^29 - 31 * q^30 - 43 * q^31 + 67 * q^32 - 49 * q^33 - 34 * q^34 - 20 * q^35 + 24 * q^36 - 77 * q^37 - 33 * q^38 - 40 * q^39 - 20 * q^40 - 50 * q^41 - 39 * q^42 - 56 * q^43 - 13 * q^44 - 48 * q^45 - 43 * q^46 - 48 * q^47 - 11 * q^48 + q^49 - 9 * q^50 - 18 * q^51 - 51 * q^52 - 91 * q^53 - 29 * q^54 - 58 * q^55 - 40 * q^56 - 65 * q^57 - 63 * q^58 - 17 * q^59 - 31 * q^60 - 45 * q^61 - 43 * q^62 - 67 * q^63 + 67 * q^64 - 65 * q^65 - 49 * q^66 - 112 * q^67 - 34 * q^68 - 57 * q^69 - 20 * q^70 - 75 * q^71 + 24 * q^72 - 79 * q^73 - 77 * q^74 - 5 * q^75 - 33 * q^76 - 85 * q^77 - 40 * q^78 - 80 * q^79 - 20 * q^80 - 77 * q^81 - 50 * q^82 - 22 * q^83 - 39 * q^84 - 134 * q^85 - 56 * q^86 - 49 * q^87 - 13 * q^88 - 77 * q^89 - 48 * q^90 - 17 * q^91 - 43 * q^92 - 97 * q^93 - 48 * q^94 - 73 * q^95 - 11 * q^96 - 87 * q^97 + q^98 - 44 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more