LMFDB - Embedded newform 8043.2.a.p.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8043,2,Mod(1,8043)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8043, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8043.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8043 = 3 \cdot 7 \cdot 383 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8043.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:	$64.2236783457$
Analytic rank:	$0$
Dimension:	$41$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	8043.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.15673 q^{2} +1.00000 q^{3} -0.661981 q^{4} +0.288867 q^{5} -1.15673 q^{6} -1.00000 q^{7} +3.07919 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.15673 q^{2} +1.00000 q^{3} -0.661981 q^{4} +0.288867 q^{5} -1.15673 q^{6} -1.00000 q^{7} +3.07919 q^{8} +1.00000 q^{9} -0.334140 q^{10} +1.95120 q^{11} -0.661981 q^{12} -4.63563 q^{13} +1.15673 q^{14} +0.288867 q^{15} -2.23782 q^{16} -4.01444 q^{17} -1.15673 q^{18} -3.80282 q^{19} -0.191224 q^{20} -1.00000 q^{21} -2.25701 q^{22} -4.00882 q^{23} +3.07919 q^{24} -4.91656 q^{25} +5.36216 q^{26} +1.00000 q^{27} +0.661981 q^{28} -9.58984 q^{29} -0.334140 q^{30} +8.91488 q^{31} -3.56983 q^{32} +1.95120 q^{33} +4.64361 q^{34} -0.288867 q^{35} -0.661981 q^{36} -6.80122 q^{37} +4.39883 q^{38} -4.63563 q^{39} +0.889475 q^{40} -0.503249 q^{41} +1.15673 q^{42} +3.40347 q^{43} -1.29166 q^{44} +0.288867 q^{45} +4.63712 q^{46} -3.40955 q^{47} -2.23782 q^{48} +1.00000 q^{49} +5.68712 q^{50} -4.01444 q^{51} +3.06870 q^{52} +12.0221 q^{53} -1.15673 q^{54} +0.563637 q^{55} -3.07919 q^{56} -3.80282 q^{57} +11.0928 q^{58} -3.17701 q^{59} -0.191224 q^{60} +8.50949 q^{61} -10.3121 q^{62} -1.00000 q^{63} +8.60496 q^{64} -1.33908 q^{65} -2.25701 q^{66} +1.98416 q^{67} +2.65748 q^{68} -4.00882 q^{69} +0.334140 q^{70} -2.98151 q^{71} +3.07919 q^{72} -6.56797 q^{73} +7.86715 q^{74} -4.91656 q^{75} +2.51740 q^{76} -1.95120 q^{77} +5.36216 q^{78} +14.1405 q^{79} -0.646432 q^{80} +1.00000 q^{81} +0.582122 q^{82} +7.14157 q^{83} +0.661981 q^{84} -1.15964 q^{85} -3.93689 q^{86} -9.58984 q^{87} +6.00811 q^{88} +16.0112 q^{89} -0.334140 q^{90} +4.63563 q^{91} +2.65376 q^{92} +8.91488 q^{93} +3.94392 q^{94} -1.09851 q^{95} -3.56983 q^{96} -7.97583 q^{97} -1.15673 q^{98} +1.95120 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 41 q + 7 q^{2} + 41 q^{3} + 45 q^{4} + 17 q^{5} + 7 q^{6} - 41 q^{7} + 12 q^{8} + 41 q^{9}+O(q^{10}) $ 41 * q + 7 * q^2 + 41 * q^3 + 45 * q^4 + 17 * q^5 + 7 * q^6 - 41 * q^7 + 12 * q^8 + 41 * q^9	\( 41 q + 7 q^{2} + 41 q^{3} + 45 q^{4} + 17 q^{5} + 7 q^{6} - 41 q^{7} + 12 q^{8} + 41 q^{9} + 18 q^{10} + 8 q^{11} + 45 q^{12} + 23 q^{13} - 7 q^{14} + 17 q^{15} + 37 q^{16} + 15 q^{17} + 7 q^{18} + 15 q^{19} + 53 q^{20} - 41 q^{21} + 13 q^{22} + 44 q^{23} + 12 q^{24} + 58 q^{25} + 9 q^{26} + 41 q^{27} - 45 q^{28} + 21 q^{29} + 18 q^{30} + 39 q^{31} + 61 q^{32} + 8 q^{33} + 9 q^{34} - 17 q^{35} + 45 q^{36} + 11 q^{37} + 44 q^{38} + 23 q^{39} + 24 q^{40} + 17 q^{41} - 7 q^{42} + 7 q^{43} + 30 q^{44} + 17 q^{45} - 12 q^{46} + 36 q^{47} + 37 q^{48} + 41 q^{49} + 28 q^{50} + 15 q^{51} + 58 q^{52} + 26 q^{53} + 7 q^{54} + 32 q^{55} - 12 q^{56} + 15 q^{57} - 4 q^{58} + 33 q^{59} + 53 q^{60} + 59 q^{61} - q^{62} - 41 q^{63} + 16 q^{64} + 72 q^{65} + 13 q^{66} + 12 q^{67} + 52 q^{68} + 44 q^{69} - 18 q^{70} + 33 q^{71} + 12 q^{72} + 18 q^{73} + 42 q^{74} + 58 q^{75} + 7 q^{76} - 8 q^{77} + 9 q^{78} + 22 q^{79} + 69 q^{80} + 41 q^{81} + 41 q^{82} + 32 q^{83} - 45 q^{84} - 44 q^{85} + 11 q^{86} + 21 q^{87} + 52 q^{88} + 63 q^{89} + 18 q^{90} - 23 q^{91} + 52 q^{92} + 39 q^{93} + 17 q^{94} + 37 q^{95} + 61 q^{96} + 8 q^{97} + 7 q^{98} + 8 q^{99}+O(q^{100}) \) 41 * q + 7 * q^2 + 41 * q^3 + 45 * q^4 + 17 * q^5 + 7 * q^6 - 41 * q^7 + 12 * q^8 + 41 * q^9 + 18 * q^10 + 8 * q^11 + 45 * q^12 + 23 * q^13 - 7 * q^14 + 17 * q^15 + 37 * q^16 + 15 * q^17 + 7 * q^18 + 15 * q^19 + 53 * q^20 - 41 * q^21 + 13 * q^22 + 44 * q^23 + 12 * q^24 + 58 * q^25 + 9 * q^26 + 41 * q^27 - 45 * q^28 + 21 * q^29 + 18 * q^30 + 39 * q^31 + 61 * q^32 + 8 * q^33 + 9 * q^34 - 17 * q^35 + 45 * q^36 + 11 * q^37 + 44 * q^38 + 23 * q^39 + 24 * q^40 + 17 * q^41 - 7 * q^42 + 7 * q^43 + 30 * q^44 + 17 * q^45 - 12 * q^46 + 36 * q^47 + 37 * q^48 + 41 * q^49 + 28 * q^50 + 15 * q^51 + 58 * q^52 + 26 * q^53 + 7 * q^54 + 32 * q^55 - 12 * q^56 + 15 * q^57 - 4 * q^58 + 33 * q^59 + 53 * q^60 + 59 * q^61 - q^62 - 41 * q^63 + 16 * q^64 + 72 * q^65 + 13 * q^66 + 12 * q^67 + 52 * q^68 + 44 * q^69 - 18 * q^70 + 33 * q^71 + 12 * q^72 + 18 * q^73 + 42 * q^74 + 58 * q^75 + 7 * q^76 - 8 * q^77 + 9 * q^78 + 22 * q^79 + 69 * q^80 + 41 * q^81 + 41 * q^82 + 32 * q^83 - 45 * q^84 - 44 * q^85 + 11 * q^86 + 21 * q^87 + 52 * q^88 + 63 * q^89 + 18 * q^90 - 23 * q^91 + 52 * q^92 + 39 * q^93 + 17 * q^94 + 37 * q^95 + 61 * q^96 + 8 * q^97 + 7 * q^98 + 8 * 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 $

$2$	−1.15673	−0.817930	−0.408965	−	0.912550i	$-0.634110\pi$
$2$	−1.15673	−0.817930	−0.408965	+	0.912550i	$0.634110\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−0.661981	−0.330990
$5$	0.288867	0.129185	0.0645926	−	0.997912i	$-0.479425\pi$
$5$	0.288867	0.129185	0.0645926	+	0.997912i	$0.479425\pi$
$6$	−1.15673	−0.472232
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	3.07919	1.08866
$9$	1.00000	0.333333
$10$	−0.334140	−0.105664
$11$	1.95120	0.588309	0.294155	−	0.955758i	$-0.404962\pi$
$11$	1.95120	0.588309	0.294155	+	0.955758i	$0.404962\pi$
$12$	−0.661981	−0.191097
$13$	−4.63563	−1.28569	−0.642846	−	0.765996i	$-0.722246\pi$
$13$	−4.63563	−1.28569	−0.642846	+	0.765996i	$0.722246\pi$
$14$	1.15673	0.309149
$15$	0.288867	0.0745851
$16$	−2.23782	−0.559455
$17$	−4.01444	−0.973644	−0.486822	−	0.873501i	$-0.661844\pi$
$17$	−4.01444	−0.973644	−0.486822	+	0.873501i	$0.661844\pi$
$18$	−1.15673	−0.272643
$19$	−3.80282	−0.872427	−0.436214	−	0.899843i	$-0.643681\pi$
$19$	−3.80282	−0.872427	−0.436214	+	0.899843i	$0.643681\pi$
$20$	−0.191224	−0.0427591
$21$	−1.00000	−0.218218
$22$	−2.25701	−0.481196
$23$	−4.00882	−0.835897	−0.417949	−	0.908471i	$-0.637251\pi$
$23$	−4.00882	−0.835897	−0.417949	+	0.908471i	$0.637251\pi$
$24$	3.07919	0.628536
$25$	−4.91656	−0.983311
$26$	5.36216	1.05161
$27$	1.00000	0.192450
$28$	0.661981	0.125103
$29$	−9.58984	−1.78079	−0.890395	−	0.455189i	$-0.849572\pi$
$29$	−9.58984	−1.78079	−0.890395	+	0.455189i	$0.849572\pi$
$30$	−0.334140	−0.0610054
$31$	8.91488	1.60116	0.800579	−	0.599227i	$-0.204525\pi$
$31$	8.91488	1.60116	0.800579	+	0.599227i	$0.204525\pi$
$32$	−3.56983	−0.631062
$33$	1.95120	0.339660
$34$	4.64361	0.796373
$35$	−0.288867	−0.0488274
$36$	−0.661981	−0.110330
$37$	−6.80122	−1.11811	−0.559056	−	0.829130i	$-0.688836\pi$
$37$	−6.80122	−1.11811	−0.559056	+	0.829130i	$0.688836\pi$
$38$	4.39883	0.713584
$39$	−4.63563	−0.742294
$40$	0.889475	0.140638
$41$	−0.503249	−0.0785943	−0.0392972	−	0.999228i	$-0.512512\pi$
$41$	−0.503249	−0.0785943	−0.0392972	+	0.999228i	$0.512512\pi$
$42$	1.15673	0.178487
$43$	3.40347	0.519025	0.259512	−	0.965740i	$-0.416438\pi$
$43$	3.40347	0.519025	0.259512	+	0.965740i	$0.416438\pi$
$44$	−1.29166	−0.194725
$45$	0.288867	0.0430617
$46$	4.63712	0.683705
$47$	−3.40955	−0.497334	−0.248667	−	0.968589i	$-0.579992\pi$
$47$	−3.40955	−0.497334	−0.248667	+	0.968589i	$0.579992\pi$
$48$	−2.23782	−0.323001
$49$	1.00000	0.142857
$50$	5.68712	0.804280
$51$	−4.01444	−0.562134
$52$	3.06870	0.425552
$53$	12.0221	1.65137	0.825684	−	0.564133i	$-0.190790\pi$
$53$	12.0221	1.65137	0.825684	+	0.564133i	$0.190790\pi$
$54$	−1.15673	−0.157411
$55$	0.563637	0.0760008
$56$	−3.07919	−0.411474
$57$	−3.80282	−0.503696
$58$	11.0928	1.45656
$59$	−3.17701	−0.413612	−0.206806	−	0.978382i	$-0.566307\pi$
$59$	−3.17701	−0.413612	−0.206806	+	0.978382i	$0.566307\pi$
$60$	−0.191224	−0.0246870
$61$	8.50949	1.08953	0.544765	−	0.838589i	$-0.316619\pi$
$61$	8.50949	1.08953	0.544765	+	0.838589i	$0.316619\pi$
$62$	−10.3121	−1.30964
$63$	−1.00000	−0.125988
$64$	8.60496	1.07562
$65$	−1.33908	−0.166092
$66$	−2.25701	−0.277818
$67$	1.98416	0.242404	0.121202	−	0.992628i	$-0.461325\pi$
$67$	1.98416	0.242404	0.121202	+	0.992628i	$0.461325\pi$
$68$	2.65748	0.322267
$69$	−4.00882	−0.482605
$70$	0.334140	0.0399374
$71$	−2.98151	−0.353840	−0.176920	−	0.984225i	$-0.556613\pi$
$71$	−2.98151	−0.353840	−0.176920	+	0.984225i	$0.556613\pi$
$72$	3.07919	0.362886
$73$	−6.56797	−0.768722	−0.384361	−	0.923183i	$-0.625578\pi$
$73$	−6.56797	−0.768722	−0.384361	+	0.923183i	$0.625578\pi$
$74$	7.86715	0.914538
$75$	−4.91656	−0.567715
$76$	2.51740	0.288765
$77$	−1.95120	−0.222360
$78$	5.36216	0.607145
$79$	14.1405	1.59093	0.795464	−	0.606000i	$-0.207227\pi$
$79$	14.1405	1.59093	0.795464	+	0.606000i	$0.207227\pi$
$80$	−0.646432	−0.0722733
$81$	1.00000	0.111111
$82$	0.582122	0.0642847
$83$	7.14157	0.783889	0.391945	−	0.919989i	$-0.371802\pi$
$83$	7.14157	0.783889	0.391945	+	0.919989i	$0.371802\pi$
$84$	0.661981	0.0722280
$85$	−1.15964	−0.125780
$86$	−3.93689	−0.424526
$87$	−9.58984	−1.02814
$88$	6.00811	0.640467
$89$	16.0112	1.69718	0.848590	−	0.529051i	$-0.177452\pi$
$89$	16.0112	1.69718	0.848590	+	0.529051i	$0.177452\pi$
$90$	−0.334140	−0.0352215
$91$	4.63563	0.485946
$92$	2.65376	0.276674
$93$	8.91488	0.924430
$94$	3.94392	0.406784
$95$	−1.09851	−0.112705
$96$	−3.56983	−0.364344
$97$	−7.97583	−0.809823	−0.404912	−	0.914356i	$-0.632698\pi$
$97$	−7.97583	−0.809823	−0.404912	+	0.914356i	$0.632698\pi$
$98$	−1.15673	−0.116847
$99$	1.95120	0.196103
$100$	3.25467	0.325467
$101$	−5.76352	−0.573492	−0.286746	−	0.958007i	$-0.592574\pi$
$101$	−5.76352	−0.573492	−0.286746	+	0.958007i	$0.592574\pi$
$102$	4.64361	0.459786
$103$	13.4974	1.32994	0.664968	−	0.746872i	$-0.268445\pi$
$103$	13.4974	1.32994	0.664968	+	0.746872i	$0.268445\pi$
$104$	−14.2740	−1.39968
$105$	−0.288867	−0.0281905
$106$	−13.9063	−1.35070
$107$	1.78267	0.172337	0.0861685	−	0.996281i	$-0.472538\pi$
$107$	1.78267	0.172337	0.0861685	+	0.996281i	$0.472538\pi$
$108$	−0.661981	−0.0636991
$109$	−0.315132	−0.0301842	−0.0150921	−	0.999886i	$-0.504804\pi$
$109$	−0.315132	−0.0301842	−0.0150921	+	0.999886i	$0.504804\pi$
$110$	−0.651975	−0.0621633
$111$	−6.80122	−0.645543
$112$	2.23782	0.211454
$113$	11.3277	1.06562	0.532810	−	0.846235i	$-0.321136\pi$
$113$	11.3277	1.06562	0.532810	+	0.846235i	$0.321136\pi$
$114$	4.39883	0.411988
$115$	−1.15802	−0.107986
$116$	6.34829	0.589424
$117$	−4.63563	−0.428564
$118$	3.67494	0.338306
$119$	4.01444	0.368003
$120$	0.889475	0.0811976
$121$	−7.19282	−0.653892
$122$	−9.84317	−0.891159
$123$	−0.503249	−0.0453765
$124$	−5.90148	−0.529968
$125$	−2.86456	−0.256214
$126$	1.15673	0.103050
$127$	14.2611	1.26547	0.632735	−	0.774368i	$-0.281932\pi$
$127$	14.2611	1.26547	0.632735	+	0.774368i	$0.281932\pi$
$128$	−2.81394	−0.248720
$129$	3.40347	0.299659
$130$	1.54895	0.135852
$131$	−0.702414	−0.0613702	−0.0306851	−	0.999529i	$-0.509769\pi$
$131$	−0.702414	−0.0613702	−0.0306851	+	0.999529i	$0.509769\pi$
$132$	−1.29166	−0.112424
$133$	3.80282	0.329747
$134$	−2.29514	−0.198270
$135$	0.288867	0.0248617
$136$	−12.3612	−1.05996
$137$	−0.0373337	−0.00318963	−0.00159482	−	0.999999i	$-0.500508\pi$
$137$	−0.0373337	−0.00318963	−0.00159482	+	0.999999i	$0.500508\pi$
$138$	4.63712	0.394737
$139$	4.73130	0.401304	0.200652	−	0.979663i	$-0.435694\pi$
$139$	4.73130	0.401304	0.200652	+	0.979663i	$0.435694\pi$
$140$	0.191224	0.0161614
$141$	−3.40955	−0.287136
$142$	3.44879	0.289416
$143$	−9.04504	−0.756384
$144$	−2.23782	−0.186485
$145$	−2.77019	−0.230052
$146$	7.59735	0.628761
$147$	1.00000	0.0824786
$148$	4.50227	0.370085
$149$	−6.92612	−0.567410	−0.283705	−	0.958912i	$-0.591564\pi$
$149$	−6.92612	−0.567410	−0.283705	+	0.958912i	$0.591564\pi$
$150$	5.68712	0.464351
$151$	18.2594	1.48593	0.742965	−	0.669330i	$-0.233419\pi$
$151$	18.2594	1.48593	0.742965	+	0.669330i	$0.233419\pi$
$152$	−11.7096	−0.949774
$153$	−4.01444	−0.324548
$154$	2.25701	0.181875
$155$	2.57521	0.206846
$156$	3.06870	0.245692
$157$	21.3258	1.70198	0.850991	−	0.525181i	$-0.176002\pi$
$157$	21.3258	1.70198	0.850991	+	0.525181i	$0.176002\pi$
$158$	−16.3567	−1.30127
$159$	12.0221	0.953417
$160$	−1.03120	−0.0815239
$161$	4.00882	0.315939
$162$	−1.15673	−0.0908811
$163$	12.0909	0.947029	0.473514	−	0.880786i	$-0.342985\pi$
$163$	12.0909	0.947029	0.473514	+	0.880786i	$0.342985\pi$
$164$	0.333141	0.0260140
$165$	0.563637	0.0438791
$166$	−8.26085	−0.641166
$167$	−11.1225	−0.860689	−0.430344	−	0.902665i	$-0.641608\pi$
$167$	−11.1225	−0.860689	−0.430344	+	0.902665i	$0.641608\pi$
$168$	−3.07919	−0.237564
$169$	8.48904	0.653003
$170$	1.34139	0.102880
$171$	−3.80282	−0.290809
$172$	−2.25303	−0.171792
$173$	18.0228	1.37025	0.685123	−	0.728428i	$-0.259749\pi$
$173$	18.0228	1.37025	0.685123	+	0.728428i	$0.259749\pi$
$174$	11.0928	0.840946
$175$	4.91656	0.371657
$176$	−4.36643	−0.329132
$177$	−3.17701	−0.238799
$178$	−18.5206	−1.38817
$179$	−4.21876	−0.315325	−0.157662	−	0.987493i	$-0.550396\pi$
$179$	−4.21876	−0.315325	−0.157662	+	0.987493i	$0.550396\pi$
$180$	−0.191224	−0.0142530
$181$	−18.2015	−1.35291	−0.676455	−	0.736484i	$-0.736485\pi$
$181$	−18.2015	−1.35291	−0.676455	+	0.736484i	$0.736485\pi$
$182$	−5.36216	−0.397470
$183$	8.50949	0.629040
$184$	−12.3439	−0.910005
$185$	−1.96465	−0.144444
$186$	−10.3121	−0.756119
$187$	−7.83297	−0.572804
$188$	2.25705	0.164613
$189$	−1.00000	−0.0727393
$190$	1.27068	0.0921845
$191$	−9.29407	−0.672496	−0.336248	−	0.941774i	$-0.609158\pi$
$191$	−9.29407	−0.672496	−0.336248	+	0.941774i	$0.609158\pi$
$192$	8.60496	0.621009
$193$	−6.40040	−0.460711	−0.230356	−	0.973107i	$-0.573989\pi$
$193$	−6.40040	−0.460711	−0.230356	+	0.973107i	$0.573989\pi$
$194$	9.22587	0.662379
$195$	−1.33908	−0.0958934
$196$	−0.661981	−0.0472843
$197$	1.03196	0.0735241	0.0367620	−	0.999324i	$-0.488296\pi$
$197$	1.03196	0.0735241	0.0367620	+	0.999324i	$0.488296\pi$
$198$	−2.25701	−0.160399
$199$	−14.3420	−1.01668	−0.508339	−	0.861157i	$-0.669740\pi$
$199$	−14.3420	−1.01668	−0.508339	+	0.861157i	$0.669740\pi$
$200$	−15.1390	−1.07049
$201$	1.98416	0.139952
$202$	6.66682	0.469076
$203$	9.58984	0.673075
$204$	2.65748	0.186061
$205$	−0.145372	−0.0101532
$206$	−15.6128	−1.08779
$207$	−4.00882	−0.278632
$208$	10.3737	0.719287
$209$	−7.42007	−0.513257
$210$	0.334140	0.0230579
$211$	−17.1778	−1.18257	−0.591285	−	0.806463i	$-0.701379\pi$
$211$	−17.1778	−1.18257	−0.591285	+	0.806463i	$0.701379\pi$
$212$	−7.95842	−0.546587
$213$	−2.98151	−0.204289
$214$	−2.06206	−0.140960
$215$	0.983150	0.0670503
$216$	3.07919	0.209512
$217$	−8.91488	−0.605181
$218$	0.364522	0.0246885
$219$	−6.56797	−0.443822
$220$	−0.373117	−0.0251555
$221$	18.6094	1.25181
$222$	7.86715	0.528009
$223$	−9.23095	−0.618150	−0.309075	−	0.951038i	$-0.600019\pi$
$223$	−9.23095	−0.618150	−0.309075	+	0.951038i	$0.600019\pi$
$224$	3.56983	0.238519
$225$	−4.91656	−0.327770
$226$	−13.1031	−0.871602
$227$	8.94694	0.593830	0.296915	−	0.954904i	$-0.404042\pi$
$227$	8.94694	0.593830	0.296915	+	0.954904i	$0.404042\pi$
$228$	2.51740	0.166719
$229$	7.63156	0.504308	0.252154	−	0.967687i	$-0.418861\pi$
$229$	7.63156	0.504308	0.252154	+	0.967687i	$0.418861\pi$
$230$	1.33951	0.0883246
$231$	−1.95120	−0.128380
$232$	−29.5289	−1.93867
$233$	5.76642	0.377771	0.188885	−	0.981999i	$-0.439512\pi$
$233$	5.76642	0.377771	0.188885	+	0.981999i	$0.439512\pi$
$234$	5.36216	0.350535
$235$	−0.984905	−0.0642481
$236$	2.10312	0.136902
$237$	14.1405	0.918523
$238$	−4.64361	−0.301001
$239$	27.1691	1.75742	0.878711	−	0.477353i	$-0.158404\pi$
$239$	27.1691	1.75742	0.878711	+	0.477353i	$0.158404\pi$
$240$	−0.646432	−0.0417270
$241$	−20.0421	−1.29102	−0.645512	−	0.763750i	$-0.723356\pi$
$241$	−20.0421	−1.29102	−0.645512	+	0.763750i	$0.723356\pi$
$242$	8.32013	0.534838
$243$	1.00000	0.0641500
$244$	−5.63312	−0.360624
$245$	0.288867	0.0184550
$246$	0.582122	0.0371148
$247$	17.6285	1.12167
$248$	27.4506	1.74311
$249$	7.14157	0.452579
$250$	3.31352	0.209565
$251$	11.8524	0.748118	0.374059	−	0.927405i	$-0.377966\pi$
$251$	11.8524	0.748118	0.374059	+	0.927405i	$0.377966\pi$
$252$	0.661981	0.0417009
$253$	−7.82201	−0.491766
$254$	−16.4962	−1.03507
$255$	−1.15964	−0.0726193
$256$	−13.9550	−0.872184
$257$	0.0890310	0.00555360	0.00277680	−	0.999996i	$-0.499116\pi$
$257$	0.0890310	0.00555360	0.00277680	+	0.999996i	$0.499116\pi$
$258$	−3.93689	−0.245100
$259$	6.80122	0.422607
$260$	0.886445	0.0549750
$261$	−9.58984	−0.593596
$262$	0.812502	0.0501965
$263$	15.7776	0.972888	0.486444	−	0.873712i	$-0.338294\pi$
$263$	15.7776	0.972888	0.486444	+	0.873712i	$0.338294\pi$
$264$	6.00811	0.369774
$265$	3.47280	0.213332
$266$	−4.39883	−0.269710
$267$	16.0112	0.979867
$268$	−1.31348	−0.0802335
$269$	−11.0386	−0.673034	−0.336517	−	0.941677i	$-0.609249\pi$
$269$	−11.0386	−0.673034	−0.336517	+	0.941677i	$0.609249\pi$
$270$	−0.334140	−0.0203351
$271$	−9.96267	−0.605189	−0.302594	−	0.953119i	$-0.597853\pi$
$271$	−9.96267	−0.605189	−0.302594	+	0.953119i	$0.597853\pi$
$272$	8.98359	0.544710
$273$	4.63563	0.280561
$274$	0.0431849	0.00260890
$275$	−9.59319	−0.578491
$276$	2.65376	0.159738
$277$	−26.0585	−1.56570	−0.782852	−	0.622208i	$-0.786236\pi$
$277$	−26.0585	−1.56570	−0.782852	+	0.622208i	$0.786236\pi$
$278$	−5.47283	−0.328239
$279$	8.91488	0.533720
$280$	−0.889475	−0.0531563
$281$	−4.73016	−0.282178	−0.141089	−	0.989997i	$-0.545060\pi$
$281$	−4.73016	−0.282178	−0.141089	+	0.989997i	$0.545060\pi$
$282$	3.94392	0.234857
$283$	23.1620	1.37684	0.688418	−	0.725314i	$-0.258305\pi$
$283$	23.1620	1.37684	0.688418	+	0.725314i	$0.258305\pi$
$284$	1.97370	0.117117
$285$	−1.09851	−0.0650701
$286$	10.4626	0.618669
$287$	0.503249	0.0297059
$288$	−3.56983	−0.210354
$289$	−0.884291	−0.0520171
$290$	3.20435	0.188166
$291$	−7.97583	−0.467552
$292$	4.34787	0.254440
$293$	−8.60384	−0.502642	−0.251321	−	0.967904i	$-0.580865\pi$
$293$	−8.60384	−0.502642	−0.251321	+	0.967904i	$0.580865\pi$
$294$	−1.15673	−0.0674617
$295$	−0.917734	−0.0534325
$296$	−20.9422	−1.21724
$297$	1.95120	0.113220
$298$	8.01163	0.464101
$299$	18.5834	1.07471
$300$	3.25467	0.187908
$301$	−3.40347	−0.196173
$302$	−21.1212	−1.21539
$303$	−5.76352	−0.331106
$304$	8.51003	0.488084
$305$	2.45811	0.140751
$306$	4.64361	0.265458
$307$	24.2224	1.38245	0.691223	−	0.722642i	$-0.257072\pi$
$307$	24.2224	1.38245	0.691223	+	0.722642i	$0.257072\pi$
$308$	1.29166	0.0735990
$309$	13.4974	0.767839
$310$	−2.97882	−0.169186
$311$	11.8456	0.671702	0.335851	−	0.941915i	$-0.390976\pi$
$311$	11.8456	0.671702	0.335851	+	0.941915i	$0.390976\pi$
$312$	−14.2740	−0.808104
$313$	−12.3037	−0.695448	−0.347724	−	0.937597i	$-0.613045\pi$
$313$	−12.3037	−0.695448	−0.347724	+	0.937597i	$0.613045\pi$
$314$	−24.6681	−1.39210
$315$	−0.288867	−0.0162758
$316$	−9.36073	−0.526582
$317$	6.10266	0.342760	0.171380	−	0.985205i	$-0.445177\pi$
$317$	6.10266	0.342760	0.171380	+	0.985205i	$0.445177\pi$
$318$	−13.9063	−0.779829
$319$	−18.7117	−1.04765
$320$	2.48569	0.138954
$321$	1.78267	0.0994988
$322$	−4.63712	−0.258416
$323$	15.2662	0.849434
$324$	−0.661981	−0.0367767
$325$	22.7913	1.26424
$326$	−13.9858	−0.774603
$327$	−0.315132	−0.0174268
$328$	−1.54960	−0.0855623
$329$	3.40955	0.187974
$330$	−0.651975	−0.0358900
$331$	16.8790	0.927752	0.463876	−	0.885900i	$-0.346458\pi$
$331$	16.8790	0.927752	0.463876	+	0.885900i	$0.346458\pi$
$332$	−4.72758	−0.259460
$333$	−6.80122	−0.372704
$334$	12.8658	0.703983
$335$	0.573159	0.0313150
$336$	2.23782	0.122083
$337$	9.24155	0.503419	0.251710	−	0.967803i	$-0.419007\pi$
$337$	9.24155	0.503419	0.251710	+	0.967803i	$0.419007\pi$
$338$	−9.81951	−0.534111
$339$	11.3277	0.615236
$340$	0.767658	0.0416321
$341$	17.3947	0.941976
$342$	4.39883	0.237861
$343$	−1.00000	−0.0539949
$344$	10.4799	0.565040
$345$	−1.15802	−0.0623455
$346$	−20.8474	−1.12076
$347$	11.2311	0.602915	0.301457	−	0.953480i	$-0.402527\pi$
$347$	11.2311	0.602915	0.301457	+	0.953480i	$0.402527\pi$
$348$	6.34829	0.340304
$349$	−15.5988	−0.834983	−0.417492	−	0.908681i	$-0.637091\pi$
$349$	−15.5988	−0.834983	−0.417492	+	0.908681i	$0.637091\pi$
$350$	−5.68712	−0.303989
$351$	−4.63563	−0.247431
$352$	−6.96545	−0.371260
$353$	37.1044	1.97487	0.987434	−	0.158034i	$-0.0505154\pi$
$353$	37.1044	1.97487	0.987434	+	0.158034i	$0.0505154\pi$
$354$	3.67494	0.195321
$355$	−0.861258	−0.0457108
$356$	−10.5991	−0.561750
$357$	4.01444	0.212467
$358$	4.87996	0.257914
$359$	3.54313	0.186999	0.0934995	−	0.995619i	$-0.470195\pi$
$359$	3.54313	0.186999	0.0934995	+	0.995619i	$0.470195\pi$
$360$	0.889475	0.0468795
$361$	−4.53854	−0.238871
$362$	21.0542	1.10659
$363$	−7.19282	−0.377525
$364$	−3.06870	−0.160843
$365$	−1.89727	−0.0993075
$366$	−9.84317	−0.514511
$367$	17.6351	0.920547	0.460273	−	0.887777i	$-0.347751\pi$
$367$	17.6351	0.920547	0.460273	+	0.887777i	$0.347751\pi$
$368$	8.97102	0.467647
$369$	−0.503249	−0.0261981
$370$	2.27256	0.118145
$371$	−12.0221	−0.624158
$372$	−5.90148	−0.305977
$373$	−27.3281	−1.41500	−0.707498	−	0.706716i	$-0.750176\pi$
$373$	−27.3281	−1.41500	−0.707498	+	0.706716i	$0.750176\pi$
$374$	9.06062	0.468513
$375$	−2.86456	−0.147925
$376$	−10.4986	−0.541426
$377$	44.4549	2.28955
$378$	1.15673	0.0594957
$379$	11.2762	0.579221	0.289610	−	0.957145i	$-0.406474\pi$
$379$	11.2762	0.579221	0.289610	+	0.957145i	$0.406474\pi$
$380$	0.727192	0.0373042
$381$	14.2611	0.730620
$382$	10.7507	0.550054
$383$	−1.00000	−0.0510976
$383$	−1.00000	−0.0510976
$384$	−2.81394	−0.143598
$385$	−0.563637	−0.0287256
$386$	7.40352	0.376829
$387$	3.40347	0.173008
$388$	5.27985	0.268044
$389$	−37.1338	−1.88276	−0.941379	−	0.337350i	$-0.890469\pi$
$389$	−37.1338	−1.88276	−0.941379	+	0.337350i	$0.890469\pi$
$390$	1.54895	0.0784341
$391$	16.0932	0.813866
$392$	3.07919	0.155522
$393$	−0.702414	−0.0354321
$394$	−1.19370	−0.0601375
$395$	4.08472	0.205524
$396$	−1.29166	−0.0649082
$397$	−2.24867	−0.112858	−0.0564288	−	0.998407i	$-0.517971\pi$
$397$	−2.24867	−0.112858	−0.0564288	+	0.998407i	$0.517971\pi$
$398$	16.5898	0.831572
$399$	3.80282	0.190379
$400$	11.0024	0.550118
$401$	3.70531	0.185034	0.0925171	−	0.995711i	$-0.470509\pi$
$401$	3.70531	0.185034	0.0925171	+	0.995711i	$0.470509\pi$
$402$	−2.29514	−0.114471
$403$	−41.3260	−2.05860
$404$	3.81534	0.189820
$405$	0.288867	0.0143539
$406$	−11.0928	−0.550528
$407$	−13.2705	−0.657796
$408$	−12.3612	−0.611971
$409$	1.62974	0.0805854	0.0402927	−	0.999188i	$-0.487171\pi$
$409$	1.62974	0.0805854	0.0402927	+	0.999188i	$0.487171\pi$
$410$	0.168156	0.00830463
$411$	−0.0373337	−0.00184153
$412$	−8.93501	−0.440196
$413$	3.17701	0.156331
$414$	4.63712	0.227902
$415$	2.06296	0.101267
$416$	16.5484	0.811351
$417$	4.73130	0.231693
$418$	8.58300	0.419808
$419$	23.2419	1.13544	0.567720	−	0.823221i	$-0.307825\pi$
$419$	23.2419	1.13544	0.567720	+	0.823221i	$0.307825\pi$
$420$	0.191224	0.00933079
$421$	17.5767	0.856637	0.428319	−	0.903628i	$-0.359106\pi$
$421$	17.5767	0.856637	0.428319	+	0.903628i	$0.359106\pi$
$422$	19.8701	0.967259
$423$	−3.40955	−0.165778
$424$	37.0184	1.79777
$425$	19.7372	0.957395
$426$	3.44879	0.167094
$427$	−8.50949	−0.411803
$428$	−1.18009	−0.0570419
$429$	−9.04504	−0.436699
$430$	−1.13724	−0.0548425
$431$	25.6624	1.23612	0.618058	−	0.786133i	$-0.287920\pi$
$431$	25.6624	1.23612	0.618058	+	0.786133i	$0.287920\pi$
$432$	−2.23782	−0.107667
$433$	6.36283	0.305778	0.152889	−	0.988243i	$-0.451142\pi$
$433$	6.36283	0.305778	0.152889	+	0.988243i	$0.451142\pi$
$434$	10.3121	0.494996
$435$	−2.77019	−0.132820
$436$	0.208611	0.00999067
$437$	15.2448	0.729259
$438$	7.59735	0.363015
$439$	−8.93760	−0.426568	−0.213284	−	0.976990i	$-0.568416\pi$
$439$	−8.93760	−0.426568	−0.213284	+	0.976990i	$0.568416\pi$
$440$	1.73554	0.0827388
$441$	1.00000	0.0476190
$442$	−21.5261	−1.02389
$443$	2.83273	0.134587	0.0672936	−	0.997733i	$-0.478564\pi$
$443$	2.83273	0.134587	0.0672936	+	0.997733i	$0.478564\pi$
$444$	4.50227	0.213668
$445$	4.62509	0.219250
$446$	10.6777	0.505604
$447$	−6.92612	−0.327594
$448$	−8.60496	−0.406546
$449$	26.0437	1.22908	0.614539	−	0.788886i	$-0.289342\pi$
$449$	26.0437	1.22908	0.614539	+	0.788886i	$0.289342\pi$
$450$	5.68712	0.268093
$451$	−0.981940	−0.0462378
$452$	−7.49871	−0.352710
$453$	18.2594	0.857902
$454$	−10.3492	−0.485711
$455$	1.33908	0.0627770
$456$	−11.7096	−0.548352
$457$	−20.2712	−0.948249	−0.474124	−	0.880458i	$-0.657235\pi$
$457$	−20.2712	−0.948249	−0.474124	+	0.880458i	$0.657235\pi$
$458$	−8.82764	−0.412488
$459$	−4.01444	−0.187378
$460$	0.766584	0.0357422
$461$	−24.2688	−1.13031	−0.565155	−	0.824985i	$-0.691184\pi$
$461$	−24.2688	−1.13031	−0.565155	+	0.824985i	$0.691184\pi$
$462$	2.25701	0.105006
$463$	−8.09160	−0.376048	−0.188024	−	0.982164i	$-0.560208\pi$
$463$	−8.09160	−0.376048	−0.188024	+	0.982164i	$0.560208\pi$
$464$	21.4603	0.996271
$465$	2.57521	0.119423
$466$	−6.67018	−0.308990
$467$	20.3363	0.941052	0.470526	−	0.882386i	$-0.344064\pi$
$467$	20.3363	0.941052	0.470526	+	0.882386i	$0.344064\pi$
$468$	3.06870	0.141851
$469$	−1.98416	−0.0916202
$470$	1.13927	0.0525505
$471$	21.3258	0.982640
$472$	−9.78262	−0.450281
$473$	6.64086	0.305347
$474$	−16.3567	−0.751288
$475$	18.6968	0.857867
$476$	−2.65748	−0.121805
$477$	12.0221	0.550456
$478$	−31.4273	−1.43745
$479$	−27.4100	−1.25240	−0.626198	−	0.779664i	$-0.715390\pi$
$479$	−27.4100	−1.25240	−0.626198	+	0.779664i	$0.715390\pi$
$480$	−1.03120	−0.0470678
$481$	31.5279	1.43755
$482$	23.1833	1.05597
$483$	4.00882	0.182408
$484$	4.76151	0.216432
$485$	−2.30395	−0.104617
$486$	−1.15673	−0.0524702
$487$	17.7532	0.804476	0.402238	−	0.915535i	$-0.368232\pi$
$487$	17.7532	0.804476	0.402238	+	0.915535i	$0.368232\pi$
$488$	26.2023	1.18612
$489$	12.0909	0.546767
$490$	−0.334140	−0.0150949
$491$	23.2170	1.04777	0.523884	−	0.851790i	$-0.324483\pi$
$491$	23.2170	1.04777	0.523884	+	0.851790i	$0.324483\pi$
$492$	0.333141	0.0150192
$493$	38.4978	1.73386
$494$	−20.3913	−0.917450
$495$	0.563637	0.0253336
$496$	−19.9499	−0.895776
$497$	2.98151	0.133739
$498$	−8.26085	−0.370178
$499$	−9.29186	−0.415961	−0.207980	−	0.978133i	$-0.566689\pi$
$499$	−9.29186	−0.415961	−0.207980	+	0.978133i	$0.566689\pi$
$500$	1.89629	0.0848045
$501$	−11.1225	−0.496919
$502$	−13.7100	−0.611908
$503$	16.9796	0.757083	0.378542	−	0.925584i	$-0.376426\pi$
$503$	16.9796	0.757083	0.378542	+	0.925584i	$0.376426\pi$
$504$	−3.07919	−0.137158
$505$	−1.66489	−0.0740866
$506$	9.04794	0.402230
$507$	8.48904	0.377012
$508$	−9.44059	−0.418859
$509$	35.8886	1.59073	0.795366	−	0.606129i	$-0.207279\pi$
$509$	35.8886	1.59073	0.795366	+	0.606129i	$0.207279\pi$
$510$	1.34139	0.0593975
$511$	6.56797	0.290550
$512$	21.7700	0.962105
$513$	−3.80282	−0.167899
$514$	−0.102985	−0.00454245
$515$	3.89895	0.171808
$516$	−2.25303	−0.0991843
$517$	−6.65271	−0.292586
$518$	−7.86715	−0.345663
$519$	18.0228	0.791111
$520$	−4.12328	−0.180818
$521$	−34.0775	−1.49296	−0.746482	−	0.665405i	$-0.768259\pi$
$521$	−34.0775	−1.49296	−0.746482	+	0.665405i	$0.768259\pi$
$522$	11.0928	0.485520
$523$	36.8354	1.61070	0.805349	−	0.592801i	$-0.201978\pi$
$523$	36.8354	1.61070	0.805349	+	0.592801i	$0.201978\pi$
$524$	0.464985	0.0203129
$525$	4.91656	0.214576
$526$	−18.2504	−0.795754
$527$	−35.7882	−1.55896
$528$	−4.36643	−0.190025
$529$	−6.92935	−0.301276
$530$	−4.01708	−0.174491
$531$	−3.17701	−0.137871
$532$	−2.51740	−0.109143
$533$	2.33288	0.101048
$534$	−18.5206	−0.801463
$535$	0.514953	0.0222634
$536$	6.10961	0.263895
$537$	−4.21876	−0.182053
$538$	12.7686	0.550495
$539$	1.95120	0.0840442
$540$	−0.191224	−0.00822898
$541$	−1.56650	−0.0673489	−0.0336745	−	0.999433i	$-0.510721\pi$
$541$	−1.56650	−0.0673489	−0.0336745	+	0.999433i	$0.510721\pi$
$542$	11.5241	0.495002
$543$	−18.2015	−0.781103
$544$	14.3308	0.614430
$545$	−0.0910312	−0.00389935
$546$	−5.36216	−0.229479
$547$	−10.7720	−0.460577	−0.230289	−	0.973122i	$-0.573967\pi$
$547$	−10.7720	−0.460577	−0.230289	+	0.973122i	$0.573967\pi$
$548$	0.0247142	0.00105574
$549$	8.50949	0.363176
$550$	11.0967	0.473165
$551$	36.4685	1.55361
$552$	−12.3439	−0.525392
$553$	−14.1405	−0.601315
$554$	30.1426	1.28064
$555$	−1.96465	−0.0833946
$556$	−3.13203	−0.132828
$557$	−16.5169	−0.699842	−0.349921	−	0.936779i	$-0.613792\pi$
$557$	−16.5169	−0.699842	−0.349921	+	0.936779i	$0.613792\pi$
$558$	−10.3121	−0.436545
$559$	−15.7772	−0.667306
$560$	0.646432	0.0273167
$561$	−7.83297	−0.330708
$562$	5.47151	0.230802
$563$	−18.0069	−0.758902	−0.379451	−	0.925212i	$-0.623887\pi$
$563$	−18.0069	−0.758902	−0.379451	+	0.925212i	$0.623887\pi$
$564$	2.25705	0.0950391
$565$	3.27219	0.137662
$566$	−26.7921	−1.12616
$567$	−1.00000	−0.0419961
$568$	−9.18061	−0.385210
$569$	20.0052	0.838660	0.419330	−	0.907834i	$-0.362265\pi$
$569$	20.0052	0.838660	0.419330	+	0.907834i	$0.362265\pi$
$570$	1.27068	0.0532228
$571$	−39.3340	−1.64608	−0.823038	−	0.567986i	$-0.807723\pi$
$571$	−39.3340	−1.64608	−0.823038	+	0.567986i	$0.807723\pi$
$572$	5.98764	0.250356
$573$	−9.29407	−0.388266
$574$	−0.582122	−0.0242973
$575$	19.7096	0.821947
$576$	8.60496	0.358540
$577$	−22.6397	−0.942502	−0.471251	−	0.881999i	$-0.656197\pi$
$577$	−22.6397	−0.942502	−0.471251	+	0.881999i	$0.656197\pi$
$578$	1.02288	0.0425464
$579$	−6.40040	−0.265992
$580$	1.83381	0.0761449
$581$	−7.14157	−0.296282
$582$	9.22587	0.382424
$583$	23.4576	0.971514
$584$	−20.2240	−0.836875
$585$	−1.33908	−0.0553641
$586$	9.95230	0.411126
$587$	7.47298	0.308443	0.154221	−	0.988036i	$-0.450713\pi$
$587$	7.47298	0.308443	0.154221	+	0.988036i	$0.450713\pi$
$588$	−0.661981	−0.0272996
$589$	−33.9017	−1.39689
$590$	1.06157	0.0437041
$591$	1.03196	0.0424491
$592$	15.2199	0.625534
$593$	11.6073	0.476653	0.238327	−	0.971185i	$-0.423401\pi$
$593$	11.6073	0.476653	0.238327	+	0.971185i	$0.423401\pi$
$594$	−2.25701	−0.0926062
$595$	1.15964	0.0475405
$596$	4.58496	0.187807
$597$	−14.3420	−0.586980
$598$	−21.4959	−0.879034
$599$	13.6301	0.556913	0.278456	−	0.960449i	$-0.410177\pi$
$599$	13.6301	0.556913	0.278456	+	0.960449i	$0.410177\pi$
$600$	−15.1390	−0.618047
$601$	−41.5575	−1.69516	−0.847582	−	0.530665i	$-0.821942\pi$
$601$	−41.5575	−1.69516	−0.847582	+	0.530665i	$0.821942\pi$
$602$	3.93689	0.160456
$603$	1.98416	0.0808014
$604$	−12.0874	−0.491829
$605$	−2.07777	−0.0844732
$606$	6.66682	0.270821
$607$	31.4811	1.27778	0.638890	−	0.769298i	$-0.279394\pi$
$607$	31.4811	1.27778	0.638890	+	0.769298i	$0.279394\pi$
$608$	13.5754	0.550556
$609$	9.58984	0.388600
$610$	−2.84336	−0.115124
$611$	15.8054	0.639418
$612$	2.65748	0.107422
$613$	42.6795	1.72381	0.861903	−	0.507072i	$-0.169272\pi$
$613$	42.6795	1.72381	0.861903	+	0.507072i	$0.169272\pi$
$614$	−28.0187	−1.13074
$615$	−0.145372	−0.00586197
$616$	−6.00811	−0.242074
$617$	−43.4709	−1.75007	−0.875037	−	0.484056i	$-0.839163\pi$
$617$	−43.4709	−1.75007	−0.875037	+	0.484056i	$0.839163\pi$
$618$	−15.6128	−0.628039
$619$	−32.9145	−1.32294	−0.661472	−	0.749970i	$-0.730068\pi$
$619$	−32.9145	−1.32294	−0.661472	+	0.749970i	$0.730068\pi$
$620$	−1.70474	−0.0684640
$621$	−4.00882	−0.160868
$622$	−13.7021	−0.549406
$623$	−16.0112	−0.641474
$624$	10.3737	0.415280
$625$	23.7553	0.950212
$626$	14.2321	0.568828
$627$	−7.42007	−0.296329
$628$	−14.1172	−0.563340
$629$	27.3031	1.08864
$630$	0.334140	0.0133125
$631$	−34.2330	−1.36279	−0.681397	−	0.731914i	$-0.738627\pi$
$631$	−34.2330	−1.36279	−0.681397	+	0.731914i	$0.738627\pi$
$632$	43.5412	1.73198
$633$	−17.1778	−0.682757
$634$	−7.05912	−0.280353
$635$	4.11957	0.163480
$636$	−7.95842	−0.315572
$637$	−4.63563	−0.183670
$638$	21.6444	0.856908
$639$	−2.98151	−0.117947
$640$	−0.812854	−0.0321309
$641$	−7.99580	−0.315815	−0.157908	−	0.987454i	$-0.550475\pi$
$641$	−7.99580	−0.315815	−0.157908	+	0.987454i	$0.550475\pi$
$642$	−2.06206	−0.0813830
$643$	32.5356	1.28308	0.641540	−	0.767089i	$-0.278296\pi$
$643$	32.5356	1.28308	0.641540	+	0.767089i	$0.278296\pi$
$644$	−2.65376	−0.104573
$645$	0.983150	0.0387115
$646$	−17.6588	−0.694777
$647$	23.4271	0.921016	0.460508	−	0.887656i	$-0.347667\pi$
$647$	23.4271	0.921016	0.460508	+	0.887656i	$0.347667\pi$
$648$	3.07919	0.120962
$649$	−6.19899	−0.243332
$650$	−26.3634	−1.03406
$651$	−8.91488	−0.349402
$652$	−8.00391	−0.313457
$653$	16.1466	0.631867	0.315933	−	0.948781i	$-0.397682\pi$
$653$	16.1466	0.631867	0.315933	+	0.948781i	$0.397682\pi$
$654$	0.364522	0.0142539
$655$	−0.202904	−0.00792812
$656$	1.12618	0.0439700
$657$	−6.56797	−0.256241
$658$	−3.94392	−0.153750
$659$	28.2238	1.09944	0.549722	−	0.835347i	$-0.314734\pi$
$659$	28.2238	1.09944	0.549722	+	0.835347i	$0.314734\pi$
$660$	−0.373117	−0.0145236
$661$	23.6888	0.921389	0.460694	−	0.887559i	$-0.347600\pi$
$661$	23.6888	0.921389	0.460694	+	0.887559i	$0.347600\pi$
$662$	−19.5244	−0.758837
$663$	18.6094	0.722731
$664$	21.9902	0.853386
$665$	1.09851	0.0425984
$666$	7.86715	0.304846
$667$	38.4440	1.48856
$668$	7.36291	0.284880
$669$	−9.23095	−0.356889
$670$	−0.662989	−0.0256135
$671$	16.6037	0.640980
$672$	3.56983	0.137709
$673$	−29.5026	−1.13724	−0.568620	−	0.822600i	$-0.692523\pi$
$673$	−29.5026	−1.13724	−0.568620	+	0.822600i	$0.692523\pi$
$674$	−10.6900	−0.411762
$675$	−4.91656	−0.189238
$676$	−5.61958	−0.216138
$677$	−1.26040	−0.0484411	−0.0242205	−	0.999707i	$-0.507710\pi$
$677$	−1.26040	−0.0484411	−0.0242205	+	0.999707i	$0.507710\pi$
$678$	−13.1031	−0.503220
$679$	7.97583	0.306084
$680$	−3.57074	−0.136932
$681$	8.94694	0.342848
$682$	−20.1209	−0.770471
$683$	−37.1665	−1.42214	−0.711069	−	0.703122i	$-0.751789\pi$
$683$	−37.1665	−1.42214	−0.711069	+	0.703122i	$0.751789\pi$
$684$	2.51740	0.0962550
$685$	−0.0107845	−0.000412053 0
$686$	1.15673	0.0441641
$687$	7.63156	0.291162
$688$	−7.61636	−0.290371
$689$	−55.7301	−2.12315
$690$	1.33951	0.0509942
$691$	−6.89639	−0.262351	−0.131176	−	0.991359i	$-0.541875\pi$
$691$	−6.89639	−0.262351	−0.131176	+	0.991359i	$0.541875\pi$
$692$	−11.9307	−0.453538
$693$	−1.95120	−0.0741200
$694$	−12.9913	−0.493142
$695$	1.36672	0.0518425
$696$	−29.5289	−1.11929
$697$	2.02026	0.0765229
$698$	18.0435	0.682958
$699$	5.76642	0.218106
$700$	−3.25467	−0.123015
$701$	−0.806247	−0.0304515	−0.0152258	−	0.999884i	$-0.504847\pi$
$701$	−0.806247	−0.0304515	−0.0152258	+	0.999884i	$0.504847\pi$
$702$	5.36216	0.202382
$703$	25.8638	0.975472
$704$	16.7900	0.632797
$705$	−0.984905	−0.0370937
$706$	−42.9197	−1.61530
$707$	5.76352	0.216760
$708$	2.10312	0.0790401
$709$	45.7137	1.71681	0.858407	−	0.512970i	$-0.171455\pi$
$709$	45.7137	1.71681	0.858407	+	0.512970i	$0.171455\pi$
$710$	0.996241	0.0373883
$711$	14.1405	0.530310
$712$	49.3014	1.84765
$713$	−35.7381	−1.33840
$714$	−4.64361	−0.173783
$715$	−2.61281	−0.0977136
$716$	2.79274	0.104369
$717$	27.1691	1.01465
$718$	−4.09843	−0.152952
$719$	13.0635	0.487187	0.243593	−	0.969877i	$-0.421674\pi$
$719$	13.0635	0.487187	0.243593	+	0.969877i	$0.421674\pi$
$720$	−0.646432	−0.0240911
$721$	−13.4974	−0.502669
$722$	5.24986	0.195380
$723$	−20.0421	−0.745374
$724$	12.0491	0.447800
$725$	47.1490	1.75107
$726$	8.32013	0.308789
$727$	44.5199	1.65115	0.825576	−	0.564292i	$-0.190851\pi$
$727$	44.5199	1.65115	0.825576	+	0.564292i	$0.190851\pi$
$728$	14.2740	0.529028
$729$	1.00000	0.0370370
$730$	2.19462	0.0812266
$731$	−13.6630	−0.505345
$732$	−5.63312	−0.208206
$733$	35.3546	1.30585	0.652927	−	0.757421i	$-0.273541\pi$
$733$	35.3546	1.30585	0.652927	+	0.757421i	$0.273541\pi$
$734$	−20.3991	−0.752943
$735$	0.288867	0.0106550
$736$	14.3108	0.527503
$737$	3.87150	0.142609
$738$	0.582122	0.0214282
$739$	−29.0801	−1.06973	−0.534864	−	0.844938i	$-0.679637\pi$
$739$	−29.0801	−1.06973	−0.534864	+	0.844938i	$0.679637\pi$
$740$	1.30056	0.0478095
$741$	17.6285	0.647598
$742$	13.9063	0.510518
$743$	−28.1069	−1.03114	−0.515571	−	0.856847i	$-0.672420\pi$
$743$	−28.1069	−1.03114	−0.515571	+	0.856847i	$0.672420\pi$
$744$	27.4506	1.00639
$745$	−2.00073	−0.0733009
$746$	31.6112	1.15737
$747$	7.14157	0.261296
$748$	5.18528	0.189593
$749$	−1.78267	−0.0651372
$750$	3.31352	0.120993
$751$	−30.7692	−1.12278	−0.561392	−	0.827550i	$-0.689734\pi$
$751$	−30.7692	−1.12278	−0.561392	+	0.827550i	$0.689734\pi$
$752$	7.62995	0.278236
$753$	11.8524	0.431926
$754$	−51.4223	−1.87269
$755$	5.27454	0.191960
$756$	0.661981	0.0240760
$757$	52.3543	1.90285	0.951425	−	0.307880i	$-0.0996195\pi$
$757$	52.3543	1.90285	0.951425	+	0.307880i	$0.0996195\pi$
$758$	−13.0435	−0.473762
$759$	−7.82201	−0.283921
$760$	−3.38252	−0.122697
$761$	20.5230	0.743959	0.371979	−	0.928241i	$-0.378679\pi$
$761$	20.5230	0.743959	0.371979	+	0.928241i	$0.378679\pi$
$762$	−16.4962	−0.597596
$763$	0.315132	0.0114085
$764$	6.15250	0.222590
$765$	−1.15964	−0.0419268
$766$	1.15673	0.0417943
$767$	14.7274	0.531777
$768$	−13.9550	−0.503556
$769$	−53.0862	−1.91434	−0.957169	−	0.289529i	$-0.906501\pi$
$769$	−53.0862	−1.91434	−0.957169	+	0.289529i	$0.906501\pi$
$770$	0.651975	0.0234955
$771$	0.0890310	0.00320637
$772$	4.23694	0.152491
$773$	−50.7763	−1.82630	−0.913149	−	0.407627i	$-0.866356\pi$
$773$	−50.7763	−1.82630	−0.913149	+	0.407627i	$0.866356\pi$
$774$	−3.93689	−0.141509
$775$	−43.8305	−1.57444
$776$	−24.5591	−0.881620
$777$	6.80122	0.243992
$778$	42.9537	1.53996
$779$	1.91377	0.0685678
$780$	0.886445	0.0317398
$781$	−5.81751	−0.208167
$782$	−18.6154	−0.665686
$783$	−9.58984	−0.342713
$784$	−2.23782	−0.0799221
$785$	6.16031	0.219871
$786$	0.812502	0.0289810
$787$	−27.7880	−0.990536	−0.495268	−	0.868740i	$-0.664930\pi$
$787$	−27.7880	−0.990536	−0.495268	+	0.868740i	$0.664930\pi$
$788$	−0.683137	−0.0243358
$789$	15.7776	0.561697
$790$	−4.72491	−0.168105
$791$	−11.3277	−0.402766
$792$	6.00811	0.213489
$793$	−39.4468	−1.40080
$794$	2.60110	0.0923096
$795$	3.47280	0.123167
$796$	9.49414	0.336511
$797$	−23.5364	−0.833703	−0.416852	−	0.908975i	$-0.636866\pi$
$797$	−23.5364	−0.833703	−0.416852	+	0.908975i	$0.636866\pi$
$798$	−4.39883	−0.155717
$799$	13.6874	0.484226
$800$	17.5512	0.620530
$801$	16.0112	0.565727
$802$	−4.28603	−0.151345
$803$	−12.8154	−0.452246
$804$	−1.31348	−0.0463228
$805$	1.15802	0.0408147
$806$	47.8030	1.68379
$807$	−11.0386	−0.388577
$808$	−17.7470	−0.624336
$809$	−6.64886	−0.233761	−0.116881	−	0.993146i	$-0.537290\pi$
$809$	−6.64886	−0.233761	−0.116881	+	0.993146i	$0.537290\pi$
$810$	−0.334140	−0.0117405
$811$	−15.3629	−0.539464	−0.269732	−	0.962935i	$-0.586935\pi$
$811$	−15.3629	−0.539464	−0.269732	+	0.962935i	$0.586935\pi$
$812$	−6.34829	−0.222781
$813$	−9.96267	−0.349406
$814$	15.3504	0.538031
$815$	3.49265	0.122342
$816$	8.98359	0.314488
$817$	−12.9428	−0.452811
$818$	−1.88516	−0.0659132
$819$	4.63563	0.161982
$820$	0.0962335	0.00336062
$821$	−26.1824	−0.913772	−0.456886	−	0.889525i	$-0.651035\pi$
$821$	−26.1824	−0.913772	−0.456886	+	0.889525i	$0.651035\pi$
$822$	0.0431849	0.00150625
$823$	25.4636	0.887605	0.443803	−	0.896125i	$-0.353629\pi$
$823$	25.4636	0.887605	0.443803	+	0.896125i	$0.353629\pi$
$824$	41.5610	1.44784
$825$	−9.59319	−0.333992
$826$	−3.67494	−0.127867
$827$	−27.0074	−0.939138	−0.469569	−	0.882896i	$-0.655591\pi$
$827$	−27.0074	−0.939138	−0.469569	+	0.882896i	$0.655591\pi$
$828$	2.65376	0.0922246
$829$	−19.4442	−0.675325	−0.337662	−	0.941267i	$-0.609636\pi$
$829$	−19.4442	−0.675325	−0.337662	+	0.941267i	$0.609636\pi$
$830$	−2.38629	−0.0828292
$831$	−26.0585	−0.903960
$832$	−39.8894	−1.38292
$833$	−4.01444	−0.139092
$834$	−5.47283	−0.189509
$835$	−3.21294	−0.111188
$836$	4.91194	0.169883
$837$	8.91488	0.308143
$838$	−26.8845	−0.928711
$839$	14.7805	0.510279	0.255139	−	0.966904i	$-0.417879\pi$
$839$	14.7805	0.510279	0.255139	+	0.966904i	$0.417879\pi$
$840$	−0.889475	−0.0306898
$841$	62.9651	2.17121
$842$	−20.3315	−0.700669
$843$	−4.73016	−0.162915
$844$	11.3714	0.391419
$845$	2.45220	0.0843584
$846$	3.94392	0.135595
$847$	7.19282	0.247148
$848$	−26.9034	−0.923866
$849$	23.1620	0.794917
$850$	−22.8306	−0.783082
$851$	27.2649	0.934627
$852$	1.97370	0.0676178
$853$	19.0689	0.652907	0.326453	−	0.945213i	$-0.394146\pi$
$853$	19.0689	0.652907	0.326453	+	0.945213i	$0.394146\pi$
$854$	9.84317	0.336826
$855$	−1.09851	−0.0375682
$856$	5.48916	0.187616
$857$	7.02097	0.239832	0.119916	−	0.992784i	$-0.461738\pi$
$857$	7.02097	0.239832	0.119916	+	0.992784i	$0.461738\pi$
$858$	10.4626	0.357189
$859$	−33.4378	−1.14088	−0.570442	−	0.821338i	$-0.693228\pi$
$859$	−33.4378	−1.14088	−0.570442	+	0.821338i	$0.693228\pi$
$860$	−0.650827	−0.0221930
$861$	0.503249	0.0171507
$862$	−29.6844	−1.01106
$863$	45.0489	1.53348	0.766741	−	0.641957i	$-0.221877\pi$
$863$	45.0489	1.53348	0.766741	+	0.641957i	$0.221877\pi$
$864$	−3.56983	−0.121448
$865$	5.20618	0.177015
$866$	−7.36006	−0.250105
$867$	−0.884291	−0.0300321
$868$	5.90148	0.200309
$869$	27.5909	0.935958
$870$	3.20435	0.108638
$871$	−9.19785	−0.311657
$872$	−0.970351	−0.0328602
$873$	−7.97583	−0.269941
$874$	−17.6341	−0.596483
$875$	2.86456	0.0968399
$876$	4.34787	0.146901
$877$	32.3638	1.09285	0.546424	−	0.837509i	$-0.315989\pi$
$877$	32.3638	1.09285	0.546424	+	0.837509i	$0.315989\pi$
$878$	10.3384	0.348903
$879$	−8.60384	−0.290200
$880$	−1.26132	−0.0425190
$881$	15.9666	0.537928	0.268964	−	0.963150i	$-0.413319\pi$
$881$	15.9666	0.537928	0.268964	+	0.963150i	$0.413319\pi$
$882$	−1.15673	−0.0389491
$883$	35.1444	1.18270	0.591351	−	0.806414i	$-0.298595\pi$
$883$	35.1444	1.18270	0.591351	+	0.806414i	$0.298595\pi$
$884$	−12.3191	−0.414336
$885$	−0.917734	−0.0308493
$886$	−3.27670	−0.110083
$887$	13.7812	0.462727	0.231364	−	0.972867i	$-0.425681\pi$
$887$	13.7812	0.462727	0.231364	+	0.972867i	$0.425681\pi$
$888$	−20.9422	−0.702775
$889$	−14.2611	−0.478303
$890$	−5.34997	−0.179332
$891$	1.95120	0.0653677
$892$	6.11071	0.204602
$893$	12.9659	0.433887
$894$	8.01163	0.267949
$895$	−1.21866	−0.0407353
$896$	2.81394	0.0940071
$897$	18.5834	0.620482
$898$	−30.1255	−1.00530
$899$	−85.4923	−2.85133
$900$	3.25467	0.108489
$901$	−48.2621	−1.60784
$902$	1.13584	0.0378193
$903$	−3.40347	−0.113260
$904$	34.8801	1.16009
$905$	−5.25782	−0.174776
$906$	−21.1212	−0.701704
$907$	−17.3254	−0.575281	−0.287640	−	0.957739i	$-0.592871\pi$
$907$	−17.3254	−0.575281	−0.287640	+	0.957739i	$0.592871\pi$
$908$	−5.92271	−0.196552
$909$	−5.76352	−0.191164
$910$	−1.54895	−0.0513472
$911$	−15.7931	−0.523250	−0.261625	−	0.965170i	$-0.584258\pi$
$911$	−15.7931	−0.523250	−0.261625	+	0.965170i	$0.584258\pi$
$912$	8.51003	0.281795
$913$	13.9346	0.461169
$914$	23.4483	0.775601
$915$	2.45811	0.0812626
$916$	−5.05195	−0.166921
$917$	0.702414	0.0231958
$918$	4.64361	0.153262
$919$	−30.2505	−0.997872	−0.498936	−	0.866639i	$-0.666276\pi$
$919$	−30.2505	−0.997872	−0.498936	+	0.866639i	$0.666276\pi$
$920$	−3.56575	−0.117559
$921$	24.2224	0.798155
$922$	28.0724	0.924515
$923$	13.8211	0.454929
$924$	1.29166	0.0424924
$925$	33.4386	1.09945
$926$	9.35977	0.307581
$927$	13.4974	0.443312
$928$	34.2341	1.12379
$929$	3.06014	0.100400	0.0502000	−	0.998739i	$-0.484014\pi$
$929$	3.06014	0.100400	0.0502000	+	0.998739i	$0.484014\pi$
$930$	−2.97882	−0.0976793
$931$	−3.80282	−0.124632
$932$	−3.81726	−0.125039
$933$	11.8456	0.387808
$934$	−23.5236	−0.769714
$935$	−2.26269	−0.0739977
$936$	−14.2740	−0.466559
$937$	15.2062	0.496764	0.248382	−	0.968662i	$-0.420101\pi$
$937$	15.2062	0.496764	0.248382	+	0.968662i	$0.420101\pi$
$938$	2.29514	0.0749389
$939$	−12.3037	−0.401517
$940$	0.651988	0.0212655
$941$	41.2498	1.34470	0.672352	−	0.740231i	$-0.265284\pi$
$941$	41.2498	1.34470	0.672352	+	0.740231i	$0.265284\pi$
$942$	−24.6681	−0.803730
$943$	2.01744	0.0656968
$944$	7.10958	0.231397
$945$	−0.288867	−0.00939684
$946$	−7.68166	−0.249752
$947$	−1.11196	−0.0361337	−0.0180669	−	0.999837i	$-0.505751\pi$
$947$	−1.11196	−0.0361337	−0.0180669	+	0.999837i	$0.505751\pi$
$948$	−9.36073	−0.304022
$949$	30.4466	0.988340
$950$	−21.6271	−0.701676
$951$	6.10266	0.197892
$952$	12.3612	0.400629
$953$	−17.0550	−0.552465	−0.276233	−	0.961091i	$-0.589086\pi$
$953$	−17.0550	−0.552465	−0.276233	+	0.961091i	$0.589086\pi$
$954$	−13.9063	−0.450234
$955$	−2.68475	−0.0868765
$956$	−17.9854	−0.581690
$957$	−18.7117	−0.604864
$958$	31.7060	1.02437
$959$	0.0373337	0.00120557
$960$	2.48569	0.0802252
$961$	48.4750	1.56371
$962$	−36.4692	−1.17581
$963$	1.78267	0.0574456
$964$	13.2675	0.427317
$965$	−1.84886	−0.0595170
$966$	−4.63712	−0.149197
$967$	−6.13462	−0.197276	−0.0986381	−	0.995123i	$-0.531449\pi$
$967$	−6.13462	−0.197276	−0.0986381	+	0.995123i	$0.531449\pi$
$968$	−22.1480	−0.711865
$969$	15.2662	0.490421
$970$	2.66505	0.0855695
$971$	50.4301	1.61838	0.809189	−	0.587549i	$-0.199907\pi$
$971$	50.4301	1.61838	0.809189	+	0.587549i	$0.199907\pi$
$972$	−0.661981	−0.0212330
$973$	−4.73130	−0.151679
$974$	−20.5357	−0.658005
$975$	22.7913	0.729906
$976$	−19.0427	−0.609542
$977$	56.5056	1.80777	0.903887	−	0.427771i	$-0.140701\pi$
$977$	56.5056	1.80777	0.903887	+	0.427771i	$0.140701\pi$
$978$	−13.9858	−0.447217
$979$	31.2410	0.998466
$980$	−0.191224	−0.00610844
$981$	−0.315132	−0.0100614
$982$	−26.8557	−0.857000
$983$	−22.6985	−0.723971	−0.361986	−	0.932184i	$-0.617901\pi$
$983$	−22.6985	−0.723971	−0.361986	+	0.932184i	$0.617901\pi$
$984$	−1.54960	−0.0493994
$985$	0.298099	0.00949822
$986$	−44.5315	−1.41817
$987$	3.40955	0.108527
$988$	−11.6697	−0.371263
$989$	−13.6439	−0.433851
$990$	−0.651975	−0.0207211
$991$	−50.2786	−1.59715	−0.798576	−	0.601894i	$-0.794413\pi$
$991$	−50.2786	−1.59715	−0.798576	+	0.601894i	$0.794413\pi$
$992$	−31.8246	−1.01043
$993$	16.8790	0.535638
$994$	−3.44879	−0.109389
$995$	−4.14293	−0.131340
$996$	−4.72758	−0.149799
$997$	20.1876	0.639348	0.319674	−	0.947528i	$-0.396427\pi$
$997$	20.1876	0.639348	0.319674	+	0.947528i	$0.396427\pi$
$998$	10.7482	0.340227
$999$	−6.80122	−0.215181

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8043.2.a.p.1.14	✓	41

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8043.2.a.p.1.14	✓	41	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:	\( N \)	\(=\)	\( 8043 = 3 \cdot 7 \cdot 383 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8043.a (trivial)

\(f(q)\)	\(=\)	\(q-1.15673 q^{2} +1.00000 q^{3} -0.661981 q^{4} +0.288867 q^{5} -1.15673 q^{6} -1.00000 q^{7} +3.07919 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.15673 q^{2} +1.00000 q^{3} -0.661981 q^{4} +0.288867 q^{5} -1.15673 q^{6} -1.00000 q^{7} +3.07919 q^{8} +1.00000 q^{9} -0.334140 q^{10} +1.95120 q^{11} -0.661981 q^{12} -4.63563 q^{13} +1.15673 q^{14} +0.288867 q^{15} -2.23782 q^{16} -4.01444 q^{17} -1.15673 q^{18} -3.80282 q^{19} -0.191224 q^{20} -1.00000 q^{21} -2.25701 q^{22} -4.00882 q^{23} +3.07919 q^{24} -4.91656 q^{25} +5.36216 q^{26} +1.00000 q^{27} +0.661981 q^{28} -9.58984 q^{29} -0.334140 q^{30} +8.91488 q^{31} -3.56983 q^{32} +1.95120 q^{33} +4.64361 q^{34} -0.288867 q^{35} -0.661981 q^{36} -6.80122 q^{37} +4.39883 q^{38} -4.63563 q^{39} +0.889475 q^{40} -0.503249 q^{41} +1.15673 q^{42} +3.40347 q^{43} -1.29166 q^{44} +0.288867 q^{45} +4.63712 q^{46} -3.40955 q^{47} -2.23782 q^{48} +1.00000 q^{49} +5.68712 q^{50} -4.01444 q^{51} +3.06870 q^{52} +12.0221 q^{53} -1.15673 q^{54} +0.563637 q^{55} -3.07919 q^{56} -3.80282 q^{57} +11.0928 q^{58} -3.17701 q^{59} -0.191224 q^{60} +8.50949 q^{61} -10.3121 q^{62} -1.00000 q^{63} +8.60496 q^{64} -1.33908 q^{65} -2.25701 q^{66} +1.98416 q^{67} +2.65748 q^{68} -4.00882 q^{69} +0.334140 q^{70} -2.98151 q^{71} +3.07919 q^{72} -6.56797 q^{73} +7.86715 q^{74} -4.91656 q^{75} +2.51740 q^{76} -1.95120 q^{77} +5.36216 q^{78} +14.1405 q^{79} -0.646432 q^{80} +1.00000 q^{81} +0.582122 q^{82} +7.14157 q^{83} +0.661981 q^{84} -1.15964 q^{85} -3.93689 q^{86} -9.58984 q^{87} +6.00811 q^{88} +16.0112 q^{89} -0.334140 q^{90} +4.63563 q^{91} +2.65376 q^{92} +8.91488 q^{93} +3.94392 q^{94} -1.09851 q^{95} -3.56983 q^{96} -7.97583 q^{97} -1.15673 q^{98} +1.95120 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 41 q + 7 q^{2} + 41 q^{3} + 45 q^{4} + 17 q^{5} + 7 q^{6} - 41 q^{7} + 12 q^{8} + 41 q^{9}+O(q^{10}) \) 41 * q + 7 * q^2 + 41 * q^3 + 45 * q^4 + 17 * q^5 + 7 * q^6 - 41 * q^7 + 12 * q^8 + 41 * q^9	\( 41 q + 7 q^{2} + 41 q^{3} + 45 q^{4} + 17 q^{5} + 7 q^{6} - 41 q^{7} + 12 q^{8} + 41 q^{9} + 18 q^{10} + 8 q^{11} + 45 q^{12} + 23 q^{13} - 7 q^{14} + 17 q^{15} + 37 q^{16} + 15 q^{17} + 7 q^{18} + 15 q^{19} + 53 q^{20} - 41 q^{21} + 13 q^{22} + 44 q^{23} + 12 q^{24} + 58 q^{25} + 9 q^{26} + 41 q^{27} - 45 q^{28} + 21 q^{29} + 18 q^{30} + 39 q^{31} + 61 q^{32} + 8 q^{33} + 9 q^{34} - 17 q^{35} + 45 q^{36} + 11 q^{37} + 44 q^{38} + 23 q^{39} + 24 q^{40} + 17 q^{41} - 7 q^{42} + 7 q^{43} + 30 q^{44} + 17 q^{45} - 12 q^{46} + 36 q^{47} + 37 q^{48} + 41 q^{49} + 28 q^{50} + 15 q^{51} + 58 q^{52} + 26 q^{53} + 7 q^{54} + 32 q^{55} - 12 q^{56} + 15 q^{57} - 4 q^{58} + 33 q^{59} + 53 q^{60} + 59 q^{61} - q^{62} - 41 q^{63} + 16 q^{64} + 72 q^{65} + 13 q^{66} + 12 q^{67} + 52 q^{68} + 44 q^{69} - 18 q^{70} + 33 q^{71} + 12 q^{72} + 18 q^{73} + 42 q^{74} + 58 q^{75} + 7 q^{76} - 8 q^{77} + 9 q^{78} + 22 q^{79} + 69 q^{80} + 41 q^{81} + 41 q^{82} + 32 q^{83} - 45 q^{84} - 44 q^{85} + 11 q^{86} + 21 q^{87} + 52 q^{88} + 63 q^{89} + 18 q^{90} - 23 q^{91} + 52 q^{92} + 39 q^{93} + 17 q^{94} + 37 q^{95} + 61 q^{96} + 8 q^{97} + 7 q^{98} + 8 q^{99}+O(q^{100}) \) 41 * q + 7 * q^2 + 41 * q^3 + 45 * q^4 + 17 * q^5 + 7 * q^6 - 41 * q^7 + 12 * q^8 + 41 * q^9 + 18 * q^10 + 8 * q^11 + 45 * q^12 + 23 * q^13 - 7 * q^14 + 17 * q^15 + 37 * q^16 + 15 * q^17 + 7 * q^18 + 15 * q^19 + 53 * q^20 - 41 * q^21 + 13 * q^22 + 44 * q^23 + 12 * q^24 + 58 * q^25 + 9 * q^26 + 41 * q^27 - 45 * q^28 + 21 * q^29 + 18 * q^30 + 39 * q^31 + 61 * q^32 + 8 * q^33 + 9 * q^34 - 17 * q^35 + 45 * q^36 + 11 * q^37 + 44 * q^38 + 23 * q^39 + 24 * q^40 + 17 * q^41 - 7 * q^42 + 7 * q^43 + 30 * q^44 + 17 * q^45 - 12 * q^46 + 36 * q^47 + 37 * q^48 + 41 * q^49 + 28 * q^50 + 15 * q^51 + 58 * q^52 + 26 * q^53 + 7 * q^54 + 32 * q^55 - 12 * q^56 + 15 * q^57 - 4 * q^58 + 33 * q^59 + 53 * q^60 + 59 * q^61 - q^62 - 41 * q^63 + 16 * q^64 + 72 * q^65 + 13 * q^66 + 12 * q^67 + 52 * q^68 + 44 * q^69 - 18 * q^70 + 33 * q^71 + 12 * q^72 + 18 * q^73 + 42 * q^74 + 58 * q^75 + 7 * q^76 - 8 * q^77 + 9 * q^78 + 22 * q^79 + 69 * q^80 + 41 * q^81 + 41 * q^82 + 32 * q^83 - 45 * q^84 - 44 * q^85 + 11 * q^86 + 21 * q^87 + 52 * q^88 + 63 * q^89 + 18 * q^90 - 23 * q^91 + 52 * q^92 + 39 * q^93 + 17 * q^94 + 37 * q^95 + 61 * q^96 + 8 * q^97 + 7 * q^98 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more