LMFDB - Embedded newform 8043.2.a.o.1.8

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:	$1$
Dimension:	$41$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
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)

$f(q)$	$=$	$q-1.93313 q^{2} -1.00000 q^{3} +1.73699 q^{4} -1.84501 q^{5} +1.93313 q^{6} +1.00000 q^{7} +0.508439 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.93313 q^{2} -1.00000 q^{3} +1.73699 q^{4} -1.84501 q^{5} +1.93313 q^{6} +1.00000 q^{7} +0.508439 q^{8} +1.00000 q^{9} +3.56663 q^{10} -2.53065 q^{11} -1.73699 q^{12} -4.11479 q^{13} -1.93313 q^{14} +1.84501 q^{15} -4.45685 q^{16} +3.49662 q^{17} -1.93313 q^{18} -0.814260 q^{19} -3.20475 q^{20} -1.00000 q^{21} +4.89207 q^{22} +1.63828 q^{23} -0.508439 q^{24} -1.59596 q^{25} +7.95441 q^{26} -1.00000 q^{27} +1.73699 q^{28} +2.11891 q^{29} -3.56663 q^{30} +6.81360 q^{31} +7.59879 q^{32} +2.53065 q^{33} -6.75942 q^{34} -1.84501 q^{35} +1.73699 q^{36} +2.95122 q^{37} +1.57407 q^{38} +4.11479 q^{39} -0.938073 q^{40} -8.64239 q^{41} +1.93313 q^{42} -8.24344 q^{43} -4.39571 q^{44} -1.84501 q^{45} -3.16701 q^{46} -9.08975 q^{47} +4.45685 q^{48} +1.00000 q^{49} +3.08519 q^{50} -3.49662 q^{51} -7.14733 q^{52} +11.3319 q^{53} +1.93313 q^{54} +4.66906 q^{55} +0.508439 q^{56} +0.814260 q^{57} -4.09612 q^{58} +3.32830 q^{59} +3.20475 q^{60} +7.44451 q^{61} -13.1716 q^{62} +1.00000 q^{63} -5.77573 q^{64} +7.59180 q^{65} -4.89207 q^{66} -5.96248 q^{67} +6.07358 q^{68} -1.63828 q^{69} +3.56663 q^{70} +12.9650 q^{71} +0.508439 q^{72} -5.66425 q^{73} -5.70509 q^{74} +1.59596 q^{75} -1.41436 q^{76} -2.53065 q^{77} -7.95441 q^{78} -0.662661 q^{79} +8.22291 q^{80} +1.00000 q^{81} +16.7068 q^{82} -0.249597 q^{83} -1.73699 q^{84} -6.45128 q^{85} +15.9356 q^{86} -2.11891 q^{87} -1.28668 q^{88} +14.7466 q^{89} +3.56663 q^{90} -4.11479 q^{91} +2.84568 q^{92} -6.81360 q^{93} +17.5717 q^{94} +1.50231 q^{95} -7.59879 q^{96} -2.36295 q^{97} -1.93313 q^{98} -2.53065 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 41 q - 4 q^{2} - 41 q^{3} + 34 q^{4} + 5 q^{5} + 4 q^{6} + 41 q^{7} - 15 q^{8} + 41 q^{9}+O(q^{10}) $ 41 * q - 4 * q^2 - 41 * q^3 + 34 * q^4 + 5 * q^5 + 4 * q^6 + 41 * q^7 - 15 * q^8 + 41 * q^9	\( 41 q - 4 q^{2} - 41 q^{3} + 34 q^{4} + 5 q^{5} + 4 q^{6} + 41 q^{7} - 15 q^{8} + 41 q^{9} - 18 q^{10} - 17 q^{11} - 34 q^{12} - 38 q^{13} - 4 q^{14} - 5 q^{15} + 16 q^{16} + 4 q^{17} - 4 q^{18} - 15 q^{19} + 23 q^{20} - 41 q^{21} - 31 q^{22} - 8 q^{23} + 15 q^{24} + 16 q^{25} + 15 q^{26} - 41 q^{27} + 34 q^{28} - 27 q^{29} + 18 q^{30} - 23 q^{31} - 24 q^{32} + 17 q^{33} - 9 q^{34} + 5 q^{35} + 34 q^{36} - 81 q^{37} + 38 q^{39} - 52 q^{40} + 29 q^{41} + 4 q^{42} - 47 q^{43} - 20 q^{44} + 5 q^{45} - 40 q^{46} + 26 q^{47} - 16 q^{48} + 41 q^{49} - 23 q^{50} - 4 q^{51} - 64 q^{52} - 66 q^{53} + 4 q^{54} - 14 q^{55} - 15 q^{56} + 15 q^{57} - 50 q^{58} + 41 q^{59} - 23 q^{60} - 47 q^{61} + 5 q^{62} + 41 q^{63} - 37 q^{64} - 24 q^{65} + 31 q^{66} - 73 q^{67} + 10 q^{68} + 8 q^{69} - 18 q^{70} + q^{71} - 15 q^{72} - 14 q^{73} + 18 q^{74} - 16 q^{75} - 37 q^{76} - 17 q^{77} - 15 q^{78} - 80 q^{79} + 63 q^{80} + 41 q^{81} - 31 q^{82} + 20 q^{83} - 34 q^{84} - 82 q^{85} - 39 q^{86} + 27 q^{87} - 132 q^{88} + 17 q^{89} - 18 q^{90} - 38 q^{91} - 26 q^{92} + 23 q^{93} - 9 q^{94} - q^{95} + 24 q^{96} - 73 q^{97} - 4 q^{98} - 17 q^{99}+O(q^{100}) \) 41 * q - 4 * q^2 - 41 * q^3 + 34 * q^4 + 5 * q^5 + 4 * q^6 + 41 * q^7 - 15 * q^8 + 41 * q^9 - 18 * q^10 - 17 * q^11 - 34 * q^12 - 38 * q^13 - 4 * q^14 - 5 * q^15 + 16 * q^16 + 4 * q^17 - 4 * q^18 - 15 * q^19 + 23 * q^20 - 41 * q^21 - 31 * q^22 - 8 * q^23 + 15 * q^24 + 16 * q^25 + 15 * q^26 - 41 * q^27 + 34 * q^28 - 27 * q^29 + 18 * q^30 - 23 * q^31 - 24 * q^32 + 17 * q^33 - 9 * q^34 + 5 * q^35 + 34 * q^36 - 81 * q^37 + 38 * q^39 - 52 * q^40 + 29 * q^41 + 4 * q^42 - 47 * q^43 - 20 * q^44 + 5 * q^45 - 40 * q^46 + 26 * q^47 - 16 * q^48 + 41 * q^49 - 23 * q^50 - 4 * q^51 - 64 * q^52 - 66 * q^53 + 4 * q^54 - 14 * q^55 - 15 * q^56 + 15 * q^57 - 50 * q^58 + 41 * q^59 - 23 * q^60 - 47 * q^61 + 5 * q^62 + 41 * q^63 - 37 * q^64 - 24 * q^65 + 31 * q^66 - 73 * q^67 + 10 * q^68 + 8 * q^69 - 18 * q^70 + q^71 - 15 * q^72 - 14 * q^73 + 18 * q^74 - 16 * q^75 - 37 * q^76 - 17 * q^77 - 15 * q^78 - 80 * q^79 + 63 * q^80 + 41 * q^81 - 31 * q^82 + 20 * q^83 - 34 * q^84 - 82 * q^85 - 39 * q^86 + 27 * q^87 - 132 * q^88 + 17 * q^89 - 18 * q^90 - 38 * q^91 - 26 * q^92 + 23 * q^93 - 9 * q^94 - q^95 + 24 * q^96 - 73 * q^97 - 4 * q^98 - 17 * 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 $

$2$	−1.93313	−1.36693	−0.683464	−	0.729984i	$-0.739527\pi$
$2$	−1.93313	−1.36693	−0.683464	+	0.729984i	$0.739527\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.73699	0.868493
$5$	−1.84501	−0.825111	−0.412556	−	0.910932i	$-0.635364\pi$
$5$	−1.84501	−0.825111	−0.412556	+	0.910932i	$0.635364\pi$
$6$	1.93313	0.789196
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0.508439	0.179760
$9$	1.00000	0.333333
$10$	3.56663	1.12787
$11$	−2.53065	−0.763020	−0.381510	−	0.924365i	$-0.624596\pi$
$11$	−2.53065	−0.763020	−0.381510	+	0.924365i	$0.624596\pi$
$12$	−1.73699	−0.501425
$13$	−4.11479	−1.14124	−0.570618	−	0.821216i	$-0.693296\pi$
$13$	−4.11479	−1.14124	−0.570618	+	0.821216i	$0.693296\pi$
$14$	−1.93313	−0.516650
$15$	1.84501	0.476378
$16$	−4.45685	−1.11421
$17$	3.49662	0.848055	0.424027	−	0.905649i	$-0.360616\pi$
$17$	3.49662	0.848055	0.424027	+	0.905649i	$0.360616\pi$
$18$	−1.93313	−0.455643
$19$	−0.814260	−0.186804	−0.0934020	−	0.995628i	$-0.529774\pi$
$19$	−0.814260	−0.186804	−0.0934020	+	0.995628i	$0.529774\pi$
$20$	−3.20475	−0.716604
$21$	−1.00000	−0.218218
$22$	4.89207	1.04299
$23$	1.63828	0.341606	0.170803	−	0.985305i	$-0.445364\pi$
$23$	1.63828	0.341606	0.170803	+	0.985305i	$0.445364\pi$
$24$	−0.508439	−0.103785
$25$	−1.59596	−0.319191
$26$	7.95441	1.55999
$27$	−1.00000	−0.192450
$28$	1.73699	0.328260
$29$	2.11891	0.393471	0.196736	−	0.980457i	$-0.436966\pi$
$29$	2.11891	0.393471	0.196736	+	0.980457i	$0.436966\pi$
$30$	−3.56663	−0.651175
$31$	6.81360	1.22376	0.611879	−	0.790951i	$-0.290414\pi$
$31$	6.81360	1.22376	0.611879	+	0.790951i	$0.290414\pi$
$32$	7.59879	1.34329
$33$	2.53065	0.440530
$34$	−6.75942	−1.15923
$35$	−1.84501	−0.311863
$36$	1.73699	0.289498
$37$	2.95122	0.485178	0.242589	−	0.970129i	$-0.422003\pi$
$37$	2.95122	0.485178	0.242589	+	0.970129i	$0.422003\pi$
$38$	1.57407	0.255348
$39$	4.11479	0.658893
$40$	−0.938073	−0.148322
$41$	−8.64239	−1.34971	−0.674857	−	0.737949i	$-0.735795\pi$
$41$	−8.64239	−1.34971	−0.674857	+	0.737949i	$0.735795\pi$
$42$	1.93313	0.298288
$43$	−8.24344	−1.25711	−0.628556	−	0.777764i	$-0.716354\pi$
$43$	−8.24344	−1.25711	−0.628556	+	0.777764i	$0.716354\pi$
$44$	−4.39571	−0.662678
$45$	−1.84501	−0.275037
$46$	−3.16701	−0.466951
$47$	−9.08975	−1.32588	−0.662938	−	0.748674i	$-0.730691\pi$
$47$	−9.08975	−1.32588	−0.662938	+	0.748674i	$0.730691\pi$
$48$	4.45685	0.643291
$49$	1.00000	0.142857
$50$	3.08519	0.436312
$51$	−3.49662	−0.489625
$52$	−7.14733	−0.991156
$53$	11.3319	1.55655	0.778275	−	0.627924i	$-0.216095\pi$
$53$	11.3319	1.55655	0.778275	+	0.627924i	$0.216095\pi$
$54$	1.93313	0.263065
$55$	4.66906	0.629577
$56$	0.508439	0.0679431
$57$	0.814260	0.107851
$58$	−4.09612	−0.537847
$59$	3.32830	0.433307	0.216654	−	0.976249i	$-0.430486\pi$
$59$	3.32830	0.433307	0.216654	+	0.976249i	$0.430486\pi$
$60$	3.20475	0.413731
$61$	7.44451	0.953172	0.476586	−	0.879128i	$-0.341874\pi$
$61$	7.44451	0.953172	0.476586	+	0.879128i	$0.341874\pi$
$62$	−13.1716	−1.67279
$63$	1.00000	0.125988
$64$	−5.77573	−0.721966
$65$	7.59180	0.941647
$66$	−4.89207	−0.602173
$67$	−5.96248	−0.728432	−0.364216	−	0.931314i	$-0.618663\pi$
$67$	−5.96248	−0.728432	−0.364216	+	0.931314i	$0.618663\pi$
$68$	6.07358	0.736530
$69$	−1.63828	−0.197226
$70$	3.56663	0.426294
$71$	12.9650	1.53866	0.769329	−	0.638853i	$-0.220591\pi$
$71$	12.9650	1.53866	0.769329	+	0.638853i	$0.220591\pi$
$72$	0.508439	0.0599202
$73$	−5.66425	−0.662951	−0.331475	−	0.943464i	$-0.607546\pi$
$73$	−5.66425	−0.662951	−0.331475	+	0.943464i	$0.607546\pi$
$74$	−5.70509	−0.663204
$75$	1.59596	0.184285
$76$	−1.41436	−0.162238
$77$	−2.53065	−0.288394
$78$	−7.95441	−0.900660
$79$	−0.662661	−0.0745552	−0.0372776	−	0.999305i	$-0.511869\pi$
$79$	−0.662661	−0.0745552	−0.0372776	+	0.999305i	$0.511869\pi$
$80$	8.22291	0.919350
$81$	1.00000	0.111111
$82$	16.7068	1.84496
$83$	−0.249597	−0.0273968	−0.0136984	−	0.999906i	$-0.504360\pi$
$83$	−0.249597	−0.0273968	−0.0136984	+	0.999906i	$0.504360\pi$
$84$	−1.73699	−0.189521
$85$	−6.45128	−0.699740
$86$	15.9356	1.71838
$87$	−2.11891	−0.227171
$88$	−1.28668	−0.137161
$89$	14.7466	1.56314	0.781570	−	0.623817i	$-0.214419\pi$
$89$	14.7466	1.56314	0.781570	+	0.623817i	$0.214419\pi$
$90$	3.56663	0.375956
$91$	−4.11479	−0.431347
$92$	2.84568	0.296682
$93$	−6.81360	−0.706537
$94$	17.5717	1.81238
$95$	1.50231	0.154134
$96$	−7.59879	−0.775548
$97$	−2.36295	−0.239921	−0.119960	−	0.992779i	$-0.538277\pi$
$97$	−2.36295	−0.239921	−0.119960	+	0.992779i	$0.538277\pi$
$98$	−1.93313	−0.195275
$99$	−2.53065	−0.254340
$100$	−2.77215	−0.277215
$101$	−11.5593	−1.15019	−0.575097	−	0.818085i	$-0.695036\pi$
$101$	−11.5593	−1.15019	−0.575097	+	0.818085i	$0.695036\pi$
$102$	6.75942	0.669282
$103$	19.6642	1.93757	0.968786	−	0.247898i	$-0.0797399\pi$
$103$	19.6642	1.93757	0.968786	+	0.247898i	$0.0797399\pi$
$104$	−2.09212	−0.205149
$105$	1.84501	0.180054
$106$	−21.9059	−2.12769
$107$	0.486586	0.0470400	0.0235200	−	0.999723i	$-0.492513\pi$
$107$	0.486586	0.0470400	0.0235200	+	0.999723i	$0.492513\pi$
$108$	−1.73699	−0.167142
$109$	−11.2785	−1.08029	−0.540144	−	0.841573i	$-0.681630\pi$
$109$	−11.2785	−1.08029	−0.540144	+	0.841573i	$0.681630\pi$
$110$	−9.02590	−0.860586
$111$	−2.95122	−0.280118
$112$	−4.45685	−0.421133
$113$	−16.8569	−1.58576	−0.792882	−	0.609375i	$-0.791420\pi$
$113$	−16.8569	−1.58576	−0.792882	+	0.609375i	$0.791420\pi$
$114$	−1.57407	−0.147425
$115$	−3.02264	−0.281863
$116$	3.68051	0.341727
$117$	−4.11479	−0.380412
$118$	−6.43403	−0.592300
$119$	3.49662	0.320535
$120$	0.938073	0.0856340
$121$	−4.59580	−0.417800
$122$	−14.3912	−1.30292
$123$	8.64239	0.779258
$124$	11.8351	1.06283
$125$	12.1696	1.08848
$126$	−1.93313	−0.172217
$127$	7.82477	0.694336	0.347168	−	0.937803i	$-0.387143\pi$
$127$	7.82477	0.694336	0.347168	+	0.937803i	$0.387143\pi$
$128$	−4.03234	−0.356412
$129$	8.24344	0.725794
$130$	−14.6759	−1.28716
$131$	17.9815	1.57105	0.785526	−	0.618828i	$-0.212392\pi$
$131$	17.9815	1.57105	0.785526	+	0.618828i	$0.212392\pi$
$132$	4.39571	0.382597
$133$	−0.814260	−0.0706053
$134$	11.5262	0.995715
$135$	1.84501	0.158793
$136$	1.77782	0.152447
$137$	3.75998	0.321237	0.160618	−	0.987017i	$-0.448651\pi$
$137$	3.75998	0.321237	0.160618	+	0.987017i	$0.448651\pi$
$138$	3.16701	0.269594
$139$	−9.88068	−0.838068	−0.419034	−	0.907971i	$-0.637631\pi$
$139$	−9.88068	−0.838068	−0.419034	+	0.907971i	$0.637631\pi$
$140$	−3.20475	−0.270851
$141$	9.08975	0.765495
$142$	−25.0629	−2.10323
$143$	10.4131	0.870786
$144$	−4.45685	−0.371404
$145$	−3.90940	−0.324658
$146$	10.9497	0.906206
$147$	−1.00000	−0.0824786
$148$	5.12623	0.421374
$149$	−3.86994	−0.317038	−0.158519	−	0.987356i	$-0.550672\pi$
$149$	−3.86994	−0.317038	−0.158519	+	0.987356i	$0.550672\pi$
$150$	−3.08519	−0.251905
$151$	−15.8058	−1.28626	−0.643128	−	0.765759i	$-0.722364\pi$
$151$	−15.8058	−1.28626	−0.643128	+	0.765759i	$0.722364\pi$
$152$	−0.414002	−0.0335800
$153$	3.49662	0.282685
$154$	4.89207	0.394215
$155$	−12.5711	−1.00974
$156$	7.14733	0.572244
$157$	3.67568	0.293351	0.146676	−	0.989185i	$-0.453143\pi$
$157$	3.67568	0.293351	0.146676	+	0.989185i	$0.453143\pi$
$158$	1.28101	0.101912
$159$	−11.3319	−0.898674
$160$	−14.0198	−1.10836
$161$	1.63828	0.129115
$162$	−1.93313	−0.151881
$163$	−21.9414	−1.71858	−0.859292	−	0.511485i	$-0.829096\pi$
$163$	−21.9414	−1.71858	−0.859292	+	0.511485i	$0.829096\pi$
$164$	−15.0117	−1.17222
$165$	−4.66906	−0.363486
$166$	0.482503	0.0374495
$167$	8.62009	0.667042	0.333521	−	0.942743i	$-0.391763\pi$
$167$	8.62009	0.667042	0.333521	+	0.942743i	$0.391763\pi$
$168$	−0.508439	−0.0392270
$169$	3.93146	0.302420
$170$	12.4712	0.956494
$171$	−0.814260	−0.0622680
$172$	−14.3187	−1.09179
$173$	21.6113	1.64308	0.821539	−	0.570152i	$-0.193116\pi$
$173$	21.6113	1.64308	0.821539	+	0.570152i	$0.193116\pi$
$174$	4.09612	0.310526
$175$	−1.59596	−0.120643
$176$	11.2787	0.850167
$177$	−3.32830	−0.250170
$178$	−28.5072	−2.13670
$179$	−11.8528	−0.885919	−0.442960	−	0.896542i	$-0.646071\pi$
$179$	−11.8528	−0.885919	−0.442960	+	0.896542i	$0.646071\pi$
$180$	−3.20475	−0.238868
$181$	−19.9768	−1.48486	−0.742432	−	0.669922i	$-0.766328\pi$
$181$	−19.9768	−1.48486	−0.742432	+	0.669922i	$0.766328\pi$
$182$	7.95441	0.589620
$183$	−7.44451	−0.550314
$184$	0.832968	0.0614072
$185$	−5.44502	−0.400326
$186$	13.1716	0.965785
$187$	−8.84873	−0.647083
$188$	−15.7888	−1.15151
$189$	−1.00000	−0.0727393
$190$	−2.90417	−0.210690
$191$	−0.628407	−0.0454699	−0.0227350	−	0.999742i	$-0.507237\pi$
$191$	−0.628407	−0.0454699	−0.0227350	+	0.999742i	$0.507237\pi$
$192$	5.77573	0.416828
$193$	17.2565	1.24215	0.621076	−	0.783750i	$-0.286696\pi$
$193$	17.2565	1.24215	0.621076	+	0.783750i	$0.286696\pi$
$194$	4.56788	0.327954
$195$	−7.59180	−0.543660
$196$	1.73699	0.124070
$197$	16.1683	1.15194	0.575971	−	0.817470i	$-0.304624\pi$
$197$	16.1683	1.15194	0.575971	+	0.817470i	$0.304624\pi$
$198$	4.89207	0.347665
$199$	−8.69162	−0.616133	−0.308066	−	0.951365i	$-0.599682\pi$
$199$	−8.69162	−0.616133	−0.308066	+	0.951365i	$0.599682\pi$
$200$	−0.811447	−0.0573780
$201$	5.96248	0.420561
$202$	22.3456	1.57223
$203$	2.11891	0.148718
$204$	−6.07358	−0.425236
$205$	15.9452	1.11366
$206$	−38.0134	−2.64852
$207$	1.63828	0.113869
$208$	18.3390	1.27158
$209$	2.06061	0.142535
$210$	−3.56663	−0.246121
$211$	8.76503	0.603410	0.301705	−	0.953401i	$-0.402444\pi$
$211$	8.76503	0.603410	0.301705	+	0.953401i	$0.402444\pi$
$212$	19.6833	1.35185
$213$	−12.9650	−0.888344
$214$	−0.940633	−0.0643004
$215$	15.2092	1.03726
$216$	−0.508439	−0.0345949
$217$	6.81360	0.462537
$218$	21.8028	1.47668
$219$	5.66425	0.382755
$220$	8.11010	0.546783
$221$	−14.3878	−0.967831
$222$	5.70509	0.382901
$223$	16.4392	1.10085	0.550426	−	0.834884i	$-0.314465\pi$
$223$	16.4392	1.10085	0.550426	+	0.834884i	$0.314465\pi$
$224$	7.59879	0.507715
$225$	−1.59596	−0.106397
$226$	32.5866	2.16763
$227$	4.23662	0.281194	0.140597	−	0.990067i	$-0.455098\pi$
$227$	4.23662	0.281194	0.140597	+	0.990067i	$0.455098\pi$
$228$	1.41436	0.0936682
$229$	−9.20539	−0.608309	−0.304155	−	0.952623i	$-0.598374\pi$
$229$	−9.20539	−0.608309	−0.304155	+	0.952623i	$0.598374\pi$
$230$	5.84316	0.385286
$231$	2.53065	0.166505
$232$	1.07734	0.0707306
$233$	28.3586	1.85783	0.928916	−	0.370290i	$-0.120742\pi$
$233$	28.3586	1.85783	0.928916	+	0.370290i	$0.120742\pi$
$234$	7.95441	0.519996
$235$	16.7706	1.09400
$236$	5.78121	0.376324
$237$	0.662661	0.0430444
$238$	−6.75942	−0.438148
$239$	19.0416	1.23170	0.615851	−	0.787863i	$-0.288813\pi$
$239$	19.0416	1.23170	0.615851	+	0.787863i	$0.288813\pi$
$240$	−8.22291	−0.530787
$241$	2.65414	0.170968	0.0854842	−	0.996340i	$-0.472756\pi$
$241$	2.65414	0.170968	0.0854842	+	0.996340i	$0.472756\pi$
$242$	8.88428	0.571103
$243$	−1.00000	−0.0641500
$244$	12.9310	0.827823
$245$	−1.84501	−0.117873
$246$	−16.7068	−1.06519
$247$	3.35051	0.213188
$248$	3.46430	0.219983
$249$	0.249597	0.0158176
$250$	−23.5253	−1.48787
$251$	9.49067	0.599046	0.299523	−	0.954089i	$-0.403172\pi$
$251$	9.49067	0.599046	0.299523	+	0.954089i	$0.403172\pi$
$252$	1.73699	0.109420
$253$	−4.14593	−0.260652
$254$	−15.1263	−0.949107
$255$	6.45128	0.403995
$256$	19.3465	1.20916
$257$	7.84837	0.489568	0.244784	−	0.969578i	$-0.421283\pi$
$257$	7.84837	0.489568	0.244784	+	0.969578i	$0.421283\pi$
$258$	−15.9356	−0.992109
$259$	2.95122	0.183380
$260$	13.1869	0.817814
$261$	2.11891	0.131157
$262$	−34.7606	−2.14752
$263$	−14.2479	−0.878563	−0.439281	−	0.898350i	$-0.644767\pi$
$263$	−14.2479	−0.878563	−0.439281	+	0.898350i	$0.644767\pi$
$264$	1.28668	0.0791899
$265$	−20.9073	−1.28433
$266$	1.57407	0.0965124
$267$	−14.7466	−0.902480
$268$	−10.3567	−0.632638
$269$	31.5484	1.92354	0.961771	−	0.273854i	$-0.0882986\pi$
$269$	31.5484	1.92354	0.961771	+	0.273854i	$0.0882986\pi$
$270$	−3.56663	−0.217058
$271$	22.2693	1.35276	0.676381	−	0.736552i	$-0.263547\pi$
$271$	22.2693	1.35276	0.676381	+	0.736552i	$0.263547\pi$
$272$	−15.5839	−0.944914
$273$	4.11479	0.249038
$274$	−7.26852	−0.439107
$275$	4.03881	0.243549
$276$	−2.84568	−0.171290
$277$	15.6701	0.941528	0.470764	−	0.882259i	$-0.343978\pi$
$277$	15.6701	0.941528	0.470764	+	0.882259i	$0.343978\pi$
$278$	19.1006	1.14558
$279$	6.81360	0.407919
$280$	−0.938073	−0.0560606
$281$	−2.53820	−0.151416	−0.0757082	−	0.997130i	$-0.524122\pi$
$281$	−2.53820	−0.151416	−0.0757082	+	0.997130i	$0.524122\pi$
$282$	−17.5717	−1.04638
$283$	−22.1456	−1.31642	−0.658208	−	0.752836i	$-0.728685\pi$
$283$	−22.1456	−1.31642	−0.658208	+	0.752836i	$0.728685\pi$
$284$	22.5200	1.33631
$285$	−1.50231	−0.0889894
$286$	−20.1298	−1.19030
$287$	−8.64239	−0.510144
$288$	7.59879	0.447763
$289$	−4.77365	−0.280803
$290$	7.55737	0.443784
$291$	2.36295	0.138518
$292$	−9.83873	−0.575768
$293$	3.19490	0.186648	0.0933241	−	0.995636i	$-0.470251\pi$
$293$	3.19490	0.186648	0.0933241	+	0.995636i	$0.470251\pi$
$294$	1.93313	0.112742
$295$	−6.14073	−0.357527
$296$	1.50052	0.0872158
$297$	2.53065	0.146843
$298$	7.48109	0.433368
$299$	−6.74119	−0.389853
$300$	2.77215	0.160050
$301$	−8.24344	−0.475144
$302$	30.5546	1.75822
$303$	11.5593	0.664065
$304$	3.62904	0.208140
$305$	−13.7352	−0.786473
$306$	−6.75942	−0.386410
$307$	9.34327	0.533248	0.266624	−	0.963801i	$-0.414092\pi$
$307$	9.34327	0.533248	0.266624	+	0.963801i	$0.414092\pi$
$308$	−4.39571	−0.250469
$309$	−19.6642	−1.11866
$310$	24.3016	1.38024
$311$	20.2084	1.14591	0.572955	−	0.819587i	$-0.305797\pi$
$311$	20.2084	1.14591	0.572955	+	0.819587i	$0.305797\pi$
$312$	2.09212	0.118443
$313$	−3.51995	−0.198959	−0.0994795	−	0.995040i	$-0.531718\pi$
$313$	−3.51995	−0.198959	−0.0994795	+	0.995040i	$0.531718\pi$
$314$	−7.10557	−0.400990
$315$	−1.84501	−0.103954
$316$	−1.15103	−0.0647507
$317$	−11.0679	−0.621638	−0.310819	−	0.950469i	$-0.600603\pi$
$317$	−11.0679	−0.621638	−0.310819	+	0.950469i	$0.600603\pi$
$318$	21.9059	1.22842
$319$	−5.36222	−0.300227
$320$	10.6563	0.595703
$321$	−0.486586	−0.0271586
$322$	−3.16701	−0.176491
$323$	−2.84716	−0.158420
$324$	1.73699	0.0964992
$325$	6.56702	0.364273
$326$	42.4156	2.34918
$327$	11.2785	0.623704
$328$	−4.39413	−0.242625
$329$	−9.08975	−0.501134
$330$	9.02590	0.496860
$331$	19.7699	1.08665	0.543325	−	0.839523i	$-0.317165\pi$
$331$	19.7699	1.08665	0.543325	+	0.839523i	$0.317165\pi$
$332$	−0.433547	−0.0237940
$333$	2.95122	0.161726
$334$	−16.6637	−0.911799
$335$	11.0008	0.601038
$336$	4.45685	0.243141
$337$	−13.9388	−0.759297	−0.379648	−	0.925131i	$-0.623955\pi$
$337$	−13.9388	−0.759297	−0.379648	+	0.925131i	$0.623955\pi$
$338$	−7.60002	−0.413386
$339$	16.8569	0.915541
$340$	−11.2058	−0.607719
$341$	−17.2428	−0.933752
$342$	1.57407	0.0851159
$343$	1.00000	0.0539949
$344$	−4.19129	−0.225979
$345$	3.02264	0.162734
$346$	−41.7775	−2.24597
$347$	−25.9435	−1.39272	−0.696361	−	0.717692i	$-0.745199\pi$
$347$	−25.9435	−1.39272	−0.696361	+	0.717692i	$0.745199\pi$
$348$	−3.68051	−0.197296
$349$	−2.39576	−0.128242	−0.0641210	−	0.997942i	$-0.520424\pi$
$349$	−2.39576	−0.128242	−0.0641210	+	0.997942i	$0.520424\pi$
$350$	3.08519	0.164910
$351$	4.11479	0.219631
$352$	−19.2299	−1.02496
$353$	20.7296	1.10332	0.551662	−	0.834068i	$-0.313994\pi$
$353$	20.7296	1.10332	0.551662	+	0.834068i	$0.313994\pi$
$354$	6.43403	0.341965
$355$	−23.9204	−1.26956
$356$	25.6147	1.35758
$357$	−3.49662	−0.185061
$358$	22.9130	1.21099
$359$	26.5556	1.40155	0.700776	−	0.713381i	$-0.252837\pi$
$359$	26.5556	1.40155	0.700776	+	0.713381i	$0.252837\pi$
$360$	−0.938073	−0.0494408
$361$	−18.3370	−0.965104
$362$	38.6177	2.02970
$363$	4.59580	0.241217
$364$	−7.14733	−0.374622
$365$	10.4506	0.547008
$366$	14.3912	0.752240
$367$	−24.2365	−1.26514	−0.632568	−	0.774505i	$-0.717999\pi$
$367$	−24.2365	−1.26514	−0.632568	+	0.774505i	$0.717999\pi$
$368$	−7.30159	−0.380622
$369$	−8.64239	−0.449905
$370$	10.5259	0.547217
$371$	11.3319	0.588321
$372$	−11.8351	−0.613623
$373$	8.44388	0.437208	0.218604	−	0.975814i	$-0.429850\pi$
$373$	8.44388	0.437208	0.218604	+	0.975814i	$0.429850\pi$
$374$	17.1057	0.884516
$375$	−12.1696	−0.628434
$376$	−4.62159	−0.238340
$377$	−8.71885	−0.449044
$378$	1.93313	0.0994294
$379$	5.57275	0.286253	0.143127	−	0.989704i	$-0.454284\pi$
$379$	5.57275	0.286253	0.143127	+	0.989704i	$0.454284\pi$
$380$	2.60950	0.133864
$381$	−7.82477	−0.400875
$382$	1.21479	0.0621541
$383$	1.00000	0.0510976
$383$	1.00000	0.0510976
$384$	4.03234	0.205775
$385$	4.66906	0.237958
$386$	−33.3591	−1.69793
$387$	−8.24344	−0.419038
$388$	−4.10440	−0.208370
$389$	−17.0201	−0.862951	−0.431476	−	0.902125i	$-0.642007\pi$
$389$	−17.0201	−0.862951	−0.431476	+	0.902125i	$0.642007\pi$
$390$	14.6759	0.743144
$391$	5.72846	0.289700
$392$	0.508439	0.0256801
$393$	−17.9815	−0.907048
$394$	−31.2553	−1.57462
$395$	1.22261	0.0615163
$396$	−4.39571	−0.220893
$397$	−27.7456	−1.39251	−0.696257	−	0.717793i	$-0.745153\pi$
$397$	−27.7456	−1.39251	−0.696257	+	0.717793i	$0.745153\pi$
$398$	16.8020	0.842209
$399$	0.814260	0.0407640
$400$	7.11294	0.355647
$401$	2.89631	0.144635	0.0723173	−	0.997382i	$-0.476961\pi$
$401$	2.89631	0.144635	0.0723173	+	0.997382i	$0.476961\pi$
$402$	−11.5262	−0.574876
$403$	−28.0365	−1.39660
$404$	−20.0784	−0.998936
$405$	−1.84501	−0.0916790
$406$	−4.09612	−0.203287
$407$	−7.46852	−0.370201
$408$	−1.77782	−0.0880152
$409$	−5.80022	−0.286802	−0.143401	−	0.989665i	$-0.545804\pi$
$409$	−5.80022	−0.286802	−0.143401	+	0.989665i	$0.545804\pi$
$410$	−30.8242	−1.52230
$411$	−3.75998	−0.185466
$412$	34.1565	1.68277
$413$	3.32830	0.163775
$414$	−3.16701	−0.155650
$415$	0.460508	0.0226054
$416$	−31.2674	−1.53301
$417$	9.88068	0.483859
$418$	−3.98342	−0.194835
$419$	2.60911	0.127463	0.0637316	−	0.997967i	$-0.479700\pi$
$419$	2.60911	0.127463	0.0637316	+	0.997967i	$0.479700\pi$
$420$	3.20475	0.156376
$421$	1.01273	0.0493577	0.0246788	−	0.999695i	$-0.492144\pi$
$421$	1.01273	0.0493577	0.0246788	+	0.999695i	$0.492144\pi$
$422$	−16.9439	−0.824818
$423$	−9.08975	−0.441959
$424$	5.76156	0.279806
$425$	−5.58045	−0.270692
$426$	25.0629	1.21430
$427$	7.44451	0.360265
$428$	0.845193	0.0408539
$429$	−10.4131	−0.502749
$430$	−29.4013	−1.41786
$431$	22.4566	1.08170	0.540849	−	0.841120i	$-0.318103\pi$
$431$	22.4566	1.08170	0.540849	+	0.841120i	$0.318103\pi$
$432$	4.45685	0.214430
$433$	−30.0977	−1.44641	−0.723203	−	0.690636i	$-0.757331\pi$
$433$	−30.0977	−1.44641	−0.723203	+	0.690636i	$0.757331\pi$
$434$	−13.1716	−0.632255
$435$	3.90940	0.187441
$436$	−19.5907	−0.938222
$437$	−1.33399	−0.0638134
$438$	−10.9497	−0.523198
$439$	19.7744	0.943779	0.471889	−	0.881658i	$-0.343572\pi$
$439$	19.7744	0.943779	0.471889	+	0.881658i	$0.343572\pi$
$440$	2.37394	0.113173
$441$	1.00000	0.0476190
$442$	27.8135	1.32296
$443$	−29.3343	−1.39371	−0.696857	−	0.717210i	$-0.745419\pi$
$443$	−29.3343	−1.39371	−0.696857	+	0.717210i	$0.745419\pi$
$444$	−5.12623	−0.243280
$445$	−27.2076	−1.28977
$446$	−31.7792	−1.50479
$447$	3.86994	0.183042
$448$	−5.77573	−0.272878
$449$	−22.1621	−1.04590	−0.522948	−	0.852365i	$-0.675168\pi$
$449$	−22.1621	−1.04590	−0.522948	+	0.852365i	$0.675168\pi$
$450$	3.08519	0.145437
$451$	21.8709	1.02986
$452$	−29.2802	−1.37723
$453$	15.8058	0.742620
$454$	−8.18993	−0.384372
$455$	7.59180	0.355909
$456$	0.414002	0.0193874
$457$	28.4595	1.33128	0.665641	−	0.746272i	$-0.268158\pi$
$457$	28.4595	1.33128	0.665641	+	0.746272i	$0.268158\pi$
$458$	17.7952	0.831515
$459$	−3.49662	−0.163208
$460$	−5.25029	−0.244796
$461$	−20.3143	−0.946129	−0.473065	−	0.881028i	$-0.656852\pi$
$461$	−20.3143	−0.946129	−0.473065	+	0.881028i	$0.656852\pi$
$462$	−4.89207	−0.227600
$463$	0.952361	0.0442600	0.0221300	−	0.999755i	$-0.492955\pi$
$463$	0.952361	0.0442600	0.0221300	+	0.999755i	$0.492955\pi$
$464$	−9.44366	−0.438411
$465$	12.5711	0.582972
$466$	−54.8208	−2.53952
$467$	−36.6798	−1.69734	−0.848669	−	0.528925i	$-0.822595\pi$
$467$	−36.6798	−1.69734	−0.848669	+	0.528925i	$0.822595\pi$
$468$	−7.14733	−0.330385
$469$	−5.96248	−0.275322
$470$	−32.4198	−1.49541
$471$	−3.67568	−0.169367
$472$	1.69224	0.0778915
$473$	20.8613	0.959202
$474$	−1.28101	−0.0588387
$475$	1.29952	0.0596262
$476$	6.07358	0.278382
$477$	11.3319	0.518850
$478$	−36.8099	−1.68365
$479$	−34.8777	−1.59360	−0.796802	−	0.604240i	$-0.793477\pi$
$479$	−34.8777	−1.59360	−0.796802	+	0.604240i	$0.793477\pi$
$480$	14.0198	0.639914
$481$	−12.1436	−0.553703
$482$	−5.13080	−0.233701
$483$	−1.63828	−0.0745445
$484$	−7.98285	−0.362857
$485$	4.35965	0.197961
$486$	1.93313	0.0876885
$487$	−26.8379	−1.21614	−0.608070	−	0.793884i	$-0.708056\pi$
$487$	−26.8379	−1.21614	−0.608070	+	0.793884i	$0.708056\pi$
$488$	3.78508	0.171343
$489$	21.9414	0.992225
$490$	3.56663	0.161124
$491$	−21.9713	−0.991550	−0.495775	−	0.868451i	$-0.665116\pi$
$491$	−21.9713	−0.991550	−0.495775	+	0.868451i	$0.665116\pi$
$492$	15.0117	0.676780
$493$	7.40902	0.333685
$494$	−6.47696	−0.291412
$495$	4.66906	0.209859
$496$	−30.3672	−1.36353
$497$	12.9650	0.581558
$498$	−0.482503	−0.0216215
$499$	33.0369	1.47893	0.739467	−	0.673193i	$-0.235078\pi$
$499$	33.0369	1.47893	0.739467	+	0.673193i	$0.235078\pi$
$500$	21.1384	0.945337
$501$	−8.62009	−0.385117
$502$	−18.3467	−0.818853
$503$	−26.4088	−1.17751	−0.588756	−	0.808311i	$-0.700382\pi$
$503$	−26.4088	−1.17751	−0.588756	+	0.808311i	$0.700382\pi$
$504$	0.508439	0.0226477
$505$	21.3270	0.949038
$506$	8.01461	0.356293
$507$	−3.93146	−0.174602
$508$	13.5915	0.603026
$509$	−10.7751	−0.477599	−0.238799	−	0.971069i	$-0.576754\pi$
$509$	−10.7751	−0.477599	−0.238799	+	0.971069i	$0.576754\pi$
$510$	−12.4712	−0.552232
$511$	−5.66425	−0.250572
$512$	−29.3346	−1.29642
$513$	0.814260	0.0359505
$514$	−15.1719	−0.669204
$515$	−36.2806	−1.59871
$516$	14.3187	0.630347
$517$	23.0030	1.01167
$518$	−5.70509	−0.250667
$519$	−21.6113	−0.948632
$520$	3.85997	0.169271
$521$	20.2604	0.887626	0.443813	−	0.896119i	$-0.353625\pi$
$521$	20.2604	0.887626	0.443813	+	0.896119i	$0.353625\pi$
$522$	−4.09612	−0.179282
$523$	26.2105	1.14611	0.573053	−	0.819519i	$-0.305759\pi$
$523$	26.2105	1.14611	0.573053	+	0.819519i	$0.305759\pi$
$524$	31.2337	1.36445
$525$	1.59596	0.0696532
$526$	27.5430	1.20093
$527$	23.8246	1.03781
$528$	−11.2787	−0.490844
$529$	−20.3160	−0.883305
$530$	40.4166	1.75558
$531$	3.32830	0.144436
$532$	−1.41436	−0.0613202
$533$	35.5616	1.54034
$534$	28.5072	1.23363
$535$	−0.897753	−0.0388133
$536$	−3.03156	−0.130943
$537$	11.8528	0.511486
$538$	−60.9872	−2.62934
$539$	−2.53065	−0.109003
$540$	3.20475	0.137910
$541$	18.5434	0.797244	0.398622	−	0.917115i	$-0.369488\pi$
$541$	18.5434	0.797244	0.398622	+	0.917115i	$0.369488\pi$
$542$	−43.0493	−1.84913
$543$	19.9768	0.857286
$544$	26.5701	1.13918
$545$	20.8089	0.891357
$546$	−7.95441	−0.340417
$547$	−5.76445	−0.246470	−0.123235	−	0.992378i	$-0.539327\pi$
$547$	−5.76445	−0.246470	−0.123235	+	0.992378i	$0.539327\pi$
$548$	6.53103	0.278992
$549$	7.44451	0.317724
$550$	−7.80754	−0.332914
$551$	−1.72534	−0.0735021
$552$	−0.832968	−0.0354535
$553$	−0.662661	−0.0281792
$554$	−30.2924	−1.28700
$555$	5.44502	0.231128
$556$	−17.1626	−0.727856
$557$	−3.42992	−0.145331	−0.0726653	−	0.997356i	$-0.523150\pi$
$557$	−3.42992	−0.145331	−0.0726653	+	0.997356i	$0.523150\pi$
$558$	−13.1716	−0.557596
$559$	33.9200	1.43466
$560$	8.22291	0.347482
$561$	8.84873	0.373593
$562$	4.90667	0.206975
$563$	29.7327	1.25308	0.626542	−	0.779388i	$-0.284470\pi$
$563$	29.7327	1.25308	0.626542	+	0.779388i	$0.284470\pi$
$564$	15.7888	0.664827
$565$	31.1011	1.30843
$566$	42.8102	1.79945
$567$	1.00000	0.0419961
$568$	6.59189	0.276590
$569$	−19.9272	−0.835392	−0.417696	−	0.908587i	$-0.637162\pi$
$569$	−19.9272	−0.835392	−0.417696	+	0.908587i	$0.637162\pi$
$570$	2.90417	0.121642
$571$	−37.5446	−1.57119	−0.785596	−	0.618740i	$-0.787644\pi$
$571$	−37.5446	−1.57119	−0.785596	+	0.618740i	$0.787644\pi$
$572$	18.0874	0.756272
$573$	0.628407	0.0262521
$574$	16.7068	0.697330
$575$	−2.61463	−0.109038
$576$	−5.77573	−0.240655
$577$	−10.4545	−0.435225	−0.217612	−	0.976035i	$-0.569827\pi$
$577$	−10.4545	−0.435225	−0.217612	+	0.976035i	$0.569827\pi$
$578$	9.22808	0.383838
$579$	−17.2565	−0.717157
$580$	−6.79057	−0.281963
$581$	−0.249597	−0.0103550
$582$	−4.56788	−0.189345
$583$	−28.6770	−1.18768
$584$	−2.87993	−0.119172
$585$	7.59180	0.313882
$586$	−6.17616	−0.255135
$587$	−34.7103	−1.43265	−0.716324	−	0.697768i	$-0.754177\pi$
$587$	−34.7103	−1.43265	−0.716324	+	0.697768i	$0.754177\pi$
$588$	−1.73699	−0.0716321
$589$	−5.54804	−0.228603
$590$	11.8708	0.488714
$591$	−16.1683	−0.665074
$592$	−13.1532	−0.540592
$593$	41.5388	1.70580	0.852898	−	0.522078i	$-0.174843\pi$
$593$	41.5388	1.70580	0.852898	+	0.522078i	$0.174843\pi$
$594$	−4.89207	−0.200724
$595$	−6.45128	−0.264477
$596$	−6.72203	−0.275345
$597$	8.69162	0.355724
$598$	13.0316	0.532901
$599$	−20.9832	−0.857351	−0.428676	−	0.903458i	$-0.641020\pi$
$599$	−20.9832	−0.857351	−0.428676	+	0.903458i	$0.641020\pi$
$600$	0.811447	0.0331272
$601$	−48.1347	−1.96346	−0.981728	−	0.190289i	$-0.939057\pi$
$601$	−48.1347	−1.96346	−0.981728	+	0.190289i	$0.939057\pi$
$602$	15.9356	0.649488
$603$	−5.96248	−0.242811
$604$	−27.4544	−1.11710
$605$	8.47928	0.344732
$606$	−22.3456	−0.907729
$607$	−2.18894	−0.0888461	−0.0444231	−	0.999013i	$-0.514145\pi$
$607$	−2.18894	−0.0888461	−0.0444231	+	0.999013i	$0.514145\pi$
$608$	−6.18739	−0.250932
$609$	−2.11891	−0.0858625
$610$	26.5518	1.07505
$611$	37.4024	1.51314
$612$	6.07358	0.245510
$613$	−22.6696	−0.915617	−0.457809	−	0.889051i	$-0.651365\pi$
$613$	−22.6696	−0.915617	−0.457809	+	0.889051i	$0.651365\pi$
$614$	−18.0617	−0.728912
$615$	−15.9452	−0.642974
$616$	−1.28668	−0.0518419
$617$	−28.7887	−1.15899	−0.579496	−	0.814975i	$-0.696750\pi$
$617$	−28.7887	−1.15899	−0.579496	+	0.814975i	$0.696750\pi$
$618$	38.0134	1.52913
$619$	−16.2840	−0.654507	−0.327254	−	0.944937i	$-0.606123\pi$
$619$	−16.2840	−0.654507	−0.327254	+	0.944937i	$0.606123\pi$
$620$	−21.8359	−0.876949
$621$	−1.63828	−0.0657421
$622$	−39.0654	−1.56638
$623$	14.7466	0.590812
$624$	−18.3390	−0.734147
$625$	−14.4731	−0.578926
$626$	6.80451	0.271963
$627$	−2.06061	−0.0822928
$628$	6.38461	0.254774
$629$	10.3193	0.411458
$630$	3.56663	0.142098
$631$	−10.9182	−0.434649	−0.217324	−	0.976099i	$-0.569733\pi$
$631$	−10.9182	−0.434649	−0.217324	+	0.976099i	$0.569733\pi$
$632$	−0.336923	−0.0134021
$633$	−8.76503	−0.348379
$634$	21.3958	0.849734
$635$	−14.4367	−0.572904
$636$	−19.6833	−0.780493
$637$	−4.11479	−0.163034
$638$	10.3659	0.410388
$639$	12.9650	0.512886
$640$	7.43969	0.294080
$641$	23.0636	0.910958	0.455479	−	0.890247i	$-0.349468\pi$
$641$	23.0636	0.910958	0.455479	+	0.890247i	$0.349468\pi$
$642$	0.940633	0.0371238
$643$	7.35089	0.289891	0.144945	−	0.989440i	$-0.453699\pi$
$643$	7.35089	0.289891	0.144945	+	0.989440i	$0.453699\pi$
$644$	2.84568	0.112135
$645$	−15.2092	−0.598861
$646$	5.50392	0.216549
$647$	−34.3844	−1.35179	−0.675895	−	0.736998i	$-0.736243\pi$
$647$	−34.3844	−1.35179	−0.675895	+	0.736998i	$0.736243\pi$
$648$	0.508439	0.0199734
$649$	−8.42276	−0.330622
$650$	−12.6949	−0.497935
$651$	−6.81360	−0.267046
$652$	−38.1119	−1.49258
$653$	−9.12787	−0.357201	−0.178601	−	0.983922i	$-0.557157\pi$
$653$	−9.12787	−0.357201	−0.178601	+	0.983922i	$0.557157\pi$
$654$	−21.8028	−0.852559
$655$	−33.1760	−1.29629
$656$	38.5178	1.50387
$657$	−5.66425	−0.220984
$658$	17.5717	0.685014
$659$	−46.6466	−1.81710	−0.908548	−	0.417781i	$-0.862808\pi$
$659$	−46.6466	−1.81710	−0.908548	+	0.417781i	$0.862808\pi$
$660$	−8.11010	−0.315685
$661$	45.6332	1.77492	0.887462	−	0.460881i	$-0.152466\pi$
$661$	45.6332	1.77492	0.887462	+	0.460881i	$0.152466\pi$
$662$	−38.2177	−1.48537
$663$	14.3878	0.558777
$664$	−0.126905	−0.00492487
$665$	1.50231	0.0582572
$666$	−5.70509	−0.221068
$667$	3.47137	0.134412
$668$	14.9730	0.579322
$669$	−16.4392	−0.635578
$670$	−21.2660	−0.821576
$671$	−18.8395	−0.727289
$672$	−7.59879	−0.293130
$673$	22.5879	0.870700	0.435350	−	0.900261i	$-0.356625\pi$
$673$	22.5879	0.870700	0.435350	+	0.900261i	$0.356625\pi$
$674$	26.9456	1.03790
$675$	1.59596	0.0614284
$676$	6.82889	0.262650
$677$	−41.2663	−1.58599	−0.792996	−	0.609227i	$-0.791480\pi$
$677$	−41.2663	−1.58599	−0.792996	+	0.609227i	$0.791480\pi$
$678$	−32.5866	−1.25148
$679$	−2.36295	−0.0906815
$680$	−3.28009	−0.125786
$681$	−4.23662	−0.162348
$682$	33.3326	1.27637
$683$	−24.1428	−0.923798	−0.461899	−	0.886933i	$-0.652832\pi$
$683$	−24.1428	−0.923798	−0.461899	+	0.886933i	$0.652832\pi$
$684$	−1.41436	−0.0540794
$685$	−6.93718	−0.265056
$686$	−1.93313	−0.0738072
$687$	9.20539	0.351207
$688$	36.7398	1.40069
$689$	−46.6281	−1.77639
$690$	−5.84316	−0.222445
$691$	−23.4356	−0.891534	−0.445767	−	0.895149i	$-0.647069\pi$
$691$	−23.4356	−0.891534	−0.445767	+	0.895149i	$0.647069\pi$
$692$	37.5386	1.42700
$693$	−2.53065	−0.0961315
$694$	50.1522	1.90375
$695$	18.2299	0.691499
$696$	−1.07734	−0.0408363
$697$	−30.2191	−1.14463
$698$	4.63131	0.175298
$699$	−28.3586	−1.07262
$700$	−2.77215	−0.104778
$701$	−47.9483	−1.81098	−0.905492	−	0.424364i	$-0.860498\pi$
$701$	−47.9483	−1.81098	−0.905492	+	0.424364i	$0.860498\pi$
$702$	−7.95441	−0.300220
$703$	−2.40306	−0.0906332
$704$	14.6164	0.550875
$705$	−16.7706	−0.631618
$706$	−40.0729	−1.50817
$707$	−11.5593	−0.434733
$708$	−5.78121	−0.217271
$709$	50.3153	1.88963	0.944815	−	0.327605i	$-0.106241\pi$
$709$	50.3153	1.88963	0.944815	+	0.327605i	$0.106241\pi$
$710$	46.2412	1.73540
$711$	−0.662661	−0.0248517
$712$	7.49777	0.280991
$713$	11.1626	0.418043
$714$	6.75942	0.252965
$715$	−19.2122	−0.718495
$716$	−20.5881	−0.769415
$717$	−19.0416	−0.711123
$718$	−51.3354	−1.91582
$719$	1.47594	0.0550433	0.0275217	−	0.999621i	$-0.491238\pi$
$719$	1.47594	0.0550433	0.0275217	+	0.999621i	$0.491238\pi$
$720$	8.22291	0.306450
$721$	19.6642	0.732333
$722$	35.4477	1.31923
$723$	−2.65414	−0.0987086
$724$	−34.6994	−1.28959
$725$	−3.38168	−0.125593
$726$	−8.88428	−0.329727
$727$	34.1884	1.26798	0.633988	−	0.773343i	$-0.281417\pi$
$727$	34.1884	1.26798	0.633988	+	0.773343i	$0.281417\pi$
$728$	−2.09212	−0.0775391
$729$	1.00000	0.0370370
$730$	−20.2023	−0.747721
$731$	−28.8242	−1.06610
$732$	−12.9310	−0.477944
$733$	−1.47941	−0.0546434	−0.0273217	−	0.999627i	$-0.508698\pi$
$733$	−1.47941	−0.0546434	−0.0273217	+	0.999627i	$0.508698\pi$
$734$	46.8523	1.72935
$735$	1.84501	0.0680540
$736$	12.4490	0.458875
$737$	15.0889	0.555808
$738$	16.7068	0.614987
$739$	−6.43228	−0.236615	−0.118308	−	0.992977i	$-0.537747\pi$
$739$	−6.43228	−0.236615	−0.118308	+	0.992977i	$0.537747\pi$
$740$	−9.45793	−0.347680
$741$	−3.35051	−0.123084
$742$	−21.9059	−0.804192
$743$	−18.6653	−0.684765	−0.342382	−	0.939561i	$-0.611234\pi$
$743$	−18.6653	−0.684765	−0.342382	+	0.939561i	$0.611234\pi$
$744$	−3.46430	−0.127007
$745$	7.14005	0.261591
$746$	−16.3231	−0.597632
$747$	−0.249597	−0.00913228
$748$	−15.3701	−0.561987
$749$	0.486586	0.0177795
$750$	23.5253	0.859024
$751$	34.3516	1.25351	0.626753	−	0.779218i	$-0.284383\pi$
$751$	34.3516	1.25351	0.626753	+	0.779218i	$0.284383\pi$
$752$	40.5117	1.47731
$753$	−9.49067	−0.345859
$754$	16.8547	0.613811
$755$	29.1617	1.06130
$756$	−1.73699	−0.0631736
$757$	7.74544	0.281513	0.140756	−	0.990044i	$-0.455047\pi$
$757$	7.74544	0.281513	0.140756	+	0.990044i	$0.455047\pi$
$758$	−10.7729	−0.391288
$759$	4.14593	0.150488
$760$	0.763836	0.0277072
$761$	−26.8844	−0.974560	−0.487280	−	0.873246i	$-0.662011\pi$
$761$	−26.8844	−0.974560	−0.487280	+	0.873246i	$0.662011\pi$
$762$	15.1263	0.547967
$763$	−11.2785	−0.408310
$764$	−1.09153	−0.0394903
$765$	−6.45128	−0.233247
$766$	−1.93313	−0.0698468
$767$	−13.6952	−0.494506
$768$	−19.3465	−0.698107
$769$	26.2122	0.945236	0.472618	−	0.881267i	$-0.343309\pi$
$769$	26.2122	0.945236	0.472618	+	0.881267i	$0.343309\pi$
$770$	−9.02590	−0.325271
$771$	−7.84837	−0.282652
$772$	29.9744	1.07880
$773$	−16.4261	−0.590805	−0.295403	−	0.955373i	$-0.595454\pi$
$773$	−16.4261	−0.590805	−0.295403	+	0.955373i	$0.595454\pi$
$774$	15.9356	0.572794
$775$	−10.8742	−0.390613
$776$	−1.20141	−0.0431283
$777$	−2.95122	−0.105875
$778$	32.9020	1.17959
$779$	7.03715	0.252132
$780$	−13.1869	−0.472165
$781$	−32.8098	−1.17403
$782$	−11.0738	−0.396000
$783$	−2.11891	−0.0757236
$784$	−4.45685	−0.159173
$785$	−6.78165	−0.242048
$786$	34.7606	1.23987
$787$	1.27077	0.0452982	0.0226491	−	0.999743i	$-0.492790\pi$
$787$	1.27077	0.0452982	0.0226491	+	0.999743i	$0.492790\pi$
$788$	28.0841	1.00045
$789$	14.2479	0.507238
$790$	−2.36347	−0.0840884
$791$	−16.8569	−0.599362
$792$	−1.28668	−0.0457203
$793$	−30.6326	−1.08779
$794$	53.6359	1.90347
$795$	20.9073	0.741506
$796$	−15.0972	−0.535107
$797$	−8.44096	−0.298994	−0.149497	−	0.988762i	$-0.547765\pi$
$797$	−8.44096	−0.298994	−0.149497	+	0.988762i	$0.547765\pi$
$798$	−1.57407	−0.0557215
$799$	−31.7834	−1.12442
$800$	−12.1273	−0.428766
$801$	14.7466	0.521047
$802$	−5.59893	−0.197705
$803$	14.3343	0.505845
$804$	10.3567	0.365254
$805$	−3.02264	−0.106534
$806$	54.1981	1.90905
$807$	−31.5484	−1.11056
$808$	−5.87721	−0.206759
$809$	32.8778	1.15592	0.577962	−	0.816064i	$-0.303848\pi$
$809$	32.8778	1.15592	0.577962	+	0.816064i	$0.303848\pi$
$810$	3.56663	0.125319
$811$	−39.8212	−1.39831	−0.699155	−	0.714970i	$-0.746440\pi$
$811$	−39.8212	−1.39831	−0.699155	+	0.714970i	$0.746440\pi$
$812$	3.68051	0.129161
$813$	−22.2693	−0.781017
$814$	14.4376	0.506038
$815$	40.4820	1.41802
$816$	15.5839	0.545546
$817$	6.71230	0.234834
$818$	11.2126	0.392038
$819$	−4.11479	−0.143782
$820$	27.6967	0.967210
$821$	−39.4683	−1.37745	−0.688727	−	0.725020i	$-0.741830\pi$
$821$	−39.4683	−1.37745	−0.688727	+	0.725020i	$0.741830\pi$
$822$	7.26852	0.253519
$823$	−38.9479	−1.35764	−0.678819	−	0.734305i	$-0.737508\pi$
$823$	−38.9479	−1.35764	−0.678819	+	0.734305i	$0.737508\pi$
$824$	9.99806	0.348299
$825$	−4.03881	−0.140613
$826$	−6.43403	−0.223868
$827$	−53.4152	−1.85743	−0.928715	−	0.370795i	$-0.879085\pi$
$827$	−53.4152	−1.85743	−0.928715	+	0.370795i	$0.879085\pi$
$828$	2.84568	0.0988941
$829$	−24.3372	−0.845268	−0.422634	−	0.906301i	$-0.638894\pi$
$829$	−24.3372	−0.845268	−0.422634	+	0.906301i	$0.638894\pi$
$830$	−0.890221	−0.0309000
$831$	−15.6701	−0.543591
$832$	23.7659	0.823934
$833$	3.49662	0.121151
$834$	−19.1006	−0.661400
$835$	−15.9041	−0.550384
$836$	3.57925	0.123791
$837$	−6.81360	−0.235512
$838$	−5.04374	−0.174233
$839$	17.4147	0.601221	0.300610	−	0.953747i	$-0.402810\pi$
$839$	17.4147	0.601221	0.300610	+	0.953747i	$0.402810\pi$
$840$	0.938073	0.0323666
$841$	−24.5102	−0.845180
$842$	−1.95775	−0.0674684
$843$	2.53820	0.0874203
$844$	15.2247	0.524057
$845$	−7.25356	−0.249530
$846$	17.5717	0.604126
$847$	−4.59580	−0.157914
$848$	−50.5044	−1.73433
$849$	22.1456	0.760033
$850$	10.7877	0.370016
$851$	4.83494	0.165740
$852$	−22.5200	−0.771521
$853$	−1.42963	−0.0489495	−0.0244747	−	0.999700i	$-0.507791\pi$
$853$	−1.42963	−0.0489495	−0.0244747	+	0.999700i	$0.507791\pi$
$854$	−14.3912	−0.492457
$855$	1.50231	0.0513781
$856$	0.247399	0.00845594
$857$	33.4477	1.14255	0.571276	−	0.820758i	$-0.306449\pi$
$857$	33.4477	1.14255	0.571276	+	0.820758i	$0.306449\pi$
$858$	20.1298	0.687221
$859$	10.8907	0.371586	0.185793	−	0.982589i	$-0.440515\pi$
$859$	10.8907	0.371586	0.185793	+	0.982589i	$0.440515\pi$
$860$	26.4182	0.900851
$861$	8.64239	0.294532
$862$	−43.4115	−1.47860
$863$	−24.4632	−0.832737	−0.416368	−	0.909196i	$-0.636697\pi$
$863$	−24.4632	−0.832737	−0.416368	+	0.909196i	$0.636697\pi$
$864$	−7.59879	−0.258516
$865$	−39.8730	−1.35572
$866$	58.1828	1.97713
$867$	4.77365	0.162122
$868$	11.8351	0.401710
$869$	1.67696	0.0568871
$870$	−7.55737	−0.256219
$871$	24.5343	0.831313
$872$	−5.73445	−0.194193
$873$	−2.36295	−0.0799736
$874$	2.57877	0.0872283
$875$	12.1696	0.411407
$876$	9.83873	0.332420
$877$	−16.0926	−0.543407	−0.271703	−	0.962381i	$-0.587587\pi$
$877$	−16.0926	−0.543407	−0.271703	+	0.962381i	$0.587587\pi$
$878$	−38.2264	−1.29008
$879$	−3.19490	−0.107761
$880$	−20.8093	−0.701482
$881$	−36.0779	−1.21549	−0.607747	−	0.794130i	$-0.707927\pi$
$881$	−36.0779	−1.21549	−0.607747	+	0.794130i	$0.707927\pi$
$882$	−1.93313	−0.0650918
$883$	18.7968	0.632561	0.316281	−	0.948666i	$-0.397566\pi$
$883$	18.7968	0.632561	0.316281	+	0.948666i	$0.397566\pi$
$884$	−24.9915	−0.840554
$885$	6.14073	0.206418
$886$	56.7069	1.90511
$887$	−13.4587	−0.451899	−0.225949	−	0.974139i	$-0.572548\pi$
$887$	−13.4587	−0.451899	−0.225949	+	0.974139i	$0.572548\pi$
$888$	−1.50052	−0.0503541
$889$	7.82477	0.262434
$890$	52.5958	1.76302
$891$	−2.53065	−0.0847800
$892$	28.5547	0.956083
$893$	7.40142	0.247679
$894$	−7.48109	−0.250205
$895$	21.8685	0.730982
$896$	−4.03234	−0.134711
$897$	6.74119	0.225082
$898$	42.8423	1.42966
$899$	14.4374	0.481514
$900$	−2.77215	−0.0924051
$901$	39.6232	1.32004
$902$	−42.2792	−1.40774
$903$	8.24344	0.274324
$904$	−8.57071	−0.285058
$905$	36.8573	1.22518
$906$	−30.5546	−1.01511
$907$	−43.0065	−1.42801	−0.714003	−	0.700143i	$-0.753120\pi$
$907$	−43.0065	−1.42801	−0.714003	+	0.700143i	$0.753120\pi$
$908$	7.35895	0.244215
$909$	−11.5593	−0.383398
$910$	−14.6759	−0.486502
$911$	40.9886	1.35801	0.679006	−	0.734133i	$-0.262411\pi$
$911$	40.9886	1.35801	0.679006	+	0.734133i	$0.262411\pi$
$912$	−3.62904	−0.120169
$913$	0.631643	0.0209043
$914$	−55.0160	−1.81977
$915$	13.7352	0.454070
$916$	−15.9896	−0.528312
$917$	17.9815	0.593802
$918$	6.75942	0.223094
$919$	17.0981	0.564013	0.282006	−	0.959413i	$-0.409000\pi$
$919$	17.0981	0.564013	0.282006	+	0.959413i	$0.409000\pi$
$920$	−1.53683	−0.0506678
$921$	−9.34327	−0.307871
$922$	39.2701	1.29329
$923$	−53.3480	−1.75597
$924$	4.39571	0.144608
$925$	−4.71002	−0.154865
$926$	−1.84104	−0.0605002
$927$	19.6642	0.645857
$928$	16.1011	0.528546
$929$	−17.2392	−0.565601	−0.282800	−	0.959179i	$-0.591263\pi$
$929$	−17.2392	−0.565601	−0.282800	+	0.959179i	$0.591263\pi$
$930$	−24.3016	−0.796881
$931$	−0.814260	−0.0266863
$932$	49.2585	1.61351
$933$	−20.2084	−0.661592
$934$	70.9067	2.32014
$935$	16.3259	0.533915
$936$	−2.09212	−0.0683831
$937$	−45.0929	−1.47312	−0.736560	−	0.676372i	$-0.763551\pi$
$937$	−45.0929	−1.47312	−0.736560	+	0.676372i	$0.763551\pi$
$938$	11.5262	0.376345
$939$	3.51995	0.114869
$940$	29.1304	0.950127
$941$	22.0967	0.720332	0.360166	−	0.932888i	$-0.382720\pi$
$941$	22.0967	0.720332	0.360166	+	0.932888i	$0.382720\pi$
$942$	7.10557	0.231512
$943$	−14.1587	−0.461070
$944$	−14.8337	−0.482797
$945$	1.84501	0.0600180
$946$	−40.3275	−1.31116
$947$	−34.0518	−1.10653	−0.553267	−	0.833004i	$-0.686619\pi$
$947$	−34.0518	−1.10653	−0.553267	+	0.833004i	$0.686619\pi$
$948$	1.15103	0.0373838
$949$	23.3072	0.756583
$950$	−2.51215	−0.0815048
$951$	11.0679	0.358903
$952$	1.77782	0.0576195
$953$	−18.7865	−0.608556	−0.304278	−	0.952583i	$-0.598415\pi$
$953$	−18.7865	−0.608556	−0.304278	+	0.952583i	$0.598415\pi$
$954$	−21.9059	−0.709231
$955$	1.15941	0.0375177
$956$	33.0751	1.06972
$957$	5.36222	0.173336
$958$	67.4231	2.17834
$959$	3.75998	0.121416
$960$	−10.6563	−0.343929
$961$	15.4251	0.497584
$962$	23.4752	0.756872
$963$	0.486586	0.0156800
$964$	4.61021	0.148485
$965$	−31.8384	−1.02491
$966$	3.16701	0.101897
$967$	−3.84442	−0.123628	−0.0618141	−	0.998088i	$-0.519689\pi$
$967$	−3.84442	−0.123628	−0.0618141	+	0.998088i	$0.519689\pi$
$968$	−2.33669	−0.0751040
$969$	2.84716	0.0914639
$970$	−8.42776	−0.270599
$971$	−2.60533	−0.0836091	−0.0418046	−	0.999126i	$-0.513311\pi$
$971$	−2.60533	−0.0836091	−0.0418046	+	0.999126i	$0.513311\pi$
$972$	−1.73699	−0.0557139
$973$	−9.88068	−0.316760
$974$	51.8810	1.66238
$975$	−6.56702	−0.210313
$976$	−33.1791	−1.06204
$977$	2.80425	0.0897159	0.0448579	−	0.998993i	$-0.485716\pi$
$977$	2.80425	0.0897159	0.0448579	+	0.998993i	$0.485716\pi$
$978$	−42.4156	−1.35630
$979$	−37.3186	−1.19271
$980$	−3.20475	−0.102372
$981$	−11.2785	−0.360096
$982$	42.4733	1.35538
$983$	−22.2231	−0.708807	−0.354404	−	0.935093i	$-0.615316\pi$
$983$	−22.2231	−0.708807	−0.354404	+	0.935093i	$0.615316\pi$
$984$	4.39413	0.140080
$985$	−29.8305	−0.950480
$986$	−14.3226	−0.456124
$987$	9.08975	0.289330
$988$	5.81978	0.185152
$989$	−13.5051	−0.429437
$990$	−9.02590	−0.286862
$991$	10.0877	0.320448	0.160224	−	0.987081i	$-0.448778\pi$
$991$	10.0877	0.320448	0.160224	+	0.987081i	$0.448778\pi$
$992$	51.7751	1.64386
$993$	−19.7699	−0.627377
$994$	−25.0629	−0.794948
$995$	16.0361	0.508378
$996$	0.433547	0.0137375
$997$	−36.0093	−1.14043	−0.570213	−	0.821497i	$-0.693139\pi$
$997$	−36.0093	−1.14043	−0.570213	+	0.821497i	$0.693139\pi$
$998$	−63.8646	−2.02160
$999$	−2.95122	−0.0933726

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8043.2.a.o.1.8	✓	41

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8043.2.a.o.1.8	✓	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)

\(f(q)\)	\(=\)	\(q-1.93313 q^{2} -1.00000 q^{3} +1.73699 q^{4} -1.84501 q^{5} +1.93313 q^{6} +1.00000 q^{7} +0.508439 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.93313 q^{2} -1.00000 q^{3} +1.73699 q^{4} -1.84501 q^{5} +1.93313 q^{6} +1.00000 q^{7} +0.508439 q^{8} +1.00000 q^{9} +3.56663 q^{10} -2.53065 q^{11} -1.73699 q^{12} -4.11479 q^{13} -1.93313 q^{14} +1.84501 q^{15} -4.45685 q^{16} +3.49662 q^{17} -1.93313 q^{18} -0.814260 q^{19} -3.20475 q^{20} -1.00000 q^{21} +4.89207 q^{22} +1.63828 q^{23} -0.508439 q^{24} -1.59596 q^{25} +7.95441 q^{26} -1.00000 q^{27} +1.73699 q^{28} +2.11891 q^{29} -3.56663 q^{30} +6.81360 q^{31} +7.59879 q^{32} +2.53065 q^{33} -6.75942 q^{34} -1.84501 q^{35} +1.73699 q^{36} +2.95122 q^{37} +1.57407 q^{38} +4.11479 q^{39} -0.938073 q^{40} -8.64239 q^{41} +1.93313 q^{42} -8.24344 q^{43} -4.39571 q^{44} -1.84501 q^{45} -3.16701 q^{46} -9.08975 q^{47} +4.45685 q^{48} +1.00000 q^{49} +3.08519 q^{50} -3.49662 q^{51} -7.14733 q^{52} +11.3319 q^{53} +1.93313 q^{54} +4.66906 q^{55} +0.508439 q^{56} +0.814260 q^{57} -4.09612 q^{58} +3.32830 q^{59} +3.20475 q^{60} +7.44451 q^{61} -13.1716 q^{62} +1.00000 q^{63} -5.77573 q^{64} +7.59180 q^{65} -4.89207 q^{66} -5.96248 q^{67} +6.07358 q^{68} -1.63828 q^{69} +3.56663 q^{70} +12.9650 q^{71} +0.508439 q^{72} -5.66425 q^{73} -5.70509 q^{74} +1.59596 q^{75} -1.41436 q^{76} -2.53065 q^{77} -7.95441 q^{78} -0.662661 q^{79} +8.22291 q^{80} +1.00000 q^{81} +16.7068 q^{82} -0.249597 q^{83} -1.73699 q^{84} -6.45128 q^{85} +15.9356 q^{86} -2.11891 q^{87} -1.28668 q^{88} +14.7466 q^{89} +3.56663 q^{90} -4.11479 q^{91} +2.84568 q^{92} -6.81360 q^{93} +17.5717 q^{94} +1.50231 q^{95} -7.59879 q^{96} -2.36295 q^{97} -1.93313 q^{98} -2.53065 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 41 q - 4 q^{2} - 41 q^{3} + 34 q^{4} + 5 q^{5} + 4 q^{6} + 41 q^{7} - 15 q^{8} + 41 q^{9}+O(q^{10}) \) 41 * q - 4 * q^2 - 41 * q^3 + 34 * q^4 + 5 * q^5 + 4 * q^6 + 41 * q^7 - 15 * q^8 + 41 * q^9	\( 41 q - 4 q^{2} - 41 q^{3} + 34 q^{4} + 5 q^{5} + 4 q^{6} + 41 q^{7} - 15 q^{8} + 41 q^{9} - 18 q^{10} - 17 q^{11} - 34 q^{12} - 38 q^{13} - 4 q^{14} - 5 q^{15} + 16 q^{16} + 4 q^{17} - 4 q^{18} - 15 q^{19} + 23 q^{20} - 41 q^{21} - 31 q^{22} - 8 q^{23} + 15 q^{24} + 16 q^{25} + 15 q^{26} - 41 q^{27} + 34 q^{28} - 27 q^{29} + 18 q^{30} - 23 q^{31} - 24 q^{32} + 17 q^{33} - 9 q^{34} + 5 q^{35} + 34 q^{36} - 81 q^{37} + 38 q^{39} - 52 q^{40} + 29 q^{41} + 4 q^{42} - 47 q^{43} - 20 q^{44} + 5 q^{45} - 40 q^{46} + 26 q^{47} - 16 q^{48} + 41 q^{49} - 23 q^{50} - 4 q^{51} - 64 q^{52} - 66 q^{53} + 4 q^{54} - 14 q^{55} - 15 q^{56} + 15 q^{57} - 50 q^{58} + 41 q^{59} - 23 q^{60} - 47 q^{61} + 5 q^{62} + 41 q^{63} - 37 q^{64} - 24 q^{65} + 31 q^{66} - 73 q^{67} + 10 q^{68} + 8 q^{69} - 18 q^{70} + q^{71} - 15 q^{72} - 14 q^{73} + 18 q^{74} - 16 q^{75} - 37 q^{76} - 17 q^{77} - 15 q^{78} - 80 q^{79} + 63 q^{80} + 41 q^{81} - 31 q^{82} + 20 q^{83} - 34 q^{84} - 82 q^{85} - 39 q^{86} + 27 q^{87} - 132 q^{88} + 17 q^{89} - 18 q^{90} - 38 q^{91} - 26 q^{92} + 23 q^{93} - 9 q^{94} - q^{95} + 24 q^{96} - 73 q^{97} - 4 q^{98} - 17 q^{99}+O(q^{100}) \) 41 * q - 4 * q^2 - 41 * q^3 + 34 * q^4 + 5 * q^5 + 4 * q^6 + 41 * q^7 - 15 * q^8 + 41 * q^9 - 18 * q^10 - 17 * q^11 - 34 * q^12 - 38 * q^13 - 4 * q^14 - 5 * q^15 + 16 * q^16 + 4 * q^17 - 4 * q^18 - 15 * q^19 + 23 * q^20 - 41 * q^21 - 31 * q^22 - 8 * q^23 + 15 * q^24 + 16 * q^25 + 15 * q^26 - 41 * q^27 + 34 * q^28 - 27 * q^29 + 18 * q^30 - 23 * q^31 - 24 * q^32 + 17 * q^33 - 9 * q^34 + 5 * q^35 + 34 * q^36 - 81 * q^37 + 38 * q^39 - 52 * q^40 + 29 * q^41 + 4 * q^42 - 47 * q^43 - 20 * q^44 + 5 * q^45 - 40 * q^46 + 26 * q^47 - 16 * q^48 + 41 * q^49 - 23 * q^50 - 4 * q^51 - 64 * q^52 - 66 * q^53 + 4 * q^54 - 14 * q^55 - 15 * q^56 + 15 * q^57 - 50 * q^58 + 41 * q^59 - 23 * q^60 - 47 * q^61 + 5 * q^62 + 41 * q^63 - 37 * q^64 - 24 * q^65 + 31 * q^66 - 73 * q^67 + 10 * q^68 + 8 * q^69 - 18 * q^70 + q^71 - 15 * q^72 - 14 * q^73 + 18 * q^74 - 16 * q^75 - 37 * q^76 - 17 * q^77 - 15 * q^78 - 80 * q^79 + 63 * q^80 + 41 * q^81 - 31 * q^82 + 20 * q^83 - 34 * q^84 - 82 * q^85 - 39 * q^86 + 27 * q^87 - 132 * q^88 + 17 * q^89 - 18 * q^90 - 38 * q^91 - 26 * q^92 + 23 * q^93 - 9 * q^94 - q^95 + 24 * q^96 - 73 * q^97 - 4 * q^98 - 17 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more