LMFDB - Embedded newform 8015.2.a.j.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8015,2,Mod(1,8015)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8015, 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("8015.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8015 = 5 \cdot 7 \cdot 229 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8015.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:	$64.0000972201$
Analytic rank:	$1$
Dimension:	$45$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	8015.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.60365 q^{2} +2.74735 q^{3} +0.571694 q^{4} -1.00000 q^{5} -4.40579 q^{6} +1.00000 q^{7} +2.29050 q^{8} +4.54792 q^{9} +O(q^{10})$	\(q-1.60365 q^{2} +2.74735 q^{3} +0.571694 q^{4} -1.00000 q^{5} -4.40579 q^{6} +1.00000 q^{7} +2.29050 q^{8} +4.54792 q^{9} +1.60365 q^{10} -2.96026 q^{11} +1.57064 q^{12} -0.939802 q^{13} -1.60365 q^{14} -2.74735 q^{15} -4.81655 q^{16} -8.10854 q^{17} -7.29327 q^{18} +3.13222 q^{19} -0.571694 q^{20} +2.74735 q^{21} +4.74722 q^{22} +5.95816 q^{23} +6.29281 q^{24} +1.00000 q^{25} +1.50711 q^{26} +4.25268 q^{27} +0.571694 q^{28} +1.13986 q^{29} +4.40579 q^{30} +2.81172 q^{31} +3.14306 q^{32} -8.13286 q^{33} +13.0033 q^{34} -1.00000 q^{35} +2.60002 q^{36} -5.15134 q^{37} -5.02299 q^{38} -2.58196 q^{39} -2.29050 q^{40} +8.49391 q^{41} -4.40579 q^{42} -7.36688 q^{43} -1.69236 q^{44} -4.54792 q^{45} -9.55481 q^{46} -11.0255 q^{47} -13.2327 q^{48} +1.00000 q^{49} -1.60365 q^{50} -22.2770 q^{51} -0.537279 q^{52} +4.01700 q^{53} -6.81980 q^{54} +2.96026 q^{55} +2.29050 q^{56} +8.60530 q^{57} -1.82794 q^{58} +3.36104 q^{59} -1.57064 q^{60} -4.48749 q^{61} -4.50901 q^{62} +4.54792 q^{63} +4.59274 q^{64} +0.939802 q^{65} +13.0423 q^{66} +8.39602 q^{67} -4.63561 q^{68} +16.3691 q^{69} +1.60365 q^{70} -2.52252 q^{71} +10.4170 q^{72} -10.6927 q^{73} +8.26095 q^{74} +2.74735 q^{75} +1.79067 q^{76} -2.96026 q^{77} +4.14056 q^{78} -5.40475 q^{79} +4.81655 q^{80} -1.96018 q^{81} -13.6213 q^{82} +11.5577 q^{83} +1.57064 q^{84} +8.10854 q^{85} +11.8139 q^{86} +3.13159 q^{87} -6.78049 q^{88} +6.58922 q^{89} +7.29327 q^{90} -0.939802 q^{91} +3.40625 q^{92} +7.72477 q^{93} +17.6810 q^{94} -3.13222 q^{95} +8.63508 q^{96} -15.6618 q^{97} -1.60365 q^{98} -13.4630 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 45 q - 6 q^{2} + 34 q^{4} - 45 q^{5} + q^{6} + 45 q^{7} - 15 q^{8} + 29 q^{9}+O(q^{10}) $ 45 * q - 6 * q^2 + 34 * q^4 - 45 * q^5 + q^6 + 45 * q^7 - 15 * q^8 + 29 * q^9	\( 45 q - 6 q^{2} + 34 q^{4} - 45 q^{5} + q^{6} + 45 q^{7} - 15 q^{8} + 29 q^{9} + 6 q^{10} - q^{11} - 3 q^{12} - 21 q^{13} - 6 q^{14} + 8 q^{16} - 7 q^{17} - 36 q^{18} - 20 q^{19} - 34 q^{20} - 34 q^{22} - 22 q^{23} - 11 q^{24} + 45 q^{25} - q^{26} + 12 q^{27} + 34 q^{28} + 10 q^{29} - q^{30} - 27 q^{31} - 26 q^{32} - 39 q^{33} - 13 q^{34} - 45 q^{35} - 3 q^{36} - 72 q^{37} + 2 q^{38} - 37 q^{39} + 15 q^{40} - 4 q^{41} + q^{42} - 49 q^{43} + 5 q^{44} - 29 q^{45} - 67 q^{46} + 2 q^{47} + 8 q^{48} + 45 q^{49} - 6 q^{50} - 49 q^{51} - 47 q^{52} - 35 q^{53} - 12 q^{54} + q^{55} - 15 q^{56} - 77 q^{57} - 50 q^{58} + 4 q^{59} + 3 q^{60} - 36 q^{61} + 17 q^{62} + 29 q^{63} + 5 q^{64} + 21 q^{65} - 8 q^{66} - 80 q^{67} + 27 q^{68} + 9 q^{69} + 6 q^{70} - 12 q^{71} - 97 q^{72} - 55 q^{73} + 32 q^{74} - 37 q^{76} - q^{77} + 20 q^{78} - 94 q^{79} - 8 q^{80} - 19 q^{81} - 36 q^{82} + 24 q^{83} - 3 q^{84} + 7 q^{85} - 3 q^{86} - 4 q^{87} - 95 q^{88} + q^{89} + 36 q^{90} - 21 q^{91} - 65 q^{92} - 71 q^{93} - 53 q^{94} + 20 q^{95} - 13 q^{96} - 110 q^{97} - 6 q^{98} - 27 q^{99}+O(q^{100}) \) 45 * q - 6 * q^2 + 34 * q^4 - 45 * q^5 + q^6 + 45 * q^7 - 15 * q^8 + 29 * q^9 + 6 * q^10 - q^11 - 3 * q^12 - 21 * q^13 - 6 * q^14 + 8 * q^16 - 7 * q^17 - 36 * q^18 - 20 * q^19 - 34 * q^20 - 34 * q^22 - 22 * q^23 - 11 * q^24 + 45 * q^25 - q^26 + 12 * q^27 + 34 * q^28 + 10 * q^29 - q^30 - 27 * q^31 - 26 * q^32 - 39 * q^33 - 13 * q^34 - 45 * q^35 - 3 * q^36 - 72 * q^37 + 2 * q^38 - 37 * q^39 + 15 * q^40 - 4 * q^41 + q^42 - 49 * q^43 + 5 * q^44 - 29 * q^45 - 67 * q^46 + 2 * q^47 + 8 * q^48 + 45 * q^49 - 6 * q^50 - 49 * q^51 - 47 * q^52 - 35 * q^53 - 12 * q^54 + q^55 - 15 * q^56 - 77 * q^57 - 50 * q^58 + 4 * q^59 + 3 * q^60 - 36 * q^61 + 17 * q^62 + 29 * q^63 + 5 * q^64 + 21 * q^65 - 8 * q^66 - 80 * q^67 + 27 * q^68 + 9 * q^69 + 6 * q^70 - 12 * q^71 - 97 * q^72 - 55 * q^73 + 32 * q^74 - 37 * q^76 - q^77 + 20 * q^78 - 94 * q^79 - 8 * q^80 - 19 * q^81 - 36 * q^82 + 24 * q^83 - 3 * q^84 + 7 * q^85 - 3 * q^86 - 4 * q^87 - 95 * q^88 + q^89 + 36 * q^90 - 21 * q^91 - 65 * q^92 - 71 * q^93 - 53 * q^94 + 20 * q^95 - 13 * q^96 - 110 * q^97 - 6 * q^98 - 27 * 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 $

$2$	−1.60365	−1.13395	−0.566976	−	0.823734i	$-0.691887\pi$
$2$	−1.60365	−1.13395	−0.566976	+	0.823734i	$0.691887\pi$
$3$	2.74735	1.58618	0.793091	−	0.609103i	$-0.208470\pi$
$3$	2.74735	1.58618	0.793091	+	0.609103i	$0.208470\pi$
$4$	0.571694	0.285847
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−4.40579	−1.79865
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.29050	0.809815
$9$	4.54792	1.51597
$10$	1.60365	0.507119
$11$	−2.96026	−0.892552	−0.446276	−	0.894895i	$-0.647250\pi$
$11$	−2.96026	−0.892552	−0.446276	+	0.894895i	$0.647250\pi$
$12$	1.57064	0.453405
$13$	−0.939802	−0.260654	−0.130327	−	0.991471i	$-0.541603\pi$
$13$	−0.939802	−0.260654	−0.130327	+	0.991471i	$0.541603\pi$
$14$	−1.60365	−0.428594
$15$	−2.74735	−0.709362
$16$	−4.81655	−1.20414
$17$	−8.10854	−1.96661	−0.983305	−	0.181962i	$-0.941755\pi$
$17$	−8.10854	−1.96661	−0.983305	+	0.181962i	$0.941755\pi$
$18$	−7.29327	−1.71904
$19$	3.13222	0.718581	0.359291	−	0.933226i	$-0.383019\pi$
$19$	3.13222	0.718581	0.359291	+	0.933226i	$0.383019\pi$
$20$	−0.571694	−0.127835
$21$	2.74735	0.599520
$22$	4.74722	1.01211
$23$	5.95816	1.24236	0.621181	−	0.783667i	$-0.286653\pi$
$23$	5.95816	1.24236	0.621181	+	0.783667i	$0.286653\pi$
$24$	6.29281	1.28451
$25$	1.00000	0.200000
$26$	1.50711	0.295569
$27$	4.25268	0.818428
$28$	0.571694	0.108040
$29$	1.13986	0.211667	0.105833	−	0.994384i	$-0.466249\pi$
$29$	1.13986	0.211667	0.105833	+	0.994384i	$0.466249\pi$
$30$	4.40579	0.804383
$31$	2.81172	0.505000	0.252500	−	0.967597i	$-0.418747\pi$
$31$	2.81172	0.505000	0.252500	+	0.967597i	$0.418747\pi$
$32$	3.14306	0.555620
$33$	−8.13286	−1.41575
$34$	13.0033	2.23004
$35$	−1.00000	−0.169031
$36$	2.60002	0.433336
$37$	−5.15134	−0.846875	−0.423438	−	0.905925i	$-0.639177\pi$
$37$	−5.15134	−0.846875	−0.423438	+	0.905925i	$0.639177\pi$
$38$	−5.02299	−0.814836
$39$	−2.58196	−0.413445
$40$	−2.29050	−0.362160
$41$	8.49391	1.32653	0.663263	−	0.748386i	$-0.269171\pi$
$41$	8.49391	1.32653	0.663263	+	0.748386i	$0.269171\pi$
$42$	−4.40579	−0.679827
$43$	−7.36688	−1.12344	−0.561719	−	0.827328i	$-0.689860\pi$
$43$	−7.36688	−1.12344	−0.561719	+	0.827328i	$0.689860\pi$
$44$	−1.69236	−0.255133
$45$	−4.54792	−0.677964
$46$	−9.55481	−1.40878
$47$	−11.0255	−1.60823	−0.804115	−	0.594474i	$-0.797360\pi$
$47$	−11.0255	−1.60823	−0.804115	+	0.594474i	$0.797360\pi$
$48$	−13.2327	−1.90998
$49$	1.00000	0.142857
$50$	−1.60365	−0.226790
$51$	−22.2770	−3.11940
$52$	−0.537279	−0.0745072
$53$	4.01700	0.551777	0.275888	−	0.961190i	$-0.411028\pi$
$53$	4.01700	0.551777	0.275888	+	0.961190i	$0.411028\pi$
$54$	−6.81980	−0.928058
$55$	2.96026	0.399161
$56$	2.29050	0.306081
$57$	8.60530	1.13980
$58$	−1.82794	−0.240020
$59$	3.36104	0.437570	0.218785	−	0.975773i	$-0.429791\pi$
$59$	3.36104	0.437570	0.218785	+	0.975773i	$0.429791\pi$
$60$	−1.57064	−0.202769
$61$	−4.48749	−0.574564	−0.287282	−	0.957846i	$-0.592752\pi$
$61$	−4.48749	−0.574564	−0.287282	+	0.957846i	$0.592752\pi$
$62$	−4.50901	−0.572645
$63$	4.54792	0.572984
$64$	4.59274	0.574092
$65$	0.939802	0.116568
$66$	13.0423	1.60539
$67$	8.39602	1.02574	0.512868	−	0.858467i	$-0.328583\pi$
$67$	8.39602	1.02574	0.512868	+	0.858467i	$0.328583\pi$
$68$	−4.63561	−0.562150
$69$	16.3691	1.97061
$70$	1.60365	0.191673
$71$	−2.52252	−0.299368	−0.149684	−	0.988734i	$-0.547826\pi$
$71$	−2.52252	−0.299368	−0.149684	+	0.988734i	$0.547826\pi$
$72$	10.4170	1.22766
$73$	−10.6927	−1.25148	−0.625742	−	0.780030i	$-0.715204\pi$
$73$	−10.6927	−1.25148	−0.625742	+	0.780030i	$0.715204\pi$
$74$	8.26095	0.960316
$75$	2.74735	0.317236
$76$	1.79067	0.205404
$77$	−2.96026	−0.337353
$78$	4.14056	0.468827
$79$	−5.40475	−0.608081	−0.304041	−	0.952659i	$-0.598336\pi$
$79$	−5.40475	−0.608081	−0.304041	+	0.952659i	$0.598336\pi$
$80$	4.81655	0.538507
$81$	−1.96018	−0.217798
$82$	−13.6213	−1.50422
$83$	11.5577	1.26862	0.634311	−	0.773078i	$-0.281284\pi$
$83$	11.5577	1.26862	0.634311	+	0.773078i	$0.281284\pi$
$84$	1.57064	0.171371
$85$	8.10854	0.879495
$86$	11.8139	1.27392
$87$	3.13159	0.335742
$88$	−6.78049	−0.722802
$89$	6.58922	0.698456	0.349228	−	0.937038i	$-0.386444\pi$
$89$	6.58922	0.698456	0.349228	+	0.937038i	$0.386444\pi$
$90$	7.29327	0.768779
$91$	−0.939802	−0.0985180
$92$	3.40625	0.355126
$93$	7.72477	0.801021
$94$	17.6810	1.82366
$95$	−3.13222	−0.321359
$96$	8.63508	0.881315
$97$	−15.6618	−1.59021	−0.795107	−	0.606470i	$-0.792585\pi$
$97$	−15.6618	−1.59021	−0.795107	+	0.606470i	$0.792585\pi$
$98$	−1.60365	−0.161993
$99$	−13.4630	−1.35309
$100$	0.571694	0.0571694
$101$	1.97338	0.196359	0.0981795	−	0.995169i	$-0.468698\pi$
$101$	1.97338	0.196359	0.0981795	+	0.995169i	$0.468698\pi$
$102$	35.7245	3.53725
$103$	−0.161889	−0.0159514	−0.00797569	−	0.999968i	$-0.502539\pi$
$103$	−0.161889	−0.0159514	−0.00797569	+	0.999968i	$0.502539\pi$
$104$	−2.15262	−0.211082
$105$	−2.74735	−0.268114
$106$	−6.44186	−0.625688
$107$	9.67151	0.934981	0.467490	−	0.883998i	$-0.345158\pi$
$107$	9.67151	0.934981	0.467490	+	0.883998i	$0.345158\pi$
$108$	2.43123	0.233945
$109$	2.64620	0.253460	0.126730	−	0.991937i	$-0.459552\pi$
$109$	2.64620	0.253460	0.126730	+	0.991937i	$0.459552\pi$
$110$	−4.74722	−0.452630
$111$	−14.1525	−1.34330
$112$	−4.81655	−0.455122
$113$	−4.39952	−0.413872	−0.206936	−	0.978355i	$-0.566349\pi$
$113$	−4.39952	−0.413872	−0.206936	+	0.978355i	$0.566349\pi$
$114$	−13.7999	−1.29248
$115$	−5.95816	−0.555601
$116$	0.651651	0.0605043
$117$	−4.27414	−0.395145
$118$	−5.38993	−0.496183
$119$	−8.10854	−0.743309
$120$	−6.29281	−0.574452
$121$	−2.23686	−0.203351
$122$	7.19636	0.651528
$123$	23.3357	2.10411
$124$	1.60744	0.144353
$125$	−1.00000	−0.0894427
$126$	−7.29327	−0.649736
$127$	−10.6845	−0.948099	−0.474049	−	0.880498i	$-0.657208\pi$
$127$	−10.6845	−0.948099	−0.474049	+	0.880498i	$0.657208\pi$
$128$	−13.6513	−1.20661
$129$	−20.2394	−1.78198
$130$	−1.50711	−0.132183
$131$	−18.1034	−1.58170	−0.790850	−	0.612010i	$-0.790361\pi$
$131$	−18.1034	−1.58170	−0.790850	+	0.612010i	$0.790361\pi$
$132$	−4.64951	−0.404688
$133$	3.13222	0.271598
$134$	−13.4643	−1.16314
$135$	−4.25268	−0.366012
$136$	−18.5726	−1.59259
$137$	−16.0468	−1.37097	−0.685487	−	0.728085i	$-0.740411\pi$
$137$	−16.0468	−1.37097	−0.685487	+	0.728085i	$0.740411\pi$
$138$	−26.2504	−2.23458
$139$	3.01494	0.255724	0.127862	−	0.991792i	$-0.459189\pi$
$139$	3.01494	0.255724	0.127862	+	0.991792i	$0.459189\pi$
$140$	−0.571694	−0.0483170
$141$	−30.2908	−2.55095
$142$	4.04524	0.339469
$143$	2.78206	0.232647
$144$	−21.9053	−1.82544
$145$	−1.13986	−0.0946602
$146$	17.1473	1.41912
$147$	2.74735	0.226597
$148$	−2.94499	−0.242077
$149$	21.1397	1.73183	0.865914	−	0.500193i	$-0.166738\pi$
$149$	21.1397	1.73183	0.865914	+	0.500193i	$0.166738\pi$
$150$	−4.40579	−0.359731
$151$	−16.0652	−1.30737	−0.653684	−	0.756767i	$-0.726778\pi$
$151$	−16.0652	−1.30737	−0.653684	+	0.756767i	$0.726778\pi$
$152$	7.17436	0.581918
$153$	−36.8770	−2.98133
$154$	4.74722	0.382542
$155$	−2.81172	−0.225843
$156$	−1.47609	−0.118182
$157$	−16.2895	−1.30004	−0.650022	−	0.759916i	$-0.725240\pi$
$157$	−16.2895	−1.30004	−0.650022	+	0.759916i	$0.725240\pi$
$158$	8.66732	0.689535
$159$	11.0361	0.875219
$160$	−3.14306	−0.248481
$161$	5.95816	0.469569
$162$	3.14345	0.246972
$163$	−4.03648	−0.316162	−0.158081	−	0.987426i	$-0.550531\pi$
$163$	−4.03648	−0.316162	−0.158081	+	0.987426i	$0.550531\pi$
$164$	4.85592	0.379184
$165$	8.13286	0.633143
$166$	−18.5345	−1.43856
$167$	19.2386	1.48873	0.744363	−	0.667775i	$-0.232753\pi$
$167$	19.2386	1.48873	0.744363	+	0.667775i	$0.232753\pi$
$168$	6.29281	0.485501
$169$	−12.1168	−0.932059
$170$	−13.0033	−0.997305
$171$	14.2451	1.08935
$172$	−4.21160	−0.321131
$173$	−19.5838	−1.48893	−0.744463	−	0.667664i	$-0.767294\pi$
$173$	−19.5838	−1.48893	−0.744463	+	0.667664i	$0.767294\pi$
$174$	−5.02198	−0.380715
$175$	1.00000	0.0755929
$176$	14.2583	1.07476
$177$	9.23394	0.694065
$178$	−10.5668	−0.792016
$179$	−11.7112	−0.875336	−0.437668	−	0.899137i	$-0.644195\pi$
$179$	−11.7112	−0.875336	−0.437668	+	0.899137i	$0.644195\pi$
$180$	−2.60002	−0.193794
$181$	−14.7386	−1.09551	−0.547755	−	0.836639i	$-0.684517\pi$
$181$	−14.7386	−1.09551	−0.547755	+	0.836639i	$0.684517\pi$
$182$	1.50711	0.111715
$183$	−12.3287	−0.911363
$184$	13.6472	1.00608
$185$	5.15134	0.378734
$186$	−12.3878	−0.908320
$187$	24.0034	1.75530
$188$	−6.30320	−0.459708
$189$	4.25268	0.309337
$190$	5.02299	0.364406
$191$	−7.11359	−0.514721	−0.257361	−	0.966315i	$-0.582853\pi$
$191$	−7.11359	−0.514721	−0.257361	+	0.966315i	$0.582853\pi$
$192$	12.6178	0.910615
$193$	−1.39667	−0.100535	−0.0502673	−	0.998736i	$-0.516007\pi$
$193$	−1.39667	−0.100535	−0.0502673	+	0.998736i	$0.516007\pi$
$194$	25.1160	1.80323
$195$	2.58196	0.184898
$196$	0.571694	0.0408353
$197$	−1.06752	−0.0760579	−0.0380289	−	0.999277i	$-0.512108\pi$
$197$	−1.06752	−0.0760579	−0.0380289	+	0.999277i	$0.512108\pi$
$198$	21.5900	1.53433
$199$	−18.4150	−1.30541	−0.652703	−	0.757614i	$-0.726365\pi$
$199$	−18.4150	−1.30541	−0.652703	+	0.757614i	$0.726365\pi$
$200$	2.29050	0.161963
$201$	23.0668	1.62701
$202$	−3.16462	−0.222662
$203$	1.13986	0.0800025
$204$	−12.7356	−0.891672
$205$	−8.49391	−0.593241
$206$	0.259613	0.0180881
$207$	27.0972	1.88339
$208$	4.52661	0.313864
$209$	−9.27219	−0.641371
$210$	4.40579	0.304028
$211$	−4.23102	−0.291275	−0.145638	−	0.989338i	$-0.546523\pi$
$211$	−4.23102	−0.291275	−0.145638	+	0.989338i	$0.546523\pi$
$212$	2.29649	0.157724
$213$	−6.93024	−0.474852
$214$	−15.5097	−1.06022
$215$	7.36688	0.502417
$216$	9.74077	0.662775
$217$	2.81172	0.190872
$218$	−4.24358	−0.287411
$219$	−29.3765	−1.98508
$220$	1.69236	0.114099
$221$	7.62042	0.512605
$222$	22.6957	1.52324
$223$	13.1294	0.879213	0.439606	−	0.898191i	$-0.355118\pi$
$223$	13.1294	0.879213	0.439606	+	0.898191i	$0.355118\pi$
$224$	3.14306	0.210005
$225$	4.54792	0.303195
$226$	7.05529	0.469311
$227$	−25.7573	−1.70957	−0.854785	−	0.518983i	$-0.826311\pi$
$227$	−25.7573	−1.70957	−0.854785	+	0.518983i	$0.826311\pi$
$228$	4.91960	0.325809
$229$	1.00000	0.0660819
$229$	1.00000	0.0660819
$230$	9.55481	0.630025
$231$	−8.13286	−0.535103
$232$	2.61085	0.171411
$233$	26.1288	1.71176	0.855878	−	0.517178i	$-0.173017\pi$
$233$	26.1288	1.71176	0.855878	+	0.517178i	$0.173017\pi$
$234$	6.85423	0.448075
$235$	11.0255	0.719222
$236$	1.92148	0.125078
$237$	−14.8487	−0.964528
$238$	13.0033	0.842877
$239$	−25.3249	−1.63813	−0.819066	−	0.573699i	$-0.805508\pi$
$239$	−25.3249	−1.63813	−0.819066	+	0.573699i	$0.805508\pi$
$240$	13.2327	0.854170
$241$	4.23234	0.272629	0.136314	−	0.990666i	$-0.456474\pi$
$241$	4.23234	0.272629	0.136314	+	0.990666i	$0.456474\pi$
$242$	3.58714	0.230590
$243$	−18.1433	−1.16390
$244$	−2.56547	−0.164237
$245$	−1.00000	−0.0638877
$246$	−37.4224	−2.38596
$247$	−2.94367	−0.187301
$248$	6.44025	0.408956
$249$	31.7530	2.01226
$250$	1.60365	0.101424
$251$	25.7954	1.62819	0.814096	−	0.580730i	$-0.197233\pi$
$251$	25.7954	1.62819	0.814096	+	0.580730i	$0.197233\pi$
$252$	2.60002	0.163786
$253$	−17.6377	−1.10887
$254$	17.1343	1.07510
$255$	22.2770	1.39504
$256$	12.7064	0.794149
$257$	19.7986	1.23500	0.617500	−	0.786571i	$-0.288145\pi$
$257$	19.7986	1.23500	0.617500	+	0.786571i	$0.288145\pi$
$258$	32.4569	2.02068
$259$	−5.15134	−0.320089
$260$	0.537279	0.0333206
$261$	5.18399	0.320881
$262$	29.0315	1.79357
$263$	1.47088	0.0906986	0.0453493	−	0.998971i	$-0.485560\pi$
$263$	1.47088	0.0906986	0.0453493	+	0.998971i	$0.485560\pi$
$264$	−18.6284	−1.14650
$265$	−4.01700	−0.246762
$266$	−5.02299	−0.307979
$267$	18.1029	1.10788
$268$	4.79995	0.293204
$269$	−14.0735	−0.858077	−0.429039	−	0.903286i	$-0.641148\pi$
$269$	−14.0735	−0.858077	−0.429039	+	0.903286i	$0.641148\pi$
$270$	6.81980	0.415040
$271$	4.46461	0.271206	0.135603	−	0.990763i	$-0.456703\pi$
$271$	4.46461	0.271206	0.135603	+	0.990763i	$0.456703\pi$
$272$	39.0552	2.36807
$273$	−2.58196	−0.156267
$274$	25.7335	1.55462
$275$	−2.96026	−0.178510
$276$	9.35814	0.563294
$277$	8.89814	0.534638	0.267319	−	0.963608i	$-0.413862\pi$
$277$	8.89814	0.534638	0.267319	+	0.963608i	$0.413862\pi$
$278$	−4.83490	−0.289978
$279$	12.7875	0.765566
$280$	−2.29050	−0.136884
$281$	−9.66254	−0.576419	−0.288209	−	0.957567i	$-0.593060\pi$
$281$	−9.66254	−0.576419	−0.288209	+	0.957567i	$0.593060\pi$
$282$	48.5759	2.89265
$283$	−1.57381	−0.0935534	−0.0467767	−	0.998905i	$-0.514895\pi$
$283$	−1.57381	−0.0935534	−0.0467767	+	0.998905i	$0.514895\pi$
$284$	−1.44211	−0.0855735
$285$	−8.60530	−0.509734
$286$	−4.46145	−0.263811
$287$	8.49391	0.501380
$288$	14.2944	0.842305
$289$	48.7485	2.86756
$290$	1.82794	0.107340
$291$	−43.0284	−2.52237
$292$	−6.11295	−0.357733
$293$	−16.1837	−0.945460	−0.472730	−	0.881207i	$-0.656731\pi$
$293$	−16.1837	−0.945460	−0.472730	+	0.881207i	$0.656731\pi$
$294$	−4.40579	−0.256951
$295$	−3.36104	−0.195687
$296$	−11.7992	−0.685813
$297$	−12.5890	−0.730489
$298$	−33.9006	−1.96381
$299$	−5.59949	−0.323827
$300$	1.57064	0.0906811
$301$	−7.36688	−0.424620
$302$	25.7630	1.48249
$303$	5.42157	0.311461
$304$	−15.0865	−0.865271
$305$	4.48749	0.256953
$306$	59.1378	3.38068
$307$	−15.8017	−0.901850	−0.450925	−	0.892562i	$-0.648906\pi$
$307$	−15.8017	−0.901850	−0.450925	+	0.892562i	$0.648906\pi$
$308$	−1.69236	−0.0964313
$309$	−0.444765	−0.0253018
$310$	4.50901	0.256095
$311$	15.2312	0.863683	0.431841	−	0.901950i	$-0.357864\pi$
$311$	15.2312	0.863683	0.431841	+	0.901950i	$0.357864\pi$
$312$	−5.91399	−0.334814
$313$	5.71890	0.323251	0.161626	−	0.986852i	$-0.448326\pi$
$313$	5.71890	0.323251	0.161626	+	0.986852i	$0.448326\pi$
$314$	26.1227	1.47419
$315$	−4.54792	−0.256246
$316$	−3.08986	−0.173818
$317$	−10.0740	−0.565810	−0.282905	−	0.959148i	$-0.591298\pi$
$317$	−10.0740	−0.565810	−0.282905	+	0.959148i	$0.591298\pi$
$318$	−17.6980	−0.992456
$319$	−3.37428	−0.188923
$320$	−4.59274	−0.256742
$321$	26.5710	1.48305
$322$	−9.55481	−0.532469
$323$	−25.3978	−1.41317
$324$	−1.12062	−0.0622569
$325$	−0.939802	−0.0521308
$326$	6.47310	0.358512
$327$	7.27003	0.402033
$328$	19.4553	1.07424
$329$	−11.0255	−0.607854
$330$	−13.0423	−0.717953
$331$	−20.5161	−1.12767	−0.563833	−	0.825889i	$-0.690674\pi$
$331$	−20.5161	−1.12767	−0.563833	+	0.825889i	$0.690674\pi$
$332$	6.60746	0.362632
$333$	−23.4279	−1.28384
$334$	−30.8520	−1.68814
$335$	−8.39602	−0.458724
$336$	−13.2327	−0.721906
$337$	−4.20533	−0.229079	−0.114540	−	0.993419i	$-0.536539\pi$
$337$	−4.20533	−0.229079	−0.114540	+	0.993419i	$0.536539\pi$
$338$	19.4311	1.05691
$339$	−12.0870	−0.656476
$340$	4.63561	0.251401
$341$	−8.32342	−0.450738
$342$	−22.8442	−1.23527
$343$	1.00000	0.0539949
$344$	−16.8739	−0.909777
$345$	−16.3691	−0.881285
$346$	31.4055	1.68837
$347$	−10.3406	−0.555114	−0.277557	−	0.960709i	$-0.589525\pi$
$347$	−10.3406	−0.555114	−0.277557	+	0.960709i	$0.589525\pi$
$348$	1.79031	0.0959708
$349$	16.8979	0.904523	0.452262	−	0.891885i	$-0.350617\pi$
$349$	16.8979	0.904523	0.452262	+	0.891885i	$0.350617\pi$
$350$	−1.60365	−0.0857187
$351$	−3.99667	−0.213327
$352$	−9.30428	−0.495920
$353$	5.40256	0.287549	0.143775	−	0.989610i	$-0.454076\pi$
$353$	5.40256	0.287549	0.143775	+	0.989610i	$0.454076\pi$
$354$	−14.8080	−0.787037
$355$	2.52252	0.133882
$356$	3.76702	0.199652
$357$	−22.2770	−1.17902
$358$	18.7807	0.992588
$359$	22.8228	1.20454	0.602270	−	0.798293i	$-0.294263\pi$
$359$	22.8228	1.20454	0.602270	+	0.798293i	$0.294263\pi$
$360$	−10.4170	−0.549025
$361$	−9.18918	−0.483641
$362$	23.6355	1.24226
$363$	−6.14543	−0.322551
$364$	−0.537279	−0.0281611
$365$	10.6927	0.559681
$366$	19.7709	1.03344
$367$	−34.9624	−1.82502	−0.912510	−	0.409054i	$-0.865859\pi$
$367$	−34.9624	−1.82502	−0.912510	+	0.409054i	$0.865859\pi$
$368$	−28.6978	−1.49598
$369$	38.6296	2.01098
$370$	−8.26095	−0.429466
$371$	4.01700	0.208552
$372$	4.41620	0.228970
$373$	−23.0746	−1.19476	−0.597379	−	0.801959i	$-0.703791\pi$
$373$	−23.0746	−1.19476	−0.597379	+	0.801959i	$0.703791\pi$
$374$	−38.4931	−1.99043
$375$	−2.74735	−0.141872
$376$	−25.2539	−1.30237
$377$	−1.07124	−0.0551718
$378$	−6.81980	−0.350773
$379$	−0.124054	−0.00637222	−0.00318611	−	0.999995i	$-0.501014\pi$
$379$	−0.124054	−0.00637222	−0.00318611	+	0.999995i	$0.501014\pi$
$380$	−1.79067	−0.0918596
$381$	−29.3541	−1.50386
$382$	11.4077	0.583669
$383$	−0.776528	−0.0396787	−0.0198394	−	0.999803i	$-0.506315\pi$
$383$	−0.776528	−0.0396787	−0.0198394	+	0.999803i	$0.506315\pi$
$384$	−37.5048	−1.91391
$385$	2.96026	0.150869
$386$	2.23977	0.114001
$387$	−33.5040	−1.70310
$388$	−8.95375	−0.454558
$389$	3.48751	0.176824	0.0884118	−	0.996084i	$-0.471821\pi$
$389$	3.48751	0.176824	0.0884118	+	0.996084i	$0.471821\pi$
$390$	−4.14056	−0.209666
$391$	−48.3120	−2.44324
$392$	2.29050	0.115688
$393$	−49.7363	−2.50887
$394$	1.71193	0.0862460
$395$	5.40475	0.271942
$396$	−7.69673	−0.386775
$397$	9.34532	0.469028	0.234514	−	0.972113i	$-0.424650\pi$
$397$	9.34532	0.469028	0.234514	+	0.972113i	$0.424650\pi$
$398$	29.5313	1.48027
$399$	8.60530	0.430804
$400$	−4.81655	−0.240828
$401$	5.83716	0.291494	0.145747	−	0.989322i	$-0.453441\pi$
$401$	5.83716	0.291494	0.145747	+	0.989322i	$0.453441\pi$
$402$	−36.9911	−1.84495
$403$	−2.64246	−0.131630
$404$	1.12817	0.0561286
$405$	1.96018	0.0974022
$406$	−1.82794	−0.0907189
$407$	15.2493	0.755880
$408$	−51.0255	−2.52614
$409$	−6.05070	−0.299188	−0.149594	−	0.988748i	$-0.547797\pi$
$409$	−6.05070	−0.299188	−0.149594	+	0.988748i	$0.547797\pi$
$410$	13.6213	0.672707
$411$	−44.0863	−2.17462
$412$	−0.0925509	−0.00455965
$413$	3.36104	0.165386
$414$	−43.4545	−2.13567
$415$	−11.5577	−0.567345
$416$	−2.95385	−0.144825
$417$	8.28308	0.405624
$418$	14.8694	0.727284
$419$	15.6671	0.765390	0.382695	−	0.923875i	$-0.374996\pi$
$419$	15.6671	0.765390	0.382695	+	0.923875i	$0.374996\pi$
$420$	−1.57064	−0.0766395
$421$	−6.08615	−0.296621	−0.148310	−	0.988941i	$-0.547383\pi$
$421$	−6.08615	−0.296621	−0.148310	+	0.988941i	$0.547383\pi$
$422$	6.78507	0.330292
$423$	−50.1430	−2.43803
$424$	9.20094	0.446837
$425$	−8.10854	−0.393322
$426$	11.1137	0.538460
$427$	−4.48749	−0.217165
$428$	5.52915	0.267261
$429$	7.64328	0.369021
$430$	−11.8139	−0.569717
$431$	7.12854	0.343370	0.171685	−	0.985152i	$-0.445079\pi$
$431$	7.12854	0.343370	0.171685	+	0.985152i	$0.445079\pi$
$432$	−20.4832	−0.985500
$433$	36.8120	1.76907	0.884536	−	0.466472i	$-0.154475\pi$
$433$	36.8120	1.76907	0.884536	+	0.466472i	$0.154475\pi$
$434$	−4.50901	−0.216440
$435$	−3.13159	−0.150148
$436$	1.51282	0.0724507
$437$	18.6623	0.892738
$438$	47.1097	2.25099
$439$	8.54599	0.407878	0.203939	−	0.978984i	$-0.434626\pi$
$439$	8.54599	0.407878	0.203939	+	0.978984i	$0.434626\pi$
$440$	6.78049	0.323247
$441$	4.54792	0.216568
$442$	−12.2205	−0.581270
$443$	−40.2729	−1.91342	−0.956711	−	0.291039i	$-0.905999\pi$
$443$	−40.2729	−1.91342	−0.956711	+	0.291039i	$0.905999\pi$
$444$	−8.09092	−0.383978
$445$	−6.58922	−0.312359
$446$	−21.0550	−0.996985
$447$	58.0780	2.74699
$448$	4.59274	0.216986
$449$	−38.2228	−1.80385	−0.901924	−	0.431895i	$-0.857845\pi$
$449$	−38.2228	−1.80385	−0.901924	+	0.431895i	$0.857845\pi$
$450$	−7.29327	−0.343808
$451$	−25.1442	−1.18399
$452$	−2.51518	−0.118304
$453$	−44.1367	−2.07373
$454$	41.3056	1.93857
$455$	0.939802	0.0440586
$456$	19.7105	0.923028
$457$	31.9614	1.49509	0.747546	−	0.664210i	$-0.231232\pi$
$457$	31.9614	1.49509	0.747546	+	0.664210i	$0.231232\pi$
$458$	−1.60365	−0.0749337
$459$	−34.4830	−1.60953
$460$	−3.40625	−0.158817
$461$	−18.4356	−0.858633	−0.429317	−	0.903154i	$-0.641246\pi$
$461$	−18.4356	−0.858633	−0.429317	+	0.903154i	$0.641246\pi$
$462$	13.0423	0.606781
$463$	−3.07070	−0.142707	−0.0713537	−	0.997451i	$-0.522732\pi$
$463$	−3.07070	−0.142707	−0.0713537	+	0.997451i	$0.522732\pi$
$464$	−5.49020	−0.254876
$465$	−7.72477	−0.358228
$466$	−41.9015	−1.94105
$467$	11.3402	0.524760	0.262380	−	0.964965i	$-0.415493\pi$
$467$	11.3402	0.524760	0.262380	+	0.964965i	$0.415493\pi$
$468$	−2.44350	−0.112951
$469$	8.39602	0.387692
$470$	−17.6810	−0.815564
$471$	−44.7529	−2.06211
$472$	7.69846	0.354351
$473$	21.8079	1.00273
$474$	23.8121	1.09373
$475$	3.13222	0.143716
$476$	−4.63561	−0.212473
$477$	18.2690	0.836479
$478$	40.6123	1.85756
$479$	16.7772	0.766571	0.383285	−	0.923630i	$-0.374793\pi$
$479$	16.7772	0.766571	0.383285	+	0.923630i	$0.374793\pi$
$480$	−8.63508	−0.394136
$481$	4.84124	0.220742
$482$	−6.78719	−0.309148
$483$	16.3691	0.744822
$484$	−1.27880	−0.0581272
$485$	15.6618	0.711165
$486$	29.0956	1.31980
$487$	13.0106	0.589567	0.294783	−	0.955564i	$-0.404753\pi$
$487$	13.0106	0.589567	0.294783	+	0.955564i	$0.404753\pi$
$488$	−10.2786	−0.465291
$489$	−11.0896	−0.501490
$490$	1.60365	0.0724455
$491$	26.6262	1.20162	0.600811	−	0.799391i	$-0.294844\pi$
$491$	26.6262	1.20162	0.600811	+	0.799391i	$0.294844\pi$
$492$	13.3409	0.601454
$493$	−9.24260	−0.416266
$494$	4.72061	0.212390
$495$	13.4630	0.605118
$496$	−13.5428	−0.608089
$497$	−2.52252	−0.113151
$498$	−50.9207	−2.28181
$499$	1.31693	0.0589537	0.0294768	−	0.999565i	$-0.490616\pi$
$499$	1.31693	0.0589537	0.0294768	+	0.999565i	$0.490616\pi$
$500$	−0.571694	−0.0255669
$501$	52.8551	2.36139
$502$	−41.3668	−1.84629
$503$	33.3440	1.48674	0.743368	−	0.668883i	$-0.233227\pi$
$503$	33.3440	1.48674	0.743368	+	0.668883i	$0.233227\pi$
$504$	10.4170	0.464011
$505$	−1.97338	−0.0878144
$506$	28.2847	1.25741
$507$	−33.2890	−1.47842
$508$	−6.10828	−0.271011
$509$	−23.5951	−1.04583	−0.522916	−	0.852384i	$-0.675156\pi$
$509$	−23.5951	−1.04583	−0.522916	+	0.852384i	$0.675156\pi$
$510$	−35.7245	−1.58191
$511$	−10.6927	−0.473017
$512$	6.92594	0.306086
$513$	13.3203	0.588107
$514$	−31.7500	−1.40043
$515$	0.161889	0.00713367
$516$	−11.5707	−0.509373
$517$	32.6383	1.43543
$518$	8.26095	0.362965
$519$	−53.8034	−2.36171
$520$	2.15262	0.0943986
$521$	28.5757	1.25192	0.625962	−	0.779854i	$-0.284706\pi$
$521$	28.5757	1.25192	0.625962	+	0.779854i	$0.284706\pi$
$522$	−8.31331	−0.363864
$523$	−28.5575	−1.24873	−0.624366	−	0.781132i	$-0.714642\pi$
$523$	−28.5575	−1.24873	−0.624366	+	0.781132i	$0.714642\pi$
$524$	−10.3496	−0.452124
$525$	2.74735	0.119904
$526$	−2.35878	−0.102848
$527$	−22.7989	−0.993138
$528$	39.1724	1.70476
$529$	12.4997	0.543465
$530$	6.44186	0.279816
$531$	15.2857	0.663344
$532$	1.79067	0.0776355
$533$	−7.98260	−0.345765
$534$	−29.0307	−1.25628
$535$	−9.67151	−0.418136
$536$	19.2311	0.830657
$537$	−32.1747	−1.38844
$538$	22.5690	0.973018
$539$	−2.96026	−0.127507
$540$	−2.43123	−0.104623
$541$	26.5436	1.14120	0.570599	−	0.821229i	$-0.306711\pi$
$541$	26.5436	1.14120	0.570599	+	0.821229i	$0.306711\pi$
$542$	−7.15967	−0.307534
$543$	−40.4920	−1.73768
$544$	−25.4857	−1.09269
$545$	−2.64620	−0.113351
$546$	4.14056	0.177200
$547$	−3.65673	−0.156351	−0.0781753	−	0.996940i	$-0.524909\pi$
$547$	−3.65673	−0.156351	−0.0781753	+	0.996940i	$0.524909\pi$
$548$	−9.17389	−0.391889
$549$	−20.4087	−0.871024
$550$	4.74722	0.202422
$551$	3.57029	0.152100
$552$	37.4936	1.59583
$553$	−5.40475	−0.229833
$554$	−14.2695	−0.606253
$555$	14.1525	0.600741
$556$	1.72362	0.0730978
$557$	−23.8937	−1.01241	−0.506203	−	0.862414i	$-0.668951\pi$
$557$	−23.8937	−1.01241	−0.506203	+	0.862414i	$0.668951\pi$
$558$	−20.5066	−0.868115
$559$	6.92340	0.292829
$560$	4.81655	0.203537
$561$	65.9457	2.78423
$562$	15.4953	0.653631
$563$	38.1597	1.60824	0.804120	−	0.594466i	$-0.202637\pi$
$563$	38.1597	1.60824	0.804120	+	0.594466i	$0.202637\pi$
$564$	−17.3171	−0.729180
$565$	4.39952	0.185089
$566$	2.52384	0.106085
$567$	−1.96018	−0.0823199
$568$	−5.77784	−0.242433
$569$	−24.9705	−1.04682	−0.523409	−	0.852081i	$-0.675340\pi$
$569$	−24.9705	−1.04682	−0.523409	+	0.852081i	$0.675340\pi$
$570$	13.7999	0.578014
$571$	−33.7206	−1.41116	−0.705581	−	0.708630i	$-0.749314\pi$
$571$	−33.7206	−1.41116	−0.705581	+	0.708630i	$0.749314\pi$
$572$	1.59049	0.0665016
$573$	−19.5435	−0.816442
$574$	−13.6213	−0.568541
$575$	5.95816	0.248473
$576$	20.8874	0.870308
$577$	−4.31157	−0.179493	−0.0897466	−	0.995965i	$-0.528606\pi$
$577$	−4.31157	−0.179493	−0.0897466	+	0.995965i	$0.528606\pi$
$578$	−78.1755	−3.25167
$579$	−3.83714	−0.159466
$580$	−0.651651	−0.0270583
$581$	11.5577	0.479494
$582$	69.0024	2.86024
$583$	−11.8914	−0.492490
$584$	−24.4916	−1.01347
$585$	4.27414	0.176714
$586$	25.9529	1.07211
$587$	36.3560	1.50057	0.750287	−	0.661112i	$-0.229915\pi$
$587$	36.3560	1.50057	0.750287	+	0.661112i	$0.229915\pi$
$588$	1.57064	0.0647722
$589$	8.80693	0.362883
$590$	5.38993	0.221900
$591$	−2.93286	−0.120642
$592$	24.8117	1.01976
$593$	−6.91071	−0.283789	−0.141895	−	0.989882i	$-0.545319\pi$
$593$	−6.91071	−0.283789	−0.141895	+	0.989882i	$0.545319\pi$
$594$	20.1884	0.828340
$595$	8.10854	0.332418
$596$	12.0854	0.495038
$597$	−50.5925	−2.07061
$598$	8.97962	0.367204
$599$	−26.9571	−1.10144	−0.550718	−	0.834691i	$-0.685646\pi$
$599$	−26.9571	−1.10144	−0.550718	+	0.834691i	$0.685646\pi$
$600$	6.29281	0.256903
$601$	19.0522	0.777155	0.388578	−	0.921416i	$-0.372966\pi$
$601$	19.0522	0.777155	0.388578	+	0.921416i	$0.372966\pi$
$602$	11.8139	0.481498
$603$	38.1844	1.55499
$604$	−9.18439	−0.373708
$605$	2.23686	0.0909413
$606$	−8.69430	−0.353182
$607$	4.82305	0.195761	0.0978807	−	0.995198i	$-0.468794\pi$
$607$	4.82305	0.195761	0.0978807	+	0.995198i	$0.468794\pi$
$608$	9.84477	0.399258
$609$	3.13159	0.126898
$610$	−7.19636	−0.291372
$611$	10.3618	0.419192
$612$	−21.0824	−0.852204
$613$	5.62056	0.227012	0.113506	−	0.993537i	$-0.463792\pi$
$613$	5.62056	0.227012	0.113506	+	0.993537i	$0.463792\pi$
$614$	25.3404	1.02265
$615$	−23.3357	−0.940988
$616$	−6.78049	−0.273194
$617$	−14.5508	−0.585792	−0.292896	−	0.956144i	$-0.594619\pi$
$617$	−14.5508	−0.585792	−0.292896	+	0.956144i	$0.594619\pi$
$618$	0.713247	0.0286910
$619$	−33.6787	−1.35366	−0.676831	−	0.736139i	$-0.736647\pi$
$619$	−33.6787	−1.35366	−0.676831	+	0.736139i	$0.736647\pi$
$620$	−1.60744	−0.0645565
$621$	25.3381	1.01678
$622$	−24.4255	−0.979375
$623$	6.58922	0.263992
$624$	12.4362	0.497845
$625$	1.00000	0.0400000
$626$	−9.17111	−0.366551
$627$	−25.4739	−1.01733
$628$	−9.31261	−0.371614
$629$	41.7699	1.66547
$630$	7.29327	0.290571
$631$	−12.4858	−0.497050	−0.248525	−	0.968625i	$-0.579946\pi$
$631$	−12.4858	−0.497050	−0.248525	+	0.968625i	$0.579946\pi$
$632$	−12.3796	−0.492434
$633$	−11.6241	−0.462016
$634$	16.1551	0.641601
$635$	10.6845	0.424003
$636$	6.30926	0.250179
$637$	−0.939802	−0.0372363
$638$	5.41117	0.214230
$639$	−11.4722	−0.453834
$640$	13.6513	0.539614
$641$	−5.02998	−0.198672	−0.0993361	−	0.995054i	$-0.531672\pi$
$641$	−5.02998	−0.198672	−0.0993361	+	0.995054i	$0.531672\pi$
$642$	−42.6106	−1.68171
$643$	−23.0519	−0.909078	−0.454539	−	0.890727i	$-0.650196\pi$
$643$	−23.0519	−0.909078	−0.454539	+	0.890727i	$0.650196\pi$
$644$	3.40625	0.134225
$645$	20.2394	0.796925
$646$	40.7291	1.60247
$647$	−14.2838	−0.561554	−0.280777	−	0.959773i	$-0.590592\pi$
$647$	−14.2838	−0.561554	−0.280777	+	0.959773i	$0.590592\pi$
$648$	−4.48980	−0.176376
$649$	−9.94954	−0.390554
$650$	1.50711	0.0591138
$651$	7.72477	0.302758
$652$	−2.30763	−0.0903738
$653$	−36.3390	−1.42206	−0.711028	−	0.703164i	$-0.751770\pi$
$653$	−36.3390	−1.42206	−0.711028	+	0.703164i	$0.751770\pi$
$654$	−11.6586	−0.455887
$655$	18.1034	0.707358
$656$	−40.9114	−1.59732
$657$	−48.6295	−1.89722
$658$	17.6810	0.689277
$659$	30.7567	1.19811	0.599056	−	0.800707i	$-0.295543\pi$
$659$	30.7567	1.19811	0.599056	+	0.800707i	$0.295543\pi$
$660$	4.64951	0.180982
$661$	25.9624	1.00982	0.504910	−	0.863172i	$-0.331526\pi$
$661$	25.9624	1.00982	0.504910	+	0.863172i	$0.331526\pi$
$662$	32.9006	1.27872
$663$	20.9360	0.813085
$664$	26.4729	1.02735
$665$	−3.13222	−0.121462
$666$	37.5701	1.45581
$667$	6.79147	0.262967
$668$	10.9986	0.425548
$669$	36.0712	1.39459
$670$	13.4643	0.520170
$671$	13.2841	0.512828
$672$	8.63508	0.333106
$673$	35.8777	1.38298	0.691491	−	0.722385i	$-0.256954\pi$
$673$	35.8777	1.38298	0.691491	+	0.722385i	$0.256954\pi$
$674$	6.74388	0.259765
$675$	4.25268	0.163686
$676$	−6.92709	−0.266426
$677$	12.0555	0.463330	0.231665	−	0.972796i	$-0.425583\pi$
$677$	12.0555	0.463330	0.231665	+	0.972796i	$0.425583\pi$
$678$	19.3833	0.744412
$679$	−15.6618	−0.601044
$680$	18.5726	0.712229
$681$	−70.7642	−2.71169
$682$	13.3479	0.511116
$683$	−43.5261	−1.66548	−0.832740	−	0.553664i	$-0.813229\pi$
$683$	−43.5261	−1.66548	−0.832740	+	0.553664i	$0.813229\pi$
$684$	8.14384	0.311387
$685$	16.0468	0.613119
$686$	−1.60365	−0.0612276
$687$	2.74735	0.104818
$688$	35.4830	1.35278
$689$	−3.77518	−0.143823
$690$	26.2504	0.999335
$691$	13.9896	0.532188	0.266094	−	0.963947i	$-0.414267\pi$
$691$	13.9896	0.532188	0.266094	+	0.963947i	$0.414267\pi$
$692$	−11.1959	−0.425605
$693$	−13.4630	−0.511418
$694$	16.5827	0.629472
$695$	−3.01494	−0.114363
$696$	7.17292	0.271889
$697$	−68.8733	−2.60876
$698$	−27.0983	−1.02569
$699$	71.7850	2.71516
$700$	0.571694	0.0216080
$701$	−6.44459	−0.243409	−0.121704	−	0.992566i	$-0.538836\pi$
$701$	−6.44459	−0.243409	−0.121704	+	0.992566i	$0.538836\pi$
$702$	6.40926	0.241902
$703$	−16.1351	−0.608549
$704$	−13.5957	−0.512407
$705$	30.2908	1.14082
$706$	−8.66381	−0.326067
$707$	1.97338	0.0742167
$708$	5.27899	0.198396
$709$	−25.7351	−0.966500	−0.483250	−	0.875482i	$-0.660544\pi$
$709$	−25.7351	−0.966500	−0.483250	+	0.875482i	$0.660544\pi$
$710$	−4.04524	−0.151815
$711$	−24.5804	−0.921835
$712$	15.0926	0.565620
$713$	16.7527	0.627393
$714$	35.7245	1.33696
$715$	−2.78206	−0.104043
$716$	−6.69522	−0.250212
$717$	−69.5764	−2.59838
$718$	−36.5997	−1.36589
$719$	−16.1014	−0.600479	−0.300240	−	0.953864i	$-0.597067\pi$
$719$	−16.1014	−0.600479	−0.300240	+	0.953864i	$0.597067\pi$
$720$	21.9053	0.816362
$721$	−0.161889	−0.00602906
$722$	14.7362	0.548426
$723$	11.6277	0.432439
$724$	−8.42596	−0.313148
$725$	1.13986	0.0423333
$726$	9.85512	0.365758
$727$	−20.0453	−0.743438	−0.371719	−	0.928345i	$-0.621232\pi$
$727$	−20.0453	−0.743438	−0.371719	+	0.928345i	$0.621232\pi$
$728$	−2.15262	−0.0797814
$729$	−43.9655	−1.62835
$730$	−17.1473	−0.634651
$731$	59.7346	2.20937
$732$	−7.04824	−0.260510
$733$	47.7238	1.76272	0.881360	−	0.472446i	$-0.156629\pi$
$733$	47.7238	1.76272	0.881360	+	0.472446i	$0.156629\pi$
$734$	56.0674	2.06948
$735$	−2.74735	−0.101337
$736$	18.7269	0.690282
$737$	−24.8544	−0.915524
$738$	−61.9484	−2.28035
$739$	4.33992	0.159646	0.0798232	−	0.996809i	$-0.474564\pi$
$739$	4.33992	0.159646	0.0798232	+	0.996809i	$0.474564\pi$
$740$	2.94499	0.108260
$741$	−8.08728	−0.297094
$742$	−6.44186	−0.236488
$743$	−24.9865	−0.916664	−0.458332	−	0.888781i	$-0.651553\pi$
$743$	−24.9865	−0.916664	−0.458332	+	0.888781i	$0.651553\pi$
$744$	17.6936	0.648679
$745$	−21.1397	−0.774497
$746$	37.0036	1.35480
$747$	52.5634	1.92320
$748$	13.7226	0.501748
$749$	9.67151	0.353389
$750$	4.40579	0.160877
$751$	47.0321	1.71623	0.858113	−	0.513462i	$-0.171637\pi$
$751$	47.0321	1.71623	0.858113	+	0.513462i	$0.171637\pi$
$752$	53.1048	1.93653
$753$	70.8690	2.58261
$754$	1.71790	0.0625621
$755$	16.0652	0.584673
$756$	2.43123	0.0884230
$757$	37.4661	1.36173	0.680865	−	0.732409i	$-0.261604\pi$
$757$	37.4661	1.36173	0.680865	+	0.732409i	$0.261604\pi$
$758$	0.198939	0.00722579
$759$	−48.4569	−1.75887
$760$	−7.17436	−0.260242
$761$	8.69862	0.315325	0.157662	−	0.987493i	$-0.449604\pi$
$761$	8.69862	0.315325	0.157662	+	0.987493i	$0.449604\pi$
$762$	47.0738	1.70530
$763$	2.64620	0.0957988
$764$	−4.06680	−0.147132
$765$	36.8770	1.33329
$766$	1.24528	0.0449938
$767$	−3.15871	−0.114054
$768$	34.9089	1.25966
$769$	−15.2420	−0.549640	−0.274820	−	0.961496i	$-0.588618\pi$
$769$	−15.2420	−0.549640	−0.274820	+	0.961496i	$0.588618\pi$
$770$	−4.74722	−0.171078
$771$	54.3935	1.95894
$772$	−0.798469	−0.0287375
$773$	−26.7997	−0.963918	−0.481959	−	0.876194i	$-0.660075\pi$
$773$	−26.7997	−0.963918	−0.481959	+	0.876194i	$0.660075\pi$
$774$	53.7286	1.93124
$775$	2.81172	0.101000
$776$	−35.8734	−1.28778
$777$	−14.1525	−0.507719
$778$	−5.59274	−0.200509
$779$	26.6048	0.953217
$780$	1.47609	0.0528526
$781$	7.46732	0.267202
$782$	77.4756	2.77052
$783$	4.84745	0.173234
$784$	−4.81655	−0.172020
$785$	16.2895	0.581397
$786$	79.7596	2.84493
$787$	−48.6728	−1.73500	−0.867499	−	0.497439i	$-0.834274\pi$
$787$	−48.6728	−1.73500	−0.867499	+	0.497439i	$0.834274\pi$
$788$	−0.610297	−0.0217409
$789$	4.04103	0.143865
$790$	−8.66732	−0.308369
$791$	−4.39952	−0.156429
$792$	−30.8371	−1.09575
$793$	4.21735	0.149762
$794$	−14.9866	−0.531855
$795$	−11.0361	−0.391410
$796$	−10.5278	−0.373147
$797$	11.0310	0.390739	0.195370	−	0.980730i	$-0.437409\pi$
$797$	11.0310	0.390739	0.195370	+	0.980730i	$0.437409\pi$
$798$	−13.7999	−0.488511
$799$	89.4005	3.16276
$800$	3.14306	0.111124
$801$	29.9673	1.05884
$802$	−9.36077	−0.330540
$803$	31.6531	1.11701
$804$	13.1871	0.465075
$805$	−5.95816	−0.209998
$806$	4.23758	0.149262
$807$	−38.6648	−1.36107
$808$	4.52004	0.159015
$809$	−5.07615	−0.178468	−0.0892339	−	0.996011i	$-0.528442\pi$
$809$	−5.07615	−0.178468	−0.0892339	+	0.996011i	$0.528442\pi$
$810$	−3.14345	−0.110449
$811$	−48.2814	−1.69539	−0.847695	−	0.530484i	$-0.822010\pi$
$811$	−48.2814	−1.69539	−0.847695	+	0.530484i	$0.822010\pi$
$812$	0.651651	0.0228685
$813$	12.2658	0.430182
$814$	−24.4546	−0.857132
$815$	4.03648	0.141392
$816$	107.298	3.75619
$817$	−23.0747	−0.807281
$818$	9.70321	0.339265
$819$	−4.27414	−0.149351
$820$	−4.85592	−0.169576
$821$	16.0692	0.560819	0.280409	−	0.959880i	$-0.409530\pi$
$821$	16.0692	0.560819	0.280409	+	0.959880i	$0.409530\pi$
$822$	70.6990	2.46591
$823$	−29.4236	−1.02564	−0.512822	−	0.858495i	$-0.671400\pi$
$823$	−29.4236	−1.02564	−0.512822	+	0.858495i	$0.671400\pi$
$824$	−0.370807	−0.0129177
$825$	−8.13286	−0.283150
$826$	−5.38993	−0.187540
$827$	10.3236	0.358988	0.179494	−	0.983759i	$-0.442554\pi$
$827$	10.3236	0.358988	0.179494	+	0.983759i	$0.442554\pi$
$828$	15.4913	0.538361
$829$	−27.8662	−0.967833	−0.483917	−	0.875114i	$-0.660786\pi$
$829$	−27.8662	−0.967833	−0.483917	+	0.875114i	$0.660786\pi$
$830$	18.5345	0.643342
$831$	24.4463	0.848033
$832$	−4.31626	−0.149639
$833$	−8.10854	−0.280944
$834$	−13.2832	−0.459958
$835$	−19.2386	−0.665779
$836$	−5.30086	−0.183334
$837$	11.9573	0.413306
$838$	−25.1246	−0.867916
$839$	−3.98388	−0.137539	−0.0687694	−	0.997633i	$-0.521907\pi$
$839$	−3.98388	−0.137539	−0.0687694	+	0.997633i	$0.521907\pi$
$840$	−6.29281	−0.217123
$841$	−27.7007	−0.955197
$842$	9.76005	0.336353
$843$	−26.5464	−0.914305
$844$	−2.41885	−0.0832602
$845$	12.1168	0.416830
$846$	80.4118	2.76461
$847$	−2.23686	−0.0768594
$848$	−19.3481	−0.664416
$849$	−4.32381	−0.148393
$850$	13.0033	0.446008
$851$	−30.6925	−1.05213
$852$	−3.96198	−0.135735
$853$	−55.9446	−1.91551	−0.957753	−	0.287591i	$-0.907146\pi$
$853$	−55.9446	−1.91551	−0.957753	+	0.287591i	$0.907146\pi$
$854$	7.19636	0.246254
$855$	−14.2451	−0.487172
$856$	22.1526	0.757161
$857$	2.00235	0.0683991	0.0341995	−	0.999415i	$-0.489112\pi$
$857$	2.00235	0.0683991	0.0341995	+	0.999415i	$0.489112\pi$
$858$	−12.2571	−0.418452
$859$	52.1920	1.78077	0.890385	−	0.455209i	$-0.150435\pi$
$859$	52.1920	1.78077	0.890385	+	0.455209i	$0.150435\pi$
$860$	4.21160	0.143614
$861$	23.3357	0.795280
$862$	−11.4317	−0.389365
$863$	34.3747	1.17013	0.585065	−	0.810987i	$-0.301069\pi$
$863$	34.3747	1.17013	0.585065	+	0.810987i	$0.301069\pi$
$864$	13.3664	0.454735
$865$	19.5838	0.665868
$866$	−59.0336	−2.00604
$867$	133.929	4.54847
$868$	1.60744	0.0545602
$869$	15.9995	0.542744
$870$	5.02198	0.170261
$871$	−7.89059	−0.267363
$872$	6.06112	0.205256
$873$	−71.2285	−2.41072
$874$	−29.9278	−1.01232
$875$	−1.00000	−0.0338062
$876$	−16.7944	−0.567430
$877$	−23.1245	−0.780859	−0.390429	−	0.920633i	$-0.627673\pi$
$877$	−23.1245	−0.780859	−0.390429	+	0.920633i	$0.627673\pi$
$878$	−13.7048	−0.462514
$879$	−44.4622	−1.49967
$880$	−14.2583	−0.480646
$881$	16.1347	0.543592	0.271796	−	0.962355i	$-0.412382\pi$
$881$	16.1347	0.543592	0.271796	+	0.962355i	$0.412382\pi$
$882$	−7.29327	−0.245577
$883$	−31.9477	−1.07512	−0.537562	−	0.843224i	$-0.680655\pi$
$883$	−31.9477	−1.07512	−0.537562	+	0.843224i	$0.680655\pi$
$884$	4.35655	0.146527
$885$	−9.23394	−0.310395
$886$	64.5836	2.16973
$887$	−31.8980	−1.07103	−0.535516	−	0.844525i	$-0.679883\pi$
$887$	−31.8980	−1.07103	−0.535516	+	0.844525i	$0.679883\pi$
$888$	−32.4164	−1.08782
$889$	−10.6845	−0.358348
$890$	10.5668	0.354200
$891$	5.80265	0.194396
$892$	7.50603	0.251320
$893$	−34.5342	−1.15564
$894$	−93.1368	−3.11496
$895$	11.7112	0.391462
$896$	−13.6513	−0.456057
$897$	−15.3837	−0.513648
$898$	61.2961	2.04548
$899$	3.20496	0.106892
$900$	2.60002	0.0866673
$901$	−32.5720	−1.08513
$902$	40.3225	1.34259
$903$	−20.2394	−0.673524
$904$	−10.0771	−0.335160
$905$	14.7386	0.489927
$906$	70.7799	2.35150
$907$	−36.3674	−1.20756	−0.603780	−	0.797151i	$-0.706340\pi$
$907$	−36.3674	−1.20756	−0.603780	+	0.797151i	$0.706340\pi$
$908$	−14.7253	−0.488675
$909$	8.97479	0.297675
$910$	−1.50711	−0.0499603
$911$	20.4502	0.677544	0.338772	−	0.940869i	$-0.389988\pi$
$911$	20.4502	0.677544	0.338772	+	0.940869i	$0.389988\pi$
$912$	−41.4479	−1.37248
$913$	−34.2138	−1.13231
$914$	−51.2550	−1.69536
$915$	12.3287	0.407574
$916$	0.571694	0.0188893
$917$	−18.1034	−0.597827
$918$	55.2987	1.82513
$919$	−26.7098	−0.881076	−0.440538	−	0.897734i	$-0.645212\pi$
$919$	−26.7098	−0.881076	−0.440538	+	0.897734i	$0.645212\pi$
$920$	−13.6472	−0.449934
$921$	−43.4127	−1.43050
$922$	29.5643	0.973649
$923$	2.37067	0.0780316
$924$	−4.64951	−0.152958
$925$	−5.15134	−0.169375
$926$	4.92432	0.161823
$927$	−0.736258	−0.0241819
$928$	3.58265	0.117606
$929$	−20.4789	−0.671892	−0.335946	−	0.941881i	$-0.609056\pi$
$929$	−20.4789	−0.671892	−0.335946	+	0.941881i	$0.609056\pi$
$930$	12.3878	0.406213
$931$	3.13222	0.102654
$932$	14.9377	0.489300
$933$	41.8454	1.36996
$934$	−18.1857	−0.595053
$935$	−24.0034	−0.784995
$936$	−9.78994	−0.319994
$937$	20.4531	0.668173	0.334087	−	0.942542i	$-0.391572\pi$
$937$	20.4531	0.668173	0.334087	+	0.942542i	$0.391572\pi$
$938$	−13.4643	−0.439624
$939$	15.7118	0.512735
$940$	6.30320	0.205588
$941$	34.8935	1.13750	0.568748	−	0.822512i	$-0.307428\pi$
$941$	34.8935	1.13750	0.568748	+	0.822512i	$0.307428\pi$
$942$	71.7680	2.33833
$943$	50.6081	1.64803
$944$	−16.1886	−0.526895
$945$	−4.25268	−0.138340
$946$	−34.9722	−1.13704
$947$	−33.6167	−1.09240	−0.546199	−	0.837656i	$-0.683926\pi$
$947$	−33.6167	−1.09240	−0.546199	+	0.837656i	$0.683926\pi$
$948$	−8.48892	−0.275707
$949$	10.0490	0.326205
$950$	−5.02299	−0.162967
$951$	−27.6767	−0.897477
$952$	−18.5726	−0.601943
$953$	42.1689	1.36598	0.682992	−	0.730426i	$-0.260678\pi$
$953$	42.1689	1.36598	0.682992	+	0.730426i	$0.260678\pi$
$954$	−29.2971	−0.948527
$955$	7.11359	0.230190
$956$	−14.4781	−0.468255
$957$	−9.27032	−0.299667
$958$	−26.9048	−0.869254
$959$	−16.0468	−0.518180
$960$	−12.6178	−0.407239
$961$	−23.0942	−0.744975
$962$	−7.76366	−0.250310
$963$	43.9853	1.41741
$964$	2.41960	0.0779301
$965$	1.39667	0.0449604
$966$	−26.2504	−0.844592
$967$	−15.3219	−0.492719	−0.246359	−	0.969179i	$-0.579234\pi$
$967$	−15.3219	−0.492719	−0.246359	+	0.969179i	$0.579234\pi$
$968$	−5.12353	−0.164677
$969$	−69.7765	−2.24154
$970$	−25.1160	−0.806427
$971$	−53.7401	−1.72460	−0.862301	−	0.506397i	$-0.830977\pi$
$971$	−53.7401	−1.72460	−0.862301	+	0.506397i	$0.830977\pi$
$972$	−10.3724	−0.332696
$973$	3.01494	0.0966544
$974$	−20.8645	−0.668540
$975$	−2.58196	−0.0826890
$976$	21.6142	0.691855
$977$	−17.0447	−0.545308	−0.272654	−	0.962112i	$-0.587901\pi$
$977$	−17.0447	−0.545308	−0.272654	+	0.962112i	$0.587901\pi$
$978$	17.7839	0.568665
$979$	−19.5058	−0.623409
$980$	−0.571694	−0.0182621
$981$	12.0347	0.384238
$982$	−42.6991	−1.36258
$983$	−2.41186	−0.0769263	−0.0384631	−	0.999260i	$-0.512246\pi$
$983$	−2.41186	−0.0769263	−0.0384631	+	0.999260i	$0.512246\pi$
$984$	53.4506	1.70394
$985$	1.06752	0.0340141
$986$	14.8219	0.472026
$987$	−30.2908	−0.964167
$988$	−1.68288	−0.0535395
$989$	−43.8930	−1.39572
$990$	−21.5900	−0.686175
$991$	7.48127	0.237650	0.118825	−	0.992915i	$-0.462087\pi$
$991$	7.48127	0.237650	0.118825	+	0.992915i	$0.462087\pi$
$992$	8.83740	0.280588
$993$	−56.3648	−1.78868
$994$	4.04524	0.128307
$995$	18.4150	0.583795
$996$	18.1530	0.575200
$997$	26.4026	0.836178	0.418089	−	0.908406i	$-0.362700\pi$
$997$	26.4026	0.836178	0.418089	+	0.908406i	$0.362700\pi$
$998$	−2.11189	−0.0668506
$999$	−21.9070	−0.693106

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8015.2.a.j.1.13	✓	45

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8015.2.a.j.1.13	✓	45	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:	\( N \)	\(=\)	\( 8015 = 5 \cdot 7 \cdot 229 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8015.a (trivial)

\(f(q)\)	\(=\)	\(q-1.60365 q^{2} +2.74735 q^{3} +0.571694 q^{4} -1.00000 q^{5} -4.40579 q^{6} +1.00000 q^{7} +2.29050 q^{8} +4.54792 q^{9} +O(q^{10})\)	\(q-1.60365 q^{2} +2.74735 q^{3} +0.571694 q^{4} -1.00000 q^{5} -4.40579 q^{6} +1.00000 q^{7} +2.29050 q^{8} +4.54792 q^{9} +1.60365 q^{10} -2.96026 q^{11} +1.57064 q^{12} -0.939802 q^{13} -1.60365 q^{14} -2.74735 q^{15} -4.81655 q^{16} -8.10854 q^{17} -7.29327 q^{18} +3.13222 q^{19} -0.571694 q^{20} +2.74735 q^{21} +4.74722 q^{22} +5.95816 q^{23} +6.29281 q^{24} +1.00000 q^{25} +1.50711 q^{26} +4.25268 q^{27} +0.571694 q^{28} +1.13986 q^{29} +4.40579 q^{30} +2.81172 q^{31} +3.14306 q^{32} -8.13286 q^{33} +13.0033 q^{34} -1.00000 q^{35} +2.60002 q^{36} -5.15134 q^{37} -5.02299 q^{38} -2.58196 q^{39} -2.29050 q^{40} +8.49391 q^{41} -4.40579 q^{42} -7.36688 q^{43} -1.69236 q^{44} -4.54792 q^{45} -9.55481 q^{46} -11.0255 q^{47} -13.2327 q^{48} +1.00000 q^{49} -1.60365 q^{50} -22.2770 q^{51} -0.537279 q^{52} +4.01700 q^{53} -6.81980 q^{54} +2.96026 q^{55} +2.29050 q^{56} +8.60530 q^{57} -1.82794 q^{58} +3.36104 q^{59} -1.57064 q^{60} -4.48749 q^{61} -4.50901 q^{62} +4.54792 q^{63} +4.59274 q^{64} +0.939802 q^{65} +13.0423 q^{66} +8.39602 q^{67} -4.63561 q^{68} +16.3691 q^{69} +1.60365 q^{70} -2.52252 q^{71} +10.4170 q^{72} -10.6927 q^{73} +8.26095 q^{74} +2.74735 q^{75} +1.79067 q^{76} -2.96026 q^{77} +4.14056 q^{78} -5.40475 q^{79} +4.81655 q^{80} -1.96018 q^{81} -13.6213 q^{82} +11.5577 q^{83} +1.57064 q^{84} +8.10854 q^{85} +11.8139 q^{86} +3.13159 q^{87} -6.78049 q^{88} +6.58922 q^{89} +7.29327 q^{90} -0.939802 q^{91} +3.40625 q^{92} +7.72477 q^{93} +17.6810 q^{94} -3.13222 q^{95} +8.63508 q^{96} -15.6618 q^{97} -1.60365 q^{98} -13.4630 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 45 q - 6 q^{2} + 34 q^{4} - 45 q^{5} + q^{6} + 45 q^{7} - 15 q^{8} + 29 q^{9}+O(q^{10}) \) 45 * q - 6 * q^2 + 34 * q^4 - 45 * q^5 + q^6 + 45 * q^7 - 15 * q^8 + 29 * q^9	\( 45 q - 6 q^{2} + 34 q^{4} - 45 q^{5} + q^{6} + 45 q^{7} - 15 q^{8} + 29 q^{9} + 6 q^{10} - q^{11} - 3 q^{12} - 21 q^{13} - 6 q^{14} + 8 q^{16} - 7 q^{17} - 36 q^{18} - 20 q^{19} - 34 q^{20} - 34 q^{22} - 22 q^{23} - 11 q^{24} + 45 q^{25} - q^{26} + 12 q^{27} + 34 q^{28} + 10 q^{29} - q^{30} - 27 q^{31} - 26 q^{32} - 39 q^{33} - 13 q^{34} - 45 q^{35} - 3 q^{36} - 72 q^{37} + 2 q^{38} - 37 q^{39} + 15 q^{40} - 4 q^{41} + q^{42} - 49 q^{43} + 5 q^{44} - 29 q^{45} - 67 q^{46} + 2 q^{47} + 8 q^{48} + 45 q^{49} - 6 q^{50} - 49 q^{51} - 47 q^{52} - 35 q^{53} - 12 q^{54} + q^{55} - 15 q^{56} - 77 q^{57} - 50 q^{58} + 4 q^{59} + 3 q^{60} - 36 q^{61} + 17 q^{62} + 29 q^{63} + 5 q^{64} + 21 q^{65} - 8 q^{66} - 80 q^{67} + 27 q^{68} + 9 q^{69} + 6 q^{70} - 12 q^{71} - 97 q^{72} - 55 q^{73} + 32 q^{74} - 37 q^{76} - q^{77} + 20 q^{78} - 94 q^{79} - 8 q^{80} - 19 q^{81} - 36 q^{82} + 24 q^{83} - 3 q^{84} + 7 q^{85} - 3 q^{86} - 4 q^{87} - 95 q^{88} + q^{89} + 36 q^{90} - 21 q^{91} - 65 q^{92} - 71 q^{93} - 53 q^{94} + 20 q^{95} - 13 q^{96} - 110 q^{97} - 6 q^{98} - 27 q^{99}+O(q^{100}) \) 45 * q - 6 * q^2 + 34 * q^4 - 45 * q^5 + q^6 + 45 * q^7 - 15 * q^8 + 29 * q^9 + 6 * q^10 - q^11 - 3 * q^12 - 21 * q^13 - 6 * q^14 + 8 * q^16 - 7 * q^17 - 36 * q^18 - 20 * q^19 - 34 * q^20 - 34 * q^22 - 22 * q^23 - 11 * q^24 + 45 * q^25 - q^26 + 12 * q^27 + 34 * q^28 + 10 * q^29 - q^30 - 27 * q^31 - 26 * q^32 - 39 * q^33 - 13 * q^34 - 45 * q^35 - 3 * q^36 - 72 * q^37 + 2 * q^38 - 37 * q^39 + 15 * q^40 - 4 * q^41 + q^42 - 49 * q^43 + 5 * q^44 - 29 * q^45 - 67 * q^46 + 2 * q^47 + 8 * q^48 + 45 * q^49 - 6 * q^50 - 49 * q^51 - 47 * q^52 - 35 * q^53 - 12 * q^54 + q^55 - 15 * q^56 - 77 * q^57 - 50 * q^58 + 4 * q^59 + 3 * q^60 - 36 * q^61 + 17 * q^62 + 29 * q^63 + 5 * q^64 + 21 * q^65 - 8 * q^66 - 80 * q^67 + 27 * q^68 + 9 * q^69 + 6 * q^70 - 12 * q^71 - 97 * q^72 - 55 * q^73 + 32 * q^74 - 37 * q^76 - q^77 + 20 * q^78 - 94 * q^79 - 8 * q^80 - 19 * q^81 - 36 * q^82 + 24 * q^83 - 3 * q^84 + 7 * q^85 - 3 * q^86 - 4 * q^87 - 95 * q^88 + q^89 + 36 * q^90 - 21 * q^91 - 65 * q^92 - 71 * q^93 - 53 * q^94 + 20 * q^95 - 13 * q^96 - 110 * q^97 - 6 * q^98 - 27 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more