LMFDB - Embedded newform 5239.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5239,2,Mod(1,5239)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5239, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5239.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5239 = 13^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5239.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:	$41.8336256189$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 17 x^{14} + 80 x^{13} + 98 x^{12} - 628 x^{11} - 158 x^{10} + 2458 x^{9} + \cdots - 147 $ x^16 - 4x^15 - 17x^14 + 80x^13 + 98x^12 - 628x^11 - 158x^10 + 2458x^9 - 510x^8 - 5030x^7 + 2318x^6 + 5112x^5 - 3154x^4 - 2086x^3 + 1542x^2 + 124*x - 147
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 403)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.45635$ of defining polynomial
Character	$\chi$	$=$	5239.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.45635 q^{2} +2.80039 q^{3} +4.03363 q^{4} +3.48535 q^{5} -6.87871 q^{6} +3.54003 q^{7} -4.99531 q^{8} +4.84216 q^{9} +O(q^{10})$	\(q-2.45635 q^{2} +2.80039 q^{3} +4.03363 q^{4} +3.48535 q^{5} -6.87871 q^{6} +3.54003 q^{7} -4.99531 q^{8} +4.84216 q^{9} -8.56123 q^{10} +5.60543 q^{11} +11.2957 q^{12} -8.69555 q^{14} +9.76032 q^{15} +4.20294 q^{16} -2.47924 q^{17} -11.8940 q^{18} +2.94408 q^{19} +14.0586 q^{20} +9.91346 q^{21} -13.7689 q^{22} +1.33162 q^{23} -13.9888 q^{24} +7.14767 q^{25} +5.15875 q^{27} +14.2792 q^{28} -6.22251 q^{29} -23.9747 q^{30} -1.00000 q^{31} -0.333245 q^{32} +15.6974 q^{33} +6.08988 q^{34} +12.3383 q^{35} +19.5315 q^{36} -7.14956 q^{37} -7.23168 q^{38} -17.4104 q^{40} +4.88228 q^{41} -24.3509 q^{42} -0.926054 q^{43} +22.6103 q^{44} +16.8766 q^{45} -3.27093 q^{46} -11.4047 q^{47} +11.7698 q^{48} +5.53184 q^{49} -17.5571 q^{50} -6.94284 q^{51} -9.27473 q^{53} -12.6717 q^{54} +19.5369 q^{55} -17.6836 q^{56} +8.24456 q^{57} +15.2846 q^{58} +6.89104 q^{59} +39.3696 q^{60} +5.72757 q^{61} +2.45635 q^{62} +17.1414 q^{63} -7.58730 q^{64} -38.5582 q^{66} -9.27764 q^{67} -10.0004 q^{68} +3.72906 q^{69} -30.3070 q^{70} +0.850046 q^{71} -24.1881 q^{72} -8.68621 q^{73} +17.5618 q^{74} +20.0162 q^{75} +11.8753 q^{76} +19.8434 q^{77} -5.69838 q^{79} +14.6487 q^{80} -0.0799826 q^{81} -11.9926 q^{82} +10.1834 q^{83} +39.9873 q^{84} -8.64103 q^{85} +2.27471 q^{86} -17.4254 q^{87} -28.0009 q^{88} +3.72778 q^{89} -41.4548 q^{90} +5.37128 q^{92} -2.80039 q^{93} +28.0139 q^{94} +10.2611 q^{95} -0.933216 q^{96} -11.4074 q^{97} -13.5881 q^{98} +27.1424 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 4 q^{2} - 2 q^{3} + 18 q^{4} + 4 q^{5} + 6 q^{6} + 2 q^{7} + 12 q^{8} + 10 q^{9}+O(q^{10}) $ 16 * q + 4 * q^2 - 2 * q^3 + 18 * q^4 + 4 * q^5 + 6 * q^6 + 2 * q^7 + 12 * q^8 + 10 * q^9	$ 16 q + 4 q^{2} - 2 q^{3} + 18 q^{4} + 4 q^{5} + 6 q^{6} + 2 q^{7} + 12 q^{8} + 10 q^{9} - 2 q^{10} + 14 q^{11} + 8 q^{12} - 8 q^{14} + 14 q^{16} + 4 q^{17} - 28 q^{18} + 22 q^{19} + 28 q^{20} + 12 q^{21} - 8 q^{22} + 4 q^{23} - 8 q^{24} - 2 q^{25} + 10 q^{27} + 16 q^{28} - 8 q^{29} - 20 q^{30} - 16 q^{31} + 48 q^{32} + 10 q^{33} + 8 q^{34} - 2 q^{35} + 22 q^{36} + 16 q^{37} - 6 q^{38} + 14 q^{40} + 44 q^{41} + 14 q^{42} + 16 q^{43} + 4 q^{44} + 56 q^{45} + 10 q^{47} + 32 q^{49} + 2 q^{50} - 6 q^{53} + 24 q^{54} + 22 q^{55} - 4 q^{56} - 8 q^{57} + 74 q^{58} + 2 q^{59} + 40 q^{60} + 8 q^{61} - 4 q^{62} + 56 q^{63} + 38 q^{64} - 34 q^{66} - 8 q^{67} + 32 q^{68} - 10 q^{69} - 108 q^{70} + 50 q^{71} - 44 q^{72} + 14 q^{73} + 8 q^{74} + 44 q^{76} + 16 q^{77} + 32 q^{79} + 68 q^{80} - 8 q^{81} - 6 q^{82} - 20 q^{83} + 136 q^{84} - 32 q^{85} + 8 q^{86} - 36 q^{87} - 40 q^{88} + 52 q^{89} - 34 q^{90} + 14 q^{92} + 2 q^{93} + 44 q^{94} - 2 q^{95} - 80 q^{96} + 18 q^{97} + 12 q^{98} + 38 q^{99}+O(q^{100}) $ 16 * q + 4 * q^2 - 2 * q^3 + 18 * q^4 + 4 * q^5 + 6 * q^6 + 2 * q^7 + 12 * q^8 + 10 * q^9 - 2 * q^10 + 14 * q^11 + 8 * q^12 - 8 * q^14 + 14 * q^16 + 4 * q^17 - 28 * q^18 + 22 * q^19 + 28 * q^20 + 12 * q^21 - 8 * q^22 + 4 * q^23 - 8 * q^24 - 2 * q^25 + 10 * q^27 + 16 * q^28 - 8 * q^29 - 20 * q^30 - 16 * q^31 + 48 * q^32 + 10 * q^33 + 8 * q^34 - 2 * q^35 + 22 * q^36 + 16 * q^37 - 6 * q^38 + 14 * q^40 + 44 * q^41 + 14 * q^42 + 16 * q^43 + 4 * q^44 + 56 * q^45 + 10 * q^47 + 32 * q^49 + 2 * q^50 - 6 * q^53 + 24 * q^54 + 22 * q^55 - 4 * q^56 - 8 * q^57 + 74 * q^58 + 2 * q^59 + 40 * q^60 + 8 * q^61 - 4 * q^62 + 56 * q^63 + 38 * q^64 - 34 * q^66 - 8 * q^67 + 32 * q^68 - 10 * q^69 - 108 * q^70 + 50 * q^71 - 44 * q^72 + 14 * q^73 + 8 * q^74 + 44 * q^76 + 16 * q^77 + 32 * q^79 + 68 * q^80 - 8 * q^81 - 6 * q^82 - 20 * q^83 + 136 * q^84 - 32 * q^85 + 8 * q^86 - 36 * q^87 - 40 * q^88 + 52 * q^89 - 34 * q^90 + 14 * q^92 + 2 * q^93 + 44 * q^94 - 2 * q^95 - 80 * q^96 + 18 * q^97 + 12 * q^98 + 38 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.45635	−1.73690	−0.868449	−	0.495778i	$-0.834883\pi$
$2$	−2.45635	−1.73690	−0.868449	+	0.495778i	$0.834883\pi$
$3$	2.80039	1.61680	0.808402	−	0.588631i	$-0.200333\pi$
$3$	2.80039	1.61680	0.808402	+	0.588631i	$0.200333\pi$
$4$	4.03363	2.01682
$5$	3.48535	1.55870	0.779348	−	0.626591i	$-0.215550\pi$
$5$	3.48535	1.55870	0.779348	+	0.626591i	$0.215550\pi$
$6$	−6.87871	−2.80822
$7$	3.54003	1.33801	0.669004	−	0.743259i	$-0.266721\pi$
$7$	3.54003	1.33801	0.669004	+	0.743259i	$0.266721\pi$
$8$	−4.99531	−1.76611
$9$	4.84216	1.61405
$10$	−8.56123	−2.70730
$11$	5.60543	1.69010	0.845051	−	0.534686i	$-0.179570\pi$
$11$	5.60543	1.69010	0.845051	+	0.534686i	$0.179570\pi$
$12$	11.2957	3.26080
$13$	0	0
$13$	0	0
$14$	−8.69555	−2.32398
$15$	9.76032	2.52010
$16$	4.20294	1.05073
$17$	−2.47924	−0.601305	−0.300652	−	0.953734i	$-0.597204\pi$
$17$	−2.47924	−0.601305	−0.300652	+	0.953734i	$0.597204\pi$
$18$	−11.8940	−2.80345
$19$	2.94408	0.675418	0.337709	−	0.941251i	$-0.390348\pi$
$19$	2.94408	0.675418	0.337709	+	0.941251i	$0.390348\pi$
$20$	14.0586	3.14360
$21$	9.91346	2.16329
$22$	−13.7689	−2.93553
$23$	1.33162	0.277663	0.138831	−	0.990316i	$-0.455665\pi$
$23$	1.33162	0.277663	0.138831	+	0.990316i	$0.455665\pi$
$24$	−13.9888	−2.85545
$25$	7.14767	1.42953
$26$	0	0
$27$	5.15875	0.992802
$28$	14.2792	2.69852
$29$	−6.22251	−1.15549	−0.577746	−	0.816217i	$-0.696067\pi$
$29$	−6.22251	−1.15549	−0.577746	+	0.816217i	$0.696067\pi$
$30$	−23.9747	−4.37717
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−0.333245	−0.0589100
$33$	15.6974	2.73256
$34$	6.08988	1.04441
$35$	12.3383	2.08555
$36$	19.5315	3.25525
$37$	−7.14956	−1.17538	−0.587691	−	0.809086i	$-0.699963\pi$
$37$	−7.14956	−1.17538	−0.587691	+	0.809086i	$0.699963\pi$
$38$	−7.23168	−1.17313
$39$	0	0
$40$	−17.4104	−2.75283
$41$	4.88228	0.762484	0.381242	−	0.924475i	$-0.375496\pi$
$41$	4.88228	0.762484	0.381242	+	0.924475i	$0.375496\pi$
$42$	−24.3509	−3.75742
$43$	−0.926054	−0.141222	−0.0706110	−	0.997504i	$-0.522495\pi$
$43$	−0.926054	−0.141222	−0.0706110	+	0.997504i	$0.522495\pi$
$44$	22.6103	3.40862
$45$	16.8766	2.51582
$46$	−3.27093	−0.482272
$47$	−11.4047	−1.66355	−0.831773	−	0.555116i	$-0.812674\pi$
$47$	−11.4047	−1.66355	−0.831773	+	0.555116i	$0.812674\pi$
$48$	11.7698	1.69883
$49$	5.53184	0.790263
$50$	−17.5571	−2.48296
$51$	−6.94284	−0.972191
$52$	0	0
$53$	−9.27473	−1.27398	−0.636991	−	0.770871i	$-0.719821\pi$
$53$	−9.27473	−1.27398	−0.636991	+	0.770871i	$0.719821\pi$
$54$	−12.6717	−1.72440
$55$	19.5369	2.63435
$56$	−17.6836	−2.36307
$57$	8.24456	1.09202
$58$	15.2846	2.00697
$59$	6.89104	0.897137	0.448568	−	0.893749i	$-0.351934\pi$
$59$	6.89104	0.897137	0.448568	+	0.893749i	$0.351934\pi$
$60$	39.3696	5.08259
$61$	5.72757	0.733340	0.366670	−	0.930351i	$-0.380498\pi$
$61$	5.72757	0.733340	0.366670	+	0.930351i	$0.380498\pi$
$62$	2.45635	0.311956
$63$	17.1414	2.15961
$64$	−7.58730	−0.948413
$65$	0	0
$66$	−38.5582	−4.74618
$67$	−9.27764	−1.13344	−0.566722	−	0.823909i	$-0.691789\pi$
$67$	−9.27764	−1.13344	−0.566722	+	0.823909i	$0.691789\pi$
$68$	−10.0004	−1.21272
$69$	3.72906	0.448926
$70$	−30.3070	−3.62238
$71$	0.850046	0.100882	0.0504410	−	0.998727i	$-0.483937\pi$
$71$	0.850046	0.100882	0.0504410	+	0.998727i	$0.483937\pi$
$72$	−24.1881	−2.85059
$73$	−8.68621	−1.01664	−0.508322	−	0.861167i	$-0.669734\pi$
$73$	−8.68621	−1.01664	−0.508322	+	0.861167i	$0.669734\pi$
$74$	17.5618	2.04152
$75$	20.0162	2.31127
$76$	11.8753	1.36219
$77$	19.8434	2.26137
$78$	0	0
$79$	−5.69838	−0.641117	−0.320559	−	0.947229i	$-0.603871\pi$
$79$	−5.69838	−0.641117	−0.320559	+	0.947229i	$0.603871\pi$
$80$	14.6487	1.63777
$81$	−0.0799826	−0.00888695
$82$	−11.9926	−1.32436
$83$	10.1834	1.11778	0.558888	−	0.829243i	$-0.311228\pi$
$83$	10.1834	1.11778	0.558888	+	0.829243i	$0.311228\pi$
$84$	39.9873	4.36297
$85$	−8.64103	−0.937251
$86$	2.27471	0.245288
$87$	−17.4254	−1.86820
$88$	−28.0009	−2.98490
$89$	3.72778	0.395144	0.197572	−	0.980288i	$-0.436694\pi$
$89$	3.72778	0.395144	0.197572	+	0.980288i	$0.436694\pi$
$90$	−41.4548	−4.36972
$91$	0	0
$92$	5.37128	0.559995
$93$	−2.80039	−0.290386
$94$	28.0139	2.88941
$95$	10.2611	1.05277
$96$	−0.933216	−0.0952459
$97$	−11.4074	−1.15824	−0.579122	−	0.815241i	$-0.696604\pi$
$97$	−11.4074	−1.15824	−0.579122	+	0.815241i	$0.696604\pi$
$98$	−13.5881	−1.37261
$99$	27.1424	2.72791
$100$	28.8311	2.88311
$101$	−6.14390	−0.611341	−0.305671	−	0.952137i	$-0.598881\pi$
$101$	−6.14390	−0.611341	−0.305671	+	0.952137i	$0.598881\pi$
$102$	17.0540	1.68860
$103$	−9.77159	−0.962823	−0.481412	−	0.876495i	$-0.659876\pi$
$103$	−9.77159	−0.962823	−0.481412	+	0.876495i	$0.659876\pi$
$104$	0	0
$105$	34.5519	3.37192
$106$	22.7819	2.21278
$107$	−0.0150159	−0.00145164	−0.000725821	−	1.00000i	$-0.500231\pi$
$107$	−0.0150159	−0.00145164	−0.000725821		1.00000i	$0.500231\pi$
$108$	20.8085	2.00230
$109$	−0.921927	−0.0883046	−0.0441523	−	0.999025i	$-0.514059\pi$
$109$	−0.921927	−0.0883046	−0.0441523	+	0.999025i	$0.514059\pi$
$110$	−47.9894	−4.57561
$111$	−20.0215	−1.90036
$112$	14.8785	1.40589
$113$	−5.96949	−0.561562	−0.280781	−	0.959772i	$-0.590594\pi$
$113$	−5.96949	−0.561562	−0.280781	+	0.959772i	$0.590594\pi$
$114$	−20.2515	−1.89673
$115$	4.64118	0.432792
$116$	−25.0993	−2.33041
$117$	0	0
$118$	−16.9268	−1.55824
$119$	−8.77660	−0.804550
$120$	−48.7558	−4.45078
$121$	20.4209	1.85644
$122$	−14.0689	−1.27374
$123$	13.6723	1.23279
$124$	−4.03363	−0.362231
$125$	7.48538	0.669512
$126$	−42.1052	−3.75103
$127$	4.63997	0.411731	0.205865	−	0.978580i	$-0.433999\pi$
$127$	4.63997	0.411731	0.205865	+	0.978580i	$0.433999\pi$
$128$	19.3035	1.70621
$129$	−2.59331	−0.228328
$130$	0	0
$131$	−12.9817	−1.13421	−0.567107	−	0.823644i	$-0.691938\pi$
$131$	−12.9817	−1.13421	−0.567107	+	0.823644i	$0.691938\pi$
$132$	63.3174	5.51108
$133$	10.4221	0.903714
$134$	22.7891	1.96868
$135$	17.9801	1.54748
$136$	12.3846	1.06197
$137$	4.66000	0.398131	0.199065	−	0.979986i	$-0.436209\pi$
$137$	4.66000	0.398131	0.199065	+	0.979986i	$0.436209\pi$
$138$	−9.15986	−0.779739
$139$	14.8173	1.25679	0.628393	−	0.777896i	$-0.283713\pi$
$139$	14.8173	1.25679	0.628393	+	0.777896i	$0.283713\pi$
$140$	49.7680	4.20617
$141$	−31.9375	−2.68963
$142$	−2.08801	−0.175222
$143$	0	0
$144$	20.3513	1.69594
$145$	−21.6876	−1.80106
$146$	21.3363	1.76581
$147$	15.4913	1.27770
$148$	−28.8387	−2.37053
$149$	−2.85800	−0.234137	−0.117068	−	0.993124i	$-0.537350\pi$
$149$	−2.85800	−0.234137	−0.117068	+	0.993124i	$0.537350\pi$
$150$	−49.1668	−4.01445
$151$	0.429054	0.0349159	0.0174580	−	0.999848i	$-0.494443\pi$
$151$	0.429054	0.0349159	0.0174580	+	0.999848i	$0.494443\pi$
$152$	−14.7066	−1.19286
$153$	−12.0049	−0.970537
$154$	−48.7423	−3.92777
$155$	−3.48535	−0.279950
$156$	0	0
$157$	−4.78512	−0.381894	−0.190947	−	0.981600i	$-0.561156\pi$
$157$	−4.78512	−0.381894	−0.190947	+	0.981600i	$0.561156\pi$
$158$	13.9972	1.11356
$159$	−25.9728	−2.05978
$160$	−1.16148	−0.0918228
$161$	4.71399	0.371515
$162$	0.196465	0.0154357
$163$	21.8453	1.71105	0.855527	−	0.517758i	$-0.173233\pi$
$163$	21.8453	1.71105	0.855527	+	0.517758i	$0.173233\pi$
$164$	19.6933	1.53779
$165$	54.7108	4.25923
$166$	−25.0140	−1.94146
$167$	3.37083	0.260843	0.130421	−	0.991459i	$-0.458367\pi$
$167$	3.37083	0.260843	0.130421	+	0.991459i	$0.458367\pi$
$168$	−49.5208	−3.82061
$169$	0	0
$170$	21.2254	1.62791
$171$	14.2557	1.09016
$172$	−3.73536	−0.284819
$173$	14.7745	1.12329	0.561643	−	0.827379i	$-0.310169\pi$
$173$	14.7745	1.12329	0.561643	+	0.827379i	$0.310169\pi$
$174$	42.8029	3.24488
$175$	25.3030	1.91273
$176$	23.5593	1.77585
$177$	19.2976	1.45049
$178$	−9.15671	−0.686325
$179$	24.6355	1.84134	0.920672	−	0.390337i	$-0.127641\pi$
$179$	24.6355	1.84134	0.920672	+	0.390337i	$0.127641\pi$
$180$	68.0741	5.07394
$181$	−5.26608	−0.391425	−0.195712	−	0.980661i	$-0.562702\pi$
$181$	−5.26608	−0.391425	−0.195712	+	0.980661i	$0.562702\pi$
$182$	0	0
$183$	16.0394	1.18567
$184$	−6.65187	−0.490383
$185$	−24.9187	−1.83206
$186$	6.87871	0.504372
$187$	−13.8972	−1.01627
$188$	−46.0024	−3.35507
$189$	18.2622	1.32838
$190$	−25.2049	−1.82856
$191$	−4.61535	−0.333955	−0.166978	−	0.985961i	$-0.553401\pi$
$191$	−4.61535	−0.333955	−0.166978	+	0.985961i	$0.553401\pi$
$192$	−21.2474	−1.53340
$193$	−1.61977	−0.116594	−0.0582969	−	0.998299i	$-0.518567\pi$
$193$	−1.61977	−0.116594	−0.0582969	+	0.998299i	$0.518567\pi$
$194$	28.0205	2.01175
$195$	0	0
$196$	22.3134	1.59382
$197$	−8.32063	−0.592820	−0.296410	−	0.955061i	$-0.595790\pi$
$197$	−8.32063	−0.592820	−0.296410	+	0.955061i	$0.595790\pi$
$198$	−66.6711	−4.73811
$199$	−23.6343	−1.67539	−0.837694	−	0.546140i	$-0.816097\pi$
$199$	−23.6343	−1.67539	−0.837694	+	0.546140i	$0.816097\pi$
$200$	−35.7048	−2.52471
$201$	−25.9810	−1.83256
$202$	15.0916	1.06184
$203$	−22.0279	−1.54606
$204$	−28.0049	−1.96073
$205$	17.0165	1.18848
$206$	24.0024	1.67233
$207$	6.44793	0.448162
$208$	0	0
$209$	16.5028	1.14153
$210$	−84.8714	−5.85668
$211$	22.5919	1.55529	0.777647	−	0.628701i	$-0.216413\pi$
$211$	22.5919	1.55529	0.777647	+	0.628701i	$0.216413\pi$
$212$	−37.4109	−2.56939
$213$	2.38046	0.163106
$214$	0.0368842	0.00252135
$215$	−3.22762	−0.220122
$216$	−25.7696	−1.75340
$217$	−3.54003	−0.240313
$218$	2.26457	0.153376
$219$	−24.3247	−1.64371
$220$	78.8047	5.31301
$221$	0	0
$222$	49.1798	3.30073
$223$	−24.0641	−1.61145	−0.805726	−	0.592288i	$-0.798225\pi$
$223$	−24.0641	−1.61145	−0.805726	+	0.592288i	$0.798225\pi$
$224$	−1.17970	−0.0788220
$225$	34.6101	2.30734
$226$	14.6631	0.975377
$227$	8.15313	0.541142	0.270571	−	0.962700i	$-0.412787\pi$
$227$	8.15313	0.541142	0.270571	+	0.962700i	$0.412787\pi$
$228$	33.2555	2.20240
$229$	−5.28147	−0.349010	−0.174505	−	0.984656i	$-0.555832\pi$
$229$	−5.28147	−0.349010	−0.174505	+	0.984656i	$0.555832\pi$
$230$	−11.4003	−0.751716
$231$	55.5692	3.65619
$232$	31.0834	2.04072
$233$	−10.6469	−0.697501	−0.348750	−	0.937216i	$-0.613394\pi$
$233$	−10.6469	−0.697501	−0.348750	+	0.937216i	$0.613394\pi$
$234$	0	0
$235$	−39.7494	−2.59296
$236$	27.7959	1.80936
$237$	−15.9576	−1.03656
$238$	21.5584	1.39742
$239$	6.31548	0.408514	0.204257	−	0.978917i	$-0.434522\pi$
$239$	6.31548	0.408514	0.204257	+	0.978917i	$0.434522\pi$
$240$	41.0220	2.64796
$241$	14.7695	0.951389	0.475695	−	0.879610i	$-0.342197\pi$
$241$	14.7695	0.951389	0.475695	+	0.879610i	$0.342197\pi$
$242$	−50.1607	−3.22445
$243$	−15.7002	−1.00717
$244$	23.1029	1.47901
$245$	19.2804	1.23178
$246$	−33.5838	−2.14123
$247$	0	0
$248$	4.99531	0.317202
$249$	28.5175	1.80723
$250$	−18.3867	−1.16288
$251$	12.3317	0.778372	0.389186	−	0.921159i	$-0.372756\pi$
$251$	12.3317	0.778372	0.389186	+	0.921159i	$0.372756\pi$
$252$	69.1421	4.35555
$253$	7.46433	0.469278
$254$	−11.3974	−0.715135
$255$	−24.1982	−1.51535
$256$	−32.2415	−2.01510
$257$	−27.5568	−1.71895	−0.859474	−	0.511180i	$-0.829209\pi$
$257$	−27.5568	−1.71895	−0.859474	+	0.511180i	$0.829209\pi$
$258$	6.37006	0.396583
$259$	−25.3097	−1.57267
$260$	0	0
$261$	−30.1304	−1.86502
$262$	31.8875	1.97002
$263$	26.3404	1.62422	0.812111	−	0.583503i	$-0.198319\pi$
$263$	26.3404	1.62422	0.812111	+	0.583503i	$0.198319\pi$
$264$	−78.4132	−4.82600
$265$	−32.3257	−1.98575
$266$	−25.6004	−1.56966
$267$	10.4392	0.638870
$268$	−37.4226	−2.28595
$269$	−18.9927	−1.15801	−0.579004	−	0.815325i	$-0.696558\pi$
$269$	−18.9927	−1.15801	−0.579004	+	0.815325i	$0.696558\pi$
$270$	−44.1652	−2.68781
$271$	−8.87920	−0.539373	−0.269686	−	0.962948i	$-0.586920\pi$
$271$	−8.87920	−0.539373	−0.269686	+	0.962948i	$0.586920\pi$
$272$	−10.4201	−0.631811
$273$	0	0
$274$	−11.4466	−0.691513
$275$	40.0658	2.41606
$276$	15.0417	0.905402
$277$	−0.263699	−0.0158442	−0.00792208	−	0.999969i	$-0.502522\pi$
$277$	−0.263699	−0.0158442	−0.00792208	+	0.999969i	$0.502522\pi$
$278$	−36.3964	−2.18291
$279$	−4.84216	−0.289892
$280$	−61.6334	−3.68330
$281$	6.28589	0.374985	0.187492	−	0.982266i	$-0.439964\pi$
$281$	6.28589	0.374985	0.187492	+	0.982266i	$0.439964\pi$
$282$	78.4497	4.67161
$283$	−11.3137	−0.672527	−0.336264	−	0.941768i	$-0.609163\pi$
$283$	−11.3137	−0.672527	−0.336264	+	0.941768i	$0.609163\pi$
$284$	3.42878	0.203460
$285$	28.7352	1.70212
$286$	0	0
$287$	17.2834	1.02021
$288$	−1.61363	−0.0950839
$289$	−10.8534	−0.638433
$290$	53.2723	3.12826
$291$	−31.9450	−1.87265
$292$	−35.0370	−2.05038
$293$	−19.5031	−1.13938	−0.569692	−	0.821858i	$-0.692938\pi$
$293$	−19.5031	−1.13938	−0.569692	+	0.821858i	$0.692938\pi$
$294$	−38.0520	−2.21924
$295$	24.0177	1.39836
$296$	35.7143	2.07585
$297$	28.9170	1.67794
$298$	7.02024	0.406672
$299$	0	0
$300$	80.7381	4.66142
$301$	−3.27826	−0.188956
$302$	−1.05390	−0.0606454
$303$	−17.2053	−0.988419
$304$	12.3738	0.709685
$305$	19.9626	1.14305
$306$	29.4881	1.68573
$307$	31.6864	1.80844	0.904219	−	0.427069i	$-0.140454\pi$
$307$	31.6864	1.80844	0.904219	+	0.427069i	$0.140454\pi$
$308$	80.0411	4.56076
$309$	−27.3642	−1.55670
$310$	8.56123	0.486245
$311$	1.23362	0.0699522	0.0349761	−	0.999388i	$-0.488864\pi$
$311$	1.23362	0.0699522	0.0349761	+	0.999388i	$0.488864\pi$
$312$	0	0
$313$	−19.5774	−1.10658	−0.553291	−	0.832988i	$-0.686628\pi$
$313$	−19.5774	−1.10658	−0.553291	+	0.832988i	$0.686628\pi$
$314$	11.7539	0.663312
$315$	59.7438	3.36618
$316$	−22.9852	−1.29302
$317$	2.05349	0.115336	0.0576678	−	0.998336i	$-0.481634\pi$
$317$	2.05349	0.115336	0.0576678	+	0.998336i	$0.481634\pi$
$318$	63.7982	3.57762
$319$	−34.8799	−1.95290
$320$	−26.4444	−1.47829
$321$	−0.0420503	−0.00234702
$322$	−11.5792	−0.645284
$323$	−7.29909	−0.406132
$324$	−0.322620	−0.0179234
$325$	0	0
$326$	−53.6595	−2.97193
$327$	−2.58175	−0.142771
$328$	−24.3885	−1.34663
$329$	−40.3730	−2.22584
$330$	−134.389	−7.39785
$331$	13.2618	0.728933	0.364467	−	0.931216i	$-0.381251\pi$
$331$	13.2618	0.728933	0.364467	+	0.931216i	$0.381251\pi$
$332$	41.0762	2.25435
$333$	−34.6193	−1.89713
$334$	−8.27993	−0.453057
$335$	−32.3358	−1.76670
$336$	41.6656	2.27305
$337$	−35.1836	−1.91657	−0.958287	−	0.285806i	$-0.907739\pi$
$337$	−35.1836	−1.91657	−0.958287	+	0.285806i	$0.907739\pi$
$338$	0	0
$339$	−16.7169	−0.907936
$340$	−34.8548	−1.89026
$341$	−5.60543	−0.303551
$342$	−35.0169	−1.89350
$343$	−5.19733	−0.280629
$344$	4.62593	0.249413
$345$	12.9971	0.699739
$346$	−36.2913	−1.95104
$347$	−12.0992	−0.649519	−0.324759	−	0.945797i	$-0.605283\pi$
$347$	−12.0992	−0.649519	−0.324759	+	0.945797i	$0.605283\pi$
$348$	−70.2878	−3.76782
$349$	−28.7745	−1.54026	−0.770131	−	0.637886i	$-0.779809\pi$
$349$	−28.7745	−1.54026	−0.770131	+	0.637886i	$0.779809\pi$
$350$	−62.1529	−3.32221
$351$	0	0
$352$	−1.86798	−0.0995639
$353$	15.0930	0.803321	0.401660	−	0.915789i	$-0.368433\pi$
$353$	15.0930	0.803321	0.401660	+	0.915789i	$0.368433\pi$
$354$	−47.4015	−2.51936
$355$	2.96271	0.157244
$356$	15.0365	0.796933
$357$	−24.5779	−1.30080
$358$	−60.5133	−3.19823
$359$	−2.92054	−0.154140	−0.0770701	−	0.997026i	$-0.524557\pi$
$359$	−2.92054	−0.154140	−0.0770701	+	0.997026i	$0.524557\pi$
$360$	−84.3039	−4.44321
$361$	−10.3324	−0.543810
$362$	12.9353	0.679865
$363$	57.1863	3.00150
$364$	0	0
$365$	−30.2745	−1.58464
$366$	−39.3983	−2.05938
$367$	7.03364	0.367153	0.183576	−	0.983005i	$-0.441233\pi$
$367$	7.03364	0.367153	0.183576	+	0.983005i	$0.441233\pi$
$368$	5.59673	0.291750
$369$	23.6408	1.23069
$370$	61.2090	3.18211
$371$	−32.8328	−1.70460
$372$	−11.2957	−0.585656
$373$	−9.23107	−0.477967	−0.238983	−	0.971024i	$-0.576814\pi$
$373$	−9.23107	−0.477967	−0.238983	+	0.971024i	$0.576814\pi$
$374$	34.1364	1.76515
$375$	20.9619	1.08247
$376$	56.9700	2.93800
$377$	0	0
$378$	−44.8582	−2.30726
$379$	−14.1434	−0.726499	−0.363249	−	0.931692i	$-0.618333\pi$
$379$	−14.1434	−0.726499	−0.363249	+	0.931692i	$0.618333\pi$
$380$	41.3897	2.12325
$381$	12.9937	0.665688
$382$	11.3369	0.580046
$383$	9.51959	0.486428	0.243214	−	0.969973i	$-0.421798\pi$
$383$	9.51959	0.486428	0.243214	+	0.969973i	$0.421798\pi$
$384$	54.0573	2.75860
$385$	69.1613	3.52478
$386$	3.97872	0.202512
$387$	−4.48410	−0.227940
$388$	−46.0132	−2.33596
$389$	−9.65776	−0.489668	−0.244834	−	0.969565i	$-0.578733\pi$
$389$	−9.65776	−0.489668	−0.244834	+	0.969565i	$0.578733\pi$
$390$	0	0
$391$	−3.30142	−0.166960
$392$	−27.6333	−1.39569
$393$	−36.3537	−1.83380
$394$	20.4383	1.02967
$395$	−19.8608	−0.999307
$396$	109.482	5.50170
$397$	−13.1443	−0.659692	−0.329846	−	0.944035i	$-0.606997\pi$
$397$	−13.1443	−0.659692	−0.329846	+	0.944035i	$0.606997\pi$
$398$	58.0539	2.90998
$399$	29.1860	1.46113
$400$	30.0412	1.50206
$401$	22.7879	1.13798	0.568988	−	0.822346i	$-0.307335\pi$
$401$	22.7879	1.13798	0.568988	+	0.822346i	$0.307335\pi$
$402$	63.8182	3.18296
$403$	0	0
$404$	−24.7823	−1.23296
$405$	−0.278767	−0.0138521
$406$	54.1081	2.68534
$407$	−40.0764	−1.98651
$408$	34.6816	1.71700
$409$	−20.1639	−0.997043	−0.498521	−	0.866877i	$-0.666123\pi$
$409$	−20.1639	−0.997043	−0.498521	+	0.866877i	$0.666123\pi$
$410$	−41.7983	−2.06427
$411$	13.0498	0.643699
$412$	−39.4150	−1.94184
$413$	24.3945	1.20038
$414$	−15.8384	−0.778413
$415$	35.4928	1.74227
$416$	0	0
$417$	41.4941	2.03198
$418$	−40.5367	−1.98271
$419$	2.09350	0.102274	0.0511371	−	0.998692i	$-0.483715\pi$
$419$	2.09350	0.102274	0.0511371	+	0.998692i	$0.483715\pi$
$420$	139.370	6.80054
$421$	−15.1244	−0.737120	−0.368560	−	0.929604i	$-0.620149\pi$
$421$	−15.1244	−0.737120	−0.368560	+	0.929604i	$0.620149\pi$
$422$	−55.4936	−2.70139
$423$	−55.2233	−2.68505
$424$	46.3301	2.24999
$425$	−17.7208	−0.859585
$426$	−5.84723	−0.283299
$427$	20.2758	0.981214
$428$	−0.0605686	−0.00292770
$429$	0	0
$430$	7.92816	0.382330
$431$	12.5939	0.606628	0.303314	−	0.952891i	$-0.401907\pi$
$431$	12.5939	0.606628	0.303314	+	0.952891i	$0.401907\pi$
$432$	21.6819	1.04317
$433$	6.05166	0.290824	0.145412	−	0.989371i	$-0.453549\pi$
$433$	6.05166	0.290824	0.145412	+	0.989371i	$0.453549\pi$
$434$	8.69555	0.417400
$435$	−60.7337	−2.91196
$436$	−3.71872	−0.178094
$437$	3.92041	0.187539
$438$	59.7499	2.85496
$439$	−2.35627	−0.112458	−0.0562292	−	0.998418i	$-0.517908\pi$
$439$	−2.35627	−0.112458	−0.0562292	+	0.998418i	$0.517908\pi$
$440$	−97.5928	−4.65255
$441$	26.7861	1.27553
$442$	0	0
$443$	7.64035	0.363004	0.181502	−	0.983391i	$-0.441904\pi$
$443$	7.64035	0.363004	0.181502	+	0.983391i	$0.441904\pi$
$444$	−80.7595	−3.83268
$445$	12.9926	0.615909
$446$	59.1098	2.79893
$447$	−8.00351	−0.378553
$448$	−26.8593	−1.26898
$449$	31.4131	1.48247	0.741237	−	0.671243i	$-0.234239\pi$
$449$	31.4131	1.48247	0.741237	+	0.671243i	$0.234239\pi$
$450$	−85.0145	−4.00762
$451$	27.3673	1.28868
$452$	−24.0787	−1.13257
$453$	1.20152	0.0564522
$454$	−20.0269	−0.939909
$455$	0	0
$456$	−41.1841	−1.92862
$457$	−14.1833	−0.663466	−0.331733	−	0.943373i	$-0.607633\pi$
$457$	−14.1833	−0.663466	−0.331733	+	0.943373i	$0.607633\pi$
$458$	12.9731	0.606194
$459$	−12.7898	−0.596977
$460$	18.7208	0.872862
$461$	0.852165	0.0396893	0.0198446	−	0.999803i	$-0.493683\pi$
$461$	0.852165	0.0396893	0.0198446	+	0.999803i	$0.493683\pi$
$462$	−136.497	−6.35043
$463$	26.5817	1.23535	0.617677	−	0.786432i	$-0.288074\pi$
$463$	26.5817	1.23535	0.617677	+	0.786432i	$0.288074\pi$
$464$	−26.1528	−1.21411
$465$	−9.76032	−0.452624
$466$	26.1524	1.21149
$467$	26.5943	1.23064	0.615320	−	0.788278i	$-0.289027\pi$
$467$	26.5943	1.23064	0.615320	+	0.788278i	$0.289027\pi$
$468$	0	0
$469$	−32.8432	−1.51656
$470$	97.6382	4.50371
$471$	−13.4002	−0.617448
$472$	−34.4228	−1.58444
$473$	−5.19093	−0.238679
$474$	39.1975	1.80040
$475$	21.0433	0.965533
$476$	−35.4016	−1.62263
$477$	−44.9097	−2.05627
$478$	−15.5130	−0.709548
$479$	28.3556	1.29560	0.647801	−	0.761809i	$-0.275689\pi$
$479$	28.3556	1.29560	0.647801	+	0.761809i	$0.275689\pi$
$480$	−3.25258	−0.148459
$481$	0	0
$482$	−36.2791	−1.65247
$483$	13.2010	0.600666
$484$	82.3703	3.74410
$485$	−39.7587	−1.80535
$486$	38.5652	1.74935
$487$	30.9730	1.40352	0.701760	−	0.712414i	$-0.252398\pi$
$487$	30.9730	1.40352	0.701760	+	0.712414i	$0.252398\pi$
$488$	−28.6110	−1.29516
$489$	61.1752	2.76644
$490$	−47.3593	−2.13948
$491$	2.33609	0.105426	0.0527131	−	0.998610i	$-0.483213\pi$
$491$	2.33609	0.105426	0.0527131	+	0.998610i	$0.483213\pi$
$492$	55.1489	2.48631
$493$	15.4271	0.694802
$494$	0	0
$495$	94.6007	4.25199
$496$	−4.20294	−0.188717
$497$	3.00919	0.134981
$498$	−70.0489	−3.13897
$499$	19.4432	0.870396	0.435198	−	0.900335i	$-0.356678\pi$
$499$	19.4432	0.870396	0.435198	+	0.900335i	$0.356678\pi$
$500$	30.1933	1.35028
$501$	9.43963	0.421731
$502$	−30.2910	−1.35195
$503$	10.6345	0.474170	0.237085	−	0.971489i	$-0.423808\pi$
$503$	10.6345	0.474170	0.237085	+	0.971489i	$0.423808\pi$
$504$	−85.6266	−3.81411
$505$	−21.4137	−0.952895
$506$	−18.3350	−0.815089
$507$	0	0
$508$	18.7159	0.830386
$509$	−43.8547	−1.94383	−0.971913	−	0.235338i	$-0.924380\pi$
$509$	−43.8547	−1.94383	−0.971913	+	0.235338i	$0.924380\pi$
$510$	59.4392	2.63201
$511$	−30.7495	−1.36028
$512$	40.5893	1.79381
$513$	15.1878	0.670557
$514$	67.6891	2.98564
$515$	−34.0574	−1.50075
$516$	−10.4605	−0.460496
$517$	−63.9283	−2.81156
$518$	62.1694	2.73157
$519$	41.3744	1.81613
$520$	0	0
$521$	34.6533	1.51819	0.759095	−	0.650980i	$-0.225642\pi$
$521$	34.6533	1.51819	0.759095	+	0.650980i	$0.225642\pi$
$522$	74.0106	3.23936
$523$	−13.5927	−0.594369	−0.297184	−	0.954820i	$-0.596048\pi$
$523$	−13.5927	−0.594369	−0.297184	+	0.954820i	$0.596048\pi$
$524$	−52.3634	−2.28750
$525$	70.8581	3.09250
$526$	−64.7012	−2.82111
$527$	2.47924	0.107998
$528$	65.9750	2.87119
$529$	−21.2268	−0.922903
$530$	79.4030	3.44905
$531$	33.3675	1.44803
$532$	42.0391	1.82263
$533$	0	0
$534$	−25.6423	−1.10965
$535$	−0.0523357	−0.00226267
$536$	46.3447	2.00179
$537$	68.9889	2.97709
$538$	46.6527	2.01134
$539$	31.0084	1.33562
$540$	72.5250	3.12098
$541$	13.9830	0.601178	0.300589	−	0.953754i	$-0.402817\pi$
$541$	13.9830	0.601178	0.300589	+	0.953754i	$0.402817\pi$
$542$	21.8104	0.936836
$543$	−14.7471	−0.632857
$544$	0.826196	0.0354229
$545$	−3.21324	−0.137640
$546$	0	0
$547$	−39.4448	−1.68654	−0.843269	−	0.537492i	$-0.819372\pi$
$547$	−39.4448	−1.68654	−0.843269	+	0.537492i	$0.819372\pi$
$548$	18.7967	0.802957
$549$	27.7338	1.18365
$550$	−98.4154	−4.19645
$551$	−18.3196	−0.780440
$552$	−18.6278	−0.792852
$553$	−20.1724	−0.857820
$554$	0.647737	0.0275197
$555$	−69.7821	−2.96208
$556$	59.7676	2.53471
$557$	27.8887	1.18168	0.590841	−	0.806788i	$-0.298796\pi$
$557$	27.8887	1.18168	0.590841	+	0.806788i	$0.298796\pi$
$558$	11.8940	0.503514
$559$	0	0
$560$	51.8569	2.19135
$561$	−38.9176	−1.64310
$562$	−15.4403	−0.651310
$563$	23.7172	0.999559	0.499780	−	0.866153i	$-0.333414\pi$
$563$	23.7172	0.999559	0.499780	+	0.866153i	$0.333414\pi$
$564$	−128.824	−5.42449
$565$	−20.8058	−0.875305
$566$	27.7903	1.16811
$567$	−0.283141	−0.0118908
$568$	−4.24624	−0.178168
$569$	15.1352	0.634501	0.317251	−	0.948342i	$-0.397240\pi$
$569$	15.1352	0.634501	0.317251	+	0.948342i	$0.397240\pi$
$570$	−70.5835	−2.95642
$571$	33.0021	1.38109	0.690547	−	0.723288i	$-0.257370\pi$
$571$	33.0021	1.38109	0.690547	+	0.723288i	$0.257370\pi$
$572$	0	0
$573$	−12.9248	−0.539940
$574$	−42.4541	−1.77200
$575$	9.51801	0.396928
$576$	−36.7389	−1.53079
$577$	15.1583	0.631049	0.315525	−	0.948917i	$-0.397819\pi$
$577$	15.1583	0.631049	0.315525	+	0.948917i	$0.397819\pi$
$578$	26.6596	1.10889
$579$	−4.53599	−0.188509
$580$	−87.4800	−3.63241
$581$	36.0497	1.49559
$582$	78.4681	3.25261
$583$	−51.9888	−2.15316
$584$	43.3903	1.79550
$585$	0	0
$586$	47.9064	1.97899
$587$	−7.05245	−0.291086	−0.145543	−	0.989352i	$-0.546493\pi$
$587$	−7.05245	−0.291086	−0.145543	+	0.989352i	$0.546493\pi$
$588$	62.4862	2.57689
$589$	−2.94408	−0.121309
$590$	−58.9957	−2.42882
$591$	−23.3010	−0.958474
$592$	−30.0492	−1.23501
$593$	21.3706	0.877584	0.438792	−	0.898589i	$-0.355407\pi$
$593$	21.3706	0.877584	0.438792	+	0.898589i	$0.355407\pi$
$594$	−71.0302	−2.91440
$595$	−30.5895	−1.25405
$596$	−11.5281	−0.472211
$597$	−66.1850	−2.70877
$598$	0	0
$599$	21.7134	0.887184	0.443592	−	0.896229i	$-0.353704\pi$
$599$	21.7134	0.887184	0.443592	+	0.896229i	$0.353704\pi$
$600$	−99.9872	−4.08196
$601$	6.23220	0.254217	0.127108	−	0.991889i	$-0.459430\pi$
$601$	6.23220	0.254217	0.127108	+	0.991889i	$0.459430\pi$
$602$	8.05255	0.328197
$603$	−44.9238	−1.82944
$604$	1.73065	0.0704190
$605$	71.1739	2.89363
$606$	42.2622	1.71678
$607$	−6.62821	−0.269031	−0.134515	−	0.990912i	$-0.542948\pi$
$607$	−6.62821	−0.269031	−0.134515	+	0.990912i	$0.542948\pi$
$608$	−0.981101	−0.0397889
$609$	−61.6866	−2.49967
$610$	−49.0350	−1.98537
$611$	0	0
$612$	−48.4233	−1.95740
$613$	15.3431	0.619700	0.309850	−	0.950785i	$-0.399721\pi$
$613$	15.3431	0.619700	0.309850	+	0.950785i	$0.399721\pi$
$614$	−77.8327	−3.14107
$615$	47.6526	1.92154
$616$	−99.1240	−3.99382
$617$	10.2734	0.413592	0.206796	−	0.978384i	$-0.433696\pi$
$617$	10.2734	0.413592	0.206796	+	0.978384i	$0.433696\pi$
$618$	67.2160	2.70382
$619$	−39.0227	−1.56845	−0.784227	−	0.620474i	$-0.786940\pi$
$619$	−39.0227	−1.56845	−0.784227	+	0.620474i	$0.786940\pi$
$620$	−14.0586	−0.564608
$621$	6.86952	0.275664
$622$	−3.03020	−0.121500
$623$	13.1965	0.528705
$624$	0	0
$625$	−9.64918	−0.385967
$626$	48.0889	1.92202
$627$	46.2143	1.84562
$628$	−19.3014	−0.770211
$629$	17.7255	0.706762
$630$	−146.751	−5.84672
$631$	−0.624321	−0.0248538	−0.0124269	−	0.999923i	$-0.503956\pi$
$631$	−0.624321	−0.0248538	−0.0124269	+	0.999923i	$0.503956\pi$
$632$	28.4651	1.13228
$633$	63.2662	2.51460
$634$	−5.04409	−0.200326
$635$	16.1719	0.641763
$636$	−104.765	−4.15419
$637$	0	0
$638$	85.6770	3.39198
$639$	4.11606	0.162829
$640$	67.2796	2.65946
$641$	15.3676	0.606983	0.303492	−	0.952834i	$-0.401848\pi$
$641$	15.3676	0.606983	0.303492	+	0.952834i	$0.401848\pi$
$642$	0.103290	0.00407653
$643$	37.6317	1.48405	0.742025	−	0.670372i	$-0.233865\pi$
$643$	37.6317	1.48405	0.742025	+	0.670372i	$0.233865\pi$
$644$	19.0145	0.749277
$645$	−9.03859	−0.355894
$646$	17.9291	0.705410
$647$	31.8747	1.25312	0.626562	−	0.779372i	$-0.284462\pi$
$647$	31.8747	1.25312	0.626562	+	0.779372i	$0.284462\pi$
$648$	0.399538	0.0156953
$649$	38.6272	1.51625
$650$	0	0
$651$	−9.91346	−0.388539
$652$	88.1158	3.45088
$653$	37.9437	1.48485	0.742426	−	0.669928i	$-0.233675\pi$
$653$	37.9437	1.48485	0.742426	+	0.669928i	$0.233675\pi$
$654$	6.34167	0.247979
$655$	−45.2457	−1.76790
$656$	20.5199	0.801168
$657$	−42.0600	−1.64092
$658$	99.1701	3.86605
$659$	34.4697	1.34275	0.671374	−	0.741119i	$-0.265705\pi$
$659$	34.4697	1.34275	0.671374	+	0.741119i	$0.265705\pi$
$660$	220.683	8.59009
$661$	−26.3201	−1.02373	−0.511867	−	0.859065i	$-0.671046\pi$
$661$	−26.3201	−1.02373	−0.511867	+	0.859065i	$0.671046\pi$
$662$	−32.5755	−1.26608
$663$	0	0
$664$	−50.8694	−1.97411
$665$	36.3248	1.40862
$666$	85.0370	3.29512
$667$	−8.28605	−0.320837
$668$	13.5967	0.526072
$669$	−67.3888	−2.60540
$670$	79.4280	3.06857
$671$	32.1055	1.23942
$672$	−3.30362	−0.127440
$673$	−22.6031	−0.871284	−0.435642	−	0.900120i	$-0.643479\pi$
$673$	−22.6031	−0.871284	−0.435642	+	0.900120i	$0.643479\pi$
$674$	86.4232	3.32890
$675$	36.8730	1.41924
$676$	0	0
$677$	22.8439	0.877962	0.438981	−	0.898496i	$-0.355339\pi$
$677$	22.8439	0.877962	0.438981	+	0.898496i	$0.355339\pi$
$678$	41.0624	1.57699
$679$	−40.3825	−1.54974
$680$	43.1646	1.65529
$681$	22.8319	0.874921
$682$	13.7689	0.527238
$683$	−19.3530	−0.740523	−0.370262	−	0.928927i	$-0.620732\pi$
$683$	−19.3530	−0.740523	−0.370262	+	0.928927i	$0.620732\pi$
$684$	57.5023	2.19865
$685$	16.2417	0.620565
$686$	12.7664	0.487425
$687$	−14.7902	−0.564280
$688$	−3.89215	−0.148387
$689$	0	0
$690$	−31.9253	−1.21538
$691$	−33.9864	−1.29290	−0.646452	−	0.762955i	$-0.723748\pi$
$691$	−33.9864	−1.29290	−0.646452	+	0.762955i	$0.723748\pi$
$692$	59.5950	2.26546
$693$	96.0850	3.64997
$694$	29.7198	1.12815
$695$	51.6435	1.95895
$696$	87.0454	3.29945
$697$	−12.1044	−0.458485
$698$	70.6800	2.67528
$699$	−29.8154	−1.12772
$700$	102.063	3.85762
$701$	14.0298	0.529898	0.264949	−	0.964262i	$-0.414645\pi$
$701$	14.0298	0.529898	0.264949	+	0.964262i	$0.414645\pi$
$702$	0	0
$703$	−21.0489	−0.793874
$704$	−42.5301	−1.60291
$705$	−111.314	−4.19231
$706$	−37.0737	−1.39529
$707$	−21.7496	−0.817979
$708$	77.8393	2.92538
$709$	6.14179	0.230660	0.115330	−	0.993327i	$-0.463207\pi$
$709$	6.14179	0.230660	0.115330	+	0.993327i	$0.463207\pi$
$710$	−7.27744	−0.273117
$711$	−27.5924	−1.03480
$712$	−18.6214	−0.697867
$713$	−1.33162	−0.0498697
$714$	60.3718	2.25936
$715$	0	0
$716$	99.3706	3.71365
$717$	17.6858	0.660487
$718$	7.17386	0.267726
$719$	8.98548	0.335102	0.167551	−	0.985863i	$-0.446414\pi$
$719$	8.98548	0.335102	0.167551	+	0.985863i	$0.446414\pi$
$720$	70.9313	2.64345
$721$	−34.5918	−1.28826
$722$	25.3799	0.944543
$723$	41.3604	1.53821
$724$	−21.2414	−0.789432
$725$	−44.4764	−1.65181
$726$	−140.469	−5.21330
$727$	−30.2413	−1.12159	−0.560795	−	0.827955i	$-0.689504\pi$
$727$	−30.2413	−1.12159	−0.560795	+	0.827955i	$0.689504\pi$
$728$	0	0
$729$	−43.7268	−1.61951
$730$	74.3646	2.75236
$731$	2.29591	0.0849174
$732$	64.6971	2.39127
$733$	51.5640	1.90456	0.952280	−	0.305227i	$-0.0987324\pi$
$733$	51.5640	1.90456	0.952280	+	0.305227i	$0.0987324\pi$
$734$	−17.2770	−0.637707
$735$	53.9926	1.99155
$736$	−0.443758	−0.0163571
$737$	−52.0052	−1.91564
$738$	−58.0699	−2.13758
$739$	10.0098	0.368216	0.184108	−	0.982906i	$-0.441060\pi$
$739$	10.0098	0.368216	0.184108	+	0.982906i	$0.441060\pi$
$740$	−100.513	−3.69493
$741$	0	0
$742$	80.6488	2.96071
$743$	16.9940	0.623449	0.311724	−	0.950173i	$-0.399093\pi$
$743$	16.9940	0.623449	0.311724	+	0.950173i	$0.399093\pi$
$744$	13.9888	0.512854
$745$	−9.96114	−0.364948
$746$	22.6747	0.830179
$747$	49.3098	1.80415
$748$	−56.0563	−2.04962
$749$	−0.0531568	−0.00194231
$750$	−51.4898	−1.88014
$751$	37.7192	1.37639	0.688196	−	0.725524i	$-0.258403\pi$
$751$	37.7192	1.37639	0.688196	+	0.725524i	$0.258403\pi$
$752$	−47.9332	−1.74794
$753$	34.5336	1.25847
$754$	0	0
$755$	1.49540	0.0544233
$756$	73.6628	2.67909
$757$	−5.31555	−0.193197	−0.0965984	−	0.995323i	$-0.530796\pi$
$757$	−5.31555	−0.193197	−0.0965984	+	0.995323i	$0.530796\pi$
$758$	34.7411	1.26186
$759$	20.9030	0.758731
$760$	−51.2576	−1.85931
$761$	−29.2832	−1.06152	−0.530758	−	0.847524i	$-0.678093\pi$
$761$	−29.2832	−1.06152	−0.530758	+	0.847524i	$0.678093\pi$
$762$	−31.9170	−1.15623
$763$	−3.26365	−0.118152
$764$	−18.6166	−0.673527
$765$	−41.8412	−1.51277
$766$	−23.3834	−0.844877
$767$	0	0
$768$	−90.2887	−3.25801
$769$	29.7376	1.07237	0.536183	−	0.844102i	$-0.319866\pi$
$769$	29.7376	1.07237	0.536183	+	0.844102i	$0.319866\pi$
$770$	−169.884	−6.12219
$771$	−77.1697	−2.77920
$772$	−6.53357	−0.235148
$773$	−1.75024	−0.0629517	−0.0314758	−	0.999505i	$-0.510021\pi$
$773$	−1.75024	−0.0629517	−0.0314758	+	0.999505i	$0.510021\pi$
$774$	11.0145	0.395908
$775$	−7.14767	−0.256752
$776$	56.9833	2.04558
$777$	−70.8769	−2.54270
$778$	23.7228	0.850503
$779$	14.3738	0.514996
$780$	0	0
$781$	4.76488	0.170501
$782$	8.10943	0.289993
$783$	−32.1004	−1.14717
$784$	23.2500	0.830356
$785$	−16.6778	−0.595257
$786$	89.2973	3.18513
$787$	1.92314	0.0685524	0.0342762	−	0.999412i	$-0.489087\pi$
$787$	1.92314	0.0685524	0.0342762	+	0.999412i	$0.489087\pi$
$788$	−33.5624	−1.19561
$789$	73.7634	2.62605
$790$	48.7851	1.73570
$791$	−21.1322	−0.751375
$792$	−135.585	−4.81779
$793$	0	0
$794$	32.2869	1.14582
$795$	−90.5243	−3.21057
$796$	−95.3320	−3.37895
$797$	5.24943	0.185944	0.0929721	−	0.995669i	$-0.470363\pi$
$797$	5.24943	0.185944	0.0929721	+	0.995669i	$0.470363\pi$
$798$	−71.6909	−2.53783
$799$	28.2750	1.00030
$800$	−2.38193	−0.0842139
$801$	18.0505	0.637783
$802$	−55.9750	−1.97655
$803$	−48.6899	−1.71823
$804$	−104.798	−3.69593
$805$	16.4299	0.579079
$806$	0	0
$807$	−53.1870	−1.87227
$808$	30.6907	1.07969
$809$	−34.3076	−1.20619	−0.603095	−	0.797670i	$-0.706066\pi$
$809$	−34.3076	−1.20619	−0.603095	+	0.797670i	$0.706066\pi$
$810$	0.684749	0.0240596
$811$	21.9495	0.770751	0.385375	−	0.922760i	$-0.374072\pi$
$811$	21.9495	0.770751	0.385375	+	0.922760i	$0.374072\pi$
$812$	−88.8525	−3.11811
$813$	−24.8652	−0.872060
$814$	98.4415	3.45037
$815$	76.1384	2.66701
$816$	−29.1803	−1.02151
$817$	−2.72638	−0.0953839
$818$	49.5296	1.73176
$819$	0	0
$820$	68.6382	2.39695
$821$	7.27538	0.253912	0.126956	−	0.991908i	$-0.459479\pi$
$821$	7.27538	0.253912	0.126956	+	0.991908i	$0.459479\pi$
$822$	−32.0548	−1.11804
$823$	42.3561	1.47644	0.738221	−	0.674559i	$-0.235666\pi$
$823$	42.3561	1.47644	0.738221	+	0.674559i	$0.235666\pi$
$824$	48.8121	1.70045
$825$	112.200	3.90629
$826$	−59.9213	−2.08493
$827$	48.1056	1.67279	0.836397	−	0.548124i	$-0.184658\pi$
$827$	48.1056	1.67279	0.836397	+	0.548124i	$0.184658\pi$
$828$	26.0086	0.903862
$829$	27.6670	0.960915	0.480458	−	0.877018i	$-0.340471\pi$
$829$	27.6670	0.960915	0.480458	+	0.877018i	$0.340471\pi$
$830$	−87.1826	−3.02615
$831$	−0.738460	−0.0256169
$832$	0	0
$833$	−13.7148	−0.475189
$834$	−101.924	−3.52934
$835$	11.7485	0.406575
$836$	66.5664	2.30225
$837$	−5.15875	−0.178313
$838$	−5.14237	−0.177640
$839$	−32.7819	−1.13176	−0.565878	−	0.824489i	$-0.691463\pi$
$839$	−32.7819	−1.13176	−0.565878	+	0.824489i	$0.691463\pi$
$840$	−172.597	−5.95517
$841$	9.71964	0.335160
$842$	37.1508	1.28030
$843$	17.6029	0.606276
$844$	91.1277	3.13674
$845$	0	0
$846$	135.648	4.66366
$847$	72.2906	2.48393
$848$	−38.9811	−1.33862
$849$	−31.6826	−1.08734
$850$	43.5284	1.49301
$851$	−9.52053	−0.326360
$852$	9.60189	0.328955
$853$	5.61212	0.192155	0.0960777	−	0.995374i	$-0.469370\pi$
$853$	5.61212	0.192155	0.0960777	+	0.995374i	$0.469370\pi$
$854$	−49.8044	−1.70427
$855$	49.6861	1.69923
$856$	0.0750090	0.00256376
$857$	−31.9107	−1.09005	−0.545024	−	0.838420i	$-0.683480\pi$
$857$	−31.9107	−1.09005	−0.545024	+	0.838420i	$0.683480\pi$
$858$	0	0
$859$	57.3951	1.95829	0.979147	−	0.203153i	$-0.0651189\pi$
$859$	57.3951	1.95829	0.979147	+	0.203153i	$0.0651189\pi$
$860$	−13.0191	−0.443946
$861$	48.4003	1.64948
$862$	−30.9350	−1.05365
$863$	31.8387	1.08380	0.541901	−	0.840442i	$-0.317705\pi$
$863$	31.8387	1.08380	0.541901	+	0.840442i	$0.317705\pi$
$864$	−1.71913	−0.0584860
$865$	51.4944	1.75086
$866$	−14.8650	−0.505133
$867$	−30.3936	−1.03222
$868$	−14.2792	−0.484668
$869$	−31.9419	−1.08355
$870$	149.183	5.05778
$871$	0	0
$872$	4.60531	0.155955
$873$	−55.2363	−1.86947
$874$	−9.62988	−0.325735
$875$	26.4985	0.895812
$876$	−98.1171	−3.31507
$877$	−11.7085	−0.395370	−0.197685	−	0.980266i	$-0.563342\pi$
$877$	−11.7085	−0.395370	−0.197685	+	0.980266i	$0.563342\pi$
$878$	5.78780	0.195329
$879$	−54.6162	−1.84216
$880$	82.1123	2.76801
$881$	−34.9127	−1.17624	−0.588120	−	0.808774i	$-0.700132\pi$
$881$	−34.9127	−1.17624	−0.588120	+	0.808774i	$0.700132\pi$
$882$	−65.7958	−2.21546
$883$	30.8289	1.03748	0.518738	−	0.854933i	$-0.326402\pi$
$883$	30.8289	1.03748	0.518738	+	0.854933i	$0.326402\pi$
$884$	0	0
$885$	67.2587	2.26088
$886$	−18.7674	−0.630501
$887$	−30.6338	−1.02858	−0.514292	−	0.857615i	$-0.671945\pi$
$887$	−30.6338	−1.02858	−0.514292	+	0.857615i	$0.671945\pi$
$888$	100.014	3.35624
$889$	16.4257	0.550899
$890$	−31.9144	−1.06977
$891$	−0.448337	−0.0150199
$892$	−97.0658	−3.25000
$893$	−33.5763	−1.12359
$894$	19.6594	0.657508
$895$	85.8634	2.87010
$896$	68.3352	2.28292
$897$	0	0
$898$	−77.1614	−2.57491
$899$	6.22251	0.207532
$900$	139.605	4.65349
$901$	22.9943	0.766051
$902$	−67.2235	−2.23830
$903$	−9.18040	−0.305505
$904$	29.8194	0.991780
$905$	−18.3541	−0.610112
$906$	−2.95134	−0.0980517
$907$	−24.1238	−0.801019	−0.400510	−	0.916293i	$-0.631167\pi$
$907$	−24.1238	−0.801019	−0.400510	+	0.916293i	$0.631167\pi$
$908$	32.8867	1.09139
$909$	−29.7498	−0.986737
$910$	0	0
$911$	−18.7893	−0.622518	−0.311259	−	0.950325i	$-0.600751\pi$
$911$	−18.7893	−0.622518	−0.311259	+	0.950325i	$0.600751\pi$
$912$	34.6513	1.14742
$913$	57.0825	1.88916
$914$	34.8391	1.15237
$915$	55.9029	1.84809
$916$	−21.3035	−0.703888
$917$	−45.9556	−1.51759
$918$	31.4162	1.03689
$919$	43.9357	1.44930	0.724652	−	0.689115i	$-0.242001\pi$
$919$	43.9357	1.44930	0.724652	+	0.689115i	$0.242001\pi$
$920$	−23.1841	−0.764357
$921$	88.7341	2.92389
$922$	−2.09321	−0.0689362
$923$	0	0
$924$	224.146	7.37386
$925$	−51.1027	−1.68025
$926$	−65.2937	−2.14569
$927$	−47.3156	−1.55405
$928$	2.07362	0.0680700
$929$	−2.75930	−0.0905299	−0.0452649	−	0.998975i	$-0.514413\pi$
$929$	−2.75930	−0.0905299	−0.0452649	+	0.998975i	$0.514413\pi$
$930$	23.9747	0.786162
$931$	16.2862	0.533758
$932$	−42.9456	−1.40673
$933$	3.45461	0.113099
$934$	−65.3249	−2.13750
$935$	−48.4367	−1.58405
$936$	0	0
$937$	38.8818	1.27021	0.635107	−	0.772424i	$-0.280956\pi$
$937$	38.8818	1.27021	0.635107	+	0.772424i	$0.280956\pi$
$938$	80.6742	2.63411
$939$	−54.8243	−1.78912
$940$	−160.334	−5.22953
$941$	36.4643	1.18870	0.594350	−	0.804206i	$-0.297409\pi$
$941$	36.4643	1.18870	0.594350	+	0.804206i	$0.297409\pi$
$942$	32.9155	1.07244
$943$	6.50136	0.211713
$944$	28.9626	0.942652
$945$	63.6500	2.07053
$946$	12.7507	0.414562
$947$	46.9659	1.52619	0.763093	−	0.646289i	$-0.223680\pi$
$947$	46.9659	1.52619	0.763093	+	0.646289i	$0.223680\pi$
$948$	−64.3673	−2.09055
$949$	0	0
$950$	−51.6896	−1.67703
$951$	5.75057	0.186475
$952$	43.8418	1.42092
$953$	−29.9638	−0.970624	−0.485312	−	0.874341i	$-0.661294\pi$
$953$	−29.9638	−0.970624	−0.485312	+	0.874341i	$0.661294\pi$
$954$	110.314	3.57154
$955$	−16.0861	−0.520535
$956$	25.4743	0.823899
$957$	−97.6770	−3.15745
$958$	−69.6513	−2.25033
$959$	16.4966	0.532702
$960$	−74.0545	−2.39010
$961$	1.00000	0.0322581
$962$	0	0
$963$	−0.0727093	−0.00234303
$964$	59.5749	1.91878
$965$	−5.64548	−0.181734
$966$	−32.4262	−1.04330
$967$	−45.1554	−1.45210	−0.726050	−	0.687641i	$-0.758646\pi$
$967$	−45.1554	−1.45210	−0.726050	+	0.687641i	$0.758646\pi$
$968$	−102.009	−3.27868
$969$	−20.4403	−0.656636
$970$	97.6611	3.13571
$971$	−44.4756	−1.42729	−0.713645	−	0.700508i	$-0.752957\pi$
$971$	−44.4756	−1.42729	−0.713645	+	0.700508i	$0.752957\pi$
$972$	−63.3290	−2.03128
$973$	52.4537	1.68159
$974$	−76.0803	−2.43777
$975$	0	0
$976$	24.0726	0.770545
$977$	−16.3554	−0.523255	−0.261628	−	0.965169i	$-0.584259\pi$
$977$	−16.3554	−0.523255	−0.261628	+	0.965169i	$0.584259\pi$
$978$	−150.267	−4.80502
$979$	20.8958	0.667833
$980$	77.7701	2.48428
$981$	−4.46412	−0.142528
$982$	−5.73824	−0.183115
$983$	54.2695	1.73093	0.865464	−	0.500971i	$-0.167024\pi$
$983$	54.2695	1.73093	0.865464	+	0.500971i	$0.167024\pi$
$984$	−68.2972	−2.17723
$985$	−29.0003	−0.924027
$986$	−37.8943	−1.20680
$987$	−113.060	−3.59874
$988$	0	0
$989$	−1.23316	−0.0392121
$990$	−232.372	−7.38527
$991$	−41.4631	−1.31712	−0.658560	−	0.752528i	$-0.728834\pi$
$991$	−41.4631	−1.31712	−0.658560	+	0.752528i	$0.728834\pi$
$992$	0.333245	0.0105806
$993$	37.1381	1.17854
$994$	−7.39162	−0.234448
$995$	−82.3737	−2.61142
$996$	115.029	3.64484
$997$	39.9735	1.26597	0.632986	−	0.774163i	$-0.281829\pi$
$997$	39.9735	1.26597	0.632986	+	0.774163i	$0.281829\pi$
$998$	−47.7592	−1.51179
$999$	−36.8828	−1.16692

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5239.2.a.l.1.1		16
13.5	odd	4		403.2.c.b.311.30	yes	32
13.8	odd	4		403.2.c.b.311.3	✓	32
13.12	even	2		5239.2.a.k.1.16		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.c.b.311.3	✓	32	13.8	odd	4
403.2.c.b.311.30	yes	32	13.5	odd	4
5239.2.a.k.1.16		16	13.12	even	2
5239.2.a.l.1.1		16	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:	\( N \)	\(=\)	\( 5239 = 13^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5239.a (trivial)

\(f(q)\)	\(=\)	\(q-2.45635 q^{2} +2.80039 q^{3} +4.03363 q^{4} +3.48535 q^{5} -6.87871 q^{6} +3.54003 q^{7} -4.99531 q^{8} +4.84216 q^{9} +O(q^{10})\)	\(q-2.45635 q^{2} +2.80039 q^{3} +4.03363 q^{4} +3.48535 q^{5} -6.87871 q^{6} +3.54003 q^{7} -4.99531 q^{8} +4.84216 q^{9} -8.56123 q^{10} +5.60543 q^{11} +11.2957 q^{12} -8.69555 q^{14} +9.76032 q^{15} +4.20294 q^{16} -2.47924 q^{17} -11.8940 q^{18} +2.94408 q^{19} +14.0586 q^{20} +9.91346 q^{21} -13.7689 q^{22} +1.33162 q^{23} -13.9888 q^{24} +7.14767 q^{25} +5.15875 q^{27} +14.2792 q^{28} -6.22251 q^{29} -23.9747 q^{30} -1.00000 q^{31} -0.333245 q^{32} +15.6974 q^{33} +6.08988 q^{34} +12.3383 q^{35} +19.5315 q^{36} -7.14956 q^{37} -7.23168 q^{38} -17.4104 q^{40} +4.88228 q^{41} -24.3509 q^{42} -0.926054 q^{43} +22.6103 q^{44} +16.8766 q^{45} -3.27093 q^{46} -11.4047 q^{47} +11.7698 q^{48} +5.53184 q^{49} -17.5571 q^{50} -6.94284 q^{51} -9.27473 q^{53} -12.6717 q^{54} +19.5369 q^{55} -17.6836 q^{56} +8.24456 q^{57} +15.2846 q^{58} +6.89104 q^{59} +39.3696 q^{60} +5.72757 q^{61} +2.45635 q^{62} +17.1414 q^{63} -7.58730 q^{64} -38.5582 q^{66} -9.27764 q^{67} -10.0004 q^{68} +3.72906 q^{69} -30.3070 q^{70} +0.850046 q^{71} -24.1881 q^{72} -8.68621 q^{73} +17.5618 q^{74} +20.0162 q^{75} +11.8753 q^{76} +19.8434 q^{77} -5.69838 q^{79} +14.6487 q^{80} -0.0799826 q^{81} -11.9926 q^{82} +10.1834 q^{83} +39.9873 q^{84} -8.64103 q^{85} +2.27471 q^{86} -17.4254 q^{87} -28.0009 q^{88} +3.72778 q^{89} -41.4548 q^{90} +5.37128 q^{92} -2.80039 q^{93} +28.0139 q^{94} +10.2611 q^{95} -0.933216 q^{96} -11.4074 q^{97} -13.5881 q^{98} +27.1424 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 4 q^{2} - 2 q^{3} + 18 q^{4} + 4 q^{5} + 6 q^{6} + 2 q^{7} + 12 q^{8} + 10 q^{9}+O(q^{10}) \) 16 * q + 4 * q^2 - 2 * q^3 + 18 * q^4 + 4 * q^5 + 6 * q^6 + 2 * q^7 + 12 * q^8 + 10 * q^9	\( 16 q + 4 q^{2} - 2 q^{3} + 18 q^{4} + 4 q^{5} + 6 q^{6} + 2 q^{7} + 12 q^{8} + 10 q^{9} - 2 q^{10} + 14 q^{11} + 8 q^{12} - 8 q^{14} + 14 q^{16} + 4 q^{17} - 28 q^{18} + 22 q^{19} + 28 q^{20} + 12 q^{21} - 8 q^{22} + 4 q^{23} - 8 q^{24} - 2 q^{25} + 10 q^{27} + 16 q^{28} - 8 q^{29} - 20 q^{30} - 16 q^{31} + 48 q^{32} + 10 q^{33} + 8 q^{34} - 2 q^{35} + 22 q^{36} + 16 q^{37} - 6 q^{38} + 14 q^{40} + 44 q^{41} + 14 q^{42} + 16 q^{43} + 4 q^{44} + 56 q^{45} + 10 q^{47} + 32 q^{49} + 2 q^{50} - 6 q^{53} + 24 q^{54} + 22 q^{55} - 4 q^{56} - 8 q^{57} + 74 q^{58} + 2 q^{59} + 40 q^{60} + 8 q^{61} - 4 q^{62} + 56 q^{63} + 38 q^{64} - 34 q^{66} - 8 q^{67} + 32 q^{68} - 10 q^{69} - 108 q^{70} + 50 q^{71} - 44 q^{72} + 14 q^{73} + 8 q^{74} + 44 q^{76} + 16 q^{77} + 32 q^{79} + 68 q^{80} - 8 q^{81} - 6 q^{82} - 20 q^{83} + 136 q^{84} - 32 q^{85} + 8 q^{86} - 36 q^{87} - 40 q^{88} + 52 q^{89} - 34 q^{90} + 14 q^{92} + 2 q^{93} + 44 q^{94} - 2 q^{95} - 80 q^{96} + 18 q^{97} + 12 q^{98} + 38 q^{99}+O(q^{100}) \) 16 * q + 4 * q^2 - 2 * q^3 + 18 * q^4 + 4 * q^5 + 6 * q^6 + 2 * q^7 + 12 * q^8 + 10 * q^9 - 2 * q^10 + 14 * q^11 + 8 * q^12 - 8 * q^14 + 14 * q^16 + 4 * q^17 - 28 * q^18 + 22 * q^19 + 28 * q^20 + 12 * q^21 - 8 * q^22 + 4 * q^23 - 8 * q^24 - 2 * q^25 + 10 * q^27 + 16 * q^28 - 8 * q^29 - 20 * q^30 - 16 * q^31 + 48 * q^32 + 10 * q^33 + 8 * q^34 - 2 * q^35 + 22 * q^36 + 16 * q^37 - 6 * q^38 + 14 * q^40 + 44 * q^41 + 14 * q^42 + 16 * q^43 + 4 * q^44 + 56 * q^45 + 10 * q^47 + 32 * q^49 + 2 * q^50 - 6 * q^53 + 24 * q^54 + 22 * q^55 - 4 * q^56 - 8 * q^57 + 74 * q^58 + 2 * q^59 + 40 * q^60 + 8 * q^61 - 4 * q^62 + 56 * q^63 + 38 * q^64 - 34 * q^66 - 8 * q^67 + 32 * q^68 - 10 * q^69 - 108 * q^70 + 50 * q^71 - 44 * q^72 + 14 * q^73 + 8 * q^74 + 44 * q^76 + 16 * q^77 + 32 * q^79 + 68 * q^80 - 8 * q^81 - 6 * q^82 - 20 * q^83 + 136 * q^84 - 32 * q^85 + 8 * q^86 - 36 * q^87 - 40 * q^88 + 52 * q^89 - 34 * q^90 + 14 * q^92 + 2 * q^93 + 44 * q^94 - 2 * q^95 - 80 * q^96 + 18 * q^97 + 12 * q^98 + 38 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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