LMFDB - Embedded newform 5239.2.a.l.1.15

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			1.15
Root			$2.67507$ 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)

$f(q)$	$=$	$q+2.67507 q^{2} -1.67043 q^{3} +5.15599 q^{4} +2.86564 q^{5} -4.46851 q^{6} -5.21058 q^{7} +8.44248 q^{8} -0.209666 q^{9} +O(q^{10})$	\(q+2.67507 q^{2} -1.67043 q^{3} +5.15599 q^{4} +2.86564 q^{5} -4.46851 q^{6} -5.21058 q^{7} +8.44248 q^{8} -0.209666 q^{9} +7.66579 q^{10} +0.107346 q^{11} -8.61271 q^{12} -13.9387 q^{14} -4.78685 q^{15} +12.2722 q^{16} +1.65501 q^{17} -0.560870 q^{18} +2.74920 q^{19} +14.7752 q^{20} +8.70391 q^{21} +0.287157 q^{22} -2.05031 q^{23} -14.1026 q^{24} +3.21190 q^{25} +5.36152 q^{27} -26.8657 q^{28} +8.54021 q^{29} -12.8052 q^{30} -1.00000 q^{31} +15.9441 q^{32} -0.179314 q^{33} +4.42728 q^{34} -14.9317 q^{35} -1.08103 q^{36} +7.15561 q^{37} +7.35430 q^{38} +24.1931 q^{40} +9.28974 q^{41} +23.2835 q^{42} +7.32217 q^{43} +0.553474 q^{44} -0.600827 q^{45} -5.48472 q^{46} +4.70074 q^{47} -20.4999 q^{48} +20.1502 q^{49} +8.59206 q^{50} -2.76459 q^{51} -3.04370 q^{53} +14.3424 q^{54} +0.307615 q^{55} -43.9902 q^{56} -4.59235 q^{57} +22.8456 q^{58} -11.4500 q^{59} -24.6809 q^{60} -5.46539 q^{61} -2.67507 q^{62} +1.09248 q^{63} +18.1070 q^{64} -0.479676 q^{66} +1.60563 q^{67} +8.53323 q^{68} +3.42490 q^{69} -39.9432 q^{70} +8.50173 q^{71} -1.77010 q^{72} +6.98159 q^{73} +19.1418 q^{74} -5.36526 q^{75} +14.1748 q^{76} -0.559334 q^{77} -1.90261 q^{79} +35.1678 q^{80} -8.32704 q^{81} +24.8507 q^{82} +9.64608 q^{83} +44.8772 q^{84} +4.74268 q^{85} +19.5873 q^{86} -14.2658 q^{87} +0.906264 q^{88} +9.70043 q^{89} -1.60725 q^{90} -10.5714 q^{92} +1.67043 q^{93} +12.5748 q^{94} +7.87823 q^{95} -26.6334 q^{96} -14.1790 q^{97} +53.9030 q^{98} -0.0225067 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.67507	1.89156	0.945779	−	0.324810i	$-0.105300\pi$
$2$	2.67507	1.89156	0.945779	+	0.324810i	$0.105300\pi$
$3$	−1.67043	−0.964423	−0.482211	−	0.876055i	$-0.660166\pi$
$3$	−1.67043	−0.964423	−0.482211	+	0.876055i	$0.660166\pi$
$4$	5.15599	2.57799
$5$	2.86564	1.28155	0.640777	−	0.767727i	$-0.278612\pi$
$5$	2.86564	1.28155	0.640777	+	0.767727i	$0.278612\pi$
$6$	−4.46851	−1.82426
$7$	−5.21058	−1.96941	−0.984707	−	0.174217i	$-0.944261\pi$
$7$	−5.21058	−1.96941	−0.984707	+	0.174217i	$0.944261\pi$
$8$	8.44248	2.98487
$9$	−0.209666	−0.0698885
$10$	7.66579	2.42413
$11$	0.107346	0.0323660	0.0161830	−	0.999869i	$-0.494849\pi$
$11$	0.107346	0.0323660	0.0161830	+	0.999869i	$0.494849\pi$
$12$	−8.61271	−2.48628
$13$	0	0
$13$	0	0
$14$	−13.9387	−3.72526
$15$	−4.78685	−1.23596
$16$	12.2722	3.06805
$17$	1.65501	0.401400	0.200700	−	0.979653i	$-0.435678\pi$
$17$	1.65501	0.401400	0.200700	+	0.979653i	$0.435678\pi$
$18$	−0.560870	−0.132198
$19$	2.74920	0.630710	0.315355	−	0.948974i	$-0.397876\pi$
$19$	2.74920	0.630710	0.315355	+	0.948974i	$0.397876\pi$
$20$	14.7752	3.30384
$21$	8.70391	1.89935
$22$	0.287157	0.0612221
$23$	−2.05031	−0.427519	−0.213760	−	0.976886i	$-0.568571\pi$
$23$	−2.05031	−0.427519	−0.213760	+	0.976886i	$0.568571\pi$
$24$	−14.1026	−2.87867
$25$	3.21190	0.642381
$26$	0	0
$27$	5.36152	1.03182
$28$	−26.8657	−5.07714
$29$	8.54021	1.58588	0.792939	−	0.609302i	$-0.208550\pi$
$29$	8.54021	1.58588	0.792939	+	0.609302i	$0.208550\pi$
$30$	−12.8052	−2.33789
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	15.9441	2.81854
$33$	−0.179314	−0.0312145
$34$	4.42728	0.759272
$35$	−14.9317	−2.52391
$36$	−1.08103	−0.180172
$37$	7.15561	1.17638	0.588188	−	0.808724i	$-0.299842\pi$
$37$	7.15561	1.17638	0.588188	+	0.808724i	$0.299842\pi$
$38$	7.35430	1.19302
$39$	0	0
$40$	24.1931	3.82527
$41$	9.28974	1.45081	0.725407	−	0.688320i	$-0.241652\pi$
$41$	9.28974	1.45081	0.725407	+	0.688320i	$0.241652\pi$
$42$	23.2835	3.59273
$43$	7.32217	1.11662	0.558311	−	0.829632i	$-0.311450\pi$
$43$	7.32217	1.11662	0.558311	+	0.829632i	$0.311450\pi$
$44$	0.553474	0.0834393
$45$	−0.600827	−0.0895659
$46$	−5.48472	−0.808678
$47$	4.70074	0.685674	0.342837	−	0.939395i	$-0.388612\pi$
$47$	4.70074	0.685674	0.342837	+	0.939395i	$0.388612\pi$
$48$	−20.4999	−2.95890
$49$	20.1502	2.87859
$50$	8.59206	1.21510
$51$	−2.76459	−0.387119
$52$	0	0
$53$	−3.04370	−0.418084	−0.209042	−	0.977907i	$-0.567035\pi$
$53$	−3.04370	−0.418084	−0.209042	+	0.977907i	$0.567035\pi$
$54$	14.3424	1.95176
$55$	0.307615	0.0414788
$56$	−43.9902	−5.87844
$57$	−4.59235	−0.608271
$58$	22.8456	2.99978
$59$	−11.4500	−1.49067	−0.745333	−	0.666692i	$-0.767710\pi$
$59$	−11.4500	−1.49067	−0.745333	+	0.666692i	$0.767710\pi$
$60$	−24.6809	−3.18630
$61$	−5.46539	−0.699772	−0.349886	−	0.936792i	$-0.613780\pi$
$61$	−5.46539	−0.699772	−0.349886	+	0.936792i	$0.613780\pi$
$62$	−2.67507	−0.339734
$63$	1.09248	0.137640
$64$	18.1070	2.26338
$65$	0	0
$66$	−0.479676	−0.0590440
$67$	1.60563	0.196159	0.0980796	−	0.995179i	$-0.468730\pi$
$67$	1.60563	0.196159	0.0980796	+	0.995179i	$0.468730\pi$
$68$	8.53323	1.03481
$69$	3.42490	0.412309
$70$	−39.9432	−4.77413
$71$	8.50173	1.00897	0.504485	−	0.863421i	$-0.331682\pi$
$71$	8.50173	1.00897	0.504485	+	0.863421i	$0.331682\pi$
$72$	−1.77010	−0.208608
$73$	6.98159	0.817133	0.408567	−	0.912729i	$-0.366029\pi$
$73$	6.98159	0.817133	0.408567	+	0.912729i	$0.366029\pi$
$74$	19.1418	2.22518
$75$	−5.36526	−0.619527
$76$	14.1748	1.62597
$77$	−0.559334	−0.0637420
$78$	0	0
$79$	−1.90261	−0.214061	−0.107030	−	0.994256i	$-0.534134\pi$
$79$	−1.90261	−0.214061	−0.107030	+	0.994256i	$0.534134\pi$
$80$	35.1678	3.93188
$81$	−8.32704	−0.925227
$82$	24.8507	2.74430
$83$	9.64608	1.05880	0.529398	−	0.848374i	$-0.322418\pi$
$83$	9.64608	1.05880	0.529398	+	0.848374i	$0.322418\pi$
$84$	44.8772	4.89651
$85$	4.74268	0.514416
$86$	19.5873	2.11215
$87$	−14.2658	−1.52946
$88$	0.906264	0.0966081
$89$	9.70043	1.02824	0.514122	−	0.857717i	$-0.328118\pi$
$89$	9.70043	1.02824	0.514122	+	0.857717i	$0.328118\pi$
$90$	−1.60725	−0.169419
$91$	0	0
$92$	−10.5714	−1.10214
$93$	1.67043	0.173215
$94$	12.5748	1.29699
$95$	7.87823	0.808289
$96$	−26.6334	−2.71826
$97$	−14.1790	−1.43966	−0.719830	−	0.694150i	$-0.755780\pi$
$97$	−14.1790	−1.43966	−0.719830	+	0.694150i	$0.755780\pi$
$98$	53.9030	5.44503
$99$	−0.0225067	−0.00226201
$100$	16.5605	1.65605
$101$	−3.99013	−0.397032	−0.198516	−	0.980098i	$-0.563612\pi$
$101$	−3.99013	−0.397032	−0.198516	+	0.980098i	$0.563612\pi$
$102$	−7.39545	−0.732259
$103$	−7.24951	−0.714316	−0.357158	−	0.934044i	$-0.616254\pi$
$103$	−7.24951	−0.714316	−0.357158	+	0.934044i	$0.616254\pi$
$104$	0	0
$105$	24.9423	2.43412
$106$	−8.14210	−0.790830
$107$	2.56526	0.247993	0.123997	−	0.992283i	$-0.460429\pi$
$107$	2.56526	0.247993	0.123997	+	0.992283i	$0.460429\pi$
$108$	27.6439	2.66004
$109$	−6.50099	−0.622682	−0.311341	−	0.950298i	$-0.600778\pi$
$109$	−6.50099	−0.622682	−0.311341	+	0.950298i	$0.600778\pi$
$110$	0.822890	0.0784595
$111$	−11.9529	−1.13452
$112$	−63.9454	−6.04227
$113$	6.19260	0.582551	0.291276	−	0.956639i	$-0.405920\pi$
$113$	6.19260	0.582551	0.291276	+	0.956639i	$0.405920\pi$
$114$	−12.2848	−1.15058
$115$	−5.87546	−0.547889
$116$	44.0332	4.08838
$117$	0	0
$118$	−30.6296	−2.81968
$119$	−8.62359	−0.790523
$120$	−40.4129	−3.68918
$121$	−10.9885	−0.998952
$122$	−14.6203	−1.32366
$123$	−15.5179	−1.39920
$124$	−5.15599	−0.463021
$125$	−5.12404	−0.458308
$126$	2.92246	0.260353
$127$	−4.47662	−0.397236	−0.198618	−	0.980077i	$-0.563645\pi$
$127$	−4.47662	−0.397236	−0.198618	+	0.980077i	$0.563645\pi$
$128$	16.5493	1.46277
$129$	−12.2312	−1.07689
$130$	0	0
$131$	8.44606	0.737935	0.368968	−	0.929442i	$-0.379711\pi$
$131$	8.44606	0.737935	0.368968	+	0.929442i	$0.379711\pi$
$132$	−0.924538	−0.0804707
$133$	−14.3249	−1.24213
$134$	4.29518	0.371047
$135$	15.3642	1.32234
$136$	13.9724	1.19813
$137$	4.93436	0.421571	0.210785	−	0.977532i	$-0.432398\pi$
$137$	4.93436	0.421571	0.210785	+	0.977532i	$0.432398\pi$
$138$	9.16184	0.779907
$139$	−9.80769	−0.831877	−0.415938	−	0.909393i	$-0.636547\pi$
$139$	−9.80769	−0.831877	−0.415938	+	0.909393i	$0.636547\pi$
$140$	−76.9874	−6.50663
$141$	−7.85226	−0.661280
$142$	22.7427	1.90852
$143$	0	0
$144$	−2.57306	−0.214422
$145$	24.4732	2.03239
$146$	18.6762	1.54565
$147$	−33.6594	−2.77618
$148$	36.8943	3.03269
$149$	1.24011	0.101593	0.0507967	−	0.998709i	$-0.483824\pi$
$149$	1.24011	0.101593	0.0507967	+	0.998709i	$0.483824\pi$
$150$	−14.3524	−1.17187
$151$	−4.52510	−0.368247	−0.184124	−	0.982903i	$-0.558945\pi$
$151$	−4.52510	−0.368247	−0.184124	+	0.982903i	$0.558945\pi$
$152$	23.2101	1.88258
$153$	−0.347000	−0.0280533
$154$	−1.49626	−0.120572
$155$	−2.86564	−0.230174
$156$	0	0
$157$	7.78510	0.621319	0.310659	−	0.950521i	$-0.399450\pi$
$157$	7.78510	0.621319	0.310659	+	0.950521i	$0.399450\pi$
$158$	−5.08961	−0.404908
$159$	5.08428	0.403210
$160$	45.6900	3.61211
$161$	10.6833	0.841963
$162$	−22.2754	−1.75012
$163$	−1.90995	−0.149599	−0.0747995	−	0.997199i	$-0.523832\pi$
$163$	−1.90995	−0.149599	−0.0747995	+	0.997199i	$0.523832\pi$
$164$	47.8978	3.74019
$165$	−0.513849	−0.0400031
$166$	25.8039	2.00277
$167$	−20.1983	−1.56299	−0.781497	−	0.623909i	$-0.785543\pi$
$167$	−20.1983	−1.56299	−0.781497	+	0.623909i	$0.785543\pi$
$168$	73.4825	5.66930
$169$	0	0
$170$	12.6870	0.973048
$171$	−0.576413	−0.0440794
$172$	37.7530	2.87864
$173$	2.87960	0.218932	0.109466	−	0.993991i	$-0.465086\pi$
$173$	2.87960	0.218932	0.109466	+	0.993991i	$0.465086\pi$
$174$	−38.1620	−2.89306
$175$	−16.7359	−1.26511
$176$	1.31737	0.0993006
$177$	19.1265	1.43763
$178$	25.9493	1.94498
$179$	7.10080	0.530739	0.265369	−	0.964147i	$-0.414506\pi$
$179$	7.10080	0.530739	0.265369	+	0.964147i	$0.414506\pi$
$180$	−3.09785	−0.230900
$181$	12.7730	0.949409	0.474704	−	0.880145i	$-0.342555\pi$
$181$	12.7730	0.949409	0.474704	+	0.880145i	$0.342555\pi$
$182$	0	0
$183$	9.12955	0.674876
$184$	−17.3097	−1.27609
$185$	20.5054	1.50759
$186$	4.46851	0.327647
$187$	0.177659	0.0129917
$188$	24.2370	1.76766
$189$	−27.9366	−2.03209
$190$	21.0748	1.52893
$191$	−15.1982	−1.09970	−0.549850	−	0.835263i	$-0.685315\pi$
$191$	−15.1982	−1.09970	−0.549850	+	0.835263i	$0.685315\pi$
$192$	−30.2465	−2.18285
$193$	−1.93144	−0.139028	−0.0695142	−	0.997581i	$-0.522145\pi$
$193$	−1.93144	−0.139028	−0.0695142	+	0.997581i	$0.522145\pi$
$194$	−37.9298	−2.72320
$195$	0	0
$196$	103.894	7.42100
$197$	−13.6030	−0.969176	−0.484588	−	0.874742i	$-0.661031\pi$
$197$	−13.6030	−0.969176	−0.484588	+	0.874742i	$0.661031\pi$
$198$	−0.0602070	−0.00427873
$199$	−11.3269	−0.802946	−0.401473	−	0.915871i	$-0.631502\pi$
$199$	−11.3269	−0.802946	−0.401473	+	0.915871i	$0.631502\pi$
$200$	27.1164	1.91742
$201$	−2.68210	−0.189180
$202$	−10.6739	−0.751010
$203$	−44.4995	−3.12325
$204$	−14.2542	−0.997991
$205$	26.6211	1.85930
$206$	−19.3929	−1.35117
$207$	0.429880	0.0298787
$208$	0	0
$209$	0.295115	0.0204136
$210$	66.7223	4.60428
$211$	−9.96107	−0.685749	−0.342874	−	0.939381i	$-0.611400\pi$
$211$	−9.96107	−0.685749	−0.342874	+	0.939381i	$0.611400\pi$
$212$	−15.6933	−1.07782
$213$	−14.2015	−0.973073
$214$	6.86225	0.469094
$215$	20.9827	1.43101
$216$	45.2645	3.07986
$217$	5.21058	0.353717
$218$	−17.3906	−1.17784
$219$	−11.6623	−0.788062
$220$	1.58606	0.106932
$221$	0	0
$222$	−31.9749	−2.14602
$223$	−17.9002	−1.19869	−0.599344	−	0.800491i	$-0.704572\pi$
$223$	−17.9002	−1.19869	−0.599344	+	0.800491i	$0.704572\pi$
$224$	−83.0779	−5.55087
$225$	−0.673426	−0.0448951
$226$	16.5656	1.10193
$227$	−1.72785	−0.114681	−0.0573406	−	0.998355i	$-0.518262\pi$
$227$	−1.72785	−0.114681	−0.0573406	+	0.998355i	$0.518262\pi$
$228$	−23.6781	−1.56812
$229$	−13.4854	−0.891141	−0.445570	−	0.895247i	$-0.646999\pi$
$229$	−13.4854	−0.891141	−0.445570	+	0.895247i	$0.646999\pi$
$230$	−15.7172	−1.03636
$231$	0.934328	0.0614743
$232$	72.1005	4.73363
$233$	−2.06862	−0.135520	−0.0677600	−	0.997702i	$-0.521585\pi$
$233$	−2.06862	−0.135520	−0.0677600	+	0.997702i	$0.521585\pi$
$234$	0	0
$235$	13.4706	0.878728
$236$	−59.0362	−3.84293
$237$	3.17818	0.206445
$238$	−23.0687	−1.49532
$239$	14.7921	0.956823	0.478411	−	0.878136i	$-0.341213\pi$
$239$	14.7921	0.956823	0.478411	+	0.878136i	$0.341213\pi$
$240$	−58.7453	−3.79199
$241$	19.9764	1.28679	0.643397	−	0.765532i	$-0.277524\pi$
$241$	19.9764	1.28679	0.643397	+	0.765532i	$0.277524\pi$
$242$	−29.3949	−1.88958
$243$	−2.17482	−0.139515
$244$	−28.1795	−1.80401
$245$	57.7431	3.68907
$246$	−41.5113	−2.64666
$247$	0	0
$248$	−8.44248	−0.536098
$249$	−16.1131	−1.02113
$250$	−13.7072	−0.866917
$251$	4.64690	0.293310	0.146655	−	0.989188i	$-0.453149\pi$
$251$	4.64690	0.293310	0.146655	+	0.989188i	$0.453149\pi$
$252$	5.63281	0.354834
$253$	−0.220092	−0.0138371
$254$	−11.9753	−0.751394
$255$	−7.92231	−0.496114
$256$	8.05659	0.503537
$257$	−9.54021	−0.595102	−0.297551	−	0.954706i	$-0.596170\pi$
$257$	−9.54021	−0.595102	−0.297551	+	0.954706i	$0.596170\pi$
$258$	−32.7192	−2.03701
$259$	−37.2849	−2.31677
$260$	0	0
$261$	−1.79059	−0.110835
$262$	22.5938	1.39585
$263$	−14.4929	−0.893672	−0.446836	−	0.894616i	$-0.647449\pi$
$263$	−14.4929	−0.893672	−0.446836	+	0.894616i	$0.647449\pi$
$264$	−1.51385	−0.0931711
$265$	−8.72215	−0.535797
$266$	−38.3202	−2.34956
$267$	−16.2039	−0.991662
$268$	8.27862	0.505697
$269$	27.3814	1.66947	0.834737	−	0.550648i	$-0.185619\pi$
$269$	27.3814	1.66947	0.834737	+	0.550648i	$0.185619\pi$
$270$	41.1003	2.50128
$271$	13.3239	0.809367	0.404683	−	0.914457i	$-0.367382\pi$
$271$	13.3239	0.809367	0.404683	+	0.914457i	$0.367382\pi$
$272$	20.3107	1.23152
$273$	0	0
$274$	13.1997	0.797426
$275$	0.344785	0.0207913
$276$	17.6587	1.06293
$277$	30.1947	1.81422	0.907112	−	0.420890i	$-0.138282\pi$
$277$	30.1947	1.81422	0.907112	+	0.420890i	$0.138282\pi$
$278$	−26.2362	−1.57354
$279$	0.209666	0.0125524
$280$	−126.060	−7.53354
$281$	−8.84908	−0.527892	−0.263946	−	0.964537i	$-0.585024\pi$
$281$	−8.84908	−0.527892	−0.263946	+	0.964537i	$0.585024\pi$
$282$	−21.0053	−1.25085
$283$	3.32142	0.197438	0.0987190	−	0.995115i	$-0.468526\pi$
$283$	3.32142	0.197438	0.0987190	+	0.995115i	$0.468526\pi$
$284$	43.8348	2.60112
$285$	−13.1600	−0.779532
$286$	0	0
$287$	−48.4049	−2.85725
$288$	−3.34292	−0.196984
$289$	−14.2609	−0.838878
$290$	65.4674	3.84438
$291$	23.6850	1.38844
$292$	35.9970	2.10656
$293$	−24.8920	−1.45420	−0.727102	−	0.686529i	$-0.759133\pi$
$293$	−24.8920	−1.45420	−0.727102	+	0.686529i	$0.759133\pi$
$294$	−90.0412	−5.25131
$295$	−32.8117	−1.91037
$296$	60.4111	3.51132
$297$	0.575537	0.0333960
$298$	3.31736	0.192170
$299$	0	0
$300$	−27.6632	−1.59714
$301$	−38.1528	−2.19909
$302$	−12.1049	−0.696561
$303$	6.66522	0.382907
$304$	33.7388	1.93505
$305$	−15.6619	−0.896796
$306$	−0.928247	−0.0530644
$307$	−4.47896	−0.255628	−0.127814	−	0.991798i	$-0.540796\pi$
$307$	−4.47896	−0.255628	−0.127814	+	0.991798i	$0.540796\pi$
$308$	−2.88392	−0.164327
$309$	12.1098	0.688902
$310$	−7.66579	−0.435387
$311$	−24.8980	−1.41184	−0.705918	−	0.708294i	$-0.749465\pi$
$311$	−24.8980	−1.41184	−0.705918	+	0.708294i	$0.749465\pi$
$312$	0	0
$313$	21.0469	1.18964	0.594819	−	0.803860i	$-0.297224\pi$
$313$	21.0469	1.18964	0.594819	+	0.803860i	$0.297224\pi$
$314$	20.8257	1.17526
$315$	3.13066	0.176392
$316$	−9.80984	−0.551847
$317$	8.99093	0.504981	0.252491	−	0.967599i	$-0.418750\pi$
$317$	8.99093	0.504981	0.252491	+	0.967599i	$0.418750\pi$
$318$	13.6008	0.762695
$319$	0.916756	0.0513285
$320$	51.8882	2.90064
$321$	−4.28509	−0.239170
$322$	28.5786	1.59262
$323$	4.54997	0.253167
$324$	−42.9341	−2.38523
$325$	0	0
$326$	−5.10925	−0.282975
$327$	10.8594	0.600529
$328$	78.4284	4.33048
$329$	−24.4936	−1.35038
$330$	−1.37458	−0.0756681
$331$	9.36893	0.514963	0.257482	−	0.966283i	$-0.417107\pi$
$331$	9.36893	0.514963	0.257482	+	0.966283i	$0.417107\pi$
$332$	49.7351	2.72957
$333$	−1.50029	−0.0822152
$334$	−54.0319	−2.95649
$335$	4.60117	0.251389
$336$	106.816	5.82731
$337$	20.3475	1.10840	0.554199	−	0.832384i	$-0.313025\pi$
$337$	20.3475	1.10840	0.554199	+	0.832384i	$0.313025\pi$
$338$	0	0
$339$	−10.3443	−0.561826
$340$	24.4532	1.32616
$341$	−0.107346	−0.00581310
$342$	−1.54194	−0.0833788
$343$	−68.5200	−3.69973
$344$	61.8173	3.33296
$345$	9.81454	0.528397
$346$	7.70312	0.414122
$347$	−12.1586	−0.652709	−0.326354	−	0.945248i	$-0.605820\pi$
$347$	−12.1586	−0.652709	−0.326354	+	0.945248i	$0.605820\pi$
$348$	−73.5544	−3.94293
$349$	−30.9550	−1.65698	−0.828492	−	0.560000i	$-0.810801\pi$
$349$	−30.9550	−1.65698	−0.828492	+	0.560000i	$0.810801\pi$
$350$	−44.7696	−2.39304
$351$	0	0
$352$	1.71153	0.0912248
$353$	5.94598	0.316472	0.158236	−	0.987401i	$-0.449419\pi$
$353$	5.94598	0.316472	0.158236	+	0.987401i	$0.449419\pi$
$354$	51.1646	2.71937
$355$	24.3629	1.29305
$356$	50.0153	2.65081
$357$	14.4051	0.762399
$358$	18.9951	1.00392
$359$	29.5605	1.56014	0.780072	−	0.625690i	$-0.215182\pi$
$359$	29.5605	1.56014	0.780072	+	0.625690i	$0.215182\pi$
$360$	−5.07246	−0.267342
$361$	−11.4419	−0.602205
$362$	34.1686	1.79586
$363$	18.3555	0.963413
$364$	0	0
$365$	20.0067	1.04720
$366$	24.4222	1.27657
$367$	26.1640	1.36575	0.682874	−	0.730536i	$-0.260730\pi$
$367$	26.1640	1.36575	0.682874	+	0.730536i	$0.260730\pi$
$368$	−25.1619	−1.31165
$369$	−1.94774	−0.101395
$370$	54.8534	2.85169
$371$	15.8594	0.823381
$372$	8.61271	0.446548
$373$	8.69976	0.450456	0.225228	−	0.974306i	$-0.427687\pi$
$373$	8.69976	0.450456	0.225228	+	0.974306i	$0.427687\pi$
$374$	0.475250	0.0245746
$375$	8.55935	0.442003
$376$	39.6859	2.04664
$377$	0	0
$378$	−74.7324	−3.84382
$379$	22.2845	1.14468	0.572338	−	0.820018i	$-0.306037\pi$
$379$	22.2845	1.14468	0.572338	+	0.820018i	$0.306037\pi$
$380$	40.6200	2.08376
$381$	7.47788	0.383103
$382$	−40.6561	−2.08015
$383$	−21.8948	−1.11877	−0.559387	−	0.828907i	$-0.688963\pi$
$383$	−21.8948	−1.11877	−0.559387	+	0.828907i	$0.688963\pi$
$384$	−27.6445	−1.41073
$385$	−1.60285	−0.0816889
$386$	−5.16674	−0.262980
$387$	−1.53521	−0.0780390
$388$	−73.1068	−3.71143
$389$	−18.6807	−0.947150	−0.473575	−	0.880754i	$-0.657037\pi$
$389$	−18.6807	−0.947150	−0.473575	+	0.880754i	$0.657037\pi$
$390$	0	0
$391$	−3.39329	−0.171606
$392$	170.117	8.59222
$393$	−14.1085	−0.711682
$394$	−36.3891	−1.83325
$395$	−5.45220	−0.274330
$396$	−0.116044	−0.00583145
$397$	−2.50967	−0.125957	−0.0629783	−	0.998015i	$-0.520060\pi$
$397$	−2.50967	−0.125957	−0.0629783	+	0.998015i	$0.520060\pi$
$398$	−30.3003	−1.51882
$399$	23.9288	1.19794
$400$	39.4172	1.97086
$401$	−1.92212	−0.0959861	−0.0479930	−	0.998848i	$-0.515283\pi$
$401$	−1.92212	−0.0959861	−0.0479930	+	0.998848i	$0.515283\pi$
$402$	−7.17479	−0.357846
$403$	0	0
$404$	−20.5730	−1.02355
$405$	−23.8623	−1.18573
$406$	−119.039	−5.90781
$407$	0.768125	0.0380746
$408$	−23.3399	−1.15550
$409$	31.4091	1.55308	0.776540	−	0.630068i	$-0.216973\pi$
$409$	31.4091	1.55308	0.776540	+	0.630068i	$0.216973\pi$
$410$	71.2132	3.51697
$411$	−8.24250	−0.406572
$412$	−37.3784	−1.84150
$413$	59.6613	2.93574
$414$	1.14996	0.0565173
$415$	27.6422	1.35690
$416$	0	0
$417$	16.3830	0.802281
$418$	0.789453	0.0386134
$419$	30.6922	1.49941	0.749707	−	0.661770i	$-0.230195\pi$
$419$	30.6922	1.49941	0.749707	+	0.661770i	$0.230195\pi$
$420$	128.602	6.27514
$421$	39.9412	1.94662	0.973308	−	0.229504i	$-0.0737104\pi$
$421$	39.9412	1.94662	0.973308	+	0.229504i	$0.0737104\pi$
$422$	−26.6465	−1.29713
$423$	−0.985584	−0.0479207
$424$	−25.6963	−1.24792
$425$	5.31575	0.257852
$426$	−37.9901	−1.84062
$427$	28.4779	1.37814
$428$	13.2265	0.639325
$429$	0	0
$430$	56.1302	2.70684
$431$	16.2333	0.781931	0.390966	−	0.920405i	$-0.372141\pi$
$431$	16.2333	0.781931	0.390966	+	0.920405i	$0.372141\pi$
$432$	65.7978	3.16570
$433$	−31.2313	−1.50088	−0.750439	−	0.660939i	$-0.770158\pi$
$433$	−31.2313	−1.50088	−0.750439	+	0.660939i	$0.770158\pi$
$434$	13.9387	0.669077
$435$	−40.8807	−1.96008
$436$	−33.5190	−1.60527
$437$	−5.63672	−0.269641
$438$	−31.1973	−1.49066
$439$	−8.10748	−0.386949	−0.193475	−	0.981105i	$-0.561976\pi$
$439$	−8.10748	−0.386949	−0.193475	+	0.981105i	$0.561976\pi$
$440$	2.59703	0.123809
$441$	−4.22480	−0.201181
$442$	0	0
$443$	−1.90009	−0.0902762	−0.0451381	−	0.998981i	$-0.514373\pi$
$443$	−1.90009	−0.0902762	−0.0451381	+	0.998981i	$0.514373\pi$
$444$	−61.6292	−2.92479
$445$	27.7980	1.31775
$446$	−47.8843	−2.26739
$447$	−2.07151	−0.0979790
$448$	−94.3481	−4.45753
$449$	7.08178	0.334210	0.167105	−	0.985939i	$-0.446558\pi$
$449$	7.08178	0.334210	0.167105	+	0.985939i	$0.446558\pi$
$450$	−1.80146	−0.0849216
$451$	0.997215	0.0469570
$452$	31.9290	1.50181
$453$	7.55885	0.355146
$454$	−4.62210	−0.216926
$455$	0	0
$456$	−38.7708	−1.81561
$457$	3.80707	0.178087	0.0890437	−	0.996028i	$-0.471619\pi$
$457$	3.80707	0.178087	0.0890437	+	0.996028i	$0.471619\pi$
$458$	−36.0744	−1.68565
$459$	8.87339	0.414175
$460$	−30.2938	−1.41245
$461$	4.09587	0.190764	0.0953818	−	0.995441i	$-0.469593\pi$
$461$	4.09587	0.190764	0.0953818	+	0.995441i	$0.469593\pi$
$462$	2.49939	0.116282
$463$	23.3795	1.08654	0.543269	−	0.839559i	$-0.317186\pi$
$463$	23.3795	1.08654	0.543269	+	0.839559i	$0.317186\pi$
$464$	104.807	4.86556
$465$	4.78685	0.221985
$466$	−5.53371	−0.256344
$467$	−12.5818	−0.582217	−0.291108	−	0.956690i	$-0.594024\pi$
$467$	−12.5818	−0.582217	−0.291108	+	0.956690i	$0.594024\pi$
$468$	0	0
$469$	−8.36628	−0.386319
$470$	36.0349	1.66217
$471$	−13.0045	−0.599214
$472$	−96.6666	−4.44944
$473$	0.786005	0.0361405
$474$	8.50184	0.390502
$475$	8.83017	0.405156
$476$	−44.4631	−2.03796
$477$	0.638159	0.0292193
$478$	39.5699	1.80989
$479$	−22.2197	−1.01524	−0.507621	−	0.861581i	$-0.669475\pi$
$479$	−22.2197	−1.01524	−0.507621	+	0.861581i	$0.669475\pi$
$480$	−76.3219	−3.48360
$481$	0	0
$482$	53.4383	2.43405
$483$	−17.8457	−0.812008
$484$	−56.6564	−2.57529
$485$	−40.6320	−1.84500
$486$	−5.81779	−0.263900
$487$	10.4466	0.473381	0.236691	−	0.971585i	$-0.423937\pi$
$487$	10.4466	0.473381	0.236691	+	0.971585i	$0.423937\pi$
$488$	−46.1415	−2.08873
$489$	3.19044	0.144277
$490$	154.467	6.97810
$491$	30.7514	1.38779	0.693895	−	0.720076i	$-0.255893\pi$
$491$	30.7514	1.38779	0.693895	+	0.720076i	$0.255893\pi$
$492$	−80.0098	−3.60712
$493$	14.1342	0.636571
$494$	0	0
$495$	−0.0644962	−0.00289889
$496$	−12.2722	−0.551039
$497$	−44.2989	−1.98708
$498$	−43.1036	−1.93152
$499$	39.9257	1.78732	0.893659	−	0.448747i	$-0.148129\pi$
$499$	39.9257	1.78732	0.893659	+	0.448747i	$0.148129\pi$
$500$	−26.4195	−1.18152
$501$	33.7399	1.50739
$502$	12.4308	0.554812
$503$	−37.8842	−1.68917	−0.844586	−	0.535420i	$-0.820153\pi$
$503$	−37.8842	−1.68917	−0.844586	+	0.535420i	$0.820153\pi$
$504$	9.22323	0.410835
$505$	−11.4343	−0.508819
$506$	−0.588762	−0.0261737
$507$	0	0
$508$	−23.0814	−1.02407
$509$	−1.38770	−0.0615089	−0.0307544	−	0.999527i	$-0.509791\pi$
$509$	−1.38770	−0.0615089	−0.0307544	+	0.999527i	$0.509791\pi$
$510$	−21.1927	−0.938429
$511$	−36.3781	−1.60927
$512$	−11.5468	−0.510299
$513$	14.7399	0.650782
$514$	−25.5207	−1.12567
$515$	−20.7745	−0.915434
$516$	−63.0638	−2.77623
$517$	0.504605	0.0221925
$518$	−99.7397	−4.38231
$519$	−4.81016	−0.211143
$520$	0	0
$521$	17.5495	0.768857	0.384428	−	0.923155i	$-0.374399\pi$
$521$	17.5495	0.768857	0.384428	+	0.923155i	$0.374399\pi$
$522$	−4.78994	−0.209650
$523$	−14.5271	−0.635224	−0.317612	−	0.948221i	$-0.602881\pi$
$523$	−14.5271	−0.635224	−0.317612	+	0.948221i	$0.602881\pi$
$524$	43.5478	1.90239
$525$	27.9561	1.22011
$526$	−38.7695	−1.69043
$527$	−1.65501	−0.0720936
$528$	−2.20058	−0.0957678
$529$	−18.7962	−0.817227
$530$	−23.3323	−1.01349
$531$	2.40068	0.104180
$532$	−73.8592	−3.20220
$533$	0	0
$534$	−43.3465	−1.87579
$535$	7.35112	0.317817
$536$	13.5555	0.585509
$537$	−11.8614	−0.511857
$538$	73.2471	3.15791
$539$	2.16304	0.0931685
$540$	79.2176	3.40898
$541$	29.8000	1.28120	0.640600	−	0.767875i	$-0.278686\pi$
$541$	29.8000	1.28120	0.640600	+	0.767875i	$0.278686\pi$
$542$	35.6422	1.53096
$543$	−21.3364	−0.915632
$544$	26.3877	1.13136
$545$	−18.6295	−0.798001
$546$	0	0
$547$	−22.7158	−0.971256	−0.485628	−	0.874166i	$-0.661409\pi$
$547$	−22.7158	−0.971256	−0.485628	+	0.874166i	$0.661409\pi$
$548$	25.4415	1.08681
$549$	1.14591	0.0489060
$550$	0.922322	0.0393279
$551$	23.4788	1.00023
$552$	28.9146	1.23069
$553$	9.91371	0.421574
$554$	80.7728	3.43171
$555$	−34.2529	−1.45395
$556$	−50.5683	−2.14457
$557$	−12.5422	−0.531428	−0.265714	−	0.964052i	$-0.585608\pi$
$557$	−12.5422	−0.531428	−0.265714	+	0.964052i	$0.585608\pi$
$558$	0.560870	0.0237435
$559$	0	0
$560$	−183.245	−7.74350
$561$	−0.296767	−0.0125295
$562$	−23.6719	−0.998538
$563$	−2.74698	−0.115771	−0.0578857	−	0.998323i	$-0.518436\pi$
$563$	−2.74698	−0.115771	−0.0578857	+	0.998323i	$0.518436\pi$
$564$	−40.4861	−1.70477
$565$	17.7458	0.746571
$566$	8.88502	0.373465
$567$	43.3887	1.82216
$568$	71.7756	3.01164
$569$	5.85158	0.245311	0.122655	−	0.992449i	$-0.460859\pi$
$569$	5.85158	0.245311	0.122655	+	0.992449i	$0.460859\pi$
$570$	−35.2039	−1.47453
$571$	−7.17387	−0.300217	−0.150109	−	0.988670i	$-0.547962\pi$
$571$	−7.17387	−0.300217	−0.150109	+	0.988670i	$0.547962\pi$
$572$	0	0
$573$	25.3874	1.06058
$574$	−129.486	−5.40466
$575$	−6.58540	−0.274630
$576$	−3.79642	−0.158184
$577$	−11.4043	−0.474768	−0.237384	−	0.971416i	$-0.576290\pi$
$577$	−11.4043	−0.474768	−0.237384	+	0.971416i	$0.576290\pi$
$578$	−38.1489	−1.58679
$579$	3.22634	0.134082
$580$	126.183	5.23948
$581$	−50.2617	−2.08521
$582$	63.3590	2.62632
$583$	−0.326728	−0.0135317
$584$	58.9419	2.43903
$585$	0	0
$586$	−66.5877	−2.75071
$587$	26.0267	1.07424	0.537119	−	0.843507i	$-0.319513\pi$
$587$	26.0267	1.07424	0.537119	+	0.843507i	$0.319513\pi$
$588$	−173.547	−7.15698
$589$	−2.74920	−0.113279
$590$	−87.7734	−3.61358
$591$	22.7229	0.934696
$592$	87.8153	3.60919
$593$	−32.6622	−1.34128	−0.670638	−	0.741785i	$-0.733980\pi$
$593$	−32.6622	−1.34128	−0.670638	+	0.741785i	$0.733980\pi$
$594$	1.53960	0.0631705
$595$	−24.7121	−1.01310
$596$	6.39396	0.261907
$597$	18.9209	0.774379
$598$	0	0
$599$	−39.3376	−1.60729	−0.803646	−	0.595108i	$-0.797109\pi$
$599$	−39.3376	−1.60729	−0.803646	+	0.595108i	$0.797109\pi$
$600$	−45.2961	−1.84920
$601$	−41.5732	−1.69580	−0.847902	−	0.530153i	$-0.822135\pi$
$601$	−41.5732	−1.69580	−0.847902	+	0.530153i	$0.822135\pi$
$602$	−102.061	−4.15971
$603$	−0.336646	−0.0137093
$604$	−23.3313	−0.949339
$605$	−31.4890	−1.28021
$606$	17.8299	0.724291
$607$	8.85069	0.359238	0.179619	−	0.983736i	$-0.442513\pi$
$607$	8.85069	0.359238	0.179619	+	0.983736i	$0.442513\pi$
$608$	43.8334	1.77768
$609$	74.3332	3.01213
$610$	−41.8965	−1.69634
$611$	0	0
$612$	−1.78913	−0.0723211
$613$	7.46986	0.301705	0.150852	−	0.988556i	$-0.451798\pi$
$613$	7.46986	0.301705	0.150852	+	0.988556i	$0.451798\pi$
$614$	−11.9815	−0.483535
$615$	−44.4686	−1.79315
$616$	−4.72216	−0.190261
$617$	−29.5623	−1.19013	−0.595067	−	0.803676i	$-0.702875\pi$
$617$	−29.5623	−1.19013	−0.595067	+	0.803676i	$0.702875\pi$
$618$	32.3945	1.30310
$619$	32.6410	1.31195	0.655976	−	0.754782i	$-0.272257\pi$
$619$	32.6410	1.31195	0.655976	+	0.754782i	$0.272257\pi$
$620$	−14.7752	−0.593387
$621$	−10.9928	−0.441125
$622$	−66.6038	−2.67057
$623$	−50.5449	−2.02504
$624$	0	0
$625$	−30.7432	−1.22973
$626$	56.3017	2.25027
$627$	−0.492969	−0.0196873
$628$	40.1399	1.60176
$629$	11.8426	0.472197
$630$	8.37472	0.333657
$631$	−1.13759	−0.0452869	−0.0226435	−	0.999744i	$-0.507208\pi$
$631$	−1.13759	−0.0452869	−0.0226435	+	0.999744i	$0.507208\pi$
$632$	−16.0628	−0.638942
$633$	16.6393	0.661352
$634$	24.0513	0.955201
$635$	−12.8284	−0.509079
$636$	26.2145	1.03947
$637$	0	0
$638$	2.45238	0.0970908
$639$	−1.78252	−0.0705154
$640$	47.4245	1.87462
$641$	14.9135	0.589049	0.294525	−	0.955644i	$-0.404839\pi$
$641$	14.9135	0.589049	0.294525	+	0.955644i	$0.404839\pi$
$642$	−11.4629	−0.452405
$643$	3.03312	0.119615	0.0598073	−	0.998210i	$-0.480951\pi$
$643$	3.03312	0.119615	0.0598073	+	0.998210i	$0.480951\pi$
$644$	55.0830	2.17057
$645$	−35.0502	−1.38010
$646$	12.1715	0.478880
$647$	−28.4704	−1.11929	−0.559643	−	0.828734i	$-0.689062\pi$
$647$	−28.4704	−1.11929	−0.559643	+	0.828734i	$0.689062\pi$
$648$	−70.3009	−2.76168
$649$	−1.22911	−0.0482469
$650$	0	0
$651$	−8.70391	−0.341133
$652$	−9.84769	−0.385665
$653$	21.9746	0.859931	0.429966	−	0.902845i	$-0.358526\pi$
$653$	21.9746	0.859931	0.429966	+	0.902845i	$0.358526\pi$
$654$	29.0498	1.13594
$655$	24.2034	0.945704
$656$	114.006	4.45118
$657$	−1.46380	−0.0571082
$658$	−65.5220	−2.55432
$659$	42.4003	1.65168	0.825841	−	0.563903i	$-0.190701\pi$
$659$	42.4003	1.65168	0.825841	+	0.563903i	$0.190701\pi$
$660$	−2.64940	−0.103128
$661$	35.4765	1.37988	0.689939	−	0.723868i	$-0.257637\pi$
$661$	35.4765	1.37988	0.689939	+	0.723868i	$0.257637\pi$
$662$	25.0625	0.974083
$663$	0	0
$664$	81.4368	3.16036
$665$	−41.0501	−1.59186
$666$	−4.01337	−0.155515
$667$	−17.5101	−0.677993
$668$	−104.142	−4.02939
$669$	29.9011	1.15604
$670$	12.3084	0.475516
$671$	−0.586687	−0.0226488
$672$	138.776	5.35339
$673$	−45.4861	−1.75336	−0.876680	−	0.481074i	$-0.840247\pi$
$673$	−45.4861	−1.75336	−0.876680	+	0.481074i	$0.840247\pi$
$674$	54.4309	2.09660
$675$	17.2207	0.662825
$676$	0	0
$677$	−11.7990	−0.453471	−0.226735	−	0.973956i	$-0.572805\pi$
$677$	−11.7990	−0.453471	−0.226735	+	0.973956i	$0.572805\pi$
$678$	−27.6717	−1.06273
$679$	73.8809	2.83529
$680$	40.0400	1.53546
$681$	2.88624	0.110601
$682$	−0.287157	−0.0109958
$683$	23.9204	0.915288	0.457644	−	0.889135i	$-0.348693\pi$
$683$	23.9204	0.915288	0.457644	+	0.889135i	$0.348693\pi$
$684$	−2.97198	−0.113636
$685$	14.1401	0.540266
$686$	−183.296	−6.99826
$687$	22.5264	0.859437
$688$	89.8593	3.42585
$689$	0	0
$690$	26.2545	0.999493
$691$	−17.3715	−0.660842	−0.330421	−	0.943834i	$-0.607191\pi$
$691$	−17.3715	−0.660842	−0.330421	+	0.943834i	$0.607191\pi$
$692$	14.8472	0.564405
$693$	0.117273	0.00445484
$694$	−32.5251	−1.23464
$695$	−28.1053	−1.06610
$696$	−120.439	−4.56522
$697$	15.3747	0.582357
$698$	−82.8068	−3.13428
$699$	3.45549	0.130699
$700$	−86.2900	−3.26146
$701$	−32.9874	−1.24592	−0.622958	−	0.782256i	$-0.714069\pi$
$701$	−32.9874	−1.24592	−0.622958	+	0.782256i	$0.714069\pi$
$702$	0	0
$703$	19.6722	0.741952
$704$	1.94371	0.0732564
$705$	−22.5018	−0.847465
$706$	15.9059	0.598626
$707$	20.7909	0.781921
$708$	98.6158	3.70621
$709$	21.3761	0.802797	0.401398	−	0.915904i	$-0.368524\pi$
$709$	21.3761	0.802797	0.401398	+	0.915904i	$0.368524\pi$
$710$	65.1724	2.44588
$711$	0.398912	0.0149604
$712$	81.8957	3.06917
$713$	2.05031	0.0767847
$714$	38.5346	1.44212
$715$	0	0
$716$	36.6116	1.36824
$717$	−24.7092	−0.922782
$718$	79.0764	2.95110
$719$	−42.6299	−1.58983	−0.794913	−	0.606723i	$-0.792484\pi$
$719$	−42.6299	−1.58983	−0.794913	+	0.606723i	$0.792484\pi$
$720$	−7.37348	−0.274793
$721$	37.7742	1.40678
$722$	−30.6078	−1.13911
$723$	−33.3692	−1.24101
$724$	65.8574	2.44757
$725$	27.4303	1.01874
$726$	49.1021	1.82235
$727$	−38.9647	−1.44512	−0.722561	−	0.691308i	$-0.757035\pi$
$727$	−38.9647	−1.44512	−0.722561	+	0.691308i	$0.757035\pi$
$728$	0	0
$729$	28.6140	1.05978
$730$	53.5194	1.98084
$731$	12.1183	0.448212
$732$	47.0719	1.73983
$733$	15.6365	0.577548	0.288774	−	0.957397i	$-0.406752\pi$
$733$	15.6365	0.577548	0.288774	+	0.957397i	$0.406752\pi$
$734$	69.9904	2.58339
$735$	−96.4558	−3.55783
$736$	−32.6903	−1.20498
$737$	0.172358	0.00634889
$738$	−5.21033	−0.191795
$739$	−3.68292	−0.135478	−0.0677392	−	0.997703i	$-0.521579\pi$
$739$	−3.68292	−0.135478	−0.0677392	+	0.997703i	$0.521579\pi$
$740$	105.726	3.88655
$741$	0	0
$742$	42.4251	1.55747
$743$	−50.2368	−1.84301	−0.921505	−	0.388367i	$-0.873039\pi$
$743$	−50.2368	−1.84301	−0.921505	+	0.388367i	$0.873039\pi$
$744$	14.1026	0.517025
$745$	3.55370	0.130197
$746$	23.2724	0.852065
$747$	−2.02245	−0.0739977
$748$	0.916007	0.0334925
$749$	−13.3665	−0.488402
$750$	22.8968	0.836074
$751$	5.96827	0.217785	0.108893	−	0.994054i	$-0.465270\pi$
$751$	5.96827	0.217785	0.108893	+	0.994054i	$0.465270\pi$
$752$	57.6885	2.10368
$753$	−7.76232	−0.282875
$754$	0	0
$755$	−12.9673	−0.471929
$756$	−144.041	−5.23872
$757$	−37.6489	−1.36837	−0.684187	−	0.729307i	$-0.739843\pi$
$757$	−37.6489	−1.36837	−0.684187	+	0.729307i	$0.739843\pi$
$758$	59.6124	2.16522
$759$	0.367649	0.0133448
$760$	66.5117	2.41263
$761$	−22.9366	−0.831451	−0.415726	−	0.909490i	$-0.636472\pi$
$761$	−22.9366	−0.831451	−0.415726	+	0.909490i	$0.636472\pi$
$762$	20.0038	0.724662
$763$	33.8739	1.22632
$764$	−78.3615	−2.83502
$765$	−0.994377	−0.0359518
$766$	−58.5702	−2.11623
$767$	0	0
$768$	−13.4580	−0.485623
$769$	−24.0231	−0.866295	−0.433148	−	0.901323i	$-0.642597\pi$
$769$	−24.0231	−0.866295	−0.433148	+	0.901323i	$0.642597\pi$
$770$	−4.28774	−0.154519
$771$	15.9362	0.573930
$772$	−9.95850	−0.358414
$773$	−24.5940	−0.884585	−0.442292	−	0.896871i	$-0.645835\pi$
$773$	−24.5940	−0.884585	−0.442292	+	0.896871i	$0.645835\pi$
$774$	−4.10679	−0.147615
$775$	−3.21190	−0.115375
$776$	−119.706	−4.29719
$777$	62.2818	2.23435
$778$	−49.9722	−1.79159
$779$	25.5394	0.915043
$780$	0	0
$781$	0.912625	0.0326563
$782$	−9.07729	−0.324603
$783$	45.7885	1.63635
$784$	247.287	8.83168
$785$	22.3093	0.796254
$786$	−37.7413	−1.34619
$787$	−40.3993	−1.44008	−0.720039	−	0.693934i	$-0.755876\pi$
$787$	−40.3993	−1.44008	−0.720039	+	0.693934i	$0.755876\pi$
$788$	−70.1371	−2.49853
$789$	24.2094	0.861878
$790$	−14.5850	−0.518911
$791$	−32.2671	−1.14728
$792$	−0.190012	−0.00675180
$793$	0	0
$794$	−6.71352	−0.238254
$795$	14.5697	0.516735
$796$	−58.4016	−2.06999
$797$	−40.4038	−1.43118	−0.715589	−	0.698522i	$-0.753841\pi$
$797$	−40.4038	−1.43118	−0.715589	+	0.698522i	$0.753841\pi$
$798$	64.0111	2.26597
$799$	7.77980	0.275229
$800$	51.2108	1.81058
$801$	−2.03385	−0.0718624
$802$	−5.14180	−0.181563
$803$	0.749444	0.0264473
$804$	−13.8289	−0.487706
$805$	30.6145	1.07902
$806$	0	0
$807$	−45.7387	−1.61008
$808$	−33.6865	−1.18509
$809$	−45.5611	−1.60184	−0.800922	−	0.598768i	$-0.795657\pi$
$809$	−45.5611	−1.60184	−0.800922	+	0.598768i	$0.795657\pi$
$810$	−63.8333	−2.24287
$811$	−21.7290	−0.763010	−0.381505	−	0.924367i	$-0.624594\pi$
$811$	−21.7290	−0.763010	−0.381505	+	0.924367i	$0.624594\pi$
$812$	−229.439	−8.05172
$813$	−22.2566	−0.780572
$814$	2.05479	0.0720203
$815$	−5.47324	−0.191719
$816$	−33.9276	−1.18770
$817$	20.1301	0.704264
$818$	84.0215	2.93774
$819$	0	0
$820$	137.258	4.79325
$821$	46.1398	1.61029	0.805146	−	0.593077i	$-0.202087\pi$
$821$	46.1398	1.61029	0.805146	+	0.593077i	$0.202087\pi$
$822$	−22.0492	−0.769055
$823$	31.2220	1.08833	0.544165	−	0.838978i	$-0.316846\pi$
$823$	31.2220	1.08833	0.544165	+	0.838978i	$0.316846\pi$
$824$	−61.2038	−2.13214
$825$	−0.575938	−0.0200516
$826$	159.598	5.55312
$827$	23.3304	0.811276	0.405638	−	0.914034i	$-0.367049\pi$
$827$	23.3304	0.811276	0.405638	+	0.914034i	$0.367049\pi$
$828$	2.21645	0.0770271
$829$	6.95045	0.241399	0.120700	−	0.992689i	$-0.461486\pi$
$829$	6.95045	0.241399	0.120700	+	0.992689i	$0.461486\pi$
$830$	73.9448	2.56666
$831$	−50.4381	−1.74968
$832$	0	0
$833$	33.3488	1.15547
$834$	43.8258	1.51756
$835$	−57.8812	−2.00306
$836$	1.52161	0.0526260
$837$	−5.36152	−0.185321
$838$	82.1038	2.83623
$839$	6.57603	0.227030	0.113515	−	0.993536i	$-0.463789\pi$
$839$	6.57603	0.227030	0.113515	+	0.993536i	$0.463789\pi$
$840$	210.575	7.26552
$841$	43.9352	1.51501
$842$	106.845	3.68214
$843$	14.7818	0.509111
$844$	−51.3592	−1.76785
$845$	0	0
$846$	−2.63650	−0.0906449
$847$	57.2564	1.96735
$848$	−37.3529	−1.28270
$849$	−5.54820	−0.190414
$850$	14.2200	0.487742
$851$	−14.6712	−0.502923
$852$	−73.2229	−2.50858
$853$	46.6898	1.59863	0.799314	−	0.600914i	$-0.205197\pi$
$853$	46.6898	1.59863	0.799314	+	0.600914i	$0.205197\pi$
$854$	76.1802	2.60683
$855$	−1.65179	−0.0564901
$856$	21.6572	0.740227
$857$	−15.1188	−0.516448	−0.258224	−	0.966085i	$-0.583137\pi$
$857$	−15.1188	−0.516448	−0.258224	+	0.966085i	$0.583137\pi$
$858$	0	0
$859$	−14.9342	−0.509547	−0.254774	−	0.967001i	$-0.582001\pi$
$859$	−14.9342	−0.509547	−0.254774	+	0.967001i	$0.582001\pi$
$860$	108.187	3.68913
$861$	80.8570	2.75560
$862$	43.4252	1.47907
$863$	−16.3095	−0.555182	−0.277591	−	0.960699i	$-0.589536\pi$
$863$	−16.3095	−0.555182	−0.277591	+	0.960699i	$0.589536\pi$
$864$	85.4844	2.90824
$865$	8.25190	0.280573
$866$	−83.5458	−2.83900
$867$	23.8219	0.809033
$868$	26.8657	0.911881
$869$	−0.204237	−0.00692828
$870$	−109.359	−3.70761
$871$	0	0
$872$	−54.8845	−1.85862
$873$	2.97285	0.100616
$874$	−15.0786	−0.510041
$875$	26.6992	0.902599
$876$	−60.1304	−2.03162
$877$	7.18224	0.242527	0.121264	−	0.992620i	$-0.461305\pi$
$877$	7.18224	0.242527	0.121264	+	0.992620i	$0.461305\pi$
$878$	−21.6881	−0.731937
$879$	41.5803	1.40247
$880$	3.77512	0.127259
$881$	28.4293	0.957806	0.478903	−	0.877868i	$-0.341035\pi$
$881$	28.4293	0.957806	0.478903	+	0.877868i	$0.341035\pi$
$882$	−11.3016	−0.380545
$883$	4.98962	0.167914	0.0839570	−	0.996469i	$-0.473244\pi$
$883$	4.98962	0.167914	0.0839570	+	0.996469i	$0.473244\pi$
$884$	0	0
$885$	54.8096	1.84240
$886$	−5.08288	−0.170763
$887$	−6.01723	−0.202039	−0.101019	−	0.994884i	$-0.532210\pi$
$887$	−6.01723	−0.202039	−0.101019	+	0.994884i	$0.532210\pi$
$888$	−100.912	−3.38640
$889$	23.3258	0.782322
$890$	74.3614	2.49260
$891$	−0.893873	−0.0299459
$892$	−92.2934	−3.09021
$893$	12.9233	0.432461
$894$	−5.54142	−0.185333
$895$	20.3484	0.680171
$896$	−86.2317	−2.88080
$897$	0	0
$898$	18.9442	0.632177
$899$	−8.54021	−0.284832
$900$	−3.47217	−0.115739
$901$	−5.03737	−0.167819
$902$	2.66762	0.0888219
$903$	63.7315	2.12085
$904$	52.2809	1.73884
$905$	36.6028	1.21672
$906$	20.2204	0.671779
$907$	−7.82026	−0.259667	−0.129834	−	0.991536i	$-0.541444\pi$
$907$	−7.82026	−0.259667	−0.129834	+	0.991536i	$0.541444\pi$
$908$	−8.90875	−0.295647
$909$	0.836592	0.0277480
$910$	0	0
$911$	28.0064	0.927894	0.463947	−	0.885863i	$-0.346433\pi$
$911$	28.0064	0.927894	0.463947	+	0.885863i	$0.346433\pi$
$912$	−56.3583	−1.86621
$913$	1.03547	0.0342689
$914$	10.1842	0.336863
$915$	26.1620	0.864890
$916$	−69.5306	−2.29736
$917$	−44.0089	−1.45330
$918$	23.7369	0.783435
$919$	37.6488	1.24192	0.620960	−	0.783842i	$-0.286743\pi$
$919$	37.6488	1.24192	0.620960	+	0.783842i	$0.286743\pi$
$920$	−49.6034	−1.63538
$921$	7.48179	0.246533
$922$	10.9567	0.360840
$923$	0	0
$924$	4.81738	0.158480
$925$	22.9832	0.755681
$926$	62.5418	2.05525
$927$	1.51997	0.0499225
$928$	136.166	4.46986
$929$	31.2651	1.02577	0.512887	−	0.858456i	$-0.328576\pi$
$929$	31.2651	1.02577	0.512887	+	0.858456i	$0.328576\pi$
$930$	12.8052	0.419898
$931$	55.3968	1.81556
$932$	−10.6658	−0.349370
$933$	41.5903	1.36161
$934$	−33.6572	−1.10130
$935$	0.509107	0.0166496
$936$	0	0
$937$	−11.2719	−0.368238	−0.184119	−	0.982904i	$-0.558943\pi$
$937$	−11.2719	−0.368238	−0.184119	+	0.982904i	$0.558943\pi$
$938$	−22.3804	−0.730745
$939$	−35.1573	−1.14731
$940$	69.4545	2.26535
$941$	−9.68665	−0.315776	−0.157888	−	0.987457i	$-0.550468\pi$
$941$	−9.68665	−0.315776	−0.157888	+	0.987457i	$0.550468\pi$
$942$	−34.7878	−1.13345
$943$	−19.0469	−0.620251
$944$	−140.517	−4.57345
$945$	−80.0564	−2.60423
$946$	2.10262	0.0683619
$947$	9.22048	0.299625	0.149813	−	0.988714i	$-0.452133\pi$
$947$	9.22048	0.299625	0.149813	+	0.988714i	$0.452133\pi$
$948$	16.3866	0.532213
$949$	0	0
$950$	23.6213	0.766376
$951$	−15.0187	−0.487015
$952$	−72.8044	−2.35961
$953$	−17.9528	−0.581548	−0.290774	−	0.956792i	$-0.593913\pi$
$953$	−17.9528	−0.581548	−0.290774	+	0.956792i	$0.593913\pi$
$954$	1.70712	0.0552700
$955$	−43.5525	−1.40933
$956$	76.2680	2.46668
$957$	−1.53138	−0.0495023
$958$	−59.4391	−1.92039
$959$	−25.7109	−0.830248
$960$	−86.6756	−2.79744
$961$	1.00000	0.0322581
$962$	0	0
$963$	−0.537847	−0.0173319
$964$	102.998	3.31735
$965$	−5.53483	−0.178172
$966$	−47.7385	−1.53596
$967$	−14.7975	−0.475855	−0.237927	−	0.971283i	$-0.576468\pi$
$967$	−14.7975	−0.475855	−0.237927	+	0.971283i	$0.576468\pi$
$968$	−92.7699	−2.98174
$969$	−7.60040	−0.244160
$970$	−108.693	−3.48993
$971$	−24.8066	−0.796082	−0.398041	−	0.917368i	$-0.630310\pi$
$971$	−24.8066	−0.796082	−0.398041	+	0.917368i	$0.630310\pi$
$972$	−11.2133	−0.359668
$973$	51.1037	1.63831
$974$	27.9454	0.895428
$975$	0	0
$976$	−67.0725	−2.14694
$977$	−10.7784	−0.344831	−0.172415	−	0.985024i	$-0.555157\pi$
$977$	−10.7784	−0.344831	−0.172415	+	0.985024i	$0.555157\pi$
$978$	8.53464	0.272908
$979$	1.04130	0.0332801
$980$	297.723	9.51041
$981$	1.36303	0.0435183
$982$	82.2620	2.62509
$983$	3.27003	0.104298	0.0521490	−	0.998639i	$-0.483393\pi$
$983$	3.27003	0.104298	0.0521490	+	0.998639i	$0.483393\pi$
$984$	−131.009	−4.17642
$985$	−38.9814	−1.24205
$986$	37.8099	1.20411
$987$	40.9148	1.30233
$988$	0	0
$989$	−15.0127	−0.477377
$990$	−0.172532	−0.00548342
$991$	14.6836	0.466440	0.233220	−	0.972424i	$-0.425074\pi$
$991$	14.6836	0.466440	0.233220	+	0.972424i	$0.425074\pi$
$992$	−15.9441	−0.506225
$993$	−15.6501	−0.496642
$994$	−118.503	−3.75868
$995$	−32.4590	−1.02902
$996$	−83.0789	−2.63246
$997$	−37.2324	−1.17916	−0.589580	−	0.807710i	$-0.700707\pi$
$997$	−37.2324	−1.17916	−0.589580	+	0.807710i	$0.700707\pi$
$998$	106.804	3.38082
$999$	38.3650	1.21381

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5239.2.a.l.1.15		16
13.5	odd	4		403.2.c.b.311.2	✓	32
13.8	odd	4		403.2.c.b.311.31	yes	32
13.12	even	2		5239.2.a.k.1.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.c.b.311.2	✓	32	13.5	odd	4
403.2.c.b.311.31	yes	32	13.8	odd	4
5239.2.a.k.1.2		16	13.12	even	2
5239.2.a.l.1.15		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)

\(f(q)\)	\(=\)	\(q+2.67507 q^{2} -1.67043 q^{3} +5.15599 q^{4} +2.86564 q^{5} -4.46851 q^{6} -5.21058 q^{7} +8.44248 q^{8} -0.209666 q^{9} +O(q^{10})\)	\(q+2.67507 q^{2} -1.67043 q^{3} +5.15599 q^{4} +2.86564 q^{5} -4.46851 q^{6} -5.21058 q^{7} +8.44248 q^{8} -0.209666 q^{9} +7.66579 q^{10} +0.107346 q^{11} -8.61271 q^{12} -13.9387 q^{14} -4.78685 q^{15} +12.2722 q^{16} +1.65501 q^{17} -0.560870 q^{18} +2.74920 q^{19} +14.7752 q^{20} +8.70391 q^{21} +0.287157 q^{22} -2.05031 q^{23} -14.1026 q^{24} +3.21190 q^{25} +5.36152 q^{27} -26.8657 q^{28} +8.54021 q^{29} -12.8052 q^{30} -1.00000 q^{31} +15.9441 q^{32} -0.179314 q^{33} +4.42728 q^{34} -14.9317 q^{35} -1.08103 q^{36} +7.15561 q^{37} +7.35430 q^{38} +24.1931 q^{40} +9.28974 q^{41} +23.2835 q^{42} +7.32217 q^{43} +0.553474 q^{44} -0.600827 q^{45} -5.48472 q^{46} +4.70074 q^{47} -20.4999 q^{48} +20.1502 q^{49} +8.59206 q^{50} -2.76459 q^{51} -3.04370 q^{53} +14.3424 q^{54} +0.307615 q^{55} -43.9902 q^{56} -4.59235 q^{57} +22.8456 q^{58} -11.4500 q^{59} -24.6809 q^{60} -5.46539 q^{61} -2.67507 q^{62} +1.09248 q^{63} +18.1070 q^{64} -0.479676 q^{66} +1.60563 q^{67} +8.53323 q^{68} +3.42490 q^{69} -39.9432 q^{70} +8.50173 q^{71} -1.77010 q^{72} +6.98159 q^{73} +19.1418 q^{74} -5.36526 q^{75} +14.1748 q^{76} -0.559334 q^{77} -1.90261 q^{79} +35.1678 q^{80} -8.32704 q^{81} +24.8507 q^{82} +9.64608 q^{83} +44.8772 q^{84} +4.74268 q^{85} +19.5873 q^{86} -14.2658 q^{87} +0.906264 q^{88} +9.70043 q^{89} -1.60725 q^{90} -10.5714 q^{92} +1.67043 q^{93} +12.5748 q^{94} +7.87823 q^{95} -26.6334 q^{96} -14.1790 q^{97} +53.9030 q^{98} -0.0225067 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 4 q^{2} - 2 q^{3} + 18 q^{4} + 4 q^{5} + 6 q^{6} + 2 q^{7} + 12 q^{8} + 10 q^{9}+O(q^{10}) \) 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

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Representations

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Database

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