LMFDB - Embedded newform 4031.2.a.b.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4031,2,Mod(1,4031)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4031, 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("4031.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4031 = 29 \cdot 139 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4031.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:	$32.1876970548$
Analytic rank:	$1$
Dimension:	$59$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Character	$\chi$	$=$	4031.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.24849 q^{2} +2.23227 q^{3} +3.05570 q^{4} -0.764282 q^{5} -5.01923 q^{6} -0.688314 q^{7} -2.37374 q^{8} +1.98301 q^{9} +O(q^{10})$	\(q-2.24849 q^{2} +2.23227 q^{3} +3.05570 q^{4} -0.764282 q^{5} -5.01923 q^{6} -0.688314 q^{7} -2.37374 q^{8} +1.98301 q^{9} +1.71848 q^{10} -1.50474 q^{11} +6.82114 q^{12} +1.32649 q^{13} +1.54767 q^{14} -1.70608 q^{15} -0.774080 q^{16} +6.99636 q^{17} -4.45878 q^{18} +3.39101 q^{19} -2.33542 q^{20} -1.53650 q^{21} +3.38340 q^{22} -6.42318 q^{23} -5.29882 q^{24} -4.41587 q^{25} -2.98261 q^{26} -2.27019 q^{27} -2.10328 q^{28} +1.00000 q^{29} +3.83610 q^{30} -5.80868 q^{31} +6.48799 q^{32} -3.35899 q^{33} -15.7312 q^{34} +0.526066 q^{35} +6.05949 q^{36} -0.624545 q^{37} -7.62464 q^{38} +2.96108 q^{39} +1.81421 q^{40} -8.39333 q^{41} +3.45480 q^{42} -9.10320 q^{43} -4.59805 q^{44} -1.51558 q^{45} +14.4425 q^{46} +3.22864 q^{47} -1.72795 q^{48} -6.52622 q^{49} +9.92904 q^{50} +15.6177 q^{51} +4.05337 q^{52} -7.09746 q^{53} +5.10451 q^{54} +1.15005 q^{55} +1.63388 q^{56} +7.56962 q^{57} -2.24849 q^{58} +3.48573 q^{59} -5.21328 q^{60} +12.7840 q^{61} +13.0608 q^{62} -1.36493 q^{63} -13.0400 q^{64} -1.01381 q^{65} +7.55265 q^{66} -2.31159 q^{67} +21.3788 q^{68} -14.3382 q^{69} -1.18285 q^{70} -1.08481 q^{71} -4.70715 q^{72} +4.96195 q^{73} +1.40428 q^{74} -9.85740 q^{75} +10.3619 q^{76} +1.03574 q^{77} -6.65797 q^{78} -2.03003 q^{79} +0.591615 q^{80} -11.0167 q^{81} +18.8723 q^{82} -0.991262 q^{83} -4.69509 q^{84} -5.34719 q^{85} +20.4685 q^{86} +2.23227 q^{87} +3.57187 q^{88} +7.13215 q^{89} +3.40776 q^{90} -0.913043 q^{91} -19.6273 q^{92} -12.9665 q^{93} -7.25957 q^{94} -2.59168 q^{95} +14.4829 q^{96} -12.6184 q^{97} +14.6741 q^{98} -2.98392 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 59 q - 5 q^{2} - 6 q^{3} + 41 q^{4} - 5 q^{5} - 7 q^{6} - 10 q^{7} - 12 q^{8} + 23 q^{9}+O(q^{10}) $ 59 * q - 5 * q^2 - 6 * q^3 + 41 * q^4 - 5 * q^5 - 7 * q^6 - 10 * q^7 - 12 * q^8 + 23 * q^9	\( 59 q - 5 q^{2} - 6 q^{3} + 41 q^{4} - 5 q^{5} - 7 q^{6} - 10 q^{7} - 12 q^{8} + 23 q^{9} - 18 q^{10} - 27 q^{11} - 8 q^{12} - 22 q^{13} - 24 q^{14} - 18 q^{15} + 5 q^{16} - 23 q^{17} + q^{18} - 32 q^{19} - 14 q^{20} - 36 q^{21} - 6 q^{22} - 3 q^{23} - 18 q^{24} - 8 q^{25} - q^{26} - 12 q^{27} - 9 q^{28} + 59 q^{29} - 18 q^{30} - 32 q^{31} - 39 q^{32} - 12 q^{33} - 18 q^{34} - 9 q^{35} + 10 q^{36} - 44 q^{37} + 5 q^{38} - 27 q^{39} - 68 q^{40} - 44 q^{41} - 25 q^{42} - 40 q^{43} - 56 q^{44} - 39 q^{45} - 40 q^{46} - 20 q^{47} - 9 q^{48} - 39 q^{49} - 21 q^{50} - 28 q^{51} - 49 q^{52} - 31 q^{53} - 32 q^{54} - 32 q^{55} - 48 q^{56} - 58 q^{57} - 5 q^{58} + 6 q^{59} - 44 q^{60} - 88 q^{61} + 35 q^{62} - 22 q^{63} - 10 q^{64} - 43 q^{65} - 31 q^{66} - 45 q^{67} - 29 q^{68} - 60 q^{69} - 14 q^{70} - 20 q^{71} - 4 q^{72} - 90 q^{73} - 25 q^{74} + 15 q^{75} - 64 q^{76} - 39 q^{77} - 28 q^{78} - 120 q^{79} + 24 q^{80} - 77 q^{81} - 71 q^{82} - 33 q^{83} - 14 q^{84} - 71 q^{85} - 61 q^{86} - 6 q^{87} - 34 q^{88} - 78 q^{89} - 88 q^{90} - 28 q^{91} - 31 q^{92} - 36 q^{93} - 4 q^{94} - 12 q^{95} - 29 q^{96} - 48 q^{97} - 4 q^{98} - 43 q^{99}+O(q^{100}) \) 59 * q - 5 * q^2 - 6 * q^3 + 41 * q^4 - 5 * q^5 - 7 * q^6 - 10 * q^7 - 12 * q^8 + 23 * q^9 - 18 * q^10 - 27 * q^11 - 8 * q^12 - 22 * q^13 - 24 * q^14 - 18 * q^15 + 5 * q^16 - 23 * q^17 + q^18 - 32 * q^19 - 14 * q^20 - 36 * q^21 - 6 * q^22 - 3 * q^23 - 18 * q^24 - 8 * q^25 - q^26 - 12 * q^27 - 9 * q^28 + 59 * q^29 - 18 * q^30 - 32 * q^31 - 39 * q^32 - 12 * q^33 - 18 * q^34 - 9 * q^35 + 10 * q^36 - 44 * q^37 + 5 * q^38 - 27 * q^39 - 68 * q^40 - 44 * q^41 - 25 * q^42 - 40 * q^43 - 56 * q^44 - 39 * q^45 - 40 * q^46 - 20 * q^47 - 9 * q^48 - 39 * q^49 - 21 * q^50 - 28 * q^51 - 49 * q^52 - 31 * q^53 - 32 * q^54 - 32 * q^55 - 48 * q^56 - 58 * q^57 - 5 * q^58 + 6 * q^59 - 44 * q^60 - 88 * q^61 + 35 * q^62 - 22 * q^63 - 10 * q^64 - 43 * q^65 - 31 * q^66 - 45 * q^67 - 29 * q^68 - 60 * q^69 - 14 * q^70 - 20 * q^71 - 4 * q^72 - 90 * q^73 - 25 * q^74 + 15 * q^75 - 64 * q^76 - 39 * q^77 - 28 * q^78 - 120 * q^79 + 24 * q^80 - 77 * q^81 - 71 * q^82 - 33 * q^83 - 14 * q^84 - 71 * q^85 - 61 * q^86 - 6 * q^87 - 34 * q^88 - 78 * q^89 - 88 * q^90 - 28 * q^91 - 31 * q^92 - 36 * q^93 - 4 * q^94 - 12 * q^95 - 29 * q^96 - 48 * q^97 - 4 * q^98 - 43 * 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 $

$2$	−2.24849	−1.58992	−0.794961	−	0.606660i	$-0.792509\pi$
$2$	−2.24849	−1.58992	−0.794961	+	0.606660i	$0.792509\pi$
$3$	2.23227	1.28880	0.644400	−	0.764689i	$-0.277107\pi$
$3$	2.23227	1.28880	0.644400	+	0.764689i	$0.277107\pi$
$4$	3.05570	1.52785
$5$	−0.764282	−0.341797	−0.170899	−	0.985289i	$-0.554667\pi$
$5$	−0.764282	−0.341797	−0.170899	+	0.985289i	$0.554667\pi$
$6$	−5.01923	−2.04909
$7$	−0.688314	−0.260158	−0.130079	−	0.991504i	$-0.541523\pi$
$7$	−0.688314	−0.260158	−0.130079	+	0.991504i	$0.541523\pi$
$8$	−2.37374	−0.839244
$9$	1.98301	0.661003
$10$	1.71848	0.543431
$11$	−1.50474	−0.453697	−0.226849	−	0.973930i	$-0.572842\pi$
$11$	−1.50474	−0.453697	−0.226849	+	0.973930i	$0.572842\pi$
$12$	6.82114	1.96909
$13$	1.32649	0.367903	0.183951	−	0.982935i	$-0.441111\pi$
$13$	1.32649	0.367903	0.183951	+	0.982935i	$0.441111\pi$
$14$	1.54767	0.413631
$15$	−1.70608	−0.440508
$16$	−0.774080	−0.193520
$17$	6.99636	1.69687	0.848433	−	0.529303i	$-0.177546\pi$
$17$	6.99636	1.69687	0.848433	+	0.529303i	$0.177546\pi$
$18$	−4.45878	−1.05094
$19$	3.39101	0.777950	0.388975	−	0.921248i	$-0.372829\pi$
$19$	3.39101	0.777950	0.388975	+	0.921248i	$0.372829\pi$
$20$	−2.33542	−0.522216
$21$	−1.53650	−0.335292
$22$	3.38340	0.721343
$23$	−6.42318	−1.33933	−0.669663	−	0.742665i	$-0.733561\pi$
$23$	−6.42318	−1.33933	−0.669663	+	0.742665i	$0.733561\pi$
$24$	−5.29882	−1.08162
$25$	−4.41587	−0.883175
$26$	−2.98261	−0.584937
$27$	−2.27019	−0.436899
$28$	−2.10328	−0.397483
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	3.83610	0.700373
$31$	−5.80868	−1.04327	−0.521635	−	0.853169i	$-0.674678\pi$
$31$	−5.80868	−1.04327	−0.521635	+	0.853169i	$0.674678\pi$
$32$	6.48799	1.14693
$33$	−3.35899	−0.584725
$34$	−15.7312	−2.69788
$35$	0.526066	0.0889213
$36$	6.05949	1.00991
$37$	−0.624545	−0.102675	−0.0513373	−	0.998681i	$-0.516348\pi$
$37$	−0.624545	−0.102675	−0.0513373	+	0.998681i	$0.516348\pi$
$38$	−7.62464	−1.23688
$39$	2.96108	0.474153
$40$	1.81421	0.286851
$41$	−8.39333	−1.31082	−0.655409	−	0.755274i	$-0.727504\pi$
$41$	−8.39333	−1.31082	−0.655409	+	0.755274i	$0.727504\pi$
$42$	3.45480	0.533087
$43$	−9.10320	−1.38823	−0.694113	−	0.719866i	$-0.744203\pi$
$43$	−9.10320	−1.38823	−0.694113	+	0.719866i	$0.744203\pi$
$44$	−4.59805	−0.693183
$45$	−1.51558	−0.225929
$46$	14.4425	2.12942
$47$	3.22864	0.470946	0.235473	−	0.971881i	$-0.424336\pi$
$47$	3.22864	0.470946	0.235473	+	0.971881i	$0.424336\pi$
$48$	−1.72795	−0.249408
$49$	−6.52622	−0.932318
$50$	9.92904	1.40418
$51$	15.6177	2.18692
$52$	4.05337	0.562101
$53$	−7.09746	−0.974911	−0.487455	−	0.873148i	$-0.662075\pi$
$53$	−7.09746	−0.974911	−0.487455	+	0.873148i	$0.662075\pi$
$54$	5.10451	0.694635
$55$	1.15005	0.155072
$56$	1.63388	0.218336
$57$	7.56962	1.00262
$58$	−2.24849	−0.295241
$59$	3.48573	0.453803	0.226902	−	0.973918i	$-0.427140\pi$
$59$	3.48573	0.453803	0.226902	+	0.973918i	$0.427140\pi$
$60$	−5.21328	−0.673031
$61$	12.7840	1.63682	0.818411	−	0.574633i	$-0.194855\pi$
$61$	12.7840	1.63682	0.818411	+	0.574633i	$0.194855\pi$
$62$	13.0608	1.65872
$63$	−1.36493	−0.171965
$64$	−13.0400	−1.63000
$65$	−1.01381	−0.125748
$66$	7.55265	0.929667
$67$	−2.31159	−0.282406	−0.141203	−	0.989981i	$-0.545097\pi$
$67$	−2.31159	−0.282406	−0.141203	+	0.989981i	$0.545097\pi$
$68$	21.3788	2.59256
$69$	−14.3382	−1.72612
$70$	−1.18285	−0.141378
$71$	−1.08481	−0.128743	−0.0643715	−	0.997926i	$-0.520504\pi$
$71$	−1.08481	−0.128743	−0.0643715	+	0.997926i	$0.520504\pi$
$72$	−4.70715	−0.554743
$73$	4.96195	0.580753	0.290376	−	0.956913i	$-0.406220\pi$
$73$	4.96195	0.580753	0.290376	+	0.956913i	$0.406220\pi$
$74$	1.40428	0.163245
$75$	−9.85740	−1.13823
$76$	10.3619	1.18859
$77$	1.03574	0.118033
$78$	−6.65797	−0.753866
$79$	−2.03003	−0.228396	−0.114198	−	0.993458i	$-0.536430\pi$
$79$	−2.03003	−0.228396	−0.114198	+	0.993458i	$0.536430\pi$
$80$	0.591615	0.0661446
$81$	−11.0167	−1.22408
$82$	18.8723	2.08410
$83$	−0.991262	−0.108805	−0.0544025	−	0.998519i	$-0.517325\pi$
$83$	−0.991262	−0.108805	−0.0544025	+	0.998519i	$0.517325\pi$
$84$	−4.69509	−0.512276
$85$	−5.34719	−0.579984
$86$	20.4685	2.20717
$87$	2.23227	0.239324
$88$	3.57187	0.380763
$89$	7.13215	0.756006	0.378003	−	0.925804i	$-0.376611\pi$
$89$	7.13215	0.756006	0.378003	+	0.925804i	$0.376611\pi$
$90$	3.40776	0.359210
$91$	−0.913043	−0.0957129
$92$	−19.6273	−2.04629
$93$	−12.9665	−1.34457
$94$	−7.25957	−0.748768
$95$	−2.59168	−0.265901
$96$	14.4829	1.47816
$97$	−12.6184	−1.28120	−0.640602	−	0.767873i	$-0.721315\pi$
$97$	−12.6184	−1.28120	−0.640602	+	0.767873i	$0.721315\pi$
$98$	14.6741	1.48231
$99$	−2.98392	−0.299895
$100$	−13.4936	−1.34936
$101$	17.2004	1.71150	0.855751	−	0.517387i	$-0.173095\pi$
$101$	17.2004	1.71150	0.855751	+	0.517387i	$0.173095\pi$
$102$	−35.1163	−3.47703
$103$	−0.380738	−0.0375152	−0.0187576	−	0.999824i	$-0.505971\pi$
$103$	−0.380738	−0.0375152	−0.0187576	+	0.999824i	$0.505971\pi$
$104$	−3.14875	−0.308760
$105$	1.17432	0.114602
$106$	15.9586	1.55003
$107$	6.69522	0.647252	0.323626	−	0.946185i	$-0.395098\pi$
$107$	6.69522	0.647252	0.323626	+	0.946185i	$0.395098\pi$
$108$	−6.93704	−0.667517
$109$	−4.95300	−0.474412	−0.237206	−	0.971459i	$-0.576232\pi$
$109$	−4.95300	−0.474412	−0.237206	+	0.971459i	$0.576232\pi$
$110$	−2.58587	−0.246553
$111$	−1.39415	−0.132327
$112$	0.532810	0.0503458
$113$	9.39411	0.883724	0.441862	−	0.897083i	$-0.354318\pi$
$113$	9.39411	0.883724	0.441862	+	0.897083i	$0.354318\pi$
$114$	−17.0202	−1.59409
$115$	4.90912	0.457778
$116$	3.05570	0.283715
$117$	2.63045	0.243185
$118$	−7.83763	−0.721512
$119$	−4.81569	−0.441453
$120$	4.04979	0.369694
$121$	−8.73575	−0.794159
$122$	−28.7447	−2.60242
$123$	−18.7361	−1.68938
$124$	−17.7496	−1.59396
$125$	7.19638	0.643664
$126$	3.06904	0.273411
$127$	−0.161955	−0.0143712	−0.00718560	−	0.999974i	$-0.502287\pi$
$127$	−0.161955	−0.0143712	−0.00718560	+	0.999974i	$0.502287\pi$
$128$	16.3444	1.44465
$129$	−20.3208	−1.78914
$130$	2.27955	0.199930
$131$	−7.30319	−0.638083	−0.319042	−	0.947741i	$-0.603361\pi$
$131$	−7.30319	−0.638083	−0.319042	+	0.947741i	$0.603361\pi$
$132$	−10.2641	−0.893373
$133$	−2.33408	−0.202390
$134$	5.19759	0.449004
$135$	1.73507	0.149331
$136$	−16.6075	−1.42408
$137$	−10.7930	−0.922108	−0.461054	−	0.887372i	$-0.652529\pi$
$137$	−10.7930	−0.922108	−0.461054	+	0.887372i	$0.652529\pi$
$138$	32.2394	2.74440
$139$	1.00000	0.0848189
$139$	1.00000	0.0848189
$140$	1.60750	0.135859
$141$	7.20719	0.606955
$142$	2.43918	0.204691
$143$	−1.99603	−0.166917
$144$	−1.53501	−0.127917
$145$	−0.764282	−0.0634701
$146$	−11.1569	−0.923351
$147$	−14.5683	−1.20157
$148$	−1.90843	−0.156872
$149$	−11.6171	−0.951706	−0.475853	−	0.879525i	$-0.657861\pi$
$149$	−11.6171	−0.951706	−0.475853	+	0.879525i	$0.657861\pi$
$150$	22.1643	1.80970
$151$	5.39440	0.438990	0.219495	−	0.975614i	$-0.429559\pi$
$151$	5.39440	0.438990	0.219495	+	0.975614i	$0.429559\pi$
$152$	−8.04936	−0.652890
$153$	13.8738	1.12163
$154$	−2.32884	−0.187663
$155$	4.43947	0.356587
$156$	9.04820	0.724436
$157$	−12.7326	−1.01617	−0.508084	−	0.861307i	$-0.669646\pi$
$157$	−12.7326	−1.01617	−0.508084	+	0.861307i	$0.669646\pi$
$158$	4.56451	0.363133
$159$	−15.8434	−1.25646
$160$	−4.95865	−0.392016
$161$	4.42116	0.348437
$162$	24.7709	1.94619
$163$	2.90550	0.227577	0.113788	−	0.993505i	$-0.463701\pi$
$163$	2.90550	0.227577	0.113788	+	0.993505i	$0.463701\pi$
$164$	−25.6475	−2.00274
$165$	2.56721	0.199857
$166$	2.22884	0.172992
$167$	−5.74386	−0.444473	−0.222237	−	0.974993i	$-0.571336\pi$
$167$	−5.74386	−0.444473	−0.222237	+	0.974993i	$0.571336\pi$
$168$	3.64725	0.281391
$169$	−11.2404	−0.864647
$170$	12.0231	0.922129
$171$	6.72439	0.514227
$172$	−27.8167	−2.12100
$173$	−8.65642	−0.658136	−0.329068	−	0.944306i	$-0.606734\pi$
$173$	−8.65642	−0.658136	−0.329068	+	0.944306i	$0.606734\pi$
$174$	−5.01923	−0.380506
$175$	3.03951	0.229765
$176$	1.16479	0.0877995
$177$	7.78107	0.584861
$178$	−16.0366	−1.20199
$179$	0.925363	0.0691649	0.0345825	−	0.999402i	$-0.488990\pi$
$179$	0.925363	0.0691649	0.0345825	+	0.999402i	$0.488990\pi$
$180$	−4.63116	−0.345186
$181$	−22.5576	−1.67669	−0.838345	−	0.545140i	$-0.816477\pi$
$181$	−22.5576	−1.67669	−0.838345	+	0.545140i	$0.816477\pi$
$182$	2.05297	0.152176
$183$	28.5373	2.10953
$184$	15.2470	1.12402
$185$	0.477329	0.0350939
$186$	29.1551	2.13775
$187$	−10.5277	−0.769864
$188$	9.86578	0.719536
$189$	1.56261	0.113663
$190$	5.82737	0.422762
$191$	14.6025	1.05660	0.528299	−	0.849059i	$-0.322830\pi$
$191$	14.6025	1.05660	0.528299	+	0.849059i	$0.322830\pi$
$192$	−29.1088	−2.10075
$193$	2.92080	0.210244	0.105122	−	0.994459i	$-0.466477\pi$
$193$	2.92080	0.210244	0.105122	+	0.994459i	$0.466477\pi$
$194$	28.3723	2.03701
$195$	−2.26310	−0.162064
$196$	−19.9422	−1.42444
$197$	−18.1871	−1.29578	−0.647890	−	0.761734i	$-0.724349\pi$
$197$	−18.1871	−1.29578	−0.647890	+	0.761734i	$0.724349\pi$
$198$	6.70932	0.476810
$199$	1.45395	0.103068	0.0515338	−	0.998671i	$-0.483589\pi$
$199$	1.45395	0.103068	0.0515338	+	0.998671i	$0.483589\pi$
$200$	10.4821	0.741199
$201$	−5.16009	−0.363965
$202$	−38.6749	−2.72116
$203$	−0.688314	−0.0483101
$204$	47.7232	3.34129
$205$	6.41487	0.448034
$206$	0.856085	0.0596463
$207$	−12.7372	−0.885299
$208$	−1.02681	−0.0711966
$209$	−5.10259	−0.352954
$210$	−2.64044	−0.182208
$211$	−20.8365	−1.43444	−0.717222	−	0.696845i	$-0.754587\pi$
$211$	−20.8365	−1.43444	−0.717222	+	0.696845i	$0.754587\pi$
$212$	−21.6877	−1.48952
$213$	−2.42158	−0.165924
$214$	−15.0541	−1.02908
$215$	6.95741	0.474492
$216$	5.38885	0.366665
$217$	3.99819	0.271415
$218$	11.1368	0.754278
$219$	11.0764	0.748473
$220$	3.51421	0.236928
$221$	9.28062	0.624282
$222$	3.13473	0.210390
$223$	−4.11008	−0.275231	−0.137616	−	0.990486i	$-0.543944\pi$
$223$	−4.11008	−0.275231	−0.137616	+	0.990486i	$0.543944\pi$
$224$	−4.46577	−0.298382
$225$	−8.75672	−0.583781
$226$	−21.1226	−1.40505
$227$	−5.15212	−0.341958	−0.170979	−	0.985275i	$-0.554693\pi$
$227$	−5.15212	−0.341958	−0.170979	+	0.985275i	$0.554693\pi$
$228$	23.1305	1.53186
$229$	−2.47284	−0.163410	−0.0817050	−	0.996657i	$-0.526037\pi$
$229$	−2.47284	−0.163410	−0.0817050	+	0.996657i	$0.526037\pi$
$230$	−11.0381	−0.727831
$231$	2.31204	0.152121
$232$	−2.37374	−0.155844
$233$	−28.7317	−1.88227	−0.941137	−	0.338024i	$-0.890241\pi$
$233$	−28.7317	−1.88227	−0.941137	+	0.338024i	$0.890241\pi$
$234$	−5.91453	−0.386645
$235$	−2.46759	−0.160968
$236$	10.6514	0.693344
$237$	−4.53157	−0.294357
$238$	10.8280	0.701877
$239$	16.2552	1.05146	0.525732	−	0.850650i	$-0.323791\pi$
$239$	16.2552	1.05146	0.525732	+	0.850650i	$0.323791\pi$
$240$	1.32064	0.0852471
$241$	−6.89768	−0.444319	−0.222159	−	0.975010i	$-0.571311\pi$
$241$	−6.89768	−0.444319	−0.222159	+	0.975010i	$0.571311\pi$
$242$	19.6422	1.26265
$243$	−17.7816	−1.14069
$244$	39.0641	2.50082
$245$	4.98787	0.318664
$246$	42.1280	2.68598
$247$	4.49815	0.286210
$248$	13.7883	0.875558
$249$	−2.21276	−0.140228
$250$	−16.1810	−1.02338
$251$	12.5812	0.794117	0.397058	−	0.917793i	$-0.370031\pi$
$251$	12.5812	0.794117	0.397058	+	0.917793i	$0.370031\pi$
$252$	−4.17083	−0.262738
$253$	9.66524	0.607649
$254$	0.364154	0.0228491
$255$	−11.9363	−0.747483
$256$	−10.6701	−0.666880
$257$	−0.991425	−0.0618434	−0.0309217	−	0.999522i	$-0.509844\pi$
$257$	−0.991425	−0.0618434	−0.0309217	+	0.999522i	$0.509844\pi$
$258$	45.6910	2.84460
$259$	0.429883	0.0267116
$260$	−3.09792	−0.192125
$261$	1.98301	0.122745
$262$	16.4212	1.01450
$263$	18.2077	1.12273	0.561367	−	0.827567i	$-0.310276\pi$
$263$	18.2077	1.12273	0.561367	+	0.827567i	$0.310276\pi$
$264$	7.97336	0.490727
$265$	5.42446	0.333222
$266$	5.24814	0.321784
$267$	15.9208	0.974340
$268$	−7.06355	−0.431475
$269$	−18.5678	−1.13210	−0.566048	−	0.824372i	$-0.691528\pi$
$269$	−18.5678	−1.13210	−0.566048	+	0.824372i	$0.691528\pi$
$270$	−3.90128	−0.237424
$271$	16.2992	0.990107	0.495053	−	0.868863i	$-0.335149\pi$
$271$	16.2992	0.990107	0.495053	+	0.868863i	$0.335149\pi$
$272$	−5.41574	−0.328377
$273$	−2.03816	−0.123355
$274$	24.2679	1.46608
$275$	6.64476	0.400694
$276$	−43.8134	−2.63726
$277$	−12.0442	−0.723664	−0.361832	−	0.932243i	$-0.617849\pi$
$277$	−12.0442	−0.723664	−0.361832	+	0.932243i	$0.617849\pi$
$278$	−2.24849	−0.134855
$279$	−11.5187	−0.689605
$280$	−1.24874	−0.0746267
$281$	9.49743	0.566569	0.283285	−	0.959036i	$-0.408576\pi$
$281$	9.49743	0.566569	0.283285	+	0.959036i	$0.408576\pi$
$282$	−16.2053	−0.965011
$283$	−10.8271	−0.643604	−0.321802	−	0.946807i	$-0.604289\pi$
$283$	−10.8271	−0.643604	−0.321802	+	0.946807i	$0.604289\pi$
$284$	−3.31485	−0.196700
$285$	−5.78533	−0.342693
$286$	4.48806	0.265384
$287$	5.77724	0.341020
$288$	12.8657	0.758121
$289$	31.9490	1.87935
$290$	1.71848	0.100913
$291$	−28.1676	−1.65121
$292$	15.1623	0.887304
$293$	9.39094	0.548625	0.274312	−	0.961641i	$-0.411550\pi$
$293$	9.39094	0.548625	0.274312	+	0.961641i	$0.411550\pi$
$294$	32.7566	1.91040
$295$	−2.66408	−0.155109
$296$	1.48251	0.0861690
$297$	3.41606	0.198220
$298$	26.1208	1.51314
$299$	−8.52031	−0.492742
$300$	−30.1213	−1.73905
$301$	6.26586	0.361158
$302$	−12.1293	−0.697960
$303$	38.3958	2.20578
$304$	−2.62491	−0.150549
$305$	−9.77057	−0.559461
$306$	−31.1952	−1.78331
$307$	24.1360	1.37751	0.688757	−	0.724992i	$-0.258157\pi$
$307$	24.1360	1.37751	0.688757	+	0.724992i	$0.258157\pi$
$308$	3.16490	0.180337
$309$	−0.849908	−0.0483496
$310$	−9.98210	−0.566945
$311$	27.4698	1.55767	0.778835	−	0.627229i	$-0.215811\pi$
$311$	27.4698	1.55767	0.778835	+	0.627229i	$0.215811\pi$
$312$	−7.02885	−0.397930
$313$	−19.1292	−1.08125	−0.540624	−	0.841264i	$-0.681812\pi$
$313$	−19.1292	−1.08125	−0.540624	+	0.841264i	$0.681812\pi$
$314$	28.6290	1.61563
$315$	1.04319	0.0587773
$316$	−6.20318	−0.348956
$317$	30.5398	1.71529	0.857643	−	0.514246i	$-0.171928\pi$
$317$	30.5398	1.71529	0.857643	+	0.514246i	$0.171928\pi$
$318$	35.6237	1.99768
$319$	−1.50474	−0.0842495
$320$	9.96625	0.557130
$321$	14.9455	0.834177
$322$	−9.94094	−0.553987
$323$	23.7247	1.32008
$324$	−33.6638	−1.87021
$325$	−5.85763	−0.324923
$326$	−6.53300	−0.361829
$327$	−11.0564	−0.611422
$328$	19.9236	1.10010
$329$	−2.22232	−0.122520
$330$	−5.77235	−0.317758
$331$	−24.8685	−1.36690	−0.683448	−	0.729999i	$-0.739521\pi$
$331$	−24.8685	−1.36690	−0.683448	+	0.729999i	$0.739521\pi$
$332$	−3.02900	−0.166238
$333$	−1.23848	−0.0678682
$334$	12.9150	0.706678
$335$	1.76671	0.0965256
$336$	1.18937	0.0648856
$337$	−14.3885	−0.783792	−0.391896	−	0.920009i	$-0.628181\pi$
$337$	−14.3885	−0.783792	−0.391896	+	0.920009i	$0.628181\pi$
$338$	25.2740	1.37472
$339$	20.9702	1.13894
$340$	−16.3394	−0.886130
$341$	8.74058	0.473329
$342$	−15.1197	−0.817581
$343$	9.31028	0.502708
$344$	21.6086	1.16506
$345$	10.9585	0.589984
$346$	19.4639	1.04638
$347$	−2.90734	−0.156074	−0.0780372	−	0.996950i	$-0.524865\pi$
$347$	−2.90734	−0.156074	−0.0780372	+	0.996950i	$0.524865\pi$
$348$	6.82114	0.365652
$349$	−23.7623	−1.27197	−0.635983	−	0.771703i	$-0.719405\pi$
$349$	−23.7623	−1.27197	−0.635983	+	0.771703i	$0.719405\pi$
$350$	−6.83430	−0.365309
$351$	−3.01140	−0.160736
$352$	−9.76276	−0.520357
$353$	−14.3184	−0.762091	−0.381045	−	0.924556i	$-0.624436\pi$
$353$	−14.3184	−0.762091	−0.381045	+	0.924556i	$0.624436\pi$
$354$	−17.4957	−0.929884
$355$	0.829099	0.0440040
$356$	21.7937	1.15507
$357$	−10.7499	−0.568945
$358$	−2.08067	−0.109967
$359$	−28.6559	−1.51240	−0.756201	−	0.654340i	$-0.772947\pi$
$359$	−28.6559	−1.51240	−0.756201	+	0.654340i	$0.772947\pi$
$360$	3.59759	0.189610
$361$	−7.50108	−0.394794
$362$	50.7204	2.66581
$363$	−19.5005	−1.02351
$364$	−2.78999	−0.146235
$365$	−3.79233	−0.198500
$366$	−64.1657	−3.35400
$367$	−0.0461160	−0.00240724	−0.00120362	−	0.999999i	$-0.500383\pi$
$367$	−0.0461160	−0.00240724	−0.00120362	+	0.999999i	$0.500383\pi$
$368$	4.97206	0.259186
$369$	−16.6440	−0.866454
$370$	−1.07327	−0.0557966
$371$	4.88528	0.253631
$372$	−39.6218	−2.05430
$373$	−9.75677	−0.505186	−0.252593	−	0.967573i	$-0.581283\pi$
$373$	−9.75677	−0.505186	−0.252593	+	0.967573i	$0.581283\pi$
$374$	23.6715	1.22402
$375$	16.0642	0.829553
$376$	−7.66396	−0.395239
$377$	1.32649	0.0683179
$378$	−3.51350	−0.180715
$379$	−8.17798	−0.420075	−0.210037	−	0.977693i	$-0.567359\pi$
$379$	−8.17798	−0.420075	−0.210037	+	0.977693i	$0.567359\pi$
$380$	−7.91942	−0.406258
$381$	−0.361527	−0.0185216
$382$	−32.8335	−1.67991
$383$	16.2824	0.831990	0.415995	−	0.909367i	$-0.363433\pi$
$383$	16.2824	0.831990	0.415995	+	0.909367i	$0.363433\pi$
$384$	36.4849	1.86186
$385$	−0.791594	−0.0403434
$386$	−6.56740	−0.334272
$387$	−18.0517	−0.917621
$388$	−38.5581	−1.95749
$389$	−19.5724	−0.992362	−0.496181	−	0.868219i	$-0.665265\pi$
$389$	−19.5724	−0.992362	−0.496181	+	0.868219i	$0.665265\pi$
$390$	5.08856	0.257669
$391$	−44.9389	−2.27266
$392$	15.4916	0.782442
$393$	−16.3027	−0.822361
$394$	40.8936	2.06019
$395$	1.55152	0.0780653
$396$	−9.11798	−0.458196
$397$	−5.92424	−0.297329	−0.148665	−	0.988888i	$-0.547497\pi$
$397$	−5.92424	−0.297329	−0.148665	+	0.988888i	$0.547497\pi$
$398$	−3.26918	−0.163869
$399$	−5.21028	−0.260840
$400$	3.41824	0.170912
$401$	−6.44664	−0.321930	−0.160965	−	0.986960i	$-0.551461\pi$
$401$	−6.44664	−0.321930	−0.160965	+	0.986960i	$0.551461\pi$
$402$	11.6024	0.578675
$403$	−7.70518	−0.383822
$404$	52.5593	2.61492
$405$	8.41986	0.418386
$406$	1.54767	0.0768094
$407$	0.939781	0.0465832
$408$	−37.0724	−1.83536
$409$	9.18313	0.454077	0.227038	−	0.973886i	$-0.427096\pi$
$409$	9.18313	0.454077	0.227038	+	0.973886i	$0.427096\pi$
$410$	−14.4238	−0.712339
$411$	−24.0928	−1.18841
$412$	−1.16342	−0.0573177
$413$	−2.39928	−0.118061
$414$	28.6395	1.40756
$415$	0.757603	0.0371893
$416$	8.60627	0.421957
$417$	2.23227	0.109315
$418$	11.4731	0.561169
$419$	−7.78234	−0.380192	−0.190096	−	0.981765i	$-0.560880\pi$
$419$	−7.78234	−0.380192	−0.190096	+	0.981765i	$0.560880\pi$
$420$	3.58837	0.175094
$421$	28.8469	1.40591	0.702957	−	0.711233i	$-0.251863\pi$
$421$	28.8469	1.40591	0.702957	+	0.711233i	$0.251863\pi$
$422$	46.8507	2.28065
$423$	6.40243	0.311297
$424$	16.8475	0.818188
$425$	−30.8950	−1.49863
$426$	5.44489	0.263806
$427$	−8.79939	−0.425833
$428$	20.4586	0.988905
$429$	−4.45567	−0.215122
$430$	−15.6437	−0.754405
$431$	10.2482	0.493637	0.246818	−	0.969062i	$-0.420615\pi$
$431$	10.2482	0.493637	0.246818	+	0.969062i	$0.420615\pi$
$432$	1.75731	0.0845487
$433$	−34.0070	−1.63427	−0.817136	−	0.576445i	$-0.804440\pi$
$433$	−34.0070	−1.63427	−0.817136	+	0.576445i	$0.804440\pi$
$434$	−8.98990	−0.431529
$435$	−1.70608	−0.0818003
$436$	−15.1349	−0.724831
$437$	−21.7810	−1.04193
$438$	−24.9052	−1.19001
$439$	2.61825	0.124962	0.0624812	−	0.998046i	$-0.480099\pi$
$439$	2.61825	0.124962	0.0624812	+	0.998046i	$0.480099\pi$
$440$	−2.72992	−0.130144
$441$	−12.9416	−0.616265
$442$	−20.8674	−0.992560
$443$	−20.0203	−0.951194	−0.475597	−	0.879663i	$-0.657768\pi$
$443$	−20.0203	−0.951194	−0.475597	+	0.879663i	$0.657768\pi$
$444$	−4.26011	−0.202176
$445$	−5.45097	−0.258401
$446$	9.24148	0.437597
$447$	−25.9323	−1.22656
$448$	8.97562	0.424058
$449$	12.2535	0.578277	0.289138	−	0.957287i	$-0.406631\pi$
$449$	12.2535	0.578277	0.289138	+	0.957287i	$0.406631\pi$
$450$	19.6894	0.928167
$451$	12.6298	0.594714
$452$	28.7056	1.35020
$453$	12.0417	0.565770
$454$	11.5845	0.543687
$455$	0.697822	0.0327144
$456$	−17.9683	−0.841444
$457$	−3.74800	−0.175324	−0.0876620	−	0.996150i	$-0.527940\pi$
$457$	−3.74800	−0.175324	−0.0876620	+	0.996150i	$0.527940\pi$
$458$	5.56016	0.259809
$459$	−15.8831	−0.741359
$460$	15.0008	0.699417
$461$	−26.6407	−1.24078	−0.620391	−	0.784293i	$-0.713026\pi$
$461$	−26.6407	−1.24078	−0.620391	+	0.784293i	$0.713026\pi$
$462$	−5.19859	−0.241860
$463$	−15.0733	−0.700516	−0.350258	−	0.936653i	$-0.613906\pi$
$463$	−15.0733	−0.700516	−0.350258	+	0.936653i	$0.613906\pi$
$464$	−0.774080	−0.0359357
$465$	9.91007	0.459569
$466$	64.6029	2.99267
$467$	2.56472	0.118681	0.0593407	−	0.998238i	$-0.481100\pi$
$467$	2.56472	0.118681	0.0593407	+	0.998238i	$0.481100\pi$
$468$	8.03787	0.371551
$469$	1.59110	0.0734702
$470$	5.54836	0.255927
$471$	−28.4225	−1.30964
$472$	−8.27422	−0.380852
$473$	13.6980	0.629834
$474$	10.1892	0.468005
$475$	−14.9742	−0.687066
$476$	−14.7153	−0.674476
$477$	−14.0743	−0.644419
$478$	−36.5497	−1.67175
$479$	2.51348	0.114844	0.0574218	−	0.998350i	$-0.481712\pi$
$479$	2.51348	0.114844	0.0574218	+	0.998350i	$0.481712\pi$
$480$	−11.0690	−0.505230
$481$	−0.828455	−0.0377743
$482$	15.5094	0.706432
$483$	9.86921	0.449065
$484$	−26.6939	−1.21336
$485$	9.64401	0.437912
$486$	39.9818	1.81361
$487$	8.72866	0.395533	0.197767	−	0.980249i	$-0.436631\pi$
$487$	8.72866	0.395533	0.197767	+	0.980249i	$0.436631\pi$
$488$	−30.3459	−1.37369
$489$	6.48586	0.293301
$490$	−11.2152	−0.506650
$491$	−36.9240	−1.66636	−0.833178	−	0.553005i	$-0.813481\pi$
$491$	−36.9240	−1.66636	−0.833178	+	0.553005i	$0.813481\pi$
$492$	−57.2521	−2.58112
$493$	6.99636	0.315100
$494$	−10.1140	−0.455052
$495$	2.28056	0.102503
$496$	4.49638	0.201894
$497$	0.746688	0.0334935
$498$	4.97537	0.222951
$499$	12.6921	0.568175	0.284088	−	0.958798i	$-0.408309\pi$
$499$	12.6921	0.568175	0.284088	+	0.958798i	$0.408309\pi$
$500$	21.9900	0.983423
$501$	−12.8218	−0.572837
$502$	−28.2886	−1.26258
$503$	27.3612	1.21998	0.609988	−	0.792410i	$-0.291174\pi$
$503$	27.3612	1.21998	0.609988	+	0.792410i	$0.291174\pi$
$504$	3.23999	0.144321
$505$	−13.1459	−0.584987
$506$	−21.7322	−0.966114
$507$	−25.0916	−1.11436
$508$	−0.494887	−0.0219571
$509$	18.1519	0.804568	0.402284	−	0.915515i	$-0.368216\pi$
$509$	18.1519	0.804568	0.402284	+	0.915515i	$0.368216\pi$
$510$	26.8387	1.18844
$511$	−3.41538	−0.151087
$512$	−8.69715	−0.384363
$513$	−7.69824	−0.339886
$514$	2.22921	0.0983262
$515$	0.290991	0.0128226
$516$	−62.0943	−2.73355
$517$	−4.85828	−0.213667
$518$	−0.966588	−0.0424694
$519$	−19.3234	−0.848205
$520$	2.40653	0.105533
$521$	20.8709	0.914372	0.457186	−	0.889371i	$-0.348857\pi$
$521$	20.8709	0.914372	0.457186	+	0.889371i	$0.348857\pi$
$522$	−4.45878	−0.195155
$523$	31.5174	1.37816	0.689081	−	0.724685i	$-0.258015\pi$
$523$	31.5174	1.37816	0.689081	+	0.724685i	$0.258015\pi$
$524$	−22.3164	−0.974897
$525$	6.78498	0.296121
$526$	−40.9398	−1.78506
$527$	−40.6396	−1.77029
$528$	2.60012	0.113156
$529$	18.2573	0.793794
$530$	−12.1968	−0.529797
$531$	6.91223	0.299965
$532$	−7.13224	−0.309222
$533$	−11.1337	−0.482254
$534$	−35.7979	−1.54912
$535$	−5.11704	−0.221229
$536$	5.48712	0.237008
$537$	2.06566	0.0891397
$538$	41.7494	1.79994
$539$	9.82030	0.422990
$540$	5.30185	0.228155
$541$	1.08405	0.0466068	0.0233034	−	0.999728i	$-0.492582\pi$
$541$	1.08405	0.0466068	0.0233034	+	0.999728i	$0.492582\pi$
$542$	−36.6486	−1.57419
$543$	−50.3545	−2.16092
$544$	45.3923	1.94618
$545$	3.78549	0.162153
$546$	4.58277	0.196124
$547$	−10.5730	−0.452071	−0.226035	−	0.974119i	$-0.572577\pi$
$547$	−10.5730	−0.452071	−0.226035	+	0.974119i	$0.572577\pi$
$548$	−32.9802	−1.40884
$549$	25.3508	1.08194
$550$	−14.9407	−0.637072
$551$	3.39101	0.144462
$552$	34.0353	1.44864
$553$	1.39730	0.0594192
$554$	27.0812	1.15057
$555$	1.06552	0.0452290
$556$	3.05570	0.129591
$557$	24.9357	1.05656	0.528279	−	0.849071i	$-0.322838\pi$
$557$	24.9357	1.05656	0.528279	+	0.849071i	$0.322838\pi$
$558$	25.8996	1.09642
$559$	−12.0753	−0.510732
$560$	−0.407217	−0.0172080
$561$	−23.5007	−0.992200
$562$	−21.3549	−0.900801
$563$	−7.93284	−0.334329	−0.167165	−	0.985929i	$-0.553461\pi$
$563$	−7.93284	−0.334329	−0.167165	+	0.985929i	$0.553461\pi$
$564$	22.0230	0.927338
$565$	−7.17975	−0.302054
$566$	24.3446	1.02328
$567$	7.58295	0.318454
$568$	2.57505	0.108047
$569$	−17.1288	−0.718078	−0.359039	−	0.933323i	$-0.616896\pi$
$569$	−17.1288	−0.718078	−0.359039	+	0.933323i	$0.616896\pi$
$570$	13.0082	0.544855
$571$	−37.7000	−1.57769	−0.788847	−	0.614589i	$-0.789322\pi$
$571$	−37.7000	−1.57769	−0.788847	+	0.614589i	$0.789322\pi$
$572$	−6.09929	−0.255024
$573$	32.5966	1.36174
$574$	−12.9901	−0.542195
$575$	28.3640	1.18286
$576$	−25.8585	−1.07744
$577$	−10.0416	−0.418036	−0.209018	−	0.977912i	$-0.567027\pi$
$577$	−10.0416	−0.418036	−0.209018	+	0.977912i	$0.567027\pi$
$578$	−71.8370	−2.98803
$579$	6.52001	0.270962
$580$	−2.33542	−0.0969730
$581$	0.682299	0.0283065
$582$	63.3346	2.62530
$583$	10.6799	0.442314
$584$	−11.7784	−0.487393
$585$	−2.01040	−0.0831200
$586$	−21.1154	−0.872271
$587$	6.49557	0.268101	0.134050	−	0.990975i	$-0.457202\pi$
$587$	6.49557	0.268101	0.134050	+	0.990975i	$0.457202\pi$
$588$	−44.5163	−1.83582
$589$	−19.6973	−0.811612
$590$	5.99016	0.246611
$591$	−40.5985	−1.67000
$592$	0.483448	0.0198696
$593$	−35.0467	−1.43920	−0.719598	−	0.694391i	$-0.755674\pi$
$593$	−35.0467	−1.43920	−0.719598	+	0.694391i	$0.755674\pi$
$594$	−7.68097	−0.315154
$595$	3.68054	0.150888
$596$	−35.4983	−1.45407
$597$	3.24559	0.132833
$598$	19.1578	0.783421
$599$	−24.8151	−1.01392	−0.506958	−	0.861971i	$-0.669230\pi$
$599$	−24.8151	−1.01392	−0.506958	+	0.861971i	$0.669230\pi$
$600$	23.3989	0.955256
$601$	16.5896	0.676703	0.338352	−	0.941020i	$-0.390131\pi$
$601$	16.5896	0.676703	0.338352	+	0.941020i	$0.390131\pi$
$602$	−14.0887	−0.574213
$603$	−4.58391	−0.186671
$604$	16.4837	0.670712
$605$	6.67657	0.271441
$606$	−86.3326	−3.50702
$607$	15.3051	0.621213	0.310606	−	0.950539i	$-0.399468\pi$
$607$	15.3051	0.621213	0.310606	+	0.950539i	$0.399468\pi$
$608$	22.0008	0.892251
$609$	−1.53650	−0.0622621
$610$	21.9690	0.889500
$611$	4.28278	0.173263
$612$	42.3944	1.71369
$613$	3.13698	0.126701	0.0633507	−	0.997991i	$-0.479821\pi$
$613$	3.13698	0.126701	0.0633507	+	0.997991i	$0.479821\pi$
$614$	−54.2696	−2.19014
$615$	14.3197	0.577425
$616$	−2.45857	−0.0990585
$617$	43.8836	1.76669	0.883343	−	0.468726i	$-0.155287\pi$
$617$	43.8836	1.76669	0.883343	+	0.468726i	$0.155287\pi$
$618$	1.91101	0.0768721
$619$	−19.0698	−0.766478	−0.383239	−	0.923649i	$-0.625191\pi$
$619$	−19.0698	−0.766478	−0.383239	+	0.923649i	$0.625191\pi$
$620$	13.5657	0.544812
$621$	14.5819	0.585150
$622$	−61.7656	−2.47657
$623$	−4.90915	−0.196681
$624$	−2.29212	−0.0917581
$625$	16.5793	0.663172
$626$	43.0119	1.71910
$627$	−11.3903	−0.454887
$628$	−38.9069	−1.55256
$629$	−4.36954	−0.174225
$630$	−2.34561	−0.0934513
$631$	12.4006	0.493661	0.246831	−	0.969059i	$-0.420611\pi$
$631$	12.4006	0.493661	0.246831	+	0.969059i	$0.420611\pi$
$632$	4.81877	0.191680
$633$	−46.5126	−1.84871
$634$	−68.6684	−2.72717
$635$	0.123779	0.00491203
$636$	−48.4128	−1.91969
$637$	−8.65699	−0.343003
$638$	3.38340	0.133950
$639$	−2.15118	−0.0850995
$640$	−12.4917	−0.493778
$641$	−16.8531	−0.665659	−0.332829	−	0.942987i	$-0.608003\pi$
$641$	−16.8531	−0.665659	−0.332829	+	0.942987i	$0.608003\pi$
$642$	−33.6048	−1.32628
$643$	−0.704089	−0.0277666	−0.0138833	−	0.999904i	$-0.504419\pi$
$643$	−0.704089	−0.0277666	−0.0138833	+	0.999904i	$0.504419\pi$
$644$	13.5098	0.532359
$645$	15.5308	0.611524
$646$	−53.3447	−2.09882
$647$	21.5117	0.845712	0.422856	−	0.906197i	$-0.361028\pi$
$647$	21.5117	0.845712	0.422856	+	0.906197i	$0.361028\pi$
$648$	26.1508	1.02730
$649$	−5.24513	−0.205889
$650$	13.1708	0.516602
$651$	8.92503	0.349800
$652$	8.87836	0.347704
$653$	14.4571	0.565749	0.282874	−	0.959157i	$-0.408712\pi$
$653$	14.4571	0.565749	0.282874	+	0.959157i	$0.408712\pi$
$654$	24.8602	0.972113
$655$	5.58170	0.218095
$656$	6.49710	0.253669
$657$	9.83960	0.383879
$658$	4.99686	0.194798
$659$	−30.3396	−1.18186	−0.590932	−	0.806722i	$-0.701240\pi$
$659$	−30.3396	−1.18186	−0.590932	+	0.806722i	$0.701240\pi$
$660$	7.84465	0.305352
$661$	−44.2046	−1.71936	−0.859680	−	0.510833i	$-0.829337\pi$
$661$	−44.2046	−1.71936	−0.859680	+	0.510833i	$0.829337\pi$
$662$	55.9166	2.17326
$663$	20.7168	0.804574
$664$	2.35300	0.0913140
$665$	1.78389	0.0691763
$666$	2.78471	0.107905
$667$	−6.42318	−0.248707
$668$	−17.5515	−0.679089
$669$	−9.17479	−0.354718
$670$	−3.97243	−0.153468
$671$	−19.2366	−0.742622
$672$	−9.96879	−0.384554
$673$	32.9763	1.27114	0.635572	−	0.772042i	$-0.280764\pi$
$673$	32.9763	1.27114	0.635572	+	0.772042i	$0.280764\pi$
$674$	32.3524	1.24617
$675$	10.0249	0.385858
$676$	−34.3474	−1.32105
$677$	−43.8280	−1.68445	−0.842224	−	0.539128i	$-0.818754\pi$
$677$	−43.8280	−1.68445	−0.842224	+	0.539128i	$0.818754\pi$
$678$	−47.1512	−1.81083
$679$	8.68541	0.333316
$680$	12.6928	0.486748
$681$	−11.5009	−0.440716
$682$	−19.6531	−0.752556
$683$	43.6322	1.66954	0.834770	−	0.550599i	$-0.185601\pi$
$683$	43.6322	1.66954	0.834770	+	0.550599i	$0.185601\pi$
$684$	20.5478	0.785663
$685$	8.24889	0.315174
$686$	−20.9341	−0.799267
$687$	−5.52004	−0.210603
$688$	7.04661	0.268649
$689$	−9.41473	−0.358673
$690$	−24.6400	−0.938028
$691$	−1.19417	−0.0454284	−0.0227142	−	0.999742i	$-0.507231\pi$
$691$	−1.19417	−0.0454284	−0.0227142	+	0.999742i	$0.507231\pi$
$692$	−26.4515	−1.00553
$693$	2.05387	0.0780202
$694$	6.53713	0.248146
$695$	−0.764282	−0.0289909
$696$	−5.29882	−0.200851
$697$	−58.7227	−2.22428
$698$	53.4292	2.02233
$699$	−64.1367	−2.42587
$700$	9.28783	0.351047
$701$	−14.9090	−0.563106	−0.281553	−	0.959546i	$-0.590850\pi$
$701$	−14.9090	−0.563106	−0.281553	+	0.959546i	$0.590850\pi$
$702$	6.77109	0.255558
$703$	−2.11784	−0.0798757
$704$	19.6219	0.739528
$705$	−5.50833	−0.207456
$706$	32.1947	1.21166
$707$	−11.8393	−0.445261
$708$	23.7767	0.893582
$709$	36.6414	1.37610	0.688048	−	0.725665i	$-0.258468\pi$
$709$	36.6414	1.37610	0.688048	+	0.725665i	$0.258468\pi$
$710$	−1.86422	−0.0699629
$711$	−4.02557	−0.150971
$712$	−16.9299	−0.634473
$713$	37.3102	1.39728
$714$	24.1710	0.904578
$715$	1.52553	0.0570516
$716$	2.82764	0.105674
$717$	36.2860	1.35513
$718$	64.4325	2.40460
$719$	12.1047	0.451428	0.225714	−	0.974194i	$-0.427528\pi$
$719$	12.1047	0.451428	0.225714	+	0.974194i	$0.427528\pi$
$720$	1.17318	0.0437218
$721$	0.262067	0.00975989
$722$	16.8661	0.627691
$723$	−15.3975	−0.572638
$724$	−68.9292	−2.56173
$725$	−4.41587	−0.164001
$726$	43.8467	1.62730
$727$	−43.2698	−1.60479	−0.802394	−	0.596795i	$-0.796440\pi$
$727$	−43.2698	−1.60479	−0.802394	+	0.596795i	$0.796440\pi$
$728$	2.16733	0.0803265
$729$	−6.64320	−0.246044
$730$	8.52701	0.315599
$731$	−63.6893	−2.35563
$732$	87.2014	3.22306
$733$	48.7992	1.80244	0.901219	−	0.433364i	$-0.142674\pi$
$733$	48.7992	1.80244	0.901219	+	0.433364i	$0.142674\pi$
$734$	0.103691	0.00382732
$735$	11.1343	0.410693
$736$	−41.6735	−1.53611
$737$	3.47836	0.128127
$738$	37.4240	1.37759
$739$	18.8163	0.692168	0.346084	−	0.938204i	$-0.387511\pi$
$739$	18.8163	0.692168	0.346084	+	0.938204i	$0.387511\pi$
$740$	1.45858	0.0536183
$741$	10.0411	0.368867
$742$	−10.9845	−0.403253
$743$	2.69815	0.0989855	0.0494928	−	0.998774i	$-0.484240\pi$
$743$	2.69815	0.0989855	0.0494928	+	0.998774i	$0.484240\pi$
$744$	30.7791	1.12842
$745$	8.87870	0.325290
$746$	21.9380	0.803207
$747$	−1.96568	−0.0719205
$748$	−32.1696	−1.17624
$749$	−4.60841	−0.168388
$750$	−36.1203	−1.31893
$751$	37.9590	1.38514	0.692572	−	0.721349i	$-0.256478\pi$
$751$	37.9590	1.38514	0.692572	+	0.721349i	$0.256478\pi$
$752$	−2.49923	−0.0911375
$753$	28.0845	1.02346
$754$	−2.98261	−0.108620
$755$	−4.12284	−0.150046
$756$	4.77486	0.173660
$757$	25.6753	0.933186	0.466593	−	0.884472i	$-0.345481\pi$
$757$	25.6753	0.933186	0.466593	+	0.884472i	$0.345481\pi$
$758$	18.3881	0.667886
$759$	21.5754	0.783137
$760$	6.15198	0.223156
$761$	−21.7994	−0.790226	−0.395113	−	0.918632i	$-0.629295\pi$
$761$	−21.7994	−0.790226	−0.395113	+	0.918632i	$0.629295\pi$
$762$	0.812889	0.0294479
$763$	3.40922	0.123422
$764$	44.6208	1.61432
$765$	−10.6035	−0.383371
$766$	−36.6107	−1.32280
$767$	4.62380	0.166956
$768$	−23.8185	−0.859475
$769$	18.3920	0.663231	0.331615	−	0.943415i	$-0.392406\pi$
$769$	18.3920	0.663231	0.331615	+	0.943415i	$0.392406\pi$
$770$	1.77989	0.0641428
$771$	−2.21312	−0.0797037
$772$	8.92511	0.321222
$773$	−24.9578	−0.897671	−0.448835	−	0.893615i	$-0.648161\pi$
$773$	−24.9578	−0.897671	−0.448835	+	0.893615i	$0.648161\pi$
$774$	40.5891	1.45895
$775$	25.6504	0.921390
$776$	29.9528	1.07524
$777$	0.959613	0.0344259
$778$	44.0084	1.57778
$779$	−28.4618	−1.01975
$780$	−6.91537	−0.247610
$781$	1.63236	0.0584104
$782$	101.045	3.61335
$783$	−2.27019	−0.0811301
$784$	5.05182	0.180422
$785$	9.73126	0.347324
$786$	36.6564	1.30749
$787$	−20.7473	−0.739562	−0.369781	−	0.929119i	$-0.620567\pi$
$787$	−20.7473	−0.739562	−0.369781	+	0.929119i	$0.620567\pi$
$788$	−55.5746	−1.97976
$789$	40.6444	1.44698
$790$	−3.48857	−0.124118
$791$	−6.46610	−0.229908
$792$	7.08305	0.251685
$793$	16.9579	0.602192
$794$	13.3206	0.472730
$795$	12.1088	0.429456
$796$	4.44283	0.157472
$797$	10.2009	0.361336	0.180668	−	0.983544i	$-0.442174\pi$
$797$	10.2009	0.361336	0.180668	+	0.983544i	$0.442174\pi$
$798$	11.7152	0.414715
$799$	22.5888	0.799133
$800$	−28.6501	−1.01294
$801$	14.1431	0.499722
$802$	14.4952	0.511843
$803$	−7.46647	−0.263486
$804$	−15.7677	−0.556084
$805$	−3.37901	−0.119095
$806$	17.3250	0.610247
$807$	−41.4481	−1.45904
$808$	−40.8293	−1.43637
$809$	37.5389	1.31980	0.659898	−	0.751355i	$-0.270599\pi$
$809$	37.5389	1.31980	0.659898	+	0.751355i	$0.270599\pi$
$810$	−18.9320	−0.665202
$811$	4.07947	0.143250	0.0716248	−	0.997432i	$-0.477182\pi$
$811$	4.07947	0.143250	0.0716248	+	0.997432i	$0.477182\pi$
$812$	−2.10328	−0.0738108
$813$	36.3842	1.27605
$814$	−2.11309	−0.0740637
$815$	−2.22062	−0.0777851
$816$	−12.0894	−0.423212
$817$	−30.8690	−1.07997
$818$	−20.6482	−0.721947
$819$	−1.81057	−0.0632665
$820$	19.6019	0.684529
$821$	−22.5354	−0.786490	−0.393245	−	0.919434i	$-0.628648\pi$
$821$	−22.5354	−0.786490	−0.393245	+	0.919434i	$0.628648\pi$
$822$	54.1725	1.88948
$823$	−49.3546	−1.72039	−0.860197	−	0.509962i	$-0.829660\pi$
$823$	−49.3546	−1.72039	−0.860197	+	0.509962i	$0.829660\pi$
$824$	0.903773	0.0314844
$825$	14.8329	0.516414
$826$	5.39475	0.187707
$827$	30.9769	1.07717	0.538585	−	0.842571i	$-0.318959\pi$
$827$	30.9769	1.07717	0.538585	+	0.842571i	$0.318959\pi$
$828$	−38.9212	−1.35261
$829$	1.60365	0.0556972	0.0278486	−	0.999612i	$-0.491134\pi$
$829$	1.60365	0.0556972	0.0278486	+	0.999612i	$0.491134\pi$
$830$	−1.70346	−0.0591281
$831$	−26.8858	−0.932657
$832$	−17.2975	−0.599683
$833$	−45.6598	−1.58202
$834$	−5.01923	−0.173802
$835$	4.38993	0.151920
$836$	−15.5920	−0.539261
$837$	13.1868	0.455804
$838$	17.4985	0.604476
$839$	12.4922	0.431279	0.215639	−	0.976473i	$-0.430816\pi$
$839$	12.4922	0.431279	0.215639	+	0.976473i	$0.430816\pi$
$840$	−2.78753	−0.0961788
$841$	1.00000	0.0344828
$842$	−64.8620	−2.23529
$843$	21.2008	0.730194
$844$	−63.6702	−2.19162
$845$	8.59085	0.295534
$846$	−14.3958	−0.494938
$847$	6.01293	0.206607
$848$	5.49400	0.188665
$849$	−24.1690	−0.829476
$850$	69.4671	2.38270
$851$	4.01157	0.137515
$852$	−7.39963	−0.253507
$853$	39.4057	1.34923	0.674613	−	0.738172i	$-0.264311\pi$
$853$	39.4057	1.34923	0.674613	+	0.738172i	$0.264311\pi$
$854$	19.7853	0.677041
$855$	−5.13933	−0.175761
$856$	−15.8927	−0.543202
$857$	48.0117	1.64005	0.820024	−	0.572329i	$-0.193960\pi$
$857$	48.0117	1.64005	0.820024	+	0.572329i	$0.193960\pi$
$858$	10.0185	0.342027
$859$	1.86333	0.0635759	0.0317879	−	0.999495i	$-0.489880\pi$
$859$	1.86333	0.0635759	0.0317879	+	0.999495i	$0.489880\pi$
$860$	21.2598	0.724953
$861$	12.8963	0.439506
$862$	−23.0429	−0.784844
$863$	22.1822	0.755091	0.377545	−	0.925991i	$-0.376768\pi$
$863$	22.1822	0.755091	0.377545	+	0.925991i	$0.376768\pi$
$864$	−14.7290	−0.501090
$865$	6.61595	0.224949
$866$	76.4644	2.59837
$867$	71.3187	2.42211
$868$	12.2173	0.414682
$869$	3.05468	0.103623
$870$	3.83610	0.130056
$871$	−3.06631	−0.103898
$872$	11.7571	0.398147
$873$	−25.0224	−0.846880
$874$	48.9744	1.65659
$875$	−4.95337	−0.167454
$876$	33.8462	1.14356
$877$	−23.0444	−0.778154	−0.389077	−	0.921205i	$-0.627206\pi$
$877$	−23.0444	−0.778154	−0.389077	+	0.921205i	$0.627206\pi$
$878$	−5.88712	−0.198681
$879$	20.9631	0.707067
$880$	−0.890229	−0.0300096
$881$	−10.7363	−0.361714	−0.180857	−	0.983509i	$-0.557887\pi$
$881$	−10.7363	−0.361714	−0.180857	+	0.983509i	$0.557887\pi$
$882$	29.0990	0.979813
$883$	23.2509	0.782455	0.391227	−	0.920294i	$-0.372051\pi$
$883$	23.2509	0.782455	0.391227	+	0.920294i	$0.372051\pi$
$884$	28.3588	0.953811
$885$	−5.94693	−0.199904
$886$	45.0155	1.51232
$887$	43.3747	1.45638	0.728189	−	0.685376i	$-0.240362\pi$
$887$	43.3747	1.45638	0.728189	+	0.685376i	$0.240362\pi$
$888$	3.30935	0.111055
$889$	0.111476	0.00373878
$890$	12.2564	0.410837
$891$	16.5773	0.555361
$892$	−12.5592	−0.420513
$893$	10.9484	0.366373
$894$	58.3086	1.95013
$895$	−0.707238	−0.0236404
$896$	−11.2500	−0.375838
$897$	−19.0196	−0.635046
$898$	−27.5518	−0.919415
$899$	−5.80868	−0.193730
$900$	−26.7579	−0.891931
$901$	−49.6563	−1.65429
$902$	−28.3980	−0.945550
$903$	13.9871	0.465460
$904$	−22.2992	−0.741660
$905$	17.2403	0.573088
$906$	−27.0757	−0.899530
$907$	51.0773	1.69599	0.847997	−	0.530001i	$-0.177808\pi$
$907$	51.0773	1.69599	0.847997	+	0.530001i	$0.177808\pi$
$908$	−15.7434	−0.522462
$909$	34.1085	1.13131
$910$	−1.56905	−0.0520134
$911$	34.6253	1.14719	0.573593	−	0.819140i	$-0.305549\pi$
$911$	34.6253	1.14719	0.573593	+	0.819140i	$0.305549\pi$
$912$	−5.85949	−0.194027
$913$	1.49159	0.0493646
$914$	8.42734	0.278752
$915$	−21.8105	−0.721033
$916$	−7.55627	−0.249666
$917$	5.02689	0.166002
$918$	35.7129	1.17870
$919$	6.11146	0.201598	0.100799	−	0.994907i	$-0.467860\pi$
$919$	6.11146	0.201598	0.100799	+	0.994907i	$0.467860\pi$
$920$	−11.6530	−0.384187
$921$	53.8780	1.77534
$922$	59.9014	1.97275
$923$	−1.43899	−0.0473649
$924$	7.06490	0.232418
$925$	2.75791	0.0906796
$926$	33.8922	1.11377
$927$	−0.755007	−0.0247977
$928$	6.48799	0.212979
$929$	8.45786	0.277493	0.138747	−	0.990328i	$-0.455693\pi$
$929$	8.45786	0.277493	0.138747	+	0.990328i	$0.455693\pi$
$930$	−22.2827	−0.730678
$931$	−22.1305	−0.725297
$932$	−87.7955	−2.87584
$933$	61.3199	2.00752
$934$	−5.76676	−0.188694
$935$	8.04615	0.263137
$936$	−6.24400	−0.204092
$937$	23.7170	0.774802	0.387401	−	0.921911i	$-0.373373\pi$
$937$	23.7170	0.774802	0.387401	+	0.921911i	$0.373373\pi$
$938$	−3.57757	−0.116812
$939$	−42.7015	−1.39351
$940$	−7.54024	−0.245935
$941$	51.4583	1.67749	0.838746	−	0.544523i	$-0.183289\pi$
$941$	51.4583	1.67749	0.838746	+	0.544523i	$0.183289\pi$
$942$	63.9076	2.08222
$943$	53.9119	1.75561
$944$	−2.69823	−0.0878200
$945$	−1.19427	−0.0388496
$946$	−30.7998	−1.00139
$947$	−27.0588	−0.879294	−0.439647	−	0.898171i	$-0.644897\pi$
$947$	−27.0588	−0.879294	−0.439647	+	0.898171i	$0.644897\pi$
$948$	−13.8471	−0.449734
$949$	6.58200	0.213661
$950$	33.6694	1.09238
$951$	68.1729	2.21066
$952$	11.4312	0.370487
$953$	−32.1618	−1.04182	−0.520911	−	0.853611i	$-0.674408\pi$
$953$	−32.1618	−1.04182	−0.520911	+	0.853611i	$0.674408\pi$
$954$	31.6460	1.02458
$955$	−11.1604	−0.361142
$956$	49.6712	1.60648
$957$	−3.35899	−0.108581
$958$	−5.65152	−0.182592
$959$	7.42897	0.239894
$960$	22.2473	0.718029
$961$	2.74078	0.0884122
$962$	1.86277	0.0600582
$963$	13.2767	0.427835
$964$	−21.0773	−0.678853
$965$	−2.23232	−0.0718608
$966$	−22.1908	−0.713978
$967$	23.1554	0.744626	0.372313	−	0.928107i	$-0.378565\pi$
$967$	23.1554	0.744626	0.372313	+	0.928107i	$0.378565\pi$
$968$	20.7364	0.666493
$969$	52.9598	1.70131
$970$	−21.6844	−0.696246
$971$	−27.7604	−0.890873	−0.445437	−	0.895314i	$-0.646952\pi$
$971$	−27.7604	−0.890873	−0.445437	+	0.895314i	$0.646952\pi$
$972$	−54.3354	−1.74281
$973$	−0.688314	−0.0220663
$974$	−19.6263	−0.628867
$975$	−13.0758	−0.418760
$976$	−9.89583	−0.316758
$977$	36.7943	1.17715	0.588577	−	0.808441i	$-0.299688\pi$
$977$	36.7943	1.17715	0.588577	+	0.808441i	$0.299688\pi$
$978$	−14.5834	−0.466325
$979$	−10.7321	−0.342998
$980$	15.2415	0.486871
$981$	−9.82185	−0.313588
$982$	83.0232	2.64938
$983$	−56.0637	−1.78815	−0.894077	−	0.447913i	$-0.852167\pi$
$983$	−56.0637	−1.78815	−0.894077	+	0.447913i	$0.852167\pi$
$984$	44.4747	1.41780
$985$	13.9001	0.442894
$986$	−15.7312	−0.500985
$987$	−4.96081	−0.157904
$988$	13.7450	0.437287
$989$	58.4715	1.85929
$990$	−5.12781	−0.162972
$991$	−48.6229	−1.54456	−0.772279	−	0.635284i	$-0.780883\pi$
$991$	−48.6229	−1.54456	−0.772279	+	0.635284i	$0.780883\pi$
$992$	−37.6867	−1.19655
$993$	−55.5131	−1.76166
$994$	−1.67892	−0.0532521
$995$	−1.11122	−0.0352282
$996$	−6.76154	−0.214248
$997$	−45.7242	−1.44810	−0.724050	−	0.689748i	$-0.757721\pi$
$997$	−45.7242	−1.44810	−0.724050	+	0.689748i	$0.757721\pi$
$998$	−28.5380	−0.903354
$999$	1.41784	0.0448584

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4031.2.a.b.1.7	✓	59

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4031.2.a.b.1.7	✓	59	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:	\( N \)	\(=\)	\( 4031 = 29 \cdot 139 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4031.a (trivial)

\(f(q)\)	\(=\)	\(q-2.24849 q^{2} +2.23227 q^{3} +3.05570 q^{4} -0.764282 q^{5} -5.01923 q^{6} -0.688314 q^{7} -2.37374 q^{8} +1.98301 q^{9} +O(q^{10})\)	\(q-2.24849 q^{2} +2.23227 q^{3} +3.05570 q^{4} -0.764282 q^{5} -5.01923 q^{6} -0.688314 q^{7} -2.37374 q^{8} +1.98301 q^{9} +1.71848 q^{10} -1.50474 q^{11} +6.82114 q^{12} +1.32649 q^{13} +1.54767 q^{14} -1.70608 q^{15} -0.774080 q^{16} +6.99636 q^{17} -4.45878 q^{18} +3.39101 q^{19} -2.33542 q^{20} -1.53650 q^{21} +3.38340 q^{22} -6.42318 q^{23} -5.29882 q^{24} -4.41587 q^{25} -2.98261 q^{26} -2.27019 q^{27} -2.10328 q^{28} +1.00000 q^{29} +3.83610 q^{30} -5.80868 q^{31} +6.48799 q^{32} -3.35899 q^{33} -15.7312 q^{34} +0.526066 q^{35} +6.05949 q^{36} -0.624545 q^{37} -7.62464 q^{38} +2.96108 q^{39} +1.81421 q^{40} -8.39333 q^{41} +3.45480 q^{42} -9.10320 q^{43} -4.59805 q^{44} -1.51558 q^{45} +14.4425 q^{46} +3.22864 q^{47} -1.72795 q^{48} -6.52622 q^{49} +9.92904 q^{50} +15.6177 q^{51} +4.05337 q^{52} -7.09746 q^{53} +5.10451 q^{54} +1.15005 q^{55} +1.63388 q^{56} +7.56962 q^{57} -2.24849 q^{58} +3.48573 q^{59} -5.21328 q^{60} +12.7840 q^{61} +13.0608 q^{62} -1.36493 q^{63} -13.0400 q^{64} -1.01381 q^{65} +7.55265 q^{66} -2.31159 q^{67} +21.3788 q^{68} -14.3382 q^{69} -1.18285 q^{70} -1.08481 q^{71} -4.70715 q^{72} +4.96195 q^{73} +1.40428 q^{74} -9.85740 q^{75} +10.3619 q^{76} +1.03574 q^{77} -6.65797 q^{78} -2.03003 q^{79} +0.591615 q^{80} -11.0167 q^{81} +18.8723 q^{82} -0.991262 q^{83} -4.69509 q^{84} -5.34719 q^{85} +20.4685 q^{86} +2.23227 q^{87} +3.57187 q^{88} +7.13215 q^{89} +3.40776 q^{90} -0.913043 q^{91} -19.6273 q^{92} -12.9665 q^{93} -7.25957 q^{94} -2.59168 q^{95} +14.4829 q^{96} -12.6184 q^{97} +14.6741 q^{98} -2.98392 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 59 q - 5 q^{2} - 6 q^{3} + 41 q^{4} - 5 q^{5} - 7 q^{6} - 10 q^{7} - 12 q^{8} + 23 q^{9}+O(q^{10}) \) 59 * q - 5 * q^2 - 6 * q^3 + 41 * q^4 - 5 * q^5 - 7 * q^6 - 10 * q^7 - 12 * q^8 + 23 * q^9	\( 59 q - 5 q^{2} - 6 q^{3} + 41 q^{4} - 5 q^{5} - 7 q^{6} - 10 q^{7} - 12 q^{8} + 23 q^{9} - 18 q^{10} - 27 q^{11} - 8 q^{12} - 22 q^{13} - 24 q^{14} - 18 q^{15} + 5 q^{16} - 23 q^{17} + q^{18} - 32 q^{19} - 14 q^{20} - 36 q^{21} - 6 q^{22} - 3 q^{23} - 18 q^{24} - 8 q^{25} - q^{26} - 12 q^{27} - 9 q^{28} + 59 q^{29} - 18 q^{30} - 32 q^{31} - 39 q^{32} - 12 q^{33} - 18 q^{34} - 9 q^{35} + 10 q^{36} - 44 q^{37} + 5 q^{38} - 27 q^{39} - 68 q^{40} - 44 q^{41} - 25 q^{42} - 40 q^{43} - 56 q^{44} - 39 q^{45} - 40 q^{46} - 20 q^{47} - 9 q^{48} - 39 q^{49} - 21 q^{50} - 28 q^{51} - 49 q^{52} - 31 q^{53} - 32 q^{54} - 32 q^{55} - 48 q^{56} - 58 q^{57} - 5 q^{58} + 6 q^{59} - 44 q^{60} - 88 q^{61} + 35 q^{62} - 22 q^{63} - 10 q^{64} - 43 q^{65} - 31 q^{66} - 45 q^{67} - 29 q^{68} - 60 q^{69} - 14 q^{70} - 20 q^{71} - 4 q^{72} - 90 q^{73} - 25 q^{74} + 15 q^{75} - 64 q^{76} - 39 q^{77} - 28 q^{78} - 120 q^{79} + 24 q^{80} - 77 q^{81} - 71 q^{82} - 33 q^{83} - 14 q^{84} - 71 q^{85} - 61 q^{86} - 6 q^{87} - 34 q^{88} - 78 q^{89} - 88 q^{90} - 28 q^{91} - 31 q^{92} - 36 q^{93} - 4 q^{94} - 12 q^{95} - 29 q^{96} - 48 q^{97} - 4 q^{98} - 43 q^{99}+O(q^{100}) \) 59 * q - 5 * q^2 - 6 * q^3 + 41 * q^4 - 5 * q^5 - 7 * q^6 - 10 * q^7 - 12 * q^8 + 23 * q^9 - 18 * q^10 - 27 * q^11 - 8 * q^12 - 22 * q^13 - 24 * q^14 - 18 * q^15 + 5 * q^16 - 23 * q^17 + q^18 - 32 * q^19 - 14 * q^20 - 36 * q^21 - 6 * q^22 - 3 * q^23 - 18 * q^24 - 8 * q^25 - q^26 - 12 * q^27 - 9 * q^28 + 59 * q^29 - 18 * q^30 - 32 * q^31 - 39 * q^32 - 12 * q^33 - 18 * q^34 - 9 * q^35 + 10 * q^36 - 44 * q^37 + 5 * q^38 - 27 * q^39 - 68 * q^40 - 44 * q^41 - 25 * q^42 - 40 * q^43 - 56 * q^44 - 39 * q^45 - 40 * q^46 - 20 * q^47 - 9 * q^48 - 39 * q^49 - 21 * q^50 - 28 * q^51 - 49 * q^52 - 31 * q^53 - 32 * q^54 - 32 * q^55 - 48 * q^56 - 58 * q^57 - 5 * q^58 + 6 * q^59 - 44 * q^60 - 88 * q^61 + 35 * q^62 - 22 * q^63 - 10 * q^64 - 43 * q^65 - 31 * q^66 - 45 * q^67 - 29 * q^68 - 60 * q^69 - 14 * q^70 - 20 * q^71 - 4 * q^72 - 90 * q^73 - 25 * q^74 + 15 * q^75 - 64 * q^76 - 39 * q^77 - 28 * q^78 - 120 * q^79 + 24 * q^80 - 77 * q^81 - 71 * q^82 - 33 * q^83 - 14 * q^84 - 71 * q^85 - 61 * q^86 - 6 * q^87 - 34 * q^88 - 78 * q^89 - 88 * q^90 - 28 * q^91 - 31 * q^92 - 36 * q^93 - 4 * q^94 - 12 * q^95 - 29 * q^96 - 48 * q^97 - 4 * q^98 - 43 * q^99

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