LMFDB - Embedded newform 1045.2.a.d.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,2,Mod(1,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1045.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:	$8.34436701122$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\Q(\zeta_{22})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 $ x^5 - x^4 - 4x^3 + 3x^2 + 3*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.30972$ of defining polynomial
Character	$\chi$	$=$	1045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.22871 q^{2} -1.47889 q^{3} +2.96714 q^{4} +1.00000 q^{5} -3.29602 q^{6} -4.42518 q^{7} +2.15546 q^{8} -0.812880 q^{9} +O(q^{10})$	\(q+2.22871 q^{2} -1.47889 q^{3} +2.96714 q^{4} +1.00000 q^{5} -3.29602 q^{6} -4.42518 q^{7} +2.15546 q^{8} -0.812880 q^{9} +2.22871 q^{10} -1.00000 q^{11} -4.38807 q^{12} -3.92613 q^{13} -9.86243 q^{14} -1.47889 q^{15} -1.13037 q^{16} -1.61520 q^{17} -1.81167 q^{18} +1.00000 q^{19} +2.96714 q^{20} +6.54436 q^{21} -2.22871 q^{22} +0.113248 q^{23} -3.18770 q^{24} +1.00000 q^{25} -8.75019 q^{26} +5.63884 q^{27} -13.1301 q^{28} +1.08612 q^{29} -3.29602 q^{30} -4.17296 q^{31} -6.83020 q^{32} +1.47889 q^{33} -3.59980 q^{34} -4.42518 q^{35} -2.41193 q^{36} -5.75406 q^{37} +2.22871 q^{38} +5.80632 q^{39} +2.15546 q^{40} +4.26750 q^{41} +14.5855 q^{42} +3.62271 q^{43} -2.96714 q^{44} -0.812880 q^{45} +0.252396 q^{46} -6.39567 q^{47} +1.67170 q^{48} +12.5822 q^{49} +2.22871 q^{50} +2.38870 q^{51} -11.6494 q^{52} +12.0154 q^{53} +12.5673 q^{54} -1.00000 q^{55} -9.53832 q^{56} -1.47889 q^{57} +2.42065 q^{58} +0.883911 q^{59} -4.38807 q^{60} +3.77712 q^{61} -9.30032 q^{62} +3.59714 q^{63} -12.9618 q^{64} -3.92613 q^{65} +3.29602 q^{66} -12.2192 q^{67} -4.79251 q^{68} -0.167481 q^{69} -9.86243 q^{70} +4.28400 q^{71} -1.75213 q^{72} -1.92676 q^{73} -12.8241 q^{74} -1.47889 q^{75} +2.96714 q^{76} +4.42518 q^{77} +12.9406 q^{78} -7.81004 q^{79} -1.13037 q^{80} -5.90059 q^{81} +9.51102 q^{82} -3.33410 q^{83} +19.4180 q^{84} -1.61520 q^{85} +8.07397 q^{86} -1.60626 q^{87} -2.15546 q^{88} -11.9145 q^{89} -1.81167 q^{90} +17.3738 q^{91} +0.336021 q^{92} +6.17136 q^{93} -14.2541 q^{94} +1.00000 q^{95} +10.1011 q^{96} -13.3478 q^{97} +28.0421 q^{98} +0.812880 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 3 q^{2} - 7 q^{3} + 5 q^{4} + 5 q^{5} + 2 q^{6} - 11 q^{7} + 3 q^{8} + 8 q^{9}+O(q^{10}) $ 5 * q - 3 * q^2 - 7 * q^3 + 5 * q^4 + 5 * q^5 + 2 * q^6 - 11 * q^7 + 3 * q^8 + 8 * q^9	\( 5 q - 3 q^{2} - 7 q^{3} + 5 q^{4} + 5 q^{5} + 2 q^{6} - 11 q^{7} + 3 q^{8} + 8 q^{9} - 3 q^{10} - 5 q^{11} - 7 q^{12} + q^{13} - 7 q^{15} - 3 q^{16} - 3 q^{17} - 7 q^{18} + 5 q^{19} + 5 q^{20} + 11 q^{21} + 3 q^{22} - 8 q^{23} + 9 q^{24} + 5 q^{25} - 16 q^{26} - 10 q^{27} - 22 q^{28} + 11 q^{29} + 2 q^{30} - 5 q^{31} - 2 q^{32} + 7 q^{33} + 4 q^{34} - 11 q^{35} - 3 q^{36} - 9 q^{37} - 3 q^{38} - 8 q^{39} + 3 q^{40} + 15 q^{41} + 11 q^{42} - 13 q^{43} - 5 q^{44} + 8 q^{45} + 18 q^{46} - 20 q^{47} - 20 q^{48} + 20 q^{49} - 3 q^{50} + 24 q^{51} + q^{52} - 5 q^{53} + 17 q^{54} - 5 q^{55} - 7 q^{57} - 33 q^{58} - 17 q^{59} - 7 q^{60} + 3 q^{61} + 14 q^{62} - 22 q^{63} - 17 q^{64} + q^{65} - 2 q^{66} - 28 q^{67} - 25 q^{68} - 2 q^{69} - 6 q^{71} - 26 q^{72} - 16 q^{73} - 21 q^{74} - 7 q^{75} + 5 q^{76} + 11 q^{77} + 29 q^{78} + 3 q^{79} - 3 q^{80} + q^{81} + 2 q^{82} - 33 q^{83} + 33 q^{84} - 3 q^{85} + 10 q^{86} - 3 q^{88} - 16 q^{89} - 7 q^{90} - 22 q^{91} - 19 q^{92} - 26 q^{93} - 10 q^{94} + 5 q^{95} + 5 q^{96} - 14 q^{97} + 10 q^{98} - 8 q^{99}+O(q^{100}) \) 5 * q - 3 * q^2 - 7 * q^3 + 5 * q^4 + 5 * q^5 + 2 * q^6 - 11 * q^7 + 3 * q^8 + 8 * q^9 - 3 * q^10 - 5 * q^11 - 7 * q^12 + q^13 - 7 * q^15 - 3 * q^16 - 3 * q^17 - 7 * q^18 + 5 * q^19 + 5 * q^20 + 11 * q^21 + 3 * q^22 - 8 * q^23 + 9 * q^24 + 5 * q^25 - 16 * q^26 - 10 * q^27 - 22 * q^28 + 11 * q^29 + 2 * q^30 - 5 * q^31 - 2 * q^32 + 7 * q^33 + 4 * q^34 - 11 * q^35 - 3 * q^36 - 9 * q^37 - 3 * q^38 - 8 * q^39 + 3 * q^40 + 15 * q^41 + 11 * q^42 - 13 * q^43 - 5 * q^44 + 8 * q^45 + 18 * q^46 - 20 * q^47 - 20 * q^48 + 20 * q^49 - 3 * q^50 + 24 * q^51 + q^52 - 5 * q^53 + 17 * q^54 - 5 * q^55 - 7 * q^57 - 33 * q^58 - 17 * q^59 - 7 * q^60 + 3 * q^61 + 14 * q^62 - 22 * q^63 - 17 * q^64 + q^65 - 2 * q^66 - 28 * q^67 - 25 * q^68 - 2 * q^69 - 6 * q^71 - 26 * q^72 - 16 * q^73 - 21 * q^74 - 7 * q^75 + 5 * q^76 + 11 * q^77 + 29 * q^78 + 3 * q^79 - 3 * q^80 + q^81 + 2 * q^82 - 33 * q^83 + 33 * q^84 - 3 * q^85 + 10 * q^86 - 3 * q^88 - 16 * q^89 - 7 * q^90 - 22 * q^91 - 19 * q^92 - 26 * q^93 - 10 * q^94 + 5 * q^95 + 5 * q^96 - 14 * q^97 + 10 * q^98 - 8 * 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 $

$2$	2.22871	1.57593	0.787967	−	0.615717i	$-0.211134\pi$
$2$	2.22871	1.57593	0.787967	+	0.615717i	$0.211134\pi$
$3$	−1.47889	−0.853838	−0.426919	−	0.904290i	$-0.640401\pi$
$3$	−1.47889	−0.853838	−0.426919	+	0.904290i	$0.640401\pi$
$4$	2.96714	1.48357
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−3.29602	−1.34559
$7$	−4.42518	−1.67256	−0.836281	−	0.548302i	$-0.815275\pi$
$7$	−4.42518	−1.67256	−0.836281	+	0.548302i	$0.815275\pi$
$8$	2.15546	0.762072
$9$	−0.812880	−0.270960
$10$	2.22871	0.704779
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−4.38807	−1.26673
$13$	−3.92613	−1.08891	−0.544456	−	0.838789i	$-0.683264\pi$
$13$	−3.92613	−1.08891	−0.544456	+	0.838789i	$0.683264\pi$
$14$	−9.86243	−2.63585
$15$	−1.47889	−0.381848
$16$	−1.13037	−0.282593
$17$	−1.61520	−0.391743	−0.195872	−	0.980630i	$-0.562754\pi$
$17$	−1.61520	−0.391743	−0.195872	+	0.980630i	$0.562754\pi$
$18$	−1.81167	−0.427015
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	2.96714	0.663472
$21$	6.54436	1.42810
$22$	−2.22871	−0.475162
$23$	0.113248	0.0236138	0.0118069	−	0.999930i	$-0.496242\pi$
$23$	0.113248	0.0236138	0.0118069	+	0.999930i	$0.496242\pi$
$24$	−3.18770	−0.650686
$25$	1.00000	0.200000
$26$	−8.75019	−1.71605
$27$	5.63884	1.08519
$28$	−13.1301	−2.48136
$29$	1.08612	0.201688	0.100844	−	0.994902i	$-0.467846\pi$
$29$	1.08612	0.201688	0.100844	+	0.994902i	$0.467846\pi$
$30$	−3.29602	−0.601767
$31$	−4.17296	−0.749487	−0.374743	−	0.927129i	$-0.622269\pi$
$31$	−4.17296	−0.749487	−0.374743	+	0.927129i	$0.622269\pi$
$32$	−6.83020	−1.20742
$33$	1.47889	0.257442
$34$	−3.59980	−0.617361
$35$	−4.42518	−0.747992
$36$	−2.41193	−0.401988
$37$	−5.75406	−0.945962	−0.472981	−	0.881073i	$-0.656822\pi$
$37$	−5.75406	−0.945962	−0.472981	+	0.881073i	$0.656822\pi$
$38$	2.22871	0.361544
$39$	5.80632	0.929755
$40$	2.15546	0.340809
$41$	4.26750	0.666472	0.333236	−	0.942843i	$-0.391859\pi$
$41$	4.26750	0.666472	0.333236	+	0.942843i	$0.391859\pi$
$42$	14.5855	2.25059
$43$	3.62271	0.552459	0.276229	−	0.961092i	$-0.410915\pi$
$43$	3.62271	0.552459	0.276229	+	0.961092i	$0.410915\pi$
$44$	−2.96714	−0.447313
$45$	−0.812880	−0.121177
$46$	0.252396	0.0372138
$47$	−6.39567	−0.932904	−0.466452	−	0.884547i	$-0.654468\pi$
$47$	−6.39567	−0.932904	−0.466452	+	0.884547i	$0.654468\pi$
$48$	1.67170	0.241289
$49$	12.5822	1.79746
$50$	2.22871	0.315187
$51$	2.38870	0.334485
$52$	−11.6494	−1.61548
$53$	12.0154	1.65044	0.825219	−	0.564812i	$-0.191051\pi$
$53$	12.0154	1.65044	0.825219	+	0.564812i	$0.191051\pi$
$54$	12.5673	1.71019
$55$	−1.00000	−0.134840
$56$	−9.53832	−1.27461
$57$	−1.47889	−0.195884
$58$	2.42065	0.317847
$59$	0.883911	0.115075	0.0575377	−	0.998343i	$-0.481675\pi$
$59$	0.883911	0.115075	0.0575377	+	0.998343i	$0.481675\pi$
$60$	−4.38807	−0.566498
$61$	3.77712	0.483611	0.241805	−	0.970325i	$-0.422260\pi$
$61$	3.77712	0.483611	0.241805	+	0.970325i	$0.422260\pi$
$62$	−9.30032	−1.18114
$63$	3.59714	0.453197
$64$	−12.9618	−1.62022
$65$	−3.92613	−0.486976
$66$	3.29602	0.405712
$67$	−12.2192	−1.49282	−0.746409	−	0.665487i	$-0.768224\pi$
$67$	−12.2192	−1.49282	−0.746409	+	0.665487i	$0.768224\pi$
$68$	−4.79251	−0.581178
$69$	−0.167481	−0.0201624
$70$	−9.86243	−1.17879
$71$	4.28400	0.508417	0.254209	−	0.967149i	$-0.418185\pi$
$71$	4.28400	0.508417	0.254209	+	0.967149i	$0.418185\pi$
$72$	−1.75213	−0.206491
$73$	−1.92676	−0.225510	−0.112755	−	0.993623i	$-0.535968\pi$
$73$	−1.92676	−0.225510	−0.112755	+	0.993623i	$0.535968\pi$
$74$	−12.8241	−1.49077
$75$	−1.47889	−0.170768
$76$	2.96714	0.340354
$77$	4.42518	0.504296
$78$	12.9406	1.46523
$79$	−7.81004	−0.878698	−0.439349	−	0.898316i	$-0.644791\pi$
$79$	−7.81004	−0.878698	−0.439349	+	0.898316i	$0.644791\pi$
$80$	−1.13037	−0.126380
$81$	−5.90059	−0.655621
$82$	9.51102	1.05032
$83$	−3.33410	−0.365964	−0.182982	−	0.983116i	$-0.558575\pi$
$83$	−3.33410	−0.365964	−0.182982	+	0.983116i	$0.558575\pi$
$84$	19.4180	2.11868
$85$	−1.61520	−0.175193
$86$	8.07397	0.870639
$87$	−1.60626	−0.172209
$88$	−2.15546	−0.229773
$89$	−11.9145	−1.26293	−0.631466	−	0.775403i	$-0.717547\pi$
$89$	−11.9145	−1.26293	−0.631466	+	0.775403i	$0.717547\pi$
$90$	−1.81167	−0.190967
$91$	17.3738	1.82127
$92$	0.336021	0.0350327
$93$	6.17136	0.639940
$94$	−14.2541	−1.47019
$95$	1.00000	0.102598
$96$	10.1011	1.03094
$97$	−13.3478	−1.35526	−0.677631	−	0.735402i	$-0.736993\pi$
$97$	−13.3478	−1.35526	−0.677631	+	0.735402i	$0.736993\pi$
$98$	28.0421	2.83268
$99$	0.812880	0.0816975
$100$	2.96714	0.296714
$101$	3.72067	0.370221	0.185110	−	0.982718i	$-0.440736\pi$
$101$	3.72067	0.370221	0.185110	+	0.982718i	$0.440736\pi$
$102$	5.32372	0.527127
$103$	−18.7975	−1.85218	−0.926088	−	0.377308i	$-0.876850\pi$
$103$	−18.7975	−1.85218	−0.926088	+	0.377308i	$0.876850\pi$
$104$	−8.46263	−0.829829
$105$	6.54436	0.638664
$106$	26.7788	2.60098
$107$	−5.33910	−0.516150	−0.258075	−	0.966125i	$-0.583088\pi$
$107$	−5.33910	−0.516150	−0.258075	+	0.966125i	$0.583088\pi$
$108$	16.7312	1.60996
$109$	16.7781	1.60706	0.803528	−	0.595268i	$-0.202954\pi$
$109$	16.7781	1.60706	0.803528	+	0.595268i	$0.202954\pi$
$110$	−2.22871	−0.212499
$111$	8.50963	0.807698
$112$	5.00211	0.472655
$113$	9.64670	0.907485	0.453742	−	0.891133i	$-0.350089\pi$
$113$	9.64670	0.907485	0.453742	+	0.891133i	$0.350089\pi$
$114$	−3.29602	−0.308700
$115$	0.113248	0.0105604
$116$	3.22268	0.299218
$117$	3.19147	0.295052
$118$	1.96998	0.181351
$119$	7.14754	0.655214
$120$	−3.18770	−0.290996
$121$	1.00000	0.0909091
$122$	8.41809	0.762138
$123$	−6.31118	−0.569060
$124$	−12.3818	−1.11191
$125$	1.00000	0.0894427
$126$	8.01698	0.714209
$127$	0.515860	0.0457752	0.0228876	−	0.999738i	$-0.492714\pi$
$127$	0.515860	0.0457752	0.0228876	+	0.999738i	$0.492714\pi$
$128$	−15.2276	−1.34594
$129$	−5.35760	−0.471710
$130$	−8.75019	−0.767442
$131$	−2.12758	−0.185888	−0.0929439	−	0.995671i	$-0.529628\pi$
$131$	−2.12758	−0.185888	−0.0929439	+	0.995671i	$0.529628\pi$
$132$	4.38807	0.381933
$133$	−4.42518	−0.383712
$134$	−27.2331	−2.35258
$135$	5.63884	0.485314
$136$	−3.48150	−0.298536
$137$	3.76917	0.322022	0.161011	−	0.986953i	$-0.448525\pi$
$137$	3.76917	0.322022	0.161011	+	0.986953i	$0.448525\pi$
$138$	−0.373266	−0.0317745
$139$	−11.2237	−0.951981	−0.475990	−	0.879451i	$-0.657910\pi$
$139$	−11.2237	−0.951981	−0.475990	+	0.879451i	$0.657910\pi$
$140$	−13.1301	−1.10970
$141$	9.45849	0.796549
$142$	9.54778	0.801232
$143$	3.92613	0.328319
$144$	0.918858	0.0765715
$145$	1.08612	0.0901976
$146$	−4.29418	−0.355389
$147$	−18.6077	−1.53474
$148$	−17.0731	−1.40340
$149$	21.0620	1.72546	0.862732	−	0.505662i	$-0.168752\pi$
$149$	21.0620	1.72546	0.862732	+	0.505662i	$0.168752\pi$
$150$	−3.29602	−0.269119
$151$	7.46759	0.607704	0.303852	−	0.952719i	$-0.401727\pi$
$151$	7.46759	0.607704	0.303852	+	0.952719i	$0.401727\pi$
$152$	2.15546	0.174831
$153$	1.31296	0.106147
$154$	9.86243	0.794738
$155$	−4.17296	−0.335181
$156$	17.2281	1.37935
$157$	−5.84405	−0.466406	−0.233203	−	0.972428i	$-0.574921\pi$
$157$	−5.84405	−0.466406	−0.233203	+	0.972428i	$0.574921\pi$
$158$	−17.4063	−1.38477
$159$	−17.7694	−1.40921
$160$	−6.83020	−0.539975
$161$	−0.501142	−0.0394955
$162$	−13.1507	−1.03321
$163$	−20.3077	−1.59062	−0.795311	−	0.606202i	$-0.792692\pi$
$163$	−20.3077	−1.59062	−0.795311	+	0.606202i	$0.792692\pi$
$164$	12.6623	0.988757
$165$	1.47889	0.115132
$166$	−7.43072	−0.576736
$167$	10.6177	0.821625	0.410812	−	0.911720i	$-0.365245\pi$
$167$	10.6177	0.821625	0.410812	+	0.911720i	$0.365245\pi$
$168$	14.1061	1.08831
$169$	2.41448	0.185729
$170$	−3.59980	−0.276092
$171$	−0.812880	−0.0621625
$172$	10.7491	0.819610
$173$	24.7343	1.88051	0.940257	−	0.340466i	$-0.110585\pi$
$173$	24.7343	1.88051	0.940257	+	0.340466i	$0.110585\pi$
$174$	−3.57988	−0.271390
$175$	−4.42518	−0.334512
$176$	1.13037	0.0852051
$177$	−1.30721	−0.0982558
$178$	−26.5539	−1.99030
$179$	−14.9732	−1.11915	−0.559574	−	0.828781i	$-0.689035\pi$
$179$	−14.9732	−1.11915	−0.559574	+	0.828781i	$0.689035\pi$
$180$	−2.41193	−0.179774
$181$	−11.2782	−0.838298	−0.419149	−	0.907917i	$-0.637672\pi$
$181$	−11.2782	−0.838298	−0.419149	+	0.907917i	$0.637672\pi$
$182$	38.7212	2.87020
$183$	−5.58595	−0.412925
$184$	0.244101	0.0179954
$185$	−5.75406	−0.423047
$186$	13.7542	1.00850
$187$	1.61520	0.118115
$188$	−18.9768	−1.38403
$189$	−24.9529	−1.81505
$190$	2.22871	0.161687
$191$	22.3038	1.61385	0.806923	−	0.590656i	$-0.201131\pi$
$191$	22.3038	1.61385	0.806923	+	0.590656i	$0.201131\pi$
$192$	19.1691	1.38341
$193$	25.1845	1.81282	0.906412	−	0.422395i	$-0.138811\pi$
$193$	25.1845	1.81282	0.906412	+	0.422395i	$0.138811\pi$
$194$	−29.7483	−2.13580
$195$	5.80632	0.415799
$196$	37.3332	2.66666
$197$	−10.4483	−0.744413	−0.372207	−	0.928150i	$-0.621399\pi$
$197$	−10.4483	−0.744413	−0.372207	+	0.928150i	$0.621399\pi$
$198$	1.81167	0.128750
$199$	−6.40716	−0.454191	−0.227096	−	0.973872i	$-0.572923\pi$
$199$	−6.40716	−0.454191	−0.227096	+	0.973872i	$0.572923\pi$
$200$	2.15546	0.152414
$201$	18.0709	1.27463
$202$	8.29229	0.583444
$203$	−4.80629	−0.337336
$204$	7.08761	0.496232
$205$	4.26750	0.298055
$206$	−41.8942	−2.91891
$207$	−0.0920568	−0.00639839
$208$	4.43799	0.307719
$209$	−1.00000	−0.0691714
$210$	14.5855	1.00649
$211$	7.96892	0.548603	0.274302	−	0.961644i	$-0.411553\pi$
$211$	7.96892	0.548603	0.274302	+	0.961644i	$0.411553\pi$
$212$	35.6513	2.44854
$213$	−6.33557	−0.434106
$214$	−11.8993	−0.813418
$215$	3.62271	0.247067
$216$	12.1543	0.826996
$217$	18.4661	1.25356
$218$	37.3936	2.53261
$219$	2.84947	0.192549
$220$	−2.96714	−0.200044
$221$	6.34147	0.426574
$222$	18.9655	1.27288
$223$	−23.9096	−1.60111	−0.800553	−	0.599262i	$-0.795461\pi$
$223$	−23.9096	−1.60111	−0.800553	+	0.599262i	$0.795461\pi$
$224$	30.2249	2.01948
$225$	−0.812880	−0.0541920
$226$	21.4997	1.43014
$227$	−10.0128	−0.664575	−0.332287	−	0.943178i	$-0.607820\pi$
$227$	−10.0128	−0.664575	−0.332287	+	0.943178i	$0.607820\pi$
$228$	−4.38807	−0.290607
$229$	10.4511	0.690629	0.345314	−	0.938487i	$-0.387772\pi$
$229$	10.4511	0.690629	0.345314	+	0.938487i	$0.387772\pi$
$230$	0.252396	0.0166425
$231$	−6.54436	−0.430587
$232$	2.34110	0.153701
$233$	−4.56505	−0.299066	−0.149533	−	0.988757i	$-0.547777\pi$
$233$	−4.56505	−0.299066	−0.149533	+	0.988757i	$0.547777\pi$
$234$	7.11286	0.464982
$235$	−6.39567	−0.417207
$236$	2.62268	0.170722
$237$	11.5502	0.750266
$238$	15.9298	1.03257
$239$	−15.3794	−0.994808	−0.497404	−	0.867519i	$-0.665713\pi$
$239$	−15.3794	−0.994808	−0.497404	+	0.867519i	$0.665713\pi$
$240$	1.67170	0.107908
$241$	17.8737	1.15135	0.575673	−	0.817680i	$-0.304740\pi$
$241$	17.8737	1.15135	0.575673	+	0.817680i	$0.304740\pi$
$242$	2.22871	0.143267
$243$	−8.19018	−0.525400
$244$	11.2072	0.717469
$245$	12.5822	0.803849
$246$	−14.0658	−0.896800
$247$	−3.92613	−0.249814
$248$	−8.99468	−0.571163
$249$	4.93076	0.312475
$250$	2.22871	0.140956
$251$	−10.0425	−0.633878	−0.316939	−	0.948446i	$-0.602655\pi$
$251$	−10.0425	−0.633878	−0.316939	+	0.948446i	$0.602655\pi$
$252$	10.6732	0.672349
$253$	−0.113248	−0.00711982
$254$	1.14970	0.0721387
$255$	2.38870	0.149586
$256$	−8.01431	−0.500895
$257$	−10.7636	−0.671416	−0.335708	−	0.941966i	$-0.608975\pi$
$257$	−10.7636	−0.671416	−0.335708	+	0.941966i	$0.608975\pi$
$258$	−11.9405	−0.743385
$259$	25.4628	1.58218
$260$	−11.6494	−0.722463
$261$	−0.882888	−0.0546494
$262$	−4.74176	−0.292947
$263$	22.9691	1.41634	0.708169	−	0.706043i	$-0.249522\pi$
$263$	22.9691	1.41634	0.708169	+	0.706043i	$0.249522\pi$
$264$	3.18770	0.196189
$265$	12.0154	0.738099
$266$	−9.86243	−0.604705
$267$	17.6202	1.07834
$268$	−36.2562	−2.21470
$269$	−20.5868	−1.25520	−0.627600	−	0.778536i	$-0.715963\pi$
$269$	−20.5868	−1.25520	−0.627600	+	0.778536i	$0.715963\pi$
$270$	12.5673	0.764822
$271$	−4.27052	−0.259416	−0.129708	−	0.991552i	$-0.541404\pi$
$271$	−4.27052	−0.259416	−0.129708	+	0.991552i	$0.541404\pi$
$272$	1.82578	0.110704
$273$	−25.6940	−1.55507
$274$	8.40038	0.507485
$275$	−1.00000	−0.0603023
$276$	−0.496939	−0.0299122
$277$	−14.6746	−0.881712	−0.440856	−	0.897578i	$-0.645325\pi$
$277$	−14.6746	−0.881712	−0.440856	+	0.897578i	$0.645325\pi$
$278$	−25.0143	−1.50026
$279$	3.39212	0.203081
$280$	−9.53832	−0.570024
$281$	−24.1329	−1.43965	−0.719824	−	0.694157i	$-0.755778\pi$
$281$	−24.1329	−1.43965	−0.719824	+	0.694157i	$0.755778\pi$
$282$	21.0802	1.25531
$283$	0.197192	0.0117218	0.00586092	−	0.999983i	$-0.498134\pi$
$283$	0.197192	0.0117218	0.00586092	+	0.999983i	$0.498134\pi$
$284$	12.7112	0.754272
$285$	−1.47889	−0.0876020
$286$	8.75019	0.517410
$287$	−18.8845	−1.11472
$288$	5.55213	0.327163
$289$	−14.3911	−0.846537
$290$	2.42065	0.142146
$291$	19.7399	1.15717
$292$	−5.71695	−0.334559
$293$	2.73471	0.159763	0.0798817	−	0.996804i	$-0.474546\pi$
$293$	2.73471	0.159763	0.0798817	+	0.996804i	$0.474546\pi$
$294$	−41.4712	−2.41865
$295$	0.883911	0.0514633
$296$	−12.4027	−0.720891
$297$	−5.63884	−0.327198
$298$	46.9410	2.71922
$299$	−0.444625	−0.0257133
$300$	−4.38807	−0.253346
$301$	−16.0312	−0.924021
$302$	16.6431	0.957702
$303$	−5.50247	−0.316109
$304$	−1.13037	−0.0648313
$305$	3.77712	0.216277
$306$	2.92621	0.167280
$307$	−16.7313	−0.954905	−0.477452	−	0.878658i	$-0.658440\pi$
$307$	−16.7313	−0.954905	−0.477452	+	0.878658i	$0.658440\pi$
$308$	13.1301	0.748158
$309$	27.7995	1.58146
$310$	−9.30032	−0.528222
$311$	−14.8941	−0.844565	−0.422283	−	0.906464i	$-0.638771\pi$
$311$	−14.8941	−0.844565	−0.422283	+	0.906464i	$0.638771\pi$
$312$	12.5153	0.708540
$313$	8.89539	0.502797	0.251399	−	0.967884i	$-0.419109\pi$
$313$	8.89539	0.502797	0.251399	+	0.967884i	$0.419109\pi$
$314$	−13.0247	−0.735026
$315$	3.59714	0.202676
$316$	−23.1735	−1.30361
$317$	6.16319	0.346159	0.173079	−	0.984908i	$-0.444628\pi$
$317$	6.16319	0.346159	0.173079	+	0.984908i	$0.444628\pi$
$318$	−39.6029	−2.22082
$319$	−1.08612	−0.0608112
$320$	−12.9618	−0.724585
$321$	7.89595	0.440709
$322$	−1.11690	−0.0622423
$323$	−1.61520	−0.0898720
$324$	−17.5078	−0.972658
$325$	−3.92613	−0.217782
$326$	−45.2599	−2.50671
$327$	−24.8131	−1.37217
$328$	9.19846	0.507900
$329$	28.3020	1.56034
$330$	3.29602	0.181440
$331$	17.5144	0.962681	0.481340	−	0.876534i	$-0.340150\pi$
$331$	17.5144	0.962681	0.481340	+	0.876534i	$0.340150\pi$
$332$	−9.89272	−0.542933
$333$	4.67736	0.256318
$334$	23.6638	1.29483
$335$	−12.2192	−0.667609
$336$	−7.39757	−0.403571
$337$	−34.0733	−1.85609	−0.928045	−	0.372467i	$-0.878512\pi$
$337$	−34.0733	−1.85609	−0.928045	+	0.372467i	$0.878512\pi$
$338$	5.38117	0.292697
$339$	−14.2664	−0.774845
$340$	−4.79251	−0.259911
$341$	4.17296	0.225979
$342$	−1.81167	−0.0979640
$343$	−24.7024	−1.33380
$344$	7.80863	0.421013
$345$	−0.167481	−0.00901688
$346$	55.1255	2.96357
$347$	14.8065	0.794856	0.397428	−	0.917633i	$-0.369903\pi$
$347$	14.8065	0.794856	0.397428	+	0.917633i	$0.369903\pi$
$348$	−4.76599	−0.255484
$349$	−20.2032	−1.08145	−0.540727	−	0.841198i	$-0.681851\pi$
$349$	−20.2032	−1.08145	−0.540727	+	0.841198i	$0.681851\pi$
$350$	−9.86243	−0.527169
$351$	−22.1388	−1.18168
$352$	6.83020	0.364051
$353$	−34.8488	−1.85481	−0.927407	−	0.374053i	$-0.877968\pi$
$353$	−34.8488	−1.85481	−0.927407	+	0.374053i	$0.877968\pi$
$354$	−2.91338	−0.154845
$355$	4.28400	0.227371
$356$	−35.3519	−1.87365
$357$	−10.5704	−0.559447
$358$	−33.3708	−1.76370
$359$	−25.8010	−1.36173	−0.680863	−	0.732410i	$-0.738395\pi$
$359$	−25.8010	−1.36173	−0.680863	+	0.732410i	$0.738395\pi$
$360$	−1.75213	−0.0923456
$361$	1.00000	0.0526316
$362$	−25.1357	−1.32110
$363$	−1.47889	−0.0776217
$364$	51.5505	2.70198
$365$	−1.92676	−0.100851
$366$	−12.4494	−0.650743
$367$	12.6344	0.659510	0.329755	−	0.944066i	$-0.393034\pi$
$367$	12.6344	0.659510	0.329755	+	0.944066i	$0.393034\pi$
$368$	−0.128012	−0.00667310
$369$	−3.46897	−0.180587
$370$	−12.8241	−0.666694
$371$	−53.1702	−2.76046
$372$	18.3113	0.949395
$373$	31.3623	1.62388	0.811940	−	0.583741i	$-0.198412\pi$
$373$	31.3623	1.62388	0.811940	+	0.583741i	$0.198412\pi$
$374$	3.59980	0.186141
$375$	−1.47889	−0.0763696
$376$	−13.7856	−0.710940
$377$	−4.26426	−0.219620
$378$	−55.6126	−2.86041
$379$	−17.2730	−0.887256	−0.443628	−	0.896211i	$-0.646309\pi$
$379$	−17.2730	−0.887256	−0.443628	+	0.896211i	$0.646309\pi$
$380$	2.96714	0.152211
$381$	−0.762901	−0.0390846
$382$	49.7086	2.54332
$383$	1.96340	0.100325	0.0501625	−	0.998741i	$-0.484026\pi$
$383$	1.96340	0.100325	0.0501625	+	0.998741i	$0.484026\pi$
$384$	22.5200	1.14922
$385$	4.42518	0.225528
$386$	56.1290	2.85689
$387$	−2.94483	−0.149694
$388$	−39.6047	−2.01062
$389$	27.2460	1.38143	0.690714	−	0.723128i	$-0.257296\pi$
$389$	27.2460	1.38143	0.690714	+	0.723128i	$0.257296\pi$
$390$	12.9406	0.655272
$391$	−0.182917	−0.00925054
$392$	27.1206	1.36979
$393$	3.14646	0.158718
$394$	−23.2863	−1.17315
$395$	−7.81004	−0.392966
$396$	2.41193	0.121204
$397$	24.1194	1.21052	0.605258	−	0.796029i	$-0.293070\pi$
$397$	24.1194	1.21052	0.605258	+	0.796029i	$0.293070\pi$
$398$	−14.2797	−0.715776
$399$	6.54436	0.327628
$400$	−1.13037	−0.0565187
$401$	−18.6773	−0.932699	−0.466349	−	0.884601i	$-0.654431\pi$
$401$	−18.6773	−0.932699	−0.466349	+	0.884601i	$0.654431\pi$
$402$	40.2748	2.00873
$403$	16.3836	0.816125
$404$	11.0397	0.549248
$405$	−5.90059	−0.293202
$406$	−10.7118	−0.531619
$407$	5.75406	0.285218
$408$	5.14876	0.254902
$409$	−19.9442	−0.986176	−0.493088	−	0.869979i	$-0.664132\pi$
$409$	−19.9442	−0.986176	−0.493088	+	0.869979i	$0.664132\pi$
$410$	9.51102	0.469716
$411$	−5.57419	−0.274955
$412$	−55.7748	−2.74783
$413$	−3.91146	−0.192471
$414$	−0.205168	−0.0100834
$415$	−3.33410	−0.163664
$416$	26.8162	1.31477
$417$	16.5986	0.812838
$418$	−2.22871	−0.109010
$419$	22.0961	1.07946	0.539731	−	0.841837i	$-0.318526\pi$
$419$	22.0961	1.07946	0.539731	+	0.841837i	$0.318526\pi$
$420$	19.4180	0.947502
$421$	−25.4045	−1.23814	−0.619070	−	0.785336i	$-0.712490\pi$
$421$	−25.4045	−1.23814	−0.619070	+	0.785336i	$0.712490\pi$
$422$	17.7604	0.864563
$423$	5.19891	0.252780
$424$	25.8987	1.25775
$425$	−1.61520	−0.0783486
$426$	−14.1201	−0.684123
$427$	−16.7144	−0.808868
$428$	−15.8418	−0.765744
$429$	−5.80632	−0.280332
$430$	8.07397	0.389361
$431$	−3.51614	−0.169367	−0.0846833	−	0.996408i	$-0.526988\pi$
$431$	−3.51614	−0.169367	−0.0846833	+	0.996408i	$0.526988\pi$
$432$	−6.37399	−0.306669
$433$	−9.71737	−0.466987	−0.233493	−	0.972358i	$-0.575016\pi$
$433$	−9.71737	−0.466987	−0.233493	+	0.972358i	$0.575016\pi$
$434$	41.1556	1.97553
$435$	−1.60626	−0.0770142
$436$	49.7831	2.38418
$437$	0.113248	0.00541737
$438$	6.35062	0.303445
$439$	0.888955	0.0424275	0.0212138	−	0.999775i	$-0.493247\pi$
$439$	0.888955	0.0424275	0.0212138	+	0.999775i	$0.493247\pi$
$440$	−2.15546	−0.102758
$441$	−10.2278	−0.487040
$442$	14.1333	0.672252
$443$	5.44113	0.258516	0.129258	−	0.991611i	$-0.458740\pi$
$443$	5.44113	0.258516	0.129258	+	0.991611i	$0.458740\pi$
$444$	25.2492	1.19828
$445$	−11.9145	−0.564801
$446$	−53.2875	−2.52324
$447$	−31.1484	−1.47327
$448$	57.3582	2.70992
$449$	21.6076	1.01972	0.509862	−	0.860256i	$-0.329697\pi$
$449$	21.6076	1.01972	0.509862	+	0.860256i	$0.329697\pi$
$450$	−1.81167	−0.0854030
$451$	−4.26750	−0.200949
$452$	28.6231	1.34632
$453$	−11.0438	−0.518881
$454$	−22.3157	−1.04733
$455$	17.3738	0.814498
$456$	−3.18770	−0.149278
$457$	−29.1996	−1.36590	−0.682950	−	0.730465i	$-0.739303\pi$
$457$	−29.1996	−1.36590	−0.682950	+	0.730465i	$0.739303\pi$
$458$	23.2925	1.08839
$459$	−9.10784	−0.425117
$460$	0.336021	0.0156671
$461$	−32.2335	−1.50126	−0.750631	−	0.660722i	$-0.770250\pi$
$461$	−32.2335	−1.50126	−0.750631	+	0.660722i	$0.770250\pi$
$462$	−14.5855	−0.678577
$463$	−11.8543	−0.550917	−0.275459	−	0.961313i	$-0.588830\pi$
$463$	−11.8543	−0.550917	−0.275459	+	0.961313i	$0.588830\pi$
$464$	−1.22772	−0.0569957
$465$	6.17136	0.286190
$466$	−10.1742	−0.471309
$467$	−31.4900	−1.45718	−0.728592	−	0.684948i	$-0.759825\pi$
$467$	−31.4900	−1.45718	−0.728592	+	0.684948i	$0.759825\pi$
$468$	9.46953	0.437729
$469$	54.0724	2.49683
$470$	−14.2541	−0.657491
$471$	8.64272	0.398236
$472$	1.90524	0.0876957
$473$	−3.62271	−0.166573
$474$	25.7420	1.18237
$475$	1.00000	0.0458831
$476$	21.2077	0.972055
$477$	−9.76706	−0.447203
$478$	−34.2761	−1.56775
$479$	15.1565	0.692519	0.346259	−	0.938139i	$-0.387452\pi$
$479$	15.1565	0.692519	0.346259	+	0.938139i	$0.387452\pi$
$480$	10.1011	0.461051
$481$	22.5912	1.03007
$482$	39.8353	1.81445
$483$	0.741134	0.0337228
$484$	2.96714	0.134870
$485$	−13.3478	−0.606092
$486$	−18.2535	−0.827996
$487$	28.0154	1.26950	0.634749	−	0.772718i	$-0.281103\pi$
$487$	28.0154	1.26950	0.634749	+	0.772718i	$0.281103\pi$
$488$	8.14145	0.368546
$489$	30.0329	1.35813
$490$	28.0421	1.26681
$491$	−1.56411	−0.0705873	−0.0352937	−	0.999377i	$-0.511237\pi$
$491$	−1.56411	−0.0705873	−0.0352937	+	0.999377i	$0.511237\pi$
$492$	−18.7261	−0.844239
$493$	−1.75430	−0.0790099
$494$	−8.75019	−0.393690
$495$	0.812880	0.0365362
$496$	4.71701	0.211800
$497$	−18.9575	−0.850359
$498$	10.9892	0.492439
$499$	15.4416	0.691263	0.345631	−	0.938370i	$-0.387665\pi$
$499$	15.4416	0.691263	0.345631	+	0.938370i	$0.387665\pi$
$500$	2.96714	0.132694
$501$	−15.7025	−0.701535
$502$	−22.3818	−0.998950
$503$	−23.3185	−1.03972	−0.519861	−	0.854251i	$-0.674016\pi$
$503$	−23.3185	−1.03972	−0.519861	+	0.854251i	$0.674016\pi$
$504$	7.75351	0.345369
$505$	3.72067	0.165568
$506$	−0.252396	−0.0112204
$507$	−3.57075	−0.158583
$508$	1.53063	0.0679106
$509$	32.7538	1.45178	0.725892	−	0.687808i	$-0.241427\pi$
$509$	32.7538	1.45178	0.725892	+	0.687808i	$0.241427\pi$
$510$	5.32372	0.235738
$511$	8.52625	0.377179
$512$	12.5936	0.556565
$513$	5.63884	0.248961
$514$	−23.9889	−1.05811
$515$	−18.7975	−0.828318
$516$	−15.8967	−0.699815
$517$	6.39567	0.281281
$518$	56.7490	2.49341
$519$	−36.5793	−1.60565
$520$	−8.46263	−0.371111
$521$	38.5714	1.68984	0.844922	−	0.534889i	$-0.179647\pi$
$521$	38.5714	1.68984	0.844922	+	0.534889i	$0.179647\pi$
$522$	−1.96770	−0.0861238
$523$	34.9926	1.53012	0.765060	−	0.643959i	$-0.222709\pi$
$523$	34.9926	1.53012	0.765060	+	0.643959i	$0.222709\pi$
$524$	−6.31283	−0.275777
$525$	6.54436	0.285619
$526$	51.1915	2.23205
$527$	6.74016	0.293606
$528$	−1.67170	−0.0727514
$529$	−22.9872	−0.999442
$530$	26.7788	1.16319
$531$	−0.718513	−0.0311808
$532$	−13.1301	−0.569263
$533$	−16.7548	−0.725730
$534$	39.2703	1.69939
$535$	−5.33910	−0.230829
$536$	−26.3382	−1.13764
$537$	22.1437	0.955571
$538$	−45.8820	−1.97811
$539$	−12.5822	−0.541955
$540$	16.7312	0.719996
$541$	35.1455	1.51102	0.755511	−	0.655136i	$-0.227389\pi$
$541$	35.1455	1.51102	0.755511	+	0.655136i	$0.227389\pi$
$542$	−9.51774	−0.408822
$543$	16.6792	0.715771
$544$	11.0321	0.472999
$545$	16.7781	0.718697
$546$	−57.2644	−2.45069
$547$	2.79933	0.119691	0.0598454	−	0.998208i	$-0.480939\pi$
$547$	2.79933	0.119691	0.0598454	+	0.998208i	$0.480939\pi$
$548$	11.1836	0.477741
$549$	−3.07035	−0.131039
$550$	−2.22871	−0.0950324
$551$	1.08612	0.0462704
$552$	−0.361000	−0.0153652
$553$	34.5608	1.46968
$554$	−32.7054	−1.38952
$555$	8.50963	0.361214
$556$	−33.3022	−1.41233
$557$	−27.8118	−1.17843	−0.589213	−	0.807978i	$-0.700562\pi$
$557$	−27.8118	−1.17843	−0.589213	+	0.807978i	$0.700562\pi$
$558$	7.56004	0.320042
$559$	−14.2232	−0.601579
$560$	5.00211	0.211378
$561$	−2.38870	−0.100851
$562$	−53.7852	−2.26879
$563$	26.4265	1.11374	0.556872	−	0.830598i	$-0.312001\pi$
$563$	26.4265	1.11374	0.556872	+	0.830598i	$0.312001\pi$
$564$	28.0646	1.18173
$565$	9.64670	0.405840
$566$	0.439483	0.0184728
$567$	26.1112	1.09657
$568$	9.23401	0.387451
$569$	15.1625	0.635643	0.317822	−	0.948151i	$-0.397049\pi$
$569$	15.1625	0.635643	0.317822	+	0.948151i	$0.397049\pi$
$570$	−3.29602	−0.138055
$571$	−11.4144	−0.477677	−0.238838	−	0.971059i	$-0.576767\pi$
$571$	−11.4144	−0.477677	−0.238838	+	0.971059i	$0.576767\pi$
$572$	11.6494	0.487084
$573$	−32.9849	−1.37796
$574$	−42.0880	−1.75672
$575$	0.113248	0.00472276
$576$	10.5364	0.439015
$577$	−9.33855	−0.388769	−0.194385	−	0.980925i	$-0.562271\pi$
$577$	−9.33855	−0.388769	−0.194385	+	0.980925i	$0.562271\pi$
$578$	−32.0736	−1.33409
$579$	−37.2452	−1.54786
$580$	3.22268	0.133814
$581$	14.7540	0.612098
$582$	43.9945	1.82363
$583$	−12.0154	−0.497626
$584$	−4.15306	−0.171855
$585$	3.19147	0.131951
$586$	6.09486	0.251776
$587$	9.38542	0.387378	0.193689	−	0.981063i	$-0.437955\pi$
$587$	9.38542	0.387378	0.193689	+	0.981063i	$0.437955\pi$
$588$	−55.2117	−2.27689
$589$	−4.17296	−0.171944
$590$	1.96998	0.0811027
$591$	15.4520	0.635609
$592$	6.50424	0.267322
$593$	−35.6523	−1.46406	−0.732032	−	0.681270i	$-0.761428\pi$
$593$	−35.6523	−1.46406	−0.732032	+	0.681270i	$0.761428\pi$
$594$	−12.5673	−0.515643
$595$	7.14754	0.293021
$596$	62.4937	2.55984
$597$	9.47549	0.387806
$598$	−0.990939	−0.0405225
$599$	37.1404	1.51751	0.758757	−	0.651373i	$-0.225807\pi$
$599$	37.1404	1.51751	0.758757	+	0.651373i	$0.225807\pi$
$600$	−3.18770	−0.130137
$601$	12.2448	0.499474	0.249737	−	0.968314i	$-0.419656\pi$
$601$	12.2448	0.499474	0.249737	+	0.968314i	$0.419656\pi$
$602$	−35.7288	−1.45620
$603$	9.93278	0.404494
$604$	22.1574	0.901571
$605$	1.00000	0.0406558
$606$	−12.2634	−0.498167
$607$	31.3652	1.27307	0.636536	−	0.771247i	$-0.280366\pi$
$607$	31.3652	1.27307	0.636536	+	0.771247i	$0.280366\pi$
$608$	−6.83020	−0.277001
$609$	7.10798	0.288030
$610$	8.41809	0.340839
$611$	25.1102	1.01585
$612$	3.89574	0.157476
$613$	−5.48484	−0.221531	−0.110765	−	0.993847i	$-0.535330\pi$
$613$	−5.48484	−0.221531	−0.110765	+	0.993847i	$0.535330\pi$
$614$	−37.2891	−1.50487
$615$	−6.31118	−0.254491
$616$	9.53832	0.384310
$617$	−32.6913	−1.31610	−0.658052	−	0.752973i	$-0.728619\pi$
$617$	−32.6913	−1.31610	−0.658052	+	0.752973i	$0.728619\pi$
$618$	61.9570	2.49227
$619$	1.71800	0.0690522	0.0345261	−	0.999404i	$-0.489008\pi$
$619$	1.71800	0.0690522	0.0345261	+	0.999404i	$0.489008\pi$
$620$	−12.3818	−0.497263
$621$	0.638585	0.0256255
$622$	−33.1945	−1.33098
$623$	52.7238	2.11233
$624$	−6.56330	−0.262742
$625$	1.00000	0.0400000
$626$	19.8252	0.792375
$627$	1.47889	0.0590612
$628$	−17.3401	−0.691946
$629$	9.29395	0.370574
$630$	8.01698	0.319404
$631$	39.9355	1.58981	0.794903	−	0.606737i	$-0.207522\pi$
$631$	39.9355	1.58981	0.794903	+	0.606737i	$0.207522\pi$
$632$	−16.8343	−0.669631
$633$	−11.7852	−0.468419
$634$	13.7359	0.545524
$635$	0.515860	0.0204713
$636$	−52.7243	−2.09066
$637$	−49.3994	−1.95728
$638$	−2.42065	−0.0958345
$639$	−3.48238	−0.137761
$640$	−15.2276	−0.601924
$641$	−3.50123	−0.138290	−0.0691452	−	0.997607i	$-0.522027\pi$
$641$	−3.50123	−0.138290	−0.0691452	+	0.997607i	$0.522027\pi$
$642$	17.5978	0.694528
$643$	6.59740	0.260176	0.130088	−	0.991502i	$-0.458474\pi$
$643$	6.59740	0.260176	0.130088	+	0.991502i	$0.458474\pi$
$644$	−1.48696	−0.0585943
$645$	−5.35760	−0.210955
$646$	−3.59980	−0.141632
$647$	−24.8389	−0.976518	−0.488259	−	0.872699i	$-0.662368\pi$
$647$	−24.8389	−0.976518	−0.488259	+	0.872699i	$0.662368\pi$
$648$	−12.7185	−0.499630
$649$	−0.883911	−0.0346965
$650$	−8.75019	−0.343211
$651$	−27.3094	−1.07034
$652$	−60.2557	−2.35980
$653$	−17.5186	−0.685555	−0.342778	−	0.939417i	$-0.611368\pi$
$653$	−17.5186	−0.685555	−0.342778	+	0.939417i	$0.611368\pi$
$654$	−55.3011	−2.16244
$655$	−2.12758	−0.0831315
$656$	−4.82387	−0.188341
$657$	1.56622	0.0611042
$658$	63.0768	2.45899
$659$	3.56748	0.138969	0.0694846	−	0.997583i	$-0.477865\pi$
$659$	3.56748	0.138969	0.0694846	+	0.997583i	$0.477865\pi$
$660$	4.38807	0.170806
$661$	−48.5242	−1.88737	−0.943687	−	0.330841i	$-0.892668\pi$
$661$	−48.5242	−1.88737	−0.943687	+	0.330841i	$0.892668\pi$
$662$	39.0346	1.51712
$663$	−9.37835	−0.364225
$664$	−7.18652	−0.278891
$665$	−4.42518	−0.171601
$666$	10.4245	0.403940
$667$	0.123001	0.00476262
$668$	31.5043	1.21894
$669$	35.3597	1.36709
$670$	−27.2331	−1.05211
$671$	−3.77712	−0.145814
$672$	−44.6993	−1.72431
$673$	−0.476829	−0.0183804	−0.00919020	−	0.999958i	$-0.502925\pi$
$673$	−0.476829	−0.0183804	−0.00919020	+	0.999958i	$0.502925\pi$
$674$	−75.9394	−2.92508
$675$	5.63884	0.217039
$676$	7.16409	0.275542
$677$	3.68052	0.141454	0.0707269	−	0.997496i	$-0.477468\pi$
$677$	3.68052	0.141454	0.0707269	+	0.997496i	$0.477468\pi$
$678$	−31.7957	−1.22111
$679$	59.0664	2.26676
$680$	−3.48150	−0.133510
$681$	14.8079	0.567439
$682$	9.30032	0.356128
$683$	0.701724	0.0268507	0.0134254	−	0.999910i	$-0.495726\pi$
$683$	0.701724	0.0268507	0.0134254	+	0.999910i	$0.495726\pi$
$684$	−2.41193	−0.0922223
$685$	3.76917	0.144013
$686$	−55.0544	−2.10199
$687$	−15.4561	−0.589685
$688$	−4.09502	−0.156121
$689$	−47.1739	−1.79718
$690$	−0.373266	−0.0142100
$691$	3.65425	0.139014	0.0695071	−	0.997581i	$-0.477857\pi$
$691$	3.65425	0.139014	0.0695071	+	0.997581i	$0.477857\pi$
$692$	73.3900	2.78987
$693$	−3.59714	−0.136644
$694$	32.9994	1.25264
$695$	−11.2237	−0.425739
$696$	−3.46223	−0.131236
$697$	−6.89287	−0.261086
$698$	−45.0271	−1.70430
$699$	6.75121	0.255354
$700$	−13.1301	−0.496272
$701$	−45.3852	−1.71417	−0.857087	−	0.515171i	$-0.827728\pi$
$701$	−45.3852	−1.71417	−0.857087	+	0.515171i	$0.827728\pi$
$702$	−49.3409	−1.86225
$703$	−5.75406	−0.217019
$704$	12.9618	0.488515
$705$	9.45849	0.356228
$706$	−77.6678	−2.92307
$707$	−16.4647	−0.619217
$708$	−3.87866	−0.145769
$709$	16.6965	0.627051	0.313525	−	0.949580i	$-0.398490\pi$
$709$	16.6965	0.627051	0.313525	+	0.949580i	$0.398490\pi$
$710$	9.54778	0.358322
$711$	6.34862	0.238092
$712$	−25.6813	−0.962446
$713$	−0.472579	−0.0176982
$714$	−23.5584	−0.881652
$715$	3.92613	0.146829
$716$	−44.4274	−1.66033
$717$	22.7444	0.849405
$718$	−57.5030	−2.14599
$719$	16.1346	0.601718	0.300859	−	0.953669i	$-0.402727\pi$
$719$	16.1346	0.601718	0.300859	+	0.953669i	$0.402727\pi$
$720$	0.918858	0.0342438
$721$	83.1825	3.09788
$722$	2.22871	0.0829439
$723$	−26.4333	−0.983064
$724$	−33.4638	−1.24367
$725$	1.08612	0.0403376
$726$	−3.29602	−0.122327
$727$	−21.9546	−0.814251	−0.407126	−	0.913372i	$-0.633469\pi$
$727$	−21.9546	−0.814251	−0.407126	+	0.913372i	$0.633469\pi$
$728$	37.4487	1.38794
$729$	29.8141	1.10423
$730$	−4.29418	−0.158935
$731$	−5.85140	−0.216422
$732$	−16.5743	−0.612603
$733$	−47.5004	−1.75447	−0.877234	−	0.480063i	$-0.840614\pi$
$733$	−47.5004	−1.75447	−0.877234	+	0.480063i	$0.840614\pi$
$734$	28.1584	1.03935
$735$	−18.6077	−0.686357
$736$	−0.773505	−0.0285118
$737$	12.2192	0.450102
$738$	−7.73132	−0.284594
$739$	29.9890	1.10316	0.551581	−	0.834122i	$-0.314025\pi$
$739$	29.9890	1.10316	0.551581	+	0.834122i	$0.314025\pi$
$740$	−17.0731	−0.627619
$741$	5.80632	0.213300
$742$	−118.501	−4.35030
$743$	2.24383	0.0823181	0.0411590	−	0.999153i	$-0.486895\pi$
$743$	2.24383	0.0823181	0.0411590	+	0.999153i	$0.486895\pi$
$744$	13.3022	0.487681
$745$	21.0620	0.771651
$746$	69.8974	2.55913
$747$	2.71022	0.0991618
$748$	4.79251	0.175232
$749$	23.6265	0.863293
$750$	−3.29602	−0.120353
$751$	29.0275	1.05923	0.529614	−	0.848239i	$-0.322337\pi$
$751$	29.0275	1.05923	0.529614	+	0.848239i	$0.322337\pi$
$752$	7.22949	0.263632
$753$	14.8518	0.541229
$754$	−9.50378	−0.346107
$755$	7.46759	0.271774
$756$	−74.0386	−2.69276
$757$	47.0009	1.70828	0.854138	−	0.520046i	$-0.174085\pi$
$757$	47.0009	1.70828	0.854138	+	0.520046i	$0.174085\pi$
$758$	−38.4965	−1.39826
$759$	0.167481	0.00607918
$760$	2.15546	0.0781869
$761$	46.6332	1.69045	0.845226	−	0.534409i	$-0.179466\pi$
$761$	46.6332	1.69045	0.845226	+	0.534409i	$0.179466\pi$
$762$	−1.70028	−0.0615948
$763$	−74.2463	−2.68790
$764$	66.1784	2.39425
$765$	1.31296	0.0474703
$766$	4.37584	0.158106
$767$	−3.47035	−0.125307
$768$	11.8523	0.427683
$769$	−22.6169	−0.815585	−0.407793	−	0.913075i	$-0.633701\pi$
$769$	−22.6169	−0.815585	−0.407793	+	0.913075i	$0.633701\pi$
$770$	9.86243	0.355417
$771$	15.9182	0.573280
$772$	74.7260	2.68945
$773$	16.1951	0.582499	0.291249	−	0.956647i	$-0.405929\pi$
$773$	16.1951	0.582499	0.291249	+	0.956647i	$0.405929\pi$
$774$	−6.56317	−0.235908
$775$	−4.17296	−0.149897
$776$	−28.7707	−1.03281
$777$	−37.6567	−1.35093
$778$	60.7234	2.17704
$779$	4.26750	0.152899
$780$	17.2281	0.616866
$781$	−4.28400	−0.153294
$782$	−0.407670	−0.0145782
$783$	6.12447	0.218871
$784$	−14.2226	−0.507950
$785$	−5.84405	−0.208583
$786$	7.01255	0.250129
$787$	−27.6814	−0.986736	−0.493368	−	0.869821i	$-0.664234\pi$
$787$	−27.6814	−0.986736	−0.493368	+	0.869821i	$0.664234\pi$
$788$	−31.0016	−1.10439
$789$	−33.9689	−1.20932
$790$	−17.4063	−0.619288
$791$	−42.6884	−1.51782
$792$	1.75213	0.0622594
$793$	−14.8295	−0.526609
$794$	53.7550	1.90769
$795$	−17.7694	−0.630217
$796$	−19.0109	−0.673824
$797$	−4.49477	−0.159213	−0.0796065	−	0.996826i	$-0.525366\pi$
$797$	−4.49477	−0.159213	−0.0796065	+	0.996826i	$0.525366\pi$
$798$	14.5855	0.516320
$799$	10.3303	0.365459
$800$	−6.83020	−0.241484
$801$	9.68505	0.342204
$802$	−41.6262	−1.46987
$803$	1.92676	0.0679938
$804$	53.6189	1.89099
$805$	−0.501142	−0.0176629
$806$	36.5142	1.28616
$807$	30.4456	1.07174
$808$	8.01978	0.282135
$809$	48.0468	1.68924	0.844618	−	0.535369i	$-0.179828\pi$
$809$	48.0468	1.68924	0.844618	+	0.535369i	$0.179828\pi$
$810$	−13.1507	−0.462068
$811$	20.9156	0.734448	0.367224	−	0.930133i	$-0.380308\pi$
$811$	20.9156	0.734448	0.367224	+	0.930133i	$0.380308\pi$
$812$	−14.2609	−0.500460
$813$	6.31564	0.221499
$814$	12.8241	0.449485
$815$	−20.3077	−0.711347
$816$	−2.70013	−0.0945233
$817$	3.62271	0.126743
$818$	−44.4497	−1.55415
$819$	−14.1228	−0.493492
$820$	12.6623	0.442186
$821$	10.7228	0.374227	0.187114	−	0.982338i	$-0.440087\pi$
$821$	10.7228	0.374227	0.187114	+	0.982338i	$0.440087\pi$
$822$	−12.4232	−0.433310
$823$	42.5971	1.48484	0.742420	−	0.669935i	$-0.233678\pi$
$823$	42.5971	1.48484	0.742420	+	0.669935i	$0.233678\pi$
$824$	−40.5174	−1.41149
$825$	1.47889	0.0514884
$826$	−8.71751	−0.303321
$827$	−15.3377	−0.533343	−0.266672	−	0.963787i	$-0.585924\pi$
$827$	−15.3377	−0.533343	−0.266672	+	0.963787i	$0.585924\pi$
$828$	−0.273145	−0.00949245
$829$	−9.78901	−0.339986	−0.169993	−	0.985445i	$-0.554375\pi$
$829$	−9.78901	−0.339986	−0.169993	+	0.985445i	$0.554375\pi$
$830$	−7.43072	−0.257924
$831$	21.7022	0.752840
$832$	50.8896	1.76428
$833$	−20.3228	−0.704143
$834$	36.9935	1.28098
$835$	10.6177	0.367442
$836$	−2.96714	−0.102621
$837$	−23.5307	−0.813339
$838$	49.2456	1.70116
$839$	−48.1417	−1.66204	−0.831019	−	0.556244i	$-0.812242\pi$
$839$	−48.1417	−1.66204	−0.831019	+	0.556244i	$0.812242\pi$
$840$	14.1061	0.486708
$841$	−27.8203	−0.959322
$842$	−56.6192	−1.95123
$843$	35.6899	1.22923
$844$	23.6449	0.813891
$845$	2.41448	0.0830606
$846$	11.5868	0.398364
$847$	−4.42518	−0.152051
$848$	−13.5819	−0.466403
$849$	−0.291625	−0.0100086
$850$	−3.59980	−0.123472
$851$	−0.651634	−0.0223377
$852$	−18.7985	−0.644026
$853$	−41.0884	−1.40684	−0.703420	−	0.710775i	$-0.748345\pi$
$853$	−41.0884	−1.40684	−0.703420	+	0.710775i	$0.748345\pi$
$854$	−37.2516	−1.27472
$855$	−0.812880	−0.0277999
$856$	−11.5082	−0.393343
$857$	−30.0628	−1.02693	−0.513464	−	0.858111i	$-0.671638\pi$
$857$	−30.0628	−1.02693	−0.513464	+	0.858111i	$0.671638\pi$
$858$	−12.9406	−0.441784
$859$	9.99982	0.341190	0.170595	−	0.985341i	$-0.445431\pi$
$859$	9.99982	0.341190	0.170595	+	0.985341i	$0.445431\pi$
$860$	10.7491	0.366541
$861$	27.9281	0.951787
$862$	−7.83646	−0.266911
$863$	18.3253	0.623800	0.311900	−	0.950115i	$-0.399035\pi$
$863$	18.3253	0.623800	0.311900	+	0.950115i	$0.399035\pi$
$864$	−38.5144	−1.31029
$865$	24.7343	0.840991
$866$	−21.6572	−0.735940
$867$	21.2829	0.722806
$868$	54.7915	1.85975
$869$	7.81004	0.264937
$870$	−3.57988	−0.121369
$871$	47.9743	1.62555
$872$	36.1647	1.22469
$873$	10.8501	0.367222
$874$	0.252396	0.00853742
$875$	−4.42518	−0.149598
$876$	8.45475	0.285660
$877$	54.2556	1.83208	0.916040	−	0.401086i	$-0.131367\pi$
$877$	54.2556	1.83208	0.916040	+	0.401086i	$0.131367\pi$
$878$	1.98122	0.0668629
$879$	−4.04434	−0.136412
$880$	1.13037	0.0381049
$881$	−20.9741	−0.706636	−0.353318	−	0.935503i	$-0.614947\pi$
$881$	−20.9741	−0.706636	−0.353318	+	0.935503i	$0.614947\pi$
$882$	−22.7949	−0.767543
$883$	−16.2916	−0.548256	−0.274128	−	0.961693i	$-0.588389\pi$
$883$	−16.2916	−0.548256	−0.274128	+	0.961693i	$0.588389\pi$
$884$	18.8160	0.632851
$885$	−1.30721	−0.0439413
$886$	12.1267	0.407404
$887$	41.2926	1.38647	0.693234	−	0.720712i	$-0.256185\pi$
$887$	41.2926	1.38647	0.693234	+	0.720712i	$0.256185\pi$
$888$	18.3422	0.615524
$889$	−2.28277	−0.0765618
$890$	−26.5539	−0.890089
$891$	5.90059	0.197677
$892$	−70.9431	−2.37535
$893$	−6.39567	−0.214023
$894$	−69.4206	−2.32177
$895$	−14.9732	−0.500498
$896$	67.3849	2.25117
$897$	0.657552	0.0219550
$898$	48.1569	1.60702
$899$	−4.53235	−0.151162
$900$	−2.41193	−0.0803975
$901$	−19.4072	−0.646548
$902$	−9.51102	−0.316682
$903$	23.7084	0.788965
$904$	20.7931	0.691569
$905$	−11.2782	−0.374898
$906$	−24.6133	−0.817722
$907$	35.5669	1.18098	0.590490	−	0.807045i	$-0.298935\pi$
$907$	35.5669	1.18098	0.590490	+	0.807045i	$0.298935\pi$
$908$	−29.7094	−0.985942
$909$	−3.02446	−0.100315
$910$	38.7212	1.28359
$911$	26.1812	0.867422	0.433711	−	0.901052i	$-0.357204\pi$
$911$	26.1812	0.867422	0.433711	+	0.901052i	$0.357204\pi$
$912$	1.67170	0.0553555
$913$	3.33410	0.110342
$914$	−65.0774	−2.15257
$915$	−5.58595	−0.184666
$916$	31.0099	1.02459
$917$	9.41494	0.310909
$918$	−20.2987	−0.669957
$919$	0.328420	0.0108336	0.00541678	−	0.999985i	$-0.498276\pi$
$919$	0.328420	0.0108336	0.00541678	+	0.999985i	$0.498276\pi$
$920$	0.244101	0.00804779
$921$	24.7438	0.815334
$922$	−71.8390	−2.36589
$923$	−16.8195	−0.553622
$924$	−19.4180	−0.638806
$925$	−5.75406	−0.189192
$926$	−26.4198	−0.868210
$927$	15.2801	0.501866
$928$	−7.41844	−0.243522
$929$	0.642511	0.0210801	0.0105401	−	0.999944i	$-0.496645\pi$
$929$	0.642511	0.0210801	0.0105401	+	0.999944i	$0.496645\pi$
$930$	13.7542	0.451017
$931$	12.5822	0.412366
$932$	−13.5451	−0.443685
$933$	22.0267	0.721122
$934$	−70.1820	−2.29643
$935$	1.61520	0.0528226
$936$	6.87910	0.224851
$937$	19.2863	0.630055	0.315027	−	0.949083i	$-0.397986\pi$
$937$	19.2863	0.630055	0.315027	+	0.949083i	$0.397986\pi$
$938$	120.512	3.93484
$939$	−13.1553	−0.429308
$940$	−18.9768	−0.618955
$941$	19.4348	0.633556	0.316778	−	0.948500i	$-0.397399\pi$
$941$	19.4348	0.633556	0.316778	+	0.948500i	$0.397399\pi$
$942$	19.2621	0.627593
$943$	0.483285	0.0157379
$944$	−0.999149	−0.0325195
$945$	−24.9529	−0.811717
$946$	−8.07397	−0.262507
$947$	−61.1924	−1.98849	−0.994243	−	0.107147i	$-0.965828\pi$
$947$	−61.1924	−1.98849	−0.994243	+	0.107147i	$0.965828\pi$
$948$	34.2710	1.11307
$949$	7.56470	0.245560
$950$	2.22871	0.0723088
$951$	−9.11468	−0.295564
$952$	15.4063	0.499320
$953$	3.19828	0.103602	0.0518012	−	0.998657i	$-0.483504\pi$
$953$	3.19828	0.103602	0.0518012	+	0.998657i	$0.483504\pi$
$954$	−21.7679	−0.704762
$955$	22.3038	0.721734
$956$	−45.6327	−1.47587
$957$	1.60626	0.0519230
$958$	33.7794	1.09136
$959$	−16.6793	−0.538601
$960$	19.1691	0.618679
$961$	−13.5864	−0.438270
$962$	50.3491	1.62332
$963$	4.34005	0.139856
$964$	53.0337	1.70810
$965$	25.1845	0.810719
$966$	1.65177	0.0531449
$967$	45.2536	1.45526	0.727629	−	0.685970i	$-0.240622\pi$
$967$	45.2536	1.45526	0.727629	+	0.685970i	$0.240622\pi$
$968$	2.15546	0.0692793
$969$	2.38870	0.0767362
$970$	−29.7483	−0.955160
$971$	−49.4103	−1.58565	−0.792826	−	0.609448i	$-0.791391\pi$
$971$	−49.4103	−1.58565	−0.792826	+	0.609448i	$0.791391\pi$
$972$	−24.3014	−0.779467
$973$	49.6668	1.59225
$974$	62.4381	2.00065
$975$	5.80632	0.185951
$976$	−4.26955	−0.136665
$977$	48.8753	1.56366	0.781830	−	0.623492i	$-0.214286\pi$
$977$	48.8753	1.56366	0.781830	+	0.623492i	$0.214286\pi$
$978$	66.9345	2.14033
$979$	11.9145	0.380789
$980$	37.3332	1.19257
$981$	−13.6386	−0.435448
$982$	−3.48594	−0.111241
$983$	16.7325	0.533684	0.266842	−	0.963740i	$-0.414020\pi$
$983$	16.7325	0.533684	0.266842	+	0.963740i	$0.414020\pi$
$984$	−13.6035	−0.433664
$985$	−10.4483	−0.332912
$986$	−3.90983	−0.124514
$987$	−41.8556	−1.33228
$988$	−11.6494	−0.370615
$989$	0.410264	0.0130456
$990$	1.81167	0.0575787
$991$	−46.6422	−1.48164	−0.740820	−	0.671704i	$-0.765563\pi$
$991$	−46.6422	−1.48164	−0.740820	+	0.671704i	$0.765563\pi$
$992$	28.5022	0.904945
$993$	−25.9020	−0.821974
$994$	−42.2507	−1.34011
$995$	−6.40716	−0.203121
$996$	14.6303	0.463577
$997$	30.3768	0.962043	0.481021	−	0.876709i	$-0.340266\pi$
$997$	30.3768	0.962043	0.481021	+	0.876709i	$0.340266\pi$
$998$	34.4149	1.08938
$999$	−32.4462	−1.02655

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.2.a.d.1.5	✓	5
3.2	odd	2		9405.2.a.v.1.1		5
5.4	even	2		5225.2.a.j.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.a.d.1.5	✓	5	1.1	even	1	trivial
5225.2.a.j.1.1		5	5.4	even	2
9405.2.a.v.1.1		5	3.2	odd	2

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:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1045.a (trivial)

\(f(q)\)	\(=\)	\(q+2.22871 q^{2} -1.47889 q^{3} +2.96714 q^{4} +1.00000 q^{5} -3.29602 q^{6} -4.42518 q^{7} +2.15546 q^{8} -0.812880 q^{9} +O(q^{10})\)	\(q+2.22871 q^{2} -1.47889 q^{3} +2.96714 q^{4} +1.00000 q^{5} -3.29602 q^{6} -4.42518 q^{7} +2.15546 q^{8} -0.812880 q^{9} +2.22871 q^{10} -1.00000 q^{11} -4.38807 q^{12} -3.92613 q^{13} -9.86243 q^{14} -1.47889 q^{15} -1.13037 q^{16} -1.61520 q^{17} -1.81167 q^{18} +1.00000 q^{19} +2.96714 q^{20} +6.54436 q^{21} -2.22871 q^{22} +0.113248 q^{23} -3.18770 q^{24} +1.00000 q^{25} -8.75019 q^{26} +5.63884 q^{27} -13.1301 q^{28} +1.08612 q^{29} -3.29602 q^{30} -4.17296 q^{31} -6.83020 q^{32} +1.47889 q^{33} -3.59980 q^{34} -4.42518 q^{35} -2.41193 q^{36} -5.75406 q^{37} +2.22871 q^{38} +5.80632 q^{39} +2.15546 q^{40} +4.26750 q^{41} +14.5855 q^{42} +3.62271 q^{43} -2.96714 q^{44} -0.812880 q^{45} +0.252396 q^{46} -6.39567 q^{47} +1.67170 q^{48} +12.5822 q^{49} +2.22871 q^{50} +2.38870 q^{51} -11.6494 q^{52} +12.0154 q^{53} +12.5673 q^{54} -1.00000 q^{55} -9.53832 q^{56} -1.47889 q^{57} +2.42065 q^{58} +0.883911 q^{59} -4.38807 q^{60} +3.77712 q^{61} -9.30032 q^{62} +3.59714 q^{63} -12.9618 q^{64} -3.92613 q^{65} +3.29602 q^{66} -12.2192 q^{67} -4.79251 q^{68} -0.167481 q^{69} -9.86243 q^{70} +4.28400 q^{71} -1.75213 q^{72} -1.92676 q^{73} -12.8241 q^{74} -1.47889 q^{75} +2.96714 q^{76} +4.42518 q^{77} +12.9406 q^{78} -7.81004 q^{79} -1.13037 q^{80} -5.90059 q^{81} +9.51102 q^{82} -3.33410 q^{83} +19.4180 q^{84} -1.61520 q^{85} +8.07397 q^{86} -1.60626 q^{87} -2.15546 q^{88} -11.9145 q^{89} -1.81167 q^{90} +17.3738 q^{91} +0.336021 q^{92} +6.17136 q^{93} -14.2541 q^{94} +1.00000 q^{95} +10.1011 q^{96} -13.3478 q^{97} +28.0421 q^{98} +0.812880 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 3 q^{2} - 7 q^{3} + 5 q^{4} + 5 q^{5} + 2 q^{6} - 11 q^{7} + 3 q^{8} + 8 q^{9}+O(q^{10}) \) 5 * q - 3 * q^2 - 7 * q^3 + 5 * q^4 + 5 * q^5 + 2 * q^6 - 11 * q^7 + 3 * q^8 + 8 * q^9	\( 5 q - 3 q^{2} - 7 q^{3} + 5 q^{4} + 5 q^{5} + 2 q^{6} - 11 q^{7} + 3 q^{8} + 8 q^{9} - 3 q^{10} - 5 q^{11} - 7 q^{12} + q^{13} - 7 q^{15} - 3 q^{16} - 3 q^{17} - 7 q^{18} + 5 q^{19} + 5 q^{20} + 11 q^{21} + 3 q^{22} - 8 q^{23} + 9 q^{24} + 5 q^{25} - 16 q^{26} - 10 q^{27} - 22 q^{28} + 11 q^{29} + 2 q^{30} - 5 q^{31} - 2 q^{32} + 7 q^{33} + 4 q^{34} - 11 q^{35} - 3 q^{36} - 9 q^{37} - 3 q^{38} - 8 q^{39} + 3 q^{40} + 15 q^{41} + 11 q^{42} - 13 q^{43} - 5 q^{44} + 8 q^{45} + 18 q^{46} - 20 q^{47} - 20 q^{48} + 20 q^{49} - 3 q^{50} + 24 q^{51} + q^{52} - 5 q^{53} + 17 q^{54} - 5 q^{55} - 7 q^{57} - 33 q^{58} - 17 q^{59} - 7 q^{60} + 3 q^{61} + 14 q^{62} - 22 q^{63} - 17 q^{64} + q^{65} - 2 q^{66} - 28 q^{67} - 25 q^{68} - 2 q^{69} - 6 q^{71} - 26 q^{72} - 16 q^{73} - 21 q^{74} - 7 q^{75} + 5 q^{76} + 11 q^{77} + 29 q^{78} + 3 q^{79} - 3 q^{80} + q^{81} + 2 q^{82} - 33 q^{83} + 33 q^{84} - 3 q^{85} + 10 q^{86} - 3 q^{88} - 16 q^{89} - 7 q^{90} - 22 q^{91} - 19 q^{92} - 26 q^{93} - 10 q^{94} + 5 q^{95} + 5 q^{96} - 14 q^{97} + 10 q^{98} - 8 q^{99}+O(q^{100}) \) 5 * q - 3 * q^2 - 7 * q^3 + 5 * q^4 + 5 * q^5 + 2 * q^6 - 11 * q^7 + 3 * q^8 + 8 * q^9 - 3 * q^10 - 5 * q^11 - 7 * q^12 + q^13 - 7 * q^15 - 3 * q^16 - 3 * q^17 - 7 * q^18 + 5 * q^19 + 5 * q^20 + 11 * q^21 + 3 * q^22 - 8 * q^23 + 9 * q^24 + 5 * q^25 - 16 * q^26 - 10 * q^27 - 22 * q^28 + 11 * q^29 + 2 * q^30 - 5 * q^31 - 2 * q^32 + 7 * q^33 + 4 * q^34 - 11 * q^35 - 3 * q^36 - 9 * q^37 - 3 * q^38 - 8 * q^39 + 3 * q^40 + 15 * q^41 + 11 * q^42 - 13 * q^43 - 5 * q^44 + 8 * q^45 + 18 * q^46 - 20 * q^47 - 20 * q^48 + 20 * q^49 - 3 * q^50 + 24 * q^51 + q^52 - 5 * q^53 + 17 * q^54 - 5 * q^55 - 7 * q^57 - 33 * q^58 - 17 * q^59 - 7 * q^60 + 3 * q^61 + 14 * q^62 - 22 * q^63 - 17 * q^64 + q^65 - 2 * q^66 - 28 * q^67 - 25 * q^68 - 2 * q^69 - 6 * q^71 - 26 * q^72 - 16 * q^73 - 21 * q^74 - 7 * q^75 + 5 * q^76 + 11 * q^77 + 29 * q^78 + 3 * q^79 - 3 * q^80 + q^81 + 2 * q^82 - 33 * q^83 + 33 * q^84 - 3 * q^85 + 10 * q^86 - 3 * q^88 - 16 * q^89 - 7 * q^90 - 22 * q^91 - 19 * q^92 - 26 * q^93 - 10 * q^94 + 5 * q^95 + 5 * q^96 - 14 * q^97 + 10 * q^98 - 8 * q^99

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