LMFDB - Embedded newform 4022.2.a.e.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4022,2,Mod(1,4022)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4022, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4022.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4022 = 2 \cdot 2011 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4022.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:	$32.1158316930$
Analytic rank:	$0$
Dimension:	$46$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	4022.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.21023 q^{3} +1.00000 q^{4} -3.23048 q^{5} +2.21023 q^{6} +2.26977 q^{7} -1.00000 q^{8} +1.88514 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.21023 q^{3} +1.00000 q^{4} -3.23048 q^{5} +2.21023 q^{6} +2.26977 q^{7} -1.00000 q^{8} +1.88514 q^{9} +3.23048 q^{10} +2.49767 q^{11} -2.21023 q^{12} +3.08731 q^{13} -2.26977 q^{14} +7.14012 q^{15} +1.00000 q^{16} -3.61192 q^{17} -1.88514 q^{18} +5.76834 q^{19} -3.23048 q^{20} -5.01673 q^{21} -2.49767 q^{22} +5.42636 q^{23} +2.21023 q^{24} +5.43600 q^{25} -3.08731 q^{26} +2.46411 q^{27} +2.26977 q^{28} -3.86102 q^{29} -7.14012 q^{30} +8.49015 q^{31} -1.00000 q^{32} -5.52044 q^{33} +3.61192 q^{34} -7.33245 q^{35} +1.88514 q^{36} -4.62119 q^{37} -5.76834 q^{38} -6.82369 q^{39} +3.23048 q^{40} +1.08900 q^{41} +5.01673 q^{42} +5.37158 q^{43} +2.49767 q^{44} -6.08990 q^{45} -5.42636 q^{46} +3.52514 q^{47} -2.21023 q^{48} -1.84814 q^{49} -5.43600 q^{50} +7.98320 q^{51} +3.08731 q^{52} -0.846882 q^{53} -2.46411 q^{54} -8.06867 q^{55} -2.26977 q^{56} -12.7494 q^{57} +3.86102 q^{58} -2.99637 q^{59} +7.14012 q^{60} +3.91010 q^{61} -8.49015 q^{62} +4.27883 q^{63} +1.00000 q^{64} -9.97350 q^{65} +5.52044 q^{66} +10.6897 q^{67} -3.61192 q^{68} -11.9935 q^{69} +7.33245 q^{70} -4.18570 q^{71} -1.88514 q^{72} -7.88420 q^{73} +4.62119 q^{74} -12.0148 q^{75} +5.76834 q^{76} +5.66914 q^{77} +6.82369 q^{78} -13.9166 q^{79} -3.23048 q^{80} -11.1017 q^{81} -1.08900 q^{82} +4.13222 q^{83} -5.01673 q^{84} +11.6682 q^{85} -5.37158 q^{86} +8.53376 q^{87} -2.49767 q^{88} -13.6897 q^{89} +6.08990 q^{90} +7.00749 q^{91} +5.42636 q^{92} -18.7652 q^{93} -3.52514 q^{94} -18.6345 q^{95} +2.21023 q^{96} +3.12403 q^{97} +1.84814 q^{98} +4.70845 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 46 q - 46 q^{2} + 8 q^{3} + 46 q^{4} + 14 q^{5} - 8 q^{6} + 28 q^{7} - 46 q^{8} + 58 q^{9}+O(q^{10}) $ 46 * q - 46 * q^2 + 8 * q^3 + 46 * q^4 + 14 * q^5 - 8 * q^6 + 28 * q^7 - 46 * q^8 + 58 * q^9	\( 46 q - 46 q^{2} + 8 q^{3} + 46 q^{4} + 14 q^{5} - 8 q^{6} + 28 q^{7} - 46 q^{8} + 58 q^{9} - 14 q^{10} - 6 q^{11} + 8 q^{12} + 37 q^{13} - 28 q^{14} + 9 q^{15} + 46 q^{16} + 6 q^{17} - 58 q^{18} + 18 q^{19} + 14 q^{20} + 19 q^{21} + 6 q^{22} - 4 q^{23} - 8 q^{24} + 86 q^{25} - 37 q^{26} + 32 q^{27} + 28 q^{28} + 15 q^{29} - 9 q^{30} + 18 q^{31} - 46 q^{32} + 37 q^{33} - 6 q^{34} - 2 q^{35} + 58 q^{36} + 74 q^{37} - 18 q^{38} - 3 q^{39} - 14 q^{40} - 18 q^{41} - 19 q^{42} + 25 q^{43} - 6 q^{44} + 94 q^{45} + 4 q^{46} + 18 q^{47} + 8 q^{48} + 92 q^{49} - 86 q^{50} - 10 q^{51} + 37 q^{52} + 17 q^{53} - 32 q^{54} + 37 q^{55} - 28 q^{56} + 43 q^{57} - 15 q^{58} - 24 q^{59} + 9 q^{60} + 46 q^{61} - 18 q^{62} + 80 q^{63} + 46 q^{64} + 24 q^{65} - 37 q^{66} + 61 q^{67} + 6 q^{68} + 59 q^{69} + 2 q^{70} - 8 q^{71} - 58 q^{72} + 101 q^{73} - 74 q^{74} + 34 q^{75} + 18 q^{76} + 40 q^{77} + 3 q^{78} + 9 q^{79} + 14 q^{80} + 58 q^{81} + 18 q^{82} + 18 q^{83} + 19 q^{84} + 60 q^{85} - 25 q^{86} + 20 q^{87} + 6 q^{88} - 25 q^{89} - 94 q^{90} + 51 q^{91} - 4 q^{92} + 63 q^{93} - 18 q^{94} - 31 q^{95} - 8 q^{96} + 76 q^{97} - 92 q^{98} + 21 q^{99}+O(q^{100}) \) 46 * q - 46 * q^2 + 8 * q^3 + 46 * q^4 + 14 * q^5 - 8 * q^6 + 28 * q^7 - 46 * q^8 + 58 * q^9 - 14 * q^10 - 6 * q^11 + 8 * q^12 + 37 * q^13 - 28 * q^14 + 9 * q^15 + 46 * q^16 + 6 * q^17 - 58 * q^18 + 18 * q^19 + 14 * q^20 + 19 * q^21 + 6 * q^22 - 4 * q^23 - 8 * q^24 + 86 * q^25 - 37 * q^26 + 32 * q^27 + 28 * q^28 + 15 * q^29 - 9 * q^30 + 18 * q^31 - 46 * q^32 + 37 * q^33 - 6 * q^34 - 2 * q^35 + 58 * q^36 + 74 * q^37 - 18 * q^38 - 3 * q^39 - 14 * q^40 - 18 * q^41 - 19 * q^42 + 25 * q^43 - 6 * q^44 + 94 * q^45 + 4 * q^46 + 18 * q^47 + 8 * q^48 + 92 * q^49 - 86 * q^50 - 10 * q^51 + 37 * q^52 + 17 * q^53 - 32 * q^54 + 37 * q^55 - 28 * q^56 + 43 * q^57 - 15 * q^58 - 24 * q^59 + 9 * q^60 + 46 * q^61 - 18 * q^62 + 80 * q^63 + 46 * q^64 + 24 * q^65 - 37 * q^66 + 61 * q^67 + 6 * q^68 + 59 * q^69 + 2 * q^70 - 8 * q^71 - 58 * q^72 + 101 * q^73 - 74 * q^74 + 34 * q^75 + 18 * q^76 + 40 * q^77 + 3 * q^78 + 9 * q^79 + 14 * q^80 + 58 * q^81 + 18 * q^82 + 18 * q^83 + 19 * q^84 + 60 * q^85 - 25 * q^86 + 20 * q^87 + 6 * q^88 - 25 * q^89 - 94 * q^90 + 51 * q^91 - 4 * q^92 + 63 * q^93 - 18 * q^94 - 31 * q^95 - 8 * q^96 + 76 * q^97 - 92 * q^98 + 21 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−2.21023	−1.27608	−0.638040	−	0.770003i	$-0.720254\pi$
$3$	−2.21023	−1.27608	−0.638040	+	0.770003i	$0.720254\pi$
$4$	1.00000	0.500000
$5$	−3.23048	−1.44471	−0.722357	−	0.691520i	$-0.756941\pi$
$5$	−3.23048	−1.44471	−0.722357	+	0.691520i	$0.756941\pi$
$6$	2.21023	0.902325
$7$	2.26977	0.857893	0.428946	−	0.903330i	$-0.358885\pi$
$7$	2.26977	0.857893	0.428946	+	0.903330i	$0.358885\pi$
$8$	−1.00000	−0.353553
$9$	1.88514	0.628379
$10$	3.23048	1.02157
$11$	2.49767	0.753076	0.376538	−	0.926401i	$-0.377114\pi$
$11$	2.49767	0.753076	0.376538	+	0.926401i	$0.377114\pi$
$12$	−2.21023	−0.638040
$13$	3.08731	0.856267	0.428133	−	0.903716i	$-0.359171\pi$
$13$	3.08731	0.856267	0.428133	+	0.903716i	$0.359171\pi$
$14$	−2.26977	−0.606622
$15$	7.14012	1.84357
$16$	1.00000	0.250000
$17$	−3.61192	−0.876020	−0.438010	−	0.898970i	$-0.644317\pi$
$17$	−3.61192	−0.876020	−0.438010	+	0.898970i	$0.644317\pi$
$18$	−1.88514	−0.444331
$19$	5.76834	1.32335	0.661674	−	0.749792i	$-0.269846\pi$
$19$	5.76834	1.32335	0.661674	+	0.749792i	$0.269846\pi$
$20$	−3.23048	−0.722357
$21$	−5.01673	−1.09474
$22$	−2.49767	−0.532505
$23$	5.42636	1.13147	0.565737	−	0.824586i	$-0.308592\pi$
$23$	5.42636	1.13147	0.565737	+	0.824586i	$0.308592\pi$
$24$	2.21023	0.451162
$25$	5.43600	1.08720
$26$	−3.08731	−0.605472
$27$	2.46411	0.474218
$28$	2.26977	0.428946
$29$	−3.86102	−0.716973	−0.358486	−	0.933535i	$-0.616707\pi$
$29$	−3.86102	−0.716973	−0.358486	+	0.933535i	$0.616707\pi$
$30$	−7.14012	−1.30360
$31$	8.49015	1.52488	0.762438	−	0.647062i	$-0.224002\pi$
$31$	8.49015	1.52488	0.762438	+	0.647062i	$0.224002\pi$
$32$	−1.00000	−0.176777
$33$	−5.52044	−0.960985
$34$	3.61192	0.619440
$35$	−7.33245	−1.23941
$36$	1.88514	0.314190
$37$	−4.62119	−0.759719	−0.379859	−	0.925044i	$-0.624028\pi$
$37$	−4.62119	−0.759719	−0.379859	+	0.925044i	$0.624028\pi$
$38$	−5.76834	−0.935748
$39$	−6.82369	−1.09266
$40$	3.23048	0.510784
$41$	1.08900	0.170073	0.0850363	−	0.996378i	$-0.472899\pi$
$41$	1.08900	0.170073	0.0850363	+	0.996378i	$0.472899\pi$
$42$	5.01673	0.774098
$43$	5.37158	0.819159	0.409579	−	0.912274i	$-0.365675\pi$
$43$	5.37158	0.819159	0.409579	+	0.912274i	$0.365675\pi$
$44$	2.49767	0.376538
$45$	−6.08990	−0.907829
$46$	−5.42636	−0.800073
$47$	3.52514	0.514194	0.257097	−	0.966386i	$-0.417234\pi$
$47$	3.52514	0.514194	0.257097	+	0.966386i	$0.417234\pi$
$48$	−2.21023	−0.319020
$49$	−1.84814	−0.264020
$50$	−5.43600	−0.768766
$51$	7.98320	1.11787
$52$	3.08731	0.428133
$53$	−0.846882	−0.116328	−0.0581641	−	0.998307i	$-0.518525\pi$
$53$	−0.846882	−0.116328	−0.0581641	+	0.998307i	$0.518525\pi$
$54$	−2.46411	−0.335323
$55$	−8.06867	−1.08798
$56$	−2.26977	−0.303311
$57$	−12.7494	−1.68870
$58$	3.86102	0.506976
$59$	−2.99637	−0.390094	−0.195047	−	0.980794i	$-0.562486\pi$
$59$	−2.99637	−0.390094	−0.195047	+	0.980794i	$0.562486\pi$
$60$	7.14012	0.921785
$61$	3.91010	0.500637	0.250319	−	0.968163i	$-0.419465\pi$
$61$	3.91010	0.500637	0.250319	+	0.968163i	$0.419465\pi$
$62$	−8.49015	−1.07825
$63$	4.27883	0.539082
$64$	1.00000	0.125000
$65$	−9.97350	−1.23706
$66$	5.52044	0.679519
$67$	10.6897	1.30596	0.652979	−	0.757376i	$-0.273519\pi$
$67$	10.6897	1.30596	0.652979	+	0.757376i	$0.273519\pi$
$68$	−3.61192	−0.438010
$69$	−11.9935	−1.44385
$70$	7.33245	0.876395
$71$	−4.18570	−0.496751	−0.248376	−	0.968664i	$-0.579897\pi$
$71$	−4.18570	−0.496751	−0.248376	+	0.968664i	$0.579897\pi$
$72$	−1.88514	−0.222166
$73$	−7.88420	−0.922776	−0.461388	−	0.887198i	$-0.652648\pi$
$73$	−7.88420	−0.922776	−0.461388	+	0.887198i	$0.652648\pi$
$74$	4.62119	0.537202
$75$	−12.0148	−1.38735
$76$	5.76834	0.661674
$77$	5.66914	0.646059
$78$	6.82369	0.772630
$79$	−13.9166	−1.56574	−0.782870	−	0.622186i	$-0.786245\pi$
$79$	−13.9166	−1.56574	−0.782870	+	0.622186i	$0.786245\pi$
$80$	−3.23048	−0.361179
$81$	−11.1017	−1.23352
$82$	−1.08900	−0.120260
$83$	4.13222	0.453570	0.226785	−	0.973945i	$-0.427178\pi$
$83$	4.13222	0.453570	0.226785	+	0.973945i	$0.427178\pi$
$84$	−5.01673	−0.547370
$85$	11.6682	1.26560
$86$	−5.37158	−0.579233
$87$	8.53376	0.914915
$88$	−2.49767	−0.266253
$89$	−13.6897	−1.45110	−0.725550	−	0.688169i	$-0.758415\pi$
$89$	−13.6897	−1.45110	−0.725550	+	0.688169i	$0.758415\pi$
$90$	6.08990	0.641932
$91$	7.00749	0.734585
$92$	5.42636	0.565737
$93$	−18.7652	−1.94586
$94$	−3.52514	−0.363590
$95$	−18.6345	−1.91186
$96$	2.21023	0.225581
$97$	3.12403	0.317197	0.158599	−	0.987343i	$-0.449302\pi$
$97$	3.12403	0.317197	0.158599	+	0.987343i	$0.449302\pi$
$98$	1.84814	0.186690
$99$	4.70845	0.473217
$100$	5.43600	0.543600
$101$	−3.16515	−0.314944	−0.157472	−	0.987523i	$-0.550334\pi$
$101$	−3.16515	−0.314944	−0.157472	+	0.987523i	$0.550334\pi$
$102$	−7.98320	−0.790454
$103$	6.11702	0.602728	0.301364	−	0.953509i	$-0.402558\pi$
$103$	6.11702	0.602728	0.301364	+	0.953509i	$0.402558\pi$
$104$	−3.08731	−0.302736
$105$	16.2064	1.58159
$106$	0.846882	0.0822565
$107$	0.0697494	0.00674293	0.00337146	−	0.999994i	$-0.498927\pi$
$107$	0.0697494	0.00674293	0.00337146	+	0.999994i	$0.498927\pi$
$108$	2.46411	0.237109
$109$	1.89821	0.181816	0.0909078	−	0.995859i	$-0.471023\pi$
$109$	1.89821	0.181816	0.0909078	+	0.995859i	$0.471023\pi$
$110$	8.06867	0.769318
$111$	10.2139	0.969462
$112$	2.26977	0.214473
$113$	15.3702	1.44590	0.722952	−	0.690898i	$-0.242785\pi$
$113$	15.3702	1.44590	0.722952	+	0.690898i	$0.242785\pi$
$114$	12.7494	1.19409
$115$	−17.5297	−1.63466
$116$	−3.86102	−0.358486
$117$	5.82001	0.538060
$118$	2.99637	0.275838
$119$	−8.19824	−0.751531
$120$	−7.14012	−0.651801
$121$	−4.76164	−0.432876
$122$	−3.91010	−0.354004
$123$	−2.40694	−0.217026
$124$	8.49015	0.762438
$125$	−1.40848	−0.125979
$126$	−4.27883	−0.381189
$127$	−1.40000	−0.124230	−0.0621148	−	0.998069i	$-0.519785\pi$
$127$	−1.40000	−0.124230	−0.0621148	+	0.998069i	$0.519785\pi$
$128$	−1.00000	−0.0883883
$129$	−11.8725	−1.04531
$130$	9.97350	0.874734
$131$	−0.343460	−0.0300083	−0.0150041	−	0.999887i	$-0.504776\pi$
$131$	−0.343460	−0.0300083	−0.0150041	+	0.999887i	$0.504776\pi$
$132$	−5.52044	−0.480493
$133$	13.0928	1.13529
$134$	−10.6897	−0.923451
$135$	−7.96025	−0.685109
$136$	3.61192	0.309720
$137$	−0.0285898	−0.00244259	−0.00122129	−	0.999999i	$-0.500389\pi$
$137$	−0.0285898	−0.00244259	−0.00122129	+	0.999999i	$0.500389\pi$
$138$	11.9935	1.02096
$139$	−7.27767	−0.617284	−0.308642	−	0.951178i	$-0.599874\pi$
$139$	−7.27767	−0.617284	−0.308642	+	0.951178i	$0.599874\pi$
$140$	−7.33245	−0.619705
$141$	−7.79139	−0.656153
$142$	4.18570	0.351256
$143$	7.71109	0.644834
$144$	1.88514	0.157095
$145$	12.4729	1.03582
$146$	7.88420	0.652501
$147$	4.08482	0.336911
$148$	−4.62119	−0.379859
$149$	4.70488	0.385439	0.192720	−	0.981254i	$-0.438269\pi$
$149$	4.70488	0.385439	0.192720	+	0.981254i	$0.438269\pi$
$150$	12.0148	0.981007
$151$	−3.30670	−0.269096	−0.134548	−	0.990907i	$-0.542958\pi$
$151$	−3.30670	−0.269096	−0.134548	+	0.990907i	$0.542958\pi$
$152$	−5.76834	−0.467874
$153$	−6.80897	−0.550473
$154$	−5.66914	−0.456832
$155$	−27.4272	−2.20301
$156$	−6.82369	−0.546332
$157$	−3.89423	−0.310793	−0.155396	−	0.987852i	$-0.549666\pi$
$157$	−3.89423	−0.310793	−0.155396	+	0.987852i	$0.549666\pi$
$158$	13.9166	1.10715
$159$	1.87181	0.148444
$160$	3.23048	0.255392
$161$	12.3166	0.970683
$162$	11.1017	0.872229
$163$	7.19565	0.563607	0.281803	−	0.959472i	$-0.409067\pi$
$163$	7.19565	0.563607	0.281803	+	0.959472i	$0.409067\pi$
$164$	1.08900	0.0850363
$165$	17.8337	1.38835
$166$	−4.13222	−0.320723
$167$	−0.207905	−0.0160882	−0.00804409	−	0.999968i	$-0.502561\pi$
$167$	−0.207905	−0.0160882	−0.00804409	+	0.999968i	$0.502561\pi$
$168$	5.01673	0.387049
$169$	−3.46850	−0.266807
$170$	−11.6682	−0.894913
$171$	10.8741	0.831564
$172$	5.37158	0.409579
$173$	−23.3512	−1.77536	−0.887681	−	0.460459i	$-0.847685\pi$
$173$	−23.3512	−1.77536	−0.887681	+	0.460459i	$0.847685\pi$
$174$	−8.53376	−0.646942
$175$	12.3385	0.932701
$176$	2.49767	0.188269
$177$	6.62268	0.497791
$178$	13.6897	1.02608
$179$	15.5484	1.16214	0.581072	−	0.813852i	$-0.302633\pi$
$179$	15.5484	1.16214	0.581072	+	0.813852i	$0.302633\pi$
$180$	−6.08990	−0.453914
$181$	8.35829	0.621267	0.310633	−	0.950530i	$-0.399459\pi$
$181$	8.35829	0.621267	0.310633	+	0.950530i	$0.399459\pi$
$182$	−7.00749	−0.519430
$183$	−8.64225	−0.638853
$184$	−5.42636	−0.400036
$185$	14.9287	1.09758
$186$	18.7652	1.37593
$187$	−9.02139	−0.659710
$188$	3.52514	0.257097
$189$	5.59296	0.406828
$190$	18.6345	1.35189
$191$	3.09447	0.223908	0.111954	−	0.993713i	$-0.464289\pi$
$191$	3.09447	0.223908	0.111954	+	0.993713i	$0.464289\pi$
$192$	−2.21023	−0.159510
$193$	−12.1847	−0.877075	−0.438538	−	0.898713i	$-0.644503\pi$
$193$	−12.1847	−0.877075	−0.438538	+	0.898713i	$0.644503\pi$
$194$	−3.12403	−0.224292
$195$	22.0438	1.57859
$196$	−1.84814	−0.132010
$197$	17.4456	1.24295	0.621475	−	0.783434i	$-0.286534\pi$
$197$	17.4456	1.24295	0.621475	+	0.783434i	$0.286534\pi$
$198$	−4.70845	−0.334615
$199$	26.4742	1.87671	0.938355	−	0.345674i	$-0.112350\pi$
$199$	26.4742	1.87671	0.938355	+	0.345674i	$0.112350\pi$
$200$	−5.43600	−0.384383
$201$	−23.6268	−1.66651
$202$	3.16515	0.222699
$203$	−8.76363	−0.615086
$204$	7.98320	0.558935
$205$	−3.51798	−0.245706
$206$	−6.11702	−0.426193
$207$	10.2294	0.710995
$208$	3.08731	0.214067
$209$	14.4074	0.996582
$210$	−16.2064	−1.11835
$211$	7.48828	0.515514	0.257757	−	0.966210i	$-0.417017\pi$
$211$	7.48828	0.515514	0.257757	+	0.966210i	$0.417017\pi$
$212$	−0.846882	−0.0581641
$213$	9.25139	0.633894
$214$	−0.0697494	−0.00476797
$215$	−17.3528	−1.18345
$216$	−2.46411	−0.167661
$217$	19.2707	1.30818
$218$	−1.89821	−0.128563
$219$	17.4259	1.17754
$220$	−8.06867	−0.543990
$221$	−11.1511	−0.750106
$222$	−10.2139	−0.685513
$223$	2.58127	0.172855	0.0864274	−	0.996258i	$-0.472455\pi$
$223$	2.58127	0.172855	0.0864274	+	0.996258i	$0.472455\pi$
$224$	−2.26977	−0.151655
$225$	10.2476	0.683174
$226$	−15.3702	−1.02241
$227$	11.6117	0.770696	0.385348	−	0.922771i	$-0.374081\pi$
$227$	11.6117	0.770696	0.385348	+	0.922771i	$0.374081\pi$
$228$	−12.7494	−0.844349
$229$	25.0755	1.65703	0.828517	−	0.559963i	$-0.189185\pi$
$229$	25.0755	1.65703	0.828517	+	0.559963i	$0.189185\pi$
$230$	17.5297	1.15588
$231$	−12.5301	−0.824422
$232$	3.86102	0.253488
$233$	8.61795	0.564580	0.282290	−	0.959329i	$-0.408906\pi$
$233$	8.61795	0.564580	0.282290	+	0.959329i	$0.408906\pi$
$234$	−5.82001	−0.380466
$235$	−11.3879	−0.742864
$236$	−2.99637	−0.195047
$237$	30.7589	1.99801
$238$	8.19824	0.531413
$239$	2.26722	0.146654	0.0733272	−	0.997308i	$-0.476638\pi$
$239$	2.26722	0.146654	0.0733272	+	0.997308i	$0.476638\pi$
$240$	7.14012	0.460893
$241$	−11.7882	−0.759345	−0.379673	−	0.925121i	$-0.623963\pi$
$241$	−11.7882	−0.759345	−0.379673	+	0.925121i	$0.623963\pi$
$242$	4.76164	0.306090
$243$	17.1450	1.09985
$244$	3.91010	0.250319
$245$	5.97038	0.381433
$246$	2.40694	0.153461
$247$	17.8087	1.13314
$248$	−8.49015	−0.539125
$249$	−9.13318	−0.578792
$250$	1.40848	0.0890804
$251$	4.64335	0.293086	0.146543	−	0.989204i	$-0.453185\pi$
$251$	4.64335	0.293086	0.146543	+	0.989204i	$0.453185\pi$
$252$	4.27883	0.269541
$253$	13.5533	0.852086
$254$	1.40000	0.0878436
$255$	−25.7896	−1.61500
$256$	1.00000	0.0625000
$257$	21.7071	1.35405	0.677026	−	0.735959i	$-0.263268\pi$
$257$	21.7071	1.35405	0.677026	+	0.735959i	$0.263268\pi$
$258$	11.8725	0.739147
$259$	−10.4890	−0.651757
$260$	−9.97350	−0.618530
$261$	−7.27855	−0.450531
$262$	0.343460	0.0212191
$263$	3.92081	0.241767	0.120884	−	0.992667i	$-0.461427\pi$
$263$	3.92081	0.241767	0.120884	+	0.992667i	$0.461427\pi$
$264$	5.52044	0.339760
$265$	2.73584	0.168061
$266$	−13.0928	−0.802772
$267$	30.2573	1.85172
$268$	10.6897	0.652979
$269$	−10.8397	−0.660909	−0.330454	−	0.943822i	$-0.607202\pi$
$269$	−10.8397	−0.660909	−0.330454	+	0.943822i	$0.607202\pi$
$270$	7.96025	0.484445
$271$	−10.1907	−0.619038	−0.309519	−	0.950893i	$-0.600168\pi$
$271$	−10.1907	−0.619038	−0.309519	+	0.950893i	$0.600168\pi$
$272$	−3.61192	−0.219005
$273$	−15.4882	−0.937389
$274$	0.0285898	0.00172717
$275$	13.5773	0.818744
$276$	−11.9935	−0.721925
$277$	−3.44077	−0.206736	−0.103368	−	0.994643i	$-0.532962\pi$
$277$	−3.44077	−0.206736	−0.103368	+	0.994643i	$0.532962\pi$
$278$	7.27767	0.436486
$279$	16.0051	0.958200
$280$	7.33245	0.438198
$281$	−5.40308	−0.322321	−0.161160	−	0.986928i	$-0.551524\pi$
$281$	−5.40308	−0.322321	−0.161160	+	0.986928i	$0.551524\pi$
$282$	7.79139	0.463970
$283$	29.2609	1.73938	0.869689	−	0.493600i	$-0.164319\pi$
$283$	29.2609	1.73938	0.869689	+	0.493600i	$0.164319\pi$
$284$	−4.18570	−0.248376
$285$	41.1866	2.43969
$286$	−7.71109	−0.455966
$287$	2.47177	0.145904
$288$	−1.88514	−0.111083
$289$	−3.95402	−0.232589
$290$	−12.4729	−0.732436
$291$	−6.90484	−0.404769
$292$	−7.88420	−0.461388
$293$	0.427125	0.0249529	0.0124765	−	0.999922i	$-0.496029\pi$
$293$	0.427125	0.0249529	0.0124765	+	0.999922i	$0.496029\pi$
$294$	−4.08482	−0.238232
$295$	9.67972	0.563575
$296$	4.62119	0.268601
$297$	6.15453	0.357122
$298$	−4.70488	−0.272547
$299$	16.7529	0.968843
$300$	−12.0148	−0.693677
$301$	12.1923	0.702750
$302$	3.30670	0.190279
$303$	6.99572	0.401894
$304$	5.76834	0.330837
$305$	−12.6315	−0.723278
$306$	6.80897	0.389243
$307$	−15.4926	−0.884209	−0.442104	−	0.896964i	$-0.645768\pi$
$307$	−15.4926	−0.884209	−0.442104	+	0.896964i	$0.645768\pi$
$308$	5.66914	0.323029
$309$	−13.5201	−0.769129
$310$	27.4272	1.55776
$311$	−1.30534	−0.0740191	−0.0370095	−	0.999315i	$-0.511783\pi$
$311$	−1.30534	−0.0740191	−0.0370095	+	0.999315i	$0.511783\pi$
$312$	6.82369	0.386315
$313$	22.0946	1.24886	0.624429	−	0.781081i	$-0.285332\pi$
$313$	22.0946	1.24886	0.624429	+	0.781081i	$0.285332\pi$
$314$	3.89423	0.219764
$315$	−13.8227	−0.778820
$316$	−13.9166	−0.782870
$317$	−2.78974	−0.156687	−0.0783437	−	0.996926i	$-0.524963\pi$
$317$	−2.78974	−0.156687	−0.0783437	+	0.996926i	$0.524963\pi$
$318$	−1.87181	−0.104966
$319$	−9.64355	−0.539935
$320$	−3.23048	−0.180589
$321$	−0.154163	−0.00860451
$322$	−12.3166	−0.686377
$323$	−20.8348	−1.15928
$324$	−11.1017	−0.616759
$325$	16.7826	0.930933
$326$	−7.19565	−0.398530
$327$	−4.19549	−0.232011
$328$	−1.08900	−0.0601298
$329$	8.00126	0.441124
$330$	−17.8337	−0.981711
$331$	−3.04211	−0.167209	−0.0836047	−	0.996499i	$-0.526643\pi$
$331$	−3.04211	−0.167209	−0.0836047	+	0.996499i	$0.526643\pi$
$332$	4.13222	0.226785
$333$	−8.71158	−0.477391
$334$	0.207905	0.0113761
$335$	−34.5329	−1.88673
$336$	−5.01673	−0.273685
$337$	−20.2651	−1.10391	−0.551955	−	0.833874i	$-0.686118\pi$
$337$	−20.2651	−1.10391	−0.551955	+	0.833874i	$0.686118\pi$
$338$	3.46850	0.188661
$339$	−33.9717	−1.84509
$340$	11.6682	0.632799
$341$	21.2056	1.14835
$342$	−10.8741	−0.588005
$343$	−20.0833	−1.08439
$344$	−5.37158	−0.289616
$345$	38.7448	2.08595
$346$	23.3512	1.25537
$347$	−10.3599	−0.556148	−0.278074	−	0.960560i	$-0.589696\pi$
$347$	−10.3599	−0.556148	−0.278074	+	0.960560i	$0.589696\pi$
$348$	8.53376	0.457457
$349$	−13.8154	−0.739523	−0.369762	−	0.929127i	$-0.620561\pi$
$349$	−13.8154	−0.739523	−0.369762	+	0.929127i	$0.620561\pi$
$350$	−12.3385	−0.659519
$351$	7.60747	0.406057
$352$	−2.49767	−0.133126
$353$	6.39431	0.340335	0.170167	−	0.985415i	$-0.445569\pi$
$353$	6.39431	0.340335	0.170167	+	0.985415i	$0.445569\pi$
$354$	−6.62268	−0.351992
$355$	13.5218	0.717664
$356$	−13.6897	−0.725550
$357$	18.1200	0.959013
$358$	−15.5484	−0.821760
$359$	−14.7946	−0.780828	−0.390414	−	0.920639i	$-0.627668\pi$
$359$	−14.7946	−0.780828	−0.390414	+	0.920639i	$0.627668\pi$
$360$	6.08990	0.320966
$361$	14.2737	0.751250
$362$	−8.35829	−0.439302
$363$	10.5243	0.552385
$364$	7.00749	0.367292
$365$	25.4698	1.33315
$366$	8.64225	0.451737
$367$	4.38257	0.228768	0.114384	−	0.993437i	$-0.463511\pi$
$367$	4.38257	0.228768	0.114384	+	0.993437i	$0.463511\pi$
$368$	5.42636	0.282868
$369$	2.05291	0.106870
$370$	−14.9287	−0.776104
$371$	−1.92223	−0.0997971
$372$	−18.7652	−0.972931
$373$	9.31738	0.482436	0.241218	−	0.970471i	$-0.422453\pi$
$373$	9.31738	0.482436	0.241218	+	0.970471i	$0.422453\pi$
$374$	9.02139	0.466485
$375$	3.11308	0.160759
$376$	−3.52514	−0.181795
$377$	−11.9202	−0.613920
$378$	−5.59296	−0.287671
$379$	26.9464	1.38414	0.692071	−	0.721829i	$-0.256698\pi$
$379$	26.9464	1.38414	0.692071	+	0.721829i	$0.256698\pi$
$380$	−18.6345	−0.955930
$381$	3.09432	0.158527
$382$	−3.09447	−0.158327
$383$	25.8748	1.32214	0.661070	−	0.750325i	$-0.270103\pi$
$383$	25.8748	1.32214	0.661070	+	0.750325i	$0.270103\pi$
$384$	2.21023	0.112791
$385$	−18.3140	−0.933370
$386$	12.1847	0.620186
$387$	10.1262	0.514742
$388$	3.12403	0.158599
$389$	10.1469	0.514467	0.257234	−	0.966349i	$-0.417189\pi$
$389$	10.1469	0.514467	0.257234	+	0.966349i	$0.417189\pi$
$390$	−22.0438	−1.11623
$391$	−19.5996	−0.991193
$392$	1.84814	0.0933452
$393$	0.759128	0.0382929
$394$	−17.4456	−0.878899
$395$	44.9573	2.26205
$396$	4.70845	0.236609
$397$	12.6819	0.636484	0.318242	−	0.948009i	$-0.396908\pi$
$397$	12.6819	0.636484	0.318242	+	0.948009i	$0.396908\pi$
$398$	−26.4742	−1.32703
$399$	−28.9382	−1.44872
$400$	5.43600	0.271800
$401$	4.84526	0.241960	0.120980	−	0.992655i	$-0.461396\pi$
$401$	4.84526	0.241960	0.120980	+	0.992655i	$0.461396\pi$
$402$	23.6268	1.17840
$403$	26.2117	1.30570
$404$	−3.16515	−0.157472
$405$	35.8637	1.78208
$406$	8.76363	0.434931
$407$	−11.5422	−0.572126
$408$	−7.98320	−0.395227
$409$	−13.7701	−0.680890	−0.340445	−	0.940265i	$-0.610578\pi$
$409$	−13.7701	−0.680890	−0.340445	+	0.940265i	$0.610578\pi$
$410$	3.51798	0.173741
$411$	0.0631901	0.00311694
$412$	6.11702	0.301364
$413$	−6.80108	−0.334659
$414$	−10.2294	−0.502749
$415$	−13.3491	−0.655279
$416$	−3.08731	−0.151368
$417$	16.0854	0.787703
$418$	−14.4074	−0.704690
$419$	3.05989	0.149485	0.0747426	−	0.997203i	$-0.476186\pi$
$419$	3.05989	0.149485	0.0747426	+	0.997203i	$0.476186\pi$
$420$	16.2064	0.790793
$421$	5.62507	0.274149	0.137074	−	0.990561i	$-0.456230\pi$
$421$	5.62507	0.274149	0.137074	+	0.990561i	$0.456230\pi$
$422$	−7.48828	−0.364524
$423$	6.64537	0.323109
$424$	0.846882	0.0411282
$425$	−19.6344	−0.952408
$426$	−9.25139	−0.448231
$427$	8.87504	0.429493
$428$	0.0697494	0.00337146
$429$	−17.0433	−0.822859
$430$	17.3528	0.836826
$431$	−2.91104	−0.140220	−0.0701098	−	0.997539i	$-0.522335\pi$
$431$	−2.91104	−0.140220	−0.0701098	+	0.997539i	$0.522335\pi$
$432$	2.46411	0.118554
$433$	−15.3307	−0.736745	−0.368372	−	0.929678i	$-0.620085\pi$
$433$	−15.3307	−0.736745	−0.368372	+	0.929678i	$0.620085\pi$
$434$	−19.2707	−0.925023
$435$	−27.5681	−1.32179
$436$	1.89821	0.0909078
$437$	31.3011	1.49733
$438$	−17.4259	−0.832643
$439$	−3.95155	−0.188597	−0.0942986	−	0.995544i	$-0.530061\pi$
$439$	−3.95155	−0.188597	−0.0942986	+	0.995544i	$0.530061\pi$
$440$	8.06867	0.384659
$441$	−3.48400	−0.165905
$442$	11.1511	0.530405
$443$	7.27431	0.345613	0.172807	−	0.984956i	$-0.444716\pi$
$443$	7.27431	0.345613	0.172807	+	0.984956i	$0.444716\pi$
$444$	10.2139	0.484731
$445$	44.2241	2.09643
$446$	−2.58127	−0.122227
$447$	−10.3989	−0.491851
$448$	2.26977	0.107237
$449$	1.23775	0.0584132	0.0292066	−	0.999573i	$-0.490702\pi$
$449$	1.23775	0.0584132	0.0292066	+	0.999573i	$0.490702\pi$
$450$	−10.2476	−0.483077
$451$	2.71996	0.128078
$452$	15.3702	0.722952
$453$	7.30859	0.343387
$454$	−11.6117	−0.544964
$455$	−22.6376	−1.06127
$456$	12.7494	0.597045
$457$	3.68260	0.172265	0.0861325	−	0.996284i	$-0.472549\pi$
$457$	3.68260	0.172265	0.0861325	+	0.996284i	$0.472549\pi$
$458$	−25.0755	−1.17170
$459$	−8.90016	−0.415424
$460$	−17.5297	−0.817328
$461$	34.0284	1.58486	0.792431	−	0.609961i	$-0.208815\pi$
$461$	34.0284	1.58486	0.792431	+	0.609961i	$0.208815\pi$
$462$	12.5301	0.582955
$463$	−10.4929	−0.487647	−0.243824	−	0.969820i	$-0.578402\pi$
$463$	−10.4929	−0.487647	−0.243824	+	0.969820i	$0.578402\pi$
$464$	−3.86102	−0.179243
$465$	60.6207	2.81122
$466$	−8.61795	−0.399219
$467$	2.00426	0.0927460	0.0463730	−	0.998924i	$-0.485234\pi$
$467$	2.00426	0.0927460	0.0463730	+	0.998924i	$0.485234\pi$
$468$	5.82001	0.269030
$469$	24.2632	1.12037
$470$	11.3879	0.525284
$471$	8.60715	0.396597
$472$	2.99637	0.137919
$473$	13.4164	0.616889
$474$	−30.7589	−1.41281
$475$	31.3567	1.43874
$476$	−8.19824	−0.375766
$477$	−1.59649	−0.0730982
$478$	−2.26722	−0.103700
$479$	−1.74400	−0.0796855	−0.0398427	−	0.999206i	$-0.512686\pi$
$479$	−1.74400	−0.0796855	−0.0398427	+	0.999206i	$0.512686\pi$
$480$	−7.14012	−0.325900
$481$	−14.2671	−0.650522
$482$	11.7882	0.536938
$483$	−27.2226	−1.23867
$484$	−4.76164	−0.216438
$485$	−10.0921	−0.458259
$486$	−17.1450	−0.777712
$487$	−8.96872	−0.406412	−0.203206	−	0.979136i	$-0.565136\pi$
$487$	−8.96872	−0.406412	−0.203206	+	0.979136i	$0.565136\pi$
$488$	−3.91010	−0.177002
$489$	−15.9041	−0.719207
$490$	−5.97038	−0.269714
$491$	−0.606682	−0.0273792	−0.0136896	−	0.999906i	$-0.504358\pi$
$491$	−0.606682	−0.0273792	−0.0136896	+	0.999906i	$0.504358\pi$
$492$	−2.40694	−0.108513
$493$	13.9457	0.628083
$494$	−17.8087	−0.801250
$495$	−15.2106	−0.683664
$496$	8.49015	0.381219
$497$	−9.50059	−0.426160
$498$	9.13318	0.409268
$499$	13.7256	0.614442	0.307221	−	0.951638i	$-0.400601\pi$
$499$	13.7256	0.614442	0.307221	+	0.951638i	$0.400601\pi$
$500$	−1.40848	−0.0629893
$501$	0.459519	0.0205298
$502$	−4.64335	−0.207243
$503$	41.2281	1.83827	0.919135	−	0.393943i	$-0.128889\pi$
$503$	41.2281	1.83827	0.919135	+	0.393943i	$0.128889\pi$
$504$	−4.27883	−0.190594
$505$	10.2250	0.455004
$506$	−13.5533	−0.602516
$507$	7.66619	0.340468
$508$	−1.40000	−0.0621148
$509$	3.74178	0.165851	0.0829257	−	0.996556i	$-0.473574\pi$
$509$	3.74178	0.165851	0.0829257	+	0.996556i	$0.473574\pi$
$510$	25.7896	1.14198
$511$	−17.8953	−0.791643
$512$	−1.00000	−0.0441942
$513$	14.2138	0.627555
$514$	−21.7071	−0.957460
$515$	−19.7609	−0.870770
$516$	−11.8725	−0.522656
$517$	8.80464	0.387228
$518$	10.4890	0.460862
$519$	51.6117	2.26550
$520$	9.97350	0.437367
$521$	35.8138	1.56903	0.784515	−	0.620110i	$-0.212912\pi$
$521$	35.8138	1.56903	0.784515	+	0.620110i	$0.212912\pi$
$522$	7.27855	0.318574
$523$	6.37395	0.278714	0.139357	−	0.990242i	$-0.455497\pi$
$523$	6.37395	0.278714	0.139357	+	0.990242i	$0.455497\pi$
$524$	−0.343460	−0.0150041
$525$	−27.2709	−1.19020
$526$	−3.92081	−0.170955
$527$	−30.6657	−1.33582
$528$	−5.52044	−0.240246
$529$	6.44536	0.280233
$530$	−2.73584	−0.118837
$531$	−5.64857	−0.245127
$532$	13.0928	0.567645
$533$	3.36207	0.145628
$534$	−30.2573	−1.30936
$535$	−0.225324	−0.00974160
$536$	−10.6897	−0.461726
$537$	−34.3657	−1.48299
$538$	10.8397	0.467333
$539$	−4.61605	−0.198827
$540$	−7.96025	−0.342555
$541$	−21.2020	−0.911547	−0.455774	−	0.890096i	$-0.650637\pi$
$541$	−21.2020	−0.911547	−0.455774	+	0.890096i	$0.650637\pi$
$542$	10.1907	0.437726
$543$	−18.4738	−0.792786
$544$	3.61192	0.154860
$545$	−6.13213	−0.262672
$546$	15.4882	0.662834
$547$	34.4206	1.47172	0.735860	−	0.677134i	$-0.236778\pi$
$547$	34.4206	1.47172	0.735860	+	0.677134i	$0.236778\pi$
$548$	−0.0285898	−0.00122129
$549$	7.37108	0.314590
$550$	−13.5773	−0.578940
$551$	−22.2717	−0.948805
$552$	11.9935	0.510478
$553$	−31.5875	−1.34324
$554$	3.44077	0.146184
$555$	−32.9958	−1.40060
$556$	−7.27767	−0.308642
$557$	−30.4901	−1.29191	−0.645954	−	0.763377i	$-0.723540\pi$
$557$	−30.4901	−1.29191	−0.645954	+	0.763377i	$0.723540\pi$
$558$	−16.0051	−0.677550
$559$	16.5838	0.701418
$560$	−7.33245	−0.309853
$561$	19.9394	0.841842
$562$	5.40308	0.227915
$563$	0.883218	0.0372232	0.0186116	−	0.999827i	$-0.494075\pi$
$563$	0.883218	0.0372232	0.0186116	+	0.999827i	$0.494075\pi$
$564$	−7.79139	−0.328077
$565$	−49.6530	−2.08892
$566$	−29.2609	−1.22993
$567$	−25.1982	−1.05823
$568$	4.18570	0.175628
$569$	−21.5542	−0.903598	−0.451799	−	0.892120i	$-0.649217\pi$
$569$	−21.5542	−0.903598	−0.451799	+	0.892120i	$0.649217\pi$
$570$	−41.1866	−1.72512
$571$	−22.4023	−0.937505	−0.468753	−	0.883330i	$-0.655296\pi$
$571$	−22.4023	−0.937505	−0.468753	+	0.883330i	$0.655296\pi$
$572$	7.71109	0.322417
$573$	−6.83951	−0.285725
$574$	−2.47177	−0.103170
$575$	29.4977	1.23014
$576$	1.88514	0.0785474
$577$	24.9024	1.03670	0.518350	−	0.855169i	$-0.326547\pi$
$577$	24.9024	1.03670	0.518350	+	0.855169i	$0.326547\pi$
$578$	3.95402	0.164466
$579$	26.9311	1.11922
$580$	12.4729	0.517911
$581$	9.37920	0.389115
$582$	6.90484	0.286215
$583$	−2.11523	−0.0876040
$584$	7.88420	0.326251
$585$	−18.8014	−0.777343
$586$	−0.427125	−0.0176444
$587$	−10.8213	−0.446643	−0.223322	−	0.974745i	$-0.571690\pi$
$587$	−10.8213	−0.446643	−0.223322	+	0.974745i	$0.571690\pi$
$588$	4.08482	0.168455
$589$	48.9741	2.01794
$590$	−9.67972	−0.398508
$591$	−38.5590	−1.58610
$592$	−4.62119	−0.189930
$593$	−34.3270	−1.40964	−0.704820	−	0.709386i	$-0.748972\pi$
$593$	−34.3270	−1.40964	−0.704820	+	0.709386i	$0.748972\pi$
$594$	−6.15453	−0.252523
$595$	26.4842	1.08575
$596$	4.70488	0.192720
$597$	−58.5143	−2.39483
$598$	−16.7529	−0.685076
$599$	−11.0133	−0.449991	−0.224995	−	0.974360i	$-0.572237\pi$
$599$	−11.0133	−0.449991	−0.224995	+	0.974360i	$0.572237\pi$
$600$	12.0148	0.490504
$601$	10.9945	0.448476	0.224238	−	0.974534i	$-0.428011\pi$
$601$	10.9945	0.448476	0.224238	+	0.974534i	$0.428011\pi$
$602$	−12.1923	−0.496920
$603$	20.1516	0.820636
$604$	−3.30670	−0.134548
$605$	15.3824	0.625383
$606$	−6.99572	−0.284182
$607$	−34.7312	−1.40970	−0.704848	−	0.709359i	$-0.748985\pi$
$607$	−34.7312	−1.40970	−0.704848	+	0.709359i	$0.748985\pi$
$608$	−5.76834	−0.233937
$609$	19.3697	0.784899
$610$	12.6315	0.511435
$611$	10.8832	0.440288
$612$	−6.80897	−0.275236
$613$	−29.0246	−1.17229	−0.586146	−	0.810205i	$-0.699356\pi$
$613$	−29.0246	−1.17229	−0.586146	+	0.810205i	$0.699356\pi$
$614$	15.4926	0.625230
$615$	7.77556	0.313541
$616$	−5.66914	−0.228416
$617$	−45.1535	−1.81781	−0.908906	−	0.417001i	$-0.863081\pi$
$617$	−45.1535	−1.81781	−0.908906	+	0.417001i	$0.863081\pi$
$618$	13.5201	0.543856
$619$	19.1996	0.771698	0.385849	−	0.922562i	$-0.373909\pi$
$619$	19.1996	0.771698	0.385849	+	0.922562i	$0.373909\pi$
$620$	−27.4272	−1.10150
$621$	13.3711	0.536565
$622$	1.30534	0.0523394
$623$	−31.0724	−1.24489
$624$	−6.82369	−0.273166
$625$	−22.6299	−0.905197
$626$	−22.0946	−0.883076
$627$	−31.8438	−1.27172
$628$	−3.89423	−0.155396
$629$	16.6914	0.665529
$630$	13.8227	0.550709
$631$	24.9281	0.992371	0.496186	−	0.868217i	$-0.334734\pi$
$631$	24.9281	0.992371	0.496186	+	0.868217i	$0.334734\pi$
$632$	13.9166	0.553573
$633$	−16.5509	−0.657837
$634$	2.78974	0.110795
$635$	4.52266	0.179476
$636$	1.87181	0.0742220
$637$	−5.70579	−0.226072
$638$	9.64355	0.381792
$639$	−7.89063	−0.312148
$640$	3.23048	0.127696
$641$	23.0920	0.912081	0.456040	−	0.889959i	$-0.349267\pi$
$641$	23.0920	0.912081	0.456040	+	0.889959i	$0.349267\pi$
$642$	0.154163	0.00608431
$643$	21.8195	0.860476	0.430238	−	0.902716i	$-0.358430\pi$
$643$	21.8195	0.860476	0.430238	+	0.902716i	$0.358430\pi$
$644$	12.3166	0.485342
$645$	38.3537	1.51018
$646$	20.8348	0.819734
$647$	−24.2294	−0.952554	−0.476277	−	0.879295i	$-0.658014\pi$
$647$	−24.2294	−0.952554	−0.476277	+	0.879295i	$0.658014\pi$
$648$	11.1017	0.436115
$649$	−7.48395	−0.293771
$650$	−16.7826	−0.658269
$651$	−42.5927	−1.66934
$652$	7.19565	0.281803
$653$	40.2515	1.57516	0.787582	−	0.616210i	$-0.211333\pi$
$653$	40.2515	1.57516	0.787582	+	0.616210i	$0.211333\pi$
$654$	4.19549	0.164057
$655$	1.10954	0.0433534
$656$	1.08900	0.0425182
$657$	−14.8628	−0.579853
$658$	−8.00126	−0.311922
$659$	−3.20247	−0.124750	−0.0623752	−	0.998053i	$-0.519868\pi$
$659$	−3.20247	−0.124750	−0.0623752	+	0.998053i	$0.519868\pi$
$660$	17.8337	0.694175
$661$	40.7696	1.58576	0.792878	−	0.609381i	$-0.208582\pi$
$661$	40.7696	1.58576	0.792878	+	0.609381i	$0.208582\pi$
$662$	3.04211	0.118235
$663$	24.6466	0.957196
$664$	−4.13222	−0.160361
$665$	−42.2961	−1.64017
$666$	8.71158	0.337567
$667$	−20.9513	−0.811236
$668$	−0.207905	−0.00804409
$669$	−5.70522	−0.220576
$670$	34.5329	1.33412
$671$	9.76615	0.377018
$672$	5.01673	0.193524
$673$	35.6520	1.37428	0.687142	−	0.726523i	$-0.258865\pi$
$673$	35.6520	1.37428	0.687142	+	0.726523i	$0.258865\pi$
$674$	20.2651	0.780583
$675$	13.3949	0.515569
$676$	−3.46850	−0.133404
$677$	12.5482	0.482265	0.241132	−	0.970492i	$-0.422481\pi$
$677$	12.5482	0.482265	0.241132	+	0.970492i	$0.422481\pi$
$678$	33.9717	1.30468
$679$	7.09083	0.272121
$680$	−11.6682	−0.447457
$681$	−25.6646	−0.983470
$682$	−21.2056	−0.812004
$683$	3.90754	0.149518	0.0747589	−	0.997202i	$-0.476181\pi$
$683$	3.90754	0.149518	0.0747589	+	0.997202i	$0.476181\pi$
$684$	10.8741	0.415782
$685$	0.0923586	0.00352884
$686$	20.0833	0.766782
$687$	−55.4227	−2.11451
$688$	5.37158	0.204790
$689$	−2.61459	−0.0996080
$690$	−38.7448	−1.47499
$691$	−27.6296	−1.05108	−0.525539	−	0.850769i	$-0.676136\pi$
$691$	−27.6296	−1.05108	−0.525539	+	0.850769i	$0.676136\pi$
$692$	−23.3512	−0.887681
$693$	10.6871	0.405970
$694$	10.3599	0.393256
$695$	23.5104	0.891799
$696$	−8.53376	−0.323471
$697$	−3.93337	−0.148987
$698$	13.8154	0.522922
$699$	−19.0477	−0.720450
$700$	12.3385	0.466350
$701$	−14.1414	−0.534114	−0.267057	−	0.963681i	$-0.586051\pi$
$701$	−14.1414	−0.534114	−0.267057	+	0.963681i	$0.586051\pi$
$702$	−7.60747	−0.287125
$703$	−26.6566	−1.00537
$704$	2.49767	0.0941345
$705$	25.1699	0.947954
$706$	−6.39431	−0.240653
$707$	−7.18416	−0.270188
$708$	6.62268	0.248896
$709$	−47.4748	−1.78295	−0.891476	−	0.453068i	$-0.850329\pi$
$709$	−47.4748	−1.78295	−0.891476	+	0.453068i	$0.850329\pi$
$710$	−13.5218	−0.507465
$711$	−26.2347	−0.983878
$712$	13.6897	0.513041
$713$	46.0706	1.72536
$714$	−18.1200	−0.678125
$715$	−24.9105	−0.931601
$716$	15.5484	0.581072
$717$	−5.01110	−0.187143
$718$	14.7946	0.552129
$719$	−34.4143	−1.28344	−0.641719	−	0.766940i	$-0.721778\pi$
$719$	−34.4143	−1.28344	−0.641719	+	0.766940i	$0.721778\pi$
$720$	−6.08990	−0.226957
$721$	13.8842	0.517076
$722$	−14.2737	−0.531214
$723$	26.0547	0.968985
$724$	8.35829	0.310633
$725$	−20.9885	−0.779493
$726$	−10.5243	−0.390595
$727$	37.5191	1.39151	0.695753	−	0.718281i	$-0.255071\pi$
$727$	37.5191	1.39151	0.695753	+	0.718281i	$0.255071\pi$
$728$	−7.00749	−0.259715
$729$	−4.58941	−0.169978
$730$	−25.4698	−0.942678
$731$	−19.4017	−0.717599
$732$	−8.64225	−0.319427
$733$	43.6965	1.61397	0.806984	−	0.590574i	$-0.201098\pi$
$733$	43.6965	1.61397	0.806984	+	0.590574i	$0.201098\pi$
$734$	−4.38257	−0.161763
$735$	−13.1959	−0.486740
$736$	−5.42636	−0.200018
$737$	26.6994	0.983485
$738$	−2.05291	−0.0755686
$739$	−42.3253	−1.55696	−0.778481	−	0.627668i	$-0.784010\pi$
$739$	−42.3253	−1.55696	−0.778481	+	0.627668i	$0.784010\pi$
$740$	14.9287	0.548788
$741$	−39.3613	−1.44598
$742$	1.92223	0.0705672
$743$	−21.2461	−0.779443	−0.389722	−	0.920933i	$-0.627429\pi$
$743$	−21.2461	−0.779443	−0.389722	+	0.920933i	$0.627429\pi$
$744$	18.7652	0.687966
$745$	−15.1990	−0.556849
$746$	−9.31738	−0.341134
$747$	7.78981	0.285014
$748$	−9.02139	−0.329855
$749$	0.158315	0.00578471
$750$	−3.11308	−0.113674
$751$	25.1737	0.918601	0.459300	−	0.888281i	$-0.348100\pi$
$751$	25.1737	0.918601	0.459300	+	0.888281i	$0.348100\pi$
$752$	3.52514	0.128549
$753$	−10.2629	−0.374001
$754$	11.9202	0.434107
$755$	10.6822	0.388766
$756$	5.59296	0.203414
$757$	51.5581	1.87391	0.936955	−	0.349449i	$-0.113631\pi$
$757$	51.5581	1.87391	0.936955	+	0.349449i	$0.113631\pi$
$758$	−26.9464	−0.978737
$759$	−29.9559	−1.08733
$760$	18.6345	0.675945
$761$	−15.7964	−0.572618	−0.286309	−	0.958137i	$-0.592428\pi$
$761$	−15.7964	−0.572618	−0.286309	+	0.958137i	$0.592428\pi$
$762$	−3.09432	−0.112095
$763$	4.30850	0.155978
$764$	3.09447	0.111954
$765$	21.9962	0.795276
$766$	−25.8748	−0.934894
$767$	−9.25074	−0.334025
$768$	−2.21023	−0.0797550
$769$	−25.0898	−0.904760	−0.452380	−	0.891825i	$-0.649425\pi$
$769$	−25.0898	−0.904760	−0.452380	+	0.891825i	$0.649425\pi$
$770$	18.3140	0.659992
$771$	−47.9778	−1.72788
$772$	−12.1847	−0.438538
$773$	−2.83900	−0.102112	−0.0510559	−	0.998696i	$-0.516259\pi$
$773$	−2.83900	−0.102112	−0.0510559	+	0.998696i	$0.516259\pi$
$774$	−10.1262	−0.363978
$775$	46.1524	1.65784
$776$	−3.12403	−0.112146
$777$	23.1832	0.831694
$778$	−10.1469	−0.363783
$779$	6.28170	0.225065
$780$	22.0438	0.789294
$781$	−10.4545	−0.374092
$782$	19.5996	0.700880
$783$	−9.51396	−0.340001
$784$	−1.84814	−0.0660050
$785$	12.5802	0.449007
$786$	−0.759128	−0.0270772
$787$	37.4787	1.33597	0.667985	−	0.744175i	$-0.267157\pi$
$787$	37.4787	1.33597	0.667985	+	0.744175i	$0.267157\pi$
$788$	17.4456	0.621475
$789$	−8.66590	−0.308514
$790$	−44.9573	−1.59951
$791$	34.8868	1.24043
$792$	−4.70845	−0.167308
$793$	12.0717	0.428679
$794$	−12.6819	−0.450062
$795$	−6.04684	−0.214459
$796$	26.4742	0.938355
$797$	−28.1173	−0.995967	−0.497983	−	0.867187i	$-0.665926\pi$
$797$	−28.1173	−0.995967	−0.497983	+	0.867187i	$0.665926\pi$
$798$	28.9382	1.02440
$799$	−12.7325	−0.450445
$800$	−5.43600	−0.192192
$801$	−25.8069	−0.911841
$802$	−4.84526	−0.171092
$803$	−19.6921	−0.694921
$804$	−23.6268	−0.833253
$805$	−39.7885	−1.40236
$806$	−26.2117	−0.923269
$807$	23.9583	0.843372
$808$	3.16515	0.111350
$809$	8.13630	0.286057	0.143029	−	0.989719i	$-0.454316\pi$
$809$	8.13630	0.286057	0.143029	+	0.989719i	$0.454316\pi$
$810$	−35.8637	−1.26012
$811$	30.0472	1.05510	0.527549	−	0.849524i	$-0.323111\pi$
$811$	30.0472	1.05510	0.527549	+	0.849524i	$0.323111\pi$
$812$	−8.76363	−0.307543
$813$	22.5237	0.789942
$814$	11.5422	0.404554
$815$	−23.2454	−0.814251
$816$	7.98320	0.279468
$817$	30.9851	1.08403
$818$	13.7701	0.481462
$819$	13.2101	0.461598
$820$	−3.51798	−0.122853
$821$	−50.2366	−1.75327	−0.876634	−	0.481157i	$-0.840217\pi$
$821$	−50.2366	−1.75327	−0.876634	+	0.481157i	$0.840217\pi$
$822$	−0.0631901	−0.00220401
$823$	29.7552	1.03720	0.518601	−	0.855017i	$-0.326453\pi$
$823$	29.7552	1.03720	0.518601	+	0.855017i	$0.326453\pi$
$824$	−6.11702	−0.213097
$825$	−30.0091	−1.04478
$826$	6.80108	0.236640
$827$	50.6440	1.76107	0.880533	−	0.473986i	$-0.157185\pi$
$827$	50.6440	1.76107	0.880533	+	0.473986i	$0.157185\pi$
$828$	10.2294	0.355497
$829$	7.12162	0.247344	0.123672	−	0.992323i	$-0.460533\pi$
$829$	7.12162	0.247344	0.123672	+	0.992323i	$0.460533\pi$
$830$	13.3491	0.463352
$831$	7.60491	0.263812
$832$	3.08731	0.107033
$833$	6.67534	0.231287
$834$	−16.0854	−0.556990
$835$	0.671633	0.0232428
$836$	14.4074	0.498291
$837$	20.9206	0.723123
$838$	−3.05989	−0.105702
$839$	33.0258	1.14018	0.570089	−	0.821583i	$-0.306909\pi$
$839$	33.0258	1.14018	0.570089	+	0.821583i	$0.306909\pi$
$840$	−16.2064	−0.559175
$841$	−14.0925	−0.485950
$842$	−5.62507	−0.193853
$843$	11.9421	0.411307
$844$	7.48828	0.257757
$845$	11.2049	0.385461
$846$	−6.64537	−0.228473
$847$	−10.8078	−0.371362
$848$	−0.846882	−0.0290821
$849$	−64.6734	−2.21958
$850$	19.6344	0.673454
$851$	−25.0762	−0.859602
$852$	9.25139	0.316947
$853$	−6.63219	−0.227082	−0.113541	−	0.993533i	$-0.536219\pi$
$853$	−6.63219	−0.227082	−0.113541	+	0.993533i	$0.536219\pi$
$854$	−8.87504	−0.303698
$855$	−35.1286	−1.20137
$856$	−0.0697494	−0.00238398
$857$	−35.1947	−1.20223	−0.601114	−	0.799164i	$-0.705276\pi$
$857$	−35.1947	−1.20223	−0.601114	+	0.799164i	$0.705276\pi$
$858$	17.0433	0.581850
$859$	−41.0426	−1.40036	−0.700178	−	0.713968i	$-0.746896\pi$
$859$	−41.0426	−1.40036	−0.700178	+	0.713968i	$0.746896\pi$
$860$	−17.3528	−0.591725
$861$	−5.46320	−0.186185
$862$	2.91104	0.0991503
$863$	32.1608	1.09477	0.547384	−	0.836882i	$-0.315624\pi$
$863$	32.1608	1.09477	0.547384	+	0.836882i	$0.315624\pi$
$864$	−2.46411	−0.0838306
$865$	75.4357	2.56489
$866$	15.3307	0.520957
$867$	8.73931	0.296803
$868$	19.2707	0.654090
$869$	−34.7591	−1.17912
$870$	27.5681	0.934647
$871$	33.0025	1.11825
$872$	−1.89821	−0.0642815
$873$	5.88923	0.199320
$874$	−31.3011	−1.05877
$875$	−3.19694	−0.108076
$876$	17.4259	0.588768
$877$	−47.2927	−1.59696	−0.798480	−	0.602022i	$-0.794362\pi$
$877$	−47.2927	−1.59696	−0.798480	+	0.602022i	$0.794362\pi$
$878$	3.95155	0.133358
$879$	−0.944047	−0.0318419
$880$	−8.06867	−0.271995
$881$	42.4394	1.42982	0.714909	−	0.699217i	$-0.246468\pi$
$881$	42.4394	1.42982	0.714909	+	0.699217i	$0.246468\pi$
$882$	3.48400	0.117312
$883$	16.8202	0.566043	0.283022	−	0.959114i	$-0.408663\pi$
$883$	16.8202	0.566043	0.283022	+	0.959114i	$0.408663\pi$
$884$	−11.1511	−0.375053
$885$	−21.3944	−0.719167
$886$	−7.27431	−0.244385
$887$	6.08835	0.204427	0.102213	−	0.994762i	$-0.467408\pi$
$887$	6.08835	0.204427	0.102213	+	0.994762i	$0.467408\pi$
$888$	−10.2139	−0.342756
$889$	−3.17767	−0.106576
$890$	−44.2241	−1.48240
$891$	−27.7283	−0.928934
$892$	2.58127	0.0864274
$893$	20.3342	0.680458
$894$	10.3989	0.347791
$895$	−50.2289	−1.67897
$896$	−2.26977	−0.0758277
$897$	−37.0278	−1.23632
$898$	−1.23775	−0.0413044
$899$	−32.7806	−1.09329
$900$	10.2476	0.341587
$901$	3.05887	0.101906
$902$	−2.71996	−0.0905646
$903$	−26.9478	−0.896765
$904$	−15.3702	−0.511204
$905$	−27.0013	−0.897553
$906$	−7.30859	−0.242812
$907$	38.7859	1.28786	0.643932	−	0.765083i	$-0.277302\pi$
$907$	38.7859	1.28786	0.643932	+	0.765083i	$0.277302\pi$
$908$	11.6117	0.385348
$909$	−5.96674	−0.197904
$910$	22.6376	0.750428
$911$	12.4665	0.413034	0.206517	−	0.978443i	$-0.433787\pi$
$911$	12.4665	0.413034	0.206517	+	0.978443i	$0.433787\pi$
$912$	−12.7494	−0.422174
$913$	10.3209	0.341573
$914$	−3.68260	−0.121810
$915$	27.9186	0.922961
$916$	25.0755	0.828517
$917$	−0.779576	−0.0257439
$918$	8.90016	0.293749
$919$	−10.6037	−0.349784	−0.174892	−	0.984588i	$-0.555958\pi$
$919$	−10.6037	−0.349784	−0.174892	+	0.984588i	$0.555958\pi$
$920$	17.5297	0.577938
$921$	34.2423	1.12832
$922$	−34.0284	−1.12067
$923$	−12.9226	−0.425352
$924$	−12.5301	−0.412211
$925$	−25.1208	−0.825966
$926$	10.4929	0.344819
$927$	11.5314	0.378742
$928$	3.86102	0.126744
$929$	14.6872	0.481870	0.240935	−	0.970541i	$-0.422546\pi$
$929$	14.6872	0.481870	0.240935	+	0.970541i	$0.422546\pi$
$930$	−60.6207	−1.98783
$931$	−10.6607	−0.349390
$932$	8.61795	0.282290
$933$	2.88511	0.0944542
$934$	−2.00426	−0.0655813
$935$	29.1434	0.953092
$936$	−5.82001	−0.190233
$937$	−6.59200	−0.215351	−0.107676	−	0.994186i	$-0.534341\pi$
$937$	−6.59200	−0.215351	−0.107676	+	0.994186i	$0.534341\pi$
$938$	−24.2632	−0.792222
$939$	−48.8342	−1.59364
$940$	−11.3879	−0.371432
$941$	−32.9895	−1.07543	−0.537714	−	0.843128i	$-0.680712\pi$
$941$	−32.9895	−1.07543	−0.537714	+	0.843128i	$0.680712\pi$
$942$	−8.60715	−0.280436
$943$	5.90929	0.192433
$944$	−2.99637	−0.0975236
$945$	−18.0679	−0.587750
$946$	−13.4164	−0.436206
$947$	−38.0836	−1.23755	−0.618775	−	0.785568i	$-0.712371\pi$
$947$	−38.0836	−1.23755	−0.618775	+	0.785568i	$0.712371\pi$
$948$	30.7589	0.999004
$949$	−24.3410	−0.790142
$950$	−31.3567	−1.01735
$951$	6.16598	0.199946
$952$	8.19824	0.265706
$953$	−5.04244	−0.163341	−0.0816704	−	0.996659i	$-0.526025\pi$
$953$	−5.04244	−0.163341	−0.0816704	+	0.996659i	$0.526025\pi$
$954$	1.59649	0.0516883
$955$	−9.99663	−0.323483
$956$	2.26722	0.0733272
$957$	21.3145	0.689000
$958$	1.74400	0.0563462
$959$	−0.0648922	−0.00209548
$960$	7.14012	0.230446
$961$	41.0826	1.32524
$962$	14.2671	0.459988
$963$	0.131487	0.00423712
$964$	−11.7882	−0.379673
$965$	39.3625	1.26712
$966$	27.2226	0.875871
$967$	11.3570	0.365215	0.182607	−	0.983186i	$-0.441546\pi$
$967$	11.3570	0.365215	0.182607	+	0.983186i	$0.441546\pi$
$968$	4.76164	0.153045
$969$	46.0498	1.47933
$970$	10.0921	0.324038
$971$	−23.0658	−0.740217	−0.370108	−	0.928989i	$-0.620679\pi$
$971$	−23.0658	−0.740217	−0.370108	+	0.928989i	$0.620679\pi$
$972$	17.1450	0.549925
$973$	−16.5186	−0.529563
$974$	8.96872	0.287376
$975$	−37.0936	−1.18794
$976$	3.91010	0.125159
$977$	−21.5874	−0.690641	−0.345321	−	0.938485i	$-0.612230\pi$
$977$	−21.5874	−0.690641	−0.345321	+	0.938485i	$0.612230\pi$
$978$	15.9041	0.508556
$979$	−34.1922	−1.09279
$980$	5.97038	0.190717
$981$	3.57839	0.114249
$982$	0.606682	0.0193600
$983$	44.4953	1.41918	0.709589	−	0.704616i	$-0.248881\pi$
$983$	44.4953	1.41918	0.709589	+	0.704616i	$0.248881\pi$
$984$	2.40694	0.0767304
$985$	−56.3578	−1.79571
$986$	−13.9457	−0.444121
$987$	−17.6847	−0.562909
$988$	17.8087	0.566569
$989$	29.1481	0.926857
$990$	15.2106	0.483423
$991$	−62.2289	−1.97677	−0.988384	−	0.151979i	$-0.951435\pi$
$991$	−62.2289	−1.97677	−0.988384	+	0.151979i	$0.951435\pi$
$992$	−8.49015	−0.269562
$993$	6.72377	0.213372
$994$	9.50059	0.301340
$995$	−85.5245	−2.71131
$996$	−9.13318	−0.289396
$997$	−39.2405	−1.24276	−0.621379	−	0.783510i	$-0.713427\pi$
$997$	−39.2405	−1.24276	−0.621379	+	0.783510i	$0.713427\pi$
$998$	−13.7256	−0.434476
$999$	−11.3871	−0.360272

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4022.2.a.e.1.8	✓	46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4022.2.a.e.1.8	✓	46	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:	\( N \)	\(=\)	\( 4022 = 2 \cdot 2011 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4022.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.21023 q^{3} +1.00000 q^{4} -3.23048 q^{5} +2.21023 q^{6} +2.26977 q^{7} -1.00000 q^{8} +1.88514 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.21023 q^{3} +1.00000 q^{4} -3.23048 q^{5} +2.21023 q^{6} +2.26977 q^{7} -1.00000 q^{8} +1.88514 q^{9} +3.23048 q^{10} +2.49767 q^{11} -2.21023 q^{12} +3.08731 q^{13} -2.26977 q^{14} +7.14012 q^{15} +1.00000 q^{16} -3.61192 q^{17} -1.88514 q^{18} +5.76834 q^{19} -3.23048 q^{20} -5.01673 q^{21} -2.49767 q^{22} +5.42636 q^{23} +2.21023 q^{24} +5.43600 q^{25} -3.08731 q^{26} +2.46411 q^{27} +2.26977 q^{28} -3.86102 q^{29} -7.14012 q^{30} +8.49015 q^{31} -1.00000 q^{32} -5.52044 q^{33} +3.61192 q^{34} -7.33245 q^{35} +1.88514 q^{36} -4.62119 q^{37} -5.76834 q^{38} -6.82369 q^{39} +3.23048 q^{40} +1.08900 q^{41} +5.01673 q^{42} +5.37158 q^{43} +2.49767 q^{44} -6.08990 q^{45} -5.42636 q^{46} +3.52514 q^{47} -2.21023 q^{48} -1.84814 q^{49} -5.43600 q^{50} +7.98320 q^{51} +3.08731 q^{52} -0.846882 q^{53} -2.46411 q^{54} -8.06867 q^{55} -2.26977 q^{56} -12.7494 q^{57} +3.86102 q^{58} -2.99637 q^{59} +7.14012 q^{60} +3.91010 q^{61} -8.49015 q^{62} +4.27883 q^{63} +1.00000 q^{64} -9.97350 q^{65} +5.52044 q^{66} +10.6897 q^{67} -3.61192 q^{68} -11.9935 q^{69} +7.33245 q^{70} -4.18570 q^{71} -1.88514 q^{72} -7.88420 q^{73} +4.62119 q^{74} -12.0148 q^{75} +5.76834 q^{76} +5.66914 q^{77} +6.82369 q^{78} -13.9166 q^{79} -3.23048 q^{80} -11.1017 q^{81} -1.08900 q^{82} +4.13222 q^{83} -5.01673 q^{84} +11.6682 q^{85} -5.37158 q^{86} +8.53376 q^{87} -2.49767 q^{88} -13.6897 q^{89} +6.08990 q^{90} +7.00749 q^{91} +5.42636 q^{92} -18.7652 q^{93} -3.52514 q^{94} -18.6345 q^{95} +2.21023 q^{96} +3.12403 q^{97} +1.84814 q^{98} +4.70845 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q - 46 q^{2} + 8 q^{3} + 46 q^{4} + 14 q^{5} - 8 q^{6} + 28 q^{7} - 46 q^{8} + 58 q^{9}+O(q^{10}) \) 46 * q - 46 * q^2 + 8 * q^3 + 46 * q^4 + 14 * q^5 - 8 * q^6 + 28 * q^7 - 46 * q^8 + 58 * q^9	\( 46 q - 46 q^{2} + 8 q^{3} + 46 q^{4} + 14 q^{5} - 8 q^{6} + 28 q^{7} - 46 q^{8} + 58 q^{9} - 14 q^{10} - 6 q^{11} + 8 q^{12} + 37 q^{13} - 28 q^{14} + 9 q^{15} + 46 q^{16} + 6 q^{17} - 58 q^{18} + 18 q^{19} + 14 q^{20} + 19 q^{21} + 6 q^{22} - 4 q^{23} - 8 q^{24} + 86 q^{25} - 37 q^{26} + 32 q^{27} + 28 q^{28} + 15 q^{29} - 9 q^{30} + 18 q^{31} - 46 q^{32} + 37 q^{33} - 6 q^{34} - 2 q^{35} + 58 q^{36} + 74 q^{37} - 18 q^{38} - 3 q^{39} - 14 q^{40} - 18 q^{41} - 19 q^{42} + 25 q^{43} - 6 q^{44} + 94 q^{45} + 4 q^{46} + 18 q^{47} + 8 q^{48} + 92 q^{49} - 86 q^{50} - 10 q^{51} + 37 q^{52} + 17 q^{53} - 32 q^{54} + 37 q^{55} - 28 q^{56} + 43 q^{57} - 15 q^{58} - 24 q^{59} + 9 q^{60} + 46 q^{61} - 18 q^{62} + 80 q^{63} + 46 q^{64} + 24 q^{65} - 37 q^{66} + 61 q^{67} + 6 q^{68} + 59 q^{69} + 2 q^{70} - 8 q^{71} - 58 q^{72} + 101 q^{73} - 74 q^{74} + 34 q^{75} + 18 q^{76} + 40 q^{77} + 3 q^{78} + 9 q^{79} + 14 q^{80} + 58 q^{81} + 18 q^{82} + 18 q^{83} + 19 q^{84} + 60 q^{85} - 25 q^{86} + 20 q^{87} + 6 q^{88} - 25 q^{89} - 94 q^{90} + 51 q^{91} - 4 q^{92} + 63 q^{93} - 18 q^{94} - 31 q^{95} - 8 q^{96} + 76 q^{97} - 92 q^{98} + 21 q^{99}+O(q^{100}) \) 46 * q - 46 * q^2 + 8 * q^3 + 46 * q^4 + 14 * q^5 - 8 * q^6 + 28 * q^7 - 46 * q^8 + 58 * q^9 - 14 * q^10 - 6 * q^11 + 8 * q^12 + 37 * q^13 - 28 * q^14 + 9 * q^15 + 46 * q^16 + 6 * q^17 - 58 * q^18 + 18 * q^19 + 14 * q^20 + 19 * q^21 + 6 * q^22 - 4 * q^23 - 8 * q^24 + 86 * q^25 - 37 * q^26 + 32 * q^27 + 28 * q^28 + 15 * q^29 - 9 * q^30 + 18 * q^31 - 46 * q^32 + 37 * q^33 - 6 * q^34 - 2 * q^35 + 58 * q^36 + 74 * q^37 - 18 * q^38 - 3 * q^39 - 14 * q^40 - 18 * q^41 - 19 * q^42 + 25 * q^43 - 6 * q^44 + 94 * q^45 + 4 * q^46 + 18 * q^47 + 8 * q^48 + 92 * q^49 - 86 * q^50 - 10 * q^51 + 37 * q^52 + 17 * q^53 - 32 * q^54 + 37 * q^55 - 28 * q^56 + 43 * q^57 - 15 * q^58 - 24 * q^59 + 9 * q^60 + 46 * q^61 - 18 * q^62 + 80 * q^63 + 46 * q^64 + 24 * q^65 - 37 * q^66 + 61 * q^67 + 6 * q^68 + 59 * q^69 + 2 * q^70 - 8 * q^71 - 58 * q^72 + 101 * q^73 - 74 * q^74 + 34 * q^75 + 18 * q^76 + 40 * q^77 + 3 * q^78 + 9 * q^79 + 14 * q^80 + 58 * q^81 + 18 * q^82 + 18 * q^83 + 19 * q^84 + 60 * q^85 - 25 * q^86 + 20 * q^87 + 6 * q^88 - 25 * q^89 - 94 * q^90 + 51 * q^91 - 4 * q^92 + 63 * q^93 - 18 * q^94 - 31 * q^95 - 8 * q^96 + 76 * q^97 - 92 * q^98 + 21 * q^99

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