LMFDB - Embedded newform 8042.2.a.a.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8042,2,Mod(1,8042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8042, 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("8042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8042 = 2 \cdot 4021 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8042.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:	$64.2156933055$
Analytic rank:	$1$
Dimension:	$67$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	8042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.46688 q^{3} +1.00000 q^{4} +0.597421 q^{5} -2.46688 q^{6} -5.04196 q^{7} +1.00000 q^{8} +3.08551 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.46688 q^{3} +1.00000 q^{4} +0.597421 q^{5} -2.46688 q^{6} -5.04196 q^{7} +1.00000 q^{8} +3.08551 q^{9} +0.597421 q^{10} -0.206237 q^{11} -2.46688 q^{12} -5.16007 q^{13} -5.04196 q^{14} -1.47377 q^{15} +1.00000 q^{16} +2.07476 q^{17} +3.08551 q^{18} +6.06110 q^{19} +0.597421 q^{20} +12.4379 q^{21} -0.206237 q^{22} +5.27834 q^{23} -2.46688 q^{24} -4.64309 q^{25} -5.16007 q^{26} -0.210949 q^{27} -5.04196 q^{28} +1.72108 q^{29} -1.47377 q^{30} +0.341745 q^{31} +1.00000 q^{32} +0.508761 q^{33} +2.07476 q^{34} -3.01217 q^{35} +3.08551 q^{36} -5.96190 q^{37} +6.06110 q^{38} +12.7293 q^{39} +0.597421 q^{40} +11.3004 q^{41} +12.4379 q^{42} -4.28605 q^{43} -0.206237 q^{44} +1.84335 q^{45} +5.27834 q^{46} -6.15907 q^{47} -2.46688 q^{48} +18.4214 q^{49} -4.64309 q^{50} -5.11820 q^{51} -5.16007 q^{52} +4.17344 q^{53} -0.210949 q^{54} -0.123210 q^{55} -5.04196 q^{56} -14.9520 q^{57} +1.72108 q^{58} +2.75819 q^{59} -1.47377 q^{60} +1.42361 q^{61} +0.341745 q^{62} -15.5570 q^{63} +1.00000 q^{64} -3.08273 q^{65} +0.508761 q^{66} +15.0329 q^{67} +2.07476 q^{68} -13.0211 q^{69} -3.01217 q^{70} -2.37465 q^{71} +3.08551 q^{72} -12.6825 q^{73} -5.96190 q^{74} +11.4540 q^{75} +6.06110 q^{76} +1.03984 q^{77} +12.7293 q^{78} +13.0518 q^{79} +0.597421 q^{80} -8.73615 q^{81} +11.3004 q^{82} +4.49973 q^{83} +12.4379 q^{84} +1.23951 q^{85} -4.28605 q^{86} -4.24570 q^{87} -0.206237 q^{88} -8.99137 q^{89} +1.84335 q^{90} +26.0169 q^{91} +5.27834 q^{92} -0.843046 q^{93} -6.15907 q^{94} +3.62103 q^{95} -2.46688 q^{96} -0.752092 q^{97} +18.4214 q^{98} -0.636345 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) $ 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9} - 20 q^{10} - 13 q^{11} - 11 q^{12} - 51 q^{13} - 40 q^{14} - 31 q^{15} + 67 q^{16} - 34 q^{17} + 24 q^{18} - 33 q^{19} - 20 q^{20} - 39 q^{21} - 13 q^{22} - 43 q^{23} - 11 q^{24} - 9 q^{25} - 51 q^{26} - 29 q^{27} - 40 q^{28} - 63 q^{29} - 31 q^{30} - 43 q^{31} + 67 q^{32} - 49 q^{33} - 34 q^{34} - 20 q^{35} + 24 q^{36} - 77 q^{37} - 33 q^{38} - 40 q^{39} - 20 q^{40} - 50 q^{41} - 39 q^{42} - 56 q^{43} - 13 q^{44} - 48 q^{45} - 43 q^{46} - 48 q^{47} - 11 q^{48} + q^{49} - 9 q^{50} - 18 q^{51} - 51 q^{52} - 91 q^{53} - 29 q^{54} - 58 q^{55} - 40 q^{56} - 65 q^{57} - 63 q^{58} - 17 q^{59} - 31 q^{60} - 45 q^{61} - 43 q^{62} - 67 q^{63} + 67 q^{64} - 65 q^{65} - 49 q^{66} - 112 q^{67} - 34 q^{68} - 57 q^{69} - 20 q^{70} - 75 q^{71} + 24 q^{72} - 79 q^{73} - 77 q^{74} - 5 q^{75} - 33 q^{76} - 85 q^{77} - 40 q^{78} - 80 q^{79} - 20 q^{80} - 77 q^{81} - 50 q^{82} - 22 q^{83} - 39 q^{84} - 134 q^{85} - 56 q^{86} - 49 q^{87} - 13 q^{88} - 77 q^{89} - 48 q^{90} - 17 q^{91} - 43 q^{92} - 97 q^{93} - 48 q^{94} - 73 q^{95} - 11 q^{96} - 87 q^{97} + q^{98} - 44 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9 - 20 * q^10 - 13 * q^11 - 11 * q^12 - 51 * q^13 - 40 * q^14 - 31 * q^15 + 67 * q^16 - 34 * q^17 + 24 * q^18 - 33 * q^19 - 20 * q^20 - 39 * q^21 - 13 * q^22 - 43 * q^23 - 11 * q^24 - 9 * q^25 - 51 * q^26 - 29 * q^27 - 40 * q^28 - 63 * q^29 - 31 * q^30 - 43 * q^31 + 67 * q^32 - 49 * q^33 - 34 * q^34 - 20 * q^35 + 24 * q^36 - 77 * q^37 - 33 * q^38 - 40 * q^39 - 20 * q^40 - 50 * q^41 - 39 * q^42 - 56 * q^43 - 13 * q^44 - 48 * q^45 - 43 * q^46 - 48 * q^47 - 11 * q^48 + q^49 - 9 * q^50 - 18 * q^51 - 51 * q^52 - 91 * q^53 - 29 * q^54 - 58 * q^55 - 40 * q^56 - 65 * q^57 - 63 * q^58 - 17 * q^59 - 31 * q^60 - 45 * q^61 - 43 * q^62 - 67 * q^63 + 67 * q^64 - 65 * q^65 - 49 * q^66 - 112 * q^67 - 34 * q^68 - 57 * q^69 - 20 * q^70 - 75 * q^71 + 24 * q^72 - 79 * q^73 - 77 * q^74 - 5 * q^75 - 33 * q^76 - 85 * q^77 - 40 * q^78 - 80 * q^79 - 20 * q^80 - 77 * q^81 - 50 * q^82 - 22 * q^83 - 39 * q^84 - 134 * q^85 - 56 * q^86 - 49 * q^87 - 13 * q^88 - 77 * q^89 - 48 * q^90 - 17 * q^91 - 43 * q^92 - 97 * q^93 - 48 * q^94 - 73 * q^95 - 11 * q^96 - 87 * q^97 + q^98 - 44 * 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
$3$	−2.46688	−1.42426	−0.712128	−	0.702050i	$-0.752268\pi$
$3$	−2.46688	−1.42426	−0.712128	+	0.702050i	$0.752268\pi$
$4$	1.00000	0.500000
$5$	0.597421	0.267175	0.133587	−	0.991037i	$-0.457350\pi$
$5$	0.597421	0.267175	0.133587	+	0.991037i	$0.457350\pi$
$6$	−2.46688	−1.00710
$7$	−5.04196	−1.90568	−0.952841	−	0.303470i	$-0.901855\pi$
$7$	−5.04196	−1.90568	−0.952841	+	0.303470i	$0.901855\pi$
$8$	1.00000	0.353553
$9$	3.08551	1.02850
$10$	0.597421	0.188921
$11$	−0.206237	−0.0621827	−0.0310913	−	0.999517i	$-0.509898\pi$
$11$	−0.206237	−0.0621827	−0.0310913	+	0.999517i	$0.509898\pi$
$12$	−2.46688	−0.712128
$13$	−5.16007	−1.43115	−0.715573	−	0.698538i	$-0.753834\pi$
$13$	−5.16007	−1.43115	−0.715573	+	0.698538i	$0.753834\pi$
$14$	−5.04196	−1.34752
$15$	−1.47377	−0.380525
$16$	1.00000	0.250000
$17$	2.07476	0.503204	0.251602	−	0.967831i	$-0.419043\pi$
$17$	2.07476	0.503204	0.251602	+	0.967831i	$0.419043\pi$
$18$	3.08551	0.727262
$19$	6.06110	1.39051	0.695256	−	0.718762i	$-0.255291\pi$
$19$	6.06110	1.39051	0.695256	+	0.718762i	$0.255291\pi$
$20$	0.597421	0.133587
$21$	12.4379	2.71418
$22$	−0.206237	−0.0439698
$23$	5.27834	1.10061	0.550305	−	0.834964i	$-0.314511\pi$
$23$	5.27834	1.10061	0.550305	+	0.834964i	$0.314511\pi$
$24$	−2.46688	−0.503550
$25$	−4.64309	−0.928618
$26$	−5.16007	−1.01197
$27$	−0.210949	−0.0405971
$28$	−5.04196	−0.952841
$29$	1.72108	0.319596	0.159798	−	0.987150i	$-0.448916\pi$
$29$	1.72108	0.319596	0.159798	+	0.987150i	$0.448916\pi$
$30$	−1.47377	−0.269072
$31$	0.341745	0.0613793	0.0306896	−	0.999529i	$-0.490230\pi$
$31$	0.341745	0.0613793	0.0306896	+	0.999529i	$0.490230\pi$
$32$	1.00000	0.176777
$33$	0.508761	0.0885640
$34$	2.07476	0.355819
$35$	−3.01217	−0.509150
$36$	3.08551	0.514252
$37$	−5.96190	−0.980130	−0.490065	−	0.871686i	$-0.663027\pi$
$37$	−5.96190	−0.980130	−0.490065	+	0.871686i	$0.663027\pi$
$38$	6.06110	0.983241
$39$	12.7293	2.03832
$40$	0.597421	0.0944605
$41$	11.3004	1.76482	0.882410	−	0.470481i	$-0.155920\pi$
$41$	11.3004	1.76482	0.882410	+	0.470481i	$0.155920\pi$
$42$	12.4379	1.91921
$43$	−4.28605	−0.653617	−0.326809	−	0.945091i	$-0.605973\pi$
$43$	−4.28605	−0.653617	−0.326809	+	0.945091i	$0.605973\pi$
$44$	−0.206237	−0.0310913
$45$	1.84335	0.274790
$46$	5.27834	0.778249
$47$	−6.15907	−0.898392	−0.449196	−	0.893433i	$-0.648290\pi$
$47$	−6.15907	−0.898392	−0.449196	+	0.893433i	$0.648290\pi$
$48$	−2.46688	−0.356064
$49$	18.4214	2.63162
$50$	−4.64309	−0.656632
$51$	−5.11820	−0.716691
$52$	−5.16007	−0.715573
$53$	4.17344	0.573266	0.286633	−	0.958040i	$-0.407464\pi$
$53$	4.17344	0.573266	0.286633	+	0.958040i	$0.407464\pi$
$54$	−0.210949	−0.0287065
$55$	−0.123210	−0.0166136
$56$	−5.04196	−0.673760
$57$	−14.9520	−1.98045
$58$	1.72108	0.225988
$59$	2.75819	0.359086	0.179543	−	0.983750i	$-0.442538\pi$
$59$	2.75819	0.359086	0.179543	+	0.983750i	$0.442538\pi$
$60$	−1.47377	−0.190263
$61$	1.42361	0.182275	0.0911375	−	0.995838i	$-0.470950\pi$
$61$	1.42361	0.182275	0.0911375	+	0.995838i	$0.470950\pi$
$62$	0.341745	0.0434017
$63$	−15.5570	−1.96000
$64$	1.00000	0.125000
$65$	−3.08273	−0.382366
$66$	0.508761	0.0626242
$67$	15.0329	1.83656	0.918280	−	0.395932i	$-0.129578\pi$
$67$	15.0329	1.83656	0.918280	+	0.395932i	$0.129578\pi$
$68$	2.07476	0.251602
$69$	−13.0211	−1.56755
$70$	−3.01217	−0.360023
$71$	−2.37465	−0.281820	−0.140910	−	0.990022i	$-0.545003\pi$
$71$	−2.37465	−0.281820	−0.140910	+	0.990022i	$0.545003\pi$
$72$	3.08551	0.363631
$73$	−12.6825	−1.48437	−0.742185	−	0.670196i	$-0.766210\pi$
$73$	−12.6825	−1.48437	−0.742185	+	0.670196i	$0.766210\pi$
$74$	−5.96190	−0.693056
$75$	11.4540	1.32259
$76$	6.06110	0.695256
$77$	1.03984	0.118500
$78$	12.7293	1.44131
$79$	13.0518	1.46844	0.734219	−	0.678913i	$-0.237549\pi$
$79$	13.0518	1.46844	0.734219	+	0.678913i	$0.237549\pi$
$80$	0.597421	0.0667937
$81$	−8.73615	−0.970683
$82$	11.3004	1.24792
$83$	4.49973	0.493909	0.246955	−	0.969027i	$-0.420570\pi$
$83$	4.49973	0.493909	0.246955	+	0.969027i	$0.420570\pi$
$84$	12.4379	1.35709
$85$	1.23951	0.134443
$86$	−4.28605	−0.462177
$87$	−4.24570	−0.455186
$88$	−0.206237	−0.0219849
$89$	−8.99137	−0.953083	−0.476542	−	0.879152i	$-0.658110\pi$
$89$	−8.99137	−0.953083	−0.476542	+	0.879152i	$0.658110\pi$
$90$	1.84335	0.194306
$91$	26.0169	2.72731
$92$	5.27834	0.550305
$93$	−0.843046	−0.0874198
$94$	−6.15907	−0.635259
$95$	3.62103	0.371510
$96$	−2.46688	−0.251775
$97$	−0.752092	−0.0763633	−0.0381817	−	0.999271i	$-0.512157\pi$
$97$	−0.752092	−0.0763633	−0.0381817	+	0.999271i	$0.512157\pi$
$98$	18.4214	1.86084
$99$	−0.636345	−0.0639551
$100$	−4.64309	−0.464309
$101$	−4.41017	−0.438828	−0.219414	−	0.975632i	$-0.570415\pi$
$101$	−4.41017	−0.438828	−0.219414	+	0.975632i	$0.570415\pi$
$102$	−5.11820	−0.506777
$103$	7.47654	0.736685	0.368343	−	0.929690i	$-0.379925\pi$
$103$	7.47654	0.736685	0.368343	+	0.929690i	$0.379925\pi$
$104$	−5.16007	−0.505987
$105$	7.43068	0.725160
$106$	4.17344	0.405360
$107$	−3.03143	−0.293059	−0.146529	−	0.989206i	$-0.546810\pi$
$107$	−3.03143	−0.293059	−0.146529	+	0.989206i	$0.546810\pi$
$108$	−0.210949	−0.0202985
$109$	−19.9203	−1.90802	−0.954009	−	0.299777i	$-0.903088\pi$
$109$	−19.9203	−1.90802	−0.954009	+	0.299777i	$0.903088\pi$
$110$	−0.123210	−0.0117476
$111$	14.7073	1.39596
$112$	−5.04196	−0.476420
$113$	6.24512	0.587492	0.293746	−	0.955884i	$-0.405098\pi$
$113$	6.24512	0.587492	0.293746	+	0.955884i	$0.405098\pi$
$114$	−14.9520	−1.40039
$115$	3.15339	0.294055
$116$	1.72108	0.159798
$117$	−15.9215	−1.47194
$118$	2.75819	0.253912
$119$	−10.4609	−0.958947
$120$	−1.47377	−0.134536
$121$	−10.9575	−0.996133
$122$	1.42361	0.128888
$123$	−27.8767	−2.51355
$124$	0.341745	0.0306896
$125$	−5.76098	−0.515278
$126$	−15.5570	−1.38593
$127$	−15.8217	−1.40395	−0.701975	−	0.712201i	$-0.747698\pi$
$127$	−15.8217	−1.40395	−0.701975	+	0.712201i	$0.747698\pi$
$128$	1.00000	0.0883883
$129$	10.5732	0.930918
$130$	−3.08273	−0.270374
$131$	−15.9499	−1.39355	−0.696775	−	0.717289i	$-0.745383\pi$
$131$	−15.9499	−1.39355	−0.696775	+	0.717289i	$0.745383\pi$
$132$	0.508761	0.0442820
$133$	−30.5598	−2.64987
$134$	15.0329	1.29864
$135$	−0.126025	−0.0108465
$136$	2.07476	0.177909
$137$	8.79086	0.751054	0.375527	−	0.926811i	$-0.377462\pi$
$137$	8.79086	0.751054	0.375527	+	0.926811i	$0.377462\pi$
$138$	−13.0211	−1.10843
$139$	5.01522	0.425386	0.212693	−	0.977119i	$-0.431777\pi$
$139$	5.01522	0.425386	0.212693	+	0.977119i	$0.431777\pi$
$140$	−3.01217	−0.254575
$141$	15.1937	1.27954
$142$	−2.37465	−0.199277
$143$	1.06420	0.0889925
$144$	3.08551	0.257126
$145$	1.02821	0.0853879
$146$	−12.6825	−1.04961
$147$	−45.4434	−3.74811
$148$	−5.96190	−0.490065
$149$	−8.33167	−0.682557	−0.341279	−	0.939962i	$-0.610860\pi$
$149$	−8.33167	−0.682557	−0.341279	+	0.939962i	$0.610860\pi$
$150$	11.4540	0.935212
$151$	−1.08744	−0.0884946	−0.0442473	−	0.999021i	$-0.514089\pi$
$151$	−1.08744	−0.0884946	−0.0442473	+	0.999021i	$0.514089\pi$
$152$	6.06110	0.491620
$153$	6.40171	0.517547
$154$	1.03984	0.0837924
$155$	0.204166	0.0163990
$156$	12.7293	1.01916
$157$	−13.7631	−1.09842	−0.549209	−	0.835685i	$-0.685071\pi$
$157$	−13.7631	−1.09842	−0.549209	+	0.835685i	$0.685071\pi$
$158$	13.0518	1.03834
$159$	−10.2954	−0.816477
$160$	0.597421	0.0472303
$161$	−26.6132	−2.09741
$162$	−8.73615	−0.686377
$163$	10.4848	0.821235	0.410618	−	0.911808i	$-0.365313\pi$
$163$	10.4848	0.821235	0.410618	+	0.911808i	$0.365313\pi$
$164$	11.3004	0.882410
$165$	0.303945	0.0236621
$166$	4.49973	0.349246
$167$	−3.90386	−0.302090	−0.151045	−	0.988527i	$-0.548264\pi$
$167$	−3.90386	−0.302090	−0.151045	+	0.988527i	$0.548264\pi$
$168$	12.4379	0.959607
$169$	13.6263	1.04818
$170$	1.23951	0.0950658
$171$	18.7016	1.43015
$172$	−4.28605	−0.326809
$173$	−9.56419	−0.727152	−0.363576	−	0.931565i	$-0.618444\pi$
$173$	−9.56419	−0.727152	−0.363576	+	0.931565i	$0.618444\pi$
$174$	−4.24570	−0.321865
$175$	23.4103	1.76965
$176$	−0.206237	−0.0155457
$177$	−6.80413	−0.511430
$178$	−8.99137	−0.673932
$179$	11.2547	0.841212	0.420606	−	0.907243i	$-0.361817\pi$
$179$	11.2547	0.841212	0.420606	+	0.907243i	$0.361817\pi$
$180$	1.84335	0.137395
$181$	25.3725	1.88592	0.942962	−	0.332900i	$-0.108027\pi$
$181$	25.3725	1.88592	0.942962	+	0.332900i	$0.108027\pi$
$182$	26.0169	1.92850
$183$	−3.51189	−0.259606
$184$	5.27834	0.389125
$185$	−3.56176	−0.261866
$186$	−0.843046	−0.0618151
$187$	−0.427892	−0.0312906
$188$	−6.15907	−0.449196
$189$	1.06359	0.0773651
$190$	3.62103	0.262697
$191$	−10.3545	−0.749226	−0.374613	−	0.927181i	$-0.622224\pi$
$191$	−10.3545	−0.749226	−0.374613	+	0.927181i	$0.622224\pi$
$192$	−2.46688	−0.178032
$193$	−0.158094	−0.0113798	−0.00568992	−	0.999984i	$-0.501811\pi$
$193$	−0.158094	−0.0113798	−0.00568992	+	0.999984i	$0.501811\pi$
$194$	−0.752092	−0.0539970
$195$	7.60475	0.544587
$196$	18.4214	1.31581
$197$	−18.1348	−1.29205	−0.646024	−	0.763317i	$-0.723569\pi$
$197$	−18.1348	−1.29205	−0.646024	+	0.763317i	$0.723569\pi$
$198$	−0.636345	−0.0452231
$199$	7.36515	0.522101	0.261051	−	0.965325i	$-0.415931\pi$
$199$	7.36515	0.522101	0.261051	+	0.965325i	$0.415931\pi$
$200$	−4.64309	−0.328316
$201$	−37.0844	−2.61573
$202$	−4.41017	−0.310298
$203$	−8.67760	−0.609048
$204$	−5.11820	−0.358346
$205$	6.75107	0.471515
$206$	7.47654	0.520915
$207$	16.2864	1.13198
$208$	−5.16007	−0.357787
$209$	−1.25002	−0.0864658
$210$	7.43068	0.512765
$211$	2.54089	0.174922	0.0874610	−	0.996168i	$-0.472125\pi$
$211$	2.54089	0.174922	0.0874610	+	0.996168i	$0.472125\pi$
$212$	4.17344	0.286633
$213$	5.85799	0.401383
$214$	−3.03143	−0.207224
$215$	−2.56058	−0.174630
$216$	−0.210949	−0.0143532
$217$	−1.72307	−0.116969
$218$	−19.9203	−1.34917
$219$	31.2861	2.11412
$220$	−0.123210	−0.00830682
$221$	−10.7059	−0.720159
$222$	14.7073	0.987089
$223$	−9.24713	−0.619233	−0.309617	−	0.950861i	$-0.600201\pi$
$223$	−9.24713	−0.619233	−0.309617	+	0.950861i	$0.600201\pi$
$224$	−5.04196	−0.336880
$225$	−14.3263	−0.955087
$226$	6.24512	0.415419
$227$	9.89379	0.656674	0.328337	−	0.944561i	$-0.393512\pi$
$227$	9.89379	0.656674	0.328337	+	0.944561i	$0.393512\pi$
$228$	−14.9520	−0.990223
$229$	−19.8534	−1.31195	−0.655974	−	0.754783i	$-0.727742\pi$
$229$	−19.8534	−1.31195	−0.655974	+	0.754783i	$0.727742\pi$
$230$	3.15339	0.207928
$231$	−2.56516	−0.168775
$232$	1.72108	0.112994
$233$	−15.2656	−1.00008	−0.500041	−	0.866002i	$-0.666682\pi$
$233$	−15.2656	−1.00008	−0.500041	+	0.866002i	$0.666682\pi$
$234$	−15.9215	−1.04082
$235$	−3.67955	−0.240028
$236$	2.75819	0.179543
$237$	−32.1972	−2.09143
$238$	−10.4609	−0.678078
$239$	1.48746	0.0962161	0.0481080	−	0.998842i	$-0.484681\pi$
$239$	1.48746	0.0962161	0.0481080	+	0.998842i	$0.484681\pi$
$240$	−1.47377	−0.0951313
$241$	−15.6219	−1.00630	−0.503149	−	0.864200i	$-0.667825\pi$
$241$	−15.6219	−1.00630	−0.503149	+	0.864200i	$0.667825\pi$
$242$	−10.9575	−0.704373
$243$	22.1839	1.42310
$244$	1.42361	0.0911375
$245$	11.0053	0.703103
$246$	−27.8767	−1.77735
$247$	−31.2757	−1.99003
$248$	0.341745	0.0217009
$249$	−11.1003	−0.703453
$250$	−5.76098	−0.364356
$251$	−22.3682	−1.41187	−0.705933	−	0.708278i	$-0.749472\pi$
$251$	−22.3682	−1.41187	−0.705933	+	0.708278i	$0.749472\pi$
$252$	−15.5570	−0.980001
$253$	−1.08859	−0.0684389
$254$	−15.8217	−0.992743
$255$	−3.05772	−0.191482
$256$	1.00000	0.0625000
$257$	23.9305	1.49274	0.746371	−	0.665530i	$-0.231794\pi$
$257$	23.9305	1.49274	0.746371	+	0.665530i	$0.231794\pi$
$258$	10.5732	0.658258
$259$	30.0596	1.86782
$260$	−3.08273	−0.191183
$261$	5.31040	0.328706
$262$	−15.9499	−0.985389
$263$	−22.0985	−1.36265	−0.681325	−	0.731981i	$-0.738596\pi$
$263$	−22.0985	−1.36265	−0.681325	+	0.731981i	$0.738596\pi$
$264$	0.508761	0.0313121
$265$	2.49330	0.153162
$266$	−30.5598	−1.87374
$267$	22.1807	1.35743
$268$	15.0329	0.918280
$269$	2.90026	0.176832	0.0884161	−	0.996084i	$-0.471819\pi$
$269$	2.90026	0.176832	0.0884161	+	0.996084i	$0.471819\pi$
$270$	−0.126025	−0.00766964
$271$	30.6715	1.86316	0.931581	−	0.363535i	$-0.118430\pi$
$271$	30.6715	1.86316	0.931581	+	0.363535i	$0.118430\pi$
$272$	2.07476	0.125801
$273$	−64.1806	−3.88439
$274$	8.79086	0.531075
$275$	0.957575	0.0577439
$276$	−13.0211	−0.783775
$277$	12.1559	0.730375	0.365188	−	0.930934i	$-0.381005\pi$
$277$	12.1559	0.730375	0.365188	+	0.930934i	$0.381005\pi$
$278$	5.01522	0.300793
$279$	1.05446	0.0631288
$280$	−3.01217	−0.180012
$281$	−5.51172	−0.328802	−0.164401	−	0.986394i	$-0.552569\pi$
$281$	−5.51172	−0.328802	−0.164401	+	0.986394i	$0.552569\pi$
$282$	15.1937	0.904772
$283$	15.5041	0.921621	0.460811	−	0.887498i	$-0.347559\pi$
$283$	15.5041	0.921621	0.460811	+	0.887498i	$0.347559\pi$
$284$	−2.37465	−0.140910
$285$	−8.93265	−0.529125
$286$	1.06420	0.0629272
$287$	−56.9760	−3.36319
$288$	3.08551	0.181816
$289$	−12.6954	−0.746786
$290$	1.02821	0.0603784
$291$	1.85532	0.108761
$292$	−12.6825	−0.742185
$293$	−1.81133	−0.105819	−0.0529095	−	0.998599i	$-0.516849\pi$
$293$	−1.81133	−0.105819	−0.0529095	+	0.998599i	$0.516849\pi$
$294$	−45.4434	−2.65031
$295$	1.64780	0.0959386
$296$	−5.96190	−0.346528
$297$	0.0435053	0.00252443
$298$	−8.33167	−0.482641
$299$	−27.2366	−1.57514
$300$	11.4540	0.661294
$301$	21.6101	1.24559
$302$	−1.08744	−0.0625751
$303$	10.8794	0.625003
$304$	6.06110	0.347628
$305$	0.850496	0.0486993
$306$	6.40171	0.365961
$307$	−5.04156	−0.287737	−0.143869	−	0.989597i	$-0.545954\pi$
$307$	−5.04156	−0.287737	−0.143869	+	0.989597i	$0.545954\pi$
$308$	1.03984	0.0592502
$309$	−18.4437	−1.04923
$310$	0.204166	0.0115958
$311$	7.76766	0.440464	0.220232	−	0.975448i	$-0.429319\pi$
$311$	7.76766	0.440464	0.220232	+	0.975448i	$0.429319\pi$
$312$	12.7293	0.720654
$313$	−6.45761	−0.365006	−0.182503	−	0.983205i	$-0.558420\pi$
$313$	−6.45761	−0.365006	−0.182503	+	0.983205i	$0.558420\pi$
$314$	−13.7631	−0.776699
$315$	−9.29409	−0.523663
$316$	13.0518	0.734219
$317$	−4.52038	−0.253890	−0.126945	−	0.991910i	$-0.540517\pi$
$317$	−4.52038	−0.253890	−0.126945	+	0.991910i	$0.540517\pi$
$318$	−10.2954	−0.577337
$319$	−0.354949	−0.0198733
$320$	0.597421	0.0333968
$321$	7.47817	0.417391
$322$	−26.6132	−1.48310
$323$	12.5754	0.699711
$324$	−8.73615	−0.485342
$325$	23.9587	1.32899
$326$	10.4848	0.580701
$327$	49.1410	2.71751
$328$	11.3004	0.623958
$329$	31.0538	1.71205
$330$	0.303945	0.0167316
$331$	−2.88305	−0.158467	−0.0792335	−	0.996856i	$-0.525247\pi$
$331$	−2.88305	−0.158467	−0.0792335	+	0.996856i	$0.525247\pi$
$332$	4.49973	0.246955
$333$	−18.3955	−1.00807
$334$	−3.90386	−0.213610
$335$	8.98096	0.490682
$336$	12.4379	0.678545
$337$	−17.2248	−0.938292	−0.469146	−	0.883121i	$-0.655438\pi$
$337$	−17.2248	−0.938292	−0.469146	+	0.883121i	$0.655438\pi$
$338$	13.6263	0.741176
$339$	−15.4060	−0.836738
$340$	1.23951	0.0672217
$341$	−0.0704804	−0.00381673
$342$	18.7016	1.01127
$343$	−57.5861	−3.10936
$344$	−4.28605	−0.231089
$345$	−7.77905	−0.418810
$346$	−9.56419	−0.514174
$347$	−17.4462	−0.936561	−0.468281	−	0.883580i	$-0.655126\pi$
$347$	−17.4462	−0.936561	−0.468281	+	0.883580i	$0.655126\pi$
$348$	−4.24570	−0.227593
$349$	17.1132	0.916047	0.458023	−	0.888940i	$-0.348558\pi$
$349$	17.1132	0.916047	0.458023	+	0.888940i	$0.348558\pi$
$350$	23.4103	1.25133
$351$	1.08851	0.0581004
$352$	−0.206237	−0.0109924
$353$	−14.9818	−0.797401	−0.398701	−	0.917081i	$-0.630539\pi$
$353$	−14.9818	−0.797401	−0.398701	+	0.917081i	$0.630539\pi$
$354$	−6.80413	−0.361635
$355$	−1.41867	−0.0752950
$356$	−8.99137	−0.476542
$357$	25.8058	1.36579
$358$	11.2547	0.594827
$359$	−11.6614	−0.615465	−0.307733	−	0.951473i	$-0.599570\pi$
$359$	−11.6614	−0.615465	−0.307733	+	0.951473i	$0.599570\pi$
$360$	1.84335	0.0971530
$361$	17.7370	0.933525
$362$	25.3725	1.33355
$363$	27.0308	1.41875
$364$	26.0169	1.36366
$365$	−7.57676	−0.396586
$366$	−3.51189	−0.183569
$367$	13.2591	0.692119	0.346060	−	0.938213i	$-0.387520\pi$
$367$	13.2591	0.692119	0.346060	+	0.938213i	$0.387520\pi$
$368$	5.27834	0.275153
$369$	34.8674	1.81512
$370$	−3.56176	−0.185167
$371$	−21.0423	−1.09246
$372$	−0.843046	−0.0437099
$373$	12.1397	0.628569	0.314285	−	0.949329i	$-0.398235\pi$
$373$	12.1397	0.628569	0.314285	+	0.949329i	$0.398235\pi$
$374$	−0.427892	−0.0221258
$375$	14.2117	0.733887
$376$	−6.15907	−0.317630
$377$	−8.88088	−0.457389
$378$	1.06359	0.0547054
$379$	−0.406652	−0.0208883	−0.0104442	−	0.999945i	$-0.503325\pi$
$379$	−0.406652	−0.0208883	−0.0104442	+	0.999945i	$0.503325\pi$
$380$	3.62103	0.185755
$381$	39.0303	1.99958
$382$	−10.3545	−0.529783
$383$	−8.16968	−0.417451	−0.208726	−	0.977974i	$-0.566932\pi$
$383$	−8.16968	−0.417451	−0.208726	+	0.977974i	$0.566932\pi$
$384$	−2.46688	−0.125888
$385$	0.621220	0.0316603
$386$	−0.158094	−0.00804676
$387$	−13.2247	−0.672248
$388$	−0.752092	−0.0381817
$389$	−10.1023	−0.512207	−0.256103	−	0.966649i	$-0.582439\pi$
$389$	−10.1023	−0.512207	−0.256103	+	0.966649i	$0.582439\pi$
$390$	7.60475	0.385081
$391$	10.9513	0.553832
$392$	18.4214	0.930420
$393$	39.3466	1.98477
$394$	−18.1348	−0.913616
$395$	7.79739	0.392329
$396$	−0.636345	−0.0319776
$397$	−37.2395	−1.86900	−0.934498	−	0.355969i	$-0.884151\pi$
$397$	−37.2395	−1.86900	−0.934498	+	0.355969i	$0.884151\pi$
$398$	7.36515	0.369181
$399$	75.3876	3.77410
$400$	−4.64309	−0.232154
$401$	27.9160	1.39406	0.697030	−	0.717042i	$-0.254504\pi$
$401$	27.9160	1.39406	0.697030	+	0.717042i	$0.254504\pi$
$402$	−37.0844	−1.84960
$403$	−1.76343	−0.0878427
$404$	−4.41017	−0.219414
$405$	−5.21916	−0.259342
$406$	−8.67760	−0.430662
$407$	1.22956	0.0609471
$408$	−5.11820	−0.253389
$409$	14.8891	0.736217	0.368108	−	0.929783i	$-0.380006\pi$
$409$	14.8891	0.736217	0.368108	+	0.929783i	$0.380006\pi$
$410$	6.75107	0.333412
$411$	−21.6860	−1.06969
$412$	7.47654	0.368343
$413$	−13.9067	−0.684303
$414$	16.2864	0.800432
$415$	2.68823	0.131960
$416$	−5.16007	−0.252993
$417$	−12.3720	−0.605858
$418$	−1.25002	−0.0611405
$419$	−15.8049	−0.772122	−0.386061	−	0.922473i	$-0.626165\pi$
$419$	−15.8049	−0.772122	−0.386061	+	0.922473i	$0.626165\pi$
$420$	7.43068	0.362580
$421$	13.5903	0.662352	0.331176	−	0.943569i	$-0.392555\pi$
$421$	13.5903	0.662352	0.331176	+	0.943569i	$0.392555\pi$
$422$	2.54089	0.123689
$423$	−19.0039	−0.924000
$424$	4.17344	0.202680
$425$	−9.63331	−0.467284
$426$	5.85799	0.283821
$427$	−7.17781	−0.347358
$428$	−3.03143	−0.146529
$429$	−2.62525	−0.126748
$430$	−2.56058	−0.123482
$431$	−23.9318	−1.15275	−0.576376	−	0.817184i	$-0.695534\pi$
$431$	−23.9318	−1.15275	−0.576376	+	0.817184i	$0.695534\pi$
$432$	−0.210949	−0.0101493
$433$	6.08276	0.292319	0.146159	−	0.989261i	$-0.453309\pi$
$433$	6.08276	0.292319	0.146159	+	0.989261i	$0.453309\pi$
$434$	−1.72307	−0.0827098
$435$	−2.53647	−0.121614
$436$	−19.9203	−0.954009
$437$	31.9926	1.53041
$438$	31.2861	1.49491
$439$	14.7184	0.702470	0.351235	−	0.936287i	$-0.385762\pi$
$439$	14.7184	0.702470	0.351235	+	0.936287i	$0.385762\pi$
$440$	−0.123210	−0.00587381
$441$	56.8394	2.70664
$442$	−10.7059	−0.509229
$443$	11.2481	0.534412	0.267206	−	0.963639i	$-0.413900\pi$
$443$	11.2481	0.534412	0.267206	+	0.963639i	$0.413900\pi$
$444$	14.7073	0.697978
$445$	−5.37163	−0.254640
$446$	−9.24713	−0.437864
$447$	20.5533	0.972136
$448$	−5.04196	−0.238210
$449$	32.3736	1.52780	0.763902	−	0.645332i	$-0.223281\pi$
$449$	32.3736	1.52780	0.763902	+	0.645332i	$0.223281\pi$
$450$	−14.3263	−0.675349
$451$	−2.33055	−0.109741
$452$	6.24512	0.293746
$453$	2.68259	0.126039
$454$	9.89379	0.464338
$455$	15.5430	0.728668
$456$	−14.9520	−0.700193
$457$	−25.1265	−1.17537	−0.587685	−	0.809090i	$-0.699961\pi$
$457$	−25.1265	−1.17537	−0.587685	+	0.809090i	$0.699961\pi$
$458$	−19.8534	−0.927688
$459$	−0.437668	−0.0204286
$460$	3.15339	0.147028
$461$	16.2603	0.757320	0.378660	−	0.925536i	$-0.376385\pi$
$461$	16.2603	0.757320	0.378660	+	0.925536i	$0.376385\pi$
$462$	−2.56516	−0.119342
$463$	−6.53072	−0.303508	−0.151754	−	0.988418i	$-0.548492\pi$
$463$	−6.53072	−0.303508	−0.151754	+	0.988418i	$0.548492\pi$
$464$	1.72108	0.0798990
$465$	−0.503653	−0.0233564
$466$	−15.2656	−0.707165
$467$	18.7351	0.866956	0.433478	−	0.901164i	$-0.357286\pi$
$467$	18.7351	0.866956	0.433478	+	0.901164i	$0.357286\pi$
$468$	−15.9215	−0.735970
$469$	−75.7952	−3.49990
$470$	−3.67955	−0.169725
$471$	33.9521	1.56443
$472$	2.75819	0.126956
$473$	0.883941	0.0406437
$474$	−32.1972	−1.47886
$475$	−28.1422	−1.29125
$476$	−10.4609	−0.479473
$477$	12.8772	0.589606
$478$	1.48746	0.0680350
$479$	−25.3736	−1.15935	−0.579675	−	0.814848i	$-0.696821\pi$
$479$	−25.3736	−1.15935	−0.579675	+	0.814848i	$0.696821\pi$
$480$	−1.47377	−0.0672680
$481$	30.7638	1.40271
$482$	−15.6219	−0.711560
$483$	65.6516	2.98725
$484$	−10.9575	−0.498067
$485$	−0.449315	−0.0204023
$486$	22.1839	1.00628
$487$	−8.14854	−0.369246	−0.184623	−	0.982809i	$-0.559106\pi$
$487$	−8.14854	−0.369246	−0.184623	+	0.982809i	$0.559106\pi$
$488$	1.42361	0.0644440
$489$	−25.8648	−1.16965
$490$	11.0053	0.497169
$491$	−9.49448	−0.428480	−0.214240	−	0.976781i	$-0.568727\pi$
$491$	−9.49448	−0.428480	−0.214240	+	0.976781i	$0.568727\pi$
$492$	−27.8767	−1.25678
$493$	3.57083	0.160822
$494$	−31.2757	−1.40716
$495$	−0.380166	−0.0170872
$496$	0.341745	0.0153448
$497$	11.9729	0.537058
$498$	−11.1003	−0.497416
$499$	−12.4639	−0.557963	−0.278981	−	0.960297i	$-0.589997\pi$
$499$	−12.4639	−0.557963	−0.278981	+	0.960297i	$0.589997\pi$
$500$	−5.76098	−0.257639
$501$	9.63036	0.430253
$502$	−22.3682	−0.998340
$503$	−15.7720	−0.703237	−0.351619	−	0.936143i	$-0.614369\pi$
$503$	−15.7720	−0.703237	−0.351619	+	0.936143i	$0.614369\pi$
$504$	−15.5570	−0.692965
$505$	−2.63473	−0.117244
$506$	−1.08859	−0.0483936
$507$	−33.6146	−1.49288
$508$	−15.8217	−0.701975
$509$	16.9849	0.752841	0.376420	−	0.926449i	$-0.377155\pi$
$509$	16.9849	0.752841	0.376420	+	0.926449i	$0.377155\pi$
$510$	−3.05772	−0.135398
$511$	63.9444	2.82874
$512$	1.00000	0.0441942
$513$	−1.27858	−0.0564507
$514$	23.9305	1.05553
$515$	4.46664	0.196824
$516$	10.5732	0.465459
$517$	1.27022	0.0558644
$518$	30.0596	1.32074
$519$	23.5937	1.03565
$520$	−3.08273	−0.135187
$521$	−31.8648	−1.39602	−0.698011	−	0.716087i	$-0.745931\pi$
$521$	−31.8648	−1.39602	−0.698011	+	0.716087i	$0.745931\pi$
$522$	5.31040	0.232430
$523$	−24.9166	−1.08953	−0.544764	−	0.838589i	$-0.683381\pi$
$523$	−24.9166	−1.08953	−0.544764	+	0.838589i	$0.683381\pi$
$524$	−15.9499	−0.696775
$525$	−57.7504	−2.52043
$526$	−22.0985	−0.963538
$527$	0.709041	0.0308863
$528$	0.508761	0.0221410
$529$	4.86090	0.211344
$530$	2.49330	0.108302
$531$	8.51043	0.369321
$532$	−30.5598	−1.32494
$533$	−58.3107	−2.52572
$534$	22.1807	0.959851
$535$	−1.81104	−0.0782979
$536$	15.0329	0.649322
$537$	−27.7639	−1.19810
$538$	2.90026	0.125039
$539$	−3.79916	−0.163641
$540$	−0.126025	−0.00542326
$541$	4.92452	0.211722	0.105861	−	0.994381i	$-0.466240\pi$
$541$	4.92452	0.211722	0.105861	+	0.994381i	$0.466240\pi$
$542$	30.6715	1.31745
$543$	−62.5910	−2.68604
$544$	2.07476	0.0889547
$545$	−11.9008	−0.509774
$546$	−64.1806	−2.74668
$547$	−17.6702	−0.755525	−0.377763	−	0.925903i	$-0.623306\pi$
$547$	−17.6702	−0.755525	−0.377763	+	0.925903i	$0.623306\pi$
$548$	8.79086	0.375527
$549$	4.39258	0.187471
$550$	0.957575	0.0408311
$551$	10.4316	0.444402
$552$	−13.0211	−0.554213
$553$	−65.8064	−2.79837
$554$	12.1559	0.516453
$555$	8.78645	0.372964
$556$	5.01522	0.212693
$557$	−41.4217	−1.75509	−0.877547	−	0.479491i	$-0.840821\pi$
$557$	−41.4217	−1.75509	−0.877547	+	0.479491i	$0.840821\pi$
$558$	1.05446	0.0446388
$559$	22.1164	0.935422
$560$	−3.01217	−0.127287
$561$	1.05556	0.0445658
$562$	−5.51172	−0.232498
$563$	16.3116	0.687452	0.343726	−	0.939070i	$-0.388311\pi$
$563$	16.3116	0.687452	0.343726	+	0.939070i	$0.388311\pi$
$564$	15.1937	0.639770
$565$	3.73097	0.156963
$566$	15.5041	0.651685
$567$	44.0473	1.84981
$568$	−2.37465	−0.0996383
$569$	23.9703	1.00489	0.502443	−	0.864611i	$-0.332435\pi$
$569$	23.9703	1.00489	0.502443	+	0.864611i	$0.332435\pi$
$570$	−8.93265	−0.374148
$571$	−21.2922	−0.891052	−0.445526	−	0.895269i	$-0.646983\pi$
$571$	−21.2922	−0.891052	−0.445526	+	0.895269i	$0.646983\pi$
$572$	1.06420	0.0444963
$573$	25.5434	1.06709
$574$	−56.9760	−2.37813
$575$	−24.5078	−1.02205
$576$	3.08551	0.128563
$577$	13.7511	0.572466	0.286233	−	0.958160i	$-0.407597\pi$
$577$	13.7511	0.572466	0.286233	+	0.958160i	$0.407597\pi$
$578$	−12.6954	−0.528057
$579$	0.389999	0.0162078
$580$	1.02821	0.0426940
$581$	−22.6874	−0.941234
$582$	1.85532	0.0769056
$583$	−0.860716	−0.0356472
$584$	−12.6825	−0.524804
$585$	−9.51181	−0.393265
$586$	−1.81133	−0.0748254
$587$	−35.9089	−1.48212	−0.741059	−	0.671440i	$-0.765676\pi$
$587$	−35.9089	−1.48212	−0.741059	+	0.671440i	$0.765676\pi$
$588$	−45.4434	−1.87405
$589$	2.07135	0.0853486
$590$	1.64780	0.0678388
$591$	44.7363	1.84021
$592$	−5.96190	−0.245032
$593$	−13.4816	−0.553623	−0.276812	−	0.960924i	$-0.589278\pi$
$593$	−13.4816	−0.553623	−0.276812	+	0.960924i	$0.589278\pi$
$594$	0.0435053	0.00178504
$595$	−6.24954	−0.256206
$596$	−8.33167	−0.341279
$597$	−18.1690	−0.743606
$598$	−27.2366	−1.11379
$599$	−25.7308	−1.05133	−0.525667	−	0.850691i	$-0.676184\pi$
$599$	−25.7308	−1.05133	−0.525667	+	0.850691i	$0.676184\pi$
$600$	11.4540	0.467606
$601$	−42.6246	−1.73869	−0.869346	−	0.494204i	$-0.835460\pi$
$601$	−42.6246	−1.73869	−0.869346	+	0.494204i	$0.835460\pi$
$602$	21.6101	0.880763
$603$	46.3842	1.88891
$604$	−1.08744	−0.0442473
$605$	−6.54622	−0.266142
$606$	10.8794	0.441944
$607$	−39.8783	−1.61861	−0.809305	−	0.587389i	$-0.800156\pi$
$607$	−39.8783	−1.61861	−0.809305	+	0.587389i	$0.800156\pi$
$608$	6.06110	0.245810
$609$	21.4066	0.867440
$610$	0.850496	0.0344356
$611$	31.7812	1.28573
$612$	6.40171	0.258774
$613$	−28.8520	−1.16532	−0.582662	−	0.812715i	$-0.697989\pi$
$613$	−28.8520	−1.16532	−0.582662	+	0.812715i	$0.697989\pi$
$614$	−5.04156	−0.203461
$615$	−16.6541	−0.671558
$616$	1.03984	0.0418962
$617$	1.17852	0.0474455	0.0237228	−	0.999719i	$-0.492448\pi$
$617$	1.17852	0.0474455	0.0237228	+	0.999719i	$0.492448\pi$
$618$	−18.4437	−0.741916
$619$	6.78632	0.272765	0.136383	−	0.990656i	$-0.456452\pi$
$619$	6.78632	0.272765	0.136383	+	0.990656i	$0.456452\pi$
$620$	0.204166	0.00819949
$621$	−1.11346	−0.0446816
$622$	7.76766	0.311455
$623$	45.3341	1.81627
$624$	12.7293	0.509580
$625$	19.7737	0.790949
$626$	−6.45761	−0.258098
$627$	3.08366	0.123149
$628$	−13.7631	−0.549209
$629$	−12.3695	−0.493205
$630$	−9.29409	−0.370286
$631$	24.0116	0.955886	0.477943	−	0.878391i	$-0.341383\pi$
$631$	24.0116	0.955886	0.477943	+	0.878391i	$0.341383\pi$
$632$	13.0518	0.519171
$633$	−6.26808	−0.249134
$634$	−4.52038	−0.179527
$635$	−9.45222	−0.375100
$636$	−10.2954	−0.408239
$637$	−95.0556	−3.76624
$638$	−0.354949	−0.0140526
$639$	−7.32702	−0.289853
$640$	0.597421	0.0236151
$641$	−45.2640	−1.78782	−0.893911	−	0.448245i	$-0.852049\pi$
$641$	−45.2640	−1.78782	−0.893911	+	0.448245i	$0.852049\pi$
$642$	7.47817	0.295140
$643$	7.84867	0.309521	0.154761	−	0.987952i	$-0.450539\pi$
$643$	7.84867	0.309521	0.154761	+	0.987952i	$0.450539\pi$
$644$	−26.6132	−1.04871
$645$	6.31665	0.248718
$646$	12.5754	0.494771
$647$	−0.373408	−0.0146802	−0.00734009	−	0.999973i	$-0.502336\pi$
$647$	−0.373408	−0.0146802	−0.00734009	+	0.999973i	$0.502336\pi$
$648$	−8.73615	−0.343188
$649$	−0.568839	−0.0223289
$650$	23.9587	0.939736
$651$	4.25060	0.166594
$652$	10.4848	0.410618
$653$	−34.5048	−1.35028	−0.675138	−	0.737691i	$-0.735916\pi$
$653$	−34.5048	−1.35028	−0.675138	+	0.737691i	$0.735916\pi$
$654$	49.1410	1.92157
$655$	−9.52881	−0.372321
$656$	11.3004	0.441205
$657$	−39.1319	−1.52668
$658$	31.0538	1.21060
$659$	34.5659	1.34650	0.673249	−	0.739416i	$-0.264898\pi$
$659$	34.5659	1.34650	0.673249	+	0.739416i	$0.264898\pi$
$660$	0.303945	0.0118310
$661$	−16.4279	−0.638970	−0.319485	−	0.947591i	$-0.603510\pi$
$661$	−16.4279	−0.638970	−0.319485	+	0.947591i	$0.603510\pi$
$662$	−2.88305	−0.112053
$663$	26.4103	1.02569
$664$	4.49973	0.174623
$665$	−18.2571	−0.707979
$666$	−18.3955	−0.712811
$667$	9.08443	0.351751
$668$	−3.90386	−0.151045
$669$	22.8116	0.881947
$670$	8.98096	0.346965
$671$	−0.293601	−0.0113343
$672$	12.4379	0.479803
$673$	−14.9764	−0.577298	−0.288649	−	0.957435i	$-0.593206\pi$
$673$	−14.9764	−0.577298	−0.288649	+	0.957435i	$0.593206\pi$
$674$	−17.2248	−0.663473
$675$	0.979453	0.0376992
$676$	13.6263	0.524090
$677$	−29.9056	−1.14936	−0.574682	−	0.818377i	$-0.694874\pi$
$677$	−29.9056	−1.14936	−0.574682	+	0.818377i	$0.694874\pi$
$678$	−15.4060	−0.591663
$679$	3.79202	0.145524
$680$	1.23951	0.0475329
$681$	−24.4068	−0.935271
$682$	−0.0704804	−0.00269883
$683$	−27.5849	−1.05551	−0.527753	−	0.849398i	$-0.676965\pi$
$683$	−27.5849	−1.05551	−0.527753	+	0.849398i	$0.676965\pi$
$684$	18.7016	0.715074
$685$	5.25184	0.200663
$686$	−57.5861	−2.19865
$687$	48.9760	1.86855
$688$	−4.28605	−0.163404
$689$	−21.5352	−0.820428
$690$	−7.77905	−0.296143
$691$	10.0637	0.382840	0.191420	−	0.981508i	$-0.438691\pi$
$691$	10.0637	0.382840	0.191420	+	0.981508i	$0.438691\pi$
$692$	−9.56419	−0.363576
$693$	3.20843	0.121878
$694$	−17.4462	−0.662249
$695$	2.99620	0.113652
$696$	−4.24570	−0.160933
$697$	23.4456	0.888064
$698$	17.1132	0.647743
$699$	37.6584	1.42437
$700$	23.4103	0.884825
$701$	13.2993	0.502309	0.251154	−	0.967947i	$-0.419190\pi$
$701$	13.2993	0.502309	0.251154	+	0.967947i	$0.419190\pi$
$702$	1.08851	0.0410832
$703$	−36.1357	−1.36288
$704$	−0.206237	−0.00777283
$705$	9.07703	0.341861
$706$	−14.9818	−0.563848
$707$	22.2359	0.836267
$708$	−6.80413	−0.255715
$709$	0.846722	0.0317993	0.0158997	−	0.999874i	$-0.494939\pi$
$709$	0.846722	0.0317993	0.0158997	+	0.999874i	$0.494939\pi$
$710$	−1.41867	−0.0532416
$711$	40.2713	1.51029
$712$	−8.99137	−0.336966
$713$	1.80385	0.0675547
$714$	25.8058	0.965756
$715$	0.635772	0.0237765
$716$	11.2547	0.420606
$717$	−3.66940	−0.137036
$718$	−11.6614	−0.435200
$719$	−12.0419	−0.449086	−0.224543	−	0.974464i	$-0.572089\pi$
$719$	−12.0419	−0.449086	−0.224543	+	0.974464i	$0.572089\pi$
$720$	1.84335	0.0686976
$721$	−37.6964	−1.40389
$722$	17.7370	0.660102
$723$	38.5375	1.43323
$724$	25.3725	0.942962
$725$	−7.99111	−0.296782
$726$	27.0308	1.00321
$727$	5.85243	0.217055	0.108527	−	0.994093i	$-0.465387\pi$
$727$	5.85243	0.217055	0.108527	+	0.994093i	$0.465387\pi$
$728$	26.0169	0.964250
$729$	−28.5167	−1.05617
$730$	−7.57676	−0.280429
$731$	−8.89255	−0.328903
$732$	−3.51189	−0.129803
$733$	−23.7870	−0.878594	−0.439297	−	0.898342i	$-0.644773\pi$
$733$	−23.7870	−0.878594	−0.439297	+	0.898342i	$0.644773\pi$
$734$	13.2591	0.489402
$735$	−27.1488	−1.00140
$736$	5.27834	0.194562
$737$	−3.10033	−0.114202
$738$	34.8674	1.28349
$739$	−24.3102	−0.894265	−0.447132	−	0.894468i	$-0.647555\pi$
$739$	−24.3102	−0.894265	−0.447132	+	0.894468i	$0.647555\pi$
$740$	−3.56176	−0.130933
$741$	77.1536	2.83431
$742$	−21.0423	−0.772488
$743$	51.2362	1.87968	0.939838	−	0.341622i	$-0.110976\pi$
$743$	51.2362	1.87968	0.939838	+	0.341622i	$0.110976\pi$
$744$	−0.843046	−0.0309076
$745$	−4.97751	−0.182362
$746$	12.1397	0.444466
$747$	13.8840	0.507988
$748$	−0.427892	−0.0156453
$749$	15.2843	0.558477
$750$	14.2117	0.518937
$751$	40.8744	1.49153	0.745764	−	0.666211i	$-0.232085\pi$
$751$	40.8744	1.49153	0.745764	+	0.666211i	$0.232085\pi$
$752$	−6.15907	−0.224598
$753$	55.1797	2.01086
$754$	−8.88088	−0.323423
$755$	−0.649659	−0.0236435
$756$	1.06359	0.0386826
$757$	17.6936	0.643086	0.321543	−	0.946895i	$-0.395799\pi$
$757$	17.6936	0.643086	0.321543	+	0.946895i	$0.395799\pi$
$758$	−0.406652	−0.0147703
$759$	2.68542	0.0974745
$760$	3.62103	0.131349
$761$	28.4541	1.03146	0.515729	−	0.856752i	$-0.327521\pi$
$761$	28.4541	1.03146	0.515729	+	0.856752i	$0.327521\pi$
$762$	39.0303	1.41392
$763$	100.437	3.63608
$764$	−10.3545	−0.374613
$765$	3.82451	0.138276
$766$	−8.16968	−0.295183
$767$	−14.2325	−0.513904
$768$	−2.46688	−0.0890160
$769$	33.6460	1.21330	0.606652	−	0.794967i	$-0.292512\pi$
$769$	33.6460	1.21330	0.606652	+	0.794967i	$0.292512\pi$
$770$	0.621220	0.0223872
$771$	−59.0337	−2.12605
$772$	−0.158094	−0.00568992
$773$	14.4416	0.519430	0.259715	−	0.965685i	$-0.416371\pi$
$773$	14.4416	0.519430	0.259715	+	0.965685i	$0.416371\pi$
$774$	−13.2247	−0.475351
$775$	−1.58675	−0.0569979
$776$	−0.752092	−0.0269985
$777$	−74.1536	−2.66025
$778$	−10.1023	−0.362185
$779$	68.4926	2.45400
$780$	7.60475	0.272294
$781$	0.489740	0.0175243
$782$	10.9513	0.391618
$783$	−0.363059	−0.0129747
$784$	18.4214	0.657906
$785$	−8.22239	−0.293470
$786$	39.3466	1.40345
$787$	32.3128	1.15183	0.575914	−	0.817510i	$-0.304646\pi$
$787$	32.3128	1.15183	0.575914	+	0.817510i	$0.304646\pi$
$788$	−18.1348	−0.646024
$789$	54.5143	1.94076
$790$	7.79739	0.277419
$791$	−31.4877	−1.11957
$792$	−0.636345	−0.0226115
$793$	−7.34595	−0.260862
$794$	−37.2395	−1.32158
$795$	−6.15068	−0.218142
$796$	7.36515	0.261051
$797$	−44.6368	−1.58112	−0.790559	−	0.612386i	$-0.790210\pi$
$797$	−44.6368	−1.58112	−0.790559	+	0.612386i	$0.790210\pi$
$798$	75.3876	2.66869
$799$	−12.7786	−0.452075
$800$	−4.64309	−0.164158
$801$	−27.7430	−0.980250
$802$	27.9160	0.985750
$803$	2.61559	0.0923020
$804$	−37.0844	−1.30787
$805$	−15.8993	−0.560376
$806$	−1.76343	−0.0621142
$807$	−7.15461	−0.251854
$808$	−4.41017	−0.155149
$809$	−35.2167	−1.23815	−0.619077	−	0.785330i	$-0.712493\pi$
$809$	−35.2167	−1.23815	−0.619077	+	0.785330i	$0.712493\pi$
$810$	−5.21916	−0.183383
$811$	19.5063	0.684960	0.342480	−	0.939525i	$-0.388733\pi$
$811$	19.5063	0.684960	0.342480	+	0.939525i	$0.388733\pi$
$812$	−8.67760	−0.304524
$813$	−75.6630	−2.65362
$814$	1.22956	0.0430961
$815$	6.26385	0.219413
$816$	−5.11820	−0.179173
$817$	−25.9782	−0.908863
$818$	14.8891	0.520584
$819$	80.2754	2.80505
$820$	6.75107	0.235758
$821$	−15.9704	−0.557370	−0.278685	−	0.960383i	$-0.589899\pi$
$821$	−15.9704	−0.557370	−0.278685	+	0.960383i	$0.589899\pi$
$822$	−21.6860	−0.756387
$823$	12.4569	0.434221	0.217110	−	0.976147i	$-0.430337\pi$
$823$	12.4569	0.434221	0.217110	+	0.976147i	$0.430337\pi$
$824$	7.47654	0.260458
$825$	−2.36222	−0.0822421
$826$	−13.9067	−0.483875
$827$	−30.7970	−1.07092	−0.535459	−	0.844561i	$-0.679861\pi$
$827$	−30.7970	−1.07092	−0.535459	+	0.844561i	$0.679861\pi$
$828$	16.2864	0.565991
$829$	−12.0695	−0.419190	−0.209595	−	0.977788i	$-0.567215\pi$
$829$	−12.0695	−0.419190	−0.209595	+	0.977788i	$0.567215\pi$
$830$	2.68823	0.0933098
$831$	−29.9871	−1.04024
$832$	−5.16007	−0.178893
$833$	38.2200	1.32424
$834$	−12.3720	−0.428406
$835$	−2.33225	−0.0807107
$836$	−1.25002	−0.0432329
$837$	−0.0720907	−0.00249182
$838$	−15.8049	−0.545973
$839$	45.2808	1.56327	0.781634	−	0.623737i	$-0.214387\pi$
$839$	45.2808	1.56327	0.781634	+	0.623737i	$0.214387\pi$
$840$	7.43068	0.256383
$841$	−26.0379	−0.897858
$842$	13.5903	0.468353
$843$	13.5968	0.468298
$844$	2.54089	0.0874610
$845$	8.14066	0.280047
$846$	−19.0039	−0.653367
$847$	55.2471	1.89831
$848$	4.17344	0.143316
$849$	−38.2467	−1.31262
$850$	−9.63331	−0.330420
$851$	−31.4689	−1.07874
$852$	5.85799	0.200692
$853$	44.2305	1.51442	0.757211	−	0.653170i	$-0.226561\pi$
$853$	44.2305	1.51442	0.757211	+	0.653170i	$0.226561\pi$
$854$	−7.17781	−0.245619
$855$	11.1727	0.382099
$856$	−3.03143	−0.103612
$857$	−16.7000	−0.570462	−0.285231	−	0.958459i	$-0.592070\pi$
$857$	−16.7000	−0.570462	−0.285231	+	0.958459i	$0.592070\pi$
$858$	−2.62525	−0.0896244
$859$	21.8236	0.744613	0.372307	−	0.928110i	$-0.378567\pi$
$859$	21.8236	0.744613	0.372307	+	0.928110i	$0.378567\pi$
$860$	−2.56058	−0.0873150
$861$	140.553	4.79004
$862$	−23.9318	−0.815119
$863$	12.7166	0.432878	0.216439	−	0.976296i	$-0.430556\pi$
$863$	12.7166	0.432878	0.216439	+	0.976296i	$0.430556\pi$
$864$	−0.210949	−0.00717662
$865$	−5.71384	−0.194276
$866$	6.08276	0.206701
$867$	31.3180	1.06361
$868$	−1.72307	−0.0584847
$869$	−2.69175	−0.0913113
$870$	−2.53647	−0.0859943
$871$	−77.5708	−2.62839
$872$	−19.9203	−0.674586
$873$	−2.32059	−0.0785400
$874$	31.9926	1.08217
$875$	29.0466	0.981956
$876$	31.2861	1.05706
$877$	19.1714	0.647371	0.323685	−	0.946165i	$-0.395078\pi$
$877$	19.1714	0.647371	0.323685	+	0.946165i	$0.395078\pi$
$878$	14.7184	0.496721
$879$	4.46834	0.150713
$880$	−0.123210	−0.00415341
$881$	39.2349	1.32186	0.660929	−	0.750449i	$-0.270163\pi$
$881$	39.2349	1.32186	0.660929	+	0.750449i	$0.270163\pi$
$882$	56.8394	1.91388
$883$	55.3914	1.86407	0.932034	−	0.362371i	$-0.118033\pi$
$883$	55.3914	1.86407	0.932034	+	0.362371i	$0.118033\pi$
$884$	−10.7059	−0.360079
$885$	−4.06493	−0.136641
$886$	11.2481	0.377887
$887$	57.1384	1.91852	0.959259	−	0.282528i	$-0.0911730\pi$
$887$	57.1384	1.91852	0.959259	+	0.282528i	$0.0911730\pi$
$888$	14.7073	0.493545
$889$	79.7725	2.67548
$890$	−5.37163	−0.180057
$891$	1.80171	0.0603597
$892$	−9.24713	−0.309617
$893$	−37.3307	−1.24923
$894$	20.5533	0.687404
$895$	6.72376	0.224751
$896$	−5.04196	−0.168440
$897$	67.1896	2.24340
$898$	32.3736	1.08032
$899$	0.588170	0.0196166
$900$	−14.3263	−0.477544
$901$	8.65890	0.288470
$902$	−2.33055	−0.0775987
$903$	−53.3096	−1.77403
$904$	6.24512	0.207710
$905$	15.1581	0.503871
$906$	2.68259	0.0891230
$907$	1.27361	0.0422896	0.0211448	−	0.999776i	$-0.493269\pi$
$907$	1.27361	0.0422896	0.0211448	+	0.999776i	$0.493269\pi$
$908$	9.89379	0.328337
$909$	−13.6076	−0.451337
$910$	15.5430	0.515246
$911$	−39.1594	−1.29741	−0.648704	−	0.761041i	$-0.724689\pi$
$911$	−39.1594	−1.29741	−0.648704	+	0.761041i	$0.724689\pi$
$912$	−14.9520	−0.495111
$913$	−0.928008	−0.0307126
$914$	−25.1265	−0.831112
$915$	−2.09808	−0.0693602
$916$	−19.8534	−0.655974
$917$	80.4189	2.65566
$918$	−0.437668	−0.0144452
$919$	15.8221	0.521923	0.260961	−	0.965349i	$-0.415961\pi$
$919$	15.8221	0.521923	0.260961	+	0.965349i	$0.415961\pi$
$920$	3.15339	0.103964
$921$	12.4369	0.409811
$922$	16.2603	0.535506
$923$	12.2534	0.403325
$924$	−2.56516	−0.0843874
$925$	27.6816	0.910166
$926$	−6.53072	−0.214613
$927$	23.0690	0.757684
$928$	1.72108	0.0564971
$929$	28.7555	0.943438	0.471719	−	0.881749i	$-0.343634\pi$
$929$	28.7555	0.943438	0.471719	+	0.881749i	$0.343634\pi$
$930$	−0.503653	−0.0165154
$931$	111.654	3.65931
$932$	−15.2656	−0.500041
$933$	−19.1619	−0.627333
$934$	18.7351	0.613030
$935$	−0.255632	−0.00836005
$936$	−15.9215	−0.520409
$937$	−8.72414	−0.285005	−0.142503	−	0.989794i	$-0.545515\pi$
$937$	−8.72414	−0.285005	−0.142503	+	0.989794i	$0.545515\pi$
$938$	−75.7952	−2.47480
$939$	15.9302	0.519862
$940$	−3.67955	−0.120014
$941$	−0.494092	−0.0161069	−0.00805347	−	0.999968i	$-0.502564\pi$
$941$	−0.494092	−0.0161069	−0.00805347	+	0.999968i	$0.502564\pi$
$942$	33.9521	1.10622
$943$	59.6472	1.94238
$944$	2.75819	0.0897714
$945$	0.635413	0.0206700
$946$	0.883941	0.0287394
$947$	−28.9003	−0.939135	−0.469567	−	0.882897i	$-0.655590\pi$
$947$	−28.9003	−0.939135	−0.469567	+	0.882897i	$0.655590\pi$
$948$	−32.1972	−1.04571
$949$	65.4424	2.12435
$950$	−28.1422	−0.913055
$951$	11.1513	0.361604
$952$	−10.4609	−0.339039
$953$	−23.3354	−0.755907	−0.377953	−	0.925825i	$-0.623372\pi$
$953$	−23.3354	−0.755907	−0.377953	+	0.925825i	$0.623372\pi$
$954$	12.8772	0.416915
$955$	−6.18600	−0.200174
$956$	1.48746	0.0481080
$957$	0.875618	0.0283047
$958$	−25.3736	−0.819785
$959$	−44.3232	−1.43127
$960$	−1.47377	−0.0475656
$961$	−30.8832	−0.996233
$962$	30.7638	0.991865
$963$	−9.35350	−0.301412
$964$	−15.6219	−0.503149
$965$	−0.0944485	−0.00304041
$966$	65.6516	2.11231
$967$	5.03580	0.161940	0.0809701	−	0.996717i	$-0.474198\pi$
$967$	5.03580	0.161940	0.0809701	+	0.996717i	$0.474198\pi$
$968$	−10.9575	−0.352186
$969$	−31.0219	−0.996568
$970$	−0.449315	−0.0144266
$971$	8.70491	0.279354	0.139677	−	0.990197i	$-0.455394\pi$
$971$	8.70491	0.279354	0.139677	+	0.990197i	$0.455394\pi$
$972$	22.1839	0.711549
$973$	−25.2866	−0.810650
$974$	−8.14854	−0.261096
$975$	−59.1032	−1.89282
$976$	1.42361	0.0455688
$977$	−5.99496	−0.191796	−0.0958979	−	0.995391i	$-0.530572\pi$
$977$	−5.99496	−0.191796	−0.0958979	+	0.995391i	$0.530572\pi$
$978$	−25.8648	−0.827067
$979$	1.85435	0.0592653
$980$	11.0053	0.351552
$981$	−61.4643	−1.96240
$982$	−9.49448	−0.302981
$983$	15.0752	0.480824	0.240412	−	0.970671i	$-0.422717\pi$
$983$	15.0752	0.480824	0.240412	+	0.970671i	$0.422717\pi$
$984$	−27.8767	−0.888676
$985$	−10.8341	−0.345202
$986$	3.57083	0.113718
$987$	−76.6060	−2.43840
$988$	−31.2757	−0.995014
$989$	−22.6233	−0.719378
$990$	−0.380166	−0.0120825
$991$	25.8087	0.819839	0.409920	−	0.912122i	$-0.365557\pi$
$991$	25.8087	0.819839	0.409920	+	0.912122i	$0.365557\pi$
$992$	0.341745	0.0108504
$993$	7.11216	0.225698
$994$	11.9729	0.379758
$995$	4.40009	0.139492
$996$	−11.1003	−0.351726
$997$	25.1232	0.795660	0.397830	−	0.917459i	$-0.369763\pi$
$997$	25.1232	0.795660	0.397830	+	0.917459i	$0.369763\pi$
$998$	−12.4639	−0.394539
$999$	1.25765	0.0397904

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.a.1.9	✓	67

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.a.1.9	✓	67	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:	\( N \)	\(=\)	\( 8042 = 2 \cdot 4021 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8042.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.46688 q^{3} +1.00000 q^{4} +0.597421 q^{5} -2.46688 q^{6} -5.04196 q^{7} +1.00000 q^{8} +3.08551 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.46688 q^{3} +1.00000 q^{4} +0.597421 q^{5} -2.46688 q^{6} -5.04196 q^{7} +1.00000 q^{8} +3.08551 q^{9} +0.597421 q^{10} -0.206237 q^{11} -2.46688 q^{12} -5.16007 q^{13} -5.04196 q^{14} -1.47377 q^{15} +1.00000 q^{16} +2.07476 q^{17} +3.08551 q^{18} +6.06110 q^{19} +0.597421 q^{20} +12.4379 q^{21} -0.206237 q^{22} +5.27834 q^{23} -2.46688 q^{24} -4.64309 q^{25} -5.16007 q^{26} -0.210949 q^{27} -5.04196 q^{28} +1.72108 q^{29} -1.47377 q^{30} +0.341745 q^{31} +1.00000 q^{32} +0.508761 q^{33} +2.07476 q^{34} -3.01217 q^{35} +3.08551 q^{36} -5.96190 q^{37} +6.06110 q^{38} +12.7293 q^{39} +0.597421 q^{40} +11.3004 q^{41} +12.4379 q^{42} -4.28605 q^{43} -0.206237 q^{44} +1.84335 q^{45} +5.27834 q^{46} -6.15907 q^{47} -2.46688 q^{48} +18.4214 q^{49} -4.64309 q^{50} -5.11820 q^{51} -5.16007 q^{52} +4.17344 q^{53} -0.210949 q^{54} -0.123210 q^{55} -5.04196 q^{56} -14.9520 q^{57} +1.72108 q^{58} +2.75819 q^{59} -1.47377 q^{60} +1.42361 q^{61} +0.341745 q^{62} -15.5570 q^{63} +1.00000 q^{64} -3.08273 q^{65} +0.508761 q^{66} +15.0329 q^{67} +2.07476 q^{68} -13.0211 q^{69} -3.01217 q^{70} -2.37465 q^{71} +3.08551 q^{72} -12.6825 q^{73} -5.96190 q^{74} +11.4540 q^{75} +6.06110 q^{76} +1.03984 q^{77} +12.7293 q^{78} +13.0518 q^{79} +0.597421 q^{80} -8.73615 q^{81} +11.3004 q^{82} +4.49973 q^{83} +12.4379 q^{84} +1.23951 q^{85} -4.28605 q^{86} -4.24570 q^{87} -0.206237 q^{88} -8.99137 q^{89} +1.84335 q^{90} +26.0169 q^{91} +5.27834 q^{92} -0.843046 q^{93} -6.15907 q^{94} +3.62103 q^{95} -2.46688 q^{96} -0.752092 q^{97} +18.4214 q^{98} -0.636345 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9} - 20 q^{10} - 13 q^{11} - 11 q^{12} - 51 q^{13} - 40 q^{14} - 31 q^{15} + 67 q^{16} - 34 q^{17} + 24 q^{18} - 33 q^{19} - 20 q^{20} - 39 q^{21} - 13 q^{22} - 43 q^{23} - 11 q^{24} - 9 q^{25} - 51 q^{26} - 29 q^{27} - 40 q^{28} - 63 q^{29} - 31 q^{30} - 43 q^{31} + 67 q^{32} - 49 q^{33} - 34 q^{34} - 20 q^{35} + 24 q^{36} - 77 q^{37} - 33 q^{38} - 40 q^{39} - 20 q^{40} - 50 q^{41} - 39 q^{42} - 56 q^{43} - 13 q^{44} - 48 q^{45} - 43 q^{46} - 48 q^{47} - 11 q^{48} + q^{49} - 9 q^{50} - 18 q^{51} - 51 q^{52} - 91 q^{53} - 29 q^{54} - 58 q^{55} - 40 q^{56} - 65 q^{57} - 63 q^{58} - 17 q^{59} - 31 q^{60} - 45 q^{61} - 43 q^{62} - 67 q^{63} + 67 q^{64} - 65 q^{65} - 49 q^{66} - 112 q^{67} - 34 q^{68} - 57 q^{69} - 20 q^{70} - 75 q^{71} + 24 q^{72} - 79 q^{73} - 77 q^{74} - 5 q^{75} - 33 q^{76} - 85 q^{77} - 40 q^{78} - 80 q^{79} - 20 q^{80} - 77 q^{81} - 50 q^{82} - 22 q^{83} - 39 q^{84} - 134 q^{85} - 56 q^{86} - 49 q^{87} - 13 q^{88} - 77 q^{89} - 48 q^{90} - 17 q^{91} - 43 q^{92} - 97 q^{93} - 48 q^{94} - 73 q^{95} - 11 q^{96} - 87 q^{97} + q^{98} - 44 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9 - 20 * q^10 - 13 * q^11 - 11 * q^12 - 51 * q^13 - 40 * q^14 - 31 * q^15 + 67 * q^16 - 34 * q^17 + 24 * q^18 - 33 * q^19 - 20 * q^20 - 39 * q^21 - 13 * q^22 - 43 * q^23 - 11 * q^24 - 9 * q^25 - 51 * q^26 - 29 * q^27 - 40 * q^28 - 63 * q^29 - 31 * q^30 - 43 * q^31 + 67 * q^32 - 49 * q^33 - 34 * q^34 - 20 * q^35 + 24 * q^36 - 77 * q^37 - 33 * q^38 - 40 * q^39 - 20 * q^40 - 50 * q^41 - 39 * q^42 - 56 * q^43 - 13 * q^44 - 48 * q^45 - 43 * q^46 - 48 * q^47 - 11 * q^48 + q^49 - 9 * q^50 - 18 * q^51 - 51 * q^52 - 91 * q^53 - 29 * q^54 - 58 * q^55 - 40 * q^56 - 65 * q^57 - 63 * q^58 - 17 * q^59 - 31 * q^60 - 45 * q^61 - 43 * q^62 - 67 * q^63 + 67 * q^64 - 65 * q^65 - 49 * q^66 - 112 * q^67 - 34 * q^68 - 57 * q^69 - 20 * q^70 - 75 * q^71 + 24 * q^72 - 79 * q^73 - 77 * q^74 - 5 * q^75 - 33 * q^76 - 85 * q^77 - 40 * q^78 - 80 * q^79 - 20 * q^80 - 77 * q^81 - 50 * q^82 - 22 * q^83 - 39 * q^84 - 134 * q^85 - 56 * q^86 - 49 * q^87 - 13 * q^88 - 77 * q^89 - 48 * q^90 - 17 * q^91 - 43 * q^92 - 97 * q^93 - 48 * q^94 - 73 * q^95 - 11 * q^96 - 87 * q^97 + q^98 - 44 * q^99

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