LMFDB - Embedded newform 8042.2.a.a.1.28

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			1.28
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)

$f(q)$	$=$	$q+1.00000 q^{2} -0.773234 q^{3} +1.00000 q^{4} +0.152858 q^{5} -0.773234 q^{6} -0.720420 q^{7} +1.00000 q^{8} -2.40211 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -0.773234 q^{3} +1.00000 q^{4} +0.152858 q^{5} -0.773234 q^{6} -0.720420 q^{7} +1.00000 q^{8} -2.40211 q^{9} +0.152858 q^{10} -1.61689 q^{11} -0.773234 q^{12} +0.412560 q^{13} -0.720420 q^{14} -0.118195 q^{15} +1.00000 q^{16} +0.962927 q^{17} -2.40211 q^{18} +4.56034 q^{19} +0.152858 q^{20} +0.557054 q^{21} -1.61689 q^{22} +0.510500 q^{23} -0.773234 q^{24} -4.97663 q^{25} +0.412560 q^{26} +4.17710 q^{27} -0.720420 q^{28} +1.89254 q^{29} -0.118195 q^{30} -6.43946 q^{31} +1.00000 q^{32} +1.25024 q^{33} +0.962927 q^{34} -0.110122 q^{35} -2.40211 q^{36} +6.40222 q^{37} +4.56034 q^{38} -0.319005 q^{39} +0.152858 q^{40} -0.251040 q^{41} +0.557054 q^{42} +0.728596 q^{43} -1.61689 q^{44} -0.367182 q^{45} +0.510500 q^{46} +4.20758 q^{47} -0.773234 q^{48} -6.48099 q^{49} -4.97663 q^{50} -0.744568 q^{51} +0.412560 q^{52} +6.90642 q^{53} +4.17710 q^{54} -0.247155 q^{55} -0.720420 q^{56} -3.52621 q^{57} +1.89254 q^{58} -3.92940 q^{59} -0.118195 q^{60} -6.20199 q^{61} -6.43946 q^{62} +1.73053 q^{63} +1.00000 q^{64} +0.0630631 q^{65} +1.25024 q^{66} -1.93421 q^{67} +0.962927 q^{68} -0.394736 q^{69} -0.110122 q^{70} -8.37880 q^{71} -2.40211 q^{72} -1.44169 q^{73} +6.40222 q^{74} +3.84810 q^{75} +4.56034 q^{76} +1.16484 q^{77} -0.319005 q^{78} -3.46527 q^{79} +0.152858 q^{80} +3.97645 q^{81} -0.251040 q^{82} -3.25465 q^{83} +0.557054 q^{84} +0.147191 q^{85} +0.728596 q^{86} -1.46338 q^{87} -1.61689 q^{88} -9.52355 q^{89} -0.367182 q^{90} -0.297216 q^{91} +0.510500 q^{92} +4.97921 q^{93} +4.20758 q^{94} +0.697086 q^{95} -0.773234 q^{96} +9.58266 q^{97} -6.48099 q^{98} +3.88395 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.773234	−0.446427	−0.223214	−	0.974770i	$-0.571655\pi$
$3$	−0.773234	−0.446427	−0.223214	+	0.974770i	$0.571655\pi$
$4$	1.00000	0.500000
$5$	0.152858	0.0683603	0.0341801	−	0.999416i	$-0.489118\pi$
$5$	0.152858	0.0683603	0.0341801	+	0.999416i	$0.489118\pi$
$6$	−0.773234	−0.315672
$7$	−0.720420	−0.272293	−0.136147	−	0.990689i	$-0.543472\pi$
$7$	−0.720420	−0.272293	−0.136147	+	0.990689i	$0.543472\pi$
$8$	1.00000	0.353553
$9$	−2.40211	−0.800703
$10$	0.152858	0.0483380
$11$	−1.61689	−0.487511	−0.243756	−	0.969837i	$-0.578380\pi$
$11$	−1.61689	−0.487511	−0.243756	+	0.969837i	$0.578380\pi$
$12$	−0.773234	−0.223214
$13$	0.412560	0.114423	0.0572117	−	0.998362i	$-0.481779\pi$
$13$	0.412560	0.114423	0.0572117	+	0.998362i	$0.481779\pi$
$14$	−0.720420	−0.192540
$15$	−0.118195	−0.0305179
$16$	1.00000	0.250000
$17$	0.962927	0.233544	0.116772	−	0.993159i	$-0.462745\pi$
$17$	0.962927	0.233544	0.116772	+	0.993159i	$0.462745\pi$
$18$	−2.40211	−0.566182
$19$	4.56034	1.04621	0.523107	−	0.852267i	$-0.324773\pi$
$19$	4.56034	1.04621	0.523107	+	0.852267i	$0.324773\pi$
$20$	0.152858	0.0341801
$21$	0.557054	0.121559
$22$	−1.61689	−0.344723
$23$	0.510500	0.106447	0.0532233	−	0.998583i	$-0.483050\pi$
$23$	0.510500	0.106447	0.0532233	+	0.998583i	$0.483050\pi$
$24$	−0.773234	−0.157836
$25$	−4.97663	−0.995327
$26$	0.412560	0.0809096
$27$	4.17710	0.803882
$28$	−0.720420	−0.136147
$29$	1.89254	0.351436	0.175718	−	0.984441i	$-0.443775\pi$
$29$	1.89254	0.351436	0.175718	+	0.984441i	$0.443775\pi$
$30$	−0.118195	−0.0215794
$31$	−6.43946	−1.15656	−0.578281	−	0.815838i	$-0.696276\pi$
$31$	−6.43946	−1.15656	−0.578281	+	0.815838i	$0.696276\pi$
$32$	1.00000	0.176777
$33$	1.25024	0.217638
$34$	0.962927	0.165141
$35$	−0.110122	−0.0186140
$36$	−2.40211	−0.400351
$37$	6.40222	1.05252	0.526259	−	0.850324i	$-0.323594\pi$
$37$	6.40222	1.05252	0.526259	+	0.850324i	$0.323594\pi$
$38$	4.56034	0.739785
$39$	−0.319005	−0.0510817
$40$	0.152858	0.0241690
$41$	−0.251040	−0.0392058	−0.0196029	−	0.999808i	$-0.506240\pi$
$41$	−0.251040	−0.0392058	−0.0196029	+	0.999808i	$0.506240\pi$
$42$	0.557054	0.0859552
$43$	0.728596	0.111110	0.0555549	−	0.998456i	$-0.482307\pi$
$43$	0.728596	0.111110	0.0555549	+	0.998456i	$0.482307\pi$
$44$	−1.61689	−0.243756
$45$	−0.367182	−0.0547363
$46$	0.510500	0.0752691
$47$	4.20758	0.613738	0.306869	−	0.951752i	$-0.400719\pi$
$47$	4.20758	0.613738	0.306869	+	0.951752i	$0.400719\pi$
$48$	−0.773234	−0.111607
$49$	−6.48099	−0.925856
$50$	−4.97663	−0.703802
$51$	−0.744568	−0.104260
$52$	0.412560	0.0572117
$53$	6.90642	0.948670	0.474335	−	0.880344i	$-0.342689\pi$
$53$	6.90642	0.948670	0.474335	+	0.880344i	$0.342689\pi$
$54$	4.17710	0.568431
$55$	−0.247155	−0.0333264
$56$	−0.720420	−0.0962702
$57$	−3.52621	−0.467058
$58$	1.89254	0.248503
$59$	−3.92940	−0.511564	−0.255782	−	0.966734i	$-0.582333\pi$
$59$	−3.92940	−0.511564	−0.255782	+	0.966734i	$0.582333\pi$
$60$	−0.118195	−0.0152589
$61$	−6.20199	−0.794083	−0.397041	−	0.917801i	$-0.629963\pi$
$61$	−6.20199	−0.794083	−0.397041	+	0.917801i	$0.629963\pi$
$62$	−6.43946	−0.817813
$63$	1.73053	0.218026
$64$	1.00000	0.125000
$65$	0.0630631	0.00782202
$66$	1.25024	0.153894
$67$	−1.93421	−0.236301	−0.118151	−	0.992996i	$-0.537697\pi$
$67$	−1.93421	−0.236301	−0.118151	+	0.992996i	$0.537697\pi$
$68$	0.962927	0.116772
$69$	−0.394736	−0.0475206
$70$	−0.110122	−0.0131621
$71$	−8.37880	−0.994381	−0.497191	−	0.867641i	$-0.665635\pi$
$71$	−8.37880	−0.994381	−0.497191	+	0.867641i	$0.665635\pi$
$72$	−2.40211	−0.283091
$73$	−1.44169	−0.168737	−0.0843683	−	0.996435i	$-0.526887\pi$
$73$	−1.44169	−0.168737	−0.0843683	+	0.996435i	$0.526887\pi$
$74$	6.40222	0.744243
$75$	3.84810	0.444341
$76$	4.56034	0.523107
$77$	1.16484	0.132746
$78$	−0.319005	−0.0361202
$79$	−3.46527	−0.389874	−0.194937	−	0.980816i	$-0.562450\pi$
$79$	−3.46527	−0.389874	−0.194937	+	0.980816i	$0.562450\pi$
$80$	0.152858	0.0170901
$81$	3.97645	0.441828
$82$	−0.251040	−0.0277227
$83$	−3.25465	−0.357245	−0.178622	−	0.983918i	$-0.557164\pi$
$83$	−3.25465	−0.357245	−0.178622	+	0.983918i	$0.557164\pi$
$84$	0.557054	0.0607795
$85$	0.147191	0.0159651
$86$	0.728596	0.0785665
$87$	−1.46338	−0.156890
$88$	−1.61689	−0.172361
$89$	−9.52355	−1.00949	−0.504747	−	0.863267i	$-0.668414\pi$
$89$	−9.52355	−1.00949	−0.504747	+	0.863267i	$0.668414\pi$
$90$	−0.367182	−0.0387044
$91$	−0.297216	−0.0311567
$92$	0.510500	0.0532233
$93$	4.97921	0.516320
$94$	4.20758	0.433979
$95$	0.697086	0.0715195
$96$	−0.773234	−0.0789179
$97$	9.58266	0.972971	0.486486	−	0.873689i	$-0.338279\pi$
$97$	9.58266	0.972971	0.486486	+	0.873689i	$0.338279\pi$
$98$	−6.48099	−0.654679
$99$	3.88395	0.390352
$100$	−4.97663	−0.497663
$101$	−10.9618	−1.09074	−0.545368	−	0.838197i	$-0.683610\pi$
$101$	−10.9618	−1.09074	−0.545368	+	0.838197i	$0.683610\pi$
$102$	−0.744568	−0.0737232
$103$	14.5473	1.43338	0.716692	−	0.697390i	$-0.245655\pi$
$103$	14.5473	1.43338	0.716692	+	0.697390i	$0.245655\pi$
$104$	0.412560	0.0404548
$105$	0.0851502	0.00830981
$106$	6.90642	0.670811
$107$	−17.4001	−1.68213	−0.841066	−	0.540933i	$-0.818071\pi$
$107$	−17.4001	−1.68213	−0.841066	+	0.540933i	$0.818071\pi$
$108$	4.17710	0.401941
$109$	−14.5673	−1.39530	−0.697649	−	0.716440i	$-0.745770\pi$
$109$	−14.5673	−1.39530	−0.697649	+	0.716440i	$0.745770\pi$
$110$	−0.247155	−0.0235653
$111$	−4.95042	−0.469873
$112$	−0.720420	−0.0680733
$113$	2.58908	0.243561	0.121780	−	0.992557i	$-0.461140\pi$
$113$	2.58908	0.243561	0.121780	+	0.992557i	$0.461140\pi$
$114$	−3.52621	−0.330260
$115$	0.0780341	0.00727672
$116$	1.89254	0.175718
$117$	−0.991013	−0.0916192
$118$	−3.92940	−0.361731
$119$	−0.693712	−0.0635925
$120$	−0.118195	−0.0107897
$121$	−8.38566	−0.762333
$122$	−6.20199	−0.561501
$123$	0.194112	0.0175025
$124$	−6.43946	−0.578281
$125$	−1.52501	−0.136401
$126$	1.73053	0.154168
$127$	−5.93917	−0.527016	−0.263508	−	0.964657i	$-0.584880\pi$
$127$	−5.93917	−0.527016	−0.263508	+	0.964657i	$0.584880\pi$
$128$	1.00000	0.0883883
$129$	−0.563375	−0.0496024
$130$	0.0630631	0.00553100
$131$	−6.71757	−0.586917	−0.293459	−	0.955972i	$-0.594806\pi$
$131$	−6.71757	−0.586917	−0.293459	+	0.955972i	$0.594806\pi$
$132$	1.25024	0.108819
$133$	−3.28536	−0.284877
$134$	−1.93421	−0.167090
$135$	0.638503	0.0549536
$136$	0.962927	0.0825703
$137$	0.820104	0.0700662	0.0350331	−	0.999386i	$-0.488846\pi$
$137$	0.820104	0.0700662	0.0350331	+	0.999386i	$0.488846\pi$
$138$	−0.394736	−0.0336022
$139$	−12.8093	−1.08647	−0.543235	−	0.839581i	$-0.682801\pi$
$139$	−12.8093	−1.08647	−0.543235	+	0.839581i	$0.682801\pi$
$140$	−0.110122	−0.00930702
$141$	−3.25344	−0.273989
$142$	−8.37880	−0.703134
$143$	−0.667065	−0.0557828
$144$	−2.40211	−0.200176
$145$	0.289290	0.0240242
$146$	−1.44169	−0.119315
$147$	5.01133	0.413327
$148$	6.40222	0.526259
$149$	6.86429	0.562345	0.281172	−	0.959657i	$-0.409277\pi$
$149$	6.86429	0.562345	0.281172	+	0.959657i	$0.409277\pi$
$150$	3.84810	0.314196
$151$	12.1919	0.992159	0.496079	−	0.868277i	$-0.334772\pi$
$151$	12.1919	0.992159	0.496079	+	0.868277i	$0.334772\pi$
$152$	4.56034	0.369892
$153$	−2.31305	−0.186999
$154$	1.16484	0.0938657
$155$	−0.984325	−0.0790629
$156$	−0.319005	−0.0255409
$157$	−18.4843	−1.47521	−0.737603	−	0.675234i	$-0.764043\pi$
$157$	−18.4843	−1.47521	−0.737603	+	0.675234i	$0.764043\pi$
$158$	−3.46527	−0.275682
$159$	−5.34028	−0.423512
$160$	0.152858	0.0120845
$161$	−0.367775	−0.0289847
$162$	3.97645	0.312420
$163$	20.9397	1.64013	0.820064	−	0.572272i	$-0.193938\pi$
$163$	20.9397	1.64013	0.820064	+	0.572272i	$0.193938\pi$
$164$	−0.251040	−0.0196029
$165$	0.191109	0.0148778
$166$	−3.25465	−0.252610
$167$	−8.90884	−0.689387	−0.344693	−	0.938715i	$-0.612017\pi$
$167$	−8.90884	−0.689387	−0.344693	+	0.938715i	$0.612017\pi$
$168$	0.557054	0.0429776
$169$	−12.8298	−0.986907
$170$	0.147191	0.0112891
$171$	−10.9544	−0.837706
$172$	0.728596	0.0555549
$173$	−10.6163	−0.807143	−0.403571	−	0.914948i	$-0.632231\pi$
$173$	−10.6163	−0.807143	−0.403571	+	0.914948i	$0.632231\pi$
$174$	−1.46338	−0.110938
$175$	3.58527	0.271021
$176$	−1.61689	−0.121878
$177$	3.03835	0.228376
$178$	−9.52355	−0.713820
$179$	−23.2148	−1.73515	−0.867577	−	0.497303i	$-0.834324\pi$
$179$	−23.2148	−1.73515	−0.867577	+	0.497303i	$0.834324\pi$
$180$	−0.367182	−0.0273681
$181$	−5.82857	−0.433234	−0.216617	−	0.976257i	$-0.569502\pi$
$181$	−5.82857	−0.433234	−0.216617	+	0.976257i	$0.569502\pi$
$182$	−0.297216	−0.0220311
$183$	4.79559	0.354500
$184$	0.510500	0.0376346
$185$	0.978632	0.0719505
$186$	4.97921	0.365094
$187$	−1.55695	−0.113855
$188$	4.20758	0.306869
$189$	−3.00926	−0.218892
$190$	0.697086	0.0505719
$191$	−1.88715	−0.136549	−0.0682747	−	0.997667i	$-0.521749\pi$
$191$	−1.88715	−0.136549	−0.0682747	+	0.997667i	$0.521749\pi$
$192$	−0.773234	−0.0558034
$193$	4.95532	0.356692	0.178346	−	0.983968i	$-0.442925\pi$
$193$	4.95532	0.356692	0.178346	+	0.983968i	$0.442925\pi$
$194$	9.58266	0.687995
$195$	−0.0487626	−0.00349196
$196$	−6.48099	−0.462928
$197$	6.81925	0.485852	0.242926	−	0.970045i	$-0.421893\pi$
$197$	6.81925	0.485852	0.242926	+	0.970045i	$0.421893\pi$
$198$	3.88395	0.276020
$199$	22.0998	1.56661	0.783307	−	0.621635i	$-0.213531\pi$
$199$	22.0998	1.56661	0.783307	+	0.621635i	$0.213531\pi$
$200$	−4.97663	−0.351901
$201$	1.49560	0.105491
$202$	−10.9618	−0.771267
$203$	−1.36342	−0.0956936
$204$	−0.744568	−0.0521302
$205$	−0.0383735	−0.00268012
$206$	14.5473	1.01356
$207$	−1.22628	−0.0852321
$208$	0.412560	0.0286059
$209$	−7.37358	−0.510041
$210$	0.0851502	0.00587592
$211$	−17.8498	−1.22883	−0.614416	−	0.788982i	$-0.710608\pi$
$211$	−17.8498	−1.22883	−0.614416	+	0.788982i	$0.710608\pi$
$212$	6.90642	0.474335
$213$	6.47878	0.443919
$214$	−17.4001	−1.18945
$215$	0.111372	0.00759550
$216$	4.17710	0.284215
$217$	4.63912	0.314924
$218$	−14.5673	−0.986624
$219$	1.11476	0.0753286
$220$	−0.247155	−0.0166632
$221$	0.397265	0.0267229
$222$	−4.95042	−0.332250
$223$	−2.46977	−0.165388	−0.0826939	−	0.996575i	$-0.526352\pi$
$223$	−2.46977	−0.165388	−0.0826939	+	0.996575i	$0.526352\pi$
$224$	−0.720420	−0.0481351
$225$	11.9544	0.796961
$226$	2.58908	0.172223
$227$	7.94717	0.527472	0.263736	−	0.964595i	$-0.415045\pi$
$227$	7.94717	0.527472	0.263736	+	0.964595i	$0.415045\pi$
$228$	−3.52621	−0.233529
$229$	11.3355	0.749074	0.374537	−	0.927212i	$-0.377802\pi$
$229$	11.3355	0.749074	0.374537	+	0.927212i	$0.377802\pi$
$230$	0.0780341	0.00514542
$231$	−0.900696	−0.0592614
$232$	1.89254	0.124251
$233$	−8.76264	−0.574060	−0.287030	−	0.957922i	$-0.592668\pi$
$233$	−8.76264	−0.574060	−0.287030	+	0.957922i	$0.592668\pi$
$234$	−0.991013	−0.0647846
$235$	0.643163	0.0419553
$236$	−3.92940	−0.255782
$237$	2.67947	0.174050
$238$	−0.693712	−0.0449667
$239$	−11.6548	−0.753885	−0.376942	−	0.926237i	$-0.623025\pi$
$239$	−11.6548	−0.753885	−0.376942	+	0.926237i	$0.623025\pi$
$240$	−0.118195	−0.00762947
$241$	12.4018	0.798871	0.399436	−	0.916761i	$-0.369206\pi$
$241$	12.4018	0.798871	0.399436	+	0.916761i	$0.369206\pi$
$242$	−8.38566	−0.539051
$243$	−15.6060	−1.00113
$244$	−6.20199	−0.397041
$245$	−0.990673	−0.0632918
$246$	0.194112	0.0123762
$247$	1.88141	0.119711
$248$	−6.43946	−0.408906
$249$	2.51661	0.159484
$250$	−1.52501	−0.0964501
$251$	−18.3227	−1.15652	−0.578260	−	0.815853i	$-0.696268\pi$
$251$	−18.3227	−1.15652	−0.578260	+	0.815853i	$0.696268\pi$
$252$	1.73053	0.109013
$253$	−0.825424	−0.0518939
$254$	−5.93917	−0.372657
$255$	−0.113813	−0.00712727
$256$	1.00000	0.0625000
$257$	2.05130	0.127957	0.0639784	−	0.997951i	$-0.479621\pi$
$257$	2.05130	0.127957	0.0639784	+	0.997951i	$0.479621\pi$
$258$	−0.563375	−0.0350742
$259$	−4.61229	−0.286594
$260$	0.0630631	0.00391101
$261$	−4.54609	−0.281396
$262$	−6.71757	−0.415013
$263$	0.848988	0.0523508	0.0261754	−	0.999657i	$-0.491667\pi$
$263$	0.848988	0.0523508	0.0261754	+	0.999657i	$0.491667\pi$
$264$	1.25024	0.0769468
$265$	1.05570	0.0648513
$266$	−3.28536	−0.201438
$267$	7.36394	0.450666
$268$	−1.93421	−0.118151
$269$	−9.51681	−0.580250	−0.290125	−	0.956989i	$-0.593697\pi$
$269$	−9.51681	−0.580250	−0.290125	+	0.956989i	$0.593697\pi$
$270$	0.638503	0.0388581
$271$	−22.2013	−1.34863	−0.674316	−	0.738443i	$-0.735562\pi$
$271$	−22.2013	−1.34863	−0.674316	+	0.738443i	$0.735562\pi$
$272$	0.962927	0.0583860
$273$	0.229818	0.0139092
$274$	0.820104	0.0495443
$275$	8.04668	0.485233
$276$	−0.394736	−0.0237603
$277$	−30.8428	−1.85316	−0.926582	−	0.376092i	$-0.877267\pi$
$277$	−30.8428	−1.85316	−0.926582	+	0.376092i	$0.877267\pi$
$278$	−12.8093	−0.768250
$279$	15.4683	0.926062
$280$	−0.110122	−0.00658106
$281$	6.82082	0.406896	0.203448	−	0.979086i	$-0.434785\pi$
$281$	6.82082	0.406896	0.203448	+	0.979086i	$0.434785\pi$
$282$	−3.25344	−0.193740
$283$	−12.0137	−0.714141	−0.357071	−	0.934077i	$-0.616224\pi$
$283$	−12.0137	−0.714141	−0.357071	+	0.934077i	$0.616224\pi$
$284$	−8.37880	−0.497191
$285$	−0.539010	−0.0319282
$286$	−0.667065	−0.0394444
$287$	0.180854	0.0106755
$288$	−2.40211	−0.141546
$289$	−16.0728	−0.945457
$290$	0.289290	0.0169877
$291$	−7.40964	−0.434361
$292$	−1.44169	−0.0843683
$293$	−6.09422	−0.356028	−0.178014	−	0.984028i	$-0.556967\pi$
$293$	−6.09422	−0.356028	−0.178014	+	0.984028i	$0.556967\pi$
$294$	5.01133	0.292267
$295$	−0.600641	−0.0349707
$296$	6.40222	0.372121
$297$	−6.75391	−0.391902
$298$	6.86429	0.397638
$299$	0.210612	0.0121800
$300$	3.84810	0.222170
$301$	−0.524895	−0.0302545
$302$	12.1919	0.701562
$303$	8.47601	0.486934
$304$	4.56034	0.261553
$305$	−0.948025	−0.0542837
$306$	−2.31305	−0.132229
$307$	9.49956	0.542169	0.271084	−	0.962556i	$-0.412618\pi$
$307$	9.49956	0.542169	0.271084	+	0.962556i	$0.412618\pi$
$308$	1.16484	0.0663730
$309$	−11.2484	−0.639901
$310$	−0.984325	−0.0559059
$311$	8.42827	0.477924	0.238962	−	0.971029i	$-0.423193\pi$
$311$	8.42827	0.477924	0.238962	+	0.971029i	$0.423193\pi$
$312$	−0.319005	−0.0180601
$313$	−26.1698	−1.47921	−0.739603	−	0.673043i	$-0.764987\pi$
$313$	−26.1698	−1.47921	−0.739603	+	0.673043i	$0.764987\pi$
$314$	−18.4843	−1.04313
$315$	0.264525	0.0149043
$316$	−3.46527	−0.194937
$317$	13.8497	0.777879	0.388940	−	0.921263i	$-0.372841\pi$
$317$	13.8497	0.777879	0.388940	+	0.921263i	$0.372841\pi$
$318$	−5.34028	−0.299468
$319$	−3.06003	−0.171329
$320$	0.152858	0.00854503
$321$	13.4544	0.750949
$322$	−0.367775	−0.0204953
$323$	4.39127	0.244337
$324$	3.97645	0.220914
$325$	−2.05316	−0.113889
$326$	20.9397	1.15974
$327$	11.2640	0.622899
$328$	−0.251040	−0.0138613
$329$	−3.03123	−0.167117
$330$	0.191109	0.0105202
$331$	10.2530	0.563553	0.281777	−	0.959480i	$-0.409076\pi$
$331$	10.2530	0.563553	0.281777	+	0.959480i	$0.409076\pi$
$332$	−3.25465	−0.178622
$333$	−15.3788	−0.842755
$334$	−8.90884	−0.487470
$335$	−0.295660	−0.0161536
$336$	0.557054	0.0303898
$337$	29.8114	1.62393	0.811967	−	0.583704i	$-0.198397\pi$
$337$	29.8114	1.62393	0.811967	+	0.583704i	$0.198397\pi$
$338$	−12.8298	−0.697849
$339$	−2.00197	−0.108732
$340$	0.147191	0.00798257
$341$	10.4119	0.563837
$342$	−10.9544	−0.592348
$343$	9.71198	0.524398
$344$	0.728596	0.0392832
$345$	−0.0603387	−0.00324852
$346$	−10.6163	−0.570736
$347$	−15.8260	−0.849585	−0.424793	−	0.905291i	$-0.639653\pi$
$347$	−15.8260	−0.849585	−0.424793	+	0.905291i	$0.639653\pi$
$348$	−1.46338	−0.0784452
$349$	−10.3989	−0.556643	−0.278321	−	0.960488i	$-0.589778\pi$
$349$	−10.3989	−0.556643	−0.278321	+	0.960488i	$0.589778\pi$
$350$	3.58527	0.191641
$351$	1.72330	0.0919830
$352$	−1.61689	−0.0861807
$353$	34.4370	1.83290	0.916449	−	0.400152i	$-0.131043\pi$
$353$	34.4370	1.83290	0.916449	+	0.400152i	$0.131043\pi$
$354$	3.03835	0.161486
$355$	−1.28077	−0.0679762
$356$	−9.52355	−0.504747
$357$	0.536402	0.0283894
$358$	−23.2148	−1.22694
$359$	26.1417	1.37970	0.689852	−	0.723950i	$-0.257675\pi$
$359$	26.1417	1.37970	0.689852	+	0.723950i	$0.257675\pi$
$360$	−0.367182	−0.0193522
$361$	1.79670	0.0945634
$362$	−5.82857	−0.306343
$363$	6.48408	0.340326
$364$	−0.297216	−0.0155784
$365$	−0.220374	−0.0115349
$366$	4.79559	0.250669
$367$	−5.34992	−0.279263	−0.139632	−	0.990204i	$-0.544592\pi$
$367$	−5.34992	−0.279263	−0.139632	+	0.990204i	$0.544592\pi$
$368$	0.510500	0.0266117
$369$	0.603025	0.0313922
$370$	0.978632	0.0508767
$371$	−4.97553	−0.258316
$372$	4.97921	0.258160
$373$	−16.9244	−0.876310	−0.438155	−	0.898899i	$-0.644368\pi$
$373$	−16.9244	−0.876310	−0.438155	+	0.898899i	$0.644368\pi$
$374$	−1.55695	−0.0805079
$375$	1.17919	0.0608931
$376$	4.20758	0.216989
$377$	0.780785	0.0402125
$378$	−3.00926	−0.154780
$379$	−9.50681	−0.488332	−0.244166	−	0.969733i	$-0.578514\pi$
$379$	−9.50681	−0.488332	−0.244166	+	0.969733i	$0.578514\pi$
$380$	0.697086	0.0357597
$381$	4.59237	0.235274
$382$	−1.88715	−0.0965550
$383$	−5.96519	−0.304807	−0.152403	−	0.988318i	$-0.548701\pi$
$383$	−5.96519	−0.304807	−0.152403	+	0.988318i	$0.548701\pi$
$384$	−0.773234	−0.0394589
$385$	0.178056	0.00907456
$386$	4.95532	0.252219
$387$	−1.75017	−0.0889659
$388$	9.58266	0.486486
$389$	−22.0044	−1.11567	−0.557833	−	0.829953i	$-0.688367\pi$
$389$	−22.0044	−1.11567	−0.557833	+	0.829953i	$0.688367\pi$
$390$	−0.0487626	−0.00246919
$391$	0.491574	0.0248600
$392$	−6.48099	−0.327340
$393$	5.19426	0.262016
$394$	6.81925	0.343549
$395$	−0.529695	−0.0266519
$396$	3.88395	0.195176
$397$	29.8358	1.49741	0.748707	−	0.662901i	$-0.230675\pi$
$397$	29.8358	1.49741	0.748707	+	0.662901i	$0.230675\pi$
$398$	22.0998	1.10776
$399$	2.54035	0.127177
$400$	−4.97663	−0.248832
$401$	6.38480	0.318842	0.159421	−	0.987211i	$-0.449037\pi$
$401$	6.38480	0.318842	0.159421	+	0.987211i	$0.449037\pi$
$402$	1.49560	0.0745936
$403$	−2.65666	−0.132338
$404$	−10.9618	−0.545368
$405$	0.607833	0.0302035
$406$	−1.36342	−0.0676656
$407$	−10.3517	−0.513115
$408$	−0.744568	−0.0368616
$409$	−21.2296	−1.04973	−0.524867	−	0.851184i	$-0.675885\pi$
$409$	−21.2296	−1.04973	−0.524867	+	0.851184i	$0.675885\pi$
$410$	−0.0383735	−0.00189513
$411$	−0.634133	−0.0312795
$412$	14.5473	0.716692
$413$	2.83082	0.139296
$414$	−1.22628	−0.0602682
$415$	−0.497501	−0.0244213
$416$	0.412560	0.0202274
$417$	9.90458	0.485029
$418$	−7.37358	−0.360654
$419$	−22.2328	−1.08614	−0.543071	−	0.839687i	$-0.682738\pi$
$419$	−22.2328	−1.08614	−0.543071	+	0.839687i	$0.682738\pi$
$420$	0.0851502	0.00415491
$421$	2.78925	0.135940	0.0679700	−	0.997687i	$-0.478348\pi$
$421$	2.78925	0.135940	0.0679700	+	0.997687i	$0.478348\pi$
$422$	−17.8498	−0.868915
$423$	−10.1071	−0.491422
$424$	6.90642	0.335406
$425$	−4.79213	−0.232453
$426$	6.47878	0.313898
$427$	4.46804	0.216223
$428$	−17.4001	−0.841066
$429$	0.515797	0.0249029
$430$	0.111372	0.00537083
$431$	19.3799	0.933496	0.466748	−	0.884390i	$-0.345425\pi$
$431$	19.3799	0.933496	0.466748	+	0.884390i	$0.345425\pi$
$432$	4.17710	0.200971
$433$	−36.0867	−1.73421	−0.867107	−	0.498122i	$-0.834023\pi$
$433$	−36.0867	−1.73421	−0.867107	+	0.498122i	$0.834023\pi$
$434$	4.63912	0.222685
$435$	−0.223689	−0.0107251
$436$	−14.5673	−0.697649
$437$	2.32805	0.111366
$438$	1.11476	0.0532654
$439$	24.0072	1.14580	0.572902	−	0.819624i	$-0.305818\pi$
$439$	24.0072	1.14580	0.572902	+	0.819624i	$0.305818\pi$
$440$	−0.247155	−0.0117827
$441$	15.5681	0.741336
$442$	0.397265	0.0188960
$443$	33.8022	1.60599	0.802995	−	0.595986i	$-0.203239\pi$
$443$	33.8022	1.60599	0.802995	+	0.595986i	$0.203239\pi$
$444$	−4.95042	−0.234936
$445$	−1.45575	−0.0690093
$446$	−2.46977	−0.116947
$447$	−5.30771	−0.251046
$448$	−0.720420	−0.0340367
$449$	5.58652	0.263644	0.131822	−	0.991273i	$-0.457917\pi$
$449$	5.58652	0.263644	0.131822	+	0.991273i	$0.457917\pi$
$450$	11.9544	0.563537
$451$	0.405904	0.0191133
$452$	2.58908	0.121780
$453$	−9.42716	−0.442927
$454$	7.94717	0.372979
$455$	−0.0454320	−0.00212988
$456$	−3.52621	−0.165130
$457$	4.21106	0.196985	0.0984924	−	0.995138i	$-0.468598\pi$
$457$	4.21106	0.196985	0.0984924	+	0.995138i	$0.468598\pi$
$458$	11.3355	0.529675
$459$	4.02224	0.187742
$460$	0.0780341	0.00363836
$461$	−22.9252	−1.06773	−0.533867	−	0.845569i	$-0.679262\pi$
$461$	−22.9252	−1.06773	−0.533867	+	0.845569i	$0.679262\pi$
$462$	−0.900696	−0.0419042
$463$	31.6413	1.47050	0.735248	−	0.677798i	$-0.237066\pi$
$463$	31.6413	1.47050	0.735248	+	0.677798i	$0.237066\pi$
$464$	1.89254	0.0878589
$465$	0.761114	0.0352958
$466$	−8.76264	−0.405922
$467$	24.6616	1.14120	0.570602	−	0.821227i	$-0.306710\pi$
$467$	24.6616	1.14120	0.570602	+	0.821227i	$0.306710\pi$
$468$	−0.991013	−0.0458096
$469$	1.39344	0.0643432
$470$	0.643163	0.0296669
$471$	14.2927	0.658572
$472$	−3.92940	−0.180865
$473$	−1.17806	−0.0541673
$474$	2.67947	0.123072
$475$	−22.6951	−1.04132
$476$	−0.693712	−0.0317962
$477$	−16.5900	−0.759603
$478$	−11.6548	−0.533077
$479$	21.9275	1.00189	0.500947	−	0.865478i	$-0.332985\pi$
$479$	21.9275	1.00189	0.500947	+	0.865478i	$0.332985\pi$
$480$	−0.118195	−0.00539485
$481$	2.64130	0.120433
$482$	12.4018	0.564887
$483$	0.284376	0.0129396
$484$	−8.38566	−0.381166
$485$	1.46479	0.0665126
$486$	−15.6060	−0.707903
$487$	8.58796	0.389158	0.194579	−	0.980887i	$-0.437666\pi$
$487$	8.58796	0.389158	0.194579	+	0.980887i	$0.437666\pi$
$488$	−6.20199	−0.280751
$489$	−16.1913	−0.732197
$490$	−0.990673	−0.0447541
$491$	32.6463	1.47331	0.736653	−	0.676271i	$-0.236405\pi$
$491$	32.6463	1.47331	0.736653	+	0.676271i	$0.236405\pi$
$492$	0.194112	0.00875127
$493$	1.82238	0.0820757
$494$	1.88141	0.0846488
$495$	0.593694	0.0266846
$496$	−6.43946	−0.289140
$497$	6.03626	0.270763
$498$	2.51661	0.112772
$499$	−6.30045	−0.282047	−0.141023	−	0.990006i	$-0.545039\pi$
$499$	−6.30045	−0.282047	−0.141023	+	0.990006i	$0.545039\pi$
$500$	−1.52501	−0.0682005
$501$	6.88862	0.307761
$502$	−18.3227	−0.817783
$503$	−8.85995	−0.395046	−0.197523	−	0.980298i	$-0.563290\pi$
$503$	−8.85995	−0.395046	−0.197523	+	0.980298i	$0.563290\pi$
$504$	1.73053	0.0770838
$505$	−1.67560	−0.0745630
$506$	−0.825424	−0.0366946
$507$	9.92044	0.440582
$508$	−5.93917	−0.263508
$509$	35.4895	1.57304	0.786521	−	0.617563i	$-0.211880\pi$
$509$	35.4895	1.57304	0.786521	+	0.617563i	$0.211880\pi$
$510$	−0.113813	−0.00503974
$511$	1.03862	0.0459458
$512$	1.00000	0.0441942
$513$	19.0490	0.841033
$514$	2.05130	0.0904792
$515$	2.22367	0.0979865
$516$	−0.563375	−0.0248012
$517$	−6.80320	−0.299205
$518$	−4.61229	−0.202652
$519$	8.20889	0.360330
$520$	0.0630631	0.00276550
$521$	−0.919339	−0.0402770	−0.0201385	−	0.999797i	$-0.506411\pi$
$521$	−0.919339	−0.0402770	−0.0201385	+	0.999797i	$0.506411\pi$
$522$	−4.54609	−0.198977
$523$	−29.9576	−1.30995	−0.654977	−	0.755649i	$-0.727322\pi$
$523$	−29.9576	−1.30995	−0.654977	+	0.755649i	$0.727322\pi$
$524$	−6.71757	−0.293459
$525$	−2.77225	−0.120991
$526$	0.848988	0.0370176
$527$	−6.20073	−0.270108
$528$	1.25024	0.0544096
$529$	−22.7394	−0.988669
$530$	1.05570	0.0458568
$531$	9.43885	0.409611
$532$	−3.28536	−0.142438
$533$	−0.103569	−0.00448606
$534$	7.36394	0.318669
$535$	−2.65975	−0.114991
$536$	−1.93421	−0.0835451
$537$	17.9505	0.774619
$538$	−9.51681	−0.410299
$539$	10.4791	0.451366
$540$	0.638503	0.0274768
$541$	−0.923040	−0.0396846	−0.0198423	−	0.999803i	$-0.506316\pi$
$541$	−0.923040	−0.0396846	−0.0198423	+	0.999803i	$0.506316\pi$
$542$	−22.2013	−0.953627
$543$	4.50685	0.193407
$544$	0.962927	0.0412851
$545$	−2.22674	−0.0953829
$546$	0.229818	0.00983530
$547$	14.5938	0.623986	0.311993	−	0.950084i	$-0.399003\pi$
$547$	14.5938	0.623986	0.311993	+	0.950084i	$0.399003\pi$
$548$	0.820104	0.0350331
$549$	14.8978	0.635825
$550$	8.04668	0.343112
$551$	8.63062	0.367677
$552$	−0.394736	−0.0168011
$553$	2.49645	0.106160
$554$	−30.8428	−1.31039
$555$	−0.756712	−0.0321206
$556$	−12.8093	−0.543235
$557$	−8.39954	−0.355900	−0.177950	−	0.984040i	$-0.556947\pi$
$557$	−8.39954	−0.355900	−0.177950	+	0.984040i	$0.556947\pi$
$558$	15.4683	0.654825
$559$	0.300589	0.0127136
$560$	−0.110122	−0.00465351
$561$	1.20389	0.0508281
$562$	6.82082	0.287719
$563$	24.9772	1.05266	0.526331	−	0.850280i	$-0.323567\pi$
$563$	24.9772	1.05266	0.526331	+	0.850280i	$0.323567\pi$
$564$	−3.25344	−0.136995
$565$	0.395763	0.0166499
$566$	−12.0137	−0.504974
$567$	−2.86472	−0.120307
$568$	−8.37880	−0.351567
$569$	−40.3105	−1.68990	−0.844952	−	0.534841i	$-0.820371\pi$
$569$	−40.3105	−1.68990	−0.844952	+	0.534841i	$0.820371\pi$
$570$	−0.539010	−0.0225767
$571$	−32.1842	−1.34687	−0.673433	−	0.739248i	$-0.735181\pi$
$571$	−32.1842	−1.34687	−0.673433	+	0.739248i	$0.735181\pi$
$572$	−0.667065	−0.0278914
$573$	1.45921	0.0609594
$574$	0.180854	0.00754870
$575$	−2.54057	−0.105949
$576$	−2.40211	−0.100088
$577$	−26.0375	−1.08396	−0.541978	−	0.840393i	$-0.682324\pi$
$577$	−26.0375	−1.08396	−0.541978	+	0.840393i	$0.682324\pi$
$578$	−16.0728	−0.668539
$579$	−3.83162	−0.159237
$580$	0.289290	0.0120121
$581$	2.34472	0.0972753
$582$	−7.40964	−0.307139
$583$	−11.1669	−0.462488
$584$	−1.44169	−0.0596574
$585$	−0.151485	−0.00626311
$586$	−6.09422	−0.251750
$587$	14.7732	0.609754	0.304877	−	0.952392i	$-0.401385\pi$
$587$	14.7732	0.609754	0.304877	+	0.952392i	$0.401385\pi$
$588$	5.01133	0.206664
$589$	−29.3661	−1.21001
$590$	−0.600641	−0.0247280
$591$	−5.27288	−0.216897
$592$	6.40222	0.263130
$593$	1.23840	0.0508551	0.0254275	−	0.999677i	$-0.491905\pi$
$593$	1.23840	0.0508551	0.0254275	+	0.999677i	$0.491905\pi$
$594$	−6.75391	−0.277116
$595$	−0.106040	−0.00434720
$596$	6.86429	0.281172
$597$	−17.0883	−0.699379
$598$	0.210612	0.00861255
$599$	−12.2334	−0.499842	−0.249921	−	0.968266i	$-0.580405\pi$
$599$	−12.2334	−0.499842	−0.249921	+	0.968266i	$0.580405\pi$
$600$	3.84810	0.157098
$601$	19.8994	0.811712	0.405856	−	0.913937i	$-0.366973\pi$
$601$	19.8994	0.811712	0.405856	+	0.913937i	$0.366973\pi$
$602$	−0.524895	−0.0213931
$603$	4.64618	0.189207
$604$	12.1919	0.496079
$605$	−1.28182	−0.0521133
$606$	8.47601	0.344315
$607$	−16.6997	−0.677820	−0.338910	−	0.940819i	$-0.610058\pi$
$607$	−16.6997	−0.677820	−0.338910	+	0.940819i	$0.610058\pi$
$608$	4.56034	0.184946
$609$	1.05425	0.0427202
$610$	−0.948025	−0.0383844
$611$	1.73588	0.0702261
$612$	−2.31305	−0.0934997
$613$	48.0013	1.93875	0.969377	−	0.245576i	$-0.0789771\pi$
$613$	48.0013	1.93875	0.969377	+	0.245576i	$0.0789771\pi$
$614$	9.49956	0.383371
$615$	0.0296717	0.00119648
$616$	1.16484	0.0469328
$617$	−0.855300	−0.0344331	−0.0172165	−	0.999852i	$-0.505480\pi$
$617$	−0.855300	−0.0344331	−0.0172165	+	0.999852i	$0.505480\pi$
$618$	−11.2484	−0.452478
$619$	2.84238	0.114245	0.0571224	−	0.998367i	$-0.481807\pi$
$619$	2.84238	0.114245	0.0571224	+	0.998367i	$0.481807\pi$
$620$	−0.984325	−0.0395314
$621$	2.13241	0.0855706
$622$	8.42827	0.337943
$623$	6.86096	0.274879
$624$	−0.319005	−0.0127704
$625$	24.6501	0.986002
$626$	−26.1698	−1.04596
$627$	5.70151	0.227696
$628$	−18.4843	−0.737603
$629$	6.16487	0.245809
$630$	0.264525	0.0105389
$631$	−5.36042	−0.213395	−0.106697	−	0.994292i	$-0.534028\pi$
$631$	−5.36042	−0.213395	−0.106697	+	0.994292i	$0.534028\pi$
$632$	−3.46527	−0.137841
$633$	13.8021	0.548584
$634$	13.8497	0.550044
$635$	−0.907851	−0.0360270
$636$	−5.34028	−0.211756
$637$	−2.67380	−0.105940
$638$	−3.06003	−0.121148
$639$	20.1268	0.796204
$640$	0.152858	0.00604225
$641$	29.6276	1.17022	0.585109	−	0.810955i	$-0.301052\pi$
$641$	29.6276	1.17022	0.585109	+	0.810955i	$0.301052\pi$
$642$	13.4544	0.531001
$643$	−9.91111	−0.390856	−0.195428	−	0.980718i	$-0.562610\pi$
$643$	−9.91111	−0.390856	−0.195428	+	0.980718i	$0.562610\pi$
$644$	−0.367775	−0.0144923
$645$	−0.0861165	−0.00339083
$646$	4.39127	0.172772
$647$	−35.7229	−1.40441	−0.702207	−	0.711973i	$-0.747802\pi$
$647$	−35.7229	−1.40441	−0.702207	+	0.711973i	$0.747802\pi$
$648$	3.97645	0.156210
$649$	6.35342	0.249393
$650$	−2.05316	−0.0805315
$651$	−3.58713	−0.140591
$652$	20.9397	0.820064
$653$	0.727882	0.0284842	0.0142421	−	0.999899i	$-0.495466\pi$
$653$	0.727882	0.0284842	0.0142421	+	0.999899i	$0.495466\pi$
$654$	11.2640	0.440456
$655$	−1.02684	−0.0401218
$656$	−0.251040	−0.00980145
$657$	3.46309	0.135108
$658$	−3.03123	−0.118169
$659$	−40.4262	−1.57478	−0.787391	−	0.616454i	$-0.788569\pi$
$659$	−40.4262	−1.57478	−0.787391	+	0.616454i	$0.788569\pi$
$660$	0.191109	0.00743891
$661$	4.50761	0.175326	0.0876628	−	0.996150i	$-0.472060\pi$
$661$	4.50761	0.175326	0.0876628	+	0.996150i	$0.472060\pi$
$662$	10.2530	0.398492
$663$	−0.307179	−0.0119298
$664$	−3.25465	−0.126305
$665$	−0.502195	−0.0194743
$666$	−15.3788	−0.595918
$667$	0.966141	0.0374091
$668$	−8.90884	−0.344693
$669$	1.90971	0.0738336
$670$	−0.295660	−0.0114223
$671$	10.0279	0.387125
$672$	0.557054	0.0214888
$673$	−1.90919	−0.0735940	−0.0367970	−	0.999323i	$-0.511715\pi$
$673$	−1.90919	−0.0735940	−0.0367970	+	0.999323i	$0.511715\pi$
$674$	29.8114	1.14829
$675$	−20.7879	−0.800126
$676$	−12.8298	−0.493454
$677$	−17.8813	−0.687233	−0.343617	−	0.939110i	$-0.611652\pi$
$677$	−17.8813	−0.687233	−0.343617	+	0.939110i	$0.611652\pi$
$678$	−2.00197	−0.0768852
$679$	−6.90354	−0.264934
$680$	0.147191	0.00564453
$681$	−6.14503	−0.235478
$682$	10.4119	0.398693
$683$	−30.4340	−1.16453	−0.582263	−	0.813001i	$-0.697832\pi$
$683$	−30.4340	−1.16453	−0.582263	+	0.813001i	$0.697832\pi$
$684$	−10.9544	−0.418853
$685$	0.125360	0.00478975
$686$	9.71198	0.370805
$687$	−8.76503	−0.334407
$688$	0.728596	0.0277775
$689$	2.84931	0.108550
$690$	−0.0603387	−0.00229705
$691$	12.9794	0.493760	0.246880	−	0.969046i	$-0.420595\pi$
$691$	12.9794	0.493760	0.246880	+	0.969046i	$0.420595\pi$
$692$	−10.6163	−0.403571
$693$	−2.79808	−0.106290
$694$	−15.8260	−0.600747
$695$	−1.95800	−0.0742713
$696$	−1.46338	−0.0554691
$697$	−0.241733	−0.00915628
$698$	−10.3989	−0.393606
$699$	6.77558	0.256276
$700$	3.58527	0.135510
$701$	−44.1379	−1.66707	−0.833534	−	0.552469i	$-0.813686\pi$
$701$	−44.1379	−1.66707	−0.833534	+	0.552469i	$0.813686\pi$
$702$	1.72330	0.0650418
$703$	29.1963	1.10116
$704$	−1.61689	−0.0609389
$705$	−0.497316	−0.0187300
$706$	34.4370	1.29605
$707$	7.89708	0.297000
$708$	3.03835	0.114188
$709$	−0.741250	−0.0278382	−0.0139191	−	0.999903i	$-0.504431\pi$
$709$	−0.741250	−0.0278382	−0.0139191	+	0.999903i	$0.504431\pi$
$710$	−1.28077	−0.0480664
$711$	8.32396	0.312173
$712$	−9.52355	−0.356910
$713$	−3.28735	−0.123112
$714$	0.536402	0.0200743
$715$	−0.101966	−0.00381332
$716$	−23.2148	−0.867577
$717$	9.01187	0.336555
$718$	26.1417	0.975598
$719$	27.8769	1.03963	0.519816	−	0.854278i	$-0.326001\pi$
$719$	27.8769	1.03963	0.519816	+	0.854278i	$0.326001\pi$
$720$	−0.367182	−0.0136841
$721$	−10.4801	−0.390301
$722$	1.79670	0.0668664
$723$	−9.58951	−0.356638
$724$	−5.82857	−0.216617
$725$	−9.41848	−0.349793
$726$	6.48408	0.240647
$727$	11.4822	0.425853	0.212926	−	0.977068i	$-0.431701\pi$
$727$	11.4822	0.425853	0.212926	+	0.977068i	$0.431701\pi$
$728$	−0.297216	−0.0110156
$729$	0.137749	0.00510181
$730$	−0.220374	−0.00815639
$731$	0.701584	0.0259490
$732$	4.79559	0.177250
$733$	−51.5913	−1.90557	−0.952784	−	0.303650i	$-0.901795\pi$
$733$	−51.5913	−1.90557	−0.952784	+	0.303650i	$0.901795\pi$
$734$	−5.34992	−0.197469
$735$	0.766023	0.0282552
$736$	0.510500	0.0188173
$737$	3.12741	0.115200
$738$	0.603025	0.0221976
$739$	12.4939	0.459597	0.229799	−	0.973238i	$-0.426193\pi$
$739$	12.4939	0.459597	0.229799	+	0.973238i	$0.426193\pi$
$740$	0.978632	0.0359752
$741$	−1.45477	−0.0534424
$742$	−4.97553	−0.182657
$743$	15.2245	0.558532	0.279266	−	0.960214i	$-0.409909\pi$
$743$	15.2245	0.558532	0.279266	+	0.960214i	$0.409909\pi$
$744$	4.97921	0.182547
$745$	1.04926	0.0384421
$746$	−16.9244	−0.619645
$747$	7.81803	0.286047
$748$	−1.55695	−0.0569277
$749$	12.5354	0.458033
$750$	1.17919	0.0430579
$751$	−19.0695	−0.695854	−0.347927	−	0.937522i	$-0.613114\pi$
$751$	−19.0695	−0.695854	−0.347927	+	0.937522i	$0.613114\pi$
$752$	4.20758	0.153435
$753$	14.1678	0.516302
$754$	0.780785	0.0284345
$755$	1.86362	0.0678242
$756$	−3.00926	−0.109446
$757$	−8.78003	−0.319116	−0.159558	−	0.987189i	$-0.551007\pi$
$757$	−8.78003	−0.319116	−0.159558	+	0.987189i	$0.551007\pi$
$758$	−9.50681	−0.345303
$759$	0.638246	0.0231669
$760$	0.697086	0.0252859
$761$	−44.0326	−1.59618	−0.798090	−	0.602539i	$-0.794156\pi$
$761$	−44.0326	−1.59618	−0.798090	+	0.602539i	$0.794156\pi$
$762$	4.59237	0.166364
$763$	10.4946	0.379930
$764$	−1.88715	−0.0682747
$765$	−0.353569	−0.0127833
$766$	−5.96519	−0.215531
$767$	−1.62111	−0.0585350
$768$	−0.773234	−0.0279017
$769$	−31.6914	−1.14282	−0.571410	−	0.820665i	$-0.693603\pi$
$769$	−31.6914	−1.14282	−0.571410	+	0.820665i	$0.693603\pi$
$770$	0.178056	0.00641668
$771$	−1.58614	−0.0571234
$772$	4.95532	0.178346
$773$	−43.1580	−1.55229	−0.776143	−	0.630557i	$-0.782826\pi$
$773$	−43.1580	−1.55229	−0.776143	+	0.630557i	$0.782826\pi$
$774$	−1.75017	−0.0629084
$775$	32.0469	1.15116
$776$	9.58266	0.343997
$777$	3.56638	0.127943
$778$	−22.0044	−0.788895
$779$	−1.14483	−0.0410177
$780$	−0.0487626	−0.00174598
$781$	13.5476	0.484772
$782$	0.491574	0.0175787
$783$	7.90532	0.282513
$784$	−6.48099	−0.231464
$785$	−2.82547	−0.100846
$786$	5.19426	0.185273
$787$	−44.6602	−1.59197	−0.795983	−	0.605319i	$-0.793045\pi$
$787$	−44.6602	−1.59197	−0.795983	+	0.605319i	$0.793045\pi$
$788$	6.81925	0.242926
$789$	−0.656466	−0.0233708
$790$	−0.529695	−0.0188457
$791$	−1.86523	−0.0663199
$792$	3.88395	0.138010
$793$	−2.55869	−0.0908617
$794$	29.8358	1.05883
$795$	−0.816306	−0.0289514
$796$	22.0998	0.783307
$797$	−30.4743	−1.07945	−0.539727	−	0.841840i	$-0.681472\pi$
$797$	−30.4743	−1.07945	−0.539727	+	0.841840i	$0.681472\pi$
$798$	2.54035	0.0899276
$799$	4.05159	0.143335
$800$	−4.97663	−0.175951
$801$	22.8766	0.808305
$802$	6.38480	0.225455
$803$	2.33105	0.0822610
$804$	1.49560	0.0527456
$805$	−0.0562174	−0.00198140
$806$	−2.65666	−0.0935770
$807$	7.35873	0.259039
$808$	−10.9618	−0.385634
$809$	11.6462	0.409459	0.204730	−	0.978819i	$-0.434368\pi$
$809$	11.6462	0.409459	0.204730	+	0.978819i	$0.434368\pi$
$810$	0.607833	0.0213571
$811$	56.3305	1.97803	0.989015	−	0.147813i	$-0.0472234\pi$
$811$	56.3305	1.97803	0.989015	+	0.147813i	$0.0472234\pi$
$812$	−1.36342	−0.0478468
$813$	17.1668	0.602066
$814$	−10.3517	−0.362827
$815$	3.20081	0.112120
$816$	−0.744568	−0.0260651
$817$	3.32264	0.116245
$818$	−21.2296	−0.742274
$819$	0.713946	0.0249473
$820$	−0.0383735	−0.00134006
$821$	29.5205	1.03027	0.515136	−	0.857108i	$-0.327741\pi$
$821$	29.5205	1.03027	0.515136	+	0.857108i	$0.327741\pi$
$822$	−0.634133	−0.0221179
$823$	7.96814	0.277752	0.138876	−	0.990310i	$-0.455651\pi$
$823$	7.96814	0.277752	0.138876	+	0.990310i	$0.455651\pi$
$824$	14.5473	0.506778
$825$	−6.22197	−0.216621
$826$	2.83082	0.0984968
$827$	25.6063	0.890418	0.445209	−	0.895427i	$-0.353129\pi$
$827$	25.6063	0.890418	0.445209	+	0.895427i	$0.353129\pi$
$828$	−1.22628	−0.0426161
$829$	32.6889	1.13533	0.567667	−	0.823258i	$-0.307846\pi$
$829$	32.6889	1.13533	0.567667	+	0.823258i	$0.307846\pi$
$830$	−0.497501	−0.0172685
$831$	23.8487	0.827303
$832$	0.412560	0.0143029
$833$	−6.24072	−0.216228
$834$	9.90458	0.342967
$835$	−1.36179	−0.0471267
$836$	−7.37358	−0.255021
$837$	−26.8983	−0.929740
$838$	−22.2328	−0.768018
$839$	8.86161	0.305937	0.152968	−	0.988231i	$-0.451117\pi$
$839$	8.86161	0.305937	0.152968	+	0.988231i	$0.451117\pi$
$840$	0.0851502	0.00293796
$841$	−25.4183	−0.876493
$842$	2.78925	0.0961240
$843$	−5.27409	−0.181649
$844$	−17.8498	−0.614416
$845$	−1.96114	−0.0674653
$846$	−10.1071	−0.347488
$847$	6.04120	0.207578
$848$	6.90642	0.237168
$849$	9.28942	0.318812
$850$	−4.79213	−0.164369
$851$	3.26833	0.112037
$852$	6.47878	0.221959
$853$	−20.1110	−0.688589	−0.344294	−	0.938862i	$-0.611882\pi$
$853$	−20.1110	−0.688589	−0.344294	+	0.938862i	$0.611882\pi$
$854$	4.46804	0.152893
$855$	−1.67448	−0.0572658
$856$	−17.4001	−0.594723
$857$	8.04065	0.274663	0.137332	−	0.990525i	$-0.456147\pi$
$857$	8.04065	0.274663	0.137332	+	0.990525i	$0.456147\pi$
$858$	0.515797	0.0176090
$859$	1.97336	0.0673303	0.0336652	−	0.999433i	$-0.489282\pi$
$859$	1.97336	0.0673303	0.0336652	+	0.999433i	$0.489282\pi$
$860$	0.111372	0.00379775
$861$	−0.139843	−0.00476582
$862$	19.3799	0.660082
$863$	45.7232	1.55644	0.778219	−	0.627993i	$-0.216123\pi$
$863$	45.7232	1.55644	0.778219	+	0.627993i	$0.216123\pi$
$864$	4.17710	0.142108
$865$	−1.62279	−0.0551765
$866$	−36.0867	−1.22627
$867$	12.4280	0.422078
$868$	4.63912	0.157462
$869$	5.60297	0.190068
$870$	−0.223689	−0.00758377
$871$	−0.797977	−0.0270384
$872$	−14.5673	−0.493312
$873$	−23.0186	−0.779061
$874$	2.32805	0.0787476
$875$	1.09865	0.0371411
$876$	1.11476	0.0376643
$877$	20.0911	0.678427	0.339213	−	0.940709i	$-0.389839\pi$
$877$	20.0911	0.678427	0.339213	+	0.940709i	$0.389839\pi$
$878$	24.0072	0.810205
$879$	4.71226	0.158941
$880$	−0.247155	−0.00833160
$881$	−43.4680	−1.46447	−0.732237	−	0.681050i	$-0.761524\pi$
$881$	−43.4680	−1.46447	−0.732237	+	0.681050i	$0.761524\pi$
$882$	15.5681	0.524204
$883$	12.0172	0.404412	0.202206	−	0.979343i	$-0.435189\pi$
$883$	12.0172	0.404412	0.202206	+	0.979343i	$0.435189\pi$
$884$	0.397265	0.0133615
$885$	0.464436	0.0156119
$886$	33.8022	1.13561
$887$	18.2721	0.613517	0.306759	−	0.951787i	$-0.400756\pi$
$887$	18.2721	0.613517	0.306759	+	0.951787i	$0.400756\pi$
$888$	−4.95042	−0.166125
$889$	4.27870	0.143503
$890$	−1.45575	−0.0487970
$891$	−6.42950	−0.215396
$892$	−2.46977	−0.0826939
$893$	19.1880	0.642102
$894$	−5.30771	−0.177516
$895$	−3.54857	−0.118616
$896$	−0.720420	−0.0240676
$897$	−0.162852	−0.00543748
$898$	5.58652	0.186425
$899$	−12.1869	−0.406457
$900$	11.9544	0.398481
$901$	6.65038	0.221556
$902$	0.405904	0.0135151
$903$	0.405867	0.0135064
$904$	2.58908	0.0861117
$905$	−0.890944	−0.0296160
$906$	−9.42716	−0.313196
$907$	11.1722	0.370966	0.185483	−	0.982647i	$-0.440615\pi$
$907$	11.1722	0.370966	0.185483	+	0.982647i	$0.440615\pi$
$908$	7.94717	0.263736
$909$	26.3314	0.873356
$910$	−0.0454320	−0.00150606
$911$	−22.8655	−0.757569	−0.378785	−	0.925485i	$-0.623658\pi$
$911$	−22.8655	−0.757569	−0.378785	+	0.925485i	$0.623658\pi$
$912$	−3.52621	−0.116765
$913$	5.26243	0.174161
$914$	4.21106	0.139289
$915$	0.733045	0.0242337
$916$	11.3355	0.374537
$917$	4.83948	0.159814
$918$	4.02224	0.132754
$919$	−37.9069	−1.25043	−0.625216	−	0.780452i	$-0.714989\pi$
$919$	−37.9069	−1.25043	−0.625216	+	0.780452i	$0.714989\pi$
$920$	0.0780341	0.00257271
$921$	−7.34539	−0.242039
$922$	−22.9252	−0.755001
$923$	−3.45676	−0.113781
$924$	−0.900696	−0.0296307
$925$	−31.8615	−1.04760
$926$	31.6413	1.03980
$927$	−34.9441	−1.14771
$928$	1.89254	0.0621256
$929$	45.9849	1.50871	0.754357	−	0.656464i	$-0.227949\pi$
$929$	45.9849	1.50871	0.754357	+	0.656464i	$0.227949\pi$
$930$	0.761114	0.0249579
$931$	−29.5555	−0.968644
$932$	−8.76264	−0.287030
$933$	−6.51703	−0.213358
$934$	24.6616	0.806952
$935$	−0.237992	−0.00778319
$936$	−0.991013	−0.0323923
$937$	−53.3261	−1.74209	−0.871043	−	0.491206i	$-0.836556\pi$
$937$	−53.3261	−1.74209	−0.871043	+	0.491206i	$0.836556\pi$
$938$	1.39344	0.0454975
$939$	20.2354	0.660357
$940$	0.643163	0.0209777
$941$	−9.44742	−0.307977	−0.153989	−	0.988073i	$-0.549212\pi$
$941$	−9.44742	−0.307977	−0.153989	+	0.988073i	$0.549212\pi$
$942$	14.2927	0.465681
$943$	−0.128156	−0.00417333
$944$	−3.92940	−0.127891
$945$	−0.459991	−0.0149635
$946$	−1.17806	−0.0383021
$947$	34.5690	1.12334	0.561671	−	0.827361i	$-0.310159\pi$
$947$	34.5690	1.12334	0.561671	+	0.827361i	$0.310159\pi$
$948$	2.67947	0.0870251
$949$	−0.594782	−0.0193074
$950$	−22.6951	−0.736328
$951$	−10.7091	−0.347266
$952$	−0.693712	−0.0224833
$953$	−35.0582	−1.13565	−0.567824	−	0.823150i	$-0.692215\pi$
$953$	−35.0582	−1.13565	−0.567824	+	0.823150i	$0.692215\pi$
$954$	−16.5900	−0.537120
$955$	−0.288467	−0.00933456
$956$	−11.6548	−0.376942
$957$	2.36612	0.0764859
$958$	21.9275	0.708446
$959$	−0.590820	−0.0190786
$960$	−0.118195	−0.00381473
$961$	10.4667	0.337635
$962$	2.64130	0.0851589
$963$	41.7969	1.34689
$964$	12.4018	0.399436
$965$	0.757461	0.0243835
$966$	0.284376	0.00914964
$967$	12.4029	0.398850	0.199425	−	0.979913i	$-0.436093\pi$
$967$	12.4029	0.398850	0.199425	+	0.979913i	$0.436093\pi$
$968$	−8.38566	−0.269525
$969$	−3.39548	−0.109079
$970$	1.46479	0.0470315
$971$	58.4535	1.87586	0.937931	−	0.346821i	$-0.112739\pi$
$971$	58.4535	1.87586	0.937931	+	0.346821i	$0.112739\pi$
$972$	−15.6060	−0.500563
$973$	9.22807	0.295838
$974$	8.58796	0.275176
$975$	1.58757	0.0508430
$976$	−6.20199	−0.198521
$977$	34.2577	1.09600	0.548001	−	0.836478i	$-0.315389\pi$
$977$	34.2577	1.09600	0.548001	+	0.836478i	$0.315389\pi$
$978$	−16.1913	−0.517741
$979$	15.3986	0.492140
$980$	−0.990673	−0.0316459
$981$	34.9923	1.11722
$982$	32.6463	1.04178
$983$	−43.0286	−1.37240	−0.686200	−	0.727413i	$-0.740723\pi$
$983$	−43.0286	−1.37240	−0.686200	+	0.727413i	$0.740723\pi$
$984$	0.194112	0.00618808
$985$	1.04238	0.0332129
$986$	1.82238	0.0580363
$987$	2.34385	0.0746055
$988$	1.88141	0.0598557
$989$	0.371948	0.0118273
$990$	0.593694	0.0188688
$991$	−21.4779	−0.682267	−0.341134	−	0.940015i	$-0.610811\pi$
$991$	−21.4779	−0.682267	−0.341134	+	0.940015i	$0.610811\pi$
$992$	−6.43946	−0.204453
$993$	−7.92793	−0.251585
$994$	6.03626	0.191459
$995$	3.37814	0.107094
$996$	2.51661	0.0797418
$997$	−24.9577	−0.790417	−0.395208	−	0.918591i	$-0.629328\pi$
$997$	−24.9577	−0.790417	−0.395208	+	0.918591i	$0.629328\pi$
$998$	−6.30045	−0.199437
$999$	26.7427	0.846101

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.a.1.28	✓	67

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.a.1.28	✓	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -0.773234 q^{3} +1.00000 q^{4} +0.152858 q^{5} -0.773234 q^{6} -0.720420 q^{7} +1.00000 q^{8} -2.40211 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -0.773234 q^{3} +1.00000 q^{4} +0.152858 q^{5} -0.773234 q^{6} -0.720420 q^{7} +1.00000 q^{8} -2.40211 q^{9} +0.152858 q^{10} -1.61689 q^{11} -0.773234 q^{12} +0.412560 q^{13} -0.720420 q^{14} -0.118195 q^{15} +1.00000 q^{16} +0.962927 q^{17} -2.40211 q^{18} +4.56034 q^{19} +0.152858 q^{20} +0.557054 q^{21} -1.61689 q^{22} +0.510500 q^{23} -0.773234 q^{24} -4.97663 q^{25} +0.412560 q^{26} +4.17710 q^{27} -0.720420 q^{28} +1.89254 q^{29} -0.118195 q^{30} -6.43946 q^{31} +1.00000 q^{32} +1.25024 q^{33} +0.962927 q^{34} -0.110122 q^{35} -2.40211 q^{36} +6.40222 q^{37} +4.56034 q^{38} -0.319005 q^{39} +0.152858 q^{40} -0.251040 q^{41} +0.557054 q^{42} +0.728596 q^{43} -1.61689 q^{44} -0.367182 q^{45} +0.510500 q^{46} +4.20758 q^{47} -0.773234 q^{48} -6.48099 q^{49} -4.97663 q^{50} -0.744568 q^{51} +0.412560 q^{52} +6.90642 q^{53} +4.17710 q^{54} -0.247155 q^{55} -0.720420 q^{56} -3.52621 q^{57} +1.89254 q^{58} -3.92940 q^{59} -0.118195 q^{60} -6.20199 q^{61} -6.43946 q^{62} +1.73053 q^{63} +1.00000 q^{64} +0.0630631 q^{65} +1.25024 q^{66} -1.93421 q^{67} +0.962927 q^{68} -0.394736 q^{69} -0.110122 q^{70} -8.37880 q^{71} -2.40211 q^{72} -1.44169 q^{73} +6.40222 q^{74} +3.84810 q^{75} +4.56034 q^{76} +1.16484 q^{77} -0.319005 q^{78} -3.46527 q^{79} +0.152858 q^{80} +3.97645 q^{81} -0.251040 q^{82} -3.25465 q^{83} +0.557054 q^{84} +0.147191 q^{85} +0.728596 q^{86} -1.46338 q^{87} -1.61689 q^{88} -9.52355 q^{89} -0.367182 q^{90} -0.297216 q^{91} +0.510500 q^{92} +4.97921 q^{93} +4.20758 q^{94} +0.697086 q^{95} -0.773234 q^{96} +9.58266 q^{97} -6.48099 q^{98} +3.88395 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) \) 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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