LMFDB - Embedded newform 8045.2.a.e.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8045,2,Mod(1,8045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8045, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8045 = 5 \cdot 1609 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8045.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:	$64.2396484261$
Analytic rank:	$0$
Dimension:	$142$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	8045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.15373 q^{2} +3.04079 q^{3} +2.63855 q^{4} +1.00000 q^{5} -6.54904 q^{6} -4.47994 q^{7} -1.37525 q^{8} +6.24641 q^{9} +O(q^{10})$	\(q-2.15373 q^{2} +3.04079 q^{3} +2.63855 q^{4} +1.00000 q^{5} -6.54904 q^{6} -4.47994 q^{7} -1.37525 q^{8} +6.24641 q^{9} -2.15373 q^{10} +0.514029 q^{11} +8.02327 q^{12} +1.16696 q^{13} +9.64858 q^{14} +3.04079 q^{15} -2.31517 q^{16} +0.00341512 q^{17} -13.4531 q^{18} +4.14200 q^{19} +2.63855 q^{20} -13.6226 q^{21} -1.10708 q^{22} +4.30110 q^{23} -4.18186 q^{24} +1.00000 q^{25} -2.51331 q^{26} +9.87167 q^{27} -11.8205 q^{28} +3.19582 q^{29} -6.54904 q^{30} -7.46464 q^{31} +7.73675 q^{32} +1.56306 q^{33} -0.00735524 q^{34} -4.47994 q^{35} +16.4814 q^{36} -0.965058 q^{37} -8.92074 q^{38} +3.54847 q^{39} -1.37525 q^{40} +2.80520 q^{41} +29.3393 q^{42} +9.37574 q^{43} +1.35629 q^{44} +6.24641 q^{45} -9.26339 q^{46} -10.0459 q^{47} -7.03994 q^{48} +13.0699 q^{49} -2.15373 q^{50} +0.0103847 q^{51} +3.07907 q^{52} -8.78450 q^{53} -21.2609 q^{54} +0.514029 q^{55} +6.16106 q^{56} +12.5950 q^{57} -6.88292 q^{58} +2.33625 q^{59} +8.02327 q^{60} -12.9264 q^{61} +16.0768 q^{62} -27.9836 q^{63} -12.0325 q^{64} +1.16696 q^{65} -3.36640 q^{66} +3.40032 q^{67} +0.00901095 q^{68} +13.0787 q^{69} +9.64858 q^{70} +11.7334 q^{71} -8.59040 q^{72} +16.0253 q^{73} +2.07847 q^{74} +3.04079 q^{75} +10.9289 q^{76} -2.30282 q^{77} -7.64244 q^{78} -15.4041 q^{79} -2.31517 q^{80} +11.2784 q^{81} -6.04163 q^{82} +15.5109 q^{83} -35.9438 q^{84} +0.00341512 q^{85} -20.1928 q^{86} +9.71781 q^{87} -0.706921 q^{88} +16.0809 q^{89} -13.4531 q^{90} -5.22789 q^{91} +11.3486 q^{92} -22.6984 q^{93} +21.6361 q^{94} +4.14200 q^{95} +23.5258 q^{96} +9.48193 q^{97} -28.1490 q^{98} +3.21084 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 142 q + 21 q^{2} + 33 q^{3} + 157 q^{4} + 142 q^{5} + 15 q^{6} + 63 q^{7} + 60 q^{8} + 157 q^{9}+O(q^{10}) $ 142 * q + 21 * q^2 + 33 * q^3 + 157 * q^4 + 142 * q^5 + 15 * q^6 + 63 * q^7 + 60 * q^8 + 157 * q^9	\( 142 q + 21 q^{2} + 33 q^{3} + 157 q^{4} + 142 q^{5} + 15 q^{6} + 63 q^{7} + 60 q^{8} + 157 q^{9} + 21 q^{10} + 36 q^{11} + 55 q^{12} + 57 q^{13} + 2 q^{14} + 33 q^{15} + 179 q^{16} + 55 q^{17} + 65 q^{18} + 130 q^{19} + 157 q^{20} + 28 q^{21} + 30 q^{22} + 117 q^{23} + 21 q^{24} + 142 q^{25} + 21 q^{26} + 120 q^{27} + 135 q^{28} + 12 q^{29} + 15 q^{30} + 74 q^{31} + 126 q^{32} + 55 q^{33} + 35 q^{34} + 63 q^{35} + 186 q^{36} + 75 q^{37} + 65 q^{38} + 23 q^{39} + 60 q^{40} + 22 q^{41} + 10 q^{42} + 190 q^{43} + 22 q^{44} + 157 q^{45} + 56 q^{46} + 102 q^{47} + 78 q^{48} + 197 q^{49} + 21 q^{50} + 30 q^{51} + 120 q^{52} + 56 q^{53} - 6 q^{54} + 36 q^{55} + 3 q^{56} + 68 q^{57} + 31 q^{58} + 55 q^{59} + 55 q^{60} + 90 q^{61} + 68 q^{62} + 167 q^{63} + 180 q^{64} + 57 q^{65} + 17 q^{66} + 151 q^{67} + 119 q^{68} + 21 q^{69} + 2 q^{70} + 4 q^{71} + 130 q^{72} + 143 q^{73} - 46 q^{74} + 33 q^{75} + 213 q^{76} + 75 q^{77} - 24 q^{78} + 47 q^{79} + 179 q^{80} + 150 q^{81} + 69 q^{82} + 201 q^{83} - 31 q^{84} + 55 q^{85} - 4 q^{86} + 153 q^{87} + 37 q^{88} + 25 q^{89} + 65 q^{90} + 132 q^{91} + 194 q^{92} + 52 q^{93} + 18 q^{94} + 130 q^{95} + 13 q^{96} + 80 q^{97} + 58 q^{98} + 103 q^{99}+O(q^{100}) \) 142 * q + 21 * q^2 + 33 * q^3 + 157 * q^4 + 142 * q^5 + 15 * q^6 + 63 * q^7 + 60 * q^8 + 157 * q^9 + 21 * q^10 + 36 * q^11 + 55 * q^12 + 57 * q^13 + 2 * q^14 + 33 * q^15 + 179 * q^16 + 55 * q^17 + 65 * q^18 + 130 * q^19 + 157 * q^20 + 28 * q^21 + 30 * q^22 + 117 * q^23 + 21 * q^24 + 142 * q^25 + 21 * q^26 + 120 * q^27 + 135 * q^28 + 12 * q^29 + 15 * q^30 + 74 * q^31 + 126 * q^32 + 55 * q^33 + 35 * q^34 + 63 * q^35 + 186 * q^36 + 75 * q^37 + 65 * q^38 + 23 * q^39 + 60 * q^40 + 22 * q^41 + 10 * q^42 + 190 * q^43 + 22 * q^44 + 157 * q^45 + 56 * q^46 + 102 * q^47 + 78 * q^48 + 197 * q^49 + 21 * q^50 + 30 * q^51 + 120 * q^52 + 56 * q^53 - 6 * q^54 + 36 * q^55 + 3 * q^56 + 68 * q^57 + 31 * q^58 + 55 * q^59 + 55 * q^60 + 90 * q^61 + 68 * q^62 + 167 * q^63 + 180 * q^64 + 57 * q^65 + 17 * q^66 + 151 * q^67 + 119 * q^68 + 21 * q^69 + 2 * q^70 + 4 * q^71 + 130 * q^72 + 143 * q^73 - 46 * q^74 + 33 * q^75 + 213 * q^76 + 75 * q^77 - 24 * q^78 + 47 * q^79 + 179 * q^80 + 150 * q^81 + 69 * q^82 + 201 * q^83 - 31 * q^84 + 55 * q^85 - 4 * q^86 + 153 * q^87 + 37 * q^88 + 25 * q^89 + 65 * q^90 + 132 * q^91 + 194 * q^92 + 52 * q^93 + 18 * q^94 + 130 * q^95 + 13 * q^96 + 80 * q^97 + 58 * q^98 + 103 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.15373	−1.52292	−0.761458	−	0.648214i	$-0.775516\pi$
$2$	−2.15373	−1.52292	−0.761458	+	0.648214i	$0.775516\pi$
$3$	3.04079	1.75560	0.877801	−	0.479026i	$-0.159010\pi$
$3$	3.04079	1.75560	0.877801	+	0.479026i	$0.159010\pi$
$4$	2.63855	1.31927
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−6.54904	−2.67363
$7$	−4.47994	−1.69326	−0.846629	−	0.532183i	$-0.821372\pi$
$7$	−4.47994	−1.69326	−0.846629	+	0.532183i	$0.821372\pi$
$8$	−1.37525	−0.486226
$9$	6.24641	2.08214
$10$	−2.15373	−0.681069
$11$	0.514029	0.154986	0.0774929	−	0.996993i	$-0.475309\pi$
$11$	0.514029	0.154986	0.0774929	+	0.996993i	$0.475309\pi$
$12$	8.02327	2.31612
$13$	1.16696	0.323655	0.161828	−	0.986819i	$-0.448261\pi$
$13$	1.16696	0.323655	0.161828	+	0.986819i	$0.448261\pi$
$14$	9.64858	2.57869
$15$	3.04079	0.785129
$16$	−2.31517	−0.578792
$17$	0.00341512	0.000828289 0	0.000414144	−	1.00000i	$-0.499868\pi$
$17$	0.00341512	0.000828289 0	0.000414144		1.00000i	$0.499868\pi$
$18$	−13.4531	−3.17092
$19$	4.14200	0.950240	0.475120	−	0.879921i	$-0.342405\pi$
$19$	4.14200	0.950240	0.475120	+	0.879921i	$0.342405\pi$
$20$	2.63855	0.589997
$21$	−13.6226	−2.97269
$22$	−1.10708	−0.236030
$23$	4.30110	0.896841	0.448420	−	0.893823i	$-0.351987\pi$
$23$	4.30110	0.896841	0.448420	+	0.893823i	$0.351987\pi$
$24$	−4.18186	−0.853619
$25$	1.00000	0.200000
$26$	−2.51331	−0.492900
$27$	9.87167	1.89980
$28$	−11.8205	−2.23387
$29$	3.19582	0.593448	0.296724	−	0.954963i	$-0.404106\pi$
$29$	3.19582	0.593448	0.296724	+	0.954963i	$0.404106\pi$
$30$	−6.54904	−1.19569
$31$	−7.46464	−1.34069	−0.670344	−	0.742050i	$-0.733854\pi$
$31$	−7.46464	−1.34069	−0.670344	+	0.742050i	$0.733854\pi$
$32$	7.73675	1.36768
$33$	1.56306	0.272093
$34$	−0.00735524	−0.00126141
$35$	−4.47994	−0.757248
$36$	16.4814	2.74691
$37$	−0.965058	−0.158654	−0.0793272	−	0.996849i	$-0.525277\pi$
$37$	−0.965058	−0.158654	−0.0793272	+	0.996849i	$0.525277\pi$
$38$	−8.92074	−1.44714
$39$	3.54847	0.568210
$40$	−1.37525	−0.217447
$41$	2.80520	0.438098	0.219049	−	0.975714i	$-0.429705\pi$
$41$	2.80520	0.438098	0.219049	+	0.975714i	$0.429705\pi$
$42$	29.3393	4.52715
$43$	9.37574	1.42979	0.714894	−	0.699233i	$-0.246475\pi$
$43$	9.37574	1.42979	0.714894	+	0.699233i	$0.246475\pi$
$44$	1.35629	0.204468
$45$	6.24641	0.931160
$46$	−9.26339	−1.36581
$47$	−10.0459	−1.46534	−0.732671	−	0.680583i	$-0.761726\pi$
$47$	−10.0459	−1.46534	−0.732671	+	0.680583i	$0.761726\pi$
$48$	−7.03994	−1.01613
$49$	13.0699	1.86713
$50$	−2.15373	−0.304583
$51$	0.0103847	0.00145415
$52$	3.07907	0.426990
$53$	−8.78450	−1.20664	−0.603322	−	0.797498i	$-0.706157\pi$
$53$	−8.78450	−1.20664	−0.603322	+	0.797498i	$0.706157\pi$
$54$	−21.2609	−2.89324
$55$	0.514029	0.0693117
$56$	6.16106	0.823306
$57$	12.5950	1.66824
$58$	−6.88292	−0.903771
$59$	2.33625	0.304154	0.152077	−	0.988369i	$-0.451404\pi$
$59$	2.33625	0.304154	0.152077	+	0.988369i	$0.451404\pi$
$60$	8.02327	1.03580
$61$	−12.9264	−1.65505	−0.827526	−	0.561427i	$-0.810253\pi$
$61$	−12.9264	−1.65505	−0.827526	+	0.561427i	$0.810253\pi$
$62$	16.0768	2.04176
$63$	−27.9836	−3.52560
$64$	−12.0325	−1.50407
$65$	1.16696	0.144743
$66$	−3.36640	−0.414375
$67$	3.40032	0.415415	0.207707	−	0.978191i	$-0.433400\pi$
$67$	3.40032	0.415415	0.207707	+	0.978191i	$0.433400\pi$
$68$	0.00901095	0.00109274
$69$	13.0787	1.57450
$70$	9.64858	1.15323
$71$	11.7334	1.39250	0.696250	−	0.717800i	$-0.254851\pi$
$71$	11.7334	1.39250	0.696250	+	0.717800i	$0.254851\pi$
$72$	−8.59040	−1.01239
$73$	16.0253	1.87562	0.937811	−	0.347145i	$-0.112849\pi$
$73$	16.0253	1.87562	0.937811	+	0.347145i	$0.112849\pi$
$74$	2.07847	0.241617
$75$	3.04079	0.351120
$76$	10.9289	1.25363
$77$	−2.30282	−0.262431
$78$	−7.64244	−0.865336
$79$	−15.4041	−1.73310	−0.866551	−	0.499089i	$-0.833668\pi$
$79$	−15.4041	−1.73310	−0.866551	+	0.499089i	$0.833668\pi$
$80$	−2.31517	−0.258844
$81$	11.2784	1.25316
$82$	−6.04163	−0.667187
$83$	15.5109	1.70254	0.851271	−	0.524726i	$-0.175832\pi$
$83$	15.5109	1.70254	0.851271	+	0.524726i	$0.175832\pi$
$84$	−35.9438	−3.92179
$85$	0.00341512	0.000370422 0
$86$	−20.1928	−2.17745
$87$	9.71781	1.04186
$88$	−0.706921	−0.0753580
$89$	16.0809	1.70457	0.852285	−	0.523078i	$-0.175216\pi$
$89$	16.0809	1.70457	0.852285	+	0.523078i	$0.175216\pi$
$90$	−13.4531	−1.41808
$91$	−5.22789	−0.548032
$92$	11.3486	1.18318
$93$	−22.6984	−2.35372
$94$	21.6361	2.23159
$95$	4.14200	0.424960
$96$	23.5258	2.40110
$97$	9.48193	0.962745	0.481372	−	0.876516i	$-0.340139\pi$
$97$	9.48193	0.962745	0.481372	+	0.876516i	$0.340139\pi$
$98$	−28.1490	−2.84348
$99$	3.21084	0.322702
$100$	2.63855	0.263855
$101$	19.1283	1.90333	0.951667	−	0.307132i	$-0.0993693\pi$
$101$	19.1283	1.90333	0.951667	+	0.307132i	$0.0993693\pi$
$102$	−0.0223658	−0.00221454
$103$	0.927050	0.0913450	0.0456725	−	0.998956i	$-0.485457\pi$
$103$	0.927050	0.0913450	0.0456725	+	0.998956i	$0.485457\pi$
$104$	−1.60486	−0.157369
$105$	−13.6226	−1.32943
$106$	18.9194	1.83762
$107$	14.6666	1.41788	0.708938	−	0.705271i	$-0.249175\pi$
$107$	14.6666	1.41788	0.708938	+	0.705271i	$0.249175\pi$
$108$	26.0468	2.50636
$109$	−11.8483	−1.13486	−0.567428	−	0.823423i	$-0.692062\pi$
$109$	−11.8483	−1.13486	−0.567428	+	0.823423i	$0.692062\pi$
$110$	−1.10708	−0.105556
$111$	−2.93454	−0.278534
$112$	10.3718	0.980045
$113$	−9.05505	−0.851828	−0.425914	−	0.904764i	$-0.640047\pi$
$113$	−9.05505	−0.851828	−0.425914	+	0.904764i	$0.640047\pi$
$114$	−27.1261	−2.54059
$115$	4.30110	0.401079
$116$	8.43230	0.782920
$117$	7.28929	0.673895
$118$	−5.03165	−0.463200
$119$	−0.0152995	−0.00140251
$120$	−4.18186	−0.381750
$121$	−10.7358	−0.975979
$122$	27.8399	2.52051
$123$	8.53002	0.769126
$124$	−19.6958	−1.76873
$125$	1.00000	0.0894427
$126$	60.2690	5.36919
$127$	0.525022	0.0465882	0.0232941	−	0.999729i	$-0.492585\pi$
$127$	0.525022	0.0465882	0.0232941	+	0.999729i	$0.492585\pi$
$128$	10.4413	0.922888
$129$	28.5097	2.51014
$130$	−2.51331	−0.220431
$131$	−15.0763	−1.31722	−0.658611	−	0.752483i	$-0.728856\pi$
$131$	−15.0763	−1.31722	−0.658611	+	0.752483i	$0.728856\pi$
$132$	4.12420	0.358965
$133$	−18.5559	−1.60900
$134$	−7.32336	−0.632642
$135$	9.87167	0.849618
$136$	−0.00469666	−0.000402735 0
$137$	−11.8307	−1.01077	−0.505383	−	0.862895i	$-0.668649\pi$
$137$	−11.8307	−1.01077	−0.505383	+	0.862895i	$0.668649\pi$
$138$	−28.1680	−2.39782
$139$	−1.45217	−0.123172	−0.0615858	−	0.998102i	$-0.519616\pi$
$139$	−1.45217	−0.123172	−0.0615858	+	0.998102i	$0.519616\pi$
$140$	−11.8205	−0.999017
$141$	−30.5474	−2.57256
$142$	−25.2706	−2.12066
$143$	0.599850	0.0501619
$144$	−14.4615	−1.20512
$145$	3.19582	0.265398
$146$	−34.5142	−2.85642
$147$	39.7428	3.27793
$148$	−2.54635	−0.209309
$149$	−7.07570	−0.579664	−0.289832	−	0.957078i	$-0.593599\pi$
$149$	−7.07570	−0.579664	−0.289832	+	0.957078i	$0.593599\pi$
$150$	−6.54904	−0.534727
$151$	−3.87246	−0.315136	−0.157568	−	0.987508i	$-0.550365\pi$
$151$	−3.87246	−0.315136	−0.157568	+	0.987508i	$0.550365\pi$
$152$	−5.69630	−0.462031
$153$	0.0213323	0.00172461
$154$	4.95965	0.399660
$155$	−7.46464	−0.599574
$156$	9.36280	0.749624
$157$	−17.2682	−1.37815	−0.689076	−	0.724689i	$-0.741983\pi$
$157$	−17.2682	−1.37815	−0.689076	+	0.724689i	$0.741983\pi$
$158$	33.1763	2.63937
$159$	−26.7118	−2.11839
$160$	7.73675	0.611644
$161$	−19.2687	−1.51858
$162$	−24.2907	−1.90846
$163$	−4.45743	−0.349133	−0.174567	−	0.984645i	$-0.555852\pi$
$163$	−4.45743	−0.349133	−0.174567	+	0.984645i	$0.555852\pi$
$164$	7.40164	0.577971
$165$	1.56306	0.121684
$166$	−33.4063	−2.59283
$167$	25.0593	1.93915	0.969575	−	0.244796i	$-0.0787210\pi$
$167$	25.0593	1.93915	0.969575	+	0.244796i	$0.0787210\pi$
$168$	18.7345	1.44540
$169$	−11.6382	−0.895247
$170$	−0.00735524	−0.000564121 0
$171$	25.8726	1.97853
$172$	24.7383	1.88628
$173$	−2.47400	−0.188095	−0.0940474	−	0.995568i	$-0.529981\pi$
$173$	−2.47400	−0.188095	−0.0940474	+	0.995568i	$0.529981\pi$
$174$	−20.9295	−1.58666
$175$	−4.47994	−0.338652
$176$	−1.19006	−0.0897045
$177$	7.10405	0.533973
$178$	−34.6338	−2.59592
$179$	15.7475	1.17702	0.588512	−	0.808489i	$-0.299714\pi$
$179$	15.7475	1.17702	0.588512	+	0.808489i	$0.299714\pi$
$180$	16.4814	1.22845
$181$	18.4803	1.37363	0.686816	−	0.726831i	$-0.259008\pi$
$181$	18.4803	1.37363	0.686816	+	0.726831i	$0.259008\pi$
$182$	11.2595	0.834607
$183$	−39.3064	−2.90561
$184$	−5.91510	−0.436067
$185$	−0.965058	−0.0709524
$186$	48.8862	3.58451
$187$	0.00175547	0.000128373 0
$188$	−26.5065	−1.93319
$189$	−44.2245	−3.21686
$190$	−8.92074	−0.647179
$191$	−0.179750	−0.0130063	−0.00650313	−	0.999979i	$-0.502070\pi$
$191$	−0.179750	−0.0130063	−0.00650313	+	0.999979i	$0.502070\pi$
$192$	−36.5884	−2.64054
$193$	18.2657	1.31480	0.657398	−	0.753543i	$-0.271657\pi$
$193$	18.2657	1.31480	0.657398	+	0.753543i	$0.271657\pi$
$194$	−20.4215	−1.46618
$195$	3.54847	0.254111
$196$	34.4855	2.46325
$197$	6.73760	0.480034	0.240017	−	0.970769i	$-0.422847\pi$
$197$	6.73760	0.480034	0.240017	+	0.970769i	$0.422847\pi$
$198$	−6.91528	−0.491447
$199$	2.09714	0.148662	0.0743312	−	0.997234i	$-0.476318\pi$
$199$	2.09714	0.148662	0.0743312	+	0.997234i	$0.476318\pi$
$200$	−1.37525	−0.0972451
$201$	10.3397	0.729303
$202$	−41.1971	−2.89862
$203$	−14.3171	−1.00486
$204$	0.0274004	0.00191841
$205$	2.80520	0.195923
$206$	−1.99661	−0.139111
$207$	26.8664	1.86735
$208$	−2.70170	−0.187329
$209$	2.12911	0.147274
$210$	29.3393	2.02461
$211$	6.85485	0.471907	0.235954	−	0.971764i	$-0.424179\pi$
$211$	6.85485	0.471907	0.235954	+	0.971764i	$0.424179\pi$
$212$	−23.1783	−1.59189
$213$	35.6789	2.44467
$214$	−31.5879	−2.15930
$215$	9.37574	0.639420
$216$	−13.5760	−0.923733
$217$	33.4411	2.27013
$218$	25.5179	1.72829
$219$	48.7297	3.29285
$220$	1.35629	0.0914411
$221$	0.00398530	0.000268080 0
$222$	6.32020	0.424184
$223$	22.2731	1.49151	0.745757	−	0.666218i	$-0.232088\pi$
$223$	22.2731	1.49151	0.745757	+	0.666218i	$0.232088\pi$
$224$	−34.6602	−2.31583
$225$	6.24641	0.416428
$226$	19.5021	1.29726
$227$	23.1135	1.53410	0.767049	−	0.641588i	$-0.221724\pi$
$227$	23.1135	1.53410	0.767049	+	0.641588i	$0.221724\pi$
$228$	33.2324	2.20087
$229$	22.6738	1.49833	0.749163	−	0.662385i	$-0.230456\pi$
$229$	22.6738	1.49833	0.749163	+	0.662385i	$0.230456\pi$
$230$	−9.26339	−0.610810
$231$	−7.00240	−0.460724
$232$	−4.39506	−0.288550
$233$	−3.67541	−0.240784	−0.120392	−	0.992726i	$-0.538415\pi$
$233$	−3.67541	−0.240784	−0.120392	+	0.992726i	$0.538415\pi$
$234$	−15.6991	−1.02629
$235$	−10.0459	−0.655321
$236$	6.16430	0.401262
$237$	−46.8408	−3.04264
$238$	0.0329511	0.00213590
$239$	−16.4370	−1.06322	−0.531611	−	0.846988i	$-0.678413\pi$
$239$	−16.4370	−1.06322	−0.531611	+	0.846988i	$0.678413\pi$
$240$	−7.03994	−0.454426
$241$	−17.7074	−1.14064	−0.570318	−	0.821424i	$-0.693180\pi$
$241$	−17.7074	−1.14064	−0.570318	+	0.821424i	$0.693180\pi$
$242$	23.1219	1.48633
$243$	4.68039	0.300247
$244$	−34.1068	−2.18347
$245$	13.0699	0.835004
$246$	−18.3713	−1.17131
$247$	4.83353	0.307550
$248$	10.2658	0.651877
$249$	47.1654	2.98899
$250$	−2.15373	−0.136214
$251$	−31.3767	−1.98048	−0.990239	−	0.139380i	$-0.955489\pi$
$251$	−31.3767	−1.98048	−0.990239	+	0.139380i	$0.955489\pi$
$252$	−73.8359	−4.65123
$253$	2.21089	0.138997
$254$	−1.13075	−0.0709499
$255$	0.0103847	0.000650313 0
$256$	1.57736	0.0985848
$257$	−5.11475	−0.319049	−0.159525	−	0.987194i	$-0.550996\pi$
$257$	−5.11475	−0.319049	−0.159525	+	0.987194i	$0.550996\pi$
$258$	−61.4021	−3.82273
$259$	4.32340	0.268643
$260$	3.07907	0.190956
$261$	19.9624	1.23564
$262$	32.4703	2.00602
$263$	5.49631	0.338917	0.169459	−	0.985537i	$-0.445798\pi$
$263$	5.49631	0.338917	0.169459	+	0.985537i	$0.445798\pi$
$264$	−2.14960	−0.132299
$265$	−8.78450	−0.539628
$266$	39.9644	2.45038
$267$	48.8986	2.99255
$268$	8.97189	0.548045
$269$	2.83134	0.172630	0.0863148	−	0.996268i	$-0.472491\pi$
$269$	2.83134	0.172630	0.0863148	+	0.996268i	$0.472491\pi$
$270$	−21.2609	−1.29390
$271$	18.7552	1.13930	0.569649	−	0.821888i	$-0.307079\pi$
$271$	18.7552	1.13930	0.569649	+	0.821888i	$0.307079\pi$
$272$	−0.00790658	−0.000479407 0
$273$	−15.8969	−0.962126
$274$	25.4801	1.53931
$275$	0.514029	0.0309971
$276$	34.5088	2.07719
$277$	3.52933	0.212057	0.106028	−	0.994363i	$-0.466187\pi$
$277$	3.52933	0.212057	0.106028	+	0.994363i	$0.466187\pi$
$278$	3.12758	0.187580
$279$	−46.6272	−2.79150
$280$	6.16106	0.368194
$281$	22.7533	1.35735	0.678673	−	0.734441i	$-0.262555\pi$
$281$	22.7533	1.35735	0.678673	+	0.734441i	$0.262555\pi$
$282$	65.7908	3.91779
$283$	10.3100	0.612865	0.306433	−	0.951892i	$-0.400865\pi$
$283$	10.3100	0.612865	0.306433	+	0.951892i	$0.400865\pi$
$284$	30.9591	1.83709
$285$	12.5950	0.746061
$286$	−1.29191	−0.0763924
$287$	−12.5671	−0.741814
$288$	48.3269	2.84769
$289$	−17.0000	−0.999999
$290$	−6.88292	−0.404179
$291$	28.8326	1.69020
$292$	42.2836	2.47446
$293$	27.5261	1.60809	0.804045	−	0.594569i	$-0.202677\pi$
$293$	27.5261	1.60809	0.804045	+	0.594569i	$0.202677\pi$
$294$	−85.5952	−4.99201
$295$	2.33625	0.136022
$296$	1.32720	0.0771419
$297$	5.07433	0.294442
$298$	15.2391	0.882779
$299$	5.01919	0.290267
$300$	8.02327	0.463224
$301$	−42.0028	−2.42100
$302$	8.34023	0.479926
$303$	58.1651	3.34150
$304$	−9.58943	−0.549991
$305$	−12.9264	−0.740162
$306$	−0.0459439	−0.00262644
$307$	−26.8023	−1.52969	−0.764843	−	0.644217i	$-0.777183\pi$
$307$	−26.8023	−1.52969	−0.764843	+	0.644217i	$0.777183\pi$
$308$	−6.07610	−0.346218
$309$	2.81897	0.160365
$310$	16.0768	0.913101
$311$	−28.9560	−1.64194	−0.820971	−	0.570970i	$-0.806567\pi$
$311$	−28.9560	−1.64194	−0.820971	+	0.570970i	$0.806567\pi$
$312$	−4.88005	−0.276278
$313$	−0.737316	−0.0416756	−0.0208378	−	0.999783i	$-0.506633\pi$
$313$	−0.737316	−0.0416756	−0.0208378	+	0.999783i	$0.506633\pi$
$314$	37.1910	2.09881
$315$	−27.9836	−1.57670
$316$	−40.6445	−2.28643
$317$	20.6312	1.15877	0.579383	−	0.815055i	$-0.303294\pi$
$317$	20.6312	1.15877	0.579383	+	0.815055i	$0.303294\pi$
$318$	57.5301	3.22613
$319$	1.64274	0.0919760
$320$	−12.0325	−0.672638
$321$	44.5981	2.48922
$322$	41.4995	2.31267
$323$	0.0141454	0.000787073 0
$324$	29.7587	1.65326
$325$	1.16696	0.0647311
$326$	9.60010	0.531700
$327$	−36.0281	−1.99236
$328$	−3.85786	−0.213015
$329$	45.0049	2.48120
$330$	−3.36640	−0.185314
$331$	29.1833	1.60406	0.802031	−	0.597283i	$-0.203753\pi$
$331$	29.1833	1.60406	0.802031	+	0.597283i	$0.203753\pi$
$332$	40.9262	2.24612
$333$	−6.02815	−0.330341
$334$	−53.9710	−2.95316
$335$	3.40032	0.185779
$336$	31.5385	1.72057
$337$	21.8886	1.19235	0.596173	−	0.802856i	$-0.296687\pi$
$337$	21.8886	1.19235	0.596173	+	0.802856i	$0.296687\pi$
$338$	25.0656	1.36339
$339$	−27.5345	−1.49547
$340$	0.00901095	0.000488688 0
$341$	−3.83704	−0.207788
$342$	−55.7226	−3.01314
$343$	−27.1927	−1.46827
$344$	−12.8940	−0.695199
$345$	13.0787	0.704136
$346$	5.32832	0.286452
$347$	−13.0222	−0.699067	−0.349533	−	0.936924i	$-0.613660\pi$
$347$	−13.0222	−0.699067	−0.349533	+	0.936924i	$0.613660\pi$
$348$	25.6409	1.37450
$349$	8.32610	0.445686	0.222843	−	0.974854i	$-0.428466\pi$
$349$	8.32610	0.445686	0.222843	+	0.974854i	$0.428466\pi$
$350$	9.64858	0.515738
$351$	11.5198	0.614881
$352$	3.97692	0.211970
$353$	6.44720	0.343150	0.171575	−	0.985171i	$-0.445114\pi$
$353$	6.44720	0.343150	0.171575	+	0.985171i	$0.445114\pi$
$354$	−15.3002	−0.813196
$355$	11.7334	0.622745
$356$	42.4301	2.24879
$357$	−0.0465227	−0.00246224
$358$	−33.9158	−1.79251
$359$	17.0810	0.901501	0.450750	−	0.892650i	$-0.351156\pi$
$359$	17.0810	0.901501	0.450750	+	0.892650i	$0.351156\pi$
$360$	−8.59040	−0.452754
$361$	−1.84384	−0.0970440
$362$	−39.8016	−2.09193
$363$	−32.6453	−1.71343
$364$	−13.7940	−0.723004
$365$	16.0253	0.838804
$366$	84.6553	4.42500
$367$	10.8894	0.568424	0.284212	−	0.958761i	$-0.408268\pi$
$367$	10.8894	0.568424	0.284212	+	0.958761i	$0.408268\pi$
$368$	−9.95776	−0.519084
$369$	17.5224	0.912181
$370$	2.07847	0.108055
$371$	39.3541	2.04316
$372$	−59.8908	−3.10519
$373$	26.0392	1.34826	0.674129	−	0.738614i	$-0.264519\pi$
$373$	26.0392	1.34826	0.674129	+	0.738614i	$0.264519\pi$
$374$	−0.00378081	−0.000195501 0
$375$	3.04079	0.157026
$376$	13.8156	0.712487
$377$	3.72937	0.192073
$378$	95.2476	4.89901
$379$	16.2347	0.833921	0.416960	−	0.908925i	$-0.363095\pi$
$379$	16.2347	0.833921	0.416960	+	0.908925i	$0.363095\pi$
$380$	10.9289	0.560638
$381$	1.59648	0.0817903
$382$	0.387133	0.0198075
$383$	−4.74563	−0.242490	−0.121245	−	0.992623i	$-0.538689\pi$
$383$	−4.74563	−0.242490	−0.121245	+	0.992623i	$0.538689\pi$
$384$	31.7498	1.62022
$385$	−2.30282	−0.117363
$386$	−39.3394	−2.00232
$387$	58.5648	2.97702
$388$	25.0185	1.27012
$389$	−5.09221	−0.258185	−0.129093	−	0.991633i	$-0.541206\pi$
$389$	−5.09221	−0.258185	−0.129093	+	0.991633i	$0.541206\pi$
$390$	−7.64244	−0.386990
$391$	0.0146888	0.000742843 0
$392$	−17.9744	−0.907845
$393$	−45.8439	−2.31252
$394$	−14.5109	−0.731051
$395$	−15.4041	−0.775067
$396$	8.47195	0.425731
$397$	−19.7021	−0.988819	−0.494410	−	0.869229i	$-0.664616\pi$
$397$	−19.7021	−0.988819	−0.494410	+	0.869229i	$0.664616\pi$
$398$	−4.51667	−0.226400
$399$	−56.4247	−2.82477
$400$	−2.31517	−0.115758
$401$	−24.2920	−1.21309	−0.606543	−	0.795051i	$-0.707444\pi$
$401$	−24.2920	−1.21309	−0.606543	+	0.795051i	$0.707444\pi$
$402$	−22.2688	−1.11067
$403$	−8.71090	−0.433921
$404$	50.4708	2.51102
$405$	11.2784	0.560430
$406$	30.8351	1.53032
$407$	−0.496068	−0.0245892
$408$	−0.0142816	−0.000707043 0
$409$	23.2495	1.14961	0.574806	−	0.818289i	$-0.305077\pi$
$409$	23.2495	1.14961	0.574806	+	0.818289i	$0.305077\pi$
$410$	−6.04163	−0.298375
$411$	−35.9747	−1.77450
$412$	2.44606	0.120509
$413$	−10.4663	−0.515011
$414$	−57.8630	−2.84381
$415$	15.5109	0.761400
$416$	9.02844	0.442656
$417$	−4.41575	−0.216240
$418$	−4.58552	−0.224285
$419$	11.9390	0.583260	0.291630	−	0.956531i	$-0.405802\pi$
$419$	11.9390	0.583260	0.291630	+	0.956531i	$0.405802\pi$
$420$	−35.9438	−1.75388
$421$	−6.70413	−0.326739	−0.163370	−	0.986565i	$-0.552236\pi$
$421$	−6.70413	−0.326739	−0.163370	+	0.986565i	$0.552236\pi$
$422$	−14.7635	−0.718675
$423$	−62.7507	−3.05104
$424$	12.0809	0.586701
$425$	0.00341512	0.000165658 0
$426$	−76.8426	−3.72303
$427$	57.9094	2.80243
$428$	38.6985	1.87056
$429$	1.82402	0.0880644
$430$	−20.1928	−0.973784
$431$	2.28637	0.110131	0.0550653	−	0.998483i	$-0.482463\pi$
$431$	2.28637	0.110131	0.0550653	+	0.998483i	$0.482463\pi$
$432$	−22.8546	−1.09959
$433$	−2.24668	−0.107969	−0.0539843	−	0.998542i	$-0.517192\pi$
$433$	−2.24668	−0.107969	−0.0539843	+	0.998542i	$0.517192\pi$
$434$	−72.0232	−3.45722
$435$	9.71781	0.465933
$436$	−31.2622	−1.49719
$437$	17.8151	0.852214
$438$	−104.951	−5.01473
$439$	−4.50891	−0.215199	−0.107599	−	0.994194i	$-0.534316\pi$
$439$	−4.50891	−0.215199	−0.107599	+	0.994194i	$0.534316\pi$
$440$	−0.706921	−0.0337011
$441$	81.6399	3.88761
$442$	−0.00858324	−0.000408263 0
$443$	−39.1776	−1.86138	−0.930691	−	0.365807i	$-0.880793\pi$
$443$	−39.1776	−1.86138	−0.930691	+	0.365807i	$0.880793\pi$
$444$	−7.74292	−0.367462
$445$	16.0809	0.762307
$446$	−47.9701	−2.27145
$447$	−21.5157	−1.01766
$448$	53.9050	2.54677
$449$	2.00698	0.0947152	0.0473576	−	0.998878i	$-0.484920\pi$
$449$	2.00698	0.0947152	0.0473576	+	0.998878i	$0.484920\pi$
$450$	−13.4531	−0.634184
$451$	1.44195	0.0678990
$452$	−23.8922	−1.12379
$453$	−11.7753	−0.553254
$454$	−49.7803	−2.33630
$455$	−5.22789	−0.245087
$456$	−17.3213	−0.811143
$457$	24.5194	1.14697	0.573484	−	0.819216i	$-0.305591\pi$
$457$	24.5194	1.14697	0.573484	+	0.819216i	$0.305591\pi$
$458$	−48.8332	−2.28183
$459$	0.0337129	0.00157359
$460$	11.3486	0.529133
$461$	−9.11146	−0.424363	−0.212182	−	0.977230i	$-0.568057\pi$
$461$	−9.11146	−0.424363	−0.212182	+	0.977230i	$0.568057\pi$
$462$	15.0813	0.701644
$463$	18.5643	0.862755	0.431377	−	0.902172i	$-0.358028\pi$
$463$	18.5643	0.862755	0.431377	+	0.902172i	$0.358028\pi$
$464$	−7.39885	−0.343483
$465$	−22.6984	−1.05261
$466$	7.91584	0.366694
$467$	−17.3358	−0.802205	−0.401103	−	0.916033i	$-0.631373\pi$
$467$	−17.3358	−0.802205	−0.401103	+	0.916033i	$0.631373\pi$
$468$	19.2331	0.889051
$469$	−15.2332	−0.703405
$470$	21.6361	0.997998
$471$	−52.5090	−2.41949
$472$	−3.21294	−0.147887
$473$	4.81941	0.221597
$474$	100.882	4.63368
$475$	4.14200	0.190048
$476$	−0.0403686	−0.00185029
$477$	−54.8716	−2.51240
$478$	35.4009	1.61920
$479$	−16.7040	−0.763227	−0.381613	−	0.924322i	$-0.624631\pi$
$479$	−16.7040	−0.763227	−0.381613	+	0.924322i	$0.624631\pi$
$480$	23.5258	1.07380
$481$	−1.12618	−0.0513494
$482$	38.1370	1.73709
$483$	−58.5920	−2.66603
$484$	−28.3268	−1.28758
$485$	9.48193	0.430552
$486$	−10.0803	−0.457251
$487$	−9.01182	−0.408365	−0.204182	−	0.978933i	$-0.565454\pi$
$487$	−9.01182	−0.408365	−0.204182	+	0.978933i	$0.565454\pi$
$488$	17.7770	0.804729
$489$	−13.5541	−0.612939
$490$	−28.1490	−1.27164
$491$	30.5117	1.37697	0.688487	−	0.725248i	$-0.258275\pi$
$491$	30.5117	1.37697	0.688487	+	0.725248i	$0.258275\pi$
$492$	22.5068	1.01469
$493$	0.0109141	0.000491546 0
$494$	−10.4101	−0.468373
$495$	3.21084	0.144317
$496$	17.2819	0.775980
$497$	−52.5650	−2.35786
$498$	−101.582	−4.55198
$499$	33.0435	1.47923	0.739614	−	0.673031i	$-0.235008\pi$
$499$	33.0435	1.47923	0.739614	+	0.673031i	$0.235008\pi$
$500$	2.63855	0.117999
$501$	76.2002	3.40437
$502$	67.5768	3.01610
$503$	34.6093	1.54315	0.771576	−	0.636137i	$-0.219469\pi$
$503$	34.6093	1.54315	0.771576	+	0.636137i	$0.219469\pi$
$504$	38.4845	1.71424
$505$	19.1283	0.851197
$506$	−4.76166	−0.211681
$507$	−35.3894	−1.57170
$508$	1.38529	0.0614625
$509$	12.0234	0.532926	0.266463	−	0.963845i	$-0.414145\pi$
$509$	12.0234	0.532926	0.266463	+	0.963845i	$0.414145\pi$
$510$	−0.0223658	−0.000990373 0
$511$	−71.7926	−3.17592
$512$	−24.2798	−1.07302
$513$	40.8884	1.80527
$514$	11.0158	0.485885
$515$	0.927050	0.0408507
$516$	75.2241	3.31156
$517$	−5.16388	−0.227107
$518$	−9.31143	−0.409121
$519$	−7.52292	−0.330219
$520$	−1.60486	−0.0703778
$521$	28.3634	1.24262	0.621311	−	0.783564i	$-0.286601\pi$
$521$	28.3634	1.24262	0.621311	+	0.783564i	$0.286601\pi$
$522$	−42.9935	−1.88178
$523$	37.2526	1.62894	0.814472	−	0.580203i	$-0.197027\pi$
$523$	37.2526	1.62894	0.814472	+	0.580203i	$0.197027\pi$
$524$	−39.7795	−1.73778
$525$	−13.6226	−0.594538
$526$	−11.8376	−0.516142
$527$	−0.0254926	−0.00111048
$528$	−3.61874	−0.157485
$529$	−4.50057	−0.195677
$530$	18.9194	0.821808
$531$	14.5932	0.633290
$532$	−48.9606	−2.12271
$533$	3.27354	0.141793
$534$	−105.314	−4.55740
$535$	14.6666	0.634093
$536$	−4.67630	−0.201985
$537$	47.8849	2.06638
$538$	−6.09793	−0.262900
$539$	6.71830	0.289378
$540$	26.0468	1.12088
$541$	16.2586	0.699012	0.349506	−	0.936934i	$-0.386349\pi$
$541$	16.2586	0.699012	0.349506	+	0.936934i	$0.386349\pi$
$542$	−40.3937	−1.73506
$543$	56.1949	2.41155
$544$	0.0264219	0.00113283
$545$	−11.8483	−0.507523
$546$	34.2377	1.46524
$547$	25.5084	1.09066	0.545331	−	0.838221i	$-0.316404\pi$
$547$	25.5084	1.09066	0.545331	+	0.838221i	$0.316404\pi$
$548$	−31.2159	−1.33348
$549$	−80.7435	−3.44605
$550$	−1.10708	−0.0472060
$551$	13.2371	0.563918
$552$	−17.9866	−0.765560
$553$	69.0097	2.93459
$554$	−7.60121	−0.322945
$555$	−2.93454	−0.124564
$556$	−3.83162	−0.162497
$557$	−17.9478	−0.760475	−0.380237	−	0.924889i	$-0.624158\pi$
$557$	−17.9478	−0.760475	−0.380237	+	0.924889i	$0.624158\pi$
$558$	100.422	4.25122
$559$	10.9411	0.462758
$560$	10.3718	0.438289
$561$	0.00533803	0.000225372 0
$562$	−49.0043	−2.06712
$563$	−28.1653	−1.18702	−0.593512	−	0.804825i	$-0.702259\pi$
$563$	−28.1653	−1.18702	−0.593512	+	0.804825i	$0.702259\pi$
$564$	−80.6007	−3.39390
$565$	−9.05505	−0.380949
$566$	−22.2049	−0.933342
$567$	−50.5268	−2.12192
$568$	−16.1364	−0.677069
$569$	−18.2443	−0.764842	−0.382421	−	0.923988i	$-0.624910\pi$
$569$	−18.2443	−0.764842	−0.382421	+	0.923988i	$0.624910\pi$
$570$	−27.1261	−1.13619
$571$	9.15693	0.383206	0.191603	−	0.981473i	$-0.438631\pi$
$571$	9.15693	0.383206	0.191603	+	0.981473i	$0.438631\pi$
$572$	1.58273	0.0661773
$573$	−0.546583	−0.0228338
$574$	27.0662	1.12972
$575$	4.30110	0.179368
$576$	−75.1601	−3.13167
$577$	−42.6744	−1.77656	−0.888279	−	0.459304i	$-0.848099\pi$
$577$	−42.6744	−1.77656	−0.888279	+	0.459304i	$0.848099\pi$
$578$	36.6134	1.52291
$579$	55.5423	2.30826
$580$	8.43230	0.350132
$581$	−69.4879	−2.88285
$582$	−62.0976	−2.57403
$583$	−4.51549	−0.187013
$584$	−22.0389	−0.911976
$585$	7.28929	0.301375
$586$	−59.2836	−2.44898
$587$	−37.1930	−1.53512	−0.767560	−	0.640978i	$-0.778529\pi$
$587$	−37.1930	−1.53512	−0.767560	+	0.640978i	$0.778529\pi$
$588$	104.863	4.32448
$589$	−30.9185	−1.27398
$590$	−5.03165	−0.207150
$591$	20.4876	0.842748
$592$	2.23427	0.0918280
$593$	−22.8540	−0.938502	−0.469251	−	0.883065i	$-0.655476\pi$
$593$	−22.8540	−0.938502	−0.469251	+	0.883065i	$0.655476\pi$
$594$	−10.9287	−0.448411
$595$	−0.0152995	−0.000627220 0
$596$	−18.6695	−0.764734
$597$	6.37697	0.260992
$598$	−10.8100	−0.442052
$599$	42.9271	1.75395	0.876977	−	0.480532i	$-0.159556\pi$
$599$	42.9271	1.75395	0.876977	+	0.480532i	$0.159556\pi$
$600$	−4.18186	−0.170724
$601$	−14.9987	−0.611810	−0.305905	−	0.952062i	$-0.598959\pi$
$601$	−14.9987	−0.611810	−0.305905	+	0.952062i	$0.598959\pi$
$602$	90.4626	3.68698
$603$	21.2398	0.864951
$604$	−10.2177	−0.415751
$605$	−10.7358	−0.436471
$606$	−125.272	−5.08882
$607$	−40.7034	−1.65210	−0.826051	−	0.563596i	$-0.809418\pi$
$607$	−40.7034	−1.65210	−0.826051	+	0.563596i	$0.809418\pi$
$608$	32.0456	1.29962
$609$	−43.5352	−1.76414
$610$	27.8399	1.12720
$611$	−11.7231	−0.474265
$612$	0.0562861	0.00227523
$613$	2.06028	0.0832138	0.0416069	−	0.999134i	$-0.486752\pi$
$613$	2.06028	0.0832138	0.0416069	+	0.999134i	$0.486752\pi$
$614$	57.7248	2.32958
$615$	8.53002	0.343964
$616$	3.16697	0.127601
$617$	−0.391425	−0.0157582	−0.00787909	−	0.999969i	$-0.502508\pi$
$617$	−0.391425	−0.0157582	−0.00787909	+	0.999969i	$0.502508\pi$
$618$	−6.07129	−0.244223
$619$	−18.1471	−0.729394	−0.364697	−	0.931126i	$-0.618827\pi$
$619$	−18.1471	−0.729394	−0.364697	+	0.931126i	$0.618827\pi$
$620$	−19.6958	−0.791002
$621$	42.4590	1.70382
$622$	62.3633	2.50054
$623$	−72.0414	−2.88628
$624$	−8.21530	−0.328875
$625$	1.00000	0.0400000
$626$	1.58798	0.0634684
$627$	6.47418	0.258554
$628$	−45.5629	−1.81816
$629$	−0.00329579	−0.000131412 0
$630$	60.2690	2.40117
$631$	3.99212	0.158924	0.0794618	−	0.996838i	$-0.474680\pi$
$631$	3.99212	0.158924	0.0794618	+	0.996838i	$0.474680\pi$
$632$	21.1846	0.842678
$633$	20.8442	0.828481
$634$	−44.4341	−1.76470
$635$	0.525022	0.0208349
$636$	−70.4804	−2.79473
$637$	15.2520	0.604305
$638$	−3.53802	−0.140072
$639$	73.2917	2.89938
$640$	10.4413	0.412728
$641$	7.10915	0.280795	0.140397	−	0.990095i	$-0.455162\pi$
$641$	7.10915	0.280795	0.140397	+	0.990095i	$0.455162\pi$
$642$	−96.0522	−3.79088
$643$	21.0749	0.831114	0.415557	−	0.909567i	$-0.363587\pi$
$643$	21.0749	0.831114	0.415557	+	0.909567i	$0.363587\pi$
$644$	−50.8412	−2.00343
$645$	28.5097	1.12257
$646$	−0.0304654	−0.00119865
$647$	−22.8323	−0.897632	−0.448816	−	0.893624i	$-0.648154\pi$
$647$	−22.8323	−0.897632	−0.448816	+	0.893624i	$0.648154\pi$
$648$	−15.5107	−0.609319
$649$	1.20090	0.0471395
$650$	−2.51331	−0.0985799
$651$	101.688	3.98545
$652$	−11.7611	−0.460602
$653$	17.2576	0.675340	0.337670	−	0.941265i	$-0.390361\pi$
$653$	17.2576	0.675340	0.337670	+	0.941265i	$0.390361\pi$
$654$	77.5947	3.03419
$655$	−15.0763	−0.589080
$656$	−6.49450	−0.253568
$657$	100.101	3.90531
$658$	−96.9284	−3.77866
$659$	−38.2435	−1.48976	−0.744878	−	0.667201i	$-0.767492\pi$
$659$	−38.2435	−1.48976	−0.744878	+	0.667201i	$0.767492\pi$
$660$	4.12420	0.160534
$661$	30.3235	1.17945	0.589723	−	0.807605i	$-0.299237\pi$
$661$	30.3235	1.17945	0.589723	+	0.807605i	$0.299237\pi$
$662$	−62.8530	−2.44285
$663$	0.0121185	0.000470642 0
$664$	−21.3314	−0.827820
$665$	−18.5559	−0.719568
$666$	12.9830	0.503081
$667$	13.7455	0.532228
$668$	66.1202	2.55827
$669$	67.7278	2.61851
$670$	−7.32336	−0.282926
$671$	−6.64454	−0.256509
$672$	−105.394	−4.06568
$673$	11.1831	0.431078	0.215539	−	0.976495i	$-0.430849\pi$
$673$	11.1831	0.431078	0.215539	+	0.976495i	$0.430849\pi$
$674$	−47.1420	−1.81584
$675$	9.87167	0.379961
$676$	−30.7080	−1.18108
$677$	−21.8316	−0.839058	−0.419529	−	0.907742i	$-0.637805\pi$
$677$	−21.8316	−0.839058	−0.419529	+	0.907742i	$0.637805\pi$
$678$	59.3019	2.27748
$679$	−42.4785	−1.63018
$680$	−0.00469666	−0.000180109 0
$681$	70.2834	2.69327
$682$	8.26395	0.316443
$683$	−8.97547	−0.343437	−0.171718	−	0.985146i	$-0.554932\pi$
$683$	−8.97547	−0.343437	−0.171718	+	0.985146i	$0.554932\pi$
$684$	68.2662	2.61022
$685$	−11.8307	−0.452028
$686$	58.5657	2.23605
$687$	68.9463	2.63046
$688$	−21.7064	−0.827550
$689$	−10.2511	−0.390537
$690$	−28.1680	−1.07234
$691$	11.7023	0.445178	0.222589	−	0.974912i	$-0.428549\pi$
$691$	11.7023	0.445178	0.222589	+	0.974912i	$0.428549\pi$
$692$	−6.52776	−0.248148
$693$	−14.3844	−0.546417
$694$	28.0462	1.06462
$695$	−1.45217	−0.0550840
$696$	−13.3645	−0.506578
$697$	0.00958009	0.000362872 0
$698$	−17.9321	−0.678742
$699$	−11.1762	−0.422721
$700$	−11.8205	−0.446774
$701$	−24.2314	−0.915208	−0.457604	−	0.889156i	$-0.651292\pi$
$701$	−24.2314	−0.915208	−0.457604	+	0.889156i	$0.651292\pi$
$702$	−24.8105	−0.936413
$703$	−3.99727	−0.150760
$704$	−6.18507	−0.233109
$705$	−30.5474	−1.15048
$706$	−13.8855	−0.522589
$707$	−85.6935	−3.22284
$708$	18.7443	0.704456
$709$	3.09913	0.116390	0.0581952	−	0.998305i	$-0.481465\pi$
$709$	3.09913	0.116390	0.0581952	+	0.998305i	$0.481465\pi$
$710$	−25.2706	−0.948388
$711$	−96.2207	−3.60856
$712$	−22.1153	−0.828806
$713$	−32.1061	−1.20238
$714$	0.100197	0.00374979
$715$	0.599850	0.0224331
$716$	41.5505	1.55281
$717$	−49.9816	−1.86660
$718$	−36.7878	−1.37291
$719$	−35.6138	−1.32817	−0.664085	−	0.747657i	$-0.731179\pi$
$719$	−35.6138	−1.32817	−0.664085	+	0.747657i	$0.731179\pi$
$720$	−14.4615	−0.538948
$721$	−4.15313	−0.154671
$722$	3.97112	0.147790
$723$	−53.8447	−2.00250
$724$	48.7612	1.81220
$725$	3.19582	0.118690
$726$	70.3090	2.60941
$727$	24.9391	0.924939	0.462469	−	0.886635i	$-0.346964\pi$
$727$	24.9391	0.924939	0.462469	+	0.886635i	$0.346964\pi$
$728$	7.18968	0.266467
$729$	−19.6032	−0.726046
$730$	−34.5142	−1.27743
$731$	0.0320193	0.00118428
$732$	−103.712	−3.83330
$733$	−16.3824	−0.605100	−0.302550	−	0.953134i	$-0.597838\pi$
$733$	−16.3824	−0.605100	−0.302550	+	0.953134i	$0.597838\pi$
$734$	−23.4529	−0.865662
$735$	39.7428	1.46593
$736$	33.2765	1.22659
$737$	1.74786	0.0643833
$738$	−37.7385	−1.38917
$739$	−18.7853	−0.691027	−0.345513	−	0.938414i	$-0.612295\pi$
$739$	−18.7853	−0.691027	−0.345513	+	0.938414i	$0.612295\pi$
$740$	−2.54635	−0.0936056
$741$	14.6978	0.539936
$742$	−84.7580	−3.11156
$743$	29.3655	1.07732	0.538659	−	0.842524i	$-0.318931\pi$
$743$	29.3655	1.07732	0.538659	+	0.842524i	$0.318931\pi$
$744$	31.2161	1.14444
$745$	−7.07570	−0.259233
$746$	−56.0813	−2.05328
$747$	96.8875	3.54493
$748$	0.00463190	0.000169359 0
$749$	−65.7056	−2.40083
$750$	−6.54904	−0.239137
$751$	3.60387	0.131507	0.0657536	−	0.997836i	$-0.479055\pi$
$751$	3.60387	0.131507	0.0657536	+	0.997836i	$0.479055\pi$
$752$	23.2579	0.848128
$753$	−95.4099	−3.47693
$754$	−8.03206	−0.292510
$755$	−3.87246	−0.140933
$756$	−116.688	−4.24391
$757$	−38.5022	−1.39939	−0.699694	−	0.714443i	$-0.746680\pi$
$757$	−38.5022	−1.39939	−0.699694	+	0.714443i	$0.746680\pi$
$758$	−34.9651	−1.26999
$759$	6.72286	0.244024
$760$	−5.69630	−0.206627
$761$	−17.1530	−0.621795	−0.310898	−	0.950443i	$-0.600630\pi$
$761$	−17.1530	−0.621795	−0.310898	+	0.950443i	$0.600630\pi$
$762$	−3.43839	−0.124560
$763$	53.0795	1.92161
$764$	−0.474279	−0.0171588
$765$	0.0213323	0.000771270 0
$766$	10.2208	0.369293
$767$	2.72630	0.0984409
$768$	4.79641	0.173076
$769$	6.86562	0.247581	0.123790	−	0.992308i	$-0.460495\pi$
$769$	6.86562	0.247581	0.123790	+	0.992308i	$0.460495\pi$
$770$	4.95965	0.178734
$771$	−15.5529	−0.560123
$772$	48.1950	1.73457
$773$	51.3375	1.84648	0.923241	−	0.384222i	$-0.125530\pi$
$773$	51.3375	1.84648	0.923241	+	0.384222i	$0.125530\pi$
$774$	−126.133	−4.53374
$775$	−7.46464	−0.268138
$776$	−13.0401	−0.468111
$777$	13.1466	0.471630
$778$	10.9672	0.393195
$779$	11.6191	0.416298
$780$	9.36280	0.335242
$781$	6.03132	0.215818
$782$	−0.0316356	−0.00113129
$783$	31.5480	1.12743
$784$	−30.2590	−1.08068
$785$	−17.2682	−0.616328
$786$	98.7353	3.52177
$787$	41.9209	1.49432	0.747159	−	0.664645i	$-0.231417\pi$
$787$	41.9209	1.49432	0.747159	+	0.664645i	$0.231417\pi$
$788$	17.7775	0.633296
$789$	16.7131	0.595004
$790$	33.1763	1.18036
$791$	40.5661	1.44236
$792$	−4.41572	−0.156906
$793$	−15.0845	−0.535666
$794$	42.4329	1.50589
$795$	−26.7118	−0.947371
$796$	5.53340	0.196126
$797$	−19.3975	−0.687095	−0.343548	−	0.939135i	$-0.611629\pi$
$797$	−19.3975	−0.687095	−0.343548	+	0.939135i	$0.611629\pi$
$798$	121.523	4.30188
$799$	−0.0343079	−0.00121373
$800$	7.73675	0.273535
$801$	100.448	3.54915
$802$	52.3184	1.84743
$803$	8.23749	0.290695
$804$	27.2816	0.962149
$805$	−19.2687	−0.679131
$806$	18.7609	0.660825
$807$	8.60950	0.303069
$808$	−26.3062	−0.925450
$809$	−29.8214	−1.04846	−0.524232	−	0.851576i	$-0.675647\pi$
$809$	−29.8214	−1.04846	−0.524232	+	0.851576i	$0.675647\pi$
$810$	−24.2907	−0.853488
$811$	−12.7644	−0.448220	−0.224110	−	0.974564i	$-0.571948\pi$
$811$	−12.7644	−0.448220	−0.224110	+	0.974564i	$0.571948\pi$
$812$	−37.7762	−1.32569
$813$	57.0307	2.00016
$814$	1.06840	0.0374473
$815$	−4.45743	−0.156137
$816$	−0.0240423	−0.000841648 0
$817$	38.8343	1.35864
$818$	−50.0731	−1.75076
$819$	−32.6556	−1.14108
$820$	7.40164	0.258477
$821$	5.44889	0.190168	0.0950838	−	0.995469i	$-0.469688\pi$
$821$	5.44889	0.190168	0.0950838	+	0.995469i	$0.469688\pi$
$822$	77.4798	2.70242
$823$	49.5694	1.72788	0.863940	−	0.503596i	$-0.167990\pi$
$823$	49.5694	1.72788	0.863940	+	0.503596i	$0.167990\pi$
$824$	−1.27493	−0.0444143
$825$	1.56306	0.0544186
$826$	22.5415	0.784318
$827$	34.8680	1.21248	0.606240	−	0.795282i	$-0.292677\pi$
$827$	34.8680	1.21248	0.606240	+	0.795282i	$0.292677\pi$
$828$	70.8883	2.46354
$829$	−45.1314	−1.56748	−0.783739	−	0.621091i	$-0.786690\pi$
$829$	−45.1314	−1.56748	−0.783739	+	0.621091i	$0.786690\pi$
$830$	−33.4063	−1.15955
$831$	10.7320	0.372287
$832$	−14.0414	−0.486799
$833$	0.0446352	0.00154652
$834$	9.51032	0.329316
$835$	25.0593	0.867214
$836$	5.61775	0.194294
$837$	−73.6884	−2.54704
$838$	−25.7134	−0.888256
$839$	−5.16571	−0.178340	−0.0891700	−	0.996016i	$-0.528421\pi$
$839$	−5.16571	−0.178340	−0.0891700	+	0.996016i	$0.528421\pi$
$840$	18.7345	0.646401
$841$	−18.7868	−0.647820
$842$	14.4389	0.497596
$843$	69.1879	2.38296
$844$	18.0868	0.622574
$845$	−11.6382	−0.400367
$846$	135.148	4.64648
$847$	48.0956	1.65259
$848$	20.3376	0.698396
$849$	31.3505	1.07595
$850$	−0.00735524	−0.000252283 0
$851$	−4.15081	−0.142288
$852$	94.1403	3.22519
$853$	25.8877	0.886378	0.443189	−	0.896428i	$-0.353847\pi$
$853$	25.8877	0.886378	0.443189	+	0.896428i	$0.353847\pi$
$854$	−124.721	−4.26787
$855$	25.8726	0.884826
$856$	−20.1703	−0.689407
$857$	−9.63051	−0.328972	−0.164486	−	0.986379i	$-0.552597\pi$
$857$	−9.63051	−0.328972	−0.164486	+	0.986379i	$0.552597\pi$
$858$	−3.92844	−0.134115
$859$	32.1248	1.09608	0.548042	−	0.836451i	$-0.315373\pi$
$859$	32.1248	1.09608	0.548042	+	0.836451i	$0.315373\pi$
$860$	24.7383	0.843570
$861$	−38.2140	−1.30233
$862$	−4.92422	−0.167720
$863$	−32.2079	−1.09637	−0.548186	−	0.836357i	$-0.684681\pi$
$863$	−32.2079	−1.09637	−0.548186	+	0.836357i	$0.684681\pi$
$864$	76.3746	2.59832
$865$	−2.47400	−0.0841185
$866$	4.83874	0.164427
$867$	−51.6934	−1.75560
$868$	88.2360	2.99493
$869$	−7.91818	−0.268606
$870$	−20.9295	−0.709577
$871$	3.96802	0.134451
$872$	16.2944	0.551797
$873$	59.2281	2.00457
$874$	−38.3690	−1.29785
$875$	−4.47994	−0.151450
$876$	128.576	4.34416
$877$	−33.7982	−1.14129	−0.570643	−	0.821199i	$-0.693306\pi$
$877$	−33.7982	−1.14129	−0.570643	+	0.821199i	$0.693306\pi$
$878$	9.71097	0.327729
$879$	83.7010	2.82316
$880$	−1.19006	−0.0401171
$881$	−18.0826	−0.609218	−0.304609	−	0.952478i	$-0.598526\pi$
$881$	−18.0826	−0.609218	−0.304609	+	0.952478i	$0.598526\pi$
$882$	−175.830	−5.92051
$883$	−54.7205	−1.84149	−0.920746	−	0.390162i	$-0.872419\pi$
$883$	−54.7205	−1.84149	−0.920746	+	0.390162i	$0.872419\pi$
$884$	0.0105154	0.000353671 0
$885$	7.10405	0.238800
$886$	84.3778	2.83473
$887$	29.4982	0.990451	0.495226	−	0.868764i	$-0.335085\pi$
$887$	29.4982	0.990451	0.495226	+	0.868764i	$0.335085\pi$
$888$	4.03574	0.135430
$889$	−2.35207	−0.0788858
$890$	−34.6338	−1.16093
$891$	5.79745	0.194222
$892$	58.7685	1.96771
$893$	−41.6100	−1.39243
$894$	46.3390	1.54981
$895$	15.7475	0.526381
$896$	−46.7763	−1.56269
$897$	15.2623	0.509594
$898$	−4.32249	−0.144243
$899$	−23.8556	−0.795629
$900$	16.4814	0.549382
$901$	−0.0300001	−0.000999450 0
$902$	−3.10558	−0.103404
$903$	−127.722	−4.25031
$904$	12.4530	0.414180
$905$	18.4803	0.614307
$906$	25.3609	0.842559
$907$	50.1703	1.66588	0.832939	−	0.553365i	$-0.186657\pi$
$907$	50.1703	1.66588	0.832939	+	0.553365i	$0.186657\pi$
$908$	60.9861	2.02389
$909$	119.483	3.96300
$910$	11.2595	0.373248
$911$	13.1382	0.435289	0.217645	−	0.976028i	$-0.430163\pi$
$911$	13.1382	0.435289	0.217645	+	0.976028i	$0.430163\pi$
$912$	−29.1594	−0.965566
$913$	7.97306	0.263870
$914$	−52.8081	−1.74674
$915$	−39.3064	−1.29943
$916$	59.8258	1.97670
$917$	67.5410	2.23040
$918$	−0.0726085	−0.00239644
$919$	14.6133	0.482049	0.241025	−	0.970519i	$-0.422517\pi$
$919$	14.6133	0.482049	0.241025	+	0.970519i	$0.422517\pi$
$920$	−5.91510	−0.195015
$921$	−81.5001	−2.68552
$922$	19.6236	0.646269
$923$	13.6924	0.450690
$924$	−18.4762	−0.607821
$925$	−0.965058	−0.0317309
$926$	−39.9824	−1.31390
$927$	5.79074	0.190193
$928$	24.7252	0.811645
$929$	−14.8462	−0.487087	−0.243544	−	0.969890i	$-0.578310\pi$
$929$	−14.8462	−0.487087	−0.243544	+	0.969890i	$0.578310\pi$
$930$	48.8862	1.60304
$931$	54.1355	1.77422
$932$	−9.69775	−0.317660
$933$	−88.0490	−2.88260
$934$	37.3366	1.22169
$935$	0.00175547	5.74101e−5 0
$936$	−10.0246	−0.327665
$937$	−44.1724	−1.44305	−0.721525	−	0.692388i	$-0.756559\pi$
$937$	−44.1724	−1.44305	−0.721525	+	0.692388i	$0.756559\pi$
$938$	32.8082	1.07123
$939$	−2.24202	−0.0731657
$940$	−26.5065	−0.864547
$941$	−1.40332	−0.0457470	−0.0228735	−	0.999738i	$-0.507282\pi$
$941$	−1.40332	−0.0457470	−0.0228735	+	0.999738i	$0.507282\pi$
$942$	113.090	3.68467
$943$	12.0654	0.392904
$944$	−5.40881	−0.176042
$945$	−44.2245	−1.43862
$946$	−10.3797	−0.337473
$947$	0.130053	0.00422615	0.00211307	−	0.999998i	$-0.499327\pi$
$947$	0.130053	0.00422615	0.00211307	+	0.999998i	$0.499327\pi$
$948$	−123.592	−4.01407
$949$	18.7008	0.607055
$950$	−8.92074	−0.289427
$951$	62.7353	2.03433
$952$	0.0210408	0.000681935 0
$953$	20.2486	0.655916	0.327958	−	0.944692i	$-0.393640\pi$
$953$	20.2486	0.655916	0.327958	+	0.944692i	$0.393640\pi$
$954$	118.179	3.82617
$955$	−0.179750	−0.00581658
$956$	−43.3698	−1.40268
$957$	4.99524	0.161473
$958$	35.9759	1.16233
$959$	53.0009	1.71149
$960$	−36.5884	−1.18089
$961$	24.7208	0.797446
$962$	2.42548	0.0782008
$963$	91.6137	2.95221
$964$	−46.7219	−1.50481
$965$	18.2657	0.587995
$966$	126.191	4.06014
$967$	22.8301	0.734166	0.367083	−	0.930188i	$-0.380356\pi$
$967$	22.8301	0.734166	0.367083	+	0.930188i	$0.380356\pi$
$968$	14.7644	0.474546
$969$	0.0430133	0.00138179
$970$	−20.4215	−0.655695
$971$	−26.7909	−0.859762	−0.429881	−	0.902885i	$-0.641445\pi$
$971$	−26.7909	−0.859762	−0.429881	+	0.902885i	$0.641445\pi$
$972$	12.3494	0.396108
$973$	6.50564	0.208561
$974$	19.4090	0.621905
$975$	3.54847	0.113642
$976$	29.9267	0.957931
$977$	19.5149	0.624336	0.312168	−	0.950027i	$-0.398945\pi$
$977$	19.5149	0.624336	0.312168	+	0.950027i	$0.398945\pi$
$978$	29.1919	0.933454
$979$	8.26605	0.264184
$980$	34.4855	1.10160
$981$	−74.0091	−2.36293
$982$	−65.7140	−2.09702
$983$	−34.5186	−1.10097	−0.550485	−	0.834845i	$-0.685557\pi$
$983$	−34.5186	−1.10097	−0.550485	+	0.834845i	$0.685557\pi$
$984$	−11.7309	−0.373969
$985$	6.73760	0.214678
$986$	−0.0235060	−0.000748584 0
$987$	136.851	4.35600
$988$	12.7535	0.405743
$989$	40.3260	1.28229
$990$	−6.91528	−0.219782
$991$	−2.73401	−0.0868487	−0.0434243	−	0.999057i	$-0.513827\pi$
$991$	−2.73401	−0.0868487	−0.0434243	+	0.999057i	$0.513827\pi$
$992$	−57.7520	−1.83363
$993$	88.7405	2.81609
$994$	113.211	3.59083
$995$	2.09714	0.0664839
$996$	124.448	3.94329
$997$	−25.3390	−0.802494	−0.401247	−	0.915970i	$-0.631423\pi$
$997$	−25.3390	−0.802494	−0.401247	+	0.915970i	$0.631423\pi$
$998$	−71.1667	−2.25274
$999$	−9.52673	−0.301412

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8045.2.a.e.1.19	✓	142

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8045.2.a.e.1.19	✓	142	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:	\( N \)	\(=\)	\( 8045 = 5 \cdot 1609 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8045.a (trivial)

\(f(q)\)	\(=\)	\(q-2.15373 q^{2} +3.04079 q^{3} +2.63855 q^{4} +1.00000 q^{5} -6.54904 q^{6} -4.47994 q^{7} -1.37525 q^{8} +6.24641 q^{9} +O(q^{10})\)	\(q-2.15373 q^{2} +3.04079 q^{3} +2.63855 q^{4} +1.00000 q^{5} -6.54904 q^{6} -4.47994 q^{7} -1.37525 q^{8} +6.24641 q^{9} -2.15373 q^{10} +0.514029 q^{11} +8.02327 q^{12} +1.16696 q^{13} +9.64858 q^{14} +3.04079 q^{15} -2.31517 q^{16} +0.00341512 q^{17} -13.4531 q^{18} +4.14200 q^{19} +2.63855 q^{20} -13.6226 q^{21} -1.10708 q^{22} +4.30110 q^{23} -4.18186 q^{24} +1.00000 q^{25} -2.51331 q^{26} +9.87167 q^{27} -11.8205 q^{28} +3.19582 q^{29} -6.54904 q^{30} -7.46464 q^{31} +7.73675 q^{32} +1.56306 q^{33} -0.00735524 q^{34} -4.47994 q^{35} +16.4814 q^{36} -0.965058 q^{37} -8.92074 q^{38} +3.54847 q^{39} -1.37525 q^{40} +2.80520 q^{41} +29.3393 q^{42} +9.37574 q^{43} +1.35629 q^{44} +6.24641 q^{45} -9.26339 q^{46} -10.0459 q^{47} -7.03994 q^{48} +13.0699 q^{49} -2.15373 q^{50} +0.0103847 q^{51} +3.07907 q^{52} -8.78450 q^{53} -21.2609 q^{54} +0.514029 q^{55} +6.16106 q^{56} +12.5950 q^{57} -6.88292 q^{58} +2.33625 q^{59} +8.02327 q^{60} -12.9264 q^{61} +16.0768 q^{62} -27.9836 q^{63} -12.0325 q^{64} +1.16696 q^{65} -3.36640 q^{66} +3.40032 q^{67} +0.00901095 q^{68} +13.0787 q^{69} +9.64858 q^{70} +11.7334 q^{71} -8.59040 q^{72} +16.0253 q^{73} +2.07847 q^{74} +3.04079 q^{75} +10.9289 q^{76} -2.30282 q^{77} -7.64244 q^{78} -15.4041 q^{79} -2.31517 q^{80} +11.2784 q^{81} -6.04163 q^{82} +15.5109 q^{83} -35.9438 q^{84} +0.00341512 q^{85} -20.1928 q^{86} +9.71781 q^{87} -0.706921 q^{88} +16.0809 q^{89} -13.4531 q^{90} -5.22789 q^{91} +11.3486 q^{92} -22.6984 q^{93} +21.6361 q^{94} +4.14200 q^{95} +23.5258 q^{96} +9.48193 q^{97} -28.1490 q^{98} +3.21084 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 142 q + 21 q^{2} + 33 q^{3} + 157 q^{4} + 142 q^{5} + 15 q^{6} + 63 q^{7} + 60 q^{8} + 157 q^{9}+O(q^{10}) \) 142 * q + 21 * q^2 + 33 * q^3 + 157 * q^4 + 142 * q^5 + 15 * q^6 + 63 * q^7 + 60 * q^8 + 157 * q^9	\( 142 q + 21 q^{2} + 33 q^{3} + 157 q^{4} + 142 q^{5} + 15 q^{6} + 63 q^{7} + 60 q^{8} + 157 q^{9} + 21 q^{10} + 36 q^{11} + 55 q^{12} + 57 q^{13} + 2 q^{14} + 33 q^{15} + 179 q^{16} + 55 q^{17} + 65 q^{18} + 130 q^{19} + 157 q^{20} + 28 q^{21} + 30 q^{22} + 117 q^{23} + 21 q^{24} + 142 q^{25} + 21 q^{26} + 120 q^{27} + 135 q^{28} + 12 q^{29} + 15 q^{30} + 74 q^{31} + 126 q^{32} + 55 q^{33} + 35 q^{34} + 63 q^{35} + 186 q^{36} + 75 q^{37} + 65 q^{38} + 23 q^{39} + 60 q^{40} + 22 q^{41} + 10 q^{42} + 190 q^{43} + 22 q^{44} + 157 q^{45} + 56 q^{46} + 102 q^{47} + 78 q^{48} + 197 q^{49} + 21 q^{50} + 30 q^{51} + 120 q^{52} + 56 q^{53} - 6 q^{54} + 36 q^{55} + 3 q^{56} + 68 q^{57} + 31 q^{58} + 55 q^{59} + 55 q^{60} + 90 q^{61} + 68 q^{62} + 167 q^{63} + 180 q^{64} + 57 q^{65} + 17 q^{66} + 151 q^{67} + 119 q^{68} + 21 q^{69} + 2 q^{70} + 4 q^{71} + 130 q^{72} + 143 q^{73} - 46 q^{74} + 33 q^{75} + 213 q^{76} + 75 q^{77} - 24 q^{78} + 47 q^{79} + 179 q^{80} + 150 q^{81} + 69 q^{82} + 201 q^{83} - 31 q^{84} + 55 q^{85} - 4 q^{86} + 153 q^{87} + 37 q^{88} + 25 q^{89} + 65 q^{90} + 132 q^{91} + 194 q^{92} + 52 q^{93} + 18 q^{94} + 130 q^{95} + 13 q^{96} + 80 q^{97} + 58 q^{98} + 103 q^{99}+O(q^{100}) \) 142 * q + 21 * q^2 + 33 * q^3 + 157 * q^4 + 142 * q^5 + 15 * q^6 + 63 * q^7 + 60 * q^8 + 157 * q^9 + 21 * q^10 + 36 * q^11 + 55 * q^12 + 57 * q^13 + 2 * q^14 + 33 * q^15 + 179 * q^16 + 55 * q^17 + 65 * q^18 + 130 * q^19 + 157 * q^20 + 28 * q^21 + 30 * q^22 + 117 * q^23 + 21 * q^24 + 142 * q^25 + 21 * q^26 + 120 * q^27 + 135 * q^28 + 12 * q^29 + 15 * q^30 + 74 * q^31 + 126 * q^32 + 55 * q^33 + 35 * q^34 + 63 * q^35 + 186 * q^36 + 75 * q^37 + 65 * q^38 + 23 * q^39 + 60 * q^40 + 22 * q^41 + 10 * q^42 + 190 * q^43 + 22 * q^44 + 157 * q^45 + 56 * q^46 + 102 * q^47 + 78 * q^48 + 197 * q^49 + 21 * q^50 + 30 * q^51 + 120 * q^52 + 56 * q^53 - 6 * q^54 + 36 * q^55 + 3 * q^56 + 68 * q^57 + 31 * q^58 + 55 * q^59 + 55 * q^60 + 90 * q^61 + 68 * q^62 + 167 * q^63 + 180 * q^64 + 57 * q^65 + 17 * q^66 + 151 * q^67 + 119 * q^68 + 21 * q^69 + 2 * q^70 + 4 * q^71 + 130 * q^72 + 143 * q^73 - 46 * q^74 + 33 * q^75 + 213 * q^76 + 75 * q^77 - 24 * q^78 + 47 * q^79 + 179 * q^80 + 150 * q^81 + 69 * q^82 + 201 * q^83 - 31 * q^84 + 55 * q^85 - 4 * q^86 + 153 * q^87 + 37 * q^88 + 25 * q^89 + 65 * q^90 + 132 * q^91 + 194 * q^92 + 52 * q^93 + 18 * q^94 + 130 * q^95 + 13 * q^96 + 80 * q^97 + 58 * q^98 + 103 * q^99

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