LMFDB - Embedded newform 8045.2.a.c.1.9

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:	$1$
Dimension:	$127$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
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)

$f(q)$	$=$	$q-2.53777 q^{2} +0.243430 q^{3} +4.44030 q^{4} +1.00000 q^{5} -0.617771 q^{6} +4.53173 q^{7} -6.19292 q^{8} -2.94074 q^{9} +O(q^{10})$	\(q-2.53777 q^{2} +0.243430 q^{3} +4.44030 q^{4} +1.00000 q^{5} -0.617771 q^{6} +4.53173 q^{7} -6.19292 q^{8} -2.94074 q^{9} -2.53777 q^{10} -2.10751 q^{11} +1.08090 q^{12} +1.57336 q^{13} -11.5005 q^{14} +0.243430 q^{15} +6.83565 q^{16} +7.01415 q^{17} +7.46294 q^{18} -6.54626 q^{19} +4.44030 q^{20} +1.10316 q^{21} +5.34839 q^{22} +3.43069 q^{23} -1.50754 q^{24} +1.00000 q^{25} -3.99284 q^{26} -1.44616 q^{27} +20.1222 q^{28} -6.05645 q^{29} -0.617771 q^{30} -5.90622 q^{31} -4.96148 q^{32} -0.513032 q^{33} -17.8003 q^{34} +4.53173 q^{35} -13.0578 q^{36} -5.45904 q^{37} +16.6129 q^{38} +0.383004 q^{39} -6.19292 q^{40} +2.37949 q^{41} -2.79957 q^{42} -12.0471 q^{43} -9.35798 q^{44} -2.94074 q^{45} -8.70631 q^{46} +11.7373 q^{47} +1.66400 q^{48} +13.5366 q^{49} -2.53777 q^{50} +1.70746 q^{51} +6.98619 q^{52} -11.4625 q^{53} +3.67002 q^{54} -2.10751 q^{55} -28.0647 q^{56} -1.59356 q^{57} +15.3699 q^{58} +6.85338 q^{59} +1.08090 q^{60} -14.7022 q^{61} +14.9886 q^{62} -13.3267 q^{63} -1.08018 q^{64} +1.57336 q^{65} +1.30196 q^{66} +13.7835 q^{67} +31.1449 q^{68} +0.835132 q^{69} -11.5005 q^{70} -4.10499 q^{71} +18.2118 q^{72} -12.3625 q^{73} +13.8538 q^{74} +0.243430 q^{75} -29.0674 q^{76} -9.55068 q^{77} -0.971976 q^{78} +4.52981 q^{79} +6.83565 q^{80} +8.47019 q^{81} -6.03861 q^{82} -5.54179 q^{83} +4.89836 q^{84} +7.01415 q^{85} +30.5727 q^{86} -1.47432 q^{87} +13.0517 q^{88} -11.6901 q^{89} +7.46294 q^{90} +7.13006 q^{91} +15.2333 q^{92} -1.43775 q^{93} -29.7866 q^{94} -6.54626 q^{95} -1.20777 q^{96} +17.6376 q^{97} -34.3529 q^{98} +6.19765 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 127 q - 20 q^{2} - 31 q^{3} + 116 q^{4} + 127 q^{5} - 17 q^{6} - 63 q^{7} - 57 q^{8} + 122 q^{9}+O(q^{10}) $ 127 * q - 20 * q^2 - 31 * q^3 + 116 * q^4 + 127 * q^5 - 17 * q^6 - 63 * q^7 - 57 * q^8 + 122 * q^9	\( 127 q - 20 q^{2} - 31 q^{3} + 116 q^{4} + 127 q^{5} - 17 q^{6} - 63 q^{7} - 57 q^{8} + 122 q^{9} - 20 q^{10} - 32 q^{11} - 65 q^{12} - 49 q^{13} - 4 q^{14} - 31 q^{15} + 98 q^{16} - 53 q^{17} - 60 q^{18} - 126 q^{19} + 116 q^{20} - 24 q^{21} - 46 q^{22} - 121 q^{23} - 51 q^{24} + 127 q^{25} - 9 q^{26} - 112 q^{27} - 123 q^{28} - 6 q^{29} - 17 q^{30} - 54 q^{31} - 119 q^{32} - 57 q^{33} - 53 q^{34} - 63 q^{35} + 133 q^{36} - 61 q^{37} - 27 q^{38} - 33 q^{39} - 57 q^{40} - 8 q^{41} - 46 q^{42} - 208 q^{43} - 54 q^{44} + 122 q^{45} - 34 q^{46} - 116 q^{47} - 106 q^{48} + 110 q^{49} - 20 q^{50} - 54 q^{51} - 142 q^{52} - 60 q^{53} - 62 q^{54} - 32 q^{55} + 33 q^{56} - 56 q^{57} - 87 q^{58} - 53 q^{59} - 65 q^{60} - 76 q^{61} - 84 q^{62} - 215 q^{63} + 67 q^{64} - 49 q^{65} + 9 q^{66} - 145 q^{67} - 133 q^{68} + q^{69} - 4 q^{70} - 4 q^{71} - 167 q^{72} - 155 q^{73} - 14 q^{74} - 31 q^{75} - 199 q^{76} - 97 q^{77} - 24 q^{78} - 73 q^{79} + 98 q^{80} + 127 q^{81} - 69 q^{82} - 225 q^{83} - 59 q^{84} - 53 q^{85} + 30 q^{86} - 179 q^{87} - 119 q^{88} - 25 q^{89} - 60 q^{90} - 160 q^{91} - 188 q^{92} - 44 q^{93} - 32 q^{94} - 126 q^{95} - 43 q^{96} - 72 q^{97} - 111 q^{98} - 141 q^{99}+O(q^{100}) \) 127 * q - 20 * q^2 - 31 * q^3 + 116 * q^4 + 127 * q^5 - 17 * q^6 - 63 * q^7 - 57 * q^8 + 122 * q^9 - 20 * q^10 - 32 * q^11 - 65 * q^12 - 49 * q^13 - 4 * q^14 - 31 * q^15 + 98 * q^16 - 53 * q^17 - 60 * q^18 - 126 * q^19 + 116 * q^20 - 24 * q^21 - 46 * q^22 - 121 * q^23 - 51 * q^24 + 127 * q^25 - 9 * q^26 - 112 * q^27 - 123 * q^28 - 6 * q^29 - 17 * q^30 - 54 * q^31 - 119 * q^32 - 57 * q^33 - 53 * q^34 - 63 * q^35 + 133 * q^36 - 61 * q^37 - 27 * q^38 - 33 * q^39 - 57 * q^40 - 8 * q^41 - 46 * q^42 - 208 * q^43 - 54 * q^44 + 122 * q^45 - 34 * q^46 - 116 * q^47 - 106 * q^48 + 110 * q^49 - 20 * q^50 - 54 * q^51 - 142 * q^52 - 60 * q^53 - 62 * q^54 - 32 * q^55 + 33 * q^56 - 56 * q^57 - 87 * q^58 - 53 * q^59 - 65 * q^60 - 76 * q^61 - 84 * q^62 - 215 * q^63 + 67 * q^64 - 49 * q^65 + 9 * q^66 - 145 * q^67 - 133 * q^68 + q^69 - 4 * q^70 - 4 * q^71 - 167 * q^72 - 155 * q^73 - 14 * q^74 - 31 * q^75 - 199 * q^76 - 97 * q^77 - 24 * q^78 - 73 * q^79 + 98 * q^80 + 127 * q^81 - 69 * q^82 - 225 * q^83 - 59 * q^84 - 53 * q^85 + 30 * q^86 - 179 * q^87 - 119 * q^88 - 25 * q^89 - 60 * q^90 - 160 * q^91 - 188 * q^92 - 44 * q^93 - 32 * q^94 - 126 * q^95 - 43 * q^96 - 72 * q^97 - 111 * q^98 - 141 * 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 $

$2$	−2.53777	−1.79448	−0.897239	−	0.441546i	$-0.854430\pi$
$2$	−2.53777	−1.79448	−0.897239	+	0.441546i	$0.854430\pi$
$3$	0.243430	0.140544	0.0702722	−	0.997528i	$-0.477613\pi$
$3$	0.243430	0.140544	0.0702722	+	0.997528i	$0.477613\pi$
$4$	4.44030	2.22015
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−0.617771	−0.252204
$7$	4.53173	1.71283	0.856417	−	0.516284i	$-0.172685\pi$
$7$	4.53173	1.71283	0.856417	+	0.516284i	$0.172685\pi$
$8$	−6.19292	−2.18953
$9$	−2.94074	−0.980247
$10$	−2.53777	−0.802515
$11$	−2.10751	−0.635439	−0.317719	−	0.948185i	$-0.602917\pi$
$11$	−2.10751	−0.635439	−0.317719	+	0.948185i	$0.602917\pi$
$12$	1.08090	0.312030
$13$	1.57336	0.436372	0.218186	−	0.975907i	$-0.429986\pi$
$13$	1.57336	0.436372	0.218186	+	0.975907i	$0.429986\pi$
$14$	−11.5005	−3.07364
$15$	0.243430	0.0628534
$16$	6.83565	1.70891
$17$	7.01415	1.70118	0.850591	−	0.525829i	$-0.176245\pi$
$17$	7.01415	1.70118	0.850591	+	0.525829i	$0.176245\pi$
$18$	7.46294	1.75903
$19$	−6.54626	−1.50182	−0.750908	−	0.660407i	$-0.770384\pi$
$19$	−6.54626	−1.50182	−0.750908	+	0.660407i	$0.770384\pi$
$20$	4.44030	0.992881
$21$	1.10316	0.240729
$22$	5.34839	1.14028
$23$	3.43069	0.715348	0.357674	−	0.933847i	$-0.383570\pi$
$23$	3.43069	0.715348	0.357674	+	0.933847i	$0.383570\pi$
$24$	−1.50754	−0.307726
$25$	1.00000	0.200000
$26$	−3.99284	−0.783060
$27$	−1.44616	−0.278313
$28$	20.1222	3.80275
$29$	−6.05645	−1.12465	−0.562327	−	0.826915i	$-0.690094\pi$
$29$	−6.05645	−1.12465	−0.562327	+	0.826915i	$0.690094\pi$
$30$	−0.617771	−0.112789
$31$	−5.90622	−1.06079	−0.530394	−	0.847751i	$-0.677956\pi$
$31$	−5.90622	−1.06079	−0.530394	+	0.847751i	$0.677956\pi$
$32$	−4.96148	−0.877074
$33$	−0.513032	−0.0893074
$34$	−17.8003	−3.05273
$35$	4.53173	0.766003
$36$	−13.0578	−2.17629
$37$	−5.45904	−0.897460	−0.448730	−	0.893667i	$-0.648124\pi$
$37$	−5.45904	−0.897460	−0.448730	+	0.893667i	$0.648124\pi$
$38$	16.6129	2.69497
$39$	0.383004	0.0613297
$40$	−6.19292	−0.979187
$41$	2.37949	0.371614	0.185807	−	0.982586i	$-0.440510\pi$
$41$	2.37949	0.371614	0.185807	+	0.982586i	$0.440510\pi$
$42$	−2.79957	−0.431983
$43$	−12.0471	−1.83716	−0.918579	−	0.395237i	$-0.870662\pi$
$43$	−12.0471	−1.83716	−0.918579	+	0.395237i	$0.870662\pi$
$44$	−9.35798	−1.41077
$45$	−2.94074	−0.438380
$46$	−8.70631	−1.28367
$47$	11.7373	1.71206	0.856031	−	0.516925i	$-0.172923\pi$
$47$	11.7373	1.71206	0.856031	+	0.516925i	$0.172923\pi$
$48$	1.66400	0.240178
$49$	13.5366	1.93380
$50$	−2.53777	−0.358895
$51$	1.70746	0.239092
$52$	6.98619	0.968811
$53$	−11.4625	−1.57449	−0.787246	−	0.616639i	$-0.788494\pi$
$53$	−11.4625	−1.57449	−0.787246	+	0.616639i	$0.788494\pi$
$54$	3.67002	0.499426
$55$	−2.10751	−0.284177
$56$	−28.0647	−3.75030
$57$	−1.59356	−0.211072
$58$	15.3699	2.01817
$59$	6.85338	0.892235	0.446117	−	0.894975i	$-0.352806\pi$
$59$	6.85338	0.892235	0.446117	+	0.894975i	$0.352806\pi$
$60$	1.08090	0.139544
$61$	−14.7022	−1.88243	−0.941214	−	0.337810i	$-0.890314\pi$
$61$	−14.7022	−1.88243	−0.941214	+	0.337810i	$0.890314\pi$
$62$	14.9886	1.90356
$63$	−13.3267	−1.67900
$64$	−1.08018	−0.135022
$65$	1.57336	0.195151
$66$	1.30196	0.160260
$67$	13.7835	1.68392	0.841961	−	0.539538i	$-0.181401\pi$
$67$	13.7835	1.68392	0.841961	+	0.539538i	$0.181401\pi$
$68$	31.1449	3.77688
$69$	0.835132	0.100538
$70$	−11.5005	−1.37457
$71$	−4.10499	−0.487173	−0.243587	−	0.969879i	$-0.578324\pi$
$71$	−4.10499	−0.487173	−0.243587	+	0.969879i	$0.578324\pi$
$72$	18.2118	2.14628
$73$	−12.3625	−1.44692	−0.723462	−	0.690364i	$-0.757450\pi$
$73$	−12.3625	−1.44692	−0.723462	+	0.690364i	$0.757450\pi$
$74$	13.8538	1.61047
$75$	0.243430	0.0281089
$76$	−29.0674	−3.33426
$77$	−9.55068	−1.08840
$78$	−0.971976	−0.110055
$79$	4.52981	0.509644	0.254822	−	0.966988i	$-0.417983\pi$
$79$	4.52981	0.509644	0.254822	+	0.966988i	$0.417983\pi$
$80$	6.83565	0.764249
$81$	8.47019	0.941132
$82$	−6.03861	−0.666853
$83$	−5.54179	−0.608291	−0.304145	−	0.952626i	$-0.598371\pi$
$83$	−5.54179	−0.608291	−0.304145	+	0.952626i	$0.598371\pi$
$84$	4.89836	0.534455
$85$	7.01415	0.760791
$86$	30.5727	3.29674
$87$	−1.47432	−0.158064
$88$	13.0517	1.39131
$89$	−11.6901	−1.23914	−0.619572	−	0.784939i	$-0.712694\pi$
$89$	−11.6901	−1.23914	−0.619572	+	0.784939i	$0.712694\pi$
$90$	7.46294	0.786663
$91$	7.13006	0.747433
$92$	15.2333	1.58818
$93$	−1.43775	−0.149088
$94$	−29.7866	−3.07226
$95$	−6.54626	−0.671633
$96$	−1.20777	−0.123268
$97$	17.6376	1.79083	0.895415	−	0.445232i	$-0.146879\pi$
$97$	17.6376	1.79083	0.895415	+	0.445232i	$0.146879\pi$
$98$	−34.3529	−3.47016
$99$	6.19765	0.622887
$100$	4.44030	0.444030
$101$	3.89735	0.387801	0.193901	−	0.981021i	$-0.437886\pi$
$101$	3.89735	0.387801	0.193901	+	0.981021i	$0.437886\pi$
$102$	−4.33314	−0.429044
$103$	−1.29740	−0.127836	−0.0639182	−	0.997955i	$-0.520360\pi$
$103$	−1.29740	−0.127836	−0.0639182	+	0.997955i	$0.520360\pi$
$104$	−9.74371	−0.955449
$105$	1.10316	0.107657
$106$	29.0892	2.82539
$107$	−11.6860	−1.12972	−0.564862	−	0.825185i	$-0.691071\pi$
$107$	−11.6860	−1.12972	−0.564862	+	0.825185i	$0.691071\pi$
$108$	−6.42136	−0.617896
$109$	18.4331	1.76558	0.882788	−	0.469772i	$-0.155664\pi$
$109$	18.4331	1.76558	0.882788	+	0.469772i	$0.155664\pi$
$110$	5.34839	0.509949
$111$	−1.32889	−0.126133
$112$	30.9773	2.92708
$113$	−0.307273	−0.0289058	−0.0144529	−	0.999896i	$-0.504601\pi$
$113$	−0.307273	−0.0289058	−0.0144529	+	0.999896i	$0.504601\pi$
$114$	4.04409	0.378764
$115$	3.43069	0.319913
$116$	−26.8924	−2.49690
$117$	−4.62685	−0.427752
$118$	−17.3923	−1.60109
$119$	31.7863	2.91384
$120$	−1.50754	−0.137619
$121$	−6.55839	−0.596218
$122$	37.3110	3.37798
$123$	0.579240	0.0522283
$124$	−26.2254	−2.35511
$125$	1.00000	0.0894427
$126$	33.8200	3.01293
$127$	−19.4688	−1.72758	−0.863790	−	0.503853i	$-0.831915\pi$
$127$	−19.4688	−1.72758	−0.863790	+	0.503853i	$0.831915\pi$
$128$	12.6642	1.11937
$129$	−2.93262	−0.258202
$130$	−3.99284	−0.350195
$131$	−5.65004	−0.493646	−0.246823	−	0.969061i	$-0.579387\pi$
$131$	−5.65004	−0.493646	−0.246823	+	0.969061i	$0.579387\pi$
$132$	−2.27801	−0.198276
$133$	−29.6659	−2.57236
$134$	−34.9794	−3.02176
$135$	−1.44616	−0.124465
$136$	−43.4381	−3.72479
$137$	−11.7065	−1.00016	−0.500078	−	0.865980i	$-0.666695\pi$
$137$	−11.7065	−1.00016	−0.500078	+	0.865980i	$0.666695\pi$
$138$	−2.11938	−0.180413
$139$	−15.1330	−1.28356	−0.641781	−	0.766888i	$-0.721804\pi$
$139$	−15.1330	−1.28356	−0.641781	+	0.766888i	$0.721804\pi$
$140$	20.1222	1.70064
$141$	2.85721	0.240621
$142$	10.4175	0.874221
$143$	−3.31588	−0.277288
$144$	−20.1019	−1.67516
$145$	−6.05645	−0.502961
$146$	31.3733	2.59647
$147$	3.29522	0.271785
$148$	−24.2398	−1.99250
$149$	−3.63410	−0.297717	−0.148859	−	0.988858i	$-0.547560\pi$
$149$	−3.63410	−0.297717	−0.148859	+	0.988858i	$0.547560\pi$
$150$	−0.617771	−0.0504408
$151$	6.77807	0.551592	0.275796	−	0.961216i	$-0.411059\pi$
$151$	6.77807	0.551592	0.275796	+	0.961216i	$0.411059\pi$
$152$	40.5405	3.28827
$153$	−20.6268	−1.66758
$154$	24.2375	1.95311
$155$	−5.90622	−0.474399
$156$	1.70065	0.136161
$157$	−6.82274	−0.544514	−0.272257	−	0.962225i	$-0.587770\pi$
$157$	−6.82274	−0.544514	−0.272257	+	0.962225i	$0.587770\pi$
$158$	−11.4956	−0.914544
$159$	−2.79031	−0.221286
$160$	−4.96148	−0.392239
$161$	15.5470	1.22527
$162$	−21.4954	−1.68884
$163$	−18.2895	−1.43254	−0.716272	−	0.697821i	$-0.754153\pi$
$163$	−18.2895	−1.43254	−0.716272	+	0.697821i	$0.754153\pi$
$164$	10.5657	0.825039
$165$	−0.513032	−0.0399395
$166$	14.0638	1.09156
$167$	−5.47331	−0.423538	−0.211769	−	0.977320i	$-0.567922\pi$
$167$	−5.47331	−0.423538	−0.211769	+	0.977320i	$0.567922\pi$
$168$	−6.83179	−0.527084
$169$	−10.5245	−0.809579
$170$	−17.8003	−1.36522
$171$	19.2509	1.47215
$172$	−53.4925	−4.07876
$173$	19.0323	1.44700	0.723499	−	0.690326i	$-0.242533\pi$
$173$	19.0323	1.44700	0.723499	+	0.690326i	$0.242533\pi$
$174$	3.74150	0.283642
$175$	4.53173	0.342567
$176$	−14.4062	−1.08591
$177$	1.66832	0.125399
$178$	29.6668	2.22362
$179$	6.12348	0.457690	0.228845	−	0.973463i	$-0.426505\pi$
$179$	6.12348	0.457690	0.228845	+	0.973463i	$0.426505\pi$
$180$	−13.0578	−0.973269
$181$	−13.4786	−1.00186	−0.500929	−	0.865488i	$-0.667008\pi$
$181$	−13.4786	−1.00186	−0.500929	+	0.865488i	$0.667008\pi$
$182$	−18.0945	−1.34125
$183$	−3.57897	−0.264565
$184$	−21.2460	−1.56627
$185$	−5.45904	−0.401356
$186$	3.64869	0.267535
$187$	−14.7824	−1.08100
$188$	52.1171	3.80103
$189$	−6.55359	−0.476704
$190$	16.6129	1.20523
$191$	−22.9772	−1.66257	−0.831284	−	0.555847i	$-0.812394\pi$
$191$	−22.9772	−1.66257	−0.831284	+	0.555847i	$0.812394\pi$
$192$	−0.262948	−0.0189766
$193$	−2.13458	−0.153651	−0.0768253	−	0.997045i	$-0.524478\pi$
$193$	−2.13458	−0.153651	−0.0768253	+	0.997045i	$0.524478\pi$
$194$	−44.7603	−3.21360
$195$	0.383004	0.0274275
$196$	60.1066	4.29333
$197$	19.2332	1.37031	0.685153	−	0.728399i	$-0.259735\pi$
$197$	19.2332	1.37031	0.685153	+	0.728399i	$0.259735\pi$
$198$	−15.7282	−1.11776
$199$	−10.0059	−0.709302	−0.354651	−	0.934999i	$-0.615400\pi$
$199$	−10.0059	−0.709302	−0.354651	+	0.934999i	$0.615400\pi$
$200$	−6.19292	−0.437906
$201$	3.35532	0.236666
$202$	−9.89061	−0.695901
$203$	−27.4462	−1.92635
$204$	7.58161	0.530819
$205$	2.37949	0.166191
$206$	3.29250	0.229400
$207$	−10.0888	−0.701217
$208$	10.7549	0.745721
$209$	13.7963	0.954312
$210$	−2.79957	−0.193189
$211$	−16.5709	−1.14079	−0.570394	−	0.821371i	$-0.693210\pi$
$211$	−16.5709	−1.14079	−0.570394	+	0.821371i	$0.693210\pi$
$212$	−50.8968	−3.49561
$213$	−0.999279	−0.0684695
$214$	29.6563	2.02727
$215$	−12.0471	−0.821602
$216$	8.95593	0.609374
$217$	−26.7654	−1.81695
$218$	−46.7792	−3.16828
$219$	−3.00941	−0.203357
$220$	−9.35798	−0.630915
$221$	11.0358	0.742348
$222$	3.37243	0.226343
$223$	3.14949	0.210905	0.105453	−	0.994424i	$-0.466371\pi$
$223$	3.14949	0.210905	0.105453	+	0.994424i	$0.466371\pi$
$224$	−22.4841	−1.50228
$225$	−2.94074	−0.196049
$226$	0.779790	0.0518708
$227$	12.5635	0.833871	0.416935	−	0.908936i	$-0.363104\pi$
$227$	12.5635	0.833871	0.416935	+	0.908936i	$0.363104\pi$
$228$	−7.07587	−0.468611
$229$	−12.9662	−0.856832	−0.428416	−	0.903582i	$-0.640928\pi$
$229$	−12.9662	−0.856832	−0.428416	+	0.903582i	$0.640928\pi$
$230$	−8.70631	−0.574077
$231$	−2.32492	−0.152969
$232$	37.5071	2.46246
$233$	−12.2781	−0.804364	−0.402182	−	0.915560i	$-0.631748\pi$
$233$	−12.2781	−0.804364	−0.402182	+	0.915560i	$0.631748\pi$
$234$	11.7419	0.767592
$235$	11.7373	0.765657
$236$	30.4311	1.98089
$237$	1.10269	0.0716276
$238$	−80.6663	−5.22882
$239$	14.2689	0.922980	0.461490	−	0.887145i	$-0.347315\pi$
$239$	14.2689	0.922980	0.461490	+	0.887145i	$0.347315\pi$
$240$	1.66400	0.107411
$241$	3.76167	0.242310	0.121155	−	0.992634i	$-0.461340\pi$
$241$	3.76167	0.242310	0.121155	+	0.992634i	$0.461340\pi$
$242$	16.6437	1.06990
$243$	6.40036	0.410584
$244$	−65.2823	−4.17927
$245$	13.5366	0.864822
$246$	−1.46998	−0.0937225
$247$	−10.2996	−0.655350
$248$	36.5768	2.32263
$249$	−1.34904	−0.0854919
$250$	−2.53777	−0.160503
$251$	5.02930	0.317446	0.158723	−	0.987323i	$-0.449262\pi$
$251$	5.02930	0.317446	0.158723	+	0.987323i	$0.449262\pi$
$252$	−59.1743	−3.72763
$253$	−7.23021	−0.454560
$254$	49.4075	3.10010
$255$	1.70746	0.106925
$256$	−29.9785	−1.87366
$257$	−7.49004	−0.467216	−0.233608	−	0.972331i	$-0.575053\pi$
$257$	−7.49004	−0.467216	−0.233608	+	0.972331i	$0.575053\pi$
$258$	7.44231	0.463338
$259$	−24.7389	−1.53720
$260$	6.98619	0.433265
$261$	17.8105	1.10244
$262$	14.3385	0.885836
$263$	9.65374	0.595275	0.297638	−	0.954679i	$-0.403801\pi$
$263$	9.65374	0.595275	0.297638	+	0.954679i	$0.403801\pi$
$264$	3.17717	0.195541
$265$	−11.4625	−0.704134
$266$	75.2854	4.61605
$267$	−2.84571	−0.174155
$268$	61.2029	3.73856
$269$	12.2921	0.749460	0.374730	−	0.927134i	$-0.377735\pi$
$269$	12.2921	0.749460	0.374730	+	0.927134i	$0.377735\pi$
$270$	3.67002	0.223350
$271$	−8.44351	−0.512907	−0.256453	−	0.966557i	$-0.582554\pi$
$271$	−8.44351	−0.512907	−0.256453	+	0.966557i	$0.582554\pi$
$272$	47.9462	2.90717
$273$	1.73567	0.105048
$274$	29.7085	1.79476
$275$	−2.10751	−0.127088
$276$	3.70824	0.223210
$277$	−8.70483	−0.523023	−0.261511	−	0.965200i	$-0.584221\pi$
$277$	−8.70483	−0.523023	−0.261511	+	0.965200i	$0.584221\pi$
$278$	38.4040	2.30332
$279$	17.3687	1.03983
$280$	−28.0647	−1.67719
$281$	−28.1285	−1.67800	−0.839002	−	0.544129i	$-0.816860\pi$
$281$	−28.1285	−1.67800	−0.839002	+	0.544129i	$0.816860\pi$
$282$	−7.25096	−0.431788
$283$	−10.3924	−0.617767	−0.308883	−	0.951100i	$-0.599955\pi$
$283$	−10.3924	−0.617767	−0.308883	+	0.951100i	$0.599955\pi$
$284$	−18.2274	−1.08160
$285$	−1.59356	−0.0943942
$286$	8.41495	0.497586
$287$	10.7832	0.636514
$288$	14.5904	0.859749
$289$	32.1983	1.89402
$290$	15.3699	0.902552
$291$	4.29353	0.251691
$292$	−54.8933	−3.21239
$293$	28.2203	1.64865	0.824324	−	0.566119i	$-0.191556\pi$
$293$	28.2203	1.64865	0.824324	+	0.566119i	$0.191556\pi$
$294$	−8.36252	−0.487712
$295$	6.85338	0.399019
$296$	33.8074	1.96502
$297$	3.04779	0.176851
$298$	9.22253	0.534247
$299$	5.39771	0.312158
$300$	1.08090	0.0624059
$301$	−54.5940	−3.14675
$302$	−17.2012	−0.989819
$303$	0.948733	0.0545033
$304$	−44.7480	−2.56647
$305$	−14.7022	−0.841848
$306$	52.3462	2.99243
$307$	−7.73474	−0.441445	−0.220722	−	0.975337i	$-0.570841\pi$
$307$	−7.73474	−0.441445	−0.220722	+	0.975337i	$0.570841\pi$
$308$	−42.4079	−2.41641
$309$	−0.315826	−0.0179667
$310$	14.9886	0.851298
$311$	20.3841	1.15588	0.577939	−	0.816080i	$-0.303857\pi$
$311$	20.3841	1.15588	0.577939	+	0.816080i	$0.303857\pi$
$312$	−2.37191	−0.134283
$313$	25.8927	1.46354	0.731770	−	0.681551i	$-0.238694\pi$
$313$	25.8927	1.46354	0.731770	+	0.681551i	$0.238694\pi$
$314$	17.3146	0.977117
$315$	−13.3267	−0.750872
$316$	20.1137	1.13148
$317$	−15.2345	−0.855658	−0.427829	−	0.903860i	$-0.640721\pi$
$317$	−15.2345	−0.855658	−0.427829	+	0.903860i	$0.640721\pi$
$318$	7.08118	0.397093
$319$	12.7640	0.714649
$320$	−1.08018	−0.0603838
$321$	−2.84472	−0.158777
$322$	−39.4547	−2.19872
$323$	−45.9165	−2.55486
$324$	37.6102	2.08945
$325$	1.57336	0.0872744
$326$	46.4146	2.57067
$327$	4.48718	0.248142
$328$	−14.7360	−0.813660
$329$	53.1903	2.93248
$330$	1.30196	0.0716705
$331$	−2.28942	−0.125838	−0.0629190	−	0.998019i	$-0.520041\pi$
$331$	−2.28942	−0.125838	−0.0629190	+	0.998019i	$0.520041\pi$
$332$	−24.6072	−1.35050
$333$	16.0536	0.879733
$334$	13.8900	0.760029
$335$	13.7835	0.753073
$336$	7.54081	0.411385
$337$	−2.73958	−0.149234	−0.0746172	−	0.997212i	$-0.523773\pi$
$337$	−2.73958	−0.149234	−0.0746172	+	0.997212i	$0.523773\pi$
$338$	26.7089	1.45277
$339$	−0.0747995	−0.00406255
$340$	31.1449	1.68907
$341$	12.4474	0.674066
$342$	−48.8544	−2.64174
$343$	29.6222	1.59945
$344$	74.6065	4.02251
$345$	0.835132	0.0449620
$346$	−48.2996	−2.59660
$347$	−12.0005	−0.644220	−0.322110	−	0.946702i	$-0.604392\pi$
$347$	−12.0005	−0.644220	−0.322110	+	0.946702i	$0.604392\pi$
$348$	−6.54643	−0.350925
$349$	20.0804	1.07488	0.537439	−	0.843302i	$-0.319392\pi$
$349$	20.0804	1.07488	0.537439	+	0.843302i	$0.319392\pi$
$350$	−11.5005	−0.614728
$351$	−2.27533	−0.121448
$352$	10.4564	0.557327
$353$	23.0003	1.22418	0.612091	−	0.790788i	$-0.290329\pi$
$353$	23.0003	1.22418	0.612091	+	0.790788i	$0.290329\pi$
$354$	−4.23382	−0.225025
$355$	−4.10499	−0.217870
$356$	−51.9074	−2.75109
$357$	7.73773	0.409524
$358$	−15.5400	−0.821315
$359$	1.85108	0.0976961	0.0488481	−	0.998806i	$-0.484445\pi$
$359$	1.85108	0.0976961	0.0488481	+	0.998806i	$0.484445\pi$
$360$	18.2118	0.959846
$361$	23.8536	1.25545
$362$	34.2057	1.79781
$363$	−1.59651	−0.0837950
$364$	31.6596	1.65941
$365$	−12.3625	−0.647084
$366$	9.08261	0.474756
$367$	−11.7661	−0.614187	−0.307094	−	0.951679i	$-0.599357\pi$
$367$	−11.7661	−0.614187	−0.307094	+	0.951679i	$0.599357\pi$
$368$	23.4510	1.22247
$369$	−6.99747	−0.364274
$370$	13.8538	0.720225
$371$	−51.9449	−2.69684
$372$	−6.38404	−0.330997
$373$	28.6090	1.48132	0.740658	−	0.671883i	$-0.234514\pi$
$373$	28.6090	1.48132	0.740658	+	0.671883i	$0.234514\pi$
$374$	37.5144	1.93982
$375$	0.243430	0.0125707
$376$	−72.6882	−3.74861
$377$	−9.52899	−0.490768
$378$	16.6315	0.855434
$379$	3.25425	0.167160	0.0835799	−	0.996501i	$-0.473365\pi$
$379$	3.25425	0.167160	0.0835799	+	0.996501i	$0.473365\pi$
$380$	−29.0674	−1.49112
$381$	−4.73930	−0.242802
$382$	58.3108	2.98344
$383$	21.4648	1.09680	0.548400	−	0.836216i	$-0.315237\pi$
$383$	21.4648	1.09680	0.548400	+	0.836216i	$0.315237\pi$
$384$	3.08285	0.157321
$385$	−9.55068	−0.486748
$386$	5.41709	0.275723
$387$	35.4273	1.80087
$388$	78.3164	3.97591
$389$	−7.44984	−0.377722	−0.188861	−	0.982004i	$-0.560480\pi$
$389$	−7.44984	−0.377722	−0.188861	+	0.982004i	$0.560480\pi$
$390$	−0.971976	−0.0492179
$391$	24.0633	1.21694
$392$	−83.8312	−4.23411
$393$	−1.37539	−0.0693792
$394$	−48.8094	−2.45898
$395$	4.52981	0.227920
$396$	27.5194	1.38290
$397$	−19.2438	−0.965820	−0.482910	−	0.875670i	$-0.660420\pi$
$397$	−19.2438	−0.965820	−0.482910	+	0.875670i	$0.660420\pi$
$398$	25.3928	1.27283
$399$	−7.22158	−0.361531
$400$	6.83565	0.341782
$401$	13.8370	0.690989	0.345494	−	0.938421i	$-0.387711\pi$
$401$	13.8370	0.690989	0.345494	+	0.938421i	$0.387711\pi$
$402$	−8.51504	−0.424692
$403$	−9.29262	−0.462898
$404$	17.3054	0.860976
$405$	8.47019	0.420887
$406$	69.6523	3.45679
$407$	11.5050	0.570281
$408$	−10.5741	−0.523498
$409$	−20.6761	−1.02237	−0.511183	−	0.859472i	$-0.670793\pi$
$409$	−20.6761	−1.02237	−0.511183	+	0.859472i	$0.670793\pi$
$410$	−6.03861	−0.298226
$411$	−2.84972	−0.140566
$412$	−5.76084	−0.283816
$413$	31.0577	1.52825
$414$	25.6030	1.25832
$415$	−5.54179	−0.272036
$416$	−7.80620	−0.382731
$417$	−3.68382	−0.180397
$418$	−35.0120	−1.71249
$419$	−10.4902	−0.512478	−0.256239	−	0.966613i	$-0.582483\pi$
$419$	−10.4902	−0.512478	−0.256239	+	0.966613i	$0.582483\pi$
$420$	4.89836	0.239015
$421$	−1.82595	−0.0889916	−0.0444958	−	0.999010i	$-0.514168\pi$
$421$	−1.82595	−0.0889916	−0.0444958	+	0.999010i	$0.514168\pi$
$422$	42.0532	2.04712
$423$	−34.5164	−1.67824
$424$	70.9863	3.44740
$425$	7.01415	0.340236
$426$	2.53594	0.122867
$427$	−66.6266	−3.22429
$428$	−51.8892	−2.50816
$429$	−0.807185	−0.0389712
$430$	30.5727	1.47435
$431$	16.6515	0.802073	0.401036	−	0.916062i	$-0.368650\pi$
$431$	16.6515	0.802073	0.401036	+	0.916062i	$0.368650\pi$
$432$	−9.88541	−0.475612
$433$	−18.7032	−0.898817	−0.449408	−	0.893326i	$-0.648365\pi$
$433$	−18.7032	−0.898817	−0.449408	+	0.893326i	$0.648365\pi$
$434$	67.9246	3.26048
$435$	−1.47432	−0.0706883
$436$	81.8487	3.91984
$437$	−22.4582	−1.07432
$438$	7.63721	0.364920
$439$	5.20716	0.248524	0.124262	−	0.992249i	$-0.460344\pi$
$439$	5.20716	0.248524	0.124262	+	0.992249i	$0.460344\pi$
$440$	13.0517	0.622214
$441$	−39.8077	−1.89560
$442$	−28.0064	−1.33213
$443$	17.2863	0.821295	0.410648	−	0.911794i	$-0.365303\pi$
$443$	17.2863	0.821295	0.410648	+	0.911794i	$0.365303\pi$
$444$	−5.90069	−0.280034
$445$	−11.6901	−0.554162
$446$	−7.99269	−0.378465
$447$	−0.884650	−0.0418425
$448$	−4.89508	−0.231271
$449$	−29.8918	−1.41068	−0.705340	−	0.708870i	$-0.749205\pi$
$449$	−29.8918	−1.41068	−0.705340	+	0.708870i	$0.749205\pi$
$450$	7.46294	0.351806
$451$	−5.01481	−0.236138
$452$	−1.36438	−0.0641752
$453$	1.64999	0.0775231
$454$	−31.8834	−1.49636
$455$	7.13006	0.334262
$456$	9.86878	0.462148
$457$	10.5552	0.493753	0.246876	−	0.969047i	$-0.420596\pi$
$457$	10.5552	0.493753	0.246876	+	0.969047i	$0.420596\pi$
$458$	32.9053	1.53757
$459$	−10.1435	−0.473460
$460$	15.2333	0.710255
$461$	−27.9466	−1.30160	−0.650801	−	0.759248i	$-0.725567\pi$
$461$	−27.9466	−1.30160	−0.650801	+	0.759248i	$0.725567\pi$
$462$	5.90013	0.274499
$463$	19.0164	0.883765	0.441882	−	0.897073i	$-0.354311\pi$
$463$	19.0164	0.883765	0.441882	+	0.897073i	$0.354311\pi$
$464$	−41.3998	−1.92194
$465$	−1.43775	−0.0666741
$466$	31.1590	1.44341
$467$	8.01788	0.371023	0.185512	−	0.982642i	$-0.440606\pi$
$467$	8.01788	0.371023	0.185512	+	0.982642i	$0.440606\pi$
$468$	−20.5446	−0.949674
$469$	62.4632	2.88428
$470$	−29.7866	−1.37395
$471$	−1.66086	−0.0765284
$472$	−42.4425	−1.95357
$473$	25.3893	1.16740
$474$	−2.79838	−0.128534
$475$	−6.54626	−0.300363
$476$	141.140	6.46916
$477$	33.7082	1.54339
$478$	−36.2113	−1.65627
$479$	−22.0584	−1.00787	−0.503937	−	0.863741i	$-0.668116\pi$
$479$	−22.0584	−1.00787	−0.503937	+	0.863741i	$0.668116\pi$
$480$	−1.20777	−0.0551271
$481$	−8.58904	−0.391627
$482$	−9.54627	−0.434820
$483$	3.78460	0.172205
$484$	−29.1212	−1.32369
$485$	17.6376	0.800884
$486$	−16.2427	−0.736783
$487$	7.70381	0.349093	0.174546	−	0.984649i	$-0.444154\pi$
$487$	7.70381	0.349093	0.174546	+	0.984649i	$0.444154\pi$
$488$	91.0498	4.12163
$489$	−4.45221	−0.201336
$490$	−34.3529	−1.55190
$491$	32.4283	1.46347	0.731734	−	0.681590i	$-0.238711\pi$
$491$	32.4283	1.46347	0.731734	+	0.681590i	$0.238711\pi$
$492$	2.57200	0.115955
$493$	−42.4809	−1.91324
$494$	26.1382	1.17601
$495$	6.19765	0.278564
$496$	−40.3728	−1.81279
$497$	−18.6027	−0.834447
$498$	3.42356	0.153413
$499$	−6.95081	−0.311161	−0.155580	−	0.987823i	$-0.549725\pi$
$499$	−6.95081	−0.311161	−0.155580	+	0.987823i	$0.549725\pi$
$500$	4.44030	0.198576
$501$	−1.33237	−0.0595258
$502$	−12.7632	−0.569650
$503$	−27.2394	−1.21455	−0.607273	−	0.794493i	$-0.707737\pi$
$503$	−27.2394	−1.21455	−0.607273	+	0.794493i	$0.707737\pi$
$504$	82.5310	3.67622
$505$	3.89735	0.173430
$506$	18.3486	0.815697
$507$	−2.56199	−0.113782
$508$	−86.4474	−3.83548
$509$	−26.3693	−1.16880	−0.584398	−	0.811467i	$-0.698669\pi$
$509$	−26.3693	−1.16880	−0.584398	+	0.811467i	$0.698669\pi$
$510$	−4.33314	−0.191874
$511$	−56.0237	−2.47834
$512$	50.7504	2.24287
$513$	9.46691	0.417975
$514$	19.0080	0.838408
$515$	−1.29740	−0.0571702
$516$	−13.0217	−0.573248
$517$	−24.7365	−1.08791
$518$	62.7818	2.75847
$519$	4.63303	0.203367
$520$	−9.74371	−0.427290
$521$	6.63707	0.290775	0.145388	−	0.989375i	$-0.453557\pi$
$521$	6.63707	0.290775	0.145388	+	0.989375i	$0.453557\pi$
$522$	−45.1989	−1.97830
$523$	13.5956	0.594496	0.297248	−	0.954800i	$-0.403931\pi$
$523$	13.5956	0.594496	0.297248	+	0.954800i	$0.403931\pi$
$524$	−25.0878	−1.09597
$525$	1.10316	0.0481459
$526$	−24.4990	−1.06821
$527$	−41.4271	−1.80459
$528$	−3.50690	−0.152618
$529$	−11.2304	−0.488278
$530$	29.0892	1.26355
$531$	−20.1540	−0.874610
$532$	−131.726	−5.71103
$533$	3.74380	0.162162
$534$	7.22178	0.312517
$535$	−11.6860	−0.505228
$536$	−85.3602	−3.68700
$537$	1.49064	0.0643258
$538$	−31.1945	−1.34489
$539$	−28.5286	−1.22881
$540$	−6.42136	−0.276331
$541$	−25.7483	−1.10701	−0.553503	−	0.832847i	$-0.686709\pi$
$541$	−25.7483	−1.10701	−0.553503	+	0.832847i	$0.686709\pi$
$542$	21.4277	0.920400
$543$	−3.28110	−0.140806
$544$	−34.8006	−1.49206
$545$	18.4331	0.789589
$546$	−4.40474	−0.188505
$547$	4.70327	0.201097	0.100549	−	0.994932i	$-0.467940\pi$
$547$	4.70327	0.201097	0.100549	+	0.994932i	$0.467940\pi$
$548$	−51.9805	−2.22050
$549$	43.2355	1.84525
$550$	5.34839	0.228056
$551$	39.6471	1.68902
$552$	−5.17191	−0.220131
$553$	20.5279	0.872935
$554$	22.0909	0.938553
$555$	−1.32889	−0.0564084
$556$	−67.1949	−2.84970
$557$	−25.3733	−1.07510	−0.537550	−	0.843232i	$-0.680650\pi$
$557$	−25.3733	−1.07510	−0.537550	+	0.843232i	$0.680650\pi$
$558$	−44.0777	−1.86596
$559$	−18.9544	−0.801684
$560$	30.9773	1.30903
$561$	−3.59848	−0.151928
$562$	71.3837	3.01114
$563$	39.3332	1.65770	0.828848	−	0.559474i	$-0.188997\pi$
$563$	39.3332	1.65770	0.828848	+	0.559474i	$0.188997\pi$
$564$	12.6869	0.534214
$565$	−0.307273	−0.0129271
$566$	26.3737	1.10857
$567$	38.3846	1.61200
$568$	25.4219	1.06668
$569$	−9.16398	−0.384174	−0.192087	−	0.981378i	$-0.561526\pi$
$569$	−9.16398	−0.384174	−0.192087	+	0.981378i	$0.561526\pi$
$570$	4.04409	0.169388
$571$	9.08005	0.379988	0.189994	−	0.981785i	$-0.439153\pi$
$571$	9.08005	0.379988	0.189994	+	0.981785i	$0.439153\pi$
$572$	−14.7235	−0.615620
$573$	−5.59333	−0.233665
$574$	−27.3654	−1.14221
$575$	3.43069	0.143070
$576$	3.17653	0.132355
$577$	18.3553	0.764141	0.382071	−	0.924133i	$-0.375211\pi$
$577$	18.3553	0.764141	0.382071	+	0.924133i	$0.375211\pi$
$578$	−81.7120	−3.39877
$579$	−0.519622	−0.0215947
$580$	−26.8924	−1.11665
$581$	−25.1139	−1.04190
$582$	−10.8960	−0.451654
$583$	24.1573	1.00049
$584$	76.5602	3.16808
$585$	−4.62685	−0.191297
$586$	−71.6168	−2.95846
$587$	−30.5945	−1.26277	−0.631386	−	0.775469i	$-0.717514\pi$
$587$	−30.5945	−1.26277	−0.631386	+	0.775469i	$0.717514\pi$
$588$	14.6317	0.603403
$589$	38.6637	1.59311
$590$	−17.3923	−0.716031
$591$	4.68193	0.192589
$592$	−37.3161	−1.53368
$593$	11.5867	0.475808	0.237904	−	0.971289i	$-0.423540\pi$
$593$	11.5867	0.475808	0.237904	+	0.971289i	$0.423540\pi$
$594$	−7.73460	−0.317355
$595$	31.7863	1.30311
$596$	−16.1365	−0.660977
$597$	−2.43575	−0.0996885
$598$	−13.6982	−0.560160
$599$	28.9328	1.18216	0.591082	−	0.806612i	$-0.298701\pi$
$599$	28.9328	1.18216	0.591082	+	0.806612i	$0.298701\pi$
$600$	−1.50754	−0.0615452
$601$	35.0249	1.42869	0.714347	−	0.699792i	$-0.246724\pi$
$601$	35.0249	1.42869	0.714347	+	0.699792i	$0.246724\pi$
$602$	138.547	5.64677
$603$	−40.5337	−1.65066
$604$	30.0967	1.22462
$605$	−6.55839	−0.266637
$606$	−2.40767	−0.0978049
$607$	12.8327	0.520862	0.260431	−	0.965492i	$-0.416135\pi$
$607$	12.8327	0.520862	0.260431	+	0.965492i	$0.416135\pi$
$608$	32.4792	1.31720
$609$	−6.68124	−0.270737
$610$	37.3110	1.51068
$611$	18.4670	0.747096
$612$	−91.5891	−3.70227
$613$	1.16448	0.0470331	0.0235165	−	0.999723i	$-0.492514\pi$
$613$	1.16448	0.0470331	0.0235165	+	0.999723i	$0.492514\pi$
$614$	19.6290	0.792163
$615$	0.579240	0.0233572
$616$	59.1467	2.38309
$617$	42.0688	1.69362	0.846812	−	0.531892i	$-0.178519\pi$
$617$	42.0688	1.69362	0.846812	+	0.531892i	$0.178519\pi$
$618$	0.801495	0.0322408
$619$	11.4203	0.459022	0.229511	−	0.973306i	$-0.426287\pi$
$619$	11.4203	0.459022	0.229511	+	0.973306i	$0.426287\pi$
$620$	−26.2254	−1.05324
$621$	−4.96131	−0.199090
$622$	−51.7303	−2.07420
$623$	−52.9763	−2.12245
$624$	2.61808	0.104807
$625$	1.00000	0.0400000
$626$	−65.7098	−2.62629
$627$	3.35844	0.134123
$628$	−30.2950	−1.20890
$629$	−38.2905	−1.52674
$630$	33.8200	1.34742
$631$	−31.6591	−1.26033	−0.630165	−	0.776462i	$-0.717013\pi$
$631$	−31.6591	−1.26033	−0.630165	+	0.776462i	$0.717013\pi$
$632$	−28.0528	−1.11588
$633$	−4.03386	−0.160331
$634$	38.6618	1.53546
$635$	−19.4688	−0.772597
$636$	−12.3898	−0.491288
$637$	21.2980	0.843857
$638$	−32.3923	−1.28242
$639$	12.0717	0.477550
$640$	12.6642	0.500597
$641$	27.0240	1.06738	0.533691	−	0.845679i	$-0.320804\pi$
$641$	27.0240	1.06738	0.533691	+	0.845679i	$0.320804\pi$
$642$	7.21925	0.284921
$643$	−27.4912	−1.08415	−0.542073	−	0.840331i	$-0.682360\pi$
$643$	−27.4912	−1.08415	−0.542073	+	0.840331i	$0.682360\pi$
$644$	69.0331	2.72029
$645$	−2.93262	−0.115472
$646$	116.526	4.58464
$647$	29.6746	1.16663	0.583315	−	0.812246i	$-0.301755\pi$
$647$	29.6746	1.16663	0.583315	+	0.812246i	$0.301755\pi$
$648$	−52.4552	−2.06064
$649$	−14.4436	−0.566960
$650$	−3.99284	−0.156612
$651$	−6.51551	−0.255363
$652$	−81.2108	−3.18046
$653$	−49.6395	−1.94254	−0.971272	−	0.237970i	$-0.923518\pi$
$653$	−49.6395	−1.94254	−0.971272	+	0.237970i	$0.923518\pi$
$654$	−11.3875	−0.445285
$655$	−5.65004	−0.220765
$656$	16.2654	0.635056
$657$	36.3550	1.41834
$658$	−134.985	−5.26226
$659$	39.7694	1.54920	0.774598	−	0.632454i	$-0.217952\pi$
$659$	39.7694	1.54920	0.774598	+	0.632454i	$0.217952\pi$
$660$	−2.27801	−0.0886716
$661$	45.1617	1.75659	0.878293	−	0.478123i	$-0.158683\pi$
$661$	45.1617	1.75659	0.878293	+	0.478123i	$0.158683\pi$
$662$	5.81003	0.225813
$663$	2.68644	0.104333
$664$	34.3199	1.33187
$665$	−29.6659	−1.15040
$666$	−40.7405	−1.57866
$667$	−20.7778	−0.804519
$668$	−24.3031	−0.940316
$669$	0.766680	0.0296416
$670$	−34.9794	−1.35137
$671$	30.9851	1.19617
$672$	−5.47331	−0.211137
$673$	5.95736	0.229639	0.114820	−	0.993386i	$-0.463371\pi$
$673$	5.95736	0.229639	0.114820	+	0.993386i	$0.463371\pi$
$674$	6.95243	0.267798
$675$	−1.44616	−0.0556625
$676$	−46.7321	−1.79739
$677$	−16.1433	−0.620436	−0.310218	−	0.950665i	$-0.600402\pi$
$677$	−16.1433	−0.620436	−0.310218	+	0.950665i	$0.600402\pi$
$678$	0.189824	0.00729016
$679$	79.9291	3.06740
$680$	−43.4381	−1.66577
$681$	3.05834	0.117196
$682$	−31.5888	−1.20960
$683$	−14.1984	−0.543287	−0.271644	−	0.962398i	$-0.587567\pi$
$683$	−14.1984	−0.543287	−0.271644	+	0.962398i	$0.587567\pi$
$684$	85.4796	3.26839
$685$	−11.7065	−0.447284
$686$	−75.1744	−2.87017
$687$	−3.15637	−0.120423
$688$	−82.3494	−3.13954
$689$	−18.0346	−0.687064
$690$	−2.11938	−0.0806833
$691$	−20.2162	−0.769061	−0.384530	−	0.923112i	$-0.625637\pi$
$691$	−20.2162	−0.769061	−0.384530	+	0.923112i	$0.625637\pi$
$692$	84.5090	3.21255
$693$	28.0861	1.06690
$694$	30.4545	1.15604
$695$	−15.1330	−0.574026
$696$	9.13037	0.346086
$697$	16.6901	0.632183
$698$	−50.9595	−1.92885
$699$	−2.98885	−0.113049
$700$	20.1222	0.760549
$701$	−40.4531	−1.52789	−0.763946	−	0.645280i	$-0.776741\pi$
$701$	−40.4531	−1.52789	−0.763946	+	0.645280i	$0.776741\pi$
$702$	5.77426	0.217935
$703$	35.7363	1.34782
$704$	2.27649	0.0857984
$705$	2.85721	0.107609
$706$	−58.3695	−2.19677
$707$	17.6618	0.664239
$708$	7.40784	0.278404
$709$	16.5764	0.622542	0.311271	−	0.950321i	$-0.399245\pi$
$709$	16.5764	0.622542	0.311271	+	0.950321i	$0.399245\pi$
$710$	10.4175	0.390964
$711$	−13.3210	−0.499577
$712$	72.3957	2.71314
$713$	−20.2624	−0.758832
$714$	−19.6366	−0.734882
$715$	−3.31588	−0.124007
$716$	27.1901	1.01614
$717$	3.47349	0.129720
$718$	−4.69762	−0.175313
$719$	1.69404	0.0631772	0.0315886	−	0.999501i	$-0.489943\pi$
$719$	1.69404	0.0631772	0.0315886	+	0.999501i	$0.489943\pi$
$720$	−20.1019	−0.749153
$721$	−5.87947	−0.218963
$722$	−60.5350	−2.25288
$723$	0.915703	0.0340554
$724$	−59.8491	−2.22427
$725$	−6.05645	−0.224931
$726$	4.05158	0.150368
$727$	8.05052	0.298577	0.149289	−	0.988794i	$-0.452302\pi$
$727$	8.05052	0.298577	0.149289	+	0.988794i	$0.452302\pi$
$728$	−44.1559	−1.63653
$729$	−23.8525	−0.883427
$730$	31.3733	1.16118
$731$	−84.4998	−3.12534
$732$	−15.8917	−0.587373
$733$	16.9679	0.626723	0.313362	−	0.949634i	$-0.398545\pi$
$733$	16.9679	0.626723	0.313362	+	0.949634i	$0.398545\pi$
$734$	29.8598	1.10215
$735$	3.29522	0.121546
$736$	−17.0213	−0.627413
$737$	−29.0489	−1.07003
$738$	17.7580	0.653681
$739$	−44.1580	−1.62438	−0.812190	−	0.583393i	$-0.801725\pi$
$739$	−44.1580	−1.62438	−0.812190	+	0.583393i	$0.801725\pi$
$740$	−24.2398	−0.891071
$741$	−2.50724	−0.0921059
$742$	131.824	4.83943
$743$	−35.9610	−1.31928	−0.659640	−	0.751582i	$-0.729291\pi$
$743$	−35.9610	−1.31928	−0.659640	+	0.751582i	$0.729291\pi$
$744$	8.90388	0.326432
$745$	−3.63410	−0.133143
$746$	−72.6031	−2.65819
$747$	16.2970	0.596275
$748$	−65.6383	−2.39997
$749$	−52.9577	−1.93503
$750$	−0.617771	−0.0225578
$751$	14.9146	0.544241	0.272121	−	0.962263i	$-0.412275\pi$
$751$	14.9146	0.544241	0.272121	+	0.962263i	$0.412275\pi$
$752$	80.2320	2.92576
$753$	1.22428	0.0446153
$754$	24.1824	0.880672
$755$	6.77807	0.246679
$756$	−29.0999	−1.05835
$757$	13.1396	0.477566	0.238783	−	0.971073i	$-0.423252\pi$
$757$	13.1396	0.477566	0.238783	+	0.971073i	$0.423252\pi$
$758$	−8.25856	−0.299964
$759$	−1.76005	−0.0638858
$760$	40.5405	1.47056
$761$	−0.714204	−0.0258899	−0.0129449	−	0.999916i	$-0.504121\pi$
$761$	−0.714204	−0.0258899	−0.0129449	+	0.999916i	$0.504121\pi$
$762$	12.0273	0.435702
$763$	83.5341	3.02414
$764$	−102.025	−3.69115
$765$	−20.6268	−0.745764
$766$	−54.4728	−1.96818
$767$	10.7829	0.389346
$768$	−7.29768	−0.263332
$769$	16.2344	0.585429	0.292714	−	0.956200i	$-0.405442\pi$
$769$	16.2344	0.585429	0.292714	+	0.956200i	$0.405442\pi$
$770$	24.2375	0.873458
$771$	−1.82330	−0.0656646
$772$	−9.47819	−0.341127
$773$	15.7925	0.568016	0.284008	−	0.958822i	$-0.408336\pi$
$773$	15.7925	0.568016	0.284008	+	0.958822i	$0.408336\pi$
$774$	−89.9064	−3.23162
$775$	−5.90622	−0.212158
$776$	−109.229	−3.92108
$777$	−6.02219	−0.216045
$778$	18.9060	0.677813
$779$	−15.5768	−0.558096
$780$	1.70065	0.0608930
$781$	8.65132	0.309569
$782$	−61.0673	−2.18376
$783$	8.75857	0.313006
$784$	92.5315	3.30470
$785$	−6.82274	−0.243514
$786$	3.49043	0.124499
$787$	−18.9231	−0.674536	−0.337268	−	0.941409i	$-0.609503\pi$
$787$	−18.9231	−0.674536	−0.337268	+	0.941409i	$0.609503\pi$
$788$	85.4010	3.04228
$789$	2.35001	0.0836626
$790$	−11.4956	−0.408997
$791$	−1.39248	−0.0495109
$792$	−38.3816	−1.36383
$793$	−23.1319	−0.821439
$794$	48.8365	1.73314
$795$	−2.79031	−0.0989622
$796$	−44.4293	−1.57476
$797$	−32.0456	−1.13511	−0.567557	−	0.823334i	$-0.692112\pi$
$797$	−32.0456	−1.13511	−0.567557	+	0.823334i	$0.692112\pi$
$798$	18.3267	0.648759
$799$	82.3272	2.91253
$800$	−4.96148	−0.175415
$801$	34.3775	1.21467
$802$	−35.1153	−1.23996
$803$	26.0542	0.919432
$804$	14.8986	0.525434
$805$	15.5470	0.547958
$806$	23.5826	0.830660
$807$	2.99226	0.105332
$808$	−24.1360	−0.849102
$809$	−18.8622	−0.663159	−0.331579	−	0.943427i	$-0.607581\pi$
$809$	−18.8622	−0.663159	−0.331579	+	0.943427i	$0.607581\pi$
$810$	−21.4954	−0.755272
$811$	−8.53254	−0.299618	−0.149809	−	0.988715i	$-0.547866\pi$
$811$	−8.53254	−0.299618	−0.149809	+	0.988715i	$0.547866\pi$
$812$	−121.869	−4.27678
$813$	−2.05540	−0.0720862
$814$	−29.1971	−1.02336
$815$	−18.2895	−0.640653
$816$	11.6716	0.408586
$817$	78.8632	2.75907
$818$	52.4713	1.83461
$819$	−20.9677	−0.732669
$820$	10.5657	0.368969
$821$	1.76882	0.0617324	0.0308662	−	0.999524i	$-0.490173\pi$
$821$	1.76882	0.0617324	0.0308662	+	0.999524i	$0.490173\pi$
$822$	7.23195	0.252243
$823$	−19.7772	−0.689391	−0.344696	−	0.938714i	$-0.612018\pi$
$823$	−19.7772	−0.689391	−0.344696	+	0.938714i	$0.612018\pi$
$824$	8.03469	0.279902
$825$	−0.513032	−0.0178615
$826$	−78.8175	−2.74241
$827$	19.3136	0.671601	0.335801	−	0.941933i	$-0.390993\pi$
$827$	19.3136	0.671601	0.335801	+	0.941933i	$0.390993\pi$
$828$	−44.7971	−1.55681
$829$	11.1500	0.387255	0.193628	−	0.981075i	$-0.437975\pi$
$829$	11.1500	0.387255	0.193628	+	0.981075i	$0.437975\pi$
$830$	14.0638	0.488162
$831$	−2.11902	−0.0735079
$832$	−1.69951	−0.0589200
$833$	94.9478	3.28975
$834$	9.34870	0.323719
$835$	−5.47331	−0.189412
$836$	61.2598	2.11872
$837$	8.54131	0.295231
$838$	26.6217	0.919630
$839$	−0.948549	−0.0327476	−0.0163738	−	0.999866i	$-0.505212\pi$
$839$	−0.948549	−0.0327476	−0.0163738	+	0.999866i	$0.505212\pi$
$840$	−6.83179	−0.235719
$841$	7.68060	0.264848
$842$	4.63386	0.159693
$843$	−6.84731	−0.235834
$844$	−73.5798	−2.53272
$845$	−10.5245	−0.362055
$846$	87.5948	3.01157
$847$	−29.7209	−1.02122
$848$	−78.3535	−2.69067
$849$	−2.52983	−0.0868237
$850$	−17.8003	−0.610546
$851$	−18.7282	−0.641996
$852$	−4.43710	−0.152012
$853$	1.82411	0.0624564	0.0312282	−	0.999512i	$-0.490058\pi$
$853$	1.82411	0.0624564	0.0312282	+	0.999512i	$0.490058\pi$
$854$	169.083	5.78591
$855$	19.2509	0.658366
$856$	72.3703	2.47357
$857$	−3.94747	−0.134843	−0.0674216	−	0.997725i	$-0.521477\pi$
$857$	−3.94747	−0.134843	−0.0674216	+	0.997725i	$0.521477\pi$
$858$	2.04845	0.0699330
$859$	−42.7098	−1.45724	−0.728620	−	0.684919i	$-0.759838\pi$
$859$	−42.7098	−1.45724	−0.728620	+	0.684919i	$0.759838\pi$
$860$	−53.4925	−1.82408
$861$	2.62496	0.0894585
$862$	−42.2577	−1.43930
$863$	25.6451	0.872968	0.436484	−	0.899712i	$-0.356224\pi$
$863$	25.6451	0.872968	0.436484	+	0.899712i	$0.356224\pi$
$864$	7.17507	0.244101
$865$	19.0323	0.647117
$866$	47.4644	1.61291
$867$	7.83803	0.266194
$868$	−118.846	−4.03391
$869$	−9.54663	−0.323847
$870$	3.74150	0.126849
$871$	21.6864	0.734817
$872$	−114.155	−3.86578
$873$	−51.8677	−1.75546
$874$	56.9938	1.92784
$875$	4.53173	0.153201
$876$	−13.3627	−0.451483
$877$	−7.82249	−0.264147	−0.132073	−	0.991240i	$-0.542163\pi$
$877$	−7.82249	−0.264147	−0.132073	+	0.991240i	$0.542163\pi$
$878$	−13.2146	−0.445971
$879$	6.86967	0.231708
$880$	−14.4062	−0.485633
$881$	4.95075	0.166795	0.0833975	−	0.996516i	$-0.473423\pi$
$881$	4.95075	0.166795	0.0833975	+	0.996516i	$0.473423\pi$
$882$	101.023	3.40162
$883$	−44.3984	−1.49413	−0.747063	−	0.664754i	$-0.768536\pi$
$883$	−44.3984	−1.49413	−0.747063	+	0.664754i	$0.768536\pi$
$884$	49.0022	1.64812
$885$	1.66832	0.0560800
$886$	−43.8686	−1.47380
$887$	−51.5010	−1.72923	−0.864616	−	0.502433i	$-0.832439\pi$
$887$	−51.5010	−1.72923	−0.864616	+	0.502433i	$0.832439\pi$
$888$	8.22974	0.276172
$889$	−88.2275	−2.95906
$890$	29.6668	0.994432
$891$	−17.8510	−0.598032
$892$	13.9847	0.468241
$893$	−76.8355	−2.57120
$894$	2.24504	0.0750854
$895$	6.12348	0.204685
$896$	57.3908	1.91729
$897$	1.31397	0.0438720
$898$	75.8585	2.53143
$899$	35.7707	1.19302
$900$	−13.0578	−0.435259
$901$	−80.3995	−2.67850
$902$	12.7265	0.423745
$903$	−13.2898	−0.442258
$904$	1.90292	0.0632901
$905$	−13.4786	−0.448044
$906$	−4.18729	−0.139113
$907$	46.2551	1.53588	0.767938	−	0.640525i	$-0.221283\pi$
$907$	46.2551	1.53588	0.767938	+	0.640525i	$0.221283\pi$
$908$	55.7858	1.85132
$909$	−11.4611	−0.380141
$910$	−18.0945	−0.599826
$911$	17.5418	0.581187	0.290593	−	0.956847i	$-0.406147\pi$
$911$	17.5418	0.581187	0.290593	+	0.956847i	$0.406147\pi$
$912$	−10.8930	−0.360703
$913$	11.6794	0.386532
$914$	−26.7868	−0.886029
$915$	−3.57897	−0.118317
$916$	−57.5739	−1.90229
$917$	−25.6045	−0.845534
$918$	25.7420	0.849614
$919$	−34.4223	−1.13549	−0.567743	−	0.823206i	$-0.692183\pi$
$919$	−34.4223	−1.13549	−0.567743	+	0.823206i	$0.692183\pi$
$920$	−21.2460	−0.700459
$921$	−1.88287	−0.0620426
$922$	70.9222	2.33570
$923$	−6.45864	−0.212589
$924$	−10.3234	−0.339613
$925$	−5.45904	−0.179492
$926$	−48.2592	−1.58590
$927$	3.81531	0.125311
$928$	30.0490	0.986405
$929$	−28.1014	−0.921976	−0.460988	−	0.887406i	$-0.652505\pi$
$929$	−28.1014	−0.921976	−0.460988	+	0.887406i	$0.652505\pi$
$930$	3.64869	0.119645
$931$	−88.6142	−2.90421
$932$	−54.5183	−1.78581
$933$	4.96211	0.162452
$934$	−20.3476	−0.665793
$935$	−14.7824	−0.483436
$936$	28.6537	0.936576
$937$	16.8992	0.552071	0.276036	−	0.961147i	$-0.410979\pi$
$937$	16.8992	0.552071	0.276036	+	0.961147i	$0.410979\pi$
$938$	−158.517	−5.17578
$939$	6.30306	0.205692
$940$	52.1171	1.69987
$941$	−34.8074	−1.13469	−0.567344	−	0.823481i	$-0.692029\pi$
$941$	−34.8074	−1.13469	−0.567344	+	0.823481i	$0.692029\pi$
$942$	4.21489	0.137328
$943$	8.16329	0.265833
$944$	46.8473	1.52475
$945$	−6.55359	−0.213188
$946$	−64.4323	−2.09488
$947$	−46.8271	−1.52168	−0.760838	−	0.648941i	$-0.775212\pi$
$947$	−46.8271	−1.52168	−0.760838	+	0.648941i	$0.775212\pi$
$948$	4.89628	0.159024
$949$	−19.4507	−0.631397
$950$	16.6129	0.538995
$951$	−3.70855	−0.120258
$952$	−196.850	−6.37994
$953$	40.9053	1.32505	0.662526	−	0.749039i	$-0.269484\pi$
$953$	40.9053	1.32505	0.662526	+	0.749039i	$0.269484\pi$
$954$	−85.5438	−2.76958
$955$	−22.9772	−0.743523
$956$	63.3583	2.04915
$957$	3.10715	0.100440
$958$	55.9792	1.80861
$959$	−53.0509	−1.71310
$960$	−0.262948	−0.00848661
$961$	3.88341	0.125271
$962$	21.7970	0.702765
$963$	34.3654	1.10741
$964$	16.7029	0.537965
$965$	−2.13458	−0.0687147
$966$	−9.60445	−0.309018
$967$	8.84789	0.284529	0.142265	−	0.989829i	$-0.454562\pi$
$967$	8.84789	0.284529	0.142265	+	0.989829i	$0.454562\pi$
$968$	40.6156	1.30544
$969$	−11.1775	−0.359072
$970$	−44.7603	−1.43717
$971$	−20.2868	−0.651033	−0.325516	−	0.945536i	$-0.605538\pi$
$971$	−20.2868	−0.651033	−0.325516	+	0.945536i	$0.605538\pi$
$972$	28.4195	0.911557
$973$	−68.5786	−2.19853
$974$	−19.5505	−0.626439
$975$	0.383004	0.0122659
$976$	−100.499	−3.21690
$977$	−17.4847	−0.559387	−0.279693	−	0.960089i	$-0.590233\pi$
$977$	−17.4847	−0.559387	−0.279693	+	0.960089i	$0.590233\pi$
$978$	11.2987	0.361293
$979$	24.6370	0.787401
$980$	60.1066	1.92003
$981$	−54.2071	−1.73070
$982$	−82.2957	−2.62616
$983$	−47.6561	−1.51999	−0.759997	−	0.649926i	$-0.774800\pi$
$983$	−47.6561	−1.51999	−0.759997	+	0.649926i	$0.774800\pi$
$984$	−3.58719	−0.114355
$985$	19.2332	0.612819
$986$	107.807	3.43327
$987$	12.9481	0.412143
$988$	−45.7335	−1.45498
$989$	−41.3297	−1.31421
$990$	−15.7282	−0.499876
$991$	−46.3291	−1.47169	−0.735845	−	0.677150i	$-0.763215\pi$
$991$	−46.3291	−1.47169	−0.735845	+	0.677150i	$0.763215\pi$
$992$	29.3036	0.930390
$993$	−0.557314	−0.0176858
$994$	47.2095	1.49740
$995$	−10.0059	−0.317210
$996$	−5.99013	−0.189805
$997$	43.5331	1.37871	0.689353	−	0.724426i	$-0.257895\pi$
$997$	43.5331	1.37871	0.689353	+	0.724426i	$0.257895\pi$
$998$	17.6396	0.558371
$999$	7.89462	0.249775

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8045.2.a.c.1.9	✓	127

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8045.2.a.c.1.9	✓	127	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)

\(f(q)\)	\(=\)	\(q-2.53777 q^{2} +0.243430 q^{3} +4.44030 q^{4} +1.00000 q^{5} -0.617771 q^{6} +4.53173 q^{7} -6.19292 q^{8} -2.94074 q^{9} +O(q^{10})\)	\(q-2.53777 q^{2} +0.243430 q^{3} +4.44030 q^{4} +1.00000 q^{5} -0.617771 q^{6} +4.53173 q^{7} -6.19292 q^{8} -2.94074 q^{9} -2.53777 q^{10} -2.10751 q^{11} +1.08090 q^{12} +1.57336 q^{13} -11.5005 q^{14} +0.243430 q^{15} +6.83565 q^{16} +7.01415 q^{17} +7.46294 q^{18} -6.54626 q^{19} +4.44030 q^{20} +1.10316 q^{21} +5.34839 q^{22} +3.43069 q^{23} -1.50754 q^{24} +1.00000 q^{25} -3.99284 q^{26} -1.44616 q^{27} +20.1222 q^{28} -6.05645 q^{29} -0.617771 q^{30} -5.90622 q^{31} -4.96148 q^{32} -0.513032 q^{33} -17.8003 q^{34} +4.53173 q^{35} -13.0578 q^{36} -5.45904 q^{37} +16.6129 q^{38} +0.383004 q^{39} -6.19292 q^{40} +2.37949 q^{41} -2.79957 q^{42} -12.0471 q^{43} -9.35798 q^{44} -2.94074 q^{45} -8.70631 q^{46} +11.7373 q^{47} +1.66400 q^{48} +13.5366 q^{49} -2.53777 q^{50} +1.70746 q^{51} +6.98619 q^{52} -11.4625 q^{53} +3.67002 q^{54} -2.10751 q^{55} -28.0647 q^{56} -1.59356 q^{57} +15.3699 q^{58} +6.85338 q^{59} +1.08090 q^{60} -14.7022 q^{61} +14.9886 q^{62} -13.3267 q^{63} -1.08018 q^{64} +1.57336 q^{65} +1.30196 q^{66} +13.7835 q^{67} +31.1449 q^{68} +0.835132 q^{69} -11.5005 q^{70} -4.10499 q^{71} +18.2118 q^{72} -12.3625 q^{73} +13.8538 q^{74} +0.243430 q^{75} -29.0674 q^{76} -9.55068 q^{77} -0.971976 q^{78} +4.52981 q^{79} +6.83565 q^{80} +8.47019 q^{81} -6.03861 q^{82} -5.54179 q^{83} +4.89836 q^{84} +7.01415 q^{85} +30.5727 q^{86} -1.47432 q^{87} +13.0517 q^{88} -11.6901 q^{89} +7.46294 q^{90} +7.13006 q^{91} +15.2333 q^{92} -1.43775 q^{93} -29.7866 q^{94} -6.54626 q^{95} -1.20777 q^{96} +17.6376 q^{97} -34.3529 q^{98} +6.19765 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 127 q - 20 q^{2} - 31 q^{3} + 116 q^{4} + 127 q^{5} - 17 q^{6} - 63 q^{7} - 57 q^{8} + 122 q^{9}+O(q^{10}) \) 127 * q - 20 * q^2 - 31 * q^3 + 116 * q^4 + 127 * q^5 - 17 * q^6 - 63 * q^7 - 57 * q^8 + 122 * q^9	\( 127 q - 20 q^{2} - 31 q^{3} + 116 q^{4} + 127 q^{5} - 17 q^{6} - 63 q^{7} - 57 q^{8} + 122 q^{9} - 20 q^{10} - 32 q^{11} - 65 q^{12} - 49 q^{13} - 4 q^{14} - 31 q^{15} + 98 q^{16} - 53 q^{17} - 60 q^{18} - 126 q^{19} + 116 q^{20} - 24 q^{21} - 46 q^{22} - 121 q^{23} - 51 q^{24} + 127 q^{25} - 9 q^{26} - 112 q^{27} - 123 q^{28} - 6 q^{29} - 17 q^{30} - 54 q^{31} - 119 q^{32} - 57 q^{33} - 53 q^{34} - 63 q^{35} + 133 q^{36} - 61 q^{37} - 27 q^{38} - 33 q^{39} - 57 q^{40} - 8 q^{41} - 46 q^{42} - 208 q^{43} - 54 q^{44} + 122 q^{45} - 34 q^{46} - 116 q^{47} - 106 q^{48} + 110 q^{49} - 20 q^{50} - 54 q^{51} - 142 q^{52} - 60 q^{53} - 62 q^{54} - 32 q^{55} + 33 q^{56} - 56 q^{57} - 87 q^{58} - 53 q^{59} - 65 q^{60} - 76 q^{61} - 84 q^{62} - 215 q^{63} + 67 q^{64} - 49 q^{65} + 9 q^{66} - 145 q^{67} - 133 q^{68} + q^{69} - 4 q^{70} - 4 q^{71} - 167 q^{72} - 155 q^{73} - 14 q^{74} - 31 q^{75} - 199 q^{76} - 97 q^{77} - 24 q^{78} - 73 q^{79} + 98 q^{80} + 127 q^{81} - 69 q^{82} - 225 q^{83} - 59 q^{84} - 53 q^{85} + 30 q^{86} - 179 q^{87} - 119 q^{88} - 25 q^{89} - 60 q^{90} - 160 q^{91} - 188 q^{92} - 44 q^{93} - 32 q^{94} - 126 q^{95} - 43 q^{96} - 72 q^{97} - 111 q^{98} - 141 q^{99}+O(q^{100}) \) 127 * q - 20 * q^2 - 31 * q^3 + 116 * q^4 + 127 * q^5 - 17 * q^6 - 63 * q^7 - 57 * q^8 + 122 * q^9 - 20 * q^10 - 32 * q^11 - 65 * q^12 - 49 * q^13 - 4 * q^14 - 31 * q^15 + 98 * q^16 - 53 * q^17 - 60 * q^18 - 126 * q^19 + 116 * q^20 - 24 * q^21 - 46 * q^22 - 121 * q^23 - 51 * q^24 + 127 * q^25 - 9 * q^26 - 112 * q^27 - 123 * q^28 - 6 * q^29 - 17 * q^30 - 54 * q^31 - 119 * q^32 - 57 * q^33 - 53 * q^34 - 63 * q^35 + 133 * q^36 - 61 * q^37 - 27 * q^38 - 33 * q^39 - 57 * q^40 - 8 * q^41 - 46 * q^42 - 208 * q^43 - 54 * q^44 + 122 * q^45 - 34 * q^46 - 116 * q^47 - 106 * q^48 + 110 * q^49 - 20 * q^50 - 54 * q^51 - 142 * q^52 - 60 * q^53 - 62 * q^54 - 32 * q^55 + 33 * q^56 - 56 * q^57 - 87 * q^58 - 53 * q^59 - 65 * q^60 - 76 * q^61 - 84 * q^62 - 215 * q^63 + 67 * q^64 - 49 * q^65 + 9 * q^66 - 145 * q^67 - 133 * q^68 + q^69 - 4 * q^70 - 4 * q^71 - 167 * q^72 - 155 * q^73 - 14 * q^74 - 31 * q^75 - 199 * q^76 - 97 * q^77 - 24 * q^78 - 73 * q^79 + 98 * q^80 + 127 * q^81 - 69 * q^82 - 225 * q^83 - 59 * q^84 - 53 * q^85 + 30 * q^86 - 179 * q^87 - 119 * q^88 - 25 * q^89 - 60 * q^90 - 160 * q^91 - 188 * q^92 - 44 * q^93 - 32 * q^94 - 126 * q^95 - 43 * q^96 - 72 * q^97 - 111 * q^98 - 141 * q^99

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