LMFDB - Embedded newform 8045.2.a.e.1.18

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			1.18
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)

$f(q)$	$=$	$q-2.15729 q^{2} -0.777397 q^{3} +2.65392 q^{4} +1.00000 q^{5} +1.67707 q^{6} +3.51600 q^{7} -1.41069 q^{8} -2.39565 q^{9} +O(q^{10})$	\(q-2.15729 q^{2} -0.777397 q^{3} +2.65392 q^{4} +1.00000 q^{5} +1.67707 q^{6} +3.51600 q^{7} -1.41069 q^{8} -2.39565 q^{9} -2.15729 q^{10} +1.41839 q^{11} -2.06315 q^{12} -0.844092 q^{13} -7.58504 q^{14} -0.777397 q^{15} -2.26456 q^{16} +3.21797 q^{17} +5.16813 q^{18} +5.80719 q^{19} +2.65392 q^{20} -2.73332 q^{21} -3.05989 q^{22} -1.17392 q^{23} +1.09666 q^{24} +1.00000 q^{25} +1.82095 q^{26} +4.19456 q^{27} +9.33116 q^{28} +10.6256 q^{29} +1.67707 q^{30} -2.24750 q^{31} +7.70670 q^{32} -1.10265 q^{33} -6.94210 q^{34} +3.51600 q^{35} -6.35787 q^{36} -10.3481 q^{37} -12.5278 q^{38} +0.656194 q^{39} -1.41069 q^{40} +4.07119 q^{41} +5.89658 q^{42} +10.2140 q^{43} +3.76430 q^{44} -2.39565 q^{45} +2.53249 q^{46} -0.920759 q^{47} +1.76046 q^{48} +5.36223 q^{49} -2.15729 q^{50} -2.50164 q^{51} -2.24015 q^{52} +3.76262 q^{53} -9.04891 q^{54} +1.41839 q^{55} -4.95997 q^{56} -4.51449 q^{57} -22.9226 q^{58} -6.81446 q^{59} -2.06315 q^{60} -5.21190 q^{61} +4.84852 q^{62} -8.42311 q^{63} -12.0965 q^{64} -0.844092 q^{65} +2.37875 q^{66} +1.18173 q^{67} +8.54021 q^{68} +0.912601 q^{69} -7.58504 q^{70} +8.62462 q^{71} +3.37952 q^{72} -0.227926 q^{73} +22.3239 q^{74} -0.777397 q^{75} +15.4118 q^{76} +4.98707 q^{77} -1.41560 q^{78} +0.255638 q^{79} -2.26456 q^{80} +3.92612 q^{81} -8.78274 q^{82} -7.54227 q^{83} -7.25401 q^{84} +3.21797 q^{85} -22.0347 q^{86} -8.26033 q^{87} -2.00091 q^{88} -1.51292 q^{89} +5.16813 q^{90} -2.96782 q^{91} -3.11548 q^{92} +1.74720 q^{93} +1.98635 q^{94} +5.80719 q^{95} -5.99116 q^{96} +6.65206 q^{97} -11.5679 q^{98} -3.39798 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.15729	−1.52544	−0.762718	−	0.646731i	$-0.776136\pi$
$2$	−2.15729	−1.52544	−0.762718	+	0.646731i	$0.776136\pi$
$3$	−0.777397	−0.448830	−0.224415	−	0.974494i	$-0.572047\pi$
$3$	−0.777397	−0.448830	−0.224415	+	0.974494i	$0.572047\pi$
$4$	2.65392	1.32696
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.67707	0.684662
$7$	3.51600	1.32892	0.664461	−	0.747323i	$-0.268661\pi$
$7$	3.51600	1.32892	0.664461	+	0.747323i	$0.268661\pi$
$8$	−1.41069	−0.498754
$9$	−2.39565	−0.798552
$10$	−2.15729	−0.682196
$11$	1.41839	0.427662	0.213831	−	0.976871i	$-0.431406\pi$
$11$	1.41839	0.427662	0.213831	+	0.976871i	$0.431406\pi$
$12$	−2.06315	−0.595579
$13$	−0.844092	−0.234109	−0.117054	−	0.993125i	$-0.537345\pi$
$13$	−0.844092	−0.234109	−0.117054	+	0.993125i	$0.537345\pi$
$14$	−7.58504	−2.02719
$15$	−0.777397	−0.200723
$16$	−2.26456	−0.566141
$17$	3.21797	0.780471	0.390236	−	0.920715i	$-0.372394\pi$
$17$	3.21797	0.780471	0.390236	+	0.920715i	$0.372394\pi$
$18$	5.16813	1.21814
$19$	5.80719	1.33226	0.666131	−	0.745835i	$-0.267949\pi$
$19$	5.80719	1.33226	0.666131	+	0.745835i	$0.267949\pi$
$20$	2.65392	0.593434
$21$	−2.73332	−0.596460
$22$	−3.05989	−0.652371
$23$	−1.17392	−0.244779	−0.122390	−	0.992482i	$-0.539056\pi$
$23$	−1.17392	−0.244779	−0.122390	+	0.992482i	$0.539056\pi$
$24$	1.09666	0.223856
$25$	1.00000	0.200000
$26$	1.82095	0.357118
$27$	4.19456	0.807244
$28$	9.33116	1.76342
$29$	10.6256	1.97313	0.986565	−	0.163372i	$-0.0522369\pi$
$29$	10.6256	1.97313	0.986565	+	0.163372i	$0.0522369\pi$
$30$	1.67707	0.306190
$31$	−2.24750	−0.403663	−0.201832	−	0.979420i	$-0.564689\pi$
$31$	−2.24750	−0.403663	−0.201832	+	0.979420i	$0.564689\pi$
$32$	7.70670	1.36237
$33$	−1.10265	−0.191948
$34$	−6.94210	−1.19056
$35$	3.51600	0.594312
$36$	−6.35787	−1.05964
$37$	−10.3481	−1.70122	−0.850609	−	0.525799i	$-0.823766\pi$
$37$	−10.3481	−1.70122	−0.850609	+	0.525799i	$0.823766\pi$
$38$	−12.5278	−2.03228
$39$	0.656194	0.105075
$40$	−1.41069	−0.223049
$41$	4.07119	0.635812	0.317906	−	0.948122i	$-0.397020\pi$
$41$	4.07119	0.635812	0.317906	+	0.948122i	$0.397020\pi$
$42$	5.89658	0.909862
$43$	10.2140	1.55763	0.778814	−	0.627255i	$-0.215822\pi$
$43$	10.2140	1.55763	0.778814	+	0.627255i	$0.215822\pi$
$44$	3.76430	0.567489
$45$	−2.39565	−0.357123
$46$	2.53249	0.373395
$47$	−0.920759	−0.134307	−0.0671533	−	0.997743i	$-0.521392\pi$
$47$	−0.920759	−0.134307	−0.0671533	+	0.997743i	$0.521392\pi$
$48$	1.76046	0.254101
$49$	5.36223	0.766032
$50$	−2.15729	−0.305087
$51$	−2.50164	−0.350299
$52$	−2.24015	−0.310653
$53$	3.76262	0.516835	0.258418	−	0.966033i	$-0.416799\pi$
$53$	3.76262	0.516835	0.258418	+	0.966033i	$0.416799\pi$
$54$	−9.04891	−1.23140
$55$	1.41839	0.191256
$56$	−4.95997	−0.662804
$57$	−4.51449	−0.597959
$58$	−22.9226	−3.00988
$59$	−6.81446	−0.887168	−0.443584	−	0.896233i	$-0.646293\pi$
$59$	−6.81446	−0.887168	−0.443584	+	0.896233i	$0.646293\pi$
$60$	−2.06315	−0.266351
$61$	−5.21190	−0.667316	−0.333658	−	0.942694i	$-0.608283\pi$
$61$	−5.21190	−0.667316	−0.333658	+	0.942694i	$0.608283\pi$
$62$	4.84852	0.615763
$63$	−8.42311	−1.06121
$64$	−12.0965	−1.51206
$65$	−0.844092	−0.104697
$66$	2.37875	0.292804
$67$	1.18173	0.144372	0.0721859	−	0.997391i	$-0.477003\pi$
$67$	1.18173	0.144372	0.0721859	+	0.997391i	$0.477003\pi$
$68$	8.54021	1.03565
$69$	0.912601	0.109864
$70$	−7.58504	−0.906585
$71$	8.62462	1.02355	0.511777	−	0.859118i	$-0.328987\pi$
$71$	8.62462	1.02355	0.511777	+	0.859118i	$0.328987\pi$
$72$	3.37952	0.398280
$73$	−0.227926	−0.0266767	−0.0133383	−	0.999911i	$-0.504246\pi$
$73$	−0.227926	−0.0266767	−0.0133383	+	0.999911i	$0.504246\pi$
$74$	22.3239	2.59510
$75$	−0.777397	−0.0897660
$76$	15.4118	1.76786
$77$	4.98707	0.568329
$78$	−1.41560	−0.160286
$79$	0.255638	0.0287615	0.0143807	−	0.999897i	$-0.495422\pi$
$79$	0.255638	0.0287615	0.0143807	+	0.999897i	$0.495422\pi$
$80$	−2.26456	−0.253186
$81$	3.92612	0.436236
$82$	−8.78274	−0.969892
$83$	−7.54227	−0.827872	−0.413936	−	0.910306i	$-0.635846\pi$
$83$	−7.54227	−0.827872	−0.413936	+	0.910306i	$0.635846\pi$
$84$	−7.25401	−0.791477
$85$	3.21797	0.349037
$86$	−22.0347	−2.37606
$87$	−8.26033	−0.885600
$88$	−2.00091	−0.213298
$89$	−1.51292	−0.160369	−0.0801846	−	0.996780i	$-0.525551\pi$
$89$	−1.51292	−0.160369	−0.0801846	+	0.996780i	$0.525551\pi$
$90$	5.16813	0.544769
$91$	−2.96782	−0.311112
$92$	−3.11548	−0.324812
$93$	1.74720	0.181176
$94$	1.98635	0.204876
$95$	5.80719	0.595806
$96$	−5.99116	−0.611471
$97$	6.65206	0.675414	0.337707	−	0.941251i	$-0.390349\pi$
$97$	6.65206	0.675414	0.337707	+	0.941251i	$0.390349\pi$
$98$	−11.5679	−1.16853
$99$	−3.39798	−0.341510
$100$	2.65392	0.265392
$101$	13.3778	1.33114	0.665572	−	0.746334i	$-0.268188\pi$
$101$	13.3778	1.33114	0.665572	+	0.746334i	$0.268188\pi$
$102$	5.39676	0.534359
$103$	11.9646	1.17890	0.589451	−	0.807804i	$-0.299344\pi$
$103$	11.9646	1.17890	0.589451	+	0.807804i	$0.299344\pi$
$104$	1.19075	0.116763
$105$	−2.73332	−0.266745
$106$	−8.11707	−0.788399
$107$	−8.56718	−0.828220	−0.414110	−	0.910227i	$-0.635907\pi$
$107$	−8.56718	−0.828220	−0.414110	+	0.910227i	$0.635907\pi$
$108$	11.1320	1.07118
$109$	12.6157	1.20837	0.604183	−	0.796845i	$-0.293499\pi$
$109$	12.6157	1.20837	0.604183	+	0.796845i	$0.293499\pi$
$110$	−3.05989	−0.291749
$111$	8.04458	0.763558
$112$	−7.96219	−0.752356
$113$	−3.83463	−0.360732	−0.180366	−	0.983600i	$-0.557728\pi$
$113$	−3.83463	−0.360732	−0.180366	+	0.983600i	$0.557728\pi$
$114$	9.73909	0.912149
$115$	−1.17392	−0.109469
$116$	28.1995	2.61826
$117$	2.02215	0.186948
$118$	14.7008	1.35332
$119$	11.3144	1.03719
$120$	1.09666	0.100111
$121$	−8.98816	−0.817105
$122$	11.2436	1.01795
$123$	−3.16493	−0.285372
$124$	−5.96468	−0.535644
$125$	1.00000	0.0894427
$126$	18.1711	1.61881
$127$	15.4908	1.37459	0.687293	−	0.726380i	$-0.258799\pi$
$127$	15.4908	1.37459	0.687293	+	0.726380i	$0.258799\pi$
$128$	10.6823	0.944190
$129$	−7.94037	−0.699110
$130$	1.82095	0.159708
$131$	−21.0257	−1.83702	−0.918511	−	0.395395i	$-0.870608\pi$
$131$	−21.0257	−1.83702	−0.918511	+	0.395395i	$0.870608\pi$
$132$	−2.92635	−0.254706
$133$	20.4181	1.77047
$134$	−2.54935	−0.220230
$135$	4.19456	0.361011
$136$	−4.53955	−0.389263
$137$	−2.00321	−0.171146	−0.0855730	−	0.996332i	$-0.527272\pi$
$137$	−2.00321	−0.171146	−0.0855730	+	0.996332i	$0.527272\pi$
$138$	−1.96875	−0.167591
$139$	0.880638	0.0746947	0.0373474	−	0.999302i	$-0.488109\pi$
$139$	0.880638	0.0746947	0.0373474	+	0.999302i	$0.488109\pi$
$140$	9.33116	0.788627
$141$	0.715795	0.0602808
$142$	−18.6058	−1.56137
$143$	−1.19725	−0.100119
$144$	5.42511	0.452092
$145$	10.6256	0.882410
$146$	0.491702	0.0406936
$147$	−4.16858	−0.343818
$148$	−27.4630	−2.25744
$149$	−3.78046	−0.309707	−0.154854	−	0.987937i	$-0.549491\pi$
$149$	−3.78046	−0.309707	−0.154854	+	0.987937i	$0.549491\pi$
$150$	1.67707	0.136932
$151$	14.3302	1.16618	0.583088	−	0.812409i	$-0.301844\pi$
$151$	14.3302	1.16618	0.583088	+	0.812409i	$0.301844\pi$
$152$	−8.19214	−0.664470
$153$	−7.70913	−0.623247
$154$	−10.7586	−0.866950
$155$	−2.24750	−0.180524
$156$	1.74148	0.139430
$157$	17.0185	1.35823	0.679113	−	0.734034i	$-0.262365\pi$
$157$	17.0185	1.35823	0.679113	+	0.734034i	$0.262365\pi$
$158$	−0.551485	−0.0438738
$159$	−2.92505	−0.231971
$160$	7.70670	0.609268
$161$	−4.12750	−0.325292
$162$	−8.46980	−0.665451
$163$	24.4538	1.91537	0.957684	−	0.287823i	$-0.0929316\pi$
$163$	24.4538	1.91537	0.957684	+	0.287823i	$0.0929316\pi$
$164$	10.8046	0.843696
$165$	−1.10265	−0.0858415
$166$	16.2709	1.26287
$167$	−4.42088	−0.342098	−0.171049	−	0.985262i	$-0.554716\pi$
$167$	−4.42088	−0.342098	−0.171049	+	0.985262i	$0.554716\pi$
$168$	3.85587	0.297487
$169$	−12.2875	−0.945193
$170$	−6.94210	−0.532435
$171$	−13.9120	−1.06388
$172$	27.1072	2.06691
$173$	1.04172	0.0792005	0.0396003	−	0.999216i	$-0.487392\pi$
$173$	1.04172	0.0792005	0.0396003	+	0.999216i	$0.487392\pi$
$174$	17.8199	1.35093
$175$	3.51600	0.265784
$176$	−3.21204	−0.242117
$177$	5.29754	0.398188
$178$	3.26381	0.244633
$179$	−22.5450	−1.68510	−0.842548	−	0.538622i	$-0.818945\pi$
$179$	−22.5450	−1.68510	−0.842548	+	0.538622i	$0.818945\pi$
$180$	−6.35787	−0.473887
$181$	8.15547	0.606191	0.303095	−	0.952960i	$-0.401980\pi$
$181$	8.15547	0.606191	0.303095	+	0.952960i	$0.401980\pi$
$182$	6.40247	0.474582
$183$	4.05171	0.299511
$184$	1.65603	0.122084
$185$	−10.3481	−0.760808
$186$	−3.76922	−0.276373
$187$	4.56434	0.333778
$188$	−2.44362	−0.178219
$189$	14.7481	1.07276
$190$	−12.5278	−0.908864
$191$	2.04221	0.147769	0.0738845	−	0.997267i	$-0.476460\pi$
$191$	2.04221	0.147769	0.0738845	+	0.997267i	$0.476460\pi$
$192$	9.40378	0.678659
$193$	24.8555	1.78914	0.894570	−	0.446928i	$-0.147482\pi$
$193$	24.8555	1.78914	0.894570	+	0.446928i	$0.147482\pi$
$194$	−14.3504	−1.03030
$195$	0.656194	0.0469910
$196$	14.2309	1.01649
$197$	−19.0199	−1.35511	−0.677556	−	0.735472i	$-0.736961\pi$
$197$	−19.0199	−1.35511	−0.677556	+	0.735472i	$0.736961\pi$
$198$	7.33044	0.520952
$199$	−7.57768	−0.537168	−0.268584	−	0.963256i	$-0.586556\pi$
$199$	−7.57768	−0.537168	−0.268584	+	0.963256i	$0.586556\pi$
$200$	−1.41069	−0.0997507
$201$	−0.918676	−0.0647984
$202$	−28.8599	−2.03058
$203$	37.3597	2.62213
$204$	−6.63913	−0.464832
$205$	4.07119	0.284344
$206$	−25.8111	−1.79834
$207$	2.81231	0.195469
$208$	1.91150	0.132539
$209$	8.23689	0.569758
$210$	5.89658	0.406903
$211$	−17.9897	−1.23846	−0.619232	−	0.785208i	$-0.712556\pi$
$211$	−17.9897	−1.23846	−0.619232	+	0.785208i	$0.712556\pi$
$212$	9.98567	0.685818
$213$	−6.70475	−0.459402
$214$	18.4819	1.26340
$215$	10.2140	0.696592
$216$	−5.91722	−0.402616
$217$	−7.90221	−0.536437
$218$	−27.2158	−1.84329
$219$	0.177189	0.0119733
$220$	3.76430	0.253789
$221$	−2.71626	−0.182715
$222$	−17.3545	−1.16476
$223$	14.9228	0.999303	0.499652	−	0.866226i	$-0.333461\pi$
$223$	14.9228	0.999303	0.499652	+	0.866226i	$0.333461\pi$
$224$	27.0967	1.81048
$225$	−2.39565	−0.159710
$226$	8.27243	0.550274
$227$	25.0715	1.66406	0.832028	−	0.554733i	$-0.187180\pi$
$227$	25.0715	1.66406	0.832028	+	0.554733i	$0.187180\pi$
$228$	−11.9811	−0.793467
$229$	−2.99543	−0.197944	−0.0989719	−	0.995090i	$-0.531555\pi$
$229$	−2.99543	−0.197944	−0.0989719	+	0.995090i	$0.531555\pi$
$230$	2.53249	0.166987
$231$	−3.87693	−0.255083
$232$	−14.9894	−0.984105
$233$	12.5000	0.818904	0.409452	−	0.912332i	$-0.365720\pi$
$233$	12.5000	0.818904	0.409452	+	0.912332i	$0.365720\pi$
$234$	−4.36238	−0.285177
$235$	−0.920759	−0.0600637
$236$	−18.0850	−1.17723
$237$	−0.198732	−0.0129090
$238$	−24.4084	−1.58216
$239$	−11.6056	−0.750703	−0.375352	−	0.926882i	$-0.622478\pi$
$239$	−11.6056	−0.750703	−0.375352	+	0.926882i	$0.622478\pi$
$240$	1.76046	0.113637
$241$	−8.49710	−0.547347	−0.273673	−	0.961823i	$-0.588239\pi$
$241$	−8.49710	−0.547347	−0.273673	+	0.961823i	$0.588239\pi$
$242$	19.3901	1.24644
$243$	−15.6358	−1.00304
$244$	−13.8319	−0.885500
$245$	5.36223	0.342580
$246$	6.82767	0.435317
$247$	−4.90181	−0.311894
$248$	3.17053	0.201329
$249$	5.86334	0.371574
$250$	−2.15729	−0.136439
$251$	−14.5832	−0.920481	−0.460241	−	0.887794i	$-0.652237\pi$
$251$	−14.5832	−0.920481	−0.460241	+	0.887794i	$0.652237\pi$
$252$	−22.3542	−1.40818
$253$	−1.66508	−0.104683
$254$	−33.4182	−2.09684
$255$	−2.50164	−0.156658
$256$	1.14816	0.0717600
$257$	−19.6367	−1.22491	−0.612453	−	0.790507i	$-0.709817\pi$
$257$	−19.6367	−1.22491	−0.612453	+	0.790507i	$0.709817\pi$
$258$	17.1297	1.06645
$259$	−36.3839	−2.26078
$260$	−2.24015	−0.138928
$261$	−25.4553	−1.57565
$262$	45.3586	2.80226
$263$	−23.9977	−1.47976	−0.739881	−	0.672738i	$-0.765118\pi$
$263$	−23.9977	−1.47976	−0.739881	+	0.672738i	$0.765118\pi$
$264$	1.55550	0.0957345
$265$	3.76262	0.231136
$266$	−44.0478	−2.70074
$267$	1.17614	0.0719785
$268$	3.13622	0.191575
$269$	16.1851	0.986825	0.493412	−	0.869795i	$-0.335749\pi$
$269$	16.1851	0.986825	0.493412	+	0.869795i	$0.335749\pi$
$270$	−9.04891	−0.550699
$271$	−12.3865	−0.752425	−0.376213	−	0.926533i	$-0.622774\pi$
$271$	−12.3865	−0.752425	−0.376213	+	0.926533i	$0.622774\pi$
$272$	−7.28728	−0.441857
$273$	2.30718	0.139637
$274$	4.32152	0.261073
$275$	1.41839	0.0855324
$276$	2.42197	0.145785
$277$	13.0869	0.786316	0.393158	−	0.919471i	$-0.371382\pi$
$277$	13.0869	0.786316	0.393158	+	0.919471i	$0.371382\pi$
$278$	−1.89979	−0.113942
$279$	5.38424	0.322346
$280$	−4.95997	−0.296415
$281$	−4.48993	−0.267847	−0.133923	−	0.990992i	$-0.542758\pi$
$281$	−4.48993	−0.267847	−0.133923	+	0.990992i	$0.542758\pi$
$282$	−1.54418	−0.0919546
$283$	17.0205	1.01177	0.505883	−	0.862602i	$-0.331167\pi$
$283$	17.0205	1.01177	0.505883	+	0.862602i	$0.331167\pi$
$284$	22.8890	1.35821
$285$	−4.51449	−0.267416
$286$	2.58283	0.152726
$287$	14.3143	0.844945
$288$	−18.4626	−1.08792
$289$	−6.64470	−0.390865
$290$	−22.9226	−1.34606
$291$	−5.17129	−0.303146
$292$	−0.604895	−0.0353988
$293$	−21.3665	−1.24824	−0.624122	−	0.781327i	$-0.714543\pi$
$293$	−21.3665	−1.24824	−0.624122	+	0.781327i	$0.714543\pi$
$294$	8.99284	0.524473
$295$	−6.81446	−0.396753
$296$	14.5979	0.848489
$297$	5.94954	0.345228
$298$	8.15556	0.472439
$299$	0.990896	0.0573050
$300$	−2.06315	−0.119116
$301$	35.9126	2.06997
$302$	−30.9145	−1.77893
$303$	−10.3999	−0.597457
$304$	−13.1508	−0.754248
$305$	−5.21190	−0.298433
$306$	16.6309	0.950723
$307$	−2.41658	−0.137921	−0.0689607	−	0.997619i	$-0.521968\pi$
$307$	−2.41658	−0.137921	−0.0689607	+	0.997619i	$0.521968\pi$
$308$	13.2353	0.754149
$309$	−9.30121	−0.529127
$310$	4.84852	0.275378
$311$	−12.1313	−0.687903	−0.343951	−	0.938987i	$-0.611766\pi$
$311$	−12.1313	−0.687903	−0.343951	+	0.938987i	$0.611766\pi$
$312$	−0.925685	−0.0524066
$313$	11.2796	0.637562	0.318781	−	0.947828i	$-0.396726\pi$
$313$	11.2796	0.637562	0.318781	+	0.947828i	$0.396726\pi$
$314$	−36.7140	−2.07189
$315$	−8.42311	−0.474589
$316$	0.678441	0.0381653
$317$	−3.12799	−0.175686	−0.0878428	−	0.996134i	$-0.527997\pi$
$317$	−3.12799	−0.175686	−0.0878428	+	0.996134i	$0.527997\pi$
$318$	6.31018	0.353857
$319$	15.0713	0.843832
$320$	−12.0965	−0.676215
$321$	6.66010	0.371730
$322$	8.90422	0.496213
$323$	18.6874	1.03979
$324$	10.4196	0.578867
$325$	−0.844092	−0.0468218
$326$	−52.7540	−2.92177
$327$	−9.80742	−0.542351
$328$	−5.74317	−0.317114
$329$	−3.23738	−0.178483
$330$	2.37875	0.130946
$331$	−20.9808	−1.15321	−0.576605	−	0.817023i	$-0.695623\pi$
$331$	−20.9808	−1.15321	−0.576605	+	0.817023i	$0.695623\pi$
$332$	−20.0166	−1.09855
$333$	24.7905	1.35851
$334$	9.53715	0.521849
$335$	1.18173	0.0645650
$336$	6.18978	0.337680
$337$	−1.81686	−0.0989704	−0.0494852	−	0.998775i	$-0.515758\pi$
$337$	−1.81686	−0.0989704	−0.0494852	+	0.998775i	$0.515758\pi$
$338$	26.5078	1.44183
$339$	2.98103	0.161907
$340$	8.54021	0.463158
$341$	−3.18784	−0.172631
$342$	30.0123	1.62288
$343$	−5.75841	−0.310925
$344$	−14.4088	−0.776873
$345$	0.912601	0.0491328
$346$	−2.24730	−0.120815
$347$	−13.0428	−0.700176	−0.350088	−	0.936717i	$-0.613848\pi$
$347$	−13.0428	−0.700176	−0.350088	+	0.936717i	$0.613848\pi$
$348$	−21.9222	−1.17515
$349$	31.1252	1.66609	0.833047	−	0.553202i	$-0.186594\pi$
$349$	31.1252	1.66609	0.833047	+	0.553202i	$0.186594\pi$
$350$	−7.58504	−0.405437
$351$	−3.54060	−0.188983
$352$	10.9311	0.582632
$353$	8.21308	0.437138	0.218569	−	0.975821i	$-0.429861\pi$
$353$	8.21308	0.437138	0.218569	+	0.975821i	$0.429861\pi$
$354$	−11.4284	−0.607410
$355$	8.62462	0.457747
$356$	−4.01516	−0.212803
$357$	−8.79574	−0.465520
$358$	48.6363	2.57051
$359$	−23.8255	−1.25746	−0.628731	−	0.777623i	$-0.716425\pi$
$359$	−23.8255	−1.25746	−0.628731	+	0.777623i	$0.716425\pi$
$360$	3.37952	0.178116
$361$	14.7235	0.774922
$362$	−17.5937	−0.924706
$363$	6.98736	0.366742
$364$	−7.87635	−0.412833
$365$	−0.227926	−0.0119302
$366$	−8.74074	−0.456886
$367$	−7.15286	−0.373376	−0.186688	−	0.982419i	$-0.559775\pi$
$367$	−7.15286	−0.373376	−0.186688	+	0.982419i	$0.559775\pi$
$368$	2.65841	0.138579
$369$	−9.75315	−0.507729
$370$	22.3239	1.16056
$371$	13.2293	0.686833
$372$	4.63692	0.240413
$373$	2.82089	0.146060	0.0730302	−	0.997330i	$-0.476733\pi$
$373$	2.82089	0.146060	0.0730302	+	0.997330i	$0.476733\pi$
$374$	−9.84663	−0.509157
$375$	−0.777397	−0.0401446
$376$	1.29890	0.0669859
$377$	−8.96900	−0.461927
$378$	−31.8159	−1.63643
$379$	−25.5434	−1.31208	−0.656038	−	0.754728i	$-0.727769\pi$
$379$	−25.5434	−1.31208	−0.656038	+	0.754728i	$0.727769\pi$
$380$	15.4118	0.790609
$381$	−12.0425	−0.616956
$382$	−4.40564	−0.225412
$383$	−16.9595	−0.866592	−0.433296	−	0.901252i	$-0.642650\pi$
$383$	−16.9595	−0.866592	−0.433296	+	0.901252i	$0.642650\pi$
$384$	−8.30437	−0.423781
$385$	4.98707	0.254164
$386$	−53.6207	−2.72922
$387$	−24.4693	−1.24385
$388$	17.6540	0.896246
$389$	3.77058	0.191176	0.0955879	−	0.995421i	$-0.469527\pi$
$389$	3.77058	0.191176	0.0955879	+	0.995421i	$0.469527\pi$
$390$	−1.41560	−0.0716819
$391$	−3.77763	−0.191043
$392$	−7.56443	−0.382061
$393$	16.3453	0.824511
$394$	41.0315	2.06714
$395$	0.255638	0.0128625
$396$	−9.01796	−0.453169
$397$	−19.0590	−0.956542	−0.478271	−	0.878212i	$-0.658736\pi$
$397$	−19.0590	−0.956542	−0.478271	+	0.878212i	$0.658736\pi$
$398$	16.3473	0.819416
$399$	−15.8729	−0.794641
$400$	−2.26456	−0.113228
$401$	9.35133	0.466983	0.233492	−	0.972359i	$-0.424985\pi$
$401$	9.35133	0.466983	0.233492	+	0.972359i	$0.424985\pi$
$402$	1.98185	0.0988458
$403$	1.89710	0.0945012
$404$	35.5036	1.76637
$405$	3.92612	0.195091
$406$	−80.5957	−3.99990
$407$	−14.6777	−0.727546
$408$	3.52903	0.174713
$409$	−34.0752	−1.68491	−0.842455	−	0.538767i	$-0.818890\pi$
$409$	−34.0752	−1.68491	−0.842455	+	0.538767i	$0.818890\pi$
$410$	−8.78274	−0.433749
$411$	1.55729	0.0768155
$412$	31.7529	1.56435
$413$	−23.9596	−1.17898
$414$	−6.06697	−0.298175
$415$	−7.54227	−0.370236
$416$	−6.50517	−0.318942
$417$	−0.684605	−0.0335252
$418$	−17.7694	−0.869129
$419$	27.6512	1.35085	0.675424	−	0.737430i	$-0.263961\pi$
$419$	27.6512	1.35085	0.675424	+	0.737430i	$0.263961\pi$
$420$	−7.25401	−0.353959
$421$	−26.2780	−1.28071	−0.640356	−	0.768078i	$-0.721213\pi$
$421$	−26.2780	−1.28071	−0.640356	+	0.768078i	$0.721213\pi$
$422$	38.8091	1.88920
$423$	2.20582	0.107251
$424$	−5.30788	−0.257773
$425$	3.21797	0.156094
$426$	14.4641	0.700789
$427$	−18.3250	−0.886810
$428$	−22.7366	−1.09901
$429$	0.930742	0.0449366
$430$	−22.0347	−1.06261
$431$	7.76790	0.374167	0.187083	−	0.982344i	$-0.440097\pi$
$431$	7.76790	0.374167	0.187083	+	0.982344i	$0.440097\pi$
$432$	−9.49885	−0.457014
$433$	11.1708	0.536835	0.268418	−	0.963303i	$-0.413499\pi$
$433$	11.1708	0.536835	0.268418	+	0.963303i	$0.413499\pi$
$434$	17.0474	0.818301
$435$	−8.26033	−0.396052
$436$	33.4811	1.60345
$437$	−6.81718	−0.326110
$438$	−0.382248	−0.0182645
$439$	−1.21333	−0.0579091	−0.0289546	−	0.999581i	$-0.509218\pi$
$439$	−1.21333	−0.0579091	−0.0289546	+	0.999581i	$0.509218\pi$
$440$	−2.00091	−0.0953897
$441$	−12.8460	−0.611716
$442$	5.85977	0.278721
$443$	4.36917	0.207586	0.103793	−	0.994599i	$-0.466902\pi$
$443$	4.36917	0.207586	0.103793	+	0.994599i	$0.466902\pi$
$444$	21.3496	1.01321
$445$	−1.51292	−0.0717193
$446$	−32.1928	−1.52437
$447$	2.93892	0.139006
$448$	−42.5312	−2.00941
$449$	−5.01909	−0.236865	−0.118433	−	0.992962i	$-0.537787\pi$
$449$	−5.01909	−0.236865	−0.118433	+	0.992962i	$0.537787\pi$
$450$	5.16813	0.243628
$451$	5.77454	0.271913
$452$	−10.1768	−0.478676
$453$	−11.1403	−0.523415
$454$	−54.0867	−2.53841
$455$	−2.96782	−0.139134
$456$	6.36854	0.298234
$457$	34.1197	1.59605	0.798025	−	0.602624i	$-0.205878\pi$
$457$	34.1197	1.59605	0.798025	+	0.602624i	$0.205878\pi$
$458$	6.46203	0.301951
$459$	13.4980	0.630031
$460$	−3.11548	−0.145260
$461$	12.4604	0.580338	0.290169	−	0.956975i	$-0.406289\pi$
$461$	12.4604	0.580338	0.290169	+	0.956975i	$0.406289\pi$
$462$	8.36367	0.389113
$463$	12.9409	0.601412	0.300706	−	0.953717i	$-0.402778\pi$
$463$	12.9409	0.601412	0.300706	+	0.953717i	$0.402778\pi$
$464$	−24.0624	−1.11707
$465$	1.74720	0.0810245
$466$	−26.9662	−1.24919
$467$	19.1201	0.884771	0.442386	−	0.896825i	$-0.354132\pi$
$467$	19.1201	0.884771	0.442386	+	0.896825i	$0.354132\pi$
$468$	5.36662	0.248072
$469$	4.15497	0.191859
$470$	1.98635	0.0916234
$471$	−13.2301	−0.609613
$472$	9.61309	0.442478
$473$	14.4875	0.666138
$474$	0.428723	0.0196919
$475$	5.80719	0.266452
$476$	30.0273	1.37630
$477$	−9.01393	−0.412719
$478$	25.0367	1.14515
$479$	19.9726	0.912572	0.456286	−	0.889833i	$-0.349179\pi$
$479$	19.9726	0.912572	0.456286	+	0.889833i	$0.349179\pi$
$480$	−5.99116	−0.273458
$481$	8.73475	0.398270
$482$	18.3308	0.834943
$483$	3.20870	0.146001
$484$	−23.8538	−1.08426
$485$	6.65206	0.302054
$486$	33.7311	1.53007
$487$	29.1550	1.32114	0.660570	−	0.750765i	$-0.270315\pi$
$487$	29.1550	1.32114	0.660570	+	0.750765i	$0.270315\pi$
$488$	7.35237	0.332826
$489$	−19.0103	−0.859674
$490$	−11.5679	−0.522584
$491$	21.7710	0.982511	0.491256	−	0.871015i	$-0.336538\pi$
$491$	21.7710	0.982511	0.491256	+	0.871015i	$0.336538\pi$
$492$	−8.39945	−0.378676
$493$	34.1929	1.53997
$494$	10.5746	0.475775
$495$	−3.39798	−0.152728
$496$	5.08961	0.228530
$497$	30.3241	1.36022
$498$	−12.6489	−0.566813
$499$	13.7436	0.615249	0.307625	−	0.951508i	$-0.400466\pi$
$499$	13.7436	0.615249	0.307625	+	0.951508i	$0.400466\pi$
$500$	2.65392	0.118687
$501$	3.43678	0.153544
$502$	31.4602	1.40414
$503$	6.67870	0.297789	0.148894	−	0.988853i	$-0.452429\pi$
$503$	6.67870	0.297789	0.148894	+	0.988853i	$0.452429\pi$
$504$	11.8824	0.529283
$505$	13.3778	0.595305
$506$	3.59207	0.159687
$507$	9.55227	0.424231
$508$	41.1113	1.82402
$509$	30.8973	1.36950	0.684749	−	0.728779i	$-0.259912\pi$
$509$	30.8973	1.36950	0.684749	+	0.728779i	$0.259912\pi$
$510$	5.39676	0.238973
$511$	−0.801385	−0.0354512
$512$	−23.8415	−1.05366
$513$	24.3586	1.07546
$514$	42.3622	1.86852
$515$	11.9646	0.527221
$516$	−21.0731	−0.927690
$517$	−1.30600	−0.0574378
$518$	78.4907	3.44868
$519$	−0.809830	−0.0355476
$520$	1.19075	0.0522179
$521$	−18.8797	−0.827136	−0.413568	−	0.910473i	$-0.635718\pi$
$521$	−18.8797	−0.827136	−0.413568	+	0.910473i	$0.635718\pi$
$522$	54.9146	2.40355
$523$	17.0514	0.745608	0.372804	−	0.927910i	$-0.378396\pi$
$523$	17.0514	0.745608	0.372804	+	0.927910i	$0.378396\pi$
$524$	−55.8004	−2.43765
$525$	−2.73332	−0.119292
$526$	51.7701	2.25728
$527$	−7.23239	−0.315048
$528$	2.49703	0.108669
$529$	−21.6219	−0.940083
$530$	−8.11707	−0.352583
$531$	16.3251	0.708449
$532$	54.1878	2.34934
$533$	−3.43645	−0.148849
$534$	−2.53728	−0.109799
$535$	−8.56718	−0.370391
$536$	−1.66706	−0.0720059
$537$	17.5264	0.756321
$538$	−34.9161	−1.50534
$539$	7.60575	0.327603
$540$	11.1320	0.479046
$541$	32.9246	1.41554	0.707769	−	0.706444i	$-0.249702\pi$
$541$	32.9246	1.41554	0.707769	+	0.706444i	$0.249702\pi$
$542$	26.7213	1.14778
$543$	−6.34003	−0.272077
$544$	24.7999	1.06329
$545$	12.6157	0.540398
$546$	−4.97726	−0.213007
$547$	−34.5471	−1.47713	−0.738563	−	0.674185i	$-0.764495\pi$
$547$	−34.5471	−1.47713	−0.738563	+	0.674185i	$0.764495\pi$
$548$	−5.31636	−0.227104
$549$	12.4859	0.532886
$550$	−3.05989	−0.130474
$551$	61.7051	2.62872
$552$	−1.28740	−0.0547952
$553$	0.898821	0.0382217
$554$	−28.2323	−1.19948
$555$	8.04458	0.341473
$556$	2.33714	0.0991167
$557$	0.584552	0.0247683	0.0123841	−	0.999923i	$-0.496058\pi$
$557$	0.584552	0.0247683	0.0123841	+	0.999923i	$0.496058\pi$
$558$	−11.6154	−0.491718
$559$	−8.62160	−0.364655
$560$	−7.96219	−0.336464
$561$	−3.54830	−0.149810
$562$	9.68610	0.408583
$563$	−27.9962	−1.17990	−0.589950	−	0.807440i	$-0.700853\pi$
$563$	−27.9962	−1.17990	−0.589950	+	0.807440i	$0.700853\pi$
$564$	1.89966	0.0799901
$565$	−3.83463	−0.161324
$566$	−36.7183	−1.54338
$567$	13.8042	0.579723
$568$	−12.1667	−0.510501
$569$	0.540800	0.0226715	0.0113358	−	0.999936i	$-0.496392\pi$
$569$	0.540800	0.0226715	0.0113358	+	0.999936i	$0.496392\pi$
$570$	9.73909	0.407926
$571$	−2.90941	−0.121755	−0.0608775	−	0.998145i	$-0.519390\pi$
$571$	−2.90941	−0.121755	−0.0608775	+	0.998145i	$0.519390\pi$
$572$	−3.17741	−0.132854
$573$	−1.58761	−0.0663232
$574$	−30.8801	−1.28891
$575$	−1.17392	−0.0489558
$576$	28.9790	1.20746
$577$	−30.9143	−1.28698	−0.643490	−	0.765455i	$-0.722514\pi$
$577$	−30.9143	−1.28698	−0.643490	+	0.765455i	$0.722514\pi$
$578$	14.3346	0.596239
$579$	−19.3226	−0.803020
$580$	28.1995	1.17092
$581$	−26.5186	−1.10018
$582$	11.1560	0.462430
$583$	5.33687	0.221031
$584$	0.321532	0.0133051
$585$	2.02215	0.0836057
$586$	46.0938	1.90412
$587$	10.5029	0.433502	0.216751	−	0.976227i	$-0.430454\pi$
$587$	10.5029	0.433502	0.216751	+	0.976227i	$0.430454\pi$
$588$	−11.0630	−0.456232
$589$	−13.0517	−0.537785
$590$	14.7008	0.605222
$591$	14.7860	0.608215
$592$	23.4339	0.963129
$593$	20.1725	0.828383	0.414192	−	0.910190i	$-0.364064\pi$
$593$	20.1725	0.828383	0.414192	+	0.910190i	$0.364064\pi$
$594$	−12.8349	−0.526623
$595$	11.3144	0.463843
$596$	−10.0330	−0.410969
$597$	5.89087	0.241097
$598$	−2.13765	−0.0874152
$599$	−29.6584	−1.21181	−0.605905	−	0.795537i	$-0.707189\pi$
$599$	−29.6584	−1.21181	−0.605905	+	0.795537i	$0.707189\pi$
$600$	1.09666	0.0447711
$601$	32.4873	1.32519	0.662593	−	0.748980i	$-0.269456\pi$
$601$	32.4873	1.32519	0.662593	+	0.748980i	$0.269456\pi$
$602$	−77.4739	−3.15760
$603$	−2.83102	−0.115288
$604$	38.0312	1.54747
$605$	−8.98816	−0.365421
$606$	22.4356	0.911383
$607$	−22.4804	−0.912449	−0.456225	−	0.889865i	$-0.650799\pi$
$607$	−22.4804	−0.912449	−0.456225	+	0.889865i	$0.650799\pi$
$608$	44.7543	1.81503
$609$	−29.0433	−1.17689
$610$	11.2436	0.455240
$611$	0.777205	0.0314424
$612$	−20.4594	−0.827022
$613$	−8.05753	−0.325441	−0.162720	−	0.986672i	$-0.552027\pi$
$613$	−8.05753	−0.325441	−0.162720	+	0.986672i	$0.552027\pi$
$614$	5.21327	0.210390
$615$	−3.16493	−0.127622
$616$	−7.03520	−0.283456
$617$	30.9641	1.24657	0.623283	−	0.781996i	$-0.285799\pi$
$617$	30.9641	1.24657	0.623283	+	0.781996i	$0.285799\pi$
$618$	20.0654	0.807150
$619$	−21.2408	−0.853739	−0.426870	−	0.904313i	$-0.640384\pi$
$619$	−21.2408	−0.853739	−0.426870	+	0.904313i	$0.640384\pi$
$620$	−5.96468	−0.239547
$621$	−4.92408	−0.197597
$622$	26.1708	1.04935
$623$	−5.31942	−0.213118
$624$	−1.48599	−0.0594873
$625$	1.00000	0.0400000
$626$	−24.3335	−0.972561
$627$	−6.40333	−0.255724
$628$	45.1657	1.80231
$629$	−33.2998	−1.32775
$630$	18.1711	0.723955
$631$	37.3437	1.48663	0.743316	−	0.668941i	$-0.233252\pi$
$631$	37.3437	1.48663	0.743316	+	0.668941i	$0.233252\pi$
$632$	−0.360625	−0.0143449
$633$	13.9851	0.555860
$634$	6.74800	0.267997
$635$	15.4908	0.614734
$636$	−7.76282	−0.307816
$637$	−4.52621	−0.179335
$638$	−32.5133	−1.28721
$639$	−20.6616	−0.817361
$640$	10.6823	0.422255
$641$	−14.8523	−0.586630	−0.293315	−	0.956016i	$-0.594758\pi$
$641$	−14.8523	−0.586630	−0.293315	+	0.956016i	$0.594758\pi$
$642$	−14.3678	−0.567051
$643$	13.4365	0.529885	0.264943	−	0.964264i	$-0.414647\pi$
$643$	13.4365	0.529885	0.264943	+	0.964264i	$0.414647\pi$
$644$	−10.9540	−0.431649
$645$	−7.94037	−0.312652
$646$	−40.3141	−1.58614
$647$	24.1928	0.951117	0.475559	−	0.879684i	$-0.342246\pi$
$647$	24.1928	0.951117	0.475559	+	0.879684i	$0.342246\pi$
$648$	−5.53854	−0.217574
$649$	−9.66559	−0.379408
$650$	1.82095	0.0714237
$651$	6.14315	0.240769
$652$	64.8982	2.54161
$653$	−25.4119	−0.994445	−0.497222	−	0.867623i	$-0.665647\pi$
$653$	−25.4119	−0.994445	−0.497222	+	0.867623i	$0.665647\pi$
$654$	21.1575	0.827323
$655$	−21.0257	−0.821541
$656$	−9.21945	−0.359959
$657$	0.546031	0.0213027
$658$	6.98399	0.272264
$659$	−28.9009	−1.12582	−0.562910	−	0.826518i	$-0.690318\pi$
$659$	−28.9009	−1.12582	−0.562910	+	0.826518i	$0.690318\pi$
$660$	−2.92635	−0.113908
$661$	3.42323	0.133148	0.0665740	−	0.997781i	$-0.478793\pi$
$661$	3.42323	0.133148	0.0665740	+	0.997781i	$0.478793\pi$
$662$	45.2618	1.75915
$663$	2.11161	0.0820081
$664$	10.6398	0.412904
$665$	20.4181	0.791779
$666$	−53.4803	−2.07232
$667$	−12.4736	−0.482981
$668$	−11.7327	−0.453950
$669$	−11.6009	−0.448517
$670$	−2.54935	−0.0984898
$671$	−7.39253	−0.285385
$672$	−21.0649	−0.812597
$673$	8.03559	0.309749	0.154875	−	0.987934i	$-0.450503\pi$
$673$	8.03559	0.309749	0.154875	+	0.987934i	$0.450503\pi$
$674$	3.91949	0.150973
$675$	4.19456	0.161449
$676$	−32.6100	−1.25423
$677$	−29.1427	−1.12004	−0.560022	−	0.828478i	$-0.689207\pi$
$677$	−29.1427	−1.12004	−0.560022	+	0.828478i	$0.689207\pi$
$678$	−6.43096	−0.246979
$679$	23.3886	0.897572
$680$	−4.53955	−0.174084
$681$	−19.4905	−0.746879
$682$	6.87711	0.263338
$683$	20.7107	0.792475	0.396237	−	0.918148i	$-0.370316\pi$
$683$	20.7107	0.792475	0.396237	+	0.918148i	$0.370316\pi$
$684$	−36.9214	−1.41172
$685$	−2.00321	−0.0765388
$686$	12.4226	0.474296
$687$	2.32864	0.0888432
$688$	−23.1304	−0.881836
$689$	−3.17599	−0.120996
$690$	−1.96875	−0.0749490
$691$	45.2796	1.72252	0.861259	−	0.508166i	$-0.169676\pi$
$691$	45.2796	1.72252	0.861259	+	0.508166i	$0.169676\pi$
$692$	2.76464	0.105096
$693$	−11.9473	−0.453840
$694$	28.1372	1.06807
$695$	0.880638	0.0334045
$696$	11.6527	0.441696
$697$	13.1009	0.496233
$698$	−67.1462	−2.54152
$699$	−9.71747	−0.367549
$700$	9.33116	0.352685
$701$	−0.162626	−0.00614230	−0.00307115	−	0.999995i	$-0.500978\pi$
$701$	−0.162626	−0.00614230	−0.00307115	+	0.999995i	$0.500978\pi$
$702$	7.63811	0.288282
$703$	−60.0935	−2.26647
$704$	−17.1576	−0.646651
$705$	0.715795	0.0269584
$706$	−17.7180	−0.666827
$707$	47.0364	1.76898
$708$	14.0592	0.528378
$709$	−36.2654	−1.36197	−0.680987	−	0.732295i	$-0.738449\pi$
$709$	−36.2654	−1.36197	−0.680987	+	0.732295i	$0.738449\pi$
$710$	−18.6058	−0.698265
$711$	−0.612419	−0.0229675
$712$	2.13426	0.0799847
$713$	2.63839	0.0988084
$714$	18.9750	0.710121
$715$	−1.19725	−0.0447748
$716$	−59.8326	−2.23605
$717$	9.02215	0.336938
$718$	51.3986	1.91818
$719$	3.05179	0.113813	0.0569064	−	0.998380i	$-0.481876\pi$
$719$	3.05179	0.113813	0.0569064	+	0.998380i	$0.481876\pi$
$720$	5.42511	0.202182
$721$	42.0673	1.56667
$722$	−31.7629	−1.18209
$723$	6.60562	0.245666
$724$	21.6439	0.804390
$725$	10.6256	0.394626
$726$	−15.0738	−0.559441
$727$	21.8333	0.809751	0.404876	−	0.914372i	$-0.367315\pi$
$727$	21.8333	0.809751	0.404876	+	0.914372i	$0.367315\pi$
$728$	4.18667	0.155168
$729$	0.376882	0.0139586
$730$	0.491702	0.0181987
$731$	32.8685	1.21568
$732$	10.7529	0.397439
$733$	12.4432	0.459602	0.229801	−	0.973238i	$-0.426193\pi$
$733$	12.4432	0.459602	0.229801	+	0.973238i	$0.426193\pi$
$734$	15.4308	0.569562
$735$	−4.16858	−0.153760
$736$	−9.04705	−0.333479
$737$	1.67616	0.0617423
$738$	21.0404	0.774508
$739$	−32.5884	−1.19878	−0.599392	−	0.800456i	$-0.704591\pi$
$739$	−32.5884	−1.19878	−0.599392	+	0.800456i	$0.704591\pi$
$740$	−27.4630	−1.00956
$741$	3.81065	0.139988
$742$	−28.5396	−1.04772
$743$	−2.39562	−0.0878867	−0.0439434	−	0.999034i	$-0.513992\pi$
$743$	−2.39562	−0.0878867	−0.0439434	+	0.999034i	$0.513992\pi$
$744$	−2.46476	−0.0903623
$745$	−3.78046	−0.138505
$746$	−6.08550	−0.222806
$747$	18.0687	0.661098
$748$	12.1134	0.442909
$749$	−30.1222	−1.10064
$750$	1.67707	0.0612380
$751$	30.0247	1.09562	0.547808	−	0.836604i	$-0.315462\pi$
$751$	30.0247	1.09562	0.547808	+	0.836604i	$0.315462\pi$
$752$	2.08512	0.0760364
$753$	11.3369	0.413140
$754$	19.3488	0.704641
$755$	14.3302	0.521530
$756$	39.1401	1.42351
$757$	−11.1719	−0.406048	−0.203024	−	0.979174i	$-0.565077\pi$
$757$	−11.1719	−0.406048	−0.203024	+	0.979174i	$0.565077\pi$
$758$	55.1046	2.00149
$759$	1.29443	0.0469848
$760$	−8.19214	−0.297160
$761$	−0.347349	−0.0125914	−0.00629570	−	0.999980i	$-0.502004\pi$
$761$	−0.347349	−0.0125914	−0.00629570	+	0.999980i	$0.502004\pi$
$762$	25.9792	0.941127
$763$	44.3568	1.60582
$764$	5.41985	0.196083
$765$	−7.70913	−0.278724
$766$	36.5867	1.32193
$767$	5.75203	0.207694
$768$	−0.892576	−0.0322081
$769$	−43.5425	−1.57018	−0.785092	−	0.619380i	$-0.787384\pi$
$769$	−43.5425	−1.57018	−0.785092	+	0.619380i	$0.787384\pi$
$770$	−10.7586	−0.387712
$771$	15.2655	0.549775
$772$	65.9645	2.37411
$773$	−16.9878	−0.611008	−0.305504	−	0.952191i	$-0.598825\pi$
$773$	−16.9878	−0.611008	−0.305504	+	0.952191i	$0.598825\pi$
$774$	52.7875	1.89741
$775$	−2.24750	−0.0807327
$776$	−9.38398	−0.336865
$777$	28.2847	1.01471
$778$	−8.13424	−0.291627
$779$	23.6422	0.847069
$780$	1.74148	0.0623551
$781$	12.2331	0.437735
$782$	8.14946	0.291424
$783$	44.5699	1.59280
$784$	−12.1431	−0.433682
$785$	17.0185	0.607417
$786$	−35.2616	−1.25774
$787$	0.727705	0.0259399	0.0129699	−	0.999916i	$-0.495871\pi$
$787$	0.727705	0.0259399	0.0129699	+	0.999916i	$0.495871\pi$
$788$	−50.4772	−1.79818
$789$	18.6557	0.664162
$790$	−0.551485	−0.0196210
$791$	−13.4826	−0.479384
$792$	4.79349	0.170329
$793$	4.39932	0.156225
$794$	41.1158	1.45914
$795$	−2.92505	−0.103741
$796$	−20.1105	−0.712799
$797$	19.3088	0.683953	0.341977	−	0.939708i	$-0.388904\pi$
$797$	19.3088	0.683953	0.341977	+	0.939708i	$0.388904\pi$
$798$	34.2426	1.21217
$799$	−2.96297	−0.104822
$800$	7.70670	0.272473
$801$	3.62443	0.128063
$802$	−20.1736	−0.712354
$803$	−0.323288	−0.0114086
$804$	−2.43809	−0.0859847
$805$	−4.12750	−0.145475
$806$	−4.09260	−0.144156
$807$	−12.5823	−0.442917
$808$	−18.8719	−0.663913
$809$	14.1642	0.497988	0.248994	−	0.968505i	$-0.419900\pi$
$809$	14.1642	0.497988	0.248994	+	0.968505i	$0.419900\pi$
$810$	−8.46980	−0.297599
$811$	−24.4197	−0.857493	−0.428746	−	0.903425i	$-0.641045\pi$
$811$	−24.4197	−0.857493	−0.428746	+	0.903425i	$0.641045\pi$
$812$	99.1494	3.47946
$813$	9.62921	0.337711
$814$	31.6641	1.10983
$815$	24.4538	0.856578
$816$	5.66511	0.198319
$817$	59.3150	2.07517
$818$	73.5102	2.57022
$819$	7.10988	0.248439
$820$	10.8046	0.377312
$821$	11.4187	0.398516	0.199258	−	0.979947i	$-0.436147\pi$
$821$	11.4187	0.398516	0.199258	+	0.979947i	$0.436147\pi$
$822$	−3.35953	−0.117177
$823$	42.0707	1.46649	0.733246	−	0.679963i	$-0.238004\pi$
$823$	42.0707	1.46649	0.733246	+	0.679963i	$0.238004\pi$
$824$	−16.8783	−0.587982
$825$	−1.10265	−0.0383895
$826$	51.6879	1.79845
$827$	−17.2977	−0.601499	−0.300749	−	0.953703i	$-0.597237\pi$
$827$	−17.2977	−0.601499	−0.300749	+	0.953703i	$0.597237\pi$
$828$	7.46362	0.259379
$829$	44.1462	1.53326	0.766631	−	0.642089i	$-0.221932\pi$
$829$	44.1462	1.53326	0.766631	+	0.642089i	$0.221932\pi$
$830$	16.2709	0.564771
$831$	−10.1737	−0.352923
$832$	10.2106	0.353987
$833$	17.2555	0.597866
$834$	1.47689	0.0511406
$835$	−4.42088	−0.152991
$836$	21.8600	0.756044
$837$	−9.42729	−0.325855
$838$	−59.6517	−2.06063
$839$	−34.5742	−1.19363	−0.596817	−	0.802377i	$-0.703568\pi$
$839$	−34.5742	−1.19363	−0.596817	+	0.802377i	$0.703568\pi$
$840$	3.85587	0.133040
$841$	83.9039	2.89324
$842$	56.6894	1.95365
$843$	3.49046	0.120218
$844$	−47.7432	−1.64339
$845$	−12.2875	−0.422703
$846$	−4.75860	−0.163604
$847$	−31.6023	−1.08587
$848$	−8.52068	−0.292601
$849$	−13.2317	−0.454111
$850$	−6.94210	−0.238112
$851$	12.1478	0.416423
$852$	−17.7938	−0.609607
$853$	18.3808	0.629346	0.314673	−	0.949200i	$-0.398105\pi$
$853$	18.3808	0.629346	0.314673	+	0.949200i	$0.398105\pi$
$854$	39.5325	1.35277
$855$	−13.9120	−0.475781
$856$	12.0856	0.413078
$857$	27.8364	0.950872	0.475436	−	0.879750i	$-0.342290\pi$
$857$	27.8364	0.950872	0.475436	+	0.879750i	$0.342290\pi$
$858$	−2.00788	−0.0685480
$859$	−5.93552	−0.202517	−0.101259	−	0.994860i	$-0.532287\pi$
$859$	−5.93552	−0.202517	−0.101259	+	0.994860i	$0.532287\pi$
$860$	27.1072	0.924349
$861$	−11.1279	−0.379237
$862$	−16.7577	−0.570768
$863$	45.7180	1.55626	0.778129	−	0.628105i	$-0.216169\pi$
$863$	45.7180	1.55626	0.778129	+	0.628105i	$0.216169\pi$
$864$	32.3263	1.09976
$865$	1.04172	0.0354195
$866$	−24.0987	−0.818908
$867$	5.16557	0.175432
$868$	−20.9718	−0.711829
$869$	0.362595	0.0123002
$870$	17.8199	0.604153
$871$	−0.997492	−0.0337987
$872$	−17.7968	−0.602677
$873$	−15.9360	−0.539353
$874$	14.7067	0.497460
$875$	3.51600	0.118862
$876$	0.470244	0.0158881
$877$	−17.6798	−0.597004	−0.298502	−	0.954409i	$-0.596487\pi$
$877$	−17.6798	−0.597004	−0.298502	+	0.954409i	$0.596487\pi$
$878$	2.61751	0.0883367
$879$	16.6103	0.560250
$880$	−3.21204	−0.108278
$881$	49.7026	1.67452	0.837262	−	0.546802i	$-0.184155\pi$
$881$	49.7026	1.67452	0.837262	+	0.546802i	$0.184155\pi$
$882$	27.7127	0.933134
$883$	36.9290	1.24276	0.621381	−	0.783509i	$-0.286572\pi$
$883$	36.9290	1.24276	0.621381	+	0.783509i	$0.286572\pi$
$884$	−7.20872	−0.242456
$885$	5.29754	0.178075
$886$	−9.42559	−0.316659
$887$	6.82050	0.229010	0.114505	−	0.993423i	$-0.463472\pi$
$887$	6.82050	0.229010	0.114505	+	0.993423i	$0.463472\pi$
$888$	−11.3484	−0.380827
$889$	54.4656	1.82672
$890$	3.26381	0.109403
$891$	5.56879	0.186561
$892$	39.6038	1.32603
$893$	−5.34703	−0.178931
$894$	−6.34011	−0.212045
$895$	−22.5450	−0.753597
$896$	37.5589	1.25475
$897$	−0.770319	−0.0257202
$898$	10.8276	0.361323
$899$	−23.8811	−0.796480
$900$	−6.35787	−0.211929
$901$	12.1080	0.403375
$902$	−12.4574	−0.414786
$903$	−27.9183	−0.929063
$904$	5.40947	0.179916
$905$	8.15547	0.271097
$906$	24.0328	0.798436
$907$	38.3604	1.27373	0.636867	−	0.770973i	$-0.280230\pi$
$907$	38.3604	1.27373	0.636867	+	0.770973i	$0.280230\pi$
$908$	66.5378	2.20813
$909$	−32.0486	−1.06299
$910$	6.40247	0.212240
$911$	34.9502	1.15795	0.578976	−	0.815345i	$-0.303453\pi$
$911$	34.9502	1.15795	0.578976	+	0.815345i	$0.303453\pi$
$912$	10.2234	0.338529
$913$	−10.6979	−0.354049
$914$	−73.6061	−2.43467
$915$	4.05171	0.133946
$916$	−7.94963	−0.262663
$917$	−73.9262	−2.44126
$918$	−29.1191	−0.961072
$919$	−49.3951	−1.62939	−0.814697	−	0.579886i	$-0.803097\pi$
$919$	−49.3951	−1.62939	−0.814697	+	0.579886i	$0.803097\pi$
$920$	1.65603	0.0545978
$921$	1.87864	0.0619033
$922$	−26.8807	−0.885269
$923$	−7.27997	−0.239623
$924$	−10.2890	−0.338485
$925$	−10.3481	−0.340244
$926$	−27.9172	−0.917417
$927$	−28.6629	−0.941415
$928$	81.8885	2.68812
$929$	0.699666	0.0229553	0.0114776	−	0.999934i	$-0.496346\pi$
$929$	0.699666	0.0229553	0.0114776	+	0.999934i	$0.496346\pi$
$930$	−3.76922	−0.123598
$931$	31.1395	1.02056
$932$	33.1740	1.08665
$933$	9.43083	0.308752
$934$	−41.2476	−1.34966
$935$	4.56434	0.149270
$936$	−2.85263	−0.0932410
$937$	37.5526	1.22679	0.613395	−	0.789776i	$-0.289803\pi$
$937$	37.5526	1.22679	0.613395	+	0.789776i	$0.289803\pi$
$938$	−8.96349	−0.292668
$939$	−8.76875	−0.286157
$940$	−2.44362	−0.0797020
$941$	−37.1722	−1.21178	−0.605890	−	0.795549i	$-0.707183\pi$
$941$	−37.1722	−1.21178	−0.605890	+	0.795549i	$0.707183\pi$
$942$	28.5413	0.929926
$943$	−4.77924	−0.155634
$944$	15.4318	0.502262
$945$	14.7481	0.479755
$946$	−31.2539	−1.01615
$947$	−10.9518	−0.355885	−0.177943	−	0.984041i	$-0.556944\pi$
$947$	−10.9518	−0.355885	−0.177943	+	0.984041i	$0.556944\pi$
$948$	−0.527418	−0.0171297
$949$	0.192390	0.00624525
$950$	−12.5278	−0.406456
$951$	2.43169	0.0788530
$952$	−15.9610	−0.517300
$953$	15.1236	0.489903	0.244951	−	0.969535i	$-0.421228\pi$
$953$	15.1236	0.489903	0.244951	+	0.969535i	$0.421228\pi$
$954$	19.4457	0.629578
$955$	2.04221	0.0660843
$956$	−30.8003	−0.996152
$957$	−11.7164	−0.378737
$958$	−43.0868	−1.39207
$959$	−7.04329	−0.227440
$960$	9.40378	0.303506
$961$	−25.9487	−0.837056
$962$	−18.8434	−0.607536
$963$	20.5240	0.661377
$964$	−22.5506	−0.726306
$965$	24.8555	0.800128
$966$	−6.92211	−0.222715
$967$	−13.6569	−0.439177	−0.219589	−	0.975593i	$-0.570472\pi$
$967$	−13.6569	−0.439177	−0.219589	+	0.975593i	$0.570472\pi$
$968$	12.6795	0.407534
$969$	−14.5275	−0.466690
$970$	−14.3504	−0.460765
$971$	40.1991	1.29005	0.645025	−	0.764161i	$-0.276847\pi$
$971$	40.1991	1.29005	0.645025	+	0.764161i	$0.276847\pi$
$972$	−41.4962	−1.33099
$973$	3.09632	0.0992634
$974$	−62.8959	−2.01531
$975$	0.656194	0.0210150
$976$	11.8027	0.377794
$977$	−31.0219	−0.992477	−0.496239	−	0.868186i	$-0.665286\pi$
$977$	−31.0219	−0.992477	−0.496239	+	0.868186i	$0.665286\pi$
$978$	41.0108	1.31138
$979$	−2.14592	−0.0685838
$980$	14.2309	0.454589
$981$	−30.2229	−0.964943
$982$	−46.9664	−1.49876
$983$	10.7922	0.344217	0.172109	−	0.985078i	$-0.444942\pi$
$983$	10.7922	0.344217	0.172109	+	0.985078i	$0.444942\pi$
$984$	4.46472	0.142330
$985$	−19.0199	−0.606024
$986$	−73.7641	−2.34913
$987$	2.51673	0.0801085
$988$	−13.0090	−0.413871
$989$	−11.9905	−0.381275
$990$	7.33044	0.232977
$991$	−2.47524	−0.0786286	−0.0393143	−	0.999227i	$-0.512517\pi$
$991$	−2.47524	−0.0786286	−0.0393143	+	0.999227i	$0.512517\pi$
$992$	−17.3208	−0.549937
$993$	16.3104	0.517595
$994$	−65.4181	−2.07493
$995$	−7.57768	−0.240229
$996$	15.5608	0.493063
$997$	21.5380	0.682114	0.341057	−	0.940043i	$-0.389215\pi$
$997$	21.5380	0.682114	0.341057	+	0.940043i	$0.389215\pi$
$998$	−29.6490	−0.938524
$999$	−43.4058	−1.37330

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8045.2.a.e.1.18	✓	142

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8045.2.a.e.1.18	✓	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)

\(f(q)\)	\(=\)	\(q-2.15729 q^{2} -0.777397 q^{3} +2.65392 q^{4} +1.00000 q^{5} +1.67707 q^{6} +3.51600 q^{7} -1.41069 q^{8} -2.39565 q^{9} +O(q^{10})\)	\(q-2.15729 q^{2} -0.777397 q^{3} +2.65392 q^{4} +1.00000 q^{5} +1.67707 q^{6} +3.51600 q^{7} -1.41069 q^{8} -2.39565 q^{9} -2.15729 q^{10} +1.41839 q^{11} -2.06315 q^{12} -0.844092 q^{13} -7.58504 q^{14} -0.777397 q^{15} -2.26456 q^{16} +3.21797 q^{17} +5.16813 q^{18} +5.80719 q^{19} +2.65392 q^{20} -2.73332 q^{21} -3.05989 q^{22} -1.17392 q^{23} +1.09666 q^{24} +1.00000 q^{25} +1.82095 q^{26} +4.19456 q^{27} +9.33116 q^{28} +10.6256 q^{29} +1.67707 q^{30} -2.24750 q^{31} +7.70670 q^{32} -1.10265 q^{33} -6.94210 q^{34} +3.51600 q^{35} -6.35787 q^{36} -10.3481 q^{37} -12.5278 q^{38} +0.656194 q^{39} -1.41069 q^{40} +4.07119 q^{41} +5.89658 q^{42} +10.2140 q^{43} +3.76430 q^{44} -2.39565 q^{45} +2.53249 q^{46} -0.920759 q^{47} +1.76046 q^{48} +5.36223 q^{49} -2.15729 q^{50} -2.50164 q^{51} -2.24015 q^{52} +3.76262 q^{53} -9.04891 q^{54} +1.41839 q^{55} -4.95997 q^{56} -4.51449 q^{57} -22.9226 q^{58} -6.81446 q^{59} -2.06315 q^{60} -5.21190 q^{61} +4.84852 q^{62} -8.42311 q^{63} -12.0965 q^{64} -0.844092 q^{65} +2.37875 q^{66} +1.18173 q^{67} +8.54021 q^{68} +0.912601 q^{69} -7.58504 q^{70} +8.62462 q^{71} +3.37952 q^{72} -0.227926 q^{73} +22.3239 q^{74} -0.777397 q^{75} +15.4118 q^{76} +4.98707 q^{77} -1.41560 q^{78} +0.255638 q^{79} -2.26456 q^{80} +3.92612 q^{81} -8.78274 q^{82} -7.54227 q^{83} -7.25401 q^{84} +3.21797 q^{85} -22.0347 q^{86} -8.26033 q^{87} -2.00091 q^{88} -1.51292 q^{89} +5.16813 q^{90} -2.96782 q^{91} -3.11548 q^{92} +1.74720 q^{93} +1.98635 q^{94} +5.80719 q^{95} -5.99116 q^{96} +6.65206 q^{97} -11.5679 q^{98} -3.39798 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 142 q + 21 q^{2} + 33 q^{3} + 157 q^{4} + 142 q^{5} + 15 q^{6} + 63 q^{7} + 60 q^{8} + 157 q^{9}+O(q^{10}) \) 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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