LMFDB - Embedded newform 8038.2.a.d.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8038,2,Mod(1,8038)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8038, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8038.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8038 = 2 \cdot 4019 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8038.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:	$64.1837531447$
Analytic rank:	$0$
Dimension:	$92$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	8038.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.40015 q^{3} +1.00000 q^{4} -4.19507 q^{5} -2.40015 q^{6} +1.33387 q^{7} +1.00000 q^{8} +2.76072 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.40015 q^{3} +1.00000 q^{4} -4.19507 q^{5} -2.40015 q^{6} +1.33387 q^{7} +1.00000 q^{8} +2.76072 q^{9} -4.19507 q^{10} +5.14951 q^{11} -2.40015 q^{12} +1.62772 q^{13} +1.33387 q^{14} +10.0688 q^{15} +1.00000 q^{16} +1.87057 q^{17} +2.76072 q^{18} +2.99832 q^{19} -4.19507 q^{20} -3.20148 q^{21} +5.14951 q^{22} -2.03990 q^{23} -2.40015 q^{24} +12.5986 q^{25} +1.62772 q^{26} +0.574317 q^{27} +1.33387 q^{28} -5.90363 q^{29} +10.0688 q^{30} +5.80616 q^{31} +1.00000 q^{32} -12.3596 q^{33} +1.87057 q^{34} -5.59566 q^{35} +2.76072 q^{36} -5.13619 q^{37} +2.99832 q^{38} -3.90678 q^{39} -4.19507 q^{40} +9.73483 q^{41} -3.20148 q^{42} -0.509499 q^{43} +5.14951 q^{44} -11.5814 q^{45} -2.03990 q^{46} -0.611929 q^{47} -2.40015 q^{48} -5.22080 q^{49} +12.5986 q^{50} -4.48965 q^{51} +1.62772 q^{52} +9.20743 q^{53} +0.574317 q^{54} -21.6026 q^{55} +1.33387 q^{56} -7.19641 q^{57} -5.90363 q^{58} +9.32025 q^{59} +10.0688 q^{60} +0.934648 q^{61} +5.80616 q^{62} +3.68243 q^{63} +1.00000 q^{64} -6.82841 q^{65} -12.3596 q^{66} -9.06045 q^{67} +1.87057 q^{68} +4.89606 q^{69} -5.59566 q^{70} -10.7495 q^{71} +2.76072 q^{72} +3.49027 q^{73} -5.13619 q^{74} -30.2385 q^{75} +2.99832 q^{76} +6.86876 q^{77} -3.90678 q^{78} -6.44152 q^{79} -4.19507 q^{80} -9.66059 q^{81} +9.73483 q^{82} -3.00764 q^{83} -3.20148 q^{84} -7.84717 q^{85} -0.509499 q^{86} +14.1696 q^{87} +5.14951 q^{88} +5.63579 q^{89} -11.5814 q^{90} +2.17117 q^{91} -2.03990 q^{92} -13.9357 q^{93} -0.611929 q^{94} -12.5782 q^{95} -2.40015 q^{96} +8.38103 q^{97} -5.22080 q^{98} +14.2163 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 92 q + 92 q^{2} + 31 q^{3} + 92 q^{4} + 28 q^{5} + 31 q^{6} + 29 q^{7} + 92 q^{8} + 113 q^{9}+O(q^{10}) $ 92 * q + 92 * q^2 + 31 * q^3 + 92 * q^4 + 28 * q^5 + 31 * q^6 + 29 * q^7 + 92 * q^8 + 113 * q^9	\( 92 q + 92 q^{2} + 31 q^{3} + 92 q^{4} + 28 q^{5} + 31 q^{6} + 29 q^{7} + 92 q^{8} + 113 q^{9} + 28 q^{10} + 37 q^{11} + 31 q^{12} + 20 q^{13} + 29 q^{14} + 30 q^{15} + 92 q^{16} + 52 q^{17} + 113 q^{18} + 61 q^{19} + 28 q^{20} + 5 q^{21} + 37 q^{22} + 71 q^{23} + 31 q^{24} + 118 q^{25} + 20 q^{26} + 112 q^{27} + 29 q^{28} + 30 q^{29} + 30 q^{30} + 89 q^{31} + 92 q^{32} + 52 q^{33} + 52 q^{34} + 58 q^{35} + 113 q^{36} + 15 q^{37} + 61 q^{38} + 43 q^{39} + 28 q^{40} + 75 q^{41} + 5 q^{42} + 46 q^{43} + 37 q^{44} + 63 q^{45} + 71 q^{46} + 92 q^{47} + 31 q^{48} + 131 q^{49} + 118 q^{50} + 45 q^{51} + 20 q^{52} + 72 q^{53} + 112 q^{54} + 86 q^{55} + 29 q^{56} + 44 q^{57} + 30 q^{58} + 95 q^{59} + 30 q^{60} - 4 q^{61} + 89 q^{62} + 67 q^{63} + 92 q^{64} + 55 q^{65} + 52 q^{66} + 40 q^{67} + 52 q^{68} + 25 q^{69} + 58 q^{70} + 84 q^{71} + 113 q^{72} + 87 q^{73} + 15 q^{74} + 132 q^{75} + 61 q^{76} + 96 q^{77} + 43 q^{78} + 68 q^{79} + 28 q^{80} + 156 q^{81} + 75 q^{82} + 120 q^{83} + 5 q^{84} - 14 q^{85} + 46 q^{86} + 73 q^{87} + 37 q^{88} + 86 q^{89} + 63 q^{90} + 93 q^{91} + 71 q^{92} + 29 q^{93} + 92 q^{94} + 67 q^{95} + 31 q^{96} + 65 q^{97} + 131 q^{98} + 94 q^{99}+O(q^{100}) \) 92 * q + 92 * q^2 + 31 * q^3 + 92 * q^4 + 28 * q^5 + 31 * q^6 + 29 * q^7 + 92 * q^8 + 113 * q^9 + 28 * q^10 + 37 * q^11 + 31 * q^12 + 20 * q^13 + 29 * q^14 + 30 * q^15 + 92 * q^16 + 52 * q^17 + 113 * q^18 + 61 * q^19 + 28 * q^20 + 5 * q^21 + 37 * q^22 + 71 * q^23 + 31 * q^24 + 118 * q^25 + 20 * q^26 + 112 * q^27 + 29 * q^28 + 30 * q^29 + 30 * q^30 + 89 * q^31 + 92 * q^32 + 52 * q^33 + 52 * q^34 + 58 * q^35 + 113 * q^36 + 15 * q^37 + 61 * q^38 + 43 * q^39 + 28 * q^40 + 75 * q^41 + 5 * q^42 + 46 * q^43 + 37 * q^44 + 63 * q^45 + 71 * q^46 + 92 * q^47 + 31 * q^48 + 131 * q^49 + 118 * q^50 + 45 * q^51 + 20 * q^52 + 72 * q^53 + 112 * q^54 + 86 * q^55 + 29 * q^56 + 44 * q^57 + 30 * q^58 + 95 * q^59 + 30 * q^60 - 4 * q^61 + 89 * q^62 + 67 * q^63 + 92 * q^64 + 55 * q^65 + 52 * q^66 + 40 * q^67 + 52 * q^68 + 25 * q^69 + 58 * q^70 + 84 * q^71 + 113 * q^72 + 87 * q^73 + 15 * q^74 + 132 * q^75 + 61 * q^76 + 96 * q^77 + 43 * q^78 + 68 * q^79 + 28 * q^80 + 156 * q^81 + 75 * q^82 + 120 * q^83 + 5 * q^84 - 14 * q^85 + 46 * q^86 + 73 * q^87 + 37 * q^88 + 86 * q^89 + 63 * q^90 + 93 * q^91 + 71 * q^92 + 29 * q^93 + 92 * q^94 + 67 * q^95 + 31 * q^96 + 65 * q^97 + 131 * q^98 + 94 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−2.40015	−1.38573	−0.692863	−	0.721069i	$-0.743651\pi$
$3$	−2.40015	−1.38573	−0.692863	+	0.721069i	$0.743651\pi$
$4$	1.00000	0.500000
$5$	−4.19507	−1.87609	−0.938046	−	0.346511i	$-0.887366\pi$
$5$	−4.19507	−1.87609	−0.938046	+	0.346511i	$0.887366\pi$
$6$	−2.40015	−0.979857
$7$	1.33387	0.504154	0.252077	−	0.967707i	$-0.418886\pi$
$7$	1.33387	0.504154	0.252077	+	0.967707i	$0.418886\pi$
$8$	1.00000	0.353553
$9$	2.76072	0.920239
$10$	−4.19507	−1.32660
$11$	5.14951	1.55264	0.776318	−	0.630341i	$-0.217085\pi$
$11$	5.14951	1.55264	0.776318	+	0.630341i	$0.217085\pi$
$12$	−2.40015	−0.692863
$13$	1.62772	0.451449	0.225725	−	0.974191i	$-0.427525\pi$
$13$	1.62772	0.451449	0.225725	+	0.974191i	$0.427525\pi$
$14$	1.33387	0.356491
$15$	10.0688	2.59975
$16$	1.00000	0.250000
$17$	1.87057	0.453680	0.226840	−	0.973932i	$-0.427161\pi$
$17$	1.87057	0.453680	0.226840	+	0.973932i	$0.427161\pi$
$18$	2.76072	0.650707
$19$	2.99832	0.687861	0.343931	−	0.938995i	$-0.388241\pi$
$19$	2.99832	0.687861	0.343931	+	0.938995i	$0.388241\pi$
$20$	−4.19507	−0.938046
$21$	−3.20148	−0.698620
$22$	5.14951	1.09788
$23$	−2.03990	−0.425348	−0.212674	−	0.977123i	$-0.568217\pi$
$23$	−2.03990	−0.425348	−0.212674	+	0.977123i	$0.568217\pi$
$24$	−2.40015	−0.489928
$25$	12.5986	2.51972
$26$	1.62772	0.319223
$27$	0.574317	0.110527
$28$	1.33387	0.252077
$29$	−5.90363	−1.09628	−0.548138	−	0.836388i	$-0.684663\pi$
$29$	−5.90363	−1.09628	−0.548138	+	0.836388i	$0.684663\pi$
$30$	10.0688	1.83830
$31$	5.80616	1.04282	0.521409	−	0.853307i	$-0.325407\pi$
$31$	5.80616	1.04282	0.521409	+	0.853307i	$0.325407\pi$
$32$	1.00000	0.176777
$33$	−12.3596	−2.15153
$34$	1.87057	0.320800
$35$	−5.59566	−0.945839
$36$	2.76072	0.460119
$37$	−5.13619	−0.844385	−0.422193	−	0.906506i	$-0.638739\pi$
$37$	−5.13619	−0.844385	−0.422193	+	0.906506i	$0.638739\pi$
$38$	2.99832	0.486391
$39$	−3.90678	−0.625585
$40$	−4.19507	−0.663299
$41$	9.73483	1.52032	0.760162	−	0.649733i	$-0.225119\pi$
$41$	9.73483	1.52032	0.760162	+	0.649733i	$0.225119\pi$
$42$	−3.20148	−0.493999
$43$	−0.509499	−0.0776979	−0.0388489	−	0.999245i	$-0.512369\pi$
$43$	−0.509499	−0.0776979	−0.0388489	+	0.999245i	$0.512369\pi$
$44$	5.14951	0.776318
$45$	−11.5814	−1.72645
$46$	−2.03990	−0.300766
$47$	−0.611929	−0.0892590	−0.0446295	−	0.999004i	$-0.514211\pi$
$47$	−0.611929	−0.0892590	−0.0446295	+	0.999004i	$0.514211\pi$
$48$	−2.40015	−0.346432
$49$	−5.22080	−0.745829
$50$	12.5986	1.78171
$51$	−4.48965	−0.628676
$52$	1.62772	0.225725
$53$	9.20743	1.26474	0.632369	−	0.774668i	$-0.282083\pi$
$53$	9.20743	1.26474	0.632369	+	0.774668i	$0.282083\pi$
$54$	0.574317	0.0781546
$55$	−21.6026	−2.91289
$56$	1.33387	0.178245
$57$	−7.19641	−0.953188
$58$	−5.90363	−0.775185
$59$	9.32025	1.21339	0.606697	−	0.794933i	$-0.292494\pi$
$59$	9.32025	1.21339	0.606697	+	0.794933i	$0.292494\pi$
$60$	10.0688	1.29988
$61$	0.934648	0.119669	0.0598347	−	0.998208i	$-0.480943\pi$
$61$	0.934648	0.119669	0.0598347	+	0.998208i	$0.480943\pi$
$62$	5.80616	0.737383
$63$	3.68243	0.463942
$64$	1.00000	0.125000
$65$	−6.82841	−0.846961
$66$	−12.3596	−1.52136
$67$	−9.06045	−1.10691	−0.553455	−	0.832879i	$-0.686691\pi$
$67$	−9.06045	−1.10691	−0.553455	+	0.832879i	$0.686691\pi$
$68$	1.87057	0.226840
$69$	4.89606	0.589416
$70$	−5.59566	−0.668809
$71$	−10.7495	−1.27573	−0.637865	−	0.770148i	$-0.720182\pi$
$71$	−10.7495	−1.27573	−0.637865	+	0.770148i	$0.720182\pi$
$72$	2.76072	0.325354
$73$	3.49027	0.408506	0.204253	−	0.978918i	$-0.434524\pi$
$73$	3.49027	0.408506	0.204253	+	0.978918i	$0.434524\pi$
$74$	−5.13619	−0.597070
$75$	−30.2385	−3.49165
$76$	2.99832	0.343931
$77$	6.86876	0.782768
$78$	−3.90678	−0.442356
$79$	−6.44152	−0.724727	−0.362364	−	0.932037i	$-0.618030\pi$
$79$	−6.44152	−0.724727	−0.362364	+	0.932037i	$0.618030\pi$
$80$	−4.19507	−0.469023
$81$	−9.66059	−1.07340
$82$	9.73483	1.07503
$83$	−3.00764	−0.330131	−0.165066	−	0.986283i	$-0.552784\pi$
$83$	−3.00764	−0.330131	−0.165066	+	0.986283i	$0.552784\pi$
$84$	−3.20148	−0.349310
$85$	−7.84717	−0.851145
$86$	−0.509499	−0.0549407
$87$	14.1696	1.51914
$88$	5.14951	0.548940
$89$	5.63579	0.597393	0.298696	−	0.954348i	$-0.403448\pi$
$89$	5.63579	0.597393	0.298696	+	0.954348i	$0.403448\pi$
$90$	−11.5814	−1.22079
$91$	2.17117	0.227600
$92$	−2.03990	−0.212674
$93$	−13.9357	−1.44506
$94$	−0.611929	−0.0631156
$95$	−12.5782	−1.29049
$96$	−2.40015	−0.244964
$97$	8.38103	0.850964	0.425482	−	0.904967i	$-0.360104\pi$
$97$	8.38103	0.850964	0.425482	+	0.904967i	$0.360104\pi$
$98$	−5.22080	−0.527381
$99$	14.2163	1.42880
$100$	12.5986	1.25986
$101$	−1.48948	−0.148209	−0.0741043	−	0.997250i	$-0.523610\pi$
$101$	−1.48948	−0.148209	−0.0741043	+	0.997250i	$0.523610\pi$
$102$	−4.48965	−0.444541
$103$	−10.2150	−1.00651	−0.503256	−	0.864137i	$-0.667865\pi$
$103$	−10.2150	−1.00651	−0.503256	+	0.864137i	$0.667865\pi$
$104$	1.62772	0.159611
$105$	13.4304	1.31067
$106$	9.20743	0.894304
$107$	−3.56194	−0.344346	−0.172173	−	0.985067i	$-0.555079\pi$
$107$	−3.56194	−0.344346	−0.172173	+	0.985067i	$0.555079\pi$
$108$	0.574317	0.0552637
$109$	−6.31799	−0.605154	−0.302577	−	0.953125i	$-0.597847\pi$
$109$	−6.31799	−0.605154	−0.302577	+	0.953125i	$0.597847\pi$
$110$	−21.6026	−2.05972
$111$	12.3276	1.17009
$112$	1.33387	0.126039
$113$	3.95333	0.371898	0.185949	−	0.982559i	$-0.440464\pi$
$113$	3.95333	0.371898	0.185949	+	0.982559i	$0.440464\pi$
$114$	−7.19641	−0.674006
$115$	8.55751	0.797992
$116$	−5.90363	−0.548138
$117$	4.49368	0.415441
$118$	9.32025	0.857999
$119$	2.49509	0.228725
$120$	10.0688	0.919151
$121$	15.5175	1.41068
$122$	0.934648	0.0846191
$123$	−23.3650	−2.10675
$124$	5.80616	0.521409
$125$	−31.8767	−2.85114
$126$	3.68243	0.328057
$127$	−7.54329	−0.669358	−0.334679	−	0.942332i	$-0.608628\pi$
$127$	−7.54329	−0.669358	−0.334679	+	0.942332i	$0.608628\pi$
$128$	1.00000	0.0883883
$129$	1.22287	0.107668
$130$	−6.82841	−0.598892
$131$	11.0865	0.968629	0.484315	−	0.874894i	$-0.339069\pi$
$131$	11.0865	0.968629	0.484315	+	0.874894i	$0.339069\pi$
$132$	−12.3596	−1.07577
$133$	3.99935	0.346788
$134$	−9.06045	−0.782704
$135$	−2.40930	−0.207360
$136$	1.87057	0.160400
$137$	19.2877	1.64786	0.823930	−	0.566691i	$-0.191777\pi$
$137$	19.2877	1.64786	0.823930	+	0.566691i	$0.191777\pi$
$138$	4.89606	0.416780
$139$	−14.1953	−1.20403	−0.602013	−	0.798486i	$-0.705635\pi$
$139$	−14.1953	−1.20403	−0.602013	+	0.798486i	$0.705635\pi$
$140$	−5.59566	−0.472920
$141$	1.46872	0.123689
$142$	−10.7495	−0.902078
$143$	8.38199	0.700937
$144$	2.76072	0.230060
$145$	24.7661	2.05672
$146$	3.49027	0.288857
$147$	12.5307	1.03351
$148$	−5.13619	−0.422193
$149$	−7.70327	−0.631076	−0.315538	−	0.948913i	$-0.602185\pi$
$149$	−7.70327	−0.631076	−0.315538	+	0.948913i	$0.602185\pi$
$150$	−30.2385	−2.46897
$151$	−9.27652	−0.754912	−0.377456	−	0.926027i	$-0.623201\pi$
$151$	−9.27652	−0.754912	−0.377456	+	0.926027i	$0.623201\pi$
$152$	2.99832	0.243196
$153$	5.16411	0.417494
$154$	6.86876	0.553501
$155$	−24.3572	−1.95642
$156$	−3.90678	−0.312793
$157$	−1.93853	−0.154712	−0.0773559	−	0.997004i	$-0.524648\pi$
$157$	−1.93853	−0.154712	−0.0773559	+	0.997004i	$0.524648\pi$
$158$	−6.44152	−0.512459
$159$	−22.0992	−1.75258
$160$	−4.19507	−0.331649
$161$	−2.72095	−0.214441
$162$	−9.66059	−0.759008
$163$	16.2933	1.27619	0.638096	−	0.769957i	$-0.279722\pi$
$163$	16.2933	1.27619	0.638096	+	0.769957i	$0.279722\pi$
$164$	9.73483	0.760162
$165$	51.8494	4.03647
$166$	−3.00764	−0.233438
$167$	19.4707	1.50669	0.753344	−	0.657627i	$-0.228440\pi$
$167$	19.4707	1.50669	0.753344	+	0.657627i	$0.228440\pi$
$168$	−3.20148	−0.246999
$169$	−10.3505	−0.796193
$170$	−7.84717	−0.601851
$171$	8.27750	0.632997
$172$	−0.509499	−0.0388489
$173$	3.69800	0.281153	0.140577	−	0.990070i	$-0.455104\pi$
$173$	3.69800	0.281153	0.140577	+	0.990070i	$0.455104\pi$
$174$	14.1696	1.07419
$175$	16.8049	1.27033
$176$	5.14951	0.388159
$177$	−22.3700	−1.68143
$178$	5.63579	0.422421
$179$	−4.20272	−0.314126	−0.157063	−	0.987589i	$-0.550203\pi$
$179$	−4.20272	−0.314126	−0.157063	+	0.987589i	$0.550203\pi$
$180$	−11.5814	−0.863226
$181$	−6.89696	−0.512647	−0.256324	−	0.966591i	$-0.582511\pi$
$181$	−6.89696	−0.512647	−0.256324	+	0.966591i	$0.582511\pi$
$182$	2.17117	0.160938
$183$	−2.24330	−0.165829
$184$	−2.03990	−0.150383
$185$	21.5467	1.58414
$186$	−13.9357	−1.02181
$187$	9.63253	0.704400
$188$	−0.611929	−0.0446295
$189$	0.766062	0.0557228
$190$	−12.5782	−0.912515
$191$	−1.11616	−0.0807621	−0.0403811	−	0.999184i	$-0.512857\pi$
$191$	−1.11616	−0.0807621	−0.0403811	+	0.999184i	$0.512857\pi$
$192$	−2.40015	−0.173216
$193$	2.20432	0.158671	0.0793353	−	0.996848i	$-0.474720\pi$
$193$	2.20432	0.158671	0.0793353	+	0.996848i	$0.474720\pi$
$194$	8.38103	0.601723
$195$	16.3892	1.17366
$196$	−5.22080	−0.372914
$197$	7.32764	0.522073	0.261037	−	0.965329i	$-0.415936\pi$
$197$	7.32764	0.522073	0.261037	+	0.965329i	$0.415936\pi$
$198$	14.2163	1.01031
$199$	15.4962	1.09850	0.549248	−	0.835660i	$-0.314914\pi$
$199$	15.4962	1.09850	0.549248	+	0.835660i	$0.314914\pi$
$200$	12.5986	0.890856
$201$	21.7464	1.53387
$202$	−1.48948	−0.104799
$203$	−7.87465	−0.552692
$204$	−4.48965	−0.314338
$205$	−40.8383	−2.85227
$206$	−10.2150	−0.711711
$207$	−5.63158	−0.391422
$208$	1.62772	0.112862
$209$	15.4399	1.06800
$210$	13.4304	0.926787
$211$	4.40134	0.303000	0.151500	−	0.988457i	$-0.451590\pi$
$211$	4.40134	0.303000	0.151500	+	0.988457i	$0.451590\pi$
$212$	9.20743	0.632369
$213$	25.8004	1.76781
$214$	−3.56194	−0.243489
$215$	2.13738	0.145768
$216$	0.574317	0.0390773
$217$	7.74464	0.525741
$218$	−6.31799	−0.427908
$219$	−8.37718	−0.566077
$220$	−21.6026	−1.45644
$221$	3.04477	0.204814
$222$	12.3276	0.827377
$223$	18.5562	1.24262	0.621308	−	0.783567i	$-0.286602\pi$
$223$	18.5562	1.24262	0.621308	+	0.783567i	$0.286602\pi$
$224$	1.33387	0.0891227
$225$	34.7812	2.31875
$226$	3.95333	0.262972
$227$	5.38584	0.357470	0.178735	−	0.983897i	$-0.442799\pi$
$227$	5.38584	0.357470	0.178735	+	0.983897i	$0.442799\pi$
$228$	−7.19641	−0.476594
$229$	−4.91922	−0.325071	−0.162536	−	0.986703i	$-0.551967\pi$
$229$	−4.91922	−0.325071	−0.162536	+	0.986703i	$0.551967\pi$
$230$	8.55751	0.564266
$231$	−16.4861	−1.08470
$232$	−5.90363	−0.387592
$233$	26.4639	1.73370	0.866852	−	0.498565i	$-0.166139\pi$
$233$	26.4639	1.73370	0.866852	+	0.498565i	$0.166139\pi$
$234$	4.49368	0.293761
$235$	2.56708	0.167458
$236$	9.32025	0.606697
$237$	15.4606	1.00427
$238$	2.49509	0.161733
$239$	9.14483	0.591530	0.295765	−	0.955261i	$-0.404426\pi$
$239$	9.14483	0.591530	0.295765	+	0.955261i	$0.404426\pi$
$240$	10.0688	0.649938
$241$	7.89595	0.508623	0.254311	−	0.967122i	$-0.418151\pi$
$241$	7.89595	0.508623	0.254311	+	0.967122i	$0.418151\pi$
$242$	15.5175	0.997502
$243$	21.4639	1.37691
$244$	0.934648	0.0598347
$245$	21.9016	1.39924
$246$	−23.3650	−1.48970
$247$	4.88043	0.310535
$248$	5.80616	0.368692
$249$	7.21878	0.457472
$250$	−31.8767	−2.01606
$251$	−25.7836	−1.62745	−0.813724	−	0.581251i	$-0.802563\pi$
$251$	−25.7836	−1.62745	−0.813724	+	0.581251i	$0.802563\pi$
$252$	3.68243	0.231971
$253$	−10.5045	−0.660411
$254$	−7.54329	−0.473308
$255$	18.8344	1.17945
$256$	1.00000	0.0625000
$257$	31.3479	1.95543	0.977713	−	0.209944i	$-0.0673283\pi$
$257$	31.3479	1.95543	0.977713	+	0.209944i	$0.0673283\pi$
$258$	1.22287	0.0761328
$259$	−6.85100	−0.425700
$260$	−6.82841	−0.423480
$261$	−16.2982	−1.00884
$262$	11.0865	0.684924
$263$	16.0129	0.987398	0.493699	−	0.869633i	$-0.335645\pi$
$263$	16.0129	0.987398	0.493699	+	0.869633i	$0.335645\pi$
$264$	−12.3596	−0.760681
$265$	−38.6258	−2.37276
$266$	3.99935	0.245216
$267$	−13.5267	−0.827823
$268$	−9.06045	−0.553455
$269$	−11.8507	−0.722547	−0.361274	−	0.932460i	$-0.617658\pi$
$269$	−11.8507	−0.722547	−0.361274	+	0.932460i	$0.617658\pi$
$270$	−2.40930	−0.146625
$271$	−17.7795	−1.08003	−0.540014	−	0.841656i	$-0.681581\pi$
$271$	−17.7795	−1.08003	−0.540014	+	0.841656i	$0.681581\pi$
$272$	1.87057	0.113420
$273$	−5.21112	−0.315391
$274$	19.2877	1.16521
$275$	64.8767	3.91221
$276$	4.89606	0.294708
$277$	31.9297	1.91847	0.959236	−	0.282606i	$-0.0911989\pi$
$277$	31.9297	1.91847	0.959236	+	0.282606i	$0.0911989\pi$
$278$	−14.1953	−0.851376
$279$	16.0292	0.959641
$280$	−5.59566	−0.334405
$281$	13.8573	0.826658	0.413329	−	0.910582i	$-0.364366\pi$
$281$	13.8573	0.826658	0.413329	+	0.910582i	$0.364366\pi$
$282$	1.46872	0.0874610
$283$	8.00812	0.476034	0.238017	−	0.971261i	$-0.423503\pi$
$283$	8.00812	0.476034	0.238017	+	0.971261i	$0.423503\pi$
$284$	−10.7495	−0.637865
$285$	30.1894	1.78827
$286$	8.38199	0.495637
$287$	12.9850	0.766478
$288$	2.76072	0.162677
$289$	−13.5010	−0.794174
$290$	24.7661	1.45432
$291$	−20.1157	−1.17920
$292$	3.49027	0.204253
$293$	−21.3014	−1.24444	−0.622219	−	0.782843i	$-0.713769\pi$
$293$	−21.3014	−1.24444	−0.622219	+	0.782843i	$0.713769\pi$
$294$	12.5307	0.730805
$295$	−39.0991	−2.27644
$296$	−5.13619	−0.298535
$297$	2.95745	0.171609
$298$	−7.70327	−0.446238
$299$	−3.32039	−0.192023
$300$	−30.2385	−1.74582
$301$	−0.679603	−0.0391717
$302$	−9.27652	−0.533804
$303$	3.57497	0.205377
$304$	2.99832	0.171965
$305$	−3.92091	−0.224511
$306$	5.16411	0.295213
$307$	−0.938982	−0.0535905	−0.0267953	−	0.999641i	$-0.508530\pi$
$307$	−0.938982	−0.0535905	−0.0267953	+	0.999641i	$0.508530\pi$
$308$	6.86876	0.391384
$309$	24.5175	1.39475
$310$	−24.3572	−1.38340
$311$	28.0508	1.59062	0.795309	−	0.606205i	$-0.207309\pi$
$311$	28.0508	1.59062	0.795309	+	0.606205i	$0.207309\pi$
$312$	−3.90678	−0.221178
$313$	17.2864	0.977087	0.488543	−	0.872540i	$-0.337528\pi$
$313$	17.2864	0.977087	0.488543	+	0.872540i	$0.337528\pi$
$314$	−1.93853	−0.109398
$315$	−15.4480	−0.870398
$316$	−6.44152	−0.362364
$317$	−6.49442	−0.364763	−0.182381	−	0.983228i	$-0.558381\pi$
$317$	−6.49442	−0.364763	−0.182381	+	0.983228i	$0.558381\pi$
$318$	−22.0992	−1.23926
$319$	−30.4008	−1.70212
$320$	−4.19507	−0.234512
$321$	8.54919	0.477169
$322$	−2.72095	−0.151633
$323$	5.60856	0.312069
$324$	−9.66059	−0.536700
$325$	20.5071	1.13753
$326$	16.2933	0.902404
$327$	15.1641	0.838578
$328$	9.73483	0.537516
$329$	−0.816231	−0.0450003
$330$	51.8494	2.85421
$331$	8.36340	0.459694	0.229847	−	0.973227i	$-0.426177\pi$
$331$	8.36340	0.459694	0.229847	+	0.973227i	$0.426177\pi$
$332$	−3.00764	−0.165066
$333$	−14.1796	−0.777036
$334$	19.4707	1.06539
$335$	38.0092	2.07667
$336$	−3.20148	−0.174655
$337$	−17.9517	−0.977892	−0.488946	−	0.872314i	$-0.662619\pi$
$337$	−17.9517	−0.977892	−0.488946	+	0.872314i	$0.662619\pi$
$338$	−10.3505	−0.562994
$339$	−9.48858	−0.515349
$340$	−7.84717	−0.425573
$341$	29.8989	1.61912
$342$	8.27750	0.447596
$343$	−16.3009	−0.880167
$344$	−0.509499	−0.0274703
$345$	−20.5393	−1.10580
$346$	3.69800	0.198805
$347$	7.01651	0.376666	0.188333	−	0.982105i	$-0.439692\pi$
$347$	7.01651	0.376666	0.188333	+	0.982105i	$0.439692\pi$
$348$	14.1696	0.759570
$349$	−0.920688	−0.0492833	−0.0246416	−	0.999696i	$-0.507844\pi$
$349$	−0.920688	−0.0492833	−0.0246416	+	0.999696i	$0.507844\pi$
$350$	16.8049	0.898257
$351$	0.934829	0.0498975
$352$	5.14951	0.274470
$353$	19.8022	1.05397	0.526983	−	0.849876i	$-0.323323\pi$
$353$	19.8022	1.05397	0.526983	+	0.849876i	$0.323323\pi$
$354$	−22.3700	−1.18895
$355$	45.0949	2.39339
$356$	5.63579	0.298696
$357$	−5.98859	−0.316950
$358$	−4.20272	−0.222121
$359$	2.35111	0.124087	0.0620435	−	0.998073i	$-0.480238\pi$
$359$	2.35111	0.124087	0.0620435	+	0.998073i	$0.480238\pi$
$360$	−11.5814	−0.610393
$361$	−10.0101	−0.526847
$362$	−6.89696	−0.362496
$363$	−37.2443	−1.95482
$364$	2.17117	0.113800
$365$	−14.6419	−0.766394
$366$	−2.24330	−0.117259
$367$	−34.0930	−1.77964	−0.889819	−	0.456314i	$-0.849169\pi$
$367$	−34.0930	−1.77964	−0.889819	+	0.456314i	$0.849169\pi$
$368$	−2.03990	−0.106337
$369$	26.8751	1.39906
$370$	21.5467	1.12016
$371$	12.2815	0.637622
$372$	−13.9357	−0.722530
$373$	11.1190	0.575719	0.287860	−	0.957673i	$-0.407056\pi$
$373$	11.1190	0.575719	0.287860	+	0.957673i	$0.407056\pi$
$374$	9.63253	0.498086
$375$	76.5088	3.95090
$376$	−0.611929	−0.0315578
$377$	−9.60948	−0.494913
$378$	0.766062	0.0394020
$379$	18.8696	0.969268	0.484634	−	0.874717i	$-0.338953\pi$
$379$	18.8696	0.969268	0.484634	+	0.874717i	$0.338953\pi$
$380$	−12.5782	−0.645246
$381$	18.1050	0.927548
$382$	−1.11616	−0.0571075
$383$	24.6948	1.26184	0.630922	−	0.775847i	$-0.282677\pi$
$383$	24.6948	1.26184	0.630922	+	0.775847i	$0.282677\pi$
$384$	−2.40015	−0.122482
$385$	−28.8149	−1.46855
$386$	2.20432	0.112197
$387$	−1.40658	−0.0715006
$388$	8.38103	0.425482
$389$	6.15586	0.312114	0.156057	−	0.987748i	$-0.450122\pi$
$389$	6.15586	0.312114	0.156057	+	0.987748i	$0.450122\pi$
$390$	16.3892	0.829900
$391$	−3.81577	−0.192972
$392$	−5.22080	−0.263690
$393$	−26.6092	−1.34226
$394$	7.32764	0.369161
$395$	27.0226	1.35965
$396$	14.2163	0.714398
$397$	19.6652	0.986966	0.493483	−	0.869755i	$-0.335723\pi$
$397$	19.6652	0.986966	0.493483	+	0.869755i	$0.335723\pi$
$398$	15.4962	0.776753
$399$	−9.59905	−0.480553
$400$	12.5986	0.629930
$401$	2.16660	0.108195	0.0540975	−	0.998536i	$-0.482772\pi$
$401$	2.16660	0.108195	0.0540975	+	0.998536i	$0.482772\pi$
$402$	21.7464	1.08461
$403$	9.45083	0.470779
$404$	−1.48948	−0.0741043
$405$	40.5269	2.01380
$406$	−7.87465	−0.390812
$407$	−26.4489	−1.31102
$408$	−4.48965	−0.222271
$409$	18.3609	0.907889	0.453945	−	0.891030i	$-0.350016\pi$
$409$	18.3609	0.907889	0.453945	+	0.891030i	$0.350016\pi$
$410$	−40.8383	−2.01686
$411$	−46.2934	−2.28348
$412$	−10.2150	−0.503256
$413$	12.4320	0.611737
$414$	−5.63158	−0.276777
$415$	12.6173	0.619357
$416$	1.62772	0.0798057
$417$	34.0708	1.66845
$418$	15.4399	0.755189
$419$	2.89933	0.141641	0.0708207	−	0.997489i	$-0.477438\pi$
$419$	2.89933	0.141641	0.0708207	+	0.997489i	$0.477438\pi$
$420$	13.4304	0.655337
$421$	−20.5218	−1.00017	−0.500085	−	0.865976i	$-0.666698\pi$
$421$	−20.5218	−1.00017	−0.500085	+	0.865976i	$0.666698\pi$
$422$	4.40134	0.214254
$423$	−1.68936	−0.0821396
$424$	9.20743	0.447152
$425$	23.5666	1.14315
$426$	25.8004	1.25003
$427$	1.24670	0.0603318
$428$	−3.56194	−0.172173
$429$	−20.1180	−0.971307
$430$	2.13738	0.103074
$431$	26.9123	1.29632	0.648161	−	0.761504i	$-0.275538\pi$
$431$	26.9123	1.29632	0.648161	+	0.761504i	$0.275538\pi$
$432$	0.574317	0.0276318
$433$	−27.3884	−1.31620	−0.658101	−	0.752929i	$-0.728640\pi$
$433$	−27.3884	−1.31620	−0.658101	+	0.752929i	$0.728640\pi$
$434$	7.74464	0.371755
$435$	−59.4424	−2.85005
$436$	−6.31799	−0.302577
$437$	−6.11626	−0.292580
$438$	−8.37718	−0.400277
$439$	−21.2812	−1.01570	−0.507848	−	0.861447i	$-0.669559\pi$
$439$	−21.2812	−1.01570	−0.507848	+	0.861447i	$0.669559\pi$
$440$	−21.6026	−1.02986
$441$	−14.4131	−0.686340
$442$	3.04477	0.144825
$443$	−13.7723	−0.654339	−0.327170	−	0.944966i	$-0.606095\pi$
$443$	−13.7723	−0.654339	−0.327170	+	0.944966i	$0.606095\pi$
$444$	12.3276	0.585044
$445$	−23.6425	−1.12076
$446$	18.5562	0.878662
$447$	18.4890	0.874499
$448$	1.33387	0.0630193
$449$	−3.02680	−0.142844	−0.0714219	−	0.997446i	$-0.522754\pi$
$449$	−3.02680	−0.142844	−0.0714219	+	0.997446i	$0.522754\pi$
$450$	34.7812	1.63960
$451$	50.1296	2.36051
$452$	3.95333	0.185949
$453$	22.2650	1.04610
$454$	5.38584	0.252770
$455$	−9.10819	−0.426999
$456$	−7.19641	−0.337003
$457$	−36.3421	−1.70001	−0.850007	−	0.526772i	$-0.823402\pi$
$457$	−36.3421	−1.70001	−0.850007	+	0.526772i	$0.823402\pi$
$458$	−4.91922	−0.229860
$459$	1.07430	0.0501440
$460$	8.55751	0.398996
$461$	−1.50869	−0.0702668	−0.0351334	−	0.999383i	$-0.511186\pi$
$461$	−1.50869	−0.0702668	−0.0351334	+	0.999383i	$0.511186\pi$
$462$	−16.4861	−0.767001
$463$	11.4740	0.533241	0.266620	−	0.963802i	$-0.414093\pi$
$463$	11.4740	0.533241	0.266620	+	0.963802i	$0.414093\pi$
$464$	−5.90363	−0.274069
$465$	58.4610	2.71107
$466$	26.4639	1.22591
$467$	−2.47676	−0.114611	−0.0573055	−	0.998357i	$-0.518251\pi$
$467$	−2.47676	−0.114611	−0.0573055	+	0.998357i	$0.518251\pi$
$468$	4.49368	0.207721
$469$	−12.0854	−0.558053
$470$	2.56708	0.118411
$471$	4.65277	0.214388
$472$	9.32025	0.428999
$473$	−2.62367	−0.120637
$474$	15.4606	0.710129
$475$	37.7746	1.73322
$476$	2.49509	0.114362
$477$	25.4191	1.16386
$478$	9.14483	0.418275
$479$	−13.3182	−0.608526	−0.304263	−	0.952588i	$-0.598410\pi$
$479$	−13.3182	−0.608526	−0.304263	+	0.952588i	$0.598410\pi$
$480$	10.0688	0.459575
$481$	−8.36031	−0.381197
$482$	7.89595	0.359651
$483$	6.53069	0.297157
$484$	15.5175	0.705341
$485$	−35.1590	−1.59649
$486$	21.4639	0.973623
$487$	16.2560	0.736629	0.368315	−	0.929701i	$-0.379935\pi$
$487$	16.2560	0.736629	0.368315	+	0.929701i	$0.379935\pi$
$488$	0.934648	0.0423095
$489$	−39.1064	−1.76845
$490$	21.9016	0.989414
$491$	−23.9341	−1.08013	−0.540066	−	0.841623i	$-0.681601\pi$
$491$	−23.9341	−1.08013	−0.540066	+	0.841623i	$0.681601\pi$
$492$	−23.3650	−1.05338
$493$	−11.0432	−0.497359
$494$	4.88043	0.219581
$495$	−59.6386	−2.68055
$496$	5.80616	0.260704
$497$	−14.3384	−0.643165
$498$	7.21878	0.323481
$499$	9.00532	0.403133	0.201567	−	0.979475i	$-0.435397\pi$
$499$	9.00532	0.403133	0.201567	+	0.979475i	$0.435397\pi$
$500$	−31.8767	−1.42557
$501$	−46.7326	−2.08786
$502$	−25.7836	−1.15078
$503$	17.2736	0.770192	0.385096	−	0.922877i	$-0.374168\pi$
$503$	17.2736	0.770192	0.385096	+	0.922877i	$0.374168\pi$
$504$	3.68243	0.164028
$505$	6.24846	0.278053
$506$	−10.5045	−0.466981
$507$	24.8428	1.10331
$508$	−7.54329	−0.334679
$509$	4.46236	0.197791	0.0988954	−	0.995098i	$-0.468469\pi$
$509$	4.46236	0.197791	0.0988954	+	0.995098i	$0.468469\pi$
$510$	18.8344	0.834001
$511$	4.65556	0.205950
$512$	1.00000	0.0441942
$513$	1.72198	0.0760275
$514$	31.3479	1.38270
$515$	42.8526	1.88831
$516$	1.22287	0.0538340
$517$	−3.15114	−0.138587
$518$	−6.85100	−0.301015
$519$	−8.87574	−0.389602
$520$	−6.82841	−0.299446
$521$	15.2295	0.667219	0.333609	−	0.942711i	$-0.391733\pi$
$521$	15.2295	0.667219	0.333609	+	0.942711i	$0.391733\pi$
$522$	−16.2982	−0.713355
$523$	−22.8319	−0.998368	−0.499184	−	0.866496i	$-0.666367\pi$
$523$	−22.8319	−0.998368	−0.499184	+	0.866496i	$0.666367\pi$
$524$	11.0865	0.484315
$525$	−40.3342	−1.76033
$526$	16.0129	0.698196
$527$	10.8608	0.473105
$528$	−12.3596	−0.537883
$529$	−18.8388	−0.819079
$530$	−38.6258	−1.67780
$531$	25.7306	1.11661
$532$	3.99935	0.173394
$533$	15.8456	0.686350
$534$	−13.5267	−0.585360
$535$	14.9426	0.646024
$536$	−9.06045	−0.391352
$537$	10.0872	0.435293
$538$	−11.8507	−0.510918
$539$	−26.8846	−1.15800
$540$	−2.40930	−0.103680
$541$	−22.2405	−0.956195	−0.478098	−	0.878307i	$-0.658674\pi$
$541$	−22.2405	−0.956195	−0.478098	+	0.878307i	$0.658674\pi$
$542$	−17.7795	−0.763695
$543$	16.5537	0.710389
$544$	1.87057	0.0802000
$545$	26.5044	1.13532
$546$	−5.21112	−0.223015
$547$	−1.58506	−0.0677724	−0.0338862	−	0.999426i	$-0.510788\pi$
$547$	−1.58506	−0.0677724	−0.0338862	+	0.999426i	$0.510788\pi$
$548$	19.2877	0.823930
$549$	2.58030	0.110124
$550$	64.8767	2.76635
$551$	−17.7010	−0.754086
$552$	4.89606	0.208390
$553$	−8.59212	−0.365374
$554$	31.9297	1.35656
$555$	−51.7153	−2.19519
$556$	−14.1953	−0.602013
$557$	−2.04736	−0.0867494	−0.0433747	−	0.999059i	$-0.513811\pi$
$557$	−2.04736	−0.0867494	−0.0433747	+	0.999059i	$0.513811\pi$
$558$	16.0292	0.678569
$559$	−0.829324	−0.0350766
$560$	−5.59566	−0.236460
$561$	−23.1195	−0.976106
$562$	13.8573	0.584535
$563$	−38.1385	−1.60735	−0.803673	−	0.595071i	$-0.797124\pi$
$563$	−38.1385	−1.60735	−0.803673	+	0.595071i	$0.797124\pi$
$564$	1.46872	0.0618443
$565$	−16.5845	−0.697715
$566$	8.00812	0.336607
$567$	−12.8859	−0.541159
$568$	−10.7495	−0.451039
$569$	12.2173	0.512175	0.256088	−	0.966654i	$-0.417566\pi$
$569$	12.2173	0.512175	0.256088	+	0.966654i	$0.417566\pi$
$570$	30.1894	1.26450
$571$	19.2479	0.805501	0.402751	−	0.915310i	$-0.368054\pi$
$571$	19.2479	0.805501	0.402751	+	0.915310i	$0.368054\pi$
$572$	8.38199	0.350468
$573$	2.67894	0.111914
$574$	12.9850	0.541982
$575$	−25.6999	−1.07176
$576$	2.76072	0.115030
$577$	−9.83452	−0.409417	−0.204708	−	0.978823i	$-0.565625\pi$
$577$	−9.83452	−0.409417	−0.204708	+	0.978823i	$0.565625\pi$
$578$	−13.5010	−0.561566
$579$	−5.29070	−0.219874
$580$	24.7661	1.02836
$581$	−4.01179	−0.166437
$582$	−20.1157	−0.833823
$583$	47.4138	1.96368
$584$	3.49027	0.144429
$585$	−18.8513	−0.779406
$586$	−21.3014	−0.879951
$587$	−15.8132	−0.652682	−0.326341	−	0.945252i	$-0.605816\pi$
$587$	−15.8132	−0.652682	−0.326341	+	0.945252i	$0.605816\pi$
$588$	12.5307	0.516757
$589$	17.4087	0.717314
$590$	−39.0991	−1.60968
$591$	−17.5874	−0.723451
$592$	−5.13619	−0.211096
$593$	−24.5731	−1.00910	−0.504549	−	0.863383i	$-0.668341\pi$
$593$	−24.5731	−1.00910	−0.504549	+	0.863383i	$0.668341\pi$
$594$	2.95745	0.121346
$595$	−10.4671	−0.429108
$596$	−7.70327	−0.315538
$597$	−37.1931	−1.52221
$598$	−3.32039	−0.135781
$599$	−19.5265	−0.797829	−0.398915	−	0.916988i	$-0.630613\pi$
$599$	−19.5265	−0.797829	−0.398915	+	0.916988i	$0.630613\pi$
$600$	−30.2385	−1.23448
$601$	21.7562	0.887452	0.443726	−	0.896163i	$-0.353656\pi$
$601$	21.7562	0.887452	0.443726	+	0.896163i	$0.353656\pi$
$602$	−0.679603	−0.0276986
$603$	−25.0133	−1.01862
$604$	−9.27652	−0.377456
$605$	−65.0970	−2.64657
$606$	3.57497	0.145223
$607$	35.1915	1.42838	0.714188	−	0.699953i	$-0.246796\pi$
$607$	35.1915	1.42838	0.714188	+	0.699953i	$0.246796\pi$
$608$	2.99832	0.121598
$609$	18.9003	0.765881
$610$	−3.92091	−0.158753
$611$	−0.996051	−0.0402959
$612$	5.16411	0.208747
$613$	5.89554	0.238119	0.119059	−	0.992887i	$-0.462012\pi$
$613$	5.89554	0.238119	0.119059	+	0.992887i	$0.462012\pi$
$614$	−0.938982	−0.0378942
$615$	98.0180	3.95247
$616$	6.86876	0.276750
$617$	16.9891	0.683955	0.341978	−	0.939708i	$-0.388903\pi$
$617$	16.9891	0.683955	0.341978	+	0.939708i	$0.388903\pi$
$618$	24.5175	0.986238
$619$	20.9641	0.842620	0.421310	−	0.906917i	$-0.361570\pi$
$619$	20.9641	0.842620	0.421310	+	0.906917i	$0.361570\pi$
$620$	−24.3572	−0.978211
$621$	−1.17155	−0.0470126
$622$	28.0508	1.12474
$623$	7.51739	0.301178
$624$	−3.90678	−0.156396
$625$	70.7319	2.82928
$626$	17.2864	0.690905
$627$	−37.0580	−1.47995
$628$	−1.93853	−0.0773559
$629$	−9.60761	−0.383081
$630$	−15.4480	−0.615464
$631$	−27.3822	−1.09007	−0.545034	−	0.838414i	$-0.683483\pi$
$631$	−27.3822	−1.09007	−0.545034	+	0.838414i	$0.683483\pi$
$632$	−6.44152	−0.256230
$633$	−10.5639	−0.419876
$634$	−6.49442	−0.257926
$635$	31.6446	1.25578
$636$	−22.0992	−0.876290
$637$	−8.49802	−0.336704
$638$	−30.4008	−1.20358
$639$	−29.6763	−1.17398
$640$	−4.19507	−0.165825
$641$	−8.26952	−0.326626	−0.163313	−	0.986574i	$-0.552218\pi$
$641$	−8.26952	−0.326626	−0.163313	+	0.986574i	$0.552218\pi$
$642$	8.54919	0.337410
$643$	−3.70139	−0.145968	−0.0729842	−	0.997333i	$-0.523252\pi$
$643$	−3.70139	−0.145968	−0.0729842	+	0.997333i	$0.523252\pi$
$644$	−2.72095	−0.107220
$645$	−5.13004	−0.201995
$646$	5.60856	0.220666
$647$	14.2533	0.560356	0.280178	−	0.959948i	$-0.409607\pi$
$647$	14.2533	0.560356	0.280178	+	0.959948i	$0.409607\pi$
$648$	−9.66059	−0.379504
$649$	47.9948	1.88396
$650$	20.5071	0.804353
$651$	−18.5883	−0.728533
$652$	16.2933	0.638096
$653$	19.5477	0.764961	0.382481	−	0.923964i	$-0.375070\pi$
$653$	19.5477	0.764961	0.382481	+	0.923964i	$0.375070\pi$
$654$	15.1641	0.592964
$655$	−46.5085	−1.81724
$656$	9.73483	0.380081
$657$	9.63565	0.375923
$658$	−0.816231	−0.0318200
$659$	9.58825	0.373505	0.186753	−	0.982407i	$-0.440204\pi$
$659$	9.58825	0.373505	0.186753	+	0.982407i	$0.440204\pi$
$660$	51.8494	2.01823
$661$	−24.5229	−0.953830	−0.476915	−	0.878949i	$-0.658245\pi$
$661$	−24.5229	−0.953830	−0.476915	+	0.878949i	$0.658245\pi$
$662$	8.36340	0.325053
$663$	−7.30791	−0.283816
$664$	−3.00764	−0.116719
$665$	−16.7776	−0.650606
$666$	−14.1796	−0.549447
$667$	12.0428	0.466299
$668$	19.4707	0.753344
$669$	−44.5377	−1.72193
$670$	38.0092	1.46842
$671$	4.81298	0.185803
$672$	−3.20148	−0.123500
$673$	16.6949	0.643539	0.321770	−	0.946818i	$-0.395722\pi$
$673$	16.6949	0.643539	0.321770	+	0.946818i	$0.395722\pi$
$674$	−17.9517	−0.691474
$675$	7.23559	0.278498
$676$	−10.3505	−0.398097
$677$	7.70324	0.296059	0.148030	−	0.988983i	$-0.452707\pi$
$677$	7.70324	0.296059	0.148030	+	0.988983i	$0.452707\pi$
$678$	−9.48858	−0.364407
$679$	11.1792	0.429017
$680$	−7.84717	−0.300925
$681$	−12.9268	−0.495356
$682$	29.8989	1.14489
$683$	−0.752349	−0.0287878	−0.0143939	−	0.999896i	$-0.504582\pi$
$683$	−0.752349	−0.0287878	−0.0143939	+	0.999896i	$0.504582\pi$
$684$	8.27750	0.316498
$685$	−80.9133	−3.09154
$686$	−16.3009	−0.622372
$687$	11.8069	0.450460
$688$	−0.509499	−0.0194245
$689$	14.9871	0.570965
$690$	−20.5393	−0.781918
$691$	28.9592	1.10166	0.550829	−	0.834618i	$-0.314312\pi$
$691$	28.9592	1.10166	0.550829	+	0.834618i	$0.314312\pi$
$692$	3.69800	0.140577
$693$	18.9627	0.720334
$694$	7.01651	0.266343
$695$	59.5501	2.25887
$696$	14.1696	0.537097
$697$	18.2097	0.689741
$698$	−0.920688	−0.0348485
$699$	−63.5172	−2.40244
$700$	16.8049	0.635164
$701$	−5.17505	−0.195459	−0.0977294	−	0.995213i	$-0.531158\pi$
$701$	−5.17505	−0.195459	−0.0977294	+	0.995213i	$0.531158\pi$
$702$	0.934829	0.0352829
$703$	−15.3999	−0.580820
$704$	5.14951	0.194080
$705$	−6.16138	−0.232051
$706$	19.8022	0.745266
$707$	−1.98676	−0.0747200
$708$	−22.3700	−0.840716
$709$	−7.46025	−0.280175	−0.140088	−	0.990139i	$-0.544738\pi$
$709$	−7.46025	−0.280175	−0.140088	+	0.990139i	$0.544738\pi$
$710$	45.0949	1.69238
$711$	−17.7832	−0.666922
$712$	5.63579	0.211210
$713$	−11.8440	−0.443560
$714$	−5.98859	−0.224117
$715$	−35.1630	−1.31502
$716$	−4.20272	−0.157063
$717$	−21.9489	−0.819699
$718$	2.35111	0.0877427
$719$	−43.9850	−1.64036	−0.820181	−	0.572104i	$-0.806127\pi$
$719$	−43.9850	−1.64036	−0.820181	+	0.572104i	$0.806127\pi$
$720$	−11.5814	−0.431613
$721$	−13.6254	−0.507437
$722$	−10.0101	−0.372537
$723$	−18.9515	−0.704812
$724$	−6.89696	−0.256324
$725$	−74.3775	−2.76231
$726$	−37.2443	−1.38227
$727$	30.2471	1.12180	0.560901	−	0.827883i	$-0.310455\pi$
$727$	30.2471	1.12180	0.560901	+	0.827883i	$0.310455\pi$
$728$	2.17117	0.0804688
$729$	−22.5348	−0.834623
$730$	−14.6419	−0.541922
$731$	−0.953054	−0.0352500
$732$	−2.24330	−0.0829146
$733$	20.0150	0.739271	0.369635	−	0.929177i	$-0.379483\pi$
$733$	20.0150	0.739271	0.369635	+	0.929177i	$0.379483\pi$
$734$	−34.0930	−1.25839
$735$	−52.5672	−1.93897
$736$	−2.03990	−0.0751916
$737$	−46.6569	−1.71863
$738$	26.8751	0.989286
$739$	46.1746	1.69856	0.849281	−	0.527942i	$-0.177036\pi$
$739$	46.1746	1.69856	0.849281	+	0.527942i	$0.177036\pi$
$740$	21.5467	0.792072
$741$	−11.7138	−0.430316
$742$	12.2815	0.450867
$743$	−49.7389	−1.82474	−0.912371	−	0.409364i	$-0.865751\pi$
$743$	−49.7389	−1.82474	−0.912371	+	0.409364i	$0.865751\pi$
$744$	−13.9357	−0.510906
$745$	32.3157	1.18396
$746$	11.1190	0.407095
$747$	−8.30324	−0.303800
$748$	9.63253	0.352200
$749$	−4.75115	−0.173603
$750$	76.5088	2.79371
$751$	33.2315	1.21264	0.606318	−	0.795222i	$-0.292646\pi$
$751$	33.2315	1.21264	0.606318	+	0.795222i	$0.292646\pi$
$752$	−0.611929	−0.0223147
$753$	61.8846	2.25520
$754$	−9.60948	−0.349957
$755$	38.9156	1.41629
$756$	0.766062	0.0278614
$757$	−43.2193	−1.57083	−0.785416	−	0.618969i	$-0.787551\pi$
$757$	−43.2193	−1.57083	−0.785416	+	0.618969i	$0.787551\pi$
$758$	18.8696	0.685376
$759$	25.2123	0.915149
$760$	−12.5782	−0.456257
$761$	4.39298	0.159245	0.0796227	−	0.996825i	$-0.474628\pi$
$761$	4.39298	0.159245	0.0796227	+	0.996825i	$0.474628\pi$
$762$	18.1050	0.655875
$763$	−8.42735	−0.305091
$764$	−1.11616	−0.0403811
$765$	−21.6638	−0.783257
$766$	24.6948	0.892258
$767$	15.1708	0.547786
$768$	−2.40015	−0.0866079
$769$	49.7710	1.79479	0.897394	−	0.441230i	$-0.145458\pi$
$769$	49.7710	1.79479	0.897394	+	0.441230i	$0.145458\pi$
$770$	−28.8149	−1.03842
$771$	−75.2396	−2.70969
$772$	2.20432	0.0793353
$773$	−33.1739	−1.19318	−0.596592	−	0.802545i	$-0.703479\pi$
$773$	−33.1739	−1.19318	−0.596592	+	0.802545i	$0.703479\pi$
$774$	−1.40658	−0.0505585
$775$	73.1495	2.62761
$776$	8.38103	0.300861
$777$	16.4434	0.589904
$778$	6.15586	0.220698
$779$	29.1881	1.04577
$780$	16.3892	0.586828
$781$	−55.3547	−1.98075
$782$	−3.81577	−0.136452
$783$	−3.39056	−0.121169
$784$	−5.22080	−0.186457
$785$	8.13229	0.290254
$786$	−26.6092	−0.949118
$787$	0.520323	0.0185475	0.00927375	−	0.999957i	$-0.497048\pi$
$787$	0.520323	0.0185475	0.00927375	+	0.999957i	$0.497048\pi$
$788$	7.32764	0.261037
$789$	−38.4333	−1.36826
$790$	27.0226	0.961421
$791$	5.27321	0.187494
$792$	14.2163	0.505156
$793$	1.52135	0.0540247
$794$	19.6652	0.697891
$795$	92.7077	3.28800
$796$	15.4962	0.549248
$797$	4.13115	0.146333	0.0731664	−	0.997320i	$-0.476690\pi$
$797$	4.13115	0.146333	0.0731664	+	0.997320i	$0.476690\pi$
$798$	−9.59905	−0.339803
$799$	−1.14466	−0.0404950
$800$	12.5986	0.445428
$801$	15.5588	0.549744
$802$	2.16660	0.0765055
$803$	17.9732	0.634261
$804$	21.7464	0.766937
$805$	11.4146	0.402311
$806$	9.45083	0.332891
$807$	28.4433	1.00125
$808$	−1.48948	−0.0523997
$809$	−30.0417	−1.05621	−0.528106	−	0.849179i	$-0.677098\pi$
$809$	−30.0417	−1.05621	−0.528106	+	0.849179i	$0.677098\pi$
$810$	40.5269	1.42397
$811$	7.05959	0.247896	0.123948	−	0.992289i	$-0.460444\pi$
$811$	7.05959	0.247896	0.123948	+	0.992289i	$0.460444\pi$
$812$	−7.87465	−0.276346
$813$	42.6735	1.49662
$814$	−26.4489	−0.927034
$815$	−68.3517	−2.39425
$816$	−4.48965	−0.157169
$817$	−1.52764	−0.0534453
$818$	18.3609	0.641975
$819$	5.99397	0.209446
$820$	−40.8383	−1.42613
$821$	5.10278	0.178088	0.0890441	−	0.996028i	$-0.471619\pi$
$821$	5.10278	0.178088	0.0890441	+	0.996028i	$0.471619\pi$
$822$	−46.2934	−1.61467
$823$	−33.9845	−1.18463	−0.592313	−	0.805708i	$-0.701785\pi$
$823$	−33.9845	−1.18463	−0.592313	+	0.805708i	$0.701785\pi$
$824$	−10.2150	−0.355856
$825$	−155.714	−5.42126
$826$	12.4320	0.432563
$827$	15.3473	0.533679	0.266840	−	0.963741i	$-0.414021\pi$
$827$	15.3473	0.533679	0.266840	+	0.963741i	$0.414021\pi$
$828$	−5.63158	−0.195711
$829$	39.6387	1.37671	0.688354	−	0.725375i	$-0.258334\pi$
$829$	39.6387	1.37671	0.688354	+	0.725375i	$0.258334\pi$
$830$	12.6173	0.437951
$831$	−76.6361	−2.65848
$832$	1.62772	0.0564312
$833$	−9.76588	−0.338368
$834$	34.0708	1.17977
$835$	−81.6809	−2.82668
$836$	15.4399	0.533999
$837$	3.33458	0.115260
$838$	2.89933	0.100156
$839$	10.5721	0.364989	0.182494	−	0.983207i	$-0.441583\pi$
$839$	10.5721	0.364989	0.182494	+	0.983207i	$0.441583\pi$
$840$	13.4304	0.463394
$841$	5.85286	0.201823
$842$	−20.5218	−0.707227
$843$	−33.2596	−1.14552
$844$	4.40134	0.151500
$845$	43.4211	1.49373
$846$	−1.68936	−0.0580814
$847$	20.6983	0.711201
$848$	9.20743	0.316184
$849$	−19.2207	−0.659652
$850$	23.5666	0.808327
$851$	10.4773	0.359158
$852$	25.8004	0.883907
$853$	14.0460	0.480927	0.240464	−	0.970658i	$-0.422701\pi$
$853$	14.0460	0.480927	0.240464	+	0.970658i	$0.422701\pi$
$854$	1.24670	0.0426610
$855$	−34.7247	−1.18756
$856$	−3.56194	−0.121745
$857$	−37.3897	−1.27721	−0.638604	−	0.769536i	$-0.720488\pi$
$857$	−37.3897	−1.27721	−0.638604	+	0.769536i	$0.720488\pi$
$858$	−20.1180	−0.686818
$859$	−33.3270	−1.13710	−0.568551	−	0.822648i	$-0.692496\pi$
$859$	−33.3270	−1.13710	−0.568551	+	0.822648i	$0.692496\pi$
$860$	2.13738	0.0728842
$861$	−31.1658	−1.06213
$862$	26.9123	0.916638
$863$	−5.90585	−0.201037	−0.100519	−	0.994935i	$-0.532050\pi$
$863$	−5.90585	−0.201037	−0.100519	+	0.994935i	$0.532050\pi$
$864$	0.574317	0.0195387
$865$	−15.5133	−0.527470
$866$	−27.3884	−0.930696
$867$	32.4043	1.10051
$868$	7.74464	0.262870
$869$	−33.1707	−1.12524
$870$	−59.4424	−2.01529
$871$	−14.7479	−0.499714
$872$	−6.31799	−0.213954
$873$	23.1376	0.783090
$874$	−6.11626	−0.206886
$875$	−42.5192	−1.43741
$876$	−8.37718	−0.283039
$877$	20.1607	0.680777	0.340389	−	0.940285i	$-0.389441\pi$
$877$	20.1607	0.680777	0.340389	+	0.940285i	$0.389441\pi$
$878$	−21.2812	−0.718205
$879$	51.1264	1.72445
$880$	−21.6026	−0.728222
$881$	36.3985	1.22630	0.613149	−	0.789968i	$-0.289903\pi$
$881$	36.3985	1.22630	0.613149	+	0.789968i	$0.289903\pi$
$882$	−14.4131	−0.485316
$883$	57.4102	1.93201	0.966004	−	0.258527i	$-0.0832372\pi$
$883$	57.4102	1.93201	0.966004	+	0.258527i	$0.0832372\pi$
$884$	3.04477	0.102407
$885$	93.8437	3.15452
$886$	−13.7723	−0.462688
$887$	30.2233	1.01480	0.507399	−	0.861711i	$-0.330607\pi$
$887$	30.2233	1.01480	0.507399	+	0.861711i	$0.330607\pi$
$888$	12.3276	0.413688
$889$	−10.0617	−0.337460
$890$	−23.6425	−0.792500
$891$	−49.7474	−1.66660
$892$	18.5562	0.621308
$893$	−1.83476	−0.0613978
$894$	18.4890	0.618364
$895$	17.6307	0.589330
$896$	1.33387	0.0445613
$897$	7.96943	0.266092
$898$	−3.02680	−0.101006
$899$	−34.2774	−1.14322
$900$	34.7812	1.15937
$901$	17.2231	0.573786
$902$	50.1296	1.66913
$903$	1.63115	0.0542813
$904$	3.95333	0.131486
$905$	28.9332	0.961773
$906$	22.2650	0.739706
$907$	−31.7292	−1.05355	−0.526775	−	0.850005i	$-0.676599\pi$
$907$	−31.7292	−1.05355	−0.526775	+	0.850005i	$0.676599\pi$
$908$	5.38584	0.178735
$909$	−4.11203	−0.136387
$910$	−9.10819	−0.301934
$911$	39.5881	1.31161	0.655806	−	0.754930i	$-0.272329\pi$
$911$	39.5881	1.31161	0.655806	+	0.754930i	$0.272329\pi$
$912$	−7.19641	−0.238297
$913$	−15.4879	−0.512574
$914$	−36.3421	−1.20209
$915$	9.41078	0.311111
$916$	−4.91922	−0.162536
$917$	14.7879	0.488338
$918$	1.07430	0.0354572
$919$	2.70578	0.0892555	0.0446278	−	0.999004i	$-0.485790\pi$
$919$	2.70578	0.0892555	0.0446278	+	0.999004i	$0.485790\pi$
$920$	8.55751	0.282133
$921$	2.25370	0.0742618
$922$	−1.50869	−0.0496861
$923$	−17.4972	−0.575928
$924$	−16.4861	−0.542351
$925$	−64.7089	−2.12762
$926$	11.4740	0.377058
$927$	−28.2007	−0.926231
$928$	−5.90363	−0.193796
$929$	5.52499	0.181269	0.0906346	−	0.995884i	$-0.471110\pi$
$929$	5.52499	0.181269	0.0906346	+	0.995884i	$0.471110\pi$
$930$	58.4610	1.91701
$931$	−15.6536	−0.513027
$932$	26.4639	0.866852
$933$	−67.3262	−2.20416
$934$	−2.47676	−0.0810422
$935$	−40.4091	−1.32152
$936$	4.49368	0.146881
$937$	9.45591	0.308911	0.154456	−	0.988000i	$-0.450638\pi$
$937$	9.45591	0.308911	0.154456	+	0.988000i	$0.450638\pi$
$938$	−12.0854	−0.394603
$939$	−41.4900	−1.35397
$940$	2.56708	0.0837290
$941$	20.1406	0.656565	0.328283	−	0.944580i	$-0.393530\pi$
$941$	20.1406	0.656565	0.328283	+	0.944580i	$0.393530\pi$
$942$	4.65277	0.151595
$943$	−19.8581	−0.646667
$944$	9.32025	0.303348
$945$	−3.21368	−0.104541
$946$	−2.62367	−0.0853029
$947$	11.8364	0.384632	0.192316	−	0.981333i	$-0.438400\pi$
$947$	11.8364	0.384632	0.192316	+	0.981333i	$0.438400\pi$
$948$	15.4606	0.502137
$949$	5.68120	0.184420
$950$	37.7746	1.22557
$951$	15.5876	0.505462
$952$	2.49509	0.0808664
$953$	2.58767	0.0838227	0.0419114	−	0.999121i	$-0.486655\pi$
$953$	2.58767	0.0838227	0.0419114	+	0.999121i	$0.486655\pi$
$954$	25.4191	0.822974
$955$	4.68235	0.151517
$956$	9.14483	0.295765
$957$	72.9665	2.35867
$958$	−13.3182	−0.430293
$959$	25.7272	0.830776
$960$	10.0688	0.324969
$961$	2.71151	0.0874680
$962$	−8.36031	−0.269547
$963$	−9.83351	−0.316880
$964$	7.89595	0.254311
$965$	−9.24728	−0.297681
$966$	6.53069	0.210121
$967$	−15.8807	−0.510690	−0.255345	−	0.966850i	$-0.582189\pi$
$967$	−15.8807	−0.510690	−0.255345	+	0.966850i	$0.582189\pi$
$968$	15.5175	0.498751
$969$	−13.4614	−0.432442
$970$	−35.1590	−1.12889
$971$	18.4571	0.592315	0.296158	−	0.955139i	$-0.404295\pi$
$971$	18.4571	0.592315	0.296158	+	0.955139i	$0.404295\pi$
$972$	21.4639	0.688455
$973$	−18.9346	−0.607015
$974$	16.2560	0.520876
$975$	−49.2200	−1.57630
$976$	0.934648	0.0299174
$977$	57.8345	1.85029	0.925145	−	0.379615i	$-0.123943\pi$
$977$	57.8345	1.85029	0.925145	+	0.379615i	$0.123943\pi$
$978$	−39.1064	−1.25049
$979$	29.0216	0.927534
$980$	21.9016	0.699622
$981$	−17.4422	−0.556886
$982$	−23.9341	−0.763769
$983$	46.0545	1.46891	0.734455	−	0.678658i	$-0.237438\pi$
$983$	46.0545	1.46891	0.734455	+	0.678658i	$0.237438\pi$
$984$	−23.3650	−0.744850
$985$	−30.7400	−0.979457
$986$	−11.0432	−0.351686
$987$	1.95908	0.0623581
$988$	4.88043	0.155267
$989$	1.03933	0.0330486
$990$	−59.6386	−1.89544
$991$	14.8647	0.472192	0.236096	−	0.971730i	$-0.424132\pi$
$991$	14.8647	0.472192	0.236096	+	0.971730i	$0.424132\pi$
$992$	5.80616	0.184346
$993$	−20.0734	−0.637011
$994$	−14.3384	−0.454786
$995$	−65.0076	−2.06088
$996$	7.21878	0.228736
$997$	−29.8051	−0.943937	−0.471968	−	0.881615i	$-0.656456\pi$
$997$	−29.8051	−0.943937	−0.471968	+	0.881615i	$0.656456\pi$
$998$	9.00532	0.285058
$999$	−2.94980	−0.0933277

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8038.2.a.d.1.13	✓	92

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8038.2.a.d.1.13	✓	92	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:	\( N \)	\(=\)	\( 8038 = 2 \cdot 4019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8038.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.40015 q^{3} +1.00000 q^{4} -4.19507 q^{5} -2.40015 q^{6} +1.33387 q^{7} +1.00000 q^{8} +2.76072 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.40015 q^{3} +1.00000 q^{4} -4.19507 q^{5} -2.40015 q^{6} +1.33387 q^{7} +1.00000 q^{8} +2.76072 q^{9} -4.19507 q^{10} +5.14951 q^{11} -2.40015 q^{12} +1.62772 q^{13} +1.33387 q^{14} +10.0688 q^{15} +1.00000 q^{16} +1.87057 q^{17} +2.76072 q^{18} +2.99832 q^{19} -4.19507 q^{20} -3.20148 q^{21} +5.14951 q^{22} -2.03990 q^{23} -2.40015 q^{24} +12.5986 q^{25} +1.62772 q^{26} +0.574317 q^{27} +1.33387 q^{28} -5.90363 q^{29} +10.0688 q^{30} +5.80616 q^{31} +1.00000 q^{32} -12.3596 q^{33} +1.87057 q^{34} -5.59566 q^{35} +2.76072 q^{36} -5.13619 q^{37} +2.99832 q^{38} -3.90678 q^{39} -4.19507 q^{40} +9.73483 q^{41} -3.20148 q^{42} -0.509499 q^{43} +5.14951 q^{44} -11.5814 q^{45} -2.03990 q^{46} -0.611929 q^{47} -2.40015 q^{48} -5.22080 q^{49} +12.5986 q^{50} -4.48965 q^{51} +1.62772 q^{52} +9.20743 q^{53} +0.574317 q^{54} -21.6026 q^{55} +1.33387 q^{56} -7.19641 q^{57} -5.90363 q^{58} +9.32025 q^{59} +10.0688 q^{60} +0.934648 q^{61} +5.80616 q^{62} +3.68243 q^{63} +1.00000 q^{64} -6.82841 q^{65} -12.3596 q^{66} -9.06045 q^{67} +1.87057 q^{68} +4.89606 q^{69} -5.59566 q^{70} -10.7495 q^{71} +2.76072 q^{72} +3.49027 q^{73} -5.13619 q^{74} -30.2385 q^{75} +2.99832 q^{76} +6.86876 q^{77} -3.90678 q^{78} -6.44152 q^{79} -4.19507 q^{80} -9.66059 q^{81} +9.73483 q^{82} -3.00764 q^{83} -3.20148 q^{84} -7.84717 q^{85} -0.509499 q^{86} +14.1696 q^{87} +5.14951 q^{88} +5.63579 q^{89} -11.5814 q^{90} +2.17117 q^{91} -2.03990 q^{92} -13.9357 q^{93} -0.611929 q^{94} -12.5782 q^{95} -2.40015 q^{96} +8.38103 q^{97} -5.22080 q^{98} +14.2163 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 92 q + 92 q^{2} + 31 q^{3} + 92 q^{4} + 28 q^{5} + 31 q^{6} + 29 q^{7} + 92 q^{8} + 113 q^{9}+O(q^{10}) \) 92 * q + 92 * q^2 + 31 * q^3 + 92 * q^4 + 28 * q^5 + 31 * q^6 + 29 * q^7 + 92 * q^8 + 113 * q^9	\( 92 q + 92 q^{2} + 31 q^{3} + 92 q^{4} + 28 q^{5} + 31 q^{6} + 29 q^{7} + 92 q^{8} + 113 q^{9} + 28 q^{10} + 37 q^{11} + 31 q^{12} + 20 q^{13} + 29 q^{14} + 30 q^{15} + 92 q^{16} + 52 q^{17} + 113 q^{18} + 61 q^{19} + 28 q^{20} + 5 q^{21} + 37 q^{22} + 71 q^{23} + 31 q^{24} + 118 q^{25} + 20 q^{26} + 112 q^{27} + 29 q^{28} + 30 q^{29} + 30 q^{30} + 89 q^{31} + 92 q^{32} + 52 q^{33} + 52 q^{34} + 58 q^{35} + 113 q^{36} + 15 q^{37} + 61 q^{38} + 43 q^{39} + 28 q^{40} + 75 q^{41} + 5 q^{42} + 46 q^{43} + 37 q^{44} + 63 q^{45} + 71 q^{46} + 92 q^{47} + 31 q^{48} + 131 q^{49} + 118 q^{50} + 45 q^{51} + 20 q^{52} + 72 q^{53} + 112 q^{54} + 86 q^{55} + 29 q^{56} + 44 q^{57} + 30 q^{58} + 95 q^{59} + 30 q^{60} - 4 q^{61} + 89 q^{62} + 67 q^{63} + 92 q^{64} + 55 q^{65} + 52 q^{66} + 40 q^{67} + 52 q^{68} + 25 q^{69} + 58 q^{70} + 84 q^{71} + 113 q^{72} + 87 q^{73} + 15 q^{74} + 132 q^{75} + 61 q^{76} + 96 q^{77} + 43 q^{78} + 68 q^{79} + 28 q^{80} + 156 q^{81} + 75 q^{82} + 120 q^{83} + 5 q^{84} - 14 q^{85} + 46 q^{86} + 73 q^{87} + 37 q^{88} + 86 q^{89} + 63 q^{90} + 93 q^{91} + 71 q^{92} + 29 q^{93} + 92 q^{94} + 67 q^{95} + 31 q^{96} + 65 q^{97} + 131 q^{98} + 94 q^{99}+O(q^{100}) \) 92 * q + 92 * q^2 + 31 * q^3 + 92 * q^4 + 28 * q^5 + 31 * q^6 + 29 * q^7 + 92 * q^8 + 113 * q^9 + 28 * q^10 + 37 * q^11 + 31 * q^12 + 20 * q^13 + 29 * q^14 + 30 * q^15 + 92 * q^16 + 52 * q^17 + 113 * q^18 + 61 * q^19 + 28 * q^20 + 5 * q^21 + 37 * q^22 + 71 * q^23 + 31 * q^24 + 118 * q^25 + 20 * q^26 + 112 * q^27 + 29 * q^28 + 30 * q^29 + 30 * q^30 + 89 * q^31 + 92 * q^32 + 52 * q^33 + 52 * q^34 + 58 * q^35 + 113 * q^36 + 15 * q^37 + 61 * q^38 + 43 * q^39 + 28 * q^40 + 75 * q^41 + 5 * q^42 + 46 * q^43 + 37 * q^44 + 63 * q^45 + 71 * q^46 + 92 * q^47 + 31 * q^48 + 131 * q^49 + 118 * q^50 + 45 * q^51 + 20 * q^52 + 72 * q^53 + 112 * q^54 + 86 * q^55 + 29 * q^56 + 44 * q^57 + 30 * q^58 + 95 * q^59 + 30 * q^60 - 4 * q^61 + 89 * q^62 + 67 * q^63 + 92 * q^64 + 55 * q^65 + 52 * q^66 + 40 * q^67 + 52 * q^68 + 25 * q^69 + 58 * q^70 + 84 * q^71 + 113 * q^72 + 87 * q^73 + 15 * q^74 + 132 * q^75 + 61 * q^76 + 96 * q^77 + 43 * q^78 + 68 * q^79 + 28 * q^80 + 156 * q^81 + 75 * q^82 + 120 * q^83 + 5 * q^84 - 14 * q^85 + 46 * q^86 + 73 * q^87 + 37 * q^88 + 86 * q^89 + 63 * q^90 + 93 * q^91 + 71 * q^92 + 29 * q^93 + 92 * q^94 + 67 * q^95 + 31 * q^96 + 65 * q^97 + 131 * q^98 + 94 * q^99

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