LMFDB - Embedded newform 8042.2.a.c.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8042,2,Mod(1,8042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8042, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8042 = 2 \cdot 4021 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8042.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$64.2156933055$
Analytic rank:	$0$
Dimension:	$86$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	8042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.75315 q^{3} +1.00000 q^{4} +3.02853 q^{5} +2.75315 q^{6} +4.50146 q^{7} -1.00000 q^{8} +4.57981 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.75315 q^{3} +1.00000 q^{4} +3.02853 q^{5} +2.75315 q^{6} +4.50146 q^{7} -1.00000 q^{8} +4.57981 q^{9} -3.02853 q^{10} +2.88661 q^{11} -2.75315 q^{12} +5.83952 q^{13} -4.50146 q^{14} -8.33799 q^{15} +1.00000 q^{16} -2.94349 q^{17} -4.57981 q^{18} +3.68725 q^{19} +3.02853 q^{20} -12.3932 q^{21} -2.88661 q^{22} +4.87572 q^{23} +2.75315 q^{24} +4.17200 q^{25} -5.83952 q^{26} -4.34944 q^{27} +4.50146 q^{28} +1.69578 q^{29} +8.33799 q^{30} -9.89201 q^{31} -1.00000 q^{32} -7.94726 q^{33} +2.94349 q^{34} +13.6328 q^{35} +4.57981 q^{36} +1.79643 q^{37} -3.68725 q^{38} -16.0771 q^{39} -3.02853 q^{40} -0.569520 q^{41} +12.3932 q^{42} -11.7069 q^{43} +2.88661 q^{44} +13.8701 q^{45} -4.87572 q^{46} +6.29159 q^{47} -2.75315 q^{48} +13.2632 q^{49} -4.17200 q^{50} +8.10386 q^{51} +5.83952 q^{52} -1.91731 q^{53} +4.34944 q^{54} +8.74219 q^{55} -4.50146 q^{56} -10.1515 q^{57} -1.69578 q^{58} +13.2185 q^{59} -8.33799 q^{60} -11.8907 q^{61} +9.89201 q^{62} +20.6158 q^{63} +1.00000 q^{64} +17.6852 q^{65} +7.94726 q^{66} +11.6665 q^{67} -2.94349 q^{68} -13.4236 q^{69} -13.6328 q^{70} +5.38045 q^{71} -4.57981 q^{72} -1.65252 q^{73} -1.79643 q^{74} -11.4861 q^{75} +3.68725 q^{76} +12.9940 q^{77} +16.0771 q^{78} +13.5744 q^{79} +3.02853 q^{80} -1.76478 q^{81} +0.569520 q^{82} -7.47975 q^{83} -12.3932 q^{84} -8.91446 q^{85} +11.7069 q^{86} -4.66873 q^{87} -2.88661 q^{88} -2.44770 q^{89} -13.8701 q^{90} +26.2864 q^{91} +4.87572 q^{92} +27.2341 q^{93} -6.29159 q^{94} +11.1669 q^{95} +2.75315 q^{96} +13.7912 q^{97} -13.2632 q^{98} +13.2201 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 86 q - 86 q^{2} + 12 q^{3} + 86 q^{4} - 4 q^{5} - 12 q^{6} + 35 q^{7} - 86 q^{8} + 72 q^{9}+O(q^{10}) $ 86 * q - 86 * q^2 + 12 * q^3 + 86 * q^4 - 4 * q^5 - 12 * q^6 + 35 * q^7 - 86 * q^8 + 72 * q^9	\( 86 q - 86 q^{2} + 12 q^{3} + 86 q^{4} - 4 q^{5} - 12 q^{6} + 35 q^{7} - 86 q^{8} + 72 q^{9} + 4 q^{10} + 13 q^{11} + 12 q^{12} + 45 q^{13} - 35 q^{14} + 17 q^{15} + 86 q^{16} + 5 q^{17} - 72 q^{18} + 47 q^{19} - 4 q^{20} + 15 q^{21} - 13 q^{22} + 6 q^{23} - 12 q^{24} + 112 q^{25} - 45 q^{26} + 51 q^{27} + 35 q^{28} - 14 q^{29} - 17 q^{30} + 24 q^{31} - 86 q^{32} + 43 q^{33} - 5 q^{34} + 42 q^{35} + 72 q^{36} + 61 q^{37} - 47 q^{38} + 20 q^{39} + 4 q^{40} - 16 q^{41} - 15 q^{42} + 72 q^{43} + 13 q^{44} + 6 q^{45} - 6 q^{46} + 11 q^{47} + 12 q^{48} + 89 q^{49} - 112 q^{50} + 56 q^{51} + 45 q^{52} - 7 q^{53} - 51 q^{54} + 48 q^{55} - 35 q^{56} + 65 q^{57} + 14 q^{58} + 24 q^{59} + 17 q^{60} + 31 q^{61} - 24 q^{62} + 98 q^{63} + 86 q^{64} - 9 q^{65} - 43 q^{66} + 157 q^{67} + 5 q^{68} + q^{69} - 42 q^{70} - 11 q^{71} - 72 q^{72} + 74 q^{73} - 61 q^{74} + 76 q^{75} + 47 q^{76} - 13 q^{77} - 20 q^{78} + 57 q^{79} - 4 q^{80} + 34 q^{81} + 16 q^{82} + 65 q^{83} + 15 q^{84} + 102 q^{85} - 72 q^{86} + 49 q^{87} - 13 q^{88} - 34 q^{89} - 6 q^{90} + 91 q^{91} + 6 q^{92} + 57 q^{93} - 11 q^{94} - 13 q^{95} - 12 q^{96} + 64 q^{97} - 89 q^{98} + 56 q^{99}+O(q^{100}) \) 86 * q - 86 * q^2 + 12 * q^3 + 86 * q^4 - 4 * q^5 - 12 * q^6 + 35 * q^7 - 86 * q^8 + 72 * q^9 + 4 * q^10 + 13 * q^11 + 12 * q^12 + 45 * q^13 - 35 * q^14 + 17 * q^15 + 86 * q^16 + 5 * q^17 - 72 * q^18 + 47 * q^19 - 4 * q^20 + 15 * q^21 - 13 * q^22 + 6 * q^23 - 12 * q^24 + 112 * q^25 - 45 * q^26 + 51 * q^27 + 35 * q^28 - 14 * q^29 - 17 * q^30 + 24 * q^31 - 86 * q^32 + 43 * q^33 - 5 * q^34 + 42 * q^35 + 72 * q^36 + 61 * q^37 - 47 * q^38 + 20 * q^39 + 4 * q^40 - 16 * q^41 - 15 * q^42 + 72 * q^43 + 13 * q^44 + 6 * q^45 - 6 * q^46 + 11 * q^47 + 12 * q^48 + 89 * q^49 - 112 * q^50 + 56 * q^51 + 45 * q^52 - 7 * q^53 - 51 * q^54 + 48 * q^55 - 35 * q^56 + 65 * q^57 + 14 * q^58 + 24 * q^59 + 17 * q^60 + 31 * q^61 - 24 * q^62 + 98 * q^63 + 86 * q^64 - 9 * q^65 - 43 * q^66 + 157 * q^67 + 5 * q^68 + q^69 - 42 * q^70 - 11 * q^71 - 72 * q^72 + 74 * q^73 - 61 * q^74 + 76 * q^75 + 47 * q^76 - 13 * q^77 - 20 * q^78 + 57 * q^79 - 4 * q^80 + 34 * q^81 + 16 * q^82 + 65 * q^83 + 15 * q^84 + 102 * q^85 - 72 * q^86 + 49 * q^87 - 13 * q^88 - 34 * q^89 - 6 * q^90 + 91 * q^91 + 6 * q^92 + 57 * q^93 - 11 * q^94 - 13 * q^95 - 12 * q^96 + 64 * q^97 - 89 * q^98 + 56 * 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
$3$	−2.75315	−1.58953	−0.794765	−	0.606918i	$-0.792406\pi$
$3$	−2.75315	−1.58953	−0.794765	+	0.606918i	$0.792406\pi$
$4$	1.00000	0.500000
$5$	3.02853	1.35440	0.677200	−	0.735799i	$-0.263193\pi$
$5$	3.02853	1.35440	0.677200	+	0.735799i	$0.263193\pi$
$6$	2.75315	1.12397
$7$	4.50146	1.70139	0.850697	−	0.525657i	$-0.176180\pi$
$7$	4.50146	1.70139	0.850697	+	0.525657i	$0.176180\pi$
$8$	−1.00000	−0.353553
$9$	4.57981	1.52660
$10$	−3.02853	−0.957706
$11$	2.88661	0.870346	0.435173	−	0.900347i	$-0.356687\pi$
$11$	2.88661	0.870346	0.435173	+	0.900347i	$0.356687\pi$
$12$	−2.75315	−0.794765
$13$	5.83952	1.61959	0.809796	−	0.586711i	$-0.199578\pi$
$13$	5.83952	1.61959	0.809796	+	0.586711i	$0.199578\pi$
$14$	−4.50146	−1.20307
$15$	−8.33799	−2.15286
$16$	1.00000	0.250000
$17$	−2.94349	−0.713902	−0.356951	−	0.934123i	$-0.616184\pi$
$17$	−2.94349	−0.713902	−0.356951	+	0.934123i	$0.616184\pi$
$18$	−4.57981	−1.07947
$19$	3.68725	0.845912	0.422956	−	0.906150i	$-0.360992\pi$
$19$	3.68725	0.845912	0.422956	+	0.906150i	$0.360992\pi$
$20$	3.02853	0.677200
$21$	−12.3932	−2.70441
$22$	−2.88661	−0.615427
$23$	4.87572	1.01666	0.508328	−	0.861163i	$-0.330264\pi$
$23$	4.87572	1.01666	0.508328	+	0.861163i	$0.330264\pi$
$24$	2.75315	0.561983
$25$	4.17200	0.834400
$26$	−5.83952	−1.14522
$27$	−4.34944	−0.837051
$28$	4.50146	0.850697
$29$	1.69578	0.314898	0.157449	−	0.987527i	$-0.449673\pi$
$29$	1.69578	0.314898	0.157449	+	0.987527i	$0.449673\pi$
$30$	8.33799	1.52230
$31$	−9.89201	−1.77666	−0.888329	−	0.459208i	$-0.848133\pi$
$31$	−9.89201	−1.77666	−0.888329	+	0.459208i	$0.848133\pi$
$32$	−1.00000	−0.176777
$33$	−7.94726	−1.38344
$34$	2.94349	0.504805
$35$	13.6328	2.30437
$36$	4.57981	0.763301
$37$	1.79643	0.295331	0.147665	−	0.989037i	$-0.452824\pi$
$37$	1.79643	0.295331	0.147665	+	0.989037i	$0.452824\pi$
$38$	−3.68725	−0.598150
$39$	−16.0771	−2.57439
$40$	−3.02853	−0.478853
$41$	−0.569520	−0.0889440	−0.0444720	−	0.999011i	$-0.514161\pi$
$41$	−0.569520	−0.0889440	−0.0444720	+	0.999011i	$0.514161\pi$
$42$	12.3932	1.91231
$43$	−11.7069	−1.78529	−0.892646	−	0.450758i	$-0.851154\pi$
$43$	−11.7069	−1.78529	−0.892646	+	0.450758i	$0.851154\pi$
$44$	2.88661	0.435173
$45$	13.8701	2.06763
$46$	−4.87572	−0.718885
$47$	6.29159	0.917723	0.458862	−	0.888508i	$-0.348257\pi$
$47$	6.29159	0.917723	0.458862	+	0.888508i	$0.348257\pi$
$48$	−2.75315	−0.397382
$49$	13.2632	1.89474
$50$	−4.17200	−0.590010
$51$	8.10386	1.13477
$52$	5.83952	0.809796
$53$	−1.91731	−0.263363	−0.131682	−	0.991292i	$-0.542038\pi$
$53$	−1.91731	−0.263363	−0.131682	+	0.991292i	$0.542038\pi$
$54$	4.34944	0.591884
$55$	8.74219	1.17880
$56$	−4.50146	−0.601533
$57$	−10.1515	−1.34460
$58$	−1.69578	−0.222667
$59$	13.2185	1.72090	0.860451	−	0.509533i	$-0.170182\pi$
$59$	13.2185	1.72090	0.860451	+	0.509533i	$0.170182\pi$
$60$	−8.33799	−1.07643
$61$	−11.8907	−1.52245	−0.761227	−	0.648486i	$-0.775403\pi$
$61$	−11.8907	−1.52245	−0.761227	+	0.648486i	$0.775403\pi$
$62$	9.89201	1.25629
$63$	20.6158	2.59735
$64$	1.00000	0.125000
$65$	17.6852	2.19358
$66$	7.94726	0.978240
$67$	11.6665	1.42529	0.712644	−	0.701526i	$-0.247498\pi$
$67$	11.6665	1.42529	0.712644	+	0.701526i	$0.247498\pi$
$68$	−2.94349	−0.356951
$69$	−13.4236	−1.61601
$70$	−13.6328	−1.62943
$71$	5.38045	0.638543	0.319271	−	0.947663i	$-0.396562\pi$
$71$	5.38045	0.638543	0.319271	+	0.947663i	$0.396562\pi$
$72$	−4.57981	−0.539736
$73$	−1.65252	−0.193413	−0.0967064	−	0.995313i	$-0.530831\pi$
$73$	−1.65252	−0.193413	−0.0967064	+	0.995313i	$0.530831\pi$
$74$	−1.79643	−0.208830
$75$	−11.4861	−1.32630
$76$	3.68725	0.422956
$77$	12.9940	1.48080
$78$	16.0771	1.82037
$79$	13.5744	1.52724	0.763621	−	0.645665i	$-0.223420\pi$
$79$	13.5744	1.52724	0.763621	+	0.645665i	$0.223420\pi$
$80$	3.02853	0.338600
$81$	−1.76478	−0.196086
$82$	0.569520	0.0628929
$83$	−7.47975	−0.821010	−0.410505	−	0.911858i	$-0.634648\pi$
$83$	−7.47975	−0.821010	−0.410505	+	0.911858i	$0.634648\pi$
$84$	−12.3932	−1.35221
$85$	−8.91446	−0.966909
$86$	11.7069	1.26239
$87$	−4.66873	−0.500540
$88$	−2.88661	−0.307714
$89$	−2.44770	−0.259456	−0.129728	−	0.991550i	$-0.541410\pi$
$89$	−2.44770	−0.259456	−0.129728	+	0.991550i	$0.541410\pi$
$90$	−13.8701	−1.46204
$91$	26.2864	2.75556
$92$	4.87572	0.508328
$93$	27.2341	2.82405
$94$	−6.29159	−0.648928
$95$	11.1669	1.14570
$96$	2.75315	0.280992
$97$	13.7912	1.40028	0.700141	−	0.714005i	$-0.253121\pi$
$97$	13.7912	1.40028	0.700141	+	0.714005i	$0.253121\pi$
$98$	−13.2632	−1.33978
$99$	13.2201	1.32867
$100$	4.17200	0.417200
$101$	17.4062	1.73198	0.865990	−	0.500061i	$-0.166689\pi$
$101$	17.4062	1.73198	0.865990	+	0.500061i	$0.166689\pi$
$102$	−8.10386	−0.802402
$103$	−12.2794	−1.20993	−0.604963	−	0.796254i	$-0.706812\pi$
$103$	−12.2794	−1.20993	−0.604963	+	0.796254i	$0.706812\pi$
$104$	−5.83952	−0.572612
$105$	−37.5331	−3.66286
$106$	1.91731	0.186226
$107$	−16.3499	−1.58060	−0.790300	−	0.612719i	$-0.790076\pi$
$107$	−16.3499	−1.58060	−0.790300	+	0.612719i	$0.790076\pi$
$108$	−4.34944	−0.418525
$109$	1.13884	0.109081	0.0545406	−	0.998512i	$-0.482631\pi$
$109$	1.13884	0.109081	0.0545406	+	0.998512i	$0.482631\pi$
$110$	−8.74219	−0.833535
$111$	−4.94582	−0.469437
$112$	4.50146	0.425348
$113$	2.68101	0.252208	0.126104	−	0.992017i	$-0.459753\pi$
$113$	2.68101	0.252208	0.126104	+	0.992017i	$0.459753\pi$
$114$	10.1515	0.950777
$115$	14.7663	1.37696
$116$	1.69578	0.157449
$117$	26.7439	2.47248
$118$	−13.2185	−1.21686
$119$	−13.2500	−1.21463
$120$	8.33799	0.761150
$121$	−2.66748	−0.242499
$122$	11.8907	1.07654
$123$	1.56797	0.141379
$124$	−9.89201	−0.888329
$125$	−2.50763	−0.224289
$126$	−20.6158	−1.83661
$127$	14.8970	1.32189	0.660946	−	0.750433i	$-0.270155\pi$
$127$	14.8970	1.32189	0.660946	+	0.750433i	$0.270155\pi$
$128$	−1.00000	−0.0883883
$129$	32.2309	2.83778
$130$	−17.6852	−1.55109
$131$	−4.79256	−0.418728	−0.209364	−	0.977838i	$-0.567139\pi$
$131$	−4.79256	−0.418728	−0.209364	+	0.977838i	$0.567139\pi$
$132$	−7.94726	−0.691720
$133$	16.5980	1.43923
$134$	−11.6665	−1.00783
$135$	−13.1724	−1.13370
$136$	2.94349	0.252402
$137$	−1.80655	−0.154344	−0.0771718	−	0.997018i	$-0.524589\pi$
$137$	−1.80655	−0.154344	−0.0771718	+	0.997018i	$0.524589\pi$
$138$	13.4236	1.14269
$139$	−0.222556	−0.0188769	−0.00943846	−	0.999955i	$-0.503004\pi$
$139$	−0.222556	−0.0188769	−0.00943846	+	0.999955i	$0.503004\pi$
$140$	13.6328	1.15218
$141$	−17.3217	−1.45875
$142$	−5.38045	−0.451518
$143$	16.8564	1.40961
$144$	4.57981	0.381651
$145$	5.13572	0.426498
$146$	1.65252	0.136764
$147$	−36.5155	−3.01174
$148$	1.79643	0.147665
$149$	−12.5431	−1.02757	−0.513787	−	0.857918i	$-0.671758\pi$
$149$	−12.5431	−1.02757	−0.513787	+	0.857918i	$0.671758\pi$
$150$	11.4861	0.937838
$151$	12.6082	1.02604	0.513019	−	0.858377i	$-0.328527\pi$
$151$	12.6082	1.02604	0.513019	+	0.858377i	$0.328527\pi$
$152$	−3.68725	−0.299075
$153$	−13.4806	−1.08984
$154$	−12.9940	−1.04708
$155$	−29.9583	−2.40631
$156$	−16.0771	−1.28719
$157$	−21.0653	−1.68119	−0.840596	−	0.541662i	$-0.817795\pi$
$157$	−21.0653	−1.68119	−0.840596	+	0.541662i	$0.817795\pi$
$158$	−13.5744	−1.07992
$159$	5.27864	0.418623
$160$	−3.02853	−0.239426
$161$	21.9479	1.72973
$162$	1.76478	0.138654
$163$	15.6421	1.22518	0.612592	−	0.790399i	$-0.290127\pi$
$163$	15.6421	1.22518	0.612592	+	0.790399i	$0.290127\pi$
$164$	−0.569520	−0.0444720
$165$	−24.0685	−1.87373
$166$	7.47975	0.580542
$167$	13.6967	1.05989	0.529943	−	0.848033i	$-0.322213\pi$
$167$	13.6967	1.05989	0.529943	+	0.848033i	$0.322213\pi$
$168$	12.3932	0.956155
$169$	21.1000	1.62308
$170$	8.91446	0.683708
$171$	16.8869	1.29137
$172$	−11.7069	−0.892646
$173$	−8.42146	−0.640272	−0.320136	−	0.947372i	$-0.603729\pi$
$173$	−8.42146	−0.640272	−0.320136	+	0.947372i	$0.603729\pi$
$174$	4.66873	0.353935
$175$	18.7801	1.41964
$176$	2.88661	0.217586
$177$	−36.3924	−2.73542
$178$	2.44770	0.183463
$179$	4.45864	0.333255	0.166627	−	0.986020i	$-0.446712\pi$
$179$	4.45864	0.333255	0.166627	+	0.986020i	$0.446712\pi$
$180$	13.8701	1.03382
$181$	6.08728	0.452464	0.226232	−	0.974073i	$-0.427359\pi$
$181$	6.08728	0.452464	0.226232	+	0.974073i	$0.427359\pi$
$182$	−26.2864	−1.94848
$183$	32.7369	2.41998
$184$	−4.87572	−0.359442
$185$	5.44053	0.399996
$186$	−27.2341	−1.99690
$187$	−8.49671	−0.621341
$188$	6.29159	0.458862
$189$	−19.5789	−1.42415
$190$	−11.1669	−0.810135
$191$	−21.5935	−1.56245	−0.781226	−	0.624248i	$-0.785406\pi$
$191$	−21.5935	−1.56245	−0.781226	+	0.624248i	$0.785406\pi$
$192$	−2.75315	−0.198691
$193$	19.9811	1.43827	0.719136	−	0.694869i	$-0.244538\pi$
$193$	19.9811	1.43827	0.719136	+	0.694869i	$0.244538\pi$
$194$	−13.7912	−0.990149
$195$	−48.6899	−3.48675
$196$	13.2632	0.947370
$197$	1.61771	0.115257	0.0576287	−	0.998338i	$-0.481646\pi$
$197$	1.61771	0.115257	0.0576287	+	0.998338i	$0.481646\pi$
$198$	−13.2201	−0.939513
$199$	7.52184	0.533209	0.266605	−	0.963806i	$-0.414098\pi$
$199$	7.52184	0.533209	0.266605	+	0.963806i	$0.414098\pi$
$200$	−4.17200	−0.295005
$201$	−32.1195	−2.26554
$202$	−17.4062	−1.22470
$203$	7.63349	0.535766
$204$	8.10386	0.567384
$205$	−1.72481	−0.120466
$206$	12.2794	0.855547
$207$	22.3298	1.55203
$208$	5.83952	0.404898
$209$	10.6436	0.736236
$210$	37.5331	2.59003
$211$	18.7559	1.29121	0.645606	−	0.763671i	$-0.276605\pi$
$211$	18.7559	1.29121	0.645606	+	0.763671i	$0.276605\pi$
$212$	−1.91731	−0.131682
$213$	−14.8132	−1.01498
$214$	16.3499	1.11765
$215$	−35.4549	−2.41800
$216$	4.34944	0.295942
$217$	−44.5285	−3.02279
$218$	−1.13884	−0.0771320
$219$	4.54963	0.307435
$220$	8.74219	0.589398
$221$	−17.1886	−1.15623
$222$	4.94582	0.331942
$223$	−9.69629	−0.649312	−0.324656	−	0.945832i	$-0.605248\pi$
$223$	−9.69629	−0.649312	−0.324656	+	0.945832i	$0.605248\pi$
$224$	−4.50146	−0.300767
$225$	19.1070	1.27380
$226$	−2.68101	−0.178338
$227$	−8.71525	−0.578452	−0.289226	−	0.957261i	$-0.593398\pi$
$227$	−8.71525	−0.578452	−0.289226	+	0.957261i	$0.593398\pi$
$228$	−10.1515	−0.672301
$229$	−27.9523	−1.84714	−0.923570	−	0.383430i	$-0.874743\pi$
$229$	−27.9523	−1.84714	−0.923570	+	0.383430i	$0.874743\pi$
$230$	−14.7663	−0.973658
$231$	−35.7743	−2.35378
$232$	−1.69578	−0.111333
$233$	−7.51877	−0.492571	−0.246286	−	0.969197i	$-0.579210\pi$
$233$	−7.51877	−0.492571	−0.246286	+	0.969197i	$0.579210\pi$
$234$	−26.7439	−1.74830
$235$	19.0543	1.24296
$236$	13.2185	0.860451
$237$	−37.3723	−2.42759
$238$	13.2500	0.858871
$239$	16.0126	1.03577	0.517885	−	0.855450i	$-0.326720\pi$
$239$	16.0126	1.03577	0.517885	+	0.855450i	$0.326720\pi$
$240$	−8.33799	−0.538215
$241$	−1.18272	−0.0761859	−0.0380930	−	0.999274i	$-0.512128\pi$
$241$	−1.18272	−0.0761859	−0.0380930	+	0.999274i	$0.512128\pi$
$242$	2.66748	0.171472
$243$	17.9070	1.14874
$244$	−11.8907	−0.761227
$245$	40.1679	2.56624
$246$	−1.56797	−0.0999701
$247$	21.5318	1.37003
$248$	9.89201	0.628143
$249$	20.5929	1.30502
$250$	2.50763	0.158596
$251$	−12.0311	−0.759394	−0.379697	−	0.925111i	$-0.623972\pi$
$251$	−12.0311	−0.759394	−0.379697	+	0.925111i	$0.623972\pi$
$252$	20.6158	1.29868
$253$	14.0743	0.884843
$254$	−14.8970	−0.934719
$255$	24.5428	1.53693
$256$	1.00000	0.0625000
$257$	−2.04290	−0.127433	−0.0637164	−	0.997968i	$-0.520295\pi$
$257$	−2.04290	−0.127433	−0.0637164	+	0.997968i	$0.520295\pi$
$258$	−32.2309	−2.00661
$259$	8.08655	0.502474
$260$	17.6852	1.09679
$261$	7.76635	0.480725
$262$	4.79256	0.296085
$263$	−29.8758	−1.84222	−0.921109	−	0.389305i	$-0.872715\pi$
$263$	−29.8758	−1.84222	−0.921109	+	0.389305i	$0.872715\pi$
$264$	7.94726	0.489120
$265$	−5.80664	−0.356699
$266$	−16.5980	−1.01769
$267$	6.73887	0.412412
$268$	11.6665	0.712644
$269$	−15.8029	−0.963523	−0.481761	−	0.876302i	$-0.660003\pi$
$269$	−15.8029	−0.963523	−0.481761	+	0.876302i	$0.660003\pi$
$270$	13.1724	0.801648
$271$	−14.3858	−0.873876	−0.436938	−	0.899492i	$-0.643937\pi$
$271$	−14.3858	−0.873876	−0.436938	+	0.899492i	$0.643937\pi$
$272$	−2.94349	−0.178475
$273$	−72.3703	−4.38005
$274$	1.80655	0.109137
$275$	12.0429	0.726216
$276$	−13.4236	−0.808003
$277$	4.98762	0.299677	0.149839	−	0.988710i	$-0.452125\pi$
$277$	4.98762	0.299677	0.149839	+	0.988710i	$0.452125\pi$
$278$	0.222556	0.0133480
$279$	−45.3035	−2.71225
$280$	−13.6328	−0.814717
$281$	−12.7563	−0.760974	−0.380487	−	0.924786i	$-0.624244\pi$
$281$	−12.7563	−0.760974	−0.380487	+	0.924786i	$0.624244\pi$
$282$	17.3217	1.03149
$283$	4.67345	0.277808	0.138904	−	0.990306i	$-0.455642\pi$
$283$	4.67345	0.277808	0.138904	+	0.990306i	$0.455642\pi$
$284$	5.38045	0.319271
$285$	−30.7442	−1.82113
$286$	−16.8564	−0.996741
$287$	−2.56367	−0.151329
$288$	−4.57981	−0.269868
$289$	−8.33586	−0.490344
$290$	−5.13572	−0.301580
$291$	−37.9691	−2.22579
$292$	−1.65252	−0.0967064
$293$	−27.0645	−1.58113	−0.790563	−	0.612380i	$-0.790212\pi$
$293$	−27.0645	−1.58113	−0.790563	+	0.612380i	$0.790212\pi$
$294$	36.5155	2.12962
$295$	40.0326	2.33079
$296$	−1.79643	−0.104415
$297$	−12.5551	−0.728523
$298$	12.5431	0.726604
$299$	28.4719	1.64657
$300$	−11.4861	−0.663151
$301$	−52.6984	−3.03749
$302$	−12.6082	−0.725518
$303$	−47.9218	−2.75303
$304$	3.68725	0.211478
$305$	−36.0115	−2.06201
$306$	13.4806	0.770636
$307$	3.03475	0.173202	0.0866012	−	0.996243i	$-0.472399\pi$
$307$	3.03475	0.173202	0.0866012	+	0.996243i	$0.472399\pi$
$308$	12.9940	0.740400
$309$	33.8070	1.92321
$310$	29.9583	1.70152
$311$	−33.4222	−1.89520	−0.947599	−	0.319462i	$-0.896498\pi$
$311$	−33.4222	−1.89520	−0.947599	+	0.319462i	$0.896498\pi$
$312$	16.0771	0.910184
$313$	10.0688	0.569123	0.284561	−	0.958658i	$-0.408152\pi$
$313$	10.0688	0.569123	0.284561	+	0.958658i	$0.408152\pi$
$314$	21.0653	1.18878
$315$	62.4357	3.51785
$316$	13.5744	0.763621
$317$	−17.1552	−0.963533	−0.481766	−	0.876300i	$-0.660005\pi$
$317$	−17.1552	−0.963533	−0.481766	+	0.876300i	$0.660005\pi$
$318$	−5.27864	−0.296011
$319$	4.89505	0.274070
$320$	3.02853	0.169300
$321$	45.0136	2.51241
$322$	−21.9479	−1.22311
$323$	−10.8534	−0.603898
$324$	−1.76478	−0.0980432
$325$	24.3625	1.35139
$326$	−15.6421	−0.866336
$327$	−3.13539	−0.173388
$328$	0.569520	0.0314465
$329$	28.3214	1.56141
$330$	24.0685	1.32493
$331$	−17.8047	−0.978634	−0.489317	−	0.872106i	$-0.662754\pi$
$331$	−17.8047	−0.978634	−0.489317	+	0.872106i	$0.662754\pi$
$332$	−7.47975	−0.410505
$333$	8.22729	0.450853
$334$	−13.6967	−0.749452
$335$	35.3323	1.93041
$336$	−12.3932	−0.676104
$337$	−0.0559972	−0.00305036	−0.00152518	−	0.999999i	$-0.500485\pi$
$337$	−0.0559972	−0.00305036	−0.00152518	+	0.999999i	$0.500485\pi$
$338$	−21.1000	−1.14769
$339$	−7.38121	−0.400892
$340$	−8.91446	−0.483454
$341$	−28.5544	−1.54631
$342$	−16.8869	−0.913138
$343$	28.1935	1.52230
$344$	11.7069	0.631196
$345$	−40.6536	−2.18872
$346$	8.42146	0.452740
$347$	14.5712	0.782221	0.391111	−	0.920344i	$-0.372091\pi$
$347$	14.5712	0.782221	0.391111	+	0.920344i	$0.372091\pi$
$348$	−4.66873	−0.250270
$349$	−15.0940	−0.807964	−0.403982	−	0.914767i	$-0.632374\pi$
$349$	−15.0940	−0.807964	−0.403982	+	0.914767i	$0.632374\pi$
$350$	−18.7801	−1.00384
$351$	−25.3987	−1.35568
$352$	−2.88661	−0.153857
$353$	−14.0657	−0.748640	−0.374320	−	0.927300i	$-0.622124\pi$
$353$	−14.0657	−0.748640	−0.374320	+	0.927300i	$0.622124\pi$
$354$	36.3924	1.93424
$355$	16.2949	0.864842
$356$	−2.44770	−0.129728
$357$	36.4792	1.93069
$358$	−4.45864	−0.235647
$359$	−36.0333	−1.90177	−0.950883	−	0.309552i	$-0.899821\pi$
$359$	−36.0333	−1.90177	−0.950883	+	0.309552i	$0.899821\pi$
$360$	−13.8701	−0.731018
$361$	−5.40422	−0.284432
$362$	−6.08728	−0.319940
$363$	7.34397	0.385459
$364$	26.2864	1.37778
$365$	−5.00471	−0.261958
$366$	−32.7369	−1.71119
$367$	29.0243	1.51506	0.757528	−	0.652802i	$-0.226407\pi$
$367$	29.0243	1.51506	0.757528	+	0.652802i	$0.226407\pi$
$368$	4.87572	0.254164
$369$	−2.60829	−0.135782
$370$	−5.44053	−0.282840
$371$	−8.63071	−0.448084
$372$	27.2341	1.41202
$373$	−5.48833	−0.284175	−0.142088	−	0.989854i	$-0.545381\pi$
$373$	−5.48833	−0.284175	−0.142088	+	0.989854i	$0.545381\pi$
$374$	8.49671	0.439355
$375$	6.90386	0.356514
$376$	−6.29159	−0.324464
$377$	9.90254	0.510007
$378$	19.5789	1.00703
$379$	23.4621	1.20517	0.602583	−	0.798056i	$-0.294138\pi$
$379$	23.4621	1.20517	0.602583	+	0.798056i	$0.294138\pi$
$380$	11.1669	0.572852
$381$	−41.0135	−2.10119
$382$	21.5935	1.10482
$383$	3.74050	0.191131	0.0955654	−	0.995423i	$-0.469534\pi$
$383$	3.74050	0.191131	0.0955654	+	0.995423i	$0.469534\pi$
$384$	2.75315	0.140496
$385$	39.3526	2.00560
$386$	−19.9811	−1.01701
$387$	−53.6156	−2.72543
$388$	13.7912	0.700141
$389$	−21.2083	−1.07530	−0.537652	−	0.843167i	$-0.680689\pi$
$389$	−21.2083	−1.07530	−0.537652	+	0.843167i	$0.680689\pi$
$390$	48.6899	2.46551
$391$	−14.3516	−0.725793
$392$	−13.2632	−0.669892
$393$	13.1946	0.665580
$394$	−1.61771	−0.0814993
$395$	41.1105	2.06850
$396$	13.2201	0.664336
$397$	24.8701	1.24819	0.624096	−	0.781347i	$-0.285467\pi$
$397$	24.8701	1.24819	0.624096	+	0.781347i	$0.285467\pi$
$398$	−7.52184	−0.377036
$399$	−45.6967	−2.28770
$400$	4.17200	0.208600
$401$	−1.88000	−0.0938826	−0.0469413	−	0.998898i	$-0.514947\pi$
$401$	−1.88000	−0.0938826	−0.0469413	+	0.998898i	$0.514947\pi$
$402$	32.1195	1.60198
$403$	−57.7647	−2.87746
$404$	17.4062	0.865990
$405$	−5.34468	−0.265579
$406$	−7.63349	−0.378844
$407$	5.18558	0.257040
$408$	−8.10386	−0.401201
$409$	24.9765	1.23501	0.617504	−	0.786567i	$-0.288144\pi$
$409$	24.9765	1.23501	0.617504	+	0.786567i	$0.288144\pi$
$410$	1.72481	0.0851822
$411$	4.97368	0.245334
$412$	−12.2794	−0.604963
$413$	59.5026	2.92793
$414$	−22.3298	−1.09745
$415$	−22.6527	−1.11198
$416$	−5.83952	−0.286306
$417$	0.612728	0.0300054
$418$	−10.6436	−0.520597
$419$	34.5174	1.68629	0.843143	−	0.537689i	$-0.180703\pi$
$419$	34.5174	1.68629	0.843143	+	0.537689i	$0.180703\pi$
$420$	−37.5331	−1.83143
$421$	11.5232	0.561605	0.280803	−	0.959766i	$-0.409399\pi$
$421$	11.5232	0.561605	0.280803	+	0.959766i	$0.409399\pi$
$422$	−18.7559	−0.913024
$423$	28.8143	1.40100
$424$	1.91731	0.0931129
$425$	−12.2802	−0.595679
$426$	14.8132	0.717701
$427$	−53.5258	−2.59029
$428$	−16.3499	−0.790300
$429$	−46.4082	−2.24061
$430$	35.4549	1.70978
$431$	−21.4825	−1.03478	−0.517388	−	0.855751i	$-0.673095\pi$
$431$	−21.4825	−1.03478	−0.517388	+	0.855751i	$0.673095\pi$
$432$	−4.34944	−0.209263
$433$	18.0303	0.866483	0.433242	−	0.901278i	$-0.357370\pi$
$433$	18.0303	0.866483	0.433242	+	0.901278i	$0.357370\pi$
$434$	44.5285	2.13744
$435$	−14.1394	−0.677932
$436$	1.13884	0.0545406
$437$	17.9780	0.860003
$438$	−4.54963	−0.217390
$439$	30.4571	1.45364	0.726820	−	0.686828i	$-0.240998\pi$
$439$	30.4571	1.45364	0.726820	+	0.686828i	$0.240998\pi$
$440$	−8.74219	−0.416767
$441$	60.7428	2.89252
$442$	17.1886	0.817578
$443$	7.20870	0.342496	0.171248	−	0.985228i	$-0.445220\pi$
$443$	7.20870	0.342496	0.171248	+	0.985228i	$0.445220\pi$
$444$	−4.94582	−0.234718
$445$	−7.41294	−0.351407
$446$	9.69629	0.459133
$447$	34.5331	1.63336
$448$	4.50146	0.212674
$449$	2.96673	0.140009	0.0700043	−	0.997547i	$-0.477699\pi$
$449$	2.96673	0.140009	0.0700043	+	0.997547i	$0.477699\pi$
$450$	−19.1070	−0.900711
$451$	−1.64398	−0.0774120
$452$	2.68101	0.126104
$453$	−34.7121	−1.63092
$454$	8.71525	0.409027
$455$	79.6092	3.73214
$456$	10.1515	0.475389
$457$	−9.33857	−0.436840	−0.218420	−	0.975855i	$-0.570090\pi$
$457$	−9.33857	−0.436840	−0.218420	+	0.975855i	$0.570090\pi$
$458$	27.9523	1.30612
$459$	12.8026	0.597572
$460$	14.7663	0.688480
$461$	26.5834	1.23811	0.619056	−	0.785347i	$-0.287515\pi$
$461$	26.5834	1.23811	0.619056	+	0.785347i	$0.287515\pi$
$462$	35.7743	1.66437
$463$	23.1414	1.07547	0.537736	−	0.843113i	$-0.319280\pi$
$463$	23.1414	1.07547	0.537736	+	0.843113i	$0.319280\pi$
$464$	1.69578	0.0787246
$465$	82.4795	3.82489
$466$	7.51877	0.348301
$467$	17.5178	0.810625	0.405312	−	0.914178i	$-0.367163\pi$
$467$	17.5178	0.810625	0.405312	+	0.914178i	$0.367163\pi$
$468$	26.7439	1.23624
$469$	52.5162	2.42497
$470$	−19.0543	−0.878909
$471$	57.9958	2.67230
$472$	−13.2185	−0.608431
$473$	−33.7934	−1.55382
$474$	37.3723	1.71657
$475$	15.3832	0.705829
$476$	−13.2500	−0.607314
$477$	−8.78092	−0.402051
$478$	−16.0126	−0.732400
$479$	−2.05590	−0.0939363	−0.0469681	−	0.998896i	$-0.514956\pi$
$479$	−2.05590	−0.0939363	−0.0469681	+	0.998896i	$0.514956\pi$
$480$	8.33799	0.380575
$481$	10.4903	0.478315
$482$	1.18272	0.0538716
$483$	−60.4256	−2.74946
$484$	−2.66748	−0.121249
$485$	41.7670	1.89654
$486$	−17.9070	−0.812279
$487$	14.8889	0.674679	0.337339	−	0.941383i	$-0.390473\pi$
$487$	14.8889	0.674679	0.337339	+	0.941383i	$0.390473\pi$
$488$	11.8907	0.538269
$489$	−43.0650	−1.94747
$490$	−40.1679	−1.81460
$491$	31.5368	1.42324	0.711618	−	0.702567i	$-0.247963\pi$
$491$	31.5368	1.42324	0.711618	+	0.702567i	$0.247963\pi$
$492$	1.56797	0.0706896
$493$	−4.99151	−0.224806
$494$	−21.5318	−0.968760
$495$	40.0375	1.79955
$496$	−9.89201	−0.444164
$497$	24.2199	1.08641
$498$	−20.5929	−0.922788
$499$	−0.563085	−0.0252071	−0.0126036	−	0.999921i	$-0.504012\pi$
$499$	−0.563085	−0.0252071	−0.0126036	+	0.999921i	$0.504012\pi$
$500$	−2.50763	−0.112144
$501$	−37.7091	−1.68472
$502$	12.0311	0.536973
$503$	9.26697	0.413194	0.206597	−	0.978426i	$-0.433761\pi$
$503$	9.26697	0.413194	0.206597	+	0.978426i	$0.433761\pi$
$504$	−20.6158	−0.918303
$505$	52.7152	2.34579
$506$	−14.0743	−0.625678
$507$	−58.0915	−2.57993
$508$	14.8970	0.660946
$509$	−37.4251	−1.65884	−0.829419	−	0.558627i	$-0.811328\pi$
$509$	−37.4251	−1.65884	−0.829419	+	0.558627i	$0.811328\pi$
$510$	−24.5428	−1.08677
$511$	−7.43876	−0.329071
$512$	−1.00000	−0.0441942
$513$	−16.0375	−0.708072
$514$	2.04290	0.0901085
$515$	−37.1886	−1.63872
$516$	32.2309	1.41889
$517$	18.1614	0.798736
$518$	−8.08655	−0.355303
$519$	23.1855	1.01773
$520$	−17.6852	−0.775546
$521$	−8.73957	−0.382887	−0.191444	−	0.981504i	$-0.561317\pi$
$521$	−8.73957	−0.382887	−0.191444	+	0.981504i	$0.561317\pi$
$522$	−7.76635	−0.339924
$523$	5.73842	0.250924	0.125462	−	0.992098i	$-0.459959\pi$
$523$	5.73842	0.250924	0.125462	+	0.992098i	$0.459959\pi$
$524$	−4.79256	−0.209364
$525$	−51.7044	−2.25656
$526$	29.8758	1.30264
$527$	29.1171	1.26836
$528$	−7.94726	−0.345860
$529$	0.772598	0.0335912
$530$	5.80664	0.252224
$531$	60.5382	2.62713
$532$	16.5980	0.719615
$533$	−3.32572	−0.144053
$534$	−6.73887	−0.291620
$535$	−49.5161	−2.14077
$536$	−11.6665	−0.503915
$537$	−12.2753	−0.529718
$538$	15.8029	0.681314
$539$	38.2856	1.64908
$540$	−13.1724	−0.566851
$541$	−27.5898	−1.18618	−0.593089	−	0.805137i	$-0.702092\pi$
$541$	−27.5898	−1.18618	−0.593089	+	0.805137i	$0.702092\pi$
$542$	14.3858	0.617924
$543$	−16.7592	−0.719204
$544$	2.94349	0.126201
$545$	3.44901	0.147740
$546$	72.3703	3.09716
$547$	27.7334	1.18580	0.592898	−	0.805278i	$-0.297984\pi$
$547$	27.7334	1.18580	0.592898	+	0.805278i	$0.297984\pi$
$548$	−1.80655	−0.0771718
$549$	−54.4573	−2.32418
$550$	−12.0429	−0.513512
$551$	6.25276	0.266376
$552$	13.4236	0.571344
$553$	61.1048	2.59844
$554$	−4.98762	−0.211904
$555$	−14.9786	−0.635805
$556$	−0.222556	−0.00943846
$557$	−33.4358	−1.41672	−0.708359	−	0.705852i	$-0.750564\pi$
$557$	−33.4358	−1.41672	−0.708359	+	0.705852i	$0.750564\pi$
$558$	45.3035	1.91785
$559$	−68.3630	−2.89145
$560$	13.6328	0.576092
$561$	23.3927	0.987640
$562$	12.7563	0.538090
$563$	15.0999	0.636384	0.318192	−	0.948026i	$-0.396924\pi$
$563$	15.0999	0.636384	0.318192	+	0.948026i	$0.396924\pi$
$564$	−17.3217	−0.729374
$565$	8.11952	0.341591
$566$	−4.67345	−0.196440
$567$	−7.94408	−0.333620
$568$	−5.38045	−0.225759
$569$	14.3409	0.601203	0.300601	−	0.953750i	$-0.402813\pi$
$569$	14.3409	0.601203	0.300601	+	0.953750i	$0.402813\pi$
$570$	30.7442	1.28773
$571$	13.8772	0.580743	0.290371	−	0.956914i	$-0.406221\pi$
$571$	13.8772	0.580743	0.290371	+	0.956914i	$0.406221\pi$
$572$	16.8564	0.704803
$573$	59.4501	2.48356
$574$	2.56367	0.107006
$575$	20.3415	0.848298
$576$	4.57981	0.190825
$577$	42.9945	1.78988	0.894942	−	0.446182i	$-0.147217\pi$
$577$	42.9945	1.78988	0.894942	+	0.446182i	$0.147217\pi$
$578$	8.33586	0.346726
$579$	−55.0109	−2.28618
$580$	5.13572	0.213249
$581$	−33.6698	−1.39686
$582$	37.9691	1.57387
$583$	−5.53453	−0.229217
$584$	1.65252	0.0683818
$585$	80.9947	3.34872
$586$	27.0645	1.11803
$587$	22.5720	0.931648	0.465824	−	0.884877i	$-0.345758\pi$
$587$	22.5720	0.931648	0.465824	+	0.884877i	$0.345758\pi$
$588$	−36.5155	−1.50587
$589$	−36.4743	−1.50290
$590$	−40.0326	−1.64812
$591$	−4.45380	−0.183205
$592$	1.79643	0.0738327
$593$	−0.103248	−0.00423988	−0.00211994	−	0.999998i	$-0.500675\pi$
$593$	−0.103248	−0.00423988	−0.00211994	+	0.999998i	$0.500675\pi$
$594$	12.5551	0.515144
$595$	−40.1281	−1.64509
$596$	−12.5431	−0.513787
$597$	−20.7087	−0.847552
$598$	−28.4719	−1.16430
$599$	−35.9175	−1.46755	−0.733775	−	0.679392i	$-0.762244\pi$
$599$	−35.9175	−1.46755	−0.733775	+	0.679392i	$0.762244\pi$
$600$	11.4861	0.468919
$601$	−20.7879	−0.847955	−0.423978	−	0.905673i	$-0.639366\pi$
$601$	−20.7879	−0.847955	−0.423978	+	0.905673i	$0.639366\pi$
$602$	52.6984	2.14783
$603$	53.4302	2.17585
$604$	12.6082	0.513019
$605$	−8.07856	−0.328440
$606$	47.9218	1.94669
$607$	−42.6480	−1.73103	−0.865514	−	0.500885i	$-0.833008\pi$
$607$	−42.6480	−1.73103	−0.865514	+	0.500885i	$0.833008\pi$
$608$	−3.68725	−0.149538
$609$	−21.0161	−0.851616
$610$	36.0115	1.45806
$611$	36.7399	1.48634
$612$	−13.4806	−0.544922
$613$	25.3915	1.02555	0.512777	−	0.858522i	$-0.328617\pi$
$613$	25.3915	1.02555	0.512777	+	0.858522i	$0.328617\pi$
$614$	−3.03475	−0.122473
$615$	4.74865	0.191484
$616$	−12.9940	−0.523542
$617$	45.0410	1.81328	0.906640	−	0.421905i	$-0.138638\pi$
$617$	45.0410	1.81328	0.906640	+	0.421905i	$0.138638\pi$
$618$	−33.8070	−1.35992
$619$	−16.4953	−0.663002	−0.331501	−	0.943455i	$-0.607555\pi$
$619$	−16.4953	−0.663002	−0.331501	+	0.943455i	$0.607555\pi$
$620$	−29.9583	−1.20315
$621$	−21.2066	−0.850993
$622$	33.4222	1.34011
$623$	−11.0182	−0.441436
$624$	−16.0771	−0.643597
$625$	−28.4544	−1.13818
$626$	−10.0688	−0.402430
$627$	−29.3035	−1.17027
$628$	−21.0653	−0.840596
$629$	−5.28777	−0.210837
$630$	−62.4357	−2.48750
$631$	−42.7056	−1.70008	−0.850042	−	0.526714i	$-0.823424\pi$
$631$	−42.7056	−1.70008	−0.850042	+	0.526714i	$0.823424\pi$
$632$	−13.5744	−0.539961
$633$	−51.6378	−2.05242
$634$	17.1552	0.681320
$635$	45.1160	1.79037
$636$	5.27864	0.209312
$637$	77.4506	3.06871
$638$	−4.89505	−0.193797
$639$	24.6415	0.974801
$640$	−3.02853	−0.119713
$641$	−29.8596	−1.17938	−0.589691	−	0.807629i	$-0.700751\pi$
$641$	−29.8596	−1.17938	−0.589691	+	0.807629i	$0.700751\pi$
$642$	−45.0136	−1.77654
$643$	−26.0198	−1.02612	−0.513060	−	0.858353i	$-0.671488\pi$
$643$	−26.0198	−1.02612	−0.513060	+	0.858353i	$0.671488\pi$
$644$	21.9479	0.864867
$645$	97.6124	3.84348
$646$	10.8534	0.427020
$647$	−0.802335	−0.0315431	−0.0157715	−	0.999876i	$-0.505020\pi$
$647$	−0.802335	−0.0315431	−0.0157715	+	0.999876i	$0.505020\pi$
$648$	1.76478	0.0693270
$649$	38.1566	1.49778
$650$	−24.3625	−0.955576
$651$	122.594	4.80482
$652$	15.6421	0.612592
$653$	−9.13808	−0.357601	−0.178800	−	0.983885i	$-0.557222\pi$
$653$	−9.13808	−0.357601	−0.178800	+	0.983885i	$0.557222\pi$
$654$	3.13539	0.122604
$655$	−14.5144	−0.567125
$656$	−0.569520	−0.0222360
$657$	−7.56823	−0.295265
$658$	−28.3214	−1.10408
$659$	3.08599	0.120213	0.0601065	−	0.998192i	$-0.480856\pi$
$659$	3.08599	0.120213	0.0601065	+	0.998192i	$0.480856\pi$
$660$	−24.0685	−0.936865
$661$	32.0236	1.24557	0.622786	−	0.782392i	$-0.286001\pi$
$661$	32.0236	1.24557	0.622786	+	0.782392i	$0.286001\pi$
$662$	17.8047	0.691999
$663$	47.3227	1.83786
$664$	7.47975	0.290271
$665$	50.2676	1.94929
$666$	−8.22729	−0.318801
$667$	8.26814	0.320144
$668$	13.6967	0.529943
$669$	26.6953	1.03210
$670$	−35.3323	−1.36501
$671$	−34.3239	−1.32506
$672$	12.3932	0.478077
$673$	−35.9097	−1.38422	−0.692109	−	0.721793i	$-0.743318\pi$
$673$	−35.9097	−1.38422	−0.692109	+	0.721793i	$0.743318\pi$
$674$	0.0559972	0.00215693
$675$	−18.1459	−0.698435
$676$	21.1000	0.811540
$677$	−10.0309	−0.385518	−0.192759	−	0.981246i	$-0.561744\pi$
$677$	−10.0309	−0.385518	−0.192759	+	0.981246i	$0.561744\pi$
$678$	7.38121	0.283474
$679$	62.0805	2.38243
$680$	8.91446	0.341854
$681$	23.9944	0.919466
$682$	28.5544	1.09340
$683$	25.8459	0.988964	0.494482	−	0.869188i	$-0.335358\pi$
$683$	25.8459	0.988964	0.494482	+	0.869188i	$0.335358\pi$
$684$	16.8869	0.645686
$685$	−5.47118	−0.209043
$686$	−28.1935	−1.07643
$687$	76.9567	2.93608
$688$	−11.7069	−0.446323
$689$	−11.1962	−0.426541
$690$	40.6536	1.54766
$691$	−18.6799	−0.710618	−0.355309	−	0.934749i	$-0.615624\pi$
$691$	−18.6799	−0.710618	−0.355309	+	0.934749i	$0.615624\pi$
$692$	−8.42146	−0.320136
$693$	59.5099	2.26059
$694$	−14.5712	−0.553114
$695$	−0.674016	−0.0255669
$696$	4.66873	0.176968
$697$	1.67638	0.0634973
$698$	15.0940	0.571317
$699$	20.7003	0.782956
$700$	18.7801	0.709821
$701$	4.13248	0.156082	0.0780408	−	0.996950i	$-0.475134\pi$
$701$	4.13248	0.156082	0.0780408	+	0.996950i	$0.475134\pi$
$702$	25.3987	0.958611
$703$	6.62387	0.249824
$704$	2.88661	0.108793
$705$	−52.4592	−1.97573
$706$	14.0657	0.529369
$707$	78.3533	2.94678
$708$	−36.3924	−1.36771
$709$	−32.3351	−1.21437	−0.607184	−	0.794561i	$-0.707701\pi$
$709$	−32.3351	−1.21437	−0.607184	+	0.794561i	$0.707701\pi$
$710$	−16.2949	−0.611536
$711$	62.1682	2.33149
$712$	2.44770	0.0917314
$713$	−48.2306	−1.80625
$714$	−36.4792	−1.36520
$715$	51.0502	1.90917
$716$	4.45864	0.166627
$717$	−44.0850	−1.64639
$718$	36.0333	1.34475
$719$	−21.0380	−0.784587	−0.392293	−	0.919840i	$-0.628318\pi$
$719$	−21.0380	−0.784587	−0.392293	+	0.919840i	$0.628318\pi$
$720$	13.8701	0.516908
$721$	−55.2753	−2.05856
$722$	5.40422	0.201124
$723$	3.25621	0.121100
$724$	6.08728	0.226232
$725$	7.07479	0.262751
$726$	−7.34397	−0.272560
$727$	21.2766	0.789104	0.394552	−	0.918874i	$-0.370900\pi$
$727$	21.2766	0.789104	0.394552	+	0.918874i	$0.370900\pi$
$728$	−26.2864	−0.974239
$729$	−44.0063	−1.62986
$730$	5.00471	0.185233
$731$	34.4593	1.27452
$732$	32.7369	1.20999
$733$	53.9883	1.99410	0.997052	−	0.0767243i	$-0.0244461\pi$
$733$	53.9883	1.99410	0.997052	+	0.0767243i	$0.0244461\pi$
$734$	−29.0243	−1.07131
$735$	−110.588	−4.07911
$736$	−4.87572	−0.179721
$737$	33.6766	1.24049
$738$	2.60829	0.0960125
$739$	8.64828	0.318132	0.159066	−	0.987268i	$-0.449152\pi$
$739$	8.64828	0.318132	0.159066	+	0.987268i	$0.449152\pi$
$740$	5.44053	0.199998
$741$	−59.2801	−2.17771
$742$	8.63071	0.316843
$743$	−21.0477	−0.772164	−0.386082	−	0.922464i	$-0.626172\pi$
$743$	−21.0477	−0.772164	−0.386082	+	0.922464i	$0.626172\pi$
$744$	−27.2341	−0.998452
$745$	−37.9873	−1.39175
$746$	5.48833	0.200942
$747$	−34.2558	−1.25336
$748$	−8.49671	−0.310671
$749$	−73.5983	−2.68922
$750$	−6.90386	−0.252093
$751$	−25.7582	−0.939931	−0.469966	−	0.882685i	$-0.655734\pi$
$751$	−25.7582	−0.939931	−0.469966	+	0.882685i	$0.655734\pi$
$752$	6.29159	0.229431
$753$	33.1233	1.20708
$754$	−9.90254	−0.360629
$755$	38.1842	1.38967
$756$	−19.5789	−0.712076
$757$	−29.3746	−1.06764	−0.533819	−	0.845599i	$-0.679244\pi$
$757$	−29.3746	−1.06764	−0.533819	+	0.845599i	$0.679244\pi$
$758$	−23.4621	−0.852181
$759$	−38.7486	−1.40648
$760$	−11.1669	−0.405067
$761$	−32.4995	−1.17811	−0.589053	−	0.808095i	$-0.700499\pi$
$761$	−32.4995	−1.17811	−0.589053	+	0.808095i	$0.700499\pi$
$762$	41.0135	1.48576
$763$	5.12645	0.185590
$764$	−21.5935	−0.781226
$765$	−40.8265	−1.47609
$766$	−3.74050	−0.135150
$767$	77.1897	2.78716
$768$	−2.75315	−0.0993456
$769$	12.5825	0.453736	0.226868	−	0.973926i	$-0.427151\pi$
$769$	12.5825	0.453736	0.226868	+	0.973926i	$0.427151\pi$
$770$	−39.3526	−1.41817
$771$	5.62441	0.202558
$772$	19.9811	0.719136
$773$	26.2665	0.944739	0.472370	−	0.881401i	$-0.343399\pi$
$773$	26.2665	0.944739	0.472370	+	0.881401i	$0.343399\pi$
$774$	53.6156	1.92717
$775$	−41.2695	−1.48244
$776$	−13.7912	−0.495074
$777$	−22.2634	−0.798697
$778$	21.2083	0.760355
$779$	−2.09996	−0.0752388
$780$	−48.6899	−1.74338
$781$	15.5313	0.555753
$782$	14.3516	0.513213
$783$	−7.37570	−0.263586
$784$	13.2632	0.473685
$785$	−63.7968	−2.27701
$786$	−13.1946	−0.470636
$787$	10.7556	0.383395	0.191697	−	0.981454i	$-0.438601\pi$
$787$	10.7556	0.383395	0.191697	+	0.981454i	$0.438601\pi$
$788$	1.61771	0.0576287
$789$	82.2523	2.92826
$790$	−41.1105	−1.46265
$791$	12.0685	0.429105
$792$	−13.2201	−0.469757
$793$	−69.4363	−2.46575
$794$	−24.8701	−0.882606
$795$	15.9865	0.566984
$796$	7.52184	0.266605
$797$	−49.3824	−1.74922	−0.874608	−	0.484832i	$-0.838881\pi$
$797$	−49.3824	−1.74922	−0.874608	+	0.484832i	$0.838881\pi$
$798$	45.6967	1.61765
$799$	−18.5193	−0.655164
$800$	−4.17200	−0.147502
$801$	−11.2100	−0.396086
$802$	1.88000	0.0663850
$803$	−4.77018	−0.168336
$804$	−32.1195	−1.13277
$805$	66.4698	2.34275
$806$	57.7647	2.03467
$807$	43.5078	1.53155
$808$	−17.4062	−0.612348
$809$	44.2939	1.55729	0.778646	−	0.627464i	$-0.215907\pi$
$809$	44.2939	1.55729	0.778646	+	0.627464i	$0.215907\pi$
$810$	5.34468	0.187793
$811$	3.30124	0.115922	0.0579610	−	0.998319i	$-0.481540\pi$
$811$	3.30124	0.115922	0.0579610	+	0.998319i	$0.481540\pi$
$812$	7.63349	0.267883
$813$	39.6063	1.38905
$814$	−5.18558	−0.181755
$815$	47.3726	1.65939
$816$	8.10386	0.283692
$817$	−43.1664	−1.51020
$818$	−24.9765	−0.873283
$819$	120.387	4.20665
$820$	−1.72481	−0.0602329
$821$	30.5554	1.06639	0.533195	−	0.845992i	$-0.320991\pi$
$821$	30.5554	1.06639	0.533195	+	0.845992i	$0.320991\pi$
$822$	−4.97368	−0.173477
$823$	−16.6829	−0.581529	−0.290765	−	0.956795i	$-0.593910\pi$
$823$	−16.6829	−0.581529	−0.290765	+	0.956795i	$0.593910\pi$
$824$	12.2794	0.427773
$825$	−33.1559	−1.15434
$826$	−59.5026	−2.07036
$827$	15.7524	0.547766	0.273883	−	0.961763i	$-0.411692\pi$
$827$	15.7524	0.547766	0.273883	+	0.961763i	$0.411692\pi$
$828$	22.3298	0.776016
$829$	−12.0254	−0.417660	−0.208830	−	0.977952i	$-0.566966\pi$
$829$	−12.0254	−0.417660	−0.208830	+	0.977952i	$0.566966\pi$
$830$	22.6527	0.786286
$831$	−13.7317	−0.476346
$832$	5.83952	0.202449
$833$	−39.0401	−1.35266
$834$	−0.612728	−0.0212170
$835$	41.4810	1.43551
$836$	10.6436	0.368118
$837$	43.0247	1.48715
$838$	−34.5174	−1.19238
$839$	10.8478	0.374507	0.187254	−	0.982312i	$-0.440041\pi$
$839$	10.8478	0.374507	0.187254	+	0.982312i	$0.440041\pi$
$840$	37.5331	1.29502
$841$	−26.1243	−0.900839
$842$	−11.5232	−0.397115
$843$	35.1198	1.20959
$844$	18.7559	0.645606
$845$	63.9021	2.19830
$846$	−28.8143	−0.990656
$847$	−12.0076	−0.412585
$848$	−1.91731	−0.0658408
$849$	−12.8667	−0.441584
$850$	12.2802	0.421209
$851$	8.75886	0.300250
$852$	−14.8132	−0.507491
$853$	−6.49724	−0.222461	−0.111231	−	0.993795i	$-0.535479\pi$
$853$	−6.49724	−0.222461	−0.111231	+	0.993795i	$0.535479\pi$
$854$	53.5258	1.83161
$855$	51.1424	1.74903
$856$	16.3499	0.558827
$857$	−19.2722	−0.658327	−0.329163	−	0.944273i	$-0.606767\pi$
$857$	−19.2722	−0.658327	−0.329163	+	0.944273i	$0.606767\pi$
$858$	46.4082	1.58435
$859$	6.56139	0.223872	0.111936	−	0.993715i	$-0.464295\pi$
$859$	6.56139	0.223872	0.111936	+	0.993715i	$0.464295\pi$
$860$	−35.4549	−1.20900
$861$	7.05816	0.240541
$862$	21.4825	0.731697
$863$	32.1228	1.09347	0.546736	−	0.837305i	$-0.315870\pi$
$863$	32.1228	1.09347	0.546736	+	0.837305i	$0.315870\pi$
$864$	4.34944	0.147971
$865$	−25.5046	−0.867184
$866$	−18.0303	−0.612696
$867$	22.9498	0.779417
$868$	−44.5285	−1.51140
$869$	39.1840	1.32923
$870$	14.1394	0.479370
$871$	68.1267	2.30838
$872$	−1.13884	−0.0385660
$873$	63.1610	2.13767
$874$	−17.9780	−0.608114
$875$	−11.2880	−0.381604
$876$	4.54963	0.153718
$877$	35.6946	1.20532	0.602661	−	0.797998i	$-0.294107\pi$
$877$	35.6946	1.20532	0.602661	+	0.797998i	$0.294107\pi$
$878$	−30.4571	−1.02788
$879$	74.5126	2.51325
$880$	8.74219	0.294699
$881$	−43.9777	−1.48165	−0.740823	−	0.671701i	$-0.765564\pi$
$881$	−43.9777	−1.48165	−0.740823	+	0.671701i	$0.765564\pi$
$882$	−60.7428	−2.04532
$883$	40.4382	1.36085	0.680427	−	0.732816i	$-0.261794\pi$
$883$	40.4382	1.36085	0.680427	+	0.732816i	$0.261794\pi$
$884$	−17.1886	−0.578115
$885$	−110.216	−3.70486
$886$	−7.20870	−0.242181
$887$	4.36711	0.146633	0.0733166	−	0.997309i	$-0.476642\pi$
$887$	4.36711	0.146633	0.0733166	+	0.997309i	$0.476642\pi$
$888$	4.94582	0.165971
$889$	67.0582	2.24906
$890$	7.41294	0.248482
$891$	−5.09422	−0.170663
$892$	−9.69629	−0.324656
$893$	23.1987	0.776313
$894$	−34.5331	−1.15496
$895$	13.5031	0.451360
$896$	−4.50146	−0.150383
$897$	−78.3872	−2.61727
$898$	−2.96673	−0.0990010
$899$	−16.7747	−0.559467
$900$	19.1070	0.636899
$901$	5.64359	0.188015
$902$	1.64398	0.0547386
$903$	145.086	4.82817
$904$	−2.68101	−0.0891691
$905$	18.4355	0.612817
$906$	34.7121	1.15323
$907$	−26.5048	−0.880078	−0.440039	−	0.897979i	$-0.645035\pi$
$907$	−26.5048	−0.880078	−0.440039	+	0.897979i	$0.645035\pi$
$908$	−8.71525	−0.289226
$909$	79.7170	2.64405
$910$	−79.6092	−2.63902
$911$	−1.17086	−0.0387923	−0.0193962	−	0.999812i	$-0.506174\pi$
$911$	−1.17086	−0.0387923	−0.0193962	+	0.999812i	$0.506174\pi$
$912$	−10.1515	−0.336151
$913$	−21.5911	−0.714562
$914$	9.33857	0.308892
$915$	99.1449	3.27763
$916$	−27.9523	−0.923570
$917$	−21.5735	−0.712421
$918$	−12.8026	−0.422547
$919$	−0.580580	−0.0191516	−0.00957578	−	0.999954i	$-0.503048\pi$
$919$	−0.580580	−0.0191516	−0.00957578	+	0.999954i	$0.503048\pi$
$920$	−14.7663	−0.486829
$921$	−8.35511	−0.275310
$922$	−26.5834	−0.875478
$923$	31.4193	1.03418
$924$	−35.7743	−1.17689
$925$	7.49469	0.246424
$926$	−23.1414	−0.760473
$927$	−56.2373	−1.84708
$928$	−1.69578	−0.0556667
$929$	18.0140	0.591020	0.295510	−	0.955340i	$-0.404510\pi$
$929$	18.0140	0.591020	0.295510	+	0.955340i	$0.404510\pi$
$930$	−82.4795	−2.70461
$931$	48.9046	1.60278
$932$	−7.51877	−0.246286
$933$	92.0161	3.01247
$934$	−17.5178	−0.573198
$935$	−25.7326	−0.841545
$936$	−26.7439	−0.874152
$937$	5.46987	0.178693	0.0893465	−	0.996001i	$-0.471522\pi$
$937$	5.46987	0.178693	0.0893465	+	0.996001i	$0.471522\pi$
$938$	−52.5162	−1.71472
$939$	−27.7209	−0.904637
$940$	19.0543	0.621482
$941$	2.06806	0.0674169	0.0337084	−	0.999432i	$-0.489268\pi$
$941$	2.06806	0.0674169	0.0337084	+	0.999432i	$0.489268\pi$
$942$	−57.9958	−1.88960
$943$	−2.77682	−0.0904256
$944$	13.2185	0.430225
$945$	−59.2952	−1.92887
$946$	33.7934	1.09872
$947$	−28.9353	−0.940270	−0.470135	−	0.882595i	$-0.655795\pi$
$947$	−28.9353	−0.940270	−0.470135	+	0.882595i	$0.655795\pi$
$948$	−37.3723	−1.21380
$949$	−9.64993	−0.313250
$950$	−15.3832	−0.499097
$951$	47.2308	1.53156
$952$	13.2500	0.429436
$953$	−34.2447	−1.10930	−0.554648	−	0.832085i	$-0.687147\pi$
$953$	−34.2447	−1.10930	−0.554648	+	0.832085i	$0.687147\pi$
$954$	8.78092	0.284293
$955$	−65.3967	−2.11619
$956$	16.0126	0.517885
$957$	−13.4768	−0.435643
$958$	2.05590	0.0664230
$959$	−8.13210	−0.262599
$960$	−8.33799	−0.269107
$961$	66.8519	2.15651
$962$	−10.4903	−0.338220
$963$	−74.8793	−2.41295
$964$	−1.18272	−0.0380930
$965$	60.5135	1.94800
$966$	60.4256	1.94416
$967$	7.27910	0.234080	0.117040	−	0.993127i	$-0.462659\pi$
$967$	7.27910	0.234080	0.117040	+	0.993127i	$0.462659\pi$
$968$	2.66748	0.0857362
$969$	29.8809	0.959914
$970$	−41.7670	−1.34106
$971$	−40.4881	−1.29932	−0.649662	−	0.760223i	$-0.725090\pi$
$971$	−40.4881	−1.29932	−0.649662	+	0.760223i	$0.725090\pi$
$972$	17.9070	0.574368
$973$	−1.00183	−0.0321171
$974$	−14.8889	−0.477070
$975$	−67.0735	−2.14807
$976$	−11.8907	−0.380613
$977$	−17.2136	−0.550713	−0.275356	−	0.961342i	$-0.588796\pi$
$977$	−17.2136	−0.550713	−0.275356	+	0.961342i	$0.588796\pi$
$978$	43.0650	1.37707
$979$	−7.06556	−0.225816
$980$	40.1679	1.28312
$981$	5.21567	0.166524
$982$	−31.5368	−1.00638
$983$	30.2200	0.963870	0.481935	−	0.876207i	$-0.339934\pi$
$983$	30.2200	0.963870	0.481935	+	0.876207i	$0.339934\pi$
$984$	−1.56797	−0.0499851
$985$	4.89930	0.156105
$986$	4.99151	0.158962
$987$	−77.9729	−2.48190
$988$	21.5318	0.685017
$989$	−57.0797	−1.81503
$990$	−40.0375	−1.27248
$991$	−45.3347	−1.44010	−0.720052	−	0.693920i	$-0.755882\pi$
$991$	−45.3347	−1.44010	−0.720052	+	0.693920i	$0.755882\pi$
$992$	9.89201	0.314072
$993$	49.0189	1.55557
$994$	−24.2199	−0.768209
$995$	22.7801	0.722179
$996$	20.5929	0.652509
$997$	−34.6550	−1.09754	−0.548768	−	0.835975i	$-0.684903\pi$
$997$	−34.6550	−1.09754	−0.548768	+	0.835975i	$0.684903\pi$
$998$	0.563085	0.0178241
$999$	−7.81346	−0.247207

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.c.1.6	✓	86

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.c.1.6	✓	86	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8042 = 2 \cdot 4021 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8042.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.75315 q^{3} +1.00000 q^{4} +3.02853 q^{5} +2.75315 q^{6} +4.50146 q^{7} -1.00000 q^{8} +4.57981 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.75315 q^{3} +1.00000 q^{4} +3.02853 q^{5} +2.75315 q^{6} +4.50146 q^{7} -1.00000 q^{8} +4.57981 q^{9} -3.02853 q^{10} +2.88661 q^{11} -2.75315 q^{12} +5.83952 q^{13} -4.50146 q^{14} -8.33799 q^{15} +1.00000 q^{16} -2.94349 q^{17} -4.57981 q^{18} +3.68725 q^{19} +3.02853 q^{20} -12.3932 q^{21} -2.88661 q^{22} +4.87572 q^{23} +2.75315 q^{24} +4.17200 q^{25} -5.83952 q^{26} -4.34944 q^{27} +4.50146 q^{28} +1.69578 q^{29} +8.33799 q^{30} -9.89201 q^{31} -1.00000 q^{32} -7.94726 q^{33} +2.94349 q^{34} +13.6328 q^{35} +4.57981 q^{36} +1.79643 q^{37} -3.68725 q^{38} -16.0771 q^{39} -3.02853 q^{40} -0.569520 q^{41} +12.3932 q^{42} -11.7069 q^{43} +2.88661 q^{44} +13.8701 q^{45} -4.87572 q^{46} +6.29159 q^{47} -2.75315 q^{48} +13.2632 q^{49} -4.17200 q^{50} +8.10386 q^{51} +5.83952 q^{52} -1.91731 q^{53} +4.34944 q^{54} +8.74219 q^{55} -4.50146 q^{56} -10.1515 q^{57} -1.69578 q^{58} +13.2185 q^{59} -8.33799 q^{60} -11.8907 q^{61} +9.89201 q^{62} +20.6158 q^{63} +1.00000 q^{64} +17.6852 q^{65} +7.94726 q^{66} +11.6665 q^{67} -2.94349 q^{68} -13.4236 q^{69} -13.6328 q^{70} +5.38045 q^{71} -4.57981 q^{72} -1.65252 q^{73} -1.79643 q^{74} -11.4861 q^{75} +3.68725 q^{76} +12.9940 q^{77} +16.0771 q^{78} +13.5744 q^{79} +3.02853 q^{80} -1.76478 q^{81} +0.569520 q^{82} -7.47975 q^{83} -12.3932 q^{84} -8.91446 q^{85} +11.7069 q^{86} -4.66873 q^{87} -2.88661 q^{88} -2.44770 q^{89} -13.8701 q^{90} +26.2864 q^{91} +4.87572 q^{92} +27.2341 q^{93} -6.29159 q^{94} +11.1669 q^{95} +2.75315 q^{96} +13.7912 q^{97} -13.2632 q^{98} +13.2201 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 86 q - 86 q^{2} + 12 q^{3} + 86 q^{4} - 4 q^{5} - 12 q^{6} + 35 q^{7} - 86 q^{8} + 72 q^{9}+O(q^{10}) \) 86 * q - 86 * q^2 + 12 * q^3 + 86 * q^4 - 4 * q^5 - 12 * q^6 + 35 * q^7 - 86 * q^8 + 72 * q^9	\( 86 q - 86 q^{2} + 12 q^{3} + 86 q^{4} - 4 q^{5} - 12 q^{6} + 35 q^{7} - 86 q^{8} + 72 q^{9} + 4 q^{10} + 13 q^{11} + 12 q^{12} + 45 q^{13} - 35 q^{14} + 17 q^{15} + 86 q^{16} + 5 q^{17} - 72 q^{18} + 47 q^{19} - 4 q^{20} + 15 q^{21} - 13 q^{22} + 6 q^{23} - 12 q^{24} + 112 q^{25} - 45 q^{26} + 51 q^{27} + 35 q^{28} - 14 q^{29} - 17 q^{30} + 24 q^{31} - 86 q^{32} + 43 q^{33} - 5 q^{34} + 42 q^{35} + 72 q^{36} + 61 q^{37} - 47 q^{38} + 20 q^{39} + 4 q^{40} - 16 q^{41} - 15 q^{42} + 72 q^{43} + 13 q^{44} + 6 q^{45} - 6 q^{46} + 11 q^{47} + 12 q^{48} + 89 q^{49} - 112 q^{50} + 56 q^{51} + 45 q^{52} - 7 q^{53} - 51 q^{54} + 48 q^{55} - 35 q^{56} + 65 q^{57} + 14 q^{58} + 24 q^{59} + 17 q^{60} + 31 q^{61} - 24 q^{62} + 98 q^{63} + 86 q^{64} - 9 q^{65} - 43 q^{66} + 157 q^{67} + 5 q^{68} + q^{69} - 42 q^{70} - 11 q^{71} - 72 q^{72} + 74 q^{73} - 61 q^{74} + 76 q^{75} + 47 q^{76} - 13 q^{77} - 20 q^{78} + 57 q^{79} - 4 q^{80} + 34 q^{81} + 16 q^{82} + 65 q^{83} + 15 q^{84} + 102 q^{85} - 72 q^{86} + 49 q^{87} - 13 q^{88} - 34 q^{89} - 6 q^{90} + 91 q^{91} + 6 q^{92} + 57 q^{93} - 11 q^{94} - 13 q^{95} - 12 q^{96} + 64 q^{97} - 89 q^{98} + 56 q^{99}+O(q^{100}) \) 86 * q - 86 * q^2 + 12 * q^3 + 86 * q^4 - 4 * q^5 - 12 * q^6 + 35 * q^7 - 86 * q^8 + 72 * q^9 + 4 * q^10 + 13 * q^11 + 12 * q^12 + 45 * q^13 - 35 * q^14 + 17 * q^15 + 86 * q^16 + 5 * q^17 - 72 * q^18 + 47 * q^19 - 4 * q^20 + 15 * q^21 - 13 * q^22 + 6 * q^23 - 12 * q^24 + 112 * q^25 - 45 * q^26 + 51 * q^27 + 35 * q^28 - 14 * q^29 - 17 * q^30 + 24 * q^31 - 86 * q^32 + 43 * q^33 - 5 * q^34 + 42 * q^35 + 72 * q^36 + 61 * q^37 - 47 * q^38 + 20 * q^39 + 4 * q^40 - 16 * q^41 - 15 * q^42 + 72 * q^43 + 13 * q^44 + 6 * q^45 - 6 * q^46 + 11 * q^47 + 12 * q^48 + 89 * q^49 - 112 * q^50 + 56 * q^51 + 45 * q^52 - 7 * q^53 - 51 * q^54 + 48 * q^55 - 35 * q^56 + 65 * q^57 + 14 * q^58 + 24 * q^59 + 17 * q^60 + 31 * q^61 - 24 * q^62 + 98 * q^63 + 86 * q^64 - 9 * q^65 - 43 * q^66 + 157 * q^67 + 5 * q^68 + q^69 - 42 * q^70 - 11 * q^71 - 72 * q^72 + 74 * q^73 - 61 * q^74 + 76 * q^75 + 47 * q^76 - 13 * q^77 - 20 * q^78 + 57 * q^79 - 4 * q^80 + 34 * q^81 + 16 * q^82 + 65 * q^83 + 15 * q^84 + 102 * q^85 - 72 * q^86 + 49 * q^87 - 13 * q^88 - 34 * q^89 - 6 * q^90 + 91 * q^91 + 6 * q^92 + 57 * q^93 - 11 * q^94 - 13 * q^95 - 12 * q^96 + 64 * q^97 - 89 * q^98 + 56 * q^99

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