LMFDB - Embedded newform 8047.2.a.c.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8047,2,Mod(1,8047)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8047, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8047.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8047 = 13 \cdot 619 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8047.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:	$64.2556185065$
Analytic rank:	$1$
Dimension:	$151$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	8047.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.65702 q^{2} +1.31176 q^{3} +5.05973 q^{4} +3.40250 q^{5} -3.48538 q^{6} +2.98608 q^{7} -8.12975 q^{8} -1.27927 q^{9} +O(q^{10})$	\(q-2.65702 q^{2} +1.31176 q^{3} +5.05973 q^{4} +3.40250 q^{5} -3.48538 q^{6} +2.98608 q^{7} -8.12975 q^{8} -1.27927 q^{9} -9.04051 q^{10} +0.0495458 q^{11} +6.63718 q^{12} -1.00000 q^{13} -7.93407 q^{14} +4.46329 q^{15} +11.4814 q^{16} -7.66270 q^{17} +3.39905 q^{18} +3.04197 q^{19} +17.2158 q^{20} +3.91704 q^{21} -0.131644 q^{22} -6.29014 q^{23} -10.6643 q^{24} +6.57704 q^{25} +2.65702 q^{26} -5.61340 q^{27} +15.1088 q^{28} +1.43269 q^{29} -11.8590 q^{30} -9.87929 q^{31} -14.2468 q^{32} +0.0649924 q^{33} +20.3599 q^{34} +10.1602 q^{35} -6.47278 q^{36} +1.23423 q^{37} -8.08255 q^{38} -1.31176 q^{39} -27.6615 q^{40} -7.37174 q^{41} -10.4076 q^{42} -0.785679 q^{43} +0.250688 q^{44} -4.35273 q^{45} +16.7130 q^{46} -7.38188 q^{47} +15.0609 q^{48} +1.91669 q^{49} -17.4753 q^{50} -10.0517 q^{51} -5.05973 q^{52} -4.47565 q^{53} +14.9149 q^{54} +0.168580 q^{55} -24.2761 q^{56} +3.99035 q^{57} -3.80667 q^{58} -1.68686 q^{59} +22.5830 q^{60} -8.76955 q^{61} +26.2494 q^{62} -3.82002 q^{63} +14.8911 q^{64} -3.40250 q^{65} -0.172686 q^{66} +11.1105 q^{67} -38.7712 q^{68} -8.25118 q^{69} -26.9957 q^{70} +15.9741 q^{71} +10.4002 q^{72} +14.2378 q^{73} -3.27938 q^{74} +8.62752 q^{75} +15.3915 q^{76} +0.147948 q^{77} +3.48538 q^{78} -1.14894 q^{79} +39.0656 q^{80} -3.52564 q^{81} +19.5868 q^{82} -9.15305 q^{83} +19.8192 q^{84} -26.0724 q^{85} +2.08756 q^{86} +1.87935 q^{87} -0.402795 q^{88} +3.72424 q^{89} +11.5653 q^{90} -2.98608 q^{91} -31.8264 q^{92} -12.9593 q^{93} +19.6138 q^{94} +10.3503 q^{95} -18.6884 q^{96} +2.99490 q^{97} -5.09268 q^{98} -0.0633826 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 151 q - 13 q^{2} - 16 q^{3} + 151 q^{4} - 43 q^{5} - 17 q^{6} - 18 q^{7} - 39 q^{8} + 135 q^{9}+O(q^{10}) $ 151 * q - 13 * q^2 - 16 * q^3 + 151 * q^4 - 43 * q^5 - 17 * q^6 - 18 * q^7 - 39 * q^8 + 135 * q^9	\( 151 q - 13 q^{2} - 16 q^{3} + 151 q^{4} - 43 q^{5} - 17 q^{6} - 18 q^{7} - 39 q^{8} + 135 q^{9} - 3 q^{10} - 27 q^{11} - 52 q^{12} - 151 q^{13} - 9 q^{14} - 14 q^{15} + 143 q^{16} - 111 q^{17} - 37 q^{18} - 17 q^{19} - 107 q^{20} - 29 q^{21} - 16 q^{22} - 47 q^{23} - 46 q^{24} + 122 q^{25} + 13 q^{26} - 55 q^{27} - 44 q^{28} + 37 q^{29} - 14 q^{30} - 27 q^{31} - 86 q^{32} - 94 q^{33} - 10 q^{34} - 47 q^{35} + 124 q^{36} - 59 q^{37} - 80 q^{38} + 16 q^{39} + 5 q^{40} - 129 q^{41} - 77 q^{42} - 11 q^{43} - 99 q^{44} - 122 q^{45} - 17 q^{46} - 130 q^{47} - 111 q^{48} + 99 q^{49} - 72 q^{50} + 15 q^{51} - 151 q^{52} - 43 q^{53} - 49 q^{54} - 40 q^{55} - 50 q^{56} - 85 q^{57} - 73 q^{58} - 74 q^{59} - 43 q^{60} - 7 q^{61} - 110 q^{62} - 70 q^{63} + 141 q^{64} + 43 q^{65} - 16 q^{66} - 39 q^{67} - 222 q^{68} + 19 q^{69} - 52 q^{70} - 72 q^{71} - 106 q^{72} - 143 q^{73} + 20 q^{74} - 73 q^{75} - 88 q^{76} - 86 q^{77} + 17 q^{78} + 10 q^{79} - 239 q^{80} + 103 q^{81} - 96 q^{82} - 96 q^{83} - 75 q^{84} - 24 q^{85} - 109 q^{86} - 65 q^{87} - 45 q^{88} - 237 q^{89} - 79 q^{90} + 18 q^{91} - 153 q^{92} - 137 q^{93} - 23 q^{94} + 10 q^{95} - 109 q^{96} - 160 q^{97} - 119 q^{98} - 86 q^{99}+O(q^{100}) \) 151 * q - 13 * q^2 - 16 * q^3 + 151 * q^4 - 43 * q^5 - 17 * q^6 - 18 * q^7 - 39 * q^8 + 135 * q^9 - 3 * q^10 - 27 * q^11 - 52 * q^12 - 151 * q^13 - 9 * q^14 - 14 * q^15 + 143 * q^16 - 111 * q^17 - 37 * q^18 - 17 * q^19 - 107 * q^20 - 29 * q^21 - 16 * q^22 - 47 * q^23 - 46 * q^24 + 122 * q^25 + 13 * q^26 - 55 * q^27 - 44 * q^28 + 37 * q^29 - 14 * q^30 - 27 * q^31 - 86 * q^32 - 94 * q^33 - 10 * q^34 - 47 * q^35 + 124 * q^36 - 59 * q^37 - 80 * q^38 + 16 * q^39 + 5 * q^40 - 129 * q^41 - 77 * q^42 - 11 * q^43 - 99 * q^44 - 122 * q^45 - 17 * q^46 - 130 * q^47 - 111 * q^48 + 99 * q^49 - 72 * q^50 + 15 * q^51 - 151 * q^52 - 43 * q^53 - 49 * q^54 - 40 * q^55 - 50 * q^56 - 85 * q^57 - 73 * q^58 - 74 * q^59 - 43 * q^60 - 7 * q^61 - 110 * q^62 - 70 * q^63 + 141 * q^64 + 43 * q^65 - 16 * q^66 - 39 * q^67 - 222 * q^68 + 19 * q^69 - 52 * q^70 - 72 * q^71 - 106 * q^72 - 143 * q^73 + 20 * q^74 - 73 * q^75 - 88 * q^76 - 86 * q^77 + 17 * q^78 + 10 * q^79 - 239 * q^80 + 103 * q^81 - 96 * q^82 - 96 * q^83 - 75 * q^84 - 24 * q^85 - 109 * q^86 - 65 * q^87 - 45 * q^88 - 237 * q^89 - 79 * q^90 + 18 * q^91 - 153 * q^92 - 137 * q^93 - 23 * q^94 + 10 * q^95 - 109 * q^96 - 160 * q^97 - 119 * q^98 - 86 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.65702	−1.87879	−0.939397	−	0.342832i	$-0.888614\pi$
$2$	−2.65702	−1.87879	−0.939397	+	0.342832i	$0.888614\pi$
$3$	1.31176	0.757348	0.378674	−	0.925530i	$-0.376380\pi$
$3$	1.31176	0.757348	0.378674	+	0.925530i	$0.376380\pi$
$4$	5.05973	2.52987
$5$	3.40250	1.52165	0.760823	−	0.648959i	$-0.224795\pi$
$5$	3.40250	1.52165	0.760823	+	0.648959i	$0.224795\pi$
$6$	−3.48538	−1.42290
$7$	2.98608	1.12863	0.564317	−	0.825558i	$-0.309140\pi$
$7$	2.98608	1.12863	0.564317	+	0.825558i	$0.309140\pi$
$8$	−8.12975	−2.87430
$9$	−1.27927	−0.426424
$10$	−9.04051	−2.85886
$11$	0.0495458	0.0149386	0.00746931	−	0.999972i	$-0.497622\pi$
$11$	0.0495458	0.0149386	0.00746931	+	0.999972i	$0.497622\pi$
$12$	6.63718	1.91599
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−7.93407	−2.12047
$15$	4.46329	1.15242
$16$	11.4814	2.87035
$17$	−7.66270	−1.85848	−0.929239	−	0.369478i	$-0.879536\pi$
$17$	−7.66270	−1.85848	−0.929239	+	0.369478i	$0.879536\pi$
$18$	3.39905	0.801163
$19$	3.04197	0.697875	0.348938	−	0.937146i	$-0.386543\pi$
$19$	3.04197	0.697875	0.348938	+	0.937146i	$0.386543\pi$
$20$	17.2158	3.84956
$21$	3.91704	0.854768
$22$	−0.131644	−0.0280666
$23$	−6.29014	−1.31158	−0.655792	−	0.754941i	$-0.727665\pi$
$23$	−6.29014	−1.31158	−0.655792	+	0.754941i	$0.727665\pi$
$24$	−10.6643	−2.17685
$25$	6.57704	1.31541
$26$	2.65702	0.521084
$27$	−5.61340	−1.08030
$28$	15.1088	2.85529
$29$	1.43269	0.266043	0.133021	−	0.991113i	$-0.457532\pi$
$29$	1.43269	0.266043	0.133021	+	0.991113i	$0.457532\pi$
$30$	−11.8590	−2.16515
$31$	−9.87929	−1.77437	−0.887186	−	0.461411i	$-0.847343\pi$
$31$	−9.87929	−1.77437	−0.887186	+	0.461411i	$0.847343\pi$
$32$	−14.2468	−2.51850
$33$	0.0649924	0.0113137
$34$	20.3599	3.49170
$35$	10.1602	1.71738
$36$	−6.47278	−1.07880
$37$	1.23423	0.202907	0.101453	−	0.994840i	$-0.467651\pi$
$37$	1.23423	0.202907	0.101453	+	0.994840i	$0.467651\pi$
$38$	−8.08255	−1.31116
$39$	−1.31176	−0.210050
$40$	−27.6615	−4.37367
$41$	−7.37174	−1.15127	−0.575636	−	0.817706i	$-0.695245\pi$
$41$	−7.37174	−1.15127	−0.575636	+	0.817706i	$0.695245\pi$
$42$	−10.4076	−1.60593
$43$	−0.785679	−0.119815	−0.0599074	−	0.998204i	$-0.519081\pi$
$43$	−0.785679	−0.119815	−0.0599074	+	0.998204i	$0.519081\pi$
$44$	0.250688	0.0377927
$45$	−4.35273	−0.648867
$46$	16.7130	2.46420
$47$	−7.38188	−1.07676	−0.538379	−	0.842703i	$-0.680963\pi$
$47$	−7.38188	−1.07676	−0.538379	+	0.842703i	$0.680963\pi$
$48$	15.0609	2.17386
$49$	1.91669	0.273813
$50$	−17.4753	−2.47138
$51$	−10.0517	−1.40751
$52$	−5.05973	−0.701658
$53$	−4.47565	−0.614778	−0.307389	−	0.951584i	$-0.599455\pi$
$53$	−4.47565	−0.614778	−0.307389	+	0.951584i	$0.599455\pi$
$54$	14.9149	2.02966
$55$	0.168580	0.0227313
$56$	−24.2761	−3.24403
$57$	3.99035	0.528534
$58$	−3.80667	−0.499840
$59$	−1.68686	−0.219610	−0.109805	−	0.993953i	$-0.535023\pi$
$59$	−1.68686	−0.219610	−0.109805	+	0.993953i	$0.535023\pi$
$60$	22.5830	2.91546
$61$	−8.76955	−1.12283	−0.561413	−	0.827536i	$-0.689742\pi$
$61$	−8.76955	−1.12283	−0.561413	+	0.827536i	$0.689742\pi$
$62$	26.2494	3.33368
$63$	−3.82002	−0.481277
$64$	14.8911	1.86139
$65$	−3.40250	−0.422029
$66$	−0.172686	−0.0212562
$67$	11.1105	1.35736	0.678679	−	0.734435i	$-0.262553\pi$
$67$	11.1105	1.35736	0.678679	+	0.734435i	$0.262553\pi$
$68$	−38.7712	−4.70170
$69$	−8.25118	−0.993325
$70$	−26.9957	−3.22660
$71$	15.9741	1.89578	0.947890	−	0.318598i	$-0.103212\pi$
$71$	15.9741	1.89578	0.947890	+	0.318598i	$0.103212\pi$
$72$	10.4002	1.22567
$73$	14.2378	1.66641	0.833205	−	0.552964i	$-0.186503\pi$
$73$	14.2378	1.66641	0.833205	+	0.552964i	$0.186503\pi$
$74$	−3.27938	−0.381220
$75$	8.62752	0.996221
$76$	15.3915	1.76553
$77$	0.147948	0.0168602
$78$	3.48538	0.394641
$79$	−1.14894	−0.129265	−0.0646327	−	0.997909i	$-0.520588\pi$
$79$	−1.14894	−0.129265	−0.0646327	+	0.997909i	$0.520588\pi$
$80$	39.0656	4.36766
$81$	−3.52564	−0.391738
$82$	19.5868	2.16300
$83$	−9.15305	−1.00468	−0.502339	−	0.864671i	$-0.667527\pi$
$83$	−9.15305	−1.00468	−0.502339	+	0.864671i	$0.667527\pi$
$84$	19.8192	2.16245
$85$	−26.0724	−2.82795
$86$	2.08756	0.225107
$87$	1.87935	0.201487
$88$	−0.402795	−0.0429381
$89$	3.72424	0.394769	0.197385	−	0.980326i	$-0.436755\pi$
$89$	3.72424	0.394769	0.197385	+	0.980326i	$0.436755\pi$
$90$	11.5653	1.21909
$91$	−2.98608	−0.313027
$92$	−31.8264	−3.31813
$93$	−12.9593	−1.34382
$94$	19.6138	2.02301
$95$	10.3503	1.06192
$96$	−18.6884	−1.90738
$97$	2.99490	0.304086	0.152043	−	0.988374i	$-0.451415\pi$
$97$	2.99490	0.304086	0.152043	+	0.988374i	$0.451415\pi$
$98$	−5.09268	−0.514438
$99$	−0.0633826	−0.00637019
$100$	33.2780	3.32780
$101$	9.73638	0.968806	0.484403	−	0.874845i	$-0.339037\pi$
$101$	9.73638	0.968806	0.484403	+	0.874845i	$0.339037\pi$
$102$	26.7074	2.64443
$103$	−3.44475	−0.339421	−0.169711	−	0.985494i	$-0.554283\pi$
$103$	−3.44475	−0.339421	−0.169711	+	0.985494i	$0.554283\pi$
$104$	8.12975	0.797188
$105$	13.3277	1.30065
$106$	11.8919	1.15504
$107$	−8.27433	−0.799910	−0.399955	−	0.916535i	$-0.630974\pi$
$107$	−8.27433	−0.799910	−0.399955	+	0.916535i	$0.630974\pi$
$108$	−28.4023	−2.73301
$109$	6.06285	0.580716	0.290358	−	0.956918i	$-0.406226\pi$
$109$	6.06285	0.580716	0.290358	+	0.956918i	$0.406226\pi$
$110$	−0.447919	−0.0427074
$111$	1.61903	0.153671
$112$	34.2844	3.23958
$113$	−16.0774	−1.51243	−0.756216	−	0.654322i	$-0.772954\pi$
$113$	−16.0774	−1.51243	−0.756216	+	0.654322i	$0.772954\pi$
$114$	−10.6024	−0.993007
$115$	−21.4022	−1.99577
$116$	7.24900	0.673053
$117$	1.27927	0.118269
$118$	4.48200	0.412602
$119$	−22.8815	−2.09754
$120$	−36.2854	−3.31239
$121$	−10.9975	−0.999777
$122$	23.3008	2.10956
$123$	−9.66998	−0.871913
$124$	−49.9865	−4.48892
$125$	5.36587	0.479938
$126$	10.1498	0.904220
$127$	15.7788	1.40014	0.700070	−	0.714074i	$-0.253152\pi$
$127$	15.7788	1.40014	0.700070	+	0.714074i	$0.253152\pi$
$128$	−11.0723	−0.978664
$129$	−1.03063	−0.0907415
$130$	9.04051	0.792905
$131$	8.80483	0.769281	0.384641	−	0.923066i	$-0.374325\pi$
$131$	8.80483	0.769281	0.384641	+	0.923066i	$0.374325\pi$
$132$	0.328844	0.0286222
$133$	9.08357	0.787645
$134$	−29.5207	−2.55020
$135$	−19.0996	−1.64383
$136$	62.2959	5.34183
$137$	−2.11771	−0.180928	−0.0904639	−	0.995900i	$-0.528835\pi$
$137$	−2.11771	−0.180928	−0.0904639	+	0.995900i	$0.528835\pi$
$138$	21.9235	1.86625
$139$	−0.629765	−0.0534159	−0.0267080	−	0.999643i	$-0.508502\pi$
$139$	−0.629765	−0.0534159	−0.0267080	+	0.999643i	$0.508502\pi$
$140$	51.4077	4.34474
$141$	−9.68329	−0.815480
$142$	−42.4435	−3.56178
$143$	−0.0495458	−0.00414323
$144$	−14.6879	−1.22399
$145$	4.87472	0.404823
$146$	−37.8301	−3.13084
$147$	2.51425	0.207372
$148$	6.24490	0.513327
$149$	8.05128	0.659587	0.329793	−	0.944053i	$-0.393021\pi$
$149$	8.05128	0.659587	0.329793	+	0.944053i	$0.393021\pi$
$150$	−22.9235	−1.87169
$151$	0.154711	0.0125902	0.00629512	−	0.999980i	$-0.497996\pi$
$151$	0.154711	0.0125902	0.00629512	+	0.999980i	$0.497996\pi$
$152$	−24.7304	−2.00590
$153$	9.80269	0.792501
$154$	−0.393100	−0.0316769
$155$	−33.6143	−2.69997
$156$	−6.63718	−0.531399
$157$	−8.81973	−0.703891	−0.351945	−	0.936021i	$-0.614480\pi$
$157$	−8.81973	−0.703891	−0.351945	+	0.936021i	$0.614480\pi$
$158$	3.05274	0.242863
$159$	−5.87100	−0.465601
$160$	−48.4748	−3.83227
$161$	−18.7829	−1.48030
$162$	9.36768	0.735995
$163$	−9.68202	−0.758354	−0.379177	−	0.925324i	$-0.623793\pi$
$163$	−9.68202	−0.758354	−0.379177	+	0.925324i	$0.623793\pi$
$164$	−37.2990	−2.91256
$165$	0.221137	0.0172155
$166$	24.3198	1.88758
$167$	−15.7842	−1.22142	−0.610709	−	0.791855i	$-0.709116\pi$
$167$	−15.7842	−1.22142	−0.610709	+	0.791855i	$0.709116\pi$
$168$	−31.8445	−2.45686
$169$	1.00000	0.0769231
$170$	69.2747	5.31313
$171$	−3.89151	−0.297591
$172$	−3.97532	−0.303116
$173$	−5.15535	−0.391954	−0.195977	−	0.980609i	$-0.562788\pi$
$173$	−5.15535	−0.391954	−0.195977	+	0.980609i	$0.562788\pi$
$174$	−4.99345	−0.378553
$175$	19.6396	1.48461
$176$	0.568856	0.0428791
$177$	−2.21276	−0.166321
$178$	−9.89538	−0.741690
$179$	20.2864	1.51628	0.758139	−	0.652093i	$-0.226109\pi$
$179$	20.2864	1.51628	0.758139	+	0.652093i	$0.226109\pi$
$180$	−22.0237	−1.64155
$181$	12.9118	0.959729	0.479864	−	0.877343i	$-0.340686\pi$
$181$	12.9118	0.959729	0.479864	+	0.877343i	$0.340686\pi$
$182$	7.93407	0.588112
$183$	−11.5036	−0.850370
$184$	51.1373	3.76989
$185$	4.19949	0.308753
$186$	34.4331	2.52476
$187$	−0.379655	−0.0277631
$188$	−37.3503	−2.72405
$189$	−16.7621	−1.21926
$190$	−27.5009	−1.99513
$191$	−7.59729	−0.549721	−0.274860	−	0.961484i	$-0.588632\pi$
$191$	−7.59729	−0.549721	−0.274860	+	0.961484i	$0.588632\pi$
$192$	19.5336	1.40972
$193$	−22.1466	−1.59415	−0.797074	−	0.603882i	$-0.793620\pi$
$193$	−22.1466	−1.59415	−0.797074	+	0.603882i	$0.793620\pi$
$194$	−7.95749	−0.571315
$195$	−4.46329	−0.319623
$196$	9.69794	0.692710
$197$	5.70328	0.406342	0.203171	−	0.979143i	$-0.434875\pi$
$197$	5.70328	0.406342	0.203171	+	0.979143i	$0.434875\pi$
$198$	0.168409	0.0119683
$199$	−5.41347	−0.383751	−0.191875	−	0.981419i	$-0.561457\pi$
$199$	−5.41347	−0.383751	−0.191875	+	0.981419i	$0.561457\pi$
$200$	−53.4697	−3.78088
$201$	14.5743	1.02799
$202$	−25.8697	−1.82019
$203$	4.27812	0.300265
$204$	−50.8587	−3.56082
$205$	−25.0824	−1.75183
$206$	9.15275	0.637703
$207$	8.04680	0.559292
$208$	−11.4814	−0.796093
$209$	0.150717	0.0104253
$210$	−35.4120	−2.44366
$211$	−13.6269	−0.938112	−0.469056	−	0.883168i	$-0.655406\pi$
$211$	−13.6269	−0.938112	−0.469056	+	0.883168i	$0.655406\pi$
$212$	−22.6456	−1.55531
$213$	20.9543	1.43576
$214$	21.9850	1.50287
$215$	−2.67328	−0.182316
$216$	45.6355	3.10511
$217$	−29.5004	−2.00262
$218$	−16.1091	−1.09105
$219$	18.6767	1.26205
$220$	0.852968	0.0575071
$221$	7.66270	0.515449
$222$	−4.30178	−0.288716
$223$	−6.62383	−0.443565	−0.221782	−	0.975096i	$-0.571187\pi$
$223$	−6.62383	−0.443565	−0.221782	+	0.975096i	$0.571187\pi$
$224$	−42.5421	−2.84246
$225$	−8.41383	−0.560922
$226$	42.7178	2.84155
$227$	−25.1253	−1.66763	−0.833813	−	0.552047i	$-0.813847\pi$
$227$	−25.1253	−1.66763	−0.833813	+	0.552047i	$0.813847\pi$
$228$	20.1901	1.33712
$229$	−27.1788	−1.79603	−0.898014	−	0.439966i	$-0.854991\pi$
$229$	−27.1788	−1.79603	−0.898014	+	0.439966i	$0.854991\pi$
$230$	56.8660	3.74963
$231$	0.194073	0.0127691
$232$	−11.6474	−0.764688
$233$	20.0651	1.31451	0.657255	−	0.753668i	$-0.271717\pi$
$233$	20.0651	1.31451	0.657255	+	0.753668i	$0.271717\pi$
$234$	−3.39905	−0.222203
$235$	−25.1169	−1.63844
$236$	−8.53503	−0.555583
$237$	−1.50713	−0.0978988
$238$	60.7964	3.94085
$239$	−19.4053	−1.25523	−0.627613	−	0.778525i	$-0.715968\pi$
$239$	−19.4053	−1.25523	−0.627613	+	0.778525i	$0.715968\pi$
$240$	51.2448	3.30784
$241$	−5.25484	−0.338494	−0.169247	−	0.985574i	$-0.554134\pi$
$241$	−5.25484	−0.338494	−0.169247	+	0.985574i	$0.554134\pi$
$242$	29.2206	1.87837
$243$	12.2154	0.783618
$244$	−44.3716	−2.84060
$245$	6.52155	0.416646
$246$	25.6933	1.63814
$247$	−3.04197	−0.193556
$248$	80.3162	5.10008
$249$	−12.0066	−0.760890
$250$	−14.2572	−0.901705
$251$	2.11041	0.133208	0.0666038	−	0.997780i	$-0.478784\pi$
$251$	2.11041	0.133208	0.0666038	+	0.997780i	$0.478784\pi$
$252$	−19.3282	−1.21757
$253$	−0.311650	−0.0195933
$254$	−41.9244	−2.63057
$255$	−34.2008	−2.14174
$256$	−0.362876	−0.0226798
$257$	−24.1293	−1.50514	−0.752571	−	0.658511i	$-0.771187\pi$
$257$	−24.1293	−1.50514	−0.752571	+	0.658511i	$0.771187\pi$
$258$	2.73839	0.170485
$259$	3.68553	0.229008
$260$	−17.2158	−1.06768
$261$	−1.83280	−0.113447
$262$	−23.3946	−1.44532
$263$	−17.7731	−1.09593	−0.547967	−	0.836500i	$-0.684598\pi$
$263$	−17.7731	−1.09593	−0.547967	+	0.836500i	$0.684598\pi$
$264$	−0.528372	−0.0325191
$265$	−15.2284	−0.935475
$266$	−24.1352	−1.47982
$267$	4.88533	0.298978
$268$	56.2159	3.43393
$269$	3.96942	0.242020	0.121010	−	0.992651i	$-0.461387\pi$
$269$	3.96942	0.242020	0.121010	+	0.992651i	$0.461387\pi$
$270$	50.7480	3.08842
$271$	2.57811	0.156609	0.0783044	−	0.996929i	$-0.475049\pi$
$271$	2.57811	0.156609	0.0783044	+	0.996929i	$0.475049\pi$
$272$	−87.9787	−5.33449
$273$	−3.91704	−0.237070
$274$	5.62678	0.339926
$275$	0.325865	0.0196504
$276$	−41.7488	−2.51298
$277$	−8.65113	−0.519796	−0.259898	−	0.965636i	$-0.583689\pi$
$277$	−8.65113	−0.519796	−0.259898	+	0.965636i	$0.583689\pi$
$278$	1.67329	0.100358
$279$	12.6383	0.756636
$280$	−82.5996	−4.93627
$281$	−14.1555	−0.844446	−0.422223	−	0.906492i	$-0.638750\pi$
$281$	−14.1555	−0.844446	−0.422223	+	0.906492i	$0.638750\pi$
$282$	25.7287	1.53212
$283$	−3.52097	−0.209300	−0.104650	−	0.994509i	$-0.533372\pi$
$283$	−3.52097	−0.209300	−0.104650	+	0.994509i	$0.533372\pi$
$284$	80.8248	4.79607
$285$	13.5772	0.804242
$286$	0.131644	0.00778427
$287$	−22.0126	−1.29936
$288$	18.2255	1.07395
$289$	41.7170	2.45394
$290$	−12.9522	−0.760579
$291$	3.92860	0.230299
$292$	72.0395	4.21579
$293$	6.93439	0.405111	0.202556	−	0.979271i	$-0.435075\pi$
$293$	6.93439	0.405111	0.202556	+	0.979271i	$0.435075\pi$
$294$	−6.68039	−0.389608
$295$	−5.73953	−0.334169
$296$	−10.0340	−0.583216
$297$	−0.278120	−0.0161382
$298$	−21.3924	−1.23923
$299$	6.29014	0.363768
$300$	43.6529	2.52030
$301$	−2.34610	−0.135227
$302$	−0.411071	−0.0236545
$303$	12.7718	0.733723
$304$	34.9261	2.00315
$305$	−29.8384	−1.70854
$306$	−26.0459	−1.48894
$307$	28.3309	1.61693	0.808465	−	0.588544i	$-0.200299\pi$
$307$	28.3309	1.61693	0.808465	+	0.588544i	$0.200299\pi$
$308$	0.748576	0.0426541
$309$	−4.51870	−0.257060
$310$	89.3138	5.07268
$311$	−8.19386	−0.464631	−0.232315	−	0.972640i	$-0.574630\pi$
$311$	−8.19386	−0.464631	−0.232315	+	0.972640i	$0.574630\pi$
$312$	10.6643	0.603748
$313$	−21.3440	−1.20644	−0.603218	−	0.797577i	$-0.706115\pi$
$313$	−21.3440	−1.20644	−0.603218	+	0.797577i	$0.706115\pi$
$314$	23.4341	1.32247
$315$	−12.9976	−0.732333
$316$	−5.81331	−0.327024
$317$	−9.88332	−0.555103	−0.277551	−	0.960711i	$-0.589523\pi$
$317$	−9.88332	−0.555103	−0.277551	+	0.960711i	$0.589523\pi$
$318$	15.5993	0.874768
$319$	0.0709835	0.00397432
$320$	50.6671	2.83237
$321$	−10.8540	−0.605810
$322$	49.9064	2.78117
$323$	−23.3097	−1.29699
$324$	−17.8388	−0.991044
$325$	−6.57704	−0.364828
$326$	25.7253	1.42479
$327$	7.95304	0.439804
$328$	59.9304	3.30910
$329$	−22.0429	−1.21526
$330$	−0.587565	−0.0323444
$331$	−26.4815	−1.45555	−0.727776	−	0.685814i	$-0.759446\pi$
$331$	−26.4815	−1.45555	−0.727776	+	0.685814i	$0.759446\pi$
$332$	−46.3119	−2.54170
$333$	−1.57892	−0.0865245
$334$	41.9389	2.29479
$335$	37.8034	2.06542
$336$	44.9731	2.45349
$337$	−7.99212	−0.435359	−0.217679	−	0.976020i	$-0.569849\pi$
$337$	−7.99212	−0.435359	−0.217679	+	0.976020i	$0.569849\pi$
$338$	−2.65702	−0.144523
$339$	−21.0897	−1.14544
$340$	−131.919	−7.15432
$341$	−0.489477	−0.0265067
$342$	10.3398	0.559112
$343$	−15.1792	−0.819599
$344$	6.38737	0.344384
$345$	−28.0747	−1.51149
$346$	13.6978	0.736400
$347$	11.9985	0.644115	0.322058	−	0.946720i	$-0.395626\pi$
$347$	11.9985	0.644115	0.322058	+	0.946720i	$0.395626\pi$
$348$	9.50898	0.509735
$349$	29.5168	1.58000	0.790000	−	0.613107i	$-0.210081\pi$
$349$	29.5168	1.58000	0.790000	+	0.613107i	$0.210081\pi$
$350$	−52.1826	−2.78928
$351$	5.61340	0.299621
$352$	−0.705868	−0.0376229
$353$	20.6622	1.09974	0.549869	−	0.835251i	$-0.314678\pi$
$353$	20.6622	1.09974	0.549869	+	0.835251i	$0.314678\pi$
$354$	5.87933	0.312483
$355$	54.3520	2.88471
$356$	18.8437	0.998713
$357$	−30.0151	−1.58857
$358$	−53.9013	−2.84877
$359$	32.6778	1.72467	0.862335	−	0.506337i	$-0.169001\pi$
$359$	32.6778	1.72467	0.862335	+	0.506337i	$0.169001\pi$
$360$	35.3866	1.86504
$361$	−9.74644	−0.512970
$362$	−34.3069	−1.80313
$363$	−14.4262	−0.757179
$364$	−15.1088	−0.791915
$365$	48.4442	2.53569
$366$	30.5652	1.59767
$367$	12.1592	0.634707	0.317353	−	0.948307i	$-0.397206\pi$
$367$	12.1592	0.634707	0.317353	+	0.948307i	$0.397206\pi$
$368$	−72.2197	−3.76471
$369$	9.43046	0.490930
$370$	−11.1581	−0.580082
$371$	−13.3647	−0.693859
$372$	−65.5706	−3.39968
$373$	19.4073	1.00487	0.502436	−	0.864615i	$-0.332437\pi$
$373$	19.4073	1.00487	0.502436	+	0.864615i	$0.332437\pi$
$374$	1.00875	0.0521611
$375$	7.03876	0.363480
$376$	60.0128	3.09493
$377$	−1.43269	−0.0737870
$378$	44.5371	2.29074
$379$	−13.6930	−0.703361	−0.351680	−	0.936120i	$-0.614390\pi$
$379$	−13.6930	−0.703361	−0.351680	+	0.936120i	$0.614390\pi$
$380$	52.3698	2.68651
$381$	20.6980	1.06039
$382$	20.1861	1.03281
$383$	−31.3356	−1.60117	−0.800586	−	0.599217i	$-0.795478\pi$
$383$	−31.3356	−1.60117	−0.800586	+	0.599217i	$0.795478\pi$
$384$	−14.5243	−0.741189
$385$	0.503393	0.0256553
$386$	58.8439	2.99507
$387$	1.00510	0.0510920
$388$	15.1534	0.769296
$389$	13.8066	0.700022	0.350011	−	0.936746i	$-0.386178\pi$
$389$	13.8066	0.700022	0.350011	+	0.936746i	$0.386178\pi$
$390$	11.8590	0.600505
$391$	48.1995	2.43755
$392$	−15.5822	−0.787021
$393$	11.5499	0.582614
$394$	−15.1537	−0.763433
$395$	−3.90926	−0.196696
$396$	−0.320699	−0.0161157
$397$	25.4092	1.27525	0.637625	−	0.770347i	$-0.279917\pi$
$397$	25.4092	1.27525	0.637625	+	0.770347i	$0.279917\pi$
$398$	14.3837	0.720988
$399$	11.9155	0.596521
$400$	75.5137	3.77568
$401$	−15.8299	−0.790508	−0.395254	−	0.918572i	$-0.629343\pi$
$401$	−15.8299	−0.790508	−0.395254	+	0.918572i	$0.629343\pi$
$402$	−38.7242	−1.93139
$403$	9.87929	0.492122
$404$	49.2635	2.45095
$405$	−11.9960	−0.596086
$406$	−11.3670	−0.564136
$407$	0.0611511	0.00303115
$408$	81.7175	4.04562
$409$	1.11974	0.0553678	0.0276839	−	0.999617i	$-0.491187\pi$
$409$	1.11974	0.0553678	0.0276839	+	0.999617i	$0.491187\pi$
$410$	66.6442	3.29132
$411$	−2.77793	−0.137025
$412$	−17.4295	−0.858690
$413$	−5.03709	−0.247859
$414$	−21.3805	−1.05079
$415$	−31.1433	−1.52876
$416$	14.2468	0.698506
$417$	−0.826103	−0.0404544
$418$	−0.400457	−0.0195870
$419$	−16.9831	−0.829679	−0.414840	−	0.909895i	$-0.636162\pi$
$419$	−16.9831	−0.829679	−0.414840	+	0.909895i	$0.636162\pi$
$420$	67.4348	3.29048
$421$	−7.24476	−0.353088	−0.176544	−	0.984293i	$-0.556492\pi$
$421$	−7.24476	−0.353088	−0.176544	+	0.984293i	$0.556492\pi$
$422$	36.2068	1.76252
$423$	9.44344	0.459156
$424$	36.3859	1.76706
$425$	−50.3979	−2.44466
$426$	−55.6759	−2.69750
$427$	−26.1866	−1.26726
$428$	−41.8659	−2.02366
$429$	−0.0649924	−0.00313786
$430$	7.10293	0.342534
$431$	16.2537	0.782912	0.391456	−	0.920197i	$-0.371972\pi$
$431$	16.2537	0.782912	0.391456	+	0.920197i	$0.371972\pi$
$432$	−64.4498	−3.10084
$433$	12.7803	0.614184	0.307092	−	0.951680i	$-0.400644\pi$
$433$	12.7803	0.614184	0.307092	+	0.951680i	$0.400644\pi$
$434$	78.3829	3.76250
$435$	6.39448	0.306592
$436$	30.6764	1.46913
$437$	−19.1344	−0.915322
$438$	−49.6242	−2.37114
$439$	16.8739	0.805350	0.402675	−	0.915343i	$-0.368081\pi$
$439$	16.8739	0.805350	0.402675	+	0.915343i	$0.368081\pi$
$440$	−1.37051	−0.0653366
$441$	−2.45197	−0.116761
$442$	−20.3599	−0.968423
$443$	2.83942	0.134905	0.0674524	−	0.997722i	$-0.478513\pi$
$443$	2.83942	0.134905	0.0674524	+	0.997722i	$0.478513\pi$
$444$	8.19183	0.388767
$445$	12.6718	0.600699
$446$	17.5996	0.833366
$447$	10.5614	0.499537
$448$	44.4661	2.10082
$449$	−29.2099	−1.37850	−0.689251	−	0.724523i	$-0.742060\pi$
$449$	−29.2099	−1.37850	−0.689251	+	0.724523i	$0.742060\pi$
$450$	22.3557	1.05386
$451$	−0.365239	−0.0171984
$452$	−81.3472	−3.82625
$453$	0.202945	0.00953519
$454$	66.7584	3.13313
$455$	−10.1602	−0.476316
$456$	−32.4405	−1.51917
$457$	7.04084	0.329357	0.164678	−	0.986347i	$-0.447341\pi$
$457$	7.04084	0.329357	0.164678	+	0.986347i	$0.447341\pi$
$458$	72.2146	3.37437
$459$	43.0138	2.00771
$460$	−108.289	−5.04902
$461$	11.2435	0.523661	0.261830	−	0.965114i	$-0.415674\pi$
$461$	11.2435	0.523661	0.261830	+	0.965114i	$0.415674\pi$
$462$	−0.515654	−0.0239904
$463$	−7.42478	−0.345059	−0.172529	−	0.985004i	$-0.555194\pi$
$463$	−7.42478	−0.345059	−0.172529	+	0.985004i	$0.555194\pi$
$464$	16.4492	0.763637
$465$	−44.0941	−2.04481
$466$	−53.3134	−2.46969
$467$	27.9627	1.29396	0.646979	−	0.762508i	$-0.276032\pi$
$467$	27.9627	1.29396	0.646979	+	0.762508i	$0.276032\pi$
$468$	6.47278	0.299204
$469$	33.1768	1.53196
$470$	66.7359	3.07830
$471$	−11.5694	−0.533090
$472$	13.7137	0.631225
$473$	−0.0389271	−0.00178987
$474$	4.00448	0.183932
$475$	20.0071	0.917990
$476$	−115.774	−5.30650
$477$	5.72558	0.262156
$478$	51.5603	2.35831
$479$	−25.3382	−1.15773	−0.578866	−	0.815422i	$-0.696505\pi$
$479$	−25.3382	−1.15773	−0.578866	+	0.815422i	$0.696505\pi$
$480$	−63.5875	−2.90236
$481$	−1.23423	−0.0562763
$482$	13.9622	0.635961
$483$	−24.6387	−1.12110
$484$	−55.6446	−2.52930
$485$	10.1902	0.462711
$486$	−32.4565	−1.47226
$487$	−1.03809	−0.0470402	−0.0235201	−	0.999723i	$-0.507487\pi$
$487$	−1.03809	−0.0470402	−0.0235201	+	0.999723i	$0.507487\pi$
$488$	71.2943	3.22734
$489$	−12.7005	−0.574338
$490$	−17.3279	−0.782793
$491$	−14.1449	−0.638350	−0.319175	−	0.947696i	$-0.603406\pi$
$491$	−14.1449	−0.638350	−0.319175	+	0.947696i	$0.603406\pi$
$492$	−48.9275	−2.20582
$493$	−10.9782	−0.494435
$494$	8.08255	0.363651
$495$	−0.215660	−0.00969318
$496$	−113.428	−5.09308
$497$	47.7001	2.13964
$498$	31.9018	1.42956
$499$	−23.8005	−1.06546	−0.532728	−	0.846286i	$-0.678833\pi$
$499$	−23.8005	−1.06546	−0.532728	+	0.846286i	$0.678833\pi$
$500$	27.1499	1.21418
$501$	−20.7052	−0.925039
$502$	−5.60738	−0.250270
$503$	−10.7406	−0.478900	−0.239450	−	0.970909i	$-0.576967\pi$
$503$	−10.7406	−0.478900	−0.239450	+	0.970909i	$0.576967\pi$
$504$	31.0558	1.38333
$505$	33.1281	1.47418
$506$	0.828059	0.0368117
$507$	1.31176	0.0582575
$508$	79.8363	3.54216
$509$	11.2272	0.497636	0.248818	−	0.968550i	$-0.419958\pi$
$509$	11.2272	0.497636	0.248818	+	0.968550i	$0.419958\pi$
$510$	90.8721	4.02389
$511$	42.5153	1.88077
$512$	23.1088	1.02128
$513$	−17.0758	−0.753914
$514$	64.1118	2.82785
$515$	−11.7208	−0.516479
$516$	−5.21469	−0.229564
$517$	−0.365741	−0.0160853
$518$	−9.79250	−0.430258
$519$	−6.76260	−0.296845
$520$	27.6615	1.21304
$521$	−17.4190	−0.763141	−0.381570	−	0.924340i	$-0.624617\pi$
$521$	−17.4190	−0.763141	−0.381570	+	0.924340i	$0.624617\pi$
$522$	4.86977	0.213144
$523$	16.8095	0.735027	0.367513	−	0.930018i	$-0.380209\pi$
$523$	16.8095	0.735027	0.367513	+	0.930018i	$0.380209\pi$
$524$	44.5501	1.94618
$525$	25.7625	1.12437
$526$	47.2233	2.05904
$527$	75.7021	3.29763
$528$	0.746205	0.0324744
$529$	16.5658	0.720253
$530$	40.4622	1.75756
$531$	2.15795	0.0936470
$532$	45.9604	1.99264
$533$	7.37174	0.319305
$534$	−12.9804	−0.561717
$535$	−28.1535	−1.21718
$536$	−90.3253	−3.90146
$537$	26.6110	1.14835
$538$	−10.5468	−0.454705
$539$	0.0949640	0.00409039
$540$	−96.6389	−4.15868
$541$	−19.4774	−0.837399	−0.418699	−	0.908125i	$-0.637514\pi$
$541$	−19.4774	−0.837399	−0.418699	+	0.908125i	$0.637514\pi$
$542$	−6.85007	−0.294236
$543$	16.9373	0.726848
$544$	109.169	4.68058
$545$	20.6289	0.883644
$546$	10.4076	0.445405
$547$	21.8372	0.933690	0.466845	−	0.884339i	$-0.345391\pi$
$547$	21.8372	0.933690	0.466845	+	0.884339i	$0.345391\pi$
$548$	−10.7150	−0.457723
$549$	11.2187	0.478800
$550$	−0.865827	−0.0369190
$551$	4.35818	0.185665
$552$	67.0800	2.85512
$553$	−3.43082	−0.145893
$554$	22.9862	0.976590
$555$	5.50874	0.233833
$556$	−3.18644	−0.135135
$557$	5.00937	0.212254	0.106127	−	0.994353i	$-0.466155\pi$
$557$	5.00937	0.212254	0.106127	+	0.994353i	$0.466155\pi$
$558$	−33.5802	−1.42156
$559$	0.785679	0.0332307
$560$	116.653	4.92949
$561$	−0.498018	−0.0210263
$562$	37.6114	1.58654
$563$	25.1208	1.05872	0.529358	−	0.848398i	$-0.322433\pi$
$563$	25.1208	1.05872	0.529358	+	0.848398i	$0.322433\pi$
$564$	−48.9948	−2.06305
$565$	−54.7033	−2.30139
$566$	9.35526	0.393231
$567$	−10.5279	−0.442128
$568$	−129.866	−5.44904
$569$	21.4705	0.900091	0.450045	−	0.893006i	$-0.351408\pi$
$569$	21.4705	0.900091	0.450045	+	0.893006i	$0.351408\pi$
$570$	−36.0747	−1.51100
$571$	−1.52935	−0.0640012	−0.0320006	−	0.999488i	$-0.510188\pi$
$571$	−1.52935	−0.0640012	−0.0320006	+	0.999488i	$0.510188\pi$
$572$	−0.250688	−0.0104818
$573$	−9.96586	−0.416330
$574$	58.4879	2.44124
$575$	−41.3705	−1.72527
$576$	−19.0498	−0.793741
$577$	−19.4407	−0.809327	−0.404663	−	0.914466i	$-0.632611\pi$
$577$	−19.4407	−0.809327	−0.404663	+	0.914466i	$0.632611\pi$
$578$	−110.843	−4.61045
$579$	−29.0511	−1.20732
$580$	24.6648	1.02415
$581$	−27.3318	−1.13391
$582$	−10.4384	−0.432684
$583$	−0.221750	−0.00918394
$584$	−115.750	−4.78976
$585$	4.35273	0.179963
$586$	−18.4248	−0.761120
$587$	34.8974	1.44037	0.720184	−	0.693783i	$-0.244057\pi$
$587$	34.8974	1.44037	0.720184	+	0.693783i	$0.244057\pi$
$588$	12.7214	0.524622
$589$	−30.0525	−1.23829
$590$	15.2500	0.627834
$591$	7.48136	0.307742
$592$	14.1708	0.582415
$593$	28.6134	1.17501	0.587507	−	0.809219i	$-0.300110\pi$
$593$	28.6134	1.17501	0.587507	+	0.809219i	$0.300110\pi$
$594$	0.738970	0.0303203
$595$	−77.8543	−3.19171
$596$	40.7373	1.66867
$597$	−7.10120	−0.290633
$598$	−16.7130	−0.683445
$599$	6.32561	0.258457	0.129229	−	0.991615i	$-0.458750\pi$
$599$	6.32561	0.258457	0.129229	+	0.991615i	$0.458750\pi$
$600$	−70.1396	−2.86344
$601$	26.7569	1.09144	0.545718	−	0.837969i	$-0.316257\pi$
$601$	26.7569	1.09144	0.545718	+	0.837969i	$0.316257\pi$
$602$	6.23363	0.254064
$603$	−14.2133	−0.578811
$604$	0.782798	0.0318516
$605$	−37.4192	−1.52131
$606$	−33.9350	−1.37851
$607$	−28.7202	−1.16572	−0.582858	−	0.812574i	$-0.698066\pi$
$607$	−28.7202	−1.16572	−0.582858	+	0.812574i	$0.698066\pi$
$608$	−43.3383	−1.75760
$609$	5.61188	0.227405
$610$	79.2812	3.21000
$611$	7.38188	0.298639
$612$	49.5990	2.00492
$613$	21.3108	0.860734	0.430367	−	0.902654i	$-0.358384\pi$
$613$	21.3108	0.860734	0.430367	+	0.902654i	$0.358384\pi$
$614$	−75.2756	−3.03788
$615$	−32.9022	−1.32674
$616$	−1.20278	−0.0484614
$617$	1.17666	0.0473705	0.0236853	−	0.999719i	$-0.492460\pi$
$617$	1.17666	0.0473705	0.0236853	+	0.999719i	$0.492460\pi$
$618$	12.0063	0.482963
$619$	−1.00000	−0.0401934
$619$	−1.00000	−0.0401934
$620$	−170.079	−6.83055
$621$	35.3091	1.41690
$622$	21.7712	0.872946
$623$	11.1209	0.445550
$624$	−15.0609	−0.602919
$625$	−14.6278	−0.585111
$626$	56.7114	2.26664
$627$	0.197705	0.00789557
$628$	−44.6254	−1.78075
$629$	−9.45757	−0.377098
$630$	34.5349	1.37590
$631$	−8.82077	−0.351150	−0.175575	−	0.984466i	$-0.556178\pi$
$631$	−8.82077	−0.351150	−0.175575	+	0.984466i	$0.556178\pi$
$632$	9.34056	0.371548
$633$	−17.8752	−0.710477
$634$	26.2601	1.04292
$635$	53.6873	2.13052
$636$	−29.7057	−1.17791
$637$	−1.91669	−0.0759420
$638$	−0.188604	−0.00746692
$639$	−20.4353	−0.808407
$640$	−37.6736	−1.48918
$641$	−3.96385	−0.156563	−0.0782814	−	0.996931i	$-0.524943\pi$
$641$	−3.96385	−0.156563	−0.0782814	+	0.996931i	$0.524943\pi$
$642$	28.8392	1.13819
$643$	−4.55908	−0.179793	−0.0898963	−	0.995951i	$-0.528654\pi$
$643$	−4.55908	−0.179793	−0.0898963	+	0.995951i	$0.528654\pi$
$644$	−95.0363	−3.74495
$645$	−3.50671	−0.138077
$646$	61.9342	2.43677
$647$	−33.5329	−1.31831	−0.659157	−	0.752005i	$-0.729087\pi$
$647$	−33.5329	−1.31831	−0.659157	+	0.752005i	$0.729087\pi$
$648$	28.6626	1.12597
$649$	−0.0835766	−0.00328067
$650$	17.4753	0.685437
$651$	−38.6976	−1.51668
$652$	−48.9884	−1.91853
$653$	24.2422	0.948670	0.474335	−	0.880344i	$-0.342689\pi$
$653$	24.2422	0.948670	0.474335	+	0.880344i	$0.342689\pi$
$654$	−21.1313	−0.826301
$655$	29.9585	1.17057
$656$	−84.6379	−3.30456
$657$	−18.2141	−0.710598
$658$	58.5683	2.28323
$659$	39.2275	1.52809	0.764044	−	0.645164i	$-0.223211\pi$
$659$	39.2275	1.52809	0.764044	+	0.645164i	$0.223211\pi$
$660$	1.11889	0.0435529
$661$	25.6966	0.999480	0.499740	−	0.866175i	$-0.333429\pi$
$661$	25.6966	0.999480	0.499740	+	0.866175i	$0.333429\pi$
$662$	70.3617	2.73468
$663$	10.0517	0.390374
$664$	74.4120	2.88774
$665$	30.9069	1.19852
$666$	4.19522	0.162562
$667$	−9.01179	−0.348938
$668$	−79.8639	−3.09003
$669$	−8.68891	−0.335933
$670$	−100.444	−3.88050
$671$	−0.434494	−0.0167735
$672$	−55.8052	−2.15273
$673$	41.3437	1.59368	0.796841	−	0.604189i	$-0.206503\pi$
$673$	41.3437	1.59368	0.796841	+	0.604189i	$0.206503\pi$
$674$	21.2352	0.817949
$675$	−36.9195	−1.42103
$676$	5.05973	0.194605
$677$	−44.7301	−1.71912	−0.859559	−	0.511037i	$-0.829261\pi$
$677$	−44.7301	−1.71912	−0.859559	+	0.511037i	$0.829261\pi$
$678$	56.0357	2.15204
$679$	8.94301	0.343201
$680$	211.962	8.12837
$681$	−32.9585	−1.26297
$682$	1.30055	0.0498006
$683$	−51.2187	−1.95983	−0.979914	−	0.199420i	$-0.936094\pi$
$683$	−51.2187	−1.95983	−0.979914	+	0.199420i	$0.936094\pi$
$684$	−19.6900	−0.752865
$685$	−7.20550	−0.275308
$686$	40.3313	1.53986
$687$	−35.6523	−1.36022
$688$	−9.02070	−0.343911
$689$	4.47565	0.170509
$690$	74.5949	2.83978
$691$	4.76156	0.181138	0.0905691	−	0.995890i	$-0.471131\pi$
$691$	4.76156	0.181138	0.0905691	+	0.995890i	$0.471131\pi$
$692$	−26.0847	−0.991590
$693$	−0.189266	−0.00718961
$694$	−31.8803	−1.21016
$695$	−2.14278	−0.0812802
$696$	−15.2786	−0.579134
$697$	56.4874	2.13961
$698$	−78.4267	−2.96849
$699$	26.3207	0.995541
$700$	99.3710	3.75587
$701$	−14.8498	−0.560870	−0.280435	−	0.959873i	$-0.590479\pi$
$701$	−14.8498	−0.560870	−0.280435	+	0.959873i	$0.590479\pi$
$702$	−14.9149	−0.562926
$703$	3.75450	0.141604
$704$	0.737792	0.0278066
$705$	−32.9474	−1.24087
$706$	−54.8998	−2.06618
$707$	29.0737	1.09343
$708$	−11.1960	−0.420770
$709$	−6.00703	−0.225599	−0.112799	−	0.993618i	$-0.535982\pi$
$709$	−6.00703	−0.225599	−0.112799	+	0.993618i	$0.535982\pi$
$710$	−144.414	−5.41977
$711$	1.46980	0.0551219
$712$	−30.2772	−1.13469
$713$	62.1421	2.32724
$714$	79.7506	2.98459
$715$	−0.168580	−0.00630453
$716$	102.644	3.83598
$717$	−25.4552	−0.950643
$718$	−86.8255	−3.24030
$719$	−9.01811	−0.336319	−0.168159	−	0.985760i	$-0.553782\pi$
$719$	−9.01811	−0.336319	−0.168159	+	0.985760i	$0.553782\pi$
$720$	−49.9755	−1.86248
$721$	−10.2863	−0.383082
$722$	25.8964	0.963765
$723$	−6.89312	−0.256358
$724$	65.3304	2.42798
$725$	9.42282	0.349955
$726$	38.3306	1.42258
$727$	27.6663	1.02608	0.513042	−	0.858363i	$-0.328518\pi$
$727$	27.6663	1.02608	0.513042	+	0.858363i	$0.328518\pi$
$728$	24.2761	0.899733
$729$	26.6006	0.985209
$730$	−128.717	−4.76403
$731$	6.02042	0.222673
$732$	−58.2051	−2.15132
$733$	−44.4702	−1.64255	−0.821273	−	0.570536i	$-0.806736\pi$
$733$	−44.4702	−1.64255	−0.821273	+	0.570536i	$0.806736\pi$
$734$	−32.3073	−1.19248
$735$	8.55474	0.315546
$736$	89.6142	3.30322
$737$	0.550477	0.0202771
$738$	−25.0569	−0.922357
$739$	−31.1474	−1.14577	−0.572887	−	0.819634i	$-0.694177\pi$
$739$	−31.1474	−1.14577	−0.572887	+	0.819634i	$0.694177\pi$
$740$	21.2483	0.781102
$741$	−3.99035	−0.146589
$742$	35.5101	1.30362
$743$	−38.1911	−1.40110	−0.700548	−	0.713605i	$-0.747061\pi$
$743$	−38.1911	−1.40110	−0.700548	+	0.713605i	$0.747061\pi$
$744$	105.356	3.86254
$745$	27.3945	1.00366
$746$	−51.5655	−1.88795
$747$	11.7092	0.428419
$748$	−1.92095	−0.0702369
$749$	−24.7078	−0.902805
$750$	−18.7021	−0.682904
$751$	47.8315	1.74540	0.872699	−	0.488259i	$-0.162368\pi$
$751$	47.8315	1.74540	0.872699	+	0.488259i	$0.162368\pi$
$752$	−84.7544	−3.09068
$753$	2.76836	0.100884
$754$	3.80667	0.138631
$755$	0.526406	0.0191579
$756$	−84.8116	−3.08457
$757$	0.157125	0.00571080	0.00285540	−	0.999996i	$-0.499091\pi$
$757$	0.157125	0.00571080	0.00285540	+	0.999996i	$0.499091\pi$
$758$	36.3824	1.32147
$759$	−0.408811	−0.0148389
$760$	−84.1454	−3.05228
$761$	6.82533	0.247418	0.123709	−	0.992319i	$-0.460521\pi$
$761$	6.82533	0.247418	0.123709	+	0.992319i	$0.460521\pi$
$762$	−54.9950	−1.99226
$763$	18.1042	0.655415
$764$	−38.4403	−1.39072
$765$	33.3537	1.20591
$766$	83.2591	3.00827
$767$	1.68686	0.0609088
$768$	−0.476008	−0.0171765
$769$	17.0441	0.614627	0.307314	−	0.951608i	$-0.400570\pi$
$769$	17.0441	0.614627	0.307314	+	0.951608i	$0.400570\pi$
$770$	−1.33752	−0.0482010
$771$	−31.6519	−1.13992
$772$	−112.056	−4.03298
$773$	−39.0622	−1.40497	−0.702485	−	0.711699i	$-0.747926\pi$
$773$	−39.0622	−1.40497	−0.702485	+	0.711699i	$0.747926\pi$
$774$	−2.67056	−0.0959913
$775$	−64.9764	−2.33402
$776$	−24.3478	−0.874034
$777$	4.83454	0.173438
$778$	−36.6844	−1.31520
$779$	−22.4246	−0.803444
$780$	−22.5830	−0.808602
$781$	0.791451	0.0283203
$782$	−128.067	−4.57966
$783$	−8.04223	−0.287406
$784$	22.0063	0.785940
$785$	−30.0092	−1.07107
$786$	−30.6882	−1.09461
$787$	−23.5061	−0.837900	−0.418950	−	0.908009i	$-0.637602\pi$
$787$	−23.5061	−0.837900	−0.418950	+	0.908009i	$0.637602\pi$
$788$	28.8571	1.02799
$789$	−23.3141	−0.830004
$790$	10.3870	0.369552
$791$	−48.0084	−1.70698
$792$	0.515285	0.0183099
$793$	8.76955	0.311416
$794$	−67.5126	−2.39593
$795$	−19.9761	−0.708480
$796$	−27.3907	−0.970837
$797$	−32.8132	−1.16230	−0.581152	−	0.813795i	$-0.697398\pi$
$797$	−32.8132	−1.16230	−0.581152	+	0.813795i	$0.697398\pi$
$798$	−31.6597	−1.12074
$799$	56.5652	2.00113
$800$	−93.7016	−3.31285
$801$	−4.76433	−0.168339
$802$	42.0603	1.48520
$803$	0.705424	0.0248939
$804$	73.7421	2.60068
$805$	−63.9088	−2.25249
$806$	−26.2494	−0.924596
$807$	5.20695	0.183293
$808$	−79.1544	−2.78464
$809$	−21.3128	−0.749317	−0.374658	−	0.927163i	$-0.622240\pi$
$809$	−21.3128	−0.749317	−0.374658	+	0.927163i	$0.622240\pi$
$810$	31.8736	1.11992
$811$	−4.01864	−0.141114	−0.0705568	−	0.997508i	$-0.522478\pi$
$811$	−4.01864	−0.141114	−0.0705568	+	0.997508i	$0.522478\pi$
$812$	21.6461	0.759630
$813$	3.38187	0.118607
$814$	−0.162480	−0.00569491
$815$	−32.9431	−1.15395
$816$	−115.407	−4.04006
$817$	−2.39001	−0.0836158
$818$	−2.97518	−0.104025
$819$	3.82002	0.133482
$820$	−126.910	−4.43189
$821$	50.5486	1.76416	0.882079	−	0.471101i	$-0.156143\pi$
$821$	50.5486	1.76416	0.882079	+	0.471101i	$0.156143\pi$
$822$	7.38101	0.257442
$823$	9.32365	0.325002	0.162501	−	0.986708i	$-0.448044\pi$
$823$	9.32365	0.325002	0.162501	+	0.986708i	$0.448044\pi$
$824$	28.0050	0.975599
$825$	0.427458	0.0148822
$826$	13.3836	0.465676
$827$	−15.2461	−0.530158	−0.265079	−	0.964227i	$-0.585398\pi$
$827$	−15.2461	−0.530158	−0.265079	+	0.964227i	$0.585398\pi$
$828$	40.7147	1.41493
$829$	28.2476	0.981079	0.490539	−	0.871419i	$-0.336800\pi$
$829$	28.2476	0.981079	0.490539	+	0.871419i	$0.336800\pi$
$830$	82.7482	2.87223
$831$	−11.3483	−0.393666
$832$	−14.8911	−0.516256
$833$	−14.6870	−0.508875
$834$	2.19497	0.0760055
$835$	−53.7058	−1.85857
$836$	0.762586	0.0263746
$837$	55.4564	1.91685
$838$	45.1244	1.55880
$839$	−34.0866	−1.17680	−0.588400	−	0.808570i	$-0.700242\pi$
$839$	−34.0866	−1.17680	−0.588400	+	0.808570i	$0.700242\pi$
$840$	−108.351	−3.73847
$841$	−26.9474	−0.929221
$842$	19.2494	0.663380
$843$	−18.5687	−0.639539
$844$	−68.9483	−2.37330
$845$	3.40250	0.117050
$846$	−25.0914	−0.862659
$847$	−32.8396	−1.12838
$848$	−51.3868	−1.76463
$849$	−4.61868	−0.158513
$850$	133.908	4.59300
$851$	−7.76351	−0.266130
$852$	106.023	3.63229
$853$	44.9902	1.54044	0.770218	−	0.637780i	$-0.220147\pi$
$853$	44.9902	1.54044	0.770218	+	0.637780i	$0.220147\pi$
$854$	69.5782	2.38092
$855$	−13.2409	−0.452828
$856$	67.2683	2.29918
$857$	42.2270	1.44245	0.721224	−	0.692702i	$-0.243580\pi$
$857$	42.2270	1.44245	0.721224	+	0.692702i	$0.243580\pi$
$858$	0.172686	0.00589540
$859$	10.8335	0.369636	0.184818	−	0.982773i	$-0.440830\pi$
$859$	10.8335	0.369636	0.184818	+	0.982773i	$0.440830\pi$
$860$	−13.5261	−0.461235
$861$	−28.8754	−0.984070
$862$	−43.1862	−1.47093
$863$	0.393779	0.0134044	0.00670219	−	0.999978i	$-0.497867\pi$
$863$	0.393779	0.0134044	0.00670219	+	0.999978i	$0.497867\pi$
$864$	79.9729	2.72073
$865$	−17.5411	−0.596415
$866$	−33.9576	−1.15392
$867$	54.7229	1.85849
$868$	−149.264	−5.06635
$869$	−0.0569250	−0.00193105
$870$	−16.9902	−0.576023
$871$	−11.1105	−0.376464
$872$	−49.2895	−1.66915
$873$	−3.83129	−0.129670
$874$	50.8404	1.71970
$875$	16.0229	0.541674
$876$	94.4989	3.19282
$877$	33.4249	1.12868	0.564339	−	0.825543i	$-0.309131\pi$
$877$	33.4249	1.12868	0.564339	+	0.825543i	$0.309131\pi$
$878$	−44.8343	−1.51309
$879$	9.09628	0.306810
$880$	1.93553	0.0652468
$881$	56.6172	1.90748	0.953740	−	0.300632i	$-0.0971976\pi$
$881$	56.6172	1.90748	0.953740	+	0.300632i	$0.0971976\pi$
$882$	6.51492	0.219369
$883$	0.370933	0.0124829	0.00624144	−	0.999981i	$-0.498013\pi$
$883$	0.370933	0.0124829	0.00624144	+	0.999981i	$0.498013\pi$
$884$	38.7712	1.30402
$885$	−7.52892	−0.253082
$886$	−7.54437	−0.253458
$887$	37.2816	1.25179	0.625896	−	0.779907i	$-0.284733\pi$
$887$	37.2816	1.25179	0.625896	+	0.779907i	$0.284733\pi$
$888$	−13.1623	−0.441697
$889$	47.1167	1.58024
$890$	−33.6691	−1.12859
$891$	−0.174681	−0.00585202
$892$	−33.5148	−1.12216
$893$	−22.4554	−0.751443
$894$	−28.0618	−0.938526
$895$	69.0246	2.30724
$896$	−33.0629	−1.10455
$897$	8.25118	0.275499
$898$	77.6112	2.58992
$899$	−14.1539	−0.472059
$900$	−42.5717	−1.41906
$901$	34.2956	1.14255
$902$	0.970445	0.0323123
$903$	−3.07753	−0.102414
$904$	130.705	4.34718
$905$	43.9326	1.46037
$906$	−0.539228	−0.0179146
$907$	9.72436	0.322892	0.161446	−	0.986882i	$-0.448384\pi$
$907$	9.72436	0.322892	0.161446	+	0.986882i	$0.448384\pi$
$908$	−127.127	−4.21887
$909$	−12.4555	−0.413123
$910$	26.9957	0.894899
$911$	34.1694	1.13208	0.566042	−	0.824377i	$-0.308474\pi$
$911$	34.1694	1.13208	0.566042	+	0.824377i	$0.308474\pi$
$912$	45.8148	1.51708
$913$	−0.453495	−0.0150085
$914$	−18.7076	−0.618793
$915$	−39.1410	−1.29396
$916$	−137.518	−4.54371
$917$	26.2919	0.868237
$918$	−114.288	−3.77208
$919$	−4.89526	−0.161480	−0.0807399	−	0.996735i	$-0.525728\pi$
$919$	−4.89526	−0.161480	−0.0807399	+	0.996735i	$0.525728\pi$
$920$	173.995	5.73644
$921$	37.1635	1.22458
$922$	−29.8741	−0.983850
$923$	−15.9741	−0.525795
$924$	0.981956	0.0323040
$925$	8.11761	0.266905
$926$	19.7278	0.648294
$927$	4.40678	0.144738
$928$	−20.4112	−0.670029
$929$	−6.34943	−0.208318	−0.104159	−	0.994561i	$-0.533215\pi$
$929$	−6.34943	−0.208318	−0.104159	+	0.994561i	$0.533215\pi$
$930$	117.159	3.84178
$931$	5.83051	0.191087
$932$	101.524	3.32553
$933$	−10.7484	−0.351887
$934$	−74.2972	−2.43108
$935$	−1.29178	−0.0422456
$936$	−10.4002	−0.339940
$937$	14.6890	0.479870	0.239935	−	0.970789i	$-0.422874\pi$
$937$	14.6890	0.479870	0.239935	+	0.970789i	$0.422874\pi$
$938$	−88.1511	−2.87824
$939$	−27.9983	−0.913691
$940$	−127.085	−4.14504
$941$	59.2539	1.93162	0.965810	−	0.259250i	$-0.0834753\pi$
$941$	59.2539	1.93162	0.965810	+	0.259250i	$0.0834753\pi$
$942$	30.7401	1.00157
$943$	46.3692	1.50999
$944$	−19.3675	−0.630358
$945$	−57.0330	−1.85528
$946$	0.103430	0.00336280
$947$	−9.64945	−0.313565	−0.156783	−	0.987633i	$-0.550112\pi$
$947$	−9.64945	−0.313565	−0.156783	+	0.987633i	$0.550112\pi$
$948$	−7.62569	−0.247671
$949$	−14.2378	−0.462179
$950$	−53.1592	−1.72471
$951$	−12.9646	−0.420406
$952$	186.021	6.02896
$953$	−5.71498	−0.185127	−0.0925633	−	0.995707i	$-0.529506\pi$
$953$	−5.71498	−0.185127	−0.0925633	+	0.995707i	$0.529506\pi$
$954$	−15.2130	−0.492538
$955$	−25.8498	−0.836481
$956$	−98.1857	−3.17555
$957$	0.0931137	0.00300994
$958$	67.3240	2.17514
$959$	−6.32365	−0.204201
$960$	66.4633	2.14509
$961$	66.6003	2.14840
$962$	3.27938	0.105731
$963$	10.5851	0.341101
$964$	−26.5881	−0.856345
$965$	−75.3539	−2.42573
$966$	65.4654	2.10632
$967$	−24.2467	−0.779720	−0.389860	−	0.920874i	$-0.627477\pi$
$967$	−24.2467	−0.779720	−0.389860	+	0.920874i	$0.627477\pi$
$968$	89.4073	2.87366
$969$	−30.5768	−0.982269
$970$	−27.0754	−0.869339
$971$	−57.9066	−1.85831	−0.929156	−	0.369689i	$-0.879464\pi$
$971$	−57.9066	−1.85831	−0.929156	+	0.369689i	$0.879464\pi$
$972$	61.8066	1.98245
$973$	−1.88053	−0.0602870
$974$	2.75821	0.0883787
$975$	−8.62752	−0.276302
$976$	−100.687	−3.22291
$977$	−4.86090	−0.155514	−0.0777569	−	0.996972i	$-0.524776\pi$
$977$	−4.86090	−0.155514	−0.0777569	+	0.996972i	$0.524776\pi$
$978$	33.7455	1.07906
$979$	0.184521	0.00589731
$980$	32.9973	1.05406
$981$	−7.75604	−0.247631
$982$	37.5832	1.19933
$983$	−19.5674	−0.624104	−0.312052	−	0.950065i	$-0.601016\pi$
$983$	−19.5674	−0.624104	−0.312052	+	0.950065i	$0.601016\pi$
$984$	78.6146	2.50614
$985$	19.4054	0.618309
$986$	29.1694	0.928942
$987$	−28.9151	−0.920378
$988$	−15.3915	−0.489670
$989$	4.94203	0.157147
$990$	0.573011	0.0182115
$991$	57.7879	1.83569	0.917846	−	0.396936i	$-0.129926\pi$
$991$	57.7879	1.83569	0.917846	+	0.396936i	$0.129926\pi$
$992$	140.748	4.46876
$993$	−34.7375	−1.10236
$994$	−126.740	−4.01994
$995$	−18.4193	−0.583933
$996$	−60.7504	−1.92495
$997$	−53.9249	−1.70782	−0.853909	−	0.520422i	$-0.825775\pi$
$997$	−53.9249	−1.70782	−0.853909	+	0.520422i	$0.825775\pi$
$998$	63.2383	2.00177
$999$	−6.92825	−0.219200

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8047.2.a.c.1.9	✓	151

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8047.2.a.c.1.9	✓	151	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:	\( N \)	\(=\)	\( 8047 = 13 \cdot 619 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8047.a (trivial)

\(f(q)\)	\(=\)	\(q-2.65702 q^{2} +1.31176 q^{3} +5.05973 q^{4} +3.40250 q^{5} -3.48538 q^{6} +2.98608 q^{7} -8.12975 q^{8} -1.27927 q^{9} +O(q^{10})\)	\(q-2.65702 q^{2} +1.31176 q^{3} +5.05973 q^{4} +3.40250 q^{5} -3.48538 q^{6} +2.98608 q^{7} -8.12975 q^{8} -1.27927 q^{9} -9.04051 q^{10} +0.0495458 q^{11} +6.63718 q^{12} -1.00000 q^{13} -7.93407 q^{14} +4.46329 q^{15} +11.4814 q^{16} -7.66270 q^{17} +3.39905 q^{18} +3.04197 q^{19} +17.2158 q^{20} +3.91704 q^{21} -0.131644 q^{22} -6.29014 q^{23} -10.6643 q^{24} +6.57704 q^{25} +2.65702 q^{26} -5.61340 q^{27} +15.1088 q^{28} +1.43269 q^{29} -11.8590 q^{30} -9.87929 q^{31} -14.2468 q^{32} +0.0649924 q^{33} +20.3599 q^{34} +10.1602 q^{35} -6.47278 q^{36} +1.23423 q^{37} -8.08255 q^{38} -1.31176 q^{39} -27.6615 q^{40} -7.37174 q^{41} -10.4076 q^{42} -0.785679 q^{43} +0.250688 q^{44} -4.35273 q^{45} +16.7130 q^{46} -7.38188 q^{47} +15.0609 q^{48} +1.91669 q^{49} -17.4753 q^{50} -10.0517 q^{51} -5.05973 q^{52} -4.47565 q^{53} +14.9149 q^{54} +0.168580 q^{55} -24.2761 q^{56} +3.99035 q^{57} -3.80667 q^{58} -1.68686 q^{59} +22.5830 q^{60} -8.76955 q^{61} +26.2494 q^{62} -3.82002 q^{63} +14.8911 q^{64} -3.40250 q^{65} -0.172686 q^{66} +11.1105 q^{67} -38.7712 q^{68} -8.25118 q^{69} -26.9957 q^{70} +15.9741 q^{71} +10.4002 q^{72} +14.2378 q^{73} -3.27938 q^{74} +8.62752 q^{75} +15.3915 q^{76} +0.147948 q^{77} +3.48538 q^{78} -1.14894 q^{79} +39.0656 q^{80} -3.52564 q^{81} +19.5868 q^{82} -9.15305 q^{83} +19.8192 q^{84} -26.0724 q^{85} +2.08756 q^{86} +1.87935 q^{87} -0.402795 q^{88} +3.72424 q^{89} +11.5653 q^{90} -2.98608 q^{91} -31.8264 q^{92} -12.9593 q^{93} +19.6138 q^{94} +10.3503 q^{95} -18.6884 q^{96} +2.99490 q^{97} -5.09268 q^{98} -0.0633826 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 151 q - 13 q^{2} - 16 q^{3} + 151 q^{4} - 43 q^{5} - 17 q^{6} - 18 q^{7} - 39 q^{8} + 135 q^{9}+O(q^{10}) \) 151 * q - 13 * q^2 - 16 * q^3 + 151 * q^4 - 43 * q^5 - 17 * q^6 - 18 * q^7 - 39 * q^8 + 135 * q^9	\( 151 q - 13 q^{2} - 16 q^{3} + 151 q^{4} - 43 q^{5} - 17 q^{6} - 18 q^{7} - 39 q^{8} + 135 q^{9} - 3 q^{10} - 27 q^{11} - 52 q^{12} - 151 q^{13} - 9 q^{14} - 14 q^{15} + 143 q^{16} - 111 q^{17} - 37 q^{18} - 17 q^{19} - 107 q^{20} - 29 q^{21} - 16 q^{22} - 47 q^{23} - 46 q^{24} + 122 q^{25} + 13 q^{26} - 55 q^{27} - 44 q^{28} + 37 q^{29} - 14 q^{30} - 27 q^{31} - 86 q^{32} - 94 q^{33} - 10 q^{34} - 47 q^{35} + 124 q^{36} - 59 q^{37} - 80 q^{38} + 16 q^{39} + 5 q^{40} - 129 q^{41} - 77 q^{42} - 11 q^{43} - 99 q^{44} - 122 q^{45} - 17 q^{46} - 130 q^{47} - 111 q^{48} + 99 q^{49} - 72 q^{50} + 15 q^{51} - 151 q^{52} - 43 q^{53} - 49 q^{54} - 40 q^{55} - 50 q^{56} - 85 q^{57} - 73 q^{58} - 74 q^{59} - 43 q^{60} - 7 q^{61} - 110 q^{62} - 70 q^{63} + 141 q^{64} + 43 q^{65} - 16 q^{66} - 39 q^{67} - 222 q^{68} + 19 q^{69} - 52 q^{70} - 72 q^{71} - 106 q^{72} - 143 q^{73} + 20 q^{74} - 73 q^{75} - 88 q^{76} - 86 q^{77} + 17 q^{78} + 10 q^{79} - 239 q^{80} + 103 q^{81} - 96 q^{82} - 96 q^{83} - 75 q^{84} - 24 q^{85} - 109 q^{86} - 65 q^{87} - 45 q^{88} - 237 q^{89} - 79 q^{90} + 18 q^{91} - 153 q^{92} - 137 q^{93} - 23 q^{94} + 10 q^{95} - 109 q^{96} - 160 q^{97} - 119 q^{98} - 86 q^{99}+O(q^{100}) \) 151 * q - 13 * q^2 - 16 * q^3 + 151 * q^4 - 43 * q^5 - 17 * q^6 - 18 * q^7 - 39 * q^8 + 135 * q^9 - 3 * q^10 - 27 * q^11 - 52 * q^12 - 151 * q^13 - 9 * q^14 - 14 * q^15 + 143 * q^16 - 111 * q^17 - 37 * q^18 - 17 * q^19 - 107 * q^20 - 29 * q^21 - 16 * q^22 - 47 * q^23 - 46 * q^24 + 122 * q^25 + 13 * q^26 - 55 * q^27 - 44 * q^28 + 37 * q^29 - 14 * q^30 - 27 * q^31 - 86 * q^32 - 94 * q^33 - 10 * q^34 - 47 * q^35 + 124 * q^36 - 59 * q^37 - 80 * q^38 + 16 * q^39 + 5 * q^40 - 129 * q^41 - 77 * q^42 - 11 * q^43 - 99 * q^44 - 122 * q^45 - 17 * q^46 - 130 * q^47 - 111 * q^48 + 99 * q^49 - 72 * q^50 + 15 * q^51 - 151 * q^52 - 43 * q^53 - 49 * q^54 - 40 * q^55 - 50 * q^56 - 85 * q^57 - 73 * q^58 - 74 * q^59 - 43 * q^60 - 7 * q^61 - 110 * q^62 - 70 * q^63 + 141 * q^64 + 43 * q^65 - 16 * q^66 - 39 * q^67 - 222 * q^68 + 19 * q^69 - 52 * q^70 - 72 * q^71 - 106 * q^72 - 143 * q^73 + 20 * q^74 - 73 * q^75 - 88 * q^76 - 86 * q^77 + 17 * q^78 + 10 * q^79 - 239 * q^80 + 103 * q^81 - 96 * q^82 - 96 * q^83 - 75 * q^84 - 24 * q^85 - 109 * q^86 - 65 * q^87 - 45 * q^88 - 237 * q^89 - 79 * q^90 + 18 * q^91 - 153 * q^92 - 137 * q^93 - 23 * q^94 + 10 * q^95 - 109 * q^96 - 160 * q^97 - 119 * q^98 - 86 * q^99

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