LMFDB - Embedded newform 8007.2.a.f.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8007,2,Mod(1,8007)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8007, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8007.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8007 = 3 \cdot 17 \cdot 157 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8007.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:	$63.9362168984$
Analytic rank:	$1$
Dimension:	$48$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	8007.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.39228 q^{2} -1.00000 q^{3} +3.72299 q^{4} +0.325110 q^{5} +2.39228 q^{6} -0.363079 q^{7} -4.12187 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.39228 q^{2} -1.00000 q^{3} +3.72299 q^{4} +0.325110 q^{5} +2.39228 q^{6} -0.363079 q^{7} -4.12187 q^{8} +1.00000 q^{9} -0.777753 q^{10} -0.861044 q^{11} -3.72299 q^{12} -0.164103 q^{13} +0.868585 q^{14} -0.325110 q^{15} +2.41467 q^{16} -1.00000 q^{17} -2.39228 q^{18} -8.07048 q^{19} +1.21038 q^{20} +0.363079 q^{21} +2.05986 q^{22} +3.99691 q^{23} +4.12187 q^{24} -4.89430 q^{25} +0.392579 q^{26} -1.00000 q^{27} -1.35174 q^{28} -3.43052 q^{29} +0.777753 q^{30} +5.64736 q^{31} +2.46717 q^{32} +0.861044 q^{33} +2.39228 q^{34} -0.118041 q^{35} +3.72299 q^{36} +5.74569 q^{37} +19.3068 q^{38} +0.164103 q^{39} -1.34006 q^{40} -3.17263 q^{41} -0.868585 q^{42} +8.78267 q^{43} -3.20566 q^{44} +0.325110 q^{45} -9.56171 q^{46} +4.83327 q^{47} -2.41467 q^{48} -6.86817 q^{49} +11.7085 q^{50} +1.00000 q^{51} -0.610953 q^{52} -4.72827 q^{53} +2.39228 q^{54} -0.279934 q^{55} +1.49656 q^{56} +8.07048 q^{57} +8.20675 q^{58} +5.82740 q^{59} -1.21038 q^{60} +11.6859 q^{61} -13.5100 q^{62} -0.363079 q^{63} -10.7315 q^{64} -0.0533514 q^{65} -2.05986 q^{66} -12.1330 q^{67} -3.72299 q^{68} -3.99691 q^{69} +0.282386 q^{70} +12.9376 q^{71} -4.12187 q^{72} +14.9024 q^{73} -13.7453 q^{74} +4.89430 q^{75} -30.0463 q^{76} +0.312627 q^{77} -0.392579 q^{78} +5.13168 q^{79} +0.785034 q^{80} +1.00000 q^{81} +7.58980 q^{82} -16.2792 q^{83} +1.35174 q^{84} -0.325110 q^{85} -21.0106 q^{86} +3.43052 q^{87} +3.54911 q^{88} +17.9831 q^{89} -0.777753 q^{90} +0.0595822 q^{91} +14.8804 q^{92} -5.64736 q^{93} -11.5625 q^{94} -2.62379 q^{95} -2.46717 q^{96} -11.0881 q^{97} +16.4306 q^{98} -0.861044 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9}+O(q^{10}) $ 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9} - 20 q^{10} + 5 q^{11} - 45 q^{12} - 8 q^{13} + 4 q^{14} - q^{15} + 39 q^{16} - 48 q^{17} - q^{18} - 6 q^{19} + 6 q^{20} + 13 q^{21} - 35 q^{22} - 8 q^{23} + 6 q^{24} + 13 q^{25} + 17 q^{26} - 48 q^{27} - 38 q^{28} + q^{29} + 20 q^{30} - 21 q^{31} - 3 q^{32} - 5 q^{33} + q^{34} + 19 q^{35} + 45 q^{36} - 58 q^{37} - 14 q^{38} + 8 q^{39} - 54 q^{40} - 3 q^{41} - 4 q^{42} - 33 q^{43} + 2 q^{44} + q^{45} - 26 q^{46} + 9 q^{47} - 39 q^{48} + 11 q^{49} + 4 q^{50} + 48 q^{51} - 31 q^{52} - 33 q^{53} + q^{54} - 21 q^{55} + 6 q^{57} - 55 q^{58} + 77 q^{59} - 6 q^{60} - 29 q^{61} - 46 q^{62} - 13 q^{63} + 24 q^{64} - 49 q^{65} + 35 q^{66} - 44 q^{67} - 45 q^{68} + 8 q^{69} + 4 q^{70} + 22 q^{71} - 6 q^{72} - 63 q^{73} - 16 q^{74} - 13 q^{75} - 46 q^{76} - 30 q^{77} - 17 q^{78} - 46 q^{79} - 14 q^{80} + 48 q^{81} - 75 q^{82} + 11 q^{83} + 38 q^{84} - q^{85} + 8 q^{86} - q^{87} - 116 q^{88} + 10 q^{89} - 20 q^{90} - 67 q^{91} - 64 q^{92} + 21 q^{93} - 16 q^{94} - 8 q^{95} + 3 q^{96} - 96 q^{97} - 46 q^{98} + 5 q^{99}+O(q^{100}) \) 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9 - 20 * q^10 + 5 * q^11 - 45 * q^12 - 8 * q^13 + 4 * q^14 - q^15 + 39 * q^16 - 48 * q^17 - q^18 - 6 * q^19 + 6 * q^20 + 13 * q^21 - 35 * q^22 - 8 * q^23 + 6 * q^24 + 13 * q^25 + 17 * q^26 - 48 * q^27 - 38 * q^28 + q^29 + 20 * q^30 - 21 * q^31 - 3 * q^32 - 5 * q^33 + q^34 + 19 * q^35 + 45 * q^36 - 58 * q^37 - 14 * q^38 + 8 * q^39 - 54 * q^40 - 3 * q^41 - 4 * q^42 - 33 * q^43 + 2 * q^44 + q^45 - 26 * q^46 + 9 * q^47 - 39 * q^48 + 11 * q^49 + 4 * q^50 + 48 * q^51 - 31 * q^52 - 33 * q^53 + q^54 - 21 * q^55 + 6 * q^57 - 55 * q^58 + 77 * q^59 - 6 * q^60 - 29 * q^61 - 46 * q^62 - 13 * q^63 + 24 * q^64 - 49 * q^65 + 35 * q^66 - 44 * q^67 - 45 * q^68 + 8 * q^69 + 4 * q^70 + 22 * q^71 - 6 * q^72 - 63 * q^73 - 16 * q^74 - 13 * q^75 - 46 * q^76 - 30 * q^77 - 17 * q^78 - 46 * q^79 - 14 * q^80 + 48 * q^81 - 75 * q^82 + 11 * q^83 + 38 * q^84 - q^85 + 8 * q^86 - q^87 - 116 * q^88 + 10 * q^89 - 20 * q^90 - 67 * q^91 - 64 * q^92 + 21 * q^93 - 16 * q^94 - 8 * q^95 + 3 * q^96 - 96 * q^97 - 46 * q^98 + 5 * 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 $

$2$	−2.39228	−1.69160	−0.845798	−	0.533504i	$-0.820875\pi$
$2$	−2.39228	−1.69160	−0.845798	+	0.533504i	$0.820875\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	3.72299	1.86149
$5$	0.325110	0.145394	0.0726968	−	0.997354i	$-0.476839\pi$
$5$	0.325110	0.145394	0.0726968	+	0.997354i	$0.476839\pi$
$6$	2.39228	0.976643
$7$	−0.363079	−0.137231	−0.0686155	−	0.997643i	$-0.521858\pi$
$7$	−0.363079	−0.137231	−0.0686155	+	0.997643i	$0.521858\pi$
$8$	−4.12187	−1.45730
$9$	1.00000	0.333333
$10$	−0.777753	−0.245947
$11$	−0.861044	−0.259614	−0.129807	−	0.991539i	$-0.541436\pi$
$11$	−0.861044	−0.259614	−0.129807	+	0.991539i	$0.541436\pi$
$12$	−3.72299	−1.07473
$13$	−0.164103	−0.0455139	−0.0227570	−	0.999741i	$-0.507244\pi$
$13$	−0.164103	−0.0455139	−0.0227570	+	0.999741i	$0.507244\pi$
$14$	0.868585	0.232139
$15$	−0.325110	−0.0839430
$16$	2.41467	0.603668
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−2.39228	−0.563865
$19$	−8.07048	−1.85149	−0.925747	−	0.378143i	$-0.876563\pi$
$19$	−8.07048	−1.85149	−0.925747	+	0.378143i	$0.876563\pi$
$20$	1.21038	0.270649
$21$	0.363079	0.0792303
$22$	2.05986	0.439163
$23$	3.99691	0.833413	0.416706	−	0.909041i	$-0.363184\pi$
$23$	3.99691	0.833413	0.416706	+	0.909041i	$0.363184\pi$
$24$	4.12187	0.841373
$25$	−4.89430	−0.978861
$26$	0.392579	0.0769911
$27$	−1.00000	−0.192450
$28$	−1.35174	−0.255455
$29$	−3.43052	−0.637031	−0.318516	−	0.947918i	$-0.603184\pi$
$29$	−3.43052	−0.637031	−0.318516	+	0.947918i	$0.603184\pi$
$30$	0.777753	0.141998
$31$	5.64736	1.01430	0.507148	−	0.861859i	$-0.330700\pi$
$31$	5.64736	1.01430	0.507148	+	0.861859i	$0.330700\pi$
$32$	2.46717	0.436138
$33$	0.861044	0.149888
$34$	2.39228	0.410272
$35$	−0.118041	−0.0199525
$36$	3.72299	0.620498
$37$	5.74569	0.944585	0.472293	−	0.881442i	$-0.343427\pi$
$37$	5.74569	0.944585	0.472293	+	0.881442i	$0.343427\pi$
$38$	19.3068	3.13198
$39$	0.164103	0.0262775
$40$	−1.34006	−0.211882
$41$	−3.17263	−0.495481	−0.247741	−	0.968826i	$-0.579688\pi$
$41$	−3.17263	−0.495481	−0.247741	+	0.968826i	$0.579688\pi$
$42$	−0.868585	−0.134026
$43$	8.78267	1.33934	0.669672	−	0.742657i	$-0.266435\pi$
$43$	8.78267	1.33934	0.669672	+	0.742657i	$0.266435\pi$
$44$	−3.20566	−0.483271
$45$	0.325110	0.0484645
$46$	−9.56171	−1.40980
$47$	4.83327	0.705005	0.352503	−	0.935811i	$-0.385331\pi$
$47$	4.83327	0.705005	0.352503	+	0.935811i	$0.385331\pi$
$48$	−2.41467	−0.348528
$49$	−6.86817	−0.981168
$50$	11.7085	1.65584
$51$	1.00000	0.140028
$52$	−0.610953	−0.0847239
$53$	−4.72827	−0.649478	−0.324739	−	0.945804i	$-0.605276\pi$
$53$	−4.72827	−0.649478	−0.324739	+	0.945804i	$0.605276\pi$
$54$	2.39228	0.325548
$55$	−0.279934	−0.0377463
$56$	1.49656	0.199987
$57$	8.07048	1.06896
$58$	8.20675	1.07760
$59$	5.82740	0.758663	0.379331	−	0.925261i	$-0.376154\pi$
$59$	5.82740	0.758663	0.379331	+	0.925261i	$0.376154\pi$
$60$	−1.21038	−0.156260
$61$	11.6859	1.49622	0.748112	−	0.663573i	$-0.230961\pi$
$61$	11.6859	1.49622	0.748112	+	0.663573i	$0.230961\pi$
$62$	−13.5100	−1.71578
$63$	−0.363079	−0.0457436
$64$	−10.7315	−1.34144
$65$	−0.0533514	−0.00661743
$66$	−2.05986	−0.253551
$67$	−12.1330	−1.48229	−0.741143	−	0.671347i	$-0.765716\pi$
$67$	−12.1330	−1.48229	−0.741143	+	0.671347i	$0.765716\pi$
$68$	−3.72299	−0.451479
$69$	−3.99691	−0.481171
$70$	0.282386	0.0337516
$71$	12.9376	1.53541	0.767703	−	0.640806i	$-0.221400\pi$
$71$	12.9376	1.53541	0.767703	+	0.640806i	$0.221400\pi$
$72$	−4.12187	−0.485767
$73$	14.9024	1.74419	0.872094	−	0.489338i	$-0.162762\pi$
$73$	14.9024	1.74419	0.872094	+	0.489338i	$0.162762\pi$
$74$	−13.7453	−1.59786
$75$	4.89430	0.565145
$76$	−30.0463	−3.44655
$77$	0.312627	0.0356271
$78$	−0.392579	−0.0444509
$79$	5.13168	0.577359	0.288679	−	0.957426i	$-0.406784\pi$
$79$	5.13168	0.577359	0.288679	+	0.957426i	$0.406784\pi$
$80$	0.785034	0.0877695
$81$	1.00000	0.111111
$82$	7.58980	0.838154
$83$	−16.2792	−1.78688	−0.893439	−	0.449185i	$-0.851714\pi$
$83$	−16.2792	−1.78688	−0.893439	+	0.449185i	$0.851714\pi$
$84$	1.35174	0.147487
$85$	−0.325110	−0.0352631
$86$	−21.0106	−2.26563
$87$	3.43052	0.367790
$88$	3.54911	0.378336
$89$	17.9831	1.90621	0.953105	−	0.302640i	$-0.0978681\pi$
$89$	17.9831	1.90621	0.953105	+	0.302640i	$0.0978681\pi$
$90$	−0.777753	−0.0819824
$91$	0.0595822	0.00624592
$92$	14.8804	1.55139
$93$	−5.64736	−0.585604
$94$	−11.5625	−1.19258
$95$	−2.62379	−0.269195
$96$	−2.46717	−0.251805
$97$	−11.0881	−1.12583	−0.562914	−	0.826516i	$-0.690320\pi$
$97$	−11.0881	−1.12583	−0.562914	+	0.826516i	$0.690320\pi$
$98$	16.4306	1.65974
$99$	−0.861044	−0.0865382
$100$	−18.2214	−1.82214
$101$	5.04213	0.501711	0.250856	−	0.968025i	$-0.419288\pi$
$101$	5.04213	0.501711	0.250856	+	0.968025i	$0.419288\pi$
$102$	−2.39228	−0.236871
$103$	−13.3677	−1.31716	−0.658578	−	0.752513i	$-0.728842\pi$
$103$	−13.3677	−1.31716	−0.658578	+	0.752513i	$0.728842\pi$
$104$	0.676410	0.0663275
$105$	0.118041	0.0115196
$106$	11.3113	1.09865
$107$	11.7694	1.13780	0.568898	−	0.822408i	$-0.307370\pi$
$107$	11.7694	1.13780	0.568898	+	0.822408i	$0.307370\pi$
$108$	−3.72299	−0.358245
$109$	−8.52725	−0.816763	−0.408381	−	0.912811i	$-0.633907\pi$
$109$	−8.52725	−0.816763	−0.408381	+	0.912811i	$0.633907\pi$
$110$	0.669680	0.0638514
$111$	−5.74569	−0.545356
$112$	−0.876717	−0.0828419
$113$	10.9586	1.03090	0.515450	−	0.856919i	$-0.327625\pi$
$113$	10.9586	1.03090	0.515450	+	0.856919i	$0.327625\pi$
$114$	−19.3068	−1.80825
$115$	1.29943	0.121173
$116$	−12.7718	−1.18583
$117$	−0.164103	−0.0151713
$118$	−13.9408	−1.28335
$119$	0.363079	0.0332834
$120$	1.34006	0.122330
$121$	−10.2586	−0.932600
$122$	−27.9559	−2.53100
$123$	3.17263	0.286066
$124$	21.0250	1.88810
$125$	−3.21674	−0.287714
$126$	0.868585	0.0773797
$127$	−17.2632	−1.53186	−0.765929	−	0.642926i	$-0.777720\pi$
$127$	−17.2632	−1.53186	−0.765929	+	0.642926i	$0.777720\pi$
$128$	20.7384	1.83303
$129$	−8.78267	−0.773271
$130$	0.127631	0.0111940
$131$	2.60700	0.227775	0.113887	−	0.993494i	$-0.463670\pi$
$131$	2.60700	0.227775	0.113887	+	0.993494i	$0.463670\pi$
$132$	3.20566	0.279017
$133$	2.93022	0.254082
$134$	29.0256	2.50743
$135$	−0.325110	−0.0279810
$136$	4.12187	0.353447
$137$	−7.69283	−0.657243	−0.328621	−	0.944462i	$-0.606584\pi$
$137$	−7.69283	−0.657243	−0.328621	+	0.944462i	$0.606584\pi$
$138$	9.56171	0.813947
$139$	−2.53629	−0.215125	−0.107562	−	0.994198i	$-0.534305\pi$
$139$	−2.53629	−0.215125	−0.107562	+	0.994198i	$0.534305\pi$
$140$	−0.439464	−0.0371415
$141$	−4.83327	−0.407035
$142$	−30.9502	−2.59729
$143$	0.141300	0.0118161
$144$	2.41467	0.201223
$145$	−1.11530	−0.0926203
$146$	−35.6506	−2.95046
$147$	6.86817	0.566477
$148$	21.3911	1.75834
$149$	5.69535	0.466582	0.233291	−	0.972407i	$-0.425051\pi$
$149$	5.69535	0.466582	0.233291	+	0.972407i	$0.425051\pi$
$150$	−11.7085	−0.955997
$151$	1.51584	0.123358	0.0616788	−	0.998096i	$-0.480355\pi$
$151$	1.51584	0.123358	0.0616788	+	0.998096i	$0.480355\pi$
$152$	33.2654	2.69818
$153$	−1.00000	−0.0808452
$154$	−0.747890	−0.0602667
$155$	1.83601	0.147472
$156$	0.610953	0.0489154
$157$	−1.00000	−0.0798087
$157$	−1.00000	−0.0798087
$158$	−12.2764	−0.976657
$159$	4.72827	0.374976
$160$	0.802102	0.0634117
$161$	−1.45119	−0.114370
$162$	−2.39228	−0.187955
$163$	−3.71552	−0.291022	−0.145511	−	0.989357i	$-0.546483\pi$
$163$	−3.71552	−0.291022	−0.145511	+	0.989357i	$0.546483\pi$
$164$	−11.8117	−0.922336
$165$	0.279934	0.0217928
$166$	38.9444	3.02267
$167$	0.109122	0.00844409	0.00422204	−	0.999991i	$-0.498656\pi$
$167$	0.109122	0.00844409	0.00422204	+	0.999991i	$0.498656\pi$
$168$	−1.49656	−0.115462
$169$	−12.9731	−0.997928
$170$	0.777753	0.0596509
$171$	−8.07048	−0.617165
$172$	32.6978	2.49318
$173$	9.41645	0.715919	0.357960	−	0.933737i	$-0.383473\pi$
$173$	9.41645	0.715919	0.357960	+	0.933737i	$0.383473\pi$
$174$	−8.20675	−0.622152
$175$	1.77702	0.134330
$176$	−2.07914	−0.156721
$177$	−5.82740	−0.438014
$178$	−43.0207	−3.22454
$179$	19.5434	1.46074	0.730371	−	0.683050i	$-0.239347\pi$
$179$	19.5434	1.46074	0.730371	+	0.683050i	$0.239347\pi$
$180$	1.21038	0.0902165
$181$	−0.0568237	−0.00422367	−0.00211184	−	0.999998i	$-0.500672\pi$
$181$	−0.0568237	−0.00422367	−0.00211184	+	0.999998i	$0.500672\pi$
$182$	−0.142537	−0.0105656
$183$	−11.6859	−0.863845
$184$	−16.4747	−1.21453
$185$	1.86798	0.137337
$186$	13.5100	0.990604
$187$	0.861044	0.0629658
$188$	17.9942	1.31236
$189$	0.363079	0.0264101
$190$	6.27684	0.455370
$191$	−19.3049	−1.39686	−0.698428	−	0.715680i	$-0.746117\pi$
$191$	−19.3049	−1.39686	−0.698428	+	0.715680i	$0.746117\pi$
$192$	10.7315	0.774479
$193$	3.06885	0.220901	0.110450	−	0.993882i	$-0.464771\pi$
$193$	3.06885	0.220901	0.110450	+	0.993882i	$0.464771\pi$
$194$	26.5258	1.90444
$195$	0.0533514	0.00382058
$196$	−25.5701	−1.82644
$197$	13.5643	0.966413	0.483207	−	0.875506i	$-0.339472\pi$
$197$	13.5643	0.966413	0.483207	+	0.875506i	$0.339472\pi$
$198$	2.05986	0.146388
$199$	16.1080	1.14187	0.570933	−	0.820997i	$-0.306582\pi$
$199$	16.1080	1.14187	0.570933	+	0.820997i	$0.306582\pi$
$200$	20.1737	1.42649
$201$	12.1330	0.855799
$202$	−12.0622	−0.848692
$203$	1.24555	0.0874204
$204$	3.72299	0.260661
$205$	−1.03145	−0.0720398
$206$	31.9792	2.22809
$207$	3.99691	0.277804
$208$	−0.396255	−0.0274753
$209$	6.94903	0.480675
$210$	−0.282386	−0.0194865
$211$	−7.51372	−0.517266	−0.258633	−	0.965976i	$-0.583272\pi$
$211$	−7.51372	−0.517266	−0.258633	+	0.965976i	$0.583272\pi$
$212$	−17.6033	−1.20900
$213$	−12.9376	−0.886467
$214$	−28.1558	−1.92469
$215$	2.85533	0.194732
$216$	4.12187	0.280458
$217$	−2.05044	−0.139193
$218$	20.3996	1.38163
$219$	−14.9024	−1.00701
$220$	−1.04219	−0.0702645
$221$	0.164103	0.0110387
$222$	13.7453	0.922522
$223$	−28.4921	−1.90797	−0.953985	−	0.299854i	$-0.903062\pi$
$223$	−28.4921	−1.90797	−0.953985	+	0.299854i	$0.903062\pi$
$224$	−0.895778	−0.0598517
$225$	−4.89430	−0.326287
$226$	−26.2161	−1.74387
$227$	0.333603	0.0221420	0.0110710	−	0.999939i	$-0.496476\pi$
$227$	0.333603	0.0221420	0.0110710	+	0.999939i	$0.496476\pi$
$228$	30.0463	1.98986
$229$	−1.04440	−0.0690159	−0.0345080	−	0.999404i	$-0.510986\pi$
$229$	−1.04440	−0.0690159	−0.0345080	+	0.999404i	$0.510986\pi$
$230$	−3.10861	−0.204975
$231$	−0.312627	−0.0205693
$232$	14.1401	0.928346
$233$	13.1069	0.858663	0.429331	−	0.903147i	$-0.358749\pi$
$233$	13.1069	0.858663	0.429331	+	0.903147i	$0.358749\pi$
$234$	0.392579	0.0256637
$235$	1.57135	0.102503
$236$	21.6953	1.41225
$237$	−5.13168	−0.333338
$238$	−0.868585	−0.0563020
$239$	−1.16543	−0.0753857	−0.0376928	−	0.999289i	$-0.512001\pi$
$239$	−1.16543	−0.0753857	−0.0376928	+	0.999289i	$0.512001\pi$
$240$	−0.785034	−0.0506738
$241$	−14.3270	−0.922881	−0.461441	−	0.887171i	$-0.652667\pi$
$241$	−14.3270	−0.922881	−0.461441	+	0.887171i	$0.652667\pi$
$242$	24.5414	1.57758
$243$	−1.00000	−0.0641500
$244$	43.5064	2.78521
$245$	−2.23291	−0.142656
$246$	−7.58980	−0.483908
$247$	1.32439	0.0842687
$248$	−23.2777	−1.47813
$249$	16.2792	1.03165
$250$	7.69533	0.486695
$251$	−4.08839	−0.258057	−0.129028	−	0.991641i	$-0.541186\pi$
$251$	−4.08839	−0.258057	−0.129028	+	0.991641i	$0.541186\pi$
$252$	−1.35174	−0.0851515
$253$	−3.44151	−0.216366
$254$	41.2982	2.59128
$255$	0.325110	0.0203592
$256$	−28.1490	−1.75931
$257$	−26.0711	−1.62627	−0.813135	−	0.582075i	$-0.802241\pi$
$257$	−26.0711	−1.62627	−0.813135	+	0.582075i	$0.802241\pi$
$258$	21.0106	1.30806
$259$	−2.08614	−0.129626
$260$	−0.198627	−0.0123183
$261$	−3.43052	−0.212344
$262$	−6.23667	−0.385303
$263$	−6.78628	−0.418460	−0.209230	−	0.977866i	$-0.567096\pi$
$263$	−6.78628	−0.418460	−0.209230	+	0.977866i	$0.567096\pi$
$264$	−3.54911	−0.218433
$265$	−1.53721	−0.0944300
$266$	−7.00990	−0.429804
$267$	−17.9831	−1.10055
$268$	−45.1712	−2.75927
$269$	0.198615	0.0121098	0.00605489	−	0.999982i	$-0.498073\pi$
$269$	0.198615	0.0121098	0.00605489	+	0.999982i	$0.498073\pi$
$270$	0.777753	0.0473326
$271$	−22.4067	−1.36111	−0.680556	−	0.732697i	$-0.738261\pi$
$271$	−22.4067	−1.36111	−0.680556	+	0.732697i	$0.738261\pi$
$272$	−2.41467	−0.146411
$273$	−0.0595822	−0.00360608
$274$	18.4034	1.11179
$275$	4.21421	0.254126
$276$	−14.8804	−0.895697
$277$	−14.8527	−0.892414	−0.446207	−	0.894930i	$-0.647226\pi$
$277$	−14.8527	−0.892414	−0.446207	+	0.894930i	$0.647226\pi$
$278$	6.06750	0.363904
$279$	5.64736	0.338098
$280$	0.486548	0.0290768
$281$	−1.14199	−0.0681253	−0.0340626	−	0.999420i	$-0.510845\pi$
$281$	−1.14199	−0.0681253	−0.0340626	+	0.999420i	$0.510845\pi$
$282$	11.5625	0.688539
$283$	11.8706	0.705633	0.352817	−	0.935692i	$-0.385224\pi$
$283$	11.8706	0.705633	0.352817	+	0.935692i	$0.385224\pi$
$284$	48.1664	2.85815
$285$	2.62379	0.155420
$286$	−0.338028	−0.0199880
$287$	1.15191	0.0679953
$288$	2.46717	0.145379
$289$	1.00000	0.0588235
$290$	2.66810	0.156676
$291$	11.0881	0.649997
$292$	55.4813	3.24680
$293$	−3.33015	−0.194550	−0.0972748	−	0.995258i	$-0.531013\pi$
$293$	−3.33015	−0.194550	−0.0972748	+	0.995258i	$0.531013\pi$
$294$	−16.4306	−0.958251
$295$	1.89455	0.110305
$296$	−23.6830	−1.37654
$297$	0.861044	0.0499628
$298$	−13.6249	−0.789267
$299$	−0.655903	−0.0379319
$300$	18.2214	1.05202
$301$	−3.18880	−0.183799
$302$	−3.62632	−0.208671
$303$	−5.04213	−0.289663
$304$	−19.4876	−1.11769
$305$	3.79920	0.217541
$306$	2.39228	0.136757
$307$	11.4073	0.651049	0.325524	−	0.945534i	$-0.394459\pi$
$307$	11.4073	0.651049	0.325524	+	0.945534i	$0.394459\pi$
$308$	1.16391	0.0663197
$309$	13.3677	0.760460
$310$	−4.39225	−0.249463
$311$	16.2862	0.923504	0.461752	−	0.887009i	$-0.347221\pi$
$311$	16.2862	0.923504	0.461752	+	0.887009i	$0.347221\pi$
$312$	−0.676410	−0.0382942
$313$	−5.08060	−0.287172	−0.143586	−	0.989638i	$-0.545863\pi$
$313$	−5.08060	−0.287172	−0.143586	+	0.989638i	$0.545863\pi$
$314$	2.39228	0.135004
$315$	−0.118041	−0.00665083
$316$	19.1052	1.07475
$317$	−28.8502	−1.62039	−0.810195	−	0.586161i	$-0.800639\pi$
$317$	−28.8502	−1.62039	−0.810195	+	0.586161i	$0.800639\pi$
$318$	−11.3113	−0.634308
$319$	2.95383	0.165383
$320$	−3.48892	−0.195036
$321$	−11.7694	−0.656907
$322$	3.47165	0.193468
$323$	8.07048	0.449053
$324$	3.72299	0.206833
$325$	0.803169	0.0445518
$326$	8.88856	0.492292
$327$	8.52725	0.471558
$328$	13.0772	0.722065
$329$	−1.75486	−0.0967485
$330$	−0.669680	−0.0368646
$331$	12.0989	0.665014	0.332507	−	0.943101i	$-0.392106\pi$
$331$	12.0989	0.665014	0.332507	+	0.943101i	$0.392106\pi$
$332$	−60.6074	−3.32626
$333$	5.74569	0.314862
$334$	−0.261049	−0.0142840
$335$	−3.94457	−0.215515
$336$	0.876717	0.0478288
$337$	−17.5789	−0.957581	−0.478791	−	0.877929i	$-0.658925\pi$
$337$	−17.5789	−0.957581	−0.478791	+	0.877929i	$0.658925\pi$
$338$	31.0352	1.68809
$339$	−10.9586	−0.595191
$340$	−1.21038	−0.0656421
$341$	−4.86262	−0.263326
$342$	19.3068	1.04399
$343$	5.03524	0.271877
$344$	−36.2010	−1.95183
$345$	−1.29943	−0.0699592
$346$	−22.5268	−1.21105
$347$	11.8556	0.636441	0.318220	−	0.948017i	$-0.396915\pi$
$347$	11.8556	0.636441	0.318220	+	0.948017i	$0.396915\pi$
$348$	12.7718	0.684640
$349$	18.1376	0.970883	0.485442	−	0.874269i	$-0.338659\pi$
$349$	18.1376	0.970883	0.485442	+	0.874269i	$0.338659\pi$
$350$	−4.25112	−0.227232
$351$	0.164103	0.00875916
$352$	−2.12434	−0.113228
$353$	7.93436	0.422303	0.211152	−	0.977453i	$-0.432279\pi$
$353$	7.93436	0.422303	0.211152	+	0.977453i	$0.432279\pi$
$354$	13.9408	0.740943
$355$	4.20613	0.223238
$356$	66.9511	3.54840
$357$	−0.363079	−0.0192162
$358$	−46.7532	−2.47099
$359$	−6.05604	−0.319626	−0.159813	−	0.987147i	$-0.551089\pi$
$359$	−6.05604	−0.319626	−0.159813	+	0.987147i	$0.551089\pi$
$360$	−1.34006	−0.0706274
$361$	46.1326	2.42803
$362$	0.135938	0.00714474
$363$	10.2586	0.538437
$364$	0.221824	0.0116267
$365$	4.84490	0.253594
$366$	27.9559	1.46128
$367$	−22.5163	−1.17534	−0.587671	−	0.809100i	$-0.699955\pi$
$367$	−22.5163	−1.17534	−0.587671	+	0.809100i	$0.699955\pi$
$368$	9.65122	0.503105
$369$	−3.17263	−0.165160
$370$	−4.46873	−0.232318
$371$	1.71674	0.0891285
$372$	−21.0250	−1.09010
$373$	35.5761	1.84206	0.921031	−	0.389489i	$-0.127348\pi$
$373$	35.5761	1.84206	0.921031	+	0.389489i	$0.127348\pi$
$374$	−2.05986	−0.106513
$375$	3.21674	0.166112
$376$	−19.9221	−1.02740
$377$	0.562958	0.0289938
$378$	−0.868585	−0.0446752
$379$	18.7211	0.961637	0.480818	−	0.876820i	$-0.340340\pi$
$379$	18.7211	0.961637	0.480818	+	0.876820i	$0.340340\pi$
$380$	−9.76835	−0.501106
$381$	17.2632	0.884418
$382$	46.1828	2.36292
$383$	4.43044	0.226385	0.113192	−	0.993573i	$-0.463892\pi$
$383$	4.43044	0.226385	0.113192	+	0.993573i	$0.463892\pi$
$384$	−20.7384	−1.05830
$385$	0.101638	0.00517996
$386$	−7.34154	−0.373675
$387$	8.78267	0.446448
$388$	−41.2809	−2.09572
$389$	−11.1334	−0.564487	−0.282244	−	0.959343i	$-0.591079\pi$
$389$	−11.1334	−0.564487	−0.282244	+	0.959343i	$0.591079\pi$
$390$	−0.127631	−0.00646287
$391$	−3.99691	−0.202132
$392$	28.3097	1.42986
$393$	−2.60700	−0.131506
$394$	−32.4495	−1.63478
$395$	1.66836	0.0839443
$396$	−3.20566	−0.161090
$397$	−28.5223	−1.43149	−0.715746	−	0.698361i	$-0.753913\pi$
$397$	−28.5223	−1.43149	−0.715746	+	0.698361i	$0.753913\pi$
$398$	−38.5348	−1.93157
$399$	−2.93022	−0.146694
$400$	−11.8181	−0.590907
$401$	−36.6755	−1.83149	−0.915744	−	0.401762i	$-0.868398\pi$
$401$	−36.6755	−1.83149	−0.915744	+	0.401762i	$0.868398\pi$
$402$	−29.0256	−1.44766
$403$	−0.926747	−0.0461645
$404$	18.7718	0.933933
$405$	0.325110	0.0161548
$406$	−2.97970	−0.147880
$407$	−4.94729	−0.245228
$408$	−4.12187	−0.204063
$409$	14.5550	0.719697	0.359849	−	0.933011i	$-0.382828\pi$
$409$	14.5550	0.719697	0.359849	+	0.933011i	$0.382828\pi$
$410$	2.46752	0.121862
$411$	7.69283	0.379459
$412$	−49.7677	−2.45188
$413$	−2.11581	−0.104112
$414$	−9.56171	−0.469932
$415$	−5.29254	−0.259801
$416$	−0.404870	−0.0198504
$417$	2.53629	0.124202
$418$	−16.6240	−0.813107
$419$	28.5433	1.39443	0.697215	−	0.716862i	$-0.254422\pi$
$419$	28.5433	1.39443	0.697215	+	0.716862i	$0.254422\pi$
$420$	0.439464	0.0214436
$421$	22.6795	1.10533	0.552666	−	0.833403i	$-0.313611\pi$
$421$	22.6795	1.10533	0.552666	+	0.833403i	$0.313611\pi$
$422$	17.9749	0.875005
$423$	4.83327	0.235002
$424$	19.4893	0.946485
$425$	4.89430	0.237409
$426$	30.9502	1.49954
$427$	−4.24290	−0.205328
$428$	43.8175	2.11800
$429$	−0.141300	−0.00682201
$430$	−6.83075	−0.329408
$431$	12.5283	0.603466	0.301733	−	0.953392i	$-0.402435\pi$
$431$	12.5283	0.603466	0.301733	+	0.953392i	$0.402435\pi$
$432$	−2.41467	−0.116176
$433$	4.52180	0.217304	0.108652	−	0.994080i	$-0.465347\pi$
$433$	4.52180	0.217304	0.108652	+	0.994080i	$0.465347\pi$
$434$	4.90521	0.235458
$435$	1.11530	0.0534744
$436$	−31.7469	−1.52040
$437$	−32.2569	−1.54306
$438$	35.6506	1.70345
$439$	−12.6756	−0.604971	−0.302486	−	0.953154i	$-0.597816\pi$
$439$	−12.6756	−0.604971	−0.302486	+	0.953154i	$0.597816\pi$
$440$	1.15385	0.0550077
$441$	−6.86817	−0.327056
$442$	−0.392579	−0.0186731
$443$	−22.1413	−1.05196	−0.525982	−	0.850496i	$-0.676302\pi$
$443$	−22.1413	−1.05196	−0.525982	+	0.850496i	$0.676302\pi$
$444$	−21.3911	−1.01518
$445$	5.84650	0.277151
$446$	68.1609	3.22751
$447$	−5.69535	−0.269381
$448$	3.89638	0.184087
$449$	−1.54679	−0.0729974	−0.0364987	−	0.999334i	$-0.511620\pi$
$449$	−1.54679	−0.0729974	−0.0364987	+	0.999334i	$0.511620\pi$
$450$	11.7085	0.551945
$451$	2.73177	0.128634
$452$	40.7989	1.91902
$453$	−1.51584	−0.0712206
$454$	−0.798070	−0.0374553
$455$	0.0193708	0.000908116 0
$456$	−33.2654	−1.55780
$457$	29.5396	1.38180	0.690902	−	0.722948i	$-0.257214\pi$
$457$	29.5396	1.38180	0.690902	+	0.722948i	$0.257214\pi$
$458$	2.49850	0.116747
$459$	1.00000	0.0466760
$460$	4.83778	0.225563
$461$	0.703831	0.0327807	0.0163903	−	0.999866i	$-0.494783\pi$
$461$	0.703831	0.0327807	0.0163903	+	0.999866i	$0.494783\pi$
$462$	0.747890	0.0347950
$463$	−24.4734	−1.13738	−0.568688	−	0.822554i	$-0.692549\pi$
$463$	−24.4734	−1.13738	−0.568688	+	0.822554i	$0.692549\pi$
$464$	−8.28358	−0.384556
$465$	−1.83601	−0.0851430
$466$	−31.3554	−1.45251
$467$	−28.9602	−1.34012	−0.670059	−	0.742308i	$-0.733731\pi$
$467$	−28.9602	−1.34012	−0.670059	+	0.742308i	$0.733731\pi$
$468$	−0.610953	−0.0282413
$469$	4.40525	0.203416
$470$	−3.75909	−0.173394
$471$	1.00000	0.0460776
$472$	−24.0198	−1.10560
$473$	−7.56226	−0.347713
$474$	12.2764	0.563873
$475$	39.4994	1.81235
$476$	1.35174	0.0619569
$477$	−4.72827	−0.216493
$478$	2.78804	0.127522
$479$	17.0037	0.776918	0.388459	−	0.921466i	$-0.373007\pi$
$479$	17.0037	0.776918	0.388459	+	0.921466i	$0.373007\pi$
$480$	−0.802102	−0.0366108
$481$	−0.942883	−0.0429918
$482$	34.2741	1.56114
$483$	1.45119	0.0660315
$484$	−38.1927	−1.73603
$485$	−3.60486	−0.163688
$486$	2.39228	0.108516
$487$	−20.5710	−0.932159	−0.466080	−	0.884743i	$-0.654334\pi$
$487$	−20.5710	−0.932159	−0.466080	+	0.884743i	$0.654334\pi$
$488$	−48.1677	−2.18045
$489$	3.71552	0.168022
$490$	5.34174	0.241315
$491$	31.8835	1.43888	0.719441	−	0.694553i	$-0.244398\pi$
$491$	31.8835	1.43888	0.719441	+	0.694553i	$0.244398\pi$
$492$	11.8117	0.532511
$493$	3.43052	0.154503
$494$	−3.16830	−0.142549
$495$	−0.279934	−0.0125821
$496$	13.6365	0.612298
$497$	−4.69736	−0.210705
$498$	−38.9444	−1.74514
$499$	−23.8861	−1.06929	−0.534645	−	0.845077i	$-0.679555\pi$
$499$	−23.8861	−1.06929	−0.534645	+	0.845077i	$0.679555\pi$
$500$	−11.9759	−0.535578
$501$	−0.109122	−0.00487520
$502$	9.78056	0.436528
$503$	42.4906	1.89456	0.947282	−	0.320402i	$-0.103818\pi$
$503$	42.4906	1.89456	0.947282	+	0.320402i	$0.103818\pi$
$504$	1.49656	0.0666622
$505$	1.63925	0.0729456
$506$	8.23305	0.366004
$507$	12.9731	0.576154
$508$	−64.2705	−2.85154
$509$	17.6220	0.781081	0.390541	−	0.920586i	$-0.372288\pi$
$509$	17.6220	0.781081	0.390541	+	0.920586i	$0.372288\pi$
$510$	−0.777753	−0.0344395
$511$	−5.41073	−0.239357
$512$	25.8633	1.14301
$513$	8.07048	0.356320
$514$	62.3693	2.75099
$515$	−4.34596	−0.191506
$516$	−32.6978	−1.43944
$517$	−4.16166	−0.183030
$518$	4.99062	0.219275
$519$	−9.41645	−0.413336
$520$	0.219908	0.00964359
$521$	−16.9983	−0.744707	−0.372353	−	0.928091i	$-0.621449\pi$
$521$	−16.9983	−0.744707	−0.372353	+	0.928091i	$0.621449\pi$
$522$	8.20675	0.359200
$523$	11.7351	0.513142	0.256571	−	0.966525i	$-0.417407\pi$
$523$	11.7351	0.513142	0.256571	+	0.966525i	$0.417407\pi$
$524$	9.70584	0.424001
$525$	−1.77702	−0.0775554
$526$	16.2347	0.707865
$527$	−5.64736	−0.246003
$528$	2.07914	0.0904829
$529$	−7.02474	−0.305423
$530$	3.67743	0.159737
$531$	5.82740	0.252888
$532$	10.9092	0.472973
$533$	0.520637	0.0225513
$534$	43.0207	1.86169
$535$	3.82637	0.165428
$536$	50.0108	2.16014
$537$	−19.5434	−0.843360
$538$	−0.475143	−0.0204848
$539$	5.91380	0.254725
$540$	−1.21038	−0.0520865
$541$	10.2845	0.442166	0.221083	−	0.975255i	$-0.429041\pi$
$541$	10.2845	0.442166	0.221083	+	0.975255i	$0.429041\pi$
$542$	53.6031	2.30245
$543$	0.0568237	0.00243854
$544$	−2.46717	−0.105779
$545$	−2.77230	−0.118752
$546$	0.142537	0.00610003
$547$	16.9824	0.726116	0.363058	−	0.931767i	$-0.381733\pi$
$547$	16.9824	0.726116	0.363058	+	0.931767i	$0.381733\pi$
$548$	−28.6403	−1.22345
$549$	11.6859	0.498741
$550$	−10.0816	−0.429879
$551$	27.6859	1.17946
$552$	16.4747	0.701211
$553$	−1.86320	−0.0792315
$554$	35.5318	1.50960
$555$	−1.86798	−0.0792913
$556$	−9.44256	−0.400454
$557$	−12.2317	−0.518276	−0.259138	−	0.965840i	$-0.583438\pi$
$557$	−12.2317	−0.518276	−0.259138	+	0.965840i	$0.583438\pi$
$558$	−13.5100	−0.571926
$559$	−1.44126	−0.0609588
$560$	−0.285029	−0.0120447
$561$	−0.861044	−0.0363533
$562$	2.73195	0.115240
$563$	12.3117	0.518877	0.259439	−	0.965760i	$-0.416462\pi$
$563$	12.3117	0.518877	0.259439	+	0.965760i	$0.416462\pi$
$564$	−17.9942	−0.757694
$565$	3.56276	0.149886
$566$	−28.3977	−1.19365
$567$	−0.363079	−0.0152479
$568$	−53.3269	−2.23755
$569$	−39.3746	−1.65067	−0.825335	−	0.564643i	$-0.809014\pi$
$569$	−39.3746	−1.65067	−0.825335	+	0.564643i	$0.809014\pi$
$570$	−6.27684	−0.262908
$571$	40.7039	1.70341	0.851703	−	0.524024i	$-0.175570\pi$
$571$	40.7039	1.70341	0.851703	+	0.524024i	$0.175570\pi$
$572$	0.526057	0.0219956
$573$	19.3049	0.806475
$574$	−2.75570	−0.115021
$575$	−19.5621	−0.815795
$576$	−10.7315	−0.447146
$577$	−2.88372	−0.120051	−0.0600254	−	0.998197i	$-0.519118\pi$
$577$	−2.88372	−0.120051	−0.0600254	+	0.998197i	$0.519118\pi$
$578$	−2.39228	−0.0995056
$579$	−3.06885	−0.127537
$580$	−4.15224	−0.172412
$581$	5.91064	0.245215
$582$	−26.5258	−1.09953
$583$	4.07125	0.168614
$584$	−61.4255	−2.54181
$585$	−0.0533514	−0.00220581
$586$	7.96665	0.329099
$587$	−8.26843	−0.341275	−0.170637	−	0.985334i	$-0.554583\pi$
$587$	−8.26843	−0.341275	−0.170637	+	0.985334i	$0.554583\pi$
$588$	25.5701	1.05449
$589$	−45.5768	−1.87796
$590$	−4.53228	−0.186591
$591$	−13.5643	−0.557959
$592$	13.8740	0.570216
$593$	−36.8666	−1.51393	−0.756964	−	0.653456i	$-0.773318\pi$
$593$	−36.8666	−1.51393	−0.756964	+	0.653456i	$0.773318\pi$
$594$	−2.05986	−0.0845169
$595$	0.118041	0.00483919
$596$	21.2037	0.868539
$597$	−16.1080	−0.659256
$598$	1.56910	0.0641654
$599$	−34.5880	−1.41323	−0.706614	−	0.707599i	$-0.749778\pi$
$599$	−34.5880	−1.41323	−0.706614	+	0.707599i	$0.749778\pi$
$600$	−20.1737	−0.823587
$601$	−13.0419	−0.531990	−0.265995	−	0.963974i	$-0.585700\pi$
$601$	−13.0419	−0.531990	−0.265995	+	0.963974i	$0.585700\pi$
$602$	7.62850	0.310914
$603$	−12.1330	−0.494096
$604$	5.64347	0.229630
$605$	−3.33517	−0.135594
$606$	12.0622	0.489993
$607$	−42.7696	−1.73597	−0.867983	−	0.496594i	$-0.834584\pi$
$607$	−42.7696	−1.73597	−0.867983	+	0.496594i	$0.834584\pi$
$608$	−19.9112	−0.807507
$609$	−1.24555	−0.0504722
$610$	−9.08873	−0.367992
$611$	−0.793154	−0.0320876
$612$	−3.72299	−0.150493
$613$	−24.8469	−1.00356	−0.501778	−	0.864996i	$-0.667321\pi$
$613$	−24.8469	−1.00356	−0.501778	+	0.864996i	$0.667321\pi$
$614$	−27.2894	−1.10131
$615$	1.03145	0.0415922
$616$	−1.28861	−0.0519194
$617$	16.6363	0.669754	0.334877	−	0.942262i	$-0.391305\pi$
$617$	16.6363	0.669754	0.334877	+	0.942262i	$0.391305\pi$
$618$	−31.9792	−1.28639
$619$	16.9519	0.681355	0.340678	−	0.940180i	$-0.389344\pi$
$619$	16.9519	0.681355	0.340678	+	0.940180i	$0.389344\pi$
$620$	6.83545	0.274518
$621$	−3.99691	−0.160390
$622$	−38.9610	−1.56219
$623$	−6.52930	−0.261591
$624$	0.396255	0.0158629
$625$	23.4257	0.937029
$626$	12.1542	0.485779
$627$	−6.94903	−0.277518
$628$	−3.72299	−0.148563
$629$	−5.74569	−0.229096
$630$	0.282386	0.0112505
$631$	7.47510	0.297579	0.148789	−	0.988869i	$-0.452462\pi$
$631$	7.47510	0.297579	0.148789	+	0.988869i	$0.452462\pi$
$632$	−21.1521	−0.841385
$633$	7.51372	0.298644
$634$	69.0177	2.74104
$635$	−5.61242	−0.222722
$636$	17.6033	0.698016
$637$	1.12709	0.0446568
$638$	−7.06637	−0.279760
$639$	12.9376	0.511802
$640$	6.74226	0.266511
$641$	−26.3900	−1.04234	−0.521171	−	0.853452i	$-0.674505\pi$
$641$	−26.3900	−1.04234	−0.521171	+	0.853452i	$0.674505\pi$
$642$	28.1558	1.11122
$643$	5.36126	0.211427	0.105714	−	0.994397i	$-0.466287\pi$
$643$	5.36126	0.211427	0.105714	+	0.994397i	$0.466287\pi$
$644$	−5.40277	−0.212899
$645$	−2.85533	−0.112429
$646$	−19.3068	−0.759616
$647$	3.32466	0.130706	0.0653530	−	0.997862i	$-0.479183\pi$
$647$	3.32466	0.130706	0.0653530	+	0.997862i	$0.479183\pi$
$648$	−4.12187	−0.161922
$649$	−5.01765	−0.196960
$650$	−1.92140	−0.0753636
$651$	2.05044	0.0803629
$652$	−13.8329	−0.541736
$653$	−29.9595	−1.17240	−0.586202	−	0.810165i	$-0.699378\pi$
$653$	−29.9595	−1.17240	−0.586202	+	0.810165i	$0.699378\pi$
$654$	−20.3996	−0.797686
$655$	0.847562	0.0331170
$656$	−7.66086	−0.299106
$657$	14.9024	0.581396
$658$	4.19811	0.163659
$659$	8.72952	0.340054	0.170027	−	0.985439i	$-0.445615\pi$
$659$	8.72952	0.340054	0.170027	+	0.985439i	$0.445615\pi$
$660$	1.04219	0.0405672
$661$	−23.9398	−0.931151	−0.465575	−	0.885008i	$-0.654153\pi$
$661$	−23.9398	−0.931151	−0.465575	+	0.885008i	$0.654153\pi$
$662$	−28.9438	−1.12493
$663$	−0.164103	−0.00637322
$664$	67.1008	2.60402
$665$	0.952644	0.0369419
$666$	−13.7453	−0.532619
$667$	−13.7115	−0.530910
$668$	0.406259	0.0157186
$669$	28.4921	1.10157
$670$	9.43651	0.364564
$671$	−10.0621	−0.388441
$672$	0.895778	0.0345554
$673$	−24.1134	−0.929502	−0.464751	−	0.885441i	$-0.653856\pi$
$673$	−24.1134	−0.929502	−0.464751	+	0.885441i	$0.653856\pi$
$674$	42.0535	1.61984
$675$	4.89430	0.188382
$676$	−48.2986	−1.85764
$677$	−35.6750	−1.37110	−0.685550	−	0.728026i	$-0.740438\pi$
$677$	−35.6750	−1.37110	−0.685550	+	0.728026i	$0.740438\pi$
$678$	26.2161	1.00682
$679$	4.02586	0.154498
$680$	1.34006	0.0513890
$681$	−0.333603	−0.0127837
$682$	11.6327	0.445440
$683$	−35.9133	−1.37418	−0.687091	−	0.726571i	$-0.741113\pi$
$683$	−35.9133	−1.37418	−0.687091	+	0.726571i	$0.741113\pi$
$684$	−30.0463	−1.14885
$685$	−2.50102	−0.0955589
$686$	−12.0457	−0.459907
$687$	1.04440	0.0398464
$688$	21.2073	0.808520
$689$	0.775922	0.0295603
$690$	3.10861	0.118343
$691$	−33.5700	−1.27706	−0.638531	−	0.769596i	$-0.720458\pi$
$691$	−33.5700	−1.27706	−0.638531	+	0.769596i	$0.720458\pi$
$692$	35.0573	1.33268
$693$	0.312627	0.0118757
$694$	−28.3618	−1.07660
$695$	−0.824572	−0.0312778
$696$	−14.1401	−0.535981
$697$	3.17263	0.120172
$698$	−43.3902	−1.64234
$699$	−13.1069	−0.495749
$700$	6.61582	0.250055
$701$	−8.13135	−0.307117	−0.153558	−	0.988140i	$-0.549073\pi$
$701$	−8.13135	−0.307117	−0.153558	+	0.988140i	$0.549073\pi$
$702$	−0.392579	−0.0148170
$703$	−46.3704	−1.74889
$704$	9.24029	0.348257
$705$	−1.57135	−0.0591803
$706$	−18.9812	−0.714366
$707$	−1.83069	−0.0688503
$708$	−21.6953	−0.815361
$709$	−6.19866	−0.232795	−0.116398	−	0.993203i	$-0.537135\pi$
$709$	−6.19866	−0.232795	−0.116398	+	0.993203i	$0.537135\pi$
$710$	−10.0622	−0.377629
$711$	5.13168	0.192453
$712$	−74.1242	−2.77792
$713$	22.5720	0.845326
$714$	0.868585	0.0325060
$715$	0.0459379	0.00171798
$716$	72.7599	2.71917
$717$	1.16543	0.0435239
$718$	14.4877	0.540677
$719$	11.2195	0.418415	0.209208	−	0.977871i	$-0.432912\pi$
$719$	11.2195	0.418415	0.209208	+	0.977871i	$0.432912\pi$
$720$	0.785034	0.0292565
$721$	4.85352	0.180754
$722$	−110.362	−4.10724
$723$	14.3270	0.532826
$724$	−0.211554	−0.00786234
$725$	16.7900	0.623565
$726$	−24.5414	−0.910818
$727$	9.23540	0.342522	0.171261	−	0.985226i	$-0.445216\pi$
$727$	9.23540	0.342522	0.171261	+	0.985226i	$0.445216\pi$
$728$	−0.245590	−0.00910218
$729$	1.00000	0.0370370
$730$	−11.5904	−0.428978
$731$	−8.78267	−0.324839
$732$	−43.5064	−1.60804
$733$	−50.1110	−1.85089	−0.925445	−	0.378882i	$-0.876309\pi$
$733$	−50.1110	−1.85089	−0.925445	+	0.378882i	$0.876309\pi$
$734$	53.8653	1.98820
$735$	2.23291	0.0823622
$736$	9.86105	0.363483
$737$	10.4471	0.384823
$738$	7.58980	0.279385
$739$	−20.1733	−0.742087	−0.371044	−	0.928615i	$-0.621000\pi$
$739$	−20.1733	−0.742087	−0.371044	+	0.928615i	$0.621000\pi$
$740$	6.95447	0.255651
$741$	−1.32439	−0.0486526
$742$	−4.10691	−0.150769
$743$	−5.84990	−0.214612	−0.107306	−	0.994226i	$-0.534222\pi$
$743$	−5.84990	−0.214612	−0.107306	+	0.994226i	$0.534222\pi$
$744$	23.2777	0.853400
$745$	1.85162	0.0678380
$746$	−85.1080	−3.11602
$747$	−16.2792	−0.595626
$748$	3.20566	0.117210
$749$	−4.27324	−0.156141
$750$	−7.69533	−0.280994
$751$	34.7838	1.26928	0.634639	−	0.772809i	$-0.281149\pi$
$751$	34.7838	1.26928	0.634639	+	0.772809i	$0.281149\pi$
$752$	11.6708	0.425589
$753$	4.08839	0.148989
$754$	−1.34675	−0.0490458
$755$	0.492816	0.0179354
$756$	1.35174	0.0491623
$757$	−14.6112	−0.531052	−0.265526	−	0.964104i	$-0.585546\pi$
$757$	−14.6112	−0.531052	−0.265526	+	0.964104i	$0.585546\pi$
$758$	−44.7860	−1.62670
$759$	3.44151	0.124919
$760$	10.8149	0.392299
$761$	−32.7708	−1.18794	−0.593970	−	0.804487i	$-0.702440\pi$
$761$	−32.7708	−1.18794	−0.593970	+	0.804487i	$0.702440\pi$
$762$	−41.2982	−1.49608
$763$	3.09607	0.112085
$764$	−71.8721	−2.60024
$765$	−0.325110	−0.0117544
$766$	−10.5988	−0.382951
$767$	−0.956292	−0.0345297
$768$	28.1490	1.01574
$769$	−6.40981	−0.231144	−0.115572	−	0.993299i	$-0.536870\pi$
$769$	−6.40981	−0.231144	−0.115572	+	0.993299i	$0.536870\pi$
$770$	−0.243146	−0.00876239
$771$	26.0711	0.938928
$772$	11.4253	0.411206
$773$	−30.8208	−1.10855	−0.554273	−	0.832335i	$-0.687004\pi$
$773$	−30.8208	−1.10855	−0.554273	+	0.832335i	$0.687004\pi$
$774$	−21.0106	−0.755209
$775$	−27.6399	−0.992854
$776$	45.7038	1.64067
$777$	2.08614	0.0748398
$778$	26.6343	0.954884
$779$	25.6046	0.917380
$780$	0.198627	0.00711198
$781$	−11.1398	−0.398614
$782$	9.56171	0.341926
$783$	3.43052	0.122597
$784$	−16.5844	−0.592300
$785$	−0.325110	−0.0116037
$786$	6.23667	0.222455
$787$	−8.77816	−0.312908	−0.156454	−	0.987685i	$-0.550006\pi$
$787$	−8.77816	−0.312908	−0.156454	+	0.987685i	$0.550006\pi$
$788$	50.4996	1.79897
$789$	6.78628	0.241598
$790$	−3.99118	−0.142000
$791$	−3.97885	−0.141471
$792$	3.54911	0.126112
$793$	−1.91768	−0.0680990
$794$	68.2331	2.42150
$795$	1.53721	0.0545192
$796$	59.9699	2.12558
$797$	−6.89139	−0.244106	−0.122053	−	0.992524i	$-0.538948\pi$
$797$	−6.89139	−0.244106	−0.122053	+	0.992524i	$0.538948\pi$
$798$	7.00990	0.248148
$799$	−4.83327	−0.170989
$800$	−12.0751	−0.426919
$801$	17.9831	0.635403
$802$	87.7380	3.09814
$803$	−12.8316	−0.452817
$804$	45.1712	1.59306
$805$	−0.471797	−0.0166287
$806$	2.21703	0.0780917
$807$	−0.198615	−0.00699159
$808$	−20.7830	−0.731144
$809$	−6.76958	−0.238006	−0.119003	−	0.992894i	$-0.537970\pi$
$809$	−6.76958	−0.238006	−0.119003	+	0.992894i	$0.537970\pi$
$810$	−0.777753	−0.0273275
$811$	49.3825	1.73405	0.867027	−	0.498261i	$-0.166028\pi$
$811$	49.3825	1.73405	0.867027	+	0.498261i	$0.166028\pi$
$812$	4.63717	0.162733
$813$	22.4067	0.785838
$814$	11.8353	0.414826
$815$	−1.20795	−0.0423128
$816$	2.41467	0.0845305
$817$	−70.8803	−2.47979
$818$	−34.8195	−1.21744
$819$	0.0595822	0.00208197
$820$	−3.84009	−0.134102
$821$	−35.1327	−1.22614	−0.613070	−	0.790028i	$-0.710066\pi$
$821$	−35.1327	−1.22614	−0.613070	+	0.790028i	$0.710066\pi$
$822$	−18.4034	−0.641891
$823$	25.5098	0.889215	0.444607	−	0.895726i	$-0.353343\pi$
$823$	25.5098	0.889215	0.444607	+	0.895726i	$0.353343\pi$
$824$	55.0998	1.91949
$825$	−4.21421	−0.146720
$826$	5.06159	0.176115
$827$	16.4263	0.571198	0.285599	−	0.958349i	$-0.407808\pi$
$827$	16.4263	0.571198	0.285599	+	0.958349i	$0.407808\pi$
$828$	14.8804	0.517131
$829$	−6.12019	−0.212563	−0.106282	−	0.994336i	$-0.533894\pi$
$829$	−6.12019	−0.212563	−0.106282	+	0.994336i	$0.533894\pi$
$830$	12.6612	0.439477
$831$	14.8527	0.515236
$832$	1.76107	0.0610541
$833$	6.86817	0.237968
$834$	−6.06750	−0.210100
$835$	0.0354765	0.00122772
$836$	25.8712	0.894773
$837$	−5.64736	−0.195201
$838$	−68.2834	−2.35881
$839$	−34.1188	−1.17791	−0.588956	−	0.808165i	$-0.700461\pi$
$839$	−34.1188	−1.17791	−0.588956	+	0.808165i	$0.700461\pi$
$840$	−0.486548	−0.0167875
$841$	−17.2315	−0.594191
$842$	−54.2557	−1.86977
$843$	1.14199	0.0393322
$844$	−27.9735	−0.962888
$845$	−4.21767	−0.145092
$846$	−11.5625	−0.397528
$847$	3.72468	0.127982
$848$	−11.4172	−0.392069
$849$	−11.8706	−0.407398
$850$	−11.7085	−0.401599
$851$	22.9650	0.787229
$852$	−48.1664	−1.65015
$853$	0.186385	0.00638169	0.00319085	−	0.999995i	$-0.498984\pi$
$853$	0.186385	0.00638169	0.00319085	+	0.999995i	$0.498984\pi$
$854$	10.1502	0.347332
$855$	−2.62379	−0.0897318
$856$	−48.5121	−1.65811
$857$	−19.6326	−0.670635	−0.335318	−	0.942105i	$-0.608844\pi$
$857$	−19.6326	−0.670635	−0.335318	+	0.942105i	$0.608844\pi$
$858$	0.338028	0.0115401
$859$	9.55139	0.325889	0.162945	−	0.986635i	$-0.447901\pi$
$859$	9.55139	0.325889	0.162945	+	0.986635i	$0.447901\pi$
$860$	10.6304	0.362493
$861$	−1.15191	−0.0392571
$862$	−29.9711	−1.02082
$863$	31.8855	1.08540	0.542698	−	0.839928i	$-0.317403\pi$
$863$	31.8855	1.08540	0.542698	+	0.839928i	$0.317403\pi$
$864$	−2.46717	−0.0839348
$865$	3.06138	0.104090
$866$	−10.8174	−0.367590
$867$	−1.00000	−0.0339618
$868$	−7.63375	−0.259106
$869$	−4.41860	−0.149891
$870$	−2.66810	−0.0904570
$871$	1.99107	0.0674647
$872$	35.1482	1.19027
$873$	−11.0881	−0.375276
$874$	77.1675	2.61023
$875$	1.16793	0.0394832
$876$	−55.4813	−1.87454
$877$	−40.7107	−1.37470	−0.687351	−	0.726325i	$-0.741227\pi$
$877$	−40.7107	−1.37470	−0.687351	+	0.726325i	$0.741227\pi$
$878$	30.3234	1.02337
$879$	3.33015	0.112323
$880$	−0.675949	−0.0227862
$881$	−29.9144	−1.00784	−0.503921	−	0.863750i	$-0.668110\pi$
$881$	−29.9144	−1.00784	−0.503921	+	0.863750i	$0.668110\pi$
$882$	16.4306	0.553246
$883$	23.6433	0.795662	0.397831	−	0.917459i	$-0.369763\pi$
$883$	23.6433	0.795662	0.397831	+	0.917459i	$0.369763\pi$
$884$	0.610953	0.0205486
$885$	−1.89455	−0.0636845
$886$	52.9681	1.77950
$887$	15.2931	0.513492	0.256746	−	0.966479i	$-0.417350\pi$
$887$	15.2931	0.513492	0.256746	+	0.966479i	$0.417350\pi$
$888$	23.6830	0.794748
$889$	6.26789	0.210218
$890$	−13.9864	−0.468827
$891$	−0.861044	−0.0288461
$892$	−106.076	−3.55168
$893$	−39.0068	−1.30531
$894$	13.6249	0.455684
$895$	6.35376	0.212383
$896$	−7.52967	−0.251549
$897$	0.655903	0.0219000
$898$	3.70035	0.123482
$899$	−19.3734	−0.646138
$900$	−18.2214	−0.607381
$901$	4.72827	0.157522
$902$	−6.53515	−0.217597
$903$	3.18880	0.106117
$904$	−45.1700	−1.50233
$905$	−0.0184739	−0.000614095 0
$906$	3.62632	0.120476
$907$	16.6982	0.554456	0.277228	−	0.960804i	$-0.410584\pi$
$907$	16.6982	0.554456	0.277228	+	0.960804i	$0.410584\pi$
$908$	1.24200	0.0412172
$909$	5.04213	0.167237
$910$	−0.0463403	−0.00153617
$911$	−26.7876	−0.887513	−0.443757	−	0.896147i	$-0.646355\pi$
$911$	−26.7876	−0.887513	−0.443757	+	0.896147i	$0.646355\pi$
$912$	19.4876	0.645298
$913$	14.0171	0.463899
$914$	−70.6669	−2.33745
$915$	−3.79920	−0.125598
$916$	−3.88829	−0.128473
$917$	−0.946547	−0.0312577
$918$	−2.39228	−0.0789569
$919$	0.941694	0.0310636	0.0155318	−	0.999879i	$-0.495056\pi$
$919$	0.941694	0.0310636	0.0155318	+	0.999879i	$0.495056\pi$
$920$	−5.35610	−0.176585
$921$	−11.4073	−0.375883
$922$	−1.68376	−0.0554517
$923$	−2.12309	−0.0698824
$924$	−1.16391	−0.0382897
$925$	−28.1211	−0.924617
$926$	58.5471	1.92398
$927$	−13.3677	−0.439052
$928$	−8.46368	−0.277834
$929$	−45.4060	−1.48972	−0.744862	−	0.667219i	$-0.767485\pi$
$929$	−45.4060	−1.48972	−0.744862	+	0.667219i	$0.767485\pi$
$930$	4.39225	0.144028
$931$	55.4294	1.81663
$932$	48.7969	1.59840
$933$	−16.2862	−0.533185
$934$	69.2808	2.26694
$935$	0.279934	0.00915482
$936$	0.676410	0.0221092
$937$	−46.8806	−1.53152	−0.765762	−	0.643125i	$-0.777638\pi$
$937$	−46.8806	−1.53152	−0.765762	+	0.643125i	$0.777638\pi$
$938$	−10.5386	−0.344097
$939$	5.08060	0.165799
$940$	5.85010	0.190809
$941$	41.5888	1.35575	0.677877	−	0.735175i	$-0.262900\pi$
$941$	41.5888	1.35575	0.677877	+	0.735175i	$0.262900\pi$
$942$	−2.39228	−0.0779446
$943$	−12.6807	−0.412940
$944$	14.0713	0.457981
$945$	0.118041	0.00383986
$946$	18.0910	0.588190
$947$	−17.3673	−0.564362	−0.282181	−	0.959361i	$-0.591058\pi$
$947$	−17.3673	−0.564362	−0.282181	+	0.959361i	$0.591058\pi$
$948$	−19.1052	−0.620507
$949$	−2.44552	−0.0793848
$950$	−94.4934	−3.06577
$951$	28.8502	0.935532
$952$	−1.49656	−0.0485039
$953$	−17.2033	−0.557269	−0.278635	−	0.960397i	$-0.589882\pi$
$953$	−17.2033	−0.557269	−0.278635	+	0.960397i	$0.589882\pi$
$954$	11.3113	0.366218
$955$	−6.27623	−0.203094
$956$	−4.33890	−0.140330
$957$	−2.95383	−0.0954837
$958$	−40.6775	−1.31423
$959$	2.79310	0.0901940
$960$	3.48892	0.112604
$961$	0.892626	0.0287944
$962$	2.25564	0.0727247
$963$	11.7694	0.379265
$964$	−53.3392	−1.71794
$965$	0.997714	0.0321176
$966$	−3.47165	−0.111699
$967$	19.4776	0.626358	0.313179	−	0.949694i	$-0.398606\pi$
$967$	19.4776	0.626358	0.313179	+	0.949694i	$0.398606\pi$
$968$	42.2846	1.35908
$969$	−8.07048	−0.259261
$970$	8.62382	0.276894
$971$	11.1417	0.357555	0.178777	−	0.983890i	$-0.442786\pi$
$971$	11.1417	0.357555	0.178777	+	0.983890i	$0.442786\pi$
$972$	−3.72299	−0.119415
$973$	0.920872	0.0295218
$974$	49.2114	1.57684
$975$	−0.803169	−0.0257220
$976$	28.2176	0.903223
$977$	4.24510	0.135813	0.0679064	−	0.997692i	$-0.478368\pi$
$977$	4.24510	0.135813	0.0679064	+	0.997692i	$0.478368\pi$
$978$	−8.88856	−0.284225
$979$	−15.4843	−0.494880
$980$	−8.31311	−0.265553
$981$	−8.52725	−0.272254
$982$	−76.2742	−2.43401
$983$	−24.8723	−0.793304	−0.396652	−	0.917969i	$-0.629828\pi$
$983$	−24.8723	−0.793304	−0.396652	+	0.917969i	$0.629828\pi$
$984$	−13.0772	−0.416884
$985$	4.40987	0.140510
$986$	−8.20675	−0.261356
$987$	1.75486	0.0558578
$988$	4.93068	0.156866
$989$	35.1035	1.11623
$990$	0.669680	0.0212838
$991$	−11.1540	−0.354318	−0.177159	−	0.984182i	$-0.556691\pi$
$991$	−11.1540	−0.354318	−0.177159	+	0.984182i	$0.556691\pi$
$992$	13.9330	0.442373
$993$	−12.0989	−0.383946
$994$	11.2374	0.356428
$995$	5.23687	0.166020
$996$	60.6074	1.92042
$997$	−48.7606	−1.54427	−0.772133	−	0.635461i	$-0.780810\pi$
$997$	−48.7606	−1.54427	−0.772133	+	0.635461i	$0.780810\pi$
$998$	57.1422	1.80880
$999$	−5.74569	−0.181785

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8007.2.a.f.1.5	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8007.2.a.f.1.5	✓	48	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:	\( N \)	\(=\)	\( 8007 = 3 \cdot 17 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8007.a (trivial)

\(f(q)\)	\(=\)	\(q-2.39228 q^{2} -1.00000 q^{3} +3.72299 q^{4} +0.325110 q^{5} +2.39228 q^{6} -0.363079 q^{7} -4.12187 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.39228 q^{2} -1.00000 q^{3} +3.72299 q^{4} +0.325110 q^{5} +2.39228 q^{6} -0.363079 q^{7} -4.12187 q^{8} +1.00000 q^{9} -0.777753 q^{10} -0.861044 q^{11} -3.72299 q^{12} -0.164103 q^{13} +0.868585 q^{14} -0.325110 q^{15} +2.41467 q^{16} -1.00000 q^{17} -2.39228 q^{18} -8.07048 q^{19} +1.21038 q^{20} +0.363079 q^{21} +2.05986 q^{22} +3.99691 q^{23} +4.12187 q^{24} -4.89430 q^{25} +0.392579 q^{26} -1.00000 q^{27} -1.35174 q^{28} -3.43052 q^{29} +0.777753 q^{30} +5.64736 q^{31} +2.46717 q^{32} +0.861044 q^{33} +2.39228 q^{34} -0.118041 q^{35} +3.72299 q^{36} +5.74569 q^{37} +19.3068 q^{38} +0.164103 q^{39} -1.34006 q^{40} -3.17263 q^{41} -0.868585 q^{42} +8.78267 q^{43} -3.20566 q^{44} +0.325110 q^{45} -9.56171 q^{46} +4.83327 q^{47} -2.41467 q^{48} -6.86817 q^{49} +11.7085 q^{50} +1.00000 q^{51} -0.610953 q^{52} -4.72827 q^{53} +2.39228 q^{54} -0.279934 q^{55} +1.49656 q^{56} +8.07048 q^{57} +8.20675 q^{58} +5.82740 q^{59} -1.21038 q^{60} +11.6859 q^{61} -13.5100 q^{62} -0.363079 q^{63} -10.7315 q^{64} -0.0533514 q^{65} -2.05986 q^{66} -12.1330 q^{67} -3.72299 q^{68} -3.99691 q^{69} +0.282386 q^{70} +12.9376 q^{71} -4.12187 q^{72} +14.9024 q^{73} -13.7453 q^{74} +4.89430 q^{75} -30.0463 q^{76} +0.312627 q^{77} -0.392579 q^{78} +5.13168 q^{79} +0.785034 q^{80} +1.00000 q^{81} +7.58980 q^{82} -16.2792 q^{83} +1.35174 q^{84} -0.325110 q^{85} -21.0106 q^{86} +3.43052 q^{87} +3.54911 q^{88} +17.9831 q^{89} -0.777753 q^{90} +0.0595822 q^{91} +14.8804 q^{92} -5.64736 q^{93} -11.5625 q^{94} -2.62379 q^{95} -2.46717 q^{96} -11.0881 q^{97} +16.4306 q^{98} -0.861044 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9}+O(q^{10}) \) 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9} - 20 q^{10} + 5 q^{11} - 45 q^{12} - 8 q^{13} + 4 q^{14} - q^{15} + 39 q^{16} - 48 q^{17} - q^{18} - 6 q^{19} + 6 q^{20} + 13 q^{21} - 35 q^{22} - 8 q^{23} + 6 q^{24} + 13 q^{25} + 17 q^{26} - 48 q^{27} - 38 q^{28} + q^{29} + 20 q^{30} - 21 q^{31} - 3 q^{32} - 5 q^{33} + q^{34} + 19 q^{35} + 45 q^{36} - 58 q^{37} - 14 q^{38} + 8 q^{39} - 54 q^{40} - 3 q^{41} - 4 q^{42} - 33 q^{43} + 2 q^{44} + q^{45} - 26 q^{46} + 9 q^{47} - 39 q^{48} + 11 q^{49} + 4 q^{50} + 48 q^{51} - 31 q^{52} - 33 q^{53} + q^{54} - 21 q^{55} + 6 q^{57} - 55 q^{58} + 77 q^{59} - 6 q^{60} - 29 q^{61} - 46 q^{62} - 13 q^{63} + 24 q^{64} - 49 q^{65} + 35 q^{66} - 44 q^{67} - 45 q^{68} + 8 q^{69} + 4 q^{70} + 22 q^{71} - 6 q^{72} - 63 q^{73} - 16 q^{74} - 13 q^{75} - 46 q^{76} - 30 q^{77} - 17 q^{78} - 46 q^{79} - 14 q^{80} + 48 q^{81} - 75 q^{82} + 11 q^{83} + 38 q^{84} - q^{85} + 8 q^{86} - q^{87} - 116 q^{88} + 10 q^{89} - 20 q^{90} - 67 q^{91} - 64 q^{92} + 21 q^{93} - 16 q^{94} - 8 q^{95} + 3 q^{96} - 96 q^{97} - 46 q^{98} + 5 q^{99}+O(q^{100}) \) 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9 - 20 * q^10 + 5 * q^11 - 45 * q^12 - 8 * q^13 + 4 * q^14 - q^15 + 39 * q^16 - 48 * q^17 - q^18 - 6 * q^19 + 6 * q^20 + 13 * q^21 - 35 * q^22 - 8 * q^23 + 6 * q^24 + 13 * q^25 + 17 * q^26 - 48 * q^27 - 38 * q^28 + q^29 + 20 * q^30 - 21 * q^31 - 3 * q^32 - 5 * q^33 + q^34 + 19 * q^35 + 45 * q^36 - 58 * q^37 - 14 * q^38 + 8 * q^39 - 54 * q^40 - 3 * q^41 - 4 * q^42 - 33 * q^43 + 2 * q^44 + q^45 - 26 * q^46 + 9 * q^47 - 39 * q^48 + 11 * q^49 + 4 * q^50 + 48 * q^51 - 31 * q^52 - 33 * q^53 + q^54 - 21 * q^55 + 6 * q^57 - 55 * q^58 + 77 * q^59 - 6 * q^60 - 29 * q^61 - 46 * q^62 - 13 * q^63 + 24 * q^64 - 49 * q^65 + 35 * q^66 - 44 * q^67 - 45 * q^68 + 8 * q^69 + 4 * q^70 + 22 * q^71 - 6 * q^72 - 63 * q^73 - 16 * q^74 - 13 * q^75 - 46 * q^76 - 30 * q^77 - 17 * q^78 - 46 * q^79 - 14 * q^80 + 48 * q^81 - 75 * q^82 + 11 * q^83 + 38 * q^84 - q^85 + 8 * q^86 - q^87 - 116 * q^88 + 10 * q^89 - 20 * q^90 - 67 * q^91 - 64 * q^92 + 21 * q^93 - 16 * q^94 - 8 * q^95 + 3 * q^96 - 96 * q^97 - 46 * q^98 + 5 * q^99

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