LMFDB - Embedded newform 8030.2.a.bf.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8030,2,Mod(1,8030)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8030, 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("8030.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8030 = 2 \cdot 5 \cdot 11 \cdot 73 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8030.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:	$64.1198728231$
Analytic rank:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 4 x^{14} - 23 x^{13} + 99 x^{12} + 191 x^{11} - 922 x^{10} - 702 x^{9} + 4108 x^{8} + 957 x^{7} + \cdots + 8 $ x^15 - 4x^14 - 23x^13 + 99x^12 + 191x^11 - 922x^10 - 702x^9 + 4108x^8 + 957x^7 - 8875x^6 + 479x^5 + 7698x^4 - 1731x^3 - 626x^2 + 46x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-0.0958822$ of defining polynomial
Character	$\chi$	$=$	8030.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +0.0958822 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.0958822 q^{6} -1.44050 q^{7} +1.00000 q^{8} -2.99081 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +0.0958822 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.0958822 q^{6} -1.44050 q^{7} +1.00000 q^{8} -2.99081 q^{9} -1.00000 q^{10} -1.00000 q^{11} +0.0958822 q^{12} +1.32450 q^{13} -1.44050 q^{14} -0.0958822 q^{15} +1.00000 q^{16} +3.53024 q^{17} -2.99081 q^{18} +2.56476 q^{19} -1.00000 q^{20} -0.138118 q^{21} -1.00000 q^{22} +2.07743 q^{23} +0.0958822 q^{24} +1.00000 q^{25} +1.32450 q^{26} -0.574412 q^{27} -1.44050 q^{28} +4.30516 q^{29} -0.0958822 q^{30} -7.80074 q^{31} +1.00000 q^{32} -0.0958822 q^{33} +3.53024 q^{34} +1.44050 q^{35} -2.99081 q^{36} +7.23181 q^{37} +2.56476 q^{38} +0.126996 q^{39} -1.00000 q^{40} -5.45499 q^{41} -0.138118 q^{42} -8.42134 q^{43} -1.00000 q^{44} +2.99081 q^{45} +2.07743 q^{46} -12.3740 q^{47} +0.0958822 q^{48} -4.92496 q^{49} +1.00000 q^{50} +0.338487 q^{51} +1.32450 q^{52} +3.14683 q^{53} -0.574412 q^{54} +1.00000 q^{55} -1.44050 q^{56} +0.245915 q^{57} +4.30516 q^{58} -10.5892 q^{59} -0.0958822 q^{60} +3.84537 q^{61} -7.80074 q^{62} +4.30825 q^{63} +1.00000 q^{64} -1.32450 q^{65} -0.0958822 q^{66} -3.40263 q^{67} +3.53024 q^{68} +0.199189 q^{69} +1.44050 q^{70} +9.28419 q^{71} -2.99081 q^{72} +1.00000 q^{73} +7.23181 q^{74} +0.0958822 q^{75} +2.56476 q^{76} +1.44050 q^{77} +0.126996 q^{78} +14.4470 q^{79} -1.00000 q^{80} +8.91734 q^{81} -5.45499 q^{82} -8.91982 q^{83} -0.138118 q^{84} -3.53024 q^{85} -8.42134 q^{86} +0.412789 q^{87} -1.00000 q^{88} -0.475579 q^{89} +2.99081 q^{90} -1.90794 q^{91} +2.07743 q^{92} -0.747953 q^{93} -12.3740 q^{94} -2.56476 q^{95} +0.0958822 q^{96} -12.5031 q^{97} -4.92496 q^{98} +2.99081 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 15 q^{2} - 4 q^{3} + 15 q^{4} - 15 q^{5} - 4 q^{6} - 6 q^{7} + 15 q^{8} + 17 q^{9}+O(q^{10}) $ 15 * q + 15 * q^2 - 4 * q^3 + 15 * q^4 - 15 * q^5 - 4 * q^6 - 6 * q^7 + 15 * q^8 + 17 * q^9	\( 15 q + 15 q^{2} - 4 q^{3} + 15 q^{4} - 15 q^{5} - 4 q^{6} - 6 q^{7} + 15 q^{8} + 17 q^{9} - 15 q^{10} - 15 q^{11} - 4 q^{12} - 6 q^{13} - 6 q^{14} + 4 q^{15} + 15 q^{16} + 2 q^{17} + 17 q^{18} - 8 q^{19} - 15 q^{20} - 17 q^{21} - 15 q^{22} - 4 q^{23} - 4 q^{24} + 15 q^{25} - 6 q^{26} - 19 q^{27} - 6 q^{28} - 13 q^{29} + 4 q^{30} - 20 q^{31} + 15 q^{32} + 4 q^{33} + 2 q^{34} + 6 q^{35} + 17 q^{36} - 15 q^{37} - 8 q^{38} - 11 q^{39} - 15 q^{40} + 2 q^{41} - 17 q^{42} - 26 q^{43} - 15 q^{44} - 17 q^{45} - 4 q^{46} - 14 q^{47} - 4 q^{48} + 11 q^{49} + 15 q^{50} - 39 q^{51} - 6 q^{52} - 21 q^{53} - 19 q^{54} + 15 q^{55} - 6 q^{56} + q^{57} - 13 q^{58} - 14 q^{59} + 4 q^{60} - 45 q^{61} - 20 q^{62} - 17 q^{63} + 15 q^{64} + 6 q^{65} + 4 q^{66} - 10 q^{67} + 2 q^{68} - 23 q^{69} + 6 q^{70} - 9 q^{71} + 17 q^{72} + 15 q^{73} - 15 q^{74} - 4 q^{75} - 8 q^{76} + 6 q^{77} - 11 q^{78} - 26 q^{79} - 15 q^{80} + 15 q^{81} + 2 q^{82} - 30 q^{83} - 17 q^{84} - 2 q^{85} - 26 q^{86} - 14 q^{87} - 15 q^{88} + 10 q^{89} - 17 q^{90} - 17 q^{91} - 4 q^{92} - 8 q^{93} - 14 q^{94} + 8 q^{95} - 4 q^{96} - 27 q^{97} + 11 q^{98} - 17 q^{99}+O(q^{100}) \) 15 * q + 15 * q^2 - 4 * q^3 + 15 * q^4 - 15 * q^5 - 4 * q^6 - 6 * q^7 + 15 * q^8 + 17 * q^9 - 15 * q^10 - 15 * q^11 - 4 * q^12 - 6 * q^13 - 6 * q^14 + 4 * q^15 + 15 * q^16 + 2 * q^17 + 17 * q^18 - 8 * q^19 - 15 * q^20 - 17 * q^21 - 15 * q^22 - 4 * q^23 - 4 * q^24 + 15 * q^25 - 6 * q^26 - 19 * q^27 - 6 * q^28 - 13 * q^29 + 4 * q^30 - 20 * q^31 + 15 * q^32 + 4 * q^33 + 2 * q^34 + 6 * q^35 + 17 * q^36 - 15 * q^37 - 8 * q^38 - 11 * q^39 - 15 * q^40 + 2 * q^41 - 17 * q^42 - 26 * q^43 - 15 * q^44 - 17 * q^45 - 4 * q^46 - 14 * q^47 - 4 * q^48 + 11 * q^49 + 15 * q^50 - 39 * q^51 - 6 * q^52 - 21 * q^53 - 19 * q^54 + 15 * q^55 - 6 * q^56 + q^57 - 13 * q^58 - 14 * q^59 + 4 * q^60 - 45 * q^61 - 20 * q^62 - 17 * q^63 + 15 * q^64 + 6 * q^65 + 4 * q^66 - 10 * q^67 + 2 * q^68 - 23 * q^69 + 6 * q^70 - 9 * q^71 + 17 * q^72 + 15 * q^73 - 15 * q^74 - 4 * q^75 - 8 * q^76 + 6 * q^77 - 11 * q^78 - 26 * q^79 - 15 * q^80 + 15 * q^81 + 2 * q^82 - 30 * q^83 - 17 * q^84 - 2 * q^85 - 26 * q^86 - 14 * q^87 - 15 * q^88 + 10 * q^89 - 17 * q^90 - 17 * q^91 - 4 * q^92 - 8 * q^93 - 14 * q^94 + 8 * q^95 - 4 * q^96 - 27 * q^97 + 11 * q^98 - 17 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0.0958822	0.0553576	0.0276788	−	0.999617i	$-0.491188\pi$
$3$	0.0958822	0.0553576	0.0276788	+	0.999617i	$0.491188\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0.0958822	0.0391438
$7$	−1.44050	−0.544457	−0.272229	−	0.962233i	$-0.587761\pi$
$7$	−1.44050	−0.544457	−0.272229	+	0.962233i	$0.587761\pi$
$8$	1.00000	0.353553
$9$	−2.99081	−0.996936
$10$	−1.00000	−0.316228
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0.0958822	0.0276788
$13$	1.32450	0.367350	0.183675	−	0.982987i	$-0.441201\pi$
$13$	1.32450	0.367350	0.183675	+	0.982987i	$0.441201\pi$
$14$	−1.44050	−0.384989
$15$	−0.0958822	−0.0247567
$16$	1.00000	0.250000
$17$	3.53024	0.856209	0.428105	−	0.903729i	$-0.359181\pi$
$17$	3.53024	0.856209	0.428105	+	0.903729i	$0.359181\pi$
$18$	−2.99081	−0.704940
$19$	2.56476	0.588396	0.294198	−	0.955745i	$-0.404948\pi$
$19$	2.56476	0.588396	0.294198	+	0.955745i	$0.404948\pi$
$20$	−1.00000	−0.223607
$21$	−0.138118	−0.0301399
$22$	−1.00000	−0.213201
$23$	2.07743	0.433175	0.216587	−	0.976263i	$-0.430507\pi$
$23$	2.07743	0.433175	0.216587	+	0.976263i	$0.430507\pi$
$24$	0.0958822	0.0195719
$25$	1.00000	0.200000
$26$	1.32450	0.259756
$27$	−0.574412	−0.110546
$28$	−1.44050	−0.272229
$29$	4.30516	0.799449	0.399724	−	0.916635i	$-0.369106\pi$
$29$	4.30516	0.799449	0.399724	+	0.916635i	$0.369106\pi$
$30$	−0.0958822	−0.0175056
$31$	−7.80074	−1.40105	−0.700527	−	0.713626i	$-0.747052\pi$
$31$	−7.80074	−1.40105	−0.700527	+	0.713626i	$0.747052\pi$
$32$	1.00000	0.176777
$33$	−0.0958822	−0.0166910
$34$	3.53024	0.605431
$35$	1.44050	0.243489
$36$	−2.99081	−0.498468
$37$	7.23181	1.18890	0.594451	−	0.804132i	$-0.297369\pi$
$37$	7.23181	1.18890	0.594451	+	0.804132i	$0.297369\pi$
$38$	2.56476	0.416059
$39$	0.126996	0.0203356
$40$	−1.00000	−0.158114
$41$	−5.45499	−0.851926	−0.425963	−	0.904741i	$-0.640065\pi$
$41$	−5.45499	−0.851926	−0.425963	+	0.904741i	$0.640065\pi$
$42$	−0.138118	−0.0213121
$43$	−8.42134	−1.28424	−0.642121	−	0.766603i	$-0.721946\pi$
$43$	−8.42134	−1.28424	−0.642121	+	0.766603i	$0.721946\pi$
$44$	−1.00000	−0.150756
$45$	2.99081	0.445843
$46$	2.07743	0.306301
$47$	−12.3740	−1.80493	−0.902464	−	0.430765i	$-0.858244\pi$
$47$	−12.3740	−1.80493	−0.902464	+	0.430765i	$0.858244\pi$
$48$	0.0958822	0.0138394
$49$	−4.92496	−0.703566
$50$	1.00000	0.141421
$51$	0.338487	0.0473977
$52$	1.32450	0.183675
$53$	3.14683	0.432251	0.216125	−	0.976366i	$-0.430658\pi$
$53$	3.14683	0.432251	0.216125	+	0.976366i	$0.430658\pi$
$54$	−0.574412	−0.0781676
$55$	1.00000	0.134840
$56$	−1.44050	−0.192495
$57$	0.245915	0.0325722
$58$	4.30516	0.565296
$59$	−10.5892	−1.37859	−0.689295	−	0.724480i	$-0.742080\pi$
$59$	−10.5892	−1.37859	−0.689295	+	0.724480i	$0.742080\pi$
$60$	−0.0958822	−0.0123783
$61$	3.84537	0.492350	0.246175	−	0.969225i	$-0.420826\pi$
$61$	3.84537	0.492350	0.246175	+	0.969225i	$0.420826\pi$
$62$	−7.80074	−0.990695
$63$	4.30825	0.542789
$64$	1.00000	0.125000
$65$	−1.32450	−0.164284
$66$	−0.0958822	−0.0118023
$67$	−3.40263	−0.415697	−0.207849	−	0.978161i	$-0.566646\pi$
$67$	−3.40263	−0.415697	−0.207849	+	0.978161i	$0.566646\pi$
$68$	3.53024	0.428105
$69$	0.199189	0.0239795
$70$	1.44050	0.172172
$71$	9.28419	1.10183	0.550915	−	0.834561i	$-0.314279\pi$
$71$	9.28419	1.10183	0.550915	+	0.834561i	$0.314279\pi$
$72$	−2.99081	−0.352470
$73$	1.00000	0.117041
$73$	1.00000	0.117041
$74$	7.23181	0.840681
$75$	0.0958822	0.0110715
$76$	2.56476	0.294198
$77$	1.44050	0.164160
$78$	0.126996	0.0143795
$79$	14.4470	1.62541	0.812707	−	0.582673i	$-0.197993\pi$
$79$	14.4470	1.62541	0.812707	+	0.582673i	$0.197993\pi$
$80$	−1.00000	−0.111803
$81$	8.91734	0.990816
$82$	−5.45499	−0.602402
$83$	−8.91982	−0.979078	−0.489539	−	0.871981i	$-0.662835\pi$
$83$	−8.91982	−0.979078	−0.489539	+	0.871981i	$0.662835\pi$
$84$	−0.138118	−0.0150699
$85$	−3.53024	−0.382908
$86$	−8.42134	−0.908097
$87$	0.412789	0.0442556
$88$	−1.00000	−0.106600
$89$	−0.475579	−0.0504113	−0.0252056	−	0.999682i	$-0.508024\pi$
$89$	−0.475579	−0.0504113	−0.0252056	+	0.999682i	$0.508024\pi$
$90$	2.99081	0.315259
$91$	−1.90794	−0.200006
$92$	2.07743	0.216587
$93$	−0.747953	−0.0775591
$94$	−12.3740	−1.27628
$95$	−2.56476	−0.263139
$96$	0.0958822	0.00978594
$97$	−12.5031	−1.26949	−0.634747	−	0.772720i	$-0.718896\pi$
$97$	−12.5031	−1.26949	−0.634747	+	0.772720i	$0.718896\pi$
$98$	−4.92496	−0.497497
$99$	2.99081	0.300587
$100$	1.00000	0.100000
$101$	1.69976	0.169133	0.0845663	−	0.996418i	$-0.473050\pi$
$101$	1.69976	0.169133	0.0845663	+	0.996418i	$0.473050\pi$
$102$	0.338487	0.0335153
$103$	−3.34194	−0.329291	−0.164645	−	0.986353i	$-0.552648\pi$
$103$	−3.34194	−0.329291	−0.164645	+	0.986353i	$0.552648\pi$
$104$	1.32450	0.129878
$105$	0.138118	0.0134790
$106$	3.14683	0.305648
$107$	12.3735	1.19619	0.598095	−	0.801425i	$-0.295924\pi$
$107$	12.3735	1.19619	0.598095	+	0.801425i	$0.295924\pi$
$108$	−0.574412	−0.0552728
$109$	6.88448	0.659414	0.329707	−	0.944083i	$-0.393050\pi$
$109$	6.88448	0.659414	0.329707	+	0.944083i	$0.393050\pi$
$110$	1.00000	0.0953463
$111$	0.693402	0.0658148
$112$	−1.44050	−0.136114
$113$	−0.833967	−0.0784530	−0.0392265	−	0.999230i	$-0.512489\pi$
$113$	−0.833967	−0.0784530	−0.0392265	+	0.999230i	$0.512489\pi$
$114$	0.245915	0.0230320
$115$	−2.07743	−0.193722
$116$	4.30516	0.399724
$117$	−3.96132	−0.366224
$118$	−10.5892	−0.974811
$119$	−5.08531	−0.466169
$120$	−0.0958822	−0.00875281
$121$	1.00000	0.0909091
$122$	3.84537	0.348144
$123$	−0.523036	−0.0471606
$124$	−7.80074	−0.700527
$125$	−1.00000	−0.0894427
$126$	4.30825	0.383810
$127$	−10.7039	−0.949815	−0.474908	−	0.880036i	$-0.657519\pi$
$127$	−10.7039	−0.949815	−0.474908	+	0.880036i	$0.657519\pi$
$128$	1.00000	0.0883883
$129$	−0.807457	−0.0710927
$130$	−1.32450	−0.116166
$131$	−7.61936	−0.665707	−0.332853	−	0.942979i	$-0.608011\pi$
$131$	−7.61936	−0.665707	−0.332853	+	0.942979i	$0.608011\pi$
$132$	−0.0958822	−0.00834548
$133$	−3.69453	−0.320356
$134$	−3.40263	−0.293942
$135$	0.574412	0.0494375
$136$	3.53024	0.302716
$137$	−13.9644	−1.19306	−0.596531	−	0.802590i	$-0.703455\pi$
$137$	−13.9644	−1.19306	−0.596531	+	0.802590i	$0.703455\pi$
$138$	0.199189	0.0169561
$139$	−13.9164	−1.18037	−0.590185	−	0.807268i	$-0.700945\pi$
$139$	−13.9164	−1.18037	−0.590185	+	0.807268i	$0.700945\pi$
$140$	1.44050	0.121744
$141$	−1.18644	−0.0999165
$142$	9.28419	0.779112
$143$	−1.32450	−0.110760
$144$	−2.99081	−0.249234
$145$	−4.30516	−0.357524
$146$	1.00000	0.0827606
$147$	−0.472217	−0.0389478
$148$	7.23181	0.594451
$149$	15.8575	1.29910	0.649548	−	0.760320i	$-0.274958\pi$
$149$	15.8575	1.29910	0.649548	+	0.760320i	$0.274958\pi$
$150$	0.0958822	0.00782875
$151$	−13.7010	−1.11497	−0.557486	−	0.830187i	$-0.688234\pi$
$151$	−13.7010	−1.11497	−0.557486	+	0.830187i	$0.688234\pi$
$152$	2.56476	0.208029
$153$	−10.5583	−0.853585
$154$	1.44050	0.116079
$155$	7.80074	0.626571
$156$	0.126996	0.0101678
$157$	−8.89200	−0.709659	−0.354829	−	0.934931i	$-0.615461\pi$
$157$	−8.89200	−0.709659	−0.354829	+	0.934931i	$0.615461\pi$
$158$	14.4470	1.14934
$159$	0.301726	0.0239284
$160$	−1.00000	−0.0790569
$161$	−2.99254	−0.235845
$162$	8.91734	0.700613
$163$	−10.5987	−0.830153	−0.415077	−	0.909786i	$-0.636245\pi$
$163$	−10.5987	−0.830153	−0.415077	+	0.909786i	$0.636245\pi$
$164$	−5.45499	−0.425963
$165$	0.0958822	0.00746442
$166$	−8.91982	−0.692313
$167$	−19.5165	−1.51023	−0.755115	−	0.655592i	$-0.772419\pi$
$167$	−19.5165	−1.51023	−0.755115	+	0.655592i	$0.772419\pi$
$168$	−0.138118	−0.0106561
$169$	−11.2457	−0.865054
$170$	−3.53024	−0.270757
$171$	−7.67069	−0.586593
$172$	−8.42134	−0.642121
$173$	−14.0203	−1.06594	−0.532972	−	0.846133i	$-0.678925\pi$
$173$	−14.0203	−1.06594	−0.532972	+	0.846133i	$0.678925\pi$
$174$	0.412789	0.0312934
$175$	−1.44050	−0.108891
$176$	−1.00000	−0.0753778
$177$	−1.01531	−0.0763155
$178$	−0.475579	−0.0356461
$179$	−25.3574	−1.89530	−0.947650	−	0.319312i	$-0.896548\pi$
$179$	−25.3574	−1.89530	−0.947650	+	0.319312i	$0.896548\pi$
$180$	2.99081	0.222922
$181$	−0.0201434	−0.00149725	−0.000748625	−	1.00000i	$-0.500238\pi$
$181$	−0.0201434	−0.00149725	−0.000748625		1.00000i	$0.500238\pi$
$182$	−1.90794	−0.141426
$183$	0.368703	0.0272553
$184$	2.07743	0.153150
$185$	−7.23181	−0.531693
$186$	−0.747953	−0.0548426
$187$	−3.53024	−0.258157
$188$	−12.3740	−0.902464
$189$	0.827440	0.0601874
$190$	−2.56476	−0.186067
$191$	−6.81372	−0.493023	−0.246512	−	0.969140i	$-0.579284\pi$
$191$	−6.81372	−0.493023	−0.246512	+	0.969140i	$0.579284\pi$
$192$	0.0958822	0.00691971
$193$	−5.64621	−0.406423	−0.203211	−	0.979135i	$-0.565138\pi$
$193$	−5.64621	−0.406423	−0.203211	+	0.979135i	$0.565138\pi$
$194$	−12.5031	−0.897668
$195$	−0.126996	−0.00909438
$196$	−4.92496	−0.351783
$197$	9.21074	0.656238	0.328119	−	0.944636i	$-0.393585\pi$
$197$	9.21074	0.656238	0.328119	+	0.944636i	$0.393585\pi$
$198$	2.99081	0.212547
$199$	−20.5838	−1.45915	−0.729574	−	0.683902i	$-0.760281\pi$
$199$	−20.5838	−1.45915	−0.729574	+	0.683902i	$0.760281\pi$
$200$	1.00000	0.0707107
$201$	−0.326251	−0.0230120
$202$	1.69976	0.119595
$203$	−6.20158	−0.435266
$204$	0.338487	0.0236989
$205$	5.45499	0.380993
$206$	−3.34194	−0.232844
$207$	−6.21320	−0.431847
$208$	1.32450	0.0918376
$209$	−2.56476	−0.177408
$210$	0.138118	0.00953106
$211$	20.6782	1.42354	0.711772	−	0.702410i	$-0.247893\pi$
$211$	20.6782	1.42354	0.711772	+	0.702410i	$0.247893\pi$
$212$	3.14683	0.216125
$213$	0.890189	0.0609947
$214$	12.3735	0.845835
$215$	8.42134	0.574331
$216$	−0.574412	−0.0390838
$217$	11.2370	0.762814
$218$	6.88448	0.466276
$219$	0.0958822	0.00647912
$220$	1.00000	0.0674200
$221$	4.67580	0.314529
$222$	0.693402	0.0465381
$223$	3.66507	0.245431	0.122716	−	0.992442i	$-0.460840\pi$
$223$	3.66507	0.245431	0.122716	+	0.992442i	$0.460840\pi$
$224$	−1.44050	−0.0962473
$225$	−2.99081	−0.199387
$226$	−0.833967	−0.0554747
$227$	11.2159	0.744427	0.372214	−	0.928147i	$-0.378599\pi$
$227$	11.2159	0.744427	0.372214	+	0.928147i	$0.378599\pi$
$228$	0.245915	0.0162861
$229$	4.64387	0.306876	0.153438	−	0.988158i	$-0.450966\pi$
$229$	4.64387	0.306876	0.153438	+	0.988158i	$0.450966\pi$
$230$	−2.07743	−0.136982
$231$	0.138118	0.00908751
$232$	4.30516	0.282648
$233$	−11.0430	−0.723452	−0.361726	−	0.932284i	$-0.617812\pi$
$233$	−11.0430	−0.723452	−0.361726	+	0.932284i	$0.617812\pi$
$234$	−3.96132	−0.258960
$235$	12.3740	0.807188
$236$	−10.5892	−0.689295
$237$	1.38521	0.0899790
$238$	−5.08531	−0.329631
$239$	1.54227	0.0997610	0.0498805	−	0.998755i	$-0.484116\pi$
$239$	1.54227	0.0997610	0.0498805	+	0.998755i	$0.484116\pi$
$240$	−0.0958822	−0.00618917
$241$	−6.61566	−0.426152	−0.213076	−	0.977036i	$-0.568348\pi$
$241$	−6.61566	−0.426152	−0.213076	+	0.977036i	$0.568348\pi$
$242$	1.00000	0.0642824
$243$	2.57825	0.165395
$244$	3.84537	0.246175
$245$	4.92496	0.314644
$246$	−0.523036	−0.0333476
$247$	3.39702	0.216147
$248$	−7.80074	−0.495348
$249$	−0.855253	−0.0541994
$250$	−1.00000	−0.0632456
$251$	−6.43147	−0.405951	−0.202976	−	0.979184i	$-0.565061\pi$
$251$	−6.43147	−0.405951	−0.202976	+	0.979184i	$0.565061\pi$
$252$	4.30825	0.271394
$253$	−2.07743	−0.130607
$254$	−10.7039	−0.671621
$255$	−0.338487	−0.0211969
$256$	1.00000	0.0625000
$257$	12.0417	0.751143	0.375571	−	0.926794i	$-0.377447\pi$
$257$	12.0417	0.751143	0.375571	+	0.926794i	$0.377447\pi$
$258$	−0.807457	−0.0502701
$259$	−10.4174	−0.647307
$260$	−1.32450	−0.0821420
$261$	−12.8759	−0.796999
$262$	−7.61936	−0.470726
$263$	−17.0967	−1.05423	−0.527114	−	0.849795i	$-0.676726\pi$
$263$	−17.0967	−1.05423	−0.527114	+	0.849795i	$0.676726\pi$
$264$	−0.0958822	−0.00590114
$265$	−3.14683	−0.193309
$266$	−3.69453	−0.226526
$267$	−0.0455996	−0.00279065
$268$	−3.40263	−0.207849
$269$	−20.2095	−1.23219	−0.616097	−	0.787670i	$-0.711287\pi$
$269$	−20.2095	−1.23219	−0.616097	+	0.787670i	$0.711287\pi$
$270$	0.574412	0.0349576
$271$	14.3186	0.869792	0.434896	−	0.900481i	$-0.356785\pi$
$271$	14.3186	0.869792	0.434896	+	0.900481i	$0.356785\pi$
$272$	3.53024	0.214052
$273$	−0.182938	−0.0110719
$274$	−13.9644	−0.843622
$275$	−1.00000	−0.0603023
$276$	0.199189	0.0119898
$277$	−7.60571	−0.456983	−0.228491	−	0.973546i	$-0.573379\pi$
$277$	−7.60571	−0.456983	−0.228491	+	0.973546i	$0.573379\pi$
$278$	−13.9164	−0.834647
$279$	23.3305	1.39676
$280$	1.44050	0.0860862
$281$	−17.8648	−1.06572	−0.532862	−	0.846202i	$-0.678883\pi$
$281$	−17.8648	−1.06572	−0.532862	+	0.846202i	$0.678883\pi$
$282$	−1.18644	−0.0706517
$283$	−1.04584	−0.0621687	−0.0310843	−	0.999517i	$-0.509896\pi$
$283$	−1.04584	−0.0621687	−0.0310843	+	0.999517i	$0.509896\pi$
$284$	9.28419	0.550915
$285$	−0.245915	−0.0145667
$286$	−1.32450	−0.0783193
$287$	7.85790	0.463837
$288$	−2.99081	−0.176235
$289$	−4.53740	−0.266906
$290$	−4.30516	−0.252808
$291$	−1.19882	−0.0702762
$292$	1.00000	0.0585206
$293$	−13.0740	−0.763792	−0.381896	−	0.924205i	$-0.624729\pi$
$293$	−13.0740	−0.763792	−0.381896	+	0.924205i	$0.624729\pi$
$294$	−0.472217	−0.0275402
$295$	10.5892	0.616525
$296$	7.23181	0.420341
$297$	0.574412	0.0333308
$298$	15.8575	0.918600
$299$	2.75156	0.159127
$300$	0.0958822	0.00553576
$301$	12.1309	0.699215
$302$	−13.7010	−0.788404
$303$	0.162977	0.00936278
$304$	2.56476	0.147099
$305$	−3.84537	−0.220186
$306$	−10.5583	−0.603576
$307$	7.44048	0.424651	0.212325	−	0.977199i	$-0.431896\pi$
$307$	7.44048	0.424651	0.212325	+	0.977199i	$0.431896\pi$
$308$	1.44050	0.0820800
$309$	−0.320433	−0.0182288
$310$	7.80074	0.443052
$311$	−2.58043	−0.146323	−0.0731615	−	0.997320i	$-0.523309\pi$
$311$	−2.58043	−0.146323	−0.0731615	+	0.997320i	$0.523309\pi$
$312$	0.126996	0.00718973
$313$	−4.03133	−0.227864	−0.113932	−	0.993489i	$-0.536345\pi$
$313$	−4.03133	−0.227864	−0.113932	+	0.993489i	$0.536345\pi$
$314$	−8.89200	−0.501804
$315$	−4.30825	−0.242742
$316$	14.4470	0.812707
$317$	9.74135	0.547129	0.273564	−	0.961854i	$-0.411797\pi$
$317$	9.74135	0.547129	0.273564	+	0.961854i	$0.411797\pi$
$318$	0.301726	0.0169199
$319$	−4.30516	−0.241043
$320$	−1.00000	−0.0559017
$321$	1.18640	0.0662183
$322$	−2.99254	−0.166768
$323$	9.05421	0.503790
$324$	8.91734	0.495408
$325$	1.32450	0.0734700
$326$	−10.5987	−0.587007
$327$	0.660100	0.0365036
$328$	−5.45499	−0.301201
$329$	17.8247	0.982706
$330$	0.0958822	0.00527814
$331$	−8.92741	−0.490695	−0.245347	−	0.969435i	$-0.578902\pi$
$331$	−8.92741	−0.490695	−0.245347	+	0.969435i	$0.578902\pi$
$332$	−8.91982	−0.489539
$333$	−21.6290	−1.18526
$334$	−19.5165	−1.06789
$335$	3.40263	0.185905
$336$	−0.138118	−0.00753497
$337$	15.5038	0.844544	0.422272	−	0.906469i	$-0.361233\pi$
$337$	15.5038	0.844544	0.422272	+	0.906469i	$0.361233\pi$
$338$	−11.2457	−0.611685
$339$	−0.0799627	−0.00434298
$340$	−3.53024	−0.191454
$341$	7.80074	0.422434
$342$	−7.67069	−0.414784
$343$	17.1779	0.927519
$344$	−8.42134	−0.454048
$345$	−0.199189	−0.0107240
$346$	−14.0203	−0.753736
$347$	17.8682	0.959217	0.479608	−	0.877483i	$-0.340779\pi$
$347$	17.8682	0.959217	0.479608	+	0.877483i	$0.340779\pi$
$348$	0.412789	0.0221278
$349$	9.18266	0.491536	0.245768	−	0.969329i	$-0.420960\pi$
$349$	9.18266	0.491536	0.245768	+	0.969329i	$0.420960\pi$
$350$	−1.44050	−0.0769979
$351$	−0.760809	−0.0406090
$352$	−1.00000	−0.0533002
$353$	−17.5887	−0.936155	−0.468077	−	0.883688i	$-0.655053\pi$
$353$	−17.5887	−0.936155	−0.468077	+	0.883688i	$0.655053\pi$
$354$	−1.01531	−0.0539632
$355$	−9.28419	−0.492754
$356$	−0.475579	−0.0252056
$357$	−0.487591	−0.0258060
$358$	−25.3574	−1.34018
$359$	−5.12767	−0.270628	−0.135314	−	0.990803i	$-0.543204\pi$
$359$	−5.12767	−0.270628	−0.135314	+	0.990803i	$0.543204\pi$
$360$	2.99081	0.157629
$361$	−12.4220	−0.653791
$362$	−0.0201434	−0.00105872
$363$	0.0958822	0.00503251
$364$	−1.90794	−0.100003
$365$	−1.00000	−0.0523424
$366$	0.368703	0.0192724
$367$	10.7767	0.562539	0.281269	−	0.959629i	$-0.409245\pi$
$367$	10.7767	0.562539	0.281269	+	0.959629i	$0.409245\pi$
$368$	2.07743	0.108294
$369$	16.3148	0.849315
$370$	−7.23181	−0.375964
$371$	−4.53301	−0.235342
$372$	−0.747953	−0.0387795
$373$	−25.2009	−1.30485	−0.652425	−	0.757853i	$-0.726248\pi$
$373$	−25.2009	−1.30485	−0.652425	+	0.757853i	$0.726248\pi$
$374$	−3.53024	−0.182544
$375$	−0.0958822	−0.00495134
$376$	−12.3740	−0.638138
$377$	5.70219	0.293678
$378$	0.827440	0.0425589
$379$	12.7580	0.655333	0.327666	−	0.944793i	$-0.393738\pi$
$379$	12.7580	0.655333	0.327666	+	0.944793i	$0.393738\pi$
$380$	−2.56476	−0.131569
$381$	−1.02631	−0.0525795
$382$	−6.81372	−0.348620
$383$	18.0362	0.921607	0.460804	−	0.887502i	$-0.347561\pi$
$383$	18.0362	0.921607	0.460804	+	0.887502i	$0.347561\pi$
$384$	0.0958822	0.00489297
$385$	−1.44050	−0.0734146
$386$	−5.64621	−0.287384
$387$	25.1866	1.28031
$388$	−12.5031	−0.634747
$389$	−16.0406	−0.813290	−0.406645	−	0.913586i	$-0.633301\pi$
$389$	−16.0406	−0.813290	−0.406645	+	0.913586i	$0.633301\pi$
$390$	−0.126996	−0.00643069
$391$	7.33384	0.370888
$392$	−4.92496	−0.248748
$393$	−0.730562	−0.0368520
$394$	9.21074	0.464030
$395$	−14.4470	−0.726907
$396$	2.99081	0.150294
$397$	−19.7435	−0.990898	−0.495449	−	0.868637i	$-0.664996\pi$
$397$	−19.7435	−0.990898	−0.495449	+	0.868637i	$0.664996\pi$
$398$	−20.5838	−1.03177
$399$	−0.354240	−0.0177342
$400$	1.00000	0.0500000
$401$	−28.4689	−1.42167	−0.710835	−	0.703359i	$-0.751683\pi$
$401$	−28.4689	−1.42167	−0.710835	+	0.703359i	$0.751683\pi$
$402$	−0.326251	−0.0162719
$403$	−10.3321	−0.514678
$404$	1.69976	0.0845663
$405$	−8.91734	−0.443106
$406$	−6.20158	−0.307779
$407$	−7.23181	−0.358468
$408$	0.338487	0.0167576
$409$	17.0882	0.844958	0.422479	−	0.906373i	$-0.361160\pi$
$409$	17.0882	0.844958	0.422479	+	0.906373i	$0.361160\pi$
$410$	5.45499	0.269403
$411$	−1.33894	−0.0660451
$412$	−3.34194	−0.164645
$413$	15.2537	0.750584
$414$	−6.21320	−0.305362
$415$	8.91982	0.437857
$416$	1.32450	0.0649390
$417$	−1.33433	−0.0653425
$418$	−2.56476	−0.125446
$419$	23.2852	1.13756	0.568779	−	0.822490i	$-0.307416\pi$
$419$	23.2852	1.13756	0.568779	+	0.822490i	$0.307416\pi$
$420$	0.138118	0.00673948
$421$	27.8801	1.35879	0.679396	−	0.733772i	$-0.262242\pi$
$421$	27.8801	1.35879	0.679396	+	0.733772i	$0.262242\pi$
$422$	20.6782	1.00660
$423$	37.0081	1.79940
$424$	3.14683	0.152824
$425$	3.53024	0.171242
$426$	0.890189	0.0431298
$427$	−5.53926	−0.268063
$428$	12.3735	0.598095
$429$	−0.126996	−0.00613143
$430$	8.42134	0.406113
$431$	29.5106	1.42148	0.710738	−	0.703457i	$-0.248361\pi$
$431$	29.5106	1.42148	0.710738	+	0.703457i	$0.248361\pi$
$432$	−0.574412	−0.0276364
$433$	−1.56969	−0.0754347	−0.0377173	−	0.999288i	$-0.512009\pi$
$433$	−1.56969	−0.0754347	−0.0377173	+	0.999288i	$0.512009\pi$
$434$	11.2370	0.539391
$435$	−0.412789	−0.0197917
$436$	6.88448	0.329707
$437$	5.32811	0.254878
$438$	0.0958822	0.00458143
$439$	−39.7839	−1.89878	−0.949390	−	0.314100i	$-0.898297\pi$
$439$	−39.7839	−1.89878	−0.949390	+	0.314100i	$0.898297\pi$
$440$	1.00000	0.0476731
$441$	14.7296	0.701410
$442$	4.67580	0.222405
$443$	−18.3085	−0.869862	−0.434931	−	0.900464i	$-0.643227\pi$
$443$	−18.3085	−0.869862	−0.434931	+	0.900464i	$0.643227\pi$
$444$	0.693402	0.0329074
$445$	0.475579	0.0225446
$446$	3.66507	0.173546
$447$	1.52045	0.0719149
$448$	−1.44050	−0.0680571
$449$	11.2912	0.532867	0.266433	−	0.963853i	$-0.414155\pi$
$449$	11.2912	0.532867	0.266433	+	0.963853i	$0.414155\pi$
$450$	−2.99081	−0.140988
$451$	5.45499	0.256865
$452$	−0.833967	−0.0392265
$453$	−1.31368	−0.0617222
$454$	11.2159	0.526390
$455$	1.90794	0.0894456
$456$	0.245915	0.0115160
$457$	16.9597	0.793344	0.396672	−	0.917960i	$-0.370165\pi$
$457$	16.9597	0.793344	0.396672	+	0.917960i	$0.370165\pi$
$458$	4.64387	0.216994
$459$	−2.02781	−0.0946502
$460$	−2.07743	−0.0968608
$461$	18.5691	0.864848	0.432424	−	0.901671i	$-0.357658\pi$
$461$	18.5691	0.864848	0.432424	+	0.901671i	$0.357658\pi$
$462$	0.138118	0.00642584
$463$	4.55711	0.211787	0.105893	−	0.994377i	$-0.466230\pi$
$463$	4.55711	0.211787	0.105893	+	0.994377i	$0.466230\pi$
$464$	4.30516	0.199862
$465$	0.747953	0.0346855
$466$	−11.0430	−0.511558
$467$	39.9216	1.84735	0.923675	−	0.383177i	$-0.125170\pi$
$467$	39.9216	1.84735	0.923675	+	0.383177i	$0.125170\pi$
$468$	−3.96132	−0.183112
$469$	4.90148	0.226329
$470$	12.3740	0.570768
$471$	−0.852585	−0.0392850
$472$	−10.5892	−0.487406
$473$	8.42134	0.387214
$474$	1.38521	0.0636248
$475$	2.56476	0.117679
$476$	−5.08531	−0.233085
$477$	−9.41157	−0.430926
$478$	1.54227	0.0705417
$479$	4.77492	0.218172	0.109086	−	0.994032i	$-0.465208\pi$
$479$	4.77492	0.218172	0.109086	+	0.994032i	$0.465208\pi$
$480$	−0.0958822	−0.00437641
$481$	9.57854	0.436744
$482$	−6.61566	−0.301335
$483$	−0.286931	−0.0130558
$484$	1.00000	0.0454545
$485$	12.5031	0.567735
$486$	2.57825	0.116952
$487$	−18.5014	−0.838379	−0.419189	−	0.907899i	$-0.637686\pi$
$487$	−18.5014	−0.838379	−0.419189	+	0.907899i	$0.637686\pi$
$488$	3.84537	0.174072
$489$	−1.01623	−0.0459553
$490$	4.92496	0.222487
$491$	−24.7563	−1.11724	−0.558618	−	0.829425i	$-0.688668\pi$
$491$	−24.7563	−1.11724	−0.558618	+	0.829425i	$0.688668\pi$
$492$	−0.523036	−0.0235803
$493$	15.1983	0.684495
$494$	3.39702	0.152839
$495$	−2.99081	−0.134427
$496$	−7.80074	−0.350264
$497$	−13.3739	−0.599900
$498$	−0.855253	−0.0383248
$499$	−36.4784	−1.63300	−0.816498	−	0.577348i	$-0.804088\pi$
$499$	−36.4784	−1.63300	−0.816498	+	0.577348i	$0.804088\pi$
$500$	−1.00000	−0.0447214
$501$	−1.87128	−0.0836028
$502$	−6.43147	−0.287051
$503$	8.27135	0.368801	0.184401	−	0.982851i	$-0.440966\pi$
$503$	8.27135	0.368801	0.184401	+	0.982851i	$0.440966\pi$
$504$	4.30825	0.191905
$505$	−1.69976	−0.0756384
$506$	−2.07743	−0.0923531
$507$	−1.07826	−0.0478873
$508$	−10.7039	−0.474908
$509$	−8.78572	−0.389420	−0.194710	−	0.980861i	$-0.562377\pi$
$509$	−8.78572	−0.389420	−0.194710	+	0.980861i	$0.562377\pi$
$510$	−0.338487	−0.0149885
$511$	−1.44050	−0.0637239
$512$	1.00000	0.0441942
$513$	−1.47323	−0.0650446
$514$	12.0417	0.531138
$515$	3.34194	0.147263
$516$	−0.807457	−0.0355463
$517$	12.3740	0.544206
$518$	−10.4174	−0.457715
$519$	−1.34430	−0.0590081
$520$	−1.32450	−0.0580832
$521$	19.1244	0.837854	0.418927	−	0.908020i	$-0.362406\pi$
$521$	19.1244	0.837854	0.418927	+	0.908020i	$0.362406\pi$
$522$	−12.8759	−0.563563
$523$	36.8468	1.61120	0.805598	−	0.592463i	$-0.201844\pi$
$523$	36.8468	1.61120	0.805598	+	0.592463i	$0.201844\pi$
$524$	−7.61936	−0.332853
$525$	−0.138118	−0.00602797
$526$	−17.0967	−0.745451
$527$	−27.5385	−1.19960
$528$	−0.0958822	−0.00417274
$529$	−18.6843	−0.812360
$530$	−3.14683	−0.136690
$531$	31.6701	1.37437
$532$	−3.69453	−0.160178
$533$	−7.22513	−0.312955
$534$	−0.0455996	−0.00197329
$535$	−12.3735	−0.534953
$536$	−3.40263	−0.146971
$537$	−2.43132	−0.104919
$538$	−20.2095	−0.871293
$539$	4.92496	0.212133
$540$	0.574412	0.0247188
$541$	18.2482	0.784550	0.392275	−	0.919848i	$-0.371688\pi$
$541$	18.2482	0.784550	0.392275	+	0.919848i	$0.371688\pi$
$542$	14.3186	0.615036
$543$	−0.00193140	−8.28842e−5 0
$544$	3.53024	0.151358
$545$	−6.88448	−0.294899
$546$	−0.182938	−0.00782901
$547$	−1.99722	−0.0853950	−0.0426975	−	0.999088i	$-0.513595\pi$
$547$	−1.99722	−0.0853950	−0.0426975	+	0.999088i	$0.513595\pi$
$548$	−13.9644	−0.596531
$549$	−11.5008	−0.490841
$550$	−1.00000	−0.0426401
$551$	11.0417	0.470392
$552$	0.199189	0.00847804
$553$	−20.8109	−0.884968
$554$	−7.60571	−0.323136
$555$	−0.693402	−0.0294333
$556$	−13.9164	−0.590185
$557$	0.260168	0.0110237	0.00551184	−	0.999985i	$-0.498246\pi$
$557$	0.260168	0.0110237	0.00551184	+	0.999985i	$0.498246\pi$
$558$	23.3305	0.987659
$559$	−11.1541	−0.471767
$560$	1.44050	0.0608722
$561$	−0.338487	−0.0142910
$562$	−17.8648	−0.753580
$563$	23.6024	0.994721	0.497361	−	0.867544i	$-0.334303\pi$
$563$	23.6024	0.994721	0.497361	+	0.867544i	$0.334303\pi$
$564$	−1.18644	−0.0499583
$565$	0.833967	0.0350853
$566$	−1.04584	−0.0439599
$567$	−12.8454	−0.539457
$568$	9.28419	0.389556
$569$	−3.83269	−0.160675	−0.0803374	−	0.996768i	$-0.525600\pi$
$569$	−3.83269	−0.160675	−0.0803374	+	0.996768i	$0.525600\pi$
$570$	−0.245915	−0.0103002
$571$	20.0363	0.838493	0.419246	−	0.907873i	$-0.362294\pi$
$571$	20.0363	0.838493	0.419246	+	0.907873i	$0.362294\pi$
$572$	−1.32450	−0.0553801
$573$	−0.653314	−0.0272926
$574$	7.85790	0.327982
$575$	2.07743	0.0866349
$576$	−2.99081	−0.124617
$577$	−6.12066	−0.254806	−0.127403	−	0.991851i	$-0.540664\pi$
$577$	−6.12066	−0.254806	−0.127403	+	0.991851i	$0.540664\pi$
$578$	−4.53740	−0.188731
$579$	−0.541371	−0.0224986
$580$	−4.30516	−0.178762
$581$	12.8490	0.533066
$582$	−1.19882	−0.0496928
$583$	−3.14683	−0.130329
$584$	1.00000	0.0413803
$585$	3.96132	0.163781
$586$	−13.0740	−0.540082
$587$	−11.1017	−0.458214	−0.229107	−	0.973401i	$-0.573581\pi$
$587$	−11.1017	−0.458214	−0.229107	+	0.973401i	$0.573581\pi$
$588$	−0.472217	−0.0194739
$589$	−20.0070	−0.824374
$590$	10.5892	0.435949
$591$	0.883146	0.0363278
$592$	7.23181	0.297226
$593$	33.2335	1.36474	0.682368	−	0.731009i	$-0.260950\pi$
$593$	33.2335	1.36474	0.682368	+	0.731009i	$0.260950\pi$
$594$	0.574412	0.0235684
$595$	5.08531	0.208477
$596$	15.8575	0.649548
$597$	−1.97362	−0.0807749
$598$	2.75156	0.112520
$599$	−16.9230	−0.691454	−0.345727	−	0.938335i	$-0.612368\pi$
$599$	−16.9230	−0.691454	−0.345727	+	0.938335i	$0.612368\pi$
$600$	0.0958822	0.00391438
$601$	−13.3540	−0.544720	−0.272360	−	0.962195i	$-0.587804\pi$
$601$	−13.3540	−0.544720	−0.272360	+	0.962195i	$0.587804\pi$
$602$	12.1309	0.494420
$603$	10.1766	0.414423
$604$	−13.7010	−0.557486
$605$	−1.00000	−0.0406558
$606$	0.162977	0.00662049
$607$	−11.8019	−0.479024	−0.239512	−	0.970893i	$-0.576987\pi$
$607$	−11.8019	−0.479024	−0.239512	+	0.970893i	$0.576987\pi$
$608$	2.56476	0.104015
$609$	−0.594621	−0.0240953
$610$	−3.84537	−0.155695
$611$	−16.3893	−0.663041
$612$	−10.5583	−0.426793
$613$	33.2603	1.34337	0.671685	−	0.740836i	$-0.265571\pi$
$613$	33.2603	1.34337	0.671685	+	0.740836i	$0.265571\pi$
$614$	7.44048	0.300273
$615$	0.523036	0.0210909
$616$	1.44050	0.0580393
$617$	31.9161	1.28489	0.642446	−	0.766331i	$-0.277920\pi$
$617$	31.9161	1.28489	0.642446	+	0.766331i	$0.277920\pi$
$618$	−0.320433	−0.0128897
$619$	0.109763	0.00441175	0.00220588	−	0.999998i	$-0.499298\pi$
$619$	0.109763	0.00441175	0.00220588	+	0.999998i	$0.499298\pi$
$620$	7.80074	0.313285
$621$	−1.19330	−0.0478856
$622$	−2.58043	−0.103466
$623$	0.685071	0.0274468
$624$	0.126996	0.00508391
$625$	1.00000	0.0400000
$626$	−4.03133	−0.161124
$627$	−0.245915	−0.00982089
$628$	−8.89200	−0.354829
$629$	25.5300	1.01795
$630$	−4.30825	−0.171645
$631$	−12.3333	−0.490983	−0.245491	−	0.969399i	$-0.578949\pi$
$631$	−12.3333	−0.490983	−0.245491	+	0.969399i	$0.578949\pi$
$632$	14.4470	0.574670
$633$	1.98267	0.0788041
$634$	9.74135	0.386879
$635$	10.7039	0.424770
$636$	0.301726	0.0119642
$637$	−6.52312	−0.258455
$638$	−4.30516	−0.170443
$639$	−27.7672	−1.09845
$640$	−1.00000	−0.0395285
$641$	33.5633	1.32567	0.662836	−	0.748765i	$-0.269353\pi$
$641$	33.5633	1.32567	0.662836	+	0.748765i	$0.269353\pi$
$642$	1.18640	0.0468234
$643$	14.1050	0.556246	0.278123	−	0.960545i	$-0.410288\pi$
$643$	14.1050	0.556246	0.278123	+	0.960545i	$0.410288\pi$
$644$	−2.99254	−0.117923
$645$	0.807457	0.0317936
$646$	9.05421	0.356233
$647$	−15.2073	−0.597861	−0.298930	−	0.954275i	$-0.596630\pi$
$647$	−15.2073	−0.597861	−0.298930	+	0.954275i	$0.596630\pi$
$648$	8.91734	0.350306
$649$	10.5892	0.415661
$650$	1.32450	0.0519512
$651$	1.07742	0.0422276
$652$	−10.5987	−0.415077
$653$	3.99948	0.156512	0.0782559	−	0.996933i	$-0.475065\pi$
$653$	3.99948	0.156512	0.0782559	+	0.996933i	$0.475065\pi$
$654$	0.660100	0.0258119
$655$	7.61936	0.297713
$656$	−5.45499	−0.212981
$657$	−2.99081	−0.116682
$658$	17.8247	0.694878
$659$	−19.0884	−0.743580	−0.371790	−	0.928317i	$-0.621256\pi$
$659$	−19.0884	−0.743580	−0.371790	+	0.928317i	$0.621256\pi$
$660$	0.0958822	0.00373221
$661$	27.0626	1.05261	0.526307	−	0.850294i	$-0.323576\pi$
$661$	27.0626	1.05261	0.526307	+	0.850294i	$0.323576\pi$
$662$	−8.92741	−0.346974
$663$	0.448327	0.0174116
$664$	−8.91982	−0.346156
$665$	3.69453	0.143268
$666$	−21.6290	−0.838105
$667$	8.94369	0.346301
$668$	−19.5165	−0.755115
$669$	0.351415	0.0135865
$670$	3.40263	0.131455
$671$	−3.84537	−0.148449
$672$	−0.138118	−0.00532803
$673$	−10.2506	−0.395132	−0.197566	−	0.980290i	$-0.563304\pi$
$673$	−10.2506	−0.395132	−0.197566	+	0.980290i	$0.563304\pi$
$674$	15.5038	0.597183
$675$	−0.574412	−0.0221091
$676$	−11.2457	−0.432527
$677$	29.0805	1.11765	0.558827	−	0.829284i	$-0.311252\pi$
$677$	29.0805	1.11765	0.558827	+	0.829284i	$0.311252\pi$
$678$	−0.0799627	−0.00307095
$679$	18.0106	0.691185
$680$	−3.53024	−0.135379
$681$	1.07541	0.0412097
$682$	7.80074	0.298706
$683$	45.1188	1.72642	0.863212	−	0.504842i	$-0.168449\pi$
$683$	45.1188	1.72642	0.863212	+	0.504842i	$0.168449\pi$
$684$	−7.67069	−0.293296
$685$	13.9644	0.533554
$686$	17.1779	0.655855
$687$	0.445265	0.0169879
$688$	−8.42134	−0.321061
$689$	4.16798	0.158787
$690$	−0.199189	−0.00758299
$691$	6.70762	0.255170	0.127585	−	0.991828i	$-0.459277\pi$
$691$	6.70762	0.255170	0.127585	+	0.991828i	$0.459277\pi$
$692$	−14.0203	−0.532972
$693$	−4.30825	−0.163657
$694$	17.8682	0.678269
$695$	13.9164	0.527877
$696$	0.412789	0.0156467
$697$	−19.2574	−0.729427
$698$	9.18266	0.347569
$699$	−1.05883	−0.0400486
$700$	−1.44050	−0.0544457
$701$	6.28780	0.237487	0.118744	−	0.992925i	$-0.462113\pi$
$701$	6.28780	0.237487	0.118744	+	0.992925i	$0.462113\pi$
$702$	−0.760809	−0.0287149
$703$	18.5478	0.699545
$704$	−1.00000	−0.0376889
$705$	1.18644	0.0446840
$706$	−17.5887	−0.661961
$707$	−2.44850	−0.0920855
$708$	−1.01531	−0.0381578
$709$	−36.4120	−1.36748	−0.683740	−	0.729726i	$-0.739648\pi$
$709$	−36.4120	−1.36748	−0.683740	+	0.729726i	$0.739648\pi$
$710$	−9.28419	−0.348429
$711$	−43.2081	−1.62043
$712$	−0.475579	−0.0178231
$713$	−16.2055	−0.606901
$714$	−0.487591	−0.0182476
$715$	1.32450	0.0495335
$716$	−25.3574	−0.947650
$717$	0.147876	0.00552253
$718$	−5.12767	−0.191363
$719$	−52.3812	−1.95349	−0.976744	−	0.214410i	$-0.931217\pi$
$719$	−52.3812	−1.95349	−0.976744	+	0.214410i	$0.931217\pi$
$720$	2.99081	0.111461
$721$	4.81406	0.179285
$722$	−12.4220	−0.462300
$723$	−0.634324	−0.0235908
$724$	−0.0201434	−0.000748625 0
$725$	4.30516	0.159890
$726$	0.0958822	0.00355852
$727$	−31.4862	−1.16776	−0.583880	−	0.811840i	$-0.698466\pi$
$727$	−31.4862	−1.16776	−0.583880	+	0.811840i	$0.698466\pi$
$728$	−1.90794	−0.0707130
$729$	−26.5048	−0.981660
$730$	−1.00000	−0.0370117
$731$	−29.7294	−1.09958
$732$	0.368703	0.0136277
$733$	−26.4233	−0.975968	−0.487984	−	0.872853i	$-0.662268\pi$
$733$	−26.4233	−0.975968	−0.487984	+	0.872853i	$0.662268\pi$
$734$	10.7767	0.397775
$735$	0.472217	0.0174180
$736$	2.07743	0.0765752
$737$	3.40263	0.125337
$738$	16.3148	0.600556
$739$	32.7807	1.20586	0.602928	−	0.797795i	$-0.294001\pi$
$739$	32.7807	1.20586	0.602928	+	0.797795i	$0.294001\pi$
$740$	−7.23181	−0.265847
$741$	0.325714	0.0119654
$742$	−4.53301	−0.166412
$743$	28.0442	1.02884	0.514421	−	0.857538i	$-0.328007\pi$
$743$	28.0442	1.02884	0.514421	+	0.857538i	$0.328007\pi$
$744$	−0.747953	−0.0274213
$745$	−15.8575	−0.580974
$746$	−25.2009	−0.922669
$747$	26.6775	0.976077
$748$	−3.53024	−0.129078
$749$	−17.8240	−0.651275
$750$	−0.0958822	−0.00350112
$751$	−1.94550	−0.0709923	−0.0354961	−	0.999370i	$-0.511301\pi$
$751$	−1.94550	−0.0709923	−0.0354961	+	0.999370i	$0.511301\pi$
$752$	−12.3740	−0.451232
$753$	−0.616664	−0.0224725
$754$	5.70219	0.207661
$755$	13.7010	0.498630
$756$	0.827440	0.0300937
$757$	32.1692	1.16921	0.584604	−	0.811319i	$-0.301250\pi$
$757$	32.1692	1.16921	0.584604	+	0.811319i	$0.301250\pi$
$758$	12.7580	0.463390
$759$	−0.199189	−0.00723010
$760$	−2.56476	−0.0930335
$761$	4.61814	0.167408	0.0837038	−	0.996491i	$-0.473325\pi$
$761$	4.61814	0.167408	0.0837038	+	0.996491i	$0.473325\pi$
$762$	−1.02631	−0.0371793
$763$	−9.91709	−0.359023
$764$	−6.81372	−0.246512
$765$	10.5583	0.381735
$766$	18.0362	0.651675
$767$	−14.0253	−0.506426
$768$	0.0958822	0.00345985
$769$	1.71310	0.0617761	0.0308881	−	0.999523i	$-0.490166\pi$
$769$	1.71310	0.0617761	0.0308881	+	0.999523i	$0.490166\pi$
$770$	−1.44050	−0.0519120
$771$	1.15459	0.0415815
$772$	−5.64621	−0.203211
$773$	46.4594	1.67103	0.835513	−	0.549470i	$-0.185170\pi$
$773$	46.4594	1.67103	0.835513	+	0.549470i	$0.185170\pi$
$774$	25.1866	0.905314
$775$	−7.80074	−0.280211
$776$	−12.5031	−0.448834
$777$	−0.998845	−0.0358334
$778$	−16.0406	−0.575083
$779$	−13.9907	−0.501269
$780$	−0.126996	−0.00454719
$781$	−9.28419	−0.332214
$782$	7.33384	0.262257
$783$	−2.47294	−0.0883756
$784$	−4.92496	−0.175892
$785$	8.89200	0.317369
$786$	−0.730562	−0.0260583
$787$	−26.0662	−0.929161	−0.464580	−	0.885531i	$-0.653795\pi$
$787$	−26.0662	−0.929161	−0.464580	+	0.885531i	$0.653795\pi$
$788$	9.21074	0.328119
$789$	−1.63927	−0.0583595
$790$	−14.4470	−0.514001
$791$	1.20133	0.0427143
$792$	2.99081	0.106274
$793$	5.09320	0.180865
$794$	−19.7435	−0.700671
$795$	−0.301726	−0.0107011
$796$	−20.5838	−0.729574
$797$	−15.0539	−0.533238	−0.266619	−	0.963802i	$-0.585906\pi$
$797$	−15.0539	−0.533238	−0.266619	+	0.963802i	$0.585906\pi$
$798$	−0.354240	−0.0125399
$799$	−43.6831	−1.54540
$800$	1.00000	0.0353553
$801$	1.42236	0.0502568
$802$	−28.4689	−1.00527
$803$	−1.00000	−0.0352892
$804$	−0.326251	−0.0115060
$805$	2.99254	0.105473
$806$	−10.3321	−0.363932
$807$	−1.93773	−0.0682114
$808$	1.69976	0.0597974
$809$	39.7522	1.39761	0.698807	−	0.715311i	$-0.253715\pi$
$809$	39.7522	1.39761	0.698807	+	0.715311i	$0.253715\pi$
$810$	−8.91734	−0.313324
$811$	12.2039	0.428537	0.214268	−	0.976775i	$-0.431263\pi$
$811$	12.2039	0.428537	0.214268	+	0.976775i	$0.431263\pi$
$812$	−6.20158	−0.217633
$813$	1.37290	0.0481496
$814$	−7.23181	−0.253475
$815$	10.5987	0.371256
$816$	0.338487	0.0118494
$817$	−21.5987	−0.755643
$818$	17.0882	0.597475
$819$	5.70628	0.199394
$820$	5.45499	0.190496
$821$	−1.65630	−0.0578051	−0.0289026	−	0.999582i	$-0.509201\pi$
$821$	−1.65630	−0.0578051	−0.0289026	+	0.999582i	$0.509201\pi$
$822$	−1.33894	−0.0467010
$823$	25.1233	0.875745	0.437872	−	0.899037i	$-0.355732\pi$
$823$	25.1233	0.875745	0.437872	+	0.899037i	$0.355732\pi$
$824$	−3.34194	−0.116422
$825$	−0.0958822	−0.00333819
$826$	15.2537	0.530743
$827$	−13.7105	−0.476760	−0.238380	−	0.971172i	$-0.576616\pi$
$827$	−13.7105	−0.476760	−0.238380	+	0.971172i	$0.576616\pi$
$828$	−6.21320	−0.215924
$829$	−0.522408	−0.0181440	−0.00907198	−	0.999959i	$-0.502888\pi$
$829$	−0.522408	−0.0181440	−0.00907198	+	0.999959i	$0.502888\pi$
$830$	8.91982	0.309612
$831$	−0.729253	−0.0252975
$832$	1.32450	0.0459188
$833$	−17.3863	−0.602400
$834$	−1.33433	−0.0462041
$835$	19.5165	0.675396
$836$	−2.56476	−0.0887040
$837$	4.48084	0.154880
$838$	23.2852	0.804375
$839$	3.88443	0.134105	0.0670527	−	0.997749i	$-0.478640\pi$
$839$	3.88443	0.134105	0.0670527	+	0.997749i	$0.478640\pi$
$840$	0.138118	0.00476553
$841$	−10.4656	−0.360882
$842$	27.8801	0.960811
$843$	−1.71292	−0.0589959
$844$	20.6782	0.711772
$845$	11.2457	0.386864
$846$	37.0081	1.27237
$847$	−1.44050	−0.0494961
$848$	3.14683	0.108063
$849$	−0.100277	−0.00344151
$850$	3.53024	0.121086
$851$	15.0236	0.515002
$852$	0.890189	0.0304974
$853$	−33.3339	−1.14133	−0.570666	−	0.821182i	$-0.693315\pi$
$853$	−33.3339	−1.14133	−0.570666	+	0.821182i	$0.693315\pi$
$854$	−5.53926	−0.189549
$855$	7.67069	0.262332
$856$	12.3735	0.422917
$857$	30.5682	1.04419	0.522095	−	0.852888i	$-0.325151\pi$
$857$	30.5682	1.04419	0.522095	+	0.852888i	$0.325151\pi$
$858$	−0.126996	−0.00433557
$859$	−13.4618	−0.459312	−0.229656	−	0.973272i	$-0.573760\pi$
$859$	−13.4618	−0.459312	−0.229656	+	0.973272i	$0.573760\pi$
$860$	8.42134	0.287165
$861$	0.753433	0.0256769
$862$	29.5106	1.00513
$863$	14.1750	0.482522	0.241261	−	0.970460i	$-0.422439\pi$
$863$	14.1750	0.482522	0.241261	+	0.970460i	$0.422439\pi$
$864$	−0.574412	−0.0195419
$865$	14.0203	0.476704
$866$	−1.56969	−0.0533404
$867$	−0.435056	−0.0147753
$868$	11.2370	0.381407
$869$	−14.4470	−0.490081
$870$	−0.412789	−0.0139948
$871$	−4.50678	−0.152706
$872$	6.88448	0.233138
$873$	37.3942	1.26560
$874$	5.32811	0.180226
$875$	1.44050	0.0486977
$876$	0.0958822	0.00323956
$877$	−5.60448	−0.189250	−0.0946250	−	0.995513i	$-0.530165\pi$
$877$	−5.60448	−0.189250	−0.0946250	+	0.995513i	$0.530165\pi$
$878$	−39.7839	−1.34264
$879$	−1.25357	−0.0422817
$880$	1.00000	0.0337100
$881$	−33.3518	−1.12365	−0.561825	−	0.827256i	$-0.689900\pi$
$881$	−33.3518	−1.12365	−0.561825	+	0.827256i	$0.689900\pi$
$882$	14.7296	0.495972
$883$	−12.1822	−0.409964	−0.204982	−	0.978766i	$-0.565714\pi$
$883$	−12.1822	−0.409964	−0.204982	+	0.978766i	$0.565714\pi$
$884$	4.67580	0.157264
$885$	1.01531	0.0341293
$886$	−18.3085	−0.615086
$887$	−19.0128	−0.638387	−0.319194	−	0.947690i	$-0.603412\pi$
$887$	−19.0128	−0.638387	−0.319194	+	0.947690i	$0.603412\pi$
$888$	0.693402	0.0232691
$889$	15.4189	0.517134
$890$	0.475579	0.0159414
$891$	−8.91734	−0.298742
$892$	3.66507	0.122716
$893$	−31.7362	−1.06201
$894$	1.52045	0.0508515
$895$	25.3574	0.847604
$896$	−1.44050	−0.0481237
$897$	0.263826	0.00880888
$898$	11.2912	0.376794
$899$	−33.5835	−1.12007
$900$	−2.99081	−0.0996936
$901$	11.1091	0.370097
$902$	5.45499	0.181631
$903$	1.16314	0.0387069
$904$	−0.833967	−0.0277373
$905$	0.0201434	0.000669590 0
$906$	−1.31368	−0.0436442
$907$	41.7262	1.38549	0.692747	−	0.721181i	$-0.256400\pi$
$907$	41.7262	1.38549	0.692747	+	0.721181i	$0.256400\pi$
$908$	11.2159	0.372214
$909$	−5.08366	−0.168614
$910$	1.90794	0.0632476
$911$	29.5734	0.979809	0.489904	−	0.871776i	$-0.337032\pi$
$911$	29.5734	0.979809	0.489904	+	0.871776i	$0.337032\pi$
$912$	0.245915	0.00814305
$913$	8.91982	0.295203
$914$	16.9597	0.560979
$915$	−0.368703	−0.0121890
$916$	4.64387	0.153438
$917$	10.9757	0.362449
$918$	−2.02781	−0.0669278
$919$	−22.9245	−0.756209	−0.378104	−	0.925763i	$-0.623424\pi$
$919$	−22.9245	−0.756209	−0.378104	+	0.925763i	$0.623424\pi$
$920$	−2.07743	−0.0684909
$921$	0.713410	0.0235077
$922$	18.5691	0.611540
$923$	12.2969	0.404758
$924$	0.138118	0.00454376
$925$	7.23181	0.237781
$926$	4.55711	0.149756
$927$	9.99509	0.328282
$928$	4.30516	0.141324
$929$	7.87628	0.258412	0.129206	−	0.991618i	$-0.458757\pi$
$929$	7.87628	0.258412	0.129206	+	0.991618i	$0.458757\pi$
$930$	0.747953	0.0245263
$931$	−12.6313	−0.413975
$932$	−11.0430	−0.361726
$933$	−0.247418	−0.00810009
$934$	39.9216	1.30627
$935$	3.53024	0.115451
$936$	−3.96132	−0.129480
$937$	41.8462	1.36706	0.683529	−	0.729924i	$-0.260444\pi$
$937$	41.8462	1.36706	0.683529	+	0.729924i	$0.260444\pi$
$938$	4.90148	0.160039
$939$	−0.386533	−0.0126140
$940$	12.3740	0.403594
$941$	4.55672	0.148545	0.0742724	−	0.997238i	$-0.476337\pi$
$941$	4.55672	0.148545	0.0742724	+	0.997238i	$0.476337\pi$
$942$	−0.852585	−0.0277787
$943$	−11.3324	−0.369033
$944$	−10.5892	−0.344648
$945$	−0.827440	−0.0269166
$946$	8.42134	0.273802
$947$	17.9981	0.584859	0.292430	−	0.956287i	$-0.405536\pi$
$947$	17.9981	0.584859	0.292430	+	0.956287i	$0.405536\pi$
$948$	1.38521	0.0449895
$949$	1.32450	0.0429951
$950$	2.56476	0.0832117
$951$	0.934023	0.0302878
$952$	−5.08531	−0.164816
$953$	−27.7927	−0.900293	−0.450147	−	0.892955i	$-0.648628\pi$
$953$	−27.7927	−0.900293	−0.450147	+	0.892955i	$0.648628\pi$
$954$	−9.41157	−0.304711
$955$	6.81372	0.220487
$956$	1.54227	0.0498805
$957$	−0.412789	−0.0133436
$958$	4.77492	0.154271
$959$	20.1157	0.649571
$960$	−0.0958822	−0.00309459
$961$	29.8516	0.962954
$962$	9.57854	0.308824
$963$	−37.0067	−1.19253
$964$	−6.61566	−0.213076
$965$	5.64621	0.181758
$966$	−0.286931	−0.00923186
$967$	−3.50953	−0.112859	−0.0564295	−	0.998407i	$-0.517972\pi$
$967$	−3.50953	−0.112859	−0.0564295	+	0.998407i	$0.517972\pi$
$968$	1.00000	0.0321412
$969$	0.868138	0.0278886
$970$	12.5031	0.401449
$971$	16.5968	0.532617	0.266308	−	0.963888i	$-0.414196\pi$
$971$	16.5968	0.532617	0.266308	+	0.963888i	$0.414196\pi$
$972$	2.57825	0.0826974
$973$	20.0465	0.642661
$974$	−18.5014	−0.592823
$975$	0.126996	0.00406713
$976$	3.84537	0.123087
$977$	23.2767	0.744688	0.372344	−	0.928095i	$-0.378554\pi$
$977$	23.2767	0.744688	0.372344	+	0.928095i	$0.378554\pi$
$978$	−1.01623	−0.0324953
$979$	0.475579	0.0151996
$980$	4.92496	0.157322
$981$	−20.5902	−0.657393
$982$	−24.7563	−0.790005
$983$	−51.2548	−1.63477	−0.817387	−	0.576089i	$-0.804578\pi$
$983$	−51.2548	−1.63477	−0.817387	+	0.576089i	$0.804578\pi$
$984$	−0.523036	−0.0166738
$985$	−9.21074	−0.293479
$986$	15.1983	0.484011
$987$	1.70907	0.0544003
$988$	3.39702	0.108074
$989$	−17.4948	−0.556301
$990$	−2.99081	−0.0950541
$991$	−61.5260	−1.95444	−0.977219	−	0.212234i	$-0.931926\pi$
$991$	−61.5260	−1.95444	−0.977219	+	0.212234i	$0.931926\pi$
$992$	−7.80074	−0.247674
$993$	−0.855980	−0.0271637
$994$	−13.3739	−0.424193
$995$	20.5838	0.652550
$996$	−0.855253	−0.0270997
$997$	15.8701	0.502611	0.251305	−	0.967908i	$-0.419140\pi$
$997$	15.8701	0.502611	0.251305	+	0.967908i	$0.419140\pi$
$998$	−36.4784	−1.15470
$999$	−4.15404	−0.131428

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8030.2.a.bf.1.9	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8030.2.a.bf.1.9	✓	15	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:	\( N \)	\(=\)	\( 8030 = 2 \cdot 5 \cdot 11 \cdot 73 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8030.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +0.0958822 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.0958822 q^{6} -1.44050 q^{7} +1.00000 q^{8} -2.99081 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +0.0958822 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.0958822 q^{6} -1.44050 q^{7} +1.00000 q^{8} -2.99081 q^{9} -1.00000 q^{10} -1.00000 q^{11} +0.0958822 q^{12} +1.32450 q^{13} -1.44050 q^{14} -0.0958822 q^{15} +1.00000 q^{16} +3.53024 q^{17} -2.99081 q^{18} +2.56476 q^{19} -1.00000 q^{20} -0.138118 q^{21} -1.00000 q^{22} +2.07743 q^{23} +0.0958822 q^{24} +1.00000 q^{25} +1.32450 q^{26} -0.574412 q^{27} -1.44050 q^{28} +4.30516 q^{29} -0.0958822 q^{30} -7.80074 q^{31} +1.00000 q^{32} -0.0958822 q^{33} +3.53024 q^{34} +1.44050 q^{35} -2.99081 q^{36} +7.23181 q^{37} +2.56476 q^{38} +0.126996 q^{39} -1.00000 q^{40} -5.45499 q^{41} -0.138118 q^{42} -8.42134 q^{43} -1.00000 q^{44} +2.99081 q^{45} +2.07743 q^{46} -12.3740 q^{47} +0.0958822 q^{48} -4.92496 q^{49} +1.00000 q^{50} +0.338487 q^{51} +1.32450 q^{52} +3.14683 q^{53} -0.574412 q^{54} +1.00000 q^{55} -1.44050 q^{56} +0.245915 q^{57} +4.30516 q^{58} -10.5892 q^{59} -0.0958822 q^{60} +3.84537 q^{61} -7.80074 q^{62} +4.30825 q^{63} +1.00000 q^{64} -1.32450 q^{65} -0.0958822 q^{66} -3.40263 q^{67} +3.53024 q^{68} +0.199189 q^{69} +1.44050 q^{70} +9.28419 q^{71} -2.99081 q^{72} +1.00000 q^{73} +7.23181 q^{74} +0.0958822 q^{75} +2.56476 q^{76} +1.44050 q^{77} +0.126996 q^{78} +14.4470 q^{79} -1.00000 q^{80} +8.91734 q^{81} -5.45499 q^{82} -8.91982 q^{83} -0.138118 q^{84} -3.53024 q^{85} -8.42134 q^{86} +0.412789 q^{87} -1.00000 q^{88} -0.475579 q^{89} +2.99081 q^{90} -1.90794 q^{91} +2.07743 q^{92} -0.747953 q^{93} -12.3740 q^{94} -2.56476 q^{95} +0.0958822 q^{96} -12.5031 q^{97} -4.92496 q^{98} +2.99081 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 15 q^{2} - 4 q^{3} + 15 q^{4} - 15 q^{5} - 4 q^{6} - 6 q^{7} + 15 q^{8} + 17 q^{9}+O(q^{10}) \) 15 * q + 15 * q^2 - 4 * q^3 + 15 * q^4 - 15 * q^5 - 4 * q^6 - 6 * q^7 + 15 * q^8 + 17 * q^9	\( 15 q + 15 q^{2} - 4 q^{3} + 15 q^{4} - 15 q^{5} - 4 q^{6} - 6 q^{7} + 15 q^{8} + 17 q^{9} - 15 q^{10} - 15 q^{11} - 4 q^{12} - 6 q^{13} - 6 q^{14} + 4 q^{15} + 15 q^{16} + 2 q^{17} + 17 q^{18} - 8 q^{19} - 15 q^{20} - 17 q^{21} - 15 q^{22} - 4 q^{23} - 4 q^{24} + 15 q^{25} - 6 q^{26} - 19 q^{27} - 6 q^{28} - 13 q^{29} + 4 q^{30} - 20 q^{31} + 15 q^{32} + 4 q^{33} + 2 q^{34} + 6 q^{35} + 17 q^{36} - 15 q^{37} - 8 q^{38} - 11 q^{39} - 15 q^{40} + 2 q^{41} - 17 q^{42} - 26 q^{43} - 15 q^{44} - 17 q^{45} - 4 q^{46} - 14 q^{47} - 4 q^{48} + 11 q^{49} + 15 q^{50} - 39 q^{51} - 6 q^{52} - 21 q^{53} - 19 q^{54} + 15 q^{55} - 6 q^{56} + q^{57} - 13 q^{58} - 14 q^{59} + 4 q^{60} - 45 q^{61} - 20 q^{62} - 17 q^{63} + 15 q^{64} + 6 q^{65} + 4 q^{66} - 10 q^{67} + 2 q^{68} - 23 q^{69} + 6 q^{70} - 9 q^{71} + 17 q^{72} + 15 q^{73} - 15 q^{74} - 4 q^{75} - 8 q^{76} + 6 q^{77} - 11 q^{78} - 26 q^{79} - 15 q^{80} + 15 q^{81} + 2 q^{82} - 30 q^{83} - 17 q^{84} - 2 q^{85} - 26 q^{86} - 14 q^{87} - 15 q^{88} + 10 q^{89} - 17 q^{90} - 17 q^{91} - 4 q^{92} - 8 q^{93} - 14 q^{94} + 8 q^{95} - 4 q^{96} - 27 q^{97} + 11 q^{98} - 17 q^{99}+O(q^{100}) \) 15 * q + 15 * q^2 - 4 * q^3 + 15 * q^4 - 15 * q^5 - 4 * q^6 - 6 * q^7 + 15 * q^8 + 17 * q^9 - 15 * q^10 - 15 * q^11 - 4 * q^12 - 6 * q^13 - 6 * q^14 + 4 * q^15 + 15 * q^16 + 2 * q^17 + 17 * q^18 - 8 * q^19 - 15 * q^20 - 17 * q^21 - 15 * q^22 - 4 * q^23 - 4 * q^24 + 15 * q^25 - 6 * q^26 - 19 * q^27 - 6 * q^28 - 13 * q^29 + 4 * q^30 - 20 * q^31 + 15 * q^32 + 4 * q^33 + 2 * q^34 + 6 * q^35 + 17 * q^36 - 15 * q^37 - 8 * q^38 - 11 * q^39 - 15 * q^40 + 2 * q^41 - 17 * q^42 - 26 * q^43 - 15 * q^44 - 17 * q^45 - 4 * q^46 - 14 * q^47 - 4 * q^48 + 11 * q^49 + 15 * q^50 - 39 * q^51 - 6 * q^52 - 21 * q^53 - 19 * q^54 + 15 * q^55 - 6 * q^56 + q^57 - 13 * q^58 - 14 * q^59 + 4 * q^60 - 45 * q^61 - 20 * q^62 - 17 * q^63 + 15 * q^64 + 6 * q^65 + 4 * q^66 - 10 * q^67 + 2 * q^68 - 23 * q^69 + 6 * q^70 - 9 * q^71 + 17 * q^72 + 15 * q^73 - 15 * q^74 - 4 * q^75 - 8 * q^76 + 6 * q^77 - 11 * q^78 - 26 * q^79 - 15 * q^80 + 15 * q^81 + 2 * q^82 - 30 * q^83 - 17 * q^84 - 2 * q^85 - 26 * q^86 - 14 * q^87 - 15 * q^88 + 10 * q^89 - 17 * q^90 - 17 * q^91 - 4 * q^92 - 8 * q^93 - 14 * q^94 + 8 * q^95 - 4 * q^96 - 27 * q^97 + 11 * q^98 - 17 * q^99

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