LMFDB - Embedded newform 8002.2.a.d.1.30

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8002,2,Mod(1,8002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8002, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8002 = 2 \cdot 4001 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8002.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:	$63.8962916974$
Analytic rank:	$1$
Dimension:	$69$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.30
Character	$\chi$	$=$	8002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -0.809311 q^{3} +1.00000 q^{4} -2.58325 q^{5} -0.809311 q^{6} +4.23385 q^{7} +1.00000 q^{8} -2.34502 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -0.809311 q^{3} +1.00000 q^{4} -2.58325 q^{5} -0.809311 q^{6} +4.23385 q^{7} +1.00000 q^{8} -2.34502 q^{9} -2.58325 q^{10} -3.30942 q^{11} -0.809311 q^{12} -3.41240 q^{13} +4.23385 q^{14} +2.09065 q^{15} +1.00000 q^{16} +5.54868 q^{17} -2.34502 q^{18} +1.21630 q^{19} -2.58325 q^{20} -3.42650 q^{21} -3.30942 q^{22} +3.43692 q^{23} -0.809311 q^{24} +1.67319 q^{25} -3.41240 q^{26} +4.32578 q^{27} +4.23385 q^{28} -6.64984 q^{29} +2.09065 q^{30} +5.40837 q^{31} +1.00000 q^{32} +2.67835 q^{33} +5.54868 q^{34} -10.9371 q^{35} -2.34502 q^{36} -10.8302 q^{37} +1.21630 q^{38} +2.76169 q^{39} -2.58325 q^{40} +10.6162 q^{41} -3.42650 q^{42} -0.720866 q^{43} -3.30942 q^{44} +6.05776 q^{45} +3.43692 q^{46} -7.28948 q^{47} -0.809311 q^{48} +10.9254 q^{49} +1.67319 q^{50} -4.49061 q^{51} -3.41240 q^{52} -5.13163 q^{53} +4.32578 q^{54} +8.54907 q^{55} +4.23385 q^{56} -0.984362 q^{57} -6.64984 q^{58} +5.73051 q^{59} +2.09065 q^{60} +2.13018 q^{61} +5.40837 q^{62} -9.92843 q^{63} +1.00000 q^{64} +8.81508 q^{65} +2.67835 q^{66} -2.44957 q^{67} +5.54868 q^{68} -2.78154 q^{69} -10.9371 q^{70} +9.47194 q^{71} -2.34502 q^{72} +10.1104 q^{73} -10.8302 q^{74} -1.35413 q^{75} +1.21630 q^{76} -14.0116 q^{77} +2.76169 q^{78} -13.3426 q^{79} -2.58325 q^{80} +3.53415 q^{81} +10.6162 q^{82} -12.1663 q^{83} -3.42650 q^{84} -14.3336 q^{85} -0.720866 q^{86} +5.38179 q^{87} -3.30942 q^{88} -9.83140 q^{89} +6.05776 q^{90} -14.4476 q^{91} +3.43692 q^{92} -4.37705 q^{93} -7.28948 q^{94} -3.14200 q^{95} -0.809311 q^{96} -12.9354 q^{97} +10.9254 q^{98} +7.76065 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9}+O(q^{10}) $ 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9} - 33 q^{10} - 30 q^{11} - 25 q^{12} - 58 q^{13} - 19 q^{14} + 2 q^{15} + 69 q^{16} - 80 q^{17} + 54 q^{18} - 40 q^{19} - 33 q^{20} - 32 q^{21} - 30 q^{22} - 45 q^{23} - 25 q^{24} + 42 q^{25} - 58 q^{26} - 76 q^{27} - 19 q^{28} - 44 q^{29} + 2 q^{30} - 12 q^{31} + 69 q^{32} - 41 q^{33} - 80 q^{34} - 49 q^{35} + 54 q^{36} - 47 q^{37} - 40 q^{38} - 14 q^{39} - 33 q^{40} - 94 q^{41} - 32 q^{42} - 10 q^{43} - 30 q^{44} - 89 q^{45} - 45 q^{46} - 85 q^{47} - 25 q^{48} + 32 q^{49} + 42 q^{50} - 10 q^{51} - 58 q^{52} - 41 q^{53} - 76 q^{54} - 27 q^{55} - 19 q^{56} - 72 q^{57} - 44 q^{58} - 75 q^{59} + 2 q^{60} - 98 q^{61} - 12 q^{62} - 61 q^{63} + 69 q^{64} - 47 q^{65} - 41 q^{66} - 22 q^{67} - 80 q^{68} - 74 q^{69} - 49 q^{70} - 22 q^{71} + 54 q^{72} - 129 q^{73} - 47 q^{74} - 106 q^{75} - 40 q^{76} - 108 q^{77} - 14 q^{78} + 21 q^{79} - 33 q^{80} + 13 q^{81} - 94 q^{82} - 111 q^{83} - 32 q^{84} - 67 q^{85} - 10 q^{86} - 38 q^{87} - 30 q^{88} - 112 q^{89} - 89 q^{90} - 55 q^{91} - 45 q^{92} - 90 q^{93} - 85 q^{94} - 38 q^{95} - 25 q^{96} - 98 q^{97} + 32 q^{98} - 51 q^{99}+O(q^{100}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9 - 33 * q^10 - 30 * q^11 - 25 * q^12 - 58 * q^13 - 19 * q^14 + 2 * q^15 + 69 * q^16 - 80 * q^17 + 54 * q^18 - 40 * q^19 - 33 * q^20 - 32 * q^21 - 30 * q^22 - 45 * q^23 - 25 * q^24 + 42 * q^25 - 58 * q^26 - 76 * q^27 - 19 * q^28 - 44 * q^29 + 2 * q^30 - 12 * q^31 + 69 * q^32 - 41 * q^33 - 80 * q^34 - 49 * q^35 + 54 * q^36 - 47 * q^37 - 40 * q^38 - 14 * q^39 - 33 * q^40 - 94 * q^41 - 32 * q^42 - 10 * q^43 - 30 * q^44 - 89 * q^45 - 45 * q^46 - 85 * q^47 - 25 * q^48 + 32 * q^49 + 42 * q^50 - 10 * q^51 - 58 * q^52 - 41 * q^53 - 76 * q^54 - 27 * q^55 - 19 * q^56 - 72 * q^57 - 44 * q^58 - 75 * q^59 + 2 * q^60 - 98 * q^61 - 12 * q^62 - 61 * q^63 + 69 * q^64 - 47 * q^65 - 41 * q^66 - 22 * q^67 - 80 * q^68 - 74 * q^69 - 49 * q^70 - 22 * q^71 + 54 * q^72 - 129 * q^73 - 47 * q^74 - 106 * q^75 - 40 * q^76 - 108 * q^77 - 14 * q^78 + 21 * q^79 - 33 * q^80 + 13 * q^81 - 94 * q^82 - 111 * q^83 - 32 * q^84 - 67 * q^85 - 10 * q^86 - 38 * q^87 - 30 * q^88 - 112 * q^89 - 89 * q^90 - 55 * q^91 - 45 * q^92 - 90 * q^93 - 85 * q^94 - 38 * q^95 - 25 * q^96 - 98 * q^97 + 32 * q^98 - 51 * 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
$3$	−0.809311	−0.467256	−0.233628	−	0.972326i	$-0.575060\pi$
$3$	−0.809311	−0.467256	−0.233628	+	0.972326i	$0.575060\pi$
$4$	1.00000	0.500000
$5$	−2.58325	−1.15526	−0.577632	−	0.816297i	$-0.696023\pi$
$5$	−2.58325	−1.15526	−0.577632	+	0.816297i	$0.696023\pi$
$6$	−0.809311	−0.330400
$7$	4.23385	1.60024	0.800122	−	0.599838i	$-0.204768\pi$
$7$	4.23385	1.60024	0.800122	+	0.599838i	$0.204768\pi$
$8$	1.00000	0.353553
$9$	−2.34502	−0.781672
$10$	−2.58325	−0.816896
$11$	−3.30942	−0.997829	−0.498914	−	0.866651i	$-0.666268\pi$
$11$	−3.30942	−0.997829	−0.498914	+	0.866651i	$0.666268\pi$
$12$	−0.809311	−0.233628
$13$	−3.41240	−0.946428	−0.473214	−	0.880947i	$-0.656906\pi$
$13$	−3.41240	−0.946428	−0.473214	+	0.880947i	$0.656906\pi$
$14$	4.23385	1.13154
$15$	2.09065	0.539804
$16$	1.00000	0.250000
$17$	5.54868	1.34575	0.672876	−	0.739755i	$-0.265059\pi$
$17$	5.54868	1.34575	0.672876	+	0.739755i	$0.265059\pi$
$18$	−2.34502	−0.552726
$19$	1.21630	0.279038	0.139519	−	0.990219i	$-0.455444\pi$
$19$	1.21630	0.279038	0.139519	+	0.990219i	$0.455444\pi$
$20$	−2.58325	−0.577632
$21$	−3.42650	−0.747723
$22$	−3.30942	−0.705572
$23$	3.43692	0.716648	0.358324	−	0.933597i	$-0.383348\pi$
$23$	3.43692	0.716648	0.358324	+	0.933597i	$0.383348\pi$
$24$	−0.809311	−0.165200
$25$	1.67319	0.334637
$26$	−3.41240	−0.669226
$27$	4.32578	0.832497
$28$	4.23385	0.800122
$29$	−6.64984	−1.23484	−0.617422	−	0.786632i	$-0.711823\pi$
$29$	−6.64984	−1.23484	−0.617422	+	0.786632i	$0.711823\pi$
$30$	2.09065	0.381699
$31$	5.40837	0.971371	0.485686	−	0.874134i	$-0.338570\pi$
$31$	5.40837	0.971371	0.485686	+	0.874134i	$0.338570\pi$
$32$	1.00000	0.176777
$33$	2.67835	0.466241
$34$	5.54868	0.951590
$35$	−10.9371	−1.84870
$36$	−2.34502	−0.390836
$37$	−10.8302	−1.78048	−0.890238	−	0.455496i	$-0.849462\pi$
$37$	−10.8302	−1.78048	−0.890238	+	0.455496i	$0.849462\pi$
$38$	1.21630	0.197309
$39$	2.76169	0.442224
$40$	−2.58325	−0.408448
$41$	10.6162	1.65797	0.828987	−	0.559268i	$-0.188918\pi$
$41$	10.6162	1.65797	0.828987	+	0.559268i	$0.188918\pi$
$42$	−3.42650	−0.528720
$43$	−0.720866	−0.109931	−0.0549656	−	0.998488i	$-0.517505\pi$
$43$	−0.720866	−0.109931	−0.0549656	+	0.998488i	$0.517505\pi$
$44$	−3.30942	−0.498914
$45$	6.05776	0.903038
$46$	3.43692	0.506747
$47$	−7.28948	−1.06328	−0.531640	−	0.846971i	$-0.678424\pi$
$47$	−7.28948	−1.06328	−0.531640	+	0.846971i	$0.678424\pi$
$48$	−0.809311	−0.116814
$49$	10.9254	1.56078
$50$	1.67319	0.236624
$51$	−4.49061	−0.628811
$52$	−3.41240	−0.473214
$53$	−5.13163	−0.704884	−0.352442	−	0.935834i	$-0.614649\pi$
$53$	−5.13163	−0.704884	−0.352442	+	0.935834i	$0.614649\pi$
$54$	4.32578	0.588664
$55$	8.54907	1.15276
$56$	4.23385	0.565771
$57$	−0.984362	−0.130382
$58$	−6.64984	−0.873167
$59$	5.73051	0.746048	0.373024	−	0.927822i	$-0.378321\pi$
$59$	5.73051	0.746048	0.373024	+	0.927822i	$0.378321\pi$
$60$	2.09065	0.269902
$61$	2.13018	0.272741	0.136371	−	0.990658i	$-0.456456\pi$
$61$	2.13018	0.272741	0.136371	+	0.990658i	$0.456456\pi$
$62$	5.40837	0.686863
$63$	−9.92843	−1.25087
$64$	1.00000	0.125000
$65$	8.81508	1.09338
$66$	2.67835	0.329682
$67$	−2.44957	−0.299263	−0.149632	−	0.988742i	$-0.547809\pi$
$67$	−2.44957	−0.299263	−0.149632	+	0.988742i	$0.547809\pi$
$68$	5.54868	0.672876
$69$	−2.78154	−0.334858
$70$	−10.9371	−1.30723
$71$	9.47194	1.12411	0.562056	−	0.827099i	$-0.310011\pi$
$71$	9.47194	1.12411	0.562056	+	0.827099i	$0.310011\pi$
$72$	−2.34502	−0.276363
$73$	10.1104	1.18333	0.591665	−	0.806184i	$-0.298471\pi$
$73$	10.1104	1.18333	0.591665	+	0.806184i	$0.298471\pi$
$74$	−10.8302	−1.25899
$75$	−1.35413	−0.156361
$76$	1.21630	0.139519
$77$	−14.0116	−1.59677
$78$	2.76169	0.312700
$79$	−13.3426	−1.50115	−0.750577	−	0.660783i	$-0.770225\pi$
$79$	−13.3426	−1.50115	−0.750577	+	0.660783i	$0.770225\pi$
$80$	−2.58325	−0.288816
$81$	3.53415	0.392683
$82$	10.6162	1.17236
$83$	−12.1663	−1.33542	−0.667712	−	0.744420i	$-0.732726\pi$
$83$	−12.1663	−1.33542	−0.667712	+	0.744420i	$0.732726\pi$
$84$	−3.42650	−0.373862
$85$	−14.3336	−1.55470
$86$	−0.720866	−0.0777330
$87$	5.38179	0.576989
$88$	−3.30942	−0.352786
$89$	−9.83140	−1.04213	−0.521063	−	0.853518i	$-0.674464\pi$
$89$	−9.83140	−1.04213	−0.521063	+	0.853518i	$0.674464\pi$
$90$	6.05776	0.638544
$91$	−14.4476	−1.51452
$92$	3.43692	0.358324
$93$	−4.37705	−0.453879
$94$	−7.28948	−0.751852
$95$	−3.14200	−0.322362
$96$	−0.809311	−0.0826000
$97$	−12.9354	−1.31340	−0.656698	−	0.754154i	$-0.728047\pi$
$97$	−12.9354	−1.31340	−0.656698	+	0.754154i	$0.728047\pi$
$98$	10.9254	1.10364
$99$	7.76065	0.779975
$100$	1.67319	0.167319
$101$	6.50406	0.647178	0.323589	−	0.946198i	$-0.395110\pi$
$101$	6.50406	0.647178	0.323589	+	0.946198i	$0.395110\pi$
$102$	−4.49061	−0.444636
$103$	−8.70264	−0.857497	−0.428748	−	0.903424i	$-0.641045\pi$
$103$	−8.70264	−0.857497	−0.428748	+	0.903424i	$0.641045\pi$
$104$	−3.41240	−0.334613
$105$	8.85150	0.863818
$106$	−5.13163	−0.498428
$107$	−16.1552	−1.56178	−0.780890	−	0.624669i	$-0.785234\pi$
$107$	−16.1552	−1.56178	−0.780890	+	0.624669i	$0.785234\pi$
$108$	4.32578	0.416248
$109$	7.42912	0.711581	0.355790	−	0.934566i	$-0.384212\pi$
$109$	7.42912	0.711581	0.355790	+	0.934566i	$0.384212\pi$
$110$	8.54907	0.815122
$111$	8.76501	0.831938
$112$	4.23385	0.400061
$113$	−3.45504	−0.325023	−0.162512	−	0.986707i	$-0.551960\pi$
$113$	−3.45504	−0.325023	−0.162512	+	0.986707i	$0.551960\pi$
$114$	−0.984362	−0.0921940
$115$	−8.87844	−0.827919
$116$	−6.64984	−0.617422
$117$	8.00212	0.739796
$118$	5.73051	0.527536
$119$	23.4922	2.15353
$120$	2.09065	0.190850
$121$	−0.0477143	−0.00433766
$122$	2.13018	0.192857
$123$	−8.59182	−0.774698
$124$	5.40837	0.485686
$125$	8.59400	0.768670
$126$	−9.92843	−0.884495
$127$	−2.97389	−0.263890	−0.131945	−	0.991257i	$-0.542122\pi$
$127$	−2.97389	−0.263890	−0.131945	+	0.991257i	$0.542122\pi$
$128$	1.00000	0.0883883
$129$	0.583405	0.0513660
$130$	8.81508	0.773133
$131$	−18.4341	−1.61060	−0.805298	−	0.592870i	$-0.797995\pi$
$131$	−18.4341	−1.61060	−0.805298	+	0.592870i	$0.797995\pi$
$132$	2.67835	0.233121
$133$	5.14961	0.446528
$134$	−2.44957	−0.211611
$135$	−11.1746	−0.961754
$136$	5.54868	0.475795
$137$	−20.2872	−1.73326	−0.866628	−	0.498954i	$-0.833718\pi$
$137$	−20.2872	−1.73326	−0.866628	+	0.498954i	$0.833718\pi$
$138$	−2.78154	−0.236780
$139$	6.67217	0.565926	0.282963	−	0.959131i	$-0.408683\pi$
$139$	6.67217	0.565926	0.282963	+	0.959131i	$0.408683\pi$
$140$	−10.9371	−0.924352
$141$	5.89945	0.496824
$142$	9.47194	0.794867
$143$	11.2931	0.944373
$144$	−2.34502	−0.195418
$145$	17.1782	1.42657
$146$	10.1104	0.836741
$147$	−8.84208	−0.729283
$148$	−10.8302	−0.890238
$149$	14.3864	1.17858	0.589291	−	0.807921i	$-0.299407\pi$
$149$	14.3864	1.17858	0.589291	+	0.807921i	$0.299407\pi$
$150$	−1.35413	−0.110564
$151$	7.27007	0.591630	0.295815	−	0.955245i	$-0.404409\pi$
$151$	7.27007	0.591630	0.295815	+	0.955245i	$0.404409\pi$
$152$	1.21630	0.0986547
$153$	−13.0117	−1.05194
$154$	−14.0116	−1.12909
$155$	−13.9712	−1.12219
$156$	2.76169	0.221112
$157$	11.6161	0.927064	0.463532	−	0.886080i	$-0.346582\pi$
$157$	11.6161	0.927064	0.463532	+	0.886080i	$0.346582\pi$
$158$	−13.3426	−1.06148
$159$	4.15309	0.329361
$160$	−2.58325	−0.204224
$161$	14.5514	1.14681
$162$	3.53415	0.277669
$163$	20.2555	1.58653	0.793267	−	0.608874i	$-0.208379\pi$
$163$	20.2555	1.58653	0.793267	+	0.608874i	$0.208379\pi$
$164$	10.6162	0.828987
$165$	−6.91886	−0.538632
$166$	−12.1663	−0.944287
$167$	−4.72686	−0.365776	−0.182888	−	0.983134i	$-0.558545\pi$
$167$	−4.72686	−0.365776	−0.182888	+	0.983134i	$0.558545\pi$
$168$	−3.42650	−0.264360
$169$	−1.35555	−0.104273
$170$	−14.3336	−1.09934
$171$	−2.85224	−0.218116
$172$	−0.720866	−0.0549656
$173$	−22.0573	−1.67699	−0.838493	−	0.544912i	$-0.816563\pi$
$173$	−22.0573	−1.67699	−0.838493	+	0.544912i	$0.816563\pi$
$174$	5.38179	0.407993
$175$	7.08401	0.535501
$176$	−3.30942	−0.249457
$177$	−4.63776	−0.348596
$178$	−9.83140	−0.736895
$179$	−16.1654	−1.20826	−0.604129	−	0.796886i	$-0.706479\pi$
$179$	−16.1654	−1.20826	−0.604129	+	0.796886i	$0.706479\pi$
$180$	6.05776	0.451519
$181$	−14.6037	−1.08548	−0.542742	−	0.839900i	$-0.682614\pi$
$181$	−14.6037	−1.08548	−0.542742	+	0.839900i	$0.682614\pi$
$182$	−14.4476	−1.07092
$183$	−1.72398	−0.127440
$184$	3.43692	0.253373
$185$	27.9772	2.05692
$186$	−4.37705	−0.320941
$187$	−18.3629	−1.34283
$188$	−7.28948	−0.531640
$189$	18.3147	1.33220
$190$	−3.14200	−0.227945
$191$	−18.8242	−1.36207	−0.681034	−	0.732252i	$-0.738470\pi$
$191$	−18.8242	−1.36207	−0.681034	+	0.732252i	$0.738470\pi$
$192$	−0.809311	−0.0584070
$193$	−0.588067	−0.0423300	−0.0211650	−	0.999776i	$-0.506738\pi$
$193$	−0.588067	−0.0423300	−0.0211650	+	0.999776i	$0.506738\pi$
$194$	−12.9354	−0.928711
$195$	−7.13414	−0.510886
$196$	10.9254	0.780389
$197$	−9.10419	−0.648647	−0.324323	−	0.945946i	$-0.605137\pi$
$197$	−9.10419	−0.648647	−0.324323	+	0.945946i	$0.605137\pi$
$198$	7.76065	0.551525
$199$	−13.0233	−0.923199	−0.461600	−	0.887088i	$-0.652724\pi$
$199$	−13.0233	−0.923199	−0.461600	+	0.887088i	$0.652724\pi$
$200$	1.67319	0.118312
$201$	1.98247	0.139832
$202$	6.50406	0.457624
$203$	−28.1544	−1.97605
$204$	−4.49061	−0.314405
$205$	−27.4243	−1.91540
$206$	−8.70264	−0.606342
$207$	−8.05964	−0.560184
$208$	−3.41240	−0.236607
$209$	−4.02524	−0.278432
$210$	8.85150	0.610812
$211$	19.8000	1.36309	0.681544	−	0.731777i	$-0.261309\pi$
$211$	19.8000	1.36309	0.681544	+	0.731777i	$0.261309\pi$
$212$	−5.13163	−0.352442
$213$	−7.66574	−0.525248
$214$	−16.1552	−1.10434
$215$	1.86218	0.127000
$216$	4.32578	0.294332
$217$	22.8982	1.55443
$218$	7.42912	0.503163
$219$	−8.18244	−0.552918
$220$	8.54907	0.576378
$221$	−18.9343	−1.27366
$222$	8.76501	0.588269
$223$	1.00228	0.0671177	0.0335588	−	0.999437i	$-0.489316\pi$
$223$	1.00228	0.0671177	0.0335588	+	0.999437i	$0.489316\pi$
$224$	4.23385	0.282886
$225$	−3.92365	−0.261577
$226$	−3.45504	−0.229826
$227$	11.8006	0.783232	0.391616	−	0.920129i	$-0.371916\pi$
$227$	11.8006	0.783232	0.391616	+	0.920129i	$0.371916\pi$
$228$	−0.984362	−0.0651910
$229$	16.0787	1.06251	0.531254	−	0.847213i	$-0.321721\pi$
$229$	16.0787	1.06251	0.531254	+	0.847213i	$0.321721\pi$
$230$	−8.87844	−0.585427
$231$	11.3397	0.746100
$232$	−6.64984	−0.436584
$233$	−26.0594	−1.70721	−0.853603	−	0.520923i	$-0.825588\pi$
$233$	−26.0594	−1.70721	−0.853603	+	0.520923i	$0.825588\pi$
$234$	8.00212	0.523115
$235$	18.8306	1.22837
$236$	5.73051	0.373024
$237$	10.7983	0.701423
$238$	23.4922	1.52278
$239$	−0.628156	−0.0406321	−0.0203160	−	0.999794i	$-0.506467\pi$
$239$	−0.628156	−0.0406321	−0.0203160	+	0.999794i	$0.506467\pi$
$240$	2.09065	0.134951
$241$	−8.50974	−0.548161	−0.274080	−	0.961707i	$-0.588373\pi$
$241$	−8.50974	−0.548161	−0.274080	+	0.961707i	$0.588373\pi$
$242$	−0.0477143	−0.00306719
$243$	−15.8376	−1.01598
$244$	2.13018	0.136371
$245$	−28.2232	−1.80311
$246$	−8.59182	−0.547794
$247$	−4.15049	−0.264089
$248$	5.40837	0.343432
$249$	9.84631	0.623984
$250$	8.59400	0.543532
$251$	−26.2134	−1.65458	−0.827288	−	0.561777i	$-0.810118\pi$
$251$	−26.2134	−1.65458	−0.827288	+	0.561777i	$0.810118\pi$
$252$	−9.92843	−0.625433
$253$	−11.3742	−0.715092
$254$	−2.97389	−0.186598
$255$	11.6004	0.726443
$256$	1.00000	0.0625000
$257$	7.30119	0.455436	0.227718	−	0.973727i	$-0.426874\pi$
$257$	7.30119	0.455436	0.227718	+	0.973727i	$0.426874\pi$
$258$	0.583405	0.0363212
$259$	−45.8534	−2.84919
$260$	8.81508	0.546688
$261$	15.5940	0.965244
$262$	−18.4341	−1.13886
$263$	5.38808	0.332243	0.166122	−	0.986105i	$-0.446876\pi$
$263$	5.38808	0.332243	0.166122	+	0.986105i	$0.446876\pi$
$264$	2.67835	0.164841
$265$	13.2563	0.814328
$266$	5.14961	0.315743
$267$	7.95666	0.486940
$268$	−2.44957	−0.149632
$269$	−14.0961	−0.859452	−0.429726	−	0.902959i	$-0.641390\pi$
$269$	−14.0961	−0.859452	−0.429726	+	0.902959i	$0.641390\pi$
$270$	−11.1746	−0.680063
$271$	−8.66597	−0.526420	−0.263210	−	0.964739i	$-0.584781\pi$
$271$	−8.66597	−0.526420	−0.263210	+	0.964739i	$0.584781\pi$
$272$	5.54868	0.336438
$273$	11.6926	0.707666
$274$	−20.2872	−1.22560
$275$	−5.53728	−0.333911
$276$	−2.78154	−0.167429
$277$	−8.86426	−0.532602	−0.266301	−	0.963890i	$-0.585801\pi$
$277$	−8.86426	−0.532602	−0.266301	+	0.963890i	$0.585801\pi$
$278$	6.67217	0.400170
$279$	−12.6827	−0.759294
$280$	−10.9371	−0.653616
$281$	13.1907	0.786890	0.393445	−	0.919348i	$-0.371283\pi$
$281$	13.1907	0.786890	0.393445	+	0.919348i	$0.371283\pi$
$282$	5.89945	0.351307
$283$	−0.0253660	−0.00150785	−0.000753926	−	1.00000i	$-0.500240\pi$
$283$	−0.0253660	−0.00150785	−0.000753926		1.00000i	$0.500240\pi$
$284$	9.47194	0.562056
$285$	2.54285	0.150626
$286$	11.2931	0.667773
$287$	44.9474	2.65316
$288$	−2.34502	−0.138181
$289$	13.7878	0.811049
$290$	17.1782	1.00874
$291$	10.4688	0.613692
$292$	10.1104	0.591665
$293$	20.4290	1.19347	0.596737	−	0.802437i	$-0.296464\pi$
$293$	20.4290	1.19347	0.596737	+	0.802437i	$0.296464\pi$
$294$	−8.84208	−0.515681
$295$	−14.8033	−0.861884
$296$	−10.8302	−0.629493
$297$	−14.3158	−0.830689
$298$	14.3864	0.833384
$299$	−11.7281	−0.678256
$300$	−1.35413	−0.0781806
$301$	−3.05204	−0.175917
$302$	7.27007	0.418345
$303$	−5.26381	−0.302398
$304$	1.21630	0.0697594
$305$	−5.50278	−0.315089
$306$	−13.0117	−0.743832
$307$	−32.5176	−1.85588	−0.927938	−	0.372735i	$-0.878420\pi$
$307$	−32.5176	−1.85588	−0.927938	+	0.372735i	$0.878420\pi$
$308$	−14.0116	−0.798384
$309$	7.04314	0.400670
$310$	−13.9712	−0.793509
$311$	20.3515	1.15403	0.577015	−	0.816734i	$-0.304218\pi$
$311$	20.3515	1.15403	0.577015	+	0.816734i	$0.304218\pi$
$312$	2.76169	0.156350
$313$	24.6415	1.39282	0.696409	−	0.717646i	$-0.254780\pi$
$313$	24.6415	1.39282	0.696409	+	0.717646i	$0.254780\pi$
$314$	11.6161	0.655533
$315$	25.6476	1.44508
$316$	−13.3426	−0.750577
$317$	−6.01236	−0.337688	−0.168844	−	0.985643i	$-0.554003\pi$
$317$	−6.01236	−0.337688	−0.168844	+	0.985643i	$0.554003\pi$
$318$	4.15309	0.232894
$319$	22.0071	1.23216
$320$	−2.58325	−0.144408
$321$	13.0746	0.729750
$322$	14.5514	0.810918
$323$	6.74884	0.375515
$324$	3.53415	0.196341
$325$	−5.70957	−0.316710
$326$	20.2555	1.12185
$327$	−6.01247	−0.332490
$328$	10.6162	0.586182
$329$	−30.8625	−1.70151
$330$	−6.91886	−0.380871
$331$	−10.0622	−0.553067	−0.276534	−	0.961004i	$-0.589186\pi$
$331$	−10.0622	−0.553067	−0.276534	+	0.961004i	$0.589186\pi$
$332$	−12.1663	−0.667712
$333$	25.3970	1.39175
$334$	−4.72686	−0.258642
$335$	6.32786	0.345728
$336$	−3.42650	−0.186931
$337$	1.28652	0.0700812	0.0350406	−	0.999386i	$-0.488844\pi$
$337$	1.28652	0.0700812	0.0350406	+	0.999386i	$0.488844\pi$
$338$	−1.35555	−0.0737324
$339$	2.79621	0.151869
$340$	−14.3336	−0.777350
$341$	−17.8986	−0.969262
$342$	−2.85224	−0.154231
$343$	16.6197	0.897382
$344$	−0.720866	−0.0388665
$345$	7.18542	0.386850
$346$	−22.0573	−1.18581
$347$	−26.5862	−1.42722	−0.713610	−	0.700544i	$-0.752941\pi$
$347$	−26.5862	−1.42722	−0.713610	+	0.700544i	$0.752941\pi$
$348$	5.38179	0.288494
$349$	23.8344	1.27582	0.637912	−	0.770109i	$-0.279798\pi$
$349$	23.8344	1.27582	0.637912	+	0.770109i	$0.279798\pi$
$350$	7.08401	0.378656
$351$	−14.7613	−0.787898
$352$	−3.30942	−0.176393
$353$	21.6078	1.15007	0.575033	−	0.818130i	$-0.304989\pi$
$353$	21.6078	1.15007	0.575033	+	0.818130i	$0.304989\pi$
$354$	−4.63776	−0.246494
$355$	−24.4684	−1.29865
$356$	−9.83140	−0.521063
$357$	−19.0125	−1.00625
$358$	−16.1654	−0.854368
$359$	−6.60774	−0.348743	−0.174371	−	0.984680i	$-0.555789\pi$
$359$	−6.60774	−0.348743	−0.174371	+	0.984680i	$0.555789\pi$
$360$	6.05776	0.319272
$361$	−17.5206	−0.922138
$362$	−14.6037	−0.767553
$363$	0.0386157	0.00202680
$364$	−14.4476	−0.757258
$365$	−26.1177	−1.36706
$366$	−1.72398	−0.0901137
$367$	−19.9687	−1.04236	−0.521180	−	0.853447i	$-0.674508\pi$
$367$	−19.9687	−1.04236	−0.521180	+	0.853447i	$0.674508\pi$
$368$	3.43692	0.179162
$369$	−24.8952	−1.29599
$370$	27.9772	1.45446
$371$	−21.7265	−1.12799
$372$	−4.37705	−0.226940
$373$	8.14640	0.421805	0.210902	−	0.977507i	$-0.432360\pi$
$373$	8.14640	0.421805	0.210902	+	0.977507i	$0.432360\pi$
$374$	−18.3629	−0.949524
$375$	−6.95521	−0.359166
$376$	−7.28948	−0.375926
$377$	22.6919	1.16869
$378$	18.3147	0.942006
$379$	2.98001	0.153073	0.0765364	−	0.997067i	$-0.475614\pi$
$379$	2.98001	0.153073	0.0765364	+	0.997067i	$0.475614\pi$
$380$	−3.14200	−0.161181
$381$	2.40680	0.123304
$382$	−18.8242	−0.963128
$383$	−10.0086	−0.511416	−0.255708	−	0.966754i	$-0.582309\pi$
$383$	−10.0086	−0.511416	−0.255708	+	0.966754i	$0.582309\pi$
$384$	−0.809311	−0.0413000
$385$	36.1955	1.84469
$386$	−0.588067	−0.0299318
$387$	1.69044	0.0859301
$388$	−12.9354	−0.656698
$389$	−4.81199	−0.243977	−0.121989	−	0.992531i	$-0.538927\pi$
$389$	−4.81199	−0.243977	−0.121989	+	0.992531i	$0.538927\pi$
$390$	−7.13414	−0.361251
$391$	19.0704	0.964431
$392$	10.9254	0.551818
$393$	14.9189	0.752561
$394$	−9.10419	−0.458663
$395$	34.4672	1.73423
$396$	7.76065	0.389987
$397$	18.3118	0.919043	0.459522	−	0.888167i	$-0.348021\pi$
$397$	18.3118	0.919043	0.459522	+	0.888167i	$0.348021\pi$
$398$	−13.0233	−0.652801
$399$	−4.16764	−0.208643
$400$	1.67319	0.0836593
$401$	−23.4496	−1.17101	−0.585507	−	0.810667i	$-0.699105\pi$
$401$	−23.4496	−1.17101	−0.585507	+	0.810667i	$0.699105\pi$
$402$	1.98247	0.0988765
$403$	−18.4555	−0.919333
$404$	6.50406	0.323589
$405$	−9.12959	−0.453653
$406$	−28.1544	−1.39728
$407$	35.8418	1.77661
$408$	−4.49061	−0.222318
$409$	−28.8880	−1.42842	−0.714209	−	0.699932i	$-0.753213\pi$
$409$	−28.8880	−1.42842	−0.714209	+	0.699932i	$0.753213\pi$
$410$	−27.4243	−1.35439
$411$	16.4187	0.809874
$412$	−8.70264	−0.428748
$413$	24.2621	1.19386
$414$	−8.05964	−0.396110
$415$	31.4286	1.54277
$416$	−3.41240	−0.167306
$417$	−5.39986	−0.264432
$418$	−4.02524	−0.196881
$419$	−4.28952	−0.209557	−0.104778	−	0.994496i	$-0.533413\pi$
$419$	−4.28952	−0.209557	−0.104778	+	0.994496i	$0.533413\pi$
$420$	8.85150	0.431909
$421$	40.0085	1.94989	0.974946	−	0.222440i	$-0.0714020\pi$
$421$	40.0085	1.94989	0.974946	+	0.222440i	$0.0714020\pi$
$422$	19.8000	0.963849
$423$	17.0939	0.831136
$424$	−5.13163	−0.249214
$425$	9.28397	0.450339
$426$	−7.66574	−0.371406
$427$	9.01884	0.436452
$428$	−16.1552	−0.780890
$429$	−9.13960	−0.441264
$430$	1.86218	0.0898023
$431$	19.9115	0.959104	0.479552	−	0.877513i	$-0.340799\pi$
$431$	19.9115	0.959104	0.479552	+	0.877513i	$0.340799\pi$
$432$	4.32578	0.208124
$433$	−37.7921	−1.81617	−0.908085	−	0.418786i	$-0.862456\pi$
$433$	−37.7921	−1.81617	−0.908085	+	0.418786i	$0.862456\pi$
$434$	22.8982	1.09915
$435$	−13.9025	−0.666575
$436$	7.42912	0.355790
$437$	4.18032	0.199972
$438$	−8.18244	−0.390972
$439$	9.25848	0.441883	0.220941	−	0.975287i	$-0.429087\pi$
$439$	9.25848	0.441883	0.220941	+	0.975287i	$0.429087\pi$
$440$	8.54907	0.407561
$441$	−25.6203	−1.22002
$442$	−18.9343	−0.900612
$443$	−25.8925	−1.23019	−0.615094	−	0.788454i	$-0.710882\pi$
$443$	−25.8925	−1.23019	−0.615094	+	0.788454i	$0.710882\pi$
$444$	8.76501	0.415969
$445$	25.3970	1.20393
$446$	1.00228	0.0474594
$447$	−11.6431	−0.550700
$448$	4.23385	0.200030
$449$	27.6830	1.30644	0.653221	−	0.757167i	$-0.273417\pi$
$449$	27.6830	1.30644	0.653221	+	0.757167i	$0.273417\pi$
$450$	−3.92365	−0.184963
$451$	−35.1335	−1.65437
$452$	−3.45504	−0.162512
$453$	−5.88374	−0.276442
$454$	11.8006	0.553829
$455$	37.3217	1.74967
$456$	−0.984362	−0.0460970
$457$	19.3994	0.907467	0.453734	−	0.891137i	$-0.350092\pi$
$457$	19.3994	0.907467	0.453734	+	0.891137i	$0.350092\pi$
$458$	16.0787	0.751307
$459$	24.0024	1.12033
$460$	−8.87844	−0.413959
$461$	−27.9407	−1.30133	−0.650665	−	0.759365i	$-0.725510\pi$
$461$	−27.9407	−1.30133	−0.650665	+	0.759365i	$0.725510\pi$
$462$	11.3397	0.527572
$463$	−18.9030	−0.878499	−0.439250	−	0.898365i	$-0.644756\pi$
$463$	−18.9030	−0.878499	−0.439250	+	0.898365i	$0.644756\pi$
$464$	−6.64984	−0.308711
$465$	11.3070	0.524351
$466$	−26.0594	−1.20718
$467$	−25.0776	−1.16045	−0.580226	−	0.814455i	$-0.697036\pi$
$467$	−25.0776	−1.16045	−0.580226	+	0.814455i	$0.697036\pi$
$468$	8.00212	0.369898
$469$	−10.3711	−0.478894
$470$	18.8306	0.868589
$471$	−9.40102	−0.433176
$472$	5.73051	0.263768
$473$	2.38565	0.109692
$474$	10.7983	0.495981
$475$	2.03509	0.0933764
$476$	23.4922	1.07677
$477$	12.0338	0.550988
$478$	−0.628156	−0.0287312
$479$	−19.9452	−0.911318	−0.455659	−	0.890154i	$-0.650596\pi$
$479$	−19.9452	−0.911318	−0.455659	+	0.890154i	$0.650596\pi$
$480$	2.09065	0.0954248
$481$	36.9570	1.68509
$482$	−8.50974	−0.387608
$483$	−11.7766	−0.535854
$484$	−0.0477143	−0.00216883
$485$	33.4155	1.51732
$486$	−15.8376	−0.718406
$487$	1.30604	0.0591824	0.0295912	−	0.999562i	$-0.490579\pi$
$487$	1.30604	0.0591824	0.0295912	+	0.999562i	$0.490579\pi$
$488$	2.13018	0.0964286
$489$	−16.3930	−0.741317
$490$	−28.2232	−1.27499
$491$	14.4570	0.652437	0.326218	−	0.945294i	$-0.394226\pi$
$491$	14.4570	0.652437	0.326218	+	0.945294i	$0.394226\pi$
$492$	−8.59182	−0.387349
$493$	−36.8978	−1.66180
$494$	−4.15049	−0.186739
$495$	−20.0477	−0.901078
$496$	5.40837	0.242843
$497$	40.1027	1.79885
$498$	9.84631	0.441224
$499$	−14.6157	−0.654288	−0.327144	−	0.944974i	$-0.606086\pi$
$499$	−14.6157	−0.654288	−0.327144	+	0.944974i	$0.606086\pi$
$500$	8.59400	0.384335
$501$	3.82550	0.170911
$502$	−26.2134	−1.16996
$503$	−21.2875	−0.949160	−0.474580	−	0.880212i	$-0.657400\pi$
$503$	−21.2875	−0.949160	−0.474580	+	0.880212i	$0.657400\pi$
$504$	−9.92843	−0.442248
$505$	−16.8016	−0.747662
$506$	−11.3742	−0.505647
$507$	1.09707	0.0487224
$508$	−2.97389	−0.131945
$509$	−36.3441	−1.61092	−0.805462	−	0.592648i	$-0.798083\pi$
$509$	−36.3441	−1.61092	−0.805462	+	0.592648i	$0.798083\pi$
$510$	11.6004	0.513673
$511$	42.8058	1.89362
$512$	1.00000	0.0441942
$513$	5.26143	0.232298
$514$	7.30119	0.322042
$515$	22.4811	0.990636
$516$	0.583405	0.0256830
$517$	24.1240	1.06097
$518$	−45.8534	−2.01468
$519$	17.8512	0.783582
$520$	8.81508	0.386567
$521$	38.5537	1.68907	0.844533	−	0.535503i	$-0.179878\pi$
$521$	38.5537	1.68907	0.844533	+	0.535503i	$0.179878\pi$
$522$	15.5940	0.682530
$523$	−21.3389	−0.933087	−0.466543	−	0.884498i	$-0.654501\pi$
$523$	−21.3389	−0.933087	−0.466543	+	0.884498i	$0.654501\pi$
$524$	−18.4341	−0.805298
$525$	−5.73317	−0.250216
$526$	5.38808	0.234932
$527$	30.0093	1.30723
$528$	2.67835	0.116560
$529$	−11.1876	−0.486415
$530$	13.2563	0.575817
$531$	−13.4381	−0.583165
$532$	5.14961	0.223264
$533$	−36.2267	−1.56915
$534$	7.95666	0.344318
$535$	41.7329	1.80427
$536$	−2.44957	−0.105805
$537$	13.0828	0.564566
$538$	−14.0961	−0.607724
$539$	−36.1569	−1.55739
$540$	−11.1746	−0.480877
$541$	−20.5760	−0.884631	−0.442316	−	0.896859i	$-0.645843\pi$
$541$	−20.5760	−0.884631	−0.442316	+	0.896859i	$0.645843\pi$
$542$	−8.66597	−0.372235
$543$	11.8189	0.507199
$544$	5.54868	0.237898
$545$	−19.1913	−0.822064
$546$	11.6926	0.500396
$547$	−6.44825	−0.275707	−0.137854	−	0.990453i	$-0.544020\pi$
$547$	−6.44825	−0.275707	−0.137854	+	0.990453i	$0.544020\pi$
$548$	−20.2872	−0.866628
$549$	−4.99530	−0.213194
$550$	−5.53728	−0.236110
$551$	−8.08818	−0.344568
$552$	−2.78154	−0.118390
$553$	−56.4903	−2.40221
$554$	−8.86426	−0.376606
$555$	−22.6422	−0.961109
$556$	6.67217	0.282963
$557$	−4.08826	−0.173225	−0.0866126	−	0.996242i	$-0.527604\pi$
$557$	−4.08826	−0.173225	−0.0866126	+	0.996242i	$0.527604\pi$
$558$	−12.6827	−0.536902
$559$	2.45988	0.104042
$560$	−10.9371	−0.462176
$561$	14.8613	0.627445
$562$	13.1907	0.556415
$563$	34.3777	1.44885	0.724425	−	0.689354i	$-0.242106\pi$
$563$	34.3777	1.44885	0.724425	+	0.689354i	$0.242106\pi$
$564$	5.89945	0.248412
$565$	8.92525	0.375488
$566$	−0.0253660	−0.00106621
$567$	14.9630	0.628388
$568$	9.47194	0.397434
$569$	−31.0508	−1.30172	−0.650859	−	0.759199i	$-0.725591\pi$
$569$	−31.0508	−1.30172	−0.650859	+	0.759199i	$0.725591\pi$
$570$	2.54285	0.106508
$571$	38.6272	1.61650	0.808250	−	0.588840i	$-0.200415\pi$
$571$	38.6272	1.61650	0.808250	+	0.588840i	$0.200415\pi$
$572$	11.2931	0.472187
$573$	15.2346	0.636435
$574$	44.9474	1.87607
$575$	5.75061	0.239817
$576$	−2.34502	−0.0977090
$577$	39.8924	1.66074	0.830372	−	0.557210i	$-0.188128\pi$
$577$	39.8924	1.66074	0.830372	+	0.557210i	$0.188128\pi$
$578$	13.7878	0.573498
$579$	0.475929	0.0197789
$580$	17.1782	0.713287
$581$	−51.5102	−2.13700
$582$	10.4688	0.433946
$583$	16.9827	0.703354
$584$	10.1104	0.418371
$585$	−20.6715	−0.854661
$586$	20.4290	0.843913
$587$	−42.6936	−1.76215	−0.881076	−	0.472974i	$-0.843180\pi$
$587$	−42.6936	−1.76215	−0.881076	+	0.472974i	$0.843180\pi$
$588$	−8.84208	−0.364641
$589$	6.57818	0.271049
$590$	−14.8033	−0.609444
$591$	7.36812	0.303084
$592$	−10.8302	−0.445119
$593$	−11.3744	−0.467091	−0.233546	−	0.972346i	$-0.575033\pi$
$593$	−11.3744	−0.467091	−0.233546	+	0.972346i	$0.575033\pi$
$594$	−14.3158	−0.587386
$595$	−60.6864	−2.48790
$596$	14.3864	0.589291
$597$	10.5399	0.431370
$598$	−11.7281	−0.479600
$599$	−24.6844	−1.00858	−0.504289	−	0.863535i	$-0.668245\pi$
$599$	−24.6844	−1.00858	−0.504289	+	0.863535i	$0.668245\pi$
$600$	−1.35413	−0.0552820
$601$	28.0064	1.14240	0.571202	−	0.820810i	$-0.306477\pi$
$601$	28.0064	1.14240	0.571202	+	0.820810i	$0.306477\pi$
$602$	−3.05204	−0.124392
$603$	5.74429	0.233926
$604$	7.27007	0.295815
$605$	0.123258	0.00501115
$606$	−5.26381	−0.213827
$607$	4.44781	0.180531	0.0902655	−	0.995918i	$-0.471228\pi$
$607$	4.44781	0.180531	0.0902655	+	0.995918i	$0.471228\pi$
$608$	1.21630	0.0493273
$609$	22.7857	0.923322
$610$	−5.50278	−0.222801
$611$	24.8746	1.00632
$612$	−13.0117	−0.525968
$613$	17.4613	0.705256	0.352628	−	0.935764i	$-0.385288\pi$
$613$	17.4613	0.705256	0.352628	+	0.935764i	$0.385288\pi$
$614$	−32.5176	−1.31230
$615$	22.1948	0.894982
$616$	−14.0116	−0.564543
$617$	15.0204	0.604699	0.302350	−	0.953197i	$-0.402229\pi$
$617$	15.0204	0.604699	0.302350	+	0.953197i	$0.402229\pi$
$618$	7.04314	0.283317
$619$	−10.2540	−0.412141	−0.206071	−	0.978537i	$-0.566068\pi$
$619$	−10.2540	−0.412141	−0.206071	+	0.978537i	$0.566068\pi$
$620$	−13.9712	−0.561096
$621$	14.8674	0.596607
$622$	20.3515	0.816022
$623$	−41.6246	−1.66766
$624$	2.76169	0.110556
$625$	−30.5664	−1.22266
$626$	24.6415	0.984870
$627$	3.25767	0.130099
$628$	11.6161	0.463532
$629$	−60.0934	−2.39608
$630$	25.6476	1.02183
$631$	43.5234	1.73264	0.866320	−	0.499489i	$-0.166479\pi$
$631$	43.5234	1.73264	0.866320	+	0.499489i	$0.166479\pi$
$632$	−13.3426	−0.530738
$633$	−16.0244	−0.636911
$634$	−6.01236	−0.238781
$635$	7.68229	0.304862
$636$	4.15309	0.164681
$637$	−37.2820	−1.47716
$638$	22.0071	0.871271
$639$	−22.2118	−0.878687
$640$	−2.58325	−0.102112
$641$	1.80397	0.0712526	0.0356263	−	0.999365i	$-0.488657\pi$
$641$	1.80397	0.0712526	0.0356263	+	0.999365i	$0.488657\pi$
$642$	13.0746	0.516011
$643$	33.2325	1.31056	0.655281	−	0.755385i	$-0.272550\pi$
$643$	33.2325	1.31056	0.655281	+	0.755385i	$0.272550\pi$
$644$	14.5514	0.573406
$645$	−1.50708	−0.0593413
$646$	6.74884	0.265530
$647$	−24.4755	−0.962231	−0.481116	−	0.876657i	$-0.659768\pi$
$647$	−24.4755	−0.962231	−0.481116	+	0.876657i	$0.659768\pi$
$648$	3.53415	0.138834
$649$	−18.9647	−0.744429
$650$	−5.70957	−0.223948
$651$	−18.5318	−0.726317
$652$	20.2555	0.793267
$653$	−33.8899	−1.32621	−0.663107	−	0.748524i	$-0.730763\pi$
$653$	−33.8899	−1.32621	−0.663107	+	0.748524i	$0.730763\pi$
$654$	−6.01247	−0.235106
$655$	47.6200	1.86067
$656$	10.6162	0.414494
$657$	−23.7090	−0.924977
$658$	−30.8625	−1.20315
$659$	21.0545	0.820168	0.410084	−	0.912048i	$-0.365499\pi$
$659$	21.0545	0.820168	0.410084	+	0.912048i	$0.365499\pi$
$660$	−6.91886	−0.269316
$661$	−2.40083	−0.0933814	−0.0466907	−	0.998909i	$-0.514868\pi$
$661$	−2.40083	−0.0933814	−0.0466907	+	0.998909i	$0.514868\pi$
$662$	−10.0622	−0.391078
$663$	15.3237	0.595124
$664$	−12.1663	−0.472143
$665$	−13.3027	−0.515858
$666$	25.3970	0.984115
$667$	−22.8550	−0.884949
$668$	−4.72686	−0.182888
$669$	−0.811157	−0.0313611
$670$	6.32786	0.244467
$671$	−7.04966	−0.272149
$672$	−3.42650	−0.132180
$673$	−8.53432	−0.328974	−0.164487	−	0.986379i	$-0.552597\pi$
$673$	−8.53432	−0.328974	−0.164487	+	0.986379i	$0.552597\pi$
$674$	1.28652	0.0495549
$675$	7.23783	0.278584
$676$	−1.35555	−0.0521367
$677$	22.3145	0.857614	0.428807	−	0.903396i	$-0.358934\pi$
$677$	22.3145	0.857614	0.428807	+	0.903396i	$0.358934\pi$
$678$	2.79621	0.107388
$679$	−54.7667	−2.10175
$680$	−14.3336	−0.549670
$681$	−9.55034	−0.365970
$682$	−17.8986	−0.685372
$683$	18.4372	0.705480	0.352740	−	0.935721i	$-0.385250\pi$
$683$	18.4372	0.705480	0.352740	+	0.935721i	$0.385250\pi$
$684$	−2.85224	−0.109058
$685$	52.4071	2.00237
$686$	16.6197	0.634545
$687$	−13.0126	−0.496463
$688$	−0.720866	−0.0274828
$689$	17.5112	0.667122
$690$	7.18542	0.273544
$691$	−28.2151	−1.07335	−0.536677	−	0.843788i	$-0.680321\pi$
$691$	−28.2151	−1.07335	−0.536677	+	0.843788i	$0.680321\pi$
$692$	−22.0573	−0.838493
$693$	32.8574	1.24815
$694$	−26.5862	−1.00920
$695$	−17.2359	−0.653794
$696$	5.38179	0.203996
$697$	58.9060	2.23122
$698$	23.8344	0.902144
$699$	21.0901	0.797702
$700$	7.08401	0.267750
$701$	−9.64351	−0.364230	−0.182115	−	0.983277i	$-0.558294\pi$
$701$	−9.64351	−0.364230	−0.182115	+	0.983277i	$0.558294\pi$
$702$	−14.7613	−0.557128
$703$	−13.1728	−0.496820
$704$	−3.30942	−0.124729
$705$	−15.2398	−0.573963
$706$	21.6078	0.813219
$707$	27.5372	1.03564
$708$	−4.63776	−0.174298
$709$	−7.65005	−0.287304	−0.143652	−	0.989628i	$-0.545885\pi$
$709$	−7.65005	−0.287304	−0.143652	+	0.989628i	$0.545885\pi$
$710$	−24.4684	−0.918282
$711$	31.2885	1.17341
$712$	−9.83140	−0.368447
$713$	18.5881	0.696132
$714$	−19.0125	−0.711526
$715$	−29.1728	−1.09100
$716$	−16.1654	−0.604129
$717$	0.508374	0.0189856
$718$	−6.60774	−0.246598
$719$	35.1376	1.31041	0.655207	−	0.755450i	$-0.272582\pi$
$719$	35.1376	1.31041	0.655207	+	0.755450i	$0.272582\pi$
$720$	6.05776	0.225760
$721$	−36.8456	−1.37220
$722$	−17.5206	−0.652050
$723$	6.88703	0.256131
$724$	−14.6037	−0.542742
$725$	−11.1264	−0.413225
$726$	0.0386157	0.00143316
$727$	41.7858	1.54975	0.774876	−	0.632114i	$-0.217812\pi$
$727$	41.7858	1.54975	0.774876	+	0.632114i	$0.217812\pi$
$728$	−14.4476	−0.535462
$729$	2.21507	0.0820398
$730$	−26.1177	−0.966658
$731$	−3.99986	−0.147940
$732$	−1.72398	−0.0637200
$733$	19.2042	0.709322	0.354661	−	0.934995i	$-0.384596\pi$
$733$	19.2042	0.709322	0.354661	+	0.934995i	$0.384596\pi$
$734$	−19.9687	−0.737059
$735$	22.8413	0.842515
$736$	3.43692	0.126687
$737$	8.10668	0.298613
$738$	−24.8952	−0.916405
$739$	−32.5324	−1.19672	−0.598361	−	0.801227i	$-0.704181\pi$
$739$	−32.5324	−1.19672	−0.598361	+	0.801227i	$0.704181\pi$
$740$	27.9772	1.02846
$741$	3.35903	0.123397
$742$	−21.7265	−0.797606
$743$	41.6646	1.52853	0.764263	−	0.644904i	$-0.223103\pi$
$743$	41.6646	1.52853	0.764263	+	0.644904i	$0.223103\pi$
$744$	−4.37705	−0.160470
$745$	−37.1638	−1.36158
$746$	8.14640	0.298261
$747$	28.5301	1.04386
$748$	−18.3629	−0.671415
$749$	−68.3985	−2.49923
$750$	−6.95521	−0.253969
$751$	−15.4031	−0.562066	−0.281033	−	0.959698i	$-0.590677\pi$
$751$	−15.4031	−0.562066	−0.281033	+	0.959698i	$0.590677\pi$
$752$	−7.28948	−0.265820
$753$	21.2148	0.773111
$754$	22.6919	0.826390
$755$	−18.7804	−0.683489
$756$	18.3147	0.666099
$757$	−13.3802	−0.486313	−0.243157	−	0.969987i	$-0.578183\pi$
$757$	−13.3802	−0.486313	−0.243157	+	0.969987i	$0.578183\pi$
$758$	2.98001	0.108239
$759$	9.20530	0.334131
$760$	−3.14200	−0.113972
$761$	4.47449	0.162200	0.0811001	−	0.996706i	$-0.474157\pi$
$761$	4.47449	0.162200	0.0811001	+	0.996706i	$0.474157\pi$
$762$	2.40680	0.0871891
$763$	31.4537	1.13870
$764$	−18.8242	−0.681034
$765$	33.6126	1.21527
$766$	−10.0086	−0.361626
$767$	−19.5548	−0.706081
$768$	−0.809311	−0.0292035
$769$	−45.2450	−1.63158	−0.815788	−	0.578352i	$-0.803696\pi$
$769$	−45.2450	−1.63158	−0.815788	+	0.578352i	$0.803696\pi$
$770$	36.1955	1.30439
$771$	−5.90893	−0.212805
$772$	−0.588067	−0.0211650
$773$	−26.2553	−0.944336	−0.472168	−	0.881509i	$-0.656528\pi$
$773$	−26.2553	−0.944336	−0.472168	+	0.881509i	$0.656528\pi$
$774$	1.69044	0.0607617
$775$	9.04920	0.325057
$776$	−12.9354	−0.464356
$777$	37.1097	1.33130
$778$	−4.81199	−0.172518
$779$	12.9125	0.462637
$780$	−7.13414	−0.255443
$781$	−31.3467	−1.12167
$782$	19.0704	0.681956
$783$	−28.7658	−1.02800
$784$	10.9254	0.390195
$785$	−30.0072	−1.07100
$786$	14.9189	0.532141
$787$	−23.5183	−0.838338	−0.419169	−	0.907908i	$-0.637679\pi$
$787$	−23.5183	−0.838338	−0.419169	+	0.907908i	$0.637679\pi$
$788$	−9.10419	−0.324323
$789$	−4.36063	−0.155243
$790$	34.4672	1.22629
$791$	−14.6281	−0.520116
$792$	7.76065	0.275763
$793$	−7.26901	−0.258130
$794$	18.3118	0.649862
$795$	−10.7285	−0.380499
$796$	−13.0233	−0.461600
$797$	−31.9205	−1.13068	−0.565341	−	0.824857i	$-0.691255\pi$
$797$	−31.9205	−1.13068	−0.565341	+	0.824857i	$0.691255\pi$
$798$	−4.16764	−0.147533
$799$	−40.4470	−1.43091
$800$	1.67319	0.0591561
$801$	23.0548	0.814601
$802$	−23.4496	−0.828032
$803$	−33.4595	−1.18076
$804$	1.98247	0.0699162
$805$	−37.5899	−1.32487
$806$	−18.4555	−0.650067
$807$	11.4081	0.401584
$808$	6.50406	0.228812
$809$	−33.6624	−1.18351	−0.591753	−	0.806119i	$-0.701564\pi$
$809$	−33.6624	−1.18351	−0.591753	+	0.806119i	$0.701564\pi$
$810$	−9.12959	−0.320781
$811$	51.7642	1.81769	0.908844	−	0.417136i	$-0.136966\pi$
$811$	51.7642	1.81769	0.908844	+	0.417136i	$0.136966\pi$
$812$	−28.1544	−0.988026
$813$	7.01347	0.245973
$814$	35.8418	1.25625
$815$	−52.3250	−1.83287
$816$	−4.49061	−0.157203
$817$	−0.876788	−0.0306749
$818$	−28.8880	−1.01004
$819$	33.8797	1.18385
$820$	−27.4243	−0.957700
$821$	−13.0435	−0.455220	−0.227610	−	0.973752i	$-0.573091\pi$
$821$	−13.0435	−0.455220	−0.227610	+	0.973752i	$0.573091\pi$
$822$	16.4187	0.572668
$823$	−6.47305	−0.225636	−0.112818	−	0.993616i	$-0.535988\pi$
$823$	−6.47305	−0.225636	−0.112818	+	0.993616i	$0.535988\pi$
$824$	−8.70264	−0.303171
$825$	4.48138	0.156022
$826$	24.2621	0.844186
$827$	43.9091	1.52687	0.763435	−	0.645885i	$-0.223512\pi$
$827$	43.9091	1.52687	0.763435	+	0.645885i	$0.223512\pi$
$828$	−8.05964	−0.280092
$829$	4.53565	0.157530	0.0787649	−	0.996893i	$-0.474902\pi$
$829$	4.53565	0.157530	0.0787649	+	0.996893i	$0.474902\pi$
$830$	31.4286	1.09090
$831$	7.17394	0.248861
$832$	−3.41240	−0.118304
$833$	60.6218	2.10042
$834$	−5.39986	−0.186982
$835$	12.2107	0.422568
$836$	−4.02524	−0.139216
$837$	23.3954	0.808663
$838$	−4.28952	−0.148179
$839$	−7.79586	−0.269143	−0.134571	−	0.990904i	$-0.542966\pi$
$839$	−7.79586	−0.269143	−0.134571	+	0.990904i	$0.542966\pi$
$840$	8.85150	0.305406
$841$	15.2204	0.524842
$842$	40.0085	1.37878
$843$	−10.6754	−0.367679
$844$	19.8000	0.681544
$845$	3.50174	0.120463
$846$	17.0939	0.587702
$847$	−0.202015	−0.00694131
$848$	−5.13163	−0.176221
$849$	0.0205290	0.000704553 0
$850$	9.28397	0.318438
$851$	−37.2226	−1.27598
$852$	−7.66574	−0.262624
$853$	41.9597	1.43667	0.718336	−	0.695697i	$-0.244904\pi$
$853$	41.9597	1.43667	0.718336	+	0.695697i	$0.244904\pi$
$854$	9.01884	0.308619
$855$	7.36804	0.251982
$856$	−16.1552	−0.552172
$857$	−28.9106	−0.987567	−0.493784	−	0.869585i	$-0.664386\pi$
$857$	−28.9106	−0.987567	−0.493784	+	0.869585i	$0.664386\pi$
$858$	−9.13960	−0.312021
$859$	−33.8800	−1.15597	−0.577985	−	0.816047i	$-0.696161\pi$
$859$	−33.8800	−1.15597	−0.577985	+	0.816047i	$0.696161\pi$
$860$	1.86218	0.0634998
$861$	−36.3764	−1.23971
$862$	19.9115	0.678189
$863$	29.2694	0.996340	0.498170	−	0.867079i	$-0.334005\pi$
$863$	29.2694	0.996340	0.498170	+	0.867079i	$0.334005\pi$
$864$	4.32578	0.147166
$865$	56.9796	1.93736
$866$	−37.7921	−1.28423
$867$	−11.1586	−0.378967
$868$	22.8982	0.777215
$869$	44.1562	1.49790
$870$	−13.9025	−0.471339
$871$	8.35892	0.283231
$872$	7.42912	0.251582
$873$	30.3338	1.02664
$874$	4.18032	0.141401
$875$	36.3857	1.23006
$876$	−8.18244	−0.276459
$877$	37.1949	1.25598	0.627990	−	0.778221i	$-0.283878\pi$
$877$	37.1949	1.25598	0.627990	+	0.778221i	$0.283878\pi$
$878$	9.25848	0.312458
$879$	−16.5334	−0.557658
$880$	8.54907	0.288189
$881$	−38.8503	−1.30890	−0.654451	−	0.756105i	$-0.727100\pi$
$881$	−38.8503	−1.30890	−0.654451	+	0.756105i	$0.727100\pi$
$882$	−25.6203	−0.862682
$883$	7.30206	0.245734	0.122867	−	0.992423i	$-0.460791\pi$
$883$	7.30206	0.245734	0.122867	+	0.992423i	$0.460791\pi$
$884$	−18.9343	−0.636829
$885$	11.9805	0.402720
$886$	−25.8925	−0.869874
$887$	−25.9506	−0.871337	−0.435669	−	0.900107i	$-0.643488\pi$
$887$	−25.9506	−0.871337	−0.435669	+	0.900107i	$0.643488\pi$
$888$	8.76501	0.294134
$889$	−12.5910	−0.422288
$890$	25.3970	0.851308
$891$	−11.6960	−0.391830
$892$	1.00228	0.0335588
$893$	−8.86617	−0.296695
$894$	−11.6431	−0.389404
$895$	41.7593	1.39586
$896$	4.23385	0.141443
$897$	9.49172	0.316919
$898$	27.6830	0.923795
$899$	−35.9648	−1.19949
$900$	−3.92365	−0.130788
$901$	−28.4738	−0.948599
$902$	−35.1335	−1.16982
$903$	2.47005	0.0821980
$904$	−3.45504	−0.114913
$905$	37.7250	1.25402
$906$	−5.88374	−0.195474
$907$	−16.2589	−0.539869	−0.269934	−	0.962879i	$-0.587002\pi$
$907$	−16.2589	−0.539869	−0.269934	+	0.962879i	$0.587002\pi$
$908$	11.8006	0.391616
$909$	−15.2521	−0.505881
$910$	37.3217	1.23720
$911$	−15.1982	−0.503539	−0.251769	−	0.967787i	$-0.581012\pi$
$911$	−15.1982	−0.503539	−0.251769	+	0.967787i	$0.581012\pi$
$912$	−0.984362	−0.0325955
$913$	40.2634	1.33252
$914$	19.3994	0.641676
$915$	4.45346	0.147227
$916$	16.0787	0.531254
$917$	−78.0472	−2.57735
$918$	24.0024	0.792196
$919$	−33.6732	−1.11078	−0.555389	−	0.831591i	$-0.687431\pi$
$919$	−33.6732	−1.11078	−0.555389	+	0.831591i	$0.687431\pi$
$920$	−8.87844	−0.292713
$921$	26.3168	0.867169
$922$	−27.9407	−0.920179
$923$	−32.3220	−1.06389
$924$	11.3397	0.373050
$925$	−18.1210	−0.595814
$926$	−18.9030	−0.621193
$927$	20.4078	0.670281
$928$	−6.64984	−0.218292
$929$	14.7462	0.483809	0.241904	−	0.970300i	$-0.422228\pi$
$929$	14.7462	0.483809	0.241904	+	0.970300i	$0.422228\pi$
$930$	11.3070	0.370772
$931$	13.2886	0.435516
$932$	−26.0594	−0.853603
$933$	−16.4707	−0.539227
$934$	−25.0776	−0.820564
$935$	47.4360	1.55132
$936$	8.00212	0.261558
$937$	17.0269	0.556246	0.278123	−	0.960546i	$-0.410288\pi$
$937$	17.0269	0.556246	0.278123	+	0.960546i	$0.410288\pi$
$938$	−10.3711	−0.338629
$939$	−19.9426	−0.650802
$940$	18.8306	0.614185
$941$	12.4355	0.405387	0.202693	−	0.979242i	$-0.435031\pi$
$941$	12.4355	0.405387	0.202693	+	0.979242i	$0.435031\pi$
$942$	−9.40102	−0.306302
$943$	36.4871	1.18818
$944$	5.73051	0.186512
$945$	−47.3114	−1.53904
$946$	2.38565	0.0775643
$947$	−10.6613	−0.346444	−0.173222	−	0.984883i	$-0.555418\pi$
$947$	−10.6613	−0.346444	−0.173222	+	0.984883i	$0.555418\pi$
$948$	10.7983	0.350712
$949$	−34.5006	−1.11994
$950$	2.03509	0.0660271
$951$	4.86587	0.157787
$952$	23.4922	0.761388
$953$	−50.9346	−1.64993	−0.824966	−	0.565182i	$-0.808806\pi$
$953$	−50.9346	−1.64993	−0.824966	+	0.565182i	$0.808806\pi$
$954$	12.0338	0.389607
$955$	48.6275	1.57355
$956$	−0.628156	−0.0203160
$957$	−17.8106	−0.575736
$958$	−19.9452	−0.644399
$959$	−85.8931	−2.77363
$960$	2.09065	0.0674755
$961$	−1.74957	−0.0564376
$962$	36.9570	1.19154
$963$	37.8841	1.22080
$964$	−8.50974	−0.274080
$965$	1.51912	0.0489023
$966$	−11.7766	−0.378906
$967$	11.5375	0.371022	0.185511	−	0.982642i	$-0.440606\pi$
$967$	11.5375	0.371022	0.185511	+	0.982642i	$0.440606\pi$
$968$	−0.0477143	−0.00153359
$969$	−5.46191	−0.175462
$970$	33.4155	1.07291
$971$	38.2092	1.22619	0.613096	−	0.790008i	$-0.289924\pi$
$971$	38.2092	1.22619	0.613096	+	0.790008i	$0.289924\pi$
$972$	−15.8376	−0.507990
$973$	28.2489	0.905619
$974$	1.30604	0.0418482
$975$	4.62082	0.147985
$976$	2.13018	0.0681853
$977$	50.5047	1.61579	0.807895	−	0.589327i	$-0.200607\pi$
$977$	50.5047	1.61579	0.807895	+	0.589327i	$0.200607\pi$
$978$	−16.3930	−0.524190
$979$	32.5363	1.03986
$980$	−28.2232	−0.901556
$981$	−17.4214	−0.556223
$982$	14.4570	0.461342
$983$	24.9732	0.796521	0.398261	−	0.917272i	$-0.369614\pi$
$983$	24.9732	0.796521	0.398261	+	0.917272i	$0.369614\pi$
$984$	−8.59182	−0.273897
$985$	23.5184	0.749359
$986$	−36.8978	−1.17507
$987$	24.9774	0.795039
$988$	−4.15049	−0.132045
$989$	−2.47756	−0.0787819
$990$	−20.0477	−0.637158
$991$	14.7155	0.467453	0.233726	−	0.972302i	$-0.424908\pi$
$991$	14.7155	0.467453	0.233726	+	0.972302i	$0.424908\pi$
$992$	5.40837	0.171716
$993$	8.14343	0.258424
$994$	40.1027	1.27198
$995$	33.6425	1.06654
$996$	9.84631	0.311992
$997$	52.9419	1.67669	0.838344	−	0.545142i	$-0.183524\pi$
$997$	52.9419	1.67669	0.838344	+	0.545142i	$0.183524\pi$
$998$	−14.6157	−0.462652
$999$	−46.8491	−1.48224

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.d.1.30	✓	69

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.d.1.30	✓	69	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:	\( N \)	\(=\)	\( 8002 = 2 \cdot 4001 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8002.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -0.809311 q^{3} +1.00000 q^{4} -2.58325 q^{5} -0.809311 q^{6} +4.23385 q^{7} +1.00000 q^{8} -2.34502 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -0.809311 q^{3} +1.00000 q^{4} -2.58325 q^{5} -0.809311 q^{6} +4.23385 q^{7} +1.00000 q^{8} -2.34502 q^{9} -2.58325 q^{10} -3.30942 q^{11} -0.809311 q^{12} -3.41240 q^{13} +4.23385 q^{14} +2.09065 q^{15} +1.00000 q^{16} +5.54868 q^{17} -2.34502 q^{18} +1.21630 q^{19} -2.58325 q^{20} -3.42650 q^{21} -3.30942 q^{22} +3.43692 q^{23} -0.809311 q^{24} +1.67319 q^{25} -3.41240 q^{26} +4.32578 q^{27} +4.23385 q^{28} -6.64984 q^{29} +2.09065 q^{30} +5.40837 q^{31} +1.00000 q^{32} +2.67835 q^{33} +5.54868 q^{34} -10.9371 q^{35} -2.34502 q^{36} -10.8302 q^{37} +1.21630 q^{38} +2.76169 q^{39} -2.58325 q^{40} +10.6162 q^{41} -3.42650 q^{42} -0.720866 q^{43} -3.30942 q^{44} +6.05776 q^{45} +3.43692 q^{46} -7.28948 q^{47} -0.809311 q^{48} +10.9254 q^{49} +1.67319 q^{50} -4.49061 q^{51} -3.41240 q^{52} -5.13163 q^{53} +4.32578 q^{54} +8.54907 q^{55} +4.23385 q^{56} -0.984362 q^{57} -6.64984 q^{58} +5.73051 q^{59} +2.09065 q^{60} +2.13018 q^{61} +5.40837 q^{62} -9.92843 q^{63} +1.00000 q^{64} +8.81508 q^{65} +2.67835 q^{66} -2.44957 q^{67} +5.54868 q^{68} -2.78154 q^{69} -10.9371 q^{70} +9.47194 q^{71} -2.34502 q^{72} +10.1104 q^{73} -10.8302 q^{74} -1.35413 q^{75} +1.21630 q^{76} -14.0116 q^{77} +2.76169 q^{78} -13.3426 q^{79} -2.58325 q^{80} +3.53415 q^{81} +10.6162 q^{82} -12.1663 q^{83} -3.42650 q^{84} -14.3336 q^{85} -0.720866 q^{86} +5.38179 q^{87} -3.30942 q^{88} -9.83140 q^{89} +6.05776 q^{90} -14.4476 q^{91} +3.43692 q^{92} -4.37705 q^{93} -7.28948 q^{94} -3.14200 q^{95} -0.809311 q^{96} -12.9354 q^{97} +10.9254 q^{98} +7.76065 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9}+O(q^{10}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9} - 33 q^{10} - 30 q^{11} - 25 q^{12} - 58 q^{13} - 19 q^{14} + 2 q^{15} + 69 q^{16} - 80 q^{17} + 54 q^{18} - 40 q^{19} - 33 q^{20} - 32 q^{21} - 30 q^{22} - 45 q^{23} - 25 q^{24} + 42 q^{25} - 58 q^{26} - 76 q^{27} - 19 q^{28} - 44 q^{29} + 2 q^{30} - 12 q^{31} + 69 q^{32} - 41 q^{33} - 80 q^{34} - 49 q^{35} + 54 q^{36} - 47 q^{37} - 40 q^{38} - 14 q^{39} - 33 q^{40} - 94 q^{41} - 32 q^{42} - 10 q^{43} - 30 q^{44} - 89 q^{45} - 45 q^{46} - 85 q^{47} - 25 q^{48} + 32 q^{49} + 42 q^{50} - 10 q^{51} - 58 q^{52} - 41 q^{53} - 76 q^{54} - 27 q^{55} - 19 q^{56} - 72 q^{57} - 44 q^{58} - 75 q^{59} + 2 q^{60} - 98 q^{61} - 12 q^{62} - 61 q^{63} + 69 q^{64} - 47 q^{65} - 41 q^{66} - 22 q^{67} - 80 q^{68} - 74 q^{69} - 49 q^{70} - 22 q^{71} + 54 q^{72} - 129 q^{73} - 47 q^{74} - 106 q^{75} - 40 q^{76} - 108 q^{77} - 14 q^{78} + 21 q^{79} - 33 q^{80} + 13 q^{81} - 94 q^{82} - 111 q^{83} - 32 q^{84} - 67 q^{85} - 10 q^{86} - 38 q^{87} - 30 q^{88} - 112 q^{89} - 89 q^{90} - 55 q^{91} - 45 q^{92} - 90 q^{93} - 85 q^{94} - 38 q^{95} - 25 q^{96} - 98 q^{97} + 32 q^{98} - 51 q^{99}+O(q^{100}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9 - 33 * q^10 - 30 * q^11 - 25 * q^12 - 58 * q^13 - 19 * q^14 + 2 * q^15 + 69 * q^16 - 80 * q^17 + 54 * q^18 - 40 * q^19 - 33 * q^20 - 32 * q^21 - 30 * q^22 - 45 * q^23 - 25 * q^24 + 42 * q^25 - 58 * q^26 - 76 * q^27 - 19 * q^28 - 44 * q^29 + 2 * q^30 - 12 * q^31 + 69 * q^32 - 41 * q^33 - 80 * q^34 - 49 * q^35 + 54 * q^36 - 47 * q^37 - 40 * q^38 - 14 * q^39 - 33 * q^40 - 94 * q^41 - 32 * q^42 - 10 * q^43 - 30 * q^44 - 89 * q^45 - 45 * q^46 - 85 * q^47 - 25 * q^48 + 32 * q^49 + 42 * q^50 - 10 * q^51 - 58 * q^52 - 41 * q^53 - 76 * q^54 - 27 * q^55 - 19 * q^56 - 72 * q^57 - 44 * q^58 - 75 * q^59 + 2 * q^60 - 98 * q^61 - 12 * q^62 - 61 * q^63 + 69 * q^64 - 47 * q^65 - 41 * q^66 - 22 * q^67 - 80 * q^68 - 74 * q^69 - 49 * q^70 - 22 * q^71 + 54 * q^72 - 129 * q^73 - 47 * q^74 - 106 * q^75 - 40 * q^76 - 108 * q^77 - 14 * q^78 + 21 * q^79 - 33 * q^80 + 13 * q^81 - 94 * q^82 - 111 * q^83 - 32 * q^84 - 67 * q^85 - 10 * q^86 - 38 * q^87 - 30 * q^88 - 112 * q^89 - 89 * q^90 - 55 * q^91 - 45 * q^92 - 90 * q^93 - 85 * q^94 - 38 * q^95 - 25 * q^96 - 98 * q^97 + 32 * q^98 - 51 * q^99

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