LMFDB - Embedded newform 8007.2.a.f.1.24

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("8007.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8007 = 3 \cdot 17 \cdot 157 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8007.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$63.9362168984$
Analytic rank:	$1$
Dimension:	$48$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.24
Character	$\chi$	$=$	8007.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.0648705 q^{2} -1.00000 q^{3} -1.99579 q^{4} +0.118616 q^{5} +0.0648705 q^{6} -4.97378 q^{7} +0.259209 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.0648705 q^{2} -1.00000 q^{3} -1.99579 q^{4} +0.118616 q^{5} +0.0648705 q^{6} -4.97378 q^{7} +0.259209 q^{8} +1.00000 q^{9} -0.00769468 q^{10} -4.34142 q^{11} +1.99579 q^{12} +3.46263 q^{13} +0.322652 q^{14} -0.118616 q^{15} +3.97477 q^{16} -1.00000 q^{17} -0.0648705 q^{18} -0.806909 q^{19} -0.236733 q^{20} +4.97378 q^{21} +0.281631 q^{22} +5.73717 q^{23} -0.259209 q^{24} -4.98593 q^{25} -0.224623 q^{26} -1.00000 q^{27} +9.92662 q^{28} -6.51973 q^{29} +0.00769468 q^{30} -2.26114 q^{31} -0.776264 q^{32} +4.34142 q^{33} +0.0648705 q^{34} -0.589969 q^{35} -1.99579 q^{36} -2.01285 q^{37} +0.0523446 q^{38} -3.46263 q^{39} +0.0307463 q^{40} +6.33093 q^{41} -0.322652 q^{42} +3.21798 q^{43} +8.66458 q^{44} +0.118616 q^{45} -0.372173 q^{46} +9.60026 q^{47} -3.97477 q^{48} +17.7385 q^{49} +0.323440 q^{50} +1.00000 q^{51} -6.91069 q^{52} -10.1285 q^{53} +0.0648705 q^{54} -0.514962 q^{55} -1.28925 q^{56} +0.806909 q^{57} +0.422939 q^{58} +2.31048 q^{59} +0.236733 q^{60} -7.66422 q^{61} +0.146681 q^{62} -4.97378 q^{63} -7.89918 q^{64} +0.410723 q^{65} -0.281631 q^{66} +8.05732 q^{67} +1.99579 q^{68} -5.73717 q^{69} +0.0382716 q^{70} +11.9634 q^{71} +0.259209 q^{72} +5.38589 q^{73} +0.130575 q^{74} +4.98593 q^{75} +1.61042 q^{76} +21.5933 q^{77} +0.224623 q^{78} +9.95475 q^{79} +0.471471 q^{80} +1.00000 q^{81} -0.410691 q^{82} +11.4537 q^{83} -9.92662 q^{84} -0.118616 q^{85} -0.208752 q^{86} +6.51973 q^{87} -1.12534 q^{88} -10.1718 q^{89} -0.00769468 q^{90} -17.2223 q^{91} -11.4502 q^{92} +2.26114 q^{93} -0.622774 q^{94} -0.0957122 q^{95} +0.776264 q^{96} -8.31464 q^{97} -1.15070 q^{98} -4.34142 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9}+O(q^{10}) $ 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9} - 20 q^{10} + 5 q^{11} - 45 q^{12} - 8 q^{13} + 4 q^{14} - q^{15} + 39 q^{16} - 48 q^{17} - q^{18} - 6 q^{19} + 6 q^{20} + 13 q^{21} - 35 q^{22} - 8 q^{23} + 6 q^{24} + 13 q^{25} + 17 q^{26} - 48 q^{27} - 38 q^{28} + q^{29} + 20 q^{30} - 21 q^{31} - 3 q^{32} - 5 q^{33} + q^{34} + 19 q^{35} + 45 q^{36} - 58 q^{37} - 14 q^{38} + 8 q^{39} - 54 q^{40} - 3 q^{41} - 4 q^{42} - 33 q^{43} + 2 q^{44} + q^{45} - 26 q^{46} + 9 q^{47} - 39 q^{48} + 11 q^{49} + 4 q^{50} + 48 q^{51} - 31 q^{52} - 33 q^{53} + q^{54} - 21 q^{55} + 6 q^{57} - 55 q^{58} + 77 q^{59} - 6 q^{60} - 29 q^{61} - 46 q^{62} - 13 q^{63} + 24 q^{64} - 49 q^{65} + 35 q^{66} - 44 q^{67} - 45 q^{68} + 8 q^{69} + 4 q^{70} + 22 q^{71} - 6 q^{72} - 63 q^{73} - 16 q^{74} - 13 q^{75} - 46 q^{76} - 30 q^{77} - 17 q^{78} - 46 q^{79} - 14 q^{80} + 48 q^{81} - 75 q^{82} + 11 q^{83} + 38 q^{84} - q^{85} + 8 q^{86} - q^{87} - 116 q^{88} + 10 q^{89} - 20 q^{90} - 67 q^{91} - 64 q^{92} + 21 q^{93} - 16 q^{94} - 8 q^{95} + 3 q^{96} - 96 q^{97} - 46 q^{98} + 5 q^{99}+O(q^{100}) \) 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9 - 20 * q^10 + 5 * q^11 - 45 * q^12 - 8 * q^13 + 4 * q^14 - q^15 + 39 * q^16 - 48 * q^17 - q^18 - 6 * q^19 + 6 * q^20 + 13 * q^21 - 35 * q^22 - 8 * q^23 + 6 * q^24 + 13 * q^25 + 17 * q^26 - 48 * q^27 - 38 * q^28 + q^29 + 20 * q^30 - 21 * q^31 - 3 * q^32 - 5 * q^33 + q^34 + 19 * q^35 + 45 * q^36 - 58 * q^37 - 14 * q^38 + 8 * q^39 - 54 * q^40 - 3 * q^41 - 4 * q^42 - 33 * q^43 + 2 * q^44 + q^45 - 26 * q^46 + 9 * q^47 - 39 * q^48 + 11 * q^49 + 4 * q^50 + 48 * q^51 - 31 * q^52 - 33 * q^53 + q^54 - 21 * q^55 + 6 * q^57 - 55 * q^58 + 77 * q^59 - 6 * q^60 - 29 * q^61 - 46 * q^62 - 13 * q^63 + 24 * q^64 - 49 * q^65 + 35 * q^66 - 44 * q^67 - 45 * q^68 + 8 * q^69 + 4 * q^70 + 22 * q^71 - 6 * q^72 - 63 * q^73 - 16 * q^74 - 13 * q^75 - 46 * q^76 - 30 * q^77 - 17 * q^78 - 46 * q^79 - 14 * q^80 + 48 * q^81 - 75 * q^82 + 11 * q^83 + 38 * q^84 - q^85 + 8 * q^86 - q^87 - 116 * q^88 + 10 * q^89 - 20 * q^90 - 67 * q^91 - 64 * q^92 + 21 * q^93 - 16 * q^94 - 8 * q^95 + 3 * q^96 - 96 * q^97 - 46 * q^98 + 5 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−0.0648705	−0.0458704	−0.0229352	−	0.999737i	$-0.507301\pi$
$2$	−0.0648705	−0.0458704	−0.0229352	+	0.999737i	$0.507301\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.99579	−0.997896
$5$	0.118616	0.0530467	0.0265233	−	0.999648i	$-0.491556\pi$
$5$	0.118616	0.0530467	0.0265233	+	0.999648i	$0.491556\pi$
$6$	0.0648705	0.0264833
$7$	−4.97378	−1.87991	−0.939956	−	0.341297i	$-0.889134\pi$
$7$	−4.97378	−1.87991	−0.939956	+	0.341297i	$0.889134\pi$
$8$	0.259209	0.0916443
$9$	1.00000	0.333333
$10$	−0.00769468	−0.00243327
$11$	−4.34142	−1.30899	−0.654494	−	0.756067i	$-0.727118\pi$
$11$	−4.34142	−1.30899	−0.654494	+	0.756067i	$0.727118\pi$
$12$	1.99579	0.576135
$13$	3.46263	0.960360	0.480180	−	0.877170i	$-0.340571\pi$
$13$	3.46263	0.960360	0.480180	+	0.877170i	$0.340571\pi$
$14$	0.322652	0.0862323
$15$	−0.118616	−0.0306265
$16$	3.97477	0.993692
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−0.0648705	−0.0152901
$19$	−0.806909	−0.185118	−0.0925588	−	0.995707i	$-0.529505\pi$
$19$	−0.806909	−0.185118	−0.0925588	+	0.995707i	$0.529505\pi$
$20$	−0.236733	−0.0529350
$21$	4.97378	1.08537
$22$	0.281631	0.0600438
$23$	5.73717	1.19628	0.598141	−	0.801391i	$-0.295906\pi$
$23$	5.73717	1.19628	0.598141	+	0.801391i	$0.295906\pi$
$24$	−0.259209	−0.0529109
$25$	−4.98593	−0.997186
$26$	−0.224623	−0.0440521
$27$	−1.00000	−0.192450
$28$	9.92662	1.87596
$29$	−6.51973	−1.21068	−0.605342	−	0.795965i	$-0.706964\pi$
$29$	−6.51973	−1.21068	−0.605342	+	0.795965i	$0.706964\pi$
$30$	0.00769468	0.00140485
$31$	−2.26114	−0.406113	−0.203056	−	0.979167i	$-0.565087\pi$
$31$	−2.26114	−0.406113	−0.203056	+	0.979167i	$0.565087\pi$
$32$	−0.776264	−0.137225
$33$	4.34142	0.755745
$34$	0.0648705	0.0111252
$35$	−0.589969	−0.0997230
$36$	−1.99579	−0.332632
$37$	−2.01285	−0.330910	−0.165455	−	0.986217i	$-0.552909\pi$
$37$	−2.01285	−0.330910	−0.165455	+	0.986217i	$0.552909\pi$
$38$	0.0523446	0.00849142
$39$	−3.46263	−0.554464
$40$	0.0307463	0.00486142
$41$	6.33093	0.988725	0.494363	−	0.869256i	$-0.335401\pi$
$41$	6.33093	0.988725	0.494363	+	0.869256i	$0.335401\pi$
$42$	−0.322652	−0.0497862
$43$	3.21798	0.490738	0.245369	−	0.969430i	$-0.421091\pi$
$43$	3.21798	0.490738	0.245369	+	0.969430i	$0.421091\pi$
$44$	8.66458	1.30623
$45$	0.118616	0.0176822
$46$	−0.372173	−0.0548740
$47$	9.60026	1.40034	0.700171	−	0.713975i	$-0.253107\pi$
$47$	9.60026	1.40034	0.700171	+	0.713975i	$0.253107\pi$
$48$	−3.97477	−0.573708
$49$	17.7385	2.53407
$50$	0.323440	0.0457413
$51$	1.00000	0.140028
$52$	−6.91069	−0.958340
$53$	−10.1285	−1.39125	−0.695627	−	0.718403i	$-0.744873\pi$
$53$	−10.1285	−1.39125	−0.695627	+	0.718403i	$0.744873\pi$
$54$	0.0648705	0.00882776
$55$	−0.514962	−0.0694375
$56$	−1.28925	−0.172283
$57$	0.806909	0.106878
$58$	0.422939	0.0555346
$59$	2.31048	0.300799	0.150399	−	0.988625i	$-0.451944\pi$
$59$	2.31048	0.300799	0.150399	+	0.988625i	$0.451944\pi$
$60$	0.236733	0.0305621
$61$	−7.66422	−0.981303	−0.490652	−	0.871356i	$-0.663241\pi$
$61$	−7.66422	−0.981303	−0.490652	+	0.871356i	$0.663241\pi$
$62$	0.146681	0.0186286
$63$	−4.97378	−0.626637
$64$	−7.89918	−0.987398
$65$	0.410723	0.0509439
$66$	−0.281631	−0.0346663
$67$	8.05732	0.984358	0.492179	−	0.870494i	$-0.336200\pi$
$67$	8.05732	0.984358	0.492179	+	0.870494i	$0.336200\pi$
$68$	1.99579	0.242025
$69$	−5.73717	−0.690674
$70$	0.0382716	0.00457433
$71$	11.9634	1.41979	0.709897	−	0.704305i	$-0.248741\pi$
$71$	11.9634	1.41979	0.709897	+	0.704305i	$0.248741\pi$
$72$	0.259209	0.0305481
$73$	5.38589	0.630371	0.315185	−	0.949030i	$-0.397933\pi$
$73$	5.38589	0.630371	0.315185	+	0.949030i	$0.397933\pi$
$74$	0.130575	0.0151790
$75$	4.98593	0.575726
$76$	1.61042	0.184728
$77$	21.5933	2.46078
$78$	0.224623	0.0254335
$79$	9.95475	1.12000	0.559998	−	0.828494i	$-0.310802\pi$
$79$	9.95475	1.12000	0.559998	+	0.828494i	$0.310802\pi$
$80$	0.471471	0.0527120
$81$	1.00000	0.111111
$82$	−0.410691	−0.0453532
$83$	11.4537	1.25721	0.628603	−	0.777726i	$-0.283627\pi$
$83$	11.4537	1.25721	0.628603	+	0.777726i	$0.283627\pi$
$84$	−9.92662	−1.08308
$85$	−0.118616	−0.0128657
$86$	−0.208752	−0.0225103
$87$	6.51973	0.698989
$88$	−1.12534	−0.119961
$89$	−10.1718	−1.07821	−0.539103	−	0.842240i	$-0.681237\pi$
$89$	−10.1718	−1.07821	−0.539103	+	0.842240i	$0.681237\pi$
$90$	−0.00769468	−0.000811090 0
$91$	−17.2223	−1.80539
$92$	−11.4502	−1.19377
$93$	2.26114	0.234469
$94$	−0.622774	−0.0642342
$95$	−0.0957122	−0.00981987
$96$	0.776264	0.0792271
$97$	−8.31464	−0.844224	−0.422112	−	0.906544i	$-0.638711\pi$
$97$	−8.31464	−0.844224	−0.422112	+	0.906544i	$0.638711\pi$
$98$	−1.15070	−0.116239
$99$	−4.34142	−0.436330
$100$	9.95088	0.995088
$101$	7.93840	0.789900	0.394950	−	0.918703i	$-0.370762\pi$
$101$	7.93840	0.789900	0.394950	+	0.918703i	$0.370762\pi$
$102$	−0.0648705	−0.00642314
$103$	12.0039	1.18277	0.591387	−	0.806388i	$-0.298580\pi$
$103$	12.0039	1.18277	0.591387	+	0.806388i	$0.298580\pi$
$104$	0.897545	0.0880115
$105$	0.589969	0.0575751
$106$	0.657040	0.0638174
$107$	−4.01437	−0.388084	−0.194042	−	0.980993i	$-0.562160\pi$
$107$	−4.01437	−0.388084	−0.194042	+	0.980993i	$0.562160\pi$
$108$	1.99579	0.192045
$109$	6.05499	0.579963	0.289982	−	0.957032i	$-0.406351\pi$
$109$	6.05499	0.579963	0.289982	+	0.957032i	$0.406351\pi$
$110$	0.0334059	0.00318512
$111$	2.01285	0.191051
$112$	−19.7696	−1.86805
$113$	−2.48475	−0.233745	−0.116873	−	0.993147i	$-0.537287\pi$
$113$	−2.48475	−0.233745	−0.116873	+	0.993147i	$0.537287\pi$
$114$	−0.0523446	−0.00490252
$115$	0.680520	0.0634588
$116$	13.0120	1.20814
$117$	3.46263	0.320120
$118$	−0.149882	−0.0137977
$119$	4.97378	0.455945
$120$	−0.0307463	−0.00280674
$121$	7.84797	0.713451
$122$	0.497182	0.0450128
$123$	−6.33093	−0.570841
$124$	4.51277	0.405258
$125$	−1.18449	−0.105944
$126$	0.322652	0.0287441
$127$	4.37521	0.388237	0.194119	−	0.980978i	$-0.437815\pi$
$127$	4.37521	0.388237	0.194119	+	0.980978i	$0.437815\pi$
$128$	2.06495	0.182518
$129$	−3.21798	−0.283328
$130$	−0.0266438	−0.00233682
$131$	1.76672	0.154359	0.0771796	−	0.997017i	$-0.475409\pi$
$131$	1.76672	0.154359	0.0771796	+	0.997017i	$0.475409\pi$
$132$	−8.66458	−0.754155
$133$	4.01338	0.348005
$134$	−0.522683	−0.0451529
$135$	−0.118616	−0.0102088
$136$	−0.259209	−0.0222270
$137$	−1.71579	−0.146590	−0.0732950	−	0.997310i	$-0.523351\pi$
$137$	−1.71579	−0.146590	−0.0732950	+	0.997310i	$0.523351\pi$
$138$	0.372173	0.0316815
$139$	15.5724	1.32083	0.660417	−	0.750899i	$-0.270380\pi$
$139$	15.5724	1.32083	0.660417	+	0.750899i	$0.270380\pi$
$140$	1.17746	0.0995132
$141$	−9.60026	−0.808488
$142$	−0.776072	−0.0651265
$143$	−15.0327	−1.25710
$144$	3.97477	0.331231
$145$	−0.773344	−0.0642228
$146$	−0.349385	−0.0289153
$147$	−17.7385	−1.46304
$148$	4.01723	0.330214
$149$	5.85511	0.479669	0.239834	−	0.970814i	$-0.422907\pi$
$149$	5.85511	0.479669	0.239834	+	0.970814i	$0.422907\pi$
$150$	−0.323440	−0.0264088
$151$	9.72038	0.791033	0.395517	−	0.918459i	$-0.370566\pi$
$151$	9.72038	0.791033	0.395517	+	0.918459i	$0.370566\pi$
$152$	−0.209158	−0.0169650
$153$	−1.00000	−0.0808452
$154$	−1.40077	−0.112877
$155$	−0.268207	−0.0215429
$156$	6.91069	0.553298
$157$	−1.00000	−0.0798087
$157$	−1.00000	−0.0798087
$158$	−0.645770	−0.0513747
$159$	10.1285	0.803241
$160$	−0.0920772	−0.00727935
$161$	−28.5354	−2.24891
$162$	−0.0648705	−0.00509671
$163$	8.84608	0.692879	0.346439	−	0.938072i	$-0.387391\pi$
$163$	8.84608	0.692879	0.346439	+	0.938072i	$0.387391\pi$
$164$	−12.6352	−0.986645
$165$	0.514962	0.0400897
$166$	−0.743007	−0.0576685
$167$	−18.1414	−1.40383	−0.701913	−	0.712263i	$-0.747670\pi$
$167$	−18.1414	−1.40383	−0.701913	+	0.712263i	$0.747670\pi$
$168$	1.28925	0.0994677
$169$	−1.01020	−0.0777078
$170$	0.00769468	0.000590155 0
$171$	−0.806909	−0.0617059
$172$	−6.42242	−0.489705
$173$	15.4822	1.17709	0.588546	−	0.808464i	$-0.299701\pi$
$173$	15.4822	1.17709	0.588546	+	0.808464i	$0.299701\pi$
$174$	−0.422939	−0.0320629
$175$	24.7989	1.87462
$176$	−17.2562	−1.30073
$177$	−2.31048	−0.173666
$178$	0.659849	0.0494578
$179$	−3.74689	−0.280055	−0.140028	−	0.990148i	$-0.544719\pi$
$179$	−3.74689	−0.280055	−0.140028	+	0.990148i	$0.544719\pi$
$180$	−0.236733	−0.0176450
$181$	−11.7797	−0.875580	−0.437790	−	0.899077i	$-0.644239\pi$
$181$	−11.7797	−0.875580	−0.437790	+	0.899077i	$0.644239\pi$
$182$	1.11722	0.0828141
$183$	7.66422	0.566556
$184$	1.48713	0.109632
$185$	−0.238756	−0.0175537
$186$	−0.146681	−0.0107552
$187$	4.34142	0.317476
$188$	−19.1601	−1.39740
$189$	4.97378	0.361789
$190$	0.00620890	0.000450441 0
$191$	−22.8104	−1.65050	−0.825251	−	0.564765i	$-0.808967\pi$
$191$	−22.8104	−1.65050	−0.825251	+	0.564765i	$0.808967\pi$
$192$	7.89918	0.570074
$193$	−7.13856	−0.513845	−0.256922	−	0.966432i	$-0.582708\pi$
$193$	−7.13856	−0.513845	−0.256922	+	0.966432i	$0.582708\pi$
$194$	0.539375	0.0387249
$195$	−0.410723	−0.0294125
$196$	−35.4023	−2.52873
$197$	1.71391	0.122111	0.0610556	−	0.998134i	$-0.480553\pi$
$197$	1.71391	0.122111	0.0610556	+	0.998134i	$0.480553\pi$
$198$	0.281631	0.0200146
$199$	−2.59523	−0.183971	−0.0919853	−	0.995760i	$-0.529321\pi$
$199$	−2.59523	−0.183971	−0.0919853	+	0.995760i	$0.529321\pi$
$200$	−1.29240	−0.0913864
$201$	−8.05732	−0.568320
$202$	−0.514968	−0.0362330
$203$	32.4277	2.27598
$204$	−1.99579	−0.139733
$205$	0.750949	0.0524486
$206$	−0.778696	−0.0542543
$207$	5.73717	0.398761
$208$	13.7631	0.954303
$209$	3.50313	0.242317
$210$	−0.0382716	−0.00264099
$211$	−13.2324	−0.910958	−0.455479	−	0.890247i	$-0.650532\pi$
$211$	−13.2324	−0.910958	−0.455479	+	0.890247i	$0.650532\pi$
$212$	20.2143	1.38833
$213$	−11.9634	−0.819719
$214$	0.260415	0.0178016
$215$	0.381704	0.0260320
$216$	−0.259209	−0.0176370
$217$	11.2464	0.763456
$218$	−0.392791	−0.0266031
$219$	−5.38589	−0.363945
$220$	1.02776	0.0692914
$221$	−3.46263	−0.232922
$222$	−0.130575	−0.00876359
$223$	−2.45175	−0.164181	−0.0820905	−	0.996625i	$-0.526160\pi$
$223$	−2.45175	−0.164181	−0.0820905	+	0.996625i	$0.526160\pi$
$224$	3.86096	0.257971
$225$	−4.98593	−0.332395
$226$	0.161187	0.0107220
$227$	−16.6359	−1.10416	−0.552082	−	0.833790i	$-0.686167\pi$
$227$	−16.6359	−1.10416	−0.552082	+	0.833790i	$0.686167\pi$
$228$	−1.61042	−0.106653
$229$	−17.1990	−1.13654	−0.568270	−	0.822842i	$-0.692387\pi$
$229$	−17.1990	−1.13654	−0.568270	+	0.822842i	$0.692387\pi$
$230$	−0.0441457	−0.00291088
$231$	−21.5933	−1.42073
$232$	−1.68997	−0.110952
$233$	−3.83870	−0.251482	−0.125741	−	0.992063i	$-0.540131\pi$
$233$	−3.83870	−0.251482	−0.125741	+	0.992063i	$0.540131\pi$
$234$	−0.224623	−0.0146840
$235$	1.13874	0.0742835
$236$	−4.61123	−0.300166
$237$	−9.95475	−0.646630
$238$	−0.322652	−0.0209144
$239$	7.17770	0.464286	0.232143	−	0.972682i	$-0.425426\pi$
$239$	7.17770	0.464286	0.232143	+	0.972682i	$0.425426\pi$
$240$	−0.471471	−0.0304333
$241$	−23.1575	−1.49171	−0.745853	−	0.666110i	$-0.767958\pi$
$241$	−23.1575	−1.49171	−0.745853	+	0.666110i	$0.767958\pi$
$242$	−0.509102	−0.0327263
$243$	−1.00000	−0.0641500
$244$	15.2962	0.979239
$245$	2.10406	0.134424
$246$	0.410691	0.0261847
$247$	−2.79403	−0.177780
$248$	−0.586108	−0.0372179
$249$	−11.4537	−0.725848
$250$	0.0768385	0.00485970
$251$	21.1381	1.33422	0.667112	−	0.744957i	$-0.267530\pi$
$251$	21.1381	1.33422	0.667112	+	0.744957i	$0.267530\pi$
$252$	9.92662	0.625319
$253$	−24.9075	−1.56592
$254$	−0.283822	−0.0178086
$255$	0.118616	0.00742802
$256$	15.6644	0.979025
$257$	−13.3372	−0.831950	−0.415975	−	0.909376i	$-0.636560\pi$
$257$	−13.3372	−0.831950	−0.415975	+	0.909376i	$0.636560\pi$
$258$	0.208752	0.0129963
$259$	10.0115	0.622082
$260$	−0.819717	−0.0508367
$261$	−6.51973	−0.403561
$262$	−0.114608	−0.00708052
$263$	−22.0305	−1.35846	−0.679228	−	0.733927i	$-0.737685\pi$
$263$	−22.0305	−1.35846	−0.679228	+	0.733927i	$0.737685\pi$
$264$	1.12534	0.0692597
$265$	−1.20140	−0.0738014
$266$	−0.260350	−0.0159631
$267$	10.1718	0.622503
$268$	−16.0807	−0.982287
$269$	−7.05221	−0.429981	−0.214990	−	0.976616i	$-0.568972\pi$
$269$	−7.05221	−0.429981	−0.214990	+	0.976616i	$0.568972\pi$
$270$	0.00769468	0.000468283 0
$271$	−11.5716	−0.702926	−0.351463	−	0.936202i	$-0.614316\pi$
$271$	−11.5716	−0.702926	−0.351463	+	0.936202i	$0.614316\pi$
$272$	−3.97477	−0.241006
$273$	17.2223	1.04234
$274$	0.111304	0.00672414
$275$	21.6460	1.30531
$276$	11.4502	0.689221
$277$	8.51107	0.511381	0.255690	−	0.966759i	$-0.417697\pi$
$277$	8.51107	0.511381	0.255690	+	0.966759i	$0.417697\pi$
$278$	−1.01019	−0.0605871
$279$	−2.26114	−0.135371
$280$	−0.152925	−0.00913904
$281$	−17.8087	−1.06238	−0.531189	−	0.847254i	$-0.678254\pi$
$281$	−17.8087	−1.06238	−0.531189	+	0.847254i	$0.678254\pi$
$282$	0.622774	0.0370857
$283$	−21.5937	−1.28361	−0.641805	−	0.766868i	$-0.721814\pi$
$283$	−21.5937	−1.28361	−0.641805	+	0.766868i	$0.721814\pi$
$284$	−23.8765	−1.41681
$285$	0.0957122	0.00566950
$286$	0.975182	0.0576637
$287$	−31.4886	−1.85872
$288$	−0.776264	−0.0457418
$289$	1.00000	0.0588235
$290$	0.0501673	0.00294592
$291$	8.31464	0.487413
$292$	−10.7491	−0.629044
$293$	−18.1737	−1.06172	−0.530860	−	0.847460i	$-0.678131\pi$
$293$	−18.1737	−1.06172	−0.530860	+	0.847460i	$0.678131\pi$
$294$	1.15070	0.0671104
$295$	0.274059	0.0159564
$296$	−0.521749	−0.0303260
$297$	4.34142	0.251915
$298$	−0.379824	−0.0220026
$299$	19.8657	1.14886
$300$	−9.95088	−0.574514
$301$	−16.0055	−0.922543
$302$	−0.630566	−0.0362850
$303$	−7.93840	−0.456049
$304$	−3.20728	−0.183950
$305$	−0.909099	−0.0520549
$306$	0.0648705	0.00370840
$307$	−3.61718	−0.206443	−0.103222	−	0.994658i	$-0.532915\pi$
$307$	−3.61718	−0.206443	−0.103222	+	0.994658i	$0.532915\pi$
$308$	−43.0957	−2.45560
$309$	−12.0039	−0.682875
$310$	0.0173988	0.000988183 0
$311$	−22.1185	−1.25423	−0.627113	−	0.778929i	$-0.715763\pi$
$311$	−22.1185	−1.25423	−0.627113	+	0.778929i	$0.715763\pi$
$312$	−0.897545	−0.0508135
$313$	11.2934	0.638338	0.319169	−	0.947698i	$-0.396596\pi$
$313$	11.2934	0.638338	0.319169	+	0.947698i	$0.396596\pi$
$314$	0.0648705	0.00366086
$315$	−0.589969	−0.0332410
$316$	−19.8676	−1.11764
$317$	−7.17556	−0.403019	−0.201510	−	0.979487i	$-0.564585\pi$
$317$	−7.17556	−0.403019	−0.201510	+	0.979487i	$0.564585\pi$
$318$	−0.657040	−0.0368450
$319$	28.3049	1.58477
$320$	−0.936969	−0.0523781
$321$	4.01437	0.224060
$322$	1.85111	0.103158
$323$	0.806909	0.0448976
$324$	−1.99579	−0.110877
$325$	−17.2644	−0.957658
$326$	−0.573850	−0.0317826
$327$	−6.05499	−0.334842
$328$	1.64104	0.0906110
$329$	−47.7496	−2.63252
$330$	−0.0334059	−0.00183893
$331$	19.5660	1.07544	0.537722	−	0.843122i	$-0.319285\pi$
$331$	19.5660	1.07544	0.537722	+	0.843122i	$0.319285\pi$
$332$	−22.8592	−1.25456
$333$	−2.01285	−0.110303
$334$	1.17684	0.0643940
$335$	0.955726	0.0522169
$336$	19.7696	1.07852
$337$	26.5021	1.44366	0.721831	−	0.692070i	$-0.243301\pi$
$337$	26.5021	1.44366	0.721831	+	0.692070i	$0.243301\pi$
$338$	0.0655323	0.00356449
$339$	2.48475	0.134953
$340$	0.236733	0.0128386
$341$	9.81657	0.531597
$342$	0.0523446	0.00283047
$343$	−53.4107	−2.88391
$344$	0.834130	0.0449733
$345$	−0.680520	−0.0366380
$346$	−1.00434	−0.0539937
$347$	16.9944	0.912306	0.456153	−	0.889901i	$-0.349227\pi$
$347$	16.9944	0.912306	0.456153	+	0.889901i	$0.349227\pi$
$348$	−13.0120	−0.697518
$349$	26.6567	1.42690	0.713451	−	0.700705i	$-0.247131\pi$
$349$	26.6567	1.42690	0.713451	+	0.700705i	$0.247131\pi$
$350$	−1.60872	−0.0859896
$351$	−3.46263	−0.184821
$352$	3.37009	0.179626
$353$	14.5904	0.776566	0.388283	−	0.921540i	$-0.373068\pi$
$353$	14.5904	0.776566	0.388283	+	0.921540i	$0.373068\pi$
$354$	0.149882	0.00796613
$355$	1.41905	0.0753153
$356$	20.3008	1.07594
$357$	−4.97378	−0.263240
$358$	0.243062	0.0128463
$359$	−15.3922	−0.812368	−0.406184	−	0.913791i	$-0.633141\pi$
$359$	−15.3922	−0.812368	−0.406184	+	0.913791i	$0.633141\pi$
$360$	0.0307463	0.00162047
$361$	−18.3489	−0.965731
$362$	0.764157	0.0401632
$363$	−7.84797	−0.411911
$364$	34.3722	1.80159
$365$	0.638852	0.0334391
$366$	−0.497182	−0.0259881
$367$	−12.3531	−0.644829	−0.322415	−	0.946599i	$-0.604494\pi$
$367$	−12.3531	−0.644829	−0.322415	+	0.946599i	$0.604494\pi$
$368$	22.8039	1.18874
$369$	6.33093	0.329575
$370$	0.0154882	0.000805195 0
$371$	50.3768	2.61543
$372$	−4.51277	−0.233976
$373$	15.3806	0.796378	0.398189	−	0.917303i	$-0.369639\pi$
$373$	15.3806	0.796378	0.398189	+	0.917303i	$0.369639\pi$
$374$	−0.281631	−0.0145628
$375$	1.18449	0.0611668
$376$	2.48848	0.128333
$377$	−22.5754	−1.16269
$378$	−0.322652	−0.0165954
$379$	−25.3664	−1.30299	−0.651493	−	0.758655i	$-0.725857\pi$
$379$	−25.3664	−1.30299	−0.651493	+	0.758655i	$0.725857\pi$
$380$	0.191022	0.00979921
$381$	−4.37521	−0.224149
$382$	1.47972	0.0757092
$383$	25.1142	1.28328	0.641638	−	0.767008i	$-0.278255\pi$
$383$	25.1142	1.28328	0.641638	+	0.767008i	$0.278255\pi$
$384$	−2.06495	−0.105377
$385$	2.56131	0.130536
$386$	0.463082	0.0235703
$387$	3.21798	0.163579
$388$	16.5943	0.842448
$389$	−13.9009	−0.704805	−0.352402	−	0.935849i	$-0.614635\pi$
$389$	−13.9009	−0.704805	−0.352402	+	0.935849i	$0.614635\pi$
$390$	0.0266438	0.00134916
$391$	−5.73717	−0.290141
$392$	4.59797	0.232233
$393$	−1.76672	−0.0891193
$394$	−0.111182	−0.00560129
$395$	1.18079	0.0594121
$396$	8.66458	0.435411
$397$	33.7127	1.69199	0.845996	−	0.533190i	$-0.179007\pi$
$397$	33.7127	1.69199	0.845996	+	0.533190i	$0.179007\pi$
$398$	0.168354	0.00843881
$399$	−4.01338	−0.200921
$400$	−19.8179	−0.990896
$401$	−8.36903	−0.417929	−0.208965	−	0.977923i	$-0.567009\pi$
$401$	−8.36903	−0.417929	−0.208965	+	0.977923i	$0.567009\pi$
$402$	0.522683	0.0260690
$403$	−7.82949	−0.390015
$404$	−15.8434	−0.788238
$405$	0.118616	0.00589407
$406$	−2.10360	−0.104400
$407$	8.73863	0.433158
$408$	0.259209	0.0128328
$409$	2.67132	0.132089	0.0660443	−	0.997817i	$-0.478962\pi$
$409$	2.67132	0.132089	0.0660443	+	0.997817i	$0.478962\pi$
$410$	−0.0487145	−0.00240584
$411$	1.71579	0.0846338
$412$	−23.9572	−1.18029
$413$	−11.4918	−0.565475
$414$	−0.372173	−0.0182913
$415$	1.35859	0.0666906
$416$	−2.68791	−0.131786
$417$	−15.5724	−0.762583
$418$	−0.227250	−0.0111152
$419$	31.8284	1.55492	0.777460	−	0.628932i	$-0.216508\pi$
$419$	31.8284	1.55492	0.777460	+	0.628932i	$0.216508\pi$
$420$	−1.17746	−0.0574540
$421$	−5.03737	−0.245506	−0.122753	−	0.992437i	$-0.539172\pi$
$421$	−5.03737	−0.245506	−0.122753	+	0.992437i	$0.539172\pi$
$422$	0.858394	0.0417860
$423$	9.60026	0.466781
$424$	−2.62539	−0.127500
$425$	4.98593	0.241853
$426$	0.776072	0.0376008
$427$	38.1201	1.84476
$428$	8.01185	0.387268
$429$	15.0327	0.725788
$430$	−0.0247613	−0.00119410
$431$	16.2108	0.780849	0.390425	−	0.920635i	$-0.372328\pi$
$431$	16.2108	0.780849	0.390425	+	0.920635i	$0.372328\pi$
$432$	−3.97477	−0.191236
$433$	19.9519	0.958829	0.479414	−	0.877589i	$-0.340849\pi$
$433$	19.9519	0.958829	0.479414	+	0.877589i	$0.340849\pi$
$434$	−0.729561	−0.0350200
$435$	0.773344	0.0370790
$436$	−12.0845	−0.578743
$437$	−4.62937	−0.221453
$438$	0.349385	0.0166943
$439$	−7.04627	−0.336300	−0.168150	−	0.985761i	$-0.553779\pi$
$439$	−7.04627	−0.336300	−0.168150	+	0.985761i	$0.553779\pi$
$440$	−0.133483	−0.00636355
$441$	17.7385	0.844689
$442$	0.224623	0.0106842
$443$	30.8886	1.46756	0.733782	−	0.679385i	$-0.237753\pi$
$443$	30.8886	1.46756	0.733782	+	0.679385i	$0.237753\pi$
$444$	−4.01723	−0.190649
$445$	−1.20654	−0.0571953
$446$	0.159046	0.00753105
$447$	−5.85511	−0.276937
$448$	39.2888	1.85622
$449$	6.88966	0.325143	0.162572	−	0.986697i	$-0.448021\pi$
$449$	6.88966	0.325143	0.162572	+	0.986697i	$0.448021\pi$
$450$	0.323440	0.0152471
$451$	−27.4853	−1.29423
$452$	4.95903	0.233253
$453$	−9.72038	−0.456703
$454$	1.07918	0.0506485
$455$	−2.04284	−0.0957700
$456$	0.209158	0.00979473
$457$	−8.68547	−0.406289	−0.203145	−	0.979149i	$-0.565116\pi$
$457$	−8.68547	−0.406289	−0.203145	+	0.979149i	$0.565116\pi$
$458$	1.11571	0.0521336
$459$	1.00000	0.0466760
$460$	−1.35818	−0.0633253
$461$	−13.6759	−0.636952	−0.318476	−	0.947931i	$-0.603171\pi$
$461$	−13.6759	−0.636952	−0.318476	+	0.947931i	$0.603171\pi$
$462$	1.40077	0.0651696
$463$	1.94603	0.0904395	0.0452197	−	0.998977i	$-0.485601\pi$
$463$	1.94603	0.0904395	0.0452197	+	0.998977i	$0.485601\pi$
$464$	−25.9144	−1.20305
$465$	0.268207	0.0124378
$466$	0.249019	0.0115356
$467$	12.0677	0.558427	0.279214	−	0.960229i	$-0.409926\pi$
$467$	12.0677	0.558427	0.279214	+	0.960229i	$0.409926\pi$
$468$	−6.91069	−0.319447
$469$	−40.0753	−1.85051
$470$	−0.0738709	−0.00340741
$471$	1.00000	0.0460776
$472$	0.598897	0.0275665
$473$	−13.9706	−0.642370
$474$	0.645770	0.0296612
$475$	4.02319	0.184597
$476$	−9.92662	−0.454986
$477$	−10.1285	−0.463751
$478$	−0.465621	−0.0212970
$479$	16.5252	0.755056	0.377528	−	0.925998i	$-0.376774\pi$
$479$	16.5252	0.755056	0.377528	+	0.925998i	$0.376774\pi$
$480$	0.0920772	0.00420273
$481$	−6.96975	−0.317793
$482$	1.50224	0.0684252
$483$	28.5354	1.29841
$484$	−15.6629	−0.711950
$485$	−0.986249	−0.0447833
$486$	0.0648705	0.00294259
$487$	−13.4957	−0.611549	−0.305774	−	0.952104i	$-0.598915\pi$
$487$	−13.4957	−0.611549	−0.305774	+	0.952104i	$0.598915\pi$
$488$	−1.98664	−0.0899308
$489$	−8.84608	−0.400034
$490$	−0.136492	−0.00616607
$491$	−0.838904	−0.0378592	−0.0189296	−	0.999821i	$-0.506026\pi$
$491$	−0.838904	−0.0378592	−0.0189296	+	0.999821i	$0.506026\pi$
$492$	12.6352	0.569640
$493$	6.51973	0.293634
$494$	0.181250	0.00815482
$495$	−0.514962	−0.0231458
$496$	−8.98751	−0.403551
$497$	−59.5033	−2.66909
$498$	0.743007	0.0332949
$499$	20.7838	0.930411	0.465206	−	0.885203i	$-0.345980\pi$
$499$	20.7838	0.930411	0.465206	+	0.885203i	$0.345980\pi$
$500$	2.36400	0.105721
$501$	18.1414	0.810499
$502$	−1.37124	−0.0612014
$503$	−22.3605	−0.997005	−0.498502	−	0.866888i	$-0.666116\pi$
$503$	−22.3605	−0.997005	−0.498502	+	0.866888i	$0.666116\pi$
$504$	−1.28925	−0.0574277
$505$	0.941620	0.0419016
$506$	1.61576	0.0718294
$507$	1.01020	0.0448646
$508$	−8.73201	−0.387420
$509$	−35.2943	−1.56439	−0.782196	−	0.623032i	$-0.785901\pi$
$509$	−35.2943	−1.56439	−0.782196	+	0.623032i	$0.785901\pi$
$510$	−0.00769468	−0.000340726 0
$511$	−26.7882	−1.18504
$512$	−5.14606	−0.227426
$513$	0.806909	0.0356259
$514$	0.865190	0.0381619
$515$	1.42385	0.0627422
$516$	6.42242	0.282731
$517$	−41.6788	−1.83303
$518$	−0.649449	−0.0285352
$519$	−15.4822	−0.679594
$520$	0.106463	0.00466872
$521$	−29.4016	−1.28811	−0.644055	−	0.764979i	$-0.722749\pi$
$521$	−29.4016	−1.28811	−0.644055	+	0.764979i	$0.722749\pi$
$522$	0.422939	0.0185115
$523$	−14.3280	−0.626522	−0.313261	−	0.949667i	$-0.601421\pi$
$523$	−14.3280	−0.626522	−0.313261	+	0.949667i	$0.601421\pi$
$524$	−3.52601	−0.154034
$525$	−24.7989	−1.08231
$526$	1.42913	0.0623130
$527$	2.26114	0.0984968
$528$	17.2562	0.750978
$529$	9.91513	0.431093
$530$	0.0779354	0.00338530
$531$	2.31048	0.100266
$532$	−8.00988	−0.347272
$533$	21.9217	0.949533
$534$	−0.659849	−0.0285545
$535$	−0.476169	−0.0205866
$536$	2.08853	0.0902108
$537$	3.74689	0.161690
$538$	0.457480	0.0197234
$539$	−77.0102	−3.31706
$540$	0.236733	0.0101874
$541$	−23.4474	−1.00808	−0.504042	−	0.863679i	$-0.668154\pi$
$541$	−23.4474	−1.00808	−0.504042	+	0.863679i	$0.668154\pi$
$542$	0.750657	0.0322435
$543$	11.7797	0.505516
$544$	0.776264	0.0332820
$545$	0.718219	0.0307651
$546$	−1.11722	−0.0478127
$547$	−35.2307	−1.50636	−0.753178	−	0.657817i	$-0.771480\pi$
$547$	−35.2307	−1.50636	−0.753178	+	0.657817i	$0.771480\pi$
$548$	3.42436	0.146282
$549$	−7.66422	−0.327101
$550$	−1.40419	−0.0598749
$551$	5.26083	0.224119
$552$	−1.48713	−0.0632963
$553$	−49.5127	−2.10549
$554$	−0.552118	−0.0234572
$555$	0.238756	0.0101346
$556$	−31.0793	−1.31805
$557$	2.37181	0.100497	0.0502484	−	0.998737i	$-0.483999\pi$
$557$	2.37181	0.100497	0.0502484	+	0.998737i	$0.483999\pi$
$558$	0.146681	0.00620952
$559$	11.1427	0.471285
$560$	−2.34499	−0.0990940
$561$	−4.34142	−0.183295
$562$	1.15526	0.0487317
$563$	16.2262	0.683854	0.341927	−	0.939727i	$-0.388920\pi$
$563$	16.2262	0.683854	0.341927	+	0.939727i	$0.388920\pi$
$564$	19.1601	0.806787
$565$	−0.294730	−0.0123994
$566$	1.40079	0.0588797
$567$	−4.97378	−0.208879
$568$	3.10102	0.130116
$569$	36.0438	1.51104	0.755518	−	0.655128i	$-0.227385\pi$
$569$	36.0438	1.51104	0.755518	+	0.655128i	$0.227385\pi$
$570$	−0.00620890	−0.000260062 0
$571$	38.1300	1.59569	0.797845	−	0.602862i	$-0.205973\pi$
$571$	38.1300	1.59569	0.797845	+	0.602862i	$0.205973\pi$
$572$	30.0022	1.25446
$573$	22.8104	0.952918
$574$	2.04269	0.0852600
$575$	−28.6051	−1.19292
$576$	−7.89918	−0.329133
$577$	−0.446766	−0.0185991	−0.00929956	−	0.999957i	$-0.502960\pi$
$577$	−0.446766	−0.0185991	−0.00929956	+	0.999957i	$0.502960\pi$
$578$	−0.0648705	−0.00269826
$579$	7.13856	0.296668
$580$	1.54343	0.0640876
$581$	−56.9681	−2.36344
$582$	−0.539375	−0.0223578
$583$	43.9720	1.82114
$584$	1.39607	0.0577699
$585$	0.410723	0.0169813
$586$	1.17894	0.0487015
$587$	24.5870	1.01481	0.507406	−	0.861707i	$-0.330604\pi$
$587$	24.5870	1.01481	0.507406	+	0.861707i	$0.330604\pi$
$588$	35.4023	1.45997
$589$	1.82453	0.0751786
$590$	−0.0177784	−0.000731924 0
$591$	−1.71391	−0.0705009
$592$	−8.00061	−0.328823
$593$	−11.0526	−0.453877	−0.226938	−	0.973909i	$-0.572872\pi$
$593$	−11.0526	−0.453877	−0.226938	+	0.973909i	$0.572872\pi$
$594$	−0.281631	−0.0115554
$595$	0.589969	0.0241864
$596$	−11.6856	−0.478660
$597$	2.59523	0.106215
$598$	−1.28870	−0.0526988
$599$	35.2735	1.44124	0.720618	−	0.693332i	$-0.243858\pi$
$599$	35.2735	1.44124	0.720618	+	0.693332i	$0.243858\pi$
$600$	1.29240	0.0527620
$601$	−42.5574	−1.73595	−0.867976	−	0.496606i	$-0.834579\pi$
$601$	−42.5574	−1.73595	−0.867976	+	0.496606i	$0.834579\pi$
$602$	1.03829	0.0423174
$603$	8.05732	0.328119
$604$	−19.3998	−0.789369
$605$	0.930894	0.0378462
$606$	0.514968	0.0209191
$607$	−7.52398	−0.305389	−0.152694	−	0.988273i	$-0.548795\pi$
$607$	−7.52398	−0.305389	−0.152694	+	0.988273i	$0.548795\pi$
$608$	0.626374	0.0254028
$609$	−32.4277	−1.31404
$610$	0.0589737	0.00238778
$611$	33.2421	1.34483
$612$	1.99579	0.0806751
$613$	48.7096	1.96736	0.983680	−	0.179924i	$-0.0575853\pi$
$613$	48.7096	1.96736	0.983680	+	0.179924i	$0.0575853\pi$
$614$	0.234648	0.00946964
$615$	−0.750949	−0.0302812
$616$	5.59718	0.225517
$617$	−46.3020	−1.86405	−0.932024	−	0.362397i	$-0.881959\pi$
$617$	−46.3020	−1.86405	−0.932024	+	0.362397i	$0.881959\pi$
$618$	0.778696	0.0313238
$619$	−2.14850	−0.0863555	−0.0431777	−	0.999067i	$-0.513748\pi$
$619$	−2.14850	−0.0863555	−0.0431777	+	0.999067i	$0.513748\pi$
$620$	0.535286	0.0214976
$621$	−5.73717	−0.230225
$622$	1.43484	0.0575318
$623$	50.5922	2.02693
$624$	−13.7631	−0.550967
$625$	24.7892	0.991566
$626$	−0.732606	−0.0292808
$627$	−3.50313	−0.139902
$628$	1.99579	0.0796408
$629$	2.01285	0.0802576
$630$	0.0382716	0.00152478
$631$	30.3936	1.20995	0.604975	−	0.796244i	$-0.293183\pi$
$631$	30.3936	1.20995	0.604975	+	0.796244i	$0.293183\pi$
$632$	2.58036	0.102641
$633$	13.2324	0.525942
$634$	0.465482	0.0184867
$635$	0.518970	0.0205947
$636$	−20.2143	−0.801551
$637$	61.4217	2.43362
$638$	−1.83616	−0.0726941
$639$	11.9634	0.473265
$640$	0.244936	0.00968195
$641$	24.6608	0.974044	0.487022	−	0.873390i	$-0.338083\pi$
$641$	24.6608	0.974044	0.487022	+	0.873390i	$0.338083\pi$
$642$	−0.260415	−0.0102777
$643$	−43.5581	−1.71776	−0.858882	−	0.512173i	$-0.828841\pi$
$643$	−43.5581	−1.71776	−0.858882	+	0.512173i	$0.828841\pi$
$644$	56.9507	2.24417
$645$	−0.381704	−0.0150296
$646$	−0.0523446	−0.00205947
$647$	10.6798	0.419867	0.209934	−	0.977716i	$-0.432675\pi$
$647$	10.6798	0.419867	0.209934	+	0.977716i	$0.432675\pi$
$648$	0.259209	0.0101827
$649$	−10.0308	−0.393742
$650$	1.11995	0.0439282
$651$	−11.2464	−0.440782
$652$	−17.6549	−0.691421
$653$	−10.7084	−0.419052	−0.209526	−	0.977803i	$-0.567192\pi$
$653$	−10.7084	−0.419052	−0.209526	+	0.977803i	$0.567192\pi$
$654$	0.392791	0.0153593
$655$	0.209561	0.00818824
$656$	25.1640	0.982489
$657$	5.38589	0.210124
$658$	3.09754	0.120755
$659$	−6.67168	−0.259892	−0.129946	−	0.991521i	$-0.541480\pi$
$659$	−6.67168	−0.259892	−0.129946	+	0.991521i	$0.541480\pi$
$660$	−1.02776	−0.0400054
$661$	13.4694	0.523899	0.261949	−	0.965082i	$-0.415635\pi$
$661$	13.4694	0.523899	0.261949	+	0.965082i	$0.415635\pi$
$662$	−1.26926	−0.0493311
$663$	3.46263	0.134477
$664$	2.96890	0.115216
$665$	0.476051	0.0184605
$666$	0.130575	0.00505966
$667$	−37.4048	−1.44832
$668$	36.2065	1.40087
$669$	2.45175	0.0947900
$670$	−0.0619985	−0.00239521
$671$	33.2737	1.28452
$672$	−3.86096	−0.148940
$673$	−24.5621	−0.946798	−0.473399	−	0.880848i	$-0.656973\pi$
$673$	−24.5621	−0.946798	−0.473399	+	0.880848i	$0.656973\pi$
$674$	−1.71921	−0.0662213
$675$	4.98593	0.191909
$676$	2.01615	0.0775443
$677$	32.6567	1.25510	0.627549	−	0.778577i	$-0.284058\pi$
$677$	32.6567	1.25510	0.627549	+	0.778577i	$0.284058\pi$
$678$	−0.161187	−0.00619034
$679$	41.3552	1.58707
$680$	−0.0307463	−0.00117907
$681$	16.6359	0.637490
$682$	−0.636806	−0.0243846
$683$	21.1801	0.810435	0.405218	−	0.914220i	$-0.367196\pi$
$683$	21.1801	0.810435	0.405218	+	0.914220i	$0.367196\pi$
$684$	1.61042	0.0615760
$685$	−0.203520	−0.00777611
$686$	3.46478	0.132286
$687$	17.1990	0.656182
$688$	12.7907	0.487642
$689$	−35.0712	−1.33611
$690$	0.0441457	0.00168060
$691$	−40.0266	−1.52268	−0.761341	−	0.648351i	$-0.775459\pi$
$691$	−40.0266	−1.52268	−0.761341	+	0.648351i	$0.775459\pi$
$692$	−30.8993	−1.17462
$693$	21.5933	0.820261
$694$	−1.10243	−0.0418478
$695$	1.84713	0.0700658
$696$	1.68997	0.0640583
$697$	−6.33093	−0.239801
$698$	−1.72924	−0.0654526
$699$	3.83870	0.145193
$700$	−49.4935	−1.87068
$701$	−15.2192	−0.574819	−0.287410	−	0.957808i	$-0.592794\pi$
$701$	−15.2192	−0.574819	−0.287410	+	0.957808i	$0.592794\pi$
$702$	0.224623	0.00847783
$703$	1.62419	0.0612573
$704$	34.2937	1.29249
$705$	−1.13874	−0.0428876
$706$	−0.946484	−0.0356214
$707$	−39.4838	−1.48494
$708$	4.61123	0.173301
$709$	42.8476	1.60917	0.804587	−	0.593834i	$-0.202387\pi$
$709$	42.8476	1.60917	0.804587	+	0.593834i	$0.202387\pi$
$710$	−0.0920545	−0.00345474
$711$	9.95475	0.373332
$712$	−2.63662	−0.0988115
$713$	−12.9725	−0.485826
$714$	0.322652	0.0120749
$715$	−1.78312	−0.0666850
$716$	7.47800	0.279466
$717$	−7.17770	−0.268056
$718$	0.998499	0.0372637
$719$	10.8590	0.404972	0.202486	−	0.979285i	$-0.435098\pi$
$719$	10.8590	0.404972	0.202486	+	0.979285i	$0.435098\pi$
$720$	0.471471	0.0175707
$721$	−59.7045	−2.22351
$722$	1.19030	0.0442985
$723$	23.1575	0.861237
$724$	23.5099	0.873738
$725$	32.5069	1.20728
$726$	0.509102	0.0188945
$727$	−22.6353	−0.839497	−0.419748	−	0.907640i	$-0.637882\pi$
$727$	−22.6353	−0.839497	−0.419748	+	0.907640i	$0.637882\pi$
$728$	−4.46419	−0.165454
$729$	1.00000	0.0370370
$730$	−0.0414427	−0.00153386
$731$	−3.21798	−0.119021
$732$	−15.2962	−0.565364
$733$	27.3952	1.01186	0.505932	−	0.862573i	$-0.331149\pi$
$733$	27.3952	1.01186	0.505932	+	0.862573i	$0.331149\pi$
$734$	0.801355	0.0295786
$735$	−2.10406	−0.0776096
$736$	−4.45356	−0.164160
$737$	−34.9802	−1.28851
$738$	−0.410691	−0.0151177
$739$	−40.0980	−1.47503	−0.737514	−	0.675332i	$-0.764000\pi$
$739$	−40.0980	−1.47503	−0.737514	+	0.675332i	$0.764000\pi$
$740$	0.476507	0.0175168
$741$	2.79403	0.102641
$742$	−3.26797	−0.119971
$743$	−5.27356	−0.193468	−0.0967341	−	0.995310i	$-0.530840\pi$
$743$	−5.27356	−0.193468	−0.0967341	+	0.995310i	$0.530840\pi$
$744$	0.586108	0.0214878
$745$	0.694509	0.0254448
$746$	−0.997749	−0.0365302
$747$	11.4537	0.419069
$748$	−8.66458	−0.316808
$749$	19.9666	0.729564
$750$	−0.0768385	−0.00280575
$751$	−21.2154	−0.774161	−0.387080	−	0.922046i	$-0.626516\pi$
$751$	−21.2154	−0.774161	−0.387080	+	0.922046i	$0.626516\pi$
$752$	38.1588	1.39151
$753$	−21.1381	−0.770315
$754$	1.46448	0.0533332
$755$	1.15299	0.0419617
$756$	−9.92662	−0.361028
$757$	−15.5366	−0.564688	−0.282344	−	0.959313i	$-0.591112\pi$
$757$	−15.5366	−0.564688	−0.282344	+	0.959313i	$0.591112\pi$
$758$	1.64553	0.0597685
$759$	24.9075	0.904085
$760$	−0.0248095	−0.000899935 0
$761$	23.2789	0.843858	0.421929	−	0.906629i	$-0.361353\pi$
$761$	23.2789	0.843858	0.421929	+	0.906629i	$0.361353\pi$
$762$	0.283822	0.0102818
$763$	−30.1162	−1.09028
$764$	45.5248	1.64703
$765$	−0.118616	−0.00428857
$766$	−1.62917	−0.0588644
$767$	8.00033	0.288875
$768$	−15.6644	−0.565241
$769$	−39.8062	−1.43545	−0.717724	−	0.696328i	$-0.754816\pi$
$769$	−39.8062	−1.43545	−0.717724	+	0.696328i	$0.754816\pi$
$770$	−0.166153	−0.00598775
$771$	13.3372	0.480327
$772$	14.2471	0.512763
$773$	0.618675	0.0222522	0.0111261	−	0.999938i	$-0.496458\pi$
$773$	0.618675	0.0222522	0.0111261	+	0.999938i	$0.496458\pi$
$774$	−0.208752	−0.00750344
$775$	11.2739	0.404970
$776$	−2.15523	−0.0773683
$777$	−10.0115	−0.359159
$778$	0.901760	0.0323297
$779$	−5.10848	−0.183030
$780$	0.819717	0.0293506
$781$	−51.9382	−1.85849
$782$	0.372173	0.0133089
$783$	6.51973	0.232996
$784$	70.5063	2.51808
$785$	−0.118616	−0.00423358
$786$	0.114608	0.00408794
$787$	−30.6959	−1.09419	−0.547096	−	0.837070i	$-0.684267\pi$
$787$	−30.6959	−1.09419	−0.547096	+	0.837070i	$0.684267\pi$
$788$	−3.42061	−0.121854
$789$	22.0305	0.784305
$790$	−0.0765986	−0.00272526
$791$	12.3586	0.439420
$792$	−1.12534	−0.0399871
$793$	−26.5384	−0.942405
$794$	−2.18696	−0.0776123
$795$	1.20140	0.0426092
$796$	5.17953	0.183584
$797$	9.23042	0.326958	0.163479	−	0.986547i	$-0.447728\pi$
$797$	9.23042	0.326958	0.163479	+	0.986547i	$0.447728\pi$
$798$	0.260350	0.00921631
$799$	−9.60026	−0.339633
$800$	3.87040	0.136839
$801$	−10.1718	−0.359402
$802$	0.542903	0.0191706
$803$	−23.3824	−0.825148
$804$	16.0807	0.567124
$805$	−3.38475	−0.119297
$806$	0.507903	0.0178901
$807$	7.05221	0.248249
$808$	2.05771	0.0723898
$809$	−3.92763	−0.138088	−0.0690440	−	0.997614i	$-0.521995\pi$
$809$	−3.92763	−0.138088	−0.0690440	+	0.997614i	$0.521995\pi$
$810$	−0.00769468	−0.000270363 0
$811$	−36.1806	−1.27047	−0.635236	−	0.772318i	$-0.719097\pi$
$811$	−36.1806	−1.27047	−0.635236	+	0.772318i	$0.719097\pi$
$812$	−64.7189	−2.27119
$813$	11.5716	0.405835
$814$	−0.566880	−0.0198691
$815$	1.04929	0.0367549
$816$	3.97477	0.139145
$817$	−2.59662	−0.0908442
$818$	−0.173290	−0.00605895
$819$	−17.2223	−0.601797
$820$	−1.49874	−0.0523382
$821$	−21.1093	−0.736720	−0.368360	−	0.929683i	$-0.620081\pi$
$821$	−21.1093	−0.736720	−0.368360	+	0.929683i	$0.620081\pi$
$822$	−0.111304	−0.00388218
$823$	−31.6091	−1.10182	−0.550912	−	0.834563i	$-0.685720\pi$
$823$	−31.6091	−1.10182	−0.550912	+	0.834563i	$0.685720\pi$
$824$	3.11151	0.108395
$825$	−21.6460	−0.753618
$826$	0.745479	0.0259385
$827$	−20.8886	−0.726367	−0.363183	−	0.931718i	$-0.618310\pi$
$827$	−20.8886	−0.726367	−0.363183	+	0.931718i	$0.618310\pi$
$828$	−11.4502	−0.397922
$829$	−31.4468	−1.09219	−0.546096	−	0.837723i	$-0.683887\pi$
$829$	−31.4468	−1.09219	−0.546096	+	0.837723i	$0.683887\pi$
$830$	−0.0881325	−0.00305912
$831$	−8.51107	−0.295246
$832$	−27.3519	−0.948258
$833$	−17.7385	−0.614601
$834$	1.01019	0.0349800
$835$	−2.15186	−0.0744682
$836$	−6.99153	−0.241807
$837$	2.26114	0.0781564
$838$	−2.06473	−0.0713248
$839$	23.1505	0.799245	0.399622	−	0.916680i	$-0.369141\pi$
$839$	23.1505	0.799245	0.399622	+	0.916680i	$0.369141\pi$
$840$	0.152925	0.00527643
$841$	13.5069	0.465756
$842$	0.326777	0.0112615
$843$	17.8087	0.613364
$844$	26.4092	0.909041
$845$	−0.119826	−0.00412214
$846$	−0.622774	−0.0214114
$847$	−39.0340	−1.34123
$848$	−40.2584	−1.38248
$849$	21.5937	0.741093
$850$	−0.323440	−0.0110939
$851$	−11.5481	−0.395862
$852$	23.8765	0.817994
$853$	−54.9840	−1.88262	−0.941308	−	0.337548i	$-0.890403\pi$
$853$	−54.9840	−1.88262	−0.941308	+	0.337548i	$0.890403\pi$
$854$	−2.47287	−0.0846200
$855$	−0.0957122	−0.00327329
$856$	−1.04056	−0.0355657
$857$	32.0628	1.09524	0.547622	−	0.836726i	$-0.315533\pi$
$857$	32.0628	1.09524	0.547622	+	0.836726i	$0.315533\pi$
$858$	−0.975182	−0.0332922
$859$	−35.0132	−1.19463	−0.597317	−	0.802005i	$-0.703767\pi$
$859$	−35.0132	−1.19463	−0.597317	+	0.802005i	$0.703767\pi$
$860$	−0.761802	−0.0259772
$861$	31.4886	1.07313
$862$	−1.05161	−0.0358179
$863$	23.9360	0.814793	0.407396	−	0.913251i	$-0.366437\pi$
$863$	23.9360	0.814793	0.407396	+	0.913251i	$0.366437\pi$
$864$	0.776264	0.0264090
$865$	1.83644	0.0624408
$866$	−1.29429	−0.0439818
$867$	−1.00000	−0.0339618
$868$	−22.4455	−0.761850
$869$	−43.2178	−1.46606
$870$	−0.0501673	−0.00170083
$871$	27.8995	0.945339
$872$	1.56951	0.0531503
$873$	−8.31464	−0.281408
$874$	0.300310	0.0101581
$875$	5.89139	0.199165
$876$	10.7491	0.363179
$877$	8.94810	0.302156	0.151078	−	0.988522i	$-0.451726\pi$
$877$	8.94810	0.302156	0.151078	+	0.988522i	$0.451726\pi$
$878$	0.457096	0.0154262
$879$	18.1737	0.612984
$880$	−2.04686	−0.0689995
$881$	40.0704	1.35001	0.675004	−	0.737814i	$-0.264142\pi$
$881$	40.0704	1.35001	0.675004	+	0.737814i	$0.264142\pi$
$882$	−1.15070	−0.0387462
$883$	41.9711	1.41244	0.706221	−	0.707992i	$-0.250399\pi$
$883$	41.9711	1.41244	0.706221	+	0.707992i	$0.250399\pi$
$884$	6.91069	0.232432
$885$	−0.274059	−0.00921241
$886$	−2.00376	−0.0673177
$887$	−12.5179	−0.420309	−0.210154	−	0.977668i	$-0.567397\pi$
$887$	−12.5179	−0.420309	−0.210154	+	0.977668i	$0.567397\pi$
$888$	0.521749	0.0175087
$889$	−21.7613	−0.729851
$890$	0.0782686	0.00262357
$891$	−4.34142	−0.145443
$892$	4.89318	0.163836
$893$	−7.74654	−0.259228
$894$	0.379824	0.0127032
$895$	−0.444440	−0.0148560
$896$	−10.2706	−0.343117
$897$	−19.8657	−0.663296
$898$	−0.446936	−0.0149144
$899$	14.7420	0.491674
$900$	9.95088	0.331696
$901$	10.1285	0.337429
$902$	1.78298	0.0593669
$903$	16.0055	0.532631
$904$	−0.644069	−0.0214214
$905$	−1.39726	−0.0464466
$906$	0.630566	0.0209492
$907$	25.4570	0.845287	0.422644	−	0.906296i	$-0.361102\pi$
$907$	25.4570	0.845287	0.422644	+	0.906296i	$0.361102\pi$
$908$	33.2018	1.10184
$909$	7.93840	0.263300
$910$	0.132520	0.00439301
$911$	−28.7376	−0.952118	−0.476059	−	0.879413i	$-0.657935\pi$
$911$	−28.7376	−0.952118	−0.476059	+	0.879413i	$0.657935\pi$
$912$	3.20728	0.106204
$913$	−49.7253	−1.64567
$914$	0.563431	0.0186366
$915$	0.909099	0.0300539
$916$	34.3256	1.13415
$917$	−8.78728	−0.290182
$918$	−0.0648705	−0.00214105
$919$	34.9113	1.15162	0.575809	−	0.817584i	$-0.304687\pi$
$919$	34.9113	1.15162	0.575809	+	0.817584i	$0.304687\pi$
$920$	0.176397	0.00581564
$921$	3.61718	0.119190
$922$	0.887165	0.0292172
$923$	41.4248	1.36351
$924$	43.0957	1.41774
$925$	10.0359	0.329979
$926$	−0.126240	−0.00414850
$927$	12.0039	0.394258
$928$	5.06103	0.166137
$929$	41.1661	1.35062	0.675308	−	0.737536i	$-0.264011\pi$
$929$	41.1661	1.35062	0.675308	+	0.737536i	$0.264011\pi$
$930$	−0.0173988	−0.000570528 0
$931$	−14.3133	−0.469100
$932$	7.66125	0.250953
$933$	22.1185	0.724127
$934$	−0.782839	−0.0256153
$935$	0.514962	0.0168411
$936$	0.897545	0.0293372
$937$	−32.2313	−1.05295	−0.526475	−	0.850191i	$-0.676487\pi$
$937$	−32.2313	−1.05295	−0.526475	+	0.850191i	$0.676487\pi$
$938$	2.59971	0.0848834
$939$	−11.2934	−0.368545
$940$	−2.27270	−0.0741272
$941$	−9.17313	−0.299035	−0.149518	−	0.988759i	$-0.547772\pi$
$941$	−9.17313	−0.299035	−0.149518	+	0.988759i	$0.547772\pi$
$942$	−0.0648705	−0.00211360
$943$	36.3216	1.18280
$944$	9.18361	0.298901
$945$	0.589969	0.0191917
$946$	0.906282	0.0294658
$947$	−28.3405	−0.920944	−0.460472	−	0.887674i	$-0.652320\pi$
$947$	−28.3405	−0.920944	−0.460472	+	0.887674i	$0.652320\pi$
$948$	19.8676	0.645270
$949$	18.6493	0.605383
$950$	−0.260987	−0.00846752
$951$	7.17556	0.232683
$952$	1.28925	0.0417848
$953$	7.59226	0.245937	0.122969	−	0.992411i	$-0.460759\pi$
$953$	7.59226	0.245937	0.122969	+	0.992411i	$0.460759\pi$
$954$	0.657040	0.0212725
$955$	−2.70568	−0.0875537
$956$	−14.3252	−0.463310
$957$	−28.3049	−0.914969
$958$	−1.07200	−0.0346347
$959$	8.53397	0.275576
$960$	0.936969	0.0302405
$961$	−25.8872	−0.835072
$962$	0.452131	0.0145773
$963$	−4.01437	−0.129361
$964$	46.2176	1.48857
$965$	−0.846747	−0.0272577
$966$	−1.85111	−0.0595584
$967$	−14.0558	−0.452005	−0.226002	−	0.974127i	$-0.572566\pi$
$967$	−14.0558	−0.452005	−0.226002	+	0.974127i	$0.572566\pi$
$968$	2.03426	0.0653837
$969$	−0.806909	−0.0259216
$970$	0.0639785	0.00205423
$971$	−19.6087	−0.629273	−0.314636	−	0.949212i	$-0.601883\pi$
$971$	−19.6087	−0.629273	−0.314636	+	0.949212i	$0.601883\pi$
$972$	1.99579	0.0640151
$973$	−77.4536	−2.48305
$974$	0.875474	0.0280520
$975$	17.2644	0.552904
$976$	−30.4635	−0.975113
$977$	33.3842	1.06806	0.534028	−	0.845467i	$-0.320678\pi$
$977$	33.3842	1.06806	0.534028	+	0.845467i	$0.320678\pi$
$978$	0.573850	0.0183497
$979$	44.1600	1.41136
$980$	−4.19927	−0.134141
$981$	6.05499	0.193321
$982$	0.0544201	0.00173662
$983$	−55.2412	−1.76192	−0.880960	−	0.473190i	$-0.843102\pi$
$983$	−55.2412	−1.76192	−0.880960	+	0.473190i	$0.843102\pi$
$984$	−1.64104	−0.0523143
$985$	0.203297	0.00647759
$986$	−0.422939	−0.0134691
$987$	47.7496	1.51989
$988$	5.57629	0.177406
$989$	18.4621	0.587061
$990$	0.0334059	0.00106171
$991$	−21.6242	−0.686916	−0.343458	−	0.939168i	$-0.611598\pi$
$991$	−21.6242	−0.686916	−0.343458	+	0.939168i	$0.611598\pi$
$992$	1.75524	0.0557290
$993$	−19.5660	−0.620908
$994$	3.86001	0.122432
$995$	−0.307835	−0.00975903
$996$	22.8592	0.724321
$997$	−34.0508	−1.07840	−0.539200	−	0.842178i	$-0.681273\pi$
$997$	−34.0508	−1.07840	−0.539200	+	0.842178i	$0.681273\pi$
$998$	−1.34826	−0.0426783
$999$	2.01285	0.0636837

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8007.2.a.f.1.24	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8007.2.a.f.1.24	✓	48	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8007 = 3 \cdot 17 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8007.a (trivial)

\(f(q)\)	\(=\)	\(q-0.0648705 q^{2} -1.00000 q^{3} -1.99579 q^{4} +0.118616 q^{5} +0.0648705 q^{6} -4.97378 q^{7} +0.259209 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.0648705 q^{2} -1.00000 q^{3} -1.99579 q^{4} +0.118616 q^{5} +0.0648705 q^{6} -4.97378 q^{7} +0.259209 q^{8} +1.00000 q^{9} -0.00769468 q^{10} -4.34142 q^{11} +1.99579 q^{12} +3.46263 q^{13} +0.322652 q^{14} -0.118616 q^{15} +3.97477 q^{16} -1.00000 q^{17} -0.0648705 q^{18} -0.806909 q^{19} -0.236733 q^{20} +4.97378 q^{21} +0.281631 q^{22} +5.73717 q^{23} -0.259209 q^{24} -4.98593 q^{25} -0.224623 q^{26} -1.00000 q^{27} +9.92662 q^{28} -6.51973 q^{29} +0.00769468 q^{30} -2.26114 q^{31} -0.776264 q^{32} +4.34142 q^{33} +0.0648705 q^{34} -0.589969 q^{35} -1.99579 q^{36} -2.01285 q^{37} +0.0523446 q^{38} -3.46263 q^{39} +0.0307463 q^{40} +6.33093 q^{41} -0.322652 q^{42} +3.21798 q^{43} +8.66458 q^{44} +0.118616 q^{45} -0.372173 q^{46} +9.60026 q^{47} -3.97477 q^{48} +17.7385 q^{49} +0.323440 q^{50} +1.00000 q^{51} -6.91069 q^{52} -10.1285 q^{53} +0.0648705 q^{54} -0.514962 q^{55} -1.28925 q^{56} +0.806909 q^{57} +0.422939 q^{58} +2.31048 q^{59} +0.236733 q^{60} -7.66422 q^{61} +0.146681 q^{62} -4.97378 q^{63} -7.89918 q^{64} +0.410723 q^{65} -0.281631 q^{66} +8.05732 q^{67} +1.99579 q^{68} -5.73717 q^{69} +0.0382716 q^{70} +11.9634 q^{71} +0.259209 q^{72} +5.38589 q^{73} +0.130575 q^{74} +4.98593 q^{75} +1.61042 q^{76} +21.5933 q^{77} +0.224623 q^{78} +9.95475 q^{79} +0.471471 q^{80} +1.00000 q^{81} -0.410691 q^{82} +11.4537 q^{83} -9.92662 q^{84} -0.118616 q^{85} -0.208752 q^{86} +6.51973 q^{87} -1.12534 q^{88} -10.1718 q^{89} -0.00769468 q^{90} -17.2223 q^{91} -11.4502 q^{92} +2.26114 q^{93} -0.622774 q^{94} -0.0957122 q^{95} +0.776264 q^{96} -8.31464 q^{97} -1.15070 q^{98} -4.34142 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9}+O(q^{10}) \) 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9} - 20 q^{10} + 5 q^{11} - 45 q^{12} - 8 q^{13} + 4 q^{14} - q^{15} + 39 q^{16} - 48 q^{17} - q^{18} - 6 q^{19} + 6 q^{20} + 13 q^{21} - 35 q^{22} - 8 q^{23} + 6 q^{24} + 13 q^{25} + 17 q^{26} - 48 q^{27} - 38 q^{28} + q^{29} + 20 q^{30} - 21 q^{31} - 3 q^{32} - 5 q^{33} + q^{34} + 19 q^{35} + 45 q^{36} - 58 q^{37} - 14 q^{38} + 8 q^{39} - 54 q^{40} - 3 q^{41} - 4 q^{42} - 33 q^{43} + 2 q^{44} + q^{45} - 26 q^{46} + 9 q^{47} - 39 q^{48} + 11 q^{49} + 4 q^{50} + 48 q^{51} - 31 q^{52} - 33 q^{53} + q^{54} - 21 q^{55} + 6 q^{57} - 55 q^{58} + 77 q^{59} - 6 q^{60} - 29 q^{61} - 46 q^{62} - 13 q^{63} + 24 q^{64} - 49 q^{65} + 35 q^{66} - 44 q^{67} - 45 q^{68} + 8 q^{69} + 4 q^{70} + 22 q^{71} - 6 q^{72} - 63 q^{73} - 16 q^{74} - 13 q^{75} - 46 q^{76} - 30 q^{77} - 17 q^{78} - 46 q^{79} - 14 q^{80} + 48 q^{81} - 75 q^{82} + 11 q^{83} + 38 q^{84} - q^{85} + 8 q^{86} - q^{87} - 116 q^{88} + 10 q^{89} - 20 q^{90} - 67 q^{91} - 64 q^{92} + 21 q^{93} - 16 q^{94} - 8 q^{95} + 3 q^{96} - 96 q^{97} - 46 q^{98} + 5 q^{99}+O(q^{100}) \) 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9 - 20 * q^10 + 5 * q^11 - 45 * q^12 - 8 * q^13 + 4 * q^14 - q^15 + 39 * q^16 - 48 * q^17 - q^18 - 6 * q^19 + 6 * q^20 + 13 * q^21 - 35 * q^22 - 8 * q^23 + 6 * q^24 + 13 * q^25 + 17 * q^26 - 48 * q^27 - 38 * q^28 + q^29 + 20 * q^30 - 21 * q^31 - 3 * q^32 - 5 * q^33 + q^34 + 19 * q^35 + 45 * q^36 - 58 * q^37 - 14 * q^38 + 8 * q^39 - 54 * q^40 - 3 * q^41 - 4 * q^42 - 33 * q^43 + 2 * q^44 + q^45 - 26 * q^46 + 9 * q^47 - 39 * q^48 + 11 * q^49 + 4 * q^50 + 48 * q^51 - 31 * q^52 - 33 * q^53 + q^54 - 21 * q^55 + 6 * q^57 - 55 * q^58 + 77 * q^59 - 6 * q^60 - 29 * q^61 - 46 * q^62 - 13 * q^63 + 24 * q^64 - 49 * q^65 + 35 * q^66 - 44 * q^67 - 45 * q^68 + 8 * q^69 + 4 * q^70 + 22 * q^71 - 6 * q^72 - 63 * q^73 - 16 * q^74 - 13 * q^75 - 46 * q^76 - 30 * q^77 - 17 * q^78 - 46 * q^79 - 14 * q^80 + 48 * q^81 - 75 * q^82 + 11 * q^83 + 38 * q^84 - q^85 + 8 * q^86 - q^87 - 116 * q^88 + 10 * q^89 - 20 * q^90 - 67 * q^91 - 64 * q^92 + 21 * q^93 - 16 * q^94 - 8 * q^95 + 3 * q^96 - 96 * q^97 - 46 * q^98 + 5 * q^99

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