LMFDB - Embedded newform 8002.2.a.e.1.2

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:	$0$
Dimension:	$77$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
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)

$f(q)$	$=$	$q-1.00000 q^{2} -3.14716 q^{3} +1.00000 q^{4} -2.17339 q^{5} +3.14716 q^{6} +4.77090 q^{7} -1.00000 q^{8} +6.90459 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -3.14716 q^{3} +1.00000 q^{4} -2.17339 q^{5} +3.14716 q^{6} +4.77090 q^{7} -1.00000 q^{8} +6.90459 q^{9} +2.17339 q^{10} +0.464609 q^{11} -3.14716 q^{12} -3.54045 q^{13} -4.77090 q^{14} +6.84001 q^{15} +1.00000 q^{16} +2.17248 q^{17} -6.90459 q^{18} -7.81420 q^{19} -2.17339 q^{20} -15.0148 q^{21} -0.464609 q^{22} -2.67609 q^{23} +3.14716 q^{24} -0.276360 q^{25} +3.54045 q^{26} -12.2884 q^{27} +4.77090 q^{28} -4.66474 q^{29} -6.84001 q^{30} -2.75066 q^{31} -1.00000 q^{32} -1.46220 q^{33} -2.17248 q^{34} -10.3691 q^{35} +6.90459 q^{36} -3.49564 q^{37} +7.81420 q^{38} +11.1423 q^{39} +2.17339 q^{40} +0.752133 q^{41} +15.0148 q^{42} +0.446218 q^{43} +0.464609 q^{44} -15.0064 q^{45} +2.67609 q^{46} +6.33329 q^{47} -3.14716 q^{48} +15.7615 q^{49} +0.276360 q^{50} -6.83713 q^{51} -3.54045 q^{52} -5.24165 q^{53} +12.2884 q^{54} -1.00978 q^{55} -4.77090 q^{56} +24.5925 q^{57} +4.66474 q^{58} +12.6820 q^{59} +6.84001 q^{60} -6.29902 q^{61} +2.75066 q^{62} +32.9411 q^{63} +1.00000 q^{64} +7.69478 q^{65} +1.46220 q^{66} -7.37205 q^{67} +2.17248 q^{68} +8.42209 q^{69} +10.3691 q^{70} -10.7392 q^{71} -6.90459 q^{72} +14.9326 q^{73} +3.49564 q^{74} +0.869748 q^{75} -7.81420 q^{76} +2.21661 q^{77} -11.1423 q^{78} -2.14485 q^{79} -2.17339 q^{80} +17.9596 q^{81} -0.752133 q^{82} -1.26263 q^{83} -15.0148 q^{84} -4.72165 q^{85} -0.446218 q^{86} +14.6807 q^{87} -0.464609 q^{88} +10.3386 q^{89} +15.0064 q^{90} -16.8911 q^{91} -2.67609 q^{92} +8.65674 q^{93} -6.33329 q^{94} +16.9833 q^{95} +3.14716 q^{96} -5.61370 q^{97} -15.7615 q^{98} +3.20794 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9}+O(q^{10}) $ 77 * q - 77 * q^2 + 10 * q^3 + 77 * q^4 + 18 * q^5 - 10 * q^6 + 21 * q^7 - 77 * q^8 + 71 * q^9	\( 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9} - 18 q^{10} + 30 q^{11} + 10 q^{12} - 2 q^{13} - 21 q^{14} + 21 q^{15} + 77 q^{16} + 60 q^{17} - 71 q^{18} - 3 q^{19} + 18 q^{20} + 10 q^{21} - 30 q^{22} + 53 q^{23} - 10 q^{24} + 59 q^{25} + 2 q^{26} + 43 q^{27} + 21 q^{28} + 30 q^{29} - 21 q^{30} + 22 q^{31} - 77 q^{32} + 31 q^{33} - 60 q^{34} + 41 q^{35} + 71 q^{36} - 3 q^{37} + 3 q^{38} + 44 q^{39} - 18 q^{40} + 48 q^{41} - 10 q^{42} + 21 q^{43} + 30 q^{44} + 33 q^{45} - 53 q^{46} + 107 q^{47} + 10 q^{48} + 24 q^{49} - 59 q^{50} + 18 q^{51} - 2 q^{52} + 42 q^{53} - 43 q^{54} + 49 q^{55} - 21 q^{56} + 32 q^{57} - 30 q^{58} + 42 q^{59} + 21 q^{60} - 31 q^{61} - 22 q^{62} + 109 q^{63} + 77 q^{64} + 39 q^{65} - 31 q^{66} - q^{67} + 60 q^{68} - 33 q^{69} - 41 q^{70} + 58 q^{71} - 71 q^{72} + 35 q^{73} + 3 q^{74} + 34 q^{75} - 3 q^{76} + 86 q^{77} - 44 q^{78} + 25 q^{79} + 18 q^{80} + 53 q^{81} - 48 q^{82} + 107 q^{83} + 10 q^{84} + 21 q^{85} - 21 q^{86} + 100 q^{87} - 30 q^{88} + 34 q^{89} - 33 q^{90} - 51 q^{91} + 53 q^{92} + 48 q^{93} - 107 q^{94} + 118 q^{95} - 10 q^{96} - 13 q^{97} - 24 q^{98} + 63 q^{99}+O(q^{100}) \) 77 * q - 77 * q^2 + 10 * q^3 + 77 * q^4 + 18 * q^5 - 10 * q^6 + 21 * q^7 - 77 * q^8 + 71 * q^9 - 18 * q^10 + 30 * q^11 + 10 * q^12 - 2 * q^13 - 21 * q^14 + 21 * q^15 + 77 * q^16 + 60 * q^17 - 71 * q^18 - 3 * q^19 + 18 * q^20 + 10 * q^21 - 30 * q^22 + 53 * q^23 - 10 * q^24 + 59 * q^25 + 2 * q^26 + 43 * q^27 + 21 * q^28 + 30 * q^29 - 21 * q^30 + 22 * q^31 - 77 * q^32 + 31 * q^33 - 60 * q^34 + 41 * q^35 + 71 * q^36 - 3 * q^37 + 3 * q^38 + 44 * q^39 - 18 * q^40 + 48 * q^41 - 10 * q^42 + 21 * q^43 + 30 * q^44 + 33 * q^45 - 53 * q^46 + 107 * q^47 + 10 * q^48 + 24 * q^49 - 59 * q^50 + 18 * q^51 - 2 * q^52 + 42 * q^53 - 43 * q^54 + 49 * q^55 - 21 * q^56 + 32 * q^57 - 30 * q^58 + 42 * q^59 + 21 * q^60 - 31 * q^61 - 22 * q^62 + 109 * q^63 + 77 * q^64 + 39 * q^65 - 31 * q^66 - q^67 + 60 * q^68 - 33 * q^69 - 41 * q^70 + 58 * q^71 - 71 * q^72 + 35 * q^73 + 3 * q^74 + 34 * q^75 - 3 * q^76 + 86 * q^77 - 44 * q^78 + 25 * q^79 + 18 * q^80 + 53 * q^81 - 48 * q^82 + 107 * q^83 + 10 * q^84 + 21 * q^85 - 21 * q^86 + 100 * q^87 - 30 * q^88 + 34 * q^89 - 33 * q^90 - 51 * q^91 + 53 * q^92 + 48 * q^93 - 107 * q^94 + 118 * q^95 - 10 * q^96 - 13 * q^97 - 24 * q^98 + 63 * 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
$3$	−3.14716	−1.81701	−0.908506	−	0.417873i	$-0.862776\pi$
$3$	−3.14716	−1.81701	−0.908506	+	0.417873i	$0.862776\pi$
$4$	1.00000	0.500000
$5$	−2.17339	−0.971971	−0.485986	−	0.873967i	$-0.661539\pi$
$5$	−2.17339	−0.971971	−0.485986	+	0.873967i	$0.661539\pi$
$6$	3.14716	1.28482
$7$	4.77090	1.80323	0.901616	−	0.432537i	$-0.142382\pi$
$7$	4.77090	1.80323	0.901616	+	0.432537i	$0.142382\pi$
$8$	−1.00000	−0.353553
$9$	6.90459	2.30153
$10$	2.17339	0.687287
$11$	0.464609	0.140085	0.0700425	−	0.997544i	$-0.477687\pi$
$11$	0.464609	0.140085	0.0700425	+	0.997544i	$0.477687\pi$
$12$	−3.14716	−0.908506
$13$	−3.54045	−0.981943	−0.490972	−	0.871176i	$-0.663358\pi$
$13$	−3.54045	−0.981943	−0.490972	+	0.871176i	$0.663358\pi$
$14$	−4.77090	−1.27508
$15$	6.84001	1.76608
$16$	1.00000	0.250000
$17$	2.17248	0.526903	0.263452	−	0.964673i	$-0.415139\pi$
$17$	2.17248	0.526903	0.263452	+	0.964673i	$0.415139\pi$
$18$	−6.90459	−1.62743
$19$	−7.81420	−1.79270	−0.896350	−	0.443347i	$-0.853791\pi$
$19$	−7.81420	−1.79270	−0.896350	+	0.443347i	$0.853791\pi$
$20$	−2.17339	−0.485986
$21$	−15.0148	−3.27649
$22$	−0.464609	−0.0990551
$23$	−2.67609	−0.558004	−0.279002	−	0.960290i	$-0.590004\pi$
$23$	−2.67609	−0.558004	−0.279002	+	0.960290i	$0.590004\pi$
$24$	3.14716	0.642410
$25$	−0.276360	−0.0552720
$26$	3.54045	0.694339
$27$	−12.2884	−2.36489
$28$	4.77090	0.901616
$29$	−4.66474	−0.866220	−0.433110	−	0.901341i	$-0.642584\pi$
$29$	−4.66474	−0.866220	−0.433110	+	0.901341i	$0.642584\pi$
$30$	−6.84001	−1.24881
$31$	−2.75066	−0.494032	−0.247016	−	0.969011i	$-0.579450\pi$
$31$	−2.75066	−0.494032	−0.247016	+	0.969011i	$0.579450\pi$
$32$	−1.00000	−0.176777
$33$	−1.46220	−0.254536
$34$	−2.17248	−0.372577
$35$	−10.3691	−1.75269
$36$	6.90459	1.15076
$37$	−3.49564	−0.574679	−0.287340	−	0.957829i	$-0.592771\pi$
$37$	−3.49564	−0.574679	−0.287340	+	0.957829i	$0.592771\pi$
$38$	7.81420	1.26763
$39$	11.1423	1.78420
$40$	2.17339	0.343644
$41$	0.752133	0.117463	0.0587317	−	0.998274i	$-0.481294\pi$
$41$	0.752133	0.117463	0.0587317	+	0.998274i	$0.481294\pi$
$42$	15.0148	2.31683
$43$	0.446218	0.0680476	0.0340238	−	0.999421i	$-0.489168\pi$
$43$	0.446218	0.0680476	0.0340238	+	0.999421i	$0.489168\pi$
$44$	0.464609	0.0700425
$45$	−15.0064	−2.23702
$46$	2.67609	0.394569
$47$	6.33329	0.923805	0.461903	−	0.886931i	$-0.347167\pi$
$47$	6.33329	0.923805	0.461903	+	0.886931i	$0.347167\pi$
$48$	−3.14716	−0.454253
$49$	15.7615	2.25165
$50$	0.276360	0.0390832
$51$	−6.83713	−0.957389
$52$	−3.54045	−0.490972
$53$	−5.24165	−0.719996	−0.359998	−	0.932953i	$-0.617223\pi$
$53$	−5.24165	−0.719996	−0.359998	+	0.932953i	$0.617223\pi$
$54$	12.2884	1.67223
$55$	−1.00978	−0.136159
$56$	−4.77090	−0.637539
$57$	24.5925	3.25736
$58$	4.66474	0.612510
$59$	12.6820	1.65105	0.825525	−	0.564365i	$-0.190879\pi$
$59$	12.6820	1.65105	0.825525	+	0.564365i	$0.190879\pi$
$60$	6.84001	0.883041
$61$	−6.29902	−0.806507	−0.403254	−	0.915088i	$-0.632121\pi$
$61$	−6.29902	−0.806507	−0.403254	+	0.915088i	$0.632121\pi$
$62$	2.75066	0.349334
$63$	32.9411	4.15019
$64$	1.00000	0.125000
$65$	7.69478	0.954421
$66$	1.46220	0.179984
$67$	−7.37205	−0.900639	−0.450320	−	0.892867i	$-0.648690\pi$
$67$	−7.37205	−0.900639	−0.450320	+	0.892867i	$0.648690\pi$
$68$	2.17248	0.263452
$69$	8.42209	1.01390
$70$	10.3691	1.23934
$71$	−10.7392	−1.27451	−0.637257	−	0.770652i	$-0.719931\pi$
$71$	−10.7392	−1.27451	−0.637257	+	0.770652i	$0.719931\pi$
$72$	−6.90459	−0.813714
$73$	14.9326	1.74773	0.873866	−	0.486167i	$-0.161605\pi$
$73$	14.9326	1.74773	0.873866	+	0.486167i	$0.161605\pi$
$74$	3.49564	0.406360
$75$	0.869748	0.100430
$76$	−7.81420	−0.896350
$77$	2.21661	0.252606
$78$	−11.1423	−1.26162
$79$	−2.14485	−0.241314	−0.120657	−	0.992694i	$-0.538500\pi$
$79$	−2.14485	−0.241314	−0.120657	+	0.992694i	$0.538500\pi$
$80$	−2.17339	−0.242993
$81$	17.9596	1.99551
$82$	−0.752133	−0.0830592
$83$	−1.26263	−0.138592	−0.0692960	−	0.997596i	$-0.522075\pi$
$83$	−1.26263	−0.138592	−0.0692960	+	0.997596i	$0.522075\pi$
$84$	−15.0148	−1.63825
$85$	−4.72165	−0.512135
$86$	−0.446218	−0.0481169
$87$	14.6807	1.57393
$88$	−0.464609	−0.0495275
$89$	10.3386	1.09589	0.547945	−	0.836514i	$-0.315410\pi$
$89$	10.3386	1.09589	0.547945	+	0.836514i	$0.315410\pi$
$90$	15.0064	1.58181
$91$	−16.8911	−1.77067
$92$	−2.67609	−0.279002
$93$	8.65674	0.897662
$94$	−6.33329	−0.653229
$95$	16.9833	1.74245
$96$	3.14716	0.321205
$97$	−5.61370	−0.569984	−0.284992	−	0.958530i	$-0.591991\pi$
$97$	−5.61370	−0.569984	−0.284992	+	0.958530i	$0.591991\pi$
$98$	−15.7615	−1.59215
$99$	3.20794	0.322410
$100$	−0.276360	−0.0276360
$101$	4.91550	0.489110	0.244555	−	0.969635i	$-0.421358\pi$
$101$	4.91550	0.489110	0.244555	+	0.969635i	$0.421358\pi$
$102$	6.83713	0.676976
$103$	−13.7917	−1.35894	−0.679468	−	0.733705i	$-0.737789\pi$
$103$	−13.7917	−1.35894	−0.679468	+	0.733705i	$0.737789\pi$
$104$	3.54045	0.347169
$105$	32.6330	3.18466
$106$	5.24165	0.509114
$107$	14.6420	1.41550	0.707750	−	0.706463i	$-0.249710\pi$
$107$	14.6420	1.41550	0.707750	+	0.706463i	$0.249710\pi$
$108$	−12.2884	−1.18245
$109$	−11.7987	−1.13011	−0.565057	−	0.825052i	$-0.691146\pi$
$109$	−11.7987	−1.13011	−0.565057	+	0.825052i	$0.691146\pi$
$110$	1.00978	0.0962787
$111$	11.0013	1.04420
$112$	4.77090	0.450808
$113$	−16.7188	−1.57277	−0.786387	−	0.617735i	$-0.788051\pi$
$113$	−16.7188	−1.57277	−0.786387	+	0.617735i	$0.788051\pi$
$114$	−24.5925	−2.30330
$115$	5.81621	0.542364
$116$	−4.66474	−0.433110
$117$	−24.4453	−2.25997
$118$	−12.6820	−1.16747
$119$	10.3647	0.950129
$120$	−6.84001	−0.624404
$121$	−10.7841	−0.980376
$122$	6.29902	0.570287
$123$	−2.36708	−0.213432
$124$	−2.75066	−0.247016
$125$	11.4676	1.02569
$126$	−32.9411	−2.93463
$127$	−6.91452	−0.613564	−0.306782	−	0.951780i	$-0.599252\pi$
$127$	−6.91452	−0.613564	−0.306782	+	0.951780i	$0.599252\pi$
$128$	−1.00000	−0.0883883
$129$	−1.40432	−0.123643
$130$	−7.69478	−0.674877
$131$	2.61568	0.228533	0.114267	−	0.993450i	$-0.463548\pi$
$131$	2.61568	0.228533	0.114267	+	0.993450i	$0.463548\pi$
$132$	−1.46220	−0.127268
$133$	−37.2808	−3.23265
$134$	7.37205	0.636848
$135$	26.7074	2.29861
$136$	−2.17248	−0.186288
$137$	−2.09352	−0.178861	−0.0894307	−	0.995993i	$-0.528505\pi$
$137$	−2.09352	−0.178861	−0.0894307	+	0.995993i	$0.528505\pi$
$138$	−8.42209	−0.716936
$139$	0.169351	0.0143642	0.00718208	−	0.999974i	$-0.497714\pi$
$139$	0.169351	0.0143642	0.00718208	+	0.999974i	$0.497714\pi$
$140$	−10.3691	−0.876345
$141$	−19.9318	−1.67856
$142$	10.7392	0.901217
$143$	−1.64493	−0.137556
$144$	6.90459	0.575382
$145$	10.1383	0.841941
$146$	−14.9326	−1.23583
$147$	−49.6040	−4.09127
$148$	−3.49564	−0.287340
$149$	16.4336	1.34629	0.673145	−	0.739511i	$-0.264943\pi$
$149$	16.4336	1.34629	0.673145	+	0.739511i	$0.264943\pi$
$150$	−0.869748	−0.0710147
$151$	8.57860	0.698116	0.349058	−	0.937101i	$-0.386502\pi$
$151$	8.57860	0.698116	0.349058	+	0.937101i	$0.386502\pi$
$152$	7.81420	0.633815
$153$	15.0001	1.21268
$154$	−2.21661	−0.178619
$155$	5.97826	0.480185
$156$	11.1423	0.892101
$157$	−10.6449	−0.849558	−0.424779	−	0.905297i	$-0.639648\pi$
$157$	−10.6449	−0.849558	−0.424779	+	0.905297i	$0.639648\pi$
$158$	2.14485	0.170635
$159$	16.4963	1.30824
$160$	2.17339	0.171822
$161$	−12.7674	−1.00621
$162$	−17.9596	−1.41104
$163$	8.77566	0.687363	0.343681	−	0.939086i	$-0.388326\pi$
$163$	8.77566	0.687363	0.343681	+	0.939086i	$0.388326\pi$
$164$	0.752133	0.0587317
$165$	3.17793	0.247402
$166$	1.26263	0.0979994
$167$	−14.8025	−1.14545	−0.572725	−	0.819748i	$-0.694114\pi$
$167$	−14.8025	−1.14545	−0.572725	+	0.819748i	$0.694114\pi$
$168$	15.0148	1.15842
$169$	−0.465237	−0.0357874
$170$	4.72165	0.362134
$171$	−53.9538	−4.12595
$172$	0.446218	0.0340238
$173$	11.2411	0.854648	0.427324	−	0.904099i	$-0.359456\pi$
$173$	11.2411	0.854648	0.427324	+	0.904099i	$0.359456\pi$
$174$	−14.6807	−1.11294
$175$	−1.31849	−0.0996683
$176$	0.464609	0.0350213
$177$	−39.9121	−2.99998
$178$	−10.3386	−0.774912
$179$	−8.63646	−0.645519	−0.322760	−	0.946481i	$-0.604611\pi$
$179$	−8.63646	−0.645519	−0.322760	+	0.946481i	$0.604611\pi$
$180$	−15.0064	−1.11851
$181$	9.03712	0.671724	0.335862	−	0.941911i	$-0.390972\pi$
$181$	9.03712	0.671724	0.335862	+	0.941911i	$0.390972\pi$
$182$	16.8911	1.25205
$183$	19.8240	1.46543
$184$	2.67609	0.197284
$185$	7.59740	0.558572
$186$	−8.65674	−0.634743
$187$	1.00935	0.0738113
$188$	6.33329	0.461903
$189$	−58.6265	−4.26445
$190$	−16.9833	−1.23210
$191$	−9.35417	−0.676844	−0.338422	−	0.940994i	$-0.609893\pi$
$191$	−9.35417	−0.676844	−0.338422	+	0.940994i	$0.609893\pi$
$192$	−3.14716	−0.227126
$193$	−4.91824	−0.354023	−0.177012	−	0.984209i	$-0.556643\pi$
$193$	−4.91824	−0.354023	−0.177012	+	0.984209i	$0.556643\pi$
$194$	5.61370	0.403040
$195$	−24.2167	−1.73419
$196$	15.7615	1.12582
$197$	14.7015	1.04744	0.523718	−	0.851892i	$-0.324544\pi$
$197$	14.7015	1.04744	0.523718	+	0.851892i	$0.324544\pi$
$198$	−3.20794	−0.227978
$199$	10.5778	0.749844	0.374922	−	0.927056i	$-0.377670\pi$
$199$	10.5778	0.749844	0.374922	+	0.927056i	$0.377670\pi$
$200$	0.276360	0.0195416
$201$	23.2010	1.63647
$202$	−4.91550	−0.345853
$203$	−22.2550	−1.56200
$204$	−6.83713	−0.478695
$205$	−1.63468	−0.114171
$206$	13.7917	0.960913
$207$	−18.4773	−1.28426
$208$	−3.54045	−0.245486
$209$	−3.63055	−0.251130
$210$	−32.6330	−2.25189
$211$	−10.8499	−0.746937	−0.373469	−	0.927643i	$-0.621832\pi$
$211$	−10.8499	−0.746937	−0.373469	+	0.927643i	$0.621832\pi$
$212$	−5.24165	−0.359998
$213$	33.7981	2.31581
$214$	−14.6420	−1.00091
$215$	−0.969807	−0.0661403
$216$	12.2884	0.836116
$217$	−13.1231	−0.890855
$218$	11.7987	0.799111
$219$	−46.9953	−3.17565
$220$	−1.00978	−0.0680793
$221$	−7.69154	−0.517389
$222$	−11.0013	−0.738360
$223$	22.6580	1.51729	0.758646	−	0.651503i	$-0.225861\pi$
$223$	22.6580	1.51729	0.758646	+	0.651503i	$0.225861\pi$
$224$	−4.77090	−0.318769
$225$	−1.90815	−0.127210
$226$	16.7188	1.11212
$227$	7.21310	0.478750	0.239375	−	0.970927i	$-0.423057\pi$
$227$	7.21310	0.478750	0.239375	+	0.970927i	$0.423057\pi$
$228$	24.5925	1.62868
$229$	22.7121	1.50086	0.750430	−	0.660950i	$-0.229847\pi$
$229$	22.7121	1.50086	0.750430	+	0.660950i	$0.229847\pi$
$230$	−5.81621	−0.383509
$231$	−6.97601	−0.458988
$232$	4.66474	0.306255
$233$	3.14012	0.205716	0.102858	−	0.994696i	$-0.467201\pi$
$233$	3.14012	0.205716	0.102858	+	0.994696i	$0.467201\pi$
$234$	24.4453	1.59804
$235$	−13.7647	−0.897912
$236$	12.6820	0.825525
$237$	6.75016	0.438470
$238$	−10.3647	−0.671843
$239$	16.9961	1.09939	0.549694	−	0.835366i	$-0.314744\pi$
$239$	16.9961	1.09939	0.549694	+	0.835366i	$0.314744\pi$
$240$	6.84001	0.441521
$241$	−23.5304	−1.51573	−0.757863	−	0.652414i	$-0.773756\pi$
$241$	−23.5304	−1.51573	−0.757863	+	0.652414i	$0.773756\pi$
$242$	10.7841	0.693231
$243$	−19.6566	−1.26097
$244$	−6.29902	−0.403254
$245$	−34.2560	−2.18853
$246$	2.36708	0.150919
$247$	27.6658	1.76033
$248$	2.75066	0.174667
$249$	3.97370	0.251823
$250$	−11.4676	−0.725275
$251$	−1.14462	−0.0722479	−0.0361239	−	0.999347i	$-0.511501\pi$
$251$	−1.14462	−0.0722479	−0.0361239	+	0.999347i	$0.511501\pi$
$252$	32.9411	2.07510
$253$	−1.24334	−0.0781680
$254$	6.91452	0.433855
$255$	14.8598	0.930555
$256$	1.00000	0.0625000
$257$	15.7954	0.985288	0.492644	−	0.870231i	$-0.336031\pi$
$257$	15.7954	0.985288	0.492644	+	0.870231i	$0.336031\pi$
$258$	1.40432	0.0874290
$259$	−16.6774	−1.03628
$260$	7.69478	0.477210
$261$	−32.2081	−1.99363
$262$	−2.61568	−0.161597
$263$	−4.88652	−0.301316	−0.150658	−	0.988586i	$-0.548139\pi$
$263$	−4.88652	−0.301316	−0.150658	+	0.988586i	$0.548139\pi$
$264$	1.46220	0.0899921
$265$	11.3922	0.699816
$266$	37.2808	2.28583
$267$	−32.5372	−1.99125
$268$	−7.37205	−0.450320
$269$	5.45781	0.332769	0.166384	−	0.986061i	$-0.446791\pi$
$269$	5.45781	0.332769	0.166384	+	0.986061i	$0.446791\pi$
$270$	−26.7074	−1.62536
$271$	−2.93812	−0.178478	−0.0892391	−	0.996010i	$-0.528444\pi$
$271$	−2.93812	−0.178478	−0.0892391	+	0.996010i	$0.528444\pi$
$272$	2.17248	0.131726
$273$	53.1590	3.21733
$274$	2.09352	0.126474
$275$	−0.128400	−0.00774278
$276$	8.42209	0.506950
$277$	9.57581	0.575354	0.287677	−	0.957727i	$-0.407117\pi$
$277$	9.57581	0.575354	0.287677	+	0.957727i	$0.407117\pi$
$278$	−0.169351	−0.0101570
$279$	−18.9921	−1.13703
$280$	10.3691	0.619669
$281$	−29.9556	−1.78700	−0.893502	−	0.449059i	$-0.851759\pi$
$281$	−29.9556	−1.78700	−0.893502	+	0.449059i	$0.851759\pi$
$282$	19.9318	1.18692
$283$	−11.7339	−0.697506	−0.348753	−	0.937215i	$-0.613395\pi$
$283$	−11.7339	−0.697506	−0.348753	+	0.937215i	$0.613395\pi$
$284$	−10.7392	−0.637257
$285$	−53.4492	−3.16606
$286$	1.64493	0.0972665
$287$	3.58835	0.211814
$288$	−6.90459	−0.406857
$289$	−12.2803	−0.722373
$290$	−10.1383	−0.595342
$291$	17.6672	1.03567
$292$	14.9326	0.873866
$293$	2.11245	0.123411	0.0617053	−	0.998094i	$-0.480346\pi$
$293$	2.11245	0.123411	0.0617053	+	0.998094i	$0.480346\pi$
$294$	49.6040	2.89296
$295$	−27.5629	−1.60477
$296$	3.49564	0.203180
$297$	−5.70928	−0.331286
$298$	−16.4336	−0.951970
$299$	9.47457	0.547928
$300$	0.869748	0.0502149
$301$	2.12886	0.122706
$302$	−8.57860	−0.493643
$303$	−15.4698	−0.888719
$304$	−7.81420	−0.448175
$305$	13.6903	0.783902
$306$	−15.0001	−0.857497
$307$	−17.9660	−1.02538	−0.512688	−	0.858575i	$-0.671350\pi$
$307$	−17.9660	−1.02538	−0.512688	+	0.858575i	$0.671350\pi$
$308$	2.21661	0.126303
$309$	43.4046	2.46920
$310$	−5.97826	−0.339542
$311$	−11.6408	−0.660089	−0.330045	−	0.943965i	$-0.607064\pi$
$311$	−11.6408	−0.660089	−0.330045	+	0.943965i	$0.607064\pi$
$312$	−11.1423	−0.630811
$313$	−4.03950	−0.228326	−0.114163	−	0.993462i	$-0.536419\pi$
$313$	−4.03950	−0.228326	−0.114163	+	0.993462i	$0.536419\pi$
$314$	10.6449	0.600728
$315$	−71.5940	−4.03387
$316$	−2.14485	−0.120657
$317$	30.2160	1.69710	0.848549	−	0.529117i	$-0.177477\pi$
$317$	30.2160	1.69710	0.848549	+	0.529117i	$0.177477\pi$
$318$	−16.4963	−0.925066
$319$	−2.16728	−0.121345
$320$	−2.17339	−0.121496
$321$	−46.0808	−2.57198
$322$	12.7674	0.711499
$323$	−16.9762	−0.944579
$324$	17.9596	0.997755
$325$	0.978438	0.0542740
$326$	−8.77566	−0.486039
$327$	37.1324	2.05343
$328$	−0.752133	−0.0415296
$329$	30.2155	1.66583
$330$	−3.17793	−0.174939
$331$	−9.39681	−0.516496	−0.258248	−	0.966079i	$-0.583145\pi$
$331$	−9.39681	−0.516496	−0.258248	+	0.966079i	$0.583145\pi$
$332$	−1.26263	−0.0692960
$333$	−24.1359	−1.32264
$334$	14.8025	0.809955
$335$	16.0224	0.875396
$336$	−15.0148	−0.819123
$337$	36.6938	1.99884	0.999419	−	0.0340823i	$-0.0108508\pi$
$337$	36.6938	1.99884	0.999419	+	0.0340823i	$0.0108508\pi$
$338$	0.465237	0.0253055
$339$	52.6167	2.85775
$340$	−4.72165	−0.256067
$341$	−1.27798	−0.0692065
$342$	53.9538	2.91749
$343$	41.8004	2.25701
$344$	−0.446218	−0.0240585
$345$	−18.3045	−0.985482
$346$	−11.2411	−0.604327
$347$	6.79205	0.364616	0.182308	−	0.983241i	$-0.441643\pi$
$347$	6.79205	0.364616	0.182308	+	0.983241i	$0.441643\pi$
$348$	14.6807	0.786966
$349$	−23.8020	−1.27409	−0.637047	−	0.770825i	$-0.719844\pi$
$349$	−23.8020	−1.27409	−0.637047	+	0.770825i	$0.719844\pi$
$350$	1.31849	0.0704761
$351$	43.5063	2.32219
$352$	−0.464609	−0.0247638
$353$	−28.7773	−1.53166	−0.765830	−	0.643043i	$-0.777672\pi$
$353$	−28.7773	−1.53166	−0.765830	+	0.643043i	$0.777672\pi$
$354$	39.9121	2.12130
$355$	23.3406	1.23879
$356$	10.3386	0.547945
$357$	−32.6193	−1.72639
$358$	8.63646	0.456451
$359$	−10.3470	−0.546093	−0.273046	−	0.962001i	$-0.588031\pi$
$359$	−10.3470	−0.546093	−0.273046	+	0.962001i	$0.588031\pi$
$360$	15.0064	0.790906
$361$	42.0617	2.21377
$362$	−9.03712	−0.474980
$363$	33.9394	1.78135
$364$	−16.8911	−0.885336
$365$	−32.4545	−1.69874
$366$	−19.8240	−1.03622
$367$	16.8307	0.878557	0.439278	−	0.898351i	$-0.355234\pi$
$367$	16.8307	0.878557	0.439278	+	0.898351i	$0.355234\pi$
$368$	−2.67609	−0.139501
$369$	5.19317	0.270345
$370$	−7.59740	−0.394970
$371$	−25.0074	−1.29832
$372$	8.65674	0.448831
$373$	−21.1791	−1.09661	−0.548307	−	0.836277i	$-0.684727\pi$
$373$	−21.1791	−1.09661	−0.548307	+	0.836277i	$0.684727\pi$
$374$	−1.00935	−0.0521924
$375$	−36.0903	−1.86370
$376$	−6.33329	−0.326614
$377$	16.5153	0.850579
$378$	58.6265	3.01542
$379$	14.5322	0.746469	0.373234	−	0.927737i	$-0.378249\pi$
$379$	14.5322	0.746469	0.373234	+	0.927737i	$0.378249\pi$
$380$	16.9833	0.871226
$381$	21.7611	1.11485
$382$	9.35417	0.478601
$383$	20.0820	1.02614	0.513070	−	0.858347i	$-0.328508\pi$
$383$	20.0820	1.02614	0.513070	+	0.858347i	$0.328508\pi$
$384$	3.14716	0.160603
$385$	−4.81756	−0.245526
$386$	4.91824	0.250332
$387$	3.08095	0.156614
$388$	−5.61370	−0.284992
$389$	−17.2326	−0.873727	−0.436864	−	0.899528i	$-0.643911\pi$
$389$	−17.2326	−0.873727	−0.436864	+	0.899528i	$0.643911\pi$
$390$	24.2167	1.22626
$391$	−5.81376	−0.294014
$392$	−15.7615	−0.796077
$393$	−8.23195	−0.415247
$394$	−14.7015	−0.740649
$395$	4.66159	0.234550
$396$	3.20794	0.161205
$397$	−4.12386	−0.206970	−0.103485	−	0.994631i	$-0.532999\pi$
$397$	−4.12386	−0.206970	−0.103485	+	0.994631i	$0.532999\pi$
$398$	−10.5778	−0.530219
$399$	117.328	5.87377
$400$	−0.276360	−0.0138180
$401$	9.40494	0.469660	0.234830	−	0.972036i	$-0.424547\pi$
$401$	9.40494	0.469660	0.234830	+	0.972036i	$0.424547\pi$
$402$	−23.2010	−1.15716
$403$	9.73855	0.485112
$404$	4.91550	0.244555
$405$	−39.0332	−1.93958
$406$	22.2550	1.10450
$407$	−1.62411	−0.0805040
$408$	6.83713	0.338488
$409$	21.4215	1.05923	0.529613	−	0.848240i	$-0.322337\pi$
$409$	21.4215	1.05923	0.529613	+	0.848240i	$0.322337\pi$
$410$	1.63468	0.0807311
$411$	6.58863	0.324993
$412$	−13.7917	−0.679468
$413$	60.5044	2.97723
$414$	18.4773	0.908111
$415$	2.74420	0.134707
$416$	3.54045	0.173585
$417$	−0.532974	−0.0260999
$418$	3.63055	0.177576
$419$	−5.82387	−0.284515	−0.142257	−	0.989830i	$-0.545436\pi$
$419$	−5.82387	−0.284515	−0.142257	+	0.989830i	$0.545436\pi$
$420$	32.6330	1.59233
$421$	28.3087	1.37968	0.689841	−	0.723961i	$-0.257680\pi$
$421$	28.3087	1.37968	0.689841	+	0.723961i	$0.257680\pi$
$422$	10.8499	0.528165
$423$	43.7288	2.12616
$424$	5.24165	0.254557
$425$	−0.600386	−0.0291230
$426$	−33.7981	−1.63752
$427$	−30.0520	−1.45432
$428$	14.6420	0.707750
$429$	5.17684	0.249940
$430$	0.969807	0.0467683
$431$	−36.3042	−1.74871	−0.874356	−	0.485285i	$-0.838716\pi$
$431$	−36.3042	−1.74871	−0.874356	+	0.485285i	$0.838716\pi$
$432$	−12.2884	−0.591224
$433$	15.3140	0.735944	0.367972	−	0.929837i	$-0.380052\pi$
$433$	15.3140	0.735944	0.367972	+	0.929837i	$0.380052\pi$
$434$	13.1231	0.629929
$435$	−31.9069	−1.52982
$436$	−11.7987	−0.565057
$437$	20.9115	1.00033
$438$	46.9953	2.24552
$439$	19.2649	0.919465	0.459732	−	0.888057i	$-0.347945\pi$
$439$	19.2649	0.919465	0.459732	+	0.888057i	$0.347945\pi$
$440$	1.00978	0.0481393
$441$	108.827	5.18223
$442$	7.69154	0.365849
$443$	15.0494	0.715019	0.357509	−	0.933910i	$-0.383626\pi$
$443$	15.0494	0.715019	0.357509	+	0.933910i	$0.383626\pi$
$444$	11.0013	0.522099
$445$	−22.4699	−1.06517
$446$	−22.6580	−1.07289
$447$	−51.7190	−2.44622
$448$	4.77090	0.225404
$449$	−27.0747	−1.27773	−0.638867	−	0.769317i	$-0.720597\pi$
$449$	−27.0747	−1.27773	−0.638867	+	0.769317i	$0.720597\pi$
$450$	1.90815	0.0899512
$451$	0.349448	0.0164549
$452$	−16.7188	−0.786387
$453$	−26.9982	−1.26848
$454$	−7.21310	−0.338528
$455$	36.7111	1.72104
$456$	−24.5925	−1.15165
$457$	−35.3447	−1.65336	−0.826678	−	0.562675i	$-0.809772\pi$
$457$	−35.3447	−1.65336	−0.826678	+	0.562675i	$0.809772\pi$
$458$	−22.7121	−1.06127
$459$	−26.6962	−1.24607
$460$	5.81621	0.271182
$461$	−28.0998	−1.30874	−0.654370	−	0.756175i	$-0.727066\pi$
$461$	−28.0998	−1.30874	−0.654370	+	0.756175i	$0.727066\pi$
$462$	6.97601	0.324553
$463$	−34.2886	−1.59353	−0.796764	−	0.604290i	$-0.793457\pi$
$463$	−34.2886	−1.59353	−0.796764	+	0.604290i	$0.793457\pi$
$464$	−4.66474	−0.216555
$465$	−18.8145	−0.872502
$466$	−3.14012	−0.145463
$467$	35.1965	1.62870	0.814350	−	0.580375i	$-0.197094\pi$
$467$	35.1965	1.62870	0.814350	+	0.580375i	$0.197094\pi$
$468$	−24.4453	−1.12999
$469$	−35.1713	−1.62406
$470$	13.7647	0.634920
$471$	33.5013	1.54366
$472$	−12.6820	−0.583734
$473$	0.207317	0.00953245
$474$	−6.75016	−0.310045
$475$	2.15953	0.0990862
$476$	10.3647	0.475064
$477$	−36.1915	−1.65709
$478$	−16.9961	−0.777384
$479$	7.85768	0.359027	0.179513	−	0.983756i	$-0.442548\pi$
$479$	7.85768	0.359027	0.179513	+	0.983756i	$0.442548\pi$
$480$	−6.84001	−0.312202
$481$	12.3761	0.564302
$482$	23.5304	1.07178
$483$	40.1810	1.82830
$484$	−10.7841	−0.490188
$485$	12.2008	0.554008
$486$	19.6566	0.891640
$487$	13.9859	0.633761	0.316880	−	0.948465i	$-0.397365\pi$
$487$	13.9859	0.633761	0.316880	+	0.948465i	$0.397365\pi$
$488$	6.29902	0.285143
$489$	−27.6184	−1.24895
$490$	34.2560	1.54753
$491$	18.0688	0.815433	0.407717	−	0.913108i	$-0.366325\pi$
$491$	18.0688	0.815433	0.407717	+	0.913108i	$0.366325\pi$
$492$	−2.36708	−0.106716
$493$	−10.1340	−0.456414
$494$	−27.6658	−1.24474
$495$	−6.97211	−0.313373
$496$	−2.75066	−0.123508
$497$	−51.2359	−2.29824
$498$	−3.97370	−0.178066
$499$	−32.1063	−1.43728	−0.718639	−	0.695384i	$-0.755234\pi$
$499$	−32.1063	−1.43728	−0.718639	+	0.695384i	$0.755234\pi$
$500$	11.4676	0.512847
$501$	46.5857	2.08129
$502$	1.14462	0.0510870
$503$	25.3731	1.13133	0.565664	−	0.824636i	$-0.308620\pi$
$503$	25.3731	1.13133	0.565664	+	0.824636i	$0.308620\pi$
$504$	−32.9411	−1.46731
$505$	−10.6833	−0.475401
$506$	1.24334	0.0552732
$507$	1.46417	0.0650262
$508$	−6.91452	−0.306782
$509$	−19.2114	−0.851530	−0.425765	−	0.904834i	$-0.639995\pi$
$509$	−19.2114	−0.851530	−0.425765	+	0.904834i	$0.639995\pi$
$510$	−14.8598	−0.658001
$511$	71.2421	3.15157
$512$	−1.00000	−0.0441942
$513$	96.0236	4.23955
$514$	−15.7954	−0.696704
$515$	29.9748	1.32085
$516$	−1.40432	−0.0618216
$517$	2.94251	0.129411
$518$	16.6774	0.732761
$519$	−35.3776	−1.55290
$520$	−7.69478	−0.337439
$521$	28.7275	1.25858	0.629288	−	0.777172i	$-0.283347\pi$
$521$	28.7275	1.25858	0.629288	+	0.777172i	$0.283347\pi$
$522$	32.2081	1.40971
$523$	0.223468	0.00977157	0.00488579	−	0.999988i	$-0.498445\pi$
$523$	0.223468	0.00977157	0.00488579	+	0.999988i	$0.498445\pi$
$524$	2.61568	0.114267
$525$	4.14949	0.181098
$526$	4.88652	0.213062
$527$	−5.97574	−0.260307
$528$	−1.46220	−0.0636340
$529$	−15.8385	−0.688631
$530$	−11.3922	−0.494844
$531$	87.5637	3.79994
$532$	−37.2808	−1.61633
$533$	−2.66289	−0.115342
$534$	32.5372	1.40802
$535$	−31.8229	−1.37582
$536$	7.37205	0.318424
$537$	27.1803	1.17292
$538$	−5.45781	−0.235303
$539$	7.32295	0.315422
$540$	26.7074	1.14930
$541$	−20.2502	−0.870626	−0.435313	−	0.900279i	$-0.643362\pi$
$541$	−20.2502	−0.870626	−0.435313	+	0.900279i	$0.643362\pi$
$542$	2.93812	0.126203
$543$	−28.4412	−1.22053
$544$	−2.17248	−0.0931442
$545$	25.6433	1.09844
$546$	−53.1590	−2.27500
$547$	35.0313	1.49783	0.748914	−	0.662667i	$-0.230575\pi$
$547$	35.0313	1.49783	0.748914	+	0.662667i	$0.230575\pi$
$548$	−2.09352	−0.0894307
$549$	−43.4922	−1.85620
$550$	0.128400	0.00547497
$551$	36.4512	1.55287
$552$	−8.42209	−0.358468
$553$	−10.2329	−0.435145
$554$	−9.57581	−0.406837
$555$	−23.9102	−1.01493
$556$	0.169351	0.00718208
$557$	25.8517	1.09537	0.547686	−	0.836684i	$-0.315509\pi$
$557$	25.8517	1.09537	0.547686	+	0.836684i	$0.315509\pi$
$558$	18.9921	0.804002
$559$	−1.57981	−0.0668189
$560$	−10.3691	−0.438172
$561$	−3.17659	−0.134116
$562$	29.9556	1.26360
$563$	7.04696	0.296994	0.148497	−	0.988913i	$-0.452556\pi$
$563$	7.04696	0.296994	0.148497	+	0.988913i	$0.452556\pi$
$564$	−19.9318	−0.839282
$565$	36.3365	1.52869
$566$	11.7339	0.493212
$567$	85.6834	3.59837
$568$	10.7392	0.450609
$569$	−21.0373	−0.881929	−0.440964	−	0.897525i	$-0.645363\pi$
$569$	−21.0373	−0.881929	−0.440964	+	0.897525i	$0.645363\pi$
$570$	53.4492	2.23874
$571$	−19.3157	−0.808336	−0.404168	−	0.914685i	$-0.632439\pi$
$571$	−19.3157	−0.808336	−0.404168	+	0.914685i	$0.632439\pi$
$572$	−1.64493	−0.0687778
$573$	29.4390	1.22983
$574$	−3.58835	−0.149775
$575$	0.739566	0.0308420
$576$	6.90459	0.287691
$577$	−24.4322	−1.01713	−0.508564	−	0.861024i	$-0.669823\pi$
$577$	−24.4322	−1.01713	−0.508564	+	0.861024i	$0.669823\pi$
$578$	12.2803	0.510795
$579$	15.4785	0.643264
$580$	10.1383	0.420971
$581$	−6.02390	−0.249914
$582$	−17.6672	−0.732328
$583$	−2.43532	−0.100861
$584$	−14.9326	−0.617916
$585$	53.1293	2.19663
$586$	−2.11245	−0.0872644
$587$	7.87924	0.325211	0.162606	−	0.986691i	$-0.448010\pi$
$587$	7.87924	0.325211	0.162606	+	0.986691i	$0.448010\pi$
$588$	−49.6040	−2.04563
$589$	21.4942	0.885652
$590$	27.5629	1.13475
$591$	−46.2678	−1.90320
$592$	−3.49564	−0.143670
$593$	25.6306	1.05252	0.526261	−	0.850323i	$-0.323594\pi$
$593$	25.6306	1.05252	0.526261	+	0.850323i	$0.323594\pi$
$594$	5.70928	0.234255
$595$	−22.5265	−0.923498
$596$	16.4336	0.673145
$597$	−33.2901	−1.36247
$598$	−9.47457	−0.387444
$599$	41.5720	1.69859	0.849294	−	0.527921i	$-0.177028\pi$
$599$	41.5720	1.69859	0.849294	+	0.527921i	$0.177028\pi$
$600$	−0.869748	−0.0355073
$601$	−31.2181	−1.27341	−0.636707	−	0.771106i	$-0.719704\pi$
$601$	−31.2181	−1.27341	−0.636707	+	0.771106i	$0.719704\pi$
$602$	−2.12886	−0.0867660
$603$	−50.9010	−2.07285
$604$	8.57860	0.349058
$605$	23.4382	0.952897
$606$	15.4698	0.628419
$607$	46.4830	1.88669	0.943344	−	0.331816i	$-0.107661\pi$
$607$	46.4830	1.88669	0.943344	+	0.331816i	$0.107661\pi$
$608$	7.81420	0.316908
$609$	70.0400	2.83816
$610$	−13.6903	−0.554302
$611$	−22.4227	−0.907124
$612$	15.0001	0.606342
$613$	24.7412	0.999288	0.499644	−	0.866231i	$-0.333464\pi$
$613$	24.7412	0.999288	0.499644	+	0.866231i	$0.333464\pi$
$614$	17.9660	0.725051
$615$	5.14459	0.207450
$616$	−2.21661	−0.0893096
$617$	17.0754	0.687428	0.343714	−	0.939074i	$-0.388315\pi$
$617$	17.0754	0.687428	0.343714	+	0.939074i	$0.388315\pi$
$618$	−43.4046	−1.74599
$619$	−18.3828	−0.738869	−0.369434	−	0.929257i	$-0.620449\pi$
$619$	−18.3828	−0.738869	−0.369434	+	0.929257i	$0.620449\pi$
$620$	5.97826	0.240093
$621$	32.8848	1.31962
$622$	11.6408	0.466753
$623$	49.3245	1.97615
$624$	11.1423	0.446050
$625$	−23.5418	−0.941673
$626$	4.03950	0.161451
$627$	11.4259	0.456307
$628$	−10.6449	−0.424779
$629$	−7.59420	−0.302800
$630$	71.5940	2.85237
$631$	37.7998	1.50479	0.752393	−	0.658714i	$-0.228899\pi$
$631$	37.7998	1.50479	0.752393	+	0.658714i	$0.228899\pi$
$632$	2.14485	0.0853174
$633$	34.1463	1.35719
$634$	−30.2160	−1.20003
$635$	15.0280	0.596367
$636$	16.4963	0.654121
$637$	−55.8028	−2.21099
$638$	2.16728	0.0858035
$639$	−74.1501	−2.93333
$640$	2.17339	0.0859109
$641$	16.1839	0.639226	0.319613	−	0.947548i	$-0.396447\pi$
$641$	16.1839	0.639226	0.319613	+	0.947548i	$0.396447\pi$
$642$	46.0808	1.81866
$643$	29.6242	1.16826	0.584132	−	0.811659i	$-0.301435\pi$
$643$	29.6242	1.16826	0.584132	+	0.811659i	$0.301435\pi$
$644$	−12.7674	−0.503106
$645$	3.05213	0.120178
$646$	16.9762	0.667919
$647$	−7.93801	−0.312075	−0.156038	−	0.987751i	$-0.549872\pi$
$647$	−7.93801	−0.312075	−0.156038	+	0.987751i	$0.549872\pi$
$648$	−17.9596	−0.705519
$649$	5.89216	0.231287
$650$	−0.978438	−0.0383775
$651$	41.3005	1.61869
$652$	8.77566	0.343681
$653$	4.28596	0.167723	0.0838613	−	0.996477i	$-0.473275\pi$
$653$	4.28596	0.167723	0.0838613	+	0.996477i	$0.473275\pi$
$654$	−37.1324	−1.45199
$655$	−5.68490	−0.222128
$656$	0.752133	0.0293658
$657$	103.104	4.02246
$658$	−30.2155	−1.17792
$659$	15.6395	0.609227	0.304614	−	0.952476i	$-0.401473\pi$
$659$	15.6395	0.609227	0.304614	+	0.952476i	$0.401473\pi$
$660$	3.17793	0.123701
$661$	−42.2031	−1.64151	−0.820755	−	0.571280i	$-0.806447\pi$
$661$	−42.2031	−1.64151	−0.820755	+	0.571280i	$0.806447\pi$
$662$	9.39681	0.365218
$663$	24.2065	0.940102
$664$	1.26263	0.0489997
$665$	81.0258	3.14205
$666$	24.1359	0.935249
$667$	12.4833	0.483355
$668$	−14.8025	−0.572725
$669$	−71.3083	−2.75694
$670$	−16.0224	−0.618998
$671$	−2.92659	−0.112980
$672$	15.0148	0.579208
$673$	−9.29626	−0.358344	−0.179172	−	0.983818i	$-0.557342\pi$
$673$	−9.29626	−0.358344	−0.179172	+	0.983818i	$0.557342\pi$
$674$	−36.6938	−1.41339
$675$	3.39601	0.130712
$676$	−0.465237	−0.0178937
$677$	30.0770	1.15595	0.577976	−	0.816054i	$-0.303843\pi$
$677$	30.0770	1.15595	0.577976	+	0.816054i	$0.303843\pi$
$678$	−52.6167	−2.02073
$679$	−26.7824	−1.02781
$680$	4.72165	0.181067
$681$	−22.7007	−0.869895
$682$	1.27798	0.0489364
$683$	−11.6634	−0.446286	−0.223143	−	0.974786i	$-0.571632\pi$
$683$	−11.6634	−0.446286	−0.223143	+	0.974786i	$0.571632\pi$
$684$	−53.9538	−2.06298
$685$	4.55004	0.173848
$686$	−41.8004	−1.59595
$687$	−71.4786	−2.72708
$688$	0.446218	0.0170119
$689$	18.5578	0.706995
$690$	18.3045	0.696841
$691$	5.73643	0.218224	0.109112	−	0.994029i	$-0.465199\pi$
$691$	5.73643	0.218224	0.109112	+	0.994029i	$0.465199\pi$
$692$	11.2411	0.427324
$693$	15.3048	0.581380
$694$	−6.79205	−0.257823
$695$	−0.368066	−0.0139616
$696$	−14.6807	−0.556469
$697$	1.63399	0.0618918
$698$	23.8020	0.900921
$699$	−9.88244	−0.373788
$700$	−1.31849	−0.0498341
$701$	20.5718	0.776986	0.388493	−	0.921452i	$-0.372996\pi$
$701$	20.5718	0.776986	0.388493	+	0.921452i	$0.372996\pi$
$702$	−43.5063	−1.64204
$703$	27.3156	1.03023
$704$	0.464609	0.0175106
$705$	43.3197	1.63152
$706$	28.7773	1.08305
$707$	23.4514	0.881979
$708$	−39.9121	−1.49999
$709$	−45.7623	−1.71864	−0.859320	−	0.511438i	$-0.829113\pi$
$709$	−45.7623	−1.71864	−0.859320	+	0.511438i	$0.829113\pi$
$710$	−23.3406	−0.875957
$711$	−14.8093	−0.555391
$712$	−10.3386	−0.387456
$713$	7.36101	0.275672
$714$	32.6193	1.22075
$715$	3.57507	0.133700
$716$	−8.63646	−0.322760
$717$	−53.4894	−1.99760
$718$	10.3470	0.386146
$719$	21.0808	0.786180	0.393090	−	0.919500i	$-0.371406\pi$
$719$	21.0808	0.786180	0.393090	+	0.919500i	$0.371406\pi$
$720$	−15.0064	−0.559255
$721$	−65.7988	−2.45048
$722$	−42.0617	−1.56537
$723$	74.0538	2.75409
$724$	9.03712	0.335862
$725$	1.28915	0.0478778
$726$	−33.9394	−1.25961
$727$	25.3964	0.941902	0.470951	−	0.882159i	$-0.343911\pi$
$727$	25.3964	0.941902	0.470951	+	0.882159i	$0.343911\pi$
$728$	16.8911	0.626027
$729$	7.98350	0.295685
$730$	32.4545	1.20119
$731$	0.969399	0.0358545
$732$	19.8240	0.732716
$733$	40.0778	1.48031	0.740153	−	0.672438i	$-0.234753\pi$
$733$	40.0778	1.48031	0.740153	+	0.672438i	$0.234753\pi$
$734$	−16.8307	−0.621233
$735$	107.809	3.97659
$736$	2.67609	0.0986421
$737$	−3.42513	−0.126166
$738$	−5.19317	−0.191163
$739$	26.6738	0.981211	0.490606	−	0.871382i	$-0.336776\pi$
$739$	26.6738	0.981211	0.490606	+	0.871382i	$0.336776\pi$
$740$	7.59740	0.279286
$741$	−87.0684	−3.19854
$742$	25.0074	0.918051
$743$	29.5772	1.08508	0.542541	−	0.840029i	$-0.317462\pi$
$743$	29.5772	1.08508	0.542541	+	0.840029i	$0.317462\pi$
$744$	−8.65674	−0.317371
$745$	−35.7166	−1.30855
$746$	21.1791	0.775423
$747$	−8.71796	−0.318974
$748$	1.00935	0.0369056
$749$	69.8558	2.55247
$750$	36.0903	1.31783
$751$	−6.76616	−0.246901	−0.123450	−	0.992351i	$-0.539396\pi$
$751$	−6.76616	−0.246901	−0.123450	+	0.992351i	$0.539396\pi$
$752$	6.33329	0.230951
$753$	3.60230	0.131275
$754$	−16.5153	−0.601450
$755$	−18.6447	−0.678549
$756$	−58.6265	−2.13223
$757$	54.1584	1.96842	0.984210	−	0.177006i	$-0.0566413\pi$
$757$	54.1584	1.96842	0.984210	+	0.177006i	$0.0566413\pi$
$758$	−14.5322	−0.527833
$759$	3.91298	0.142032
$760$	−16.9833	−0.616050
$761$	−9.32399	−0.337995	−0.168997	−	0.985617i	$-0.554053\pi$
$761$	−9.32399	−0.337995	−0.168997	+	0.985617i	$0.554053\pi$
$762$	−21.7611	−0.788320
$763$	−56.2906	−2.03786
$764$	−9.35417	−0.338422
$765$	−32.6010	−1.17869
$766$	−20.0820	−0.725591
$767$	−44.8998	−1.62124
$768$	−3.14716	−0.113563
$769$	−32.8772	−1.18558	−0.592791	−	0.805356i	$-0.701974\pi$
$769$	−32.8772	−1.18558	−0.592791	+	0.805356i	$0.701974\pi$
$770$	4.81756	0.173613
$771$	−49.7105	−1.79028
$772$	−4.91824	−0.177012
$773$	16.3598	0.588423	0.294211	−	0.955740i	$-0.404943\pi$
$773$	16.3598	0.588423	0.294211	+	0.955740i	$0.404943\pi$
$774$	−3.08095	−0.110743
$775$	0.760171	0.0273062
$776$	5.61370	0.201520
$777$	52.4862	1.88293
$778$	17.2326	0.617818
$779$	−5.87731	−0.210577
$780$	−24.2167	−0.867096
$781$	−4.98955	−0.178540
$782$	5.81376	0.207899
$783$	57.3220	2.04852
$784$	15.7615	0.562911
$785$	23.1356	0.825746
$786$	8.23195	0.293624
$787$	−3.75331	−0.133791	−0.0668956	−	0.997760i	$-0.521309\pi$
$787$	−3.75331	−0.133791	−0.0668956	+	0.997760i	$0.521309\pi$
$788$	14.7015	0.523718
$789$	15.3786	0.547494
$790$	−4.66159	−0.165852
$791$	−79.7638	−2.83607
$792$	−3.20794	−0.113989
$793$	22.3014	0.791944
$794$	4.12386	0.146350
$795$	−35.8529	−1.27157
$796$	10.5778	0.374922
$797$	−22.0905	−0.782485	−0.391243	−	0.920288i	$-0.627955\pi$
$797$	−22.0905	−0.782485	−0.391243	+	0.920288i	$0.627955\pi$
$798$	−117.328	−4.15338
$799$	13.7589	0.486756
$800$	0.276360	0.00977081
$801$	71.3839	2.52223
$802$	−9.40494	−0.332100
$803$	6.93784	0.244831
$804$	23.2010	0.818236
$805$	27.7486	0.978008
$806$	−9.73855	−0.343026
$807$	−17.1766	−0.604644
$808$	−4.91550	−0.172927
$809$	−1.98014	−0.0696181	−0.0348090	−	0.999394i	$-0.511082\pi$
$809$	−1.98014	−0.0696181	−0.0348090	+	0.999394i	$0.511082\pi$
$810$	39.0332	1.37149
$811$	−7.90446	−0.277563	−0.138782	−	0.990323i	$-0.544319\pi$
$811$	−7.90446	−0.277563	−0.138782	+	0.990323i	$0.544319\pi$
$812$	−22.2550	−0.780998
$813$	9.24672	0.324297
$814$	1.62411	0.0569249
$815$	−19.0730	−0.668097
$816$	−6.83713	−0.239347
$817$	−3.48684	−0.121989
$818$	−21.4215	−0.748986
$819$	−116.626	−4.07525
$820$	−1.63468	−0.0570855
$821$	−6.71468	−0.234344	−0.117172	−	0.993112i	$-0.537383\pi$
$821$	−6.71468	−0.234344	−0.117172	+	0.993112i	$0.537383\pi$
$822$	−6.58863	−0.229805
$823$	−0.808931	−0.0281976	−0.0140988	−	0.999901i	$-0.504488\pi$
$823$	−0.808931	−0.0281976	−0.0140988	+	0.999901i	$0.504488\pi$
$824$	13.7917	0.480456
$825$	0.404093	0.0140687
$826$	−60.5044	−2.10522
$827$	17.7160	0.616045	0.308022	−	0.951379i	$-0.400333\pi$
$827$	17.7160	0.616045	0.308022	+	0.951379i	$0.400333\pi$
$828$	−18.4773	−0.642132
$829$	−23.6672	−0.821994	−0.410997	−	0.911637i	$-0.634819\pi$
$829$	−23.6672	−0.821994	−0.410997	+	0.911637i	$0.634819\pi$
$830$	−2.74420	−0.0952525
$831$	−30.1366	−1.04543
$832$	−3.54045	−0.122743
$833$	34.2415	1.18640
$834$	0.532974	0.0184554
$835$	32.1716	1.11334
$836$	−3.63055	−0.125565
$837$	33.8010	1.16833
$838$	5.82387	0.201182
$839$	−19.5659	−0.675491	−0.337746	−	0.941237i	$-0.609664\pi$
$839$	−19.5659	−0.675491	−0.337746	+	0.941237i	$0.609664\pi$
$840$	−32.6330	−1.12595
$841$	−7.24020	−0.249662
$842$	−28.3087	−0.975583
$843$	94.2751	3.24701
$844$	−10.8499	−0.373469
$845$	1.01114	0.0347844
$846$	−43.7288	−1.50343
$847$	−51.4501	−1.76785
$848$	−5.24165	−0.179999
$849$	36.9283	1.26738
$850$	0.600386	0.0205931
$851$	9.35466	0.320674
$852$	33.7981	1.15790
$853$	28.7187	0.983309	0.491655	−	0.870790i	$-0.336392\pi$
$853$	28.7187	0.983309	0.491655	+	0.870790i	$0.336392\pi$
$854$	30.0520	1.02836
$855$	117.263	4.01031
$856$	−14.6420	−0.500455
$857$	−32.5695	−1.11255	−0.556276	−	0.830997i	$-0.687770\pi$
$857$	−32.5695	−1.11255	−0.556276	+	0.830997i	$0.687770\pi$
$858$	−5.17684	−0.176734
$859$	13.7296	0.468448	0.234224	−	0.972183i	$-0.424745\pi$
$859$	13.7296	0.468448	0.234224	+	0.972183i	$0.424745\pi$
$860$	−0.969807	−0.0330702
$861$	−11.2931	−0.384868
$862$	36.3042	1.23653
$863$	−16.3660	−0.557104	−0.278552	−	0.960421i	$-0.589855\pi$
$863$	−16.3660	−0.557104	−0.278552	+	0.960421i	$0.589855\pi$
$864$	12.2884	0.418058
$865$	−24.4314	−0.830693
$866$	−15.3140	−0.520391
$867$	38.6481	1.31256
$868$	−13.1231	−0.445427
$869$	−0.996516	−0.0338045
$870$	31.9069	1.08174
$871$	26.1004	0.884377
$872$	11.7987	0.399555
$873$	−38.7603	−1.31184
$874$	−20.9115	−0.707343
$875$	54.7108	1.84956
$876$	−46.9953	−1.58782
$877$	−41.0199	−1.38514	−0.692572	−	0.721349i	$-0.743522\pi$
$877$	−41.0199	−1.38514	−0.692572	+	0.721349i	$0.743522\pi$
$878$	−19.2649	−0.650160
$879$	−6.64820	−0.224238
$880$	−1.00978	−0.0340397
$881$	−37.5027	−1.26350	−0.631749	−	0.775173i	$-0.717663\pi$
$881$	−37.5027	−1.26350	−0.631749	+	0.775173i	$0.717663\pi$
$882$	−108.827	−3.66439
$883$	28.2312	0.950057	0.475028	−	0.879970i	$-0.342438\pi$
$883$	28.2312	0.950057	0.475028	+	0.879970i	$0.342438\pi$
$884$	−7.69154	−0.258695
$885$	86.7447	2.91589
$886$	−15.0494	−0.505595
$887$	−35.0722	−1.17761	−0.588804	−	0.808276i	$-0.700401\pi$
$887$	−35.0722	−1.17761	−0.588804	+	0.808276i	$0.700401\pi$
$888$	−11.0013	−0.369180
$889$	−32.9885	−1.10640
$890$	22.4699	0.753192
$891$	8.34419	0.279541
$892$	22.6580	0.758646
$893$	−49.4896	−1.65611
$894$	51.7190	1.72974
$895$	18.7704	0.627426
$896$	−4.77090	−0.159385
$897$	−29.8179	−0.995592
$898$	27.0747	0.903495
$899$	12.8311	0.427941
$900$	−1.90815	−0.0636051
$901$	−11.3874	−0.379368
$902$	−0.349448	−0.0116353
$903$	−6.69986	−0.222957
$904$	16.7188	0.556059
$905$	−19.6412	−0.652896
$906$	26.9982	0.896954
$907$	−5.26669	−0.174877	−0.0874387	−	0.996170i	$-0.527868\pi$
$907$	−5.26669	−0.174877	−0.0874387	+	0.996170i	$0.527868\pi$
$908$	7.21310	0.239375
$909$	33.9395	1.12570
$910$	−36.7111	−1.21696
$911$	57.5271	1.90596	0.952980	−	0.303035i	$-0.0979998\pi$
$911$	57.5271	1.90596	0.952980	+	0.303035i	$0.0979998\pi$
$912$	24.5925	0.814339
$913$	−0.586631	−0.0194147
$914$	35.3447	1.16910
$915$	−43.0854	−1.42436
$916$	22.7121	0.750430
$917$	12.4792	0.412098
$918$	26.6962	0.881105
$919$	33.4468	1.10331	0.551655	−	0.834073i	$-0.313997\pi$
$919$	33.4468	1.10331	0.551655	+	0.834073i	$0.313997\pi$
$920$	−5.81621	−0.191755
$921$	56.5419	1.86312
$922$	28.0998	0.925418
$923$	38.0217	1.25150
$924$	−6.97601	−0.229494
$925$	0.966055	0.0317637
$926$	34.2886	1.12679
$927$	−95.2260	−3.12763
$928$	4.66474	0.153128
$929$	11.5284	0.378233	0.189117	−	0.981955i	$-0.439438\pi$
$929$	11.5284	0.378233	0.189117	+	0.981955i	$0.439438\pi$
$930$	18.8145	0.616952
$931$	−123.164	−4.03652
$932$	3.14012	0.102858
$933$	36.6354	1.19939
$934$	−35.1965	−1.15166
$935$	−2.19372	−0.0717424
$936$	24.4453	0.799021
$937$	3.19135	0.104257	0.0521284	−	0.998640i	$-0.483399\pi$
$937$	3.19135	0.104257	0.0521284	+	0.998640i	$0.483399\pi$
$938$	35.1713	1.14839
$939$	12.7129	0.414871
$940$	−13.7647	−0.448956
$941$	12.9938	0.423586	0.211793	−	0.977315i	$-0.432070\pi$
$941$	12.9938	0.423586	0.211793	+	0.977315i	$0.432070\pi$
$942$	−33.5013	−1.09153
$943$	−2.01278	−0.0655451
$944$	12.6820	0.412763
$945$	127.419	4.14493
$946$	−0.207317	−0.00674046
$947$	31.2482	1.01543	0.507715	−	0.861525i	$-0.330490\pi$
$947$	31.2482	1.01543	0.507715	+	0.861525i	$0.330490\pi$
$948$	6.75016	0.219235
$949$	−52.8682	−1.71617
$950$	−2.15953	−0.0700645
$951$	−95.0943	−3.08364
$952$	−10.3647	−0.335921
$953$	−0.739162	−0.0239438	−0.0119719	−	0.999928i	$-0.503811\pi$
$953$	−0.739162	−0.0239438	−0.0119719	+	0.999928i	$0.503811\pi$
$954$	36.1915	1.17174
$955$	20.3303	0.657873
$956$	16.9961	0.549694
$957$	6.82077	0.220484
$958$	−7.85768	−0.253870
$959$	−9.98798	−0.322529
$960$	6.84001	0.220760
$961$	−23.4339	−0.755932
$962$	−12.3761	−0.399022
$963$	101.097	3.25781
$964$	−23.5304	−0.757863
$965$	10.6893	0.344100
$966$	−40.1810	−1.29280
$967$	5.28271	0.169881	0.0849403	−	0.996386i	$-0.472930\pi$
$967$	5.28271	0.169881	0.0849403	+	0.996386i	$0.472930\pi$
$968$	10.7841	0.346615
$969$	53.4267	1.71631
$970$	−12.2008	−0.391743
$971$	19.9107	0.638963	0.319482	−	0.947592i	$-0.396491\pi$
$971$	19.9107	0.638963	0.319482	+	0.947592i	$0.396491\pi$
$972$	−19.6566	−0.630484
$973$	0.807957	0.0259019
$974$	−13.9859	−0.448137
$975$	−3.07930	−0.0986165
$976$	−6.29902	−0.201627
$977$	−33.3537	−1.06708	−0.533540	−	0.845775i	$-0.679139\pi$
$977$	−33.3537	−1.06708	−0.533540	+	0.845775i	$0.679139\pi$
$978$	27.6184	0.883138
$979$	4.80342	0.153518
$980$	−34.2560	−1.09427
$981$	−81.4654	−2.60099
$982$	−18.0688	−0.576598
$983$	−10.1301	−0.323099	−0.161550	−	0.986865i	$-0.551649\pi$
$983$	−10.1301	−0.323099	−0.161550	+	0.986865i	$0.551649\pi$
$984$	2.36708	0.0754597
$985$	−31.9521	−1.01808
$986$	10.1340	0.322734
$987$	−95.0929	−3.02684
$988$	27.6658	0.880165
$989$	−1.19412	−0.0379709
$990$	6.97211	0.221588
$991$	11.6121	0.368870	0.184435	−	0.982845i	$-0.440954\pi$
$991$	11.6121	0.368870	0.184435	+	0.982845i	$0.440954\pi$
$992$	2.75066	0.0873334
$993$	29.5732	0.938478
$994$	51.2359	1.62510
$995$	−22.9898	−0.728826
$996$	3.97370	0.125912
$997$	7.08944	0.224525	0.112262	−	0.993679i	$-0.464190\pi$
$997$	7.08944	0.224525	0.112262	+	0.993679i	$0.464190\pi$
$998$	32.1063	1.01631
$999$	42.9556	1.35906

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.e.1.2	✓	77

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.e.1.2	✓	77	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -3.14716 q^{3} +1.00000 q^{4} -2.17339 q^{5} +3.14716 q^{6} +4.77090 q^{7} -1.00000 q^{8} +6.90459 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -3.14716 q^{3} +1.00000 q^{4} -2.17339 q^{5} +3.14716 q^{6} +4.77090 q^{7} -1.00000 q^{8} +6.90459 q^{9} +2.17339 q^{10} +0.464609 q^{11} -3.14716 q^{12} -3.54045 q^{13} -4.77090 q^{14} +6.84001 q^{15} +1.00000 q^{16} +2.17248 q^{17} -6.90459 q^{18} -7.81420 q^{19} -2.17339 q^{20} -15.0148 q^{21} -0.464609 q^{22} -2.67609 q^{23} +3.14716 q^{24} -0.276360 q^{25} +3.54045 q^{26} -12.2884 q^{27} +4.77090 q^{28} -4.66474 q^{29} -6.84001 q^{30} -2.75066 q^{31} -1.00000 q^{32} -1.46220 q^{33} -2.17248 q^{34} -10.3691 q^{35} +6.90459 q^{36} -3.49564 q^{37} +7.81420 q^{38} +11.1423 q^{39} +2.17339 q^{40} +0.752133 q^{41} +15.0148 q^{42} +0.446218 q^{43} +0.464609 q^{44} -15.0064 q^{45} +2.67609 q^{46} +6.33329 q^{47} -3.14716 q^{48} +15.7615 q^{49} +0.276360 q^{50} -6.83713 q^{51} -3.54045 q^{52} -5.24165 q^{53} +12.2884 q^{54} -1.00978 q^{55} -4.77090 q^{56} +24.5925 q^{57} +4.66474 q^{58} +12.6820 q^{59} +6.84001 q^{60} -6.29902 q^{61} +2.75066 q^{62} +32.9411 q^{63} +1.00000 q^{64} +7.69478 q^{65} +1.46220 q^{66} -7.37205 q^{67} +2.17248 q^{68} +8.42209 q^{69} +10.3691 q^{70} -10.7392 q^{71} -6.90459 q^{72} +14.9326 q^{73} +3.49564 q^{74} +0.869748 q^{75} -7.81420 q^{76} +2.21661 q^{77} -11.1423 q^{78} -2.14485 q^{79} -2.17339 q^{80} +17.9596 q^{81} -0.752133 q^{82} -1.26263 q^{83} -15.0148 q^{84} -4.72165 q^{85} -0.446218 q^{86} +14.6807 q^{87} -0.464609 q^{88} +10.3386 q^{89} +15.0064 q^{90} -16.8911 q^{91} -2.67609 q^{92} +8.65674 q^{93} -6.33329 q^{94} +16.9833 q^{95} +3.14716 q^{96} -5.61370 q^{97} -15.7615 q^{98} +3.20794 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9}+O(q^{10}) \) 77 * q - 77 * q^2 + 10 * q^3 + 77 * q^4 + 18 * q^5 - 10 * q^6 + 21 * q^7 - 77 * q^8 + 71 * q^9	\( 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9} - 18 q^{10} + 30 q^{11} + 10 q^{12} - 2 q^{13} - 21 q^{14} + 21 q^{15} + 77 q^{16} + 60 q^{17} - 71 q^{18} - 3 q^{19} + 18 q^{20} + 10 q^{21} - 30 q^{22} + 53 q^{23} - 10 q^{24} + 59 q^{25} + 2 q^{26} + 43 q^{27} + 21 q^{28} + 30 q^{29} - 21 q^{30} + 22 q^{31} - 77 q^{32} + 31 q^{33} - 60 q^{34} + 41 q^{35} + 71 q^{36} - 3 q^{37} + 3 q^{38} + 44 q^{39} - 18 q^{40} + 48 q^{41} - 10 q^{42} + 21 q^{43} + 30 q^{44} + 33 q^{45} - 53 q^{46} + 107 q^{47} + 10 q^{48} + 24 q^{49} - 59 q^{50} + 18 q^{51} - 2 q^{52} + 42 q^{53} - 43 q^{54} + 49 q^{55} - 21 q^{56} + 32 q^{57} - 30 q^{58} + 42 q^{59} + 21 q^{60} - 31 q^{61} - 22 q^{62} + 109 q^{63} + 77 q^{64} + 39 q^{65} - 31 q^{66} - q^{67} + 60 q^{68} - 33 q^{69} - 41 q^{70} + 58 q^{71} - 71 q^{72} + 35 q^{73} + 3 q^{74} + 34 q^{75} - 3 q^{76} + 86 q^{77} - 44 q^{78} + 25 q^{79} + 18 q^{80} + 53 q^{81} - 48 q^{82} + 107 q^{83} + 10 q^{84} + 21 q^{85} - 21 q^{86} + 100 q^{87} - 30 q^{88} + 34 q^{89} - 33 q^{90} - 51 q^{91} + 53 q^{92} + 48 q^{93} - 107 q^{94} + 118 q^{95} - 10 q^{96} - 13 q^{97} - 24 q^{98} + 63 q^{99}+O(q^{100}) \) 77 * q - 77 * q^2 + 10 * q^3 + 77 * q^4 + 18 * q^5 - 10 * q^6 + 21 * q^7 - 77 * q^8 + 71 * q^9 - 18 * q^10 + 30 * q^11 + 10 * q^12 - 2 * q^13 - 21 * q^14 + 21 * q^15 + 77 * q^16 + 60 * q^17 - 71 * q^18 - 3 * q^19 + 18 * q^20 + 10 * q^21 - 30 * q^22 + 53 * q^23 - 10 * q^24 + 59 * q^25 + 2 * q^26 + 43 * q^27 + 21 * q^28 + 30 * q^29 - 21 * q^30 + 22 * q^31 - 77 * q^32 + 31 * q^33 - 60 * q^34 + 41 * q^35 + 71 * q^36 - 3 * q^37 + 3 * q^38 + 44 * q^39 - 18 * q^40 + 48 * q^41 - 10 * q^42 + 21 * q^43 + 30 * q^44 + 33 * q^45 - 53 * q^46 + 107 * q^47 + 10 * q^48 + 24 * q^49 - 59 * q^50 + 18 * q^51 - 2 * q^52 + 42 * q^53 - 43 * q^54 + 49 * q^55 - 21 * q^56 + 32 * q^57 - 30 * q^58 + 42 * q^59 + 21 * q^60 - 31 * q^61 - 22 * q^62 + 109 * q^63 + 77 * q^64 + 39 * q^65 - 31 * q^66 - q^67 + 60 * q^68 - 33 * q^69 - 41 * q^70 + 58 * q^71 - 71 * q^72 + 35 * q^73 + 3 * q^74 + 34 * q^75 - 3 * q^76 + 86 * q^77 - 44 * q^78 + 25 * q^79 + 18 * q^80 + 53 * q^81 - 48 * q^82 + 107 * q^83 + 10 * q^84 + 21 * q^85 - 21 * q^86 + 100 * q^87 - 30 * q^88 + 34 * q^89 - 33 * q^90 - 51 * q^91 + 53 * q^92 + 48 * q^93 - 107 * q^94 + 118 * q^95 - 10 * q^96 - 13 * q^97 - 24 * q^98 + 63 * q^99

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