LMFDB - Embedded newform 8002.2.a.e.1.46

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			1.46
Character	$\chi$	$=$	8002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +0.866718 q^{3} +1.00000 q^{4} +0.141140 q^{5} -0.866718 q^{6} -3.99131 q^{7} -1.00000 q^{8} -2.24880 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +0.866718 q^{3} +1.00000 q^{4} +0.141140 q^{5} -0.866718 q^{6} -3.99131 q^{7} -1.00000 q^{8} -2.24880 q^{9} -0.141140 q^{10} -3.27521 q^{11} +0.866718 q^{12} -3.84268 q^{13} +3.99131 q^{14} +0.122329 q^{15} +1.00000 q^{16} +5.60219 q^{17} +2.24880 q^{18} -5.08106 q^{19} +0.141140 q^{20} -3.45934 q^{21} +3.27521 q^{22} -8.27079 q^{23} -0.866718 q^{24} -4.98008 q^{25} +3.84268 q^{26} -4.54923 q^{27} -3.99131 q^{28} -9.92324 q^{29} -0.122329 q^{30} +0.0745375 q^{31} -1.00000 q^{32} -2.83868 q^{33} -5.60219 q^{34} -0.563335 q^{35} -2.24880 q^{36} -1.49849 q^{37} +5.08106 q^{38} -3.33052 q^{39} -0.141140 q^{40} +0.746641 q^{41} +3.45934 q^{42} -6.19471 q^{43} -3.27521 q^{44} -0.317396 q^{45} +8.27079 q^{46} +7.57172 q^{47} +0.866718 q^{48} +8.93059 q^{49} +4.98008 q^{50} +4.85552 q^{51} -3.84268 q^{52} -9.49613 q^{53} +4.54923 q^{54} -0.462264 q^{55} +3.99131 q^{56} -4.40385 q^{57} +9.92324 q^{58} +2.89542 q^{59} +0.122329 q^{60} +10.9659 q^{61} -0.0745375 q^{62} +8.97567 q^{63} +1.00000 q^{64} -0.542357 q^{65} +2.83868 q^{66} +6.62889 q^{67} +5.60219 q^{68} -7.16844 q^{69} +0.563335 q^{70} +8.35977 q^{71} +2.24880 q^{72} +3.67131 q^{73} +1.49849 q^{74} -4.31632 q^{75} -5.08106 q^{76} +13.0724 q^{77} +3.33052 q^{78} +1.10726 q^{79} +0.141140 q^{80} +2.80351 q^{81} -0.746641 q^{82} -6.18093 q^{83} -3.45934 q^{84} +0.790696 q^{85} +6.19471 q^{86} -8.60065 q^{87} +3.27521 q^{88} -12.7702 q^{89} +0.317396 q^{90} +15.3374 q^{91} -8.27079 q^{92} +0.0646029 q^{93} -7.57172 q^{94} -0.717143 q^{95} -0.866718 q^{96} -4.13442 q^{97} -8.93059 q^{98} +7.36530 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.866718	0.500400	0.250200	−	0.968194i	$-0.419504\pi$
$3$	0.866718	0.500400	0.250200	+	0.968194i	$0.419504\pi$
$4$	1.00000	0.500000
$5$	0.141140	0.0631199	0.0315599	−	0.999502i	$-0.489952\pi$
$5$	0.141140	0.0631199	0.0315599	+	0.999502i	$0.489952\pi$
$6$	−0.866718	−0.353836
$7$	−3.99131	−1.50858	−0.754288	−	0.656544i	$-0.772018\pi$
$7$	−3.99131	−1.50858	−0.754288	+	0.656544i	$0.772018\pi$
$8$	−1.00000	−0.353553
$9$	−2.24880	−0.749600
$10$	−0.141140	−0.0446325
$11$	−3.27521	−0.987513	−0.493757	−	0.869600i	$-0.664377\pi$
$11$	−3.27521	−0.987513	−0.493757	+	0.869600i	$0.664377\pi$
$12$	0.866718	0.250200
$13$	−3.84268	−1.06577	−0.532884	−	0.846188i	$-0.678892\pi$
$13$	−3.84268	−1.06577	−0.532884	+	0.846188i	$0.678892\pi$
$14$	3.99131	1.06672
$15$	0.122329	0.0315852
$16$	1.00000	0.250000
$17$	5.60219	1.35873	0.679366	−	0.733800i	$-0.262255\pi$
$17$	5.60219	1.35873	0.679366	+	0.733800i	$0.262255\pi$
$18$	2.24880	0.530047
$19$	−5.08106	−1.16568	−0.582838	−	0.812589i	$-0.698058\pi$
$19$	−5.08106	−1.16568	−0.582838	+	0.812589i	$0.698058\pi$
$20$	0.141140	0.0315599
$21$	−3.45934	−0.754890
$22$	3.27521	0.698277
$23$	−8.27079	−1.72458	−0.862290	−	0.506415i	$-0.830970\pi$
$23$	−8.27079	−1.72458	−0.862290	+	0.506415i	$0.830970\pi$
$24$	−0.866718	−0.176918
$25$	−4.98008	−0.996016
$26$	3.84268	0.753612
$27$	−4.54923	−0.875499
$28$	−3.99131	−0.754288
$29$	−9.92324	−1.84270	−0.921350	−	0.388734i	$-0.872912\pi$
$29$	−9.92324	−1.84270	−0.921350	+	0.388734i	$0.872912\pi$
$30$	−0.122329	−0.0223341
$31$	0.0745375	0.0133873	0.00669366	−	0.999978i	$-0.497869\pi$
$31$	0.0745375	0.0133873	0.00669366	+	0.999978i	$0.497869\pi$
$32$	−1.00000	−0.176777
$33$	−2.83868	−0.494151
$34$	−5.60219	−0.960768
$35$	−0.563335	−0.0952211
$36$	−2.24880	−0.374800
$37$	−1.49849	−0.246351	−0.123175	−	0.992385i	$-0.539308\pi$
$37$	−1.49849	−0.246351	−0.123175	+	0.992385i	$0.539308\pi$
$38$	5.08106	0.824257
$39$	−3.33052	−0.533310
$40$	−0.141140	−0.0223162
$41$	0.746641	0.116606	0.0583029	−	0.998299i	$-0.481431\pi$
$41$	0.746641	0.116606	0.0583029	+	0.998299i	$0.481431\pi$
$42$	3.45934	0.533788
$43$	−6.19471	−0.944685	−0.472343	−	0.881415i	$-0.656591\pi$
$43$	−6.19471	−0.944685	−0.472343	+	0.881415i	$0.656591\pi$
$44$	−3.27521	−0.493757
$45$	−0.317396	−0.0473147
$46$	8.27079	1.21946
$47$	7.57172	1.10445	0.552224	−	0.833696i	$-0.313779\pi$
$47$	7.57172	1.10445	0.552224	+	0.833696i	$0.313779\pi$
$48$	0.866718	0.125100
$49$	8.93059	1.27580
$50$	4.98008	0.704290
$51$	4.85552	0.679909
$52$	−3.84268	−0.532884
$53$	−9.49613	−1.30439	−0.652197	−	0.758050i	$-0.726152\pi$
$53$	−9.49613	−1.30439	−0.652197	+	0.758050i	$0.726152\pi$
$54$	4.54923	0.619072
$55$	−0.462264	−0.0623317
$56$	3.99131	0.533362
$57$	−4.40385	−0.583304
$58$	9.92324	1.30299
$59$	2.89542	0.376951	0.188475	−	0.982078i	$-0.439645\pi$
$59$	2.89542	0.376951	0.188475	+	0.982078i	$0.439645\pi$
$60$	0.122329	0.0157926
$61$	10.9659	1.40403	0.702017	−	0.712160i	$-0.252283\pi$
$61$	10.9659	1.40403	0.702017	+	0.712160i	$0.252283\pi$
$62$	−0.0745375	−0.00946627
$63$	8.97567	1.13083
$64$	1.00000	0.125000
$65$	−0.542357	−0.0672712
$66$	2.83868	0.349418
$67$	6.62889	0.809847	0.404924	−	0.914351i	$-0.367298\pi$
$67$	6.62889	0.809847	0.404924	+	0.914351i	$0.367298\pi$
$68$	5.60219	0.679366
$69$	−7.16844	−0.862979
$70$	0.563335	0.0673315
$71$	8.35977	0.992123	0.496061	−	0.868287i	$-0.334779\pi$
$71$	8.35977	0.992123	0.496061	+	0.868287i	$0.334779\pi$
$72$	2.24880	0.265024
$73$	3.67131	0.429694	0.214847	−	0.976648i	$-0.431075\pi$
$73$	3.67131	0.429694	0.214847	+	0.976648i	$0.431075\pi$
$74$	1.49849	0.174196
$75$	−4.31632	−0.498406
$76$	−5.08106	−0.582838
$77$	13.0724	1.48974
$78$	3.33052	0.377107
$79$	1.10726	0.124576	0.0622880	−	0.998058i	$-0.480160\pi$
$79$	1.10726	0.124576	0.0622880	+	0.998058i	$0.480160\pi$
$80$	0.141140	0.0157800
$81$	2.80351	0.311501
$82$	−0.746641	−0.0824527
$83$	−6.18093	−0.678445	−0.339222	−	0.940706i	$-0.610164\pi$
$83$	−6.18093	−0.678445	−0.339222	+	0.940706i	$0.610164\pi$
$84$	−3.45934	−0.377445
$85$	0.790696	0.0857630
$86$	6.19471	0.667993
$87$	−8.60065	−0.922086
$88$	3.27521	0.349139
$89$	−12.7702	−1.35364	−0.676819	−	0.736149i	$-0.736642\pi$
$89$	−12.7702	−1.35364	−0.676819	+	0.736149i	$0.736642\pi$
$90$	0.317396	0.0334565
$91$	15.3374	1.60779
$92$	−8.27079	−0.862290
$93$	0.0646029	0.00669901
$94$	−7.57172	−0.780963
$95$	−0.717143	−0.0735773
$96$	−0.866718	−0.0884590
$97$	−4.13442	−0.419787	−0.209893	−	0.977724i	$-0.567312\pi$
$97$	−4.13442	−0.419787	−0.209893	+	0.977724i	$0.567312\pi$
$98$	−8.93059	−0.902126
$99$	7.36530	0.740240
$100$	−4.98008	−0.498008
$101$	8.33819	0.829681	0.414840	−	0.909894i	$-0.363837\pi$
$101$	8.33819	0.829681	0.414840	+	0.909894i	$0.363837\pi$
$102$	−4.85552	−0.480768
$103$	12.8290	1.26408	0.632039	−	0.774936i	$-0.282218\pi$
$103$	12.8290	1.26408	0.632039	+	0.774936i	$0.282218\pi$
$104$	3.84268	0.376806
$105$	−0.488253	−0.0476486
$106$	9.49613	0.922345
$107$	−9.34186	−0.903112	−0.451556	−	0.892243i	$-0.649131\pi$
$107$	−9.34186	−0.903112	−0.451556	+	0.892243i	$0.649131\pi$
$108$	−4.54923	−0.437750
$109$	−15.4318	−1.47810	−0.739051	−	0.673650i	$-0.764726\pi$
$109$	−15.4318	−1.47810	−0.739051	+	0.673650i	$0.764726\pi$
$110$	0.462264	0.0440752
$111$	−1.29877	−0.123274
$112$	−3.99131	−0.377144
$113$	0.429369	0.0403916	0.0201958	−	0.999796i	$-0.493571\pi$
$113$	0.429369	0.0403916	0.0201958	+	0.999796i	$0.493571\pi$
$114$	4.40385	0.412458
$115$	−1.16734	−0.108855
$116$	−9.92324	−0.921350
$117$	8.64143	0.798900
$118$	−2.89542	−0.266545
$119$	−22.3601	−2.04975
$120$	−0.122329	−0.0111670
$121$	−0.272992	−0.0248174
$122$	−10.9659	−0.992802
$123$	0.647127	0.0583495
$124$	0.0745375	0.00669366
$125$	−1.40859	−0.125988
$126$	−8.97567	−0.799616
$127$	9.56785	0.849009	0.424505	−	0.905426i	$-0.360448\pi$
$127$	9.56785	0.849009	0.424505	+	0.905426i	$0.360448\pi$
$128$	−1.00000	−0.0883883
$129$	−5.36907	−0.472720
$130$	0.542357	0.0475679
$131$	−2.36538	−0.206664	−0.103332	−	0.994647i	$-0.532950\pi$
$131$	−2.36538	−0.206664	−0.103332	+	0.994647i	$0.532950\pi$
$132$	−2.83868	−0.247076
$133$	20.2801	1.75851
$134$	−6.62889	−0.572648
$135$	−0.642080	−0.0552614
$136$	−5.60219	−0.480384
$137$	−2.84546	−0.243104	−0.121552	−	0.992585i	$-0.538787\pi$
$137$	−2.84546	−0.243104	−0.121552	+	0.992585i	$0.538787\pi$
$138$	7.16844	0.610218
$139$	4.71650	0.400048	0.200024	−	0.979791i	$-0.435898\pi$
$139$	4.71650	0.400048	0.200024	+	0.979791i	$0.435898\pi$
$140$	−0.563335	−0.0476105
$141$	6.56254	0.552666
$142$	−8.35977	−0.701537
$143$	12.5856	1.05246
$144$	−2.24880	−0.187400
$145$	−1.40057	−0.116311
$146$	−3.67131	−0.303840
$147$	7.74030	0.638409
$148$	−1.49849	−0.123175
$149$	−18.9945	−1.55609	−0.778044	−	0.628210i	$-0.783788\pi$
$149$	−18.9945	−1.55609	−0.778044	+	0.628210i	$0.783788\pi$
$150$	4.31632	0.352426
$151$	12.6935	1.03298	0.516489	−	0.856294i	$-0.327239\pi$
$151$	12.6935	1.03298	0.516489	+	0.856294i	$0.327239\pi$
$152$	5.08106	0.412129
$153$	−12.5982	−1.01851
$154$	−13.0724	−1.05340
$155$	0.0105202	0.000845006 0
$156$	−3.33052	−0.266655
$157$	1.53000	0.122108	0.0610538	−	0.998134i	$-0.480554\pi$
$157$	1.53000	0.122108	0.0610538	+	0.998134i	$0.480554\pi$
$158$	−1.10726	−0.0880886
$159$	−8.23046	−0.652718
$160$	−0.141140	−0.0111581
$161$	33.0113	2.60166
$162$	−2.80351	−0.220264
$163$	−20.4784	−1.60399	−0.801997	−	0.597328i	$-0.796229\pi$
$163$	−20.4784	−1.60399	−0.801997	+	0.597328i	$0.796229\pi$
$164$	0.746641	0.0583029
$165$	−0.400653	−0.0311908
$166$	6.18093	0.479733
$167$	9.32085	0.721269	0.360634	−	0.932707i	$-0.382560\pi$
$167$	9.32085	0.721269	0.360634	+	0.932707i	$0.382560\pi$
$168$	3.45934	0.266894
$169$	1.76621	0.135862
$170$	−0.790696	−0.0606436
$171$	11.4263	0.873791
$172$	−6.19471	−0.472343
$173$	11.5891	0.881104	0.440552	−	0.897727i	$-0.354783\pi$
$173$	11.5891	0.881104	0.440552	+	0.897727i	$0.354783\pi$
$174$	8.60065	0.652013
$175$	19.8771	1.50256
$176$	−3.27521	−0.246878
$177$	2.50951	0.188626
$178$	12.7702	0.957167
$179$	11.8135	0.882982	0.441491	−	0.897266i	$-0.354450\pi$
$179$	11.8135	0.882982	0.441491	+	0.897266i	$0.354450\pi$
$180$	−0.317396	−0.0236573
$181$	−22.2799	−1.65605	−0.828025	−	0.560692i	$-0.810535\pi$
$181$	−22.2799	−1.65605	−0.828025	+	0.560692i	$0.810535\pi$
$182$	−15.3374	−1.13688
$183$	9.50430	0.702578
$184$	8.27079	0.609731
$185$	−0.211498	−0.0155496
$186$	−0.0646029	−0.00473692
$187$	−18.3484	−1.34177
$188$	7.57172	0.552224
$189$	18.1574	1.32076
$190$	0.717143	0.0520270
$191$	−21.2338	−1.53643	−0.768213	−	0.640194i	$-0.778854\pi$
$191$	−21.2338	−1.53643	−0.768213	+	0.640194i	$0.778854\pi$
$192$	0.866718	0.0625500
$193$	12.7121	0.915035	0.457517	−	0.889201i	$-0.348739\pi$
$193$	12.7121	0.915035	0.457517	+	0.889201i	$0.348739\pi$
$194$	4.13442	0.296834
$195$	−0.470071	−0.0336625
$196$	8.93059	0.637899
$197$	−12.2669	−0.873980	−0.436990	−	0.899466i	$-0.643956\pi$
$197$	−12.2669	−0.873980	−0.436990	+	0.899466i	$0.643956\pi$
$198$	−7.36530	−0.523429
$199$	−3.32959	−0.236028	−0.118014	−	0.993012i	$-0.537653\pi$
$199$	−3.32959	−0.236028	−0.118014	+	0.993012i	$0.537653\pi$
$200$	4.98008	0.352145
$201$	5.74537	0.405247
$202$	−8.33819	−0.586673
$203$	39.6068	2.77985
$204$	4.85552	0.339954
$205$	0.105381	0.00736014
$206$	−12.8290	−0.893838
$207$	18.5994	1.29275
$208$	−3.84268	−0.266442
$209$	16.6416	1.15112
$210$	0.488253	0.0336926
$211$	24.5244	1.68833	0.844166	−	0.536082i	$-0.180096\pi$
$211$	24.5244	1.68833	0.844166	+	0.536082i	$0.180096\pi$
$212$	−9.49613	−0.652197
$213$	7.24556	0.496458
$214$	9.34186	0.638597
$215$	−0.874324	−0.0596284
$216$	4.54923	0.309536
$217$	−0.297503	−0.0201958
$218$	15.4318	1.04518
$219$	3.18199	0.215019
$220$	−0.462264	−0.0311659
$221$	−21.5275	−1.44809
$222$	1.29877	0.0871677
$223$	16.8789	1.13029	0.565146	−	0.824991i	$-0.308820\pi$
$223$	16.8789	1.13029	0.565146	+	0.824991i	$0.308820\pi$
$224$	3.99131	0.266681
$225$	11.1992	0.746614
$226$	−0.429369	−0.0285612
$227$	−13.2745	−0.881062	−0.440531	−	0.897737i	$-0.645210\pi$
$227$	−13.2745	−0.881062	−0.440531	+	0.897737i	$0.645210\pi$
$228$	−4.40385	−0.291652
$229$	−18.9593	−1.25287	−0.626433	−	0.779475i	$-0.715486\pi$
$229$	−18.9593	−1.25287	−0.626433	+	0.779475i	$0.715486\pi$
$230$	1.16734	0.0769723
$231$	11.3301	0.745464
$232$	9.92324	0.651493
$233$	−9.58021	−0.627621	−0.313810	−	0.949486i	$-0.601606\pi$
$233$	−9.58021	−0.627621	−0.313810	+	0.949486i	$0.601606\pi$
$234$	−8.64143	−0.564908
$235$	1.06867	0.0697126
$236$	2.89542	0.188475
$237$	0.959678	0.0623378
$238$	22.3601	1.44939
$239$	19.5558	1.26496	0.632479	−	0.774578i	$-0.282038\pi$
$239$	19.5558	1.26496	0.632479	+	0.774578i	$0.282038\pi$
$240$	0.122329	0.00789629
$241$	−7.93943	−0.511424	−0.255712	−	0.966753i	$-0.582310\pi$
$241$	−7.93943	−0.511424	−0.255712	+	0.966753i	$0.582310\pi$
$242$	0.272992	0.0175486
$243$	16.0775	1.03137
$244$	10.9659	0.702017
$245$	1.26047	0.0805282
$246$	−0.647127	−0.0412593
$247$	19.5249	1.24234
$248$	−0.0745375	−0.00473313
$249$	−5.35712	−0.339494
$250$	1.40859	0.0890872
$251$	−15.0136	−0.947651	−0.473825	−	0.880619i	$-0.657127\pi$
$251$	−15.0136	−0.947651	−0.473825	+	0.880619i	$0.657127\pi$
$252$	8.97567	0.565414
$253$	27.0886	1.70305
$254$	−9.56785	−0.600340
$255$	0.685310	0.0429158
$256$	1.00000	0.0625000
$257$	−8.51641	−0.531239	−0.265620	−	0.964078i	$-0.585576\pi$
$257$	−8.51641	−0.531239	−0.265620	+	0.964078i	$0.585576\pi$
$258$	5.36907	0.334264
$259$	5.98095	0.371638
$260$	−0.542357	−0.0336356
$261$	22.3154	1.38129
$262$	2.36538	0.146134
$263$	24.5400	1.51320	0.756601	−	0.653877i	$-0.226859\pi$
$263$	24.5400	1.51320	0.756601	+	0.653877i	$0.226859\pi$
$264$	2.83868	0.174709
$265$	−1.34029	−0.0823332
$266$	−20.2801	−1.24345
$267$	−11.0682	−0.677360
$268$	6.62889	0.404924
$269$	7.95184	0.484832	0.242416	−	0.970172i	$-0.422060\pi$
$269$	7.95184	0.484832	0.242416	+	0.970172i	$0.422060\pi$
$270$	0.642080	0.0390757
$271$	−10.8137	−0.656884	−0.328442	−	0.944524i	$-0.606524\pi$
$271$	−10.8137	−0.656884	−0.328442	+	0.944524i	$0.606524\pi$
$272$	5.60219	0.339683
$273$	13.2932	0.804538
$274$	2.84546	0.171901
$275$	16.3108	0.983579
$276$	−7.16844	−0.431490
$277$	−7.01059	−0.421225	−0.210613	−	0.977570i	$-0.567546\pi$
$277$	−7.01059	−0.421225	−0.210613	+	0.977570i	$0.567546\pi$
$278$	−4.71650	−0.282877
$279$	−0.167620	−0.0100351
$280$	0.563335	0.0336657
$281$	−14.0044	−0.835435	−0.417717	−	0.908577i	$-0.637170\pi$
$281$	−14.0044	−0.835435	−0.417717	+	0.908577i	$0.637170\pi$
$282$	−6.56254	−0.390794
$283$	−18.6465	−1.10842	−0.554211	−	0.832376i	$-0.686980\pi$
$283$	−18.6465	−1.10842	−0.554211	+	0.832376i	$0.686980\pi$
$284$	8.35977	0.496061
$285$	−0.621560	−0.0368181
$286$	−12.5856	−0.744202
$287$	−2.98008	−0.175909
$288$	2.24880	0.132512
$289$	14.3846	0.846152
$290$	1.40057	0.0822443
$291$	−3.58337	−0.210061
$292$	3.67131	0.214847
$293$	−9.21904	−0.538582	−0.269291	−	0.963059i	$-0.586789\pi$
$293$	−9.21904	−0.538582	−0.269291	+	0.963059i	$0.586789\pi$
$294$	−7.74030	−0.451423
$295$	0.408660	0.0237931
$296$	1.49849	0.0870981
$297$	14.8997	0.864567
$298$	18.9945	1.10032
$299$	31.7820	1.83800
$300$	−4.31632	−0.249203
$301$	24.7251	1.42513
$302$	−12.6935	−0.730426
$303$	7.22686	0.415172
$304$	−5.08106	−0.291419
$305$	1.54772	0.0886224
$306$	12.5982	0.720192
$307$	−5.52048	−0.315071	−0.157535	−	0.987513i	$-0.550355\pi$
$307$	−5.52048	−0.315071	−0.157535	+	0.987513i	$0.550355\pi$
$308$	13.0724	0.744869
$309$	11.1191	0.632544
$310$	−0.0105202	−0.000597510 0
$311$	5.88297	0.333593	0.166796	−	0.985991i	$-0.446658\pi$
$311$	5.88297	0.333593	0.166796	+	0.985991i	$0.446658\pi$
$312$	3.33052	0.188554
$313$	−34.4206	−1.94557	−0.972785	−	0.231710i	$-0.925568\pi$
$313$	−34.4206	−1.94557	−0.972785	+	0.231710i	$0.925568\pi$
$314$	−1.53000	−0.0863431
$315$	1.26683	0.0713777
$316$	1.10726	0.0622880
$317$	−1.37854	−0.0774265	−0.0387132	−	0.999250i	$-0.512326\pi$
$317$	−1.37854	−0.0774265	−0.0387132	+	0.999250i	$0.512326\pi$
$318$	8.23046	0.461541
$319$	32.5007	1.81969
$320$	0.141140	0.00788998
$321$	−8.09676	−0.451917
$322$	−33.0113	−1.83965
$323$	−28.4651	−1.58384
$324$	2.80351	0.155750
$325$	19.1369	1.06152
$326$	20.4784	1.13420
$327$	−13.3750	−0.739642
$328$	−0.746641	−0.0412264
$329$	−30.2211	−1.66614
$330$	0.400653	0.0220552
$331$	1.57012	0.0863016	0.0431508	−	0.999069i	$-0.486260\pi$
$331$	1.57012	0.0863016	0.0431508	+	0.999069i	$0.486260\pi$
$332$	−6.18093	−0.339222
$333$	3.36981	0.184665
$334$	−9.32085	−0.510014
$335$	0.935603	0.0511175
$336$	−3.45934	−0.188723
$337$	8.08636	0.440492	0.220246	−	0.975444i	$-0.429314\pi$
$337$	8.08636	0.440492	0.220246	+	0.975444i	$0.429314\pi$
$338$	−1.76621	−0.0960690
$339$	0.372142	0.0202120
$340$	0.790696	0.0428815
$341$	−0.244126	−0.0132202
$342$	−11.4263	−0.617863
$343$	−7.70559	−0.416063
$344$	6.19471	0.333997
$345$	−1.01176	−0.0544711
$346$	−11.5891	−0.623035
$347$	25.4951	1.36865	0.684325	−	0.729177i	$-0.260097\pi$
$347$	25.4951	1.36865	0.684325	+	0.729177i	$0.260097\pi$
$348$	−8.60065	−0.461043
$349$	−21.2961	−1.13995	−0.569977	−	0.821661i	$-0.693048\pi$
$349$	−21.2961	−1.13995	−0.569977	+	0.821661i	$0.693048\pi$
$350$	−19.8771	−1.06247
$351$	17.4812	0.933079
$352$	3.27521	0.174569
$353$	11.2208	0.597221	0.298611	−	0.954375i	$-0.403477\pi$
$353$	11.2208	0.597221	0.298611	+	0.954375i	$0.403477\pi$
$354$	−2.50951	−0.133379
$355$	1.17990	0.0626227
$356$	−12.7702	−0.676819
$357$	−19.3799	−1.02569
$358$	−11.8135	−0.624363
$359$	−13.9016	−0.733700	−0.366850	−	0.930280i	$-0.619564\pi$
$359$	−13.9016	−0.733700	−0.366850	+	0.930280i	$0.619564\pi$
$360$	0.317396	0.0167283
$361$	6.81719	0.358800
$362$	22.2799	1.17100
$363$	−0.236607	−0.0124186
$364$	15.3374	0.803896
$365$	0.518170	0.0271223
$366$	−9.50430	−0.496798
$367$	5.82865	0.304253	0.152127	−	0.988361i	$-0.451388\pi$
$367$	5.82865	0.304253	0.152127	+	0.988361i	$0.451388\pi$
$368$	−8.27079	−0.431145
$369$	−1.67905	−0.0874077
$370$	0.211498	0.0109952
$371$	37.9020	1.96778
$372$	0.0646029	0.00334951
$373$	−18.6312	−0.964688	−0.482344	−	0.875982i	$-0.660214\pi$
$373$	−18.6312	−0.964688	−0.482344	+	0.875982i	$0.660214\pi$
$374$	18.3484	0.948772
$375$	−1.22085	−0.0630445
$376$	−7.57172	−0.390482
$377$	38.1319	1.96389
$378$	−18.1574	−0.933916
$379$	−34.1518	−1.75426	−0.877130	−	0.480253i	$-0.840545\pi$
$379$	−34.1518	−1.75426	−0.877130	+	0.480253i	$0.840545\pi$
$380$	−0.717143	−0.0367886
$381$	8.29262	0.424844
$382$	21.2338	1.08642
$383$	16.3788	0.836916	0.418458	−	0.908236i	$-0.362571\pi$
$383$	16.3788	0.836916	0.418458	+	0.908236i	$0.362571\pi$
$384$	−0.866718	−0.0442295
$385$	1.84504	0.0940321
$386$	−12.7121	−0.647027
$387$	13.9307	0.708136
$388$	−4.13442	−0.209893
$389$	−7.75851	−0.393372	−0.196686	−	0.980467i	$-0.563018\pi$
$389$	−7.75851	−0.393372	−0.196686	+	0.980467i	$0.563018\pi$
$390$	0.470071	0.0238030
$391$	−46.3346	−2.34324
$392$	−8.93059	−0.451063
$393$	−2.05012	−0.103415
$394$	12.2669	0.617997
$395$	0.156278	0.00786322
$396$	7.36530	0.370120
$397$	−35.4676	−1.78007	−0.890034	−	0.455893i	$-0.849320\pi$
$397$	−35.4676	−1.78007	−0.890034	+	0.455893i	$0.849320\pi$
$398$	3.32959	0.166897
$399$	17.5771	0.879957
$400$	−4.98008	−0.249004
$401$	−5.05558	−0.252464	−0.126232	−	0.992001i	$-0.540288\pi$
$401$	−5.05558	−0.252464	−0.126232	+	0.992001i	$0.540288\pi$
$402$	−5.74537	−0.286553
$403$	−0.286424	−0.0142678
$404$	8.33819	0.414840
$405$	0.395688	0.0196619
$406$	−39.6068	−1.96565
$407$	4.90788	0.243275
$408$	−4.85552	−0.240384
$409$	29.8907	1.47800	0.738999	−	0.673706i	$-0.235299\pi$
$409$	29.8907	1.47800	0.738999	+	0.673706i	$0.235299\pi$
$410$	−0.105381	−0.00520441
$411$	−2.46621	−0.121649
$412$	12.8290	0.632039
$413$	−11.5565	−0.568659
$414$	−18.5994	−0.914109
$415$	−0.872378	−0.0428234
$416$	3.84268	0.188403
$417$	4.08787	0.200184
$418$	−16.6416	−0.813965
$419$	−28.0315	−1.36943	−0.684713	−	0.728813i	$-0.740072\pi$
$419$	−28.0315	−1.36943	−0.684713	+	0.728813i	$0.740072\pi$
$420$	−0.488253	−0.0238243
$421$	22.6989	1.10628	0.553139	−	0.833089i	$-0.313430\pi$
$421$	22.6989	1.10628	0.553139	+	0.833089i	$0.313430\pi$
$422$	−24.5244	−1.19383
$423$	−17.0273	−0.827895
$424$	9.49613	0.461173
$425$	−27.8994	−1.35332
$426$	−7.24556	−0.351049
$427$	−43.7682	−2.11809
$428$	−9.34186	−0.451556
$429$	10.9082	0.526651
$430$	0.874324	0.0421636
$431$	15.2857	0.736284	0.368142	−	0.929770i	$-0.379994\pi$
$431$	15.2857	0.736284	0.368142	+	0.929770i	$0.379994\pi$
$432$	−4.54923	−0.218875
$433$	4.81038	0.231172	0.115586	−	0.993297i	$-0.463125\pi$
$433$	4.81038	0.231172	0.115586	+	0.993297i	$0.463125\pi$
$434$	0.297503	0.0142806
$435$	−1.21390	−0.0582020
$436$	−15.4318	−0.739051
$437$	42.0244	2.01030
$438$	−3.18199	−0.152041
$439$	−9.32921	−0.445259	−0.222629	−	0.974903i	$-0.571464\pi$
$439$	−9.32921	−0.445259	−0.222629	+	0.974903i	$0.571464\pi$
$440$	0.462264	0.0220376
$441$	−20.0831	−0.956339
$442$	21.5275	1.02396
$443$	−5.17427	−0.245837	−0.122918	−	0.992417i	$-0.539225\pi$
$443$	−5.17427	−0.245837	−0.122918	+	0.992417i	$0.539225\pi$
$444$	−1.29877	−0.0616369
$445$	−1.80239	−0.0854415
$446$	−16.8789	−0.799237
$447$	−16.4628	−0.778665
$448$	−3.99131	−0.188572
$449$	−5.15276	−0.243174	−0.121587	−	0.992581i	$-0.538798\pi$
$449$	−5.15276	−0.243174	−0.121587	+	0.992581i	$0.538798\pi$
$450$	−11.1992	−0.527936
$451$	−2.44541	−0.115150
$452$	0.429369	0.0201958
$453$	11.0016	0.516902
$454$	13.2745	0.623005
$455$	2.16472	0.101484
$456$	4.40385	0.206229
$457$	−18.6393	−0.871909	−0.435955	−	0.899969i	$-0.643589\pi$
$457$	−18.6393	−0.871909	−0.435955	+	0.899969i	$0.643589\pi$
$458$	18.9593	0.885911
$459$	−25.4857	−1.18957
$460$	−1.16734	−0.0544276
$461$	−6.87073	−0.320002	−0.160001	−	0.987117i	$-0.551150\pi$
$461$	−6.87073	−0.320002	−0.160001	+	0.987117i	$0.551150\pi$
$462$	−11.3301	−0.527123
$463$	10.1695	0.472615	0.236308	−	0.971678i	$-0.424063\pi$
$463$	10.1695	0.472615	0.236308	+	0.971678i	$0.424063\pi$
$464$	−9.92324	−0.460675
$465$	0.00911808	0.000422841 0
$466$	9.58021	0.443795
$467$	35.5275	1.64402	0.822008	−	0.569476i	$-0.192854\pi$
$467$	35.5275	1.64402	0.822008	+	0.569476i	$0.192854\pi$
$468$	8.64143	0.399450
$469$	−26.4580	−1.22172
$470$	−1.06867	−0.0492943
$471$	1.32608	0.0611026
$472$	−2.89542	−0.133272
$473$	20.2890	0.932889
$474$	−0.959678	−0.0440795
$475$	25.3041	1.16103
$476$	−22.3601	−1.02487
$477$	21.3549	0.977774
$478$	−19.5558	−0.894460
$479$	23.4649	1.07214	0.536068	−	0.844175i	$-0.319909\pi$
$479$	23.4649	1.07214	0.536068	+	0.844175i	$0.319909\pi$
$480$	−0.122329	−0.00558352
$481$	5.75823	0.262553
$482$	7.93943	0.361631
$483$	28.6115	1.30187
$484$	−0.272992	−0.0124087
$485$	−0.583533	−0.0264969
$486$	−16.0775	−0.729292
$487$	24.4635	1.10855	0.554275	−	0.832334i	$-0.312996\pi$
$487$	24.4635	1.10855	0.554275	+	0.832334i	$0.312996\pi$
$488$	−10.9659	−0.496401
$489$	−17.7490	−0.802638
$490$	−1.26047	−0.0569421
$491$	−42.1499	−1.90220	−0.951099	−	0.308885i	$-0.900044\pi$
$491$	−42.1499	−1.90220	−0.951099	+	0.308885i	$0.900044\pi$
$492$	0.647127	0.0291748
$493$	−55.5919	−2.50373
$494$	−19.5249	−0.878467
$495$	1.03954	0.0467239
$496$	0.0745375	0.00334683
$497$	−33.3665	−1.49669
$498$	5.35712	0.240058
$499$	−11.2495	−0.503599	−0.251799	−	0.967779i	$-0.581022\pi$
$499$	−11.2495	−0.503599	−0.251799	+	0.967779i	$0.581022\pi$
$500$	−1.40859	−0.0629941
$501$	8.07854	0.360923
$502$	15.0136	0.670090
$503$	−1.58436	−0.0706430	−0.0353215	−	0.999376i	$-0.511246\pi$
$503$	−1.58436	−0.0706430	−0.0353215	+	0.999376i	$0.511246\pi$
$504$	−8.97567	−0.399808
$505$	1.17685	0.0523694
$506$	−27.0886	−1.20423
$507$	1.53080	0.0679853
$508$	9.56785	0.424505
$509$	32.6957	1.44921	0.724606	−	0.689163i	$-0.242022\pi$
$509$	32.6957	1.44921	0.724606	+	0.689163i	$0.242022\pi$
$510$	−0.685310	−0.0303460
$511$	−14.6534	−0.648226
$512$	−1.00000	−0.0441942
$513$	23.1149	1.02055
$514$	8.51641	0.375643
$515$	1.81069	0.0797885
$516$	−5.36907	−0.236360
$517$	−24.7990	−1.09066
$518$	−5.98095	−0.262788
$519$	10.0445	0.440904
$520$	0.542357	0.0237839
$521$	−20.9647	−0.918481	−0.459240	−	0.888312i	$-0.651878\pi$
$521$	−20.9647	−0.918481	−0.459240	+	0.888312i	$0.651878\pi$
$522$	−22.3154	−0.976718
$523$	−42.9951	−1.88005	−0.940023	−	0.341111i	$-0.889197\pi$
$523$	−42.9951	−1.88005	−0.940023	+	0.341111i	$0.889197\pi$
$524$	−2.36538	−0.103332
$525$	17.2278	0.751883
$526$	−24.5400	−1.07000
$527$	0.417573	0.0181898
$528$	−2.83868	−0.123538
$529$	45.4060	1.97418
$530$	1.34029	0.0582183
$531$	−6.51121	−0.282562
$532$	20.2801	0.879255
$533$	−2.86911	−0.124275
$534$	11.0682	0.478966
$535$	−1.31851	−0.0570043
$536$	−6.62889	−0.286324
$537$	10.2390	0.441844
$538$	−7.95184	−0.342828
$539$	−29.2496	−1.25987
$540$	−0.642080	−0.0276307
$541$	30.6454	1.31755	0.658775	−	0.752340i	$-0.271075\pi$
$541$	30.6454	1.31755	0.658775	+	0.752340i	$0.271075\pi$
$542$	10.8137	0.464487
$543$	−19.3103	−0.828686
$544$	−5.60219	−0.240192
$545$	−2.17805	−0.0932976
$546$	−13.2932	−0.568894
$547$	1.35164	0.0577920	0.0288960	−	0.999582i	$-0.490801\pi$
$547$	1.35164	0.0577920	0.0288960	+	0.999582i	$0.490801\pi$
$548$	−2.84546	−0.121552
$549$	−24.6600	−1.05246
$550$	−16.3108	−0.695495
$551$	50.4206	2.14799
$552$	7.16844	0.305109
$553$	−4.41941	−0.187932
$554$	7.01059	0.297851
$555$	−0.183309	−0.00778103
$556$	4.71650	0.200024
$557$	3.28198	0.139062	0.0695310	−	0.997580i	$-0.477850\pi$
$557$	3.28198	0.139062	0.0695310	+	0.997580i	$0.477850\pi$
$558$	0.167620	0.00709592
$559$	23.8043	1.00682
$560$	−0.563335	−0.0238053
$561$	−15.9029	−0.671419
$562$	14.0044	0.590742
$563$	−6.46700	−0.272552	−0.136276	−	0.990671i	$-0.543513\pi$
$563$	−6.46700	−0.272552	−0.136276	+	0.990671i	$0.543513\pi$
$564$	6.56254	0.276333
$565$	0.0606013	0.00254951
$566$	18.6465	0.783772
$567$	−11.1897	−0.469922
$568$	−8.35977	−0.350768
$569$	22.6225	0.948383	0.474192	−	0.880422i	$-0.342740\pi$
$569$	22.6225	0.948383	0.474192	+	0.880422i	$0.342740\pi$
$570$	0.621560	0.0260343
$571$	18.6274	0.779531	0.389766	−	0.920914i	$-0.372556\pi$
$571$	18.6274	0.779531	0.389766	+	0.920914i	$0.372556\pi$
$572$	12.5856	0.526230
$573$	−18.4037	−0.768827
$574$	2.98008	0.124386
$575$	41.1892	1.71771
$576$	−2.24880	−0.0937000
$577$	6.00584	0.250026	0.125013	−	0.992155i	$-0.460103\pi$
$577$	6.00584	0.250026	0.125013	+	0.992155i	$0.460103\pi$
$578$	−14.3846	−0.598320
$579$	11.0178	0.457883
$580$	−1.40057	−0.0581555
$581$	24.6700	1.02348
$582$	3.58337	0.148536
$583$	31.1018	1.28811
$584$	−3.67131	−0.151920
$585$	1.21965	0.0504265
$586$	9.21904	0.380835
$587$	9.89376	0.408359	0.204180	−	0.978933i	$-0.434547\pi$
$587$	9.89376	0.408359	0.204180	+	0.978933i	$0.434547\pi$
$588$	7.74030	0.319205
$589$	−0.378730	−0.0156053
$590$	−0.408660	−0.0168243
$591$	−10.6319	−0.437339
$592$	−1.49849	−0.0615877
$593$	−10.8276	−0.444636	−0.222318	−	0.974974i	$-0.571362\pi$
$593$	−10.8276	−0.444636	−0.222318	+	0.974974i	$0.571362\pi$
$594$	−14.8997	−0.611341
$595$	−3.15591	−0.129380
$596$	−18.9945	−0.778044
$597$	−2.88581	−0.118108
$598$	−31.7820	−1.29966
$599$	21.4584	0.876768	0.438384	−	0.898788i	$-0.355551\pi$
$599$	21.4584	0.876768	0.438384	+	0.898788i	$0.355551\pi$
$600$	4.31632	0.176213
$601$	32.3765	1.32066	0.660332	−	0.750974i	$-0.270416\pi$
$601$	32.3765	1.32066	0.660332	+	0.750974i	$0.270416\pi$
$602$	−24.7251	−1.00772
$603$	−14.9070	−0.607062
$604$	12.6935	0.516489
$605$	−0.0385301	−0.00156647
$606$	−7.22686	−0.293571
$607$	−25.8123	−1.04769	−0.523844	−	0.851814i	$-0.675503\pi$
$607$	−25.8123	−1.04769	−0.523844	+	0.851814i	$0.675503\pi$
$608$	5.08106	0.206064
$609$	34.3279	1.39104
$610$	−1.54772	−0.0626655
$611$	−29.0957	−1.17709
$612$	−12.5982	−0.509253
$613$	−5.83142	−0.235529	−0.117764	−	0.993042i	$-0.537573\pi$
$613$	−5.83142	−0.235529	−0.117764	+	0.993042i	$0.537573\pi$
$614$	5.52048	0.222788
$615$	0.0913358	0.00368301
$616$	−13.0724	−0.526702
$617$	−29.2702	−1.17838	−0.589188	−	0.807996i	$-0.700552\pi$
$617$	−29.2702	−1.17838	−0.589188	+	0.807996i	$0.700552\pi$
$618$	−11.1191	−0.447276
$619$	3.99793	0.160690	0.0803451	−	0.996767i	$-0.474398\pi$
$619$	3.99793	0.160690	0.0803451	+	0.996767i	$0.474398\pi$
$620$	0.0105202	0.000422503 0
$621$	37.6257	1.50987
$622$	−5.88297	−0.235886
$623$	50.9699	2.04206
$624$	−3.33052	−0.133328
$625$	24.7016	0.988064
$626$	34.4206	1.37573
$627$	14.4235	0.576020
$628$	1.53000	0.0610538
$629$	−8.39485	−0.334724
$630$	−1.26683	−0.0504717
$631$	13.4759	0.536468	0.268234	−	0.963354i	$-0.413560\pi$
$631$	13.4759	0.536468	0.268234	+	0.963354i	$0.413560\pi$
$632$	−1.10726	−0.0440443
$633$	21.2558	0.844841
$634$	1.37854	0.0547488
$635$	1.35041	0.0535894
$636$	−8.23046	−0.326359
$637$	−34.3174	−1.35971
$638$	−32.5007	−1.28672
$639$	−18.7995	−0.743695
$640$	−0.141140	−0.00557906
$641$	−28.4896	−1.12527	−0.562636	−	0.826705i	$-0.690213\pi$
$641$	−28.4896	−1.12527	−0.562636	+	0.826705i	$0.690213\pi$
$642$	8.09676	0.319554
$643$	4.35191	0.171623	0.0858113	−	0.996311i	$-0.472652\pi$
$643$	4.35191	0.171623	0.0858113	+	0.996311i	$0.472652\pi$
$644$	33.0113	1.30083
$645$	−0.757792	−0.0298380
$646$	28.4651	1.11994
$647$	−20.5108	−0.806364	−0.403182	−	0.915120i	$-0.632096\pi$
$647$	−20.5108	−0.806364	−0.403182	+	0.915120i	$0.632096\pi$
$648$	−2.80351	−0.110132
$649$	−9.48310	−0.372244
$650$	−19.1369	−0.750609
$651$	−0.257851	−0.0101060
$652$	−20.4784	−0.801997
$653$	13.6350	0.533578	0.266789	−	0.963755i	$-0.414037\pi$
$653$	13.6350	0.533578	0.266789	+	0.963755i	$0.414037\pi$
$654$	13.3750	0.523006
$655$	−0.333851	−0.0130446
$656$	0.746641	0.0291514
$657$	−8.25605	−0.322099
$658$	30.2211	1.17814
$659$	−48.6976	−1.89699	−0.948494	−	0.316795i	$-0.897393\pi$
$659$	−48.6976	−1.89699	−0.948494	+	0.316795i	$0.897393\pi$
$660$	−0.400653	−0.0155954
$661$	4.32949	0.168398	0.0841989	−	0.996449i	$-0.473167\pi$
$661$	4.32949	0.168398	0.0841989	+	0.996449i	$0.473167\pi$
$662$	−1.57012	−0.0610244
$663$	−18.6582	−0.724625
$664$	6.18093	0.239866
$665$	2.86234	0.110997
$666$	−3.36981	−0.130578
$667$	82.0731	3.17788
$668$	9.32085	0.360634
$669$	14.6292	0.565598
$670$	−0.935603	−0.0361455
$671$	−35.9155	−1.38650
$672$	3.45934	0.133447
$673$	−13.4173	−0.517198	−0.258599	−	0.965985i	$-0.583261\pi$
$673$	−13.4173	−0.517198	−0.258599	+	0.965985i	$0.583261\pi$
$674$	−8.08636	−0.311475
$675$	22.6555	0.872011
$676$	1.76621	0.0679310
$677$	−3.43175	−0.131893	−0.0659465	−	0.997823i	$-0.521007\pi$
$677$	−3.43175	−0.131893	−0.0659465	+	0.997823i	$0.521007\pi$
$678$	−0.372142	−0.0142920
$679$	16.5018	0.633280
$680$	−0.790696	−0.0303218
$681$	−11.5053	−0.440883
$682$	0.244126	0.00934807
$683$	−4.90524	−0.187694	−0.0938469	−	0.995587i	$-0.529916\pi$
$683$	−4.90524	−0.187694	−0.0938469	+	0.995587i	$0.529916\pi$
$684$	11.4263	0.436895
$685$	−0.401609	−0.0153447
$686$	7.70559	0.294201
$687$	−16.4324	−0.626934
$688$	−6.19471	−0.236171
$689$	36.4906	1.39018
$690$	1.01176	0.0385169
$691$	15.9616	0.607208	0.303604	−	0.952798i	$-0.401810\pi$
$691$	15.9616	0.607208	0.303604	+	0.952798i	$0.401810\pi$
$692$	11.5891	0.440552
$693$	−29.3972	−1.11671
$694$	−25.4951	−0.967781
$695$	0.665688	0.0252510
$696$	8.60065	0.326007
$697$	4.18283	0.158436
$698$	21.2961	0.806069
$699$	−8.30334	−0.314061
$700$	19.8771	0.751282
$701$	47.3828	1.78962	0.894812	−	0.446444i	$-0.147310\pi$
$701$	47.3828	1.78962	0.894812	+	0.446444i	$0.147310\pi$
$702$	−17.4812	−0.659787
$703$	7.61393	0.287165
$704$	−3.27521	−0.123439
$705$	0.926239	0.0348842
$706$	−11.2208	−0.422299
$707$	−33.2803	−1.25164
$708$	2.50951	0.0943131
$709$	−25.2644	−0.948823	−0.474411	−	0.880303i	$-0.657339\pi$
$709$	−25.2644	−0.948823	−0.474411	+	0.880303i	$0.657339\pi$
$710$	−1.17990	−0.0442809
$711$	−2.49000	−0.0933822
$712$	12.7702	0.478583
$713$	−0.616484	−0.0230875
$714$	19.3799	0.725275
$715$	1.77634	0.0664312
$716$	11.8135	0.441491
$717$	16.9493	0.632984
$718$	13.9016	0.518804
$719$	6.03160	0.224941	0.112470	−	0.993655i	$-0.464124\pi$
$719$	6.03160	0.224941	0.112470	+	0.993655i	$0.464124\pi$
$720$	−0.317396	−0.0118287
$721$	−51.2046	−1.90696
$722$	−6.81719	−0.253710
$723$	−6.88124	−0.255916
$724$	−22.2799	−0.828025
$725$	49.4185	1.83536
$726$	0.236607	0.00878129
$727$	2.47318	0.0917251	0.0458625	−	0.998948i	$-0.485396\pi$
$727$	2.47318	0.0917251	0.0458625	+	0.998948i	$0.485396\pi$
$728$	−15.3374	−0.568440
$729$	5.52416	0.204599
$730$	−0.518170	−0.0191783
$731$	−34.7040	−1.28357
$732$	9.50430	0.351289
$733$	13.3018	0.491312	0.245656	−	0.969357i	$-0.420997\pi$
$733$	13.3018	0.491312	0.245656	+	0.969357i	$0.420997\pi$
$734$	−5.82865	−0.215140
$735$	1.09247	0.0402963
$736$	8.27079	0.304865
$737$	−21.7110	−0.799735
$738$	1.67905	0.0618066
$739$	11.3989	0.419315	0.209658	−	0.977775i	$-0.432765\pi$
$739$	11.3989	0.419315	0.209658	+	0.977775i	$0.432765\pi$
$740$	−0.211498	−0.00777481
$741$	16.9226	0.621667
$742$	−37.9020	−1.39143
$743$	52.9130	1.94119	0.970595	−	0.240717i	$-0.0773827\pi$
$743$	52.9130	1.94119	0.970595	+	0.240717i	$0.0773827\pi$
$744$	−0.0646029	−0.00236846
$745$	−2.68088	−0.0982200
$746$	18.6312	0.682137
$747$	13.8997	0.508562
$748$	−18.3484	−0.670883
$749$	37.2863	1.36241
$750$	1.22085	0.0445792
$751$	−11.0636	−0.403717	−0.201859	−	0.979415i	$-0.564698\pi$
$751$	−11.0636	−0.403717	−0.201859	+	0.979415i	$0.564698\pi$
$752$	7.57172	0.276112
$753$	−13.0126	−0.474204
$754$	−38.1319	−1.38868
$755$	1.79156	0.0652015
$756$	18.1574	0.660378
$757$	−51.3656	−1.86691	−0.933457	−	0.358690i	$-0.883223\pi$
$757$	−51.3656	−1.86691	−0.933457	+	0.358690i	$0.883223\pi$
$758$	34.1518	1.24045
$759$	23.4782	0.852203
$760$	0.717143	0.0260135
$761$	−12.3326	−0.447058	−0.223529	−	0.974697i	$-0.571758\pi$
$761$	−12.3326	−0.447058	−0.223529	+	0.974697i	$0.571758\pi$
$762$	−8.29262	−0.300410
$763$	61.5933	2.22983
$764$	−21.2338	−0.768213
$765$	−1.77812	−0.0642879
$766$	−16.3788	−0.591789
$767$	−11.1262	−0.401742
$768$	0.866718	0.0312750
$769$	−37.1203	−1.33859	−0.669296	−	0.742996i	$-0.733404\pi$
$769$	−37.1203	−1.33859	−0.669296	+	0.742996i	$0.733404\pi$
$770$	−1.84504	−0.0664907
$771$	−7.38132	−0.265832
$772$	12.7121	0.457517
$773$	4.94419	0.177830	0.0889151	−	0.996039i	$-0.471660\pi$
$773$	4.94419	0.177830	0.0889151	+	0.996039i	$0.471660\pi$
$774$	−13.9307	−0.500728
$775$	−0.371203	−0.0133340
$776$	4.13442	0.148417
$777$	5.18380	0.185968
$778$	7.75851	0.278156
$779$	−3.79373	−0.135925
$780$	−0.470071	−0.0168312
$781$	−27.3800	−0.979734
$782$	46.3346	1.65692
$783$	45.1431	1.61328
$784$	8.93059	0.318950
$785$	0.215945	0.00770742
$786$	2.05012	0.0731253
$787$	−19.7437	−0.703786	−0.351893	−	0.936040i	$-0.614462\pi$
$787$	−19.7437	−0.703786	−0.351893	+	0.936040i	$0.614462\pi$
$788$	−12.2669	−0.436990
$789$	21.2693	0.757206
$790$	−0.156278	−0.00556014
$791$	−1.71375	−0.0609338
$792$	−7.36530	−0.261714
$793$	−42.1383	−1.49637
$794$	35.4676	1.25870
$795$	−1.16165	−0.0411995
$796$	−3.32959	−0.118014
$797$	−31.9102	−1.13032	−0.565158	−	0.824983i	$-0.691185\pi$
$797$	−31.9102	−1.13032	−0.565158	+	0.824983i	$0.691185\pi$
$798$	−17.5771	−0.622224
$799$	42.4182	1.50065
$800$	4.98008	0.176072
$801$	28.7176	1.01469
$802$	5.05558	0.178519
$803$	−12.0243	−0.424329
$804$	5.74537	0.202624
$805$	4.65923	0.164216
$806$	0.286424	0.0100888
$807$	6.89200	0.242610
$808$	−8.33819	−0.293337
$809$	44.2919	1.55722	0.778610	−	0.627509i	$-0.215925\pi$
$809$	44.2919	1.55722	0.778610	+	0.627509i	$0.215925\pi$
$810$	−0.395688	−0.0139030
$811$	−14.1035	−0.495241	−0.247620	−	0.968857i	$-0.579649\pi$
$811$	−14.1035	−0.495241	−0.247620	+	0.968857i	$0.579649\pi$
$812$	39.6068	1.38993
$813$	−9.37240	−0.328704
$814$	−4.90788	−0.172021
$815$	−2.89033	−0.101244
$816$	4.85552	0.169977
$817$	31.4757	1.10120
$818$	−29.8907	−1.04510
$819$	−34.4906	−1.20520
$820$	0.105381	0.00368007
$821$	45.1165	1.57458	0.787288	−	0.616586i	$-0.211485\pi$
$821$	45.1165	1.57458	0.787288	+	0.616586i	$0.211485\pi$
$822$	2.46621	0.0860190
$823$	−4.19781	−0.146326	−0.0731632	−	0.997320i	$-0.523309\pi$
$823$	−4.19781	−0.146326	−0.0731632	+	0.997320i	$0.523309\pi$
$824$	−12.8290	−0.446919
$825$	14.1369	0.492183
$826$	11.5565	0.402102
$827$	27.2184	0.946478	0.473239	−	0.880934i	$-0.343085\pi$
$827$	27.2184	0.946478	0.473239	+	0.880934i	$0.343085\pi$
$828$	18.5994	0.646373
$829$	−44.1592	−1.53371	−0.766857	−	0.641818i	$-0.778180\pi$
$829$	−44.1592	−1.53371	−0.766857	+	0.641818i	$0.778180\pi$
$830$	0.872378	0.0302807
$831$	−6.07620	−0.210781
$832$	−3.84268	−0.133221
$833$	50.0309	1.73347
$834$	−4.08787	−0.141551
$835$	1.31555	0.0455264
$836$	16.6416	0.575560
$837$	−0.339088	−0.0117206
$838$	28.0315	0.968331
$839$	30.2653	1.04488	0.522438	−	0.852678i	$-0.325023\pi$
$839$	30.2653	1.04488	0.522438	+	0.852678i	$0.325023\pi$
$840$	0.488253	0.0168463
$841$	69.4707	2.39554
$842$	−22.6989	−0.782256
$843$	−12.1379	−0.418051
$844$	24.5244	0.844166
$845$	0.249283	0.00857559
$846$	17.0273	0.585410
$847$	1.08959	0.0374389
$848$	−9.49613	−0.326098
$849$	−16.1613	−0.554654
$850$	27.8994	0.956941
$851$	12.3937	0.424851
$852$	7.24556	0.248229
$853$	30.1396	1.03196	0.515980	−	0.856600i	$-0.327428\pi$
$853$	30.1396	1.03196	0.515980	+	0.856600i	$0.327428\pi$
$854$	43.7682	1.49772
$855$	1.61271	0.0551536
$856$	9.34186	0.319298
$857$	16.1046	0.550123	0.275061	−	0.961427i	$-0.411302\pi$
$857$	16.1046	0.550123	0.275061	+	0.961427i	$0.411302\pi$
$858$	−10.9082	−0.372398
$859$	−34.7256	−1.18482	−0.592411	−	0.805636i	$-0.701824\pi$
$859$	−34.7256	−1.18482	−0.592411	+	0.805636i	$0.701824\pi$
$860$	−0.874324	−0.0298142
$861$	−2.58289	−0.0880246
$862$	−15.2857	−0.520632
$863$	1.37714	0.0468784	0.0234392	−	0.999725i	$-0.492538\pi$
$863$	1.37714	0.0468784	0.0234392	+	0.999725i	$0.492538\pi$
$864$	4.54923	0.154768
$865$	1.63569	0.0556152
$866$	−4.81038	−0.163463
$867$	12.4674	0.423414
$868$	−0.297503	−0.0100979
$869$	−3.62650	−0.123020
$870$	1.21390	0.0411550
$871$	−25.4727	−0.863110
$872$	15.4318	0.522588
$873$	9.29748	0.314672
$874$	−42.0244	−1.42150
$875$	5.62213	0.190063
$876$	3.18199	0.107509
$877$	35.1123	1.18566	0.592830	−	0.805328i	$-0.298011\pi$
$877$	35.1123	1.18566	0.592830	+	0.805328i	$0.298011\pi$
$878$	9.32921	0.314846
$879$	−7.99030	−0.269506
$880$	−0.462264	−0.0155829
$881$	39.5492	1.33245	0.666223	−	0.745753i	$-0.267910\pi$
$881$	39.5492	1.33245	0.666223	+	0.745753i	$0.267910\pi$
$882$	20.0831	0.676234
$883$	−35.1323	−1.18230	−0.591148	−	0.806563i	$-0.701325\pi$
$883$	−35.1323	−1.18230	−0.591148	+	0.806563i	$0.701325\pi$
$884$	−21.5275	−0.724047
$885$	0.354193	0.0119061
$886$	5.17427	0.173833
$887$	−1.74303	−0.0585252	−0.0292626	−	0.999572i	$-0.509316\pi$
$887$	−1.74303	−0.0585252	−0.0292626	+	0.999572i	$0.509316\pi$
$888$	1.29877	0.0435839
$889$	−38.1883	−1.28079
$890$	1.80239	0.0604162
$891$	−9.18207	−0.307611
$892$	16.8789	0.565146
$893$	−38.4724	−1.28743
$894$	16.4628	0.550600
$895$	1.66736	0.0557337
$896$	3.99131	0.133340
$897$	27.5460	0.919736
$898$	5.15276	0.171950
$899$	−0.739653	−0.0246688
$900$	11.1992	0.373307
$901$	−53.1992	−1.77232
$902$	2.44541	0.0814232
$903$	21.4296	0.713134
$904$	−0.429369	−0.0142806
$905$	−3.14459	−0.104530
$906$	−11.0016	−0.365505
$907$	−3.46939	−0.115199	−0.0575996	−	0.998340i	$-0.518345\pi$
$907$	−3.46939	−0.115199	−0.0575996	+	0.998340i	$0.518345\pi$
$908$	−13.2745	−0.440531
$909$	−18.7509	−0.621929
$910$	−2.16472	−0.0717597
$911$	32.1336	1.06463	0.532317	−	0.846545i	$-0.321322\pi$
$911$	32.1336	1.06463	0.532317	+	0.846545i	$0.321322\pi$
$912$	−4.40385	−0.145826
$913$	20.2438	0.669973
$914$	18.6393	0.616533
$915$	1.34144	0.0443466
$916$	−18.9593	−0.626433
$917$	9.44098	0.311769
$918$	25.4857	0.841152
$919$	−17.0304	−0.561782	−0.280891	−	0.959740i	$-0.590630\pi$
$919$	−17.0304	−0.561782	−0.280891	+	0.959740i	$0.590630\pi$
$920$	1.16734	0.0384861
$921$	−4.78470	−0.157661
$922$	6.87073	0.226275
$923$	−32.1240	−1.05737
$924$	11.3301	0.372732
$925$	7.46261	0.245369
$926$	−10.1695	−0.334189
$927$	−28.8499	−0.947553
$928$	9.92324	0.325746
$929$	−7.00178	−0.229721	−0.114861	−	0.993382i	$-0.536642\pi$
$929$	−7.00178	−0.229721	−0.114861	+	0.993382i	$0.536642\pi$
$930$	−0.00911808	−0.000298994 0
$931$	−45.3769	−1.48717
$932$	−9.58021	−0.313810
$933$	5.09888	0.166930
$934$	−35.5275	−1.16249
$935$	−2.58970	−0.0846921
$936$	−8.64143	−0.282454
$937$	−22.7671	−0.743768	−0.371884	−	0.928279i	$-0.621288\pi$
$937$	−22.7671	−0.743768	−0.371884	+	0.928279i	$0.621288\pi$
$938$	26.4580	0.863883
$939$	−29.8330	−0.973562
$940$	1.06867	0.0348563
$941$	−5.05815	−0.164891	−0.0824454	−	0.996596i	$-0.526273\pi$
$941$	−5.05815	−0.164891	−0.0824454	+	0.996596i	$0.526273\pi$
$942$	−1.32608	−0.0432061
$943$	−6.17532	−0.201096
$944$	2.89542	0.0942377
$945$	2.56274	0.0833660
$946$	−20.2890	−0.659652
$947$	−24.3063	−0.789849	−0.394924	−	0.918714i	$-0.629229\pi$
$947$	−24.3063	−0.789849	−0.394924	+	0.918714i	$0.629229\pi$
$948$	0.959678	0.0311689
$949$	−14.1077	−0.457955
$950$	−25.3041	−0.820973
$951$	−1.19480	−0.0387442
$952$	22.3601	0.724696
$953$	8.40085	0.272130	0.136065	−	0.990700i	$-0.456554\pi$
$953$	8.40085	0.272130	0.136065	+	0.990700i	$0.456554\pi$
$954$	−21.3549	−0.691390
$955$	−2.99695	−0.0969790
$956$	19.5558	0.632479
$957$	28.1689	0.910573
$958$	−23.4649	−0.758115
$959$	11.3571	0.366741
$960$	0.122329	0.00394815
$961$	−30.9944	−0.999821
$962$	−5.75823	−0.185653
$963$	21.0080	0.676973
$964$	−7.93943	−0.255712
$965$	1.79419	0.0577569
$966$	−28.6115	−0.920560
$967$	−27.8764	−0.896445	−0.448223	−	0.893922i	$-0.647943\pi$
$967$	−27.8764	−0.896445	−0.448223	+	0.893922i	$0.647943\pi$
$968$	0.272992	0.00877428
$969$	−24.6712	−0.792553
$970$	0.583533	0.0187361
$971$	−54.1930	−1.73914	−0.869569	−	0.493812i	$-0.835603\pi$
$971$	−54.1930	−1.73914	−0.869569	+	0.493812i	$0.835603\pi$
$972$	16.0775	0.515687
$973$	−18.8250	−0.603503
$974$	−24.4635	−0.783863
$975$	16.5863	0.531185
$976$	10.9659	0.351008
$977$	−51.6187	−1.65143	−0.825714	−	0.564088i	$-0.809228\pi$
$977$	−51.6187	−1.65143	−0.825714	+	0.564088i	$0.809228\pi$
$978$	17.7490	0.567551
$979$	41.8251	1.33674
$980$	1.26047	0.0402641
$981$	34.7031	1.10799
$982$	42.1499	1.34506
$983$	−28.0023	−0.893135	−0.446568	−	0.894750i	$-0.647354\pi$
$983$	−28.0023	−0.893135	−0.446568	+	0.894750i	$0.647354\pi$
$984$	−0.647127	−0.0206297
$985$	−1.73135	−0.0551655
$986$	55.5919	1.77041
$987$	−26.1932	−0.833738
$988$	19.5249	0.621170
$989$	51.2352	1.62918
$990$	−1.03954	−0.0330388
$991$	12.5546	0.398809	0.199404	−	0.979917i	$-0.436099\pi$
$991$	12.5546	0.398809	0.199404	+	0.979917i	$0.436099\pi$
$992$	−0.0745375	−0.00236657
$993$	1.36085	0.0431853
$994$	33.3665	1.05832
$995$	−0.469939	−0.0148981
$996$	−5.35712	−0.169747
$997$	32.6949	1.03546	0.517729	−	0.855544i	$-0.326777\pi$
$997$	32.6949	1.03546	0.517729	+	0.855544i	$0.326777\pi$
$998$	11.2495	0.356098
$999$	6.81698	0.215680

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.e.1.46	✓	77

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.e.1.46	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +0.866718 q^{3} +1.00000 q^{4} +0.141140 q^{5} -0.866718 q^{6} -3.99131 q^{7} -1.00000 q^{8} -2.24880 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +0.866718 q^{3} +1.00000 q^{4} +0.141140 q^{5} -0.866718 q^{6} -3.99131 q^{7} -1.00000 q^{8} -2.24880 q^{9} -0.141140 q^{10} -3.27521 q^{11} +0.866718 q^{12} -3.84268 q^{13} +3.99131 q^{14} +0.122329 q^{15} +1.00000 q^{16} +5.60219 q^{17} +2.24880 q^{18} -5.08106 q^{19} +0.141140 q^{20} -3.45934 q^{21} +3.27521 q^{22} -8.27079 q^{23} -0.866718 q^{24} -4.98008 q^{25} +3.84268 q^{26} -4.54923 q^{27} -3.99131 q^{28} -9.92324 q^{29} -0.122329 q^{30} +0.0745375 q^{31} -1.00000 q^{32} -2.83868 q^{33} -5.60219 q^{34} -0.563335 q^{35} -2.24880 q^{36} -1.49849 q^{37} +5.08106 q^{38} -3.33052 q^{39} -0.141140 q^{40} +0.746641 q^{41} +3.45934 q^{42} -6.19471 q^{43} -3.27521 q^{44} -0.317396 q^{45} +8.27079 q^{46} +7.57172 q^{47} +0.866718 q^{48} +8.93059 q^{49} +4.98008 q^{50} +4.85552 q^{51} -3.84268 q^{52} -9.49613 q^{53} +4.54923 q^{54} -0.462264 q^{55} +3.99131 q^{56} -4.40385 q^{57} +9.92324 q^{58} +2.89542 q^{59} +0.122329 q^{60} +10.9659 q^{61} -0.0745375 q^{62} +8.97567 q^{63} +1.00000 q^{64} -0.542357 q^{65} +2.83868 q^{66} +6.62889 q^{67} +5.60219 q^{68} -7.16844 q^{69} +0.563335 q^{70} +8.35977 q^{71} +2.24880 q^{72} +3.67131 q^{73} +1.49849 q^{74} -4.31632 q^{75} -5.08106 q^{76} +13.0724 q^{77} +3.33052 q^{78} +1.10726 q^{79} +0.141140 q^{80} +2.80351 q^{81} -0.746641 q^{82} -6.18093 q^{83} -3.45934 q^{84} +0.790696 q^{85} +6.19471 q^{86} -8.60065 q^{87} +3.27521 q^{88} -12.7702 q^{89} +0.317396 q^{90} +15.3374 q^{91} -8.27079 q^{92} +0.0646029 q^{93} -7.57172 q^{94} -0.717143 q^{95} -0.866718 q^{96} -4.13442 q^{97} -8.93059 q^{98} +7.36530 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9}+O(q^{10}) \) 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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