LMFDB - Embedded newform 4153.2.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4153,2,Mod(1,4153)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4153, 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("4153.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4153 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4153.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:	$33.1618719594$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	4153.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.23607 q^{3} -1.00000 q^{4} -3.23607 q^{5} -1.23607 q^{6} -2.00000 q^{7} +3.00000 q^{8} -1.47214 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.23607 q^{3} -1.00000 q^{4} -3.23607 q^{5} -1.23607 q^{6} -2.00000 q^{7} +3.00000 q^{8} -1.47214 q^{9} +3.23607 q^{10} +5.61803 q^{11} -1.23607 q^{12} -6.47214 q^{13} +2.00000 q^{14} -4.00000 q^{15} -1.00000 q^{16} +5.23607 q^{17} +1.47214 q^{18} -0.763932 q^{19} +3.23607 q^{20} -2.47214 q^{21} -5.61803 q^{22} -2.76393 q^{23} +3.70820 q^{24} +5.47214 q^{25} +6.47214 q^{26} -5.52786 q^{27} +2.00000 q^{28} +6.38197 q^{29} +4.00000 q^{30} +6.61803 q^{31} -5.00000 q^{32} +6.94427 q^{33} -5.23607 q^{34} +6.47214 q^{35} +1.47214 q^{36} +1.85410 q^{37} +0.763932 q^{38} -8.00000 q^{39} -9.70820 q^{40} +11.7082 q^{41} +2.47214 q^{42} -11.2361 q^{43} -5.61803 q^{44} +4.76393 q^{45} +2.76393 q^{46} +13.7082 q^{47} -1.23607 q^{48} -3.00000 q^{49} -5.47214 q^{50} +6.47214 q^{51} +6.47214 q^{52} +6.76393 q^{53} +5.52786 q^{54} -18.1803 q^{55} -6.00000 q^{56} -0.944272 q^{57} -6.38197 q^{58} -4.94427 q^{59} +4.00000 q^{60} -12.3262 q^{61} -6.61803 q^{62} +2.94427 q^{63} +7.00000 q^{64} +20.9443 q^{65} -6.94427 q^{66} +2.32624 q^{67} -5.23607 q^{68} -3.41641 q^{69} -6.47214 q^{70} -12.9443 q^{71} -4.41641 q^{72} +4.85410 q^{73} -1.85410 q^{74} +6.76393 q^{75} +0.763932 q^{76} -11.2361 q^{77} +8.00000 q^{78} -4.00000 q^{79} +3.23607 q^{80} -2.41641 q^{81} -11.7082 q^{82} +4.47214 q^{83} +2.47214 q^{84} -16.9443 q^{85} +11.2361 q^{86} +7.88854 q^{87} +16.8541 q^{88} +10.0000 q^{89} -4.76393 q^{90} +12.9443 q^{91} +2.76393 q^{92} +8.18034 q^{93} -13.7082 q^{94} +2.47214 q^{95} -6.18034 q^{96} +3.38197 q^{97} +3.00000 q^{98} -8.27051 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{3} - 2 q^{4} - 2 q^{5} + 2 q^{6} - 4 q^{7} + 6 q^{8} + 6 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 - 2 * q^3 - 2 * q^4 - 2 * q^5 + 2 * q^6 - 4 * q^7 + 6 * q^8 + 6 * q^9	\( 2 q - 2 q^{2} - 2 q^{3} - 2 q^{4} - 2 q^{5} + 2 q^{6} - 4 q^{7} + 6 q^{8} + 6 q^{9} + 2 q^{10} + 9 q^{11} + 2 q^{12} - 4 q^{13} + 4 q^{14} - 8 q^{15} - 2 q^{16} + 6 q^{17} - 6 q^{18} - 6 q^{19} + 2 q^{20} + 4 q^{21} - 9 q^{22} - 10 q^{23} - 6 q^{24} + 2 q^{25} + 4 q^{26} - 20 q^{27} + 4 q^{28} + 15 q^{29} + 8 q^{30} + 11 q^{31} - 10 q^{32} - 4 q^{33} - 6 q^{34} + 4 q^{35} - 6 q^{36} - 3 q^{37} + 6 q^{38} - 16 q^{39} - 6 q^{40} + 10 q^{41} - 4 q^{42} - 18 q^{43} - 9 q^{44} + 14 q^{45} + 10 q^{46} + 14 q^{47} + 2 q^{48} - 6 q^{49} - 2 q^{50} + 4 q^{51} + 4 q^{52} + 18 q^{53} + 20 q^{54} - 14 q^{55} - 12 q^{56} + 16 q^{57} - 15 q^{58} + 8 q^{59} + 8 q^{60} - 9 q^{61} - 11 q^{62} - 12 q^{63} + 14 q^{64} + 24 q^{65} + 4 q^{66} - 11 q^{67} - 6 q^{68} + 20 q^{69} - 4 q^{70} - 8 q^{71} + 18 q^{72} + 3 q^{73} + 3 q^{74} + 18 q^{75} + 6 q^{76} - 18 q^{77} + 16 q^{78} - 8 q^{79} + 2 q^{80} + 22 q^{81} - 10 q^{82} - 4 q^{84} - 16 q^{85} + 18 q^{86} - 20 q^{87} + 27 q^{88} + 20 q^{89} - 14 q^{90} + 8 q^{91} + 10 q^{92} - 6 q^{93} - 14 q^{94} - 4 q^{95} + 10 q^{96} + 9 q^{97} + 6 q^{98} + 17 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 - 2 * q^4 - 2 * q^5 + 2 * q^6 - 4 * q^7 + 6 * q^8 + 6 * q^9 + 2 * q^10 + 9 * q^11 + 2 * q^12 - 4 * q^13 + 4 * q^14 - 8 * q^15 - 2 * q^16 + 6 * q^17 - 6 * q^18 - 6 * q^19 + 2 * q^20 + 4 * q^21 - 9 * q^22 - 10 * q^23 - 6 * q^24 + 2 * q^25 + 4 * q^26 - 20 * q^27 + 4 * q^28 + 15 * q^29 + 8 * q^30 + 11 * q^31 - 10 * q^32 - 4 * q^33 - 6 * q^34 + 4 * q^35 - 6 * q^36 - 3 * q^37 + 6 * q^38 - 16 * q^39 - 6 * q^40 + 10 * q^41 - 4 * q^42 - 18 * q^43 - 9 * q^44 + 14 * q^45 + 10 * q^46 + 14 * q^47 + 2 * q^48 - 6 * q^49 - 2 * q^50 + 4 * q^51 + 4 * q^52 + 18 * q^53 + 20 * q^54 - 14 * q^55 - 12 * q^56 + 16 * q^57 - 15 * q^58 + 8 * q^59 + 8 * q^60 - 9 * q^61 - 11 * q^62 - 12 * q^63 + 14 * q^64 + 24 * q^65 + 4 * q^66 - 11 * q^67 - 6 * q^68 + 20 * q^69 - 4 * q^70 - 8 * q^71 + 18 * q^72 + 3 * q^73 + 3 * q^74 + 18 * q^75 + 6 * q^76 - 18 * q^77 + 16 * q^78 - 8 * q^79 + 2 * q^80 + 22 * q^81 - 10 * q^82 - 4 * q^84 - 16 * q^85 + 18 * q^86 - 20 * q^87 + 27 * q^88 + 20 * q^89 - 14 * q^90 + 8 * q^91 + 10 * q^92 - 6 * q^93 - 14 * q^94 - 4 * q^95 + 10 * q^96 + 9 * q^97 + 6 * q^98 + 17 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	1.23607	0.713644	0.356822	−	0.934172i	$-0.383860\pi$
$3$	1.23607	0.713644	0.356822	+	0.934172i	$0.383860\pi$
$4$	−1.00000	−0.500000
$5$	−3.23607	−1.44721	−0.723607	−	0.690212i	$-0.757517\pi$
$5$	−3.23607	−1.44721	−0.723607	+	0.690212i	$0.757517\pi$
$6$	−1.23607	−0.504623
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	3.00000	1.06066
$9$	−1.47214	−0.490712
$10$	3.23607	1.02333
$11$	5.61803	1.69390	0.846950	−	0.531672i	$-0.178436\pi$
$11$	5.61803	1.69390	0.846950	+	0.531672i	$0.178436\pi$
$12$	−1.23607	−0.356822
$13$	−6.47214	−1.79505	−0.897524	−	0.440966i	$-0.854636\pi$
$13$	−6.47214	−1.79505	−0.897524	+	0.440966i	$0.854636\pi$
$14$	2.00000	0.534522
$15$	−4.00000	−1.03280
$16$	−1.00000	−0.250000
$17$	5.23607	1.26993	0.634967	−	0.772540i	$-0.281014\pi$
$17$	5.23607	1.26993	0.634967	+	0.772540i	$0.281014\pi$
$18$	1.47214	0.346986
$19$	−0.763932	−0.175258	−0.0876290	−	0.996153i	$-0.527929\pi$
$19$	−0.763932	−0.175258	−0.0876290	+	0.996153i	$0.527929\pi$
$20$	3.23607	0.723607
$21$	−2.47214	−0.539464
$22$	−5.61803	−1.19777
$23$	−2.76393	−0.576320	−0.288160	−	0.957582i	$-0.593043\pi$
$23$	−2.76393	−0.576320	−0.288160	+	0.957582i	$0.593043\pi$
$24$	3.70820	0.756934
$25$	5.47214	1.09443
$26$	6.47214	1.26929
$27$	−5.52786	−1.06384
$28$	2.00000	0.377964
$29$	6.38197	1.18510	0.592551	−	0.805533i	$-0.298121\pi$
$29$	6.38197	1.18510	0.592551	+	0.805533i	$0.298121\pi$
$30$	4.00000	0.730297
$31$	6.61803	1.18863	0.594317	−	0.804231i	$-0.297422\pi$
$31$	6.61803	1.18863	0.594317	+	0.804231i	$0.297422\pi$
$32$	−5.00000	−0.883883
$33$	6.94427	1.20884
$34$	−5.23607	−0.897978
$35$	6.47214	1.09399
$36$	1.47214	0.245356
$37$	1.85410	0.304812	0.152406	−	0.988318i	$-0.451298\pi$
$37$	1.85410	0.304812	0.152406	+	0.988318i	$0.451298\pi$
$38$	0.763932	0.123926
$39$	−8.00000	−1.28103
$40$	−9.70820	−1.53500
$41$	11.7082	1.82851	0.914257	−	0.405134i	$-0.132775\pi$
$41$	11.7082	1.82851	0.914257	+	0.405134i	$0.132775\pi$
$42$	2.47214	0.381459
$43$	−11.2361	−1.71348	−0.856742	−	0.515745i	$-0.827515\pi$
$43$	−11.2361	−1.71348	−0.856742	+	0.515745i	$0.827515\pi$
$44$	−5.61803	−0.846950
$45$	4.76393	0.710165
$46$	2.76393	0.407520
$47$	13.7082	1.99955	0.999774	−	0.0212814i	$-0.00677460\pi$
$47$	13.7082	1.99955	0.999774	+	0.0212814i	$0.00677460\pi$
$48$	−1.23607	−0.178411
$49$	−3.00000	−0.428571
$50$	−5.47214	−0.773877
$51$	6.47214	0.906280
$52$	6.47214	0.897524
$53$	6.76393	0.929098	0.464549	−	0.885548i	$-0.346217\pi$
$53$	6.76393	0.929098	0.464549	+	0.885548i	$0.346217\pi$
$54$	5.52786	0.752247
$55$	−18.1803	−2.45144
$56$	−6.00000	−0.801784
$57$	−0.944272	−0.125072
$58$	−6.38197	−0.837993
$59$	−4.94427	−0.643689	−0.321845	−	0.946792i	$-0.604303\pi$
$59$	−4.94427	−0.643689	−0.321845	+	0.946792i	$0.604303\pi$
$60$	4.00000	0.516398
$61$	−12.3262	−1.57821	−0.789107	−	0.614256i	$-0.789456\pi$
$61$	−12.3262	−1.57821	−0.789107	+	0.614256i	$0.789456\pi$
$62$	−6.61803	−0.840491
$63$	2.94427	0.370943
$64$	7.00000	0.875000
$65$	20.9443	2.59782
$66$	−6.94427	−0.854781
$67$	2.32624	0.284195	0.142098	−	0.989853i	$-0.454615\pi$
$67$	2.32624	0.284195	0.142098	+	0.989853i	$0.454615\pi$
$68$	−5.23607	−0.634967
$69$	−3.41641	−0.411287
$70$	−6.47214	−0.773568
$71$	−12.9443	−1.53620	−0.768101	−	0.640328i	$-0.778798\pi$
$71$	−12.9443	−1.53620	−0.768101	+	0.640328i	$0.778798\pi$
$72$	−4.41641	−0.520479
$73$	4.85410	0.568130	0.284065	−	0.958805i	$-0.408317\pi$
$73$	4.85410	0.568130	0.284065	+	0.958805i	$0.408317\pi$
$74$	−1.85410	−0.215535
$75$	6.76393	0.781032
$76$	0.763932	0.0876290
$77$	−11.2361	−1.28047
$78$	8.00000	0.905822
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	3.23607	0.361803
$81$	−2.41641	−0.268490
$82$	−11.7082	−1.29295
$83$	4.47214	0.490881	0.245440	−	0.969412i	$-0.421067\pi$
$83$	4.47214	0.490881	0.245440	+	0.969412i	$0.421067\pi$
$84$	2.47214	0.269732
$85$	−16.9443	−1.83786
$86$	11.2361	1.21162
$87$	7.88854	0.845741
$88$	16.8541	1.79665
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	−4.76393	−0.502163
$91$	12.9443	1.35693
$92$	2.76393	0.288160
$93$	8.18034	0.848262
$94$	−13.7082	−1.41389
$95$	2.47214	0.253636
$96$	−6.18034	−0.630778
$97$	3.38197	0.343387	0.171693	−	0.985150i	$-0.445076\pi$
$97$	3.38197	0.343387	0.171693	+	0.985150i	$0.445076\pi$
$98$	3.00000	0.303046
$99$	−8.27051	−0.831218
$100$	−5.47214	−0.547214
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	−6.47214	−0.640837
$103$	−18.6525	−1.83788	−0.918942	−	0.394394i	$-0.870955\pi$
$103$	−18.6525	−1.83788	−0.918942	+	0.394394i	$0.870955\pi$
$104$	−19.4164	−1.90394
$105$	8.00000	0.780720
$106$	−6.76393	−0.656971
$107$	6.00000	0.580042	0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000	0.580042	0.290021	+	0.957020i	$0.406338\pi$
$108$	5.52786	0.531919
$109$	10.9443	1.04827	0.524136	−	0.851635i	$-0.324389\pi$
$109$	10.9443	1.04827	0.524136	+	0.851635i	$0.324389\pi$
$110$	18.1803	1.73343
$111$	2.29180	0.217528
$112$	2.00000	0.188982
$113$	−16.0902	−1.51364	−0.756818	−	0.653626i	$-0.773247\pi$
$113$	−16.0902	−1.51364	−0.756818	+	0.653626i	$0.773247\pi$
$114$	0.944272	0.0884392
$115$	8.94427	0.834058
$116$	−6.38197	−0.592551
$117$	9.52786	0.880851
$118$	4.94427	0.455157
$119$	−10.4721	−0.959979
$120$	−12.0000	−1.09545
$121$	20.5623	1.86930
$122$	12.3262	1.11597
$123$	14.4721	1.30491
$124$	−6.61803	−0.594317
$125$	−1.52786	−0.136656
$126$	−2.94427	−0.262297
$127$	−18.3262	−1.62619	−0.813095	−	0.582131i	$-0.802219\pi$
$127$	−18.3262	−1.62619	−0.813095	+	0.582131i	$0.802219\pi$
$128$	3.00000	0.265165
$129$	−13.8885	−1.22282
$130$	−20.9443	−1.83693
$131$	20.7984	1.81716	0.908581	−	0.417708i	$-0.137167\pi$
$131$	20.7984	1.81716	0.908581	+	0.417708i	$0.137167\pi$
$132$	−6.94427	−0.604421
$133$	1.52786	0.132483
$134$	−2.32624	−0.200956
$135$	17.8885	1.53960
$136$	15.7082	1.34697
$137$	−4.18034	−0.357151	−0.178575	−	0.983926i	$-0.557149\pi$
$137$	−4.18034	−0.357151	−0.178575	+	0.983926i	$0.557149\pi$
$138$	3.41641	0.290824
$139$	−3.52786	−0.299230	−0.149615	−	0.988744i	$-0.547803\pi$
$139$	−3.52786	−0.299230	−0.149615	+	0.988744i	$0.547803\pi$
$140$	−6.47214	−0.546995
$141$	16.9443	1.42697
$142$	12.9443	1.08626
$143$	−36.3607	−3.04063
$144$	1.47214	0.122678
$145$	−20.6525	−1.71509
$146$	−4.85410	−0.401728
$147$	−3.70820	−0.305848
$148$	−1.85410	−0.152406
$149$	−5.61803	−0.460247	−0.230124	−	0.973161i	$-0.573913\pi$
$149$	−5.61803	−0.460247	−0.230124	+	0.973161i	$0.573913\pi$
$150$	−6.76393	−0.552273
$151$	11.2361	0.914378	0.457189	−	0.889369i	$-0.348856\pi$
$151$	11.2361	0.914378	0.457189	+	0.889369i	$0.348856\pi$
$152$	−2.29180	−0.185889
$153$	−7.70820	−0.623171
$154$	11.2361	0.905428
$155$	−21.4164	−1.72021
$156$	8.00000	0.640513
$157$	−15.3262	−1.22317	−0.611583	−	0.791180i	$-0.709467\pi$
$157$	−15.3262	−1.22317	−0.611583	+	0.791180i	$0.709467\pi$
$158$	4.00000	0.318223
$159$	8.36068	0.663045
$160$	16.1803	1.27917
$161$	5.52786	0.435657
$162$	2.41641	0.189851
$163$	−9.67376	−0.757708	−0.378854	−	0.925457i	$-0.623682\pi$
$163$	−9.67376	−0.757708	−0.378854	+	0.925457i	$0.623682\pi$
$164$	−11.7082	−0.914257
$165$	−22.4721	−1.74945
$166$	−4.47214	−0.347105
$167$	−15.5623	−1.20425	−0.602124	−	0.798403i	$-0.705679\pi$
$167$	−15.5623	−1.20425	−0.602124	+	0.798403i	$0.705679\pi$
$168$	−7.41641	−0.572188
$169$	28.8885	2.22220
$170$	16.9443	1.29957
$171$	1.12461	0.0860012
$172$	11.2361	0.856742
$173$	−4.61803	−0.351103	−0.175551	−	0.984470i	$-0.556171\pi$
$173$	−4.61803	−0.351103	−0.175551	+	0.984470i	$0.556171\pi$
$174$	−7.88854	−0.598029
$175$	−10.9443	−0.827309
$176$	−5.61803	−0.423475
$177$	−6.11146	−0.459365
$178$	−10.0000	−0.749532
$179$	8.00000	0.597948	0.298974	−	0.954261i	$-0.403356\pi$
$179$	8.00000	0.597948	0.298974	+	0.954261i	$0.403356\pi$
$180$	−4.76393	−0.355083
$181$	−1.05573	−0.0784717	−0.0392358	−	0.999230i	$-0.512492\pi$
$181$	−1.05573	−0.0784717	−0.0392358	+	0.999230i	$0.512492\pi$
$182$	−12.9443	−0.959493
$183$	−15.2361	−1.12628
$184$	−8.29180	−0.611279
$185$	−6.00000	−0.441129
$186$	−8.18034	−0.599812
$187$	29.4164	2.15114
$188$	−13.7082	−0.999774
$189$	11.0557	0.804186
$190$	−2.47214	−0.179348
$191$	1.43769	0.104028	0.0520139	−	0.998646i	$-0.483436\pi$
$191$	1.43769	0.104028	0.0520139	+	0.998646i	$0.483436\pi$
$192$	8.65248	0.624439
$193$	−12.2705	−0.883251	−0.441625	−	0.897200i	$-0.645598\pi$
$193$	−12.2705	−0.883251	−0.441625	+	0.897200i	$0.645598\pi$
$194$	−3.38197	−0.242811
$195$	25.8885	1.85392
$196$	3.00000	0.214286
$197$	9.85410	0.702076	0.351038	−	0.936361i	$-0.385829\pi$
$197$	9.85410	0.702076	0.351038	+	0.936361i	$0.385829\pi$
$198$	8.27051	0.587760
$199$	−15.3262	−1.08645	−0.543224	−	0.839588i	$-0.682797\pi$
$199$	−15.3262	−1.08645	−0.543224	+	0.839588i	$0.682797\pi$
$200$	16.4164	1.16082
$201$	2.87539	0.202814
$202$	10.0000	0.703598
$203$	−12.7639	−0.895852
$204$	−6.47214	−0.453140
$205$	−37.8885	−2.64625
$206$	18.6525	1.29958
$207$	4.06888	0.282807
$208$	6.47214	0.448762
$209$	−4.29180	−0.296870
$210$	−8.00000	−0.552052
$211$	−26.6525	−1.83483	−0.917416	−	0.397929i	$-0.869729\pi$
$211$	−26.6525	−1.83483	−0.917416	+	0.397929i	$0.869729\pi$
$212$	−6.76393	−0.464549
$213$	−16.0000	−1.09630
$214$	−6.00000	−0.410152
$215$	36.3607	2.47978
$216$	−16.5836	−1.12837
$217$	−13.2361	−0.898523
$218$	−10.9443	−0.741240
$219$	6.00000	0.405442
$220$	18.1803	1.22572
$221$	−33.8885	−2.27959
$222$	−2.29180	−0.153815
$223$	−13.2361	−0.886353	−0.443176	−	0.896434i	$-0.646148\pi$
$223$	−13.2361	−0.886353	−0.443176	+	0.896434i	$0.646148\pi$
$224$	10.0000	0.668153
$225$	−8.05573	−0.537049
$226$	16.0902	1.07030
$227$	−12.2705	−0.814422	−0.407211	−	0.913334i	$-0.633499\pi$
$227$	−12.2705	−0.814422	−0.407211	+	0.913334i	$0.633499\pi$
$228$	0.944272	0.0625359
$229$	−11.0557	−0.730583	−0.365292	−	0.930893i	$-0.619031\pi$
$229$	−11.0557	−0.730583	−0.365292	+	0.930893i	$0.619031\pi$
$230$	−8.94427	−0.589768
$231$	−13.8885	−0.913799
$232$	19.1459	1.25699
$233$	−24.9443	−1.63415	−0.817077	−	0.576529i	$-0.804407\pi$
$233$	−24.9443	−1.63415	−0.817077	+	0.576529i	$0.804407\pi$
$234$	−9.52786	−0.622856
$235$	−44.3607	−2.89377
$236$	4.94427	0.321845
$237$	−4.94427	−0.321165
$238$	10.4721	0.678808
$239$	16.9443	1.09603	0.548017	−	0.836467i	$-0.315383\pi$
$239$	16.9443	1.09603	0.548017	+	0.836467i	$0.315383\pi$
$240$	4.00000	0.258199
$241$	1.05573	0.0680054	0.0340027	−	0.999422i	$-0.489175\pi$
$241$	1.05573	0.0680054	0.0340027	+	0.999422i	$0.489175\pi$
$242$	−20.5623	−1.32180
$243$	13.5967	0.872232
$244$	12.3262	0.789107
$245$	9.70820	0.620234
$246$	−14.4721	−0.922710
$247$	4.94427	0.314596
$248$	19.8541	1.26074
$249$	5.52786	0.350314
$250$	1.52786	0.0966306
$251$	8.09017	0.510647	0.255323	−	0.966856i	$-0.417818\pi$
$251$	8.09017	0.510647	0.255323	+	0.966856i	$0.417818\pi$
$252$	−2.94427	−0.185472
$253$	−15.5279	−0.976228
$254$	18.3262	1.14989
$255$	−20.9443	−1.31158
$256$	−17.0000	−1.06250
$257$	−12.7639	−0.796192	−0.398096	−	0.917344i	$-0.630329\pi$
$257$	−12.7639	−0.796192	−0.398096	+	0.917344i	$0.630329\pi$
$258$	13.8885	0.864663
$259$	−3.70820	−0.230417
$260$	−20.9443	−1.29891
$261$	−9.39512	−0.581543
$262$	−20.7984	−1.28493
$263$	−17.5279	−1.08081	−0.540407	−	0.841404i	$-0.681730\pi$
$263$	−17.5279	−1.08081	−0.540407	+	0.841404i	$0.681730\pi$
$264$	20.8328	1.28217
$265$	−21.8885	−1.34460
$266$	−1.52786	−0.0936794
$267$	12.3607	0.756461
$268$	−2.32624	−0.142098
$269$	2.20163	0.134236	0.0671178	−	0.997745i	$-0.478620\pi$
$269$	2.20163	0.134236	0.0671178	+	0.997745i	$0.478620\pi$
$270$	−17.8885	−1.08866
$271$	−10.4721	−0.636137	−0.318068	−	0.948068i	$-0.603034\pi$
$271$	−10.4721	−0.636137	−0.318068	+	0.948068i	$0.603034\pi$
$272$	−5.23607	−0.317483
$273$	16.0000	0.968364
$274$	4.18034	0.252544
$275$	30.7426	1.85385
$276$	3.41641	0.205644
$277$	10.1803	0.611677	0.305839	−	0.952083i	$-0.401063\pi$
$277$	10.1803	0.611677	0.305839	+	0.952083i	$0.401063\pi$
$278$	3.52786	0.211587
$279$	−9.74265	−0.583277
$280$	19.4164	1.16035
$281$	−12.0344	−0.717915	−0.358957	−	0.933354i	$-0.616868\pi$
$281$	−12.0344	−0.717915	−0.358957	+	0.933354i	$0.616868\pi$
$282$	−16.9443	−1.00902
$283$	1.52786	0.0908221	0.0454110	−	0.998968i	$-0.485540\pi$
$283$	1.52786	0.0908221	0.0454110	+	0.998968i	$0.485540\pi$
$284$	12.9443	0.768101
$285$	3.05573	0.181006
$286$	36.3607	2.15005
$287$	−23.4164	−1.38223
$288$	7.36068	0.433732
$289$	10.4164	0.612730
$290$	20.6525	1.21276
$291$	4.18034	0.245056
$292$	−4.85410	−0.284065
$293$	16.7639	0.979359	0.489680	−	0.871902i	$-0.337114\pi$
$293$	16.7639	0.979359	0.489680	+	0.871902i	$0.337114\pi$
$294$	3.70820	0.216267
$295$	16.0000	0.931556
$296$	5.56231	0.323302
$297$	−31.0557	−1.80204
$298$	5.61803	0.325444
$299$	17.8885	1.03452
$300$	−6.76393	−0.390516
$301$	22.4721	1.29527
$302$	−11.2361	−0.646563
$303$	−12.3607	−0.710102
$304$	0.763932	0.0438145
$305$	39.8885	2.28401
$306$	7.70820	0.440649
$307$	−11.1459	−0.636130	−0.318065	−	0.948069i	$-0.603033\pi$
$307$	−11.1459	−0.636130	−0.318065	+	0.948069i	$0.603033\pi$
$308$	11.2361	0.640234
$309$	−23.0557	−1.31159
$310$	21.4164	1.21637
$311$	−31.9230	−1.81019	−0.905093	−	0.425213i	$-0.860199\pi$
$311$	−31.9230	−1.81019	−0.905093	+	0.425213i	$0.860199\pi$
$312$	−24.0000	−1.35873
$313$	11.4164	0.645294	0.322647	−	0.946519i	$-0.395427\pi$
$313$	11.4164	0.645294	0.322647	+	0.946519i	$0.395427\pi$
$314$	15.3262	0.864910
$315$	−9.52786	−0.536834
$316$	4.00000	0.225018
$317$	6.29180	0.353382	0.176691	−	0.984266i	$-0.443461\pi$
$317$	6.29180	0.353382	0.176691	+	0.984266i	$0.443461\pi$
$318$	−8.36068	−0.468844
$319$	35.8541	2.00744
$320$	−22.6525	−1.26631
$321$	7.41641	0.413944
$322$	−5.52786	−0.308056
$323$	−4.00000	−0.222566
$324$	2.41641	0.134245
$325$	−35.4164	−1.96455
$326$	9.67376	0.535780
$327$	13.5279	0.748093
$328$	35.1246	1.93943
$329$	−27.4164	−1.51152
$330$	22.4721	1.23705
$331$	30.4721	1.67490	0.837450	−	0.546514i	$-0.184045\pi$
$331$	30.4721	1.67490	0.837450	+	0.546514i	$0.184045\pi$
$332$	−4.47214	−0.245440
$333$	−2.72949	−0.149575
$334$	15.5623	0.851531
$335$	−7.52786	−0.411291
$336$	2.47214	0.134866
$337$	−15.7082	−0.855680	−0.427840	−	0.903854i	$-0.640725\pi$
$337$	−15.7082	−0.855680	−0.427840	+	0.903854i	$0.640725\pi$
$338$	−28.8885	−1.57133
$339$	−19.8885	−1.08020
$340$	16.9443	0.918932
$341$	37.1803	2.01343
$342$	−1.12461	−0.0608120
$343$	20.0000	1.07990
$344$	−33.7082	−1.81742
$345$	11.0557	0.595220
$346$	4.61803	0.248267
$347$	28.8541	1.54897	0.774485	−	0.632593i	$-0.218009\pi$
$347$	28.8541	1.54897	0.774485	+	0.632593i	$0.218009\pi$
$348$	−7.88854	−0.422870
$349$	−19.2361	−1.02968	−0.514842	−	0.857285i	$-0.672149\pi$
$349$	−19.2361	−1.02968	−0.514842	+	0.857285i	$0.672149\pi$
$350$	10.9443	0.584996
$351$	35.7771	1.90964
$352$	−28.0902	−1.49721
$353$	0.854102	0.0454593	0.0227296	−	0.999742i	$-0.492764\pi$
$353$	0.854102	0.0454593	0.0227296	+	0.999742i	$0.492764\pi$
$354$	6.11146	0.324820
$355$	41.8885	2.22321
$356$	−10.0000	−0.529999
$357$	−12.9443	−0.685084
$358$	−8.00000	−0.422813
$359$	−15.2361	−0.804129	−0.402064	−	0.915611i	$-0.631707\pi$
$359$	−15.2361	−0.804129	−0.402064	+	0.915611i	$0.631707\pi$
$360$	14.2918	0.753244
$361$	−18.4164	−0.969285
$362$	1.05573	0.0554878
$363$	25.4164	1.33402
$364$	−12.9443	−0.678464
$365$	−15.7082	−0.822205
$366$	15.2361	0.796402
$367$	9.88854	0.516178	0.258089	−	0.966121i	$-0.416907\pi$
$367$	9.88854	0.516178	0.258089	+	0.966121i	$0.416907\pi$
$368$	2.76393	0.144080
$369$	−17.2361	−0.897274
$370$	6.00000	0.311925
$371$	−13.5279	−0.702332
$372$	−8.18034	−0.424131
$373$	7.23607	0.374669	0.187335	−	0.982296i	$-0.440015\pi$
$373$	7.23607	0.374669	0.187335	+	0.982296i	$0.440015\pi$
$374$	−29.4164	−1.52109
$375$	−1.88854	−0.0975240
$376$	41.1246	2.12084
$377$	−41.3050	−2.12731
$378$	−11.0557	−0.568645
$379$	8.27051	0.424828	0.212414	−	0.977180i	$-0.431868\pi$
$379$	8.27051	0.424828	0.212414	+	0.977180i	$0.431868\pi$
$380$	−2.47214	−0.126818
$381$	−22.6525	−1.16052
$382$	−1.43769	−0.0735588
$383$	−22.9443	−1.17240	−0.586199	−	0.810167i	$-0.699376\pi$
$383$	−22.9443	−1.17240	−0.586199	+	0.810167i	$0.699376\pi$
$384$	3.70820	0.189233
$385$	36.3607	1.85311
$386$	12.2705	0.624553
$387$	16.5410	0.840827
$388$	−3.38197	−0.171693
$389$	−3.70820	−0.188013	−0.0940067	−	0.995572i	$-0.529968\pi$
$389$	−3.70820	−0.188013	−0.0940067	+	0.995572i	$0.529968\pi$
$390$	−25.8885	−1.31092
$391$	−14.4721	−0.731887
$392$	−9.00000	−0.454569
$393$	25.7082	1.29681
$394$	−9.85410	−0.496442
$395$	12.9443	0.651297
$396$	8.27051	0.415609
$397$	−13.5279	−0.678944	−0.339472	−	0.940616i	$-0.610248\pi$
$397$	−13.5279	−0.678944	−0.339472	+	0.940616i	$0.610248\pi$
$398$	15.3262	0.768235
$399$	1.88854	0.0945454
$400$	−5.47214	−0.273607
$401$	−28.0000	−1.39825	−0.699127	−	0.714998i	$-0.746428\pi$
$401$	−28.0000	−1.39825	−0.699127	+	0.714998i	$0.746428\pi$
$402$	−2.87539	−0.143411
$403$	−42.8328	−2.13365
$404$	10.0000	0.497519
$405$	7.81966	0.388562
$406$	12.7639	0.633463
$407$	10.4164	0.516322
$408$	19.4164	0.961255
$409$	30.8328	1.52458	0.762292	−	0.647233i	$-0.224074\pi$
$409$	30.8328	1.52458	0.762292	+	0.647233i	$0.224074\pi$
$410$	37.8885	1.87118
$411$	−5.16718	−0.254878
$412$	18.6525	0.918942
$413$	9.88854	0.486583
$414$	−4.06888	−0.199975
$415$	−14.4721	−0.710409
$416$	32.3607	1.58661
$417$	−4.36068	−0.213543
$418$	4.29180	0.209919
$419$	30.9787	1.51341	0.756705	−	0.653757i	$-0.226808\pi$
$419$	30.9787	1.51341	0.756705	+	0.653757i	$0.226808\pi$
$420$	−8.00000	−0.390360
$421$	0.763932	0.0372318	0.0186159	−	0.999827i	$-0.494074\pi$
$421$	0.763932	0.0372318	0.0186159	+	0.999827i	$0.494074\pi$
$422$	26.6525	1.29742
$423$	−20.1803	−0.981202
$424$	20.2918	0.985457
$425$	28.6525	1.38985
$426$	16.0000	0.775203
$427$	24.6525	1.19302
$428$	−6.00000	−0.290021
$429$	−44.9443	−2.16993
$430$	−36.3607	−1.75347
$431$	22.7426	1.09547	0.547737	−	0.836650i	$-0.315489\pi$
$431$	22.7426	1.09547	0.547737	+	0.836650i	$0.315489\pi$
$432$	5.52786	0.265959
$433$	−4.90983	−0.235951	−0.117976	−	0.993016i	$-0.537640\pi$
$433$	−4.90983	−0.235951	−0.117976	+	0.993016i	$0.537640\pi$
$434$	13.2361	0.635352
$435$	−25.5279	−1.22397
$436$	−10.9443	−0.524136
$437$	2.11146	0.101005
$438$	−6.00000	−0.286691
$439$	9.79837	0.467651	0.233825	−	0.972279i	$-0.424876\pi$
$439$	9.79837	0.467651	0.233825	+	0.972279i	$0.424876\pi$
$440$	−54.5410	−2.60014
$441$	4.41641	0.210305
$442$	33.8885	1.61191
$443$	2.00000	0.0950229	0.0475114	−	0.998871i	$-0.484871\pi$
$443$	2.00000	0.0950229	0.0475114	+	0.998871i	$0.484871\pi$
$444$	−2.29180	−0.108764
$445$	−32.3607	−1.53404
$446$	13.2361	0.626746
$447$	−6.94427	−0.328453
$448$	−14.0000	−0.661438
$449$	0.0344419	0.00162541	0.000812706	−	1.00000i	$-0.499741\pi$
$449$	0.0344419	0.00162541	0.000812706		1.00000i	$0.499741\pi$
$450$	8.05573	0.379751
$451$	65.7771	3.09732
$452$	16.0902	0.756818
$453$	13.8885	0.652541
$454$	12.2705	0.575884
$455$	−41.8885	−1.96377
$456$	−2.83282	−0.132659
$457$	−25.1246	−1.17528	−0.587640	−	0.809123i	$-0.699943\pi$
$457$	−25.1246	−1.17528	−0.587640	+	0.809123i	$0.699943\pi$
$458$	11.0557	0.516600
$459$	−28.9443	−1.35100
$460$	−8.94427	−0.417029
$461$	23.3262	1.08641	0.543206	−	0.839600i	$-0.317210\pi$
$461$	23.3262	1.08641	0.543206	+	0.839600i	$0.317210\pi$
$462$	13.8885	0.646154
$463$	28.7639	1.33677	0.668387	−	0.743814i	$-0.266985\pi$
$463$	28.7639	1.33677	0.668387	+	0.743814i	$0.266985\pi$
$464$	−6.38197	−0.296275
$465$	−26.4721	−1.22762
$466$	24.9443	1.15552
$467$	9.41641	0.435739	0.217870	−	0.975978i	$-0.430089\pi$
$467$	9.41641	0.435739	0.217870	+	0.975978i	$0.430089\pi$
$468$	−9.52786	−0.440426
$469$	−4.65248	−0.214831
$470$	44.3607	2.04621
$471$	−18.9443	−0.872906
$472$	−14.8328	−0.682736
$473$	−63.1246	−2.90247
$474$	4.94427	0.227098
$475$	−4.18034	−0.191807
$476$	10.4721	0.479990
$477$	−9.95743	−0.455919
$478$	−16.9443	−0.775013
$479$	−22.4721	−1.02678	−0.513389	−	0.858156i	$-0.671610\pi$
$479$	−22.4721	−1.02678	−0.513389	+	0.858156i	$0.671610\pi$
$480$	20.0000	0.912871
$481$	−12.0000	−0.547153
$482$	−1.05573	−0.0480871
$483$	6.83282	0.310904
$484$	−20.5623	−0.934650
$485$	−10.9443	−0.496954
$486$	−13.5967	−0.616761
$487$	−6.20163	−0.281023	−0.140511	−	0.990079i	$-0.544875\pi$
$487$	−6.20163	−0.281023	−0.140511	+	0.990079i	$0.544875\pi$
$488$	−36.9787	−1.67395
$489$	−11.9574	−0.540734
$490$	−9.70820	−0.438572
$491$	−5.52786	−0.249469	−0.124735	−	0.992190i	$-0.539808\pi$
$491$	−5.52786	−0.249469	−0.124735	+	0.992190i	$0.539808\pi$
$492$	−14.4721	−0.652454
$493$	33.4164	1.50500
$494$	−4.94427	−0.222453
$495$	26.7639	1.20295
$496$	−6.61803	−0.297158
$497$	25.8885	1.16126
$498$	−5.52786	−0.247710
$499$	4.94427	0.221336	0.110668	−	0.993857i	$-0.464701\pi$
$499$	4.94427	0.221336	0.110668	+	0.993857i	$0.464701\pi$
$500$	1.52786	0.0683282
$501$	−19.2361	−0.859404
$502$	−8.09017	−0.361082
$503$	0.111456	0.00496959	0.00248479	−	0.999997i	$-0.499209\pi$
$503$	0.111456	0.00496959	0.00248479	+	0.999997i	$0.499209\pi$
$504$	8.83282	0.393445
$505$	32.3607	1.44003
$506$	15.5279	0.690298
$507$	35.7082	1.58586
$508$	18.3262	0.813095
$509$	−27.0902	−1.20075	−0.600375	−	0.799718i	$-0.704982\pi$
$509$	−27.0902	−1.20075	−0.600375	+	0.799718i	$0.704982\pi$
$510$	20.9443	0.927428
$511$	−9.70820	−0.429466
$512$	11.0000	0.486136
$513$	4.22291	0.186446
$514$	12.7639	0.562993
$515$	60.3607	2.65981
$516$	13.8885	0.611409
$517$	77.0132	3.38703
$518$	3.70820	0.162929
$519$	−5.70820	−0.250562
$520$	62.8328	2.75540
$521$	21.4164	0.938270	0.469135	−	0.883127i	$-0.344566\pi$
$521$	21.4164	0.938270	0.469135	+	0.883127i	$0.344566\pi$
$522$	9.39512	0.411213
$523$	−19.3262	−0.845077	−0.422539	−	0.906345i	$-0.638861\pi$
$523$	−19.3262	−0.845077	−0.422539	+	0.906345i	$0.638861\pi$
$524$	−20.7984	−0.908581
$525$	−13.5279	−0.590404
$526$	17.5279	0.764251
$527$	34.6525	1.50949
$528$	−6.94427	−0.302211
$529$	−15.3607	−0.667856
$530$	21.8885	0.950778
$531$	7.27864	0.315866
$532$	−1.52786	−0.0662413
$533$	−75.7771	−3.28227
$534$	−12.3607	−0.534899
$535$	−19.4164	−0.839445
$536$	6.97871	0.301435
$537$	9.88854	0.426722
$538$	−2.20163	−0.0949188
$539$	−16.8541	−0.725958
$540$	−17.8885	−0.769800
$541$	−36.6312	−1.57490	−0.787449	−	0.616380i	$-0.788599\pi$
$541$	−36.6312	−1.57490	−0.787449	+	0.616380i	$0.788599\pi$
$542$	10.4721	0.449817
$543$	−1.30495	−0.0560008
$544$	−26.1803	−1.12247
$545$	−35.4164	−1.51707
$546$	−16.0000	−0.684737
$547$	20.6525	0.883036	0.441518	−	0.897252i	$-0.354440\pi$
$547$	20.6525	0.883036	0.441518	+	0.897252i	$0.354440\pi$
$548$	4.18034	0.178575
$549$	18.1459	0.774448
$550$	−30.7426	−1.31087
$551$	−4.87539	−0.207699
$552$	−10.2492	−0.436236
$553$	8.00000	0.340195
$554$	−10.1803	−0.432521
$555$	−7.41641	−0.314809
$556$	3.52786	0.149615
$557$	17.8885	0.757962	0.378981	−	0.925404i	$-0.376275\pi$
$557$	17.8885	0.757962	0.378981	+	0.925404i	$0.376275\pi$
$558$	9.74265	0.412439
$559$	72.7214	3.07579
$560$	−6.47214	−0.273498
$561$	36.3607	1.53515
$562$	12.0344	0.507642
$563$	−8.43769	−0.355606	−0.177803	−	0.984066i	$-0.556899\pi$
$563$	−8.43769	−0.355606	−0.177803	+	0.984066i	$0.556899\pi$
$564$	−16.9443	−0.713483
$565$	52.0689	2.19055
$566$	−1.52786	−0.0642209
$567$	4.83282	0.202959
$568$	−38.8328	−1.62939
$569$	6.50658	0.272770	0.136385	−	0.990656i	$-0.456452\pi$
$569$	6.50658	0.272770	0.136385	+	0.990656i	$0.456452\pi$
$570$	−3.05573	−0.127990
$571$	−11.5623	−0.483867	−0.241934	−	0.970293i	$-0.577782\pi$
$571$	−11.5623	−0.483867	−0.241934	+	0.970293i	$0.577782\pi$
$572$	36.3607	1.52032
$573$	1.77709	0.0742389
$574$	23.4164	0.977382
$575$	−15.1246	−0.630740
$576$	−10.3050	−0.429373
$577$	−14.2016	−0.591221	−0.295611	−	0.955308i	$-0.595523\pi$
$577$	−14.2016	−0.591221	−0.295611	+	0.955308i	$0.595523\pi$
$578$	−10.4164	−0.433265
$579$	−15.1672	−0.630327
$580$	20.6525	0.857547
$581$	−8.94427	−0.371071
$582$	−4.18034	−0.173281
$583$	38.0000	1.57380
$584$	14.5623	0.602593
$585$	−30.8328	−1.27478
$586$	−16.7639	−0.692512
$587$	14.7426	0.608494	0.304247	−	0.952593i	$-0.401595\pi$
$587$	14.7426	0.608494	0.304247	+	0.952593i	$0.401595\pi$
$588$	3.70820	0.152924
$589$	−5.05573	−0.208318
$590$	−16.0000	−0.658710
$591$	12.1803	0.501032
$592$	−1.85410	−0.0762031
$593$	0.618034	0.0253796	0.0126898	−	0.999919i	$-0.495961\pi$
$593$	0.618034	0.0253796	0.0126898	+	0.999919i	$0.495961\pi$
$594$	31.0557	1.27423
$595$	33.8885	1.38929
$596$	5.61803	0.230124
$597$	−18.9443	−0.775337
$598$	−17.8885	−0.731517
$599$	24.4508	0.999035	0.499517	−	0.866304i	$-0.333511\pi$
$599$	24.4508	0.999035	0.499517	+	0.866304i	$0.333511\pi$
$600$	20.2918	0.828409
$601$	22.5066	0.918062	0.459031	−	0.888420i	$-0.348197\pi$
$601$	22.5066	0.918062	0.459031	+	0.888420i	$0.348197\pi$
$602$	−22.4721	−0.915896
$603$	−3.42454	−0.139458
$604$	−11.2361	−0.457189
$605$	−66.5410	−2.70528
$606$	12.3607	0.502118
$607$	26.0689	1.05810	0.529052	−	0.848590i	$-0.322548\pi$
$607$	26.0689	1.05810	0.529052	+	0.848590i	$0.322548\pi$
$608$	3.81966	0.154908
$609$	−15.7771	−0.639320
$610$	−39.8885	−1.61504
$611$	−88.7214	−3.58928
$612$	7.70820	0.311586
$613$	5.20163	0.210092	0.105046	−	0.994467i	$-0.466501\pi$
$613$	5.20163	0.210092	0.105046	+	0.994467i	$0.466501\pi$
$614$	11.1459	0.449812
$615$	−46.8328	−1.88848
$616$	−33.7082	−1.35814
$617$	−43.7426	−1.76101	−0.880506	−	0.474034i	$-0.842797\pi$
$617$	−43.7426	−1.76101	−0.880506	+	0.474034i	$0.842797\pi$
$618$	23.0557	0.927437
$619$	−37.7426	−1.51701	−0.758503	−	0.651670i	$-0.774069\pi$
$619$	−37.7426	−1.51701	−0.758503	+	0.651670i	$0.774069\pi$
$620$	21.4164	0.860104
$621$	15.2786	0.613111
$622$	31.9230	1.28000
$623$	−20.0000	−0.801283
$624$	8.00000	0.320256
$625$	−22.4164	−0.896656
$626$	−11.4164	−0.456291
$627$	−5.30495	−0.211859
$628$	15.3262	0.611583
$629$	9.70820	0.387091
$630$	9.52786	0.379599
$631$	−33.1591	−1.32004	−0.660021	−	0.751248i	$-0.729452\pi$
$631$	−33.1591	−1.32004	−0.660021	+	0.751248i	$0.729452\pi$
$632$	−12.0000	−0.477334
$633$	−32.9443	−1.30942
$634$	−6.29180	−0.249879
$635$	59.3050	2.35345
$636$	−8.36068	−0.331523
$637$	19.4164	0.769306
$638$	−35.8541	−1.41948
$639$	19.0557	0.753833
$640$	−9.70820	−0.383750
$641$	−8.58359	−0.339032	−0.169516	−	0.985527i	$-0.554220\pi$
$641$	−8.58359	−0.339032	−0.169516	+	0.985527i	$0.554220\pi$
$642$	−7.41641	−0.292702
$643$	−28.0689	−1.10693	−0.553464	−	0.832873i	$-0.686694\pi$
$643$	−28.0689	−1.10693	−0.553464	+	0.832873i	$0.686694\pi$
$644$	−5.52786	−0.217828
$645$	44.9443	1.76968
$646$	4.00000	0.157378
$647$	−46.6525	−1.83410	−0.917049	−	0.398774i	$-0.869436\pi$
$647$	−46.6525	−1.83410	−0.917049	+	0.398774i	$0.869436\pi$
$648$	−7.24922	−0.284776
$649$	−27.7771	−1.09035
$650$	35.4164	1.38915
$651$	−16.3607	−0.641226
$652$	9.67376	0.378854
$653$	34.9443	1.36748	0.683738	−	0.729728i	$-0.260353\pi$
$653$	34.9443	1.36748	0.683738	+	0.729728i	$0.260353\pi$
$654$	−13.5279	−0.528981
$655$	−67.3050	−2.62982
$656$	−11.7082	−0.457129
$657$	−7.14590	−0.278788
$658$	27.4164	1.06880
$659$	−14.3262	−0.558071	−0.279035	−	0.960281i	$-0.590015\pi$
$659$	−14.3262	−0.558071	−0.279035	+	0.960281i	$0.590015\pi$
$660$	22.4721	0.874727
$661$	−33.0344	−1.28489	−0.642445	−	0.766331i	$-0.722080\pi$
$661$	−33.0344	−1.28489	−0.642445	+	0.766331i	$0.722080\pi$
$662$	−30.4721	−1.18433
$663$	−41.8885	−1.62682
$664$	13.4164	0.520658
$665$	−4.94427	−0.191731
$666$	2.72949	0.105766
$667$	−17.6393	−0.682997
$668$	15.5623	0.602124
$669$	−16.3607	−0.632540
$670$	7.52786	0.290827
$671$	−69.2492	−2.67334
$672$	12.3607	0.476824
$673$	−22.6525	−0.873189	−0.436594	−	0.899658i	$-0.643816\pi$
$673$	−22.6525	−0.873189	−0.436594	+	0.899658i	$0.643816\pi$
$674$	15.7082	0.605057
$675$	−30.2492	−1.16429
$676$	−28.8885	−1.11110
$677$	−27.0557	−1.03984	−0.519918	−	0.854216i	$-0.674038\pi$
$677$	−27.0557	−1.03984	−0.519918	+	0.854216i	$0.674038\pi$
$678$	19.8885	0.763815
$679$	−6.76393	−0.259576
$680$	−50.8328	−1.94935
$681$	−15.1672	−0.581208
$682$	−37.1803	−1.42371
$683$	−0.111456	−0.00426475	−0.00213238	−	0.999998i	$-0.500679\pi$
$683$	−0.111456	−0.00426475	−0.00213238	+	0.999998i	$0.500679\pi$
$684$	−1.12461	−0.0430006
$685$	13.5279	0.516873
$686$	−20.0000	−0.763604
$687$	−13.6656	−0.521376
$688$	11.2361	0.428371
$689$	−43.7771	−1.66777
$690$	−11.0557	−0.420884
$691$	17.1246	0.651451	0.325725	−	0.945464i	$-0.394391\pi$
$691$	17.1246	0.651451	0.325725	+	0.945464i	$0.394391\pi$
$692$	4.61803	0.175551
$693$	16.5410	0.628341
$694$	−28.8541	−1.09529
$695$	11.4164	0.433049
$696$	23.6656	0.897043
$697$	61.3050	2.32209
$698$	19.2361	0.728096
$699$	−30.8328	−1.16620
$700$	10.9443	0.413655
$701$	−16.0689	−0.606913	−0.303457	−	0.952845i	$-0.598141\pi$
$701$	−16.0689	−0.606913	−0.303457	+	0.952845i	$0.598141\pi$
$702$	−35.7771	−1.35032
$703$	−1.41641	−0.0534208
$704$	39.3262	1.48216
$705$	−54.8328	−2.06512
$706$	−0.854102	−0.0321446
$707$	20.0000	0.752177
$708$	6.11146	0.229683
$709$	−0.652476	−0.0245042	−0.0122521	−	0.999925i	$-0.503900\pi$
$709$	−0.652476	−0.0245042	−0.0122521	+	0.999925i	$0.503900\pi$
$710$	−41.8885	−1.57205
$711$	5.88854	0.220838
$712$	30.0000	1.12430
$713$	−18.2918	−0.685033
$714$	12.9443	0.484427
$715$	117.666	4.40045
$716$	−8.00000	−0.298974
$717$	20.9443	0.782178
$718$	15.2361	0.568605
$719$	−15.8885	−0.592543	−0.296271	−	0.955104i	$-0.595743\pi$
$719$	−15.8885	−0.592543	−0.296271	+	0.955104i	$0.595743\pi$
$720$	−4.76393	−0.177541
$721$	37.3050	1.38931
$722$	18.4164	0.685388
$723$	1.30495	0.0485317
$724$	1.05573	0.0392358
$725$	34.9230	1.29701
$726$	−25.4164	−0.943291
$727$	33.2148	1.23187	0.615934	−	0.787798i	$-0.288779\pi$
$727$	33.2148	1.23187	0.615934	+	0.787798i	$0.288779\pi$
$728$	38.8328	1.43924
$729$	24.0557	0.890953
$730$	15.7082	0.581387
$731$	−58.8328	−2.17601
$732$	15.2361	0.563141
$733$	17.1246	0.632512	0.316256	−	0.948674i	$-0.397574\pi$
$733$	17.1246	0.632512	0.316256	+	0.948674i	$0.397574\pi$
$734$	−9.88854	−0.364993
$735$	12.0000	0.442627
$736$	13.8197	0.509399
$737$	13.0689	0.481399
$738$	17.2361	0.634468
$739$	−22.5410	−0.829185	−0.414592	−	0.910007i	$-0.636076\pi$
$739$	−22.5410	−0.829185	−0.414592	+	0.910007i	$0.636076\pi$
$740$	6.00000	0.220564
$741$	6.11146	0.224510
$742$	13.5279	0.496624
$743$	−10.4721	−0.384185	−0.192093	−	0.981377i	$-0.561527\pi$
$743$	−10.4721	−0.384185	−0.192093	+	0.981377i	$0.561527\pi$
$744$	24.5410	0.899717
$745$	18.1803	0.666076
$746$	−7.23607	−0.264931
$747$	−6.58359	−0.240881
$748$	−29.4164	−1.07557
$749$	−12.0000	−0.438470
$750$	1.88854	0.0689599
$751$	16.9443	0.618305	0.309153	−	0.951012i	$-0.399955\pi$
$751$	16.9443	0.618305	0.309153	+	0.951012i	$0.399955\pi$
$752$	−13.7082	−0.499887
$753$	10.0000	0.364420
$754$	41.3050	1.50424
$755$	−36.3607	−1.32330
$756$	−11.0557	−0.402093
$757$	24.4721	0.889455	0.444727	−	0.895666i	$-0.353301\pi$
$757$	24.4721	0.889455	0.444727	+	0.895666i	$0.353301\pi$
$758$	−8.27051	−0.300398
$759$	−19.1935	−0.696680
$760$	7.41641	0.269021
$761$	4.65248	0.168652	0.0843261	−	0.996438i	$-0.473126\pi$
$761$	4.65248	0.168652	0.0843261	+	0.996438i	$0.473126\pi$
$762$	22.6525	0.820613
$763$	−21.8885	−0.792418
$764$	−1.43769	−0.0520139
$765$	24.9443	0.901862
$766$	22.9443	0.829010
$767$	32.0000	1.15545
$768$	−21.0132	−0.758247
$769$	−1.97871	−0.0713542	−0.0356771	−	0.999363i	$-0.511359\pi$
$769$	−1.97871	−0.0713542	−0.0356771	+	0.999363i	$0.511359\pi$
$770$	−36.3607	−1.31035
$771$	−15.7771	−0.568198
$772$	12.2705	0.441625
$773$	−11.1246	−0.400124	−0.200062	−	0.979783i	$-0.564114\pi$
$773$	−11.1246	−0.400124	−0.200062	+	0.979783i	$0.564114\pi$
$774$	−16.5410	−0.594555
$775$	36.2148	1.30087
$776$	10.1459	0.364217
$777$	−4.58359	−0.164435
$778$	3.70820	0.132946
$779$	−8.94427	−0.320462
$780$	−25.8885	−0.926959
$781$	−72.7214	−2.60217
$782$	14.4721	0.517523
$783$	−35.2786	−1.26076
$784$	3.00000	0.107143
$785$	49.5967	1.77018
$786$	−25.7082	−0.916981
$787$	24.6525	0.878766	0.439383	−	0.898300i	$-0.355197\pi$
$787$	24.6525	0.878766	0.439383	+	0.898300i	$0.355197\pi$
$788$	−9.85410	−0.351038
$789$	−21.6656	−0.771317
$790$	−12.9443	−0.460537
$791$	32.1803	1.14420
$792$	−24.8115	−0.881639
$793$	79.7771	2.83297
$794$	13.5279	0.480086
$795$	−27.0557	−0.959568
$796$	15.3262	0.543224
$797$	22.5410	0.798444	0.399222	−	0.916854i	$-0.369280\pi$
$797$	22.5410	0.798444	0.399222	+	0.916854i	$0.369280\pi$
$798$	−1.88854	−0.0668537
$799$	71.7771	2.53929
$800$	−27.3607	−0.967346
$801$	−14.7214	−0.520154
$802$	28.0000	0.988714
$803$	27.2705	0.962355
$804$	−2.87539	−0.101407
$805$	−17.8885	−0.630488
$806$	42.8328	1.50872
$807$	2.72136	0.0957964
$808$	−30.0000	−1.05540
$809$	43.1246	1.51618	0.758090	−	0.652150i	$-0.226133\pi$
$809$	43.1246	1.51618	0.758090	+	0.652150i	$0.226133\pi$
$810$	−7.81966	−0.274755
$811$	−55.5066	−1.94910	−0.974550	−	0.224171i	$-0.928033\pi$
$811$	−55.5066	−1.94910	−0.974550	+	0.224171i	$0.928033\pi$
$812$	12.7639	0.447926
$813$	−12.9443	−0.453975
$814$	−10.4164	−0.365095
$815$	31.3050	1.09656
$816$	−6.47214	−0.226570
$817$	8.58359	0.300302
$818$	−30.8328	−1.07804
$819$	−19.0557	−0.665861
$820$	37.8885	1.32313
$821$	27.5967	0.963133	0.481567	−	0.876410i	$-0.340068\pi$
$821$	27.5967	0.963133	0.481567	+	0.876410i	$0.340068\pi$
$822$	5.16718	0.180226
$823$	−18.5623	−0.647041	−0.323521	−	0.946221i	$-0.604867\pi$
$823$	−18.5623	−0.647041	−0.323521	+	0.946221i	$0.604867\pi$
$824$	−55.9574	−1.94937
$825$	38.0000	1.32299
$826$	−9.88854	−0.344066
$827$	−23.5623	−0.819342	−0.409671	−	0.912233i	$-0.634356\pi$
$827$	−23.5623	−0.819342	−0.409671	+	0.912233i	$0.634356\pi$
$828$	−4.06888	−0.141403
$829$	2.87539	0.0998664	0.0499332	−	0.998753i	$-0.484099\pi$
$829$	2.87539	0.0998664	0.0499332	+	0.998753i	$0.484099\pi$
$830$	14.4721	0.502335
$831$	12.5836	0.436520
$832$	−45.3050	−1.57067
$833$	−15.7082	−0.544257
$834$	4.36068	0.150998
$835$	50.3607	1.74280
$836$	4.29180	0.148435
$837$	−36.5836	−1.26451
$838$	−30.9787	−1.07014
$839$	−55.0476	−1.90045	−0.950227	−	0.311558i	$-0.899149\pi$
$839$	−55.0476	−1.90045	−0.950227	+	0.311558i	$0.899149\pi$
$840$	24.0000	0.828079
$841$	11.7295	0.404465
$842$	−0.763932	−0.0263268
$843$	−14.8754	−0.512336
$844$	26.6525	0.917416
$845$	−93.4853	−3.21599
$846$	20.1803	0.693814
$847$	−41.1246	−1.41306
$848$	−6.76393	−0.232274
$849$	1.88854	0.0648147
$850$	−28.6525	−0.982772
$851$	−5.12461	−0.175669
$852$	16.0000	0.548151
$853$	1.88854	0.0646625	0.0323313	−	0.999477i	$-0.489707\pi$
$853$	1.88854	0.0646625	0.0323313	+	0.999477i	$0.489707\pi$
$854$	−24.6525	−0.843590
$855$	−3.63932	−0.124462
$856$	18.0000	0.615227
$857$	2.14590	0.0733025	0.0366512	−	0.999328i	$-0.488331\pi$
$857$	2.14590	0.0733025	0.0366512	+	0.999328i	$0.488331\pi$
$858$	44.9443	1.53437
$859$	14.1803	0.483827	0.241913	−	0.970298i	$-0.422225\pi$
$859$	14.1803	0.483827	0.241913	+	0.970298i	$0.422225\pi$
$860$	−36.3607	−1.23989
$861$	−28.9443	−0.986418
$862$	−22.7426	−0.774618
$863$	22.9098	0.779860	0.389930	−	0.920845i	$-0.372499\pi$
$863$	22.9098	0.779860	0.389930	+	0.920845i	$0.372499\pi$
$864$	27.6393	0.940309
$865$	14.9443	0.508120
$866$	4.90983	0.166843
$867$	12.8754	0.437271
$868$	13.2361	0.449261
$869$	−22.4721	−0.762315
$870$	25.5279	0.865476
$871$	−15.0557	−0.510144
$872$	32.8328	1.11186
$873$	−4.97871	−0.168504
$874$	−2.11146	−0.0714211
$875$	3.05573	0.103302
$876$	−6.00000	−0.202721
$877$	−9.41641	−0.317970	−0.158985	−	0.987281i	$-0.550822\pi$
$877$	−9.41641	−0.317970	−0.158985	+	0.987281i	$0.550822\pi$
$878$	−9.79837	−0.330679
$879$	20.7214	0.698914
$880$	18.1803	0.612859
$881$	26.2492	0.884359	0.442179	−	0.896927i	$-0.354205\pi$
$881$	26.2492	0.884359	0.442179	+	0.896927i	$0.354205\pi$
$882$	−4.41641	−0.148708
$883$	48.6525	1.63729	0.818643	−	0.574303i	$-0.194727\pi$
$883$	48.6525	1.63729	0.818643	+	0.574303i	$0.194727\pi$
$884$	33.8885	1.13980
$885$	19.7771	0.664800
$886$	−2.00000	−0.0671913
$887$	−31.0557	−1.04275	−0.521375	−	0.853328i	$-0.674581\pi$
$887$	−31.0557	−1.04275	−0.521375	+	0.853328i	$0.674581\pi$
$888$	6.87539	0.230723
$889$	36.6525	1.22928
$890$	32.3607	1.08473
$891$	−13.5755	−0.454795
$892$	13.2361	0.443176
$893$	−10.4721	−0.350437
$894$	6.94427	0.232251
$895$	−25.8885	−0.865359
$896$	−6.00000	−0.200446
$897$	22.1115	0.738280
$898$	−0.0344419	−0.00114934
$899$	42.2361	1.40865
$900$	8.05573	0.268524
$901$	35.4164	1.17989
$902$	−65.7771	−2.19014
$903$	27.7771	0.924364
$904$	−48.2705	−1.60545
$905$	3.41641	0.113565
$906$	−13.8885	−0.461416
$907$	40.1803	1.33417	0.667083	−	0.744983i	$-0.267543\pi$
$907$	40.1803	1.33417	0.667083	+	0.744983i	$0.267543\pi$
$908$	12.2705	0.407211
$909$	14.7214	0.488277
$910$	41.8885	1.38859
$911$	57.8885	1.91793	0.958967	−	0.283519i	$-0.0915020\pi$
$911$	57.8885	1.91793	0.958967	+	0.283519i	$0.0915020\pi$
$912$	0.944272	0.0312680
$913$	25.1246	0.831503
$914$	25.1246	0.831048
$915$	49.3050	1.62997
$916$	11.0557	0.365292
$917$	−41.5967	−1.37365
$918$	28.9443	0.955303
$919$	−17.7426	−0.585276	−0.292638	−	0.956223i	$-0.594533\pi$
$919$	−17.7426	−0.585276	−0.292638	+	0.956223i	$0.594533\pi$
$920$	26.8328	0.884652
$921$	−13.7771	−0.453970
$922$	−23.3262	−0.768209
$923$	83.7771	2.75756
$924$	13.8885	0.456900
$925$	10.1459	0.333595
$926$	−28.7639	−0.945241
$927$	27.4590	0.901871
$928$	−31.9098	−1.04749
$929$	−14.2016	−0.465940	−0.232970	−	0.972484i	$-0.574844\pi$
$929$	−14.2016	−0.465940	−0.232970	+	0.972484i	$0.574844\pi$
$930$	26.4721	0.868056
$931$	2.29180	0.0751106
$932$	24.9443	0.817077
$933$	−39.4590	−1.29183
$934$	−9.41641	−0.308114
$935$	−95.1935	−3.11316
$936$	28.5836	0.934284
$937$	−8.40325	−0.274522	−0.137261	−	0.990535i	$-0.543830\pi$
$937$	−8.40325	−0.274522	−0.137261	+	0.990535i	$0.543830\pi$
$938$	4.65248	0.151909
$939$	14.1115	0.460510
$940$	44.3607	1.44689
$941$	57.7771	1.88348	0.941740	−	0.336343i	$-0.109190\pi$
$941$	57.7771	1.88348	0.941740	+	0.336343i	$0.109190\pi$
$942$	18.9443	0.617238
$943$	−32.3607	−1.05381
$944$	4.94427	0.160922
$945$	−35.7771	−1.16383
$946$	63.1246	2.05236
$947$	32.9443	1.07054	0.535272	−	0.844679i	$-0.320209\pi$
$947$	32.9443	1.07054	0.535272	+	0.844679i	$0.320209\pi$
$948$	4.94427	0.160582
$949$	−31.4164	−1.01982
$950$	4.18034	0.135628
$951$	7.77709	0.252189
$952$	−31.4164	−1.01821
$953$	−41.9787	−1.35982	−0.679912	−	0.733294i	$-0.737982\pi$
$953$	−41.9787	−1.35982	−0.679912	+	0.733294i	$0.737982\pi$
$954$	9.95743	0.322384
$955$	−4.65248	−0.150551
$956$	−16.9443	−0.548017
$957$	44.3181	1.43260
$958$	22.4721	0.726042
$959$	8.36068	0.269980
$960$	−28.0000	−0.903696
$961$	12.7984	0.412851
$962$	12.0000	0.386896
$963$	−8.83282	−0.284634
$964$	−1.05573	−0.0340027
$965$	39.7082	1.27825
$966$	−6.83282	−0.219842
$967$	−46.8328	−1.50604	−0.753021	−	0.657997i	$-0.771404\pi$
$967$	−46.8328	−1.50604	−0.753021	+	0.657997i	$0.771404\pi$
$968$	61.6869	1.98269
$969$	−4.94427	−0.158833
$970$	10.9443	0.351399
$971$	21.0557	0.675710	0.337855	−	0.941198i	$-0.390299\pi$
$971$	21.0557	0.675710	0.337855	+	0.941198i	$0.390299\pi$
$972$	−13.5967	−0.436116
$973$	7.05573	0.226196
$974$	6.20163	0.198713
$975$	−43.7771	−1.40199
$976$	12.3262	0.394553
$977$	34.7214	1.11083	0.555417	−	0.831572i	$-0.312559\pi$
$977$	34.7214	1.11083	0.555417	+	0.831572i	$0.312559\pi$
$978$	11.9574	0.382356
$979$	56.1803	1.79553
$980$	−9.70820	−0.310117
$981$	−16.1115	−0.514399
$982$	5.52786	0.176401
$983$	−13.0557	−0.416413	−0.208207	−	0.978085i	$-0.566763\pi$
$983$	−13.0557	−0.416413	−0.208207	+	0.978085i	$0.566763\pi$
$984$	43.4164	1.38406
$985$	−31.8885	−1.01605
$986$	−33.4164	−1.06420
$987$	−33.8885	−1.07868
$988$	−4.94427	−0.157298
$989$	31.0557	0.987515
$990$	−26.7639	−0.850614
$991$	48.8328	1.55123	0.775613	−	0.631209i	$-0.217441\pi$
$991$	48.8328	1.55123	0.775613	+	0.631209i	$0.217441\pi$
$992$	−33.0902	−1.05061
$993$	37.6656	1.19528
$994$	−25.8885	−0.821135
$995$	49.5967	1.57232
$996$	−5.52786	−0.175157
$997$	50.9230	1.61275	0.806374	−	0.591407i	$-0.201427\pi$
$997$	50.9230	1.61275	0.806374	+	0.591407i	$0.201427\pi$
$998$	−4.94427	−0.156508
$999$	−10.2492	−0.324271

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4153.2.a.a.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4153.2.a.a.1.2	✓	2	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:	\( N \)	\(=\)	\( 4153 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4153.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.23607 q^{3} -1.00000 q^{4} -3.23607 q^{5} -1.23607 q^{6} -2.00000 q^{7} +3.00000 q^{8} -1.47214 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.23607 q^{3} -1.00000 q^{4} -3.23607 q^{5} -1.23607 q^{6} -2.00000 q^{7} +3.00000 q^{8} -1.47214 q^{9} +3.23607 q^{10} +5.61803 q^{11} -1.23607 q^{12} -6.47214 q^{13} +2.00000 q^{14} -4.00000 q^{15} -1.00000 q^{16} +5.23607 q^{17} +1.47214 q^{18} -0.763932 q^{19} +3.23607 q^{20} -2.47214 q^{21} -5.61803 q^{22} -2.76393 q^{23} +3.70820 q^{24} +5.47214 q^{25} +6.47214 q^{26} -5.52786 q^{27} +2.00000 q^{28} +6.38197 q^{29} +4.00000 q^{30} +6.61803 q^{31} -5.00000 q^{32} +6.94427 q^{33} -5.23607 q^{34} +6.47214 q^{35} +1.47214 q^{36} +1.85410 q^{37} +0.763932 q^{38} -8.00000 q^{39} -9.70820 q^{40} +11.7082 q^{41} +2.47214 q^{42} -11.2361 q^{43} -5.61803 q^{44} +4.76393 q^{45} +2.76393 q^{46} +13.7082 q^{47} -1.23607 q^{48} -3.00000 q^{49} -5.47214 q^{50} +6.47214 q^{51} +6.47214 q^{52} +6.76393 q^{53} +5.52786 q^{54} -18.1803 q^{55} -6.00000 q^{56} -0.944272 q^{57} -6.38197 q^{58} -4.94427 q^{59} +4.00000 q^{60} -12.3262 q^{61} -6.61803 q^{62} +2.94427 q^{63} +7.00000 q^{64} +20.9443 q^{65} -6.94427 q^{66} +2.32624 q^{67} -5.23607 q^{68} -3.41641 q^{69} -6.47214 q^{70} -12.9443 q^{71} -4.41641 q^{72} +4.85410 q^{73} -1.85410 q^{74} +6.76393 q^{75} +0.763932 q^{76} -11.2361 q^{77} +8.00000 q^{78} -4.00000 q^{79} +3.23607 q^{80} -2.41641 q^{81} -11.7082 q^{82} +4.47214 q^{83} +2.47214 q^{84} -16.9443 q^{85} +11.2361 q^{86} +7.88854 q^{87} +16.8541 q^{88} +10.0000 q^{89} -4.76393 q^{90} +12.9443 q^{91} +2.76393 q^{92} +8.18034 q^{93} -13.7082 q^{94} +2.47214 q^{95} -6.18034 q^{96} +3.38197 q^{97} +3.00000 q^{98} -8.27051 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} - 2 q^{4} - 2 q^{5} + 2 q^{6} - 4 q^{7} + 6 q^{8} + 6 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 - 2 * q^3 - 2 * q^4 - 2 * q^5 + 2 * q^6 - 4 * q^7 + 6 * q^8 + 6 * q^9	\( 2 q - 2 q^{2} - 2 q^{3} - 2 q^{4} - 2 q^{5} + 2 q^{6} - 4 q^{7} + 6 q^{8} + 6 q^{9} + 2 q^{10} + 9 q^{11} + 2 q^{12} - 4 q^{13} + 4 q^{14} - 8 q^{15} - 2 q^{16} + 6 q^{17} - 6 q^{18} - 6 q^{19} + 2 q^{20} + 4 q^{21} - 9 q^{22} - 10 q^{23} - 6 q^{24} + 2 q^{25} + 4 q^{26} - 20 q^{27} + 4 q^{28} + 15 q^{29} + 8 q^{30} + 11 q^{31} - 10 q^{32} - 4 q^{33} - 6 q^{34} + 4 q^{35} - 6 q^{36} - 3 q^{37} + 6 q^{38} - 16 q^{39} - 6 q^{40} + 10 q^{41} - 4 q^{42} - 18 q^{43} - 9 q^{44} + 14 q^{45} + 10 q^{46} + 14 q^{47} + 2 q^{48} - 6 q^{49} - 2 q^{50} + 4 q^{51} + 4 q^{52} + 18 q^{53} + 20 q^{54} - 14 q^{55} - 12 q^{56} + 16 q^{57} - 15 q^{58} + 8 q^{59} + 8 q^{60} - 9 q^{61} - 11 q^{62} - 12 q^{63} + 14 q^{64} + 24 q^{65} + 4 q^{66} - 11 q^{67} - 6 q^{68} + 20 q^{69} - 4 q^{70} - 8 q^{71} + 18 q^{72} + 3 q^{73} + 3 q^{74} + 18 q^{75} + 6 q^{76} - 18 q^{77} + 16 q^{78} - 8 q^{79} + 2 q^{80} + 22 q^{81} - 10 q^{82} - 4 q^{84} - 16 q^{85} + 18 q^{86} - 20 q^{87} + 27 q^{88} + 20 q^{89} - 14 q^{90} + 8 q^{91} + 10 q^{92} - 6 q^{93} - 14 q^{94} - 4 q^{95} + 10 q^{96} + 9 q^{97} + 6 q^{98} + 17 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 - 2 * q^4 - 2 * q^5 + 2 * q^6 - 4 * q^7 + 6 * q^8 + 6 * q^9 + 2 * q^10 + 9 * q^11 + 2 * q^12 - 4 * q^13 + 4 * q^14 - 8 * q^15 - 2 * q^16 + 6 * q^17 - 6 * q^18 - 6 * q^19 + 2 * q^20 + 4 * q^21 - 9 * q^22 - 10 * q^23 - 6 * q^24 + 2 * q^25 + 4 * q^26 - 20 * q^27 + 4 * q^28 + 15 * q^29 + 8 * q^30 + 11 * q^31 - 10 * q^32 - 4 * q^33 - 6 * q^34 + 4 * q^35 - 6 * q^36 - 3 * q^37 + 6 * q^38 - 16 * q^39 - 6 * q^40 + 10 * q^41 - 4 * q^42 - 18 * q^43 - 9 * q^44 + 14 * q^45 + 10 * q^46 + 14 * q^47 + 2 * q^48 - 6 * q^49 - 2 * q^50 + 4 * q^51 + 4 * q^52 + 18 * q^53 + 20 * q^54 - 14 * q^55 - 12 * q^56 + 16 * q^57 - 15 * q^58 + 8 * q^59 + 8 * q^60 - 9 * q^61 - 11 * q^62 - 12 * q^63 + 14 * q^64 + 24 * q^65 + 4 * q^66 - 11 * q^67 - 6 * q^68 + 20 * q^69 - 4 * q^70 - 8 * q^71 + 18 * q^72 + 3 * q^73 + 3 * q^74 + 18 * q^75 + 6 * q^76 - 18 * q^77 + 16 * q^78 - 8 * q^79 + 2 * q^80 + 22 * q^81 - 10 * q^82 - 4 * q^84 - 16 * q^85 + 18 * q^86 - 20 * q^87 + 27 * q^88 + 20 * q^89 - 14 * q^90 + 8 * q^91 + 10 * q^92 - 6 * q^93 - 14 * q^94 - 4 * q^95 + 10 * q^96 + 9 * q^97 + 6 * q^98 + 17 * q^99

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