LMFDB - Embedded newform 1334.2.a.i.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1334,2,Mod(1,1334)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1334, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1334.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1334 = 2 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1334.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:	$10.6520436296$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 22x^{6} + 151x^{4} - 332x^{2} + 6 $ x^8 - 22x^6 + 151x^4 - 332*x^2 + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.32381$ of defining polynomial
Character	$\chi$	$=$	1334.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -0.153561 q^{3} +1.00000 q^{4} -3.32381 q^{5} -0.153561 q^{6} -3.68051 q^{7} +1.00000 q^{8} -2.97642 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -0.153561 q^{3} +1.00000 q^{4} -3.32381 q^{5} -0.153561 q^{6} -3.68051 q^{7} +1.00000 q^{8} -2.97642 q^{9} -3.32381 q^{10} +3.79005 q^{11} -0.153561 q^{12} +4.76277 q^{13} -3.68051 q^{14} +0.510408 q^{15} +1.00000 q^{16} +3.34112 q^{17} -2.97642 q^{18} +6.46686 q^{19} -3.32381 q^{20} +0.565183 q^{21} +3.79005 q^{22} -1.00000 q^{23} -0.153561 q^{24} +6.04772 q^{25} +4.76277 q^{26} +0.917745 q^{27} -3.68051 q^{28} -1.00000 q^{29} +0.510408 q^{30} -7.96366 q^{31} +1.00000 q^{32} -0.582004 q^{33} +3.34112 q^{34} +12.2333 q^{35} -2.97642 q^{36} +5.25321 q^{37} +6.46686 q^{38} -0.731375 q^{39} -3.32381 q^{40} +2.92040 q^{41} +0.565183 q^{42} +11.8781 q^{43} +3.79005 q^{44} +9.89306 q^{45} -1.00000 q^{46} +12.7448 q^{47} -0.153561 q^{48} +6.54617 q^{49} +6.04772 q^{50} -0.513066 q^{51} +4.76277 q^{52} -1.82898 q^{53} +0.917745 q^{54} -12.5974 q^{55} -3.68051 q^{56} -0.993057 q^{57} -1.00000 q^{58} -13.3206 q^{59} +0.510408 q^{60} +8.11991 q^{61} -7.96366 q^{62} +10.9547 q^{63} +1.00000 q^{64} -15.8305 q^{65} -0.582004 q^{66} +9.13383 q^{67} +3.34112 q^{68} +0.153561 q^{69} +12.2333 q^{70} -3.83171 q^{71} -2.97642 q^{72} -14.2989 q^{73} +5.25321 q^{74} -0.928694 q^{75} +6.46686 q^{76} -13.9493 q^{77} -0.731375 q^{78} -10.9981 q^{79} -3.32381 q^{80} +8.78833 q^{81} +2.92040 q^{82} +10.3489 q^{83} +0.565183 q^{84} -11.1053 q^{85} +11.8781 q^{86} +0.153561 q^{87} +3.79005 q^{88} +3.55410 q^{89} +9.89306 q^{90} -17.5294 q^{91} -1.00000 q^{92} +1.22291 q^{93} +12.7448 q^{94} -21.4946 q^{95} -0.153561 q^{96} +12.9528 q^{97} +6.54617 q^{98} -11.2808 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} + 4 q^{3} + 8 q^{4} + 4 q^{6} + 8 q^{7} + 8 q^{8} + 12 q^{9}+O(q^{10}) $ 8 * q + 8 * q^2 + 4 * q^3 + 8 * q^4 + 4 * q^6 + 8 * q^7 + 8 * q^8 + 12 * q^9	$ 8 q + 8 q^{2} + 4 q^{3} + 8 q^{4} + 4 q^{6} + 8 q^{7} + 8 q^{8} + 12 q^{9} + 8 q^{11} + 4 q^{12} + 4 q^{13} + 8 q^{14} + 8 q^{16} + 8 q^{17} + 12 q^{18} + 16 q^{19} - 4 q^{21} + 8 q^{22} - 8 q^{23} + 4 q^{24} + 4 q^{25} + 4 q^{26} + 4 q^{27} + 8 q^{28} - 8 q^{29} + 8 q^{31} + 8 q^{32} + 12 q^{33} + 8 q^{34} + 4 q^{35} + 12 q^{36} + 8 q^{37} + 16 q^{38} - 16 q^{39} + 8 q^{41} - 4 q^{42} + 32 q^{43} + 8 q^{44} - 8 q^{46} + 16 q^{47} + 4 q^{48} + 4 q^{49} + 4 q^{50} + 12 q^{51} + 4 q^{52} + 4 q^{54} + 8 q^{56} + 8 q^{57} - 8 q^{58} - 4 q^{59} + 8 q^{61} + 8 q^{62} + 24 q^{63} + 8 q^{64} - 4 q^{65} + 12 q^{66} + 20 q^{67} + 8 q^{68} - 4 q^{69} + 4 q^{70} - 24 q^{71} + 12 q^{72} - 4 q^{73} + 8 q^{74} - 16 q^{75} + 16 q^{76} - 16 q^{77} - 16 q^{78} + 4 q^{79} - 8 q^{81} + 8 q^{82} - 4 q^{83} - 4 q^{84} - 44 q^{85} + 32 q^{86} - 4 q^{87} + 8 q^{88} + 20 q^{89} - 8 q^{92} - 4 q^{93} + 16 q^{94} - 8 q^{95} + 4 q^{96} + 4 q^{97} + 4 q^{98}+O(q^{100}) $ 8 * q + 8 * q^2 + 4 * q^3 + 8 * q^4 + 4 * q^6 + 8 * q^7 + 8 * q^8 + 12 * q^9 + 8 * q^11 + 4 * q^12 + 4 * q^13 + 8 * q^14 + 8 * q^16 + 8 * q^17 + 12 * q^18 + 16 * q^19 - 4 * q^21 + 8 * q^22 - 8 * q^23 + 4 * q^24 + 4 * q^25 + 4 * q^26 + 4 * q^27 + 8 * q^28 - 8 * q^29 + 8 * q^31 + 8 * q^32 + 12 * q^33 + 8 * q^34 + 4 * q^35 + 12 * q^36 + 8 * q^37 + 16 * q^38 - 16 * q^39 + 8 * q^41 - 4 * q^42 + 32 * q^43 + 8 * q^44 - 8 * q^46 + 16 * q^47 + 4 * q^48 + 4 * q^49 + 4 * q^50 + 12 * q^51 + 4 * q^52 + 4 * q^54 + 8 * q^56 + 8 * q^57 - 8 * q^58 - 4 * q^59 + 8 * q^61 + 8 * q^62 + 24 * q^63 + 8 * q^64 - 4 * q^65 + 12 * q^66 + 20 * q^67 + 8 * q^68 - 4 * q^69 + 4 * q^70 - 24 * q^71 + 12 * q^72 - 4 * q^73 + 8 * q^74 - 16 * q^75 + 16 * q^76 - 16 * q^77 - 16 * q^78 + 4 * q^79 - 8 * q^81 + 8 * q^82 - 4 * q^83 - 4 * q^84 - 44 * q^85 + 32 * q^86 - 4 * q^87 + 8 * q^88 + 20 * q^89 - 8 * q^92 - 4 * q^93 + 16 * q^94 - 8 * q^95 + 4 * q^96 + 4 * q^97 + 4 * q^98

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
$3$	−0.153561	−0.0886585	−0.0443292	−	0.999017i	$-0.514115\pi$
$3$	−0.153561	−0.0886585	−0.0443292	+	0.999017i	$0.514115\pi$
$4$	1.00000	0.500000
$5$	−3.32381	−1.48645	−0.743227	−	0.669039i	$-0.766706\pi$
$5$	−3.32381	−1.48645	−0.743227	+	0.669039i	$0.766706\pi$
$6$	−0.153561	−0.0626910
$7$	−3.68051	−1.39110	−0.695551	−	0.718476i	$-0.744840\pi$
$7$	−3.68051	−1.39110	−0.695551	+	0.718476i	$0.744840\pi$
$8$	1.00000	0.353553
$9$	−2.97642	−0.992140
$10$	−3.32381	−1.05108
$11$	3.79005	1.14274	0.571371	−	0.820692i	$-0.306412\pi$
$11$	3.79005	1.14274	0.571371	+	0.820692i	$0.306412\pi$
$12$	−0.153561	−0.0443292
$13$	4.76277	1.32095	0.660477	−	0.750846i	$-0.270354\pi$
$13$	4.76277	1.32095	0.660477	+	0.750846i	$0.270354\pi$
$14$	−3.68051	−0.983658
$15$	0.510408	0.131787
$16$	1.00000	0.250000
$17$	3.34112	0.810341	0.405170	−	0.914241i	$-0.367212\pi$
$17$	3.34112	0.810341	0.405170	+	0.914241i	$0.367212\pi$
$18$	−2.97642	−0.701549
$19$	6.46686	1.48360	0.741800	−	0.670621i	$-0.233972\pi$
$19$	6.46686	1.48360	0.741800	+	0.670621i	$0.233972\pi$
$20$	−3.32381	−0.743227
$21$	0.565183	0.123333
$22$	3.79005	0.808041
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−0.153561	−0.0313455
$25$	6.04772	1.20954
$26$	4.76277	0.934055
$27$	0.917745	0.176620
$28$	−3.68051	−0.695551
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0.510408	0.0931873
$31$	−7.96366	−1.43032	−0.715158	−	0.698963i	$-0.753645\pi$
$31$	−7.96366	−1.43032	−0.715158	+	0.698963i	$0.753645\pi$
$32$	1.00000	0.176777
$33$	−0.582004	−0.101314
$34$	3.34112	0.572997
$35$	12.2333	2.06781
$36$	−2.97642	−0.496070
$37$	5.25321	0.863622	0.431811	−	0.901964i	$-0.357875\pi$
$37$	5.25321	0.863622	0.431811	+	0.901964i	$0.357875\pi$
$38$	6.46686	1.04906
$39$	−0.731375	−0.117114
$40$	−3.32381	−0.525541
$41$	2.92040	0.456090	0.228045	−	0.973651i	$-0.426767\pi$
$41$	2.92040	0.456090	0.228045	+	0.973651i	$0.426767\pi$
$42$	0.565183	0.0872096
$43$	11.8781	1.81139	0.905696	−	0.423927i	$-0.139349\pi$
$43$	11.8781	1.81139	0.905696	+	0.423927i	$0.139349\pi$
$44$	3.79005	0.571371
$45$	9.89306	1.47477
$46$	−1.00000	−0.147442
$47$	12.7448	1.85902	0.929510	−	0.368798i	$-0.120230\pi$
$47$	12.7448	1.85902	0.929510	+	0.368798i	$0.120230\pi$
$48$	−0.153561	−0.0221646
$49$	6.54617	0.935167
$50$	6.04772	0.855277
$51$	−0.513066	−0.0718436
$52$	4.76277	0.660477
$53$	−1.82898	−0.251230	−0.125615	−	0.992079i	$-0.540090\pi$
$53$	−1.82898	−0.251230	−0.125615	+	0.992079i	$0.540090\pi$
$54$	0.917745	0.124889
$55$	−12.5974	−1.69863
$56$	−3.68051	−0.491829
$57$	−0.993057	−0.131534
$58$	−1.00000	−0.131306
$59$	−13.3206	−1.73420	−0.867099	−	0.498136i	$-0.834018\pi$
$59$	−13.3206	−1.73420	−0.867099	+	0.498136i	$0.834018\pi$
$60$	0.510408	0.0658934
$61$	8.11991	1.03965	0.519824	−	0.854274i	$-0.325998\pi$
$61$	8.11991	1.03965	0.519824	+	0.854274i	$0.325998\pi$
$62$	−7.96366	−1.01139
$63$	10.9547	1.38017
$64$	1.00000	0.125000
$65$	−15.8305	−1.96354
$66$	−0.582004	−0.0716397
$67$	9.13383	1.11587	0.557937	−	0.829883i	$-0.311593\pi$
$67$	9.13383	1.11587	0.557937	+	0.829883i	$0.311593\pi$
$68$	3.34112	0.405170
$69$	0.153561	0.0184866
$70$	12.2333	1.46216
$71$	−3.83171	−0.454740	−0.227370	−	0.973808i	$-0.573013\pi$
$71$	−3.83171	−0.454740	−0.227370	+	0.973808i	$0.573013\pi$
$72$	−2.97642	−0.350774
$73$	−14.2989	−1.67357	−0.836783	−	0.547535i	$-0.815566\pi$
$73$	−14.2989	−1.67357	−0.836783	+	0.547535i	$0.815566\pi$
$74$	5.25321	0.610673
$75$	−0.928694	−0.107236
$76$	6.46686	0.741800
$77$	−13.9493	−1.58967
$78$	−0.731375	−0.0828119
$79$	−10.9981	−1.23738	−0.618689	−	0.785636i	$-0.712336\pi$
$79$	−10.9981	−1.23738	−0.618689	+	0.785636i	$0.712336\pi$
$80$	−3.32381	−0.371613
$81$	8.78833	0.976481
$82$	2.92040	0.322505
$83$	10.3489	1.13594	0.567969	−	0.823050i	$-0.307729\pi$
$83$	10.3489	1.13594	0.567969	+	0.823050i	$0.307729\pi$
$84$	0.565183	0.0616665
$85$	−11.1053	−1.20453
$86$	11.8781	1.28085
$87$	0.153561	0.0164635
$88$	3.79005	0.404021
$89$	3.55410	0.376734	0.188367	−	0.982099i	$-0.439681\pi$
$89$	3.55410	0.376734	0.188367	+	0.982099i	$0.439681\pi$
$90$	9.89306	1.04282
$91$	−17.5294	−1.83758
$92$	−1.00000	−0.104257
$93$	1.22291	0.126810
$94$	12.7448	1.31453
$95$	−21.4946	−2.20530
$96$	−0.153561	−0.0156727
$97$	12.9528	1.31516	0.657578	−	0.753386i	$-0.271581\pi$
$97$	12.9528	1.31516	0.657578	+	0.753386i	$0.271581\pi$
$98$	6.54617	0.661263
$99$	−11.2808	−1.13376
$100$	6.04772	0.604772
$101$	3.04676	0.303164	0.151582	−	0.988445i	$-0.451563\pi$
$101$	3.04676	0.303164	0.151582	+	0.988445i	$0.451563\pi$
$102$	−0.513066	−0.0508011
$103$	−12.1329	−1.19549	−0.597746	−	0.801686i	$-0.703937\pi$
$103$	−12.1329	−1.19549	−0.597746	+	0.801686i	$0.703937\pi$
$104$	4.76277	0.467028
$105$	−1.87856	−0.183329
$106$	−1.82898	−0.177647
$107$	−0.179653	−0.0173677	−0.00868386	−	0.999962i	$-0.502764\pi$
$107$	−0.179653	−0.0173677	−0.00868386	+	0.999962i	$0.502764\pi$
$108$	0.917745	0.0883100
$109$	13.5834	1.30105	0.650526	−	0.759484i	$-0.274549\pi$
$109$	13.5834	1.30105	0.650526	+	0.759484i	$0.274549\pi$
$110$	−12.5974	−1.20112
$111$	−0.806688	−0.0765674
$112$	−3.68051	−0.347776
$113$	−19.6485	−1.84838	−0.924188	−	0.381937i	$-0.875257\pi$
$113$	−19.6485	−1.84838	−0.924188	+	0.381937i	$0.875257\pi$
$114$	−0.993057	−0.0930083
$115$	3.32381	0.309947
$116$	−1.00000	−0.0928477
$117$	−14.1760	−1.31057
$118$	−13.3206	−1.22626
$119$	−12.2970	−1.12727
$120$	0.510408	0.0465936
$121$	3.36448	0.305861
$122$	8.11991	0.735142
$123$	−0.448460	−0.0404363
$124$	−7.96366	−0.715158
$125$	−3.48244	−0.311479
$126$	10.9547	0.975926
$127$	−6.76011	−0.599863	−0.299931	−	0.953961i	$-0.596964\pi$
$127$	−6.76011	−0.599863	−0.299931	+	0.953961i	$0.596964\pi$
$128$	1.00000	0.0883883
$129$	−1.82401	−0.160595
$130$	−15.8305	−1.38843
$131$	13.6416	1.19187	0.595937	−	0.803031i	$-0.296781\pi$
$131$	13.6416	1.19187	0.595937	+	0.803031i	$0.296781\pi$
$132$	−0.582004	−0.0506569
$133$	−23.8014	−2.06384
$134$	9.13383	0.789043
$135$	−3.05041	−0.262538
$136$	3.34112	0.286499
$137$	14.4545	1.23494	0.617468	−	0.786596i	$-0.288159\pi$
$137$	14.4545	1.23494	0.617468	+	0.786596i	$0.288159\pi$
$138$	0.153561	0.0130720
$139$	15.1686	1.28658	0.643291	−	0.765622i	$-0.277569\pi$
$139$	15.1686	1.28658	0.643291	+	0.765622i	$0.277569\pi$
$140$	12.2333	1.03390
$141$	−1.95710	−0.164818
$142$	−3.83171	−0.321550
$143$	18.0511	1.50951
$144$	−2.97642	−0.248035
$145$	3.32381	0.276028
$146$	−14.2989	−1.18339
$147$	−1.00524	−0.0829104
$148$	5.25321	0.431811
$149$	−13.9262	−1.14088	−0.570438	−	0.821340i	$-0.693227\pi$
$149$	−13.9262	−1.14088	−0.570438	+	0.821340i	$0.693227\pi$
$150$	−0.928694	−0.0758276
$151$	12.3548	1.00542	0.502708	−	0.864456i	$-0.332337\pi$
$151$	12.3548	1.00542	0.502708	+	0.864456i	$0.332337\pi$
$152$	6.46686	0.524532
$153$	−9.94457	−0.803971
$154$	−13.9493	−1.12407
$155$	26.4697	2.12610
$156$	−0.731375	−0.0585569
$157$	5.83169	0.465420	0.232710	−	0.972546i	$-0.425241\pi$
$157$	5.83169	0.465420	0.232710	+	0.972546i	$0.425241\pi$
$158$	−10.9981	−0.874958
$159$	0.280860	0.0222737
$160$	−3.32381	−0.262770
$161$	3.68051	0.290065
$162$	8.78833	0.690476
$163$	−15.1244	−1.18464	−0.592319	−	0.805704i	$-0.701787\pi$
$163$	−15.1244	−1.18464	−0.592319	+	0.805704i	$0.701787\pi$
$164$	2.92040	0.228045
$165$	1.93447	0.150598
$166$	10.3489	0.803229
$167$	−2.19370	−0.169754	−0.0848768	−	0.996391i	$-0.527050\pi$
$167$	−2.19370	−0.169754	−0.0848768	+	0.996391i	$0.527050\pi$
$168$	0.565183	0.0436048
$169$	9.68395	0.744919
$170$	−11.1053	−0.851734
$171$	−19.2481	−1.47194
$172$	11.8781	0.905696
$173$	−0.0210255	−0.00159854	−0.000799269	−	1.00000i	$-0.500254\pi$
$173$	−0.0210255	−0.00159854	−0.000799269		1.00000i	$0.500254\pi$
$174$	0.153561	0.0116414
$175$	−22.2587	−1.68260
$176$	3.79005	0.285686
$177$	2.04553	0.153751
$178$	3.55410	0.266391
$179$	16.9121	1.26407	0.632035	−	0.774940i	$-0.282220\pi$
$179$	16.9121	1.26407	0.632035	+	0.774940i	$0.282220\pi$
$180$	9.89306	0.737385
$181$	1.19011	0.0884605	0.0442303	−	0.999021i	$-0.485916\pi$
$181$	1.19011	0.0884605	0.0442303	+	0.999021i	$0.485916\pi$
$182$	−17.5294	−1.29937
$183$	−1.24690	−0.0921736
$184$	−1.00000	−0.0737210
$185$	−17.4607	−1.28373
$186$	1.22291	0.0896679
$187$	12.6630	0.926011
$188$	12.7448	0.929510
$189$	−3.37777	−0.245697
$190$	−21.4946	−1.55938
$191$	1.94022	0.140389	0.0701946	−	0.997533i	$-0.477638\pi$
$191$	1.94022	0.140389	0.0701946	+	0.997533i	$0.477638\pi$
$192$	−0.153561	−0.0110823
$193$	−16.4817	−1.18638	−0.593188	−	0.805064i	$-0.702131\pi$
$193$	−16.4817	−1.18638	−0.593188	+	0.805064i	$0.702131\pi$
$194$	12.9528	0.929956
$195$	2.43095	0.174084
$196$	6.54617	0.467583
$197$	−16.0611	−1.14431	−0.572154	−	0.820146i	$-0.693892\pi$
$197$	−16.0611	−1.14431	−0.572154	+	0.820146i	$0.693892\pi$
$198$	−11.2808	−0.801690
$199$	12.3706	0.876931	0.438465	−	0.898748i	$-0.355522\pi$
$199$	12.3706	0.876931	0.438465	+	0.898748i	$0.355522\pi$
$200$	6.04772	0.427639
$201$	−1.40260	−0.0989317
$202$	3.04676	0.214369
$203$	3.68051	0.258321
$204$	−0.513066	−0.0359218
$205$	−9.70687	−0.677957
$206$	−12.1329	−0.845340
$207$	2.97642	0.206875
$208$	4.76277	0.330238
$209$	24.5097	1.69537
$210$	−1.87856	−0.129633
$211$	19.2843	1.32759	0.663793	−	0.747916i	$-0.268946\pi$
$211$	19.2843	1.32759	0.663793	+	0.747916i	$0.268946\pi$
$212$	−1.82898	−0.125615
$213$	0.588401	0.0403166
$214$	−0.179653	−0.0122808
$215$	−39.4806	−2.69255
$216$	0.917745	0.0624446
$217$	29.3103	1.98972
$218$	13.5834	0.919983
$219$	2.19576	0.148376
$220$	−12.5974	−0.849317
$221$	15.9130	1.07042
$222$	−0.806688	−0.0541413
$223$	−0.721402	−0.0483086	−0.0241543	−	0.999708i	$-0.507689\pi$
$223$	−0.721402	−0.0483086	−0.0241543	+	0.999708i	$0.507689\pi$
$224$	−3.68051	−0.245915
$225$	−18.0006	−1.20004
$226$	−19.6485	−1.30700
$227$	−4.02683	−0.267270	−0.133635	−	0.991031i	$-0.542665\pi$
$227$	−4.02683	−0.267270	−0.133635	+	0.991031i	$0.542665\pi$
$228$	−0.993057	−0.0657668
$229$	6.91497	0.456954	0.228477	−	0.973549i	$-0.426625\pi$
$229$	6.91497	0.456954	0.228477	+	0.973549i	$0.426625\pi$
$230$	3.32381	0.219166
$231$	2.14207	0.140938
$232$	−1.00000	−0.0656532
$233$	−10.7646	−0.705212	−0.352606	−	0.935772i	$-0.614704\pi$
$233$	−10.7646	−0.705212	−0.352606	+	0.935772i	$0.614704\pi$
$234$	−14.1760	−0.926714
$235$	−42.3613	−2.76335
$236$	−13.3206	−0.867099
$237$	1.68887	0.109704
$238$	−12.2970	−0.797098
$239$	11.3832	0.736317	0.368159	−	0.929763i	$-0.379988\pi$
$239$	11.3832	0.736317	0.368159	+	0.929763i	$0.379988\pi$
$240$	0.510408	0.0329467
$241$	12.9513	0.834267	0.417133	−	0.908845i	$-0.363035\pi$
$241$	12.9513	0.834267	0.417133	+	0.908845i	$0.363035\pi$
$242$	3.36448	0.216277
$243$	−4.10278	−0.263193
$244$	8.11991	0.519824
$245$	−21.7582	−1.39008
$246$	−0.448460	−0.0285928
$247$	30.8001	1.95977
$248$	−7.96366	−0.505693
$249$	−1.58918	−0.100710
$250$	−3.48244	−0.220249
$251$	−1.01351	−0.0639725	−0.0319862	−	0.999488i	$-0.510183\pi$
$251$	−1.01351	−0.0639725	−0.0319862	+	0.999488i	$0.510183\pi$
$252$	10.9547	0.690084
$253$	−3.79005	−0.238278
$254$	−6.76011	−0.424167
$255$	1.70533	0.106792
$256$	1.00000	0.0625000
$257$	−2.93468	−0.183060	−0.0915302	−	0.995802i	$-0.529176\pi$
$257$	−2.93468	−0.183060	−0.0915302	+	0.995802i	$0.529176\pi$
$258$	−1.82401	−0.113558
$259$	−19.3345	−1.20139
$260$	−15.8305	−0.981769
$261$	2.97642	0.184236
$262$	13.6416	0.842782
$263$	−22.9519	−1.41527	−0.707637	−	0.706576i	$-0.750239\pi$
$263$	−22.9519	−1.41527	−0.707637	+	0.706576i	$0.750239\pi$
$264$	−0.582004	−0.0358198
$265$	6.07920	0.373442
$266$	−23.8014	−1.45935
$267$	−0.545771	−0.0334007
$268$	9.13383	0.557937
$269$	5.13971	0.313374	0.156687	−	0.987648i	$-0.449919\pi$
$269$	5.13971	0.313374	0.156687	+	0.987648i	$0.449919\pi$
$270$	−3.05041	−0.185642
$271$	17.9844	1.09247	0.546237	−	0.837631i	$-0.316060\pi$
$271$	17.9844	1.09247	0.546237	+	0.837631i	$0.316060\pi$
$272$	3.34112	0.202585
$273$	2.69183	0.162917
$274$	14.4545	0.873231
$275$	22.9212	1.38220
$276$	0.153561	0.00924328
$277$	7.11200	0.427319	0.213659	−	0.976908i	$-0.431462\pi$
$277$	7.11200	0.427319	0.213659	+	0.976908i	$0.431462\pi$
$278$	15.1686	0.909751
$279$	23.7032	1.41907
$280$	12.2333	0.731081
$281$	−11.3336	−0.676103	−0.338052	−	0.941128i	$-0.609768\pi$
$281$	−11.3336	−0.676103	−0.338052	+	0.941128i	$0.609768\pi$
$282$	−1.95710	−0.116544
$283$	17.0841	1.01555	0.507773	−	0.861491i	$-0.330469\pi$
$283$	17.0841	1.01555	0.507773	+	0.861491i	$0.330469\pi$
$284$	−3.83171	−0.227370
$285$	3.30074	0.195519
$286$	18.0511	1.06739
$287$	−10.7486	−0.634468
$288$	−2.97642	−0.175387
$289$	−5.83691	−0.343348
$290$	3.32381	0.195181
$291$	−1.98904	−0.116600
$292$	−14.2989	−0.836783
$293$	−12.9898	−0.758872	−0.379436	−	0.925218i	$-0.623882\pi$
$293$	−12.9898	−0.758872	−0.379436	+	0.925218i	$0.623882\pi$
$294$	−1.00524	−0.0586265
$295$	44.2752	2.57780
$296$	5.25321	0.305337
$297$	3.47830	0.201831
$298$	−13.9262	−0.806722
$299$	−4.76277	−0.275438
$300$	−0.928694	−0.0536182
$301$	−43.7175	−2.51983
$302$	12.3548	0.710937
$303$	−0.467864	−0.0268781
$304$	6.46686	0.370900
$305$	−26.9890	−1.54539
$306$	−9.94457	−0.568494
$307$	−0.0639224	−0.00364824	−0.00182412	−	0.999998i	$-0.500581\pi$
$307$	−0.0639224	−0.00364824	−0.00182412	+	0.999998i	$0.500581\pi$
$308$	−13.9493	−0.794836
$309$	1.86314	0.105990
$310$	26.4697	1.50338
$311$	4.74977	0.269335	0.134667	−	0.990891i	$-0.457003\pi$
$311$	4.74977	0.269335	0.134667	+	0.990891i	$0.457003\pi$
$312$	−0.731375	−0.0414060
$313$	6.13966	0.347034	0.173517	−	0.984831i	$-0.444487\pi$
$313$	6.13966	0.347034	0.173517	+	0.984831i	$0.444487\pi$
$314$	5.83169	0.329101
$315$	−36.4115	−2.05156
$316$	−10.9981	−0.618689
$317$	−32.5622	−1.82888	−0.914438	−	0.404727i	$-0.867367\pi$
$317$	−32.5622	−1.82888	−0.914438	+	0.404727i	$0.867367\pi$
$318$	0.280860	0.0157499
$319$	−3.79005	−0.212202
$320$	−3.32381	−0.185807
$321$	0.0275877	0.00153980
$322$	3.68051	0.205107
$323$	21.6066	1.20222
$324$	8.78833	0.488240
$325$	28.8039	1.59775
$326$	−15.1244	−0.837665
$327$	−2.08588	−0.115349
$328$	2.92040	0.161252
$329$	−46.9074	−2.58609
$330$	1.93447	0.106489
$331$	23.3467	1.28325	0.641625	−	0.767019i	$-0.278261\pi$
$331$	23.3467	1.28325	0.641625	+	0.767019i	$0.278261\pi$
$332$	10.3489	0.567969
$333$	−15.6357	−0.856834
$334$	−2.19370	−0.120034
$335$	−30.3591	−1.65870
$336$	0.565183	0.0308333
$337$	−3.93242	−0.214212	−0.107106	−	0.994248i	$-0.534158\pi$
$337$	−3.93242	−0.214212	−0.107106	+	0.994248i	$0.534158\pi$
$338$	9.68395	0.526738
$339$	3.01724	0.163874
$340$	−11.1053	−0.602267
$341$	−30.1827	−1.63448
$342$	−19.2481	−1.04082
$343$	1.67034	0.0901897
$344$	11.8781	0.640424
$345$	−0.510408	−0.0274794
$346$	−0.0210255	−0.00113034
$347$	26.5041	1.42281	0.711407	−	0.702780i	$-0.248058\pi$
$347$	26.5041	1.42281	0.711407	+	0.702780i	$0.248058\pi$
$348$	0.153561	0.00823173
$349$	15.6322	0.836772	0.418386	−	0.908269i	$-0.362596\pi$
$349$	15.6322	0.836772	0.418386	+	0.908269i	$0.362596\pi$
$350$	−22.2587	−1.18978
$351$	4.37100	0.233307
$352$	3.79005	0.202010
$353$	−32.2076	−1.71424	−0.857119	−	0.515118i	$-0.827748\pi$
$353$	−32.2076	−1.71424	−0.857119	+	0.515118i	$0.827748\pi$
$354$	2.04553	0.108719
$355$	12.7359	0.675950
$356$	3.55410	0.188367
$357$	1.88834	0.0999418
$358$	16.9121	0.893833
$359$	−18.9741	−1.00141	−0.500707	−	0.865617i	$-0.666927\pi$
$359$	−18.9741	−1.00141	−0.500707	+	0.865617i	$0.666927\pi$
$360$	9.89306	0.521410
$361$	22.8203	1.20107
$362$	1.19011	0.0625510
$363$	−0.516652	−0.0271172
$364$	−17.5294	−0.918791
$365$	47.5270	2.48768
$366$	−1.24690	−0.0651765
$367$	36.4422	1.90227	0.951133	−	0.308781i	$-0.0999210\pi$
$367$	36.4422	1.90227	0.951133	+	0.308781i	$0.0999210\pi$
$368$	−1.00000	−0.0521286
$369$	−8.69234	−0.452505
$370$	−17.4607	−0.907737
$371$	6.73160	0.349487
$372$	1.22291	0.0634048
$373$	3.59585	0.186186	0.0930931	−	0.995657i	$-0.470325\pi$
$373$	3.59585	0.186186	0.0930931	+	0.995657i	$0.470325\pi$
$374$	12.6630	0.654789
$375$	0.534767	0.0276152
$376$	12.7448	0.657263
$377$	−4.76277	−0.245295
$378$	−3.37777	−0.173734
$379$	−31.2247	−1.60390	−0.801952	−	0.597388i	$-0.796205\pi$
$379$	−31.2247	−1.60390	−0.801952	+	0.597388i	$0.796205\pi$
$380$	−21.4946	−1.10265
$381$	1.03809	0.0531829
$382$	1.94022	0.0992702
$383$	6.86962	0.351021	0.175511	−	0.984478i	$-0.443842\pi$
$383$	6.86962	0.351021	0.175511	+	0.984478i	$0.443842\pi$
$384$	−0.153561	−0.00783637
$385$	46.3649	2.36298
$386$	−16.4817	−0.838895
$387$	−35.3542	−1.79715
$388$	12.9528	0.657578
$389$	4.86917	0.246877	0.123438	−	0.992352i	$-0.460608\pi$
$389$	4.86917	0.246877	0.123438	+	0.992352i	$0.460608\pi$
$390$	2.43095	0.123096
$391$	−3.34112	−0.168968
$392$	6.54617	0.330631
$393$	−2.09482	−0.105670
$394$	−16.0611	−0.809148
$395$	36.5555	1.83930
$396$	−11.2808	−0.566880
$397$	−27.7701	−1.39374	−0.696870	−	0.717197i	$-0.745425\pi$
$397$	−27.7701	−1.39374	−0.696870	+	0.717197i	$0.745425\pi$
$398$	12.3706	0.620084
$399$	3.65496	0.182977
$400$	6.04772	0.302386
$401$	12.6095	0.629688	0.314844	−	0.949143i	$-0.398048\pi$
$401$	12.6095	0.629688	0.314844	+	0.949143i	$0.398048\pi$
$402$	−1.40260	−0.0699553
$403$	−37.9291	−1.88938
$404$	3.04676	0.151582
$405$	−29.2107	−1.45149
$406$	3.68051	0.182661
$407$	19.9099	0.986898
$408$	−0.513066	−0.0254005
$409$	17.5417	0.867384	0.433692	−	0.901061i	$-0.357211\pi$
$409$	17.5417	0.867384	0.433692	+	0.901061i	$0.357211\pi$
$410$	−9.70687	−0.479388
$411$	−2.21965	−0.109487
$412$	−12.1329	−0.597746
$413$	49.0267	2.41245
$414$	2.97642	0.146283
$415$	−34.3977	−1.68852
$416$	4.76277	0.233514
$417$	−2.32930	−0.114066
$418$	24.5097	1.19881
$419$	5.63168	0.275125	0.137563	−	0.990493i	$-0.456073\pi$
$419$	5.63168	0.275125	0.137563	+	0.990493i	$0.456073\pi$
$420$	−1.87856	−0.0916644
$421$	27.9853	1.36392	0.681960	−	0.731389i	$-0.261128\pi$
$421$	27.9853	1.36392	0.681960	+	0.731389i	$0.261128\pi$
$422$	19.2843	0.938745
$423$	−37.9339	−1.84441
$424$	−1.82898	−0.0888233
$425$	20.2062	0.980144
$426$	0.588401	0.0285081
$427$	−29.8854	−1.44626
$428$	−0.179653	−0.00868386
$429$	−2.77195	−0.133831
$430$	−39.4806	−1.90392
$431$	1.93279	0.0930991	0.0465496	−	0.998916i	$-0.485177\pi$
$431$	1.93279	0.0930991	0.0465496	+	0.998916i	$0.485177\pi$
$432$	0.917745	0.0441550
$433$	−6.58356	−0.316386	−0.158193	−	0.987408i	$-0.550567\pi$
$433$	−6.58356	−0.316386	−0.158193	+	0.987408i	$0.550567\pi$
$434$	29.3103	1.40694
$435$	−0.510408	−0.0244722
$436$	13.5834	0.650526
$437$	−6.46686	−0.309352
$438$	2.19576	0.104917
$439$	−25.6888	−1.22606	−0.613029	−	0.790060i	$-0.710049\pi$
$439$	−25.6888	−1.22606	−0.613029	+	0.790060i	$0.710049\pi$
$440$	−12.5974	−0.600558
$441$	−19.4841	−0.927816
$442$	15.9130	0.756903
$443$	−21.3438	−1.01408	−0.507038	−	0.861923i	$-0.669260\pi$
$443$	−21.3438	−1.01408	−0.507038	+	0.861923i	$0.669260\pi$
$444$	−0.806688	−0.0382837
$445$	−11.8132	−0.559998
$446$	−0.721402	−0.0341594
$447$	2.13852	0.101148
$448$	−3.68051	−0.173888
$449$	−9.94264	−0.469222	−0.234611	−	0.972089i	$-0.575382\pi$
$449$	−9.94264	−0.469222	−0.234611	+	0.972089i	$0.575382\pi$
$450$	−18.0006	−0.848555
$451$	11.0685	0.521194
$452$	−19.6485	−0.924188
$453$	−1.89721	−0.0891387
$454$	−4.02683	−0.188988
$455$	58.2645	2.73148
$456$	−0.993057	−0.0465042
$457$	−36.8227	−1.72249	−0.861247	−	0.508187i	$-0.830316\pi$
$457$	−36.8227	−1.72249	−0.861247	+	0.508187i	$0.830316\pi$
$458$	6.91497	0.323116
$459$	3.06630	0.143122
$460$	3.32381	0.154974
$461$	−28.4810	−1.32649	−0.663246	−	0.748401i	$-0.730822\pi$
$461$	−28.4810	−1.32649	−0.663246	+	0.748401i	$0.730822\pi$
$462$	2.14207	0.0996582
$463$	34.9277	1.62323	0.811613	−	0.584195i	$-0.198590\pi$
$463$	34.9277	1.62323	0.811613	+	0.584195i	$0.198590\pi$
$464$	−1.00000	−0.0464238
$465$	−4.06471	−0.188497
$466$	−10.7646	−0.498660
$467$	−32.7985	−1.51773	−0.758866	−	0.651247i	$-0.774246\pi$
$467$	−32.7985	−1.51773	−0.758866	+	0.651247i	$0.774246\pi$
$468$	−14.1760	−0.655285
$469$	−33.6172	−1.55230
$470$	−42.3613	−1.95398
$471$	−0.895520	−0.0412634
$472$	−13.3206	−0.613131
$473$	45.0186	2.06996
$474$	1.68887	0.0775724
$475$	39.1098	1.79448
$476$	−12.2970	−0.563634
$477$	5.44382	0.249255
$478$	11.3832	0.520655
$479$	−13.8848	−0.634415	−0.317207	−	0.948356i	$-0.602745\pi$
$479$	−13.8848	−0.634415	−0.317207	+	0.948356i	$0.602745\pi$
$480$	0.510408	0.0232968
$481$	25.0198	1.14081
$482$	12.9513	0.589916
$483$	−0.565183	−0.0257167
$484$	3.36448	0.152931
$485$	−43.0526	−1.95492
$486$	−4.10278	−0.186106
$487$	−5.84024	−0.264646	−0.132323	−	0.991207i	$-0.542244\pi$
$487$	−5.84024	−0.264646	−0.132323	+	0.991207i	$0.542244\pi$
$488$	8.11991	0.367571
$489$	2.32252	0.105028
$490$	−21.7582	−0.982937
$491$	−0.890073	−0.0401684	−0.0200842	−	0.999798i	$-0.506393\pi$
$491$	−0.890073	−0.0401684	−0.0200842	+	0.999798i	$0.506393\pi$
$492$	−0.448460	−0.0202181
$493$	−3.34112	−0.150477
$494$	30.8001	1.38576
$495$	37.4952	1.68528
$496$	−7.96366	−0.357579
$497$	14.1026	0.632590
$498$	−1.58918	−0.0712131
$499$	17.4571	0.781488	0.390744	−	0.920499i	$-0.372218\pi$
$499$	17.4571	0.781488	0.390744	+	0.920499i	$0.372218\pi$
$500$	−3.48244	−0.155739
$501$	0.336867	0.0150501
$502$	−1.01351	−0.0452354
$503$	28.1825	1.25660	0.628299	−	0.777972i	$-0.283752\pi$
$503$	28.1825	1.25660	0.628299	+	0.777972i	$0.283752\pi$
$504$	10.9547	0.487963
$505$	−10.1269	−0.450639
$506$	−3.79005	−0.168488
$507$	−1.48708	−0.0660434
$508$	−6.76011	−0.299931
$509$	33.2079	1.47192	0.735958	−	0.677027i	$-0.236732\pi$
$509$	33.2079	1.47192	0.735958	+	0.677027i	$0.236732\pi$
$510$	1.70533	0.0755134
$511$	52.6274	2.32810
$512$	1.00000	0.0441942
$513$	5.93493	0.262033
$514$	−2.93468	−0.129443
$515$	40.3275	1.77704
$516$	−1.82401	−0.0802976
$517$	48.3034	2.12438
$518$	−19.3345	−0.849509
$519$	0.00322869	0.000141724 0
$520$	−15.8305	−0.694215
$521$	16.9604	0.743047	0.371524	−	0.928424i	$-0.378836\pi$
$521$	16.9604	0.743047	0.371524	+	0.928424i	$0.378836\pi$
$522$	2.97642	0.130274
$523$	−22.5309	−0.985208	−0.492604	−	0.870254i	$-0.663955\pi$
$523$	−22.5309	−0.985208	−0.492604	+	0.870254i	$0.663955\pi$
$524$	13.6416	0.595937
$525$	3.41807	0.149177
$526$	−22.9519	−1.00075
$527$	−26.6075	−1.15904
$528$	−0.582004	−0.0253285
$529$	1.00000	0.0434783
$530$	6.07920	0.264063
$531$	39.6478	1.72057
$532$	−23.8014	−1.03192
$533$	13.9092	0.602474
$534$	−0.545771	−0.0236178
$535$	0.597133	0.0258163
$536$	9.13383	0.394521
$537$	−2.59704	−0.112071
$538$	5.13971	0.221589
$539$	24.8103	1.06866
$540$	−3.05041	−0.131269
$541$	−39.4315	−1.69529	−0.847647	−	0.530561i	$-0.821981\pi$
$541$	−39.4315	−1.69529	−0.847647	+	0.530561i	$0.821981\pi$
$542$	17.9844	0.772496
$543$	−0.182755	−0.00784277
$544$	3.34112	0.143249
$545$	−45.1486	−1.93395
$546$	2.69183	0.115200
$547$	30.1010	1.28702	0.643512	−	0.765436i	$-0.277477\pi$
$547$	30.1010	1.28702	0.643512	+	0.765436i	$0.277477\pi$
$548$	14.4545	0.617468
$549$	−24.1682	−1.03148
$550$	22.9212	0.977362
$551$	−6.46686	−0.275497
$552$	0.153561	0.00653599
$553$	40.4785	1.72132
$554$	7.11200	0.302160
$555$	2.68128	0.113814
$556$	15.1686	0.643291
$557$	−36.8338	−1.56070	−0.780349	−	0.625344i	$-0.784959\pi$
$557$	−36.8338	−1.56070	−0.780349	+	0.625344i	$0.784959\pi$
$558$	23.7032	1.00344
$559$	56.5726	2.39277
$560$	12.2333	0.516952
$561$	−1.94454	−0.0820987
$562$	−11.3336	−0.478077
$563$	20.9055	0.881061	0.440531	−	0.897738i	$-0.354790\pi$
$563$	20.9055	0.881061	0.440531	+	0.897738i	$0.354790\pi$
$564$	−1.95710	−0.0824089
$565$	65.3080	2.74753
$566$	17.0841	0.718099
$567$	−32.3455	−1.35839
$568$	−3.83171	−0.160775
$569$	−21.8688	−0.916790	−0.458395	−	0.888749i	$-0.651575\pi$
$569$	−21.8688	−0.916790	−0.458395	+	0.888749i	$0.651575\pi$
$570$	3.30074	0.138253
$571$	−14.3165	−0.599127	−0.299563	−	0.954076i	$-0.596841\pi$
$571$	−14.3165	−0.599127	−0.299563	+	0.954076i	$0.596841\pi$
$572$	18.0511	0.754755
$573$	−0.297942	−0.0124467
$574$	−10.7486	−0.448637
$575$	−6.04772	−0.252208
$576$	−2.97642	−0.124017
$577$	−0.0422136	−0.00175738	−0.000878688	−	1.00000i	$-0.500280\pi$
$577$	−0.0422136	−0.00175738	−0.000878688		1.00000i	$0.500280\pi$
$578$	−5.83691	−0.242784
$579$	2.53094	0.105182
$580$	3.32381	0.138014
$581$	−38.0892	−1.58021
$582$	−1.98904	−0.0824485
$583$	−6.93194	−0.287092
$584$	−14.2989	−0.591695
$585$	47.1183	1.94810
$586$	−12.9898	−0.536604
$587$	−27.8498	−1.14948	−0.574742	−	0.818335i	$-0.694898\pi$
$587$	−27.8498	−1.14948	−0.574742	+	0.818335i	$0.694898\pi$
$588$	−1.00524	−0.0414552
$589$	−51.4999	−2.12202
$590$	44.2752	1.82278
$591$	2.46636	0.101453
$592$	5.25321	0.215906
$593$	−11.4931	−0.471965	−0.235982	−	0.971757i	$-0.575831\pi$
$593$	−11.4931	−0.471965	−0.235982	+	0.971757i	$0.575831\pi$
$594$	3.47830	0.142716
$595$	40.8730	1.67563
$596$	−13.9262	−0.570438
$597$	−1.89965	−0.0777473
$598$	−4.76277	−0.194764
$599$	21.6975	0.886537	0.443268	−	0.896389i	$-0.353819\pi$
$599$	21.6975	0.886537	0.443268	+	0.896389i	$0.353819\pi$
$600$	−0.928694	−0.0379138
$601$	17.5454	0.715691	0.357845	−	0.933781i	$-0.383511\pi$
$601$	17.5454	0.715691	0.357845	+	0.933781i	$0.383511\pi$
$602$	−43.7175	−1.78179
$603$	−27.1861	−1.10710
$604$	12.3548	0.502708
$605$	−11.1829	−0.454649
$606$	−0.467864	−0.0190057
$607$	−30.3344	−1.23124	−0.615618	−	0.788045i	$-0.711093\pi$
$607$	−30.3344	−1.23124	−0.615618	+	0.788045i	$0.711093\pi$
$608$	6.46686	0.262266
$609$	−0.565183	−0.0229024
$610$	−26.9890	−1.09275
$611$	60.7005	2.45568
$612$	−9.94457	−0.401986
$613$	−20.6585	−0.834391	−0.417195	−	0.908817i	$-0.636987\pi$
$613$	−20.6585	−0.834391	−0.417195	+	0.908817i	$0.636987\pi$
$614$	−0.0639224	−0.00257970
$615$	1.49060	0.0601066
$616$	−13.9493	−0.562034
$617$	26.2894	1.05837	0.529185	−	0.848506i	$-0.322498\pi$
$617$	26.2894	1.05837	0.529185	+	0.848506i	$0.322498\pi$
$618$	1.86314	0.0749466
$619$	−15.5629	−0.625526	−0.312763	−	0.949831i	$-0.601255\pi$
$619$	−15.5629	−0.625526	−0.312763	+	0.949831i	$0.601255\pi$
$620$	26.4697	1.06305
$621$	−0.917745	−0.0368278
$622$	4.74977	0.190448
$623$	−13.0809	−0.524076
$624$	−0.731375	−0.0292784
$625$	−18.6636	−0.746546
$626$	6.13966	0.245390
$627$	−3.76374	−0.150309
$628$	5.83169	0.232710
$629$	17.5516	0.699828
$630$	−36.4115	−1.45067
$631$	30.5328	1.21549	0.607747	−	0.794131i	$-0.292074\pi$
$631$	30.5328	1.21549	0.607747	+	0.794131i	$0.292074\pi$
$632$	−10.9981	−0.437479
$633$	−2.96132	−0.117702
$634$	−32.5622	−1.29321
$635$	22.4693	0.891668
$636$	0.280860	0.0111368
$637$	31.1779	1.23531
$638$	−3.79005	−0.150049
$639$	11.4048	0.451166
$640$	−3.32381	−0.131385
$641$	−23.4658	−0.926842	−0.463421	−	0.886138i	$-0.653378\pi$
$641$	−23.4658	−0.926842	−0.463421	+	0.886138i	$0.653378\pi$
$642$	0.0275877	0.00108880
$643$	41.1081	1.62115	0.810573	−	0.585638i	$-0.199156\pi$
$643$	41.1081	1.62115	0.810573	+	0.585638i	$0.199156\pi$
$644$	3.68051	0.145032
$645$	6.06267	0.238717
$646$	21.6066	0.850099
$647$	−13.7646	−0.541144	−0.270572	−	0.962700i	$-0.587213\pi$
$647$	−13.7646	−0.541144	−0.270572	+	0.962700i	$0.587213\pi$
$648$	8.78833	0.345238
$649$	−50.4858	−1.98174
$650$	28.8039	1.12978
$651$	−4.50092	−0.176405
$652$	−15.1244	−0.592319
$653$	−49.3565	−1.93147	−0.965734	−	0.259534i	$-0.916431\pi$
$653$	−49.3565	−1.93147	−0.965734	+	0.259534i	$0.916431\pi$
$654$	−2.08588	−0.0815642
$655$	−45.3422	−1.77167
$656$	2.92040	0.114023
$657$	42.5597	1.66041
$658$	−46.9074	−1.82864
$659$	−18.7854	−0.731773	−0.365887	−	0.930659i	$-0.619234\pi$
$659$	−18.7854	−0.731773	−0.365887	+	0.930659i	$0.619234\pi$
$660$	1.93447	0.0752992
$661$	11.5786	0.450355	0.225178	−	0.974318i	$-0.427704\pi$
$661$	11.5786	0.450355	0.225178	+	0.974318i	$0.427704\pi$
$662$	23.3467	0.907395
$663$	−2.44361	−0.0949020
$664$	10.3489	0.401615
$665$	79.1112	3.06780
$666$	−15.6357	−0.605873
$667$	1.00000	0.0387202
$668$	−2.19370	−0.0848768
$669$	0.110779	0.00428297
$670$	−30.3591	−1.17288
$671$	30.7748	1.18805
$672$	0.565183	0.0218024
$673$	−34.9754	−1.34820	−0.674101	−	0.738639i	$-0.735469\pi$
$673$	−34.9754	−1.34820	−0.674101	+	0.738639i	$0.735469\pi$
$674$	−3.93242	−0.151471
$675$	5.55027	0.213630
$676$	9.68395	0.372460
$677$	−1.98303	−0.0762142	−0.0381071	−	0.999274i	$-0.512133\pi$
$677$	−1.98303	−0.0762142	−0.0381071	+	0.999274i	$0.512133\pi$
$678$	3.01724	0.115877
$679$	−47.6729	−1.82952
$680$	−11.1053	−0.425867
$681$	0.618364	0.0236958
$682$	−30.1827	−1.15575
$683$	33.1474	1.26835	0.634175	−	0.773189i	$-0.281340\pi$
$683$	33.1474	1.26835	0.634175	+	0.773189i	$0.281340\pi$
$684$	−19.2481	−0.735969
$685$	−48.0442	−1.83567
$686$	1.67034	0.0637737
$687$	−1.06187	−0.0405129
$688$	11.8781	0.452848
$689$	−8.71102	−0.331864
$690$	−0.510408	−0.0194309
$691$	34.8012	1.32390	0.661950	−	0.749548i	$-0.269729\pi$
$691$	34.8012	1.32390	0.661950	+	0.749548i	$0.269729\pi$
$692$	−0.0210255	−0.000799269 0
$693$	41.5190	1.57718
$694$	26.5041	1.00608
$695$	−50.4175	−1.91244
$696$	0.153561	0.00582071
$697$	9.75742	0.369589
$698$	15.6322	0.591687
$699$	1.65302	0.0625230
$700$	−22.2587	−0.841301
$701$	−12.4652	−0.470805	−0.235403	−	0.971898i	$-0.575641\pi$
$701$	−12.4652	−0.470805	−0.235403	+	0.971898i	$0.575641\pi$
$702$	4.37100	0.164973
$703$	33.9718	1.28127
$704$	3.79005	0.142843
$705$	6.50504	0.244994
$706$	−32.2076	−1.21215
$707$	−11.2136	−0.421732
$708$	2.04553	0.0768756
$709$	−18.2980	−0.687196	−0.343598	−	0.939117i	$-0.611646\pi$
$709$	−18.2980	−0.687196	−0.343598	+	0.939117i	$0.611646\pi$
$710$	12.7359	0.477969
$711$	32.7348	1.22765
$712$	3.55410	0.133196
$713$	7.96366	0.298241
$714$	1.88834	0.0706695
$715$	−59.9985	−2.24382
$716$	16.9121	0.632035
$717$	−1.74801	−0.0652808
$718$	−18.9741	−0.708107
$719$	2.05230	0.0765377	0.0382688	−	0.999267i	$-0.487816\pi$
$719$	2.05230	0.0765377	0.0382688	+	0.999267i	$0.487816\pi$
$720$	9.89306	0.368692
$721$	44.6553	1.66305
$722$	22.8203	0.849283
$723$	−1.98881	−0.0739648
$724$	1.19011	0.0442303
$725$	−6.04772	−0.224607
$726$	−0.516652	−0.0191748
$727$	1.73331	0.0642850	0.0321425	−	0.999483i	$-0.489767\pi$
$727$	1.73331	0.0642850	0.0321425	+	0.999483i	$0.489767\pi$
$728$	−17.5294	−0.649684
$729$	−25.7350	−0.953147
$730$	47.5270	1.75905
$731$	39.6862	1.46785
$732$	−1.24690	−0.0460868
$733$	−9.33965	−0.344968	−0.172484	−	0.985012i	$-0.555179\pi$
$733$	−9.33965	−0.344968	−0.172484	+	0.985012i	$0.555179\pi$
$734$	36.4422	1.34511
$735$	3.34121	0.123243
$736$	−1.00000	−0.0368605
$737$	34.6177	1.27516
$738$	−8.69234	−0.319970
$739$	−37.0230	−1.36191	−0.680956	−	0.732324i	$-0.738436\pi$
$739$	−37.0230	−1.36191	−0.680956	+	0.732324i	$0.738436\pi$
$740$	−17.4607	−0.641867
$741$	−4.72970	−0.173750
$742$	6.73160	0.247125
$743$	−7.99656	−0.293365	−0.146683	−	0.989184i	$-0.546860\pi$
$743$	−7.99656	−0.293365	−0.146683	+	0.989184i	$0.546860\pi$
$744$	1.22291	0.0448340
$745$	46.2880	1.69586
$746$	3.59585	0.131654
$747$	−30.8026	−1.12701
$748$	12.6630	0.463006
$749$	0.661216	0.0241603
$750$	0.534767	0.0195269
$751$	−13.7945	−0.503369	−0.251685	−	0.967809i	$-0.580985\pi$
$751$	−13.7945	−0.503369	−0.251685	+	0.967809i	$0.580985\pi$
$752$	12.7448	0.464755
$753$	0.155636	0.00567170
$754$	−4.76277	−0.173450
$755$	−41.0649	−1.49451
$756$	−3.37777	−0.122848
$757$	−29.7815	−1.08243	−0.541213	−	0.840885i	$-0.682035\pi$
$757$	−29.7815	−1.08243	−0.541213	+	0.840885i	$0.682035\pi$
$758$	−31.2247	−1.13413
$759$	0.582004	0.0211254
$760$	−21.4946	−0.779692
$761$	7.08875	0.256967	0.128483	−	0.991712i	$-0.458989\pi$
$761$	7.08875	0.256967	0.128483	+	0.991712i	$0.458989\pi$
$762$	1.03809	0.0376060
$763$	−49.9938	−1.80990
$764$	1.94022	0.0701946
$765$	33.0539	1.19507
$766$	6.86962	0.248210
$767$	−63.4430	−2.29079
$768$	−0.153561	−0.00554115
$769$	34.5246	1.24499	0.622494	−	0.782624i	$-0.286119\pi$
$769$	34.5246	1.24499	0.622494	+	0.782624i	$0.286119\pi$
$770$	46.3649	1.67088
$771$	0.450652	0.0162298
$772$	−16.4817	−0.593188
$773$	6.13756	0.220753	0.110376	−	0.993890i	$-0.464794\pi$
$773$	6.13756	0.220753	0.110376	+	0.993890i	$0.464794\pi$
$774$	−35.3542	−1.27078
$775$	−48.1620	−1.73003
$776$	12.9528	0.464978
$777$	2.96902	0.106513
$778$	4.86917	0.174568
$779$	18.8858	0.676655
$780$	2.43095	0.0870421
$781$	−14.5224	−0.519651
$782$	−3.34112	−0.119478
$783$	−0.917745	−0.0327975
$784$	6.54617	0.233792
$785$	−19.3835	−0.691825
$786$	−2.09482	−0.0747197
$787$	−10.3650	−0.369473	−0.184736	−	0.982788i	$-0.559143\pi$
$787$	−10.3650	−0.369473	−0.184736	+	0.982788i	$0.559143\pi$
$788$	−16.0611	−0.572154
$789$	3.52451	0.125476
$790$	36.5555	1.30058
$791$	72.3166	2.57128
$792$	−11.2808	−0.400845
$793$	38.6732	1.37333
$794$	−27.7701	−0.985523
$795$	−0.933527	−0.0331088
$796$	12.3706	0.438465
$797$	−42.4187	−1.50255	−0.751274	−	0.659990i	$-0.770561\pi$
$797$	−42.4187	−1.50255	−0.751274	+	0.659990i	$0.770561\pi$
$798$	3.65496	0.129384
$799$	42.5819	1.50644
$800$	6.04772	0.213819
$801$	−10.5785	−0.373773
$802$	12.6095	0.445256
$803$	−54.1937	−1.91245
$804$	−1.40260	−0.0494659
$805$	−12.2333	−0.431168
$806$	−37.9291	−1.33599
$807$	−0.789259	−0.0277832
$808$	3.04676	0.107185
$809$	−27.8144	−0.977901	−0.488950	−	0.872312i	$-0.662620\pi$
$809$	−27.8144	−0.977901	−0.488950	+	0.872312i	$0.662620\pi$
$810$	−29.2107	−1.02636
$811$	45.7690	1.60717	0.803584	−	0.595191i	$-0.202924\pi$
$811$	45.7690	1.60717	0.803584	+	0.595191i	$0.202924\pi$
$812$	3.68051	0.129161
$813$	−2.76170	−0.0968571
$814$	19.9099	0.697842
$815$	50.2708	1.76091
$816$	−0.513066	−0.0179609
$817$	76.8140	2.68738
$818$	17.5417	0.613333
$819$	52.1749	1.82314
$820$	−9.70687	−0.338979
$821$	−10.1337	−0.353668	−0.176834	−	0.984241i	$-0.556586\pi$
$821$	−10.1337	−0.353668	−0.176834	+	0.984241i	$0.556586\pi$
$822$	−2.21965	−0.0774193
$823$	−40.1213	−1.39854	−0.699269	−	0.714858i	$-0.746491\pi$
$823$	−40.1213	−1.39854	−0.699269	+	0.714858i	$0.746491\pi$
$824$	−12.1329	−0.422670
$825$	−3.51980	−0.122544
$826$	49.0267	1.70586
$827$	11.1517	0.387781	0.193891	−	0.981023i	$-0.437889\pi$
$827$	11.1517	0.387781	0.193891	+	0.981023i	$0.437889\pi$
$828$	2.97642	0.103438
$829$	26.5261	0.921290	0.460645	−	0.887584i	$-0.347618\pi$
$829$	26.5261	0.921290	0.460645	+	0.887584i	$0.347618\pi$
$830$	−34.3977	−1.19396
$831$	−1.09213	−0.0378854
$832$	4.76277	0.165119
$833$	21.8715	0.757804
$834$	−2.32930	−0.0806571
$835$	7.29145	0.252331
$836$	24.5097	0.847686
$837$	−7.30861	−0.252622
$838$	5.63168	0.194543
$839$	48.7994	1.68474	0.842372	−	0.538897i	$-0.181159\pi$
$839$	48.7994	1.68474	0.842372	+	0.538897i	$0.181159\pi$
$840$	−1.87856	−0.0648165
$841$	1.00000	0.0344828
$842$	27.9853	0.964437
$843$	1.74039	0.0599423
$844$	19.2843	0.663793
$845$	−32.1876	−1.10729
$846$	−37.9339	−1.30419
$847$	−12.3830	−0.425485
$848$	−1.82898	−0.0628076
$849$	−2.62345	−0.0900367
$850$	20.2062	0.693066
$851$	−5.25321	−0.180078
$852$	0.588401	0.0201583
$853$	28.4382	0.973707	0.486853	−	0.873484i	$-0.338145\pi$
$853$	28.4382	0.973707	0.486853	+	0.873484i	$0.338145\pi$
$854$	−29.8854	−1.02266
$855$	63.9770	2.18797
$856$	−0.179653	−0.00614042
$857$	−12.0798	−0.412639	−0.206320	−	0.978485i	$-0.566149\pi$
$857$	−12.0798	−0.412639	−0.206320	+	0.978485i	$0.566149\pi$
$858$	−2.77195	−0.0946327
$859$	0.00832020	0.000283882 0	0.000141941	−	1.00000i	$-0.499955\pi$
$859$	0.00832020	0.000283882 0	0.000141941		1.00000i	$0.499955\pi$
$860$	−39.4806	−1.34628
$861$	1.65056	0.0562510
$862$	1.93279	0.0658310
$863$	22.9933	0.782700	0.391350	−	0.920242i	$-0.372008\pi$
$863$	22.9933	0.782700	0.391350	+	0.920242i	$0.372008\pi$
$864$	0.917745	0.0312223
$865$	0.0698848	0.00237615
$866$	−6.58356	−0.223718
$867$	0.896322	0.0304407
$868$	29.3103	0.994858
$869$	−41.6832	−1.41400
$870$	−0.510408	−0.0173044
$871$	43.5023	1.47402
$872$	13.5834	0.459991
$873$	−38.5529	−1.30482
$874$	−6.46686	−0.218745
$875$	12.8172	0.433299
$876$	2.19576	0.0741878
$877$	−20.1234	−0.679518	−0.339759	−	0.940513i	$-0.610345\pi$
$877$	−20.1234	−0.679518	−0.339759	+	0.940513i	$0.610345\pi$
$878$	−25.6888	−0.866954
$879$	1.99473	0.0672805
$880$	−12.5974	−0.424659
$881$	−6.53590	−0.220200	−0.110100	−	0.993921i	$-0.535117\pi$
$881$	−6.53590	−0.220200	−0.110100	+	0.993921i	$0.535117\pi$
$882$	−19.4841	−0.656065
$883$	38.6301	1.30001	0.650004	−	0.759931i	$-0.274767\pi$
$883$	38.6301	1.30001	0.650004	+	0.759931i	$0.274767\pi$
$884$	15.9130	0.535211
$885$	−6.79895	−0.228544
$886$	−21.3438	−0.717061
$887$	−1.32965	−0.0446452	−0.0223226	−	0.999751i	$-0.507106\pi$
$887$	−1.32965	−0.0446452	−0.0223226	+	0.999751i	$0.507106\pi$
$888$	−0.806688	−0.0270707
$889$	24.8807	0.834471
$890$	−11.8132	−0.395978
$891$	33.3082	1.11587
$892$	−0.721402	−0.0241543
$893$	82.4188	2.75804
$894$	2.13852	0.0715227
$895$	−56.2127	−1.87898
$896$	−3.68051	−0.122957
$897$	0.731375	0.0244199
$898$	−9.94264	−0.331790
$899$	7.96366	0.265603
$900$	−18.0006	−0.600019
$901$	−6.11085	−0.203582
$902$	11.0685	0.368540
$903$	6.71330	0.223405
$904$	−19.6485	−0.653500
$905$	−3.95572	−0.131492
$906$	−1.89721	−0.0630306
$907$	−42.2897	−1.40421	−0.702104	−	0.712074i	$-0.747756\pi$
$907$	−42.2897	−1.40421	−0.702104	+	0.712074i	$0.747756\pi$
$908$	−4.02683	−0.133635
$909$	−9.06844	−0.300781
$910$	58.2645	1.93145
$911$	51.1577	1.69493	0.847465	−	0.530851i	$-0.178128\pi$
$911$	51.1577	1.69493	0.847465	+	0.530851i	$0.178128\pi$
$912$	−0.993057	−0.0328834
$913$	39.2228	1.29808
$914$	−36.8227	−1.21799
$915$	4.14446	0.137012
$916$	6.91497	0.228477
$917$	−50.2081	−1.65802
$918$	3.06630	0.101203
$919$	−6.76651	−0.223206	−0.111603	−	0.993753i	$-0.535599\pi$
$919$	−6.76651	−0.223206	−0.111603	+	0.993753i	$0.535599\pi$
$920$	3.32381	0.109583
$921$	0.00981598	0.000323448 0
$922$	−28.4810	−0.937972
$923$	−18.2495	−0.600691
$924$	2.14207	0.0704690
$925$	31.7700	1.04459
$926$	34.9277	1.14779
$927$	36.1126	1.18609
$928$	−1.00000	−0.0328266
$929$	1.05212	0.0345189	0.0172594	−	0.999851i	$-0.494506\pi$
$929$	1.05212	0.0345189	0.0172594	+	0.999851i	$0.494506\pi$
$930$	−4.06471	−0.133287
$931$	42.3331	1.38741
$932$	−10.7646	−0.352606
$933$	−0.729379	−0.0238788
$934$	−32.7985	−1.07320
$935$	−42.0895	−1.37647
$936$	−14.1760	−0.463357
$937$	2.71757	0.0887790	0.0443895	−	0.999014i	$-0.485866\pi$
$937$	2.71757	0.0887790	0.0443895	+	0.999014i	$0.485866\pi$
$938$	−33.6172	−1.09764
$939$	−0.942812	−0.0307675
$940$	−42.3613	−1.38167
$941$	−6.69800	−0.218348	−0.109174	−	0.994023i	$-0.534821\pi$
$941$	−6.69800	−0.218348	−0.109174	+	0.994023i	$0.534821\pi$
$942$	−0.895520	−0.0291776
$943$	−2.92040	−0.0951014
$944$	−13.3206	−0.433549
$945$	11.2271	0.365217
$946$	45.0186	1.46368
$947$	15.0604	0.489396	0.244698	−	0.969599i	$-0.421311\pi$
$947$	15.0604	0.489396	0.244698	+	0.969599i	$0.421311\pi$
$948$	1.68887	0.0548520
$949$	−68.1025	−2.21070
$950$	39.1098	1.26889
$951$	5.00028	0.162145
$952$	−12.2970	−0.398549
$953$	7.01525	0.227246	0.113623	−	0.993524i	$-0.463754\pi$
$953$	7.01525	0.227246	0.113623	+	0.993524i	$0.463754\pi$
$954$	5.44382	0.176250
$955$	−6.44892	−0.208682
$956$	11.3832	0.368159
$957$	0.582004	0.0188135
$958$	−13.8848	−0.448599
$959$	−53.2001	−1.71792
$960$	0.510408	0.0164733
$961$	32.4199	1.04580
$962$	25.0198	0.806671
$963$	0.534723	0.0172312
$964$	12.9513	0.417133
$965$	54.7820	1.76349
$966$	−0.565183	−0.0181845
$967$	10.2729	0.330355	0.165178	−	0.986264i	$-0.447180\pi$
$967$	10.2729	0.330355	0.165178	+	0.986264i	$0.447180\pi$
$968$	3.36448	0.108138
$969$	−3.31792	−0.106587
$970$	−43.0526	−1.38234
$971$	−51.1377	−1.64109	−0.820543	−	0.571585i	$-0.806329\pi$
$971$	−51.1377	−1.64109	−0.820543	+	0.571585i	$0.806329\pi$
$972$	−4.10278	−0.131597
$973$	−55.8281	−1.78977
$974$	−5.84024	−0.187133
$975$	−4.42316	−0.141654
$976$	8.11991	0.259912
$977$	−6.74364	−0.215748	−0.107874	−	0.994165i	$-0.534404\pi$
$977$	−6.74364	−0.215748	−0.107874	+	0.994165i	$0.534404\pi$
$978$	2.32252	0.0742661
$979$	13.4702	0.430510
$980$	−21.7582	−0.695041
$981$	−40.4298	−1.29083
$982$	−0.890073	−0.0284034
$983$	17.3268	0.552638	0.276319	−	0.961066i	$-0.410885\pi$
$983$	17.3268	0.552638	0.276319	+	0.961066i	$0.410885\pi$
$984$	−0.448460	−0.0142964
$985$	53.3842	1.70096
$986$	−3.34112	−0.106403
$987$	7.20314	0.229278
$988$	30.8001	0.979883
$989$	−11.8781	−0.377701
$990$	37.4952	1.19167
$991$	39.0886	1.24169	0.620845	−	0.783934i	$-0.286790\pi$
$991$	39.0886	1.24169	0.620845	+	0.783934i	$0.286790\pi$
$992$	−7.96366	−0.252846
$993$	−3.58514	−0.113771
$994$	14.1026	0.447309
$995$	−41.1176	−1.30352
$996$	−1.58918	−0.0503552
$997$	32.1515	1.01825	0.509124	−	0.860693i	$-0.329970\pi$
$997$	32.1515	1.01825	0.509124	+	0.860693i	$0.329970\pi$
$998$	17.4571	0.552596
$999$	4.82110	0.152533

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1334.2.a.i.1.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1334.2.a.i.1.3	✓	8	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:	\( N \)	\(=\)	\( 1334 = 2 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1334.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -0.153561 q^{3} +1.00000 q^{4} -3.32381 q^{5} -0.153561 q^{6} -3.68051 q^{7} +1.00000 q^{8} -2.97642 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -0.153561 q^{3} +1.00000 q^{4} -3.32381 q^{5} -0.153561 q^{6} -3.68051 q^{7} +1.00000 q^{8} -2.97642 q^{9} -3.32381 q^{10} +3.79005 q^{11} -0.153561 q^{12} +4.76277 q^{13} -3.68051 q^{14} +0.510408 q^{15} +1.00000 q^{16} +3.34112 q^{17} -2.97642 q^{18} +6.46686 q^{19} -3.32381 q^{20} +0.565183 q^{21} +3.79005 q^{22} -1.00000 q^{23} -0.153561 q^{24} +6.04772 q^{25} +4.76277 q^{26} +0.917745 q^{27} -3.68051 q^{28} -1.00000 q^{29} +0.510408 q^{30} -7.96366 q^{31} +1.00000 q^{32} -0.582004 q^{33} +3.34112 q^{34} +12.2333 q^{35} -2.97642 q^{36} +5.25321 q^{37} +6.46686 q^{38} -0.731375 q^{39} -3.32381 q^{40} +2.92040 q^{41} +0.565183 q^{42} +11.8781 q^{43} +3.79005 q^{44} +9.89306 q^{45} -1.00000 q^{46} +12.7448 q^{47} -0.153561 q^{48} +6.54617 q^{49} +6.04772 q^{50} -0.513066 q^{51} +4.76277 q^{52} -1.82898 q^{53} +0.917745 q^{54} -12.5974 q^{55} -3.68051 q^{56} -0.993057 q^{57} -1.00000 q^{58} -13.3206 q^{59} +0.510408 q^{60} +8.11991 q^{61} -7.96366 q^{62} +10.9547 q^{63} +1.00000 q^{64} -15.8305 q^{65} -0.582004 q^{66} +9.13383 q^{67} +3.34112 q^{68} +0.153561 q^{69} +12.2333 q^{70} -3.83171 q^{71} -2.97642 q^{72} -14.2989 q^{73} +5.25321 q^{74} -0.928694 q^{75} +6.46686 q^{76} -13.9493 q^{77} -0.731375 q^{78} -10.9981 q^{79} -3.32381 q^{80} +8.78833 q^{81} +2.92040 q^{82} +10.3489 q^{83} +0.565183 q^{84} -11.1053 q^{85} +11.8781 q^{86} +0.153561 q^{87} +3.79005 q^{88} +3.55410 q^{89} +9.89306 q^{90} -17.5294 q^{91} -1.00000 q^{92} +1.22291 q^{93} +12.7448 q^{94} -21.4946 q^{95} -0.153561 q^{96} +12.9528 q^{97} +6.54617 q^{98} -11.2808 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} + 4 q^{3} + 8 q^{4} + 4 q^{6} + 8 q^{7} + 8 q^{8} + 12 q^{9}+O(q^{10}) \) 8 * q + 8 * q^2 + 4 * q^3 + 8 * q^4 + 4 * q^6 + 8 * q^7 + 8 * q^8 + 12 * q^9	\( 8 q + 8 q^{2} + 4 q^{3} + 8 q^{4} + 4 q^{6} + 8 q^{7} + 8 q^{8} + 12 q^{9} + 8 q^{11} + 4 q^{12} + 4 q^{13} + 8 q^{14} + 8 q^{16} + 8 q^{17} + 12 q^{18} + 16 q^{19} - 4 q^{21} + 8 q^{22} - 8 q^{23} + 4 q^{24} + 4 q^{25} + 4 q^{26} + 4 q^{27} + 8 q^{28} - 8 q^{29} + 8 q^{31} + 8 q^{32} + 12 q^{33} + 8 q^{34} + 4 q^{35} + 12 q^{36} + 8 q^{37} + 16 q^{38} - 16 q^{39} + 8 q^{41} - 4 q^{42} + 32 q^{43} + 8 q^{44} - 8 q^{46} + 16 q^{47} + 4 q^{48} + 4 q^{49} + 4 q^{50} + 12 q^{51} + 4 q^{52} + 4 q^{54} + 8 q^{56} + 8 q^{57} - 8 q^{58} - 4 q^{59} + 8 q^{61} + 8 q^{62} + 24 q^{63} + 8 q^{64} - 4 q^{65} + 12 q^{66} + 20 q^{67} + 8 q^{68} - 4 q^{69} + 4 q^{70} - 24 q^{71} + 12 q^{72} - 4 q^{73} + 8 q^{74} - 16 q^{75} + 16 q^{76} - 16 q^{77} - 16 q^{78} + 4 q^{79} - 8 q^{81} + 8 q^{82} - 4 q^{83} - 4 q^{84} - 44 q^{85} + 32 q^{86} - 4 q^{87} + 8 q^{88} + 20 q^{89} - 8 q^{92} - 4 q^{93} + 16 q^{94} - 8 q^{95} + 4 q^{96} + 4 q^{97} + 4 q^{98}+O(q^{100}) \) 8 * q + 8 * q^2 + 4 * q^3 + 8 * q^4 + 4 * q^6 + 8 * q^7 + 8 * q^8 + 12 * q^9 + 8 * q^11 + 4 * q^12 + 4 * q^13 + 8 * q^14 + 8 * q^16 + 8 * q^17 + 12 * q^18 + 16 * q^19 - 4 * q^21 + 8 * q^22 - 8 * q^23 + 4 * q^24 + 4 * q^25 + 4 * q^26 + 4 * q^27 + 8 * q^28 - 8 * q^29 + 8 * q^31 + 8 * q^32 + 12 * q^33 + 8 * q^34 + 4 * q^35 + 12 * q^36 + 8 * q^37 + 16 * q^38 - 16 * q^39 + 8 * q^41 - 4 * q^42 + 32 * q^43 + 8 * q^44 - 8 * q^46 + 16 * q^47 + 4 * q^48 + 4 * q^49 + 4 * q^50 + 12 * q^51 + 4 * q^52 + 4 * q^54 + 8 * q^56 + 8 * q^57 - 8 * q^58 - 4 * q^59 + 8 * q^61 + 8 * q^62 + 24 * q^63 + 8 * q^64 - 4 * q^65 + 12 * q^66 + 20 * q^67 + 8 * q^68 - 4 * q^69 + 4 * q^70 - 24 * q^71 + 12 * q^72 - 4 * q^73 + 8 * q^74 - 16 * q^75 + 16 * q^76 - 16 * q^77 - 16 * q^78 + 4 * q^79 - 8 * q^81 + 8 * q^82 - 4 * q^83 - 4 * q^84 - 44 * q^85 + 32 * q^86 - 4 * q^87 + 8 * q^88 + 20 * q^89 - 8 * q^92 - 4 * q^93 + 16 * q^94 - 8 * q^95 + 4 * q^96 + 4 * q^97 + 4 * q^98

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