LMFDB - Embedded newform 861.2.a.h.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [861,2,Mod(1,861)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(861, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("861.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 861 = 3 \cdot 7 \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	861.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:	$6.87511961403$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.48119$ of defining polynomial
Character	$\chi$	$=$	861.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.48119 q^{2} -1.00000 q^{3} +0.193937 q^{4} +0.193937 q^{5} -1.48119 q^{6} +1.00000 q^{7} -2.67513 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.48119 q^{2} -1.00000 q^{3} +0.193937 q^{4} +0.193937 q^{5} -1.48119 q^{6} +1.00000 q^{7} -2.67513 q^{8} +1.00000 q^{9} +0.287258 q^{10} -5.35026 q^{11} -0.193937 q^{12} -3.86907 q^{13} +1.48119 q^{14} -0.193937 q^{15} -4.35026 q^{16} -0.156325 q^{17} +1.48119 q^{18} -1.84367 q^{19} +0.0376114 q^{20} -1.00000 q^{21} -7.92478 q^{22} +7.79384 q^{23} +2.67513 q^{24} -4.96239 q^{25} -5.73084 q^{26} -1.00000 q^{27} +0.193937 q^{28} -4.28726 q^{29} -0.287258 q^{30} +0.806063 q^{31} -1.09332 q^{32} +5.35026 q^{33} -0.231548 q^{34} +0.193937 q^{35} +0.193937 q^{36} -5.80606 q^{37} -2.73084 q^{38} +3.86907 q^{39} -0.518806 q^{40} +1.00000 q^{41} -1.48119 q^{42} -6.41819 q^{43} -1.03761 q^{44} +0.193937 q^{45} +11.5442 q^{46} -3.15633 q^{47} +4.35026 q^{48} +1.00000 q^{49} -7.35026 q^{50} +0.156325 q^{51} -0.750354 q^{52} -1.86907 q^{53} -1.48119 q^{54} -1.03761 q^{55} -2.67513 q^{56} +1.84367 q^{57} -6.35026 q^{58} -1.96968 q^{59} -0.0376114 q^{60} +6.15633 q^{61} +1.19394 q^{62} +1.00000 q^{63} +7.08110 q^{64} -0.750354 q^{65} +7.92478 q^{66} -10.8945 q^{67} -0.0303172 q^{68} -7.79384 q^{69} +0.287258 q^{70} -10.3430 q^{71} -2.67513 q^{72} +11.8192 q^{73} -8.59991 q^{74} +4.96239 q^{75} -0.357556 q^{76} -5.35026 q^{77} +5.73084 q^{78} -1.61942 q^{79} -0.843675 q^{80} +1.00000 q^{81} +1.48119 q^{82} +6.12601 q^{83} -0.193937 q^{84} -0.0303172 q^{85} -9.50659 q^{86} +4.28726 q^{87} +14.3127 q^{88} +16.6253 q^{89} +0.287258 q^{90} -3.86907 q^{91} +1.51151 q^{92} -0.806063 q^{93} -4.67513 q^{94} -0.357556 q^{95} +1.09332 q^{96} -2.28726 q^{97} +1.48119 q^{98} -5.35026 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} - 3 q^{3} + q^{4} + q^{5} + q^{6} + 3 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q - q^2 - 3 * q^3 + q^4 + q^5 + q^6 + 3 * q^7 - 3 * q^8 + 3 * q^9	\( 3 q - q^{2} - 3 q^{3} + q^{4} + q^{5} + q^{6} + 3 q^{7} - 3 q^{8} + 3 q^{9} - 5 q^{10} - 6 q^{11} - q^{12} - 7 q^{13} - q^{14} - q^{15} - 3 q^{16} + 10 q^{17} - q^{18} - 16 q^{19} + 11 q^{20} - 3 q^{21} - 2 q^{22} - 3 q^{23} + 3 q^{24} - 4 q^{25} + 5 q^{26} - 3 q^{27} + q^{28} - 7 q^{29} + 5 q^{30} + 2 q^{31} + 3 q^{32} + 6 q^{33} - 12 q^{34} + q^{35} + q^{36} - 17 q^{37} + 14 q^{38} + 7 q^{39} - 7 q^{40} + 3 q^{41} + q^{42} - 18 q^{43} - 14 q^{44} + q^{45} + 25 q^{46} + q^{47} + 3 q^{48} + 3 q^{49} - 12 q^{50} - 10 q^{51} - 19 q^{52} - q^{53} + q^{54} - 14 q^{55} - 3 q^{56} + 16 q^{57} - 9 q^{58} - 8 q^{59} - 11 q^{60} + 8 q^{61} + 4 q^{62} + 3 q^{63} - 11 q^{64} - 19 q^{65} + 2 q^{66} - 13 q^{67} + 2 q^{68} + 3 q^{69} - 5 q^{70} - 8 q^{71} - 3 q^{72} - 6 q^{73} + q^{74} + 4 q^{75} - 4 q^{76} - 6 q^{77} - 5 q^{78} - 17 q^{79} - 13 q^{80} + 3 q^{81} - q^{82} + 10 q^{83} - q^{84} + 2 q^{85} - 8 q^{86} + 7 q^{87} + 22 q^{88} + 8 q^{89} - 5 q^{90} - 7 q^{91} - 3 q^{92} - 2 q^{93} - 9 q^{94} - 4 q^{95} - 3 q^{96} - q^{97} - q^{98} - 6 q^{99}+O(q^{100}) \) 3 * q - q^2 - 3 * q^3 + q^4 + q^5 + q^6 + 3 * q^7 - 3 * q^8 + 3 * q^9 - 5 * q^10 - 6 * q^11 - q^12 - 7 * q^13 - q^14 - q^15 - 3 * q^16 + 10 * q^17 - q^18 - 16 * q^19 + 11 * q^20 - 3 * q^21 - 2 * q^22 - 3 * q^23 + 3 * q^24 - 4 * q^25 + 5 * q^26 - 3 * q^27 + q^28 - 7 * q^29 + 5 * q^30 + 2 * q^31 + 3 * q^32 + 6 * q^33 - 12 * q^34 + q^35 + q^36 - 17 * q^37 + 14 * q^38 + 7 * q^39 - 7 * q^40 + 3 * q^41 + q^42 - 18 * q^43 - 14 * q^44 + q^45 + 25 * q^46 + q^47 + 3 * q^48 + 3 * q^49 - 12 * q^50 - 10 * q^51 - 19 * q^52 - q^53 + q^54 - 14 * q^55 - 3 * q^56 + 16 * q^57 - 9 * q^58 - 8 * q^59 - 11 * q^60 + 8 * q^61 + 4 * q^62 + 3 * q^63 - 11 * q^64 - 19 * q^65 + 2 * q^66 - 13 * q^67 + 2 * q^68 + 3 * q^69 - 5 * q^70 - 8 * q^71 - 3 * q^72 - 6 * q^73 + q^74 + 4 * q^75 - 4 * q^76 - 6 * q^77 - 5 * q^78 - 17 * q^79 - 13 * q^80 + 3 * q^81 - q^82 + 10 * q^83 - q^84 + 2 * q^85 - 8 * q^86 + 7 * q^87 + 22 * q^88 + 8 * q^89 - 5 * q^90 - 7 * q^91 - 3 * q^92 - 2 * q^93 - 9 * q^94 - 4 * q^95 - 3 * q^96 - q^97 - q^98 - 6 * 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 $

$2$	1.48119	1.04736	0.523681	−	0.851914i	$-0.324558\pi$
$2$	1.48119	1.04736	0.523681	+	0.851914i	$0.324558\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0.193937	0.0969683
$5$	0.193937	0.0867311	0.0433655	−	0.999059i	$-0.486192\pi$
$5$	0.193937	0.0867311	0.0433655	+	0.999059i	$0.486192\pi$
$6$	−1.48119	−0.604695
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.67513	−0.945802
$9$	1.00000	0.333333
$10$	0.287258	0.0908389
$11$	−5.35026	−1.61316	−0.806582	−	0.591122i	$-0.798685\pi$
$11$	−5.35026	−1.61316	−0.806582	+	0.591122i	$0.798685\pi$
$12$	−0.193937	−0.0559847
$13$	−3.86907	−1.07309	−0.536543	−	0.843873i	$-0.680270\pi$
$13$	−3.86907	−1.07309	−0.536543	+	0.843873i	$0.680270\pi$
$14$	1.48119	0.395866
$15$	−0.193937	−0.0500742
$16$	−4.35026	−1.08757
$17$	−0.156325	−0.0379144	−0.0189572	−	0.999820i	$-0.506035\pi$
$17$	−0.156325	−0.0379144	−0.0189572	+	0.999820i	$0.506035\pi$
$18$	1.48119	0.349121
$19$	−1.84367	−0.422968	−0.211484	−	0.977381i	$-0.567830\pi$
$19$	−1.84367	−0.422968	−0.211484	+	0.977381i	$0.567830\pi$
$20$	0.0376114	0.00841016
$21$	−1.00000	−0.218218
$22$	−7.92478	−1.68957
$23$	7.79384	1.62513	0.812564	−	0.582871i	$-0.198071\pi$
$23$	7.79384	1.62513	0.812564	+	0.582871i	$0.198071\pi$
$24$	2.67513	0.546059
$25$	−4.96239	−0.992478
$26$	−5.73084	−1.12391
$27$	−1.00000	−0.192450
$28$	0.193937	0.0366506
$29$	−4.28726	−0.796124	−0.398062	−	0.917359i	$-0.630317\pi$
$29$	−4.28726	−0.796124	−0.398062	+	0.917359i	$0.630317\pi$
$30$	−0.287258	−0.0524458
$31$	0.806063	0.144773	0.0723866	−	0.997377i	$-0.476938\pi$
$31$	0.806063	0.144773	0.0723866	+	0.997377i	$0.476938\pi$
$32$	−1.09332	−0.193274
$33$	5.35026	0.931361
$34$	−0.231548	−0.0397101
$35$	0.193937	0.0327813
$36$	0.193937	0.0323228
$37$	−5.80606	−0.954511	−0.477255	−	0.878765i	$-0.658368\pi$
$37$	−5.80606	−0.954511	−0.477255	+	0.878765i	$0.658368\pi$
$38$	−2.73084	−0.443001
$39$	3.86907	0.619547
$40$	−0.518806	−0.0820304
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	−1.48119	−0.228553
$43$	−6.41819	−0.978765	−0.489382	−	0.872069i	$-0.662778\pi$
$43$	−6.41819	−0.978765	−0.489382	+	0.872069i	$0.662778\pi$
$44$	−1.03761	−0.156426
$45$	0.193937	0.0289104
$46$	11.5442	1.70210
$47$	−3.15633	−0.460397	−0.230199	−	0.973144i	$-0.573938\pi$
$47$	−3.15633	−0.460397	−0.230199	+	0.973144i	$0.573938\pi$
$48$	4.35026	0.627906
$49$	1.00000	0.142857
$50$	−7.35026	−1.03948
$51$	0.156325	0.0218899
$52$	−0.750354	−0.104055
$53$	−1.86907	−0.256736	−0.128368	−	0.991727i	$-0.540974\pi$
$53$	−1.86907	−0.256736	−0.128368	+	0.991727i	$0.540974\pi$
$54$	−1.48119	−0.201565
$55$	−1.03761	−0.139911
$56$	−2.67513	−0.357479
$57$	1.84367	0.244201
$58$	−6.35026	−0.833830
$59$	−1.96968	−0.256431	−0.128215	−	0.991746i	$-0.540925\pi$
$59$	−1.96968	−0.256431	−0.128215	+	0.991746i	$0.540925\pi$
$60$	−0.0376114	−0.00485561
$61$	6.15633	0.788237	0.394118	−	0.919060i	$-0.371050\pi$
$61$	6.15633	0.788237	0.394118	+	0.919060i	$0.371050\pi$
$62$	1.19394	0.151630
$63$	1.00000	0.125988
$64$	7.08110	0.885138
$65$	−0.750354	−0.0930699
$66$	7.92478	0.975473
$67$	−10.8945	−1.33097	−0.665485	−	0.746411i	$-0.731775\pi$
$67$	−10.8945	−1.33097	−0.665485	+	0.746411i	$0.731775\pi$
$68$	−0.0303172	−0.00367650
$69$	−7.79384	−0.938269
$70$	0.287258	0.0343339
$71$	−10.3430	−1.22748	−0.613742	−	0.789506i	$-0.710337\pi$
$71$	−10.3430	−1.22748	−0.613742	+	0.789506i	$0.710337\pi$
$72$	−2.67513	−0.315267
$73$	11.8192	1.38334	0.691669	−	0.722215i	$-0.256876\pi$
$73$	11.8192	1.38334	0.691669	+	0.722215i	$0.256876\pi$
$74$	−8.59991	−0.999719
$75$	4.96239	0.573007
$76$	−0.357556	−0.0410145
$77$	−5.35026	−0.609719
$78$	5.73084	0.648890
$79$	−1.61942	−0.182199	−0.0910996	−	0.995842i	$-0.529038\pi$
$79$	−1.61942	−0.182199	−0.0910996	+	0.995842i	$0.529038\pi$
$80$	−0.843675	−0.0943257
$81$	1.00000	0.111111
$82$	1.48119	0.163571
$83$	6.12601	0.672417	0.336208	−	0.941788i	$-0.390855\pi$
$83$	6.12601	0.672417	0.336208	+	0.941788i	$0.390855\pi$
$84$	−0.193937	−0.0211602
$85$	−0.0303172	−0.00328836
$86$	−9.50659	−1.02512
$87$	4.28726	0.459642
$88$	14.3127	1.52573
$89$	16.6253	1.76228	0.881139	−	0.472857i	$-0.156777\pi$
$89$	16.6253	1.76228	0.881139	+	0.472857i	$0.156777\pi$
$90$	0.287258	0.0302796
$91$	−3.86907	−0.405588
$92$	1.51151	0.157586
$93$	−0.806063	−0.0835849
$94$	−4.67513	−0.482203
$95$	−0.357556	−0.0366845
$96$	1.09332	0.111587
$97$	−2.28726	−0.232236	−0.116118	−	0.993235i	$-0.537045\pi$
$97$	−2.28726	−0.232236	−0.116118	+	0.993235i	$0.537045\pi$
$98$	1.48119	0.149623
$99$	−5.35026	−0.537722
$100$	−0.962389	−0.0962389
$101$	14.4993	1.44273	0.721367	−	0.692553i	$-0.243514\pi$
$101$	14.4993	1.44273	0.721367	+	0.692553i	$0.243514\pi$
$102$	0.231548	0.0229267
$103$	−5.63752	−0.555481	−0.277741	−	0.960656i	$-0.589586\pi$
$103$	−5.63752	−0.555481	−0.277741	+	0.960656i	$0.589586\pi$
$104$	10.3503	1.01493
$105$	−0.193937	−0.0189263
$106$	−2.76845	−0.268896
$107$	−2.93700	−0.283930	−0.141965	−	0.989872i	$-0.545342\pi$
$107$	−2.93700	−0.283930	−0.141965	+	0.989872i	$0.545342\pi$
$108$	−0.193937	−0.0186616
$109$	5.89446	0.564587	0.282293	−	0.959328i	$-0.408905\pi$
$109$	5.89446	0.564587	0.282293	+	0.959328i	$0.408905\pi$
$110$	−1.53690	−0.146538
$111$	5.80606	0.551087
$112$	−4.35026	−0.411061
$113$	15.5369	1.46159	0.730794	−	0.682598i	$-0.239150\pi$
$113$	15.5369	1.46159	0.730794	+	0.682598i	$0.239150\pi$
$114$	2.73084	0.255767
$115$	1.51151	0.140949
$116$	−0.831456	−0.0771988
$117$	−3.86907	−0.357695
$118$	−2.91748	−0.268576
$119$	−0.156325	−0.0143303
$120$	0.518806	0.0473603
$121$	17.6253	1.60230
$122$	9.11871	0.825570
$123$	−1.00000	−0.0901670
$124$	0.156325	0.0140384
$125$	−1.93207	−0.172810
$126$	1.48119	0.131955
$127$	−16.4993	−1.46408	−0.732038	−	0.681264i	$-0.761431\pi$
$127$	−16.4993	−1.46408	−0.732038	+	0.681264i	$0.761431\pi$
$128$	12.6751	1.12033
$129$	6.41819	0.565090
$130$	−1.11142	−0.0974779
$131$	4.46898	0.390456	0.195228	−	0.980758i	$-0.437455\pi$
$131$	4.46898	0.390456	0.195228	+	0.980758i	$0.437455\pi$
$132$	1.03761	0.0903125
$133$	−1.84367	−0.159867
$134$	−16.1368	−1.39401
$135$	−0.193937	−0.0166914
$136$	0.418190	0.0358595
$137$	−5.56230	−0.475219	−0.237610	−	0.971361i	$-0.576364\pi$
$137$	−5.56230	−0.475219	−0.237610	+	0.971361i	$0.576364\pi$
$138$	−11.5442	−0.982707
$139$	−1.55642	−0.132014	−0.0660068	−	0.997819i	$-0.521026\pi$
$139$	−1.55642	−0.132014	−0.0660068	+	0.997819i	$0.521026\pi$
$140$	0.0376114	0.00317874
$141$	3.15633	0.265811
$142$	−15.3199	−1.28562
$143$	20.7005	1.73106
$144$	−4.35026	−0.362522
$145$	−0.831456	−0.0690487
$146$	17.5066	1.44886
$147$	−1.00000	−0.0824786
$148$	−1.12601	−0.0925573
$149$	5.69323	0.466408	0.233204	−	0.972428i	$-0.425079\pi$
$149$	5.69323	0.466408	0.233204	+	0.972428i	$0.425079\pi$
$150$	7.35026	0.600146
$151$	9.58769	0.780235	0.390118	−	0.920765i	$-0.372434\pi$
$151$	9.58769	0.780235	0.390118	+	0.920765i	$0.372434\pi$
$152$	4.93207	0.400044
$153$	−0.156325	−0.0126381
$154$	−7.92478	−0.638597
$155$	0.156325	0.0125563
$156$	0.750354	0.0600764
$157$	−2.68006	−0.213892	−0.106946	−	0.994265i	$-0.534107\pi$
$157$	−2.68006	−0.213892	−0.106946	+	0.994265i	$0.534107\pi$
$158$	−2.39868	−0.190829
$159$	1.86907	0.148227
$160$	−0.212035	−0.0167628
$161$	7.79384	0.614241
$162$	1.48119	0.116374
$163$	−11.9551	−0.936395	−0.468198	−	0.883624i	$-0.655096\pi$
$163$	−11.9551	−0.936395	−0.468198	+	0.883624i	$0.655096\pi$
$164$	0.193937	0.0151439
$165$	1.03761	0.0807779
$166$	9.07381	0.704264
$167$	18.6121	1.44025	0.720125	−	0.693845i	$-0.244085\pi$
$167$	18.6121	1.44025	0.720125	+	0.693845i	$0.244085\pi$
$168$	2.67513	0.206391
$169$	1.96968	0.151514
$170$	−0.0449056	−0.00344410
$171$	−1.84367	−0.140989
$172$	−1.24472	−0.0949091
$173$	−9.00000	−0.684257	−0.342129	−	0.939653i	$-0.611148\pi$
$173$	−9.00000	−0.684257	−0.342129	+	0.939653i	$0.611148\pi$
$174$	6.35026	0.481412
$175$	−4.96239	−0.375121
$176$	23.2750	1.75442
$177$	1.96968	0.148050
$178$	24.6253	1.84574
$179$	−17.5066	−1.30850	−0.654252	−	0.756277i	$-0.727016\pi$
$179$	−17.5066	−1.30850	−0.654252	+	0.756277i	$0.727016\pi$
$180$	0.0376114	0.00280339
$181$	−11.2447	−0.835814	−0.417907	−	0.908490i	$-0.637236\pi$
$181$	−11.2447	−0.835814	−0.417907	+	0.908490i	$0.637236\pi$
$182$	−5.73084	−0.424798
$183$	−6.15633	−0.455089
$184$	−20.8496	−1.53705
$185$	−1.12601	−0.0827858
$186$	−1.19394	−0.0875437
$187$	0.836381	0.0611622
$188$	−0.612127	−0.0446439
$189$	−1.00000	−0.0727393
$190$	−0.529610	−0.0384219
$191$	−16.3938	−1.18621	−0.593105	−	0.805125i	$-0.702098\pi$
$191$	−16.3938	−1.18621	−0.593105	+	0.805125i	$0.702098\pi$
$192$	−7.08110	−0.511035
$193$	16.6556	1.19890	0.599449	−	0.800413i	$-0.295386\pi$
$193$	16.6556	1.19890	0.599449	+	0.800413i	$0.295386\pi$
$194$	−3.38787	−0.243235
$195$	0.750354	0.0537339
$196$	0.193937	0.0138526
$197$	−9.41090	−0.670499	−0.335249	−	0.942129i	$-0.608821\pi$
$197$	−9.41090	−0.670499	−0.335249	+	0.942129i	$0.608821\pi$
$198$	−7.92478	−0.563189
$199$	−21.2301	−1.50496	−0.752482	−	0.658613i	$-0.771144\pi$
$199$	−21.2301	−1.50496	−0.752482	+	0.658613i	$0.771144\pi$
$200$	13.2750	0.938687
$201$	10.8945	0.768436
$202$	21.4763	1.51107
$203$	−4.28726	−0.300907
$204$	0.0303172	0.00212263
$205$	0.193937	0.0135451
$206$	−8.35026	−0.581790
$207$	7.79384	0.541710
$208$	16.8315	1.16705
$209$	9.86414	0.682317
$210$	−0.287258	−0.0198227
$211$	0.380579	0.0262001	0.0131001	−	0.999914i	$-0.495830\pi$
$211$	0.380579	0.0262001	0.0131001	+	0.999914i	$0.495830\pi$
$212$	−0.362481	−0.0248953
$213$	10.3430	0.708688
$214$	−4.35026	−0.297378
$215$	−1.24472	−0.0848893
$216$	2.67513	0.182020
$217$	0.806063	0.0547192
$218$	8.73084	0.591327
$219$	−11.8192	−0.798670
$220$	−0.201231	−0.0135670
$221$	0.604833	0.0406854
$222$	8.59991	0.577188
$223$	−14.8070	−0.991551	−0.495776	−	0.868451i	$-0.665116\pi$
$223$	−14.8070	−0.991551	−0.495776	+	0.868451i	$0.665116\pi$
$224$	−1.09332	−0.0730506
$225$	−4.96239	−0.330826
$226$	23.0132	1.53081
$227$	9.39375	0.623485	0.311743	−	0.950167i	$-0.399087\pi$
$227$	9.39375	0.623485	0.311743	+	0.950167i	$0.399087\pi$
$228$	0.357556	0.0236797
$229$	−25.2252	−1.66693	−0.833464	−	0.552573i	$-0.813646\pi$
$229$	−25.2252	−1.66693	−0.833464	+	0.552573i	$0.813646\pi$
$230$	2.23884	0.147625
$231$	5.35026	0.352021
$232$	11.4690	0.752975
$233$	1.06793	0.0699623	0.0349812	−	0.999388i	$-0.488863\pi$
$233$	1.06793	0.0699623	0.0349812	+	0.999388i	$0.488863\pi$
$234$	−5.73084	−0.374637
$235$	−0.612127	−0.0399308
$236$	−0.381994	−0.0248657
$237$	1.61942	0.105193
$238$	−0.231548	−0.0150090
$239$	−9.21440	−0.596030	−0.298015	−	0.954561i	$-0.596325\pi$
$239$	−9.21440	−0.596030	−0.298015	+	0.954561i	$0.596325\pi$
$240$	0.843675	0.0544590
$241$	−7.79877	−0.502363	−0.251181	−	0.967940i	$-0.580819\pi$
$241$	−7.79877	−0.502363	−0.251181	+	0.967940i	$0.580819\pi$
$242$	26.1065	1.67819
$243$	−1.00000	−0.0641500
$244$	1.19394	0.0764340
$245$	0.193937	0.0123902
$246$	−1.48119	−0.0944375
$247$	7.13330	0.453881
$248$	−2.15633	−0.136927
$249$	−6.12601	−0.388220
$250$	−2.86177	−0.180994
$251$	21.9756	1.38709	0.693543	−	0.720416i	$-0.256049\pi$
$251$	21.9756	1.38709	0.693543	+	0.720416i	$0.256049\pi$
$252$	0.193937	0.0122169
$253$	−41.6991	−2.62160
$254$	−24.4387	−1.53342
$255$	0.0303172	0.00189853
$256$	4.61213	0.288258
$257$	−20.2882	−1.26554	−0.632772	−	0.774338i	$-0.718083\pi$
$257$	−20.2882	−1.26554	−0.632772	+	0.774338i	$0.718083\pi$
$258$	9.50659	0.591854
$259$	−5.80606	−0.360771
$260$	−0.145521	−0.00902483
$261$	−4.28726	−0.265375
$262$	6.61942	0.408949
$263$	14.1622	0.873279	0.436639	−	0.899637i	$-0.356169\pi$
$263$	14.1622	0.873279	0.436639	+	0.899637i	$0.356169\pi$
$264$	−14.3127	−0.880883
$265$	−0.362481	−0.0222670
$266$	−2.73084	−0.167439
$267$	−16.6253	−1.01745
$268$	−2.11283	−0.129062
$269$	−1.49929	−0.0914135	−0.0457067	−	0.998955i	$-0.514554\pi$
$269$	−1.49929	−0.0914135	−0.0457067	+	0.998955i	$0.514554\pi$
$270$	−0.287258	−0.0174819
$271$	−23.6072	−1.43404	−0.717018	−	0.697055i	$-0.754493\pi$
$271$	−23.6072	−1.43404	−0.717018	+	0.697055i	$0.754493\pi$
$272$	0.680055	0.0412344
$273$	3.86907	0.234167
$274$	−8.23884	−0.497727
$275$	26.5501	1.60103
$276$	−1.51151	−0.0909823
$277$	6.94921	0.417538	0.208769	−	0.977965i	$-0.433054\pi$
$277$	6.94921	0.417538	0.208769	+	0.977965i	$0.433054\pi$
$278$	−2.30536	−0.138266
$279$	0.806063	0.0482578
$280$	−0.518806	−0.0310046
$281$	−22.8315	−1.36201	−0.681005	−	0.732279i	$-0.738457\pi$
$281$	−22.8315	−1.36201	−0.681005	+	0.732279i	$0.738457\pi$
$282$	4.67513	0.278400
$283$	0.992706	0.0590102	0.0295051	−	0.999565i	$-0.490607\pi$
$283$	0.992706	0.0590102	0.0295051	+	0.999565i	$0.490607\pi$
$284$	−2.00588	−0.119027
$285$	0.357556	0.0211798
$286$	30.6615	1.81305
$287$	1.00000	0.0590281
$288$	−1.09332	−0.0644246
$289$	−16.9756	−0.998562
$290$	−1.23155	−0.0723190
$291$	2.28726	0.134081
$292$	2.29218	0.134140
$293$	−12.4083	−0.724903	−0.362452	−	0.932003i	$-0.618060\pi$
$293$	−12.4083	−0.724903	−0.362452	+	0.932003i	$0.618060\pi$
$294$	−1.48119	−0.0863850
$295$	−0.381994	−0.0222405
$296$	15.5320	0.902778
$297$	5.35026	0.310454
$298$	8.43278	0.488498
$299$	−30.1549	−1.74390
$300$	0.962389	0.0555635
$301$	−6.41819	−0.369938
$302$	14.2012	0.817189
$303$	−14.4993	−0.832963
$304$	8.02047	0.460005
$305$	1.19394	0.0683646
$306$	−0.231548	−0.0132367
$307$	−33.4626	−1.90981	−0.954907	−	0.296906i	$-0.904045\pi$
$307$	−33.4626	−1.90981	−0.954907	+	0.296906i	$0.904045\pi$
$308$	−1.03761	−0.0591234
$309$	5.63752	0.320707
$310$	0.231548	0.0131510
$311$	−17.3430	−0.983429	−0.491715	−	0.870756i	$-0.663630\pi$
$311$	−17.3430	−0.983429	−0.491715	+	0.870756i	$0.663630\pi$
$312$	−10.3503	−0.585968
$313$	1.04254	0.0589276	0.0294638	−	0.999566i	$-0.490620\pi$
$313$	1.04254	0.0589276	0.0294638	+	0.999566i	$0.490620\pi$
$314$	−3.96968	−0.224022
$315$	0.193937	0.0109271
$316$	−0.314065	−0.0176675
$317$	3.92970	0.220714	0.110357	−	0.993892i	$-0.464801\pi$
$317$	3.92970	0.220714	0.110357	+	0.993892i	$0.464801\pi$
$318$	2.76845	0.155247
$319$	22.9380	1.28428
$320$	1.37328	0.0767689
$321$	2.93700	0.163927
$322$	11.5442	0.643333
$323$	0.288213	0.0160366
$324$	0.193937	0.0107743
$325$	19.1998	1.06501
$326$	−17.7078	−0.980745
$327$	−5.89446	−0.325964
$328$	−2.67513	−0.147709
$329$	−3.15633	−0.174014
$330$	1.53690	0.0846038
$331$	17.4993	0.961848	0.480924	−	0.876762i	$-0.340301\pi$
$331$	17.4993	0.961848	0.480924	+	0.876762i	$0.340301\pi$
$332$	1.18806	0.0652031
$333$	−5.80606	−0.318170
$334$	27.5682	1.50846
$335$	−2.11283	−0.115436
$336$	4.35026	0.237326
$337$	−10.3199	−0.562163	−0.281082	−	0.959684i	$-0.590693\pi$
$337$	−10.3199	−0.562163	−0.281082	+	0.959684i	$0.590693\pi$
$338$	2.91748	0.158690
$339$	−15.5369	−0.843849
$340$	−0.00587961	−0.000318866 0
$341$	−4.31265	−0.233543
$342$	−2.73084	−0.147667
$343$	1.00000	0.0539949
$344$	17.1695	0.925717
$345$	−1.51151	−0.0813770
$346$	−13.3307	−0.716665
$347$	−3.32979	−0.178753	−0.0893763	−	0.995998i	$-0.528487\pi$
$347$	−3.32979	−0.178753	−0.0893763	+	0.995998i	$0.528487\pi$
$348$	0.831456	0.0445707
$349$	22.6702	1.21351	0.606754	−	0.794890i	$-0.292471\pi$
$349$	22.6702	1.21351	0.606754	+	0.794890i	$0.292471\pi$
$350$	−7.35026	−0.392888
$351$	3.86907	0.206516
$352$	5.84955	0.311782
$353$	35.5271	1.89091	0.945457	−	0.325746i	$-0.105616\pi$
$353$	35.5271	1.89091	0.945457	+	0.325746i	$0.105616\pi$
$354$	2.91748	0.155062
$355$	−2.00588	−0.106461
$356$	3.22425	0.170885
$357$	0.156325	0.00827361
$358$	−25.9307	−1.37048
$359$	23.8505	1.25878	0.629391	−	0.777089i	$-0.283304\pi$
$359$	23.8505	1.25878	0.629391	+	0.777089i	$0.283304\pi$
$360$	−0.518806	−0.0273435
$361$	−15.6009	−0.821098
$362$	−16.6556	−0.875400
$363$	−17.6253	−0.925088
$364$	−0.750354	−0.0393292
$365$	2.29218	0.119978
$366$	−9.11871	−0.476643
$367$	27.0240	1.41064	0.705320	−	0.708889i	$-0.250803\pi$
$367$	27.0240	1.41064	0.705320	+	0.708889i	$0.250803\pi$
$368$	−33.9053	−1.76743
$369$	1.00000	0.0520579
$370$	−1.66784	−0.0867067
$371$	−1.86907	−0.0970371
$372$	−0.156325	−0.00810508
$373$	26.9452	1.39517	0.697586	−	0.716501i	$-0.254258\pi$
$373$	26.9452	1.39517	0.697586	+	0.716501i	$0.254258\pi$
$374$	1.23884	0.0640590
$375$	1.93207	0.0997717
$376$	8.44358	0.435445
$377$	16.5877	0.854309
$378$	−1.48119	−0.0761844
$379$	−22.5804	−1.15988	−0.579938	−	0.814660i	$-0.696923\pi$
$379$	−22.5804	−1.15988	−0.579938	+	0.814660i	$0.696923\pi$
$380$	−0.0693432	−0.00355723
$381$	16.4993	0.845284
$382$	−24.2823	−1.24239
$383$	−23.7308	−1.21259	−0.606295	−	0.795240i	$-0.707345\pi$
$383$	−23.7308	−1.21259	−0.606295	+	0.795240i	$0.707345\pi$
$384$	−12.6751	−0.646825
$385$	−1.03761	−0.0528816
$386$	24.6702	1.25568
$387$	−6.41819	−0.326255
$388$	−0.443583	−0.0225195
$389$	23.2447	1.17855	0.589277	−	0.807931i	$-0.299413\pi$
$389$	23.2447	1.17855	0.589277	+	0.807931i	$0.299413\pi$
$390$	1.11142	0.0562789
$391$	−1.21837	−0.0616158
$392$	−2.67513	−0.135115
$393$	−4.46898	−0.225430
$394$	−13.9394	−0.702255
$395$	−0.314065	−0.0158023
$396$	−1.03761	−0.0521419
$397$	10.1563	0.509731	0.254866	−	0.966976i	$-0.417969\pi$
$397$	10.1563	0.509731	0.254866	+	0.966976i	$0.417969\pi$
$398$	−31.4460	−1.57624
$399$	1.84367	0.0922992
$400$	21.5877	1.07938
$401$	22.1768	1.10746	0.553728	−	0.832698i	$-0.313205\pi$
$401$	22.1768	1.10746	0.553728	+	0.832698i	$0.313205\pi$
$402$	16.1368	0.804831
$403$	−3.11871	−0.155354
$404$	2.81194	0.139899
$405$	0.193937	0.00963679
$406$	−6.35026	−0.315158
$407$	31.0640	1.53978
$408$	−0.418190	−0.0207035
$409$	−36.2228	−1.79110	−0.895552	−	0.444957i	$-0.853219\pi$
$409$	−36.2228	−1.79110	−0.895552	+	0.444957i	$0.853219\pi$
$410$	0.287258	0.0141866
$411$	5.56230	0.274368
$412$	−1.09332	−0.0538641
$413$	−1.96968	−0.0969218
$414$	11.5442	0.567366
$415$	1.18806	0.0583194
$416$	4.23013	0.207399
$417$	1.55642	0.0762181
$418$	14.6107	0.714633
$419$	38.7064	1.89093	0.945466	−	0.325721i	$-0.105607\pi$
$419$	38.7064	1.89093	0.945466	+	0.325721i	$0.105607\pi$
$420$	−0.0376114	−0.00183525
$421$	−2.04491	−0.0996626	−0.0498313	−	0.998758i	$-0.515868\pi$
$421$	−2.04491	−0.0996626	−0.0498313	+	0.998758i	$0.515868\pi$
$422$	0.563711	0.0274410
$423$	−3.15633	−0.153466
$424$	5.00000	0.242821
$425$	0.775746	0.0376292
$426$	15.3199	0.742254
$427$	6.15633	0.297925
$428$	−0.569591	−0.0275322
$429$	−20.7005	−0.999431
$430$	−1.84367	−0.0889099
$431$	−27.8799	−1.34293	−0.671463	−	0.741038i	$-0.734334\pi$
$431$	−27.8799	−1.34293	−0.671463	+	0.741038i	$0.734334\pi$
$432$	4.35026	0.209302
$433$	−15.5271	−0.746183	−0.373091	−	0.927795i	$-0.621702\pi$
$433$	−15.5271	−0.746183	−0.373091	+	0.927795i	$0.621702\pi$
$434$	1.19394	0.0573108
$435$	0.831456	0.0398653
$436$	1.14315	0.0547470
$437$	−14.3693	−0.687378
$438$	−17.5066	−0.836497
$439$	1.74401	0.0832373	0.0416186	−	0.999134i	$-0.486749\pi$
$439$	1.74401	0.0832373	0.0416186	+	0.999134i	$0.486749\pi$
$440$	2.77575	0.132329
$441$	1.00000	0.0476190
$442$	0.895875	0.0426124
$443$	10.9829	0.521811	0.260906	−	0.965364i	$-0.415979\pi$
$443$	10.9829	0.521811	0.260906	+	0.965364i	$0.415979\pi$
$444$	1.12601	0.0534380
$445$	3.22425	0.152844
$446$	−21.9321	−1.03851
$447$	−5.69323	−0.269281
$448$	7.08110	0.334551
$449$	22.8568	1.07868	0.539341	−	0.842088i	$-0.318674\pi$
$449$	22.8568	1.07868	0.539341	+	0.842088i	$0.318674\pi$
$450$	−7.35026	−0.346495
$451$	−5.35026	−0.251934
$452$	3.01317	0.141728
$453$	−9.58769	−0.450469
$454$	13.9140	0.653015
$455$	−0.750354	−0.0351771
$456$	−4.93207	−0.230965
$457$	−0.252016	−0.0117888	−0.00589441	−	0.999983i	$-0.501876\pi$
$457$	−0.252016	−0.0117888	−0.00589441	+	0.999983i	$0.501876\pi$
$458$	−37.3634	−1.74588
$459$	0.156325	0.00729663
$460$	0.293137	0.0136676
$461$	2.56134	0.119294	0.0596468	−	0.998220i	$-0.481003\pi$
$461$	2.56134	0.119294	0.0596468	+	0.998220i	$0.481003\pi$
$462$	7.92478	0.368694
$463$	−11.1465	−0.518021	−0.259010	−	0.965875i	$-0.583396\pi$
$463$	−11.1465	−0.518021	−0.259010	+	0.965875i	$0.583396\pi$
$464$	18.6507	0.865837
$465$	−0.156325	−0.00724941
$466$	1.58181	0.0732759
$467$	−3.87987	−0.179539	−0.0897695	−	0.995963i	$-0.528613\pi$
$467$	−3.87987	−0.179539	−0.0897695	+	0.995963i	$0.528613\pi$
$468$	−0.750354	−0.0346851
$469$	−10.8945	−0.503059
$470$	−0.906679	−0.0418220
$471$	2.68006	0.123490
$472$	5.26916	0.242533
$473$	34.3390	1.57891
$474$	2.39868	0.110175
$475$	9.14903	0.419786
$476$	−0.0303172	−0.00138959
$477$	−1.86907	−0.0855787
$478$	−13.6483	−0.624260
$479$	−39.6747	−1.81278	−0.906391	−	0.422440i	$-0.861174\pi$
$479$	−39.6747	−1.81278	−0.906391	+	0.422440i	$0.861174\pi$
$480$	0.212035	0.00967803
$481$	22.4641	1.02427
$482$	−11.5515	−0.526156
$483$	−7.79384	−0.354632
$484$	3.41819	0.155372
$485$	−0.443583	−0.0201421
$486$	−1.48119	−0.0671883
$487$	15.6385	0.708647	0.354323	−	0.935123i	$-0.384711\pi$
$487$	15.6385	0.708647	0.354323	+	0.935123i	$0.384711\pi$
$488$	−16.4690	−0.745515
$489$	11.9551	0.540628
$490$	0.287258	0.0129770
$491$	−19.7539	−0.891479	−0.445740	−	0.895163i	$-0.647059\pi$
$491$	−19.7539	−0.891479	−0.445740	+	0.895163i	$0.647059\pi$
$492$	−0.193937	−0.00874334
$493$	0.670206	0.0301846
$494$	10.5658	0.475378
$495$	−1.03761	−0.0466372
$496$	−3.50659	−0.157450
$497$	−10.3430	−0.463945
$498$	−9.07381	−0.406607
$499$	−35.3620	−1.58302	−0.791511	−	0.611155i	$-0.790705\pi$
$499$	−35.3620	−1.58302	−0.791511	+	0.611155i	$0.790705\pi$
$500$	−0.374699	−0.0167571
$501$	−18.6121	−0.831529
$502$	32.5501	1.45278
$503$	−1.68594	−0.0751721	−0.0375861	−	0.999293i	$-0.511967\pi$
$503$	−1.68594	−0.0751721	−0.0375861	+	0.999293i	$0.511967\pi$
$504$	−2.67513	−0.119160
$505$	2.81194	0.125130
$506$	−61.7645	−2.74577
$507$	−1.96968	−0.0874767
$508$	−3.19982	−0.141969
$509$	24.2433	1.07457	0.537283	−	0.843402i	$-0.319451\pi$
$509$	24.2433	1.07457	0.537283	+	0.843402i	$0.319451\pi$
$510$	0.0449056	0.00198845
$511$	11.8192	0.522852
$512$	−18.5188	−0.818423
$513$	1.84367	0.0814002
$514$	−30.0508	−1.32548
$515$	−1.09332	−0.0481775
$516$	1.24472	0.0547958
$517$	16.8872	0.742697
$518$	−8.59991	−0.377858
$519$	9.00000	0.395056
$520$	2.00729	0.0880257
$521$	−1.32582	−0.0580854	−0.0290427	−	0.999578i	$-0.509246\pi$
$521$	−1.32582	−0.0580854	−0.0290427	+	0.999578i	$0.509246\pi$
$522$	−6.35026	−0.277943
$523$	−18.1055	−0.791700	−0.395850	−	0.918315i	$-0.629550\pi$
$523$	−18.1055	−0.791700	−0.395850	+	0.918315i	$0.629550\pi$
$524$	0.866698	0.0378619
$525$	4.96239	0.216576
$526$	20.9770	0.914640
$527$	−0.126008	−0.00548899
$528$	−23.2750	−1.01292
$529$	37.7440	1.64104
$530$	−0.536904	−0.0233216
$531$	−1.96968	−0.0854770
$532$	−0.357556	−0.0155020
$533$	−3.86907	−0.167588
$534$	−24.6253	−1.06564
$535$	−0.569591	−0.0246256
$536$	29.1441	1.25883
$537$	17.5066	0.755465
$538$	−2.22074	−0.0957431
$539$	−5.35026	−0.230452
$540$	−0.0376114	−0.00161854
$541$	−7.59498	−0.326534	−0.163267	−	0.986582i	$-0.552203\pi$
$541$	−7.59498	−0.326534	−0.163267	+	0.986582i	$0.552203\pi$
$542$	−34.9669	−1.50195
$543$	11.2447	0.482557
$544$	0.170914	0.00732786
$545$	1.14315	0.0489672
$546$	5.73084	0.245257
$547$	39.1886	1.67558	0.837791	−	0.545991i	$-0.183847\pi$
$547$	39.1886	1.67558	0.837791	+	0.545991i	$0.183847\pi$
$548$	−1.07873	−0.0460812
$549$	6.15633	0.262746
$550$	39.3258	1.67686
$551$	7.90431	0.336735
$552$	20.8496	0.887416
$553$	−1.61942	−0.0688648
$554$	10.2931	0.437314
$555$	1.12601	0.0477964
$556$	−0.301846	−0.0128011
$557$	−20.3331	−0.861542	−0.430771	−	0.902461i	$-0.641758\pi$
$557$	−20.3331	−0.861542	−0.430771	+	0.902461i	$0.641758\pi$
$558$	1.19394	0.0505434
$559$	24.8324	1.05030
$560$	−0.843675	−0.0356518
$561$	−0.836381	−0.0353120
$562$	−33.8178	−1.42652
$563$	−15.7367	−0.663224	−0.331612	−	0.943416i	$-0.607592\pi$
$563$	−15.7367	−0.663224	−0.331612	+	0.943416i	$0.607592\pi$
$564$	0.612127	0.0257752
$565$	3.01317	0.126765
$566$	1.47039	0.0618051
$567$	1.00000	0.0419961
$568$	27.6688	1.16096
$569$	−36.2823	−1.52103	−0.760517	−	0.649318i	$-0.775055\pi$
$569$	−36.2823	−1.52103	−0.760517	+	0.649318i	$0.775055\pi$
$570$	0.529610	0.0221829
$571$	−3.81336	−0.159584	−0.0797920	−	0.996812i	$-0.525426\pi$
$571$	−3.81336	−0.159584	−0.0797920	+	0.996812i	$0.525426\pi$
$572$	4.01459	0.167858
$573$	16.3938	0.684859
$574$	1.48119	0.0618239
$575$	−38.6761	−1.61290
$576$	7.08110	0.295046
$577$	20.7064	0.862019	0.431009	−	0.902347i	$-0.358158\pi$
$577$	20.7064	0.862019	0.431009	+	0.902347i	$0.358158\pi$
$578$	−25.1441	−1.04586
$579$	−16.6556	−0.692184
$580$	−0.161250	−0.00669553
$581$	6.12601	0.254150
$582$	3.38787	0.140432
$583$	10.0000	0.414158
$584$	−31.6180	−1.30836
$585$	−0.750354	−0.0310233
$586$	−18.3792	−0.759236
$587$	36.8773	1.52209	0.761045	−	0.648699i	$-0.224687\pi$
$587$	36.8773	1.52209	0.761045	+	0.648699i	$0.224687\pi$
$588$	−0.193937	−0.00799781
$589$	−1.48612	−0.0612345
$590$	−0.565807	−0.0232939
$591$	9.41090	0.387113
$592$	25.2579	1.03809
$593$	18.6399	0.765449	0.382724	−	0.923863i	$-0.374986\pi$
$593$	18.6399	0.765449	0.382724	+	0.923863i	$0.374986\pi$
$594$	7.92478	0.325158
$595$	−0.0303172	−0.00124288
$596$	1.10413	0.0452267
$597$	21.2301	0.868892
$598$	−44.6653	−1.82650
$599$	−32.0167	−1.30817	−0.654083	−	0.756423i	$-0.726945\pi$
$599$	−32.0167	−1.30817	−0.654083	+	0.756423i	$0.726945\pi$
$600$	−13.2750	−0.541951
$601$	−14.0254	−0.572108	−0.286054	−	0.958214i	$-0.592344\pi$
$601$	−14.0254	−0.572108	−0.286054	+	0.958214i	$0.592344\pi$
$602$	−9.50659	−0.387460
$603$	−10.8945	−0.443657
$604$	1.85940	0.0756581
$605$	3.41819	0.138969
$606$	−21.4763	−0.872414
$607$	−29.9062	−1.21386	−0.606928	−	0.794757i	$-0.707598\pi$
$607$	−29.9062	−1.21386	−0.606928	+	0.794757i	$0.707598\pi$
$608$	2.01573	0.0817486
$609$	4.28726	0.173728
$610$	1.76845	0.0716025
$611$	12.2120	0.494046
$612$	−0.0303172	−0.00122550
$613$	11.3733	0.459363	0.229681	−	0.973266i	$-0.426232\pi$
$613$	11.3733	0.459363	0.229681	+	0.973266i	$0.426232\pi$
$614$	−49.5647	−2.00027
$615$	−0.193937	−0.00782028
$616$	14.3127	0.576673
$617$	−24.4142	−0.982880	−0.491440	−	0.870912i	$-0.663529\pi$
$617$	−24.4142	−0.982880	−0.491440	+	0.870912i	$0.663529\pi$
$618$	8.35026	0.335897
$619$	−28.0372	−1.12691	−0.563454	−	0.826147i	$-0.690528\pi$
$619$	−28.0372	−1.12691	−0.563454	+	0.826147i	$0.690528\pi$
$620$	0.0303172	0.00121757
$621$	−7.79384	−0.312756
$622$	−25.6883	−1.03001
$623$	16.6253	0.666079
$624$	−16.8315	−0.673797
$625$	24.4372	0.977490
$626$	1.54420	0.0617186
$627$	−9.86414	−0.393936
$628$	−0.519761	−0.0207407
$629$	0.907634	0.0361897
$630$	0.287258	0.0114446
$631$	18.5745	0.739440	0.369720	−	0.929143i	$-0.379454\pi$
$631$	18.5745	0.739440	0.369720	+	0.929143i	$0.379454\pi$
$632$	4.33216	0.172324
$633$	−0.380579	−0.0151267
$634$	5.82065	0.231168
$635$	−3.19982	−0.126981
$636$	0.362481	0.0143733
$637$	−3.86907	−0.153298
$638$	33.9756	1.34511
$639$	−10.3430	−0.409161
$640$	2.45817	0.0971678
$641$	−43.7802	−1.72921	−0.864607	−	0.502448i	$-0.832433\pi$
$641$	−43.7802	−1.72921	−0.864607	+	0.502448i	$0.832433\pi$
$642$	4.35026	0.171691
$643$	−22.2677	−0.878154	−0.439077	−	0.898449i	$-0.644694\pi$
$643$	−22.2677	−0.878154	−0.439077	+	0.898449i	$0.644694\pi$
$644$	1.51151	0.0595619
$645$	1.24472	0.0490109
$646$	0.426899	0.0167961
$647$	−45.6589	−1.79504	−0.897519	−	0.440976i	$-0.854633\pi$
$647$	−45.6589	−1.79504	−0.897519	+	0.440976i	$0.854633\pi$
$648$	−2.67513	−0.105089
$649$	10.5383	0.413665
$650$	28.4387	1.11546
$651$	−0.806063	−0.0315921
$652$	−2.31853	−0.0908006
$653$	7.11871	0.278577	0.139288	−	0.990252i	$-0.455518\pi$
$653$	7.11871	0.278577	0.139288	+	0.990252i	$0.455518\pi$
$654$	−8.73084	−0.341403
$655$	0.866698	0.0338647
$656$	−4.35026	−0.169849
$657$	11.8192	0.461112
$658$	−4.67513	−0.182256
$659$	−15.6531	−0.609757	−0.304878	−	0.952391i	$-0.598616\pi$
$659$	−15.6531	−0.609757	−0.304878	+	0.952391i	$0.598616\pi$
$660$	0.201231	0.00783290
$661$	30.2981	1.17846	0.589229	−	0.807966i	$-0.299432\pi$
$661$	30.2981	1.17846	0.589229	+	0.807966i	$0.299432\pi$
$662$	25.9199	1.00740
$663$	−0.604833	−0.0234898
$664$	−16.3879	−0.635973
$665$	−0.357556	−0.0138654
$666$	−8.59991	−0.333240
$667$	−33.4142	−1.29380
$668$	3.60957	0.139659
$669$	14.8070	0.572472
$670$	−3.12952	−0.120904
$671$	−32.9380	−1.27156
$672$	1.09332	0.0421758
$673$	−39.6688	−1.52912	−0.764560	−	0.644553i	$-0.777044\pi$
$673$	−39.6688	−1.52912	−0.764560	+	0.644553i	$0.777044\pi$
$674$	−15.2858	−0.588789
$675$	4.96239	0.191002
$676$	0.381994	0.0146921
$677$	29.3503	1.12802	0.564011	−	0.825767i	$-0.309258\pi$
$677$	29.3503	1.12802	0.564011	+	0.825767i	$0.309258\pi$
$678$	−23.0132	−0.883816
$679$	−2.28726	−0.0877769
$680$	0.0811024	0.00311013
$681$	−9.39375	−0.359969
$682$	−6.38787	−0.244604
$683$	−13.7626	−0.526610	−0.263305	−	0.964713i	$-0.584813\pi$
$683$	−13.7626	−0.526610	−0.263305	+	0.964713i	$0.584813\pi$
$684$	−0.357556	−0.0136715
$685$	−1.07873	−0.0412163
$686$	1.48119	0.0565523
$687$	25.2252	0.962402
$688$	27.9208	1.06447
$689$	7.23155	0.275500
$690$	−2.23884	−0.0852313
$691$	51.4274	1.95639	0.978195	−	0.207688i	$-0.0665938\pi$
$691$	51.4274	1.95639	0.978195	+	0.207688i	$0.0665938\pi$
$692$	−1.74543	−0.0663513
$693$	−5.35026	−0.203240
$694$	−4.93207	−0.187219
$695$	−0.301846	−0.0114497
$696$	−11.4690	−0.434730
$697$	−0.156325	−0.00592124
$698$	33.5790	1.27098
$699$	−1.06793	−0.0403928
$700$	−0.962389	−0.0363749
$701$	−22.9478	−0.866726	−0.433363	−	0.901219i	$-0.642673\pi$
$701$	−22.9478	−0.866726	−0.433363	+	0.901219i	$0.642673\pi$
$702$	5.73084	0.216297
$703$	10.7045	0.403728
$704$	−37.8858	−1.42787
$705$	0.612127	0.0230540
$706$	52.6225	1.98047
$707$	14.4993	0.545302
$708$	0.381994	0.0143562
$709$	−41.9511	−1.57551	−0.787754	−	0.615990i	$-0.788756\pi$
$709$	−41.9511	−1.57551	−0.787754	+	0.615990i	$0.788756\pi$
$710$	−2.97110	−0.111503
$711$	−1.61942	−0.0607330
$712$	−44.4749	−1.66677
$713$	6.28233	0.235275
$714$	0.231548	0.00866546
$715$	4.01459	0.150137
$716$	−3.39517	−0.126883
$717$	9.21440	0.344118
$718$	35.3272	1.31840
$719$	−34.8265	−1.29881	−0.649405	−	0.760443i	$-0.724982\pi$
$719$	−34.8265	−1.29881	−0.649405	+	0.760443i	$0.724982\pi$
$720$	−0.843675	−0.0314419
$721$	−5.63752	−0.209952
$722$	−23.1079	−0.859987
$723$	7.79877	0.290039
$724$	−2.18076	−0.0810474
$725$	21.2750	0.790135
$726$	−26.1065	−0.968903
$727$	9.21440	0.341743	0.170872	−	0.985293i	$-0.445342\pi$
$727$	9.21440	0.341743	0.170872	+	0.985293i	$0.445342\pi$
$728$	10.3503	0.383606
$729$	1.00000	0.0370370
$730$	3.39517	0.125661
$731$	1.00332	0.0371093
$732$	−1.19394	−0.0441292
$733$	−22.2374	−0.821358	−0.410679	−	0.911780i	$-0.634708\pi$
$733$	−22.2374	−0.821358	−0.410679	+	0.911780i	$0.634708\pi$
$734$	40.0278	1.47745
$735$	−0.193937	−0.00715346
$736$	−8.52118	−0.314095
$737$	58.2882	2.14707
$738$	1.48119	0.0545235
$739$	20.0910	0.739058	0.369529	−	0.929219i	$-0.379519\pi$
$739$	20.0910	0.739058	0.369529	+	0.929219i	$0.379519\pi$
$740$	−0.218374	−0.00802759
$741$	−7.13330	−0.262048
$742$	−2.76845	−0.101633
$743$	35.3561	1.29709	0.648546	−	0.761176i	$-0.275378\pi$
$743$	35.3561	1.29709	0.648546	+	0.761176i	$0.275378\pi$
$744$	2.15633	0.0790547
$745$	1.10413	0.0404520
$746$	39.9111	1.46125
$747$	6.12601	0.224139
$748$	0.162205	0.00593079
$749$	−2.93700	−0.107315
$750$	2.86177	0.104497
$751$	35.9829	1.31303	0.656517	−	0.754312i	$-0.272029\pi$
$751$	35.9829	1.31303	0.656517	+	0.754312i	$0.272029\pi$
$752$	13.7308	0.500712
$753$	−21.9756	−0.800834
$754$	24.5696	0.894772
$755$	1.85940	0.0676706
$756$	−0.193937	−0.00705340
$757$	1.66291	0.0604396	0.0302198	−	0.999543i	$-0.490379\pi$
$757$	1.66291	0.0604396	0.0302198	+	0.999543i	$0.490379\pi$
$758$	−33.4460	−1.21481
$759$	41.6991	1.51358
$760$	0.956509	0.0346962
$761$	39.0059	1.41396	0.706981	−	0.707233i	$-0.250057\pi$
$761$	39.0059	1.41396	0.706981	+	0.707233i	$0.250057\pi$
$762$	24.4387	0.885319
$763$	5.89446	0.213394
$764$	−3.17935	−0.115025
$765$	−0.0303172	−0.00109612
$766$	−35.1500	−1.27002
$767$	7.62084	0.275172
$768$	−4.61213	−0.166426
$769$	41.5271	1.49750	0.748752	−	0.662850i	$-0.230653\pi$
$769$	41.5271	1.49750	0.748752	+	0.662850i	$0.230653\pi$
$770$	−1.53690	−0.0553862
$771$	20.2882	0.730662
$772$	3.23013	0.116255
$773$	46.4894	1.67211	0.836055	−	0.548646i	$-0.184857\pi$
$773$	46.4894	1.67211	0.836055	+	0.548646i	$0.184857\pi$
$774$	−9.50659	−0.341707
$775$	−4.00000	−0.143684
$776$	6.11871	0.219649
$777$	5.80606	0.208291
$778$	34.4299	1.23437
$779$	−1.84367	−0.0660565
$780$	0.145521	0.00521049
$781$	55.3376	1.98013
$782$	−1.80465	−0.0645341
$783$	4.28726	0.153214
$784$	−4.35026	−0.155366
$785$	−0.519761	−0.0185511
$786$	−6.61942	−0.236107
$787$	24.6058	0.877102	0.438551	−	0.898706i	$-0.355492\pi$
$787$	24.6058	0.877102	0.438551	+	0.898706i	$0.355492\pi$
$788$	−1.82512	−0.0650171
$789$	−14.1622	−0.504188
$790$	−0.465191	−0.0165508
$791$	15.5369	0.552429
$792$	14.3127	0.508578
$793$	−23.8192	−0.845846
$794$	15.0435	0.533874
$795$	0.362481	0.0128559
$796$	−4.11730	−0.145934
$797$	−40.7123	−1.44210	−0.721052	−	0.692881i	$-0.756341\pi$
$797$	−40.7123	−1.44210	−0.721052	+	0.692881i	$0.756341\pi$
$798$	2.73084	0.0966707
$799$	0.493413	0.0174557
$800$	5.42548	0.191820
$801$	16.6253	0.587426
$802$	32.8481	1.15991
$803$	−63.2360	−2.23155
$804$	2.11283	0.0745139
$805$	1.51151	0.0532738
$806$	−4.61942	−0.162712
$807$	1.49929	0.0527776
$808$	−38.7875	−1.36454
$809$	−12.1916	−0.428633	−0.214316	−	0.976764i	$-0.568752\pi$
$809$	−12.1916	−0.428633	−0.214316	+	0.976764i	$0.568752\pi$
$810$	0.287258	0.0100932
$811$	−39.7997	−1.39756	−0.698779	−	0.715338i	$-0.746273\pi$
$811$	−39.7997	−1.39756	−0.698779	+	0.715338i	$0.746273\pi$
$812$	−0.831456	−0.0291784
$813$	23.6072	0.827941
$814$	46.0118	1.61271
$815$	−2.31853	−0.0812146
$816$	−0.680055	−0.0238067
$817$	11.8331	0.413986
$818$	−53.6531	−1.87594
$819$	−3.86907	−0.135196
$820$	0.0376114	0.00131345
$821$	4.47882	0.156312	0.0781560	−	0.996941i	$-0.475097\pi$
$821$	4.47882	0.156312	0.0781560	+	0.996941i	$0.475097\pi$
$822$	8.23884	0.287363
$823$	23.4471	0.817314	0.408657	−	0.912688i	$-0.365997\pi$
$823$	23.4471	0.817314	0.408657	+	0.912688i	$0.365997\pi$
$824$	15.0811	0.525375
$825$	−26.5501	−0.924355
$826$	−2.91748	−0.101512
$827$	−18.1524	−0.631219	−0.315610	−	0.948889i	$-0.602209\pi$
$827$	−18.1524	−0.631219	−0.315610	+	0.948889i	$0.602209\pi$
$828$	1.51151	0.0525287
$829$	−6.37328	−0.221353	−0.110677	−	0.993856i	$-0.535302\pi$
$829$	−6.37328	−0.221353	−0.110677	+	0.993856i	$0.535302\pi$
$830$	1.75974	0.0610816
$831$	−6.94921	−0.241066
$832$	−27.3973	−0.949829
$833$	−0.156325	−0.00541635
$834$	2.30536	0.0798280
$835$	3.60957	0.124914
$836$	1.91302	0.0661631
$837$	−0.806063	−0.0278616
$838$	57.3317	1.98049
$839$	−7.63515	−0.263595	−0.131797	−	0.991277i	$-0.542075\pi$
$839$	−7.63515	−0.263595	−0.131797	+	0.991277i	$0.542075\pi$
$840$	0.518806	0.0179005
$841$	−10.6194	−0.366187
$842$	−3.02890	−0.104383
$843$	22.8315	0.786357
$844$	0.0738082	0.00254058
$845$	0.381994	0.0131410
$846$	−4.67513	−0.160734
$847$	17.6253	0.605612
$848$	8.13093	0.279217
$849$	−0.992706	−0.0340696
$850$	1.14903	0.0394114
$851$	−45.2516	−1.55120
$852$	2.00588	0.0687203
$853$	−11.0884	−0.379659	−0.189830	−	0.981817i	$-0.560794\pi$
$853$	−11.0884	−0.379659	−0.189830	+	0.981817i	$0.560794\pi$
$854$	9.11871	0.312036
$855$	−0.357556	−0.0122282
$856$	7.85685	0.268542
$857$	10.1286	0.345985	0.172993	−	0.984923i	$-0.444656\pi$
$857$	10.1286	0.345985	0.172993	+	0.984923i	$0.444656\pi$
$858$	−30.6615	−1.04677
$859$	47.5221	1.62143	0.810717	−	0.585438i	$-0.199077\pi$
$859$	47.5221	1.62143	0.810717	+	0.585438i	$0.199077\pi$
$860$	−0.241397	−0.00823157
$861$	−1.00000	−0.0340799
$862$	−41.2955	−1.40653
$863$	48.3498	1.64585	0.822923	−	0.568153i	$-0.192342\pi$
$863$	48.3498	1.64585	0.822923	+	0.568153i	$0.192342\pi$
$864$	1.09332	0.0371955
$865$	−1.74543	−0.0593464
$866$	−22.9986	−0.781524
$867$	16.9756	0.576520
$868$	0.156325	0.00530602
$869$	8.66433	0.293917
$870$	1.23155	0.0417534
$871$	42.1514	1.42825
$872$	−15.7685	−0.533987
$873$	−2.28726	−0.0774119
$874$	−21.2837	−0.719934
$875$	−1.93207	−0.0653159
$876$	−2.29218	−0.0774457
$877$	47.7137	1.61118	0.805589	−	0.592475i	$-0.201849\pi$
$877$	47.7137	1.61118	0.805589	+	0.592475i	$0.201849\pi$
$878$	2.58322	0.0871796
$879$	12.4083	0.418523
$880$	4.51388	0.152163
$881$	38.3660	1.29258	0.646292	−	0.763091i	$-0.276319\pi$
$881$	38.3660	1.29258	0.646292	+	0.763091i	$0.276319\pi$
$882$	1.48119	0.0498744
$883$	36.4372	1.22621	0.613105	−	0.790001i	$-0.289920\pi$
$883$	36.4372	1.22621	0.613105	+	0.790001i	$0.289920\pi$
$884$	0.117299	0.00394520
$885$	0.381994	0.0128406
$886$	16.2677	0.546526
$887$	−10.9149	−0.366487	−0.183244	−	0.983068i	$-0.558660\pi$
$887$	−10.9149	−0.366487	−0.183244	+	0.983068i	$0.558660\pi$
$888$	−15.5320	−0.521219
$889$	−16.4993	−0.553369
$890$	4.77575	0.160083
$891$	−5.35026	−0.179241
$892$	−2.87162	−0.0961490
$893$	5.81924	0.194733
$894$	−8.43278	−0.282034
$895$	−3.39517	−0.113488
$896$	12.6751	0.423446
$897$	30.1549	1.00684
$898$	33.8554	1.12977
$899$	−3.45580	−0.115257
$900$	−0.962389	−0.0320796
$901$	0.292182	0.00973400
$902$	−7.92478	−0.263866
$903$	6.41819	0.213584
$904$	−41.5633	−1.38237
$905$	−2.18076	−0.0724910
$906$	−14.2012	−0.471804
$907$	42.4993	1.41117	0.705583	−	0.708627i	$-0.250685\pi$
$907$	42.4993	1.41117	0.705583	+	0.708627i	$0.250685\pi$
$908$	1.82179	0.0604583
$909$	14.4993	0.480911
$910$	−1.11142	−0.0368432
$911$	−4.93207	−0.163407	−0.0817034	−	0.996657i	$-0.526036\pi$
$911$	−4.93207	−0.163407	−0.0817034	+	0.996657i	$0.526036\pi$
$912$	−8.02047	−0.265584
$913$	−32.7757	−1.08472
$914$	−0.373285	−0.0123472
$915$	−1.19394	−0.0394703
$916$	−4.89209	−0.161639
$917$	4.46898	0.147579
$918$	0.231548	0.00764222
$919$	−4.02776	−0.132864	−0.0664318	−	0.997791i	$-0.521161\pi$
$919$	−4.02776	−0.132864	−0.0664318	+	0.997791i	$0.521161\pi$
$920$	−4.04349	−0.133310
$921$	33.4626	1.10263
$922$	3.79384	0.124944
$923$	40.0176	1.31720
$924$	1.03761	0.0341349
$925$	28.8119	0.947331
$926$	−16.5101	−0.542555
$927$	−5.63752	−0.185160
$928$	4.68735	0.153870
$929$	14.9076	0.489104	0.244552	−	0.969636i	$-0.421359\pi$
$929$	14.9076	0.489104	0.244552	+	0.969636i	$0.421359\pi$
$930$	−0.231548	−0.00759276
$931$	−1.84367	−0.0604240
$932$	0.207110	0.00678413
$933$	17.3430	0.567783
$934$	−5.74684	−0.188042
$935$	0.162205	0.00530466
$936$	10.3503	0.338309
$937$	−22.0957	−0.721835	−0.360917	−	0.932598i	$-0.617536\pi$
$937$	−22.0957	−0.721835	−0.360917	+	0.932598i	$0.617536\pi$
$938$	−16.1368	−0.526886
$939$	−1.04254	−0.0340219
$940$	−0.118714	−0.00387202
$941$	34.0395	1.10966	0.554828	−	0.831965i	$-0.312784\pi$
$941$	34.0395	1.10966	0.554828	+	0.831965i	$0.312784\pi$
$942$	3.96968	0.129339
$943$	7.79384	0.253803
$944$	8.56864	0.278885
$945$	−0.193937	−0.00630876
$946$	50.8627	1.65369
$947$	27.4363	0.891560	0.445780	−	0.895143i	$-0.352926\pi$
$947$	27.4363	0.891560	0.445780	+	0.895143i	$0.352926\pi$
$948$	0.314065	0.0102004
$949$	−45.7294	−1.48444
$950$	13.5515	0.439668
$951$	−3.92970	−0.127429
$952$	0.418190	0.0135536
$953$	43.4676	1.40805	0.704026	−	0.710174i	$-0.251384\pi$
$953$	43.4676	1.40805	0.704026	+	0.710174i	$0.251384\pi$
$954$	−2.76845	−0.0896319
$955$	−3.17935	−0.102881
$956$	−1.78701	−0.0577960
$957$	−22.9380	−0.741479
$958$	−58.7659	−1.89864
$959$	−5.56230	−0.179616
$960$	−1.37328	−0.0443226
$961$	−30.3503	−0.979041
$962$	33.2736	1.07278
$963$	−2.93700	−0.0946434
$964$	−1.51247	−0.0487133
$965$	3.23013	0.103982
$966$	−11.5442	−0.371428
$967$	24.5026	0.787951	0.393976	−	0.919121i	$-0.371099\pi$
$967$	24.5026	0.787951	0.393976	+	0.919121i	$0.371099\pi$
$968$	−47.1500	−1.51546
$969$	−0.288213	−0.00925873
$970$	−0.657032	−0.0210960
$971$	−14.2184	−0.456289	−0.228145	−	0.973627i	$-0.573266\pi$
$971$	−14.2184	−0.456289	−0.228145	+	0.973627i	$0.573266\pi$
$972$	−0.193937	−0.00622052
$973$	−1.55642	−0.0498964
$974$	23.1636	0.742210
$975$	−19.1998	−0.614886
$976$	−26.7816	−0.857259
$977$	42.4201	1.35714	0.678570	−	0.734536i	$-0.262600\pi$
$977$	42.4201	1.35714	0.678570	+	0.734536i	$0.262600\pi$
$978$	17.7078	0.566234
$979$	−88.9497	−2.84285
$980$	0.0376114	0.00120145
$981$	5.89446	0.188196
$982$	−29.2593	−0.933702
$983$	9.16950	0.292462	0.146231	−	0.989251i	$-0.453286\pi$
$983$	9.16950	0.292462	0.146231	+	0.989251i	$0.453286\pi$
$984$	2.67513	0.0852801
$985$	−1.82512	−0.0581531
$986$	0.992706	0.0316142
$987$	3.15633	0.100467
$988$	1.38341	0.0440121
$989$	−50.0224	−1.59062
$990$	−1.53690	−0.0488460
$991$	13.0870	0.415722	0.207861	−	0.978158i	$-0.433350\pi$
$991$	13.0870	0.415722	0.207861	+	0.978158i	$0.433350\pi$
$992$	−0.881286	−0.0279809
$993$	−17.4993	−0.555323
$994$	−15.3199	−0.485919
$995$	−4.11730	−0.130527
$996$	−1.18806	−0.0376450
$997$	−49.5280	−1.56857	−0.784284	−	0.620402i	$-0.786969\pi$
$997$	−49.5280	−1.56857	−0.784284	+	0.620402i	$0.786969\pi$
$998$	−52.3780	−1.65800
$999$	5.80606	0.183696

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	861.2.a.h.1.3	✓	3
3.2	odd	2		2583.2.a.n.1.1		3
7.6	odd	2		6027.2.a.o.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
861.2.a.h.1.3	✓	3	1.1	even	1	trivial
2583.2.a.n.1.1		3	3.2	odd	2
6027.2.a.o.1.3		3	7.6	odd	2

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:	\( N \)	\(=\)	\( 861 = 3 \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	861.a (trivial)

\(f(q)\)	\(=\)	\(q+1.48119 q^{2} -1.00000 q^{3} +0.193937 q^{4} +0.193937 q^{5} -1.48119 q^{6} +1.00000 q^{7} -2.67513 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.48119 q^{2} -1.00000 q^{3} +0.193937 q^{4} +0.193937 q^{5} -1.48119 q^{6} +1.00000 q^{7} -2.67513 q^{8} +1.00000 q^{9} +0.287258 q^{10} -5.35026 q^{11} -0.193937 q^{12} -3.86907 q^{13} +1.48119 q^{14} -0.193937 q^{15} -4.35026 q^{16} -0.156325 q^{17} +1.48119 q^{18} -1.84367 q^{19} +0.0376114 q^{20} -1.00000 q^{21} -7.92478 q^{22} +7.79384 q^{23} +2.67513 q^{24} -4.96239 q^{25} -5.73084 q^{26} -1.00000 q^{27} +0.193937 q^{28} -4.28726 q^{29} -0.287258 q^{30} +0.806063 q^{31} -1.09332 q^{32} +5.35026 q^{33} -0.231548 q^{34} +0.193937 q^{35} +0.193937 q^{36} -5.80606 q^{37} -2.73084 q^{38} +3.86907 q^{39} -0.518806 q^{40} +1.00000 q^{41} -1.48119 q^{42} -6.41819 q^{43} -1.03761 q^{44} +0.193937 q^{45} +11.5442 q^{46} -3.15633 q^{47} +4.35026 q^{48} +1.00000 q^{49} -7.35026 q^{50} +0.156325 q^{51} -0.750354 q^{52} -1.86907 q^{53} -1.48119 q^{54} -1.03761 q^{55} -2.67513 q^{56} +1.84367 q^{57} -6.35026 q^{58} -1.96968 q^{59} -0.0376114 q^{60} +6.15633 q^{61} +1.19394 q^{62} +1.00000 q^{63} +7.08110 q^{64} -0.750354 q^{65} +7.92478 q^{66} -10.8945 q^{67} -0.0303172 q^{68} -7.79384 q^{69} +0.287258 q^{70} -10.3430 q^{71} -2.67513 q^{72} +11.8192 q^{73} -8.59991 q^{74} +4.96239 q^{75} -0.357556 q^{76} -5.35026 q^{77} +5.73084 q^{78} -1.61942 q^{79} -0.843675 q^{80} +1.00000 q^{81} +1.48119 q^{82} +6.12601 q^{83} -0.193937 q^{84} -0.0303172 q^{85} -9.50659 q^{86} +4.28726 q^{87} +14.3127 q^{88} +16.6253 q^{89} +0.287258 q^{90} -3.86907 q^{91} +1.51151 q^{92} -0.806063 q^{93} -4.67513 q^{94} -0.357556 q^{95} +1.09332 q^{96} -2.28726 q^{97} +1.48119 q^{98} -5.35026 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} - 3 q^{3} + q^{4} + q^{5} + q^{6} + 3 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q - q^2 - 3 * q^3 + q^4 + q^5 + q^6 + 3 * q^7 - 3 * q^8 + 3 * q^9	\( 3 q - q^{2} - 3 q^{3} + q^{4} + q^{5} + q^{6} + 3 q^{7} - 3 q^{8} + 3 q^{9} - 5 q^{10} - 6 q^{11} - q^{12} - 7 q^{13} - q^{14} - q^{15} - 3 q^{16} + 10 q^{17} - q^{18} - 16 q^{19} + 11 q^{20} - 3 q^{21} - 2 q^{22} - 3 q^{23} + 3 q^{24} - 4 q^{25} + 5 q^{26} - 3 q^{27} + q^{28} - 7 q^{29} + 5 q^{30} + 2 q^{31} + 3 q^{32} + 6 q^{33} - 12 q^{34} + q^{35} + q^{36} - 17 q^{37} + 14 q^{38} + 7 q^{39} - 7 q^{40} + 3 q^{41} + q^{42} - 18 q^{43} - 14 q^{44} + q^{45} + 25 q^{46} + q^{47} + 3 q^{48} + 3 q^{49} - 12 q^{50} - 10 q^{51} - 19 q^{52} - q^{53} + q^{54} - 14 q^{55} - 3 q^{56} + 16 q^{57} - 9 q^{58} - 8 q^{59} - 11 q^{60} + 8 q^{61} + 4 q^{62} + 3 q^{63} - 11 q^{64} - 19 q^{65} + 2 q^{66} - 13 q^{67} + 2 q^{68} + 3 q^{69} - 5 q^{70} - 8 q^{71} - 3 q^{72} - 6 q^{73} + q^{74} + 4 q^{75} - 4 q^{76} - 6 q^{77} - 5 q^{78} - 17 q^{79} - 13 q^{80} + 3 q^{81} - q^{82} + 10 q^{83} - q^{84} + 2 q^{85} - 8 q^{86} + 7 q^{87} + 22 q^{88} + 8 q^{89} - 5 q^{90} - 7 q^{91} - 3 q^{92} - 2 q^{93} - 9 q^{94} - 4 q^{95} - 3 q^{96} - q^{97} - q^{98} - 6 q^{99}+O(q^{100}) \) 3 * q - q^2 - 3 * q^3 + q^4 + q^5 + q^6 + 3 * q^7 - 3 * q^8 + 3 * q^9 - 5 * q^10 - 6 * q^11 - q^12 - 7 * q^13 - q^14 - q^15 - 3 * q^16 + 10 * q^17 - q^18 - 16 * q^19 + 11 * q^20 - 3 * q^21 - 2 * q^22 - 3 * q^23 + 3 * q^24 - 4 * q^25 + 5 * q^26 - 3 * q^27 + q^28 - 7 * q^29 + 5 * q^30 + 2 * q^31 + 3 * q^32 + 6 * q^33 - 12 * q^34 + q^35 + q^36 - 17 * q^37 + 14 * q^38 + 7 * q^39 - 7 * q^40 + 3 * q^41 + q^42 - 18 * q^43 - 14 * q^44 + q^45 + 25 * q^46 + q^47 + 3 * q^48 + 3 * q^49 - 12 * q^50 - 10 * q^51 - 19 * q^52 - q^53 + q^54 - 14 * q^55 - 3 * q^56 + 16 * q^57 - 9 * q^58 - 8 * q^59 - 11 * q^60 + 8 * q^61 + 4 * q^62 + 3 * q^63 - 11 * q^64 - 19 * q^65 + 2 * q^66 - 13 * q^67 + 2 * q^68 + 3 * q^69 - 5 * q^70 - 8 * q^71 - 3 * q^72 - 6 * q^73 + q^74 + 4 * q^75 - 4 * q^76 - 6 * q^77 - 5 * q^78 - 17 * q^79 - 13 * q^80 + 3 * q^81 - q^82 + 10 * q^83 - q^84 + 2 * q^85 - 8 * q^86 + 7 * q^87 + 22 * q^88 + 8 * q^89 - 5 * q^90 - 7 * q^91 - 3 * q^92 - 2 * q^93 - 9 * q^94 - 4 * q^95 - 3 * q^96 - q^97 - q^98 - 6 * q^99

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