LMFDB - Embedded newform 8034.2.a.p.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8034,2,Mod(1,8034)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8034, 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("8034.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8034 = 2 \cdot 3 \cdot 13 \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8034.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:	$64.1518129839$
Analytic rank:	$1$
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} - x^{7} - 12x^{6} + 12x^{5} + 43x^{4} - 38x^{3} - 49x^{2} + 23x + 20 $ x^8 - x^7 - 12x^6 + 12x^5 + 43x^4 - 38x^3 - 49x^2 + 23x + 20
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			1.4
Root			$2.41269$ of defining polynomial
Character	$\chi$	$=$	8034.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.15050 q^{5} +1.00000 q^{6} -2.62035 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.15050 q^{5} +1.00000 q^{6} -2.62035 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.15050 q^{10} -1.79292 q^{11} +1.00000 q^{12} +1.00000 q^{13} -2.62035 q^{14} -1.15050 q^{15} +1.00000 q^{16} +3.85728 q^{17} +1.00000 q^{18} +1.79827 q^{19} -1.15050 q^{20} -2.62035 q^{21} -1.79292 q^{22} -1.50203 q^{23} +1.00000 q^{24} -3.67636 q^{25} +1.00000 q^{26} +1.00000 q^{27} -2.62035 q^{28} -4.09888 q^{29} -1.15050 q^{30} -7.08604 q^{31} +1.00000 q^{32} -1.79292 q^{33} +3.85728 q^{34} +3.01470 q^{35} +1.00000 q^{36} +0.639684 q^{37} +1.79827 q^{38} +1.00000 q^{39} -1.15050 q^{40} +8.05054 q^{41} -2.62035 q^{42} -5.61939 q^{43} -1.79292 q^{44} -1.15050 q^{45} -1.50203 q^{46} +8.43678 q^{47} +1.00000 q^{48} -0.133785 q^{49} -3.67636 q^{50} +3.85728 q^{51} +1.00000 q^{52} -4.93059 q^{53} +1.00000 q^{54} +2.06275 q^{55} -2.62035 q^{56} +1.79827 q^{57} -4.09888 q^{58} +1.09119 q^{59} -1.15050 q^{60} -2.73413 q^{61} -7.08604 q^{62} -2.62035 q^{63} +1.00000 q^{64} -1.15050 q^{65} -1.79292 q^{66} -2.82294 q^{67} +3.85728 q^{68} -1.50203 q^{69} +3.01470 q^{70} +0.437093 q^{71} +1.00000 q^{72} -14.4632 q^{73} +0.639684 q^{74} -3.67636 q^{75} +1.79827 q^{76} +4.69807 q^{77} +1.00000 q^{78} -3.04943 q^{79} -1.15050 q^{80} +1.00000 q^{81} +8.05054 q^{82} -14.1654 q^{83} -2.62035 q^{84} -4.43779 q^{85} -5.61939 q^{86} -4.09888 q^{87} -1.79292 q^{88} -7.12626 q^{89} -1.15050 q^{90} -2.62035 q^{91} -1.50203 q^{92} -7.08604 q^{93} +8.43678 q^{94} -2.06891 q^{95} +1.00000 q^{96} +2.70016 q^{97} -0.133785 q^{98} -1.79292 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} - 8 q^{5} + 8 q^{6} - 6 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^2 + 8 * q^3 + 8 * q^4 - 8 * q^5 + 8 * q^6 - 6 * q^7 + 8 * q^8 + 8 * q^9	\( 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} - 8 q^{5} + 8 q^{6} - 6 q^{7} + 8 q^{8} + 8 q^{9} - 8 q^{10} - 7 q^{11} + 8 q^{12} + 8 q^{13} - 6 q^{14} - 8 q^{15} + 8 q^{16} - 20 q^{17} + 8 q^{18} - 12 q^{19} - 8 q^{20} - 6 q^{21} - 7 q^{22} - 14 q^{23} + 8 q^{24} - 2 q^{25} + 8 q^{26} + 8 q^{27} - 6 q^{28} - 25 q^{29} - 8 q^{30} - 12 q^{31} + 8 q^{32} - 7 q^{33} - 20 q^{34} - 18 q^{35} + 8 q^{36} - 15 q^{37} - 12 q^{38} + 8 q^{39} - 8 q^{40} - 18 q^{41} - 6 q^{42} - 8 q^{43} - 7 q^{44} - 8 q^{45} - 14 q^{46} - 12 q^{47} + 8 q^{48} - 8 q^{49} - 2 q^{50} - 20 q^{51} + 8 q^{52} - 25 q^{53} + 8 q^{54} - 8 q^{55} - 6 q^{56} - 12 q^{57} - 25 q^{58} - 9 q^{59} - 8 q^{60} - 2 q^{61} - 12 q^{62} - 6 q^{63} + 8 q^{64} - 8 q^{65} - 7 q^{66} - 8 q^{67} - 20 q^{68} - 14 q^{69} - 18 q^{70} - 13 q^{71} + 8 q^{72} - 2 q^{73} - 15 q^{74} - 2 q^{75} - 12 q^{76} - 5 q^{77} + 8 q^{78} + q^{79} - 8 q^{80} + 8 q^{81} - 18 q^{82} - 6 q^{83} - 6 q^{84} + 5 q^{85} - 8 q^{86} - 25 q^{87} - 7 q^{88} - 17 q^{89} - 8 q^{90} - 6 q^{91} - 14 q^{92} - 12 q^{93} - 12 q^{94} + 10 q^{95} + 8 q^{96} + 19 q^{97} - 8 q^{98} - 7 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 + 8 * q^3 + 8 * q^4 - 8 * q^5 + 8 * q^6 - 6 * q^7 + 8 * q^8 + 8 * q^9 - 8 * q^10 - 7 * q^11 + 8 * q^12 + 8 * q^13 - 6 * q^14 - 8 * q^15 + 8 * q^16 - 20 * q^17 + 8 * q^18 - 12 * q^19 - 8 * q^20 - 6 * q^21 - 7 * q^22 - 14 * q^23 + 8 * q^24 - 2 * q^25 + 8 * q^26 + 8 * q^27 - 6 * q^28 - 25 * q^29 - 8 * q^30 - 12 * q^31 + 8 * q^32 - 7 * q^33 - 20 * q^34 - 18 * q^35 + 8 * q^36 - 15 * q^37 - 12 * q^38 + 8 * q^39 - 8 * q^40 - 18 * q^41 - 6 * q^42 - 8 * q^43 - 7 * q^44 - 8 * q^45 - 14 * q^46 - 12 * q^47 + 8 * q^48 - 8 * q^49 - 2 * q^50 - 20 * q^51 + 8 * q^52 - 25 * q^53 + 8 * q^54 - 8 * q^55 - 6 * q^56 - 12 * q^57 - 25 * q^58 - 9 * q^59 - 8 * q^60 - 2 * q^61 - 12 * q^62 - 6 * q^63 + 8 * q^64 - 8 * q^65 - 7 * q^66 - 8 * q^67 - 20 * q^68 - 14 * q^69 - 18 * q^70 - 13 * q^71 + 8 * q^72 - 2 * q^73 - 15 * q^74 - 2 * q^75 - 12 * q^76 - 5 * q^77 + 8 * q^78 + q^79 - 8 * q^80 + 8 * q^81 - 18 * q^82 - 6 * q^83 - 6 * q^84 + 5 * q^85 - 8 * q^86 - 25 * q^87 - 7 * q^88 - 17 * q^89 - 8 * q^90 - 6 * q^91 - 14 * q^92 - 12 * q^93 - 12 * q^94 + 10 * q^95 + 8 * q^96 + 19 * q^97 - 8 * q^98 - 7 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	−1.15050	−0.514518	−0.257259	−	0.966343i	$-0.582819\pi$
$5$	−1.15050	−0.514518	−0.257259	+	0.966343i	$0.582819\pi$
$6$	1.00000	0.408248
$7$	−2.62035	−0.990398	−0.495199	−	0.868780i	$-0.664905\pi$
$7$	−2.62035	−0.990398	−0.495199	+	0.868780i	$0.664905\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−1.15050	−0.363819
$11$	−1.79292	−0.540586	−0.270293	−	0.962778i	$-0.587121\pi$
$11$	−1.79292	−0.540586	−0.270293	+	0.962778i	$0.587121\pi$
$12$	1.00000	0.288675
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−2.62035	−0.700317
$15$	−1.15050	−0.297057
$16$	1.00000	0.250000
$17$	3.85728	0.935528	0.467764	−	0.883853i	$-0.345060\pi$
$17$	3.85728	0.935528	0.467764	+	0.883853i	$0.345060\pi$
$18$	1.00000	0.235702
$19$	1.79827	0.412552	0.206276	−	0.978494i	$-0.433866\pi$
$19$	1.79827	0.412552	0.206276	+	0.978494i	$0.433866\pi$
$20$	−1.15050	−0.257259
$21$	−2.62035	−0.571806
$22$	−1.79292	−0.382252
$23$	−1.50203	−0.313196	−0.156598	−	0.987662i	$-0.550053\pi$
$23$	−1.50203	−0.313196	−0.156598	+	0.987662i	$0.550053\pi$
$24$	1.00000	0.204124
$25$	−3.67636	−0.735272
$26$	1.00000	0.196116
$27$	1.00000	0.192450
$28$	−2.62035	−0.495199
$29$	−4.09888	−0.761143	−0.380572	−	0.924751i	$-0.624273\pi$
$29$	−4.09888	−0.761143	−0.380572	+	0.924751i	$0.624273\pi$
$30$	−1.15050	−0.210051
$31$	−7.08604	−1.27269	−0.636345	−	0.771404i	$-0.719555\pi$
$31$	−7.08604	−1.27269	−0.636345	+	0.771404i	$0.719555\pi$
$32$	1.00000	0.176777
$33$	−1.79292	−0.312107
$34$	3.85728	0.661518
$35$	3.01470	0.509577
$36$	1.00000	0.166667
$37$	0.639684	0.105163	0.0525817	−	0.998617i	$-0.483255\pi$
$37$	0.639684	0.105163	0.0525817	+	0.998617i	$0.483255\pi$
$38$	1.79827	0.291718
$39$	1.00000	0.160128
$40$	−1.15050	−0.181909
$41$	8.05054	1.25728	0.628642	−	0.777695i	$-0.283611\pi$
$41$	8.05054	1.25728	0.628642	+	0.777695i	$0.283611\pi$
$42$	−2.62035	−0.404328
$43$	−5.61939	−0.856949	−0.428475	−	0.903554i	$-0.640949\pi$
$43$	−5.61939	−0.856949	−0.428475	+	0.903554i	$0.640949\pi$
$44$	−1.79292	−0.270293
$45$	−1.15050	−0.171506
$46$	−1.50203	−0.221463
$47$	8.43678	1.23063	0.615315	−	0.788281i	$-0.289029\pi$
$47$	8.43678	1.23063	0.615315	+	0.788281i	$0.289029\pi$
$48$	1.00000	0.144338
$49$	−0.133785	−0.0191122
$50$	−3.67636	−0.519916
$51$	3.85728	0.540127
$52$	1.00000	0.138675
$53$	−4.93059	−0.677269	−0.338634	−	0.940918i	$-0.609965\pi$
$53$	−4.93059	−0.677269	−0.338634	+	0.940918i	$0.609965\pi$
$54$	1.00000	0.136083
$55$	2.06275	0.278141
$56$	−2.62035	−0.350159
$57$	1.79827	0.238187
$58$	−4.09888	−0.538210
$59$	1.09119	0.142061	0.0710305	−	0.997474i	$-0.477371\pi$
$59$	1.09119	0.142061	0.0710305	+	0.997474i	$0.477371\pi$
$60$	−1.15050	−0.148528
$61$	−2.73413	−0.350069	−0.175035	−	0.984562i	$-0.556004\pi$
$61$	−2.73413	−0.350069	−0.175035	+	0.984562i	$0.556004\pi$
$62$	−7.08604	−0.899928
$63$	−2.62035	−0.330133
$64$	1.00000	0.125000
$65$	−1.15050	−0.142702
$66$	−1.79292	−0.220693
$67$	−2.82294	−0.344877	−0.172438	−	0.985020i	$-0.555165\pi$
$67$	−2.82294	−0.344877	−0.172438	+	0.985020i	$0.555165\pi$
$68$	3.85728	0.467764
$69$	−1.50203	−0.180824
$70$	3.01470	0.360325
$71$	0.437093	0.0518734	0.0259367	−	0.999664i	$-0.491743\pi$
$71$	0.437093	0.0518734	0.0259367	+	0.999664i	$0.491743\pi$
$72$	1.00000	0.117851
$73$	−14.4632	−1.69279	−0.846394	−	0.532557i	$-0.821231\pi$
$73$	−14.4632	−1.69279	−0.846394	+	0.532557i	$0.821231\pi$
$74$	0.639684	0.0743617
$75$	−3.67636	−0.424509
$76$	1.79827	0.206276
$77$	4.69807	0.535395
$78$	1.00000	0.113228
$79$	−3.04943	−0.343088	−0.171544	−	0.985176i	$-0.554876\pi$
$79$	−3.04943	−0.343088	−0.171544	+	0.985176i	$0.554876\pi$
$80$	−1.15050	−0.128629
$81$	1.00000	0.111111
$82$	8.05054	0.889034
$83$	−14.1654	−1.55485	−0.777427	−	0.628973i	$-0.783476\pi$
$83$	−14.1654	−1.55485	−0.777427	+	0.628973i	$0.783476\pi$
$84$	−2.62035	−0.285903
$85$	−4.43779	−0.481346
$86$	−5.61939	−0.605955
$87$	−4.09888	−0.439446
$88$	−1.79292	−0.191126
$89$	−7.12626	−0.755382	−0.377691	−	0.925932i	$-0.623282\pi$
$89$	−7.12626	−0.755382	−0.377691	+	0.925932i	$0.623282\pi$
$90$	−1.15050	−0.121273
$91$	−2.62035	−0.274687
$92$	−1.50203	−0.156598
$93$	−7.08604	−0.734788
$94$	8.43678	0.870187
$95$	−2.06891	−0.212265
$96$	1.00000	0.102062
$97$	2.70016	0.274160	0.137080	−	0.990560i	$-0.456228\pi$
$97$	2.70016	0.274160	0.137080	+	0.990560i	$0.456228\pi$
$98$	−0.133785	−0.0135143
$99$	−1.79292	−0.180195
$100$	−3.67636	−0.367636
$101$	−6.93641	−0.690198	−0.345099	−	0.938566i	$-0.612155\pi$
$101$	−6.93641	−0.690198	−0.345099	+	0.938566i	$0.612155\pi$
$102$	3.85728	0.381928
$103$	1.00000	0.0985329
$103$	1.00000	0.0985329
$104$	1.00000	0.0980581
$105$	3.01470	0.294204
$106$	−4.93059	−0.478901
$107$	5.51064	0.532734	0.266367	−	0.963872i	$-0.414177\pi$
$107$	5.51064	0.532734	0.266367	+	0.963872i	$0.414177\pi$
$108$	1.00000	0.0962250
$109$	12.8695	1.23268	0.616339	−	0.787481i	$-0.288615\pi$
$109$	12.8695	1.23268	0.616339	+	0.787481i	$0.288615\pi$
$110$	2.06275	0.196675
$111$	0.639684	0.0607161
$112$	−2.62035	−0.247599
$113$	−19.3310	−1.81851	−0.909254	−	0.416241i	$-0.863347\pi$
$113$	−19.3310	−1.81851	−0.909254	+	0.416241i	$0.863347\pi$
$114$	1.79827	0.168424
$115$	1.72808	0.161145
$116$	−4.09888	−0.380572
$117$	1.00000	0.0924500
$118$	1.09119	0.100452
$119$	−10.1074	−0.926545
$120$	−1.15050	−0.105025
$121$	−7.78544	−0.707767
$122$	−2.73413	−0.247536
$123$	8.05054	0.725893
$124$	−7.08604	−0.636345
$125$	9.98212	0.892828
$126$	−2.62035	−0.233439
$127$	10.5725	0.938158	0.469079	−	0.883156i	$-0.344586\pi$
$127$	10.5725	0.938158	0.469079	+	0.883156i	$0.344586\pi$
$128$	1.00000	0.0883883
$129$	−5.61939	−0.494760
$130$	−1.15050	−0.100905
$131$	−18.4698	−1.61372	−0.806858	−	0.590745i	$-0.798834\pi$
$131$	−18.4698	−1.61372	−0.806858	+	0.590745i	$0.798834\pi$
$132$	−1.79292	−0.156054
$133$	−4.71210	−0.408591
$134$	−2.82294	−0.243865
$135$	−1.15050	−0.0990190
$136$	3.85728	0.330759
$137$	−14.2723	−1.21937	−0.609683	−	0.792645i	$-0.708703\pi$
$137$	−14.2723	−1.21937	−0.609683	+	0.792645i	$0.708703\pi$
$138$	−1.50203	−0.127862
$139$	3.41157	0.289366	0.144683	−	0.989478i	$-0.453784\pi$
$139$	3.41157	0.289366	0.144683	+	0.989478i	$0.453784\pi$
$140$	3.01470	0.254789
$141$	8.43678	0.710505
$142$	0.437093	0.0366801
$143$	−1.79292	−0.149932
$144$	1.00000	0.0833333
$145$	4.71575	0.391622
$146$	−14.4632	−1.19698
$147$	−0.133785	−0.0110344
$148$	0.639684	0.0525817
$149$	12.0073	0.983679	0.491840	−	0.870686i	$-0.336325\pi$
$149$	12.0073	0.983679	0.491840	+	0.870686i	$0.336325\pi$
$150$	−3.67636	−0.300173
$151$	−0.734564	−0.0597780	−0.0298890	−	0.999553i	$-0.509515\pi$
$151$	−0.734564	−0.0597780	−0.0298890	+	0.999553i	$0.509515\pi$
$152$	1.79827	0.145859
$153$	3.85728	0.311843
$154$	4.69807	0.378581
$155$	8.15246	0.654822
$156$	1.00000	0.0800641
$157$	5.28140	0.421502	0.210751	−	0.977540i	$-0.432409\pi$
$157$	5.28140	0.421502	0.210751	+	0.977540i	$0.432409\pi$
$158$	−3.04943	−0.242600
$159$	−4.93059	−0.391021
$160$	−1.15050	−0.0909547
$161$	3.93585	0.310188
$162$	1.00000	0.0785674
$163$	15.0028	1.17511	0.587555	−	0.809184i	$-0.300091\pi$
$163$	15.0028	1.17511	0.587555	+	0.809184i	$0.300091\pi$
$164$	8.05054	0.628642
$165$	2.06275	0.160585
$166$	−14.1654	−1.09945
$167$	3.98078	0.308042	0.154021	−	0.988068i	$-0.450778\pi$
$167$	3.98078	0.308042	0.154021	+	0.988068i	$0.450778\pi$
$168$	−2.62035	−0.202164
$169$	1.00000	0.0769231
$170$	−4.43779	−0.340363
$171$	1.79827	0.137517
$172$	−5.61939	−0.428475
$173$	−19.2577	−1.46414	−0.732069	−	0.681231i	$-0.761445\pi$
$173$	−19.2577	−1.46414	−0.732069	+	0.681231i	$0.761445\pi$
$174$	−4.09888	−0.310735
$175$	9.63333	0.728211
$176$	−1.79292	−0.135146
$177$	1.09119	0.0820189
$178$	−7.12626	−0.534136
$179$	10.8022	0.807395	0.403698	−	0.914892i	$-0.367725\pi$
$179$	10.8022	0.807395	0.403698	+	0.914892i	$0.367725\pi$
$180$	−1.15050	−0.0857529
$181$	−24.9820	−1.85690	−0.928449	−	0.371461i	$-0.878857\pi$
$181$	−24.9820	−1.85690	−0.928449	+	0.371461i	$0.878857\pi$
$182$	−2.62035	−0.194233
$183$	−2.73413	−0.202113
$184$	−1.50203	−0.110731
$185$	−0.735954	−0.0541084
$186$	−7.08604	−0.519574
$187$	−6.91580	−0.505733
$188$	8.43678	0.615315
$189$	−2.62035	−0.190602
$190$	−2.06891	−0.150094
$191$	−0.180399	−0.0130532	−0.00652661	−	0.999979i	$-0.502077\pi$
$191$	−0.180399	−0.0130532	−0.00652661	+	0.999979i	$0.502077\pi$
$192$	1.00000	0.0721688
$193$	−8.09799	−0.582906	−0.291453	−	0.956585i	$-0.594139\pi$
$193$	−8.09799	−0.582906	−0.291453	+	0.956585i	$0.594139\pi$
$194$	2.70016	0.193860
$195$	−1.15050	−0.0823888
$196$	−0.133785	−0.00955609
$197$	2.85360	0.203310	0.101655	−	0.994820i	$-0.467586\pi$
$197$	2.85360	0.203310	0.101655	+	0.994820i	$0.467586\pi$
$198$	−1.79292	−0.127417
$199$	4.86762	0.345056	0.172528	−	0.985005i	$-0.444806\pi$
$199$	4.86762	0.345056	0.172528	+	0.985005i	$0.444806\pi$
$200$	−3.67636	−0.259958
$201$	−2.82294	−0.199115
$202$	−6.93641	−0.488044
$203$	10.7405	0.753835
$204$	3.85728	0.270064
$205$	−9.26212	−0.646894
$206$	1.00000	0.0696733
$207$	−1.50203	−0.104399
$208$	1.00000	0.0693375
$209$	−3.22416	−0.223020
$210$	3.01470	0.208034
$211$	−7.32644	−0.504373	−0.252187	−	0.967679i	$-0.581150\pi$
$211$	−7.32644	−0.504373	−0.252187	+	0.967679i	$0.581150\pi$
$212$	−4.93059	−0.338634
$213$	0.437093	0.0299491
$214$	5.51064	0.376700
$215$	6.46509	0.440916
$216$	1.00000	0.0680414
$217$	18.5679	1.26047
$218$	12.8695	0.871635
$219$	−14.4632	−0.977331
$220$	2.06275	0.139070
$221$	3.85728	0.259469
$222$	0.639684	0.0429328
$223$	1.36950	0.0917084	0.0458542	−	0.998948i	$-0.485399\pi$
$223$	1.36950	0.0917084	0.0458542	+	0.998948i	$0.485399\pi$
$224$	−2.62035	−0.175079
$225$	−3.67636	−0.245091
$226$	−19.3310	−1.28588
$227$	−20.3973	−1.35382	−0.676909	−	0.736067i	$-0.736681\pi$
$227$	−20.3973	−1.35382	−0.676909	+	0.736067i	$0.736681\pi$
$228$	1.79827	0.119093
$229$	8.49841	0.561591	0.280796	−	0.959768i	$-0.409402\pi$
$229$	8.49841	0.561591	0.280796	+	0.959768i	$0.409402\pi$
$230$	1.72808	0.113946
$231$	4.69807	0.309110
$232$	−4.09888	−0.269105
$233$	−11.8650	−0.777304	−0.388652	−	0.921385i	$-0.627059\pi$
$233$	−11.8650	−0.777304	−0.388652	+	0.921385i	$0.627059\pi$
$234$	1.00000	0.0653720
$235$	−9.70648	−0.633181
$236$	1.09119	0.0710305
$237$	−3.04943	−0.198082
$238$	−10.1074	−0.655166
$239$	−15.1702	−0.981279	−0.490640	−	0.871363i	$-0.663237\pi$
$239$	−15.1702	−0.981279	−0.490640	+	0.871363i	$0.663237\pi$
$240$	−1.15050	−0.0742642
$241$	−3.78276	−0.243669	−0.121834	−	0.992550i	$-0.538878\pi$
$241$	−3.78276	−0.243669	−0.121834	+	0.992550i	$0.538878\pi$
$242$	−7.78544	−0.500467
$243$	1.00000	0.0641500
$244$	−2.73413	−0.175035
$245$	0.153919	0.00983355
$246$	8.05054	0.513284
$247$	1.79827	0.114421
$248$	−7.08604	−0.449964
$249$	−14.1654	−0.897696
$250$	9.98212	0.631325
$251$	−20.8748	−1.31760	−0.658802	−	0.752317i	$-0.728936\pi$
$251$	−20.8748	−1.31760	−0.658802	+	0.752317i	$0.728936\pi$
$252$	−2.62035	−0.165066
$253$	2.69303	0.169309
$254$	10.5725	0.663378
$255$	−4.43779	−0.277905
$256$	1.00000	0.0625000
$257$	−6.29727	−0.392813	−0.196407	−	0.980523i	$-0.562927\pi$
$257$	−6.29727	−0.392813	−0.196407	+	0.980523i	$0.562927\pi$
$258$	−5.61939	−0.349848
$259$	−1.67619	−0.104154
$260$	−1.15050	−0.0713508
$261$	−4.09888	−0.253714
$262$	−18.4698	−1.14107
$263$	21.3171	1.31447	0.657234	−	0.753687i	$-0.271726\pi$
$263$	21.3171	1.31447	0.657234	+	0.753687i	$0.271726\pi$
$264$	−1.79292	−0.110347
$265$	5.67263	0.348467
$266$	−4.71210	−0.288917
$267$	−7.12626	−0.436120
$268$	−2.82294	−0.172438
$269$	−26.2291	−1.59921	−0.799607	−	0.600524i	$-0.794959\pi$
$269$	−26.2291	−1.59921	−0.799607	+	0.600524i	$0.794959\pi$
$270$	−1.15050	−0.0700170
$271$	−19.0576	−1.15767	−0.578834	−	0.815445i	$-0.696492\pi$
$271$	−19.0576	−1.15767	−0.578834	+	0.815445i	$0.696492\pi$
$272$	3.85728	0.233882
$273$	−2.62035	−0.158591
$274$	−14.2723	−0.862222
$275$	6.59142	0.397477
$276$	−1.50203	−0.0904118
$277$	21.8703	1.31406	0.657030	−	0.753865i	$-0.271813\pi$
$277$	21.8703	1.31406	0.657030	+	0.753865i	$0.271813\pi$
$278$	3.41157	0.204613
$279$	−7.08604	−0.424230
$280$	3.01470	0.180163
$281$	1.35116	0.0806036	0.0403018	−	0.999188i	$-0.487168\pi$
$281$	1.35116	0.0806036	0.0403018	+	0.999188i	$0.487168\pi$
$282$	8.43678	0.502403
$283$	8.23219	0.489353	0.244676	−	0.969605i	$-0.421318\pi$
$283$	8.23219	0.489353	0.244676	+	0.969605i	$0.421318\pi$
$284$	0.437093	0.0259367
$285$	−2.06891	−0.122551
$286$	−1.79292	−0.106018
$287$	−21.0952	−1.24521
$288$	1.00000	0.0589256
$289$	−2.12138	−0.124787
$290$	4.71575	0.276918
$291$	2.70016	0.158286
$292$	−14.4632	−0.846394
$293$	−16.0630	−0.938408	−0.469204	−	0.883090i	$-0.655459\pi$
$293$	−16.0630	−0.938408	−0.469204	+	0.883090i	$0.655459\pi$
$294$	−0.133785	−0.00780251
$295$	−1.25541	−0.0730929
$296$	0.639684	0.0371809
$297$	−1.79292	−0.104036
$298$	12.0073	0.695566
$299$	−1.50203	−0.0868648
$300$	−3.67636	−0.212255
$301$	14.7248	0.848721
$302$	−0.734564	−0.0422694
$303$	−6.93641	−0.398486
$304$	1.79827	0.103138
$305$	3.14560	0.180117
$306$	3.85728	0.220506
$307$	−10.4274	−0.595122	−0.297561	−	0.954703i	$-0.596173\pi$
$307$	−10.4274	−0.595122	−0.297561	+	0.954703i	$0.596173\pi$
$308$	4.69807	0.267698
$309$	1.00000	0.0568880
$310$	8.15246	0.463029
$311$	17.5788	0.996805	0.498402	−	0.866946i	$-0.333920\pi$
$311$	17.5788	0.996805	0.498402	+	0.866946i	$0.333920\pi$
$312$	1.00000	0.0566139
$313$	−28.6533	−1.61958	−0.809791	−	0.586718i	$-0.800420\pi$
$313$	−28.6533	−1.61958	−0.809791	+	0.586718i	$0.800420\pi$
$314$	5.28140	0.298047
$315$	3.01470	0.169859
$316$	−3.04943	−0.171544
$317$	−29.3446	−1.64816	−0.824079	−	0.566475i	$-0.808307\pi$
$317$	−29.3446	−1.64816	−0.824079	+	0.566475i	$0.808307\pi$
$318$	−4.93059	−0.276494
$319$	7.34897	0.411463
$320$	−1.15050	−0.0643147
$321$	5.51064	0.307574
$322$	3.93585	0.219336
$323$	6.93644	0.385954
$324$	1.00000	0.0555556
$325$	−3.67636	−0.203928
$326$	15.0028	0.830928
$327$	12.8695	0.711687
$328$	8.05054	0.444517
$329$	−22.1073	−1.21881
$330$	2.06275	0.113551
$331$	−5.09020	−0.279783	−0.139891	−	0.990167i	$-0.544675\pi$
$331$	−5.09020	−0.279783	−0.139891	+	0.990167i	$0.544675\pi$
$332$	−14.1654	−0.777427
$333$	0.639684	0.0350544
$334$	3.98078	0.217819
$335$	3.24778	0.177445
$336$	−2.62035	−0.142952
$337$	13.2539	0.721986	0.360993	−	0.932569i	$-0.382438\pi$
$337$	13.2539	0.721986	0.360993	+	0.932569i	$0.382438\pi$
$338$	1.00000	0.0543928
$339$	−19.3310	−1.04992
$340$	−4.43779	−0.240673
$341$	12.7047	0.687998
$342$	1.79827	0.0972394
$343$	18.6930	1.00933
$344$	−5.61939	−0.302977
$345$	1.72808	0.0930369
$346$	−19.2577	−1.03530
$347$	32.1612	1.72650	0.863252	−	0.504774i	$-0.168424\pi$
$347$	32.1612	1.72650	0.863252	+	0.504774i	$0.168424\pi$
$348$	−4.09888	−0.219723
$349$	−15.4460	−0.826803	−0.413402	−	0.910549i	$-0.635659\pi$
$349$	−15.4460	−0.826803	−0.413402	+	0.910549i	$0.635659\pi$
$350$	9.63333	0.514923
$351$	1.00000	0.0533761
$352$	−1.79292	−0.0955630
$353$	3.13266	0.166735	0.0833674	−	0.996519i	$-0.473432\pi$
$353$	3.13266	0.166735	0.0833674	+	0.996519i	$0.473432\pi$
$354$	1.09119	0.0579962
$355$	−0.502874	−0.0266898
$356$	−7.12626	−0.377691
$357$	−10.1074	−0.534941
$358$	10.8022	0.570915
$359$	10.8503	0.572659	0.286330	−	0.958131i	$-0.407565\pi$
$359$	10.8503	0.572659	0.286330	+	0.958131i	$0.407565\pi$
$360$	−1.15050	−0.0606365
$361$	−15.7662	−0.829801
$362$	−24.9820	−1.31302
$363$	−7.78544	−0.408629
$364$	−2.62035	−0.137343
$365$	16.6398	0.870969
$366$	−2.73413	−0.142915
$367$	5.90800	0.308395	0.154198	−	0.988040i	$-0.450721\pi$
$367$	5.90800	0.308395	0.154198	+	0.988040i	$0.450721\pi$
$368$	−1.50203	−0.0782989
$369$	8.05054	0.419094
$370$	−0.735954	−0.0382604
$371$	12.9199	0.670765
$372$	−7.08604	−0.367394
$373$	−12.8464	−0.665162	−0.332581	−	0.943075i	$-0.607919\pi$
$373$	−12.8464	−0.665162	−0.332581	+	0.943075i	$0.607919\pi$
$374$	−6.91580	−0.357607
$375$	9.98212	0.515474
$376$	8.43678	0.435094
$377$	−4.09888	−0.211103
$378$	−2.62035	−0.134776
$379$	−15.3285	−0.787373	−0.393687	−	0.919245i	$-0.628800\pi$
$379$	−15.3285	−0.787373	−0.393687	+	0.919245i	$0.628800\pi$
$380$	−2.06891	−0.106133
$381$	10.5725	0.541646
$382$	−0.180399	−0.00923002
$383$	12.8640	0.657318	0.328659	−	0.944449i	$-0.393403\pi$
$383$	12.8640	0.657318	0.328659	+	0.944449i	$0.393403\pi$
$384$	1.00000	0.0510310
$385$	−5.40512	−0.275470
$386$	−8.09799	−0.412177
$387$	−5.61939	−0.285650
$388$	2.70016	0.137080
$389$	18.8181	0.954114	0.477057	−	0.878872i	$-0.341704\pi$
$389$	18.8181	0.954114	0.477057	+	0.878872i	$0.341704\pi$
$390$	−1.15050	−0.0582577
$391$	−5.79377	−0.293003
$392$	−0.133785	−0.00675717
$393$	−18.4698	−0.931680
$394$	2.85360	0.143762
$395$	3.50836	0.176525
$396$	−1.79292	−0.0900976
$397$	−19.1279	−0.960002	−0.480001	−	0.877268i	$-0.659364\pi$
$397$	−19.1279	−0.960002	−0.480001	+	0.877268i	$0.659364\pi$
$398$	4.86762	0.243992
$399$	−4.71210	−0.235900
$400$	−3.67636	−0.183818
$401$	−7.79062	−0.389045	−0.194523	−	0.980898i	$-0.562316\pi$
$401$	−7.79062	−0.389045	−0.194523	+	0.980898i	$0.562316\pi$
$402$	−2.82294	−0.140795
$403$	−7.08604	−0.352981
$404$	−6.93641	−0.345099
$405$	−1.15050	−0.0571686
$406$	10.7405	0.533042
$407$	−1.14690	−0.0568498
$408$	3.85728	0.190964
$409$	7.16032	0.354055	0.177027	−	0.984206i	$-0.443352\pi$
$409$	7.16032	0.354055	0.177027	+	0.984206i	$0.443352\pi$
$410$	−9.26212	−0.457423
$411$	−14.2723	−0.704002
$412$	1.00000	0.0492665
$413$	−2.85930	−0.140697
$414$	−1.50203	−0.0738209
$415$	16.2972	0.800000
$416$	1.00000	0.0490290
$417$	3.41157	0.167066
$418$	−3.22416	−0.157699
$419$	8.59778	0.420029	0.210015	−	0.977698i	$-0.432649\pi$
$419$	8.59778	0.420029	0.210015	+	0.977698i	$0.432649\pi$
$420$	3.01470	0.147102
$421$	10.9638	0.534344	0.267172	−	0.963649i	$-0.413911\pi$
$421$	10.9638	0.534344	0.267172	+	0.963649i	$0.413911\pi$
$422$	−7.32644	−0.356646
$423$	8.43678	0.410210
$424$	−4.93059	−0.239451
$425$	−14.1807	−0.687867
$426$	0.437093	0.0211772
$427$	7.16436	0.346708
$428$	5.51064	0.266367
$429$	−1.79292	−0.0865630
$430$	6.46509	0.311774
$431$	28.2135	1.35900	0.679499	−	0.733676i	$-0.262197\pi$
$431$	28.2135	1.35900	0.679499	+	0.733676i	$0.262197\pi$
$432$	1.00000	0.0481125
$433$	37.4720	1.80079	0.900394	−	0.435075i	$-0.143278\pi$
$433$	37.4720	1.80079	0.900394	+	0.435075i	$0.143278\pi$
$434$	18.5679	0.891287
$435$	4.71575	0.226103
$436$	12.8695	0.616339
$437$	−2.70106	−0.129209
$438$	−14.4632	−0.691078
$439$	27.4396	1.30962	0.654809	−	0.755794i	$-0.272749\pi$
$439$	27.4396	1.30962	0.654809	+	0.755794i	$0.272749\pi$
$440$	2.06275	0.0983377
$441$	−0.133785	−0.00637072
$442$	3.85728	0.183472
$443$	−13.1662	−0.625546	−0.312773	−	0.949828i	$-0.601258\pi$
$443$	−13.1662	−0.625546	−0.312773	+	0.949828i	$0.601258\pi$
$444$	0.639684	0.0303580
$445$	8.19874	0.388657
$446$	1.36950	0.0648477
$447$	12.0073	0.567927
$448$	−2.62035	−0.123800
$449$	14.8065	0.698763	0.349381	−	0.936981i	$-0.386392\pi$
$449$	14.8065	0.698763	0.349381	+	0.936981i	$0.386392\pi$
$450$	−3.67636	−0.173305
$451$	−14.4340	−0.679670
$452$	−19.3310	−0.909254
$453$	−0.734564	−0.0345128
$454$	−20.3973	−0.957294
$455$	3.01470	0.141331
$456$	1.79827	0.0842118
$457$	35.1413	1.64384	0.821920	−	0.569603i	$-0.192903\pi$
$457$	35.1413	1.64384	0.821920	+	0.569603i	$0.192903\pi$
$458$	8.49841	0.397105
$459$	3.85728	0.180042
$460$	1.72808	0.0805723
$461$	37.7601	1.75866	0.879331	−	0.476211i	$-0.157990\pi$
$461$	37.7601	1.75866	0.879331	+	0.476211i	$0.157990\pi$
$462$	4.69807	0.218574
$463$	32.2057	1.49673	0.748364	−	0.663288i	$-0.230840\pi$
$463$	32.2057	1.49673	0.748364	+	0.663288i	$0.230840\pi$
$464$	−4.09888	−0.190286
$465$	8.15246	0.378061
$466$	−11.8650	−0.549637
$467$	9.92123	0.459100	0.229550	−	0.973297i	$-0.426275\pi$
$467$	9.92123	0.459100	0.229550	+	0.973297i	$0.426275\pi$
$468$	1.00000	0.0462250
$469$	7.39708	0.341565
$470$	−9.70648	−0.447727
$471$	5.28140	0.243354
$472$	1.09119	0.0502261
$473$	10.0751	0.463255
$474$	−3.04943	−0.140065
$475$	−6.61109	−0.303338
$476$	−10.1074	−0.463273
$477$	−4.93059	−0.225756
$478$	−15.1702	−0.693869
$479$	34.9768	1.59813	0.799065	−	0.601244i	$-0.205328\pi$
$479$	34.9768	1.59813	0.799065	+	0.601244i	$0.205328\pi$
$480$	−1.15050	−0.0525127
$481$	0.639684	0.0291671
$482$	−3.78276	−0.172300
$483$	3.93585	0.179087
$484$	−7.78544	−0.353883
$485$	−3.10653	−0.141060
$486$	1.00000	0.0453609
$487$	−6.27601	−0.284393	−0.142197	−	0.989838i	$-0.545417\pi$
$487$	−6.27601	−0.284393	−0.142197	+	0.989838i	$0.545417\pi$
$488$	−2.73413	−0.123768
$489$	15.0028	0.678450
$490$	0.153919	0.00695337
$491$	−21.7909	−0.983411	−0.491705	−	0.870762i	$-0.663626\pi$
$491$	−21.7909	−0.983411	−0.491705	+	0.870762i	$0.663626\pi$
$492$	8.05054	0.362946
$493$	−15.8105	−0.712071
$494$	1.79827	0.0809081
$495$	2.06275	0.0927136
$496$	−7.08604	−0.318173
$497$	−1.14534	−0.0513753
$498$	−14.1654	−0.634767
$499$	20.6973	0.926541	0.463270	−	0.886217i	$-0.346676\pi$
$499$	20.6973	0.926541	0.463270	+	0.886217i	$0.346676\pi$
$500$	9.98212	0.446414
$501$	3.98078	0.177848
$502$	−20.8748	−0.931686
$503$	−6.46015	−0.288044	−0.144022	−	0.989575i	$-0.546004\pi$
$503$	−6.46015	−0.288044	−0.144022	+	0.989575i	$0.546004\pi$
$504$	−2.62035	−0.116720
$505$	7.98031	0.355119
$506$	2.69303	0.119720
$507$	1.00000	0.0444116
$508$	10.5725	0.469079
$509$	5.06274	0.224402	0.112201	−	0.993686i	$-0.464210\pi$
$509$	5.06274	0.224402	0.112201	+	0.993686i	$0.464210\pi$
$510$	−4.43779	−0.196509
$511$	37.8986	1.67653
$512$	1.00000	0.0441942
$513$	1.79827	0.0793957
$514$	−6.29727	−0.277761
$515$	−1.15050	−0.0506969
$516$	−5.61939	−0.247380
$517$	−15.1265	−0.665261
$518$	−1.67619	−0.0736477
$519$	−19.2577	−0.845320
$520$	−1.15050	−0.0504526
$521$	−23.2912	−1.02041	−0.510203	−	0.860054i	$-0.670430\pi$
$521$	−23.2912	−1.02041	−0.510203	+	0.860054i	$0.670430\pi$
$522$	−4.09888	−0.179403
$523$	17.4548	0.763243	0.381622	−	0.924319i	$-0.375366\pi$
$523$	17.4548	0.763243	0.381622	+	0.924319i	$0.375366\pi$
$524$	−18.4698	−0.806858
$525$	9.63333	0.420433
$526$	21.3171	0.929469
$527$	−27.3329	−1.19064
$528$	−1.79292	−0.0780268
$529$	−20.7439	−0.901909
$530$	5.67263	0.246403
$531$	1.09119	0.0473537
$532$	−4.71210	−0.204295
$533$	8.05054	0.348708
$534$	−7.12626	−0.308383
$535$	−6.33998	−0.274101
$536$	−2.82294	−0.121932
$537$	10.8022	0.466150
$538$	−26.2291	−1.13081
$539$	0.239866	0.0103318
$540$	−1.15050	−0.0495095
$541$	31.8351	1.36870	0.684350	−	0.729154i	$-0.260086\pi$
$541$	31.8351	1.36870	0.684350	+	0.729154i	$0.260086\pi$
$542$	−19.0576	−0.818596
$543$	−24.9820	−1.07208
$544$	3.85728	0.165380
$545$	−14.8064	−0.634235
$546$	−2.62035	−0.112140
$547$	−31.2250	−1.33508	−0.667542	−	0.744572i	$-0.732654\pi$
$547$	−31.2250	−1.33508	−0.667542	+	0.744572i	$0.732654\pi$
$548$	−14.2723	−0.609683
$549$	−2.73413	−0.116690
$550$	6.59142	0.281059
$551$	−7.37091	−0.314011
$552$	−1.50203	−0.0639308
$553$	7.99057	0.339794
$554$	21.8703	0.929180
$555$	−0.735954	−0.0312395
$556$	3.41157	0.144683
$557$	17.8896	0.758006	0.379003	−	0.925395i	$-0.376267\pi$
$557$	17.8896	0.758006	0.379003	+	0.925395i	$0.376267\pi$
$558$	−7.08604	−0.299976
$559$	−5.61939	−0.237675
$560$	3.01470	0.127394
$561$	−6.91580	−0.291985
$562$	1.35116	0.0569954
$563$	−6.93859	−0.292427	−0.146213	−	0.989253i	$-0.546709\pi$
$563$	−6.93859	−0.292427	−0.146213	+	0.989253i	$0.546709\pi$
$564$	8.43678	0.355252
$565$	22.2403	0.935655
$566$	8.23219	0.346025
$567$	−2.62035	−0.110044
$568$	0.437093	0.0183400
$569$	−19.5069	−0.817771	−0.408886	−	0.912586i	$-0.634083\pi$
$569$	−19.5069	−0.817771	−0.408886	+	0.912586i	$0.634083\pi$
$570$	−2.06891	−0.0866569
$571$	23.6498	0.989712	0.494856	−	0.868975i	$-0.335221\pi$
$571$	23.6498	0.989712	0.494856	+	0.868975i	$0.335221\pi$
$572$	−1.79292	−0.0749658
$573$	−0.180399	−0.00753628
$574$	−21.0952	−0.880497
$575$	5.52201	0.230284
$576$	1.00000	0.0416667
$577$	8.20271	0.341483	0.170742	−	0.985316i	$-0.445384\pi$
$577$	8.20271	0.341483	0.170742	+	0.985316i	$0.445384\pi$
$578$	−2.12138	−0.0882376
$579$	−8.09799	−0.336541
$580$	4.71575	0.195811
$581$	37.1183	1.53992
$582$	2.70016	0.111925
$583$	8.84016	0.366122
$584$	−14.4632	−0.598491
$585$	−1.15050	−0.0475672
$586$	−16.0630	−0.663554
$587$	−7.28493	−0.300681	−0.150341	−	0.988634i	$-0.548037\pi$
$587$	−7.28493	−0.300681	−0.150341	+	0.988634i	$0.548037\pi$
$588$	−0.133785	−0.00551721
$589$	−12.7426	−0.525051
$590$	−1.25541	−0.0516845
$591$	2.85360	0.117381
$592$	0.639684	0.0262908
$593$	−41.5026	−1.70431	−0.852154	−	0.523291i	$-0.824704\pi$
$593$	−41.5026	−1.70431	−0.852154	+	0.523291i	$0.824704\pi$
$594$	−1.79292	−0.0735644
$595$	11.6285	0.476724
$596$	12.0073	0.491840
$597$	4.86762	0.199218
$598$	−1.50203	−0.0614227
$599$	−26.8059	−1.09526	−0.547630	−	0.836721i	$-0.684470\pi$
$599$	−26.8059	−1.09526	−0.547630	+	0.836721i	$0.684470\pi$
$600$	−3.67636	−0.150087
$601$	−9.01887	−0.367888	−0.183944	−	0.982937i	$-0.558886\pi$
$601$	−9.01887	−0.367888	−0.183944	+	0.982937i	$0.558886\pi$
$602$	14.7248	0.600136
$603$	−2.82294	−0.114959
$604$	−0.734564	−0.0298890
$605$	8.95712	0.364159
$606$	−6.93641	−0.281772
$607$	3.74487	0.151999	0.0759997	−	0.997108i	$-0.475785\pi$
$607$	3.74487	0.151999	0.0759997	+	0.997108i	$0.475785\pi$
$608$	1.79827	0.0729296
$609$	10.7405	0.435227
$610$	3.14560	0.127362
$611$	8.43678	0.341315
$612$	3.85728	0.155921
$613$	37.8394	1.52832	0.764159	−	0.645028i	$-0.223154\pi$
$613$	37.8394	1.52832	0.764159	+	0.645028i	$0.223154\pi$
$614$	−10.4274	−0.420815
$615$	−9.26212	−0.373485
$616$	4.69807	0.189291
$617$	−0.710445	−0.0286014	−0.0143007	−	0.999898i	$-0.504552\pi$
$617$	−0.710445	−0.0286014	−0.0143007	+	0.999898i	$0.504552\pi$
$618$	1.00000	0.0402259
$619$	−10.5501	−0.424043	−0.212022	−	0.977265i	$-0.568005\pi$
$619$	−10.5501	−0.424043	−0.212022	+	0.977265i	$0.568005\pi$
$620$	8.15246	0.327411
$621$	−1.50203	−0.0602745
$622$	17.5788	0.704847
$623$	18.6733	0.748129
$624$	1.00000	0.0400320
$625$	6.89740	0.275896
$626$	−28.6533	−1.14522
$627$	−3.22416	−0.128761
$628$	5.28140	0.210751
$629$	2.46744	0.0983833
$630$	3.01470	0.120108
$631$	6.67187	0.265603	0.132802	−	0.991143i	$-0.457603\pi$
$631$	6.67187	0.265603	0.132802	+	0.991143i	$0.457603\pi$
$632$	−3.04943	−0.121300
$633$	−7.32644	−0.291200
$634$	−29.3446	−1.16542
$635$	−12.1636	−0.482699
$636$	−4.93059	−0.195511
$637$	−0.133785	−0.00530076
$638$	7.34897	0.290948
$639$	0.437093	0.0172911
$640$	−1.15050	−0.0454774
$641$	25.8636	1.02155	0.510775	−	0.859714i	$-0.329359\pi$
$641$	25.8636	1.02155	0.510775	+	0.859714i	$0.329359\pi$
$642$	5.51064	0.217488
$643$	12.1423	0.478847	0.239423	−	0.970915i	$-0.423042\pi$
$643$	12.1423	0.478847	0.239423	+	0.970915i	$0.423042\pi$
$644$	3.93585	0.155094
$645$	6.46509	0.254563
$646$	6.93644	0.272911
$647$	24.3522	0.957382	0.478691	−	0.877983i	$-0.341111\pi$
$647$	24.3522	0.957382	0.478691	+	0.877983i	$0.341111\pi$
$648$	1.00000	0.0392837
$649$	−1.95642	−0.0767962
$650$	−3.67636	−0.144199
$651$	18.5679	0.727733
$652$	15.0028	0.587555
$653$	−5.29412	−0.207175	−0.103587	−	0.994620i	$-0.533032\pi$
$653$	−5.29412	−0.207175	−0.103587	+	0.994620i	$0.533032\pi$
$654$	12.8695	0.503239
$655$	21.2495	0.830286
$656$	8.05054	0.314321
$657$	−14.4632	−0.564263
$658$	−22.1073	−0.861831
$659$	−39.7350	−1.54786	−0.773928	−	0.633273i	$-0.781711\pi$
$659$	−39.7350	−1.54786	−0.773928	+	0.633273i	$0.781711\pi$
$660$	2.06275	0.0802924
$661$	4.99686	0.194355	0.0971776	−	0.995267i	$-0.469019\pi$
$661$	4.99686	0.194355	0.0971776	+	0.995267i	$0.469019\pi$
$662$	−5.09020	−0.197836
$663$	3.85728	0.149804
$664$	−14.1654	−0.549724
$665$	5.42125	0.210227
$666$	0.639684	0.0247872
$667$	6.15666	0.238387
$668$	3.98078	0.154021
$669$	1.36950	0.0529479
$670$	3.24778	0.125473
$671$	4.90207	0.189242
$672$	−2.62035	−0.101082
$673$	−44.1027	−1.70003	−0.850017	−	0.526756i	$-0.823408\pi$
$673$	−44.1027	−1.70003	−0.850017	+	0.526756i	$0.823408\pi$
$674$	13.2539	0.510521
$675$	−3.67636	−0.141503
$676$	1.00000	0.0384615
$677$	−26.7178	−1.02685	−0.513424	−	0.858135i	$-0.671623\pi$
$677$	−26.7178	−1.02685	−0.513424	+	0.858135i	$0.671623\pi$
$678$	−19.3310	−0.742403
$679$	−7.07536	−0.271527
$680$	−4.43779	−0.170181
$681$	−20.3973	−0.781627
$682$	12.7047	0.486488
$683$	35.5611	1.36071	0.680354	−	0.732883i	$-0.261826\pi$
$683$	35.5611	1.36071	0.680354	+	0.732883i	$0.261826\pi$
$684$	1.79827	0.0687587
$685$	16.4203	0.627386
$686$	18.6930	0.713702
$687$	8.49841	0.324235
$688$	−5.61939	−0.214237
$689$	−4.93059	−0.187841
$690$	1.72808	0.0657870
$691$	−28.8625	−1.09798	−0.548991	−	0.835828i	$-0.684988\pi$
$691$	−28.8625	−1.09798	−0.548991	+	0.835828i	$0.684988\pi$
$692$	−19.2577	−0.732069
$693$	4.69807	0.178465
$694$	32.1612	1.22082
$695$	−3.92500	−0.148884
$696$	−4.09888	−0.155368
$697$	31.0532	1.17622
$698$	−15.4460	−0.584638
$699$	−11.8650	−0.448777
$700$	9.63333	0.364106
$701$	7.03988	0.265893	0.132946	−	0.991123i	$-0.457556\pi$
$701$	7.03988	0.265893	0.132946	+	0.991123i	$0.457556\pi$
$702$	1.00000	0.0377426
$703$	1.15033	0.0433853
$704$	−1.79292	−0.0675732
$705$	−9.70648	−0.365567
$706$	3.13266	0.117899
$707$	18.1758	0.683571
$708$	1.09119	0.0410095
$709$	−31.9295	−1.19914	−0.599568	−	0.800324i	$-0.704661\pi$
$709$	−31.9295	−1.19914	−0.599568	+	0.800324i	$0.704661\pi$
$710$	−0.502874	−0.0188725
$711$	−3.04943	−0.114363
$712$	−7.12626	−0.267068
$713$	10.6435	0.398601
$714$	−10.1074	−0.378260
$715$	2.06275	0.0771424
$716$	10.8022	0.403698
$717$	−15.1702	−0.566542
$718$	10.8503	0.404931
$719$	−4.21451	−0.157175	−0.0785873	−	0.996907i	$-0.525041\pi$
$719$	−4.21451	−0.157175	−0.0785873	+	0.996907i	$0.525041\pi$
$720$	−1.15050	−0.0428765
$721$	−2.62035	−0.0975868
$722$	−15.7662	−0.586758
$723$	−3.78276	−0.140682
$724$	−24.9820	−0.928449
$725$	15.0690	0.559647
$726$	−7.78544	−0.288945
$727$	43.3322	1.60710	0.803552	−	0.595235i	$-0.202941\pi$
$727$	43.3322	1.60710	0.803552	+	0.595235i	$0.202941\pi$
$728$	−2.62035	−0.0971165
$729$	1.00000	0.0370370
$730$	16.6398	0.615868
$731$	−21.6756	−0.801700
$732$	−2.73413	−0.101056
$733$	7.01040	0.258935	0.129468	−	0.991584i	$-0.458673\pi$
$733$	7.01040	0.258935	0.129468	+	0.991584i	$0.458673\pi$
$734$	5.90800	0.218068
$735$	0.153919	0.00567740
$736$	−1.50203	−0.0553657
$737$	5.06131	0.186436
$738$	8.05054	0.296345
$739$	15.8916	0.584582	0.292291	−	0.956329i	$-0.405582\pi$
$739$	15.8916	0.584582	0.292291	+	0.956329i	$0.405582\pi$
$740$	−0.735954	−0.0270542
$741$	1.79827	0.0660612
$742$	12.9199	0.474303
$743$	−45.5860	−1.67239	−0.836195	−	0.548432i	$-0.815225\pi$
$743$	−45.5860	−1.67239	−0.836195	+	0.548432i	$0.815225\pi$
$744$	−7.08604	−0.259787
$745$	−13.8144	−0.506120
$746$	−12.8464	−0.470340
$747$	−14.1654	−0.518285
$748$	−6.91580	−0.252867
$749$	−14.4398	−0.527619
$750$	9.98212	0.364495
$751$	12.1928	0.444920	0.222460	−	0.974942i	$-0.428591\pi$
$751$	12.1928	0.444920	0.222460	+	0.974942i	$0.428591\pi$
$752$	8.43678	0.307658
$753$	−20.8748	−0.760719
$754$	−4.09888	−0.149272
$755$	0.845113	0.0307568
$756$	−2.62035	−0.0953011
$757$	34.9938	1.27187	0.635936	−	0.771742i	$-0.280614\pi$
$757$	34.9938	1.27187	0.635936	+	0.771742i	$0.280614\pi$
$758$	−15.3285	−0.556757
$759$	2.69303	0.0977506
$760$	−2.06891	−0.0750471
$761$	12.7188	0.461057	0.230529	−	0.973066i	$-0.425954\pi$
$761$	12.7188	0.461057	0.230529	+	0.973066i	$0.425954\pi$
$762$	10.5725	0.383001
$763$	−33.7226	−1.22084
$764$	−0.180399	−0.00652661
$765$	−4.43779	−0.160449
$766$	12.8640	0.464794
$767$	1.09119	0.0394006
$768$	1.00000	0.0360844
$769$	50.6941	1.82807	0.914037	−	0.405631i	$-0.132948\pi$
$769$	50.6941	1.82807	0.914037	+	0.405631i	$0.132948\pi$
$770$	−5.40512	−0.194787
$771$	−6.29727	−0.226791
$772$	−8.09799	−0.291453
$773$	9.45067	0.339917	0.169958	−	0.985451i	$-0.445637\pi$
$773$	9.45067	0.339917	0.169958	+	0.985451i	$0.445637\pi$
$774$	−5.61939	−0.201985
$775$	26.0508	0.935773
$776$	2.70016	0.0969302
$777$	−1.67619	−0.0601331
$778$	18.8181	0.674660
$779$	14.4771	0.518695
$780$	−1.15050	−0.0411944
$781$	−0.783673	−0.0280420
$782$	−5.79377	−0.207185
$783$	−4.09888	−0.146482
$784$	−0.133785	−0.00477804
$785$	−6.07623	−0.216870
$786$	−18.4698	−0.658797
$787$	22.5047	0.802205	0.401102	−	0.916033i	$-0.368627\pi$
$787$	22.5047	0.802205	0.401102	+	0.916033i	$0.368627\pi$
$788$	2.85360	0.101655
$789$	21.3171	0.758908
$790$	3.50836	0.124822
$791$	50.6539	1.80105
$792$	−1.79292	−0.0637087
$793$	−2.73413	−0.0970917
$794$	−19.1279	−0.678824
$795$	5.67263	0.201187
$796$	4.86762	0.172528
$797$	−25.6180	−0.907435	−0.453717	−	0.891146i	$-0.649902\pi$
$797$	−25.6180	−0.907435	−0.453717	+	0.891146i	$0.649902\pi$
$798$	−4.71210	−0.166806
$799$	32.5430	1.15129
$800$	−3.67636	−0.129979
$801$	−7.12626	−0.251794
$802$	−7.79062	−0.275096
$803$	25.9313	0.915097
$804$	−2.82294	−0.0995574
$805$	−4.52818	−0.159597
$806$	−7.08604	−0.249595
$807$	−26.2291	−0.923307
$808$	−6.93641	−0.244022
$809$	−11.8814	−0.417726	−0.208863	−	0.977945i	$-0.566976\pi$
$809$	−11.8814	−0.417726	−0.208863	+	0.977945i	$0.566976\pi$
$810$	−1.15050	−0.0404243
$811$	28.1927	0.989978	0.494989	−	0.868899i	$-0.335172\pi$
$811$	28.1927	0.989978	0.494989	+	0.868899i	$0.335172\pi$
$812$	10.7405	0.376917
$813$	−19.0576	−0.668380
$814$	−1.14690	−0.0401989
$815$	−17.2607	−0.604615
$816$	3.85728	0.135032
$817$	−10.1052	−0.353536
$818$	7.16032	0.250355
$819$	−2.62035	−0.0915623
$820$	−9.26212	−0.323447
$821$	−0.563831	−0.0196778	−0.00983892	−	0.999952i	$-0.503132\pi$
$821$	−0.563831	−0.0196778	−0.00983892	+	0.999952i	$0.503132\pi$
$822$	−14.2723	−0.497804
$823$	−33.7443	−1.17625	−0.588127	−	0.808769i	$-0.700134\pi$
$823$	−33.7443	−1.17625	−0.588127	+	0.808769i	$0.700134\pi$
$824$	1.00000	0.0348367
$825$	6.59142	0.229484
$826$	−2.85930	−0.0994877
$827$	−13.6650	−0.475178	−0.237589	−	0.971366i	$-0.576357\pi$
$827$	−13.6650	−0.475178	−0.237589	+	0.971366i	$0.576357\pi$
$828$	−1.50203	−0.0521993
$829$	47.0957	1.63570	0.817850	−	0.575431i	$-0.195166\pi$
$829$	47.0957	1.63570	0.817850	+	0.575431i	$0.195166\pi$
$830$	16.2972	0.565686
$831$	21.8703	0.758672
$832$	1.00000	0.0346688
$833$	−0.516047	−0.0178800
$834$	3.41157	0.118133
$835$	−4.57988	−0.158493
$836$	−3.22416	−0.111510
$837$	−7.08604	−0.244929
$838$	8.59778	0.297005
$839$	10.3373	0.356885	0.178442	−	0.983950i	$-0.442894\pi$
$839$	10.3373	0.356885	0.178442	+	0.983950i	$0.442894\pi$
$840$	3.01470	0.104017
$841$	−12.1992	−0.420661
$842$	10.9638	0.377838
$843$	1.35116	0.0465365
$844$	−7.32644	−0.252187
$845$	−1.15050	−0.0395783
$846$	8.43678	0.290062
$847$	20.4005	0.700971
$848$	−4.93059	−0.169317
$849$	8.23219	0.282528
$850$	−14.1807	−0.486396
$851$	−0.960826	−0.0329367
$852$	0.437093	0.0149746
$853$	−20.2833	−0.694486	−0.347243	−	0.937775i	$-0.612882\pi$
$853$	−20.2833	−0.694486	−0.347243	+	0.937775i	$0.612882\pi$
$854$	7.16436	0.245159
$855$	−2.06891	−0.0707551
$856$	5.51064	0.188350
$857$	−47.9237	−1.63704	−0.818521	−	0.574476i	$-0.805206\pi$
$857$	−47.9237	−1.63704	−0.818521	+	0.574476i	$0.805206\pi$
$858$	−1.79292	−0.0612093
$859$	8.18718	0.279343	0.139671	−	0.990198i	$-0.455395\pi$
$859$	8.18718	0.279343	0.139671	+	0.990198i	$0.455395\pi$
$860$	6.46509	0.220458
$861$	−21.0952	−0.718923
$862$	28.2135	0.960957
$863$	−15.5847	−0.530509	−0.265254	−	0.964178i	$-0.585456\pi$
$863$	−15.5847	−0.530509	−0.265254	+	0.964178i	$0.585456\pi$
$864$	1.00000	0.0340207
$865$	22.1559	0.753325
$866$	37.4720	1.27335
$867$	−2.12138	−0.0720457
$868$	18.5679	0.630235
$869$	5.46739	0.185469
$870$	4.71575	0.159879
$871$	−2.82294	−0.0956516
$872$	12.8695	0.435817
$873$	2.70016	0.0913866
$874$	−2.70106	−0.0913649
$875$	−26.1566	−0.884255
$876$	−14.4632	−0.488666
$877$	23.8151	0.804180	0.402090	−	0.915600i	$-0.368284\pi$
$877$	23.8151	0.804180	0.402090	+	0.915600i	$0.368284\pi$
$878$	27.4396	0.926040
$879$	−16.0630	−0.541790
$880$	2.06275	0.0695352
$881$	2.79154	0.0940494	0.0470247	−	0.998894i	$-0.485026\pi$
$881$	2.79154	0.0940494	0.0470247	+	0.998894i	$0.485026\pi$
$882$	−0.133785	−0.00450478
$883$	26.1357	0.879536	0.439768	−	0.898111i	$-0.355061\pi$
$883$	26.1357	0.879536	0.439768	+	0.898111i	$0.355061\pi$
$884$	3.85728	0.129734
$885$	−1.25541	−0.0422002
$886$	−13.1662	−0.442328
$887$	−24.0286	−0.806800	−0.403400	−	0.915024i	$-0.632172\pi$
$887$	−24.0286	−0.806800	−0.403400	+	0.915024i	$0.632172\pi$
$888$	0.639684	0.0214664
$889$	−27.7036	−0.929149
$890$	8.19874	0.274822
$891$	−1.79292	−0.0600651
$892$	1.36950	0.0458542
$893$	15.1716	0.507699
$894$	12.0073	0.401585
$895$	−12.4279	−0.415419
$896$	−2.62035	−0.0875396
$897$	−1.50203	−0.0501514
$898$	14.8065	0.494100
$899$	29.0448	0.968700
$900$	−3.67636	−0.122545
$901$	−19.0187	−0.633604
$902$	−14.4340	−0.480599
$903$	14.7248	0.490009
$904$	−19.3310	−0.642940
$905$	28.7417	0.955406
$906$	−0.734564	−0.0244042
$907$	−50.5963	−1.68002	−0.840012	−	0.542568i	$-0.817452\pi$
$907$	−50.5963	−1.68002	−0.840012	+	0.542568i	$0.817452\pi$
$908$	−20.3973	−0.676909
$909$	−6.93641	−0.230066
$910$	3.01470	0.0999363
$911$	23.0663	0.764219	0.382110	−	0.924117i	$-0.375198\pi$
$911$	23.0663	0.764219	0.382110	+	0.924117i	$0.375198\pi$
$912$	1.79827	0.0595467
$913$	25.3974	0.840532
$914$	35.1413	1.16237
$915$	3.14560	0.103990
$916$	8.49841	0.280796
$917$	48.3974	1.59822
$918$	3.85728	0.127309
$919$	−3.77710	−0.124595	−0.0622974	−	0.998058i	$-0.519843\pi$
$919$	−3.77710	−0.124595	−0.0622974	+	0.998058i	$0.519843\pi$
$920$	1.72808	0.0569732
$921$	−10.4274	−0.343594
$922$	37.7601	1.24356
$923$	0.437093	0.0143871
$924$	4.69807	0.154555
$925$	−2.35171	−0.0773236
$926$	32.2057	1.05835
$927$	1.00000	0.0328443
$928$	−4.09888	−0.134552
$929$	5.55966	0.182406	0.0912032	−	0.995832i	$-0.470929\pi$
$929$	5.55966	0.182406	0.0912032	+	0.995832i	$0.470929\pi$
$930$	8.15246	0.267330
$931$	−0.240582	−0.00788476
$932$	−11.8650	−0.388652
$933$	17.5788	0.575506
$934$	9.92123	0.324632
$935$	7.95660	0.260209
$936$	1.00000	0.0326860
$937$	9.74423	0.318330	0.159165	−	0.987252i	$-0.449120\pi$
$937$	9.74423	0.318330	0.159165	+	0.987252i	$0.449120\pi$
$938$	7.39708	0.241523
$939$	−28.6533	−0.935066
$940$	−9.70648	−0.316590
$941$	−1.84027	−0.0599911	−0.0299956	−	0.999550i	$-0.509549\pi$
$941$	−1.84027	−0.0599911	−0.0299956	+	0.999550i	$0.509549\pi$
$942$	5.28140	0.172077
$943$	−12.0922	−0.393776
$944$	1.09119	0.0355152
$945$	3.01470	0.0980682
$946$	10.0751	0.327571
$947$	−16.7275	−0.543571	−0.271785	−	0.962358i	$-0.587614\pi$
$947$	−16.7275	−0.543571	−0.271785	+	0.962358i	$0.587614\pi$
$948$	−3.04943	−0.0990410
$949$	−14.4632	−0.469495
$950$	−6.61109	−0.214492
$951$	−29.3446	−0.951564
$952$	−10.1074	−0.327583
$953$	21.5348	0.697579	0.348790	−	0.937201i	$-0.386593\pi$
$953$	21.5348	0.697579	0.348790	+	0.937201i	$0.386593\pi$
$954$	−4.93059	−0.159634
$955$	0.207548	0.00671611
$956$	−15.1702	−0.490640
$957$	7.34897	0.237558
$958$	34.9768	1.13005
$959$	37.3984	1.20766
$960$	−1.15050	−0.0371321
$961$	19.2120	0.619741
$962$	0.639684	0.0206242
$963$	5.51064	0.177578
$964$	−3.78276	−0.121834
$965$	9.31671	0.299915
$966$	3.93585	0.126634
$967$	−3.26712	−0.105064	−0.0525318	−	0.998619i	$-0.516729\pi$
$967$	−3.26712	−0.105064	−0.0525318	+	0.998619i	$0.516729\pi$
$968$	−7.78544	−0.250233
$969$	6.93644	0.222831
$970$	−3.10653	−0.0997446
$971$	11.6355	0.373402	0.186701	−	0.982417i	$-0.440220\pi$
$971$	11.6355	0.373402	0.186701	+	0.982417i	$0.440220\pi$
$972$	1.00000	0.0320750
$973$	−8.93951	−0.286587
$974$	−6.27601	−0.201096
$975$	−3.67636	−0.117738
$976$	−2.73413	−0.0875173
$977$	−61.4172	−1.96491	−0.982455	−	0.186501i	$-0.940285\pi$
$977$	−61.4172	−1.96491	−0.982455	+	0.186501i	$0.940285\pi$
$978$	15.0028	0.479737
$979$	12.7768	0.408349
$980$	0.153919	0.00491677
$981$	12.8695	0.410893
$982$	−21.7909	−0.695376
$983$	22.1881	0.707690	0.353845	−	0.935304i	$-0.384874\pi$
$983$	22.1881	0.707690	0.353845	+	0.935304i	$0.384874\pi$
$984$	8.05054	0.256642
$985$	−3.28305	−0.104607
$986$	−15.8105	−0.503510
$987$	−22.1073	−0.703682
$988$	1.79827	0.0572107
$989$	8.44051	0.268393
$990$	2.06275	0.0655584
$991$	2.09942	0.0666903	0.0333451	−	0.999444i	$-0.489384\pi$
$991$	2.09942	0.0666903	0.0333451	+	0.999444i	$0.489384\pi$
$992$	−7.08604	−0.224982
$993$	−5.09020	−0.161533
$994$	−1.14534	−0.0363278
$995$	−5.60018	−0.177538
$996$	−14.1654	−0.448848
$997$	4.56968	0.144723	0.0723616	−	0.997378i	$-0.476946\pi$
$997$	4.56968	0.144723	0.0723616	+	0.997378i	$0.476946\pi$
$998$	20.6973	0.655163
$999$	0.639684	0.0202387

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8034.2.a.p.1.4	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8034.2.a.p.1.4	✓	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:	\( N \)	\(=\)	\( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8034.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.15050 q^{5} +1.00000 q^{6} -2.62035 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.15050 q^{5} +1.00000 q^{6} -2.62035 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.15050 q^{10} -1.79292 q^{11} +1.00000 q^{12} +1.00000 q^{13} -2.62035 q^{14} -1.15050 q^{15} +1.00000 q^{16} +3.85728 q^{17} +1.00000 q^{18} +1.79827 q^{19} -1.15050 q^{20} -2.62035 q^{21} -1.79292 q^{22} -1.50203 q^{23} +1.00000 q^{24} -3.67636 q^{25} +1.00000 q^{26} +1.00000 q^{27} -2.62035 q^{28} -4.09888 q^{29} -1.15050 q^{30} -7.08604 q^{31} +1.00000 q^{32} -1.79292 q^{33} +3.85728 q^{34} +3.01470 q^{35} +1.00000 q^{36} +0.639684 q^{37} +1.79827 q^{38} +1.00000 q^{39} -1.15050 q^{40} +8.05054 q^{41} -2.62035 q^{42} -5.61939 q^{43} -1.79292 q^{44} -1.15050 q^{45} -1.50203 q^{46} +8.43678 q^{47} +1.00000 q^{48} -0.133785 q^{49} -3.67636 q^{50} +3.85728 q^{51} +1.00000 q^{52} -4.93059 q^{53} +1.00000 q^{54} +2.06275 q^{55} -2.62035 q^{56} +1.79827 q^{57} -4.09888 q^{58} +1.09119 q^{59} -1.15050 q^{60} -2.73413 q^{61} -7.08604 q^{62} -2.62035 q^{63} +1.00000 q^{64} -1.15050 q^{65} -1.79292 q^{66} -2.82294 q^{67} +3.85728 q^{68} -1.50203 q^{69} +3.01470 q^{70} +0.437093 q^{71} +1.00000 q^{72} -14.4632 q^{73} +0.639684 q^{74} -3.67636 q^{75} +1.79827 q^{76} +4.69807 q^{77} +1.00000 q^{78} -3.04943 q^{79} -1.15050 q^{80} +1.00000 q^{81} +8.05054 q^{82} -14.1654 q^{83} -2.62035 q^{84} -4.43779 q^{85} -5.61939 q^{86} -4.09888 q^{87} -1.79292 q^{88} -7.12626 q^{89} -1.15050 q^{90} -2.62035 q^{91} -1.50203 q^{92} -7.08604 q^{93} +8.43678 q^{94} -2.06891 q^{95} +1.00000 q^{96} +2.70016 q^{97} -0.133785 q^{98} -1.79292 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} - 8 q^{5} + 8 q^{6} - 6 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^2 + 8 * q^3 + 8 * q^4 - 8 * q^5 + 8 * q^6 - 6 * q^7 + 8 * q^8 + 8 * q^9	\( 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} - 8 q^{5} + 8 q^{6} - 6 q^{7} + 8 q^{8} + 8 q^{9} - 8 q^{10} - 7 q^{11} + 8 q^{12} + 8 q^{13} - 6 q^{14} - 8 q^{15} + 8 q^{16} - 20 q^{17} + 8 q^{18} - 12 q^{19} - 8 q^{20} - 6 q^{21} - 7 q^{22} - 14 q^{23} + 8 q^{24} - 2 q^{25} + 8 q^{26} + 8 q^{27} - 6 q^{28} - 25 q^{29} - 8 q^{30} - 12 q^{31} + 8 q^{32} - 7 q^{33} - 20 q^{34} - 18 q^{35} + 8 q^{36} - 15 q^{37} - 12 q^{38} + 8 q^{39} - 8 q^{40} - 18 q^{41} - 6 q^{42} - 8 q^{43} - 7 q^{44} - 8 q^{45} - 14 q^{46} - 12 q^{47} + 8 q^{48} - 8 q^{49} - 2 q^{50} - 20 q^{51} + 8 q^{52} - 25 q^{53} + 8 q^{54} - 8 q^{55} - 6 q^{56} - 12 q^{57} - 25 q^{58} - 9 q^{59} - 8 q^{60} - 2 q^{61} - 12 q^{62} - 6 q^{63} + 8 q^{64} - 8 q^{65} - 7 q^{66} - 8 q^{67} - 20 q^{68} - 14 q^{69} - 18 q^{70} - 13 q^{71} + 8 q^{72} - 2 q^{73} - 15 q^{74} - 2 q^{75} - 12 q^{76} - 5 q^{77} + 8 q^{78} + q^{79} - 8 q^{80} + 8 q^{81} - 18 q^{82} - 6 q^{83} - 6 q^{84} + 5 q^{85} - 8 q^{86} - 25 q^{87} - 7 q^{88} - 17 q^{89} - 8 q^{90} - 6 q^{91} - 14 q^{92} - 12 q^{93} - 12 q^{94} + 10 q^{95} + 8 q^{96} + 19 q^{97} - 8 q^{98} - 7 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 + 8 * q^3 + 8 * q^4 - 8 * q^5 + 8 * q^6 - 6 * q^7 + 8 * q^8 + 8 * q^9 - 8 * q^10 - 7 * q^11 + 8 * q^12 + 8 * q^13 - 6 * q^14 - 8 * q^15 + 8 * q^16 - 20 * q^17 + 8 * q^18 - 12 * q^19 - 8 * q^20 - 6 * q^21 - 7 * q^22 - 14 * q^23 + 8 * q^24 - 2 * q^25 + 8 * q^26 + 8 * q^27 - 6 * q^28 - 25 * q^29 - 8 * q^30 - 12 * q^31 + 8 * q^32 - 7 * q^33 - 20 * q^34 - 18 * q^35 + 8 * q^36 - 15 * q^37 - 12 * q^38 + 8 * q^39 - 8 * q^40 - 18 * q^41 - 6 * q^42 - 8 * q^43 - 7 * q^44 - 8 * q^45 - 14 * q^46 - 12 * q^47 + 8 * q^48 - 8 * q^49 - 2 * q^50 - 20 * q^51 + 8 * q^52 - 25 * q^53 + 8 * q^54 - 8 * q^55 - 6 * q^56 - 12 * q^57 - 25 * q^58 - 9 * q^59 - 8 * q^60 - 2 * q^61 - 12 * q^62 - 6 * q^63 + 8 * q^64 - 8 * q^65 - 7 * q^66 - 8 * q^67 - 20 * q^68 - 14 * q^69 - 18 * q^70 - 13 * q^71 + 8 * q^72 - 2 * q^73 - 15 * q^74 - 2 * q^75 - 12 * q^76 - 5 * q^77 + 8 * q^78 + q^79 - 8 * q^80 + 8 * q^81 - 18 * q^82 - 6 * q^83 - 6 * q^84 + 5 * q^85 - 8 * q^86 - 25 * q^87 - 7 * q^88 - 17 * q^89 - 8 * q^90 - 6 * q^91 - 14 * q^92 - 12 * q^93 - 12 * q^94 + 10 * q^95 + 8 * q^96 + 19 * q^97 - 8 * q^98 - 7 * q^99

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