LMFDB - Embedded newform 6171.2.a.bk.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6171,2,Mod(1,6171)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6171, 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("6171.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6171 = 3 \cdot 11^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6171.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:	$49.2756830873$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} - 16 x^{10} + 15 x^{9} + 89 x^{8} - 78 x^{7} - 201 x^{6} + 157 x^{5} + 159 x^{4} + \cdots - 1 $ x^12 - x^11 - 16x^10 + 15x^9 + 89x^8 - 78x^7 - 201x^6 + 157x^5 + 159x^4 - 92x^3 - 29x^2 + 17x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 561)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-0.578221$ of defining polynomial
Character	$\chi$	$=$	6171.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.578221 q^{2} +1.00000 q^{3} -1.66566 q^{4} +0.295515 q^{5} +0.578221 q^{6} -0.675716 q^{7} -2.11956 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.578221 q^{2} +1.00000 q^{3} -1.66566 q^{4} +0.295515 q^{5} +0.578221 q^{6} -0.675716 q^{7} -2.11956 q^{8} +1.00000 q^{9} +0.170873 q^{10} -1.66566 q^{12} -1.30008 q^{13} -0.390713 q^{14} +0.295515 q^{15} +2.10575 q^{16} +1.00000 q^{17} +0.578221 q^{18} +3.78178 q^{19} -0.492227 q^{20} -0.675716 q^{21} +3.89916 q^{23} -2.11956 q^{24} -4.91267 q^{25} -0.751731 q^{26} +1.00000 q^{27} +1.12551 q^{28} -3.63988 q^{29} +0.170873 q^{30} -10.2196 q^{31} +5.45671 q^{32} +0.578221 q^{34} -0.199684 q^{35} -1.66566 q^{36} +0.661263 q^{37} +2.18670 q^{38} -1.30008 q^{39} -0.626361 q^{40} +4.03059 q^{41} -0.390713 q^{42} -1.32094 q^{43} +0.295515 q^{45} +2.25458 q^{46} -7.60771 q^{47} +2.10575 q^{48} -6.54341 q^{49} -2.84061 q^{50} +1.00000 q^{51} +2.16548 q^{52} +2.64220 q^{53} +0.578221 q^{54} +1.43222 q^{56} +3.78178 q^{57} -2.10466 q^{58} +5.46963 q^{59} -0.492227 q^{60} +1.46761 q^{61} -5.90918 q^{62} -0.675716 q^{63} -1.05630 q^{64} -0.384191 q^{65} +7.18223 q^{67} -1.66566 q^{68} +3.89916 q^{69} -0.115461 q^{70} -10.6555 q^{71} -2.11956 q^{72} +6.38038 q^{73} +0.382356 q^{74} -4.91267 q^{75} -6.29916 q^{76} -0.751731 q^{78} -7.13648 q^{79} +0.622278 q^{80} +1.00000 q^{81} +2.33057 q^{82} -4.69050 q^{83} +1.12551 q^{84} +0.295515 q^{85} -0.763797 q^{86} -3.63988 q^{87} -1.61617 q^{89} +0.170873 q^{90} +0.878482 q^{91} -6.49468 q^{92} -10.2196 q^{93} -4.39894 q^{94} +1.11757 q^{95} +5.45671 q^{96} +0.776216 q^{97} -3.78354 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - q^{2} + 12 q^{3} + 9 q^{4} - 14 q^{5} - q^{6} - 5 q^{7} + 12 q^{9}+O(q^{10}) $ 12 * q - q^2 + 12 * q^3 + 9 * q^4 - 14 * q^5 - q^6 - 5 * q^7 + 12 * q^9	$ 12 q - q^{2} + 12 q^{3} + 9 q^{4} - 14 q^{5} - q^{6} - 5 q^{7} + 12 q^{9} + 8 q^{10} + 9 q^{12} - 7 q^{13} - 13 q^{14} - 14 q^{15} + 11 q^{16} + 12 q^{17} - q^{18} + 3 q^{19} - 36 q^{20} - 5 q^{21} - 39 q^{23} + 8 q^{25} - 25 q^{26} + 12 q^{27} - 9 q^{28} + 10 q^{29} + 8 q^{30} - 25 q^{31} - 11 q^{32} - q^{34} + 12 q^{35} + 9 q^{36} - 10 q^{37} - 34 q^{38} - 7 q^{39} - 13 q^{40} + 17 q^{41} - 13 q^{42} - 18 q^{43} - 14 q^{45} + 19 q^{46} - 38 q^{47} + 11 q^{48} - 15 q^{49} - 18 q^{50} + 12 q^{51} + 32 q^{52} - 40 q^{53} - q^{54} - 25 q^{56} + 3 q^{57} + 3 q^{58} - 18 q^{59} - 36 q^{60} - 22 q^{61} + 3 q^{62} - 5 q^{63} - 20 q^{64} + 3 q^{65} - 9 q^{67} + 9 q^{68} - 39 q^{69} + 13 q^{70} - 24 q^{71} + q^{73} - 4 q^{74} + 8 q^{75} + 29 q^{76} - 25 q^{78} - 3 q^{79} - 43 q^{80} + 12 q^{81} + q^{82} + 4 q^{83} - 9 q^{84} - 14 q^{85} - 4 q^{86} + 10 q^{87} - 62 q^{89} + 8 q^{90} - 5 q^{91} - 52 q^{92} - 25 q^{93} - 16 q^{94} - 4 q^{95} - 11 q^{96} - 5 q^{97} - 7 q^{98}+O(q^{100}) $ 12 * q - q^2 + 12 * q^3 + 9 * q^4 - 14 * q^5 - q^6 - 5 * q^7 + 12 * q^9 + 8 * q^10 + 9 * q^12 - 7 * q^13 - 13 * q^14 - 14 * q^15 + 11 * q^16 + 12 * q^17 - q^18 + 3 * q^19 - 36 * q^20 - 5 * q^21 - 39 * q^23 + 8 * q^25 - 25 * q^26 + 12 * q^27 - 9 * q^28 + 10 * q^29 + 8 * q^30 - 25 * q^31 - 11 * q^32 - q^34 + 12 * q^35 + 9 * q^36 - 10 * q^37 - 34 * q^38 - 7 * q^39 - 13 * q^40 + 17 * q^41 - 13 * q^42 - 18 * q^43 - 14 * q^45 + 19 * q^46 - 38 * q^47 + 11 * q^48 - 15 * q^49 - 18 * q^50 + 12 * q^51 + 32 * q^52 - 40 * q^53 - q^54 - 25 * q^56 + 3 * q^57 + 3 * q^58 - 18 * q^59 - 36 * q^60 - 22 * q^61 + 3 * q^62 - 5 * q^63 - 20 * q^64 + 3 * q^65 - 9 * q^67 + 9 * q^68 - 39 * q^69 + 13 * q^70 - 24 * q^71 + q^73 - 4 * q^74 + 8 * q^75 + 29 * q^76 - 25 * q^78 - 3 * q^79 - 43 * q^80 + 12 * q^81 + q^82 + 4 * q^83 - 9 * q^84 - 14 * q^85 - 4 * q^86 + 10 * q^87 - 62 * q^89 + 8 * q^90 - 5 * q^91 - 52 * q^92 - 25 * q^93 - 16 * q^94 - 4 * q^95 - 11 * q^96 - 5 * q^97 - 7 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.578221	0.408864	0.204432	−	0.978881i	$-0.434465\pi$
$2$	0.578221	0.408864	0.204432	+	0.978881i	$0.434465\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.66566	−0.832830
$5$	0.295515	0.132158	0.0660791	−	0.997814i	$-0.478951\pi$
$5$	0.295515	0.132158	0.0660791	+	0.997814i	$0.478951\pi$
$6$	0.578221	0.236058
$7$	−0.675716	−0.255397	−0.127698	−	0.991813i	$-0.540759\pi$
$7$	−0.675716	−0.255397	−0.127698	+	0.991813i	$0.540759\pi$
$8$	−2.11956	−0.749378
$9$	1.00000	0.333333
$10$	0.170873	0.0540347
$11$	0	0
$11$	0	0
$12$	−1.66566	−0.480835
$13$	−1.30008	−0.360576	−0.180288	−	0.983614i	$-0.557703\pi$
$13$	−1.30008	−0.360576	−0.180288	+	0.983614i	$0.557703\pi$
$14$	−0.390713	−0.104423
$15$	0.295515	0.0763015
$16$	2.10575	0.526436
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0.578221	0.136288
$19$	3.78178	0.867599	0.433800	−	0.901009i	$-0.357173\pi$
$19$	3.78178	0.867599	0.433800	+	0.901009i	$0.357173\pi$
$20$	−0.492227	−0.110065
$21$	−0.675716	−0.147453
$22$	0	0
$23$	3.89916	0.813031	0.406516	−	0.913644i	$-0.366744\pi$
$23$	3.89916	0.813031	0.406516	+	0.913644i	$0.366744\pi$
$24$	−2.11956	−0.432654
$25$	−4.91267	−0.982534
$26$	−0.751731	−0.147427
$27$	1.00000	0.192450
$28$	1.12551	0.212702
$29$	−3.63988	−0.675909	−0.337955	−	0.941162i	$-0.609735\pi$
$29$	−3.63988	−0.675909	−0.337955	+	0.941162i	$0.609735\pi$
$30$	0.170873	0.0311970
$31$	−10.2196	−1.83549	−0.917747	−	0.397167i	$-0.869994\pi$
$31$	−10.2196	−1.83549	−0.917747	+	0.397167i	$0.869994\pi$
$32$	5.45671	0.964619
$33$	0	0
$34$	0.578221	0.0991641
$35$	−0.199684	−0.0337527
$36$	−1.66566	−0.277610
$37$	0.661263	0.108711	0.0543555	−	0.998522i	$-0.482690\pi$
$37$	0.661263	0.108711	0.0543555	+	0.998522i	$0.482690\pi$
$38$	2.18670	0.354730
$39$	−1.30008	−0.208179
$40$	−0.626361	−0.0990364
$41$	4.03059	0.629473	0.314736	−	0.949179i	$-0.398084\pi$
$41$	4.03059	0.629473	0.314736	+	0.949179i	$0.398084\pi$
$42$	−0.390713	−0.0602884
$43$	−1.32094	−0.201442	−0.100721	−	0.994915i	$-0.532115\pi$
$43$	−1.32094	−0.201442	−0.100721	+	0.994915i	$0.532115\pi$
$44$	0	0
$45$	0.295515	0.0440527
$46$	2.25458	0.332419
$47$	−7.60771	−1.10970	−0.554849	−	0.831951i	$-0.687224\pi$
$47$	−7.60771	−1.10970	−0.554849	+	0.831951i	$0.687224\pi$
$48$	2.10575	0.303938
$49$	−6.54341	−0.934773
$50$	−2.84061	−0.401723
$51$	1.00000	0.140028
$52$	2.16548	0.300299
$53$	2.64220	0.362934	0.181467	−	0.983397i	$-0.441915\pi$
$53$	2.64220	0.362934	0.181467	+	0.983397i	$0.441915\pi$
$54$	0.578221	0.0786859
$55$	0	0
$56$	1.43222	0.191389
$57$	3.78178	0.500909
$58$	−2.10466	−0.276355
$59$	5.46963	0.712085	0.356042	−	0.934470i	$-0.384126\pi$
$59$	5.46963	0.712085	0.356042	+	0.934470i	$0.384126\pi$
$60$	−0.492227	−0.0635462
$61$	1.46761	0.187909	0.0939543	−	0.995577i	$-0.470049\pi$
$61$	1.46761	0.187909	0.0939543	+	0.995577i	$0.470049\pi$
$62$	−5.90918	−0.750467
$63$	−0.675716	−0.0851322
$64$	−1.05630	−0.132038
$65$	−0.384191	−0.0476530
$66$	0	0
$67$	7.18223	0.877449	0.438724	−	0.898622i	$-0.355430\pi$
$67$	7.18223	0.877449	0.438724	+	0.898622i	$0.355430\pi$
$68$	−1.66566	−0.201991
$69$	3.89916	0.469404
$70$	−0.115461	−0.0138003
$71$	−10.6555	−1.26457	−0.632285	−	0.774736i	$-0.717883\pi$
$71$	−10.6555	−1.26457	−0.632285	+	0.774736i	$0.717883\pi$
$72$	−2.11956	−0.249793
$73$	6.38038	0.746767	0.373383	−	0.927677i	$-0.378198\pi$
$73$	6.38038	0.746767	0.373383	+	0.927677i	$0.378198\pi$
$74$	0.382356	0.0444480
$75$	−4.91267	−0.567266
$76$	−6.29916	−0.722563
$77$	0	0
$78$	−0.751731	−0.0851168
$79$	−7.13648	−0.802916	−0.401458	−	0.915877i	$-0.631496\pi$
$79$	−7.13648	−0.802916	−0.401458	+	0.915877i	$0.631496\pi$
$80$	0.622278	0.0695728
$81$	1.00000	0.111111
$82$	2.33057	0.257369
$83$	−4.69050	−0.514849	−0.257425	−	0.966298i	$-0.582874\pi$
$83$	−4.69050	−0.514849	−0.257425	+	0.966298i	$0.582874\pi$
$84$	1.12551	0.122804
$85$	0.295515	0.0320531
$86$	−0.763797	−0.0823623
$87$	−3.63988	−0.390236
$88$	0	0
$89$	−1.61617	−0.171313	−0.0856567	−	0.996325i	$-0.527299\pi$
$89$	−1.61617	−0.171313	−0.0856567	+	0.996325i	$0.527299\pi$
$90$	0.170873	0.0180116
$91$	0.878482	0.0920899
$92$	−6.49468	−0.677117
$93$	−10.2196	−1.05972
$94$	−4.39894	−0.453716
$95$	1.11757	0.114660
$96$	5.45671	0.556923
$97$	0.776216	0.0788128	0.0394064	−	0.999223i	$-0.487453\pi$
$97$	0.776216	0.0788128	0.0394064	+	0.999223i	$0.487453\pi$
$98$	−3.78354	−0.382195
$99$	0	0
$100$	8.18284	0.818284
$101$	5.35155	0.532499	0.266249	−	0.963904i	$-0.414216\pi$
$101$	5.35155	0.532499	0.266249	+	0.963904i	$0.414216\pi$
$102$	0.578221	0.0572524
$103$	−0.594384	−0.0585664	−0.0292832	−	0.999571i	$-0.509322\pi$
$103$	−0.594384	−0.0585664	−0.0292832	+	0.999571i	$0.509322\pi$
$104$	2.75559	0.270208
$105$	−0.199684	−0.0194872
$106$	1.52778	0.148391
$107$	15.0520	1.45513	0.727565	−	0.686039i	$-0.240652\pi$
$107$	15.0520	1.45513	0.727565	+	0.686039i	$0.240652\pi$
$108$	−1.66566	−0.160278
$109$	−15.0353	−1.44012	−0.720059	−	0.693913i	$-0.755885\pi$
$109$	−15.0353	−1.44012	−0.720059	+	0.693913i	$0.755885\pi$
$110$	0	0
$111$	0.661263	0.0627643
$112$	−1.42289	−0.134450
$113$	−0.349642	−0.0328916	−0.0164458	−	0.999865i	$-0.505235\pi$
$113$	−0.349642	−0.0328916	−0.0164458	+	0.999865i	$0.505235\pi$
$114$	2.18670	0.204804
$115$	1.15226	0.107449
$116$	6.06281	0.562918
$117$	−1.30008	−0.120192
$118$	3.16265	0.291146
$119$	−0.675716	−0.0619428
$120$	−0.626361	−0.0571787
$121$	0	0
$122$	0.848604	0.0768290
$123$	4.03059	0.363426
$124$	17.0224	1.52865
$125$	−2.92934	−0.262008
$126$	−0.390713	−0.0348075
$127$	−11.2686	−0.999924	−0.499962	−	0.866047i	$-0.666653\pi$
$127$	−11.2686	−0.999924	−0.499962	+	0.866047i	$0.666653\pi$
$128$	−11.5242	−1.01860
$129$	−1.32094	−0.116302
$130$	−0.222147	−0.0194836
$131$	−10.5531	−0.922030	−0.461015	−	0.887392i	$-0.652515\pi$
$131$	−10.5531	−0.922030	−0.461015	+	0.887392i	$0.652515\pi$
$132$	0	0
$133$	−2.55541	−0.221582
$134$	4.15292	0.358757
$135$	0.295515	0.0254338
$136$	−2.11956	−0.181751
$137$	−22.3847	−1.91245	−0.956227	−	0.292625i	$-0.905471\pi$
$137$	−22.3847	−1.91245	−0.956227	+	0.292625i	$0.905471\pi$
$138$	2.25458	0.191922
$139$	16.3147	1.38379	0.691896	−	0.721998i	$-0.256776\pi$
$139$	16.3147	1.38379	0.691896	+	0.721998i	$0.256776\pi$
$140$	0.332606	0.0281103
$141$	−7.60771	−0.640685
$142$	−6.16121	−0.517037
$143$	0	0
$144$	2.10575	0.175479
$145$	−1.07564	−0.0893269
$146$	3.68927	0.305326
$147$	−6.54341	−0.539691
$148$	−1.10144	−0.0905378
$149$	−13.8579	−1.13528	−0.567640	−	0.823277i	$-0.692143\pi$
$149$	−13.8579	−1.13528	−0.567640	+	0.823277i	$0.692143\pi$
$150$	−2.84061	−0.231935
$151$	6.53073	0.531463	0.265731	−	0.964047i	$-0.414387\pi$
$151$	6.53073	0.531463	0.265731	+	0.964047i	$0.414387\pi$
$152$	−8.01571	−0.650160
$153$	1.00000	0.0808452
$154$	0	0
$155$	−3.02004	−0.242575
$156$	2.16548	0.173377
$157$	−14.3597	−1.14602	−0.573012	−	0.819547i	$-0.694225\pi$
$157$	−14.3597	−1.14602	−0.573012	+	0.819547i	$0.694225\pi$
$158$	−4.12646	−0.328284
$159$	2.64220	0.209540
$160$	1.61254	0.127482
$161$	−2.63473	−0.207645
$162$	0.578221	0.0454293
$163$	−25.1415	−1.96924	−0.984618	−	0.174721i	$-0.944098\pi$
$163$	−25.1415	−1.96924	−0.984618	+	0.174721i	$0.944098\pi$
$164$	−6.71360	−0.524244
$165$	0	0
$166$	−2.71215	−0.210503
$167$	−2.88020	−0.222877	−0.111438	−	0.993771i	$-0.535546\pi$
$167$	−2.88020	−0.222877	−0.111438	+	0.993771i	$0.535546\pi$
$168$	1.43222	0.110498
$169$	−11.3098	−0.869985
$170$	0.170873	0.0131053
$171$	3.78178	0.289200
$172$	2.20024	0.167767
$173$	−17.9028	−1.36112	−0.680562	−	0.732690i	$-0.738264\pi$
$173$	−17.9028	−1.36112	−0.680562	+	0.732690i	$0.738264\pi$
$174$	−2.10466	−0.159554
$175$	3.31957	0.250936
$176$	0	0
$177$	5.46963	0.411122
$178$	−0.934502	−0.0700439
$179$	8.06537	0.602834	0.301417	−	0.953492i	$-0.402540\pi$
$179$	8.06537	0.602834	0.301417	+	0.953492i	$0.402540\pi$
$180$	−0.492227	−0.0366884
$181$	−1.13459	−0.0843338	−0.0421669	−	0.999111i	$-0.513426\pi$
$181$	−1.13459	−0.0843338	−0.0421669	+	0.999111i	$0.513426\pi$
$182$	0.507957	0.0376523
$183$	1.46761	0.108489
$184$	−8.26451	−0.609268
$185$	0.195413	0.0143670
$186$	−5.90918	−0.433282
$187$	0	0
$188$	12.6719	0.924190
$189$	−0.675716	−0.0491511
$190$	0.646203	0.0468805
$191$	−18.3971	−1.33116	−0.665582	−	0.746325i	$-0.731817\pi$
$191$	−18.3971	−1.33116	−0.665582	+	0.746325i	$0.731817\pi$
$192$	−1.05630	−0.0762322
$193$	−16.3887	−1.17968	−0.589842	−	0.807519i	$-0.700810\pi$
$193$	−16.3887	−1.17968	−0.589842	+	0.807519i	$0.700810\pi$
$194$	0.448824	0.0322237
$195$	−0.384191	−0.0275125
$196$	10.8991	0.778507
$197$	−2.73159	−0.194618	−0.0973088	−	0.995254i	$-0.531023\pi$
$197$	−2.73159	−0.194618	−0.0973088	+	0.995254i	$0.531023\pi$
$198$	0	0
$199$	−5.58625	−0.395999	−0.197999	−	0.980202i	$-0.563444\pi$
$199$	−5.58625	−0.395999	−0.197999	+	0.980202i	$0.563444\pi$
$200$	10.4127	0.736290
$201$	7.18223	0.506595
$202$	3.09438	0.217720
$203$	2.45953	0.172625
$204$	−1.66566	−0.116620
$205$	1.19110	0.0831899
$206$	−0.343685	−0.0239457
$207$	3.89916	0.271010
$208$	−2.73763	−0.189820
$209$	0	0
$210$	−0.115461	−0.00796760
$211$	−13.2282	−0.910666	−0.455333	−	0.890321i	$-0.650480\pi$
$211$	−13.2282	−0.910666	−0.455333	+	0.890321i	$0.650480\pi$
$212$	−4.40101	−0.302263
$213$	−10.6555	−0.730100
$214$	8.70338	0.594950
$215$	−0.390358	−0.0266222
$216$	−2.11956	−0.144218
$217$	6.90554	0.468779
$218$	−8.69371	−0.588812
$219$	6.38038	0.431146
$220$	0	0
$221$	−1.30008	−0.0874525
$222$	0.382356	0.0256621
$223$	−19.1046	−1.27934	−0.639669	−	0.768650i	$-0.720929\pi$
$223$	−19.1046	−1.27934	−0.639669	+	0.768650i	$0.720929\pi$
$224$	−3.68719	−0.246361
$225$	−4.91267	−0.327511
$226$	−0.202171	−0.0134482
$227$	−5.63298	−0.373874	−0.186937	−	0.982372i	$-0.559856\pi$
$227$	−5.63298	−0.373874	−0.186937	+	0.982372i	$0.559856\pi$
$228$	−6.29916	−0.417172
$229$	3.47425	0.229585	0.114792	−	0.993389i	$-0.463380\pi$
$229$	3.47425	0.229585	0.114792	+	0.993389i	$0.463380\pi$
$230$	0.666260	0.0439319
$231$	0	0
$232$	7.71496	0.506512
$233$	23.8538	1.56272	0.781358	−	0.624083i	$-0.214527\pi$
$233$	23.8538	1.56272	0.781358	+	0.624083i	$0.214527\pi$
$234$	−0.751731	−0.0491422
$235$	−2.24819	−0.146656
$236$	−9.11054	−0.593046
$237$	−7.13648	−0.463564
$238$	−0.390713	−0.0253262
$239$	9.79938	0.633869	0.316935	−	0.948447i	$-0.397346\pi$
$239$	9.79938	0.633869	0.316935	+	0.948447i	$0.397346\pi$
$240$	0.622278	0.0401679
$241$	18.4928	1.19123	0.595614	−	0.803270i	$-0.296909\pi$
$241$	18.4928	1.19123	0.595614	+	0.803270i	$0.296909\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	−2.44454	−0.156496
$245$	−1.93367	−0.123538
$246$	2.33057	0.148592
$247$	−4.91659	−0.312835
$248$	21.6611	1.37548
$249$	−4.69050	−0.297248
$250$	−1.69381	−0.107126
$251$	2.68951	0.169761	0.0848803	−	0.996391i	$-0.472949\pi$
$251$	2.68951	0.169761	0.0848803	+	0.996391i	$0.472949\pi$
$252$	1.12551	0.0709007
$253$	0	0
$254$	−6.51572	−0.408833
$255$	0.295515	0.0185058
$256$	−4.55093	−0.284433
$257$	0.898777	0.0560641	0.0280321	−	0.999607i	$-0.491076\pi$
$257$	0.898777	0.0560641	0.0280321	+	0.999607i	$0.491076\pi$
$258$	−0.763797	−0.0475519
$259$	−0.446826	−0.0277644
$260$	0.639932	0.0396869
$261$	−3.63988	−0.225303
$262$	−6.10203	−0.376985
$263$	−18.2970	−1.12824	−0.564119	−	0.825693i	$-0.690784\pi$
$263$	−18.2970	−1.12824	−0.564119	+	0.825693i	$0.690784\pi$
$264$	0	0
$265$	0.780809	0.0479647
$266$	−1.47759	−0.0905969
$267$	−1.61617	−0.0989078
$268$	−11.9632	−0.730766
$269$	−17.3033	−1.05500	−0.527499	−	0.849556i	$-0.676870\pi$
$269$	−17.3033	−1.05500	−0.527499	+	0.849556i	$0.676870\pi$
$270$	0.170873	0.0103990
$271$	−29.0442	−1.76431	−0.882155	−	0.470959i	$-0.843908\pi$
$271$	−29.0442	−1.76431	−0.882155	+	0.470959i	$0.843908\pi$
$272$	2.10575	0.127680
$273$	0.878482	0.0531681
$274$	−12.9433	−0.781934
$275$	0	0
$276$	−6.49468	−0.390934
$277$	8.68398	0.521770	0.260885	−	0.965370i	$-0.415986\pi$
$277$	8.68398	0.521770	0.260885	+	0.965370i	$0.415986\pi$
$278$	9.43348	0.565783
$279$	−10.2196	−0.611831
$280$	0.423242	0.0252936
$281$	25.7488	1.53604	0.768021	−	0.640424i	$-0.221242\pi$
$281$	25.7488	1.53604	0.768021	+	0.640424i	$0.221242\pi$
$282$	−4.39894	−0.261953
$283$	−5.27292	−0.313442	−0.156721	−	0.987643i	$-0.550092\pi$
$283$	−5.27292	−0.313442	−0.156721	+	0.987643i	$0.550092\pi$
$284$	17.7484	1.05317
$285$	1.11757	0.0661991
$286$	0	0
$287$	−2.72354	−0.160765
$288$	5.45671	0.321540
$289$	1.00000	0.0588235
$290$	−0.621957	−0.0365226
$291$	0.776216	0.0455026
$292$	−10.6275	−0.621930
$293$	24.9621	1.45830	0.729152	−	0.684352i	$-0.239915\pi$
$293$	24.9621	1.45830	0.729152	+	0.684352i	$0.239915\pi$
$294$	−3.78354	−0.220660
$295$	1.61635	0.0941078
$296$	−1.40159	−0.0814657
$297$	0	0
$298$	−8.01291	−0.464175
$299$	−5.06920	−0.293159
$300$	8.18284	0.472437
$301$	0.892582	0.0514476
$302$	3.77620	0.217296
$303$	5.35155	0.307438
$304$	7.96346	0.456736
$305$	0.433701	0.0248336
$306$	0.578221	0.0330547
$307$	−4.55687	−0.260075	−0.130037	−	0.991509i	$-0.541510\pi$
$307$	−4.55687	−0.260075	−0.130037	+	0.991509i	$0.541510\pi$
$308$	0	0
$309$	−0.594384	−0.0338133
$310$	−1.74625	−0.0991803
$311$	10.9031	0.618256	0.309128	−	0.951021i	$-0.399963\pi$
$311$	10.9031	0.618256	0.309128	+	0.951021i	$0.399963\pi$
$312$	2.75559	0.156005
$313$	−9.08484	−0.513505	−0.256753	−	0.966477i	$-0.582653\pi$
$313$	−9.08484	−0.513505	−0.256753	+	0.966477i	$0.582653\pi$
$314$	−8.30305	−0.468568
$315$	−0.199684	−0.0112509
$316$	11.8869	0.668693
$317$	−7.53172	−0.423023	−0.211512	−	0.977375i	$-0.567839\pi$
$317$	−7.53172	−0.423023	−0.211512	+	0.977375i	$0.567839\pi$
$318$	1.52778	0.0856735
$319$	0	0
$320$	−0.312153	−0.0174499
$321$	15.0520	0.840120
$322$	−1.52345	−0.0848988
$323$	3.78178	0.210424
$324$	−1.66566	−0.0925367
$325$	6.38684	0.354278
$326$	−14.5374	−0.805150
$327$	−15.0353	−0.831452
$328$	−8.54309	−0.471713
$329$	5.14065	0.283413
$330$	0	0
$331$	14.1498	0.777745	0.388873	−	0.921292i	$-0.372865\pi$
$331$	14.1498	0.777745	0.388873	+	0.921292i	$0.372865\pi$
$332$	7.81278	0.428782
$333$	0.661263	0.0362370
$334$	−1.66539	−0.0911263
$335$	2.12245	0.115962
$336$	−1.42289	−0.0776248
$337$	−8.95116	−0.487601	−0.243800	−	0.969825i	$-0.578394\pi$
$337$	−8.95116	−0.487601	−0.243800	+	0.969825i	$0.578394\pi$
$338$	−6.53957	−0.355706
$339$	−0.349642	−0.0189900
$340$	−0.492227	−0.0266947
$341$	0	0
$342$	2.18670	0.118243
$343$	9.15150	0.494134
$344$	2.79982	0.150956
$345$	1.15226	0.0620355
$346$	−10.3518	−0.556515
$347$	17.8972	0.960771	0.480386	−	0.877057i	$-0.340497\pi$
$347$	17.8972	0.960771	0.480386	+	0.877057i	$0.340497\pi$
$348$	6.06281	0.325001
$349$	31.7749	1.70087	0.850435	−	0.526080i	$-0.176339\pi$
$349$	31.7749	1.70087	0.850435	+	0.526080i	$0.176339\pi$
$350$	1.91945	0.102599
$351$	−1.30008	−0.0693929
$352$	0	0
$353$	−26.8576	−1.42948	−0.714742	−	0.699388i	$-0.753456\pi$
$353$	−26.8576	−1.42948	−0.714742	+	0.699388i	$0.753456\pi$
$354$	3.16265	0.168093
$355$	−3.14884	−0.167123
$356$	2.69199	0.142675
$357$	−0.675716	−0.0357627
$358$	4.66357	0.246477
$359$	−1.75084	−0.0924059	−0.0462030	−	0.998932i	$-0.514712\pi$
$359$	−1.75084	−0.0924059	−0.0462030	+	0.998932i	$0.514712\pi$
$360$	−0.626361	−0.0330121
$361$	−4.69816	−0.247272
$362$	−0.656047	−0.0344810
$363$	0	0
$364$	−1.46325	−0.0766953
$365$	1.88549	0.0986913
$366$	0.848604	0.0443573
$367$	−32.9315	−1.71901	−0.859504	−	0.511129i	$-0.829228\pi$
$367$	−32.9315	−1.71901	−0.859504	+	0.511129i	$0.829228\pi$
$368$	8.21064	0.428009
$369$	4.03059	0.209824
$370$	0.112992	0.00587417
$371$	−1.78538	−0.0926922
$372$	17.0224	0.882569
$373$	8.98754	0.465357	0.232679	−	0.972554i	$-0.425251\pi$
$373$	8.98754	0.465357	0.232679	+	0.972554i	$0.425251\pi$
$374$	0	0
$375$	−2.92934	−0.151270
$376$	16.1250	0.831584
$377$	4.73212	0.243717
$378$	−0.390713	−0.0200961
$379$	28.1973	1.44840	0.724198	−	0.689592i	$-0.242210\pi$
$379$	28.1973	1.44840	0.724198	+	0.689592i	$0.242210\pi$
$380$	−1.86149	−0.0954925
$381$	−11.2686	−0.577306
$382$	−10.6376	−0.544265
$383$	−30.7660	−1.57207	−0.786033	−	0.618184i	$-0.787869\pi$
$383$	−30.7660	−1.57207	−0.786033	+	0.618184i	$0.787869\pi$
$384$	−11.5242	−0.588092
$385$	0	0
$386$	−9.47629	−0.482331
$387$	−1.32094	−0.0671473
$388$	−1.29291	−0.0656376
$389$	24.8648	1.26069	0.630347	−	0.776313i	$-0.282913\pi$
$389$	24.8648	1.26069	0.630347	+	0.776313i	$0.282913\pi$
$390$	−0.222147	−0.0112489
$391$	3.89916	0.197189
$392$	13.8692	0.700498
$393$	−10.5531	−0.532334
$394$	−1.57946	−0.0795722
$395$	−2.10893	−0.106112
$396$	0	0
$397$	35.3979	1.77657	0.888286	−	0.459291i	$-0.151897\pi$
$397$	35.3979	1.77657	0.888286	+	0.459291i	$0.151897\pi$
$398$	−3.23009	−0.161910
$399$	−2.55541	−0.127930
$400$	−10.3448	−0.517242
$401$	3.56983	0.178269	0.0891343	−	0.996020i	$-0.471590\pi$
$401$	3.56983	0.178269	0.0891343	+	0.996020i	$0.471590\pi$
$402$	4.15292	0.207129
$403$	13.2862	0.661835
$404$	−8.91386	−0.443481
$405$	0.295515	0.0146842
$406$	1.42215	0.0705802
$407$	0	0
$408$	−2.11956	−0.104934
$409$	17.0101	0.841094	0.420547	−	0.907271i	$-0.361838\pi$
$409$	17.0101	0.841094	0.420547	+	0.907271i	$0.361838\pi$
$410$	0.688718	0.0340134
$411$	−22.3847	−1.10416
$412$	0.990042	0.0487759
$413$	−3.69592	−0.181864
$414$	2.25458	0.110806
$415$	−1.38611	−0.0680415
$416$	−7.09413	−0.347819
$417$	16.3147	0.798932
$418$	0	0
$419$	−21.0925	−1.03044	−0.515218	−	0.857059i	$-0.672289\pi$
$419$	−21.0925	−1.03044	−0.515218	+	0.857059i	$0.672289\pi$
$420$	0.332606	0.0162295
$421$	18.1661	0.885361	0.442680	−	0.896680i	$-0.354028\pi$
$421$	18.1661	0.885361	0.442680	+	0.896680i	$0.354028\pi$
$422$	−7.64882	−0.372339
$423$	−7.60771	−0.369899
$424$	−5.60031	−0.271975
$425$	−4.91267	−0.238300
$426$	−6.16121	−0.298512
$427$	−0.991689	−0.0479912
$428$	−25.0715	−1.21188
$429$	0	0
$430$	−0.225713	−0.0108848
$431$	34.5967	1.66647	0.833233	−	0.552922i	$-0.186487\pi$
$431$	34.5967	1.66647	0.833233	+	0.552922i	$0.186487\pi$
$432$	2.10575	0.101313
$433$	−17.4537	−0.838770	−0.419385	−	0.907809i	$-0.637754\pi$
$433$	−17.4537	−0.838770	−0.419385	+	0.907809i	$0.637754\pi$
$434$	3.99293	0.191667
$435$	−1.07564	−0.0515729
$436$	25.0437	1.19937
$437$	14.7458	0.705385
$438$	3.68927	0.176280
$439$	−15.5593	−0.742607	−0.371304	−	0.928512i	$-0.621089\pi$
$439$	−15.5593	−0.742607	−0.371304	+	0.928512i	$0.621089\pi$
$440$	0	0
$441$	−6.54341	−0.311591
$442$	−0.751731	−0.0357562
$443$	−4.88933	−0.232299	−0.116150	−	0.993232i	$-0.537055\pi$
$443$	−4.88933	−0.232299	−0.116150	+	0.993232i	$0.537055\pi$
$444$	−1.10144	−0.0522720
$445$	−0.477601	−0.0226405
$446$	−11.0467	−0.523076
$447$	−13.8579	−0.655454
$448$	0.713762	0.0337221
$449$	−31.0403	−1.46488	−0.732441	−	0.680831i	$-0.761619\pi$
$449$	−31.0403	−1.46488	−0.732441	+	0.680831i	$0.761619\pi$
$450$	−2.84061	−0.133908
$451$	0	0
$452$	0.582385	0.0273931
$453$	6.53073	0.306840
$454$	−3.25711	−0.152864
$455$	0.259604	0.0121704
$456$	−8.01571	−0.375370
$457$	−27.9289	−1.30646	−0.653230	−	0.757159i	$-0.726587\pi$
$457$	−27.9289	−1.30646	−0.653230	+	0.757159i	$0.726587\pi$
$458$	2.00889	0.0938690
$459$	1.00000	0.0466760
$460$	−1.91927	−0.0894865
$461$	0.985322	0.0458910	0.0229455	−	0.999737i	$-0.492696\pi$
$461$	0.985322	0.0458910	0.0229455	+	0.999737i	$0.492696\pi$
$462$	0	0
$463$	10.4666	0.486426	0.243213	−	0.969973i	$-0.421799\pi$
$463$	10.4666	0.486426	0.243213	+	0.969973i	$0.421799\pi$
$464$	−7.66467	−0.355823
$465$	−3.02004	−0.140051
$466$	13.7928	0.638939
$467$	−12.7740	−0.591112	−0.295556	−	0.955325i	$-0.595505\pi$
$467$	−12.7740	−0.591112	−0.295556	+	0.955325i	$0.595505\pi$
$468$	2.16548	0.100100
$469$	−4.85315	−0.224097
$470$	−1.29995	−0.0599622
$471$	−14.3597	−0.661658
$472$	−11.5932	−0.533621
$473$	0	0
$474$	−4.12646	−0.189535
$475$	−18.5786	−0.852446
$476$	1.12551	0.0515878
$477$	2.64220	0.120978
$478$	5.66621	0.259166
$479$	−6.22844	−0.284585	−0.142292	−	0.989825i	$-0.545447\pi$
$479$	−6.22844	−0.284585	−0.142292	+	0.989825i	$0.545447\pi$
$480$	1.61254	0.0736019
$481$	−0.859692	−0.0391986
$482$	10.6930	0.487051
$483$	−2.63473	−0.119884
$484$	0	0
$485$	0.229383	0.0104157
$486$	0.578221	0.0262286
$487$	−2.50449	−0.113489	−0.0567445	−	0.998389i	$-0.518072\pi$
$487$	−2.50449	−0.113489	−0.0567445	+	0.998389i	$0.518072\pi$
$488$	−3.11070	−0.140815
$489$	−25.1415	−1.13694
$490$	−1.11809	−0.0505102
$491$	1.16504	0.0525776	0.0262888	−	0.999654i	$-0.491631\pi$
$491$	1.16504	0.0525776	0.0262888	+	0.999654i	$0.491631\pi$
$492$	−6.71360	−0.302672
$493$	−3.63988	−0.163932
$494$	−2.84288	−0.127907
$495$	0	0
$496$	−21.5199	−0.966270
$497$	7.20006	0.322967
$498$	−2.71215	−0.121534
$499$	27.3400	1.22391	0.611954	−	0.790893i	$-0.290384\pi$
$499$	27.3400	1.22391	0.611954	+	0.790893i	$0.290384\pi$
$500$	4.87928	0.218208
$501$	−2.88020	−0.128678
$502$	1.55513	0.0694090
$503$	31.8150	1.41856	0.709281	−	0.704925i	$-0.249020\pi$
$503$	31.8150	1.41856	0.709281	+	0.704925i	$0.249020\pi$
$504$	1.43222	0.0637963
$505$	1.58146	0.0703740
$506$	0	0
$507$	−11.3098	−0.502286
$508$	18.7696	0.832767
$509$	25.8529	1.14591	0.572956	−	0.819586i	$-0.305797\pi$
$509$	25.8529	1.14591	0.572956	+	0.819586i	$0.305797\pi$
$510$	0.170873	0.00756637
$511$	−4.31132	−0.190722
$512$	20.4170	0.902311
$513$	3.78178	0.166970
$514$	0.519692	0.0229226
$515$	−0.175649	−0.00774002
$516$	2.20024	0.0968602
$517$	0	0
$518$	−0.258364	−0.0113519
$519$	−17.9028	−0.785846
$520$	0.814317	0.0357102
$521$	−22.1630	−0.970977	−0.485488	−	0.874243i	$-0.661358\pi$
$521$	−22.1630	−0.970977	−0.485488	+	0.874243i	$0.661358\pi$
$522$	−2.10466	−0.0921183
$523$	35.8663	1.56832	0.784161	−	0.620557i	$-0.213093\pi$
$523$	35.8663	1.56832	0.784161	+	0.620557i	$0.213093\pi$
$524$	17.5779	0.767894
$525$	3.31957	0.144878
$526$	−10.5797	−0.461296
$527$	−10.2196	−0.445172
$528$	0	0
$529$	−7.79655	−0.338980
$530$	0.451480	0.0196110
$531$	5.46963	0.237362
$532$	4.25644	0.184540
$533$	−5.24007	−0.226973
$534$	−0.934502	−0.0404399
$535$	4.44808	0.192307
$536$	−15.2232	−0.657541
$537$	8.06537	0.348046
$538$	−10.0051	−0.431351
$539$	0	0
$540$	−0.492227	−0.0211821
$541$	−35.0618	−1.50742	−0.753712	−	0.657204i	$-0.771739\pi$
$541$	−35.0618	−1.50742	−0.753712	+	0.657204i	$0.771739\pi$
$542$	−16.7940	−0.721363
$543$	−1.13459	−0.0486901
$544$	5.45671	0.233955
$545$	−4.44314	−0.190323
$546$	0.507957	0.0217385
$547$	4.75924	0.203490	0.101745	−	0.994810i	$-0.467557\pi$
$547$	4.75924	0.203490	0.101745	+	0.994810i	$0.467557\pi$
$548$	37.2853	1.59275
$549$	1.46761	0.0626362
$550$	0	0
$551$	−13.7652	−0.586418
$552$	−8.26451	−0.351761
$553$	4.82223	0.205062
$554$	5.02126	0.213333
$555$	0.195413	0.00829482
$556$	−27.1747	−1.15246
$557$	−12.4294	−0.526648	−0.263324	−	0.964707i	$-0.584819\pi$
$557$	−12.4294	−0.526648	−0.263324	+	0.964707i	$0.584819\pi$
$558$	−5.90918	−0.250156
$559$	1.71732	0.0726351
$560$	−0.420483	−0.0177687
$561$	0	0
$562$	14.8885	0.628033
$563$	−2.79895	−0.117962	−0.0589808	−	0.998259i	$-0.518785\pi$
$563$	−2.79895	−0.117962	−0.0589808	+	0.998259i	$0.518785\pi$
$564$	12.6719	0.533581
$565$	−0.103324	−0.00434689
$566$	−3.04891	−0.128155
$567$	−0.675716	−0.0283774
$568$	22.5849	0.947642
$569$	−0.558994	−0.0234342	−0.0117171	−	0.999931i	$-0.503730\pi$
$569$	−0.558994	−0.0234342	−0.0117171	+	0.999931i	$0.503730\pi$
$570$	0.646203	0.0270664
$571$	12.0788	0.505481	0.252741	−	0.967534i	$-0.418668\pi$
$571$	12.0788	0.505481	0.252741	+	0.967534i	$0.418668\pi$
$572$	0	0
$573$	−18.3971	−0.768548
$574$	−1.57481	−0.0657311
$575$	−19.1553	−0.798831
$576$	−1.05630	−0.0440127
$577$	25.7252	1.07096	0.535478	−	0.844549i	$-0.320132\pi$
$577$	25.7252	1.07096	0.535478	+	0.844549i	$0.320132\pi$
$578$	0.578221	0.0240508
$579$	−16.3887	−0.681091
$580$	1.79165	0.0743941
$581$	3.16945	0.131491
$582$	0.448824	0.0186044
$583$	0	0
$584$	−13.5236	−0.559611
$585$	−0.384191	−0.0158843
$586$	14.4336	0.596248
$587$	6.77198	0.279510	0.139755	−	0.990186i	$-0.455369\pi$
$587$	6.77198	0.279510	0.139755	+	0.990186i	$0.455369\pi$
$588$	10.8991	0.449471
$589$	−38.6482	−1.59247
$590$	0.934610	0.0384773
$591$	−2.73159	−0.112363
$592$	1.39245	0.0572294
$593$	−34.5327	−1.41809	−0.709044	−	0.705164i	$-0.750873\pi$
$593$	−34.5327	−1.41809	−0.709044	+	0.705164i	$0.750873\pi$
$594$	0	0
$595$	−0.199684	−0.00818624
$596$	23.0825	0.945495
$597$	−5.58625	−0.228630
$598$	−2.93112	−0.119862
$599$	−37.7651	−1.54304	−0.771519	−	0.636206i	$-0.780503\pi$
$599$	−37.7651	−1.54304	−0.771519	+	0.636206i	$0.780503\pi$
$600$	10.4127	0.425097
$601$	9.55603	0.389799	0.194899	−	0.980823i	$-0.437562\pi$
$601$	9.55603	0.389799	0.194899	+	0.980823i	$0.437562\pi$
$602$	0.516110	0.0210351
$603$	7.18223	0.292483
$604$	−10.8780	−0.442618
$605$	0	0
$606$	3.09438	0.125701
$607$	35.0898	1.42425	0.712126	−	0.702052i	$-0.247733\pi$
$607$	35.0898	1.42425	0.712126	+	0.702052i	$0.247733\pi$
$608$	20.6361	0.836903
$609$	2.45953	0.0996651
$610$	0.250775	0.0101536
$611$	9.89059	0.400131
$612$	−1.66566	−0.0673303
$613$	12.1780	0.491863	0.245931	−	0.969287i	$-0.420906\pi$
$613$	12.1780	0.491863	0.245931	+	0.969287i	$0.420906\pi$
$614$	−2.63488	−0.106335
$615$	1.19110	0.0480297
$616$	0	0
$617$	25.7045	1.03482	0.517411	−	0.855737i	$-0.326896\pi$
$617$	25.7045	1.03482	0.517411	+	0.855737i	$0.326896\pi$
$618$	−0.343685	−0.0138251
$619$	14.5687	0.585566	0.292783	−	0.956179i	$-0.405419\pi$
$619$	14.5687	0.585566	0.292783	+	0.956179i	$0.405419\pi$
$620$	5.03036	0.202024
$621$	3.89916	0.156468
$622$	6.30438	0.252782
$623$	1.09207	0.0437529
$624$	−2.73763	−0.109593
$625$	23.6977	0.947908
$626$	−5.25305	−0.209954
$627$	0	0
$628$	23.9183	0.954444
$629$	0.661263	0.0263663
$630$	−0.115461	−0.00460009
$631$	35.6847	1.42058	0.710292	−	0.703907i	$-0.248563\pi$
$631$	35.6847	1.42058	0.710292	+	0.703907i	$0.248563\pi$
$632$	15.1262	0.601688
$633$	−13.2282	−0.525773
$634$	−4.35500	−0.172959
$635$	−3.33003	−0.132148
$636$	−4.40101	−0.174511
$637$	8.50692	0.337057
$638$	0	0
$639$	−10.6555	−0.421523
$640$	−3.40557	−0.134617
$641$	−38.8493	−1.53445	−0.767227	−	0.641375i	$-0.778364\pi$
$641$	−38.8493	−1.53445	−0.767227	+	0.641375i	$0.778364\pi$
$642$	8.70338	0.343495
$643$	27.0458	1.06658	0.533291	−	0.845932i	$-0.320955\pi$
$643$	27.0458	1.06658	0.533291	+	0.845932i	$0.320955\pi$
$644$	4.38856	0.172933
$645$	−0.390358	−0.0153703
$646$	2.18670	0.0860347
$647$	15.7435	0.618939	0.309470	−	0.950909i	$-0.399848\pi$
$647$	15.7435	0.618939	0.309470	+	0.950909i	$0.399848\pi$
$648$	−2.11956	−0.0832643
$649$	0	0
$650$	3.69301	0.144852
$651$	6.90554	0.270650
$652$	41.8772	1.64004
$653$	−24.8760	−0.973474	−0.486737	−	0.873549i	$-0.661813\pi$
$653$	−24.8760	−0.973474	−0.486737	+	0.873549i	$0.661813\pi$
$654$	−8.69371	−0.339951
$655$	−3.11860	−0.121854
$656$	8.48740	0.331377
$657$	6.38038	0.248922
$658$	2.97243	0.115878
$659$	36.3046	1.41423	0.707114	−	0.707100i	$-0.249997\pi$
$659$	36.3046	1.41423	0.707114	+	0.707100i	$0.249997\pi$
$660$	0	0
$661$	7.19715	0.279937	0.139968	−	0.990156i	$-0.455300\pi$
$661$	7.19715	0.279937	0.139968	+	0.990156i	$0.455300\pi$
$662$	8.18173	0.317992
$663$	−1.30008	−0.0504907
$664$	9.94180	0.385817
$665$	−0.755160	−0.0292839
$666$	0.382356	0.0148160
$667$	−14.1925	−0.549535
$668$	4.79744	0.185618
$669$	−19.1046	−0.738626
$670$	1.22725	0.0474127
$671$	0	0
$672$	−3.68719	−0.142236
$673$	−20.5216	−0.791050	−0.395525	−	0.918455i	$-0.629437\pi$
$673$	−20.5216	−0.791050	−0.395525	+	0.918455i	$0.629437\pi$
$674$	−5.17575	−0.199362
$675$	−4.91267	−0.189089
$676$	18.8383	0.724550
$677$	−11.8481	−0.455359	−0.227680	−	0.973736i	$-0.573114\pi$
$677$	−11.8481	−0.455359	−0.227680	+	0.973736i	$0.573114\pi$
$678$	−0.202171	−0.00776431
$679$	−0.524501	−0.0201285
$680$	−0.626361	−0.0240199
$681$	−5.63298	−0.215856
$682$	0	0
$683$	1.97507	0.0755741	0.0377871	−	0.999286i	$-0.487969\pi$
$683$	1.97507	0.0755741	0.0377871	+	0.999286i	$0.487969\pi$
$684$	−6.29916	−0.240854
$685$	−6.61501	−0.252746
$686$	5.29159	0.202034
$687$	3.47425	0.132551
$688$	−2.78157	−0.106046
$689$	−3.43506	−0.130865
$690$	0.666260	0.0253641
$691$	19.7687	0.752036	0.376018	−	0.926612i	$-0.377293\pi$
$691$	19.7687	0.752036	0.376018	+	0.926612i	$0.377293\pi$
$692$	29.8200	1.13359
$693$	0	0
$694$	10.3485	0.392825
$695$	4.82122	0.182879
$696$	7.71496	0.292435
$697$	4.03059	0.152670
$698$	18.3729	0.695425
$699$	23.8538	0.902235
$700$	−5.52928	−0.208987
$701$	−10.0101	−0.378078	−0.189039	−	0.981970i	$-0.560537\pi$
$701$	−10.0101	−0.378078	−0.189039	+	0.981970i	$0.560537\pi$
$702$	−0.751731	−0.0283723
$703$	2.50075	0.0943176
$704$	0	0
$705$	−2.24819	−0.0846717
$706$	−15.5296	−0.584465
$707$	−3.61613	−0.135998
$708$	−9.11054	−0.342395
$709$	9.60252	0.360630	0.180315	−	0.983609i	$-0.442288\pi$
$709$	9.60252	0.360630	0.180315	+	0.983609i	$0.442288\pi$
$710$	−1.82073	−0.0683307
$711$	−7.13648	−0.267639
$712$	3.42557	0.128379
$713$	−39.8478	−1.49231
$714$	−0.390713	−0.0146221
$715$	0	0
$716$	−13.4342	−0.502058
$717$	9.79938	0.365965
$718$	−1.01237	−0.0377815
$719$	−29.3200	−1.09345	−0.546725	−	0.837312i	$-0.684126\pi$
$719$	−29.3200	−1.09345	−0.546725	+	0.837312i	$0.684126\pi$
$720$	0.622278	0.0231909
$721$	0.401635	0.0149577
$722$	−2.71658	−0.101101
$723$	18.4928	0.687756
$724$	1.88985	0.0702357
$725$	17.8815	0.664104
$726$	0	0
$727$	41.9106	1.55438	0.777189	−	0.629268i	$-0.216645\pi$
$727$	41.9106	1.55438	0.777189	+	0.629268i	$0.216645\pi$
$728$	−1.86200	−0.0690102
$729$	1.00000	0.0370370
$730$	1.09023	0.0403513
$731$	−1.32094	−0.0488568
$732$	−2.44454	−0.0903529
$733$	22.6393	0.836203	0.418101	−	0.908400i	$-0.362696\pi$
$733$	22.6393	0.836203	0.418101	+	0.908400i	$0.362696\pi$
$734$	−19.0417	−0.702841
$735$	−1.93367	−0.0713246
$736$	21.2766	0.784265
$737$	0	0
$738$	2.33057	0.0857896
$739$	−51.9055	−1.90938	−0.954688	−	0.297608i	$-0.903811\pi$
$739$	−51.9055	−1.90938	−0.954688	+	0.297608i	$0.903811\pi$
$740$	−0.325492	−0.0119653
$741$	−4.91659	−0.180616
$742$	−1.03234	−0.0378985
$743$	28.3093	1.03857	0.519284	−	0.854601i	$-0.326199\pi$
$743$	28.3093	1.03857	0.519284	+	0.854601i	$0.326199\pi$
$744$	21.6611	0.794133
$745$	−4.09520	−0.150036
$746$	5.19679	0.190268
$747$	−4.69050	−0.171616
$748$	0	0
$749$	−10.1709	−0.371635
$750$	−1.69381	−0.0618490
$751$	−6.06832	−0.221436	−0.110718	−	0.993852i	$-0.535315\pi$
$751$	−6.06832	−0.221436	−0.110718	+	0.993852i	$0.535315\pi$
$752$	−16.0199	−0.584185
$753$	2.68951	0.0980113
$754$	2.73621	0.0996470
$755$	1.92992	0.0702371
$756$	1.12551	0.0409345
$757$	−13.2417	−0.481278	−0.240639	−	0.970615i	$-0.577357\pi$
$757$	−13.2417	−0.481278	−0.240639	+	0.970615i	$0.577357\pi$
$758$	16.3043	0.592197
$759$	0	0
$760$	−2.36876	−0.0859239
$761$	22.4131	0.812475	0.406238	−	0.913767i	$-0.366841\pi$
$761$	22.4131	0.812475	0.406238	+	0.913767i	$0.366841\pi$
$762$	−6.51572	−0.236040
$763$	10.1596	0.367801
$764$	30.6432	1.10863
$765$	0.295515	0.0106844
$766$	−17.7895	−0.642762
$767$	−7.11093	−0.256761
$768$	−4.55093	−0.164217
$769$	32.7094	1.17953	0.589766	−	0.807574i	$-0.299220\pi$
$769$	32.7094	1.17953	0.589766	+	0.807574i	$0.299220\pi$
$770$	0	0
$771$	0.898777	0.0323686
$772$	27.2980	0.982477
$773$	−26.7230	−0.961159	−0.480580	−	0.876951i	$-0.659574\pi$
$773$	−26.7230	−0.961159	−0.480580	+	0.876951i	$0.659574\pi$
$774$	−0.763797	−0.0274541
$775$	50.2055	1.80343
$776$	−1.64524	−0.0590606
$777$	−0.446826	−0.0160298
$778$	14.3773	0.515453
$779$	15.2428	0.546130
$780$	0.639932	0.0229132
$781$	0	0
$782$	2.25458	0.0806235
$783$	−3.63988	−0.130079
$784$	−13.7787	−0.492098
$785$	−4.24349	−0.151456
$786$	−6.10203	−0.217652
$787$	−54.7292	−1.95089	−0.975443	−	0.220251i	$-0.929312\pi$
$787$	−54.7292	−1.95089	−0.975443	+	0.220251i	$0.929312\pi$
$788$	4.54990	0.162083
$789$	−18.2970	−0.651389
$790$	−1.21943	−0.0433853
$791$	0.236259	0.00840040
$792$	0	0
$793$	−1.90801	−0.0677553
$794$	20.4678	0.726376
$795$	0.780809	0.0276924
$796$	9.30479	0.329800
$797$	22.6545	0.802462	0.401231	−	0.915977i	$-0.368582\pi$
$797$	22.6545	0.802462	0.401231	+	0.915977i	$0.368582\pi$
$798$	−1.47759	−0.0523061
$799$	−7.60771	−0.269141
$800$	−26.8070	−0.947771
$801$	−1.61617	−0.0571045
$802$	2.06415	0.0728877
$803$	0	0
$804$	−11.9632	−0.421908
$805$	−0.778600	−0.0274420
$806$	7.68238	0.270600
$807$	−17.3033	−0.609104
$808$	−11.3429	−0.399043
$809$	13.2059	0.464296	0.232148	−	0.972680i	$-0.425425\pi$
$809$	13.2059	0.464296	0.232148	+	0.972680i	$0.425425\pi$
$810$	0.170873	0.00600386
$811$	42.4601	1.49098	0.745488	−	0.666519i	$-0.232216\pi$
$811$	42.4601	1.49098	0.745488	+	0.666519i	$0.232216\pi$
$812$	−4.09674	−0.143767
$813$	−29.0442	−1.01862
$814$	0	0
$815$	−7.42969	−0.260251
$816$	2.10575	0.0737158
$817$	−4.99551	−0.174771
$818$	9.83558	0.343893
$819$	0.878482	0.0306966
$820$	−1.98397	−0.0692831
$821$	−15.8506	−0.553189	−0.276594	−	0.960987i	$-0.589206\pi$
$821$	−15.8506	−0.553189	−0.276594	+	0.960987i	$0.589206\pi$
$822$	−12.9433	−0.451450
$823$	13.1200	0.457334	0.228667	−	0.973505i	$-0.426563\pi$
$823$	13.1200	0.457334	0.228667	+	0.973505i	$0.426563\pi$
$824$	1.25983	0.0438884
$825$	0	0
$826$	−2.13706	−0.0743577
$827$	−42.9856	−1.49476	−0.747378	−	0.664399i	$-0.768688\pi$
$827$	−42.9856	−1.49476	−0.747378	+	0.664399i	$0.768688\pi$
$828$	−6.49468	−0.225706
$829$	−4.32908	−0.150355	−0.0751775	−	0.997170i	$-0.523952\pi$
$829$	−4.32908	−0.150355	−0.0751775	+	0.997170i	$0.523952\pi$
$830$	−0.801478	−0.0278197
$831$	8.68398	0.301244
$832$	1.37328	0.0476098
$833$	−6.54341	−0.226716
$834$	9.43348	0.326655
$835$	−0.851142	−0.0294550
$836$	0	0
$837$	−10.2196	−0.353241
$838$	−12.1961	−0.421308
$839$	−12.9261	−0.446259	−0.223130	−	0.974789i	$-0.571627\pi$
$839$	−12.9261	−0.446259	−0.223130	+	0.974789i	$0.571627\pi$
$840$	0.423242	0.0146033
$841$	−15.7513	−0.543147
$842$	10.5040	0.361992
$843$	25.7488	0.886834
$844$	22.0337	0.758430
$845$	−3.34221	−0.114976
$846$	−4.39894	−0.151239
$847$	0	0
$848$	5.56380	0.191062
$849$	−5.27292	−0.180966
$850$	−2.84061	−0.0974321
$851$	2.57837	0.0883855
$852$	17.7484	0.608049
$853$	−46.4066	−1.58893	−0.794466	−	0.607309i	$-0.792249\pi$
$853$	−46.4066	−1.58893	−0.794466	+	0.607309i	$0.792249\pi$
$854$	−0.573416	−0.0196219
$855$	1.11757	0.0382201
$856$	−31.9036	−1.09044
$857$	47.6426	1.62744	0.813720	−	0.581257i	$-0.197439\pi$
$857$	47.6426	1.62744	0.813720	+	0.581257i	$0.197439\pi$
$858$	0	0
$859$	24.7107	0.843119	0.421559	−	0.906801i	$-0.361483\pi$
$859$	24.7107	0.843119	0.421559	+	0.906801i	$0.361483\pi$
$860$	0.650203	0.0221717
$861$	−2.72354	−0.0928179
$862$	20.0046	0.681358
$863$	−24.3475	−0.828799	−0.414400	−	0.910095i	$-0.636008\pi$
$863$	−24.3475	−0.828799	−0.414400	+	0.910095i	$0.636008\pi$
$864$	5.45671	0.185641
$865$	−5.29054	−0.179884
$866$	−10.0921	−0.342943
$867$	1.00000	0.0339618
$868$	−11.5023	−0.390413
$869$	0	0
$870$	−0.621957	−0.0210863
$871$	−9.33743	−0.316387
$872$	31.8682	1.07919
$873$	0.776216	0.0262709
$874$	8.52631	0.288407
$875$	1.97940	0.0669160
$876$	−10.6275	−0.359071
$877$	49.6635	1.67702	0.838509	−	0.544887i	$-0.183428\pi$
$877$	49.6635	1.67702	0.838509	+	0.544887i	$0.183428\pi$
$878$	−8.99674	−0.303625
$879$	24.9621	0.841952
$880$	0	0
$881$	42.9423	1.44676	0.723381	−	0.690449i	$-0.242587\pi$
$881$	42.9423	1.44676	0.723381	+	0.690449i	$0.242587\pi$
$882$	−3.78354	−0.127398
$883$	20.9624	0.705440	0.352720	−	0.935729i	$-0.385257\pi$
$883$	20.9624	0.705440	0.352720	+	0.935729i	$0.385257\pi$
$884$	2.16548	0.0728331
$885$	1.61635	0.0543332
$886$	−2.82712	−0.0949788
$887$	−20.7678	−0.697315	−0.348658	−	0.937250i	$-0.613362\pi$
$887$	−20.7678	−0.697315	−0.348658	+	0.937250i	$0.613362\pi$
$888$	−1.40159	−0.0470342
$889$	7.61435	0.255377
$890$	−0.276159	−0.00925687
$891$	0	0
$892$	31.8218	1.06547
$893$	−28.7707	−0.962773
$894$	−8.01291	−0.267992
$895$	2.38343	0.0796694
$896$	7.78709	0.260148
$897$	−5.06920	−0.169256
$898$	−17.9482	−0.598938
$899$	37.1981	1.24063
$900$	8.18284	0.272761
$901$	2.64220	0.0880245
$902$	0	0
$903$	0.892582	0.0297033
$904$	0.741089	0.0246482
$905$	−0.335289	−0.0111454
$906$	3.77620	0.125456
$907$	−6.40291	−0.212605	−0.106302	−	0.994334i	$-0.533901\pi$
$907$	−6.40291	−0.212605	−0.106302	+	0.994334i	$0.533901\pi$
$908$	9.38263	0.311373
$909$	5.35155	0.177500
$910$	0.150109	0.00497605
$911$	39.1247	1.29626	0.648130	−	0.761529i	$-0.275551\pi$
$911$	39.1247	1.29626	0.648130	+	0.761529i	$0.275551\pi$
$912$	7.96346	0.263696
$913$	0	0
$914$	−16.1491	−0.534165
$915$	0.433701	0.0143377
$916$	−5.78692	−0.191205
$917$	7.13091	0.235483
$918$	0.578221	0.0190841
$919$	−36.4352	−1.20188	−0.600942	−	0.799292i	$-0.705208\pi$
$919$	−36.4352	−1.20188	−0.600942	+	0.799292i	$0.705208\pi$
$920$	−2.44228	−0.0805197
$921$	−4.55687	−0.150154
$922$	0.569734	0.0187632
$923$	13.8529	0.455974
$924$	0	0
$925$	−3.24857	−0.106812
$926$	6.05203	0.198882
$927$	−0.594384	−0.0195221
$928$	−19.8618	−0.651995
$929$	4.72020	0.154865	0.0774323	−	0.996998i	$-0.475328\pi$
$929$	4.72020	0.154865	0.0774323	+	0.996998i	$0.475328\pi$
$930$	−1.74625	−0.0572618
$931$	−24.7457	−0.811008
$932$	−39.7324	−1.30148
$933$	10.9031	0.356950
$934$	−7.38622	−0.241685
$935$	0	0
$936$	2.75559	0.0900693
$937$	7.05051	0.230330	0.115165	−	0.993346i	$-0.463260\pi$
$937$	7.05051	0.230330	0.115165	+	0.993346i	$0.463260\pi$
$938$	−2.80619	−0.0916254
$939$	−9.08484	−0.296473
$940$	3.74472	0.122139
$941$	−4.42677	−0.144309	−0.0721543	−	0.997393i	$-0.522987\pi$
$941$	−4.42677	−0.144309	−0.0721543	+	0.997393i	$0.522987\pi$
$942$	−8.30305	−0.270528
$943$	15.7159	0.511781
$944$	11.5176	0.374867
$945$	−0.199684	−0.00649572
$946$	0	0
$947$	−8.07211	−0.262308	−0.131154	−	0.991362i	$-0.541868\pi$
$947$	−8.07211	−0.262308	−0.131154	+	0.991362i	$0.541868\pi$
$948$	11.8869	0.386070
$949$	−8.29497	−0.269266
$950$	−10.7426	−0.348534
$951$	−7.53172	−0.244233
$952$	1.43222	0.0464186
$953$	−27.7725	−0.899640	−0.449820	−	0.893119i	$-0.648512\pi$
$953$	−27.7725	−0.899640	−0.449820	+	0.893119i	$0.648512\pi$
$954$	1.52778	0.0494636
$955$	−5.43660	−0.175924
$956$	−16.3224	−0.527905
$957$	0	0
$958$	−3.60142	−0.116357
$959$	15.1257	0.488434
$960$	−0.312153	−0.0100747
$961$	73.4401	2.36903
$962$	−0.497092	−0.0160269
$963$	15.0520	0.485043
$964$	−30.8028	−0.992091
$965$	−4.84310	−0.155905
$966$	−1.52345	−0.0490163
$967$	40.7837	1.31151	0.655757	−	0.754972i	$-0.272349\pi$
$967$	40.7837	1.31151	0.655757	+	0.754972i	$0.272349\pi$
$968$	0	0
$969$	3.78178	0.121488
$970$	0.132634	0.00425862
$971$	−33.0089	−1.05931	−0.529653	−	0.848214i	$-0.677678\pi$
$971$	−33.0089	−1.05931	−0.529653	+	0.848214i	$0.677678\pi$
$972$	−1.66566	−0.0534261
$973$	−11.0241	−0.353416
$974$	−1.44815	−0.0464016
$975$	6.38684	0.204543
$976$	3.09042	0.0989219
$977$	10.7592	0.344218	0.172109	−	0.985078i	$-0.444942\pi$
$977$	10.7592	0.344218	0.172109	+	0.985078i	$0.444942\pi$
$978$	−14.5374	−0.464853
$979$	0	0
$980$	3.22084	0.102886
$981$	−15.0353	−0.480039
$982$	0.673652	0.0214971
$983$	28.7702	0.917627	0.458814	−	0.888533i	$-0.348275\pi$
$983$	28.7702	0.917627	0.458814	+	0.888533i	$0.348275\pi$
$984$	−8.54309	−0.272344
$985$	−0.807224	−0.0257203
$986$	−2.10466	−0.0670259
$987$	5.14065	0.163629
$988$	8.18938	0.260539
$989$	−5.15057	−0.163778
$990$	0	0
$991$	19.6869	0.625373	0.312687	−	0.949856i	$-0.398771\pi$
$991$	19.6869	0.625373	0.312687	+	0.949856i	$0.398771\pi$
$992$	−55.7654	−1.77055
$993$	14.1498	0.449031
$994$	4.16323	0.132050
$995$	−1.65082	−0.0523344
$996$	7.81278	0.247557
$997$	29.5313	0.935265	0.467633	−	0.883923i	$-0.345107\pi$
$997$	29.5313	0.935265	0.467633	+	0.883923i	$0.345107\pi$
$998$	15.8086	0.500412
$999$	0.661263	0.0209214

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6171.2.a.bk.1.8		12
11.3	even	5		561.2.m.d.460.4	✓	24
11.4	even	5		561.2.m.d.511.4	yes	24
11.10	odd	2		6171.2.a.bl.1.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
561.2.m.d.460.4	✓	24	11.3	even	5
561.2.m.d.511.4	yes	24	11.4	even	5
6171.2.a.bk.1.8		12	1.1	even	1	trivial
6171.2.a.bl.1.5		12	11.10	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:	\( N \)	\(=\)	\( 6171 = 3 \cdot 11^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6171.a (trivial)

\(f(q)\)	\(=\)	\(q+0.578221 q^{2} +1.00000 q^{3} -1.66566 q^{4} +0.295515 q^{5} +0.578221 q^{6} -0.675716 q^{7} -2.11956 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.578221 q^{2} +1.00000 q^{3} -1.66566 q^{4} +0.295515 q^{5} +0.578221 q^{6} -0.675716 q^{7} -2.11956 q^{8} +1.00000 q^{9} +0.170873 q^{10} -1.66566 q^{12} -1.30008 q^{13} -0.390713 q^{14} +0.295515 q^{15} +2.10575 q^{16} +1.00000 q^{17} +0.578221 q^{18} +3.78178 q^{19} -0.492227 q^{20} -0.675716 q^{21} +3.89916 q^{23} -2.11956 q^{24} -4.91267 q^{25} -0.751731 q^{26} +1.00000 q^{27} +1.12551 q^{28} -3.63988 q^{29} +0.170873 q^{30} -10.2196 q^{31} +5.45671 q^{32} +0.578221 q^{34} -0.199684 q^{35} -1.66566 q^{36} +0.661263 q^{37} +2.18670 q^{38} -1.30008 q^{39} -0.626361 q^{40} +4.03059 q^{41} -0.390713 q^{42} -1.32094 q^{43} +0.295515 q^{45} +2.25458 q^{46} -7.60771 q^{47} +2.10575 q^{48} -6.54341 q^{49} -2.84061 q^{50} +1.00000 q^{51} +2.16548 q^{52} +2.64220 q^{53} +0.578221 q^{54} +1.43222 q^{56} +3.78178 q^{57} -2.10466 q^{58} +5.46963 q^{59} -0.492227 q^{60} +1.46761 q^{61} -5.90918 q^{62} -0.675716 q^{63} -1.05630 q^{64} -0.384191 q^{65} +7.18223 q^{67} -1.66566 q^{68} +3.89916 q^{69} -0.115461 q^{70} -10.6555 q^{71} -2.11956 q^{72} +6.38038 q^{73} +0.382356 q^{74} -4.91267 q^{75} -6.29916 q^{76} -0.751731 q^{78} -7.13648 q^{79} +0.622278 q^{80} +1.00000 q^{81} +2.33057 q^{82} -4.69050 q^{83} +1.12551 q^{84} +0.295515 q^{85} -0.763797 q^{86} -3.63988 q^{87} -1.61617 q^{89} +0.170873 q^{90} +0.878482 q^{91} -6.49468 q^{92} -10.2196 q^{93} -4.39894 q^{94} +1.11757 q^{95} +5.45671 q^{96} +0.776216 q^{97} -3.78354 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - q^{2} + 12 q^{3} + 9 q^{4} - 14 q^{5} - q^{6} - 5 q^{7} + 12 q^{9}+O(q^{10}) \) 12 * q - q^2 + 12 * q^3 + 9 * q^4 - 14 * q^5 - q^6 - 5 * q^7 + 12 * q^9	\( 12 q - q^{2} + 12 q^{3} + 9 q^{4} - 14 q^{5} - q^{6} - 5 q^{7} + 12 q^{9} + 8 q^{10} + 9 q^{12} - 7 q^{13} - 13 q^{14} - 14 q^{15} + 11 q^{16} + 12 q^{17} - q^{18} + 3 q^{19} - 36 q^{20} - 5 q^{21} - 39 q^{23} + 8 q^{25} - 25 q^{26} + 12 q^{27} - 9 q^{28} + 10 q^{29} + 8 q^{30} - 25 q^{31} - 11 q^{32} - q^{34} + 12 q^{35} + 9 q^{36} - 10 q^{37} - 34 q^{38} - 7 q^{39} - 13 q^{40} + 17 q^{41} - 13 q^{42} - 18 q^{43} - 14 q^{45} + 19 q^{46} - 38 q^{47} + 11 q^{48} - 15 q^{49} - 18 q^{50} + 12 q^{51} + 32 q^{52} - 40 q^{53} - q^{54} - 25 q^{56} + 3 q^{57} + 3 q^{58} - 18 q^{59} - 36 q^{60} - 22 q^{61} + 3 q^{62} - 5 q^{63} - 20 q^{64} + 3 q^{65} - 9 q^{67} + 9 q^{68} - 39 q^{69} + 13 q^{70} - 24 q^{71} + q^{73} - 4 q^{74} + 8 q^{75} + 29 q^{76} - 25 q^{78} - 3 q^{79} - 43 q^{80} + 12 q^{81} + q^{82} + 4 q^{83} - 9 q^{84} - 14 q^{85} - 4 q^{86} + 10 q^{87} - 62 q^{89} + 8 q^{90} - 5 q^{91} - 52 q^{92} - 25 q^{93} - 16 q^{94} - 4 q^{95} - 11 q^{96} - 5 q^{97} - 7 q^{98}+O(q^{100}) \) 12 * q - q^2 + 12 * q^3 + 9 * q^4 - 14 * q^5 - q^6 - 5 * q^7 + 12 * q^9 + 8 * q^10 + 9 * q^12 - 7 * q^13 - 13 * q^14 - 14 * q^15 + 11 * q^16 + 12 * q^17 - q^18 + 3 * q^19 - 36 * q^20 - 5 * q^21 - 39 * q^23 + 8 * q^25 - 25 * q^26 + 12 * q^27 - 9 * q^28 + 10 * q^29 + 8 * q^30 - 25 * q^31 - 11 * q^32 - q^34 + 12 * q^35 + 9 * q^36 - 10 * q^37 - 34 * q^38 - 7 * q^39 - 13 * q^40 + 17 * q^41 - 13 * q^42 - 18 * q^43 - 14 * q^45 + 19 * q^46 - 38 * q^47 + 11 * q^48 - 15 * q^49 - 18 * q^50 + 12 * q^51 + 32 * q^52 - 40 * q^53 - q^54 - 25 * q^56 + 3 * q^57 + 3 * q^58 - 18 * q^59 - 36 * q^60 - 22 * q^61 + 3 * q^62 - 5 * q^63 - 20 * q^64 + 3 * q^65 - 9 * q^67 + 9 * q^68 - 39 * q^69 + 13 * q^70 - 24 * q^71 + q^73 - 4 * q^74 + 8 * q^75 + 29 * q^76 - 25 * q^78 - 3 * q^79 - 43 * q^80 + 12 * q^81 + q^82 + 4 * q^83 - 9 * q^84 - 14 * q^85 - 4 * q^86 + 10 * q^87 - 62 * q^89 + 8 * q^90 - 5 * q^91 - 52 * q^92 - 25 * q^93 - 16 * q^94 - 4 * q^95 - 11 * q^96 - 5 * q^97 - 7 * q^98

Introduction

L-functions

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