LMFDB - Embedded newform 177.5.c.a.58.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [177,5,Mod(58,177)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(177, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("177.58");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	177.c (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$18.2964834658$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			58.3
Character	$\chi$	$=$	177.58
Dual form			177.5.c.a.58.38

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.65331i q^{2} -5.19615 q^{3} -42.5731 q^{4} -38.1687 q^{5} +39.7678i q^{6} -35.1454 q^{7} +203.372i q^{8} +27.0000 q^{9} +O(q^{10})$	\(q-7.65331i q^{2} -5.19615 q^{3} -42.5731 q^{4} -38.1687 q^{5} +39.7678i q^{6} -35.1454 q^{7} +203.372i q^{8} +27.0000 q^{9} +292.117i q^{10} -147.824i q^{11} +221.217 q^{12} -17.0918i q^{13} +268.979i q^{14} +198.331 q^{15} +875.302 q^{16} -354.287 q^{17} -206.639i q^{18} -647.468 q^{19} +1624.96 q^{20} +182.621 q^{21} -1131.34 q^{22} -862.278i q^{23} -1056.75i q^{24} +831.852 q^{25} -130.809 q^{26} -140.296 q^{27} +1496.25 q^{28} +440.875 q^{29} -1517.88i q^{30} +347.019i q^{31} -3445.00i q^{32} +768.116i q^{33} +2711.47i q^{34} +1341.45 q^{35} -1149.47 q^{36} +972.830i q^{37} +4955.27i q^{38} +88.8114i q^{39} -7762.47i q^{40} +2783.36 q^{41} -1397.65i q^{42} -2837.80i q^{43} +6293.33i q^{44} -1030.56 q^{45} -6599.28 q^{46} +1287.07i q^{47} -4548.20 q^{48} -1165.80 q^{49} -6366.42i q^{50} +1840.93 q^{51} +727.650i q^{52} -1103.72 q^{53} +1073.73i q^{54} +5642.25i q^{55} -7147.61i q^{56} +3364.34 q^{57} -3374.15i q^{58} +(1692.03 + 3042.10i) q^{59} -8443.55 q^{60} -5980.77i q^{61} +2655.84 q^{62} -948.926 q^{63} -12360.8 q^{64} +652.371i q^{65} +5878.63 q^{66} +3352.44i q^{67} +15083.1 q^{68} +4480.53i q^{69} -10266.6i q^{70} -1396.21 q^{71} +5491.06i q^{72} -1649.77i q^{73} +7445.37 q^{74} -4322.43 q^{75} +27564.7 q^{76} +5195.33i q^{77} +679.701 q^{78} -8357.83 q^{79} -33409.2 q^{80} +729.000 q^{81} -21301.9i q^{82} -5025.77i q^{83} -7774.74 q^{84} +13522.7 q^{85} -21718.6 q^{86} -2290.85 q^{87} +30063.3 q^{88} +5292.92i q^{89} +7887.16i q^{90} +600.697i q^{91} +36709.9i q^{92} -1803.16i q^{93} +9850.36 q^{94} +24713.0 q^{95} +17900.7i q^{96} +14051.7i q^{97} +8922.24i q^{98} -3991.25i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 320 q^{4} + 80 q^{7} + 1080 q^{9}+O(q^{10}) $ 40 * q - 320 * q^4 + 80 * q^7 + 1080 * q^9	$ 40 q - 320 q^{4} + 80 q^{7} + 1080 q^{9} + 360 q^{12} + 144 q^{15} + 3944 q^{16} - 528 q^{17} + 444 q^{19} + 444 q^{20} + 1304 q^{22} + 4880 q^{25} - 1452 q^{26} - 1160 q^{28} - 996 q^{29} + 10320 q^{35} - 8640 q^{36} - 5196 q^{41} - 10476 q^{46} + 576 q^{48} + 5104 q^{49} + 936 q^{51} - 2184 q^{53} - 2520 q^{57} - 11736 q^{59} - 11448 q^{60} + 15240 q^{62} + 2160 q^{63} - 81012 q^{64} + 17352 q^{66} + 29568 q^{68} - 5964 q^{71} + 14376 q^{74} - 2736 q^{75} + 3480 q^{76} + 37692 q^{78} + 19020 q^{79} + 33096 q^{80} + 29160 q^{81} + 25128 q^{84} + 20220 q^{85} - 65880 q^{86} + 1512 q^{87} - 14932 q^{88} - 17864 q^{94} + 11004 q^{95}+O(q^{100}) $ 40 * q - 320 * q^4 + 80 * q^7 + 1080 * q^9 + 360 * q^12 + 144 * q^15 + 3944 * q^16 - 528 * q^17 + 444 * q^19 + 444 * q^20 + 1304 * q^22 + 4880 * q^25 - 1452 * q^26 - 1160 * q^28 - 996 * q^29 + 10320 * q^35 - 8640 * q^36 - 5196 * q^41 - 10476 * q^46 + 576 * q^48 + 5104 * q^49 + 936 * q^51 - 2184 * q^53 - 2520 * q^57 - 11736 * q^59 - 11448 * q^60 + 15240 * q^62 + 2160 * q^63 - 81012 * q^64 + 17352 * q^66 + 29568 * q^68 - 5964 * q^71 + 14376 * q^74 - 2736 * q^75 + 3480 * q^76 + 37692 * q^78 + 19020 * q^79 + 33096 * q^80 + 29160 * q^81 + 25128 * q^84 + 20220 * q^85 - 65880 * q^86 + 1512 * q^87 - 14932 * q^88 - 17864 * q^94 + 11004 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/177\mathbb{Z}\right)^\times$.

$n$	$61$	$119$
$\chi(n)$	$-1$	$1$

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 $

$2$		−	7.65331i		−	1.91333i	−0.291195	−	0.956664i	$-0.594053\pi$
$2$		−	7.65331i		−	1.91333i	0.291195	−	0.956664i	$-0.405947\pi$
$3$	−5.19615			−0.577350
$3$	−5.19615			−0.577350
$4$	−42.5731			−2.66082
$5$	−38.1687			−1.52675			−0.763375	−	0.645956i	$-0.776459\pi$
$5$	−38.1687			−1.52675			−0.763375	+	0.645956i	$0.776459\pi$
$6$			39.7678i			1.10466i
$7$	−35.1454			−0.717253			−0.358626	−	0.933481i	$-0.616755\pi$
$7$	−35.1454			−0.717253			−0.358626	+	0.933481i	$0.616755\pi$
$8$			203.372i			3.17769i
$9$	27.0000			0.333333
$10$			292.117i			2.92117i
$11$		−	147.824i		−	1.22169i	−0.791752	−	0.610843i	$-0.790831\pi$
$11$		−	147.824i		−	1.22169i	0.791752	−	0.610843i	$-0.209169\pi$
$12$	221.217			1.53623
$13$		−	17.0918i		−	0.101135i	−0.998721	−	0.0505674i	$-0.983897\pi$
$13$		−	17.0918i		−	0.101135i	0.998721	−	0.0505674i	$-0.0161030\pi$
$14$			268.979i			1.37234i
$15$	198.331			0.881469
$16$	875.302			3.41915
$17$	−354.287			−1.22591			−0.612953	−	0.790119i	$-0.710019\pi$
$17$	−354.287			−1.22591			−0.612953	+	0.790119i	$0.710019\pi$
$18$		−	206.639i		−	0.637776i
$19$	−647.468			−1.79354			−0.896770	−	0.442498i	$-0.854092\pi$
$19$	−647.468			−1.79354			−0.896770	+	0.442498i	$0.854092\pi$
$20$	1624.96			4.06241
$21$	182.621			0.414106
$22$	−1131.34			−2.33748
$23$		−	862.278i		−	1.63001i	−0.579451	−	0.815007i	$-0.696733\pi$
$23$		−	862.278i		−	1.63001i	0.579451	−	0.815007i	$-0.303267\pi$
$24$		−	1056.75i		−	1.83464i
$25$	831.852			1.33096
$26$	−130.809			−0.193504
$27$	−140.296			−0.192450
$28$	1496.25			1.90848
$29$	440.875			0.524227			0.262113	−	0.965037i	$-0.415581\pi$
$29$	440.875			0.524227			0.262113	+	0.965037i	$0.415581\pi$
$30$		−	1517.88i		−	1.68654i
$31$			347.019i			0.361101i	0.983566	+	0.180551i	$0.0577880\pi$
$31$			347.019i			0.361101i	−0.983566	+	0.180551i	$0.942212\pi$
$32$		−	3445.00i		−	3.36426i
$33$			768.116i			0.705341i
$34$			2711.47i			2.34556i
$35$	1341.45			1.09507
$36$	−1149.47			−0.886940
$37$			972.830i			0.710613i	0.934750	+	0.355307i	$0.115624\pi$
$37$			972.830i			0.710613i	−0.934750	+	0.355307i	$0.884376\pi$
$38$			4955.27i			3.43163i
$39$			88.8114i			0.0583902i
$40$		−	7762.47i		−	4.85154i
$41$	2783.36			1.65578			0.827889	−	0.560892i	$-0.189542\pi$
$41$	2783.36			1.65578			0.827889	+	0.560892i	$0.189542\pi$
$42$		−	1397.65i		−	0.792321i
$43$		−	2837.80i		−	1.53478i	−0.641183	−	0.767388i	$-0.721556\pi$
$43$		−	2837.80i		−	1.53478i	0.641183	−	0.767388i	$-0.278444\pi$
$44$			6293.33i			3.25069i
$45$	−1030.56			−0.508916
$46$	−6599.28			−3.11875
$47$			1287.07i			0.582649i	0.956624	+	0.291324i	$0.0940959\pi$
$47$			1287.07i			0.582649i	−0.956624	+	0.291324i	$0.905904\pi$
$48$	−4548.20			−1.97405
$49$	−1165.80			−0.485548
$50$		−	6366.42i		−	2.54657i
$51$	1840.93			0.707777
$52$			727.650i			0.269101i
$53$	−1103.72			−0.392923			−0.196462	−	0.980511i	$-0.562945\pi$
$53$	−1103.72			−0.392923			−0.196462	+	0.980511i	$0.562945\pi$
$54$			1073.73i			0.368220i
$55$			5642.25i			1.86521i
$56$		−	7147.61i		−	2.27921i
$57$	3364.34			1.03550
$58$		−	3374.15i		−	1.00302i
$59$	1692.03	+	3042.10i	0.486076	+	0.873917i
$59$	1692.03	+	3042.10i	0.486076	+	0.873917i
$60$	−8443.55			−2.34543
$61$		−	5980.77i		−	1.60730i	−0.595100	−	0.803651i	$-0.702888\pi$
$61$		−	5980.77i		−	1.60730i	0.595100	−	0.803651i	$-0.297112\pi$
$62$	2655.84			0.690905
$63$	−948.926			−0.239084
$64$	−12360.8			−3.01777
$65$			652.371i			0.154407i
$66$	5878.63			1.34955
$67$			3352.44i			0.746812i	0.927668	+	0.373406i	$0.121810\pi$
$67$			3352.44i			0.746812i	−0.927668	+	0.373406i	$0.878190\pi$
$68$	15083.1			3.26192
$69$			4480.53i			0.941089i
$70$		−	10266.6i		−	2.09522i
$71$	−1396.21			−0.276971			−0.138486	−	0.990364i	$-0.544223\pi$
$71$	−1396.21			−0.276971			−0.138486	+	0.990364i	$0.544223\pi$
$72$			5491.06i			1.05923i
$73$		−	1649.77i		−	0.309584i	−0.987947	−	0.154792i	$-0.950529\pi$
$73$		−	1649.77i		−	0.309584i	0.987947	−	0.154792i	$-0.0494707\pi$
$74$	7445.37			1.35964
$75$	−4322.43			−0.768432
$76$	27564.7			4.77229
$77$			5195.33i			0.876258i
$78$	679.701			0.111719
$79$	−8357.83			−1.33918			−0.669591	−	0.742730i	$-0.733530\pi$
$79$	−8357.83			−1.33918			−0.669591	+	0.742730i	$0.733530\pi$
$80$	−33409.2			−5.22018
$81$	729.000			0.111111
$82$		−	21301.9i		−	3.16805i
$83$		−	5025.77i		−	0.729536i	−0.931098	−	0.364768i	$-0.881148\pi$
$83$		−	5025.77i		−	0.729536i	0.931098	−	0.364768i	$-0.118852\pi$
$84$	−7774.74			−1.10186
$85$	13522.7			1.87165
$86$	−21718.6			−2.93653
$87$	−2290.85			−0.302662
$88$	30063.3			3.88214
$89$			5292.92i			0.668213i	0.942535	+	0.334107i	$0.108435\pi$
$89$			5292.92i			0.668213i	−0.942535	+	0.334107i	$0.891565\pi$
$90$			7887.16i			0.973724i
$91$			600.697i			0.0725392i
$92$			36709.9i			4.33718i
$93$		−	1803.16i		−	0.208482i
$94$	9850.36			1.11480
$95$	24713.0			2.73828
$96$			17900.7i			1.94235i
$97$			14051.7i			1.49343i	0.665142	+	0.746716i	$0.268371\pi$
$97$			14051.7i			1.49343i	−0.665142	+	0.746716i	$0.731629\pi$
$98$			8922.24i			0.929013i
$99$		−	3991.25i		−	0.407229i
$100$	−35414.5			−3.54145
$101$			623.626i			0.0611338i	0.999533	+	0.0305669i	$0.00973126\pi$
$101$			623.626i			0.0611338i	−0.999533	+	0.0305669i	$0.990269\pi$
$102$		−	14089.2i		−	1.35421i
$103$			8032.20i			0.757112i	0.925578	+	0.378556i	$0.123579\pi$
$103$			8032.20i			0.757112i	−0.925578	+	0.378556i	$0.876421\pi$
$104$	3475.99			0.321375
$105$	−6970.40			−0.632236
$106$			8447.13i			0.751791i
$107$	11098.7			0.969402			0.484701	−	0.874680i	$-0.338928\pi$
$107$	11098.7			0.969402			0.484701	+	0.874680i	$0.338928\pi$
$108$	5972.85			0.512075
$109$		−	7809.17i		−	0.657283i	−0.944455	−	0.328641i	$-0.893409\pi$
$109$		−	7809.17i		−	0.657283i	0.944455	−	0.328641i	$-0.106591\pi$
$110$	43181.9			3.56875
$111$		−	5054.97i		−	0.410273i
$112$	−30762.8			−2.45239
$113$			3130.68i			0.245178i	0.992457	+	0.122589i	$0.0391198\pi$
$113$			3130.68i			0.245178i	−0.992457	+	0.122589i	$0.960880\pi$
$114$		−	25748.3i		−	1.98125i
$115$			32912.0i			2.48862i
$116$	−18769.4			−1.39487
$117$		−	461.478i		−	0.0337116i
$118$	23282.2	−	12949.6i	1.67209	−	0.930022i
$119$	12451.6			0.879285
$120$			40335.0i			2.80104i
$121$	−7210.93			−0.492516
$122$	−45772.7			−3.07530
$123$	−14462.8			−0.955964
$124$		−	14773.7i		−	0.960827i
$125$	−7895.26			−0.505297
$126$			7262.42i			0.457447i
$127$	−18711.5			−1.16011			−0.580056	−	0.814576i	$-0.696969\pi$
$127$	−18711.5			−1.16011			−0.580056	+	0.814576i	$0.696969\pi$
$128$			39481.1i			2.40973i
$129$			14745.6i			0.886103i
$130$	4992.80			0.295432
$131$		−	12408.9i		−	0.723089i	−0.932355	−	0.361545i	$-0.882249\pi$
$131$		−	12408.9i		−	0.723089i	0.932355	−	0.361545i	$-0.117751\pi$
$132$		−	32701.1i		−	1.87679i
$133$	22755.5			1.28642
$134$	25657.2			1.42889
$135$	5354.92			0.293823
$136$		−	72052.2i		−	3.89556i
$137$	−24025.7			−1.28007			−0.640037	−	0.768344i	$-0.721081\pi$
$137$	−24025.7			−1.28007			−0.640037	+	0.768344i	$0.721081\pi$
$138$	34290.9			1.80061
$139$	−3014.36			−0.156014			−0.0780072	−	0.996953i	$-0.524856\pi$
$139$	−3014.36			−0.156014			−0.0780072	+	0.996953i	$0.524856\pi$
$140$	−57109.9			−2.91377
$141$		−	6687.82i		−	0.336393i
$142$			10685.6i			0.529937i
$143$	−2526.57			−0.123555
$144$	23633.2			1.13972
$145$	−16827.6			−0.800362
$146$	−12626.2			−0.592335
$147$	6057.68			0.280331
$148$		−	41416.4i		−	1.89082i
$149$		−	10398.6i		−	0.468383i	−0.972190	−	0.234192i	$-0.924756\pi$
$149$		−	10398.6i		−	0.468383i	0.972190	−	0.234192i	$-0.0752443\pi$
$150$			33080.9i			1.47026i
$151$		−	35440.0i		−	1.55432i	−0.629305	−	0.777159i	$-0.716660\pi$
$151$		−	35440.0i		−	1.55432i	0.629305	−	0.777159i	$-0.283340\pi$
$152$		−	131677.i		−	5.69932i
$153$	−9565.75			−0.408635
$154$	39761.5			1.67657
$155$		−	13245.3i		−	0.551311i
$156$		−	3780.98i		−	0.155366i
$157$			23302.5i			0.945373i	0.881231	+	0.472686i	$0.156716\pi$
$157$			23302.5i			0.945373i	−0.881231	+	0.472686i	$0.843284\pi$
$158$			63965.0i			2.56229i
$159$	5735.11			0.226854
$160$			131491.i			5.13637i
$161$			30305.1i			1.16913i
$162$		−	5579.26i		−	0.212592i
$163$	−8996.62			−0.338613			−0.169307	−	0.985563i	$-0.554153\pi$
$163$	−8996.62			−0.338613			−0.169307	+	0.985563i	$0.554153\pi$
$164$	−118497.			−4.40573
$165$		−	29318.0i		−	1.07688i
$166$	−38463.8			−1.39584
$167$	−441.902			−0.0158450			−0.00792252	−	0.999969i	$-0.502522\pi$
$167$	−441.902			−0.0158450			−0.00792252	+	0.999969i	$0.502522\pi$
$168$			37140.0i			1.31590i
$169$	28268.9			0.989772
$170$		−	103493.i		−	3.58108i
$171$	−17481.6			−0.597846
$172$			120814.i			4.08376i
$173$		−	2947.42i		−	0.0984803i	−0.998787	−	0.0492402i	$-0.984320\pi$
$173$		−	2947.42i		−	0.0984803i	0.998787	−	0.0492402i	$-0.0156800\pi$
$174$			17532.6i			0.579092i
$175$	−29235.8			−0.954637
$176$		−	129391.i		−	4.17713i
$177$	−8792.04	−	15807.2i	−0.280636	−	0.504556i
$178$	40508.3			1.27851
$179$			17326.2i			0.540750i	0.962755	+	0.270375i	$0.0871477\pi$
$179$			17326.2i			0.540750i	−0.962755	+	0.270375i	$0.912852\pi$
$180$	43874.0			1.35414
$181$	−9276.44			−0.283155			−0.141578	−	0.989927i	$-0.545217\pi$
$181$	−9276.44			−0.283155			−0.141578	+	0.989927i	$0.545217\pi$
$182$	4597.32			0.138791
$183$			31077.0i			0.927977i
$184$	175364.			5.17969
$185$		−	37131.7i		−	1.08493i
$186$	−13800.2			−0.398894
$187$			52372.1i			1.49767i
$188$		−	54794.7i		−	1.55032i
$189$	4930.76			0.138035
$190$		−	189136.i		−	5.23923i
$191$			14709.0i			0.403195i	0.979468	+	0.201598i	$0.0646134\pi$
$191$			14709.0i			0.403195i	−0.979468	+	0.201598i	$0.935387\pi$
$192$	64228.6			1.74231
$193$	−34249.4			−0.919472			−0.459736	−	0.888056i	$-0.652056\pi$
$193$	−34249.4			−0.919472			−0.459736	+	0.888056i	$0.652056\pi$
$194$	107542.			2.85743
$195$		−	3389.82i		−	0.0891471i
$196$	49631.8			1.29196
$197$	7911.51			0.203858			0.101929	−	0.994792i	$-0.467499\pi$
$197$	7911.51			0.203858			0.101929	+	0.994792i	$0.467499\pi$
$198$	−30546.3			−0.779162
$199$	−57454.5			−1.45084			−0.725418	−	0.688309i	$-0.758353\pi$
$199$	−57454.5			−1.45084			−0.725418	+	0.688309i	$0.758353\pi$
$200$			169176.i			4.22939i
$201$		−	17419.8i		−	0.431172i
$202$	4772.80			0.116969
$203$	−15494.7			−0.376003
$204$	−78374.1			−1.88327
$205$	−106237.			−2.52796
$206$	61472.9			1.44860
$207$		−	23281.5i		−	0.543338i
$208$		−	14960.5i		−	0.345795i
$209$			95711.3i			2.19114i
$210$			53346.7i			1.20967i
$211$			49004.2i			1.10070i	0.834935	+	0.550349i	$0.185505\pi$
$211$			49004.2i			1.10070i	−0.834935	+	0.550349i	$0.814495\pi$
$212$	46988.9			1.04550
$213$	7254.93			0.159909
$214$		−	84941.6i		−	1.85478i
$215$			108315.i			2.34322i
$216$		−	28532.4i		−	0.611548i
$217$		−	12196.1i		−	0.259001i
$218$	−59766.0			−1.25760
$219$			8572.47i			0.178738i
$220$		−	240208.i		−	4.96298i
$221$			6055.39i			0.123982i
$222$	−38687.3			−0.784986
$223$	94322.8			1.89674			0.948368	−	0.317171i	$-0.102733\pi$
$223$	94322.8			1.89674			0.948368	+	0.317171i	$0.102733\pi$
$224$			121076.i			2.41302i
$225$	22460.0			0.443654
$226$	23960.1			0.469107
$227$		−	73496.8i		−	1.42632i	−0.701002	−	0.713159i	$-0.747264\pi$
$227$		−	73496.8i		−	1.42632i	0.701002	−	0.713159i	$-0.252736\pi$
$228$	−143231.			−2.75528
$229$		−	71268.4i		−	1.35902i	−0.733666	−	0.679510i	$-0.762192\pi$
$229$		−	71268.4i		−	1.35902i	0.733666	−	0.679510i	$-0.237808\pi$
$230$	251886.			4.76155
$231$		−	26995.7i		−	0.505908i
$232$			89661.7i			1.66583i
$233$		−	21675.4i		−	0.399259i	−0.979871	−	0.199630i	$-0.936026\pi$
$233$		−	21675.4i		−	0.399259i	0.979871	−	0.199630i	$-0.0639739\pi$
$234$	−3531.83			−0.0645013
$235$		−	49125.9i		−	0.889559i
$236$	−72035.0	−	129512.i	−1.29336	−	2.32534i
$237$	43428.6			0.773177
$238$		−	95295.6i		−	1.68236i
$239$	−556.432			−0.00974129			−0.00487065	−	0.999988i	$-0.501550\pi$
$239$	−556.432			−0.00974129			−0.00487065	+	0.999988i	$0.501550\pi$
$240$	173599.			3.01387
$241$	34167.1			0.588266			0.294133	−	0.955764i	$-0.404969\pi$
$241$	34167.1			0.588266			0.294133	+	0.955764i	$0.404969\pi$
$242$			55187.5i			0.942345i
$243$	−3788.00			−0.0641500
$244$			254620.i			4.27675i
$245$	44497.2			0.741310
$246$			110688.i			1.82907i
$247$			11066.4i			0.181389i
$248$	−70574.0			−1.14747
$249$			26114.7i			0.421198i
$250$			60424.9i			0.966798i
$251$	57018.6			0.905043			0.452522	−	0.891753i	$-0.350525\pi$
$251$	57018.6			0.905043			0.452522	+	0.891753i	$0.350525\pi$
$252$	40398.7			0.636161
$253$	−127465.			−1.99137
$254$			143205.i			2.21967i
$255$	−70265.9			−1.08060
$256$	104388.			1.59283
$257$	−3529.50			−0.0534375			−0.0267188	−	0.999643i	$-0.508506\pi$
$257$	−3529.50			−0.0534375			−0.0267188	+	0.999643i	$0.508506\pi$
$258$	112853.			1.69540
$259$		−	34190.5i		−	0.509689i
$260$		−	27773.5i		−	0.410850i
$261$	11903.6			0.174742
$262$	−94969.4			−1.38351
$263$	39797.7			0.575369			0.287684	−	0.957725i	$-0.407115\pi$
$263$	39797.7			0.575369			0.287684	+	0.957725i	$0.407115\pi$
$264$	−156214.			−2.24136
$265$	42127.7			0.599896
$266$		−	174155.i		−	2.46134i
$267$		−	27502.8i		−	0.385793i
$268$		−	142724.i		−	1.98713i
$269$			99994.5i			1.38188i	0.722910	+	0.690942i	$0.242804\pi$
$269$			99994.5i			1.38188i	−0.722910	+	0.690942i	$0.757196\pi$
$270$		−	40982.9i		−	0.562180i
$271$	−70921.8			−0.965698			−0.482849	−	0.875704i	$-0.660398\pi$
$271$	−70921.8			−0.965698			−0.482849	+	0.875704i	$0.660398\pi$
$272$	−310108.			−4.19156
$273$		−	3121.31i		−	0.0418805i
$274$			183876.i			2.44920i
$275$		−	122968.i		−	1.62602i
$276$		−	190750.i		−	2.50407i
$277$	−27566.1			−0.359265			−0.179633	−	0.983734i	$-0.557491\pi$
$277$	−27566.1			−0.359265			−0.179633	+	0.983734i	$0.557491\pi$
$278$			23069.8i			0.298507i
$279$			9369.50i			0.120367i
$280$			272815.i			3.47978i
$281$	116989.			1.48160			0.740802	−	0.671723i	$-0.234446\pi$
$281$	116989.			1.48160			0.740802	+	0.671723i	$0.234446\pi$
$282$	−51183.9			−0.643629
$283$		−	124680.i		−	1.55676i	−0.627791	−	0.778382i	$-0.716041\pi$
$283$		−	124680.i		−	1.55676i	0.627791	−	0.778382i	$-0.283959\pi$
$284$	59441.1			0.736971
$285$	−128413.			−1.58095
$286$			19336.6i			0.236401i
$287$	−97822.4			−1.18761
$288$		−	93014.9i		−	1.12142i
$289$	41998.2			0.502846
$290$			128787.i			1.53136i
$291$		−	73014.8i		−	0.862234i
$292$			70236.0i			0.823747i
$293$	−117066.			−1.36362			−0.681812	−	0.731528i	$-0.738808\pi$
$293$	−117066.			−1.36362			−0.681812	+	0.731528i	$0.738808\pi$
$294$		−	46361.3i		−	0.536366i
$295$	−64582.6	−	116113.i	−0.742116	−	1.33425i
$296$	−197847.			−2.25811
$297$			20739.1i			0.235114i
$298$	−79583.5			−0.896170
$299$	−14737.8			−0.164851
$300$	184019.			2.04466
$301$			99735.6i			1.10082i
$302$	−271233.			−2.97392
$303$		−	3240.45i		−	0.0352956i
$304$	−566730.			−6.13238
$305$			228278.i			2.45395i
$306$			73209.6i			0.781853i
$307$	−74586.4			−0.791376			−0.395688	−	0.918385i	$-0.629494\pi$
$307$	−74586.4			−0.791376			−0.395688	+	0.918385i	$0.629494\pi$
$308$		−	221182.i		−	2.33157i
$309$		−	41736.5i		−	0.437119i
$310$	−101370.			−1.05484
$311$	164539.			1.70117			0.850587	−	0.525834i	$-0.176247\pi$
$311$	164539.			1.70117			0.850587	+	0.525834i	$0.176247\pi$
$312$	−18061.8			−0.185546
$313$			135789.i			1.38604i	0.720919	+	0.693019i	$0.243720\pi$
$313$			135789.i			1.38604i	−0.720919	+	0.693019i	$0.756280\pi$
$314$	178341.			1.80881
$315$	36219.3			0.365022
$316$	355819.			3.56332
$317$	−92280.2			−0.918311			−0.459156	−	0.888356i	$-0.651848\pi$
$317$	−92280.2			−0.918311			−0.459156	+	0.888356i	$0.651848\pi$
$318$		−	43892.6i		−	0.434047i
$319$		−	65171.8i		−	0.640440i
$320$	471796.			4.60738
$321$	−57670.4			−0.559684
$322$	231934.			2.23693
$323$	229389.			2.19871
$324$	−31035.8			−0.295647
$325$		−	14217.8i		−	0.134607i
$326$			68853.9i			0.647878i
$327$			40577.7i			0.379482i
$328$			566059.i			5.26156i
$329$		−	45234.6i		−	0.417907i
$330$	−224380.			−2.06042
$331$	10830.5			0.0988536			0.0494268	−	0.998778i	$-0.484261\pi$
$331$	10830.5			0.0988536			0.0494268	+	0.998778i	$0.484261\pi$
$332$			213963.i			1.94116i
$333$			26266.4i			0.236871i
$334$			3382.02i			0.0303167i
$335$		−	127958.i		−	1.14019i
$336$	159848.			1.41589
$337$		−	124414.i		−	1.09549i	−0.836645	−	0.547746i	$-0.815486\pi$
$337$		−	124414.i		−	1.09549i	0.836645	−	0.547746i	$-0.184514\pi$
$338$		−	216350.i		−	1.89376i
$339$		−	16267.5i		−	0.141554i
$340$	−575703.			−4.98013
$341$	51297.7			0.441153
$342$			133792.i			1.14388i
$343$	125357.			1.06551
$344$	577130.			4.87705
$345$		−	171016.i		−	1.43681i
$346$	−22557.5			−0.188425
$347$		−	226048.i		−	1.87733i	−0.344829	−	0.938665i	$-0.612063\pi$
$347$		−	226048.i		−	1.87733i	0.344829	−	0.938665i	$-0.387937\pi$
$348$	97528.7			0.805330
$349$			223773.i			1.83720i	0.395185	+	0.918602i	$0.370681\pi$
$349$			223773.i			1.83720i	−0.395185	+	0.918602i	$0.629319\pi$
$350$			223750.i			1.82653i
$351$			2397.91i			0.0194634i
$352$	−509253.			−4.11006
$353$			132923.i			1.06672i	0.845889	+	0.533358i	$0.179070\pi$
$353$			132923.i			1.06672i	−0.845889	+	0.533358i	$0.820930\pi$
$354$	−120978.	+	67288.2i	−0.965381	+	0.536948i
$355$	53291.6			0.422866
$356$		−	225336.i		−	1.77800i
$357$	−64700.2			−0.507655
$358$	132603.			1.03463
$359$	52427.0			0.406786			0.203393	−	0.979097i	$-0.434803\pi$
$359$	52427.0			0.406786			0.203393	+	0.979097i	$0.434803\pi$
$360$		−	209587.i		−	1.61718i
$361$	288893.			2.21678
$362$			70995.5i			0.541768i
$363$	37469.1			0.284354
$364$		−	25573.6i		−	0.193014i
$365$			62969.7i			0.472657i
$366$	237842.			1.77552
$367$			30282.9i			0.224835i	0.993661	+	0.112418i	$0.0358595\pi$
$367$			30282.9i			0.224835i	−0.993661	+	0.112418i	$0.964141\pi$
$368$		−	754753.i		−	5.57326i
$369$	75150.8			0.551926
$370$	−284180.			−2.07582
$371$	38790.7			0.281826
$372$			76766.2i			0.554733i
$373$	47523.7			0.341580			0.170790	−	0.985307i	$-0.445368\pi$
$373$	47523.7			0.341580			0.170790	+	0.985307i	$0.445368\pi$
$374$	400820.			2.86554
$375$	41025.0			0.291733
$376$	−261755.			−1.85148
$377$		−	7535.33i		−	0.0530175i
$378$		−	37736.6i		−	0.264107i
$379$	185993.			1.29485			0.647425	−	0.762130i	$-0.275846\pi$
$379$	185993.			1.29485			0.647425	+	0.762130i	$0.275846\pi$
$380$	−1.05211e6			−7.28609
$381$	97227.6			0.669791
$382$	112572.			0.771445
$383$	−141142.			−0.962188			−0.481094	−	0.876669i	$-0.659760\pi$
$383$	−141142.			−0.962188			−0.481094	+	0.876669i	$0.659760\pi$
$384$		−	205150.i		−	1.39126i
$385$		−	198299.i		−	1.33783i
$386$			262121.i			1.75925i
$387$		−	76620.6i		−	0.511592i
$388$		−	598225.i		−	3.97376i
$389$	58679.8			0.387784			0.193892	−	0.981023i	$-0.437889\pi$
$389$	58679.8			0.387784			0.193892	+	0.981023i	$0.437889\pi$
$390$	−25943.3			−0.170568
$391$			305494.i			1.99825i
$392$		−	237092.i		−	1.54292i
$393$			64478.7i			0.417476i
$394$		−	60549.3i		−	0.390046i
$395$	319008.			2.04459
$396$			169920.i			1.08356i
$397$			277665.i			1.76173i	0.473365	+	0.880866i	$0.343039\pi$
$397$			277665.i			1.76173i	−0.473365	+	0.880866i	$0.656961\pi$
$398$			439717.i			2.77592i
$399$	−118241.			−0.742716
$400$	728121.			4.55076
$401$		−	59794.3i		−	0.371853i	−0.982564	−	0.185927i	$-0.940471\pi$
$401$		−	59794.3i		−	0.371853i	0.982564	−	0.185927i	$-0.0595286\pi$
$402$	−133319.			−0.824973
$403$	5931.16			0.0365199
$404$		−	26549.7i		−	0.162666i
$405$	−27825.0			−0.169639
$406$			118586.i			0.719417i
$407$	143808.			0.868146
$408$			374394.i			2.24910i
$409$		−	92288.3i		−	0.551696i	−0.961201	−	0.275848i	$-0.911041\pi$
$409$		−	92288.3i		−	0.551696i	0.961201	−	0.275848i	$-0.0889587\pi$
$410$			813068.i			4.83681i
$411$	124841.			0.739051
$412$		−	341956.i		−	2.01454i
$413$	−59467.0	−	106916.i	−0.348639	−	0.626819i
$414$	−178180.			−1.03958
$415$			191827.i			1.11382i
$416$	−58881.1			−0.340243
$417$	15663.1			0.0900750
$418$	732508.			4.19237
$419$		−	100145.i		−	0.570429i	−0.958464	−	0.285214i	$-0.907935\pi$
$419$		−	100145.i		−	0.570429i	0.958464	−	0.285214i	$-0.0920648\pi$
$420$	296752.			1.68227
$421$			102049.i			0.575762i	0.957666	+	0.287881i	$0.0929507\pi$
$421$			102049.i			0.575762i	−0.957666	+	0.287881i	$0.907049\pi$
$422$	375044.			2.10600
$423$			34750.9i			0.194216i
$424$		−	224467.i		−	1.24859i
$425$	−294714.			−1.63164
$426$		−	55524.2i		−	0.305959i
$427$			210197.i			1.15284i
$428$	−472506.			−2.57941
$429$	13128.5			0.0713344
$430$	828970.			4.48334
$431$		−	259464.i		−	1.39676i	−0.715725	−	0.698382i	$-0.753904\pi$
$431$		−	259464.i		−	1.39676i	0.715725	−	0.698382i	$-0.246096\pi$
$432$	−122801.			−0.658015
$433$	−332067.			−1.77113			−0.885565	−	0.464516i	$-0.846228\pi$
$433$	−332067.			−1.77113			−0.885565	+	0.464516i	$0.846228\pi$
$434$	−93340.5			−0.495554
$435$	87438.9			0.462089
$436$			332461.i			1.74891i
$437$			558297.i			2.92350i
$438$	65607.8			0.341985
$439$	70862.7			0.367696			0.183848	−	0.982955i	$-0.441145\pi$
$439$	70862.7			0.367696			0.183848	+	0.982955i	$0.441145\pi$
$440$	−1.14748e6			−5.92706
$441$	−31476.6			−0.161849
$442$	46343.8			0.237218
$443$		−	85593.5i		−	0.436147i	−0.975932	−	0.218074i	$-0.930023\pi$
$443$		−	85593.5i		−	0.436147i	0.975932	−	0.218074i	$-0.0699773\pi$
$444$			215206.i			1.09166i
$445$		−	202024.i		−	1.02019i
$446$		−	721882.i		−	3.62908i
$447$			54032.6i			0.270421i
$448$	434425.			2.16451
$449$	70020.7			0.347323			0.173662	−	0.984805i	$-0.444440\pi$
$449$	70020.7			0.347323			0.173662	+	0.984805i	$0.444440\pi$
$450$		−	171893.i		−	0.848856i
$451$		−	411448.i		−	2.02284i
$452$		−	133283.i		−	0.652376i
$453$			184152.i			0.897386i
$454$	−562494.			−2.72901
$455$		−	22927.8i		−	0.110749i
$456$			684214.i			3.29050i
$457$		−	97704.0i		−	0.467821i	−0.972258	−	0.233911i	$-0.924848\pi$
$457$		−	97704.0i		−	0.467821i	0.972258	−	0.233911i	$-0.0751523\pi$
$458$	−545439.			−2.60025
$459$	49705.1			0.235926
$460$		−	1.40117e6i		−	6.62178i
$461$	149231.			0.702194			0.351097	−	0.936339i	$-0.385809\pi$
$461$	149231.			0.702194			0.351097	+	0.936339i	$0.385809\pi$
$462$	−206607.			−0.967967
$463$			350096.i			1.63315i	0.577242	+	0.816573i	$0.304129\pi$
$463$			350096.i			1.63315i	−0.577242	+	0.816573i	$0.695871\pi$
$464$	385898.			1.79241
$465$			68824.4i			0.318300i
$466$	−165888.			−0.763914
$467$		−	370985.i		−	1.70107i	−0.525917	−	0.850536i	$-0.676278\pi$
$467$		−	370985.i		−	1.70107i	0.525917	−	0.850536i	$-0.323722\pi$
$468$			19646.6i			0.0897005i
$469$		−	117823.i		−	0.535653i
$470$	−375976.			−1.70202
$471$		−	121083.i		−	0.545811i
$472$	−618680.	+	344112.i	−2.77704	+	1.54460i
$473$	−419495.			−1.87501
$474$		−	332372.i		−	1.47934i
$475$	−538597.			−2.38713
$476$	−530102.			−2.33962
$477$	−29800.5			−0.130974
$478$			4258.55i			0.0186383i
$479$	−64617.2			−0.281629			−0.140814	−	0.990036i	$-0.544972\pi$
$479$	−64617.2			−0.281629			−0.140814	+	0.990036i	$0.544972\pi$
$480$		−	683248.i		−	2.96549i
$481$	16627.4			0.0718677
$482$		−	261491.i		−	1.12555i
$483$		−	157470.i		−	0.674999i
$484$	306992.			1.31050
$485$		−	536336.i		−	2.28010i
$486$			28990.7i			0.122740i
$487$	177831.			0.749808			0.374904	−	0.927064i	$-0.377676\pi$
$487$	177831.			0.749808			0.374904	+	0.927064i	$0.377676\pi$
$488$	1.21632e6			5.10752
$489$	46747.8			0.195499
$490$		−	340550.i		−	1.41837i
$491$	−146359.			−0.607094			−0.303547	−	0.952817i	$-0.598171\pi$
$491$	−146359.			−0.607094			−0.303547	+	0.952817i	$0.598171\pi$
$492$	615726.			2.54365
$493$	−156196.			−0.642653
$494$	84694.3			0.347057
$495$			152341.i			0.621736i
$496$			303746.i			1.23466i
$497$	49070.4			0.198658
$498$	199864.			0.805889
$499$	297990.			1.19674			0.598372	−	0.801218i	$-0.295815\pi$
$499$	297990.			1.19674			0.598372	+	0.801218i	$0.295815\pi$
$500$	336126.			1.34450
$501$	2296.19			0.00914814
$502$		−	436381.i		−	1.73164i
$503$			379938.i			1.50168i	0.660486	+	0.750839i	$0.270350\pi$
$503$			379938.i			1.50168i	−0.660486	+	0.750839i	$0.729650\pi$
$504$		−	192985.i		−	0.759737i
$505$		−	23803.0i		−	0.0933359i
$506$			975532.i			3.81013i
$507$	−146889.			−0.571445
$508$	796605.			3.08685
$509$		−	68834.8i		−	0.265688i	−0.991137	−	0.132844i	$-0.957589\pi$
$509$		−	68834.8i		−	0.265688i	0.991137	−	0.132844i	$-0.0424110\pi$
$510$			537767.i			2.06754i
$511$			57981.9i			0.222050i
$512$		−	167216.i		−	0.637880i
$513$	90837.2			0.345167
$514$			27012.3i			0.102243i
$515$		−	306579.i		−	1.15592i
$516$		−	627768.i		−	2.35776i
$517$	190260.			0.711814
$518$	−261670.			−0.975203
$519$			15315.2i			0.0568576i
$520$	−132674.			−0.490659
$521$	450862.			1.66099			0.830497	−	0.557023i	$-0.188057\pi$
$521$	450862.			1.66099			0.830497	+	0.557023i	$0.188057\pi$
$522$		−	91102.0i		−	0.334339i
$523$	−157180.			−0.574639			−0.287319	−	0.957835i	$-0.592764\pi$
$523$	−157180.			−0.574639			−0.287319	+	0.957835i	$0.592764\pi$
$524$			528287.i			1.92401i
$525$	151913.			0.551160
$526$		−	304584.i		−	1.10087i
$527$		−	122944.i		−	0.442677i
$528$			672334.i			2.41166i
$529$	−463682.			−1.65695
$530$		−	322416.i		−	1.14780i
$531$	45684.8	+	82136.8i	0.162025	+	0.291306i
$532$	−968773.			−3.42294
$533$		−	47572.6i		−	0.167457i
$534$	−210487.			−0.738149
$535$	−423623.			−1.48003
$536$	−681793.			−2.37314
$537$		−	90029.5i		−	0.312202i
$538$	765289.			2.64400
$539$			172333.i			0.593187i
$540$	−227976.			−0.781810
$541$			272034.i			0.929455i	0.885454	+	0.464727i	$0.153848\pi$
$541$			272034.i			0.929455i	−0.885454	+	0.464727i	$0.846152\pi$
$542$			542787.i			1.84770i
$543$	48201.8			0.163480
$544$			1.22052e6i			4.12426i
$545$			298066.i			1.00351i
$546$	−23888.4			−0.0801311
$547$	351369.			1.17433			0.587163	−	0.809469i	$-0.300245\pi$
$547$	351369.			1.17433			0.587163	+	0.809469i	$0.300245\pi$
$548$	1.02285e6			3.40605
$549$		−	161481.i		−	0.535768i
$550$	−941109.			−3.11110
$551$	−285452.			−0.940221
$552$	−911216.			−2.99049
$553$	293739.			0.960532
$554$			210972.i			0.687392i
$555$			192942.i			0.626384i
$556$	128331.			0.415127
$557$	88233.8			0.284397			0.142198	−	0.989838i	$-0.454583\pi$
$557$	88233.8			0.284397			0.142198	+	0.989838i	$0.454583\pi$
$558$	71707.7			0.230302
$559$	−48503.0			−0.155219
$560$	1.17418e6			3.74419
$561$		−	272133.i		−	0.864682i
$562$		−	895352.i		−	2.83479i
$563$		−	203437.i		−	0.641819i	−0.947110	−	0.320910i	$-0.896011\pi$
$563$		−	203437.i		−	0.641819i	0.947110	−	0.320910i	$-0.103989\pi$
$564$			284721.i			0.895080i
$565$		−	119494.i		−	0.374326i
$566$	−954212.			−2.97860
$567$	−25621.0			−0.0796948
$568$		−	283951.i		−	0.880130i
$569$		−	20875.8i		−	0.0644791i	−0.999480	−	0.0322395i	$-0.989736\pi$
$569$		−	20875.8i		−	0.0644791i	0.999480	−	0.0322395i	$-0.0102639\pi$
$570$			982781.i			3.02487i
$571$			308480.i			0.946139i	0.881025	+	0.473069i	$0.156854\pi$
$571$			308480.i			0.946139i	−0.881025	+	0.473069i	$0.843146\pi$
$572$	107564.			0.328757
$573$		−	76430.1i		−	0.232785i
$574$			748665.i			2.27229i
$575$		−	717287.i		−	2.16949i
$576$	−333742.			−1.00592
$577$	−332414.			−0.998454			−0.499227	−	0.866471i	$-0.666383\pi$
$577$	−332414.			−0.998454			−0.499227	+	0.866471i	$0.666383\pi$
$578$		−	321425.i		−	0.962110i
$579$	177965.			0.530857
$580$	716405.			2.12962
$581$			176633.i			0.523262i
$582$	−558805.			−1.64974
$583$			163157.i			0.480029i
$584$	335518.			0.983763
$585$			17614.0i			0.0514691i
$586$			895940.i			2.60906i
$587$		−	398675.i		−	1.15703i	−0.815673	−	0.578513i	$-0.803633\pi$
$587$		−	398675.i		−	1.15703i	0.815673	−	0.578513i	$-0.196367\pi$
$588$	−257895.			−0.745912
$589$		−	224683.i		−	0.647650i
$590$	−888650.	+	494271.i	−2.55286	+	1.41991i
$591$	−41109.4			−0.117697
$592$			851520.i			2.42969i
$593$	150216.			0.427177			0.213589	−	0.976924i	$-0.431485\pi$
$593$	150216.			0.427177			0.213589	+	0.976924i	$0.431485\pi$
$594$	158723.			0.449849
$595$	−475260.			−1.34245
$596$			442700.i			1.24628i
$597$	298543.			0.837640
$598$			112793.i			0.315414i
$599$	−79272.8			−0.220938			−0.110469	−	0.993880i	$-0.535235\pi$
$599$	−79272.8			−0.220938			−0.110469	+	0.993880i	$0.535235\pi$
$600$		−	879063.i		−	2.44184i
$601$			510334.i			1.41288i	0.707772	+	0.706441i	$0.249700\pi$
$601$			510334.i			1.41288i	−0.707772	+	0.706441i	$0.750300\pi$
$602$	763307.			2.10623
$603$			90515.8i			0.248937i
$604$			1.50879e6i			4.13576i
$605$	275232.			0.751949
$606$	−24800.2			−0.0675320
$607$	−108872.			−0.295487			−0.147743	−	0.989026i	$-0.547201\pi$
$607$	−108872.			−0.295487			−0.147743	+	0.989026i	$0.547201\pi$
$608$			2.23052e6i			6.03392i
$609$	80512.9			0.217085
$610$	1.74709e6			4.69521
$611$	21998.3			0.0589260
$612$	407244.			1.08731
$613$			71190.4i			0.189453i	0.995503	+	0.0947263i	$0.0301976\pi$
$613$			71190.4i			0.189453i	−0.995503	+	0.0947263i	$0.969802\pi$
$614$			570833.i			1.51416i
$615$	552026.			1.45952
$616$	−1.05659e6			−2.78448
$617$	−463811.			−1.21835			−0.609173	−	0.793037i	$-0.708498\pi$
$617$	−463811.			−1.21835			−0.609173	+	0.793037i	$0.708498\pi$
$618$	−319423.			−0.836351
$619$	192587.			0.502627			0.251313	−	0.967906i	$-0.419138\pi$
$619$	192587.			0.502627			0.251313	+	0.967906i	$0.419138\pi$
$620$			563892.i			1.46694i
$621$			120974.i			0.313696i
$622$		−	1.25927e6i		−	3.25490i
$623$		−	186022.i		−	0.479278i
$624$			77736.8i			0.199645i
$625$	−218555.			−0.559501
$626$	1.03923e6			2.65194
$627$		−	497330.i		−	1.26506i
$628$		−	992060.i		−	2.51547i
$629$		−	344661.i		−	0.871145i
$630$		−	277197.i		−	0.698406i
$631$	−13953.1			−0.0350438			−0.0175219	−	0.999846i	$-0.505578\pi$
$631$	−13953.1			−0.0350438			−0.0175219	+	0.999846i	$0.505578\pi$
$632$		−	1.69975e6i		−	4.25551i
$633$		−	254633.i		−	0.635489i
$634$			706249.i			1.75703i
$635$	714192.			1.77120
$636$	−244162.			−0.603619
$637$			19925.6i			0.0491058i
$638$	−498780.			−1.22537
$639$	−37697.7			−0.0923238
$640$		−	1.50694e6i		−	3.67906i
$641$	−416645.			−1.01403			−0.507014	−	0.861938i	$-0.669251\pi$
$641$	−416645.			−1.01403			−0.507014	+	0.861938i	$0.669251\pi$
$642$			441370.i			1.07086i
$643$	79601.2			0.192530			0.0962648	−	0.995356i	$-0.469310\pi$
$643$	79601.2			0.192530			0.0962648	+	0.995356i	$0.469310\pi$
$644$		−	1.29018e6i		−	3.11085i
$645$		−	562822.i		−	1.35286i
$646$		−	1.75559e6i		−	4.20685i
$647$	−681530.			−1.62808			−0.814041	−	0.580807i	$-0.802737\pi$
$647$	−681530.			−1.62808			−0.814041	+	0.580807i	$0.802737\pi$
$648$			148259.i			0.353077i
$649$	449696.	−	250123.i	1.06765	−	0.593832i
$650$	−108813.			−0.257546
$651$			63372.8i			0.149534i
$652$	383014.			0.900990
$653$	−383016.			−0.898237			−0.449118	−	0.893472i	$-0.648262\pi$
$653$	−383016.			−0.898237			−0.449118	+	0.893472i	$0.648262\pi$
$654$	310553.			0.726074
$655$			473633.i			1.10398i
$656$	2.43628e6			5.66135
$657$		−	44543.9i		−	0.103195i
$658$	−346195.			−0.799592
$659$		−	527534.i		−	1.21473i	−0.794423	−	0.607365i	$-0.792227\pi$
$659$		−	527534.i		−	1.21473i	0.794423	−	0.607365i	$-0.207773\pi$
$660$			1.24816e6i			2.86538i
$661$	282774.			0.647197			0.323599	−	0.946194i	$-0.395107\pi$
$661$	282774.			0.647197			0.323599	+	0.946194i	$0.395107\pi$
$662$		−	82889.2i		−	0.189139i
$663$		−	31464.7i		−	0.0715809i
$664$	1.02210e6			2.31824
$665$	−868549.			−1.96404
$666$	201025.			0.453212
$667$		−	380156.i		−	0.854497i
$668$	18813.2			0.0421608
$669$	−490116.			−1.09508
$670$	−979304.			−2.18156
$671$	−884102.			−1.96362
$672$		−	629128.i		−	1.39316i
$673$			709243.i			1.56590i	0.622083	+	0.782952i	$0.286287\pi$
$673$			709243.i			1.56590i	−0.622083	+	0.782952i	$0.713713\pi$
$674$	−952178.			−2.09604
$675$	−116706.			−0.256144
$676$	−1.20349e6			−2.63361
$677$	−419269.			−0.914777			−0.457389	−	0.889267i	$-0.651215\pi$
$677$	−419269.			−0.914777			−0.457389	+	0.889267i	$0.651215\pi$
$678$	−124500.			−0.270839
$679$		−	493853.i		−	1.07117i
$680$			2.75014e6i			5.94754i
$681$			381900.i			0.823486i
$682$		−	392597.i		−	0.844069i
$683$			341293.i			0.731620i	0.930690	+	0.365810i	$0.119208\pi$
$683$			341293.i			0.731620i	−0.930690	+	0.365810i	$0.880792\pi$
$684$	744248.			1.59076
$685$	917030.			1.95435
$686$		−	959393.i		−	2.03868i
$687$			370322.i			0.784631i
$688$		−	2.48393e6i		−	5.24762i
$689$			18864.6i			0.0397382i
$690$	−1.30884e6			−2.74908
$691$		−	57041.7i		−	0.119464i	−0.998214	−	0.0597319i	$-0.980975\pi$
$691$		−	57041.7i		−	0.119464i	0.998214	−	0.0597319i	$-0.0190246\pi$
$692$			125481.i			0.262039i
$693$			140274.i			0.292086i
$694$	−1.73001e6			−3.59195
$695$	115054.			0.238195
$696$		−	465896.i		−	0.961769i
$697$	−986109.			−2.02983
$698$	1.71261e6			3.51517
$699$			112629.i			0.230512i
$700$	1.24466e6			2.54012
$701$			292521.i			0.595279i	0.954678	+	0.297640i	$0.0961994\pi$
$701$			292521.i			0.595279i	−0.954678	+	0.297640i	$0.903801\pi$
$702$	18351.9			0.0372398
$703$		−	629876.i		−	1.27451i
$704$			1.82722e6i			3.68677i
$705$			255266.i			0.513587i
$706$	1.01730e6			2.04098
$707$		−	21917.6i		−	0.0438484i
$708$	374305.	+	672964.i	0.746722	+	1.34253i
$709$	265410.			0.527988			0.263994	−	0.964524i	$-0.414960\pi$
$709$	265410.			0.527988			0.263994	+	0.964524i	$0.414960\pi$
$710$		−	407857.i		−	0.809080i
$711$	−225661.			−0.446394
$712$	−1.07643e6			−2.12338
$713$	299226.			0.588601
$714$			495170.i			0.971311i
$715$	96436.1			0.188637
$716$		−	737630.i		−	1.43884i
$717$	2891.31			0.00562414
$718$		−	401240.i		−	0.778315i
$719$			843328.i			1.63132i	0.578532	+	0.815659i	$0.303626\pi$
$719$			843328.i			1.63132i	−0.578532	+	0.815659i	$0.696374\pi$
$720$	−902047.			−1.74006
$721$		−	282295.i		−	0.543041i
$722$		−	2.21099e6i		−	4.24143i
$723$	−177537.			−0.339636
$724$	394927.			0.753425
$725$	366742.			0.697726
$726$		−	286763.i		−	0.544063i
$727$	361987.			0.684895			0.342448	−	0.939537i	$-0.388744\pi$
$727$	361987.			0.684895			0.342448	+	0.939537i	$0.388744\pi$
$728$	−122165.			−0.230507
$729$	19683.0			0.0370370
$730$	481927.			0.904347
$731$			1.00540e6i			1.88149i
$732$		−	1.32305e6i		−	2.46918i
$733$	364992.			0.679322			0.339661	−	0.940548i	$-0.389688\pi$
$733$	364992.			0.679322			0.339661	+	0.940548i	$0.389688\pi$
$734$	231764.			0.430184
$735$	−231214.			−0.427996
$736$	−2.97054e6			−5.48379
$737$	495571.			0.912369
$738$		−	575152.i		−	1.05602i
$739$		−	397416.i		−	0.727707i	−0.931456	−	0.363854i	$-0.881461\pi$
$739$		−	397416.i		−	0.727707i	0.931456	−	0.363854i	$-0.118539\pi$
$740$			1.58081e6i			2.88680i
$741$		−	57502.5i		−	0.104725i
$742$		−	296878.i		−	0.539224i
$743$	208889.			0.378389			0.189195	−	0.981940i	$-0.439412\pi$
$743$	208889.			0.378389			0.189195	+	0.981940i	$0.439412\pi$
$744$	366713.			0.662492
$745$			396900.i			0.715104i
$746$		−	363714.i		−	0.653555i
$747$		−	135696.i		−	0.243179i
$748$		−	2.22965e6i		−	3.98504i
$749$	−390068.			−0.695306
$750$		−	313977.i		−	0.558181i
$751$			1.12660e6i			1.99751i	0.0498694	+	0.998756i	$0.484120\pi$
$751$			1.12660e6i			1.99751i	−0.0498694	+	0.998756i	$0.515880\pi$
$752$			1.12658e6i			1.99216i
$753$	−296277.			−0.522527
$754$	−57670.2			−0.101440
$755$			1.35270e6i			2.37305i
$756$	−209918.			−0.367288
$757$	−287250.			−0.501266			−0.250633	−	0.968082i	$-0.580639\pi$
$757$	−287250.			−0.501266			−0.250633	+	0.968082i	$0.580639\pi$
$758$		−	1.42347e6i		−	2.47747i
$759$	662329.			1.14972
$760$			5.02595e6i			8.70143i
$761$	−39472.2			−0.0681587			−0.0340794	−	0.999419i	$-0.510850\pi$
$761$	−39472.2			−0.0681587			−0.0340794	+	0.999419i	$0.510850\pi$
$762$		−	744113.i		−	1.28153i
$763$			274456.i			0.471438i
$764$		−	626207.i		−	1.07283i
$765$	365112.			0.623884
$766$			1.08021e6i			1.84098i
$767$	51994.9	−	28919.8i	0.0883833	−	0.0491591i
$768$	−542416.			−0.919623
$769$			696042.i			1.17702i	0.808491	+	0.588509i	$0.200285\pi$
$769$			696042.i			1.17702i	−0.808491	+	0.588509i	$0.799715\pi$
$770$	−1.51765e6			−2.55970
$771$	18339.8			0.0308522
$772$	1.45810e6			2.44655
$773$			245846.i			0.411438i	0.978611	+	0.205719i	$0.0659533\pi$
$773$			245846.i			0.411438i	−0.978611	+	0.205719i	$0.934047\pi$
$774$	−586401.			−0.978843
$775$			288668.i			0.480613i
$776$	−2.85773e6			−4.74567
$777$			177659.i			0.294269i
$778$		−	449095.i		−	0.741957i
$779$	−1.80214e6			−2.96970
$780$			144315.i			0.237205i
$781$			206394.i			0.338372i
$782$	2.33804e6			3.82330
$783$	−61853.0			−0.100887
$784$	−1.02043e6			−1.66016
$785$		−	889426.i		−	1.44335i
$786$	493476.			0.798768
$787$	437284.			0.706016			0.353008	−	0.935620i	$-0.385159\pi$
$787$	437284.			0.706016			0.353008	+	0.935620i	$0.385159\pi$
$788$	−336818.			−0.542429
$789$	−206795.			−0.332189
$790$		−	2.44146e6i		−	3.91198i
$791$		−	110029.i		−	0.175855i
$792$	811710.			1.29405
$793$	−102222.			−0.162554
$794$	2.12506e6			3.37077
$795$	−218902.			−0.346350
$796$	2.44602e6			3.86041
$797$			377368.i			0.594085i	0.954864	+	0.297042i	$0.0960003\pi$
$797$			377368.i			0.594085i	−0.954864	+	0.297042i	$0.904000\pi$
$798$			904936.i			1.42106i
$799$		−	455993.i		−	0.714273i
$800$		−	2.86573e6i		−	4.47770i
$801$			142909.i			0.222738i
$802$	−457625.			−0.711477
$803$	−243876.			−0.378214
$804$			741614.i			1.14727i
$805$		−	1.15671e6i		−	1.78497i
$806$		−	45393.0i		−	0.0698745i
$807$		−	519587.i		−	0.797831i
$808$	−126828.			−0.194264
$809$		−	84575.4i		−	0.129225i	−0.997910	−	0.0646126i	$-0.979419\pi$
$809$		−	84575.4i		−	0.129225i	0.997910	−	0.0646126i	$-0.0205812\pi$
$810$			212953.i			0.324575i
$811$		−	280073.i		−	0.425823i	−0.977071	−	0.212911i	$-0.931705\pi$
$811$		−	280073.i		−	0.425823i	0.977071	−	0.212911i	$-0.0682946\pi$
$812$	659658.			1.00048
$813$	368521.			0.557546
$814$		−	1.10060e6i		−	1.66105i
$815$	343390.			0.516978
$816$	1.61137e6			2.42000
$817$			1.83738e6i			2.75268i
$818$	−706311.			−1.05558
$819$			16218.8i			0.0241797i
$820$	4.52286e6			6.72644
$821$		−	475894.i		−	0.706031i	−0.935617	−	0.353016i	$-0.885156\pi$
$821$		−	475894.i		−	0.706031i	0.935617	−	0.353016i	$-0.114844\pi$
$822$		−	955448.i		−	1.41405i
$823$			448500.i			0.662160i	0.943603	+	0.331080i	$0.107413\pi$
$823$			448500.i			0.662160i	−0.943603	+	0.331080i	$0.892587\pi$
$824$	−1.63353e6			−2.40587
$825$			638959.i			0.938782i
$826$	−818261.	+	455120.i	−1.19931	+	0.667061i
$827$	794086.			1.16107			0.580533	−	0.814237i	$-0.302844\pi$
$827$	794086.			1.16107			0.580533	+	0.814237i	$0.302844\pi$
$828$			991166.i			1.44573i
$829$	351649.			0.511682			0.255841	−	0.966719i	$-0.417648\pi$
$829$	351649.			0.511682			0.255841	+	0.966719i	$0.417648\pi$
$830$	1.46811e6			2.13110
$831$	143238.			0.207422
$832$			211268.i			0.305202i
$833$	413028.			0.595237
$834$		−	119874.i		−	0.172343i
$835$	16866.8			0.0241914
$836$		−	4.07473e6i		−	5.83024i
$837$		−	48685.4i		−	0.0694940i
$838$	−766441.			−1.09142
$839$			249257.i			0.354098i	0.984202	+	0.177049i	$0.0566551\pi$
$839$			249257.i			0.354098i	−0.984202	+	0.177049i	$0.943345\pi$
$840$		−	1.41759e6i		−	2.00905i
$841$	−512911.			−0.725186
$842$	781009.			1.10162
$843$	−607892.			−0.855404
$844$		−	2.08626e6i		−	2.92876i
$845$	−1.07899e6			−1.51113
$846$	265960.			0.371599
$847$	253431.			0.353259
$848$	−966090.			−1.34346
$849$			647854.i			0.898798i
$850$			2.25554e6i			3.12185i
$851$	838849.			1.15831
$852$	−308865.			−0.425490
$853$	−526565.			−0.723692			−0.361846	−	0.932238i	$-0.617853\pi$
$853$	−526565.			−0.723692			−0.361846	+	0.932238i	$0.617853\pi$
$854$	1.60870e6			2.20577
$855$	667251.			0.912761
$856$			2.25717e6i			3.08046i
$857$			972800.i			1.32453i	0.749269	+	0.662265i	$0.230405\pi$
$857$			972800.i			1.32453i	−0.749269	+	0.662265i	$0.769595\pi$
$858$		−	100476.i		−	0.136486i
$859$		−	1.07599e6i		−	1.45822i	−0.684396	−	0.729110i	$-0.739934\pi$
$859$		−	1.07599e6i		−	1.45822i	0.684396	−	0.729110i	$-0.260066\pi$
$860$		−	4.61132e6i		−	6.23488i
$861$	508300.			0.685668
$862$	−1.98576e6			−2.67247
$863$		−	119816.i		−	0.160876i	−0.996760	−	0.0804381i	$-0.974368\pi$
$863$		−	119816.i		−	0.160876i	0.996760	−	0.0804381i	$-0.0256320\pi$
$864$			483320.i			0.647451i
$865$			112499.i			0.150355i
$866$			2.54141e6i			3.38875i
$867$	−218229.			−0.290319
$868$			519226.i			0.689156i
$869$			1.23549e6i			1.63606i
$870$		−	669197.i		−	0.884128i
$871$	57299.1			0.0755286
$872$	1.58817e6			2.08864
$873$			379396.i			0.497811i
$874$	4.27282e6			5.59360
$875$	277482.			0.362426
$876$		−	364957.i		−	0.475591i
$877$	−1.46716e6			−1.90756			−0.953778	−	0.300512i	$-0.902843\pi$
$877$	−1.46716e6			−1.90756			−0.953778	+	0.300512i	$0.902843\pi$
$878$		−	542334.i		−	0.703522i
$879$	608291.			0.787289
$880$			4.93868e6i			6.37742i
$881$		−	220168.i		−	0.283663i	−0.989891	−	0.141831i	$-0.954701\pi$
$881$		−	220168.i		−	0.283663i	0.989891	−	0.141831i	$-0.0452991\pi$
$882$			240900.i			0.309671i
$883$	1.11202e6			1.42624			0.713119	−	0.701044i	$-0.247282\pi$
$883$	1.11202e6			1.42624			0.713119	+	0.701044i	$0.247282\pi$
$884$		−	257797.i		−	0.329893i
$885$	335581.	+	603342.i	0.428461	+	0.770330i
$886$	−655073.			−0.834493
$887$		−	23264.8i		−	0.0295701i	−0.999891	−	0.0147851i	$-0.995294\pi$
$887$		−	23264.8i		−	0.0295701i	0.999891	−	0.0147851i	$-0.00470640\pi$
$888$	1.02804e6			1.30372
$889$	657621.			0.832094
$890$	−1.54615e6			−1.95197
$891$		−	107764.i		−	0.135743i
$892$	−4.01562e6			−5.04688
$893$		−	833337.i		−	1.04500i
$894$	413528.			0.517404
$895$		−	661318.i		−	0.825590i
$896$		−	1.38758e6i		−	1.72839i
$897$	76580.1			0.0951768
$898$		−	535890.i		−	0.664543i
$899$			152992.i			0.189299i
$900$	−956192.			−1.18048
$901$	391034.			0.481687
$902$	−3.14894e6			−3.87036
$903$		−	518241.i		−	0.635560i
$904$	−636695.			−0.779102
$905$	354070.			0.432307
$906$	1.40937e6			1.71699
$907$	−1.05698e6			−1.28485			−0.642425	−	0.766348i	$-0.722072\pi$
$907$	−1.05698e6			−1.28485			−0.642425	+	0.766348i	$0.722072\pi$
$908$			3.12899e6i			3.79518i
$909$			16837.9i			0.0203779i
$910$	−175474.			−0.211899
$911$	673892.			0.811995			0.405997	−	0.913874i	$-0.366924\pi$
$911$	673892.			0.811995			0.405997	+	0.913874i	$0.366924\pi$
$912$	2.94481e6			3.54053
$913$	−742930.			−0.891264
$914$	−747759.			−0.895095
$915$		−	1.18617e6i		−	1.41679i
$916$			3.03412e6i			3.61611i
$917$			436117.i			0.518638i
$918$		−	380408.i		−	0.451403i
$919$			1.40839e6i			1.66760i	0.552070	+	0.833798i	$0.313838\pi$
$919$			1.40839e6i			1.66760i	−0.552070	+	0.833798i	$0.686162\pi$
$920$	−6.69340e6			−7.90808
$921$	387562.			0.456901
$922$		−	1.14211e6i		−	1.34353i
$923$			23863.7i			0.0280114i
$924$			1.14929e6i			1.34613i
$925$			809250.i			0.945800i
$926$	2.67939e6			3.12474
$927$			216869.i			0.252371i
$928$		−	1.51881e6i		−	1.76363i
$929$		−	1.48700e6i		−	1.72298i	−0.507773	−	0.861491i	$-0.669531\pi$
$929$		−	1.48700e6i		−	1.72298i	0.507773	−	0.861491i	$-0.330469\pi$
$930$	526734.			0.609012
$931$	754819.			0.870850
$932$			922789.i			1.06236i
$933$	−854972.			−0.982174
$934$	−2.83926e6			−3.25471
$935$		−	1.99898e6i		−	2.28657i
$936$	93851.9			0.107125
$937$			715876.i			0.815378i	0.913121	+	0.407689i	$0.133665\pi$
$937$			715876.i			0.815378i	−0.913121	+	0.407689i	$0.866335\pi$
$938$	−901734.			−1.02488
$939$		−	705579.i		−	0.800230i
$940$			2.09144e6i			2.36696i
$941$		−	351558.i		−	0.397025i	−0.980098	−	0.198512i	$-0.936389\pi$
$941$		−	351558.i		−	0.397025i	0.980098	−	0.198512i	$-0.0636110\pi$
$942$	−926688.			−1.04432
$943$		−	2.40003e6i		−	2.69894i
$944$	1.48104e6	+	2.66276e6i	1.66197	+	2.98805i
$945$	−188201.			−0.210745
$946$			3.21052e6i			3.58751i
$947$	894274.			0.997174			0.498587	−	0.866840i	$-0.333852\pi$
$947$	894274.			0.997174			0.498587	+	0.866840i	$0.333852\pi$
$948$	−1.84889e6			−2.05728
$949$	−28197.5			−0.0313097
$950$			4.12205e6i			4.56737i
$951$	479502.			0.530187
$952$			2.53230e6i			2.79410i
$953$	−868548.			−0.956330			−0.478165	−	0.878270i	$-0.658698\pi$
$953$	−868548.			−0.956330			−0.478165	+	0.878270i	$0.658698\pi$
$954$			228072.i			0.250597i
$955$		−	561423.i		−	0.615578i
$956$	23689.1			0.0259198
$957$			338643.i			0.369758i
$958$			494535.i			0.538848i
$959$	844393.			0.918136
$960$	−2.45152e6			−2.66007
$961$	803099.			0.869606
$962$		−	127254.i		−	0.137506i
$963$	299664.			0.323134
$964$	−1.45460e6			−1.56527
$965$	1.30726e6			1.40380
$966$	−1.20517e6			−1.29149
$967$		−	27414.2i		−	0.0293172i	−0.999893	−	0.0146586i	$-0.995334\pi$
$967$		−	27414.2i		−	0.0293172i	0.999893	−	0.0146586i	$-0.00466615\pi$
$968$		−	1.46651e6i		−	1.56507i
$969$	−1.19194e6			−1.26943
$970$	−4.10474e6			−4.36257
$971$	−202062.			−0.214312			−0.107156	−	0.994242i	$-0.534174\pi$
$971$	−202062.			−0.214312			−0.107156	+	0.994242i	$0.534174\pi$
$972$	161267.			0.170692
$973$	105941.			0.111902
$974$		−	1.36100e6i		−	1.43463i
$975$			73877.9i			0.0777151i
$976$		−	5.23498e6i		−	5.49561i
$977$			231436.i			0.242461i	0.992624	+	0.121230i	$0.0386840\pi$
$977$			231436.i			0.242461i	−0.992624	+	0.121230i	$0.961316\pi$
$978$		−	357775.i		−	0.374053i
$979$	782420.			0.816347
$980$	−1.89438e6			−1.97249
$981$		−	210848.i		−	0.219094i
$982$			1.12013e6i			1.16157i
$983$			492235.i			0.509407i	0.967019	+	0.254704i	$0.0819779\pi$
$983$			492235.i			0.509407i	−0.967019	+	0.254704i	$0.918022\pi$
$984$		−	2.94133e6i		−	3.03776i
$985$	−301972.			−0.311240
$986$			1.19542e6i			1.22961i
$987$			235046.i			0.241279i
$988$		−	471130.i		−	0.482644i
$989$	−2.44697e6			−2.50171
$990$	1.16591e6			1.18958
$991$			861884.i			0.877610i	0.898582	+	0.438805i	$0.144598\pi$
$991$			861884.i			0.877610i	−0.898582	+	0.438805i	$0.855402\pi$
$992$	1.19548e6			1.21484
$993$	−56276.9			−0.0570732
$994$		−	375551.i		−	0.380099i
$995$	2.19297e6			2.21506
$996$		−	1.11178e6i		−	1.12073i
$997$	740627.			0.745091			0.372546	−	0.928014i	$-0.378485\pi$
$997$	740627.			0.745091			0.372546	+	0.928014i	$0.378485\pi$
$998$		−	2.28061e6i		−	2.28976i
$999$		−	136484.i		−	0.136758i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.5.c.a.58.3	✓	40
3.2	odd	2		531.5.c.d.235.38		40
59.58	odd	2	inner	177.5.c.a.58.38	yes	40
177.176	even	2		531.5.c.d.235.3		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.5.c.a.58.3	✓	40	1.1	even	1	trivial
177.5.c.a.58.38	yes	40	59.58	odd	2	inner
531.5.c.d.235.3		40	177.176	even	2
531.5.c.d.235.38		40	3.2	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:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	177.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-7.65331i q^{2} -5.19615 q^{3} -42.5731 q^{4} -38.1687 q^{5} +39.7678i q^{6} -35.1454 q^{7} +203.372i q^{8} +27.0000 q^{9} +O(q^{10})\)	\(q-7.65331i q^{2} -5.19615 q^{3} -42.5731 q^{4} -38.1687 q^{5} +39.7678i q^{6} -35.1454 q^{7} +203.372i q^{8} +27.0000 q^{9} +292.117i q^{10} -147.824i q^{11} +221.217 q^{12} -17.0918i q^{13} +268.979i q^{14} +198.331 q^{15} +875.302 q^{16} -354.287 q^{17} -206.639i q^{18} -647.468 q^{19} +1624.96 q^{20} +182.621 q^{21} -1131.34 q^{22} -862.278i q^{23} -1056.75i q^{24} +831.852 q^{25} -130.809 q^{26} -140.296 q^{27} +1496.25 q^{28} +440.875 q^{29} -1517.88i q^{30} +347.019i q^{31} -3445.00i q^{32} +768.116i q^{33} +2711.47i q^{34} +1341.45 q^{35} -1149.47 q^{36} +972.830i q^{37} +4955.27i q^{38} +88.8114i q^{39} -7762.47i q^{40} +2783.36 q^{41} -1397.65i q^{42} -2837.80i q^{43} +6293.33i q^{44} -1030.56 q^{45} -6599.28 q^{46} +1287.07i q^{47} -4548.20 q^{48} -1165.80 q^{49} -6366.42i q^{50} +1840.93 q^{51} +727.650i q^{52} -1103.72 q^{53} +1073.73i q^{54} +5642.25i q^{55} -7147.61i q^{56} +3364.34 q^{57} -3374.15i q^{58} +(1692.03 + 3042.10i) q^{59} -8443.55 q^{60} -5980.77i q^{61} +2655.84 q^{62} -948.926 q^{63} -12360.8 q^{64} +652.371i q^{65} +5878.63 q^{66} +3352.44i q^{67} +15083.1 q^{68} +4480.53i q^{69} -10266.6i q^{70} -1396.21 q^{71} +5491.06i q^{72} -1649.77i q^{73} +7445.37 q^{74} -4322.43 q^{75} +27564.7 q^{76} +5195.33i q^{77} +679.701 q^{78} -8357.83 q^{79} -33409.2 q^{80} +729.000 q^{81} -21301.9i q^{82} -5025.77i q^{83} -7774.74 q^{84} +13522.7 q^{85} -21718.6 q^{86} -2290.85 q^{87} +30063.3 q^{88} +5292.92i q^{89} +7887.16i q^{90} +600.697i q^{91} +36709.9i q^{92} -1803.16i q^{93} +9850.36 q^{94} +24713.0 q^{95} +17900.7i q^{96} +14051.7i q^{97} +8922.24i q^{98} -3991.25i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 320 q^{4} + 80 q^{7} + 1080 q^{9}+O(q^{10}) \) 40 * q - 320 * q^4 + 80 * q^7 + 1080 * q^9	\( 40 q - 320 q^{4} + 80 q^{7} + 1080 q^{9} + 360 q^{12} + 144 q^{15} + 3944 q^{16} - 528 q^{17} + 444 q^{19} + 444 q^{20} + 1304 q^{22} + 4880 q^{25} - 1452 q^{26} - 1160 q^{28} - 996 q^{29} + 10320 q^{35} - 8640 q^{36} - 5196 q^{41} - 10476 q^{46} + 576 q^{48} + 5104 q^{49} + 936 q^{51} - 2184 q^{53} - 2520 q^{57} - 11736 q^{59} - 11448 q^{60} + 15240 q^{62} + 2160 q^{63} - 81012 q^{64} + 17352 q^{66} + 29568 q^{68} - 5964 q^{71} + 14376 q^{74} - 2736 q^{75} + 3480 q^{76} + 37692 q^{78} + 19020 q^{79} + 33096 q^{80} + 29160 q^{81} + 25128 q^{84} + 20220 q^{85} - 65880 q^{86} + 1512 q^{87} - 14932 q^{88} - 17864 q^{94} + 11004 q^{95}+O(q^{100}) \) 40 * q - 320 * q^4 + 80 * q^7 + 1080 * q^9 + 360 * q^12 + 144 * q^15 + 3944 * q^16 - 528 * q^17 + 444 * q^19 + 444 * q^20 + 1304 * q^22 + 4880 * q^25 - 1452 * q^26 - 1160 * q^28 - 996 * q^29 + 10320 * q^35 - 8640 * q^36 - 5196 * q^41 - 10476 * q^46 + 576 * q^48 + 5104 * q^49 + 936 * q^51 - 2184 * q^53 - 2520 * q^57 - 11736 * q^59 - 11448 * q^60 + 15240 * q^62 + 2160 * q^63 - 81012 * q^64 + 17352 * q^66 + 29568 * q^68 - 5964 * q^71 + 14376 * q^74 - 2736 * q^75 + 3480 * q^76 + 37692 * q^78 + 19020 * q^79 + 33096 * q^80 + 29160 * q^81 + 25128 * q^84 + 20220 * q^85 - 65880 * q^86 + 1512 * q^87 - 14932 * q^88 - 17864 * q^94 + 11004 * q^95

Introduction

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