LMFDB - Embedded newform 177.5.c.a.58.16

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			58.16
Character	$\chi$	$=$	177.58
Dual form			177.5.c.a.58.25

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.31914i q^{2} +5.19615 q^{3} +10.6216 q^{4} -38.6371 q^{5} -12.0506i q^{6} +9.48964 q^{7} -61.7392i q^{8} +27.0000 q^{9} +O(q^{10})$	\(q-2.31914i q^{2} +5.19615 q^{3} +10.6216 q^{4} -38.6371 q^{5} -12.0506i q^{6} +9.48964 q^{7} -61.7392i q^{8} +27.0000 q^{9} +89.6048i q^{10} +58.4880i q^{11} +55.1914 q^{12} -290.914i q^{13} -22.0078i q^{14} -200.764 q^{15} +26.7637 q^{16} -467.291 q^{17} -62.6168i q^{18} -261.109 q^{19} -410.387 q^{20} +49.3096 q^{21} +135.642 q^{22} -190.226i q^{23} -320.806i q^{24} +867.825 q^{25} -674.670 q^{26} +140.296 q^{27} +100.795 q^{28} -407.645 q^{29} +465.600i q^{30} -1893.18i q^{31} -1049.90i q^{32} +303.913i q^{33} +1083.71i q^{34} -366.652 q^{35} +286.783 q^{36} +306.386i q^{37} +605.549i q^{38} -1511.63i q^{39} +2385.42i q^{40} -486.816 q^{41} -114.356i q^{42} +1830.69i q^{43} +621.236i q^{44} -1043.20 q^{45} -441.160 q^{46} +1576.43i q^{47} +139.068 q^{48} -2310.95 q^{49} -2012.61i q^{50} -2428.12 q^{51} -3089.97i q^{52} -2715.01 q^{53} -325.366i q^{54} -2259.81i q^{55} -585.883i q^{56} -1356.76 q^{57} +945.384i q^{58} +(3310.90 - 1074.84i) q^{59} -2132.44 q^{60} -3956.15i q^{61} -4390.55 q^{62} +256.220 q^{63} -2006.63 q^{64} +11240.1i q^{65} +704.816 q^{66} -5269.11i q^{67} -4963.37 q^{68} -988.442i q^{69} +850.317i q^{70} +7353.03 q^{71} -1666.96i q^{72} +8254.16i q^{73} +710.553 q^{74} +4509.35 q^{75} -2773.40 q^{76} +555.030i q^{77} -3505.69 q^{78} +8388.54 q^{79} -1034.07 q^{80} +729.000 q^{81} +1128.99i q^{82} +8211.40i q^{83} +523.747 q^{84} +18054.8 q^{85} +4245.63 q^{86} -2118.18 q^{87} +3611.00 q^{88} -13635.4i q^{89} +2419.33i q^{90} -2760.67i q^{91} -2020.50i q^{92} -9837.26i q^{93} +3655.95 q^{94} +10088.5 q^{95} -5455.42i q^{96} -5348.42i q^{97} +5359.41i q^{98} +1579.18i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	2.31914i		−	0.579785i	−0.957059	−	0.289892i	$-0.906380\pi$
$2$		−	2.31914i		−	0.579785i	0.957059	−	0.289892i	$-0.0936195\pi$
$3$	5.19615			0.577350
$3$	5.19615			0.577350
$4$	10.6216			0.663850
$5$	−38.6371			−1.54548			−0.772742	−	0.634720i	$-0.781115\pi$
$5$	−38.6371			−1.54548			−0.772742	+	0.634720i	$0.781115\pi$
$6$		−	12.0506i		−	0.334739i
$7$	9.48964			0.193666			0.0968331	−	0.995301i	$-0.469129\pi$
$7$	9.48964			0.193666			0.0968331	+	0.995301i	$0.469129\pi$
$8$		−	61.7392i		−	0.964675i
$9$	27.0000			0.333333
$10$			89.6048i			0.896048i
$11$			58.4880i			0.483372i	0.970355	+	0.241686i	$0.0777004\pi$
$11$			58.4880i			0.483372i	−0.970355	+	0.241686i	$0.922300\pi$
$12$	55.1914			0.383274
$13$		−	290.914i		−	1.72138i	−0.509125	−	0.860692i	$-0.670031\pi$
$13$		−	290.914i		−	1.72138i	0.509125	−	0.860692i	$-0.329969\pi$
$14$		−	22.0078i		−	0.112285i
$15$	−200.764			−0.892285
$16$	26.7637			0.104546
$17$	−467.291			−1.61692			−0.808462	−	0.588548i	$-0.799700\pi$
$17$	−467.291			−1.61692			−0.808462	+	0.588548i	$0.799700\pi$
$18$		−	62.6168i		−	0.193262i
$19$	−261.109			−0.723294			−0.361647	−	0.932315i	$-0.617785\pi$
$19$	−261.109			−0.723294			−0.361647	+	0.932315i	$0.617785\pi$
$20$	−410.387			−1.02597
$21$	49.3096			0.111813
$22$	135.642			0.280252
$23$		−	190.226i		−	0.359595i	−0.983704	−	0.179798i	$-0.942456\pi$
$23$		−	190.226i		−	0.359595i	0.983704	−	0.179798i	$-0.0575443\pi$
$24$		−	320.806i		−	0.556955i
$25$	867.825			1.38852
$26$	−674.670			−0.998033
$27$	140.296			0.192450
$28$	100.795			0.128565
$29$	−407.645			−0.484714			−0.242357	−	0.970187i	$-0.577921\pi$
$29$	−407.645			−0.484714			−0.242357	+	0.970187i	$0.577921\pi$
$30$			465.600i			0.517334i
$31$		−	1893.18i		−	1.97001i	−0.172518	−	0.985006i	$-0.555190\pi$
$31$		−	1893.18i		−	1.97001i	0.172518	−	0.985006i	$-0.444810\pi$
$32$		−	1049.90i		−	1.02529i
$33$			303.913i			0.279075i
$34$			1083.71i			0.937468i
$35$	−366.652			−0.299308
$36$	286.783			0.221283
$37$			306.386i			0.223803i	0.993719	+	0.111902i	$0.0356941\pi$
$37$			306.386i			0.223803i	−0.993719	+	0.111902i	$0.964306\pi$
$38$			605.549i			0.419355i
$39$		−	1511.63i		−	0.993842i
$40$			2385.42i			1.49089i
$41$	−486.816			−0.289599			−0.144799	−	0.989461i	$-0.546254\pi$
$41$	−486.816			−0.289599			−0.144799	+	0.989461i	$0.546254\pi$
$42$		−	114.356i		−	0.0648276i
$43$			1830.69i			0.990098i	0.868865	+	0.495049i	$0.164850\pi$
$43$			1830.69i			0.990098i	−0.868865	+	0.495049i	$0.835150\pi$
$44$			621.236i			0.320886i
$45$	−1043.20			−0.515161
$46$	−441.160			−0.208488
$47$			1576.43i			0.713637i	0.934174	+	0.356819i	$0.116139\pi$
$47$			1576.43i			0.713637i	−0.934174	+	0.356819i	$0.883861\pi$
$48$	139.068			0.0603595
$49$	−2310.95			−0.962493
$50$		−	2012.61i		−	0.805043i
$51$	−2428.12			−0.933531
$52$		−	3089.97i		−	1.14274i
$53$	−2715.01			−0.966541			−0.483271	−	0.875471i	$-0.660551\pi$
$53$	−2715.01			−0.966541			−0.483271	+	0.875471i	$0.660551\pi$
$54$		−	325.366i		−	0.111580i
$55$		−	2259.81i		−	0.747043i
$56$		−	585.883i		−	0.186825i
$57$	−1356.76			−0.417594
$58$			945.384i			0.281030i
$59$	3310.90	−	1074.84i	0.951135	−	0.308774i
$59$	3310.90	−	1074.84i	0.951135	−	0.308774i
$60$	−2132.44			−0.592343
$61$		−	3956.15i		−	1.06320i	−0.846997	−	0.531598i	$-0.821592\pi$
$61$		−	3956.15i		−	1.06320i	0.846997	−	0.531598i	$-0.178408\pi$
$62$	−4390.55			−1.14218
$63$	256.220			0.0645554
$64$	−2006.63			−0.489901
$65$			11240.1i			2.66037i
$66$	704.816			0.161803
$67$		−	5269.11i		−	1.17378i	−0.809666	−	0.586891i	$-0.800352\pi$
$67$		−	5269.11i		−	1.17378i	0.809666	−	0.586891i	$-0.199648\pi$
$68$	−4963.37			−1.07339
$69$		−	988.442i		−	0.207612i
$70$			850.317i			0.173534i
$71$	7353.03			1.45865			0.729323	−	0.684169i	$-0.239835\pi$
$71$	7353.03			1.45865			0.729323	+	0.684169i	$0.239835\pi$
$72$		−	1666.96i		−	0.321558i
$73$			8254.16i			1.54891i	0.632627	+	0.774457i	$0.281977\pi$
$73$			8254.16i			1.54891i	−0.632627	+	0.774457i	$0.718023\pi$
$74$	710.553			0.129758
$75$	4509.35			0.801662
$76$	−2773.40			−0.480158
$77$			555.030i			0.0936128i
$78$	−3505.69			−0.576214
$79$	8388.54			1.34410			0.672051	−	0.740505i	$-0.265414\pi$
$79$	8388.54			1.34410			0.672051	+	0.740505i	$0.265414\pi$
$80$	−1034.07			−0.161574
$81$	729.000			0.111111
$82$			1128.99i			0.167905i
$83$			8211.40i			1.19196i	0.803000	+	0.595979i	$0.203236\pi$
$83$			8211.40i			1.19196i	−0.803000	+	0.595979i	$0.796764\pi$
$84$	523.747			0.0742271
$85$	18054.8			2.49893
$86$	4245.63			0.574044
$87$	−2118.18			−0.279850
$88$	3611.00			0.466297
$89$		−	13635.4i		−	1.72142i	−0.509092	−	0.860712i	$-0.670019\pi$
$89$		−	13635.4i		−	1.72142i	0.509092	−	0.860712i	$-0.329981\pi$
$90$			2419.33i			0.298683i
$91$		−	2760.67i		−	0.333374i
$92$		−	2020.50i		−	0.238717i
$93$		−	9837.26i		−	1.13739i
$94$	3655.95			0.413756
$95$	10088.5			1.11784
$96$		−	5455.42i		−	0.591951i
$97$		−	5348.42i		−	0.568436i	−0.958760	−	0.284218i	$-0.908266\pi$
$97$		−	5348.42i		−	0.568436i	0.958760	−	0.284218i	$-0.0917340\pi$
$98$			5359.41i			0.558039i
$99$			1579.18i			0.161124i
$100$	9217.68			0.921768
$101$		−	11578.9i		−	1.13508i	−0.823347	−	0.567538i	$-0.807896\pi$
$101$		−	11578.9i		−	1.13508i	0.823347	−	0.567538i	$-0.192104\pi$
$102$			5631.14i			0.541247i
$103$			18902.9i			1.78178i	0.454222	+	0.890888i	$0.349917\pi$
$103$			18902.9i			1.78178i	−0.454222	+	0.890888i	$0.650083\pi$
$104$	−17960.8			−1.66058
$105$	−1905.18			−0.172805
$106$			6296.50i			0.560386i
$107$	15473.8			1.35155			0.675773	−	0.737110i	$-0.263810\pi$
$107$	15473.8			1.35155			0.675773	+	0.737110i	$0.263810\pi$
$108$	1490.17			0.127758
$109$			10396.5i			0.875049i	0.899206	+	0.437525i	$0.144145\pi$
$109$			10396.5i			0.875049i	−0.899206	+	0.437525i	$0.855855\pi$
$110$	−5240.80			−0.433124
$111$			1592.03i			0.129213i
$112$	253.978			0.0202470
$113$		−	3655.52i		−	0.286281i	−0.989702	−	0.143141i	$-0.954280\pi$
$113$		−	3655.52i		−	0.286281i	0.989702	−	0.143141i	$-0.0457201\pi$
$114$			3146.52i			0.242115i
$115$			7349.77i			0.555748i
$116$	−4329.83			−0.321777
$117$		−	7854.68i		−	0.573795i
$118$	−2492.71	−	7678.44i	−0.179022	−	0.551454i
$119$	−4434.42			−0.313143
$120$			12395.0i			0.860765i
$121$	11220.2			0.766352
$122$	−9174.87			−0.616425
$123$	−2529.57			−0.167200
$124$		−	20108.6i		−	1.30779i
$125$	−9382.04			−0.600451
$126$		−	594.211i		−	0.0374282i
$127$	8960.25			0.555536			0.277768	−	0.960648i	$-0.410405\pi$
$127$	8960.25			0.555536			0.277768	+	0.960648i	$0.410405\pi$
$128$		−	12144.7i		−	0.741252i
$129$			9512.55i			0.571633i
$130$	26067.3			1.54244
$131$		−	12620.5i		−	0.735417i	−0.929941	−	0.367708i	$-0.880142\pi$
$131$		−	12620.5i		−	0.735417i	0.929941	−	0.367708i	$-0.119858\pi$
$132$			3228.04i			0.185264i
$133$	−2477.83			−0.140078
$134$	−12219.8			−0.680541
$135$	−5420.63			−0.297428
$136$			28850.2i			1.55981i
$137$	−21067.5			−1.12246			−0.561232	−	0.827659i	$-0.689672\pi$
$137$	−21067.5			−1.12246			−0.561232	+	0.827659i	$0.689672\pi$
$138$	−2292.34			−0.120370
$139$	23705.1			1.22691			0.613455	−	0.789729i	$-0.289779\pi$
$139$	23705.1			1.22691			0.613455	+	0.789729i	$0.289779\pi$
$140$	−3894.43			−0.198695
$141$			8191.34i			0.412019i
$142$		−	17052.7i		−	0.845701i
$143$	17015.0			0.832069
$144$	722.620			0.0348486
$145$	15750.2			0.749118
$146$	19142.5			0.898037
$147$	−12008.0			−0.555696
$148$			3254.31i			0.148572i
$149$		−	5346.43i		−	0.240819i	−0.992724	−	0.120410i	$-0.961579\pi$
$149$		−	5346.43i		−	0.240819i	0.992724	−	0.120410i	$-0.0384208\pi$
$150$		−	10457.8i		−	0.464792i
$151$		−	27531.3i		−	1.20746i	−0.797188	−	0.603731i	$-0.793680\pi$
$151$		−	27531.3i		−	1.20746i	0.797188	−	0.603731i	$-0.206320\pi$
$152$			16120.7i			0.697744i
$153$	−12616.9			−0.538975
$154$	1287.19			0.0542753
$155$			73147.1i			3.04462i
$156$		−	16056.0i		−	0.659761i
$157$			29614.0i			1.20143i	0.799465	+	0.600713i	$0.205117\pi$
$157$			29614.0i			1.20143i	−0.799465	+	0.600713i	$0.794883\pi$
$158$		−	19454.2i		−	0.779289i
$159$	−14107.6			−0.558033
$160$			40564.9i			1.58457i
$161$		−	1805.17i		−	0.0696414i
$162$		−	1690.65i		−	0.0644205i
$163$	523.410			0.0197000			0.00985001	−	0.999951i	$-0.496865\pi$
$163$	523.410			0.0197000			0.00985001	+	0.999951i	$0.496865\pi$
$164$	−5170.76			−0.192250
$165$		−	11742.3i		−	0.431306i
$166$	19043.4			0.691079
$167$	8380.72			0.300503			0.150251	−	0.988648i	$-0.451992\pi$
$167$	8380.72			0.300503			0.150251	+	0.988648i	$0.451992\pi$
$168$		−	3044.34i		−	0.107863i
$169$	−56070.0			−1.96317
$170$		−	41871.5i		−	1.44884i
$171$	−7049.95			−0.241098
$172$			19444.9i			0.657276i
$173$		−	31624.9i		−	1.05666i	−0.849038	−	0.528331i	$-0.822818\pi$
$173$		−	31624.9i		−	1.05666i	0.849038	−	0.528331i	$-0.177182\pi$
$174$			4912.36i			0.162253i
$175$	8235.35			0.268909
$176$			1565.36i			0.0505345i
$177$	17204.0	−	5585.04i	0.549138	−	0.178271i
$178$	−31622.4			−0.998055
$179$			14600.1i			0.455668i	0.973700	+	0.227834i	$0.0731644\pi$
$179$			14600.1i			0.455668i	−0.973700	+	0.227834i	$0.926836\pi$
$180$	−11080.5			−0.341990
$181$	−48410.5			−1.47769			−0.738843	−	0.673877i	$-0.764628\pi$
$181$	−48410.5			−1.47769			−0.738843	+	0.673877i	$0.764628\pi$
$182$	−6402.38			−0.193285
$183$		−	20556.8i		−	0.613837i
$184$	−11744.4			−0.346892
$185$		−	11837.9i		−	0.345884i
$186$	−22814.0			−0.659440
$187$		−	27330.9i		−	0.781575i
$188$			16744.1i			0.473748i
$189$	1331.36			0.0372711
$190$		−	23396.6i		−	0.648106i
$191$		−	11986.4i		−	0.328566i	−0.986413	−	0.164283i	$-0.947469\pi$
$191$		−	11986.4i		−	0.328566i	0.986413	−	0.164283i	$-0.0525310\pi$
$192$	−10426.8			−0.282845
$193$	19496.9			0.523420			0.261710	−	0.965147i	$-0.415714\pi$
$193$	19496.9			0.523420			0.261710	+	0.965147i	$0.415714\pi$
$194$	−12403.7			−0.329571
$195$			58405.1i			1.53597i
$196$	−24545.9			−0.638951
$197$	−42645.4			−1.09885			−0.549426	−	0.835542i	$-0.685154\pi$
$197$	−42645.4			−1.09885			−0.549426	+	0.835542i	$0.685154\pi$
$198$	3662.33			0.0934172
$199$	10669.4			0.269421			0.134711	−	0.990885i	$-0.456990\pi$
$199$	10669.4			0.269421			0.134711	+	0.990885i	$0.456990\pi$
$200$		−	53578.8i		−	1.33947i
$201$		−	27379.1i		−	0.677683i
$202$	−26853.1			−0.658100
$203$	−3868.40			−0.0938727
$204$	−25790.5			−0.619724
$205$	18809.1			0.447570
$206$	43838.4			1.03305
$207$		−	5136.10i		−	0.119865i
$208$		−	7785.94i		−	0.179963i
$209$		−	15271.8i		−	0.349620i
$210$			4418.38i			0.100190i
$211$			15256.3i			0.342677i	0.985212	+	0.171339i	$0.0548092\pi$
$211$			15256.3i			0.342677i	−0.985212	+	0.171339i	$0.945191\pi$
$212$	−28837.8			−0.641638
$213$	38207.5			0.842150
$214$		−	35886.0i		−	0.783605i
$215$		−	70732.6i		−	1.53018i
$216$		−	8661.77i		−	0.185652i
$217$		−	17965.6i		−	0.381525i
$218$	24110.8			0.507340
$219$			42889.9i			0.894266i
$220$		−	24002.7i		−	0.495924i
$221$			135941.i			2.78335i
$222$	3692.14			0.0749156
$223$	17062.9			0.343117			0.171558	−	0.985174i	$-0.445120\pi$
$223$	17062.9			0.343117			0.171558	+	0.985174i	$0.445120\pi$
$224$		−	9963.13i		−	0.198564i
$225$	23431.3			0.462840
$226$	−8477.67			−0.165981
$227$		−	15718.2i		−	0.305037i	−0.988301	−	0.152518i	$-0.951262\pi$
$227$		−	15718.2i		−	0.305037i	0.988301	−	0.152518i	$-0.0487383\pi$
$228$	−14411.0			−0.277220
$229$		−	10725.1i		−	0.204518i	−0.994758	−	0.102259i	$-0.967393\pi$
$229$		−	10725.1i		−	0.204518i	0.994758	−	0.102259i	$-0.0326071\pi$
$230$	17045.1			0.322214
$231$			2884.02i			0.0540474i
$232$			25167.6i			0.467591i
$233$			34953.2i			0.643836i	0.946768	+	0.321918i	$0.104327\pi$
$233$			34953.2i			0.643836i	−0.946768	+	0.321918i	$0.895673\pi$
$234$	−18216.1			−0.332678
$235$		−	60908.5i		−	1.10291i
$236$	35167.1	−	11416.5i	0.631411	−	0.204979i
$237$	43588.1			0.776017
$238$			10284.0i			0.181556i
$239$	43682.3			0.764732			0.382366	−	0.924011i	$-0.375109\pi$
$239$	43682.3			0.764732			0.382366	+	0.924011i	$0.375109\pi$
$240$	−5373.20			−0.0932847
$241$	−37284.2			−0.641935			−0.320968	−	0.947090i	$-0.604008\pi$
$241$	−37284.2			−0.641935			−0.320968	+	0.947090i	$0.604008\pi$
$242$		−	26021.1i		−	0.444319i
$243$	3788.00			0.0641500
$244$		−	42020.7i		−	0.705802i
$245$	89288.3			1.48752
$246$			5866.42i			0.0969400i
$247$			75960.3i			1.24507i
$248$	−116884.			−1.90042
$249$			42667.7i			0.688177i
$250$			21758.3i			0.348132i
$251$	−42855.1			−0.680229			−0.340114	−	0.940384i	$-0.610466\pi$
$251$	−42855.1			−0.680229			−0.340114	+	0.940384i	$0.610466\pi$
$252$	2721.47			0.0428551
$253$	11125.9			0.173818
$254$		−	20780.1i		−	0.322092i
$255$	93815.3			1.44276
$256$	−60271.3			−0.919668
$257$	−67944.9			−1.02870			−0.514352	−	0.857579i	$-0.671967\pi$
$257$	−67944.9			−1.02870			−0.514352	+	0.857579i	$0.671967\pi$
$258$	22060.9			0.331424
$259$			2907.50i			0.0433431i
$260$			119387.i			1.76609i
$261$	−11006.4			−0.161571
$262$	−29268.7			−0.426383
$263$	6967.85			0.100737			0.0503683	−	0.998731i	$-0.483960\pi$
$263$	6967.85			0.100737			0.0503683	+	0.998731i	$0.483960\pi$
$264$	18763.3			0.269216
$265$	104900.			1.49377
$266$			5746.44i			0.0812149i
$267$		−	70851.6i		−	0.993865i
$268$		−	55966.3i		−	0.779215i
$269$			19627.0i			0.271238i	0.990761	+	0.135619i	$0.0433023\pi$
$269$			19627.0i			0.271238i	−0.990761	+	0.135619i	$0.956698\pi$
$270$			12571.2i			0.172445i
$271$	−25914.9			−0.352866			−0.176433	−	0.984313i	$-0.556456\pi$
$271$	−25914.9			−0.352866			−0.176433	+	0.984313i	$0.556456\pi$
$272$	−12506.4			−0.169043
$273$		−	14344.9i		−	0.192474i
$274$			48858.5i			0.650787i
$275$			50757.3i			0.671171i
$276$		−	10498.8i		−	0.137823i
$277$	−104861.			−1.36664			−0.683318	−	0.730121i	$-0.739464\pi$
$277$	−104861.			−1.36664			−0.683318	+	0.730121i	$0.739464\pi$
$278$		−	54975.5i		−	0.711344i
$279$		−	51115.9i		−	0.656671i
$280$			22636.8i			0.288735i
$281$	−14867.0			−0.188283			−0.0941417	−	0.995559i	$-0.530011\pi$
$281$	−14867.0			−0.188283			−0.0941417	+	0.995559i	$0.530011\pi$
$282$	18996.9			0.238882
$283$		−	72926.7i		−	0.910570i	−0.890346	−	0.455285i	$-0.849537\pi$
$283$		−	72926.7i		−	0.910570i	0.890346	−	0.455285i	$-0.150463\pi$
$284$	78100.9			0.968322
$285$	52421.4			0.645385
$286$		−	39460.1i		−	0.482421i
$287$	−4619.71			−0.0560855
$288$		−	28347.2i		−	0.341763i
$289$	134840.			1.61444
$290$		−	36526.9i		−	0.434327i
$291$		−	27791.2i		−	0.328187i
$292$			87672.3i			1.02825i
$293$	69720.2			0.812125			0.406063	−	0.913845i	$-0.366901\pi$
$293$	69720.2			0.812125			0.406063	+	0.913845i	$0.366901\pi$
$294$			27848.3i			0.322184i
$295$	−127924.	+	41528.8i	−1.46996	+	0.477205i
$296$	18916.1			0.215897
$297$			8205.64i			0.0930250i
$298$	−12399.1			−0.139623
$299$	−55339.4			−0.619002
$300$	47896.5			0.532183
$301$			17372.6i			0.191748i
$302$	−63849.0			−0.700068
$303$		−	60165.8i		−	0.655336i
$304$	−6988.25			−0.0756173
$305$			152854.i			1.64315i
$306$			29260.2i			0.312489i
$307$	−140429.			−1.48997			−0.744987	−	0.667079i	$-0.767544\pi$
$307$	−140429.			−1.48997			−0.744987	+	0.667079i	$0.767544\pi$
$308$			5895.30i			0.0621448i
$309$			98222.2i			1.02871i
$310$	169638.			1.76523
$311$	−106994.			−1.10621			−0.553105	−	0.833112i	$-0.686557\pi$
$311$	−106994.			−1.10621			−0.553105	+	0.833112i	$0.686557\pi$
$312$	−93327.0			−0.958734
$313$			111408.i			1.13717i	0.822623	+	0.568587i	$0.192510\pi$
$313$			111408.i			1.13717i	−0.822623	+	0.568587i	$0.807490\pi$
$314$	68678.9			0.696569
$315$	−9899.61			−0.0997693
$316$	89099.6			0.892281
$317$	139639.			1.38960			0.694799	−	0.719204i	$-0.255493\pi$
$317$	139639.			1.38960			0.694799	+	0.719204i	$0.255493\pi$
$318$			32717.6i			0.323539i
$319$		−	23842.3i		−	0.234297i
$320$	77530.5			0.757134
$321$	80404.4			0.780315
$322$	−4186.45			−0.0403770
$323$	122014.			1.16951
$324$	7743.14			0.0737611
$325$		−	252462.i		−	2.39018i
$326$		−	1213.86i		−	0.0114218i
$327$			54021.6i			0.505210i
$328$			30055.6i			0.279369i
$329$			14959.7i			0.138207i
$330$	−27232.0			−0.250064
$331$	54555.5			0.497947			0.248973	−	0.968510i	$-0.419907\pi$
$331$	54555.5			0.497947			0.248973	+	0.968510i	$0.419907\pi$
$332$			87218.1i			0.791281i
$333$			8272.44i			0.0746010i
$334$		−	19436.1i		−	0.174227i
$335$			203583.i			1.81406i
$336$	1319.71			0.0116896
$337$		−	48646.0i		−	0.428339i	−0.976797	−	0.214169i	$-0.931296\pi$
$337$		−	48646.0i		−	0.428339i	0.976797	−	0.214169i	$-0.0687045\pi$
$338$			130034.i			1.13821i
$339$		−	18994.7i		−	0.165284i
$340$	191770.			1.65891
$341$	110728.			0.952249
$342$			16349.8i			0.139785i
$343$	−44714.7			−0.380069
$344$	113025.			0.955123
$345$			38190.5i			0.320861i
$346$	−73342.5			−0.612637
$347$		−	115161.i		−	0.956413i	−0.878248	−	0.478206i	$-0.841287\pi$
$347$		−	115161.i		−	0.956413i	0.878248	−	0.478206i	$-0.158713\pi$
$348$	−22498.5			−0.185778
$349$			45753.9i			0.375645i	0.982203	+	0.187822i	$0.0601429\pi$
$349$			45753.9i			0.375645i	−0.982203	+	0.187822i	$0.939857\pi$
$350$		−	19098.9i		−	0.155909i
$351$		−	40814.1i		−	0.331281i
$352$	61406.3			0.495596
$353$		−	118207.i		−	0.948622i	−0.880357	−	0.474311i	$-0.842697\pi$
$353$		−	118207.i		−	0.948622i	0.880357	−	0.474311i	$-0.157303\pi$
$354$	−12952.5	−	39898.4i	−0.103359	−	0.318382i
$355$	−284100.			−2.25431
$356$		−	144830.i		−	1.14277i
$357$	−23041.9			−0.180793
$358$	33859.6			0.264189
$359$	210839.			1.63592			0.817961	−	0.575274i	$-0.195104\pi$
$359$	210839.			1.63592			0.817961	+	0.575274i	$0.195104\pi$
$360$			64406.4i			0.496963i
$361$	−62143.0			−0.476846
$362$			112271.i			0.856741i
$363$	58301.6			0.442453
$364$		−	29322.7i		−	0.221310i
$365$		−	318917.i		−	2.39382i
$366$	−47674.0			−0.355893
$367$		−	29773.6i		−	0.221054i	−0.993873	−	0.110527i	$-0.964746\pi$
$367$		−	29773.6i		−	0.221054i	0.993873	−	0.110527i	$-0.0352539\pi$
$368$		−	5091.15i		−	0.0375942i
$369$	−13144.0			−0.0965329
$370$	−27453.7			−0.200538
$371$	−25764.5			−0.187186
$372$		−	104487.i		−	0.755054i
$373$	−25947.3			−0.186498			−0.0932492	−	0.995643i	$-0.529725\pi$
$373$	−25947.3			−0.186498			−0.0932492	+	0.995643i	$0.529725\pi$
$374$	−63384.2			−0.453146
$375$	−48750.5			−0.346670
$376$	97327.2			0.688428
$377$			118590.i			0.834379i
$378$		−	3087.61i		−	0.0216092i
$379$	−268474.			−1.86906			−0.934530	−	0.355884i	$-0.884180\pi$
$379$	−268474.			−1.86906			−0.934530	+	0.355884i	$0.884180\pi$
$380$	107156.			0.742077
$381$	46558.8			0.320739
$382$	−27798.2			−0.190498
$383$	164228.			1.11956			0.559782	−	0.828640i	$-0.310885\pi$
$383$	164228.			1.11956			0.559782	+	0.828640i	$0.310885\pi$
$384$		−	63105.5i		−	0.427962i
$385$		−	21444.7i		−	0.144677i
$386$		−	45216.0i		−	0.303471i
$387$			49428.7i			0.330033i
$388$		−	56808.7i		−	0.377356i
$389$	−231786.			−1.53175			−0.765876	−	0.642988i	$-0.777695\pi$
$389$	−231786.			−1.53175			−0.765876	+	0.642988i	$0.777695\pi$
$390$	135450.			0.890530
$391$			88890.8i			0.581438i
$392$			142676.i			0.928493i
$393$		−	65578.0i		−	0.424593i
$394$			98900.6i			0.637098i
$395$	−324109.			−2.07729
$396$			16773.4i			0.106962i
$397$		−	66558.9i		−	0.422304i	−0.977453	−	0.211152i	$-0.932278\pi$
$397$		−	66558.9i		−	0.422304i	0.977453	−	0.211152i	$-0.0677215\pi$
$398$		−	24743.7i		−	0.156206i
$399$	−12875.2			−0.0808738
$400$	23226.2			0.145164
$401$		−	22759.8i		−	0.141540i	−0.997493	−	0.0707702i	$-0.977454\pi$
$401$		−	22759.8i		−	0.141540i	0.997493	−	0.0707702i	$-0.0225457\pi$
$402$	−63495.9			−0.392911
$403$	−550753.			−3.39115
$404$		−	122986.i		−	0.753520i
$405$	−28166.4			−0.171720
$406$			8971.36i			0.0544260i
$407$	−17919.9			−0.108180
$408$			149910.i			0.900554i
$409$		−	312184.i		−	1.86623i	−0.359583	−	0.933113i	$-0.617081\pi$
$409$		−	312184.i		−	1.86623i	0.359583	−	0.933113i	$-0.382919\pi$
$410$		−	43621.0i		−	0.259494i
$411$	−109470.			−0.648055
$412$			200779.i			1.18283i
$413$	31419.3	−	10199.9i	0.184203	−	0.0597991i
$414$	−11911.3			−0.0694959
$415$		−	317264.i		−	1.84215i
$416$	−305429.			−1.76492
$417$	123176.			0.708357
$418$	−35417.3			−0.202704
$419$		−	314728.i		−	1.79270i	−0.443346	−	0.896351i	$-0.646209\pi$
$419$		−	314728.i		−	1.79270i	0.443346	−	0.896351i	$-0.353791\pi$
$420$	−20236.0			−0.114717
$421$			25224.4i			0.142317i	0.997465	+	0.0711585i	$0.0226696\pi$
$421$			25224.4i			0.142317i	−0.997465	+	0.0711585i	$0.977330\pi$
$422$	35381.6			0.198679
$423$			42563.5i			0.237879i
$424$			167623.i			0.932398i
$425$	−405527.			−2.24513
$426$		−	88608.5i		−	0.488266i
$427$		−	37542.5i		−	0.205905i
$428$	164357.			0.897223
$429$	88412.4			0.480395
$430$	−164039.			−0.887175
$431$		−	58420.2i		−	0.314491i	−0.987560	−	0.157246i	$-0.949739\pi$
$431$		−	58420.2i		−	0.314491i	0.987560	−	0.157246i	$-0.0502614\pi$
$432$	3754.85			0.0201198
$433$	59683.1			0.318328			0.159164	−	0.987252i	$-0.449120\pi$
$433$	59683.1			0.318328			0.159164	+	0.987252i	$0.449120\pi$
$434$	−41664.8			−0.221202
$435$	81840.4			0.432503
$436$			110427.i			0.580901i
$437$			49669.7i			0.260093i
$438$	99467.6			0.518482
$439$	200004.			1.03779			0.518895	−	0.854838i	$-0.326344\pi$
$439$	200004.			1.03779			0.518895	+	0.854838i	$0.326344\pi$
$440$	−139519.			−0.720654
$441$	−62395.6			−0.320831
$442$	315267.			1.61374
$443$			299059.i			1.52388i	0.647649	+	0.761939i	$0.275752\pi$
$443$			299059.i			1.52388i	−0.647649	+	0.761939i	$0.724248\pi$
$444$			16909.9i			0.0857779i
$445$			526832.i			2.66043i
$446$		−	39571.1i		−	0.198934i
$447$		−	27780.9i		−	0.139037i
$448$	−19042.2			−0.0948773
$449$	308677.			1.53113			0.765565	−	0.643359i	$-0.222460\pi$
$449$	308677.			1.53113			0.765565	+	0.643359i	$0.222460\pi$
$450$		−	54340.4i		−	0.268348i
$451$		−	28472.9i		−	0.139984i
$452$		−	38827.5i		−	0.190048i
$453$		−	143057.i		−	0.697128i
$454$	−36452.8			−0.176856
$455$			106664.i			0.515224i
$456$			83765.4i			0.402842i
$457$		−	224815.i		−	1.07645i	−0.842803	−	0.538223i	$-0.819096\pi$
$457$		−	224815.i		−	1.07645i	0.842803	−	0.538223i	$-0.180904\pi$
$458$	−24873.1			−0.118577
$459$	−65559.1			−0.311177
$460$			78066.3i			0.368933i
$461$	186398.			0.877079			0.438539	−	0.898712i	$-0.355496\pi$
$461$	186398.			0.877079			0.438539	+	0.898712i	$0.355496\pi$
$462$	6688.45			0.0313358
$463$			45013.2i			0.209980i	0.994473	+	0.104990i	$0.0334810\pi$
$463$			45013.2i			0.209980i	−0.994473	+	0.104990i	$0.966519\pi$
$464$	−10910.1			−0.0506748
$465$			380083.i			1.75781i
$466$	81061.3			0.373286
$467$		−	243089.i		−	1.11463i	−0.830300	−	0.557316i	$-0.811831\pi$
$467$		−	243089.i		−	1.11463i	0.830300	−	0.557316i	$-0.188169\pi$
$468$		−	83429.2i		−	0.380913i
$469$		−	50001.9i		−	0.227322i
$470$	−141255.			−0.639453
$471$			153879.i			0.693644i
$472$	−66359.9	−	204412.i	−0.297866	−	0.917536i
$473$	−107073.			−0.478586
$474$		−	101087.i		−	0.449923i
$475$	−226597.			−1.00431
$476$	−47100.6			−0.207880
$477$	−73305.4			−0.322180
$478$		−	101305.i		−	0.443380i
$479$	115564.			0.503677			0.251839	−	0.967769i	$-0.418965\pi$
$479$	115564.			0.503677			0.251839	+	0.967769i	$0.418965\pi$
$480$			210781.i			0.914850i
$481$	89132.1			0.385251
$482$			86467.3i			0.372184i
$483$		−	9379.96i		−	0.0402075i
$484$	119176.			0.508742
$485$			206647.i			0.878509i
$486$		−	8784.89i		−	0.0371932i
$487$	151048.			0.636880			0.318440	−	0.947943i	$-0.396841\pi$
$487$	151048.			0.636880			0.318440	+	0.947943i	$0.396841\pi$
$488$	−244250.			−1.02564
$489$	2719.72			0.0113738
$490$		−	207072.i		−	0.862440i
$491$	−20445.7			−0.0848086			−0.0424043	−	0.999101i	$-0.513502\pi$
$491$	−20445.7			−0.0848086			−0.0424043	+	0.999101i	$0.513502\pi$
$492$	−26868.0			−0.110996
$493$	190489.			0.783746
$494$	176163.			0.721871
$495$		−	61014.8i		−	0.249014i
$496$		−	50668.6i		−	0.205957i
$497$	69777.7			0.282490
$498$	98952.3			0.398995
$499$	−59385.2			−0.238494			−0.119247	−	0.992865i	$-0.538048\pi$
$499$	−59385.2			−0.238494			−0.119247	+	0.992865i	$0.538048\pi$
$500$	−99652.2			−0.398609
$501$	43547.5			0.173495
$502$			99386.9i			0.394386i
$503$			156531.i			0.618677i	0.950952	+	0.309339i	$0.100108\pi$
$503$			156531.i			0.618677i	−0.950952	+	0.309339i	$0.899892\pi$
$504$		−	15818.8i		−	0.0622749i
$505$			447375.i			1.75424i
$506$		−	25802.6i		−	0.100777i
$507$	−291348.			−1.13343
$508$	95172.1			0.368793
$509$		−	250897.i		−	0.968410i	−0.874954	−	0.484205i	$-0.839109\pi$
$509$		−	250897.i		−	0.968410i	0.874954	−	0.484205i	$-0.160891\pi$
$510$		−	217571.i		−	0.836489i
$511$			78329.0i			0.299972i
$512$		−	54537.0i		−	0.208042i
$513$	−36632.6			−0.139198
$514$			157574.i			0.596427i
$515$		−	730352.i		−	2.75371i
$516$			101038.i			0.379479i
$517$	−92201.9			−0.344952
$518$	6742.89			0.0251297
$519$		−	164328.i		−	0.610065i
$520$	693953.			2.56639
$521$	−149931.			−0.552351			−0.276175	−	0.961107i	$-0.589067\pi$
$521$	−149931.			−0.552351			−0.276175	+	0.961107i	$0.589067\pi$
$522$			25525.4i			0.0936766i
$523$	−90744.5			−0.331754			−0.165877	−	0.986146i	$-0.553046\pi$
$523$	−90744.5			−0.331754			−0.165877	+	0.986146i	$0.553046\pi$
$524$		−	134050.i		−	0.488206i
$525$	42792.1			0.155255
$526$		−	16159.4i		−	0.0584055i
$527$			884667.i			3.18536i
$528$			8133.83i			0.0291761i
$529$	243655.			0.870691
$530$		−	243278.i		−	0.866067i
$531$	89394.4	−	29020.7i	0.317045	−	0.102925i
$532$	−26318.5			−0.0929904
$533$			141621.i			0.498511i
$534$	−164315.			−0.576228
$535$	−597864.			−2.08879
$536$	−325310.			−1.13232
$537$			75864.1i			0.263080i
$538$	45517.8			0.157259
$539$		−	135163.i		−	0.465242i
$540$	−57575.8			−0.197448
$541$			293061.i			1.00130i	0.865650	+	0.500650i	$0.166906\pi$
$541$			293061.i			1.00130i	−0.865650	+	0.500650i	$0.833094\pi$
$542$			60100.2i			0.204587i
$543$	−251548.			−0.853143
$544$			490607.i			1.65781i
$545$		−	401689.i		−	1.35237i
$546$	−33267.7			−0.111593
$547$	−264316.			−0.883384			−0.441692	−	0.897167i	$-0.645622\pi$
$547$	−264316.			−0.883384			−0.441692	+	0.897167i	$0.645622\pi$
$548$	−223771.			−0.745147
$549$		−	106816.i		−	0.354399i
$550$	117713.			0.389135
$551$	106440.			0.350591
$552$	−61025.6			−0.200278
$553$	79604.2			0.260307
$554$			243187.i			0.792355i
$555$		−	61511.4i		−	0.199696i
$556$	251786.			0.814484
$557$	349573.			1.12675			0.563375	−	0.826202i	$-0.309503\pi$
$557$	349573.			1.12675			0.563375	+	0.826202i	$0.309503\pi$
$558$	−118545.			−0.380728
$559$	532574.			1.70434
$560$	−9812.97			−0.0312914
$561$		−	142016.i		−	0.451243i
$562$			34478.7i			0.109164i
$563$		−	87644.3i		−	0.276508i	−0.990397	−	0.138254i	$-0.955851\pi$
$563$		−	87644.3i		−	0.276508i	0.990397	−	0.138254i	$-0.0441490\pi$
$564$			87005.1i			0.273518i
$565$			141239.i			0.442443i
$566$	−169127.			−0.527935
$567$	6917.95			0.0215185
$568$		−	453970.i		−	1.40712i
$569$			81258.9i			0.250984i	0.992095	+	0.125492i	$0.0400510\pi$
$569$			81258.9i			0.250984i	−0.992095	+	0.125492i	$0.959949\pi$
$570$		−	121572.i		−	0.374184i
$571$		−	490531.i		−	1.50451i	−0.658875	−	0.752253i	$-0.728967\pi$
$571$		−	490531.i		−	1.50451i	0.658875	−	0.752253i	$-0.271033\pi$
$572$	180726.			0.552369
$573$		−	62283.3i		−	0.189698i
$574$			10713.7i			0.0325175i
$575$		−	165083.i		−	0.499305i
$576$	−54179.1			−0.163300
$577$	458744.			1.37790			0.688951	−	0.724807i	$-0.258071\pi$
$577$	458744.			1.37790			0.688951	+	0.724807i	$0.258071\pi$
$578$		−	312712.i		−	0.936029i
$579$	101309.			0.302197
$580$	167292.			0.497301
$581$			77923.2i			0.230842i
$582$	−64451.6			−0.190278
$583$		−	158796.i		−	0.467199i
$584$	509605.			1.49420
$585$			303482.i			0.886791i
$586$		−	161691.i		−	0.470858i
$587$			346247.i			1.00487i	0.864615	+	0.502435i	$0.167562\pi$
$587$			346247.i			1.00487i	−0.864615	+	0.502435i	$0.832438\pi$
$588$	−127544.			−0.368898
$589$			494327.i			1.42490i
$590$	96311.0	+	296673.i	0.276676	+	0.852263i
$591$	−221592.			−0.634423
$592$			8200.04i			0.0233977i
$593$	507006.			1.44180			0.720898	−	0.693041i	$-0.243729\pi$
$593$	507006.			1.44180			0.720898	+	0.693041i	$0.243729\pi$
$594$	19030.0			0.0539345
$595$	171333.			0.483958
$596$		−	56787.6i		−	0.159868i
$597$	55439.6			0.155550
$598$			128340.i			0.358888i
$599$	571461.			1.59270			0.796348	−	0.604838i	$-0.206762\pi$
$599$	571461.			1.59270			0.796348	+	0.604838i	$0.206762\pi$
$600$		−	278404.i		−	0.773343i
$601$		−	393314.i		−	1.08891i	−0.838792	−	0.544453i	$-0.816737\pi$
$601$		−	393314.i		−	1.08891i	0.838792	−	0.544453i	$-0.183263\pi$
$602$	40289.5			0.111173
$603$		−	142266.i		−	0.391261i
$604$		−	292427.i		−	0.801573i
$605$	−433514.			−1.18438
$606$	−139533.			−0.379954
$607$	−130471.			−0.354108			−0.177054	−	0.984201i	$-0.556657\pi$
$607$	−130471.			−0.354108			−0.177054	+	0.984201i	$0.556657\pi$
$608$			274137.i			0.741585i
$609$	−20100.8			−0.0541974
$610$	354490.			0.952675
$611$	458604.			1.22844
$612$	−134011.			−0.357798
$613$		−	512655.i		−	1.36428i	−0.731220	−	0.682142i	$-0.761049\pi$
$613$		−	512655.i		−	1.36428i	0.731220	−	0.682142i	$-0.238951\pi$
$614$			325673.i			0.863864i
$615$	97735.2			0.258405
$616$	34267.1			0.0903059
$617$	120180.			0.315692			0.157846	−	0.987464i	$-0.449545\pi$
$617$	120180.			0.315692			0.157846	+	0.987464i	$0.449545\pi$
$618$	227791.			0.596430
$619$	460899.			1.20289			0.601443	−	0.798916i	$-0.294593\pi$
$619$	460899.			1.20289			0.601443	+	0.798916i	$0.294593\pi$
$620$			776938.i			2.02117i
$621$		−	26687.9i		−	0.0692041i
$622$			248133.i			0.641364i
$623$		−	129395.i		−	0.333382i
$624$		−	40456.9i		−	0.103902i
$625$	−179896.			−0.460533
$626$	258370.			0.659316
$627$		−	79354.3i		−	0.201853i
$628$			314547.i			0.797566i
$629$		−	143172.i		−	0.361873i
$630$			22958.6i			0.0578447i
$631$	218188.			0.547989			0.273995	−	0.961731i	$-0.411655\pi$
$631$	218188.			0.547989			0.273995	+	0.961731i	$0.411655\pi$
$632$		−	517901.i		−	1.29662i
$633$			79274.3i			0.197845i
$634$		−	323843.i		−	0.805668i
$635$	−346198.			−0.858572
$636$	−149845.			−0.370450
$637$			672287.i			1.65682i
$638$	−55293.6			−0.135842
$639$	198532.			0.486215
$640$			469235.i			1.14559i
$641$	−761314.			−1.85288			−0.926441	−	0.376440i	$-0.877148\pi$
$641$	−761314.			−1.85288			−0.926441	+	0.376440i	$0.877148\pi$
$642$		−	186469.i		−	0.452415i
$643$	−131827.			−0.318848			−0.159424	−	0.987210i	$-0.550964\pi$
$643$	−131827.			−0.318848			−0.159424	+	0.987210i	$0.550964\pi$
$644$		−	19173.8i		−	0.0462314i
$645$		−	367537.i		−	0.883450i
$646$		−	282967.i		−	0.678065i
$647$	−673769.			−1.60954			−0.804771	−	0.593585i	$-0.797712\pi$
$647$	−673769.			−1.60954			−0.804771	+	0.593585i	$0.797712\pi$
$648$		−	45007.9i		−	0.107186i
$649$	62865.4	+	193648.i	0.149253	+	0.459752i
$650$	−585495.			−1.38579
$651$		−	93352.1i		−	0.220273i
$652$	5559.45			0.0130779
$653$	−310717.			−0.728684			−0.364342	−	0.931265i	$-0.618706\pi$
$653$	−310717.			−0.728684			−0.364342	+	0.931265i	$0.618706\pi$
$654$	125284.			0.292913
$655$			487619.i			1.13657i
$656$	−13029.0			−0.0302763
$657$			222862.i			0.516305i
$658$	34693.6			0.0801306
$659$			459674.i			1.05847i	0.848475	+	0.529235i	$0.177521\pi$
$659$			459674.i			1.05847i	−0.848475	+	0.529235i	$0.822479\pi$
$660$		−	124722.i		−	0.286322i
$661$	−70113.4			−0.160472			−0.0802358	−	0.996776i	$-0.525567\pi$
$661$	−70113.4			−0.160472			−0.0802358	+	0.996776i	$0.525567\pi$
$662$		−	126522.i		−	0.288702i
$663$			706373.i			1.60697i
$664$	506965.			1.14985
$665$	95736.2			0.216488
$666$	19184.9			0.0432526
$667$			77544.5i			0.174301i
$668$	89016.6			0.199489
$669$	88661.2			0.198099
$670$	472137.			1.05177
$671$	231388.			0.513919
$672$		−	51770.0i		−	0.114641i
$673$		−	832203.i		−	1.83738i	−0.394979	−	0.918690i	$-0.629248\pi$
$673$		−	832203.i		−	1.83738i	0.394979	−	0.918690i	$-0.370752\pi$
$674$	−112817.			−0.248344
$675$	121752.			0.267221
$676$	−595552.			−1.30325
$677$	372770.			0.813323			0.406661	−	0.913579i	$-0.366693\pi$
$677$	372770.			0.813323			0.406661	+	0.913579i	$0.366693\pi$
$678$	−44051.2			−0.0958294
$679$		−	50754.6i		−	0.110087i
$680$		−	1.11469e6i		−	2.41065i
$681$		−	81674.4i		−	0.176113i
$682$		−	256795.i		−	0.552099i
$683$			231892.i			0.497102i	0.968619	+	0.248551i	$0.0799543\pi$
$683$			231892.i			0.497102i	−0.968619	+	0.248551i	$0.920046\pi$
$684$	−74881.7			−0.160053
$685$	813988.			1.73475
$686$			103700.i			0.220358i
$687$		−	55729.4i		−	0.118079i
$688$			48996.1i			0.103511i
$689$			789836.i			1.66379i
$690$	88569.2			0.186031
$691$			177978.i			0.372744i	0.982479	+	0.186372i	$0.0596730\pi$
$691$			177978.i			0.372744i	−0.982479	+	0.186372i	$0.940327\pi$
$692$		−	335906.i		−	0.701465i
$693$			14985.8i			0.0312043i
$694$	−267074.			−0.554514
$695$	−915898.			−1.89617
$696$			130775.i			0.269964i
$697$	227485.			0.468259
$698$	106110.			0.217793
$699$			181622.i			0.371719i
$700$	87472.5			0.178515
$701$			316317.i			0.643705i	0.946790	+	0.321852i	$0.104305\pi$
$701$			316317.i			0.643705i	−0.946790	+	0.321852i	$0.895695\pi$
$702$	−94653.6			−0.192071
$703$		−	80000.3i		−	0.161875i
$704$		−	117364.i		−	0.236804i
$705$		−	316490.i		−	0.636768i
$706$	−274138.			−0.549997
$707$		−	109880.i		−	0.219826i
$708$	182733.	−	59322.0i	0.364545	−	0.118345i
$709$	288012.			0.572952			0.286476	−	0.958087i	$-0.407516\pi$
$709$	288012.			0.572952			0.286476	+	0.958087i	$0.407516\pi$
$710$			658867.i			1.30702i
$711$	226490.			0.448034
$712$	−841838.			−1.66061
$713$	−360132.			−0.708407
$714$			53437.5i			0.104821i
$715$	−657409.			−1.28595
$716$			155076.i			0.302495i
$717$	226980.			0.441518
$718$		−	488966.i		−	0.948483i
$719$		−	622044.i		−	1.20327i	−0.798771	−	0.601636i	$-0.794516\pi$
$719$		−	622044.i		−	1.20327i	0.798771	−	0.601636i	$-0.205484\pi$
$720$	−27919.9			−0.0538579
$721$			179381.i			0.345070i
$722$			144118.i			0.276468i
$723$	−193735.			−0.370621
$724$	−514197.			−0.980962
$725$	−353764.			−0.673035
$726$		−	135210.i		−	0.256528i
$727$	−323221.			−0.611548			−0.305774	−	0.952104i	$-0.598915\pi$
$727$	−323221.			−0.611548			−0.305774	+	0.952104i	$0.598915\pi$
$728$	−170441.			−0.321597
$729$	19683.0			0.0370370
$730$	−739612.			−1.38790
$731$		−	855466.i		−	1.60091i
$732$		−	218346.i		−	0.407495i
$733$	−314696.			−0.585711			−0.292856	−	0.956157i	$-0.594606\pi$
$733$	−314696.			−0.585711			−0.292856	+	0.956157i	$0.594606\pi$
$734$	−69049.0			−0.128164
$735$	463955.			0.858819
$736$	−199717.			−0.368689
$737$	308180.			0.567373
$738$			30482.8i			0.0559683i
$739$			703004.i			1.28727i	0.765334	+	0.643634i	$0.222574\pi$
$739$			703004.i			1.28727i	−0.765334	+	0.643634i	$0.777426\pi$
$740$		−	125737.i		−	0.229615i
$741$			394701.i			0.718840i
$742$			59751.5i			0.108528i
$743$	−128013.			−0.231886			−0.115943	−	0.993256i	$-0.536989\pi$
$743$	−128013.			−0.231886			−0.115943	+	0.993256i	$0.536989\pi$
$744$	−607345.			−1.09721
$745$			206571.i			0.372182i
$746$			60175.5i			0.108129i
$747$			221708.i			0.397319i
$748$		−	290298.i		−	0.518849i
$749$	146841.			0.261749
$750$			113059.i			0.200994i
$751$			905740.i			1.60592i	0.596034	+	0.802959i	$0.296742\pi$
$751$			905740.i			1.60592i	−0.596034	+	0.802959i	$0.703258\pi$
$752$			42191.0i			0.0746078i
$753$	−222682.			−0.392730
$754$	275026.			0.483760
$755$			1.06373e6i			1.86611i
$756$	14141.2			0.0247424
$757$	302200.			0.527355			0.263678	−	0.964611i	$-0.415065\pi$
$757$	302200.			0.527355			0.263678	+	0.964611i	$0.415065\pi$
$758$			622628.i			1.08365i
$759$	57812.0			0.100354
$760$		−	622856.i		−	1.07835i
$761$	844330.			1.45795			0.728976	−	0.684539i	$-0.239997\pi$
$761$	844330.			1.45795			0.728976	+	0.684539i	$0.239997\pi$
$762$		−	107976.i		−	0.185960i
$763$			98658.6i			0.169467i
$764$		−	127315.i		−	0.218118i
$765$	487479.			0.832976
$766$		−	380867.i		−	0.649107i
$767$	−312687.	−	963188.i	−0.531519	−	1.63727i
$768$	−313179.			−0.530970
$769$		−	232294.i		−	0.392813i	−0.980523	−	0.196406i	$-0.937073\pi$
$769$		−	232294.i		−	0.392813i	0.980523	−	0.196406i	$-0.0629272\pi$
$770$	−49733.4			−0.0838815
$771$	−353052.			−0.593922
$772$	207088.			0.347472
$773$		−	340313.i		−	0.569534i	−0.958597	−	0.284767i	$-0.908084\pi$
$773$		−	340313.i		−	0.569534i	0.958597	−	0.284767i	$-0.0919162\pi$
$774$	114632.			0.191348
$775$		−	1.64295e6i		−	2.73540i
$776$	−330207.			−0.548356
$777$			15107.8i			0.0250241i
$778$			537545.i			0.888087i
$779$	127112.			0.209465
$780$			620355.i			1.01965i
$781$			430064.i			0.705068i
$782$	206150.			0.337109
$783$	−57190.9			−0.0932833
$784$	−61849.5			−0.100625
$785$		−	1.14420e6i		−	1.85679i
$786$	−152084.			−0.246173
$787$	−18969.9			−0.0306278			−0.0153139	−	0.999883i	$-0.504875\pi$
$787$	−18969.9			−0.0306278			−0.0153139	+	0.999883i	$0.504875\pi$
$788$	−452962.			−0.729473
$789$	36206.0			0.0581603
$790$			751653.i			1.20438i
$791$		−	34689.6i		−	0.0554429i
$792$	97497.0			0.155432
$793$	−1.15090e6			−1.83017
$794$	−154359.			−0.244846
$795$	545078.			0.862431
$796$	113326.			0.178855
$797$			845431.i			1.33095i	0.746421	+	0.665474i	$0.231771\pi$
$797$			845431.i			1.33095i	−0.746421	+	0.665474i	$0.768229\pi$
$798$			29859.4i			0.0468894i
$799$		−	736649.i		−	1.15390i
$800$		−	911125.i		−	1.42363i
$801$		−	368156.i		−	0.573808i
$802$	−52783.2			−0.0820629
$803$	−482769.			−0.748701
$804$		−	290810.i		−	0.449880i
$805$			69746.7i			0.107630i
$806$			1.27727e6i			1.96614i
$807$			101985.i			0.156599i
$808$	−714872.			−1.09498
$809$		−	146619.i		−	0.224023i	−0.993707	−	0.112012i	$-0.964271\pi$
$809$		−	146619.i		−	0.224023i	0.993707	−	0.112012i	$-0.0357294\pi$
$810$			65321.9i			0.0995609i
$811$		−	807106.i		−	1.22712i	−0.789646	−	0.613562i	$-0.789736\pi$
$811$		−	807106.i		−	1.22712i	0.789646	−	0.613562i	$-0.210264\pi$
$812$	−41088.6			−0.0623173
$813$	−134658.			−0.203727
$814$			41558.8i			0.0627212i
$815$	−20223.0			−0.0304461
$816$	−64985.4			−0.0975968
$817$		−	478010.i		−	0.716132i
$818$	−723999.			−1.08201
$819$		−	74538.1i		−	0.111125i
$820$	199783.			0.297119
$821$		−	1.32897e6i		−	1.97164i	−0.167797	−	0.985822i	$-0.553665\pi$
$821$		−	1.32897e6i		−	1.97164i	0.167797	−	0.985822i	$-0.446335\pi$
$822$			253876.i			0.375732i
$823$			49834.3i			0.0735748i	0.999323	+	0.0367874i	$0.0117124\pi$
$823$			49834.3i			0.0735748i	−0.999323	+	0.0367874i	$0.988288\pi$
$824$	1.16705e6			1.71884
$825$			263743.i			0.387501i
$826$	−23654.9	−	72865.7i	−0.0346706	−	0.106798i
$827$	40785.1			0.0596335			0.0298167	−	0.999555i	$-0.490508\pi$
$827$	40785.1			0.0596335			0.0298167	+	0.999555i	$0.490508\pi$
$828$		−	54553.5i		−	0.0795724i
$829$	333941.			0.485916			0.242958	−	0.970037i	$-0.421882\pi$
$829$	333941.			0.485916			0.242958	+	0.970037i	$0.421882\pi$
$830$	−735780.			−1.06805
$831$	−544872.			−0.789028
$832$			583758.i			0.843308i
$833$	1.07988e6			1.55628
$834$		−	285661.i		−	0.410695i
$835$	−323807.			−0.464422
$836$		−	162210.i		−	0.232095i
$837$		−	265606.i		−	0.379129i
$838$	−729899.			−1.03938
$839$		−	183481.i		−	0.260656i	−0.991471	−	0.130328i	$-0.958397\pi$
$839$		−	183481.i		−	0.260656i	0.991471	−	0.130328i	$-0.0416030\pi$
$840$			117624.i			0.166701i
$841$	−541107.			−0.765052
$842$	58498.9			0.0825133
$843$	−77251.4			−0.108705
$844$			162047.i			0.227486i
$845$	2.16638e6			3.03404
$846$	98710.6			0.137919
$847$	106475.			0.148416
$848$	−72663.9			−0.101048
$849$		−	378938.i		−	0.525718i
$850$			940473.i			1.30169i
$851$	58282.6			0.0804785
$852$	405824.			0.559061
$853$	−179610.			−0.246850			−0.123425	−	0.992354i	$-0.539388\pi$
$853$	−179610.			−0.246850			−0.123425	+	0.992354i	$0.539388\pi$
$854$	−87066.2			−0.119381
$855$	272389.			0.372613
$856$		−	955342.i		−	1.30380i
$857$			787527.i			1.07227i	0.844133	+	0.536134i	$0.180116\pi$
$857$			787527.i			1.07227i	−0.844133	+	0.536134i	$0.819884\pi$
$858$		−	205041.i		−	0.278526i
$859$		−	345592.i		−	0.468357i	−0.972194	−	0.234179i	$-0.924760\pi$
$859$		−	345592.i		−	0.468357i	0.972194	−	0.234179i	$-0.0752401\pi$
$860$		−	751293.i		−	1.01581i
$861$	−24004.7			−0.0323810
$862$	−135485.			−0.182337
$863$			1.32129e6i			1.77410i	0.461677	+	0.887048i	$0.347248\pi$
$863$			1.32129e6i			1.77410i	−0.461677	+	0.887048i	$0.652752\pi$
$864$		−	147296.i		−	0.197317i
$865$			1.22189e6i			1.63306i
$866$		−	138413.i		−	0.184562i
$867$	700649.			0.932099
$868$		−	190824.i		−	0.253275i
$869$			490629.i			0.649701i
$870$		−	189799.i		−	0.250759i
$871$	−1.53286e6			−2.02053
$872$	641869.			0.844138
$873$		−	144407.i		−	0.189479i
$874$	115191.			0.150798
$875$	−89032.2			−0.116287
$876$			455559.i			0.593658i
$877$	1.10906e6			1.44197			0.720984	−	0.692951i	$-0.243690\pi$
$877$	1.10906e6			1.44197			0.720984	+	0.692951i	$0.243690\pi$
$878$		−	463837.i		−	0.601695i
$879$	362277.			0.468881
$880$		−	60480.8i		−	0.0781002i
$881$			17042.2i			0.0219571i	0.999940	+	0.0109786i	$0.00349465\pi$
$881$			17042.2i			0.0219571i	−0.999940	+	0.0109786i	$0.996505\pi$
$882$			144704.i			0.186013i
$883$	−554450.			−0.711117			−0.355559	−	0.934654i	$-0.615709\pi$
$883$	−554450.			−0.711117			−0.355559	+	0.934654i	$0.615709\pi$
$884$			1.44392e6i			1.84772i
$885$	−664711.	+	215790.i	−0.848684	+	0.275514i
$886$	693561.			0.883521
$887$		−	563820.i		−	0.716627i	−0.933601	−	0.358313i	$-0.883352\pi$
$887$		−	563820.i		−	0.716627i	0.933601	−	0.358313i	$-0.116648\pi$
$888$	98290.7			0.124648
$889$	85029.5			0.107589
$890$	1.22180e6			1.54248
$891$			42637.7i			0.0537080i
$892$	181235.			0.227778
$893$		−	411619.i		−	0.516170i
$894$	−64427.7			−0.0806116
$895$		−	564104.i		−	0.704227i
$896$		−	115249.i		−	0.143555i
$897$	−287552.			−0.357381
$898$		−	715865.i		−	0.887726i
$899$			771745.i			0.954893i
$900$	248877.			0.307256
$901$	1.26870e6			1.56282
$902$	−66032.6			−0.0811606
$903$			90270.7i			0.110706i
$904$	−225689.			−0.276168
$905$	1.87044e6			2.28374
$906$	−331769.			−0.404184
$907$	−220065.			−0.267508			−0.133754	−	0.991015i	$-0.542703\pi$
$907$	−220065.			−0.267508			−0.133754	+	0.991015i	$0.542703\pi$
$908$		−	166953.i		−	0.202499i
$909$		−	312631.i		−	0.378359i
$910$	247369.			0.298719
$911$	631121.			0.760459			0.380230	−	0.924892i	$-0.375845\pi$
$911$	631121.			0.760459			0.380230	+	0.924892i	$0.375845\pi$
$912$	−36312.0			−0.0436577
$913$	−480268.			−0.576159
$914$	−521376.			−0.624107
$915$			794254.i			0.948675i
$916$		−	113918.i		−	0.135769i
$917$		−	119764.i		−	0.142425i
$918$			152041.i			0.180416i
$919$			655325.i			0.775936i	0.921673	+	0.387968i	$0.126823\pi$
$919$			655325.i			0.775936i	−0.921673	+	0.387968i	$0.873177\pi$
$920$	453769.			0.536116
$921$	−729688.			−0.860237
$922$		−	432282.i		−	0.508517i
$923$		−	2.13910e6i		−	2.51089i
$924$			30632.9i			0.0358793i
$925$			265890.i			0.310755i
$926$	104392.			0.121743
$927$			510378.i			0.593926i
$928$			427984.i			0.496972i
$929$			1.56736e6i			1.81609i	0.418874	+	0.908044i	$0.362425\pi$
$929$			1.56736e6i			1.81609i	−0.418874	+	0.908044i	$0.637575\pi$
$930$	881466.			1.01915
$931$	603409.			0.696166
$932$			371259.i			0.427410i
$933$	−555956.			−0.638671
$934$	−563757.			−0.646247
$935$			1.05599e6i			1.20791i
$936$	−484941.			−0.553525
$937$			287024.i			0.326918i	0.986550	+	0.163459i	$0.0522651\pi$
$937$			287024.i			0.326918i	−0.986550	+	0.163459i	$0.947735\pi$
$938$	−115961.			−0.131798
$939$			578891.i			0.656547i
$940$		−	646945.i		−	0.732170i
$941$		−	865888.i		−	0.977873i	−0.872319	−	0.488937i	$-0.837385\pi$
$941$		−	865888.i		−	0.977873i	0.872319	−	0.488937i	$-0.162615\pi$
$942$	356866.			0.402164
$943$			92604.9i			0.104138i
$944$	88612.1	−	28766.8i	0.0994372	−	0.0322810i
$945$	−51439.9			−0.0576018
$946$			248318.i			0.277477i
$947$	−1.35112e6			−1.50659			−0.753294	−	0.657684i	$-0.771536\pi$
$947$	−1.35112e6			−1.50659			−0.753294	+	0.657684i	$0.771536\pi$
$948$	462975.			0.515159
$949$	2.40125e6			2.66628
$950$			525510.i			0.582283i
$951$	725587.			0.802285
$952$			273778.i			0.302082i
$953$	−851986.			−0.938095			−0.469047	−	0.883173i	$-0.655403\pi$
$953$	−851986.			−0.938095			−0.469047	+	0.883173i	$0.655403\pi$
$954$			170005.i			0.186795i
$955$			463120.i			0.507793i
$956$	463975.			0.507667
$957$		−	123888.i		−	0.135272i
$958$		−	268009.i		−	0.292024i
$959$	−199923.			−0.217383
$960$	402860.			0.437132
$961$	−2.66062e6			−2.88095
$962$		−	206710.i		−	0.223363i
$963$	417794.			0.450515
$964$	−396018.			−0.426148
$965$	−753302.			−0.808937
$966$	−21753.4			−0.0233117
$967$		−	1.19167e6i		−	1.27439i	−0.770703	−	0.637194i	$-0.780095\pi$
$967$		−	1.19167e6i		−	1.27439i	0.770703	−	0.637194i	$-0.219905\pi$
$968$		−	692723.i		−	0.739280i
$969$	634003.			0.675218
$970$	479244.			0.509346
$971$	336441.			0.356837			0.178419	−	0.983955i	$-0.442902\pi$
$971$	336441.			0.356837			0.178419	+	0.983955i	$0.442902\pi$
$972$	40234.5			0.0425860
$973$	224953.			0.237611
$974$		−	350302.i		−	0.369254i
$975$		−	1.31183e6i		−	1.37997i
$976$		−	105881.i		−	0.111153i
$977$			1.10642e6i			1.15912i	0.814929	+	0.579561i	$0.196776\pi$
$977$			1.10642e6i			1.15912i	−0.814929	+	0.579561i	$0.803224\pi$
$978$		−	6307.41i		−	0.00659437i
$979$	797507.			0.832088
$980$	948384.			0.987488
$981$			280704.i			0.291683i
$982$			47416.5i			0.0491707i
$983$		−	1.26199e6i		−	1.30602i	−0.757351	−	0.653008i	$-0.773507\pi$
$983$		−	1.26199e6i		−	1.30602i	0.757351	−	0.653008i	$-0.226493\pi$
$984$			156173.i			0.161294i
$985$	1.64769e6			1.69826
$986$		−	441770.i		−	0.454404i
$987$			77732.9i			0.0797941i
$988$			806820.i			0.826537i
$989$	348245.			0.356034
$990$	−141502.			−0.144375
$991$			1.38128e6i			1.40649i	0.710950	+	0.703243i	$0.248265\pi$
$991$			1.38128e6i			1.40649i	−0.710950	+	0.703243i	$0.751735\pi$
$992$	−1.98764e6			−2.01983
$993$	283479.			0.287490
$994$		−	161824.i		−	0.163784i
$995$	−412233.			−0.416386
$996$			453199.i			0.456846i
$997$	−1.91430e6			−1.92584			−0.962918	−	0.269795i	$-0.913044\pi$
$997$	−1.91430e6			−1.92584			−0.962918	+	0.269795i	$0.913044\pi$
$998$			137722.i			0.138275i
$999$			42984.8i			0.0430709i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.5.c.a.58.16	✓	40
3.2	odd	2		531.5.c.d.235.25		40
59.58	odd	2	inner	177.5.c.a.58.25	yes	40
177.176	even	2		531.5.c.d.235.16		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.5.c.a.58.16	✓	40	1.1	even	1	trivial
177.5.c.a.58.25	yes	40	59.58	odd	2	inner
531.5.c.d.235.16		40	177.176	even	2
531.5.c.d.235.25		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)

\(f(q)\)	\(=\)	\(q-2.31914i q^{2} +5.19615 q^{3} +10.6216 q^{4} -38.6371 q^{5} -12.0506i q^{6} +9.48964 q^{7} -61.7392i q^{8} +27.0000 q^{9} +O(q^{10})\)	\(q-2.31914i q^{2} +5.19615 q^{3} +10.6216 q^{4} -38.6371 q^{5} -12.0506i q^{6} +9.48964 q^{7} -61.7392i q^{8} +27.0000 q^{9} +89.6048i q^{10} +58.4880i q^{11} +55.1914 q^{12} -290.914i q^{13} -22.0078i q^{14} -200.764 q^{15} +26.7637 q^{16} -467.291 q^{17} -62.6168i q^{18} -261.109 q^{19} -410.387 q^{20} +49.3096 q^{21} +135.642 q^{22} -190.226i q^{23} -320.806i q^{24} +867.825 q^{25} -674.670 q^{26} +140.296 q^{27} +100.795 q^{28} -407.645 q^{29} +465.600i q^{30} -1893.18i q^{31} -1049.90i q^{32} +303.913i q^{33} +1083.71i q^{34} -366.652 q^{35} +286.783 q^{36} +306.386i q^{37} +605.549i q^{38} -1511.63i q^{39} +2385.42i q^{40} -486.816 q^{41} -114.356i q^{42} +1830.69i q^{43} +621.236i q^{44} -1043.20 q^{45} -441.160 q^{46} +1576.43i q^{47} +139.068 q^{48} -2310.95 q^{49} -2012.61i q^{50} -2428.12 q^{51} -3089.97i q^{52} -2715.01 q^{53} -325.366i q^{54} -2259.81i q^{55} -585.883i q^{56} -1356.76 q^{57} +945.384i q^{58} +(3310.90 - 1074.84i) q^{59} -2132.44 q^{60} -3956.15i q^{61} -4390.55 q^{62} +256.220 q^{63} -2006.63 q^{64} +11240.1i q^{65} +704.816 q^{66} -5269.11i q^{67} -4963.37 q^{68} -988.442i q^{69} +850.317i q^{70} +7353.03 q^{71} -1666.96i q^{72} +8254.16i q^{73} +710.553 q^{74} +4509.35 q^{75} -2773.40 q^{76} +555.030i q^{77} -3505.69 q^{78} +8388.54 q^{79} -1034.07 q^{80} +729.000 q^{81} +1128.99i q^{82} +8211.40i q^{83} +523.747 q^{84} +18054.8 q^{85} +4245.63 q^{86} -2118.18 q^{87} +3611.00 q^{88} -13635.4i q^{89} +2419.33i q^{90} -2760.67i q^{91} -2020.50i q^{92} -9837.26i q^{93} +3655.95 q^{94} +10088.5 q^{95} -5455.42i q^{96} -5348.42i q^{97} +5359.41i q^{98} +1579.18i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 320 q^{4} + 80 q^{7} + 1080 q^{9}+O(q^{10}) \) 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

L-functions

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