LMFDB - Embedded newform 177.5.c.a.58.27

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			58.27
Character	$\chi$	$=$	177.58
Dual form			177.5.c.a.58.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.43119i q^{2} +5.19615 q^{3} +10.0893 q^{4} +25.9319 q^{5} +12.6328i q^{6} -17.4595 q^{7} +63.4281i q^{8} +27.0000 q^{9} +O(q^{10})$	\(q+2.43119i q^{2} +5.19615 q^{3} +10.0893 q^{4} +25.9319 q^{5} +12.6328i q^{6} -17.4595 q^{7} +63.4281i q^{8} +27.0000 q^{9} +63.0454i q^{10} +44.7152i q^{11} +52.4255 q^{12} +157.364i q^{13} -42.4475i q^{14} +134.746 q^{15} +7.22282 q^{16} -328.532 q^{17} +65.6422i q^{18} +678.379 q^{19} +261.635 q^{20} -90.7223 q^{21} -108.711 q^{22} -424.331i q^{23} +329.582i q^{24} +47.4624 q^{25} -382.582 q^{26} +140.296 q^{27} -176.154 q^{28} +870.549 q^{29} +327.594i q^{30} -1380.41i q^{31} +1032.41i q^{32} +232.347i q^{33} -798.725i q^{34} -452.758 q^{35} +272.411 q^{36} +2568.09i q^{37} +1649.27i q^{38} +817.687i q^{39} +1644.81i q^{40} +503.324 q^{41} -220.563i q^{42} -2003.40i q^{43} +451.145i q^{44} +700.161 q^{45} +1031.63 q^{46} +2422.01i q^{47} +37.5309 q^{48} -2096.17 q^{49} +115.390i q^{50} -1707.10 q^{51} +1587.69i q^{52} -1526.26 q^{53} +341.087i q^{54} +1159.55i q^{55} -1107.42i q^{56} +3524.96 q^{57} +2116.47i q^{58} +(-1105.23 + 3300.88i) q^{59} +1359.49 q^{60} -5708.35i q^{61} +3356.05 q^{62} -471.407 q^{63} -2394.42 q^{64} +4080.74i q^{65} -564.880 q^{66} -4773.92i q^{67} -3314.66 q^{68} -2204.89i q^{69} -1100.74i q^{70} -2795.15 q^{71} +1712.56i q^{72} -4719.76i q^{73} -6243.53 q^{74} +246.622 q^{75} +6844.37 q^{76} -780.705i q^{77} -1987.96 q^{78} +8669.75 q^{79} +187.301 q^{80} +729.000 q^{81} +1223.68i q^{82} +7458.93i q^{83} -915.325 q^{84} -8519.45 q^{85} +4870.65 q^{86} +4523.51 q^{87} -2836.20 q^{88} -4001.99i q^{89} +1702.23i q^{90} -2747.50i q^{91} -4281.21i q^{92} -7172.83i q^{93} -5888.37 q^{94} +17591.6 q^{95} +5364.56i q^{96} -5912.30i q^{97} -5096.18i q^{98} +1207.31i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$			2.43119i			0.607798i	0.952704	+	0.303899i	$0.0982886\pi$
$2$			2.43119i			0.607798i	−0.952704	+	0.303899i	$0.901711\pi$
$3$	5.19615			0.577350
$3$	5.19615			0.577350
$4$	10.0893			0.630581
$5$	25.9319			1.03728			0.518638	−	0.854994i	$-0.326439\pi$
$5$	25.9319			1.03728			0.518638	+	0.854994i	$0.326439\pi$
$6$			12.6328i			0.350912i
$7$	−17.4595			−0.356317			−0.178158	−	0.984002i	$-0.557014\pi$
$7$	−17.4595			−0.356317			−0.178158	+	0.984002i	$0.557014\pi$
$8$			63.4281i			0.991064i
$9$	27.0000			0.333333
$10$			63.0454i			0.630454i
$11$			44.7152i			0.369547i	0.982781	+	0.184773i	$0.0591551\pi$
$11$			44.7152i			0.369547i	−0.982781	+	0.184773i	$0.940845\pi$
$12$	52.4255			0.364066
$13$			157.364i			0.931148i	0.885009	+	0.465574i	$0.154152\pi$
$13$			157.364i			0.931148i	−0.885009	+	0.465574i	$0.845848\pi$
$14$		−	42.4475i		−	0.216569i
$15$	134.746			0.598871
$16$	7.22282			0.0282141
$17$	−328.532			−1.13679			−0.568395	−	0.822756i	$-0.692435\pi$
$17$	−328.532			−1.13679			−0.568395	+	0.822756i	$0.692435\pi$
$18$			65.6422i			0.202599i
$19$	678.379			1.87917			0.939583	−	0.342322i	$-0.111213\pi$
$19$	678.379			1.87917			0.939583	+	0.342322i	$0.111213\pi$
$20$	261.635			0.654086
$21$	−90.7223			−0.205720
$22$	−108.711			−0.224610
$23$		−	424.331i		−	0.802139i	−0.916048	−	0.401069i	$-0.868639\pi$
$23$		−	424.331i		−	0.802139i	0.916048	−	0.401069i	$-0.131361\pi$
$24$			329.582i			0.572191i
$25$	47.4624			0.0759398
$26$	−382.582			−0.565950
$27$	140.296			0.192450
$28$	−176.154			−0.224687
$29$	870.549			1.03514			0.517568	−	0.855642i	$-0.326837\pi$
$29$	870.549			1.03514			0.517568	+	0.855642i	$0.326837\pi$
$30$			327.594i			0.363993i
$31$		−	1380.41i		−	1.43643i	−0.695820	−	0.718216i	$-0.744959\pi$
$31$		−	1380.41i		−	1.43643i	0.695820	−	0.718216i	$-0.255041\pi$
$32$			1032.41i			1.00821i
$33$			232.347i			0.213358i
$34$		−	798.725i		−	0.690938i
$35$	−452.758			−0.369599
$36$	272.411			0.210194
$37$			2568.09i			1.87589i	0.346784	+	0.937945i	$0.387274\pi$
$37$			2568.09i			1.87589i	−0.346784	+	0.937945i	$0.612726\pi$
$38$			1649.27i			1.14215i
$39$			817.687i			0.537598i
$40$			1644.81i			1.02801i
$41$	503.324			0.299419			0.149710	−	0.988730i	$-0.452166\pi$
$41$	503.324			0.299419			0.149710	+	0.988730i	$0.452166\pi$
$42$		−	220.563i		−	0.125036i
$43$		−	2003.40i		−	1.08350i	−0.840538	−	0.541752i	$-0.817761\pi$
$43$		−	2003.40i		−	1.08350i	0.840538	−	0.541752i	$-0.182239\pi$
$44$			451.145i			0.233029i
$45$	700.161			0.345758
$46$	1031.63			0.487538
$47$			2422.01i			1.09643i	0.836338	+	0.548214i	$0.184692\pi$
$47$			2422.01i			1.09643i	−0.836338	+	0.548214i	$0.815308\pi$
$48$	37.5309			0.0162894
$49$	−2096.17			−0.873038
$50$			115.390i			0.0461561i
$51$	−1707.10			−0.656325
$52$			1587.69i			0.587164i
$53$	−1526.26			−0.543347			−0.271674	−	0.962389i	$-0.587577\pi$
$53$	−1526.26			−0.543347			−0.271674	+	0.962389i	$0.587577\pi$
$54$			341.087i			0.116971i
$55$			1159.55i			0.383322i
$56$		−	1107.42i		−	0.353133i
$57$	3524.96			1.08494
$58$			2116.47i			0.629154i
$59$	−1105.23	+	3300.88i	−0.317504	+	0.948257i
$59$	−1105.23	+	3300.88i	−0.317504	+	0.948257i
$60$	1359.49			0.377637
$61$		−	5708.35i		−	1.53409i	−0.641594	−	0.767045i	$-0.721726\pi$
$61$		−	5708.35i		−	1.53409i	0.641594	−	0.767045i	$-0.278274\pi$
$62$	3356.05			0.873061
$63$	−471.407			−0.118772
$64$	−2394.42			−0.584576
$65$			4080.74i			0.965857i
$66$	−564.880			−0.129679
$67$		−	4773.92i		−	1.06347i	−0.846911	−	0.531735i	$-0.821540\pi$
$67$		−	4773.92i		−	1.06347i	0.846911	−	0.531735i	$-0.178460\pi$
$68$	−3314.66			−0.716838
$69$		−	2204.89i		−	0.463115i
$70$		−	1100.74i		−	0.224641i
$71$	−2795.15			−0.554484			−0.277242	−	0.960800i	$-0.589420\pi$
$71$	−2795.15			−0.554484			−0.277242	+	0.960800i	$0.589420\pi$
$72$			1712.56i			0.330355i
$73$		−	4719.76i		−	0.885674i	−0.896602	−	0.442837i	$-0.853972\pi$
$73$		−	4719.76i		−	0.885674i	0.896602	−	0.442837i	$-0.146028\pi$
$74$	−6243.53			−1.14016
$75$	246.622			0.0438439
$76$	6844.37			1.18497
$77$		−	780.705i		−	0.131676i
$78$	−1987.96			−0.326751
$79$	8669.75			1.38916			0.694580	−	0.719415i	$-0.255590\pi$
$79$	8669.75			1.38916			0.694580	+	0.719415i	$0.255590\pi$
$80$	187.301			0.0292658
$81$	729.000			0.111111
$82$			1223.68i			0.181987i
$83$			7458.93i			1.08273i	0.840787	+	0.541365i	$0.182092\pi$
$83$			7458.93i			1.08273i	−0.840787	+	0.541365i	$0.817908\pi$
$84$	−915.325			−0.129723
$85$	−8519.45			−1.17916
$86$	4870.65			0.658552
$87$	4523.51			0.597636
$88$	−2836.20			−0.366245
$89$		−	4001.99i		−	0.505238i	−0.967566	−	0.252619i	$-0.918708\pi$
$89$		−	4001.99i		−	0.505238i	0.967566	−	0.252619i	$-0.0812920\pi$
$90$			1702.23i			0.210151i
$91$		−	2747.50i		−	0.331784i
$92$		−	4281.21i		−	0.505814i
$93$		−	7172.83i		−	0.829325i
$94$	−5888.37			−0.666407
$95$	17591.6			1.94921
$96$			5364.56i			0.582092i
$97$		−	5912.30i		−	0.628367i	−0.949362	−	0.314183i	$-0.898269\pi$
$97$		−	5912.30i		−	0.628367i	0.949362	−	0.314183i	$-0.101731\pi$
$98$		−	5096.18i		−	0.530631i
$99$			1207.31i			0.123182i
$100$	478.862			0.0478862
$101$		−	14483.7i		−	1.41983i	−0.704286	−	0.709916i	$-0.748733\pi$
$101$		−	14483.7i		−	1.41983i	0.704286	−	0.709916i	$-0.251267\pi$
$102$		−	4150.30i		−	0.398913i
$103$			211.537i			0.0199394i	0.999950	+	0.00996968i	$0.00317350\pi$
$103$			211.537i			0.0199394i	−0.999950	+	0.00996968i	$0.996826\pi$
$104$	−9981.30			−0.922828
$105$	−2352.60			−0.213388
$106$		−	3710.64i		−	0.330246i
$107$	−12751.0			−1.11373			−0.556863	−	0.830605i	$-0.687995\pi$
$107$	−12751.0			−1.11373			−0.556863	+	0.830605i	$0.687995\pi$
$108$	1415.49			0.121355
$109$		−	18050.8i		−	1.51930i	−0.650334	−	0.759648i	$-0.725371\pi$
$109$		−	18050.8i		−	1.51930i	0.650334	−	0.759648i	$-0.274629\pi$
$110$	−2819.09			−0.232982
$111$			13344.2i			1.08305i
$112$	−126.107			−0.0100532
$113$		−	9631.35i		−	0.754276i	−0.926157	−	0.377138i	$-0.876908\pi$
$113$		−	9631.35i		−	0.754276i	0.926157	−	0.377138i	$-0.123092\pi$
$114$			8569.86i			0.659423i
$115$		−	11003.7i		−	0.832039i
$116$	8783.24			0.652737
$117$			4248.83i			0.310383i
$118$	−8025.08	−	2687.03i	−0.576349	−	0.192979i
$119$	5736.01			0.405057
$120$			8546.69i			0.593520i
$121$	12641.6			0.863435
$122$	13878.1			0.932417
$123$	2615.35			0.172870
$124$		−	13927.4i		−	0.905788i
$125$	−14976.6			−0.958505
$126$		−	1146.08i		−	0.0721896i
$127$	−25023.8			−1.55148			−0.775741	−	0.631052i	$-0.782624\pi$
$127$	−25023.8			−1.55148			−0.775741	+	0.631052i	$0.782624\pi$
$128$			10697.3i			0.652909i
$129$		−	10410.0i		−	0.625562i
$130$	−9921.08			−0.587046
$131$			2066.15i			0.120398i	0.998186	+	0.0601990i	$0.0191735\pi$
$131$			2066.15i			0.120398i	−0.998186	+	0.0601990i	$0.980826\pi$
$132$			2344.22i			0.134540i
$133$	−11844.2			−0.669578
$134$	11606.3			0.646375
$135$	3638.14			0.199624
$136$		−	20838.2i		−	1.12663i
$137$	8060.06			0.429434			0.214717	−	0.976676i	$-0.431117\pi$
$137$	8060.06			0.429434			0.214717	+	0.976676i	$0.431117\pi$
$138$	5360.51			0.281480
$139$	−8208.19			−0.424833			−0.212416	−	0.977179i	$-0.568133\pi$
$139$	−8208.19			−0.424833			−0.212416	+	0.977179i	$0.568133\pi$
$140$	−4568.01			−0.233062
$141$			12585.1i			0.633023i
$142$		−	6795.56i		−	0.337014i
$143$	−7036.56			−0.344103
$144$	195.016			0.00940471
$145$	22575.0			1.07372
$146$	11474.6			0.538311
$147$	−10892.0			−0.504049
$148$			25910.3i			1.18290i
$149$			37339.6i			1.68189i	0.541122	+	0.840944i	$0.318000\pi$
$149$			37339.6i			1.68189i	−0.541122	+	0.840944i	$0.682000\pi$
$150$			599.585i			0.0266482i
$151$		−	10313.8i		−	0.452340i	−0.974088	−	0.226170i	$-0.927379\pi$
$151$		−	10313.8i		−	0.452340i	0.974088	−	0.226170i	$-0.0726205\pi$
$152$			43028.3i			1.86237i
$153$	−8870.37			−0.378930
$154$	1898.04			0.0800323
$155$		−	35796.7i		−	1.48998i
$156$			8249.89i			0.339000i
$157$		−	10989.2i		−	0.445826i	−0.974838	−	0.222913i	$-0.928443\pi$
$157$		−	10989.2i		−	0.445826i	0.974838	−	0.222913i	$-0.0715566\pi$
$158$			21077.8i			0.844329i
$159$	−7930.69			−0.313702
$160$			26772.3i			1.04579i
$161$			7408.62i			0.285815i
$162$			1772.34i			0.0675331i
$163$	1659.70			0.0624676			0.0312338	−	0.999512i	$-0.490056\pi$
$163$	1659.70			0.0624676			0.0312338	+	0.999512i	$0.490056\pi$
$164$	5078.19			0.188808
$165$			6025.19i			0.221311i
$166$	−18134.1			−0.658082
$167$	−5744.24			−0.205968			−0.102984	−	0.994683i	$-0.532839\pi$
$167$	−5744.24			−0.205968			−0.102984	+	0.994683i	$0.532839\pi$
$168$		−	5754.35i		−	0.203881i
$169$	3797.58			0.132964
$170$		−	20712.4i		−	0.716693i
$171$	18316.2			0.626389
$172$		−	20212.9i		−	0.683238i
$173$		−	48337.4i		−	1.61507i	−0.589821	−	0.807534i	$-0.700802\pi$
$173$		−	48337.4i		−	1.61507i	0.589821	−	0.807534i	$-0.299198\pi$
$174$			10997.5i			0.363242i
$175$	−828.671			−0.0270586
$176$			322.969i			0.0104264i
$177$	−5742.96	+	17151.9i	−0.183311	+	0.547476i
$178$	9729.62			0.307083
$179$			8489.09i			0.264945i	0.991187	+	0.132472i	$0.0422916\pi$
$179$			8489.09i			0.264945i	−0.991187	+	0.132472i	$0.957708\pi$
$180$	7064.13			0.218029
$181$	−63579.4			−1.94070			−0.970352	−	0.241696i	$-0.922296\pi$
$181$	−63579.4			−1.94070			−0.970352	+	0.241696i	$0.922296\pi$
$182$	6679.70			0.201657
$183$		−	29661.4i		−	0.885707i
$184$	26914.5			0.794971
$185$			66595.5i			1.94581i
$186$	17438.5			0.504062
$187$		−	14690.4i		−	0.420097i
$188$			24436.4i			0.691387i
$189$	−2449.50			−0.0685732
$190$			42768.7i			1.18473i
$191$			36006.4i			0.986991i	0.869748	+	0.493495i	$0.164281\pi$
$191$			36006.4i			0.986991i	−0.869748	+	0.493495i	$0.835719\pi$
$192$	−12441.8			−0.337505
$193$	7371.06			0.197886			0.0989431	−	0.995093i	$-0.468454\pi$
$193$	7371.06			0.197886			0.0989431	+	0.995093i	$0.468454\pi$
$194$	14373.9			0.381920
$195$			21204.2i			0.557638i
$196$	−21148.8			−0.550522
$197$	62886.8			1.62042			0.810209	−	0.586141i	$-0.199353\pi$
$197$	62886.8			1.62042			0.810209	+	0.586141i	$0.199353\pi$
$198$	−2935.20			−0.0748700
$199$	24801.5			0.626285			0.313143	−	0.949706i	$-0.398618\pi$
$199$	24801.5			0.626285			0.313143	+	0.949706i	$0.398618\pi$
$200$			3010.45i			0.0752613i
$201$		−	24806.0i		−	0.613995i
$202$	35212.7			0.862972
$203$	−15199.4			−0.368836
$204$	−17223.5			−0.413867
$205$	13052.1			0.310580
$206$	−514.286			−0.0121191
$207$		−	11456.9i		−	0.267380i
$208$			1136.61i			0.0262715i
$209$			30333.8i			0.694440i
$210$		−	5719.63i		−	0.129697i
$211$		−	17502.7i		−	0.393134i	−0.980490	−	0.196567i	$-0.937021\pi$
$211$		−	17502.7i		−	0.393134i	0.980490	−	0.196567i	$-0.0629794\pi$
$212$	−15398.9			−0.342625
$213$	−14524.0			−0.320131
$214$		−	31000.2i		−	0.676920i
$215$		−	51951.9i		−	1.12389i
$216$			8898.72i			0.190730i
$217$			24101.3i			0.511825i
$218$	43884.9			0.923426
$219$		−	24524.6i		−	0.511344i
$220$			11699.0i			0.241716i
$221$		−	51699.1i		−	1.05852i
$222$	−32442.3			−0.658273
$223$	54042.6			1.08674			0.543371	−	0.839493i	$-0.317148\pi$
$223$	54042.6			1.08674			0.543371	+	0.839493i	$0.317148\pi$
$224$		−	18025.4i		−	0.359243i
$225$	1281.48			0.0253133
$226$	23415.7			0.458448
$227$			24892.4i			0.483076i	0.970391	+	0.241538i	$0.0776518\pi$
$227$			24892.4i			0.483076i	−0.970391	+	0.241538i	$0.922348\pi$
$228$	35564.4			0.684141
$229$			81045.6i			1.54546i	0.634733	+	0.772731i	$0.281110\pi$
$229$			81045.6i			1.54546i	−0.634733	+	0.772731i	$0.718890\pi$
$230$	26752.1			0.505712
$231$		−	4056.66i		−	0.0760230i
$232$			55217.3i			1.02589i
$233$		−	92463.9i		−	1.70318i	−0.524209	−	0.851589i	$-0.675639\pi$
$233$		−	92463.9i		−	1.70318i	0.524209	−	0.851589i	$-0.324361\pi$
$234$	−10329.7			−0.188650
$235$			62807.2i			1.13730i
$236$	−11151.0	+	33303.6i	−0.200212	+	0.597953i
$237$	45049.3			0.802032
$238$			13945.4i			0.246193i
$239$	−66752.2			−1.16861			−0.584305	−	0.811534i	$-0.698633\pi$
$239$	−66752.2			−1.16861			−0.584305	+	0.811534i	$0.698633\pi$
$240$	973.246			0.0168966
$241$	−57886.0			−0.996643			−0.498322	−	0.866992i	$-0.666050\pi$
$241$	−57886.0			−0.996643			−0.498322	+	0.866992i	$0.666050\pi$
$242$			30734.1i			0.524794i
$243$	3788.00			0.0641500
$244$		−	57593.2i		−	0.967368i
$245$	−54357.5			−0.905581
$246$			6358.42i			0.105070i
$247$			106752.i			1.74978i
$248$	87556.9			1.42360
$249$			38757.8i			0.625115i
$250$		−	36411.1i		−	0.582577i
$251$	−1482.64			−0.0235336			−0.0117668	−	0.999931i	$-0.503746\pi$
$251$	−1482.64			−0.0235336			−0.0117668	+	0.999931i	$0.503746\pi$
$252$	−4756.17			−0.0748956
$253$	18974.0			0.296428
$254$		−	60837.8i		−	0.942988i
$255$	−44268.4			−0.680790
$256$	−64317.9			−0.981413
$257$	−63595.3			−0.962850			−0.481425	−	0.876487i	$-0.659881\pi$
$257$	−63595.3			−0.962850			−0.481425	+	0.876487i	$0.659881\pi$
$258$	25308.6			0.380215
$259$		−	44837.7i		−	0.668411i
$260$			41171.9i			0.609051i
$261$	23504.8			0.345045
$262$	−5023.21			−0.0731777
$263$	−79407.3			−1.14802			−0.574010	−	0.818849i	$-0.694613\pi$
$263$	−79407.3			−1.14802			−0.574010	+	0.818849i	$0.694613\pi$
$264$	−14737.3			−0.211451
$265$	−39578.9			−0.563601
$266$		−	28795.5i		−	0.406968i
$267$		−	20795.0i		−	0.291699i
$268$		−	48165.5i		−	0.670605i
$269$		−	103489.i		−	1.43018i	−0.699031	−	0.715091i	$-0.746385\pi$
$269$		−	103489.i		−	1.43018i	0.699031	−	0.715091i	$-0.253615\pi$
$270$			8845.02i			0.121331i
$271$	144360.			1.96566			0.982830	−	0.184513i	$-0.0590709\pi$
$271$	144360.			1.96566			0.982830	+	0.184513i	$0.0590709\pi$
$272$	−2372.93			−0.0320735
$273$		−	14276.4i		−	0.191555i
$274$			19595.5i			0.261010i
$275$			2122.29i			0.0280633i
$276$		−	22245.8i		−	0.292032i
$277$	47254.9			0.615867			0.307934	−	0.951408i	$-0.400363\pi$
$277$	47254.9			0.615867			0.307934	+	0.951408i	$0.400363\pi$
$278$		−	19955.7i		−	0.258213i
$279$		−	37271.1i		−	0.478811i
$280$		−	28717.6i		−	0.366296i
$281$	−31731.5			−0.401863			−0.200931	−	0.979605i	$-0.564397\pi$
$281$	−31731.5			−0.401863			−0.200931	+	0.979605i	$0.564397\pi$
$282$	−30596.9			−0.384750
$283$		−	12999.5i		−	0.162313i	−0.996701	−	0.0811563i	$-0.974139\pi$
$283$		−	12999.5i		−	0.162313i	0.996701	−	0.0811563i	$-0.0258613\pi$
$284$	−28201.1			−0.349647
$285$	91408.8			1.12538
$286$		−	17107.2i		−	0.209145i
$287$	−8787.80			−0.106688
$288$			27875.1i			0.336071i
$289$	24412.3			0.292289
$290$			54884.1i			0.652606i
$291$		−	30721.2i		−	0.362788i
$292$		−	47619.0i		−	0.558489i
$293$	−27627.7			−0.321817			−0.160909	−	0.986969i	$-0.551442\pi$
$293$	−27627.7			−0.321817			−0.160909	+	0.986969i	$0.551442\pi$
$294$		−	26480.5i		−	0.306360i
$295$	−28660.8	+	85598.1i	−0.329339	+	0.983603i
$296$	−162889.			−1.85913
$297$			6273.36i			0.0711193i
$298$	−90779.8			−1.02225
$299$	66774.5			0.746910
$300$	2488.24			0.0276471
$301$			34978.4i			0.386071i
$302$	25074.9			0.274932
$303$		−	75259.6i		−	0.819741i
$304$	4899.81			0.0530190
$305$		−	148028.i		−	1.59127i
$306$		−	21565.6i		−	0.230313i
$307$	−25754.0			−0.273255			−0.136627	−	0.990623i	$-0.543626\pi$
$307$	−25754.0			−0.273255			−0.136627	+	0.990623i	$0.543626\pi$
$308$		−	7876.77i		−	0.0830322i
$309$			1099.18i			0.0115120i
$310$	87028.6			0.905605
$311$	123728.			1.27923			0.639613	−	0.768697i	$-0.279095\pi$
$311$	123728.			1.27923			0.639613	+	0.768697i	$0.279095\pi$
$312$	−51864.4			−0.532795
$313$		−	12824.2i		−	0.130900i	−0.997856	−	0.0654502i	$-0.979152\pi$
$313$		−	12824.2i		−	0.130900i	0.997856	−	0.0654502i	$-0.0208484\pi$
$314$	26716.8			0.270972
$315$	−12224.5			−0.123200
$316$	87471.7			0.875978
$317$	−136329.			−1.35666			−0.678329	−	0.734759i	$-0.737295\pi$
$317$	−136329.			−1.35666			−0.678329	+	0.734759i	$0.737295\pi$
$318$		−	19281.0i		−	0.190667i
$319$			38926.8i			0.382531i
$320$	−62091.9			−0.606366
$321$	−66256.4			−0.643010
$322$	−18011.8			−0.173718
$323$	−222869.			−2.13622
$324$	7355.10			0.0700646
$325$			7468.87i			0.0707112i
$326$			4035.05i			0.0379677i
$327$		−	93794.5i		−	0.877166i
$328$			31924.9i			0.296744i
$329$		−	42287.1i		−	0.390676i
$330$	−14648.4			−0.134512
$331$	−153145.			−1.39781			−0.698904	−	0.715216i	$-0.746328\pi$
$331$	−153145.			−1.39781			−0.698904	+	0.715216i	$0.746328\pi$
$332$			75255.4i			0.682750i
$333$			69338.5i			0.625297i
$334$		−	13965.4i		−	0.125187i
$335$		−	123797.i		−	1.10311i
$336$	−655.271			−0.00580420
$337$		−	60949.0i		−	0.536669i	−0.963326	−	0.268335i	$-0.913527\pi$
$337$		−	60949.0i		−	0.536669i	0.963326	−	0.268335i	$-0.0864733\pi$
$338$			9232.64i			0.0808151i
$339$		−	50046.0i		−	0.435482i
$340$	−85955.3			−0.743558
$341$	61725.3			0.530829
$342$			44530.3i			0.380718i
$343$	78518.3			0.667395
$344$	127072.			1.07382
$345$		−	57177.0i		−	0.480378i
$346$	117517.			0.981635
$347$			150274.i			1.24803i	0.781412	+	0.624015i	$0.214500\pi$
$347$			150274.i			1.24803i	−0.781412	+	0.624015i	$0.785500\pi$
$348$	45639.0			0.376858
$349$			4128.34i			0.0338942i	0.999856	+	0.0169471i	$0.00539468\pi$
$349$			4128.34i			0.0338942i	−0.999856	+	0.0169471i	$0.994605\pi$
$350$		−	2014.66i		−	0.0164462i
$351$			22077.6i			0.179199i
$352$	−46164.4			−0.372582
$353$			126346.i			1.01394i	0.861964	+	0.506969i	$0.169234\pi$
$353$			126346.i			1.01394i	−0.861964	+	0.506969i	$0.830766\pi$
$354$	−41699.5	−	13962.2i	−0.332755	−	0.111416i
$355$	−72483.6			−0.575152
$356$		−	40377.3i		−	0.318594i
$357$	29805.2			0.233860
$358$	−20638.6			−0.161033
$359$	182628.			1.41703			0.708514	−	0.705697i	$-0.249366\pi$
$359$	182628.			1.41703			0.708514	+	0.705697i	$0.249366\pi$
$360$			44409.9i			0.342669i
$361$	329877.			2.53126
$362$		−	154574.i		−	1.17956i
$363$	65687.4			0.498505
$364$		−	27720.4i		−	0.209217i
$365$		−	122392.i		−	0.918688i
$366$	72112.7			0.538331
$367$			8500.43i			0.0631116i	0.999502	+	0.0315558i	$0.0100462\pi$
$367$			8500.43i			0.0631116i	−0.999502	+	0.0315558i	$0.989954\pi$
$368$		−	3064.87i		−	0.0226316i
$369$	13589.7			0.0998065
$370$	−161907.			−1.18266
$371$	26647.8			0.193604
$372$		−	72368.8i		−	0.522957i
$373$	82283.7			0.591420			0.295710	−	0.955278i	$-0.404444\pi$
$373$	82283.7			0.591420			0.295710	+	0.955278i	$0.404444\pi$
$374$	35715.1			0.255334
$375$	−77820.9			−0.553393
$376$	−153623.			−1.08663
$377$			136993.i			0.963865i
$378$		−	5955.21i		−	0.0416787i
$379$	2637.31			0.0183604			0.00918022	−	0.999958i	$-0.497078\pi$
$379$	2637.31			0.0183604			0.00918022	+	0.999958i	$0.497078\pi$
$380$	177487.			1.22914
$381$	−130028.			−0.895748
$382$	−87538.5			−0.599891
$383$	45438.5			0.309761			0.154880	−	0.987933i	$-0.450501\pi$
$383$	45438.5			0.309761			0.154880	+	0.987933i	$0.450501\pi$
$384$			55584.6i			0.376957i
$385$		−	20245.2i		−	0.136584i
$386$			17920.5i			0.120275i
$387$		−	54091.8i		−	0.361168i
$388$		−	59651.0i		−	0.396236i
$389$	−36809.8			−0.243257			−0.121628	−	0.992576i	$-0.538812\pi$
$389$	−36809.8			−0.243257			−0.121628	+	0.992576i	$0.538812\pi$
$390$	−51551.4			−0.338931
$391$			139406.i			0.911862i
$392$		−	132956.i		−	0.865237i
$393$			10736.0i			0.0695118i
$394$			152890.i			0.984888i
$395$	224823.			1.44094
$396$			12180.9i			0.0776764i
$397$			184237.i			1.16895i	0.811412	+	0.584475i	$0.198700\pi$
$397$			184237.i			1.16895i	−0.811412	+	0.584475i	$0.801300\pi$
$398$			60297.3i			0.380655i
$399$	−61544.1			−0.386581
$400$	342.812			0.00214258
$401$			243995.i			1.51738i	0.651455	+	0.758688i	$0.274159\pi$
$401$			243995.i			1.51738i	−0.651455	+	0.758688i	$0.725841\pi$
$402$	60308.2			0.373185
$403$	217227.			1.33753
$404$		−	146131.i		−	0.895320i
$405$	18904.3			0.115253
$406$		−	36952.6i		−	0.224178i
$407$	−114833.			−0.693229
$408$		−	108278.i		−	0.650461i
$409$			39142.7i			0.233994i	0.993132	+	0.116997i	$0.0373268\pi$
$409$			39142.7i			0.233994i	−0.993132	+	0.116997i	$0.962673\pi$
$410$			31732.3i			0.188770i
$411$	41881.3			0.247934
$412$			2134.26i			0.0125734i
$413$	19296.8	−	57631.8i	0.113132	−	0.337880i
$414$	27854.0			0.162513
$415$			193424.i			1.12309i
$416$	−162464.			−0.938795
$417$	−42651.0			−0.245277
$418$	−73747.4			−0.422079
$419$			148939.i			0.848362i	0.905577	+	0.424181i	$0.139438\pi$
$419$			148939.i			0.848362i	−0.905577	+	0.424181i	$0.860562\pi$
$420$	−23736.1			−0.134558
$421$			258237.i			1.45698i	0.685055	+	0.728492i	$0.259778\pi$
$421$			258237.i			1.45698i	−0.685055	+	0.728492i	$0.740222\pi$
$422$	42552.5			0.238946
$423$			65394.2i			0.365476i
$424$		−	96808.0i		−	0.538492i
$425$	−15592.9			−0.0863276
$426$		−	35310.8i		−	0.194575i
$427$			99665.0i			0.546622i
$428$	−128649.			−0.702295
$429$	−36563.0			−0.198668
$430$	126305.			0.683100
$431$			260135.i			1.40037i	0.713960	+	0.700187i	$0.246900\pi$
$431$			260135.i			1.40037i	−0.713960	+	0.700187i	$0.753100\pi$
$432$	1013.33			0.00542981
$433$	281301.			1.50036			0.750181	−	0.661232i	$-0.229966\pi$
$433$	281301.			1.50036			0.750181	+	0.661232i	$0.229966\pi$
$434$	−58595.0			−0.311086
$435$	117303.			0.619913
$436$		−	182120.i		−	0.958040i
$437$		−	287857.i		−	1.50735i
$438$	59624.0			0.310794
$439$	−226376.			−1.17463			−0.587316	−	0.809357i	$-0.699816\pi$
$439$	−226376.			−1.17463			−0.587316	+	0.809357i	$0.699816\pi$
$440$	−73548.0			−0.379897
$441$	−56596.5			−0.291013
$442$	125691.			0.643366
$443$			191214.i			0.974346i	0.873305	+	0.487173i	$0.161972\pi$
$443$			191214.i			0.974346i	−0.873305	+	0.487173i	$0.838028\pi$
$444$			134634.i			0.682948i
$445$		−	103779.i		−	0.524071i
$446$			131388.i			0.660519i
$447$			194022.i			0.971039i
$448$	41805.5			0.208294
$449$	205380.			1.01874			0.509372	−	0.860546i	$-0.329878\pi$
$449$	205380.			1.01874			0.509372	+	0.860546i	$0.329878\pi$
$450$			3115.54i			0.0153854i
$451$			22506.2i			0.110649i
$452$		−	97173.6i		−	0.475633i
$453$		−	53592.1i		−	0.261159i
$454$	−60518.3			−0.293613
$455$		−	71247.8i		−	0.344151i
$456$			223582.i			1.07524i
$457$			145715.i			0.697703i	0.937178	+	0.348852i	$0.113428\pi$
$457$			145715.i			0.697703i	−0.937178	+	0.348852i	$0.886572\pi$
$458$	−197038.			−0.939329
$459$	−46091.8			−0.218775
$460$		−	111020.i		−	0.524668i
$461$	−91731.3			−0.431634			−0.215817	−	0.976434i	$-0.569241\pi$
$461$	−91731.3			−0.431634			−0.215817	+	0.976434i	$0.569241\pi$
$462$	9862.53			0.0462066
$463$			363576.i			1.69603i	0.529974	+	0.848014i	$0.322202\pi$
$463$			363576.i			1.69603i	−0.529974	+	0.848014i	$0.677798\pi$
$464$	6287.82			0.0292055
$465$		−	186005.i		−	0.860238i
$466$	224797.			1.03519
$467$		−	107740.i		−	0.494019i	−0.969013	−	0.247009i	$-0.920552\pi$
$467$		−	107740.i		−	0.494019i	0.969013	−	0.247009i	$-0.0794478\pi$
$468$			42867.7i			0.195721i
$469$			83350.3i			0.378932i
$470$	−152697.			−0.691247
$471$		−	57101.4i		−	0.257398i
$472$	−209369.	−	70102.9i	−0.939784	−	0.314667i
$473$	89582.3			0.400406
$474$			109524.i			0.487474i
$475$	32197.5			0.142704
$476$	57872.4			0.255421
$477$	−41209.1			−0.181116
$478$		−	162288.i		−	0.710280i
$479$	148486.			0.647165			0.323582	−	0.946200i	$-0.395113\pi$
$479$	148486.			0.647165			0.323582	+	0.946200i	$0.395113\pi$
$480$			139113.i			0.603790i
$481$	−404125.			−1.74673
$482$		−	140732.i		−	0.605758i
$483$			38496.3i			0.165016i
$484$	127544.			0.544466
$485$		−	153317.i		−	0.651789i
$486$			9209.35i			0.0389903i
$487$	−3404.16			−0.0143533			−0.00717666	−	0.999974i	$-0.502284\pi$
$487$	−3404.16			−0.0143533			−0.00717666	+	0.999974i	$0.502284\pi$
$488$	362070.			1.52038
$489$	8624.06			0.0360657
$490$		−	132154.i		−	0.550411i
$491$	326718.			1.35522			0.677611	−	0.735420i	$-0.263015\pi$
$491$	326718.			1.35522			0.677611	+	0.735420i	$0.263015\pi$
$492$	26387.0			0.109009
$493$	−286003.			−1.17673
$494$	−259536.			−1.06351
$495$			31307.8i			0.127774i
$496$		−	9970.46i		−	0.0405277i
$497$	48802.0			0.197572
$498$	−94227.6			−0.379944
$499$	170255.			0.683754			0.341877	−	0.939745i	$-0.388937\pi$
$499$	170255.			0.683754			0.341877	+	0.939745i	$0.388937\pi$
$500$	−151104.			−0.604415
$501$	−29847.9			−0.118916
$502$		−	3604.58i		−	0.0143037i
$503$			198224.i			0.783466i	0.920079	+	0.391733i	$0.128124\pi$
$503$			198224.i			0.783466i	−0.920079	+	0.391733i	$0.871876\pi$
$504$		−	29900.5i		−	0.117711i
$505$		−	375590.i		−	1.47276i
$506$			46129.6i			0.180168i
$507$	19732.8			0.0767666
$508$	−252473.			−0.978335
$509$		−	252370.i		−	0.974099i	−0.873374	−	0.487049i	$-0.838073\pi$
$509$		−	252370.i		−	0.974099i	0.873374	−	0.487049i	$-0.161927\pi$
$510$		−	107625.i		−	0.413783i
$511$			82404.7i			0.315580i
$512$			14787.0i			0.0564079i
$513$	95173.9			0.361646
$514$		−	154612.i		−	0.585219i
$515$			5485.54i			0.0206826i
$516$		−	105029.i		−	0.394467i
$517$	−108301.			−0.405181
$518$	109009.			0.406259
$519$		−	251168.i		−	0.932460i
$520$	−258834.			−0.957226
$521$	−372104.			−1.37085			−0.685424	−	0.728144i	$-0.740383\pi$
$521$	−372104.			−1.37085			−0.685424	+	0.728144i	$0.740383\pi$
$522$			57144.8i			0.209718i
$523$	−68686.7			−0.251113			−0.125557	−	0.992086i	$-0.540072\pi$
$523$	−68686.7			−0.251113			−0.125557	+	0.992086i	$0.540072\pi$
$524$			20846.0i			0.0759207i
$525$	−4305.90			−0.0156223
$526$		−	193055.i		−	0.697764i
$527$			453510.i			1.63292i
$528$			1678.20i			0.00601971i
$529$	99783.9			0.356574
$530$		−	96223.8i		−	0.342556i
$531$	−29841.3	+	89123.8i	−0.105835	+	0.316086i
$532$	−119499.			−0.422223
$533$			79205.1i			0.278804i
$534$	50556.6			0.177294
$535$	−330659.			−1.15524
$536$	302801.			1.05397
$537$			44110.6i			0.152966i
$538$	251603.			0.869262
$539$		−	93730.4i		−	0.322629i
$540$	36706.3			0.125879
$541$		−	51022.3i		−	0.174327i	−0.996194	−	0.0871637i	$-0.972220\pi$
$541$		−	51022.3i		−	0.174327i	0.996194	−	0.0871637i	$-0.0277803\pi$
$542$			350967.i			1.19472i
$543$	−330368.			−1.12047
$544$		−	339180.i		−	1.14613i
$545$		−	468090.i		−	1.57593i
$546$	34708.7			0.116427
$547$	116201.			0.388362			0.194181	−	0.980966i	$-0.437795\pi$
$547$	116201.			0.388362			0.194181	+	0.980966i	$0.437795\pi$
$548$	81320.3			0.270793
$549$		−	154125.i		−	0.511363i
$550$	−5159.69			−0.0170568
$551$	590562.			1.94519
$552$	139852.			0.458977
$553$	−151370.			−0.494981
$554$			114886.i			0.374323i
$555$			346040.i			1.12342i
$556$	−82814.9			−0.267892
$557$	−428570.			−1.38137			−0.690687	−	0.723154i	$-0.742692\pi$
$557$	−428570.			−1.38137			−0.690687	+	0.723154i	$0.742692\pi$
$558$	90613.3			0.291020
$559$	315263.			1.00890
$560$	−3270.19			−0.0104279
$561$		−	76333.4i		−	0.242543i
$562$		−	77145.4i		−	0.244251i
$563$		−	131577.i		−	0.415110i	−0.978223	−	0.207555i	$-0.933449\pi$
$563$		−	131577.i		−	0.415110i	0.978223	−	0.207555i	$-0.0665506\pi$
$564$			126975.i			0.399172i
$565$		−	249759.i		−	0.782392i
$566$	31604.2			0.0986533
$567$	−12728.0			−0.0395907
$568$		−	177291.i		−	0.549529i
$569$			399439.i			1.23375i	0.787063	+	0.616873i	$0.211601\pi$
$569$			399439.i			1.23375i	−0.787063	+	0.616873i	$0.788399\pi$
$570$			222233.i			0.684003i
$571$		−	131744.i		−	0.404072i	−0.979378	−	0.202036i	$-0.935244\pi$
$571$		−	131744.i		−	0.404072i	0.979378	−	0.202036i	$-0.0647558\pi$
$572$	−70993.9			−0.216985
$573$			187095.i			0.569839i
$574$		−	21364.8i		−	0.0648449i
$575$		−	20139.8i		−	0.0609143i
$576$	−64649.4			−0.194859
$577$	−160918.			−0.483339			−0.241670	−	0.970359i	$-0.577695\pi$
$577$	−160918.			−0.483339			−0.241670	+	0.970359i	$0.577695\pi$
$578$			59351.0i			0.177653i
$579$	38301.2			0.114250
$580$	227766.			0.677068
$581$		−	130229.i		−	0.385795i
$582$	74689.2			0.220502
$583$		−	68247.1i		−	0.200792i
$584$	299365.			0.877760
$585$			110180.i			0.321952i
$586$		−	67168.2i		−	0.195600i
$587$			171759.i			0.498474i	0.968443	+	0.249237i	$0.0801798\pi$
$587$			171759.i			0.498474i	−0.968443	+	0.249237i	$0.919820\pi$
$588$	−109893.			−0.317844
$589$		−	936442.i		−	2.69929i
$590$	−208105.	−	69679.9i	−0.597832	−	0.200172i
$591$	326770.			0.935549
$592$			18548.9i			0.0529266i
$593$	457675.			1.30151			0.650755	−	0.759288i	$-0.274453\pi$
$593$	457675.			1.30151			0.650755	+	0.759288i	$0.274453\pi$
$594$	−15251.8			−0.0432262
$595$	148746.			0.420156
$596$			376731.i			1.06057i
$597$	128872.			0.361586
$598$			162342.i			0.453970i
$599$	−450231.			−1.25482			−0.627410	−	0.778689i	$-0.715885\pi$
$599$	−450231.			−1.25482			−0.627410	+	0.778689i	$0.715885\pi$
$600$			15642.8i			0.0434521i
$601$		−	188572.i		−	0.522071i	−0.965329	−	0.261035i	$-0.915936\pi$
$601$		−	188572.i		−	0.522071i	0.965329	−	0.261035i	$-0.0840639\pi$
$602$	−85039.2			−0.234653
$603$		−	128896.i		−	0.354490i
$604$		−	104059.i		−	0.285237i
$605$	327819.			0.895620
$606$	182971.			0.498237
$607$	266272.			0.722682			0.361341	−	0.932434i	$-0.382319\pi$
$607$	266272.			0.722682			0.361341	+	0.932434i	$0.382319\pi$
$608$			700365.i			1.89460i
$609$	−78978.3			−0.212948
$610$	359885.			0.967173
$611$	−381137.			−1.02094
$612$	−89495.8			−0.238946
$613$		−	331432.i		−	0.882010i	−0.897505	−	0.441005i	$-0.854622\pi$
$613$		−	331432.i		−	0.882010i	0.897505	−	0.441005i	$-0.145378\pi$
$614$		−	62612.9i		−	0.166084i
$615$	67820.9			0.179314
$616$	49518.7			0.130499
$617$	−341099.			−0.896005			−0.448002	−	0.894032i	$-0.647864\pi$
$617$	−341099.			−0.896005			−0.448002	+	0.894032i	$0.647864\pi$
$618$	−2672.31			−0.00699697
$619$	−101548.			−0.265027			−0.132513	−	0.991181i	$-0.542305\pi$
$619$	−101548.			−0.265027			−0.132513	+	0.991181i	$0.542305\pi$
$620$		−	361163.i		−	0.939551i
$621$		−	59532.0i		−	0.154372i
$622$			300807.i			0.777511i
$623$			69872.9i			0.180025i
$624$			5906.01i			0.0151679i
$625$	−418036.			−1.07017
$626$	31178.1			0.0795611
$627$			157619.i			0.400935i
$628$		−	110873.i		−	0.281130i
$629$		−	843701.i		−	2.13249i
$630$		−	29720.0i		−	0.0748804i
$631$	−278409.			−0.699238			−0.349619	−	0.936892i	$-0.613689\pi$
$631$	−278409.			−0.699238			−0.349619	+	0.936892i	$0.613689\pi$
$632$			549906.i			1.37675i
$633$		−	90946.9i		−	0.226976i
$634$		−	331442.i		−	0.824574i
$635$	−648915.			−1.60931
$636$	−80015.2			−0.197814
$637$		−	329861.i		−	0.812928i
$638$	−94638.5			−0.232502
$639$	−75469.1			−0.184828
$640$			277400.i			0.677246i
$641$	267843.			0.651874			0.325937	−	0.945391i	$-0.394320\pi$
$641$	267843.			0.651874			0.325937	+	0.945391i	$0.394320\pi$
$642$		−	161082.i		−	0.390820i
$643$	247371.			0.598311			0.299155	−	0.954204i	$-0.403295\pi$
$643$	247371.			0.598311			0.299155	+	0.954204i	$0.403295\pi$
$644$			74747.8i			0.180230i
$645$		−	269950.i		−	0.648880i
$646$		−	541838.i		−	1.29839i
$647$	−66763.5			−0.159489			−0.0797445	−	0.996815i	$-0.525410\pi$
$647$	−66763.5			−0.159489			−0.0797445	+	0.996815i	$0.525410\pi$
$648$			46239.1i			0.110118i
$649$	−147599.	−	49420.7i	−0.350425	−	0.117333i
$650$	−18158.3			−0.0429781
$651$			125234.i			0.295502i
$652$	16745.2			0.0393909
$653$	−585647.			−1.37344			−0.686719	−	0.726923i	$-0.740950\pi$
$653$	−585647.			−1.37344			−0.686719	+	0.726923i	$0.740950\pi$
$654$	228033.			0.533140
$655$			53579.2i			0.124886i
$656$	3635.42			0.00844786
$657$		−	127433.i		−	0.295225i
$658$	102808.			0.237452
$659$			34391.7i			0.0791922i	0.999216	+	0.0395961i	$0.0126071\pi$
$659$			34391.7i			0.0791922i	−0.999216	+	0.0395961i	$0.987393\pi$
$660$			60789.9i			0.139555i
$661$	−151480.			−0.346698			−0.173349	−	0.984860i	$-0.555459\pi$
$661$	−151480.			−0.346698			−0.173349	+	0.984860i	$0.555459\pi$
$662$		−	372325.i		−	0.849585i
$663$		−	268636.i		−	0.611136i
$664$	−473106.			−1.07306
$665$	−307142.			−0.694537
$666$	−168575.			−0.380054
$667$		−	369401.i		−	0.830323i
$668$	−57955.4			−0.129880
$669$	280813.			0.627430
$670$	300974.			0.670469
$671$	255250.			0.566918
$672$		−	93662.6i		−	0.207409i
$673$			355378.i			0.784622i	0.919833	+	0.392311i	$0.128324\pi$
$673$			355378.i			0.784622i	−0.919833	+	0.392311i	$0.871676\pi$
$674$	148179.			0.326187
$675$	6658.79			0.0146146
$676$	38314.9			0.0838444
$677$	670210.			1.46229			0.731145	−	0.682222i	$-0.238986\pi$
$677$	670210.			1.46229			0.731145	+	0.682222i	$0.238986\pi$
$678$	121671.			0.264685
$679$			103226.i			0.223898i
$680$		−	540373.i		−	1.16863i
$681$			129345.i			0.278904i
$682$			150066.i			0.322637i
$683$		−	237295.i		−	0.508683i	−0.967114	−	0.254341i	$-0.918141\pi$
$683$		−	237295.i		−	0.508683i	0.967114	−	0.254341i	$-0.0818587\pi$
$684$	184798.			0.394989
$685$	209012.			0.445442
$686$			190893.i			0.405641i
$687$			421125.i			0.892273i
$688$		−	14470.2i		−	0.0305701i
$689$		−	240179.i		−	0.505937i
$690$	139008.			0.291973
$691$			40265.6i			0.0843293i	0.999111	+	0.0421646i	$0.0134254\pi$
$691$			40265.6i			0.0843293i	−0.999111	+	0.0421646i	$0.986575\pi$
$692$		−	487690.i		−	1.01843i
$693$		−	21079.0i		−	0.0438919i
$694$	−365345.			−0.758551
$695$	−212854.			−0.440668
$696$			286918.i			0.592296i
$697$	−165358.			−0.340377
$698$	−10036.8			−0.0206008
$699$		−	480456.i		−	0.983331i
$700$	−8360.71			−0.0170627
$701$			55287.6i			0.112510i	0.998416	+	0.0562551i	$0.0179160\pi$
$701$			55287.6i			0.112510i	−0.998416	+	0.0562551i	$0.982084\pi$
$702$	−53674.8			−0.108917
$703$			1.74214e6i			3.52511i
$704$		−	107067.i		−	0.216028i
$705$			326356.i			0.656619i
$706$	−307171.			−0.616270
$707$			252879.i			0.505910i
$708$	−57942.4	+	173051.i	−0.115593	+	0.345228i
$709$	91346.8			0.181719			0.0908596	−	0.995864i	$-0.471039\pi$
$709$	91346.8			0.181719			0.0908596	+	0.995864i	$0.471039\pi$
$710$		−	176222.i		−	0.349577i
$711$	234083.			0.463053
$712$	253839.			0.500724
$713$	−585752.			−1.15222
$714$			72462.2i			0.142140i
$715$	−182471.			−0.356929
$716$			85649.0i			0.167069i
$717$	−346855.			−0.674698
$718$			444004.i			0.861267i
$719$		−	25854.5i		−	0.0500124i	−0.999687	−	0.0250062i	$-0.992039\pi$
$719$		−	25854.5i		−	0.0500124i	0.999687	−	0.0250062i	$-0.00796055\pi$
$720$	5057.13			0.00975527
$721$		−	3693.33i		−	0.00710473i
$722$			801994.i			1.53850i
$723$	−300785.			−0.575412
$724$	−641472.			−1.22377
$725$	41318.4			0.0786081
$726$			159699.i			0.302990i
$727$	441000.			0.834391			0.417196	−	0.908817i	$-0.363013\pi$
$727$	441000.			0.834391			0.417196	+	0.908817i	$0.363013\pi$
$728$	174269.			0.328819
$729$	19683.0			0.0370370
$730$	297559.			0.558377
$731$			658181.i			1.23172i
$732$		−	299263.i		−	0.558510i
$733$	426201.			0.793244			0.396622	−	0.917982i	$-0.370182\pi$
$733$	426201.			0.793244			0.396622	+	0.917982i	$0.370182\pi$
$734$	−20666.2			−0.0383591
$735$	−282450.			−0.522837
$736$	438084.			0.808727
$737$	213467.			0.393002
$738$			33039.3i			0.0606622i
$739$		−	91383.8i		−	0.167333i	−0.996494	−	0.0836663i	$-0.973337\pi$
$739$		−	91383.8i		−	0.167333i	0.996494	−	0.0836663i	$-0.0266630\pi$
$740$			671902.i			1.22699i
$741$			554702.i			1.01024i
$742$			64786.0i			0.117672i
$743$	309686.			0.560975			0.280488	−	0.959858i	$-0.409504\pi$
$743$	309686.			0.560975			0.280488	+	0.959858i	$0.409504\pi$
$744$	454959.			0.821914
$745$			968286.i			1.74458i
$746$			200048.i			0.359464i
$747$			201391.i			0.360910i
$748$		−	148215.i		−	0.264905i
$749$	222627.			0.396839
$750$		−	189198.i		−	0.336351i
$751$			218166.i			0.386819i	0.981118	+	0.193409i	$0.0619545\pi$
$751$			218166.i			0.386819i	−0.981118	+	0.193409i	$0.938045\pi$
$752$			17493.7i			0.0309348i
$753$	−7704.02			−0.0135871
$754$	−333057.			−0.585835
$755$		−	267457.i		−	0.469201i
$756$	−24713.8			−0.0432410
$757$	48000.1			0.0837627			0.0418813	−	0.999123i	$-0.486665\pi$
$757$	48000.1			0.0837627			0.0418813	+	0.999123i	$0.486665\pi$
$758$			6411.81i			0.0111594i
$759$	98592.0			0.171143
$760$			1.11580e6i			1.93179i
$761$	236294.			0.408021			0.204011	−	0.978969i	$-0.434602\pi$
$761$	236294.			0.408021			0.204011	+	0.978969i	$0.434602\pi$
$762$		−	316122.i		−	0.544434i
$763$			315158.i			0.541351i
$764$			363279.i			0.622378i
$765$	−230025.			−0.393054
$766$			110470.i			0.188272i
$767$	−519440.	−	173924.i	−0.882967	−	0.295644i
$768$	−334205.			−0.566619
$769$		−	110994.i		−	0.187692i	−0.995587	−	0.0938459i	$-0.970084\pi$
$769$		−	110994.i		−	0.187692i	0.995587	−	0.0938459i	$-0.0299161\pi$
$770$	49219.9			0.0830155
$771$	−330451.			−0.555902
$772$	74368.9			0.124783
$773$			164246.i			0.274875i	0.990510	+	0.137437i	$0.0438866\pi$
$773$			164246.i			0.274875i	−0.990510	+	0.137437i	$0.956113\pi$
$774$	131508.			0.219517
$775$		−	65517.7i		−	0.109082i
$776$	375006.			0.622752
$777$		−	232983.i		−	0.385907i
$778$		−	89491.8i		−	0.147851i
$779$	341444.			0.562659
$780$			213935.i			0.351636i
$781$		−	124986.i		−	0.204908i
$782$	−338924.			−0.554228
$783$	122135.			0.199212
$784$	−15140.2			−0.0246320
$785$		−	284970.i		−	0.462445i
$786$	−26101.4			−0.0422492
$787$	909477.			1.46839			0.734196	−	0.678937i	$-0.237559\pi$
$787$	909477.			1.46839			0.734196	+	0.678937i	$0.237559\pi$
$788$	634484.			1.02181
$789$	−412613.			−0.662809
$790$			546588.i			0.875802i
$791$			168159.i			0.268761i
$792$	−76577.4			−0.122082
$793$	898288.			1.42846
$794$	−447916.			−0.710486
$795$	−205658.			−0.325395
$796$	250230.			0.394924
$797$		−	686063.i		−	1.08006i	−0.841646	−	0.540029i	$-0.818413\pi$
$797$		−	686063.i		−	1.08006i	0.841646	−	0.540029i	$-0.181587\pi$
$798$		−	149626.i		−	0.234963i
$799$		−	795708.i		−	1.24641i
$800$			49000.7i			0.0765635i
$801$		−	108054.i		−	0.168413i
$802$	−593200.			−0.922258
$803$	211045.			0.327298
$804$		−	250275.i		−	0.387174i
$805$			192119.i			0.296469i
$806$			528121.i			0.812949i
$807$		−	537747.i		−	0.825716i
$808$	918675.			1.40715
$809$		−	187550.i		−	0.286563i	−0.989682	−	0.143282i	$-0.954235\pi$
$809$		−	187550.i		−	0.286563i	0.989682	−	0.143282i	$-0.0457654\pi$
$810$			45960.1i			0.0700504i
$811$			752282.i			1.14377i	0.820333	+	0.571886i	$0.193788\pi$
$811$			752282.i			1.14377i	−0.820333	+	0.571886i	$0.806212\pi$
$812$	−153351.			−0.232581
$813$	750117.			1.13487
$814$		−	279181.i		−	0.421343i
$815$	43039.2			0.0647961
$816$	−12330.1			−0.0185177
$817$		−	1.35906e6i		−	2.03608i
$818$	−95163.6			−0.142221
$819$		−	74182.5i		−	0.110595i
$820$	131687.			0.195846
$821$		−	763503.i		−	1.13272i	−0.824156	−	0.566362i	$-0.808350\pi$
$821$		−	763503.i		−	1.13272i	0.824156	−	0.566362i	$-0.191650\pi$
$822$			101821.i			0.150694i
$823$			1.04357e6i			1.54071i	0.637616	+	0.770355i	$0.279921\pi$
$823$			1.04357e6i			1.54071i	−0.637616	+	0.770355i	$0.720079\pi$
$824$	−13417.4			−0.0197612
$825$			11027.7i			0.0162024i
$826$	140114.	+	46914.3i	0.205363	+	0.0687615i
$827$	−291033.			−0.425531			−0.212766	−	0.977103i	$-0.568247\pi$
$827$	−291033.			−0.425531			−0.212766	+	0.977103i	$0.568247\pi$
$828$		−	115593.i		−	0.168605i
$829$	−1.12966e6			−1.64376			−0.821882	−	0.569658i	$-0.807076\pi$
$829$	−1.12966e6			−1.64376			−0.821882	+	0.569658i	$0.807076\pi$
$830$	−470251.			−0.682612
$831$	245543.			0.355571
$832$		−	376796.i		−	0.544327i
$833$	688657.			0.992461
$834$		−	103693.i		−	0.149079i
$835$	−148959.			−0.213645
$836$			306047.i			0.437901i
$837$		−	193666.i		−	0.276442i
$838$	−362100.			−0.515633
$839$		−	415299.i		−	0.589979i	−0.955500	−	0.294990i	$-0.904684\pi$
$839$		−	415299.i		−	0.589979i	0.955500	−	0.294990i	$-0.0953162\pi$
$840$		−	149221.i		−	0.211481i
$841$	50575.2			0.0715066
$842$	−627824.			−0.885552
$843$	−164882.			−0.232016
$844$		−	176590.i		−	0.247903i
$845$	98478.3			0.137920
$846$	−158986.			−0.222136
$847$	−220715.			−0.307656
$848$	−11023.9			−0.0153301
$849$		−	67547.1i		−	0.0937112i
$850$		−	37909.4i		−	0.0524697i
$851$	1.08972e6			1.50472
$852$	−146537.			−0.201869
$853$	−1.12470e6			−1.54575			−0.772876	−	0.634558i	$-0.781182\pi$
$853$	−1.12470e6			−1.54575			−0.772876	+	0.634558i	$0.781182\pi$
$854$	−242305.			−0.332236
$855$	474974.			0.649737
$856$		−	808775.i		−	1.10377i
$857$			1.37093e6i			1.86661i	0.359085	+	0.933305i	$0.383089\pi$
$857$			1.37093e6i			1.86661i	−0.359085	+	0.933305i	$0.616911\pi$
$858$		−	88891.7i		−	0.120750i
$859$			763286.i			1.03443i	0.855855	+	0.517215i	$0.173031\pi$
$859$			763286.i			1.03443i	−0.855855	+	0.517215i	$0.826969\pi$
$860$		−	524159.i		−	0.708706i
$861$	−45662.7			−0.0615964
$862$	−632438.			−0.851145
$863$		−	850435.i		−	1.14188i	−0.820992	−	0.570939i	$-0.806579\pi$
$863$		−	850435.i		−	1.14188i	0.820992	−	0.570939i	$-0.193421\pi$
$864$			144843.i			0.194031i
$865$		−	1.25348e6i		−	1.67527i
$866$			683898.i			0.911918i
$867$	126850.			0.168753
$868$			243166.i			0.322747i
$869$			387669.i			0.513360i
$870$			285186.i			0.376782i
$871$	751243.			0.990248
$872$	1.14493e6			1.50572
$873$		−	159632.i		−	0.209456i
$874$	699837.			0.916166
$875$	261485.			0.341531
$876$		−	247436.i		−	0.322444i
$877$	−1.17892e6			−1.53280			−0.766400	−	0.642364i	$-0.777954\pi$
$877$	−1.17892e6			−1.53280			−0.766400	+	0.642364i	$0.777954\pi$
$878$		−	550365.i		−	0.713940i
$879$	−143558.			−0.185801
$880$			8375.21i			0.0108151i
$881$		−	932263.i		−	1.20112i	−0.799579	−	0.600560i	$-0.794944\pi$
$881$		−	932263.i		−	1.20112i	0.799579	−	0.600560i	$-0.205056\pi$
$882$		−	137597.i		−	0.176877i
$883$	569874.			0.730900			0.365450	−	0.930831i	$-0.380915\pi$
$883$	569874.			0.730900			0.365450	+	0.930831i	$0.380915\pi$
$884$		−	521608.i		−	0.667482i
$885$	−148926.	+	444781.i	−0.190144	+	0.567884i
$886$	−464879.			−0.592206
$887$			147269.i			0.187183i	0.995611	+	0.0935913i	$0.0298347\pi$
$887$			147269.i			0.187183i	−0.995611	+	0.0935913i	$0.970165\pi$
$888$	−846398.			−1.07337
$889$	436904.			0.552819
$890$	252307.			0.318530
$891$			32597.4i			0.0410608i
$892$	545252.			0.685279
$893$			1.64304e6i			2.06037i
$894$	−471706.			−0.590196
$895$			220138.i			0.274820i
$896$		−	186769.i		−	0.232642i
$897$	346970.			0.431229
$898$			499318.i			0.619191i
$899$		−	1.20172e6i		−	1.48690i
$900$	12929.3			0.0159621
$901$	501426.			0.617671
$902$	−54716.9			−0.0672526
$903$			181753.i			0.222898i
$904$	610899.			0.747536
$905$	−1.64873e6			−2.01304
$906$	130293.			0.158732
$907$	−546243.			−0.664005			−0.332003	−	0.943278i	$-0.607724\pi$
$907$	−546243.			−0.664005			−0.332003	+	0.943278i	$0.607724\pi$
$908$			251147.i			0.304619i
$909$		−	391060.i		−	0.473277i
$910$	173217.			0.209174
$911$	819476.			0.987414			0.493707	−	0.869628i	$-0.335641\pi$
$911$	819476.			0.987414			0.493707	+	0.869628i	$0.335641\pi$
$912$	25460.1			0.0306106
$913$	−333527.			−0.400120
$914$	−354260.			−0.424063
$915$		−	769177.i		−	0.918722i
$916$			817694.i			0.974540i
$917$		−	36074.0i		−	0.0428998i
$918$		−	112058.i		−	0.132971i
$919$		−	1.44513e6i		−	1.71111i	−0.517715	−	0.855553i	$-0.673217\pi$
$919$		−	1.44513e6i		−	1.71111i	0.517715	−	0.855553i	$-0.326783\pi$
$920$	697945.			0.824604
$921$	−133822.			−0.157764
$922$		−	223016.i		−	0.262346i
$923$		−	439857.i		−	0.516307i
$924$		−	40928.9i		−	0.0479387i
$925$			121888.i			0.142455i
$926$	−883923.			−1.03084
$927$			5711.49i			0.00664645i
$928$			898764.i			1.04364i
$929$			69295.6i			0.0802923i	0.999194	+	0.0401462i	$0.0127824\pi$
$929$			69295.6i			0.0802923i	−0.999194	+	0.0401462i	$0.987218\pi$
$930$	452214.			0.522851
$931$	−1.42199e6			−1.64058
$932$		−	932896.i		−	1.07399i
$933$	642909.			0.738561
$934$	261937.			0.300264
$935$		−	380949.i		−	0.435756i
$936$	−269495.			−0.307609
$937$			403625.i			0.459726i	0.973223	+	0.229863i	$0.0738278\pi$
$937$			403625.i			0.459726i	−0.973223	+	0.229863i	$0.926172\pi$
$938$	−202641.			−0.230314
$939$		−	66636.4i		−	0.0755754i
$940$			633681.i			0.717159i
$941$		−	496072.i		−	0.560229i	−0.959967	−	0.280114i	$-0.909628\pi$
$941$		−	496072.i		−	0.560229i	0.959967	−	0.280114i	$-0.0903724\pi$
$942$	138825.			0.156446
$943$		−	213576.i		−	0.240176i
$944$	−7982.90	+	23841.7i	−0.00895811	+	0.0267542i
$945$	−63520.2			−0.0711293
$946$			217792.i			0.243366i
$947$	145308.			0.162028			0.0810139	−	0.996713i	$-0.474184\pi$
$947$	145308.			0.162028			0.0810139	+	0.996713i	$0.474184\pi$
$948$	454516.			0.505746
$949$	742720.			0.824693
$950$			78278.3i			0.0867350i
$951$	−708387.			−0.783267
$952$			363824.i			0.401438i
$953$	1.62459e6			1.78879			0.894394	−	0.447280i	$-0.147607\pi$
$953$	1.62459e6			1.78879			0.894394	+	0.447280i	$0.147607\pi$
$954$		−	100187.i		−	0.110082i
$955$			933714.i			1.02378i
$956$	−673483.			−0.736904
$957$			202269.i			0.220854i
$958$			360998.i			0.393346i
$959$	−140725.			−0.153015
$960$	−322639.			−0.350086
$961$	−982016.			−1.06334
$962$		−	982507.i		−	1.06166i
$963$	−344278.			−0.371242
$964$	−584030.			−0.628465
$965$	191146.			0.205262
$966$	−93592.0			−0.100296
$967$		−	54787.5i		−	0.0585907i	−0.999571	−	0.0292953i	$-0.990674\pi$
$967$		−	54787.5i		−	0.0585907i	0.999571	−	0.0292953i	$-0.00932633\pi$
$968$			801830.i			0.855720i
$969$	−1.15806e6			−1.23334
$970$	372743.			0.396156
$971$	−1.29608e6			−1.37466			−0.687328	−	0.726347i	$-0.741216\pi$
$971$	−1.29608e6			−1.37466			−0.687328	+	0.726347i	$0.741216\pi$
$972$	38218.2			0.0404518
$973$	143311.			0.151375
$974$		−	8276.18i		−	0.00872393i
$975$			38809.4i			0.0408251i
$976$		−	41230.3i		−	0.0432830i
$977$		−	944291.i		−	0.989275i	−0.869099	−	0.494637i	$-0.835301\pi$
$977$		−	944291.i		−	0.989275i	0.869099	−	0.494637i	$-0.164699\pi$
$978$			20966.8i			0.0219207i
$979$	178950.			0.186709
$980$	−548429.			−0.571043
$981$		−	487370.i		−	0.506432i
$982$			794315.i			0.823702i
$983$		−	274967.i		−	0.284560i	−0.989826	−	0.142280i	$-0.954557\pi$
$983$		−	274967.i		−	0.284560i	0.989826	−	0.142280i	$-0.0454433\pi$
$984$			165887.i			0.171325i
$985$	1.63077e6			1.68082
$986$		−	695329.i		−	0.715215i
$987$		−	219730.i		−	0.225557i
$988$			1.07706e6i			1.10338i
$989$	−850105.			−0.869121
$990$	−76115.3			−0.0776608
$991$		−	1.10811e6i		−	1.12833i	−0.825661	−	0.564166i	$-0.809198\pi$
$991$		−	1.10811e6i		−	1.12833i	0.825661	−	0.564166i	$-0.190802\pi$
$992$	1.42515e6			1.44823
$993$	−795766.			−0.807024
$994$			118647.i			0.120084i
$995$	643150.			0.649630
$996$			391039.i			0.394186i
$997$	34272.3			0.0344788			0.0172394	−	0.999851i	$-0.494512\pi$
$997$	34272.3			0.0344788			0.0172394	+	0.999851i	$0.494512\pi$
$998$			413924.i			0.415584i
$999$			360294.i			0.361015i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.5.c.a.58.27	yes	40
3.2	odd	2		531.5.c.d.235.14		40
59.58	odd	2	inner	177.5.c.a.58.14	✓	40
177.176	even	2		531.5.c.d.235.27		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.5.c.a.58.14	✓	40	59.58	odd	2	inner
177.5.c.a.58.27	yes	40	1.1	even	1	trivial
531.5.c.d.235.14		40	3.2	odd	2
531.5.c.d.235.27		40	177.176	even	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)

\(f(q)\)	\(=\)	\(q+2.43119i q^{2} +5.19615 q^{3} +10.0893 q^{4} +25.9319 q^{5} +12.6328i q^{6} -17.4595 q^{7} +63.4281i q^{8} +27.0000 q^{9} +O(q^{10})\)	\(q+2.43119i q^{2} +5.19615 q^{3} +10.0893 q^{4} +25.9319 q^{5} +12.6328i q^{6} -17.4595 q^{7} +63.4281i q^{8} +27.0000 q^{9} +63.0454i q^{10} +44.7152i q^{11} +52.4255 q^{12} +157.364i q^{13} -42.4475i q^{14} +134.746 q^{15} +7.22282 q^{16} -328.532 q^{17} +65.6422i q^{18} +678.379 q^{19} +261.635 q^{20} -90.7223 q^{21} -108.711 q^{22} -424.331i q^{23} +329.582i q^{24} +47.4624 q^{25} -382.582 q^{26} +140.296 q^{27} -176.154 q^{28} +870.549 q^{29} +327.594i q^{30} -1380.41i q^{31} +1032.41i q^{32} +232.347i q^{33} -798.725i q^{34} -452.758 q^{35} +272.411 q^{36} +2568.09i q^{37} +1649.27i q^{38} +817.687i q^{39} +1644.81i q^{40} +503.324 q^{41} -220.563i q^{42} -2003.40i q^{43} +451.145i q^{44} +700.161 q^{45} +1031.63 q^{46} +2422.01i q^{47} +37.5309 q^{48} -2096.17 q^{49} +115.390i q^{50} -1707.10 q^{51} +1587.69i q^{52} -1526.26 q^{53} +341.087i q^{54} +1159.55i q^{55} -1107.42i q^{56} +3524.96 q^{57} +2116.47i q^{58} +(-1105.23 + 3300.88i) q^{59} +1359.49 q^{60} -5708.35i q^{61} +3356.05 q^{62} -471.407 q^{63} -2394.42 q^{64} +4080.74i q^{65} -564.880 q^{66} -4773.92i q^{67} -3314.66 q^{68} -2204.89i q^{69} -1100.74i q^{70} -2795.15 q^{71} +1712.56i q^{72} -4719.76i q^{73} -6243.53 q^{74} +246.622 q^{75} +6844.37 q^{76} -780.705i q^{77} -1987.96 q^{78} +8669.75 q^{79} +187.301 q^{80} +729.000 q^{81} +1223.68i q^{82} +7458.93i q^{83} -915.325 q^{84} -8519.45 q^{85} +4870.65 q^{86} +4523.51 q^{87} -2836.20 q^{88} -4001.99i q^{89} +1702.23i q^{90} -2747.50i q^{91} -4281.21i q^{92} -7172.83i q^{93} -5888.37 q^{94} +17591.6 q^{95} +5364.56i q^{96} -5912.30i q^{97} -5096.18i q^{98} +1207.31i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 320 q^{4} + 80 q^{7} + 1080 q^{9}+O(q^{10}) \) 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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