LMFDB - Embedded newform 177.5.c.a.58.11

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			58.11
Character	$\chi$	$=$	177.58
Dual form			177.5.c.a.58.30

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.10319i q^{2} -5.19615 q^{3} -0.836195 q^{4} -39.6276 q^{5} +21.3208i q^{6} -85.8400 q^{7} -62.2200i q^{8} +27.0000 q^{9} +O(q^{10})$	\(q-4.10319i q^{2} -5.19615 q^{3} -0.836195 q^{4} -39.6276 q^{5} +21.3208i q^{6} -85.8400 q^{7} -62.2200i q^{8} +27.0000 q^{9} +162.600i q^{10} +68.0678i q^{11} +4.34500 q^{12} -254.341i q^{13} +352.218i q^{14} +205.911 q^{15} -268.680 q^{16} +229.593 q^{17} -110.786i q^{18} +425.861 q^{19} +33.1364 q^{20} +446.038 q^{21} +279.295 q^{22} +954.505i q^{23} +323.305i q^{24} +945.343 q^{25} -1043.61 q^{26} -140.296 q^{27} +71.7790 q^{28} -396.928 q^{29} -844.892i q^{30} +1421.82i q^{31} +106.925i q^{32} -353.691i q^{33} -942.065i q^{34} +3401.63 q^{35} -22.5773 q^{36} -996.969i q^{37} -1747.39i q^{38} +1321.60i q^{39} +2465.63i q^{40} -1020.06 q^{41} -1830.18i q^{42} -2587.59i q^{43} -56.9180i q^{44} -1069.94 q^{45} +3916.52 q^{46} +1477.61i q^{47} +1396.10 q^{48} +4967.50 q^{49} -3878.93i q^{50} -1193.00 q^{51} +212.679i q^{52} -3156.03 q^{53} +575.662i q^{54} -2697.36i q^{55} +5340.97i q^{56} -2212.84 q^{57} +1628.67i q^{58} +(3480.96 - 16.5981i) q^{59} -172.182 q^{60} -2170.20i q^{61} +5834.00 q^{62} -2317.68 q^{63} -3860.14 q^{64} +10078.9i q^{65} -1451.26 q^{66} -667.653i q^{67} -191.985 q^{68} -4959.75i q^{69} -13957.5i q^{70} -2586.30 q^{71} -1679.94i q^{72} -1200.66i q^{73} -4090.76 q^{74} -4912.15 q^{75} -356.103 q^{76} -5842.94i q^{77} +5422.76 q^{78} -1243.99 q^{79} +10647.1 q^{80} +729.000 q^{81} +4185.50i q^{82} +6111.54i q^{83} -372.975 q^{84} -9098.21 q^{85} -10617.4 q^{86} +2062.50 q^{87} +4235.18 q^{88} +12034.3i q^{89} +4390.19i q^{90} +21832.7i q^{91} -798.152i q^{92} -7387.98i q^{93} +6062.92 q^{94} -16875.8 q^{95} -555.600i q^{96} +16943.5i q^{97} -20382.6i q^{98} +1837.83i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	4.10319i		−	1.02580i	−0.858449	−	0.512899i	$-0.828571\pi$
$2$		−	4.10319i		−	1.02580i	0.858449	−	0.512899i	$-0.171429\pi$
$3$	−5.19615			−0.577350
$3$	−5.19615			−0.577350
$4$	−0.836195			−0.0522622
$5$	−39.6276			−1.58510			−0.792551	−	0.609805i	$-0.791248\pi$
$5$	−39.6276			−1.58510			−0.792551	+	0.609805i	$0.791248\pi$
$6$			21.3208i			0.592245i
$7$	−85.8400			−1.75184			−0.875918	−	0.482459i	$-0.839744\pi$
$7$	−85.8400			−1.75184			−0.875918	+	0.482459i	$0.839744\pi$
$8$		−	62.2200i		−	0.972188i
$9$	27.0000			0.333333
$10$			162.600i			1.62600i
$11$			68.0678i			0.562544i	0.959628	+	0.281272i	$0.0907563\pi$
$11$			68.0678i			0.562544i	−0.959628	+	0.281272i	$0.909244\pi$
$12$	4.34500			0.0301736
$13$		−	254.341i		−	1.50498i	−0.658605	−	0.752489i	$-0.728853\pi$
$13$		−	254.341i		−	1.50498i	0.658605	−	0.752489i	$-0.271147\pi$
$14$			352.218i			1.79703i
$15$	205.911			0.915159
$16$	−268.680			−1.04953
$17$	229.593			0.794440			0.397220	−	0.917724i	$-0.369975\pi$
$17$	229.593			0.794440			0.397220	+	0.917724i	$0.369975\pi$
$18$		−	110.786i		−	0.341933i
$19$	425.861			1.17967			0.589836	−	0.807523i	$-0.299193\pi$
$19$	425.861			1.17967			0.589836	+	0.807523i	$0.299193\pi$
$20$	33.1364			0.0828409
$21$	446.038			1.01142
$22$	279.295			0.577056
$23$			954.505i			1.80436i	0.431363	+	0.902178i	$0.358033\pi$
$23$			954.505i			1.80436i	−0.431363	+	0.902178i	$0.641967\pi$
$24$			323.305i			0.561293i
$25$	945.343			1.51255
$26$	−1043.61			−1.54380
$27$	−140.296			−0.192450
$28$	71.7790			0.0915548
$29$	−396.928			−0.471971			−0.235986	−	0.971757i	$-0.575832\pi$
$29$	−396.928			−0.471971			−0.235986	+	0.971757i	$0.575832\pi$
$30$		−	844.892i		−	0.938769i
$31$			1421.82i			1.47952i	0.672871	+	0.739760i	$0.265061\pi$
$31$			1421.82i			1.47952i	−0.672871	+	0.739760i	$0.734939\pi$
$32$			106.925i			0.104419i
$33$		−	353.691i		−	0.324785i
$34$		−	942.065i		−	0.814935i
$35$	3401.63			2.77684
$36$	−22.5773			−0.0174207
$37$		−	996.969i		−	0.728246i	−0.931351	−	0.364123i	$-0.881369\pi$
$37$		−	996.969i		−	0.728246i	0.931351	−	0.364123i	$-0.118631\pi$
$38$		−	1747.39i		−	1.21010i
$39$			1321.60i			0.868899i
$40$			2465.63i			1.54102i
$41$	−1020.06			−0.606817			−0.303408	−	0.952861i	$-0.598125\pi$
$41$	−1020.06			−0.606817			−0.303408	+	0.952861i	$0.598125\pi$
$42$		−	1830.18i		−	1.03752i
$43$		−	2587.59i		−	1.39945i	−0.714410	−	0.699727i	$-0.753305\pi$
$43$		−	2587.59i		−	1.39945i	0.714410	−	0.699727i	$-0.246695\pi$
$44$		−	56.9180i		−	0.0293998i
$45$	−1069.94			−0.528367
$46$	3916.52			1.85091
$47$			1477.61i			0.668905i	0.942413	+	0.334452i	$0.108551\pi$
$47$			1477.61i			0.668905i	−0.942413	+	0.334452i	$0.891449\pi$
$48$	1396.10			0.605947
$49$	4967.50			2.06893
$50$		−	3878.93i		−	1.55157i
$51$	−1193.00			−0.458670
$52$			212.679i			0.0786535i
$53$	−3156.03			−1.12354			−0.561770	−	0.827293i	$-0.689880\pi$
$53$	−3156.03			−1.12354			−0.561770	+	0.827293i	$0.689880\pi$
$54$			575.662i			0.197415i
$55$		−	2697.36i		−	0.891689i
$56$			5340.97i			1.70311i
$57$	−2212.84			−0.681083
$58$			1628.67i			0.484147i
$59$	3480.96	−	16.5981i	0.999989	−	0.00476820i
$59$	3480.96	−	16.5981i	0.999989	−	0.00476820i
$60$	−172.182			−0.0478282
$61$		−	2170.20i		−	0.583230i	−0.956536	−	0.291615i	$-0.905807\pi$
$61$		−	2170.20i		−	0.583230i	0.956536	−	0.291615i	$-0.0941927\pi$
$62$	5834.00			1.51769
$63$	−2317.68			−0.583946
$64$	−3860.14			−0.942418
$65$			10078.9i			2.38554i
$66$	−1451.26			−0.333164
$67$		−	667.653i		−	0.148731i	−0.997231	−	0.0743655i	$-0.976307\pi$
$67$		−	667.653i		−	0.148731i	0.997231	−	0.0743655i	$-0.0236931\pi$
$68$	−191.985			−0.0415192
$69$		−	4959.75i		−	1.04175i
$70$		−	13957.5i		−	2.84848i
$71$	−2586.30			−0.513054			−0.256527	−	0.966537i	$-0.582578\pi$
$71$	−2586.30			−0.513054			−0.256527	+	0.966537i	$0.582578\pi$
$72$		−	1679.94i		−	0.324063i
$73$		−	1200.66i		−	0.225306i	−0.993634	−	0.112653i	$-0.964065\pi$
$73$		−	1200.66i		−	0.225306i	0.993634	−	0.112653i	$-0.0359349\pi$
$74$	−4090.76			−0.747034
$75$	−4912.15			−0.873271
$76$	−356.103			−0.0616522
$77$		−	5842.94i		−	0.985485i
$78$	5422.76			0.891316
$79$	−1243.99			−0.199325			−0.0996626	−	0.995021i	$-0.531776\pi$
$79$	−1243.99			−0.199325			−0.0996626	+	0.995021i	$0.531776\pi$
$80$	10647.1			1.66361
$81$	729.000			0.111111
$82$			4185.50i			0.622471i
$83$			6111.54i			0.887145i	0.896239	+	0.443572i	$0.146289\pi$
$83$			6111.54i			0.887145i	−0.896239	+	0.443572i	$0.853711\pi$
$84$	−372.975			−0.0528592
$85$	−9098.21			−1.25927
$86$	−10617.4			−1.43556
$87$	2062.50			0.272493
$88$	4235.18			0.546898
$89$			12034.3i			1.51929i	0.650340	+	0.759643i	$0.274626\pi$
$89$			12034.3i			1.51929i	−0.650340	+	0.759643i	$0.725374\pi$
$90$			4390.19i			0.541998i
$91$			21832.7i			2.63648i
$92$		−	798.152i		−	0.0942997i
$93$		−	7387.98i		−	0.854201i
$94$	6062.92			0.686161
$95$	−16875.8			−1.86990
$96$		−	555.600i		−	0.0602864i
$97$			16943.5i			1.80078i	0.435085	+	0.900389i	$0.356718\pi$
$97$			16943.5i			1.80078i	−0.435085	+	0.900389i	$0.643282\pi$
$98$		−	20382.6i		−	2.12231i
$99$			1837.83i			0.187515i
$100$	−790.492			−0.0790492
$101$			8659.11i			0.848849i	0.905463	+	0.424424i	$0.139524\pi$
$101$			8659.11i			0.848849i	−0.905463	+	0.424424i	$0.860476\pi$
$102$			4895.11i			0.470503i
$103$		−	1183.70i		−	0.111575i	−0.998443	−	0.0557873i	$-0.982233\pi$
$103$		−	1183.70i		−	0.111575i	0.998443	−	0.0557873i	$-0.0177669\pi$
$104$	−15825.1			−1.46312
$105$	−17675.4			−1.60321
$106$			12949.8i			1.15253i
$107$	11910.3			1.04029			0.520145	−	0.854078i	$-0.325878\pi$
$107$	11910.3			1.04029			0.520145	+	0.854078i	$0.325878\pi$
$108$	117.315			0.0100579
$109$			5882.90i			0.495152i	0.968868	+	0.247576i	$0.0796340\pi$
$109$			5882.90i			0.495152i	−0.968868	+	0.247576i	$0.920366\pi$
$110$	−11067.8			−0.914693
$111$			5180.40i			0.420453i
$112$	23063.5			1.83861
$113$		−	21194.2i		−	1.65982i	−0.557900	−	0.829908i	$-0.688393\pi$
$113$		−	21194.2i		−	1.65982i	0.557900	−	0.829908i	$-0.311607\pi$
$114$			9079.71i			0.698654i
$115$		−	37824.7i		−	2.86009i
$116$	331.909			0.0246663
$117$		−	6867.21i		−	0.501659i
$118$	−68.1052	−	14283.1i	−0.00489121	−	1.02579i
$119$	−19708.3			−1.39173
$120$		−	12811.8i		−	0.889707i
$121$	10007.8			0.683545
$122$	−8904.75			−0.598277
$123$	5300.38			0.350346
$124$		−	1188.92i		−	0.0773230i
$125$	−12694.4			−0.812443
$126$			9509.89i			0.599010i
$127$	26800.6			1.66164			0.830820	−	0.556541i	$-0.187872\pi$
$127$	26800.6			1.66164			0.830820	+	0.556541i	$0.187872\pi$
$128$			17549.7i			1.07115i
$129$			13445.5i			0.807976i
$130$	41355.8			2.44709
$131$			1959.64i			0.114192i	0.998369	+	0.0570958i	$0.0181840\pi$
$131$			1959.64i			0.114192i	−0.998369	+	0.0570958i	$0.981816\pi$
$132$			295.754i			0.0169740i
$133$	−36555.9			−2.06659
$134$	−2739.51			−0.152568
$135$	5559.59			0.305053
$136$		−	14285.3i		−	0.772345i
$137$	−3510.97			−0.187062			−0.0935310	−	0.995616i	$-0.529815\pi$
$137$	−3510.97			−0.187062			−0.0935310	+	0.995616i	$0.529815\pi$
$138$	−20350.8			−1.06862
$139$	−8435.85			−0.436616			−0.218308	−	0.975880i	$-0.570054\pi$
$139$	−8435.85			−0.436616			−0.218308	+	0.975880i	$0.570054\pi$
$140$	−2844.43			−0.145124
$141$		−	7677.89i		−	0.386192i
$142$			10612.1i			0.526290i
$143$	17312.4			0.846616
$144$	−7254.36			−0.349844
$145$	15729.3			0.748123
$146$	−4926.52			−0.231119
$147$	−25811.9			−1.19450
$148$			833.661i			0.0380597i
$149$			25735.7i			1.15921i	0.814896	+	0.579607i	$0.196794\pi$
$149$			25735.7i			1.15921i	−0.814896	+	0.579607i	$0.803206\pi$
$150$			20155.5i			0.895800i
$151$			13763.3i			0.603628i	0.953367	+	0.301814i	$0.0975921\pi$
$151$			13763.3i			0.603628i	−0.953367	+	0.301814i	$0.902408\pi$
$152$		−	26497.1i		−	1.14686i
$153$	6199.01			0.264813
$154$	−23974.7			−1.01091
$155$		−	56343.2i		−	2.34519i
$156$		−	1105.11i		−	0.0454106i
$157$			5484.96i			0.222523i	0.993791	+	0.111261i	$0.0354891\pi$
$157$			5484.96i			0.222523i	−0.993791	+	0.111261i	$0.964511\pi$
$158$			5104.33i			0.204467i
$159$	16399.2			0.648676
$160$		−	4237.19i		−	0.165515i
$161$		−	81934.7i		−	3.16094i
$162$		−	2991.23i		−	0.113978i
$163$	137.440			0.00517295			0.00258648	−	0.999997i	$-0.499177\pi$
$163$	137.440			0.00517295			0.00258648	+	0.999997i	$0.499177\pi$
$164$	852.968			0.0317136
$165$			14015.9i			0.514817i
$166$	25076.8			0.910032
$167$	−4707.17			−0.168782			−0.0843911	−	0.996433i	$-0.526895\pi$
$167$	−4707.17			−0.168782			−0.0843911	+	0.996433i	$0.526895\pi$
$168$		−	27752.5i		−	0.983293i
$169$	−36128.5			−1.26496
$170$			37331.7i			1.29176i
$171$	11498.3			0.393224
$172$			2163.73i			0.0731386i
$173$			25132.5i			0.839737i	0.907585	+	0.419869i	$0.137924\pi$
$173$			25132.5i			0.839737i	−0.907585	+	0.419869i	$0.862076\pi$
$174$		−	8462.83i		−	0.279523i
$175$	−81148.3			−2.64974
$176$		−	18288.4i		−	0.590407i
$177$	−18087.6	+	86.2463i	−0.577344	+	0.00275292i
$178$	49378.9			1.55848
$179$			16585.2i			0.517623i	0.965928	+	0.258811i	$0.0833308\pi$
$179$			16585.2i			0.517623i	−0.965928	+	0.258811i	$0.916669\pi$
$180$	894.682			0.0276136
$181$	51985.4			1.58681			0.793404	−	0.608695i	$-0.208307\pi$
$181$	51985.4			1.58681			0.793404	+	0.608695i	$0.208307\pi$
$182$	89583.6			2.70449
$183$			11276.7i			0.336728i
$184$	59389.3			1.75417
$185$			39507.4i			1.15434i
$186$	−30314.3			−0.876238
$187$			15627.9i			0.446907i
$188$		−	1235.57i		−	0.0349584i
$189$	12043.0			0.337141
$190$			69244.8i			1.91814i
$191$		−	39194.8i		−	1.07439i	−0.843458	−	0.537195i	$-0.819484\pi$
$191$		−	39194.8i		−	1.07439i	0.843458	−	0.537195i	$-0.180516\pi$
$192$	20057.9			0.544105
$193$	36220.3			0.972384			0.486192	−	0.873852i	$-0.338386\pi$
$193$	36220.3			0.972384			0.486192	+	0.873852i	$0.338386\pi$
$194$	69522.6			1.84724
$195$		−	52371.6i		−	1.37729i
$196$	−4153.80			−0.108127
$197$	37466.1			0.965396			0.482698	−	0.875787i	$-0.339657\pi$
$197$	37466.1			0.965396			0.482698	+	0.875787i	$0.339657\pi$
$198$	7540.97			0.192352
$199$	−76403.0			−1.92932			−0.964660	−	0.263497i	$-0.915124\pi$
$199$	−76403.0			−1.92932			−0.964660	+	0.263497i	$0.915124\pi$
$200$		−	58819.3i		−	1.47048i
$201$			3469.23i			0.0858698i
$202$	35530.0			0.870748
$203$	34072.3			0.826817
$204$	997.581			0.0239711
$205$	40422.4			0.961866
$206$	−4856.93			−0.114453
$207$			25771.6i			0.601452i
$208$			68336.4i			1.57952i
$209$			28987.4i			0.663616i
$210$			72525.5i			1.64457i
$211$		−	24399.1i		−	0.548036i	−0.961725	−	0.274018i	$-0.911647\pi$
$211$		−	24399.1i		−	0.548036i	0.961725	−	0.274018i	$-0.0883527\pi$
$212$	2639.05			0.0587187
$213$	13438.8			0.296212
$214$		−	48870.2i		−	1.06713i
$215$			102540.i			2.21828i
$216$			8729.23i			0.187098i
$217$		−	122049.i		−	2.59188i
$218$	24138.7			0.507926
$219$			6238.79i			0.130081i
$220$			2255.52i			0.0466016i
$221$		−	58395.0i		−	1.19561i
$222$	21256.2			0.431300
$223$	−26520.0			−0.533290			−0.266645	−	0.963795i	$-0.585915\pi$
$223$	−26520.0			−0.533290			−0.266645	+	0.963795i	$0.585915\pi$
$224$		−	9178.46i		−	0.182925i
$225$	25524.3			0.504183
$226$	−86963.9			−1.70264
$227$			71618.1i			1.38986i	0.719077	+	0.694930i	$0.244565\pi$
$227$			71618.1i			1.38986i	−0.719077	+	0.694930i	$0.755435\pi$
$228$	1850.37			0.0355949
$229$			42669.2i			0.813660i	0.913504	+	0.406830i	$0.133366\pi$
$229$			42669.2i			0.813660i	−0.913504	+	0.406830i	$0.866634\pi$
$230$	−155202.			−2.93388
$231$			30360.8i			0.568970i
$232$			24696.9i			0.458845i
$233$			68586.1i			1.26335i	0.775233	+	0.631676i	$0.217633\pi$
$233$			68586.1i			1.26335i	−0.775233	+	0.631676i	$0.782367\pi$
$234$	−28177.5			−0.514601
$235$		−	58554.1i		−	1.06028i
$236$	−2910.76	+	13.8793i	−0.0522616	+	0.000249197i
$237$	6463.95			0.115080
$238$			80866.8i			1.42763i
$239$	963.908			0.0168749			0.00843743	−	0.999964i	$-0.497314\pi$
$239$	963.908			0.0168749			0.00843743	+	0.999964i	$0.497314\pi$
$240$	−55324.1			−0.960488
$241$	16329.4			0.281149			0.140574	−	0.990070i	$-0.455105\pi$
$241$	16329.4			0.281149			0.140574	+	0.990070i	$0.455105\pi$
$242$		−	41063.8i		−	0.701179i
$243$	−3788.00			−0.0641500
$244$			1814.71i			0.0304809i
$245$	−196850.			−3.27947
$246$		−	21748.5i		−	0.359384i
$247$		−	108314.i		−	1.77538i
$248$	88465.6			1.43837
$249$		−	31756.5i		−	0.512193i
$250$			52087.7i			0.833403i
$251$	57856.7			0.918346			0.459173	−	0.888347i	$-0.348146\pi$
$251$	57856.7			0.918346			0.459173	+	0.888347i	$0.348146\pi$
$252$	1938.03			0.0305183
$253$	−64971.0			−1.01503
$254$		−	109968.i		−	1.70451i
$255$	47275.7			0.727039
$256$	10247.6			0.156366
$257$	−94257.4			−1.42708			−0.713542	−	0.700613i	$-0.752910\pi$
$257$	−94257.4			−1.42708			−0.713542	+	0.700613i	$0.752910\pi$
$258$	55169.6			0.828820
$259$			85579.8i			1.27577i
$260$		−	8427.95i		−	0.124674i
$261$	−10717.1			−0.157324
$262$	8040.79			0.117137
$263$	6388.77			0.0923646			0.0461823	−	0.998933i	$-0.485294\pi$
$263$	6388.77			0.0923646			0.0461823	+	0.998933i	$0.485294\pi$
$264$	−22006.6			−0.315752
$265$	125066.			1.78093
$266$			149996.i			2.11991i
$267$		−	62531.9i		−	0.877160i
$268$			558.288i			0.00777301i
$269$		−	50975.4i		−	0.704460i	−0.935914	−	0.352230i	$-0.885424\pi$
$269$		−	50975.4i		−	0.704460i	0.935914	−	0.352230i	$-0.114576\pi$
$270$		−	22812.1i		−	0.312923i
$271$	103419.			1.40820			0.704098	−	0.710103i	$-0.251352\pi$
$271$	103419.			1.40820			0.704098	+	0.710103i	$0.251352\pi$
$272$	−61687.0			−0.833789
$273$		−	113446.i		−	1.52217i
$274$			14406.2i			0.191888i
$275$			64347.4i			0.850875i
$276$			4147.32i			0.0544439i
$277$	−118682.			−1.54676			−0.773381	−	0.633941i	$-0.781436\pi$
$277$	−118682.			−1.54676			−0.773381	+	0.633941i	$0.781436\pi$
$278$			34613.9i			0.447879i
$279$			38389.1i			0.493173i
$280$		−	211649.i		−	2.69961i
$281$	−63690.0			−0.806600			−0.403300	−	0.915068i	$-0.632137\pi$
$281$	−63690.0			−0.806600			−0.403300	+	0.915068i	$0.632137\pi$
$282$	−31503.9			−0.396155
$283$			7522.96i			0.0939325i	0.998896	+	0.0469662i	$0.0149553\pi$
$283$			7522.96i			0.0939325i	−0.998896	+	0.0469662i	$0.985045\pi$
$284$	2162.66			0.0268133
$285$	87689.4			1.07959
$286$		−	71036.3i		−	0.868457i
$287$	87561.8			1.06304
$288$			2886.98i			0.0348064i
$289$	−30808.0			−0.368866
$290$		−	64540.3i		−	0.767423i
$291$		−	88041.1i		−	1.03968i
$292$			1003.98i			0.0117750i
$293$	−32964.4			−0.383981			−0.191990	−	0.981397i	$-0.561494\pi$
$293$	−32964.4			−0.383981			−0.191990	+	0.981397i	$0.561494\pi$
$294$			105911.i			1.22531i
$295$	−137942.	+	657.742i	−1.58508	+	0.00755809i
$296$	−62031.4			−0.707992
$297$		−	9549.65i		−	0.108262i
$298$	105599.			1.18912
$299$	242770.			2.71552
$300$	4107.52			0.0456391
$301$			222119.i			2.45162i
$302$	56473.5			0.619200
$303$		−	44994.0i		−	0.490083i
$304$	−114420.			−1.23810
$305$			85999.7i			0.924480i
$306$		−	25435.7i		−	0.271645i
$307$	−85864.5			−0.911039			−0.455520	−	0.890226i	$-0.650547\pi$
$307$	−85864.5			−0.911039			−0.455520	+	0.890226i	$0.650547\pi$
$308$			4885.84i			0.0515036i
$309$			6150.66i			0.0644177i
$310$	−231187.			−2.40569
$311$	−20927.4			−0.216368			−0.108184	−	0.994131i	$-0.534504\pi$
$311$	−20927.4			−0.216368			−0.108184	+	0.994131i	$0.534504\pi$
$312$	82229.7			0.844733
$313$		−	132516.i		−	1.35264i	−0.736610	−	0.676318i	$-0.763575\pi$
$313$		−	132516.i		−	1.35264i	0.736610	−	0.676318i	$-0.236425\pi$
$314$	22505.9			0.228263
$315$	91844.0			0.925613
$316$	1040.22			0.0104172
$317$	−108659.			−1.08130			−0.540649	−	0.841248i	$-0.681821\pi$
$317$	−108659.			−1.08130			−0.540649	+	0.841248i	$0.681821\pi$
$318$		−	67289.0i		−	0.665411i
$319$		−	27018.0i		−	0.265504i
$320$	152968.			1.49383
$321$	−61887.6			−0.600611
$322$	−336194.			−3.24249
$323$	97774.8			0.937177
$324$	−609.586			−0.00580691
$325$		−	240440.i		−	2.27635i
$326$		−	563.944i		−	0.00530641i
$327$		−	30568.5i		−	0.285876i
$328$			63468.1i			0.589940i
$329$		−	126838.i		−	1.17181i
$330$	57509.9			0.528098
$331$	−101850.			−0.929619			−0.464809	−	0.885411i	$-0.653877\pi$
$331$	−101850.			−0.929619			−0.464809	+	0.885411i	$0.653877\pi$
$332$		−	5110.44i		−	0.0463641i
$333$		−	26918.2i		−	0.242749i
$334$			19314.4i			0.173137i
$335$			26457.5i			0.235754i
$336$	−119841.			−1.06152
$337$		−	74871.2i		−	0.659258i	−0.944111	−	0.329629i	$-0.893076\pi$
$337$		−	74871.2i		−	0.659258i	0.944111	−	0.329629i	$-0.106924\pi$
$338$			148242.i			1.29759i
$339$			110128.i			0.958295i
$340$	7607.88			0.0658121
$341$	−96780.0			−0.832294
$342$		−	47179.6i		−	0.403368i
$343$	−220309.			−1.87259
$344$	−161000.			−1.36053
$345$			196543.i			1.65127i
$346$	103123.			0.861401
$347$		−	90583.5i		−	0.752298i	−0.926559	−	0.376149i	$-0.877248\pi$
$347$		−	90583.5i		−	0.752298i	0.926559	−	0.376149i	$-0.122752\pi$
$348$	−1724.65			−0.0142411
$349$			134492.i			1.10420i	0.833779	+	0.552098i	$0.186172\pi$
$349$			134492.i			1.10420i	−0.833779	+	0.552098i	$0.813828\pi$
$350$			332967.i			2.71810i
$351$			35683.1i			0.289633i
$352$	−7278.16			−0.0587403
$353$			81414.3i			0.653358i	0.945135	+	0.326679i	$0.105930\pi$
$353$			81414.3i			0.653358i	−0.945135	+	0.326679i	$0.894070\pi$
$354$	353.885	+	74216.9i	0.00282394	+	0.592238i
$355$	102489.			0.813243
$356$		−	10063.0i		−	0.0794013i
$357$	102407.			0.803515
$358$	68052.1			0.530977
$359$	64385.0			0.499570			0.249785	−	0.968301i	$-0.419640\pi$
$359$	64385.0			0.499570			0.249785	+	0.968301i	$0.419640\pi$
$360$			66571.9i			0.513672i
$361$	51036.8			0.391624
$362$		−	213306.i		−	1.62775i
$363$	−52001.9			−0.394645
$364$		−	18256.4i		−	0.137788i
$365$			47579.1i			0.357133i
$366$	46270.4			0.345415
$367$			129178.i			0.959083i	0.877519	+	0.479541i	$0.159197\pi$
$367$			129178.i			0.959083i	−0.877519	+	0.479541i	$0.840803\pi$
$368$		−	256456.i		−	1.89373i
$369$	−27541.6			−0.202272
$370$	162107.			1.18412
$371$	270913.			1.96826
$372$			6177.80i			0.0446424i
$373$	−71866.8			−0.516548			−0.258274	−	0.966072i	$-0.583154\pi$
$373$	−71866.8			−0.516548			−0.258274	+	0.966072i	$0.583154\pi$
$374$	64124.3			0.458436
$375$	65962.2			0.469064
$376$	91936.9			0.650301
$377$			100955.i			0.710306i
$378$		−	49414.8i		−	0.345839i
$379$	85467.7			0.595009			0.297504	−	0.954720i	$-0.403846\pi$
$379$	85467.7			0.595009			0.297504	+	0.954720i	$0.403846\pi$
$380$	14111.5			0.0977251
$381$	−139260.			−0.959349
$382$	−160824.			−1.10211
$383$	278371.			1.89769			0.948846	−	0.315739i	$-0.102253\pi$
$383$	278371.			1.89769			0.948846	+	0.315739i	$0.102253\pi$
$384$		−	91191.0i		−	0.618429i
$385$			231541.i			1.56209i
$386$		−	148619.i		−	0.997470i
$387$		−	69865.0i		−	0.466485i
$388$		−	14168.1i		−	0.0941127i
$389$	−21226.2			−0.140273			−0.0701364	−	0.997537i	$-0.522343\pi$
$389$	−21226.2			−0.140273			−0.0701364	+	0.997537i	$0.522343\pi$
$390$	−214891.			−1.41283
$391$			219148.i			1.43345i
$392$		−	309078.i		−	2.01139i
$393$		−	10182.6i		−	0.0659285i
$394$		−	153731.i		−	0.990302i
$395$	49296.2			0.315951
$396$		−	1536.78i		−	0.00979992i
$397$			45611.1i			0.289394i	0.989476	+	0.144697i	$0.0462208\pi$
$397$			45611.1i			0.289394i	−0.989476	+	0.144697i	$0.953779\pi$
$398$			313496.i			1.97909i
$399$	189950.			1.19315
$400$	−253995.			−1.58747
$401$			56817.4i			0.353340i	0.984270	+	0.176670i	$0.0565325\pi$
$401$			56817.4i			0.353340i	−0.984270	+	0.176670i	$0.943468\pi$
$402$	14234.9			0.0880851
$403$	361627.			2.22664
$404$		−	7240.70i		−	0.0443627i
$405$	−28888.5			−0.176122
$406$		−	139805.i		−	0.848147i
$407$	67861.5			0.409670
$408$			74228.5i			0.445913i
$409$			152246.i			0.910123i	0.890460	+	0.455061i	$0.150383\pi$
$409$			152246.i			0.910123i	−0.890460	+	0.455061i	$0.849617\pi$
$410$		−	165861.i		−	0.986681i
$411$	18243.5			0.108000
$412$			989.801i			0.00583114i
$413$	−298806.	+	1424.78i	−1.75182	+	0.00835311i
$414$	105746.			0.616969
$415$		−	242185.i		−	1.40622i
$416$	27195.5			0.157149
$417$	43834.0			0.252080
$418$	118941.			0.680737
$419$			14956.8i			0.0851945i	0.999092	+	0.0425972i	$0.0135632\pi$
$419$			14956.8i			0.0851945i	−0.999092	+	0.0425972i	$0.986437\pi$
$420$	14780.1			0.0837873
$421$			192146.i			1.08410i	0.840347	+	0.542048i	$0.182351\pi$
$421$			192146.i			1.08410i	−0.840347	+	0.542048i	$0.817649\pi$
$422$	−100114.			−0.562174
$423$			39895.5i			0.222968i
$424$			196368.i			1.09229i
$425$	217044.			1.20163
$426$		−	55142.1i		−	0.303854i
$427$			186290.i			1.02172i
$428$	−9959.31			−0.0543678
$429$	−89958.1			−0.488794
$430$	420741.			2.27551
$431$			134989.i			0.726679i	0.931657	+	0.363340i	$0.118364\pi$
$431$			134989.i			0.726679i	−0.931657	+	0.363340i	$0.881636\pi$
$432$	37694.7			0.201982
$433$	−27732.6			−0.147916			−0.0739580	−	0.997261i	$-0.523563\pi$
$433$	−27732.6			−0.147916			−0.0739580	+	0.997261i	$0.523563\pi$
$434$	−500790.			−2.65874
$435$	−81731.7			−0.431929
$436$		−	4919.25i		−	0.0258777i
$437$			406487.i			2.12855i
$438$	25599.0			0.133436
$439$	−256732.			−1.33215			−0.666073	−	0.745887i	$-0.732026\pi$
$439$	−256732.			−1.33215			−0.666073	+	0.745887i	$0.732026\pi$
$440$	−167830.			−0.866889
$441$	134123.			0.689644
$442$	−239606.			−1.22646
$443$			104213.i			0.531022i	0.964108	+	0.265511i	$0.0855407\pi$
$443$			104213.i			0.531022i	−0.964108	+	0.265511i	$0.914459\pi$
$444$		−	4331.83i		−	0.0219738i
$445$		−	476889.i		−	2.40822i
$446$			108817.i			0.547048i
$447$		−	133727.i		−	0.669272i
$448$	331355.			1.65096
$449$	15237.3			0.0755814			0.0377907	−	0.999286i	$-0.487968\pi$
$449$	15237.3			0.0755814			0.0377907	+	0.999286i	$0.487968\pi$
$450$		−	104731.i		−	0.517190i
$451$		−	69433.1i		−	0.341361i
$452$			17722.5i			0.0867457i
$453$		−	71516.3i		−	0.348505i
$454$	293863.			1.42572
$455$		−	865175.i		−	4.17908i
$456$			137683.i			0.662141i
$457$		−	351744.i		−	1.68420i	−0.539318	−	0.842102i	$-0.681318\pi$
$457$		−	351744.i		−	1.68420i	0.539318	−	0.842102i	$-0.318682\pi$
$458$	175080.			0.834651
$459$	−32211.0			−0.152890
$460$			31628.8i			0.149475i
$461$	−91900.9			−0.432432			−0.216216	−	0.976346i	$-0.569372\pi$
$461$	−91900.9			−0.432432			−0.216216	+	0.976346i	$0.569372\pi$
$462$	124576.			0.583648
$463$			284633.i			1.32777i	0.747835	+	0.663885i	$0.231093\pi$
$463$			284633.i			1.32777i	−0.747835	+	0.663885i	$0.768907\pi$
$464$	106647.			0.495348
$465$			292768.i			1.35400i
$466$	281422.			1.29594
$467$		−	304437.i		−	1.39593i	−0.716132	−	0.697965i	$-0.754089\pi$
$467$		−	304437.i		−	1.39593i	0.716132	−	0.697965i	$-0.245911\pi$
$468$			5742.33i			0.0262178i
$469$			57311.3i			0.260552i
$470$	−240259.			−1.08764
$471$		−	28500.7i		−	0.128474i
$472$	−1032.73	−	216585.i	−0.00463559	−	0.972177i
$473$	176132.			0.787254
$474$		−	26522.9i		−	0.118049i
$475$	402585.			1.78431
$476$	16480.0			0.0727348
$477$	−85212.7			−0.374514
$478$		−	3955.10i		−	0.0173102i
$479$	397464.			1.73231			0.866157	−	0.499772i	$-0.166583\pi$
$479$	397464.			1.73231			0.866157	+	0.499772i	$0.166583\pi$
$480$			22017.1i			0.0955602i
$481$	−253570.			−1.09599
$482$		−	67002.7i		−	0.288402i
$483$			425745.i			1.82497i
$484$	−8368.46			−0.0357235
$485$		−	671431.i		−	2.85442i
$486$			15542.9i			0.0658050i
$487$	−157762.			−0.665187			−0.332593	−	0.943070i	$-0.607924\pi$
$487$	−157762.			−0.665187			−0.332593	+	0.943070i	$0.607924\pi$
$488$	−135030.			−0.567010
$489$	−714.160			−0.00298660
$490$			807714.i			3.36407i
$491$	76943.9			0.319162			0.159581	−	0.987185i	$-0.448986\pi$
$491$	76943.9			0.319162			0.159581	+	0.987185i	$0.448986\pi$
$492$	−4432.15			−0.0183098
$493$	−91131.9			−0.374953
$494$	−444434.			−1.82118
$495$		−	72828.7i		−	0.297230i
$496$		−	382014.i		−	1.55280i
$497$	222008.			0.898787
$498$	−130303.			−0.525407
$499$	1118.26			0.00449100			0.00224550	−	0.999997i	$-0.499285\pi$
$499$	1118.26			0.00449100			0.00224550	+	0.999997i	$0.499285\pi$
$500$	10615.0			0.0424601
$501$	24459.2			0.0974465
$502$		−	237397.i		−	0.942038i
$503$		−	143292.i		−	0.566351i	−0.959068	−	0.283176i	$-0.908612\pi$
$503$		−	143292.i		−	0.566351i	0.959068	−	0.283176i	$-0.0913879\pi$
$504$			144206.i			0.567705i
$505$		−	343139.i		−	1.34551i
$506$			266589.i			1.04122i
$507$	187729.			0.730324
$508$	−22410.5			−0.0868410
$509$			39802.5i			0.153630i	0.997045	+	0.0768148i	$0.0244750\pi$
$509$			39802.5i			0.153630i	−0.997045	+	0.0768148i	$0.975525\pi$
$510$		−	193981.i		−	0.745795i
$511$			103064.i			0.394699i
$512$			238748.i			0.910750i
$513$	−59746.7			−0.227028
$514$			386756.i			1.46390i
$515$			46907.0i			0.176857i
$516$		−	11243.1i		−	0.0422266i
$517$	−100578.			−0.376288
$518$	351150.			1.30868
$519$		−	130592.i		−	0.484823i
$520$	627111.			2.31920
$521$	−228531.			−0.841917			−0.420958	−	0.907080i	$-0.638306\pi$
$521$	−228531.			−0.841917			−0.420958	+	0.907080i	$0.638306\pi$
$522$			43974.1i			0.161382i
$523$	508793.			1.86011			0.930054	−	0.367423i	$-0.119760\pi$
$523$	508793.			1.86011			0.930054	+	0.367423i	$0.119760\pi$
$524$		−	1638.64i		−	0.00596790i
$525$	421659.			1.52983
$526$		−	26214.4i		−	0.0947475i
$527$			326440.i			1.17539i
$528$			95029.5i			0.340872i
$529$	−631238.			−2.25570
$530$		−	513168.i		−	1.82687i
$531$	93985.9	−	448.149i	0.333330	−	0.00158940i
$532$	30567.9			0.108005
$533$			259443.i			0.913246i
$534$	−256580.			−0.899790
$535$	−471975.			−1.64897
$536$	−41541.4			−0.144594
$537$		−	86179.0i		−	0.298850i
$538$	−209162.			−0.722634
$539$			338127.i			1.16386i
$540$	−4648.90			−0.0159427
$541$		−	560174.i		−	1.91394i	−0.290187	−	0.956970i	$-0.593718\pi$
$541$		−	560174.i		−	1.91394i	0.290187	−	0.956970i	$-0.406282\pi$
$542$		−	424349.i		−	1.44452i
$543$	−270124.			−0.916144
$544$			24549.3i			0.0829547i
$545$		−	233125.i		−	0.784867i
$546$	−465490.			−1.56144
$547$	−273979.			−0.915679			−0.457839	−	0.889035i	$-0.651377\pi$
$547$	−273979.			−0.915679			−0.457839	+	0.889035i	$0.651377\pi$
$548$	2935.85			0.00977627
$549$		−	58595.4i		−	0.194410i
$550$	264030.			0.872826
$551$	−169036.			−0.556771
$552$	−308596.			−1.01277
$553$	106784.			0.349185
$554$			486973.i			1.58667i
$555$		−	205287.i		−	0.666461i
$556$	7054.02			0.0228185
$557$	339532.			1.09439			0.547193	−	0.837006i	$-0.315696\pi$
$557$	339532.			1.09439			0.547193	+	0.837006i	$0.315696\pi$
$558$	157518.			0.505896
$559$	−658131.			−2.10615
$560$	−913949.			−2.91438
$561$		−	81204.9i		−	0.258022i
$562$			261332.i			0.827409i
$563$			92097.7i			0.290558i	0.989391	+	0.145279i	$0.0464079\pi$
$563$			92097.7i			0.290558i	−0.989391	+	0.145279i	$0.953592\pi$
$564$			6420.21i			0.0201833i
$565$			839874.i			2.63098i
$566$	30868.1			0.0963558
$567$	−62577.4			−0.194649
$568$			160920.i			0.498785i
$569$		−	443913.i		−	1.37111i	−0.728019	−	0.685557i	$-0.759559\pi$
$569$		−	443913.i		−	1.37111i	0.728019	−	0.685557i	$-0.240441\pi$
$570$		−	359807.i		−	1.10744i
$571$		−	254417.i		−	0.780321i	−0.920747	−	0.390161i	$-0.872419\pi$
$571$		−	254417.i		−	0.780321i	0.920747	−	0.390161i	$-0.127581\pi$
$572$	−14476.6			−0.0442460
$573$			203662.i			0.620299i
$574$		−	359283.i		−	1.09047i
$575$			902335.i			2.72918i
$576$	−104224.			−0.314139
$577$	311437.			0.935445			0.467723	−	0.883875i	$-0.345075\pi$
$577$	311437.			0.935445			0.467723	+	0.883875i	$0.345075\pi$
$578$			126411.i			0.378382i
$579$	−188206.			−0.561406
$580$	−13152.8			−0.0390985
$581$		−	524615.i		−	1.55413i
$582$	−361250.			−1.06650
$583$		−	214824.i		−	0.632041i
$584$	−74704.8			−0.219040
$585$			272131.i			0.795181i
$586$			135259.i			0.393887i
$587$			581088.i			1.68642i	0.537584	+	0.843210i	$0.319337\pi$
$587$			581088.i			1.68642i	−0.537584	+	0.843210i	$0.680663\pi$
$588$	21583.8			0.0624271
$589$			605497.i			1.74535i
$590$	2698.84	+	566003.i	0.00775307	+	1.62598i
$591$	−194679.			−0.557372
$592$			267866.i			0.764317i
$593$	−218284.			−0.620743			−0.310371	−	0.950615i	$-0.600453\pi$
$593$	−218284.			−0.620743			−0.310371	+	0.950615i	$0.600453\pi$
$594$	−39184.0			−0.111055
$595$	780990.			2.20603
$596$		−	21520.1i		−	0.0605830i
$597$	397002.			1.11389
$598$		−	996132.i		−	2.78557i
$599$	−549532.			−1.53158			−0.765789	−	0.643091i	$-0.777651\pi$
$599$	−549532.			−1.53158			−0.765789	+	0.643091i	$0.777651\pi$
$600$			305634.i			0.848983i
$601$			611697.i			1.69351i	0.531984	+	0.846755i	$0.321447\pi$
$601$			611697.i			1.69351i	−0.531984	+	0.846755i	$0.678553\pi$
$602$	911397.			2.51486
$603$		−	18026.6i		−	0.0495770i
$604$		−	11508.8i		−	0.0315469i
$605$	−396584.			−1.08349
$606$	−184619.			−0.502726
$607$	1086.59			0.00294908			0.00147454	−	0.999999i	$-0.499531\pi$
$607$	1086.59			0.00294908			0.00147454	+	0.999999i	$0.499531\pi$
$608$			45535.3i			0.123180i
$609$	−177045.			−0.477363
$610$	352874.			0.948330
$611$	375817.			1.00669
$612$	−5183.58			−0.0138397
$613$			331052.i			0.880999i	0.897753	+	0.440500i	$0.145199\pi$
$613$			331052.i			0.880999i	−0.897753	+	0.440500i	$0.854801\pi$
$614$			352319.i			0.934543i
$615$	−210041.			−0.555334
$616$	−363548.			−0.958076
$617$	−247255.			−0.649492			−0.324746	−	0.945801i	$-0.605279\pi$
$617$	−247255.			−0.649492			−0.324746	+	0.945801i	$0.605279\pi$
$618$	25237.4			0.0660795
$619$	307344.			0.802127			0.401064	−	0.916050i	$-0.368641\pi$
$619$	307344.			0.802127			0.401064	+	0.916050i	$0.368641\pi$
$620$			47113.9i			0.122565i
$621$		−	133913.i		−	0.347249i
$622$			85869.1i			0.221950i
$623$		−	1.03302e6i		−	2.66154i
$624$		−	355086.i		−	0.911937i
$625$	−87790.4			−0.224744
$626$	−543740.			−1.38753
$627$		−	150623.i		−	0.383139i
$628$		−	4586.50i		−	0.0116295i
$629$		−	228897.i		−	0.578548i
$630$		−	376854.i		−	0.949493i
$631$	469556.			1.17931			0.589657	−	0.807654i	$-0.299263\pi$
$631$	469556.			1.17931			0.589657	+	0.807654i	$0.299263\pi$
$632$			77401.0i			0.193782i
$633$			126781.i			0.316409i
$634$			445847.i			1.10919i
$635$	−1.06204e6			−2.63387
$636$	−13712.9			−0.0339013
$637$		−	1.26344e6i		−	3.11370i
$638$	−110860.			−0.272354
$639$	−69830.2			−0.171018
$640$		−	695453.i		−	1.69788i
$641$	244837.			0.595882			0.297941	−	0.954584i	$-0.403700\pi$
$641$	244837.			0.595882			0.297941	+	0.954584i	$0.403700\pi$
$642$			253937.i			0.616106i
$643$	−199751.			−0.483133			−0.241567	−	0.970384i	$-0.577661\pi$
$643$	−199751.			−0.483133			−0.241567	+	0.970384i	$0.577661\pi$
$644$			68513.4i			0.165198i
$645$		−	532813.i		−	1.28072i
$646$		−	401189.i		−	0.961355i
$647$	406276.			0.970538			0.485269	−	0.874365i	$-0.338722\pi$
$647$	406276.			0.970538			0.485269	+	0.874365i	$0.338722\pi$
$648$		−	45358.4i		−	0.108021i
$649$	1129.80	+	236941.i	0.00268232	+	0.562537i
$650$	−986571.			−2.33508
$651$			634185.i			1.49642i
$652$	−114.927			−0.000270350
$653$	334544.			0.784562			0.392281	−	0.919845i	$-0.371686\pi$
$653$	334544.			0.784562			0.392281	+	0.919845i	$0.371686\pi$
$654$	−125428.			−0.293251
$655$		−	77655.8i		−	0.181005i
$656$	274069.			0.636873
$657$		−	32417.7i		−	0.0751020i
$658$	−520441.			−1.20204
$659$			286185.i			0.658986i	0.944158	+	0.329493i	$0.106878\pi$
$659$			286185.i			0.658986i	−0.944158	+	0.329493i	$0.893122\pi$
$660$		−	11720.0i		−	0.0269055i
$661$	759053.			1.73728			0.868639	−	0.495446i	$-0.164995\pi$
$661$	759053.			1.73728			0.868639	+	0.495446i	$0.164995\pi$
$662$			417910.i			0.953601i
$663$			303429.i			0.690288i
$664$	380260.			0.862471
$665$	1.44862e6			3.27576
$666$	−110450.			−0.249011
$667$		−	378870.i		−	0.851605i
$668$	3936.11			0.00882093
$669$	137802.			0.307895
$670$	108560.			0.241836
$671$	147721.			0.328093
$672$			47692.7i			0.105612i
$673$			664503.i			1.46712i	0.679622	+	0.733562i	$0.262144\pi$
$673$			664503.i			1.46712i	−0.679622	+	0.733562i	$0.737856\pi$
$674$	−307211.			−0.676265
$675$	−132628.			−0.291090
$676$	30210.5			0.0661095
$677$	−252653.			−0.551249			−0.275624	−	0.961265i	$-0.588885\pi$
$677$	−252653.			−0.551249			−0.275624	+	0.961265i	$0.588885\pi$
$678$	451878.			0.983018
$679$		−	1.45443e6i		−	3.15467i
$680$			566091.i			1.22425i
$681$		−	372139.i		−	0.802436i
$682$			397107.i			0.853766i
$683$		−	506823.i		−	1.08646i	−0.839583	−	0.543232i	$-0.817200\pi$
$683$		−	506823.i		−	1.08646i	0.839583	−	0.543232i	$-0.182800\pi$
$684$	−9614.79			−0.0205507
$685$	139131.			0.296512
$686$			903969.i			1.92090i
$687$		−	221715.i		−	0.469767i
$688$			695234.i			1.46877i
$689$			802708.i			1.69090i
$690$	806453.			1.69387
$691$			789140.i			1.65271i	0.563146	+	0.826357i	$0.309591\pi$
$691$			789140.i			1.65271i	−0.563146	+	0.826357i	$0.690409\pi$
$692$		−	21015.7i		−	0.0438865i
$693$		−	157759.i		−	0.328495i
$694$	−371682.			−0.771706
$695$	334292.			0.692080
$696$		−	128329.i		−	0.264914i
$697$	−234198.			−0.482079
$698$	551847.			1.13268
$699$		−	356384.i		−	0.729396i
$700$	67855.8			0.138481
$701$		−	259759.i		−	0.528608i	−0.964439	−	0.264304i	$-0.914858\pi$
$701$		−	259759.i		−	0.528608i	0.964439	−	0.264304i	$-0.0851423\pi$
$702$	146415.			0.297105
$703$		−	424570.i		−	0.859091i
$704$		−	262751.i		−	0.530151i
$705$			304256.i			0.612154i
$706$	334059.			0.670213
$707$		−	743298.i		−	1.48704i
$708$	15124.8	−	72.1187i	0.0301733	−	0.000143874i
$709$	171973.			0.342112			0.171056	−	0.985261i	$-0.445282\pi$
$709$	171973.			0.342112			0.171056	+	0.985261i	$0.445282\pi$
$710$		−	420532.i		−	0.834223i
$711$	−33587.7			−0.0664417
$712$	748772.			1.47703
$713$	−1.35713e6			−2.66958
$714$		−	420196.i		−	0.824244i
$715$	−686050.			−1.34197
$716$		−	13868.4i		−	0.0270521i
$717$	−5008.62			−0.00974270
$718$		−	264184.i		−	0.512458i
$719$			823365.i			1.59270i	0.604835	+	0.796351i	$0.293239\pi$
$719$			823365.i			1.59270i	−0.604835	+	0.796351i	$0.706761\pi$
$720$	287472.			0.554538
$721$			101608.i			0.195461i
$722$		−	209414.i		−	0.401727i
$723$	−84850.0			−0.162321
$724$	−43470.0			−0.0829301
$725$	−375233.			−0.713880
$726$			213374.i			0.404826i
$727$	−403704.			−0.763825			−0.381912	−	0.924198i	$-0.624734\pi$
$727$	−403704.			−0.763825			−0.381912	+	0.924198i	$0.624734\pi$
$728$	1.35843e6			2.56315
$729$	19683.0			0.0370370
$730$	195226.			0.366347
$731$		−	594093.i		−	1.11178i
$732$		−	9429.52i		−	0.0175982i
$733$	−472977.			−0.880302			−0.440151	−	0.897924i	$-0.645075\pi$
$733$	−472977.			−0.880302			−0.440151	+	0.897924i	$0.645075\pi$
$734$	530042.			0.983825
$735$	1.02286e6			1.89340
$736$	−102061.			−0.188409
$737$	45445.7			0.0836676
$738$			113008.i			0.207490i
$739$		−	107013.i		−	0.195951i	−0.995189	−	0.0979757i	$-0.968763\pi$
$739$		−	107013.i		−	0.195951i	0.995189	−	0.0979757i	$-0.0312368\pi$
$740$		−	33035.9i		−	0.0603286i
$741$			562817.i			1.02502i
$742$		−	1.11161e6i		−	2.01904i
$743$	−915468.			−1.65831			−0.829155	−	0.559019i	$-0.811178\pi$
$743$	−915468.			−1.65831			−0.829155	+	0.559019i	$0.811178\pi$
$744$	−459681.			−0.830444
$745$		−	1.01984e6i		−	1.83747i
$746$			294883.i			0.529874i
$747$			165012.i			0.295715i
$748$		−	13068.0i		−	0.0233563i
$749$	−1.02238e6			−1.82242
$750$		−	270656.i		−	0.481165i
$751$			832564.i			1.47617i	0.674705	+	0.738087i	$0.264271\pi$
$751$			832564.i			1.47617i	−0.674705	+	0.738087i	$0.735729\pi$
$752$		−	397004.i		−	0.702036i
$753$	−300632.			−0.530208
$754$	414238.			0.728631
$755$		−	545406.i		−	0.956811i
$756$	−10070.3			−0.0176197
$757$	247540.			0.431970			0.215985	−	0.976397i	$-0.430704\pi$
$757$	247540.			0.431970			0.215985	+	0.976397i	$0.430704\pi$
$758$		−	350690.i		−	0.610359i
$759$	337599.			0.586028
$760$			1.05002e6i			1.81789i
$761$	−539545.			−0.931662			−0.465831	−	0.884874i	$-0.654244\pi$
$761$	−539545.			−0.931662			−0.465831	+	0.884874i	$0.654244\pi$
$762$			571411.i			0.984098i
$763$		−	504988.i		−	0.867425i
$764$			32774.5i			0.0561500i
$765$	−245652.			−0.419756
$766$		−	1.14221e6i		−	1.94665i
$767$	−4221.58	−	885352.i	−0.00717604	−	1.50496i
$768$	−53248.0			−0.0902779
$769$		−	97270.6i		−	0.164486i	−0.996612	−	0.0822430i	$-0.973792\pi$
$769$		−	97270.6i		−	0.164486i	0.996612	−	0.0822430i	$-0.0262084\pi$
$770$	950059.			1.60239
$771$	489776.			0.823927
$772$	−30287.3			−0.0508189
$773$			234313.i			0.392136i	0.980590	+	0.196068i	$0.0628173\pi$
$773$			234313.i			0.392136i	−0.980590	+	0.196068i	$0.937183\pi$
$774$	−286670.			−0.478520
$775$			1.34411e6i			2.23785i
$776$	1.05423e6			1.75069
$777$		−	444686.i		−	0.736565i
$778$			87095.3i			0.143892i
$779$	−434403.			−0.715844
$780$			43792.9i			0.0719804i
$781$		−	176044.i		−	0.288615i
$782$	899205.			1.47043
$783$	55687.4			0.0908309
$784$	−1.33467e6			−2.17141
$785$		−	217356.i		−	0.352721i
$786$	−41781.1			−0.0676294
$787$	773520.			1.24888			0.624442	−	0.781071i	$-0.285327\pi$
$787$	773520.			1.24888			0.624442	+	0.781071i	$0.285327\pi$
$788$	−31328.9			−0.0504537
$789$	−33197.0			−0.0533267
$790$		−	202272.i		−	0.324102i
$791$			1.81931e6i			2.90773i
$792$	114350.			0.182299
$793$	−551972.			−0.877749
$794$	187151.			0.296860
$795$	−649860.			−1.02822
$796$	63887.8			0.100831
$797$			254532.i			0.400705i	0.979724	+	0.200353i	$0.0642088\pi$
$797$			254532.i			0.400705i	−0.979724	+	0.200353i	$0.935791\pi$
$798$		−	779402.i		−	1.22393i
$799$			339249.i			0.531404i
$800$			101081.i			0.157939i
$801$			324925.i			0.506429i
$802$	233133.			0.362455
$803$	81726.0			0.126744
$804$		−	2900.95i		−	0.00448775i
$805$			3.24687e6i			5.01041i
$806$		−	1.48383e6i		−	2.28409i
$807$			264876.i			0.406720i
$808$	538770.			0.825240
$809$			3663.27i			0.00559721i	0.999996	+	0.00279861i	$0.000890825\pi$
$809$			3663.27i			0.00559721i	−0.999996	+	0.00279861i	$0.999109\pi$
$810$			118535.i			0.180666i
$811$		−	137551.i		−	0.209133i	−0.994518	−	0.104567i	$-0.966654\pi$
$811$		−	137551.i		−	0.209133i	0.994518	−	0.104567i	$-0.0333456\pi$
$812$	−28491.1			−0.0432113
$813$	−537383.			−0.813022
$814$		−	278449.i		−	0.420239i
$815$	−5446.42			−0.00819966
$816$	320535.			0.481388
$817$		−	1.10196e6i		−	1.65090i
$818$	624696.			0.933602
$819$			589482.i			0.878825i
$820$	−33801.0			−0.0502693
$821$		−	336436.i		−	0.499132i	−0.968358	−	0.249566i	$-0.919712\pi$
$821$		−	336436.i		−	0.499132i	0.968358	−	0.249566i	$-0.0802880\pi$
$822$		−	74856.7i		−	0.110787i
$823$			197126.i			0.291035i	0.989356	+	0.145517i	$0.0464847\pi$
$823$			197126.i			0.291035i	−0.989356	+	0.145517i	$0.953515\pi$
$824$	−73649.6			−0.108472
$825$		−	334359.i		−	0.491253i
$826$	5846.15	+	1.22606e6i	0.00856860	+	1.79701i
$827$	−1.17052e6			−1.71146			−0.855730	−	0.517422i	$-0.826892\pi$
$827$	−1.17052e6			−1.71146			−0.855730	+	0.517422i	$0.826892\pi$
$828$		−	21550.1i		−	0.0314332i
$829$	−596609.			−0.868122			−0.434061	−	0.900884i	$-0.642920\pi$
$829$	−596609.			−0.868122			−0.434061	+	0.900884i	$0.642920\pi$
$830$	−993734.			−1.44249
$831$	616688.			0.893024
$832$			981794.i			1.41832i
$833$	1.14050e6			1.64364
$834$		−	179859.i		−	0.258583i
$835$	186534.			0.267537
$836$		−	24239.2i		−	0.0346821i
$837$		−	199476.i		−	0.284734i
$838$	61370.7			0.0873923
$839$		−	763822.i		−	1.08510i	−0.840025	−	0.542548i	$-0.817460\pi$
$839$		−	763822.i		−	1.08510i	0.840025	−	0.542548i	$-0.182540\pi$
$840$			1.09976e6i			1.55862i
$841$	−549729.			−0.777243
$842$	788414.			1.11206
$843$	330943.			0.465691
$844$			20402.4i			0.0286416i
$845$	1.43168e6			2.00509
$846$	163699.			0.228720
$847$	−859068.			−1.19746
$848$	847961.			1.17919
$849$		−	39090.4i		−	0.0542319i
$850$		−	890575.i		−	1.23263i
$851$	951612.			1.31402
$852$	−11237.5			−0.0154807
$853$	262734.			0.361092			0.180546	−	0.983567i	$-0.442214\pi$
$853$	262734.			0.361092			0.180546	+	0.983567i	$0.442214\pi$
$854$	764384.			1.04808
$855$	−455648.			−0.623300
$856$		−	741057.i		−	1.01136i
$857$			1.18773e6i			1.61717i	0.588382	+	0.808583i	$0.299765\pi$
$857$			1.18773e6i			1.61717i	−0.588382	+	0.808583i	$0.700235\pi$
$858$			369116.i			0.501404i
$859$			403204.i			0.546435i	0.961952	+	0.273217i	$0.0880878\pi$
$859$			403204.i			0.546435i	−0.961952	+	0.273217i	$0.911912\pi$
$860$		−	85743.4i		−	0.115932i
$861$	−454985.			−0.613748
$862$	553885.			0.745426
$863$			709539.i			0.952697i	0.879257	+	0.476348i	$0.158040\pi$
$863$			709539.i			0.952697i	−0.879257	+	0.476348i	$0.841960\pi$
$864$		−	15001.2i		−	0.0200955i
$865$		−	995939.i		−	1.33107i
$866$			113792.i			0.151732i
$867$	160083.			0.212965
$868$			102057.i			0.135457i
$869$		−	84675.5i		−	0.112129i
$870$			335361.i			0.443072i
$871$	−169812.			−0.223837
$872$	366034.			0.481381
$873$			457475.i			0.600259i
$874$	1.66789e6			2.18346
$875$	1.08969e6			1.42327
$876$		−	5216.85i		−	0.00679829i
$877$	557208.			0.724466			0.362233	−	0.932087i	$-0.382014\pi$
$877$	557208.			0.724466			0.362233	+	0.932087i	$0.382014\pi$
$878$			1.05342e6i			1.36651i
$879$	171288.			0.221691
$880$			724726.i			0.935855i
$881$		−	511735.i		−	0.659316i	−0.944101	−	0.329658i	$-0.893067\pi$
$881$		−	511735.i		−	0.659316i	0.944101	−	0.329658i	$-0.106933\pi$
$882$		−	550331.i		−	0.707436i
$883$	1.39799e6			1.79301			0.896507	−	0.443029i	$-0.146096\pi$
$883$	1.39799e6			1.79301			0.896507	+	0.443029i	$0.146096\pi$
$884$			48829.6i			0.0624854i
$885$	716767.	−	3417.73i	0.915149	−	0.00436366i
$886$	427605.			0.544722
$887$			526850.i			0.669638i	0.942282	+	0.334819i	$0.108675\pi$
$887$			526850.i			0.669638i	−0.942282	+	0.334819i	$0.891325\pi$
$888$	322325.			0.408759
$889$	−2.30056e6			−2.91092
$890$	−1.95677e6			−2.47035
$891$			49621.4i			0.0625048i
$892$	22175.9			0.0278709
$893$			629257.i			0.789087i
$894$	−548706.			−0.686538
$895$		−	657229.i		−	0.820485i
$896$		−	1.50647e6i		−	1.87648i
$897$	−1.26147e6			−1.56780
$898$		−	62521.5i		−	0.0775313i
$899$		−	564359.i		−	0.698291i
$900$	−21343.3			−0.0263497
$901$	−724602.			−0.892585
$902$	−284898.			−0.350167
$903$		−	1.15416e6i		−	1.41544i
$904$	−1.31870e6			−1.61365
$905$	−2.06006e6			−2.51525
$906$	−293445.			−0.357495
$907$	−836151.			−1.01641			−0.508206	−	0.861235i	$-0.669691\pi$
$907$	−836151.			−1.01641			−0.508206	+	0.861235i	$0.669691\pi$
$908$		−	59886.7i		−	0.0726371i
$909$			233796.i			0.282950i
$910$	−3.54998e6			−4.28690
$911$	527285.			0.635343			0.317672	−	0.948201i	$-0.397099\pi$
$911$	527285.			0.635343			0.317672	+	0.948201i	$0.397099\pi$
$912$	594546.			0.714818
$913$	−415999.			−0.499058
$914$	−1.44328e6			−1.72765
$915$		−	446868.i		−	0.533749i
$916$		−	35679.7i		−	0.0425237i
$917$		−	168216.i		−	0.200045i
$918$			132168.i			0.156834i
$919$		−	156081.i		−	0.184807i	−0.995722	−	0.0924036i	$-0.970545\pi$
$919$		−	156081.i		−	0.184807i	0.995722	−	0.0924036i	$-0.0294550\pi$
$920$	−2.35345e6			−2.78055
$921$	446165.			0.525989
$922$			377087.i			0.443588i
$923$			657804.i			0.772135i
$924$		−	25387.6i		−	0.0297356i
$925$		−	942478.i		−	1.10151i
$926$	1.16790e6			1.36202
$927$		−	31959.8i		−	0.0371916i
$928$		−	42441.6i		−	0.0492828i
$929$			443250.i			0.513591i	0.966466	+	0.256795i	$0.0826667\pi$
$929$			443250.i			0.513591i	−0.966466	+	0.256795i	$0.917333\pi$
$930$	1.20128e6			1.38893
$931$	2.11547e6			2.44066
$932$		−	57351.3i		−	0.0660255i
$933$	108742.			0.124920
$934$	−1.24916e6			−1.43194
$935$		−	619295.i		−	0.708393i
$936$	−427278.			−0.487707
$937$		−	176637.i		−	0.201188i	−0.994928	−	0.100594i	$-0.967926\pi$
$937$		−	176637.i		−	0.201188i	0.994928	−	0.100594i	$-0.0320743\pi$
$938$	235160.			0.267274
$939$			688575.i			0.780944i
$940$			48962.6i			0.0554127i
$941$		−	1.41132e6i		−	1.59385i	−0.604078	−	0.796925i	$-0.706458\pi$
$941$		−	1.41132e6i		−	1.59385i	0.604078	−	0.796925i	$-0.293542\pi$
$942$	−116944.			−0.131788
$943$		−	973651.i		−	1.09491i
$944$	−935264.	+	4459.58i	−1.04952	+	0.00500437i
$945$	−477235.			−0.534403
$946$		−	722702.i		−	0.807564i
$947$	774655.			0.863791			0.431895	−	0.901924i	$-0.357845\pi$
$947$	774655.			0.863791			0.431895	+	0.901924i	$0.357845\pi$
$948$	−5405.13			−0.00601436
$949$	−305376.			−0.339081
$950$		−	1.65188e6i		−	1.83034i
$951$	564606.			0.624288
$952$			1.22625e6i			1.35302i
$953$	443289.			0.488091			0.244046	−	0.969764i	$-0.421525\pi$
$953$	443289.			0.488091			0.244046	+	0.969764i	$0.421525\pi$
$954$			349644.i			0.384175i
$955$			1.55320e6i			1.70302i
$956$	−806.016			−0.000881917
$957$			140390.i			0.153289i
$958$		−	1.63087e6i		−	1.77701i
$959$	301381.			0.327702
$960$	−794845.			−0.862462
$961$	−1.09805e6			−1.18898
$962$			1.04045e6i			1.12427i
$963$	321577.			0.346763
$964$	−13654.6			−0.0146934
$965$	−1.43532e6			−1.54133
$966$	1.74691e6			1.87205
$967$			376709.i			0.402859i	0.979503	+	0.201429i	$0.0645587\pi$
$967$			376709.i			0.402859i	−0.979503	+	0.201429i	$0.935441\pi$
$968$		−	622684.i		−	0.664534i
$969$	−508053.			−0.541080
$970$	−2.75501e6			−2.92806
$971$	563582.			0.597749			0.298875	−	0.954292i	$-0.403389\pi$
$971$	563582.			0.597749			0.298875	+	0.954292i	$0.403389\pi$
$972$	3167.50			0.00335262
$973$	724133.			0.764879
$974$			647327.i			0.682347i
$975$			1.24936e6i			1.31425i
$976$			583089.i			0.612118i
$977$		−	242271.i		−	0.253812i	−0.991915	−	0.126906i	$-0.959495\pi$
$977$		−	242271.i		−	0.253812i	0.991915	−	0.126906i	$-0.0405046\pi$
$978$			2930.34i			0.00306365i
$979$	−819146.			−0.854665
$980$	164605.			0.171392
$981$			158838.i			0.165051i
$982$		−	315716.i		−	0.327396i
$983$		−	1.65719e6i		−	1.71501i	−0.514478	−	0.857504i	$-0.672014\pi$
$983$		−	1.65719e6i		−	1.71501i	0.514478	−	0.857504i	$-0.327986\pi$
$984$		−	329790.i		−	0.340602i
$985$	−1.48469e6			−1.53025
$986$			373932.i			0.384626i
$987$			659070.i			0.676546i
$988$			90571.7i			0.0927852i
$989$	2.46987e6			2.52512
$990$	−298830.			−0.304898
$991$			1.43312e6i			1.45927i	0.683835	+	0.729636i	$0.260311\pi$
$991$			1.43312e6i			1.45927i	−0.683835	+	0.729636i	$0.739689\pi$
$992$	−152028.			−0.154490
$993$	529228.			0.536716
$994$		−	910943.i		−	0.921974i
$995$	3.02766e6			3.05817
$996$			26554.6i			0.0267683i
$997$	1.54766e6			1.55699			0.778495	−	0.627650i	$-0.215983\pi$
$997$	1.54766e6			1.55699			0.778495	+	0.627650i	$0.215983\pi$
$998$		−	4588.45i		−	0.00460686i
$999$			139871.i			0.140151i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.5.c.a.58.11	✓	40
3.2	odd	2		531.5.c.d.235.30		40
59.58	odd	2	inner	177.5.c.a.58.30	yes	40
177.176	even	2		531.5.c.d.235.11		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.5.c.a.58.11	✓	40	1.1	even	1	trivial
177.5.c.a.58.30	yes	40	59.58	odd	2	inner
531.5.c.d.235.11		40	177.176	even	2
531.5.c.d.235.30		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)

\(f(q)\)	\(=\)	\(q-4.10319i q^{2} -5.19615 q^{3} -0.836195 q^{4} -39.6276 q^{5} +21.3208i q^{6} -85.8400 q^{7} -62.2200i q^{8} +27.0000 q^{9} +O(q^{10})\)	\(q-4.10319i q^{2} -5.19615 q^{3} -0.836195 q^{4} -39.6276 q^{5} +21.3208i q^{6} -85.8400 q^{7} -62.2200i q^{8} +27.0000 q^{9} +162.600i q^{10} +68.0678i q^{11} +4.34500 q^{12} -254.341i q^{13} +352.218i q^{14} +205.911 q^{15} -268.680 q^{16} +229.593 q^{17} -110.786i q^{18} +425.861 q^{19} +33.1364 q^{20} +446.038 q^{21} +279.295 q^{22} +954.505i q^{23} +323.305i q^{24} +945.343 q^{25} -1043.61 q^{26} -140.296 q^{27} +71.7790 q^{28} -396.928 q^{29} -844.892i q^{30} +1421.82i q^{31} +106.925i q^{32} -353.691i q^{33} -942.065i q^{34} +3401.63 q^{35} -22.5773 q^{36} -996.969i q^{37} -1747.39i q^{38} +1321.60i q^{39} +2465.63i q^{40} -1020.06 q^{41} -1830.18i q^{42} -2587.59i q^{43} -56.9180i q^{44} -1069.94 q^{45} +3916.52 q^{46} +1477.61i q^{47} +1396.10 q^{48} +4967.50 q^{49} -3878.93i q^{50} -1193.00 q^{51} +212.679i q^{52} -3156.03 q^{53} +575.662i q^{54} -2697.36i q^{55} +5340.97i q^{56} -2212.84 q^{57} +1628.67i q^{58} +(3480.96 - 16.5981i) q^{59} -172.182 q^{60} -2170.20i q^{61} +5834.00 q^{62} -2317.68 q^{63} -3860.14 q^{64} +10078.9i q^{65} -1451.26 q^{66} -667.653i q^{67} -191.985 q^{68} -4959.75i q^{69} -13957.5i q^{70} -2586.30 q^{71} -1679.94i q^{72} -1200.66i q^{73} -4090.76 q^{74} -4912.15 q^{75} -356.103 q^{76} -5842.94i q^{77} +5422.76 q^{78} -1243.99 q^{79} +10647.1 q^{80} +729.000 q^{81} +4185.50i q^{82} +6111.54i q^{83} -372.975 q^{84} -9098.21 q^{85} -10617.4 q^{86} +2062.50 q^{87} +4235.18 q^{88} +12034.3i q^{89} +4390.19i q^{90} +21832.7i q^{91} -798.152i q^{92} -7387.98i q^{93} +6062.92 q^{94} -16875.8 q^{95} -555.600i q^{96} +16943.5i q^{97} -20382.6i q^{98} +1837.83i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 320 q^{4} + 80 q^{7} + 1080 q^{9}+O(q^{10}) \) 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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