LMFDB - Embedded newform 177.5.c.a.58.39

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			58.39
Character	$\chi$	$=$	177.58
Dual form			177.5.c.a.58.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+7.77798i q^{2} -5.19615 q^{3} -44.4970 q^{4} +28.5836 q^{5} -40.4156i q^{6} +72.7001 q^{7} -221.650i q^{8} +27.0000 q^{9} +O(q^{10})$	\(q+7.77798i q^{2} -5.19615 q^{3} -44.4970 q^{4} +28.5836 q^{5} -40.4156i q^{6} +72.7001 q^{7} -221.650i q^{8} +27.0000 q^{9} +222.323i q^{10} +9.33885i q^{11} +231.213 q^{12} +106.784i q^{13} +565.461i q^{14} -148.525 q^{15} +1012.03 q^{16} +337.515 q^{17} +210.006i q^{18} +546.806 q^{19} -1271.89 q^{20} -377.761 q^{21} -72.6374 q^{22} -39.3036i q^{23} +1151.72i q^{24} +192.023 q^{25} -830.565 q^{26} -140.296 q^{27} -3234.94 q^{28} -444.446 q^{29} -1155.22i q^{30} -1230.69i q^{31} +4325.19i q^{32} -48.5261i q^{33} +2625.19i q^{34} +2078.03 q^{35} -1201.42 q^{36} -1943.27i q^{37} +4253.05i q^{38} -554.866i q^{39} -6335.54i q^{40} -835.576 q^{41} -2938.22i q^{42} +2242.91i q^{43} -415.551i q^{44} +771.758 q^{45} +305.703 q^{46} +3914.09i q^{47} -5258.68 q^{48} +2884.31 q^{49} +1493.55i q^{50} -1753.78 q^{51} -4751.57i q^{52} +3820.28 q^{53} -1091.22i q^{54} +266.938i q^{55} -16113.9i q^{56} -2841.29 q^{57} -3456.89i q^{58} +(-3000.50 - 1764.76i) q^{59} +6608.91 q^{60} +2344.07i q^{61} +9572.29 q^{62} +1962.90 q^{63} -17448.7 q^{64} +3052.27i q^{65} +377.435 q^{66} +3650.67i q^{67} -15018.4 q^{68} +204.228i q^{69} +16162.9i q^{70} -2630.49 q^{71} -5984.54i q^{72} +8924.21i q^{73} +15114.7 q^{74} -997.781 q^{75} -24331.3 q^{76} +678.935i q^{77} +4315.74 q^{78} +1497.53 q^{79} +28927.6 q^{80} +729.000 q^{81} -6499.09i q^{82} +3985.30i q^{83} +16809.2 q^{84} +9647.40 q^{85} -17445.3 q^{86} +2309.41 q^{87} +2069.95 q^{88} +2531.49i q^{89} +6002.72i q^{90} +7763.21i q^{91} +1748.89i q^{92} +6394.85i q^{93} -30443.7 q^{94} +15629.7 q^{95} -22474.3i q^{96} -9210.22i q^{97} +22434.1i q^{98} +252.149i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 320 q^{4} + 80 q^{7} + 1080 q^{9}+O(q^{10}) $ 40 * q - 320 * q^4 + 80 * q^7 + 1080 * q^9	$ 40 q - 320 q^{4} + 80 q^{7} + 1080 q^{9} + 360 q^{12} + 144 q^{15} + 3944 q^{16} - 528 q^{17} + 444 q^{19} + 444 q^{20} + 1304 q^{22} + 4880 q^{25} - 1452 q^{26} - 1160 q^{28} - 996 q^{29} + 10320 q^{35} - 8640 q^{36} - 5196 q^{41} - 10476 q^{46} + 576 q^{48} + 5104 q^{49} + 936 q^{51} - 2184 q^{53} - 2520 q^{57} - 11736 q^{59} - 11448 q^{60} + 15240 q^{62} + 2160 q^{63} - 81012 q^{64} + 17352 q^{66} + 29568 q^{68} - 5964 q^{71} + 14376 q^{74} - 2736 q^{75} + 3480 q^{76} + 37692 q^{78} + 19020 q^{79} + 33096 q^{80} + 29160 q^{81} + 25128 q^{84} + 20220 q^{85} - 65880 q^{86} + 1512 q^{87} - 14932 q^{88} - 17864 q^{94} + 11004 q^{95}+O(q^{100}) $ 40 * q - 320 * q^4 + 80 * q^7 + 1080 * q^9 + 360 * q^12 + 144 * q^15 + 3944 * q^16 - 528 * q^17 + 444 * q^19 + 444 * q^20 + 1304 * q^22 + 4880 * q^25 - 1452 * q^26 - 1160 * q^28 - 996 * q^29 + 10320 * q^35 - 8640 * q^36 - 5196 * q^41 - 10476 * q^46 + 576 * q^48 + 5104 * q^49 + 936 * q^51 - 2184 * q^53 - 2520 * q^57 - 11736 * q^59 - 11448 * q^60 + 15240 * q^62 + 2160 * q^63 - 81012 * q^64 + 17352 * q^66 + 29568 * q^68 - 5964 * q^71 + 14376 * q^74 - 2736 * q^75 + 3480 * q^76 + 37692 * q^78 + 19020 * q^79 + 33096 * q^80 + 29160 * q^81 + 25128 * q^84 + 20220 * q^85 - 65880 * q^86 + 1512 * q^87 - 14932 * q^88 - 17864 * q^94 + 11004 * q^95

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$			7.77798i			1.94450i	0.233953	+	0.972248i	$0.424834\pi$
$2$			7.77798i			1.94450i	−0.233953	+	0.972248i	$0.575166\pi$
$3$	−5.19615			−0.577350
$3$	−5.19615			−0.577350
$4$	−44.4970			−2.78106
$5$	28.5836			1.14334			0.571672	−	0.820482i	$-0.306295\pi$
$5$	28.5836			1.14334			0.571672	+	0.820482i	$0.306295\pi$
$6$		−	40.4156i		−	1.12266i
$7$	72.7001			1.48368			0.741838	−	0.670579i	$-0.233954\pi$
$7$	72.7001			1.48368			0.741838	+	0.670579i	$0.233954\pi$
$8$		−	221.650i		−	3.46327i
$9$	27.0000			0.333333
$10$			222.323i			2.22323i
$11$			9.33885i			0.0771806i	0.999255	+	0.0385903i	$0.0122867\pi$
$11$			9.33885i			0.0771806i	−0.999255	+	0.0385903i	$0.987713\pi$
$12$	231.213			1.60565
$13$			106.784i			0.631858i	0.948783	+	0.315929i	$0.102316\pi$
$13$			106.784i			0.631858i	−0.948783	+	0.315929i	$0.897684\pi$
$14$			565.461i			2.88500i
$15$	−148.525			−0.660110
$16$	1012.03			3.95326
$17$	337.515			1.16787			0.583936	−	0.811800i	$-0.301512\pi$
$17$	337.515			1.16787			0.583936	+	0.811800i	$0.301512\pi$
$18$			210.006i			0.648165i
$19$	546.806			1.51470			0.757349	−	0.653010i	$-0.226494\pi$
$19$	546.806			1.51470			0.757349	+	0.653010i	$0.226494\pi$
$20$	−1271.89			−3.17972
$21$	−377.761			−0.856601
$22$	−72.6374			−0.150077
$23$		−	39.3036i		−	0.0742979i	−0.999310	−	0.0371490i	$-0.988172\pi$
$23$		−	39.3036i		−	0.0742979i	0.999310	−	0.0371490i	$-0.0118276\pi$
$24$			1151.72i			1.99952i
$25$	192.023			0.307237
$26$	−830.565			−1.22865
$27$	−140.296			−0.192450
$28$	−3234.94			−4.12620
$29$	−444.446			−0.528473			−0.264236	−	0.964458i	$-0.585120\pi$
$29$	−444.446			−0.528473			−0.264236	+	0.964458i	$0.585120\pi$
$30$		−	1155.22i		−	1.28358i
$31$		−	1230.69i		−	1.28063i	−0.768110	−	0.640317i	$-0.778803\pi$
$31$		−	1230.69i		−	1.28063i	0.768110	−	0.640317i	$-0.221197\pi$
$32$			4325.19i			4.22382i
$33$		−	48.5261i		−	0.0445602i
$34$			2625.19i			2.27092i
$35$	2078.03			1.69635
$36$	−1201.42			−0.927022
$37$		−	1943.27i		−	1.41948i	−0.704464	−	0.709740i	$-0.748812\pi$
$37$		−	1943.27i		−	1.41948i	0.704464	−	0.709740i	$-0.251188\pi$
$38$			4253.05i			2.94532i
$39$		−	554.866i		−	0.364804i
$40$		−	6335.54i		−	3.95971i
$41$	−835.576			−0.497071			−0.248535	−	0.968623i	$-0.579949\pi$
$41$	−835.576			−0.497071			−0.248535	+	0.968623i	$0.579949\pi$
$42$		−	2938.22i		−	1.66566i
$43$			2242.91i			1.21304i	0.795069	+	0.606519i	$0.207435\pi$
$43$			2242.91i			1.21304i	−0.795069	+	0.606519i	$0.792565\pi$
$44$		−	415.551i		−	0.214644i
$45$	771.758			0.381115
$46$	305.703			0.144472
$47$			3914.09i			1.77188i	0.463797	+	0.885942i	$0.346487\pi$
$47$			3914.09i			1.77188i	−0.463797	+	0.885942i	$0.653513\pi$
$48$	−5258.68			−2.28241
$49$	2884.31			1.20130
$50$			1493.55i			0.597421i
$51$	−1753.78			−0.674271
$52$		−	4751.57i		−	1.75724i
$53$	3820.28			1.36001			0.680006	−	0.733206i	$-0.261977\pi$
$53$	3820.28			1.36001			0.680006	+	0.733206i	$0.261977\pi$
$54$		−	1091.22i		−	0.374218i
$55$			266.938i			0.0882440i
$56$		−	16113.9i		−	5.13838i
$57$	−2841.29			−0.874511
$58$		−	3456.89i		−	1.02761i
$59$	−3000.50	−	1764.76i	−0.861964	−	0.506970i
$59$	−3000.50	−	1764.76i	−0.861964	−	0.506970i
$60$	6608.91			1.83581
$61$			2344.07i			0.629958i	0.949099	+	0.314979i	$0.101997\pi$
$61$			2344.07i			0.629958i	−0.949099	+	0.314979i	$0.898003\pi$
$62$	9572.29			2.49019
$63$	1962.90			0.494559
$64$	−17448.7			−4.25994
$65$			3052.27i			0.722432i
$66$	377.435			0.0866472
$67$			3650.67i			0.813249i	0.913595	+	0.406624i	$0.133294\pi$
$67$			3650.67i			0.813249i	−0.913595	+	0.406624i	$0.866706\pi$
$68$	−15018.4			−3.24793
$69$			204.228i			0.0428959i
$70$			16162.9i			3.29855i
$71$	−2630.49			−0.521819			−0.260909	−	0.965363i	$-0.584022\pi$
$71$	−2630.49			−0.521819			−0.260909	+	0.965363i	$0.584022\pi$
$72$		−	5984.54i		−	1.15442i
$73$			8924.21i			1.67465i	0.546705	+	0.837325i	$0.315882\pi$
$73$			8924.21i			1.67465i	−0.546705	+	0.837325i	$0.684118\pi$
$74$	15114.7			2.76017
$75$	−997.781			−0.177383
$76$	−24331.3			−4.21247
$77$			678.935i			0.114511i
$78$	4315.74			0.709359
$79$	1497.53			0.239951			0.119976	−	0.992777i	$-0.461718\pi$
$79$	1497.53			0.239951			0.119976	+	0.992777i	$0.461718\pi$
$80$	28927.6			4.51993
$81$	729.000			0.111111
$82$		−	6499.09i		−	0.966552i
$83$			3985.30i			0.578503i	0.957253	+	0.289251i	$0.0934063\pi$
$83$			3985.30i			0.578503i	−0.957253	+	0.289251i	$0.906594\pi$
$84$	16809.2			2.38226
$85$	9647.40			1.33528
$86$	−17445.3			−2.35875
$87$	2309.41			0.305114
$88$	2069.95			0.267297
$89$			2531.49i			0.319593i	0.987150	+	0.159796i	$0.0510838\pi$
$89$			2531.49i			0.319593i	−0.987150	+	0.159796i	$0.948916\pi$
$90$			6002.72i			0.741076i
$91$			7763.21i			0.937473i
$92$			1748.89i			0.206627i
$93$			6394.85i			0.739375i
$94$	−30443.7			−3.44542
$95$	15629.7			1.73182
$96$		−	22474.3i		−	2.43862i
$97$		−	9210.22i		−	0.978874i	−0.872039	−	0.489437i	$-0.837202\pi$
$97$		−	9210.22i		−	0.978874i	0.872039	−	0.489437i	$-0.162798\pi$
$98$			22434.1i			2.33591i
$99$			252.149i			0.0257269i
$100$	−8544.45			−0.854445
$101$		−	13290.3i		−	1.30284i	−0.758717	−	0.651420i	$-0.774174\pi$
$101$		−	13290.3i		−	1.30284i	0.758717	−	0.651420i	$-0.225826\pi$
$102$		−	13640.9i		−	1.31112i
$103$		−	14436.1i		−	1.36074i	−0.732867	−	0.680372i	$-0.761818\pi$
$103$		−	14436.1i		−	1.36074i	0.732867	−	0.680372i	$-0.238182\pi$
$104$	23668.6			2.18830
$105$	−10797.8			−0.979390
$106$			29714.0i			2.64454i
$107$	−986.412			−0.0861570			−0.0430785	−	0.999072i	$-0.513717\pi$
$107$	−986.412			−0.0861570			−0.0430785	+	0.999072i	$0.513717\pi$
$108$	6242.76			0.535216
$109$		−	14934.0i		−	1.25697i	−0.777824	−	0.628483i	$-0.783676\pi$
$109$		−	14934.0i		−	1.25697i	0.777824	−	0.628483i	$-0.216324\pi$
$110$	−2076.24			−0.171590
$111$			10097.5i			0.819537i
$112$	73575.0			5.86535
$113$			19908.8i			1.55915i	0.626308	+	0.779576i	$0.284565\pi$
$113$			19908.8i			1.55915i	−0.626308	+	0.779576i	$0.715435\pi$
$114$		−	22099.5i		−	1.70048i
$115$		−	1123.44i		−	0.0849481i
$116$	19776.5			1.46972
$117$			2883.17i			0.210619i
$118$	13726.3	−	23337.8i	0.985802	−	1.67608i
$119$	24537.4			1.73274
$120$			32920.5i			2.28614i
$121$	14553.8			0.994043
$122$	−18232.2			−1.22495
$123$	4341.78			0.286984
$124$			54762.1i			3.56153i
$125$	−12376.0			−0.792067
$126$			15267.4i			0.961668i
$127$	−15531.0			−0.962922			−0.481461	−	0.876468i	$-0.659894\pi$
$127$	−15531.0			−0.962922			−0.481461	+	0.876468i	$0.659894\pi$
$128$		−	66512.8i		−	4.05962i
$129$		−	11654.5i		−	0.700348i
$130$	−23740.5			−1.40477
$131$			1246.21i			0.0726188i	0.999341	+	0.0363094i	$0.0115602\pi$
$131$			1246.21i			0.0726188i	−0.999341	+	0.0363094i	$0.988440\pi$
$132$			2159.27i			0.123925i
$133$	39752.9			2.24732
$134$	−28394.9			−1.58136
$135$	−4010.17			−0.220037
$136$		−	74810.0i		−	4.04466i
$137$	16792.4			0.894689			0.447344	−	0.894362i	$-0.352370\pi$
$137$	16792.4			0.894689			0.447344	+	0.894362i	$0.352370\pi$
$138$	−1588.48			−0.0834110
$139$	−35922.2			−1.85923			−0.929616	−	0.368530i	$-0.879861\pi$
$139$	−35922.2			−1.85923			−0.929616	+	0.368530i	$0.879861\pi$
$140$	−92466.3			−4.71767
$141$		−	20338.2i		−	1.02300i
$142$		−	20459.9i		−	1.01467i
$143$	−997.240			−0.0487672
$144$	27324.9			1.31775
$145$	−12703.9			−0.604227
$146$	−69412.4			−3.25635
$147$	−14987.3			−0.693568
$148$			86469.7i			3.94767i
$149$			25955.6i			1.16912i	0.811351	+	0.584559i	$0.198732\pi$
$149$			25955.6i			1.16912i	−0.811351	+	0.584559i	$0.801268\pi$
$150$		−	7760.72i		−	0.344921i
$151$		−	18565.1i		−	0.814222i	−0.913379	−	0.407111i	$-0.866536\pi$
$151$		−	18565.1i		−	0.814222i	0.913379	−	0.407111i	$-0.133464\pi$
$152$		−	121199.i		−	5.24581i
$153$	9112.91			0.389291
$154$	−5280.75			−0.222666
$155$		−	35177.6i		−	1.46421i
$156$			24689.9i			1.01454i
$157$		−	46527.1i		−	1.88759i	−0.330537	−	0.943793i	$-0.607230\pi$
$157$		−	46527.1i		−	1.88759i	0.330537	−	0.943793i	$-0.392770\pi$
$158$			11647.8i			0.466584i
$159$	−19850.7			−0.785204
$160$			123630.i			4.82928i
$161$		−	2857.38i		−	0.110234i
$162$			5670.15i			0.216055i
$163$	−437.670			−0.0164730			−0.00823648	−	0.999966i	$-0.502622\pi$
$163$	−437.670			−0.0164730			−0.00823648	+	0.999966i	$0.502622\pi$
$164$	37180.6			1.38239
$165$		−	1387.05i		−	0.0509477i
$166$	−30997.6			−1.12490
$167$	49721.0			1.78282			0.891408	−	0.453201i	$-0.149718\pi$
$167$	49721.0			1.78282			0.891408	+	0.453201i	$0.149718\pi$
$168$			83730.5i			2.96664i
$169$	17158.2			0.600755
$170$			75037.3i			2.59645i
$171$	14763.8			0.504899
$172$		−	99802.8i		−	3.37354i
$173$			30766.7i			1.02799i	0.857793	+	0.513995i	$0.171835\pi$
$173$			30766.7i			1.02799i	−0.857793	+	0.513995i	$0.828165\pi$
$174$			17962.5i			0.593293i
$175$	13960.1			0.455840
$176$			9451.23i			0.305115i
$177$	15591.0	+	9169.98i	0.497655	+	0.292699i
$178$	−19689.9			−0.621447
$179$			20767.6i			0.648156i	0.946030	+	0.324078i	$0.105054\pi$
$179$			20767.6i			0.648156i	−0.946030	+	0.324078i	$0.894946\pi$
$180$	−34340.9			−1.05991
$181$	27789.6			0.848252			0.424126	−	0.905603i	$-0.360581\pi$
$181$	27789.6			0.848252			0.424126	+	0.905603i	$0.360581\pi$
$182$	−60382.2			−1.82291
$183$		−	12180.2i		−	0.363706i
$184$	−8711.63			−0.257314
$185$		−	55545.6i		−	1.62295i
$186$	−49739.1			−1.43771
$187$			3152.00i			0.0901370i
$188$		−	174165.i		−	4.92772i
$189$	−10199.5			−0.285534
$190$			121568.i			3.36752i
$191$			22222.5i			0.609153i	0.952488	+	0.304576i	$0.0985149\pi$
$191$			22222.5i			0.609153i	−0.952488	+	0.304576i	$0.901485\pi$
$192$	90666.2			2.45948
$193$	22533.4			0.604941			0.302470	−	0.953159i	$-0.402189\pi$
$193$	22533.4			0.604941			0.302470	+	0.953159i	$0.402189\pi$
$194$	71637.0			1.90342
$195$		−	15860.1i		−	0.417096i
$196$	−128343.			−3.34088
$197$	11138.9			0.287018			0.143509	−	0.989649i	$-0.454161\pi$
$197$	11138.9			0.287018			0.143509	+	0.989649i	$0.454161\pi$
$198$	−1961.21			−0.0500258
$199$	−27923.7			−0.705126			−0.352563	−	0.935788i	$-0.614690\pi$
$199$	−27923.7			−0.705126			−0.352563	+	0.935788i	$0.614690\pi$
$200$		−	42561.8i		−	1.06404i
$201$		−	18969.5i		−	0.469529i
$202$	103372.			2.53337
$203$	−32311.3			−0.784083
$204$	78038.0			1.87519
$205$	−23883.8			−0.568323
$206$	112284.			2.64596
$207$		−	1061.20i		−	0.0247660i
$208$			108069.i			2.49790i
$209$			5106.54i			0.116905i
$210$		−	83984.9i		−	1.90442i
$211$		−	4163.53i		−	0.0935182i	−0.998906	−	0.0467591i	$-0.985111\pi$
$211$		−	4163.53i		−	0.0935182i	0.998906	−	0.0467591i	$-0.0148893\pi$
$212$	−169991.			−3.78228
$213$	13668.4			0.301272
$214$		−	7672.29i		−	0.167532i
$215$			64110.4i			1.38692i
$216$			31096.6i			0.666507i
$217$		−	89471.3i		−	1.90005i
$218$	116156.			2.44416
$219$		−	46371.6i		−	0.966860i
$220$		−	11878.0i		−	0.245412i
$221$			36041.2i			0.737930i
$222$	−78538.3			−1.59359
$223$	−77429.7			−1.55703			−0.778517	−	0.627624i	$-0.784028\pi$
$223$	−77429.7			−1.55703			−0.778517	+	0.627624i	$0.784028\pi$
$224$			314442.i			6.26678i
$225$	5184.62			0.102412
$226$	−154850.			−3.03176
$227$			6994.10i			0.135731i	0.997694	+	0.0678656i	$0.0216189\pi$
$227$			6994.10i			0.135731i	−0.997694	+	0.0678656i	$0.978381\pi$
$228$	126429.			2.43207
$229$		−	51473.3i		−	0.981547i	−0.871287	−	0.490773i	$-0.836714\pi$
$229$		−	51473.3i		−	0.981547i	0.871287	−	0.490773i	$-0.163286\pi$
$230$	8738.09			0.165181
$231$		−	3527.85i		−	0.0661129i
$232$			98511.2i			1.83025i
$233$		−	79635.1i		−	1.46687i	−0.679758	−	0.733437i	$-0.737915\pi$
$233$		−	79635.1i		−	1.46687i	0.679758	−	0.733437i	$-0.262085\pi$
$234$	−22425.2			−0.409549
$235$			111879.i			2.02587i
$236$	133513.	+	78526.7i	2.39718	+	1.40992i
$237$	−7781.42			−0.138536
$238$			190851.i			3.36931i
$239$	−27642.4			−0.483927			−0.241964	−	0.970285i	$-0.577792\pi$
$239$	−27642.4			−0.483927			−0.241964	+	0.970285i	$0.577792\pi$
$240$	−150312.			−2.60959
$241$	11767.3			0.202602			0.101301	−	0.994856i	$-0.467700\pi$
$241$	11767.3			0.202602			0.101301	+	0.994856i	$0.467700\pi$
$242$			113199.i			1.93291i
$243$	−3788.00			−0.0641500
$244$		−	104304.i		−	1.75195i
$245$	82444.0			1.37349
$246$			33770.3i			0.558039i
$247$			58390.2i			0.957075i
$248$	−272782.			−4.43519
$249$		−	20708.2i		−	0.333999i
$250$		−	96260.7i		−	1.54017i
$251$	−98763.5			−1.56765			−0.783825	−	0.620982i	$-0.786734\pi$
$251$	−98763.5			−1.56765			−0.783825	+	0.620982i	$0.786734\pi$
$252$	−87343.4			−1.37540
$253$	367.050			0.00573436
$254$		−	120800.i		−	1.87240i
$255$	−50129.4			−0.770924
$256$	238156.			3.63398
$257$	36087.2			0.546370			0.273185	−	0.961961i	$-0.411923\pi$
$257$	36087.2			0.546370			0.273185	+	0.961961i	$0.411923\pi$
$258$	90648.5			1.36182
$259$		−	141276.i		−	2.10605i
$260$		−	135817.i		−	2.00913i
$261$	−12000.0			−0.176158
$262$	−9693.01			−0.141207
$263$	−16587.3			−0.239808			−0.119904	−	0.992786i	$-0.538259\pi$
$263$	−16587.3			−0.239808			−0.119904	+	0.992786i	$0.538259\pi$
$264$	−10755.8			−0.154324
$265$	109197.			1.55496
$266$			309197.i			4.36991i
$267$		−	13154.0i		−	0.184517i
$268$		−	162444.i		−	2.26170i
$269$		−	47175.2i		−	0.651943i	−0.945380	−	0.325971i	$-0.894309\pi$
$269$		−	47175.2i		−	0.651943i	0.945380	−	0.325971i	$-0.105691\pi$
$270$		−	31191.0i		−	0.427861i
$271$	37006.4			0.503893			0.251946	−	0.967741i	$-0.418929\pi$
$271$	37006.4			0.503893			0.251946	+	0.967741i	$0.418929\pi$
$272$	341577.			4.61690
$273$		−	40338.8i		−	0.541250i
$274$			130611.i			1.73972i
$275$			1793.27i			0.0237127i
$276$		−	9087.52i		−	0.119296i
$277$	−37499.2			−0.488722			−0.244361	−	0.969684i	$-0.578578\pi$
$277$	−37499.2			−0.488722			−0.244361	+	0.969684i	$0.578578\pi$
$278$		−	279402.i		−	3.61527i
$279$		−	33228.6i		−	0.426878i
$280$		−	460595.i		−	5.87494i
$281$	−8246.03			−0.104432			−0.0522158	−	0.998636i	$-0.516628\pi$
$281$	−8246.03			−0.104432			−0.0522158	+	0.998636i	$0.516628\pi$
$282$	158190.			1.98921
$283$			3380.85i			0.0422137i	0.999777	+	0.0211069i	$0.00671902\pi$
$283$			3380.85i			0.0422137i	−0.999777	+	0.0211069i	$0.993281\pi$
$284$	117049.			1.45121
$285$	−81214.3			−0.999868
$286$		−	7756.52i		−	0.0948276i
$287$	−60746.5			−0.737492
$288$			116780.i			1.40794i
$289$	30395.4			0.363925
$290$		−	98810.4i		−	1.17492i
$291$			47857.7i			0.565153i
$292$		−	397101.i		−	4.65731i
$293$	−146875.			−1.71085			−0.855427	−	0.517923i	$-0.826705\pi$
$293$	−146875.			−1.71085			−0.855427	+	0.517923i	$0.826705\pi$
$294$		−	116571.i		−	1.34864i
$295$	−85765.0	−	50443.3i	−0.985521	−	0.579642i
$296$	−430724.			−4.91605
$297$		−	1310.20i		−	0.0148534i
$298$	−201882.			−2.27334
$299$	4197.00			0.0469458
$300$	44398.3			0.493314
$301$			163060.i			1.79976i
$302$	144399.			1.58325
$303$			69058.3i			0.752195i
$304$	553386.			5.98799
$305$			67002.1i			0.720259i
$306$			70880.0i			0.756974i
$307$	92101.3			0.977213			0.488606	−	0.872504i	$-0.337505\pi$
$307$	92101.3			0.977213			0.488606	+	0.872504i	$0.337505\pi$
$308$		−	30210.6i		−	0.318462i
$309$			75012.4i			0.785626i
$310$	273611.			2.84714
$311$	−32978.0			−0.340960			−0.170480	−	0.985361i	$-0.554532\pi$
$311$	−32978.0			−0.340960			−0.170480	+	0.985361i	$0.554532\pi$
$312$	−122986.			−1.26341
$313$		−	40714.0i		−	0.415581i	−0.978173	−	0.207790i	$-0.933373\pi$
$313$		−	40714.0i		−	0.415581i	0.978173	−	0.207790i	$-0.0666272\pi$
$314$	361887.			3.67040
$315$	56106.9			0.565451
$316$	−66635.9			−0.667320
$317$	5097.47			0.0507266			0.0253633	−	0.999678i	$-0.491926\pi$
$317$	5097.47			0.0507266			0.0253633	+	0.999678i	$0.491926\pi$
$318$		−	154399.i		−	1.52683i
$319$		−	4150.61i		−	0.0407878i
$320$	−498748.			−4.87058
$321$	5125.54			0.0497428
$322$	22224.6			0.214350
$323$	184555.			1.76897
$324$	−32438.3			−0.309007
$325$			20505.0i			0.194130i
$326$		−	3404.19i		−	0.0320316i
$327$			77599.4i			0.725709i
$328$			185205.i			1.72149i
$329$			284555.i			2.62890i
$330$	10788.5			0.0990676
$331$	−18523.1			−0.169067			−0.0845334	−	0.996421i	$-0.526940\pi$
$331$	−18523.1			−0.169067			−0.0845334	+	0.996421i	$0.526940\pi$
$332$		−	177334.i		−	1.60885i
$333$		−	52468.2i		−	0.473160i
$334$			386729.i			3.46668i
$335$			104349.i			0.929823i
$336$	−382307.			−3.38636
$337$		−	137124.i		−	1.20741i	−0.797209	−	0.603704i	$-0.793691\pi$
$337$		−	137124.i		−	1.20741i	0.797209	−	0.603704i	$-0.206309\pi$
$338$			133456.i			1.16817i
$339$		−	103449.i		−	0.900176i
$340$	−429281.			−3.71350
$341$	11493.2			0.0988401
$342$			114832.i			0.981775i
$343$	35136.7			0.298657
$344$	497140.			4.20108
$345$			5837.56i			0.0490448i
$346$	−239303.			−1.99892
$347$			92388.3i			0.767287i	0.923481	+	0.383643i	$0.125331\pi$
$347$			92388.3i			0.767287i	−0.923481	+	0.383643i	$0.874669\pi$
$348$	−102762.			−0.848542
$349$		−	192179.i		−	1.57781i	−0.614513	−	0.788907i	$-0.710648\pi$
$349$		−	192179.i		−	1.57781i	0.614513	−	0.788907i	$-0.289352\pi$
$350$			108581.i			0.886379i
$351$		−	14981.4i		−	0.121601i
$352$	−40392.3			−0.325997
$353$			133742.i			1.07329i	0.843808	+	0.536645i	$0.180308\pi$
$353$			133742.i			1.07329i	−0.843808	+	0.536645i	$0.819692\pi$
$354$	−71323.9	+	121267.i	−0.569153	+	0.967688i
$355$	−75188.9			−0.596619
$356$		−	112644.i		−	0.888808i
$357$	−127500.			−1.00040
$358$	−161530.			−1.26034
$359$	−63441.2			−0.492246			−0.246123	−	0.969239i	$-0.579157\pi$
$359$	−63441.2			−0.492246			−0.246123	+	0.969239i	$0.579157\pi$
$360$		−	171060.i		−	1.31990i
$361$	168676.			1.29431
$362$			216147.i			1.64942i
$363$	−75623.7			−0.573911
$364$		−	345440.i		−	2.60717i
$365$			255086.i			1.91470i
$366$	94737.1			0.707225
$367$		−	62757.7i		−	0.465946i	−0.972483	−	0.232973i	$-0.925155\pi$
$367$		−	62757.7i		−	0.465946i	0.972483	−	0.232973i	$-0.0748453\pi$
$368$		−	39776.6i		−	0.293719i
$369$	−22560.5			−0.165690
$370$	432033.			3.15583
$371$	277735.			2.01782
$372$		−	284552.i		−	2.05625i
$373$	−102419.			−0.736148			−0.368074	−	0.929797i	$-0.619983\pi$
$373$	−102419.			−0.736148			−0.368074	+	0.929797i	$0.619983\pi$
$374$	−24516.2			−0.175271
$375$	64307.8			0.457300
$376$	867556.			6.13652
$377$		−	47459.7i		−	0.333920i
$378$		−	79331.9i		−	0.555219i
$379$	−115222.			−0.802151			−0.401076	−	0.916045i	$-0.631364\pi$
$379$	−115222.			−0.802151			−0.401076	+	0.916045i	$0.631364\pi$
$380$	−695475.			−4.81631
$381$	80701.3			0.555943
$382$	−172846.			−1.18450
$383$	68700.2			0.468339			0.234170	−	0.972196i	$-0.424763\pi$
$383$	68700.2			0.468339			0.234170	+	0.972196i	$0.424763\pi$
$384$			345611.i			2.34382i
$385$			19406.4i			0.130925i
$386$			175265.i			1.17631i
$387$			60558.5i			0.404346i
$388$			409828.i			2.72231i
$389$	1971.29			0.0130272			0.00651360	−	0.999979i	$-0.497927\pi$
$389$	1971.29			0.0130272			0.00651360	+	0.999979i	$0.497927\pi$
$390$	123359.			0.811042
$391$		−	13265.6i		−	0.0867705i
$392$		−	639306.i		−	4.16041i
$393$		−	6475.50i		−	0.0419265i
$394$			86638.0i			0.558105i
$395$	42805.0			0.274347
$396$		−	11219.9i		−	0.0715480i
$397$			42440.9i			0.269280i	0.990895	+	0.134640i	$0.0429877\pi$
$397$			42440.9i			0.269280i	−0.990895	+	0.134640i	$0.957012\pi$
$398$		−	217190.i		−	1.37111i
$399$	−206562.			−1.29749
$400$	194334.			1.21459
$401$		−	103710.i		−	0.644959i	−0.946576	−	0.322480i	$-0.895484\pi$
$401$		−	103710.i		−	0.644959i	0.946576	−	0.322480i	$-0.104516\pi$
$402$	147544.			0.912998
$403$	131418.			0.809180
$404$			591378.i			3.62328i
$405$	20837.5			0.127038
$406$		−	251316.i		−	1.52465i
$407$	18147.9			0.109556
$408$			388724.i			2.33519i
$409$			29250.4i			0.174858i	0.996171	+	0.0874289i	$0.0278651\pi$
$409$			29250.4i			0.174858i	−0.996171	+	0.0874289i	$0.972135\pi$
$410$		−	185768.i		−	1.10510i
$411$	−87255.9			−0.516549
$412$			642365.i			3.78432i
$413$	−218136.	−	128299.i	−1.27887	−	0.752180i
$414$	8253.98			0.0481573
$415$			113914.i			0.661428i
$416$	−461861.			−2.66885
$417$	186657.			1.07343
$418$	−39718.6			−0.227322
$419$			54737.0i			0.311784i	0.987774	+	0.155892i	$0.0498251\pi$
$419$			54737.0i			0.311784i	−0.987774	+	0.155892i	$0.950175\pi$
$420$	480469.			2.72375
$421$			135111.i			0.762303i	0.924513	+	0.381151i	$0.124472\pi$
$421$			135111.i			0.762303i	−0.924513	+	0.381151i	$0.875528\pi$
$422$	32383.8			0.181846
$423$			105680.i			0.590628i
$424$		−	846762.i		−	4.71010i
$425$	64810.6			0.358813
$426$			106313.i			0.585823i
$427$			170414.i			0.934653i
$428$	43892.4			0.239608
$429$	5181.81			0.0281557
$430$	−498650.			−2.69686
$431$		−	332324.i		−	1.78899i	−0.447081	−	0.894493i	$-0.647537\pi$
$431$		−	332324.i		−	1.78899i	0.447081	−	0.894493i	$-0.352463\pi$
$432$	−141984.			−0.760805
$433$	245001.			1.30675			0.653375	−	0.757035i	$-0.273353\pi$
$433$	245001.			1.30675			0.653375	+	0.757035i	$0.273353\pi$
$434$	695907.			3.69463
$435$	66011.2			0.348850
$436$			664519.i			3.49570i
$437$		−	21491.5i		−	0.112539i
$438$	360677.			1.88005
$439$	−181499.			−0.941771			−0.470885	−	0.882194i	$-0.656066\pi$
$439$	−181499.			−0.941771			−0.470885	+	0.882194i	$0.656066\pi$
$440$	59166.7			0.305613
$441$	77876.4			0.400432
$442$	−280328.			−1.43490
$443$			57457.9i			0.292781i	0.989227	+	0.146390i	$0.0467655\pi$
$443$			57457.9i			0.292781i	−0.989227	+	0.146390i	$0.953234\pi$
$444$		−	449310.i		−	2.27919i
$445$			72359.3i			0.365405i
$446$		−	602247.i		−	3.02765i
$447$		−	134869.i		−	0.674990i
$448$	−1.26852e6			−6.32038
$449$	26596.2			0.131925			0.0659624	−	0.997822i	$-0.478988\pi$
$449$	26596.2			0.131925			0.0659624	+	0.997822i	$0.478988\pi$
$450$			40325.9i			0.199140i
$451$		−	7803.31i		−	0.0383642i
$452$		−	885883.i		−	4.33610i
$453$			96466.9i			0.470091i
$454$	−54400.0			−0.263929
$455$			221901.i			1.07185i
$456$			629770.i			3.02867i
$457$		−	300518.i		−	1.43892i	−0.694532	−	0.719462i	$-0.744388\pi$
$457$		−	300518.i		−	1.43892i	0.694532	−	0.719462i	$-0.255612\pi$
$458$	400358.			1.90861
$459$	−47352.0			−0.224757
$460$			49989.7i			0.236246i
$461$	−140059.			−0.659038			−0.329519	−	0.944149i	$-0.606887\pi$
$461$	−140059.			−0.659038			−0.329519	+	0.944149i	$0.606887\pi$
$462$	27439.6			0.128556
$463$			127338.i			0.594014i	0.954875	+	0.297007i	$0.0959885\pi$
$463$			127338.i			0.594014i	−0.954875	+	0.297007i	$0.904012\pi$
$464$	−449794.			−2.08919
$465$			182788.i			0.845360i
$466$	619400.			2.85233
$467$		−	27447.8i		−	0.125856i	−0.998018	−	0.0629280i	$-0.979956\pi$
$467$		−	27447.8i		−	0.125856i	0.998018	−	0.0629280i	$-0.0200439\pi$
$468$		−	128292.i		−	0.585746i
$469$			265404.i			1.20660i
$470$	−870192.			−3.93930
$471$			241762.i			1.08980i
$472$	−391159.	+	665058.i	−1.75578	+	2.98522i
$473$	−20946.2			−0.0936230
$474$		−	60523.7i		−	0.269382i
$475$	104999.			0.465371
$476$	−1.09184e6			−4.81887
$477$	103147.			0.453338
$478$		−	215002.i		−	0.940995i
$479$	−403672.			−1.75937			−0.879687	−	0.475554i	$-0.842248\pi$
$479$	−403672.			−1.75937			−0.879687	+	0.475554i	$0.842248\pi$
$480$		−	642398.i		−	2.78819i
$481$	207510.			0.896910
$482$			91526.0i			0.393959i
$483$			14847.4i			0.0636437i
$484$	−647600.			−2.76450
$485$		−	263261.i		−	1.11919i
$486$		−	29463.0i		−	0.124739i
$487$	160164.			0.675316			0.337658	−	0.941269i	$-0.390365\pi$
$487$	160164.			0.675316			0.337658	+	0.941269i	$0.390365\pi$
$488$	519563.			2.18172
$489$	2274.20			0.00951067
$490$			641248.i			2.67075i
$491$	−27802.8			−0.115326			−0.0576628	−	0.998336i	$-0.518365\pi$
$491$	−27802.8			−0.115326			−0.0576628	+	0.998336i	$0.518365\pi$
$492$	−193196.			−0.798121
$493$	−150007.			−0.617189
$494$	−454158.			−1.86103
$495$			7207.33i			0.0294147i
$496$		−	1.24550e6i		−	5.06268i
$497$	−191237.			−0.774210
$498$	161068.			0.649459
$499$	136913.			0.549848			0.274924	−	0.961466i	$-0.411347\pi$
$499$	136913.			0.549848			0.274924	+	0.961466i	$0.411347\pi$
$500$	550697.			2.20279
$501$	−258358.			−1.02931
$502$		−	768181.i		−	3.04829i
$503$			45212.1i			0.178698i	0.996000	+	0.0893489i	$0.0284786\pi$
$503$			45212.1i			0.178698i	−0.996000	+	0.0893489i	$0.971521\pi$
$504$		−	435077.i		−	1.71279i
$505$		−	379884.i		−	1.48960i
$506$			2854.91i			0.0111504i
$507$	−89156.5			−0.346846
$508$	691082.			2.67795
$509$			420229.i			1.62200i	0.585048	+	0.810999i	$0.301076\pi$
$509$			420229.i			1.62200i	−0.585048	+	0.810999i	$0.698924\pi$
$510$		−	389905.i		−	1.49906i
$511$			648791.i			2.48464i
$512$			788170.i			3.00663i
$513$	−76714.8			−0.291504
$514$			280686.i			1.06241i
$515$		−	412637.i		−	1.55580i
$516$			518590.i			1.94771i
$517$	−36553.1			−0.136755
$518$	1.09884e6			4.09520
$519$		−	159869.i		−	0.593510i
$520$	676535.			2.50198
$521$	176873.			0.651608			0.325804	−	0.945437i	$-0.394365\pi$
$521$	176873.			0.651608			0.325804	+	0.945437i	$0.394365\pi$
$522$		−	93336.1i		−	0.342538i
$523$	−361972.			−1.32334			−0.661670	−	0.749795i	$-0.730152\pi$
$523$	−361972.			−1.32334			−0.661670	+	0.749795i	$0.730152\pi$
$524$		−	55452.7i		−	0.201958i
$525$	−72538.8			−0.263179
$526$		−	129015.i		−	0.466305i
$527$		−	415376.i		−	1.49562i
$528$		−	49110.0i		−	0.176158i
$529$	278296.			0.994480
$530$			849335.i			3.02362i
$531$	−81013.4	−	47648.6i	−0.287321	−	0.168990i
$532$	−1.76889e6			−6.24995
$533$		−	89226.1i		−	0.314078i
$534$	102312.			0.358793
$535$	−28195.2			−0.0985071
$536$	809170.			2.81650
$537$		−	107912.i		−	0.374213i
$538$	366928.			1.26770
$539$			26936.1i			0.0927166i
$540$	178441.			0.611937
$541$		−	276519.i		−	0.944779i	−0.881390	−	0.472390i	$-0.843392\pi$
$541$		−	276519.i		−	0.944779i	0.881390	−	0.472390i	$-0.156608\pi$
$542$			287835.i			0.979818i
$543$	−144399.			−0.489738
$544$			1.45982e6i			4.93288i
$545$		−	426868.i		−	1.43714i
$546$	313755.			1.05246
$547$	−424332.			−1.41818			−0.709090	−	0.705118i	$-0.750894\pi$
$547$	−424332.			−1.41818			−0.709090	+	0.705118i	$0.750894\pi$
$548$	−747213.			−2.48819
$549$			63290.0i			0.209986i
$550$	−13948.1			−0.0461093
$551$	−243026.			−0.800477
$552$	45266.9			0.148560
$553$	108871.			0.356010
$554$		−	291668.i		−	0.950318i
$555$			288624.i			0.937013i
$556$	1.59843e6			5.17064
$557$	−408914.			−1.31802			−0.659010	−	0.752134i	$-0.729024\pi$
$557$	−408914.			−1.31802			−0.659010	+	0.752134i	$0.729024\pi$
$558$	258452.			0.830063
$559$	−239507.			−0.766468
$560$	2.10304e6			6.70612
$561$		−	16378.3i		−	0.0520406i
$562$		−	64137.5i		−	0.203067i
$563$		−	165648.i		−	0.522600i	−0.965258	−	0.261300i	$-0.915849\pi$
$563$		−	165648.i		−	0.522600i	0.965258	−	0.261300i	$-0.0841512\pi$
$564$			904990.i			2.84502i
$565$			569065.i			1.78265i
$566$	−26296.2			−0.0820844
$567$	52998.4			0.164853
$568$			583047.i			1.80720i
$569$		−	413247.i		−	1.27639i	−0.769873	−	0.638197i	$-0.779680\pi$
$569$		−	413247.i		−	1.27639i	0.769873	−	0.638197i	$-0.220320\pi$
$570$		−	631683.i		−	1.94424i
$571$		−	531700.i		−	1.63078i	−0.578914	−	0.815388i	$-0.696523\pi$
$571$		−	531700.i		−	1.63078i	0.578914	−	0.815388i	$-0.303477\pi$
$572$	44374.2			0.135625
$573$		−	115472.i		−	0.351695i
$574$		−	472485.i		−	1.43405i
$575$		−	7547.20i		−	0.0228271i
$576$	−471116.			−1.41998
$577$	−94373.0			−0.283463			−0.141731	−	0.989905i	$-0.545267\pi$
$577$	−94373.0			−0.283463			−0.141731	+	0.989905i	$0.545267\pi$
$578$			236415.i			0.707651i
$579$	−117087.			−0.349263
$580$	565284.			1.68039
$581$			289732.i			0.858310i
$582$	−372237.			−1.09894
$583$			35677.0i			0.104967i
$584$	1.97805e6			5.79977
$585$			82411.4i			0.240811i
$586$		−	1.14239e6i		−	3.32675i
$587$		−	16716.5i		−	0.0485142i	−0.999706	−	0.0242571i	$-0.992278\pi$
$587$		−	16716.5i		−	0.0485142i	0.999706	−	0.0242571i	$-0.00772203\pi$
$588$	666891.			1.92886
$589$		−	672949.i		−	1.93978i
$590$	392347.	−	667079.i	1.12711	−	1.91634i
$591$	−57879.3			−0.165710
$592$		−	1.96665e6i		−	5.61157i
$593$	−406858.			−1.15700			−0.578500	−	0.815683i	$-0.696362\pi$
$593$	−406858.			−1.15700			−0.578500	+	0.815683i	$0.696362\pi$
$594$	10190.7			0.0288824
$595$	701367.			1.98112
$596$		−	1.15495e6i		−	3.25139i
$597$	145096.			0.407105
$598$			32644.2i			0.0912858i
$599$	−216469.			−0.603313			−0.301656	−	0.953417i	$-0.597540\pi$
$599$	−216469.			−0.603313			−0.301656	+	0.953417i	$0.597540\pi$
$600$			221158.i			0.614327i
$601$			114850.i			0.317966i	0.987281	+	0.158983i	$0.0508214\pi$
$601$			114850.i			0.317966i	−0.987281	+	0.158983i	$0.949179\pi$
$602$	−1.26828e6			−3.49962
$603$			98568.2i			0.271083i
$604$			826091.i			2.26440i
$605$	416000.			1.13653
$606$	−537134.			−1.46264
$607$	460090.			1.24872			0.624360	−	0.781136i	$-0.285360\pi$
$607$	460090.			1.24872			0.624360	+	0.781136i	$0.285360\pi$
$608$			2.36504e6i			6.39781i
$609$	167894.			0.452690
$610$	−521141.			−1.40054
$611$	−417962.			−1.11958
$612$	−405497.			−1.08264
$613$			54204.8i			0.144250i	0.997396	+	0.0721252i	$0.0229781\pi$
$613$			54204.8i			0.144250i	−0.997396	+	0.0721252i	$0.977022\pi$
$614$			716363.i			1.90019i
$615$	124104.			0.328121
$616$	150486.			0.396583
$617$	−446671.			−1.17332			−0.586662	−	0.809832i	$-0.699558\pi$
$617$	−446671.			−1.17332			−0.586662	+	0.809832i	$0.699558\pi$
$618$	−583445.			−1.52765
$619$	608019.			1.58685			0.793425	−	0.608668i	$-0.208296\pi$
$619$	608019.			1.58685			0.793425	+	0.608668i	$0.208296\pi$
$620$			1.56530e6i			4.07205i
$621$			5514.14i			0.0142986i
$622$		−	256503.i		−	0.662996i
$623$			184040.i			0.474172i
$624$		−	561543.i		−	1.44216i
$625$	−473767.			−1.21284
$626$	316673.			0.808095
$627$		−	26534.4i		−	0.0674953i
$628$			2.07032e6i			5.24950i
$629$		−	655882.i		−	1.65777i
$630$			436398.i			1.09952i
$631$	−89180.6			−0.223981			−0.111991	−	0.993709i	$-0.535723\pi$
$631$	−89180.6			−0.223981			−0.111991	+	0.993709i	$0.535723\pi$
$632$		−	331928.i		−	0.831016i
$633$			21634.3i			0.0539928i
$634$			39648.0i			0.0986377i
$635$	−443931.			−1.10095
$636$	883299.			2.18370
$637$			307998.i			0.759048i
$638$	32283.4			0.0793118
$639$	−71023.2			−0.173940
$640$		−	1.90118e6i		−	4.64155i
$641$	496660.			1.20877			0.604384	−	0.796693i	$-0.293419\pi$
$641$	496660.			1.20877			0.604384	+	0.796693i	$0.293419\pi$
$642$			39866.4i			0.0967246i
$643$	−477619.			−1.15521			−0.577603	−	0.816318i	$-0.696012\pi$
$643$	−477619.			−1.15521			−0.577603	+	0.816318i	$0.696012\pi$
$644$			127145.i			0.306568i
$645$		−	333128.i		−	0.800739i
$646$			1.43547e6i			3.43976i
$647$	255266.			0.609796			0.304898	−	0.952385i	$-0.401378\pi$
$647$	255266.			0.609796			0.304898	+	0.952385i	$0.401378\pi$
$648$		−	161582.i		−	0.384808i
$649$	16480.9	−	28021.2i	0.0391282	−	0.0665268i
$650$	−159487.			−0.377485
$651$			464907.i			1.09699i
$652$	19475.0			0.0458124
$653$	12720.0			0.0298305			0.0149152	−	0.999889i	$-0.495252\pi$
$653$	12720.0			0.0298305			0.0149152	+	0.999889i	$0.495252\pi$
$654$	−603567.			−1.41114
$655$			35621.2i			0.0830283i
$656$	−845631.			−1.96505
$657$			240954.i			0.558217i
$658$	−2.21326e6			−5.11189
$659$			732103.i			1.68578i	0.538084	+	0.842891i	$0.319148\pi$
$659$			732103.i			1.68578i	−0.538084	+	0.842891i	$0.680852\pi$
$660$			61719.6i			0.141689i
$661$	601813.			1.37740			0.688698	−	0.725049i	$-0.258183\pi$
$661$	601813.			1.37740			0.688698	+	0.725049i	$0.258183\pi$
$662$		−	144073.i		−	0.328750i
$663$		−	187276.i		−	0.426044i
$664$	883341.			2.00351
$665$	1.13628e6			2.56946
$666$	408097.			0.920058
$667$			17468.3i			0.0392644i
$668$	−2.21244e6			−4.95813
$669$	402337.			0.898954
$670$	−811628.			−1.80804
$671$	−21890.9			−0.0486205
$672$		−	1.63389e6i		−	3.61813i
$673$		−	177803.i		−	0.392563i	−0.980548	−	0.196282i	$-0.937113\pi$
$673$		−	177803.i		−	0.392563i	0.980548	−	0.196282i	$-0.0628867\pi$
$674$	1.06655e6			2.34780
$675$	−26940.1			−0.0591277
$676$	−763488.			−1.67074
$677$	238228.			0.519774			0.259887	−	0.965639i	$-0.416315\pi$
$677$	238228.			0.519774			0.259887	+	0.965639i	$0.416315\pi$
$678$	804626.			1.75039
$679$		−	669585.i		−	1.45233i
$680$		−	2.13834e6i		−	4.62444i
$681$		−	36342.4i		−	0.0783645i
$682$			89394.1i			0.192194i
$683$			660787.i			1.41651i	0.705956	+	0.708256i	$0.250518\pi$
$683$			660787.i			1.41651i	−0.705956	+	0.708256i	$0.749482\pi$
$684$	−656944.			−1.40416
$685$	479988.			1.02294
$686$			273293.i			0.580737i
$687$			267463.i			0.566696i
$688$			2.26990e6i			4.79545i
$689$			407945.i			0.859335i
$690$	−45404.5			−0.0953675
$691$			252072.i			0.527919i	0.964534	+	0.263960i	$0.0850286\pi$
$691$			252072.i			0.527919i	−0.964534	+	0.263960i	$0.914971\pi$
$692$		−	1.36903e6i		−	2.85891i
$693$			18331.3i			0.0381703i
$694$	−718594.			−1.49199
$695$	−1.02679e6			−2.12574
$696$		−	511879.i		−	1.05669i
$697$	−282019.			−0.580515
$698$	1.49477e6			3.06805
$699$			413796.i			0.846900i
$700$	−621183.			−1.26772
$701$		−	601181.i		−	1.22340i	−0.791089	−	0.611701i	$-0.790486\pi$
$701$		−	601181.i		−	1.22340i	0.791089	−	0.611701i	$-0.209514\pi$
$702$	116525.			0.236453
$703$		−	1.06259e6i		−	2.15008i
$704$		−	162951.i		−	0.328785i
$705$		−	581340.i		−	1.16964i
$706$	−1.04024e6			−2.08701
$707$		−	966205.i		−	1.93299i
$708$	−693755.	−	408037.i	−1.38401	−	0.814016i
$709$	769466.			1.53072			0.765362	−	0.643600i	$-0.222560\pi$
$709$	769466.			1.53072			0.765362	+	0.643600i	$0.222560\pi$
$710$		−	584818.i		−	1.16012i
$711$	40433.4			0.0799837
$712$	561105.			1.10684
$713$	−48370.6			−0.0951485
$714$		−	991693.i		−	1.94527i
$715$	−28504.7			−0.0557577
$716$		−	924096.i		−	1.80256i
$717$	143634.			0.279396
$718$		−	493444.i		−	0.957171i
$719$			412760.i			0.798435i	0.916856	+	0.399218i	$0.130718\pi$
$719$			412760.i			0.798435i	−0.916856	+	0.399218i	$0.869282\pi$
$720$	781045.			1.50664
$721$		−	1.04951e6i		−	2.01890i
$722$			1.31196e6i			2.51678i
$723$	−61144.8			−0.116972
$724$	−1.23655e6			−2.35904
$725$	−85343.8			−0.162366
$726$		−	588200.i		−	1.11597i
$727$	−352870.			−0.667645			−0.333822	−	0.942636i	$-0.608339\pi$
$727$	−352870.			−0.667645			−0.333822	+	0.942636i	$0.608339\pi$
$728$	1.72071e6			3.24673
$729$	19683.0			0.0370370
$730$	−1.98406e6			−3.72313
$731$			757015.i			1.41667i
$732$			541981.i			1.01149i
$733$	566886.			1.05509			0.527543	−	0.849528i	$-0.323113\pi$
$733$	566886.			1.05509			0.527543	+	0.849528i	$0.323113\pi$
$734$	488129.			0.906029
$735$	−428392.			−0.792987
$736$	169996.			0.313821
$737$	−34093.1			−0.0627670
$738$		−	175476.i		−	0.322184i
$739$		−	110960.i		−	0.203179i	−0.994826	−	0.101589i	$-0.967607\pi$
$739$		−	110960.i		−	0.203179i	0.994826	−	0.101589i	$-0.0323928\pi$
$740$			2.47162e6i			4.51354i
$741$		−	303404.i		−	0.552567i
$742$			2.16022e6i			3.92364i
$743$	−565178.			−1.02378			−0.511891	−	0.859050i	$-0.671055\pi$
$743$	−565178.			−1.02378			−0.511891	+	0.859050i	$0.671055\pi$
$744$	1.41742e6			2.56066
$745$			741904.i			1.33670i
$746$		−	796617.i		−	1.43144i
$747$			107603.i			0.192834i
$748$		−	140255.i		−	0.250677i
$749$	−71712.3			−0.127829
$750$			500185.i			0.889218i
$751$		−	217676.i		−	0.385950i	−0.981204	−	0.192975i	$-0.938186\pi$
$751$		−	217676.i		−	0.385950i	0.981204	−	0.192975i	$-0.0618136\pi$
$752$			3.96119e6i			7.00471i
$753$	513190.			0.905083
$754$	369141.			0.649306
$755$		−	530657.i		−	0.930936i
$756$	453850.			0.794088
$757$	169980.			0.296623			0.148312	−	0.988941i	$-0.452616\pi$
$757$	169980.			0.296623			0.148312	+	0.988941i	$0.452616\pi$
$758$		−	896194.i		−	1.55978i
$759$	−1907.25			−0.00331073
$760$		−	3.46431e6i		−	5.99777i
$761$	880285.			1.52004			0.760019	−	0.649901i	$-0.225190\pi$
$761$	880285.			1.52004			0.760019	+	0.649901i	$0.225190\pi$
$762$			627693.i			1.08103i
$763$		−	1.08570e6i		−	1.86493i
$764$		−	988836.i		−	1.69409i
$765$	260480.			0.445093
$766$			534349.i			0.910684i
$767$	188449.	−	320405.i	0.320333	−	0.544639i
$768$	−1.23750e6			−2.09808
$769$			619749.i			1.04801i	0.851717	+	0.524003i	$0.175562\pi$
$769$			619749.i			1.04801i	−0.851717	+	0.524003i	$0.824438\pi$
$770$	−150943.			−0.254584
$771$	−187515.			−0.315447
$772$	−1.00267e6			−1.68238
$773$		−	1.11076e6i		−	1.85892i	−0.368917	−	0.929462i	$-0.620271\pi$
$773$		−	1.11076e6i		−	1.85892i	0.368917	−	0.929462i	$-0.379729\pi$
$774$	−471023.			−0.786250
$775$		−	236321.i		−	0.393458i
$776$	−2.04144e6			−3.39011
$777$			734091.i			1.21593i
$778$			15332.7i			0.0253313i
$779$	−456898.			−0.752912
$780$			705727.i			1.15997i
$781$		−	24565.7i		−	0.0402743i
$782$	103179.			0.168725
$783$	62354.0			0.101705
$784$	2.91902e6			4.74903
$785$		−	1.32991e6i		−	2.15816i
$786$	50366.4			0.0815259
$787$	243033.			0.392389			0.196194	−	0.980565i	$-0.437142\pi$
$787$	243033.			0.392389			0.196194	+	0.980565i	$0.437142\pi$
$788$	−495647.			−0.798216
$789$	86190.0			0.138453
$790$			332936.i			0.533466i
$791$			1.44737e6i			2.31328i
$792$	55888.7			0.0890991
$793$	−250310.			−0.398044
$794$	−330105.			−0.523613
$795$	−567406.			−0.897758
$796$	1.24252e6			1.96100
$797$		−	419997.i		−	0.661195i	−0.943772	−	0.330597i	$-0.892750\pi$
$797$		−	419997.i		−	0.661195i	0.943772	−	0.330597i	$-0.107250\pi$
$798$		−	1.60664e6i		−	2.52297i
$799$			1.32106e6i			2.06933i
$800$			830536.i			1.29771i
$801$			68350.4i			0.106531i
$802$	806655.			1.25412
$803$	−83341.8			−0.129250
$804$			844085.i			1.30579i
$805$		−	81674.2i		−	0.126036i
$806$			1.02217e6i			1.57345i
$807$			245130.i			0.376399i
$808$	−2.94578e6			−4.51209
$809$			573138.i			0.875713i	0.899045	+	0.437857i	$0.144262\pi$
$809$			573138.i			0.875713i	−0.899045	+	0.437857i	$0.855738\pi$
$810$			162073.i			0.247025i
$811$			133816.i			0.203454i	0.994812	+	0.101727i	$0.0324368\pi$
$811$			133816.i			0.203454i	−0.994812	+	0.101727i	$0.967563\pi$
$812$	1.43776e6			2.18058
$813$	−192291.			−0.290923
$814$			141154.i			0.213032i
$815$	−12510.2			−0.0188343
$816$	−1.77488e6			−2.66557
$817$			1.22644e6i			1.83739i
$818$	−227509.			−0.340010
$819$			209607.i			0.312491i
$820$	1.06276e6			1.58054
$821$		−	197462.i		−	0.292952i	−0.989214	−	0.146476i	$-0.953207\pi$
$821$		−	197462.i		−	0.292952i	0.989214	−	0.146476i	$-0.0467931\pi$
$822$		−	678675.i		−	1.00443i
$823$			562096.i			0.829871i	0.909851	+	0.414936i	$0.136196\pi$
$823$			562096.i			0.829871i	−0.909851	+	0.414936i	$0.863804\pi$
$824$	−3.19976e6			−4.71263
$825$		−	9318.12i		−	0.0136905i
$826$	997904.	−	1.69666e6i	1.46261	−	2.48677i
$827$	−567007.			−0.829043			−0.414522	−	0.910039i	$-0.636051\pi$
$827$	−567007.			−0.829043			−0.414522	+	0.910039i	$0.636051\pi$
$828$			47220.1i			0.0688758i
$829$	−430710.			−0.626723			−0.313362	−	0.949634i	$-0.601455\pi$
$829$	−430710.			−0.626723			−0.313362	+	0.949634i	$0.601455\pi$
$830$	−886024.			−1.28614
$831$	194851.			0.282164
$832$		−	1.86325e6i		−	2.69168i
$833$	973498.			1.40296
$834$			1.45182e6i			2.08728i
$835$	1.42120e6			2.03837
$836$		−	227226.i		−	0.325121i
$837$			172661.i			0.246458i
$838$	−425744.			−0.606262
$839$		−	53824.6i		−	0.0764640i	−0.999269	−	0.0382320i	$-0.987827\pi$
$839$		−	53824.6i		−	0.0764640i	0.999269	−	0.0382320i	$-0.0121726\pi$
$840$			2.39332e6i			3.39190i
$841$	−509749.			−0.720716
$842$	−1.05089e6			−1.48229
$843$	42847.6			0.0602936
$844$			185265.i			0.260080i
$845$	490442.			0.686870
$846$	−821981.			−1.14847
$847$	1.05806e6			1.47484
$848$	3.86625e6			5.37648
$849$		−	17567.4i		−	0.0243721i
$850$			504096.i			0.697711i
$851$	−76377.4			−0.105464
$852$	−608204.			−0.837858
$853$	−418548.			−0.575237			−0.287619	−	0.957745i	$-0.592864\pi$
$853$	−418548.			−0.575237			−0.287619	+	0.957745i	$0.592864\pi$
$854$	−1.32548e6			−1.81743
$855$	422002.			0.577274
$856$			218638.i			0.298385i
$857$		−	1.10791e6i		−	1.50849i	−0.656596	−	0.754243i	$-0.728004\pi$
$857$		−	1.10791e6i		−	1.50849i	0.656596	−	0.754243i	$-0.271996\pi$
$858$			40304.0i			0.0547487i
$859$			226154.i			0.306491i	0.988188	+	0.153245i	$0.0489725\pi$
$859$			226154.i			0.306491i	−0.988188	+	0.153245i	$0.951028\pi$
$860$		−	2.85272e6i		−	3.85712i
$861$	315648.			0.425791
$862$	2.58481e6			3.47868
$863$		−	539905.i		−	0.724929i	−0.931998	−	0.362464i	$-0.881935\pi$
$863$		−	539905.i		−	0.724929i	0.931998	−	0.362464i	$-0.118065\pi$
$864$		−	606807.i		−	0.812874i
$865$			879424.i			1.17535i
$866$			1.90561e6i			2.54097i
$867$	−157939.			−0.210112
$868$			3.98121e6i			5.28416i
$869$			13985.2i			0.0185196i
$870$			513434.i			0.678338i
$871$	−389834.			−0.513858
$872$	−3.31011e6			−4.35321
$873$		−	248676.i		−	0.326291i
$874$	167160.			0.218832
$875$	−899740.			−1.17517
$876$			2.06340e6i			2.68890i
$877$	−288067.			−0.374537			−0.187269	−	0.982309i	$-0.559963\pi$
$877$	−288067.			−0.374537			−0.187269	+	0.982309i	$0.559963\pi$
$878$		−	1.41170e6i		−	1.83127i
$879$	763186.			0.987762
$880$			270150.i			0.348851i
$881$		−	80450.5i		−	0.103652i	−0.998656	−	0.0518259i	$-0.983496\pi$
$881$		−	80450.5i		−	0.103652i	0.998656	−	0.0518259i	$-0.0165041\pi$
$882$			605721.i			0.778638i
$883$	210993.			0.270611			0.135306	−	0.990804i	$-0.456798\pi$
$883$	210993.			0.270611			0.135306	+	0.990804i	$0.456798\pi$
$884$		−	1.60373e6i		−	2.05223i
$885$	445648.	+	262111.i	0.568991	+	0.334656i
$886$	−446907.			−0.569311
$887$			428946.i			0.545199i	0.962128	+	0.272600i	$0.0878834\pi$
$887$			428946.i			0.545199i	−0.962128	+	0.272600i	$0.912117\pi$
$888$	2.23811e6			2.83828
$889$	−1.12910e6			−1.42866
$890$	−562809.			−0.710528
$891$			6808.02i			0.00857562i
$892$	3.44539e6			4.33021
$893$			2.14025e6i			2.68387i
$894$	1.04901e6			1.31252
$895$			593612.i			0.741066i
$896$		−	4.83549e6i		−	6.02316i
$897$	−21808.2			−0.0271041
$898$			206865.i			0.256527i
$899$			546975.i			0.676781i
$900$	−230700.			−0.284815
$901$	1.28940e6			1.58832
$902$	60694.0			0.0745990
$903$		−	847283.i		−	1.03909i
$904$	4.41278e6			5.39977
$905$	794327.			0.969844
$906$	−750318.			−0.914090
$907$	1.21610e6			1.47827			0.739134	−	0.673558i	$-0.235235\pi$
$907$	1.21610e6			1.47827			0.739134	+	0.673558i	$0.235235\pi$
$908$		−	311217.i		−	0.377477i
$909$		−	358837.i		−	0.434280i
$910$	−1.72594e6			−2.08422
$911$	789820.			0.951681			0.475841	−	0.879532i	$-0.342144\pi$
$911$	789820.			0.951681			0.475841	+	0.879532i	$0.342144\pi$
$912$	−2.87548e6			−3.45717
$913$	−37218.1			−0.0446491
$914$	2.33742e6			2.79798
$915$		−	348153.i		−	0.415842i
$916$			2.29041e6i			2.72975i
$917$			90599.7i			0.107743i
$918$		−	368303.i		−	0.437039i
$919$			1.09607e6i			1.29780i	0.760872	+	0.648902i	$0.224772\pi$
$919$			1.09607e6i			1.29780i	−0.760872	+	0.648902i	$0.775228\pi$
$920$	−249010.			−0.294199
$921$	−478573.			−0.564194
$922$		−	1.08938e6i		−	1.28150i
$923$		−	280894.i		−	0.329716i
$924$			156979.i			0.183864i
$925$		−	373152.i		−	0.436116i
$926$	−990435.			−1.15506
$927$		−	389776.i		−	0.453581i
$928$		−	1.92231e6i		−	2.23217i
$929$		−	346274.i		−	0.401225i	−0.979671	−	0.200612i	$-0.935707\pi$
$929$		−	346274.i		−	0.401225i	0.979671	−	0.200612i	$-0.0642932\pi$
$930$	−1.42172e6			−1.64380
$931$	1.57716e6			1.81960
$932$			3.54353e6i			4.07947i
$933$	171359.			0.196854
$934$	213489.			0.244727
$935$			90095.6i			0.103058i
$936$	639053.			0.729433
$937$		−	1.06972e6i		−	1.21840i	−0.793017	−	0.609200i	$-0.791491\pi$
$937$		−	1.06972e6i		−	1.21840i	0.793017	−	0.609200i	$-0.208509\pi$
$938$	−2.06431e6			−2.34622
$939$			211556.i			0.239936i
$940$		−	4.97828e6i		−	5.63408i
$941$			1.10921e6i			1.25267i	0.779555	+	0.626333i	$0.215445\pi$
$941$			1.10921e6i			1.25267i	−0.779555	+	0.626333i	$0.784555\pi$
$942$	−1.88042e6			−2.11911
$943$			32841.1i			0.0369313i
$944$	−3.03660e6	−	1.78600e6i	−3.40756	−	2.00418i
$945$	−291540.			−0.326463
$946$		−	162919.i		−	0.182050i
$947$	−502348.			−0.560151			−0.280075	−	0.959978i	$-0.590359\pi$
$947$	−502348.			−0.560151			−0.280075	+	0.959978i	$0.590359\pi$
$948$	346250.			0.385277
$949$	−952963.			−1.05814
$950$			816683.i			0.904912i
$951$	−26487.2			−0.0292870
$952$		−	5.43870e6i		−	6.00097i
$953$	498865.			0.549284			0.274642	−	0.961547i	$-0.411441\pi$
$953$	498865.			0.549284			0.274642	+	0.961547i	$0.411441\pi$
$954$			802279.i			0.881513i
$955$			635200.i			0.696472i
$956$	1.23001e6			1.34583
$957$			21567.2i			0.0235489i
$958$		−	3.13976e6i		−	3.42109i
$959$	1.22081e6			1.32743
$960$	2.59157e6			2.81203
$961$	−591077.			−0.640026
$962$			1.61401e6i			1.74404i
$963$	−26633.1			−0.0287190
$964$	−523611.			−0.563449
$965$	644087.			0.691656
$966$	−115483.			−0.123755
$967$		−	790210.i		−	0.845064i	−0.906348	−	0.422532i	$-0.861141\pi$
$967$		−	790210.i		−	0.845064i	0.906348	−	0.422532i	$-0.138859\pi$
$968$		−	3.22584e6i		−	3.44264i
$969$	−958977.			−1.02132
$970$	2.04764e6			2.17626
$971$	546860.			0.580013			0.290006	−	0.957025i	$-0.406343\pi$
$971$	546860.			0.580013			0.290006	+	0.957025i	$0.406343\pi$
$972$	168555.			0.178405
$973$	−2.61155e6			−2.75850
$974$			1.24575e6i			1.31315i
$975$		−	106547.i		−	0.112081i
$976$			2.37228e6i			2.49038i
$977$			710431.i			0.744274i	0.928178	+	0.372137i	$0.121375\pi$
$977$			710431.i			0.744274i	−0.928178	+	0.372137i	$0.878625\pi$
$978$			17688.7i			0.0184935i
$979$	−23641.2			−0.0246664
$980$	−3.66851e6			−3.81978
$981$		−	403218.i		−	0.418988i
$982$		−	216250.i		−	0.224250i
$983$		−	1.52173e6i		−	1.57482i	−0.616429	−	0.787410i	$-0.711421\pi$
$983$		−	1.52173e6i		−	1.57482i	0.616429	−	0.787410i	$-0.288579\pi$
$984$		−	962353.i		−	0.993903i
$985$	318389.			0.328160
$986$		−	1.16675e6i		−	1.20012i
$987$		−	1.47859e6i		−	1.51780i
$988$		−	2.59819e6i		−	2.66169i
$989$	88154.4			0.0901263
$990$	−56058.5			−0.0571967
$991$		−	845241.i		−	0.860664i	−0.902671	−	0.430332i	$-0.858397\pi$
$991$		−	845241.i		−	0.860664i	0.902671	−	0.430332i	$-0.141603\pi$
$992$	5.32297e6			5.40917
$993$	96249.0			0.0976108
$994$		−	1.48744e6i		−	1.50545i
$995$	−798160.			−0.806202
$996$			921456.i			0.928872i
$997$	−168798.			−0.169816			−0.0849079	−	0.996389i	$-0.527060\pi$
$997$	−168798.			−0.169816			−0.0849079	+	0.996389i	$0.527060\pi$
$998$			1.06491e6i			1.06918i
$999$			272633.i			0.273179i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.5.c.a.58.39	yes	40
3.2	odd	2		531.5.c.d.235.2		40
59.58	odd	2	inner	177.5.c.a.58.2	✓	40
177.176	even	2		531.5.c.d.235.39		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.5.c.a.58.2	✓	40	59.58	odd	2	inner
177.5.c.a.58.39	yes	40	1.1	even	1	trivial
531.5.c.d.235.2		40	3.2	odd	2
531.5.c.d.235.39		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)

\(f(q)\)	\(=\)	\(q+7.77798i q^{2} -5.19615 q^{3} -44.4970 q^{4} +28.5836 q^{5} -40.4156i q^{6} +72.7001 q^{7} -221.650i q^{8} +27.0000 q^{9} +O(q^{10})\)	\(q+7.77798i q^{2} -5.19615 q^{3} -44.4970 q^{4} +28.5836 q^{5} -40.4156i q^{6} +72.7001 q^{7} -221.650i q^{8} +27.0000 q^{9} +222.323i q^{10} +9.33885i q^{11} +231.213 q^{12} +106.784i q^{13} +565.461i q^{14} -148.525 q^{15} +1012.03 q^{16} +337.515 q^{17} +210.006i q^{18} +546.806 q^{19} -1271.89 q^{20} -377.761 q^{21} -72.6374 q^{22} -39.3036i q^{23} +1151.72i q^{24} +192.023 q^{25} -830.565 q^{26} -140.296 q^{27} -3234.94 q^{28} -444.446 q^{29} -1155.22i q^{30} -1230.69i q^{31} +4325.19i q^{32} -48.5261i q^{33} +2625.19i q^{34} +2078.03 q^{35} -1201.42 q^{36} -1943.27i q^{37} +4253.05i q^{38} -554.866i q^{39} -6335.54i q^{40} -835.576 q^{41} -2938.22i q^{42} +2242.91i q^{43} -415.551i q^{44} +771.758 q^{45} +305.703 q^{46} +3914.09i q^{47} -5258.68 q^{48} +2884.31 q^{49} +1493.55i q^{50} -1753.78 q^{51} -4751.57i q^{52} +3820.28 q^{53} -1091.22i q^{54} +266.938i q^{55} -16113.9i q^{56} -2841.29 q^{57} -3456.89i q^{58} +(-3000.50 - 1764.76i) q^{59} +6608.91 q^{60} +2344.07i q^{61} +9572.29 q^{62} +1962.90 q^{63} -17448.7 q^{64} +3052.27i q^{65} +377.435 q^{66} +3650.67i q^{67} -15018.4 q^{68} +204.228i q^{69} +16162.9i q^{70} -2630.49 q^{71} -5984.54i q^{72} +8924.21i q^{73} +15114.7 q^{74} -997.781 q^{75} -24331.3 q^{76} +678.935i q^{77} +4315.74 q^{78} +1497.53 q^{79} +28927.6 q^{80} +729.000 q^{81} -6499.09i q^{82} +3985.30i q^{83} +16809.2 q^{84} +9647.40 q^{85} -17445.3 q^{86} +2309.41 q^{87} +2069.95 q^{88} +2531.49i q^{89} +6002.72i q^{90} +7763.21i q^{91} +1748.89i q^{92} +6394.85i q^{93} -30443.7 q^{94} +15629.7 q^{95} -22474.3i q^{96} -9210.22i q^{97} +22434.1i q^{98} +252.149i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 320 q^{4} + 80 q^{7} + 1080 q^{9}+O(q^{10}) \) 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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