LMFDB - Embedded newform 177.5.c.a.58.18

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			58.18
Character	$\chi$	$=$	177.58
Dual form			177.5.c.a.58.23

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.07792i q^{2} +5.19615 q^{3} +14.8381 q^{4} +26.1400 q^{5} -5.60104i q^{6} +49.1298 q^{7} -33.2410i q^{8} +27.0000 q^{9} +O(q^{10})$	\(q-1.07792i q^{2} +5.19615 q^{3} +14.8381 q^{4} +26.1400 q^{5} -5.60104i q^{6} +49.1298 q^{7} -33.2410i q^{8} +27.0000 q^{9} -28.1769i q^{10} -190.415i q^{11} +77.1010 q^{12} +198.277i q^{13} -52.9580i q^{14} +135.828 q^{15} +201.578 q^{16} -81.6050 q^{17} -29.1038i q^{18} -590.362 q^{19} +387.868 q^{20} +255.286 q^{21} -205.252 q^{22} +808.730i q^{23} -172.725i q^{24} +58.3017 q^{25} +213.726 q^{26} +140.296 q^{27} +728.992 q^{28} -714.690 q^{29} -146.411i q^{30} -1370.46i q^{31} -749.141i q^{32} -989.426i q^{33} +87.9637i q^{34} +1284.26 q^{35} +400.628 q^{36} +2106.01i q^{37} +636.363i q^{38} +1030.28i q^{39} -868.921i q^{40} +1132.48 q^{41} -275.178i q^{42} +349.669i q^{43} -2825.40i q^{44} +705.781 q^{45} +871.746 q^{46} -2257.08i q^{47} +1047.43 q^{48} +12.7381 q^{49} -62.8446i q^{50} -424.032 q^{51} +2942.05i q^{52} +4733.28 q^{53} -151.228i q^{54} -4977.46i q^{55} -1633.12i q^{56} -3067.61 q^{57} +770.378i q^{58} +(-3019.48 - 1732.08i) q^{59} +2015.42 q^{60} -1500.86i q^{61} -1477.24 q^{62} +1326.50 q^{63} +2417.74 q^{64} +5182.96i q^{65} -1066.52 q^{66} -233.568i q^{67} -1210.86 q^{68} +4202.28i q^{69} -1384.32i q^{70} +2664.24 q^{71} -897.507i q^{72} +9123.79i q^{73} +2270.11 q^{74} +302.945 q^{75} -8759.85 q^{76} -9355.06i q^{77} +1110.55 q^{78} -8286.52 q^{79} +5269.26 q^{80} +729.000 q^{81} -1220.72i q^{82} +7478.73i q^{83} +3787.96 q^{84} -2133.16 q^{85} +376.915 q^{86} -3713.64 q^{87} -6329.59 q^{88} -10590.0i q^{89} -760.776i q^{90} +9741.29i q^{91} +12000.0i q^{92} -7121.10i q^{93} -2432.95 q^{94} -15432.1 q^{95} -3892.65i q^{96} +15056.8i q^{97} -13.7306i q^{98} -5141.21i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	1.07792i		−	0.269480i	−0.990881	−	0.134740i	$-0.956980\pi$
$2$		−	1.07792i		−	0.269480i	0.990881	−	0.134740i	$-0.0430199\pi$
$3$	5.19615			0.577350
$3$	5.19615			0.577350
$4$	14.8381			0.927380
$5$	26.1400			1.04560			0.522801	−	0.852455i	$-0.324887\pi$
$5$	26.1400			1.04560			0.522801	+	0.852455i	$0.324887\pi$
$6$		−	5.60104i		−	0.155584i
$7$	49.1298			1.00265			0.501325	−	0.865259i	$-0.332846\pi$
$7$	49.1298			1.00265			0.501325	+	0.865259i	$0.332846\pi$
$8$		−	33.2410i		−	0.519391i
$9$	27.0000			0.333333
$10$		−	28.1769i		−	0.281769i
$11$		−	190.415i		−	1.57368i	−0.617158	−	0.786839i	$-0.711716\pi$
$11$		−	190.415i		−	1.57368i	0.617158	−	0.786839i	$-0.288284\pi$
$12$	77.1010			0.535423
$13$			198.277i			1.17323i	0.809864	+	0.586617i	$0.199541\pi$
$13$			198.277i			1.17323i	−0.809864	+	0.586617i	$0.800459\pi$
$14$		−	52.9580i		−	0.270194i
$15$	135.828			0.603678
$16$	201.578			0.787415
$17$	−81.6050			−0.282370			−0.141185	−	0.989983i	$-0.545091\pi$
$17$	−81.6050			−0.282370			−0.141185	+	0.989983i	$0.545091\pi$
$18$		−	29.1038i		−	0.0898267i
$19$	−590.362			−1.63535			−0.817676	−	0.575678i	$-0.804738\pi$
$19$	−590.362			−1.63535			−0.817676	+	0.575678i	$0.804738\pi$
$20$	387.868			0.969671
$21$	255.286			0.578880
$22$	−205.252			−0.424075
$23$			808.730i			1.52879i	0.644748	+	0.764395i	$0.276962\pi$
$23$			808.730i			1.52879i	−0.644748	+	0.764395i	$0.723038\pi$
$24$		−	172.725i		−	0.299870i
$25$	58.3017			0.0932828
$26$	213.726			0.316163
$27$	140.296			0.192450
$28$	728.992			0.929837
$29$	−714.690			−0.849809			−0.424905	−	0.905238i	$-0.639692\pi$
$29$	−714.690			−0.849809			−0.424905	+	0.905238i	$0.639692\pi$
$30$		−	146.411i		−	0.162679i
$31$		−	1370.46i		−	1.42607i	−0.701126	−	0.713037i	$-0.747319\pi$
$31$		−	1370.46i		−	1.42607i	0.701126	−	0.713037i	$-0.252681\pi$
$32$		−	749.141i		−	0.731583i
$33$		−	989.426i		−	0.908564i
$34$			87.9637i			0.0760932i
$35$	1284.26			1.04837
$36$	400.628			0.309127
$37$			2106.01i			1.53835i	0.639036	+	0.769177i	$0.279334\pi$
$37$			2106.01i			1.53835i	−0.639036	+	0.769177i	$0.720666\pi$
$38$			636.363i			0.440695i
$39$			1030.28i			0.677367i
$40$		−	868.921i		−	0.543076i
$41$	1132.48			0.673694			0.336847	−	0.941559i	$-0.390640\pi$
$41$	1132.48			0.673694			0.336847	+	0.941559i	$0.390640\pi$
$42$		−	275.178i		−	0.155997i
$43$			349.669i			0.189113i	0.995520	+	0.0945563i	$0.0301432\pi$
$43$			349.669i			0.189113i	−0.995520	+	0.0945563i	$0.969857\pi$
$44$		−	2825.40i		−	1.45940i
$45$	705.781			0.348534
$46$	871.746			0.411978
$47$		−	2257.08i		−	1.02176i	−0.859651	−	0.510882i	$-0.829319\pi$
$47$		−	2257.08i		−	1.02176i	0.859651	−	0.510882i	$-0.170681\pi$
$48$	1047.43			0.454614
$49$	12.7381			0.00530533
$50$		−	62.8446i		−	0.0251378i
$51$	−424.032			−0.163027
$52$			2942.05i			1.08803i
$53$	4733.28			1.68504			0.842520	−	0.538665i	$-0.181071\pi$
$53$	4733.28			1.68504			0.842520	+	0.538665i	$0.181071\pi$
$54$		−	151.228i		−	0.0518615i
$55$		−	4977.46i		−	1.64544i
$56$		−	1633.12i		−	0.520767i
$57$	−3067.61			−0.944171
$58$			770.378i			0.229007i
$59$	−3019.48	−	1732.08i	−0.867417	−	0.497582i
$59$	−3019.48	−	1732.08i	−0.867417	−	0.497582i
$60$	2015.42			0.559840
$61$		−	1500.86i		−	0.403349i	−0.979453	−	0.201675i	$-0.935362\pi$
$61$		−	1500.86i		−	0.403349i	0.979453	−	0.201675i	$-0.0646384\pi$
$62$	−1477.24			−0.384298
$63$	1326.50			0.334216
$64$	2417.74			0.590268
$65$			5182.96i			1.22674i
$66$	−1066.52			−0.244840
$67$		−	233.568i		−	0.0520312i	−0.999662	−	0.0260156i	$-0.991718\pi$
$67$		−	233.568i		−	0.0520312i	0.999662	−	0.0260156i	$-0.00828195\pi$
$68$	−1210.86			−0.261865
$69$			4202.28i			0.882647i
$70$		−	1384.32i		−	0.282515i
$71$	2664.24			0.528515			0.264258	−	0.964452i	$-0.414873\pi$
$71$	2664.24			0.528515			0.264258	+	0.964452i	$0.414873\pi$
$72$		−	897.507i		−	0.173130i
$73$			9123.79i			1.71210i	0.516892	+	0.856051i	$0.327089\pi$
$73$			9123.79i			1.71210i	−0.516892	+	0.856051i	$0.672911\pi$
$74$	2270.11			0.414556
$75$	302.945			0.0538568
$76$	−8759.85			−1.51659
$77$		−	9355.06i		−	1.57785i
$78$	1110.55			0.182537
$79$	−8286.52			−1.32776			−0.663878	−	0.747841i	$-0.731090\pi$
$79$	−8286.52			−1.32776			−0.663878	+	0.747841i	$0.731090\pi$
$80$	5269.26			0.823323
$81$	729.000			0.111111
$82$		−	1220.72i		−	0.181547i
$83$			7478.73i			1.08560i	0.839860	+	0.542802i	$0.182637\pi$
$83$			7478.73i			1.08560i	−0.839860	+	0.542802i	$0.817363\pi$
$84$	3787.96			0.536842
$85$	−2133.16			−0.295247
$86$	376.915			0.0509621
$87$	−3713.64			−0.490638
$88$	−6329.59			−0.817354
$89$		−	10590.0i		−	1.33696i	−0.743732	−	0.668478i	$-0.766946\pi$
$89$		−	10590.0i		−	1.33696i	0.743732	−	0.668478i	$-0.233054\pi$
$90$		−	760.776i		−	0.0939229i
$91$			9741.29i			1.17634i
$92$			12000.0i			1.41777i
$93$		−	7121.10i		−	0.823344i
$94$	−2432.95			−0.275345
$95$	−15432.1			−1.70993
$96$		−	3892.65i		−	0.422380i
$97$			15056.8i			1.60025i	0.599833	+	0.800125i	$0.295234\pi$
$97$			15056.8i			1.60025i	−0.599833	+	0.800125i	$0.704766\pi$
$98$		−	13.7306i		−	0.00142968i
$99$		−	5141.21i		−	0.524559i
$100$	865.086			0.0865086
$101$			7157.42i			0.701639i	0.936443	+	0.350819i	$0.114097\pi$
$101$			7157.42i			0.701639i	−0.936443	+	0.350819i	$0.885903\pi$
$102$			457.073i			0.0439324i
$103$		−	10141.1i		−	0.955896i	−0.878388	−	0.477948i	$-0.841381\pi$
$103$		−	10141.1i		−	0.955896i	0.878388	−	0.477948i	$-0.158619\pi$
$104$	6590.91			0.609367
$105$	6673.19			0.605278
$106$		−	5102.10i		−	0.454085i
$107$	3578.48			0.312558			0.156279	−	0.987713i	$-0.450050\pi$
$107$	3578.48			0.312558			0.156279	+	0.987713i	$0.450050\pi$
$108$	2081.73			0.178474
$109$			2420.34i			0.203716i	0.994799	+	0.101858i	$0.0324787\pi$
$109$			2420.34i			0.203716i	−0.994799	+	0.101858i	$0.967521\pi$
$110$	−5365.30			−0.443413
$111$			10943.1i			0.888169i
$112$	9903.50			0.789501
$113$			1468.79i			0.115028i	0.998345	+	0.0575140i	$0.0183174\pi$
$113$			1468.79i			0.115028i	−0.998345	+	0.0575140i	$0.981683\pi$
$114$			3306.64i			0.254435i
$115$			21140.2i			1.59851i
$116$	−10604.6			−0.788097
$117$			5353.47i			0.391078i
$118$	−1867.05	+	3254.76i	−0.134089	+	0.233752i
$119$	−4009.24			−0.283118
$120$		−	4515.05i		−	0.313545i
$121$	−21616.9			−1.47646
$122$	−1617.81			−0.108695
$123$	5884.53			0.388957
$124$		−	20335.0i		−	1.32251i
$125$	−14813.5			−0.948065
$126$		−	1429.87i		−	0.0900647i
$127$	−23012.3			−1.42677			−0.713383	−	0.700774i	$-0.752838\pi$
$127$	−23012.3			−1.42677			−0.713383	+	0.700774i	$0.752838\pi$
$128$		−	14592.4i		−	0.890649i
$129$			1816.93i			0.109184i
$130$	5586.82			0.330581
$131$			16089.2i			0.937543i	0.883319	+	0.468772i	$0.155303\pi$
$131$			16089.2i			0.937543i	−0.883319	+	0.468772i	$0.844697\pi$
$132$		−	14681.2i		−	0.842584i
$133$	−29004.4			−1.63968
$134$	−251.768			−0.0140214
$135$	3667.35			0.201226
$136$			2712.63i			0.146661i
$137$	−4170.90			−0.222223			−0.111111	−	0.993808i	$-0.535441\pi$
$137$	−4170.90			−0.222223			−0.111111	+	0.993808i	$0.535441\pi$
$138$	4529.73			0.237856
$139$	22317.2			1.15508			0.577538	−	0.816364i	$-0.304014\pi$
$139$	22317.2			1.15508			0.577538	+	0.816364i	$0.304014\pi$
$140$	19055.9			0.972239
$141$		−	11728.1i		−	0.589916i
$142$		−	2871.84i		−	0.142424i
$143$	37754.9			1.84629
$144$	5442.61			0.262472
$145$	−18682.0			−0.888562
$146$	9834.72			0.461377
$147$	66.1890			0.00306303
$148$			31249.1i			1.42664i
$149$		−	20688.1i		−	0.931853i	−0.884823	−	0.465927i	$-0.845721\pi$
$149$		−	20688.1i		−	0.931853i	0.884823	−	0.465927i	$-0.154279\pi$
$150$		−	326.550i		−	0.0145133i
$151$			30416.0i			1.33397i	0.745069	+	0.666987i	$0.232417\pi$
$151$			30416.0i			1.33397i	−0.745069	+	0.666987i	$0.767583\pi$
$152$			19624.2i			0.849387i
$153$	−2203.34			−0.0941235
$154$	−10084.0			−0.425198
$155$		−	35823.8i		−	1.49111i
$156$			15287.3i			0.628177i
$157$		−	37967.4i		−	1.54032i	−0.637849	−	0.770162i	$-0.720175\pi$
$157$		−	37967.4i		−	1.54032i	0.637849	−	0.770162i	$-0.279825\pi$
$158$			8932.21i			0.357804i
$159$	24594.8			0.972859
$160$		−	19582.6i		−	0.764945i
$161$			39732.7i			1.53284i
$162$		−	785.804i		−	0.0299422i
$163$	−25366.6			−0.954745			−0.477372	−	0.878701i	$-0.658411\pi$
$163$	−25366.6			−0.954745			−0.477372	+	0.878701i	$0.658411\pi$
$164$	16803.8			0.624770
$165$		−	25863.6i		−	0.949996i
$166$	8061.48			0.292549
$167$	33472.9			1.20022			0.600110	−	0.799917i	$-0.295123\pi$
$167$	33472.9			1.20022			0.600110	+	0.799917i	$0.295123\pi$
$168$		−	8485.96i		−	0.300665i
$169$	−10752.6			−0.376479
$170$			2299.38i			0.0795632i
$171$	−15939.8			−0.545118
$172$			5188.42i			0.175379i
$173$		−	23093.0i		−	0.771594i	−0.922584	−	0.385797i	$-0.873927\pi$
$173$		−	23093.0i		−	0.771594i	0.922584	−	0.385797i	$-0.126073\pi$
$174$			4003.00i			0.132217i
$175$	2864.35			0.0935299
$176$		−	38383.5i		−	1.23914i
$177$	−15689.7	−	9000.17i	−0.500803	−	0.287279i
$178$	−11415.2			−0.360283
$179$		−	49281.4i		−	1.53807i	−0.639205	−	0.769036i	$-0.720736\pi$
$179$		−	49281.4i		−	1.53807i	0.639205	−	0.769036i	$-0.279264\pi$
$180$	10472.4			0.323224
$181$	52984.7			1.61731			0.808656	−	0.588282i	$-0.200196\pi$
$181$	52984.7			1.61731			0.808656	+	0.588282i	$0.200196\pi$
$182$	10500.3			0.317001
$183$		−	7798.71i		−	0.232874i
$184$	26883.0			0.794039
$185$			55051.1i			1.60851i
$186$	−7675.98			−0.221875
$187$			15538.8i			0.444360i
$188$		−	33490.7i		−	0.947565i
$189$	6892.72			0.192960
$190$			16634.6i			0.460791i
$191$		−	31600.3i		−	0.866213i	−0.901343	−	0.433106i	$-0.857417\pi$
$191$		−	31600.3i		−	0.866213i	0.901343	−	0.433106i	$-0.142583\pi$
$192$	12562.9			0.340791
$193$	−6662.77			−0.178871			−0.0894355	−	0.995993i	$-0.528506\pi$
$193$	−6662.77			−0.178871			−0.0894355	+	0.995993i	$0.528506\pi$
$194$	16230.0			0.431236
$195$			26931.4i			0.708256i
$196$	189.009			0.00492006
$197$	−3249.83			−0.0837391			−0.0418696	−	0.999123i	$-0.513331\pi$
$197$	−3249.83			−0.0837391			−0.0418696	+	0.999123i	$0.513331\pi$
$198$	−5541.81			−0.141358
$199$	−39419.7			−0.995421			−0.497710	−	0.867343i	$-0.665826\pi$
$199$	−39419.7			−0.995421			−0.497710	+	0.867343i	$0.665826\pi$
$200$		−	1938.01i		−	0.0484502i
$201$		−	1213.65i		−	0.0300402i
$202$	7715.12			0.189078
$203$	−35112.6			−0.852061
$204$	−6291.83			−0.151188
$205$	29603.0			0.704415
$206$	−10931.3			−0.257595
$207$			21835.7i			0.509597i
$208$			39968.3i			0.923822i
$209$			112414.i			2.57352i
$210$		−	7193.16i		−	0.163110i
$211$			8215.58i			0.184533i	0.995734	+	0.0922663i	$0.0294111\pi$
$211$			8215.58i			0.184533i	−0.995734	+	0.0922663i	$0.970589\pi$
$212$	70232.8			1.56267
$213$	13843.8			0.305138
$214$		−	3857.31i		−	0.0842281i
$215$			9140.37i			0.197736i
$216$		−	4663.58i		−	0.0999568i
$217$		−	67330.3i		−	1.42985i
$218$	2608.94			0.0548973
$219$			47408.6i			0.988482i
$220$		−	73856.0i		−	1.52595i
$221$		−	16180.4i		−	0.331287i
$222$	11795.8			0.239344
$223$	24717.2			0.497037			0.248519	−	0.968627i	$-0.420056\pi$
$223$	24717.2			0.497037			0.248519	+	0.968627i	$0.420056\pi$
$224$		−	36805.2i		−	0.733521i
$225$	1574.15			0.0310943
$226$	1583.24			0.0309978
$227$			22950.1i			0.445382i	0.974889	+	0.222691i	$0.0714842\pi$
$227$			22950.1i			0.445382i	−0.974889	+	0.222691i	$0.928516\pi$
$228$	−45517.5			−0.875606
$229$			26740.2i			0.509911i	0.966953	+	0.254955i	$0.0820608\pi$
$229$			26740.2i			0.509911i	−0.966953	+	0.254955i	$0.917939\pi$
$230$	22787.5			0.430765
$231$		−	48610.3i		−	0.910970i
$232$			23757.0i			0.441383i
$233$		−	29499.8i		−	0.543385i	−0.962384	−	0.271693i	$-0.912417\pi$
$233$		−	29499.8i		−	0.543385i	0.962384	−	0.271693i	$-0.0875834\pi$
$234$	5770.61			0.105388
$235$		−	59000.1i		−	1.06836i
$236$	−44803.3	−	25700.8i	−0.804425	−	0.461448i
$237$	−43058.0			−0.766580
$238$			4321.64i			0.0762948i
$239$	37689.9			0.659825			0.329913	−	0.944011i	$-0.392981\pi$
$239$	37689.9			0.659825			0.329913	+	0.944011i	$0.392981\pi$
$240$	27379.9			0.475345
$241$	49193.4			0.846979			0.423489	−	0.905901i	$-0.360805\pi$
$241$	49193.4			0.846979			0.423489	+	0.905901i	$0.360805\pi$
$242$			23301.3i			0.397877i
$243$	3788.00			0.0641500
$244$		−	22269.9i		−	0.374058i
$245$	332.974			0.00554726
$246$		−	6343.06i		−	0.104816i
$247$		−	117055.i		−	1.91865i
$248$	−45555.4			−0.740689
$249$			38860.6i			0.626774i
$250$			15967.8i			0.255485i
$251$	−67494.9			−1.07133			−0.535666	−	0.844430i	$-0.679939\pi$
$251$	−67494.9			−1.07133			−0.535666	+	0.844430i	$0.679939\pi$
$252$	19682.8			0.309946
$253$	153994.			2.40582
$254$			24805.4i			0.384485i
$255$	−11084.2			−0.170461
$256$	22954.4			0.350256
$257$	−94378.9			−1.42892			−0.714461	−	0.699675i	$-0.753328\pi$
$257$	−94378.9			−1.42892			−0.714461	+	0.699675i	$0.753328\pi$
$258$	1958.51			0.0294230
$259$			103468.i			1.54243i
$260$			76905.2i			1.13765i
$261$	−19296.6			−0.283270
$262$	17342.8			0.252649
$263$	−96817.8			−1.39973			−0.699864	−	0.714276i	$-0.746756\pi$
$263$	−96817.8			−1.39973			−0.699864	+	0.714276i	$0.746756\pi$
$264$	−32889.5			−0.471899
$265$	123728.			1.76188
$266$			31264.4i			0.441862i
$267$		−	55027.4i		−	0.771892i
$268$		−	3465.70i		−	0.0482527i
$269$			56566.9i			0.781732i	0.920448	+	0.390866i	$0.127824\pi$
$269$			56566.9i			0.781732i	−0.920448	+	0.390866i	$0.872176\pi$
$270$		−	3953.11i		−	0.0542264i
$271$	−17865.0			−0.243257			−0.121628	−	0.992576i	$-0.538812\pi$
$271$	−17865.0			−0.243257			−0.121628	+	0.992576i	$0.538812\pi$
$272$	−16449.8			−0.222343
$273$			50617.2i			0.679162i
$274$			4495.90i			0.0598846i
$275$		−	11101.5i		−	0.146797i
$276$			62353.9i			0.818550i
$277$	−86629.0			−1.12902			−0.564512	−	0.825425i	$-0.690936\pi$
$277$	−86629.0			−1.12902			−0.564512	+	0.825425i	$0.690936\pi$
$278$		−	24056.2i		−	0.311270i
$279$		−	37002.3i		−	0.475358i
$280$		−	42689.9i		−	0.544514i
$281$	−84063.8			−1.06462			−0.532312	−	0.846548i	$-0.678677\pi$
$281$	−84063.8			−1.06462			−0.532312	+	0.846548i	$0.678677\pi$
$282$	−12642.0			−0.158971
$283$		−	10026.8i		−	0.125195i	−0.998039	−	0.0625977i	$-0.980062\pi$
$283$		−	10026.8i		−	0.125195i	0.998039	−	0.0625977i	$-0.0199385\pi$
$284$	39532.3			0.490135
$285$	−80187.5			−0.987227
$286$		−	40696.7i		−	0.497539i
$287$	55638.5			0.675478
$288$		−	20226.8i		−	0.243861i
$289$	−76861.6			−0.920267
$290$			20137.7i			0.239450i
$291$			78237.2i			0.923905i
$292$			135380.i			1.58777i
$293$	−68620.8			−0.799319			−0.399660	−	0.916664i	$-0.630872\pi$
$293$	−68620.8			−0.799319			−0.399660	+	0.916664i	$0.630872\pi$
$294$		−	71.3465i		−	0.000825426i
$295$	−78929.3	−	45276.8i	−0.906972	−	0.520273i
$296$	70005.8			0.799007
$297$		−	26714.5i		−	0.302855i
$298$	−22300.1			−0.251116
$299$	−160352.			−1.79363
$300$	4495.12			0.0499458
$301$			17179.2i			0.189614i
$302$	32786.0			0.359480
$303$			37191.0i			0.405091i
$304$	−119004.			−1.28770
$305$		−	39232.6i		−	0.421743i
$306$			2375.02i			0.0253644i
$307$	158352.			1.68014			0.840072	−	0.542475i	$-0.182513\pi$
$307$	158352.			1.68014			0.840072	+	0.542475i	$0.182513\pi$
$308$		−	138811.i		−	1.46326i
$309$		−	52694.7i		−	0.551887i
$310$	−38615.2			−0.401823
$311$	58229.0			0.602030			0.301015	−	0.953619i	$-0.402675\pi$
$311$	58229.0			0.602030			0.301015	+	0.953619i	$0.402675\pi$
$312$	34247.4			0.351818
$313$			38385.1i			0.391808i	0.980623	+	0.195904i	$0.0627642\pi$
$313$			38385.1i			0.391808i	−0.980623	+	0.195904i	$0.937236\pi$
$314$	−40925.9			−0.415086
$315$	34674.9			0.349457
$316$	−122956.			−1.23133
$317$	25846.8			0.257210			0.128605	−	0.991696i	$-0.458950\pi$
$317$	25846.8			0.257210			0.128605	+	0.991696i	$0.458950\pi$
$318$		−	26511.3i		−	0.262166i
$319$			136088.i			1.33733i
$320$	63199.8			0.617185
$321$	18594.3			0.180455
$322$	42828.7			0.413070
$323$	48176.5			0.461775
$324$	10817.0			0.103042
$325$			11559.9i			0.109443i
$326$			27343.2i			0.257285i
$327$			12576.5i			0.117615i
$328$		−	37644.7i		−	0.349910i
$329$		−	110890.i		−	1.02447i
$330$	−27878.9			−0.256005
$331$	127603.			1.16467			0.582337	−	0.812947i	$-0.302138\pi$
$331$	127603.			1.16467			0.582337	+	0.812947i	$0.302138\pi$
$332$			110970.i			1.00677i
$333$			56862.2i			0.512785i
$334$		−	36081.2i		−	0.323435i
$335$		−	6105.48i		−	0.0544039i
$336$	51460.1			0.455819
$337$		−	55504.5i		−	0.488730i	−0.969683	−	0.244365i	$-0.921421\pi$
$337$		−	55504.5i		−	0.488730i	0.969683	−	0.244365i	$-0.0785794\pi$
$338$			11590.5i			0.101454i
$339$			7632.07i			0.0664115i
$340$	−31652.0			−0.273806
$341$	−260956.			−2.24418
$342$			17181.8i			0.146898i
$343$	−117335.			−0.997330
$344$	11623.4			0.0982233
$345$			109848.i			0.922897i
$346$	−24892.5			−0.207929
$347$		−	84690.6i		−	0.703358i	−0.936121	−	0.351679i	$-0.885611\pi$
$347$		−	84690.6i		−	0.703358i	0.936121	−	0.351679i	$-0.114389\pi$
$348$	−55103.3			−0.455008
$349$		−	74255.3i		−	0.609645i	−0.952409	−	0.304822i	$-0.901403\pi$
$349$		−	74255.3i		−	0.609645i	0.952409	−	0.304822i	$-0.0985971\pi$
$350$		−	3087.54i		−	0.0252044i
$351$			27817.4i			0.225789i
$352$	−142648.			−1.15128
$353$			221926.i			1.78098i	0.455001	+	0.890491i	$0.349639\pi$
$353$			221926.i			1.78098i	−0.455001	+	0.890491i	$0.650361\pi$
$354$	−9701.47	+	16912.2i	−0.0774160	+	0.134956i
$355$	69643.5			0.552616
$356$		−	157136.i		−	1.23987i
$357$	−20832.6			−0.163459
$358$	−53121.4			−0.414480
$359$	55048.5			0.427127			0.213563	−	0.976929i	$-0.431493\pi$
$359$	55048.5			0.427127			0.213563	+	0.976929i	$0.431493\pi$
$360$		−	23460.9i		−	0.181025i
$361$	218207.			1.67438
$362$		−	57113.3i		−	0.435833i
$363$	−112325.			−0.852436
$364$			144542.i			1.09092i
$365$			238496.i			1.79018i
$366$	−8406.39			−0.0627549
$367$		−	146662.i		−	1.08889i	−0.838796	−	0.544446i	$-0.816740\pi$
$367$		−	146662.i		−	1.08889i	0.838796	−	0.544446i	$-0.183260\pi$
$368$			163022.i			1.20379i
$369$	30576.9			0.224565
$370$	59340.7			0.433460
$371$	232545.			1.68950
$372$		−	105664.i		−	0.763553i
$373$	166279.			1.19515			0.597573	−	0.801815i	$-0.296132\pi$
$373$	166279.			1.19515			0.597573	+	0.801815i	$0.296132\pi$
$374$	16749.6			0.119746
$375$	−76973.3			−0.547366
$376$	−75027.5			−0.530695
$377$		−	141706.i		−	0.997025i
$378$		−	7429.80i		−	0.0519989i
$379$	60758.9			0.422991			0.211496	−	0.977379i	$-0.432167\pi$
$379$	60758.9			0.422991			0.211496	+	0.977379i	$0.432167\pi$
$380$	−228983.			−1.58575
$381$	−119575.			−0.823744
$382$	−34062.6			−0.233427
$383$	120080.			0.818601			0.409300	−	0.912400i	$-0.365773\pi$
$383$	120080.			0.818601			0.409300	+	0.912400i	$0.365773\pi$
$384$		−	75824.3i		−	0.514216i
$385$		−	244542.i		−	1.64980i
$386$			7181.93i			0.0482022i
$387$			9441.07i			0.0630375i
$388$			223413.i			1.48404i
$389$	184834.			1.22147			0.610735	−	0.791835i	$-0.290874\pi$
$389$	184834.			1.22147			0.610735	+	0.791835i	$0.290874\pi$
$390$	29029.9			0.190861
$391$		−	65996.4i		−	0.431685i
$392$		−	423.427i		−	0.00275554i
$393$			83601.8i			0.541291i
$394$			3503.06i			0.0225660i
$395$	−216610.			−1.38830
$396$		−	76285.7i		−	0.486466i
$397$			170314.i			1.08061i	0.841469	+	0.540306i	$0.181692\pi$
$397$			170314.i			1.08061i	−0.841469	+	0.540306i	$0.818308\pi$
$398$			42491.2i			0.268246i
$399$	−150711.			−0.946672
$400$	11752.4			0.0734523
$401$		−	32629.9i		−	0.202921i	−0.994840	−	0.101461i	$-0.967648\pi$
$401$		−	32629.9i		−	0.202921i	0.994840	−	0.101461i	$-0.0323516\pi$
$402$	−1308.22			−0.00809524
$403$	271730.			1.67312
$404$			106202.i			0.650686i
$405$	19056.1			0.116178
$406$			37848.5i			0.229613i
$407$	401015.			2.42087
$408$			14095.3i			0.0846745i
$409$		−	278424.i		−	1.66441i	−0.554468	−	0.832205i	$-0.687078\pi$
$409$		−	278424.i		−	1.66441i	0.554468	−	0.832205i	$-0.312922\pi$
$410$		−	31909.7i		−	0.189826i
$411$	−21672.6			−0.128300
$412$		−	150475.i		−	0.886480i
$413$	−148346.	−	85097.0i	−0.869715	−	0.498901i
$414$	23537.2			0.137326
$415$			195494.i			1.13511i
$416$	148537.			0.858319
$417$	115964.			0.666883
$418$	121173.			0.693512
$419$			99149.5i			0.564758i	0.959303	+	0.282379i	$0.0911236\pi$
$419$			99149.5i			0.564758i	−0.959303	+	0.282379i	$0.908876\pi$
$420$	99017.3			0.561323
$421$		−	124249.i		−	0.701015i	−0.936560	−	0.350508i	$-0.886009\pi$
$421$		−	124249.i		−	0.701015i	0.936560	−	0.350508i	$-0.113991\pi$
$422$	8855.74			0.0497279
$423$		−	60941.1i		−	0.340588i
$424$		−	157339.i		−	0.875194i
$425$	−4757.72			−0.0263403
$426$		−	14922.5i		−	0.0822287i
$427$		−	73737.1i		−	0.404418i
$428$	53097.7			0.289860
$429$	196180.			1.06596
$430$	9852.59			0.0532860
$431$			80751.0i			0.434704i	0.976093	+	0.217352i	$0.0697419\pi$
$431$			80751.0i			0.434704i	−0.976093	+	0.217352i	$0.930258\pi$
$432$	28280.6			0.151538
$433$	267478.			1.42663			0.713317	−	0.700841i	$-0.247192\pi$
$433$	267478.			1.42663			0.713317	+	0.700841i	$0.247192\pi$
$434$	−72576.7			−0.385317
$435$	−97074.6			−0.513011
$436$			35913.3i			0.188922i
$437$		−	477444.i		−	2.50011i
$438$	51102.7			0.266376
$439$	244540.			1.26888			0.634441	−	0.772971i	$-0.281230\pi$
$439$	244540.			1.26888			0.634441	+	0.772971i	$0.281230\pi$
$440$	−165456.			−0.854626
$441$	343.928			0.00176844
$442$	−17441.1			−0.0892751
$443$		−	303375.i		−	1.54587i	−0.634487	−	0.772933i	$-0.718789\pi$
$443$		−	303375.i		−	1.54587i	0.634487	−	0.772933i	$-0.281211\pi$
$444$			162375.i			0.823671i
$445$		−	276824.i		−	1.39792i
$446$		−	26643.1i		−	0.133942i
$447$		−	107498.i		−	0.538006i
$448$	118783.			0.591832
$449$	−265110.			−1.31503			−0.657513	−	0.753444i	$-0.728391\pi$
$449$	−265110.			−1.31503			−0.657513	+	0.753444i	$0.728391\pi$
$450$		−	1696.80i		−	0.00837928i
$451$		−	215641.i		−	1.06018i
$452$			21794.1i			0.106675i
$453$			158046.i			0.770171i
$454$	24738.4			0.120022
$455$			254638.i			1.22999i
$456$			101970.i			0.490394i
$457$			351393.i			1.68252i	0.540628	+	0.841262i	$0.318187\pi$
$457$			351393.i			1.68252i	−0.540628	+	0.841262i	$0.681813\pi$
$458$	28823.8			0.137411
$459$	−11448.9			−0.0543422
$460$			313681.i			1.48242i
$461$	96578.1			0.454440			0.227220	−	0.973843i	$-0.427036\pi$
$461$	96578.1			0.454440			0.227220	+	0.973843i	$0.427036\pi$
$462$	−52398.0			−0.245488
$463$		−	247348.i		−	1.15384i	−0.816800	−	0.576922i	$-0.804254\pi$
$463$		−	247348.i		−	1.15384i	0.816800	−	0.576922i	$-0.195746\pi$
$464$	−144066.			−0.669153
$465$		−	186146.i		−	0.860890i
$466$	−31798.5			−0.146432
$467$			233363.i			1.07004i	0.844841	+	0.535018i	$0.179695\pi$
$467$			233363.i			1.07004i	−0.844841	+	0.535018i	$0.820305\pi$
$468$			79435.2i			0.362678i
$469$		−	11475.1i		−	0.0521690i
$470$	−63597.4			−0.287901
$471$		−	197285.i		−	0.889306i
$472$	−57576.2	+	100370.i	−0.258440	+	0.450528i
$473$	66582.3			0.297602
$474$			46413.1i			0.206578i
$475$	−34419.1			−0.152550
$476$	−59489.5			−0.262559
$477$	127799.			0.561680
$478$		−	40626.7i		−	0.177810i
$479$	254275.			1.10824			0.554119	−	0.832438i	$-0.313055\pi$
$479$	254275.			1.10824			0.554119	+	0.832438i	$0.313055\pi$
$480$		−	101754.i		−	0.441641i
$481$	−417572.			−1.80485
$482$		−	53026.5i		−	0.228244i
$483$			206457.i			0.884986i
$484$	−320753.			−1.36924
$485$			393584.i			1.67322i
$486$		−	4083.16i		−	0.0172872i
$487$	221313.			0.933146			0.466573	−	0.884483i	$-0.345488\pi$
$487$	221313.			0.933146			0.466573	+	0.884483i	$0.345488\pi$
$488$	−49890.2			−0.209496
$489$	−131809.			−0.551222
$490$		−	358.920i		−	0.00149488i
$491$	20143.2			0.0835537			0.0417769	−	0.999127i	$-0.486698\pi$
$491$	20143.2			0.0835537			0.0417769	+	0.999127i	$0.486698\pi$
$492$	87315.2			0.360711
$493$	58322.3			0.239961
$494$	−126176.			−0.517038
$495$		−	134391.i		−	0.548480i
$496$		−	276254.i		−	1.12291i
$497$	130894.			0.529915
$498$	41888.7			0.168903
$499$	−142382.			−0.571813			−0.285907	−	0.958258i	$-0.592295\pi$
$499$	−142382.			−0.571813			−0.285907	+	0.958258i	$0.592295\pi$
$500$	−219804.			−0.879217
$501$	173930.			0.692947
$502$			72754.1i			0.288702i
$503$			61029.3i			0.241214i	0.992700	+	0.120607i	$0.0384841\pi$
$503$			61029.3i			0.241214i	−0.992700	+	0.120607i	$0.961516\pi$
$504$		−	44094.3i		−	0.173589i
$505$			187095.i			0.733635i
$506$		−	165994.i		−	0.648321i
$507$	−55872.2			−0.217360
$508$	−341459.			−1.32315
$509$		−	119724.i		−	0.462111i	−0.972941	−	0.231056i	$-0.925782\pi$
$509$		−	119724.i		−	0.462111i	0.972941	−	0.231056i	$-0.0742179\pi$
$510$			11947.9i			0.0459358i
$511$			448250.i			1.71664i
$512$		−	258221.i		−	0.985036i
$513$	−82825.5			−0.314724
$514$			101733.i			0.385066i
$515$		−	265089.i		−	0.999487i
$516$			26959.8i			0.101255i
$517$	−429782.			−1.60793
$518$	111530.			0.415654
$519$		−	119995.i		−	0.445480i
$520$	172287.			0.637155
$521$	526245.			1.93871			0.969355	−	0.245663i	$-0.0790058\pi$
$521$	526245.			1.93871			0.969355	+	0.245663i	$0.0790058\pi$
$522$			20800.2i			0.0763356i
$523$	43580.1			0.159325			0.0796626	−	0.996822i	$-0.474616\pi$
$523$	43580.1			0.159325			0.0796626	+	0.996822i	$0.474616\pi$
$524$			238733.i			0.869459i
$525$	14883.6			0.0539995
$526$			104362.i			0.377199i
$527$			111836.i			0.402681i
$528$		−	199447.i		−	0.715417i
$529$	−374203.			−1.33720
$530$		−	133369.i		−	0.474792i
$531$	−81525.9	−	46766.3i	−0.289139	−	0.165861i
$532$	−430370.			−1.52061
$533$			224544.i			0.790400i
$534$	−59315.1			−0.208009
$535$	93541.5			0.326811
$536$	−7764.03			−0.0270245
$537$		−	256074.i		−	0.888007i
$538$	60974.6			0.210661
$539$		−	2425.52i		−	0.00834888i
$540$	54416.4			0.186613
$541$			318207.i			1.08721i	0.839340	+	0.543606i	$0.182942\pi$
$541$			318207.i			1.08721i	−0.839340	+	0.543606i	$0.817058\pi$
$542$			19257.1i			0.0655529i
$543$	275317.			0.933755
$544$			61133.7i			0.206577i
$545$			63267.9i			0.213005i
$546$	54561.3			0.183021
$547$	−151057.			−0.504856			−0.252428	−	0.967616i	$-0.581229\pi$
$547$	−151057.			−0.504856			−0.252428	+	0.967616i	$0.581229\pi$
$548$	−61888.2			−0.206085
$549$		−	40523.3i		−	0.134450i
$550$	−11966.6			−0.0395589
$551$	421926.			1.38974
$552$	139688.			0.458439
$553$	−407115.			−1.33127
$554$			93379.1i			0.304250i
$555$			286054.i			0.928671i
$556$	331145.			1.07119
$557$	−439612.			−1.41697			−0.708483	−	0.705728i	$-0.750620\pi$
$557$	−439612.			−1.41697			−0.708483	+	0.705728i	$0.750620\pi$
$558$	−39885.6			−0.128099
$559$	−69331.2			−0.221873
$560$	258878.			0.825504
$561$			80742.1i			0.256551i
$562$			90614.0i			0.286895i
$563$		−	468132.i		−	1.47690i	−0.674308	−	0.738450i	$-0.735558\pi$
$563$		−	468132.i		−	1.47690i	0.674308	−	0.738450i	$-0.264442\pi$
$564$		−	174023.i		−	0.547077i
$565$			38394.3i			0.120274i
$566$	−10808.1			−0.0337376
$567$	35815.6			0.111405
$568$		−	88562.2i		−	0.274506i
$569$		−	156283.i		−	0.482711i	−0.970437	−	0.241356i	$-0.922408\pi$
$569$		−	156283.i		−	0.482711i	0.970437	−	0.241356i	$-0.0775920\pi$
$570$			86435.7i			0.266038i
$571$		−	214333.i		−	0.657379i	−0.944438	−	0.328690i	$-0.893393\pi$
$571$		−	214333.i		−	0.657379i	0.944438	−	0.328690i	$-0.106607\pi$
$572$	560210.			1.71222
$573$		−	164200.i		−	0.500108i
$574$		−	59973.8i		−	0.182028i
$575$			47150.4i			0.142610i
$576$	65278.9			0.196756
$577$	−83530.0			−0.250894			−0.125447	−	0.992100i	$-0.540037\pi$
$577$	−83530.0			−0.250894			−0.125447	+	0.992100i	$0.540037\pi$
$578$			82850.7i			0.247994i
$579$	−34620.8			−0.103271
$580$	−277205.			−0.824035
$581$			367429.i			1.08848i
$582$	84333.5			0.248974
$583$		−	901288.i		−	2.65171i
$584$	303284.			0.889249
$585$			139940.i			0.408912i
$586$			73967.7i			0.215401i
$587$		−	65804.0i		−	0.190975i	−0.995431	−	0.0954873i	$-0.969559\pi$
$587$		−	65804.0i		−	0.190975i	0.995431	−	0.0954873i	$-0.0304409\pi$
$588$	982.119			0.00284060
$589$			809066.i			2.33213i
$590$	−48804.7	+	85079.5i	−0.140203	+	0.244411i
$591$	−16886.6			−0.0483468
$592$			424525.i			1.21132i
$593$	−180576.			−0.513511			−0.256756	−	0.966476i	$-0.582654\pi$
$593$	−180576.			−0.513511			−0.256756	+	0.966476i	$0.582654\pi$
$594$	−28796.1			−0.0816133
$595$	−104802.			−0.296029
$596$		−	306971.i		−	0.864183i
$597$	−204831.			−0.574706
$598$			172847.i			0.483347i
$599$	−121500.			−0.338629			−0.169314	−	0.985562i	$-0.554155\pi$
$599$	−121500.			−0.338629			−0.169314	+	0.985562i	$0.554155\pi$
$600$		−	10070.2i		−	0.0279727i
$601$			246152.i			0.681483i	0.940157	+	0.340742i	$0.110678\pi$
$601$			246152.i			0.681483i	−0.940157	+	0.340742i	$0.889322\pi$
$602$	18517.8			0.0510971
$603$		−	6306.33i		−	0.0173437i
$604$			451315.i			1.23710i
$605$	−565067.			−1.54379
$606$	40089.0			0.109164
$607$	91016.6			0.247026			0.123513	−	0.992343i	$-0.460584\pi$
$607$	91016.6			0.247026			0.123513	+	0.992343i	$0.460584\pi$
$608$			442265.i			1.19640i
$609$	−182450.			−0.491937
$610$	−42289.6			−0.113651
$611$	447526.			1.19877
$612$	−32693.3			−0.0872883
$613$			165519.i			0.440481i	0.975446	+	0.220240i	$0.0706842\pi$
$613$			165519.i			0.440481i	−0.975446	+	0.220240i	$0.929316\pi$
$614$		−	170691.i		−	0.452765i
$615$	153822.			0.406694
$616$	−310971.			−0.819519
$617$	−337650.			−0.886945			−0.443472	−	0.896288i	$-0.646254\pi$
$617$	−337650.			−0.886945			−0.443472	+	0.896288i	$0.646254\pi$
$618$	−56800.7			−0.148723
$619$	−158426.			−0.413470			−0.206735	−	0.978397i	$-0.566284\pi$
$619$	−158426.			−0.413470			−0.206735	+	0.978397i	$0.566284\pi$
$620$		−	531557.i		−	1.38282i
$621$			113462.i			0.294216i
$622$		−	62766.2i		−	0.162235i
$623$		−	520286.i		−	1.34050i
$624$			207681.i			0.533369i
$625$	−423664.			−1.08458
$626$	41376.1			0.105585
$627$			584120.i			1.48582i
$628$		−	563364.i		−	1.42847i
$629$		−	171861.i		−	0.434386i
$630$		−	37376.8i		−	0.0941717i
$631$	563902.			1.41627			0.708133	−	0.706080i	$-0.249538\pi$
$631$	563902.			1.41627			0.708133	+	0.706080i	$0.249538\pi$
$632$			275452.i			0.689624i
$633$			42689.4i			0.106540i
$634$		−	27860.7i		−	0.0693129i
$635$	−601543.			−1.49183
$636$	364940.			0.902210
$637$			2525.66i			0.00622439i
$638$	146692.			0.360383
$639$	71934.6			0.176172
$640$		−	381446.i		−	0.931264i
$641$	−46995.0			−0.114376			−0.0571881	−	0.998363i	$-0.518213\pi$
$641$	−46995.0			−0.114376			−0.0571881	+	0.998363i	$0.518213\pi$
$642$		−	20043.2i		−	0.0486291i
$643$	357241.			0.864050			0.432025	−	0.901862i	$-0.357799\pi$
$643$	357241.			0.864050			0.432025	+	0.901862i	$0.357799\pi$
$644$			589558.i			1.42153i
$645$			47494.7i			0.114163i
$646$		−	51930.5i		−	0.124439i
$647$	−146425.			−0.349790			−0.174895	−	0.984587i	$-0.555959\pi$
$647$	−146425.			−0.349790			−0.174895	+	0.984587i	$0.555959\pi$
$648$		−	24232.7i		−	0.0577101i
$649$	−329815.	+	574954.i	−0.783035	+	1.36503i
$650$	12460.6			0.0294926
$651$		−	349858.i		−	0.825525i
$652$	−376392.			−0.885412
$653$	−149895.			−0.351529			−0.175764	−	0.984432i	$-0.556240\pi$
$653$	−149895.			−0.351529			−0.175764	+	0.984432i	$0.556240\pi$
$654$	13556.4			0.0316950
$655$			420572.i			0.980297i
$656$	228283.			0.530477
$657$			246342.i			0.570700i
$658$	−119530.			−0.276075
$659$		−	561285.i		−	1.29245i	−0.763148	−	0.646223i	$-0.776347\pi$
$659$		−	561285.i		−	1.29245i	0.763148	−	0.646223i	$-0.223653\pi$
$660$		−	383767.i		−	0.881007i
$661$	−158811.			−0.363478			−0.181739	−	0.983347i	$-0.558173\pi$
$661$	−158811.			−0.363478			−0.181739	+	0.983347i	$0.558173\pi$
$662$		−	137546.i		−	0.313856i
$663$		−	84075.7i		−	0.191268i
$664$	248600.			0.563853
$665$	−758176.			−1.71446
$666$	61292.9			0.138185
$667$		−	577991.i		−	1.29918i
$668$	496674.			1.11306
$669$	128434.			0.286964
$670$	−6581.22			−0.0146608
$671$	−285787.			−0.634742
$672$		−	191245.i		−	0.423499i
$673$			468994.i			1.03547i	0.855541	+	0.517735i	$0.173225\pi$
$673$			468994.i			1.03547i	−0.855541	+	0.517735i	$0.826775\pi$
$674$	−59829.5			−0.131703
$675$	8179.51			0.0179523
$676$	−159548.			−0.349139
$677$	−560613.			−1.22317			−0.611584	−	0.791180i	$-0.709467\pi$
$677$	−560613.			−1.22317			−0.611584	+	0.791180i	$0.709467\pi$
$678$	8226.77			0.0178966
$679$			739736.i			1.60449i
$680$			70908.3i			0.153348i
$681$			119252.i			0.257142i
$682$			281289.i			0.604762i
$683$			39053.9i			0.0837188i	0.999124	+	0.0418594i	$0.0133282\pi$
$683$			39053.9i			0.0837188i	−0.999124	+	0.0418594i	$0.986672\pi$
$684$	−236516.			−0.505531
$685$	−109027.			−0.232356
$686$			126478.i			0.268760i
$687$			138946.i			0.294397i
$688$			70485.7i			0.148910i
$689$			938499.i			1.97695i
$690$	118407.			0.248702
$691$		−	138810.i		−	0.290714i	−0.989379	−	0.145357i	$-0.953567\pi$
$691$		−	138810.i		−	0.290714i	0.989379	−	0.145357i	$-0.0464330\pi$
$692$		−	342657.i		−	0.715561i
$693$		−	252587.i		−	0.525949i
$694$	−91289.7			−0.189541
$695$	583373.			1.20775
$696$			123445.i			0.254833i
$697$	−92416.0			−0.190231
$698$	−80041.3			−0.164287
$699$		−	153286.i		−	0.313724i
$700$	42501.5			0.0867378
$701$			399850.i			0.813695i	0.913496	+	0.406847i	$0.133372\pi$
$701$			399850.i			0.813695i	−0.913496	+	0.406847i	$0.866628\pi$
$702$	29985.0			0.0608456
$703$		−	1.24331e6i		−	2.51575i
$704$		−	460374.i		−	0.928892i
$705$		−	306574.i		−	0.616817i
$706$	239219.			0.479939
$707$			351642.i			0.703497i
$708$	−232805.	−	133545.i	−0.464435	−	0.266417i
$709$	−572973.			−1.13983			−0.569917	−	0.821702i	$-0.693025\pi$
$709$	−572973.			−1.13983			−0.569917	+	0.821702i	$0.693025\pi$
$710$		−	75070.1i		−	0.148919i
$711$	−223736.			−0.442585
$712$	−352023.			−0.694402
$713$	1.10833e6			2.18017
$714$			22455.9i			0.0440488i
$715$	986913.			1.93049
$716$		−	731242.i		−	1.42638i
$717$	195842.			0.380950
$718$		−	59337.9i		−	0.115102i
$719$			655458.i			1.26791i	0.773372	+	0.633953i	$0.218569\pi$
$719$			655458.i			1.26791i	−0.773372	+	0.633953i	$0.781431\pi$
$720$	142270.			0.274441
$721$		−	498231.i		−	0.958429i
$722$		−	235209.i		−	0.451211i
$723$	255616.			0.489003
$724$	786192.			1.49986
$725$	−41667.6			−0.0792726
$726$			121077.i			0.229715i
$727$	438565.			0.829785			0.414892	−	0.909870i	$-0.363819\pi$
$727$	438565.			0.829785			0.414892	+	0.909870i	$0.363819\pi$
$728$	323810.			0.610981
$729$	19683.0			0.0370370
$730$	257080.			0.482417
$731$		−	28534.8i		−	0.0533998i
$732$		−	115718.i		−	0.215963i
$733$	524612.			0.976406			0.488203	−	0.872730i	$-0.337653\pi$
$733$	524612.			0.976406			0.488203	+	0.872730i	$0.337653\pi$
$734$	−158090.			−0.293435
$735$	1730.18			0.00320271
$736$	605853.			1.11844
$737$	−44474.9			−0.0818803
$738$		−	32959.5i		−	0.0605157i
$739$		−	354679.i		−	0.649451i	−0.945808	−	0.324725i	$-0.894728\pi$
$739$		−	354679.i		−	0.649451i	0.945808	−	0.324725i	$-0.105272\pi$
$740$			816853.i			1.49170i
$741$		−	608236.i		−	1.10773i
$742$		−	250665.i		−	0.455288i
$743$	−9241.58			−0.0167405			−0.00837025	−	0.999965i	$-0.502664\pi$
$743$	−9241.58			−0.0167405			−0.00837025	+	0.999965i	$0.502664\pi$
$744$	−236713.			−0.427637
$745$		−	540787.i		−	0.974347i
$746$		−	179236.i		−	0.322068i
$747$			201926.i			0.361868i
$748$			230567.i			0.412091i
$749$	175810.			0.313386
$750$			82971.1i			0.147504i
$751$		−	1.07245e6i		−	1.90150i	−0.309962	−	0.950749i	$-0.600316\pi$
$751$		−	1.07245e6i		−	1.90150i	0.309962	−	0.950749i	$-0.399684\pi$
$752$		−	454978.i		−	0.804553i
$753$	−350714.			−0.618533
$754$	−152748.			−0.268678
$755$			795074.i			1.39481i
$756$	102275.			0.178947
$757$	513889.			0.896762			0.448381	−	0.893842i	$-0.352001\pi$
$757$	513889.			0.896762			0.448381	+	0.893842i	$0.352001\pi$
$758$		−	65493.2i		−	0.113988i
$759$	800178.			1.38900
$760$			512978.i			0.888120i
$761$	300411.			0.518736			0.259368	−	0.965779i	$-0.416486\pi$
$761$	300411.			0.518736			0.259368	+	0.965779i	$0.416486\pi$
$762$			128893.i			0.221983i
$763$			118911.i			0.204255i
$764$		−	468888.i		−	0.803309i
$765$	−57595.3			−0.0984156
$766$		−	129436.i		−	0.220597i
$767$	343432.	−	598692.i	0.583781	−	1.01768i
$768$	119274.			0.202220
$769$		−	690998.i		−	1.16849i	−0.811579	−	0.584243i	$-0.801391\pi$
$769$		−	690998.i		−	1.16849i	0.811579	−	0.584243i	$-0.198609\pi$
$770$	−263596.			−0.444588
$771$	−490407.			−0.824988
$772$	−98862.7			−0.165882
$773$		−	312549.i		−	0.523069i	−0.965194	−	0.261534i	$-0.915771\pi$
$773$		−	312549.i		−	0.523069i	0.965194	−	0.261534i	$-0.0842285\pi$
$774$	10176.7			0.0169874
$775$		−	79900.0i		−	0.133028i
$776$	500502.			0.831155
$777$			537634.i			0.890522i
$778$		−	199236.i		−	0.329162i
$779$	−668573.			−1.10173
$780$			399611.i			0.656823i
$781$		−	507312.i		−	0.831713i
$782$	−71138.9			−0.116331
$783$	−100268.			−0.163546
$784$	2567.72			0.00417749
$785$		−	992470.i		−	1.61056i
$786$	90116.1			0.145867
$787$	858686.			1.38639			0.693194	−	0.720751i	$-0.256203\pi$
$787$	858686.			1.38639			0.693194	+	0.720751i	$0.256203\pi$
$788$	−48221.3			−0.0776580
$789$	−503080.			−0.808134
$790$			233488.i			0.374120i
$791$			72161.5i			0.115333i
$792$	−170899.			−0.272451
$793$	297586.			0.473223
$794$	183585.			0.291203
$795$	642910.			1.01722
$796$	−584912.			−0.923134
$797$			795885.i			1.25295i	0.779442	+	0.626475i	$0.215503\pi$
$797$			795885.i			1.25295i	−0.779442	+	0.626475i	$0.784497\pi$
$798$			162455.i			0.255109i
$799$			184189.i			0.288516i
$800$		−	43676.2i		−	0.0682441i
$801$		−	285931.i		−	0.445652i
$802$	−35172.5			−0.0546832
$803$	1.73731e6			2.69430
$804$		−	18008.3i		−	0.0278587i
$805$			1.03862e6i			1.60274i
$806$		−	292903.i		−	0.450872i
$807$			293930.i			0.451333i
$808$	237920.			0.364425
$809$		−	610886.i		−	0.933390i	−0.884418	−	0.466695i	$-0.845445\pi$
$809$		−	610886.i		−	0.933390i	0.884418	−	0.466695i	$-0.154555\pi$
$810$		−	20540.9i		−	0.0313076i
$811$			1.12919e6i			1.71682i	0.512967	+	0.858408i	$0.328546\pi$
$811$			1.12919e6i			1.71682i	−0.512967	+	0.858408i	$0.671454\pi$
$812$	−521003.			−0.790184
$813$	−92829.5			−0.140444
$814$		−	432263.i		−	0.652377i
$815$	−663084.			−0.998283
$816$	−85475.7			−0.128370
$817$		−	206431.i		−	0.309266i
$818$	−300119.			−0.448525
$819$			263015.i			0.392114i
$820$	439253.			0.653261
$821$		−	458260.i		−	0.679869i	−0.940449	−	0.339935i	$-0.889595\pi$
$821$		−	458260.i		−	0.679869i	0.940449	−	0.339935i	$-0.110405\pi$
$822$			23361.4i			0.0345744i
$823$			749847.i			1.10707i	0.832828	+	0.553533i	$0.186721\pi$
$823$			749847.i			1.10707i	−0.832828	+	0.553533i	$0.813279\pi$
$824$	−337100.			−0.496484
$825$		−	57685.2i		−	0.0847533i
$826$	−91727.7	+	159906.i	−0.134444	+	0.234371i
$827$	−241049.			−0.352447			−0.176223	−	0.984350i	$-0.556388\pi$
$827$	−241049.			−0.352447			−0.176223	+	0.984350i	$0.556388\pi$
$828$			324000.i			0.472590i
$829$	−459844.			−0.669116			−0.334558	−	0.942375i	$-0.608587\pi$
$829$	−459844.			−0.669116			−0.334558	+	0.942375i	$0.608587\pi$
$830$	210727.			0.305890
$831$	−450137.			−0.651843
$832$			479381.i			0.692523i
$833$	−1039.49			−0.00149807
$834$		−	125000.i		−	0.179712i
$835$	874984.			1.25495
$836$			1.66801e6i			2.38663i
$837$		−	192270.i		−	0.274448i
$838$	106875.			0.152191
$839$			614502.i			0.872970i	0.899712	+	0.436485i	$0.143777\pi$
$839$			614502.i			0.872970i	−0.899712	+	0.436485i	$0.856223\pi$
$840$		−	221823.i		−	0.314376i
$841$	−196500.			−0.277824
$842$	−133930.			−0.188910
$843$	−436808.			−0.614661
$844$			121903.i			0.171132i
$845$	−281074.			−0.393647
$846$	−65689.7			−0.0917817
$847$	−1.06203e6			−1.48037
$848$	954126.			1.32683
$849$		−	52100.6i		−	0.0722815i
$850$			5128.44i			0.00709818i
$851$	−1.70319e6			−2.35182
$852$	205416.			0.282979
$853$	124975.			0.171761			0.0858805	−	0.996305i	$-0.472630\pi$
$853$	124975.			0.171761			0.0858805	+	0.996305i	$0.472630\pi$
$854$	−79482.7			−0.108983
$855$	−416667.			−0.569976
$856$		−	118952.i		−	0.162340i
$857$		−	250279.i		−	0.340771i	−0.985377	−	0.170386i	$-0.945499\pi$
$857$		−	250279.i		−	0.340771i	0.985377	−	0.170386i	$-0.0545014\pi$
$858$		−	211466.i		−	0.287254i
$859$			223143.i			0.302411i	0.988502	+	0.151206i	$0.0483155\pi$
$859$			223143.i			0.302411i	−0.988502	+	0.151206i	$0.951684\pi$
$860$			135626.i			0.183377i
$861$	289106.			0.389988
$862$	87043.1			0.117144
$863$		−	719881.i		−	0.966583i	−0.875459	−	0.483292i	$-0.839441\pi$
$863$		−	719881.i		−	0.966583i	0.875459	−	0.483292i	$-0.160559\pi$
$864$		−	105102.i		−	0.140793i
$865$		−	603653.i		−	0.806780i
$866$		−	288320.i		−	0.384450i
$867$	−399385.			−0.531316
$868$		−	999053.i		−	1.32602i
$869$			1.57788e6i			2.08946i
$870$			104639.i			0.138246i
$871$	46311.1			0.0610448
$872$	80454.7			0.105808
$873$			406532.i			0.533417i
$874$	−514646.			−0.673730
$875$	−727785.			−0.950577
$876$			703453.i			0.916699i
$877$	−191233.			−0.248636			−0.124318	−	0.992242i	$-0.539674\pi$
$877$	−191233.			−0.248636			−0.124318	+	0.992242i	$0.539674\pi$
$878$		−	263595.i		−	0.341939i
$879$	−356564.			−0.461487
$880$		−	1.00335e6i		−	1.29564i
$881$			253450.i			0.326543i	0.986581	+	0.163272i	$0.0522047\pi$
$881$			253450.i			0.326543i	−0.986581	+	0.163272i	$0.947795\pi$
$882$		−	370.727i		−	0.000476560i
$883$	499438.			0.640560			0.320280	−	0.947323i	$-0.396223\pi$
$883$	499438.			0.640560			0.320280	+	0.947323i	$0.396223\pi$
$884$		−	240086.i		−	0.307229i
$885$	−410129.	−	235265.i	−0.523641	−	0.300380i
$886$	−327014.			−0.416580
$887$		−	941595.i		−	1.19679i	−0.801202	−	0.598393i	$-0.795806\pi$
$887$		−	941595.i		−	1.19679i	0.801202	−	0.598393i	$-0.204194\pi$
$888$	363761.			0.461307
$889$	−1.13059e6			−1.43055
$890$	−298394.			−0.376712
$891$		−	138813.i		−	0.174853i
$892$	366755.			0.460942
$893$			1.33249e6i			1.67095i
$894$	−115875.			−0.144982
$895$		−	1.28822e6i		−	1.60821i
$896$		−	716921.i		−	0.893008i
$897$	−833215.			−1.03555
$898$			285768.i			0.354373i
$899$			979451.i			1.21189i
$900$	23357.3			0.0288362
$901$	−386259.			−0.475806
$902$	−232444.			−0.285697
$903$			89265.6i			0.109473i
$904$	48824.2			0.0597445
$905$	1.38502e6			1.69106
$906$	170361.			0.207546
$907$	965170.			1.17325			0.586623	−	0.809860i	$-0.300457\pi$
$907$	965170.			1.17325			0.586623	+	0.809860i	$0.300457\pi$
$908$			340536.i			0.413039i
$909$			193250.i			0.233880i
$910$	274479.			0.331457
$911$	720113.			0.867689			0.433845	−	0.900988i	$-0.357157\pi$
$911$	720113.			0.867689			0.433845	+	0.900988i	$0.357157\pi$
$912$	−618364.			−0.743455
$913$	1.42406e6			1.70839
$914$	378774.			0.453406
$915$		−	203859.i		−	0.243493i
$916$			396774.i			0.472881i
$917$			790458.i			0.940027i
$918$			12341.0i			0.0146441i
$919$			1.34142e6i			1.58831i	0.607716	+	0.794155i	$0.292086\pi$
$919$			1.34142e6i			1.58831i	−0.607716	+	0.794155i	$0.707914\pi$
$920$	702722.			0.830249
$921$	822820.			0.970031
$922$		−	104104.i		−	0.122463i
$923$			528257.i			0.620072i
$924$		−	721284.i		−	0.844816i
$925$			122784.i			0.143502i
$926$	−266622.			−0.310938
$927$		−	273810.i		−	0.318632i
$928$			535403.i			0.621706i
$929$		−	1.26115e6i		−	1.46128i	−0.682761	−	0.730642i	$-0.739221\pi$
$929$		−	1.26115e6i		−	1.46128i	0.682761	−	0.730642i	$-0.260779\pi$
$930$	−200650.			−0.231993
$931$	−7520.09			−0.00867608
$932$		−	437721.i		−	0.503925i
$933$	302567.			0.347582
$934$	251547.			0.288353
$935$			406186.i			0.464624i
$936$	177955.			0.203122
$937$			251396.i			0.286338i	0.989698	+	0.143169i	$0.0457293\pi$
$937$			251396.i			0.286338i	−0.989698	+	0.143169i	$0.954271\pi$
$938$	−12369.3			−0.0140585
$939$			199455.i			0.226211i
$940$		−	875449.i		−	0.990775i
$941$		−	766362.i		−	0.865475i	−0.901520	−	0.432737i	$-0.857548\pi$
$941$		−	766362.i		−	0.865475i	0.901520	−	0.432737i	$-0.142452\pi$
$942$	−212657.			−0.239650
$943$			915870.i			1.02994i
$944$	−608661.	−	349151.i	−0.683017	−	0.391804i
$945$	180176.			0.201759
$946$		−	71770.4i		−	0.0801979i
$947$	−2912.67			−0.00324782			−0.00162391	−	0.999999i	$-0.500517\pi$
$947$	−2912.67			−0.00324782			−0.00162391	+	0.999999i	$0.500517\pi$
$948$	−638899.			−0.710911
$949$	−1.80903e6			−2.00870
$950$			37101.1i			0.0411092i
$951$	134304.			0.148500
$952$			133271.i			0.147049i
$953$	−1.10603e6			−1.21782			−0.608910	−	0.793240i	$-0.708393\pi$
$953$	−1.10603e6			−1.21782			−0.608910	+	0.793240i	$0.708393\pi$
$954$		−	137757.i		−	0.151362i
$955$		−	826033.i		−	0.905714i
$956$	559246.			0.611909
$957$			707132.i			0.772106i
$958$		−	274088.i		−	0.298648i
$959$	−204915.			−0.222811
$960$	328396.			0.356332
$961$	−954632.			−1.03369
$962$			450109.i			0.486371i
$963$	96618.9			0.104186
$964$	729936.			0.785472
$965$	−174165.			−0.187028
$966$	222545.			0.238486
$967$			165920.i			0.177438i	0.996057	+	0.0887188i	$0.0282772\pi$
$967$			165920.i			0.177438i	−0.996057	+	0.0887188i	$0.971723\pi$
$968$			718567.i			0.766861i
$969$	250333.			0.266606
$970$	424252.			0.450901
$971$	236926.			0.251290			0.125645	−	0.992075i	$-0.459900\pi$
$971$	236926.			0.251290			0.125645	+	0.992075i	$0.459900\pi$
$972$	56206.6			0.0594915
$973$	1.09644e6			1.15814
$974$		−	238558.i		−	0.251464i
$975$			60066.8i			0.0631867i
$976$		−	302541.i		−	0.317603i
$977$			904625.i			0.947719i	0.880600	+	0.473860i	$0.157140\pi$
$977$			904625.i			0.947719i	−0.880600	+	0.473860i	$0.842860\pi$
$978$			142079.i			0.148543i
$979$	−2.01650e6			−2.10394
$980$	4940.70			0.00514442
$981$			65349.3i			0.0679052i
$982$		−	21712.8i		−	0.0225161i
$983$			902770.i			0.934265i	0.884187	+	0.467133i	$0.154713\pi$
$983$			902770.i			0.934265i	−0.884187	+	0.467133i	$0.845287\pi$
$984$		−	195608.i		−	0.202021i
$985$	−84950.7			−0.0875578
$986$		−	62866.8i		−	0.0646647i
$987$		−	576200.i		−	0.591479i
$988$		−	1.73687e6i		−	1.77932i
$989$	−282788.			−0.289113
$990$	−144863.			−0.147804
$991$		−	480106.i		−	0.488866i	−0.969666	−	0.244433i	$-0.921398\pi$
$991$		−	480106.i		−	0.488866i	0.969666	−	0.244433i	$-0.0786018\pi$
$992$	−1.02667e6			−1.04329
$993$	663044.			0.672425
$994$		−	141093.i		−	0.142802i
$995$	−1.03043e6			−1.04081
$996$			576617.i			0.581258i
$997$	−144762.			−0.145635			−0.0728173	−	0.997345i	$-0.523199\pi$
$997$	−144762.			−0.145635			−0.0728173	+	0.997345i	$0.523199\pi$
$998$			153476.i			0.154092i
$999$			295465.i			0.296056i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.5.c.a.58.18	✓	40
3.2	odd	2		531.5.c.d.235.23		40
59.58	odd	2	inner	177.5.c.a.58.23	yes	40
177.176	even	2		531.5.c.d.235.18		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.5.c.a.58.18	✓	40	1.1	even	1	trivial
177.5.c.a.58.23	yes	40	59.58	odd	2	inner
531.5.c.d.235.18		40	177.176	even	2
531.5.c.d.235.23		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)

\(f(q)\)	\(=\)	\(q-1.07792i q^{2} +5.19615 q^{3} +14.8381 q^{4} +26.1400 q^{5} -5.60104i q^{6} +49.1298 q^{7} -33.2410i q^{8} +27.0000 q^{9} +O(q^{10})\)	\(q-1.07792i q^{2} +5.19615 q^{3} +14.8381 q^{4} +26.1400 q^{5} -5.60104i q^{6} +49.1298 q^{7} -33.2410i q^{8} +27.0000 q^{9} -28.1769i q^{10} -190.415i q^{11} +77.1010 q^{12} +198.277i q^{13} -52.9580i q^{14} +135.828 q^{15} +201.578 q^{16} -81.6050 q^{17} -29.1038i q^{18} -590.362 q^{19} +387.868 q^{20} +255.286 q^{21} -205.252 q^{22} +808.730i q^{23} -172.725i q^{24} +58.3017 q^{25} +213.726 q^{26} +140.296 q^{27} +728.992 q^{28} -714.690 q^{29} -146.411i q^{30} -1370.46i q^{31} -749.141i q^{32} -989.426i q^{33} +87.9637i q^{34} +1284.26 q^{35} +400.628 q^{36} +2106.01i q^{37} +636.363i q^{38} +1030.28i q^{39} -868.921i q^{40} +1132.48 q^{41} -275.178i q^{42} +349.669i q^{43} -2825.40i q^{44} +705.781 q^{45} +871.746 q^{46} -2257.08i q^{47} +1047.43 q^{48} +12.7381 q^{49} -62.8446i q^{50} -424.032 q^{51} +2942.05i q^{52} +4733.28 q^{53} -151.228i q^{54} -4977.46i q^{55} -1633.12i q^{56} -3067.61 q^{57} +770.378i q^{58} +(-3019.48 - 1732.08i) q^{59} +2015.42 q^{60} -1500.86i q^{61} -1477.24 q^{62} +1326.50 q^{63} +2417.74 q^{64} +5182.96i q^{65} -1066.52 q^{66} -233.568i q^{67} -1210.86 q^{68} +4202.28i q^{69} -1384.32i q^{70} +2664.24 q^{71} -897.507i q^{72} +9123.79i q^{73} +2270.11 q^{74} +302.945 q^{75} -8759.85 q^{76} -9355.06i q^{77} +1110.55 q^{78} -8286.52 q^{79} +5269.26 q^{80} +729.000 q^{81} -1220.72i q^{82} +7478.73i q^{83} +3787.96 q^{84} -2133.16 q^{85} +376.915 q^{86} -3713.64 q^{87} -6329.59 q^{88} -10590.0i q^{89} -760.776i q^{90} +9741.29i q^{91} +12000.0i q^{92} -7121.10i q^{93} -2432.95 q^{94} -15432.1 q^{95} -3892.65i q^{96} +15056.8i q^{97} -13.7306i q^{98} -5141.21i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 320 q^{4} + 80 q^{7} + 1080 q^{9}+O(q^{10}) \) 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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