LMFDB - Embedded newform 177.3.c.a.58.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [177,3,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, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("177.58");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 3 $
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:	$4.82290067918$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 60 x^{18} + 1522 x^{16} + 21286 x^{14} + 179593 x^{12} + 941588 x^{10} + 3057025 x^{8} + \cdots + 570861 $ x^20 + 60x^18 + 1522x^16 + 21286x^14 + 179593x^12 + 941588x^10 + 3057025x^8 + 5962230x^6 + 6517782x^4 + 3428244*x^2 + 570861
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{7}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			58.2
Root			$-3.38526i$ of defining polynomial
Character	$\chi$	$=$	177.58
Dual form			177.3.c.a.58.19

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.38526i q^{2} +1.73205 q^{3} -7.45997 q^{4} +9.27487 q^{5} -5.86344i q^{6} -9.96030 q^{7} +11.7129i q^{8} +3.00000 q^{9} +O(q^{10})$	\(q-3.38526i q^{2} +1.73205 q^{3} -7.45997 q^{4} +9.27487 q^{5} -5.86344i q^{6} -9.96030 q^{7} +11.7129i q^{8} +3.00000 q^{9} -31.3978i q^{10} -20.5213i q^{11} -12.9211 q^{12} -4.19899i q^{13} +33.7182i q^{14} +16.0645 q^{15} +9.81130 q^{16} +0.362285 q^{17} -10.1558i q^{18} +11.7217 q^{19} -69.1903 q^{20} -17.2518 q^{21} -69.4698 q^{22} +21.5797i q^{23} +20.2873i q^{24} +61.0232 q^{25} -14.2147 q^{26} +5.19615 q^{27} +74.3036 q^{28} +18.1338 q^{29} -54.3826i q^{30} +42.6196i q^{31} +13.6378i q^{32} -35.5439i q^{33} -1.22643i q^{34} -92.3805 q^{35} -22.3799 q^{36} +46.4991i q^{37} -39.6809i q^{38} -7.27287i q^{39} +108.636i q^{40} +16.1117 q^{41} +58.4016i q^{42} -30.4120i q^{43} +153.088i q^{44} +27.8246 q^{45} +73.0530 q^{46} +5.06360i q^{47} +16.9937 q^{48} +50.2077 q^{49} -206.579i q^{50} +0.627496 q^{51} +31.3244i q^{52} +36.6918 q^{53} -17.5903i q^{54} -190.332i q^{55} -116.664i q^{56} +20.3025 q^{57} -61.3875i q^{58} +(-52.1348 + 27.6217i) q^{59} -119.841 q^{60} +32.0930i q^{61} +144.279 q^{62} -29.8809 q^{63} +85.4127 q^{64} -38.9451i q^{65} -120.325 q^{66} -81.4723i q^{67} -2.70263 q^{68} +37.3772i q^{69} +312.732i q^{70} +9.92600 q^{71} +35.1387i q^{72} -43.0824i q^{73} +157.412 q^{74} +105.695 q^{75} -87.4433 q^{76} +204.398i q^{77} -24.6205 q^{78} -104.049 q^{79} +90.9985 q^{80} +9.00000 q^{81} -54.5422i q^{82} +118.124i q^{83} +128.698 q^{84} +3.36014 q^{85} -102.953 q^{86} +31.4086 q^{87} +240.364 q^{88} -48.1069i q^{89} -94.1935i q^{90} +41.8233i q^{91} -160.984i q^{92} +73.8194i q^{93} +17.1416 q^{94} +108.717 q^{95} +23.6214i q^{96} +125.543i q^{97} -169.966i q^{98} -61.5638i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 40 q^{4} - 8 q^{7} + 60 q^{9}+O(q^{10}) $ 20 * q - 40 * q^4 - 8 * q^7 + 60 * q^9	$ 20 q - 40 q^{4} - 8 q^{7} + 60 q^{9} - 24 q^{12} + 24 q^{15} - 8 q^{16} + 16 q^{17} - 60 q^{19} - 164 q^{20} + 40 q^{22} + 100 q^{25} - 156 q^{26} + 200 q^{28} - 60 q^{29} - 32 q^{35} - 120 q^{36} + 28 q^{41} + 180 q^{46} - 96 q^{48} + 284 q^{49} - 24 q^{51} - 8 q^{53} + 24 q^{57} - 152 q^{59} - 72 q^{60} - 8 q^{62} - 24 q^{63} + 204 q^{64} - 120 q^{66} + 384 q^{68} + 92 q^{71} + 104 q^{74} + 240 q^{75} + 120 q^{76} - 468 q^{78} - 420 q^{79} + 376 q^{80} + 180 q^{81} + 168 q^{84} - 348 q^{85} + 232 q^{86} + 144 q^{87} - 212 q^{88} + 152 q^{94} + 788 q^{95}+O(q^{100}) $ 20 * q - 40 * q^4 - 8 * q^7 + 60 * q^9 - 24 * q^12 + 24 * q^15 - 8 * q^16 + 16 * q^17 - 60 * q^19 - 164 * q^20 + 40 * q^22 + 100 * q^25 - 156 * q^26 + 200 * q^28 - 60 * q^29 - 32 * q^35 - 120 * q^36 + 28 * q^41 + 180 * q^46 - 96 * q^48 + 284 * q^49 - 24 * q^51 - 8 * q^53 + 24 * q^57 - 152 * q^59 - 72 * q^60 - 8 * q^62 - 24 * q^63 + 204 * q^64 - 120 * q^66 + 384 * q^68 + 92 * q^71 + 104 * q^74 + 240 * q^75 + 120 * q^76 - 468 * q^78 - 420 * q^79 + 376 * q^80 + 180 * q^81 + 168 * q^84 - 348 * q^85 + 232 * q^86 + 144 * q^87 - 212 * q^88 + 152 * q^94 + 788 * 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 $

$2$		−	3.38526i		−	1.69263i	−0.532684	−	0.846315i	$-0.678816\pi$
$2$		−	3.38526i		−	1.69263i	0.532684	−	0.846315i	$-0.321184\pi$
$3$	1.73205			0.577350
$3$	1.73205			0.577350
$4$	−7.45997			−1.86499
$5$	9.27487			1.85497			0.927487	−	0.373855i	$-0.121964\pi$
$5$	9.27487			1.85497			0.927487	+	0.373855i	$0.121964\pi$
$6$		−	5.86344i		−	0.977240i
$7$	−9.96030			−1.42290			−0.711450	−	0.702736i	$-0.751961\pi$
$7$	−9.96030			−1.42290			−0.711450	+	0.702736i	$0.751961\pi$
$8$			11.7129i			1.46411i
$9$	3.00000			0.333333
$10$		−	31.3978i		−	3.13978i
$11$		−	20.5213i		−	1.86557i	−0.360434	−	0.932785i	$-0.617371\pi$
$11$		−	20.5213i		−	1.86557i	0.360434	−	0.932785i	$-0.382629\pi$
$12$	−12.9211			−1.07675
$13$		−	4.19899i		−	0.323000i	−0.986873	−	0.161500i	$-0.948367\pi$
$13$		−	4.19899i		−	0.323000i	0.986873	−	0.161500i	$-0.0516331\pi$
$14$			33.7182i			2.40844i
$15$	16.0645			1.07097
$16$	9.81130			0.613206
$17$	0.362285			0.0213109			0.0106554	−	0.999943i	$-0.496608\pi$
$17$	0.362285			0.0213109			0.0106554	+	0.999943i	$0.496608\pi$
$18$		−	10.1558i		−	0.564210i
$19$	11.7217			0.616930			0.308465	−	0.951236i	$-0.400185\pi$
$19$	11.7217			0.616930			0.308465	+	0.951236i	$0.400185\pi$
$20$	−69.1903			−3.45951
$21$	−17.2518			−0.821512
$22$	−69.4698			−3.15772
$23$			21.5797i			0.938249i	0.883132	+	0.469125i	$0.155431\pi$
$23$			21.5797i			0.938249i	−0.883132	+	0.469125i	$0.844569\pi$
$24$			20.2873i			0.845306i
$25$	61.0232			2.44093
$26$	−14.2147			−0.546718
$27$	5.19615			0.192450
$28$	74.3036			2.65370
$29$	18.1338			0.625303			0.312651	−	0.949868i	$-0.398783\pi$
$29$	18.1338			0.625303			0.312651	+	0.949868i	$0.398783\pi$
$30$		−	54.3826i		−	1.81275i
$31$			42.6196i			1.37483i	0.726266	+	0.687414i	$0.241254\pi$
$31$			42.6196i			1.37483i	−0.726266	+	0.687414i	$0.758746\pi$
$32$			13.6378i			0.426182i
$33$		−	35.5439i		−	1.07709i
$34$		−	1.22643i		−	0.0360714i
$35$	−92.3805			−2.63944
$36$	−22.3799			−0.621664
$37$			46.4991i			1.25673i	0.777917	+	0.628367i	$0.216276\pi$
$37$			46.4991i			1.25673i	−0.777917	+	0.628367i	$0.783724\pi$
$38$		−	39.6809i		−	1.04423i
$39$		−	7.27287i		−	0.186484i
$40$			108.636i			2.71589i
$41$	16.1117			0.392968			0.196484	−	0.980507i	$-0.437048\pi$
$41$	16.1117			0.392968			0.196484	+	0.980507i	$0.437048\pi$
$42$			58.4016i			1.39052i
$43$		−	30.4120i		−	0.707257i	−0.935386	−	0.353628i	$-0.884948\pi$
$43$		−	30.4120i		−	0.707257i	0.935386	−	0.353628i	$-0.115052\pi$
$44$			153.088i			3.47927i
$45$	27.8246			0.618325
$46$	73.0530			1.58811
$47$			5.06360i			0.107736i	0.998548	+	0.0538681i	$0.0171551\pi$
$47$			5.06360i			0.107736i	−0.998548	+	0.0538681i	$0.982845\pi$
$48$	16.9937			0.354035
$49$	50.2077			1.02465
$50$		−	206.579i		−	4.13159i
$51$	0.627496			0.0123038
$52$			31.3244i			0.602392i
$53$	36.6918			0.692299			0.346149	−	0.938179i	$-0.387489\pi$
$53$	36.6918			0.692299			0.346149	+	0.938179i	$0.387489\pi$
$54$		−	17.5903i		−	0.325747i
$55$		−	190.332i		−	3.46058i
$56$		−	116.664i		−	2.08329i
$57$	20.3025			0.356185
$58$		−	61.3875i		−	1.05841i
$59$	−52.1348	+	27.6217i	−0.883641	+	0.468164i
$59$	−52.1348	+	27.6217i	−0.883641	+	0.468164i
$60$	−119.841			−1.99735
$61$			32.0930i			0.526115i	0.964780	+	0.263058i	$0.0847309\pi$
$61$			32.0930i			0.526115i	−0.964780	+	0.263058i	$0.915269\pi$
$62$	144.279			2.32707
$63$	−29.8809			−0.474300
$64$	85.4127			1.33457
$65$		−	38.9451i		−	0.599156i
$66$	−120.325			−1.82311
$67$		−	81.4723i		−	1.21600i	−0.793936	−	0.608002i	$-0.791971\pi$
$67$		−	81.4723i		−	1.21600i	0.793936	−	0.608002i	$-0.208029\pi$
$68$	−2.70263			−0.0397446
$69$			37.3772i			0.541699i
$70$			312.732i			4.46760i
$71$	9.92600			0.139803			0.0699014	−	0.997554i	$-0.477732\pi$
$71$	9.92600			0.139803			0.0699014	+	0.997554i	$0.477732\pi$
$72$			35.1387i			0.488038i
$73$		−	43.0824i		−	0.590170i	−0.955471	−	0.295085i	$-0.904652\pi$
$73$		−	43.0824i		−	0.590170i	0.955471	−	0.295085i	$-0.0953480\pi$
$74$	157.412			2.12718
$75$	105.695			1.40927
$76$	−87.4433			−1.15057
$77$			204.398i			2.65452i
$78$	−24.6205			−0.315648
$79$	−104.049			−1.31708			−0.658541	−	0.752545i	$-0.728826\pi$
$79$	−104.049			−1.31708			−0.658541	+	0.752545i	$0.728826\pi$
$80$	90.9985			1.13748
$81$	9.00000			0.111111
$82$		−	54.5422i		−	0.665149i
$83$			118.124i			1.42318i	0.702597	+	0.711588i	$0.252023\pi$
$83$			118.124i			1.42318i	−0.702597	+	0.711588i	$0.747977\pi$
$84$	128.698			1.53211
$85$	3.36014			0.0395311
$86$	−102.953			−1.19712
$87$	31.4086			0.361019
$88$	240.364			2.73140
$89$		−	48.1069i		−	0.540527i	−0.962786	−	0.270264i	$-0.912889\pi$
$89$		−	48.1069i		−	0.540527i	0.962786	−	0.270264i	$-0.0871108\pi$
$90$		−	94.1935i		−	1.04659i
$91$			41.8233i			0.459596i
$92$		−	160.984i		−	1.74983i
$93$			73.8194i			0.793757i
$94$	17.1416			0.182358
$95$	108.717			1.14439
$96$			23.6214i			0.246056i
$97$			125.543i			1.29426i	0.762379	+	0.647130i	$0.224031\pi$
$97$			125.543i			1.29426i	−0.762379	+	0.647130i	$0.775969\pi$
$98$		−	169.966i		−	1.73435i
$99$		−	61.5638i		−	0.621856i
$100$	−455.232			−4.55232
$101$			34.1908i			0.338523i	0.985571	+	0.169261i	$0.0541382\pi$
$101$			34.1908i			0.338523i	−0.985571	+	0.169261i	$0.945862\pi$
$102$		−	2.12424i		−	0.0208258i
$103$			119.958i			1.16464i	0.812959	+	0.582321i	$0.197855\pi$
$103$			119.958i			1.16464i	−0.812959	+	0.582321i	$0.802145\pi$
$104$	49.1824			0.472908
$105$	−160.008			−1.52388
$106$		−	124.211i		−	1.17180i
$107$	39.8153			0.372106			0.186053	−	0.982540i	$-0.440430\pi$
$107$	39.8153			0.372106			0.186053	+	0.982540i	$0.440430\pi$
$108$	−38.7632			−0.358918
$109$		−	168.447i		−	1.54538i	−0.634781	−	0.772692i	$-0.718910\pi$
$109$		−	168.447i		−	1.54538i	0.634781	−	0.772692i	$-0.281090\pi$
$110$	−644.323			−5.85748
$111$			80.5389i			0.725575i
$112$	−97.7235			−0.872532
$113$		−	23.4960i		−	0.207929i	−0.994581	−	0.103965i	$-0.966847\pi$
$113$		−	23.4960i		−	0.207929i	0.994581	−	0.103965i	$-0.0331529\pi$
$114$		−	68.7293i		−	0.602888i
$115$			200.149i			1.74043i
$116$	−135.278			−1.16619
$117$		−	12.5970i		−	0.107667i
$118$	93.5065	+	176.490i	0.792428	+	1.49568i
$119$	−3.60847			−0.0303233
$120$			188.162i			1.56802i
$121$	−300.122			−2.48035
$122$	108.643			0.890518
$123$	27.9062			0.226880
$124$		−	317.941i		−	2.56404i
$125$	334.111			2.67289
$126$			101.155i			0.802814i
$127$	−40.6699			−0.320236			−0.160118	−	0.987098i	$-0.551187\pi$
$127$	−40.6699			−0.320236			−0.160118	+	0.987098i	$0.551187\pi$
$128$		−	234.593i		−	1.83276i
$129$		−	52.6752i		−	0.408335i
$130$	−131.839			−1.01415
$131$		−	121.173i		−	0.924988i	−0.886622	−	0.462494i	$-0.846955\pi$
$131$		−	121.173i		−	0.924988i	0.886622	−	0.462494i	$-0.153045\pi$
$132$			265.156i			2.00876i
$133$	−116.751			−0.877830
$134$	−275.805			−2.05824
$135$	48.1936			0.356990
$136$			4.24341i			0.0312015i
$137$	−198.510			−1.44898			−0.724489	−	0.689286i	$-0.757924\pi$
$137$	−198.510			−1.44898			−0.724489	+	0.689286i	$0.757924\pi$
$138$	126.531			0.916895
$139$	228.034			1.64053			0.820267	−	0.571981i	$-0.193825\pi$
$139$	228.034			1.64053			0.820267	+	0.571981i	$0.193825\pi$
$140$	689.156			4.92254
$141$			8.77042i			0.0622016i
$142$		−	33.6021i		−	0.236634i
$143$	−86.1687			−0.602578
$144$	29.4339			0.204402
$145$	168.188			1.15992
$146$	−145.845			−0.998940
$147$	86.9622			0.591580
$148$		−	346.882i		−	2.34380i
$149$			95.3536i			0.639957i	0.947425	+	0.319979i	$0.103676\pi$
$149$			95.3536i			0.639957i	−0.947425	+	0.319979i	$0.896324\pi$
$150$		−	357.806i		−	2.38537i
$151$		−	262.692i		−	1.73968i	−0.493333	−	0.869841i	$-0.664221\pi$
$151$		−	262.692i		−	1.73968i	0.493333	−	0.869841i	$-0.335779\pi$
$152$			137.295i			0.903255i
$153$	1.08685			0.00710362
$154$	691.940			4.49312
$155$			395.292i			2.55027i
$156$			54.2554i			0.347791i
$157$			23.3960i			0.149019i	0.997220	+	0.0745096i	$0.0237392\pi$
$157$			23.3960i			0.149019i	−0.997220	+	0.0745096i	$0.976261\pi$
$158$			352.234i			2.22933i
$159$	63.5521			0.399699
$160$			126.489i			0.790556i
$161$		−	214.941i		−	1.33504i
$162$		−	30.4673i		−	0.188070i
$163$	−29.9339			−0.183644			−0.0918218	−	0.995775i	$-0.529269\pi$
$163$	−29.9339			−0.183644			−0.0918218	+	0.995775i	$0.529269\pi$
$164$	−120.193			−0.732882
$165$		−	329.665i		−	1.99797i
$166$	399.879			2.40891
$167$	−140.361			−0.840485			−0.420242	−	0.907412i	$-0.638055\pi$
$167$	−140.361			−0.840485			−0.420242	+	0.907412i	$0.638055\pi$
$168$		−	202.068i		−	1.20279i
$169$	151.368			0.895671
$170$		−	11.3750i		−	0.0669115i
$171$	35.1650			0.205643
$172$			226.873i			1.31903i
$173$			122.872i			0.710245i	0.934820	+	0.355123i	$0.115561\pi$
$173$			122.872i			0.710245i	−0.934820	+	0.355123i	$0.884439\pi$
$174$		−	106.326i		−	0.611071i
$175$	−607.810			−3.47320
$176$		−	201.340i		−	1.14398i
$177$	−90.3002	+	47.8422i	−0.510171	+	0.270295i
$178$	−162.854			−0.914912
$179$			13.1743i			0.0735996i	0.999323	+	0.0367998i	$0.0117164\pi$
$179$			13.1743i			0.0735996i	−0.999323	+	0.0367998i	$0.988284\pi$
$180$	−207.571			−1.15317
$181$	−194.551			−1.07487			−0.537433	−	0.843307i	$-0.680606\pi$
$181$	−194.551			−1.07487			−0.537433	+	0.843307i	$0.680606\pi$
$182$	141.583			0.777926
$183$			55.5867i			0.303753i
$184$	−252.761			−1.37370
$185$			431.274i			2.33121i
$186$	249.898			1.34354
$187$		−	7.43454i		−	0.0397569i
$188$		−	37.7743i		−	0.200927i
$189$	−51.7553			−0.273837
$190$		−	368.035i		−	1.93703i
$191$		−	299.916i		−	1.57024i	−0.619342	−	0.785121i	$-0.712601\pi$
$191$		−	299.916i		−	1.57024i	0.619342	−	0.785121i	$-0.287399\pi$
$192$	147.939			0.770517
$193$	−27.4449			−0.142202			−0.0711008	−	0.997469i	$-0.522651\pi$
$193$	−27.4449			−0.142202			−0.0711008	+	0.997469i	$0.522651\pi$
$194$	424.996			2.19070
$195$		−	67.4549i		−	0.345923i
$196$	−374.548			−1.91096
$197$	82.5430			0.419000			0.209500	−	0.977809i	$-0.432816\pi$
$197$	82.5430			0.419000			0.209500	+	0.977809i	$0.432816\pi$
$198$	−208.409			−1.05257
$199$	−41.4291			−0.208186			−0.104093	−	0.994568i	$-0.533194\pi$
$199$	−41.4291			−0.208186			−0.104093	+	0.994568i	$0.533194\pi$
$200$			714.759i			3.57379i
$201$		−	141.114i		−	0.702060i
$202$	115.745			0.572994
$203$	−180.618			−0.889744
$204$	−4.68110			−0.0229466
$205$	149.434			0.728945
$206$	406.089			1.97131
$207$			64.7392i			0.312750i
$208$		−	41.1976i		−	0.198065i
$209$		−	240.543i		−	1.15093i
$210$			541.668i			2.57937i
$211$		−	241.695i		−	1.14547i	−0.819740	−	0.572736i	$-0.805882\pi$
$211$		−	241.695i		−	1.14547i	0.819740	−	0.572736i	$-0.194118\pi$
$212$	−273.720			−1.29113
$213$	17.1923			0.0807152
$214$		−	134.785i		−	0.629837i
$215$		−	282.068i		−	1.31194i
$216$			60.8620i			0.281769i
$217$		−	424.505i		−	1.95624i
$218$	−570.236			−2.61576
$219$		−	74.6210i		−	0.340735i
$220$			1419.87i			6.45396i
$221$		−	1.52123i		−	0.00688340i
$222$	272.645			1.22813
$223$	200.733			0.900148			0.450074	−	0.892991i	$-0.351398\pi$
$223$	200.733			0.900148			0.450074	+	0.892991i	$0.351398\pi$
$224$		−	135.837i		−	0.606414i
$225$	183.070			0.813643
$226$	−79.5400			−0.351947
$227$			221.780i			0.977002i	0.872563	+	0.488501i	$0.162456\pi$
$227$			221.780i			0.977002i	−0.872563	+	0.488501i	$0.837544\pi$
$228$	−151.456			−0.664282
$229$			169.854i			0.741720i	0.928689	+	0.370860i	$0.120937\pi$
$229$			169.854i			0.741720i	−0.928689	+	0.370860i	$0.879063\pi$
$230$	677.557			2.94590
$231$			354.028i			1.53259i
$232$			212.399i			0.915514i
$233$			56.1420i			0.240953i	0.992716	+	0.120476i	$0.0384422\pi$
$233$			56.1420i			0.240953i	−0.992716	+	0.120476i	$0.961558\pi$
$234$	−42.6440			−0.182239
$235$			46.9643i			0.199848i
$236$	388.925	−	206.057i	1.64799	−	0.873123i
$237$	−180.219			−0.760417
$238$			12.2156i			0.0513260i
$239$	−282.703			−1.18286			−0.591429	−	0.806357i	$-0.701436\pi$
$239$	−282.703			−1.18286			−0.591429	+	0.806357i	$0.701436\pi$
$240$	157.614			0.656725
$241$	90.4885			0.375471			0.187736	−	0.982220i	$-0.439885\pi$
$241$	90.4885			0.375471			0.187736	+	0.982220i	$0.439885\pi$
$242$			1015.99i			4.19831i
$243$	15.5885			0.0641500
$244$		−	239.413i		−	0.981201i
$245$	465.669			1.90069
$246$		−	94.4698i		−	0.384024i
$247$		−	49.2192i		−	0.199268i
$248$	−499.200			−2.01290
$249$			204.596i			0.821671i
$250$		−	1131.05i		−	4.52420i
$251$	90.8848			0.362091			0.181045	−	0.983475i	$-0.442052\pi$
$251$	90.8848			0.362091			0.181045	+	0.983475i	$0.442052\pi$
$252$	222.911			0.884567
$253$	442.843			1.75037
$254$			137.678i			0.542040i
$255$	5.81994			0.0228233
$256$	−452.507			−1.76760
$257$	−75.3044			−0.293013			−0.146507	−	0.989210i	$-0.546803\pi$
$257$	−75.3044			−0.293013			−0.146507	+	0.989210i	$0.546803\pi$
$258$	−178.319			−0.691160
$259$		−	463.146i		−	1.78821i
$260$			290.530i			1.11742i
$261$	54.4013			0.208434
$262$	−410.203			−1.56566
$263$	366.374			1.39306			0.696528	−	0.717530i	$-0.254727\pi$
$263$	366.374			1.39306			0.696528	+	0.717530i	$0.254727\pi$
$264$	416.322			1.57698
$265$	340.312			1.28420
$266$			395.234i			1.48584i
$267$		−	83.3236i		−	0.312073i
$268$			607.781i			2.26784i
$269$		−	0.373353i		−	0.00138793i	−1.00000		0.000693966i	$-0.999779\pi$
$269$		−	0.373353i		−	0.00138793i	1.00000		0.000693966i	$-0.000220896\pi$
$270$		−	163.148i		−	0.604252i
$271$	−375.056			−1.38397			−0.691986	−	0.721911i	$-0.743264\pi$
$271$	−375.056			−1.38397			−0.691986	+	0.721911i	$0.743264\pi$
$272$	3.55449			0.0130680
$273$			72.4400i			0.265348i
$274$			672.008i			2.45258i
$275$		−	1252.27i		−	4.55372i
$276$		−	278.833i		−	1.01026i
$277$	87.9624			0.317554			0.158777	−	0.987314i	$-0.449245\pi$
$277$	87.9624			0.317554			0.158777	+	0.987314i	$0.449245\pi$
$278$		−	771.955i		−	2.77682i
$279$			127.859i			0.458276i
$280$		−	1082.04i		−	3.86444i
$281$	−358.335			−1.27521			−0.637607	−	0.770362i	$-0.720076\pi$
$281$	−358.335			−1.27521			−0.637607	+	0.770362i	$0.720076\pi$
$282$	29.6901			0.105284
$283$			372.886i			1.31762i	0.752311	+	0.658808i	$0.228939\pi$
$283$			372.886i			1.31762i	−0.752311	+	0.658808i	$0.771061\pi$
$284$	−74.0477			−0.260731
$285$	188.303			0.660713
$286$			291.703i			1.01994i
$287$	−160.477			−0.559154
$288$			40.9135i			0.142061i
$289$	−288.869			−0.999546
$290$		−	569.361i		−	1.96332i
$291$			217.447i			0.747242i
$292$			321.394i			1.10066i
$293$	−82.4199			−0.281297			−0.140648	−	0.990060i	$-0.544919\pi$
$293$	−82.4199			−0.281297			−0.140648	+	0.990060i	$0.544919\pi$
$294$		−	294.390i		−	1.00132i
$295$	−483.544	+	256.188i	−1.63913	+	0.868432i
$296$	−544.640			−1.84000
$297$		−	106.632i		−	0.359029i
$298$	322.797			1.08321
$299$	90.6132			0.303054
$300$	−788.484			−2.62828
$301$			302.913i			1.00636i
$302$	−889.280			−2.94464
$303$			59.2202i			0.195446i
$304$	115.005			0.378305
$305$			297.659i			0.975930i
$306$		−	3.67928i		−	0.0120238i
$307$	−469.709			−1.53000			−0.764999	−	0.644032i	$-0.777260\pi$
$307$	−469.709			−1.53000			−0.764999	+	0.644032i	$0.777260\pi$
$308$		−	1524.80i		−	4.95066i
$309$			207.774i			0.672406i
$310$	1338.16			4.31666
$311$	53.4334			0.171812			0.0859058	−	0.996303i	$-0.472622\pi$
$311$	53.4334			0.171812			0.0859058	+	0.996303i	$0.472622\pi$
$312$	85.1864			0.273033
$313$		−	8.38179i		−	0.0267789i	−0.999910	−	0.0133894i	$-0.995738\pi$
$313$		−	8.38179i		−	0.0267789i	0.999910	−	0.0133894i	$-0.00426212\pi$
$314$	79.2016			0.252234
$315$	−277.142			−0.879815
$316$	776.206			2.45635
$317$	−400.493			−1.26338			−0.631692	−	0.775219i	$-0.717639\pi$
$317$	−400.493			−1.26338			−0.631692	+	0.775219i	$0.717639\pi$
$318$		−	215.140i		−	0.676542i
$319$		−	372.128i		−	1.16655i
$320$	792.192			2.47560
$321$	68.9622			0.214836
$322$	−727.630			−2.25972
$323$	4.24658			0.0131473
$324$	−67.1398			−0.207221
$325$		−	256.236i		−	0.788419i
$326$			101.334i			0.310840i
$327$		−	291.759i		−	0.892228i
$328$			188.714i			0.575349i
$329$		−	50.4350i		−	0.153298i
$330$	−1116.00			−3.38182
$331$	−29.6370			−0.0895377			−0.0447688	−	0.998997i	$-0.514255\pi$
$331$	−29.6370			−0.0895377			−0.0447688	+	0.998997i	$0.514255\pi$
$332$		−	881.198i		−	2.65421i
$333$			139.497i			0.418911i
$334$			475.158i			1.42263i
$335$		−	755.645i		−	2.25566i
$336$	−169.262			−0.503756
$337$		−	112.535i		−	0.333932i	−0.985963	−	0.166966i	$-0.946603\pi$
$337$		−	112.535i		−	0.333932i	0.985963	−	0.166966i	$-0.0533971\pi$
$338$		−	512.421i		−	1.51604i
$339$		−	40.6963i		−	0.120048i
$340$	−25.0666			−0.0737253
$341$	874.609			2.56484
$342$		−	119.043i		−	0.348078i
$343$	−12.0286			−0.0350687
$344$	356.213			1.03550
$345$			346.669i			1.00484i
$346$	415.955			1.20218
$347$			388.058i			1.11832i	0.829059	+	0.559161i	$0.188877\pi$
$347$			388.058i			1.11832i	−0.829059	+	0.559161i	$0.811123\pi$
$348$	−234.308			−0.673297
$349$			542.368i			1.55406i	0.629462	+	0.777031i	$0.283275\pi$
$349$			542.368i			1.55406i	−0.629462	+	0.777031i	$0.716725\pi$
$350$			2057.59i			5.87884i
$351$		−	21.8186i		−	0.0621613i
$352$	279.865			0.795072
$353$		−	102.916i		−	0.291546i	−0.989318	−	0.145773i	$-0.953433\pi$
$353$		−	102.916i		−	0.291546i	0.989318	−	0.145773i	$-0.0465669\pi$
$354$	161.958	+	305.690i	0.457509	+	0.863530i
$355$	92.0624			0.259331
$356$			358.876i			1.00808i
$357$	−6.25005			−0.0175071
$358$	44.5985			0.124577
$359$	281.215			0.783328			0.391664	−	0.920108i	$-0.371899\pi$
$359$	281.215			0.783328			0.391664	+	0.920108i	$0.371899\pi$
$360$			325.907i			0.905297i
$361$	−223.603			−0.619398
$362$			658.604i			1.81935i
$363$	−519.827			−1.43203
$364$		−	312.000i		−	0.857144i
$365$		−	399.584i		−	1.09475i
$366$	188.175			0.514141
$367$			22.4570i			0.0611907i	0.999532	+	0.0305953i	$0.00974032\pi$
$367$			22.4570i			0.0611907i	−0.999532	+	0.0305953i	$0.990260\pi$
$368$			211.725i			0.575340i
$369$	48.3350			0.130989
$370$	1459.97			3.94587
$371$	−365.462			−0.985072
$372$		−	550.691i		−	1.48035i
$373$	70.8071			0.189831			0.0949157	−	0.995485i	$-0.469742\pi$
$373$	70.8071			0.189831			0.0949157	+	0.995485i	$0.469742\pi$
$374$	−25.1678			−0.0672937
$375$	578.697			1.54319
$376$	−59.3095			−0.157738
$377$		−	76.1436i		−	0.201973i
$378$			175.205i			0.463505i
$379$	−249.514			−0.658349			−0.329174	−	0.944269i	$-0.606770\pi$
$379$	−249.514			−0.658349			−0.329174	+	0.944269i	$0.606770\pi$
$380$	−811.025			−2.13428
$381$	−70.4424			−0.184888
$382$	−1015.29			−2.65784
$383$	−674.681			−1.76157			−0.880785	−	0.473516i	$-0.842985\pi$
$383$	−674.681			−1.76157			−0.880785	+	0.473516i	$0.842985\pi$
$384$		−	406.327i		−	1.05814i
$385$			1895.77i			4.92407i
$386$			92.9081i			0.240694i
$387$		−	91.2361i		−	0.235752i
$388$		−	936.549i		−	2.41379i
$389$	493.759			1.26930			0.634651	−	0.772799i	$-0.281144\pi$
$389$	493.759			1.26930			0.634651	+	0.772799i	$0.281144\pi$
$390$	−228.352			−0.585519
$391$			7.81801i			0.0199949i
$392$			588.077i			1.50020i
$393$		−	209.879i		−	0.534042i
$394$		−	279.429i		−	0.709212i
$395$	−965.045			−2.44315
$396$			459.264i			1.15976i
$397$		−	648.987i		−	1.63473i	−0.576122	−	0.817364i	$-0.695435\pi$
$397$		−	648.987i		−	1.63473i	0.576122	−	0.817364i	$-0.304565\pi$
$398$			140.248i			0.352382i
$399$	−202.219			−0.506815
$400$	598.717			1.49679
$401$			59.8224i			0.149183i	0.997214	+	0.0745915i	$0.0237653\pi$
$401$			59.8224i			0.149183i	−0.997214	+	0.0745915i	$0.976235\pi$
$402$	−477.708			−1.18833
$403$	178.960			0.444069
$404$		−	255.063i		−	0.631343i
$405$	83.4738			0.206108
$406$			611.438i			1.50601i
$407$	954.221			2.34452
$408$			7.34979i			0.0180142i
$409$		−	279.180i		−	0.682591i	−0.939956	−	0.341296i	$-0.889134\pi$
$409$		−	279.180i		−	0.682591i	0.939956	−	0.341296i	$-0.110866\pi$
$410$		−	505.872i		−	1.23383i
$411$	−343.830			−0.836568
$412$		−	894.884i		−	2.17205i
$413$	519.279	−	275.120i	1.25733	−	0.666151i
$414$	219.159			0.529369
$415$			1095.58i			2.63995i
$416$	57.2651			0.137657
$417$	394.967			0.947163
$418$	−814.302			−1.94809
$419$			442.679i			1.05651i	0.849085	+	0.528256i	$0.177154\pi$
$419$			442.679i			1.05651i	−0.849085	+	0.528256i	$0.822846\pi$
$420$	1193.65			2.84203
$421$			73.7045i			0.175070i	0.996161	+	0.0875350i	$0.0278990\pi$
$421$			73.7045i			0.175070i	−0.996161	+	0.0875350i	$0.972101\pi$
$422$	−818.198			−1.93886
$423$			15.1908i			0.0359121i
$424$			429.768i			1.01360i
$425$	22.1078			0.0520183
$426$		−	58.2005i		−	0.136621i
$427$		−	319.656i		−	0.748609i
$428$	−297.021			−0.693975
$429$	−149.249			−0.347899
$430$	−954.872			−2.22063
$431$			25.3259i			0.0587607i	0.999568	+	0.0293804i	$0.00935340\pi$
$431$			25.3259i			0.0587607i	−0.999568	+	0.0293804i	$0.990647\pi$
$432$	50.9810			0.118012
$433$	374.398			0.864659			0.432330	−	0.901716i	$-0.357692\pi$
$433$	374.398			0.864659			0.432330	+	0.901716i	$0.357692\pi$
$434$	−1437.06			−3.31119
$435$	291.311			0.669680
$436$			1256.61i			2.88213i
$437$			252.950i			0.578834i
$438$	−252.611			−0.576738
$439$	−282.618			−0.643776			−0.321888	−	0.946778i	$-0.604318\pi$
$439$	−282.618			−0.643776			−0.321888	+	0.946778i	$0.604318\pi$
$440$	2229.34			5.06668
$441$	150.623			0.341549
$442$	−5.14976			−0.0116510
$443$			543.714i			1.22735i	0.789560	+	0.613673i	$0.210309\pi$
$443$			543.714i			1.22735i	−0.789560	+	0.613673i	$0.789691\pi$
$444$		−	600.818i		−	1.35319i
$445$		−	446.185i		−	1.00266i
$446$		−	679.533i		−	1.52362i
$447$			165.157i			0.369479i
$448$	−850.737			−1.89897
$449$	369.058			0.821955			0.410977	−	0.911646i	$-0.365188\pi$
$449$	369.058			0.821955			0.410977	+	0.911646i	$0.365188\pi$
$450$		−	619.738i		−	1.37720i
$451$		−	330.632i		−	0.733109i
$452$			175.279i			0.387786i
$453$		−	454.996i		−	1.00441i
$454$	750.781			1.65370
$455$			387.905i			0.852539i
$456$			237.801i			0.521494i
$457$		−	529.971i		−	1.15967i	−0.814732	−	0.579837i	$-0.803116\pi$
$457$		−	529.971i		−	1.15967i	0.814732	−	0.579837i	$-0.196884\pi$
$458$	574.999			1.25546
$459$	1.88249			0.00410128
$460$		−	1493.11i		−	3.24589i
$461$	−131.591			−0.285448			−0.142724	−	0.989763i	$-0.545586\pi$
$461$	−131.591			−0.285448			−0.142724	+	0.989763i	$0.545586\pi$
$462$	1198.48			2.59410
$463$		−	489.009i		−	1.05618i	−0.849190	−	0.528088i	$-0.822909\pi$
$463$		−	489.009i		−	1.05618i	0.849190	−	0.528088i	$-0.177091\pi$
$464$	177.916			0.383440
$465$			684.665i			1.47240i
$466$	190.055			0.407844
$467$		−	58.5903i		−	0.125461i	−0.998031	−	0.0627305i	$-0.980019\pi$
$467$		−	58.5903i		−	0.125461i	0.998031	−	0.0627305i	$-0.0199808\pi$
$468$			93.9732i			0.200797i
$469$			811.488i			1.73025i
$470$	158.986			0.338268
$471$			40.5231i			0.0860363i
$472$	−323.530	−	610.650i	−0.685445	−	1.29375i
$473$	−624.094			−1.31944
$474$			610.087i			1.28710i
$475$	715.294			1.50588
$476$	26.9191			0.0565527
$477$	110.075			0.230766
$478$			957.023i			2.00214i
$479$	571.506			1.19312			0.596562	−	0.802567i	$-0.296533\pi$
$479$	571.506			1.19312			0.596562	+	0.802567i	$0.296533\pi$
$480$			219.085i			0.456428i
$481$	195.250			0.405924
$482$		−	306.327i		−	0.635533i
$483$		−	372.288i		−	0.770783i
$484$	2238.90			4.62583
$485$			1164.40i			2.40082i
$486$		−	52.7710i		−	0.108582i
$487$	420.688			0.863836			0.431918	−	0.901913i	$-0.357837\pi$
$487$	420.688			0.863836			0.431918	+	0.901913i	$0.357837\pi$
$488$	−375.902			−0.770292
$489$	−51.8470			−0.106027
$490$		−	1576.41i		−	3.21717i
$491$	880.226			1.79272			0.896360	−	0.443326i	$-0.146202\pi$
$491$	880.226			1.79272			0.896360	+	0.443326i	$0.146202\pi$
$492$	−208.180			−0.423130
$493$	6.56959			0.0133257
$494$	−166.620			−0.337287
$495$		−	570.996i		−	1.15353i
$496$			418.154i			0.843053i
$497$	−98.8660			−0.198925
$498$	692.610			1.39078
$499$	−232.423			−0.465778			−0.232889	−	0.972503i	$-0.574818\pi$
$499$	−232.423			−0.465778			−0.232889	+	0.972503i	$0.574818\pi$
$500$	−2492.46			−4.98491
$501$	−243.112			−0.485254
$502$		−	307.668i		−	0.612885i
$503$			769.859i			1.53053i	0.643713	+	0.765267i	$0.277393\pi$
$503$			769.859i			1.53053i	−0.643713	+	0.765267i	$0.722607\pi$
$504$		−	349.992i		−	0.694429i
$505$			317.115i			0.627951i
$506$		−	1499.14i		−	2.96273i
$507$	262.178			0.517116
$508$	303.397			0.597237
$509$		−	237.299i		−	0.466207i	−0.972452	−	0.233103i	$-0.925112\pi$
$509$		−	237.299i		−	0.466207i	0.972452	−	0.233103i	$-0.0748881\pi$
$510$		−	19.7020i		−	0.0386314i
$511$			429.114i			0.839754i
$512$			593.480i			1.15914i
$513$	60.9076			0.118728
$514$			254.925i			0.495963i
$515$			1112.60i			2.16038i
$516$			392.956i			0.761542i
$517$	103.912			0.200989
$518$	−1567.87			−3.02677
$519$			212.821i			0.410060i
$520$	456.160			0.877232
$521$	−28.9394			−0.0555459			−0.0277730	−	0.999614i	$-0.508842\pi$
$521$	−28.9394			−0.0555459			−0.0277730	+	0.999614i	$0.508842\pi$
$522$		−	184.163i		−	0.352802i
$523$	−390.540			−0.746730			−0.373365	−	0.927684i	$-0.621796\pi$
$523$	−390.540			−0.746730			−0.373365	+	0.927684i	$0.621796\pi$
$524$			903.951i			1.72510i
$525$	−1052.76			−2.00525
$526$		−	1240.27i		−	2.35793i
$527$			15.4405i			0.0292988i
$528$		−	348.732i		−	0.660476i
$529$	63.3150			0.119688
$530$		−	1152.04i		−	2.17367i
$531$	−156.405	+	82.8651i	−0.294547	+	0.156055i
$532$	870.962			1.63715
$533$		−	67.6528i		−	0.126928i
$534$	−282.072			−0.528225
$535$	369.282			0.690247
$536$	954.276			1.78037
$537$			22.8186i			0.0424927i
$538$	−1.26390			−0.00234925
$539$		−	1030.32i		−	1.91155i
$540$	−359.523			−0.665784
$541$		−	639.799i		−	1.18262i	−0.806443	−	0.591311i	$-0.798610\pi$
$541$		−	639.799i		−	1.18262i	0.806443	−	0.591311i	$-0.201390\pi$
$542$			1269.66i			2.34255i
$543$	−336.971			−0.620574
$544$			4.94077i			0.00908231i
$545$		−	1562.32i		−	2.86665i
$546$	245.228			0.449136
$547$	−33.9867			−0.0621329			−0.0310664	−	0.999517i	$-0.509890\pi$
$547$	−33.9867			−0.0621329			−0.0310664	+	0.999517i	$0.509890\pi$
$548$	1480.88			2.70234
$549$			96.2791i			0.175372i
$550$	−4239.27			−7.70776
$551$	212.558			0.385768
$552$	−437.795			−0.793108
$553$	1036.36			1.87408
$554$		−	297.776i		−	0.537501i
$555$			746.988i			1.34592i
$556$	−1701.13			−3.05958
$557$	−587.282			−1.05437			−0.527183	−	0.849752i	$-0.676752\pi$
$557$	−587.282			−1.05437			−0.527183	+	0.849752i	$0.676752\pi$
$558$	432.836			0.775691
$559$	−127.700			−0.228444
$560$	−906.373			−1.61852
$561$		−	12.8770i		−	0.0229537i
$562$			1213.06i			2.15846i
$563$			416.727i			0.740190i	0.928994	+	0.370095i	$0.120675\pi$
$563$			416.727i			0.740190i	−0.928994	+	0.370095i	$0.879325\pi$
$564$		−	65.4271i		−	0.116005i
$565$		−	217.922i		−	0.385703i
$566$	1262.31			2.23024
$567$	−89.6427			−0.158100
$568$			116.262i			0.204687i
$569$		−	429.645i		−	0.755089i	−0.925992	−	0.377544i	$-0.876769\pi$
$569$		−	429.645i		−	0.755089i	0.925992	−	0.377544i	$-0.123231\pi$
$570$		−	637.455i		−	1.11834i
$571$		−	783.544i		−	1.37223i	−0.727492	−	0.686116i	$-0.759314\pi$
$571$		−	783.544i		−	1.37223i	0.727492	−	0.686116i	$-0.240686\pi$
$572$	642.816			1.12380
$573$		−	519.470i		−	0.906580i
$574$			543.257i			0.946440i
$575$			1316.87i			2.29020i
$576$	256.238			0.444858
$577$	14.6938			0.0254659			0.0127330	−	0.999919i	$-0.495947\pi$
$577$	14.6938			0.0254659			0.0127330	+	0.999919i	$0.495947\pi$
$578$			977.895i			1.69186i
$579$	−47.5360			−0.0821001
$580$	−1254.68			−2.16324
$581$		−	1176.55i		−	2.02504i
$582$	736.115			1.26480
$583$		−	752.963i		−	1.29153i
$584$	504.620			0.864076
$585$		−	116.835i		−	0.199719i
$586$			279.013i			0.476131i
$587$		−	847.153i		−	1.44319i	−0.692315	−	0.721596i	$-0.743409\pi$
$587$		−	847.153i		−	1.44319i	0.692315	−	0.721596i	$-0.256591\pi$
$588$	−648.736			−1.10329
$589$			499.573i			0.848172i
$590$	867.261	+	1636.92i	1.46993	+	2.77444i
$591$	142.969			0.241910
$592$			456.217i			0.770637i
$593$	−669.594			−1.12916			−0.564582	−	0.825377i	$-0.690963\pi$
$593$	−669.594			−1.12916			−0.564582	+	0.825377i	$0.690963\pi$
$594$	−360.976			−0.607703
$595$	−33.4681			−0.0562488
$596$		−	711.335i		−	1.19352i
$597$	−71.7573			−0.120196
$598$		−	306.749i		−	0.512958i
$599$	666.722			1.11306			0.556529	−	0.830828i	$-0.312133\pi$
$599$	666.722			1.11306			0.556529	+	0.830828i	$0.312133\pi$
$600$			1238.00i			2.06333i
$601$			341.243i			0.567792i	0.958855	+	0.283896i	$0.0916270\pi$
$601$			341.243i			0.567792i	−0.958855	+	0.283896i	$0.908373\pi$
$602$	1025.44			1.70339
$603$		−	244.417i		−	0.405335i
$604$			1959.67i			3.24449i
$605$	−2783.59			−4.60098
$606$	200.476			0.330818
$607$	−137.196			−0.226023			−0.113011	−	0.993594i	$-0.536050\pi$
$607$	−137.196			−0.226023			−0.113011	+	0.993594i	$0.536050\pi$
$608$			159.858i			0.262924i
$609$	−312.839			−0.513694
$610$	1007.65			1.65189
$611$	21.2620			0.0347988
$612$	−8.10790			−0.0132482
$613$			583.317i			0.951577i	0.879560	+	0.475788i	$0.157837\pi$
$613$			583.317i			0.951577i	−0.879560	+	0.475788i	$0.842163\pi$
$614$			1590.09i			2.58972i
$615$	258.827			0.420857
$616$	−2394.09			−3.88652
$617$	775.180			1.25637			0.628185	−	0.778064i	$-0.283798\pi$
$617$	775.180			1.25637			0.628185	+	0.778064i	$0.283798\pi$
$618$	703.367			1.13813
$619$	297.409			0.480467			0.240233	−	0.970715i	$-0.422776\pi$
$619$	297.409			0.480467			0.240233	+	0.970715i	$0.422776\pi$
$620$		−	2948.87i		−	4.75623i
$621$			112.132i			0.180566i
$622$		−	180.886i		−	0.290813i
$623$			479.159i			0.769116i
$624$		−	71.3563i		−	0.114353i
$625$	1573.25			2.51721
$626$	−28.3745			−0.0453267
$627$		−	416.633i		−	0.664487i
$628$		−	174.534i		−	0.277920i
$629$			16.8459i			0.0267821i
$630$			938.196i			1.48920i
$631$	172.413			0.273238			0.136619	−	0.990624i	$-0.456376\pi$
$631$	172.413			0.273238			0.136619	+	0.990624i	$0.456376\pi$
$632$		−	1218.72i		−	1.92836i
$633$		−	418.627i		−	0.661338i
$634$			1355.77i			2.13844i
$635$	−377.208			−0.594029
$636$	−474.097			−0.745435
$637$		−	210.822i		−	0.330960i
$638$	−1259.75			−1.97453
$639$	29.7780			0.0466009
$640$		−	2175.82i		−	3.39972i
$641$	219.920			0.343089			0.171544	−	0.985176i	$-0.445124\pi$
$641$	219.920			0.343089			0.171544	+	0.985176i	$0.445124\pi$
$642$		−	233.455i		−	0.363637i
$643$	−930.457			−1.44706			−0.723528	−	0.690295i	$-0.757481\pi$
$643$	−930.457			−1.44706			−0.723528	+	0.690295i	$0.757481\pi$
$644$			1603.45i			2.48983i
$645$		−	488.556i		−	0.757451i
$646$		−	14.3758i		−	0.0222535i
$647$	748.437			1.15678			0.578390	−	0.815760i	$-0.303681\pi$
$647$	748.437			1.15678			0.578390	+	0.815760i	$0.303681\pi$
$648$			105.416i			0.162679i
$649$	566.832	+	1069.87i	0.873393	+	1.64849i
$650$	−867.426			−1.33450
$651$		−	735.264i		−	1.12944i
$652$	223.306			0.342494
$653$	−172.727			−0.264514			−0.132257	−	0.991215i	$-0.542222\pi$
$653$	−172.727			−0.264514			−0.132257	+	0.991215i	$0.542222\pi$
$654$	−987.678			−1.51021
$655$		−	1123.87i		−	1.71583i
$656$	158.077			0.240970
$657$		−	129.247i		−	0.196723i
$658$	−170.736			−0.259477
$659$		−	552.763i		−	0.838790i	−0.907804	−	0.419395i	$-0.862242\pi$
$659$		−	552.763i		−	0.838790i	0.907804	−	0.419395i	$-0.137758\pi$
$660$			2459.29i			3.72620i
$661$	890.466			1.34715			0.673575	−	0.739119i	$-0.264758\pi$
$661$	890.466			1.34715			0.673575	+	0.739119i	$0.264758\pi$
$662$			100.329i			0.151554i
$663$		−	2.63485i		−	0.00397413i
$664$	−1383.57			−2.08369
$665$	−1082.85			−1.62835
$666$	472.235			0.709061
$667$			391.322i			0.586690i
$668$	1047.09			1.56750
$669$	347.680			0.519701
$670$	−2558.05			−3.81799
$671$	658.589			0.981504
$672$		−	235.276i		−	0.350113i
$673$			136.255i			0.202459i	0.994863	+	0.101230i	$0.0322777\pi$
$673$			136.255i			0.202459i	−0.994863	+	0.101230i	$0.967722\pi$
$674$	−380.961			−0.565223
$675$	317.086			0.469757
$676$	−1129.20			−1.67042
$677$	265.723			0.392501			0.196251	−	0.980554i	$-0.437123\pi$
$677$	265.723			0.392501			0.196251	+	0.980554i	$0.437123\pi$
$678$	−137.767			−0.203197
$679$		−	1250.45i		−	1.84160i
$680$			39.3570i			0.0578780i
$681$			384.133i			0.564073i
$682$		−	2960.78i		−	4.34132i
$683$			445.533i			0.652318i	0.945315	+	0.326159i	$0.105754\pi$
$683$			445.533i			0.652318i	−0.945315	+	0.326159i	$0.894246\pi$
$684$	−262.330			−0.383523
$685$	−1841.16			−2.68782
$686$			40.7198i			0.0593583i
$687$			294.196i			0.428232i
$688$		−	298.382i		−	0.433694i
$689$		−	154.069i		−	0.223612i
$690$	1173.56			1.70082
$691$			370.939i			0.536815i	0.963306	+	0.268407i	$0.0864973\pi$
$691$			370.939i			0.536815i	−0.963306	+	0.268407i	$0.913503\pi$
$692$		−	916.625i		−	1.32460i
$693$			613.194i			0.884840i
$694$	1313.68			1.89290
$695$	2114.99			3.04315
$696$			367.886i			0.528572i
$697$	5.83702			0.00837449
$698$	1836.05			2.63045
$699$			97.2409i			0.139114i
$700$	4534.24			6.47749
$701$		−	416.398i		−	0.594006i	−0.954877	−	0.297003i	$-0.904013\pi$
$701$		−	416.398i		−	0.594006i	0.954877	−	0.297003i	$-0.0959871\pi$
$702$	−73.8616			−0.105216
$703$			545.047i			0.775316i
$704$		−	1752.78i		−	2.48974i
$705$			81.3445i			0.115382i
$706$	−348.397			−0.493480
$707$		−	340.551i		−	0.481684i
$708$	673.637	−	356.901i	0.951465	−	0.504098i
$709$	−751.459			−1.05989			−0.529943	−	0.848033i	$-0.677787\pi$
$709$	−751.459			−1.05989			−0.529943	+	0.848033i	$0.677787\pi$
$710$		−	311.655i		−	0.438951i
$711$	−312.148			−0.439027
$712$	563.471			0.791392
$713$	−919.721			−1.28993
$714$			21.1580i			0.0296331i
$715$	−799.203			−1.11777
$716$		−	98.2801i		−	0.137263i
$717$	−489.656			−0.682923
$718$		−	951.985i		−	1.32588i
$719$			869.723i			1.20963i	0.796367	+	0.604814i	$0.206753\pi$
$719$			869.723i			1.20963i	−0.796367	+	0.604814i	$0.793247\pi$
$720$	272.996			0.379161
$721$		−	1194.82i		−	1.65717i
$722$			756.952i			1.04841i
$723$	156.731			0.216778
$724$	1451.34			2.00462
$725$	1106.58			1.52632
$726$			1759.75i			2.42390i
$727$	1392.73			1.91572			0.957859	−	0.287238i	$-0.0927370\pi$
$727$	1392.73			1.91572			0.957859	+	0.287238i	$0.0927370\pi$
$728$	−489.872			−0.672901
$729$	27.0000			0.0370370
$730$	−1352.70			−1.85301
$731$		−	11.0178i		−	0.0150723i
$732$		−	414.676i		−	0.566497i
$733$	−57.5400			−0.0784993			−0.0392497	−	0.999229i	$-0.512497\pi$
$733$	−57.5400			−0.0784993			−0.0392497	+	0.999229i	$0.512497\pi$
$734$	76.0226			0.103573
$735$	806.563			1.09736
$736$	−294.300			−0.399865
$737$	−1671.91			−2.26854
$738$		−	163.627i		−	0.221716i
$739$		−	194.224i		−	0.262820i	−0.991328	−	0.131410i	$-0.958049\pi$
$739$		−	194.224i		−	0.262820i	0.991328	−	0.131410i	$-0.0419505\pi$
$740$		−	3217.29i		−	4.34769i
$741$		−	85.2502i		−	0.115047i
$742$			1237.18i			1.66736i
$743$	−365.950			−0.492531			−0.246265	−	0.969202i	$-0.579203\pi$
$743$	−365.950			−0.492531			−0.246265	+	0.969202i	$0.579203\pi$
$744$	−864.639			−1.16215
$745$			884.392i			1.18710i
$746$		−	239.700i		−	0.321314i
$747$			354.371i			0.474392i
$748$			55.4615i			0.0741464i
$749$	−396.573			−0.529470
$750$		−	1959.04i		−	2.61205i
$751$		−	196.887i		−	0.262166i	−0.991371	−	0.131083i	$-0.958155\pi$
$751$		−	196.887i		−	0.262166i	0.991371	−	0.131083i	$-0.0418455\pi$
$752$			49.6805i			0.0660645i
$753$	157.417			0.209053
$754$	−257.766			−0.341865
$755$		−	2436.43i		−	3.22706i
$756$	386.093			0.510705
$757$	581.124			0.767667			0.383834	−	0.923402i	$-0.374604\pi$
$757$	581.124			0.767667			0.383834	+	0.923402i	$0.374604\pi$
$758$			844.670i			1.11434i
$759$	767.027			1.01058
$760$			1273.39i			1.67551i
$761$	−1257.55			−1.65249			−0.826246	−	0.563310i	$-0.809528\pi$
$761$	−1257.55			−1.65249			−0.826246	+	0.563310i	$0.809528\pi$
$762$			238.466i			0.312947i
$763$			1677.78i			2.19893i
$764$			2237.37i			2.92849i
$765$	10.0804			0.0131770
$766$			2283.97i			2.98168i
$767$	115.983	+	218.914i	0.151217	+	0.285416i
$768$	−783.764			−1.02053
$769$		−	865.286i		−	1.12521i	−0.826726	−	0.562605i	$-0.809799\pi$
$769$		−	865.286i		−	1.12521i	0.826726	−	0.562605i	$-0.190201\pi$
$770$	6417.65			8.33462
$771$	−130.431			−0.169171
$772$	204.738			0.265205
$773$			1118.01i			1.44633i	0.690675	+	0.723165i	$0.257313\pi$
$773$			1118.01i			1.44633i	−0.690675	+	0.723165i	$0.742687\pi$
$774$	−308.858			−0.399041
$775$			2600.79i			3.35586i
$776$	−1470.48			−1.89494
$777$		−	802.192i		−	1.03242i
$778$		−	1671.50i		−	2.14846i
$779$	188.856			0.242434
$780$			503.212i			0.645144i
$781$		−	203.694i		−	0.260812i
$782$	26.4660			0.0338440
$783$	94.2259			0.120340
$784$	492.602			0.628319
$785$			216.995i			0.276427i
$786$	−710.493			−0.903935
$787$	−1066.39			−1.35501			−0.677506	−	0.735517i	$-0.736939\pi$
$787$	−1066.39			−1.35501			−0.677506	+	0.735517i	$0.736939\pi$
$788$	−615.769			−0.781432
$789$	634.578			0.804281
$790$			3266.93i			4.13535i
$791$			234.027i			0.295862i
$792$	721.091			0.910468
$793$	134.758			0.169935
$794$	−2196.99			−2.76699
$795$	589.438			0.741431
$796$	309.060			0.388266
$797$			464.166i			0.582392i	0.956663	+	0.291196i	$0.0940531\pi$
$797$			464.166i			0.582392i	−0.956663	+	0.291196i	$0.905947\pi$
$798$			684.565i			0.857850i
$799$			1.83447i			0.00229595i
$800$			832.224i			1.04028i
$801$		−	144.321i		−	0.180176i
$802$	202.514			0.252512
$803$	−884.106			−1.10100
$804$			1052.71i			1.30934i
$805$		−	1993.55i		−	2.47646i
$806$		−	605.825i		−	0.751644i
$807$		−	0.646667i		−	0.000801322i
$808$	−400.474			−0.495636
$809$			376.680i			0.465612i	0.972523	+	0.232806i	$0.0747907\pi$
$809$			376.680i			0.465612i	−0.972523	+	0.232806i	$0.925209\pi$
$810$		−	282.580i		−	0.348865i
$811$			356.067i			0.439047i	0.975607	+	0.219524i	$0.0704503\pi$
$811$			356.067i			0.439047i	−0.975607	+	0.219524i	$0.929550\pi$
$812$	1347.41			1.65937
$813$	−649.617			−0.799037
$814$		−	3230.28i		−	3.96841i
$815$	−277.633			−0.340654
$816$	6.15655			0.00754479
$817$		−	356.480i		−	0.436328i
$818$	−945.096			−1.15537
$819$			125.470i			0.153199i
$820$	−1114.77			−1.35948
$821$			7.62665i			0.00928946i	0.999989	+	0.00464473i	$0.00147847\pi$
$821$			7.62665i			0.00928946i	−0.999989	+	0.00464473i	$0.998522\pi$
$822$			1163.95i			1.41600i
$823$		−	846.755i		−	1.02886i	−0.857531	−	0.514432i	$-0.828003\pi$
$823$		−	846.755i		−	1.02886i	0.857531	−	0.514432i	$-0.171997\pi$
$824$	−1405.06			−1.70517
$825$		−	2169.00i		−	2.62909i
$826$	−931.354	−	1757.89i	−1.12755	−	2.12820i
$827$	1039.36			1.25679			0.628393	−	0.777896i	$-0.283713\pi$
$827$	1039.36			1.25679			0.628393	+	0.777896i	$0.283713\pi$
$828$		−	482.953i		−	0.583276i
$829$	−528.130			−0.637069			−0.318535	−	0.947911i	$-0.603191\pi$
$829$	−528.130			−0.637069			−0.318535	+	0.947911i	$0.603191\pi$
$830$	3708.82			4.46846
$831$	152.355			0.183340
$832$		−	358.648i		−	0.431067i
$833$	18.1895			0.0218361
$834$		−	1337.06i		−	1.60320i
$835$	−1301.83			−1.55908
$836$			1794.45i			2.14647i
$837$			221.458i			0.264586i
$838$	1498.58			1.78828
$839$			814.378i			0.970653i	0.874333	+	0.485327i	$0.161299\pi$
$839$			814.378i			0.970653i	−0.874333	+	0.485327i	$0.838701\pi$
$840$		−	1874.16i		−	2.23114i
$841$	−512.166			−0.608996
$842$	249.509			0.296329
$843$	−620.655			−0.736245
$844$			1803.03i			2.13630i
$845$	1403.92			1.66145
$846$	51.4248			0.0607858
$847$	2989.31			3.52929
$848$	359.995			0.424522
$849$			645.857i			0.760726i
$850$		−	74.8406i		−	0.0880477i
$851$	−1003.44			−1.17913
$852$	−128.254			−0.150533
$853$	186.085			0.218153			0.109077	−	0.994033i	$-0.465211\pi$
$853$	186.085			0.218153			0.109077	+	0.994033i	$0.465211\pi$
$854$	−1082.12			−1.26712
$855$	326.151			0.381463
$856$			466.353i			0.544805i
$857$		−	1618.09i		−	1.88809i	−0.329816	−	0.944045i	$-0.606987\pi$
$857$		−	1618.09i		−	1.88809i	0.329816	−	0.944045i	$-0.393013\pi$
$858$			505.245i			0.588863i
$859$		−	718.693i		−	0.836662i	−0.908295	−	0.418331i	$-0.862615\pi$
$859$		−	718.693i		−	0.836662i	0.908295	−	0.418331i	$-0.137385\pi$
$860$			2104.22i			2.44677i
$861$	−277.955			−0.322828
$862$	85.7346			0.0994601
$863$			1217.22i			1.41045i	0.708981	+	0.705227i	$0.249155\pi$
$863$			1217.22i			1.41045i	−0.708981	+	0.705227i	$0.750845\pi$
$864$			70.8642i			0.0820187i
$865$			1139.63i			1.31749i
$866$		−	1267.43i		−	1.46355i
$867$	−500.335			−0.577088
$868$			3166.79i			3.64838i
$869$			2135.23i			2.45711i
$870$		−	986.163i		−	1.13352i
$871$	−342.102			−0.392769
$872$	1973.00			2.26262
$873$			376.630i			0.431420i
$874$	856.303			0.979751
$875$	−3327.84			−3.80325
$876$			556.670i			0.635468i
$877$	1635.00			1.86431			0.932154	−	0.362061i	$-0.117927\pi$
$877$	1635.00			1.86431			0.932154	+	0.362061i	$0.117927\pi$
$878$			956.734i			1.08967i
$879$	−142.755			−0.162407
$880$		−	1867.40i		−	2.12205i
$881$		−	842.026i		−	0.955762i	−0.878425	−	0.477881i	$-0.841405\pi$
$881$		−	842.026i		−	0.955762i	0.878425	−	0.477881i	$-0.158595\pi$
$882$		−	509.898i		−	0.578115i
$883$	−306.069			−0.346624			−0.173312	−	0.984867i	$-0.555447\pi$
$883$	−306.069			−0.346624			−0.173312	+	0.984867i	$0.555447\pi$
$884$			11.3483i			0.0128375i
$885$	−837.523	+	443.730i	−0.946353	+	0.501390i
$886$	1840.61			2.07744
$887$			910.611i			1.02662i	0.858204	+	0.513309i	$0.171581\pi$
$887$			910.611i			1.02662i	−0.858204	+	0.513309i	$0.828419\pi$
$888$	−943.344			−1.06232
$889$	405.085			0.455663
$890$	−1510.45			−1.69714
$891$		−	184.691i		−	0.207285i
$892$	−1497.46			−1.67877
$893$			59.3539i			0.0664657i
$894$	559.100			0.625392
$895$			122.190i			0.136525i
$896$			2336.62i			2.60783i
$897$	156.947			0.174968
$898$		−	1249.36i		−	1.39126i
$899$			772.855i			0.859683i
$900$	−1365.69			−1.51744
$901$	13.2929			0.0147535
$902$	−1119.27			−1.24088
$903$			524.661i			0.581020i
$904$	275.206			0.304432
$905$	−1804.43			−1.99385
$906$	−1540.28			−1.70009
$907$	−1445.12			−1.59330			−0.796648	−	0.604444i	$-0.793395\pi$
$907$	−1445.12			−1.59330			−0.796648	+	0.604444i	$0.793395\pi$
$908$		−	1654.47i		−	1.82210i
$909$			102.572i			0.112841i
$910$	1313.16			1.44303
$911$	−884.175			−0.970554			−0.485277	−	0.874360i	$-0.661281\pi$
$911$	−884.175			−0.970554			−0.485277	+	0.874360i	$0.661281\pi$
$912$	199.194			0.218415
$913$	2424.04			2.65503
$914$	−1794.09			−1.96290
$915$			515.560i			0.563453i
$916$		−	1267.11i		−	1.38330i
$917$			1206.92i			1.31617i
$918$		−	6.37271i		−	0.00694194i
$919$		−	1285.28i		−	1.39856i	−0.714848	−	0.699280i	$-0.753504\pi$
$919$		−	1285.28i		−	1.39856i	0.714848	−	0.699280i	$-0.246496\pi$
$920$	−2344.33			−2.54818
$921$	−813.560			−0.883344
$922$			445.471i			0.483157i
$923$		−	41.6792i		−	0.0451563i
$924$		−	2641.04i		−	2.85827i
$925$			2837.53i			3.06760i
$926$	−1655.42			−1.78771
$927$			359.874i			0.388214i
$928$			247.305i			0.266493i
$929$		−	1119.46i		−	1.20502i	−0.798113	−	0.602508i	$-0.794168\pi$
$929$		−	1119.46i		−	1.20502i	0.798113	−	0.602508i	$-0.205832\pi$
$930$	2317.77			2.49222
$931$	588.517			0.632135
$932$		−	418.818i		−	0.449376i
$933$	92.5493			0.0991954
$934$	−198.343			−0.212359
$935$		−	68.9544i		−	0.0737480i
$936$	147.547			0.157636
$937$		−	1417.82i		−	1.51315i	−0.653908	−	0.756574i	$-0.726872\pi$
$937$		−	1417.82i		−	1.51315i	0.653908	−	0.756574i	$-0.273128\pi$
$938$	2747.10			2.92868
$939$		−	14.5177i		−	0.0154608i
$940$		−	350.352i		−	0.372715i
$941$		−	698.423i		−	0.742214i	−0.928590	−	0.371107i	$-0.878978\pi$
$941$		−	698.423i		−	0.742214i	0.928590	−	0.371107i	$-0.121022\pi$
$942$	137.181			0.145628
$943$			347.686i			0.368702i
$944$	−511.511	+	271.005i	−0.541855	+	0.287081i
$945$	−480.023			−0.507961
$946$			2112.72i			2.23332i
$947$	210.515			0.222297			0.111149	−	0.993804i	$-0.464547\pi$
$947$	210.515			0.222297			0.111149	+	0.993804i	$0.464547\pi$
$948$	1344.43			1.41817
$949$	−180.903			−0.190625
$950$		−	2421.45i		−	2.54890i
$951$	−693.674			−0.729416
$952$		−	42.2656i		−	0.0443967i
$953$	−681.588			−0.715202			−0.357601	−	0.933874i	$-0.616405\pi$
$953$	−681.588			−0.715202			−0.357601	+	0.933874i	$0.616405\pi$
$954$		−	372.634i		−	0.390602i
$955$		−	2781.69i		−	2.91276i
$956$	2108.96			2.20602
$957$		−	644.545i		−	0.673505i
$958$		−	1934.70i		−	2.01952i
$959$	1977.22			2.06175
$960$	1372.12			1.42929
$961$	−855.434			−0.890150
$962$		−	660.970i		−	0.687080i
$963$	119.446			0.124035
$964$	−675.042			−0.700251
$965$	−254.548			−0.263780
$966$	−1260.29			−1.30465
$967$			1165.11i			1.20487i	0.798168	+	0.602435i	$0.205803\pi$
$967$			1165.11i			1.20487i	−0.798168	+	0.602435i	$0.794197\pi$
$968$		−	3515.30i		−	3.63151i
$969$	7.35530			0.00759060
$970$	3941.79			4.06370
$971$	−393.936			−0.405701			−0.202850	−	0.979210i	$-0.565021\pi$
$971$	−393.936			−0.405701			−0.202850	+	0.979210i	$0.565021\pi$
$972$	−116.289			−0.119639
$973$	−2271.29			−2.33432
$974$		−	1424.14i		−	1.46215i
$975$		−	443.814i		−	0.455194i
$976$			314.874i			0.322617i
$977$			1020.38i			1.04440i	0.852822	+	0.522202i	$0.174889\pi$
$977$			1020.38i			1.04440i	−0.852822	+	0.522202i	$0.825111\pi$
$978$			175.516i			0.179464i
$979$	−987.214			−1.00839
$980$	−3473.88			−3.54478
$981$		−	505.341i		−	0.515128i
$982$		−	2979.79i		−	3.03441i
$983$		−	464.597i		−	0.472632i	−0.971676	−	0.236316i	$-0.924060\pi$
$983$		−	464.597i		−	0.472632i	0.971676	−	0.236316i	$-0.0759401\pi$
$984$			326.863i			0.332178i
$985$	765.576			0.777234
$986$		−	22.2398i		−	0.0225555i
$987$		−	87.3560i		−	0.0885066i
$988$			367.174i			0.371634i
$989$	656.284			0.663583
$990$	−1932.97			−1.95249
$991$			512.343i			0.516996i	0.966012	+	0.258498i	$0.0832276\pi$
$991$			512.343i			0.516996i	−0.966012	+	0.258498i	$0.916772\pi$
$992$	−581.239			−0.585926
$993$	−51.3327			−0.0516946
$994$			334.687i			0.336707i
$995$	−384.249			−0.386180
$996$		−	1526.28i		−	1.53241i
$997$	−174.174			−0.174698			−0.0873491	−	0.996178i	$-0.527840\pi$
$997$	−174.174			−0.174698			−0.0873491	+	0.996178i	$0.527840\pi$
$998$			786.813i			0.788390i
$999$			241.617i			0.241858i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.3.c.a.58.2	✓	20
3.2	odd	2		531.3.c.c.235.19		20
59.58	odd	2	inner	177.3.c.a.58.19	yes	20
177.176	even	2		531.3.c.c.235.2		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.c.a.58.2	✓	20	1.1	even	1	trivial
177.3.c.a.58.19	yes	20	59.58	odd	2	inner
531.3.c.c.235.2		20	177.176	even	2
531.3.c.c.235.19		20	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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	177.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.38526i q^{2} +1.73205 q^{3} -7.45997 q^{4} +9.27487 q^{5} -5.86344i q^{6} -9.96030 q^{7} +11.7129i q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q-3.38526i q^{2} +1.73205 q^{3} -7.45997 q^{4} +9.27487 q^{5} -5.86344i q^{6} -9.96030 q^{7} +11.7129i q^{8} +3.00000 q^{9} -31.3978i q^{10} -20.5213i q^{11} -12.9211 q^{12} -4.19899i q^{13} +33.7182i q^{14} +16.0645 q^{15} +9.81130 q^{16} +0.362285 q^{17} -10.1558i q^{18} +11.7217 q^{19} -69.1903 q^{20} -17.2518 q^{21} -69.4698 q^{22} +21.5797i q^{23} +20.2873i q^{24} +61.0232 q^{25} -14.2147 q^{26} +5.19615 q^{27} +74.3036 q^{28} +18.1338 q^{29} -54.3826i q^{30} +42.6196i q^{31} +13.6378i q^{32} -35.5439i q^{33} -1.22643i q^{34} -92.3805 q^{35} -22.3799 q^{36} +46.4991i q^{37} -39.6809i q^{38} -7.27287i q^{39} +108.636i q^{40} +16.1117 q^{41} +58.4016i q^{42} -30.4120i q^{43} +153.088i q^{44} +27.8246 q^{45} +73.0530 q^{46} +5.06360i q^{47} +16.9937 q^{48} +50.2077 q^{49} -206.579i q^{50} +0.627496 q^{51} +31.3244i q^{52} +36.6918 q^{53} -17.5903i q^{54} -190.332i q^{55} -116.664i q^{56} +20.3025 q^{57} -61.3875i q^{58} +(-52.1348 + 27.6217i) q^{59} -119.841 q^{60} +32.0930i q^{61} +144.279 q^{62} -29.8809 q^{63} +85.4127 q^{64} -38.9451i q^{65} -120.325 q^{66} -81.4723i q^{67} -2.70263 q^{68} +37.3772i q^{69} +312.732i q^{70} +9.92600 q^{71} +35.1387i q^{72} -43.0824i q^{73} +157.412 q^{74} +105.695 q^{75} -87.4433 q^{76} +204.398i q^{77} -24.6205 q^{78} -104.049 q^{79} +90.9985 q^{80} +9.00000 q^{81} -54.5422i q^{82} +118.124i q^{83} +128.698 q^{84} +3.36014 q^{85} -102.953 q^{86} +31.4086 q^{87} +240.364 q^{88} -48.1069i q^{89} -94.1935i q^{90} +41.8233i q^{91} -160.984i q^{92} +73.8194i q^{93} +17.1416 q^{94} +108.717 q^{95} +23.6214i q^{96} +125.543i q^{97} -169.966i q^{98} -61.5638i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 40 q^{4} - 8 q^{7} + 60 q^{9}+O(q^{10}) \) 20 * q - 40 * q^4 - 8 * q^7 + 60 * q^9	\( 20 q - 40 q^{4} - 8 q^{7} + 60 q^{9} - 24 q^{12} + 24 q^{15} - 8 q^{16} + 16 q^{17} - 60 q^{19} - 164 q^{20} + 40 q^{22} + 100 q^{25} - 156 q^{26} + 200 q^{28} - 60 q^{29} - 32 q^{35} - 120 q^{36} + 28 q^{41} + 180 q^{46} - 96 q^{48} + 284 q^{49} - 24 q^{51} - 8 q^{53} + 24 q^{57} - 152 q^{59} - 72 q^{60} - 8 q^{62} - 24 q^{63} + 204 q^{64} - 120 q^{66} + 384 q^{68} + 92 q^{71} + 104 q^{74} + 240 q^{75} + 120 q^{76} - 468 q^{78} - 420 q^{79} + 376 q^{80} + 180 q^{81} + 168 q^{84} - 348 q^{85} + 232 q^{86} + 144 q^{87} - 212 q^{88} + 152 q^{94} + 788 q^{95}+O(q^{100}) \) 20 * q - 40 * q^4 - 8 * q^7 + 60 * q^9 - 24 * q^12 + 24 * q^15 - 8 * q^16 + 16 * q^17 - 60 * q^19 - 164 * q^20 + 40 * q^22 + 100 * q^25 - 156 * q^26 + 200 * q^28 - 60 * q^29 - 32 * q^35 - 120 * q^36 + 28 * q^41 + 180 * q^46 - 96 * q^48 + 284 * q^49 - 24 * q^51 - 8 * q^53 + 24 * q^57 - 152 * q^59 - 72 * q^60 - 8 * q^62 - 24 * q^63 + 204 * q^64 - 120 * q^66 + 384 * q^68 + 92 * q^71 + 104 * q^74 + 240 * q^75 + 120 * q^76 - 468 * q^78 - 420 * q^79 + 376 * q^80 + 180 * q^81 + 168 * q^84 - 348 * q^85 + 232 * q^86 + 144 * q^87 - 212 * q^88 + 152 * q^94 + 788 * q^95

Introduction

L-functions

Modular forms

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Database

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