Embedded newform 177.4.e.b.4.9

• Label: 177.4.e.b.4.9
• Level: $177$
• Weight: $4$
• Character: 177.4
• Analytic conductor: $10.443$
• Analytic rank: $0$
• Dimension: $420$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	177.e (of order \(29\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.4433380710\)
Analytic rank:	\(0\)
Dimension:	\(420\)
Relative dimension:	\(15\) over \(\Q(\zeta_{29})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{29}]$

Embedding invariants

Embedding label			4.9
Character	\(\chi\)	\(=\)	177.4
Dual form			177.4.e.b.133.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.24705 + 0.135624i) q^{2} +(1.94216 - 2.28649i) q^{3} +(-6.27623 - 1.38150i) q^{4} +(16.4405 + 12.4978i) q^{5} +(2.73207 - 2.58795i) q^{6} +(-11.4144 + 28.6480i) q^{7} +(-17.1493 - 5.77826i) q^{8} +(-1.45604 - 8.88144i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q + 2 q^{2} - 45 q^{3} - 56 q^{4} - 14 q^{5} + 6 q^{6} + 6 q^{7} + 66 q^{8} - 135 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{1}{29}\right)\)	\(1\)

Coefficient data

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

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.24705	+	0.135624i	0.440897	+	0.0479505i	0.325876	−	0.945413i	\(-0.394341\pi\)
							0.115021	+	0.993363i	\(0.463306\pi\)
\(3\)	1.94216	−	2.28649i	0.373769	−	0.440034i
							\(5\)	16.4405	+	12.4978i	1.47048	+	1.11783i	0.967294	+	0.253658i	\(0.0816339\pi\)
0.503191	+	0.864175i	\(0.332159\pi\)
\(7\)	−11.4144	+	28.6480i	−0.616321	+	1.54685i	0.206439	+	0.978459i	\(0.433812\pi\)
							−0.822760	+	0.568389i	\(0.807567\pi\)
\(11\)	−49.2402	+	22.7810i	−1.34968	+	0.624429i	−0.955605	−	0.294650i	\(-0.904797\pi\)
							−0.394075	+	0.919078i	\(0.628935\pi\)
\(13\)	0.101299	−	0.617895i	0.00216117	−	0.0131826i	−0.985724	−	0.168370i	\(-0.946150\pi\)
							0.987885	+	0.155187i	\(0.0495980\pi\)
\(17\)	−27.7891	−	69.7455i	−0.396462	−	0.995045i	−0.982750	−	0.184942i	\(-0.940790\pi\)
							0.586288	−	0.810103i	\(-0.300589\pi\)
\(19\)	45.0158	+	66.3933i	0.543544	+	0.801667i	0.995928	−	0.0901563i	\(-0.0287367\pi\)
							−0.452384	+	0.891823i	\(0.649426\pi\)
\(23\)	−8.73367	+	161.083i	−0.0791781	+	1.46035i	0.641521	+	0.767106i	\(0.278304\pi\)
							−0.720699	+	0.693248i	\(0.756179\pi\)
\(29\)	213.866	−	23.2594i	1.36945	−	0.148936i	0.606339	−	0.795206i	\(-0.292638\pi\)
							0.763109	+	0.646270i	\(0.223672\pi\)
\(31\)	120.151	−	177.210i	0.696124	−	1.02671i	−0.301198	−	0.953562i	\(-0.597386\pi\)
							0.997321	−	0.0731439i	\(-0.0233032\pi\)
\(37\)	−166.688	+	56.1636i	−0.740629	+	0.249547i	−0.664228	−	0.747530i	\(-0.731240\pi\)
							−0.0764011	+	0.997077i	\(0.524343\pi\)
\(41\)	−12.6394	−	233.120i	−0.0481449	−	0.887980i	−0.918467	−	0.395497i	\(-0.870572\pi\)
							0.870322	−	0.492483i	\(-0.163910\pi\)
\(43\)	−31.6715	−	14.6528i	−0.112322	−	0.0519657i	0.362923	−	0.931819i	\(-0.381779\pi\)
							−0.475245	+	0.879853i	\(0.657641\pi\)
\(47\)	97.4073	−	74.0471i	0.302304	−	0.229806i	−0.443014	−	0.896515i	\(-0.646091\pi\)
							0.745318	+	0.666709i	\(0.232298\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.e.b.4.9	✓	420
59.15	even	29	inner	177.4.e.b.133.9	yes	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.e.b.4.9	✓	420	1.1	even	1	trivial
177.4.e.b.133.9	yes	420	59.15	even	29	inner