Embedded newform 153.4.n.a.106.41

• Label: 153.4.n.a.106.41
• Level: $153$
• Weight: $4$
• Character: 153.106
• Analytic conductor: $9.027$
• Analytic rank: $0$
• Dimension: $208$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 153 = 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	153.n (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.02729223088\)
Analytic rank:	\(0\)
Dimension:	\(208\)
Relative dimension:	\(52\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			106.41
Character	\(\chi\)	\(=\)	153.106
Dual form			153.4.n.a.13.41

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.00334 + 1.73398i) q^{2} +(-5.19611 - 0.0205841i) q^{3} +(2.01337 + 3.48726i) q^{4} +(-6.93548 - 1.85836i) q^{5} +(-15.5700 - 9.07177i) q^{6} +(16.2216 - 4.34656i) q^{7} -13.7791i q^{8} +(26.9992 + 0.213915i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 208 q - 6 q^{3} + 396 q^{4} - 2 q^{5} - 40 q^{6} - 2 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/153\mathbb{Z}\right)^\times\).

\(n\)	\(37\)	\(137\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.00334	+	1.73398i	1.06184	+	0.613054i	0.925941	−	0.377668i	\(-0.123274\pi\)
							0.135900	+	0.990723i	\(0.456607\pi\)
\(3\)	−5.19611	−	0.0205841i	−0.999992	−	0.00396142i
							\(5\)	−6.93548	−	1.85836i	−0.620328	−	0.166217i	−0.0650514	−	0.997882i	\(-0.520721\pi\)
−0.555277	+	0.831665i	\(0.687388\pi\)
\(7\)	16.2216	−	4.34656i	0.875883	−	0.234692i	0.207253	−	0.978287i	\(-0.433548\pi\)
							0.668630	+	0.743595i	\(0.266881\pi\)
\(11\)	57.9859	−	15.5373i	1.58940	−	0.425878i	0.647580	−	0.761997i	\(-0.275781\pi\)
							0.941820	+	0.336119i	\(0.109114\pi\)
\(13\)	8.10113	+	14.0316i	0.172835	+	0.299358i	0.939410	−	0.342796i	\(-0.111374\pi\)
							−0.766575	+	0.642155i	\(0.778041\pi\)
\(17\)	−8.20945	+	69.6104i	−0.117123	+	0.993117i
							\(19\)		−	89.3143i		−	1.07843i	−0.842169	−	0.539213i	\(-0.818722\pi\)
0.842169	−	0.539213i	\(-0.181278\pi\)
\(23\)	15.6110	−	58.2611i	0.141527	−	0.528186i	−0.858358	−	0.513050i	\(-0.828515\pi\)
							0.999885	−	0.0151355i	\(-0.00481796\pi\)
\(29\)	−51.1701	−	190.969i	−0.327657	−	1.22283i	−0.911614	−	0.411047i	\(-0.865163\pi\)
							0.583957	−	0.811784i	\(-0.301503\pi\)
\(31\)	209.725	+	56.1957i	1.21509	+	0.325582i	0.808756	−	0.588144i	\(-0.200141\pi\)
							0.406332	+	0.913726i	\(0.366808\pi\)
\(37\)	149.730	−	149.730i	0.665285	−	0.665285i	−0.291336	−	0.956621i	\(-0.594100\pi\)
							0.956621	+	0.291336i	\(0.0940997\pi\)
\(41\)	11.0690	−	41.3101i	0.0421631	−	0.157355i	−0.941635	−	0.336636i	\(-0.890711\pi\)
							0.983798	+	0.179282i	\(0.0573773\pi\)
\(43\)	148.680	+	85.8406i	0.527291	+	0.304432i	0.739913	−	0.672703i	\(-0.234867\pi\)
							−0.212621	+	0.977135i	\(0.568200\pi\)
\(47\)	−241.458	+	418.217i	−0.749367	+	1.29794i	0.198760	+	0.980048i	\(0.436309\pi\)
							−0.948126	+	0.317893i	\(0.897025\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	153.4.n.a.106.41	yes	208
9.4	even	3	inner	153.4.n.a.4.12	✓	208
17.13	even	4	inner	153.4.n.a.115.12	yes	208
153.13	even	12	inner	153.4.n.a.13.41	yes	208

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
153.4.n.a.4.12	✓	208	9.4	even	3	inner
153.4.n.a.13.41	yes	208	153.13	even	12	inner
153.4.n.a.106.41	yes	208	1.1	even	1	trivial
153.4.n.a.115.12	yes	208	17.13	even	4	inner