Embedded newform 153.4.n.a.106.28

• Label: 153.4.n.a.106.28
• 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			106.28
Character	\(\chi\)	\(=\)	153.106
Dual form			153.4.n.a.13.28

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.560645 + 0.323689i) q^{2} +(-3.92528 - 3.40473i) q^{3} +(-3.79045 - 6.56525i) q^{4} +(-14.5119 - 3.88845i) q^{5} +(-1.09862 - 3.17941i) q^{6} +(2.63545 - 0.706168i) q^{7} -10.0867i q^{8} +(3.81567 + 26.7290i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 208 q - 6 q^{3} + 396 q^{4} - 2 q^{5} - 40 q^{6} - 2 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.560645	+	0.323689i	0.198218	+	0.114441i	0.595824	−	0.803115i	\(-0.296826\pi\)
							−0.397606	+	0.917556i	\(0.630159\pi\)
\(3\)	−3.92528	−	3.40473i	−0.755421	−	0.655240i
							\(5\)	−14.5119	−	3.88845i	−1.29798	−	0.347794i	−0.457296	−	0.889314i	\(-0.651182\pi\)
−0.840687	+	0.541521i	\(0.817849\pi\)
\(7\)	2.63545	−	0.706168i	0.142301	−	0.0381295i	−0.186965	−	0.982367i	\(-0.559865\pi\)
							0.329266	+	0.944237i	\(0.393199\pi\)
\(11\)	9.41669	−	2.52319i	0.258113	−	0.0691611i	−0.127442	−	0.991846i	\(-0.540677\pi\)
							0.385555	+	0.922685i	\(0.374010\pi\)
\(13\)	24.6218	+	42.6462i	0.525296	+	0.909840i	0.999566	+	0.0294600i	\(0.00937877\pi\)
							−0.474270	+	0.880380i	\(0.657288\pi\)
\(17\)	2.46884	−	70.0493i	0.0352225	−	0.999379i
							\(19\)			55.6926i			0.672461i	0.941780	+	0.336231i	\(0.109152\pi\)
−0.941780	+	0.336231i	\(0.890848\pi\)
\(23\)	−11.0350	+	41.1830i	−0.100041	+	0.373359i	−0.997735	−	0.0672601i	\(-0.978574\pi\)
							0.897694	+	0.440619i	\(0.145241\pi\)
\(29\)	−19.9401	−	74.4176i	−0.127682	−	0.476517i	0.872239	−	0.489081i	\(-0.162668\pi\)
							−0.999921	+	0.0125631i	\(0.996001\pi\)
\(31\)	−146.957	−	39.3771i	−0.851429	−	0.228140i	−0.193389	−	0.981122i	\(-0.561948\pi\)
							−0.658040	+	0.752983i	\(0.728614\pi\)
\(37\)	−221.663	+	221.663i	−0.984898	+	0.984898i	−0.999888	−	0.0149895i	\(-0.995229\pi\)
							0.0149895	+	0.999888i	\(0.495229\pi\)
\(41\)	−30.7733	+	114.847i	−0.117219	+	0.437467i	−0.999443	−	0.0333621i	\(-0.989379\pi\)
							0.882224	+	0.470829i	\(0.156045\pi\)
\(43\)	258.897	+	149.474i	0.918172	+	0.530107i	0.883051	−	0.469276i	\(-0.155485\pi\)
							0.0351206	+	0.999383i	\(0.488818\pi\)
\(47\)	−121.301	+	210.100i	−0.376460	+	0.652047i	−0.990544	−	0.137193i	\(-0.956192\pi\)
							0.614085	+	0.789240i	\(0.289525\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	153.4.n.a.106.28	yes	208
9.4	even	3	inner	153.4.n.a.4.25	✓	208
17.13	even	4	inner	153.4.n.a.115.25	yes	208
153.13	even	12	inner	153.4.n.a.13.28	yes	208

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
153.4.n.a.4.25	✓	208	9.4	even	3	inner
153.4.n.a.13.28	yes	208	153.13	even	12	inner
153.4.n.a.106.28	yes	208	1.1	even	1	trivial
153.4.n.a.115.25	yes	208	17.13	even	4	inner