Embedded newform 153.4.n.a.106.7

• Label: 153.4.n.a.106.7
• 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			106.7
Character	\(\chi\)	\(=\)	153.106
Dual form			153.4.n.a.13.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.94348 - 2.27677i) q^{2} +(-4.87688 + 1.79332i) q^{3} +(6.36736 + 11.0286i) q^{4} +(18.6477 + 4.99664i) q^{5} +(23.3149 + 4.03160i) q^{6} +(-3.38429 + 0.906817i) q^{7} -21.5598i q^{8} +(20.5680 - 17.4917i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 208 q - 6 q^{3} + 396 q^{4} - 2 q^{5} - 40 q^{6} - 2 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.94348	−	2.27677i	−1.39423	−	0.804960i	−0.400451	−	0.916318i	\(-0.631146\pi\)
							−0.993780	+	0.111358i	\(0.964480\pi\)
\(3\)	−4.87688	+	1.79332i	−0.938557	+	0.345125i
							\(5\)	18.6477	+	4.99664i	1.66790	+	0.446913i	0.964544	−	0.263921i	\(-0.0850160\pi\)
0.703359	+	0.710835i	\(0.251683\pi\)
\(7\)	−3.38429	+	0.906817i	−0.182734	+	0.0489635i	−0.349026	−	0.937113i	\(-0.613487\pi\)
							0.166291	+	0.986077i	\(0.446821\pi\)
\(11\)	−44.8551	+	12.0189i	−1.22948	+	0.329439i	−0.814378	−	0.580334i	\(-0.802922\pi\)
							−0.415105	+	0.909773i	\(0.636255\pi\)
\(13\)	2.05215	+	3.55443i	0.0437819	+	0.0758324i	0.887086	−	0.461604i	\(-0.152726\pi\)
							−0.843304	+	0.537437i	\(0.819393\pi\)
\(17\)	37.5191	+	59.2057i	0.535278	+	0.844676i
							\(19\)		−	84.9624i		−	1.02588i	−0.858425	−	0.512940i	\(-0.828556\pi\)
0.858425	−	0.512940i	\(-0.171444\pi\)
\(23\)	−15.5199	+	57.9212i	−0.140701	+	0.525105i	0.859208	+	0.511627i	\(0.170957\pi\)
							−0.999909	+	0.0134779i	\(0.995710\pi\)
\(29\)	24.2746	+	90.5940i	0.155437	+	0.580099i	0.999068	+	0.0431749i	\(0.0137473\pi\)
							−0.843630	+	0.536924i	\(0.819586\pi\)
\(31\)	−174.742	−	46.8220i	−1.01241	−	0.271273i	−0.285772	−	0.958298i	\(-0.592250\pi\)
							−0.726635	+	0.687024i	\(0.758917\pi\)
\(37\)	−13.9602	+	13.9602i	−0.0620280	+	0.0620280i	−0.737440	−	0.675412i	\(-0.763966\pi\)
							0.675412	+	0.737440i	\(0.263966\pi\)
\(41\)	−117.273	+	437.668i	−0.446705	+	1.66713i	0.264689	+	0.964334i	\(0.414731\pi\)
							−0.711394	+	0.702793i	\(0.751936\pi\)
\(43\)	426.882	+	246.461i	1.51393	+	0.874067i	0.999867	+	0.0163141i	\(0.00519317\pi\)
							0.514062	+	0.857753i	\(0.328140\pi\)
\(47\)	−1.06175	+	1.83900i	−0.00329515	+	0.00570736i	−0.867668	−	0.497144i	\(-0.834382\pi\)
							0.864373	+	0.502851i	\(0.167716\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	153.4.n.a.106.7	yes	208
9.4	even	3	inner	153.4.n.a.4.46	✓	208
17.13	even	4	inner	153.4.n.a.115.46	yes	208
153.13	even	12	inner	153.4.n.a.13.7	yes	208

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
153.4.n.a.4.46	✓	208	9.4	even	3	inner
153.4.n.a.13.7	yes	208	153.13	even	12	inner
153.4.n.a.106.7	yes	208	1.1	even	1	trivial
153.4.n.a.115.46	yes	208	17.13	even	4	inner