Embedded newform 153.4.n.a.115.46

• Label: 153.4.n.a.115.46
• Level: $153$
• Weight: $4$
• Character: 153.115
• 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			115.46
Character	\(\chi\)	\(=\)	153.115
Dual form			153.4.n.a.4.46

$q$-expansion

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

Character values

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

\(n\)	\(37\)	\(137\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{2}{3}\right)\)

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	0.993780	−	0.111358i	\(-0.0355202\pi\)
							0.400451	+	0.916318i	\(0.368854\pi\)
\(3\)	−1.79332	−	4.87688i	−0.345125	−	0.938557i
							\(5\)	−4.99664	+	18.6477i	−0.446913	+	1.66790i	0.263921	+	0.964544i	\(0.414984\pi\)
−0.710835	+	0.703359i	\(0.751683\pi\)
\(7\)	0.906817	+	3.38429i	0.0489635	+	0.182734i	0.986077	−	0.166291i	\(-0.0531792\pi\)
							−0.937113	+	0.349026i	\(0.886513\pi\)
\(11\)	12.0189	+	44.8551i	0.329439	+	1.22948i	0.909773	+	0.415105i	\(0.136255\pi\)
							−0.580334	+	0.814378i	\(0.697078\pi\)
\(13\)	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	\(0.171444\pi\)
−0.858425	+	0.512940i	\(0.828556\pi\)
\(23\)	57.9212	+	15.5199i	0.525105	+	0.140701i	0.511627	−	0.859208i	\(-0.329043\pi\)
							0.0134779	+	0.999909i	\(0.495710\pi\)
\(29\)	−90.5940	+	24.2746i	−0.580099	+	0.155437i	−0.536924	−	0.843630i	\(-0.680414\pi\)
							−0.0431749	+	0.999068i	\(0.513747\pi\)
\(31\)	46.8220	−	174.742i	0.271273	−	1.01241i	−0.687024	−	0.726635i	\(-0.741083\pi\)
							0.958298	−	0.285772i	\(-0.0922499\pi\)
\(37\)	−13.9602	−	13.9602i	−0.0620280	−	0.0620280i	0.675412	−	0.737440i	\(-0.263966\pi\)
							−0.737440	+	0.675412i	\(0.763966\pi\)
\(41\)	437.668	+	117.273i	1.66713	+	0.446705i	0.964334	−	0.264689i	\(-0.0852692\pi\)
							0.702793	+	0.711394i	\(0.251936\pi\)
\(43\)	−426.882	−	246.461i	−1.51393	−	0.874067i	−0.999867	−	0.0163141i	\(-0.994807\pi\)
							−0.514062	−	0.857753i	\(-0.671860\pi\)
\(47\)	−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.115.46	yes	208
9.4	even	3	inner	153.4.n.a.13.7	yes	208
17.4	even	4	inner	153.4.n.a.106.7	yes	208
153.4	even	12	inner	153.4.n.a.4.46	✓	208

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
153.4.n.a.4.46	✓	208	153.4	even	12	inner
153.4.n.a.13.7	yes	208	9.4	even	3	inner
153.4.n.a.106.7	yes	208	17.4	even	4	inner
153.4.n.a.115.46	yes	208	1.1	even	1	trivial