Embedded newform 153.4.n.a.106.9

• Label: 153.4.n.a.106.9
• 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			106.9
Character	\(\chi\)	\(=\)	153.106
Dual form			153.4.n.a.13.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.64692 - 2.10555i) q^{2} +(5.16412 - 0.576070i) q^{3} +(4.86669 + 8.42935i) q^{4} +(9.48034 + 2.54025i) q^{5} +(-20.0461 - 8.77243i) q^{6} +(10.8786 - 2.91492i) q^{7} -7.29941i q^{8} +(26.3363 - 5.94979i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 208 q - 6 q^{3} + 396 q^{4} - 2 q^{5} - 40 q^{6} - 2 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.64692	−	2.10555i	−1.28938	−	0.744425i	−0.310837	−	0.950463i	\(-0.600609\pi\)
							−0.978544	+	0.206039i	\(0.933943\pi\)
\(3\)	5.16412	−	0.576070i	0.993835	−	0.110865i
							\(5\)	9.48034	+	2.54025i	0.847948	+	0.227207i	0.656528	−	0.754301i	\(-0.272024\pi\)
0.191419	+	0.981508i	\(0.438691\pi\)
\(7\)	10.8786	−	2.91492i	0.587392	−	0.157391i	0.0471308	−	0.998889i	\(-0.484992\pi\)
							0.540261	+	0.841498i	\(0.318326\pi\)
\(11\)	−23.0072	+	6.16475i	−0.630629	+	0.168977i	−0.559955	−	0.828523i	\(-0.689182\pi\)
							−0.0706740	+	0.997499i	\(0.522515\pi\)
\(13\)	45.6557	+	79.0779i	0.974046	+	1.68710i	0.683047	+	0.730374i	\(0.260654\pi\)
							0.290999	+	0.956723i	\(0.406012\pi\)
\(17\)	−36.6837	+	59.7269i	−0.523359	+	0.852112i
							\(19\)			78.5169i			0.948053i	0.880511	+	0.474027i	\(0.157200\pi\)
−0.880511	+	0.474027i	\(0.842800\pi\)
\(23\)	−0.702186	+	2.62059i	−0.00636591	+	0.0237579i	−0.969036	−	0.246921i	\(-0.920581\pi\)
							0.962670	+	0.270678i	\(0.0872480\pi\)
\(29\)	−60.2683	−	224.924i	−0.385915	−	1.44025i	−0.836719	−	0.547632i	\(-0.815529\pi\)
							0.450804	−	0.892623i	\(-0.351137\pi\)
\(31\)	294.379	+	78.8787i	1.70555	+	0.457001i	0.974327	−	0.225137i	\(-0.0722830\pi\)
							0.731224	+	0.682138i	\(0.238950\pi\)
\(37\)	213.099	−	213.099i	0.946845	−	0.946845i	−0.0518122	−	0.998657i	\(-0.516500\pi\)
							0.998657	+	0.0518122i	\(0.0164997\pi\)
\(41\)	96.1012	−	358.655i	0.366061	−	1.36616i	−0.499917	−	0.866074i	\(-0.666636\pi\)
							0.865977	−	0.500083i	\(-0.166697\pi\)
\(43\)	−141.117	−	81.4737i	−0.500467	−	0.288945i	0.228439	−	0.973558i	\(-0.426638\pi\)
							−0.728906	+	0.684613i	\(0.759971\pi\)
\(47\)	34.0546	−	58.9843i	0.105689	−	0.183058i	−0.808331	−	0.588729i	\(-0.799629\pi\)
							0.914019	+	0.405670i	\(0.132962\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	153.4.n.a.106.9	yes	208
9.4	even	3	inner	153.4.n.a.4.44	✓	208
17.13	even	4	inner	153.4.n.a.115.44	yes	208
153.13	even	12	inner	153.4.n.a.13.9	yes	208

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
153.4.n.a.4.44	✓	208	9.4	even	3	inner
153.4.n.a.13.9	yes	208	153.13	even	12	inner
153.4.n.a.106.9	yes	208	1.1	even	1	trivial
153.4.n.a.115.44	yes	208	17.13	even	4	inner