Embedded newform 153.4.n.a.106.21

• Label: 153.4.n.a.106.21
• 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			106.21
Character	\(\chi\)	\(=\)	153.106
Dual form			153.4.n.a.13.21

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.40561 - 0.811526i) q^{2} +(-2.42173 + 4.59731i) q^{3} +(-2.68285 - 4.64683i) q^{4} +(7.52225 + 2.01558i) q^{5} +(7.13483 - 4.49670i) q^{6} +(-6.20904 + 1.66371i) q^{7} +21.6932i q^{8} +(-15.2704 - 22.2669i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 208 q - 6 q^{3} + 396 q^{4} - 2 q^{5} - 40 q^{6} - 2 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.40561	−	0.811526i	−0.496956	−	0.286918i	0.230499	−	0.973072i	\(-0.425964\pi\)
							−0.727456	+	0.686155i	\(0.759297\pi\)
\(3\)	−2.42173	+	4.59731i	−0.466062	+	0.884752i
							\(5\)	7.52225	+	2.01558i	0.672810	+	0.180279i	0.579021	−	0.815313i	\(-0.303435\pi\)
0.0937900	+	0.995592i	\(0.470102\pi\)
\(7\)	−6.20904	+	1.66371i	−0.335257	+	0.0898317i	−0.422520	−	0.906354i	\(-0.638854\pi\)
							0.0872636	+	0.996185i	\(0.472188\pi\)
\(11\)	−11.2122	+	3.00429i	−0.307327	+	0.0823479i	−0.409186	−	0.912451i	\(-0.634187\pi\)
							0.101860	+	0.994799i	\(0.467521\pi\)
\(13\)	10.5825	+	18.3295i	0.225775	+	0.391053i	0.956552	−	0.291563i	\(-0.0941754\pi\)
							−0.730777	+	0.682616i	\(0.760842\pi\)
\(17\)	38.1959	−	58.7714i	0.544933	−	0.838480i
							\(19\)		−	153.582i		−	1.85443i	−0.374532	−	0.927214i	\(-0.622196\pi\)
0.374532	−	0.927214i	\(-0.377804\pi\)
\(23\)	18.3142	−	68.3496i	0.166034	−	0.619647i	−0.831872	−	0.554967i	\(-0.812731\pi\)
							0.997906	−	0.0646797i	\(-0.0206026\pi\)
\(29\)	−76.9266	−	287.094i	−0.492583	−	1.83835i	−0.543166	−	0.839625i	\(-0.682775\pi\)
							0.0505829	−	0.998720i	\(-0.483892\pi\)
\(31\)	220.845	+	59.1753i	1.27952	+	0.342845i	0.833669	−	0.552265i	\(-0.186236\pi\)
							0.445846	+	0.895110i	\(0.352903\pi\)
\(37\)	−235.362	+	235.362i	−1.04577	+	1.04577i	−0.0468640	+	0.998901i	\(0.514923\pi\)
							−0.998901	+	0.0468640i	\(0.985077\pi\)
\(41\)	124.752	−	465.582i	0.475196	−	1.77346i	−0.145449	−	0.989366i	\(-0.546463\pi\)
							0.620645	−	0.784091i	\(-0.286871\pi\)
\(43\)	−34.8778	−	20.1367i	−0.123693	−	0.0714144i	0.436877	−	0.899521i	\(-0.356085\pi\)
							−0.560570	+	0.828107i	\(0.689418\pi\)
\(47\)	−72.5809	+	125.714i	−0.225256	+	0.390154i	−0.956396	−	0.292073i	\(-0.905655\pi\)
							0.731140	+	0.682227i	\(0.238988\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	153.4.n.a.106.21	yes	208
9.4	even	3	inner	153.4.n.a.4.32	✓	208
17.13	even	4	inner	153.4.n.a.115.32	yes	208
153.13	even	12	inner	153.4.n.a.13.21	yes	208

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
153.4.n.a.4.32	✓	208	9.4	even	3	inner
153.4.n.a.13.21	yes	208	153.13	even	12	inner
153.4.n.a.106.21	yes	208	1.1	even	1	trivial
153.4.n.a.115.32	yes	208	17.13	even	4	inner