Embedded newform 69.7.f.a.34.15

• Label: 69.7.f.a.34.15
• Level: $69$
• Weight: $7$
• Character: 69.34
• Analytic conductor: $15.874$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	69.f (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(15.8737317698\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(24\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			34.15
Character	\(\chi\)	\(=\)	69.34
Dual form			69.7.f.a.67.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.28868 - 2.11350i) q^{2} +(2.21847 - 15.4298i) q^{3} +(-20.2381 + 44.3152i) q^{4} +(-61.7592 - 210.333i) q^{5} +(-25.3151 - 55.4323i) q^{6} +(204.316 - 177.041i) q^{7} +(62.7100 + 436.157i) q^{8} +(-233.157 - 68.4610i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 20 q^{2} - 816 q^{4} + 324 q^{6} + 940 q^{8} - 5832 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/69\mathbb{Z}\right)^\times\).

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(e\left(\frac{9}{22}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.28868	−	2.11350i	0.411085	−	0.264188i	−0.318713	−	0.947851i	\(-0.603251\pi\)
							0.729798	+	0.683663i	\(0.239614\pi\)
\(3\)	2.21847	−	15.4298i	0.0821655	−	0.571474i
							\(5\)	−61.7592	−	210.333i	−0.494074	−	1.68266i	−0.708328	−	0.705883i	\(-0.750550\pi\)
0.214254	−	0.976778i	\(-0.431268\pi\)
\(7\)	204.316	−	177.041i	0.595674	−	0.516155i	−0.304026	−	0.952664i	\(-0.598331\pi\)
							0.899700	+	0.436509i	\(0.143785\pi\)
\(11\)	−177.705	+	276.514i	−0.133512	+	0.207749i	−0.901572	−	0.432629i	\(-0.857586\pi\)
							0.768060	+	0.640378i	\(0.221222\pi\)
\(13\)	−1741.71	+	2010.04i	−0.792767	+	0.914902i	−0.997961	−	0.0638192i	\(-0.979672\pi\)
							0.205194	+	0.978721i	\(0.434217\pi\)
\(17\)	−3505.08	+	1600.72i	−0.713430	+	0.325812i	−0.738855	−	0.673864i	\(-0.764633\pi\)
							0.0254254	+	0.999677i	\(0.491906\pi\)
\(19\)	−10760.1	−	4913.95i	−1.56875	−	0.716424i	−0.574009	−	0.818849i	\(-0.694613\pi\)
							−0.994741	+	0.102425i	\(0.967340\pi\)
\(23\)	5794.48	−	10698.6i	0.476246	−	0.879312i
							\(29\)	6411.05	+	14038.2i	0.262866	+	0.575597i	0.994337	−	0.106276i	\(-0.0338928\pi\)
−0.731470	+	0.681873i	\(0.761166\pi\)
\(31\)	2898.22	+	20157.6i	0.0972852	+	0.676634i	0.978851	+	0.204573i	\(0.0655806\pi\)
							−0.881566	+	0.472061i	\(0.843510\pi\)
\(37\)	21658.4	−	73761.7i	0.427583	−	1.45622i	−0.411097	−	0.911592i	\(-0.634854\pi\)
							0.838680	−	0.544624i	\(-0.183328\pi\)
\(41\)	−19150.6	+	5623.11i	−0.277863	+	0.0815878i	−0.417695	−	0.908587i	\(-0.637162\pi\)
							0.139832	+	0.990175i	\(0.455344\pi\)
\(43\)	−76724.2	−	11031.3i	−0.964999	−	0.138746i	−0.358238	−	0.933630i	\(-0.616622\pi\)
							−0.606761	+	0.794884i	\(0.707531\pi\)
\(47\)	54575.6			0.525660			0.262830	−	0.964842i	\(-0.415344\pi\)
							0.262830	+	0.964842i	\(0.415344\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.7.f.a.34.15	✓	240
23.21	odd	22	inner	69.7.f.a.67.15	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.7.f.a.34.15	✓	240	1.1	even	1	trivial
69.7.f.a.67.15	yes	240	23.21	odd	22	inner