Embedded newform 69.7.f.a.37.1

• Label: 69.7.f.a.37.1
• Level: $69$
• Weight: $7$
• Character: 69.37
• 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			37.1
Character	\(\chi\)	\(=\)	69.37
Dual form			69.7.f.a.28.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-15.1151 + 4.43818i) q^{2} +(10.2083 + 11.7810i) q^{3} +(154.927 - 99.5656i) q^{4} +(-155.791 + 22.3993i) q^{5} +(-206.585 - 132.764i) q^{6} +(234.606 + 107.141i) q^{7} +(-1239.61 + 1430.59i) q^{8} +(-34.5825 + 240.527i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 20 q^{2} - 816 q^{4} + 324 q^{6} + 940 q^{8} - 5832 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(e\left(\frac{21}{22}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−15.1151	+	4.43818i	−1.88938	+	0.554773i	−0.895395	+	0.445273i	\(0.853107\pi\)
							−0.993987	+	0.109499i	\(0.965075\pi\)
\(3\)	10.2083	+	11.7810i	0.378084	+	0.436332i
							\(5\)	−155.791	+	22.3993i	−1.24633	+	0.179195i	−0.733734	−	0.679437i	\(-0.762224\pi\)
−0.512593	+	0.858632i	\(0.671315\pi\)
\(7\)	234.606	+	107.141i	0.683981	+	0.312364i	0.726928	−	0.686714i	\(-0.240947\pi\)
							−0.0429469	+	0.999077i	\(0.513675\pi\)
\(11\)	259.765	−	884.680i	0.195166	−	0.664673i	−0.802517	−	0.596630i	\(-0.796506\pi\)
							0.997682	−	0.0680435i	\(-0.0216757\pi\)
\(13\)	−1016.91	−	2226.72i	−0.462864	−	1.01353i	−0.986826	−	0.161788i	\(-0.948274\pi\)
							0.523962	−	0.851742i	\(-0.324453\pi\)
\(17\)	2389.72	−	3718.47i	0.486407	−	0.756864i	−0.508126	−	0.861283i	\(-0.669662\pi\)
							0.994533	+	0.104418i	\(0.0332980\pi\)
\(19\)	3920.51	+	6100.43i	0.571586	+	0.889405i	0.999899	−	0.0142253i	\(-0.00452820\pi\)
							−0.428313	+	0.903631i	\(0.640892\pi\)
\(23\)	10695.2	−	5800.76i	0.879032	−	0.476762i
							\(29\)	−1504.05	−	966.595i	−0.0616692	−	0.0396324i	0.509443	−	0.860504i	\(-0.329851\pi\)
−0.571112	+	0.820872i	\(0.693488\pi\)
\(31\)	−27117.5	+	31295.3i	−0.910259	+	1.05050i	0.0882599	+	0.996097i	\(0.471869\pi\)
							−0.998519	+	0.0543980i	\(0.982676\pi\)
\(37\)	47987.2	+	6899.52i	0.947372	+	0.136212i	0.598647	−	0.801013i	\(-0.295705\pi\)
							0.348726	+	0.937225i	\(0.386615\pi\)
\(41\)	12024.0	+	83629.0i	0.174461	+	1.21340i	0.869317	+	0.494255i	\(0.164559\pi\)
							−0.694856	+	0.719149i	\(0.744532\pi\)
\(43\)	−24067.9	+	20854.9i	−0.302714	+	0.262303i	−0.792950	−	0.609287i	\(-0.791456\pi\)
							0.490236	+	0.871589i	\(0.336910\pi\)
\(47\)	120683.			1.16239			0.581197	−	0.813763i	\(-0.302585\pi\)
							0.581197	+	0.813763i	\(0.302585\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.7.f.a.37.1	yes	240
23.5	odd	22	inner	69.7.f.a.28.1	✓	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.7.f.a.28.1	✓	240	23.5	odd	22	inner
69.7.f.a.37.1	yes	240	1.1	even	1	trivial