Embedded newform 69.7.f.a.10.23

• Label: 69.7.f.a.10.23
• Level: $69$
• Weight: $7$
• Character: 69.10
• 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			10.23
Character	\(\chi\)	\(=\)	69.10
Dual form			69.7.f.a.7.23

$q$-expansion

\(f(q)\)	\(=\)	\(q+(8.99478 + 10.3805i) q^{2} +(-13.1138 - 8.42776i) q^{3} +(-17.7412 + 123.392i) q^{4} +(191.587 + 87.4948i) q^{5} +(-30.4716 - 211.934i) q^{6} +(-4.97165 + 16.9319i) q^{7} +(-700.939 + 450.466i) q^{8} +(100.946 + 221.041i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 20 q^{2} - 816 q^{4} + 324 q^{6} + 940 q^{8} - 5832 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(e\left(\frac{3}{22}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	8.99478	+	10.3805i	1.12435	+	1.29757i	0.949779	+	0.312921i	\(0.101307\pi\)
							0.174568	+	0.984645i	\(0.444147\pi\)
\(3\)	−13.1138	−	8.42776i	−0.485698	−	0.312139i
							\(5\)	191.587	+	87.4948i	1.53269	+	0.699958i	0.990142	−	0.140064i	\(-0.0447308\pi\)
0.542552	+	0.840022i	\(0.317458\pi\)
\(7\)	−4.97165	+	16.9319i	−0.0144946	+	0.0493641i	−0.966423	−	0.256957i	\(-0.917280\pi\)
							0.951928	+	0.306321i	\(0.0990982\pi\)
\(11\)	771.276	+	668.315i	0.579471	+	0.502115i	0.894560	−	0.446947i	\(-0.147489\pi\)
							−0.315089	+	0.949062i	\(0.602034\pi\)
\(13\)	−497.903	+	146.198i	−0.226629	+	0.0665442i	−0.393075	−	0.919506i	\(-0.628589\pi\)
							0.166447	+	0.986050i	\(0.446771\pi\)
\(17\)	−7069.94	+	1016.50i	−1.43903	+	0.206901i	−0.817246	−	0.576289i	\(-0.804500\pi\)
							−0.621782	+	0.783190i	\(0.713591\pi\)
\(19\)	2117.98	+	304.520i	0.308789	+	0.0443971i	0.294967	−	0.955507i	\(-0.404691\pi\)
							0.0138216	+	0.999904i	\(0.495600\pi\)
\(23\)	−10461.9	−	6211.64i	−0.859859	−	0.510532i
							\(29\)	−1621.93	−	11280.8i	−0.0665026	−	0.462536i	−0.995676	−	0.0928922i	\(-0.970389\pi\)
0.929174	−	0.369644i	\(-0.120520\pi\)
\(31\)	−8352.10	+	5367.57i	−0.280357	+	0.180174i	−0.673261	−	0.739405i	\(-0.735107\pi\)
							0.392904	+	0.919579i	\(0.371470\pi\)
\(37\)	77609.2	−	35442.9i	1.53217	−	0.699721i	0.542106	−	0.840310i	\(-0.317627\pi\)
							0.990068	+	0.140589i	\(0.0448998\pi\)
\(41\)	7177.46	−	15716.4i	0.104140	−	0.228036i	−0.850388	−	0.526157i	\(-0.823633\pi\)
							0.954528	+	0.298121i	\(0.0963599\pi\)
\(43\)	32344.4	−	50328.9i	0.406812	−	0.633012i	−0.576037	−	0.817424i	\(-0.695402\pi\)
							0.982849	+	0.184412i	\(0.0590379\pi\)
\(47\)	148461.			1.42994			0.714972	−	0.699153i	\(-0.246440\pi\)
							0.714972	+	0.699153i	\(0.246440\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.7.f.a.10.23	yes	240
23.7	odd	22	inner	69.7.f.a.7.23	✓	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.7.f.a.7.23	✓	240	23.7	odd	22	inner
69.7.f.a.10.23	yes	240	1.1	even	1	trivial