Embedded newform 69.7.f.a.10.3

• Label: 69.7.f.a.10.3
• 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			10.3
Character	\(\chi\)	\(=\)	69.10
Dual form			69.7.f.a.7.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.55085 - 8.71415i) q^{2} +(-13.1138 - 8.42776i) q^{3} +(-9.81285 + 68.2498i) q^{4} +(-14.6117 - 6.67294i) q^{5} +(25.5800 + 177.913i) q^{6} +(9.29392 - 31.6522i) q^{7} +(48.0315 - 30.8680i) q^{8} +(100.946 + 221.041i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 20 q^{2} - 816 q^{4} + 324 q^{6} + 940 q^{8} - 5832 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.55085	−	8.71415i	−0.943856	−	1.08927i	−0.995885	−	0.0906227i	\(-0.971114\pi\)
							0.0520290	−	0.998646i	\(-0.483431\pi\)
\(3\)	−13.1138	−	8.42776i	−0.485698	−	0.312139i
							\(5\)	−14.6117	−	6.67294i	−0.116894	−	0.0533835i	0.356110	−	0.934444i	\(-0.384103\pi\)
−0.473003	+	0.881061i	\(0.656830\pi\)
\(7\)	9.29392	−	31.6522i	0.0270960	−	0.0922804i	−0.944835	−	0.327547i	\(-0.893778\pi\)
							0.971931	+	0.235266i	\(0.0755963\pi\)
\(11\)	1471.87	+	1275.38i	1.10584	+	0.958215i	0.999348	−	0.0360989i	\(-0.0114931\pi\)
							0.106491	+	0.994314i	\(0.466039\pi\)
\(13\)	1286.50	−	377.752i	0.585573	−	0.171940i	0.0244913	−	0.999700i	\(-0.492203\pi\)
							0.561082	+	0.827760i	\(0.310385\pi\)
\(17\)	7509.14	−	1079.65i	1.52842	−	0.219754i	0.673765	−	0.738946i	\(-0.264676\pi\)
							0.854658	+	0.519192i	\(0.173767\pi\)
\(19\)	−12754.2	−	1833.77i	−1.85948	−	0.267353i	−0.880912	−	0.473279i	\(-0.843070\pi\)
							−0.978567	+	0.205927i	\(0.933979\pi\)
\(23\)	−199.935	+	12165.4i	−0.0164326	+	0.999865i
							\(29\)	−293.494	−	2041.29i	−0.0120339	−	0.0836973i	0.982919	−	0.184036i	\(-0.0589164\pi\)
−0.994953	+	0.100339i	\(0.968007\pi\)
\(31\)	3043.92	−	1956.21i	0.102176	−	0.0656645i	−0.488555	−	0.872533i	\(-0.662476\pi\)
							0.590731	+	0.806869i	\(0.298839\pi\)
\(37\)	13608.7	−	6214.90i	0.268666	−	0.122696i	−0.276527	−	0.961006i	\(-0.589184\pi\)
							0.545193	+	0.838311i	\(0.316456\pi\)
\(41\)	27206.4	−	59573.7i	0.394748	−	0.864377i	−0.603028	−	0.797720i	\(-0.706039\pi\)
							0.997776	−	0.0666571i	\(-0.0212334\pi\)
\(43\)	54087.3	−	84161.5i	0.680283	−	1.05854i	−0.313754	−	0.949504i	\(-0.601587\pi\)
							0.994038	−	0.109037i	\(-0.0347768\pi\)
\(47\)	92871.2			0.894514			0.447257	−	0.894405i	\(-0.352401\pi\)
							0.447257	+	0.894405i	\(0.352401\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.7.f.a.10.3	yes	240
23.7	odd	22	inner	69.7.f.a.7.3	✓	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.7.f.a.7.3	✓	240	23.7	odd	22	inner
69.7.f.a.10.3	yes	240	1.1	even	1	trivial