Embedded newform 69.7.f.a.7.7

• Label: 69.7.f.a.7.7
• Level: $69$
• Weight: $7$
• Character: 69.7
• 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			7.7
Character	\(\chi\)	\(=\)	69.7
Dual form			69.7.f.a.10.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.49904 + 5.19217i) q^{2} +(13.1138 - 8.42776i) q^{3} +(2.39089 + 16.6290i) q^{4} +(160.721 - 73.3986i) q^{5} +(-15.2414 + 106.006i) q^{6} +(-169.038 - 575.691i) q^{7} +(-466.992 - 300.117i) q^{8} +(100.946 - 221.041i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 20 q^{2} - 816 q^{4} + 324 q^{6} + 940 q^{8} - 5832 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(e\left(\frac{19}{22}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.49904	+	5.19217i	−0.562380	+	0.649021i	−0.963722	−	0.266907i	\(-0.913998\pi\)
							0.401342	+	0.915928i	\(0.368544\pi\)
\(3\)	13.1138	−	8.42776i	0.485698	−	0.312139i
							\(5\)	160.721	−	73.3986i	1.28576	−	0.587189i	0.348991	−	0.937126i	\(-0.386524\pi\)
0.936773	+	0.349937i	\(0.113797\pi\)
\(7\)	−169.038	−	575.691i	−0.492823	−	1.67840i	−0.711567	−	0.702618i	\(-0.752014\pi\)
							0.218745	−	0.975782i	\(-0.429804\pi\)
\(11\)	−1019.39	+	883.307i	−0.765883	+	0.663642i	−0.947510	−	0.319726i	\(-0.896409\pi\)
							0.181627	+	0.983368i	\(0.441864\pi\)
\(13\)	−1349.88	−	396.361i	−0.614420	−	0.180410i	−0.0403088	−	0.999187i	\(-0.512834\pi\)
							−0.574111	+	0.818777i	\(0.694652\pi\)
\(17\)	−5658.08	−	813.510i	−1.15166	−	0.165583i	−0.460073	−	0.887881i	\(-0.652177\pi\)
							−0.691582	+	0.722298i	\(0.743086\pi\)
\(19\)	−9413.27	+	1353.42i	−1.37240	+	0.197321i	−0.788764	−	0.614696i	\(-0.789279\pi\)
							−0.583633	+	0.812017i	\(0.698370\pi\)
\(23\)	2586.66	−	11888.9i	0.212597	−	0.977140i
							\(29\)	1462.05	−	10168.8i	0.0599472	−	0.416942i	−0.937645	−	0.347594i	\(-0.886999\pi\)
0.997593	−	0.0693484i	\(-0.0220920\pi\)
\(31\)	13883.0	+	8922.09i	0.466015	+	0.299489i	0.752496	−	0.658596i	\(-0.228849\pi\)
							−0.286482	+	0.958086i	\(0.592486\pi\)
\(37\)	25882.9	+	11820.3i	0.510984	+	0.233358i	0.654184	−	0.756335i	\(-0.273012\pi\)
							−0.143200	+	0.989694i	\(0.545739\pi\)
\(41\)	−35813.4	−	78420.4i	−0.519630	−	1.13783i	−0.969580	−	0.244777i	\(-0.921285\pi\)
							0.449950	−	0.893054i	\(-0.351442\pi\)
\(43\)	63252.2	+	98422.4i	0.795555	+	1.23791i	0.967517	+	0.252804i	\(0.0813529\pi\)
							−0.171962	+	0.985104i	\(0.555011\pi\)
\(47\)	160134.			1.54238			0.771188	−	0.636607i	\(-0.219663\pi\)
							0.771188	+	0.636607i	\(0.219663\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.7.f.a.7.7	✓	240
23.10	odd	22	inner	69.7.f.a.10.7	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.7.f.a.7.7	✓	240	1.1	even	1	trivial
69.7.f.a.10.7	yes	240	23.10	odd	22	inner