Embedded newform 69.7.f.a.37.19

• Label: 69.7.f.a.37.19
• 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			37.19
Character	\(\chi\)	\(=\)	69.37
Dual form			69.7.f.a.28.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(10.1493 - 2.98010i) q^{2} +(-10.2083 - 11.7810i) q^{3} +(40.2866 - 25.8907i) q^{4} +(-60.9323 + 8.76075i) q^{5} +(-138.715 - 89.1467i) q^{6} +(-246.699 - 112.664i) q^{7} +(-111.601 + 128.795i) q^{8} +(-34.5825 + 240.527i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 20 q^{2} - 816 q^{4} + 324 q^{6} + 940 q^{8} - 5832 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	10.1493	−	2.98010i	1.26866	−	0.372512i	0.422949	−	0.906154i	\(-0.360995\pi\)
							0.845711	+	0.533641i	\(0.179177\pi\)
\(3\)	−10.2083	−	11.7810i	−0.378084	−	0.436332i
							\(5\)	−60.9323	+	8.76075i	−0.487459	+	0.0700860i	−0.381662	−	0.924302i	\(-0.624648\pi\)
−0.105796	+	0.994388i	\(0.533739\pi\)
\(7\)	−246.699	−	112.664i	−0.719240	−	0.328466i	0.0219521	−	0.999759i	\(-0.493012\pi\)
							−0.741192	+	0.671293i	\(0.765739\pi\)
\(11\)	−492.943	+	1678.81i	−0.370356	+	1.26132i	0.537939	+	0.842984i	\(0.319203\pi\)
							−0.908294	+	0.418332i	\(0.862615\pi\)
\(13\)	−236.378	−	517.595i	−0.107591	−	0.235592i	0.848177	−	0.529713i	\(-0.177700\pi\)
							−0.955768	+	0.294121i	\(0.904973\pi\)
\(17\)	343.519	−	534.526i	0.0699204	−	0.108798i	−0.804533	−	0.593908i	\(-0.797584\pi\)
							0.874453	+	0.485110i	\(0.161221\pi\)
\(19\)	−5927.08	−	9222.72i	−0.864132	−	1.34462i	−0.937743	−	0.347329i	\(-0.887089\pi\)
							0.0736112	−	0.997287i	\(-0.476548\pi\)
\(23\)	1014.84	+	12124.6i	0.0834094	+	0.996515i
							\(29\)	−23330.0	−	14993.3i	−0.956580	−	0.614757i	−0.0335304	−	0.999438i	\(-0.510675\pi\)
−0.923050	+	0.384681i	\(0.874311\pi\)
\(31\)	11698.9	−	13501.3i	0.392700	−	0.453200i	−0.524628	−	0.851331i	\(-0.675796\pi\)
							0.917328	+	0.398131i	\(0.130341\pi\)
\(37\)	71976.5	+	10348.7i	1.42097	+	0.204305i	0.809574	−	0.587018i	\(-0.199698\pi\)
							0.611399	+	0.791323i	\(0.290607\pi\)
\(41\)	−13708.3	−	95343.5i	−0.198899	−	1.38337i	−0.807486	−	0.589886i	\(-0.799173\pi\)
							0.608587	−	0.793487i	\(-0.291737\pi\)
\(43\)	−56997.1	+	49388.3i	−0.716881	+	0.621181i	−0.934961	−	0.354750i	\(-0.884566\pi\)
							0.218080	+	0.975931i	\(0.430021\pi\)
\(47\)	−29912.6			−0.288112			−0.144056	−	0.989570i	\(-0.546015\pi\)
							−0.144056	+	0.989570i	\(0.546015\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.7.f.a.37.19	yes	240
23.5	odd	22	inner	69.7.f.a.28.19	✓	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.7.f.a.28.19	✓	240	23.5	odd	22	inner
69.7.f.a.37.19	yes	240	1.1	even	1	trivial