Embedded newform 69.5.h.a.2.7

• Label: 69.5.h.a.2.7
• Level: $69$
• Weight: $5$
• Character: 69.2
• Analytic conductor: $7.133$
• Analytic rank: $0$
• Dimension: $300$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	69.h (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.13252745279\)
Analytic rank:	\(0\)
Dimension:	\(300\)
Relative dimension:	\(30\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			2.7
Character	\(\chi\)	\(=\)	69.2
Dual form			69.5.h.a.35.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.52768 + 3.93314i) q^{2} +(7.84620 + 4.40876i) q^{3} +(-2.43382 - 5.32931i) q^{4} +(-7.84851 + 26.7296i) q^{5} +(-37.1729 + 19.7163i) q^{6} +(16.6015 - 19.1591i) q^{7} +(-46.9311 - 6.74767i) q^{8} +(42.1256 + 69.1841i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 300 q - 19 q^{3} + 234 q^{4} - 10 q^{6} - 18 q^{7} - 111 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(e\left(\frac{1}{11}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.52768	+	3.93314i	−0.631919	+	0.983286i	0.366679	+	0.930348i	\(0.380495\pi\)
							−0.998598	+	0.0529378i	\(0.983141\pi\)
\(3\)	7.84620	+	4.40876i	0.871800	+	0.489863i
							\(5\)	−7.84851	+	26.7296i	−0.313941	+	1.06918i	0.639797	+	0.768544i	\(0.279018\pi\)
−0.953738	+	0.300640i	\(0.902800\pi\)
\(7\)	16.6015	−	19.1591i	0.338806	−	0.391003i	−0.560622	−	0.828072i	\(-0.689438\pi\)
							0.899428	+	0.437069i	\(0.143983\pi\)
\(11\)	95.0842	+	147.954i	0.785820	+	1.22276i	0.970770	+	0.240012i	\(0.0771512\pi\)
							−0.184950	+	0.982748i	\(0.559212\pi\)
\(13\)	−71.6340	−	82.6701i	−0.423870	−	0.489172i	0.503142	−	0.864204i	\(-0.332177\pi\)
							−0.927012	+	0.375032i	\(0.877632\pi\)
\(17\)	−270.172	−	123.383i	−0.934851	−	0.426932i	−0.111052	−	0.993815i	\(-0.535422\pi\)
							−0.823799	+	0.566883i	\(0.808149\pi\)
\(19\)	−79.6501	−	174.409i	−0.220637	−	0.483128i	0.766652	−	0.642063i	\(-0.221921\pi\)
							−0.987289	+	0.158935i	\(0.949194\pi\)
\(23\)	−405.879	−	339.269i	−0.767257	−	0.641340i
							\(29\)	1049.67	+	479.367i	1.24812	+	0.569996i	0.926293	−	0.376803i	\(-0.122977\pi\)
0.321824	+	0.946799i	\(0.395704\pi\)
\(31\)	−172.134	+	1197.22i	−0.179120	+	1.24581i	0.679685	+	0.733504i	\(0.262117\pi\)
							−0.858805	+	0.512303i	\(0.828793\pi\)
\(37\)	1756.84	−	515.855i	1.28330	−	0.376812i	0.432185	−	0.901785i	\(-0.357743\pi\)
							0.851119	+	0.524973i	\(0.175925\pi\)
\(41\)	14.6021	−	49.7302i	0.00868655	−	0.0295837i	−0.955036	−	0.296491i	\(-0.904184\pi\)
							0.963722	+	0.266907i	\(0.0860017\pi\)
\(43\)	201.689	+	1402.78i	0.109080	+	0.758670i	0.968789	+	0.247886i	\(0.0797358\pi\)
							−0.859709	+	0.510784i	\(0.829355\pi\)
\(47\)		−	1792.56i		−	0.811482i	−0.913988	−	0.405741i	\(-0.867013\pi\)
							0.913988	−	0.405741i	\(-0.132987\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.5.h.a.2.7	✓	300
3.2	odd	2	inner	69.5.h.a.2.24	yes	300
23.12	even	11	inner	69.5.h.a.35.24	yes	300
69.35	odd	22	inner	69.5.h.a.35.7	yes	300

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.5.h.a.2.7	✓	300	1.1	even	1	trivial
69.5.h.a.2.24	yes	300	3.2	odd	2	inner
69.5.h.a.35.7	yes	300	69.35	odd	22	inner
69.5.h.a.35.24	yes	300	23.12	even	11	inner