Embedded newform 69.8.g.a.5.6

• Label: 69.8.g.a.5.6
• Level: $69$
• Weight: $8$
• Character: 69.5
• Analytic conductor: $21.555$
• Analytic rank: $0$
• Dimension: $540$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	69.g (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(21.5545667584\)
Analytic rank:	\(0\)
Dimension:	\(540\)
Relative dimension:	\(54\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			5.6
Character	\(\chi\)	\(=\)	69.5
Dual form			69.8.g.a.14.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.35232 + 18.2283i) q^{2} +(37.5882 - 27.8231i) q^{3} +(-195.945 - 125.926i) q^{4} +(46.1891 - 321.252i) q^{5} +(305.985 + 834.089i) q^{6} +(-583.063 + 266.276i) q^{7} +(1506.40 - 1305.30i) q^{8} +(638.749 - 2091.64i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 540 q - 35 q^{3} + 3362 q^{4} - 1280 q^{6} - 22 q^{7} - 659 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.35232	+	18.2283i	−0.473083	+	1.61117i	0.284666	+	0.958627i	\(0.408117\pi\)
							−0.757749	+	0.652546i	\(0.773701\pi\)
\(3\)	37.5882	−	27.8231i	0.803762	−	0.594951i
							\(5\)	46.1891	−	321.252i	0.165251	−	1.14935i	−0.723288	−	0.690546i	\(-0.757370\pi\)
0.888539	−	0.458801i	\(-0.151721\pi\)
\(7\)	−583.063	+	266.276i	−0.642499	+	0.293419i	−0.709896	−	0.704307i	\(-0.751258\pi\)
							0.0673969	+	0.997726i	\(0.478531\pi\)
\(11\)	1844.09	−	541.475i	0.417743	−	0.122660i	−0.0661048	−	0.997813i	\(-0.521057\pi\)
							0.483847	+	0.875152i	\(0.339239\pi\)
\(13\)	−5084.76	+	11134.1i	−0.641903	+	1.40557i	0.256564	+	0.966527i	\(0.417410\pi\)
							−0.898466	+	0.439043i	\(0.855318\pi\)
\(17\)	−13782.5	+	8857.48i	−0.680388	+	0.437259i	−0.834657	−	0.550770i	\(-0.814334\pi\)
							0.154269	+	0.988029i	\(0.450698\pi\)
\(19\)	−30968.1	+	48187.3i	−1.03580	+	1.61174i	−0.276488	+	0.961017i	\(0.589171\pi\)
							−0.759315	+	0.650724i	\(0.774466\pi\)
\(23\)	−17722.4	+	55594.4i	−0.303722	+	0.952761i
							\(29\)	94232.5	+	146629.i	0.717477	+	1.11642i	0.988112	+	0.153734i	\(0.0491299\pi\)
−0.270635	+	0.962682i	\(0.587234\pi\)
\(31\)	−155038.	−	178924.i	−0.934701	−	1.07870i	−0.996744	−	0.0806289i	\(-0.974307\pi\)
							0.0620432	−	0.998073i	\(-0.480238\pi\)
\(37\)	41634.5	−	5986.13i	0.135128	−	0.0194285i	−0.0744187	−	0.997227i	\(-0.523710\pi\)
							0.209547	+	0.977799i	\(0.432801\pi\)
\(41\)	102503.	+	14737.7i	0.232270	+	0.0333954i	0.257467	−	0.966287i	\(-0.417112\pi\)
							−0.0251969	+	0.999683i	\(0.508021\pi\)
\(43\)	109441.	+	94831.2i	0.209914	+	0.181891i	0.753472	−	0.657480i	\(-0.228378\pi\)
							−0.543558	+	0.839372i	\(0.682923\pi\)
\(47\)		−	53126.2i		−	0.0746391i	−0.999303	−	0.0373195i	\(-0.988118\pi\)
							0.999303	−	0.0373195i	\(-0.0118819\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.8.g.a.5.6	✓	540
3.2	odd	2	inner	69.8.g.a.5.49	yes	540
23.14	odd	22	inner	69.8.g.a.14.49	yes	540
69.14	even	22	inner	69.8.g.a.14.6	yes	540

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.8.g.a.5.6	✓	540	1.1	even	1	trivial
69.8.g.a.5.49	yes	540	3.2	odd	2	inner
69.8.g.a.14.6	yes	540	69.14	even	22	inner
69.8.g.a.14.49	yes	540	23.14	odd	22	inner