Embedded newform 69.8.e.a.58.14

• Label: 69.8.e.a.58.14
• Level: $69$
• Weight: $8$
• Character: 69.58
• Analytic conductor: $21.555$
• Analytic rank: $0$
• Dimension: $140$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	69.e (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(21.5545667584\)
Analytic rank:	\(0\)
Dimension:	\(140\)
Relative dimension:	\(14\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			58.14
Character	\(\chi\)	\(=\)	69.58
Dual form			69.8.e.a.25.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(17.6769 - 11.3603i) q^{2} +(-3.84250 + 26.7252i) q^{3} +(130.244 - 285.195i) q^{4} +(311.285 - 91.4016i) q^{5} +(235.681 + 516.070i) q^{6} +(1148.31 + 1325.22i) q^{7} +(-554.803 - 3858.74i) q^{8} +(-699.470 - 205.383i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - 16 q^{2} - 378 q^{3} - 740 q^{4} - 6 q^{5} + 1647 q^{6} + 978 q^{7} - 824 q^{8} - 10206 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(e\left(\frac{10}{11}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	17.6769	−	11.3603i	1.56243	−	1.00411i	0.580638	−	0.814162i	\(-0.302803\pi\)
							0.981793	−	0.189952i	\(-0.0608333\pi\)
\(3\)	−3.84250	+	26.7252i	−0.0821655	+	0.571474i
							\(5\)	311.285	−	91.4016i	1.11369	−	0.327008i	0.327411	−	0.944882i	\(-0.393824\pi\)
0.786277	+	0.617874i	\(0.212006\pi\)
\(7\)	1148.31	+	1325.22i	1.26537	+	1.46031i	0.827692	+	0.561182i	\(0.189653\pi\)
							0.437677	+	0.899132i	\(0.355801\pi\)
\(11\)	2251.90	+	1447.21i	0.510122	+	0.327835i	0.770254	−	0.637738i	\(-0.220130\pi\)
							−0.260132	+	0.965573i	\(0.583766\pi\)
\(13\)	−7435.87	+	8581.45i	−0.938707	+	1.08333i	0.0576750	+	0.998335i	\(0.481631\pi\)
							−0.996382	+	0.0849899i	\(0.972914\pi\)
\(17\)	−7072.11	−	15485.8i	−0.349122	−	0.764471i	−0.999986	−	0.00527677i	\(-0.998320\pi\)
							0.650864	−	0.759195i	\(-0.274407\pi\)
\(19\)	13816.5	−	30253.9i	0.462126	−	1.01191i	−0.524872	−	0.851181i	\(-0.675887\pi\)
							0.986998	−	0.160733i	\(-0.0513858\pi\)
\(23\)	−21881.2	−	54092.9i	−0.374994	−	0.927027i
							\(29\)	−63175.1	−	138334.i	−0.481009	−	1.05326i	−0.982185	−	0.187917i	\(-0.939826\pi\)
0.501176	−	0.865345i	\(-0.332901\pi\)
\(31\)	16241.9	+	112965.i	0.0979198	+	0.681047i	0.978363	+	0.206896i	\(0.0663361\pi\)
							−0.880443	+	0.474152i	\(0.842755\pi\)
\(37\)	108562.	+	31876.5i	0.352346	+	0.103458i	0.453116	−	0.891452i	\(-0.350312\pi\)
							−0.100770	+	0.994910i	\(0.532130\pi\)
\(41\)	−520553.	+	152848.i	−1.17956	+	0.346351i	−0.812005	−	0.583650i	\(-0.801624\pi\)
							−0.367557	+	0.930001i	\(0.619806\pi\)
\(43\)	−49217.2	+	342313.i	−0.0944012	+	0.656575i	0.886595	+	0.462547i	\(0.153064\pi\)
							−0.980996	+	0.194028i	\(0.937845\pi\)
\(47\)	−640666.			−0.900096			−0.450048	−	0.893004i	\(-0.648593\pi\)
							−0.450048	+	0.893004i	\(0.648593\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.8.e.a.58.14	yes	140
23.2	even	11	inner	69.8.e.a.25.14	✓	140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.8.e.a.25.14	✓	140	23.2	even	11	inner
69.8.e.a.58.14	yes	140	1.1	even	1	trivial