Embedded newform 69.8.e.a.4.13

• Label: 69.8.e.a.4.13
• Level: $69$
• Weight: $8$
• Character: 69.4
• 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			4.13
Character	\(\chi\)	\(=\)	69.4
Dual form			69.8.e.a.52.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(7.08490 + 15.5138i) q^{2} +(-25.9063 + 7.60678i) q^{3} +(-106.659 + 123.091i) q^{4} +(333.068 + 214.050i) q^{5} +(-301.553 - 348.011i) q^{6} +(19.0468 + 132.473i) q^{7} +(-570.665 - 167.562i) q^{8} +(613.274 - 394.127i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - 16 q^{2} - 378 q^{3} - 740 q^{4} - 6 q^{5} + 1647 q^{6} + 978 q^{7} - 824 q^{8} - 10206 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(e\left(\frac{2}{11}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	7.08490	+	15.5138i	0.626223	+	1.37124i	0.910906	+	0.412615i	\(0.135384\pi\)
							−0.284683	+	0.958622i	\(0.591888\pi\)
\(3\)	−25.9063	+	7.60678i	−0.553964	+	0.162658i
							\(5\)	333.068	+	214.050i	1.19162	+	0.765809i	0.977487	−	0.210995i	\(-0.0676705\pi\)
0.214135	+	0.976804i	\(0.431307\pi\)
\(7\)	19.0468	+	132.473i	0.0209884	+	0.145977i	0.997621	−	0.0689388i	\(-0.0219613\pi\)
							−0.976633	+	0.214916i	\(0.931052\pi\)
\(11\)	−3066.40	+	6714.49i	−0.694632	+	1.52103i	0.151728	+	0.988422i	\(0.451516\pi\)
							−0.846360	+	0.532610i	\(0.821211\pi\)
\(13\)	−773.212	+	5377.81i	−0.0976106	+	0.678897i	0.880991	+	0.473133i	\(0.156877\pi\)
							−0.978602	+	0.205764i	\(0.934032\pi\)
\(17\)	−9012.63	−	10401.1i	−0.444918	−	0.513463i	0.488348	−	0.872649i	\(-0.337600\pi\)
							−0.933266	+	0.359186i	\(0.883054\pi\)
\(19\)	38912.3	−	44907.2i	1.30152	−	1.50203i	0.566288	−	0.824207i	\(-0.308379\pi\)
							0.735227	−	0.677821i	\(-0.237076\pi\)
\(23\)	−57766.1	+	8240.53i	−0.989978	+	0.141224i
							\(29\)	−92390.7	−	106625.i	−0.703453	−	0.811828i	0.285762	−	0.958301i	\(-0.407753\pi\)
−0.989215	+	0.146473i	\(0.953208\pi\)
\(31\)	162331.	+	47664.6i	0.978667	+	0.287363i	0.731674	−	0.681655i	\(-0.238739\pi\)
							0.246993	+	0.969017i	\(0.420558\pi\)
\(37\)	−789.354	+	507.287i	−0.00256192	+	0.00164645i	−0.541921	−	0.840429i	\(-0.682303\pi\)
							0.539359	+	0.842076i	\(0.318667\pi\)
\(41\)	−364217.	−	234068.i	−0.825309	−	0.530394i	0.0584752	−	0.998289i	\(-0.481376\pi\)
							−0.883784	+	0.467895i	\(0.845013\pi\)
\(43\)	−343082.	+	100738.i	−0.658049	+	0.193221i	−0.593675	−	0.804705i	\(-0.702323\pi\)
							−0.0643745	+	0.997926i	\(0.520505\pi\)
\(47\)	236676.			0.332515			0.166258	−	0.986082i	\(-0.446832\pi\)
							0.166258	+	0.986082i	\(0.446832\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.8.e.a.4.13	✓	140
23.6	even	11	inner	69.8.e.a.52.13	yes	140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.8.e.a.4.13	✓	140	1.1	even	1	trivial
69.8.e.a.52.13	yes	140	23.6	even	11	inner