Embedded newform 69.8.e.b.58.13

• Label: 69.8.e.b.58.13
• 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			58.13
Character	\(\chi\)	\(=\)	69.58
Dual form			69.8.e.b.25.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(16.7743 - 10.7802i) q^{2} +(3.84250 - 26.7252i) q^{3} +(111.991 - 245.227i) q^{4} +(292.972 - 86.0245i) q^{5} +(-223.647 - 489.719i) q^{6} +(164.828 + 190.222i) q^{7} +(-401.787 - 2794.49i) q^{8} +(-699.470 - 205.383i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - 12 q^{2} + 378 q^{3} - 740 q^{4} + 2 q^{5} - 1755 q^{6} - 394 q^{7} - 1612 q^{8} - 10206 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	16.7743	−	10.7802i	1.48265	−	0.952842i	0.485756	−	0.874094i	\(-0.338544\pi\)
							0.996895	−	0.0787477i	\(-0.0250922\pi\)
\(3\)	3.84250	−	26.7252i	0.0821655	−	0.571474i
							\(5\)	292.972	−	86.0245i	1.04817	−	0.307770i	0.288092	−	0.957603i	\(-0.406979\pi\)
0.760078	+	0.649832i	\(0.225161\pi\)
\(7\)	164.828	+	190.222i	0.181630	+	0.209613i	0.839262	−	0.543727i	\(-0.182987\pi\)
							−0.657632	+	0.753339i	\(0.728442\pi\)
\(11\)	−1823.04	−	1171.60i	−0.412974	−	0.265402i	0.317615	−	0.948220i	\(-0.397118\pi\)
							−0.730589	+	0.682817i	\(0.760754\pi\)
\(13\)	6189.59	−	7143.16i	0.781376	−	0.901755i	−0.215832	−	0.976430i	\(-0.569246\pi\)
							0.997208	+	0.0746750i	\(0.0237919\pi\)
\(17\)	8286.87	+	18145.7i	0.409091	+	0.895783i	0.996267	+	0.0863245i	\(0.0275122\pi\)
							−0.587177	+	0.809459i	\(0.699761\pi\)
\(19\)	−11192.4	+	24507.9i	−0.374356	+	0.819726i	0.624883	+	0.780719i	\(0.285147\pi\)
							−0.999239	+	0.0390072i	\(0.987580\pi\)
\(23\)	−48936.8	+	31780.7i	−0.838665	+	0.544647i
							\(29\)	−71472.3	−	156502.i	−0.544182	−	1.19159i	−0.959446	−	0.281892i	\(-0.909038\pi\)
0.415264	−	0.909701i	\(-0.363689\pi\)
\(31\)	10268.7	+	71420.6i	0.0619086	+	0.430584i	0.997078	+	0.0763841i	\(0.0243375\pi\)
							−0.935170	+	0.354200i	\(0.884753\pi\)
\(37\)	−104264.	−	30614.6i	−0.338398	−	0.0993626i	0.108119	−	0.994138i	\(-0.465517\pi\)
							−0.446516	+	0.894775i	\(0.647336\pi\)
\(41\)	357555.	−	104988.i	0.810213	−	0.237900i	0.149716	−	0.988729i	\(-0.452164\pi\)
							0.660496	+	0.750829i	\(0.270346\pi\)
\(43\)	−102866.	+	715452.i	−0.197303	+	1.37227i	0.614764	+	0.788711i	\(0.289251\pi\)
							−0.812067	+	0.583564i	\(0.801658\pi\)
\(47\)	934574.			1.31302			0.656510	−	0.754318i	\(-0.272032\pi\)
							0.656510	+	0.754318i	\(0.272032\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.8.e.b.58.13	yes	140
23.2	even	11	inner	69.8.e.b.25.13	✓	140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.8.e.b.25.13	✓	140	23.2	even	11	inner
69.8.e.b.58.13	yes	140	1.1	even	1	trivial