Embedded newform 69.8.e.a.4.7

• Label: 69.8.e.a.4.7
• 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			4.7
Character	\(\chi\)	\(=\)	69.4
Dual form			69.8.e.a.52.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.464659 - 1.01746i) q^{2} +(-25.9063 + 7.60678i) q^{3} +(83.0029 - 95.7904i) q^{4} +(-317.877 - 204.287i) q^{5} +(19.7772 + 22.8241i) q^{6} +(233.295 + 1622.60i) q^{7} +(-273.405 - 80.2789i) q^{8} +(613.274 - 394.127i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - 16 q^{2} - 378 q^{3} - 740 q^{4} - 6 q^{5} + 1647 q^{6} + 978 q^{7} - 824 q^{8} - 10206 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.464659	−	1.01746i	−0.0410705	−	0.0899318i	0.887985	−	0.459873i	\(-0.152105\pi\)
							−0.929055	+	0.369941i	\(0.879378\pi\)
\(3\)	−25.9063	+	7.60678i	−0.553964	+	0.162658i
							\(5\)	−317.877	−	204.287i	−1.13727	−	0.730879i	−0.170205	−	0.985409i	\(-0.554443\pi\)
−0.967065	+	0.254530i	\(0.918079\pi\)
\(7\)	233.295	+	1622.60i	0.257077	+	1.78801i	0.553396	+	0.832919i	\(0.313332\pi\)
							−0.296319	+	0.955089i	\(0.595759\pi\)
\(11\)	−2621.15	+	5739.52i	−0.593769	+	1.30017i	0.339368	+	0.940654i	\(0.389787\pi\)
							−0.933138	+	0.359520i	\(0.882941\pi\)
\(13\)	1621.96	−	11281.0i	0.204757	−	1.42412i	−0.585166	−	0.810913i	\(-0.698971\pi\)
							0.789923	−	0.613205i	\(-0.210120\pi\)
\(17\)	8133.59	+	9386.66i	0.401523	+	0.463383i	0.920120	−	0.391636i	\(-0.128091\pi\)
							−0.518597	+	0.855019i	\(0.673545\pi\)
\(19\)	23413.4	−	27020.6i	0.783119	−	0.903767i	−0.214212	−	0.976787i	\(-0.568718\pi\)
							0.997330	+	0.0730199i	\(0.0232637\pi\)
\(23\)	53850.9	+	22470.0i	0.922881	+	0.385084i
							\(29\)	56881.3	+	65644.5i	0.433088	+	0.499811i	0.929780	−	0.368117i	\(-0.119997\pi\)
−0.496691	+	0.867927i	\(0.665452\pi\)
\(31\)	189313.	+	55587.3i	1.14134	+	0.335127i	0.797155	−	0.603775i	\(-0.206337\pi\)
							0.344184	+	0.938902i	\(0.388156\pi\)
\(37\)	274048.	−	176120.i	0.889448	−	0.571614i	−0.0141951	−	0.999899i	\(-0.504519\pi\)
							0.903643	+	0.428286i	\(0.140882\pi\)
\(41\)	−379496.	−	243888.i	−0.859932	−	0.552645i	0.0347257	−	0.999397i	\(-0.488944\pi\)
							−0.894658	+	0.446752i	\(0.852581\pi\)
\(43\)	282089.	−	82828.8i	0.541061	−	0.158870i	0.000228269	−	1.00000i	\(-0.499927\pi\)
							0.540833	+	0.841130i	\(0.318109\pi\)
\(47\)	987257.			1.38704			0.693518	−	0.720439i	\(-0.256060\pi\)
							0.693518	+	0.720439i	\(0.256060\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.8.e.a.4.7	✓	140
23.6	even	11	inner	69.8.e.a.52.7	yes	140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.8.e.a.4.7	✓	140	1.1	even	1	trivial
69.8.e.a.52.7	yes	140	23.6	even	11	inner