Embedded newform 69.2.g.a.14.4

• Label: 69.2.g.a.14.4
• Level: $69$
• Weight: $2$
• Character: 69.14
• Analytic conductor: $0.551$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	69.g (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.550967773947\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(6\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			14.4
Character	\(\chi\)	\(=\)	69.14
Dual form			69.2.g.a.5.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.298617 + 1.01700i) q^{2} +(-0.239057 - 1.71547i) q^{3} +(0.737395 - 0.473895i) q^{4} +(0.228939 + 1.59231i) q^{5} +(1.67325 - 0.755391i) q^{6} +(-1.28446 - 0.586593i) q^{7} +(2.30424 + 1.99663i) q^{8} +(-2.88570 + 0.820191i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 11 q^{3} - 10 q^{4} - 14 q^{6} - 22 q^{7} - 11 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(e\left(\frac{21}{22}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.298617	+	1.01700i	0.211154	+	0.719126i	0.995151	+	0.0983605i	\(0.0313598\pi\)
							−0.783996	+	0.620765i	\(0.786822\pi\)
\(3\)	−0.239057	−	1.71547i	−0.138019	−	0.990430i
							\(5\)	0.228939	+	1.59231i	0.102385	+	0.712102i	0.974758	+	0.223262i	\(0.0716707\pi\)
−0.872374	+	0.488840i	\(0.837420\pi\)
\(7\)	−1.28446	−	0.586593i	−0.485480	−	0.221711i	0.157611	−	0.987501i	\(-0.449621\pi\)
							−0.643091	+	0.765790i	\(0.722348\pi\)
\(11\)	−4.31116	−	1.26587i	−1.29986	−	0.381674i	−0.442677	−	0.896681i	\(-0.645971\pi\)
							−0.857186	+	0.515007i	\(0.827789\pi\)
\(13\)	1.54864	+	3.39104i	0.429514	+	0.940505i	0.993405	+	0.114656i	\(0.0365765\pi\)
							−0.563891	+	0.825849i	\(0.690696\pi\)
\(17\)	−3.34349	−	2.14873i	−0.810915	−	0.521143i	0.0682462	−	0.997669i	\(-0.478260\pi\)
							−0.879161	+	0.476525i	\(0.841896\pi\)
\(19\)	−1.74911	−	2.72167i	−0.401273	−	0.624394i	0.580542	−	0.814230i	\(-0.302841\pi\)
							−0.981816	+	0.189836i	\(0.939204\pi\)
\(23\)	4.78334	−	0.345872i	0.997396	−	0.0721194i
							\(29\)	−0.915281	+	1.42420i	−0.169963	+	0.264468i	−0.915777	−	0.401686i	\(-0.868424\pi\)
0.745814	+	0.666154i	\(0.232061\pi\)
\(31\)	−5.36145	+	6.18744i	−0.962945	+	1.11130i	0.0307889	+	0.999526i	\(0.490198\pi\)
							−0.993734	+	0.111772i	\(0.964347\pi\)
\(37\)	3.88889	+	0.559137i	0.639329	+	0.0919216i	0.454353	−	0.890822i	\(-0.349870\pi\)
							0.184975	+	0.982743i	\(0.440779\pi\)
\(41\)	1.40816	−	0.202463i	0.219917	−	0.0316193i	−0.0314753	−	0.999505i	\(-0.510021\pi\)
							0.251393	+	0.967885i	\(0.419111\pi\)
\(43\)	7.30732	−	6.33183i	1.11436	−	0.965595i	0.114742	−	0.993395i	\(-0.463396\pi\)
							0.999613	+	0.0278003i	\(0.00885026\pi\)
\(47\)		−	9.42076i		−	1.37416i	−0.726582	−	0.687080i	\(-0.758892\pi\)
							0.726582	−	0.687080i	\(-0.241108\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.2.g.a.14.4	yes	60
3.2	odd	2	inner	69.2.g.a.14.3	yes	60
23.5	odd	22	inner	69.2.g.a.5.3	✓	60
69.5	even	22	inner	69.2.g.a.5.4	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.2.g.a.5.3	✓	60	23.5	odd	22	inner
69.2.g.a.5.4	yes	60	69.5	even	22	inner
69.2.g.a.14.3	yes	60	3.2	odd	2	inner
69.2.g.a.14.4	yes	60	1.1	even	1	trivial