Embedded newform 69.2.g.a.11.1

• Label: 69.2.g.a.11.1
• Level: $69$
• Weight: $2$
• Character: 69.11
• 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			11.1
Character	\(\chi\)	\(=\)	69.11
Dual form			69.2.g.a.44.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.13923 - 1.77267i) q^{2} +(0.311887 - 1.70374i) q^{3} +(-1.01370 + 2.21969i) q^{4} +(1.10513 - 0.324494i) q^{5} +(-3.37548 + 1.38807i) q^{6} +(-1.21996 + 1.05710i) q^{7} +(0.918158 - 0.132011i) q^{8} +(-2.80545 - 1.06275i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 11 q^{3} - 10 q^{4} - 14 q^{6} - 22 q^{7} - 11 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(e\left(\frac{9}{22}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.13923	−	1.77267i	−0.805556	−	1.25347i	−0.963947	−	0.266093i	\(-0.914267\pi\)
							0.158392	−	0.987376i	\(-0.449369\pi\)
\(3\)	0.311887	−	1.70374i	0.180068	−	0.983654i
							\(5\)	1.10513	−	0.324494i	0.494227	−	0.145118i	−0.0251149	−	0.999685i	\(-0.507995\pi\)
0.519342	+	0.854566i	\(0.326177\pi\)
\(7\)	−1.21996	+	1.05710i	−0.461102	+	0.399547i	−0.854196	−	0.519951i	\(-0.825950\pi\)
							0.393094	+	0.919498i	\(0.371405\pi\)
\(11\)	4.10123	+	2.63570i	1.23657	+	0.794694i	0.984901	−	0.173120i	\(-0.0553849\pi\)
							0.251667	+	0.967814i	\(0.419021\pi\)
\(13\)	2.30778	−	2.66332i	0.640064	−	0.738673i	−0.339322	−	0.940670i	\(-0.610198\pi\)
							0.979386	+	0.201997i	\(0.0647432\pi\)
\(17\)	−2.44661	−	5.35733i	−0.593390	−	1.29934i	−0.933372	−	0.358910i	\(-0.883148\pi\)
							0.339982	−	0.940432i	\(-0.389579\pi\)
\(19\)	3.45437	+	1.57756i	0.792488	+	0.361917i	0.770174	−	0.637834i	\(-0.220169\pi\)
							0.0223137	+	0.999751i	\(0.492897\pi\)
\(23\)	2.74840	+	3.93018i	0.573082	+	0.819498i
							\(29\)	−4.87704	+	2.22727i	−0.905643	+	0.413593i	−0.813106	−	0.582116i	\(-0.802225\pi\)
−0.0925373	+	0.995709i	\(0.529498\pi\)
\(31\)	−0.200482	−	1.39439i	−0.0360077	−	0.250439i	0.963865	−	0.266391i	\(-0.0858312\pi\)
							−0.999873	+	0.0159520i	\(0.994922\pi\)
\(37\)	−0.670140	+	2.28229i	−0.110170	+	0.375206i	−0.996059	−	0.0886921i	\(-0.971731\pi\)
							0.885889	+	0.463898i	\(0.153549\pi\)
\(41\)	−0.848837	−	2.89087i	−0.132566	−	0.451479i	0.866277	−	0.499564i	\(-0.166506\pi\)
							−0.998843	+	0.0480851i	\(0.984688\pi\)
\(43\)	1.65948	+	0.238597i	0.253068	+	0.0363857i	0.267681	−	0.963508i	\(-0.413743\pi\)
							−0.0146131	+	0.999893i	\(0.504652\pi\)
\(47\)			11.0600i			1.61327i	0.591049	+	0.806636i	\(0.298714\pi\)
							−0.591049	+	0.806636i	\(0.701286\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.2.g.a.11.1	✓	60
3.2	odd	2	inner	69.2.g.a.11.6	yes	60
23.21	odd	22	inner	69.2.g.a.44.6	yes	60
69.44	even	22	inner	69.2.g.a.44.1	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.2.g.a.11.1	✓	60	1.1	even	1	trivial
69.2.g.a.11.6	yes	60	3.2	odd	2	inner
69.2.g.a.44.1	yes	60	69.44	even	22	inner
69.2.g.a.44.6	yes	60	23.21	odd	22	inner