Embedded newform 69.2.g.a.11.5

• Label: 69.2.g.a.11.5
• 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			11.5
Character	\(\chi\)	\(=\)	69.11
Dual form			69.2.g.a.44.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.848533 + 1.32034i) q^{2} +(-1.00593 + 1.41000i) q^{3} +(-0.192468 + 0.421446i) q^{4} +(-1.50619 + 0.442257i) q^{5} +(-2.71525 - 0.131731i) q^{6} +(1.83527 - 1.59027i) q^{7} +(2.38727 - 0.343238i) q^{8} +(-0.976228 - 2.83672i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 11 q^{3} - 10 q^{4} - 14 q^{6} - 22 q^{7} - 11 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.848533	+	1.32034i	0.600004	+	0.933624i	0.999855	+	0.0170437i	\(0.00542544\pi\)
							−0.399851	+	0.916580i	\(0.630938\pi\)
\(3\)	−1.00593	+	1.41000i	−0.580771	+	0.814067i
							\(5\)	−1.50619	+	0.442257i	−0.673589	+	0.197783i	−0.600600	−	0.799549i	\(-0.705072\pi\)
−0.0729882	+	0.997333i	\(0.523254\pi\)
\(7\)	1.83527	−	1.59027i	0.693666	−	0.601065i	−0.234994	−	0.971997i	\(-0.575507\pi\)
							0.928660	+	0.370932i	\(0.120962\pi\)
\(11\)	0.834250	+	0.536140i	0.251536	+	0.161652i	0.660333	−	0.750973i	\(-0.270415\pi\)
							−0.408797	+	0.912625i	\(0.634052\pi\)
\(13\)	−1.51758	+	1.75138i	−0.420901	+	0.485745i	−0.926111	−	0.377251i	\(-0.876869\pi\)
							0.505210	+	0.862996i	\(0.331415\pi\)
\(17\)	−1.86290	−	4.07918i	−0.451820	−	0.989347i	−0.989276	−	0.146056i	\(-0.953342\pi\)
							0.537456	−	0.843291i	\(-0.319385\pi\)
\(19\)	−5.75160	−	2.62667i	−1.31951	−	0.602599i	−0.373767	−	0.927522i	\(-0.621934\pi\)
							−0.945740	+	0.324924i	\(0.894661\pi\)
\(23\)	4.71382	−	0.883116i	0.982900	−	0.184142i
							\(29\)	−7.93286	+	3.62282i	−1.47310	+	0.672740i	−0.980320	−	0.197416i	\(-0.936745\pi\)
−0.492776	+	0.870156i	\(0.664018\pi\)
\(31\)	−0.409065	−	2.84511i	−0.0734702	−	0.510996i	−0.993013	−	0.118006i	\(-0.962350\pi\)
							0.919543	−	0.392990i	\(-0.128559\pi\)
\(37\)	−0.564099	+	1.92114i	−0.0927373	+	0.315834i	−0.992777	−	0.119974i	\(-0.961719\pi\)
							0.900040	+	0.435808i	\(0.143537\pi\)
\(41\)	2.89123	+	9.84661i	0.451533	+	1.53778i	0.799730	+	0.600360i	\(0.204976\pi\)
							−0.348196	+	0.937422i	\(0.613206\pi\)
\(43\)	10.8860	+	1.56517i	1.66010	+	0.238687i	0.907580	−	0.419880i	\(-0.137928\pi\)
							0.752523	+	0.658566i	\(0.228837\pi\)
\(47\)		−	0.223176i		−	0.0325535i	−0.999868	−	0.0162768i	\(-0.994819\pi\)
							0.999868	−	0.0162768i	\(-0.00518128\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.2.g.a.11.5	yes	60
3.2	odd	2	inner	69.2.g.a.11.2	✓	60
23.21	odd	22	inner	69.2.g.a.44.2	yes	60
69.44	even	22	inner	69.2.g.a.44.5	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.2.g.a.11.2	✓	60	3.2	odd	2	inner
69.2.g.a.11.5	yes	60	1.1	even	1	trivial
69.2.g.a.44.2	yes	60	23.21	odd	22	inner
69.2.g.a.44.5	yes	60	69.44	even	22	inner