Embedded newform 69.5.h.a.2.13

• Label: 69.5.h.a.2.13
• Level: $69$
• Weight: $5$
• Character: 69.2
• Analytic conductor: $7.133$
• Analytic rank: $0$
• Dimension: $300$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	69.h (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.13252745279\)
Analytic rank:	\(0\)
Dimension:	\(300\)
Relative dimension:	\(30\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			2.13
Character	\(\chi\)	\(=\)	69.2
Dual form			69.5.h.a.35.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.842718 + 1.31130i) q^{2} +(7.53602 - 4.92020i) q^{3} +(5.63732 + 12.3440i) q^{4} +(-5.53880 + 18.8634i) q^{5} +(0.101095 + 14.0283i) q^{6} +(-41.7057 + 48.1309i) q^{7} +(-45.6233 - 6.55964i) q^{8} +(32.5832 - 74.1575i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 300 q - 19 q^{3} + 234 q^{4} - 10 q^{6} - 18 q^{7} - 111 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(e\left(\frac{1}{11}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.842718	+	1.31130i	−0.210680	+	0.327824i	−0.930470	−	0.366367i	\(-0.880601\pi\)
							0.719791	+	0.694191i	\(0.244238\pi\)
\(3\)	7.53602	−	4.92020i	0.837336	−	0.546689i
							\(5\)	−5.53880	+	18.8634i	−0.221552	+	0.754537i	0.771434	+	0.636309i	\(0.219540\pi\)
−0.992986	+	0.118228i	\(0.962279\pi\)
\(7\)	−41.7057	+	48.1309i	−0.851136	+	0.982264i	−0.999978	−	0.00659724i	\(-0.997900\pi\)
							0.148842	+	0.988861i	\(0.452445\pi\)
\(11\)	−9.66976	−	15.0464i	−0.0799154	−	0.124351i	0.798995	−	0.601338i	\(-0.205365\pi\)
							−0.878910	+	0.476987i	\(0.841729\pi\)
\(13\)	195.223	+	225.299i	1.15517	+	1.33313i	0.933738	+	0.357956i	\(0.116526\pi\)
							0.221428	+	0.975177i	\(0.428928\pi\)
\(17\)	−299.178	−	136.630i	−1.03522	−	0.472768i	−0.176006	−	0.984389i	\(-0.556318\pi\)
							−0.859211	+	0.511621i	\(0.829045\pi\)
\(19\)	14.3826	+	31.4934i	0.0398409	+	0.0872393i	0.928506	−	0.371317i	\(-0.121094\pi\)
							−0.888665	+	0.458557i	\(0.848367\pi\)
\(23\)	528.813	+	14.0712i	0.999646	+	0.0265995i
							\(29\)	1073.86	+	490.415i	1.27688	+	0.583133i	0.934345	−	0.356371i	\(-0.115986\pi\)
0.342539	+	0.939504i	\(0.388713\pi\)
\(31\)	−47.7329	+	331.990i	−0.0496700	+	0.345463i	0.949798	+	0.312862i	\(0.101288\pi\)
							−0.999468	+	0.0326002i	\(0.989621\pi\)
\(37\)	327.672	−	96.2131i	0.239351	−	0.0702798i	−0.159857	−	0.987140i	\(-0.551103\pi\)
							0.399208	+	0.916860i	\(0.369285\pi\)
\(41\)	692.657	−	2358.97i	0.412050	−	1.40331i	−0.448424	−	0.893821i	\(-0.648015\pi\)
							0.860474	−	0.509494i	\(-0.170167\pi\)
\(43\)	−383.975	−	2670.60i	−0.207666	−	1.44435i	−0.780745	−	0.624849i	\(-0.785160\pi\)
							0.573079	−	0.819500i	\(-0.305749\pi\)
\(47\)			1017.33i			0.460537i	0.973127	+	0.230268i	\(0.0739604\pi\)
							−0.973127	+	0.230268i	\(0.926040\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.5.h.a.2.13	✓	300
3.2	odd	2	inner	69.5.h.a.2.18	yes	300
23.12	even	11	inner	69.5.h.a.35.18	yes	300
69.35	odd	22	inner	69.5.h.a.35.13	yes	300

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.5.h.a.2.13	✓	300	1.1	even	1	trivial
69.5.h.a.2.18	yes	300	3.2	odd	2	inner
69.5.h.a.35.13	yes	300	69.35	odd	22	inner
69.5.h.a.35.18	yes	300	23.12	even	11	inner