Embedded newform 69.7.h.a.2.10

• Label: 69.7.h.a.2.10
• Level: $69$
• Weight: $7$
• Character: 69.2
• Analytic conductor: $15.874$
• Analytic rank: $0$
• Dimension: $460$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	69.h (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(15.8737317698\)
Analytic rank:	\(0\)
Dimension:	\(460\)
Relative dimension:	\(46\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			2.10
Character	\(\chi\)	\(=\)	69.2
Dual form			69.7.h.a.35.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.60625 + 8.72349i) q^{2} +(-16.9633 + 21.0059i) q^{3} +(-18.0827 - 39.5956i) q^{4} +(-33.0589 + 112.588i) q^{5} +(-88.1443 - 265.743i) q^{6} +(124.109 - 143.230i) q^{7} +(-210.113 - 30.2097i) q^{8} +(-153.494 - 712.657i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 460 q + 23 q^{3} + 1258 q^{4} - 106 q^{6} - 18 q^{7} - 921 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.60625	+	8.72349i	−0.700781	+	1.09044i	0.290269	+	0.956945i	\(0.406255\pi\)
							−0.991050	+	0.133491i	\(0.957381\pi\)
\(3\)	−16.9633	+	21.0059i	−0.628270	+	0.777996i
							\(5\)	−33.0589	+	112.588i	−0.264471	+	0.900706i	0.715000	+	0.699124i	\(0.246427\pi\)
−0.979472	+	0.201582i	\(0.935392\pi\)
\(7\)	124.109	−	143.230i	0.361834	−	0.417579i	−0.545419	−	0.838163i	\(-0.683629\pi\)
							0.907253	+	0.420584i	\(0.138175\pi\)
\(11\)	−134.779	−	209.721i	−0.101262	−	0.157566i	0.786926	−	0.617048i	\(-0.211672\pi\)
							−0.888187	+	0.459482i	\(0.848035\pi\)
\(13\)	−819.025	−	945.206i	−0.372793	−	0.430226i	0.538092	−	0.842886i	\(-0.319145\pi\)
							−0.910885	+	0.412660i	\(0.864600\pi\)
\(17\)	−1549.92	−	707.825i	−0.315473	−	0.144072i	0.251385	−	0.967887i	\(-0.419114\pi\)
							−0.566858	+	0.823815i	\(0.691841\pi\)
\(19\)	−1292.03	−	2829.16i	−0.188371	−	0.412474i	0.791759	−	0.610834i	\(-0.209166\pi\)
							−0.980129	+	0.198360i	\(0.936438\pi\)
\(23\)	−4952.97	+	11113.2i	−0.407083	+	0.913391i
							\(29\)	16824.3	+	7683.41i	0.689832	+	0.315036i	0.729309	−	0.684185i	\(-0.239842\pi\)
−0.0394768	+	0.999220i	\(0.512569\pi\)
\(31\)	−762.365	+	5302.37i	−0.0255905	+	0.177986i	−0.998608	−	0.0527457i	\(-0.983203\pi\)
							0.973018	+	0.230731i	\(0.0741118\pi\)
\(37\)	−56679.8	+	16642.7i	−1.11898	+	0.328563i	−0.788368	−	0.615204i	\(-0.789074\pi\)
							−0.330613	+	0.943766i	\(0.607256\pi\)
\(41\)	19346.9	−	65889.4i	0.280711	−	0.956013i	−0.691592	−	0.722288i	\(-0.743090\pi\)
							0.972303	−	0.233725i	\(-0.0750914\pi\)
\(43\)	−3893.76	−	27081.7i	−0.0489738	−	0.340620i	−0.999546	−	0.0301177i	\(-0.990412\pi\)
							0.950573	−	0.310502i	\(-0.100497\pi\)
\(47\)		−	182096.i		−	1.75391i	−0.480571	−	0.876956i	\(-0.659570\pi\)
							0.480571	−	0.876956i	\(-0.340430\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.7.h.a.2.10	✓	460
3.2	odd	2	inner	69.7.h.a.2.37	yes	460
23.12	even	11	inner	69.7.h.a.35.37	yes	460
69.35	odd	22	inner	69.7.h.a.35.10	yes	460

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.7.h.a.2.10	✓	460	1.1	even	1	trivial
69.7.h.a.2.37	yes	460	3.2	odd	2	inner
69.7.h.a.35.10	yes	460	69.35	odd	22	inner
69.7.h.a.35.37	yes	460	23.12	even	11	inner