Embedded newform 69.7.f.a.7.8

• Label: 69.7.f.a.7.8
• Level: $69$
• Weight: $7$
• Character: 69.7
• Analytic conductor: $15.874$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(15.8737317698\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(24\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			7.8
Character	\(\chi\)	\(=\)	69.7
Dual form			69.7.f.a.10.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.02561 + 4.64580i) q^{2} +(13.1138 - 8.42776i) q^{3} +(3.73022 + 25.9443i) q^{4} +(-145.258 + 66.3370i) q^{5} +(-13.6375 + 94.8511i) q^{6} +(94.3193 + 321.222i) q^{7} +(-466.519 - 299.813i) q^{8} +(100.946 - 221.041i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 20 q^{2} - 816 q^{4} + 324 q^{6} + 940 q^{8} - 5832 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(e\left(\frac{19}{22}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.02561	+	4.64580i	−0.503201	+	0.580725i	−0.949345	−	0.314237i	\(-0.898251\pi\)
							0.446144	+	0.894961i	\(0.352797\pi\)
\(3\)	13.1138	−	8.42776i	0.485698	−	0.312139i
							\(5\)	−145.258	+	66.3370i	−1.16206	+	0.530696i	−0.900656	−	0.434533i	\(-0.856913\pi\)
−0.261406	+	0.965229i	\(0.584186\pi\)
\(7\)	94.3193	+	321.222i	0.274983	+	0.936508i	0.974967	+	0.222349i	\(0.0713724\pi\)
							−0.699984	+	0.714159i	\(0.746809\pi\)
\(11\)	438.217	−	379.717i	0.329239	−	0.285287i	−0.474517	−	0.880246i	\(-0.657377\pi\)
							0.803756	+	0.594959i	\(0.202832\pi\)
\(13\)	431.986	+	126.843i	0.196626	+	0.0577345i	0.378562	−	0.925576i	\(-0.376419\pi\)
							−0.181937	+	0.983310i	\(0.558237\pi\)
\(17\)	−6024.22	−	866.152i	−1.22618	−	0.176298i	−0.501361	−	0.865238i	\(-0.667167\pi\)
							−0.724818	+	0.688940i	\(0.758076\pi\)
\(19\)	−2864.47	+	411.849i	−0.417622	+	0.0600450i	−0.347923	−	0.937523i	\(-0.613113\pi\)
							−0.0696991	+	0.997568i	\(0.522204\pi\)
\(23\)	6293.72	−	10412.7i	0.517278	−	0.855817i
							\(29\)	3453.02	−	24016.3i	0.141581	−	0.984718i	−0.787888	−	0.615818i	\(-0.788826\pi\)
0.929469	−	0.368900i	\(-0.120265\pi\)
\(31\)	−35675.6	−	22927.3i	−1.19753	−	0.769605i	−0.219002	−	0.975724i	\(-0.570280\pi\)
							−0.978527	+	0.206119i	\(0.933916\pi\)
\(37\)	−34097.8	−	15571.9i	−0.673164	−	0.307424i	0.0493482	−	0.998782i	\(-0.484286\pi\)
							−0.722513	+	0.691358i	\(0.757013\pi\)
\(41\)	1092.74	+	2392.76i	0.0158549	+	0.0347174i	0.917394	−	0.397981i	\(-0.130289\pi\)
							−0.901539	+	0.432698i	\(0.857562\pi\)
\(43\)	36192.8	+	56317.1i	0.455216	+	0.708329i	0.990677	−	0.136230i	\(-0.0434987\pi\)
							−0.535462	+	0.844560i	\(0.679862\pi\)
\(47\)	−123916.			−1.19353			−0.596766	−	0.802416i	\(-0.703548\pi\)
							−0.596766	+	0.802416i	\(0.703548\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.7.f.a.7.8	✓	240
23.10	odd	22	inner	69.7.f.a.10.8	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.7.f.a.7.8	✓	240	1.1	even	1	trivial
69.7.f.a.10.8	yes	240	23.10	odd	22	inner