Embedded newform 69.7.f.a.10.18

• Label: 69.7.f.a.10.18
• Level: $69$
• Weight: $7$
• Character: 69.10
• 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			10.18
Character	\(\chi\)	\(=\)	69.10
Dual form			69.7.f.a.7.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(5.80964 + 6.70469i) q^{2} +(13.1138 + 8.42776i) q^{3} +(-2.09271 + 14.5551i) q^{4} +(-138.264 - 63.1430i) q^{5} +(19.6813 + 136.886i) q^{6} +(130.352 - 443.939i) q^{7} +(367.902 - 236.436i) q^{8} +(100.946 + 221.041i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 20 q^{2} - 816 q^{4} + 324 q^{6} + 940 q^{8} - 5832 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(e\left(\frac{3}{22}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	5.80964	+	6.70469i	0.726206	+	0.838086i	0.992039	−	0.125933i	\(-0.0401925\pi\)
							−0.265833	+	0.964019i	\(0.585647\pi\)
\(3\)	13.1138	+	8.42776i	0.485698	+	0.312139i
							\(5\)	−138.264	−	63.1430i	−1.10611	−	0.505144i	−0.223242	−	0.974763i	\(-0.571664\pi\)
−0.882869	+	0.469619i	\(0.844391\pi\)
\(7\)	130.352	−	443.939i	0.380036	−	1.29428i	−0.518384	−	0.855148i	\(-0.673466\pi\)
							0.898420	−	0.439136i	\(-0.144715\pi\)
\(11\)	208.448	+	180.622i	0.156610	+	0.135704i	0.729641	−	0.683830i	\(-0.239687\pi\)
							−0.573031	+	0.819534i	\(0.694233\pi\)
\(13\)	3359.88	−	986.551i	1.52931	−	0.449045i	0.594468	−	0.804120i	\(-0.297363\pi\)
							0.934838	+	0.355075i	\(0.115545\pi\)
\(17\)	3866.98	−	555.988i	0.787092	−	0.113167i	0.262963	−	0.964806i	\(-0.415300\pi\)
							0.524129	+	0.851639i	\(0.324391\pi\)
\(19\)	−11871.7	−	1706.89i	−1.73081	−	0.248854i	−0.796331	−	0.604860i	\(-0.793229\pi\)
							−0.934483	+	0.356007i	\(0.884138\pi\)
\(23\)	−12128.5	−	967.045i	−0.996836	−	0.0794810i
							\(29\)	−984.475	−	6847.18i	−0.0403655	−	0.280749i	0.959634	−	0.281251i	\(-0.0907492\pi\)
−1.00000		0.000502158i	\(0.999840\pi\)
\(31\)	25657.8	−	16489.3i	0.861260	−	0.553498i	−0.0338078	−	0.999428i	\(-0.510763\pi\)
							0.895068	+	0.445930i	\(0.147127\pi\)
\(37\)	78214.7	−	35719.4i	1.54413	−	0.705179i	0.552404	−	0.833577i	\(-0.313711\pi\)
							0.991723	+	0.128397i	\(0.0409833\pi\)
\(41\)	−27261.8	+	59695.0i	−0.395551	+	0.866136i	0.602151	+	0.798382i	\(0.294311\pi\)
							−0.997702	+	0.0677541i	\(0.978417\pi\)
\(43\)	6763.07	−	10523.5i	0.0850626	−	0.132360i	−0.796115	−	0.605145i	\(-0.793115\pi\)
							0.881178	+	0.472785i	\(0.156751\pi\)
\(47\)	27980.0			0.269497			0.134749	−	0.990880i	\(-0.456977\pi\)
							0.134749	+	0.990880i	\(0.456977\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.7.f.a.10.18	yes	240
23.7	odd	22	inner	69.7.f.a.7.18	✓	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.7.f.a.7.18	✓	240	23.7	odd	22	inner
69.7.f.a.10.18	yes	240	1.1	even	1	trivial