Embedded newform 69.7.f.a.34.13

• Label: 69.7.f.a.34.13
• Level: $69$
• Weight: $7$
• Character: 69.34
• 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			34.13
Character	\(\chi\)	\(=\)	69.34
Dual form			69.7.f.a.67.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0859252 + 0.0552208i) q^{2} +(-2.21847 + 15.4298i) q^{3} +(-26.5822 + 58.2070i) q^{4} +(-42.2344 - 143.837i) q^{5} +(-0.661423 - 1.44831i) q^{6} +(-186.782 + 161.848i) q^{7} +(-1.86045 - 12.9397i) q^{8} +(-233.157 - 68.4610i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 20 q^{2} - 816 q^{4} + 324 q^{6} + 940 q^{8} - 5832 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(e\left(\frac{9}{22}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0859252	+	0.0552208i	−0.0107407	+	0.00690260i	−0.546000	−	0.837785i	\(-0.683850\pi\)
							0.535259	+	0.844688i	\(0.320214\pi\)
\(3\)	−2.21847	+	15.4298i	−0.0821655	+	0.571474i
							\(5\)	−42.2344	−	143.837i	−0.337875	−	1.15070i	−0.936792	−	0.349886i	\(-0.886220\pi\)
0.598917	−	0.800811i	\(-0.295598\pi\)
\(7\)	−186.782	+	161.848i	−0.544555	+	0.471860i	−0.883162	−	0.469068i	\(-0.844590\pi\)
							0.338607	+	0.940928i	\(0.390044\pi\)
\(11\)	1196.41	−	1861.65i	0.898882	−	1.39869i	−0.0181339	−	0.999836i	\(-0.505773\pi\)
							0.917015	−	0.398852i	\(-0.130591\pi\)
\(13\)	−1100.66	+	1270.23i	−0.500985	+	0.578168i	−0.948768	−	0.315975i	\(-0.897668\pi\)
							0.447783	+	0.894143i	\(0.352214\pi\)
\(17\)	1968.78	−	899.111i	0.400728	−	0.183006i	−0.204848	−	0.978794i	\(-0.565670\pi\)
							0.605576	+	0.795787i	\(0.292943\pi\)
\(19\)	10106.6	+	4615.53i	1.47348	+	0.672917i	0.980391	−	0.197061i	\(-0.0631399\pi\)
							0.493091	+	0.869978i	\(0.335867\pi\)
\(23\)	10586.2	−	5997.36i	0.870075	−	0.492920i
							\(29\)	−18617.0	−	40765.6i	−0.763337	−	1.67147i	−0.740798	−	0.671728i	\(-0.765552\pi\)
−0.0225386	−	0.999746i	\(-0.507175\pi\)
\(31\)	−2188.44	−	15220.9i	−0.0734597	−	0.510923i	−0.993017	−	0.117968i	\(-0.962362\pi\)
							0.919558	−	0.392955i	\(-0.128547\pi\)
\(37\)	24051.4	−	81911.4i	0.474826	−	1.61711i	−0.279178	−	0.960239i	\(-0.590062\pi\)
							0.754004	−	0.656870i	\(-0.228120\pi\)
\(41\)	−66232.1	+	19447.5i	−0.960985	+	0.282171i	−0.724354	−	0.689429i	\(-0.757862\pi\)
							−0.236632	+	0.971599i	\(0.576043\pi\)
\(43\)	88725.7	+	12756.8i	1.11595	+	0.160449i	0.675522	−	0.737340i	\(-0.263918\pi\)
							0.440426	+	0.897789i	\(0.354827\pi\)
\(47\)	−84616.8			−0.815010			−0.407505	−	0.913203i	\(-0.633601\pi\)
							−0.407505	+	0.913203i	\(0.633601\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.7.f.a.34.13	✓	240
23.21	odd	22	inner	69.7.f.a.67.13	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.7.f.a.34.13	✓	240	1.1	even	1	trivial
69.7.f.a.67.13	yes	240	23.21	odd	22	inner