Embedded newform 69.7.f.a.34.23

• Label: 69.7.f.a.34.23
• 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			34.23
Character	\(\chi\)	\(=\)	69.34
Dual form			69.7.f.a.67.23

$q$-expansion

\(f(q)\)	\(=\)	\(q+(11.8244 - 7.59911i) q^{2} +(2.21847 - 15.4298i) q^{3} +(55.4845 - 121.494i) q^{4} +(-0.763800 - 2.60126i) q^{5} +(-91.0205 - 199.307i) q^{6} +(353.764 - 306.538i) q^{7} +(-139.152 - 967.822i) q^{8} +(-233.157 - 68.4610i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 20 q^{2} - 816 q^{4} + 324 q^{6} + 940 q^{8} - 5832 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	11.8244	−	7.59911i	1.47806	−	0.949889i	0.480725	−	0.876871i	\(-0.340373\pi\)
							0.997331	−	0.0730175i	\(-0.0232629\pi\)
\(3\)	2.21847	−	15.4298i	0.0821655	−	0.571474i
							\(5\)	−0.763800	−	2.60126i	−0.00611040	−	0.0208101i	0.956381	−	0.292121i	\(-0.0943610\pi\)
−0.962492	+	0.271311i	\(0.912543\pi\)
\(7\)	353.764	−	306.538i	1.03138	−	0.893698i	0.0369754	−	0.999316i	\(-0.488228\pi\)
							0.994407	+	0.105618i	\(0.0336822\pi\)
\(11\)	−1169.73	+	1820.13i	−0.878835	+	1.36749i	0.0506822	+	0.998715i	\(0.483860\pi\)
							−0.929517	+	0.368779i	\(0.879776\pi\)
\(13\)	1992.67	−	2299.67i	0.906998	−	1.04673i	−0.0917036	−	0.995786i	\(-0.529231\pi\)
							0.998701	−	0.0509450i	\(-0.0162233\pi\)
\(17\)	−6422.54	+	2933.08i	−1.30725	+	0.597003i	−0.942533	−	0.334113i	\(-0.891563\pi\)
							−0.364722	+	0.931117i	\(0.618836\pi\)
\(19\)	2156.60	+	984.887i	0.314419	+	0.143590i	0.566373	−	0.824149i	\(-0.308346\pi\)
							−0.251954	+	0.967739i	\(0.581073\pi\)
\(23\)	−6045.47	−	10558.8i	−0.496874	−	0.867823i
							\(29\)	19859.6	+	43486.4i	0.814284	+	1.78303i	0.587775	+	0.809025i	\(0.300004\pi\)
0.226510	+	0.974009i	\(0.427269\pi\)
\(31\)	1406.44	+	9781.99i	0.0472102	+	0.328354i	0.999716	+	0.0238334i	\(0.00758714\pi\)
							−0.952506	+	0.304521i	\(0.901504\pi\)
\(37\)	8352.79	−	28447.0i	0.164902	−	0.561605i	−0.835033	−	0.550200i	\(-0.814551\pi\)
							0.999935	−	0.0114050i	\(-0.00363040\pi\)
\(41\)	52021.0	−	15274.7i	0.754792	−	0.221627i	0.118372	−	0.992969i	\(-0.462233\pi\)
							0.636420	+	0.771342i	\(0.280414\pi\)
\(43\)	−12588.9	−	1810.00i	−0.158336	−	0.0227653i	0.0626907	−	0.998033i	\(-0.480032\pi\)
							−0.221027	+	0.975268i	\(0.570941\pi\)
\(47\)	83994.5			0.809017			0.404508	−	0.914534i	\(-0.367443\pi\)
							0.404508	+	0.914534i	\(0.367443\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.7.f.a.34.23	✓	240
23.21	odd	22	inner	69.7.f.a.67.23	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.7.f.a.34.23	✓	240	1.1	even	1	trivial
69.7.f.a.67.23	yes	240	23.21	odd	22	inner