Embedded newform 69.7.f.a.34.9

• Label: 69.7.f.a.34.9
• 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			34.9
Character	\(\chi\)	\(=\)	69.34
Dual form			69.7.f.a.67.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.31688 + 3.41695i) q^{2} +(-2.21847 + 15.4298i) q^{3} +(-9.99289 + 21.8814i) q^{4} +(13.2258 + 45.0429i) q^{5} +(-40.9275 - 89.6188i) q^{6} +(429.072 - 371.793i) q^{7} +(-79.2018 - 550.861i) q^{8} +(-233.157 - 68.4610i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 20 q^{2} - 816 q^{4} + 324 q^{6} + 940 q^{8} - 5832 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.31688	+	3.41695i	−0.664610	+	0.427119i	−0.828979	−	0.559280i	\(-0.811078\pi\)
							0.164369	+	0.986399i	\(0.447441\pi\)
\(3\)	−2.21847	+	15.4298i	−0.0821655	+	0.571474i
							\(5\)	13.2258	+	45.0429i	0.105806	+	0.360343i	0.995328	−	0.0965493i	\(-0.0307805\pi\)
−0.889522	+	0.456893i	\(0.848962\pi\)
\(7\)	429.072	−	371.793i	1.25094	−	1.08394i	0.257898	−	0.966172i	\(-0.416970\pi\)
							0.993041	−	0.117772i	\(-0.0375753\pi\)
\(11\)	322.123	−	501.234i	0.242016	−	0.376584i	−0.698903	−	0.715217i	\(-0.746328\pi\)
							0.940919	+	0.338633i	\(0.109964\pi\)
\(13\)	895.686	−	1033.68i	0.407686	−	0.470495i	−0.514360	−	0.857574i	\(-0.671971\pi\)
							0.922046	+	0.387079i	\(0.126516\pi\)
\(17\)	1894.50	−	865.189i	0.385609	−	0.176102i	−0.213171	−	0.977015i	\(-0.568379\pi\)
							0.598781	+	0.800913i	\(0.295652\pi\)
\(19\)	634.935	+	289.965i	0.0925696	+	0.0422751i	0.461162	−	0.887316i	\(-0.347433\pi\)
							−0.368592	+	0.929591i	\(0.620160\pi\)
\(23\)	−4287.07	−	11386.7i	−0.352352	−	0.935868i
							\(29\)	16785.5	+	36755.1i	0.688240	+	1.50704i	0.853670	+	0.520814i	\(0.174372\pi\)
−0.165430	+	0.986222i	\(0.552901\pi\)
\(31\)	5081.25	+	35340.9i	0.170563	+	1.18629i	0.877698	+	0.479215i	\(0.159078\pi\)
							−0.707134	+	0.707079i	\(0.750012\pi\)
\(37\)	25164.6	−	85702.7i	0.496804	−	1.69196i	−0.204309	−	0.978906i	\(-0.565495\pi\)
							0.701112	−	0.713051i	\(-0.252687\pi\)
\(41\)	−89682.8	+	26333.3i	−1.30124	+	0.382079i	−0.857688	−	0.514170i	\(-0.828100\pi\)
							−0.443552	+	0.896248i	\(0.646282\pi\)
\(43\)	95615.6	+	13747.5i	1.20261	+	0.172909i	0.714348	−	0.699790i	\(-0.246723\pi\)
							0.488258	+	0.872699i	\(0.337632\pi\)
\(47\)	−133376.			−1.28465			−0.642323	−	0.766434i	\(-0.722029\pi\)
							−0.642323	+	0.766434i	\(0.722029\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.7.f.a.34.9	✓	240
23.21	odd	22	inner	69.7.f.a.67.9	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.7.f.a.34.9	✓	240	1.1	even	1	trivial
69.7.f.a.67.9	yes	240	23.21	odd	22	inner