Embedded newform 288.3.be.a.5.17

• Label: 288.3.be.a.5.17
• Level: $288$
• Weight: $3$
• Character: 288.5
• Analytic conductor: $7.847$
• Analytic rank: $0$
• Dimension: $752$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 288 = 2^{5} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	288.be (of order \(24\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.84743161358\)
Analytic rank:	\(0\)
Dimension:	\(752\)
Relative dimension:	\(94\) over \(\Q(\zeta_{24})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{24}]$

Embedding invariants

Embedding label			5.17
Character	\(\chi\)	\(=\)	288.5
Dual form			288.3.be.a.173.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.76073 - 0.948594i) q^{2} +(2.97865 - 0.357260i) q^{3} +(2.20034 + 3.34044i) q^{4} +(-4.50483 - 5.87081i) q^{5} +(-5.58350 - 2.19649i) q^{6} +(0.930710 - 3.47346i) q^{7} +(-0.705487 - 7.96883i) q^{8} +(8.74473 - 2.12831i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 752 q - 12 q^{2} - 8 q^{3} - 4 q^{4} - 12 q^{5} - 8 q^{6} - 4 q^{7} - 8 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/288\mathbb{Z}\right)^\times\).

\(n\)	\(37\)	\(65\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.76073	−	0.948594i	−0.880365	−	0.474297i
							\(3\)	2.97865	−	0.357260i	0.992884	−	0.119087i
\(5\)	−4.50483	−	5.87081i	−0.900966	−	1.17416i								−0.984099	−	0.177624i	\(-0.943159\pi\)
							0.0831323	−	0.996539i	\(-0.473508\pi\)
\(7\)	0.930710	−	3.47346i	0.132959	−	0.496208i	−0.867039	−	0.498240i	\(-0.833980\pi\)
							0.999998	+	0.00203154i	\(0.000646658\pi\)
\(11\)	−0.466426	−	3.54286i	−0.0424023	−	0.322078i	−0.999480	−	0.0322460i	\(-0.989734\pi\)
							0.957078	−	0.289832i	\(-0.0935993\pi\)
\(13\)	−0.219083	+	1.66410i	−0.0168525	+	0.128008i	−0.997803	−	0.0662502i	\(-0.978896\pi\)
							0.980951	+	0.194258i	\(0.0622298\pi\)
\(17\)	−24.0662			−1.41566			−0.707828	−	0.706385i	\(-0.750325\pi\)
							−0.707828	+	0.706385i	\(0.750325\pi\)
\(19\)	−12.2360	+	5.06834i	−0.644003	+	0.266755i	−0.680689	−	0.732572i	\(-0.738320\pi\)
							0.0366867	+	0.999327i	\(0.488320\pi\)
\(23\)	16.6696	−	4.46661i	0.724766	−	0.194200i	0.122469	−	0.992472i	\(-0.460919\pi\)
							0.602297	+	0.798272i	\(0.294252\pi\)
\(29\)	−23.8153	−	18.2741i	−0.821218	−	0.630143i	0.110275	−	0.993901i	\(-0.464827\pi\)
							−0.931493	+	0.363758i	\(0.881493\pi\)
\(31\)	26.0767	−	45.1661i	0.841183	−	1.45697i	−0.0477129	−	0.998861i	\(-0.515193\pi\)
							0.888896	−	0.458110i	\(-0.151473\pi\)
\(37\)	−28.7044	−	11.8898i	−0.775795	−	0.321345i	−0.0405775	−	0.999176i	\(-0.512920\pi\)
							−0.735217	+	0.677832i	\(0.762920\pi\)
\(41\)	−4.38117	+	1.17393i	−0.106858	+	0.0286324i	−0.311852	−	0.950131i	\(-0.600949\pi\)
							0.204994	+	0.978763i	\(0.434283\pi\)
\(43\)	−68.5988	+	9.03120i	−1.59532	+	0.210028i	−0.875179	−	0.483799i	\(-0.839257\pi\)
							−0.720142	+	0.693827i	\(0.755923\pi\)
\(47\)	−31.0122	−	53.7147i	−0.659835	−	1.14287i	−0.980658	−	0.195728i	\(-0.937293\pi\)
							0.320824	−	0.947139i	\(-0.396040\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	288.3.be.a.5.17	✓	752
9.2	odd	6	inner	288.3.be.a.101.47	yes	752
32.13	even	8	inner	288.3.be.a.77.47	yes	752
288.173	odd	24	inner	288.3.be.a.173.17	yes	752

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
288.3.be.a.5.17	✓	752	1.1	even	1	trivial
288.3.be.a.77.47	yes	752	32.13	even	8	inner
288.3.be.a.101.47	yes	752	9.2	odd	6	inner
288.3.be.a.173.17	yes	752	288.173	odd	24	inner