Embedded newform 287.3.v.a.44.1

• Label: 287.3.v.a.44.1
• Level: $287$
• Weight: $3$
• Character: 287.44
• Analytic conductor: $7.820$
• Analytic rank: $0$
• Dimension: $432$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	287.v (of order \(24\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82018358714\)
Analytic rank:	\(0\)
Dimension:	\(432\)
Relative dimension:	\(54\) over \(\Q(\zeta_{24})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{24}]$

Embedding invariants

Embedding label			44.1
Character	\(\chi\)	\(=\)	287.44
Dual form			287.3.v.a.137.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.00113 - 3.73626i) q^{2} +(4.33433 - 0.570626i) q^{3} +(-9.49326 + 5.48094i) q^{4} +(6.12199 - 1.64038i) q^{5} +(-6.47122 - 15.6229i) q^{6} +(-6.59685 - 2.34128i) q^{7} +(19.0416 + 19.0416i) q^{8} +(9.76749 - 2.61719i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 432 q - 4 q^{2} - 4 q^{3} - 4 q^{5} - 16 q^{6} - 8 q^{7} - 48 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/287\mathbb{Z}\right)^\times\).

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{3}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.00113	−	3.73626i	−0.500564	−	1.86813i	−0.496323	−	0.868138i	\(-0.665317\pi\)
							−0.00424042	−	0.999991i	\(-0.501350\pi\)
\(3\)	4.33433	−	0.570626i	1.44478	−	0.190209i	0.632989	−	0.774161i	\(-0.281828\pi\)
							0.811788	+	0.583952i	\(0.198494\pi\)
\(5\)	6.12199	−	1.64038i	1.22440	−	0.328077i	0.412002	−	0.911183i	\(-0.364830\pi\)
							0.812396	+	0.583106i	\(0.198163\pi\)
\(7\)	−6.59685	−	2.34128i	−0.942407	−	0.334468i
							\(11\)	−1.30113	−	9.88309i	−0.118285	−	0.898463i	−0.942972	−	0.332873i	\(-0.891982\pi\)
0.824687	−	0.565590i	\(-0.191351\pi\)
\(13\)	5.74547	−	13.8708i	0.441960	−	1.06698i	−0.533301	−	0.845926i	\(-0.679049\pi\)
							0.975260	−	0.221059i	\(-0.0709514\pi\)
\(17\)	−4.39171	+	5.72339i	−0.258336	+	0.336670i	−0.904457	−	0.426565i	\(-0.859724\pi\)
							0.646121	+	0.763235i	\(0.276390\pi\)
\(19\)	−12.0340	−	1.58431i	−0.633371	−	0.0833848i	−0.192997	−	0.981199i	\(-0.561821\pi\)
							−0.440374	+	0.897815i	\(0.645154\pi\)
\(23\)	−6.23342	−	3.59887i	−0.271018	−	0.156472i	0.358332	−	0.933594i	\(-0.383346\pi\)
							−0.629350	+	0.777122i	\(0.716679\pi\)
\(29\)	10.3605	−	25.0124i	0.357258	−	0.862498i	−0.638425	−	0.769684i	\(-0.720414\pi\)
							0.995683	−	0.0928142i	\(-0.0295863\pi\)
\(31\)	47.9772	−	27.6996i	1.54765	−	0.893537i	0.549331	−	0.835605i	\(-0.314883\pi\)
							0.998321	−	0.0579319i	\(-0.0184506\pi\)
\(37\)	−16.2343	+	28.1187i	−0.438766	+	0.759965i	−0.997595	−	0.0693184i	\(-0.977918\pi\)
							0.558829	+	0.829283i	\(0.311251\pi\)
\(41\)	1.71066	+	40.9643i	0.0417233	+	0.999129i
							\(43\)	−11.8475	+	11.8475i	−0.275524	+	0.275524i	−0.831319	−	0.555796i	\(-0.812414\pi\)
0.555796	+	0.831319i	\(0.312414\pi\)
\(47\)	67.0611	+	8.82876i	1.42683	+	0.187846i	0.804055	−	0.594554i	\(-0.202671\pi\)
							0.622776	+	0.782400i	\(0.286005\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.v.a.44.1	✓	432
7.4	even	3	inner	287.3.v.a.249.54	yes	432
41.14	odd	8	inner	287.3.v.a.219.54	yes	432
287.137	odd	24	inner	287.3.v.a.137.1	yes	432

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.v.a.44.1	✓	432	1.1	even	1	trivial
287.3.v.a.137.1	yes	432	287.137	odd	24	inner
287.3.v.a.219.54	yes	432	41.14	odd	8	inner
287.3.v.a.249.54	yes	432	7.4	even	3	inner