Embedded newform 287.3.bd.a.5.18

• Label: 287.3.bd.a.5.18
• Level: $287$
• Weight: $3$
• Character: 287.5
• Analytic conductor: $7.820$
• Analytic rank: $0$
• Dimension: $864$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	287.bd (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82018358714\)
Analytic rank:	\(0\)
Dimension:	\(864\)
Relative dimension:	\(54\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			5.18
Character	\(\chi\)	\(=\)	287.5
Dual form			287.3.bd.a.115.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.80705 + 0.189928i) q^{2} +(-2.98183 - 0.798979i) q^{3} +(-0.683242 + 0.145228i) q^{4} +(0.710415 + 0.788996i) q^{5} +(5.54006 + 0.877459i) q^{6} +(-2.34483 + 6.59559i) q^{7} +(8.11935 - 2.63814i) q^{8} +(0.458720 + 0.264842i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 864 q - 10 q^{2} - 24 q^{3} - 214 q^{4} - 30 q^{5} - 16 q^{7} - 40 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{11}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.80705	+	0.189928i	−0.903524	+	0.0949642i	−0.544884	−	0.838512i	\(-0.683426\pi\)
							−0.358640	+	0.933476i	\(0.616760\pi\)
\(3\)	−2.98183	−	0.798979i	−0.993944	−	0.266326i	−0.275037	−	0.961434i	\(-0.588690\pi\)
							−0.718906	+	0.695107i	\(0.755357\pi\)
\(5\)	0.710415	+	0.788996i	0.142083	+	0.157799i	0.809986	−	0.586449i	\(-0.199475\pi\)
							−0.667903	+	0.744248i	\(0.732808\pi\)
\(7\)	−2.34483	+	6.59559i	−0.334976	+	0.942227i
							\(11\)	7.97962	−	0.418194i	0.725420	−	0.0380176i	0.313948	−	0.949440i	\(-0.398348\pi\)
0.411472	+	0.911423i	\(0.365015\pi\)
\(13\)	−3.64162	+	22.9923i	−0.280124	+	1.76864i	0.299835	+	0.953991i	\(0.403068\pi\)
							−0.579960	+	0.814645i	\(0.696932\pi\)
\(17\)	9.20264	−	0.482290i	0.541332	−	0.0283700i	0.220289	−	0.975435i	\(-0.429300\pi\)
							0.321043	+	0.947065i	\(0.395967\pi\)
\(19\)	−6.90769	−	2.65161i	−0.363563	−	0.139559i	0.169728	−	0.985491i	\(-0.445711\pi\)
							−0.533290	+	0.845932i	\(0.679045\pi\)
\(23\)	−2.20143	−	20.9452i	−0.0957142	−	0.910660i	−0.932022	−	0.362401i	\(-0.881957\pi\)
							0.836308	−	0.548260i	\(-0.184709\pi\)
\(29\)	−45.5002	+	23.1835i	−1.56897	+	0.799431i	−0.999741	−	0.0227471i	\(-0.992759\pi\)
							−0.569230	+	0.822178i	\(0.692759\pi\)
\(31\)	−16.3815	−	14.7500i	−0.528436	−	0.475806i	0.361192	−	0.932491i	\(-0.382370\pi\)
							−0.889628	+	0.456685i	\(0.849037\pi\)
\(37\)	−38.6905	−	42.9702i	−1.04569	−	1.16136i	−0.986609	−	0.163104i	\(-0.947849\pi\)
							−0.0590815	−	0.998253i	\(-0.518817\pi\)
\(41\)	−18.7300	−	36.4717i	−0.456830	−	0.889554i
							\(43\)	15.5916	+	21.4600i	0.362595	+	0.499070i	0.950870	−	0.309592i	\(-0.100192\pi\)
−0.588274	+	0.808662i	\(0.700192\pi\)
\(47\)	34.6060	−	42.7348i	0.736297	−	0.909251i	−0.262146	−	0.965028i	\(-0.584430\pi\)
							0.998443	+	0.0557773i	\(0.0177637\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.bd.a.5.18	✓	864
7.3	odd	6	inner	287.3.bd.a.87.37	yes	864
41.33	even	20	inner	287.3.bd.a.33.37	yes	864
287.115	odd	60	inner	287.3.bd.a.115.18	yes	864

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.bd.a.5.18	✓	864	1.1	even	1	trivial
287.3.bd.a.33.37	yes	864	41.33	even	20	inner
287.3.bd.a.87.37	yes	864	7.3	odd	6	inner
287.3.bd.a.115.18	yes	864	287.115	odd	60	inner