Embedded newform 287.3.bd.a.5.2

• Label: 287.3.bd.a.5.2
• 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			5.2
Character	\(\chi\)	\(=\)	287.5
Dual form			287.3.bd.a.115.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.78322 + 0.397633i) q^{2} +(-2.36963 - 0.634941i) q^{3} +(10.2421 - 2.17702i) q^{4} +(-4.89261 - 5.43380i) q^{5} +(9.21731 + 1.45988i) q^{6} +(-4.50728 + 5.35578i) q^{7} +(-23.4109 + 7.60666i) q^{8} +(-2.58223 - 1.49085i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 864 q - 10 q^{2} - 24 q^{3} - 214 q^{4} - 30 q^{5} - 16 q^{7} - 40 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.78322	+	0.397633i	−1.89161	+	0.198816i	−0.978120	−	0.208042i	\(-0.933291\pi\)
							−0.913491	+	0.406858i	\(0.866624\pi\)
\(3\)	−2.36963	−	0.634941i	−0.789877	−	0.211647i	−0.158742	−	0.987320i	\(-0.550744\pi\)
							−0.631135	+	0.775673i	\(0.717411\pi\)
\(5\)	−4.89261	−	5.43380i	−0.978522	−	1.08676i	−0.996216	−	0.0869156i	\(-0.972299\pi\)
							0.0176934	−	0.999843i	\(-0.494368\pi\)
\(7\)	−4.50728	+	5.35578i	−0.643898	+	0.765112i
							\(11\)	1.25628	−	0.0658387i	0.114207	−	0.00598534i	0.00485319	−	0.999988i	\(-0.498455\pi\)
0.109354	+	0.994003i	\(0.465122\pi\)
\(13\)	−1.19930	+	7.57209i	−0.0922539	+	0.582468i	0.897648	+	0.440713i	\(0.145274\pi\)
							−0.989902	+	0.141755i	\(0.954726\pi\)
\(17\)	−29.5543	+	1.54887i	−1.73849	+	0.0911103i	−0.895008	−	0.446051i	\(-0.852830\pi\)
							−0.843480	+	0.537161i	\(0.819497\pi\)
\(19\)	−30.0385	−	11.5307i	−1.58098	−	0.606880i	−0.599826	−	0.800131i	\(-0.704764\pi\)
							−0.981149	+	0.193251i	\(0.938097\pi\)
\(23\)	−3.34584	−	31.8335i	−0.145471	−	1.38407i	−0.786993	−	0.616962i	\(-0.788363\pi\)
							0.641522	−	0.767105i	\(-0.278303\pi\)
\(29\)	15.4244	−	7.85914i	0.531877	−	0.271005i	−0.167359	−	0.985896i	\(-0.553524\pi\)
							0.699236	+	0.714891i	\(0.253524\pi\)
\(31\)	6.51344	+	5.86472i	0.210111	+	0.189185i	0.767466	−	0.641089i	\(-0.221517\pi\)
							−0.557355	+	0.830274i	\(0.688184\pi\)
\(37\)	34.5306	+	38.3501i	0.933259	+	1.03649i	0.999252	+	0.0386690i	\(0.0123118\pi\)
							−0.0659931	+	0.997820i	\(0.521022\pi\)
\(41\)	11.9960	+	39.2058i	0.292584	+	0.956240i
							\(43\)	−23.6154	−	32.5039i	−0.549196	−	0.755904i	0.440707	−	0.897651i	\(-0.354728\pi\)
−0.989903	+	0.141748i	\(0.954728\pi\)
\(47\)	−9.99761	+	12.3460i	−0.212715	+	0.262681i	−0.872325	−	0.488927i	\(-0.837388\pi\)
							0.659610	+	0.751608i	\(0.270722\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.bd.a.5.2	✓	864
7.3	odd	6	inner	287.3.bd.a.87.53	yes	864
41.33	even	20	inner	287.3.bd.a.33.53	yes	864
287.115	odd	60	inner	287.3.bd.a.115.2	yes	864

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.bd.a.5.2	✓	864	1.1	even	1	trivial
287.3.bd.a.33.53	yes	864	41.33	even	20	inner
287.3.bd.a.87.53	yes	864	7.3	odd	6	inner
287.3.bd.a.115.2	yes	864	287.115	odd	60	inner