Embedded newform 287.2.be.a.12.2

• Label: 287.2.be.a.12.2
• Level: $287$
• Weight: $2$
• Character: 287.12
• Analytic conductor: $2.292$
• Analytic rank: $0$
• Dimension: $832$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	287.be (of order \(120\), degree \(32\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.29170653801\)
Analytic rank:	\(0\)
Dimension:	\(832\)
Relative dimension:	\(26\) over \(\Q(\zeta_{120})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{120}]$

Embedding invariants

Embedding label			12.2
Character	\(\chi\)	\(=\)	287.12
Dual form			287.2.be.a.24.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.93631 - 1.56800i) q^{2} +(-2.94781 + 0.388087i) q^{3} +(0.874875 + 4.11596i) q^{4} +(0.221276 + 0.0115966i) q^{5} +(6.31640 + 3.87070i) q^{6} +(-0.519127 + 2.59432i) q^{7} +(2.49748 - 4.90158i) q^{8} +(5.64121 - 1.51156i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 832 q - 16 q^{2} - 48 q^{3} - 20 q^{4} - 48 q^{5} - 32 q^{7} - 48 q^{8} - 24 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{27}{40}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.93631	−	1.56800i	−1.36918	−	1.10874i	−0.982446	−	0.186549i	\(-0.940270\pi\)
							−0.386734	−	0.922191i	\(-0.626397\pi\)
\(3\)	−2.94781	+	0.388087i	−1.70192	+	0.224062i	−0.917939	−	0.396722i	\(-0.870148\pi\)
							−0.783982	+	0.620784i	\(0.786814\pi\)
\(5\)	0.221276	+	0.0115966i	0.0989579	+	0.00518616i	0.101751	−	0.994810i	\(-0.467556\pi\)
							−0.00279275	+	0.999996i	\(0.500889\pi\)
\(7\)	−0.519127	+	2.59432i	−0.196211	+	0.980562i
							\(11\)	−1.69000	−	3.54317i	−0.509555	−	1.06830i	−0.982336	−	0.187125i	\(-0.940083\pi\)
0.472781	−	0.881180i	\(-0.343250\pi\)
\(13\)	0.818719	+	3.41021i	0.227072	+	0.945822i	0.963158	+	0.268934i	\(0.0866715\pi\)
							−0.736087	+	0.676887i	\(0.763329\pi\)
\(17\)	3.06494	+	1.08535i	0.743358	+	0.263237i	0.678442	−	0.734654i	\(-0.262655\pi\)
							0.0649160	+	0.997891i	\(0.479322\pi\)
\(19\)	−1.42280	−	1.35019i	−0.326413	−	0.309754i	0.507498	−	0.861653i	\(-0.330570\pi\)
							−0.833911	+	0.551898i	\(0.813904\pi\)
\(23\)	−7.46902	+	0.785025i	−1.55740	+	0.163689i	−0.843719	−	0.536785i	\(-0.819639\pi\)
							−0.713678	+	0.700474i	\(0.752972\pi\)
\(29\)	−0.0897382	+	0.105070i	−0.0166640	+	0.0195110i	−0.768675	−	0.639639i	\(-0.779084\pi\)
							0.752011	+	0.659150i	\(0.229084\pi\)
\(31\)	3.36876	−	3.74138i	0.605047	−	0.671972i	−0.360332	−	0.932824i	\(-0.617337\pi\)
							0.965379	+	0.260852i	\(0.0840033\pi\)
\(37\)	−1.18420	−	1.31519i	−0.194682	−	0.216216i	0.637899	−	0.770120i	\(-0.279804\pi\)
							−0.832580	+	0.553905i	\(0.813137\pi\)
\(41\)	4.84190	−	4.18999i	0.756178	−	0.654366i
							\(43\)	1.96704	−	12.4194i	0.299970	−	1.89394i	−0.130690	−	0.991423i	\(-0.541719\pi\)
0.430660	−	0.902514i	\(-0.358281\pi\)
\(47\)	0.124269	−	0.0674724i	0.0181265	−	0.00984187i	−0.468070	−	0.883691i	\(-0.655051\pi\)
							0.486197	+	0.873849i	\(0.338384\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.be.a.12.2	✓	832
7.3	odd	6	inner	287.2.be.a.94.25	yes	832
41.24	odd	40	inner	287.2.be.a.229.25	yes	832
287.24	even	120	inner	287.2.be.a.24.2	yes	832

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.be.a.12.2	✓	832	1.1	even	1	trivial
287.2.be.a.24.2	yes	832	287.24	even	120	inner
287.2.be.a.94.25	yes	832	7.3	odd	6	inner
287.2.be.a.229.25	yes	832	41.24	odd	40	inner