Embedded newform 385.2.bg.a.16.4

• Label: 385.2.bg.a.16.4
• Level: $385$
• Weight: $2$
• Character: 385.16
• Analytic conductor: $3.074$
• Analytic rank: $0$
• Dimension: $128$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 385 = 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	385.bg (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.07424047782\)
Analytic rank:	\(0\)
Dimension:	\(128\)
Relative dimension:	\(16\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			16.4
Character	\(\chi\)	\(=\)	385.16
Dual form			385.2.bg.a.361.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.73586 - 0.368968i) q^{2} +(0.324493 - 3.08734i) q^{3} +(1.04998 + 0.467481i) q^{4} +(-0.669131 - 0.743145i) q^{5} +(-1.70240 + 5.23946i) q^{6} +(0.427265 + 2.61102i) q^{7} +(1.22130 + 0.887324i) q^{8} +(-6.49193 - 1.37990i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q + 3 q^{3} + 20 q^{4} - 16 q^{5} - 6 q^{7} + 20 q^{8} + 13 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/385\mathbb{Z}\right)^\times\).

\(n\)	\(211\)	\(232\)	\(276\)
\(\chi(n)\)	\(e\left(\frac{2}{5}\right)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.73586	−	0.368968i	−1.22744	−	0.260900i	−0.451824	−	0.892107i	\(-0.649226\pi\)
							−0.775614	+	0.631207i	\(0.782560\pi\)
\(3\)	0.324493	−	3.08734i	0.187346	−	1.78248i	−0.347652	−	0.937623i	\(-0.613021\pi\)
							0.534998	−	0.844853i	\(-0.320312\pi\)
\(5\)	−0.669131	−	0.743145i	−0.299244	−	0.332344i
							\(7\)	0.427265	+	2.61102i	0.161491	+	0.986874i
\(11\)	−2.85108	+	1.69451i	−0.859632	+	0.510913i
							\(13\)	1.56545	+	4.81797i	0.434179	+	1.33627i	0.893926	+	0.448215i	\(0.147940\pi\)
−0.459747	+	0.888050i	\(0.652060\pi\)
\(17\)	−6.63974	+	1.41132i	−1.61037	+	0.342296i	−0.923237	−	0.384231i	\(-0.874466\pi\)
							−0.687138	+	0.726527i	\(0.741133\pi\)
\(19\)	−4.11279	+	1.83113i	−0.943539	+	0.420091i	−0.820074	−	0.572258i	\(-0.806068\pi\)
							−0.123466	+	0.992349i	\(0.539401\pi\)
\(23\)	1.37086	−	2.37440i	0.285844	−	0.495096i	−0.686969	−	0.726686i	\(-0.741059\pi\)
							0.972814	+	0.231590i	\(0.0743927\pi\)
\(29\)	1.53934	−	1.11840i	0.285849	−	0.207681i	−0.435616	−	0.900133i	\(-0.643469\pi\)
							0.721464	+	0.692452i	\(0.243469\pi\)
\(31\)	1.69731	−	1.88506i	0.304846	−	0.338566i	−0.571184	−	0.820822i	\(-0.693516\pi\)
							0.876030	+	0.482256i	\(0.160182\pi\)
\(37\)	0.227490	+	2.16442i	0.0373991	+	0.355829i	0.997178	+	0.0750720i	\(0.0239187\pi\)
							−0.959779	+	0.280757i	\(0.909415\pi\)
\(41\)	−7.04414	−	5.11787i	−1.10011	−	0.799277i	−0.119032	−	0.992890i	\(-0.537979\pi\)
							−0.981078	+	0.193614i	\(0.937979\pi\)
\(43\)	−4.20393			−0.641093			−0.320546	−	0.947233i	\(-0.603867\pi\)
							−0.320546	+	0.947233i	\(0.603867\pi\)
\(47\)	2.48297	−	1.10549i	0.362179	−	0.161252i	−0.217576	−	0.976043i	\(-0.569815\pi\)
							0.579755	+	0.814791i	\(0.303148\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	385.2.bg.a.16.4	✓	128
7.4	even	3	inner	385.2.bg.a.291.13	yes	128
11.9	even	5	inner	385.2.bg.a.86.13	yes	128
77.53	even	15	inner	385.2.bg.a.361.4	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
385.2.bg.a.16.4	✓	128	1.1	even	1	trivial
385.2.bg.a.86.13	yes	128	11.9	even	5	inner
385.2.bg.a.291.13	yes	128	7.4	even	3	inner
385.2.bg.a.361.4	yes	128	77.53	even	15	inner