Embedded newform 475.2.bc.a.396.1

• Label: 475.2.bc.a.396.1
• Level: $475$
• Weight: $2$
• Character: 475.396
• Analytic conductor: $3.793$
• Analytic rank: $0$
• Dimension: $1152$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	475.bc (of order \(45\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.79289409601\)
Analytic rank:	\(0\)
Dimension:	\(1152\)
Relative dimension:	\(48\) over \(\Q(\zeta_{45})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{45}]$

Embedding invariants

Embedding label			396.1
Character	\(\chi\)	\(=\)	475.396
Dual form			475.2.bc.a.6.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.29973 + 1.43703i) q^{2} +(2.30476 + 0.660880i) q^{3} +(2.34697 - 4.81199i) q^{4} +(2.22377 + 0.234179i) q^{5} +(-6.25005 + 1.79217i) q^{6} +(1.23280 + 2.13527i) q^{7} +(0.950677 + 9.04509i) q^{8} +(2.33103 + 1.45659i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1152 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 24 q^{5} - 18 q^{6} - 30 q^{7} - 9 q^{8} - 18 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/475\mathbb{Z}\right)^\times\).

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{2}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.29973	+	1.43703i	−1.62616	+	1.01614i	−0.662474	+	0.749085i	\(0.730494\pi\)
							−0.963682	+	0.267051i	\(0.913951\pi\)
\(3\)	2.30476	+	0.660880i	1.33066	+	0.381559i	0.864273	−	0.503023i	\(-0.167779\pi\)
							0.466383	+	0.884583i	\(0.345557\pi\)
\(5\)	2.22377	+	0.234179i	0.994501	+	0.104728i
							\(7\)	1.23280	+	2.13527i	0.465955	+	0.807058i	0.999244	−	0.0388752i	\(-0.0123775\pi\)
−0.533289	+	0.845933i	\(0.679044\pi\)
\(11\)	−2.92924	−	3.25325i	−0.883199	−	0.980892i	0.116726	−	0.993164i	\(-0.462760\pi\)
							−0.999925	+	0.0122725i	\(0.996093\pi\)
\(13\)	−0.173547	+	4.96974i	−0.0481333	+	1.37836i	0.699683	+	0.714453i	\(0.253325\pi\)
							−0.747816	+	0.663905i	\(0.768898\pi\)
\(17\)	−4.87508	−	0.340899i	−1.18238	−	0.0826801i	−0.535003	−	0.844850i	\(-0.679690\pi\)
							−0.647377	+	0.762170i	\(0.724134\pi\)
\(19\)	0.388345	+	4.34157i	0.0890925	+	0.996023i
							\(23\)	3.20232	+	0.450057i	0.667730	+	0.0938433i	0.464890	−	0.885369i	\(-0.346094\pi\)
0.202840	+	0.979212i	\(0.434983\pi\)
\(29\)	5.33818	−	0.373282i	0.991275	−	0.0693167i	0.435100	−	0.900382i	\(-0.356713\pi\)
							0.556175	+	0.831065i	\(0.312268\pi\)
\(31\)	7.59354	+	3.38086i	1.36384	+	0.607221i	0.952578	−	0.304296i	\(-0.0984211\pi\)
							0.411263	+	0.911517i	\(0.365088\pi\)
\(37\)	0.586232	+	1.80424i	0.0963759	+	0.296615i	0.987610	−	0.156929i	\(-0.0501595\pi\)
							−0.891234	+	0.453544i	\(0.850160\pi\)
\(41\)	−4.18856	−	5.36111i	−0.654143	−	0.837265i	0.340542	−	0.940229i	\(-0.389390\pi\)
							−0.994685	+	0.102964i	\(0.967167\pi\)
\(43\)	0.722383	−	4.09683i	0.110162	−	0.624761i	−0.878870	−	0.477062i	\(-0.841702\pi\)
							0.989032	−	0.147700i	\(-0.0471870\pi\)
\(47\)	0.529117	−	0.0369994i	0.0771796	−	0.00539692i	−0.0311145	−	0.999516i	\(-0.509906\pi\)
							0.108294	+	0.994119i	\(0.465461\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.bc.a.396.1	yes	1152
19.6	even	9	inner	475.2.bc.a.196.48	yes	1152
25.6	even	5	inner	475.2.bc.a.206.48	yes	1152
475.6	even	45	inner	475.2.bc.a.6.1	✓	1152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.bc.a.6.1	✓	1152	475.6	even	45	inner
475.2.bc.a.196.48	yes	1152	19.6	even	9	inner
475.2.bc.a.206.48	yes	1152	25.6	even	5	inner
475.2.bc.a.396.1	yes	1152	1.1	even	1	trivial