Embedded newform 475.2.bi.a.117.46

• Label: 475.2.bi.a.117.46
• Level: $475$
• Weight: $2$
• Character: 475.117
• Analytic conductor: $3.793$
• Analytic rank: $0$
• Dimension: $2304$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	475.bi (of order \(180\), degree \(48\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.79289409601\)
Analytic rank:	\(0\)
Dimension:	\(2304\)
Relative dimension:	\(48\) over \(\Q(\zeta_{180})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{180}]$

Embedding invariants

Embedding label			117.46
Character	\(\chi\)	\(=\)	475.117
Dual form			475.2.bi.a.203.46

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.02238 + 2.40858i) q^{2} +(-2.61303 + 0.507921i) q^{3} +(-3.36668 + 3.48630i) q^{4} +(-2.13894 - 0.651867i) q^{5} +(-3.89488 - 5.77439i) q^{6} +(-1.09276 + 4.07822i) q^{7} +(-6.95347 - 2.66919i) q^{8} +(3.78837 - 1.53060i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2304 q - 48 q^{2} - 48 q^{3} - 60 q^{4} - 48 q^{5} - 36 q^{6} - 30 q^{7} - 72 q^{8} - 60 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{13}{20}\right)\)	\(e\left(\frac{13}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.02238	+	2.40858i	0.722933	+	1.70312i	0.710997	+	0.703195i	\(0.248244\pi\)
							0.0119356	+	0.999929i	\(0.496201\pi\)
\(3\)	−2.61303	+	0.507921i	−1.50863	+	0.293248i	−0.876420	−	0.481548i	\(-0.840075\pi\)
							−0.632211	+	0.774796i	\(0.717853\pi\)
\(5\)	−2.13894	−	0.651867i	−0.956564	−	0.291524i
							\(7\)	−1.09276	+	4.07822i	−0.413023	+	1.54142i	0.375739	+	0.926725i	\(0.377389\pi\)
−0.788763	+	0.614698i	\(0.789278\pi\)
\(11\)	3.20081	+	0.680353i	0.965080	+	0.205134i	0.663392	−	0.748272i	\(-0.269116\pi\)
							0.301689	+	0.953407i	\(0.402450\pi\)
\(13\)	1.12128	−	0.137675i	0.310986	−	0.0381843i	0.0351285	−	0.999383i	\(-0.488816\pi\)
							0.275858	+	0.961199i	\(0.411038\pi\)
\(17\)	1.62579	−	2.70577i	0.394312	−	0.656245i	−0.594723	−	0.803931i	\(-0.702738\pi\)
							0.989034	+	0.147686i	\(0.0471825\pi\)
\(19\)	−4.07989	+	1.53444i	−0.935990	+	0.352026i
							\(23\)	−1.04400	+	3.41476i	−0.217689	+	0.712027i	0.778722	+	0.627370i	\(0.215868\pi\)
−0.996410	+	0.0846575i	\(0.973020\pi\)
\(29\)	2.20289	−	8.83532i	0.409067	−	1.64068i	−0.313654	−	0.949537i	\(-0.601553\pi\)
							0.722721	−	0.691140i	\(-0.242891\pi\)
\(31\)	−6.77803	−	0.712400i	−1.21737	−	0.127951i	−0.526021	−	0.850472i	\(-0.676317\pi\)
							−0.691349	+	0.722521i	\(0.742983\pi\)
\(37\)	2.75789	+	5.41267i	0.453394	+	0.889837i	0.998669	+	0.0515791i	\(0.0164254\pi\)
							−0.545274	+	0.838258i	\(0.683575\pi\)
\(41\)	−9.56524	+	0.334025i	−1.49384	+	0.0521660i	−0.770107	−	0.637915i	\(-0.779797\pi\)
							−0.723732	+	0.690081i	\(0.757575\pi\)
\(43\)	−1.00961	−	1.44187i	−0.153964	−	0.219883i	0.734759	−	0.678328i	\(-0.237295\pi\)
							−0.888723	+	0.458446i	\(0.848406\pi\)
\(47\)	−6.64401	+	3.99212i	−0.969129	+	0.582311i	−0.909409	−	0.415904i	\(-0.863465\pi\)
							−0.0597201	+	0.998215i	\(0.519021\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.bi.a.117.46	yes	2304
19.13	odd	18	inner	475.2.bi.a.317.3	yes	2304
25.3	odd	20	inner	475.2.bi.a.3.3	✓	2304
475.203	even	180	inner	475.2.bi.a.203.46	yes	2304

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.bi.a.3.3	✓	2304	25.3	odd	20	inner
475.2.bi.a.117.46	yes	2304	1.1	even	1	trivial
475.2.bi.a.203.46	yes	2304	475.203	even	180	inner
475.2.bi.a.317.3	yes	2304	19.13	odd	18	inner