Embedded newform 475.2.bc.a.396.17

• Label: 475.2.bc.a.396.17
• 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			396.17
Character	\(\chi\)	\(=\)	475.396
Dual form			475.2.bc.a.6.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.918651 + 0.574037i) q^{2} +(-1.08630 - 0.311492i) q^{3} +(-0.362341 + 0.742910i) q^{4} +(-0.569939 + 2.16221i) q^{5} +(1.17674 - 0.337424i) q^{6} +(-1.35582 - 2.34835i) q^{7} +(-0.320054 - 3.04511i) q^{8} +(-1.46112 - 0.913011i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1152 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 24 q^{5} - 18 q^{6} - 30 q^{7} - 9 q^{8} - 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.918651	+	0.574037i	−0.649584	+	0.405905i	−0.814307	−	0.580435i	\(-0.802883\pi\)
							0.164723	+	0.986340i	\(0.447327\pi\)
\(3\)	−1.08630	−	0.311492i	−0.627176	−	0.179840i	−0.0530300	−	0.998593i	\(-0.516888\pi\)
							−0.574146	+	0.818753i	\(0.694666\pi\)
\(5\)	−0.569939	+	2.16221i	−0.254884	+	0.966972i
							\(7\)	−1.35582	−	2.34835i	−0.512451	−	0.887592i	−0.999896	−	0.0144378i	\(-0.995404\pi\)
0.487444	−	0.873154i	\(-0.337929\pi\)
\(11\)	0.234143	+	0.260042i	0.0705967	+	0.0784056i	0.777408	−	0.628996i	\(-0.216534\pi\)
							−0.706812	+	0.707402i	\(0.749867\pi\)
\(13\)	0.0306491	−	0.877676i	0.00850054	−	0.243424i	−0.987701	−	0.156352i	\(-0.950027\pi\)
							0.996202	−	0.0870721i	\(-0.0277510\pi\)
\(17\)	5.64801	+	0.394948i	1.36984	+	0.0957889i	0.735638	−	0.677375i	\(-0.236882\pi\)
							0.634207	+	0.773164i	\(0.281327\pi\)
\(19\)	4.32744	+	0.522754i	0.992783	+	0.119928i
							\(23\)	−4.73422	−	0.665352i	−0.987154	−	0.138735i	−0.372923	−	0.927862i	\(-0.621645\pi\)
−0.614230	+	0.789127i	\(0.710533\pi\)
\(29\)	−3.35920	+	0.234898i	−0.623787	+	0.0436194i	−0.378153	−	0.925743i	\(-0.623441\pi\)
							−0.245634	+	0.969363i	\(0.578996\pi\)
\(31\)	8.11144	+	3.61145i	1.45686	+	0.648635i	0.973893	−	0.227005i	\(-0.0728935\pi\)
							0.482964	+	0.875640i	\(0.339560\pi\)
\(37\)	−2.08017	−	6.40210i	−0.341977	−	1.05250i	−0.963182	−	0.268850i	\(-0.913356\pi\)
							0.621205	−	0.783648i	\(-0.286644\pi\)
\(41\)	3.28618	+	4.20612i	0.513215	+	0.656886i	0.972908	−	0.231192i	\(-0.0742627\pi\)
							−0.459693	+	0.888078i	\(0.652040\pi\)
\(43\)	2.07113	−	11.7459i	0.315844	−	1.79124i	−0.251605	−	0.967830i	\(-0.580958\pi\)
							0.567449	−	0.823409i	\(-0.307930\pi\)
\(47\)	13.0163	−	0.910185i	1.89862	−	0.132764i	0.927875	−	0.372892i	\(-0.121634\pi\)
							0.970741	+	0.240127i	\(0.0771892\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.bc.a.396.17	yes	1152
19.6	even	9	inner	475.2.bc.a.196.32	yes	1152
25.6	even	5	inner	475.2.bc.a.206.32	yes	1152
475.6	even	45	inner	475.2.bc.a.6.17	✓	1152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.bc.a.6.17	✓	1152	475.6	even	45	inner
475.2.bc.a.196.32	yes	1152	19.6	even	9	inner
475.2.bc.a.206.32	yes	1152	25.6	even	5	inner
475.2.bc.a.396.17	yes	1152	1.1	even	1	trivial