Embedded newform 475.2.bc.a.6.2

• Label: 475.2.bc.a.6.2
• Level: $475$
• Weight: $2$
• Character: 475.6
• 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			6.2
Character	\(\chi\)	\(=\)	475.6
Dual form			475.2.bc.a.396.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.29691 - 1.43527i) q^{2} +(-2.18186 + 0.625639i) q^{3} +(2.33907 + 4.79580i) q^{4} +(1.19746 + 1.88841i) q^{5} +(5.90951 + 1.69453i) q^{6} +(0.140645 - 0.243604i) q^{7} +(0.944410 - 8.98546i) q^{8} +(1.82496 - 1.14036i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1152 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 24 q^{5} - 18 q^{6} - 30 q^{7} - 9 q^{8} - 18 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{7}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.29691	−	1.43527i	−1.62416	−	1.01489i	−0.964771	−	0.263091i	\(-0.915258\pi\)
							−0.659392	−	0.751799i	\(-0.729186\pi\)
\(3\)	−2.18186	+	0.625639i	−1.25970	+	0.361213i	−0.838074	−	0.545557i	\(-0.816318\pi\)
							−0.421626	+	0.906770i	\(0.638540\pi\)
\(5\)	1.19746	+	1.88841i	0.535521	+	0.844522i
							\(7\)	0.140645	−	0.243604i	0.0531588	−	0.0920738i	−0.838221	−	0.545330i	\(-0.816404\pi\)
0.891380	+	0.453256i	\(0.149738\pi\)
\(11\)	2.37271	−	2.63516i	0.715399	−	0.794531i	−0.270348	−	0.962763i	\(-0.587139\pi\)
							0.985748	+	0.168231i	\(0.0538055\pi\)
\(13\)	0.0497317	+	1.42413i	0.0137931	+	0.394983i	0.986580	+	0.163280i	\(0.0522074\pi\)
							−0.972787	+	0.231703i	\(0.925570\pi\)
\(17\)	−2.57858	+	0.180312i	−0.625397	+	0.0437320i	−0.378940	−	0.925421i	\(-0.623711\pi\)
							−0.246458	+	0.969153i	\(0.579267\pi\)
\(19\)	4.29120	+	0.765232i	0.984469	+	0.175556i
							\(23\)	6.72332	−	0.944901i	1.40191	−	0.197025i	0.602647	−	0.798008i	\(-0.294113\pi\)
0.799262	+	0.600983i	\(0.205224\pi\)
\(29\)	−3.22577	−	0.225568i	−0.599010	−	0.0418868i	−0.232967	−	0.972485i	\(-0.574843\pi\)
							−0.366043	+	0.930598i	\(0.619288\pi\)
\(31\)	−5.63863	+	2.51048i	−1.01273	+	0.450895i	−0.844902	−	0.534921i	\(-0.820341\pi\)
							−0.167825	+	0.985817i	\(0.553675\pi\)
\(37\)	0.698388	−	2.14942i	0.114814	−	0.353362i	−0.877094	−	0.480319i	\(-0.840521\pi\)
							0.991908	+	0.126957i	\(0.0405210\pi\)
\(41\)	6.29873	−	8.06201i	0.983697	−	1.25907i	0.0178820	−	0.999840i	\(-0.494308\pi\)
							0.965815	−	0.259234i	\(-0.0834701\pi\)
\(43\)	0.849231	+	4.81623i	0.129506	+	0.734468i	0.978529	+	0.206111i	\(0.0660806\pi\)
							−0.849022	+	0.528357i	\(0.822808\pi\)
\(47\)	−4.27564	−	0.298982i	−0.623667	−	0.0436110i	−0.245573	−	0.969378i	\(-0.578976\pi\)
							−0.378094	+	0.925767i	\(0.623420\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.bc.a.6.2	✓	1152
19.16	even	9	inner	475.2.bc.a.206.47	yes	1152
25.21	even	5	inner	475.2.bc.a.196.47	yes	1152
475.396	even	45	inner	475.2.bc.a.396.2	yes	1152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.bc.a.6.2	✓	1152	1.1	even	1	trivial
475.2.bc.a.196.47	yes	1152	25.21	even	5	inner
475.2.bc.a.206.47	yes	1152	19.16	even	9	inner
475.2.bc.a.396.2	yes	1152	475.396	even	45	inner