Embedded newform 475.2.bc.a.111.10

• Label: 475.2.bc.a.111.10
• Level: $475$
• Weight: $2$
• Character: 475.111
• 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			111.10
Character	\(\chi\)	\(=\)	475.111
Dual form			475.2.bc.a.291.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.84706 - 0.129159i) q^{2} +(-0.0736767 - 2.10983i) q^{3} +(1.41441 + 0.198783i) q^{4} +(-1.96937 - 1.05905i) q^{5} +(-0.136418 + 3.90649i) q^{6} +(-2.35929 - 4.08640i) q^{7} +(1.03539 + 0.220078i) q^{8} +(-1.45324 + 0.101621i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1152 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 24 q^{5} - 18 q^{6} - 30 q^{7} - 9 q^{8} - 18 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{2}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.84706	−	0.129159i	−1.30607	−	0.0913292i	−0.600302	−	0.799773i	\(-0.704953\pi\)
							−0.705767	+	0.708444i	\(0.749397\pi\)
\(3\)	−0.0736767	−	2.10983i	−0.0425373	−	1.21811i	−0.813629	−	0.581384i	\(-0.802511\pi\)
							0.771092	−	0.636724i	\(-0.219711\pi\)
\(5\)	−1.96937	−	1.05905i	−0.880727	−	0.473623i
							\(7\)	−2.35929	−	4.08640i	−0.891726	−	1.54452i	−0.837804	−	0.545970i	\(-0.816161\pi\)
−0.0539221	−	0.998545i	\(-0.517172\pi\)
\(11\)	−0.430600	−	4.09689i	−0.129831	−	1.23526i	−0.844408	−	0.535701i	\(-0.820048\pi\)
							0.714577	−	0.699557i	\(-0.246619\pi\)
\(13\)	−0.277808	−	0.411867i	−0.0770501	−	0.114231i	0.788543	−	0.614980i	\(-0.210836\pi\)
							−0.865593	+	0.500749i	\(0.833058\pi\)
\(17\)	2.73459	−	6.76836i	0.663236	−	1.64157i	−0.0996403	−	0.995024i	\(-0.531769\pi\)
							0.762877	−	0.646544i	\(-0.223786\pi\)
\(19\)	−2.38417	+	3.64907i	−0.546966	+	0.837155i
							\(23\)	0.906872	+	0.875756i	0.189096	+	0.182608i	0.783180	−	0.621795i	\(-0.213596\pi\)
−0.594084	+	0.804403i	\(0.702485\pi\)
\(29\)	0.273146	+	0.676061i	0.0507220	+	0.125541i	0.950418	−	0.310976i	\(-0.100656\pi\)
							−0.899696	+	0.436518i	\(0.856212\pi\)
\(31\)	2.13136	−	2.36711i	0.382803	−	0.425146i	−0.520692	−	0.853745i	\(-0.674326\pi\)
							0.903495	+	0.428599i	\(0.140993\pi\)
\(37\)	4.68518	+	3.40398i	0.770239	+	0.559611i	0.902034	−	0.431666i	\(-0.142074\pi\)
							−0.131795	+	0.991277i	\(0.542074\pi\)
\(41\)	7.88574	+	2.26120i	1.23155	+	0.353140i	0.827430	−	0.561568i	\(-0.189802\pi\)
							0.404115	+	0.914708i	\(0.367580\pi\)
\(43\)	−1.20872	+	6.85502i	−0.184329	+	1.04538i	0.742486	+	0.669861i	\(0.233647\pi\)
							−0.926815	+	0.375519i	\(0.877465\pi\)
\(47\)	−2.41048	−	5.96615i	−0.351605	−	0.870253i	−0.994542	−	0.104334i	\(-0.966729\pi\)
							0.642937	−	0.765919i	\(-0.277716\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.bc.a.111.10	yes	1152
19.6	even	9	inner	475.2.bc.a.386.10	yes	1152
25.16	even	5	inner	475.2.bc.a.16.10	✓	1152
475.291	even	45	inner	475.2.bc.a.291.10	yes	1152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.bc.a.16.10	✓	1152	25.16	even	5	inner
475.2.bc.a.111.10	yes	1152	1.1	even	1	trivial
475.2.bc.a.291.10	yes	1152	475.291	even	45	inner
475.2.bc.a.386.10	yes	1152	19.6	even	9	inner