Embedded newform 483.2.y.b.25.8

• Label: 483.2.y.b.25.8
• Level: $483$
• Weight: $2$
• Character: 483.25
• Analytic conductor: $3.857$
• Analytic rank: $0$
• Dimension: $320$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 483 = 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	483.y (of order \(33\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.85677441763\)
Analytic rank:	\(0\)
Dimension:	\(320\)
Relative dimension:	\(16\) over \(\Q(\zeta_{33})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{33}]$

Embedding invariants

Embedding label			25.8
Character	\(\chi\)	\(=\)	483.25
Dual form			483.2.y.b.58.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0101495 + 0.213065i) q^{2} +(0.928368 + 0.371662i) q^{3} +(1.94565 + 0.185787i) q^{4} +(0.752201 + 3.10061i) q^{5} +(-0.0886107 + 0.194030i) q^{6} +(1.41895 - 2.23306i) q^{7} +(-0.120045 + 0.834935i) q^{8} +(0.723734 + 0.690079i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} + 16 q^{3} + 18 q^{4} - 2 q^{5} + 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/483\mathbb{Z}\right)^\times\).

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0101495	+	0.213065i	−0.00717680	+	0.150660i	0.992434	+	0.122776i	\(0.0391797\pi\)
							−0.999611	+	0.0278834i	\(0.991123\pi\)
\(3\)	0.928368	+	0.371662i	0.535993	+	0.214579i
							\(5\)	0.752201	+	3.10061i	0.336394	+	1.38664i	0.849173	+	0.528115i	\(0.177101\pi\)
−0.512778	+	0.858521i	\(0.671384\pi\)
\(7\)	1.41895	−	2.23306i	0.536313	−	0.844019i
							\(11\)	0.0594422	+	1.24784i	0.0179225	+	0.376239i	0.989954	+	0.141389i	\(0.0451569\pi\)
−0.972032	+	0.234850i	\(0.924540\pi\)
\(13\)	−1.60717	−	1.85477i	−0.445748	−	0.514420i	0.487760	−	0.872978i	\(-0.337814\pi\)
							−0.933508	+	0.358558i	\(0.883269\pi\)
\(17\)	−3.32741	−	4.67269i	−0.807015	−	1.13329i	−0.988696	−	0.149936i	\(-0.952093\pi\)
							0.181681	−	0.983358i	\(-0.441846\pi\)
\(19\)	0.0934860	−	0.131283i	0.0214472	−	0.0301183i	−0.803713	−	0.595017i	\(-0.797145\pi\)
							0.825160	+	0.564899i	\(0.191085\pi\)
\(23\)	−4.51567	+	1.61516i	−0.941582	+	0.336785i
							\(29\)	0.470414	−	1.03006i	0.0873537	−	0.191278i	−0.860913	−	0.508752i	\(-0.830107\pi\)
0.948267	+	0.317474i	\(0.102834\pi\)
\(31\)	−7.33334	−	5.76700i	−1.31711	−	1.03578i	−0.996056	−	0.0887222i	\(-0.971722\pi\)
							−0.321050	−	0.947062i	\(-0.604036\pi\)
\(37\)	0.521993	+	0.497720i	0.0858152	+	0.0818246i	0.731773	−	0.681549i	\(-0.238693\pi\)
							−0.645958	+	0.763373i	\(0.723542\pi\)
\(41\)	4.70839	+	1.38251i	0.735326	+	0.215911i	0.627892	−	0.778301i	\(-0.283918\pi\)
							0.107435	+	0.994212i	\(0.465736\pi\)
\(43\)	1.06050	+	7.37596i	0.161725	+	1.12482i	0.895380	+	0.445302i	\(0.146904\pi\)
							−0.733655	+	0.679522i	\(0.762187\pi\)
\(47\)	−2.15646	−	3.73510i	−0.314552	−	0.544821i	0.664790	−	0.747030i	\(-0.268521\pi\)
							−0.979342	+	0.202210i	\(0.935188\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.b.25.8	✓	320
7.2	even	3	inner	483.2.y.b.163.9	yes	320
23.12	even	11	inner	483.2.y.b.403.9	yes	320
161.58	even	33	inner	483.2.y.b.58.8	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.b.25.8	✓	320	1.1	even	1	trivial
483.2.y.b.58.8	yes	320	161.58	even	33	inner
483.2.y.b.163.9	yes	320	7.2	even	3	inner
483.2.y.b.403.9	yes	320	23.12	even	11	inner