Embedded newform 483.2.bc.a.11.9

• Label: 483.2.bc.a.11.9
• Level: $483$
• Weight: $2$
• Character: 483.11
• Analytic conductor: $3.857$
• Analytic rank: $0$
• Dimension: $1200$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 483 = 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	483.bc (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.85677441763\)
Analytic rank:	\(0\)
Dimension:	\(1200\)
Relative dimension:	\(60\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			11.9
Character	\(\chi\)	\(=\)	483.11
Dual form			483.2.bc.a.44.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.970052 + 1.88164i) q^{2} +(-0.673848 + 1.59560i) q^{3} +(-1.43945 - 2.02142i) q^{4} +(2.06335 + 1.96740i) q^{5} +(-2.34867 - 2.81575i) q^{6} +(2.64569 - 0.0181926i) q^{7} +(1.00908 - 0.145084i) q^{8} +(-2.09186 - 2.15038i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1200 q - 9 q^{3} - 74 q^{4} - 20 q^{6} - 44 q^{7} - 13 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{9}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.970052	+	1.88164i	−0.685930	+	1.33052i	0.246490	+	0.969145i	\(0.420723\pi\)
							−0.932421	+	0.361374i	\(0.882308\pi\)
\(3\)	−0.673848	+	1.59560i	−0.389046	+	0.921218i
							\(5\)	2.06335	+	1.96740i	0.922758	+	0.879848i	0.993052	−	0.117679i	\(-0.0375453\pi\)
−0.0702940	+	0.997526i	\(0.522394\pi\)
\(7\)	2.64569	−	0.0181926i	0.999976	−	0.00687617i
							\(11\)	0.954322	−	0.491987i	0.287739	−	0.148340i	−0.308314	−	0.951284i	\(-0.599765\pi\)
0.596054	+	0.802945i	\(0.296735\pi\)
\(13\)	−4.19299	+	4.83897i	−1.16293	+	1.34209i	−0.233820	+	0.972280i	\(0.575123\pi\)
							−0.929107	+	0.369810i	\(0.879423\pi\)
\(17\)	−4.52671	−	0.432249i	−1.09789	−	0.104836i	−0.469647	−	0.882854i	\(-0.655619\pi\)
							−0.628242	+	0.778018i	\(0.716225\pi\)
\(19\)	0.487717	+	5.10760i	0.111890	+	1.17176i	0.860747	+	0.509033i	\(0.169997\pi\)
							−0.748857	+	0.662731i	\(0.769397\pi\)
\(23\)	−4.58108	+	1.41904i	−0.955222	+	0.295890i
							\(29\)	6.92914	−	3.16443i	1.28671	−	0.587620i	0.349679	−	0.936870i	\(-0.386291\pi\)
0.937030	+	0.349249i	\(0.113563\pi\)
\(31\)	−3.16042	−	1.26524i	−0.567627	−	0.227244i	0.0700550	−	0.997543i	\(-0.477683\pi\)
							−0.637682	+	0.770299i	\(0.720107\pi\)
\(37\)	1.54900	−	0.375783i	0.254654	−	0.0617783i	−0.106400	−	0.994323i	\(-0.533932\pi\)
							0.361053	+	0.932545i	\(0.382417\pi\)
\(41\)	−0.511184	−	1.74093i	−0.0798335	−	0.271888i	0.909897	−	0.414834i	\(-0.136160\pi\)
							−0.989730	+	0.142947i	\(0.954342\pi\)
\(43\)	2.85194	+	0.410046i	0.434916	+	0.0625315i	0.356296	−	0.934373i	\(-0.384039\pi\)
							0.0786202	+	0.996905i	\(0.474949\pi\)
\(47\)	−7.98378	+	4.60944i	−1.16455	+	0.672355i	−0.952391	−	0.304879i	\(-0.901384\pi\)
							−0.212162	+	0.977234i	\(0.568051\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bc.a.11.9	✓	1200
3.2	odd	2	inner	483.2.bc.a.11.52	yes	1200
7.2	even	3	inner	483.2.bc.a.149.52	yes	1200
21.2	odd	6	inner	483.2.bc.a.149.9	yes	1200
23.21	odd	22	inner	483.2.bc.a.389.9	yes	1200
69.44	even	22	inner	483.2.bc.a.389.52	yes	1200
161.44	odd	66	inner	483.2.bc.a.44.52	yes	1200
483.44	even	66	inner	483.2.bc.a.44.9	yes	1200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bc.a.11.9	✓	1200	1.1	even	1	trivial
483.2.bc.a.11.52	yes	1200	3.2	odd	2	inner
483.2.bc.a.44.9	yes	1200	483.44	even	66	inner
483.2.bc.a.44.52	yes	1200	161.44	odd	66	inner
483.2.bc.a.149.9	yes	1200	21.2	odd	6	inner
483.2.bc.a.149.52	yes	1200	7.2	even	3	inner
483.2.bc.a.389.9	yes	1200	23.21	odd	22	inner
483.2.bc.a.389.52	yes	1200	69.44	even	22	inner