Embedded newform 84.3.j.a.47.12

• Label: 84.3.j.a.47.12
• Level: $84$
• Weight: $3$
• Character: 84.47
• Analytic conductor: $2.289$
• Analytic rank: $0$
• Dimension: $56$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	84.j (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.28883422063\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(28\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			47.12
Character	\(\chi\)	\(=\)	84.47
Dual form			84.3.j.a.59.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.604384 + 1.90649i) q^{2} +(2.13041 - 2.11219i) q^{3} +(-3.26944 - 2.30451i) q^{4} +(3.91160 - 6.77509i) q^{5} +(2.73930 + 5.33818i) q^{6} +(-6.35163 - 2.94225i) q^{7} +(6.36953 - 4.84036i) q^{8} +(0.0772646 - 8.99967i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 2 q^{4} - 2 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/84\mathbb{Z}\right)^\times\).

\(n\)	\(29\)	\(43\)	\(73\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.604384	+	1.90649i	−0.302192	+	0.953247i
							\(3\)	2.13041	−	2.11219i	0.710136	−	0.704065i
\(5\)	3.91160	−	6.77509i	0.782320	−	1.35502i								−0.148268	−	0.988947i	\(-0.547370\pi\)
							0.930587	−	0.366070i	\(-0.119297\pi\)
\(7\)	−6.35163	−	2.94225i	−0.907375	−	0.420322i
							\(11\)	3.99744	+	6.92378i	0.363404	+	0.629434i	0.988519	−	0.151099i	\(-0.0482811\pi\)
−0.625115	+	0.780533i	\(0.714948\pi\)
\(13\)			12.2401i			0.941547i	0.882254	+	0.470773i	\(0.156025\pi\)
							−0.882254	+	0.470773i	\(0.843975\pi\)
\(17\)	12.9750	+	22.4734i	0.763238	+	1.32197i	0.941173	+	0.337924i	\(0.109725\pi\)
							−0.177936	+	0.984042i	\(0.556942\pi\)
\(19\)	−0.176417	+	0.305564i	−0.00928512	+	0.0160823i	−0.870631	−	0.491937i	\(-0.836289\pi\)
							0.861346	+	0.508020i	\(0.169622\pi\)
\(23\)	12.2582	−	21.2318i	0.532965	−	0.923122i	−0.466294	−	0.884630i	\(-0.654411\pi\)
							0.999259	−	0.0384922i	\(-0.0122555\pi\)
\(29\)			8.90744i			0.307153i	0.988137	+	0.153576i	\(0.0490791\pi\)
							−0.988137	+	0.153576i	\(0.950921\pi\)
\(31\)	2.12769	+	3.68526i	0.0686350	+	0.118879i	0.898301	−	0.439381i	\(-0.144802\pi\)
							−0.829666	+	0.558261i	\(0.811469\pi\)
\(37\)	4.82452	−	8.35632i	0.130393	−	0.225847i	−0.793435	−	0.608654i	\(-0.791710\pi\)
							0.923828	+	0.382808i	\(0.125043\pi\)
\(41\)	10.8181			0.263856			0.131928	−	0.991259i	\(-0.457883\pi\)
							0.131928	+	0.991259i	\(0.457883\pi\)
\(43\)			45.9410i			1.06839i	0.845360	+	0.534197i	\(0.179386\pi\)
							−0.845360	+	0.534197i	\(0.820614\pi\)
\(47\)	−2.70440	−	1.56139i	−0.0575405	−	0.0332210i	0.470954	−	0.882158i	\(-0.343910\pi\)
							−0.528494	+	0.848937i	\(0.677243\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.3.j.a.47.12	yes	56
3.2	odd	2	inner	84.3.j.a.47.17	yes	56
4.3	odd	2	inner	84.3.j.a.47.21	yes	56
7.3	odd	6	inner	84.3.j.a.59.8	yes	56
12.11	even	2	inner	84.3.j.a.47.8	✓	56
21.17	even	6	inner	84.3.j.a.59.21	yes	56
28.3	even	6	inner	84.3.j.a.59.17	yes	56
84.59	odd	6	inner	84.3.j.a.59.12	yes	56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.3.j.a.47.8	✓	56	12.11	even	2	inner
84.3.j.a.47.12	yes	56	1.1	even	1	trivial
84.3.j.a.47.17	yes	56	3.2	odd	2	inner
84.3.j.a.47.21	yes	56	4.3	odd	2	inner
84.3.j.a.59.8	yes	56	7.3	odd	6	inner
84.3.j.a.59.12	yes	56	84.59	odd	6	inner
84.3.j.a.59.17	yes	56	28.3	even	6	inner
84.3.j.a.59.21	yes	56	21.17	even	6	inner