Embedded newform 84.3.j.a.59.17

• Label: 84.3.j.a.59.17
• Level: $84$
• Weight: $3$
• Character: 84.59
• 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			59.17
Character	\(\chi\)	\(=\)	84.59
Dual form			84.3.j.a.47.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.604384 + 1.90649i) q^{2} +(-2.89442 - 0.788889i) q^{3} +(-3.26944 + 2.30451i) q^{4} +(-3.91160 - 6.77509i) q^{5} +(-0.245328 - 5.99498i) q^{6} +(-6.35163 + 2.94225i) q^{7} +(-6.36953 - 4.84036i) q^{8} +(7.75531 + 4.56675i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 2 q^{4} - 2 q^{9}+O(q^{10}) \)

Character values

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

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

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
							\(3\)	−2.89442	−	0.788889i	−0.964806	−	0.262963i
\(5\)	−3.91160	−	6.77509i	−0.782320	−	1.35502i								−0.930587	−	0.366070i	\(-0.880703\pi\)
							0.148268	−	0.988947i	\(-0.452630\pi\)
\(7\)	−6.35163	+	2.94225i	−0.907375	+	0.420322i
							\(11\)	−3.99744	+	6.92378i	−0.363404	+	0.629434i	−0.988519	−	0.151099i	\(-0.951719\pi\)
0.625115	+	0.780533i	\(0.285052\pi\)
\(13\)		−	12.2401i		−	0.941547i	−0.882254	−	0.470773i	\(-0.843975\pi\)
							0.882254	−	0.470773i	\(-0.156025\pi\)
\(17\)	−12.9750	+	22.4734i	−0.763238	+	1.32197i	0.177936	+	0.984042i	\(0.443058\pi\)
							−0.941173	+	0.337924i	\(0.890275\pi\)
\(19\)	−0.176417	−	0.305564i	−0.00928512	−	0.0160823i	0.861346	−	0.508020i	\(-0.169622\pi\)
							−0.870631	+	0.491937i	\(0.836289\pi\)
\(23\)	−12.2582	−	21.2318i	−0.532965	−	0.923122i	−0.999259	−	0.0384922i	\(-0.987745\pi\)
							0.466294	−	0.884630i	\(-0.345589\pi\)
\(29\)			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	−0.829666	−	0.558261i	\(-0.811469\pi\)
							0.898301	+	0.439381i	\(0.144802\pi\)
\(37\)	4.82452	+	8.35632i	0.130393	+	0.225847i	0.923828	−	0.382808i	\(-0.125043\pi\)
							−0.793435	+	0.608654i	\(0.791710\pi\)
\(41\)	−10.8181			−0.263856			−0.131928	−	0.991259i	\(-0.542117\pi\)
							−0.131928	+	0.991259i	\(0.542117\pi\)
\(43\)		−	45.9410i		−	1.06839i	−0.845360	−	0.534197i	\(-0.820614\pi\)
							0.845360	−	0.534197i	\(-0.179386\pi\)
\(47\)	2.70440	−	1.56139i	0.0575405	−	0.0332210i	−0.470954	−	0.882158i	\(-0.656090\pi\)
							0.528494	+	0.848937i	\(0.322757\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.3.j.a.47.8	✓	56	21.5	even	6	inner
84.3.j.a.47.12	yes	56	28.19	even	6	inner
84.3.j.a.47.17	yes	56	84.47	odd	6	inner
84.3.j.a.47.21	yes	56	7.5	odd	6	inner
84.3.j.a.59.8	yes	56	4.3	odd	2	inner
84.3.j.a.59.12	yes	56	3.2	odd	2	inner
84.3.j.a.59.17	yes	56	1.1	even	1	trivial
84.3.j.a.59.21	yes	56	12.11	even	2	inner