Embedded newform 84.9.l.a.67.22

• Label: 84.9.l.a.67.22
• Level: $84$
• Weight: $9$
• Character: 84.67
• Analytic conductor: $34.220$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	84.l (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(34.2198032451\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(32\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			67.22
Character	\(\chi\)	\(=\)	84.67
Dual form			84.9.l.a.79.22

$q$-expansion

\(f(q)\)	\(=\)	\(q+(7.68020 + 14.0362i) q^{2} +(-40.5000 - 23.3827i) q^{3} +(-138.029 + 215.601i) q^{4} +(178.709 + 309.534i) q^{5} +(17.1558 - 748.049i) q^{6} +(-1021.22 + 2173.00i) q^{7} +(-4086.31 - 281.542i) q^{8} +(1093.50 + 1894.00i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 3 q^{2} - 2592 q^{3} - 93 q^{4} - 2256 q^{7} + 4500 q^{8} + 69984 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(29\)	\(43\)	\(73\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	7.68020	+	14.0362i	0.480012	+	0.877262i
							\(3\)	−40.5000	−	23.3827i	−0.500000	−	0.288675i
\(5\)	178.709	+	309.534i	0.285935	+	0.495254i								0.972836	−	0.231497i	\(-0.0743625\pi\)
							−0.686900	+	0.726752i	\(0.741029\pi\)
\(7\)	−1021.22	+	2173.00i	−0.425330	+	0.905038i
							\(11\)	21973.5	+	12686.4i	1.50082	+	0.866498i	1.00000		0.000945873i	\(0.000301081\pi\)
0.500819	+	0.865552i	\(0.333032\pi\)
\(13\)	−24586.5			−0.860842			−0.430421	−	0.902628i	\(-0.641635\pi\)
							−0.430421	+	0.902628i	\(0.641635\pi\)
\(17\)	−62089.1	+	107541.i	−0.743395	+	1.28760i	0.207546	+	0.978225i	\(0.433452\pi\)
							−0.950941	+	0.309373i	\(0.899881\pi\)
\(19\)	101161.	−	58405.2i	0.776243	−	0.448164i	−0.0588539	−	0.998267i	\(-0.518745\pi\)
							0.835097	+	0.550102i	\(0.185411\pi\)
\(23\)	−57834.6	+	33390.8i	−0.206669	+	0.119321i	−0.599763	−	0.800178i	\(-0.704738\pi\)
							0.393093	+	0.919499i	\(0.371405\pi\)
\(29\)	−1.13061e6			−1.59854			−0.799268	−	0.600974i	\(-0.794779\pi\)
							−0.799268	+	0.600974i	\(0.794779\pi\)
\(31\)	−506104.	−	292199.i	−0.548016	−	0.316397i	0.200306	−	0.979733i	\(-0.435806\pi\)
							−0.748321	+	0.663337i	\(0.769140\pi\)
\(37\)	328438.	+	568871.i	0.175245	+	0.303534i	0.940246	−	0.340496i	\(-0.110595\pi\)
							−0.765001	+	0.644029i	\(0.777261\pi\)
\(41\)	−1.54495e6			−0.546739			−0.273369	−	0.961909i	\(-0.588138\pi\)
							−0.273369	+	0.961909i	\(0.588138\pi\)
\(43\)		−	5.54890e6i		−	1.62305i	−0.584314	−	0.811527i	\(-0.698637\pi\)
							0.584314	−	0.811527i	\(-0.301363\pi\)
\(47\)	−2.29100e6	+	1.32271e6i	−0.469498	+	0.271065i	−0.716030	−	0.698070i	\(-0.754042\pi\)
							0.246532	+	0.969135i	\(0.420709\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.9.l.a.67.22	✓	64
4.3	odd	2		84.9.l.b.67.23	yes	64
7.2	even	3		84.9.l.b.79.23	yes	64
28.23	odd	6	inner	84.9.l.a.79.22	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.9.l.a.67.22	✓	64	1.1	even	1	trivial
84.9.l.a.79.22	yes	64	28.23	odd	6	inner
84.9.l.b.67.23	yes	64	4.3	odd	2
84.9.l.b.79.23	yes	64	7.2	even	3