Embedded newform 84.9.l.a.67.12

• Label: 84.9.l.a.67.12
• 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			67.12
Character	\(\chi\)	\(=\)	84.67
Dual form			84.9.l.a.79.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.43741 - 14.6479i) q^{2} +(-40.5000 - 23.3827i) q^{3} +(-173.120 + 188.589i) q^{4} +(478.743 + 829.208i) q^{5} +(-81.7912 + 743.762i) q^{6} +(544.790 - 2338.38i) q^{7} +(3876.86 + 1321.81i) q^{8} +(1093.50 + 1894.00i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 3 q^{2} - 2592 q^{3} - 93 q^{4} - 2256 q^{7} + 4500 q^{8} + 69984 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.43741	−	14.6479i	−0.402338	−	0.915491i
							\(3\)	−40.5000	−	23.3827i	−0.500000	−	0.288675i
\(5\)	478.743	+	829.208i	0.765989	+	1.32673i								0.939722	+	0.341940i	\(0.111084\pi\)
							−0.173733	+	0.984793i	\(0.555583\pi\)
\(7\)	544.790	−	2338.38i	0.226901	−	0.973918i
							\(11\)	−12967.6	−	7486.86i	−0.885706	−	0.511363i	−0.0131706	−	0.999913i	\(-0.504192\pi\)
−0.872536	+	0.488551i	\(0.837526\pi\)
\(13\)	28268.5			0.989759			0.494879	−	0.868962i	\(-0.335212\pi\)
							0.494879	+	0.868962i	\(0.335212\pi\)
\(17\)	−30364.8	+	52593.3i	−0.363558	+	0.629701i	−0.988544	−	0.150935i	\(-0.951772\pi\)
							0.624985	+	0.780636i	\(0.285105\pi\)
\(19\)	−95178.3	+	54951.2i	−0.730338	+	0.421661i	−0.818546	−	0.574441i	\(-0.805219\pi\)
							0.0882081	+	0.996102i	\(0.471886\pi\)
\(23\)	34621.2	−	19988.6i	0.123718	−	0.0714283i	−0.436864	−	0.899528i	\(-0.643911\pi\)
							0.560582	+	0.828099i	\(0.310578\pi\)
\(29\)	−568718.			−0.804091			−0.402046	−	0.915620i	\(-0.631701\pi\)
							−0.402046	+	0.915620i	\(0.631701\pi\)
\(31\)	615152.	+	355158.i	0.666094	+	0.384570i	0.794595	−	0.607140i	\(-0.207683\pi\)
							−0.128501	+	0.991709i	\(0.541016\pi\)
\(37\)	−413972.	−	717021.i	−0.220884	−	0.382582i	0.734193	−	0.678941i	\(-0.237561\pi\)
							−0.955077	+	0.296359i	\(0.904228\pi\)
\(41\)	4.10445e6			1.45251			0.726255	−	0.687425i	\(-0.241259\pi\)
							0.726255	+	0.687425i	\(0.241259\pi\)
\(43\)		−	1.55049e6i		−	0.453517i	−0.973951	−	0.226759i	\(-0.927187\pi\)
							0.973951	−	0.226759i	\(-0.0728129\pi\)
\(47\)	−6.70274e6	+	3.86983e6i	−1.37360	+	0.793050i	−0.991380	−	0.131020i	\(-0.958175\pi\)
							−0.382223	+	0.924070i	\(0.624841\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.9.l.a.67.12	✓	64
4.3	odd	2		84.9.l.b.67.11	yes	64
7.2	even	3		84.9.l.b.79.11	yes	64
28.23	odd	6	inner	84.9.l.a.79.12	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.9.l.a.67.12	✓	64	1.1	even	1	trivial
84.9.l.a.79.12	yes	64	28.23	odd	6	inner
84.9.l.b.67.11	yes	64	4.3	odd	2
84.9.l.b.79.11	yes	64	7.2	even	3