Embedded newform 63.6.i.a.38.7

• Label: 63.6.i.a.38.7
• Level: $63$
• Weight: $6$
• Character: 63.38
• Analytic conductor: $10.104$
• Analytic rank: $0$
• Dimension: $76$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 63 = 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	63.i (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.1041806482\)
Analytic rank:	\(0\)
Dimension:	\(76\)
Relative dimension:	\(38\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			38.7
Character	\(\chi\)	\(=\)	63.38
Dual form			63.6.i.a.5.32

$q$-expansion

\(f(q)\)	\(=\)	\(q-8.30014i q^{2} +(4.89674 + 14.7994i) q^{3} -36.8923 q^{4} +(-6.73589 + 11.6669i) q^{5} +(122.837 - 40.6437i) q^{6} +(-99.7296 - 82.8312i) q^{7} +40.6067i q^{8} +(-195.044 + 144.938i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q - 3 q^{3} - 1154 q^{4} - 3 q^{5} + 96 q^{6} - 30 q^{7} + 489 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).

\(n\)	\(10\)	\(29\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	8.30014i		−	1.46727i	−0.679543	−	0.733636i	\(-0.737822\pi\)
							0.679543	−	0.733636i	\(-0.262178\pi\)
\(3\)	4.89674	+	14.7994i	0.314126	+	0.949381i
							\(5\)	−6.73589	+	11.6669i	−0.120495	+	0.208704i	−0.919963	−	0.392005i	\(-0.871782\pi\)
0.799468	+	0.600709i	\(0.205115\pi\)
\(7\)	−99.7296	−	82.8312i	−0.769270	−	0.638924i
							\(11\)	−524.423	+	302.776i	−1.30677	+	0.754465i	−0.981556	−	0.191174i	\(-0.938770\pi\)
−0.325216	+	0.945640i	\(0.605437\pi\)
\(13\)	−362.675	+	209.390i	−0.595195	+	0.343636i	−0.767149	−	0.641469i	\(-0.778325\pi\)
							0.171954	+	0.985105i	\(0.444992\pi\)
\(17\)	−165.403	+	286.486i	−0.138810	+	0.240426i	−0.927046	−	0.374947i	\(-0.877661\pi\)
							0.788237	+	0.615372i	\(0.210994\pi\)
\(19\)	1722.48	−	994.474i	1.09464	−	0.631989i	0.159830	−	0.987145i	\(-0.448906\pi\)
							0.934807	+	0.355156i	\(0.115572\pi\)
\(23\)	−2700.13	−	1558.92i	−1.06430	−	0.614475i	−0.137683	−	0.990476i	\(-0.543966\pi\)
							−0.926619	+	0.376001i	\(0.877299\pi\)
\(29\)	−83.1858	−	48.0273i	−0.0183677	−	0.0106046i	0.490788	−	0.871279i	\(-0.336709\pi\)
							−0.509156	+	0.860674i	\(0.670042\pi\)
\(31\)			4795.53i			0.896257i	0.893969	+	0.448128i	\(0.147909\pi\)
							−0.893969	+	0.448128i	\(0.852091\pi\)
\(37\)	58.2940	+	100.968i	0.00700034	+	0.0121249i	0.869504	−	0.493925i	\(-0.164438\pi\)
							−0.862504	+	0.506050i	\(0.831105\pi\)
\(41\)	−7297.17	−	12639.1i	−0.677945	−	1.17424i	−0.975599	−	0.219562i	\(-0.929537\pi\)
							0.297653	−	0.954674i	\(-0.403796\pi\)
\(43\)	10129.6	−	17544.9i	0.835449	−	1.44704i	−0.0582155	−	0.998304i	\(-0.518541\pi\)
							0.893664	−	0.448736i	\(-0.148126\pi\)
\(47\)	−13744.4			−0.907574			−0.453787	−	0.891110i	\(-0.649927\pi\)
							−0.453787	+	0.891110i	\(0.649927\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.6.i.a.38.7	yes	76
3.2	odd	2		189.6.i.a.143.32		76
7.5	odd	6		63.6.s.a.47.7	yes	76
9.4	even	3		189.6.s.a.17.32		76
9.5	odd	6		63.6.s.a.59.7	yes	76
21.5	even	6		189.6.s.a.89.32		76
63.5	even	6	inner	63.6.i.a.5.32	✓	76
63.40	odd	6		189.6.i.a.152.7		76

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.6.i.a.5.32	✓	76	63.5	even	6	inner
63.6.i.a.38.7	yes	76	1.1	even	1	trivial
63.6.s.a.47.7	yes	76	7.5	odd	6
63.6.s.a.59.7	yes	76	9.5	odd	6
189.6.i.a.143.32		76	3.2	odd	2
189.6.i.a.152.7		76	63.40	odd	6
189.6.s.a.17.32		76	9.4	even	3
189.6.s.a.89.32		76	21.5	even	6