Embedded newform 63.4.s.a.47.3

• Label: 63.4.s.a.47.3
• Level: $63$
• Weight: $4$
• Character: 63.47
• Analytic conductor: $3.717$
• Analytic rank: $0$
• Dimension: $44$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 63 = 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	63.s (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.71712033036\)
Analytic rank:	\(0\)
Dimension:	\(44\)
Relative dimension:	\(22\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			47.3
Character	\(\chi\)	\(=\)	63.47
Dual form			63.4.s.a.59.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.26829 + 2.46430i) q^{2} +(-1.44219 - 4.99200i) q^{3} +(8.14552 - 14.1085i) q^{4} -9.24869 q^{5} +(18.4575 + 17.7533i) q^{6} +(17.8986 + 4.75801i) q^{7} +40.8632i q^{8} +(-22.8402 + 14.3988i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 3 q^{2} - 3 q^{3} + 81 q^{4} - 6 q^{5} - 24 q^{6} + 5 q^{7} - 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(10\)	\(29\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.26829	+	2.46430i	−1.50907	+	0.871261i	−0.509124	+	0.860693i	\(0.670030\pi\)
							−0.999944	+	0.0105673i	\(0.996636\pi\)
\(3\)	−1.44219	−	4.99200i	−0.277549	−	0.960711i
							\(5\)	−9.24869			−0.827228			−0.413614	−	0.910452i	\(-0.635734\pi\)
−0.413614	+	0.910452i	\(0.635734\pi\)
\(7\)	17.8986	+	4.75801i	0.966436	+	0.256908i
							\(11\)			15.2595i			0.418265i	0.977887	+	0.209132i	\(0.0670639\pi\)
−0.977887	+	0.209132i	\(0.932936\pi\)
\(13\)	−61.0927	+	35.2719i	−1.30339	+	0.752512i	−0.980984	−	0.194090i	\(-0.937825\pi\)
							−0.322405	+	0.946602i	\(0.604491\pi\)
\(17\)	17.8844	+	30.9767i	0.255154	+	0.441939i	0.964937	−	0.262481i	\(-0.0845406\pi\)
							−0.709784	+	0.704420i	\(0.751207\pi\)
\(19\)	91.6472	+	52.9125i	1.10660	+	0.638893i	0.937945	−	0.346783i	\(-0.112726\pi\)
							0.168650	+	0.985676i	\(0.446059\pi\)
\(23\)			92.4427i			0.838071i	0.907970	+	0.419035i	\(0.137632\pi\)
							−0.907970	+	0.419035i	\(0.862368\pi\)
\(29\)	222.341	+	128.369i	1.42372	+	0.821983i	0.996614	−	0.0822202i	\(-0.0262011\pi\)
							0.427102	+	0.904203i	\(0.359534\pi\)
\(31\)	−123.551	−	71.3321i	−0.715819	−	0.413278i	0.0973929	−	0.995246i	\(-0.468950\pi\)
							−0.813212	+	0.581968i	\(0.802283\pi\)
\(37\)	−100.112	+	173.399i	−0.444818	+	0.770447i	−0.998040	−	0.0625870i	\(-0.980065\pi\)
							0.553222	+	0.833034i	\(0.313398\pi\)
\(41\)	−162.085	−	280.739i	−0.617401	−	1.06937i	−0.989958	−	0.141360i	\(-0.954852\pi\)
							0.372558	−	0.928009i	\(-0.378481\pi\)
\(43\)	−92.5782	+	160.350i	−0.328327	+	0.568679i	−0.982180	−	0.187943i	\(-0.939818\pi\)
							0.653853	+	0.756621i	\(0.273151\pi\)
\(47\)	−235.747	−	408.326i	−0.731643	−	1.26724i	−0.956181	−	0.292778i	\(-0.905420\pi\)
							0.224538	−	0.974465i	\(-0.427913\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.4.s.a.47.3	yes	44
3.2	odd	2		189.4.s.a.89.20		44
7.3	odd	6		63.4.i.a.38.3	yes	44
9.4	even	3		189.4.i.a.152.3		44
9.5	odd	6		63.4.i.a.5.20	✓	44
21.17	even	6		189.4.i.a.143.20		44
63.31	odd	6		189.4.s.a.17.20		44
63.59	even	6	inner	63.4.s.a.59.3	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.i.a.5.20	✓	44	9.5	odd	6
63.4.i.a.38.3	yes	44	7.3	odd	6
63.4.s.a.47.3	yes	44	1.1	even	1	trivial
63.4.s.a.59.3	yes	44	63.59	even	6	inner
189.4.i.a.143.20		44	21.17	even	6
189.4.i.a.152.3		44	9.4	even	3
189.4.s.a.17.20		44	63.31	odd	6
189.4.s.a.89.20		44	3.2	odd	2