Embedded newform 63.4.s.a.59.11

• Label: 63.4.s.a.59.11
• Level: $63$
• Weight: $4$
• Character: 63.59
• 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			59.11
Character	\(\chi\)	\(=\)	63.59
Dual form			63.4.s.a.47.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.223110 - 0.128812i) q^{2} +(-3.39738 + 3.93164i) q^{3} +(-3.96681 - 6.87072i) q^{4} +6.38772 q^{5} +(1.26443 - 0.439563i) q^{6} +(-1.54394 - 18.4558i) q^{7} +4.10490i q^{8} +(-3.91563 - 26.7146i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 3 q^{2} - 3 q^{3} + 81 q^{4} - 6 q^{5} - 24 q^{6} + 5 q^{7} - 3 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.223110	−	0.128812i	−0.0788812	−	0.0455421i	0.460041	−	0.887898i	\(-0.347835\pi\)
							−0.538922	+	0.842356i	\(0.681168\pi\)
\(3\)	−3.39738	+	3.93164i	−0.653826	+	0.756645i
							\(5\)	6.38772			0.571335			0.285668	−	0.958329i	\(-0.407785\pi\)
0.285668	+	0.958329i	\(0.407785\pi\)
\(7\)	−1.54394	−	18.4558i	−0.0833649	−	0.996519i
							\(11\)		−	61.0818i		−	1.67426i	−0.547004	−	0.837130i	\(-0.684232\pi\)
0.547004	−	0.837130i	\(-0.315768\pi\)
\(13\)	8.78677	+	5.07304i	0.187462	+	0.108231i	0.590794	−	0.806822i	\(-0.298815\pi\)
							−0.403332	+	0.915054i	\(0.632148\pi\)
\(17\)	22.5082	−	38.9853i	0.321120	−	0.556196i	−0.659599	−	0.751617i	\(-0.729274\pi\)
							0.980719	+	0.195421i	\(0.0626074\pi\)
\(19\)	−69.6373	+	40.2051i	−0.840837	+	0.485457i	−0.857549	−	0.514403i	\(-0.828014\pi\)
							0.0167118	+	0.999860i	\(0.494680\pi\)
\(23\)			27.6800i			0.250943i	0.992097	+	0.125471i	\(0.0400444\pi\)
							−0.992097	+	0.125471i	\(0.959956\pi\)
\(29\)	48.9383	−	28.2545i	0.313366	−	0.180922i	−0.335066	−	0.942195i	\(-0.608759\pi\)
							0.648432	+	0.761273i	\(0.275425\pi\)
\(31\)	92.1526	−	53.2043i	0.533906	−	0.308251i	−0.208699	−	0.977980i	\(-0.566923\pi\)
							0.742606	+	0.669729i	\(0.233590\pi\)
\(37\)	95.6403	+	165.654i	0.424951	+	0.736036i	0.996416	−	0.0845902i	\(-0.0269581\pi\)
							−0.571465	+	0.820626i	\(0.693625\pi\)
\(41\)	−15.0856	+	26.1289i	−0.0574626	+	0.0995282i	−0.893326	−	0.449410i	\(-0.851634\pi\)
							0.835863	+	0.548938i	\(0.184968\pi\)
\(43\)	−185.062	−	320.536i	−0.656317	−	1.13678i	−0.981562	−	0.191145i	\(-0.938780\pi\)
							0.325244	−	0.945630i	\(-0.394553\pi\)
\(47\)	248.041	−	429.620i	0.769798	−	1.33333i	−0.167874	−	0.985808i	\(-0.553690\pi\)
							0.937672	−	0.347521i	\(-0.112976\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.4.s.a.59.11	yes	44
3.2	odd	2		189.4.s.a.17.12		44
7.5	odd	6		63.4.i.a.5.12	✓	44
9.2	odd	6		63.4.i.a.38.11	yes	44
9.7	even	3		189.4.i.a.143.12		44
21.5	even	6		189.4.i.a.152.11		44
63.47	even	6	inner	63.4.s.a.47.11	yes	44
63.61	odd	6		189.4.s.a.89.12		44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.i.a.5.12	✓	44	7.5	odd	6
63.4.i.a.38.11	yes	44	9.2	odd	6
63.4.s.a.47.11	yes	44	63.47	even	6	inner
63.4.s.a.59.11	yes	44	1.1	even	1	trivial
189.4.i.a.143.12		44	9.7	even	3
189.4.i.a.152.11		44	21.5	even	6
189.4.s.a.17.12		44	3.2	odd	2
189.4.s.a.89.12		44	63.61	odd	6