Embedded newform 63.4.h.a.25.8

• Label: 63.4.h.a.25.8
• Level: $63$
• Weight: $4$
• Character: 63.25
• 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.h (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.71712033036\)
Analytic rank:	\(0\)
Dimension:	\(44\)
Relative dimension:	\(22\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			25.8
Character	\(\chi\)	\(=\)	63.25
Dual form			63.4.h.a.58.8

$q$-expansion

\(f(q)\)	\(=\)	\(q-2.37882 q^{2} +(3.09078 - 4.17697i) q^{3} -2.34120 q^{4} +(-9.23374 + 15.9933i) q^{5} +(-7.35242 + 9.93628i) q^{6} +(15.8733 + 9.54133i) q^{7} +24.5999 q^{8} +(-7.89419 - 25.8202i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 2 q^{2} - q^{3} + 158 q^{4} - 19 q^{5} - 20 q^{6} - 7 q^{7} - 24 q^{8} + 11 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(10\)	\(29\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.37882			−0.841041			−0.420521	−	0.907283i	\(-0.638153\pi\)
							−0.420521	+	0.907283i	\(0.638153\pi\)
\(3\)	3.09078	−	4.17697i	0.594820	−	0.803859i
							\(5\)	−9.23374	+	15.9933i	−0.825891	+	1.43048i	0.0753458	+	0.997157i	\(0.475994\pi\)
−0.901237	+	0.433327i	\(0.857339\pi\)
\(7\)	15.8733	+	9.54133i	0.857080	+	0.515183i
							\(11\)	28.3801	+	49.1557i	0.777901	+	1.34736i	0.933150	+	0.359488i	\(0.117049\pi\)
−0.155249	+	0.987875i	\(0.549618\pi\)
\(13\)	3.61910	+	6.26847i	0.0772121	+	0.133735i	0.902046	−	0.431640i	\(-0.142065\pi\)
							−0.824834	+	0.565375i	\(0.808731\pi\)
\(17\)	−42.2739	+	73.2205i	−0.603113	+	1.04462i	0.389234	+	0.921139i	\(0.372740\pi\)
							−0.992347	+	0.123483i	\(0.960594\pi\)
\(19\)	−1.77532	−	3.07495i	−0.0214362	−	0.0371286i	0.855108	−	0.518449i	\(-0.173491\pi\)
							−0.876544	+	0.481321i	\(0.840157\pi\)
\(23\)	−45.3428	+	78.5361i	−0.411071	+	0.711996i	−0.995007	−	0.0998036i	\(-0.968179\pi\)
							0.583936	+	0.811800i	\(0.301512\pi\)
\(29\)	25.6893	−	44.4952i	0.164496	−	0.284915i	−0.771980	−	0.635647i	\(-0.780734\pi\)
							0.936476	+	0.350731i	\(0.114067\pi\)
\(31\)	156.464			0.906508			0.453254	−	0.891381i	\(-0.350263\pi\)
							0.453254	+	0.891381i	\(0.350263\pi\)
\(37\)	28.9522	+	50.1467i	0.128641	+	0.222813i	0.923150	−	0.384439i	\(-0.125605\pi\)
							−0.794509	+	0.607252i	\(0.792272\pi\)
\(41\)	−79.8397	−	138.286i	−0.304119	−	0.526749i	0.672946	−	0.739692i	\(-0.265029\pi\)
							−0.977065	+	0.212943i	\(0.931695\pi\)
\(43\)	−20.6544	+	35.7745i	−0.0732505	+	0.126874i	−0.900324	−	0.435220i	\(-0.856671\pi\)
							0.827074	+	0.562094i	\(0.190004\pi\)
\(47\)	144.985			0.449962			0.224981	−	0.974363i	\(-0.427768\pi\)
							0.224981	+	0.974363i	\(0.427768\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.4.h.a.25.8	yes	44
3.2	odd	2		189.4.h.a.46.15		44
7.2	even	3		63.4.g.a.16.15	yes	44
9.4	even	3		63.4.g.a.4.15	✓	44
9.5	odd	6		189.4.g.a.172.8		44
21.2	odd	6		189.4.g.a.100.8		44
63.23	odd	6		189.4.h.a.37.15		44
63.58	even	3	inner	63.4.h.a.58.8	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.g.a.4.15	✓	44	9.4	even	3
63.4.g.a.16.15	yes	44	7.2	even	3
63.4.h.a.25.8	yes	44	1.1	even	1	trivial
63.4.h.a.58.8	yes	44	63.58	even	3	inner
189.4.g.a.100.8		44	21.2	odd	6
189.4.g.a.172.8		44	9.5	odd	6
189.4.h.a.37.15		44	63.23	odd	6
189.4.h.a.46.15		44	3.2	odd	2