Embedded newform 63.4.h.a.58.15

• Label: 63.4.h.a.58.15
• Level: $63$
• Weight: $4$
• Character: 63.58
• 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			58.15
Character	\(\chi\)	\(=\)	63.58
Dual form			63.4.h.a.25.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+1.44809 q^{2} +(0.196108 - 5.19245i) q^{3} -5.90304 q^{4} +(-2.21638 - 3.83887i) q^{5} +(0.283981 - 7.51913i) q^{6} +(9.71690 - 15.7665i) q^{7} -20.1328 q^{8} +(-26.9231 - 2.03656i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 2 q^{2} - q^{3} + 158 q^{4} - 19 q^{5} - 20 q^{6} - 7 q^{7} - 24 q^{8} + 11 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(10\)	\(29\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.44809			0.511977			0.255988	−	0.966680i	\(-0.417599\pi\)
							0.255988	+	0.966680i	\(0.417599\pi\)
\(3\)	0.196108	−	5.19245i	0.0377409	−	0.999288i
							\(5\)	−2.21638	−	3.83887i	−0.198239	−	0.343359i	0.749719	−	0.661757i	\(-0.230189\pi\)
−0.947957	+	0.318397i	\(0.896855\pi\)
\(7\)	9.71690	−	15.7665i	0.524664	−	0.851310i
							\(11\)	26.2410	−	45.4507i	0.719269	−	1.24581i	−0.242021	−	0.970271i	\(-0.577810\pi\)
0.961290	−	0.275539i	\(-0.0888564\pi\)
\(13\)	−21.3388	+	36.9599i	−0.455256	+	0.788526i	−0.998703	−	0.0509172i	\(-0.983786\pi\)
							0.543447	+	0.839443i	\(0.317119\pi\)
\(17\)	54.8281	+	94.9651i	0.782222	+	1.35485i	0.930644	+	0.365925i	\(0.119247\pi\)
							−0.148422	+	0.988924i	\(0.547419\pi\)
\(19\)	72.6470	−	125.828i	0.877178	−	1.51932i	0.0227527	−	0.999741i	\(-0.492757\pi\)
							0.854425	−	0.519575i	\(-0.173910\pi\)
\(23\)	18.8228	+	32.6021i	0.170645	+	0.295566i	0.938645	−	0.344884i	\(-0.112082\pi\)
							−0.768001	+	0.640449i	\(0.778748\pi\)
\(29\)	97.0157	+	168.036i	0.621220	+	1.07598i	0.989259	+	0.146174i	\(0.0466958\pi\)
							−0.368039	+	0.929810i	\(0.619971\pi\)
\(31\)	−202.266			−1.17187			−0.585936	−	0.810357i	\(-0.699273\pi\)
							−0.585936	+	0.810357i	\(0.699273\pi\)
\(37\)	81.1070	−	140.481i	0.360376	−	0.624189i	−0.627647	−	0.778498i	\(-0.715982\pi\)
							0.988023	+	0.154309i	\(0.0493151\pi\)
\(41\)	−52.5349	+	90.9932i	−0.200112	+	0.346603i	−0.948564	−	0.316585i	\(-0.897464\pi\)
							0.748453	+	0.663188i	\(0.230797\pi\)
\(43\)	−103.864	−	179.898i	−0.368352	−	0.638004i	0.620956	−	0.783845i	\(-0.286745\pi\)
							−0.989308	+	0.145841i	\(0.953411\pi\)
\(47\)	75.0172			0.232817			0.116408	−	0.993201i	\(-0.462862\pi\)
							0.116408	+	0.993201i	\(0.462862\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.4.h.a.58.15	yes	44
3.2	odd	2		189.4.h.a.37.8		44
7.4	even	3		63.4.g.a.4.8	✓	44
9.2	odd	6		189.4.g.a.100.15		44
9.7	even	3		63.4.g.a.16.8	yes	44
21.11	odd	6		189.4.g.a.172.15		44
63.11	odd	6		189.4.h.a.46.8		44
63.25	even	3	inner	63.4.h.a.25.15	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.g.a.4.8	✓	44	7.4	even	3
63.4.g.a.16.8	yes	44	9.7	even	3
63.4.h.a.25.15	yes	44	63.25	even	3	inner
63.4.h.a.58.15	yes	44	1.1	even	1	trivial
189.4.g.a.100.15		44	9.2	odd	6
189.4.g.a.172.15		44	21.11	odd	6
189.4.h.a.37.8		44	3.2	odd	2
189.4.h.a.46.8		44	63.11	odd	6