Embedded newform 63.4.h.a.25.11

• Label: 63.4.h.a.25.11
• 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			25.11
Character	\(\chi\)	\(=\)	63.25
Dual form			63.4.h.a.58.11

$q$-expansion

\(f(q)\)	\(=\)	\(q-0.438515 q^{2} +(4.98047 + 1.48153i) q^{3} -7.80770 q^{4} +(8.04659 - 13.9371i) q^{5} +(-2.18401 - 0.649674i) q^{6} +(16.9337 - 7.49990i) q^{7} +6.93192 q^{8} +(22.6101 + 14.7574i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 2 q^{2} - q^{3} + 158 q^{4} - 19 q^{5} - 20 q^{6} - 7 q^{7} - 24 q^{8} + 11 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.438515			−0.155038			−0.0775192	−	0.996991i	\(-0.524700\pi\)
							−0.0775192	+	0.996991i	\(0.524700\pi\)
\(3\)	4.98047	+	1.48153i	0.958492	+	0.285121i
							\(5\)	8.04659	−	13.9371i	0.719709	−	1.24657i	−0.241407	−	0.970424i	\(-0.577609\pi\)
0.961115	−	0.276148i	\(-0.0890579\pi\)
\(7\)	16.9337	−	7.49990i	0.914336	−	0.404957i
							\(11\)	13.2763	+	22.9953i	0.363906	+	0.630304i	0.988600	−	0.150566i	\(-0.0481096\pi\)
−0.624694	+	0.780870i	\(0.714776\pi\)
\(13\)	−7.29221	−	12.6305i	−0.155577	−	0.269467i	0.777692	−	0.628645i	\(-0.216390\pi\)
							−0.933269	+	0.359179i	\(0.883057\pi\)
\(17\)	−17.3158	+	29.9918i	−0.247041	+	0.427887i	−0.962703	−	0.270559i	\(-0.912791\pi\)
							0.715663	+	0.698446i	\(0.246125\pi\)
\(19\)	−65.3508	−	113.191i	−0.789079	−	1.36673i	−0.926532	−	0.376217i	\(-0.877225\pi\)
							0.137453	−	0.990508i	\(-0.456109\pi\)
\(23\)	−33.0087	+	57.1728i	−0.299252	+	0.518320i	−0.975965	−	0.217927i	\(-0.930070\pi\)
							0.676713	+	0.736247i	\(0.263404\pi\)
\(29\)	−102.374	+	177.316i	−0.655528	+	1.13541i	0.326234	+	0.945289i	\(0.394220\pi\)
							−0.981761	+	0.190118i	\(0.939113\pi\)
\(31\)	−253.184			−1.46688			−0.733438	−	0.679757i	\(-0.762085\pi\)
							−0.733438	+	0.679757i	\(0.762085\pi\)
\(37\)	111.689	+	193.451i	0.496258	+	0.859545i	0.999991	−	0.00431513i	\(-0.00137355\pi\)
							−0.503732	+	0.863860i	\(0.668040\pi\)
\(41\)	42.4951	+	73.6037i	0.161869	+	0.280365i	0.935539	−	0.353224i	\(-0.114915\pi\)
							−0.773670	+	0.633589i	\(0.781581\pi\)
\(43\)	−16.6287	+	28.8017i	−0.0589733	+	0.102145i	−0.894005	−	0.448057i	\(-0.852116\pi\)
							0.835031	+	0.550202i	\(0.185449\pi\)
\(47\)	525.268			1.63018			0.815088	−	0.579337i	\(-0.196688\pi\)
							0.815088	+	0.579337i	\(0.196688\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.4.h.a.25.11	yes	44
3.2	odd	2		189.4.h.a.46.12		44
7.2	even	3		63.4.g.a.16.12	yes	44
9.4	even	3		63.4.g.a.4.12	✓	44
9.5	odd	6		189.4.g.a.172.11		44
21.2	odd	6		189.4.g.a.100.11		44
63.23	odd	6		189.4.h.a.37.12		44
63.58	even	3	inner	63.4.h.a.58.11	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.g.a.4.12	✓	44	9.4	even	3
63.4.g.a.16.12	yes	44	7.2	even	3
63.4.h.a.25.11	yes	44	1.1	even	1	trivial
63.4.h.a.58.11	yes	44	63.58	even	3	inner
189.4.g.a.100.11		44	21.2	odd	6
189.4.g.a.172.11		44	9.5	odd	6
189.4.h.a.37.12		44	63.23	odd	6
189.4.h.a.46.12		44	3.2	odd	2