Embedded newform 63.4.f.c.43.4

• Label: 63.4.f.c.43.4
• Level: $63$
• Weight: $4$
• Character: 63.43
• Analytic conductor: $3.717$
• Analytic rank: $0$
• Dimension: $18$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(3.71712033036\)
Analytic rank:	\(0\)
Dimension:	\(18\)
Relative dimension:	\(9\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Defining polynomial:	x^18 - 3*x^17 + 6*x^16 - 23*x^15 - 6*x^14 + 255*x^13 - 56*x^12 - 81*x^11 + 5832*x^10 - 32373*x^9 + 52488*x^8 - 6561*x^7 - 40824*x^6 + 1673055*x^5 - 354294*x^4 - 12223143*x^3 + 28697814*x^2 - 129140163*x + 387420489
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 3^{10} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			43.4
Root			\(-0.831471 - 2.88247i\) of defining polynomial
Character	\(\chi\)	\(=\)	63.43
Dual form			63.4.f.c.22.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.231183 + 0.400421i) q^{2} +(1.24909 + 5.04379i) q^{3} +(3.89311 + 6.74306i) q^{4} +(-2.26496 - 3.92303i) q^{5} +(-2.30841 - 0.665877i) q^{6} +(-3.50000 + 6.06218i) q^{7} -7.29902 q^{8} +(-23.8796 + 12.6003i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 6 q^{2} + 9 q^{3} - 36 q^{4} + 24 q^{5} - 63 q^{7} - 150 q^{8} + 63 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(10\)	\(29\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.231183	+	0.400421i	−0.0817357	+	0.141570i	−0.903996	−	0.427542i	\(-0.859380\pi\)
							0.822260	+	0.569112i	\(0.192713\pi\)
\(3\)	1.24909	+	5.04379i	0.240387	+	0.970677i
							\(5\)	−2.26496	−	3.92303i	−0.202585	−	0.350887i	0.746776	−	0.665076i	\(-0.231601\pi\)
−0.949360	+	0.314189i	\(0.898267\pi\)
\(7\)	−3.50000	+	6.06218i	−0.188982	+	0.327327i
							\(11\)	3.34049	−	5.78590i	0.0915633	−	0.158592i	−0.816606	−	0.577196i	\(-0.804147\pi\)
0.908169	+	0.418603i	\(0.137480\pi\)
\(13\)	17.5338	+	30.3695i	0.374078	+	0.647921i	0.990189	−	0.139738i	\(-0.0446260\pi\)
							−0.616111	+	0.787659i	\(0.711293\pi\)
\(17\)	94.5148			1.34842			0.674212	−	0.738538i	\(-0.264484\pi\)
							0.674212	+	0.738538i	\(0.264484\pi\)
\(19\)	−0.0567941			−0.000685761			−0.000342881	−	1.00000i	\(-0.500109\pi\)
							−0.000342881		1.00000i	\(0.500109\pi\)
\(23\)	−22.0791	−	38.2422i	−0.200166	−	0.346697i	0.748416	−	0.663230i	\(-0.230815\pi\)
							−0.948582	+	0.316532i	\(0.897481\pi\)
\(29\)	122.186	−	211.633i	0.782394	−	1.35515i	−0.148149	−	0.988965i	\(-0.547331\pi\)
							0.930543	−	0.366182i	\(-0.119335\pi\)
\(31\)	113.702	+	196.937i	0.658756	+	1.14100i	0.980938	+	0.194321i	\(0.0622504\pi\)
							−0.322182	+	0.946678i	\(0.604416\pi\)
\(37\)	264.710			1.17616			0.588081	−	0.808802i	\(-0.299884\pi\)
							0.588081	+	0.808802i	\(0.299884\pi\)
\(41\)	8.06639	+	13.9714i	0.0307258	+	0.0532187i	0.880979	−	0.473155i	\(-0.156885\pi\)
							−0.850254	+	0.526373i	\(0.823551\pi\)
\(43\)	−33.4417	+	57.9228i	−0.118600	+	0.205422i	−0.919213	−	0.393760i	\(-0.871174\pi\)
							0.800613	+	0.599182i	\(0.204507\pi\)
\(47\)	59.0207	−	102.227i	0.183171	−	0.317262i	−0.759787	−	0.650172i	\(-0.774697\pi\)
							0.942959	+	0.332910i	\(0.108030\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.4.f.c.43.4	yes	18
3.2	odd	2		189.4.f.c.127.6		18
9.2	odd	6		567.4.a.k.1.4		9
9.4	even	3	inner	63.4.f.c.22.4	✓	18
9.5	odd	6		189.4.f.c.64.6		18
9.7	even	3		567.4.a.j.1.6		9

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.f.c.22.4	✓	18	9.4	even	3	inner
63.4.f.c.43.4	yes	18	1.1	even	1	trivial
189.4.f.c.64.6		18	9.5	odd	6
189.4.f.c.127.6		18	3.2	odd	2
567.4.a.j.1.6		9	9.7	even	3
567.4.a.k.1.4		9	9.2	odd	6