Newform orbit 104.4.i.a

• Label: 104.4.i.a
• Level: $104$
• Weight: $4$
• Character orbit: 104.i
• Analytic conductor: $6.136$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 104 = 2^{3} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	104.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.13619864060\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + (8 \zeta_{6} - 8) q^{3} - 9 q^{5} + 4 \zeta_{6} q^{7} - 37 \zeta_{6} q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{3} - 18 q^{5} + 4 q^{7} - 37 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/104\mathbb{Z}\right)^\times\).

\(n\)	\(41\)	\(53\)	\(79\)
\(\chi(n)\)	\(-\zeta_{6}\)	\(1\)	\(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

9.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

−4.00000

+

6.92820i

0

−9.00000

0

2.00000

+

3.46410i

0

−18.5000

−

32.0429i

0

81.1

0

−4.00000

−

6.92820i

0

−9.00000

0

2.00000

−

3.46410i

0

−18.5000

+

32.0429i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	104.4.i.a	✓	2
4.b	odd	2	1		208.4.i.c		2
13.c	even	3	1	inner	104.4.i.a	✓	2
13.c	even	3	1		1352.4.a.d		1
13.e	even	6	1		1352.4.a.e		1
52.j	odd	6	1		208.4.i.c		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
104.4.i.a	✓	2	1.a	even	1	1	trivial
104.4.i.a	✓	2	13.c	even	3	1	inner
208.4.i.c		2	4.b	odd	2	1
208.4.i.c		2	52.j	odd	6	1
1352.4.a.d		1	13.c	even	3	1
1352.4.a.e		1	13.e	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 8T_{3} + 64 \) acting on \(S_{4}^{\mathrm{new}}(104, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} + 8T + 64 \)
$5$	\( (T + 9)^{2} \)
$7$	\( T^{2} - 4T + 16 \)
$11$	\( T^{2} - 20T + 400 \)