Newform orbit 84.3.p.c

• Label: 84.3.p.c
• Level: $84$
• Weight: $3$
• Character orbit: 84.p
• Analytic conductor: $2.289$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	84.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.28883422063\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\Q(\sqrt{-3}, \sqrt{-35})\)
Defining polynomial:	\( x^{4} - x^{3} - 8x^{2} - 9x + 81 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + \beta_1 q^{3} + (2 \beta_{3} + \beta_{2}) q^{5} - 7 q^{7} + (\beta_{3} + 9 \beta_{2}) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{3} - 28 q^{7} + 17 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 8x^{2} - 9x + 81 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{3} + 8\nu^{2} - 8\nu - 9 ) / 72 \)
\(\beta_{3}\)	\(=\)	\( ( -\nu^{3} + 8\nu + 9 ) / 8 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/84\mathbb{Z}\right)^\times\).

\(n\)	\(29\)	\(43\)	\(73\)
\(\chi(n)\)	\(-1\)	\(1\)	\(-\beta_{2}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

53.1

−2.31174	−	1.91203i
2.81174	+	1.04601i
−2.31174	+	1.91203i
2.81174	−	1.04601i

0

−2.31174

−

1.91203i

0

−5.12348

+

2.95804i

0

−7.00000

0

1.68826

+

8.84024i

0

53.2

0

2.81174

+

1.04601i

0

5.12348

−

2.95804i

0

−7.00000

0

6.81174

+

5.88220i

0

65.1

0

−2.31174

+

1.91203i

0

−5.12348

−

2.95804i

0

−7.00000

0

1.68826

−

8.84024i

0

65.2

0

2.81174

−

1.04601i

0

5.12348

+

2.95804i

0

−7.00000

0

6.81174

−

5.88220i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
7.c	even	3	1	inner
21.h	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	84.3.p.c	✓	4
3.b	odd	2	1	inner	84.3.p.c	✓	4
4.b	odd	2	1		336.3.bn.d		4
7.b	odd	2	1		588.3.p.d		4
7.c	even	3	1	inner	84.3.p.c	✓	4
7.c	even	3	1		588.3.c.e		2
7.d	odd	6	1		588.3.c.f		2
7.d	odd	6	1		588.3.p.d		4
12.b	even	2	1		336.3.bn.d		4
21.c	even	2	1		588.3.p.d		4
21.g	even	6	1		588.3.c.f		2
21.g	even	6	1		588.3.p.d		4
21.h	odd	6	1	inner	84.3.p.c	✓	4
21.h	odd	6	1		588.3.c.e		2
28.g	odd	6	1		336.3.bn.d		4
84.n	even	6	1		336.3.bn.d		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
84.3.p.c	✓	4	1.a	even	1	1	trivial
84.3.p.c	✓	4	3.b	odd	2	1	inner
84.3.p.c	✓	4	7.c	even	3	1	inner
84.3.p.c	✓	4	21.h	odd	6	1	inner
336.3.bn.d		4	4.b	odd	2	1
336.3.bn.d		4	12.b	even	2	1
336.3.bn.d		4	28.g	odd	6	1
336.3.bn.d		4	84.n	even	6	1
588.3.c.e		2	7.c	even	3	1
588.3.c.e		2	21.h	odd	6	1
588.3.c.f		2	7.d	odd	6	1
588.3.c.f		2	21.g	even	6	1
588.3.p.d		4	7.b	odd	2	1
588.3.p.d		4	7.d	odd	6	1
588.3.p.d		4	21.c	even	2	1
588.3.p.d		4	21.g	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 35T_{5}^{2} + 1225 \) acting on \(S_{3}^{\mathrm{new}}(84, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{4} \)
$3$	T^4 - T^3 - 8*T^2 - 9*T + 81
$5$	\( T^{4} - 35T^{2} + 1225 \)
$7$	\( (T + 7)^{4} \)
$11$	\( T^{4} - 35T^{2} + 1225 \)