Newform orbit 3136.2.f.b

• Label: 3136.2.f.b
• Level: $3136$
• Weight: $2$
• Character orbit: 3136.f
• Analytic conductor: $25.041$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3136 = 2^{6} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3136.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(25.0410860739\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + q^{3} + \beta q^{5} - 2 q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 4 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3136\mathbb{Z}\right)^\times\).

\(n\)	\(197\)	\(1471\)	\(1473\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

3135.1

0.500000	−	0.866025i
0.500000	+	0.866025i

0

1.00000

0

−

1.73205i

0

0

0

−2.00000

0

3135.2

0

1.00000

0

1.73205i

0

0

0

−2.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
28.d	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3136.2.f.b		2
4.b	odd	2	1		3136.2.f.a		2
7.b	odd	2	1		3136.2.f.a		2
7.c	even	3	1		448.2.p.a		2
7.d	odd	6	1		448.2.p.b		2
8.b	even	2	1		784.2.f.a		2
8.d	odd	2	1		784.2.f.b		2
24.f	even	2	1		7056.2.b.m		2
24.h	odd	2	1		7056.2.b.b		2
28.d	even	2	1	inner	3136.2.f.b		2
28.f	even	6	1		448.2.p.a		2
28.g	odd	6	1		448.2.p.b		2
56.e	even	2	1		784.2.f.a		2
56.h	odd	2	1		784.2.f.b		2
56.j	odd	6	1		112.2.p.a	✓	2
56.j	odd	6	1		784.2.p.c		2
56.k	odd	6	1		112.2.p.a	✓	2
56.k	odd	6	1		784.2.p.c		2
56.m	even	6	1		112.2.p.b	yes	2
56.m	even	6	1		784.2.p.d		2
56.p	even	6	1		112.2.p.b	yes	2
56.p	even	6	1		784.2.p.d		2
168.e	odd	2	1		7056.2.b.b		2
168.i	even	2	1		7056.2.b.m		2
168.s	odd	6	1		1008.2.cs.c		2
168.v	even	6	1		1008.2.cs.f		2
168.ba	even	6	1		1008.2.cs.f		2
168.be	odd	6	1		1008.2.cs.c		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
112.2.p.a	✓	2	56.j	odd	6	1
112.2.p.a	✓	2	56.k	odd	6	1
112.2.p.b	yes	2	56.m	even	6	1
112.2.p.b	yes	2	56.p	even	6	1
448.2.p.a		2	7.c	even	3	1
448.2.p.a		2	28.f	even	6	1
448.2.p.b		2	7.d	odd	6	1
448.2.p.b		2	28.g	odd	6	1
784.2.f.a		2	8.b	even	2	1
784.2.f.a		2	56.e	even	2	1
784.2.f.b		2	8.d	odd	2	1
784.2.f.b		2	56.h	odd	2	1
784.2.p.c		2	56.j	odd	6	1
784.2.p.c		2	56.k	odd	6	1
784.2.p.d		2	56.m	even	6	1
784.2.p.d		2	56.p	even	6	1
1008.2.cs.c		2	168.s	odd	6	1
1008.2.cs.c		2	168.be	odd	6	1
1008.2.cs.f		2	168.v	even	6	1
1008.2.cs.f		2	168.ba	even	6	1
3136.2.f.a		2	4.b	odd	2	1
3136.2.f.a		2	7.b	odd	2	1
3136.2.f.b		2	1.a	even	1	1	trivial
3136.2.f.b		2	28.d	even	2	1	inner
7056.2.b.b		2	24.h	odd	2	1
7056.2.b.b		2	168.e	odd	2	1
7056.2.b.m		2	24.f	even	2	1
7056.2.b.m		2	168.i	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3136, [\chi])\):

\( T_{3} - 1 \)

\( T_{5}^{2} + 3 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( (T - 1)^{2} \)
$5$	\( T^{2} + 3 \)
$7$	\( T^{2} \)
$11$	\( T^{2} + 3 \)