Newform orbit 3136.1.d.b

• Label: 3136.1.d.b
• Level: $3136$
• Weight: $1$
• Character orbit: 3136.d
• Analytic conductor: $1.565$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $A_{4}$
• CM/RM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.56506787962\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(i)\)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 224)
Projective image:	\(A_{4}\)
Projective field:	Galois closure of 4.0.3136.1
Artin image:	$\SL(2,3):C_2$
Artin field:	Galois closure of \(\mathbb{Q}[x]/(x^{16} - \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\)	\(=\)	q + z * q^3 - q^5 - z * q^11 - z * q^15 + q^17 + z * q^19 + z * q^23 + z * q^27 + z * q^31 + q^33 + q^37 - z * q^47 + z * q^51 + q^53 + z * q^55 - q^57 + z * q^59 + q^61 + z * q^67 - q^69 - q^73 + z * q^79 - q^81 - q^85 - q^89 - q^93 - z * q^95
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 2 q^{17} + 2 q^{33} + 2 q^{37} + 2 q^{53} - 2 q^{57} + 2 q^{61} - 2 q^{69} - 2 q^{73} - 2 q^{81} - 2 q^{85} - 2 q^{89} - 2 q^{93}+O(q^{100}) \)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1471.1

	−	1.00000i
		1.00000i

0

−

1.00000i

0

−1.00000

0

0

0

0

0

1471.2

0

1.00000i

0

−1.00000

0

0

0

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3136.1.d.b		2
4.b	odd	2	1	inner	3136.1.d.b		2
7.b	odd	2	1		3136.1.d.d		2
7.c	even	3	2		448.1.r.a		4
7.d	odd	6	2		3136.1.r.b		4
8.b	even	2	1		1568.1.d.b		2
8.d	odd	2	1		1568.1.d.b		2
28.d	even	2	1		3136.1.d.d		2
28.f	even	6	2		3136.1.r.b		4
28.g	odd	6	2		448.1.r.a		4
56.e	even	2	1		1568.1.d.a		2
56.h	odd	2	1		1568.1.d.a		2
56.j	odd	6	2		1568.1.r.a		4
56.k	odd	6	2		224.1.r.a	✓	4
56.m	even	6	2		1568.1.r.a		4
56.p	even	6	2		224.1.r.a	✓	4
112.u	odd	12	2		1792.1.o.a		4
112.u	odd	12	2		1792.1.o.b		4
112.w	even	12	2		1792.1.o.a		4
112.w	even	12	2		1792.1.o.b		4
168.s	odd	6	2		2016.1.cd.a		4
168.v	even	6	2		2016.1.cd.a		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
224.1.r.a	✓	4	56.k	odd	6	2
224.1.r.a	✓	4	56.p	even	6	2
448.1.r.a		4	7.c	even	3	2
448.1.r.a		4	28.g	odd	6	2
1568.1.d.a		2	56.e	even	2	1
1568.1.d.a		2	56.h	odd	2	1
1568.1.d.b		2	8.b	even	2	1
1568.1.d.b		2	8.d	odd	2	1
1568.1.r.a		4	56.j	odd	6	2
1568.1.r.a		4	56.m	even	6	2
1792.1.o.a		4	112.u	odd	12	2
1792.1.o.a		4	112.w	even	12	2
1792.1.o.b		4	112.u	odd	12	2
1792.1.o.b		4	112.w	even	12	2
2016.1.cd.a		4	168.s	odd	6	2
2016.1.cd.a		4	168.v	even	6	2
3136.1.d.b		2	1.a	even	1	1	trivial
3136.1.d.b		2	4.b	odd	2	1	inner
3136.1.d.d		2	7.b	odd	2	1
3136.1.d.d		2	28.d	even	2	1
3136.1.r.b		4	7.d	odd	6	2
3136.1.r.b		4	28.f	even	6	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3136, [\chi])\):

\( T_{3}^{2} + 1 \)

\( T_{5} + 1 \)

Hecke characteristic polynomials

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