Newform orbit 448.7.c.a

• Label: 448.7.c.a
• Level: $448$
• Weight: $7$
• Character orbit: 448.c
• Self dual: yes
• Analytic conductor: $103.064$
• Analytic rank: $0$
• Dimension: $1$
• CM discriminant: -7
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	448.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(103.064229462\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 7)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\)

\(=\)

\( q - 343 q^{7} + 729 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/448\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(129\)	\(197\)
\(\chi(n)\)	\(1\)	\(-1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

321.1

0

0

0

0

0

0

−343.000

0

729.000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by \(\Q(\sqrt{-7}) \)

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	448.7.c.a		1
4.b	odd	2	1		448.7.c.b		1
7.b	odd	2	1	CM	448.7.c.a		1
8.b	even	2	1		7.7.b.a	✓	1
8.d	odd	2	1		112.7.c.a		1
24.h	odd	2	1		63.7.d.a		1
28.d	even	2	1		448.7.c.b		1
40.f	even	2	1		175.7.d.a		1
40.i	odd	4	2		175.7.c.a		2
56.e	even	2	1		112.7.c.a		1
56.h	odd	2	1		7.7.b.a	✓	1
56.j	odd	6	2		49.7.d.a		2
56.p	even	6	2		49.7.d.a		2
168.i	even	2	1		63.7.d.a		1
280.c	odd	2	1		175.7.d.a		1
280.s	even	4	2		175.7.c.a		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
7.7.b.a	✓	1	8.b	even	2	1
7.7.b.a	✓	1	56.h	odd	2	1
49.7.d.a		2	56.j	odd	6	2
49.7.d.a		2	56.p	even	6	2
63.7.d.a		1	24.h	odd	2	1
63.7.d.a		1	168.i	even	2	1
112.7.c.a		1	8.d	odd	2	1
112.7.c.a		1	56.e	even	2	1
175.7.c.a		2	40.i	odd	4	2
175.7.c.a		2	280.s	even	4	2
175.7.d.a		1	40.f	even	2	1
175.7.d.a		1	280.c	odd	2	1
448.7.c.a		1	1.a	even	1	1	trivial
448.7.c.a		1	7.b	odd	2	1	CM
448.7.c.b		1	4.b	odd	2	1
448.7.c.b		1	28.d	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{7}^{\mathrm{new}}(448, [\chi])\):

\( T_{3} \)

\( T_{11} + 1962 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T \)
$3$	\( T \)
$5$	\( T \)
$7$	\( T + 343 \)
$11$	\( T + 1962 \)