Newform orbit 448.4.i.b

• Label: 448.4.i.b
• Level: $448$
• Weight: $4$
• Character orbit: 448.i
• Analytic conductor: $26.433$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	448.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(26.4328556826\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 14)
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.

\(f(q)\)	\(=\)	\( q + (5 \zeta_{6} - 5) q^{3} - 9 \zeta_{6} q^{5} + (14 \zeta_{6} - 21) q^{7} + 2 \zeta_{6} q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 5 q^{3} - 9 q^{5} - 28 q^{7} + 2 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(129\)	\(197\)
\(\chi(n)\)	\(1\)	\(-\zeta_{6}\)	\(1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

65.1

0.500000	−	0.866025i
0.500000	+	0.866025i

0

−2.50000

−

4.33013i

0

−4.50000

+

7.79423i

0

−14.0000

−

12.1244i

0

1.00000

−

1.73205i

0

193.1

0

−2.50000

+

4.33013i

0

−4.50000

−

7.79423i

0

−14.0000

+

12.1244i

0

1.00000

+

1.73205i

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	448.4.i.b		2
4.b	odd	2	1		448.4.i.e		2
7.c	even	3	1	inner	448.4.i.b		2
8.b	even	2	1		14.4.c.a	✓	2
8.d	odd	2	1		112.4.i.a		2
24.h	odd	2	1		126.4.g.d		2
28.g	odd	6	1		448.4.i.e		2
40.f	even	2	1		350.4.e.e		2
40.i	odd	4	2		350.4.j.b		4
56.h	odd	2	1		98.4.c.a		2
56.j	odd	6	1		98.4.a.f		1
56.j	odd	6	1		98.4.c.a		2
56.k	odd	6	1		112.4.i.a		2
56.k	odd	6	1		784.4.a.p		1
56.m	even	6	1		784.4.a.c		1
56.p	even	6	1		14.4.c.a	✓	2
56.p	even	6	1		98.4.a.d		1
168.i	even	2	1		882.4.g.u		2
168.s	odd	6	1		126.4.g.d		2
168.s	odd	6	1		882.4.a.f		1
168.ba	even	6	1		882.4.a.c		1
168.ba	even	6	1		882.4.g.u		2
280.bf	even	6	1		350.4.e.e		2
280.bf	even	6	1		2450.4.a.q		1
280.bk	odd	6	1		2450.4.a.d		1
280.bt	odd	12	2		350.4.j.b		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
14.4.c.a	✓	2	8.b	even	2	1
14.4.c.a	✓	2	56.p	even	6	1
98.4.a.d		1	56.p	even	6	1
98.4.a.f		1	56.j	odd	6	1
98.4.c.a		2	56.h	odd	2	1
98.4.c.a		2	56.j	odd	6	1
112.4.i.a		2	8.d	odd	2	1
112.4.i.a		2	56.k	odd	6	1
126.4.g.d		2	24.h	odd	2	1
126.4.g.d		2	168.s	odd	6	1
350.4.e.e		2	40.f	even	2	1
350.4.e.e		2	280.bf	even	6	1
350.4.j.b		4	40.i	odd	4	2
350.4.j.b		4	280.bt	odd	12	2
448.4.i.b		2	1.a	even	1	1	trivial
448.4.i.b		2	7.c	even	3	1	inner
448.4.i.e		2	4.b	odd	2	1
448.4.i.e		2	28.g	odd	6	1
784.4.a.c		1	56.m	even	6	1
784.4.a.p		1	56.k	odd	6	1
882.4.a.c		1	168.ba	even	6	1
882.4.a.f		1	168.s	odd	6	1
882.4.g.u		2	168.i	even	2	1
882.4.g.u		2	168.ba	even	6	1
2450.4.a.d		1	280.bk	odd	6	1
2450.4.a.q		1	280.bf	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(448, [\chi])\):

\( T_{3}^{2} + 5T_{3} + 25 \)

\( T_{11}^{2} + 57T_{11} + 3249 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} + 5T + 25 \)
$5$	\( T^{2} + 9T + 81 \)
$7$	\( T^{2} + 28T + 343 \)
$11$	\( T^{2} + 57T + 3249 \)