Newform orbit 448.2.i.a

• Label: 448.2.i.a
• Level: $448$
• Weight: $2$
• Character orbit: 448.i
• Analytic conductor: $3.577$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(3.57729801055\)
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 56)
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.

\(f(q)\)	\(=\)	\( q + (3 \zeta_{6} - 3) q^{3} - \zeta_{6} q^{5} + (2 \zeta_{6} - 3) q^{7} - 6 \zeta_{6} q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{3} - q^{5} - 4 q^{7} - 6 q^{9}+O(q^{10}) \)

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

−1.50000

−

2.59808i

0

−0.500000

+

0.866025i

0

−2.00000

−

1.73205i

0

−3.00000

+

5.19615i

0

193.1

0

−1.50000

+

2.59808i

0

−0.500000

−

0.866025i

0

−2.00000

+

1.73205i

0

−3.00000

−

5.19615i

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	448.2.i.a		2
4.b	odd	2	1		448.2.i.f		2
7.c	even	3	1	inner	448.2.i.a		2
7.c	even	3	1		3136.2.a.bc		1
7.d	odd	6	1		3136.2.a.a		1
8.b	even	2	1		112.2.i.c		2
8.d	odd	2	1		56.2.i.a	✓	2
24.f	even	2	1		504.2.s.e		2
24.h	odd	2	1		1008.2.s.e		2
28.f	even	6	1		3136.2.a.bb		1
28.g	odd	6	1		448.2.i.f		2
28.g	odd	6	1		3136.2.a.b		1
40.e	odd	2	1		1400.2.q.g		2
40.k	even	4	2		1400.2.bh.f		4
56.e	even	2	1		392.2.i.f		2
56.h	odd	2	1		784.2.i.a		2
56.j	odd	6	1		784.2.a.j		1
56.j	odd	6	1		784.2.i.a		2
56.k	odd	6	1		56.2.i.a	✓	2
56.k	odd	6	1		392.2.a.f		1
56.m	even	6	1		392.2.a.a		1
56.m	even	6	1		392.2.i.f		2
56.p	even	6	1		112.2.i.c		2
56.p	even	6	1		784.2.a.a		1
168.e	odd	2	1		3528.2.s.o		2
168.s	odd	6	1		1008.2.s.e		2
168.s	odd	6	1		7056.2.a.bi		1
168.v	even	6	1		504.2.s.e		2
168.v	even	6	1		3528.2.a.r		1
168.ba	even	6	1		7056.2.a.s		1
168.be	odd	6	1		3528.2.a.k		1
168.be	odd	6	1		3528.2.s.o		2
280.ba	even	6	1		9800.2.a.bp		1
280.bi	odd	6	1		1400.2.q.g		2
280.bi	odd	6	1		9800.2.a.b		1
280.br	even	12	2		1400.2.bh.f		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
56.2.i.a	✓	2	8.d	odd	2	1
56.2.i.a	✓	2	56.k	odd	6	1
112.2.i.c		2	8.b	even	2	1
112.2.i.c		2	56.p	even	6	1
392.2.a.a		1	56.m	even	6	1
392.2.a.f		1	56.k	odd	6	1
392.2.i.f		2	56.e	even	2	1
392.2.i.f		2	56.m	even	6	1
448.2.i.a		2	1.a	even	1	1	trivial
448.2.i.a		2	7.c	even	3	1	inner
448.2.i.f		2	4.b	odd	2	1
448.2.i.f		2	28.g	odd	6	1
504.2.s.e		2	24.f	even	2	1
504.2.s.e		2	168.v	even	6	1
784.2.a.a		1	56.p	even	6	1
784.2.a.j		1	56.j	odd	6	1
784.2.i.a		2	56.h	odd	2	1
784.2.i.a		2	56.j	odd	6	1
1008.2.s.e		2	24.h	odd	2	1
1008.2.s.e		2	168.s	odd	6	1
1400.2.q.g		2	40.e	odd	2	1
1400.2.q.g		2	280.bi	odd	6	1
1400.2.bh.f		4	40.k	even	4	2
1400.2.bh.f		4	280.br	even	12	2
3136.2.a.a		1	7.d	odd	6	1
3136.2.a.b		1	28.g	odd	6	1
3136.2.a.bb		1	28.f	even	6	1
3136.2.a.bc		1	7.c	even	3	1
3528.2.a.k		1	168.be	odd	6	1
3528.2.a.r		1	168.v	even	6	1
3528.2.s.o		2	168.e	odd	2	1
3528.2.s.o		2	168.be	odd	6	1
7056.2.a.s		1	168.ba	even	6	1
7056.2.a.bi		1	168.s	odd	6	1
9800.2.a.b		1	280.bi	odd	6	1
9800.2.a.bp		1	280.ba	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(448, [\chi])\):

\( T_{3}^{2} + 3T_{3} + 9 \)

\( T_{5}^{2} + T_{5} + 1 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} + 3T + 9 \)
$5$	\( T^{2} + T + 1 \)
$7$	\( T^{2} + 4T + 7 \)
$11$	\( T^{2} - T + 1 \)