Newform orbit 448.2.p.a

• Label: 448.2.p.a
• Level: $448$
• Weight: $2$
• Character orbit: 448.p
• 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.p (of order \(6\), 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:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\)	\(=\)	\( q + (\zeta_{6} - 1) q^{3} + (\zeta_{6} - 2) q^{5} + ( - 2 \zeta_{6} - 1) q^{7} + 2 \zeta_{6} q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - 3 q^{5} - 4 q^{7} + 2 q^{9}+O(q^{10}) \)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

255.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

−0.500000

+

0.866025i

0

−1.50000

+

0.866025i

0

−2.00000

−

1.73205i

0

1.00000

+

1.73205i

0

383.1

0

−0.500000

−

0.866025i

0

−1.50000

−

0.866025i

0

−2.00000

+

1.73205i

0

1.00000

−

1.73205i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
28.f	even	6	1	inner

Twists

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

    

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(448, [\chi])\).

Hecke characteristic polynomials

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