Newform orbit 448.4.i.f

• Label: 448.4.i.f
• Level: $448$
• Weight: $4$
• Character orbit: 448.i
• Analytic conductor: $26.433$
• Analytic rank: $0$
• 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:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 7)
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.

\(f(q)\)	\(=\)	\( q + ( - 7 \zeta_{6} + 7) q^{3} + 7 \zeta_{6} q^{5} + (14 \zeta_{6} + 7) q^{7} - 22 \zeta_{6} q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 7 q^{3} + 7 q^{5} + 28 q^{7} - 22 q^{9}+O(q^{10}) \)

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

3.50000

+

6.06218i

0

3.50000

−

6.06218i

0

14.0000

−

12.1244i

0

−11.0000

+

19.0526i

0

193.1

0

3.50000

−

6.06218i

0

3.50000

+

6.06218i

0

14.0000

+

12.1244i

0

−11.0000

−

19.0526i

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.f		2
4.b	odd	2	1		448.4.i.a		2
7.c	even	3	1	inner	448.4.i.f		2
8.b	even	2	1		7.4.c.a	✓	2
8.d	odd	2	1		112.4.i.c		2
24.h	odd	2	1		63.4.e.b		2
28.g	odd	6	1		448.4.i.a		2
40.f	even	2	1		175.4.e.a		2
40.i	odd	4	2		175.4.k.a		4
56.h	odd	2	1		49.4.c.a		2
56.j	odd	6	1		49.4.a.c		1
56.j	odd	6	1		49.4.c.a		2
56.k	odd	6	1		112.4.i.c		2
56.k	odd	6	1		784.4.a.b		1
56.m	even	6	1		784.4.a.r		1
56.p	even	6	1		7.4.c.a	✓	2
56.p	even	6	1		49.4.a.d		1
168.i	even	2	1		441.4.e.k		2
168.s	odd	6	1		63.4.e.b		2
168.s	odd	6	1		441.4.a.d		1
168.ba	even	6	1		441.4.a.e		1
168.ba	even	6	1		441.4.e.k		2
280.bf	even	6	1		175.4.e.a		2
280.bf	even	6	1		1225.4.a.c		1
280.bk	odd	6	1		1225.4.a.d		1
280.bt	odd	12	2		175.4.k.a		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
7.4.c.a	✓	2	8.b	even	2	1
7.4.c.a	✓	2	56.p	even	6	1
49.4.a.c		1	56.j	odd	6	1
49.4.a.d		1	56.p	even	6	1
49.4.c.a		2	56.h	odd	2	1
49.4.c.a		2	56.j	odd	6	1
63.4.e.b		2	24.h	odd	2	1
63.4.e.b		2	168.s	odd	6	1
112.4.i.c		2	8.d	odd	2	1
112.4.i.c		2	56.k	odd	6	1
175.4.e.a		2	40.f	even	2	1
175.4.e.a		2	280.bf	even	6	1
175.4.k.a		4	40.i	odd	4	2
175.4.k.a		4	280.bt	odd	12	2
441.4.a.d		1	168.s	odd	6	1
441.4.a.e		1	168.ba	even	6	1
441.4.e.k		2	168.i	even	2	1
441.4.e.k		2	168.ba	even	6	1
448.4.i.a		2	4.b	odd	2	1
448.4.i.a		2	28.g	odd	6	1
448.4.i.f		2	1.a	even	1	1	trivial
448.4.i.f		2	7.c	even	3	1	inner
784.4.a.b		1	56.k	odd	6	1
784.4.a.r		1	56.m	even	6	1
1225.4.a.c		1	280.bf	even	6	1
1225.4.a.d		1	280.bk	odd	6	1

Hecke kernels

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

\( T_{3}^{2} - 7T_{3} + 49 \)

\( T_{11}^{2} + 5T_{11} + 25 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} - 7T + 49 \)
$5$	\( T^{2} - 7T + 49 \)
$7$	\( T^{2} - 28T + 343 \)
$11$	\( T^{2} + 5T + 25 \)