Newform orbit 448.3.s.a

• Label: 448.3.s.a
• Level: $448$
• Weight: $3$
• Character orbit: 448.s
• Analytic conductor: $12.207$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(12.2071158433\)
Analytic rank:	\(0\)
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 28)
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.

\(f(q)\)	\(=\)	\( q + ( - \zeta_{6} - 1) q^{3} + (\zeta_{6} - 2) q^{5} - 7 q^{7} - 6 \zeta_{6} q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{3} - 3 q^{5} - 14 q^{7} - 6 q^{9}+O(q^{10}) \)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

129.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

−1.50000

−

0.866025i

0

−1.50000

+

0.866025i

0

−7.00000

0

−3.00000

−

5.19615i

0

257.1

0

−1.50000

+

0.866025i

0

−1.50000

−

0.866025i

0

−7.00000

0

−3.00000

+

5.19615i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.d	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	448.3.s.a		2
4.b	odd	2	1		448.3.s.b		2
7.d	odd	6	1	inner	448.3.s.a		2
8.b	even	2	1		28.3.h.a	✓	2
8.d	odd	2	1		112.3.s.a		2
24.f	even	2	1		1008.3.cg.c		2
24.h	odd	2	1		252.3.z.a		2
28.f	even	6	1		448.3.s.b		2
40.f	even	2	1		700.3.s.a		2
40.i	odd	4	2		700.3.o.a		4
56.e	even	2	1		784.3.s.b		2
56.h	odd	2	1		196.3.h.a		2
56.j	odd	6	1		28.3.h.a	✓	2
56.j	odd	6	1		196.3.b.a		2
56.k	odd	6	1		784.3.c.a		2
56.k	odd	6	1		784.3.s.b		2
56.m	even	6	1		112.3.s.a		2
56.m	even	6	1		784.3.c.a		2
56.p	even	6	1		196.3.b.a		2
56.p	even	6	1		196.3.h.a		2
168.i	even	2	1		1764.3.z.f		2
168.s	odd	6	1		1764.3.d.a		2
168.s	odd	6	1		1764.3.z.f		2
168.ba	even	6	1		252.3.z.a		2
168.ba	even	6	1		1764.3.d.a		2
168.be	odd	6	1		1008.3.cg.c		2
280.bk	odd	6	1		700.3.s.a		2
280.bv	even	12	2		700.3.o.a		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
28.3.h.a	✓	2	8.b	even	2	1
28.3.h.a	✓	2	56.j	odd	6	1
112.3.s.a		2	8.d	odd	2	1
112.3.s.a		2	56.m	even	6	1
196.3.b.a		2	56.j	odd	6	1
196.3.b.a		2	56.p	even	6	1
196.3.h.a		2	56.h	odd	2	1
196.3.h.a		2	56.p	even	6	1
252.3.z.a		2	24.h	odd	2	1
252.3.z.a		2	168.ba	even	6	1
448.3.s.a		2	1.a	even	1	1	trivial
448.3.s.a		2	7.d	odd	6	1	inner
448.3.s.b		2	4.b	odd	2	1
448.3.s.b		2	28.f	even	6	1
700.3.o.a		4	40.i	odd	4	2
700.3.o.a		4	280.bv	even	12	2
700.3.s.a		2	40.f	even	2	1
700.3.s.a		2	280.bk	odd	6	1
784.3.c.a		2	56.k	odd	6	1
784.3.c.a		2	56.m	even	6	1
784.3.s.b		2	56.e	even	2	1
784.3.s.b		2	56.k	odd	6	1
1008.3.cg.c		2	24.f	even	2	1
1008.3.cg.c		2	168.be	odd	6	1
1764.3.d.a		2	168.s	odd	6	1
1764.3.d.a		2	168.ba	even	6	1
1764.3.z.f		2	168.i	even	2	1
1764.3.z.f		2	168.s	odd	6	1

Hecke kernels

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

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} + 3T + 3 \)
$5$	\( T^{2} + 3T + 3 \)
$7$	\( (T + 7)^{2} \)
$11$	\( T^{2} - 15T + 225 \)