# Properties

 Label 448.1.l.a Level $448$ Weight $1$ Character orbit 448.l Analytic conductor $0.224$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ CM discriminant -7 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$448 = 2^{6} \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 448.l (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.223581125660$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 112) Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.14336.1 Artin image: $C_4\wr C_2$ Artin field: Galois closure of 8.0.78675968.2

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q -i q^{7} -i q^{9} +O(q^{10})$$ $$q -i q^{7} -i q^{9} + ( 1 + i ) q^{11} + i q^{25} + ( -1 + i ) q^{29} + ( -1 - i ) q^{37} + ( -1 - i ) q^{43} - q^{49} + ( 1 + i ) q^{53} - q^{63} + ( -1 + i ) q^{67} + 2 i q^{71} + ( 1 - i ) q^{77} - q^{81} + ( 1 - i ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + O(q^{10})$$ $$2 q + 2 q^{11} - 2 q^{29} - 2 q^{37} - 2 q^{43} - 2 q^{49} + 2 q^{53} - 2 q^{63} - 2 q^{67} + 2 q^{77} - 2 q^{81} + 2 q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$127$$ $$129$$ $$197$$ $$\chi(n)$$ $$1$$ $$-1$$ $$i$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
209.1
 − 1.00000i 1.00000i
0 0 0 0 0 1.00000i 0 1.00000i 0
433.1 0 0 0 0 0 1.00000i 0 1.00000i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
16.e even 4 1 inner
112.l odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.1.l.a 2
4.b odd 2 1 112.1.l.a 2
7.b odd 2 1 CM 448.1.l.a 2
7.c even 3 2 3136.1.bc.a 4
7.d odd 6 2 3136.1.bc.a 4
8.b even 2 1 896.1.l.a 2
8.d odd 2 1 896.1.l.b 2
12.b even 2 1 1008.1.u.b 2
16.e even 4 1 inner 448.1.l.a 2
16.e even 4 1 896.1.l.a 2
16.f odd 4 1 112.1.l.a 2
16.f odd 4 1 896.1.l.b 2
20.d odd 2 1 2800.1.z.a 2
20.e even 4 1 2800.1.bf.a 2
20.e even 4 1 2800.1.bf.b 2
28.d even 2 1 112.1.l.a 2
28.f even 6 2 784.1.y.a 4
28.g odd 6 2 784.1.y.a 4
48.k even 4 1 1008.1.u.b 2
56.e even 2 1 896.1.l.b 2
56.h odd 2 1 896.1.l.a 2
80.j even 4 1 2800.1.bf.b 2
80.k odd 4 1 2800.1.z.a 2
80.s even 4 1 2800.1.bf.a 2
84.h odd 2 1 1008.1.u.b 2
112.j even 4 1 112.1.l.a 2
112.j even 4 1 896.1.l.b 2
112.l odd 4 1 inner 448.1.l.a 2
112.l odd 4 1 896.1.l.a 2
112.u odd 12 2 784.1.y.a 4
112.v even 12 2 784.1.y.a 4
112.w even 12 2 3136.1.bc.a 4
112.x odd 12 2 3136.1.bc.a 4
140.c even 2 1 2800.1.z.a 2
140.j odd 4 1 2800.1.bf.a 2
140.j odd 4 1 2800.1.bf.b 2
336.v odd 4 1 1008.1.u.b 2
560.u odd 4 1 2800.1.bf.a 2
560.be even 4 1 2800.1.z.a 2
560.bm odd 4 1 2800.1.bf.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.1.l.a 2 4.b odd 2 1
112.1.l.a 2 16.f odd 4 1
112.1.l.a 2 28.d even 2 1
112.1.l.a 2 112.j even 4 1
448.1.l.a 2 1.a even 1 1 trivial
448.1.l.a 2 7.b odd 2 1 CM
448.1.l.a 2 16.e even 4 1 inner
448.1.l.a 2 112.l odd 4 1 inner
784.1.y.a 4 28.f even 6 2
784.1.y.a 4 28.g odd 6 2
784.1.y.a 4 112.u odd 12 2
784.1.y.a 4 112.v even 12 2
896.1.l.a 2 8.b even 2 1
896.1.l.a 2 16.e even 4 1
896.1.l.a 2 56.h odd 2 1
896.1.l.a 2 112.l odd 4 1
896.1.l.b 2 8.d odd 2 1
896.1.l.b 2 16.f odd 4 1
896.1.l.b 2 56.e even 2 1
896.1.l.b 2 112.j even 4 1
1008.1.u.b 2 12.b even 2 1
1008.1.u.b 2 48.k even 4 1
1008.1.u.b 2 84.h odd 2 1
1008.1.u.b 2 336.v odd 4 1
2800.1.z.a 2 20.d odd 2 1
2800.1.z.a 2 80.k odd 4 1
2800.1.z.a 2 140.c even 2 1
2800.1.z.a 2 560.be even 4 1
2800.1.bf.a 2 20.e even 4 1
2800.1.bf.a 2 80.s even 4 1
2800.1.bf.a 2 140.j odd 4 1
2800.1.bf.a 2 560.u odd 4 1
2800.1.bf.b 2 20.e even 4 1
2800.1.bf.b 2 80.j even 4 1
2800.1.bf.b 2 140.j odd 4 1
2800.1.bf.b 2 560.bm odd 4 1
3136.1.bc.a 4 7.c even 3 2
3136.1.bc.a 4 7.d odd 6 2
3136.1.bc.a 4 112.w even 12 2
3136.1.bc.a 4 112.x odd 12 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(448, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$2 - 2 T + T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$2 + 2 T + T^{2}$$
$31$ $$T^{2}$$
$37$ $$2 + 2 T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$2 + 2 T + T^{2}$$
$47$ $$T^{2}$$
$53$ $$2 - 2 T + T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2}$$
$67$ $$2 + 2 T + T^{2}$$
$71$ $$4 + T^{2}$$
$73$ $$T^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$