# Properties

 Label 448.1.r.a Level $448$ Weight $1$ Character orbit 448.r Analytic conductor $0.224$ Analytic rank $0$ Dimension $4$ Projective image $A_{4}$ CM/RM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$0.223581125660$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 224) Projective image: $$A_{4}$$ Projective field: Galois closure of 4.0.3136.1 Artin image: $\SL(2,3):C_2$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{16} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{12} q^{3} + \zeta_{12}^{2} q^{5} -\zeta_{12}^{3} q^{7} +O(q^{10})$$ $$q + \zeta_{12} q^{3} + \zeta_{12}^{2} q^{5} -\zeta_{12}^{3} q^{7} -\zeta_{12} q^{11} + \zeta_{12}^{3} q^{15} + \zeta_{12}^{4} q^{17} + \zeta_{12}^{5} q^{19} -\zeta_{12}^{4} q^{21} + \zeta_{12}^{5} q^{23} -\zeta_{12}^{3} q^{27} + \zeta_{12} q^{31} -\zeta_{12}^{2} q^{33} -\zeta_{12}^{5} q^{35} -\zeta_{12}^{2} q^{37} -\zeta_{12}^{5} q^{47} - q^{49} + \zeta_{12}^{5} q^{51} + \zeta_{12}^{4} q^{53} -\zeta_{12}^{3} q^{55} - q^{57} + \zeta_{12} q^{59} -\zeta_{12}^{2} q^{61} + \zeta_{12} q^{67} - q^{69} -\zeta_{12}^{4} q^{73} + \zeta_{12}^{4} q^{77} + \zeta_{12}^{5} q^{79} -\zeta_{12}^{4} q^{81} - q^{85} + \zeta_{12}^{2} q^{89} + \zeta_{12}^{2} q^{93} -\zeta_{12} q^{95} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{5} + O(q^{10})$$ $$4 q + 2 q^{5} - 2 q^{17} + 2 q^{21} - 2 q^{33} - 2 q^{37} - 4 q^{49} - 2 q^{53} - 4 q^{57} - 2 q^{61} - 4 q^{69} + 2 q^{73} - 2 q^{77} + 2 q^{81} - 4 q^{85} + 2 q^{89} + 2 q^{93} + O(q^{100})$$

## Character values

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

 $$n$$ $$127$$ $$129$$ $$197$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{12}^{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
191.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
0 −0.866025 + 0.500000i 0 0.500000 0.866025i 0 1.00000i 0 0 0
191.2 0 0.866025 0.500000i 0 0.500000 0.866025i 0 1.00000i 0 0 0
319.1 0 −0.866025 0.500000i 0 0.500000 + 0.866025i 0 1.00000i 0 0 0
319.2 0 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0 1.00000i 0 0 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
4.b odd 2 1 inner
7.c even 3 1 inner
28.g odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.1.r.a 4
4.b odd 2 1 inner 448.1.r.a 4
7.b odd 2 1 3136.1.r.b 4
7.c even 3 1 inner 448.1.r.a 4
7.c even 3 1 3136.1.d.b 2
7.d odd 6 1 3136.1.d.d 2
7.d odd 6 1 3136.1.r.b 4
8.b even 2 1 224.1.r.a 4
8.d odd 2 1 224.1.r.a 4
16.e even 4 1 1792.1.o.a 4
16.e even 4 1 1792.1.o.b 4
16.f odd 4 1 1792.1.o.a 4
16.f odd 4 1 1792.1.o.b 4
24.f even 2 1 2016.1.cd.a 4
24.h odd 2 1 2016.1.cd.a 4
28.d even 2 1 3136.1.r.b 4
28.f even 6 1 3136.1.d.d 2
28.f even 6 1 3136.1.r.b 4
28.g odd 6 1 inner 448.1.r.a 4
28.g odd 6 1 3136.1.d.b 2
56.e even 2 1 1568.1.r.a 4
56.h odd 2 1 1568.1.r.a 4
56.j odd 6 1 1568.1.d.a 2
56.j odd 6 1 1568.1.r.a 4
56.k odd 6 1 224.1.r.a 4
56.k odd 6 1 1568.1.d.b 2
56.m even 6 1 1568.1.d.a 2
56.m even 6 1 1568.1.r.a 4
56.p even 6 1 224.1.r.a 4
56.p even 6 1 1568.1.d.b 2
112.u odd 12 1 1792.1.o.a 4
112.u odd 12 1 1792.1.o.b 4
112.w even 12 1 1792.1.o.a 4
112.w even 12 1 1792.1.o.b 4
168.s odd 6 1 2016.1.cd.a 4
168.v even 6 1 2016.1.cd.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.1.r.a 4 8.b even 2 1
224.1.r.a 4 8.d odd 2 1
224.1.r.a 4 56.k odd 6 1
224.1.r.a 4 56.p even 6 1
448.1.r.a 4 1.a even 1 1 trivial
448.1.r.a 4 4.b odd 2 1 inner
448.1.r.a 4 7.c even 3 1 inner
448.1.r.a 4 28.g odd 6 1 inner
1568.1.d.a 2 56.j odd 6 1
1568.1.d.a 2 56.m even 6 1
1568.1.d.b 2 56.k odd 6 1
1568.1.d.b 2 56.p even 6 1
1568.1.r.a 4 56.e even 2 1
1568.1.r.a 4 56.h odd 2 1
1568.1.r.a 4 56.j odd 6 1
1568.1.r.a 4 56.m even 6 1
1792.1.o.a 4 16.e even 4 1
1792.1.o.a 4 16.f odd 4 1
1792.1.o.a 4 112.u odd 12 1
1792.1.o.a 4 112.w even 12 1
1792.1.o.b 4 16.e even 4 1
1792.1.o.b 4 16.f odd 4 1
1792.1.o.b 4 112.u odd 12 1
1792.1.o.b 4 112.w even 12 1
2016.1.cd.a 4 24.f even 2 1
2016.1.cd.a 4 24.h odd 2 1
2016.1.cd.a 4 168.s odd 6 1
2016.1.cd.a 4 168.v even 6 1
3136.1.d.b 2 7.c even 3 1
3136.1.d.b 2 28.g odd 6 1
3136.1.d.d 2 7.d odd 6 1
3136.1.d.d 2 28.f even 6 1
3136.1.r.b 4 7.b odd 2 1
3136.1.r.b 4 7.d odd 6 1
3136.1.r.b 4 28.d even 2 1
3136.1.r.b 4 28.f even 6 1

## Hecke kernels

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

## Hecke characteristic polynomials

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