# Properties

 Label 448.6.a.c Level $448$ Weight $6$ Character orbit 448.a Self dual yes Analytic conductor $71.852$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,6,Mod(1,448)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("448.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$448 = 2^{6} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 448.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$71.8519512762$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 7) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - 14 q^{3} + 56 q^{5} + 49 q^{7} - 47 q^{9}+O(q^{10})$$ q - 14 * q^3 + 56 * q^5 + 49 * q^7 - 47 * q^9 $$q - 14 q^{3} + 56 q^{5} + 49 q^{7} - 47 q^{9} + 232 q^{11} + 140 q^{13} - 784 q^{15} - 1722 q^{17} - 98 q^{19} - 686 q^{21} - 1824 q^{23} + 11 q^{25} + 4060 q^{27} - 3418 q^{29} + 7644 q^{31} - 3248 q^{33} + 2744 q^{35} + 10398 q^{37} - 1960 q^{39} - 17962 q^{41} + 10880 q^{43} - 2632 q^{45} - 9324 q^{47} + 2401 q^{49} + 24108 q^{51} - 2262 q^{53} + 12992 q^{55} + 1372 q^{57} - 2730 q^{59} - 25648 q^{61} - 2303 q^{63} + 7840 q^{65} - 48404 q^{67} + 25536 q^{69} + 58560 q^{71} + 68082 q^{73} - 154 q^{75} + 11368 q^{77} - 31784 q^{79} - 45419 q^{81} - 20538 q^{83} - 96432 q^{85} + 47852 q^{87} - 50582 q^{89} + 6860 q^{91} - 107016 q^{93} - 5488 q^{95} - 58506 q^{97} - 10904 q^{99}+O(q^{100})$$ q - 14 * q^3 + 56 * q^5 + 49 * q^7 - 47 * q^9 + 232 * q^11 + 140 * q^13 - 784 * q^15 - 1722 * q^17 - 98 * q^19 - 686 * q^21 - 1824 * q^23 + 11 * q^25 + 4060 * q^27 - 3418 * q^29 + 7644 * q^31 - 3248 * q^33 + 2744 * q^35 + 10398 * q^37 - 1960 * q^39 - 17962 * q^41 + 10880 * q^43 - 2632 * q^45 - 9324 * q^47 + 2401 * q^49 + 24108 * q^51 - 2262 * q^53 + 12992 * q^55 + 1372 * q^57 - 2730 * q^59 - 25648 * q^61 - 2303 * q^63 + 7840 * q^65 - 48404 * q^67 + 25536 * q^69 + 58560 * q^71 + 68082 * q^73 - 154 * q^75 + 11368 * q^77 - 31784 * q^79 - 45419 * q^81 - 20538 * q^83 - 96432 * q^85 + 47852 * q^87 - 50582 * q^89 + 6860 * q^91 - 107016 * q^93 - 5488 * q^95 - 58506 * q^97 - 10904 * q^99

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 −14.0000 0 56.0000 0 49.0000 0 −47.0000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$7$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.6.a.c 1
4.b odd 2 1 448.6.a.m 1
8.b even 2 1 112.6.a.g 1
8.d odd 2 1 7.6.a.a 1
24.f even 2 1 63.6.a.e 1
24.h odd 2 1 1008.6.a.y 1
40.e odd 2 1 175.6.a.b 1
40.k even 4 2 175.6.b.a 2
56.e even 2 1 49.6.a.a 1
56.h odd 2 1 784.6.a.c 1
56.k odd 6 2 49.6.c.c 2
56.m even 6 2 49.6.c.b 2
88.g even 2 1 847.6.a.b 1
168.e odd 2 1 441.6.a.k 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.6.a.a 1 8.d odd 2 1
49.6.a.a 1 56.e even 2 1
49.6.c.b 2 56.m even 6 2
49.6.c.c 2 56.k odd 6 2
63.6.a.e 1 24.f even 2 1
112.6.a.g 1 8.b even 2 1
175.6.a.b 1 40.e odd 2 1
175.6.b.a 2 40.k even 4 2
441.6.a.k 1 168.e odd 2 1
448.6.a.c 1 1.a even 1 1 trivial
448.6.a.m 1 4.b odd 2 1
784.6.a.c 1 56.h odd 2 1
847.6.a.b 1 88.g even 2 1
1008.6.a.y 1 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(448))$$:

 $$T_{3} + 14$$ T3 + 14 $$T_{5} - 56$$ T5 - 56

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 14$$
$5$ $$T - 56$$
$7$ $$T - 49$$
$11$ $$T - 232$$
$13$ $$T - 140$$
$17$ $$T + 1722$$
$19$ $$T + 98$$
$23$ $$T + 1824$$
$29$ $$T + 3418$$
$31$ $$T - 7644$$
$37$ $$T - 10398$$
$41$ $$T + 17962$$
$43$ $$T - 10880$$
$47$ $$T + 9324$$
$53$ $$T + 2262$$
$59$ $$T + 2730$$
$61$ $$T + 25648$$
$67$ $$T + 48404$$
$71$ $$T - 58560$$
$73$ $$T - 68082$$
$79$ $$T + 31784$$
$83$ $$T + 20538$$
$89$ $$T + 50582$$
$97$ $$T + 58506$$