# Properties

 Label 448.6.a.n 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 224) 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)

 $$f(q)$$ $$=$$ $$q + 14 q^{3} + 64 q^{5} + 49 q^{7} - 47 q^{9}+O(q^{10})$$ q + 14 * q^3 + 64 * q^5 + 49 * q^7 - 47 * q^9 $$q + 14 q^{3} + 64 q^{5} + 49 q^{7} - 47 q^{9} - 420 q^{11} - 860 q^{13} + 896 q^{15} + 830 q^{17} + 490 q^{19} + 686 q^{21} - 4872 q^{23} + 971 q^{25} - 4060 q^{27} - 8754 q^{29} - 5628 q^{31} - 5880 q^{33} + 3136 q^{35} - 1434 q^{37} - 12040 q^{39} - 9258 q^{41} + 14756 q^{43} - 3008 q^{45} + 10108 q^{47} + 2401 q^{49} + 11620 q^{51} + 23058 q^{53} - 26880 q^{55} + 6860 q^{57} - 13734 q^{59} + 25352 q^{61} - 2303 q^{63} - 55040 q^{65} + 19768 q^{67} - 68208 q^{69} - 1792 q^{71} + 37914 q^{73} + 13594 q^{75} - 20580 q^{77} - 95984 q^{79} - 45419 q^{81} + 88242 q^{83} + 53120 q^{85} - 122556 q^{87} + 43762 q^{89} - 42140 q^{91} - 78792 q^{93} + 31360 q^{95} + 65790 q^{97} + 19740 q^{99}+O(q^{100})$$ q + 14 * q^3 + 64 * q^5 + 49 * q^7 - 47 * q^9 - 420 * q^11 - 860 * q^13 + 896 * q^15 + 830 * q^17 + 490 * q^19 + 686 * q^21 - 4872 * q^23 + 971 * q^25 - 4060 * q^27 - 8754 * q^29 - 5628 * q^31 - 5880 * q^33 + 3136 * q^35 - 1434 * q^37 - 12040 * q^39 - 9258 * q^41 + 14756 * q^43 - 3008 * q^45 + 10108 * q^47 + 2401 * q^49 + 11620 * q^51 + 23058 * q^53 - 26880 * q^55 + 6860 * q^57 - 13734 * q^59 + 25352 * q^61 - 2303 * q^63 - 55040 * q^65 + 19768 * q^67 - 68208 * q^69 - 1792 * q^71 + 37914 * q^73 + 13594 * q^75 - 20580 * q^77 - 95984 * q^79 - 45419 * q^81 + 88242 * q^83 + 53120 * q^85 - 122556 * q^87 + 43762 * q^89 - 42140 * q^91 - 78792 * q^93 + 31360 * q^95 + 65790 * q^97 + 19740 * 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 64.0000 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.n 1
4.b odd 2 1 448.6.a.d 1
8.b even 2 1 224.6.a.a 1
8.d odd 2 1 224.6.a.b yes 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.6.a.a 1 8.b even 2 1
224.6.a.b yes 1 8.d odd 2 1
448.6.a.d 1 4.b odd 2 1
448.6.a.n 1 1.a even 1 1 trivial

## 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 $$T_{5} - 64$$ T5 - 64

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 14$$
$5$ $$T - 64$$
$7$ $$T - 49$$
$11$ $$T + 420$$
$13$ $$T + 860$$
$17$ $$T - 830$$
$19$ $$T - 490$$
$23$ $$T + 4872$$
$29$ $$T + 8754$$
$31$ $$T + 5628$$
$37$ $$T + 1434$$
$41$ $$T + 9258$$
$43$ $$T - 14756$$
$47$ $$T - 10108$$
$53$ $$T - 23058$$
$59$ $$T + 13734$$
$61$ $$T - 25352$$
$67$ $$T - 19768$$
$71$ $$T + 1792$$
$73$ $$T - 37914$$
$79$ $$T + 95984$$
$83$ $$T - 88242$$
$89$ $$T - 43762$$
$97$ $$T - 65790$$