# Properties

 Label 224.2.a.b Level $224$ Weight $2$ Character orbit 224.a Self dual yes Analytic conductor $1.789$ Analytic rank $0$ 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] = [224,2,Mod(1,224)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(224, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("224.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$224 = 2^{5} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 224.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: $$1.78864900528$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes 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)

 $$f(q)$$ $$=$$ $$q + 2 q^{3} + q^{7} + q^{9}+O(q^{10})$$ q + 2 * q^3 + q^7 + q^9 $$q + 2 q^{3} + q^{7} + q^{9} + 4 q^{11} - 4 q^{13} - 2 q^{17} + 6 q^{19} + 2 q^{21} - 8 q^{23} - 5 q^{25} - 4 q^{27} + 2 q^{29} + 4 q^{31} + 8 q^{33} + 10 q^{37} - 8 q^{39} - 10 q^{41} - 4 q^{43} - 4 q^{47} + q^{49} - 4 q^{51} - 2 q^{53} + 12 q^{57} - 10 q^{59} - 8 q^{61} + q^{63} + 8 q^{67} - 16 q^{69} - 6 q^{73} - 10 q^{75} + 4 q^{77} + 16 q^{79} - 11 q^{81} - 2 q^{83} + 4 q^{87} + 18 q^{89} - 4 q^{91} + 8 q^{93} - 2 q^{97} + 4 q^{99}+O(q^{100})$$ q + 2 * q^3 + q^7 + q^9 + 4 * q^11 - 4 * q^13 - 2 * q^17 + 6 * q^19 + 2 * q^21 - 8 * q^23 - 5 * q^25 - 4 * q^27 + 2 * q^29 + 4 * q^31 + 8 * q^33 + 10 * q^37 - 8 * q^39 - 10 * q^41 - 4 * q^43 - 4 * q^47 + q^49 - 4 * q^51 - 2 * q^53 + 12 * q^57 - 10 * q^59 - 8 * q^61 + q^63 + 8 * q^67 - 16 * q^69 - 6 * q^73 - 10 * q^75 + 4 * q^77 + 16 * q^79 - 11 * q^81 - 2 * q^83 + 4 * q^87 + 18 * q^89 - 4 * q^91 + 8 * q^93 - 2 * q^97 + 4 * 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 2.00000 0 0 0 1.00000 0 1.00000 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 224.2.a.b yes 1
3.b odd 2 1 2016.2.a.g 1
4.b odd 2 1 224.2.a.a 1
5.b even 2 1 5600.2.a.c 1
7.b odd 2 1 1568.2.a.b 1
7.c even 3 2 1568.2.i.b 2
7.d odd 6 2 1568.2.i.j 2
8.b even 2 1 448.2.a.b 1
8.d odd 2 1 448.2.a.f 1
12.b even 2 1 2016.2.a.e 1
16.e even 4 2 1792.2.b.b 2
16.f odd 4 2 1792.2.b.f 2
20.d odd 2 1 5600.2.a.t 1
24.f even 2 1 4032.2.a.p 1
24.h odd 2 1 4032.2.a.z 1
28.d even 2 1 1568.2.a.h 1
28.f even 6 2 1568.2.i.c 2
28.g odd 6 2 1568.2.i.k 2
56.e even 2 1 3136.2.a.f 1
56.h odd 2 1 3136.2.a.y 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.a.a 1 4.b odd 2 1
224.2.a.b yes 1 1.a even 1 1 trivial
448.2.a.b 1 8.b even 2 1
448.2.a.f 1 8.d odd 2 1
1568.2.a.b 1 7.b odd 2 1
1568.2.a.h 1 28.d even 2 1
1568.2.i.b 2 7.c even 3 2
1568.2.i.c 2 28.f even 6 2
1568.2.i.j 2 7.d odd 6 2
1568.2.i.k 2 28.g odd 6 2
1792.2.b.b 2 16.e even 4 2
1792.2.b.f 2 16.f odd 4 2
2016.2.a.e 1 12.b even 2 1
2016.2.a.g 1 3.b odd 2 1
3136.2.a.f 1 56.e even 2 1
3136.2.a.y 1 56.h odd 2 1
4032.2.a.p 1 24.f even 2 1
4032.2.a.z 1 24.h odd 2 1
5600.2.a.c 1 5.b even 2 1
5600.2.a.t 1 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 2$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(224))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 2$$
$5$ $$T$$
$7$ $$T - 1$$
$11$ $$T - 4$$
$13$ $$T + 4$$
$17$ $$T + 2$$
$19$ $$T - 6$$
$23$ $$T + 8$$
$29$ $$T - 2$$
$31$ $$T - 4$$
$37$ $$T - 10$$
$41$ $$T + 10$$
$43$ $$T + 4$$
$47$ $$T + 4$$
$53$ $$T + 2$$
$59$ $$T + 10$$
$61$ $$T + 8$$
$67$ $$T - 8$$
$71$ $$T$$
$73$ $$T + 6$$
$79$ $$T - 16$$
$83$ $$T + 2$$
$89$ $$T - 18$$
$97$ $$T + 2$$