# Properties

 Label 224.1.v.a Level $224$ Weight $1$ Character orbit 224.v Analytic conductor $0.112$ Analytic rank $0$ Dimension $4$ Projective image $D_{8}$ CM discriminant -7 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [224,1,Mod(13,224)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(224, base_ring=CyclotomicField(8))

chi = DirichletCharacter(H, H._module([0, 7, 4]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("224.13");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$224 = 2^{5} \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 224.v (of order $$8$$, degree $$4$$, minimal)

## 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: no Analytic conductor: $$0.111790562830$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{8}$$ Projective field: Galois closure of 8.0.5156108238848.3

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

 $$f(q)$$ $$=$$ $$q + \zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} - \zeta_{8} q^{7} + \zeta_{8}^{3} q^{8} + \zeta_{8}^{3} q^{9} +O(q^{10})$$ q + z * q^2 + z^2 * q^4 - z * q^7 + z^3 * q^8 + z^3 * q^9 $$q + \zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} - \zeta_{8} q^{7} + \zeta_{8}^{3} q^{8} + \zeta_{8}^{3} q^{9} + ( - \zeta_{8}^{3} - \zeta_{8}^{2}) q^{11} - \zeta_{8}^{2} q^{14} - q^{16} - q^{18} + ( - \zeta_{8}^{3} + 1) q^{22} + (\zeta_{8}^{2} - 1) q^{23} + \zeta_{8} q^{25} - \zeta_{8}^{3} q^{28} + ( - \zeta_{8}^{2} - \zeta_{8}) q^{29} - \zeta_{8} q^{32} - \zeta_{8} q^{36} + ( - \zeta_{8}^{3} + \zeta_{8}^{2}) q^{37} + (\zeta_{8} - 1) q^{43} + (\zeta_{8} + 1) q^{44} + (\zeta_{8}^{3} - \zeta_{8}) q^{46} + \zeta_{8}^{2} q^{49} + \zeta_{8}^{2} q^{50} + (\zeta_{8} - 1) q^{53} + q^{56} + ( - \zeta_{8}^{3} - \zeta_{8}^{2}) q^{58} + q^{63} - \zeta_{8}^{2} q^{64} + (\zeta_{8}^{3} + 1) q^{67} - \zeta_{8}^{2} q^{72} + (\zeta_{8}^{3} + 1) q^{74} + (\zeta_{8}^{3} - 1) q^{77} + (\zeta_{8}^{3} + \zeta_{8}) q^{79} - \zeta_{8}^{2} q^{81} + (\zeta_{8}^{2} - \zeta_{8}) q^{86} + (\zeta_{8}^{2} + \zeta_{8}) q^{88} + ( - \zeta_{8}^{2} - 1) q^{92} + \zeta_{8}^{3} q^{98} + (\zeta_{8}^{2} + \zeta_{8}) q^{99} +O(q^{100})$$ q + z * q^2 + z^2 * q^4 - z * q^7 + z^3 * q^8 + z^3 * q^9 + (-z^3 - z^2) * q^11 - z^2 * q^14 - q^16 - q^18 + (-z^3 + 1) * q^22 + (z^2 - 1) * q^23 + z * q^25 - z^3 * q^28 + (-z^2 - z) * q^29 - z * q^32 - z * q^36 + (-z^3 + z^2) * q^37 + (z - 1) * q^43 + (z + 1) * q^44 + (z^3 - z) * q^46 + z^2 * q^49 + z^2 * q^50 + (z - 1) * q^53 + q^56 + (-z^3 - z^2) * q^58 + q^63 - z^2 * q^64 + (z^3 + 1) * q^67 - z^2 * q^72 + (z^3 + 1) * q^74 + (z^3 - 1) * q^77 + (z^3 + z) * q^79 - z^2 * q^81 + (z^2 - z) * q^86 + (z^2 + z) * q^88 + (-z^2 - 1) * q^92 + z^3 * q^98 + (z^2 + z) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q - 4 q^{16} - 4 q^{18} + 4 q^{22} - 4 q^{23} - 4 q^{43} + 4 q^{44} - 4 q^{53} + 4 q^{56} + 4 q^{63} + 4 q^{67} + 4 q^{74} - 4 q^{77} - 4 q^{92}+O(q^{100})$$ 4 * q - 4 * q^16 - 4 * q^18 + 4 * q^22 - 4 * q^23 - 4 * q^43 + 4 * q^44 - 4 * q^53 + 4 * q^56 + 4 * q^63 + 4 * q^67 + 4 * q^74 - 4 * q^77 - 4 * q^92

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/224\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$129$$ $$197$$ $$\chi(n)$$ $$1$$ $$-1$$ $$\zeta_{8}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1
 0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i −0.707107 − 0.707107i
0.707107 0.707107i 0 1.00000i 0 0 −0.707107 + 0.707107i −0.707107 0.707107i −0.707107 0.707107i 0
69.1 0.707107 + 0.707107i 0 1.00000i 0 0 −0.707107 0.707107i −0.707107 + 0.707107i −0.707107 + 0.707107i 0
125.1 −0.707107 + 0.707107i 0 1.00000i 0 0 0.707107 0.707107i 0.707107 + 0.707107i 0.707107 + 0.707107i 0
181.1 −0.707107 0.707107i 0 1.00000i 0 0 0.707107 + 0.707107i 0.707107 0.707107i 0.707107 0.707107i 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 $$\Q(\sqrt{-7})$$
32.g even 8 1 inner
224.v odd 8 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.1.v.a 4
3.b odd 2 1 2016.1.dp.b 4
4.b odd 2 1 896.1.v.a 4
7.b odd 2 1 CM 224.1.v.a 4
7.c even 3 2 1568.1.bl.a 8
7.d odd 6 2 1568.1.bl.a 8
8.b even 2 1 1792.1.v.a 4
8.d odd 2 1 1792.1.v.b 4
16.e even 4 1 3584.1.v.b 4
16.e even 4 1 3584.1.v.d 4
16.f odd 4 1 3584.1.v.a 4
16.f odd 4 1 3584.1.v.c 4
21.c even 2 1 2016.1.dp.b 4
28.d even 2 1 896.1.v.a 4
32.g even 8 1 inner 224.1.v.a 4
32.g even 8 1 1792.1.v.a 4
32.g even 8 1 3584.1.v.b 4
32.g even 8 1 3584.1.v.d 4
32.h odd 8 1 896.1.v.a 4
32.h odd 8 1 1792.1.v.b 4
32.h odd 8 1 3584.1.v.a 4
32.h odd 8 1 3584.1.v.c 4
56.e even 2 1 1792.1.v.b 4
56.h odd 2 1 1792.1.v.a 4
96.p odd 8 1 2016.1.dp.b 4
112.j even 4 1 3584.1.v.a 4
112.j even 4 1 3584.1.v.c 4
112.l odd 4 1 3584.1.v.b 4
112.l odd 4 1 3584.1.v.d 4
224.v odd 8 1 inner 224.1.v.a 4
224.v odd 8 1 1792.1.v.a 4
224.v odd 8 1 3584.1.v.b 4
224.v odd 8 1 3584.1.v.d 4
224.x even 8 1 896.1.v.a 4
224.x even 8 1 1792.1.v.b 4
224.x even 8 1 3584.1.v.a 4
224.x even 8 1 3584.1.v.c 4
224.bc odd 24 2 1568.1.bl.a 8
224.bd even 24 2 1568.1.bl.a 8
672.bo even 8 1 2016.1.dp.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.1.v.a 4 1.a even 1 1 trivial
224.1.v.a 4 7.b odd 2 1 CM
224.1.v.a 4 32.g even 8 1 inner
224.1.v.a 4 224.v odd 8 1 inner
896.1.v.a 4 4.b odd 2 1
896.1.v.a 4 28.d even 2 1
896.1.v.a 4 32.h odd 8 1
896.1.v.a 4 224.x even 8 1
1568.1.bl.a 8 7.c even 3 2
1568.1.bl.a 8 7.d odd 6 2
1568.1.bl.a 8 224.bc odd 24 2
1568.1.bl.a 8 224.bd even 24 2
1792.1.v.a 4 8.b even 2 1
1792.1.v.a 4 32.g even 8 1
1792.1.v.a 4 56.h odd 2 1
1792.1.v.a 4 224.v odd 8 1
1792.1.v.b 4 8.d odd 2 1
1792.1.v.b 4 32.h odd 8 1
1792.1.v.b 4 56.e even 2 1
1792.1.v.b 4 224.x even 8 1
2016.1.dp.b 4 3.b odd 2 1
2016.1.dp.b 4 21.c even 2 1
2016.1.dp.b 4 96.p odd 8 1
2016.1.dp.b 4 672.bo even 8 1
3584.1.v.a 4 16.f odd 4 1
3584.1.v.a 4 32.h odd 8 1
3584.1.v.a 4 112.j even 4 1
3584.1.v.a 4 224.x even 8 1
3584.1.v.b 4 16.e even 4 1
3584.1.v.b 4 32.g even 8 1
3584.1.v.b 4 112.l odd 4 1
3584.1.v.b 4 224.v odd 8 1
3584.1.v.c 4 16.f odd 4 1
3584.1.v.c 4 32.h odd 8 1
3584.1.v.c 4 112.j even 4 1
3584.1.v.c 4 224.x even 8 1
3584.1.v.d 4 16.e even 4 1
3584.1.v.d 4 32.g even 8 1
3584.1.v.d 4 112.l odd 4 1
3584.1.v.d 4 224.v odd 8 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(224, [\chi])$$.

## Hecke characteristic polynomials

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