# Properties

 Label 224.2.b.a Level $224$ Weight $2$ Character orbit 224.b Analytic conductor $1.789$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [224,2,Mod(113,224)]

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, 1, 0]))

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

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

chi := DirichletCharacter("224.113");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$224 = 2^{5} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 224.b (of order $$2$$, degree $$1$$, not 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: $$1.78864900528$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 2$$ x^2 + 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 56) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + \beta q^{5} - q^{7} + q^{9}+O(q^{10})$$ q + b * q^3 + b * q^5 - q^7 + q^9 $$q + \beta q^{3} + \beta q^{5} - q^{7} + q^{9} + 2 \beta q^{11} + 3 \beta q^{13} - 2 q^{15} - 6 q^{17} - 3 \beta q^{19} - \beta q^{21} + 6 q^{23} + 3 q^{25} + 4 \beta q^{27} - 2 \beta q^{29} + 4 q^{31} - 4 q^{33} - \beta q^{35} - 6 \beta q^{37} - 6 q^{39} + 6 q^{41} - 6 \beta q^{43} + \beta q^{45} + q^{49} - 6 \beta q^{51} + 4 \beta q^{53} - 4 q^{55} + 6 q^{57} - \beta q^{59} - 9 \beta q^{61} - q^{63} - 6 q^{65} + 6 \beta q^{69} + 2 q^{73} + 3 \beta q^{75} - 2 \beta q^{77} - 8 q^{79} - 5 q^{81} + 11 \beta q^{83} - 6 \beta q^{85} + 4 q^{87} + 6 q^{89} - 3 \beta q^{91} + 4 \beta q^{93} + 6 q^{95} - 10 q^{97} + 2 \beta q^{99} +O(q^{100})$$ q + b * q^3 + b * q^5 - q^7 + q^9 + 2*b * q^11 + 3*b * q^13 - 2 * q^15 - 6 * q^17 - 3*b * q^19 - b * q^21 + 6 * q^23 + 3 * q^25 + 4*b * q^27 - 2*b * q^29 + 4 * q^31 - 4 * q^33 - b * q^35 - 6*b * q^37 - 6 * q^39 + 6 * q^41 - 6*b * q^43 + b * q^45 + q^49 - 6*b * q^51 + 4*b * q^53 - 4 * q^55 + 6 * q^57 - b * q^59 - 9*b * q^61 - q^63 - 6 * q^65 + 6*b * q^69 + 2 * q^73 + 3*b * q^75 - 2*b * q^77 - 8 * q^79 - 5 * q^81 + 11*b * q^83 - 6*b * q^85 + 4 * q^87 + 6 * q^89 - 3*b * q^91 + 4*b * q^93 + 6 * q^95 - 10 * q^97 + 2*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^7 + 2 * q^9 $$2 q - 2 q^{7} + 2 q^{9} - 4 q^{15} - 12 q^{17} + 12 q^{23} + 6 q^{25} + 8 q^{31} - 8 q^{33} - 12 q^{39} + 12 q^{41} + 2 q^{49} - 8 q^{55} + 12 q^{57} - 2 q^{63} - 12 q^{65} + 4 q^{73} - 16 q^{79} - 10 q^{81} + 8 q^{87} + 12 q^{89} + 12 q^{95} - 20 q^{97}+O(q^{100})$$ 2 * q - 2 * q^7 + 2 * q^9 - 4 * q^15 - 12 * q^17 + 12 * q^23 + 6 * q^25 + 8 * q^31 - 8 * q^33 - 12 * q^39 + 12 * q^41 + 2 * q^49 - 8 * q^55 + 12 * q^57 - 2 * q^63 - 12 * q^65 + 4 * q^73 - 16 * q^79 - 10 * q^81 + 8 * q^87 + 12 * q^89 + 12 * q^95 - 20 * q^97

## Character values

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

 $$n$$ $$127$$ $$129$$ $$197$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
113.1
 − 1.41421i 1.41421i
0 1.41421i 0 1.41421i 0 −1.00000 0 1.00000 0
113.2 0 1.41421i 0 1.41421i 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

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.2.b.a 2
3.b odd 2 1 2016.2.c.a 2
4.b odd 2 1 56.2.b.a 2
7.b odd 2 1 1568.2.b.a 2
7.c even 3 2 1568.2.t.c 4
7.d odd 6 2 1568.2.t.b 4
8.b even 2 1 inner 224.2.b.a 2
8.d odd 2 1 56.2.b.a 2
12.b even 2 1 504.2.c.a 2
16.e even 4 2 1792.2.a.p 2
16.f odd 4 2 1792.2.a.n 2
24.f even 2 1 504.2.c.a 2
24.h odd 2 1 2016.2.c.a 2
28.d even 2 1 392.2.b.b 2
28.f even 6 2 392.2.p.b 4
28.g odd 6 2 392.2.p.a 4
56.e even 2 1 392.2.b.b 2
56.h odd 2 1 1568.2.b.a 2
56.j odd 6 2 1568.2.t.b 4
56.k odd 6 2 392.2.p.a 4
56.m even 6 2 392.2.p.b 4
56.p even 6 2 1568.2.t.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.b.a 2 4.b odd 2 1
56.2.b.a 2 8.d odd 2 1
224.2.b.a 2 1.a even 1 1 trivial
224.2.b.a 2 8.b even 2 1 inner
392.2.b.b 2 28.d even 2 1
392.2.b.b 2 56.e even 2 1
392.2.p.a 4 28.g odd 6 2
392.2.p.a 4 56.k odd 6 2
392.2.p.b 4 28.f even 6 2
392.2.p.b 4 56.m even 6 2
504.2.c.a 2 12.b even 2 1
504.2.c.a 2 24.f even 2 1
1568.2.b.a 2 7.b odd 2 1
1568.2.b.a 2 56.h odd 2 1
1568.2.t.b 4 7.d odd 6 2
1568.2.t.b 4 56.j odd 6 2
1568.2.t.c 4 7.c even 3 2
1568.2.t.c 4 56.p even 6 2
1792.2.a.n 2 16.f odd 4 2
1792.2.a.p 2 16.e even 4 2
2016.2.c.a 2 3.b odd 2 1
2016.2.c.a 2 24.h odd 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 2$$
$5$ $$T^{2} + 2$$
$7$ $$(T + 1)^{2}$$
$11$ $$T^{2} + 8$$
$13$ $$T^{2} + 18$$
$17$ $$(T + 6)^{2}$$
$19$ $$T^{2} + 18$$
$23$ $$(T - 6)^{2}$$
$29$ $$T^{2} + 8$$
$31$ $$(T - 4)^{2}$$
$37$ $$T^{2} + 72$$
$41$ $$(T - 6)^{2}$$
$43$ $$T^{2} + 72$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 32$$
$59$ $$T^{2} + 2$$
$61$ $$T^{2} + 162$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$(T - 2)^{2}$$
$79$ $$(T + 8)^{2}$$
$83$ $$T^{2} + 242$$
$89$ $$(T - 6)^{2}$$
$97$ $$(T + 10)^{2}$$