# Properties

 Label 224.2.i.b Level $224$ Weight $2$ Character orbit 224.i Analytic conductor $1.789$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [224,2,Mod(65,224)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(224, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

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

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

chi := DirichletCharacter("224.65");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$224 = 2^{5} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 224.i (of order $$3$$, degree $$2$$, 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + (\zeta_{12}^{2} - 1) q^{5} + ( - 3 \zeta_{12}^{3} + 2 \zeta_{12}) q^{7}+O(q^{10})$$ q + (-z^3 - z) * q^3 + (z^2 - 1) * q^5 + (-3*z^3 + 2*z) * q^7 $$q + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + (\zeta_{12}^{2} - 1) q^{5} + ( - 3 \zeta_{12}^{3} + 2 \zeta_{12}) q^{7} + ( - 3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{11} + ( - \zeta_{12}^{3} + 2 \zeta_{12}) q^{15} - 5 \zeta_{12}^{2} q^{17} + (2 \zeta_{12}^{3} - \zeta_{12}) q^{19} + ( - \zeta_{12}^{2} - 4) q^{21} + ( - 2 \zeta_{12}^{3} + \zeta_{12}) q^{23} + 4 \zeta_{12}^{2} q^{25} + (3 \zeta_{12}^{3} - 6 \zeta_{12}) q^{27} + 8 q^{29} + (5 \zeta_{12}^{3} + 5 \zeta_{12}) q^{31} + (9 \zeta_{12}^{2} - 9) q^{33} + (2 \zeta_{12}^{3} + \zeta_{12}) q^{35} + ( - 5 \zeta_{12}^{2} + 5) q^{37} + 4 q^{41} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12}) q^{43} + (10 \zeta_{12}^{3} - 5 \zeta_{12}) q^{47} + ( - 8 \zeta_{12}^{2} + 3) q^{49} + (10 \zeta_{12}^{3} - 5 \zeta_{12}) q^{51} + \zeta_{12}^{2} q^{53} + ( - 3 \zeta_{12}^{3} + 6 \zeta_{12}) q^{55} + 3 q^{57} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{59} + (11 \zeta_{12}^{2} - 11) q^{61} + (7 \zeta_{12}^{3} + 7 \zeta_{12}) q^{67} - 3 q^{69} + (8 \zeta_{12}^{3} - 16 \zeta_{12}) q^{71} - 15 \zeta_{12}^{2} q^{73} + ( - 8 \zeta_{12}^{3} + 4 \zeta_{12}) q^{75} + ( - 3 \zeta_{12}^{2} - 12) q^{77} + ( - 2 \zeta_{12}^{3} + \zeta_{12}) q^{79} + 9 \zeta_{12}^{2} q^{81} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12}) q^{83} + 5 q^{85} + ( - 8 \zeta_{12}^{3} - 8 \zeta_{12}) q^{87} + (7 \zeta_{12}^{2} - 7) q^{89} + ( - 15 \zeta_{12}^{2} + 15) q^{93} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{95} + 12 q^{97} +O(q^{100})$$ q + (-z^3 - z) * q^3 + (z^2 - 1) * q^5 + (-3*z^3 + 2*z) * q^7 + (-3*z^3 - 3*z) * q^11 + (-z^3 + 2*z) * q^15 - 5*z^2 * q^17 + (2*z^3 - z) * q^19 + (-z^2 - 4) * q^21 + (-2*z^3 + z) * q^23 + 4*z^2 * q^25 + (3*z^3 - 6*z) * q^27 + 8 * q^29 + (5*z^3 + 5*z) * q^31 + (9*z^2 - 9) * q^33 + (2*z^3 + z) * q^35 + (-5*z^2 + 5) * q^37 + 4 * q^41 + (-4*z^3 + 8*z) * q^43 + (10*z^3 - 5*z) * q^47 + (-8*z^2 + 3) * q^49 + (10*z^3 - 5*z) * q^51 + z^2 * q^53 + (-3*z^3 + 6*z) * q^55 + 3 * q^57 + (-z^3 - z) * q^59 + (11*z^2 - 11) * q^61 + (7*z^3 + 7*z) * q^67 - 3 * q^69 + (8*z^3 - 16*z) * q^71 - 15*z^2 * q^73 + (-8*z^3 + 4*z) * q^75 + (-3*z^2 - 12) * q^77 + (-2*z^3 + z) * q^79 + 9*z^2 * q^81 + (-4*z^3 + 8*z) * q^83 + 5 * q^85 + (-8*z^3 - 8*z) * q^87 + (7*z^2 - 7) * q^89 + (-15*z^2 + 15) * q^93 + (-z^3 - z) * q^95 + 12 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{5}+O(q^{10})$$ 4 * q - 2 * q^5 $$4 q - 2 q^{5} - 10 q^{17} - 18 q^{21} + 8 q^{25} + 32 q^{29} - 18 q^{33} + 10 q^{37} + 16 q^{41} - 4 q^{49} + 2 q^{53} + 12 q^{57} - 22 q^{61} - 12 q^{69} - 30 q^{73} - 54 q^{77} + 18 q^{81} + 20 q^{85} - 14 q^{89} + 30 q^{93} + 48 q^{97}+O(q^{100})$$ 4 * q - 2 * q^5 - 10 * q^17 - 18 * q^21 + 8 * q^25 + 32 * q^29 - 18 * q^33 + 10 * q^37 + 16 * q^41 - 4 * q^49 + 2 * q^53 + 12 * q^57 - 22 * q^61 - 12 * q^69 - 30 * q^73 - 54 * q^77 + 18 * q^81 + 20 * q^85 - 14 * q^89 + 30 * q^93 + 48 * q^97

## Character values

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

 $$n$$ $$127$$ $$129$$ $$197$$ $$\chi(n)$$ $$1$$ $$-1 + \zeta_{12}^{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
65.1
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
0 −0.866025 1.50000i 0 −0.500000 + 0.866025i 0 1.73205 2.00000i 0 0 0
65.2 0 0.866025 + 1.50000i 0 −0.500000 + 0.866025i 0 −1.73205 + 2.00000i 0 0 0
193.1 0 −0.866025 + 1.50000i 0 −0.500000 0.866025i 0 1.73205 + 2.00000i 0 0 0
193.2 0 0.866025 1.50000i 0 −0.500000 0.866025i 0 −1.73205 2.00000i 0 0 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
4.b odd 2 1 inner
7.c even 3 1 inner
28.g odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.2.i.b 4
3.b odd 2 1 2016.2.s.r 4
4.b odd 2 1 inner 224.2.i.b 4
7.b odd 2 1 1568.2.i.u 4
7.c even 3 1 inner 224.2.i.b 4
7.c even 3 1 1568.2.a.s 2
7.d odd 6 1 1568.2.a.n 2
7.d odd 6 1 1568.2.i.u 4
8.b even 2 1 448.2.i.i 4
8.d odd 2 1 448.2.i.i 4
12.b even 2 1 2016.2.s.r 4
21.h odd 6 1 2016.2.s.r 4
28.d even 2 1 1568.2.i.u 4
28.f even 6 1 1568.2.a.n 2
28.f even 6 1 1568.2.i.u 4
28.g odd 6 1 inner 224.2.i.b 4
28.g odd 6 1 1568.2.a.s 2
56.j odd 6 1 3136.2.a.bu 2
56.k odd 6 1 448.2.i.i 4
56.k odd 6 1 3136.2.a.bh 2
56.m even 6 1 3136.2.a.bu 2
56.p even 6 1 448.2.i.i 4
56.p even 6 1 3136.2.a.bh 2
84.n even 6 1 2016.2.s.r 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.i.b 4 1.a even 1 1 trivial
224.2.i.b 4 4.b odd 2 1 inner
224.2.i.b 4 7.c even 3 1 inner
224.2.i.b 4 28.g odd 6 1 inner
448.2.i.i 4 8.b even 2 1
448.2.i.i 4 8.d odd 2 1
448.2.i.i 4 56.k odd 6 1
448.2.i.i 4 56.p even 6 1
1568.2.a.n 2 7.d odd 6 1
1568.2.a.n 2 28.f even 6 1
1568.2.a.s 2 7.c even 3 1
1568.2.a.s 2 28.g odd 6 1
1568.2.i.u 4 7.b odd 2 1
1568.2.i.u 4 7.d odd 6 1
1568.2.i.u 4 28.d even 2 1
1568.2.i.u 4 28.f even 6 1
2016.2.s.r 4 3.b odd 2 1
2016.2.s.r 4 12.b even 2 1
2016.2.s.r 4 21.h odd 6 1
2016.2.s.r 4 84.n even 6 1
3136.2.a.bh 2 56.k odd 6 1
3136.2.a.bh 2 56.p even 6 1
3136.2.a.bu 2 56.j odd 6 1
3136.2.a.bu 2 56.m even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 3T_{3}^{2} + 9$$ acting on $$S_{2}^{\mathrm{new}}(224, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 3T^{2} + 9$$
$5$ $$(T^{2} + T + 1)^{2}$$
$7$ $$T^{4} + 2T^{2} + 49$$
$11$ $$T^{4} + 27T^{2} + 729$$
$13$ $$T^{4}$$
$17$ $$(T^{2} + 5 T + 25)^{2}$$
$19$ $$T^{4} + 3T^{2} + 9$$
$23$ $$T^{4} + 3T^{2} + 9$$
$29$ $$(T - 8)^{4}$$
$31$ $$T^{4} + 75T^{2} + 5625$$
$37$ $$(T^{2} - 5 T + 25)^{2}$$
$41$ $$(T - 4)^{4}$$
$43$ $$(T^{2} - 48)^{2}$$
$47$ $$T^{4} + 75T^{2} + 5625$$
$53$ $$(T^{2} - T + 1)^{2}$$
$59$ $$T^{4} + 3T^{2} + 9$$
$61$ $$(T^{2} + 11 T + 121)^{2}$$
$67$ $$T^{4} + 147 T^{2} + 21609$$
$71$ $$(T^{2} - 192)^{2}$$
$73$ $$(T^{2} + 15 T + 225)^{2}$$
$79$ $$T^{4} + 3T^{2} + 9$$
$83$ $$(T^{2} - 48)^{2}$$
$89$ $$(T^{2} + 7 T + 49)^{2}$$
$97$ $$(T - 12)^{4}$$