# Properties

 Label 224.2.i.c 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: $$\Q(\sqrt{-3}, \sqrt{7})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 7x^{2} + 49$$ x^4 + 7*x^2 + 49 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} - 3 \beta_{2} q^{5} - \beta_{3} q^{7} + 4 \beta_{2} q^{9}+O(q^{10})$$ q + b1 * q^3 - 3*b2 * q^5 - b3 * q^7 + 4*b2 * q^9 $$q + \beta_1 q^{3} - 3 \beta_{2} q^{5} - \beta_{3} q^{7} + 4 \beta_{2} q^{9} - \beta_1 q^{11} - 4 q^{13} - 3 \beta_{3} q^{15} + ( - \beta_{2} - 1) q^{17} + (3 \beta_{3} + 3 \beta_1) q^{19} + (7 \beta_{2} + 7) q^{21} + (\beta_{3} + \beta_1) q^{23} + ( - 4 \beta_{2} - 4) q^{25} + \beta_{3} q^{27} - 4 q^{29} - \beta_1 q^{31} - 7 \beta_{2} q^{33} + ( - 3 \beta_{3} - 3 \beta_1) q^{35} - 5 \beta_{2} q^{37} - 4 \beta_1 q^{39} + 8 q^{41} + 4 \beta_{3} q^{43} + (12 \beta_{2} + 12) q^{45} + ( - \beta_{3} - \beta_1) q^{47} + 7 q^{49} + ( - \beta_{3} - \beta_1) q^{51} + ( - 7 \beta_{2} - 7) q^{53} + 3 \beta_{3} q^{55} - 21 q^{57} + \beta_1 q^{59} - 5 \beta_{2} q^{61} + (4 \beta_{3} + 4 \beta_1) q^{63} + 12 \beta_{2} q^{65} + \beta_1 q^{67} - 7 q^{69} + (9 \beta_{2} + 9) q^{73} + ( - 4 \beta_{3} - 4 \beta_1) q^{75} + ( - 7 \beta_{2} - 7) q^{77} + (\beta_{3} + \beta_1) q^{79} + (5 \beta_{2} + 5) q^{81} - 4 \beta_{3} q^{83} - 3 q^{85} - 4 \beta_1 q^{87} - 9 \beta_{2} q^{89} + 4 \beta_{3} q^{91} - 7 \beta_{2} q^{93} + 9 \beta_1 q^{95} - 8 q^{97} - 4 \beta_{3} q^{99}+O(q^{100})$$ q + b1 * q^3 - 3*b2 * q^5 - b3 * q^7 + 4*b2 * q^9 - b1 * q^11 - 4 * q^13 - 3*b3 * q^15 + (-b2 - 1) * q^17 + (3*b3 + 3*b1) * q^19 + (7*b2 + 7) * q^21 + (b3 + b1) * q^23 + (-4*b2 - 4) * q^25 + b3 * q^27 - 4 * q^29 - b1 * q^31 - 7*b2 * q^33 + (-3*b3 - 3*b1) * q^35 - 5*b2 * q^37 - 4*b1 * q^39 + 8 * q^41 + 4*b3 * q^43 + (12*b2 + 12) * q^45 + (-b3 - b1) * q^47 + 7 * q^49 + (-b3 - b1) * q^51 + (-7*b2 - 7) * q^53 + 3*b3 * q^55 - 21 * q^57 + b1 * q^59 - 5*b2 * q^61 + (4*b3 + 4*b1) * q^63 + 12*b2 * q^65 + b1 * q^67 - 7 * q^69 + (9*b2 + 9) * q^73 + (-4*b3 - 4*b1) * q^75 + (-7*b2 - 7) * q^77 + (b3 + b1) * q^79 + (5*b2 + 5) * q^81 - 4*b3 * q^83 - 3 * q^85 - 4*b1 * q^87 - 9*b2 * q^89 + 4*b3 * q^91 - 7*b2 * q^93 + 9*b1 * q^95 - 8 * q^97 - 4*b3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 6 q^{5} - 8 q^{9}+O(q^{10})$$ 4 * q + 6 * q^5 - 8 * q^9 $$4 q + 6 q^{5} - 8 q^{9} - 16 q^{13} - 2 q^{17} + 14 q^{21} - 8 q^{25} - 16 q^{29} + 14 q^{33} + 10 q^{37} + 32 q^{41} + 24 q^{45} + 28 q^{49} - 14 q^{53} - 84 q^{57} + 10 q^{61} - 24 q^{65} - 28 q^{69} + 18 q^{73} - 14 q^{77} + 10 q^{81} - 12 q^{85} + 18 q^{89} + 14 q^{93} - 32 q^{97}+O(q^{100})$$ 4 * q + 6 * q^5 - 8 * q^9 - 16 * q^13 - 2 * q^17 + 14 * q^21 - 8 * q^25 - 16 * q^29 + 14 * q^33 + 10 * q^37 + 32 * q^41 + 24 * q^45 + 28 * q^49 - 14 * q^53 - 84 * q^57 + 10 * q^61 - 24 * q^65 - 28 * q^69 + 18 * q^73 - 14 * q^77 + 10 * q^81 - 12 * q^85 + 18 * q^89 + 14 * q^93 - 32 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7x^{2} + 49$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 7$$ (v^2) / 7 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 7$$ (v^3) / 7
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$7\beta_{2}$$ 7*b2 $$\nu^{3}$$ $$=$$ $$7\beta_{3}$$ 7*b3

## Character values

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

 $$n$$ $$127$$ $$129$$ $$197$$ $$\chi(n)$$ $$1$$ $$\beta_{2}$$ $$1$$

## 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
 −1.32288 − 2.29129i 1.32288 + 2.29129i −1.32288 + 2.29129i 1.32288 − 2.29129i
0 −1.32288 2.29129i 0 1.50000 2.59808i 0 −2.64575 0 −2.00000 + 3.46410i 0
65.2 0 1.32288 + 2.29129i 0 1.50000 2.59808i 0 2.64575 0 −2.00000 + 3.46410i 0
193.1 0 −1.32288 + 2.29129i 0 1.50000 + 2.59808i 0 −2.64575 0 −2.00000 3.46410i 0
193.2 0 1.32288 2.29129i 0 1.50000 + 2.59808i 0 2.64575 0 −2.00000 3.46410i 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.c 4
3.b odd 2 1 2016.2.s.o 4
4.b odd 2 1 inner 224.2.i.c 4
7.b odd 2 1 1568.2.i.p 4
7.c even 3 1 inner 224.2.i.c 4
7.c even 3 1 1568.2.a.m 2
7.d odd 6 1 1568.2.a.t 2
7.d odd 6 1 1568.2.i.p 4
8.b even 2 1 448.2.i.h 4
8.d odd 2 1 448.2.i.h 4
12.b even 2 1 2016.2.s.o 4
21.h odd 6 1 2016.2.s.o 4
28.d even 2 1 1568.2.i.p 4
28.f even 6 1 1568.2.a.t 2
28.f even 6 1 1568.2.i.p 4
28.g odd 6 1 inner 224.2.i.c 4
28.g odd 6 1 1568.2.a.m 2
56.j odd 6 1 3136.2.a.bg 2
56.k odd 6 1 448.2.i.h 4
56.k odd 6 1 3136.2.a.bv 2
56.m even 6 1 3136.2.a.bg 2
56.p even 6 1 448.2.i.h 4
56.p even 6 1 3136.2.a.bv 2
84.n even 6 1 2016.2.s.o 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 7T_{3}^{2} + 49$$ acting on $$S_{2}^{\mathrm{new}}(224, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 7T^{2} + 49$$
$5$ $$(T^{2} - 3 T + 9)^{2}$$
$7$ $$(T^{2} - 7)^{2}$$
$11$ $$T^{4} + 7T^{2} + 49$$
$13$ $$(T + 4)^{4}$$
$17$ $$(T^{2} + T + 1)^{2}$$
$19$ $$T^{4} + 63T^{2} + 3969$$
$23$ $$T^{4} + 7T^{2} + 49$$
$29$ $$(T + 4)^{4}$$
$31$ $$T^{4} + 7T^{2} + 49$$
$37$ $$(T^{2} - 5 T + 25)^{2}$$
$41$ $$(T - 8)^{4}$$
$43$ $$(T^{2} - 112)^{2}$$
$47$ $$T^{4} + 7T^{2} + 49$$
$53$ $$(T^{2} + 7 T + 49)^{2}$$
$59$ $$T^{4} + 7T^{2} + 49$$
$61$ $$(T^{2} - 5 T + 25)^{2}$$
$67$ $$T^{4} + 7T^{2} + 49$$
$71$ $$T^{4}$$
$73$ $$(T^{2} - 9 T + 81)^{2}$$
$79$ $$T^{4} + 7T^{2} + 49$$
$83$ $$(T^{2} - 112)^{2}$$
$89$ $$(T^{2} - 9 T + 81)^{2}$$
$97$ $$(T + 8)^{4}$$