# Properties

 Label 224.2.i.a Level $224$ Weight $2$ Character orbit 224.i Analytic conductor $1.789$ Analytic rank $0$ Dimension $4$ 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(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: $$\Q(\sqrt{2}, \sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 2x^{2} + 4$$ x^4 + 2*x^2 + 4 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} + \beta_1 - 1) q^{3} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{5} + ( - \beta_{3} - 2 \beta_1 - 1) q^{7} + ( - 2 \beta_{3} - 2 \beta_1) q^{9}+O(q^{10})$$ q + (-b2 + b1 - 1) * q^3 + (-2*b3 + b2 - 2*b1) * q^5 + (-b3 - 2*b1 - 1) * q^7 + (-2*b3 - 2*b1) * q^9 $$q + ( - \beta_{2} + \beta_1 - 1) q^{3} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{5} + ( - \beta_{3} - 2 \beta_1 - 1) q^{7} + ( - 2 \beta_{3} - 2 \beta_1) q^{9} + (\beta_{2} + \beta_1 + 1) q^{11} - 2 \beta_{3} q^{13} + (3 \beta_{3} + 5) q^{15} + (3 \beta_{2} - 2 \beta_1 + 3) q^{17} + (\beta_{3} + 5 \beta_{2} + \beta_1) q^{19} + (2 \beta_{3} - \beta_{2} + 3) q^{21} + ( - 3 \beta_{3} - \beta_{2} - 3 \beta_1) q^{23} + ( - 4 \beta_{2} + 4 \beta_1 - 4) q^{25} + ( - \beta_{3} + 1) q^{27} + 2 \beta_{3} q^{29} + ( - 7 \beta_{2} + \beta_1 - 7) q^{31} + \beta_{2} q^{33} + (\beta_{3} - 5 \beta_{2} + 3 \beta_1 - 8) q^{35} + (4 \beta_{3} + 3 \beta_{2} + 4 \beta_1) q^{37} + (4 \beta_{2} - 2 \beta_1 + 4) q^{39} + (2 \beta_{3} - 4) q^{41} + ( - 4 \beta_{3} + 4) q^{43} + ( - 8 \beta_{2} + 2 \beta_1 - 8) q^{45} + (\beta_{3} + 9 \beta_{2} + \beta_1) q^{47} + (2 \beta_{3} + 4 \beta_1 - 5) q^{49} + (5 \beta_{3} - 7 \beta_{2} + 5 \beta_1) q^{51} + (\beta_{2} + 1) q^{53} + ( - \beta_{3} + 3) q^{55} + (4 \beta_{3} + 3) q^{57} + ( - \beta_{2} - 7 \beta_1 - 1) q^{59} + (4 \beta_{3} + 3 \beta_{2} + 4 \beta_1) q^{61} + (2 \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 8) q^{63} + (2 \beta_{3} - 8 \beta_{2} + 2 \beta_1) q^{65} + (7 \beta_{2} - 3 \beta_1 + 7) q^{67} + (2 \beta_{3} + 5) q^{69} + ( - 4 \beta_{3} + 8) q^{71} + (9 \beta_{2} + 4 \beta_1 + 9) q^{73} + ( - 8 \beta_{3} + 12 \beta_{2} - 8 \beta_1) q^{75} + ( - 2 \beta_{3} - 3 \beta_{2} - 2 \beta_1 + 1) q^{77} + (5 \beta_{3} - \beta_{2} + 5 \beta_1) q^{79} + (\beta_{2} - 6 \beta_1 + 1) q^{81} + (8 \beta_{3} + 4) q^{83} + ( - 8 \beta_{3} - 11) q^{85} + ( - 4 \beta_{2} + 2 \beta_1 - 4) q^{87} - 9 \beta_{2} q^{89} + (2 \beta_{3} - 8 \beta_{2} - 4) q^{91} + ( - 8 \beta_{3} + 9 \beta_{2} - 8 \beta_1) q^{93} + ( - \beta_{2} + 9 \beta_1 - 1) q^{95} + ( - 2 \beta_{3} - 4) q^{97} + ( - 2 \beta_{3} + 4) q^{99}+O(q^{100})$$ q + (-b2 + b1 - 1) * q^3 + (-2*b3 + b2 - 2*b1) * q^5 + (-b3 - 2*b1 - 1) * q^7 + (-2*b3 - 2*b1) * q^9 + (b2 + b1 + 1) * q^11 - 2*b3 * q^13 + (3*b3 + 5) * q^15 + (3*b2 - 2*b1 + 3) * q^17 + (b3 + 5*b2 + b1) * q^19 + (2*b3 - b2 + 3) * q^21 + (-3*b3 - b2 - 3*b1) * q^23 + (-4*b2 + 4*b1 - 4) * q^25 + (-b3 + 1) * q^27 + 2*b3 * q^29 + (-7*b2 + b1 - 7) * q^31 + b2 * q^33 + (b3 - 5*b2 + 3*b1 - 8) * q^35 + (4*b3 + 3*b2 + 4*b1) * q^37 + (4*b2 - 2*b1 + 4) * q^39 + (2*b3 - 4) * q^41 + (-4*b3 + 4) * q^43 + (-8*b2 + 2*b1 - 8) * q^45 + (b3 + 9*b2 + b1) * q^47 + (2*b3 + 4*b1 - 5) * q^49 + (5*b3 - 7*b2 + 5*b1) * q^51 + (b2 + 1) * q^53 + (-b3 + 3) * q^55 + (4*b3 + 3) * q^57 + (-b2 - 7*b1 - 1) * q^59 + (4*b3 + 3*b2 + 4*b1) * q^61 + (2*b3 - 4*b2 + 2*b1 - 8) * q^63 + (2*b3 - 8*b2 + 2*b1) * q^65 + (7*b2 - 3*b1 + 7) * q^67 + (2*b3 + 5) * q^69 + (-4*b3 + 8) * q^71 + (9*b2 + 4*b1 + 9) * q^73 + (-8*b3 + 12*b2 - 8*b1) * q^75 + (-2*b3 - 3*b2 - 2*b1 + 1) * q^77 + (5*b3 - b2 + 5*b1) * q^79 + (b2 - 6*b1 + 1) * q^81 + (8*b3 + 4) * q^83 + (-8*b3 - 11) * q^85 + (-4*b2 + 2*b1 - 4) * q^87 - 9*b2 * q^89 + (2*b3 - 8*b2 - 4) * q^91 + (-8*b3 + 9*b2 - 8*b1) * q^93 + (-b2 + 9*b1 - 1) * q^95 + (-2*b3 - 4) * q^97 + (-2*b3 + 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{3} - 2 q^{5} - 4 q^{7}+O(q^{10})$$ 4 * q - 2 * q^3 - 2 * q^5 - 4 * q^7 $$4 q - 2 q^{3} - 2 q^{5} - 4 q^{7} + 2 q^{11} + 20 q^{15} + 6 q^{17} - 10 q^{19} + 14 q^{21} + 2 q^{23} - 8 q^{25} + 4 q^{27} - 14 q^{31} - 2 q^{33} - 22 q^{35} - 6 q^{37} + 8 q^{39} - 16 q^{41} + 16 q^{43} - 16 q^{45} - 18 q^{47} - 20 q^{49} + 14 q^{51} + 2 q^{53} + 12 q^{55} + 12 q^{57} - 2 q^{59} - 6 q^{61} - 24 q^{63} + 16 q^{65} + 14 q^{67} + 20 q^{69} + 32 q^{71} + 18 q^{73} - 24 q^{75} + 10 q^{77} + 2 q^{79} + 2 q^{81} + 16 q^{83} - 44 q^{85} - 8 q^{87} + 18 q^{89} - 18 q^{93} - 2 q^{95} - 16 q^{97} + 16 q^{99}+O(q^{100})$$ 4 * q - 2 * q^3 - 2 * q^5 - 4 * q^7 + 2 * q^11 + 20 * q^15 + 6 * q^17 - 10 * q^19 + 14 * q^21 + 2 * q^23 - 8 * q^25 + 4 * q^27 - 14 * q^31 - 2 * q^33 - 22 * q^35 - 6 * q^37 + 8 * q^39 - 16 * q^41 + 16 * q^43 - 16 * q^45 - 18 * q^47 - 20 * q^49 + 14 * q^51 + 2 * q^53 + 12 * q^55 + 12 * q^57 - 2 * q^59 - 6 * q^61 - 24 * q^63 + 16 * q^65 + 14 * q^67 + 20 * q^69 + 32 * q^71 + 18 * q^73 - 24 * q^75 + 10 * q^77 + 2 * q^79 + 2 * q^81 + 16 * q^83 - 44 * q^85 - 8 * q^87 + 18 * q^89 - 18 * q^93 - 2 * q^95 - 16 * q^97 + 16 * q^99

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*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
 −0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i
0 −1.20711 2.09077i 0 −1.91421 + 3.31552i 0 −1.00000 + 2.44949i 0 −1.41421 + 2.44949i 0
65.2 0 0.207107 + 0.358719i 0 0.914214 1.58346i 0 −1.00000 2.44949i 0 1.41421 2.44949i 0
193.1 0 −1.20711 + 2.09077i 0 −1.91421 3.31552i 0 −1.00000 2.44949i 0 −1.41421 2.44949i 0
193.2 0 0.207107 0.358719i 0 0.914214 + 1.58346i 0 −1.00000 + 2.44949i 0 1.41421 + 2.44949i 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.2.i.a 4
3.b odd 2 1 2016.2.s.q 4
4.b odd 2 1 224.2.i.d yes 4
7.b odd 2 1 1568.2.i.x 4
7.c even 3 1 inner 224.2.i.a 4
7.c even 3 1 1568.2.a.w 2
7.d odd 6 1 1568.2.a.j 2
7.d odd 6 1 1568.2.i.x 4
8.b even 2 1 448.2.i.j 4
8.d odd 2 1 448.2.i.g 4
12.b even 2 1 2016.2.s.s 4
21.h odd 6 1 2016.2.s.q 4
28.d even 2 1 1568.2.i.o 4
28.f even 6 1 1568.2.a.u 2
28.f even 6 1 1568.2.i.o 4
28.g odd 6 1 224.2.i.d yes 4
28.g odd 6 1 1568.2.a.l 2
56.j odd 6 1 3136.2.a.bx 2
56.k odd 6 1 448.2.i.g 4
56.k odd 6 1 3136.2.a.bw 2
56.m even 6 1 3136.2.a.be 2
56.p even 6 1 448.2.i.j 4
56.p even 6 1 3136.2.a.bd 2
84.n even 6 1 2016.2.s.s 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.i.a 4 1.a even 1 1 trivial
224.2.i.a 4 7.c even 3 1 inner
224.2.i.d yes 4 4.b odd 2 1
224.2.i.d yes 4 28.g odd 6 1
448.2.i.g 4 8.d odd 2 1
448.2.i.g 4 56.k odd 6 1
448.2.i.j 4 8.b even 2 1
448.2.i.j 4 56.p even 6 1
1568.2.a.j 2 7.d odd 6 1
1568.2.a.l 2 28.g odd 6 1
1568.2.a.u 2 28.f even 6 1
1568.2.a.w 2 7.c even 3 1
1568.2.i.o 4 28.d even 2 1
1568.2.i.o 4 28.f even 6 1
1568.2.i.x 4 7.b odd 2 1
1568.2.i.x 4 7.d odd 6 1
2016.2.s.q 4 3.b odd 2 1
2016.2.s.q 4 21.h odd 6 1
2016.2.s.s 4 12.b even 2 1
2016.2.s.s 4 84.n even 6 1
3136.2.a.bd 2 56.p even 6 1
3136.2.a.be 2 56.m even 6 1
3136.2.a.bw 2 56.k odd 6 1
3136.2.a.bx 2 56.j odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 2T_{3}^{3} + 5T_{3}^{2} - 2T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(224, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 2 T^{3} + 5 T^{2} - 2 T + 1$$
$5$ $$T^{4} + 2 T^{3} + 11 T^{2} - 14 T + 49$$
$7$ $$(T^{2} + 2 T + 7)^{2}$$
$11$ $$T^{4} - 2 T^{3} + 5 T^{2} + 2 T + 1$$
$13$ $$(T^{2} - 8)^{2}$$
$17$ $$T^{4} - 6 T^{3} + 35 T^{2} - 6 T + 1$$
$19$ $$T^{4} + 10 T^{3} + 77 T^{2} + \cdots + 529$$
$23$ $$T^{4} - 2 T^{3} + 21 T^{2} + 34 T + 289$$
$29$ $$(T^{2} - 8)^{2}$$
$31$ $$T^{4} + 14 T^{3} + 149 T^{2} + \cdots + 2209$$
$37$ $$T^{4} + 6 T^{3} + 59 T^{2} - 138 T + 529$$
$41$ $$(T^{2} + 8 T + 8)^{2}$$
$43$ $$(T^{2} - 8 T - 16)^{2}$$
$47$ $$T^{4} + 18 T^{3} + 245 T^{2} + \cdots + 6241$$
$53$ $$(T^{2} - T + 1)^{2}$$
$59$ $$T^{4} + 2 T^{3} + 101 T^{2} + \cdots + 9409$$
$61$ $$T^{4} + 6 T^{3} + 59 T^{2} - 138 T + 529$$
$67$ $$T^{4} - 14 T^{3} + 165 T^{2} + \cdots + 961$$
$71$ $$(T^{2} - 16 T + 32)^{2}$$
$73$ $$T^{4} - 18 T^{3} + 275 T^{2} + \cdots + 2401$$
$79$ $$T^{4} - 2 T^{3} + 53 T^{2} + \cdots + 2401$$
$83$ $$(T^{2} - 8 T - 112)^{2}$$
$89$ $$(T^{2} - 9 T + 81)^{2}$$
$97$ $$(T^{2} + 8 T + 8)^{2}$$