# Properties

 Label 288.2.i.d Level $288$ Weight $2$ Character orbit 288.i Analytic conductor $2.300$ 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] = [288,2,Mod(97,288)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(288, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("288.97");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$288 = 2^{5} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 288.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: $$2.29969157821$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{3} + ( - \beta_1 + 1) q^{5} + \beta_{2} q^{7} + 3 q^{9}+O(q^{10})$$ q - b3 * q^3 + (-b1 + 1) * q^5 + b2 * q^7 + 3 * q^9 $$q - \beta_{3} q^{3} + ( - \beta_1 + 1) q^{5} + \beta_{2} q^{7} + 3 q^{9} - \beta_{2} q^{11} + ( - 3 \beta_1 + 3) q^{13} + ( - \beta_{3} + \beta_{2}) q^{15} + 4 q^{17} + 4 \beta_{3} q^{19} - 3 \beta_1 q^{21} + (5 \beta_{3} - 5 \beta_{2}) q^{23} + 4 \beta_1 q^{25} - 3 \beta_{3} q^{27} - \beta_1 q^{29} + ( - 3 \beta_{3} + 3 \beta_{2}) q^{31} + 3 \beta_1 q^{33} + \beta_{3} q^{35} - 8 q^{37} + ( - 3 \beta_{3} + 3 \beta_{2}) q^{39} + (5 \beta_1 - 5) q^{41} + 5 \beta_{2} q^{43} + ( - 3 \beta_1 + 3) q^{45} - 7 \beta_{2} q^{47} + ( - 4 \beta_1 + 4) q^{49} - 4 \beta_{3} q^{51} - 8 q^{53} - \beta_{3} q^{55} - 12 q^{57} + ( - \beta_{3} + \beta_{2}) q^{59} + 7 \beta_1 q^{61} + 3 \beta_{2} q^{63} - 3 \beta_1 q^{65} + (5 \beta_{3} - 5 \beta_{2}) q^{67} + (15 \beta_1 - 15) q^{69} + 2 \beta_{3} q^{71} - 12 q^{73} - 4 \beta_{2} q^{75} + ( - 3 \beta_1 + 3) q^{77} + 3 \beta_{2} q^{79} + 9 q^{81} + 5 \beta_{2} q^{83} + ( - 4 \beta_1 + 4) q^{85} + \beta_{2} q^{87} - 4 q^{89} + 3 \beta_{3} q^{91} + ( - 9 \beta_1 + 9) q^{93} + (4 \beta_{3} - 4 \beta_{2}) q^{95} + 3 \beta_1 q^{97} - 3 \beta_{2} q^{99}+O(q^{100})$$ q - b3 * q^3 + (-b1 + 1) * q^5 + b2 * q^7 + 3 * q^9 - b2 * q^11 + (-3*b1 + 3) * q^13 + (-b3 + b2) * q^15 + 4 * q^17 + 4*b3 * q^19 - 3*b1 * q^21 + (5*b3 - 5*b2) * q^23 + 4*b1 * q^25 - 3*b3 * q^27 - b1 * q^29 + (-3*b3 + 3*b2) * q^31 + 3*b1 * q^33 + b3 * q^35 - 8 * q^37 + (-3*b3 + 3*b2) * q^39 + (5*b1 - 5) * q^41 + 5*b2 * q^43 + (-3*b1 + 3) * q^45 - 7*b2 * q^47 + (-4*b1 + 4) * q^49 - 4*b3 * q^51 - 8 * q^53 - b3 * q^55 - 12 * q^57 + (-b3 + b2) * q^59 + 7*b1 * q^61 + 3*b2 * q^63 - 3*b1 * q^65 + (5*b3 - 5*b2) * q^67 + (15*b1 - 15) * q^69 + 2*b3 * q^71 - 12 * q^73 - 4*b2 * q^75 + (-3*b1 + 3) * q^77 + 3*b2 * q^79 + 9 * q^81 + 5*b2 * q^83 + (-4*b1 + 4) * q^85 + b2 * q^87 - 4 * q^89 + 3*b3 * q^91 + (-9*b1 + 9) * q^93 + (4*b3 - 4*b2) * q^95 + 3*b1 * q^97 - 3*b2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{5} + 12 q^{9}+O(q^{10})$$ 4 * q + 2 * q^5 + 12 * q^9 $$4 q + 2 q^{5} + 12 q^{9} + 6 q^{13} + 16 q^{17} - 6 q^{21} + 8 q^{25} - 2 q^{29} + 6 q^{33} - 32 q^{37} - 10 q^{41} + 6 q^{45} + 8 q^{49} - 32 q^{53} - 48 q^{57} + 14 q^{61} - 6 q^{65} - 30 q^{69} - 48 q^{73} + 6 q^{77} + 36 q^{81} + 8 q^{85} - 16 q^{89} + 18 q^{93} + 6 q^{97}+O(q^{100})$$ 4 * q + 2 * q^5 + 12 * q^9 + 6 * q^13 + 16 * q^17 - 6 * q^21 + 8 * q^25 - 2 * q^29 + 6 * q^33 - 32 * q^37 - 10 * q^41 + 6 * q^45 + 8 * q^49 - 32 * q^53 - 48 * q^57 + 14 * q^61 - 6 * q^65 - 30 * q^69 - 48 * q^73 + 6 * q^77 + 36 * q^81 + 8 * q^85 - 16 * q^89 + 18 * q^93 + 6 * q^97

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{12}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{12}^{3} + \zeta_{12}$$ v^3 + v $$\beta_{3}$$ $$=$$ $$-\zeta_{12}^{3} + 2\zeta_{12}$$ -v^3 + 2*v
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 3$$ (b3 + b2) / 3 $$\zeta_{12}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{12}^{3}$$ $$=$$ $$( -\beta_{3} + 2\beta_{2} ) / 3$$ (-b3 + 2*b2) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/288\mathbb{Z}\right)^\times$$.

 $$n$$ $$37$$ $$65$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{1}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
97.1
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
0 −1.73205 0 0.500000 + 0.866025i 0 0.866025 1.50000i 0 3.00000 0
97.2 0 1.73205 0 0.500000 + 0.866025i 0 −0.866025 + 1.50000i 0 3.00000 0
193.1 0 −1.73205 0 0.500000 0.866025i 0 0.866025 + 1.50000i 0 3.00000 0
193.2 0 1.73205 0 0.500000 0.866025i 0 −0.866025 1.50000i 0 3.00000 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
4.b odd 2 1 inner
9.c even 3 1 inner
36.f odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.2.i.d 4
3.b odd 2 1 864.2.i.d 4
4.b odd 2 1 inner 288.2.i.d 4
8.b even 2 1 576.2.i.k 4
8.d odd 2 1 576.2.i.k 4
9.c even 3 1 inner 288.2.i.d 4
9.c even 3 1 2592.2.a.l 2
9.d odd 6 1 864.2.i.d 4
9.d odd 6 1 2592.2.a.p 2
12.b even 2 1 864.2.i.d 4
24.f even 2 1 1728.2.i.l 4
24.h odd 2 1 1728.2.i.l 4
36.f odd 6 1 inner 288.2.i.d 4
36.f odd 6 1 2592.2.a.l 2
36.h even 6 1 864.2.i.d 4
36.h even 6 1 2592.2.a.p 2
72.j odd 6 1 1728.2.i.l 4
72.j odd 6 1 5184.2.a.bl 2
72.l even 6 1 1728.2.i.l 4
72.l even 6 1 5184.2.a.bl 2
72.n even 6 1 576.2.i.k 4
72.n even 6 1 5184.2.a.bx 2
72.p odd 6 1 576.2.i.k 4
72.p odd 6 1 5184.2.a.bx 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.i.d 4 1.a even 1 1 trivial
288.2.i.d 4 4.b odd 2 1 inner
288.2.i.d 4 9.c even 3 1 inner
288.2.i.d 4 36.f odd 6 1 inner
576.2.i.k 4 8.b even 2 1
576.2.i.k 4 8.d odd 2 1
576.2.i.k 4 72.n even 6 1
576.2.i.k 4 72.p odd 6 1
864.2.i.d 4 3.b odd 2 1
864.2.i.d 4 9.d odd 6 1
864.2.i.d 4 12.b even 2 1
864.2.i.d 4 36.h even 6 1
1728.2.i.l 4 24.f even 2 1
1728.2.i.l 4 24.h odd 2 1
1728.2.i.l 4 72.j odd 6 1
1728.2.i.l 4 72.l even 6 1
2592.2.a.l 2 9.c even 3 1
2592.2.a.l 2 36.f odd 6 1
2592.2.a.p 2 9.d odd 6 1
2592.2.a.p 2 36.h even 6 1
5184.2.a.bl 2 72.j odd 6 1
5184.2.a.bl 2 72.l even 6 1
5184.2.a.bx 2 72.n even 6 1
5184.2.a.bx 2 72.p odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(288, [\chi])$$:

 $$T_{5}^{2} - T_{5} + 1$$ T5^2 - T5 + 1 $$T_{7}^{4} + 3T_{7}^{2} + 9$$ T7^4 + 3*T7^2 + 9

## Hecke characteristic polynomials

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