# Properties

 Label 288.2.i.c Level $288$ Weight $2$ Character orbit 288.i Analytic conductor $2.300$ 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] = [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: $$\Q(\sqrt{-2}, \sqrt{-3})$$ 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} - 1) q^{3} + \beta_{2} q^{5} + (2 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{7} + ( - 2 \beta_{3} - 1) q^{9}+O(q^{10})$$ q + (b3 - 1) * q^3 + b2 * q^5 + (2*b3 - b2 - b1 + 1) * q^7 + (-2*b3 - 1) * q^9 $$q + (\beta_{3} - 1) q^{3} + \beta_{2} q^{5} + (2 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{7} + ( - 2 \beta_{3} - 1) q^{9} + (2 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{11} + (2 \beta_{3} - \beta_{2} + 2 \beta_1) q^{13} + (\beta_{3} - \beta_{2} - \beta_1) q^{15} + (2 \beta_{3} - 4 \beta_1) q^{17} - 4 q^{19} + ( - 2 \beta_{3} - \beta_{2} + 2 \beta_1 - 3) q^{21} + (\beta_{3} - 3 \beta_{2} + \beta_1) q^{23} + ( - 4 \beta_{2} + 4) q^{25} + (\beta_{3} + 5) q^{27} + (4 \beta_{3} + 5 \beta_{2} - 2 \beta_1 - 5) q^{29} + (\beta_{3} + 5 \beta_{2} + \beta_1) q^{31} + ( - 2 \beta_{3} - 3 \beta_{2} - 1) q^{33} + (\beta_{3} - 2 \beta_1 + 1) q^{35} + ( - 2 \beta_{3} + 4 \beta_1 + 4) q^{37} + ( - 3 \beta_{3} + 5 \beta_{2} - \beta_1 - 8) q^{39} + ( - 2 \beta_{3} + 7 \beta_{2} - 2 \beta_1) q^{41} + ( - 6 \beta_{3} - 5 \beta_{2} + 3 \beta_1 + 5) q^{43} + ( - 2 \beta_{3} - \beta_{2} + 2 \beta_1) q^{45} + ( - 6 \beta_{3} - \beta_{2} + 3 \beta_1 + 1) q^{47} + (2 \beta_{3} + 2 \beta_1) q^{49} + ( - 2 \beta_{3} - 8 \beta_{2} + 4 \beta_1 + 4) q^{51} + (2 \beta_{3} - 4 \beta_1 + 4) q^{53} + (\beta_{3} - 2 \beta_1 - 1) q^{55} + ( - 4 \beta_{3} + 4) q^{57} + ( - 3 \beta_{3} + 7 \beta_{2} - 3 \beta_1) q^{59} + (4 \beta_{3} - 3 \beta_{2} - 2 \beta_1 + 3) q^{61} + ( - 2 \beta_{3} + 5 \beta_{2} - \beta_1 + 3) q^{63} + (4 \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{65} + ( - 3 \beta_{3} + 5 \beta_{2} - 3 \beta_1) q^{67} + ( - 4 \beta_{3} + 5 \beta_{2} + 2 \beta_1 - 4) q^{69} + ( - 4 \beta_{3} + 8 \beta_1 + 2) q^{71} + ( - 2 \beta_{3} + 4 \beta_1) q^{73} + (4 \beta_{2} + 4 \beta_1 - 4) q^{75} - 5 \beta_{2} q^{77} + (2 \beta_{3} - 11 \beta_{2} - \beta_1 + 11) q^{79} + (4 \beta_{3} - 7) q^{81} + (2 \beta_{3} + 3 \beta_{2} - \beta_1 - 3) q^{83} + ( - 2 \beta_{3} - 2 \beta_1) q^{85} + ( - 4 \beta_{3} - 9 \beta_{2} - 3 \beta_1 + 1) q^{87} + ( - 2 \beta_{3} + 4 \beta_1 - 8) q^{89} + ( - 3 \beta_{3} + 6 \beta_1 - 13) q^{91} + (4 \beta_{3} - 3 \beta_{2} - 6 \beta_1 - 4) q^{93} - 4 \beta_{2} q^{95} + (4 \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{97} + ( - 2 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 5) q^{99}+O(q^{100})$$ q + (b3 - 1) * q^3 + b2 * q^5 + (2*b3 - b2 - b1 + 1) * q^7 + (-2*b3 - 1) * q^9 + (2*b3 + b2 - b1 - 1) * q^11 + (2*b3 - b2 + 2*b1) * q^13 + (b3 - b2 - b1) * q^15 + (2*b3 - 4*b1) * q^17 - 4 * q^19 + (-2*b3 - b2 + 2*b1 - 3) * q^21 + (b3 - 3*b2 + b1) * q^23 + (-4*b2 + 4) * q^25 + (b3 + 5) * q^27 + (4*b3 + 5*b2 - 2*b1 - 5) * q^29 + (b3 + 5*b2 + b1) * q^31 + (-2*b3 - 3*b2 - 1) * q^33 + (b3 - 2*b1 + 1) * q^35 + (-2*b3 + 4*b1 + 4) * q^37 + (-3*b3 + 5*b2 - b1 - 8) * q^39 + (-2*b3 + 7*b2 - 2*b1) * q^41 + (-6*b3 - 5*b2 + 3*b1 + 5) * q^43 + (-2*b3 - b2 + 2*b1) * q^45 + (-6*b3 - b2 + 3*b1 + 1) * q^47 + (2*b3 + 2*b1) * q^49 + (-2*b3 - 8*b2 + 4*b1 + 4) * q^51 + (2*b3 - 4*b1 + 4) * q^53 + (b3 - 2*b1 - 1) * q^55 + (-4*b3 + 4) * q^57 + (-3*b3 + 7*b2 - 3*b1) * q^59 + (4*b3 - 3*b2 - 2*b1 + 3) * q^61 + (-2*b3 + 5*b2 - b1 + 3) * q^63 + (4*b3 - b2 - 2*b1 + 1) * q^65 + (-3*b3 + 5*b2 - 3*b1) * q^67 + (-4*b3 + 5*b2 + 2*b1 - 4) * q^69 + (-4*b3 + 8*b1 + 2) * q^71 + (-2*b3 + 4*b1) * q^73 + (4*b2 + 4*b1 - 4) * q^75 - 5*b2 * q^77 + (2*b3 - 11*b2 - b1 + 11) * q^79 + (4*b3 - 7) * q^81 + (2*b3 + 3*b2 - b1 - 3) * q^83 + (-2*b3 - 2*b1) * q^85 + (-4*b3 - 9*b2 - 3*b1 + 1) * q^87 + (-2*b3 + 4*b1 - 8) * q^89 + (-3*b3 + 6*b1 - 13) * q^91 + (4*b3 - 3*b2 - 6*b1 - 4) * q^93 - 4*b2 * q^95 + (4*b3 + b2 - 2*b1 - 1) * q^97 + (-2*b3 + 3*b2 + 3*b1 + 5) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{3} + 2 q^{5} + 2 q^{7} - 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^3 + 2 * q^5 + 2 * q^7 - 4 * q^9 $$4 q - 4 q^{3} + 2 q^{5} + 2 q^{7} - 4 q^{9} - 2 q^{11} - 2 q^{13} - 2 q^{15} - 16 q^{19} - 14 q^{21} - 6 q^{23} + 8 q^{25} + 20 q^{27} - 10 q^{29} + 10 q^{31} - 10 q^{33} + 4 q^{35} + 16 q^{37} - 22 q^{39} + 14 q^{41} + 10 q^{43} - 2 q^{45} + 2 q^{47} + 16 q^{53} - 4 q^{55} + 16 q^{57} + 14 q^{59} + 6 q^{61} + 22 q^{63} + 2 q^{65} + 10 q^{67} - 6 q^{69} + 8 q^{71} - 8 q^{75} - 10 q^{77} + 22 q^{79} - 28 q^{81} - 6 q^{83} - 14 q^{87} - 32 q^{89} - 52 q^{91} - 22 q^{93} - 8 q^{95} - 2 q^{97} + 26 q^{99}+O(q^{100})$$ 4 * q - 4 * q^3 + 2 * q^5 + 2 * q^7 - 4 * q^9 - 2 * q^11 - 2 * q^13 - 2 * q^15 - 16 * q^19 - 14 * q^21 - 6 * q^23 + 8 * q^25 + 20 * q^27 - 10 * q^29 + 10 * q^31 - 10 * q^33 + 4 * q^35 + 16 * q^37 - 22 * q^39 + 14 * q^41 + 10 * q^43 - 2 * q^45 + 2 * q^47 + 16 * q^53 - 4 * q^55 + 16 * q^57 + 14 * q^59 + 6 * q^61 + 22 * q^63 + 2 * q^65 + 10 * q^67 - 6 * q^69 + 8 * q^71 - 8 * q^75 - 10 * q^77 + 22 * q^79 - 28 * q^81 - 6 * q^83 - 14 * q^87 - 32 * q^89 - 52 * q^91 - 22 * q^93 - 8 * q^95 - 2 * q^97 + 26 * 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 $$\left(\mathbb{Z}/288\mathbb{Z}\right)^\times$$.

 $$n$$ $$37$$ $$65$$ $$127$$ $$\chi(n)$$ $$1$$ $$-\beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
97.1
 −1.22474 − 0.707107i 1.22474 + 0.707107i 1.22474 − 0.707107i −1.22474 + 0.707107i
0 −1.00000 1.41421i 0 0.500000 + 0.866025i 0 1.72474 2.98735i 0 −1.00000 + 2.82843i 0
97.2 0 −1.00000 + 1.41421i 0 0.500000 + 0.866025i 0 −0.724745 + 1.25529i 0 −1.00000 2.82843i 0
193.1 0 −1.00000 1.41421i 0 0.500000 0.866025i 0 −0.724745 1.25529i 0 −1.00000 + 2.82843i 0
193.2 0 −1.00000 + 1.41421i 0 0.500000 0.866025i 0 1.72474 + 2.98735i 0 −1.00000 2.82843i 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
9.c even 3 1 inner

## Twists

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.i.c 4 1.a even 1 1 trivial
288.2.i.c 4 9.c even 3 1 inner
288.2.i.e yes 4 4.b odd 2 1
288.2.i.e yes 4 36.f odd 6 1
576.2.i.i 4 8.d odd 2 1
576.2.i.i 4 72.p odd 6 1
576.2.i.m 4 8.b even 2 1
576.2.i.m 4 72.n even 6 1
864.2.i.c 4 12.b even 2 1
864.2.i.c 4 36.h even 6 1
864.2.i.e 4 3.b odd 2 1
864.2.i.e 4 9.d odd 6 1
1728.2.i.k 4 24.f even 2 1
1728.2.i.k 4 72.l even 6 1
1728.2.i.m 4 24.h odd 2 1
1728.2.i.m 4 72.j odd 6 1
2592.2.a.j 2 9.c even 3 1
2592.2.a.n 2 36.f odd 6 1
2592.2.a.o 2 9.d odd 6 1
2592.2.a.s 2 36.h even 6 1
5184.2.a.bj 2 72.j odd 6 1
5184.2.a.bn 2 72.l even 6 1
5184.2.a.bu 2 72.n even 6 1
5184.2.a.by 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 $$T_{7}^{4} - 2T_{7}^{3} + 9T_{7}^{2} + 10T_{7} + 25$$ T7^4 - 2*T7^3 + 9*T7^2 + 10*T7 + 25

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + 2 T + 3)^{2}$$
$5$ $$(T^{2} - T + 1)^{2}$$
$7$ $$T^{4} - 2 T^{3} + 9 T^{2} + 10 T + 25$$
$11$ $$T^{4} + 2 T^{3} + 9 T^{2} - 10 T + 25$$
$13$ $$T^{4} + 2 T^{3} + 27 T^{2} - 46 T + 529$$
$17$ $$(T^{2} - 24)^{2}$$
$19$ $$(T + 4)^{4}$$
$23$ $$T^{4} + 6 T^{3} + 33 T^{2} + 18 T + 9$$
$29$ $$T^{4} + 10 T^{3} + 99 T^{2} + 10 T + 1$$
$31$ $$T^{4} - 10 T^{3} + 81 T^{2} + \cdots + 361$$
$37$ $$(T^{2} - 8 T - 8)^{2}$$
$41$ $$T^{4} - 14 T^{3} + 171 T^{2} + \cdots + 625$$
$43$ $$T^{4} - 10 T^{3} + 129 T^{2} + \cdots + 841$$
$47$ $$T^{4} - 2 T^{3} + 57 T^{2} + \cdots + 2809$$
$53$ $$(T^{2} - 8 T - 8)^{2}$$
$59$ $$T^{4} - 14 T^{3} + 201 T^{2} + \cdots + 25$$
$61$ $$T^{4} - 6 T^{3} + 51 T^{2} + 90 T + 225$$
$67$ $$T^{4} - 10 T^{3} + 129 T^{2} + \cdots + 841$$
$71$ $$(T^{2} - 4 T - 92)^{2}$$
$73$ $$(T^{2} - 24)^{2}$$
$79$ $$T^{4} - 22 T^{3} + 369 T^{2} + \cdots + 13225$$
$83$ $$T^{4} + 6 T^{3} + 33 T^{2} + 18 T + 9$$
$89$ $$(T^{2} + 16 T + 40)^{2}$$
$97$ $$T^{4} + 2 T^{3} + 27 T^{2} - 46 T + 529$$