# Properties

 Label 288.2.i.a Level $288$ Weight $2$ Character orbit 288.i Analytic conductor $2.300$ Analytic rank $1$ Dimension $2$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} - 1) q^{3} - 4 \zeta_{6} q^{5} + (2 \zeta_{6} - 2) q^{7} + 3 \zeta_{6} q^{9} +O(q^{10})$$ q + (-z - 1) * q^3 - 4*z * q^5 + (2*z - 2) * q^7 + 3*z * q^9 $$q + ( - \zeta_{6} - 1) q^{3} - 4 \zeta_{6} q^{5} + (2 \zeta_{6} - 2) q^{7} + 3 \zeta_{6} q^{9} + (5 \zeta_{6} - 5) q^{11} + 2 \zeta_{6} q^{13} + (8 \zeta_{6} - 4) q^{15} - 3 q^{17} - q^{19} + ( - 2 \zeta_{6} + 4) q^{21} - 6 \zeta_{6} q^{23} + (11 \zeta_{6} - 11) q^{25} + ( - 6 \zeta_{6} + 3) q^{27} + ( - 2 \zeta_{6} + 2) q^{29} - 4 \zeta_{6} q^{31} + ( - 5 \zeta_{6} + 10) q^{33} + 8 q^{35} - 8 q^{37} + ( - 4 \zeta_{6} + 2) q^{39} - \zeta_{6} q^{41} + (7 \zeta_{6} - 7) q^{43} + ( - 12 \zeta_{6} + 12) q^{45} + ( - 2 \zeta_{6} + 2) q^{47} + 3 \zeta_{6} q^{49} + (3 \zeta_{6} + 3) q^{51} - 4 q^{53} + 20 q^{55} + (\zeta_{6} + 1) q^{57} + 5 \zeta_{6} q^{59} - 6 q^{63} + ( - 8 \zeta_{6} + 8) q^{65} - 13 \zeta_{6} q^{67} + (12 \zeta_{6} - 6) q^{69} - 8 q^{71} + 3 q^{73} + ( - 11 \zeta_{6} + 22) q^{75} - 10 \zeta_{6} q^{77} + ( - 8 \zeta_{6} + 8) q^{79} + (9 \zeta_{6} - 9) q^{81} + ( - 12 \zeta_{6} + 12) q^{83} + 12 \zeta_{6} q^{85} + (2 \zeta_{6} - 4) q^{87} - 10 q^{89} - 4 q^{91} + (8 \zeta_{6} - 4) q^{93} + 4 \zeta_{6} q^{95} + ( - 11 \zeta_{6} + 11) q^{97} - 15 q^{99} +O(q^{100})$$ q + (-z - 1) * q^3 - 4*z * q^5 + (2*z - 2) * q^7 + 3*z * q^9 + (5*z - 5) * q^11 + 2*z * q^13 + (8*z - 4) * q^15 - 3 * q^17 - q^19 + (-2*z + 4) * q^21 - 6*z * q^23 + (11*z - 11) * q^25 + (-6*z + 3) * q^27 + (-2*z + 2) * q^29 - 4*z * q^31 + (-5*z + 10) * q^33 + 8 * q^35 - 8 * q^37 + (-4*z + 2) * q^39 - z * q^41 + (7*z - 7) * q^43 + (-12*z + 12) * q^45 + (-2*z + 2) * q^47 + 3*z * q^49 + (3*z + 3) * q^51 - 4 * q^53 + 20 * q^55 + (z + 1) * q^57 + 5*z * q^59 - 6 * q^63 + (-8*z + 8) * q^65 - 13*z * q^67 + (12*z - 6) * q^69 - 8 * q^71 + 3 * q^73 + (-11*z + 22) * q^75 - 10*z * q^77 + (-8*z + 8) * q^79 + (9*z - 9) * q^81 + (-12*z + 12) * q^83 + 12*z * q^85 + (2*z - 4) * q^87 - 10 * q^89 - 4 * q^91 + (8*z - 4) * q^93 + 4*z * q^95 + (-11*z + 11) * q^97 - 15 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{3} - 4 q^{5} - 2 q^{7} + 3 q^{9}+O(q^{10})$$ 2 * q - 3 * q^3 - 4 * q^5 - 2 * q^7 + 3 * q^9 $$2 q - 3 q^{3} - 4 q^{5} - 2 q^{7} + 3 q^{9} - 5 q^{11} + 2 q^{13} - 6 q^{17} - 2 q^{19} + 6 q^{21} - 6 q^{23} - 11 q^{25} + 2 q^{29} - 4 q^{31} + 15 q^{33} + 16 q^{35} - 16 q^{37} - q^{41} - 7 q^{43} + 12 q^{45} + 2 q^{47} + 3 q^{49} + 9 q^{51} - 8 q^{53} + 40 q^{55} + 3 q^{57} + 5 q^{59} - 12 q^{63} + 8 q^{65} - 13 q^{67} - 16 q^{71} + 6 q^{73} + 33 q^{75} - 10 q^{77} + 8 q^{79} - 9 q^{81} + 12 q^{83} + 12 q^{85} - 6 q^{87} - 20 q^{89} - 8 q^{91} + 4 q^{95} + 11 q^{97} - 30 q^{99}+O(q^{100})$$ 2 * q - 3 * q^3 - 4 * q^5 - 2 * q^7 + 3 * q^9 - 5 * q^11 + 2 * q^13 - 6 * q^17 - 2 * q^19 + 6 * q^21 - 6 * q^23 - 11 * q^25 + 2 * q^29 - 4 * q^31 + 15 * q^33 + 16 * q^35 - 16 * q^37 - q^41 - 7 * q^43 + 12 * q^45 + 2 * q^47 + 3 * q^49 + 9 * q^51 - 8 * q^53 + 40 * q^55 + 3 * q^57 + 5 * q^59 - 12 * q^63 + 8 * q^65 - 13 * q^67 - 16 * q^71 + 6 * q^73 + 33 * q^75 - 10 * q^77 + 8 * q^79 - 9 * q^81 + 12 * q^83 + 12 * q^85 - 6 * q^87 - 20 * q^89 - 8 * q^91 + 4 * q^95 + 11 * q^97 - 30 * q^99

## Character values

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

 $$n$$ $$37$$ $$65$$ $$127$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$

## 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
 0.5 + 0.866025i 0.5 − 0.866025i
0 −1.50000 0.866025i 0 −2.00000 3.46410i 0 −1.00000 + 1.73205i 0 1.50000 + 2.59808i 0
193.1 0 −1.50000 + 0.866025i 0 −2.00000 + 3.46410i 0 −1.00000 1.73205i 0 1.50000 2.59808i 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.a 2
3.b odd 2 1 864.2.i.a 2
4.b odd 2 1 288.2.i.b yes 2
8.b even 2 1 576.2.i.h 2
8.d odd 2 1 576.2.i.b 2
9.c even 3 1 inner 288.2.i.a 2
9.c even 3 1 2592.2.a.h 1
9.d odd 6 1 864.2.i.a 2
9.d odd 6 1 2592.2.a.b 1
12.b even 2 1 864.2.i.b 2
24.f even 2 1 1728.2.i.b 2
24.h odd 2 1 1728.2.i.a 2
36.f odd 6 1 288.2.i.b yes 2
36.f odd 6 1 2592.2.a.g 1
36.h even 6 1 864.2.i.b 2
36.h even 6 1 2592.2.a.a 1
72.j odd 6 1 1728.2.i.a 2
72.j odd 6 1 5184.2.a.bf 1
72.l even 6 1 1728.2.i.b 2
72.l even 6 1 5184.2.a.be 1
72.n even 6 1 576.2.i.h 2
72.n even 6 1 5184.2.a.b 1
72.p odd 6 1 576.2.i.b 2
72.p odd 6 1 5184.2.a.a 1

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

## Hecke kernels

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

 $$T_{5}^{2} + 4T_{5} + 16$$ T5^2 + 4*T5 + 16 $$T_{7}^{2} + 2T_{7} + 4$$ T7^2 + 2*T7 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 3T + 3$$
$5$ $$T^{2} + 4T + 16$$
$7$ $$T^{2} + 2T + 4$$
$11$ $$T^{2} + 5T + 25$$
$13$ $$T^{2} - 2T + 4$$
$17$ $$(T + 3)^{2}$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2} + 6T + 36$$
$29$ $$T^{2} - 2T + 4$$
$31$ $$T^{2} + 4T + 16$$
$37$ $$(T + 8)^{2}$$
$41$ $$T^{2} + T + 1$$
$43$ $$T^{2} + 7T + 49$$
$47$ $$T^{2} - 2T + 4$$
$53$ $$(T + 4)^{2}$$
$59$ $$T^{2} - 5T + 25$$
$61$ $$T^{2}$$
$67$ $$T^{2} + 13T + 169$$
$71$ $$(T + 8)^{2}$$
$73$ $$(T - 3)^{2}$$
$79$ $$T^{2} - 8T + 64$$
$83$ $$T^{2} - 12T + 144$$
$89$ $$(T + 10)^{2}$$
$97$ $$T^{2} - 11T + 121$$