# Properties

 Label 720.3.bh.b Level $720$ Weight $3$ Character orbit 720.bh Analytic conductor $19.619$ Analytic rank $0$ 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] = [720,3,Mod(433,720)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(720, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 0, 3]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("720.433");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$720 = 2^{4} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 720.bh (of order $$4$$, degree $$2$$, not 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: $$19.6185790339$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 90) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (4 i - 3) q^{5} + ( - 8 i - 8) q^{7}+O(q^{10})$$ q + (4*i - 3) * q^5 + (-8*i - 8) * q^7 $$q + (4 i - 3) q^{5} + ( - 8 i - 8) q^{7} - 4 q^{11} + (3 i - 3) q^{13} + (19 i + 19) q^{17} + 8 i q^{19} + ( - 20 i + 20) q^{23} + ( - 24 i - 7) q^{25} - 38 i q^{29} + 44 q^{31} + ( - 8 i + 56) q^{35} + ( - 3 i - 3) q^{37} + 70 q^{41} + (36 i - 36) q^{43} + 79 i q^{49} + ( - 17 i + 17) q^{53} + ( - 16 i + 12) q^{55} + 92 i q^{59} + 72 q^{61} + ( - 21 i - 3) q^{65} + ( - 44 i - 44) q^{67} + 88 q^{71} + ( - 55 i + 55) q^{73} + (32 i + 32) q^{77} - 12 i q^{79} + ( - 24 i + 24) q^{83} + (19 i - 133) q^{85} + 26 i q^{89} + 48 q^{91} + ( - 24 i - 32) q^{95} + ( - 57 i - 57) q^{97} +O(q^{100})$$ q + (4*i - 3) * q^5 + (-8*i - 8) * q^7 - 4 * q^11 + (3*i - 3) * q^13 + (19*i + 19) * q^17 + 8*i * q^19 + (-20*i + 20) * q^23 + (-24*i - 7) * q^25 - 38*i * q^29 + 44 * q^31 + (-8*i + 56) * q^35 + (-3*i - 3) * q^37 + 70 * q^41 + (36*i - 36) * q^43 + 79*i * q^49 + (-17*i + 17) * q^53 + (-16*i + 12) * q^55 + 92*i * q^59 + 72 * q^61 + (-21*i - 3) * q^65 + (-44*i - 44) * q^67 + 88 * q^71 + (-55*i + 55) * q^73 + (32*i + 32) * q^77 - 12*i * q^79 + (-24*i + 24) * q^83 + (19*i - 133) * q^85 + 26*i * q^89 + 48 * q^91 + (-24*i - 32) * q^95 + (-57*i - 57) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{5} - 16 q^{7}+O(q^{10})$$ 2 * q - 6 * q^5 - 16 * q^7 $$2 q - 6 q^{5} - 16 q^{7} - 8 q^{11} - 6 q^{13} + 38 q^{17} + 40 q^{23} - 14 q^{25} + 88 q^{31} + 112 q^{35} - 6 q^{37} + 140 q^{41} - 72 q^{43} + 34 q^{53} + 24 q^{55} + 144 q^{61} - 6 q^{65} - 88 q^{67} + 176 q^{71} + 110 q^{73} + 64 q^{77} + 48 q^{83} - 266 q^{85} + 96 q^{91} - 64 q^{95} - 114 q^{97}+O(q^{100})$$ 2 * q - 6 * q^5 - 16 * q^7 - 8 * q^11 - 6 * q^13 + 38 * q^17 + 40 * q^23 - 14 * q^25 + 88 * q^31 + 112 * q^35 - 6 * q^37 + 140 * q^41 - 72 * q^43 + 34 * q^53 + 24 * q^55 + 144 * q^61 - 6 * q^65 - 88 * q^67 + 176 * q^71 + 110 * q^73 + 64 * q^77 + 48 * q^83 - 266 * q^85 + 96 * q^91 - 64 * q^95 - 114 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/720\mathbb{Z}\right)^\times$$.

 $$n$$ $$181$$ $$271$$ $$577$$ $$641$$ $$\chi(n)$$ $$1$$ $$1$$ $$i$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
433.1
 − 1.00000i 1.00000i
0 0 0 −3.00000 4.00000i 0 −8.00000 + 8.00000i 0 0 0
577.1 0 0 0 −3.00000 + 4.00000i 0 −8.00000 8.00000i 0 0 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
5.c odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.3.bh.b 2
3.b odd 2 1 720.3.bh.d 2
4.b odd 2 1 90.3.g.a 2
5.c odd 4 1 inner 720.3.bh.b 2
12.b even 2 1 90.3.g.c yes 2
15.e even 4 1 720.3.bh.d 2
20.d odd 2 1 450.3.g.d 2
20.e even 4 1 90.3.g.a 2
20.e even 4 1 450.3.g.d 2
60.h even 2 1 450.3.g.a 2
60.l odd 4 1 90.3.g.c yes 2
60.l odd 4 1 450.3.g.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.3.g.a 2 4.b odd 2 1
90.3.g.a 2 20.e even 4 1
90.3.g.c yes 2 12.b even 2 1
90.3.g.c yes 2 60.l odd 4 1
450.3.g.a 2 60.h even 2 1
450.3.g.a 2 60.l odd 4 1
450.3.g.d 2 20.d odd 2 1
450.3.g.d 2 20.e even 4 1
720.3.bh.b 2 1.a even 1 1 trivial
720.3.bh.b 2 5.c odd 4 1 inner
720.3.bh.d 2 3.b odd 2 1
720.3.bh.d 2 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(720, [\chi])$$:

 $$T_{7}^{2} + 16T_{7} + 128$$ T7^2 + 16*T7 + 128 $$T_{11} + 4$$ T11 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 6T + 25$$
$7$ $$T^{2} + 16T + 128$$
$11$ $$(T + 4)^{2}$$
$13$ $$T^{2} + 6T + 18$$
$17$ $$T^{2} - 38T + 722$$
$19$ $$T^{2} + 64$$
$23$ $$T^{2} - 40T + 800$$
$29$ $$T^{2} + 1444$$
$31$ $$(T - 44)^{2}$$
$37$ $$T^{2} + 6T + 18$$
$41$ $$(T - 70)^{2}$$
$43$ $$T^{2} + 72T + 2592$$
$47$ $$T^{2}$$
$53$ $$T^{2} - 34T + 578$$
$59$ $$T^{2} + 8464$$
$61$ $$(T - 72)^{2}$$
$67$ $$T^{2} + 88T + 3872$$
$71$ $$(T - 88)^{2}$$
$73$ $$T^{2} - 110T + 6050$$
$79$ $$T^{2} + 144$$
$83$ $$T^{2} - 48T + 1152$$
$89$ $$T^{2} + 676$$
$97$ $$T^{2} + 114T + 6498$$