# Properties

 Label 720.2.x.a Level $720$ Weight $2$ Character orbit 720.x Analytic conductor $5.749$ Analytic rank $0$ Dimension $2$ CM discriminant -4 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [720,2,Mod(127,720)]

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([2, 0, 0, 1]))

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

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

chi := DirichletCharacter("720.127");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$720 = 2^{4} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 720.x (of order $$4$$, 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: $$5.74922894553$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 80) Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - i - 2) q^{5}+O(q^{10})$$ q + (-i - 2) * q^5 $$q + ( - i - 2) q^{5} + (5 i - 5) q^{13} + (5 i + 5) q^{17} + (4 i + 3) q^{25} + 4 i q^{29} + (5 i + 5) q^{37} - 8 q^{41} + 7 i q^{49} + (5 i - 5) q^{53} - 12 q^{61} + ( - 5 i + 15) q^{65} + ( - 5 i + 5) q^{73} + ( - 15 i - 5) q^{85} - 16 i q^{89} + (5 i + 5) q^{97} +O(q^{100})$$ q + (-i - 2) * q^5 + (5*i - 5) * q^13 + (5*i + 5) * q^17 + (4*i + 3) * q^25 + 4*i * q^29 + (5*i + 5) * q^37 - 8 * q^41 + 7*i * q^49 + (5*i - 5) * q^53 - 12 * q^61 + (-5*i + 15) * q^65 + (-5*i + 5) * q^73 + (-15*i - 5) * q^85 - 16*i * q^89 + (5*i + 5) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{5}+O(q^{10})$$ 2 * q - 4 * q^5 $$2 q - 4 q^{5} - 10 q^{13} + 10 q^{17} + 6 q^{25} + 10 q^{37} - 16 q^{41} - 10 q^{53} - 24 q^{61} + 30 q^{65} + 10 q^{73} - 10 q^{85} + 10 q^{97}+O(q^{100})$$ 2 * q - 4 * q^5 - 10 * q^13 + 10 * q^17 + 6 * q^25 + 10 * q^37 - 16 * q^41 - 10 * q^53 - 24 * q^61 + 30 * q^65 + 10 * q^73 - 10 * q^85 + 10 * 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 1.00000i − 1.00000i
0 0 0 −2.00000 1.00000i 0 0 0 0 0
703.1 0 0 0 −2.00000 + 1.00000i 0 0 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
5.c odd 4 1 inner
20.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.2.x.a 2
3.b odd 2 1 80.2.n.a 2
4.b odd 2 1 CM 720.2.x.a 2
5.b even 2 1 3600.2.x.c 2
5.c odd 4 1 inner 720.2.x.a 2
5.c odd 4 1 3600.2.x.c 2
12.b even 2 1 80.2.n.a 2
15.d odd 2 1 400.2.n.a 2
15.e even 4 1 80.2.n.a 2
15.e even 4 1 400.2.n.a 2
20.d odd 2 1 3600.2.x.c 2
20.e even 4 1 inner 720.2.x.a 2
20.e even 4 1 3600.2.x.c 2
24.f even 2 1 320.2.n.d 2
24.h odd 2 1 320.2.n.d 2
48.i odd 4 1 1280.2.o.h 2
48.i odd 4 1 1280.2.o.i 2
48.k even 4 1 1280.2.o.h 2
48.k even 4 1 1280.2.o.i 2
60.h even 2 1 400.2.n.a 2
60.l odd 4 1 80.2.n.a 2
60.l odd 4 1 400.2.n.a 2
120.i odd 2 1 1600.2.n.g 2
120.m even 2 1 1600.2.n.g 2
120.q odd 4 1 320.2.n.d 2
120.q odd 4 1 1600.2.n.g 2
120.w even 4 1 320.2.n.d 2
120.w even 4 1 1600.2.n.g 2
240.z odd 4 1 1280.2.o.h 2
240.bb even 4 1 1280.2.o.h 2
240.bd odd 4 1 1280.2.o.i 2
240.bf even 4 1 1280.2.o.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.2.n.a 2 3.b odd 2 1
80.2.n.a 2 12.b even 2 1
80.2.n.a 2 15.e even 4 1
80.2.n.a 2 60.l odd 4 1
320.2.n.d 2 24.f even 2 1
320.2.n.d 2 24.h odd 2 1
320.2.n.d 2 120.q odd 4 1
320.2.n.d 2 120.w even 4 1
400.2.n.a 2 15.d odd 2 1
400.2.n.a 2 15.e even 4 1
400.2.n.a 2 60.h even 2 1
400.2.n.a 2 60.l odd 4 1
720.2.x.a 2 1.a even 1 1 trivial
720.2.x.a 2 4.b odd 2 1 CM
720.2.x.a 2 5.c odd 4 1 inner
720.2.x.a 2 20.e even 4 1 inner
1280.2.o.h 2 48.i odd 4 1
1280.2.o.h 2 48.k even 4 1
1280.2.o.h 2 240.z odd 4 1
1280.2.o.h 2 240.bb even 4 1
1280.2.o.i 2 48.i odd 4 1
1280.2.o.i 2 48.k even 4 1
1280.2.o.i 2 240.bd odd 4 1
1280.2.o.i 2 240.bf even 4 1
1600.2.n.g 2 120.i odd 2 1
1600.2.n.g 2 120.m even 2 1
1600.2.n.g 2 120.q odd 4 1
1600.2.n.g 2 120.w even 4 1
3600.2.x.c 2 5.b even 2 1
3600.2.x.c 2 5.c odd 4 1
3600.2.x.c 2 20.d odd 2 1
3600.2.x.c 2 20.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 $$S_{2}^{\mathrm{new}}(720, [\chi])$$:

 $$T_{7}$$ T7 $$T_{11}$$ T11 $$T_{13}^{2} + 10T_{13} + 50$$ T13^2 + 10*T13 + 50 $$T_{17}^{2} - 10T_{17} + 50$$ T17^2 - 10*T17 + 50

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 4T + 5$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 10T + 50$$
$17$ $$T^{2} - 10T + 50$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2} + 16$$
$31$ $$T^{2}$$
$37$ $$T^{2} - 10T + 50$$
$41$ $$(T + 8)^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 10T + 50$$
$59$ $$T^{2}$$
$61$ $$(T + 12)^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2} - 10T + 50$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2} + 256$$
$97$ $$T^{2} - 10T + 50$$