# Properties

 Label 720.4.f.j Level $720$ Weight $4$ Character orbit 720.f Analytic conductor $42.481$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [720,4,Mod(289,720)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(720, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("720.289");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$720 = 2^{4} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 720.f (of order $$2$$, degree $$1$$, 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: $$42.4813752041$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{41})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 21x^{2} + 100$$ x^4 + 21*x^2 + 100 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 15) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} - \beta_{2} - 1) q^{5} + ( - \beta_{2} + 2 \beta_1) q^{7}+O(q^{10})$$ q + (-b3 - b2 - 1) * q^5 + (-b2 + 2*b1) * q^7 $$q + ( - \beta_{3} - \beta_{2} - 1) q^{5} + ( - \beta_{2} + 2 \beta_1) q^{7} + ( - 2 \beta_{3} - \beta_{2} - 20) q^{11} + ( - 4 \beta_{2} - \beta_1) q^{13} + ( - 3 \beta_{2} - 8 \beta_1) q^{17} + ( - 8 \beta_{3} - 4 \beta_{2} + 32) q^{19} + (3 \beta_{2} - \beta_1) q^{23} + (6 \beta_{3} - 4 \beta_{2} + 5 \beta_1 + 61) q^{25} + (14 \beta_{3} + 7 \beta_{2} - 166) q^{29} + (4 \beta_{3} + 2 \beta_{2} - 28) q^{31} + (10 \beta_{3} - 10 \beta_{2} + \cdots - 80) q^{35}+ \cdots + ( - 2 \beta_{2} - 122 \beta_1) q^{97}+O(q^{100})$$ q + (-b3 - b2 - 1) * q^5 + (-b2 + 2*b1) * q^7 + (-2*b3 - b2 - 20) * q^11 + (-4*b2 - b1) * q^13 + (-3*b2 - 8*b1) * q^17 + (-8*b3 - 4*b2 + 32) * q^19 + (3*b2 - b1) * q^23 + (6*b3 - 4*b2 + 5*b1 + 61) * q^25 + (14*b3 + 7*b2 - 166) * q^29 + (4*b3 + 2*b2 - 28) * q^31 + (10*b3 - 10*b2 - 15*b1 - 80) * q^35 + (6*b2 + 33*b1) * q^37 + (4*b3 + 2*b2 + 202) * q^41 + (-7*b2 + 41*b1) * q^43 + (19*b2 - 27*b1) * q^47 + (12*b3 + 6*b2 - 41) * q^49 + (27*b2 - 14*b1) * q^53 + (24*b3 + 14*b2 + 5*b1 + 204) * q^55 + (2*b3 + b2 + 92) * q^59 + (32*b3 + 16*b2 + 154) * q^61 + (22*b3 - 13*b2 + 30*b1 - 248) * q^65 + (-23*b2 - 53*b1) * q^67 + (60*b3 + 30*b2 - 48) * q^71 + (-42*b2 - 96*b1) * q^73 + (-12*b2 - 66*b1) * q^77 + (100*b3 + 50*b2 + 140) * q^79 + (15*b2 + 119*b1) * q^83 + (2*b3 + 12*b2 + 95*b1 - 128) * q^85 + (-48*b3 - 24*b2 + 582) * q^89 + (84*b3 + 42*b2 - 528) * q^91 + (-16*b3 - 56*b2 + 20*b1 + 704) * q^95 + (-2*b2 - 122*b1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 6 q^{5}+O(q^{10})$$ 4 * q - 6 * q^5 $$4 q - 6 q^{5} - 84 q^{11} + 112 q^{19} + 256 q^{25} - 636 q^{29} - 104 q^{31} - 300 q^{35} + 816 q^{41} - 140 q^{49} + 864 q^{55} + 372 q^{59} + 680 q^{61} - 948 q^{65} - 72 q^{71} + 760 q^{79} - 508 q^{85} + 2232 q^{89} - 1944 q^{91} + 2784 q^{95}+O(q^{100})$$ 4 * q - 6 * q^5 - 84 * q^11 + 112 * q^19 + 256 * q^25 - 636 * q^29 - 104 * q^31 - 300 * q^35 + 816 * q^41 - 140 * q^49 + 864 * q^55 + 372 * q^59 + 680 * q^61 - 948 * q^65 - 72 * q^71 + 760 * q^79 - 508 * q^85 + 2232 * q^89 - 1944 * q^91 + 2784 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 21x^{2} + 100$$ :

 $$\beta_{1}$$ $$=$$ $$( 2\nu^{3} + 32\nu ) / 5$$ (2*v^3 + 32*v) / 5 $$\beta_{2}$$ $$=$$ $$( 4\nu^{3} + 34\nu ) / 5$$ (4*v^3 + 34*v) / 5 $$\beta_{3}$$ $$=$$ $$( -2\nu^{3} + 15\nu^{2} - 17\nu + 160 ) / 5$$ (-2*v^3 + 15*v^2 - 17*v + 160) / 5
 $$\nu$$ $$=$$ $$( -\beta_{2} + 2\beta_1 ) / 6$$ (-b2 + 2*b1) / 6 $$\nu^{2}$$ $$=$$ $$( 2\beta_{3} + \beta_{2} - 64 ) / 6$$ (2*b3 + b2 - 64) / 6 $$\nu^{3}$$ $$=$$ $$( 16\beta_{2} - 17\beta_1 ) / 6$$ (16*b2 - 17*b1) / 6

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 2.70156i − 2.70156i − 3.70156i 3.70156i
0 0 0 −11.1047 1.29844i 0 16.2094i 0 0 0
289.2 0 0 0 −11.1047 + 1.29844i 0 16.2094i 0 0 0
289.3 0 0 0 8.10469 7.70156i 0 22.2094i 0 0 0
289.4 0 0 0 8.10469 + 7.70156i 0 22.2094i 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.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.4.f.j 4
3.b odd 2 1 240.4.f.f 4
4.b odd 2 1 45.4.b.b 4
5.b even 2 1 inner 720.4.f.j 4
12.b even 2 1 15.4.b.a 4
15.d odd 2 1 240.4.f.f 4
15.e even 4 1 1200.4.a.bn 2
15.e even 4 1 1200.4.a.bt 2
20.d odd 2 1 45.4.b.b 4
20.e even 4 1 225.4.a.i 2
20.e even 4 1 225.4.a.o 2
24.f even 2 1 960.4.f.q 4
24.h odd 2 1 960.4.f.p 4
60.h even 2 1 15.4.b.a 4
60.l odd 4 1 75.4.a.c 2
60.l odd 4 1 75.4.a.f 2
120.i odd 2 1 960.4.f.p 4
120.m even 2 1 960.4.f.q 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.b.a 4 12.b even 2 1
15.4.b.a 4 60.h even 2 1
45.4.b.b 4 4.b odd 2 1
45.4.b.b 4 20.d odd 2 1
75.4.a.c 2 60.l odd 4 1
75.4.a.f 2 60.l odd 4 1
225.4.a.i 2 20.e even 4 1
225.4.a.o 2 20.e even 4 1
240.4.f.f 4 3.b odd 2 1
240.4.f.f 4 15.d odd 2 1
720.4.f.j 4 1.a even 1 1 trivial
720.4.f.j 4 5.b even 2 1 inner
960.4.f.p 4 24.h odd 2 1
960.4.f.p 4 120.i odd 2 1
960.4.f.q 4 24.f even 2 1
960.4.f.q 4 120.m even 2 1
1200.4.a.bn 2 15.e even 4 1
1200.4.a.bt 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 $$S_{4}^{\mathrm{new}}(720, [\chi])$$:

 $$T_{7}^{4} + 756T_{7}^{2} + 129600$$ T7^4 + 756*T7^2 + 129600 $$T_{11}^{2} + 42T_{11} + 72$$ T11^2 + 42*T11 + 72

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 6 T^{3} + \cdots + 15625$$
$7$ $$T^{4} + 756 T^{2} + 129600$$
$11$ $$(T^{2} + 42 T + 72)^{2}$$
$13$ $$T^{4} + 3780 T^{2} + 1327104$$
$17$ $$T^{4} + 7252 T^{2} + 2483776$$
$19$ $$(T^{2} - 56 T - 5120)^{2}$$
$23$ $$T^{4} + 2464 T^{2} + 6400$$
$29$ $$(T^{2} + 318 T + 7200)^{2}$$
$31$ $$(T^{2} + 52 T - 800)^{2}$$
$37$ $$T^{4} + 106596 T^{2} + 41990400$$
$41$ $$(T^{2} - 408 T + 40140)^{2}$$
$43$ $$T^{4} + \cdots + 8256266496$$
$47$ $$T^{4} + \cdots + 6186766336$$
$53$ $$T^{4} + 218644 T^{2} + 813390400$$
$59$ $$(T^{2} - 186 T + 8280)^{2}$$
$61$ $$(T^{2} - 340 T - 65564)^{2}$$
$67$ $$T^{4} + \cdots + 9419867136$$
$71$ $$(T^{2} + 36 T - 331776)^{2}$$
$73$ $$T^{4} + \cdots + 104976000000$$
$79$ $$(T^{2} - 380 T - 886400)^{2}$$
$83$ $$T^{4} + \cdots + 40558737664$$
$89$ $$(T^{2} - 1116 T + 98820)^{2}$$
$97$ $$T^{4} + \cdots + 196199387136$$