# Properties

 Label 720.4.f.f Level $720$ Weight $4$ Character orbit 720.f Analytic conductor $42.481$ Analytic rank $0$ Dimension $2$ 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: $$2$$ 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 10) 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 $$\beta = 2i$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - 5 \beta + 5) q^{5} + 13 \beta q^{7}+O(q^{10})$$ q + (-5*b + 5) * q^5 + 13*b * q^7 $$q + ( - 5 \beta + 5) q^{5} + 13 \beta q^{7} - 28 q^{11} + 6 \beta q^{13} - 32 \beta q^{17} - 60 q^{19} - 29 \beta q^{23} + ( - 50 \beta - 75) q^{25} + 90 q^{29} + 128 q^{31} + (65 \beta + 260) q^{35} - 118 \beta q^{37} - 242 q^{41} - 181 \beta q^{43} - 113 \beta q^{47} - 333 q^{49} + 54 \beta q^{53} + (140 \beta - 140) q^{55} + 20 q^{59} + 542 q^{61} + (30 \beta + 120) q^{65} - 217 \beta q^{67} - 1128 q^{71} + 316 \beta q^{73} - 364 \beta q^{77} - 720 q^{79} - 239 \beta q^{83} + ( - 160 \beta - 640) q^{85} - 490 q^{89} - 312 q^{91} + (300 \beta - 300) q^{95} - 728 \beta q^{97} +O(q^{100})$$ q + (-5*b + 5) * q^5 + 13*b * q^7 - 28 * q^11 + 6*b * q^13 - 32*b * q^17 - 60 * q^19 - 29*b * q^23 + (-50*b - 75) * q^25 + 90 * q^29 + 128 * q^31 + (65*b + 260) * q^35 - 118*b * q^37 - 242 * q^41 - 181*b * q^43 - 113*b * q^47 - 333 * q^49 + 54*b * q^53 + (140*b - 140) * q^55 + 20 * q^59 + 542 * q^61 + (30*b + 120) * q^65 - 217*b * q^67 - 1128 * q^71 + 316*b * q^73 - 364*b * q^77 - 720 * q^79 - 239*b * q^83 + (-160*b - 640) * q^85 - 490 * q^89 - 312 * q^91 + (300*b - 300) * q^95 - 728*b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 10 q^{5}+O(q^{10})$$ 2 * q + 10 * q^5 $$2 q + 10 q^{5} - 56 q^{11} - 120 q^{19} - 150 q^{25} + 180 q^{29} + 256 q^{31} + 520 q^{35} - 484 q^{41} - 666 q^{49} - 280 q^{55} + 40 q^{59} + 1084 q^{61} + 240 q^{65} - 2256 q^{71} - 1440 q^{79} - 1280 q^{85} - 980 q^{89} - 624 q^{91} - 600 q^{95}+O(q^{100})$$ 2 * q + 10 * q^5 - 56 * q^11 - 120 * q^19 - 150 * q^25 + 180 * q^29 + 256 * q^31 + 520 * q^35 - 484 * q^41 - 666 * q^49 - 280 * q^55 + 40 * q^59 + 1084 * q^61 + 240 * q^65 - 2256 * q^71 - 1440 * q^79 - 1280 * q^85 - 980 * q^89 - 624 * q^91 - 600 * q^95

## 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
 1.00000i − 1.00000i
0 0 0 5.00000 10.0000i 0 26.0000i 0 0 0
289.2 0 0 0 5.00000 + 10.0000i 0 26.0000i 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.f 2
3.b odd 2 1 80.4.c.a 2
4.b odd 2 1 90.4.c.b 2
5.b even 2 1 inner 720.4.f.f 2
12.b even 2 1 10.4.b.a 2
15.d odd 2 1 80.4.c.a 2
15.e even 4 1 400.4.a.h 1
15.e even 4 1 400.4.a.n 1
20.d odd 2 1 90.4.c.b 2
20.e even 4 1 450.4.a.j 1
20.e even 4 1 450.4.a.k 1
24.f even 2 1 320.4.c.d 2
24.h odd 2 1 320.4.c.c 2
60.h even 2 1 10.4.b.a 2
60.l odd 4 1 50.4.a.b 1
60.l odd 4 1 50.4.a.d 1
84.h odd 2 1 490.4.c.b 2
120.i odd 2 1 320.4.c.c 2
120.m even 2 1 320.4.c.d 2
120.q odd 4 1 1600.4.a.u 1
120.q odd 4 1 1600.4.a.bh 1
120.w even 4 1 1600.4.a.t 1
120.w even 4 1 1600.4.a.bg 1
420.o odd 2 1 490.4.c.b 2
420.w even 4 1 2450.4.a.o 1
420.w even 4 1 2450.4.a.bb 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.4.b.a 2 12.b even 2 1
10.4.b.a 2 60.h even 2 1
50.4.a.b 1 60.l odd 4 1
50.4.a.d 1 60.l odd 4 1
80.4.c.a 2 3.b odd 2 1
80.4.c.a 2 15.d odd 2 1
90.4.c.b 2 4.b odd 2 1
90.4.c.b 2 20.d odd 2 1
320.4.c.c 2 24.h odd 2 1
320.4.c.c 2 120.i odd 2 1
320.4.c.d 2 24.f even 2 1
320.4.c.d 2 120.m even 2 1
400.4.a.h 1 15.e even 4 1
400.4.a.n 1 15.e even 4 1
450.4.a.j 1 20.e even 4 1
450.4.a.k 1 20.e even 4 1
490.4.c.b 2 84.h odd 2 1
490.4.c.b 2 420.o odd 2 1
720.4.f.f 2 1.a even 1 1 trivial
720.4.f.f 2 5.b even 2 1 inner
1600.4.a.t 1 120.w even 4 1
1600.4.a.u 1 120.q odd 4 1
1600.4.a.bg 1 120.w even 4 1
1600.4.a.bh 1 120.q odd 4 1
2450.4.a.o 1 420.w even 4 1
2450.4.a.bb 1 420.w even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(720, [\chi])$$:

 $$T_{7}^{2} + 676$$ T7^2 + 676 $$T_{11} + 28$$ T11 + 28

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 10T + 125$$
$7$ $$T^{2} + 676$$
$11$ $$(T + 28)^{2}$$
$13$ $$T^{2} + 144$$
$17$ $$T^{2} + 4096$$
$19$ $$(T + 60)^{2}$$
$23$ $$T^{2} + 3364$$
$29$ $$(T - 90)^{2}$$
$31$ $$(T - 128)^{2}$$
$37$ $$T^{2} + 55696$$
$41$ $$(T + 242)^{2}$$
$43$ $$T^{2} + 131044$$
$47$ $$T^{2} + 51076$$
$53$ $$T^{2} + 11664$$
$59$ $$(T - 20)^{2}$$
$61$ $$(T - 542)^{2}$$
$67$ $$T^{2} + 188356$$
$71$ $$(T + 1128)^{2}$$
$73$ $$T^{2} + 399424$$
$79$ $$(T + 720)^{2}$$
$83$ $$T^{2} + 228484$$
$89$ $$(T + 490)^{2}$$
$97$ $$T^{2} + 2119936$$