# Properties

 Label 720.4.f.g 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: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 120) 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - 5 i + 10) q^{5} + 10 i q^{7}+O(q^{10})$$ q + (-5*i + 10) * q^5 + 10*i * q^7 $$q + ( - 5 i + 10) q^{5} + 10 i q^{7} - 46 q^{11} - 34 i q^{13} + 66 i q^{17} + 104 q^{19} + 164 i q^{23} + ( - 100 i + 75) q^{25} + 224 q^{29} + 72 q^{31} + (100 i + 50) q^{35} + 22 i q^{37} - 194 q^{41} - 108 i q^{43} + 480 i q^{47} + 243 q^{49} - 286 i q^{53} + (230 i - 460) q^{55} - 426 q^{59} + 698 q^{61} + ( - 340 i - 170) q^{65} + 328 i q^{67} + 188 q^{71} - 740 i q^{73} - 460 i q^{77} + 1168 q^{79} + 412 i q^{83} + (660 i + 330) q^{85} + 1206 q^{89} + 340 q^{91} + ( - 520 i + 1040) q^{95} + 1384 i q^{97} +O(q^{100})$$ q + (-5*i + 10) * q^5 + 10*i * q^7 - 46 * q^11 - 34*i * q^13 + 66*i * q^17 + 104 * q^19 + 164*i * q^23 + (-100*i + 75) * q^25 + 224 * q^29 + 72 * q^31 + (100*i + 50) * q^35 + 22*i * q^37 - 194 * q^41 - 108*i * q^43 + 480*i * q^47 + 243 * q^49 - 286*i * q^53 + (230*i - 460) * q^55 - 426 * q^59 + 698 * q^61 + (-340*i - 170) * q^65 + 328*i * q^67 + 188 * q^71 - 740*i * q^73 - 460*i * q^77 + 1168 * q^79 + 412*i * q^83 + (660*i + 330) * q^85 + 1206 * q^89 + 340 * q^91 + (-520*i + 1040) * q^95 + 1384*i * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 20 q^{5}+O(q^{10})$$ 2 * q + 20 * q^5 $$2 q + 20 q^{5} - 92 q^{11} + 208 q^{19} + 150 q^{25} + 448 q^{29} + 144 q^{31} + 100 q^{35} - 388 q^{41} + 486 q^{49} - 920 q^{55} - 852 q^{59} + 1396 q^{61} - 340 q^{65} + 376 q^{71} + 2336 q^{79} + 660 q^{85} + 2412 q^{89} + 680 q^{91} + 2080 q^{95}+O(q^{100})$$ 2 * q + 20 * q^5 - 92 * q^11 + 208 * q^19 + 150 * q^25 + 448 * q^29 + 144 * q^31 + 100 * q^35 - 388 * q^41 + 486 * q^49 - 920 * q^55 - 852 * q^59 + 1396 * q^61 - 340 * q^65 + 376 * q^71 + 2336 * q^79 + 660 * q^85 + 2412 * q^89 + 680 * q^91 + 2080 * 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 10.0000 5.00000i 0 10.0000i 0 0 0
289.2 0 0 0 10.0000 + 5.00000i 0 10.0000i 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.g 2
3.b odd 2 1 240.4.f.b 2
4.b odd 2 1 360.4.f.c 2
5.b even 2 1 inner 720.4.f.g 2
12.b even 2 1 120.4.f.a 2
15.d odd 2 1 240.4.f.b 2
15.e even 4 1 1200.4.a.f 1
15.e even 4 1 1200.4.a.bh 1
20.d odd 2 1 360.4.f.c 2
20.e even 4 1 1800.4.a.j 1
20.e even 4 1 1800.4.a.y 1
24.f even 2 1 960.4.f.l 2
24.h odd 2 1 960.4.f.g 2
60.h even 2 1 120.4.f.a 2
60.l odd 4 1 600.4.a.b 1
60.l odd 4 1 600.4.a.o 1
120.i odd 2 1 960.4.f.g 2
120.m even 2 1 960.4.f.l 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.4.f.a 2 12.b even 2 1
120.4.f.a 2 60.h even 2 1
240.4.f.b 2 3.b odd 2 1
240.4.f.b 2 15.d odd 2 1
360.4.f.c 2 4.b odd 2 1
360.4.f.c 2 20.d odd 2 1
600.4.a.b 1 60.l odd 4 1
600.4.a.o 1 60.l odd 4 1
720.4.f.g 2 1.a even 1 1 trivial
720.4.f.g 2 5.b even 2 1 inner
960.4.f.g 2 24.h odd 2 1
960.4.f.g 2 120.i odd 2 1
960.4.f.l 2 24.f even 2 1
960.4.f.l 2 120.m even 2 1
1200.4.a.f 1 15.e even 4 1
1200.4.a.bh 1 15.e even 4 1
1800.4.a.j 1 20.e even 4 1
1800.4.a.y 1 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 $$S_{4}^{\mathrm{new}}(720, [\chi])$$:

 $$T_{7}^{2} + 100$$ T7^2 + 100 $$T_{11} + 46$$ T11 + 46

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 20T + 125$$
$7$ $$T^{2} + 100$$
$11$ $$(T + 46)^{2}$$
$13$ $$T^{2} + 1156$$
$17$ $$T^{2} + 4356$$
$19$ $$(T - 104)^{2}$$
$23$ $$T^{2} + 26896$$
$29$ $$(T - 224)^{2}$$
$31$ $$(T - 72)^{2}$$
$37$ $$T^{2} + 484$$
$41$ $$(T + 194)^{2}$$
$43$ $$T^{2} + 11664$$
$47$ $$T^{2} + 230400$$
$53$ $$T^{2} + 81796$$
$59$ $$(T + 426)^{2}$$
$61$ $$(T - 698)^{2}$$
$67$ $$T^{2} + 107584$$
$71$ $$(T - 188)^{2}$$
$73$ $$T^{2} + 547600$$
$79$ $$(T - 1168)^{2}$$
$83$ $$T^{2} + 169744$$
$89$ $$(T - 1206)^{2}$$
$97$ $$T^{2} + 1915456$$