# Properties

 Label 5880.2.a.p Level $5880$ Weight $2$ Character orbit 5880.a Self dual yes Analytic conductor $46.952$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5880,2,Mod(1,5880)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5880, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5880.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5880 = 2^{3} \cdot 3 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5880.a (trivial)

## 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: yes Analytic conductor: $$46.9520363885$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 120) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{5} + q^{9}+O(q^{10})$$ q - q^3 + q^5 + q^9 $$q - q^{3} + q^{5} + q^{9} + 6 q^{13} - q^{15} + 2 q^{17} - 4 q^{19} - 8 q^{23} + q^{25} - q^{27} - 6 q^{29} - 6 q^{37} - 6 q^{39} - 10 q^{41} - 4 q^{43} + q^{45} - 8 q^{47} - 2 q^{51} + 10 q^{53} + 4 q^{57} - 6 q^{61} + 6 q^{65} - 4 q^{67} + 8 q^{69} + 14 q^{73} - q^{75} + 16 q^{79} + q^{81} - 12 q^{83} + 2 q^{85} + 6 q^{87} - 2 q^{89} - 4 q^{95} - 2 q^{97}+O(q^{100})$$ q - q^3 + q^5 + q^9 + 6 * q^13 - q^15 + 2 * q^17 - 4 * q^19 - 8 * q^23 + q^25 - q^27 - 6 * q^29 - 6 * q^37 - 6 * q^39 - 10 * q^41 - 4 * q^43 + q^45 - 8 * q^47 - 2 * q^51 + 10 * q^53 + 4 * q^57 - 6 * q^61 + 6 * q^65 - 4 * q^67 + 8 * q^69 + 14 * q^73 - q^75 + 16 * q^79 + q^81 - 12 * q^83 + 2 * q^85 + 6 * q^87 - 2 * q^89 - 4 * q^95 - 2 * q^97

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 −1.00000 0 1.00000 0 0 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$+1$$
$$3$$ $$+1$$
$$5$$ $$-1$$
$$7$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5880.2.a.p 1
7.b odd 2 1 120.2.a.a 1
21.c even 2 1 360.2.a.e 1
28.d even 2 1 240.2.a.a 1
35.c odd 2 1 600.2.a.a 1
35.f even 4 2 600.2.f.c 2
56.e even 2 1 960.2.a.n 1
56.h odd 2 1 960.2.a.g 1
63.l odd 6 2 3240.2.q.m 2
63.o even 6 2 3240.2.q.a 2
84.h odd 2 1 720.2.a.f 1
105.g even 2 1 1800.2.a.c 1
105.k odd 4 2 1800.2.f.g 2
112.j even 4 2 3840.2.k.z 2
112.l odd 4 2 3840.2.k.a 2
140.c even 2 1 1200.2.a.r 1
140.j odd 4 2 1200.2.f.f 2
168.e odd 2 1 2880.2.a.b 1
168.i even 2 1 2880.2.a.r 1
280.c odd 2 1 4800.2.a.bl 1
280.n even 2 1 4800.2.a.bh 1
280.s even 4 2 4800.2.f.u 2
280.y odd 4 2 4800.2.f.n 2
420.o odd 2 1 3600.2.a.bo 1
420.w even 4 2 3600.2.f.l 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.a.a 1 7.b odd 2 1
240.2.a.a 1 28.d even 2 1
360.2.a.e 1 21.c even 2 1
600.2.a.a 1 35.c odd 2 1
600.2.f.c 2 35.f even 4 2
720.2.a.f 1 84.h odd 2 1
960.2.a.g 1 56.h odd 2 1
960.2.a.n 1 56.e even 2 1
1200.2.a.r 1 140.c even 2 1
1200.2.f.f 2 140.j odd 4 2
1800.2.a.c 1 105.g even 2 1
1800.2.f.g 2 105.k odd 4 2
2880.2.a.b 1 168.e odd 2 1
2880.2.a.r 1 168.i even 2 1
3240.2.q.a 2 63.o even 6 2
3240.2.q.m 2 63.l odd 6 2
3600.2.a.bo 1 420.o odd 2 1
3600.2.f.l 2 420.w even 4 2
3840.2.k.a 2 112.l odd 4 2
3840.2.k.z 2 112.j even 4 2
4800.2.a.bh 1 280.n even 2 1
4800.2.a.bl 1 280.c odd 2 1
4800.2.f.n 2 280.y odd 4 2
4800.2.f.u 2 280.s even 4 2
5880.2.a.p 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5880))$$:

 $$T_{11}$$ T11 $$T_{13} - 6$$ T13 - 6 $$T_{17} - 2$$ T17 - 2 $$T_{19} + 4$$ T19 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 1$$
$5$ $$T - 1$$
$7$ $$T$$
$11$ $$T$$
$13$ $$T - 6$$
$17$ $$T - 2$$
$19$ $$T + 4$$
$23$ $$T + 8$$
$29$ $$T + 6$$
$31$ $$T$$
$37$ $$T + 6$$
$41$ $$T + 10$$
$43$ $$T + 4$$
$47$ $$T + 8$$
$53$ $$T - 10$$
$59$ $$T$$
$61$ $$T + 6$$
$67$ $$T + 4$$
$71$ $$T$$
$73$ $$T - 14$$
$79$ $$T - 16$$
$83$ $$T + 12$$
$89$ $$T + 2$$
$97$ $$T + 2$$