# Properties

 Label 8880.2.a.z Level $8880$ Weight $2$ Character orbit 8880.a Self dual yes Analytic conductor $70.907$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8880,2,Mod(1,8880)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8880, 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("8880.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$8880 = 2^{4} \cdot 3 \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8880.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: $$70.9071569949$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1110) 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)

 $$f(q)$$ $$=$$ $$q + q^{3} + q^{5} - 3 q^{7} + q^{9}+O(q^{10})$$ q + q^3 + q^5 - 3 * q^7 + q^9 $$q + q^{3} + q^{5} - 3 q^{7} + q^{9} + 5 q^{11} - 2 q^{13} + q^{15} - 7 q^{17} + 2 q^{19} - 3 q^{21} - 4 q^{23} + q^{25} + q^{27} - 5 q^{29} + 7 q^{31} + 5 q^{33} - 3 q^{35} + q^{37} - 2 q^{39} + q^{41} - 9 q^{43} + q^{45} + 2 q^{49} - 7 q^{51} + 3 q^{53} + 5 q^{55} + 2 q^{57} + 14 q^{59} - 7 q^{61} - 3 q^{63} - 2 q^{65} + 4 q^{67} - 4 q^{69} - 8 q^{71} - 12 q^{73} + q^{75} - 15 q^{77} - 4 q^{79} + q^{81} + 10 q^{83} - 7 q^{85} - 5 q^{87} - 14 q^{89} + 6 q^{91} + 7 q^{93} + 2 q^{95} + q^{97} + 5 q^{99}+O(q^{100})$$ q + q^3 + q^5 - 3 * q^7 + q^9 + 5 * q^11 - 2 * q^13 + q^15 - 7 * q^17 + 2 * q^19 - 3 * q^21 - 4 * q^23 + q^25 + q^27 - 5 * q^29 + 7 * q^31 + 5 * q^33 - 3 * q^35 + q^37 - 2 * q^39 + q^41 - 9 * q^43 + q^45 + 2 * q^49 - 7 * q^51 + 3 * q^53 + 5 * q^55 + 2 * q^57 + 14 * q^59 - 7 * q^61 - 3 * q^63 - 2 * q^65 + 4 * q^67 - 4 * q^69 - 8 * q^71 - 12 * q^73 + q^75 - 15 * q^77 - 4 * q^79 + q^81 + 10 * q^83 - 7 * q^85 - 5 * q^87 - 14 * q^89 + 6 * q^91 + 7 * q^93 + 2 * q^95 + q^97 + 5 * q^99

## 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 −3.00000 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$37$$ $$-1$$

## 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 8880.2.a.z 1
4.b odd 2 1 1110.2.a.d 1
12.b even 2 1 3330.2.a.q 1
20.d odd 2 1 5550.2.a.bf 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.a.d 1 4.b odd 2 1
3330.2.a.q 1 12.b even 2 1
5550.2.a.bf 1 20.d odd 2 1
8880.2.a.z 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8880))$$:

 $$T_{7} + 3$$ T7 + 3 $$T_{11} - 5$$ T11 - 5 $$T_{13} + 2$$ T13 + 2 $$T_{23} + 4$$ T23 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T - 1$$
$7$ $$T + 3$$
$11$ $$T - 5$$
$13$ $$T + 2$$
$17$ $$T + 7$$
$19$ $$T - 2$$
$23$ $$T + 4$$
$29$ $$T + 5$$
$31$ $$T - 7$$
$37$ $$T - 1$$
$41$ $$T - 1$$
$43$ $$T + 9$$
$47$ $$T$$
$53$ $$T - 3$$
$59$ $$T - 14$$
$61$ $$T + 7$$
$67$ $$T - 4$$
$71$ $$T + 8$$
$73$ $$T + 12$$
$79$ $$T + 4$$
$83$ $$T - 10$$
$89$ $$T + 14$$
$97$ $$T - 1$$
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