# Properties

 Label 3120.2.a.g Level $3120$ Weight $2$ Character orbit 3120.a Self dual yes Analytic conductor $24.913$ Analytic rank $0$ 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] = [3120,2,Mod(1,3120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3120, 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("3120.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3120.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: $$24.9133254306$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 780) 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)

 $$f(q)$$ $$=$$ $$q - q^{3} - q^{5} + 2 q^{7} + q^{9}+O(q^{10})$$ q - q^3 - q^5 + 2 * q^7 + q^9 $$q - q^{3} - q^{5} + 2 q^{7} + q^{9} + 6 q^{11} - q^{13} + q^{15} - 2 q^{17} + 2 q^{19} - 2 q^{21} + 4 q^{23} + q^{25} - q^{27} + 2 q^{29} + 2 q^{31} - 6 q^{33} - 2 q^{35} + 2 q^{37} + q^{39} - 6 q^{41} - q^{45} - 6 q^{47} - 3 q^{49} + 2 q^{51} - 2 q^{53} - 6 q^{55} - 2 q^{57} + 6 q^{59} + 14 q^{61} + 2 q^{63} + q^{65} - 2 q^{67} - 4 q^{69} + 10 q^{71} - 6 q^{73} - q^{75} + 12 q^{77} - 4 q^{79} + q^{81} - 2 q^{83} + 2 q^{85} - 2 q^{87} - 14 q^{89} - 2 q^{91} - 2 q^{93} - 2 q^{95} + 18 q^{97} + 6 q^{99}+O(q^{100})$$ q - q^3 - q^5 + 2 * q^7 + q^9 + 6 * q^11 - q^13 + q^15 - 2 * q^17 + 2 * q^19 - 2 * q^21 + 4 * q^23 + q^25 - q^27 + 2 * q^29 + 2 * q^31 - 6 * q^33 - 2 * q^35 + 2 * q^37 + q^39 - 6 * q^41 - q^45 - 6 * q^47 - 3 * q^49 + 2 * q^51 - 2 * q^53 - 6 * q^55 - 2 * q^57 + 6 * q^59 + 14 * q^61 + 2 * q^63 + q^65 - 2 * q^67 - 4 * q^69 + 10 * q^71 - 6 * q^73 - q^75 + 12 * q^77 - 4 * q^79 + q^81 - 2 * q^83 + 2 * q^85 - 2 * q^87 - 14 * q^89 - 2 * q^91 - 2 * q^93 - 2 * q^95 + 18 * q^97 + 6 * 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 2.00000 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
$$2$$ $$-1$$
$$3$$ $$1$$
$$5$$ $$1$$
$$13$$ $$1$$

## 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 3120.2.a.g 1
3.b odd 2 1 9360.2.a.bw 1
4.b odd 2 1 780.2.a.c 1
12.b even 2 1 2340.2.a.e 1
20.d odd 2 1 3900.2.a.e 1
20.e even 4 2 3900.2.h.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
780.2.a.c 1 4.b odd 2 1
2340.2.a.e 1 12.b even 2 1
3120.2.a.g 1 1.a even 1 1 trivial
3900.2.a.e 1 20.d odd 2 1
3900.2.h.a 2 20.e even 4 2
9360.2.a.bw 1 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3120))$$:

 $$T_{7} - 2$$ T7 - 2 $$T_{11} - 6$$ T11 - 6 $$T_{17} + 2$$ T17 + 2 $$T_{19} - 2$$ T19 - 2 $$T_{31} - 2$$ T31 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 1$$
$5$ $$T + 1$$
$7$ $$T - 2$$
$11$ $$T - 6$$
$13$ $$T + 1$$
$17$ $$T + 2$$
$19$ $$T - 2$$
$23$ $$T - 4$$
$29$ $$T - 2$$
$31$ $$T - 2$$
$37$ $$T - 2$$
$41$ $$T + 6$$
$43$ $$T$$
$47$ $$T + 6$$
$53$ $$T + 2$$
$59$ $$T - 6$$
$61$ $$T - 14$$
$67$ $$T + 2$$
$71$ $$T - 10$$
$73$ $$T + 6$$
$79$ $$T + 4$$
$83$ $$T + 2$$
$89$ $$T + 14$$
$97$ $$T - 18$$