# Properties

 Label 2960.2.a.h Level $2960$ Weight $2$ Character orbit 2960.a Self dual yes Analytic conductor $23.636$ 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] = [2960,2,Mod(1,2960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2960, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2960.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2960 = 2^{4} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2960.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: $$23.6357189983$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 740) 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)

 $$f(q)$$ $$=$$ $$q + q^{3} + q^{5} - q^{7} - 2 q^{9}+O(q^{10})$$ q + q^3 + q^5 - q^7 - 2 * q^9 $$q + q^{3} + q^{5} - q^{7} - 2 q^{9} + 3 q^{11} - 6 q^{13} + q^{15} - q^{21} - 2 q^{23} + q^{25} - 5 q^{27} - 6 q^{29} + 3 q^{33} - q^{35} - q^{37} - 6 q^{39} - 9 q^{41} + 10 q^{43} - 2 q^{45} - q^{47} - 6 q^{49} + q^{53} + 3 q^{55} - 12 q^{61} + 2 q^{63} - 6 q^{65} - 2 q^{69} + 5 q^{71} + 3 q^{73} + q^{75} - 3 q^{77} - 16 q^{79} + q^{81} - 11 q^{83} - 6 q^{87} + 6 q^{91} + 8 q^{97} - 6 q^{99}+O(q^{100})$$ q + q^3 + q^5 - q^7 - 2 * q^9 + 3 * q^11 - 6 * q^13 + q^15 - q^21 - 2 * q^23 + q^25 - 5 * q^27 - 6 * q^29 + 3 * q^33 - q^35 - q^37 - 6 * q^39 - 9 * q^41 + 10 * q^43 - 2 * q^45 - q^47 - 6 * q^49 + q^53 + 3 * q^55 - 12 * q^61 + 2 * q^63 - 6 * q^65 - 2 * q^69 + 5 * q^71 + 3 * q^73 + q^75 - 3 * q^77 - 16 * q^79 + q^81 - 11 * q^83 - 6 * q^87 + 6 * q^91 + 8 * 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 −1.00000 0 −2.00000 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
$$2$$ $$-1$$
$$5$$ $$-1$$
$$37$$ $$1$$

## 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 2960.2.a.h 1
4.b odd 2 1 740.2.a.a 1
12.b even 2 1 6660.2.a.a 1
20.d odd 2 1 3700.2.a.d 1
20.e even 4 2 3700.2.d.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.a.a 1 4.b odd 2 1
2960.2.a.h 1 1.a even 1 1 trivial
3700.2.a.d 1 20.d odd 2 1
3700.2.d.c 2 20.e even 4 2
6660.2.a.a 1 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2960))$$:

 $$T_{3} - 1$$ T3 - 1 $$T_{7} + 1$$ T7 + 1 $$T_{13} + 6$$ T13 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T - 1$$
$7$ $$T + 1$$
$11$ $$T - 3$$
$13$ $$T + 6$$
$17$ $$T$$
$19$ $$T$$
$23$ $$T + 2$$
$29$ $$T + 6$$
$31$ $$T$$
$37$ $$T + 1$$
$41$ $$T + 9$$
$43$ $$T - 10$$
$47$ $$T + 1$$
$53$ $$T - 1$$
$59$ $$T$$
$61$ $$T + 12$$
$67$ $$T$$
$71$ $$T - 5$$
$73$ $$T - 3$$
$79$ $$T + 16$$
$83$ $$T + 11$$
$89$ $$T$$
$97$ $$T - 8$$