# Properties

 Label 2960.2.a.v Level $2960$ Weight $2$ Character orbit 2960.a Self dual yes Analytic conductor $23.636$ Analytic rank $0$ Dimension $4$ 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: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.286164.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 11x^{2} - 3x + 12$$ x^4 - x^3 - 11*x^2 - 3*x + 12 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 1) q^{3} + q^{5} + (\beta_1 + 1) q^{7} + (\beta_{2} + 3) q^{9}+O(q^{10})$$ q + (b1 - 1) * q^3 + q^5 + (b1 + 1) * q^7 + (b2 + 3) * q^9 $$q + (\beta_1 - 1) q^{3} + q^{5} + (\beta_1 + 1) q^{7} + (\beta_{2} + 3) q^{9} + ( - \beta_{3} + \beta_{2} - 2) q^{11} + ( - \beta_{3} - 2) q^{13} + (\beta_1 - 1) q^{15} + ( - \beta_{3} + 2) q^{17} + ( - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{19} + (\beta_{2} + 2 \beta_1 + 4) q^{21} - 2 \beta_1 q^{23} + q^{25} + (\beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{27} + 2 \beta_1 q^{29} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{31} + (3 \beta_{3} - 3 \beta_{2} - 2 \beta_1) q^{33} + (\beta_1 + 1) q^{35} + q^{37} + (2 \beta_{3} - 2 \beta_{2} - 4 \beta_1 + 2) q^{39} + (\beta_{3} - \beta_{2} + 8) q^{41} + ( - 2 \beta_{2} - 2 \beta_1) q^{43} + (\beta_{2} + 3) q^{45} + ( - 2 \beta_{3} + \beta_1 + 1) q^{47} + (\beta_{2} + 4 \beta_1 - 1) q^{49} + (2 \beta_{3} - 2 \beta_{2} - 2) q^{51} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 - 2) q^{53} + ( - \beta_{3} + \beta_{2} - 2) q^{55} + (3 \beta_{3} - 2 \beta_{2} + 4) q^{57} + (\beta_{3} + \beta_{2} - \beta_1 + 3) q^{59} + 2 q^{61} + (\beta_{3} + \beta_{2} + 5 \beta_1 + 1) q^{63} + ( - \beta_{3} - 2) q^{65} + (\beta_{3} - \beta_{2} + \beta_1 - 3) q^{67} + ( - 2 \beta_{2} - 2 \beta_1 - 10) q^{69} + ( - \beta_{3} + \beta_{2} - 2 \beta_1 + 4) q^{71} + ( - \beta_{3} + \beta_{2} + 4 \beta_1) q^{73} + (\beta_1 - 1) q^{75} + (\beta_{3} - \beta_{2} - 2 \beta_1 - 4) q^{77} + ( - \beta_{3} + \beta_{2} + \beta_1 + 5) q^{79} + ( - 3 \beta_{3} + 2 \beta_{2} + 5) q^{81} + ( - 2 \beta_{2} - \beta_1 - 7) q^{83} + ( - \beta_{3} + 2) q^{85} + (2 \beta_{2} + 2 \beta_1 + 10) q^{87} + ( - 2 \beta_{3} - 2) q^{89} + ( - 2 \beta_{2} - 4 \beta_1 - 2) q^{91} + ( - \beta_{3} + 2 \beta_{2} + 6 \beta_1 + 2) q^{93} + ( - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{95} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{97} + ( - 6 \beta_{3} + 4 \beta_{2} - 2 \beta_1 + 2) q^{99}+O(q^{100})$$ q + (b1 - 1) * q^3 + q^5 + (b1 + 1) * q^7 + (b2 + 3) * q^9 + (-b3 + b2 - 2) * q^11 + (-b3 - 2) * q^13 + (b1 - 1) * q^15 + (-b3 + 2) * q^17 + (-b3 + b2 + b1 - 1) * q^19 + (b2 + 2*b1 + 4) * q^21 - 2*b1 * q^23 + q^25 + (b3 - b2 + 2*b1 - 2) * q^27 + 2*b1 * q^29 + (b3 + b2 + b1 + 1) * q^31 + (3*b3 - 3*b2 - 2*b1) * q^33 + (b1 + 1) * q^35 + q^37 + (2*b3 - 2*b2 - 4*b1 + 2) * q^39 + (b3 - b2 + 8) * q^41 + (-2*b2 - 2*b1) * q^43 + (b2 + 3) * q^45 + (-2*b3 + b1 + 1) * q^47 + (b2 + 4*b1 - 1) * q^49 + (2*b3 - 2*b2 - 2) * q^51 + (-b3 + b2 + 2*b1 - 2) * q^53 + (-b3 + b2 - 2) * q^55 + (3*b3 - 2*b2 + 4) * q^57 + (b3 + b2 - b1 + 3) * q^59 + 2 * q^61 + (b3 + b2 + 5*b1 + 1) * q^63 + (-b3 - 2) * q^65 + (b3 - b2 + b1 - 3) * q^67 + (-2*b2 - 2*b1 - 10) * q^69 + (-b3 + b2 - 2*b1 + 4) * q^71 + (-b3 + b2 + 4*b1) * q^73 + (b1 - 1) * q^75 + (b3 - b2 - 2*b1 - 4) * q^77 + (-b3 + b2 + b1 + 5) * q^79 + (-3*b3 + 2*b2 + 5) * q^81 + (-2*b2 - b1 - 7) * q^83 + (-b3 + 2) * q^85 + (2*b2 + 2*b1 + 10) * q^87 + (-2*b3 - 2) * q^89 + (-2*b2 - 4*b1 - 2) * q^91 + (-b3 + 2*b2 + 6*b1 + 2) * q^93 + (-b3 + b2 + b1 - 1) * q^95 + (-2*b2 - 2*b1 + 4) * q^97 + (-6*b3 + 4*b2 - 2*b1 + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 3 q^{3} + 4 q^{5} + 5 q^{7} + 13 q^{9}+O(q^{10})$$ 4 * q - 3 * q^3 + 4 * q^5 + 5 * q^7 + 13 * q^9 $$4 q - 3 q^{3} + 4 q^{5} + 5 q^{7} + 13 q^{9} - 5 q^{11} - 6 q^{13} - 3 q^{15} + 10 q^{17} + 19 q^{21} - 2 q^{23} + 4 q^{25} - 9 q^{27} + 2 q^{29} + 4 q^{31} - 11 q^{33} + 5 q^{35} + 4 q^{37} - 2 q^{39} + 29 q^{41} - 4 q^{43} + 13 q^{45} + 9 q^{47} + q^{49} - 14 q^{51} - 3 q^{53} - 5 q^{55} + 8 q^{57} + 10 q^{59} + 8 q^{61} + 8 q^{63} - 6 q^{65} - 14 q^{67} - 44 q^{69} + 17 q^{71} + 7 q^{73} - 3 q^{75} - 21 q^{77} + 24 q^{79} + 28 q^{81} - 31 q^{83} + 10 q^{85} + 44 q^{87} - 4 q^{89} - 14 q^{91} + 18 q^{93} + 12 q^{97} + 22 q^{99}+O(q^{100})$$ 4 * q - 3 * q^3 + 4 * q^5 + 5 * q^7 + 13 * q^9 - 5 * q^11 - 6 * q^13 - 3 * q^15 + 10 * q^17 + 19 * q^21 - 2 * q^23 + 4 * q^25 - 9 * q^27 + 2 * q^29 + 4 * q^31 - 11 * q^33 + 5 * q^35 + 4 * q^37 - 2 * q^39 + 29 * q^41 - 4 * q^43 + 13 * q^45 + 9 * q^47 + q^49 - 14 * q^51 - 3 * q^53 - 5 * q^55 + 8 * q^57 + 10 * q^59 + 8 * q^61 + 8 * q^63 - 6 * q^65 - 14 * q^67 - 44 * q^69 + 17 * q^71 + 7 * q^73 - 3 * q^75 - 21 * q^77 + 24 * q^79 + 28 * q^81 - 31 * q^83 + 10 * q^85 + 44 * q^87 - 4 * q^89 - 14 * q^91 + 18 * q^93 + 12 * q^97 + 22 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 11x^{2} - 3x + 12$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2\nu - 5$$ v^2 - 2*v - 5 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 2\nu^{2} - 7\nu + 2$$ v^3 - 2*v^2 - 7*v + 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2\beta _1 + 5$$ b2 + 2*b1 + 5 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 2\beta_{2} + 11\beta _1 + 8$$ b3 + 2*b2 + 11*b1 + 8

## 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
 −2.24418 −1.51951 0.914132 3.84956
0 −3.24418 0 1.00000 0 −1.24418 0 7.52471 0
1.2 0 −2.51951 0 1.00000 0 −0.519509 0 3.34793 0
1.3 0 −0.0858680 0 1.00000 0 1.91413 0 −2.99263 0
1.4 0 2.84956 0 1.00000 0 4.84956 0 5.11999 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.v 4
4.b odd 2 1 740.2.a.f 4
12.b even 2 1 6660.2.a.r 4
20.d odd 2 1 3700.2.a.j 4
20.e even 4 2 3700.2.d.i 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.a.f 4 4.b odd 2 1
2960.2.a.v 4 1.a even 1 1 trivial
3700.2.a.j 4 20.d odd 2 1
3700.2.d.i 8 20.e even 4 2
6660.2.a.r 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2960))$$:

 $$T_{3}^{4} + 3T_{3}^{3} - 8T_{3}^{2} - 24T_{3} - 2$$ T3^4 + 3*T3^3 - 8*T3^2 - 24*T3 - 2 $$T_{7}^{4} - 5T_{7}^{3} - 2T_{7}^{2} + 12T_{7} + 6$$ T7^4 - 5*T7^3 - 2*T7^2 + 12*T7 + 6 $$T_{13}^{4} + 6T_{13}^{3} - 20T_{13}^{2} - 84T_{13} + 160$$ T13^4 + 6*T13^3 - 20*T13^2 - 84*T13 + 160

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 3 T^{3} - 8 T^{2} - 24 T - 2$$
$5$ $$(T - 1)^{4}$$
$7$ $$T^{4} - 5 T^{3} - 2 T^{2} + 12 T + 6$$
$11$ $$T^{4} + 5 T^{3} - 32 T^{2} - 192 T - 240$$
$13$ $$T^{4} + 6 T^{3} - 20 T^{2} - 84 T + 160$$
$17$ $$T^{4} - 10 T^{3} + 4 T^{2} + 108 T + 48$$
$19$ $$T^{4} - 38 T^{2} + 54 T + 64$$
$23$ $$T^{4} + 2 T^{3} - 44 T^{2} + 24 T + 192$$
$29$ $$T^{4} - 2 T^{3} - 44 T^{2} - 24 T + 192$$
$31$ $$T^{4} - 4 T^{3} - 90 T^{2} + 350 T + 148$$
$37$ $$(T - 1)^{4}$$
$41$ $$T^{4} - 29 T^{3} + 274 T^{2} + \cdots - 168$$
$43$ $$T^{4} + 4 T^{3} - 128 T^{2} + \cdots + 1296$$
$47$ $$T^{4} - 9 T^{3} - 102 T^{2} + 720 T + 54$$
$53$ $$T^{4} + 3 T^{3} - 54 T^{2} + 36 T + 72$$
$59$ $$T^{4} - 10 T^{3} - 62 T^{2} + \cdots - 1968$$
$61$ $$(T - 2)^{4}$$
$67$ $$T^{4} + 14 T^{3} + 6 T^{2} - 22 T - 8$$
$71$ $$T^{4} - 17 T^{3} - 8 T^{2} + 240 T - 288$$
$73$ $$T^{4} - 7 T^{3} - 146 T^{2} + \cdots + 360$$
$79$ $$T^{4} - 24 T^{3} + 178 T^{2} + \cdots - 332$$
$83$ $$T^{4} + 31 T^{3} + 244 T^{2} + \cdots - 5238$$
$89$ $$T^{4} + 4 T^{3} - 128 T^{2} + \cdots + 3504$$
$97$ $$T^{4} - 12 T^{3} - 80 T^{2} + \cdots + 400$$