# Properties

 Label 2960.2.a.t Level $2960$ Weight $2$ Character orbit 2960.a Self dual yes Analytic conductor $23.636$ Analytic rank $0$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.148.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} - q^{5} + (\beta_{2} - 2 \beta_1) q^{7} + ( - \beta_{2} - \beta_1) q^{9}+O(q^{10})$$ q + b2 * q^3 - q^5 + (b2 - 2*b1) * q^7 + (-b2 - b1) * q^9 $$q + \beta_{2} q^{3} - q^{5} + (\beta_{2} - 2 \beta_1) q^{7} + ( - \beta_{2} - \beta_1) q^{9} + ( - 2 \beta_1 + 2) q^{11} + ( - \beta_{2} - 3 \beta_1 + 3) q^{13} - \beta_{2} q^{15} + (3 \beta_{2} + \beta_1 + 3) q^{17} + (\beta_{2} - 6) q^{19} + ( - \beta_{2} - 3 \beta_1 + 5) q^{21} + 2 \beta_1 q^{23} + q^{25} + ( - 2 \beta_{2} - 2) q^{27} + (2 \beta_{2} + 4 \beta_1 - 2) q^{29} + 3 \beta_{2} q^{31} + (2 \beta_{2} - 2 \beta_1 + 2) q^{33} + ( - \beta_{2} + 2 \beta_1) q^{35} - q^{37} + (4 \beta_{2} - 2 \beta_1) q^{39} + ( - 2 \beta_{2} - 2) q^{41} + ( - 2 \beta_{2} - 4 \beta_1 - 2) q^{43} + (\beta_{2} + \beta_1) q^{45} + ( - \beta_{2} + 4 \beta_1 - 6) q^{47} + (3 \beta_{2} - \beta_1 + 8) q^{49} + ( - 2 \beta_1 + 8) q^{51} + (2 \beta_{2} + 6 \beta_1) q^{53} + (2 \beta_1 - 2) q^{55} + ( - 7 \beta_{2} - \beta_1 + 3) q^{57} + (5 \beta_{2} + 4 \beta_1 - 4) q^{59} + (8 \beta_{2} + 4 \beta_1 + 2) q^{61} + (3 \beta_{2} + 4 \beta_1) q^{63} + (\beta_{2} + 3 \beta_1 - 3) q^{65} + ( - 7 \beta_{2} - 2 \beta_1 - 6) q^{67} + (2 \beta_1 - 2) q^{69} + ( - 2 \beta_{2} + 2 \beta_1 + 10) q^{71} + 2 \beta_1 q^{73} + \beta_{2} q^{75} + (6 \beta_{2} - 2 \beta_1 + 10) q^{77} + ( - 3 \beta_{2} + 2 \beta_1 + 2) q^{79} + (3 \beta_{2} + 5 \beta_1 - 6) q^{81} + (\beta_{2} - 2 \beta_1 + 6) q^{83} + ( - 3 \beta_{2} - \beta_1 - 3) q^{85} + ( - 4 \beta_{2} + 2 \beta_1 + 2) q^{87} + ( - 6 \beta_{2} - 6 \beta_1) q^{89} + (10 \beta_{2} + 10) q^{91} + ( - 3 \beta_{2} - 3 \beta_1 + 9) q^{93} + ( - \beta_{2} + 6) q^{95} + ( - 2 \beta_{2} + 6) q^{97} + (2 \beta_1 + 2) q^{99}+O(q^{100})$$ q + b2 * q^3 - q^5 + (b2 - 2*b1) * q^7 + (-b2 - b1) * q^9 + (-2*b1 + 2) * q^11 + (-b2 - 3*b1 + 3) * q^13 - b2 * q^15 + (3*b2 + b1 + 3) * q^17 + (b2 - 6) * q^19 + (-b2 - 3*b1 + 5) * q^21 + 2*b1 * q^23 + q^25 + (-2*b2 - 2) * q^27 + (2*b2 + 4*b1 - 2) * q^29 + 3*b2 * q^31 + (2*b2 - 2*b1 + 2) * q^33 + (-b2 + 2*b1) * q^35 - q^37 + (4*b2 - 2*b1) * q^39 + (-2*b2 - 2) * q^41 + (-2*b2 - 4*b1 - 2) * q^43 + (b2 + b1) * q^45 + (-b2 + 4*b1 - 6) * q^47 + (3*b2 - b1 + 8) * q^49 + (-2*b1 + 8) * q^51 + (2*b2 + 6*b1) * q^53 + (2*b1 - 2) * q^55 + (-7*b2 - b1 + 3) * q^57 + (5*b2 + 4*b1 - 4) * q^59 + (8*b2 + 4*b1 + 2) * q^61 + (3*b2 + 4*b1) * q^63 + (b2 + 3*b1 - 3) * q^65 + (-7*b2 - 2*b1 - 6) * q^67 + (2*b1 - 2) * q^69 + (-2*b2 + 2*b1 + 10) * q^71 + 2*b1 * q^73 + b2 * q^75 + (6*b2 - 2*b1 + 10) * q^77 + (-3*b2 + 2*b1 + 2) * q^79 + (3*b2 + 5*b1 - 6) * q^81 + (b2 - 2*b1 + 6) * q^83 + (-3*b2 - b1 - 3) * q^85 + (-4*b2 + 2*b1 + 2) * q^87 + (-6*b2 - 6*b1) * q^89 + (10*b2 + 10) * q^91 + (-3*b2 - 3*b1 + 9) * q^93 + (-b2 + 6) * q^95 + (-2*b2 + 6) * q^97 + (2*b1 + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{5} - 2 q^{7} - q^{9}+O(q^{10})$$ 3 * q - 3 * q^5 - 2 * q^7 - q^9 $$3 q - 3 q^{5} - 2 q^{7} - q^{9} + 4 q^{11} + 6 q^{13} + 10 q^{17} - 18 q^{19} + 12 q^{21} + 2 q^{23} + 3 q^{25} - 6 q^{27} - 2 q^{29} + 4 q^{33} + 2 q^{35} - 3 q^{37} - 2 q^{39} - 6 q^{41} - 10 q^{43} + q^{45} - 14 q^{47} + 23 q^{49} + 22 q^{51} + 6 q^{53} - 4 q^{55} + 8 q^{57} - 8 q^{59} + 10 q^{61} + 4 q^{63} - 6 q^{65} - 20 q^{67} - 4 q^{69} + 32 q^{71} + 2 q^{73} + 28 q^{77} + 8 q^{79} - 13 q^{81} + 16 q^{83} - 10 q^{85} + 8 q^{87} - 6 q^{89} + 30 q^{91} + 24 q^{93} + 18 q^{95} + 18 q^{97} + 8 q^{99}+O(q^{100})$$ 3 * q - 3 * q^5 - 2 * q^7 - q^9 + 4 * q^11 + 6 * q^13 + 10 * q^17 - 18 * q^19 + 12 * q^21 + 2 * q^23 + 3 * q^25 - 6 * q^27 - 2 * q^29 + 4 * q^33 + 2 * q^35 - 3 * q^37 - 2 * q^39 - 6 * q^41 - 10 * q^43 + q^45 - 14 * q^47 + 23 * q^49 + 22 * q^51 + 6 * q^53 - 4 * q^55 + 8 * q^57 - 8 * q^59 + 10 * q^61 + 4 * q^63 - 6 * q^65 - 20 * q^67 - 4 * q^69 + 32 * q^71 + 2 * q^73 + 28 * q^77 + 8 * q^79 - 13 * q^81 + 16 * q^83 - 10 * q^85 + 8 * q^87 - 6 * q^89 + 30 * q^91 + 24 * q^93 + 18 * q^95 + 18 * q^97 + 8 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 3x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$ v^2 - v - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 2$$ b2 + b1 + 2

## 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
 0.311108 2.17009 −1.48119
0 −2.21432 0 −1.00000 0 −2.83654 0 1.90321 0
1.2 0 0.539189 0 −1.00000 0 −3.80098 0 −2.70928 0
1.3 0 1.67513 0 −1.00000 0 4.63752 0 −0.193937 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.t 3
4.b odd 2 1 740.2.a.e 3
12.b even 2 1 6660.2.a.q 3
20.d odd 2 1 3700.2.a.i 3
20.e even 4 2 3700.2.d.h 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.a.e 3 4.b odd 2 1
2960.2.a.t 3 1.a even 1 1 trivial
3700.2.a.i 3 20.d odd 2 1
3700.2.d.h 6 20.e even 4 2
6660.2.a.q 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2960))$$:

 $$T_{3}^{3} - 4T_{3} + 2$$ T3^3 - 4*T3 + 2 $$T_{7}^{3} + 2T_{7}^{2} - 20T_{7} - 50$$ T7^3 + 2*T7^2 - 20*T7 - 50 $$T_{13}^{3} - 6T_{13}^{2} - 16T_{13} + 100$$ T13^3 - 6*T13^2 - 16*T13 + 100

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} - 4T + 2$$
$5$ $$(T + 1)^{3}$$
$7$ $$T^{3} + 2 T^{2} - 20 T - 50$$
$11$ $$T^{3} - 4 T^{2} - 8 T + 16$$
$13$ $$T^{3} - 6 T^{2} - 16 T + 100$$
$17$ $$T^{3} - 10T^{2} + 148$$
$19$ $$T^{3} + 18 T^{2} + 104 T + 194$$
$23$ $$T^{3} - 2 T^{2} - 12 T + 8$$
$29$ $$T^{3} + 2 T^{2} - 52 T - 184$$
$31$ $$T^{3} - 36T + 54$$
$37$ $$(T + 1)^{3}$$
$41$ $$T^{3} + 6 T^{2} - 4 T - 40$$
$43$ $$T^{3} + 10 T^{2} - 20 T + 8$$
$47$ $$T^{3} + 14T^{2} - 74$$
$53$ $$T^{3} - 6 T^{2} - 100 T - 200$$
$59$ $$T^{3} + 8 T^{2} - 92 T - 158$$
$61$ $$T^{3} - 10 T^{2} - 212 T + 2056$$
$67$ $$T^{3} + 20 T^{2} - 48 T - 1850$$
$71$ $$T^{3} - 32 T^{2} + 304 T - 736$$
$73$ $$T^{3} - 2 T^{2} - 12 T + 8$$
$79$ $$T^{3} - 8 T^{2} - 40 T + 262$$
$83$ $$T^{3} - 16 T^{2} + 64 T - 74$$
$89$ $$T^{3} + 6 T^{2} - 180 T - 216$$
$97$ $$T^{3} - 18 T^{2} + 92 T - 136$$