# Properties

 Label 2960.2.a.r Level $2960$ Weight $2$ Character orbit 2960.a Self dual yes Analytic conductor $23.636$ Analytic rank $1$ 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: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.316.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 4x + 2$$ x^3 - x^2 - 4*x + 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1480) 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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{3} - q^{5} + \beta_1 q^{7} + \beta_{2} q^{9}+O(q^{10})$$ q - b1 * q^3 - q^5 + b1 * q^7 + b2 * q^9 $$q - \beta_1 q^{3} - q^{5} + \beta_1 q^{7} + \beta_{2} q^{9} + (\beta_{2} + 1) q^{11} - 2 \beta_1 q^{13} + \beta_1 q^{15} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{17} + ( - \beta_{2} + \beta_1 - 1) q^{19} + ( - \beta_{2} - 3) q^{21} + ( - 2 \beta_{2} + 2) q^{23} + q^{25} + ( - \beta_{2} + 2 \beta_1 - 1) q^{27} - 2 q^{29} + (\beta_{2} + 3 \beta_1 + 1) q^{31} + ( - \beta_{2} - 2 \beta_1 - 1) q^{33} - \beta_1 q^{35} - q^{37} + (2 \beta_{2} + 6) q^{39} + (\beta_{2} + 2 \beta_1 - 5) q^{41} + ( - 2 \beta_{2} - 2) q^{43} - \beta_{2} q^{45} + \beta_1 q^{47} + (\beta_{2} - 4) q^{49} + (4 \beta_1 - 4) q^{51} + (3 \beta_{2} - 2 \beta_1 - 3) q^{53} + ( - \beta_{2} - 1) q^{55} + (2 \beta_1 - 2) q^{57} + (\beta_{2} - 5 \beta_1 + 5) q^{59} + (2 \beta_{2} - 4) q^{61} + (\beta_{2} + \beta_1 + 1) q^{63} + 2 \beta_1 q^{65} + (\beta_{2} - \beta_1 - 3) q^{67} + (2 \beta_{2} + 2) q^{69} + (\beta_{2} + 2 \beta_1 + 5) q^{71} + (\beta_{2} + 2 \beta_1 - 9) q^{73} - \beta_1 q^{75} + (\beta_{2} + 2 \beta_1 + 1) q^{77} + (\beta_{2} - 5 \beta_1 + 1) q^{79} + ( - 4 \beta_{2} + 2 \beta_1 - 5) q^{81} + (6 \beta_{2} - 3 \beta_1 + 6) q^{83} + (2 \beta_{2} - 2 \beta_1 + 2) q^{85} + 2 \beta_1 q^{87} + ( - 2 \beta_{2} + 4 \beta_1 - 4) q^{89} + ( - 2 \beta_{2} - 6) q^{91} + ( - 4 \beta_{2} - 2 \beta_1 - 10) q^{93} + (\beta_{2} - \beta_1 + 1) q^{95} + (2 \beta_{2} + 4 \beta_1 + 4) q^{97} + (2 \beta_1 + 4) q^{99}+O(q^{100})$$ q - b1 * q^3 - q^5 + b1 * q^7 + b2 * q^9 + (b2 + 1) * q^11 - 2*b1 * q^13 + b1 * q^15 + (-2*b2 + 2*b1 - 2) * q^17 + (-b2 + b1 - 1) * q^19 + (-b2 - 3) * q^21 + (-2*b2 + 2) * q^23 + q^25 + (-b2 + 2*b1 - 1) * q^27 - 2 * q^29 + (b2 + 3*b1 + 1) * q^31 + (-b2 - 2*b1 - 1) * q^33 - b1 * q^35 - q^37 + (2*b2 + 6) * q^39 + (b2 + 2*b1 - 5) * q^41 + (-2*b2 - 2) * q^43 - b2 * q^45 + b1 * q^47 + (b2 - 4) * q^49 + (4*b1 - 4) * q^51 + (3*b2 - 2*b1 - 3) * q^53 + (-b2 - 1) * q^55 + (2*b1 - 2) * q^57 + (b2 - 5*b1 + 5) * q^59 + (2*b2 - 4) * q^61 + (b2 + b1 + 1) * q^63 + 2*b1 * q^65 + (b2 - b1 - 3) * q^67 + (2*b2 + 2) * q^69 + (b2 + 2*b1 + 5) * q^71 + (b2 + 2*b1 - 9) * q^73 - b1 * q^75 + (b2 + 2*b1 + 1) * q^77 + (b2 - 5*b1 + 1) * q^79 + (-4*b2 + 2*b1 - 5) * q^81 + (6*b2 - 3*b1 + 6) * q^83 + (2*b2 - 2*b1 + 2) * q^85 + 2*b1 * q^87 + (-2*b2 + 4*b1 - 4) * q^89 + (-2*b2 - 6) * q^91 + (-4*b2 - 2*b1 - 10) * q^93 + (b2 - b1 + 1) * q^95 + (2*b2 + 4*b1 + 4) * q^97 + (2*b1 + 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{3} - 3 q^{5} + q^{7}+O(q^{10})$$ 3 * q - q^3 - 3 * q^5 + q^7 $$3 q - q^{3} - 3 q^{5} + q^{7} + 3 q^{11} - 2 q^{13} + q^{15} - 4 q^{17} - 2 q^{19} - 9 q^{21} + 6 q^{23} + 3 q^{25} - q^{27} - 6 q^{29} + 6 q^{31} - 5 q^{33} - q^{35} - 3 q^{37} + 18 q^{39} - 13 q^{41} - 6 q^{43} + q^{47} - 12 q^{49} - 8 q^{51} - 11 q^{53} - 3 q^{55} - 4 q^{57} + 10 q^{59} - 12 q^{61} + 4 q^{63} + 2 q^{65} - 10 q^{67} + 6 q^{69} + 17 q^{71} - 25 q^{73} - q^{75} + 5 q^{77} - 2 q^{79} - 13 q^{81} + 15 q^{83} + 4 q^{85} + 2 q^{87} - 8 q^{89} - 18 q^{91} - 32 q^{93} + 2 q^{95} + 16 q^{97} + 14 q^{99}+O(q^{100})$$ 3 * q - q^3 - 3 * q^5 + q^7 + 3 * q^11 - 2 * q^13 + q^15 - 4 * q^17 - 2 * q^19 - 9 * q^21 + 6 * q^23 + 3 * q^25 - q^27 - 6 * q^29 + 6 * q^31 - 5 * q^33 - q^35 - 3 * q^37 + 18 * q^39 - 13 * q^41 - 6 * q^43 + q^47 - 12 * q^49 - 8 * q^51 - 11 * q^53 - 3 * q^55 - 4 * q^57 + 10 * q^59 - 12 * q^61 + 4 * q^63 + 2 * q^65 - 10 * q^67 + 6 * q^69 + 17 * q^71 - 25 * q^73 - q^75 + 5 * q^77 - 2 * q^79 - 13 * q^81 + 15 * q^83 + 4 * q^85 + 2 * q^87 - 8 * q^89 - 18 * q^91 - 32 * q^93 + 2 * q^95 + 16 * q^97 + 14 * q^99

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3

## 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
 2.34292 0.470683 −1.81361
0 −2.34292 0 −1.00000 0 2.34292 0 2.48929 0
1.2 0 −0.470683 0 −1.00000 0 0.470683 0 −2.77846 0
1.3 0 1.81361 0 −1.00000 0 −1.81361 0 0.289169 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.r 3
4.b odd 2 1 1480.2.a.e 3
20.d odd 2 1 7400.2.a.l 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1480.2.a.e 3 4.b odd 2 1
2960.2.a.r 3 1.a even 1 1 trivial
7400.2.a.l 3 20.d odd 2 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} + T^{2} - 4T - 2$$
$5$ $$(T + 1)^{3}$$
$7$ $$T^{3} - T^{2} - 4T + 2$$
$11$ $$T^{3} - 3 T^{2} - 4 T + 8$$
$13$ $$T^{3} + 2 T^{2} - 16 T - 16$$
$17$ $$T^{3} + 4 T^{2} - 24 T - 64$$
$19$ $$T^{3} + 2 T^{2} - 6 T - 8$$
$23$ $$T^{3} - 6 T^{2} - 16 T + 32$$
$29$ $$(T + 2)^{3}$$
$31$ $$T^{3} - 6 T^{2} - 46 T - 16$$
$37$ $$(T + 1)^{3}$$
$41$ $$T^{3} + 13 T^{2} + 24 T - 124$$
$43$ $$T^{3} + 6 T^{2} - 16 T - 64$$
$47$ $$T^{3} - T^{2} - 4T + 2$$
$53$ $$T^{3} + 11 T^{2} - 16 T - 4$$
$59$ $$T^{3} - 10 T^{2} - 62 T - 8$$
$61$ $$T^{3} + 12 T^{2} + 20 T - 32$$
$67$ $$T^{3} + 10 T^{2} + 26 T + 16$$
$71$ $$T^{3} - 17 T^{2} + 64 T - 64$$
$73$ $$T^{3} + 25 T^{2} + 176 T + 244$$
$79$ $$T^{3} + 2 T^{2} - 94 T - 352$$
$83$ $$T^{3} - 15 T^{2} - 144 T + 2214$$
$89$ $$T^{3} + 8 T^{2} - 44 T + 16$$
$97$ $$T^{3} - 16 T^{2} - 44 T + 16$$