# Properties

 Label 2960.2.a.y Level $2960$ Weight $2$ Character orbit 2960.a Self dual yes Analytic conductor $23.636$ Analytic rank $0$ Dimension $5$ 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: $$5$$ Coefficient field: 5.5.6397264.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - 10x^{3} - 2x^{2} + 14x - 4$$ x^5 - 10*x^3 - 2*x^2 + 14*x - 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} - q^{5} + (\beta_{3} - \beta_1 - 1) q^{7} + ( - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{9}+O(q^{10})$$ q + b1 * q^3 - q^5 + (b3 - b1 - 1) * q^7 + (-b3 + b2 + b1 + 1) * q^9 $$q + \beta_1 q^{3} - q^{5} + (\beta_{3} - \beta_1 - 1) q^{7} + ( - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{9} + ( - \beta_{4} - 1) q^{11} + (\beta_{3} + \beta_{2} - \beta_1) q^{13} - \beta_1 q^{15} + ( - \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{17} + (2 \beta_{2} + \beta_1) q^{19} + (\beta_{3} - \beta_{2} - 3 \beta_1 - 2) q^{21} + (2 \beta_1 - 2) q^{23} + q^{25} + (\beta_{4} - \beta_{3} + 2 \beta_1 + 2) q^{27} + (\beta_{4} - 2 \beta_{2} + 2 \beta_1 + 5) q^{29} + (\beta_{3} + \beta_1 - 3) q^{31} + ( - 2 \beta_{2} - 2 \beta_1 + 2) q^{33} + ( - \beta_{3} + \beta_1 + 1) q^{35} + q^{37} + (\beta_{4} + \beta_{3} - 2 \beta_{2} - 2) q^{39} + ( - \beta_{4} + 2 \beta_{3} + 1) q^{41} + ( - \beta_{4} - 2 \beta_{3} + 3) q^{43} + (\beta_{3} - \beta_{2} - \beta_1 - 1) q^{45} + (\beta_{4} + \beta_{3} + 3 \beta_1) q^{47} + (\beta_{4} + \beta_{3} + \beta_{2} + 3 \beta_1) q^{49} + (\beta_{4} - \beta_{3} - 2 \beta_{2} + 4 \beta_1 + 4) q^{51} + ( - \beta_{4} - 2 \beta_1 + 5) q^{53} + (\beta_{4} + 1) q^{55} + (2 \beta_{4} - \beta_{3} - \beta_{2} + 5 \beta_1 + 4) q^{57} + ( - \beta_{4} - \beta_{3} + 2 \beta_{2} - \beta_1 - 4) q^{59} + (\beta_{4} + 2 \beta_{2} + 5) q^{61} + ( - \beta_{4} - 2 \beta_{2} - 5 \beta_1 - 7) q^{63} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{65} + ( - 2 \beta_{2} - \beta_1 + 2) q^{67} + ( - 2 \beta_{3} + 2 \beta_{2} + 8) q^{69} + ( - 2 \beta_{2} - 2 \beta_1 - 6) q^{71} + ( - 2 \beta_{2} + 4) q^{73} + \beta_1 q^{75} + (\beta_{4} - 2 \beta_{3} + 2 \beta_1 + 3) q^{77} + ( - \beta_{4} + \beta_{3} - 3 \beta_1 - 4) q^{79} + (\beta_{3} + \beta_{2} + 3 \beta_1 + 1) q^{81} + (2 \beta_{2} + 3 \beta_1 + 2) q^{83} + (\beta_{4} + \beta_{3} - \beta_{2} - \beta_1 - 1) q^{85} + ( - 2 \beta_{4} - 2 \beta_{3} + 6 \beta_{2} + 4 \beta_1 + 6) q^{87} + ( - 2 \beta_{4} + 2 \beta_{2} + 2 \beta_1 + 6) q^{89} + (\beta_{4} + 3 \beta_{3} + 6) q^{91} + ( - \beta_{3} + \beta_{2} - 3 \beta_1 + 6) q^{93} + ( - 2 \beta_{2} - \beta_1) q^{95} + ( - \beta_{4} + 4 \beta_{2} - 1) q^{97} + (\beta_{4} + 2 \beta_{3} - 4 \beta_1 - 5) q^{99}+O(q^{100})$$ q + b1 * q^3 - q^5 + (b3 - b1 - 1) * q^7 + (-b3 + b2 + b1 + 1) * q^9 + (-b4 - 1) * q^11 + (b3 + b2 - b1) * q^13 - b1 * q^15 + (-b4 - b3 + b2 + b1 + 1) * q^17 + (2*b2 + b1) * q^19 + (b3 - b2 - 3*b1 - 2) * q^21 + (2*b1 - 2) * q^23 + q^25 + (b4 - b3 + 2*b1 + 2) * q^27 + (b4 - 2*b2 + 2*b1 + 5) * q^29 + (b3 + b1 - 3) * q^31 + (-2*b2 - 2*b1 + 2) * q^33 + (-b3 + b1 + 1) * q^35 + q^37 + (b4 + b3 - 2*b2 - 2) * q^39 + (-b4 + 2*b3 + 1) * q^41 + (-b4 - 2*b3 + 3) * q^43 + (b3 - b2 - b1 - 1) * q^45 + (b4 + b3 + 3*b1) * q^47 + (b4 + b3 + b2 + 3*b1) * q^49 + (b4 - b3 - 2*b2 + 4*b1 + 4) * q^51 + (-b4 - 2*b1 + 5) * q^53 + (b4 + 1) * q^55 + (2*b4 - b3 - b2 + 5*b1 + 4) * q^57 + (-b4 - b3 + 2*b2 - b1 - 4) * q^59 + (b4 + 2*b2 + 5) * q^61 + (-b4 - 2*b2 - 5*b1 - 7) * q^63 + (-b3 - b2 + b1) * q^65 + (-2*b2 - b1 + 2) * q^67 + (-2*b3 + 2*b2 + 8) * q^69 + (-2*b2 - 2*b1 - 6) * q^71 + (-2*b2 + 4) * q^73 + b1 * q^75 + (b4 - 2*b3 + 2*b1 + 3) * q^77 + (-b4 + b3 - 3*b1 - 4) * q^79 + (b3 + b2 + 3*b1 + 1) * q^81 + (2*b2 + 3*b1 + 2) * q^83 + (b4 + b3 - b2 - b1 - 1) * q^85 + (-2*b4 - 2*b3 + 6*b2 + 4*b1 + 6) * q^87 + (-2*b4 + 2*b2 + 2*b1 + 6) * q^89 + (b4 + 3*b3 + 6) * q^91 + (-b3 + b2 - 3*b1 + 6) * q^93 + (-2*b2 - b1) * q^95 + (-b4 + 4*b2 - 1) * q^97 + (b4 + 2*b3 - 4*b1 - 5) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - 5 q^{5} - 3 q^{7} + 5 q^{9}+O(q^{10})$$ 5 * q - 5 * q^5 - 3 * q^7 + 5 * q^9 $$5 q - 5 q^{5} - 3 q^{7} + 5 q^{9} - 3 q^{11} + 4 q^{13} + 7 q^{17} + 4 q^{19} - 10 q^{21} - 10 q^{23} + 5 q^{25} + 6 q^{27} + 19 q^{29} - 13 q^{31} + 6 q^{33} + 3 q^{35} + 5 q^{37} - 14 q^{39} + 11 q^{41} + 13 q^{43} - 5 q^{45} + 2 q^{49} + 12 q^{51} + 27 q^{53} + 3 q^{55} + 12 q^{57} - 16 q^{59} + 27 q^{61} - 37 q^{63} - 4 q^{65} + 6 q^{67} + 40 q^{69} - 34 q^{71} + 16 q^{73} + 9 q^{77} - 16 q^{79} + 9 q^{81} + 14 q^{83} - 7 q^{85} + 42 q^{87} + 38 q^{89} + 34 q^{91} + 30 q^{93} - 4 q^{95} + 5 q^{97} - 23 q^{99}+O(q^{100})$$ 5 * q - 5 * q^5 - 3 * q^7 + 5 * q^9 - 3 * q^11 + 4 * q^13 + 7 * q^17 + 4 * q^19 - 10 * q^21 - 10 * q^23 + 5 * q^25 + 6 * q^27 + 19 * q^29 - 13 * q^31 + 6 * q^33 + 3 * q^35 + 5 * q^37 - 14 * q^39 + 11 * q^41 + 13 * q^43 - 5 * q^45 + 2 * q^49 + 12 * q^51 + 27 * q^53 + 3 * q^55 + 12 * q^57 - 16 * q^59 + 27 * q^61 - 37 * q^63 - 4 * q^65 + 6 * q^67 + 40 * q^69 - 34 * q^71 + 16 * q^73 + 9 * q^77 - 16 * q^79 + 9 * q^81 + 14 * q^83 - 7 * q^85 + 42 * q^87 + 38 * q^89 + 34 * q^91 + 30 * q^93 - 4 * q^95 + 5 * q^97 - 23 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 10x^{3} - 2x^{2} + 14x - 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{4} - 8\nu^{2} - 4\nu + 4 ) / 2$$ (v^4 - 8*v^2 - 4*v + 4) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{4} - 10\nu^{2} - 2\nu + 12 ) / 2$$ (v^4 - 10*v^2 - 2*v + 12) / 2 $$\beta_{4}$$ $$=$$ $$( \nu^{4} + 2\nu^{3} - 10\nu^{2} - 18\nu + 8 ) / 2$$ (v^4 + 2*v^3 - 10*v^2 - 18*v + 8) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{3} + \beta_{2} + \beta _1 + 4$$ -b3 + b2 + b1 + 4 $$\nu^{3}$$ $$=$$ $$\beta_{4} - \beta_{3} + 8\beta _1 + 2$$ b4 - b3 + 8*b1 + 2 $$\nu^{4}$$ $$=$$ $$-8\beta_{3} + 10\beta_{2} + 12\beta _1 + 28$$ -8*b3 + 10*b2 + 12*b1 + 28

## 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
 −2.65435 −1.64390 0.325094 0.925120 3.04803
0 −2.65435 0 −1.00000 0 −0.0991320 0 4.04557 0
1.2 0 −1.64390 0 −1.00000 0 −1.57272 0 −0.297599 0
1.3 0 0.325094 0 −1.00000 0 3.82696 0 −2.89431 0
1.4 0 0.925120 0 −1.00000 0 −0.763238 0 −2.14415 0
1.5 0 3.04803 0 −1.00000 0 −4.39187 0 6.29050 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 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.y 5
4.b odd 2 1 1480.2.a.i 5
20.d odd 2 1 7400.2.a.p 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1480.2.a.i 5 4.b odd 2 1
2960.2.a.y 5 1.a even 1 1 trivial
7400.2.a.p 5 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2960))$$:

 $$T_{3}^{5} - 10T_{3}^{3} - 2T_{3}^{2} + 14T_{3} - 4$$ T3^5 - 10*T3^3 - 2*T3^2 + 14*T3 - 4 $$T_{7}^{5} + 3T_{7}^{4} - 14T_{7}^{3} - 40T_{7}^{2} - 24T_{7} - 2$$ T7^5 + 3*T7^4 - 14*T7^3 - 40*T7^2 - 24*T7 - 2 $$T_{13}^{5} - 4T_{13}^{4} - 28T_{13}^{3} + 44T_{13}^{2} + 292T_{13} + 272$$ T13^5 - 4*T13^4 - 28*T13^3 + 44*T13^2 + 292*T13 + 272

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5}$$
$3$ $$T^{5} - 10 T^{3} - 2 T^{2} + 14 T - 4$$
$5$ $$(T + 1)^{5}$$
$7$ $$T^{5} + 3 T^{4} - 14 T^{3} - 40 T^{2} + \cdots - 2$$
$11$ $$T^{5} + 3 T^{4} - 36 T^{3} - 144 T^{2} + \cdots + 32$$
$13$ $$T^{5} - 4 T^{4} - 28 T^{3} + 44 T^{2} + \cdots + 272$$
$17$ $$T^{5} - 7 T^{4} - 32 T^{3} + \cdots - 2188$$
$19$ $$T^{5} - 4 T^{4} - 58 T^{3} + \cdots - 2068$$
$23$ $$T^{5} + 10 T^{4} - 176 T^{2} + \cdots - 32$$
$29$ $$T^{5} - 19 T^{4} + 24 T^{3} + \cdots + 18992$$
$31$ $$T^{5} + 13 T^{4} + 30 T^{3} - 104 T^{2} + \cdots - 34$$
$37$ $$(T - 1)^{5}$$
$41$ $$T^{5} - 11 T^{4} - 88 T^{3} + \cdots + 1504$$
$43$ $$T^{5} - 13 T^{4} - 16 T^{3} + \cdots + 3056$$
$47$ $$T^{5} - 146 T^{3} - 18 T^{2} + \cdots + 7112$$
$53$ $$T^{5} - 27 T^{4} + 224 T^{3} + \cdots + 4688$$
$59$ $$T^{5} + 16 T^{4} - 6 T^{3} + \cdots - 13088$$
$61$ $$T^{5} - 27 T^{4} + 164 T^{3} + \cdots + 42928$$
$67$ $$T^{5} - 6 T^{4} - 50 T^{3} + 178 T^{2} + \cdots + 152$$
$71$ $$T^{5} + 34 T^{4} + 376 T^{3} + \cdots - 2432$$
$73$ $$T^{5} - 16 T^{4} + 40 T^{3} + \cdots + 128$$
$79$ $$T^{5} + 16 T^{4} - 10 T^{3} + \cdots + 224$$
$83$ $$T^{5} - 14 T^{4} - 50 T^{3} + \cdots - 968$$
$89$ $$T^{5} - 38 T^{4} + 360 T^{3} + \cdots + 174496$$
$97$ $$T^{5} - 5 T^{4} - 228 T^{3} + \cdots - 11552$$