Properties

 Label 2960.2.a.z 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: 5.5.935504.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - x^{4} - 8x^{3} + 4x^{2} + 8x + 2$$ x^5 - x^4 - 8*x^3 + 4*x^2 + 8*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 $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} - q^{5} - \beta_{3} q^{7} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{9}+O(q^{10})$$ q + b1 * q^3 - q^5 - b3 * q^7 + (-b3 + b2 + b1) * q^9 $$q + \beta_1 q^{3} - q^{5} - \beta_{3} q^{7} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{9} + (\beta_{4} + \beta_{2} - \beta_1 + 2) q^{11} + (\beta_{4} - \beta_{3} + 1) q^{13} - \beta_1 q^{15} + ( - \beta_{4} - \beta_{3} + 2 \beta_1 - 1) q^{17} + ( - 3 \beta_{4} - 2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{19} + (\beta_{3} - \beta_{2} + \beta_1 - 1) q^{21} + ( - 2 \beta_1 + 2) q^{23} + q^{25} + (\beta_{4} + \beta_{3} + \beta_{2} + 2) q^{27} + ( - 2 \beta_{3} - 2) q^{29} + (3 \beta_{4} + \beta_{3} - 3 \beta_{2} - \beta_1 + 2) q^{31} + (\beta_{4} + 2 \beta_{3} + \beta_{2} + 3 \beta_1 - 2) q^{33} + \beta_{3} q^{35} + q^{37} + (\beta_{3} + 3 \beta_1) q^{39} + (\beta_{4} + 2 \beta_{3} + \beta_{2} + \beta_1 - 2) q^{41} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{43} + (\beta_{3} - \beta_{2} - \beta_1) q^{45} + (\beta_{3} + 2 \beta_{2} + 2) q^{47} + ( - 2 \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 - 2) q^{49} + ( - \beta_{3} + \beta_1 + 4) q^{51} + ( - 3 \beta_{4} + \beta_{2} + \beta_1 - 4) q^{53} + ( - \beta_{4} - \beta_{2} + \beta_1 - 2) q^{55} + (\beta_{4} + \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 1) q^{57} + (3 \beta_{4} + \beta_{3} - 3 \beta_{2} + \beta_1 + 6) q^{59} + (4 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1 + 2) q^{61} + ( - \beta_{4} + \beta_{2} - 2 \beta_1 + 4) q^{63} + ( - \beta_{4} + \beta_{3} - 1) q^{65} + ( - 3 \beta_{4} - 2 \beta_{3} + \beta_{2} + 4 \beta_1 + 2) q^{67} + (2 \beta_{3} - 2 \beta_{2} - 6) q^{69} + ( - 3 \beta_{4} - 2 \beta_{3} + \beta_{2} + 3 \beta_1 + 2) q^{71} + ( - 3 \beta_{4} - 2 \beta_{3} + \beta_{2} + \beta_1 + 4) q^{73} + \beta_1 q^{75} + ( - \beta_{4} - 2 \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 2) q^{77} + ( - \beta_{4} - \beta_{3} - \beta_{2} + 3 \beta_1) q^{79} + (\beta_{4} + 3 \beta_{3} + 2) q^{81} + ( - 2 \beta_{4} - \beta_1 + 6) q^{83} + (\beta_{4} + \beta_{3} - 2 \beta_1 + 1) q^{85} + (2 \beta_{3} - 2 \beta_{2} - 2) q^{87} + (2 \beta_{4} - 4 \beta_{2} - 2 \beta_1 - 4) q^{89} + ( - 4 \beta_{4} - \beta_{3} + 2 \beta_{2} - \beta_1 + 6) q^{91} + ( - 3 \beta_{4} - 3 \beta_{3} + 1) q^{93} + (3 \beta_{4} + 2 \beta_{3} - \beta_{2} - 2 \beta_1) q^{95} + (2 \beta_{4} + 2 \beta_{3} + 2 \beta_1 + 6) q^{97} + ( - 2 \beta_{4} - 4 \beta_{3} + 4 \beta_{2} + 4 \beta_1 + 6) q^{99}+O(q^{100})$$ q + b1 * q^3 - q^5 - b3 * q^7 + (-b3 + b2 + b1) * q^9 + (b4 + b2 - b1 + 2) * q^11 + (b4 - b3 + 1) * q^13 - b1 * q^15 + (-b4 - b3 + 2*b1 - 1) * q^17 + (-3*b4 - 2*b3 + b2 + 2*b1) * q^19 + (b3 - b2 + b1 - 1) * q^21 + (-2*b1 + 2) * q^23 + q^25 + (b4 + b3 + b2 + 2) * q^27 + (-2*b3 - 2) * q^29 + (3*b4 + b3 - 3*b2 - b1 + 2) * q^31 + (b4 + 2*b3 + b2 + 3*b1 - 2) * q^33 + b3 * q^35 + q^37 + (b3 + 3*b1) * q^39 + (b4 + 2*b3 + b2 + b1 - 2) * q^41 + (-2*b2 - 2*b1 + 4) * q^43 + (b3 - b2 - b1) * q^45 + (b3 + 2*b2 + 2) * q^47 + (-2*b4 - b3 + b2 - b1 - 2) * q^49 + (-b3 + b1 + 4) * q^51 + (-3*b4 + b2 + b1 - 4) * q^53 + (-b4 - b2 + b1 - 2) * q^55 + (b4 + b3 - 2*b2 + 2*b1 + 1) * q^57 + (3*b4 + b3 - 3*b2 + b1 + 6) * q^59 + (4*b4 + 2*b3 - 4*b2 - 2*b1 + 2) * q^61 + (-b4 + b2 - 2*b1 + 4) * q^63 + (-b4 + b3 - 1) * q^65 + (-3*b4 - 2*b3 + b2 + 4*b1 + 2) * q^67 + (2*b3 - 2*b2 - 6) * q^69 + (-3*b4 - 2*b3 + b2 + 3*b1 + 2) * q^71 + (-3*b4 - 2*b3 + b2 + b1 + 4) * q^73 + b1 * q^75 + (-b4 - 2*b3 + 3*b2 - 3*b1 + 2) * q^77 + (-b4 - b3 - b2 + 3*b1) * q^79 + (b4 + 3*b3 + 2) * q^81 + (-2*b4 - b1 + 6) * q^83 + (b4 + b3 - 2*b1 + 1) * q^85 + (2*b3 - 2*b2 - 2) * q^87 + (2*b4 - 4*b2 - 2*b1 - 4) * q^89 + (-4*b4 - b3 + 2*b2 - b1 + 6) * q^91 + (-3*b4 - 3*b3 + 1) * q^93 + (3*b4 + 2*b3 - b2 - 2*b1) * q^95 + (2*b4 + 2*b3 + 2*b1 + 6) * q^97 + (-2*b4 - 4*b3 + 4*b2 + 4*b1 + 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + q^{3} - 5 q^{5} + q^{7} + 2 q^{9}+O(q^{10})$$ 5 * q + q^3 - 5 * q^5 + q^7 + 2 * q^9 $$5 q + q^{3} - 5 q^{5} + q^{7} + 2 q^{9} + 7 q^{11} + 4 q^{13} - q^{15} + 10 q^{19} - 5 q^{21} + 8 q^{23} + 5 q^{25} + 7 q^{27} - 8 q^{29} + 2 q^{31} - 11 q^{33} - q^{35} + 5 q^{37} + 2 q^{39} - 13 q^{41} + 18 q^{43} - 2 q^{45} + 9 q^{47} - 6 q^{49} + 22 q^{51} - 13 q^{53} - 7 q^{55} + 4 q^{57} + 24 q^{59} - 2 q^{61} + 20 q^{63} - 4 q^{65} + 22 q^{67} - 32 q^{69} + 21 q^{71} + 29 q^{73} + q^{75} + 11 q^{77} + 6 q^{79} + 5 q^{81} + 33 q^{83} - 12 q^{87} - 26 q^{89} + 38 q^{91} + 14 q^{93} - 10 q^{95} + 26 q^{97} + 42 q^{99}+O(q^{100})$$ 5 * q + q^3 - 5 * q^5 + q^7 + 2 * q^9 + 7 * q^11 + 4 * q^13 - q^15 + 10 * q^19 - 5 * q^21 + 8 * q^23 + 5 * q^25 + 7 * q^27 - 8 * q^29 + 2 * q^31 - 11 * q^33 - q^35 + 5 * q^37 + 2 * q^39 - 13 * q^41 + 18 * q^43 - 2 * q^45 + 9 * q^47 - 6 * q^49 + 22 * q^51 - 13 * q^53 - 7 * q^55 + 4 * q^57 + 24 * q^59 - 2 * q^61 + 20 * q^63 - 4 * q^65 + 22 * q^67 - 32 * q^69 + 21 * q^71 + 29 * q^73 + q^75 + 11 * q^77 + 6 * q^79 + 5 * q^81 + 33 * q^83 - 12 * q^87 - 26 * q^89 + 38 * q^91 + 14 * q^93 - 10 * q^95 + 26 * q^97 + 42 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - x^{4} - 8x^{3} + 4x^{2} + 8x + 2$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{4} - \nu^{3} - 7\nu^{2} + 4\nu + 3$$ v^4 - v^3 - 7*v^2 + 4*v + 3 $$\beta_{3}$$ $$=$$ $$\nu^{4} - \nu^{3} - 8\nu^{2} + 5\nu + 6$$ v^4 - v^3 - 8*v^2 + 5*v + 6 $$\beta_{4}$$ $$=$$ $$-2\nu^{4} + 3\nu^{3} + 15\nu^{2} - 15\nu - 11$$ -2*v^4 + 3*v^3 + 15*v^2 - 15*v - 11
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{3} + \beta_{2} + \beta _1 + 3$$ -b3 + b2 + b1 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{4} + \beta_{3} + \beta_{2} + 6\beta _1 + 2$$ b4 + b3 + b2 + 6*b1 + 2 $$\nu^{4}$$ $$=$$ $$\beta_{4} - 6\beta_{3} + 9\beta_{2} + 9\beta _1 + 20$$ b4 - 6*b3 + 9*b2 + 9*b1 + 20

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
 −2.44705 −0.550328 −0.355205 1.43118 2.92141
0 −2.44705 0 −1.00000 0 3.62974 0 2.98807 0
1.2 0 −0.550328 0 −1.00000 0 −1.08387 0 −2.69714 0
1.3 0 −0.355205 0 −1.00000 0 −3.27534 0 −2.87383 0
1.4 0 1.43118 0 −1.00000 0 1.96628 0 −0.951735 0
1.5 0 2.92141 0 −1.00000 0 −0.236809 0 5.53464 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.z 5
4.b odd 2 1 1480.2.a.h 5
20.d odd 2 1 7400.2.a.q 5

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

 $$T_{3}^{5} - T_{3}^{4} - 8T_{3}^{3} + 4T_{3}^{2} + 8T_{3} + 2$$ T3^5 - T3^4 - 8*T3^3 + 4*T3^2 + 8*T3 + 2 $$T_{7}^{5} - T_{7}^{4} - 14T_{7}^{3} + 8T_{7}^{2} + 28T_{7} + 6$$ T7^5 - T7^4 - 14*T7^3 + 8*T7^2 + 28*T7 + 6 $$T_{13}^{5} - 4T_{13}^{4} - 28T_{13}^{3} + 172T_{13}^{2} - 284T_{13} + 144$$ T13^5 - 4*T13^4 - 28*T13^3 + 172*T13^2 - 284*T13 + 144

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5}$$
$3$ $$T^{5} - T^{4} - 8 T^{3} + 4 T^{2} + 8 T + 2$$
$5$ $$(T + 1)^{5}$$
$7$ $$T^{5} - T^{4} - 14 T^{3} + 8 T^{2} + \cdots + 6$$
$11$ $$T^{5} - 7 T^{4} - 20 T^{3} + 192 T^{2} + \cdots - 288$$
$13$ $$T^{5} - 4 T^{4} - 28 T^{3} + 172 T^{2} + \cdots + 144$$
$17$ $$T^{5} - 28 T^{3} - 20 T^{2} + \cdots + 112$$
$19$ $$T^{5} - 10 T^{4} - 26 T^{3} + \cdots + 1544$$
$23$ $$T^{5} - 8 T^{4} - 8 T^{3} + 128 T^{2} + \cdots - 192$$
$29$ $$T^{5} + 8 T^{4} - 32 T^{3} - 240 T^{2} + \cdots + 896$$
$31$ $$T^{5} - 2 T^{4} - 130 T^{3} + \cdots + 7556$$
$37$ $$(T - 1)^{5}$$
$41$ $$T^{5} + 13 T^{4} - 40 T^{3} + \cdots + 224$$
$43$ $$T^{5} - 18 T^{4} + 40 T^{3} + \cdots + 9056$$
$47$ $$T^{5} - 9 T^{4} - 54 T^{3} + 672 T^{2} + \cdots + 886$$
$53$ $$T^{5} + 13 T^{4} - 16 T^{3} + \cdots - 4656$$
$59$ $$T^{5} - 24 T^{4} + 38 T^{3} + \cdots - 44056$$
$61$ $$T^{5} + 2 T^{4} - 216 T^{3} + \cdots + 43424$$
$67$ $$T^{5} - 22 T^{4} + 90 T^{3} + \cdots + 7944$$
$71$ $$T^{5} - 21 T^{4} + 100 T^{3} + \cdots + 11072$$
$73$ $$T^{5} - 29 T^{4} + 264 T^{3} + \cdots + 7088$$
$79$ $$T^{5} - 6 T^{4} - 78 T^{3} + 18 T^{2} + \cdots + 692$$
$83$ $$T^{5} - 33 T^{4} + 368 T^{3} + \cdots + 6074$$
$89$ $$T^{5} + 26 T^{4} + 40 T^{3} + \cdots + 43936$$
$97$ $$T^{5} - 26 T^{4} + 120 T^{3} + \cdots + 9024$$