# Properties

 Label 2960.2.a.bb 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: 5.5.583504.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - 8x^{3} - 2x^{2} + 15x + 8$$ x^5 - 8*x^3 - 2*x^2 + 15*x + 8 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 8x^{3} - 2x^{2} + 15x + 8$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 4\nu - 1$$ v^3 - 4*v - 1 $$\beta_{4}$$ $$=$$ $$\nu^{4} - \nu^{3} - 6\nu^{2} + 4\nu + 7$$ v^4 - v^3 - 6*v^2 + 4*v + 7
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 4\beta _1 + 1$$ b3 + 4*b1 + 1 $$\nu^{4}$$ $$=$$ $$\beta_{4} + \beta_{3} + 6\beta_{2} + 12$$ b4 + b3 + 6*b2 + 12

## 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
 2.34794 1.84514 −0.592644 −1.44679 −2.15365
0 −1.34794 0 1.00000 0 2.75056 0 −1.18306 0
1.2 0 −0.845141 0 1.00000 0 2.14219 0 −2.28574 0
1.3 0 1.59264 0 1.00000 0 −4.50235 0 −0.463486 0
1.4 0 2.44679 0 1.00000 0 4.02963 0 2.98676 0
1.5 0 3.15365 0 1.00000 0 0.579974 0 6.94552 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.bb 5
4.b odd 2 1 1480.2.a.g 5
20.d odd 2 1 7400.2.a.r 5

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

 $$T_{3}^{5} - 5T_{3}^{4} + 2T_{3}^{3} + 16T_{3}^{2} - 8T_{3} - 14$$ T3^5 - 5*T3^4 + 2*T3^3 + 16*T3^2 - 8*T3 - 14 $$T_{7}^{5} - 5T_{7}^{4} - 12T_{7}^{3} + 100T_{7}^{2} - 160T_{7} + 62$$ T7^5 - 5*T7^4 - 12*T7^3 + 100*T7^2 - 160*T7 + 62 $$T_{13}^{5} + 6T_{13}^{4} - 28T_{13}^{3} - 172T_{13}^{2} - 196T_{13} - 8$$ T13^5 + 6*T13^4 - 28*T13^3 - 172*T13^2 - 196*T13 - 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5}$$
$3$ $$T^{5} - 5 T^{4} + \cdots - 14$$
$5$ $$(T - 1)^{5}$$
$7$ $$T^{5} - 5 T^{4} + \cdots + 62$$
$11$ $$T^{5} - 7 T^{4} + \cdots - 32$$
$13$ $$T^{5} + 6 T^{4} + \cdots - 8$$
$17$ $$T^{5} - 76 T^{3} + \cdots + 2896$$
$19$ $$T^{5} - 12 T^{4} + \cdots - 32$$
$23$ $$T^{5} - 6 T^{4} + \cdots + 224$$
$29$ $$T^{5} - 6 T^{4} + \cdots + 224$$
$31$ $$T^{5} - 10 T^{4} + \cdots + 1112$$
$37$ $$(T + 1)^{5}$$
$41$ $$T^{5} - 7 T^{4} + \cdots - 32$$
$43$ $$T^{5} - 22 T^{4} + \cdots + 32$$
$47$ $$T^{5} - 13 T^{4} + \cdots + 1238$$
$53$ $$T^{5} + 11 T^{4} + \cdots + 58544$$
$59$ $$T^{5} - 10 T^{4} + \cdots - 1448$$
$61$ $$T^{5} - 128 T^{3} + \cdots - 10816$$
$67$ $$T^{5} - 24 T^{4} + \cdots - 992$$
$71$ $$T^{5} - 13 T^{4} + \cdots - 99136$$
$73$ $$T^{5} + 13 T^{4} + \cdots + 11632$$
$79$ $$T^{5} + 2 T^{4} + \cdots + 11456$$
$83$ $$T^{5} - 13 T^{4} + \cdots + 35434$$
$89$ $$T^{5} - 8 T^{4} + \cdots + 1984$$
$97$ $$T^{5} + 20 T^{4} + \cdots - 31744$$