# Properties

 Label 2960.2.a.x Level $2960$ Weight $2$ Character orbit 2960.a Self dual yes Analytic conductor $23.636$ Analytic rank $1$ Dimension $5$ CM no Inner twists $1$

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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: $$5$$ Coefficient field: 5.5.998068.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - 2x^{4} - 6x^{3} + 10x^{2} + 3x - 2$$ x^5 - 2*x^4 - 6*x^3 + 10*x^2 + 3*x - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_{4} q^{3} - q^{5} + \beta_{2} q^{7} + (\beta_{3} - \beta_{2} - \beta_1 + 2) q^{9}+O(q^{10})$$ q - b4 * q^3 - q^5 + b2 * q^7 + (b3 - b2 - b1 + 2) * q^9 $$q - \beta_{4} q^{3} - q^{5} + \beta_{2} q^{7} + (\beta_{3} - \beta_{2} - \beta_1 + 2) q^{9} + (\beta_1 - 2) q^{11} + (\beta_{4} - \beta_{3} - 1) q^{13} + \beta_{4} q^{15} + (2 \beta_{4} - \beta_{3} + \beta_{2} - 1) q^{17} + ( - \beta_{2} + \beta_1 - 2) q^{19} + (\beta_{3} - \beta_{2} + \beta_1 + 1) q^{21} - 2 \beta_1 q^{23} + q^{25} + ( - \beta_{4} - 2 \beta_{3} + \cdots - \beta_1) q^{27}+ \cdots + (3 \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + \cdots - 10) q^{99}+O(q^{100})$$ q - b4 * q^3 - q^5 + b2 * q^7 + (b3 - b2 - b1 + 2) * q^9 + (b1 - 2) * q^11 + (b4 - b3 - 1) * q^13 + b4 * q^15 + (2*b4 - b3 + b2 - 1) * q^17 + (-b2 + b1 - 2) * q^19 + (b3 - b2 + b1 + 1) * q^21 - 2*b1 * q^23 + q^25 + (-b4 - 2*b3 + b2 - b1) * q^27 + (-b4 - b2 + 2*b1 + 2) * q^29 + (-b4 - 2*b2 - b1 + 2) * q^31 + (3*b4 + b2 + b1 - 2) * q^33 - b2 * q^35 - q^37 + (2*b4 - 4) * q^39 + (2*b4 - 3*b1 + 4) * q^41 + (b4 + 2*b3 - b2 - 6) * q^43 + (-b3 + b2 + b1 - 2) * q^45 + (b4 + 2*b3 + 2*b1 - 2) * q^47 + (b4 - b3 + 2*b2 + b1) * q^49 + (2*b4 + 2*b1 - 8) * q^51 + (-2*b3 - b1 + 2) * q^53 + (-b1 + 2) * q^55 + (3*b4 - b3 + 2*b2 - 3) * q^57 + (2*b4 + b2 + b1 - 6) * q^59 + (3*b4 + 2*b3 + b2 - 4) * q^61 + (-b4 - 2*b3 + b1 - 4) * q^63 + (-b4 + b3 + 1) * q^65 + (4*b4 - 2*b3 + 3*b2 - b1 - 4) * q^67 + (-2*b4 - 2*b2 - 2*b1 + 4) * q^69 + (-b4 + 2*b3 - b2 - b1) * q^71 + (-b4 + 2*b3 - b2 - b1 + 2) * q^73 - b4 * q^75 + (-2*b4 + 2*b3 - 2*b2 + b1) * q^77 + (2*b3 - 3*b2 + b1) * q^79 + (b4 + b3 - 2*b2 + 4) * q^81 + (-3*b4 - 2*b1 - 4) * q^83 + (-2*b4 + b3 - b2 + 1) * q^85 + 2*b2 * q^87 + (-4*b4 + 2*b3 - 2*b2 + 2*b1 + 4) * q^89 + (2*b4 - 4*b1 - 4) * q^91 + (-3*b4 - b3 - 4*b1 + 5) * q^93 + (b2 - b1 + 2) * q^95 + (-b4 - 2*b3 + b2 - 4) * q^97 + (3*b4 - 2*b3 + 3*b2 + 2*b1 - 10) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - q^{3} - 5 q^{5} + 2 q^{7} + 6 q^{9}+O(q^{10})$$ 5 * q - q^3 - 5 * q^5 + 2 * q^7 + 6 * q^9 $$5 q - q^{3} - 5 q^{5} + 2 q^{7} + 6 q^{9} - 8 q^{11} - 4 q^{13} + q^{15} - q^{17} - 10 q^{19} + 5 q^{21} - 4 q^{23} + 5 q^{25} - q^{27} + 11 q^{29} + 3 q^{31} - 3 q^{33} - 2 q^{35} - 5 q^{37} - 18 q^{39} + 16 q^{41} - 31 q^{43} - 6 q^{45} - 5 q^{47} + 7 q^{49} - 34 q^{51} + 8 q^{53} + 8 q^{55} - 8 q^{57} - 24 q^{59} - 15 q^{61} - 19 q^{63} + 4 q^{65} - 12 q^{67} + 10 q^{69} - 5 q^{71} + 5 q^{73} - q^{75} - 4 q^{77} - 4 q^{79} + 17 q^{81} - 27 q^{83} + q^{85} + 4 q^{87} + 16 q^{89} - 26 q^{91} + 14 q^{93} + 10 q^{95} - 19 q^{97} - 37 q^{99}+O(q^{100})$$ 5 * q - q^3 - 5 * q^5 + 2 * q^7 + 6 * q^9 - 8 * q^11 - 4 * q^13 + q^15 - q^17 - 10 * q^19 + 5 * q^21 - 4 * q^23 + 5 * q^25 - q^27 + 11 * q^29 + 3 * q^31 - 3 * q^33 - 2 * q^35 - 5 * q^37 - 18 * q^39 + 16 * q^41 - 31 * q^43 - 6 * q^45 - 5 * q^47 + 7 * q^49 - 34 * q^51 + 8 * q^53 + 8 * q^55 - 8 * q^57 - 24 * q^59 - 15 * q^61 - 19 * q^63 + 4 * q^65 - 12 * q^67 + 10 * q^69 - 5 * q^71 + 5 * q^73 - q^75 - 4 * q^77 - 4 * q^79 + 17 * q^81 - 27 * q^83 + q^85 + 4 * q^87 + 16 * q^89 - 26 * q^91 + 14 * q^93 + 10 * q^95 - 19 * q^97 - 37 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 2x^{4} - 6x^{3} + 10x^{2} + 3x - 2$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{4} - \nu^{3} - 5\nu^{2} + 3\nu ) / 2$$ (v^4 - v^3 - 5*v^2 + 3*v) / 2 $$\beta_{2}$$ $$=$$ $$( -\nu^{4} + 3\nu^{3} + 5\nu^{2} - 13\nu ) / 2$$ (-v^4 + 3*v^3 + 5*v^2 - 13*v) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{4} - \nu^{3} - 7\nu^{2} + 7\nu + 4 ) / 2$$ (v^4 - v^3 - 7*v^2 + 7*v + 4) / 2 $$\beta_{4}$$ $$=$$ $$( \nu^{4} - \nu^{3} - 7\nu^{2} + 3\nu + 6 ) / 2$$ (v^4 - v^3 - 7*v^2 + 3*v + 6) / 2
 $$\nu$$ $$=$$ $$( -\beta_{4} + \beta_{3} + 1 ) / 2$$ (-b4 + b3 + 1) / 2 $$\nu^{2}$$ $$=$$ $$-\beta_{4} + \beta _1 + 3$$ -b4 + b1 + 3 $$\nu^{3}$$ $$=$$ $$( -5\beta_{4} + 5\beta_{3} + 2\beta_{2} + 2\beta _1 + 5 ) / 2$$ (-5*b4 + 5*b3 + 2*b2 + 2*b1 + 5) / 2 $$\nu^{4}$$ $$=$$ $$-6\beta_{4} + \beta_{3} + \beta_{2} + 8\beta _1 + 16$$ -6*b4 + b3 + b2 + 8*b1 + 16

## 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
 0.350931 −0.537811 −2.26473 2.75031 1.70130
0 −3.08134 0 −1.00000 0 −1.91593 0 6.49464 0
1.2 0 −1.30055 0 −1.00000 0 3.94371 0 −1.30857 0
1.3 0 −0.612608 0 −1.00000 0 −3.03374 0 −2.62471 0
1.4 0 1.14266 0 −1.00000 0 3.63076 0 −1.69433 0
1.5 0 2.85184 0 −1.00000 0 −0.624804 0 5.13298 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.x 5
4.b odd 2 1 1480.2.a.j 5
20.d odd 2 1 7400.2.a.o 5

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

 $$T_{3}^{5} + T_{3}^{4} - 10T_{3}^{3} - 8T_{3}^{2} + 12T_{3} + 8$$ T3^5 + T3^4 - 10*T3^3 - 8*T3^2 + 12*T3 + 8 $$T_{7}^{5} - 2T_{7}^{4} - 19T_{7}^{3} + 16T_{7}^{2} + 100T_{7} + 52$$ T7^5 - 2*T7^4 - 19*T7^3 + 16*T7^2 + 100*T7 + 52 $$T_{13}^{5} + 4T_{13}^{4} - 24T_{13}^{3} - 80T_{13}^{2} + 48T_{13} + 64$$ T13^5 + 4*T13^4 - 24*T13^3 - 80*T13^2 + 48*T13 + 64

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5}$$
$3$ $$T^{5} + T^{4} - 10 T^{3} + \cdots + 8$$
$5$ $$(T + 1)^{5}$$
$7$ $$T^{5} - 2 T^{4} + \cdots + 52$$
$11$ $$T^{5} + 8 T^{4} + \cdots + 32$$
$13$ $$T^{5} + 4 T^{4} + \cdots + 64$$
$17$ $$T^{5} + T^{4} + \cdots - 128$$
$19$ $$T^{5} + 10 T^{4} + \cdots + 32$$
$23$ $$T^{5} + 4 T^{4} + \cdots + 256$$
$29$ $$T^{5} - 11 T^{4} + \cdots - 208$$
$31$ $$T^{5} - 3 T^{4} + \cdots - 4736$$
$37$ $$(T + 1)^{5}$$
$41$ $$T^{5} - 16 T^{4} + \cdots - 19016$$
$43$ $$T^{5} + 31 T^{4} + \cdots - 20992$$
$47$ $$T^{5} + 5 T^{4} + \cdots + 24944$$
$53$ $$T^{5} - 8 T^{4} + \cdots - 8104$$
$59$ $$T^{5} + 24 T^{4} + \cdots + 128$$
$61$ $$T^{5} + 15 T^{4} + \cdots - 12608$$
$67$ $$T^{5} + 12 T^{4} + \cdots - 19072$$
$71$ $$T^{5} + 5 T^{4} + \cdots - 1024$$
$73$ $$T^{5} - 5 T^{4} + \cdots - 3616$$
$79$ $$T^{5} + 4 T^{4} + \cdots + 512$$
$83$ $$T^{5} + 27 T^{4} + \cdots - 60976$$
$89$ $$T^{5} - 16 T^{4} + \cdots + 9536$$
$97$ $$T^{5} + 19 T^{4} + \cdots + 32$$
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