# Properties

 Label 6960.2.a.cl Level $6960$ Weight $2$ Character orbit 6960.a Self dual yes Analytic conductor $55.576$ Analytic rank $0$ 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] = [6960,2,Mod(1,6960)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6960, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6960.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6960 = 2^{4} \cdot 3 \cdot 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6960.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: $$55.5758798068$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.469.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 5x + 4$$ x^3 - x^2 - 5*x + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 435) 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.

 $$f(q)$$ $$=$$ $$q + q^{3} + q^{5} + (\beta_{2} + 1) q^{7} + q^{9}+O(q^{10})$$ q + q^3 + q^5 + (b2 + 1) * q^7 + q^9 $$q + q^{3} + q^{5} + (\beta_{2} + 1) q^{7} + q^{9} - 3 q^{11} + (\beta_1 + 2) q^{13} + q^{15} + ( - \beta_{2} + 1) q^{17} + (\beta_{2} + \beta_1 + 1) q^{19} + (\beta_{2} + 1) q^{21} + ( - \beta_{2} + 2 \beta_1) q^{23} + q^{25} + q^{27} - q^{29} - 3 q^{33} + (\beta_{2} + 1) q^{35} + (\beta_{2} - 2 \beta_1 + 4) q^{37} + (\beta_1 + 2) q^{39} + (\beta_{2} + 4) q^{41} + ( - 2 \beta_{2} + \beta_1 + 5) q^{43} + q^{45} + ( - 2 \beta_{2} - \beta_1) q^{47} + (3 \beta_{2} - 2 \beta_1 + 2) q^{49} + ( - \beta_{2} + 1) q^{51} + ( - 3 \beta_1 - 1) q^{53} - 3 q^{55} + (\beta_{2} + \beta_1 + 1) q^{57} + ( - \beta_{2} + \beta_1 - 7) q^{59} + (\beta_{2} + \beta_1 + 3) q^{61} + (\beta_{2} + 1) q^{63} + (\beta_1 + 2) q^{65} + (\beta_{2} - 2 \beta_1 + 9) q^{67} + ( - \beta_{2} + 2 \beta_1) q^{69} + (3 \beta_{2} - 3 \beta_1 - 1) q^{71} + (3 \beta_{2} + 2 \beta_1) q^{73} + q^{75} + ( - 3 \beta_{2} - 3) q^{77} + ( - \beta_{2} - 3 \beta_1 + 1) q^{79} + q^{81} + ( - \beta_1 + 5) q^{83} + ( - \beta_{2} + 1) q^{85} - q^{87} + ( - 3 \beta_1 + 10) q^{89} + (\beta_1 + 4) q^{91} + (\beta_{2} + \beta_1 + 1) q^{95} + (2 \beta_{2} - 3 \beta_1 - 1) q^{97} - 3 q^{99}+O(q^{100})$$ q + q^3 + q^5 + (b2 + 1) * q^7 + q^9 - 3 * q^11 + (b1 + 2) * q^13 + q^15 + (-b2 + 1) * q^17 + (b2 + b1 + 1) * q^19 + (b2 + 1) * q^21 + (-b2 + 2*b1) * q^23 + q^25 + q^27 - q^29 - 3 * q^33 + (b2 + 1) * q^35 + (b2 - 2*b1 + 4) * q^37 + (b1 + 2) * q^39 + (b2 + 4) * q^41 + (-2*b2 + b1 + 5) * q^43 + q^45 + (-2*b2 - b1) * q^47 + (3*b2 - 2*b1 + 2) * q^49 + (-b2 + 1) * q^51 + (-3*b1 - 1) * q^53 - 3 * q^55 + (b2 + b1 + 1) * q^57 + (-b2 + b1 - 7) * q^59 + (b2 + b1 + 3) * q^61 + (b2 + 1) * q^63 + (b1 + 2) * q^65 + (b2 - 2*b1 + 9) * q^67 + (-b2 + 2*b1) * q^69 + (3*b2 - 3*b1 - 1) * q^71 + (3*b2 + 2*b1) * q^73 + q^75 + (-3*b2 - 3) * q^77 + (-b2 - 3*b1 + 1) * q^79 + q^81 + (-b1 + 5) * q^83 + (-b2 + 1) * q^85 - q^87 + (-3*b1 + 10) * q^89 + (b1 + 4) * q^91 + (b2 + b1 + 1) * q^95 + (2*b2 - 3*b1 - 1) * q^97 - 3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{3} + 3 q^{5} + 4 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^3 + 3 * q^5 + 4 * q^7 + 3 * q^9 $$3 q + 3 q^{3} + 3 q^{5} + 4 q^{7} + 3 q^{9} - 9 q^{11} + 6 q^{13} + 3 q^{15} + 2 q^{17} + 4 q^{19} + 4 q^{21} - q^{23} + 3 q^{25} + 3 q^{27} - 3 q^{29} - 9 q^{33} + 4 q^{35} + 13 q^{37} + 6 q^{39} + 13 q^{41} + 13 q^{43} + 3 q^{45} - 2 q^{47} + 9 q^{49} + 2 q^{51} - 3 q^{53} - 9 q^{55} + 4 q^{57} - 22 q^{59} + 10 q^{61} + 4 q^{63} + 6 q^{65} + 28 q^{67} - q^{69} + 3 q^{73} + 3 q^{75} - 12 q^{77} + 2 q^{79} + 3 q^{81} + 15 q^{83} + 2 q^{85} - 3 q^{87} + 30 q^{89} + 12 q^{91} + 4 q^{95} - q^{97} - 9 q^{99}+O(q^{100})$$ 3 * q + 3 * q^3 + 3 * q^5 + 4 * q^7 + 3 * q^9 - 9 * q^11 + 6 * q^13 + 3 * q^15 + 2 * q^17 + 4 * q^19 + 4 * q^21 - q^23 + 3 * q^25 + 3 * q^27 - 3 * q^29 - 9 * q^33 + 4 * q^35 + 13 * q^37 + 6 * q^39 + 13 * q^41 + 13 * q^43 + 3 * q^45 - 2 * q^47 + 9 * q^49 + 2 * q^51 - 3 * q^53 - 9 * q^55 + 4 * q^57 - 22 * q^59 + 10 * q^61 + 4 * q^63 + 6 * q^65 + 28 * q^67 - q^69 + 3 * q^73 + 3 * q^75 - 12 * q^77 + 2 * q^79 + 3 * q^81 + 15 * q^83 + 2 * q^85 - 3 * q^87 + 30 * q^89 + 12 * q^91 + 4 * q^95 - q^97 - 9 * q^99

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

 $$\beta_{1}$$ $$=$$ $$\nu^{2} + \nu - 4$$ v^2 + v - 4 $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ v^2 - v - 3
 $$\nu$$ $$=$$ $$( -\beta_{2} + \beta _1 + 1 ) / 2$$ (-b2 + b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{2} + \beta _1 + 7 ) / 2$$ (b2 + b1 + 7) / 2

## 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
 0.772866 2.39138 −2.16425
0 1.00000 0 1.00000 0 −2.17554 0 1.00000 0
1.2 0 1.00000 0 1.00000 0 1.32733 0 1.00000 0
1.3 0 1.00000 0 1.00000 0 4.84822 0 1.00000 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$29$$ $$1$$

## 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 6960.2.a.cl 3
4.b odd 2 1 435.2.a.i 3
12.b even 2 1 1305.2.a.q 3
20.d odd 2 1 2175.2.a.u 3
20.e even 4 2 2175.2.c.m 6
60.h even 2 1 6525.2.a.bf 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.2.a.i 3 4.b odd 2 1
1305.2.a.q 3 12.b even 2 1
2175.2.a.u 3 20.d odd 2 1
2175.2.c.m 6 20.e even 4 2
6525.2.a.bf 3 60.h even 2 1
6960.2.a.cl 3 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6960))$$:

 $$T_{7}^{3} - 4T_{7}^{2} - 7T_{7} + 14$$ T7^3 - 4*T7^2 - 7*T7 + 14 $$T_{11} + 3$$ T11 + 3 $$T_{13}^{3} - 6T_{13}^{2} - T_{13} + 2$$ T13^3 - 6*T13^2 - T13 + 2 $$T_{17}^{3} - 2T_{17}^{2} - 11T_{17} + 8$$ T17^3 - 2*T17^2 - 11*T17 + 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$(T - 1)^{3}$$
$5$ $$(T - 1)^{3}$$
$7$ $$T^{3} - 4 T^{2} + \cdots + 14$$
$11$ $$(T + 3)^{3}$$
$13$ $$T^{3} - 6T^{2} - T + 2$$
$17$ $$T^{3} - 2 T^{2} + \cdots + 8$$
$19$ $$T^{3} - 4 T^{2} + \cdots + 88$$
$23$ $$T^{3} + T^{2} + \cdots - 112$$
$29$ $$(T + 1)^{3}$$
$31$ $$T^{3}$$
$37$ $$T^{3} - 13T^{2} + 256$$
$41$ $$T^{3} - 13 T^{2} + \cdots - 28$$
$43$ $$T^{3} - 13 T^{2} + \cdots + 308$$
$47$ $$T^{3} + 2 T^{2} + \cdots - 266$$
$53$ $$T^{3} + 3 T^{2} + \cdots + 316$$
$59$ $$T^{3} + 22 T^{2} + \cdots + 256$$
$61$ $$T^{3} - 10 T^{2} + \cdots + 112$$
$67$ $$T^{3} - 28 T^{2} + \cdots - 194$$
$71$ $$T^{3} - 192T - 488$$
$73$ $$T^{3} - 3 T^{2} + \cdots + 1168$$
$79$ $$T^{3} - 2 T^{2} + \cdots + 224$$
$83$ $$T^{3} - 15 T^{2} + \cdots - 44$$
$89$ $$T^{3} - 30 T^{2} + \cdots + 602$$
$97$ $$T^{3} + T^{2} + \cdots + 76$$