# Properties

 Label 6960.2.a.co Level $6960$ Weight $2$ Character orbit 6960.a Self dual yes Analytic conductor $55.576$ Analytic rank $1$ Dimension $4$ 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: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.2225.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 5x^{2} + 2x + 4$$ x^4 - x^3 - 5*x^2 + 2*x + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{3}$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} - q^{5} + (\beta_1 - 1) q^{7} + q^{9}+O(q^{10})$$ q + q^3 - q^5 + (b1 - 1) * q^7 + q^9 $$q + q^{3} - q^{5} + (\beta_1 - 1) q^{7} + q^{9} + ( - \beta_1 + 1) q^{11} + ( - \beta_{3} - 2) q^{13} - q^{15} + ( - \beta_{2} - 2) q^{17} + (\beta_{3} - \beta_1 + 1) q^{19} + (\beta_1 - 1) q^{21} + (2 \beta_{2} + 2) q^{23} + q^{25} + q^{27} - q^{29} + ( - 2 \beta_{2} + 2) q^{31} + ( - \beta_1 + 1) q^{33} + ( - \beta_1 + 1) q^{35} + (2 \beta_{3} + \beta_{2} + \beta_1 - 5) q^{37} + ( - \beta_{3} - 2) q^{39} + (2 \beta_{3} + 2 \beta_1 - 4) q^{41} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 - 1) q^{43} - q^{45} + ( - \beta_{3} - 2 \beta_{2} + 4) q^{47} + (2 \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 2) q^{49} + ( - \beta_{2} - 2) q^{51} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 - 3) q^{53} + (\beta_1 - 1) q^{55} + (\beta_{3} - \beta_1 + 1) q^{57} + (\beta_{3} - 2 \beta_{2} + 3 \beta_1 - 1) q^{59} + (\beta_{3} - 3 \beta_1 - 5) q^{61} + (\beta_1 - 1) q^{63} + (\beta_{3} + 2) q^{65} + ( - 2 \beta_{3} - 2 \beta_{2} + \cdots + 1) q^{67}+ \cdots + ( - \beta_1 + 1) q^{99}+O(q^{100})$$ q + q^3 - q^5 + (b1 - 1) * q^7 + q^9 + (-b1 + 1) * q^11 + (-b3 - 2) * q^13 - q^15 + (-b2 - 2) * q^17 + (b3 - b1 + 1) * q^19 + (b1 - 1) * q^21 + (2*b2 + 2) * q^23 + q^25 + q^27 - q^29 + (-2*b2 + 2) * q^31 + (-b1 + 1) * q^33 + (-b1 + 1) * q^35 + (2*b3 + b2 + b1 - 5) * q^37 + (-b3 - 2) * q^39 + (2*b3 + 2*b1 - 4) * q^41 + (-b3 + 2*b2 - b1 - 1) * q^43 - q^45 + (-b3 - 2*b2 + 4) * q^47 + (2*b3 + 2*b2 - 3*b1 + 2) * q^49 + (-b2 - 2) * q^51 + (-b3 + 2*b2 - b1 - 3) * q^53 + (b1 - 1) * q^55 + (b3 - b1 + 1) * q^57 + (b3 - 2*b2 + 3*b1 - 1) * q^59 + (b3 - 3*b1 - 5) * q^61 + (b1 - 1) * q^63 + (b3 + 2) * q^65 + (-2*b3 - 2*b2 - b1 + 1) * q^67 + (2*b2 + 2) * q^69 + (b3 + 2*b2 + b1 + 1) * q^71 + (-2*b3 - b2 - b1 + 1) * q^73 + q^75 + (-2*b3 - 2*b2 + 3*b1 - 9) * q^77 + (-3*b3 - 2*b2 + b1 - 5) * q^79 + q^81 + (b3 + 2*b2 - 3*b1 + 3) * q^83 + (b2 + 2) * q^85 - q^87 + (-3*b3 - 4*b2 + 2*b1) * q^89 + (-b3 - 4*b1 + 4) * q^91 + (-2*b2 + 2) * q^93 + (-b3 + b1 - 1) * q^95 + (b3 + 3*b2 + 2*b1 - 8) * q^97 + (-b1 + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{3} - 4 q^{5} - 2 q^{7} + 4 q^{9}+O(q^{10})$$ 4 * q + 4 * q^3 - 4 * q^5 - 2 * q^7 + 4 * q^9 $$4 q + 4 q^{3} - 4 q^{5} - 2 q^{7} + 4 q^{9} + 2 q^{11} - 8 q^{13} - 4 q^{15} - 10 q^{17} + 2 q^{19} - 2 q^{21} + 12 q^{23} + 4 q^{25} + 4 q^{27} - 4 q^{29} + 4 q^{31} + 2 q^{33} + 2 q^{35} - 16 q^{37} - 8 q^{39} - 12 q^{41} - 2 q^{43} - 4 q^{45} + 12 q^{47} + 6 q^{49} - 10 q^{51} - 10 q^{53} - 2 q^{55} + 2 q^{57} - 2 q^{59} - 26 q^{61} - 2 q^{63} + 8 q^{65} - 2 q^{67} + 12 q^{69} + 10 q^{71} + 4 q^{75} - 34 q^{77} - 22 q^{79} + 4 q^{81} + 10 q^{83} + 10 q^{85} - 4 q^{87} - 4 q^{89} + 8 q^{91} + 4 q^{93} - 2 q^{95} - 22 q^{97} + 2 q^{99}+O(q^{100})$$ 4 * q + 4 * q^3 - 4 * q^5 - 2 * q^7 + 4 * q^9 + 2 * q^11 - 8 * q^13 - 4 * q^15 - 10 * q^17 + 2 * q^19 - 2 * q^21 + 12 * q^23 + 4 * q^25 + 4 * q^27 - 4 * q^29 + 4 * q^31 + 2 * q^33 + 2 * q^35 - 16 * q^37 - 8 * q^39 - 12 * q^41 - 2 * q^43 - 4 * q^45 + 12 * q^47 + 6 * q^49 - 10 * q^51 - 10 * q^53 - 2 * q^55 + 2 * q^57 - 2 * q^59 - 26 * q^61 - 2 * q^63 + 8 * q^65 - 2 * q^67 + 12 * q^69 + 10 * q^71 + 4 * q^75 - 34 * q^77 - 22 * q^79 + 4 * q^81 + 10 * q^83 + 10 * q^85 - 4 * q^87 - 4 * q^89 + 8 * q^91 + 4 * q^93 - 2 * q^95 - 22 * q^97 + 2 * q^99

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

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

## 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
 1.13856 −1.75660 −0.820249 2.43828
0 1.00000 0 −1.00000 0 −5.07830 0 1.00000 0
1.2 0 1.00000 0 −1.00000 0 −0.393832 0 1.00000 0
1.3 0 1.00000 0 −1.00000 0 0.729126 0 1.00000 0
1.4 0 1.00000 0 −1.00000 0 2.74301 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.co 4
4.b odd 2 1 435.2.a.j 4
12.b even 2 1 1305.2.a.r 4
20.d odd 2 1 2175.2.a.v 4
20.e even 4 2 2175.2.c.n 8
60.h even 2 1 6525.2.a.bi 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.2.a.j 4 4.b odd 2 1
1305.2.a.r 4 12.b even 2 1
2175.2.a.v 4 20.d odd 2 1
2175.2.c.n 8 20.e even 4 2
6525.2.a.bi 4 60.h even 2 1
6960.2.a.co 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6960))$$:

 $$T_{7}^{4} + 2T_{7}^{3} - 15T_{7}^{2} + 4T_{7} + 4$$ T7^4 + 2*T7^3 - 15*T7^2 + 4*T7 + 4 $$T_{11}^{4} - 2T_{11}^{3} - 15T_{11}^{2} - 4T_{11} + 4$$ T11^4 - 2*T11^3 - 15*T11^2 - 4*T11 + 4 $$T_{13}^{4} + 8T_{13}^{3} + 3T_{13}^{2} - 92T_{13} - 164$$ T13^4 + 8*T13^3 + 3*T13^2 - 92*T13 - 164 $$T_{17}^{4} + 10T_{17}^{3} + 21T_{17}^{2} - 40T_{17} - 116$$ T17^4 + 10*T17^3 + 21*T17^2 - 40*T17 - 116

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T - 1)^{4}$$
$5$ $$(T + 1)^{4}$$
$7$ $$T^{4} + 2 T^{3} + \cdots + 4$$
$11$ $$T^{4} - 2 T^{3} + \cdots + 4$$
$13$ $$T^{4} + 8 T^{3} + \cdots - 164$$
$17$ $$T^{4} + 10 T^{3} + \cdots - 116$$
$19$ $$T^{4} - 2 T^{3} + \cdots - 16$$
$23$ $$T^{4} - 12 T^{3} + \cdots - 1024$$
$29$ $$(T + 1)^{4}$$
$31$ $$T^{4} - 4 T^{3} + \cdots + 64$$
$37$ $$T^{4} + 16 T^{3} + \cdots - 2624$$
$41$ $$T^{4} + 12 T^{3} + \cdots - 1616$$
$43$ $$T^{4} + 2 T^{3} + \cdots + 1216$$
$47$ $$T^{4} - 12 T^{3} + \cdots - 64$$
$53$ $$T^{4} + 10 T^{3} + \cdots + 400$$
$59$ $$T^{4} + 2 T^{3} + \cdots + 10496$$
$61$ $$T^{4} + 26 T^{3} + \cdots - 11344$$
$67$ $$T^{4} + 2 T^{3} + \cdots - 124$$
$71$ $$T^{4} - 10 T^{3} + \cdots + 64$$
$73$ $$T^{4} - 84 T^{2} + \cdots - 256$$
$79$ $$T^{4} + 22 T^{3} + \cdots + 2416$$
$83$ $$T^{4} - 10 T^{3} + \cdots - 16$$
$89$ $$T^{4} + 4 T^{3} + \cdots + 10156$$
$97$ $$T^{4} + 22 T^{3} + \cdots + 2384$$