# Properties

 Label 6960.2.a.bw Level $6960$ Weight $2$ Character orbit 6960.a Self dual yes Analytic conductor $55.576$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{21})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 5$$ x^2 - x - 5 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 $$\beta = \sqrt{21}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −1.79129 2.79129
0 −1.00000 0 1.00000 0 −1.00000 0 1.00000 0
1.2 0 −1.00000 0 1.00000 0 −1.00000 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.bw 2
4.b odd 2 1 435.2.a.f 2
12.b even 2 1 1305.2.a.m 2
20.d odd 2 1 2175.2.a.r 2
20.e even 4 2 2175.2.c.f 4
60.h even 2 1 6525.2.a.t 2

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

 $$T_{7} + 1$$ T7 + 1 $$T_{11} + 5$$ T11 + 5 $$T_{13}^{2} - 21$$ T13^2 - 21 $$T_{17} + 3$$ T17 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$(T + 1)^{2}$$
$11$ $$(T + 5)^{2}$$
$13$ $$T^{2} - 21$$
$17$ $$(T + 3)^{2}$$
$19$ $$T^{2} - 2T - 20$$
$23$ $$(T - 4)^{2}$$
$29$ $$(T - 1)^{2}$$
$31$ $$(T + 4)^{2}$$
$37$ $$(T + 4)^{2}$$
$41$ $$T^{2} - 84$$
$43$ $$T^{2} - 10T + 4$$
$47$ $$T^{2} + 12T + 15$$
$53$ $$T^{2} - 10T + 4$$
$59$ $$T^{2} - 6T - 12$$
$61$ $$T^{2} + 2T - 188$$
$67$ $$T^{2} + 10T - 59$$
$71$ $$T^{2} - 10T + 4$$
$73$ $$(T - 4)^{2}$$
$79$ $$T^{2} + 6T - 12$$
$83$ $$T^{2} - 14T + 28$$
$89$ $$T^{2} - 12T + 15$$
$97$ $$T^{2} - 14T + 28$$