# Properties

 Label 6960.2.a.bx 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: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{5} - 2 \beta q^{7} + q^{9} +O(q^{10})$$ q - q^3 + q^5 - 2*b * q^7 + q^9 $$q - q^{3} + q^{5} - 2 \beta q^{7} + q^{9} + ( - \beta + 4) q^{11} - 2 q^{13} - q^{15} + ( - 2 \beta - 2) q^{17} - 2 \beta q^{19} + 2 \beta q^{21} + ( - \beta - 4) q^{23} + q^{25} - q^{27} + q^{29} - 4 q^{31} + (\beta - 4) q^{33} - 2 \beta q^{35} + ( - 3 \beta + 6) q^{37} + 2 q^{39} + ( - 3 \beta + 6) q^{41} - 3 \beta q^{43} + q^{45} + (2 \beta + 8) q^{47} + (4 \beta + 9) q^{49} + (2 \beta + 2) q^{51} + (\beta - 6) q^{53} + ( - \beta + 4) q^{55} + 2 \beta q^{57} - 12 q^{59} + ( - 2 \beta + 6) q^{61} - 2 \beta q^{63} - 2 q^{65} + (6 \beta - 4) q^{67} + (\beta + 4) q^{69} + ( - 2 \beta + 8) q^{71} + (3 \beta - 6) q^{73} - q^{75} + ( - 6 \beta + 8) q^{77} + 12 q^{79} + q^{81} + \beta q^{83} + ( - 2 \beta - 2) q^{85} - q^{87} + (4 \beta + 2) q^{89} + 4 \beta q^{91} + 4 q^{93} - 2 \beta q^{95} + ( - 3 \beta + 2) q^{97} + ( - \beta + 4) q^{99} +O(q^{100})$$ q - q^3 + q^5 - 2*b * q^7 + q^9 + (-b + 4) * q^11 - 2 * q^13 - q^15 + (-2*b - 2) * q^17 - 2*b * q^19 + 2*b * q^21 + (-b - 4) * q^23 + q^25 - q^27 + q^29 - 4 * q^31 + (b - 4) * q^33 - 2*b * q^35 + (-3*b + 6) * q^37 + 2 * q^39 + (-3*b + 6) * q^41 - 3*b * q^43 + q^45 + (2*b + 8) * q^47 + (4*b + 9) * q^49 + (2*b + 2) * q^51 + (b - 6) * q^53 + (-b + 4) * q^55 + 2*b * q^57 - 12 * q^59 + (-2*b + 6) * q^61 - 2*b * q^63 - 2 * q^65 + (6*b - 4) * q^67 + (b + 4) * q^69 + (-2*b + 8) * q^71 + (3*b - 6) * q^73 - q^75 + (-6*b + 8) * q^77 + 12 * q^79 + q^81 + b * q^83 + (-2*b - 2) * q^85 - q^87 + (4*b + 2) * q^89 + 4*b * q^91 + 4 * q^93 - 2*b * q^95 + (-3*b + 2) * q^97 + (-b + 4) * 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 $$2 q - 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} + 7 q^{11} - 4 q^{13} - 2 q^{15} - 6 q^{17} - 2 q^{19} + 2 q^{21} - 9 q^{23} + 2 q^{25} - 2 q^{27} + 2 q^{29} - 8 q^{31} - 7 q^{33} - 2 q^{35} + 9 q^{37} + 4 q^{39} + 9 q^{41} - 3 q^{43} + 2 q^{45} + 18 q^{47} + 22 q^{49} + 6 q^{51} - 11 q^{53} + 7 q^{55} + 2 q^{57} - 24 q^{59} + 10 q^{61} - 2 q^{63} - 4 q^{65} - 2 q^{67} + 9 q^{69} + 14 q^{71} - 9 q^{73} - 2 q^{75} + 10 q^{77} + 24 q^{79} + 2 q^{81} + q^{83} - 6 q^{85} - 2 q^{87} + 8 q^{89} + 4 q^{91} + 8 q^{93} - 2 q^{95} + q^{97} + 7 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9 + 7 * q^11 - 4 * q^13 - 2 * q^15 - 6 * q^17 - 2 * q^19 + 2 * q^21 - 9 * q^23 + 2 * q^25 - 2 * q^27 + 2 * q^29 - 8 * q^31 - 7 * q^33 - 2 * q^35 + 9 * q^37 + 4 * q^39 + 9 * q^41 - 3 * q^43 + 2 * q^45 + 18 * q^47 + 22 * q^49 + 6 * q^51 - 11 * q^53 + 7 * q^55 + 2 * q^57 - 24 * q^59 + 10 * q^61 - 2 * q^63 - 4 * q^65 - 2 * q^67 + 9 * q^69 + 14 * q^71 - 9 * q^73 - 2 * q^75 + 10 * q^77 + 24 * q^79 + 2 * q^81 + q^83 - 6 * q^85 - 2 * q^87 + 8 * q^89 + 4 * q^91 + 8 * q^93 - 2 * q^95 + q^97 + 7 * 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
 2.56155 −1.56155
0 −1.00000 0 1.00000 0 −5.12311 0 1.00000 0
1.2 0 −1.00000 0 1.00000 0 3.12311 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.bx 2
4.b odd 2 1 435.2.a.h 2
12.b even 2 1 1305.2.a.i 2
20.d odd 2 1 2175.2.a.m 2
20.e even 4 2 2175.2.c.h 4
60.h even 2 1 6525.2.a.bc 2

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

 $$T_{7}^{2} + 2T_{7} - 16$$ T7^2 + 2*T7 - 16 $$T_{11}^{2} - 7T_{11} + 8$$ T11^2 - 7*T11 + 8 $$T_{13} + 2$$ T13 + 2 $$T_{17}^{2} + 6T_{17} - 8$$ T17^2 + 6*T17 - 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} + 2T - 16$$
$11$ $$T^{2} - 7T + 8$$
$13$ $$(T + 2)^{2}$$
$17$ $$T^{2} + 6T - 8$$
$19$ $$T^{2} + 2T - 16$$
$23$ $$T^{2} + 9T + 16$$
$29$ $$(T - 1)^{2}$$
$31$ $$(T + 4)^{2}$$
$37$ $$T^{2} - 9T - 18$$
$41$ $$T^{2} - 9T - 18$$
$43$ $$T^{2} + 3T - 36$$
$47$ $$T^{2} - 18T + 64$$
$53$ $$T^{2} + 11T + 26$$
$59$ $$(T + 12)^{2}$$
$61$ $$T^{2} - 10T + 8$$
$67$ $$T^{2} + 2T - 152$$
$71$ $$T^{2} - 14T + 32$$
$73$ $$T^{2} + 9T - 18$$
$79$ $$(T - 12)^{2}$$
$83$ $$T^{2} - T - 4$$
$89$ $$T^{2} - 8T - 52$$
$97$ $$T^{2} - T - 38$$