# Properties

 Label 9576.2.a.bc Level $9576$ Weight $2$ Character orbit 9576.a Self dual yes Analytic conductor $76.465$ 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] = [9576,2,Mod(1,9576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9576, 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("9576.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9576.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: $$76.4647449756$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1064) 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 $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta - 2) q^{5} + q^{7}+O(q^{10})$$ q + (-b - 2) * q^5 + q^7 $$q + ( - \beta - 2) q^{5} + q^{7} + (3 \beta + 1) q^{11} + (2 \beta + 2) q^{13} + (4 \beta - 4) q^{17} - q^{19} + (6 \beta - 4) q^{23} + 5 \beta q^{25} + (3 \beta - 4) q^{29} + (4 \beta - 2) q^{31} + ( - \beta - 2) q^{35} + ( - 5 \beta + 8) q^{37} + (5 \beta - 2) q^{41} + (3 \beta + 2) q^{43} + (3 \beta + 7) q^{47} + q^{49} + ( - \beta + 5) q^{53} + ( - 10 \beta - 5) q^{55} + ( - 5 \beta + 4) q^{59} + ( - 3 \beta - 3) q^{61} + ( - 8 \beta - 6) q^{65} + ( - 8 \beta - 2) q^{67} + ( - 9 \beta + 1) q^{71} + ( - 2 \beta + 12) q^{73} + (3 \beta + 1) q^{77} + ( - 3 \beta - 2) q^{79} - 4 q^{83} + ( - 8 \beta + 4) q^{85} + ( - 5 \beta + 7) q^{89} + (2 \beta + 2) q^{91} + (\beta + 2) q^{95} + ( - 9 \beta + 11) q^{97}+O(q^{100})$$ q + (-b - 2) * q^5 + q^7 + (3*b + 1) * q^11 + (2*b + 2) * q^13 + (4*b - 4) * q^17 - q^19 + (6*b - 4) * q^23 + 5*b * q^25 + (3*b - 4) * q^29 + (4*b - 2) * q^31 + (-b - 2) * q^35 + (-5*b + 8) * q^37 + (5*b - 2) * q^41 + (3*b + 2) * q^43 + (3*b + 7) * q^47 + q^49 + (-b + 5) * q^53 + (-10*b - 5) * q^55 + (-5*b + 4) * q^59 + (-3*b - 3) * q^61 + (-8*b - 6) * q^65 + (-8*b - 2) * q^67 + (-9*b + 1) * q^71 + (-2*b + 12) * q^73 + (3*b + 1) * q^77 + (-3*b - 2) * q^79 - 4 * q^83 + (-8*b + 4) * q^85 + (-5*b + 7) * q^89 + (2*b + 2) * q^91 + (b + 2) * q^95 + (-9*b + 11) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 5 q^{5} + 2 q^{7}+O(q^{10})$$ 2 * q - 5 * q^5 + 2 * q^7 $$2 q - 5 q^{5} + 2 q^{7} + 5 q^{11} + 6 q^{13} - 4 q^{17} - 2 q^{19} - 2 q^{23} + 5 q^{25} - 5 q^{29} - 5 q^{35} + 11 q^{37} + q^{41} + 7 q^{43} + 17 q^{47} + 2 q^{49} + 9 q^{53} - 20 q^{55} + 3 q^{59} - 9 q^{61} - 20 q^{65} - 12 q^{67} - 7 q^{71} + 22 q^{73} + 5 q^{77} - 7 q^{79} - 8 q^{83} + 9 q^{89} + 6 q^{91} + 5 q^{95} + 13 q^{97}+O(q^{100})$$ 2 * q - 5 * q^5 + 2 * q^7 + 5 * q^11 + 6 * q^13 - 4 * q^17 - 2 * q^19 - 2 * q^23 + 5 * q^25 - 5 * q^29 - 5 * q^35 + 11 * q^37 + q^41 + 7 * q^43 + 17 * q^47 + 2 * q^49 + 9 * q^53 - 20 * q^55 + 3 * q^59 - 9 * q^61 - 20 * q^65 - 12 * q^67 - 7 * q^71 + 22 * q^73 + 5 * q^77 - 7 * q^79 - 8 * q^83 + 9 * q^89 + 6 * q^91 + 5 * q^95 + 13 * q^97

## 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
 1.61803 −0.618034
0 0 0 −3.61803 0 1.00000 0 0 0
1.2 0 0 0 −1.38197 0 1.00000 0 0 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$7$$ $$-1$$
$$19$$ $$+1$$

## 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 9576.2.a.bc 2
3.b odd 2 1 1064.2.a.e 2
12.b even 2 1 2128.2.a.d 2
21.c even 2 1 7448.2.a.y 2
24.f even 2 1 8512.2.a.ba 2
24.h odd 2 1 8512.2.a.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1064.2.a.e 2 3.b odd 2 1
2128.2.a.d 2 12.b even 2 1
7448.2.a.y 2 21.c even 2 1
8512.2.a.e 2 24.h odd 2 1
8512.2.a.ba 2 24.f even 2 1
9576.2.a.bc 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9576))$$:

 $$T_{5}^{2} + 5T_{5} + 5$$ T5^2 + 5*T5 + 5 $$T_{11}^{2} - 5T_{11} - 5$$ T11^2 - 5*T11 - 5 $$T_{13}^{2} - 6T_{13} + 4$$ T13^2 - 6*T13 + 4 $$T_{17}^{2} + 4T_{17} - 16$$ T17^2 + 4*T17 - 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 5T + 5$$
$7$ $$(T - 1)^{2}$$
$11$ $$T^{2} - 5T - 5$$
$13$ $$T^{2} - 6T + 4$$
$17$ $$T^{2} + 4T - 16$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2} + 2T - 44$$
$29$ $$T^{2} + 5T - 5$$
$31$ $$T^{2} - 20$$
$37$ $$T^{2} - 11T - 1$$
$41$ $$T^{2} - T - 31$$
$43$ $$T^{2} - 7T + 1$$
$47$ $$T^{2} - 17T + 61$$
$53$ $$T^{2} - 9T + 19$$
$59$ $$T^{2} - 3T - 29$$
$61$ $$T^{2} + 9T + 9$$
$67$ $$T^{2} + 12T - 44$$
$71$ $$T^{2} + 7T - 89$$
$73$ $$T^{2} - 22T + 116$$
$79$ $$T^{2} + 7T + 1$$
$83$ $$(T + 4)^{2}$$
$89$ $$T^{2} - 9T - 11$$
$97$ $$T^{2} - 13T - 59$$