# Properties

 Label 9576.2.a.by 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: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 3192) 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 $$\beta = \sqrt{2}$$. 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 $$q + (\beta + 2) q^{5} - q^{7} + 2 \beta q^{11} + 4 \beta q^{13} + ( - 3 \beta + 2) q^{17} + q^{19} + 6 \beta q^{23} + (4 \beta + 1) q^{25} + ( - 3 \beta - 6) q^{29} + (2 \beta + 6) q^{31} + ( - \beta - 2) q^{35} - 6 q^{37} + (2 \beta + 6) q^{41} + ( - 2 \beta - 6) q^{43} + ( - \beta + 12) q^{47} + q^{49} + (5 \beta - 2) q^{53} + (4 \beta + 4) q^{55} + ( - 4 \beta + 8) q^{59} + 2 q^{61} + (8 \beta + 8) q^{65} + (2 \beta - 12) q^{67} - 9 \beta q^{71} + ( - 2 \beta - 6) q^{73} - 2 \beta q^{77} + ( - 4 \beta + 8) q^{79} + (3 \beta + 8) q^{83} + ( - 4 \beta - 2) q^{85} + (2 \beta + 14) q^{89} - 4 \beta q^{91} + (\beta + 2) q^{95} + (4 \beta - 6) q^{97} +O(q^{100})$$ q + (b + 2) * q^5 - q^7 + 2*b * q^11 + 4*b * q^13 + (-3*b + 2) * q^17 + q^19 + 6*b * q^23 + (4*b + 1) * q^25 + (-3*b - 6) * q^29 + (2*b + 6) * q^31 + (-b - 2) * q^35 - 6 * q^37 + (2*b + 6) * q^41 + (-2*b - 6) * q^43 + (-b + 12) * q^47 + q^49 + (5*b - 2) * q^53 + (4*b + 4) * q^55 + (-4*b + 8) * q^59 + 2 * q^61 + (8*b + 8) * q^65 + (2*b - 12) * q^67 - 9*b * q^71 + (-2*b - 6) * q^73 - 2*b * q^77 + (-4*b + 8) * q^79 + (3*b + 8) * q^83 + (-4*b - 2) * q^85 + (2*b + 14) * q^89 - 4*b * q^91 + (b + 2) * q^95 + (4*b - 6) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{5} - 2 q^{7}+O(q^{10})$$ 2 * q + 4 * q^5 - 2 * q^7 $$2 q + 4 q^{5} - 2 q^{7} + 4 q^{17} + 2 q^{19} + 2 q^{25} - 12 q^{29} + 12 q^{31} - 4 q^{35} - 12 q^{37} + 12 q^{41} - 12 q^{43} + 24 q^{47} + 2 q^{49} - 4 q^{53} + 8 q^{55} + 16 q^{59} + 4 q^{61} + 16 q^{65} - 24 q^{67} - 12 q^{73} + 16 q^{79} + 16 q^{83} - 4 q^{85} + 28 q^{89} + 4 q^{95} - 12 q^{97}+O(q^{100})$$ 2 * q + 4 * q^5 - 2 * q^7 + 4 * q^17 + 2 * q^19 + 2 * q^25 - 12 * q^29 + 12 * q^31 - 4 * q^35 - 12 * q^37 + 12 * q^41 - 12 * q^43 + 24 * q^47 + 2 * q^49 - 4 * q^53 + 8 * q^55 + 16 * q^59 + 4 * q^61 + 16 * q^65 - 24 * q^67 - 12 * q^73 + 16 * q^79 + 16 * q^83 - 4 * q^85 + 28 * q^89 + 4 * q^95 - 12 * 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
 −1.41421 1.41421
0 0 0 0.585786 0 −1.00000 0 0 0
1.2 0 0 0 3.41421 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.by 2
3.b odd 2 1 3192.2.a.r 2
12.b even 2 1 6384.2.a.bh 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3192.2.a.r 2 3.b odd 2 1
6384.2.a.bh 2 12.b even 2 1
9576.2.a.by 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9576))$$:

 $$T_{5}^{2} - 4T_{5} + 2$$ T5^2 - 4*T5 + 2 $$T_{11}^{2} - 8$$ T11^2 - 8 $$T_{13}^{2} - 32$$ T13^2 - 32 $$T_{17}^{2} - 4T_{17} - 14$$ T17^2 - 4*T17 - 14

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 4T + 2$$
$7$ $$(T + 1)^{2}$$
$11$ $$T^{2} - 8$$
$13$ $$T^{2} - 32$$
$17$ $$T^{2} - 4T - 14$$
$19$ $$(T - 1)^{2}$$
$23$ $$T^{2} - 72$$
$29$ $$T^{2} + 12T + 18$$
$31$ $$T^{2} - 12T + 28$$
$37$ $$(T + 6)^{2}$$
$41$ $$T^{2} - 12T + 28$$
$43$ $$T^{2} + 12T + 28$$
$47$ $$T^{2} - 24T + 142$$
$53$ $$T^{2} + 4T - 46$$
$59$ $$T^{2} - 16T + 32$$
$61$ $$(T - 2)^{2}$$
$67$ $$T^{2} + 24T + 136$$
$71$ $$T^{2} - 162$$
$73$ $$T^{2} + 12T + 28$$
$79$ $$T^{2} - 16T + 32$$
$83$ $$T^{2} - 16T + 46$$
$89$ $$T^{2} - 28T + 188$$
$97$ $$T^{2} + 12T + 4$$