Properties

 Label 9576.2.a.bt Level $9576$ Weight $2$ Character orbit 9576.a Self dual yes Analytic conductor $76.465$ Analytic rank $1$ 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: $$1$$ 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: $$\Z[a_1, \ldots, a_{13}]$$ 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.

 $$f(q)$$ $$=$$ $$q + q^{5} - q^{7}+O(q^{10})$$ q + q^5 - q^7 $$q + q^{5} - q^{7} + (3 \beta - 1) q^{11} + ( - 4 \beta + 3) q^{13} + ( - \beta + 3) q^{17} - q^{19} + (2 \beta - 1) q^{23} - 4 q^{25} + ( - 5 \beta + 3) q^{29} + (5 \beta - 7) q^{31} - q^{35} + (4 \beta - 7) q^{37} + (\beta + 1) q^{41} - 2 q^{43} + ( - 6 \beta - 3) q^{47} + q^{49} + ( - 5 \beta + 8) q^{53} + (3 \beta - 1) q^{55} + (8 \beta - 3) q^{59} - 5 q^{61} + ( - 4 \beta + 3) q^{65} + (7 \beta - 11) q^{67} - 9 q^{71} + (3 \beta - 4) q^{73} + ( - 3 \beta + 1) q^{77} + ( - 8 \beta + 10) q^{79} + ( - \beta - 6) q^{83} + ( - \beta + 3) q^{85} + ( - 2 \beta + 6) q^{89} + (4 \beta - 3) q^{91} - q^{95} + ( - 4 \beta - 7) q^{97}+O(q^{100})$$ q + q^5 - q^7 + (3*b - 1) * q^11 + (-4*b + 3) * q^13 + (-b + 3) * q^17 - q^19 + (2*b - 1) * q^23 - 4 * q^25 + (-5*b + 3) * q^29 + (5*b - 7) * q^31 - q^35 + (4*b - 7) * q^37 + (b + 1) * q^41 - 2 * q^43 + (-6*b - 3) * q^47 + q^49 + (-5*b + 8) * q^53 + (3*b - 1) * q^55 + (8*b - 3) * q^59 - 5 * q^61 + (-4*b + 3) * q^65 + (7*b - 11) * q^67 - 9 * q^71 + (3*b - 4) * q^73 + (-3*b + 1) * q^77 + (-8*b + 10) * q^79 + (-b - 6) * q^83 + (-b + 3) * q^85 + (-2*b + 6) * q^89 + (4*b - 3) * q^91 - q^95 + (-4*b - 7) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5} - 2 q^{7}+O(q^{10})$$ 2 * q + 2 * q^5 - 2 * q^7 $$2 q + 2 q^{5} - 2 q^{7} + q^{11} + 2 q^{13} + 5 q^{17} - 2 q^{19} - 8 q^{25} + q^{29} - 9 q^{31} - 2 q^{35} - 10 q^{37} + 3 q^{41} - 4 q^{43} - 12 q^{47} + 2 q^{49} + 11 q^{53} + q^{55} + 2 q^{59} - 10 q^{61} + 2 q^{65} - 15 q^{67} - 18 q^{71} - 5 q^{73} - q^{77} + 12 q^{79} - 13 q^{83} + 5 q^{85} + 10 q^{89} - 2 q^{91} - 2 q^{95} - 18 q^{97}+O(q^{100})$$ 2 * q + 2 * q^5 - 2 * q^7 + q^11 + 2 * q^13 + 5 * q^17 - 2 * q^19 - 8 * q^25 + q^29 - 9 * q^31 - 2 * q^35 - 10 * q^37 + 3 * q^41 - 4 * q^43 - 12 * q^47 + 2 * q^49 + 11 * q^53 + q^55 + 2 * q^59 - 10 * q^61 + 2 * q^65 - 15 * q^67 - 18 * q^71 - 5 * q^73 - q^77 + 12 * q^79 - 13 * q^83 + 5 * q^85 + 10 * q^89 - 2 * q^91 - 2 * q^95 - 18 * 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
 −0.618034 1.61803
0 0 0 1.00000 0 −1.00000 0 0 0
1.2 0 0 0 1.00000 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.bt 2
3.b odd 2 1 1064.2.a.d 2
12.b even 2 1 2128.2.a.f 2
21.c even 2 1 7448.2.a.z 2
24.f even 2 1 8512.2.a.y 2
24.h odd 2 1 8512.2.a.q 2

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

 $$T_{5} - 1$$ T5 - 1 $$T_{11}^{2} - T_{11} - 11$$ T11^2 - T11 - 11 $$T_{13}^{2} - 2T_{13} - 19$$ T13^2 - 2*T13 - 19 $$T_{17}^{2} - 5T_{17} + 5$$ T17^2 - 5*T17 + 5

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$(T + 1)^{2}$$
$11$ $$T^{2} - T - 11$$
$13$ $$T^{2} - 2T - 19$$
$17$ $$T^{2} - 5T + 5$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2} - 5$$
$29$ $$T^{2} - T - 31$$
$31$ $$T^{2} + 9T - 11$$
$37$ $$T^{2} + 10T + 5$$
$41$ $$T^{2} - 3T + 1$$
$43$ $$(T + 2)^{2}$$
$47$ $$T^{2} + 12T - 9$$
$53$ $$T^{2} - 11T - 1$$
$59$ $$T^{2} - 2T - 79$$
$61$ $$(T + 5)^{2}$$
$67$ $$T^{2} + 15T - 5$$
$71$ $$(T + 9)^{2}$$
$73$ $$T^{2} + 5T - 5$$
$79$ $$T^{2} - 12T - 44$$
$83$ $$T^{2} + 13T + 41$$
$89$ $$T^{2} - 10T + 20$$
$97$ $$T^{2} + 18T + 61$$