# Properties

 Label 3744.2.a.z Level $3744$ Weight $2$ Character orbit 3744.a Self dual yes Analytic conductor $29.896$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3744,2,Mod(1,3744)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3744, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3744.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3744 = 2^{5} \cdot 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3744.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: $$29.8959905168$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 1248) 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 - 1) q^{5} - \beta_{2} q^{7}+O(q^{10})$$ q + (-b1 - 1) * q^5 - b2 * q^7 $$q + ( - \beta_1 - 1) q^{5} - \beta_{2} q^{7} + (\beta_{2} + \beta_1 - 1) q^{11} + q^{13} - 2 q^{17} + \beta_{2} q^{19} - 2 \beta_{2} q^{23} + (2 \beta_{2} + 2 \beta_1 + 5) q^{25} - 2 q^{29} + (\beta_{2} + 2 \beta_1 - 2) q^{31} + (2 \beta_1 - 2) q^{35} - 2 \beta_1 q^{37} + (\beta_1 - 3) q^{41} + (2 \beta_{2} + 2 \beta_1 + 2) q^{43} + (\beta_{2} + \beta_1 + 3) q^{47} + ( - 2 \beta_{2} - 2 \beta_1 + 3) q^{49} + (2 \beta_1 - 4) q^{53} + ( - 2 \beta_{2} - 2 \beta_1 - 6) q^{55} + (\beta_{2} - \beta_1 - 7) q^{59} + (2 \beta_{2} - 2 \beta_1) q^{61} + ( - \beta_1 - 1) q^{65} + ( - 3 \beta_{2} - 2 \beta_1 - 6) q^{67} + ( - \beta_{2} + \beta_1 + 3) q^{71} + (2 \beta_1 + 8) q^{73} + (4 \beta_{2} - 8) q^{77} - 12 q^{79} + ( - \beta_{2} + \beta_1 - 1) q^{83} + (2 \beta_1 + 2) q^{85} + (3 \beta_1 + 3) q^{89} - \beta_{2} q^{91} + ( - 2 \beta_1 + 2) q^{95} + 2 \beta_1 q^{97}+O(q^{100})$$ q + (-b1 - 1) * q^5 - b2 * q^7 + (b2 + b1 - 1) * q^11 + q^13 - 2 * q^17 + b2 * q^19 - 2*b2 * q^23 + (2*b2 + 2*b1 + 5) * q^25 - 2 * q^29 + (b2 + 2*b1 - 2) * q^31 + (2*b1 - 2) * q^35 - 2*b1 * q^37 + (b1 - 3) * q^41 + (2*b2 + 2*b1 + 2) * q^43 + (b2 + b1 + 3) * q^47 + (-2*b2 - 2*b1 + 3) * q^49 + (2*b1 - 4) * q^53 + (-2*b2 - 2*b1 - 6) * q^55 + (b2 - b1 - 7) * q^59 + (2*b2 - 2*b1) * q^61 + (-b1 - 1) * q^65 + (-3*b2 - 2*b1 - 6) * q^67 + (-b2 + b1 + 3) * q^71 + (2*b1 + 8) * q^73 + (4*b2 - 8) * q^77 - 12 * q^79 + (-b2 + b1 - 1) * q^83 + (2*b1 + 2) * q^85 + (3*b1 + 3) * q^89 - b2 * q^91 + (-2*b1 + 2) * q^95 + 2*b1 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{5}+O(q^{10})$$ 3 * q - 2 * q^5 $$3 q - 2 q^{5} - 4 q^{11} + 3 q^{13} - 6 q^{17} + 13 q^{25} - 6 q^{29} - 8 q^{31} - 8 q^{35} + 2 q^{37} - 10 q^{41} + 4 q^{43} + 8 q^{47} + 11 q^{49} - 14 q^{53} - 16 q^{55} - 20 q^{59} + 2 q^{61} - 2 q^{65} - 16 q^{67} + 8 q^{71} + 22 q^{73} - 24 q^{77} - 36 q^{79} - 4 q^{83} + 4 q^{85} + 6 q^{89} + 8 q^{95} - 2 q^{97}+O(q^{100})$$ 3 * q - 2 * q^5 - 4 * q^11 + 3 * q^13 - 6 * q^17 + 13 * q^25 - 6 * q^29 - 8 * q^31 - 8 * q^35 + 2 * q^37 - 10 * q^41 + 4 * q^43 + 8 * q^47 + 11 * q^49 - 14 * q^53 - 16 * q^55 - 20 * q^59 + 2 * q^61 - 2 * q^65 - 16 * q^67 + 8 * q^71 + 22 * q^73 - 24 * q^77 - 36 * q^79 - 4 * q^83 + 4 * q^85 + 6 * q^89 + 8 * q^95 - 2 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 3x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu - 1$$ 2*v - 1 $$\beta_{2}$$ $$=$$ $$2\nu^{2} - 2\nu - 4$$ 2*v^2 - 2*v - 4
 $$\nu$$ $$=$$ $$( \beta _1 + 1 ) / 2$$ (b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{2} + \beta _1 + 5 ) / 2$$ (b2 + b1 + 5) / 2

## 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
 2.17009 0.311108 −1.48119
0 0 0 −4.34017 0 −1.07838 0 0 0
1.2 0 0 0 −0.622216 0 4.42864 0 0 0
1.3 0 0 0 2.96239 0 −3.35026 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
$$2$$ $$+1$$
$$3$$ $$-1$$
$$13$$ $$-1$$

## 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 3744.2.a.z 3
3.b odd 2 1 1248.2.a.p yes 3
4.b odd 2 1 3744.2.a.ba 3
8.b even 2 1 7488.2.a.cy 3
8.d odd 2 1 7488.2.a.cx 3
12.b even 2 1 1248.2.a.o 3
24.f even 2 1 2496.2.a.bl 3
24.h odd 2 1 2496.2.a.bk 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1248.2.a.o 3 12.b even 2 1
1248.2.a.p yes 3 3.b odd 2 1
2496.2.a.bk 3 24.h odd 2 1
2496.2.a.bl 3 24.f even 2 1
3744.2.a.z 3 1.a even 1 1 trivial
3744.2.a.ba 3 4.b odd 2 1
7488.2.a.cx 3 8.d odd 2 1
7488.2.a.cy 3 8.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3744))$$:

 $$T_{5}^{3} + 2T_{5}^{2} - 12T_{5} - 8$$ T5^3 + 2*T5^2 - 12*T5 - 8 $$T_{7}^{3} - 16T_{7} - 16$$ T7^3 - 16*T7 - 16 $$T_{11}^{3} + 4T_{11}^{2} - 16T_{11} - 32$$ T11^3 + 4*T11^2 - 16*T11 - 32 $$T_{29} + 2$$ T29 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3}$$
$5$ $$T^{3} + 2 T^{2} + \cdots - 8$$
$7$ $$T^{3} - 16T - 16$$
$11$ $$T^{3} + 4 T^{2} + \cdots - 32$$
$13$ $$(T - 1)^{3}$$
$17$ $$(T + 2)^{3}$$
$19$ $$T^{3} - 16T + 16$$
$23$ $$T^{3} - 64T - 128$$
$29$ $$(T + 2)^{3}$$
$31$ $$T^{3} + 8 T^{2} + \cdots - 272$$
$37$ $$T^{3} - 2 T^{2} + \cdots + 40$$
$41$ $$T^{3} + 10 T^{2} + \cdots - 8$$
$43$ $$T^{3} - 4 T^{2} + \cdots + 64$$
$47$ $$T^{3} - 8T^{2} + 32$$
$53$ $$T^{3} + 14 T^{2} + \cdots - 152$$
$59$ $$T^{3} + 20 T^{2} + \cdots - 32$$
$61$ $$T^{3} - 2 T^{2} + \cdots - 536$$
$67$ $$T^{3} + 16 T^{2} + \cdots - 1040$$
$71$ $$T^{3} - 8 T^{2} + \cdots + 160$$
$73$ $$T^{3} - 22 T^{2} + \cdots - 8$$
$79$ $$(T + 12)^{3}$$
$83$ $$T^{3} + 4 T^{2} + \cdots + 32$$
$89$ $$T^{3} - 6 T^{2} + \cdots + 216$$
$97$ $$T^{3} + 2 T^{2} + \cdots - 40$$