# Properties

 Label 3744.2.a.v Level $3744$ Weight $2$ Character orbit 3744.a Self dual yes Analytic conductor $29.896$ 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] = [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: $$0$$ 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: $$2$$ 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 $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{5} + (\beta - 3) q^{7}+O(q^{10})$$ q + (b + 1) * q^5 + (b - 3) * q^7 $$q + (\beta + 1) q^{5} + (\beta - 3) q^{7} - 2 \beta q^{11} - q^{13} + 2 \beta q^{17} + ( - \beta - 1) q^{19} + (2 \beta + 2) q^{23} + (2 \beta + 1) q^{25} + ( - 2 \beta + 4) q^{29} + (\beta - 7) q^{31} + ( - 2 \beta + 2) q^{35} + 2 \beta q^{37} + ( - \beta + 7) q^{41} + (2 \beta - 2) q^{43} + (2 \beta + 4) q^{47} + ( - 6 \beta + 7) q^{49} + (2 \beta + 4) q^{53} + ( - 2 \beta - 10) q^{55} + (4 \beta + 2) q^{59} + (2 \beta + 8) q^{61} + ( - \beta - 1) q^{65} + ( - 3 \beta + 1) q^{67} + 10 q^{71} + 2 \beta q^{73} + (6 \beta - 10) q^{77} + 4 \beta q^{79} + (4 \beta + 6) q^{83} + (2 \beta + 10) q^{85} + ( - 3 \beta + 9) q^{89} + ( - \beta + 3) q^{91} + ( - 2 \beta - 6) q^{95} + ( - 2 \beta + 12) q^{97} +O(q^{100})$$ q + (b + 1) * q^5 + (b - 3) * q^7 - 2*b * q^11 - q^13 + 2*b * q^17 + (-b - 1) * q^19 + (2*b + 2) * q^23 + (2*b + 1) * q^25 + (-2*b + 4) * q^29 + (b - 7) * q^31 + (-2*b + 2) * q^35 + 2*b * q^37 + (-b + 7) * q^41 + (2*b - 2) * q^43 + (2*b + 4) * q^47 + (-6*b + 7) * q^49 + (2*b + 4) * q^53 + (-2*b - 10) * q^55 + (4*b + 2) * q^59 + (2*b + 8) * q^61 + (-b - 1) * q^65 + (-3*b + 1) * q^67 + 10 * q^71 + 2*b * q^73 + (6*b - 10) * q^77 + 4*b * q^79 + (4*b + 6) * q^83 + (2*b + 10) * q^85 + (-3*b + 9) * q^89 + (-b + 3) * q^91 + (-2*b - 6) * q^95 + (-2*b + 12) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5} - 6 q^{7}+O(q^{10})$$ 2 * q + 2 * q^5 - 6 * q^7 $$2 q + 2 q^{5} - 6 q^{7} - 2 q^{13} - 2 q^{19} + 4 q^{23} + 2 q^{25} + 8 q^{29} - 14 q^{31} + 4 q^{35} + 14 q^{41} - 4 q^{43} + 8 q^{47} + 14 q^{49} + 8 q^{53} - 20 q^{55} + 4 q^{59} + 16 q^{61} - 2 q^{65} + 2 q^{67} + 20 q^{71} - 20 q^{77} + 12 q^{83} + 20 q^{85} + 18 q^{89} + 6 q^{91} - 12 q^{95} + 24 q^{97}+O(q^{100})$$ 2 * q + 2 * q^5 - 6 * q^7 - 2 * q^13 - 2 * q^19 + 4 * q^23 + 2 * q^25 + 8 * q^29 - 14 * q^31 + 4 * q^35 + 14 * q^41 - 4 * q^43 + 8 * q^47 + 14 * q^49 + 8 * q^53 - 20 * q^55 + 4 * q^59 + 16 * q^61 - 2 * q^65 + 2 * q^67 + 20 * q^71 - 20 * q^77 + 12 * q^83 + 20 * q^85 + 18 * q^89 + 6 * q^91 - 12 * q^95 + 24 * 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 −1.23607 0 −5.23607 0 0 0
1.2 0 0 0 3.23607 0 −0.763932 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.v 2
3.b odd 2 1 1248.2.a.m yes 2
4.b odd 2 1 3744.2.a.w 2
8.b even 2 1 7488.2.a.cg 2
8.d odd 2 1 7488.2.a.ch 2
12.b even 2 1 1248.2.a.k 2
24.f even 2 1 2496.2.a.bj 2
24.h odd 2 1 2496.2.a.bg 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1248.2.a.k 2 12.b even 2 1
1248.2.a.m yes 2 3.b odd 2 1
2496.2.a.bg 2 24.h odd 2 1
2496.2.a.bj 2 24.f even 2 1
3744.2.a.v 2 1.a even 1 1 trivial
3744.2.a.w 2 4.b odd 2 1
7488.2.a.cg 2 8.b even 2 1
7488.2.a.ch 2 8.d odd 2 1

## Hecke kernels

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

 $$T_{5}^{2} - 2T_{5} - 4$$ T5^2 - 2*T5 - 4 $$T_{7}^{2} + 6T_{7} + 4$$ T7^2 + 6*T7 + 4 $$T_{11}^{2} - 20$$ T11^2 - 20 $$T_{29}^{2} - 8T_{29} - 4$$ T29^2 - 8*T29 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 2T - 4$$
$7$ $$T^{2} + 6T + 4$$
$11$ $$T^{2} - 20$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2} - 20$$
$19$ $$T^{2} + 2T - 4$$
$23$ $$T^{2} - 4T - 16$$
$29$ $$T^{2} - 8T - 4$$
$31$ $$T^{2} + 14T + 44$$
$37$ $$T^{2} - 20$$
$41$ $$T^{2} - 14T + 44$$
$43$ $$T^{2} + 4T - 16$$
$47$ $$T^{2} - 8T - 4$$
$53$ $$T^{2} - 8T - 4$$
$59$ $$T^{2} - 4T - 76$$
$61$ $$T^{2} - 16T + 44$$
$67$ $$T^{2} - 2T - 44$$
$71$ $$(T - 10)^{2}$$
$73$ $$T^{2} - 20$$
$79$ $$T^{2} - 80$$
$83$ $$T^{2} - 12T - 44$$
$89$ $$T^{2} - 18T + 36$$
$97$ $$T^{2} - 24T + 124$$