# Properties

 Label 3724.2.a.e Level $3724$ Weight $2$ Character orbit 3724.a Self dual yes Analytic conductor $29.736$ 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] = [3724,2,Mod(1,3724)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3724, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3724.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3724 = 2^{2} \cdot 7^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3724.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: $$29.7362897127$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{21})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 5$$ x^2 - x - 5 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 532) 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 $$\beta = \frac{1}{2}(1 + \sqrt{21})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} - 3 q^{5} + (\beta + 2) q^{9}+O(q^{10})$$ q + b * q^3 - 3 * q^5 + (b + 2) * q^9 $$q + \beta q^{3} - 3 q^{5} + (\beta + 2) q^{9} + ( - \beta - 1) q^{11} + q^{13} - 3 \beta q^{15} + ( - \beta - 1) q^{17} - q^{19} + (2 \beta - 1) q^{23} + 4 q^{25} + 5 q^{27} + (\beta + 1) q^{29} + ( - 3 \beta + 1) q^{31} + ( - 2 \beta - 5) q^{33} + 5 q^{37} + \beta q^{39} + (\beta + 1) q^{41} + 2 q^{43} + ( - 3 \beta - 6) q^{45} + ( - 2 \beta - 5) q^{47} + ( - 2 \beta - 5) q^{51} - 3 \beta q^{53} + (3 \beta + 3) q^{55} - \beta q^{57} + ( - 4 \beta - 1) q^{59} + q^{61} - 3 q^{65} + ( - 3 \beta - 1) q^{67} + (\beta + 10) q^{69} + ( - 4 \beta - 1) q^{71} + ( - 3 \beta - 8) q^{73} + 4 \beta q^{75} - 10 q^{79} + (2 \beta - 6) q^{81} + ( - 3 \beta - 6) q^{83} + (3 \beta + 3) q^{85} + (2 \beta + 5) q^{87} + ( - 2 \beta - 2) q^{89} + ( - 2 \beta - 15) q^{93} + 3 q^{95} + 7 q^{97} + ( - 4 \beta - 7) q^{99} +O(q^{100})$$ q + b * q^3 - 3 * q^5 + (b + 2) * q^9 + (-b - 1) * q^11 + q^13 - 3*b * q^15 + (-b - 1) * q^17 - q^19 + (2*b - 1) * q^23 + 4 * q^25 + 5 * q^27 + (b + 1) * q^29 + (-3*b + 1) * q^31 + (-2*b - 5) * q^33 + 5 * q^37 + b * q^39 + (b + 1) * q^41 + 2 * q^43 + (-3*b - 6) * q^45 + (-2*b - 5) * q^47 + (-2*b - 5) * q^51 - 3*b * q^53 + (3*b + 3) * q^55 - b * q^57 + (-4*b - 1) * q^59 + q^61 - 3 * q^65 + (-3*b - 1) * q^67 + (b + 10) * q^69 + (-4*b - 1) * q^71 + (-3*b - 8) * q^73 + 4*b * q^75 - 10 * q^79 + (2*b - 6) * q^81 + (-3*b - 6) * q^83 + (3*b + 3) * q^85 + (2*b + 5) * q^87 + (-2*b - 2) * q^89 + (-2*b - 15) * q^93 + 3 * q^95 + 7 * q^97 + (-4*b - 7) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} - 6 q^{5} + 5 q^{9}+O(q^{10})$$ 2 * q + q^3 - 6 * q^5 + 5 * q^9 $$2 q + q^{3} - 6 q^{5} + 5 q^{9} - 3 q^{11} + 2 q^{13} - 3 q^{15} - 3 q^{17} - 2 q^{19} + 8 q^{25} + 10 q^{27} + 3 q^{29} - q^{31} - 12 q^{33} + 10 q^{37} + q^{39} + 3 q^{41} + 4 q^{43} - 15 q^{45} - 12 q^{47} - 12 q^{51} - 3 q^{53} + 9 q^{55} - q^{57} - 6 q^{59} + 2 q^{61} - 6 q^{65} - 5 q^{67} + 21 q^{69} - 6 q^{71} - 19 q^{73} + 4 q^{75} - 20 q^{79} - 10 q^{81} - 15 q^{83} + 9 q^{85} + 12 q^{87} - 6 q^{89} - 32 q^{93} + 6 q^{95} + 14 q^{97} - 18 q^{99}+O(q^{100})$$ 2 * q + q^3 - 6 * q^5 + 5 * q^9 - 3 * q^11 + 2 * q^13 - 3 * q^15 - 3 * q^17 - 2 * q^19 + 8 * q^25 + 10 * q^27 + 3 * q^29 - q^31 - 12 * q^33 + 10 * q^37 + q^39 + 3 * q^41 + 4 * q^43 - 15 * q^45 - 12 * q^47 - 12 * q^51 - 3 * q^53 + 9 * q^55 - q^57 - 6 * q^59 + 2 * q^61 - 6 * q^65 - 5 * q^67 + 21 * q^69 - 6 * q^71 - 19 * q^73 + 4 * q^75 - 20 * q^79 - 10 * q^81 - 15 * q^83 + 9 * q^85 + 12 * q^87 - 6 * q^89 - 32 * q^93 + 6 * q^95 + 14 * q^97 - 18 * q^99

## 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
 −1.79129 2.79129
0 −1.79129 0 −3.00000 0 0 0 0.208712 0
1.2 0 2.79129 0 −3.00000 0 0 0 4.79129 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
$$2$$ $$-1$$
$$7$$ $$-1$$
$$19$$ $$+1$$

## 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 3724.2.a.e 2
7.b odd 2 1 532.2.a.c 2
21.c even 2 1 4788.2.a.g 2
28.d even 2 1 2128.2.a.k 2
56.e even 2 1 8512.2.a.m 2
56.h odd 2 1 8512.2.a.t 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
532.2.a.c 2 7.b odd 2 1
2128.2.a.k 2 28.d even 2 1
3724.2.a.e 2 1.a even 1 1 trivial
4788.2.a.g 2 21.c even 2 1
8512.2.a.m 2 56.e even 2 1
8512.2.a.t 2 56.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - T_{3} - 5$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3724))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T - 5$$
$5$ $$(T + 3)^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 3T - 3$$
$13$ $$(T - 1)^{2}$$
$17$ $$T^{2} + 3T - 3$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2} - 21$$
$29$ $$T^{2} - 3T - 3$$
$31$ $$T^{2} + T - 47$$
$37$ $$(T - 5)^{2}$$
$41$ $$T^{2} - 3T - 3$$
$43$ $$(T - 2)^{2}$$
$47$ $$T^{2} + 12T + 15$$
$53$ $$T^{2} + 3T - 45$$
$59$ $$T^{2} + 6T - 75$$
$61$ $$(T - 1)^{2}$$
$67$ $$T^{2} + 5T - 41$$
$71$ $$T^{2} + 6T - 75$$
$73$ $$T^{2} + 19T + 43$$
$79$ $$(T + 10)^{2}$$
$83$ $$T^{2} + 15T + 9$$
$89$ $$T^{2} + 6T - 12$$
$97$ $$(T - 7)^{2}$$