# Properties

 Label 3762.2.a.bd Level $3762$ Weight $2$ Character orbit 3762.a Self dual yes Analytic conductor $30.040$ 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] = [3762,2,Mod(1,3762)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3762, 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("3762.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3762 = 2 \cdot 3^{2} \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3762.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: $$30.0397212404$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.469.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 5x + 4$$ x^3 - x^2 - 5*x + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 418) 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + (\beta_1 - 2) q^{5} - \beta_{2} q^{7} - q^{8}+O(q^{10})$$ q - q^2 + q^4 + (b1 - 2) * q^5 - b2 * q^7 - q^8 $$q - q^{2} + q^{4} + (\beta_1 - 2) q^{5} - \beta_{2} q^{7} - q^{8} + ( - \beta_1 + 2) q^{10} + q^{11} + (\beta_{2} + 2) q^{13} + \beta_{2} q^{14} + q^{16} + ( - 2 \beta_{2} - 2) q^{17} - q^{19} + (\beta_1 - 2) q^{20} - q^{22} + 2 \beta_{2} q^{23} + (\beta_{2} - 4 \beta_1 + 3) q^{25} + ( - \beta_{2} - 2) q^{26} - \beta_{2} q^{28} + (2 \beta_{2} + \beta_1 - 2) q^{29} + (\beta_{2} - 2 \beta_1) q^{31} - q^{32} + (2 \beta_{2} + 2) q^{34} + (\beta_{2} - \beta_1) q^{35} + ( - 2 \beta_{2} + 2 \beta_1 - 6) q^{37} + q^{38} + ( - \beta_1 + 2) q^{40} + (\beta_{2} - 2 \beta_1 + 2) q^{41} + (2 \beta_{2} - 3 \beta_1) q^{43} + q^{44} - 2 \beta_{2} q^{46} + (2 \beta_{2} + 2 \beta_1) q^{47} + ( - 2 \beta_{2} + \beta_1 - 3) q^{49} + ( - \beta_{2} + 4 \beta_1 - 3) q^{50} + (\beta_{2} + 2) q^{52} + ( - 2 \beta_1 + 6) q^{53} + (\beta_1 - 2) q^{55} + \beta_{2} q^{56} + ( - 2 \beta_{2} - \beta_1 + 2) q^{58} + ( - 4 \beta_{2} - 4 \beta_1 + 4) q^{59} + 2 q^{61} + ( - \beta_{2} + 2 \beta_1) q^{62} + q^{64} + ( - \beta_{2} + 3 \beta_1 - 4) q^{65} - \beta_{2} q^{67} + ( - 2 \beta_{2} - 2) q^{68} + ( - \beta_{2} + \beta_1) q^{70} + ( - 2 \beta_{2} + \beta_1) q^{71} + (4 \beta_{2} + 2 \beta_1 + 2) q^{73} + (2 \beta_{2} - 2 \beta_1 + 6) q^{74} - q^{76} - \beta_{2} q^{77} + (2 \beta_{2} + 4 \beta_1) q^{79} + (\beta_1 - 2) q^{80} + ( - \beta_{2} + 2 \beta_1 - 2) q^{82} + (2 \beta_{2} - 5 \beta_1) q^{83} + (2 \beta_{2} - 4 \beta_1 + 4) q^{85} + ( - 2 \beta_{2} + 3 \beta_1) q^{86} - q^{88} + (2 \beta_{2} - 8 \beta_1 - 2) q^{89} + ( - \beta_1 - 4) q^{91} + 2 \beta_{2} q^{92} + ( - 2 \beta_{2} - 2 \beta_1) q^{94} + ( - \beta_1 + 2) q^{95} + ( - 2 \beta_{2} + 2 \beta_1 - 6) q^{97} + (2 \beta_{2} - \beta_1 + 3) q^{98}+O(q^{100})$$ q - q^2 + q^4 + (b1 - 2) * q^5 - b2 * q^7 - q^8 + (-b1 + 2) * q^10 + q^11 + (b2 + 2) * q^13 + b2 * q^14 + q^16 + (-2*b2 - 2) * q^17 - q^19 + (b1 - 2) * q^20 - q^22 + 2*b2 * q^23 + (b2 - 4*b1 + 3) * q^25 + (-b2 - 2) * q^26 - b2 * q^28 + (2*b2 + b1 - 2) * q^29 + (b2 - 2*b1) * q^31 - q^32 + (2*b2 + 2) * q^34 + (b2 - b1) * q^35 + (-2*b2 + 2*b1 - 6) * q^37 + q^38 + (-b1 + 2) * q^40 + (b2 - 2*b1 + 2) * q^41 + (2*b2 - 3*b1) * q^43 + q^44 - 2*b2 * q^46 + (2*b2 + 2*b1) * q^47 + (-2*b2 + b1 - 3) * q^49 + (-b2 + 4*b1 - 3) * q^50 + (b2 + 2) * q^52 + (-2*b1 + 6) * q^53 + (b1 - 2) * q^55 + b2 * q^56 + (-2*b2 - b1 + 2) * q^58 + (-4*b2 - 4*b1 + 4) * q^59 + 2 * q^61 + (-b2 + 2*b1) * q^62 + q^64 + (-b2 + 3*b1 - 4) * q^65 - b2 * q^67 + (-2*b2 - 2) * q^68 + (-b2 + b1) * q^70 + (-2*b2 + b1) * q^71 + (4*b2 + 2*b1 + 2) * q^73 + (2*b2 - 2*b1 + 6) * q^74 - q^76 - b2 * q^77 + (2*b2 + 4*b1) * q^79 + (b1 - 2) * q^80 + (-b2 + 2*b1 - 2) * q^82 + (2*b2 - 5*b1) * q^83 + (2*b2 - 4*b1 + 4) * q^85 + (-2*b2 + 3*b1) * q^86 - q^88 + (2*b2 - 8*b1 - 2) * q^89 + (-b1 - 4) * q^91 + 2*b2 * q^92 + (-2*b2 - 2*b1) * q^94 + (-b1 + 2) * q^95 + (-2*b2 + 2*b1 - 6) * q^97 + (2*b2 - b1 + 3) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{2} + 3 q^{4} - 5 q^{5} + q^{7} - 3 q^{8}+O(q^{10})$$ 3 * q - 3 * q^2 + 3 * q^4 - 5 * q^5 + q^7 - 3 * q^8 $$3 q - 3 q^{2} + 3 q^{4} - 5 q^{5} + q^{7} - 3 q^{8} + 5 q^{10} + 3 q^{11} + 5 q^{13} - q^{14} + 3 q^{16} - 4 q^{17} - 3 q^{19} - 5 q^{20} - 3 q^{22} - 2 q^{23} + 4 q^{25} - 5 q^{26} + q^{28} - 7 q^{29} - 3 q^{31} - 3 q^{32} + 4 q^{34} - 2 q^{35} - 14 q^{37} + 3 q^{38} + 5 q^{40} + 3 q^{41} - 5 q^{43} + 3 q^{44} + 2 q^{46} - 6 q^{49} - 4 q^{50} + 5 q^{52} + 16 q^{53} - 5 q^{55} - q^{56} + 7 q^{58} + 12 q^{59} + 6 q^{61} + 3 q^{62} + 3 q^{64} - 8 q^{65} + q^{67} - 4 q^{68} + 2 q^{70} + 3 q^{71} + 4 q^{73} + 14 q^{74} - 3 q^{76} + q^{77} + 2 q^{79} - 5 q^{80} - 3 q^{82} - 7 q^{83} + 6 q^{85} + 5 q^{86} - 3 q^{88} - 16 q^{89} - 13 q^{91} - 2 q^{92} + 5 q^{95} - 14 q^{97} + 6 q^{98}+O(q^{100})$$ 3 * q - 3 * q^2 + 3 * q^4 - 5 * q^5 + q^7 - 3 * q^8 + 5 * q^10 + 3 * q^11 + 5 * q^13 - q^14 + 3 * q^16 - 4 * q^17 - 3 * q^19 - 5 * q^20 - 3 * q^22 - 2 * q^23 + 4 * q^25 - 5 * q^26 + q^28 - 7 * q^29 - 3 * q^31 - 3 * q^32 + 4 * q^34 - 2 * q^35 - 14 * q^37 + 3 * q^38 + 5 * q^40 + 3 * q^41 - 5 * q^43 + 3 * q^44 + 2 * q^46 - 6 * q^49 - 4 * q^50 + 5 * q^52 + 16 * q^53 - 5 * q^55 - q^56 + 7 * q^58 + 12 * q^59 + 6 * q^61 + 3 * q^62 + 3 * q^64 - 8 * q^65 + q^67 - 4 * q^68 + 2 * q^70 + 3 * q^71 + 4 * q^73 + 14 * q^74 - 3 * q^76 + q^77 + 2 * q^79 - 5 * q^80 - 3 * q^82 - 7 * q^83 + 6 * q^85 + 5 * q^86 - 3 * q^88 - 16 * q^89 - 13 * q^91 - 2 * q^92 + 5 * q^95 - 14 * q^97 + 6 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 5x + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ v^2 - 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ b2 + 4

## 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
 −2.16425 0.772866 2.39138
−1.00000 0 1.00000 −4.16425 0 −0.683969 −1.00000 0 4.16425
1.2 −1.00000 0 1.00000 −1.22713 0 3.40268 −1.00000 0 1.22713
1.3 −1.00000 0 1.00000 0.391382 0 −1.71871 −1.00000 0 −0.391382
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$11$$ $$-1$$
$$19$$ $$1$$

## 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 3762.2.a.bd 3
3.b odd 2 1 418.2.a.h 3
12.b even 2 1 3344.2.a.p 3
33.d even 2 1 4598.2.a.bm 3
57.d even 2 1 7942.2.a.bc 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.a.h 3 3.b odd 2 1
3344.2.a.p 3 12.b even 2 1
3762.2.a.bd 3 1.a even 1 1 trivial
4598.2.a.bm 3 33.d even 2 1
7942.2.a.bc 3 57.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3762))$$:

 $$T_{5}^{3} + 5T_{5}^{2} + 3T_{5} - 2$$ T5^3 + 5*T5^2 + 3*T5 - 2 $$T_{7}^{3} - T_{7}^{2} - 7T_{7} - 4$$ T7^3 - T7^2 - 7*T7 - 4 $$T_{13}^{3} - 5T_{13}^{2} + T_{13} + 14$$ T13^3 - 5*T13^2 + T13 + 14 $$T_{17}^{3} + 4T_{17}^{2} - 24T_{17} - 88$$ T17^3 + 4*T17^2 - 24*T17 - 88

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{3}$$
$3$ $$T^{3}$$
$5$ $$T^{3} + 5 T^{2} + \cdots - 2$$
$7$ $$T^{3} - T^{2} - 7T - 4$$
$11$ $$(T - 1)^{3}$$
$13$ $$T^{3} - 5T^{2} + T + 14$$
$17$ $$T^{3} + 4 T^{2} + \cdots - 88$$
$19$ $$(T + 1)^{3}$$
$23$ $$T^{3} + 2 T^{2} + \cdots + 32$$
$29$ $$T^{3} + 7 T^{2} + \cdots - 86$$
$31$ $$T^{3} + 3 T^{2} + \cdots - 76$$
$37$ $$T^{3} + 14 T^{2} + \cdots - 128$$
$41$ $$T^{3} - 3 T^{2} + \cdots - 22$$
$43$ $$T^{3} + 5 T^{2} + \cdots - 268$$
$47$ $$T^{3} - 52T - 128$$
$53$ $$T^{3} - 16 T^{2} + \cdots - 56$$
$59$ $$T^{3} - 12 T^{2} + \cdots + 1792$$
$61$ $$(T - 2)^{3}$$
$67$ $$T^{3} - T^{2} - 7T - 4$$
$71$ $$T^{3} - 3 T^{2} + \cdots - 28$$
$73$ $$T^{3} - 4 T^{2} + \cdots + 56$$
$79$ $$T^{3} - 2 T^{2} + \cdots - 352$$
$83$ $$T^{3} + 7 T^{2} + \cdots - 1108$$
$89$ $$T^{3} + 16 T^{2} + \cdots - 4424$$
$97$ $$T^{3} + 14 T^{2} + \cdots - 128$$