# Properties

 Label 740.2.bg.a Level $740$ Weight $2$ Character orbit 740.bg Analytic conductor $5.909$ Analytic rank $0$ Dimension $4$ CM discriminant -4 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [740,2,Mod(47,740)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(740, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([6, 3, 8]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("740.47");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$740 = 2^{2} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 740.bg (of order $$12$$, degree $$4$$, minimal)

## 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: no Analytic conductor: $$5.90892974957$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{12}]$

## $q$-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 primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12}) q^{2} - 2 \zeta_{12} q^{4} + (\zeta_{12}^{3} - 2) q^{5} + ( - 2 \zeta_{12}^{3} + 2) q^{8} + (3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{9} +O(q^{10})$$ q + (z^3 + z^2 - z) * q^2 - 2*z * q^4 + (z^3 - 2) * q^5 + (-2*z^3 + 2) * q^8 + (3*z^3 - 3*z) * q^9 $$q + (\zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12}) q^{2} - 2 \zeta_{12} q^{4} + (\zeta_{12}^{3} - 2) q^{5} + ( - 2 \zeta_{12}^{3} + 2) q^{8} + (3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{9} + ( - \zeta_{12}^{3} - 3 \zeta_{12}^{2} + \zeta_{12}) q^{10} + ( - \zeta_{12}^{2} - \zeta_{12} + 1) q^{13} + 4 \zeta_{12}^{2} q^{16} + ( - \zeta_{12}^{3} - \zeta_{12}^{2} + \cdots + 5) q^{17} + \cdots + (7 \zeta_{12}^{2} + 7 \zeta_{12} - 7) q^{98} +O(q^{100})$$ q + (z^3 + z^2 - z) * q^2 - 2*z * q^4 + (z^3 - 2) * q^5 + (-2*z^3 + 2) * q^8 + (3*z^3 - 3*z) * q^9 + (-z^3 - 3*z^2 + z) * q^10 + (-z^2 - z + 1) * q^13 + 4*z^2 * q^16 + (-z^3 - z^2 - 4*z + 5) * q^17 + (-3*z^2 - 3*z + 3) * q^18 + (-2*z^2 + 4*z + 2) * q^20 + (-4*z^3 + 3) * q^25 + 2 * q^26 + (-2*z^3 - 10*z^2 + 5) * q^29 + (4*z^2 - 4*z - 4) * q^32 + (5*z^2 - 3*z + 5) * q^34 + 6 * q^36 + (-z^2 + 6*z + 1) * q^37 + (6*z^3 - 2) * q^40 + (-10*z^3 + 4*z^2 + 5*z - 4) * q^41 + (-6*z^3 - 3*z^2 + 6*z) * q^45 + (-7*z^3 + 7*z) * q^49 + (-z^3 + 7*z^2 + z) * q^50 + (2*z^3 + 2*z^2 - 2*z) * q^52 + (9*z^3 - 9*z^2 - 9*z) * q^53 + (3*z^3 - 3*z^2 + 7*z + 10) * q^58 + (-10*z^3 - 6*z^2 + 5*z + 6) * q^61 - 8*z^3 * q^64 + (z^2 + 3*z - 1) * q^65 + (2*z^3 + 10*z^2 - 10*z - 2) * q^68 + (6*z^3 + 6*z^2 - 6*z) * q^72 + (11*z^3 - 11) * q^73 + (7*z^3 - 5) * q^74 + (4*z^3 - 8*z^2 - 4*z) * q^80 + (-9*z^2 + 9) * q^81 + (-9*z^3 + 10*z^2 + 10*z - 9) * q^82 + (6*z^3 - 2*z^2 + 9*z - 5) * q^85 + (-8*z^3 + 5*z^2 + 8*z - 10) * q^89 + (3*z^2 + 9*z - 3) * q^90 + (-9*z^3 + 5*z^2 + 5*z - 9) * q^97 + (7*z^2 + 7*z - 7) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} - 8 q^{5} + 8 q^{8}+O(q^{10})$$ 4 * q + 2 * q^2 - 8 * q^5 + 8 * q^8 $$4 q + 2 q^{2} - 8 q^{5} + 8 q^{8} - 6 q^{10} + 2 q^{13} + 8 q^{16} + 18 q^{17} + 6 q^{18} + 4 q^{20} + 12 q^{25} + 8 q^{26} - 8 q^{32} + 30 q^{34} + 24 q^{36} + 2 q^{37} - 8 q^{40} - 8 q^{41} - 6 q^{45} + 14 q^{50} + 4 q^{52} - 18 q^{53} + 34 q^{58} + 12 q^{61} - 2 q^{65} + 12 q^{68} + 12 q^{72} - 44 q^{73} - 20 q^{74} - 16 q^{80} + 18 q^{81} - 16 q^{82} - 24 q^{85} - 30 q^{89} - 6 q^{90} - 26 q^{97} - 14 q^{98}+O(q^{100})$$ 4 * q + 2 * q^2 - 8 * q^5 + 8 * q^8 - 6 * q^10 + 2 * q^13 + 8 * q^16 + 18 * q^17 + 6 * q^18 + 4 * q^20 + 12 * q^25 + 8 * q^26 - 8 * q^32 + 30 * q^34 + 24 * q^36 + 2 * q^37 - 8 * q^40 - 8 * q^41 - 6 * q^45 + 14 * q^50 + 4 * q^52 - 18 * q^53 + 34 * q^58 + 12 * q^61 - 2 * q^65 + 12 * q^68 + 12 * q^72 - 44 * q^73 - 20 * q^74 - 16 * q^80 + 18 * q^81 - 16 * q^82 - 24 * q^85 - 30 * q^89 - 6 * q^90 - 26 * q^97 - 14 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/740\mathbb{Z}\right)^\times$$.

 $$n$$ $$261$$ $$297$$ $$371$$ $$\chi(n)$$ $$-1 + \zeta_{12}^{2}$$ $$\zeta_{12}^{3}$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1
 −0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i 0.866025 + 0.500000i
1.36603 0.366025i 0 1.73205 1.00000i −2.00000 + 1.00000i 0 0 2.00000 2.00000i 2.59808 + 1.50000i −2.36603 + 2.09808i
63.1 1.36603 + 0.366025i 0 1.73205 + 1.00000i −2.00000 1.00000i 0 0 2.00000 + 2.00000i 2.59808 1.50000i −2.36603 2.09808i
343.1 −0.366025 1.36603i 0 −1.73205 + 1.00000i −2.00000 1.00000i 0 0 2.00000 + 2.00000i −2.59808 1.50000i −0.633975 + 3.09808i
507.1 −0.366025 + 1.36603i 0 −1.73205 1.00000i −2.00000 + 1.00000i 0 0 2.00000 2.00000i −2.59808 + 1.50000i −0.633975 3.09808i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
185.s odd 12 1 inner
740.bg even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 740.2.bg.a 4
4.b odd 2 1 CM 740.2.bg.a 4
5.c odd 4 1 740.2.bg.b yes 4
20.e even 4 1 740.2.bg.b yes 4
37.c even 3 1 740.2.bg.b yes 4
148.i odd 6 1 740.2.bg.b yes 4
185.s odd 12 1 inner 740.2.bg.a 4
740.bg even 12 1 inner 740.2.bg.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.bg.a 4 1.a even 1 1 trivial
740.2.bg.a 4 4.b odd 2 1 CM
740.2.bg.a 4 185.s odd 12 1 inner
740.2.bg.a 4 740.bg even 12 1 inner
740.2.bg.b yes 4 5.c odd 4 1
740.2.bg.b yes 4 20.e even 4 1
740.2.bg.b yes 4 37.c even 3 1
740.2.bg.b yes 4 148.i odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(740, [\chi])$$:

 $$T_{3}$$ T3 $$T_{17}^{4} - 18T_{17}^{3} + 117T_{17}^{2} - 396T_{17} + 1089$$ T17^4 - 18*T17^3 + 117*T17^2 - 396*T17 + 1089

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2 T^{3} + \cdots + 4$$
$3$ $$T^{4}$$
$5$ $$(T^{2} + 4 T + 5)^{2}$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4} - 2 T^{3} + \cdots + 4$$
$17$ $$T^{4} - 18 T^{3} + \cdots + 1089$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$T^{4} + 158T^{2} + 5041$$
$31$ $$T^{4}$$
$37$ $$T^{4} - 2 T^{3} + \cdots + 1369$$
$41$ $$T^{4} + 8 T^{3} + \cdots + 3481$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$T^{4} + 18 T^{3} + \cdots + 26244$$
$59$ $$T^{4}$$
$61$ $$T^{4} - 12 T^{3} + \cdots + 1521$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$(T^{2} + 22 T + 242)^{2}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$T^{4} + 30 T^{3} + \cdots + 121$$
$97$ $$T^{4} + 26 T^{3} + \cdots + 2209$$