# Properties

 Label 3700.1.bu.a Level $3700$ Weight $1$ Character orbit 3700.bu Analytic conductor $1.847$ Analytic rank $0$ Dimension $4$ Projective image $D_{12}$ CM discriminant -4 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3700,1,Mod(2043,3700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3700, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([6, 9, 1]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("3700.2043");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3700 = 2^{2} \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3700.bu (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: $$1.84654054674$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 740) Projective image: $$D_{12}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{12} - \cdots)$$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q - \zeta_{12}^{5} q^{2} - \zeta_{12}^{4} q^{4} - \zeta_{12}^{3} q^{8} - \zeta_{12}^{5} q^{9} +O(q^{10})$$ q - z^5 * q^2 - z^4 * q^4 - z^3 * q^8 - z^5 * q^9 $$q - \zeta_{12}^{5} q^{2} - \zeta_{12}^{4} q^{4} - \zeta_{12}^{3} q^{8} - \zeta_{12}^{5} q^{9} - \zeta_{12}^{2} q^{16} - \zeta_{12}^{2} q^{17} - \zeta_{12}^{4} q^{18} + (\zeta_{12}^{2} - \zeta_{12}) q^{29} - \zeta_{12} q^{32} - \zeta_{12} q^{34} - \zeta_{12}^{3} q^{36} - \zeta_{12} q^{37} + (\zeta_{12}^{2} + 1) q^{41} - \zeta_{12}^{5} q^{49} + ( - \zeta_{12}^{5} + \zeta_{12}^{2}) q^{53} + (\zeta_{12} - 1) q^{58} + (\zeta_{12}^{5} - 1) q^{61} - q^{64} - q^{68} - \zeta_{12}^{2} q^{72} + (\zeta_{12}^{3} + 1) q^{73} - q^{74} - \zeta_{12}^{4} q^{81} + ( - \zeta_{12}^{5} + \zeta_{12}) q^{82} + (\zeta_{12}^{4} - \zeta_{12}^{3}) q^{89} + (\zeta_{12}^{5} - \zeta_{12}) q^{97} - \zeta_{12}^{4} q^{98} +O(q^{100})$$ q - z^5 * q^2 - z^4 * q^4 - z^3 * q^8 - z^5 * q^9 - z^2 * q^16 - z^2 * q^17 - z^4 * q^18 + (z^2 - z) * q^29 - z * q^32 - z * q^34 - z^3 * q^36 - z * q^37 + (z^2 + 1) * q^41 - z^5 * q^49 + (-z^5 + z^2) * q^53 + (z - 1) * q^58 + (z^5 - 1) * q^61 - q^64 - q^68 - z^2 * q^72 + (z^3 + 1) * q^73 - q^74 - z^4 * q^81 + (-z^5 + z) * q^82 + (z^4 - z^3) * q^89 + (z^5 - z) * q^97 - z^4 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{4}+O(q^{10})$$ 4 * q + 2 * q^4 $$4 q + 2 q^{4} - 2 q^{16} - 2 q^{17} + 2 q^{18} + 2 q^{29} + 6 q^{41} + 2 q^{53} - 4 q^{58} - 4 q^{61} - 4 q^{64} - 4 q^{68} - 2 q^{72} + 4 q^{73} - 4 q^{74} + 2 q^{81} - 2 q^{89} + 2 q^{98}+O(q^{100})$$ 4 * q + 2 * q^4 - 2 * q^16 - 2 * q^17 + 2 * q^18 + 2 * q^29 + 6 * q^41 + 2 * q^53 - 4 * q^58 - 4 * q^61 - 4 * q^64 - 4 * q^68 - 2 * q^72 + 4 * q^73 - 4 * q^74 + 2 * q^81 - 2 * q^89 + 2 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3700\mathbb{Z}\right)^\times$$.

 $$n$$ $$1001$$ $$1777$$ $$1851$$ $$\chi(n)$$ $$-\zeta_{12}$$ $$\zeta_{12}^{3}$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2043.1
 −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i
−0.866025 + 0.500000i 0 0.500000 0.866025i 0 0 0 1.00000i −0.866025 + 0.500000i 0
2243.1 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0 0 0 1.00000i 0.866025 + 0.500000i 0
3307.1 −0.866025 0.500000i 0 0.500000 + 0.866025i 0 0 0 1.00000i −0.866025 0.500000i 0
3507.1 0.866025 0.500000i 0 0.500000 0.866025i 0 0 0 1.00000i 0.866025 0.500000i 0
 $$n$$: 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.u even 12 1 inner
740.bm odd 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3700.1.bu.a 4
4.b odd 2 1 CM 3700.1.bu.a 4
5.b even 2 1 740.1.bm.a yes 4
5.c odd 4 1 740.1.bj.a 4
5.c odd 4 1 3700.1.br.a 4
20.d odd 2 1 740.1.bm.a yes 4
20.e even 4 1 740.1.bj.a 4
20.e even 4 1 3700.1.br.a 4
37.g odd 12 1 3700.1.br.a 4
148.l even 12 1 3700.1.br.a 4
185.p even 12 1 740.1.bm.a yes 4
185.q odd 12 1 740.1.bj.a 4
185.u even 12 1 inner 3700.1.bu.a 4
740.bj odd 12 1 740.1.bm.a yes 4
740.bm odd 12 1 inner 3700.1.bu.a 4
740.bo even 12 1 740.1.bj.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.1.bj.a 4 5.c odd 4 1
740.1.bj.a 4 20.e even 4 1
740.1.bj.a 4 185.q odd 12 1
740.1.bj.a 4 740.bo even 12 1
740.1.bm.a yes 4 5.b even 2 1
740.1.bm.a yes 4 20.d odd 2 1
740.1.bm.a yes 4 185.p even 12 1
740.1.bm.a yes 4 740.bj odd 12 1
3700.1.br.a 4 5.c odd 4 1
3700.1.br.a 4 20.e even 4 1
3700.1.br.a 4 37.g odd 12 1
3700.1.br.a 4 148.l even 12 1
3700.1.bu.a 4 1.a even 1 1 trivial
3700.1.bu.a 4 4.b odd 2 1 CM
3700.1.bu.a 4 185.u even 12 1 inner
3700.1.bu.a 4 740.bm odd 12 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(3700, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{2} + 1$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$(T^{2} + T + 1)^{2}$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$T^{4} - 2 T^{3} + \cdots + 1$$
$31$ $$T^{4}$$
$37$ $$T^{4} - T^{2} + 1$$
$41$ $$(T^{2} - 3 T + 3)^{2}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$T^{4} - 2 T^{3} + \cdots + 4$$
$59$ $$T^{4}$$
$61$ $$T^{4} + 4 T^{3} + \cdots + 1$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$(T^{2} - 2 T + 2)^{2}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$T^{4} + 2 T^{3} + \cdots + 1$$
$97$ $$(T^{2} - 3)^{2}$$