# Properties

 Label 1900.1.bf.a Level $1900$ Weight $1$ Character orbit 1900.bf Analytic conductor $0.948$ Analytic rank $0$ Dimension $4$ Projective image $S_{4}$ CM/RM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1900,1,Mod(657,1900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1900, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([0, 3, 8]))

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

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

chi := DirichletCharacter("1900.657");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1900 = 2^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1900.bf (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: $$0.948223524003$$ 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: no (minimal twist has level 380) Projective image: $$S_{4}$$ Projective field: Galois closure of 4.0.180500.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{12}^{4} + \zeta_{12}) q^{3} - \zeta_{12}^{5} q^{9} +O(q^{10})$$ q + (-z^4 + z) * q^3 - z^5 * q^9 $$q + ( - \zeta_{12}^{4} + \zeta_{12}) q^{3} - \zeta_{12}^{5} q^{9} + q^{11} + (\zeta_{12}^{5} + \zeta_{12}^{2}) q^{13} + ( - \zeta_{12}^{4} - \zeta_{12}) q^{17} - \zeta_{12}^{5} q^{19} + (\zeta_{12}^{3} - 1) q^{27} + \zeta_{12}^{5} q^{29} - q^{31} + ( - \zeta_{12}^{4} + \zeta_{12}) q^{33} + \zeta_{12}^{3} q^{39} + ( - \zeta_{12}^{5} + \zeta_{12}^{2}) q^{47} + \zeta_{12}^{3} q^{49} - \zeta_{12}^{2} q^{51} + ( - \zeta_{12}^{5} - \zeta_{12}^{2}) q^{53} + ( - \zeta_{12}^{3} + 1) q^{57} - \zeta_{12} q^{59} - \zeta_{12}^{2} q^{61} + (\zeta_{12}^{5} - \zeta_{12}^{2}) q^{67} + \zeta_{12}^{4} q^{71} + \zeta_{12} q^{79} + \zeta_{12}^{4} q^{81} + (\zeta_{12}^{3} - 1) q^{87} - \zeta_{12}^{5} q^{89} + (\zeta_{12}^{4} - \zeta_{12}) q^{93} - \zeta_{12}^{5} q^{99} +O(q^{100})$$ q + (-z^4 + z) * q^3 - z^5 * q^9 + q^11 + (z^5 + z^2) * q^13 + (-z^4 - z) * q^17 - z^5 * q^19 + (z^3 - 1) * q^27 + z^5 * q^29 - q^31 + (-z^4 + z) * q^33 + z^3 * q^39 + (-z^5 + z^2) * q^47 + z^3 * q^49 - z^2 * q^51 + (-z^5 - z^2) * q^53 + (-z^3 + 1) * q^57 - z * q^59 - z^2 * q^61 + (z^5 - z^2) * q^67 + z^4 * q^71 + z * q^79 + z^4 * q^81 + (z^3 - 1) * q^87 - z^5 * q^89 + (z^4 - z) * q^93 - z^5 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{3}+O(q^{10})$$ 4 * q + 2 * q^3 $$4 q + 2 q^{3} + 4 q^{11} + 2 q^{13} + 2 q^{17} - 4 q^{31} + 2 q^{33} + 2 q^{47} - 4 q^{51} - 2 q^{53} + 4 q^{57} - 2 q^{61} - 2 q^{67} - 2 q^{71} - 2 q^{81} - 4 q^{87} - 2 q^{93}+O(q^{100})$$ 4 * q + 2 * q^3 + 4 * q^11 + 2 * q^13 + 2 * q^17 - 4 * q^31 + 2 * q^33 + 2 * q^47 - 4 * q^51 - 2 * q^53 + 4 * q^57 - 2 * q^61 - 2 * q^67 - 2 * q^71 - 2 * q^81 - 4 * q^87 - 2 * q^93

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$951$$ $$\chi(n)$$ $$-\zeta_{12}^{3}$$ $$-\zeta_{12}^{2}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
657.1
 −0.866025 − 0.500000i 0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i
0 −0.366025 1.36603i 0 0 0 0 0 −0.866025 + 0.500000i 0
957.1 0 1.36603 + 0.366025i 0 0 0 0 0 0.866025 + 0.500000i 0
1493.1 0 1.36603 0.366025i 0 0 0 0 0 0.866025 0.500000i 0
1793.1 0 −0.366025 + 1.36603i 0 0 0 0 0 −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
5.c odd 4 1 inner
19.c even 3 1 inner
95.m odd 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1900.1.bf.a 4
5.b even 2 1 380.1.x.a 4
5.c odd 4 1 380.1.x.a 4
5.c odd 4 1 inner 1900.1.bf.a 4
15.d odd 2 1 3420.1.es.b 4
15.e even 4 1 3420.1.es.b 4
19.c even 3 1 inner 1900.1.bf.a 4
20.d odd 2 1 1520.1.cr.b 4
20.e even 4 1 1520.1.cr.b 4
95.i even 6 1 380.1.x.a 4
95.m odd 12 1 380.1.x.a 4
95.m odd 12 1 inner 1900.1.bf.a 4
285.n odd 6 1 3420.1.es.b 4
285.v even 12 1 3420.1.es.b 4
380.p odd 6 1 1520.1.cr.b 4
380.v even 12 1 1520.1.cr.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.1.x.a 4 5.b even 2 1
380.1.x.a 4 5.c odd 4 1
380.1.x.a 4 95.i even 6 1
380.1.x.a 4 95.m odd 12 1
1520.1.cr.b 4 20.d odd 2 1
1520.1.cr.b 4 20.e even 4 1
1520.1.cr.b 4 380.p odd 6 1
1520.1.cr.b 4 380.v even 12 1
1900.1.bf.a 4 1.a even 1 1 trivial
1900.1.bf.a 4 5.c odd 4 1 inner
1900.1.bf.a 4 19.c even 3 1 inner
1900.1.bf.a 4 95.m odd 12 1 inner
3420.1.es.b 4 15.d odd 2 1
3420.1.es.b 4 15.e even 4 1
3420.1.es.b 4 285.n odd 6 1
3420.1.es.b 4 285.v even 12 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1900, [\chi])$$.

## Hecke characteristic polynomials

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