# Properties

 Label 2100.1.bh.a Level $2100$ Weight $1$ Character orbit 2100.bh Analytic conductor $1.048$ Analytic rank $0$ Dimension $4$ Projective image $D_{3}$ CM discriminant -3 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2100,1,Mod(149,2100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2100, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 3, 2]))

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

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

chi := DirichletCharacter("2100.149");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2100.bh (of order $$6$$, degree $$2$$, not 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: $$1.04803652653$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 84) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.588.1 Artin image: $S_3\times C_{12}$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{24} - \cdots)$$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \zeta_{12} q^{3} + \zeta_{12}^{5} q^{7} + \zeta_{12}^{2} q^{9} +O(q^{10})$$ q - z * q^3 + z^5 * q^7 + z^2 * q^9 $$q - \zeta_{12} q^{3} + \zeta_{12}^{5} q^{7} + \zeta_{12}^{2} q^{9} - \zeta_{12}^{3} q^{13} - \zeta_{12}^{2} q^{19} + q^{21} - \zeta_{12}^{3} q^{27} - \zeta_{12}^{4} q^{31} - \zeta_{12}^{5} q^{37} + \zeta_{12}^{4} q^{39} - \zeta_{12}^{3} q^{43} - \zeta_{12}^{4} q^{49} + \zeta_{12}^{3} q^{57} - \zeta_{12}^{2} q^{61} - \zeta_{12} q^{63} - \zeta_{12} q^{67} + \zeta_{12} q^{73} - \zeta_{12}^{2} q^{79} + \zeta_{12}^{4} q^{81} + \zeta_{12}^{2} q^{91} + \zeta_{12}^{5} q^{93} - \zeta_{12}^{3} q^{97} +O(q^{100})$$ q - z * q^3 + z^5 * q^7 + z^2 * q^9 - z^3 * q^13 - z^2 * q^19 + q^21 - z^3 * q^27 - z^4 * q^31 - z^5 * q^37 + z^4 * q^39 - z^3 * q^43 - z^4 * q^49 + z^3 * q^57 - z^2 * q^61 - z * q^63 - z * q^67 + z * q^73 - z^2 * q^79 + z^4 * q^81 + z^2 * q^91 + z^5 * q^93 - z^3 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{9}+O(q^{10})$$ 4 * q + 2 * q^9 $$4 q + 2 q^{9} - 2 q^{19} + 4 q^{21} + 2 q^{31} - 2 q^{39} + 2 q^{49} - 4 q^{61} - 2 q^{79} - 2 q^{81} + 2 q^{91}+O(q^{100})$$ 4 * q + 2 * q^9 - 2 * q^19 + 4 * q^21 + 2 * q^31 - 2 * q^39 + 2 * q^49 - 4 * q^61 - 2 * q^79 - 2 * q^81 + 2 * q^91

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times$$.

 $$n$$ $$701$$ $$1051$$ $$1177$$ $$1501$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$-\zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
149.1
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
0 −0.866025 + 0.500000i 0 0 0 −0.866025 0.500000i 0 0.500000 0.866025i 0
149.2 0 0.866025 0.500000i 0 0 0 0.866025 + 0.500000i 0 0.500000 0.866025i 0
1649.1 0 −0.866025 0.500000i 0 0 0 −0.866025 + 0.500000i 0 0.500000 + 0.866025i 0
1649.2 0 0.866025 + 0.500000i 0 0 0 0.866025 0.500000i 0 0.500000 + 0.866025i 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
5.b even 2 1 inner
7.c even 3 1 inner
15.d odd 2 1 inner
21.h odd 6 1 inner
35.j even 6 1 inner
105.o odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2100.1.bh.a 4
3.b odd 2 1 CM 2100.1.bh.a 4
5.b even 2 1 inner 2100.1.bh.a 4
5.c odd 4 1 84.1.p.a 2
5.c odd 4 1 2100.1.bn.c 2
7.c even 3 1 inner 2100.1.bh.a 4
15.d odd 2 1 inner 2100.1.bh.a 4
15.e even 4 1 84.1.p.a 2
15.e even 4 1 2100.1.bn.c 2
20.e even 4 1 336.1.bn.a 2
21.h odd 6 1 inner 2100.1.bh.a 4
35.f even 4 1 588.1.p.a 2
35.j even 6 1 inner 2100.1.bh.a 4
35.k even 12 1 588.1.c.a 1
35.k even 12 1 588.1.p.a 2
35.l odd 12 1 84.1.p.a 2
35.l odd 12 1 588.1.c.b 1
35.l odd 12 1 2100.1.bn.c 2
40.i odd 4 1 1344.1.bn.b 2
40.k even 4 1 1344.1.bn.a 2
45.k odd 12 1 2268.1.m.a 2
45.k odd 12 1 2268.1.bh.b 2
45.l even 12 1 2268.1.m.a 2
45.l even 12 1 2268.1.bh.b 2
60.l odd 4 1 336.1.bn.a 2
105.k odd 4 1 588.1.p.a 2
105.o odd 6 1 inner 2100.1.bh.a 4
105.w odd 12 1 588.1.c.a 1
105.w odd 12 1 588.1.p.a 2
105.x even 12 1 84.1.p.a 2
105.x even 12 1 588.1.c.b 1
105.x even 12 1 2100.1.bn.c 2
120.q odd 4 1 1344.1.bn.a 2
120.w even 4 1 1344.1.bn.b 2
140.j odd 4 1 2352.1.bn.a 2
140.w even 12 1 336.1.bn.a 2
140.w even 12 1 2352.1.d.a 1
140.x odd 12 1 2352.1.d.b 1
140.x odd 12 1 2352.1.bn.a 2
280.br even 12 1 1344.1.bn.a 2
280.bt odd 12 1 1344.1.bn.b 2
315.bt odd 12 1 2268.1.m.a 2
315.bv even 12 1 2268.1.m.a 2
315.bx even 12 1 2268.1.bh.b 2
315.ch odd 12 1 2268.1.bh.b 2
420.w even 4 1 2352.1.bn.a 2
420.bp odd 12 1 336.1.bn.a 2
420.bp odd 12 1 2352.1.d.a 1
420.br even 12 1 2352.1.d.b 1
420.br even 12 1 2352.1.bn.a 2
840.dc even 12 1 1344.1.bn.b 2
840.dp odd 12 1 1344.1.bn.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.1.p.a 2 5.c odd 4 1
84.1.p.a 2 15.e even 4 1
84.1.p.a 2 35.l odd 12 1
84.1.p.a 2 105.x even 12 1
336.1.bn.a 2 20.e even 4 1
336.1.bn.a 2 60.l odd 4 1
336.1.bn.a 2 140.w even 12 1
336.1.bn.a 2 420.bp odd 12 1
588.1.c.a 1 35.k even 12 1
588.1.c.a 1 105.w odd 12 1
588.1.c.b 1 35.l odd 12 1
588.1.c.b 1 105.x even 12 1
588.1.p.a 2 35.f even 4 1
588.1.p.a 2 35.k even 12 1
588.1.p.a 2 105.k odd 4 1
588.1.p.a 2 105.w odd 12 1
1344.1.bn.a 2 40.k even 4 1
1344.1.bn.a 2 120.q odd 4 1
1344.1.bn.a 2 280.br even 12 1
1344.1.bn.a 2 840.dp odd 12 1
1344.1.bn.b 2 40.i odd 4 1
1344.1.bn.b 2 120.w even 4 1
1344.1.bn.b 2 280.bt odd 12 1
1344.1.bn.b 2 840.dc even 12 1
2100.1.bh.a 4 1.a even 1 1 trivial
2100.1.bh.a 4 3.b odd 2 1 CM
2100.1.bh.a 4 5.b even 2 1 inner
2100.1.bh.a 4 7.c even 3 1 inner
2100.1.bh.a 4 15.d odd 2 1 inner
2100.1.bh.a 4 21.h odd 6 1 inner
2100.1.bh.a 4 35.j even 6 1 inner
2100.1.bh.a 4 105.o odd 6 1 inner
2100.1.bn.c 2 5.c odd 4 1
2100.1.bn.c 2 15.e even 4 1
2100.1.bn.c 2 35.l odd 12 1
2100.1.bn.c 2 105.x even 12 1
2268.1.m.a 2 45.k odd 12 1
2268.1.m.a 2 45.l even 12 1
2268.1.m.a 2 315.bt odd 12 1
2268.1.m.a 2 315.bv even 12 1
2268.1.bh.b 2 45.k odd 12 1
2268.1.bh.b 2 45.l even 12 1
2268.1.bh.b 2 315.bx even 12 1
2268.1.bh.b 2 315.ch odd 12 1
2352.1.d.a 1 140.w even 12 1
2352.1.d.a 1 420.bp odd 12 1
2352.1.d.b 1 140.x odd 12 1
2352.1.d.b 1 420.br even 12 1
2352.1.bn.a 2 140.j odd 4 1
2352.1.bn.a 2 140.x odd 12 1
2352.1.bn.a 2 420.w even 4 1
2352.1.bn.a 2 420.br even 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{13}^{2} + 1$$ acting on $$S_{1}^{\mathrm{new}}(2100, [\chi])$$.

## Hecke characteristic polynomials

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