# Properties

 Label 2304.1.o.c Level $2304$ Weight $1$ Character orbit 2304.o Analytic conductor $1.150$ Analytic rank $0$ Dimension $4$ Projective image $D_{3}$ CM discriminant -8 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,1,Mod(511,2304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2304, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 4]))

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

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

chi := DirichletCharacter("2304.511");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2304 = 2^{8} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2304.o (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: $$1.14984578911$$ 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 72) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.648.1 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{12}^{5} q^{3} - \zeta_{12}^{4} q^{9} +O(q^{10})$$ q + z^5 * q^3 - z^4 * q^9 $$q + \zeta_{12}^{5} q^{3} - \zeta_{12}^{4} q^{9} + \zeta_{12}^{5} q^{11} - q^{17} + \zeta_{12}^{3} q^{19} + \zeta_{12}^{2} q^{25} + \zeta_{12}^{3} q^{27} - \zeta_{12}^{4} q^{33} + \zeta_{12}^{4} q^{41} + \zeta_{12}^{5} q^{43} + \zeta_{12}^{4} q^{49} - \zeta_{12}^{5} q^{51} - \zeta_{12}^{2} q^{57} + \zeta_{12} q^{59} - \zeta_{12} q^{67} + q^{73} - \zeta_{12} q^{75} - \zeta_{12}^{2} q^{81} + \zeta_{12}^{5} q^{83} - q^{89} + \zeta_{12}^{2} q^{97} + \zeta_{12}^{3} q^{99} +O(q^{100})$$ q + z^5 * q^3 - z^4 * q^9 + z^5 * q^11 - q^17 + z^3 * q^19 + z^2 * q^25 + z^3 * q^27 - z^4 * q^33 + z^4 * q^41 + z^5 * q^43 + z^4 * q^49 - z^5 * q^51 - z^2 * q^57 + z * q^59 - z * q^67 + q^73 - z * q^75 - z^2 * q^81 + z^5 * q^83 - q^89 + z^2 * q^97 + z^3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{9}+O(q^{10})$$ 4 * q + 2 * q^9 $$4 q + 2 q^{9} - 4 q^{17} + 2 q^{25} + 2 q^{33} - 2 q^{41} - 2 q^{49} - 2 q^{57} + 4 q^{73} - 2 q^{81} - 8 q^{89} + 2 q^{97}+O(q^{100})$$ 4 * q + 2 * q^9 - 4 * q^17 + 2 * q^25 + 2 * q^33 - 2 * q^41 - 2 * q^49 - 2 * q^57 + 4 * q^73 - 2 * q^81 - 8 * q^89 + 2 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times$$.

 $$n$$ $$1279$$ $$1793$$ $$2053$$ $$\chi(n)$$ $$-1$$ $$\zeta_{12}^{4}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
511.1
 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 0 0.500000 + 0.866025i 0
511.2 0 0.866025 + 0.500000i 0 0 0 0 0 0.500000 + 0.866025i 0
2047.1 0 −0.866025 + 0.500000i 0 0 0 0 0 0.500000 0.866025i 0
2047.2 0 0.866025 0.500000i 0 0 0 0 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
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
4.b odd 2 1 inner
8.b even 2 1 inner
9.c even 3 1 inner
36.f odd 6 1 inner
72.n even 6 1 inner
72.p odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.1.o.c 4
4.b odd 2 1 inner 2304.1.o.c 4
8.b even 2 1 inner 2304.1.o.c 4
8.d odd 2 1 CM 2304.1.o.c 4
9.c even 3 1 inner 2304.1.o.c 4
16.e even 4 1 72.1.p.a 2
16.e even 4 1 288.1.t.a 2
16.f odd 4 1 72.1.p.a 2
16.f odd 4 1 288.1.t.a 2
36.f odd 6 1 inner 2304.1.o.c 4
48.i odd 4 1 216.1.p.a 2
48.i odd 4 1 864.1.t.a 2
48.k even 4 1 216.1.p.a 2
48.k even 4 1 864.1.t.a 2
72.n even 6 1 inner 2304.1.o.c 4
72.p odd 6 1 inner 2304.1.o.c 4
80.i odd 4 1 1800.1.ba.b 4
80.j even 4 1 1800.1.ba.b 4
80.k odd 4 1 1800.1.bk.d 2
80.q even 4 1 1800.1.bk.d 2
80.s even 4 1 1800.1.ba.b 4
80.t odd 4 1 1800.1.ba.b 4
112.j even 4 1 3528.1.cg.a 2
112.l odd 4 1 3528.1.cg.a 2
112.u odd 12 1 3528.1.ba.b 2
112.u odd 12 1 3528.1.ce.a 2
112.v even 12 1 3528.1.ba.a 2
112.v even 12 1 3528.1.ce.b 2
112.w even 12 1 3528.1.ba.b 2
112.w even 12 1 3528.1.ce.a 2
112.x odd 12 1 3528.1.ba.a 2
112.x odd 12 1 3528.1.ce.b 2
144.u even 12 1 216.1.p.a 2
144.u even 12 1 648.1.b.a 1
144.u even 12 1 864.1.t.a 2
144.u even 12 1 2592.1.b.a 1
144.v odd 12 1 72.1.p.a 2
144.v odd 12 1 288.1.t.a 2
144.v odd 12 1 648.1.b.b 1
144.v odd 12 1 2592.1.b.b 1
144.w odd 12 1 216.1.p.a 2
144.w odd 12 1 648.1.b.a 1
144.w odd 12 1 864.1.t.a 2
144.w odd 12 1 2592.1.b.a 1
144.x even 12 1 72.1.p.a 2
144.x even 12 1 288.1.t.a 2
144.x even 12 1 648.1.b.b 1
144.x even 12 1 2592.1.b.b 1
720.ce even 12 1 1800.1.bk.d 2
720.cn odd 12 1 1800.1.ba.b 4
720.cp even 12 1 1800.1.ba.b 4
720.cr odd 12 1 1800.1.ba.b 4
720.ct even 12 1 1800.1.ba.b 4
720.cz odd 12 1 1800.1.bk.d 2
1008.dk even 12 1 3528.1.cg.a 2
1008.dn odd 12 1 3528.1.cg.a 2
1008.do even 12 1 3528.1.ce.a 2
1008.dq odd 12 1 3528.1.ba.b 2
1008.dx even 12 1 3528.1.ba.b 2
1008.dz odd 12 1 3528.1.ce.a 2
1008.eb odd 12 1 3528.1.ce.b 2
1008.ef even 12 1 3528.1.ba.a 2
1008.eg odd 12 1 3528.1.ba.a 2
1008.ek even 12 1 3528.1.ce.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.1.p.a 2 16.e even 4 1
72.1.p.a 2 16.f odd 4 1
72.1.p.a 2 144.v odd 12 1
72.1.p.a 2 144.x even 12 1
216.1.p.a 2 48.i odd 4 1
216.1.p.a 2 48.k even 4 1
216.1.p.a 2 144.u even 12 1
216.1.p.a 2 144.w odd 12 1
288.1.t.a 2 16.e even 4 1
288.1.t.a 2 16.f odd 4 1
288.1.t.a 2 144.v odd 12 1
288.1.t.a 2 144.x even 12 1
648.1.b.a 1 144.u even 12 1
648.1.b.a 1 144.w odd 12 1
648.1.b.b 1 144.v odd 12 1
648.1.b.b 1 144.x even 12 1
864.1.t.a 2 48.i odd 4 1
864.1.t.a 2 48.k even 4 1
864.1.t.a 2 144.u even 12 1
864.1.t.a 2 144.w odd 12 1
1800.1.ba.b 4 80.i odd 4 1
1800.1.ba.b 4 80.j even 4 1
1800.1.ba.b 4 80.s even 4 1
1800.1.ba.b 4 80.t odd 4 1
1800.1.ba.b 4 720.cn odd 12 1
1800.1.ba.b 4 720.cp even 12 1
1800.1.ba.b 4 720.cr odd 12 1
1800.1.ba.b 4 720.ct even 12 1
1800.1.bk.d 2 80.k odd 4 1
1800.1.bk.d 2 80.q even 4 1
1800.1.bk.d 2 720.ce even 12 1
1800.1.bk.d 2 720.cz odd 12 1
2304.1.o.c 4 1.a even 1 1 trivial
2304.1.o.c 4 4.b odd 2 1 inner
2304.1.o.c 4 8.b even 2 1 inner
2304.1.o.c 4 8.d odd 2 1 CM
2304.1.o.c 4 9.c even 3 1 inner
2304.1.o.c 4 36.f odd 6 1 inner
2304.1.o.c 4 72.n even 6 1 inner
2304.1.o.c 4 72.p odd 6 1 inner
2592.1.b.a 1 144.u even 12 1
2592.1.b.a 1 144.w odd 12 1
2592.1.b.b 1 144.v odd 12 1
2592.1.b.b 1 144.x even 12 1
3528.1.ba.a 2 112.v even 12 1
3528.1.ba.a 2 112.x odd 12 1
3528.1.ba.a 2 1008.ef even 12 1
3528.1.ba.a 2 1008.eg odd 12 1
3528.1.ba.b 2 112.u odd 12 1
3528.1.ba.b 2 112.w even 12 1
3528.1.ba.b 2 1008.dq odd 12 1
3528.1.ba.b 2 1008.dx even 12 1
3528.1.ce.a 2 112.u odd 12 1
3528.1.ce.a 2 112.w even 12 1
3528.1.ce.a 2 1008.do even 12 1
3528.1.ce.a 2 1008.dz odd 12 1
3528.1.ce.b 2 112.v even 12 1
3528.1.ce.b 2 112.x odd 12 1
3528.1.ce.b 2 1008.eb odd 12 1
3528.1.ce.b 2 1008.ek even 12 1
3528.1.cg.a 2 112.j even 4 1
3528.1.cg.a 2 112.l odd 4 1
3528.1.cg.a 2 1008.dk even 12 1
3528.1.cg.a 2 1008.dn odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{4} - T_{11}^{2} + 1$$ acting on $$S_{1}^{\mathrm{new}}(2304, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - T^{2} + 1$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$T^{4} - T^{2} + 1$$
$13$ $$T^{4}$$
$17$ $$(T + 1)^{4}$$
$19$ $$(T^{2} + 1)^{2}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$(T^{2} + T + 1)^{2}$$
$43$ $$T^{4} - T^{2} + 1$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4} - T^{2} + 1$$
$61$ $$T^{4}$$
$67$ $$T^{4} - T^{2} + 1$$
$71$ $$T^{4}$$
$73$ $$(T - 1)^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4} - 4T^{2} + 16$$
$89$ $$(T + 2)^{4}$$
$97$ $$(T^{2} - T + 1)^{2}$$