# Properties

 Label 1872.2.by.d Level $1872$ Weight $2$ Character orbit 1872.by Analytic conductor $14.948$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1872,2,Mod(433,1872)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1872, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

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

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

chi := DirichletCharacter("1872.433");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1872 = 2^{4} \cdot 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1872.by (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: $$14.9479952584$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 13) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (2 \zeta_{6} - 1) q^{5}+O(q^{10})$$ q + (2*z - 1) * q^5 $$q + (2 \zeta_{6} - 1) q^{5} + ( - 3 \zeta_{6} - 1) q^{13} - 3 \zeta_{6} q^{17} + ( - 2 \zeta_{6} + 4) q^{19} + ( - 6 \zeta_{6} + 6) q^{23} + 2 q^{25} + ( - 3 \zeta_{6} + 3) q^{29} + ( - 4 \zeta_{6} + 2) q^{31} + (5 \zeta_{6} + 5) q^{37} + (3 \zeta_{6} + 3) q^{41} + 8 \zeta_{6} q^{43} + ( - 4 \zeta_{6} + 2) q^{47} + (7 \zeta_{6} - 7) q^{49} + 3 q^{53} + ( - 4 \zeta_{6} + 8) q^{59} - \zeta_{6} q^{61} + ( - 5 \zeta_{6} + 7) q^{65} + ( - 2 \zeta_{6} - 2) q^{67} + ( - 2 \zeta_{6} + 4) q^{71} + (2 \zeta_{6} - 1) q^{73} - 4 q^{79} + (16 \zeta_{6} - 8) q^{83} + ( - 3 \zeta_{6} + 6) q^{85} + (4 \zeta_{6} + 4) q^{89} + 6 \zeta_{6} q^{95} + ( - 4 \zeta_{6} + 8) q^{97} +O(q^{100})$$ q + (2*z - 1) * q^5 + (-3*z - 1) * q^13 - 3*z * q^17 + (-2*z + 4) * q^19 + (-6*z + 6) * q^23 + 2 * q^25 + (-3*z + 3) * q^29 + (-4*z + 2) * q^31 + (5*z + 5) * q^37 + (3*z + 3) * q^41 + 8*z * q^43 + (-4*z + 2) * q^47 + (7*z - 7) * q^49 + 3 * q^53 + (-4*z + 8) * q^59 - z * q^61 + (-5*z + 7) * q^65 + (-2*z - 2) * q^67 + (-2*z + 4) * q^71 + (2*z - 1) * q^73 - 4 * q^79 + (16*z - 8) * q^83 + (-3*z + 6) * q^85 + (4*z + 4) * q^89 + 6*z * q^95 + (-4*z + 8) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q - 5 q^{13} - 3 q^{17} + 6 q^{19} + 6 q^{23} + 4 q^{25} + 3 q^{29} + 15 q^{37} + 9 q^{41} + 8 q^{43} - 7 q^{49} + 6 q^{53} + 12 q^{59} - q^{61} + 9 q^{65} - 6 q^{67} + 6 q^{71} - 8 q^{79} + 9 q^{85} + 12 q^{89} + 6 q^{95} + 12 q^{97}+O(q^{100})$$ 2 * q - 5 * q^13 - 3 * q^17 + 6 * q^19 + 6 * q^23 + 4 * q^25 + 3 * q^29 + 15 * q^37 + 9 * q^41 + 8 * q^43 - 7 * q^49 + 6 * q^53 + 12 * q^59 - q^61 + 9 * q^65 - 6 * q^67 + 6 * q^71 - 8 * q^79 + 9 * q^85 + 12 * q^89 + 6 * q^95 + 12 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1872\mathbb{Z}\right)^\times$$.

 $$n$$ $$145$$ $$209$$ $$469$$ $$703$$ $$\chi(n)$$ $$\zeta_{6}$$ $$1$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
433.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 1.73205i 0 0 0 0 0
1297.1 0 0 0 1.73205i 0 0 0 0 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
13.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1872.2.by.d 2
3.b odd 2 1 208.2.w.b 2
4.b odd 2 1 117.2.q.c 2
12.b even 2 1 13.2.e.a 2
13.e even 6 1 inner 1872.2.by.d 2
24.f even 2 1 832.2.w.d 2
24.h odd 2 1 832.2.w.a 2
39.h odd 6 1 208.2.w.b 2
39.h odd 6 1 2704.2.f.b 2
39.i odd 6 1 2704.2.f.b 2
39.k even 12 2 2704.2.a.o 2
52.i odd 6 1 117.2.q.c 2
52.i odd 6 1 1521.2.b.a 2
52.j odd 6 1 1521.2.b.a 2
52.l even 12 2 1521.2.a.k 2
60.h even 2 1 325.2.n.a 2
60.l odd 4 2 325.2.m.a 4
84.h odd 2 1 637.2.q.a 2
84.j odd 6 1 637.2.k.c 2
84.j odd 6 1 637.2.u.b 2
84.n even 6 1 637.2.k.a 2
84.n even 6 1 637.2.u.c 2
156.h even 2 1 169.2.e.a 2
156.l odd 4 2 169.2.c.a 4
156.p even 6 1 169.2.b.a 2
156.p even 6 1 169.2.e.a 2
156.r even 6 1 13.2.e.a 2
156.r even 6 1 169.2.b.a 2
156.v odd 12 2 169.2.a.a 2
156.v odd 12 2 169.2.c.a 4
312.ba even 6 1 832.2.w.d 2
312.bg odd 6 1 832.2.w.a 2
780.cb even 6 1 325.2.n.a 2
780.cr odd 12 2 4225.2.a.v 2
780.cw odd 12 2 325.2.m.a 4
1092.bh odd 6 1 637.2.q.a 2
1092.bn even 6 1 637.2.k.a 2
1092.bv even 6 1 637.2.u.c 2
1092.cg odd 6 1 637.2.k.c 2
1092.cy odd 6 1 637.2.u.b 2
1092.eh even 12 2 8281.2.a.q 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.2.e.a 2 12.b even 2 1
13.2.e.a 2 156.r even 6 1
117.2.q.c 2 4.b odd 2 1
117.2.q.c 2 52.i odd 6 1
169.2.a.a 2 156.v odd 12 2
169.2.b.a 2 156.p even 6 1
169.2.b.a 2 156.r even 6 1
169.2.c.a 4 156.l odd 4 2
169.2.c.a 4 156.v odd 12 2
169.2.e.a 2 156.h even 2 1
169.2.e.a 2 156.p even 6 1
208.2.w.b 2 3.b odd 2 1
208.2.w.b 2 39.h odd 6 1
325.2.m.a 4 60.l odd 4 2
325.2.m.a 4 780.cw odd 12 2
325.2.n.a 2 60.h even 2 1
325.2.n.a 2 780.cb even 6 1
637.2.k.a 2 84.n even 6 1
637.2.k.a 2 1092.bn even 6 1
637.2.k.c 2 84.j odd 6 1
637.2.k.c 2 1092.cg odd 6 1
637.2.q.a 2 84.h odd 2 1
637.2.q.a 2 1092.bh odd 6 1
637.2.u.b 2 84.j odd 6 1
637.2.u.b 2 1092.cy odd 6 1
637.2.u.c 2 84.n even 6 1
637.2.u.c 2 1092.bv even 6 1
832.2.w.a 2 24.h odd 2 1
832.2.w.a 2 312.bg odd 6 1
832.2.w.d 2 24.f even 2 1
832.2.w.d 2 312.ba even 6 1
1521.2.a.k 2 52.l even 12 2
1521.2.b.a 2 52.i odd 6 1
1521.2.b.a 2 52.j odd 6 1
1872.2.by.d 2 1.a even 1 1 trivial
1872.2.by.d 2 13.e even 6 1 inner
2704.2.a.o 2 39.k even 12 2
2704.2.f.b 2 39.h odd 6 1
2704.2.f.b 2 39.i odd 6 1
4225.2.a.v 2 780.cr odd 12 2
8281.2.a.q 2 1092.eh even 12 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1872, [\chi])$$:

 $$T_{5}^{2} + 3$$ T5^2 + 3 $$T_{7}$$ T7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 3$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 5T + 13$$
$17$ $$T^{2} + 3T + 9$$
$19$ $$T^{2} - 6T + 12$$
$23$ $$T^{2} - 6T + 36$$
$29$ $$T^{2} - 3T + 9$$
$31$ $$T^{2} + 12$$
$37$ $$T^{2} - 15T + 75$$
$41$ $$T^{2} - 9T + 27$$
$43$ $$T^{2} - 8T + 64$$
$47$ $$T^{2} + 12$$
$53$ $$(T - 3)^{2}$$
$59$ $$T^{2} - 12T + 48$$
$61$ $$T^{2} + T + 1$$
$67$ $$T^{2} + 6T + 12$$
$71$ $$T^{2} - 6T + 12$$
$73$ $$T^{2} + 3$$
$79$ $$(T + 4)^{2}$$
$83$ $$T^{2} + 192$$
$89$ $$T^{2} - 12T + 48$$
$97$ $$T^{2} - 12T + 48$$