# Properties

 Label 1872.2.by.f 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) 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.

 $$f(q)$$ $$=$$ $$q + ( - 4 \zeta_{6} + 2) q^{5} + ( - \zeta_{6} + 2) q^{7}+O(q^{10})$$ q + (-4*z + 2) * q^5 + (-z + 2) * q^7 $$q + ( - 4 \zeta_{6} + 2) q^{5} + ( - \zeta_{6} + 2) q^{7} + ( - 2 \zeta_{6} - 2) q^{11} + ( - \zeta_{6} + 4) q^{13} + (2 \zeta_{6} - 4) q^{19} + ( - 6 \zeta_{6} + 6) q^{23} - 7 q^{25} + ( - 6 \zeta_{6} + 6) q^{29} + ( - 2 \zeta_{6} + 1) q^{31} - 6 \zeta_{6} q^{35} + (4 \zeta_{6} + 4) q^{41} - \zeta_{6} q^{43} + (4 \zeta_{6} - 2) q^{47} + (4 \zeta_{6} - 4) q^{49} - 12 q^{53} + (12 \zeta_{6} - 12) q^{55} + (2 \zeta_{6} - 4) q^{59} - \zeta_{6} q^{61} + ( - 14 \zeta_{6} + 4) q^{65} + ( - 5 \zeta_{6} - 5) q^{67} + (6 \zeta_{6} - 12) q^{71} + (2 \zeta_{6} - 1) q^{73} - 6 q^{77} + 11 q^{79} + ( - 16 \zeta_{6} + 8) q^{83} + ( - 4 \zeta_{6} - 4) q^{89} + ( - 5 \zeta_{6} + 7) q^{91} + 12 \zeta_{6} q^{95} + (3 \zeta_{6} - 6) q^{97} +O(q^{100})$$ q + (-4*z + 2) * q^5 + (-z + 2) * q^7 + (-2*z - 2) * q^11 + (-z + 4) * q^13 + (2*z - 4) * q^19 + (-6*z + 6) * q^23 - 7 * q^25 + (-6*z + 6) * q^29 + (-2*z + 1) * q^31 - 6*z * q^35 + (4*z + 4) * q^41 - z * q^43 + (4*z - 2) * q^47 + (4*z - 4) * q^49 - 12 * q^53 + (12*z - 12) * q^55 + (2*z - 4) * q^59 - z * q^61 + (-14*z + 4) * q^65 + (-5*z - 5) * q^67 + (6*z - 12) * q^71 + (2*z - 1) * q^73 - 6 * q^77 + 11 * q^79 + (-16*z + 8) * q^83 + (-4*z - 4) * q^89 + (-5*z + 7) * q^91 + 12*z * q^95 + (3*z - 6) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{7}+O(q^{10})$$ 2 * q + 3 * q^7 $$2 q + 3 q^{7} - 6 q^{11} + 7 q^{13} - 6 q^{19} + 6 q^{23} - 14 q^{25} + 6 q^{29} - 6 q^{35} + 12 q^{41} - q^{43} - 4 q^{49} - 24 q^{53} - 12 q^{55} - 6 q^{59} - q^{61} - 6 q^{65} - 15 q^{67} - 18 q^{71} - 12 q^{77} + 22 q^{79} - 12 q^{89} + 9 q^{91} + 12 q^{95} - 9 q^{97}+O(q^{100})$$ 2 * q + 3 * q^7 - 6 * q^11 + 7 * q^13 - 6 * q^19 + 6 * q^23 - 14 * q^25 + 6 * q^29 - 6 * q^35 + 12 * q^41 - q^43 - 4 * q^49 - 24 * q^53 - 12 * q^55 - 6 * q^59 - q^61 - 6 * q^65 - 15 * q^67 - 18 * q^71 - 12 * q^77 + 22 * q^79 - 12 * q^89 + 9 * q^91 + 12 * q^95 - 9 * 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 3.46410i 0 1.50000 0.866025i 0 0 0
1297.1 0 0 0 3.46410i 0 1.50000 + 0.866025i 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.f 2
3.b odd 2 1 624.2.bv.b 2
4.b odd 2 1 117.2.q.a 2
12.b even 2 1 39.2.j.a 2
13.e even 6 1 inner 1872.2.by.f 2
39.h odd 6 1 624.2.bv.b 2
39.k even 12 2 8112.2.a.bu 2
52.i odd 6 1 117.2.q.a 2
52.i odd 6 1 1521.2.b.f 2
52.j odd 6 1 1521.2.b.f 2
52.l even 12 2 1521.2.a.h 2
60.h even 2 1 975.2.bc.c 2
60.l odd 4 2 975.2.w.d 4
156.h even 2 1 507.2.j.b 2
156.l odd 4 2 507.2.e.f 4
156.p even 6 1 507.2.b.c 2
156.p even 6 1 507.2.j.b 2
156.r even 6 1 39.2.j.a 2
156.r even 6 1 507.2.b.c 2
156.v odd 12 2 507.2.a.e 2
156.v odd 12 2 507.2.e.f 4
780.cb even 6 1 975.2.bc.c 2
780.cw odd 12 2 975.2.w.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.j.a 2 12.b even 2 1
39.2.j.a 2 156.r even 6 1
117.2.q.a 2 4.b odd 2 1
117.2.q.a 2 52.i odd 6 1
507.2.a.e 2 156.v odd 12 2
507.2.b.c 2 156.p even 6 1
507.2.b.c 2 156.r even 6 1
507.2.e.f 4 156.l odd 4 2
507.2.e.f 4 156.v odd 12 2
507.2.j.b 2 156.h even 2 1
507.2.j.b 2 156.p even 6 1
624.2.bv.b 2 3.b odd 2 1
624.2.bv.b 2 39.h odd 6 1
975.2.w.d 4 60.l odd 4 2
975.2.w.d 4 780.cw odd 12 2
975.2.bc.c 2 60.h even 2 1
975.2.bc.c 2 780.cb even 6 1
1521.2.a.h 2 52.l even 12 2
1521.2.b.f 2 52.i odd 6 1
1521.2.b.f 2 52.j odd 6 1
1872.2.by.f 2 1.a even 1 1 trivial
1872.2.by.f 2 13.e even 6 1 inner
8112.2.a.bu 2 39.k even 12 2

## Hecke kernels

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

 $$T_{5}^{2} + 12$$ T5^2 + 12 $$T_{7}^{2} - 3T_{7} + 3$$ T7^2 - 3*T7 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 12$$
$7$ $$T^{2} - 3T + 3$$
$11$ $$T^{2} + 6T + 12$$
$13$ $$T^{2} - 7T + 13$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 6T + 12$$
$23$ $$T^{2} - 6T + 36$$
$29$ $$T^{2} - 6T + 36$$
$31$ $$T^{2} + 3$$
$37$ $$T^{2}$$
$41$ $$T^{2} - 12T + 48$$
$43$ $$T^{2} + T + 1$$
$47$ $$T^{2} + 12$$
$53$ $$(T + 12)^{2}$$
$59$ $$T^{2} + 6T + 12$$
$61$ $$T^{2} + T + 1$$
$67$ $$T^{2} + 15T + 75$$
$71$ $$T^{2} + 18T + 108$$
$73$ $$T^{2} + 3$$
$79$ $$(T - 11)^{2}$$
$83$ $$T^{2} + 192$$
$89$ $$T^{2} + 12T + 48$$
$97$ $$T^{2} + 9T + 27$$