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 $2$

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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: \( x^{2} - x + 1 \) Copy content Toggle raw display
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.

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

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
0.500000 + 0.866025i
0.500000 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:

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 \(S_{2}^{\mathrm{new}}(1872, [\chi])\):

\( T_{5}^{2} + 3 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display

Hecke characteristic polynomials

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