Properties

Label 1872.2.t.i
Level $1872$
Weight $2$
Character orbit 1872.t
Analytic conductor $14.948$
Analytic rank $0$
Dimension $2$
CM no
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(289,1872)]
 
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, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1872.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1872.t (of order \(3\), 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 78)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + q^{5} - 2 \zeta_{6} q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q + q^{5} - 2 \zeta_{6} q^{7} + (2 \zeta_{6} - 2) q^{11} + (3 \zeta_{6} + 1) q^{13} + 5 \zeta_{6} q^{17} - 2 \zeta_{6} q^{19} + (6 \zeta_{6} - 6) q^{23} - 4 q^{25} + (9 \zeta_{6} - 9) q^{29} + 4 q^{31} - 2 \zeta_{6} q^{35} + ( - 11 \zeta_{6} + 11) q^{37} + ( - 5 \zeta_{6} + 5) q^{41} + 10 \zeta_{6} q^{43} + 2 q^{47} + ( - 3 \zeta_{6} + 3) q^{49} + q^{53} + (2 \zeta_{6} - 2) q^{55} + 8 \zeta_{6} q^{59} + 11 \zeta_{6} q^{61} + (3 \zeta_{6} + 1) q^{65} + ( - 2 \zeta_{6} + 2) q^{67} + 14 \zeta_{6} q^{71} - 13 q^{73} + 4 q^{77} + 4 q^{79} + 6 q^{83} + 5 \zeta_{6} q^{85} + ( - 2 \zeta_{6} + 2) q^{89} + ( - 8 \zeta_{6} + 6) q^{91} - 2 \zeta_{6} q^{95} + 2 \zeta_{6} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} - 2 q^{7} - 2 q^{11} + 5 q^{13} + 5 q^{17} - 2 q^{19} - 6 q^{23} - 8 q^{25} - 9 q^{29} + 8 q^{31} - 2 q^{35} + 11 q^{37} + 5 q^{41} + 10 q^{43} + 4 q^{47} + 3 q^{49} + 2 q^{53} - 2 q^{55} + 8 q^{59} + 11 q^{61} + 5 q^{65} + 2 q^{67} + 14 q^{71} - 26 q^{73} + 8 q^{77} + 8 q^{79} + 12 q^{83} + 5 q^{85} + 2 q^{89} + 4 q^{91} - 2 q^{95} + 2 q^{97}+O(q^{100}) \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 0 1.00000 0 −1.00000 + 1.73205i 0 0 0
1153.1 0 0 0 1.00000 0 −1.00000 1.73205i 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.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1872.2.t.i 2
3.b odd 2 1 624.2.q.b 2
4.b odd 2 1 234.2.h.b 2
12.b even 2 1 78.2.e.b 2
13.c even 3 1 inner 1872.2.t.i 2
39.h odd 6 1 8112.2.a.bb 1
39.i odd 6 1 624.2.q.b 2
39.i odd 6 1 8112.2.a.x 1
52.i odd 6 1 3042.2.a.d 1
52.j odd 6 1 234.2.h.b 2
52.j odd 6 1 3042.2.a.m 1
52.l even 12 2 3042.2.b.d 2
60.h even 2 1 1950.2.i.b 2
60.l odd 4 2 1950.2.z.b 4
156.h even 2 1 1014.2.e.d 2
156.l odd 4 2 1014.2.i.e 4
156.p even 6 1 78.2.e.b 2
156.p even 6 1 1014.2.a.a 1
156.r even 6 1 1014.2.a.e 1
156.r even 6 1 1014.2.e.d 2
156.v odd 12 2 1014.2.b.a 2
156.v odd 12 2 1014.2.i.e 4
780.br even 6 1 1950.2.i.b 2
780.cj odd 12 2 1950.2.z.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.e.b 2 12.b even 2 1
78.2.e.b 2 156.p even 6 1
234.2.h.b 2 4.b odd 2 1
234.2.h.b 2 52.j odd 6 1
624.2.q.b 2 3.b odd 2 1
624.2.q.b 2 39.i odd 6 1
1014.2.a.a 1 156.p even 6 1
1014.2.a.e 1 156.r even 6 1
1014.2.b.a 2 156.v odd 12 2
1014.2.e.d 2 156.h even 2 1
1014.2.e.d 2 156.r even 6 1
1014.2.i.e 4 156.l odd 4 2
1014.2.i.e 4 156.v odd 12 2
1872.2.t.i 2 1.a even 1 1 trivial
1872.2.t.i 2 13.c even 3 1 inner
1950.2.i.b 2 60.h even 2 1
1950.2.i.b 2 780.br even 6 1
1950.2.z.b 4 60.l odd 4 2
1950.2.z.b 4 780.cj odd 12 2
3042.2.a.d 1 52.i odd 6 1
3042.2.a.m 1 52.j odd 6 1
3042.2.b.d 2 52.l even 12 2
8112.2.a.x 1 39.i odd 6 1
8112.2.a.bb 1 39.h odd 6 1

Hecke kernels

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

\( T_{5} - 1 \) Copy content Toggle raw display
\( T_{7}^{2} + 2T_{7} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} + 2T_{11} + 4 \) 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
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$11$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$13$ \( T^{2} - 5T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$19$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$23$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$29$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$31$ \( (T - 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 11T + 121 \) Copy content Toggle raw display
$41$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$43$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$47$ \( (T - 2)^{2} \) Copy content Toggle raw display
$53$ \( (T - 1)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$61$ \( T^{2} - 11T + 121 \) Copy content Toggle raw display
$67$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$71$ \( T^{2} - 14T + 196 \) Copy content Toggle raw display
$73$ \( (T + 13)^{2} \) Copy content Toggle raw display
$79$ \( (T - 4)^{2} \) Copy content Toggle raw display
$83$ \( (T - 6)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$97$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
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