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Properties

Label 5292.2.i.g
Level $5292$
Weight $2$
Character orbit 5292.i
Analytic conductor $42.257$
Analytic rank $0$
Dimension $6$
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] = [5292,2,Mod(1549,5292)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5292, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5292.1549");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5292.i (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: \(42.2568327497\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.309123.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: no (minimal twist has level 252)
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 basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - \beta_1 + 1) q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - \beta_1 + 1) q^{5} + ( - 2 \beta_{3} - \beta_{2}) q^{11} - \beta_{3} q^{13} + ( - \beta_{5} + \beta_{4} + \beta_1) q^{17} + (\beta_{5} + \beta_{3} - \beta_{2}) q^{19} + ( - \beta_{5} + \beta_{4} + 2 \beta_{3} - 2) q^{23} + (\beta_{5} - 2 \beta_{3} + 2 \beta_{2}) q^{25} + ( - \beta_{5} + \beta_{4} + 5 \beta_{3} - 5) q^{29} + (3 \beta_{2} + 3 \beta_1 + 1) q^{31} + (\beta_{3} - 3 \beta_{2}) q^{37} + (2 \beta_{3} + \beta_{2}) q^{41} + ( - 2 \beta_{5} + 2 \beta_{4} + \cdots - 1) q^{43}+ \cdots + ( - \beta_{5} + \beta_{4} + 5 \beta_{3} + \cdots - 5) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{5} - 6 q^{11} - 3 q^{13} + 3 q^{19} - 6 q^{23} - 6 q^{25} - 15 q^{29} + 6 q^{31} + 3 q^{37} + 6 q^{41} - 3 q^{43} - 30 q^{47} - 18 q^{53} + 24 q^{55} - 6 q^{59} + 12 q^{61} - 6 q^{65} - 12 q^{67} + 30 q^{71} + 9 q^{73} + 6 q^{79} + 18 q^{83} + 15 q^{85} - 6 q^{89} + 48 q^{95} - 15 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} - \nu^{4} + 5\nu^{3} - 2\nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 4\nu^{4} - 11\nu^{3} + 17\nu^{2} - 12\nu + 3 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{5} - 5\nu^{4} + 16\nu^{3} - 19\nu^{2} + 21\nu - 6 ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 2\nu^{3} + 8\nu^{2} - 7\nu + 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7\nu^{5} - 16\nu^{4} + 62\nu^{3} - 68\nu^{2} + 99\nu - 30 ) / 3 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} + \beta_{3} - 2\beta _1 - 6 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{5} + \beta_{4} - 6\beta_{3} - 4\beta_{2} + \beta _1 - 6 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{5} - 3\beta_{4} - 13\beta_{3} - \beta_{2} + 11\beta _1 + 19 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -3\beta_{5} - 6\beta_{4} + 19\beta_{3} + 19\beta_{2} + 11\beta _1 + 37 ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5292\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\) \(2647\)
\(\chi(n)\) \(-\beta_{3}\) \(-\beta_{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1549.1
0.500000 − 2.05195i
0.500000 − 0.224437i
0.500000 + 1.41036i
0.500000 + 2.05195i
0.500000 + 0.224437i
0.500000 − 1.41036i
0 0 0 −1.02704 − 1.77889i 0 0 0 0 0
1549.2 0 0 0 0.555632 + 0.962383i 0 0 0 0 0
1549.3 0 0 0 1.97141 + 3.41458i 0 0 0 0 0
2125.1 0 0 0 −1.02704 + 1.77889i 0 0 0 0 0
2125.2 0 0 0 0.555632 − 0.962383i 0 0 0 0 0
2125.3 0 0 0 1.97141 − 3.41458i 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1549.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5292.2.i.g 6
3.b odd 2 1 1764.2.i.e 6
7.b odd 2 1 5292.2.i.d 6
7.c even 3 1 5292.2.j.e 6
7.c even 3 1 5292.2.l.d 6
7.d odd 6 1 756.2.j.a 6
7.d odd 6 1 5292.2.l.g 6
9.c even 3 1 5292.2.l.d 6
9.d odd 6 1 1764.2.l.g 6
21.c even 2 1 1764.2.i.f 6
21.g even 6 1 252.2.j.b ✓ 6
21.g even 6 1 1764.2.l.d 6
21.h odd 6 1 1764.2.j.d 6
21.h odd 6 1 1764.2.l.g 6
28.f even 6 1 3024.2.r.i 6
63.g even 3 1 5292.2.j.e 6
63.h even 3 1 inner 5292.2.i.g 6
63.i even 6 1 1764.2.i.f 6
63.i even 6 1 2268.2.a.g 3
63.j odd 6 1 1764.2.i.e 6
63.k odd 6 1 756.2.j.a 6
63.l odd 6 1 5292.2.l.g 6
63.n odd 6 1 1764.2.j.d 6
63.o even 6 1 1764.2.l.d 6
63.s even 6 1 252.2.j.b ✓ 6
63.t odd 6 1 2268.2.a.j 3
63.t odd 6 1 5292.2.i.d 6
84.j odd 6 1 1008.2.r.g 6
252.n even 6 1 3024.2.r.i 6
252.r odd 6 1 9072.2.a.bt 3
252.bj even 6 1 9072.2.a.bz 3
252.bn odd 6 1 1008.2.r.g 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.j.b ✓ 6 21.g even 6 1
252.2.j.b ✓ 6 63.s even 6 1
756.2.j.a 6 7.d odd 6 1
756.2.j.a 6 63.k odd 6 1
1008.2.r.g 6 84.j odd 6 1
1008.2.r.g 6 252.bn odd 6 1
1764.2.i.e 6 3.b odd 2 1
1764.2.i.e 6 63.j odd 6 1
1764.2.i.f 6 21.c even 2 1
1764.2.i.f 6 63.i even 6 1
1764.2.j.d 6 21.h odd 6 1
1764.2.j.d 6 63.n odd 6 1
1764.2.l.d 6 21.g even 6 1
1764.2.l.d 6 63.o even 6 1
1764.2.l.g 6 9.d odd 6 1
1764.2.l.g 6 21.h odd 6 1
2268.2.a.g 3 63.i even 6 1
2268.2.a.j 3 63.t odd 6 1
3024.2.r.i 6 28.f even 6 1
3024.2.r.i 6 252.n even 6 1
5292.2.i.d 6 7.b odd 2 1
5292.2.i.d 6 63.t odd 6 1
5292.2.i.g 6 1.a even 1 1 trivial
5292.2.i.g 6 63.h even 3 1 inner
5292.2.j.e 6 7.c even 3 1
5292.2.j.e 6 63.g even 3 1
5292.2.l.d 6 7.c even 3 1
5292.2.l.d 6 9.c even 3 1
5292.2.l.g 6 7.d odd 6 1
5292.2.l.g 6 63.l odd 6 1
9072.2.a.bt 3 252.r odd 6 1
9072.2.a.bz 3 252.bj even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} - 3T_{5}^{5} + 15T_{5}^{4} + 63T_{5}^{2} - 54T_{5} + 81 \) acting on \(S_{2}^{\mathrm{new}}(5292, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - 3 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} + 6 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$17$ \( T^{6} + 33 T^{4} + \cdots + 81 \) Copy content Toggle raw display
$19$ \( T^{6} - 3 T^{5} + \cdots + 2401 \) Copy content Toggle raw display
$23$ \( T^{6} + 6 T^{5} + \cdots + 9801 \) Copy content Toggle raw display
$29$ \( T^{6} + 15 T^{5} + \cdots + 3969 \) Copy content Toggle raw display
$31$ \( (T^{3} - 3 T^{2} - 78 T + 53)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} - 3 T^{5} + \cdots + 11449 \) Copy content Toggle raw display
$41$ \( T^{6} - 6 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$43$ \( T^{6} + 3 T^{5} + \cdots + 214369 \) Copy content Toggle raw display
$47$ \( (T^{3} + 15 T^{2} + \cdots - 81)^{2} \) Copy content Toggle raw display
$53$ \( T^{6} + 18 T^{5} + \cdots + 149769 \) Copy content Toggle raw display
$59$ \( (T^{3} + 3 T^{2} - 54 T + 81)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 6 T^{2} - 45 T - 31)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} + 6 T^{2} - 27 T - 59)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} - 15 T^{2} + \cdots + 297)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} - 9 T^{5} + \cdots + 6241 \) Copy content Toggle raw display
$79$ \( (T^{3} - 3 T^{2} + \cdots + 1363)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} - 18 T^{5} + \cdots + 729 \) Copy content Toggle raw display
$89$ \( T^{6} + 6 T^{5} + \cdots + 1185921 \) Copy content Toggle raw display
$97$ \( T^{6} + 15 T^{5} + \cdots + 529 \) Copy content Toggle raw display
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