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Properties

Label 1764.2.l.e
Level $1764$
Weight $2$
Character orbit 1764.l
Analytic conductor $14.086$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1764,2,Mod(949,1764)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1764, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1764.949");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1764.l (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: \(14.0856109166\)
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: \( 1 \)
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.

\(f(q)\) \(=\) \( q + (\beta_{5} - \beta_{2} - \beta_1) q^{3} + ( - \beta_{3} - \beta_{2}) q^{5} + ( - \beta_{5} + 2 \beta_{4} + \cdots + 2 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} - \beta_{2} - \beta_1) q^{3} + ( - \beta_{3} - \beta_{2}) q^{5} + ( - \beta_{5} + 2 \beta_{4} + \cdots + 2 \beta_1) q^{9}+ \cdots + ( - 8 \beta_{5} - 8 \beta_{4} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{3} + 2 q^{5} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{3} + 2 q^{5} - 4 q^{9} + 4 q^{11} - 3 q^{13} + q^{15} - 2 q^{17} - 3 q^{19} + 28 q^{23} - 12 q^{25} + 7 q^{27} - q^{29} + 3 q^{31} - 25 q^{33} + 3 q^{37} + 18 q^{39} - 3 q^{43} - 10 q^{45} - 21 q^{47} - 13 q^{51} - 6 q^{53} + 12 q^{55} - 37 q^{57} - 31 q^{59} - 6 q^{61} - 15 q^{65} - 6 q^{67} + 5 q^{69} + 34 q^{71} + 3 q^{73} - 7 q^{75} + 9 q^{79} - 40 q^{81} - 20 q^{83} + 15 q^{85} + 16 q^{87} - 12 q^{89} + 33 q^{93} - 20 q^{95} + 9 q^{97} - 8 q^{99}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - \nu^{4} + 5\nu^{3} + \nu^{2} + 6 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} + \nu^{4} - 5\nu^{3} + 2\nu^{2} - 3\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{5} + 5\nu^{4} - 16\nu^{3} + 19\nu^{2} - 21\nu + 6 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{5} - 5\nu^{4} + 19\nu^{3} - 22\nu^{2} + 33\nu - 9 ) / 3 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 3\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} + 3\beta_{4} - 5\beta_{3} - 3\beta_{2} - 6\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -3\beta_{5} - 2\beta_{4} - 11\beta_{3} - 6\beta_{2} + 8\beta _1 + 7 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(-1 - \beta_{4}\) \(1\) \(\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
949.1
0.500000 + 0.224437i
0.500000 + 2.05195i
0.500000 − 1.41036i
0.500000 − 0.224437i
0.500000 − 2.05195i
0.500000 + 1.41036i
0 −1.29418 + 1.15113i 0 −1.69963 0 0 0 0.349814 − 2.97954i 0
949.2 0 −0.796790 − 1.53790i 0 2.46050 0 0 0 −1.73025 + 2.45076i 0
949.3 0 1.09097 − 1.34528i 0 0.239123 0 0 0 −0.619562 − 2.93533i 0
961.1 0 −1.29418 − 1.15113i 0 −1.69963 0 0 0 0.349814 + 2.97954i 0
961.2 0 −0.796790 + 1.53790i 0 2.46050 0 0 0 −1.73025 − 2.45076i 0
961.3 0 1.09097 + 1.34528i 0 0.239123 0 0 0 −0.619562 + 2.93533i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 949.3
Significant digits:
Format:

Inner twists

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

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{3} - T_{5}^{2} - 4T_{5} + 1 \) acting on \(S_{2}^{\mathrm{new}}(1764, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 2 T^{5} + \cdots + 27 \) Copy content Toggle raw display
$5$ \( (T^{3} - T^{2} - 4 T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( (T^{3} - 2 T^{2} - 25 T + 59)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 3 T^{5} + \cdots + 729 \) Copy content Toggle raw display
$17$ \( T^{6} + 2 T^{5} + \cdots + 2209 \) Copy content Toggle raw display
$19$ \( T^{6} + 3 T^{5} + \cdots + 6241 \) Copy content Toggle raw display
$23$ \( (T^{3} - 14 T^{2} + \cdots - 31)^{2} \) Copy content Toggle raw display
$29$ \( T^{6} + T^{5} + \cdots + 11881 \) Copy content Toggle raw display
$31$ \( T^{6} - 3 T^{5} + \cdots + 729 \) Copy content Toggle raw display
$37$ \( T^{6} - 3 T^{5} + \cdots + 529 \) Copy content Toggle raw display
$41$ \( T^{6} + 33 T^{4} + \cdots + 81 \) Copy content Toggle raw display
$43$ \( T^{6} + 3 T^{5} + \cdots + 78961 \) Copy content Toggle raw display
$47$ \( T^{6} + 21 T^{5} + \cdots + 42849 \) Copy content Toggle raw display
$53$ \( T^{6} + 6 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$59$ \( T^{6} + 31 T^{5} + \cdots + 978121 \) Copy content Toggle raw display
$61$ \( T^{6} + 6 T^{5} + \cdots + 1185921 \) Copy content Toggle raw display
$67$ \( T^{6} + 6 T^{5} + \cdots + 49 \) Copy content Toggle raw display
$71$ \( (T^{3} - 17 T^{2} + \cdots + 1907)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} - 3 T^{5} + \cdots + 6241 \) Copy content Toggle raw display
$79$ \( T^{6} - 9 T^{5} + \cdots + 136161 \) Copy content Toggle raw display
$83$ \( T^{6} + 20 T^{5} + \cdots + 16129 \) Copy content Toggle raw display
$89$ \( T^{6} + 12 T^{5} + \cdots + 505521 \) Copy content Toggle raw display
$97$ \( T^{6} - 9 T^{5} + \cdots + 5329 \) Copy content Toggle raw display
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