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Properties

Label 1764.2.i.h
Level $1764$
Weight $2$
Character orbit 1764.i
Analytic conductor $14.086$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1764,2,Mod(373,1764)]
 
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, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1764.373");
 
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.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: \(14.0856109166\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2x^{10} + 16x^{8} - 39x^{6} + 144x^{4} - 162x^{2} + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{10} + \beta_1) q^{3} + (\beta_{11} - \beta_{8}) q^{5} + (\beta_{6} - \beta_{3} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{10} + \beta_1) q^{3} + (\beta_{11} - \beta_{8}) q^{5} + (\beta_{6} - \beta_{3} - 1) q^{9} + ( - \beta_{5} - \beta_{4}) q^{11} + ( - \beta_{10} - \beta_{9} + \cdots - \beta_{2}) q^{13}+ \cdots + ( - \beta_{6} + 3 \beta_{5} - 2 \beta_{3} + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 8 q^{9} + 8 q^{11} - 10 q^{15} - 12 q^{25} + 2 q^{29} + 6 q^{37} - 28 q^{39} + 6 q^{43} - 12 q^{51} + 20 q^{53} + 26 q^{57} - 92 q^{65} + 24 q^{67} + 44 q^{71} - 12 q^{79} + 40 q^{81} - 18 q^{85} + 52 q^{93} - 56 q^{95} + 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2x^{10} + 16x^{8} - 39x^{6} + 144x^{4} - 162x^{2} + 729 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{9} + 4\nu^{7} - 5\nu^{5} + 24\nu^{3} - 18\nu ) / 135 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{10} - 4\nu^{8} + 5\nu^{6} - 24\nu^{4} + 423\nu^{2} ) / 405 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{10} - 4\nu^{8} + 5\nu^{6} - 24\nu^{4} + 18\nu^{2} ) / 405 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{10} + 5\nu^{8} - 13\nu^{6} + 201\nu^{4} - 90\nu^{2} + 1296 ) / 405 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2\nu^{10} + 23\nu^{8} - 49\nu^{6} + 246\nu^{4} - 306\nu^{2} + 1863 ) / 405 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{10} - 2\nu^{8} + 16\nu^{6} - 39\nu^{4} + 63\nu^{2} - 81 ) / 81 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{11} + 11\nu^{9} - 43\nu^{7} + 147\nu^{5} - 207\nu^{3} + 621\nu ) / 405 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -5\nu^{11} + 28\nu^{9} - 116\nu^{7} + 240\nu^{5} - 936\nu^{3} + 972\nu ) / 1215 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -2\nu^{11} + 4\nu^{9} - 5\nu^{7} + 24\nu^{5} - 18\nu^{3} - 405\nu ) / 405 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -7\nu^{11} + 5\nu^{9} - 67\nu^{7} - 6\nu^{5} + 180\nu^{3} - 1296\nu ) / 1215 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{4} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} - 2\beta_{9} + \beta_{8} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{6} + 3\beta_{5} - 2\beta_{4} - 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{11} + 2\beta_{10} - 2\beta_{9} + 4\beta_{8} + 5\beta_{2} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 6\beta_{7} - 2\beta_{6} + 6\beta_{5} - 15\beta_{4} - 4\beta_{3} - 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -14\beta_{11} + 19\beta_{10} - 2\beta_{9} - 2\beta_{8} + 2\beta_{2} + 9\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 12\beta_{7} + 21\beta_{6} - 18\beta_{5} - 37\beta_{4} + 4\beta_{3} - 27 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -11\beta_{11} + 33\beta_{10} - 23\beta_{9} - 2\beta_{8} - 88\beta_{2} + 31\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 9\beta_{7} + 35\beta_{6} - 15\beta_{5} + 151\beta_{4} + 9\beta_{3} - 104 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -20\beta_{11} - 160\beta_{10} - 47\beta_{9} + 40\beta_{8} - 112\beta_{2} - 220\beta_1 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
373.1
−1.19475 − 1.25402i
0.965975 − 1.43767i
−1.62464 + 0.600450i
1.62464 − 0.600450i
−0.965975 + 1.43767i
1.19475 + 1.25402i
−1.19475 + 1.25402i
0.965975 + 1.43767i
−1.62464 − 0.600450i
1.62464 + 0.600450i
−0.965975 − 1.43767i
1.19475 − 1.25402i
0 −1.68339 + 0.407675i 0 0.749222 + 1.29769i 0 0 0 2.66760 − 1.37255i 0
373.2 0 −0.762070 − 1.55539i 0 −1.95014 − 3.37774i 0 0 0 −1.83850 + 2.37064i 0
373.3 0 −0.292315 + 1.70721i 0 −0.941081 − 1.63000i 0 0 0 −2.82910 − 0.998085i 0
373.4 0 0.292315 − 1.70721i 0 0.941081 + 1.63000i 0 0 0 −2.82910 − 0.998085i 0
373.5 0 0.762070 + 1.55539i 0 1.95014 + 3.37774i 0 0 0 −1.83850 + 2.37064i 0
373.6 0 1.68339 − 0.407675i 0 −0.749222 − 1.29769i 0 0 0 2.66760 − 1.37255i 0
1537.1 0 −1.68339 − 0.407675i 0 0.749222 − 1.29769i 0 0 0 2.66760 + 1.37255i 0
1537.2 0 −0.762070 + 1.55539i 0 −1.95014 + 3.37774i 0 0 0 −1.83850 − 2.37064i 0
1537.3 0 −0.292315 − 1.70721i 0 −0.941081 + 1.63000i 0 0 0 −2.82910 + 0.998085i 0
1537.4 0 0.292315 + 1.70721i 0 0.941081 − 1.63000i 0 0 0 −2.82910 + 0.998085i 0
1537.5 0 0.762070 − 1.55539i 0 1.95014 − 3.37774i 0 0 0 −1.83850 − 2.37064i 0
1537.6 0 1.68339 + 0.407675i 0 −0.749222 + 1.29769i 0 0 0 2.66760 + 1.37255i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 373.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
63.h even 3 1 inner
63.t odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.2.i.h 12
3.b odd 2 1 5292.2.i.h 12
7.b odd 2 1 inner 1764.2.i.h 12
7.c even 3 1 1764.2.j.f ✓ 12
7.c even 3 1 1764.2.l.h 12
7.d odd 6 1 1764.2.j.f ✓ 12
7.d odd 6 1 1764.2.l.h 12
9.c even 3 1 1764.2.l.h 12
9.d odd 6 1 5292.2.l.h 12
21.c even 2 1 5292.2.i.h 12
21.g even 6 1 5292.2.j.f 12
21.g even 6 1 5292.2.l.h 12
21.h odd 6 1 5292.2.j.f 12
21.h odd 6 1 5292.2.l.h 12
63.g even 3 1 1764.2.j.f ✓ 12
63.h even 3 1 inner 1764.2.i.h 12
63.i even 6 1 5292.2.i.h 12
63.j odd 6 1 5292.2.i.h 12
63.k odd 6 1 1764.2.j.f ✓ 12
63.l odd 6 1 1764.2.l.h 12
63.n odd 6 1 5292.2.j.f 12
63.o even 6 1 5292.2.l.h 12
63.s even 6 1 5292.2.j.f 12
63.t odd 6 1 inner 1764.2.i.h 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1764.2.i.h 12 1.a even 1 1 trivial
1764.2.i.h 12 7.b odd 2 1 inner
1764.2.i.h 12 63.h even 3 1 inner
1764.2.i.h 12 63.t odd 6 1 inner
1764.2.j.f ✓ 12 7.c even 3 1
1764.2.j.f ✓ 12 7.d odd 6 1
1764.2.j.f ✓ 12 63.g even 3 1
1764.2.j.f ✓ 12 63.k odd 6 1
1764.2.l.h 12 7.c even 3 1
1764.2.l.h 12 7.d odd 6 1
1764.2.l.h 12 9.c even 3 1
1764.2.l.h 12 63.l odd 6 1
5292.2.i.h 12 3.b odd 2 1
5292.2.i.h 12 21.c even 2 1
5292.2.i.h 12 63.i even 6 1
5292.2.i.h 12 63.j odd 6 1
5292.2.j.f 12 21.g even 6 1
5292.2.j.f 12 21.h odd 6 1
5292.2.j.f 12 63.n odd 6 1
5292.2.j.f 12 63.s even 6 1
5292.2.l.h 12 9.d odd 6 1
5292.2.l.h 12 21.g even 6 1
5292.2.l.h 12 21.h odd 6 1
5292.2.l.h 12 63.o even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 21T_{5}^{10} + 345T_{5}^{8} + 1774T_{5}^{6} + 6675T_{5}^{4} + 11616T_{5}^{2} + 14641 \) acting on \(S_{2}^{\mathrm{new}}(1764, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 4 T^{10} + \cdots + 729 \) Copy content Toggle raw display
$5$ \( T^{12} + 21 T^{10} + \cdots + 14641 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( (T^{6} - 4 T^{5} + \cdots + 625)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + 39 T^{10} + \cdots + 390625 \) Copy content Toggle raw display
$17$ \( T^{12} + 48 T^{10} + \cdots + 4100625 \) Copy content Toggle raw display
$19$ \( T^{12} + 27 T^{10} + \cdots + 625 \) Copy content Toggle raw display
$23$ \( (T^{6} + 51 T^{4} + \cdots + 18225)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - T^{5} + 35 T^{4} + \cdots + 6241)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 69 T^{4} + \cdots - 25)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 3 T^{5} + \cdots + 164025)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + 66 T^{10} + \cdots + 4100625 \) Copy content Toggle raw display
$43$ \( (T^{6} - 3 T^{5} + \cdots + 319225)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 63 T^{4} + \cdots - 7225)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} - 10 T^{5} + \cdots + 21025)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 171 T^{4} + \cdots - 15625)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - 132 T^{4} + \cdots - 625)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} - 6 T^{2} + \cdots + 155)^{4} \) Copy content Toggle raw display
$71$ \( (T^{3} - 11 T^{2} + \cdots + 11)^{4} \) Copy content Toggle raw display
$73$ \( T^{12} + 375 T^{10} + \cdots + 12117361 \) Copy content Toggle raw display
$79$ \( (T^{3} + 3 T^{2} - 60 T - 25)^{4} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 18539817921 \) Copy content Toggle raw display
$89$ \( T^{12} + 102 T^{10} + \cdots + 1874161 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 9704626800625 \) Copy content Toggle raw display
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