Properties

Label 1638.2.j.q
Level $1638$
Weight $2$
Character orbit 1638.j
Analytic conductor $13.079$
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] = [1638,2,Mod(235,1638)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1638, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1638.235");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1638.j (of order \(3\), degree \(2\), 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: \(13.0794958511\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.21870000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 24x^{4} - 43x^{3} + 138x^{2} - 117x + 73 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( - \beta_{2} - 1) q^{4} + (\beta_{4} - \beta_{2} - \beta_1) q^{5} + (\beta_{5} - \beta_{4} + \beta_{3} + \cdots + 1) q^{7}+ \cdots + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + ( - \beta_{2} - 1) q^{4} + (\beta_{4} - \beta_{2} - \beta_1) q^{5} + (\beta_{5} - \beta_{4} + \beta_{3} + \cdots + 1) q^{7}+ \cdots + (3 \beta_{5} - \beta_{4} + 2 \beta_{2} + \cdots - 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{2} - 3 q^{4} + 3 q^{5} + 3 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{2} - 3 q^{4} + 3 q^{5} + 3 q^{7} + 6 q^{8} + 3 q^{10} - 3 q^{11} + 6 q^{13} - 3 q^{16} + 6 q^{17} - 6 q^{19} - 6 q^{20} + 6 q^{22} - 18 q^{25} - 3 q^{26} - 3 q^{28} + 18 q^{29} + 15 q^{31} - 3 q^{32} - 12 q^{34} - 30 q^{35} - 6 q^{37} - 6 q^{38} + 3 q^{40} - 36 q^{41} - 24 q^{43} - 3 q^{44} - 6 q^{47} + 21 q^{49} + 36 q^{50} - 3 q^{52} + 3 q^{53} + 54 q^{55} + 3 q^{56} - 9 q^{58} - 3 q^{59} - 30 q^{62} + 6 q^{64} + 3 q^{65} - 6 q^{67} + 6 q^{68} + 3 q^{70} + 72 q^{71} - 24 q^{73} - 6 q^{74} + 12 q^{76} - 33 q^{77} - 15 q^{79} + 3 q^{80} + 18 q^{82} - 18 q^{83} - 48 q^{85} + 12 q^{86} - 3 q^{88} + 30 q^{89} + 3 q^{91} - 6 q^{94} + 6 q^{95} + 6 q^{97} - 9 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} + 24x^{4} - 43x^{3} + 138x^{2} - 117x + 73 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{5} + 5\nu^{4} - 30\nu^{3} + 40\nu^{2} - 70\nu + 13 ) / 31 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} + 18\nu^{4} - 15\nu^{3} + 144\nu^{2} + 151\nu - 164 ) / 62 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} - 13\nu^{4} + 47\nu^{3} - 197\nu^{2} + 461\nu - 133 ) / 62 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{5} - 13\nu^{4} + 16\nu^{3} - 166\nu^{2} + 151\nu - 102 ) / 31 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -2\beta_{5} + 2\beta_{4} - 2\beta_{3} + \beta_{2} + 2\beta _1 - 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{5} + 4\beta_{4} - 2\beta_{3} + \beta_{2} - 8\beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 16\beta_{5} - 16\beta_{4} + 20\beta_{3} - 9\beta_{2} - 28\beta _1 + 75 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 45\beta_{5} - 60\beta_{4} + 40\beta_{3} - 33\beta_{2} + 55\beta _1 + 139 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1638\mathbb{Z}\right)^\times\).

\(n\) \(379\) \(703\) \(911\)
\(\chi(n)\) \(1\) \(\beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
235.1
0.500000 + 3.23735i
0.500000 3.05087i
0.500000 + 0.679547i
0.500000 3.23735i
0.500000 + 3.05087i
0.500000 0.679547i
−0.500000 0.866025i 0 −0.500000 + 0.866025i −1.70942 2.96080i 0 2.36521 1.18566i 1.00000 0 −1.70942 + 2.96080i
235.2 −0.500000 0.866025i 0 −0.500000 + 0.866025i 1.30661 + 2.26312i 0 1.77890 + 1.95845i 1.00000 0 1.30661 2.26312i
235.3 −0.500000 0.866025i 0 −0.500000 + 0.866025i 1.90280 + 3.29575i 0 −2.64411 + 0.0932392i 1.00000 0 1.90280 3.29575i
1171.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i −1.70942 + 2.96080i 0 2.36521 + 1.18566i 1.00000 0 −1.70942 2.96080i
1171.2 −0.500000 + 0.866025i 0 −0.500000 0.866025i 1.30661 2.26312i 0 1.77890 1.95845i 1.00000 0 1.30661 + 2.26312i
1171.3 −0.500000 + 0.866025i 0 −0.500000 0.866025i 1.90280 3.29575i 0 −2.64411 0.0932392i 1.00000 0 1.90280 + 3.29575i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 235.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1638.2.j.q 6
3.b odd 2 1 546.2.i.k 6
7.c even 3 1 inner 1638.2.j.q 6
21.g even 6 1 3822.2.a.bw 3
21.h odd 6 1 546.2.i.k 6
21.h odd 6 1 3822.2.a.bv 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.i.k 6 3.b odd 2 1
546.2.i.k 6 21.h odd 6 1
1638.2.j.q 6 1.a even 1 1 trivial
1638.2.j.q 6 7.c even 3 1 inner
3822.2.a.bv 3 21.h odd 6 1
3822.2.a.bw 3 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1638, [\chi])\):

\( T_{5}^{6} - 3T_{5}^{5} + 21T_{5}^{4} - 32T_{5}^{3} + 246T_{5}^{2} - 408T_{5} + 1156 \) Copy content Toggle raw display
\( T_{11}^{6} + 3T_{11}^{5} + 36T_{11}^{4} + 97T_{11}^{3} + 996T_{11}^{2} + 2403T_{11} + 7921 \) Copy content Toggle raw display
\( T_{17}^{6} - 6T_{17}^{5} + 99T_{17}^{4} - 386T_{17}^{3} + 6261T_{17}^{2} - 24066T_{17} + 145924 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - 3 T^{5} + \cdots + 1156 \) Copy content Toggle raw display
$7$ \( T^{6} - 3 T^{5} + \cdots + 343 \) Copy content Toggle raw display
$11$ \( T^{6} + 3 T^{5} + \cdots + 7921 \) Copy content Toggle raw display
$13$ \( (T - 1)^{6} \) Copy content Toggle raw display
$17$ \( T^{6} - 6 T^{5} + \cdots + 145924 \) Copy content Toggle raw display
$19$ \( T^{6} + 6 T^{5} + \cdots + 1024 \) Copy content Toggle raw display
$23$ \( T^{6} + 30 T^{4} + \cdots + 3600 \) Copy content Toggle raw display
$29$ \( (T - 3)^{6} \) Copy content Toggle raw display
$31$ \( T^{6} - 15 T^{5} + \cdots + 1600 \) Copy content Toggle raw display
$37$ \( T^{6} + 6 T^{5} + \cdots + 1024 \) Copy content Toggle raw display
$41$ \( (T^{3} + 18 T^{2} + \cdots + 56)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} + 12 T^{2} + \cdots - 16)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 6 T^{5} + \cdots + 44944 \) Copy content Toggle raw display
$53$ \( T^{6} - 3 T^{5} + \cdots + 1681 \) Copy content Toggle raw display
$59$ \( T^{6} + 3 T^{5} + \cdots + 2401 \) Copy content Toggle raw display
$61$ \( T^{6} + 75 T^{4} + \cdots + 40000 \) Copy content Toggle raw display
$67$ \( T^{6} + 6 T^{5} + \cdots + 64 \) Copy content Toggle raw display
$71$ \( (T^{3} - 36 T^{2} + \cdots - 1538)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 24 T^{5} + \cdots + 44944 \) Copy content Toggle raw display
$79$ \( T^{6} + 15 T^{5} + \cdots + 10000 \) Copy content Toggle raw display
$83$ \( (T^{3} + 9 T^{2} + \cdots - 108)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} - 30 T^{5} + \cdots + 462400 \) Copy content Toggle raw display
$97$ \( (T^{3} - 3 T^{2} + \cdots - 166)^{2} \) Copy content Toggle raw display
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