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Properties

Label 1890.2.j.i
Level $1890$
Weight $2$
Character orbit 1890.j
Analytic conductor $15.092$
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] = [1890,2,Mod(631,1890)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1890, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1890.631");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1890 = 2 \cdot 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1890.j (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: \(15.0917259820\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.954288.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: no (minimal twist has level 630)
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.

\(f(q)\) \(=\) \( q + (\beta_{2} + 1) q^{2} + \beta_{2} q^{4} + \beta_{2} q^{5} + ( - \beta_{2} - 1) q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 1) q^{2} + \beta_{2} q^{4} + \beta_{2} q^{5} + ( - \beta_{2} - 1) q^{7} - q^{8} - q^{10} + (\beta_{4} - \beta_{2} + \beta_1 - 1) q^{11} - \beta_{4} q^{13} - \beta_{2} q^{14} + ( - \beta_{2} - 1) q^{16} + ( - \beta_{3} + 3) q^{17} + ( - \beta_{3} - 1) q^{19} + ( - \beta_{2} - 1) q^{20} + (\beta_{4} - \beta_{2}) q^{22} + (\beta_{4} + 2 \beta_{2}) q^{23} + ( - \beta_{2} - 1) q^{25} + \beta_1 q^{26} + q^{28} + (\beta_{4} - 4 \beta_{2} + \beta_1 - 4) q^{29} - 4 \beta_{2} q^{31} - \beta_{2} q^{32} + ( - \beta_{5} + 3 \beta_{2} + 3) q^{34} + q^{35} + ( - \beta_{3} - \beta_1) q^{37} + ( - \beta_{5} - \beta_{2} - 1) q^{38} - \beta_{2} q^{40} + (\beta_{4} + 5 \beta_{2}) q^{41} + (\beta_{2} + 1) q^{43} + ( - \beta_1 + 1) q^{44} + ( - \beta_1 - 2) q^{46} + ( - 2 \beta_{5} + 2 \beta_{4} + \cdots - 2) q^{47}+ \cdots - q^{98}+O(q^{100}) \) 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 \( 6 q + 3 q^{2} - 3 q^{4} - 3 q^{5} - 3 q^{7} - 6 q^{8} - 6 q^{10} - 3 q^{11} + 3 q^{14} - 3 q^{16} + 18 q^{17} - 6 q^{19} - 3 q^{20} + 3 q^{22} - 6 q^{23} - 3 q^{25} + 6 q^{28} - 12 q^{29} + 12 q^{31} + 3 q^{32} + 9 q^{34} + 6 q^{35} - 3 q^{38} + 3 q^{40} - 15 q^{41} + 3 q^{43} + 6 q^{44} - 12 q^{46} - 6 q^{47} - 3 q^{49} + 3 q^{50} + 24 q^{53} + 6 q^{55} + 3 q^{56} + 12 q^{58} - 3 q^{59} + 24 q^{62} + 6 q^{64} + 9 q^{67} - 9 q^{68} + 3 q^{70} - 24 q^{71} - 6 q^{73} + 3 q^{76} - 3 q^{77} - 6 q^{79} + 6 q^{80} - 30 q^{82} - 24 q^{83} - 9 q^{85} - 3 q^{86} + 3 q^{88} + 36 q^{89} - 6 q^{92} + 6 q^{94} + 3 q^{95} + 9 q^{97} - 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} - 5\nu^{4} + 8\nu^{3} + 27\nu^{2} - 21\nu + 18 ) / 27 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{5} + \nu^{4} + 2\nu^{3} - 12\nu - 36 ) / 27 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{5} + 2\nu^{4} + 13\nu^{3} + 57\nu + 36 ) / 27 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7\nu^{5} - \nu^{4} - 11\nu^{3} + 12\nu - 72 ) / 27 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 8\nu^{5} + 13\nu^{4} - 10\nu^{3} + 27\nu^{2} + 6\nu - 171 ) / 27 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} + \beta_{3} - \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - 3\beta_{2} + 2\beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{5} + 3\beta_{3} + 12\beta_{2} + \beta _1 + 5 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4\beta_{5} - 6\beta_{4} + 3\beta_{2} - 4\beta _1 + 16 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -\beta_{5} + 9\beta_{4} + 3\beta_{3} + 21\beta_{2} + \beta _1 + 41 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1890\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(1081\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
631.1
0.403374 − 1.68443i
1.71903 + 0.211943i
−1.62241 + 0.606458i
0.403374 + 1.68443i
1.71903 − 0.211943i
−1.62241 − 0.606458i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.500000 + 0.866025i 0 −0.500000 − 0.866025i −1.00000 0 −1.00000
631.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.500000 + 0.866025i 0 −0.500000 − 0.866025i −1.00000 0 −1.00000
631.3 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.500000 + 0.866025i 0 −0.500000 − 0.866025i −1.00000 0 −1.00000
1261.1 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −0.500000 − 0.866025i 0 −0.500000 + 0.866025i −1.00000 0 −1.00000
1261.2 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −0.500000 − 0.866025i 0 −0.500000 + 0.866025i −1.00000 0 −1.00000
1261.3 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −0.500000 − 0.866025i 0 −0.500000 + 0.866025i −1.00000 0 −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 631.3
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1890.2.j.i 6
3.b odd 2 1 630.2.j.j ✓ 6
9.c even 3 1 inner 1890.2.j.i 6
9.c even 3 1 5670.2.a.bq 3
9.d odd 6 1 630.2.j.j ✓ 6
9.d odd 6 1 5670.2.a.br 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.j.j ✓ 6 3.b odd 2 1
630.2.j.j ✓ 6 9.d odd 6 1
1890.2.j.i 6 1.a even 1 1 trivial
1890.2.j.i 6 9.c even 3 1 inner
5670.2.a.bq 3 9.c even 3 1
5670.2.a.br 3 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1890, [\chi])\):

\( T_{11}^{6} + 3T_{11}^{5} + 24T_{11}^{4} - 63T_{11}^{3} + 198T_{11}^{2} - 135T_{11} + 81 \) Copy content Toggle raw display
\( T_{13}^{6} + 18T_{13}^{4} - 52T_{13}^{3} + 324T_{13}^{2} - 468T_{13} + 676 \) 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
$5$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$7$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{6} + 3 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( T^{6} + 18 T^{4} + \cdots + 676 \) Copy content Toggle raw display
$17$ \( (T^{3} - 9 T^{2} + 3 T + 81)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} + 3 T^{2} - 21 T + 13)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 6 T^{5} + \cdots + 2916 \) Copy content Toggle raw display
$29$ \( T^{6} + 12 T^{5} + \cdots + 324 \) Copy content Toggle raw display
$31$ \( (T^{2} - 4 T + 16)^{3} \) Copy content Toggle raw display
$37$ \( (T^{3} - 48 T + 34)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + 15 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$43$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$47$ \( T^{6} + 6 T^{5} + \cdots + 876096 \) Copy content Toggle raw display
$53$ \( (T^{3} - 12 T^{2} + 162)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 3 T^{5} + \cdots + 6561 \) Copy content Toggle raw display
$61$ \( T^{6} + 36 T^{4} + \cdots + 6724 \) Copy content Toggle raw display
$67$ \( T^{6} - 9 T^{5} + \cdots + 2809 \) Copy content Toggle raw display
$71$ \( (T^{3} + 12 T^{2} - 162)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + 3 T^{2} + \cdots - 383)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + 6 T^{5} + \cdots + 10816 \) Copy content Toggle raw display
$83$ \( T^{6} + 24 T^{5} + \cdots + 1679616 \) Copy content Toggle raw display
$89$ \( (T^{3} - 18 T^{2} + \cdots + 648)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} - 9 T^{5} + \cdots + 361 \) Copy content Toggle raw display
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