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Properties

Label 1890.2.j.j
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: \( 3 \)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + (\beta_{3} - 1) q^{4} + ( - \beta_{3} + 1) q^{5} - \beta_{3} q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + (\beta_{3} - 1) q^{4} + ( - \beta_{3} + 1) q^{5} - \beta_{3} q^{7} - q^{8} + q^{10} + ( - \beta_{5} + \beta_{4}) q^{11} + (\beta_{5} - \beta_{3} + 1) q^{13} + ( - \beta_{3} + 1) q^{14} - \beta_{3} q^{16} + (\beta_{2} - 2) q^{17} + (2 \beta_{4} + \beta_{2}) q^{19} + \beta_{3} q^{20} - \beta_{5} q^{22} + ( - \beta_{5} + \beta_{3} + 2 \beta_{2} + \cdots - 1) q^{23}+ \cdots - q^{98}+O(q^{100}) \) 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 \( 6 q + 3 q^{2} - 3 q^{4} + 3 q^{5} - 3 q^{7} - 6 q^{8} + 6 q^{10} - q^{11} + 2 q^{13} + 3 q^{14} - 3 q^{16} - 14 q^{17} - 6 q^{19} + 3 q^{20} + q^{22} - 4 q^{23} - 3 q^{25} + 4 q^{26} + 6 q^{28} + 6 q^{29} - 4 q^{31} + 3 q^{32} - 7 q^{34} - 6 q^{35} + 20 q^{37} - 3 q^{38} - 3 q^{40} + 7 q^{41} - 17 q^{43} + 2 q^{44} - 8 q^{46} + 14 q^{47} - 3 q^{49} + 3 q^{50} + 2 q^{52} + 12 q^{53} - 2 q^{55} + 3 q^{56} - 6 q^{58} + 29 q^{59} + 2 q^{61} - 8 q^{62} + 6 q^{64} - 2 q^{65} + q^{67} + 7 q^{68} - 3 q^{70} - 20 q^{71} + 2 q^{73} + 10 q^{74} + 3 q^{76} - q^{77} - 2 q^{79} - 6 q^{80} + 14 q^{82} + 12 q^{83} - 7 q^{85} + 17 q^{86} + q^{88} - 44 q^{89} - 4 q^{91} - 4 q^{92} - 14 q^{94} - 3 q^{95} + 5 q^{97} - 6 q^{98}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + 4\nu^{4} - \nu^{3} + 9\nu^{2} - 21\nu - 9 ) / 27 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 4\nu^{4} - \nu^{3} - 18\nu^{2} + 33\nu - 9 ) / 27 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{5} - \nu^{4} - 2\nu^{3} + 12\nu + 36 ) / 27 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{5} - \nu^{4} + 7\nu^{3} + 9\nu^{2} + 12\nu + 9 ) / 27 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4\nu^{5} + 2\nu^{4} - 5\nu^{3} + 18\nu^{2} + 3\nu - 72 ) / 27 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{5} + 2\beta_{4} + 2\beta_{3} - \beta_{2} + \beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{5} + 7\beta_{4} - 11\beta_{3} + \beta_{2} - \beta _1 + 7 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{5} + 2\beta_{4} - 7\beta_{3} + 8\beta_{2} + 10\beta _1 + 20 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 7\beta_{5} - 2\beta_{4} - 20\beta_{3} + \beta_{2} - 10\beta _1 + 43 ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(757\) \(1081\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(-1 + \beta_{3}\)

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
−1.62241 − 0.606458i
1.71903 − 0.211943i
0.403374 − 1.68443i
−1.62241 + 0.606458i
1.71903 + 0.211943i
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.j 6
3.b odd 2 1 630.2.j.k ✓ 6
9.c even 3 1 inner 1890.2.j.j 6
9.c even 3 1 5670.2.a.bp 3
9.d odd 6 1 630.2.j.k ✓ 6
9.d odd 6 1 5670.2.a.bt 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.j.k ✓ 6 3.b odd 2 1
630.2.j.k ✓ 6 9.d odd 6 1
1890.2.j.j 6 1.a even 1 1 trivial
1890.2.j.j 6 9.c even 3 1 inner
5670.2.a.bp 3 9.c even 3 1
5670.2.a.bt 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 \(S_{2}^{\mathrm{new}}(1890, [\chi])\):

\( T_{11}^{6} + T_{11}^{5} + 6T_{11}^{4} + T_{11}^{3} + 28T_{11}^{2} + 15T_{11} + 9 \) Copy content Toggle raw display
\( T_{13}^{6} - 2T_{13}^{5} + 8T_{13}^{4} + 4T_{13}^{3} + 20T_{13}^{2} - 8T_{13} + 4 \) 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
$11$ \( T^{6} + T^{5} + 6 T^{4} + \cdots + 9 \) Copy content Toggle raw display
$13$ \( T^{6} - 2 T^{5} + \cdots + 4 \) Copy content Toggle raw display
$17$ \( (T^{3} + 7 T^{2} + 7 T - 3)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} + 3 T^{2} - 21 T - 59)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 4 T^{5} + \cdots + 12996 \) Copy content Toggle raw display
$29$ \( T^{6} - 6 T^{5} + \cdots + 324 \) Copy content Toggle raw display
$31$ \( T^{6} + 4 T^{5} + \cdots + 256 \) Copy content Toggle raw display
$37$ \( (T^{3} - 10 T^{2} + \cdots + 346)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} - 7 T^{5} + \cdots + 826281 \) Copy content Toggle raw display
$43$ \( T^{6} + 17 T^{5} + \cdots + 2601 \) Copy content Toggle raw display
$47$ \( T^{6} - 14 T^{5} + \cdots + 576 \) Copy content Toggle raw display
$53$ \( (T^{3} - 6 T^{2} - 6 T + 54)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} - 29 T^{5} + \cdots + 651249 \) Copy content Toggle raw display
$61$ \( T^{6} - 2 T^{5} + \cdots + 4 \) Copy content Toggle raw display
$67$ \( T^{6} - T^{5} + \cdots + 19881 \) Copy content Toggle raw display
$71$ \( (T^{3} + 10 T^{2} + \cdots - 2502)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - T^{2} - 37 T + 61)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + 2 T^{5} + \cdots + 576 \) Copy content Toggle raw display
$83$ \( T^{6} - 12 T^{5} + \cdots + 186624 \) Copy content Toggle raw display
$89$ \( (T^{3} + 22 T^{2} + \cdots - 312)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} - 5 T^{5} + \cdots + 12769 \) Copy content Toggle raw display
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