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Properties

Label 1890.2.bf.b
Level $1890$
Weight $2$
Character orbit 1890.bf
Analytic conductor $15.092$
Analytic rank $0$
Dimension $8$
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(629,1890)]
 
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([1, 3, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1890.629");
 
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.bf (of order \(6\), 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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.856615824.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 11x^{6} + 36x^{4} + 32x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 630)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} - 1) q^{2} - \beta_{2} q^{4} + (\beta_{5} - \beta_{2} + \beta_1 + 1) q^{5} + ( - \beta_{7} + \beta_{4} + \cdots - 2 \beta_1) q^{7}+ \cdots + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - 1) q^{2} - \beta_{2} q^{4} + (\beta_{5} - \beta_{2} + \beta_1 + 1) q^{5} + ( - \beta_{7} + \beta_{4} + \cdots - 2 \beta_1) q^{7}+ \cdots + ( - 2 \beta_{6} - 2 \beta_{5} + \cdots + 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} - 4 q^{4} + 2 q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{2} - 4 q^{4} + 2 q^{7} + 8 q^{8} + 6 q^{10} - 9 q^{11} + 4 q^{13} - q^{14} - 4 q^{16} - 6 q^{20} + 9 q^{22} + 9 q^{23} - 16 q^{25} - 8 q^{26} - q^{28} - 21 q^{29} + 42 q^{31} - 4 q^{32} + 9 q^{34} + 33 q^{35} - 21 q^{38} + 24 q^{41} - 27 q^{43} - 18 q^{46} - 15 q^{47} - 4 q^{49} + 20 q^{50} + 4 q^{52} + 12 q^{53} - 20 q^{55} + 2 q^{56} + 21 q^{58} + 3 q^{59} + 21 q^{61} + 8 q^{64} - 18 q^{65} - 9 q^{68} - 24 q^{70} + 82 q^{73} + 6 q^{74} + 21 q^{76} + 9 q^{77} - 8 q^{79} + 6 q^{80} - 48 q^{82} - 15 q^{83} + 19 q^{85} + 27 q^{86} - 9 q^{88} + 42 q^{89} - 8 q^{91} + 9 q^{92} + 15 q^{94} - 9 q^{95} - 11 q^{97} + 29 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 11x^{6} + 36x^{4} + 32x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 7\nu^{3} + 10\nu + 2 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + 7\nu^{4} + 10\nu^{2} - 2\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} + 9\nu^{4} + 2\nu^{3} + 22\nu^{2} + 10\nu + 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} + 9\nu^{4} - 2\nu^{3} + 22\nu^{2} - 10\nu + 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} - \nu^{6} - 10\nu^{5} - 9\nu^{4} - 29\nu^{3} - 20\nu^{2} - 20\nu - 6 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} - \nu^{6} + 10\nu^{5} - 9\nu^{4} + 29\nu^{3} - 20\nu^{2} + 20\nu - 6 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{5} + \beta_{4} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{7} - 6\beta_{6} - 5\beta_{5} - 5\beta_{4} - 2\beta_{3} - \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 7\beta_{5} - 7\beta_{4} + 4\beta_{2} + 25\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 32\beta_{7} + 32\beta_{6} + 25\beta_{5} + 25\beta_{4} + 18\beta_{3} + 9\beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2\beta_{7} - 2\beta_{6} - 41\beta_{5} + 41\beta_{4} - 40\beta_{2} - 125\beta _1 + 20 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
629.1
1.07834i
2.33086i
− 2.06288i
0.385731i
− 1.07834i
− 2.33086i
2.06288i
− 0.385731i
−0.500000 + 0.866025i 0 −0.500000 − 0.866025i −1.24624 − 1.85658i 0 −2.47720 + 0.929227i 1.00000 0 2.23096 − 0.150985i
629.2 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i −1.05903 + 1.96938i 0 1.11699 − 2.39840i 1.00000 0 −1.17602 − 1.90184i
629.3 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 0.574618 − 2.16098i 0 −0.00953166 + 2.64573i 1.00000 0 1.58415 + 1.57812i
629.4 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 1.73065 − 1.41593i 0 2.36975 − 1.17656i 1.00000 0 0.360902 + 2.20675i
1259.1 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i −1.24624 + 1.85658i 0 −2.47720 − 0.929227i 1.00000 0 2.23096 + 0.150985i
1259.2 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i −1.05903 − 1.96938i 0 1.11699 + 2.39840i 1.00000 0 −1.17602 + 1.90184i
1259.3 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i 0.574618 + 2.16098i 0 −0.00953166 − 2.64573i 1.00000 0 1.58415 − 1.57812i
1259.4 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i 1.73065 + 1.41593i 0 2.36975 + 1.17656i 1.00000 0 0.360902 − 2.20675i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 629.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
315.z even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1890.2.bf.b 8
3.b odd 2 1 630.2.bf.c yes 8
5.b even 2 1 1890.2.bf.c 8
7.b odd 2 1 1890.2.bf.a 8
9.c even 3 1 630.2.bf.a ✓ 8
9.d odd 6 1 1890.2.bf.d 8
15.d odd 2 1 630.2.bf.b yes 8
21.c even 2 1 630.2.bf.d yes 8
35.c odd 2 1 1890.2.bf.d 8
45.h odd 6 1 1890.2.bf.a 8
45.j even 6 1 630.2.bf.d yes 8
63.l odd 6 1 630.2.bf.b yes 8
63.o even 6 1 1890.2.bf.c 8
105.g even 2 1 630.2.bf.a ✓ 8
315.z even 6 1 inner 1890.2.bf.b 8
315.bg odd 6 1 630.2.bf.c yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.bf.a ✓ 8 9.c even 3 1
630.2.bf.a ✓ 8 105.g even 2 1
630.2.bf.b yes 8 15.d odd 2 1
630.2.bf.b yes 8 63.l odd 6 1
630.2.bf.c yes 8 3.b odd 2 1
630.2.bf.c yes 8 315.bg odd 6 1
630.2.bf.d yes 8 21.c even 2 1
630.2.bf.d yes 8 45.j even 6 1
1890.2.bf.a 8 7.b odd 2 1
1890.2.bf.a 8 45.h odd 6 1
1890.2.bf.b 8 1.a even 1 1 trivial
1890.2.bf.b 8 315.z even 6 1 inner
1890.2.bf.c 8 5.b even 2 1
1890.2.bf.c 8 63.o even 6 1
1890.2.bf.d 8 9.d odd 6 1
1890.2.bf.d 8 35.c odd 2 1

Hecke kernels

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

\( T_{11}^{8} + 9T_{11}^{7} + 32T_{11}^{6} + 45T_{11}^{5} + 18T_{11}^{4} - 15T_{11}^{3} - 7T_{11}^{2} + 6T_{11} + 4 \) Copy content Toggle raw display
\( T_{13}^{8} - 4T_{13}^{7} + 22T_{13}^{6} - 28T_{13}^{5} + 136T_{13}^{4} - 124T_{13}^{3} + 700T_{13}^{2} + 104T_{13} + 16 \) Copy content Toggle raw display
\( T_{23}^{8} - 9T_{23}^{7} + 81T_{23}^{6} - 108T_{23}^{5} + 540T_{23}^{4} - 972T_{23}^{3} + 2916T_{23}^{2} - 2916T_{23} + 2916 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 8 T^{6} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{8} - 2 T^{7} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{8} + 9 T^{7} + \cdots + 4 \) Copy content Toggle raw display
$13$ \( T^{8} - 4 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$17$ \( T^{8} + 29 T^{6} + \cdots + 4 \) Copy content Toggle raw display
$19$ \( T^{8} + 93 T^{6} + \cdots + 324 \) Copy content Toggle raw display
$23$ \( T^{8} - 9 T^{7} + \cdots + 2916 \) Copy content Toggle raw display
$29$ \( T^{8} + 21 T^{7} + \cdots + 4 \) Copy content Toggle raw display
$31$ \( T^{8} - 42 T^{7} + \cdots + 1065024 \) Copy content Toggle raw display
$37$ \( T^{8} + 24 T^{6} + \cdots + 144 \) Copy content Toggle raw display
$41$ \( T^{8} - 24 T^{7} + \cdots + 848241 \) Copy content Toggle raw display
$43$ \( T^{8} + 27 T^{7} + \cdots + 2916 \) Copy content Toggle raw display
$47$ \( T^{8} + 15 T^{7} + \cdots + 868624 \) Copy content Toggle raw display
$53$ \( (T^{4} - 6 T^{3} + \cdots + 2064)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 3 T^{7} + \cdots + 46457856 \) Copy content Toggle raw display
$61$ \( T^{8} - 21 T^{7} + \cdots + 2022084 \) Copy content Toggle raw display
$67$ \( T^{8} - 42 T^{6} + \cdots + 56169 \) Copy content Toggle raw display
$71$ \( T^{8} + 404 T^{6} + \cdots + 17272336 \) Copy content Toggle raw display
$73$ \( (T^{4} - 41 T^{3} + \cdots + 7588)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 8 T^{7} + \cdots + 614656 \) Copy content Toggle raw display
$83$ \( T^{8} + 15 T^{7} + \cdots + 1024 \) Copy content Toggle raw display
$89$ \( (T^{4} - 21 T^{3} + \cdots - 40272)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 11 T^{7} + \cdots + 78074896 \) Copy content Toggle raw display
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