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Properties

Label 1890.2.b.a
Level $1890$
Weight $2$
Character orbit 1890.b
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(1511,1890)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1890, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1890.1511");
 
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.b (of order \(2\), degree \(1\), 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: \(8\)
Coefficient field: 8.0.303595776.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} - q^{4} - q^{5} + ( - \beta_{6} - \beta_{5} + \beta_1) q^{7} + \beta_{4} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} - q^{4} - q^{5} + ( - \beta_{6} - \beta_{5} + \beta_1) q^{7} + \beta_{4} q^{8} + \beta_{4} q^{10} + \beta_1 q^{11} + (\beta_{5} - \beta_{4} - \beta_{2} - \beta_1) q^{13} + (\beta_{6} - \beta_{3} + \beta_1) q^{14} + q^{16} + ( - \beta_{6} - \beta_{3}) q^{17} + ( - 2 \beta_{4} - \beta_1) q^{19} + q^{20} + \beta_{6} q^{22} + ( - 2 \beta_{5} + \beta_{2}) q^{23} + q^{25} + ( - \beta_{7} - \beta_{6} + \beta_{3} - 1) q^{26} + (\beta_{6} + \beta_{5} - \beta_1) q^{28} + \beta_{2} q^{29} + ( - \beta_{5} - 2 \beta_{2} + \beta_1) q^{31} - \beta_{4} q^{32} + (\beta_{5} + \beta_1) q^{34} + (\beta_{6} + \beta_{5} - \beta_1) q^{35} + ( - 2 \beta_{7} + 2 \beta_{6} + \cdots + 2) q^{37}+ \cdots + (\beta_{4} + 2 \beta_{2} - 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4} - 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} - 8 q^{5} + 8 q^{16} + 8 q^{20} + 8 q^{25} - 4 q^{26} + 24 q^{37} - 16 q^{38} - 4 q^{41} + 20 q^{43} - 4 q^{46} - 8 q^{47} - 4 q^{58} - 24 q^{59} + 8 q^{62} - 8 q^{64} - 12 q^{67} - 16 q^{77} - 24 q^{79} - 8 q^{80} - 64 q^{83} - 52 q^{89} + 28 q^{91} - 32 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{6} + 4\nu^{4} + 20\nu^{2} + 27 ) / 36 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 8\nu^{5} + 40\nu^{3} + 165\nu ) / 72 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 4\nu^{5} + 2\nu^{3} - 9\nu ) / 54 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5\nu^{7} + 16\nu^{5} + 8\nu^{3} + 81\nu ) / 216 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5\nu^{6} + 16\nu^{4} + 80\nu^{2} + 153 ) / 72 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} + 5\nu^{5} + 16\nu^{3} + 18\nu ) / 27 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{6} + 5\nu^{4} + 7\nu^{2} + 18 ) / 9 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{6} + \beta_{4} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} + 2\beta_{5} + \beta _1 - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{6} - 4\beta_{4} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 5\beta_{7} - 6\beta_{5} + 5\beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -\beta_{6} + 15\beta_{4} + 16\beta_{3} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{5} - 16\beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 13\beta_{6} + 35\beta_{4} - 48\beta_{3} - 13\beta_{2} ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(757\) \(1081\) \(1541\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1511.1
−1.26217 − 1.18614i
−0.396143 + 1.68614i
0.396143 + 1.68614i
1.26217 − 1.18614i
−1.26217 + 1.18614i
−0.396143 − 1.68614i
0.396143 − 1.68614i
1.26217 + 1.18614i
− 1.00000i 0 −1.00000 −1.00000 0 −2.52434 + 0.792287i 1.00000i 0 1.00000i
1511.2 − 1.00000i 0 −1.00000 −1.00000 0 −0.792287 + 2.52434i 1.00000i 0 1.00000i
1511.3 − 1.00000i 0 −1.00000 −1.00000 0 0.792287 − 2.52434i 1.00000i 0 1.00000i
1511.4 − 1.00000i 0 −1.00000 −1.00000 0 2.52434 − 0.792287i 1.00000i 0 1.00000i
1511.5 1.00000i 0 −1.00000 −1.00000 0 −2.52434 − 0.792287i − 1.00000i 0 − 1.00000i
1511.6 1.00000i 0 −1.00000 −1.00000 0 −0.792287 − 2.52434i − 1.00000i 0 − 1.00000i
1511.7 1.00000i 0 −1.00000 −1.00000 0 0.792287 + 2.52434i − 1.00000i 0 − 1.00000i
1511.8 1.00000i 0 −1.00000 −1.00000 0 2.52434 + 0.792287i − 1.00000i 0 − 1.00000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1511.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1890.2.b.a ✓ 8
3.b odd 2 1 1890.2.b.b yes 8
7.b odd 2 1 1890.2.b.b yes 8
21.c even 2 1 inner 1890.2.b.a ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1890.2.b.a ✓ 8 1.a even 1 1 trivial
1890.2.b.a ✓ 8 21.c even 2 1 inner
1890.2.b.b yes 8 3.b odd 2 1
1890.2.b.b yes 8 7.b odd 2 1

Hecke kernels

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

\( T_{11}^{4} + 7T_{11}^{2} + 4 \) Copy content Toggle raw display
\( T_{17}^{4} - 19T_{17}^{2} + 16 \) Copy content Toggle raw display
\( T_{41}^{4} + 2T_{41}^{3} - 43T_{41}^{2} - 176T_{41} - 176 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T + 1)^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 34T^{4} + 2401 \) Copy content Toggle raw display
$11$ \( (T^{4} + 7 T^{2} + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 48 T^{6} + \cdots + 625 \) Copy content Toggle raw display
$17$ \( (T^{4} - 19 T^{2} + 16)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 30 T^{6} + \cdots + 64 \) Copy content Toggle raw display
$23$ \( T^{8} + 82 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$29$ \( (T^{4} + 17 T^{2} + 64)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 150 T^{6} + \cdots + 524176 \) Copy content Toggle raw display
$37$ \( (T^{4} - 12 T^{3} + \cdots - 1328)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 2 T^{3} + \cdots - 176)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 10 T^{3} + \cdots - 563)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 4 T^{3} + \cdots + 592)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 204 T^{6} + \cdots + 434281 \) Copy content Toggle raw display
$59$ \( (T^{4} + 12 T^{3} + \cdots - 800)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + 360 T^{6} + \cdots + 7929856 \) Copy content Toggle raw display
$67$ \( (T^{4} + 6 T^{3} + \cdots - 747)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 474 T^{6} + \cdots + 26337424 \) Copy content Toggle raw display
$73$ \( T^{8} + 338 T^{6} + \cdots + 19360000 \) Copy content Toggle raw display
$79$ \( (T^{4} + 12 T^{3} + \cdots - 1184)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 32 T^{3} + \cdots + 3652)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 26 T^{3} + \cdots - 5843)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 296 T^{6} + \cdots + 1401856 \) Copy content Toggle raw display
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