Properties

Label 230.6.a.c
Level $230$
Weight $6$
Character orbit 230.a
Self dual yes
Analytic conductor $36.888$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [230,6,Mod(1,230)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(230, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("230.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 230.a (trivial)

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: yes
Analytic conductor: \(36.8882785570\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.27980.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 47x - 106 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 4 q^{2} + (\beta_{2} - 9) q^{3} + 16 q^{4} + 25 q^{5} + ( - 4 \beta_{2} + 36) q^{6} + (4 \beta_{2} + 7 \beta_1 - 1) q^{7} - 64 q^{8} + ( - 17 \beta_{2} - 6 \beta_1 - 18) q^{9} - 100 q^{10} + (7 \beta_{2} - 19 \beta_1 + 57) q^{11}+ \cdots + ( - 3560 \beta_{2} + 2082 \beta_1 - 1530) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 12 q^{2} - 26 q^{3} + 48 q^{4} + 75 q^{5} + 104 q^{6} + q^{7} - 192 q^{8} - 71 q^{9} - 300 q^{10} + 178 q^{11} - 416 q^{12} - 236 q^{13} - 4 q^{14} - 650 q^{15} + 768 q^{16} - 1683 q^{17} + 284 q^{18}+ \cdots - 8150 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 47x - 106 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 3\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4\nu - 31 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{2} + 4\beta _1 + 93 ) / 3 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.65230
7.78556
−5.13326
−4.00000 −22.3561 16.0000 25.0000 89.4244 −110.123 −64.0000 256.795 −100.000
1.2 −4.00000 −10.5273 16.0000 25.0000 42.1092 156.388 −64.0000 −132.176 −100.000
1.3 −4.00000 6.88339 16.0000 25.0000 −27.5336 −45.2649 −64.0000 −195.619 −100.000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(5\) \( -1 \)
\(23\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 230.6.a.c 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.6.a.c 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} + 26T_{3}^{2} + 9T_{3} - 1620 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(230))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 4)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + 26 T^{2} + \cdots - 1620 \) Copy content Toggle raw display
$5$ \( (T - 25)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - T^{2} + \cdots - 779544 \) Copy content Toggle raw display
$11$ \( T^{3} - 178 T^{2} + \cdots + 21004632 \) Copy content Toggle raw display
$13$ \( T^{3} + 236 T^{2} + \cdots - 16482042 \) Copy content Toggle raw display
$17$ \( T^{3} + 1683 T^{2} + \cdots + 73501360 \) Copy content Toggle raw display
$19$ \( T^{3} + 194 T^{2} + \cdots - 841047768 \) Copy content Toggle raw display
$23$ \( (T - 529)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} - 6901 T^{2} + \cdots + 809182077 \) Copy content Toggle raw display
$31$ \( T^{3} + \cdots + 30813892995 \) Copy content Toggle raw display
$37$ \( T^{3} + \cdots + 881126317804 \) Copy content Toggle raw display
$41$ \( T^{3} + \cdots + 1050649591 \) Copy content Toggle raw display
$43$ \( T^{3} + \cdots - 1900451896320 \) Copy content Toggle raw display
$47$ \( T^{3} + \cdots - 98633296700 \) Copy content Toggle raw display
$53$ \( T^{3} + \cdots + 1549341371484 \) Copy content Toggle raw display
$59$ \( T^{3} + \cdots - 22979507348832 \) Copy content Toggle raw display
$61$ \( T^{3} + \cdots - 4638543486768 \) Copy content Toggle raw display
$67$ \( T^{3} + \cdots + 4102954370460 \) Copy content Toggle raw display
$71$ \( T^{3} + \cdots - 1521596424555 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots + 4522640531874 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots - 11537777158848 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots - 70485885180896 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots - 650267509666176 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots - 42022925072144 \) Copy content Toggle raw display
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