Properties

Label 230.6.a.b
Level $230$
Weight $6$
Character orbit 230.a
Self dual yes
Analytic conductor $36.888$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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 \(\beta = 2\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 4 q^{2} + (6 \beta + 3) q^{3} + 16 q^{4} - 25 q^{5} + ( - 24 \beta - 12) q^{6} + ( - 31 \beta - 82) q^{7} - 64 q^{8} + (36 \beta + 54) q^{9} + 100 q^{10} + (45 \beta + 244) q^{11} + (96 \beta + 48) q^{12}+ \cdots + (11214 \beta + 26136) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{2} + 6 q^{3} + 32 q^{4} - 50 q^{5} - 24 q^{6} - 164 q^{7} - 128 q^{8} + 108 q^{9} + 200 q^{10} + 488 q^{11} + 96 q^{12} + 738 q^{13} + 656 q^{14} - 150 q^{15} + 512 q^{16} + 1112 q^{17} - 432 q^{18}+ \cdots + 52272 q^{99}+O(q^{100}) \) 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
−1.41421
1.41421
−4.00000 −13.9706 16.0000 −25.0000 55.8823 5.68124 −64.0000 −47.8234 100.000
1.2 −4.00000 19.9706 16.0000 −25.0000 −79.8823 −169.681 −64.0000 155.823 100.000
\(n\): 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.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.6.a.b 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 6T_{3} - 279 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(230))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 6T - 279 \) Copy content Toggle raw display
$5$ \( (T + 25)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 164T - 964 \) Copy content Toggle raw display
$11$ \( T^{2} - 488T + 43336 \) Copy content Toggle raw display
$13$ \( T^{2} - 738T + 106393 \) Copy content Toggle raw display
$17$ \( T^{2} - 1112T - 30352 \) Copy content Toggle raw display
$19$ \( T^{2} + 1584 T + 481464 \) Copy content Toggle raw display
$23$ \( (T + 529)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 2158 T - 33247567 \) Copy content Toggle raw display
$31$ \( T^{2} - 3418 T - 48408031 \) Copy content Toggle raw display
$37$ \( T^{2} + 152T + 3464 \) Copy content Toggle raw display
$41$ \( T^{2} + 31482 T + 237235849 \) Copy content Toggle raw display
$43$ \( T^{2} - 468T - 308196 \) Copy content Toggle raw display
$47$ \( T^{2} - 10062 T - 33009039 \) Copy content Toggle raw display
$53$ \( T^{2} - 3380 T - 256334812 \) Copy content Toggle raw display
$59$ \( T^{2} + 42512 T - 375613664 \) Copy content Toggle raw display
$61$ \( T^{2} + 44472 T + 264688648 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 3527283128 \) Copy content Toggle raw display
$71$ \( T^{2} + 36126 T + 277583257 \) Copy content Toggle raw display
$73$ \( T^{2} + 39514 T - 556133983 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 4014464456 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 3075302432 \) Copy content Toggle raw display
$89$ \( T^{2} + 4356 T - 440349764 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 4764185692 \) Copy content Toggle raw display
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