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Properties

Label 2254.4.a.o
Level $2254$
Weight $4$
Character orbit 2254.a
Self dual yes
Analytic conductor $132.990$
Analytic rank $1$
Dimension $8$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2254,4,Mod(1,2254)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2254, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2254.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2254 = 2 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2254.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: \(132.990305153\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 141x^{6} + 4946x^{4} - 36656x^{2} + 22472 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{2} + \beta_1 q^{3} + 4 q^{4} + \beta_{6} q^{5} - 2 \beta_1 q^{6} - 8 q^{8} + (\beta_{2} + 8) q^{9} - 2 \beta_{6} q^{10} + (\beta_{4} - \beta_{2} - 16) q^{11} + 4 \beta_1 q^{12} - \beta_{5} q^{13}+ \cdots + ( - 23 \beta_{4} + 15 \beta_{3} + \cdots - 1259) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{2} + 32 q^{4} - 64 q^{8} + 66 q^{9} - 128 q^{11} + 116 q^{15} + 128 q^{16} - 132 q^{18} + 256 q^{22} - 184 q^{23} + 180 q^{25} - 278 q^{29} - 232 q^{30} - 256 q^{32} + 264 q^{36} - 252 q^{37}+ \cdots - 10164 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 141x^{6} + 4946x^{4} - 36656x^{2} + 22472 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 35 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} + 29\nu^{4} + 5637\nu^{2} - 65500 ) / 1467 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -14\nu^{6} + 1873\nu^{4} - 56046\nu^{2} + 172981 ) / 4401 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 77\nu^{7} - 8101\nu^{5} + 145416\nu^{3} - 2185876\nu ) / 466506 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -190\nu^{7} + 26048\nu^{5} - 840471\nu^{3} + 4227455\nu ) / 233253 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -398\nu^{7} + 57019\nu^{5} - 2072388\nu^{3} + 16429990\nu ) / 233253 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 35 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{7} + 5\beta_{6} + 4\beta_{5} + 69\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{4} - 14\beta_{3} + 92\beta_{2} + 2477 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -159\beta_{7} + 436\beta_{6} + 508\beta_{5} + 5678\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 87\beta_{4} - 1873\beta_{3} + 8305\beta_{2} + 203628 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -12951\beta_{7} + 36428\beta_{6} + 51950\beta_{5} + 495450\beta_1 \) Copy content Toggle raw display

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

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
−9.54412
−6.33199
−3.02370
−0.820364
0.820364
3.02370
6.33199
9.54412
−2.00000 −9.54412 4.00000 −7.99660 19.0882 0 −8.00000 64.0902 15.9932
1.2 −2.00000 −6.33199 4.00000 −4.25151 12.6640 0 −8.00000 13.0941 8.50303
1.3 −2.00000 −3.02370 4.00000 18.4675 6.04739 0 −8.00000 −17.8573 −36.9351
1.4 −2.00000 −0.820364 4.00000 −12.9201 1.64073 0 −8.00000 −26.3270 25.8402
1.5 −2.00000 0.820364 4.00000 12.9201 −1.64073 0 −8.00000 −26.3270 −25.8402
1.6 −2.00000 3.02370 4.00000 −18.4675 −6.04739 0 −8.00000 −17.8573 36.9351
1.7 −2.00000 6.33199 4.00000 4.25151 −12.6640 0 −8.00000 13.0941 −8.50303
1.8 −2.00000 9.54412 4.00000 7.99660 −19.0882 0 −8.00000 64.0902 −15.9932
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(7\) \( -1 \)
\(23\) \( +1 \)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2254.4.a.o ✓ 8
7.b odd 2 1 inner 2254.4.a.o ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2254.4.a.o ✓ 8 1.a even 1 1 trivial
2254.4.a.o ✓ 8 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 141T_{3}^{6} + 4946T_{3}^{4} - 36656T_{3}^{2} + 22472 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2254))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 2)^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 141 T^{6} + \cdots + 22472 \) Copy content Toggle raw display
$5$ \( T^{8} - 590 T^{6} + \cdots + 65803392 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} + 64 T^{3} + \cdots - 1468688)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 7735689728 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 943534176080000 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 48673871659008 \) Copy content Toggle raw display
$23$ \( (T + 23)^{8} \) Copy content Toggle raw display
$29$ \( (T^{4} + 139 T^{3} + \cdots + 89274472)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 521460906245000 \) Copy content Toggle raw display
$37$ \( (T^{4} + 126 T^{3} + \cdots + 978327104)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 82\!\cdots\!68 \) Copy content Toggle raw display
$43$ \( (T^{4} - 128 T^{3} + \cdots + 2021015296)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 52\!\cdots\!08 \) Copy content Toggle raw display
$53$ \( (T^{4} - 686 T^{3} + \cdots + 2600010528)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 47\!\cdots\!92 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 41\!\cdots\!72 \) Copy content Toggle raw display
$67$ \( (T^{4} - 486 T^{3} + \cdots + 37069560736)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 2121 T^{3} + \cdots + 14363210304)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 92\!\cdots\!92 \) Copy content Toggle raw display
$79$ \( (T^{4} - 1210 T^{3} + \cdots - 14586523200)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 26\!\cdots\!48 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 31\!\cdots\!72 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 47\!\cdots\!28 \) Copy content Toggle raw display
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