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Properties

Label 2254.4.a.bb
Level $2254$
Weight $4$
Character orbit 2254.a
Self dual yes
Analytic conductor $132.990$
Analytic rank $0$
Dimension $18$
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: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 344 x^{16} + 46921 x^{14} - 3221102 x^{12} + 116812804 x^{10} - 2177115434 x^{8} + \cdots - 512512128 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index: \( 2^{6}\cdot 7^{4} \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{2} + \beta_1 q^{3} + 4 q^{4} + ( - \beta_{11} + \beta_1) q^{5} - 2 \beta_1 q^{6} - 8 q^{8} + (\beta_{2} + 11) q^{9} + (2 \beta_{11} - 2 \beta_1) q^{10} + ( - \beta_{9} + 4) q^{11} + 4 \beta_1 q^{12}+ \cdots + ( - 7 \beta_{9} + 24 \beta_{8} + \cdots - 207) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 36 q^{2} + 72 q^{4} - 144 q^{8} + 202 q^{9} + 76 q^{11} + 348 q^{15} + 288 q^{16} - 404 q^{18} - 152 q^{22} - 414 q^{23} + 662 q^{25} + 1164 q^{29} - 696 q^{30} - 576 q^{32} + 808 q^{36} + 188 q^{37}+ \cdots - 3864 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 344 x^{16} + 46921 x^{14} - 3221102 x^{12} + 116812804 x^{10} - 2177115434 x^{8} + \cdots - 512512128 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 38 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 85333555717 \nu^{16} + 34333605670192 \nu^{14} + \cdots - 16\!\cdots\!72 ) / 24\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 24269532241 \nu^{16} + 13637433885880 \nu^{14} + \cdots - 24\!\cdots\!08 ) / 48\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 31430501293 \nu^{16} + 7736579886458 \nu^{14} - 644611133885589 \nu^{12} + \cdots - 29\!\cdots\!28 ) / 40\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 209367159229 \nu^{16} + 62503260509164 \nu^{14} + \cdots + 46\!\cdots\!76 ) / 24\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 95420480321 \nu^{16} - 31818676957964 \nu^{14} + \cdots - 62\!\cdots\!52 ) / 48\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 300307810631 \nu^{16} - 106545843668436 \nu^{14} + \cdots + 29\!\cdots\!76 ) / 80\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 220755198757 \nu^{16} - 71638335257524 \nu^{14} + \cdots + 28\!\cdots\!48 ) / 48\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 141414707879707 \nu^{17} + \cdots + 17\!\cdots\!12 \nu ) / 16\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 412513190007683 \nu^{17} + \cdots + 31\!\cdots\!28 \nu ) / 32\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 412513190007683 \nu^{17} + \cdots + 23\!\cdots\!48 \nu ) / 32\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 380808892 \nu^{17} + 131525829835 \nu^{15} - 18007660215048 \nu^{13} + \cdots - 26\!\cdots\!28 \nu ) / 21\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 114342816325367 \nu^{17} + \cdots + 21\!\cdots\!08 \nu ) / 46\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 18\!\cdots\!47 \nu^{17} + \cdots - 11\!\cdots\!32 \nu ) / 32\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 168717027690221 \nu^{17} + \cdots + 13\!\cdots\!96 \nu ) / 17\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 29\!\cdots\!73 \nu^{17} + \cdots - 22\!\cdots\!08 \nu ) / 16\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 38 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{12} - 3\beta_{11} + 73\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{9} - 6\beta_{8} - 3\beta_{7} - 3\beta_{6} - 9\beta_{5} - 12\beta_{4} + 3\beta_{3} + 91\beta_{2} + 2699 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 15 \beta_{17} + 75 \beta_{16} + 30 \beta_{15} - 24 \beta_{14} + 171 \beta_{13} + 339 \beta_{12} + \cdots + 5890 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 366 \beta_{9} - 957 \beta_{8} - 150 \beta_{7} - 222 \beta_{6} - 1539 \beta_{5} - 1842 \beta_{4} + \cdots + 214439 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2874 \beta_{17} + 13623 \beta_{16} + 5046 \beta_{15} - 4332 \beta_{14} + 28485 \beta_{13} + \cdots + 496330 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 34191 \beta_{9} - 114228 \beta_{8} + 963 \beta_{7} - 12618 \beta_{6} - 201204 \beta_{5} + \cdots + 17785244 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 368136 \beta_{17} + 1781766 \beta_{16} + 651033 \beta_{15} - 564723 \beta_{14} + 3637089 \beta_{13} + \cdots + 43061653 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2763408 \beta_{9} - 12206751 \beta_{8} + 1289589 \beta_{7} - 664239 \beta_{6} - 23661189 \beta_{5} + \cdots + 1518598115 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 40389819 \beta_{17} + 205238505 \beta_{16} + 75789201 \beta_{15} - 65154714 \beta_{14} + \cdots + 3824543938 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 195289215 \beta_{9} - 1235684190 \beta_{8} + 212548917 \beta_{7} - 35901726 \beta_{6} + \cdots + 132820588778 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 4115405586 \beta_{17} + 22185378276 \beta_{16} + 8346892029 \beta_{15} - 7070503449 \beta_{14} + \cdots + 346230151081 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 11322144108 \beta_{9} - 121498871085 \beta_{8} + 26977391667 \beta_{7} - 2350212507 \beta_{6} + \cdots + 11855074198979 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 403318148571 \beta_{17} + 2312811369333 \beta_{16} + 887985264609 \beta_{15} + \cdots + 31827053380420 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 376284511821 \beta_{9} - 11755951549206 \beta_{8} + 3081498985653 \beta_{7} - 213012932484 \beta_{6} + \cdots + 10\!\cdots\!70 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 38697802529496 \beta_{17} + 235774858256292 \beta_{16} + 92287079451489 \beta_{15} + \cdots + 29\!\cdots\!95 \beta_1 \) 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
−9.84229
−8.86398
−8.06273
−8.05029
−4.15009
−3.50406
−2.80026
−1.18080
−0.0831437
0.0831437
1.18080
2.80026
3.50406
4.15009
8.05029
8.06273
8.86398
9.84229
−2.00000 −9.84229 4.00000 −20.7232 19.6846 0 −8.00000 69.8707 41.4465
1.2 −2.00000 −8.86398 4.00000 3.89774 17.7280 0 −8.00000 51.5701 −7.79547
1.3 −2.00000 −8.06273 4.00000 5.73327 16.1255 0 −8.00000 38.0077 −11.4665
1.4 −2.00000 −8.05029 4.00000 −7.60000 16.1006 0 −8.00000 37.8072 15.2000
1.5 −2.00000 −4.15009 4.00000 5.16794 8.30017 0 −8.00000 −9.77679 −10.3359
1.6 −2.00000 −3.50406 4.00000 −19.4419 7.00812 0 −8.00000 −14.7216 38.8838
1.7 −2.00000 −2.80026 4.00000 13.0761 5.60051 0 −8.00000 −19.1586 −26.1522
1.8 −2.00000 −1.18080 4.00000 16.7368 2.36161 0 −8.00000 −25.6057 −33.4736
1.9 −2.00000 −0.0831437 4.00000 8.05749 0.166287 0 −8.00000 −26.9931 −16.1150
1.10 −2.00000 0.0831437 4.00000 −8.05749 −0.166287 0 −8.00000 −26.9931 16.1150
1.11 −2.00000 1.18080 4.00000 −16.7368 −2.36161 0 −8.00000 −25.6057 33.4736
1.12 −2.00000 2.80026 4.00000 −13.0761 −5.60051 0 −8.00000 −19.1586 26.1522
1.13 −2.00000 3.50406 4.00000 19.4419 −7.00812 0 −8.00000 −14.7216 −38.8838
1.14 −2.00000 4.15009 4.00000 −5.16794 −8.30017 0 −8.00000 −9.77679 10.3359
1.15 −2.00000 8.05029 4.00000 7.60000 −16.1006 0 −8.00000 37.8072 −15.2000
1.16 −2.00000 8.06273 4.00000 −5.73327 −16.1255 0 −8.00000 38.0077 11.4665
1.17 −2.00000 8.86398 4.00000 −3.89774 −17.7280 0 −8.00000 51.5701 7.79547
1.18 −2.00000 9.84229 4.00000 20.7232 −19.6846 0 −8.00000 69.8707 −41.4465
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
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.bb ✓ 18
7.b odd 2 1 inner 2254.4.a.bb ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2254.4.a.bb ✓ 18 1.a even 1 1 trivial
2254.4.a.bb ✓ 18 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{18} - 344 T_{3}^{16} + 46921 T_{3}^{14} - 3221102 T_{3}^{12} + 116812804 T_{3}^{10} + \cdots - 512512128 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2254))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 2)^{18} \) Copy content Toggle raw display
$3$ \( T^{18} + \cdots - 512512128 \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots - 38\!\cdots\!32 \) Copy content Toggle raw display
$7$ \( T^{18} \) Copy content Toggle raw display
$11$ \( (T^{9} + \cdots + 11683671792128)^{2} \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots - 32\!\cdots\!92 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots - 89\!\cdots\!48 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots - 32\!\cdots\!88 \) Copy content Toggle raw display
$23$ \( (T + 23)^{18} \) Copy content Toggle raw display
$29$ \( (T^{9} + \cdots + 13\!\cdots\!88)^{2} \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots - 36\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{9} + \cdots - 22\!\cdots\!68)^{2} \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots - 58\!\cdots\!68 \) Copy content Toggle raw display
$43$ \( (T^{9} + \cdots - 40\!\cdots\!24)^{2} \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots - 14\!\cdots\!92 \) Copy content Toggle raw display
$53$ \( (T^{9} + \cdots + 98\!\cdots\!32)^{2} \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots - 30\!\cdots\!28 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots - 15\!\cdots\!68 \) Copy content Toggle raw display
$67$ \( (T^{9} + \cdots - 32\!\cdots\!36)^{2} \) Copy content Toggle raw display
$71$ \( (T^{9} + \cdots + 47\!\cdots\!28)^{2} \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots - 10\!\cdots\!88 \) Copy content Toggle raw display
$79$ \( (T^{9} + \cdots - 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots - 20\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots - 69\!\cdots\!12 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots - 14\!\cdots\!68 \) Copy content Toggle raw display
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