↑

Properties

Label 1058.4.a.s
Level $1058$
Weight $4$
Character orbit 1058.a
Self dual yes
Analytic conductor $62.424$
Analytic rank $0$
Dimension $12$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1058,4,Mod(1,1058)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1058, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1058.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1058 = 2 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1058.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: \(62.4240207861\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 188x^{10} + 13557x^{8} - 463784x^{6} + 7423888x^{4} - 45948504x^{2} + 91814724 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{2} + (\beta_{2} - \beta_1) q^{3} + 4 q^{4} + (\beta_{11} + \beta_{10} + \cdots + \beta_{7}) q^{5} + ( - 2 \beta_{2} + 2 \beta_1) q^{6} + (\beta_{10} + \beta_{9} + \cdots + 3 \beta_{7}) q^{7}+ \cdots + ( - 56 \beta_{11} + \cdots - 34 \beta_{6}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 24 q^{2} + 48 q^{4} - 96 q^{8} + 72 q^{9} + 112 q^{13} + 192 q^{16} - 144 q^{18} + 644 q^{25} - 224 q^{26} - 552 q^{27} - 592 q^{29} + 1288 q^{31} - 384 q^{32} + 464 q^{35} + 288 q^{36} - 520 q^{39}+ \cdots - 1080 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 188x^{10} + 13557x^{8} - 463784x^{6} + 7423888x^{4} - 45948504x^{2} + 91814724 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2387\nu^{10} + 361315\nu^{8} - 19159977\nu^{6} + 410644183\nu^{4} - 2963358764\nu^{2} + 6553305300 ) / 55970454 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 59404234 \nu^{10} - 8905565807 \nu^{8} + 464779003422 \nu^{6} - 9648993383279 \nu^{4} + \cdots - 126627140691252 ) / 420506020902 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 21025288 \nu^{10} + 3164790245 \nu^{8} - 166268011970 \nu^{6} + 3494010368711 \nu^{4} + \cdots + 47273395509346 ) / 140168673634 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 70895593 \nu^{10} + 10565030042 \nu^{8} - 548086584537 \nu^{6} + 11338444883510 \nu^{4} + \cdots + 150078363053952 ) / 420506020902 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -224\nu^{10} + 33673\nu^{8} - 1770354\nu^{6} + 37475317\nu^{4} - 263582876\nu^{2} + 565366746 ) / 979422 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1090870675 \nu^{11} - 280611751352 \nu^{9} + 23412402826605 \nu^{7} + \cdots - 12\!\cdots\!28 \nu ) / 40\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1042221247 \nu^{11} - 159055158911 \nu^{9} + 8555398025481 \nu^{7} + \cdots - 37\!\cdots\!98 \nu ) / 366298972025724 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1260724846 \nu^{11} - 190123817165 \nu^{9} + 10072107108054 \nu^{7} + \cdots - 35\!\cdots\!82 \nu ) / 366298972025724 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 57324254767 \nu^{11} + 8945391780815 \nu^{9} - 498070664358957 \nu^{7} + \cdots + 29\!\cdots\!46 \nu ) / 40\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 33548955323 \nu^{11} - 4991385423220 \nu^{9} + 256818195417477 \nu^{7} + \cdots - 46\!\cdots\!96 \nu ) / 13\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 114514839275 \nu^{11} - 17065518258475 \nu^{9} + 881247764441025 \nu^{7} + \cdots - 18\!\cdots\!18 \nu ) / 40\!\cdots\!64 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{11} + \beta_{10} + \beta_{8} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{4} + 2\beta_{3} + 2\beta_{2} + \beta _1 + 31 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -22\beta_{11} + 21\beta_{10} + 2\beta_{9} + 8\beta_{8} + 36\beta_{7} - 3\beta_{6} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -68\beta_{5} + 67\beta_{4} + 97\beta_{3} + 110\beta_{2} + 123\beta _1 + 1349 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -1048\beta_{11} + 942\beta_{10} + 253\beta_{9} + 385\beta_{8} + 3004\beta_{7} - 67\beta_{6} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -4327\beta_{5} + 4030\beta_{4} + 4434\beta_{3} + 5477\beta_{2} + 9820\beta _1 + 63537 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -52218\beta_{11} + 43538\beta_{10} + 20653\beta_{9} + 28320\beta_{8} + 208277\beta_{7} + 1074\beta_{6} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -270056\beta_{5} + 235217\beta_{4} + 208943\beta_{3} + 273452\beta_{2} + 689269\beta _1 + 3113051 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 2689275 \beta_{11} + 2066229 \beta_{10} + 1452637 \beta_{9} + 2115836 \beta_{8} + \cdots + 240965 \beta_{6} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -16602643\beta_{5} + 13541066\beta_{4} + 10240750\beta_{3} + 13869819\beta_{2} + 45405178\beta _1 + 157550377 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 142266274 \beta_{11} + 100659144 \beta_{10} + 95123093 \beta_{9} + 150636160 \beta_{8} + \cdots + 20051684 \beta_{6} \) 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
6.10491
−6.10491
2.48311
−2.48311
7.60599
−7.60599
−2.01872
2.01872
−6.69770
6.69770
−6.14645
6.14645
−2.00000 −7.80838 4.00000 −18.6572 15.6168 2.20422 −8.00000 33.9708 37.3143
1.2 −2.00000 −7.80838 4.00000 18.6572 15.6168 −2.20422 −8.00000 33.9708 −37.3143
1.3 −2.00000 −7.00599 4.00000 −0.188661 14.0120 2.20851 −8.00000 22.0840 0.377322
1.4 −2.00000 −7.00599 4.00000 0.188661 14.0120 −2.20851 −8.00000 22.0840 −0.377322
1.5 −2.00000 −0.551908 4.00000 −5.39834 1.10382 34.6396 −8.00000 −26.6954 10.7967
1.6 −2.00000 −0.551908 4.00000 5.39834 1.10382 −34.6396 −8.00000 −26.6954 −10.7967
1.7 −2.00000 3.16414 4.00000 −5.53141 −6.32827 −25.2627 −8.00000 −16.9882 11.0628
1.8 −2.00000 3.16414 4.00000 5.53141 −6.32827 25.2627 −8.00000 −16.9882 −11.0628
1.9 −2.00000 4.83959 4.00000 −19.1394 −9.67918 2.83116 −8.00000 −3.57836 38.2787
1.10 −2.00000 4.83959 4.00000 19.1394 −9.67918 −2.83116 −8.00000 −3.57836 −38.2787
1.11 −2.00000 7.36256 4.00000 −17.2575 −14.7251 −21.7289 −8.00000 27.2072 34.5150
1.12 −2.00000 7.36256 4.00000 17.2575 −14.7251 21.7289 −8.00000 27.2072 −34.5150
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1058.4.a.s ✓ 12
23.b odd 2 1 inner 1058.4.a.s ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1058.4.a.s ✓ 12 1.a even 1 1 trivial
1058.4.a.s ✓ 12 23.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1058))\):

\( T_{3}^{6} - 99T_{3}^{4} + 92T_{3}^{3} + 2472T_{3}^{2} - 4848T_{3} - 3404 \) Copy content Toggle raw display
\( T_{5}^{12} - 1072T_{5}^{10} + 401674T_{5}^{8} - 59219712T_{5}^{6} + 2574104169T_{5}^{4} - 33952090608T_{5}^{2} + 1205200656 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 2)^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} - 99 T^{4} + \cdots - 3404)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 1205200656 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 68677540096 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$13$ \( (T^{6} - 56 T^{5} + \cdots - 26554844784)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 23\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 52\!\cdots\!64 \) Copy content Toggle raw display
$23$ \( T^{12} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 39800872207272)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots - 2054281367232)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 18\!\cdots\!04 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 43000133853312)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 17\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots - 9871339637424)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 88\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 31626584278116)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 19\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 55\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 670147989793488)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 41\!\cdots\!33)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 34\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 35\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 64\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
show more
show less