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Properties

Label 8028.2.a.p
Level $8028$
Weight $2$
Character orbit 8028.a
Self dual yes
Analytic conductor $64.104$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8028,2,Mod(1,8028)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8028, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8028.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8028 = 2^{2} \cdot 3^{2} \cdot 223 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8028.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: \(64.1039027427\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 45x^{14} + 782x^{12} - 6932x^{10} + 34345x^{8} - 96535x^{6} + 146904x^{4} - 106416x^{2} + 28224 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 7 \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} + ( - \beta_{7} + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{5} + ( - \beta_{7} + 1) q^{7} + \beta_{9} q^{11} + ( - \beta_{10} + 1) q^{13} + (\beta_{14} + \beta_{3}) q^{17} + (\beta_{11} + 1) q^{19} - \beta_{13} q^{23} + ( - \beta_{12} + \beta_{11} + \beta_{8} + \cdots + 1) q^{25}+ \cdots + (\beta_{12} + \beta_{10} - \beta_{8} + \cdots + 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 22 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 22 q^{7} + 12 q^{13} + 10 q^{19} + 10 q^{25} + 26 q^{31} + 30 q^{37} + 28 q^{43} + 26 q^{49} + 6 q^{55} + 40 q^{61} + 66 q^{67} + 36 q^{73} + 30 q^{79} - 2 q^{85} + 52 q^{91} + 38 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 45x^{14} + 782x^{12} - 6932x^{10} + 34345x^{8} - 96535x^{6} + 146904x^{4} - 106416x^{2} + 28224 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 3751 \nu^{15} + 152979 \nu^{13} - 2284706 \nu^{11} + 16230524 \nu^{9} - 58484431 \nu^{7} + \cdots + 23806896 \nu ) / 558624 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 19709 \nu^{15} + 814585 \nu^{13} - 12419470 \nu^{11} + 90887652 \nu^{9} - 340909189 \nu^{7} + \cdots + 136260960 \nu ) / 558624 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 12521 \nu^{14} - 518217 \nu^{12} + 7918732 \nu^{10} - 58154932 \nu^{8} + 219333533 \nu^{6} + \cdots - 93359736 ) / 139656 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 94 \nu^{15} + 3893 \nu^{13} - 59525 \nu^{11} + 437114 \nu^{9} - 1644646 \nu^{7} + \cdots + 647184 \nu ) / 2024 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 8542 \nu^{14} + 353928 \nu^{12} - 5418011 \nu^{10} + 39903512 \nu^{8} - 151217860 \nu^{6} + \cdots + 67325208 ) / 69828 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 39137 \nu^{14} + 1623681 \nu^{12} - 24905626 \nu^{10} + 183976924 \nu^{8} - 700282505 \nu^{6} + \cdots + 323872224 ) / 279312 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11252 \nu^{14} - 467421 \nu^{12} + 7182661 \nu^{10} - 53168350 \nu^{8} + 202737200 \nu^{6} + \cdots - 92656560 ) / 69828 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 11161 \nu^{15} + 463951 \nu^{13} - 7136558 \nu^{11} + 52903516 \nu^{9} - 202150497 \nu^{7} + \cdots + 93195984 \nu ) / 139656 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 5875 \nu^{14} + 243807 \nu^{12} - 3740771 \nu^{10} + 27632195 \nu^{8} - 105077656 \nu^{6} + \cdots + 47187954 ) / 34914 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 16917 \nu^{14} - 701145 \nu^{12} + 10736958 \nu^{10} - 79096028 \nu^{8} + 299701893 \nu^{6} + \cdots - 133583968 ) / 93104 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 61591 \nu^{14} - 2557407 \nu^{12} + 39269474 \nu^{10} - 290347436 \nu^{8} + 1105183039 \nu^{6} + \cdots - 500401440 ) / 279312 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 8575 \nu^{15} + 355384 \nu^{13} - 5440694 \nu^{11} + 40044382 \nu^{9} - 151367745 \nu^{7} + \cdots + 64789692 \nu ) / 69828 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 39137 \nu^{15} - 1623681 \nu^{13} + 24905626 \nu^{11} - 183976924 \nu^{9} + 700282505 \nu^{7} + \cdots - 324151536 \nu ) / 279312 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 84277 \nu^{15} + 3499189 \nu^{13} - 53727002 \nu^{11} + 397213804 \nu^{9} - 1511830845 \nu^{7} + \cdots + 685473648 \nu ) / 558624 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{12} + \beta_{11} + \beta_{8} + \beta_{6} + 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{15} - \beta_{14} + 2\beta_{13} - 2\beta_{3} - 2\beta_{2} + 11\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -21\beta_{12} + 19\beta_{11} - 2\beta_{10} + 21\beta_{8} + 4\beta_{7} + 14\beta_{6} - 3\beta_{4} + 65 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -35\beta_{15} - 21\beta_{14} + 45\beta_{13} - 13\beta_{9} - 7\beta_{5} - 42\beta_{3} - 51\beta_{2} + 155\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -377\beta_{12} + 334\beta_{11} - 29\beta_{10} + 390\beta_{8} + 103\beta_{7} + 195\beta_{6} - 78\beta_{4} + 913 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 571 \beta_{15} - 376 \beta_{14} + 845 \beta_{13} - 351 \beta_{9} - 209 \beta_{5} - 726 \beta_{3} + \cdots + 2439 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 6591 \beta_{12} + 5802 \beta_{11} - 329 \beta_{10} + 7008 \beta_{8} + 2071 \beta_{7} + 2912 \beta_{6} + \cdots + 14388 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 9460 \beta_{15} - 6534 \beta_{14} + 15150 \beta_{13} - 7198 \beta_{9} - 4556 \beta_{5} + \cdots + 40389 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 114721 \beta_{12} + 100635 \beta_{11} - 3594 \beta_{10} + 124047 \beta_{8} + 38394 \beta_{7} + \cdots + 238706 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 159762 \beta_{15} - 113039 \beta_{14} + 267206 \beta_{13} - 134716 \beta_{9} - 88332 \beta_{5} + \cdots + 685391 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1995561 \beta_{12} + 1746567 \beta_{11} - 40854 \beta_{10} + 2178073 \beta_{8} + 688160 \beta_{7} + \cdots + 4055971 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 2733249 \beta_{15} - 1957377 \beta_{14} + 4679535 \beta_{13} - 2426559 \beta_{9} + \cdots + 11772267 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 34709885 \beta_{12} + 30336024 \beta_{11} - 503647 \beta_{10} + 38074632 \beta_{8} + 12145757 \beta_{7} + \cdots + 69717369 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 47115843 \beta_{15} - 33946054 \beta_{14} + 81673389 \beta_{13} - 42940951 \beta_{9} + \cdots + 203434135 \beta_1 \) 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
−4.17030
−3.00352
−2.40835
−2.05123
−1.87633
−1.86656
−0.962456
−0.805463
0.805463
0.962456
1.86656
1.87633
2.05123
2.40835
3.00352
4.17030
0 0 0 −4.17030 0 1.66521 0 0 0
1.2 0 0 0 −3.00352 0 1.79184 0 0 0
1.3 0 0 0 −2.40835 0 4.08421 0 0 0
1.4 0 0 0 −2.05123 0 −2.18104 0 0 0
1.5 0 0 0 −1.87633 0 3.83443 0 0 0
1.6 0 0 0 −1.86656 0 −0.561849 0 0 0
1.7 0 0 0 −0.962456 0 4.63021 0 0 0
1.8 0 0 0 −0.805463 0 −2.26302 0 0 0
1.9 0 0 0 0.805463 0 −2.26302 0 0 0
1.10 0 0 0 0.962456 0 4.63021 0 0 0
1.11 0 0 0 1.86656 0 −0.561849 0 0 0
1.12 0 0 0 1.87633 0 3.83443 0 0 0
1.13 0 0 0 2.05123 0 −2.18104 0 0 0
1.14 0 0 0 2.40835 0 4.08421 0 0 0
1.15 0 0 0 3.00352 0 1.79184 0 0 0
1.16 0 0 0 4.17030 0 1.66521 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(223\) \(1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8028.2.a.p ✓ 16
3.b odd 2 1 inner 8028.2.a.p ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8028.2.a.p ✓ 16 1.a even 1 1 trivial
8028.2.a.p ✓ 16 3.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8028))\):

\( T_{5}^{16} - 45 T_{5}^{14} + 782 T_{5}^{12} - 6932 T_{5}^{10} + 34345 T_{5}^{8} - 96535 T_{5}^{6} + \cdots + 28224 \) Copy content Toggle raw display
\( T_{7}^{8} - 11T_{7}^{7} + 26T_{7}^{6} + 87T_{7}^{5} - 360T_{7}^{4} - 68T_{7}^{3} + 1134T_{7}^{2} - 480T_{7} - 600 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} - 45 T^{14} + \cdots + 28224 \) Copy content Toggle raw display
$7$ \( (T^{8} - 11 T^{7} + \cdots - 600)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} - 101 T^{14} + \cdots + 39337984 \) Copy content Toggle raw display
$13$ \( (T^{8} - 6 T^{7} + \cdots + 7168)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 138627076 \) Copy content Toggle raw display
$19$ \( (T^{8} - 5 T^{7} + \cdots + 2422)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} - 147 T^{14} + \cdots + 21086464 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 5134009104 \) Copy content Toggle raw display
$31$ \( (T^{8} - 13 T^{7} + \cdots - 5554)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 15 T^{7} + \cdots - 501)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 65925697600 \) Copy content Toggle raw display
$43$ \( (T^{8} - 14 T^{7} + \cdots + 4494)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 1050042880656 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 75593669469444 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 5903156224 \) Copy content Toggle raw display
$61$ \( (T^{8} - 20 T^{7} + \cdots + 125888)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 33 T^{7} + \cdots - 58669056)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 1876856160256 \) Copy content Toggle raw display
$73$ \( (T^{8} - 18 T^{7} + \cdots + 1981908)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} - 15 T^{7} + \cdots - 10368)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 16097265625 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 1440595261504 \) Copy content Toggle raw display
$97$ \( (T^{8} - 19 T^{7} + \cdots - 1084160)^{2} \) Copy content Toggle raw display
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