Properties

Label 80.20.a.g
Level $80$
Weight $20$
Character orbit 80.a
Self dual yes
Analytic conductor $183.053$
Analytic rank $1$
Dimension $4$
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] = [80,20,Mod(1,80)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(80, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 20, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("80.1");
 
S:= CuspForms(chi, 20);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 20 \)
Character orbit: \([\chi]\) \(=\) 80.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: \(183.053357245\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 323561x^{2} - 26738538x + 10870990650 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{24}\cdot 3^{2}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 5)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 770) q^{3} - 1953125 q^{5} + ( - 16 \beta_{3} + 29 \beta_{2} + 350 \beta_1 - 53505350) q^{7} + (375 \beta_{3} + 33 \beta_{2} - 967 \beta_1 + 38516137) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 770) q^{3} - 1953125 q^{5} + ( - 16 \beta_{3} + 29 \beta_{2} + 350 \beta_1 - 53505350) q^{7} + (375 \beta_{3} + 33 \beta_{2} - 967 \beta_1 + 38516137) q^{9} + (1460 \beta_{3} - 1055 \beta_{2} - 73595 \beta_1 - 2896428192) q^{11} + (7033 \beta_{3} + 5843 \beta_{2} - 714013 \beta_1 + 3703083290) q^{13} + (1953125 \beta_1 + 1503906250) q^{15} + ( - 114665 \beta_{3} - 20995 \beta_{2} + \cdots + 212258435610) q^{17}+ \cdots + ( - 723376795980 \beta_{3} + \cdots + 63\!\cdots\!96) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3080 q^{3} - 7812500 q^{5} - 214021400 q^{7} + 154064548 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 3080 q^{3} - 7812500 q^{5} - 214021400 q^{7} + 154064548 q^{9} - 11585712768 q^{11} + 14812333160 q^{13} + 6015625000 q^{15} + 849033742440 q^{17} - 1978167708560 q^{19} - 1487020185552 q^{21} + 26569906952760 q^{23} + 15258789062500 q^{25} + 7557605929360 q^{27} + 116267174339640 q^{29} - 251049672388688 q^{31} + 359905680636160 q^{33} + 418010546875000 q^{35} + 53471657716520 q^{37} + 34\!\cdots\!04 q^{39}+ \cdots + 25\!\cdots\!84 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 323561x^{2} - 26738538x + 10870990650 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -400\nu^{3} + 101696\nu^{2} - 24245536\nu - 8224709856 ) / 1360737 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -156880\nu^{3} + 5050304\nu^{2} + 41558816096\nu + 2384371904736 ) / 1360737 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -9440\nu^{3} + 14011648\nu^{2} + 983762752\nu - 2073426820608 ) / 453579 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - 463\beta _1 + 20480 ) / 40960 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 733\beta_{3} - 67\beta_{2} - 25619\beta _1 + 3313285120 ) / 20480 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 156051\beta_{3} - 47341\beta_{2} - 62151005\beta _1 + 420643778560 ) / 20480 \) 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
−462.664
−267.923
150.720
581.867
0 −48079.9 0 −1.95312e6 0 −9.13351e7 0 1.14942e9 0
1.2 0 −10517.7 0 −1.95312e6 0 −1.23996e8 0 −1.05164e9 0
1.3 0 7268.55 0 −1.95312e6 0 1.76809e8 0 −1.10943e9 0
1.4 0 48249.1 0 −1.95312e6 0 −1.75500e8 0 1.16572e9 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)

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 80.20.a.g 4
4.b odd 2 1 5.20.a.b 4
12.b even 2 1 45.20.a.f 4
20.d odd 2 1 25.20.a.c 4
20.e even 4 2 25.20.b.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.20.a.b 4 4.b odd 2 1
25.20.a.c 4 20.d odd 2 1
25.20.b.c 8 20.e even 4 2
45.20.a.f 4 12.b even 2 1
80.20.a.g 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 3080T_{3}^{3} - 2396812008T_{3}^{2} - 7524612119520T_{3} + 177346815108838416 \) acting on \(S_{20}^{\mathrm{new}}(\Gamma_0(80))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 3080 T^{3} + \cdots + 17\!\cdots\!16 \) Copy content Toggle raw display
$5$ \( (T + 1953125)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 214021400 T^{3} + \cdots - 35\!\cdots\!04 \) Copy content Toggle raw display
$11$ \( T^{4} + 11585712768 T^{3} + \cdots - 53\!\cdots\!04 \) Copy content Toggle raw display
$13$ \( T^{4} - 14812333160 T^{3} + \cdots + 38\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( T^{4} - 849033742440 T^{3} + \cdots - 65\!\cdots\!04 \) Copy content Toggle raw display
$19$ \( T^{4} + 1978167708560 T^{3} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{4} - 26569906952760 T^{3} + \cdots - 42\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( T^{4} - 116267174339640 T^{3} + \cdots - 13\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{4} + 251049672388688 T^{3} + \cdots - 30\!\cdots\!44 \) Copy content Toggle raw display
$37$ \( T^{4} - 53471657716520 T^{3} + \cdots + 35\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 54\!\cdots\!36 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 10\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots - 43\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 52\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 59\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 53\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots - 98\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots - 18\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots - 22\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 27\!\cdots\!96 \) Copy content Toggle raw display
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