Properties

Label 80.16.a.a
Level $80$
Weight $16$
Character orbit 80.a
Self dual yes
Analytic conductor $114.155$
Analytic rank $1$
Dimension $1$
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] = [80,16,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, 16, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("80.1");
 
S:= CuspForms(chi, 16);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 16 \)
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: \(114.154804080\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 10)
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)
 
\(f(q)\) \(=\) \( q + 918 q^{3} - 78125 q^{5} + 953554 q^{7} - 13506183 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 918 q^{3} - 78125 q^{5} + 953554 q^{7} - 13506183 q^{9} - 17783232 q^{11} + 140533322 q^{13} - 71718750 q^{15} + 2998870746 q^{17} - 3255852500 q^{19} + 875362572 q^{21} - 6774812202 q^{23} + 6103515625 q^{25} - 25570972620 q^{27} - 7340322690 q^{29} + 115428411388 q^{31} - 16325006976 q^{33} - 74496406250 q^{35} + 150300986906 q^{37} + 129009589596 q^{39} + 1841603525142 q^{41} - 1510018315682 q^{43} + 1055170546875 q^{45} - 6093750843366 q^{47} - 3838296279027 q^{49} + 2752963344828 q^{51} - 8267412829038 q^{53} + 1389315000000 q^{55} - 2988872595000 q^{57} + 23516883061980 q^{59} - 3135369104278 q^{61} - 12878874824382 q^{63} - 10979165781250 q^{65} + 36030983954794 q^{67} - 6219277601436 q^{69} - 52169735384172 q^{71} + 69977143684082 q^{73} + 5603027343750 q^{75} - 16957272006528 q^{77} + 135317670906760 q^{79} + 170324810926821 q^{81} - 427456158822882 q^{83} - 234286777031250 q^{85} - 6738416229420 q^{87} - 446581617299190 q^{89} + 134006111326388 q^{91} + 105963281654184 q^{93} + 254363476562500 q^{95} + 181247411845826 q^{97} + 240183585723456 q^{99}+O(q^{100}) \) 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
0
0 918.000 0 −78125.0 0 953554. 0 −1.35062e7 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 80.16.a.a 1
4.b odd 2 1 10.16.a.b 1
12.b even 2 1 90.16.a.h 1
20.d odd 2 1 50.16.a.c 1
20.e even 4 2 50.16.b.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.16.a.b 1 4.b odd 2 1
50.16.a.c 1 20.d odd 2 1
50.16.b.c 2 20.e even 4 2
80.16.a.a 1 1.a even 1 1 trivial
90.16.a.h 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 918 \) acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(80))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 918 \) Copy content Toggle raw display
$5$ \( T + 78125 \) Copy content Toggle raw display
$7$ \( T - 953554 \) Copy content Toggle raw display
$11$ \( T + 17783232 \) Copy content Toggle raw display
$13$ \( T - 140533322 \) Copy content Toggle raw display
$17$ \( T - 2998870746 \) Copy content Toggle raw display
$19$ \( T + 3255852500 \) Copy content Toggle raw display
$23$ \( T + 6774812202 \) Copy content Toggle raw display
$29$ \( T + 7340322690 \) Copy content Toggle raw display
$31$ \( T - 115428411388 \) Copy content Toggle raw display
$37$ \( T - 150300986906 \) Copy content Toggle raw display
$41$ \( T - 1841603525142 \) Copy content Toggle raw display
$43$ \( T + 1510018315682 \) Copy content Toggle raw display
$47$ \( T + 6093750843366 \) Copy content Toggle raw display
$53$ \( T + 8267412829038 \) Copy content Toggle raw display
$59$ \( T - 23516883061980 \) Copy content Toggle raw display
$61$ \( T + 3135369104278 \) Copy content Toggle raw display
$67$ \( T - 36030983954794 \) Copy content Toggle raw display
$71$ \( T + 52169735384172 \) Copy content Toggle raw display
$73$ \( T - 69977143684082 \) Copy content Toggle raw display
$79$ \( T - 135317670906760 \) Copy content Toggle raw display
$83$ \( T + 427456158822882 \) Copy content Toggle raw display
$89$ \( T + 446581617299190 \) Copy content Toggle raw display
$97$ \( T - 181247411845826 \) Copy content Toggle raw display
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