Properties

Label 8.16.a.a
Level $8$
Weight $16$
Character orbit 8.a
Self dual yes
Analytic conductor $11.415$
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] = [8,16,Mod(1,8)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("8.1");
 
S:= CuspForms(chi, 16);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 8.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: \(11.4154804080\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
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)
 
\(f(q)\) \(=\) \( q - 3444 q^{3} + 313358 q^{5} - 2324616 q^{7} - 2487771 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3444 q^{3} + 313358 q^{5} - 2324616 q^{7} - 2487771 q^{9} - 55249084 q^{11} - 110259578 q^{13} - 1079204952 q^{15} - 2601428750 q^{17} + 1952124284 q^{19} + 8005977504 q^{21} - 25430340376 q^{23} + 67675658039 q^{25} + 57985519032 q^{27} - 2277224202 q^{29} - 190667257120 q^{31} + 190277845296 q^{33} - 728437020528 q^{35} - 288229450002 q^{37} + 379733986632 q^{39} + 756412456602 q^{41} - 354186592988 q^{43} - 779562945018 q^{45} + 6035922573648 q^{47} + 656278037513 q^{49} + 8959320615000 q^{51} - 12198920684962 q^{53} - 17312742464072 q^{55} - 6723116034096 q^{57} - 4090911936748 q^{59} + 17565907389910 q^{61} + 5783112270936 q^{63} - 34550720842924 q^{65} - 3931246965172 q^{67} + 87582092254944 q^{69} + 58825436072248 q^{71} + 107571519617914 q^{73} - 233074966286316 q^{75} + 128432904651744 q^{77} + 61543860115504 q^{79} - 164005332829911 q^{81} + 13432070277436 q^{83} - 815178510242500 q^{85} + 7842760151688 q^{87} + 269696339030634 q^{89} + 256311179172048 q^{91} + 656658033521280 q^{93} + 611713761385672 q^{95} - 793796744596318 q^{97} + 137447068951764 q^{99}+O(q^{100}) \) 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 −3444.00 0 313358. 0 −2.32462e6 0 −2.48777e6 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 8.16.a.a 1
3.b odd 2 1 72.16.a.a 1
4.b odd 2 1 16.16.a.e 1
8.b even 2 1 64.16.a.j 1
8.d odd 2 1 64.16.a.b 1
12.b even 2 1 144.16.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.16.a.a 1 1.a even 1 1 trivial
16.16.a.e 1 4.b odd 2 1
64.16.a.b 1 8.d odd 2 1
64.16.a.j 1 8.b even 2 1
72.16.a.a 1 3.b odd 2 1
144.16.a.a 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 3444 \) acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(8))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 3444 \) Copy content Toggle raw display
$5$ \( T - 313358 \) Copy content Toggle raw display
$7$ \( T + 2324616 \) Copy content Toggle raw display
$11$ \( T + 55249084 \) Copy content Toggle raw display
$13$ \( T + 110259578 \) Copy content Toggle raw display
$17$ \( T + 2601428750 \) Copy content Toggle raw display
$19$ \( T - 1952124284 \) Copy content Toggle raw display
$23$ \( T + 25430340376 \) Copy content Toggle raw display
$29$ \( T + 2277224202 \) Copy content Toggle raw display
$31$ \( T + 190667257120 \) Copy content Toggle raw display
$37$ \( T + 288229450002 \) Copy content Toggle raw display
$41$ \( T - 756412456602 \) Copy content Toggle raw display
$43$ \( T + 354186592988 \) Copy content Toggle raw display
$47$ \( T - 6035922573648 \) Copy content Toggle raw display
$53$ \( T + 12198920684962 \) Copy content Toggle raw display
$59$ \( T + 4090911936748 \) Copy content Toggle raw display
$61$ \( T - 17565907389910 \) Copy content Toggle raw display
$67$ \( T + 3931246965172 \) Copy content Toggle raw display
$71$ \( T - 58825436072248 \) Copy content Toggle raw display
$73$ \( T - 107571519617914 \) Copy content Toggle raw display
$79$ \( T - 61543860115504 \) Copy content Toggle raw display
$83$ \( T - 13432070277436 \) Copy content Toggle raw display
$89$ \( T - 269696339030634 \) Copy content Toggle raw display
$97$ \( T + 793796744596318 \) Copy content Toggle raw display
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