Properties

Label 8.16.a.b
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)
 
\(f(q)\) \(=\) \( q + 2700 q^{3} - 251890 q^{5} + 1374072 q^{7} - 7058907 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2700 q^{3} - 251890 q^{5} + 1374072 q^{7} - 7058907 q^{9} - 43286716 q^{11} - 323161466 q^{13} - 680103000 q^{15} - 191653646 q^{17} - 6515456644 q^{19} + 3709994400 q^{21} + 23880801512 q^{23} + 32930993975 q^{25} - 57801097800 q^{27} + 176820596982 q^{29} - 152007193888 q^{31} - 116874133200 q^{33} - 346114996080 q^{35} + 21581233902 q^{37} - 872535958200 q^{39} - 245334499686 q^{41} + 2769961534756 q^{43} + 1778068084230 q^{45} + 2811771943248 q^{47} - 2859487648759 q^{49} - 517464844200 q^{51} - 3491413730722 q^{53} + 10903490893240 q^{55} - 17591732938800 q^{57} - 15827800893676 q^{59} - 24609047974442 q^{61} - 9699446459304 q^{63} + 81401141670740 q^{65} - 20706233653684 q^{67} + 64478164082400 q^{69} - 719982528200 q^{71} + 29883036220282 q^{73} + 88913683732500 q^{75} - 59479064427552 q^{77} - 148100908648400 q^{79} - 54775363995351 q^{81} - 302806756982468 q^{83} + 48275636890940 q^{85} + 477415611851400 q^{87} - 496150966996374 q^{89} - 444047121909552 q^{91} - 410419423497600 q^{93} + 16\!\cdots\!60 q^{95}+ \cdots + 305556902579412 q^{99}+O(q^{100}) \) 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 2700.00 0 −251890. 0 1.37407e6 0 −7.05891e6 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.b 1
3.b odd 2 1 72.16.a.c 1
4.b odd 2 1 16.16.a.b 1
8.b even 2 1 64.16.a.d 1
8.d odd 2 1 64.16.a.h 1
12.b even 2 1 144.16.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.16.a.b 1 1.a even 1 1 trivial
16.16.a.b 1 4.b odd 2 1
64.16.a.d 1 8.b even 2 1
64.16.a.h 1 8.d odd 2 1
72.16.a.c 1 3.b odd 2 1
144.16.a.m 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 2700 \) 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
$3$ \( T - 2700 \) Copy content Toggle raw display
$5$ \( T + 251890 \) Copy content Toggle raw display
$7$ \( T - 1374072 \) Copy content Toggle raw display
$11$ \( T + 43286716 \) Copy content Toggle raw display
$13$ \( T + 323161466 \) Copy content Toggle raw display
$17$ \( T + 191653646 \) Copy content Toggle raw display
$19$ \( T + 6515456644 \) Copy content Toggle raw display
$23$ \( T - 23880801512 \) Copy content Toggle raw display
$29$ \( T - 176820596982 \) Copy content Toggle raw display
$31$ \( T + 152007193888 \) Copy content Toggle raw display
$37$ \( T - 21581233902 \) Copy content Toggle raw display
$41$ \( T + 245334499686 \) Copy content Toggle raw display
$43$ \( T - 2769961534756 \) Copy content Toggle raw display
$47$ \( T - 2811771943248 \) Copy content Toggle raw display
$53$ \( T + 3491413730722 \) Copy content Toggle raw display
$59$ \( T + 15827800893676 \) Copy content Toggle raw display
$61$ \( T + 24609047974442 \) Copy content Toggle raw display
$67$ \( T + 20706233653684 \) Copy content Toggle raw display
$71$ \( T + 719982528200 \) Copy content Toggle raw display
$73$ \( T - 29883036220282 \) Copy content Toggle raw display
$79$ \( T + 148100908648400 \) Copy content Toggle raw display
$83$ \( T + 302806756982468 \) Copy content Toggle raw display
$89$ \( T + 496150966996374 \) Copy content Toggle raw display
$97$ \( T - 309183128990882 \) Copy content Toggle raw display
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