Properties

Label 8.10.a.a
Level $8$
Weight $10$
Character orbit 8.a
Self dual yes
Analytic conductor $4.120$
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] = [8,10,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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 10 \)
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: \(4.12028668931\)
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)
 
\(f(q)\) \(=\) \( q - 60 q^{3} - 2074 q^{5} - 4344 q^{7} - 16083 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 60 q^{3} - 2074 q^{5} - 4344 q^{7} - 16083 q^{9} + 93644 q^{11} - 12242 q^{13} + 124440 q^{15} - 319598 q^{17} - 553516 q^{19} + 260640 q^{21} - 712936 q^{23} + 2348351 q^{25} + 2145960 q^{27} + 2075838 q^{29} - 6420448 q^{31} - 5618640 q^{33} + 9009456 q^{35} - 18197754 q^{37} + 734520 q^{39} + 9033834 q^{41} + 19594732 q^{43} + 33356142 q^{45} - 18484176 q^{47} - 21483271 q^{49} + 19175880 q^{51} + 10255766 q^{53} - 194217656 q^{55} + 33210960 q^{57} + 121666556 q^{59} - 45948962 q^{61} + 69864552 q^{63} + 25389908 q^{65} + 50535428 q^{67} + 42776160 q^{69} + 267044680 q^{71} - 176213366 q^{73} - 140901060 q^{75} - 406789536 q^{77} - 269685680 q^{79} + 187804089 q^{81} - 227032556 q^{83} + 662846252 q^{85} - 124550280 q^{87} + 72141594 q^{89} + 53179248 q^{91} + 385226880 q^{93} + 1147992184 q^{95} + 228776546 q^{97} - 1506076452 q^{99}+O(q^{100}) \) 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 −60.0000 0 −2074.00 0 −4344.00 0 −16083.0 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 8.10.a.a 1
3.b odd 2 1 72.10.a.e 1
4.b odd 2 1 16.10.a.c 1
5.b even 2 1 200.10.a.b 1
5.c odd 4 2 200.10.c.b 2
7.b odd 2 1 392.10.a.b 1
8.b even 2 1 64.10.a.f 1
8.d odd 2 1 64.10.a.d 1
12.b even 2 1 144.10.a.n 1
16.e even 4 2 256.10.b.i 2
16.f odd 4 2 256.10.b.c 2
20.d odd 2 1 400.10.a.d 1
20.e even 4 2 400.10.c.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.10.a.a 1 1.a even 1 1 trivial
16.10.a.c 1 4.b odd 2 1
64.10.a.d 1 8.d odd 2 1
64.10.a.f 1 8.b even 2 1
72.10.a.e 1 3.b odd 2 1
144.10.a.n 1 12.b even 2 1
200.10.a.b 1 5.b even 2 1
200.10.c.b 2 5.c odd 4 2
256.10.b.c 2 16.f odd 4 2
256.10.b.i 2 16.e even 4 2
392.10.a.b 1 7.b odd 2 1
400.10.a.d 1 20.d odd 2 1
400.10.c.g 2 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 60 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(8))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 60 \) Copy content Toggle raw display
$5$ \( T + 2074 \) Copy content Toggle raw display
$7$ \( T + 4344 \) Copy content Toggle raw display
$11$ \( T - 93644 \) Copy content Toggle raw display
$13$ \( T + 12242 \) Copy content Toggle raw display
$17$ \( T + 319598 \) Copy content Toggle raw display
$19$ \( T + 553516 \) Copy content Toggle raw display
$23$ \( T + 712936 \) Copy content Toggle raw display
$29$ \( T - 2075838 \) Copy content Toggle raw display
$31$ \( T + 6420448 \) Copy content Toggle raw display
$37$ \( T + 18197754 \) Copy content Toggle raw display
$41$ \( T - 9033834 \) Copy content Toggle raw display
$43$ \( T - 19594732 \) Copy content Toggle raw display
$47$ \( T + 18484176 \) Copy content Toggle raw display
$53$ \( T - 10255766 \) Copy content Toggle raw display
$59$ \( T - 121666556 \) Copy content Toggle raw display
$61$ \( T + 45948962 \) Copy content Toggle raw display
$67$ \( T - 50535428 \) Copy content Toggle raw display
$71$ \( T - 267044680 \) Copy content Toggle raw display
$73$ \( T + 176213366 \) Copy content Toggle raw display
$79$ \( T + 269685680 \) Copy content Toggle raw display
$83$ \( T + 227032556 \) Copy content Toggle raw display
$89$ \( T - 72141594 \) Copy content Toggle raw display
$97$ \( T - 228776546 \) Copy content Toggle raw display
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