Properties

Label 84.8.a.a
Level $84$
Weight $8$
Character orbit 84.a
Self dual yes
Analytic conductor $26.240$
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] = [84,8,Mod(1,84)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(84, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("84.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 84.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: \(26.2403421407\)
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)
 
\(f(q)\) \(=\) \( q - 27 q^{3} + 100 q^{5} - 343 q^{7} + 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 27 q^{3} + 100 q^{5} - 343 q^{7} + 729 q^{9} + 2774 q^{11} - 3294 q^{13} - 2700 q^{15} + 5900 q^{17} + 6644 q^{19} + 9261 q^{21} + 1982 q^{23} - 68125 q^{25} - 19683 q^{27} - 208106 q^{29} - 117792 q^{31} - 74898 q^{33} - 34300 q^{35} - 335686 q^{37} + 88938 q^{39} - 265488 q^{41} - 93292 q^{43} + 72900 q^{45} - 657516 q^{47} + 117649 q^{49} - 159300 q^{51} - 608718 q^{53} + 277400 q^{55} - 179388 q^{57} - 536120 q^{59} - 1797090 q^{61} - 250047 q^{63} - 329400 q^{65} + 2123176 q^{67} - 53514 q^{69} - 1191214 q^{71} + 1056430 q^{73} + 1839375 q^{75} - 951482 q^{77} + 998484 q^{79} + 531441 q^{81} + 3898004 q^{83} + 590000 q^{85} + 5618862 q^{87} - 4622352 q^{89} + 1129842 q^{91} + 3180384 q^{93} + 664400 q^{95} + 15287710 q^{97} + 2022246 q^{99}+O(q^{100}) \) 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 −27.0000 0 100.000 0 −343.000 0 729.000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( +1 \)
\(7\) \( +1 \)

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 84.8.a.a 1
3.b odd 2 1 252.8.a.a 1
4.b odd 2 1 336.8.a.j 1
7.b odd 2 1 588.8.a.c 1
7.c even 3 2 588.8.i.f 2
7.d odd 6 2 588.8.i.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.8.a.a 1 1.a even 1 1 trivial
252.8.a.a 1 3.b odd 2 1
336.8.a.j 1 4.b odd 2 1
588.8.a.c 1 7.b odd 2 1
588.8.i.c 2 7.d odd 6 2
588.8.i.f 2 7.c even 3 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 100 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(84))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 27 \) Copy content Toggle raw display
$5$ \( T - 100 \) Copy content Toggle raw display
$7$ \( T + 343 \) Copy content Toggle raw display
$11$ \( T - 2774 \) Copy content Toggle raw display
$13$ \( T + 3294 \) Copy content Toggle raw display
$17$ \( T - 5900 \) Copy content Toggle raw display
$19$ \( T - 6644 \) Copy content Toggle raw display
$23$ \( T - 1982 \) Copy content Toggle raw display
$29$ \( T + 208106 \) Copy content Toggle raw display
$31$ \( T + 117792 \) Copy content Toggle raw display
$37$ \( T + 335686 \) Copy content Toggle raw display
$41$ \( T + 265488 \) Copy content Toggle raw display
$43$ \( T + 93292 \) Copy content Toggle raw display
$47$ \( T + 657516 \) Copy content Toggle raw display
$53$ \( T + 608718 \) Copy content Toggle raw display
$59$ \( T + 536120 \) Copy content Toggle raw display
$61$ \( T + 1797090 \) Copy content Toggle raw display
$67$ \( T - 2123176 \) Copy content Toggle raw display
$71$ \( T + 1191214 \) Copy content Toggle raw display
$73$ \( T - 1056430 \) Copy content Toggle raw display
$79$ \( T - 998484 \) Copy content Toggle raw display
$83$ \( T - 3898004 \) Copy content Toggle raw display
$89$ \( T + 4622352 \) Copy content Toggle raw display
$97$ \( T - 15287710 \) Copy content Toggle raw display
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