Properties

Label 84.4.a.a
Level $84$
Weight $4$
Character orbit 84.a
Self dual yes
Analytic conductor $4.956$
Analytic rank $0$
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] = [84,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("84.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(4.95616044048\)
Analytic rank: \(0\)
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)
 
\(f(q)\) \(=\) \( q - 3 q^{3} + 6 q^{5} + 7 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} + 6 q^{5} + 7 q^{7} + 9 q^{9} + 36 q^{11} + 62 q^{13} - 18 q^{15} + 114 q^{17} - 76 q^{19} - 21 q^{21} - 24 q^{23} - 89 q^{25} - 27 q^{27} + 54 q^{29} - 112 q^{31} - 108 q^{33} + 42 q^{35} - 178 q^{37} - 186 q^{39} + 378 q^{41} - 172 q^{43} + 54 q^{45} - 192 q^{47} + 49 q^{49} - 342 q^{51} - 402 q^{53} + 216 q^{55} + 228 q^{57} + 396 q^{59} + 254 q^{61} + 63 q^{63} + 372 q^{65} - 1012 q^{67} + 72 q^{69} + 840 q^{71} + 890 q^{73} + 267 q^{75} + 252 q^{77} + 80 q^{79} + 81 q^{81} - 108 q^{83} + 684 q^{85} - 162 q^{87} - 1638 q^{89} + 434 q^{91} + 336 q^{93} - 456 q^{95} + 1010 q^{97} + 324 q^{99}+O(q^{100}) \) 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 −3.00000 0 6.00000 0 7.00000 0 9.00000 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 84.4.a.a 1
3.b odd 2 1 252.4.a.b 1
4.b odd 2 1 336.4.a.k 1
5.b even 2 1 2100.4.a.l 1
5.c odd 4 2 2100.4.k.j 2
7.b odd 2 1 588.4.a.d 1
7.c even 3 2 588.4.i.f 2
7.d odd 6 2 588.4.i.c 2
8.b even 2 1 1344.4.a.q 1
8.d odd 2 1 1344.4.a.d 1
12.b even 2 1 1008.4.a.h 1
21.c even 2 1 1764.4.a.j 1
21.g even 6 2 1764.4.k.f 2
21.h odd 6 2 1764.4.k.l 2
28.d even 2 1 2352.4.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.a.a 1 1.a even 1 1 trivial
252.4.a.b 1 3.b odd 2 1
336.4.a.k 1 4.b odd 2 1
588.4.a.d 1 7.b odd 2 1
588.4.i.c 2 7.d odd 6 2
588.4.i.f 2 7.c even 3 2
1008.4.a.h 1 12.b even 2 1
1344.4.a.d 1 8.d odd 2 1
1344.4.a.q 1 8.b even 2 1
1764.4.a.j 1 21.c even 2 1
1764.4.k.f 2 21.g even 6 2
1764.4.k.l 2 21.h odd 6 2
2100.4.a.l 1 5.b even 2 1
2100.4.k.j 2 5.c odd 4 2
2352.4.a.d 1 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 6 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(84))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 3 \) Copy content Toggle raw display
$5$ \( T - 6 \) Copy content Toggle raw display
$7$ \( T - 7 \) Copy content Toggle raw display
$11$ \( T - 36 \) Copy content Toggle raw display
$13$ \( T - 62 \) Copy content Toggle raw display
$17$ \( T - 114 \) Copy content Toggle raw display
$19$ \( T + 76 \) Copy content Toggle raw display
$23$ \( T + 24 \) Copy content Toggle raw display
$29$ \( T - 54 \) Copy content Toggle raw display
$31$ \( T + 112 \) Copy content Toggle raw display
$37$ \( T + 178 \) Copy content Toggle raw display
$41$ \( T - 378 \) Copy content Toggle raw display
$43$ \( T + 172 \) Copy content Toggle raw display
$47$ \( T + 192 \) Copy content Toggle raw display
$53$ \( T + 402 \) Copy content Toggle raw display
$59$ \( T - 396 \) Copy content Toggle raw display
$61$ \( T - 254 \) Copy content Toggle raw display
$67$ \( T + 1012 \) Copy content Toggle raw display
$71$ \( T - 840 \) Copy content Toggle raw display
$73$ \( T - 890 \) Copy content Toggle raw display
$79$ \( T - 80 \) Copy content Toggle raw display
$83$ \( T + 108 \) Copy content Toggle raw display
$89$ \( T + 1638 \) Copy content Toggle raw display
$97$ \( T - 1010 \) Copy content Toggle raw display
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