Properties

Label 63.6.a.d
Level $63$
Weight $6$
Character orbit 63.a
Self dual yes
Analytic conductor $10.104$
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] = [63,6,Mod(1,63)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(63, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("63.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 63.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: \(10.1041806482\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
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)
 
\(f(q)\) \(=\) \( q + 6 q^{2} + 4 q^{4} - 78 q^{5} + 49 q^{7} - 168 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 6 q^{2} + 4 q^{4} - 78 q^{5} + 49 q^{7} - 168 q^{8} - 468 q^{10} - 444 q^{11} - 442 q^{13} + 294 q^{14} - 1136 q^{16} + 126 q^{17} + 2684 q^{19} - 312 q^{20} - 2664 q^{22} - 4200 q^{23} + 2959 q^{25} - 2652 q^{26} + 196 q^{28} + 5442 q^{29} + 80 q^{31} - 1440 q^{32} + 756 q^{34} - 3822 q^{35} - 5434 q^{37} + 16104 q^{38} + 13104 q^{40} - 7962 q^{41} - 11524 q^{43} - 1776 q^{44} - 25200 q^{46} + 13920 q^{47} + 2401 q^{49} + 17754 q^{50} - 1768 q^{52} + 9594 q^{53} + 34632 q^{55} - 8232 q^{56} + 32652 q^{58} - 27492 q^{59} + 49478 q^{61} + 480 q^{62} + 27712 q^{64} + 34476 q^{65} - 59356 q^{67} + 504 q^{68} - 22932 q^{70} - 32040 q^{71} - 61846 q^{73} - 32604 q^{74} + 10736 q^{76} - 21756 q^{77} - 65776 q^{79} + 88608 q^{80} - 47772 q^{82} - 40188 q^{83} - 9828 q^{85} - 69144 q^{86} + 74592 q^{88} + 7974 q^{89} - 21658 q^{91} - 16800 q^{92} + 83520 q^{94} - 209352 q^{95} - 143662 q^{97} + 14406 q^{98}+O(q^{100}) \) 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
6.00000 0 4.00000 −78.0000 0 49.0000 −168.000 0 −468.000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 63.6.a.d 1
3.b odd 2 1 21.6.a.a 1
4.b odd 2 1 1008.6.a.c 1
7.b odd 2 1 441.6.a.j 1
12.b even 2 1 336.6.a.r 1
15.d odd 2 1 525.6.a.d 1
15.e even 4 2 525.6.d.b 2
21.c even 2 1 147.6.a.b 1
21.g even 6 2 147.6.e.i 2
21.h odd 6 2 147.6.e.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.a.a 1 3.b odd 2 1
63.6.a.d 1 1.a even 1 1 trivial
147.6.a.b 1 21.c even 2 1
147.6.e.i 2 21.g even 6 2
147.6.e.j 2 21.h odd 6 2
336.6.a.r 1 12.b even 2 1
441.6.a.j 1 7.b odd 2 1
525.6.a.d 1 15.d odd 2 1
525.6.d.b 2 15.e even 4 2
1008.6.a.c 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 6 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(63))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 6 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 78 \) Copy content Toggle raw display
$7$ \( T - 49 \) Copy content Toggle raw display
$11$ \( T + 444 \) Copy content Toggle raw display
$13$ \( T + 442 \) Copy content Toggle raw display
$17$ \( T - 126 \) Copy content Toggle raw display
$19$ \( T - 2684 \) Copy content Toggle raw display
$23$ \( T + 4200 \) Copy content Toggle raw display
$29$ \( T - 5442 \) Copy content Toggle raw display
$31$ \( T - 80 \) Copy content Toggle raw display
$37$ \( T + 5434 \) Copy content Toggle raw display
$41$ \( T + 7962 \) Copy content Toggle raw display
$43$ \( T + 11524 \) Copy content Toggle raw display
$47$ \( T - 13920 \) Copy content Toggle raw display
$53$ \( T - 9594 \) Copy content Toggle raw display
$59$ \( T + 27492 \) Copy content Toggle raw display
$61$ \( T - 49478 \) Copy content Toggle raw display
$67$ \( T + 59356 \) Copy content Toggle raw display
$71$ \( T + 32040 \) Copy content Toggle raw display
$73$ \( T + 61846 \) Copy content Toggle raw display
$79$ \( T + 65776 \) Copy content Toggle raw display
$83$ \( T + 40188 \) Copy content Toggle raw display
$89$ \( T - 7974 \) Copy content Toggle raw display
$97$ \( T + 143662 \) Copy content Toggle raw display
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