Properties

Label 63.6.a.b
Level $63$
Weight $6$
Character orbit 63.a
Self dual yes
Analytic conductor $10.104$
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] = [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: \(0\)
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)
 
\(f(q)\) \(=\) \( q - 5 q^{2} - 7 q^{4} - 94 q^{5} - 49 q^{7} + 195 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 5 q^{2} - 7 q^{4} - 94 q^{5} - 49 q^{7} + 195 q^{8} + 470 q^{10} - 52 q^{11} - 770 q^{13} + 245 q^{14} - 751 q^{16} + 2022 q^{17} + 1732 q^{19} + 658 q^{20} + 260 q^{22} + 576 q^{23} + 5711 q^{25} + 3850 q^{26} + 343 q^{28} - 5518 q^{29} + 6336 q^{31} - 2485 q^{32} - 10110 q^{34} + 4606 q^{35} - 7338 q^{37} - 8660 q^{38} - 18330 q^{40} + 3262 q^{41} + 5420 q^{43} + 364 q^{44} - 2880 q^{46} - 864 q^{47} + 2401 q^{49} - 28555 q^{50} + 5390 q^{52} - 4182 q^{53} + 4888 q^{55} - 9555 q^{56} + 27590 q^{58} + 11220 q^{59} - 45602 q^{61} - 31680 q^{62} + 36457 q^{64} + 72380 q^{65} + 1396 q^{67} - 14154 q^{68} - 23030 q^{70} - 18720 q^{71} + 46362 q^{73} + 36690 q^{74} - 12124 q^{76} + 2548 q^{77} + 97424 q^{79} + 70594 q^{80} - 16310 q^{82} + 81228 q^{83} - 190068 q^{85} - 27100 q^{86} - 10140 q^{88} + 3182 q^{89} + 37730 q^{91} - 4032 q^{92} + 4320 q^{94} - 162808 q^{95} + 4914 q^{97} - 12005 q^{98}+O(q^{100}) \) 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
−5.00000 0 −7.00000 −94.0000 0 −49.0000 195.000 0 470.000
\(n\): 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.b 1
3.b odd 2 1 21.6.a.c 1
4.b odd 2 1 1008.6.a.a 1
7.b odd 2 1 441.6.a.c 1
12.b even 2 1 336.6.a.i 1
15.d odd 2 1 525.6.a.b 1
15.e even 4 2 525.6.d.c 2
21.c even 2 1 147.6.a.f 1
21.g even 6 2 147.6.e.d 2
21.h odd 6 2 147.6.e.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.a.c 1 3.b odd 2 1
63.6.a.b 1 1.a even 1 1 trivial
147.6.a.f 1 21.c even 2 1
147.6.e.c 2 21.h odd 6 2
147.6.e.d 2 21.g even 6 2
336.6.a.i 1 12.b even 2 1
441.6.a.c 1 7.b odd 2 1
525.6.a.b 1 15.d odd 2 1
525.6.d.c 2 15.e even 4 2
1008.6.a.a 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 5 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(63))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 5 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 94 \) Copy content Toggle raw display
$7$ \( T + 49 \) Copy content Toggle raw display
$11$ \( T + 52 \) Copy content Toggle raw display
$13$ \( T + 770 \) Copy content Toggle raw display
$17$ \( T - 2022 \) Copy content Toggle raw display
$19$ \( T - 1732 \) Copy content Toggle raw display
$23$ \( T - 576 \) Copy content Toggle raw display
$29$ \( T + 5518 \) Copy content Toggle raw display
$31$ \( T - 6336 \) Copy content Toggle raw display
$37$ \( T + 7338 \) Copy content Toggle raw display
$41$ \( T - 3262 \) Copy content Toggle raw display
$43$ \( T - 5420 \) Copy content Toggle raw display
$47$ \( T + 864 \) Copy content Toggle raw display
$53$ \( T + 4182 \) Copy content Toggle raw display
$59$ \( T - 11220 \) Copy content Toggle raw display
$61$ \( T + 45602 \) Copy content Toggle raw display
$67$ \( T - 1396 \) Copy content Toggle raw display
$71$ \( T + 18720 \) Copy content Toggle raw display
$73$ \( T - 46362 \) Copy content Toggle raw display
$79$ \( T - 97424 \) Copy content Toggle raw display
$83$ \( T - 81228 \) Copy content Toggle raw display
$89$ \( T - 3182 \) Copy content Toggle raw display
$97$ \( T - 4914 \) Copy content Toggle raw display
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