Properties

Label 147.6.a.e
Level $147$
Weight $6$
Character orbit 147.a
Self dual yes
Analytic conductor $23.576$
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] = [147,6,Mod(1,147)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(147, 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("147.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 147.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: \(23.5764215125\)
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)
 
\(f(q)\) \(=\) \( q + q^{2} + 9 q^{3} - 31 q^{4} + 34 q^{5} + 9 q^{6} - 63 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + 9 q^{3} - 31 q^{4} + 34 q^{5} + 9 q^{6} - 63 q^{8} + 81 q^{9} + 34 q^{10} - 340 q^{11} - 279 q^{12} - 454 q^{13} + 306 q^{15} + 929 q^{16} + 798 q^{17} + 81 q^{18} - 892 q^{19} - 1054 q^{20} - 340 q^{22} - 3192 q^{23} - 567 q^{24} - 1969 q^{25} - 454 q^{26} + 729 q^{27} - 8242 q^{29} + 306 q^{30} + 2496 q^{31} + 2945 q^{32} - 3060 q^{33} + 798 q^{34} - 2511 q^{36} + 9798 q^{37} - 892 q^{38} - 4086 q^{39} - 2142 q^{40} - 19834 q^{41} - 17236 q^{43} + 10540 q^{44} + 2754 q^{45} - 3192 q^{46} - 8928 q^{47} + 8361 q^{48} - 1969 q^{50} + 7182 q^{51} + 14074 q^{52} + 150 q^{53} + 729 q^{54} - 11560 q^{55} - 8028 q^{57} - 8242 q^{58} + 42396 q^{59} - 9486 q^{60} - 14758 q^{61} + 2496 q^{62} - 26783 q^{64} - 15436 q^{65} - 3060 q^{66} - 1676 q^{67} - 24738 q^{68} - 28728 q^{69} + 14568 q^{71} - 5103 q^{72} - 78378 q^{73} + 9798 q^{74} - 17721 q^{75} + 27652 q^{76} - 4086 q^{78} - 2272 q^{79} + 31586 q^{80} + 6561 q^{81} - 19834 q^{82} + 37764 q^{83} + 27132 q^{85} - 17236 q^{86} - 74178 q^{87} + 21420 q^{88} + 117286 q^{89} + 2754 q^{90} + 98952 q^{92} + 22464 q^{93} - 8928 q^{94} - 30328 q^{95} + 26505 q^{96} - 10002 q^{97} - 27540 q^{99}+O(q^{100}) \) 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
1.00000 9.00000 −31.0000 34.0000 9.00000 0 −63.0000 81.0000 34.0000
\(n\): 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 147.6.a.e 1
3.b odd 2 1 441.6.a.d 1
7.b odd 2 1 21.6.a.b 1
7.c even 3 2 147.6.e.e 2
7.d odd 6 2 147.6.e.f 2
21.c even 2 1 63.6.a.c 1
28.d even 2 1 336.6.a.l 1
35.c odd 2 1 525.6.a.c 1
35.f even 4 2 525.6.d.d 2
84.h odd 2 1 1008.6.a.t 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.a.b 1 7.b odd 2 1
63.6.a.c 1 21.c even 2 1
147.6.a.e 1 1.a even 1 1 trivial
147.6.e.e 2 7.c even 3 2
147.6.e.f 2 7.d odd 6 2
336.6.a.l 1 28.d even 2 1
441.6.a.d 1 3.b odd 2 1
525.6.a.c 1 35.c odd 2 1
525.6.d.d 2 35.f even 4 2
1008.6.a.t 1 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(147))\):

\( T_{2} - 1 \) Copy content Toggle raw display
\( T_{5} - 34 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1 \) Copy content Toggle raw display
$3$ \( T - 9 \) Copy content Toggle raw display
$5$ \( T - 34 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 340 \) Copy content Toggle raw display
$13$ \( T + 454 \) Copy content Toggle raw display
$17$ \( T - 798 \) Copy content Toggle raw display
$19$ \( T + 892 \) Copy content Toggle raw display
$23$ \( T + 3192 \) Copy content Toggle raw display
$29$ \( T + 8242 \) Copy content Toggle raw display
$31$ \( T - 2496 \) Copy content Toggle raw display
$37$ \( T - 9798 \) Copy content Toggle raw display
$41$ \( T + 19834 \) Copy content Toggle raw display
$43$ \( T + 17236 \) Copy content Toggle raw display
$47$ \( T + 8928 \) Copy content Toggle raw display
$53$ \( T - 150 \) Copy content Toggle raw display
$59$ \( T - 42396 \) Copy content Toggle raw display
$61$ \( T + 14758 \) Copy content Toggle raw display
$67$ \( T + 1676 \) Copy content Toggle raw display
$71$ \( T - 14568 \) Copy content Toggle raw display
$73$ \( T + 78378 \) Copy content Toggle raw display
$79$ \( T + 2272 \) Copy content Toggle raw display
$83$ \( T - 37764 \) Copy content Toggle raw display
$89$ \( T - 117286 \) Copy content Toggle raw display
$97$ \( T + 10002 \) Copy content Toggle raw display
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