Properties

Label 546.6.a.b
Level $546$
Weight $6$
Character orbit 546.a
Self dual yes
Analytic conductor $87.570$
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] = [546,6,Mod(1,546)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(546, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("546.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 546.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: \(87.5695656179\)
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)
 
\(f(q)\) \(=\) \( q - 4 q^{2} + 9 q^{3} + 16 q^{4} + 99 q^{5} - 36 q^{6} + 49 q^{7} - 64 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{2} + 9 q^{3} + 16 q^{4} + 99 q^{5} - 36 q^{6} + 49 q^{7} - 64 q^{8} + 81 q^{9} - 396 q^{10} + 783 q^{11} + 144 q^{12} + 169 q^{13} - 196 q^{14} + 891 q^{15} + 256 q^{16} + 33 q^{17} - 324 q^{18} + 1433 q^{19} + 1584 q^{20} + 441 q^{21} - 3132 q^{22} - 3087 q^{23} - 576 q^{24} + 6676 q^{25} - 676 q^{26} + 729 q^{27} + 784 q^{28} + 1515 q^{29} - 3564 q^{30} + 2780 q^{31} - 1024 q^{32} + 7047 q^{33} - 132 q^{34} + 4851 q^{35} + 1296 q^{36} - 6595 q^{37} - 5732 q^{38} + 1521 q^{39} - 6336 q^{40} + 4380 q^{41} - 1764 q^{42} + 16043 q^{43} + 12528 q^{44} + 8019 q^{45} + 12348 q^{46} + 3480 q^{47} + 2304 q^{48} + 2401 q^{49} - 26704 q^{50} + 297 q^{51} + 2704 q^{52} + 618 q^{53} - 2916 q^{54} + 77517 q^{55} - 3136 q^{56} + 12897 q^{57} - 6060 q^{58} - 52116 q^{59} + 14256 q^{60} + 887 q^{61} - 11120 q^{62} + 3969 q^{63} + 4096 q^{64} + 16731 q^{65} - 28188 q^{66} + 854 q^{67} + 528 q^{68} - 27783 q^{69} - 19404 q^{70} - 47340 q^{71} - 5184 q^{72} - 64915 q^{73} + 26380 q^{74} + 60084 q^{75} + 22928 q^{76} + 38367 q^{77} - 6084 q^{78} - 73282 q^{79} + 25344 q^{80} + 6561 q^{81} - 17520 q^{82} - 97062 q^{83} + 7056 q^{84} + 3267 q^{85} - 64172 q^{86} + 13635 q^{87} - 50112 q^{88} + 118218 q^{89} - 32076 q^{90} + 8281 q^{91} - 49392 q^{92} + 25020 q^{93} - 13920 q^{94} + 141867 q^{95} - 9216 q^{96} - 184522 q^{97} - 9604 q^{98} + 63423 q^{99}+O(q^{100}) \) 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
−4.00000 9.00000 16.0000 99.0000 −36.0000 49.0000 −64.0000 81.0000 −396.000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(7\) \(-1\)
\(13\) \(-1\)

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 546.6.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.6.a.b 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(546))\):

\( T_{5} - 99 \) Copy content Toggle raw display
\( T_{11} - 783 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 4 \) Copy content Toggle raw display
$3$ \( T - 9 \) Copy content Toggle raw display
$5$ \( T - 99 \) Copy content Toggle raw display
$7$ \( T - 49 \) Copy content Toggle raw display
$11$ \( T - 783 \) Copy content Toggle raw display
$13$ \( T - 169 \) Copy content Toggle raw display
$17$ \( T - 33 \) Copy content Toggle raw display
$19$ \( T - 1433 \) Copy content Toggle raw display
$23$ \( T + 3087 \) Copy content Toggle raw display
$29$ \( T - 1515 \) Copy content Toggle raw display
$31$ \( T - 2780 \) Copy content Toggle raw display
$37$ \( T + 6595 \) Copy content Toggle raw display
$41$ \( T - 4380 \) Copy content Toggle raw display
$43$ \( T - 16043 \) Copy content Toggle raw display
$47$ \( T - 3480 \) Copy content Toggle raw display
$53$ \( T - 618 \) Copy content Toggle raw display
$59$ \( T + 52116 \) Copy content Toggle raw display
$61$ \( T - 887 \) Copy content Toggle raw display
$67$ \( T - 854 \) Copy content Toggle raw display
$71$ \( T + 47340 \) Copy content Toggle raw display
$73$ \( T + 64915 \) Copy content Toggle raw display
$79$ \( T + 73282 \) Copy content Toggle raw display
$83$ \( T + 97062 \) Copy content Toggle raw display
$89$ \( T - 118218 \) Copy content Toggle raw display
$97$ \( T + 184522 \) Copy content Toggle raw display
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