Properties

Label 546.6.a.g
Level $546$
Weight $6$
Character orbit 546.a
Self dual yes
Analytic conductor $87.570$
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] = [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: \(1\)
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)
 
\(f(q)\) \(=\) \( q + 4 q^{2} + 9 q^{3} + 16 q^{4} - 21 q^{5} + 36 q^{6} + 49 q^{7} + 64 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{2} + 9 q^{3} + 16 q^{4} - 21 q^{5} + 36 q^{6} + 49 q^{7} + 64 q^{8} + 81 q^{9} - 84 q^{10} - 654 q^{11} + 144 q^{12} + 169 q^{13} + 196 q^{14} - 189 q^{15} + 256 q^{16} + 756 q^{17} + 324 q^{18} - 2905 q^{19} - 336 q^{20} + 441 q^{21} - 2616 q^{22} + 3039 q^{23} + 576 q^{24} - 2684 q^{25} + 676 q^{26} + 729 q^{27} + 784 q^{28} - 3957 q^{29} - 756 q^{30} - 3241 q^{31} + 1024 q^{32} - 5886 q^{33} + 3024 q^{34} - 1029 q^{35} + 1296 q^{36} - 628 q^{37} - 11620 q^{38} + 1521 q^{39} - 1344 q^{40} + 3678 q^{41} + 1764 q^{42} - 8797 q^{43} - 10464 q^{44} - 1701 q^{45} + 12156 q^{46} - 14163 q^{47} + 2304 q^{48} + 2401 q^{49} - 10736 q^{50} + 6804 q^{51} + 2704 q^{52} - 16497 q^{53} + 2916 q^{54} + 13734 q^{55} + 3136 q^{56} - 26145 q^{57} - 15828 q^{58} - 24528 q^{59} - 3024 q^{60} + 3686 q^{61} - 12964 q^{62} + 3969 q^{63} + 4096 q^{64} - 3549 q^{65} - 23544 q^{66} + 63818 q^{67} + 12096 q^{68} + 27351 q^{69} - 4116 q^{70} - 14040 q^{71} + 5184 q^{72} - 4579 q^{73} - 2512 q^{74} - 24156 q^{75} - 46480 q^{76} - 32046 q^{77} + 6084 q^{78} - 20713 q^{79} - 5376 q^{80} + 6561 q^{81} + 14712 q^{82} + 59421 q^{83} + 7056 q^{84} - 15876 q^{85} - 35188 q^{86} - 35613 q^{87} - 41856 q^{88} - 112995 q^{89} - 6804 q^{90} + 8281 q^{91} + 48624 q^{92} - 29169 q^{93} - 56652 q^{94} + 61005 q^{95} + 9216 q^{96} - 78511 q^{97} + 9604 q^{98} - 52974 q^{99}+O(q^{100}) \) 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 −21.0000 36.0000 49.0000 64.0000 81.0000 −84.0000
\(n\): 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.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.6.a.g 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(546))\):

\( T_{5} + 21 \) Copy content Toggle raw display
\( T_{11} + 654 \) 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
$5$ \( T + 21 \) Copy content Toggle raw display
$7$ \( T - 49 \) Copy content Toggle raw display
$11$ \( T + 654 \) Copy content Toggle raw display
$13$ \( T - 169 \) Copy content Toggle raw display
$17$ \( T - 756 \) Copy content Toggle raw display
$19$ \( T + 2905 \) Copy content Toggle raw display
$23$ \( T - 3039 \) Copy content Toggle raw display
$29$ \( T + 3957 \) Copy content Toggle raw display
$31$ \( T + 3241 \) Copy content Toggle raw display
$37$ \( T + 628 \) Copy content Toggle raw display
$41$ \( T - 3678 \) Copy content Toggle raw display
$43$ \( T + 8797 \) Copy content Toggle raw display
$47$ \( T + 14163 \) Copy content Toggle raw display
$53$ \( T + 16497 \) Copy content Toggle raw display
$59$ \( T + 24528 \) Copy content Toggle raw display
$61$ \( T - 3686 \) Copy content Toggle raw display
$67$ \( T - 63818 \) Copy content Toggle raw display
$71$ \( T + 14040 \) Copy content Toggle raw display
$73$ \( T + 4579 \) Copy content Toggle raw display
$79$ \( T + 20713 \) Copy content Toggle raw display
$83$ \( T - 59421 \) Copy content Toggle raw display
$89$ \( T + 112995 \) Copy content Toggle raw display
$97$ \( T + 78511 \) Copy content Toggle raw display
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