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Properties

Label 546.2.bx.b
Level $546$
Weight $2$
Character orbit 546.bx
Analytic conductor $4.360$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [546,2,Mod(97,546)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(546, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 6, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("546.97");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 546.bx (of order \(12\), degree \(4\), minimal)

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: no
Analytic conductor: \(4.35983195036\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(10\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 8 q^{7} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 8 q^{7} + 20 q^{9} + 8 q^{11} - 40 q^{12} - 8 q^{14} + 20 q^{16} - 16 q^{17} - 16 q^{19} + 4 q^{21} + 8 q^{22} - 32 q^{26} - 8 q^{28} + 8 q^{33} - 16 q^{34} + 40 q^{35} + 40 q^{37} + 16 q^{38} - 16 q^{39} + 4 q^{41} - 12 q^{42} + 24 q^{43} + 8 q^{44} - 4 q^{46} - 16 q^{47} - 20 q^{49} - 16 q^{50} + 8 q^{52} + 32 q^{53} + 72 q^{55} + 12 q^{56} + 8 q^{57} - 36 q^{58} - 84 q^{59} + 48 q^{61} + 4 q^{62} + 4 q^{63} - 16 q^{65} - 12 q^{68} + 8 q^{69} - 20 q^{70} - 40 q^{71} + 48 q^{73} + 8 q^{74} - 36 q^{75} - 16 q^{76} + 24 q^{77} - 8 q^{78} - 48 q^{79} - 20 q^{81} - 36 q^{83} - 8 q^{84} + 8 q^{85} - 8 q^{86} - 12 q^{89} + 32 q^{91} - 16 q^{92} - 24 q^{93} - 96 q^{95} - 8 q^{98} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1 −0.258819 − 0.965926i 0.866025 + 0.500000i −0.866025 + 0.500000i −2.52974 − 2.52974i 0.258819 − 0.965926i 2.60750 + 0.448249i 0.707107 + 0.707107i 0.500000 + 0.866025i −1.78880 + 3.09829i
97.2 −0.258819 − 0.965926i 0.866025 + 0.500000i −0.866025 + 0.500000i −0.413783 − 0.413783i 0.258819 − 0.965926i 1.23233 + 2.34123i 0.707107 + 0.707107i 0.500000 + 0.866025i −0.292588 + 0.506778i
97.3 −0.258819 − 0.965926i 0.866025 + 0.500000i −0.866025 + 0.500000i −0.301627 − 0.301627i 0.258819 − 0.965926i −2.51185 − 0.831019i 0.707107 + 0.707107i 0.500000 + 0.866025i −0.213282 + 0.369416i
97.4 −0.258819 − 0.965926i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.201763 + 0.201763i 0.258819 − 0.965926i −0.864369 − 2.50057i 0.707107 + 0.707107i 0.500000 + 0.866025i 0.142668 − 0.247108i
97.5 −0.258819 − 0.965926i 0.866025 + 0.500000i −0.866025 + 0.500000i 3.04339 + 3.04339i 0.258819 − 0.965926i 2.10952 − 1.59685i 0.707107 + 0.707107i 0.500000 + 0.866025i 2.15200 − 3.72738i
97.6 0.258819 + 0.965926i 0.866025 + 0.500000i −0.866025 + 0.500000i −2.46294 − 2.46294i −0.258819 + 0.965926i 0.0731721 + 2.64474i −0.707107 − 0.707107i 0.500000 + 0.866025i 1.74156 − 3.01647i
97.7 0.258819 + 0.965926i 0.866025 + 0.500000i −0.866025 + 0.500000i −1.65224 − 1.65224i −0.258819 + 0.965926i 0.918006 − 2.48138i −0.707107 − 0.707107i 0.500000 + 0.866025i 1.16831 − 2.02357i
97.8 0.258819 + 0.965926i 0.866025 + 0.500000i −0.866025 + 0.500000i −0.0374438 − 0.0374438i −0.258819 + 0.965926i 2.56811 + 0.636228i −0.707107 − 0.707107i 0.500000 + 0.866025i 0.0264768 − 0.0458592i
97.9 0.258819 + 0.965926i 0.866025 + 0.500000i −0.866025 + 0.500000i 1.52143 + 1.52143i −0.258819 + 0.965926i −2.62909 − 0.296473i −0.707107 − 0.707107i 0.500000 + 0.866025i −1.07582 + 1.86337i
97.10 0.258819 + 0.965926i 0.866025 + 0.500000i −0.866025 + 0.500000i 2.63119 + 2.63119i −0.258819 + 0.965926i 0.228714 + 2.63585i −0.707107 − 0.707107i 0.500000 + 0.866025i −1.86053 + 3.22253i
223.1 −0.965926 − 0.258819i −0.866025 + 0.500000i 0.866025 + 0.500000i −2.53071 − 2.53071i 0.965926 − 0.258819i 0.289566 + 2.62986i −0.707107 − 0.707107i 0.500000 − 0.866025i 1.78948 + 3.09948i
223.2 −0.965926 − 0.258819i −0.866025 + 0.500000i 0.866025 + 0.500000i −1.23174 − 1.23174i 0.965926 − 0.258819i −2.50370 − 0.855258i −0.707107 − 0.707107i 0.500000 − 0.866025i 0.870974 + 1.50857i
223.3 −0.965926 − 0.258819i −0.866025 + 0.500000i 0.866025 + 0.500000i 0.781970 + 0.781970i 0.965926 − 0.258819i −2.22139 − 1.43716i −0.707107 − 0.707107i 0.500000 − 0.866025i −0.552937 − 0.957714i
223.4 −0.965926 − 0.258819i −0.866025 + 0.500000i 0.866025 + 0.500000i 1.20611 + 1.20611i 0.965926 − 0.258819i 2.09456 − 1.61642i −0.707107 − 0.707107i 0.500000 − 0.866025i −0.852845 − 1.47717i
223.5 −0.965926 − 0.258819i −0.866025 + 0.500000i 0.866025 + 0.500000i 1.77438 + 1.77438i 0.965926 − 0.258819i 1.76783 + 1.96844i −0.707107 − 0.707107i 0.500000 − 0.866025i −1.25468 − 2.17316i
223.6 0.965926 + 0.258819i −0.866025 + 0.500000i 0.866025 + 0.500000i −2.50346 − 2.50346i −0.965926 + 0.258819i −0.691933 + 2.55367i 0.707107 + 0.707107i 0.500000 − 0.866025i −1.77021 − 3.06610i
223.7 0.965926 + 0.258819i −0.866025 + 0.500000i 0.866025 + 0.500000i −1.14071 − 1.14071i −0.965926 + 0.258819i 2.54651 − 0.717822i 0.707107 + 0.707107i 0.500000 − 0.866025i −0.806606 − 1.39708i
223.8 0.965926 + 0.258819i −0.866025 + 0.500000i 0.866025 + 0.500000i −1.10131 − 1.10131i −0.965926 + 0.258819i −2.04402 − 1.67988i 0.707107 + 0.707107i 0.500000 − 0.866025i −0.778741 − 1.34882i
223.9 0.965926 + 0.258819i −0.866025 + 0.500000i 0.866025 + 0.500000i 1.73144 + 1.73144i −0.965926 + 0.258819i 0.130508 + 2.64253i 0.707107 + 0.707107i 0.500000 − 0.866025i 1.22431 + 2.12057i
223.10 0.965926 + 0.258819i −0.866025 + 0.500000i 0.866025 + 0.500000i 3.01404 + 3.01404i −0.965926 + 0.258819i 0.900010 − 2.48797i 0.707107 + 0.707107i 0.500000 − 0.866025i 2.13125 + 3.69143i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.10
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.bc even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.2.bx.b yes 40
7.b odd 2 1 546.2.bx.a ✓ 40
13.f odd 12 1 546.2.bx.a ✓ 40
91.bc even 12 1 inner 546.2.bx.b yes 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.bx.a ✓ 40 7.b odd 2 1
546.2.bx.a ✓ 40 13.f odd 12 1
546.2.bx.b yes 40 1.a even 1 1 trivial
546.2.bx.b yes 40 91.bc even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{40} + 24 T_{5}^{37} + 828 T_{5}^{36} + 424 T_{5}^{35} + 288 T_{5}^{34} + 12696 T_{5}^{33} + \cdots + 107495424 \) acting on \(S_{2}^{\mathrm{new}}(546, [\chi])\). Copy content Toggle raw display