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Properties

Label 546.2.bx.a
Level $546$
Weight $2$
Character orbit 546.bx
Analytic conductor $4.360$
Analytic rank $0$
Dimension $40$
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 algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 8 q^{7} + 20 q^{9} + 8 q^{11} + 40 q^{12} - 8 q^{14} + 20 q^{16} + 16 q^{17} + 16 q^{19} + 4 q^{21} + 8 q^{22} + 32 q^{26} + 8 q^{28} - 8 q^{33} + 16 q^{34} - 32 q^{35} + 40 q^{37} - 16 q^{38} - 16 q^{39}+ \cdots - 8 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1 −0.258819 − 0.965926i −0.866025 − 0.500000i −0.866025 + 0.500000i −3.04339 − 3.04339i −0.258819 + 0.965926i −1.02848 − 2.43767i 0.707107 + 0.707107i 0.500000 + 0.866025i −2.15200 + 3.72738i
97.2 −0.258819 − 0.965926i −0.866025 − 0.500000i −0.866025 + 0.500000i −0.201763 − 0.201763i −0.258819 + 0.965926i 1.99885 − 1.73338i 0.707107 + 0.707107i 0.500000 + 0.866025i −0.142668 + 0.247108i
97.3 −0.258819 − 0.965926i −0.866025 − 0.500000i −0.866025 + 0.500000i 0.301627 + 0.301627i −0.258819 + 0.965926i 2.59084 + 0.536243i 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 0.413783 + 0.413783i −0.258819 + 0.965926i −2.23784 + 1.41140i 0.707107 + 0.707107i 0.500000 + 0.866025i 0.292588 − 0.506778i
97.5 −0.258819 − 0.965926i −0.866025 − 0.500000i −0.866025 + 0.500000i 2.52974 + 2.52974i −0.258819 + 0.965926i −2.48229 − 0.915557i 0.707107 + 0.707107i 0.500000 + 0.866025i 1.78880 − 3.09829i
97.6 0.258819 + 0.965926i −0.866025 − 0.500000i −0.866025 + 0.500000i −2.63119 − 2.63119i 0.258819 − 0.965926i −1.51600 + 2.16835i −0.707107 − 0.707107i 0.500000 + 0.866025i 1.86053 − 3.22253i
97.7 0.258819 + 0.965926i −0.866025 − 0.500000i −0.866025 + 0.500000i −1.52143 − 1.52143i 0.258819 − 0.965926i 2.42509 + 1.05779i −0.707107 − 0.707107i 0.500000 + 0.866025i 1.07582 − 1.86337i
97.8 0.258819 + 0.965926i −0.866025 − 0.500000i −0.866025 + 0.500000i 0.0374438 + 0.0374438i 0.258819 − 0.965926i −2.54217 − 0.733067i −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 1.65224 + 1.65224i 0.258819 − 0.965926i 0.445675 − 2.60794i −0.707107 − 0.707107i 0.500000 + 0.866025i −1.16831 + 2.02357i
97.10 0.258819 + 0.965926i −0.866025 − 0.500000i −0.866025 + 0.500000i 2.46294 + 2.46294i 0.258819 − 0.965926i −1.38574 + 2.25383i −0.707107 − 0.707107i 0.500000 + 0.866025i −1.74156 + 3.01647i
223.1 −0.965926 − 0.258819i 0.866025 − 0.500000i 0.866025 + 0.500000i −1.77438 − 1.77438i −0.965926 + 0.258819i 0.546766 − 2.58864i −0.707107 − 0.707107i 0.500000 − 0.866025i 1.25468 + 2.17316i
223.2 −0.965926 − 0.258819i 0.866025 − 0.500000i 0.866025 + 0.500000i −1.20611 − 1.20611i −0.965926 + 0.258819i 2.62215 + 0.352575i −0.707107 − 0.707107i 0.500000 − 0.866025i 0.852845 + 1.47717i
223.3 −0.965926 − 0.258819i 0.866025 − 0.500000i 0.866025 + 0.500000i −0.781970 − 0.781970i −0.965926 + 0.258819i −1.20520 + 2.35531i −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 1.23174 + 1.23174i −0.965926 + 0.258819i −1.74064 + 1.99253i −0.707107 − 0.707107i 0.500000 − 0.866025i −0.870974 − 1.50857i
223.5 −0.965926 − 0.258819i 0.866025 − 0.500000i 0.866025 + 0.500000i 2.53071 + 2.53071i −0.965926 + 0.258819i −1.06416 − 2.42231i −0.707107 − 0.707107i 0.500000 − 0.866025i −1.78948 − 3.09948i
223.6 0.965926 + 0.258819i 0.866025 − 0.500000i 0.866025 + 0.500000i −3.01404 − 3.01404i 0.965926 − 0.258819i 2.02342 + 1.70464i 0.707107 + 0.707107i 0.500000 − 0.866025i −2.13125 − 3.69143i
223.7 0.965926 + 0.258819i 0.866025 − 0.500000i 0.866025 + 0.500000i −1.73144 − 1.73144i 0.965926 − 0.258819i −1.20824 − 2.35375i 0.707107 + 0.707107i 0.500000 − 0.866025i −1.22431 − 2.12057i
223.8 0.965926 + 0.258819i 0.866025 − 0.500000i 0.866025 + 0.500000i 1.10131 + 1.10131i 0.965926 − 0.258819i −0.930231 + 2.47683i 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 1.14071 + 1.14071i 0.965926 − 0.258819i 2.56426 − 0.651604i 0.707107 + 0.707107i 0.500000 − 0.866025i 0.806606 + 1.39708i
223.10 0.965926 + 0.258819i 0.866025 − 0.500000i 0.866025 + 0.500000i 2.50346 + 2.50346i 0.965926 − 0.258819i −1.87607 − 1.86558i 0.707107 + 0.707107i 0.500000 − 0.866025i 1.77021 + 3.06610i
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.a ✓ 40
7.b odd 2 1 546.2.bx.b yes 40
13.f odd 12 1 546.2.bx.b yes 40
91.bc even 12 1 inner 546.2.bx.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.bx.a ✓ 40 1.a even 1 1 trivial
546.2.bx.a ✓ 40 91.bc even 12 1 inner
546.2.bx.b yes 40 7.b odd 2 1
546.2.bx.b yes 40 13.f odd 12 1

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