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Properties

Label 546.4.s.d
Level $546$
Weight $4$
Character orbit 546.s
Analytic conductor $32.215$
Analytic rank $0$
Dimension $24$
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] = [546,4,Mod(43,546)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(546, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("546.43");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 546.s (of order \(6\), degree \(2\), 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: \(32.2150428631\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 36 q^{3} + 48 q^{4} - 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 36 q^{3} + 48 q^{4} - 108 q^{9} - 24 q^{10} - 174 q^{11} + 288 q^{12} - 96 q^{13} - 336 q^{14} - 90 q^{15} - 192 q^{16} + 140 q^{17} - 60 q^{19} + 120 q^{20} - 128 q^{22} + 34 q^{23} - 1004 q^{25} - 144 q^{26} - 648 q^{27} + 54 q^{29} + 72 q^{30} - 522 q^{33} - 84 q^{35} + 432 q^{36} + 1416 q^{37} + 1056 q^{38} + 108 q^{39} - 192 q^{40} - 942 q^{41} - 504 q^{42} + 762 q^{43} - 270 q^{45} + 636 q^{46} + 576 q^{48} + 588 q^{49} + 720 q^{50} + 840 q^{51} - 528 q^{52} + 184 q^{53} - 256 q^{55} - 672 q^{56} - 2088 q^{58} + 2322 q^{59} - 1274 q^{61} - 232 q^{62} - 1536 q^{64} - 1298 q^{65} - 768 q^{66} - 2472 q^{67} - 560 q^{68} - 102 q^{69} - 2196 q^{71} - 348 q^{74} - 1506 q^{75} - 240 q^{76} + 896 q^{77} - 468 q^{78} - 520 q^{79} + 480 q^{80} - 972 q^{81} - 244 q^{82} + 7974 q^{85} - 162 q^{87} + 512 q^{88} + 1032 q^{89} + 432 q^{90} + 546 q^{91} + 272 q^{92} - 2880 q^{93} - 1104 q^{94} - 2574 q^{95} - 3936 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −1.73205 − 1.00000i 1.50000 − 2.59808i 2.00000 + 3.46410i − 20.8505i −5.19615 + 3.00000i 6.06218 − 3.50000i − 8.00000i −4.50000 − 7.79423i −20.8505 + 36.1141i
43.2 −1.73205 − 1.00000i 1.50000 − 2.59808i 2.00000 + 3.46410i − 9.97677i −5.19615 + 3.00000i 6.06218 − 3.50000i − 8.00000i −4.50000 − 7.79423i −9.97677 + 17.2803i
43.3 −1.73205 − 1.00000i 1.50000 − 2.59808i 2.00000 + 3.46410i − 10.6352i −5.19615 + 3.00000i 6.06218 − 3.50000i − 8.00000i −4.50000 − 7.79423i −10.6352 + 18.4208i
43.4 −1.73205 − 1.00000i 1.50000 − 2.59808i 2.00000 + 3.46410i − 2.48320i −5.19615 + 3.00000i 6.06218 − 3.50000i − 8.00000i −4.50000 − 7.79423i −2.48320 + 4.30102i
43.5 −1.73205 − 1.00000i 1.50000 − 2.59808i 2.00000 + 3.46410i 11.2847i −5.19615 + 3.00000i 6.06218 − 3.50000i − 8.00000i −4.50000 − 7.79423i 11.2847 − 19.5457i
43.6 −1.73205 − 1.00000i 1.50000 − 2.59808i 2.00000 + 3.46410i 18.0007i −5.19615 + 3.00000i 6.06218 − 3.50000i − 8.00000i −4.50000 − 7.79423i 18.0007 − 31.1782i
43.7 1.73205 + 1.00000i 1.50000 − 2.59808i 2.00000 + 3.46410i − 22.1146i 5.19615 − 3.00000i −6.06218 + 3.50000i 8.00000i −4.50000 − 7.79423i 22.1146 − 38.3036i
43.8 1.73205 + 1.00000i 1.50000 − 2.59808i 2.00000 + 3.46410i − 3.23892i 5.19615 − 3.00000i −6.06218 + 3.50000i 8.00000i −4.50000 − 7.79423i 3.23892 − 5.60997i
43.9 1.73205 + 1.00000i 1.50000 − 2.59808i 2.00000 + 3.46410i − 0.0691052i 5.19615 − 3.00000i −6.06218 + 3.50000i 8.00000i −4.50000 − 7.79423i 0.0691052 − 0.119694i
43.10 1.73205 + 1.00000i 1.50000 − 2.59808i 2.00000 + 3.46410i − 1.79706i 5.19615 − 3.00000i −6.06218 + 3.50000i 8.00000i −4.50000 − 7.79423i 1.79706 − 3.11260i
43.11 1.73205 + 1.00000i 1.50000 − 2.59808i 2.00000 + 3.46410i 5.47106i 5.19615 − 3.00000i −6.06218 + 3.50000i 8.00000i −4.50000 − 7.79423i −5.47106 + 9.47616i
43.12 1.73205 + 1.00000i 1.50000 − 2.59808i 2.00000 + 3.46410i 19.0883i 5.19615 − 3.00000i −6.06218 + 3.50000i 8.00000i −4.50000 − 7.79423i −19.0883 + 33.0620i
127.1 −1.73205 + 1.00000i 1.50000 + 2.59808i 2.00000 − 3.46410i 20.8505i −5.19615 − 3.00000i 6.06218 + 3.50000i 8.00000i −4.50000 + 7.79423i −20.8505 − 36.1141i
127.2 −1.73205 + 1.00000i 1.50000 + 2.59808i 2.00000 − 3.46410i 9.97677i −5.19615 − 3.00000i 6.06218 + 3.50000i 8.00000i −4.50000 + 7.79423i −9.97677 − 17.2803i
127.3 −1.73205 + 1.00000i 1.50000 + 2.59808i 2.00000 − 3.46410i 10.6352i −5.19615 − 3.00000i 6.06218 + 3.50000i 8.00000i −4.50000 + 7.79423i −10.6352 − 18.4208i
127.4 −1.73205 + 1.00000i 1.50000 + 2.59808i 2.00000 − 3.46410i 2.48320i −5.19615 − 3.00000i 6.06218 + 3.50000i 8.00000i −4.50000 + 7.79423i −2.48320 − 4.30102i
127.5 −1.73205 + 1.00000i 1.50000 + 2.59808i 2.00000 − 3.46410i − 11.2847i −5.19615 − 3.00000i 6.06218 + 3.50000i 8.00000i −4.50000 + 7.79423i 11.2847 + 19.5457i
127.6 −1.73205 + 1.00000i 1.50000 + 2.59808i 2.00000 − 3.46410i − 18.0007i −5.19615 − 3.00000i 6.06218 + 3.50000i 8.00000i −4.50000 + 7.79423i 18.0007 + 31.1782i
127.7 1.73205 − 1.00000i 1.50000 + 2.59808i 2.00000 − 3.46410i 22.1146i 5.19615 + 3.00000i −6.06218 − 3.50000i − 8.00000i −4.50000 + 7.79423i 22.1146 + 38.3036i
127.8 1.73205 − 1.00000i 1.50000 + 2.59808i 2.00000 − 3.46410i 3.23892i 5.19615 + 3.00000i −6.06218 − 3.50000i − 8.00000i −4.50000 + 7.79423i 3.23892 + 5.60997i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.4.s.d ✓ 24
13.e even 6 1 inner 546.4.s.d ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.4.s.d ✓ 24 1.a even 1 1 trivial
546.4.s.d ✓ 24 13.e even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} + 2002 T_{5}^{22} + 1651051 T_{5}^{20} + 726081706 T_{5}^{18} + 184680126175 T_{5}^{16} + \cdots + 10\!\cdots\!64 \) acting on \(S_{4}^{\mathrm{new}}(546, [\chi])\). Copy content Toggle raw display