Properties

Label 546.8.c.a
Level $546$
Weight $8$
Character orbit 546.c
Analytic conductor $170.562$
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] = [546,8,Mod(337,546)]
 
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, 1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("546.337");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 546.c (of order \(2\), degree \(1\), 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: \(170.562223914\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 648 q^{3} - 1536 q^{4} + 17496 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 648 q^{3} - 1536 q^{4} + 17496 q^{9} + 1520 q^{10} + 41472 q^{12} + 13718 q^{13} + 65856 q^{14} + 98304 q^{16} + 13586 q^{17} - 11824 q^{22} + 76574 q^{23} - 357850 q^{25} + 132848 q^{26} - 472392 q^{27} + 85198 q^{29} - 41040 q^{30} - 65170 q^{35} - 1119744 q^{36} + 295696 q^{38} - 370386 q^{39} - 97280 q^{40} - 1778112 q^{42} - 91370 q^{43} - 2654208 q^{48} - 2823576 q^{49} - 366822 q^{51} - 877952 q^{52} - 360000 q^{53} - 2519718 q^{55} - 4214784 q^{56} + 874734 q^{61} + 1984480 q^{62} - 6291456 q^{64} - 12450388 q^{65} + 319248 q^{66} - 869504 q^{68} - 2067498 q^{69} - 1405616 q^{74} + 9661950 q^{75} + 506954 q^{77} - 3586896 q^{78} - 2141176 q^{79} + 12754584 q^{81} + 14087488 q^{82} - 2300346 q^{87} + 756736 q^{88} + 1108080 q^{90} - 5695858 q^{91} - 4900736 q^{92} + 8123808 q^{94} + 19191246 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1 8.00000i −27.0000 −64.0000 438.711i 216.000i 343.000i 512.000i 729.000 −3509.69
337.2 8.00000i −27.0000 −64.0000 367.879i 216.000i 343.000i 512.000i 729.000 −2943.03
337.3 8.00000i −27.0000 −64.0000 309.586i 216.000i 343.000i 512.000i 729.000 −2476.69
337.4 8.00000i −27.0000 −64.0000 268.352i 216.000i 343.000i 512.000i 729.000 −2146.82
337.5 8.00000i −27.0000 −64.0000 208.684i 216.000i 343.000i 512.000i 729.000 −1669.47
337.6 8.00000i −27.0000 −64.0000 40.5290i 216.000i 343.000i 512.000i 729.000 324.232
337.7 8.00000i −27.0000 −64.0000 96.2247i 216.000i 343.000i 512.000i 729.000 769.797
337.8 8.00000i −27.0000 −64.0000 144.475i 216.000i 343.000i 512.000i 729.000 1155.80
337.9 8.00000i −27.0000 −64.0000 176.597i 216.000i 343.000i 512.000i 729.000 1412.78
337.10 8.00000i −27.0000 −64.0000 342.315i 216.000i 343.000i 512.000i 729.000 2738.52
337.11 8.00000i −27.0000 −64.0000 407.004i 216.000i 343.000i 512.000i 729.000 3256.03
337.12 8.00000i −27.0000 −64.0000 481.066i 216.000i 343.000i 512.000i 729.000 3848.53
337.13 8.00000i −27.0000 −64.0000 481.066i 216.000i 343.000i 512.000i 729.000 3848.53
337.14 8.00000i −27.0000 −64.0000 407.004i 216.000i 343.000i 512.000i 729.000 3256.03
337.15 8.00000i −27.0000 −64.0000 342.315i 216.000i 343.000i 512.000i 729.000 2738.52
337.16 8.00000i −27.0000 −64.0000 176.597i 216.000i 343.000i 512.000i 729.000 1412.78
337.17 8.00000i −27.0000 −64.0000 144.475i 216.000i 343.000i 512.000i 729.000 1155.80
337.18 8.00000i −27.0000 −64.0000 96.2247i 216.000i 343.000i 512.000i 729.000 769.797
337.19 8.00000i −27.0000 −64.0000 40.5290i 216.000i 343.000i 512.000i 729.000 324.232
337.20 8.00000i −27.0000 −64.0000 208.684i 216.000i 343.000i 512.000i 729.000 −1669.47
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.8.c.a 24
13.b even 2 1 inner 546.8.c.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.8.c.a 24 1.a even 1 1 trivial
546.8.c.a 24 13.b even 2 1 inner