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Properties

Label 600.2.bc.c
Level $600$
Weight $2$
Character orbit 600.bc
Analytic conductor $4.791$
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] = [600,2,Mod(169,600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(600, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("600.169");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 600.bc (of order \(10\), 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: \(4.79102412128\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 6 q^{9} + 4 q^{11} + 20 q^{13} + 20 q^{17} + 2 q^{19} + 2 q^{21} + 30 q^{23} - 20 q^{25} + 2 q^{29} + 18 q^{31} - 10 q^{33} + 20 q^{35} + 30 q^{37} - 4 q^{39} + 4 q^{41} - 10 q^{47} + 40 q^{49} + 12 q^{51} + 20 q^{55} + 16 q^{59} + 16 q^{61} - 10 q^{65} + 30 q^{67} - 14 q^{69} - 12 q^{71} - 40 q^{73} - 20 q^{75} - 60 q^{77} - 20 q^{79} - 6 q^{81} - 80 q^{83} + 20 q^{85} - 10 q^{87} + 20 q^{89} + 38 q^{91} - 30 q^{95} - 60 q^{97} + 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
169.1 0 −0.587785 − 0.809017i 0 −1.25610 − 1.84992i 0 4.48696i 0 −0.309017 + 0.951057i 0
169.2 0 −0.587785 − 0.809017i 0 −0.374438 + 2.20449i 0 − 1.00681i 0 −0.309017 + 0.951057i 0
169.3 0 −0.587785 − 0.809017i 0 1.63054 − 1.53014i 0 − 2.48015i 0 −0.309017 + 0.951057i 0
169.4 0 0.587785 + 0.809017i 0 −2.11138 − 0.736270i 0 1.66088i 0 −0.309017 + 0.951057i 0
169.5 0 0.587785 + 0.809017i 0 0.656658 − 2.13748i 0 − 2.52958i 0 −0.309017 + 0.951057i 0
169.6 0 0.587785 + 0.809017i 0 1.45472 + 1.69817i 0 − 0.131295i 0 −0.309017 + 0.951057i 0
289.1 0 −0.951057 − 0.309017i 0 −2.16755 + 0.549306i 0 0.939284i 0 0.809017 + 0.587785i 0
289.2 0 −0.951057 − 0.309017i 0 0.112113 + 2.23326i 0 − 1.23761i 0 0.809017 + 0.587785i 0
289.3 0 −0.951057 − 0.309017i 0 2.05543 − 0.880448i 0 − 0.701672i 0 0.809017 + 0.587785i 0
289.4 0 0.951057 + 0.309017i 0 −1.67047 + 1.48645i 0 − 3.67784i 0 0.809017 + 0.587785i 0
289.5 0 0.951057 + 0.309017i 0 0.336899 + 2.21054i 0 2.06150i 0 0.809017 + 0.587785i 0
289.6 0 0.951057 + 0.309017i 0 1.33357 − 1.79488i 0 2.61634i 0 0.809017 + 0.587785i 0
409.1 0 −0.951057 + 0.309017i 0 −2.16755 − 0.549306i 0 − 0.939284i 0 0.809017 − 0.587785i 0
409.2 0 −0.951057 + 0.309017i 0 0.112113 − 2.23326i 0 1.23761i 0 0.809017 − 0.587785i 0
409.3 0 −0.951057 + 0.309017i 0 2.05543 + 0.880448i 0 0.701672i 0 0.809017 − 0.587785i 0
409.4 0 0.951057 − 0.309017i 0 −1.67047 − 1.48645i 0 3.67784i 0 0.809017 − 0.587785i 0
409.5 0 0.951057 − 0.309017i 0 0.336899 − 2.21054i 0 − 2.06150i 0 0.809017 − 0.587785i 0
409.6 0 0.951057 − 0.309017i 0 1.33357 + 1.79488i 0 − 2.61634i 0 0.809017 − 0.587785i 0
529.1 0 −0.587785 + 0.809017i 0 −1.25610 + 1.84992i 0 − 4.48696i 0 −0.309017 − 0.951057i 0
529.2 0 −0.587785 + 0.809017i 0 −0.374438 − 2.20449i 0 1.00681i 0 −0.309017 − 0.951057i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 169.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 600.2.bc.c ✓ 24
25.e even 10 1 inner 600.2.bc.c ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.2.bc.c ✓ 24 1.a even 1 1 trivial
600.2.bc.c ✓ 24 25.e even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{24} + 64 T_{7}^{22} + 1676 T_{7}^{20} + 23734 T_{7}^{18} + 202006 T_{7}^{16} + 1081940 T_{7}^{14} + \cdots + 10000 \) acting on \(S_{2}^{\mathrm{new}}(600, [\chi])\). Copy content Toggle raw display