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Properties

Label 1638.2.cf.d
Level $1638$
Weight $2$
Character orbit 1638.cf
Analytic conductor $13.079$
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] = [1638,2,Mod(521,1638)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1638, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1638.521");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1638.cf (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: \(13.0794958511\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 12 q^{4} + 4 q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 12 q^{4} + 4 q^{5} - 4 q^{7} - 12 q^{11} + 8 q^{14} - 12 q^{16} - 12 q^{19} + 8 q^{20} + 12 q^{23} - 8 q^{25} - 12 q^{26} + 4 q^{28} + 12 q^{31} + 4 q^{35} + 20 q^{37} + 20 q^{38} + 8 q^{41} - 16 q^{43} - 12 q^{44} - 12 q^{46} - 16 q^{47} + 4 q^{49} - 60 q^{53} + 16 q^{56} - 24 q^{58} + 4 q^{59} + 12 q^{61} + 32 q^{62} - 24 q^{64} - 20 q^{67} + 28 q^{70} + 60 q^{73} - 12 q^{74} + 128 q^{77} + 12 q^{79} + 4 q^{80} - 12 q^{82} + 32 q^{83} + 72 q^{85} + 12 q^{89} + 16 q^{91} - 156 q^{95} + 32 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
521.1 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −1.73119 − 2.99850i 0 −2.54567 − 0.720790i 1.00000i 0 2.99850 + 1.73119i
521.2 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −0.954776 − 1.65372i 0 1.84503 − 1.89628i 1.00000i 0 1.65372 + 0.954776i
521.3 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −0.0316635 − 0.0548427i 0 0.562976 + 2.58516i 1.00000i 0 0.0548427 + 0.0316635i
521.4 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 0.194984 + 0.337722i 0 −1.05885 − 2.42463i 1.00000i 0 −0.337722 − 0.194984i
521.5 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 1.56142 + 2.70447i 0 1.92773 − 1.81214i 1.00000i 0 −2.70447 − 1.56142i
521.6 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 1.96122 + 3.39693i 0 −1.73120 + 2.00073i 1.00000i 0 −3.39693 − 1.96122i
521.7 0.866025 − 0.500000i 0 0.500000 − 0.866025i −1.54739 − 2.68016i 0 −2.58114 − 0.581114i − 1.00000i 0 −2.68016 − 1.54739i
521.8 0.866025 − 0.500000i 0 0.500000 − 0.866025i 0.891798 + 1.54464i 0 2.45996 − 0.973964i − 1.00000i 0 1.54464 + 0.891798i
521.9 0.866025 − 0.500000i 0 0.500000 − 0.866025i −0.461498 − 0.799337i 0 0.389745 + 2.61689i − 1.00000i 0 −0.799337 − 0.461498i
521.10 0.866025 − 0.500000i 0 0.500000 − 0.866025i 0.396474 + 0.686713i 0 −2.55519 + 0.686287i − 1.00000i 0 0.686713 + 0.396474i
521.11 0.866025 − 0.500000i 0 0.500000 − 0.866025i −0.0691516 − 0.119774i 0 −0.896452 + 2.48925i − 1.00000i 0 −0.119774 − 0.0691516i
521.12 0.866025 − 0.500000i 0 0.500000 − 0.866025i 1.78977 + 3.09997i 0 2.18309 + 1.49470i − 1.00000i 0 3.09997 + 1.78977i
1223.1 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −1.73119 + 2.99850i 0 −2.54567 + 0.720790i − 1.00000i 0 2.99850 − 1.73119i
1223.2 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −0.954776 + 1.65372i 0 1.84503 + 1.89628i − 1.00000i 0 1.65372 − 0.954776i
1223.3 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −0.0316635 + 0.0548427i 0 0.562976 − 2.58516i − 1.00000i 0 0.0548427 − 0.0316635i
1223.4 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 0.194984 − 0.337722i 0 −1.05885 + 2.42463i − 1.00000i 0 −0.337722 + 0.194984i
1223.5 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 1.56142 − 2.70447i 0 1.92773 + 1.81214i − 1.00000i 0 −2.70447 + 1.56142i
1223.6 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 1.96122 − 3.39693i 0 −1.73120 − 2.00073i − 1.00000i 0 −3.39693 + 1.96122i
1223.7 0.866025 + 0.500000i 0 0.500000 + 0.866025i −1.54739 + 2.68016i 0 −2.58114 + 0.581114i 1.00000i 0 −2.68016 + 1.54739i
1223.8 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.891798 − 1.54464i 0 2.45996 + 0.973964i 1.00000i 0 1.54464 − 0.891798i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 521.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1638.2.cf.d yes 24
3.b odd 2 1 1638.2.cf.c ✓ 24
7.d odd 6 1 1638.2.cf.c ✓ 24
21.g even 6 1 inner 1638.2.cf.d yes 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1638.2.cf.c ✓ 24 3.b odd 2 1
1638.2.cf.c ✓ 24 7.d odd 6 1
1638.2.cf.d yes 24 1.a even 1 1 trivial
1638.2.cf.d yes 24 21.g even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} - 4 T_{5}^{23} + 42 T_{5}^{22} - 104 T_{5}^{21} + 879 T_{5}^{20} - 1852 T_{5}^{19} + \cdots + 16 \) acting on \(S_{2}^{\mathrm{new}}(1638, [\chi])\). Copy content Toggle raw display