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Properties

Label 714.2.ba.e
Level $714$
Weight $2$
Character orbit 714.ba
Analytic conductor $5.701$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [714,2,Mod(319,714)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(714, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 8, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("714.319");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 714 = 2 \cdot 3 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 714.ba (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: \(5.70131870432\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 12 q^{4} + 12 q^{5} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 12 q^{4} + 12 q^{5} - 8 q^{7} + 12 q^{10} + 8 q^{11} + 8 q^{13} - 4 q^{14} - 12 q^{16} + 20 q^{17} - 12 q^{18} + 24 q^{20} - 4 q^{21} + 16 q^{22} - 4 q^{28} + 24 q^{29} + 12 q^{30} - 4 q^{31} + 8 q^{33} + 8 q^{34} - 4 q^{35} + 20 q^{37} - 8 q^{38} - 4 q^{39} - 12 q^{40} + 32 q^{41} - 8 q^{44} - 12 q^{45} - 12 q^{47} - 48 q^{50} - 4 q^{51} + 4 q^{52} + 48 q^{55} - 8 q^{56} - 12 q^{58} + 24 q^{61} + 8 q^{62} + 4 q^{63} - 24 q^{64} + 8 q^{65} + 16 q^{67} - 20 q^{68} - 80 q^{69} + 48 q^{71} + 12 q^{72} - 32 q^{73} + 20 q^{74} + 24 q^{75} + 8 q^{78} + 12 q^{80} + 12 q^{81} + 16 q^{82} + 4 q^{84} + 40 q^{85} - 8 q^{86} + 8 q^{88} - 32 q^{89} - 24 q^{90} + 72 q^{91} - 8 q^{95} - 48 q^{97} + 20 q^{98} - 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
319.1 −0.866025 + 0.500000i −0.965926 + 0.258819i 0.500000 − 0.866025i −0.624844 + 2.33195i 0.707107 − 0.707107i −2.04649 + 1.67687i 1.00000i 0.866025 − 0.500000i −0.624844 − 2.33195i
319.2 −0.866025 + 0.500000i −0.965926 + 0.258819i 0.500000 − 0.866025i −0.624844 + 2.33195i 0.707107 − 0.707107i −0.609074 − 2.57469i 1.00000i 0.866025 − 0.500000i −0.624844 − 2.33195i
319.3 −0.866025 + 0.500000i −0.965926 + 0.258819i 0.500000 − 0.866025i −0.624844 + 2.33195i 0.707107 − 0.707107i 2.36267 + 1.19071i 1.00000i 0.866025 − 0.500000i −0.624844 − 2.33195i
319.4 −0.866025 + 0.500000i 0.965926 − 0.258819i 0.500000 − 0.866025i −0.107206 + 0.400100i −0.707107 + 0.707107i −2.44445 + 1.01225i 1.00000i 0.866025 − 0.500000i −0.107206 − 0.400100i
319.5 −0.866025 + 0.500000i 0.965926 − 0.258819i 0.500000 − 0.866025i −0.107206 + 0.400100i −0.707107 + 0.707107i −1.41228 + 2.23729i 1.00000i 0.866025 − 0.500000i −0.107206 − 0.400100i
319.6 −0.866025 + 0.500000i 0.965926 − 0.258819i 0.500000 − 0.866025i −0.107206 + 0.400100i −0.707107 + 0.707107i 2.14963 − 1.54244i 1.00000i 0.866025 − 0.500000i −0.107206 − 0.400100i
361.1 0.866025 − 0.500000i −0.258819 − 0.965926i 0.500000 − 0.866025i 0.400100 + 0.107206i −0.707107 − 0.707107i −2.23729 − 1.41228i − 1.00000i −0.866025 + 0.500000i 0.400100 − 0.107206i
361.2 0.866025 − 0.500000i −0.258819 − 0.965926i 0.500000 − 0.866025i 0.400100 + 0.107206i −0.707107 − 0.707107i −1.01225 − 2.44445i − 1.00000i −0.866025 + 0.500000i 0.400100 − 0.107206i
361.3 0.866025 − 0.500000i −0.258819 − 0.965926i 0.500000 − 0.866025i 0.400100 + 0.107206i −0.707107 − 0.707107i 1.54244 + 2.14963i − 1.00000i −0.866025 + 0.500000i 0.400100 − 0.107206i
361.4 0.866025 − 0.500000i 0.258819 + 0.965926i 0.500000 − 0.866025i 2.33195 + 0.624844i 0.707107 + 0.707107i −1.67687 − 2.04649i − 1.00000i −0.866025 + 0.500000i 2.33195 − 0.624844i
361.5 0.866025 − 0.500000i 0.258819 + 0.965926i 0.500000 − 0.866025i 2.33195 + 0.624844i 0.707107 + 0.707107i −1.19071 + 2.36267i − 1.00000i −0.866025 + 0.500000i 2.33195 − 0.624844i
361.6 0.866025 − 0.500000i 0.258819 + 0.965926i 0.500000 − 0.866025i 2.33195 + 0.624844i 0.707107 + 0.707107i 2.57469 − 0.609074i − 1.00000i −0.866025 + 0.500000i 2.33195 − 0.624844i
625.1 0.866025 + 0.500000i −0.258819 + 0.965926i 0.500000 + 0.866025i 0.400100 − 0.107206i −0.707107 + 0.707107i −2.23729 + 1.41228i 1.00000i −0.866025 − 0.500000i 0.400100 + 0.107206i
625.2 0.866025 + 0.500000i −0.258819 + 0.965926i 0.500000 + 0.866025i 0.400100 − 0.107206i −0.707107 + 0.707107i −1.01225 + 2.44445i 1.00000i −0.866025 − 0.500000i 0.400100 + 0.107206i
625.3 0.866025 + 0.500000i −0.258819 + 0.965926i 0.500000 + 0.866025i 0.400100 − 0.107206i −0.707107 + 0.707107i 1.54244 − 2.14963i 1.00000i −0.866025 − 0.500000i 0.400100 + 0.107206i
625.4 0.866025 + 0.500000i 0.258819 − 0.965926i 0.500000 + 0.866025i 2.33195 − 0.624844i 0.707107 − 0.707107i −1.67687 + 2.04649i 1.00000i −0.866025 − 0.500000i 2.33195 + 0.624844i
625.5 0.866025 + 0.500000i 0.258819 − 0.965926i 0.500000 + 0.866025i 2.33195 − 0.624844i 0.707107 − 0.707107i −1.19071 − 2.36267i 1.00000i −0.866025 − 0.500000i 2.33195 + 0.624844i
625.6 0.866025 + 0.500000i 0.258819 − 0.965926i 0.500000 + 0.866025i 2.33195 − 0.624844i 0.707107 − 0.707107i 2.57469 + 0.609074i 1.00000i −0.866025 − 0.500000i 2.33195 + 0.624844i
667.1 −0.866025 − 0.500000i −0.965926 − 0.258819i 0.500000 + 0.866025i −0.624844 − 2.33195i 0.707107 + 0.707107i −2.04649 − 1.67687i − 1.00000i 0.866025 + 0.500000i −0.624844 + 2.33195i
667.2 −0.866025 − 0.500000i −0.965926 − 0.258819i 0.500000 + 0.866025i −0.624844 − 2.33195i 0.707107 + 0.707107i −0.609074 + 2.57469i − 1.00000i 0.866025 + 0.500000i −0.624844 + 2.33195i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 319.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
17.c even 4 1 inner
119.n even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 714.2.ba.e ✓ 24
7.c even 3 1 inner 714.2.ba.e ✓ 24
17.c even 4 1 inner 714.2.ba.e ✓ 24
119.n even 12 1 inner 714.2.ba.e ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
714.2.ba.e ✓ 24 1.a even 1 1 trivial
714.2.ba.e ✓ 24 7.c even 3 1 inner
714.2.ba.e ✓ 24 17.c even 4 1 inner
714.2.ba.e ✓ 24 119.n even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 4T_{5}^{7} + 8T_{5}^{6} - 24T_{5}^{5} + 47T_{5}^{4} - 24T_{5}^{3} + 8T_{5}^{2} - 4T_{5} + 1 \) acting on \(S_{2}^{\mathrm{new}}(714, [\chi])\). Copy content Toggle raw display