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Properties

Label 1428.2.cc.c
Level $1428$
Weight $2$
Character orbit 1428.cc
Analytic conductor $11.403$
Analytic rank $0$
Dimension $72$
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] = [1428,2,Mod(361,1428)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1428, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 0, 8, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1428.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1428 = 2^{2} \cdot 3 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1428.cc (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: \(11.4026374086\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(18\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 12 q^{5} - 20 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q + 12 q^{5} - 20 q^{7} + 8 q^{11} + 24 q^{13} - 16 q^{17} + 20 q^{21} - 28 q^{23} + 48 q^{29} - 28 q^{31} + 24 q^{33} + 68 q^{35} + 20 q^{37} + 32 q^{41} - 12 q^{45} + 44 q^{47} - 4 q^{51} + 48 q^{55} - 16 q^{61} + 16 q^{63} - 16 q^{65} - 16 q^{67} - 48 q^{69} + 8 q^{71} + 40 q^{73} + 24 q^{75} + 20 q^{79} + 36 q^{81} + 8 q^{85} - 16 q^{89} - 20 q^{91} - 68 q^{95} + 120 q^{97} - 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1 0 −0.258819 − 0.965926i 0 −2.11820 − 0.567569i 0 −1.17334 − 2.37135i 0 −0.866025 + 0.500000i 0
361.2 0 −0.258819 − 0.965926i 0 −1.98788 − 0.532652i 0 0.734227 + 2.54183i 0 −0.866025 + 0.500000i 0
361.3 0 −0.258819 − 0.965926i 0 −0.0637398 − 0.0170790i 0 −2.59449 − 0.518291i 0 −0.866025 + 0.500000i 0
361.4 0 −0.258819 − 0.965926i 0 0.509687 + 0.136570i 0 0.427202 − 2.61103i 0 −0.866025 + 0.500000i 0
361.5 0 −0.258819 − 0.965926i 0 1.53263 + 0.410668i 0 2.49728 − 0.873838i 0 −0.866025 + 0.500000i 0
361.6 0 −0.258819 − 0.965926i 0 −2.66869 − 0.715072i 0 −0.235760 + 2.63523i 0 −0.866025 + 0.500000i 0
361.7 0 −0.258819 − 0.965926i 0 3.00868 + 0.806174i 0 −2.22929 − 1.42487i 0 −0.866025 + 0.500000i 0
361.8 0 −0.258819 − 0.965926i 0 3.13761 + 0.840719i 0 1.57572 + 2.12535i 0 −0.866025 + 0.500000i 0
361.9 0 −0.258819 − 0.965926i 0 3.71390 + 0.995136i 0 −2.17810 + 1.50195i 0 −0.866025 + 0.500000i 0
361.10 0 0.258819 + 0.965926i 0 −1.42415 − 0.381600i 0 −1.51957 + 2.16585i 0 −0.866025 + 0.500000i 0
361.11 0 0.258819 + 0.965926i 0 −0.176417 − 0.0472707i 0 0.590075 − 2.57911i 0 −0.866025 + 0.500000i 0
361.12 0 0.258819 + 0.965926i 0 0.0547953 + 0.0146824i 0 2.60033 − 0.488127i 0 −0.866025 + 0.500000i 0
361.13 0 0.258819 + 0.965926i 0 1.70802 + 0.457664i 0 −1.88501 − 1.85654i 0 −0.866025 + 0.500000i 0
361.14 0 0.258819 + 0.965926i 0 1.94381 + 0.520841i 0 −2.48739 − 0.901603i 0 −0.866025 + 0.500000i 0
361.15 0 0.258819 + 0.965926i 0 −3.36399 − 0.901377i 0 −2.41732 + 1.07544i 0 −0.866025 + 0.500000i 0
361.16 0 0.258819 + 0.965926i 0 −3.71286 − 0.994859i 0 −1.50816 − 2.17381i 0 −0.866025 + 0.500000i 0
361.17 0 0.258819 + 0.965926i 0 3.98819 + 1.06863i 0 1.86963 − 1.87203i 0 −0.866025 + 0.500000i 0
361.18 0 0.258819 + 0.965926i 0 4.11475 + 1.10254i 0 1.20191 + 2.35699i 0 −0.866025 + 0.500000i 0
625.1 0 −0.258819 + 0.965926i 0 −2.11820 + 0.567569i 0 −1.17334 + 2.37135i 0 −0.866025 − 0.500000i 0
625.2 0 −0.258819 + 0.965926i 0 −1.98788 + 0.532652i 0 0.734227 − 2.54183i 0 −0.866025 − 0.500000i 0
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 361.18
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 1428.2.cc.c ✓ 72
7.c even 3 1 inner 1428.2.cc.c ✓ 72
17.c even 4 1 inner 1428.2.cc.c ✓ 72
119.n even 12 1 inner 1428.2.cc.c ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1428.2.cc.c ✓ 72 1.a even 1 1 trivial
1428.2.cc.c ✓ 72 7.c even 3 1 inner
1428.2.cc.c ✓ 72 17.c even 4 1 inner
1428.2.cc.c ✓ 72 119.n even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{72} - 12 T_{5}^{71} + 72 T_{5}^{70} - 424 T_{5}^{69} + 1723 T_{5}^{68} - 1868 T_{5}^{67} + \cdots + 13841287201 \) acting on \(S_{2}^{\mathrm{new}}(1428, [\chi])\). Copy content Toggle raw display