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Properties

Label 1344.4.b.i
Level $1344$
Weight $4$
Character orbit 1344.b
Analytic conductor $79.299$
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] = [1344,4,Mod(895,1344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1344, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1344.895");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1344.b (of order \(2\), degree \(1\), not 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: \(79.2985670477\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 672)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 72 q^{3} + 20 q^{7} + 216 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 72 q^{3} + 20 q^{7} + 216 q^{9} + 56 q^{19} - 60 q^{21} - 432 q^{25} - 648 q^{27} - 464 q^{31} - 568 q^{35} - 504 q^{37} - 560 q^{47} - 128 q^{49} + 784 q^{53} - 424 q^{55} - 168 q^{57} - 800 q^{59} + 180 q^{63} + 560 q^{65} + 1296 q^{75} + 1568 q^{77} + 1944 q^{81} - 1936 q^{83} - 3000 q^{85} + 496 q^{91} + 1392 q^{93}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
895.1 0 −3.00000 0 − 19.8256i 0 −13.1775 − 13.0136i 0 9.00000 0
895.2 0 −3.00000 0 − 15.7575i 0 −15.0622 + 10.7764i 0 9.00000 0
895.3 0 −3.00000 0 − 15.0520i 0 17.2151 − 6.82941i 0 9.00000 0
895.4 0 −3.00000 0 − 14.9540i 0 18.3556 + 2.46445i 0 9.00000 0
895.5 0 −3.00000 0 − 13.1053i 0 9.87442 − 15.6683i 0 9.00000 0
895.6 0 −3.00000 0 − 12.7173i 0 2.54263 + 18.3449i 0 9.00000 0
895.7 0 −3.00000 0 − 12.4907i 0 −8.98608 − 16.1941i 0 9.00000 0
895.8 0 −3.00000 0 − 8.63807i 0 18.2735 − 3.01325i 0 9.00000 0
895.9 0 −3.00000 0 − 5.77465i 0 3.77600 + 18.1312i 0 9.00000 0
895.10 0 −3.00000 0 − 4.15525i 0 −10.5863 − 15.1964i 0 9.00000 0
895.11 0 −3.00000 0 − 2.26482i 0 5.56016 + 17.6659i 0 9.00000 0
895.12 0 −3.00000 0 − 2.14586i 0 −17.7853 − 5.16554i 0 9.00000 0
895.13 0 −3.00000 0 2.14586i 0 −17.7853 + 5.16554i 0 9.00000 0
895.14 0 −3.00000 0 2.26482i 0 5.56016 − 17.6659i 0 9.00000 0
895.15 0 −3.00000 0 4.15525i 0 −10.5863 + 15.1964i 0 9.00000 0
895.16 0 −3.00000 0 5.77465i 0 3.77600 − 18.1312i 0 9.00000 0
895.17 0 −3.00000 0 8.63807i 0 18.2735 + 3.01325i 0 9.00000 0
895.18 0 −3.00000 0 12.4907i 0 −8.98608 + 16.1941i 0 9.00000 0
895.19 0 −3.00000 0 12.7173i 0 2.54263 − 18.3449i 0 9.00000 0
895.20 0 −3.00000 0 13.1053i 0 9.87442 + 15.6683i 0 9.00000 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 895.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1344.4.b.i 24
4.b odd 2 1 1344.4.b.j 24
7.b odd 2 1 1344.4.b.j 24
8.b even 2 1 672.4.b.b yes 24
8.d odd 2 1 672.4.b.a ✓ 24
28.d even 2 1 inner 1344.4.b.i 24
56.e even 2 1 672.4.b.b yes 24
56.h odd 2 1 672.4.b.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.4.b.a ✓ 24 8.d odd 2 1
672.4.b.a ✓ 24 56.h odd 2 1
672.4.b.b yes 24 8.b even 2 1
672.4.b.b yes 24 56.e even 2 1
1344.4.b.i 24 1.a even 1 1 trivial
1344.4.b.i 24 28.d even 2 1 inner
1344.4.b.j 24 4.b odd 2 1
1344.4.b.j 24 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1344, [\chi])\):

\( T_{5}^{24} + 1716 T_{5}^{22} + 1270076 T_{5}^{20} + 532572832 T_{5}^{18} + 139508402672 T_{5}^{16} + \cdots + 21\!\cdots\!24 \) Copy content Toggle raw display
\( T_{19}^{12} - 28 T_{19}^{11} - 46324 T_{19}^{10} + 1990400 T_{19}^{9} + 692579632 T_{19}^{8} + \cdots + 80\!\cdots\!08 \) Copy content Toggle raw display