Properties

Label 1344.4.b.j
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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Newspace parameters

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

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.

\(\operatorname{Tr}(f)(q) = \) \( 24q + 72q^{3} - 20q^{7} + 216q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 24q + 72q^{3} - 20q^{7} + 216q^{9} - 56q^{19} - 60q^{21} - 432q^{25} + 648q^{27} + 464q^{31} + 568q^{35} - 504q^{37} + 560q^{47} - 128q^{49} + 784q^{53} + 424q^{55} - 168q^{57} + 800q^{59} - 180q^{63} + 560q^{65} - 1296q^{75} + 1568q^{77} + 1944q^{81} + 1936q^{83} - 3000q^{85} - 496q^{91} + 1392q^{93} + O(q^{100}) \)

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.

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

Hecke kernels

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

\(23\!\cdots\!40\)\( T_{5}^{14} + \)\(26\!\cdots\!24\)\( T_{5}^{12} + \)\(18\!\cdots\!12\)\( T_{5}^{10} + \)\(79\!\cdots\!96\)\( T_{5}^{8} + \)\(18\!\cdots\!88\)\( T_{5}^{6} + \)\(22\!\cdots\!84\)\( T_{5}^{4} + \)\(11\!\cdots\!56\)\( T_{5}^{2} + \)\(21\!\cdots\!24\)\( \)">\(T_{5}^{24} + \cdots\)
\(32\!\cdots\!16\)\( T_{19}^{6} - \)\(23\!\cdots\!32\)\( T_{19}^{5} + \)\(82\!\cdots\!96\)\( T_{19}^{4} + \)\(28\!\cdots\!40\)\( T_{19}^{3} + \)\(59\!\cdots\!48\)\( T_{19}^{2} + \)\(39\!\cdots\!20\)\( T_{19} + \)\(80\!\cdots\!08\)\( \)">\(T_{19}^{12} + \cdots\)

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database