# 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 $2$

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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