Properties

Label 1568.3.h.a
Level 1568
Weight 3
Character orbit 1568.h
Analytic conductor 42.725
Analytic rank 0
Dimension 28
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 1568 = 2^{5} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1568.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(42.7249054517\)
Analytic rank: \(0\)
Dimension: \(28\)
Twist minimal: no (minimal twist has level 56)
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.

\(\operatorname{Tr}(f)(q) = \) \( 28q + 64q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 28q + 64q^{9} - 28q^{15} + 60q^{23} + 64q^{25} - 40q^{39} + 124q^{57} + 104q^{65} + 136q^{71} - 324q^{79} + 36q^{81} + 580q^{95} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
881.1 0 −5.56010 0 −3.05841 0 0 0 21.9148 0
881.2 0 −5.56010 0 −3.05841 0 0 0 21.9148 0
881.3 0 −3.89635 0 8.85969 0 0 0 6.18155 0
881.4 0 −3.89635 0 8.85969 0 0 0 6.18155 0
881.5 0 −3.86988 0 4.67764 0 0 0 5.97596 0
881.6 0 −3.86988 0 4.67764 0 0 0 5.97596 0
881.7 0 −3.40276 0 −4.31716 0 0 0 2.57876 0
881.8 0 −3.40276 0 −4.31716 0 0 0 2.57876 0
881.9 0 −2.33563 0 −3.10110 0 0 0 −3.54484 0
881.10 0 −2.33563 0 −3.10110 0 0 0 −3.54484 0
881.11 0 −0.910863 0 −6.34503 0 0 0 −8.17033 0
881.12 0 −0.910863 0 −6.34503 0 0 0 −8.17033 0
881.13 0 −0.253256 0 −3.57178 0 0 0 −8.93586 0
881.14 0 −0.253256 0 −3.57178 0 0 0 −8.93586 0
881.15 0 0.253256 0 3.57178 0 0 0 −8.93586 0
881.16 0 0.253256 0 3.57178 0 0 0 −8.93586 0
881.17 0 0.910863 0 6.34503 0 0 0 −8.17033 0
881.18 0 0.910863 0 6.34503 0 0 0 −8.17033 0
881.19 0 2.33563 0 3.10110 0 0 0 −3.54484 0
881.20 0 2.33563 0 3.10110 0 0 0 −3.54484 0
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 881.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
8.b even 2 1 inner
56.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1568.3.h.a 28
4.b odd 2 1 392.3.h.a 28
7.b odd 2 1 inner 1568.3.h.a 28
7.c even 3 1 224.3.n.a 28
7.d odd 6 1 224.3.n.a 28
8.b even 2 1 inner 1568.3.h.a 28
8.d odd 2 1 392.3.h.a 28
28.d even 2 1 392.3.h.a 28
28.f even 6 1 56.3.j.a 28
28.f even 6 1 392.3.j.e 28
28.g odd 6 1 56.3.j.a 28
28.g odd 6 1 392.3.j.e 28
56.e even 2 1 392.3.h.a 28
56.h odd 2 1 inner 1568.3.h.a 28
56.j odd 6 1 224.3.n.a 28
56.k odd 6 1 56.3.j.a 28
56.k odd 6 1 392.3.j.e 28
56.m even 6 1 56.3.j.a 28
56.m even 6 1 392.3.j.e 28
56.p even 6 1 224.3.n.a 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.j.a 28 28.f even 6 1
56.3.j.a 28 28.g odd 6 1
56.3.j.a 28 56.k odd 6 1
56.3.j.a 28 56.m even 6 1
224.3.n.a 28 7.c even 3 1
224.3.n.a 28 7.d odd 6 1
224.3.n.a 28 56.j odd 6 1
224.3.n.a 28 56.p even 6 1
392.3.h.a 28 4.b odd 2 1
392.3.h.a 28 8.d odd 2 1
392.3.h.a 28 28.d even 2 1
392.3.h.a 28 56.e even 2 1
392.3.j.e 28 28.f even 6 1
392.3.j.e 28 28.g odd 6 1
392.3.j.e 28 56.k odd 6 1
392.3.j.e 28 56.m even 6 1
1568.3.h.a 28 1.a even 1 1 trivial
1568.3.h.a 28 7.b odd 2 1 inner
1568.3.h.a 28 8.b even 2 1 inner
1568.3.h.a 28 56.h odd 2 1 inner

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database