Properties

Label 224.3.n.a
Level 224
Weight 3
Character orbit 224.n
Analytic conductor 6.104
Analytic rank 0
Dimension 28
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 224.n (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.10355792167\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) over \(\Q(\zeta_{6})\)
Coefficient ring index: multiple of None
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 28q + 4q^{7} - 32q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 28q + 4q^{7} - 32q^{9} - 28q^{15} - 6q^{17} - 30q^{23} - 32q^{25} + 6q^{31} - 6q^{33} + 20q^{39} + 294q^{47} - 20q^{49} + 124q^{57} - 432q^{63} - 52q^{65} + 136q^{71} + 234q^{73} + 162q^{79} - 18q^{81} - 48q^{87} - 150q^{89} - 290q^{95} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 0 −2.78005 + 4.81519i 0 −1.52921 2.64866i 0 −0.608243 6.97352i 0 −10.9574 18.9787i 0
17.2 0 −1.94818 + 3.37434i 0 4.42985 + 7.67272i 0 6.92329 + 1.03347i 0 −3.09078 5.35338i 0
17.3 0 −1.93494 + 3.35141i 0 2.33882 + 4.05096i 0 −6.95505 + 0.792023i 0 −2.98798 5.17534i 0
17.4 0 −1.70138 + 2.94687i 0 −2.15858 3.73877i 0 1.43197 + 6.85197i 0 −1.28938 2.23327i 0
17.5 0 −1.16781 + 2.02271i 0 −1.55055 2.68563i 0 6.89374 1.21502i 0 1.77242 + 3.06992i 0
17.6 0 −0.455431 + 0.788830i 0 −3.17251 5.49495i 0 −3.79106 + 5.88455i 0 4.08516 + 7.07571i 0
17.7 0 −0.126628 + 0.219326i 0 −1.78589 3.09325i 0 −2.89466 6.37346i 0 4.46793 + 7.73868i 0
17.8 0 0.126628 0.219326i 0 1.78589 + 3.09325i 0 −2.89466 6.37346i 0 4.46793 + 7.73868i 0
17.9 0 0.455431 0.788830i 0 3.17251 + 5.49495i 0 −3.79106 + 5.88455i 0 4.08516 + 7.07571i 0
17.10 0 1.16781 2.02271i 0 1.55055 + 2.68563i 0 6.89374 1.21502i 0 1.77242 + 3.06992i 0
17.11 0 1.70138 2.94687i 0 2.15858 + 3.73877i 0 1.43197 + 6.85197i 0 −1.28938 2.23327i 0
17.12 0 1.93494 3.35141i 0 −2.33882 4.05096i 0 −6.95505 + 0.792023i 0 −2.98798 5.17534i 0
17.13 0 1.94818 3.37434i 0 −4.42985 7.67272i 0 6.92329 + 1.03347i 0 −3.09078 5.35338i 0
17.14 0 2.78005 4.81519i 0 1.52921 + 2.64866i 0 −0.608243 6.97352i 0 −10.9574 18.9787i 0
145.1 0 −2.78005 4.81519i 0 −1.52921 + 2.64866i 0 −0.608243 + 6.97352i 0 −10.9574 + 18.9787i 0
145.2 0 −1.94818 3.37434i 0 4.42985 7.67272i 0 6.92329 1.03347i 0 −3.09078 + 5.35338i 0
145.3 0 −1.93494 3.35141i 0 2.33882 4.05096i 0 −6.95505 0.792023i 0 −2.98798 + 5.17534i 0
145.4 0 −1.70138 2.94687i 0 −2.15858 + 3.73877i 0 1.43197 6.85197i 0 −1.28938 + 2.23327i 0
145.5 0 −1.16781 2.02271i 0 −1.55055 + 2.68563i 0 6.89374 + 1.21502i 0 1.77242 3.06992i 0
145.6 0 −0.455431 0.788830i 0 −3.17251 + 5.49495i 0 −3.79106 5.88455i 0 4.08516 7.07571i 0
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 145.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
8.b even 2 1 inner
56.j odd 6 1 inner

Twists

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

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(224, [\chi])\).

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database