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Properties

Label 574.2.j.a
Level $574$
Weight $2$
Character orbit 574.j
Analytic conductor $4.583$
Analytic rank $0$
Dimension $28$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [574,2,Mod(81,574)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(574, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("574.81");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 574 = 2 \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 574.j (of order \(6\), degree \(2\), 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: \(4.58341307602\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 14 q^{2} - 14 q^{4} + 28 q^{8} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - 14 q^{2} - 14 q^{4} + 28 q^{8} + 16 q^{9} - 14 q^{16} + 16 q^{18} + 16 q^{21} - 10 q^{23} - 18 q^{25} + 6 q^{31} - 14 q^{32} - 10 q^{33} - 32 q^{36} + 24 q^{37} - 6 q^{39} - 32 q^{41} - 20 q^{42} + 44 q^{43} + 28 q^{45} - 10 q^{46} - 28 q^{49} + 36 q^{50} + 14 q^{51} - 20 q^{57} + 14 q^{59} - 36 q^{61} - 12 q^{62} + 28 q^{64} - 10 q^{66} + 16 q^{72} - 4 q^{73} + 24 q^{74} + 10 q^{77} + 12 q^{78} + 2 q^{81} + 16 q^{82} + 80 q^{83} + 4 q^{84} - 22 q^{86} - 44 q^{87} - 56 q^{90} - 24 q^{91} + 20 q^{92} + 8 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1 −0.500000 − 0.866025i −2.52339 − 1.45688i −0.500000 + 0.866025i −1.72861 − 2.99404i 2.91376i 0.765788 + 2.53250i 1.00000 2.74500 + 4.75449i −1.72861 + 2.99404i
81.2 −0.500000 − 0.866025i −2.34516 − 1.35398i −0.500000 + 0.866025i 0.815102 + 1.41180i 2.70795i 1.07408 + 2.41792i 1.00000 2.16650 + 3.75249i 0.815102 − 1.41180i
81.3 −0.500000 − 0.866025i −2.22935 − 1.28712i −0.500000 + 0.866025i 0.977882 + 1.69374i 2.57423i −2.55146 − 0.700038i 1.00000 1.81333 + 3.14079i 0.977882 − 1.69374i
81.4 −0.500000 − 0.866025i −1.96169 − 1.13258i −0.500000 + 0.866025i −0.773227 − 1.33927i 2.26516i 0.699483 − 2.55161i 1.00000 1.06549 + 1.84547i −0.773227 + 1.33927i
81.5 −0.500000 − 0.866025i −0.776228 − 0.448155i −0.500000 + 0.866025i −1.30685 − 2.26354i 0.896311i −2.59330 − 0.524221i 1.00000 −1.09831 − 1.90233i −1.30685 + 2.26354i
81.6 −0.500000 − 0.866025i −0.611065 − 0.352798i −0.500000 + 0.866025i −0.00555225 − 0.00961678i 0.705597i 1.50062 − 2.17902i 1.00000 −1.25107 − 2.16691i −0.00555225 + 0.00961678i
81.7 −0.500000 − 0.866025i −0.297627 − 0.171835i −0.500000 + 0.866025i 2.02126 + 3.50092i 0.343671i −1.81208 + 1.92779i 1.00000 −1.44095 − 2.49579i 2.02126 − 3.50092i
81.8 −0.500000 − 0.866025i 0.297627 + 0.171835i −0.500000 + 0.866025i 2.02126 + 3.50092i − 0.343671i 1.81208 − 1.92779i 1.00000 −1.44095 − 2.49579i 2.02126 − 3.50092i
81.9 −0.500000 − 0.866025i 0.611065 + 0.352798i −0.500000 + 0.866025i −0.00555225 − 0.00961678i − 0.705597i −1.50062 + 2.17902i 1.00000 −1.25107 − 2.16691i −0.00555225 + 0.00961678i
81.10 −0.500000 − 0.866025i 0.776228 + 0.448155i −0.500000 + 0.866025i −1.30685 − 2.26354i − 0.896311i 2.59330 + 0.524221i 1.00000 −1.09831 − 1.90233i −1.30685 + 2.26354i
81.11 −0.500000 − 0.866025i 1.96169 + 1.13258i −0.500000 + 0.866025i −0.773227 − 1.33927i − 2.26516i −0.699483 + 2.55161i 1.00000 1.06549 + 1.84547i −0.773227 + 1.33927i
81.12 −0.500000 − 0.866025i 2.22935 + 1.28712i −0.500000 + 0.866025i 0.977882 + 1.69374i − 2.57423i 2.55146 + 0.700038i 1.00000 1.81333 + 3.14079i 0.977882 − 1.69374i
81.13 −0.500000 − 0.866025i 2.34516 + 1.35398i −0.500000 + 0.866025i 0.815102 + 1.41180i − 2.70795i −1.07408 − 2.41792i 1.00000 2.16650 + 3.75249i 0.815102 − 1.41180i
81.14 −0.500000 − 0.866025i 2.52339 + 1.45688i −0.500000 + 0.866025i −1.72861 − 2.99404i − 2.91376i −0.765788 − 2.53250i 1.00000 2.74500 + 4.75449i −1.72861 + 2.99404i
163.1 −0.500000 + 0.866025i −2.52339 + 1.45688i −0.500000 − 0.866025i −1.72861 + 2.99404i − 2.91376i 0.765788 − 2.53250i 1.00000 2.74500 − 4.75449i −1.72861 − 2.99404i
163.2 −0.500000 + 0.866025i −2.34516 + 1.35398i −0.500000 − 0.866025i 0.815102 − 1.41180i − 2.70795i 1.07408 − 2.41792i 1.00000 2.16650 − 3.75249i 0.815102 + 1.41180i
163.3 −0.500000 + 0.866025i −2.22935 + 1.28712i −0.500000 − 0.866025i 0.977882 − 1.69374i − 2.57423i −2.55146 + 0.700038i 1.00000 1.81333 − 3.14079i 0.977882 + 1.69374i
163.4 −0.500000 + 0.866025i −1.96169 + 1.13258i −0.500000 − 0.866025i −0.773227 + 1.33927i − 2.26516i 0.699483 + 2.55161i 1.00000 1.06549 − 1.84547i −0.773227 − 1.33927i
163.5 −0.500000 + 0.866025i −0.776228 + 0.448155i −0.500000 − 0.866025i −1.30685 + 2.26354i − 0.896311i −2.59330 + 0.524221i 1.00000 −1.09831 + 1.90233i −1.30685 − 2.26354i
163.6 −0.500000 + 0.866025i −0.611065 + 0.352798i −0.500000 − 0.866025i −0.00555225 + 0.00961678i − 0.705597i 1.50062 + 2.17902i 1.00000 −1.25107 + 2.16691i −0.00555225 − 0.00961678i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
41.b even 2 1 inner
287.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 574.2.j.a ✓ 28
7.c even 3 1 inner 574.2.j.a ✓ 28
41.b even 2 1 inner 574.2.j.a ✓ 28
287.j even 6 1 inner 574.2.j.a ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
574.2.j.a ✓ 28 1.a even 1 1 trivial
574.2.j.a ✓ 28 7.c even 3 1 inner
574.2.j.a ✓ 28 41.b even 2 1 inner
574.2.j.a ✓ 28 287.j even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{28} - 29 T_{3}^{26} + 519 T_{3}^{24} - 5966 T_{3}^{22} + 50713 T_{3}^{20} - 310147 T_{3}^{18} + \cdots + 10000 \) acting on \(S_{2}^{\mathrm{new}}(574, [\chi])\). Copy content Toggle raw display