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Properties

Label 574.2.j.b
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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 14 q^{2} - 14 q^{4} + 4 q^{5} - 28 q^{8} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 14 q^{2} - 14 q^{4} + 4 q^{5} - 28 q^{8} + 16 q^{9} - 4 q^{10} - 14 q^{16} - 16 q^{18} - 8 q^{20} + 32 q^{21} + 6 q^{23} + 6 q^{25} + 2 q^{31} + 14 q^{32} + 14 q^{33} - 32 q^{36} + 4 q^{37} - 14 q^{39} - 4 q^{40} - 8 q^{41} + 28 q^{42} - 12 q^{43} - 8 q^{45} - 6 q^{46} - 24 q^{49} + 12 q^{50} - 26 q^{51} - 36 q^{57} + 6 q^{59} + 4 q^{61} + 4 q^{62} + 28 q^{64} - 14 q^{66} - 16 q^{72} + 24 q^{73} - 4 q^{74} - 6 q^{77} - 28 q^{78} + 4 q^{80} - 6 q^{81} - 4 q^{82} + 64 q^{83} - 4 q^{84} - 6 q^{86} + 60 q^{87} - 16 q^{90} - 12 q^{91} - 12 q^{92} - 24 q^{98}+O(q^{100}) \) 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 −2.80575 − 1.61990i −0.500000 + 0.866025i −0.981980 − 1.70084i − 3.23980i −1.10509 + 2.40391i −1.00000 3.74814 + 6.49197i 0.981980 − 1.70084i
81.2 0.500000 + 0.866025i −2.12975 − 1.22961i −0.500000 + 0.866025i 0.805135 + 1.39454i − 2.45922i −1.03929 − 2.43308i −1.00000 1.52389 + 2.63946i −0.805135 + 1.39454i
81.3 0.500000 + 0.866025i −2.09612 − 1.21020i −0.500000 + 0.866025i 0.324130 + 0.561409i − 2.42039i −2.56253 + 0.658350i −1.00000 1.42915 + 2.47537i −0.324130 + 0.561409i
81.4 0.500000 + 0.866025i −1.72562 − 0.996289i −0.500000 + 0.866025i 2.11242 + 3.65882i − 1.99258i 1.64641 + 2.07107i −1.00000 0.485182 + 0.840359i −2.11242 + 3.65882i
81.5 0.500000 + 0.866025i −1.21447 − 0.701173i −0.500000 + 0.866025i 0.428346 + 0.741916i − 1.40235i 1.80890 − 1.93077i −1.00000 −0.516714 − 0.894975i −0.428346 + 0.741916i
81.6 0.500000 + 0.866025i −0.633075 − 0.365506i −0.500000 + 0.866025i −1.16949 − 2.02561i − 0.731013i 0.592388 + 2.57858i −1.00000 −1.23281 − 2.13529i 1.16949 − 2.02561i
81.7 0.500000 + 0.866025i −0.307795 − 0.177706i −0.500000 + 0.866025i −0.518563 − 0.898177i − 0.355411i 2.50965 + 0.837638i −1.00000 −1.43684 − 2.48868i 0.518563 − 0.898177i
81.8 0.500000 + 0.866025i 0.307795 + 0.177706i −0.500000 + 0.866025i −0.518563 − 0.898177i 0.355411i −2.50965 − 0.837638i −1.00000 −1.43684 − 2.48868i 0.518563 − 0.898177i
81.9 0.500000 + 0.866025i 0.633075 + 0.365506i −0.500000 + 0.866025i −1.16949 − 2.02561i 0.731013i −0.592388 − 2.57858i −1.00000 −1.23281 − 2.13529i 1.16949 − 2.02561i
81.10 0.500000 + 0.866025i 1.21447 + 0.701173i −0.500000 + 0.866025i 0.428346 + 0.741916i 1.40235i −1.80890 + 1.93077i −1.00000 −0.516714 − 0.894975i −0.428346 + 0.741916i
81.11 0.500000 + 0.866025i 1.72562 + 0.996289i −0.500000 + 0.866025i 2.11242 + 3.65882i 1.99258i −1.64641 − 2.07107i −1.00000 0.485182 + 0.840359i −2.11242 + 3.65882i
81.12 0.500000 + 0.866025i 2.09612 + 1.21020i −0.500000 + 0.866025i 0.324130 + 0.561409i 2.42039i 2.56253 − 0.658350i −1.00000 1.42915 + 2.47537i −0.324130 + 0.561409i
81.13 0.500000 + 0.866025i 2.12975 + 1.22961i −0.500000 + 0.866025i 0.805135 + 1.39454i 2.45922i 1.03929 + 2.43308i −1.00000 1.52389 + 2.63946i −0.805135 + 1.39454i
81.14 0.500000 + 0.866025i 2.80575 + 1.61990i −0.500000 + 0.866025i −0.981980 − 1.70084i 3.23980i 1.10509 − 2.40391i −1.00000 3.74814 + 6.49197i 0.981980 − 1.70084i
163.1 0.500000 − 0.866025i −2.80575 + 1.61990i −0.500000 − 0.866025i −0.981980 + 1.70084i 3.23980i −1.10509 − 2.40391i −1.00000 3.74814 − 6.49197i 0.981980 + 1.70084i
163.2 0.500000 − 0.866025i −2.12975 + 1.22961i −0.500000 − 0.866025i 0.805135 − 1.39454i 2.45922i −1.03929 + 2.43308i −1.00000 1.52389 − 2.63946i −0.805135 − 1.39454i
163.3 0.500000 − 0.866025i −2.09612 + 1.21020i −0.500000 − 0.866025i 0.324130 − 0.561409i 2.42039i −2.56253 − 0.658350i −1.00000 1.42915 − 2.47537i −0.324130 − 0.561409i
163.4 0.500000 − 0.866025i −1.72562 + 0.996289i −0.500000 − 0.866025i 2.11242 − 3.65882i 1.99258i 1.64641 − 2.07107i −1.00000 0.485182 − 0.840359i −2.11242 − 3.65882i
163.5 0.500000 − 0.866025i −1.21447 + 0.701173i −0.500000 − 0.866025i 0.428346 − 0.741916i 1.40235i 1.80890 + 1.93077i −1.00000 −0.516714 + 0.894975i −0.428346 − 0.741916i
163.6 0.500000 − 0.866025i −0.633075 + 0.365506i −0.500000 − 0.866025i −1.16949 + 2.02561i 0.731013i 0.592388 − 2.57858i −1.00000 −1.23281 + 2.13529i 1.16949 + 2.02561i
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.b ✓ 28
7.c even 3 1 inner 574.2.j.b ✓ 28
41.b even 2 1 inner 574.2.j.b ✓ 28
287.j even 6 1 inner 574.2.j.b ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
574.2.j.b ✓ 28 1.a even 1 1 trivial
574.2.j.b ✓ 28 7.c even 3 1 inner
574.2.j.b ✓ 28 41.b even 2 1 inner
574.2.j.b ✓ 28 287.j even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{28} - 29 T_{3}^{26} + 521 T_{3}^{24} - 5880 T_{3}^{22} + 48629 T_{3}^{20} - 289973 T_{3}^{18} + \cdots + 38416 \) acting on \(S_{2}^{\mathrm{new}}(574, [\chi])\). Copy content Toggle raw display