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Properties

Label 572.2.bg.a
Level $572$
Weight $2$
Character orbit 572.bg
Analytic conductor $4.567$
Analytic rank $0$
Dimension $112$
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] = [572,2,Mod(9,572)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(572, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([0, 18, 20]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("572.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 572 = 2^{2} \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 572.bg (of order \(15\), degree \(8\), 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: \(4.56744299562\)
Analytic rank: \(0\)
Dimension: \(112\)
Relative dimension: \(14\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 112 q + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 112 q + 8 q^{9} - 10 q^{11} + 11 q^{13} - 2 q^{15} + 4 q^{17} - 12 q^{19} - 40 q^{21} + 10 q^{23} - 16 q^{25} - 12 q^{27} + q^{29} + 4 q^{31} + 35 q^{33} - 5 q^{35} - 12 q^{37} + 21 q^{39} - 10 q^{41} - 32 q^{43} + 34 q^{45} + 70 q^{47} + 16 q^{49} - 48 q^{51} - 26 q^{53} + 10 q^{55} - 12 q^{57} - 5 q^{59} + 28 q^{61} + 34 q^{63} + 22 q^{65} - 68 q^{67} - 58 q^{69} + 44 q^{71} + 42 q^{73} - 24 q^{75} + 46 q^{77} - 24 q^{79} + 64 q^{81} - 114 q^{83} + 4 q^{85} - 30 q^{87} - 6 q^{89} + 77 q^{91} - 5 q^{93} - 36 q^{95} - 15 q^{97} - 40 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 0 −0.351299 − 3.34239i 0 −0.419191 − 1.29014i 0 −0.149591 + 1.42326i 0 −8.11369 + 1.72462i 0
9.2 0 −0.263130 − 2.50351i 0 1.21356 + 3.73495i 0 0.197160 − 1.87585i 0 −3.26389 + 0.693762i 0
9.3 0 −0.229534 − 2.18387i 0 −0.547837 − 1.68607i 0 −0.0708172 + 0.673781i 0 −1.78216 + 0.378809i 0
9.4 0 −0.165782 − 1.57731i 0 −1.14854 − 3.53484i 0 0.493072 − 4.69127i 0 0.474019 − 0.100756i 0
9.5 0 −0.153632 − 1.46171i 0 0.347814 + 1.07046i 0 0.456258 − 4.34101i 0 0.821453 − 0.174605i 0
9.6 0 −0.0694093 − 0.660385i 0 −0.596743 − 1.83659i 0 −0.373949 + 3.55789i 0 2.50315 − 0.532061i 0
9.7 0 −0.0692906 − 0.659256i 0 0.701430 + 2.15878i 0 −0.337098 + 3.20727i 0 2.50463 − 0.532375i 0
9.8 0 0.0211820 + 0.201533i 0 −0.370496 − 1.14027i 0 −0.131751 + 1.25353i 0 2.89428 − 0.615197i 0
9.9 0 0.0911637 + 0.867364i 0 0.671959 + 2.06808i 0 0.257245 − 2.44753i 0 2.19043 − 0.465591i 0
9.10 0 0.151007 + 1.43674i 0 −0.310916 − 0.956900i 0 0.0904890 − 0.860946i 0 0.893027 − 0.189819i 0
9.11 0 0.161126 + 1.53301i 0 1.30886 + 4.02825i 0 −0.295810 + 2.81444i 0 0.610280 − 0.129719i 0
9.12 0 0.251214 + 2.39014i 0 −0.900429 − 2.77123i 0 0.251542 − 2.39326i 0 −2.71524 + 0.577141i 0
9.13 0 0.277675 + 2.64191i 0 0.389171 + 1.19775i 0 0.0870458 − 0.828185i 0 −3.96812 + 0.843450i 0
9.14 0 0.348708 + 3.31773i 0 −0.338638 − 1.04222i 0 −0.473796 + 4.50786i 0 −7.95129 + 1.69010i 0
81.1 0 −2.59967 + 0.552578i 0 −1.82163 + 1.32349i 0 −4.40077 − 0.935412i 0 3.71233 − 1.65283i 0
81.2 0 −2.59692 + 0.551993i 0 1.71674 − 1.24728i 0 −1.20270 − 0.255642i 0 3.69867 − 1.64675i 0
81.3 0 −2.50220 + 0.531859i 0 −0.767747 + 0.557801i 0 3.31034 + 0.703635i 0 3.23748 − 1.44142i 0
81.4 0 −2.03406 + 0.432352i 0 3.08678 − 2.24268i 0 1.74512 + 0.370936i 0 1.20982 − 0.538647i 0
81.5 0 −0.978574 + 0.208002i 0 −3.20054 + 2.32533i 0 0.357037 + 0.0758905i 0 −1.82629 + 0.813119i 0
81.6 0 −0.807770 + 0.171697i 0 0.437086 − 0.317562i 0 −2.23753 − 0.475601i 0 −2.11762 + 0.942827i 0
See next 80 embeddings (of 112 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner
13.c even 3 1 inner
143.q even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 572.2.bg.a ✓ 112
11.c even 5 1 inner 572.2.bg.a ✓ 112
13.c even 3 1 inner 572.2.bg.a ✓ 112
143.q even 15 1 inner 572.2.bg.a ✓ 112
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
572.2.bg.a ✓ 112 1.a even 1 1 trivial
572.2.bg.a ✓ 112 11.c even 5 1 inner
572.2.bg.a ✓ 112 13.c even 3 1 inner
572.2.bg.a ✓ 112 143.q even 15 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(572, [\chi])\).