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Properties

Label 585.2.bt.b
Level $585$
Weight $2$
Character orbit 585.bt
Analytic conductor $4.671$
Analytic rank $0$
Dimension $108$
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] = [585,2,Mod(376,585)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(585, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("585.376");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.bt (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: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(108\)
Relative dimension: \(54\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 108 q - 6 q^{3} + 52 q^{4} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 108 q - 6 q^{3} + 52 q^{4} - 14 q^{9} - 8 q^{10} - 36 q^{12} - 4 q^{13} - 8 q^{14} - 64 q^{16} + 36 q^{17} + 24 q^{22} - 22 q^{23} + 54 q^{25} + 40 q^{26} + 48 q^{27} - 16 q^{29} + 20 q^{30} - 40 q^{35} - 76 q^{36} - 12 q^{38} + 26 q^{39} - 52 q^{42} - 6 q^{43} - 68 q^{48} + 58 q^{49} + 34 q^{51} + 22 q^{52} - 116 q^{53} + 24 q^{55} + 40 q^{56} + 18 q^{61} + 152 q^{62} - 216 q^{64} + 2 q^{65} - 72 q^{66} - 134 q^{69} + 32 q^{74} + 40 q^{77} - 34 q^{78} + 18 q^{79} + 34 q^{81} + 88 q^{82} + 60 q^{87} + 8 q^{90} + 16 q^{91} + 176 q^{92} - 76 q^{94} - 56 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
376.1 −2.39334 − 1.38179i −0.969577 − 1.43524i 2.81871 + 4.88215i 0.866025 − 0.500000i 0.337317 + 4.77477i −0.814209 − 0.470084i − 10.0523i −1.11984 + 2.78316i −2.76359
376.2 −2.36351 − 1.36457i 1.68089 + 0.417855i 2.72413 + 4.71833i −0.866025 + 0.500000i −3.40262 − 3.28131i 3.99906 + 2.30886i − 9.41082i 2.65080 + 1.40474i 2.72915
376.3 −2.26372 − 1.30696i −0.538321 + 1.64627i 2.41629 + 4.18514i 0.866025 − 0.500000i 3.37022 − 3.02314i 3.14827 + 1.81766i − 7.40414i −2.42042 − 1.77245i −2.61392
376.4 −2.24228 − 1.29458i −0.927160 + 1.46300i 2.35187 + 4.07355i −0.866025 + 0.500000i 3.97292 − 2.08017i −1.32705 − 0.766174i − 7.00039i −1.28075 − 2.71287i 2.58916
376.5 −2.20318 − 1.27200i 0.350451 − 1.69623i 2.23599 + 3.87285i −0.866025 + 0.500000i −2.92971 + 3.29131i −0.586224 − 0.338456i − 6.28874i −2.75437 − 1.18889i 2.54401
376.6 −2.03176 − 1.17304i 0.447750 + 1.67318i 1.75203 + 3.03461i 0.866025 − 0.500000i 1.05298 − 3.92472i −2.60767 − 1.50554i − 3.52864i −2.59904 + 1.49833i −2.34607
376.7 −1.91643 − 1.10645i 1.62047 + 0.611626i 1.44848 + 2.50883i 0.866025 − 0.500000i −2.42878 − 2.96511i 0.916933 + 0.529392i − 1.98487i 2.25183 + 1.98224i −2.21291
376.8 −1.85413 − 1.07048i −1.70628 + 0.297682i 1.29187 + 2.23759i 0.866025 − 0.500000i 3.48233 + 1.27460i −4.18669 − 2.41719i − 1.24978i 2.82277 − 1.01586i −2.14097
376.9 −1.71224 − 0.988562i −1.13208 − 1.31087i 0.954509 + 1.65326i −0.866025 + 0.500000i 0.642514 + 3.36366i −3.18919 − 1.84128i 0.179881i −0.436783 + 2.96803i 1.97712
376.10 −1.71151 − 0.988142i 1.03985 + 1.38517i 0.952848 + 1.65038i −0.866025 + 0.500000i −0.410976 − 3.39826i −1.22619 − 0.707941i 0.186373i −0.837406 + 2.88076i 1.97628
376.11 −1.64129 − 0.947602i −1.39724 − 1.02358i 0.795899 + 1.37854i 0.866025 − 0.500000i 1.32334 + 3.00402i 2.82539 + 1.63124i 0.773626i 0.904573 + 2.86038i −1.89520
376.12 −1.62679 − 0.939229i −1.72455 + 0.161066i 0.764302 + 1.32381i −0.866025 + 0.500000i 2.95676 + 1.35772i 1.80641 + 1.04293i 0.885499i 2.94812 − 0.555532i 1.87846
376.13 −1.39901 − 0.807720i 0.0517001 − 1.73128i 0.304825 + 0.527972i 0.866025 − 0.500000i −1.47072 + 2.38032i 1.67328 + 0.966071i 2.24603i −2.99465 − 0.179015i −1.61544
376.14 −1.23573 − 0.713450i −1.00032 + 1.41399i 0.0180219 + 0.0312149i −0.866025 + 0.500000i 2.24493 − 1.03364i 2.69830 + 1.55786i 2.80237i −0.998738 − 2.82887i 1.42690
376.15 −1.22618 − 0.707937i 1.65800 − 0.501035i 0.00235097 + 0.00407201i 0.866025 − 0.500000i −2.38771 − 0.559399i −0.00992873 − 0.00573236i 2.82509i 2.49793 − 1.66143i −1.41587
376.16 −1.16682 − 0.673666i 0.619932 − 1.61731i −0.0923480 − 0.159951i 0.866025 − 0.500000i −1.81288 + 1.46949i −4.07352 − 2.35185i 2.94351i −2.23137 − 2.00524i −1.34733
376.17 −1.09899 − 0.634500i 1.58392 − 0.700848i −0.194820 − 0.337439i −0.866025 + 0.500000i −2.18540 − 0.234776i 4.16228 + 2.40310i 3.03245i 2.01762 − 2.22018i 1.26900
376.18 −1.07284 − 0.619404i −1.48219 + 0.896168i −0.232678 − 0.403011i 0.866025 − 0.500000i 2.14524 − 0.0433707i 0.166629 + 0.0962033i 3.05410i 1.39377 − 2.65658i −1.23881
376.19 −0.883008 − 0.509805i −0.597446 + 1.62575i −0.480198 − 0.831727i −0.866025 + 0.500000i 1.35636 − 1.13097i −3.32453 − 1.91942i 3.01845i −2.28612 − 1.94260i 1.01961
376.20 −0.774258 − 0.447018i −0.513274 − 1.65425i −0.600350 − 1.03984i −0.866025 + 0.500000i −0.342074 + 1.51026i 1.76954 + 1.02164i 2.86154i −2.47310 + 1.69817i 0.894036
See next 80 embeddings (of 108 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 376.54
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
13.b even 2 1 inner
117.t even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.2.bt.b ✓ 108
9.c even 3 1 inner 585.2.bt.b ✓ 108
13.b even 2 1 inner 585.2.bt.b ✓ 108
117.t even 6 1 inner 585.2.bt.b ✓ 108
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.bt.b ✓ 108 1.a even 1 1 trivial
585.2.bt.b ✓ 108 9.c even 3 1 inner
585.2.bt.b ✓ 108 13.b even 2 1 inner
585.2.bt.b ✓ 108 117.t even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{108} - 80 T_{2}^{106} + 3402 T_{2}^{104} - 99900 T_{2}^{102} + 2249817 T_{2}^{100} + \cdots + 1679616 \) acting on \(S_{2}^{\mathrm{new}}(585, [\chi])\). Copy content Toggle raw display