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Properties

Label 585.2.cr.a
Level $585$
Weight $2$
Character orbit 585.cr
Analytic conductor $4.671$
Analytic rank $0$
Dimension $320$
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(23,585)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(585, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([10, 9, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("585.23");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.cr (of order \(12\), degree \(4\), 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: \(320\)
Relative dimension: \(80\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 320 q - 6 q^{2} - 2 q^{3} - 12 q^{6} - 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 320 q - 6 q^{2} - 2 q^{3} - 12 q^{6} - 24 q^{8} - 4 q^{10} - 12 q^{11} - 6 q^{12} - 2 q^{13} - 6 q^{15} + 140 q^{16} - 36 q^{18} - 12 q^{20} + 12 q^{21} + 18 q^{22} - 4 q^{25} - 48 q^{26} + 4 q^{27} + 30 q^{30} + 18 q^{32} - 42 q^{33} - 36 q^{35} - 100 q^{36} - 12 q^{37} + 36 q^{38} - 12 q^{40} - 72 q^{41} + 36 q^{42} - 4 q^{43} + 24 q^{45} - 24 q^{46} - 30 q^{47} - 52 q^{48} - 12 q^{50} + 8 q^{51} - 10 q^{52} + 6 q^{55} + 72 q^{57} - 6 q^{58} - 24 q^{60} - 8 q^{61} - 24 q^{62} + 12 q^{63} + 66 q^{65} - 120 q^{66} - 48 q^{71} + 114 q^{72} + 28 q^{75} - 60 q^{77} - 150 q^{78} + 30 q^{80} + 32 q^{81} - 36 q^{82} - 78 q^{85} + 108 q^{86} - 26 q^{87} - 42 q^{88} - 56 q^{90} - 28 q^{91} - 96 q^{92} - 60 q^{93} - 36 q^{95} + 72 q^{96} - 42 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1 −0.717003 − 2.67589i 1.47117 − 0.914150i −4.91426 + 2.83725i 1.07814 + 1.95898i −3.50100 − 3.28124i 0.0237576 + 0.0237576i 7.19794 + 7.19794i 1.32866 − 2.68973i 4.46899 − 4.28959i
23.2 −0.697931 − 2.60471i −1.72376 + 0.169273i −4.56537 + 2.63582i −1.48899 + 1.66821i 1.64397 + 4.37176i 0.276332 + 0.276332i 6.23830 + 6.23830i 2.94269 − 0.583571i 5.38441 + 2.71410i
23.3 −0.675438 − 2.52077i −0.574188 + 1.63411i −4.16601 + 2.40525i 2.23251 − 0.126023i 4.50704 + 0.343657i 1.12854 + 1.12854i 5.18630 + 5.18630i −2.34062 − 1.87657i −1.82560 − 5.54253i
23.4 −0.662779 − 2.47352i 1.52516 + 0.820900i −3.94699 + 2.27880i −1.51317 − 1.64630i 1.01967 − 4.31660i 2.84408 + 2.84408i 4.63116 + 4.63116i 1.65225 + 2.50401i −3.06926 + 4.83401i
23.5 −0.656064 − 2.44846i 0.446623 − 1.67348i −3.83251 + 2.21270i −2.15538 − 0.595272i −4.39046 + 0.00436903i 0.477907 + 0.477907i 4.34728 + 4.34728i −2.60106 − 1.49483i −0.0434358 + 5.66790i
23.6 −0.637945 − 2.38084i −1.30563 − 1.13812i −3.52939 + 2.03769i 2.06620 + 0.854868i −1.87677 + 3.83456i −3.11455 − 3.11455i 3.61719 + 3.61719i 0.409348 + 2.97194i 0.717182 − 5.46467i
23.7 −0.630882 − 2.35448i −1.70082 + 0.327447i −3.41354 + 1.97081i 1.11243 − 1.93972i 1.84398 + 3.79797i 1.26720 + 1.26720i 3.34657 + 3.34657i 2.78556 − 1.11385i −5.26884 − 1.39547i
23.8 −0.616238 − 2.29983i 1.57740 − 0.715415i −3.17742 + 1.83449i 0.767792 − 2.10012i −2.61739 − 3.18688i −2.24398 − 2.24398i 2.80987 + 2.80987i 1.97636 − 2.25699i −5.30306 − 0.471619i
23.9 −0.590239 − 2.20280i 1.45461 + 0.940274i −2.77190 + 1.60036i −1.80520 + 1.31957i 1.21267 − 3.75920i −1.86183 − 1.86183i 1.93623 + 1.93623i 1.23177 + 2.73546i 3.97224 + 3.19763i
23.10 −0.574766 − 2.14506i 1.26394 + 1.18426i −2.53886 + 1.46581i 2.12619 + 0.692328i 1.81383 − 3.39188i 0.705021 + 0.705021i 1.46291 + 1.46291i 0.195068 + 2.99365i 0.263021 − 4.95872i
23.11 −0.570904 − 2.13064i −0.0448509 − 1.73147i −2.48165 + 1.43278i 1.29752 − 1.82111i −3.66354 + 1.08406i −0.363504 − 0.363504i 1.35006 + 1.35006i −2.99598 + 0.155316i −4.62090 − 1.72486i
23.12 −0.558344 − 2.08377i −0.912550 − 1.47216i −2.29829 + 1.32692i −1.38639 + 1.75440i −2.55812 + 2.72351i 1.57433 + 1.57433i 0.997375 + 0.997375i −1.33451 + 2.68684i 4.42985 + 1.90935i
23.13 −0.556364 − 2.07638i −0.322964 + 1.70167i −2.26976 + 1.31045i −0.131201 + 2.23222i 3.71301 − 0.276155i −2.38170 − 2.38170i 0.943772 + 0.943772i −2.79139 − 1.09916i 4.70792 − 0.969501i
23.14 −0.527430 − 1.96839i −1.56568 + 0.740704i −1.86434 + 1.07638i −1.80426 − 1.32085i 2.28378 + 2.69121i −2.17087 − 2.17087i 0.220120 + 0.220120i 1.90272 − 2.31941i −1.64833 + 4.24815i
23.15 −0.449836 − 1.67881i −1.47087 − 0.914631i −0.884005 + 0.510381i −0.597525 − 2.15475i −0.873844 + 2.88074i 2.98284 + 2.98284i −1.20346 − 1.20346i 1.32690 + 2.69060i −3.34864 + 1.97242i
23.16 −0.448282 − 1.67301i 0.113601 + 1.72832i −0.865957 + 0.499960i 0.378500 − 2.20380i 2.84057 − 0.964831i 0.658169 + 0.658169i −1.22483 − 1.22483i −2.97419 + 0.392679i −3.85666 + 0.354689i
23.17 −0.427870 − 1.59683i −1.23261 − 1.21683i −0.634750 + 0.366473i −2.05492 − 0.881639i −1.41567 + 2.48892i −2.80098 − 2.80098i −1.48114 − 1.48114i 0.0386560 + 2.99975i −0.528589 + 3.65860i
23.18 −0.424954 − 1.58595i 1.60059 − 0.661912i −0.602600 + 0.347911i −1.56830 + 1.59388i −1.72993 − 2.25717i 2.82427 + 2.82427i −1.51414 − 1.51414i 2.12375 − 2.11889i 3.19426 + 1.80991i
23.19 −0.420429 − 1.56906i 1.35916 − 1.07363i −0.553141 + 0.319356i 2.13643 − 0.660058i −2.25603 − 1.68121i 3.25589 + 3.25589i −1.56362 − 1.56362i 0.694620 − 2.91848i −1.93389 − 3.07468i
23.20 −0.419386 − 1.56517i −1.37907 + 1.04793i −0.541824 + 0.312822i 0.825659 + 2.07805i 2.21855 + 1.71900i 1.38799 + 1.38799i −1.57472 − 1.57472i 0.803685 − 2.89034i 2.90623 − 2.16380i
See next 80 embeddings (of 320 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.80
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
117.m odd 6 1 inner
585.cr even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.2.cr.a yes 320
5.c odd 4 1 inner 585.2.cr.a yes 320
9.d odd 6 1 585.2.co.a ✓ 320
13.e even 6 1 585.2.co.a ✓ 320
45.l even 12 1 585.2.co.a ✓ 320
65.r odd 12 1 585.2.co.a ✓ 320
117.m odd 6 1 inner 585.2.cr.a yes 320
585.cr even 12 1 inner 585.2.cr.a yes 320
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.co.a ✓ 320 9.d odd 6 1
585.2.co.a ✓ 320 13.e even 6 1
585.2.co.a ✓ 320 45.l even 12 1
585.2.co.a ✓ 320 65.r odd 12 1
585.2.cr.a yes 320 1.a even 1 1 trivial
585.2.cr.a yes 320 5.c odd 4 1 inner
585.2.cr.a yes 320 117.m odd 6 1 inner
585.2.cr.a yes 320 585.cr even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(585, [\chi])\).