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Properties

Label 585.2.k.a
Level $585$
Weight $2$
Character orbit 585.k
Analytic conductor $4.671$
Analytic rank $0$
Dimension $56$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [585,2,Mod(61,585)]
 
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, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("585.61");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.k (of order \(3\), 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: \(56\)
Relative dimension: \(28\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56 q - q^{3} - 28 q^{4} + 28 q^{5} + 2 q^{6} - 12 q^{7} + 6 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q - q^{3} - 28 q^{4} + 28 q^{5} + 2 q^{6} - 12 q^{7} + 6 q^{8} + 5 q^{9} + 2 q^{11} - 10 q^{12} - 2 q^{13} - 2 q^{15} - 36 q^{16} - 5 q^{17} - 7 q^{18} + 19 q^{19} - 56 q^{20} + 17 q^{21} + 24 q^{23} + 12 q^{24} - 28 q^{25} + 2 q^{26} - 16 q^{27} + 21 q^{28} - q^{29} + q^{30} + 20 q^{31} - 6 q^{32} - 12 q^{33} + 2 q^{34} - 6 q^{35} - 20 q^{36} + 17 q^{37} - 2 q^{38} + 3 q^{39} + 3 q^{40} + 4 q^{41} - 11 q^{42} - 8 q^{43} - 56 q^{44} + 7 q^{45} + 20 q^{46} - 15 q^{47} + 86 q^{48} + 48 q^{49} + 10 q^{51} + 43 q^{52} + 32 q^{53} + 50 q^{54} - 2 q^{55} + 40 q^{56} + 5 q^{57} + 48 q^{58} + 10 q^{59} + q^{60} - 14 q^{61} - 25 q^{62} - 51 q^{63} + 106 q^{64} - q^{65} + 7 q^{66} - 96 q^{67} + 50 q^{68} + 14 q^{69} - 6 q^{71} - 21 q^{72} - 62 q^{73} + 20 q^{74} - q^{75} - 88 q^{76} - 34 q^{77} - 2 q^{78} + 10 q^{79} + 36 q^{80} - 7 q^{81} - q^{82} + 9 q^{83} - 51 q^{84} + 5 q^{85} - 69 q^{86} - 61 q^{87} + 19 q^{88} + 10 q^{89} + 19 q^{90} - 26 q^{91} + 14 q^{92} - 59 q^{93} + 52 q^{94} - 19 q^{95} - 76 q^{96} - 34 q^{97} + 24 q^{98} + q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1 −1.38023 − 2.39064i −1.38841 + 1.03552i −2.81010 + 4.86723i 0.500000 + 0.866025i 4.39189 + 1.88993i −0.293232 9.99342 0.855386 − 2.87547i 1.38023 − 2.39064i
61.2 −1.30568 − 2.26150i 1.68776 − 0.389194i −2.40958 + 4.17352i 0.500000 + 0.866025i −3.08383 − 3.30870i −1.51841 7.36184 2.69706 − 1.31373i 1.30568 − 2.26150i
61.3 −1.25132 − 2.16735i 0.699351 + 1.58458i −2.13161 + 3.69205i 0.500000 + 0.866025i 2.55924 − 3.49856i 1.60411 5.66401 −2.02182 + 2.21636i 1.25132 − 2.16735i
61.4 −1.14132 − 1.97683i −1.46357 − 0.926262i −1.60524 + 2.78036i 0.500000 + 0.866025i −0.160655 + 3.95040i 2.08368 2.76309 1.28408 + 2.71130i 1.14132 − 1.97683i
61.5 −1.02562 − 1.77643i 0.793706 − 1.53949i −1.10380 + 1.91184i 0.500000 + 0.866025i −3.54884 + 0.168973i −0.630057 0.425844 −1.74006 − 2.44381i 1.02562 − 1.77643i
61.6 −0.974447 − 1.68779i −1.69149 + 0.372651i −0.899093 + 1.55727i 0.500000 + 0.866025i 2.27722 + 2.49175i −3.88796 −0.393315 2.72226 − 1.26067i 0.974447 − 1.68779i
61.7 −0.855108 − 1.48109i 1.42084 + 0.990566i −0.462418 + 0.800931i 0.500000 + 0.866025i 0.252149 − 2.95143i −3.42791 −1.83876 1.03756 + 2.81487i 0.855108 − 1.48109i
61.8 −0.664436 − 1.15084i 1.66714 + 0.469717i 0.117049 − 0.202735i 0.500000 + 0.866025i −0.567143 − 2.23071i 4.79646 −2.96883 2.55873 + 1.56617i 0.664436 − 1.15084i
61.9 −0.655386 − 1.13516i −1.02315 − 1.39756i 0.140940 − 0.244115i 0.500000 + 0.866025i −0.915893 + 2.07738i −3.53103 −2.99102 −0.906325 + 2.85982i 0.655386 − 1.13516i
61.10 −0.414825 − 0.718497i 1.30514 − 1.13869i 0.655841 − 1.13595i 0.500000 + 0.866025i −1.35955 − 0.465380i 2.23827 −2.74753 0.406764 − 2.97230i 0.414825 − 0.718497i
61.11 −0.404809 − 0.701150i −0.587178 + 1.62949i 0.672259 − 1.16439i 0.500000 + 0.866025i 1.38021 − 0.247931i −1.32494 −2.70778 −2.31044 − 1.91359i 0.404809 − 0.701150i
61.12 −0.394277 − 0.682908i −1.68092 − 0.417748i 0.689092 − 1.19354i 0.500000 + 0.866025i 0.377464 + 1.31262i 2.80420 −2.66388 2.65097 + 1.40440i 0.394277 − 0.682908i
61.13 −0.139959 − 0.242417i 0.0434776 − 1.73151i 0.960823 − 1.66419i 0.500000 + 0.866025i −0.425831 + 0.231801i −0.918766 −1.09774 −2.99622 − 0.150563i 0.139959 − 0.242417i
61.14 −0.00716709 − 0.0124138i 0.359964 + 1.69423i 0.999897 − 1.73187i 0.500000 + 0.866025i 0.0184519 − 0.0166112i −0.599190 −0.0573338 −2.74085 + 1.21973i 0.00716709 − 0.0124138i
61.15 0.0458987 + 0.0794988i −1.67396 + 0.444806i 0.995787 − 1.72475i 0.500000 + 0.866025i −0.112194 − 0.112662i −4.29371 0.366416 2.60430 − 1.48918i −0.0458987 + 0.0794988i
61.16 0.257758 + 0.446450i −1.08489 + 1.35019i 0.867122 − 1.50190i 0.500000 + 0.866025i −0.882431 − 0.136327i 4.50750 1.92506 −0.646022 − 2.92962i −0.257758 + 0.446450i
61.17 0.328543 + 0.569053i 1.33677 − 1.10139i 0.784119 − 1.35813i 0.500000 + 0.866025i 1.06593 + 0.398837i −3.77886 2.34464 0.573884 − 2.94460i −0.328543 + 0.569053i
61.18 0.335869 + 0.581742i 1.54059 + 0.791566i 0.774384 − 1.34127i 0.500000 + 0.866025i 0.0569498 + 1.16209i 0.0948640 2.38384 1.74685 + 2.43896i −0.335869 + 0.581742i
61.19 0.381893 + 0.661458i −0.340179 − 1.69832i 0.708316 − 1.22684i 0.500000 + 0.866025i 0.993453 − 0.873589i 4.69076 2.60957 −2.76856 + 1.15546i −0.381893 + 0.661458i
61.20 0.709529 + 1.22894i −1.58681 + 0.694283i −0.00686234 + 0.0118859i 0.500000 + 0.866025i −1.97912 − 1.45748i 0.144528 2.81864 2.03594 − 2.20339i −0.709529 + 1.22894i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 61.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
117.f even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.2.k.a ✓ 56
9.c even 3 1 585.2.l.a yes 56
13.c even 3 1 585.2.l.a yes 56
117.f even 3 1 inner 585.2.k.a ✓ 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.k.a ✓ 56 1.a even 1 1 trivial
585.2.k.a ✓ 56 117.f even 3 1 inner
585.2.l.a yes 56 9.c even 3 1
585.2.l.a yes 56 13.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{56} + 42 T_{2}^{54} - 2 T_{2}^{53} + 989 T_{2}^{52} - 78 T_{2}^{51} + 15997 T_{2}^{50} + \cdots + 81 \) acting on \(S_{2}^{\mathrm{new}}(585, [\chi])\). Copy content Toggle raw display