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Properties

Label 585.2.df.a
Level $585$
Weight $2$
Character orbit 585.df
Analytic conductor $4.671$
Analytic rank $0$
Dimension $224$
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(86,585)]
 
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, 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("585.86");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.df (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: \(224\)
Relative dimension: \(56\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 224 q - 12 q^{6} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 224 q - 12 q^{6} + 4 q^{7} + 12 q^{11} - 4 q^{15} + 112 q^{16} - 4 q^{18} - 8 q^{19} - 36 q^{21} + 64 q^{24} - 48 q^{27} + 32 q^{28} - 16 q^{31} - 72 q^{32} + 4 q^{33} + 8 q^{37} - 40 q^{39} - 24 q^{41} - 8 q^{42} + 8 q^{45} + 72 q^{47} - 8 q^{48} - 52 q^{52} - 40 q^{54} - 36 q^{57} + 36 q^{58} + 72 q^{59} - 16 q^{60} - 80 q^{63} + 144 q^{66} + 28 q^{67} + 36 q^{72} - 56 q^{73} - 384 q^{74} + 16 q^{76} + 12 q^{78} + 24 q^{79} - 48 q^{81} + 36 q^{83} - 172 q^{84} - 12 q^{85} + 12 q^{86} - 160 q^{87} - 32 q^{91} - 120 q^{92} + 168 q^{93} - 92 q^{96} - 4 q^{97} + 44 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
86.1 −2.71361 − 0.727110i 1.55043 − 0.772109i 5.10294 + 2.94619i −0.258819 − 0.965926i −4.76868 + 0.967866i 0.123185 + 0.0330073i −7.73220 − 7.73220i 1.80770 − 2.39421i 2.80934i
86.2 −2.65344 − 0.710987i 1.10147 + 1.33670i 4.80319 + 2.77312i 0.258819 + 0.965926i −1.97230 − 4.32998i 2.48831 + 0.666741i −6.88840 − 6.88840i −0.573541 + 2.94466i − 2.74704i
86.3 −2.58735 − 0.693279i −0.336600 + 1.69903i 4.48171 + 2.58752i −0.258819 − 0.965926i 2.04881 − 4.16263i −3.66280 − 0.981444i −6.01375 − 6.01375i −2.77340 − 1.14379i 2.67863i
86.4 −2.57479 − 0.689913i −1.71187 − 0.263610i 4.42151 + 2.55276i 0.258819 + 0.965926i 4.22585 + 1.85978i 2.06711 + 0.553881i −5.85353 − 5.85353i 2.86102 + 0.902533i − 2.66562i
86.5 −2.37099 − 0.635305i −0.616399 − 1.61866i 3.48592 + 2.01260i −0.258819 − 0.965926i 0.433135 + 4.22942i 1.88469 + 0.505001i −3.51511 − 3.51511i −2.24010 + 1.99548i 2.45463i
86.6 −2.25935 − 0.605391i −1.31200 + 1.13078i 3.00611 + 1.73558i 0.258819 + 0.965926i 3.64882 − 1.76056i −1.65975 − 0.444728i −2.43323 − 2.43323i 0.442664 − 2.96716i − 2.33905i
86.7 −2.20260 − 0.590184i 1.12348 − 1.31826i 2.77106 + 1.59987i 0.258819 + 0.965926i −3.25258 + 2.24053i −3.47792 − 0.931905i −1.93448 − 1.93448i −0.475603 − 2.96206i − 2.28029i
86.8 −2.15366 − 0.577070i −1.45400 + 0.941215i 2.57317 + 1.48562i −0.258819 − 0.965926i 3.67456 − 1.18799i 4.81233 + 1.28946i −1.53125 − 1.53125i 1.22823 − 2.73705i 2.22963i
86.9 −2.11700 − 0.567249i 1.03665 + 1.38757i 2.42787 + 1.40173i −0.258819 − 0.965926i −1.40749 − 3.52553i 1.89714 + 0.508338i −1.24517 − 1.24517i −0.850717 + 2.87685i 2.19168i
86.10 −2.10805 − 0.564851i 1.30004 + 1.14451i 2.39278 + 1.38147i 0.258819 + 0.965926i −2.09407 − 3.14701i −3.94037 − 1.05582i −1.17737 − 1.17737i 0.380195 + 2.97581i − 2.18242i
86.11 −1.76029 − 0.471669i −1.72344 + 0.172469i 1.14410 + 0.660548i −0.258819 − 0.965926i 3.11511 + 0.509298i −3.03139 − 0.812259i 0.874850 + 0.874850i 2.94051 − 0.594481i 1.82239i
86.12 −1.68652 − 0.451902i −0.604274 − 1.62322i 0.908089 + 0.524286i 0.258819 + 0.965926i 0.285583 + 3.01067i 2.16195 + 0.579292i 1.17465 + 1.17465i −2.26971 + 1.96174i − 1.74602i
86.13 −1.59337 − 0.426942i −0.197867 + 1.72071i 0.624491 + 0.360550i −0.258819 − 0.965926i 1.04992 − 2.65725i −0.135135 − 0.0362094i 1.49174 + 1.49174i −2.92170 − 0.680945i 1.64958i
86.14 −1.53711 − 0.411866i 0.742682 − 1.56474i 0.461008 + 0.266163i −0.258819 − 0.965926i −1.78605 + 2.09929i −3.20377 − 0.858447i 1.65148 + 1.65148i −1.89685 − 2.32422i 1.59133i
86.15 −1.50813 − 0.404103i 1.72793 + 0.119423i 0.379115 + 0.218882i 0.258819 + 0.965926i −2.55769 − 0.878367i 2.97555 + 0.797295i 1.72475 + 1.72475i 2.97148 + 0.412709i − 1.56133i
86.16 −1.43270 − 0.383891i −0.988555 − 1.42224i 0.173204 + 0.0999995i 0.258819 + 0.965926i 0.870318 + 2.41714i −2.84396 − 0.762036i 1.88786 + 1.88786i −1.04552 + 2.81192i − 1.48324i
86.17 −1.41119 − 0.378128i 0.150854 + 1.72547i 0.116433 + 0.0672226i 0.258819 + 0.965926i 0.439565 − 2.49201i 4.69385 + 1.25771i 1.92724 + 1.92724i −2.95449 + 0.520586i − 1.46097i
86.18 −1.38926 − 0.372250i −1.25892 + 1.18959i 0.0594100 + 0.0343004i 0.258819 + 0.965926i 2.19178 − 1.18402i −0.173072 − 0.0463746i 1.96424 + 1.96424i 0.169740 − 2.99519i − 1.43826i
86.19 −1.19154 − 0.319272i 1.35936 − 1.07338i −0.414217 − 0.239148i −0.258819 − 0.965926i −1.96243 + 0.844973i 2.73057 + 0.731654i 2.16174 + 2.16174i 0.695702 − 2.91822i 1.23357i
86.20 −0.916028 − 0.245449i −0.105668 − 1.72882i −0.953189 − 0.550324i −0.258819 − 0.965926i −0.327543 + 1.60959i 0.444005 + 0.118971i 2.07923 + 2.07923i −2.97767 + 0.365364i 0.948342i
See next 80 embeddings (of 224 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 86.56
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
13.d odd 4 1 inner
117.z 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.df.a ✓ 224
9.d odd 6 1 inner 585.2.df.a ✓ 224
13.d odd 4 1 inner 585.2.df.a ✓ 224
117.z even 12 1 inner 585.2.df.a ✓ 224
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.df.a ✓ 224 1.a even 1 1 trivial
585.2.df.a ✓ 224 9.d odd 6 1 inner
585.2.df.a ✓ 224 13.d odd 4 1 inner
585.2.df.a ✓ 224 117.z even 12 1 inner

Hecke kernels

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