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Properties

Label 405.2.q.b
Level $405$
Weight $2$
Character orbit 405.q
Analytic conductor $3.234$
Analytic rank $0$
Dimension $342$
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] = [405,2,Mod(16,405)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(405, base_ring=CyclotomicField(54))
 
chi = DirichletCharacter(H, H._module([4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("405.16");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 405.q (of order \(27\), degree \(18\), 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: \(3.23394128186\)
Analytic rank: \(0\)
Dimension: \(342\)
Relative dimension: \(19\) over \(\Q(\zeta_{27})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{27}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 342 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 342 q - 9 q^{15} - 9 q^{18} + 72 q^{21} + 126 q^{22} - 27 q^{23} - 54 q^{24} - 63 q^{27} - 90 q^{28} - 27 q^{29} - 54 q^{32} - 27 q^{33} - 90 q^{34} - 36 q^{36} - 9 q^{38} - 90 q^{41} + 99 q^{42} - 36 q^{44} - 54 q^{47} - 225 q^{48} - 54 q^{49} - 9 q^{51} - 54 q^{53} - 117 q^{54} - 63 q^{56} - 54 q^{57} - 9 q^{59} + 72 q^{61} - 99 q^{62} + 36 q^{63} + 18 q^{65} + 36 q^{66} - 54 q^{67} + 81 q^{68} - 108 q^{69} - 36 q^{70} + 72 q^{71} + 72 q^{74} - 243 q^{76} + 144 q^{78} - 27 q^{79} + 450 q^{80} + 216 q^{81} + 144 q^{82} + 288 q^{83} + 180 q^{84} - 27 q^{85} + 144 q^{86} - 27 q^{87} - 252 q^{88} - 27 q^{89} + 27 q^{90} - 90 q^{92} + 72 q^{93} - 90 q^{94} + 36 q^{95} - 171 q^{96} - 108 q^{97} + 72 q^{98} - 90 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −2.60037 − 0.616300i −0.829132 + 1.52070i 4.59485 + 2.30762i 0.597159 − 0.802123i 3.09326 − 3.44340i −0.766411 − 0.504077i −6.43176 − 5.39689i −1.62508 − 2.52173i −2.04718 + 1.71779i
16.2 −2.55110 − 0.604621i 1.72848 + 0.111168i 4.35526 + 2.18729i 0.597159 − 0.802123i −4.34230 − 1.32868i 1.54085 + 1.01343i −5.77143 − 4.84280i 2.97528 + 0.384304i −2.00839 + 1.68524i
16.3 −1.84951 − 0.438342i −0.667384 − 1.59831i 1.44127 + 0.723835i 0.597159 − 0.802123i 0.533726 + 3.24863i 1.37107 + 0.901768i 0.563745 + 0.473039i −2.10920 + 2.13337i −1.45605 + 1.22177i
16.4 −1.74631 − 0.413882i 1.12852 + 1.31394i 1.09102 + 0.547930i 0.597159 − 0.802123i −1.42693 − 2.76162i −3.33215 − 2.19159i 1.07114 + 0.898791i −0.452878 + 2.96562i −1.37481 + 1.15360i
16.5 −1.70295 − 0.403606i −1.62961 − 0.586821i 0.949870 + 0.477043i 0.597159 − 0.802123i 2.53830 + 1.65705i −1.43799 − 0.945780i 1.25630 + 1.05416i 2.31128 + 1.91258i −1.34067 + 1.12496i
16.6 −1.39478 − 0.330570i 0.745073 − 1.56361i 0.0488775 + 0.0245472i 0.597159 − 0.802123i −1.55610 + 1.93459i 0.543768 + 0.357642i 2.13607 + 1.79237i −1.88973 − 2.33000i −1.09806 + 0.921385i
16.7 −0.941041 − 0.223031i 1.73050 − 0.0733800i −0.951450 − 0.477836i 0.597159 − 0.802123i −1.64483 − 0.316900i 3.34472 + 2.19986i 2.27048 + 1.90516i 2.98923 − 0.253968i −0.740849 + 0.621646i
16.8 −0.863104 − 0.204559i −0.300289 + 1.70582i −1.08416 − 0.544486i 0.597159 − 0.802123i 0.608123 − 1.41087i −0.637139 − 0.419053i 2.18335 + 1.83205i −2.81965 − 1.02448i −0.679492 + 0.570161i
16.9 −0.192862 − 0.0457091i −1.53295 + 0.806262i −1.75216 − 0.879967i 0.597159 − 0.802123i 0.332501 − 0.0854273i −0.0754003 − 0.0495915i 0.601369 + 0.504608i 1.69988 − 2.47192i −0.151833 + 0.127403i
16.10 0.162872 + 0.0386014i 1.68623 + 0.395782i −1.76223 − 0.885024i 0.597159 − 0.802123i 0.259361 + 0.129552i −4.24161 − 2.78975i −0.509302 − 0.427355i 2.68671 + 1.33475i 0.128224 − 0.107592i
16.11 0.173459 + 0.0411105i 0.920812 − 1.46701i −1.75887 − 0.883336i 0.597159 − 0.802123i 0.220032 − 0.216610i −1.18432 − 0.778940i −0.541893 − 0.454702i −1.30421 − 2.70167i 0.136558 − 0.114586i
16.12 0.562156 + 0.133234i 0.555292 + 1.64063i −1.48900 − 0.747802i 0.597159 − 0.802123i 0.0935748 + 0.996272i 2.51890 + 1.65671i −1.62255 − 1.36148i −2.38330 + 1.82205i 0.442566 − 0.371357i
16.13 1.42493 + 0.337714i −1.46177 − 0.929104i 0.129097 + 0.0648348i 0.597159 − 0.802123i −1.76914 − 1.81756i −2.78897 − 1.83433i −2.08153 − 1.74661i 1.27353 + 2.71627i 1.12179 − 0.941297i
16.14 1.47567 + 0.349741i −1.72794 − 0.119206i 0.268025 + 0.134607i 0.597159 − 0.802123i −2.50819 − 0.780241i 0.995671 + 0.654863i −1.97505 − 1.65726i 2.97158 + 0.411962i 1.16175 − 0.974820i
16.15 1.62524 + 0.385188i 1.72599 − 0.144796i 0.705755 + 0.354444i 0.597159 − 0.802123i 2.86091 + 0.429501i 0.770991 + 0.507089i −1.54849 − 1.29934i 2.95807 − 0.499834i 1.27949 − 1.07362i
16.16 1.93599 + 0.458838i 0.0631018 − 1.73090i 1.75027 + 0.879016i 0.597159 − 0.802123i 0.916368 − 3.32206i 4.03024 + 2.65073i −0.0631055 − 0.0529518i −2.99204 − 0.218446i 1.52414 − 1.27890i
16.17 2.24130 + 0.531198i 0.614269 + 1.61947i 2.95400 + 1.48355i 0.597159 − 0.802123i 0.516503 + 3.95601i −0.828157 − 0.544687i 2.30375 + 1.93307i −2.24535 + 1.98958i 1.76450 − 1.48059i
16.18 2.49463 + 0.591239i 0.0536210 − 1.73122i 4.08637 + 2.05225i 0.597159 − 0.802123i 1.15733 − 4.28706i −3.73613 − 2.45729i 5.05275 + 4.23976i −2.99425 − 0.185659i 1.96394 − 1.64794i
16.19 2.71882 + 0.644371i −1.64067 + 0.555154i 5.18948 + 2.60626i 0.597159 − 0.802123i −4.81841 + 0.452161i 0.570120 + 0.374974i 8.14900 + 6.83782i 2.38361 − 1.82165i 2.14043 − 1.79603i
31.1 −1.04408 − 2.42045i 1.65535 + 0.509724i −3.39598 + 3.59953i −0.0581448 + 0.998308i −0.494556 − 4.53888i −2.70933 + 0.642124i 7.30403 + 2.65845i 2.48036 + 1.68754i 2.47706 − 0.901576i
See next 80 embeddings (of 342 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.19
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
81.g even 27 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 405.2.q.b ✓ 342
81.g even 27 1 inner 405.2.q.b ✓ 342
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.2.q.b ✓ 342 1.a even 1 1 trivial
405.2.q.b ✓ 342 81.g even 27 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{342} + 54 T_{2}^{337} + 2851 T_{2}^{333} + 5823 T_{2}^{332} + 1188 T_{2}^{331} + \cdots + 38\!\cdots\!81 \) acting on \(S_{2}^{\mathrm{new}}(405, [\chi])\). Copy content Toggle raw display