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Properties

Label 605.2.u.a
Level $605$
Weight $2$
Character orbit 605.u
Analytic conductor $4.831$
Analytic rank $0$
Dimension $2560$
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] = [605,2,Mod(4,605)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(605, base_ring=CyclotomicField(110))
 
chi = DirichletCharacter(H, H._module([55, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("605.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 605.u (of order \(110\), degree \(40\), 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: \(4.83094932229\)
Analytic rank: \(0\)
Dimension: \(2560\)
Relative dimension: \(64\) over \(\Q(\zeta_{110})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{110}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 2560 q - 144 q^{4} - 43 q^{5} - 114 q^{6} + 534 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 2560 q - 144 q^{4} - 43 q^{5} - 114 q^{6} + 534 q^{9} - 22 q^{10} - 104 q^{11} - 80 q^{14} - 58 q^{15} - 40 q^{16} - 94 q^{19} - 32 q^{20} - 162 q^{21} - 248 q^{24} - 25 q^{25} - 124 q^{26} - 90 q^{29} - 103 q^{30} - 102 q^{31} - 56 q^{34} - 66 q^{35} - 24 q^{36} - 184 q^{39} - 45 q^{40} - 36 q^{41} - 92 q^{44} - 2 q^{45} - 26 q^{46} - 182 q^{49} - 94 q^{50} + 108 q^{51} - 180 q^{54} + 52 q^{55} + 16 q^{56} - 96 q^{59} - 63 q^{60} - 48 q^{61} - 124 q^{64} - 37 q^{65} - 212 q^{66} - 170 q^{69} + 30 q^{70} - 194 q^{71} - 136 q^{74} - 46 q^{75} - 188 q^{76} - 170 q^{79} + 65 q^{80} - 730 q^{81} - 166 q^{84} - 58 q^{85} - 126 q^{86} + 38 q^{89} - 144 q^{90} - 10 q^{91} + 8 q^{94} - 180 q^{95} - 68 q^{96} + 226 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −0.159369 + 2.78705i 1.09686 + 0.356392i −5.75530 − 0.660360i −0.217398 − 2.22547i −1.16809 + 3.00022i −1.19567 + 2.82929i 1.80562 − 10.4337i −1.35096 − 0.981530i 6.23716 − 0.251227i
4.2 −0.158138 + 2.76551i −1.72481 − 0.560425i −5.63610 − 0.646682i 2.21911 − 0.274891i 1.82262 − 4.68137i 0.760205 − 1.79886i 1.73499 − 10.0255i 0.233849 + 0.169902i 0.409289 + 6.18044i
4.3 −0.153290 + 2.68074i 3.01326 + 0.979066i −5.17593 − 0.593882i −0.0647326 + 2.23513i −3.08653 + 7.92768i −0.638035 + 1.50977i 1.46972 − 8.49271i 5.69409 + 4.13700i −5.98189 − 0.516156i
4.4 −0.148331 + 2.59401i −0.797799 − 0.259221i −4.71991 − 0.541559i −1.50336 + 1.65527i 0.790758 − 2.03105i 0.372801 − 0.882154i 1.21880 − 7.04277i −1.85776 − 1.34974i −4.07078 − 4.14525i
4.5 −0.138346 + 2.41940i 1.32326 + 0.429952i −3.84739 − 0.441447i −1.60632 − 1.55555i −1.22329 + 3.14201i 1.89653 − 4.48773i 0.773838 − 4.47158i −0.860900 − 0.625481i 3.98572 − 3.67113i
4.6 −0.133785 + 2.33963i −1.75629 − 0.570655i −3.46900 − 0.398031i −2.20930 + 0.344984i 1.57009 − 4.03273i −1.66518 + 3.94028i 0.596124 − 3.44467i 0.331874 + 0.241121i −0.511565 − 5.21509i
4.7 −0.133191 + 2.32924i −2.85319 − 0.927057i −3.42066 − 0.392484i −0.141192 − 2.23161i 2.53936 − 6.52229i −0.386813 + 0.915309i 0.574118 − 3.31751i 4.85420 + 3.52678i 5.21675 − 0.0316400i
4.8 −0.132849 + 2.32326i 2.12234 + 0.689590i −3.39293 − 0.389302i 2.06750 − 0.851734i −1.88405 + 4.83913i 1.43745 − 3.40141i 0.561567 − 3.24498i 1.60174 + 1.16373i 1.70413 + 4.91649i
4.9 −0.127550 + 2.23059i 0.100823 + 0.0327593i −2.97228 − 0.341037i 1.47600 + 1.67971i −0.0859325 + 0.220716i −0.678322 + 1.60510i 0.377855 − 2.18341i −2.41796 − 1.75675i −3.93499 + 3.07811i
4.10 −0.120999 + 2.11603i 2.08525 + 0.677538i −2.47598 − 0.284092i −2.23590 + 0.0274065i −1.68600 + 4.33047i −0.239400 + 0.566489i 0.177899 − 1.02798i 1.46215 + 1.06231i 0.212549 − 4.73455i
4.11 −0.120780 + 2.11219i 1.16323 + 0.377956i −2.45981 − 0.282237i 2.04224 + 0.910624i −0.938810 + 2.41131i −1.54461 + 3.65497i 0.171705 − 0.992189i −1.21680 − 0.884057i −2.17008 + 4.20363i
4.12 −0.112439 + 1.96633i −3.09691 − 1.00625i −1.86684 − 0.214200i 0.326385 + 2.21212i 2.32682 − 5.97640i −0.209014 + 0.494586i −0.0406069 + 0.234645i 6.15126 + 4.46915i −4.38645 + 0.393052i
4.13 −0.111823 + 1.95556i −2.00057 − 0.650024i −1.82475 − 0.209370i 1.72287 + 1.42539i 1.49487 − 3.83954i 0.963092 − 2.27895i −0.0545378 + 0.315144i 1.15269 + 0.837480i −2.98008 + 3.20978i
4.14 −0.109164 + 1.90906i −0.681776 − 0.221522i −1.64564 − 0.188819i 1.44293 − 1.70820i 0.497325 − 1.27737i 0.922133 − 2.18203i −0.112026 + 0.647336i −2.01131 − 1.46130i 3.10355 + 2.94111i
4.15 −0.0908431 + 1.58866i −0.905007 − 0.294055i −0.528640 − 0.0606558i −0.647343 − 2.14031i 0.549368 − 1.41104i 0.322482 − 0.763085i −0.398305 + 2.30159i −1.69448 − 1.23111i 3.45905 − 0.833978i
4.16 −0.0893404 + 1.56238i 2.68142 + 0.871245i −0.446099 − 0.0511851i 2.14234 − 0.640594i −1.60078 + 4.11156i −0.235648 + 0.557610i −0.413887 + 2.39163i 4.00387 + 2.90898i 0.809456 + 3.40440i
4.17 −0.0872065 + 1.52507i −0.0606761 − 0.0197148i −0.331261 − 0.0380087i −2.10361 − 0.758176i 0.0353578 − 0.0908158i −0.188162 + 0.445244i −0.434111 + 2.50849i −2.42376 − 1.76096i 1.33972 − 3.14203i
4.18 −0.0843987 + 1.47596i −1.88212 − 0.611538i −0.184383 − 0.0211560i −1.88871 + 1.19699i 1.06146 − 2.72633i 1.72780 − 4.08846i −0.457404 + 2.64309i 0.741350 + 0.538623i −1.60731 − 2.88869i
4.19 −0.0814111 + 1.42372i 2.78227 + 0.904016i −0.0333779 − 0.00382975i −0.483711 + 2.18312i −1.51357 + 3.88757i 1.42689 − 3.37643i −0.478174 + 2.76310i 4.49676 + 3.26708i −3.06877 − 0.866398i
4.20 −0.0802644 + 1.40366i 2.27059 + 0.737760i 0.0231347 + 0.00265446i −0.445272 − 2.19129i −1.21781 + 3.12793i −1.04870 + 2.48153i −0.485076 + 2.80299i 2.18425 + 1.58695i 3.11157 − 0.449130i
See next 80 embeddings (of 2560 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.64
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
121.g even 55 1 inner
605.u even 110 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.u.a ✓ 2560
5.b even 2 1 inner 605.2.u.a ✓ 2560
121.g even 55 1 inner 605.2.u.a ✓ 2560
605.u even 110 1 inner 605.2.u.a ✓ 2560
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.u.a ✓ 2560 1.a even 1 1 trivial
605.2.u.a ✓ 2560 5.b even 2 1 inner
605.2.u.a ✓ 2560 121.g even 55 1 inner
605.2.u.a ✓ 2560 605.u even 110 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(605, [\chi])\).