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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Newspace parameters

Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 605.u (of order \(110\), degree \(40\), minimal)

Newform invariants

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.

\(\operatorname{Tr}(f)(q) = \) \( 2560q - 144q^{4} - 43q^{5} - 114q^{6} + 534q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 2560q - 144q^{4} - 43q^{5} - 114q^{6} + 534q^{9} - 22q^{10} - 104q^{11} - 80q^{14} - 58q^{15} - 40q^{16} - 94q^{19} - 32q^{20} - 162q^{21} - 248q^{24} - 25q^{25} - 124q^{26} - 90q^{29} - 103q^{30} - 102q^{31} - 56q^{34} - 66q^{35} - 24q^{36} - 184q^{39} - 45q^{40} - 36q^{41} - 92q^{44} - 2q^{45} - 26q^{46} - 182q^{49} - 94q^{50} + 108q^{51} - 180q^{54} + 52q^{55} + 16q^{56} - 96q^{59} - 63q^{60} - 48q^{61} - 124q^{64} - 37q^{65} - 212q^{66} - 170q^{69} + 30q^{70} - 194q^{71} - 136q^{74} - 46q^{75} - 188q^{76} - 170q^{79} + 65q^{80} - 730q^{81} - 166q^{84} - 58q^{85} - 126q^{86} + 38q^{89} - 144q^{90} - 10q^{91} + 8q^{94} - 180q^{95} - 68q^{96} + 226q^{99} + O(q^{100}) \)

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.

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 599.64
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 \(S_{2}^{\mathrm{new}}(605, [\chi])\).