Properties

Label 555.2.bb.a
Level $555$
Weight $2$
Character orbit 555.bb
Analytic conductor $4.432$
Analytic rank $0$
Dimension $72$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 555 = 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 555.bb (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.43169731218\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(36\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72q + 32q^{4} - 2q^{5} + 36q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 72q + 32q^{4} - 2q^{5} + 36q^{9} - 8q^{11} - 16q^{14} - 24q^{16} + 4q^{19} + 4q^{20} + 12q^{21} + 10q^{25} + 24q^{26} + 16q^{29} + 12q^{30} + 24q^{31} - 44q^{34} - 2q^{35} + 64q^{36} - 8q^{39} + 46q^{40} - 48q^{41} - 8q^{44} - 4q^{45} - 64q^{46} + 32q^{49} - 46q^{50} - 8q^{51} - 12q^{55} - 72q^{56} + 32q^{59} - 68q^{60} - 8q^{61} - 136q^{64} - 42q^{65} - 32q^{66} - 4q^{69} + 4q^{70} + 12q^{71} - 96q^{74} - 8q^{75} + 8q^{76} - 8q^{79} - 44q^{80} - 36q^{81} + 136q^{84} + 4q^{85} + 8q^{86} + 8q^{91} - 28q^{94} + 18q^{95} - 20q^{96} - 4q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
454.1 −2.37811 1.37300i 0.866025 0.500000i 2.77028 + 4.79826i −2.00871 + 0.982395i −2.74601 4.01945 2.32063i 9.72238i 0.500000 0.866025i 6.12576 + 0.421716i
454.2 −2.28225 1.31765i −0.866025 + 0.500000i 2.47243 + 4.28237i 2.17287 0.527852i 2.63531 −2.57365 + 1.48590i 7.76061i 0.500000 0.866025i −5.65455 1.65841i
454.3 −2.16333 1.24900i −0.866025 + 0.500000i 2.11999 + 3.67194i −2.13789 0.655312i 2.49800 −0.174209 + 0.100580i 5.59548i 0.500000 0.866025i 3.80647 + 4.08787i
454.4 −1.98150 1.14402i 0.866025 0.500000i 1.61755 + 2.80168i 2.23334 + 0.110439i −2.28803 1.71703 0.991326i 2.82596i 0.500000 0.866025i −4.29901 2.77381i
454.5 −1.96173 1.13261i −0.866025 + 0.500000i 1.56559 + 2.71168i 0.414545 + 2.19731i 2.26521 1.72326 0.994927i 2.56237i 0.500000 0.866025i 1.67546 4.78004i
454.6 −1.73475 1.00156i 0.866025 0.500000i 1.00624 + 1.74286i −1.32535 1.80096i −2.00312 −3.12135 + 1.80211i 0.0249930i 0.500000 0.866025i 0.495375 + 4.45163i
454.7 −1.63447 0.943665i 0.866025 0.500000i 0.781006 + 1.35274i −1.89133 + 1.19284i −1.88733 −1.09596 + 0.632750i 0.826629i 0.500000 0.866025i 4.21697 0.164890i
454.8 −1.45470 0.839870i 0.866025 0.500000i 0.410763 + 0.711463i −0.906648 2.04401i −1.67974 3.05983 1.76660i 1.97953i 0.500000 0.866025i −0.397806 + 3.73489i
454.9 −1.34809 0.778318i −0.866025 + 0.500000i 0.211559 + 0.366431i 0.197466 2.22733i 1.55664 0.151651 0.0875559i 2.45463i 0.500000 0.866025i −1.99977 + 2.84895i
454.10 −1.22378 0.706551i −0.866025 + 0.500000i −0.00157058 0.00272032i 2.18051 + 0.495360i 1.41310 −1.52412 + 0.879952i 2.83064i 0.500000 0.866025i −2.31847 2.14685i
454.11 −1.07810 0.622440i 0.866025 0.500000i −0.225137 0.389949i 1.69944 1.45324i −1.24488 −2.38473 + 1.37683i 3.05030i 0.500000 0.866025i −2.73671 + 0.508942i
454.12 −1.04703 0.604502i −0.866025 + 0.500000i −0.269155 0.466190i −2.14730 0.623789i 1.20900 0.944369 0.545232i 3.06883i 0.500000 0.866025i 1.87120 + 1.95117i
454.13 −0.950972 0.549044i 0.866025 0.500000i −0.397102 0.687801i 0.837757 + 2.07320i −1.09809 −1.35262 + 0.780936i 3.06828i 0.500000 0.866025i 0.341594 2.43152i
454.14 −0.931961 0.538068i −0.866025 + 0.500000i −0.420965 0.729134i −0.374645 + 2.20446i 1.07614 −3.54514 + 2.04679i 3.05830i 0.500000 0.866025i 1.53530 1.85289i
454.15 −0.625647 0.361218i −0.866025 + 0.500000i −0.739044 1.28006i 2.23591 0.0267130i 0.722435 4.45380 2.57140i 2.51269i 0.500000 0.866025i −1.40854 0.790937i
454.16 −0.331339 0.191298i 0.866025 0.500000i −0.926810 1.60528i 1.22222 1.87248i −0.382597 3.21815 1.85800i 1.47438i 0.500000 0.866025i −0.763172 + 0.386614i
454.17 −0.153667 0.0887196i 0.866025 0.500000i −0.984258 1.70478i 2.22191 + 0.251234i −0.177439 −1.07771 + 0.622215i 0.704171i 0.500000 0.866025i −0.319145 0.235733i
454.18 −0.113794 0.0656990i −0.866025 + 0.500000i −0.991367 1.71710i 0.217380 2.22548i 0.131398 −1.67003 + 0.964193i 0.523323i 0.500000 0.866025i −0.170948 + 0.238964i
454.19 0.113794 + 0.0656990i 0.866025 0.500000i −0.991367 1.71710i −2.03601 0.924481i 0.131398 1.67003 0.964193i 0.523323i 0.500000 0.866025i −0.170948 0.238964i
454.20 0.153667 + 0.0887196i −0.866025 + 0.500000i −0.984258 1.70478i −0.893380 + 2.04985i −0.177439 1.07771 0.622215i 0.704171i 0.500000 0.866025i −0.319145 + 0.235733i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 544.36
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
37.c even 3 1 inner
185.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 555.2.bb.a 72
5.b even 2 1 inner 555.2.bb.a 72
37.c even 3 1 inner 555.2.bb.a 72
185.n even 6 1 inner 555.2.bb.a 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
555.2.bb.a 72 1.a even 1 1 trivial
555.2.bb.a 72 5.b even 2 1 inner
555.2.bb.a 72 37.c even 3 1 inner
555.2.bb.a 72 185.n even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(555, [\chi])\).