Properties

Label 360.2.bm.a
Level $360$
Weight $2$
Character orbit 360.bm
Analytic conductor $2.875$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 360.bm (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.87461447277\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 48q - 24q^{5} + 7q^{6} - 6q^{8} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 48q - 24q^{5} + 7q^{6} - 6q^{8} - 15q^{12} + 15q^{14} + 12q^{16} - 11q^{18} - 4q^{21} + 21q^{22} - 24q^{24} - 24q^{25} + 12q^{27} - 14q^{30} - 8q^{33} + 33q^{34} - 23q^{36} - 33q^{38} + 16q^{39} - 6q^{40} + 12q^{41} - 53q^{42} - 24q^{44} - 6q^{46} + 12q^{47} + 45q^{48} + 24q^{49} - 20q^{51} - 36q^{52} - 36q^{54} + 21q^{56} + 4q^{57} - 51q^{58} - 36q^{59} + 15q^{60} + 12q^{61} + 42q^{62} + 56q^{63} - 12q^{64} + 24q^{66} + 57q^{68} - 40q^{69} - 15q^{70} + 46q^{72} + 30q^{74} + 57q^{76} + 78q^{78} - 8q^{81} - 18q^{82} - 60q^{83} - 31q^{84} + 27q^{86} + 36q^{87} + 57q^{88} + 4q^{90} - 51q^{92} + 57q^{94} - 119q^{96} + 42q^{98} - 16q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −1.40729 + 0.139770i 1.36390 + 1.06761i 1.96093 0.393394i −0.500000 0.866025i −2.06862 1.31180i 1.02179 + 0.589933i −2.70461 + 0.827699i 0.720424 + 2.91221i 0.824689 + 1.14886i
11.2 −1.39929 0.204905i 0.875353 1.49458i 1.91603 + 0.573443i −0.500000 0.866025i −1.53112 + 1.91198i −1.05351 0.608247i −2.56358 1.19502i −1.46751 2.61656i 0.522192 + 1.31427i
11.3 −1.29220 + 0.574647i −0.857820 + 1.50471i 1.33956 1.48512i −0.500000 0.866025i 0.243800 2.43733i 0.661806 + 0.382094i −0.877566 + 2.68884i −1.52829 2.58154i 1.14376 + 0.831755i
11.4 −1.28964 0.580361i −0.952268 1.44678i 1.32636 + 1.49692i −0.500000 0.866025i 0.388429 + 2.41850i 2.56287 + 1.47968i −0.841782 2.70026i −1.18637 + 2.75545i 0.142215 + 1.40704i
11.5 −1.27893 + 0.603598i −1.41981 0.992033i 1.27134 1.54392i −0.500000 0.866025i 2.41464 + 0.411747i −4.17077 2.40800i −0.694050 + 2.74195i 1.03174 + 2.81700i 1.16220 + 0.805790i
11.6 −1.10684 0.880287i 0.424591 + 1.67920i 0.450191 + 1.94867i −0.500000 0.866025i 1.00823 2.23237i −3.88456 2.24275i 1.21710 2.55317i −2.63944 + 1.42595i −0.208931 + 1.39870i
11.7 −1.06985 0.924887i −1.29784 + 1.14700i 0.289169 + 1.97898i −0.500000 0.866025i 2.44934 0.0267591i 3.14000 + 1.81288i 1.52097 2.38467i 0.368796 2.97725i −0.266049 + 1.38896i
11.8 −0.537760 1.30798i 1.67034 0.458221i −1.42163 + 1.40676i −0.500000 0.866025i −1.49759 1.93836i 3.22730 + 1.86328i 2.60451 + 1.10297i 2.58007 1.53077i −0.863865 + 1.11970i
11.9 −0.513744 + 1.31760i −1.18636 1.26196i −1.47213 1.35382i −0.500000 0.866025i 2.27224 0.914825i 2.20775 + 1.27465i 2.54009 1.24417i −0.185091 + 2.99428i 1.39795 0.213885i
11.10 −0.477617 + 1.33112i −1.45808 + 0.934888i −1.54376 1.27153i −0.500000 0.866025i −0.548047 2.38739i −1.98473 1.14588i 2.42989 1.44763i 1.25197 2.72627i 1.39159 0.251932i
11.11 −0.325988 + 1.37613i 1.62132 0.609351i −1.78746 0.897203i −0.500000 0.866025i 0.310014 + 2.42979i 0.518944 + 0.299612i 1.81736 2.16730i 2.25738 1.97591i 1.35476 0.405751i
11.12 0.0218455 1.41404i −0.452164 + 1.67199i −1.99905 0.0617811i −0.500000 0.866025i 2.35439 + 0.675906i 0.550736 + 0.317967i −0.131031 + 2.82539i −2.59109 1.51203i −1.23552 + 0.688104i
11.13 0.102357 1.41050i 1.72054 0.199393i −1.97905 0.288749i −0.500000 0.866025i −0.105136 2.44723i −3.97204 2.29326i −0.609850 + 2.76190i 2.92049 0.686124i −1.27271 + 0.616609i
11.14 0.158141 + 1.40534i 0.284002 + 1.70861i −1.94998 + 0.444484i −0.500000 0.866025i −2.35627 + 0.669321i −2.24682 1.29720i −0.933024 2.67011i −2.83869 + 0.970497i 1.13799 0.839626i
11.15 0.524388 + 1.31340i −0.0300586 1.73179i −1.45003 + 1.37746i −0.500000 0.866025i 2.25877 0.947610i −3.45090 1.99238i −2.56954 1.18215i −2.99819 + 0.104110i 0.875243 1.11083i
11.16 0.652259 1.25481i −1.71670 + 0.230091i −1.14912 1.63693i −0.500000 0.866025i −0.831012 + 2.30422i −1.06767 0.616420i −2.80356 + 0.374228i 2.89412 0.789993i −1.41283 + 0.0625343i
11.17 0.767212 1.18802i 0.801968 1.53520i −0.822770 1.82292i −0.500000 0.866025i −1.20857 2.13058i 4.07138 + 2.35061i −2.79690 0.421104i −1.71369 2.46237i −1.41246 0.0704168i
11.18 0.893891 + 1.09588i 1.59734 + 0.669699i −0.401917 + 1.95920i −0.500000 0.866025i 0.693939 + 2.34914i 2.53202 + 1.46186i −2.50632 + 1.31086i 2.10301 + 2.13948i 0.502116 1.32207i
11.19 0.962641 + 1.03601i −1.57528 0.720074i −0.146644 + 1.99462i −0.500000 0.866025i −0.770419 2.32518i 1.68164 + 0.970893i −2.20761 + 1.76818i 1.96299 + 2.26863i 0.415893 1.35168i
11.20 1.20329 0.743030i 1.46420 + 0.925262i 0.895814 1.78816i −0.500000 0.866025i 2.44936 + 0.0254133i −0.947055 0.546782i −0.250732 2.81729i 1.28778 + 2.70954i −1.24513 0.670565i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 131.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
72.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 360.2.bm.a 48
3.b odd 2 1 1080.2.bm.b 48
4.b odd 2 1 1440.2.cc.a 48
8.b even 2 1 1440.2.cc.b 48
8.d odd 2 1 360.2.bm.b yes 48
9.c even 3 1 1080.2.bm.a 48
9.d odd 6 1 360.2.bm.b yes 48
12.b even 2 1 4320.2.cc.b 48
24.f even 2 1 1080.2.bm.a 48
24.h odd 2 1 4320.2.cc.a 48
36.f odd 6 1 4320.2.cc.a 48
36.h even 6 1 1440.2.cc.b 48
72.j odd 6 1 1440.2.cc.a 48
72.l even 6 1 inner 360.2.bm.a 48
72.n even 6 1 4320.2.cc.b 48
72.p odd 6 1 1080.2.bm.b 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.bm.a 48 1.a even 1 1 trivial
360.2.bm.a 48 72.l even 6 1 inner
360.2.bm.b yes 48 8.d odd 2 1
360.2.bm.b yes 48 9.d odd 6 1
1080.2.bm.a 48 9.c even 3 1
1080.2.bm.a 48 24.f even 2 1
1080.2.bm.b 48 3.b odd 2 1
1080.2.bm.b 48 72.p odd 6 1
1440.2.cc.a 48 4.b odd 2 1
1440.2.cc.a 48 72.j odd 6 1
1440.2.cc.b 48 8.b even 2 1
1440.2.cc.b 48 36.h even 6 1
4320.2.cc.a 48 24.h odd 2 1
4320.2.cc.a 48 36.f odd 6 1
4320.2.cc.b 48 12.b even 2 1
4320.2.cc.b 48 72.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(25\!\cdots\!60\)\( T_{7}^{30} + \)\(62\!\cdots\!88\)\( T_{7}^{29} + \)\(16\!\cdots\!99\)\( T_{7}^{28} - \)\(50\!\cdots\!76\)\( T_{7}^{27} - \)\(89\!\cdots\!64\)\( T_{7}^{26} + \)\(31\!\cdots\!60\)\( T_{7}^{25} + \)\(36\!\cdots\!69\)\( T_{7}^{24} - \)\(16\!\cdots\!80\)\( T_{7}^{23} - \)\(10\!\cdots\!76\)\( T_{7}^{22} + \)\(64\!\cdots\!24\)\( T_{7}^{21} + \)\(21\!\cdots\!90\)\( T_{7}^{20} - \)\(20\!\cdots\!52\)\( T_{7}^{19} - \)\(14\!\cdots\!12\)\( T_{7}^{18} + \)\(51\!\cdots\!32\)\( T_{7}^{17} - \)\(66\!\cdots\!65\)\( T_{7}^{16} - \)\(10\!\cdots\!52\)\( T_{7}^{15} + \)\(30\!\cdots\!88\)\( T_{7}^{14} + \)\(15\!\cdots\!16\)\( T_{7}^{13} - \)\(69\!\cdots\!39\)\( T_{7}^{12} - \)\(18\!\cdots\!40\)\( T_{7}^{11} + \)\(11\!\cdots\!88\)\( T_{7}^{10} + \)\(15\!\cdots\!92\)\( T_{7}^{9} - \)\(12\!\cdots\!07\)\( T_{7}^{8} - \)\(89\!\cdots\!24\)\( T_{7}^{7} + \)\(10\!\cdots\!88\)\( T_{7}^{6} + \)\(21\!\cdots\!64\)\( T_{7}^{5} - \)\(54\!\cdots\!24\)\( T_{7}^{4} + \)\(76\!\cdots\!68\)\( T_{7}^{3} + \)\(16\!\cdots\!00\)\( T_{7}^{2} - \)\(91\!\cdots\!44\)\( T_{7} + \)\(15\!\cdots\!64\)\( \)">\(T_{7}^{48} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(360, [\chi])\).