Properties

Label 360.2.bm.b
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.

\(\operatorname{Tr}(f)(q) = \) \( 48q + 24q^{5} - q^{6} + 6q^{8} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 48q + 24q^{5} - q^{6} + 6q^{8} + 13q^{12} + 15q^{14} - 12q^{16} + 7q^{18} + 4q^{21} - 21q^{22} - 4q^{24} - 24q^{25} + 12q^{27} - 2q^{30} - 8q^{33} - 27q^{34} - 31q^{36} - 27q^{38} - 16q^{39} + 12q^{40} + 12q^{41} - 9q^{42} + 24q^{44} - 6q^{46} - 12q^{47} + 7q^{48} + 24q^{49} - 20q^{51} + 54q^{52} - 32q^{54} + 21q^{56} + 4q^{57} + 33q^{58} - 36q^{59} - q^{60} - 12q^{61} - 42q^{62} - 56q^{63} - 12q^{64} - 32q^{66} + 51q^{68} + 40q^{69} + 15q^{70} + 6q^{72} + 54q^{74} - 51q^{76} - 24q^{78} - 8q^{81} - 18q^{82} - 60q^{83} + 41q^{84} + 27q^{86} - 36q^{87} - 57q^{88} - 22q^{90} - 9q^{92} - 75q^{94} + 13q^{96} - 42q^{98} - 16q^{99} + O(q^{100}) \)

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 −1.40162 + 0.188277i 1.67034 0.458221i 1.92910 0.527787i 0.500000 + 0.866025i −2.25492 + 0.956740i −3.22730 1.86328i −2.60451 + 1.10297i 2.58007 1.53077i −0.863865 1.11970i
11.2 −1.33590 0.464076i −1.29784 + 1.14700i 1.56927 + 1.23992i 0.500000 + 0.866025i 2.26609 0.929977i −3.14000 1.81288i −1.52097 2.38467i 0.368796 2.97725i −0.266049 1.38896i
11.3 −1.31577 0.518408i 0.424591 + 1.67920i 1.46251 + 1.36421i 0.500000 + 0.866025i 0.311848 2.42956i 3.88456 + 2.24275i −1.21710 2.55317i −2.63944 + 1.42595i −0.208931 1.39870i
11.4 −1.21368 + 0.725941i −0.452164 + 1.67199i 0.946019 1.76211i 0.500000 + 0.866025i −0.664985 2.35750i −0.550736 0.317967i 0.131031 + 2.82539i −2.59109 1.51203i −1.23552 0.688104i
11.5 −1.17035 + 0.793896i 1.72054 0.199393i 0.739459 1.85828i 0.500000 + 0.866025i −1.85534 + 1.59929i 3.97204 + 2.29326i 0.609850 + 2.76190i 2.92049 0.686124i −1.27271 0.616609i
11.6 −1.14743 0.826684i −0.952268 1.44678i 0.633188 + 1.89712i 0.500000 + 0.866025i −0.103374 + 2.44731i −2.56287 1.47968i 0.841782 2.70026i −1.18637 + 2.75545i 0.142215 1.40704i
11.7 −0.877098 1.10937i 0.875353 1.49458i −0.461398 + 1.94605i 0.500000 + 0.866025i −2.42581 + 0.339801i 1.05351 + 0.608247i 2.56358 1.19502i −1.46751 2.61656i 0.522192 1.31427i
11.8 −0.760571 + 1.19228i −1.71670 + 0.230091i −0.843062 1.81363i 0.500000 + 0.866025i 1.03134 2.22179i 1.06767 + 0.616420i 2.80356 + 0.374228i 2.89412 0.789993i −1.41283 0.0625343i
11.9 −0.645247 + 1.25843i 0.801968 1.53520i −1.16731 1.62400i 0.500000 + 0.866025i 1.41448 + 1.99981i −4.07138 2.35061i 2.79690 0.421104i −1.71369 2.46237i −1.41246 + 0.0704168i
11.10 −0.582600 1.28863i 1.36390 + 1.06761i −1.32115 + 1.50152i 0.500000 + 0.866025i 0.581150 2.37955i −1.02179 0.589933i 2.70461 + 0.827699i 0.720424 + 2.91221i 0.824689 1.14886i
11.11 −0.148442 1.40640i −0.857820 + 1.50471i −1.95593 + 0.417537i 0.500000 + 0.866025i 2.24356 + 0.983078i −0.661806 0.382094i 0.877566 + 2.68884i −1.52829 2.58154i 1.14376 0.831755i
11.12 −0.116736 1.40939i −1.41981 0.992033i −1.97275 + 0.329051i 0.500000 + 0.866025i −1.23242 + 2.11687i 4.17077 + 2.40800i 0.694050 + 2.74195i 1.03174 + 2.81700i 1.16220 0.805790i
11.13 −0.0418375 + 1.41359i 1.46420 + 0.925262i −1.99650 0.118282i 0.500000 + 0.866025i −1.36920 + 2.03108i 0.947055 + 0.546782i 0.250732 2.81729i 1.28778 + 2.70954i −1.24513 + 0.670565i
11.14 0.0747682 + 1.41224i −0.478797 1.66456i −1.98882 + 0.211181i 0.500000 + 0.866025i 2.31495 0.800631i 1.88846 + 1.09030i −0.446938 2.79289i −2.54151 + 1.59397i −1.18565 + 0.770869i
11.15 0.475189 + 1.33199i −1.73205 + 0.00397907i −1.54839 + 1.26589i 0.500000 + 0.866025i −0.828349 2.30518i −2.40441 1.38819i −2.42193 1.46090i 2.99997 0.0137839i −0.915942 + 1.07752i
11.16 0.884203 1.10371i −1.18636 1.26196i −0.436372 1.95181i 0.500000 + 0.866025i −2.44183 + 0.193576i −2.20775 1.27465i −2.54009 1.24417i −0.185091 + 2.99428i 1.39795 + 0.213885i
11.17 0.913976 1.07919i −1.45808 + 0.934888i −0.329296 1.97270i 0.500000 + 0.866025i −0.323726 + 2.42800i 1.98473 + 1.14588i −2.42989 1.44763i 1.25197 2.72627i 1.39159 + 0.251932i
11.18 0.951037 + 1.04667i 1.28580 1.16048i −0.191057 + 1.99085i 0.500000 + 0.866025i 2.43749 + 0.242161i 1.13105 + 0.653010i −2.26548 + 1.69340i 0.306582 2.98429i −0.430929 + 1.34696i
11.19 0.959041 + 1.03935i 0.0478668 + 1.73139i −0.160481 + 1.99355i 0.500000 + 0.866025i −1.75361 + 1.71022i −1.21691 0.702581i −2.22590 + 1.74510i −2.99542 + 0.165752i −0.420580 + 1.35023i
11.20 1.02877 0.970379i 1.62132 0.609351i 0.116731 1.99659i 0.500000 + 0.866025i 1.07667 2.20018i −0.518944 0.299612i −1.81736 2.16730i 2.25738 1.97591i 1.35476 + 0.405751i
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.b yes 48
3.b odd 2 1 1080.2.bm.a 48
4.b odd 2 1 1440.2.cc.b 48
8.b even 2 1 1440.2.cc.a 48
8.d odd 2 1 360.2.bm.a 48
9.c even 3 1 1080.2.bm.b 48
9.d odd 6 1 360.2.bm.a 48
12.b even 2 1 4320.2.cc.a 48
24.f even 2 1 1080.2.bm.b 48
24.h odd 2 1 4320.2.cc.b 48
36.f odd 6 1 4320.2.cc.b 48
36.h even 6 1 1440.2.cc.a 48
72.j odd 6 1 1440.2.cc.b 48
72.l even 6 1 inner 360.2.bm.b yes 48
72.n even 6 1 4320.2.cc.a 48
72.p odd 6 1 1080.2.bm.a 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.bm.a 48 8.d odd 2 1
360.2.bm.a 48 9.d odd 6 1
360.2.bm.b yes 48 1.a even 1 1 trivial
360.2.bm.b yes 48 72.l even 6 1 inner
1080.2.bm.a 48 3.b odd 2 1
1080.2.bm.a 48 72.p odd 6 1
1080.2.bm.b 48 9.c even 3 1
1080.2.bm.b 48 24.f even 2 1
1440.2.cc.a 48 8.b even 2 1
1440.2.cc.a 48 36.h even 6 1
1440.2.cc.b 48 4.b odd 2 1
1440.2.cc.b 48 72.j odd 6 1
4320.2.cc.a 48 12.b even 2 1
4320.2.cc.a 48 72.n even 6 1
4320.2.cc.b 48 24.h odd 2 1
4320.2.cc.b 48 36.f odd 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 \(S_{2}^{\mathrm{new}}(360, [\chi])\).