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Properties

Label 360.2.bf.b
Level $360$
Weight $2$
Character orbit 360.bf
Analytic conductor $2.875$
Analytic rank $0$
Dimension $92$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(2.87461447277\)
Analytic rank: \(0\)
Dimension: \(92\)
Relative dimension: \(46\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 92q + 2q^{2} - 12q^{6} - 8q^{7} + 20q^{8} + 6q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 92q + 2q^{2} - 12q^{6} - 8q^{7} + 20q^{8} + 6q^{9} + 4q^{10} + 22q^{12} - 18q^{14} - 8q^{16} + 12q^{17} - 16q^{18} - 4q^{20} - 10q^{22} + 28q^{23} + 24q^{24} + 46q^{25} - 64q^{26} - 10q^{30} + 8q^{31} - 28q^{32} - 14q^{33} - 38q^{36} - 6q^{38} - 24q^{39} + 2q^{40} - 18q^{41} + 22q^{42} - 36q^{44} + 4q^{46} - 36q^{47} - 76q^{48} - 30q^{49} - 2q^{50} + 6q^{52} + 42q^{54} + 20q^{55} - 38q^{56} - 34q^{57} + 18q^{58} - 14q^{60} + 60q^{62} - 32q^{63} - 48q^{64} + 12q^{65} + 36q^{66} + 16q^{68} + 8q^{70} - 8q^{71} + 92q^{72} - 4q^{73} + 52q^{74} + 22q^{76} + 58q^{78} + 20q^{79} - 16q^{80} + 18q^{81} - 16q^{82} + 38q^{84} - 60q^{86} - 48q^{87} - 20q^{88} - 24q^{89} - 24q^{90} + 36q^{92} - 14q^{94} - 18q^{95} - 38q^{96} - 14q^{97} - 60q^{98} + O(q^{100}) \)

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.

Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1 −1.41336 + 0.0492552i 1.32635 + 1.11391i 1.99515 − 0.139230i 0.866025 − 0.500000i −1.92947 − 1.50902i 0.206718 − 0.358047i −2.81300 + 0.295053i 0.518424 + 2.95487i −1.19937 + 0.749334i
61.2 −1.41185 − 0.0816689i −1.65748 − 0.502749i 1.98666 + 0.230609i −0.866025 + 0.500000i 2.29906 + 0.845173i 0.375568 − 0.650502i −2.78604 − 0.487834i 2.49449 + 1.66660i 1.26354 − 0.635199i
61.3 −1.34949 − 0.422946i 0.162796 − 1.72438i 1.64223 + 1.14152i 0.866025 − 0.500000i −0.949013 + 2.25818i 0.543298 − 0.941020i −1.73337 − 2.23505i −2.94699 − 0.561447i −1.38016 + 0.308462i
61.4 −1.32896 + 0.483599i 0.351155 − 1.69608i 1.53226 − 1.28537i −0.866025 + 0.500000i 0.353554 + 2.42384i −1.83657 + 3.18103i −1.41471 + 2.44920i −2.75338 − 1.19117i 0.909112 − 1.08329i
61.5 −1.32877 − 0.484123i 1.19560 + 1.25321i 1.53125 + 1.28658i −0.866025 + 0.500000i −0.981963 − 2.24405i −1.81917 + 3.15090i −1.41181 − 2.45087i −0.141089 + 2.99668i 1.39281 − 0.245121i
61.6 −1.28966 + 0.580335i 1.73199 + 0.0140878i 1.32642 − 1.49686i −0.866025 + 0.500000i −2.24185 + 0.986968i 1.38868 − 2.40527i −0.841945 + 2.70021i 2.99960 + 0.0488000i 0.826707 − 1.14741i
61.7 −1.24962 − 0.662158i −0.443022 + 1.67443i 1.12309 + 1.65489i −0.866025 + 0.500000i 1.66235 − 1.79905i 2.04026 − 3.53383i −0.307637 − 2.81165i −2.60746 − 1.48362i 1.41328 − 0.0513631i
61.8 −1.24717 − 0.666764i −1.64287 + 0.548626i 1.11085 + 1.66313i 0.866025 − 0.500000i 2.41473 + 0.411176i −0.740799 + 1.28310i −0.276500 − 2.81488i 2.39802 − 1.80264i −1.41346 + 0.0461488i
61.9 −1.12757 + 0.853578i 0.994723 − 1.41793i 0.542810 − 1.92493i 0.866025 − 0.500000i 0.0886978 + 2.44788i −0.293997 + 0.509218i 1.03102 + 2.63382i −1.02105 − 2.82090i −0.549712 + 1.30300i
61.10 −1.04483 − 0.953069i −0.410386 − 1.68273i 0.183320 + 1.99158i −0.866025 + 0.500000i −1.17498 + 2.14929i −1.05202 + 1.82216i 1.70658 − 2.25557i −2.66317 + 1.38114i 1.38138 + 0.302969i
61.11 −0.941863 + 1.05494i 0.904267 + 1.47726i −0.225788 − 1.98721i 0.866025 − 0.500000i −2.41011 − 0.437433i −0.852557 + 1.47667i 2.30905 + 1.63349i −1.36460 + 2.67168i −0.288208 + 1.38453i
61.12 −0.911534 + 1.08125i −1.64095 + 0.554332i −0.338210 − 1.97120i −0.866025 + 0.500000i 0.896409 − 2.27957i 1.02186 − 1.76992i 2.43965 + 1.43112i 2.38543 − 1.81926i 0.248786 − 1.39216i
61.13 −0.884696 + 1.10332i −1.72914 − 0.100320i −0.434624 − 1.95220i 0.866025 − 0.500000i 1.64045 − 1.81904i −2.03225 + 3.51996i 2.53841 + 1.24758i 2.97987 + 0.346937i −0.214510 + 1.39785i
61.14 −0.860361 − 1.12240i 1.69766 − 0.343430i −0.519559 + 1.93134i 0.866025 − 0.500000i −1.84607 − 1.60998i −2.54751 + 4.41241i 2.61474 − 1.07849i 2.76411 − 1.16605i −1.30629 − 0.541846i
61.15 −0.785855 − 1.17577i −1.36869 − 1.06146i −0.764864 + 1.84797i 0.866025 − 0.500000i −0.172445 + 2.44341i 2.36602 − 4.09807i 2.77385 − 0.552930i 0.746600 + 2.90561i −1.26845 − 0.625318i
61.16 −0.705423 − 1.22572i 1.15336 − 1.29220i −1.00476 + 1.72930i −0.866025 + 0.500000i −2.39747 − 0.502143i 0.807871 − 1.39927i 2.82840 + 0.0116626i −0.339538 − 2.98072i 1.22377 + 0.708789i
61.17 −0.513154 + 1.31783i 1.72914 + 0.100320i −1.47335 − 1.35250i −0.866025 + 0.500000i −1.01952 + 2.22723i −2.03225 + 3.51996i 2.53841 − 1.24758i 2.97987 + 0.346937i −0.214510 − 1.39785i
61.18 −0.480624 + 1.33004i 1.64095 − 0.554332i −1.53800 − 1.27850i 0.866025 − 0.500000i −0.0513970 + 2.44895i 1.02186 − 1.76992i 2.43965 − 1.43112i 2.38543 − 1.81926i 0.248786 + 1.39216i
61.19 −0.442672 + 1.34315i −0.904267 − 1.47726i −1.60808 − 1.18915i −0.866025 + 0.500000i 2.38447 − 0.560621i −0.852557 + 1.47667i 2.30905 − 1.63349i −1.36460 + 2.67168i −0.288208 − 1.38453i
61.20 −0.332127 − 1.37466i 1.42280 + 0.987751i −1.77938 + 0.913123i −0.866025 + 0.500000i 0.885274 − 2.28392i 0.647320 − 1.12119i 1.84622 + 2.14278i 1.04869 + 2.81074i 0.974960 + 1.02443i
See all 92 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 301.46
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
9.c even 3 1 inner
72.n 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.bf.b ✓ 92
3.b odd 2 1 1080.2.bf.b 92
4.b odd 2 1 1440.2.bv.b 92
8.b even 2 1 inner 360.2.bf.b ✓ 92
8.d odd 2 1 1440.2.bv.b 92
9.c even 3 1 inner 360.2.bf.b ✓ 92
9.d odd 6 1 1080.2.bf.b 92
12.b even 2 1 4320.2.bv.b 92
24.f even 2 1 4320.2.bv.b 92
24.h odd 2 1 1080.2.bf.b 92
36.f odd 6 1 1440.2.bv.b 92
36.h even 6 1 4320.2.bv.b 92
72.j odd 6 1 1080.2.bf.b 92
72.l even 6 1 4320.2.bv.b 92
72.n even 6 1 inner 360.2.bf.b ✓ 92
72.p odd 6 1 1440.2.bv.b 92
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.bf.b ✓ 92 1.a even 1 1 trivial
360.2.bf.b ✓ 92 8.b even 2 1 inner
360.2.bf.b ✓ 92 9.c even 3 1 inner
360.2.bf.b ✓ 92 72.n even 6 1 inner
1080.2.bf.b 92 3.b odd 2 1
1080.2.bf.b 92 9.d odd 6 1
1080.2.bf.b 92 24.h odd 2 1
1080.2.bf.b 92 72.j odd 6 1
1440.2.bv.b 92 4.b odd 2 1
1440.2.bv.b 92 8.d odd 2 1
1440.2.bv.b 92 36.f odd 6 1
1440.2.bv.b 92 72.p odd 6 1
4320.2.bv.b 92 12.b even 2 1
4320.2.bv.b 92 24.f even 2 1
4320.2.bv.b 92 36.h even 6 1
4320.2.bv.b 92 72.l even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(13\!\cdots\!44\)\( T_{7}^{28} + \)\(18\!\cdots\!24\)\( T_{7}^{27} + \)\(87\!\cdots\!85\)\( T_{7}^{26} + \)\(10\!\cdots\!56\)\( T_{7}^{25} + \)\(45\!\cdots\!52\)\( T_{7}^{24} + \)\(43\!\cdots\!68\)\( T_{7}^{23} + \)\(18\!\cdots\!01\)\( T_{7}^{22} + \)\(14\!\cdots\!16\)\( T_{7}^{21} + \)\(62\!\cdots\!20\)\( T_{7}^{20} + \)\(38\!\cdots\!84\)\( T_{7}^{19} + \)\(16\!\cdots\!18\)\( T_{7}^{18} + \)\(74\!\cdots\!44\)\( T_{7}^{17} + \)\(35\!\cdots\!20\)\( T_{7}^{16} + \)\(10\!\cdots\!28\)\( T_{7}^{15} + \)\(59\!\cdots\!91\)\( T_{7}^{14} + \)\(87\!\cdots\!16\)\( T_{7}^{13} + \)\(77\!\cdots\!48\)\( T_{7}^{12} + \)\(27\!\cdots\!92\)\( T_{7}^{11} + \)\(74\!\cdots\!99\)\( T_{7}^{10} - \)\(28\!\cdots\!60\)\( T_{7}^{9} + \)\(50\!\cdots\!52\)\( T_{7}^{8} - \)\(20\!\cdots\!84\)\( T_{7}^{7} + \)\(20\!\cdots\!69\)\( T_{7}^{6} + \)\(29\!\cdots\!32\)\( T_{7}^{5} + \)\(55\!\cdots\!48\)\( T_{7}^{4} + \)\(48\!\cdots\!76\)\( T_{7}^{3} + \)\(68\!\cdots\!20\)\( T_{7}^{2} + \)\(49\!\cdots\!28\)\( T_{7} + \)\(55\!\cdots\!24\)\( \)">\(T_{7}^{46} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(360, [\chi])\).