Properties

Label 360.2.bd.b
Level $360$
Weight $2$
Character orbit 360.bd
Analytic conductor $2.875$
Analytic rank $0$
Dimension $128$
CM no
Inner twists $8$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(2.87461447277\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(64\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 128q - 10q^{4} + 4q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 128q - 10q^{4} + 4q^{9} - 48q^{11} - 30q^{14} + 14q^{16} + 24q^{19} - 24q^{20} - 10q^{24} + 14q^{25} + 26q^{30} + 8q^{36} + 12q^{40} - 24q^{41} - 36q^{46} - 52q^{49} - 42q^{50} - 68q^{51} - 62q^{54} - 54q^{56} - 24q^{60} + 68q^{64} - 18q^{65} + 44q^{66} - 60q^{74} + 10q^{75} + 24q^{76} + 36q^{81} + 94q^{84} - 74q^{90} + 88q^{91} - 26q^{94} - 22q^{96} - 76q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
59.1 −1.41125 + 0.0914777i 1.32971 + 1.10989i 1.98326 0.258196i −1.67067 + 1.48622i −1.97809 1.44470i −2.22153 3.84780i −2.77527 + 0.545804i 0.536278 + 2.95168i 2.22179 2.25026i
59.2 −1.40842 + 0.127848i −0.679246 1.59331i 1.96731 0.360128i −2.06621 0.854849i 1.16037 + 2.15721i 1.70997 + 2.96175i −2.72476 + 0.758729i −2.07725 + 2.16449i 3.01939 + 0.939829i
59.3 −1.40586 0.153453i 1.71600 0.235231i 1.95290 + 0.431469i 2.23606 0.00425117i −2.44856 + 0.0673764i 1.34736 + 2.33369i −2.67931 0.906266i 2.88933 0.807315i −3.14425 0.337155i
59.4 −1.39143 0.252849i −0.573395 + 1.63439i 1.87213 + 0.703642i −0.0677451 2.23504i 1.21109 2.12915i −0.0861243 0.149172i −2.42702 1.45243i −2.34244 1.87430i −0.470866 + 3.12702i
59.5 −1.38690 + 0.276612i −1.70706 0.293175i 1.84697 0.767264i 2.22251 + 0.245870i 2.44861 0.0655888i 0.193132 + 0.334514i −2.34933 + 1.57501i 2.82810 + 1.00093i −3.15040 + 0.273776i
59.6 −1.37971 + 0.310484i −0.0381767 + 1.73163i 1.80720 0.856756i 0.289425 + 2.21726i −0.484971 2.40100i 2.06690 + 3.57998i −2.22740 + 1.74318i −2.99709 0.132216i −1.08775 2.96931i
59.7 −1.37227 0.341873i 1.16746 1.27946i 1.76625 + 0.938283i −0.996511 + 2.00174i −2.03949 + 1.35665i −0.0301683 0.0522531i −2.10299 1.89141i −0.274059 2.98746i 2.05182 2.40625i
59.8 −1.29321 + 0.572373i 1.41725 0.995694i 1.34478 1.48040i −0.776540 2.09690i −1.26289 + 2.09884i −0.720108 1.24726i −0.891741 + 2.68418i 1.01719 2.82229i 2.20444 + 2.26726i
59.9 −1.27900 0.603460i −1.13393 + 1.30928i 1.27167 + 1.54365i 1.63855 + 1.52157i 2.24039 0.990283i −2.19953 3.80970i −0.694935 2.74173i −0.428416 2.96925i −1.17749 2.93488i
59.10 −1.24345 0.673664i 1.46348 + 0.926406i 1.09235 + 1.67534i −1.40051 1.74315i −1.19568 2.13784i 0.618137 + 1.07065i −0.229672 2.81909i 1.28354 + 2.71155i 0.567178 + 3.11100i
59.11 −1.24141 0.677426i −1.65025 + 0.525984i 1.08219 + 1.68192i −1.85825 + 1.24374i 2.40495 + 0.464965i 1.64197 + 2.84397i −0.204056 2.82106i 2.44668 1.73602i 3.14939 0.285165i
59.12 −1.17126 0.792553i −0.681584 1.59231i 0.743721 + 1.85658i 1.18672 + 1.89518i −0.463672 + 2.40520i −0.113645 0.196839i 0.600342 2.76398i −2.07089 + 2.17058i 0.112070 3.16029i
59.13 −1.14229 + 0.833765i −1.41725 + 0.995694i 0.609671 1.90481i −2.20424 + 0.375946i 0.788739 2.31903i −0.720108 1.24726i 0.891741 + 2.68418i 1.01719 2.82229i 2.20444 2.26726i
59.14 −1.02286 0.976600i −1.56092 0.750684i 0.0925056 + 1.99786i 1.04334 1.97774i 0.863494 + 2.29224i 0.753496 + 1.30509i 1.85649 2.13388i 1.87295 + 2.34352i −2.99865 + 1.00404i
59.15 −0.958742 + 1.03962i 0.0381767 1.73163i −0.161627 1.99346i 2.06491 0.857979i 1.76364 + 1.69988i 2.06690 + 3.57998i 2.22740 + 1.74318i −2.99709 0.132216i −1.08775 + 2.96931i
59.16 −0.933002 + 1.06278i 1.70706 + 0.293175i −0.259016 1.98316i 1.32418 + 1.80181i −1.90427 + 1.54070i 0.193132 + 0.334514i 2.34933 + 1.57501i 2.82810 + 1.00093i −3.15040 0.273776i
59.17 −0.912258 1.08064i 1.72276 0.179111i −0.335569 + 1.97165i 1.97813 1.04260i −1.76516 1.69829i −2.24901 3.89541i 2.43677 1.43602i 2.93584 0.617133i −2.93124 1.18653i
59.18 −0.903164 1.08825i 0.662421 1.60037i −0.368591 + 1.96574i −1.94669 1.10018i −2.33989 + 0.724518i −1.20226 2.08238i 2.47212 1.37427i −2.12240 2.12024i 0.560906 + 3.11214i
59.19 −0.814931 + 1.15581i 0.679246 + 1.59331i −0.671775 1.88380i −1.77343 1.36197i −2.39509 0.513357i 1.70997 + 2.96175i 2.72476 + 0.758729i −2.07725 + 2.16449i 3.01939 0.939829i
59.20 −0.784848 + 1.17644i −1.32971 1.10989i −0.768027 1.84665i 0.451766 2.18996i 2.34935 0.693233i −2.22153 3.84780i 2.77527 + 0.545804i 0.536278 + 2.95168i 2.22179 + 2.25026i
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 299.64
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
8.d odd 2 1 inner
9.d odd 6 1 inner
40.e odd 2 1 inner
45.h odd 6 1 inner
72.l even 6 1 inner
360.bd 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.bd.b 128
5.b even 2 1 inner 360.2.bd.b 128
8.d odd 2 1 inner 360.2.bd.b 128
9.d odd 6 1 inner 360.2.bd.b 128
40.e odd 2 1 inner 360.2.bd.b 128
45.h odd 6 1 inner 360.2.bd.b 128
72.l even 6 1 inner 360.2.bd.b 128
360.bd even 6 1 inner 360.2.bd.b 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.bd.b 128 1.a even 1 1 trivial
360.2.bd.b 128 5.b even 2 1 inner
360.2.bd.b 128 8.d odd 2 1 inner
360.2.bd.b 128 9.d odd 6 1 inner
360.2.bd.b 128 40.e odd 2 1 inner
360.2.bd.b 128 45.h odd 6 1 inner
360.2.bd.b 128 72.l even 6 1 inner
360.2.bd.b 128 360.bd even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(34\!\cdots\!96\)\( T_{7}^{48} + \)\(49\!\cdots\!26\)\( T_{7}^{46} + \)\(59\!\cdots\!16\)\( T_{7}^{44} + \)\(60\!\cdots\!54\)\( T_{7}^{42} + \)\(53\!\cdots\!43\)\( T_{7}^{40} + \)\(39\!\cdots\!97\)\( T_{7}^{38} + \)\(24\!\cdots\!57\)\( T_{7}^{36} + \)\(13\!\cdots\!16\)\( T_{7}^{34} + \)\(59\!\cdots\!96\)\( T_{7}^{32} + \)\(21\!\cdots\!34\)\( T_{7}^{30} + \)\(65\!\cdots\!44\)\( T_{7}^{28} + \)\(15\!\cdots\!62\)\( T_{7}^{26} + \)\(30\!\cdots\!71\)\( T_{7}^{24} + \)\(46\!\cdots\!41\)\( T_{7}^{22} + \)\(52\!\cdots\!57\)\( T_{7}^{20} + \)\(41\!\cdots\!40\)\( T_{7}^{18} + \)\(21\!\cdots\!31\)\( T_{7}^{16} + \)\(41\!\cdots\!05\)\( T_{7}^{14} + \)\(58\!\cdots\!31\)\( T_{7}^{12} + \)\(39\!\cdots\!98\)\( T_{7}^{10} + \)\(18\!\cdots\!04\)\( T_{7}^{8} + \)\(46\!\cdots\!00\)\( T_{7}^{6} + \)\(80\!\cdots\!20\)\( T_{7}^{4} + 286442762400 T_{7}^{2} + 850305600 \)">\(T_{7}^{64} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(360, [\chi])\).