Properties

Label 360.2.br.e
Level $360$
Weight $2$
Character orbit 360.br
Analytic conductor $2.875$
Analytic rank $0$
Dimension $256$
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.br (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.87461447277\)
Analytic rank: \(0\)
Dimension: \(256\)
Relative dimension: \(64\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 256q + 6q^{2} - 8q^{6} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 256q + 6q^{2} - 8q^{6} - 10q^{12} + 28q^{15} + 12q^{16} - 28q^{18} - 54q^{20} + 14q^{22} - 28q^{25} - 32q^{28} + 14q^{30} - 32q^{31} - 114q^{32} + 4q^{33} - 40q^{36} - 30q^{38} + 46q^{40} + 24q^{41} - 10q^{42} - 16q^{46} + 24q^{47} - 2q^{48} + 78q^{50} + 38q^{52} - 8q^{55} - 96q^{56} - 80q^{57} - 18q^{58} - 2q^{60} - 144q^{63} - 84q^{65} - 4q^{66} - 30q^{68} - 30q^{70} - 86q^{72} + 64q^{73} + 16q^{76} - 82q^{78} + 72q^{81} - 64q^{82} + 48q^{86} - 4q^{87} + 38q^{88} + 78q^{90} - 108q^{92} - 24q^{95} - 116q^{96} + 36q^{97} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
77.1 −1.41351 0.0447368i 0.131650 + 1.72704i 1.99600 + 0.126472i −0.531426 2.17200i −0.108826 2.44707i 0.193986 0.723964i −2.81570 0.268063i −2.96534 + 0.454729i 0.654006 + 3.09391i
77.2 −1.40492 + 0.161860i 1.19421 1.25454i 1.94760 0.454802i −1.72814 + 1.41899i −1.47470 + 1.95582i 0.183802 0.685960i −2.66261 + 0.954200i −0.147746 2.99636i 2.19822 2.27328i
77.3 −1.40115 + 0.191780i 0.925695 + 1.46393i 1.92644 0.537425i −0.274171 + 2.21920i −1.57779 1.87365i 0.362508 1.35290i −2.59616 + 1.12247i −1.28618 + 2.71030i −0.0414427 3.16201i
77.4 −1.38126 0.303534i 1.72383 + 0.168545i 1.81573 + 0.838517i −0.461790 2.18786i −2.32989 0.756046i −0.976425 + 3.64407i −2.25347 1.70934i 2.94318 + 0.581088i −0.0262424 + 3.16217i
77.5 −1.38022 + 0.308205i −1.08549 1.34971i 1.81002 0.850781i −2.00950 0.980772i 1.91421 + 1.52834i −0.563716 + 2.10382i −2.23601 + 1.73212i −0.643411 + 2.93019i 3.07583 + 0.734345i
77.6 −1.37718 0.321532i 1.63984 0.557615i 1.79323 + 0.885613i 2.22426 + 0.229476i −2.43764 + 0.240675i 0.712972 2.66085i −2.18485 1.79623i 2.37813 1.82880i −2.98942 1.03120i
77.7 −1.35344 0.410134i −1.63984 + 0.557615i 1.66358 + 1.11018i −2.22426 0.229476i 2.44811 0.0821442i 0.712972 2.66085i −1.79623 2.18485i 2.37813 1.82880i 2.91628 + 1.22283i
77.8 −1.34797 0.427759i −1.72383 0.168545i 1.63404 + 1.15321i 0.461790 + 2.18786i 2.25157 + 0.964579i −0.976425 + 3.64407i −1.70934 2.25347i 2.94318 + 0.581088i 0.313401 3.14671i
77.9 −1.24650 0.668010i −0.131650 1.72704i 1.10753 + 1.66535i 0.531426 + 2.17200i −0.989578 + 2.24070i 0.193986 0.723964i −0.268063 2.81570i −2.96534 + 0.454729i 0.788494 3.06240i
77.10 −1.15671 + 0.813649i 0.674078 1.59550i 0.675951 1.88231i 0.576180 2.16056i 0.518465 + 2.39399i 0.331473 1.23707i 0.749662 + 2.72727i −2.09124 2.15098i 1.09146 + 2.96795i
77.11 −1.14969 + 0.823530i −1.51198 + 0.844934i 0.643596 1.89362i −0.944749 + 2.02668i 1.04249 2.21658i 0.661416 2.46844i 0.819513 + 2.70710i 1.57217 2.55505i −0.582864 3.10810i
77.12 −1.13577 0.842635i −1.19421 + 1.25454i 0.579931 + 1.91407i 1.72814 1.41899i 2.41346 0.418586i 0.183802 0.685960i 0.954200 2.66261i −0.147746 2.99636i −3.15845 + 0.155449i
77.13 −1.11797 + 0.866111i 1.51581 0.838041i 0.499705 1.93657i 1.47125 + 1.68387i −0.968794 + 2.24976i −1.06946 + 3.99129i 1.11863 + 2.59782i 1.59537 2.54063i −3.10323 0.608245i
77.14 −1.11754 0.866661i −0.925695 1.46393i 0.497797 + 1.93706i 0.274171 2.21920i −0.234228 + 2.43827i 0.362508 1.35290i 1.12247 2.59616i −1.28618 + 2.71030i −2.22969 + 2.24243i
77.15 −1.04120 0.957024i 1.08549 + 1.34971i 0.168211 + 1.99291i 2.00950 + 0.980772i 0.161481 2.44416i −0.563716 + 2.10382i 1.73212 2.23601i −0.643411 + 2.93019i −1.15368 2.94432i
77.16 −0.967822 + 1.03117i 1.22306 + 1.22643i −0.126641 1.99599i −2.20381 + 0.378455i −2.44836 + 0.0742232i −1.11197 + 4.14991i 2.18078 + 1.80117i −0.00825140 + 2.99999i 1.74264 2.63879i
77.17 −0.922014 + 1.07233i −1.71827 + 0.218017i −0.299780 1.97741i 0.420988 2.19608i 1.35049 2.04357i −0.688838 + 2.57078i 2.39683 + 1.50173i 2.90494 0.749227i 1.96676 + 2.47626i
77.18 −0.890307 + 1.09880i −0.176815 + 1.72300i −0.414706 1.95653i −1.48043 1.67581i −1.73581 1.72829i 0.827385 3.08784i 2.51905 + 1.28624i −2.93747 0.609306i 3.15941 0.134701i
77.19 −0.745090 + 1.20202i −0.272060 1.71055i −0.889682 1.79122i −1.63914 + 1.52092i 2.25882 + 0.947494i 0.619613 2.31243i 2.81597 + 0.265208i −2.85197 + 0.930744i −0.606863 3.10350i
77.20 −0.613736 + 1.27410i −0.853956 1.50690i −1.24666 1.56392i 2.16728 + 0.550360i 2.44405 0.163184i −0.556805 + 2.07802i 2.75771 0.628530i −1.54152 + 2.57366i −2.03135 + 2.42355i
See next 80 embeddings (of 256 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 317.64
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
8.b even 2 1 inner
9.d odd 6 1 inner
40.i odd 4 1 inner
45.l even 12 1 inner
72.j odd 6 1 inner
360.br even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 360.2.br.e 256
5.c odd 4 1 inner 360.2.br.e 256
8.b even 2 1 inner 360.2.br.e 256
9.d odd 6 1 inner 360.2.br.e 256
40.i odd 4 1 inner 360.2.br.e 256
45.l even 12 1 inner 360.2.br.e 256
72.j odd 6 1 inner 360.2.br.e 256
360.br even 12 1 inner 360.2.br.e 256
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.br.e 256 1.a even 1 1 trivial
360.2.br.e 256 5.c odd 4 1 inner
360.2.br.e 256 8.b even 2 1 inner
360.2.br.e 256 9.d odd 6 1 inner
360.2.br.e 256 40.i odd 4 1 inner
360.2.br.e 256 45.l even 12 1 inner
360.2.br.e 256 72.j odd 6 1 inner
360.2.br.e 256 360.br even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(360, [\chi])\):

\(11\!\cdots\!28\)\( T_{7}^{109} - \)\(17\!\cdots\!48\)\( T_{7}^{108} + \)\(35\!\cdots\!68\)\( T_{7}^{107} + \)\(14\!\cdots\!64\)\( T_{7}^{106} + \)\(42\!\cdots\!12\)\( T_{7}^{105} + \)\(49\!\cdots\!02\)\( T_{7}^{104} - \)\(12\!\cdots\!52\)\( T_{7}^{103} - \)\(51\!\cdots\!40\)\( T_{7}^{102} - \)\(12\!\cdots\!36\)\( T_{7}^{101} - \)\(11\!\cdots\!96\)\( T_{7}^{100} + \)\(35\!\cdots\!52\)\( T_{7}^{99} + \)\(14\!\cdots\!76\)\( T_{7}^{98} + \)\(26\!\cdots\!40\)\( T_{7}^{97} + \)\(19\!\cdots\!66\)\( T_{7}^{96} - \)\(73\!\cdots\!96\)\( T_{7}^{95} - \)\(30\!\cdots\!28\)\( T_{7}^{94} - \)\(45\!\cdots\!60\)\( T_{7}^{93} - \)\(25\!\cdots\!36\)\( T_{7}^{92} + \)\(11\!\cdots\!96\)\( T_{7}^{91} + \)\(50\!\cdots\!40\)\( T_{7}^{90} + \)\(58\!\cdots\!12\)\( T_{7}^{89} + \)\(27\!\cdots\!74\)\( T_{7}^{88} - \)\(12\!\cdots\!92\)\( T_{7}^{87} - \)\(61\!\cdots\!68\)\( T_{7}^{86} - \)\(57\!\cdots\!80\)\( T_{7}^{85} - \)\(22\!\cdots\!28\)\( T_{7}^{84} + \)\(11\!\cdots\!00\)\( T_{7}^{83} + \)\(57\!\cdots\!60\)\( T_{7}^{82} + \)\(43\!\cdots\!60\)\( T_{7}^{81} + \)\(13\!\cdots\!65\)\( T_{7}^{80} - \)\(74\!\cdots\!84\)\( T_{7}^{79} - \)\(40\!\cdots\!24\)\( T_{7}^{78} - \)\(25\!\cdots\!36\)\( T_{7}^{77} - \)\(67\!\cdots\!80\)\( T_{7}^{76} + \)\(40\!\cdots\!04\)\( T_{7}^{75} + \)\(21\!\cdots\!92\)\( T_{7}^{74} + \)\(11\!\cdots\!64\)\( T_{7}^{73} + \)\(25\!\cdots\!50\)\( T_{7}^{72} - \)\(17\!\cdots\!52\)\( T_{7}^{71} - \)\(91\!\cdots\!12\)\( T_{7}^{70} - \)\(40\!\cdots\!24\)\( T_{7}^{69} - \)\(73\!\cdots\!52\)\( T_{7}^{68} + \)\(57\!\cdots\!76\)\( T_{7}^{67} + \)\(29\!\cdots\!92\)\( T_{7}^{66} + \)\(10\!\cdots\!92\)\( T_{7}^{65} + \)\(16\!\cdots\!58\)\( T_{7}^{64} - \)\(14\!\cdots\!52\)\( T_{7}^{63} - \)\(69\!\cdots\!00\)\( T_{7}^{62} - \)\(21\!\cdots\!48\)\( T_{7}^{61} - \)\(27\!\cdots\!24\)\( T_{7}^{60} + \)\(26\!\cdots\!04\)\( T_{7}^{59} + \)\(12\!\cdots\!60\)\( T_{7}^{58} + \)\(30\!\cdots\!56\)\( T_{7}^{57} + \)\(35\!\cdots\!78\)\( T_{7}^{56} - \)\(29\!\cdots\!24\)\( T_{7}^{55} - \)\(13\!\cdots\!88\)\( T_{7}^{54} - \)\(32\!\cdots\!28\)\( T_{7}^{53} - \)\(39\!\cdots\!28\)\( T_{7}^{52} + \)\(11\!\cdots\!72\)\( T_{7}^{51} + \)\(10\!\cdots\!56\)\( T_{7}^{50} + \)\(25\!\cdots\!08\)\( T_{7}^{49} + \)\(36\!\cdots\!01\)\( T_{7}^{48} + \)\(14\!\cdots\!44\)\( T_{7}^{47} - \)\(38\!\cdots\!28\)\( T_{7}^{46} - \)\(12\!\cdots\!24\)\( T_{7}^{45} - \)\(23\!\cdots\!00\)\( T_{7}^{44} - \)\(22\!\cdots\!68\)\( T_{7}^{43} - \)\(58\!\cdots\!52\)\( T_{7}^{42} + \)\(26\!\cdots\!76\)\( T_{7}^{41} + \)\(79\!\cdots\!98\)\( T_{7}^{40} + \)\(13\!\cdots\!48\)\( T_{7}^{39} + \)\(16\!\cdots\!44\)\( T_{7}^{38} + \)\(16\!\cdots\!24\)\( T_{7}^{37} + \)\(13\!\cdots\!04\)\( T_{7}^{36} + \)\(74\!\cdots\!88\)\( T_{7}^{35} + \)\(14\!\cdots\!44\)\( T_{7}^{34} - \)\(30\!\cdots\!40\)\( T_{7}^{33} - \)\(50\!\cdots\!67\)\( T_{7}^{32} - \)\(40\!\cdots\!96\)\( T_{7}^{31} - \)\(16\!\cdots\!64\)\( T_{7}^{30} + \)\(75\!\cdots\!20\)\( T_{7}^{29} + \)\(24\!\cdots\!44\)\( T_{7}^{28} + \)\(27\!\cdots\!68\)\( T_{7}^{27} + \)\(21\!\cdots\!28\)\( T_{7}^{26} + \)\(12\!\cdots\!12\)\( T_{7}^{25} + \)\(38\!\cdots\!60\)\( T_{7}^{24} - \)\(11\!\cdots\!96\)\( T_{7}^{23} - \)\(24\!\cdots\!88\)\( T_{7}^{22} - \)\(20\!\cdots\!68\)\( T_{7}^{21} - \)\(95\!\cdots\!16\)\( T_{7}^{20} - \)\(25\!\cdots\!64\)\( T_{7}^{19} + \)\(31\!\cdots\!20\)\( T_{7}^{18} + \)\(33\!\cdots\!08\)\( T_{7}^{17} + \)\(21\!\cdots\!80\)\( T_{7}^{16} + \)\(82\!\cdots\!92\)\( T_{7}^{15} + \)\(18\!\cdots\!52\)\( T_{7}^{14} - \)\(43\!\cdots\!36\)\( T_{7}^{13} - \)\(70\!\cdots\!64\)\( T_{7}^{12} - \)\(26\!\cdots\!64\)\( T_{7}^{11} - \)\(24\!\cdots\!84\)\( T_{7}^{10} + \)\(21\!\cdots\!64\)\( T_{7}^{9} + \)\(15\!\cdots\!72\)\( T_{7}^{8} + \)\(37\!\cdots\!80\)\( T_{7}^{7} + \)\(60\!\cdots\!76\)\( T_{7}^{6} + \)\(11\!\cdots\!16\)\( T_{7}^{5} + \)\(12\!\cdots\!32\)\( T_{7}^{4} + \)\(42\!\cdots\!04\)\( T_{7}^{3} + \)\(10\!\cdots\!88\)\( T_{7}^{2} + \)\(33\!\cdots\!24\)\( T_{7} + \)\(55\!\cdots\!76\)\( \)">\(T_{7}^{128} - \cdots\)
\(72\!\cdots\!87\)\( T_{11}^{116} + \)\(45\!\cdots\!98\)\( T_{11}^{114} + \)\(25\!\cdots\!21\)\( T_{11}^{112} + \)\(12\!\cdots\!00\)\( T_{11}^{110} + \)\(53\!\cdots\!26\)\( T_{11}^{108} + \)\(21\!\cdots\!92\)\( T_{11}^{106} + \)\(74\!\cdots\!33\)\( T_{11}^{104} + \)\(24\!\cdots\!62\)\( T_{11}^{102} + \)\(72\!\cdots\!67\)\( T_{11}^{100} + \)\(19\!\cdots\!98\)\( T_{11}^{98} + \)\(49\!\cdots\!37\)\( T_{11}^{96} + \)\(11\!\cdots\!32\)\( T_{11}^{94} + \)\(24\!\cdots\!02\)\( T_{11}^{92} + \)\(47\!\cdots\!96\)\( T_{11}^{90} + \)\(86\!\cdots\!05\)\( T_{11}^{88} + \)\(14\!\cdots\!98\)\( T_{11}^{86} + \)\(22\!\cdots\!95\)\( T_{11}^{84} + \)\(32\!\cdots\!14\)\( T_{11}^{82} + \)\(43\!\cdots\!48\)\( T_{11}^{80} + \)\(52\!\cdots\!94\)\( T_{11}^{78} + \)\(59\!\cdots\!17\)\( T_{11}^{76} + \)\(62\!\cdots\!90\)\( T_{11}^{74} + \)\(59\!\cdots\!01\)\( T_{11}^{72} + \)\(52\!\cdots\!24\)\( T_{11}^{70} + \)\(42\!\cdots\!04\)\( T_{11}^{68} + \)\(30\!\cdots\!04\)\( T_{11}^{66} + \)\(20\!\cdots\!20\)\( T_{11}^{64} + \)\(12\!\cdots\!88\)\( T_{11}^{62} + \)\(67\!\cdots\!36\)\( T_{11}^{60} + \)\(32\!\cdots\!76\)\( T_{11}^{58} + \)\(14\!\cdots\!40\)\( T_{11}^{56} + \)\(54\!\cdots\!32\)\( T_{11}^{54} + \)\(18\!\cdots\!00\)\( T_{11}^{52} + \)\(53\!\cdots\!20\)\( T_{11}^{50} + \)\(13\!\cdots\!68\)\( T_{11}^{48} + \)\(29\!\cdots\!32\)\( T_{11}^{46} + \)\(55\!\cdots\!56\)\( T_{11}^{44} + \)\(86\!\cdots\!96\)\( T_{11}^{42} + \)\(11\!\cdots\!76\)\( T_{11}^{40} + \)\(12\!\cdots\!20\)\( T_{11}^{38} + \)\(11\!\cdots\!64\)\( T_{11}^{36} + \)\(83\!\cdots\!84\)\( T_{11}^{34} + \)\(50\!\cdots\!96\)\( T_{11}^{32} + \)\(25\!\cdots\!68\)\( T_{11}^{30} + \)\(98\!\cdots\!12\)\( T_{11}^{28} + \)\(30\!\cdots\!84\)\( T_{11}^{26} + \)\(69\!\cdots\!72\)\( T_{11}^{24} + \)\(11\!\cdots\!84\)\( T_{11}^{22} + \)\(13\!\cdots\!12\)\( T_{11}^{20} + \)\(98\!\cdots\!96\)\( T_{11}^{18} + \)\(49\!\cdots\!60\)\( T_{11}^{16} + \)\(11\!\cdots\!68\)\( T_{11}^{14} + \)\(18\!\cdots\!48\)\( T_{11}^{12} + \)\(18\!\cdots\!36\)\( T_{11}^{10} + \)\(13\!\cdots\!84\)\( T_{11}^{8} + \)\(57\!\cdots\!88\)\( T_{11}^{6} + \)\(16\!\cdots\!68\)\( T_{11}^{4} + \)\(45\!\cdots\!12\)\( T_{11}^{2} + \)\(11\!\cdots\!16\)\( \)">\(T_{11}^{128} + \cdots\)