# Properties

 Label 288.2.bf.a Level $288$ Weight $2$ Character orbit 288.bf Analytic conductor $2.300$ Analytic rank $0$ Dimension $368$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$288 = 2^{5} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 288.bf (of order $$24$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.29969157821$$ Analytic rank: $$0$$ Dimension: $$368$$ Relative dimension: $$46$$ over $$\Q(\zeta_{24})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{24}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$368q - 12q^{2} - 8q^{3} - 4q^{4} - 12q^{5} - 8q^{6} - 4q^{7} - 8q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$368q - 12q^{2} - 8q^{3} - 4q^{4} - 12q^{5} - 8q^{6} - 4q^{7} - 8q^{9} - 16q^{10} - 12q^{11} - 8q^{12} - 4q^{13} - 12q^{14} - 16q^{15} - 4q^{16} - 8q^{18} - 16q^{19} - 12q^{20} - 8q^{21} - 4q^{22} - 12q^{23} + 32q^{24} - 4q^{25} + 16q^{27} - 16q^{28} - 12q^{29} - 56q^{30} - 12q^{32} - 16q^{33} - 12q^{34} - 60q^{36} - 16q^{37} - 12q^{38} + 16q^{39} - 4q^{40} - 12q^{41} - 8q^{42} - 4q^{43} - 8q^{45} - 16q^{46} - 24q^{47} - 60q^{48} - 168q^{50} - 32q^{51} - 4q^{52} - 52q^{54} - 16q^{55} - 12q^{56} - 8q^{57} + 32q^{58} - 12q^{59} - 20q^{60} - 4q^{61} - 16q^{64} - 24q^{65} - 80q^{66} - 4q^{67} - 60q^{68} - 8q^{69} - 4q^{70} + 52q^{72} - 16q^{73} - 12q^{74} - 28q^{75} - 28q^{76} - 12q^{77} + 80q^{78} - 8q^{79} - 16q^{82} - 132q^{83} - 104q^{84} - 24q^{85} - 12q^{86} - 64q^{87} - 4q^{88} + 124q^{90} - 16q^{91} + 216q^{92} - 20q^{93} - 20q^{94} + 92q^{96} - 8q^{97} - 72q^{99} + O(q^{100})$$

## 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 −1.41395 + 0.0275526i 1.72268 0.179925i 1.99848 0.0779158i −0.211536 1.60678i −2.43082 + 0.301868i 0.419720 + 1.56642i −2.82360 + 0.165232i 2.93525 0.619906i 0.343371 + 2.26607i
11.2 −1.41252 + 0.0691722i −0.126594 + 1.72742i 1.99043 0.195414i 0.386920 + 2.93895i 0.0593276 2.44877i 0.887390 + 3.31179i −2.79801 + 0.413709i −2.96795 0.437362i −0.749827 4.12457i
11.3 −1.40115 + 0.191775i −1.72586 + 0.146277i 1.92644 0.537410i 0.234629 + 1.78218i 2.39014 0.535934i −0.927052 3.45980i −2.59618 + 1.12244i 2.95721 0.504910i −0.670528 2.45211i
11.4 −1.34246 0.444750i −0.318231 1.70257i 1.60439 + 1.19412i 0.201801 + 1.53283i −0.330005 + 2.42716i 0.765564 + 2.85712i −1.62275 2.31661i −2.79746 + 1.08362i 0.410817 2.14751i
11.5 −1.31749 0.514013i −1.60170 0.659214i 1.47158 + 1.35442i −0.503783 3.82661i 1.77138 + 1.69180i −0.191129 0.713303i −1.24261 2.54085i 2.13087 + 2.11172i −1.30320 + 5.30049i
11.6 −1.31237 0.526966i 1.34553 1.09066i 1.44461 + 1.38314i 0.197197 + 1.49786i −2.34057 + 0.722304i −0.749169 2.79594i −1.16699 2.57646i 0.620902 2.93504i 0.530525 2.06965i
11.7 −1.28187 + 0.597341i −1.23148 + 1.21798i 1.28637 1.53142i −0.451380 3.42857i 0.851043 2.29689i 0.762324 + 2.84503i −0.734171 + 2.73148i 0.0330693 2.99982i 2.62663 + 4.12534i
11.8 −1.26591 + 0.630463i −1.05404 1.37441i 1.20503 1.59621i 0.00886053 + 0.0673024i 2.20083 + 1.07534i 0.666754 + 2.48836i −0.519104 + 2.78038i −0.778009 + 2.89736i −0.0536483 0.0796122i
11.9 −1.18241 0.775832i −0.200716 + 1.72038i 0.796170 + 1.83470i −0.263676 2.00282i 1.57205 1.87847i −0.327090 1.22072i 0.482020 2.78705i −2.91943 0.690617i −1.24208 + 2.57271i
11.10 −1.14354 + 0.832059i 0.897985 1.48109i 0.615355 1.90298i −0.289571 2.19951i 0.205474 + 2.44086i −0.477791 1.78314i 0.879712 + 2.68814i −1.38725 2.65999i 2.16126 + 2.27428i
11.11 −1.13651 + 0.841635i 1.61183 + 0.634044i 0.583300 1.91305i 0.335718 + 2.55003i −2.36549 + 0.635975i −0.254371 0.949325i 0.947165 + 2.66512i 2.19598 + 2.04394i −2.52774 2.61558i
11.12 −0.979102 1.02047i 1.47070 + 0.914907i −0.0827170 + 1.99829i −0.257196 1.95359i −0.506327 2.39659i 1.34022 + 5.00178i 2.12018 1.87212i 1.32589 + 2.69110i −1.74176 + 2.17523i
11.13 −0.939466 + 1.05707i −0.494102 + 1.66008i −0.234807 1.98617i −0.0359748 0.273256i −1.29063 2.08189i −1.13217 4.22533i 2.32012 + 1.61773i −2.51173 1.64050i 0.322648 + 0.218686i
11.14 −0.931130 1.06442i −1.32517 + 1.11531i −0.265992 + 1.98223i 0.240456 + 1.82644i 2.42107 + 0.372039i −0.327296 1.22149i 2.35761 1.56259i 0.512151 2.95596i 1.72021 1.95660i
11.15 −0.749302 1.19939i 0.475486 1.66551i −0.877093 + 1.79742i −0.384557 2.92100i −2.35388 + 0.677673i −0.163982 0.611989i 2.81302 0.294828i −2.54783 1.58385i −3.21528 + 2.64995i
11.16 −0.747170 1.20072i −1.18400 1.26417i −0.883475 + 1.79429i 0.396975 + 3.01532i −0.633273 + 2.36621i −0.375906 1.40290i 2.81455 0.279828i −0.196274 + 2.99357i 3.32396 2.72962i
11.17 −0.564220 + 1.29679i 0.888753 1.48665i −1.36331 1.46335i 0.132037 + 1.00292i 1.42641 + 1.99132i 1.09866 + 4.10026i 2.66685 0.942277i −1.42024 2.64252i −1.37507 0.394642i
11.18 −0.514835 + 1.31717i −0.894226 1.48336i −1.46989 1.35625i 0.446317 + 3.39012i 2.41422 0.414163i −0.836733 3.12273i 2.54317 1.23785i −1.40072 + 2.65292i −4.69515 1.15747i
11.19 −0.484847 + 1.32850i 0.622180 + 1.61644i −1.52985 1.28824i −0.0345958 0.262781i −2.44912 + 0.0428399i 0.779563 + 2.90937i 2.45318 1.40781i −2.22579 + 2.01144i 0.365879 + 0.0814479i
11.20 −0.370153 1.36491i 1.60090 + 0.661153i −1.72597 + 1.01045i −0.155947 1.18454i 0.309839 2.42981i −1.23982 4.62708i 2.01805 + 1.98178i 2.12575 + 2.11688i −1.55907 + 0.651315i
See next 80 embeddings (of 368 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 275.46 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
32.h odd 8 1 inner
288.bf even 24 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.2.bf.a 368
3.b odd 2 1 864.2.bn.a 368
9.c even 3 1 864.2.bn.a 368
9.d odd 6 1 inner 288.2.bf.a 368
32.h odd 8 1 inner 288.2.bf.a 368
96.o even 8 1 864.2.bn.a 368
288.bd odd 24 1 864.2.bn.a 368
288.bf even 24 1 inner 288.2.bf.a 368

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.bf.a 368 1.a even 1 1 trivial
288.2.bf.a 368 9.d odd 6 1 inner
288.2.bf.a 368 32.h odd 8 1 inner
288.2.bf.a 368 288.bf even 24 1 inner
864.2.bn.a 368 3.b odd 2 1
864.2.bn.a 368 9.c even 3 1
864.2.bn.a 368 96.o even 8 1
864.2.bn.a 368 288.bd odd 24 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(288, [\chi])$$.