# Properties

 Label 864.2.bt.a Level 864 Weight 2 Character orbit 864.bt Analytic conductor 6.899 Analytic rank 0 Dimension 3408 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$864 = 2^{5} \cdot 3^{3}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 864.bt (of order $$72$$, degree $$24$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.89907473464$$ Analytic rank: $$0$$ Dimension: $$3408$$ Relative dimension: $$142$$ over $$\Q(\zeta_{72})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{72}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$3408q - 24q^{2} - 24q^{3} - 24q^{4} - 24q^{5} - 24q^{6} - 24q^{7} - 36q^{8} - 24q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$3408q - 24q^{2} - 24q^{3} - 24q^{4} - 24q^{5} - 24q^{6} - 24q^{7} - 36q^{8} - 24q^{9} - 12q^{10} - 24q^{11} - 24q^{12} - 24q^{13} - 24q^{14} - 48q^{15} - 24q^{16} - 24q^{18} - 12q^{19} - 24q^{20} - 24q^{21} - 24q^{22} - 24q^{23} - 84q^{24} - 24q^{25} - 24q^{27} - 48q^{28} - 24q^{29} - 24q^{30} - 24q^{32} - 48q^{33} - 36q^{35} - 24q^{36} - 12q^{37} - 24q^{38} - 24q^{39} - 24q^{40} - 24q^{41} + 96q^{42} - 24q^{43} - 36q^{44} - 24q^{45} - 12q^{46} - 48q^{47} - 24q^{48} - 336q^{50} + 12q^{51} - 24q^{52} - 24q^{54} - 48q^{55} - 24q^{56} - 24q^{57} - 132q^{58} - 24q^{59} - 24q^{60} - 24q^{61} - 36q^{62} - 12q^{64} - 48q^{65} - 216q^{66} - 24q^{67} + 24q^{68} - 24q^{69} - 24q^{70} - 36q^{71} - 24q^{72} - 12q^{73} - 24q^{74} - 24q^{75} - 24q^{76} - 24q^{77} - 24q^{78} - 48q^{79} - 48q^{82} + 96q^{83} + 204q^{84} + 36q^{85} - 24q^{86} - 24q^{87} - 24q^{88} - 36q^{89} - 24q^{90} - 12q^{91} - 252q^{92} - 24q^{93} - 24q^{94} - 24q^{96} - 48q^{97} - 36q^{98} - 24q^{99} + O(q^{100})$$

## 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 −1.41414 0.0139384i 0.185106 1.72213i 1.99961 + 0.0394217i 3.08870 + 2.83028i −0.285770 + 2.43276i 1.76195 + 2.51633i −2.82719 0.0836194i −2.93147 0.637552i −4.32843 4.04547i
11.2 −1.41363 + 0.0407906i 0.204734 1.71991i 1.99667 0.115325i −1.35883 1.24514i −0.219261 + 2.43966i 1.85804 + 2.65356i −2.81784 + 0.244472i −2.91617 0.704247i 1.97166 + 1.70473i
11.3 −1.41271 + 0.0652429i −1.34471 + 1.09168i 1.99149 0.184338i 1.03243 + 0.946048i 1.82845 1.62996i −1.95935 2.79824i −2.80136 + 0.390347i 0.616467 2.93598i −1.52025 1.26913i
11.4 −1.41165 + 0.0851035i 1.59667 + 0.671291i 1.98551 0.240273i 2.40984 + 2.20821i −2.31108 0.811745i 0.164322 + 0.234676i −2.78241 + 0.508155i 2.09874 + 2.14366i −3.58978 2.91214i
11.5 −1.41018 + 0.106692i −0.243550 + 1.71484i 1.97723 0.300912i −0.0643323 0.0589497i 0.160489 2.44423i 2.92052 + 4.17093i −2.75616 + 0.635297i −2.88137 0.835299i 0.0970098 + 0.0762661i
11.6 −1.40429 0.167206i −0.631256 1.61292i 1.94408 + 0.469613i −2.68720 2.46236i 0.616779 + 2.37057i −2.07740 2.96684i −2.65154 0.984538i −2.20303 + 2.03633i 3.36189 + 3.90719i
11.7 −1.40277 + 0.179519i 1.46202 0.928701i 1.93555 0.503650i 0.162424 + 0.148834i −1.88417 + 1.56522i 0.0602653 + 0.0860678i −2.62472 + 1.05398i 1.27503 2.71557i −0.254563 0.179622i
11.8 −1.39800 + 0.213530i −1.67838 0.427824i 1.90881 0.597031i −0.187368 0.171692i 2.43773 + 0.239713i −0.568924 0.812508i −2.54103 + 1.24224i 2.63393 + 1.43611i 0.298602 + 0.200016i
11.9 −1.39297 + 0.244177i −0.927439 + 1.46282i 1.88075 0.680266i −2.98399 2.73432i 0.934710 2.26414i −0.441286 0.630221i −2.45374 + 1.40683i −1.27971 2.71336i 4.82428 + 3.08022i
11.10 −1.38679 0.277148i 0.934968 + 1.45802i 1.84638 + 0.768693i −1.28907 1.18122i −0.892516 2.28110i −0.239361 0.341842i −2.34750 1.57774i −1.25167 + 2.72641i 1.46030 + 1.99537i
11.11 −1.38529 0.284573i 0.873142 1.49587i 1.83804 + 0.788430i −0.201543 0.184680i −1.63524 + 1.82373i −1.85768 2.65304i −2.32184 1.61526i −1.47525 2.61221i 0.226640 + 0.313189i
11.12 −1.36394 + 0.373712i −1.63885 + 0.560498i 1.72068 1.01944i 1.13561 + 1.04059i 2.02584 1.37695i 2.47485 + 3.53445i −1.96593 + 2.03350i 2.37168 1.83715i −1.93778 0.994916i
11.13 −1.35632 0.400507i −1.37398 1.05460i 1.67919 + 1.08643i 0.982807 + 0.900577i 1.44117 + 1.98066i 0.399771 + 0.570933i −1.84239 2.14607i 0.775631 + 2.89800i −0.972311 1.61509i
11.14 −1.35346 0.410054i −0.153915 + 1.72520i 1.66371 + 1.10998i 2.53456 + 2.32250i 0.915743 2.27187i 0.366857 + 0.523926i −1.79661 2.18453i −2.95262 0.531068i −2.47808 4.18272i
11.15 −1.35292 0.411830i 1.69193 + 0.370633i 1.66079 + 1.11435i −2.86206 2.62259i −2.13641 1.19823i 2.22100 + 3.17192i −1.78800 2.19159i 2.72526 + 1.25417i 2.79207 + 4.72684i
11.16 −1.33659 + 0.462092i −1.08282 1.35185i 1.57294 1.23525i 1.59945 + 1.46563i 2.07196 + 1.30651i −1.32812 1.89675i −1.53158 + 2.37787i −0.655021 + 2.92762i −2.81507 1.21985i
11.17 −1.32493 + 0.494525i 1.65521 + 0.510185i 1.51089 1.31042i −1.77522 1.62669i −2.44534 + 0.142581i −2.52489 3.60592i −1.35379 + 2.48340i 2.47942 + 1.68892i 3.15648 + 1.27736i
11.18 −1.31349 0.524166i −1.41051 + 1.00522i 1.45050 + 1.37697i 1.01410 + 0.929248i 2.37959 0.581007i −0.562323 0.803081i −1.18345 2.56894i 0.979057 2.83574i −0.844922 1.75211i
11.19 −1.30373 0.547978i 1.71908 + 0.211552i 1.39944 + 1.42883i 0.540899 + 0.495643i −2.12530 1.21783i −1.09679 1.56638i −1.04153 2.62968i 2.91049 + 0.727352i −0.433587 0.942587i
11.20 −1.29919 + 0.558675i 1.58138 0.706576i 1.37577 1.45164i −2.22832 2.04188i −1.65975 + 1.80145i 1.01305 + 1.44679i −0.976377 + 2.65456i 2.00150 2.23473i 4.03574 + 1.40787i
See next 80 embeddings (of 3408 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 851.142 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.f odd 18 1 inner
32.h odd 8 1 inner
864.bt even 72 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 864.2.bt.a 3408
27.f odd 18 1 inner 864.2.bt.a 3408
32.h odd 8 1 inner 864.2.bt.a 3408
864.bt even 72 1 inner 864.2.bt.a 3408

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.2.bt.a 3408 1.a even 1 1 trivial
864.2.bt.a 3408 27.f odd 18 1 inner
864.2.bt.a 3408 32.h odd 8 1 inner
864.2.bt.a 3408 864.bt even 72 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(864, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database