# Properties

 Label 864.2.bu.a Level 864 Weight 2 Character orbit 864.bu 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.bu (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.

 $$\operatorname{Tr}(f)(q) =$$ $$3408q - 24q^{2} - 24q^{3} - 24q^{4} - 24q^{5} - 24q^{6} - 24q^{7} - 12q^{8} - 24q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$3408q - 24q^{2} - 24q^{3} - 24q^{4} - 24q^{5} - 24q^{6} - 24q^{7} - 12q^{8} - 24q^{9} - 12q^{10} - 24q^{11} - 24q^{12} - 24q^{13} - 24q^{14} - 24q^{16} - 24q^{18} - 12q^{19} - 24q^{20} - 24q^{21} - 24q^{22} - 24q^{23} + 36q^{24} - 24q^{25} - 48q^{26} - 24q^{27} - 48q^{28} - 24q^{29} - 24q^{30} - 48q^{31} - 24q^{32} - 48q^{33} - 48q^{34} - 12q^{35} - 24q^{36} - 12q^{37} - 24q^{38} - 24q^{39} - 24q^{40} - 24q^{41} - 144q^{42} - 24q^{43} - 12q^{44} - 24q^{45} - 12q^{46} - 24q^{48} + 288q^{50} - 60q^{51} - 24q^{52} - 48q^{53} - 24q^{54} - 48q^{55} - 24q^{56} - 24q^{57} + 84q^{58} - 24q^{59} - 24q^{60} - 24q^{61} - 12q^{62} - 48q^{63} - 12q^{64} - 48q^{65} + 168q^{66} - 24q^{67} - 72q^{68} - 24q^{69} - 24q^{70} - 12q^{71} - 24q^{72} - 12q^{73} - 24q^{74} - 24q^{75} - 24q^{76} - 24q^{77} - 24q^{78} - 48q^{80} - 48q^{82} - 144q^{83} - 252q^{84} - 84q^{85} - 24q^{86} - 24q^{87} - 24q^{88} - 12q^{89} - 24q^{90} - 12q^{91} + 204q^{92} - 24q^{93} - 24q^{94} - 48q^{95} - 24q^{96} - 48q^{97} - 12q^{98} - 24q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1 −1.41381 + 0.0337132i 1.72171 + 0.189028i 1.99773 0.0953282i 3.68351 + 1.16141i −2.44054 0.209206i 2.44754 1.14131i −2.82120 + 0.202126i 2.92854 + 0.650902i −5.24694 1.51783i
13.2 −1.41221 + 0.0752681i −1.68627 + 0.395578i 1.98867 0.212588i 0.0172778 + 0.00544766i 2.35160 0.685561i 3.98127 1.85650i −2.79242 + 0.449903i 2.68704 1.33411i −0.0248099 0.00639277i
13.3 −1.41215 0.0762974i 1.09754 1.33992i 1.98836 + 0.215487i −2.03835 0.642689i −1.65213 + 1.80844i −2.07741 + 0.968715i −2.79143 0.456008i −0.590794 2.94125i 2.82943 + 1.06310i
13.4 −1.40641 0.148327i −0.561294 + 1.63858i 1.95600 + 0.417219i −3.27422 1.03236i 1.03246 2.22127i −1.49563 + 0.697422i −2.68906 0.876911i −2.36990 1.83945i 4.45178 + 1.93758i
13.5 −1.40628 + 0.149622i −1.68874 + 0.384932i 1.95523 0.420821i 2.95245 + 0.930903i 2.31723 0.793993i −4.02937 + 1.87893i −2.68662 + 0.884337i 2.70366 1.30010i −4.29124 0.867354i
13.6 −1.40602 + 0.152001i −0.406383 1.68370i 1.95379 0.427432i 2.49034 + 0.785202i 0.827306 + 2.30555i −1.45250 + 0.677311i −2.68210 + 0.897956i −2.66971 + 1.36845i −3.62082 0.725477i
13.7 −1.40459 + 0.164681i 0.625973 + 1.61498i 1.94576 0.462618i 0.0622953 + 0.0196416i −1.14519 2.16530i −3.55538 + 1.65790i −2.65682 + 0.970219i −2.21632 + 2.02187i −0.0907341 0.0173297i
13.8 −1.39315 + 0.243197i 1.64970 + 0.527711i 1.88171 0.677616i −0.224370 0.0707437i −2.42661 0.333976i 2.93505 1.36864i −2.45670 + 1.40164i 2.44304 + 1.74113i 0.329785 + 0.0439902i
13.9 −1.38819 + 0.270068i −1.71796 + 0.220516i 1.85413 0.749809i −3.40435 1.07339i 2.32529 0.770083i 0.509100 0.237397i −2.37138 + 1.54162i 2.90275 0.757675i 5.01576 + 0.570657i
13.10 −1.38811 0.270444i 0.417493 1.68098i 1.85372 + 0.750815i 0.0983780 + 0.0310185i −1.03414 + 2.22049i 2.27483 1.06077i −2.37012 1.54354i −2.65140 1.40360i −0.128171 0.0696629i
13.11 −1.37712 0.321786i 0.390709 + 1.68741i 1.79291 + 0.886276i 3.68959 + 1.16332i 0.00493340 2.44948i −0.687024 + 0.320365i −2.18385 1.79744i −2.69469 + 1.31857i −4.70666 2.78929i
13.12 −1.36689 0.362787i −0.738111 + 1.56691i 1.73677 + 0.991780i 0.0389079 + 0.0122676i 1.57737 1.87401i 2.13798 0.996959i −2.01417 1.98573i −1.91039 2.31310i −0.0487323 0.0308838i
13.13 −1.36276 0.378009i −1.60615 0.648285i 1.71422 + 1.03027i 1.57723 + 0.497299i 1.94374 + 1.49060i 1.30846 0.610147i −1.94661 2.05200i 2.15945 + 2.08249i −1.96140 1.27391i
13.14 −1.35215 + 0.414342i 1.46175 + 0.929134i 1.65664 1.12051i −2.56205 0.807812i −2.36149 0.650669i 0.176479 0.0822935i −1.77576 + 2.20152i 1.27342 + 2.71632i 3.79900 + 0.0307205i
13.15 −1.34354 0.441482i −1.19271 1.25596i 1.61019 + 1.18630i −2.66785 0.841171i 1.04797 + 2.21399i −3.54003 + 1.65074i −1.63962 2.30470i −0.154877 + 2.99600i 3.21300 + 2.30796i
13.16 −1.34305 + 0.442964i 1.68579 0.397645i 1.60757 1.18985i −0.412367 0.130019i −2.08795 + 1.28080i −2.48268 + 1.15769i −1.63198 + 2.31012i 2.68376 1.34069i 0.611423 0.00804205i
13.17 −1.33912 0.454707i 1.64935 0.528821i 1.58648 + 1.21781i 0.966263 + 0.304661i −2.44913 0.0418154i −2.27200 + 1.05945i −1.57074 2.35218i 2.44070 1.74442i −1.15541 0.847344i
13.18 −1.33888 0.455412i 0.688830 + 1.58919i 1.58520 + 1.21948i −0.558202 0.176001i −0.198526 2.44143i 3.77341 1.75957i −1.56702 2.35466i −2.05103 + 2.18936i 0.667213 + 0.489856i
13.19 −1.31022 + 0.532279i −0.981940 1.42681i 1.43336 1.39481i −1.65608 0.522159i 2.04602 + 1.34677i 1.21774 0.567841i −1.13559 + 2.59045i −1.07159 + 2.80209i 2.44776 0.197351i
13.20 −1.27166 0.618782i 1.45331 + 0.942280i 1.23422 + 1.57375i −3.36929 1.06233i −1.26504 2.09754i −0.580055 + 0.270484i −0.595692 2.76499i 1.22422 + 2.73885i 3.62722 + 3.43577i
See next 80 embeddings (of 3408 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 853.142 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.e even 9 1 inner
32.g even 8 1 inner
864.bu 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.bu.a 3408
27.e even 9 1 inner 864.2.bu.a 3408
32.g even 8 1 inner 864.2.bu.a 3408
864.bu even 72 1 inner 864.2.bu.a 3408

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.2.bu.a 3408 1.a even 1 1 trivial
864.2.bu.a 3408 27.e even 9 1 inner
864.2.bu.a 3408 32.g even 8 1 inner
864.2.bu.a 3408 864.bu 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