# Properties

 Label 864.2.bk.a Level $864$ Weight $2$ Character orbit 864.bk Analytic conductor $6.899$ Analytic rank $0$ Dimension $368$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$864 = 2^{5} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 864.bk (of order $$24$$, degree $$8$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.89907473464$$ Analytic rank: $$0$$ Dimension: $$368$$ Relative dimension: $$46$$ over $$\Q(\zeta_{24})$$ Twist minimal: no (minimal twist has level 288) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$368 q + 4 q^{2} - 4 q^{4} + 4 q^{5} - 4 q^{7} + 16 q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$368 q + 4 q^{2} - 4 q^{4} + 4 q^{5} - 4 q^{7} + 16 q^{8} - 16 q^{10} + 4 q^{11} - 4 q^{13} + 4 q^{14} - 4 q^{16} - 16 q^{19} + 4 q^{20} - 4 q^{22} + 4 q^{23} - 4 q^{25} + 16 q^{26} - 16 q^{28} + 4 q^{29} - 8 q^{31} + 4 q^{32} + 4 q^{34} + 16 q^{35} - 16 q^{37} + 60 q^{38} - 4 q^{40} + 4 q^{41} - 4 q^{43} + 104 q^{44} - 16 q^{46} - 48 q^{50} - 4 q^{52} + 16 q^{53} - 16 q^{55} + 84 q^{56} - 40 q^{58} + 4 q^{59} - 4 q^{61} + 24 q^{62} - 16 q^{64} + 8 q^{65} - 4 q^{67} - 12 q^{68} - 4 q^{70} + 16 q^{71} - 16 q^{73} + 4 q^{74} + 20 q^{76} + 4 q^{77} - 48 q^{80} - 16 q^{82} - 36 q^{83} + 16 q^{85} - 100 q^{86} - 4 q^{88} + 16 q^{89} - 16 q^{91} + 80 q^{92} - 20 q^{94} + 136 q^{95} - 8 q^{97} - 104 q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1 −1.41330 + 0.0509169i 0 1.99481 0.143921i −2.15664 + 1.65485i 0 −0.251218 + 0.937559i −2.81194 + 0.304973i 0 2.96372 2.44861i
37.2 −1.39820 0.212242i 0 1.90991 + 0.593513i 0.306163 0.234927i 0 0.593754 2.21592i −2.54446 1.23521i 0 −0.477938 + 0.263494i
37.3 −1.39517 + 0.231288i 0 1.89301 0.645374i 2.21674 1.70096i 0 −0.400471 + 1.49458i −2.49181 + 1.33824i 0 −2.69932 + 2.88584i
37.4 −1.34055 + 0.450480i 0 1.59413 1.20778i −1.37652 + 1.05624i 0 1.04771 3.91012i −1.59293 + 2.33721i 0 1.36948 2.03604i
37.5 −1.32629 0.490865i 0 1.51810 + 1.30206i −0.489423 + 0.375547i 0 −0.584477 + 2.18130i −1.37431 2.47210i 0 0.833461 0.257845i
37.6 −1.32317 0.499220i 0 1.50156 + 1.32111i 1.16198 0.891617i 0 0.778386 2.90498i −1.32729 2.49766i 0 −1.98261 + 0.599678i
37.7 −1.31129 + 0.529637i 0 1.43897 1.38902i 0.931920 0.715087i 0 −0.366881 + 1.36922i −1.15123 + 2.58354i 0 −0.843281 + 1.43127i
37.8 −1.16868 0.796364i 0 0.731610 + 1.86138i 3.18925 2.44720i 0 −0.890421 + 3.32310i 0.627322 2.75798i 0 −5.67606 + 0.320180i
37.9 −1.15525 + 0.815721i 0 0.669198 1.88472i −1.28782 + 0.988180i 0 −0.0841003 + 0.313867i 0.764318 + 2.72320i 0 0.681674 2.19210i
37.10 −1.04770 + 0.949902i 0 0.195371 1.99043i 2.93870 2.25494i 0 1.16359 4.34257i 1.68603 + 2.27097i 0 −0.936914 + 5.15400i
37.11 −0.930084 1.06534i 0 −0.269888 + 1.98171i −2.36155 + 1.81208i 0 0.338610 1.26371i 2.36221 1.55563i 0 4.12692 + 0.830461i
37.12 −0.923453 + 1.07109i 0 −0.294470 1.97820i −2.37671 + 1.82371i 0 −0.989396 + 3.69248i 2.39076 + 1.51137i 0 0.241417 4.22979i
37.13 −0.917969 + 1.07579i 0 −0.314664 1.97509i 2.56852 1.97090i 0 −0.846806 + 3.16032i 2.41364 + 1.47456i 0 −0.237547 + 4.57243i
37.14 −0.874211 1.11165i 0 −0.471511 + 1.94362i −1.36482 + 1.04726i 0 1.09705 4.09423i 2.57282 1.17498i 0 2.35732 + 0.601666i
37.15 −0.830777 1.14447i 0 −0.619620 + 1.90160i 1.83769 1.41011i 0 0.454899 1.69771i 2.69109 0.870667i 0 −3.14054 0.931695i
37.16 −0.685362 1.23704i 0 −1.06056 + 1.69565i −2.94194 + 2.25743i 0 −1.22999 + 4.59037i 2.82446 + 0.149828i 0 4.80884 + 2.09216i
37.17 −0.683019 + 1.23834i 0 −1.06697 1.69162i 0.526137 0.403719i 0 0.911495 3.40174i 2.82356 0.165864i 0 0.140580 + 0.927284i
37.18 −0.624817 1.26870i 0 −1.21921 + 1.58541i 1.33786 1.02658i 0 −0.437818 + 1.63396i 2.77320 + 0.556221i 0 −2.13834 1.05592i
37.19 −0.532138 + 1.31028i 0 −1.43366 1.39450i −0.794880 + 0.609933i 0 0.0316510 0.118123i 2.59008 1.13642i 0 −0.376196 1.36608i
37.20 −0.218918 + 1.39717i 0 −1.90415 0.611729i 0.538426 0.413148i 0 −0.988129 + 3.68775i 1.27154 2.52650i 0 0.459366 + 0.842716i
See next 80 embeddings (of 368 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 829.46 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.c even 3 1 inner
32.g even 8 1 inner
288.bc even 24 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 864.2.bk.a 368
3.b odd 2 1 288.2.bc.a 368
9.c even 3 1 inner 864.2.bk.a 368
9.d odd 6 1 288.2.bc.a 368
32.g even 8 1 inner 864.2.bk.a 368
96.p odd 8 1 288.2.bc.a 368
288.bc even 24 1 inner 864.2.bk.a 368
288.be odd 24 1 288.2.bc.a 368

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.bc.a 368 3.b odd 2 1
288.2.bc.a 368 9.d odd 6 1
288.2.bc.a 368 96.p odd 8 1
288.2.bc.a 368 288.be odd 24 1
864.2.bk.a 368 1.a even 1 1 trivial
864.2.bk.a 368 9.c even 3 1 inner
864.2.bk.a 368 32.g even 8 1 inner
864.2.bk.a 368 288.bc even 24 1 inner

## Hecke kernels

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