# Properties

 Label 864.2.bn.a Level 864 Weight 2 Character orbit 864.bn 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.bn (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 $$\Q(\zeta_{24})$$ Coefficient ring index: multiple of None 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.

 $$\operatorname{Tr}(f)(q) =$$ $$368q + 12q^{2} - 4q^{4} + 12q^{5} - 4q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$368q + 12q^{2} - 4q^{4} + 12q^{5} - 4q^{7} - 16q^{10} + 12q^{11} - 4q^{13} + 12q^{14} - 4q^{16} - 16q^{19} + 12q^{20} - 4q^{22} + 12q^{23} - 4q^{25} - 16q^{28} + 12q^{29} + 12q^{32} - 12q^{34} - 16q^{37} + 12q^{38} - 4q^{40} + 12q^{41} - 4q^{43} - 16q^{46} + 24q^{47} + 168q^{50} - 4q^{52} - 16q^{55} + 12q^{56} + 32q^{58} + 12q^{59} - 4q^{61} - 16q^{64} + 24q^{65} - 4q^{67} + 60q^{68} - 4q^{70} - 16q^{73} + 12q^{74} - 28q^{76} + 12q^{77} - 8q^{79} - 16q^{82} + 132q^{83} - 24q^{85} + 12q^{86} - 4q^{88} - 16q^{91} - 216q^{92} - 20q^{94} - 8q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
35.1 −1.41409 0.0185843i 0 1.99931 + 0.0525599i −0.473408 0.363259i 0 −2.79519 + 0.748968i −2.82623 0.111480i 0 0.662692 + 0.522479i
35.2 −1.40263 0.180610i 0 1.93476 + 0.506659i 0.0140517 + 0.0107822i 0 0.714379 0.191417i −2.62225 1.06009i 0 −0.0177620 0.0176614i
35.3 −1.36587 + 0.366606i 0 1.73120 1.00147i −0.924826 0.709644i 0 1.85343 0.496625i −1.99745 + 2.00255i 0 1.52335 + 0.630235i
35.4 −1.34272 + 0.443967i 0 1.60579 1.19224i 3.31478 + 2.54352i 0 −0.681945 + 0.182727i −1.62680 + 2.31376i 0 −5.58006 1.94358i
35.5 −1.33268 + 0.473259i 0 1.55205 1.26140i 2.25126 + 1.72746i 0 −0.665173 + 0.178233i −1.47141 + 2.41556i 0 −3.81774 1.23671i
35.6 −1.31542 0.519298i 0 1.46066 + 1.36619i −2.54801 1.95516i 0 −3.63481 + 0.973946i −1.21192 2.55563i 0 2.33640 + 3.89503i
35.7 −1.24919 0.662973i 0 1.12093 + 1.65635i 1.36141 + 1.04465i 0 4.23968 1.13602i −0.302134 2.81224i 0 −1.00808 2.20753i
35.8 −1.16912 0.795718i 0 0.733665 + 1.86057i 0.274775 + 0.210843i 0 −1.33748 + 0.358377i 0.622752 2.75902i 0 −0.153473 0.465143i
35.9 −1.12313 + 0.859413i 0 0.522820 1.93046i −3.20104 2.45624i 0 2.32357 0.622598i 1.07187 + 2.61746i 0 5.70609 + 0.00765608i
35.10 −1.07440 + 0.919596i 0 0.308687 1.97603i −1.05601 0.810304i 0 −0.165787 + 0.0444224i 1.48550 + 2.40693i 0 1.87973 0.100507i
35.11 −1.00355 0.996435i 0 0.0142344 + 1.99995i −2.06818 1.58697i 0 3.83711 1.02815i 1.97853 2.02124i 0 0.494213 + 3.65341i
35.12 −0.908095 + 1.08414i 0 −0.350728 1.96901i −0.210277 0.161351i 0 −2.90937 + 0.779563i 2.45318 + 1.40781i 0 0.365879 0.0814479i
35.13 −0.886889 1.10156i 0 −0.426855 + 1.95392i 1.69953 + 1.30410i 0 −0.980387 + 0.262694i 2.53092 1.26270i 0 −0.0707598 3.02872i
35.14 −0.883288 + 1.10445i 0 −0.439606 1.95109i 2.71277 + 2.08158i 0 3.12273 0.836733i 2.54317 + 1.23785i 0 −4.69515 + 1.15747i
35.15 −0.840941 + 1.13702i 0 −0.585638 1.91234i 0.802533 + 0.615805i 0 −4.10026 + 1.09866i 2.66685 + 0.942277i 0 −1.37507 + 0.394642i
35.16 −0.728560 1.21211i 0 −0.938401 + 1.76618i −2.86669 2.19969i 0 −0.326320 + 0.0874372i 2.82448 0.149329i 0 −0.577700 + 5.07734i
35.17 −0.488984 1.32699i 0 −1.52179 + 1.29775i 1.11620 + 0.856491i 0 −1.82515 + 0.489046i 2.46623 + 1.38481i 0 0.590748 1.90000i
35.18 −0.445719 + 1.34214i 0 −1.60267 1.19643i −0.218659 0.167783i 0 4.22533 1.13217i 2.32012 1.61773i 0 0.322648 0.218686i
35.19 −0.426698 1.34831i 0 −1.63586 + 1.15064i 3.24976 + 2.49363i 0 4.30845 1.15445i 2.24943 + 1.71466i 0 1.97551 5.44570i
35.20 −0.195262 1.40067i 0 −1.92375 + 0.546996i −0.729399 0.559687i 0 1.92647 0.516196i 1.14179 + 2.58772i 0 −0.641512 + 1.13093i
See next 80 embeddings (of 368 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 827.46 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 864.2.bn.a 368
3.b odd 2 1 288.2.bf.a 368
9.c even 3 1 288.2.bf.a 368
9.d odd 6 1 inner 864.2.bn.a 368
32.h odd 8 1 inner 864.2.bn.a 368
96.o even 8 1 288.2.bf.a 368
288.bd odd 24 1 288.2.bf.a 368
288.bf even 24 1 inner 864.2.bn.a 368

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