# Properties

 Label 288.2.bc.a Level $288$ Weight $2$ Character orbit 288.bc 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.bc (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.

 $$\operatorname{Tr}(f)(q) =$$ $$368q - 4q^{2} - 8q^{3} - 4q^{4} - 4q^{5} - 8q^{6} - 4q^{7} - 16q^{8} - 8q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$368q - 4q^{2} - 8q^{3} - 4q^{4} - 4q^{5} - 8q^{6} - 4q^{7} - 16q^{8} - 8q^{9} - 16q^{10} - 4q^{11} - 8q^{12} - 4q^{13} - 4q^{14} - 4q^{16} - 8q^{18} - 16q^{19} - 4q^{20} - 8q^{21} - 4q^{22} - 4q^{23} - 48q^{24} - 4q^{25} - 16q^{26} - 32q^{27} - 16q^{28} - 4q^{29} + 24q^{30} - 8q^{31} - 4q^{32} - 16q^{33} + 4q^{34} - 16q^{35} + 20q^{36} - 16q^{37} - 60q^{38} - 32q^{39} - 4q^{40} - 4q^{41} - 88q^{42} - 4q^{43} - 104q^{44} - 8q^{45} - 16q^{46} - 60q^{48} + 48q^{50} + 16q^{51} - 4q^{52} - 16q^{53} - 52q^{54} - 16q^{55} - 84q^{56} - 8q^{57} - 40q^{58} - 4q^{59} - 52q^{60} - 4q^{61} - 24q^{62} - 16q^{63} - 16q^{64} - 8q^{65} + 64q^{66} - 4q^{67} + 12q^{68} - 8q^{69} - 4q^{70} - 16q^{71} - 68q^{72} - 16q^{73} - 4q^{74} - 28q^{75} + 20q^{76} - 4q^{77} - 72q^{78} + 48q^{80} - 16q^{82} + 36q^{83} + 200q^{84} + 16q^{85} + 100q^{86} - 64q^{87} - 4q^{88} - 16q^{89} + 4q^{90} - 16q^{91} - 80q^{92} + 4q^{93} - 20q^{94} - 136q^{95} + 28q^{96} - 8q^{97} + 104q^{98} - 72q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1 −1.41394 0.0276581i −0.219683 1.71806i 1.99847 + 0.0782141i −0.0865625 + 0.657507i 0.263101 + 2.43532i −3.40174 0.911495i −2.82356 0.165864i −2.90348 + 0.754859i 0.140580 0.927284i
13.2 −1.40080 0.194294i −1.43289 + 0.973053i 1.92450 + 0.544335i 0.130777 0.993353i 2.19625 1.08465i −0.118123 0.0316510i −2.59008 1.13642i 1.10634 2.78855i −0.376196 + 1.36608i
13.3 −1.39065 + 0.257088i 1.59735 0.669693i 1.86781 0.715038i −0.422585 + 3.20986i −2.04918 + 1.34197i 3.16032 + 0.846806i −2.41364 + 1.47456i 2.10302 2.13946i −0.237547 4.57243i
13.4 −1.38932 + 0.264188i −1.06043 1.36949i 1.86041 0.734083i 0.391027 2.97015i 1.83507 + 1.62250i 3.69248 + 0.989396i −2.39076 + 1.51137i −0.750983 + 2.90448i 0.241417 + 4.22979i
13.5 −1.34649 + 0.432388i −0.112087 + 1.72842i 1.62608 1.16441i −0.483489 + 3.67246i −0.596424 2.37577i −4.34257 1.16359i −1.68603 + 2.27097i −2.97487 0.387465i −0.936914 5.15400i
13.6 −1.31944 0.508995i 0.552463 + 1.64158i 1.48185 + 1.34318i −0.0885843 + 0.672864i 0.106613 2.44717i 3.68775 + 0.988129i −1.27154 2.52650i −2.38957 + 1.81383i 0.459366 0.842716i
13.7 −1.28406 + 0.592614i 1.54046 + 0.791819i 1.29762 1.52190i 0.211878 1.60938i −2.44729 0.103844i 0.313867 + 0.0841003i −0.764318 + 2.72320i 1.74604 + 2.43953i 0.681674 + 2.19210i
13.8 −1.25694 0.648153i 1.41818 0.994373i 1.15979 + 1.62938i 0.579315 4.40033i −2.42707 + 0.330671i 1.20832 + 0.323768i −0.401705 2.79976i 1.02245 2.82039i −3.58025 + 5.15547i
13.9 −1.11432 + 0.870793i −1.69615 + 0.350802i 0.483439 1.94069i −0.153324 + 1.16461i 1.58459 1.86791i 1.36922 + 0.366881i 1.15123 + 2.58354i 2.75388 1.19003i −0.843281 1.43127i
13.10 −1.10333 0.884682i −1.59251 0.681109i 0.434674 + 1.95219i 0.108084 0.820982i 1.15450 + 2.16035i −3.31582 0.888470i 1.24748 2.53846i 2.07218 + 2.16935i −0.845561 + 0.810194i
13.11 −1.09136 0.899411i 0.499198 1.65855i 0.382121 + 1.96316i −0.209198 + 1.58902i −2.03652 + 1.36109i 1.05886 + 0.283722i 1.34865 2.48619i −2.50160 1.65589i 1.65749 1.54603i
13.12 −1.06040 + 0.935708i 1.12868 1.31380i 0.248901 1.98445i 0.226472 1.72022i 0.0324819 + 2.44927i −3.91012 1.04771i 1.59293 + 2.33721i −0.452160 2.96573i 1.36948 + 2.03604i
13.13 −0.992813 1.00714i 1.67119 + 0.455112i −0.0286462 + 1.99979i −0.393225 + 2.98684i −1.20082 2.13496i −2.61511 0.700716i 2.04251 1.95657i 2.58575 + 1.52116i 3.39856 2.56934i
13.14 −0.897888 + 1.09261i −0.186796 1.72195i −0.387596 1.96208i −0.364708 + 2.77023i 2.04914 + 1.34202i 1.49458 + 0.400471i 2.49181 + 1.33824i −2.93021 + 0.643306i −2.69932 2.88584i
13.15 −0.835352 1.14113i 0.0993087 + 1.72920i −0.604373 + 1.90650i 0.387153 2.94072i 1.89029 1.55782i −3.87349 1.03790i 2.68043 0.902926i −2.98028 + 0.343450i −3.67916 + 2.01474i
13.16 −0.750744 + 1.19849i −0.667307 + 1.59834i −0.872768 1.79952i 0.354821 2.69513i −1.41463 1.99971i 0.937559 + 0.251218i 2.81194 + 0.304973i −2.10940 2.13317i 2.96372 + 2.44861i
13.17 −0.721590 1.21627i −1.43301 0.972870i −0.958617 + 1.75529i −0.00931719 + 0.0707711i −0.149225 + 2.44494i 4.24636 + 1.13781i 2.82664 0.100666i 1.10705 + 2.78827i 0.0927998 0.0397355i
13.18 −0.515291 + 1.31699i 0.925920 + 1.46379i −1.46895 1.35727i −0.0503714 + 0.382609i −2.40492 + 0.465155i −2.21592 0.593754i 2.54446 1.23521i −1.28535 + 2.71070i −0.477938 0.263494i
13.19 −0.338290 1.37316i 1.72791 + 0.119758i −1.77112 + 0.929050i 0.148961 1.13147i −0.420086 2.41320i 1.82706 + 0.489558i 1.87488 + 2.11774i 2.97132 + 0.413862i −1.60408 + 0.178219i
13.20 −0.238045 + 1.39404i 1.61798 0.618165i −1.88667 0.663686i 0.0805221 0.611626i 0.476591 + 2.40268i 2.18130 + 0.584477i 1.37431 2.47210i 2.23575 2.00036i 0.833461 + 0.257845i
See next 80 embeddings (of 368 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 277.46 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 288.2.bc.a 368
3.b odd 2 1 864.2.bk.a 368
9.c even 3 1 inner 288.2.bc.a 368
9.d odd 6 1 864.2.bk.a 368
32.g even 8 1 inner 288.2.bc.a 368
96.p odd 8 1 864.2.bk.a 368
288.bc even 24 1 inner 288.2.bc.a 368
288.be odd 24 1 864.2.bk.a 368

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

## Hecke kernels

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