# Properties

 Label 576.2.bd.a Level $576$ Weight $2$ Character orbit 576.bd Analytic conductor $4.599$ Analytic rank $0$ Dimension $56$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$576 = 2^{6} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 576.bd (of order $$16$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.59938315643$$ Analytic rank: $$0$$ Dimension: $$56$$ Relative dimension: $$7$$ over $$\Q(\zeta_{16})$$ Twist minimal: no (minimal twist has level 64) Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$56q + 8q^{2} - 8q^{4} + 8q^{5} - 8q^{7} + 8q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$56q + 8q^{2} - 8q^{4} + 8q^{5} - 8q^{7} + 8q^{8} - 8q^{10} + 8q^{11} - 8q^{13} + 8q^{14} - 8q^{16} + 8q^{17} - 8q^{19} + 8q^{20} + 8q^{23} - 8q^{25} - 32q^{26} + 32q^{28} + 8q^{29} - 32q^{32} + 32q^{34} + 8q^{35} - 8q^{37} - 32q^{38} + 32q^{40} + 8q^{41} - 8q^{43} - 8q^{46} + 8q^{47} - 8q^{49} + 32q^{50} - 56q^{52} + 8q^{53} + 56q^{55} + 64q^{56} - 80q^{58} - 56q^{59} - 8q^{61} + 40q^{62} - 104q^{64} + 16q^{65} + 72q^{67} + 56q^{68} - 104q^{70} - 56q^{71} - 8q^{73} + 64q^{74} - 72q^{76} + 8q^{77} + 24q^{79} - 32q^{80} + 72q^{82} + 8q^{83} - 8q^{85} - 96q^{86} + 72q^{88} + 8q^{89} - 8q^{91} - 144q^{92} + 88q^{94} - 128q^{98} + O(q^{100})$$

## 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 −0.998819 + 1.00118i 0 −0.00472188 1.99999i −0.0517508 + 0.260169i 0 −0.515195 + 1.24379i 2.00707 + 1.99290i 0 −0.208786 0.311673i
37.2 −0.797087 1.16818i 0 −0.729306 + 1.86229i −0.631428 + 3.17440i 0 −0.127129 + 0.306917i 2.75681 0.632441i 0 4.21159 1.79265i
37.3 0.166454 + 1.40438i 0 −1.94459 + 0.467530i 0.573974 2.88556i 0 −0.410118 + 0.990113i −0.980274 2.65312i 0 4.14798 + 0.325767i
37.4 0.466807 1.33495i 0 −1.56418 1.24633i 0.690473 3.47124i 0 0.983337 2.37399i −2.39396 + 1.50631i 0 −4.31162 2.54215i
37.5 0.809383 + 1.15970i 0 −0.689800 + 1.87728i 0.159018 0.799435i 0 −0.742008 + 1.79137i −2.73539 + 0.719478i 0 1.05581 0.462637i
37.6 0.919278 1.07468i 0 −0.309855 1.97585i −0.509835 + 2.56311i 0 −1.78664 + 4.31333i −2.40824 1.48336i 0 2.28583 + 2.90412i
37.7 1.35786 + 0.395231i 0 1.68758 + 1.07334i −0.154331 + 0.775873i 0 1.53949 3.71667i 1.86729 + 2.12443i 0 −0.516209 + 0.992533i
109.1 −0.998819 1.00118i 0 −0.00472188 + 1.99999i −0.0517508 0.260169i 0 −0.515195 1.24379i 2.00707 1.99290i 0 −0.208786 + 0.311673i
109.2 −0.797087 + 1.16818i 0 −0.729306 1.86229i −0.631428 3.17440i 0 −0.127129 0.306917i 2.75681 + 0.632441i 0 4.21159 + 1.79265i
109.3 0.166454 1.40438i 0 −1.94459 0.467530i 0.573974 + 2.88556i 0 −0.410118 0.990113i −0.980274 + 2.65312i 0 4.14798 0.325767i
109.4 0.466807 + 1.33495i 0 −1.56418 + 1.24633i 0.690473 + 3.47124i 0 0.983337 + 2.37399i −2.39396 1.50631i 0 −4.31162 + 2.54215i
109.5 0.809383 1.15970i 0 −0.689800 1.87728i 0.159018 + 0.799435i 0 −0.742008 1.79137i −2.73539 0.719478i 0 1.05581 + 0.462637i
109.6 0.919278 + 1.07468i 0 −0.309855 + 1.97585i −0.509835 2.56311i 0 −1.78664 4.31333i −2.40824 + 1.48336i 0 2.28583 2.90412i
109.7 1.35786 0.395231i 0 1.68758 1.07334i −0.154331 0.775873i 0 1.53949 + 3.71667i 1.86729 2.12443i 0 −0.516209 0.992533i
181.1 −1.28112 + 0.598933i 0 1.28256 1.53462i −0.884671 1.32400i 0 −2.40727 0.997123i −0.723982 + 2.73420i 0 1.92636 + 1.16635i
181.2 −0.468362 1.33441i 0 −1.56127 + 1.24997i −0.914126 1.36809i 0 2.65574 + 1.10004i 2.39921 + 1.49793i 0 −1.39744 + 1.86057i
181.3 −0.325482 + 1.37625i 0 −1.78812 0.895889i 0.967135 + 1.44742i 0 4.53283 + 1.87756i 1.81497 2.16931i 0 −2.30680 + 0.859909i
181.4 −0.182356 1.40241i 0 −1.93349 + 0.511475i 0.787711 + 1.17889i 0 −2.16489 0.896725i 1.06988 + 2.61827i 0 1.50964 1.31967i
181.5 1.11482 + 0.870162i 0 0.485635 + 1.94014i 0.153107 + 0.229142i 0 −0.843108 0.349227i −1.14685 + 2.58549i 0 −0.0287035 + 0.388679i
181.6 1.12131 0.861781i 0 0.514666 1.93265i −1.82421 2.73012i 0 0.00395016 + 0.00163621i −1.08842 2.61062i 0 −4.39827 1.48924i
See all 56 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 541.7 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
64.i even 16 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.2.bd.a 56
3.b odd 2 1 64.2.i.a 56
12.b even 2 1 256.2.i.a 56
24.f even 2 1 512.2.i.a 56
24.h odd 2 1 512.2.i.b 56
64.i even 16 1 inner 576.2.bd.a 56
192.q odd 16 1 64.2.i.a 56
192.q odd 16 1 512.2.i.b 56
192.s even 16 1 256.2.i.a 56
192.s even 16 1 512.2.i.a 56

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
64.2.i.a 56 3.b odd 2 1
64.2.i.a 56 192.q odd 16 1
256.2.i.a 56 12.b even 2 1
256.2.i.a 56 192.s even 16 1
512.2.i.a 56 24.f even 2 1
512.2.i.a 56 192.s even 16 1
512.2.i.b 56 24.h odd 2 1
512.2.i.b 56 192.q odd 16 1
576.2.bd.a 56 1.a even 1 1 trivial
576.2.bd.a 56 64.i even 16 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$37\!\cdots\!60$$$$T_{5}^{30} +$$$$12\!\cdots\!16$$$$T_{5}^{29} -$$$$11\!\cdots\!64$$$$T_{5}^{28} -$$$$11\!\cdots\!00$$$$T_{5}^{27} +$$$$41\!\cdots\!16$$$$T_{5}^{26} +$$$$52\!\cdots\!84$$$$T_{5}^{25} -$$$$13\!\cdots\!04$$$$T_{5}^{24} -$$$$78\!\cdots\!28$$$$T_{5}^{23} +$$$$38\!\cdots\!76$$$$T_{5}^{22} +$$$$38\!\cdots\!28$$$$T_{5}^{21} -$$$$49\!\cdots\!00$$$$T_{5}^{20} -$$$$96\!\cdots\!12$$$$T_{5}^{19} +$$$$64\!\cdots\!32$$$$T_{5}^{18} +$$$$26\!\cdots\!92$$$$T_{5}^{17} -$$$$17\!\cdots\!52$$$$T_{5}^{16} -$$$$12\!\cdots\!44$$$$T_{5}^{15} +$$$$14\!\cdots\!44$$$$T_{5}^{14} +$$$$41\!\cdots\!20$$$$T_{5}^{13} +$$$$63\!\cdots\!16$$$$T_{5}^{12} +$$$$62\!\cdots\!88$$$$T_{5}^{11} +$$$$42\!\cdots\!20$$$$T_{5}^{10} +$$$$19\!\cdots\!68$$$$T_{5}^{9} +$$$$40\!\cdots\!92$$$$T_{5}^{8} -$$$$28\!\cdots\!40$$$$T_{5}^{7} -$$$$17\!\cdots\!52$$$$T_{5}^{6} +$$$$11\!\cdots\!00$$$$T_{5}^{5} +$$$$10\!\cdots\!20$$$$T_{5}^{4} -$$$$62\!\cdots\!36$$$$T_{5}^{3} +$$$$25\!\cdots\!24$$$$T_{5}^{2} - 524142598656 T_{5} + 191120240768$$">$$T_{5}^{56} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(576, [\chi])$$.