Properties

Label 576.2.bm.a
Level $576$
Weight $2$
Character orbit 576.bm
Analytic conductor $4.599$
Analytic rank $0$
Dimension $1504$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 576.bm (of order \(48\), degree \(16\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.59938315643\)
Analytic rank: \(0\)
Dimension: \(1504\)
Relative dimension: \(94\) over \(\Q(\zeta_{48})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{48}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1504q - 8q^{2} - 16q^{3} - 8q^{4} - 8q^{5} - 16q^{6} - 8q^{7} - 32q^{8} - 16q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 1504q - 8q^{2} - 16q^{3} - 8q^{4} - 8q^{5} - 16q^{6} - 8q^{7} - 32q^{8} - 16q^{9} - 32q^{10} - 8q^{11} - 16q^{12} - 8q^{13} - 8q^{14} - 16q^{15} - 8q^{16} - 32q^{17} - 16q^{18} - 32q^{19} - 8q^{20} - 16q^{21} - 8q^{22} - 8q^{23} + 64q^{24} - 8q^{25} - 32q^{26} - 16q^{27} - 32q^{28} - 8q^{29} - 96q^{30} - 8q^{32} - 8q^{34} - 32q^{35} - 96q^{36} - 32q^{37} - 8q^{38} - 16q^{39} - 8q^{40} - 8q^{41} + 64q^{42} - 8q^{43} - 32q^{44} - 16q^{45} - 32q^{46} - 8q^{47} - 16q^{48} - 8q^{49} - 8q^{50} - 16q^{51} - 8q^{52} - 32q^{53} - 16q^{54} - 32q^{55} - 8q^{56} - 16q^{57} - 80q^{58} - 8q^{59} - 16q^{60} - 8q^{61} - 64q^{62} - 32q^{63} - 32q^{64} - 16q^{65} - 32q^{66} - 8q^{67} - 8q^{68} - 16q^{69} - 8q^{70} - 32q^{71} - 16q^{72} - 32q^{73} - 8q^{74} - 16q^{75} + 40q^{76} - 8q^{77} - 40q^{78} - 8q^{79} - 160q^{80} - 16q^{81} - 32q^{82} - 8q^{83} - 128q^{84} - 8q^{85} - 216q^{86} - 16q^{87} - 8q^{88} - 32q^{89} - 160q^{90} - 32q^{91} + 144q^{92} - 40q^{93} - 8q^{94} - 152q^{96} - 304q^{98} - 16q^{99} + O(q^{100}) \)

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 −1.41382 0.0331870i −0.374386 + 1.69110i 1.99780 + 0.0938413i −0.000471804 0.00138989i 0.585439 2.37850i 1.11009 1.44669i −2.82142 0.198976i −2.71967 1.26625i 0.000713174 0.00194940i
13.2 −1.41279 0.0635185i 1.66301 0.484141i 1.99193 + 0.179476i 0.131392 0.387068i −2.38023 + 0.578356i −2.81879 + 3.67352i −2.80277 0.380086i 2.53122 1.61026i −0.210215 + 0.538499i
13.3 −1.40857 + 0.126197i −1.06415 1.36659i 1.96815 0.355516i −0.795912 + 2.34468i 1.67140 + 1.79065i 2.79810 3.64656i −2.72741 + 0.749145i −0.735151 + 2.90853i 0.825206 3.40309i
13.4 −1.40823 0.129985i 1.52000 + 0.830426i 1.96621 + 0.366096i 1.17841 3.47147i −2.03256 1.36700i 0.673012 0.877086i −2.72128 0.771123i 1.62079 + 2.52449i −2.11070 + 4.73545i
13.5 −1.40630 0.149384i −1.38997 + 1.03343i 1.95537 + 0.420158i −0.631028 + 1.85895i 2.10910 1.24568i 0.0625219 0.0814801i −2.68707 0.882971i 0.864037 2.87288i 1.16511 2.51998i
13.6 −1.38632 + 0.279491i −1.72237 0.182914i 1.84377 0.774927i 0.355498 1.04726i 2.43887 0.227808i −0.746195 + 0.972460i −2.33947 + 1.58961i 2.93309 + 0.630089i −0.200134 + 1.55120i
13.7 −1.35456 + 0.406401i 0.757412 1.55767i 1.66968 1.10099i −0.620708 + 1.82855i −0.392925 + 2.41777i −1.17718 + 1.53413i −1.81424 + 2.16992i −1.85265 2.35959i 0.0976643 2.72914i
13.8 −1.35449 0.406653i 0.905946 + 1.47623i 1.66927 + 1.10161i −1.35493 + 3.99150i −0.626776 2.36794i 0.639597 0.833539i −1.81302 2.17093i −1.35852 + 2.67477i 3.45839 4.85544i
13.9 −1.35331 0.410562i −0.328391 1.70063i 1.66288 + 1.11123i 0.981205 2.89054i −0.253802 + 2.43631i 0.185279 0.241461i −1.79416 2.18655i −2.78432 + 1.11695i −2.51461 + 3.50894i
13.10 −1.33912 + 0.454702i 0.902433 1.47838i 1.58649 1.21780i 0.807782 2.37965i −0.536243 + 2.39007i 2.02026 2.63285i −1.57077 + 2.35217i −1.37123 2.66828i 0.000313823 3.55394i
13.11 −1.32727 + 0.488210i 1.62021 + 0.612310i 1.52330 1.29597i −1.15808 + 3.41160i −2.44939 0.0217003i −0.367980 + 0.479561i −1.38913 + 2.46380i 2.25015 + 1.98414i −0.128487 5.09351i
13.12 −1.31640 0.516800i 1.64727 0.535253i 1.46584 + 1.36063i −0.299490 + 0.882268i −2.44509 0.146700i 1.60558 2.09243i −1.22646 2.54869i 2.42701 1.76341i 0.850205 1.00664i
13.13 −1.28635 + 0.587628i −0.713799 + 1.57813i 1.30939 1.51179i 1.42869 4.20879i −0.00915756 2.44947i −1.84067 + 2.39881i −0.795959 + 2.71412i −1.98098 2.25293i 0.635407 + 6.25352i
13.14 −1.27578 0.610228i −1.60869 0.641966i 1.25524 + 1.55704i −1.11222 + 3.27649i 1.66059 + 1.80068i −2.57039 + 3.34980i −0.651268 2.75243i 2.17576 + 2.06545i 3.41836 3.50138i
13.15 −1.27256 0.616911i −1.71120 + 0.267919i 1.23884 + 1.57012i 1.18675 3.49605i 2.34290 + 0.714716i 2.38213 3.10445i −0.607882 2.76233i 2.85644 0.916929i −3.66696 + 3.71683i
13.16 −1.25673 0.648556i 0.0897512 1.72972i 1.15875 + 1.63012i −0.591531 + 1.74259i −1.23462 + 2.11559i −0.895036 + 1.16643i −0.399010 2.80014i −2.98389 0.310490i 1.87357 1.80633i
13.17 −1.18741 + 0.768145i 0.146556 + 1.72584i 0.819907 1.82421i −0.0104531 + 0.0307939i −1.49972 1.93671i 1.55545 2.02711i 0.427690 + 2.79590i −2.95704 + 0.505864i −0.0112420 0.0445947i
13.18 −1.18500 + 0.771868i −0.449175 1.67279i 0.808438 1.82932i 0.169627 0.499705i 1.82345 + 1.63555i −1.57668 + 2.05477i 0.454001 + 2.79175i −2.59648 + 1.50275i 0.184699 + 0.723078i
13.19 −1.17354 0.789181i 0.665269 + 1.59919i 0.754388 + 1.85227i 0.464376 1.36801i 0.481332 2.40173i −0.878947 + 1.14547i 0.576471 2.76906i −2.11483 + 2.12779i −1.62457 + 1.23893i
13.20 −1.16694 0.798899i −1.18950 + 1.25900i 0.723520 + 1.86454i 0.262679 0.773826i 2.39390 0.518891i −1.66626 + 2.17151i 0.645274 2.75384i −0.170166 2.99517i −0.924740 + 0.693158i
See next 80 embeddings (of 1504 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 565.94
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
64.i even 16 1 inner
576.bm even 48 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.2.bm.a 1504
9.c even 3 1 inner 576.2.bm.a 1504
64.i even 16 1 inner 576.2.bm.a 1504
576.bm even 48 1 inner 576.2.bm.a 1504
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
576.2.bm.a 1504 1.a even 1 1 trivial
576.2.bm.a 1504 9.c even 3 1 inner
576.2.bm.a 1504 64.i even 16 1 inner
576.2.bm.a 1504 576.bm even 48 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(576, [\chi])\).