Properties

Label 360.2.bo.a
Level $360$
Weight $2$
Character orbit 360.bo
Analytic conductor $2.875$
Analytic rank $0$
Dimension $272$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 360.bo (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.87461447277\)
Analytic rank: \(0\)
Dimension: \(272\)
Relative dimension: \(68\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 272q - 2q^{2} - 8q^{3} - 8q^{6} - 8q^{8} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 272q - 2q^{2} - 8q^{3} - 8q^{6} - 8q^{8} - 8q^{10} - 8q^{11} - 10q^{12} - 4q^{16} - 16q^{17} + 20q^{18} + 14q^{20} + 6q^{22} - 4q^{25} - 48q^{26} - 8q^{27} + 8q^{28} - 34q^{30} - 22q^{32} + 4q^{33} - 16q^{35} - 8q^{36} - 26q^{38} - 2q^{40} - 8q^{41} - 66q^{42} - 4q^{43} - 40q^{46} - 38q^{48} - 42q^{50} - 16q^{51} + 14q^{52} + 24q^{56} + 16q^{57} + 6q^{58} + 14q^{60} - 76q^{62} - 4q^{65} - 44q^{66} - 4q^{67} - 46q^{68} + 18q^{70} + 38q^{72} - 16q^{73} - 120q^{75} - 38q^{78} + 92q^{80} - 32q^{81} - 4q^{83} - 40q^{86} - 42q^{88} - 14q^{90} - 32q^{91} + 52q^{92} + 108q^{96} - 4q^{97} - 140q^{98} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −1.41408 0.0194149i −0.626202 1.61489i 1.99925 + 0.0549086i 1.27437 + 1.83738i 0.854146 + 2.29574i −1.73854 + 0.465842i −2.82603 0.116460i −2.21574 + 2.02249i −1.76639 2.62295i
43.2 −1.40680 + 0.144634i −1.28498 + 1.16139i 1.95816 0.406943i −1.91503 + 1.15441i 1.63973 1.81969i −0.414804 + 0.111147i −2.69588 + 0.855704i 0.302361 2.98472i 2.52709 1.90100i
43.3 −1.40635 + 0.148909i 0.312010 1.70372i 1.95565 0.418836i 0.0247121 2.23593i −0.185098 + 2.44249i −0.146504 + 0.0392556i −2.68797 + 0.880244i −2.80530 1.06315i 0.298195 + 3.14819i
43.4 −1.37392 + 0.335162i 0.954643 + 1.44522i 1.77533 0.920974i −1.61515 1.54638i −1.79599 1.66566i 3.82269 1.02429i −2.13050 + 1.86037i −1.17731 + 2.75934i 2.73738 + 1.58327i
43.5 −1.37266 0.340317i −1.62317 0.604421i 1.76837 + 0.934277i 1.13051 1.92924i 2.02236 + 1.38205i 3.44877 0.924095i −2.10941 1.88425i 2.26935 + 1.96215i −2.20835 + 2.26345i
43.6 −1.36313 0.376650i −0.727225 + 1.57199i 1.71627 + 1.02685i 2.06459 + 0.858752i 1.58339 1.86892i 2.73850 0.733779i −1.95274 2.04617i −1.94229 2.28638i −2.49087 1.94822i
43.7 −1.35823 0.393964i 0.305508 + 1.70489i 1.68959 + 1.07019i 0.853647 2.06671i 0.256716 2.43600i −4.24587 + 1.13768i −1.87323 2.11920i −2.81333 + 1.04172i −1.97366 + 2.47076i
43.8 −1.33317 + 0.471880i 1.70913 + 0.280859i 1.55466 1.25819i 2.00296 0.994061i −2.41108 + 0.432071i −1.80031 + 0.482392i −1.47890 + 2.41098i 2.84224 + 0.960049i −2.20120 + 2.27040i
43.9 −1.32591 0.491884i 1.50731 + 0.853236i 1.51610 + 1.30439i −1.08846 + 1.95327i −1.57887 1.87274i −0.188817 + 0.0505935i −1.36861 2.47526i 1.54398 + 2.57218i 2.40398 2.05448i
43.10 −1.27550 + 0.610814i 1.53896 0.794731i 1.25381 1.55819i 0.533101 + 2.17159i −1.47752 + 1.95370i 4.60452 1.23378i −0.647478 + 2.75332i 1.73680 2.44612i −2.00641 2.44424i
43.11 −1.26873 + 0.624750i −1.72137 + 0.192067i 1.21937 1.58528i 2.20640 + 0.363051i 2.06397 1.31911i −2.71026 + 0.726213i −0.556655 + 2.77311i 2.92622 0.661238i −3.02615 + 0.917833i
43.12 −1.26562 0.631044i −1.52201 0.826733i 1.20357 + 1.59732i −2.21053 0.336960i 1.40457 + 2.00678i −4.27697 + 1.14601i −0.515274 2.78110i 1.63303 + 2.51659i 2.58505 + 1.82141i
43.13 −1.21505 0.723644i 1.31796 1.12382i 0.952679 + 1.75852i 2.19583 + 0.422281i −2.41463 + 0.411761i −0.968556 + 0.259524i 0.114995 2.82609i 0.474050 2.96231i −2.36246 2.10209i
43.14 −1.14261 + 0.833334i −0.0193408 1.73194i 0.611108 1.90435i −2.10858 + 0.744241i 1.46539 + 1.96282i −0.993552 + 0.266221i 0.888702 + 2.68518i −2.99925 + 0.0669942i 1.78908 2.60753i
43.15 −1.13786 0.839807i 0.0696109 1.73065i 0.589450 + 1.91116i −2.07285 + 0.838627i −1.53262 + 1.91078i 4.92620 1.31997i 0.934297 2.66966i −2.99031 0.240945i 3.06290 + 0.786552i
43.16 −1.09088 + 0.899994i −1.66761 0.468039i 0.380022 1.96356i −1.69375 1.45986i 2.24039 0.990270i 1.60107 0.429006i 1.35264 + 2.48402i 2.56188 + 1.56102i 3.16154 + 0.0681632i
43.17 −1.07322 + 0.920972i 0.380005 + 1.68985i 0.303622 1.97682i 0.0281379 + 2.23589i −1.96414 1.46362i −1.94379 + 0.520837i 1.49474 + 2.40120i −2.71119 + 1.28430i −2.08939 2.37370i
43.18 −0.931362 1.06422i −0.953544 + 1.44594i −0.265129 + 1.98235i −1.70462 1.44716i 2.42690 0.331917i 2.12579 0.569602i 2.35659 1.56413i −1.18151 2.75754i 0.0475269 + 3.16192i
43.19 −0.871892 1.11347i 1.72671 + 0.135893i −0.479609 + 1.94164i −0.893456 2.04981i −1.35419 2.04112i −0.371305 + 0.0994908i 2.58012 1.15887i 2.96307 + 0.469295i −1.50340 + 2.78205i
43.20 −0.771662 1.18513i 1.04238 + 1.38327i −0.809074 + 1.82904i 2.10881 + 0.743585i 0.834993 2.30278i 3.18343 0.852996i 2.79199 0.452544i −0.826881 + 2.88379i −0.746043 3.07301i
See next 80 embeddings (of 272 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 283.68
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
8.d odd 2 1 inner
9.c even 3 1 inner
40.k even 4 1 inner
45.k odd 12 1 inner
72.p odd 6 1 inner
360.bo even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 360.2.bo.a 272
5.c odd 4 1 inner 360.2.bo.a 272
8.d odd 2 1 inner 360.2.bo.a 272
9.c even 3 1 inner 360.2.bo.a 272
40.k even 4 1 inner 360.2.bo.a 272
45.k odd 12 1 inner 360.2.bo.a 272
72.p odd 6 1 inner 360.2.bo.a 272
360.bo even 12 1 inner 360.2.bo.a 272
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.bo.a 272 1.a even 1 1 trivial
360.2.bo.a 272 5.c odd 4 1 inner
360.2.bo.a 272 8.d odd 2 1 inner
360.2.bo.a 272 9.c even 3 1 inner
360.2.bo.a 272 40.k even 4 1 inner
360.2.bo.a 272 45.k odd 12 1 inner
360.2.bo.a 272 72.p odd 6 1 inner
360.2.bo.a 272 360.bo even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(360, [\chi])\).