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Properties

Label 288.3.be.a
Level $288$
Weight $3$
Character orbit 288.be
Analytic conductor $7.847$
Analytic rank $0$
Dimension $752$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [288,3,Mod(5,288)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(288, base_ring=CyclotomicField(24))
 
chi = DirichletCharacter(H, H._module([0, 3, 20]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("288.5");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 288.be (of order \(24\), degree \(8\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.84743161358\)
Analytic rank: \(0\)
Dimension: \(752\)
Relative dimension: \(94\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 752 q - 12 q^{2} - 8 q^{3} - 4 q^{4} - 12 q^{5} - 8 q^{6} - 4 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 752 q - 12 q^{2} - 8 q^{3} - 4 q^{4} - 12 q^{5} - 8 q^{6} - 4 q^{7} - 8 q^{9} - 16 q^{10} - 12 q^{11} - 8 q^{12} - 4 q^{13} - 12 q^{14} - 4 q^{16} - 8 q^{18} - 16 q^{19} - 12 q^{20} - 8 q^{21} - 4 q^{22} - 12 q^{23} - 288 q^{24} - 4 q^{25} - 104 q^{27} - 16 q^{28} - 12 q^{29} + 48 q^{30} - 8 q^{31} - 12 q^{32} - 16 q^{33} - 20 q^{34} - 76 q^{36} - 16 q^{37} - 12 q^{38} + 184 q^{39} - 4 q^{40} - 12 q^{41} + 632 q^{42} - 4 q^{43} - 8 q^{45} - 16 q^{46} - 60 q^{48} - 480 q^{50} - 80 q^{51} - 4 q^{52} - 316 q^{54} - 16 q^{55} - 12 q^{56} - 8 q^{57} - 184 q^{58} - 12 q^{59} - 436 q^{60} - 4 q^{61} - 16 q^{63} - 16 q^{64} - 24 q^{65} + 808 q^{66} - 4 q^{67} - 204 q^{68} - 8 q^{69} - 4 q^{70} - 428 q^{72} - 16 q^{73} - 12 q^{74} + 92 q^{75} - 316 q^{76} - 12 q^{77} - 288 q^{78} - 16 q^{82} + 1428 q^{83} - 1384 q^{84} - 104 q^{85} - 12 q^{86} + 440 q^{87} - 4 q^{88} + 292 q^{90} - 16 q^{91} - 696 q^{92} + 28 q^{93} + 28 q^{94} - 24 q^{95} + 604 q^{96} - 8 q^{97} - 264 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −2.00000 + 0.00235621i −0.563414 − 2.94662i 3.99999 − 0.00942482i 1.86031 + 2.42440i 1.13377 + 5.89191i −1.48152 + 5.52909i −7.99995 + 0.0282744i −8.36513 + 3.32033i −3.72632 − 4.84441i
5.2 −1.99968 − 0.0357915i 2.99921 − 0.0687119i 3.99744 + 0.143143i 3.60813 + 4.70220i −5.99992 + 0.0300553i 2.52412 − 9.42015i −7.98847 − 0.429315i 8.99056 − 0.412163i −7.04680 − 9.53204i
5.3 −1.99393 + 0.155661i 0.245038 − 2.98998i 3.95154 − 0.620753i −3.55624 − 4.63459i −0.0231679 + 5.99996i 3.44020 − 12.8390i −7.78248 + 1.85284i −8.87991 − 1.46531i 7.81233 + 8.68749i
5.4 −1.99046 + 0.195158i −2.86129 − 0.901664i 3.92383 − 0.776907i −1.61940 − 2.11045i 5.87125 + 1.23632i −1.24291 + 4.63860i −7.65858 + 2.31216i 7.37401 + 5.15985i 3.63522 + 3.88472i
5.5 −1.98892 + 0.210201i 2.67712 − 1.35389i 3.91163 − 0.836149i 1.62613 + 2.11922i −5.04000 + 3.25552i −0.999859 + 3.73152i −7.60417 + 2.48527i 5.33395 − 7.24907i −3.67972 − 3.87315i
5.6 −1.98262 − 0.263109i −1.94864 + 2.28097i 3.86155 + 1.04329i −5.13122 − 6.68713i 4.46355 − 4.00958i −1.82464 + 6.80964i −7.38147 − 3.08445i −1.40561 − 8.88956i 8.41380 + 14.6081i
5.7 −1.98179 + 0.269246i −0.576000 + 2.94418i 3.85501 − 1.06718i 4.69904 + 6.12390i 0.348803 − 5.98985i −3.06639 + 11.4439i −7.35251 + 3.15288i −8.33645 − 3.39170i −10.9614 − 10.8711i
5.8 −1.93240 − 0.515596i −2.83578 − 0.978943i 3.46832 + 1.99267i −0.770194 − 1.00374i 4.97512 + 3.35383i 1.61397 − 6.02341i −5.67476 − 5.63889i 7.08334 + 5.55214i 0.970798 + 2.33673i
5.9 −1.92234 − 0.551927i 1.46461 + 2.61819i 3.39075 + 2.12198i −1.33122 − 1.73488i −1.37041 − 5.84140i −0.502291 + 1.87458i −5.34699 − 5.95061i −4.70986 + 7.66924i 1.60153 + 4.06976i
5.10 −1.90389 + 0.612532i 2.53246 + 1.60831i 3.24961 − 2.33239i −3.06252 − 3.99116i −5.80666 − 1.51085i 0.815106 − 3.04202i −4.75824 + 6.43110i 3.82666 + 8.14596i 8.27542 + 5.72284i
5.11 −1.89843 − 0.629270i −2.06988 + 2.17154i 3.20804 + 2.38924i 2.87703 + 3.74942i 5.29600 − 2.82000i 1.48033 − 5.52467i −4.58675 − 6.55452i −0.431199 − 8.98966i −3.10243 − 8.92842i
5.12 −1.82905 + 0.809056i −2.11069 + 2.13190i 2.69086 − 2.95961i −0.510822 − 0.665716i 2.13574 − 5.60702i 2.69863 − 10.0714i −2.52723 + 7.59033i −0.0899683 − 8.99955i 1.47292 + 0.804346i
5.13 −1.81607 + 0.837779i −2.88848 − 0.810352i 2.59625 − 3.04294i 5.59148 + 7.28696i 5.92459 − 0.948253i 0.984758 − 3.67517i −2.16567 + 7.70129i 7.68666 + 4.68137i −16.2594 − 8.54923i
5.14 −1.80222 + 0.867179i 0.210021 + 2.99264i 2.49600 − 3.12570i −0.205717 − 0.268095i −2.97366 − 5.21127i 0.447938 − 1.67173i −1.78781 + 7.79767i −8.91178 + 1.25703i 0.603234 + 0.304774i
5.15 −1.79472 + 0.882595i 2.24442 − 1.99062i 2.44205 − 3.16802i −4.88979 − 6.37250i −2.27120 + 5.55353i −2.67411 + 9.97990i −1.58673 + 7.84107i 1.07484 − 8.93559i 14.4001 + 7.12116i
5.16 −1.77339 − 0.924701i 1.98646 − 2.24810i 2.28986 + 3.27972i 1.61949 + 2.11056i −5.60161 + 2.14988i −1.58106 + 5.90061i −1.02806 − 7.93367i −1.10792 − 8.93155i −0.920358 − 5.24040i
5.17 −1.76073 − 0.948594i 2.97865 − 0.357260i 2.20034 + 3.34044i −4.50483 − 5.87081i −5.58350 − 2.19649i 0.930710 − 3.47346i −0.705487 − 7.96883i 8.74473 − 2.12831i 2.36278 + 14.6102i
5.18 −1.70533 − 1.04491i −0.771784 − 2.89903i 1.81631 + 3.56385i 5.43191 + 7.07900i −1.71309 + 5.75025i 2.11474 − 7.89232i 0.626515 − 7.97543i −7.80870 + 4.47484i −1.86625 − 17.7479i
5.19 −1.59103 + 1.21186i 2.80923 + 1.05272i 1.06277 − 3.85623i 3.39559 + 4.42522i −5.74533 + 1.72950i −1.87452 + 6.99581i 2.98231 + 7.42333i 6.78358 + 5.91465i −10.7653 − 2.92568i
5.20 −1.58369 + 1.22144i −0.963305 − 2.84113i 1.01615 − 3.86878i 0.119341 + 0.155529i 4.99586 + 3.32286i −0.779013 + 2.90732i 3.11621 + 7.36812i −7.14409 + 5.47376i −0.378969 − 0.100541i
See next 80 embeddings (of 752 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.94
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
32.g even 8 1 inner
288.be odd 24 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.3.be.a ✓ 752
9.d odd 6 1 inner 288.3.be.a ✓ 752
32.g even 8 1 inner 288.3.be.a ✓ 752
288.be odd 24 1 inner 288.3.be.a ✓ 752
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.3.be.a ✓ 752 1.a even 1 1 trivial
288.3.be.a ✓ 752 9.d odd 6 1 inner
288.3.be.a ✓ 752 32.g even 8 1 inner
288.3.be.a ✓ 752 288.be odd 24 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(288, [\chi])\).