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Properties

Label 288.3.u.b
Level $288$
Weight $3$
Character orbit 288.u
Analytic conductor $7.847$
Analytic rank $0$
Dimension $64$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [288,3,Mod(19,288)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(288, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([4, 7, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("288.19");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 288.u (of order \(8\), degree \(4\), 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: \(64\)
Relative dimension: \(16\) over \(\Q(\zeta_{8})\)
Twist minimal: no (minimal twist has level 96)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q + 40 q^{10} - 32 q^{14} - 8 q^{16} + 160 q^{20} - 184 q^{22} - 128 q^{23} + 200 q^{26} - 120 q^{28} - 40 q^{32} + 120 q^{34} + 192 q^{35} - 280 q^{38} + 584 q^{40} - 192 q^{43} - 104 q^{44} + 32 q^{46} + 312 q^{50} - 424 q^{52} - 320 q^{53} - 256 q^{55} + 392 q^{56} - 352 q^{58} + 256 q^{59} + 64 q^{61} + 48 q^{62} + 408 q^{64} + 64 q^{67} - 856 q^{68} + 984 q^{70} - 512 q^{71} - 1056 q^{74} + 296 q^{76} + 448 q^{77} + 512 q^{79} - 328 q^{80} - 760 q^{82} + 448 q^{86} - 1072 q^{88} + 192 q^{91} + 784 q^{92} - 480 q^{94} - 272 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −1.98432 − 0.249937i 0 3.87506 + 0.991913i 1.26028 − 0.522023i 0 −2.90283 − 2.90283i −7.44145 − 2.93680i 0 −2.63126 + 0.720872i
19.2 −1.92938 + 0.526758i 0 3.44505 − 2.03264i −3.51382 + 1.45547i 0 −1.77216 − 1.77216i −5.57613 + 5.73645i 0 6.01282 − 4.65909i
19.3 −1.64362 − 1.13952i 0 1.40297 + 3.74589i −0.838363 + 0.347262i 0 8.29065 + 8.29065i 1.96258 − 7.75553i 0 1.77366 + 0.384569i
19.4 −1.32170 + 1.50104i 0 −0.506212 − 3.96784i −2.17599 + 0.901324i 0 0.825541 + 0.825541i 6.62493 + 4.48446i 0 1.52309 − 4.45751i
19.5 −1.02518 − 1.71727i 0 −1.89801 + 3.52102i 7.00900 − 2.90322i 0 2.35957 + 2.35957i 7.99233 − 0.350291i 0 −12.1711 − 9.05999i
19.6 −0.948970 + 1.76053i 0 −2.19891 − 3.34138i 4.89089 − 2.02587i 0 −6.40097 − 6.40097i 7.96928 − 0.700377i 0 −1.07471 + 10.5330i
19.7 −0.729010 − 1.86240i 0 −2.93709 + 2.71542i −8.33737 + 3.45345i 0 1.00716 + 1.00716i 7.19837 + 3.49048i 0 12.5098 + 13.0099i
19.8 −0.408652 − 1.95781i 0 −3.66601 + 1.60012i 4.62667 − 1.91643i 0 −2.45923 − 2.45923i 4.63085 + 6.52344i 0 −5.64269 − 8.27497i
19.9 0.712728 + 1.86869i 0 −2.98404 + 2.66374i 4.75081 − 1.96785i 0 5.89664 + 5.89664i −7.10452 − 3.67773i 0 7.06335 + 7.47527i
19.10 0.795895 − 1.83482i 0 −2.73310 − 2.92064i −6.54910 + 2.71273i 0 1.95699 + 1.95699i −7.53410 + 2.69022i 0 −0.235042 + 14.1754i
19.11 1.19507 + 1.60369i 0 −1.14363 + 3.83303i −2.44597 + 1.01315i 0 3.89082 + 3.89082i −7.51370 + 2.74670i 0 −4.54788 − 2.71179i
19.12 1.46158 − 1.36521i 0 0.272417 − 3.99071i 1.99171 − 0.824993i 0 8.28172 + 8.28172i −5.04999 − 6.20464i 0 1.78475 − 3.92489i
19.13 1.49565 + 1.32779i 0 0.473939 + 3.97182i −5.66092 + 2.34483i 0 −8.77775 − 8.77775i −4.56491 + 6.56975i 0 −11.5802 − 4.00947i
19.14 1.84914 − 0.762031i 0 2.83862 − 2.81820i −6.06733 + 2.51317i 0 −9.25703 − 9.25703i 3.10143 − 7.37436i 0 −9.30421 + 9.27069i
19.15 1.92437 + 0.544810i 0 3.40636 + 2.09683i 8.76568 − 3.63086i 0 −4.20494 − 4.20494i 5.41272 + 5.89088i 0 18.8465 − 2.21148i
19.16 1.97063 + 0.341475i 0 3.76679 + 1.34584i 2.29382 − 0.950132i 0 3.26583 + 3.26583i 6.96339 + 3.93843i 0 4.84473 − 1.08908i
91.1 −1.98432 + 0.249937i 0 3.87506 − 0.991913i 1.26028 + 0.522023i 0 −2.90283 + 2.90283i −7.44145 + 2.93680i 0 −2.63126 − 0.720872i
91.2 −1.92938 − 0.526758i 0 3.44505 + 2.03264i −3.51382 − 1.45547i 0 −1.77216 + 1.77216i −5.57613 − 5.73645i 0 6.01282 + 4.65909i
91.3 −1.64362 + 1.13952i 0 1.40297 − 3.74589i −0.838363 − 0.347262i 0 8.29065 − 8.29065i 1.96258 + 7.75553i 0 1.77366 − 0.384569i
91.4 −1.32170 − 1.50104i 0 −0.506212 + 3.96784i −2.17599 − 0.901324i 0 0.825541 − 0.825541i 6.62493 − 4.48446i 0 1.52309 + 4.45751i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.16
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
32.h odd 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.3.u.b 64
3.b odd 2 1 96.3.m.a ✓ 64
12.b even 2 1 384.3.m.a 64
32.h odd 8 1 inner 288.3.u.b 64
96.o even 8 1 96.3.m.a ✓ 64
96.p odd 8 1 384.3.m.a 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.3.m.a ✓ 64 3.b odd 2 1
96.3.m.a ✓ 64 96.o even 8 1
288.3.u.b 64 1.a even 1 1 trivial
288.3.u.b 64 32.h odd 8 1 inner
384.3.m.a 64 12.b even 2 1
384.3.m.a 64 96.p odd 8 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{64} - 64 T_{5}^{61} - 2304 T_{5}^{59} + 14912 T_{5}^{58} + 750336 T_{5}^{57} + \cdots + 11\!\cdots\!24 \) acting on \(S_{3}^{\mathrm{new}}(288, [\chi])\). Copy content Toggle raw display