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Properties

Label 288.3.u.a
Level $288$
Weight $3$
Character orbit 288.u
Analytic conductor $7.847$
Analytic rank $0$
Dimension $28$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [288,3,Mod(19,288)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://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);
 
Copy content sage: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")
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 288.u (of order \(8\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [28,4,0,-4,4,0,-4,4,0,-44,4,0,-4,20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.84743161358\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(7\) over \(\Q(\zeta_{8})\)
Twist minimal: no (minimal twist has level 32)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 4 q^{2} - 4 q^{4} + 4 q^{5} - 4 q^{7} + 4 q^{8} - 44 q^{10} + 4 q^{11} - 4 q^{13} + 20 q^{14} + 16 q^{16} - 4 q^{19} - 76 q^{20} + 144 q^{22} + 68 q^{23} - 4 q^{25} - 96 q^{26} + 56 q^{28} + 4 q^{29}+ \cdots + 328 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −1.82416 + 0.820030i 0 2.65510 − 2.99173i 7.60625 − 3.15061i 0 6.84161 + 6.84161i −2.39002 + 7.63465i 0 −11.2914 + 11.9846i
19.2 −1.61758 − 1.17620i 0 1.23313 + 3.80518i −2.28872 + 0.948019i 0 −6.37744 − 6.37744i 2.48095 − 7.60558i 0 4.81725 + 1.15849i
19.3 −0.345994 + 1.96984i 0 −3.76058 − 1.36311i −7.20074 + 2.98264i 0 4.26150 + 4.26150i 3.98625 − 6.93612i 0 −3.38393 − 15.2163i
19.4 0.108191 + 1.99707i 0 −3.97659 + 0.432130i 2.81639 − 1.16659i 0 −6.23443 − 6.23443i −1.29322 − 7.89478i 0 2.63447 + 5.49832i
19.5 0.360897 − 1.96717i 0 −3.73951 − 1.41989i 0.452310 − 0.187353i 0 0.429965 + 0.429965i −4.14274 + 6.84381i 0 −0.205317 − 0.957385i
19.6 1.62478 − 1.16623i 0 1.27980 − 3.78974i 4.51028 − 1.86822i 0 −3.85317 − 3.85317i −2.34032 − 7.65003i 0 5.14942 − 8.29547i
19.7 1.98676 + 0.229757i 0 3.89442 + 0.912943i −4.18866 + 1.73500i 0 3.93197 + 3.93197i 7.52753 + 2.70857i 0 −8.72048 + 2.48465i
91.1 −1.82416 − 0.820030i 0 2.65510 + 2.99173i 7.60625 + 3.15061i 0 6.84161 − 6.84161i −2.39002 − 7.63465i 0 −11.2914 − 11.9846i
91.2 −1.61758 + 1.17620i 0 1.23313 − 3.80518i −2.28872 − 0.948019i 0 −6.37744 + 6.37744i 2.48095 + 7.60558i 0 4.81725 − 1.15849i
91.3 −0.345994 − 1.96984i 0 −3.76058 + 1.36311i −7.20074 − 2.98264i 0 4.26150 − 4.26150i 3.98625 + 6.93612i 0 −3.38393 + 15.2163i
91.4 0.108191 − 1.99707i 0 −3.97659 − 0.432130i 2.81639 + 1.16659i 0 −6.23443 + 6.23443i −1.29322 + 7.89478i 0 2.63447 − 5.49832i
91.5 0.360897 + 1.96717i 0 −3.73951 + 1.41989i 0.452310 + 0.187353i 0 0.429965 − 0.429965i −4.14274 − 6.84381i 0 −0.205317 + 0.957385i
91.6 1.62478 + 1.16623i 0 1.27980 + 3.78974i 4.51028 + 1.86822i 0 −3.85317 + 3.85317i −2.34032 + 7.65003i 0 5.14942 + 8.29547i
91.7 1.98676 − 0.229757i 0 3.89442 − 0.912943i −4.18866 − 1.73500i 0 3.93197 − 3.93197i 7.52753 − 2.70857i 0 −8.72048 − 2.48465i
163.1 −1.93931 + 0.488972i 0 3.52181 − 1.89653i 1.85856 + 4.48696i 0 −5.27676 − 5.27676i −5.90252 + 5.40002i 0 −5.79831 − 7.79280i
163.2 −1.20513 − 1.59614i 0 −1.09531 + 3.84712i 0.642823 + 1.55191i 0 −4.95044 − 4.95044i 7.46052 − 2.88803i 0 1.70238 − 2.89629i
163.3 −0.682385 + 1.87999i 0 −3.06870 − 2.56575i −1.34740 − 3.25291i 0 0.583225 + 0.583225i 6.91761 − 4.01829i 0 7.03487 − 0.313357i
163.4 0.658450 − 1.88850i 0 −3.13289 − 2.48697i 0.659338 + 1.59178i 0 9.54718 + 9.54718i −6.75950 + 4.27892i 0 3.44023 − 0.197052i
163.5 1.44490 + 1.38284i 0 0.175499 + 3.99615i 3.18221 + 7.68254i 0 3.67370 + 3.67370i −5.27246 + 6.01674i 0 −6.02574 + 15.5010i
163.6 1.46783 − 1.35848i 0 0.309042 − 3.98804i −1.74699 − 4.21761i 0 −0.392379 − 0.392379i −4.96407 − 6.27359i 0 −8.29385 − 3.81747i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.7
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.a 28
3.b odd 2 1 32.3.h.a ✓ 28
12.b even 2 1 128.3.h.a 28
24.f even 2 1 256.3.h.a 28
24.h odd 2 1 256.3.h.b 28
32.h odd 8 1 inner 288.3.u.a 28
96.o even 8 1 32.3.h.a ✓ 28
96.o even 8 1 256.3.h.b 28
96.p odd 8 1 128.3.h.a 28
96.p odd 8 1 256.3.h.a 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.3.h.a ✓ 28 3.b odd 2 1
32.3.h.a ✓ 28 96.o even 8 1
128.3.h.a 28 12.b even 2 1
128.3.h.a 28 96.p odd 8 1
256.3.h.a 28 24.f even 2 1
256.3.h.a 28 96.p odd 8 1
256.3.h.b 28 24.h odd 2 1
256.3.h.b 28 96.o even 8 1
288.3.u.a 28 1.a even 1 1 trivial
288.3.u.a 28 32.h odd 8 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{28} - 4 T_{5}^{27} + 10 T_{5}^{26} + 12 T_{5}^{25} - 94 T_{5}^{24} - 2800 T_{5}^{23} + \cdots + 58\!\cdots\!28 \) acting on \(S_{3}^{\mathrm{new}}(288, [\chi])\). Copy content Toggle raw display