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: | \(28\) |
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 dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.
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 | −1.82416 | + | 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 |
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 \(S_{3}^{\mathrm{new}}(288, [\chi])\).