Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [90,5,Mod(11,90)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(90, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("90.11");
S:= CuspForms(chi, 5);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 90 = 2 \cdot 3^{2} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 5 \) |
Character orbit: | \([\chi]\) | \(=\) | 90.h (of order \(6\), degree \(2\), 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: | \(9.30329667755\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(16\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
11.1 | −2.44949 | − | 1.41421i | −8.23651 | − | 3.62767i | 4.00000 | + | 6.92820i | 9.68246 | − | 5.59017i | 15.0449 | + | 20.5341i | 30.9376 | − | 53.5855i | − | 22.6274i | 54.6801 | + | 59.7586i | −31.6228 | |||
11.2 | −2.44949 | − | 1.41421i | −7.00963 | − | 5.64492i | 4.00000 | + | 6.92820i | −9.68246 | + | 5.59017i | 9.18690 | + | 23.7403i | −10.8913 | + | 18.8643i | − | 22.6274i | 17.2698 | + | 79.1376i | 31.6228 | |||
11.3 | −2.44949 | − | 1.41421i | −1.37920 | + | 8.89369i | 4.00000 | + | 6.92820i | −9.68246 | + | 5.59017i | 15.9559 | − | 19.8345i | −23.8583 | + | 41.3239i | − | 22.6274i | −77.1956 | − | 24.5324i | 31.6228 | |||
11.4 | −2.44949 | − | 1.41421i | −1.32770 | + | 8.90153i | 4.00000 | + | 6.92820i | 9.68246 | − | 5.59017i | 15.8409 | − | 19.9266i | 27.8508 | − | 48.2391i | − | 22.6274i | −77.4744 | − | 23.6371i | −31.6228 | |||
11.5 | −2.44949 | − | 1.41421i | −0.611821 | − | 8.97918i | 4.00000 | + | 6.92820i | 9.68246 | − | 5.59017i | −11.1998 | + | 22.8597i | −45.2607 | + | 78.3939i | − | 22.6274i | −80.2513 | + | 10.9873i | −31.6228 | |||
11.6 | −2.44949 | − | 1.41421i | 4.21210 | − | 7.95350i | 4.00000 | + | 6.92820i | −9.68246 | + | 5.59017i | −21.5655 | + | 13.5252i | 12.4476 | − | 21.5599i | − | 22.6274i | −45.5164 | − | 67.0020i | 31.6228 | |||
11.7 | −2.44949 | − | 1.41421i | 8.01375 | − | 4.09632i | 4.00000 | + | 6.92820i | 9.68246 | − | 5.59017i | −25.4227 | − | 1.29927i | 9.80759 | − | 16.9872i | − | 22.6274i | 47.4404 | − | 65.6537i | −31.6228 | |||
11.8 | −2.44949 | − | 1.41421i | 8.33901 | + | 3.38540i | 4.00000 | + | 6.92820i | −9.68246 | + | 5.59017i | −15.6386 | − | 20.0856i | 26.6637 | − | 46.1828i | − | 22.6274i | 58.0781 | + | 56.4618i | 31.6228 | |||
11.9 | 2.44949 | + | 1.41421i | −8.95263 | + | 0.922181i | 4.00000 | + | 6.92820i | −9.68246 | + | 5.59017i | −23.2335 | − | 10.4021i | 4.22228 | − | 7.31320i | 22.6274i | 79.2992 | − | 16.5119i | −31.6228 | ||||
11.10 | 2.44949 | + | 1.41421i | −8.93009 | − | 1.11963i | 4.00000 | + | 6.92820i | 9.68246 | − | 5.59017i | −20.2908 | − | 15.3716i | 13.1727 | − | 22.8158i | 22.6274i | 78.4929 | + | 19.9968i | 31.6228 | ||||
11.11 | 2.44949 | + | 1.41421i | −4.63058 | + | 7.71736i | 4.00000 | + | 6.92820i | 9.68246 | − | 5.59017i | −22.2566 | + | 12.3550i | −36.2886 | + | 62.8537i | 22.6274i | −38.1154 | − | 71.4718i | 31.6228 | ||||
11.12 | 2.44949 | + | 1.41421i | −4.34936 | − | 7.87928i | 4.00000 | + | 6.92820i | −9.68246 | + | 5.59017i | 0.489266 | − | 25.4511i | −24.5976 | + | 42.6042i | 22.6274i | −43.1661 | + | 68.5397i | −31.6228 | ||||
11.13 | 2.44949 | + | 1.41421i | 5.35414 | + | 7.23417i | 4.00000 | + | 6.92820i | −9.68246 | + | 5.59017i | 2.88424 | + | 25.2919i | −10.7341 | + | 18.5921i | 22.6274i | −23.6664 | + | 77.4655i | −31.6228 | ||||
11.14 | 2.44949 | + | 1.41421i | 5.78558 | − | 6.89399i | 4.00000 | + | 6.92820i | −9.68246 | + | 5.59017i | 23.9213 | − | 8.70471i | 39.7478 | − | 68.8452i | 22.6274i | −14.0542 | − | 79.7714i | −31.6228 | ||||
11.15 | 2.44949 | + | 1.41421i | 8.72295 | + | 2.21590i | 4.00000 | + | 6.92820i | 9.68246 | − | 5.59017i | 18.2330 | + | 17.7639i | 45.5675 | − | 78.9253i | 22.6274i | 71.1795 | + | 38.6584i | 31.6228 | ||||
11.16 | 2.44949 | + | 1.41421i | 9.00000 | − | 0.00397022i | 4.00000 | + | 6.92820i | 9.68246 | − | 5.59017i | 22.0510 | + | 12.7182i | −32.7869 | + | 56.7886i | 22.6274i | 81.0000 | − | 0.0714639i | 31.6228 | ||||
41.1 | −2.44949 | + | 1.41421i | −8.23651 | + | 3.62767i | 4.00000 | − | 6.92820i | 9.68246 | + | 5.59017i | 15.0449 | − | 20.5341i | 30.9376 | + | 53.5855i | 22.6274i | 54.6801 | − | 59.7586i | −31.6228 | ||||
41.2 | −2.44949 | + | 1.41421i | −7.00963 | + | 5.64492i | 4.00000 | − | 6.92820i | −9.68246 | − | 5.59017i | 9.18690 | − | 23.7403i | −10.8913 | − | 18.8643i | 22.6274i | 17.2698 | − | 79.1376i | 31.6228 | ||||
41.3 | −2.44949 | + | 1.41421i | −1.37920 | − | 8.89369i | 4.00000 | − | 6.92820i | −9.68246 | − | 5.59017i | 15.9559 | + | 19.8345i | −23.8583 | − | 41.3239i | 22.6274i | −77.1956 | + | 24.5324i | 31.6228 | ||||
41.4 | −2.44949 | + | 1.41421i | −1.32770 | − | 8.90153i | 4.00000 | − | 6.92820i | 9.68246 | + | 5.59017i | 15.8409 | + | 19.9266i | 27.8508 | + | 48.2391i | 22.6274i | −77.4744 | + | 23.6371i | −31.6228 | ||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.d | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 90.5.h.a | ✓ | 32 |
3.b | odd | 2 | 1 | 270.5.h.a | 32 | ||
9.c | even | 3 | 1 | 270.5.h.a | 32 | ||
9.c | even | 3 | 1 | 810.5.d.c | 32 | ||
9.d | odd | 6 | 1 | inner | 90.5.h.a | ✓ | 32 |
9.d | odd | 6 | 1 | 810.5.d.c | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
90.5.h.a | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
90.5.h.a | ✓ | 32 | 9.d | odd | 6 | 1 | inner |
270.5.h.a | 32 | 3.b | odd | 2 | 1 | ||
270.5.h.a | 32 | 9.c | even | 3 | 1 | ||
810.5.d.c | 32 | 9.c | even | 3 | 1 | ||
810.5.d.c | 32 | 9.d | odd | 6 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(90, [\chi])\).