Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [99,5,Mod(19,99)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(99, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 3]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("99.19");
S:= CuspForms(chi, 5);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 99 = 3^{2} \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 5 \) |
Character orbit: | \([\chi]\) | \(=\) | 99.k (of order \(10\), 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: | \(10.2336263453\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(8\) over \(\Q(\zeta_{10})\) |
Twist minimal: | no (minimal twist has level 33) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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 | −3.38915 | − | 4.66477i | 0 | −5.32944 | + | 16.4023i | −4.81744 | − | 3.50008i | 0 | −13.8978 | − | 4.51567i | 6.83522 | − | 2.22090i | 0 | 34.3346i | ||||||||
19.2 | −1.95429 | − | 2.68985i | 0 | 1.52822 | − | 4.70338i | −35.4362 | − | 25.7459i | 0 | 8.04612 | + | 2.61434i | −66.2318 | + | 21.5200i | 0 | 145.633i | ||||||||
19.3 | −1.01888 | − | 1.40237i | 0 | 4.01574 | − | 12.3592i | 32.0081 | + | 23.2552i | 0 | 27.3116 | + | 8.87408i | −47.8012 | + | 15.5316i | 0 | − | 68.5817i | |||||||
19.4 | 0.0886329 | + | 0.121993i | 0 | 4.93725 | − | 15.1953i | −11.7622 | − | 8.54577i | 0 | 66.7007 | + | 21.6724i | 4.58589 | − | 1.49005i | 0 | − | 2.19234i | |||||||
19.5 | 1.95593 | + | 2.69210i | 0 | 1.52251 | − | 4.68579i | −6.63359 | − | 4.81958i | 0 | −71.1959 | − | 23.1329i | 66.2286 | − | 21.5190i | 0 | − | 27.2851i | |||||||
19.6 | 2.44632 | + | 3.36708i | 0 | −0.408428 | + | 1.25701i | 26.0050 | + | 18.8937i | 0 | −15.8191 | − | 5.13994i | 58.1002 | − | 18.8779i | 0 | 133.781i | ||||||||
19.7 | 4.20830 | + | 5.79223i | 0 | −10.8958 | + | 33.5339i | −33.2604 | − | 24.1651i | 0 | −32.2093 | − | 10.4654i | −131.142 | + | 42.6108i | 0 | − | 294.346i | |||||||
19.8 | 4.37135 | + | 6.01665i | 0 | −12.1471 | + | 37.3849i | 24.8968 | + | 18.0886i | 0 | 40.6127 | + | 13.1959i | −164.863 | + | 53.5673i | 0 | 228.867i | ||||||||
28.1 | −6.48369 | + | 2.10668i | 0 | 24.6559 | − | 17.9136i | −2.10789 | + | 6.48741i | 0 | 12.4454 | + | 17.1296i | −58.0090 | + | 79.8425i | 0 | − | 46.5030i | |||||||
28.2 | −6.41780 | + | 2.08527i | 0 | 23.8955 | − | 17.3611i | 9.80980 | − | 30.1915i | 0 | −46.3989 | − | 63.8626i | −53.6911 | + | 73.8995i | 0 | 214.219i | ||||||||
28.3 | −4.40557 | + | 1.43146i | 0 | 4.41572 | − | 3.20821i | −12.7557 | + | 39.2579i | 0 | 37.7759 | + | 51.9941i | 28.7033 | − | 39.5067i | 0 | − | 191.213i | |||||||
28.4 | −0.743344 | + | 0.241527i | 0 | −12.4500 | + | 9.04549i | 9.71709 | − | 29.9061i | 0 | 16.8776 | + | 23.2301i | 14.4205 | − | 19.8482i | 0 | 24.5775i | ||||||||
28.5 | −0.619915 | + | 0.201423i | 0 | −12.6005 | + | 9.15483i | −12.5302 | + | 38.5641i | 0 | −44.8315 | − | 61.7053i | 12.0973 | − | 16.6506i | 0 | − | 26.4303i | |||||||
28.6 | 0.173259 | − | 0.0562952i | 0 | −12.9174 | + | 9.38506i | 6.40673 | − | 19.7179i | 0 | 12.6762 | + | 17.4473i | −3.42300 | + | 4.71136i | 0 | − | 3.77697i | |||||||
28.7 | 4.49285 | − | 1.45982i | 0 | 5.11038 | − | 3.71291i | −5.35863 | + | 16.4922i | 0 | 48.2339 | + | 66.3883i | −26.8877 | + | 37.0078i | 0 | 81.9194i | ||||||||
28.8 | 7.29601 | − | 2.37062i | 0 | 34.6676 | − | 25.1875i | −2.18122 | + | 6.71310i | 0 | 28.6722 | + | 39.4639i | 121.078 | − | 166.650i | 0 | 54.1497i | ||||||||
46.1 | −6.48369 | − | 2.10668i | 0 | 24.6559 | + | 17.9136i | −2.10789 | − | 6.48741i | 0 | 12.4454 | − | 17.1296i | −58.0090 | − | 79.8425i | 0 | 46.5030i | ||||||||
46.2 | −6.41780 | − | 2.08527i | 0 | 23.8955 | + | 17.3611i | 9.80980 | + | 30.1915i | 0 | −46.3989 | + | 63.8626i | −53.6911 | − | 73.8995i | 0 | − | 214.219i | |||||||
46.3 | −4.40557 | − | 1.43146i | 0 | 4.41572 | + | 3.20821i | −12.7557 | − | 39.2579i | 0 | 37.7759 | − | 51.9941i | 28.7033 | + | 39.5067i | 0 | 191.213i | ||||||||
46.4 | −0.743344 | − | 0.241527i | 0 | −12.4500 | − | 9.04549i | 9.71709 | + | 29.9061i | 0 | 16.8776 | − | 23.2301i | 14.4205 | + | 19.8482i | 0 | − | 24.5775i | |||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.d | odd | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 99.5.k.c | 32 | |
3.b | odd | 2 | 1 | 33.5.g.a | ✓ | 32 | |
11.d | odd | 10 | 1 | inner | 99.5.k.c | 32 | |
33.f | even | 10 | 1 | 33.5.g.a | ✓ | 32 | |
33.f | even | 10 | 1 | 363.5.c.e | 32 | ||
33.h | odd | 10 | 1 | 363.5.c.e | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
33.5.g.a | ✓ | 32 | 3.b | odd | 2 | 1 | |
33.5.g.a | ✓ | 32 | 33.f | even | 10 | 1 | |
99.5.k.c | 32 | 1.a | even | 1 | 1 | trivial | |
99.5.k.c | 32 | 11.d | odd | 10 | 1 | inner | |
363.5.c.e | 32 | 33.f | even | 10 | 1 | ||
363.5.c.e | 32 | 33.h | odd | 10 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{32} - 102 T_{2}^{30} + 480 T_{2}^{29} + 6836 T_{2}^{28} - 48960 T_{2}^{27} - 279108 T_{2}^{26} + 2770830 T_{2}^{25} + 10746133 T_{2}^{24} - 97459650 T_{2}^{23} - 216432384 T_{2}^{22} + \cdots + 7015378606336 \)
acting on \(S_{5}^{\mathrm{new}}(99, [\chi])\).