Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [38,10,Mod(5,38)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(38, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([16]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("38.5");
S:= CuspForms(chi, 10);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 38 = 2 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 10 \) |
Character orbit: | \([\chi]\) | \(=\) | 38.e (of order \(9\), degree \(6\), 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: | \(19.5713617742\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Relative dimension: | \(8\) over \(\Q(\zeta_{9})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{9}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
5.1 | −15.0351 | + | 5.47232i | −40.3563 | + | 228.872i | 196.107 | − | 164.554i | 931.504 | + | 781.625i | −645.701 | − | 3661.95i | −2748.77 | − | 4761.01i | −2048.00 | + | 3547.24i | −32257.8 | − | 11740.9i | −18282.5 | − | 6654.30i |
5.2 | −15.0351 | + | 5.47232i | −25.7718 | + | 146.159i | 196.107 | − | 164.554i | −400.333 | − | 335.920i | −412.348 | − | 2338.54i | 3638.56 | + | 6302.17i | −2048.00 | + | 3547.24i | −2202.31 | − | 801.574i | 7857.31 | + | 2859.83i |
5.3 | −15.0351 | + | 5.47232i | −13.8002 | + | 78.2646i | 196.107 | − | 164.554i | 1219.82 | + | 1023.55i | −220.802 | − | 1252.23i | 851.666 | + | 1475.13i | −2048.00 | + | 3547.24i | 12561.1 | + | 4571.86i | −23941.4 | − | 8713.94i |
5.4 | −15.0351 | + | 5.47232i | −11.5377 | + | 65.4333i | 196.107 | − | 164.554i | −2027.32 | − | 1701.12i | −184.602 | − | 1046.93i | −2305.11 | − | 3992.57i | −2048.00 | + | 3547.24i | 14347.6 | + | 5222.09i | 39790.0 | + | 14482.4i |
5.5 | −15.0351 | + | 5.47232i | 7.21020 | − | 40.8911i | 196.107 | − | 164.554i | 488.978 | + | 410.301i | 115.363 | + | 654.257i | −4413.16 | − | 7643.82i | −2048.00 | + | 3547.24i | 16875.9 | + | 6142.32i | −9597.12 | − | 3493.06i |
5.6 | −15.0351 | + | 5.47232i | 19.6083 | − | 111.204i | 196.107 | − | 164.554i | −757.043 | − | 635.234i | 313.732 | + | 1779.26i | 3786.65 | + | 6558.68i | −2048.00 | + | 3547.24i | 6514.11 | + | 2370.94i | 14858.4 | + | 5408.02i |
5.7 | −15.0351 | + | 5.47232i | 34.1990 | − | 193.952i | 196.107 | − | 164.554i | 1838.42 | + | 1542.62i | 547.184 | + | 3103.23i | 977.539 | + | 1693.15i | −2048.00 | + | 3547.24i | −17951.8 | − | 6533.94i | −36082.6 | − | 13133.0i |
5.8 | −15.0351 | + | 5.47232i | 48.1038 | − | 272.810i | 196.107 | − | 164.554i | −1032.82 | − | 866.635i | 769.661 | + | 4364.97i | −4220.81 | − | 7310.67i | −2048.00 | + | 3547.24i | −53615.6 | − | 19514.5i | 20271.0 | + | 7378.03i |
9.1 | 2.77837 | + | 15.7569i | −172.140 | + | 144.442i | −240.561 | + | 87.5572i | 432.938 | + | 157.576i | −2754.24 | − | 2311.08i | 4035.63 | − | 6989.91i | −2048.00 | − | 3547.24i | 5350.58 | − | 30344.6i | −1280.06 | + | 7259.57i |
9.2 | 2.77837 | + | 15.7569i | −127.396 | + | 106.898i | −240.561 | + | 87.5572i | 2138.47 | + | 778.340i | −2038.34 | − | 1710.37i | −5339.17 | + | 9247.71i | −2048.00 | − | 3547.24i | 1384.68 | − | 7852.94i | −6322.78 | + | 35858.3i |
9.3 | 2.77837 | + | 15.7569i | −127.296 | + | 106.814i | −240.561 | + | 87.5572i | −1972.56 | − | 717.951i | −2036.73 | − | 1709.02i | −5187.34 | + | 8984.74i | −2048.00 | − | 3547.24i | 1377.11 | − | 7809.96i | 5832.22 | − | 33076.1i |
9.4 | 2.77837 | + | 15.7569i | 15.1406 | − | 12.7045i | −240.561 | + | 87.5572i | −8.62619 | − | 3.13968i | 242.250 | + | 203.272i | 80.6742 | − | 139.732i | −2048.00 | − | 3547.24i | −3350.08 | + | 18999.3i | 25.5049 | − | 144.645i |
9.5 | 2.77837 | + | 15.7569i | 30.0038 | − | 25.1762i | −240.561 | + | 87.5572i | −2542.70 | − | 925.467i | 480.061 | + | 402.819i | 3969.67 | − | 6875.67i | −2048.00 | − | 3547.24i | −3151.53 | + | 17873.2i | 7517.95 | − | 42636.4i |
9.6 | 2.77837 | + | 15.7569i | 99.1452 | − | 83.1927i | −240.561 | + | 87.5572i | 750.555 | + | 273.180i | 1586.32 | + | 1331.08i | −2034.77 | + | 3524.32i | −2048.00 | − | 3547.24i | −509.171 | + | 2887.65i | −2219.15 | + | 12585.4i |
9.7 | 2.77837 | + | 15.7569i | 147.173 | − | 123.493i | −240.561 | + | 87.5572i | 2321.00 | + | 844.777i | 2354.77 | + | 1975.89i | 2792.61 | − | 4836.95i | −2048.00 | − | 3547.24i | 2991.52 | − | 16965.8i | −6862.47 | + | 38919.0i |
9.8 | 2.77837 | + | 15.7569i | 194.492 | − | 163.198i | −240.561 | + | 87.5572i | −1439.52 | − | 523.944i | 3111.88 | + | 2611.17i | −2289.54 | + | 3965.60i | −2048.00 | − | 3547.24i | 7775.61 | − | 44097.7i | 4256.21 | − | 24138.2i |
17.1 | 2.77837 | − | 15.7569i | −172.140 | − | 144.442i | −240.561 | − | 87.5572i | 432.938 | − | 157.576i | −2754.24 | + | 2311.08i | 4035.63 | + | 6989.91i | −2048.00 | + | 3547.24i | 5350.58 | + | 30344.6i | −1280.06 | − | 7259.57i |
17.2 | 2.77837 | − | 15.7569i | −127.396 | − | 106.898i | −240.561 | − | 87.5572i | 2138.47 | − | 778.340i | −2038.34 | + | 1710.37i | −5339.17 | − | 9247.71i | −2048.00 | + | 3547.24i | 1384.68 | + | 7852.94i | −6322.78 | − | 35858.3i |
17.3 | 2.77837 | − | 15.7569i | −127.296 | − | 106.814i | −240.561 | − | 87.5572i | −1972.56 | + | 717.951i | −2036.73 | + | 1709.02i | −5187.34 | − | 8984.74i | −2048.00 | + | 3547.24i | 1377.11 | + | 7809.96i | 5832.22 | + | 33076.1i |
17.4 | 2.77837 | − | 15.7569i | 15.1406 | + | 12.7045i | −240.561 | − | 87.5572i | −8.62619 | + | 3.13968i | 242.250 | − | 203.272i | 80.6742 | + | 139.732i | −2048.00 | + | 3547.24i | −3350.08 | − | 18999.3i | 25.5049 | + | 144.645i |
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
19.e | even | 9 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 38.10.e.b | ✓ | 48 |
19.e | even | 9 | 1 | inner | 38.10.e.b | ✓ | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
38.10.e.b | ✓ | 48 | 1.a | even | 1 | 1 | trivial |
38.10.e.b | ✓ | 48 | 19.e | even | 9 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{48} - 33 T_{3}^{47} - 3432 T_{3}^{46} + 7223768 T_{3}^{45} - 764354085 T_{3}^{44} + \cdots + 34\!\cdots\!21 \) acting on \(S_{10}^{\mathrm{new}}(38, [\chi])\).