Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [150,6,Mod(31,150)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(150, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 4]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("150.31");
S:= CuspForms(chi, 6);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 150 = 2 \cdot 3 \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 150.g (of order \(5\), 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: | \(24.0575729719\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(6\) over \(\Q(\zeta_{5})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
31.1 | 1.23607 | + | 3.80423i | −7.28115 | + | 5.29007i | −12.9443 | + | 9.40456i | −55.5966 | − | 5.83204i | −29.1246 | − | 21.1603i | −184.364 | −51.7771 | − | 37.6183i | 25.0304 | − | 77.0356i | −46.5348 | − | 218.711i | ||
31.2 | 1.23607 | + | 3.80423i | −7.28115 | + | 5.29007i | −12.9443 | + | 9.40456i | −55.5791 | + | 5.99730i | −29.1246 | − | 21.1603i | 161.028 | −51.7771 | − | 37.6183i | 25.0304 | − | 77.0356i | −91.5146 | − | 204.022i | ||
31.3 | 1.23607 | + | 3.80423i | −7.28115 | + | 5.29007i | −12.9443 | + | 9.40456i | −10.1537 | + | 54.9718i | −29.1246 | − | 21.1603i | 197.545 | −51.7771 | − | 37.6183i | 25.0304 | − | 77.0356i | −221.676 | + | 29.3220i | ||
31.4 | 1.23607 | + | 3.80423i | −7.28115 | + | 5.29007i | −12.9443 | + | 9.40456i | 37.7606 | + | 41.2206i | −29.1246 | − | 21.1603i | −165.301 | −51.7771 | − | 37.6183i | 25.0304 | − | 77.0356i | −110.138 | + | 194.601i | ||
31.5 | 1.23607 | + | 3.80423i | −7.28115 | + | 5.29007i | −12.9443 | + | 9.40456i | 38.8459 | − | 40.1995i | −29.1246 | − | 21.1603i | −37.1005 | −51.7771 | − | 37.6183i | 25.0304 | − | 77.0356i | 200.944 | + | 98.0892i | ||
31.6 | 1.23607 | + | 3.80423i | −7.28115 | + | 5.29007i | −12.9443 | + | 9.40456i | 55.5770 | + | 6.01600i | −29.1246 | − | 21.1603i | 98.1755 | −51.7771 | − | 37.6183i | 25.0304 | − | 77.0356i | 45.8108 | + | 218.864i | ||
61.1 | −3.23607 | + | 2.35114i | 2.78115 | − | 8.55951i | 4.94427 | − | 15.2169i | −55.1199 | + | 9.31618i | 11.1246 | + | 34.2380i | 58.3864 | 19.7771 | + | 60.8676i | −65.5304 | − | 47.6106i | 156.468 | − | 159.743i | ||
61.2 | −3.23607 | + | 2.35114i | 2.78115 | − | 8.55951i | 4.94427 | − | 15.2169i | −44.9762 | − | 33.1985i | 11.1246 | + | 34.2380i | −54.5329 | 19.7771 | + | 60.8676i | −65.5304 | − | 47.6106i | 223.600 | + | 1.68726i | ||
61.3 | −3.23607 | + | 2.35114i | 2.78115 | − | 8.55951i | 4.94427 | − | 15.2169i | −8.34027 | + | 55.2760i | 11.1246 | + | 34.2380i | −188.522 | 19.7771 | + | 60.8676i | −65.5304 | − | 47.6106i | −102.972 | − | 198.486i | ||
61.4 | −3.23607 | + | 2.35114i | 2.78115 | − | 8.55951i | 4.94427 | − | 15.2169i | 10.3231 | − | 54.9403i | 11.1246 | + | 34.2380i | 175.692 | 19.7771 | + | 60.8676i | −65.5304 | − | 47.6106i | 95.7661 | + | 202.061i | ||
61.5 | −3.23607 | + | 2.35114i | 2.78115 | − | 8.55951i | 4.94427 | − | 15.2169i | 49.1876 | − | 26.5627i | 11.1246 | + | 34.2380i | −149.411 | 19.7771 | + | 60.8676i | −65.5304 | − | 47.6106i | −96.7217 | + | 201.606i | ||
61.6 | −3.23607 | + | 2.35114i | 2.78115 | − | 8.55951i | 4.94427 | − | 15.2169i | 53.0716 | + | 17.5614i | 11.1246 | + | 34.2380i | −17.5965 | 19.7771 | + | 60.8676i | −65.5304 | − | 47.6106i | −213.033 | + | 67.9490i | ||
91.1 | −3.23607 | − | 2.35114i | 2.78115 | + | 8.55951i | 4.94427 | + | 15.2169i | −55.1199 | − | 9.31618i | 11.1246 | − | 34.2380i | 58.3864 | 19.7771 | − | 60.8676i | −65.5304 | + | 47.6106i | 156.468 | + | 159.743i | ||
91.2 | −3.23607 | − | 2.35114i | 2.78115 | + | 8.55951i | 4.94427 | + | 15.2169i | −44.9762 | + | 33.1985i | 11.1246 | − | 34.2380i | −54.5329 | 19.7771 | − | 60.8676i | −65.5304 | + | 47.6106i | 223.600 | − | 1.68726i | ||
91.3 | −3.23607 | − | 2.35114i | 2.78115 | + | 8.55951i | 4.94427 | + | 15.2169i | −8.34027 | − | 55.2760i | 11.1246 | − | 34.2380i | −188.522 | 19.7771 | − | 60.8676i | −65.5304 | + | 47.6106i | −102.972 | + | 198.486i | ||
91.4 | −3.23607 | − | 2.35114i | 2.78115 | + | 8.55951i | 4.94427 | + | 15.2169i | 10.3231 | + | 54.9403i | 11.1246 | − | 34.2380i | 175.692 | 19.7771 | − | 60.8676i | −65.5304 | + | 47.6106i | 95.7661 | − | 202.061i | ||
91.5 | −3.23607 | − | 2.35114i | 2.78115 | + | 8.55951i | 4.94427 | + | 15.2169i | 49.1876 | + | 26.5627i | 11.1246 | − | 34.2380i | −149.411 | 19.7771 | − | 60.8676i | −65.5304 | + | 47.6106i | −96.7217 | − | 201.606i | ||
91.6 | −3.23607 | − | 2.35114i | 2.78115 | + | 8.55951i | 4.94427 | + | 15.2169i | 53.0716 | − | 17.5614i | 11.1246 | − | 34.2380i | −17.5965 | 19.7771 | − | 60.8676i | −65.5304 | + | 47.6106i | −213.033 | − | 67.9490i | ||
121.1 | 1.23607 | − | 3.80423i | −7.28115 | − | 5.29007i | −12.9443 | − | 9.40456i | −55.5966 | + | 5.83204i | −29.1246 | + | 21.1603i | −184.364 | −51.7771 | + | 37.6183i | 25.0304 | + | 77.0356i | −46.5348 | + | 218.711i | ||
121.2 | 1.23607 | − | 3.80423i | −7.28115 | − | 5.29007i | −12.9443 | − | 9.40456i | −55.5791 | − | 5.99730i | −29.1246 | + | 21.1603i | 161.028 | −51.7771 | + | 37.6183i | 25.0304 | + | 77.0356i | −91.5146 | + | 204.022i | ||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
25.d | even | 5 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 150.6.g.b | ✓ | 24 |
25.d | even | 5 | 1 | inner | 150.6.g.b | ✓ | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
150.6.g.b | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
150.6.g.b | ✓ | 24 | 25.d | even | 5 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{12} + 106 T_{7}^{11} - 110736 T_{7}^{10} - 11273520 T_{7}^{9} + 4480317260 T_{7}^{8} + 428230842606 T_{7}^{7} - 79253486789869 T_{7}^{6} + \cdots - 97\!\cdots\!24 \)
acting on \(S_{6}^{\mathrm{new}}(150, [\chi])\).