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 | −49.3905 | + | 26.1836i | 29.1246 | + | 21.1603i | −91.4091 | 51.7771 | + | 37.6183i | 25.0304 | − | 77.0356i | 160.658 | + | 155.528i | ||
31.2 | −1.23607 | − | 3.80423i | −7.28115 | + | 5.29007i | −12.9443 | + | 9.40456i | −44.8368 | − | 33.3866i | 29.1246 | + | 21.1603i | 131.563 | 51.7771 | + | 37.6183i | 25.0304 | − | 77.0356i | −71.5889 | + | 211.837i | ||
31.3 | −1.23607 | − | 3.80423i | −7.28115 | + | 5.29007i | −12.9443 | + | 9.40456i | −5.54795 | − | 55.6257i | 29.1246 | + | 21.1603i | −223.096 | 51.7771 | + | 37.6183i | 25.0304 | − | 77.0356i | −204.755 | + | 89.8628i | ||
31.4 | −1.23607 | − | 3.80423i | −7.28115 | + | 5.29007i | −12.9443 | + | 9.40456i | 22.4321 | + | 51.2035i | 29.1246 | + | 21.1603i | 146.638 | 51.7771 | + | 37.6183i | 25.0304 | − | 77.0356i | 167.062 | − | 148.628i | ||
31.5 | −1.23607 | − | 3.80423i | −7.28115 | + | 5.29007i | −12.9443 | + | 9.40456i | 43.0312 | − | 35.6835i | 29.1246 | + | 21.1603i | 32.5493 | 51.7771 | + | 37.6183i | 25.0304 | − | 77.0356i | −188.938 | − | 119.593i | ||
31.6 | −1.23607 | − | 3.80423i | −7.28115 | + | 5.29007i | −12.9443 | + | 9.40456i | 50.9577 | + | 22.9850i | 29.1246 | + | 21.1603i | −122.044 | 51.7771 | + | 37.6183i | 25.0304 | − | 77.0356i | 24.4527 | − | 222.266i | ||
61.1 | 3.23607 | − | 2.35114i | 2.78115 | − | 8.55951i | 4.94427 | − | 15.2169i | −41.7488 | + | 37.1757i | −11.1246 | − | 34.2380i | −164.912 | −19.7771 | − | 60.8676i | −65.5304 | − | 47.6106i | −47.6968 | + | 218.461i | ||
61.2 | 3.23607 | − | 2.35114i | 2.78115 | − | 8.55951i | 4.94427 | − | 15.2169i | −21.1242 | − | 51.7568i | −11.1246 | − | 34.2380i | 92.5726 | −19.7771 | − | 60.8676i | −65.5304 | − | 47.6106i | −190.047 | − | 117.822i | ||
61.3 | 3.23607 | − | 2.35114i | 2.78115 | − | 8.55951i | 4.94427 | − | 15.2169i | −10.7648 | + | 54.8554i | −11.1246 | − | 34.2380i | 36.7606 | −19.7771 | − | 60.8676i | −65.5304 | − | 47.6106i | 94.1374 | + | 202.825i | ||
61.4 | 3.23607 | − | 2.35114i | 2.78115 | − | 8.55951i | 4.94427 | − | 15.2169i | 6.45828 | − | 55.5274i | −11.1246 | − | 34.2380i | −133.388 | −19.7771 | − | 60.8676i | −65.5304 | − | 47.6106i | −109.653 | − | 194.875i | ||
61.5 | 3.23607 | − | 2.35114i | 2.78115 | − | 8.55951i | 4.94427 | − | 15.2169i | 36.5224 | + | 42.3216i | −11.1246 | − | 34.2380i | −114.252 | −19.7771 | − | 60.8676i | −65.5304 | − | 47.6106i | 217.693 | + | 51.0861i | ||
61.6 | 3.23607 | − | 2.35114i | 2.78115 | − | 8.55951i | 4.94427 | − | 15.2169i | 54.0113 | + | 14.4147i | −11.1246 | − | 34.2380i | 182.018 | −19.7771 | − | 60.8676i | −65.5304 | − | 47.6106i | 208.675 | − | 80.3412i | ||
91.1 | 3.23607 | + | 2.35114i | 2.78115 | + | 8.55951i | 4.94427 | + | 15.2169i | −41.7488 | − | 37.1757i | −11.1246 | + | 34.2380i | −164.912 | −19.7771 | + | 60.8676i | −65.5304 | + | 47.6106i | −47.6968 | − | 218.461i | ||
91.2 | 3.23607 | + | 2.35114i | 2.78115 | + | 8.55951i | 4.94427 | + | 15.2169i | −21.1242 | + | 51.7568i | −11.1246 | + | 34.2380i | 92.5726 | −19.7771 | + | 60.8676i | −65.5304 | + | 47.6106i | −190.047 | + | 117.822i | ||
91.3 | 3.23607 | + | 2.35114i | 2.78115 | + | 8.55951i | 4.94427 | + | 15.2169i | −10.7648 | − | 54.8554i | −11.1246 | + | 34.2380i | 36.7606 | −19.7771 | + | 60.8676i | −65.5304 | + | 47.6106i | 94.1374 | − | 202.825i | ||
91.4 | 3.23607 | + | 2.35114i | 2.78115 | + | 8.55951i | 4.94427 | + | 15.2169i | 6.45828 | + | 55.5274i | −11.1246 | + | 34.2380i | −133.388 | −19.7771 | + | 60.8676i | −65.5304 | + | 47.6106i | −109.653 | + | 194.875i | ||
91.5 | 3.23607 | + | 2.35114i | 2.78115 | + | 8.55951i | 4.94427 | + | 15.2169i | 36.5224 | − | 42.3216i | −11.1246 | + | 34.2380i | −114.252 | −19.7771 | + | 60.8676i | −65.5304 | + | 47.6106i | 217.693 | − | 51.0861i | ||
91.6 | 3.23607 | + | 2.35114i | 2.78115 | + | 8.55951i | 4.94427 | + | 15.2169i | 54.0113 | − | 14.4147i | −11.1246 | + | 34.2380i | 182.018 | −19.7771 | + | 60.8676i | −65.5304 | + | 47.6106i | 208.675 | + | 80.3412i | ||
121.1 | −1.23607 | + | 3.80423i | −7.28115 | − | 5.29007i | −12.9443 | − | 9.40456i | −49.3905 | − | 26.1836i | 29.1246 | − | 21.1603i | −91.4091 | 51.7771 | − | 37.6183i | 25.0304 | + | 77.0356i | 160.658 | − | 155.528i | ||
121.2 | −1.23607 | + | 3.80423i | −7.28115 | − | 5.29007i | −12.9443 | − | 9.40456i | −44.8368 | + | 33.3866i | 29.1246 | − | 21.1603i | 131.563 | 51.7771 | − | 37.6183i | 25.0304 | + | 77.0356i | −71.5889 | − | 211.837i | ||
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.c | ✓ | 24 |
25.d | even | 5 | 1 | inner | 150.6.g.c | ✓ | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
150.6.g.c | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
150.6.g.c | ✓ | 24 | 25.d | even | 5 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{12} + 227 T_{7}^{11} - 81229 T_{7}^{10} - 19105670 T_{7}^{9} + 2328673420 T_{7}^{8} + 587745624282 T_{7}^{7} - 28690653762651 T_{7}^{6} + \cdots + 24\!\cdots\!76 \)
acting on \(S_{6}^{\mathrm{new}}(150, [\chi])\).