Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [240,4,Mod(17,240)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(240, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 0, 2, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("240.17");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 240 = 2^{4} \cdot 3 \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 240.v (of order \(4\), degree \(2\), not 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: | \(14.1604584014\) |
Analytic rank: | \(0\) |
Dimension: | \(36\) |
Relative dimension: | \(18\) over \(\Q(i)\) |
Twist minimal: | no (minimal twist has level 120) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
17.1 | 0 | −5.08308 | − | 1.07808i | 0 | 10.2050 | − | 4.56698i | 0 | −1.68165 | − | 1.68165i | 0 | 24.6755 | + | 10.9599i | 0 | ||||||||||
17.2 | 0 | −4.99399 | − | 1.43528i | 0 | 0.901387 | + | 11.1439i | 0 | −0.0607868 | − | 0.0607868i | 0 | 22.8800 | + | 14.3355i | 0 | ||||||||||
17.3 | 0 | −4.83368 | + | 1.90672i | 0 | −6.19347 | − | 9.30811i | 0 | −20.8011 | − | 20.8011i | 0 | 19.7289 | − | 18.4329i | 0 | ||||||||||
17.4 | 0 | −4.20571 | + | 3.05156i | 0 | −9.28746 | + | 6.22439i | 0 | 21.8991 | + | 21.8991i | 0 | 8.37595 | − | 25.6679i | 0 | ||||||||||
17.5 | 0 | −3.51445 | − | 3.82735i | 0 | −11.0949 | − | 1.37965i | 0 | −7.07013 | − | 7.07013i | 0 | −2.29721 | + | 26.9021i | 0 | ||||||||||
17.6 | 0 | −3.05156 | + | 4.20571i | 0 | 9.28746 | − | 6.22439i | 0 | 21.8991 | + | 21.8991i | 0 | −8.37595 | − | 25.6679i | 0 | ||||||||||
17.7 | 0 | −1.90672 | + | 4.83368i | 0 | 6.19347 | + | 9.30811i | 0 | −20.8011 | − | 20.8011i | 0 | −19.7289 | − | 18.4329i | 0 | ||||||||||
17.8 | 0 | −1.89532 | − | 4.83816i | 0 | −8.68358 | − | 7.04240i | 0 | 20.0016 | + | 20.0016i | 0 | −19.8155 | + | 18.3397i | 0 | ||||||||||
17.9 | 0 | −0.394983 | − | 5.18112i | 0 | 11.1795 | − | 0.136761i | 0 | −10.5113 | − | 10.5113i | 0 | −26.6880 | + | 4.09291i | 0 | ||||||||||
17.10 | 0 | 1.07808 | + | 5.08308i | 0 | −10.2050 | + | 4.56698i | 0 | −1.68165 | − | 1.68165i | 0 | −24.6755 | + | 10.9599i | 0 | ||||||||||
17.11 | 0 | 1.43528 | + | 4.99399i | 0 | −0.901387 | − | 11.1439i | 0 | −0.0607868 | − | 0.0607868i | 0 | −22.8800 | + | 14.3355i | 0 | ||||||||||
17.12 | 0 | 1.52818 | − | 4.96635i | 0 | 1.11114 | + | 11.1250i | 0 | −9.09927 | − | 9.09927i | 0 | −22.3293 | − | 15.1790i | 0 | ||||||||||
17.13 | 0 | 2.43434 | − | 4.59064i | 0 | 4.14870 | − | 10.3821i | 0 | 10.3236 | + | 10.3236i | 0 | −15.1480 | − | 22.3504i | 0 | ||||||||||
17.14 | 0 | 3.82735 | + | 3.51445i | 0 | 11.0949 | + | 1.37965i | 0 | −7.07013 | − | 7.07013i | 0 | 2.29721 | + | 26.9021i | 0 | ||||||||||
17.15 | 0 | 4.59064 | − | 2.43434i | 0 | −4.14870 | + | 10.3821i | 0 | 10.3236 | + | 10.3236i | 0 | 15.1480 | − | 22.3504i | 0 | ||||||||||
17.16 | 0 | 4.83816 | + | 1.89532i | 0 | 8.68358 | + | 7.04240i | 0 | 20.0016 | + | 20.0016i | 0 | 19.8155 | + | 18.3397i | 0 | ||||||||||
17.17 | 0 | 4.96635 | − | 1.52818i | 0 | −1.11114 | − | 11.1250i | 0 | −9.09927 | − | 9.09927i | 0 | 22.3293 | − | 15.1790i | 0 | ||||||||||
17.18 | 0 | 5.18112 | + | 0.394983i | 0 | −11.1795 | + | 0.136761i | 0 | −10.5113 | − | 10.5113i | 0 | 26.6880 | + | 4.09291i | 0 | ||||||||||
113.1 | 0 | −5.08308 | + | 1.07808i | 0 | 10.2050 | + | 4.56698i | 0 | −1.68165 | + | 1.68165i | 0 | 24.6755 | − | 10.9599i | 0 | ||||||||||
113.2 | 0 | −4.99399 | + | 1.43528i | 0 | 0.901387 | − | 11.1439i | 0 | −0.0607868 | + | 0.0607868i | 0 | 22.8800 | − | 14.3355i | 0 | ||||||||||
See all 36 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
5.c | odd | 4 | 1 | inner |
15.e | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 240.4.v.e | 36 | |
3.b | odd | 2 | 1 | inner | 240.4.v.e | 36 | |
4.b | odd | 2 | 1 | 120.4.r.a | ✓ | 36 | |
5.c | odd | 4 | 1 | inner | 240.4.v.e | 36 | |
12.b | even | 2 | 1 | 120.4.r.a | ✓ | 36 | |
15.e | even | 4 | 1 | inner | 240.4.v.e | 36 | |
20.e | even | 4 | 1 | 120.4.r.a | ✓ | 36 | |
60.l | odd | 4 | 1 | 120.4.r.a | ✓ | 36 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
120.4.r.a | ✓ | 36 | 4.b | odd | 2 | 1 | |
120.4.r.a | ✓ | 36 | 12.b | even | 2 | 1 | |
120.4.r.a | ✓ | 36 | 20.e | even | 4 | 1 | |
120.4.r.a | ✓ | 36 | 60.l | odd | 4 | 1 | |
240.4.v.e | 36 | 1.a | even | 1 | 1 | trivial | |
240.4.v.e | 36 | 3.b | odd | 2 | 1 | inner | |
240.4.v.e | 36 | 5.c | odd | 4 | 1 | inner | |
240.4.v.e | 36 | 15.e | even | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{18} - 6 T_{7}^{17} + 18 T_{7}^{16} + 11072 T_{7}^{15} + 1153776 T_{7}^{14} + \cdots + 21\!\cdots\!12 \) acting on \(S_{4}^{\mathrm{new}}(240, [\chi])\).