Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [156,3,Mod(79,156)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(156, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("156.79");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 156 = 2^{2} \cdot 3 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 156.f (of order \(2\), degree \(1\), 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: | \(4.25069212402\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
79.1 | −1.99963 | − | 0.0384798i | 1.73205i | 3.99704 | + | 0.153891i | −4.98564 | 0.0666489 | − | 3.46346i | − | 3.34189i | −7.98668 | − | 0.461529i | −3.00000 | 9.96944 | + | 0.191846i | |||||||
79.2 | −1.99963 | + | 0.0384798i | − | 1.73205i | 3.99704 | − | 0.153891i | −4.98564 | 0.0666489 | + | 3.46346i | 3.34189i | −7.98668 | + | 0.461529i | −3.00000 | 9.96944 | − | 0.191846i | |||||||
79.3 | −1.89675 | − | 0.634294i | 1.73205i | 3.19534 | + | 2.40620i | 6.10663 | 1.09863 | − | 3.28527i | 5.33206i | −4.53454 | − | 6.59075i | −3.00000 | −11.5828 | − | 3.87340i | ||||||||
79.4 | −1.89675 | + | 0.634294i | − | 1.73205i | 3.19534 | − | 2.40620i | 6.10663 | 1.09863 | + | 3.28527i | − | 5.33206i | −4.53454 | + | 6.59075i | −3.00000 | −11.5828 | + | 3.87340i | ||||||
79.5 | −1.66943 | − | 1.10137i | − | 1.73205i | 1.57397 | + | 3.67731i | −0.251667 | −1.90763 | + | 2.89153i | − | 2.33352i | 1.42246 | − | 7.87252i | −3.00000 | 0.420139 | + | 0.277178i | ||||||
79.6 | −1.66943 | + | 1.10137i | 1.73205i | 1.57397 | − | 3.67731i | −0.251667 | −1.90763 | − | 2.89153i | 2.33352i | 1.42246 | + | 7.87252i | −3.00000 | 0.420139 | − | 0.277178i | ||||||||
79.7 | −1.25145 | − | 1.56009i | − | 1.73205i | −0.867741 | + | 3.90474i | 6.81265 | −2.70215 | + | 2.16758i | 11.2335i | 7.17767 | − | 3.53285i | −3.00000 | −8.52570 | − | 10.6283i | |||||||
79.8 | −1.25145 | + | 1.56009i | 1.73205i | −0.867741 | − | 3.90474i | 6.81265 | −2.70215 | − | 2.16758i | − | 11.2335i | 7.17767 | + | 3.53285i | −3.00000 | −8.52570 | + | 10.6283i | |||||||
79.9 | −0.253155 | − | 1.98391i | − | 1.73205i | −3.87182 | + | 1.00448i | −5.54446 | −3.43624 | + | 0.438478i | − | 3.35464i | 2.97297 | + | 7.42708i | −3.00000 | 1.40361 | + | 10.9997i | ||||||
79.10 | −0.253155 | + | 1.98391i | 1.73205i | −3.87182 | − | 1.00448i | −5.54446 | −3.43624 | − | 0.438478i | 3.35464i | 2.97297 | − | 7.42708i | −3.00000 | 1.40361 | − | 10.9997i | ||||||||
79.11 | 0.337137 | − | 1.97138i | 1.73205i | −3.77268 | − | 1.32925i | −3.28562 | 3.41453 | + | 0.583939i | 11.8734i | −3.89237 | + | 6.98924i | −3.00000 | −1.10771 | + | 6.47720i | ||||||||
79.12 | 0.337137 | + | 1.97138i | − | 1.73205i | −3.77268 | + | 1.32925i | −3.28562 | 3.41453 | − | 0.583939i | − | 11.8734i | −3.89237 | − | 6.98924i | −3.00000 | −1.10771 | − | 6.47720i | ||||||
79.13 | 0.805717 | − | 1.83052i | − | 1.73205i | −2.70164 | − | 2.94977i | 5.48615 | −3.17056 | − | 1.39554i | − | 6.35783i | −7.57638 | + | 2.56874i | −3.00000 | 4.42028 | − | 10.0425i | ||||||
79.14 | 0.805717 | + | 1.83052i | 1.73205i | −2.70164 | + | 2.94977i | 5.48615 | −3.17056 | + | 1.39554i | 6.35783i | −7.57638 | − | 2.56874i | −3.00000 | 4.42028 | + | 10.0425i | ||||||||
79.15 | 0.969763 | − | 1.74916i | 1.73205i | −2.11912 | − | 3.39254i | −4.25416 | 3.02963 | + | 1.67968i | − | 10.6496i | −7.98914 | + | 0.416717i | −3.00000 | −4.12553 | + | 7.44120i | |||||||
79.16 | 0.969763 | + | 1.74916i | − | 1.73205i | −2.11912 | + | 3.39254i | −4.25416 | 3.02963 | − | 1.67968i | 10.6496i | −7.98914 | − | 0.416717i | −3.00000 | −4.12553 | − | 7.44120i | |||||||
79.17 | 1.41412 | − | 1.41431i | − | 1.73205i | −0.000546494 | − | 4.00000i | −9.54425 | −2.44966 | − | 2.44932i | 4.66845i | −5.65801 | − | 5.65569i | −3.00000 | −13.4967 | + | 13.4985i | |||||||
79.18 | 1.41412 | + | 1.41431i | 1.73205i | −0.000546494 | 4.00000i | −9.54425 | −2.44966 | + | 2.44932i | − | 4.66845i | −5.65801 | + | 5.65569i | −3.00000 | −13.4967 | − | 13.4985i | ||||||||
79.19 | 1.69411 | − | 1.06301i | 1.73205i | 1.74001 | − | 3.60171i | 5.51671 | 1.84119 | + | 2.93428i | 1.01261i | −0.880886 | − | 7.95135i | −3.00000 | 9.34592 | − | 5.86433i | ||||||||
79.20 | 1.69411 | + | 1.06301i | − | 1.73205i | 1.74001 | + | 3.60171i | 5.51671 | 1.84119 | − | 2.93428i | − | 1.01261i | −0.880886 | + | 7.95135i | −3.00000 | 9.34592 | + | 5.86433i | ||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 156.3.f.a | ✓ | 24 |
3.b | odd | 2 | 1 | 468.3.f.b | 24 | ||
4.b | odd | 2 | 1 | inner | 156.3.f.a | ✓ | 24 |
8.b | even | 2 | 1 | 2496.3.k.e | 24 | ||
8.d | odd | 2 | 1 | 2496.3.k.e | 24 | ||
12.b | even | 2 | 1 | 468.3.f.b | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
156.3.f.a | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
156.3.f.a | ✓ | 24 | 4.b | odd | 2 | 1 | inner |
468.3.f.b | 24 | 3.b | odd | 2 | 1 | ||
468.3.f.b | 24 | 12.b | even | 2 | 1 | ||
2496.3.k.e | 24 | 8.b | even | 2 | 1 | ||
2496.3.k.e | 24 | 8.d | odd | 2 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(156, [\chi])\).