Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [348,3,Mod(53,348)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(348, base_ring=CyclotomicField(14))
chi = DirichletCharacter(H, H._module([0, 7, 4]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("348.53");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 348 = 2^{2} \cdot 3 \cdot 29 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 348.r (of order \(14\), 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: | \(9.48231319974\) |
Analytic rank: | \(0\) |
Dimension: | \(120\) |
Relative dimension: | \(20\) over \(\Q(\zeta_{14})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{14}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
53.1 | 0 | −2.98396 | − | 0.309769i | 0 | −5.06274 | + | 4.03740i | 0 | 0.629032 | + | 2.75597i | 0 | 8.80809 | + | 1.84868i | 0 | ||||||||||
53.2 | 0 | −2.97989 | + | 0.346754i | 0 | 7.46308 | − | 5.95161i | 0 | −1.85676 | − | 8.13499i | 0 | 8.75952 | − | 2.06658i | 0 | ||||||||||
53.3 | 0 | −2.83401 | + | 0.984062i | 0 | 1.79198 | − | 1.42906i | 0 | 2.72567 | + | 11.9419i | 0 | 7.06324 | − | 5.57769i | 0 | ||||||||||
53.4 | 0 | −2.38273 | + | 1.82280i | 0 | −3.02183 | + | 2.40983i | 0 | −1.77361 | − | 7.77068i | 0 | 2.35477 | − | 8.68649i | 0 | ||||||||||
53.5 | 0 | −2.23026 | − | 2.00647i | 0 | 4.19426 | − | 3.34481i | 0 | 0.912105 | + | 3.99619i | 0 | 0.948145 | + | 8.94992i | 0 | ||||||||||
53.6 | 0 | −1.93357 | − | 2.29376i | 0 | −4.35455 | + | 3.47263i | 0 | −2.75840 | − | 12.0854i | 0 | −1.52265 | + | 8.87026i | 0 | ||||||||||
53.7 | 0 | −1.88476 | + | 2.33403i | 0 | 2.02360 | − | 1.61377i | 0 | 0.397329 | + | 1.74081i | 0 | −1.89539 | − | 8.79815i | 0 | ||||||||||
53.8 | 0 | −1.73632 | − | 2.44646i | 0 | 0.698785 | − | 0.557262i | 0 | 0.563272 | + | 2.46786i | 0 | −2.97037 | + | 8.49570i | 0 | ||||||||||
53.9 | 0 | −0.343594 | + | 2.98026i | 0 | 1.96181 | − | 1.56449i | 0 | −1.30069 | − | 5.69871i | 0 | −8.76389 | − | 2.04800i | 0 | ||||||||||
53.10 | 0 | −0.0184643 | + | 2.99994i | 0 | −6.38586 | + | 5.09256i | 0 | 1.90710 | + | 8.35555i | 0 | −8.99932 | − | 0.110783i | 0 | ||||||||||
53.11 | 0 | 0.502892 | − | 2.95755i | 0 | −0.698785 | + | 0.557262i | 0 | 0.563272 | + | 2.46786i | 0 | −8.49420 | − | 2.97466i | 0 | ||||||||||
53.12 | 0 | 0.746858 | − | 2.90555i | 0 | 4.35455 | − | 3.47263i | 0 | −2.75840 | − | 12.0854i | 0 | −7.88441 | − | 4.34006i | 0 | ||||||||||
53.13 | 0 | 1.13882 | − | 2.77544i | 0 | −4.19426 | + | 3.34481i | 0 | 0.912105 | + | 3.99619i | 0 | −6.40617 | − | 6.32147i | 0 | ||||||||||
53.14 | 0 | 1.31826 | + | 2.69484i | 0 | 6.38586 | − | 5.09256i | 0 | 1.90710 | + | 8.35555i | 0 | −5.52437 | + | 7.10502i | 0 | ||||||||||
53.15 | 0 | 1.60265 | + | 2.53604i | 0 | −1.96181 | + | 1.56449i | 0 | −1.30069 | − | 5.69871i | 0 | −3.86300 | + | 8.12879i | 0 | ||||||||||
53.16 | 0 | 2.55406 | − | 1.57379i | 0 | 5.06274 | − | 4.03740i | 0 | 0.629032 | + | 2.75597i | 0 | 4.04640 | − | 8.03907i | 0 | ||||||||||
53.17 | 0 | 2.71080 | + | 1.28512i | 0 | −2.02360 | + | 1.61377i | 0 | 0.397329 | + | 1.74081i | 0 | 5.69692 | + | 6.96743i | 0 | ||||||||||
53.18 | 0 | 2.83524 | − | 0.980513i | 0 | −7.46308 | + | 5.95161i | 0 | −1.85676 | − | 8.13499i | 0 | 7.07719 | − | 5.55998i | 0 | ||||||||||
53.19 | 0 | 2.93765 | + | 0.608463i | 0 | 3.02183 | − | 2.40983i | 0 | −1.77361 | − | 7.77068i | 0 | 8.25954 | + | 3.57490i | 0 | ||||||||||
53.20 | 0 | 2.98032 | − | 0.343023i | 0 | −1.79198 | + | 1.42906i | 0 | 2.72567 | + | 11.9419i | 0 | 8.76467 | − | 2.04464i | 0 | ||||||||||
See next 80 embeddings (of 120 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
29.d | even | 7 | 1 | inner |
87.j | odd | 14 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 348.3.r.a | ✓ | 120 |
3.b | odd | 2 | 1 | inner | 348.3.r.a | ✓ | 120 |
29.d | even | 7 | 1 | inner | 348.3.r.a | ✓ | 120 |
87.j | odd | 14 | 1 | inner | 348.3.r.a | ✓ | 120 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
348.3.r.a | ✓ | 120 | 1.a | even | 1 | 1 | trivial |
348.3.r.a | ✓ | 120 | 3.b | odd | 2 | 1 | inner |
348.3.r.a | ✓ | 120 | 29.d | even | 7 | 1 | inner |
348.3.r.a | ✓ | 120 | 87.j | odd | 14 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(348, [\chi])\).