Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [430,3,Mod(51,430)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(430, base_ring=CyclotomicField(14))
chi = DirichletCharacter(H, H._module([0, 13]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("430.51");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 430 = 2 \cdot 5 \cdot 43 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 430.o (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: | \(11.7166513675\) |
Analytic rank: | \(0\) |
Dimension: | \(192\) |
Relative dimension: | \(32\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
51.1 | −1.37876 | − | 0.314692i | −4.91798 | + | 1.12250i | 1.80194 | + | 0.867767i | −1.74823 | − | 1.39417i | 7.13393 | 2.11341i | −2.21135 | − | 1.76350i | 14.8178 | − | 7.13587i | 1.97165 | + | 2.47237i | ||||
51.2 | −1.37876 | − | 0.314692i | −4.62040 | + | 1.05458i | 1.80194 | + | 0.867767i | 1.74823 | + | 1.39417i | 6.70227 | 6.45803i | −2.21135 | − | 1.76350i | 12.1272 | − | 5.84017i | −1.97165 | − | 2.47237i | ||||
51.3 | −1.37876 | − | 0.314692i | −3.65955 | + | 0.835268i | 1.80194 | + | 0.867767i | 1.74823 | + | 1.39417i | 5.30848 | − | 1.97483i | −2.21135 | − | 1.76350i | 4.58591 | − | 2.20846i | −1.97165 | − | 2.47237i | |||
51.4 | −1.37876 | − | 0.314692i | −2.51285 | + | 0.573542i | 1.80194 | + | 0.867767i | −1.74823 | − | 1.39417i | 3.64510 | − | 0.264072i | −2.21135 | − | 1.76350i | −2.12325 | + | 1.02250i | 1.97165 | + | 2.47237i | |||
51.5 | −1.37876 | − | 0.314692i | −2.44063 | + | 0.557057i | 1.80194 | + | 0.867767i | −1.74823 | − | 1.39417i | 3.54033 | − | 12.4602i | −2.21135 | − | 1.76350i | −2.46237 | + | 1.18581i | 1.97165 | + | 2.47237i | |||
51.6 | −1.37876 | − | 0.314692i | −0.877549 | + | 0.200295i | 1.80194 | + | 0.867767i | 1.74823 | + | 1.39417i | 1.27296 | − | 9.83826i | −2.21135 | − | 1.76350i | −7.37875 | + | 3.55342i | −1.97165 | − | 2.47237i | |||
51.7 | −1.37876 | − | 0.314692i | −0.776351 | + | 0.177197i | 1.80194 | + | 0.867767i | 1.74823 | + | 1.39417i | 1.12616 | 11.5147i | −2.21135 | − | 1.76350i | −7.53740 | + | 3.62982i | −1.97165 | − | 2.47237i | ||||
51.8 | −1.37876 | − | 0.314692i | 0.0560067 | − | 0.0127832i | 1.80194 | + | 0.867767i | 1.74823 | + | 1.39417i | −0.0812423 | − | 0.183399i | −2.21135 | − | 1.76350i | −8.10575 | + | 3.90352i | −1.97165 | − | 2.47237i | |||
51.9 | −1.37876 | − | 0.314692i | 0.355051 | − | 0.0810381i | 1.80194 | + | 0.867767i | −1.74823 | − | 1.39417i | −0.515031 | 3.35827i | −2.21135 | − | 1.76350i | −7.98923 | + | 3.84741i | 1.97165 | + | 2.47237i | ||||
51.10 | −1.37876 | − | 0.314692i | 1.84541 | − | 0.421203i | 1.80194 | + | 0.867767i | −1.74823 | − | 1.39417i | −2.67692 | − | 0.339121i | −2.21135 | − | 1.76350i | −4.88059 | + | 2.35037i | 1.97165 | + | 2.47237i | |||
51.11 | −1.37876 | − | 0.314692i | 2.32327 | − | 0.530272i | 1.80194 | + | 0.867767i | 1.74823 | + | 1.39417i | −3.37010 | − | 5.80669i | −2.21135 | − | 1.76350i | −2.99230 | + | 1.44102i | −1.97165 | − | 2.47237i | |||
51.12 | −1.37876 | − | 0.314692i | 3.13226 | − | 0.714918i | 1.80194 | + | 0.867767i | −1.74823 | − | 1.39417i | −4.54360 | − | 6.42690i | −2.21135 | − | 1.76350i | 1.19123 | − | 0.573664i | 1.97165 | + | 2.47237i | |||
51.13 | −1.37876 | − | 0.314692i | 3.38564 | − | 0.772751i | 1.80194 | + | 0.867767i | 1.74823 | + | 1.39417i | −4.91116 | 13.2826i | −2.21135 | − | 1.76350i | 2.75673 | − | 1.32757i | −1.97165 | − | 2.47237i | ||||
51.14 | −1.37876 | − | 0.314692i | 4.18687 | − | 0.955625i | 1.80194 | + | 0.867767i | 1.74823 | + | 1.39417i | −6.07340 | − | 3.82766i | −2.21135 | − | 1.76350i | 8.50793 | − | 4.09720i | −1.97165 | − | 2.47237i | |||
51.15 | −1.37876 | − | 0.314692i | 5.03773 | − | 1.14983i | 1.80194 | + | 0.867767i | −1.74823 | − | 1.39417i | −7.30765 | 12.8599i | −2.21135 | − | 1.76350i | 15.9479 | − | 7.68011i | 1.97165 | + | 2.47237i | ||||
51.16 | −1.37876 | − | 0.314692i | 5.53034 | − | 1.26226i | 1.80194 | + | 0.867767i | −1.74823 | − | 1.39417i | −8.02222 | − | 9.70495i | −2.21135 | − | 1.76350i | 20.8826 | − | 10.0566i | 1.97165 | + | 2.47237i | |||
51.17 | 1.37876 | + | 0.314692i | −5.74102 | + | 1.31035i | 1.80194 | + | 0.867767i | −1.74823 | − | 1.39417i | −8.32782 | − | 4.81175i | 2.21135 | + | 1.76350i | 23.1335 | − | 11.1405i | −1.97165 | − | 2.47237i | |||
51.18 | 1.37876 | + | 0.314692i | −4.91296 | + | 1.12135i | 1.80194 | + | 0.867767i | 1.74823 | + | 1.39417i | −7.12665 | 1.26363i | 2.21135 | + | 1.76350i | 14.7710 | − | 7.11335i | 1.97165 | + | 2.47237i | ||||
51.19 | 1.37876 | + | 0.314692i | −4.29562 | + | 0.980447i | 1.80194 | + | 0.867767i | −1.74823 | − | 1.39417i | −6.23115 | 6.77593i | 2.21135 | + | 1.76350i | 9.38234 | − | 4.51829i | −1.97165 | − | 2.47237i | ||||
51.20 | 1.37876 | + | 0.314692i | −3.48189 | + | 0.794718i | 1.80194 | + | 0.867767i | 1.74823 | + | 1.39417i | −5.05077 | − | 9.36450i | 2.21135 | + | 1.76350i | 3.38324 | − | 1.62928i | 1.97165 | + | 2.47237i | |||
See next 80 embeddings (of 192 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
43.f | odd | 14 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 430.3.o.a | ✓ | 192 |
43.f | odd | 14 | 1 | inner | 430.3.o.a | ✓ | 192 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
430.3.o.a | ✓ | 192 | 1.a | even | 1 | 1 | trivial |
430.3.o.a | ✓ | 192 | 43.f | odd | 14 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(430, [\chi])\).