Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [231,2,Mod(13,231)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(231, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 5, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("231.13");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 231 = 3 \cdot 7 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 231.w (of order \(10\), 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: | \(1.84454428669\) |
Analytic rank: | \(0\) |
Dimension: | \(64\) |
Relative dimension: | \(16\) over \(\Q(\zeta_{10})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
13.1 | −2.36445 | − | 0.768257i | −0.587785 | − | 0.809017i | 3.38238 | + | 2.45745i | 1.43953 | − | 0.467733i | 0.768257 | + | 2.36445i | −0.764664 | + | 2.53284i | −3.18691 | − | 4.38641i | −0.309017 | + | 0.951057i | −3.76305 | ||
13.2 | −2.36445 | − | 0.768257i | 0.587785 | + | 0.809017i | 3.38238 | + | 2.45745i | −1.43953 | + | 0.467733i | −0.768257 | − | 2.36445i | −2.17258 | + | 1.50993i | −3.18691 | − | 4.38641i | −0.309017 | + | 0.951057i | 3.76305 | ||
13.3 | −1.82917 | − | 0.594332i | −0.587785 | − | 0.809017i | 1.37458 | + | 0.998694i | −3.42067 | + | 1.11144i | 0.594332 | + | 1.82917i | 2.15042 | + | 1.54133i | 0.340186 | + | 0.468226i | −0.309017 | + | 0.951057i | 6.91755 | ||
13.4 | −1.82917 | − | 0.594332i | 0.587785 | + | 0.809017i | 1.37458 | + | 0.998694i | 3.42067 | − | 1.11144i | −0.594332 | − | 1.82917i | −2.13041 | − | 1.56887i | 0.340186 | + | 0.468226i | −0.309017 | + | 0.951057i | −6.91755 | ||
13.5 | −1.50185 | − | 0.487979i | −0.587785 | − | 0.809017i | 0.399382 | + | 0.290168i | 2.27762 | − | 0.740044i | 0.487979 | + | 1.50185i | 1.84555 | − | 1.89577i | 1.39817 | + | 1.92441i | −0.309017 | + | 0.951057i | −3.78176 | ||
13.6 | −1.50185 | − | 0.487979i | 0.587785 | + | 0.809017i | 0.399382 | + | 0.290168i | −2.27762 | + | 0.740044i | −0.487979 | − | 1.50185i | 1.23267 | − | 2.34105i | 1.39817 | + | 1.92441i | −0.309017 | + | 0.951057i | 3.78176 | ||
13.7 | −0.725588 | − | 0.235758i | −0.587785 | − | 0.809017i | −1.14714 | − | 0.833444i | −0.666245 | + | 0.216476i | 0.235758 | + | 0.725588i | −2.63718 | + | 0.212765i | 1.53273 | + | 2.10963i | −0.309017 | + | 0.951057i | 0.534456 | ||
13.8 | −0.725588 | − | 0.235758i | 0.587785 | + | 0.809017i | −1.14714 | − | 0.833444i | 0.666245 | − | 0.216476i | −0.235758 | − | 0.725588i | 0.612583 | + | 2.57386i | 1.53273 | + | 2.10963i | −0.309017 | + | 0.951057i | −0.534456 | ||
13.9 | 0.332589 | + | 0.108065i | −0.587785 | − | 0.809017i | −1.51910 | − | 1.10369i | 3.96320 | − | 1.28772i | −0.108065 | − | 0.332589i | −2.37010 | + | 1.17584i | −0.797068 | − | 1.09707i | −0.309017 | + | 0.951057i | 1.45728 | ||
13.10 | 0.332589 | + | 0.108065i | 0.587785 | + | 0.809017i | −1.51910 | − | 1.10369i | −3.96320 | + | 1.28772i | 0.108065 | + | 0.332589i | −0.385893 | + | 2.61746i | −0.797068 | − | 1.09707i | −0.309017 | + | 0.951057i | −1.45728 | ||
13.11 | 0.819861 | + | 0.266389i | −0.587785 | − | 0.809017i | −1.01682 | − | 0.738766i | −2.84938 | + | 0.925820i | −0.266389 | − | 0.819861i | 0.656605 | − | 2.56298i | −1.65026 | − | 2.27139i | −0.309017 | + | 0.951057i | −2.58273 | ||
13.12 | 0.819861 | + | 0.266389i | 0.587785 | + | 0.809017i | −1.01682 | − | 0.738766i | 2.84938 | − | 0.925820i | 0.266389 | + | 0.819861i | 2.23464 | − | 1.41647i | −1.65026 | − | 2.27139i | −0.309017 | + | 0.951057i | 2.58273 | ||
13.13 | 1.94435 | + | 0.631758i | −0.587785 | − | 0.809017i | 1.76335 | + | 1.28115i | 2.31266 | − | 0.751430i | −0.631758 | − | 1.94435i | −1.00115 | − | 2.44902i | 0.215840 | + | 0.297079i | −0.309017 | + | 0.951057i | 4.97135 | ||
13.14 | 1.94435 | + | 0.631758i | 0.587785 | + | 0.809017i | 1.76335 | + | 1.28115i | −2.31266 | + | 0.751430i | 0.631758 | + | 1.94435i | 2.63853 | + | 0.195359i | 0.215840 | + | 0.297079i | −0.309017 | + | 0.951057i | −4.97135 | ||
13.15 | 2.20622 | + | 0.716844i | −0.587785 | − | 0.809017i | 2.73550 | + | 1.98746i | 0.0209622 | − | 0.00681102i | −0.716844 | − | 2.20622i | 1.69347 | + | 2.03277i | 1.88338 | + | 2.59224i | −0.309017 | + | 0.951057i | 0.0511296 | ||
13.16 | 2.20622 | + | 0.716844i | 0.587785 | + | 0.809017i | 2.73550 | + | 1.98746i | −0.0209622 | + | 0.00681102i | 0.716844 | + | 2.20622i | −2.45659 | − | 0.982422i | 1.88338 | + | 2.59224i | −0.309017 | + | 0.951057i | −0.0511296 | ||
118.1 | −1.44554 | − | 1.98962i | −0.951057 | + | 0.309017i | −1.25096 | + | 3.85006i | 1.57908 | − | 2.17341i | 1.98962 | + | 1.44554i | 2.53314 | − | 0.763690i | 4.79060 | − | 1.55656i | 0.809017 | − | 0.587785i | −6.60689 | ||
118.2 | −1.44554 | − | 1.98962i | 0.951057 | − | 0.309017i | −1.25096 | + | 3.85006i | −1.57908 | + | 2.17341i | −1.98962 | − | 1.44554i | 1.60046 | + | 2.10678i | 4.79060 | − | 1.55656i | 0.809017 | − | 0.587785i | 6.60689 | ||
118.3 | −0.917545 | − | 1.26289i | −0.951057 | + | 0.309017i | −0.134974 | + | 0.415407i | 0.118521 | − | 0.163130i | 1.26289 | + | 0.917545i | −2.40543 | + | 1.10177i | −2.32078 | + | 0.754067i | 0.809017 | − | 0.587785i | −0.314765 | ||
118.4 | −0.917545 | − | 1.26289i | 0.951057 | − | 0.309017i | −0.134974 | + | 0.415407i | −0.118521 | + | 0.163130i | −1.26289 | − | 0.917545i | −1.29844 | − | 2.30523i | −2.32078 | + | 0.754067i | 0.809017 | − | 0.587785i | 0.314765 | ||
See all 64 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | inner |
11.d | odd | 10 | 1 | inner |
77.l | even | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 231.2.w.a | ✓ | 64 |
3.b | odd | 2 | 1 | 693.2.bu.f | 64 | ||
7.b | odd | 2 | 1 | inner | 231.2.w.a | ✓ | 64 |
11.d | odd | 10 | 1 | inner | 231.2.w.a | ✓ | 64 |
21.c | even | 2 | 1 | 693.2.bu.f | 64 | ||
33.f | even | 10 | 1 | 693.2.bu.f | 64 | ||
77.l | even | 10 | 1 | inner | 231.2.w.a | ✓ | 64 |
231.r | odd | 10 | 1 | 693.2.bu.f | 64 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
231.2.w.a | ✓ | 64 | 1.a | even | 1 | 1 | trivial |
231.2.w.a | ✓ | 64 | 7.b | odd | 2 | 1 | inner |
231.2.w.a | ✓ | 64 | 11.d | odd | 10 | 1 | inner |
231.2.w.a | ✓ | 64 | 77.l | even | 10 | 1 | inner |
693.2.bu.f | 64 | 3.b | odd | 2 | 1 | ||
693.2.bu.f | 64 | 21.c | even | 2 | 1 | ||
693.2.bu.f | 64 | 33.f | even | 10 | 1 | ||
693.2.bu.f | 64 | 231.r | odd | 10 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(231, [\chi])\).