Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [650,2,Mod(181,650)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(650, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([4, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("650.181");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 650 = 2 \cdot 5^{2} \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 650.p (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: | \(5.19027613138\) |
Analytic rank: | \(0\) |
Dimension: | \(136\) |
Relative dimension: | \(34\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
181.1 | −0.951057 | + | 0.309017i | −2.77229 | + | 2.01419i | 0.809017 | − | 0.587785i | −2.17604 | + | 0.514625i | 2.01419 | − | 2.77229i | 1.58290i | −0.587785 | + | 0.809017i | 2.70160 | − | 8.31467i | 1.91051 | − | 1.16187i | ||
181.2 | −0.951057 | + | 0.309017i | −2.39624 | + | 1.74097i | 0.809017 | − | 0.587785i | 2.00896 | + | 0.981871i | 1.74097 | − | 2.39624i | 0.812549i | −0.587785 | + | 0.809017i | 1.78393 | − | 5.49037i | −2.21405 | − | 0.313011i | ||
181.3 | −0.951057 | + | 0.309017i | −1.64902 | + | 1.19809i | 0.809017 | − | 0.587785i | 1.65401 | + | 1.50474i | 1.19809 | − | 1.64902i | − | 3.17306i | −0.587785 | + | 0.809017i | 0.356819 | − | 1.09818i | −2.03805 | − | 0.919973i | |
181.4 | −0.951057 | + | 0.309017i | −1.63416 | + | 1.18728i | 0.809017 | − | 0.587785i | −0.970013 | − | 2.01471i | 1.18728 | − | 1.63416i | 1.37615i | −0.587785 | + | 0.809017i | 0.333771 | − | 1.02724i | 1.54512 | + | 1.61636i | ||
181.5 | −0.951057 | + | 0.309017i | −1.03852 | + | 0.754528i | 0.809017 | − | 0.587785i | −1.01800 | + | 1.99090i | 0.754528 | − | 1.03852i | − | 0.731554i | −0.587785 | + | 0.809017i | −0.417843 | + | 1.28599i | 0.352954 | − | 2.20804i | |
181.6 | −0.951057 | + | 0.309017i | −0.802302 | + | 0.582906i | 0.809017 | − | 0.587785i | 0.515358 | + | 2.17587i | 0.582906 | − | 0.802302i | 4.67023i | −0.587785 | + | 0.809017i | −0.623143 | + | 1.91784i | −1.16251 | − | 1.91012i | ||
181.7 | −0.951057 | + | 0.309017i | −0.757142 | + | 0.550096i | 0.809017 | − | 0.587785i | −2.06002 | − | 0.869669i | 0.550096 | − | 0.757142i | − | 2.56855i | −0.587785 | + | 0.809017i | −0.656392 | + | 2.02017i | 2.22794 | + | 0.190524i | |
181.8 | −0.951057 | + | 0.309017i | −0.454733 | + | 0.330383i | 0.809017 | − | 0.587785i | 2.02575 | − | 0.946747i | 0.330383 | − | 0.454733i | 3.59180i | −0.587785 | + | 0.809017i | −0.829422 | + | 2.55270i | −1.63404 | + | 1.52640i | ||
181.9 | −0.951057 | + | 0.309017i | −0.271435 | + | 0.197209i | 0.809017 | − | 0.587785i | −2.19946 | + | 0.402964i | 0.197209 | − | 0.271435i | − | 3.44579i | −0.587785 | + | 0.809017i | −0.892265 | + | 2.74611i | 1.96729 | − | 1.06291i | |
181.10 | −0.951057 | + | 0.309017i | −0.0856483 | + | 0.0622271i | 0.809017 | − | 0.587785i | 1.86289 | − | 1.23679i | 0.0622271 | − | 0.0856483i | − | 1.22434i | −0.587785 | + | 0.809017i | −0.923588 | + | 2.84251i | −1.38952 | + | 1.75192i | |
181.11 | −0.951057 | + | 0.309017i | 1.08542 | − | 0.788606i | 0.809017 | − | 0.587785i | −1.26746 | − | 1.84216i | −0.788606 | + | 1.08542i | 1.88803i | −0.587785 | + | 0.809017i | −0.370807 | + | 1.14123i | 1.77468 | + | 1.36033i | ||
181.12 | −0.951057 | + | 0.309017i | 1.23469 | − | 0.897052i | 0.809017 | − | 0.587785i | −2.23531 | + | 0.0581791i | −0.897052 | + | 1.23469i | 2.12559i | −0.587785 | + | 0.809017i | −0.207303 | + | 0.638013i | 2.10793 | − | 0.746081i | ||
181.13 | −0.951057 | + | 0.309017i | 1.37663 | − | 1.00018i | 0.809017 | − | 0.587785i | 0.672142 | − | 2.13266i | −1.00018 | + | 1.37663i | − | 3.88027i | −0.587785 | + | 0.809017i | −0.0322965 | + | 0.0993983i | 0.0197820 | + | 2.23598i | |
181.14 | −0.951057 | + | 0.309017i | 1.45190 | − | 1.05487i | 0.809017 | − | 0.587785i | 0.184771 | + | 2.22842i | −1.05487 | + | 1.45190i | 1.28692i | −0.587785 | + | 0.809017i | 0.0682224 | − | 0.209967i | −0.864348 | − | 2.06226i | ||
181.15 | −0.951057 | + | 0.309017i | 2.12082 | − | 1.54087i | 0.809017 | − | 0.587785i | −1.15648 | + | 1.91378i | −1.54087 | + | 2.12082i | − | 3.25927i | −0.587785 | + | 0.809017i | 1.19656 | − | 3.68265i | 0.508489 | − | 2.17748i | |
181.16 | −0.951057 | + | 0.309017i | 2.13781 | − | 1.55321i | 0.809017 | − | 0.587785i | 2.03958 | + | 0.916581i | −1.55321 | + | 2.13781i | − | 0.170643i | −0.587785 | + | 0.809017i | 1.23072 | − | 3.78776i | −2.22299 | − | 0.241456i | |
181.17 | −0.951057 | + | 0.309017i | 2.45421 | − | 1.78309i | 0.809017 | − | 0.587785i | 0.943747 | − | 2.02715i | −1.78309 | + | 2.45421i | 5.11933i | −0.587785 | + | 0.809017i | 1.91669 | − | 5.89896i | −0.271133 | + | 2.21957i | ||
181.18 | 0.951057 | − | 0.309017i | −2.77229 | + | 2.01419i | 0.809017 | − | 0.587785i | 2.17604 | − | 0.514625i | −2.01419 | + | 2.77229i | − | 1.58290i | 0.587785 | − | 0.809017i | 2.70160 | − | 8.31467i | 1.91051 | − | 1.16187i | |
181.19 | 0.951057 | − | 0.309017i | −2.39624 | + | 1.74097i | 0.809017 | − | 0.587785i | −2.00896 | − | 0.981871i | −1.74097 | + | 2.39624i | − | 0.812549i | 0.587785 | − | 0.809017i | 1.78393 | − | 5.49037i | −2.21405 | − | 0.313011i | |
181.20 | 0.951057 | − | 0.309017i | −1.64902 | + | 1.19809i | 0.809017 | − | 0.587785i | −1.65401 | − | 1.50474i | −1.19809 | + | 1.64902i | 3.17306i | 0.587785 | − | 0.809017i | 0.356819 | − | 1.09818i | −2.03805 | − | 0.919973i | ||
See next 80 embeddings (of 136 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.b | even | 2 | 1 | inner |
25.d | even | 5 | 1 | inner |
325.q | even | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 650.2.p.a | ✓ | 136 |
13.b | even | 2 | 1 | inner | 650.2.p.a | ✓ | 136 |
25.d | even | 5 | 1 | inner | 650.2.p.a | ✓ | 136 |
325.q | even | 10 | 1 | inner | 650.2.p.a | ✓ | 136 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
650.2.p.a | ✓ | 136 | 1.a | even | 1 | 1 | trivial |
650.2.p.a | ✓ | 136 | 13.b | even | 2 | 1 | inner |
650.2.p.a | ✓ | 136 | 25.d | even | 5 | 1 | inner |
650.2.p.a | ✓ | 136 | 325.q | even | 10 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(650, [\chi])\).