Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [240,2,Mod(109,240)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(240, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3, 0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("240.109");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 240 = 2^{4} \cdot 3 \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 240.bl (of order \(4\), degree \(2\), 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: | \(1.91640964851\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Relative dimension: | \(24\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
109.1 | −1.40976 | − | 0.112110i | −0.707107 | + | 0.707107i | 1.97486 | + | 0.316097i | 2.18678 | + | 0.466917i | 1.07613 | − | 0.917579i | 1.00010 | −2.74865 | − | 0.667024i | − | 1.00000i | −3.03049 | − | 0.903402i | |||
109.2 | −1.39033 | − | 0.258835i | 0.707107 | − | 0.707107i | 1.86601 | + | 0.719729i | 0.404088 | − | 2.19925i | −1.16613 | + | 0.800085i | 1.81567 | −2.40807 | − | 1.48364i | − | 1.00000i | −1.13106 | + | 2.95308i | |||
109.3 | −1.36038 | + | 0.386472i | 0.707107 | − | 0.707107i | 1.70128 | − | 1.05150i | −1.98421 | + | 1.03097i | −0.688658 | + | 1.23521i | −3.91927 | −1.90801 | + | 2.08794i | − | 1.00000i | 2.30085 | − | 2.16936i | |||
109.4 | −1.22294 | − | 0.710230i | −0.707107 | + | 0.707107i | 0.991146 | + | 1.73713i | −2.15195 | − | 0.607542i | 1.36696 | − | 0.362538i | 2.25286 | 0.0216568 | − | 2.82834i | − | 1.00000i | 2.20020 | + | 2.27137i | |||
109.5 | −1.20386 | + | 0.742113i | −0.707107 | + | 0.707107i | 0.898535 | − | 1.78679i | −2.06370 | − | 0.860885i | 0.326501 | − | 1.37601i | 0.707398 | 0.244298 | + | 2.81786i | − | 1.00000i | 3.12328 | − | 0.495122i | |||
109.6 | −1.06224 | − | 0.933621i | 0.707107 | − | 0.707107i | 0.256702 | + | 1.98346i | −1.86022 | + | 1.24079i | −1.41129 | + | 0.0909462i | 1.58988 | 1.57912 | − | 2.34657i | − | 1.00000i | 3.13443 | + | 0.418728i | |||
109.7 | −0.903247 | + | 1.08818i | −0.707107 | + | 0.707107i | −0.368289 | − | 1.96580i | 0.770325 | + | 2.09919i | −0.130770 | − | 1.40815i | −3.05002 | 2.47181 | + | 1.37484i | − | 1.00000i | −2.98010 | − | 1.05783i | |||
109.8 | −0.750333 | − | 1.19875i | −0.707107 | + | 0.707107i | −0.874002 | + | 1.79892i | 1.07735 | − | 1.95942i | 1.37821 | + | 0.317079i | −1.22137 | 2.81225 | − | 0.302081i | − | 1.00000i | −3.15722 | + | 0.178735i | |||
109.9 | −0.550383 | + | 1.30272i | 0.707107 | − | 0.707107i | −1.39416 | − | 1.43399i | 0.162008 | + | 2.23019i | 0.531983 | + | 1.31034i | 2.93661 | 2.63541 | − | 1.02695i | − | 1.00000i | −2.99448 | − | 1.01641i | |||
109.10 | −0.456856 | + | 1.33839i | 0.707107 | − | 0.707107i | −1.58257 | − | 1.22290i | −1.50085 | − | 1.65754i | 0.623338 | + | 1.26943i | −2.58977 | 2.35972 | − | 1.55940i | − | 1.00000i | 2.90411 | − | 1.25146i | |||
109.11 | −0.382275 | + | 1.36157i | −0.707107 | + | 0.707107i | −1.70773 | − | 1.04099i | 1.38805 | − | 1.75308i | −0.692464 | − | 1.23308i | 4.66030 | 2.07020 | − | 1.92725i | − | 1.00000i | 1.85633 | + | 2.56009i | |||
109.12 | −0.345118 | − | 1.37146i | 0.707107 | − | 0.707107i | −1.76179 | + | 0.946629i | 2.16437 | + | 0.561697i | −1.21380 | − | 0.725731i | 4.51614 | 1.90629 | + | 2.08952i | − | 1.00000i | 0.0233792 | − | 3.16219i | |||
109.13 | 0.345118 | + | 1.37146i | −0.707107 | + | 0.707107i | −1.76179 | + | 0.946629i | −0.561697 | − | 2.16437i | −1.21380 | − | 0.725731i | −4.51614 | −1.90629 | − | 2.08952i | − | 1.00000i | 2.77449 | − | 1.51731i | |||
109.14 | 0.382275 | − | 1.36157i | 0.707107 | − | 0.707107i | −1.70773 | − | 1.04099i | 1.75308 | − | 1.38805i | −0.692464 | − | 1.23308i | −4.66030 | −2.07020 | + | 1.92725i | − | 1.00000i | −1.21977 | − | 2.91756i | |||
109.15 | 0.456856 | − | 1.33839i | −0.707107 | + | 0.707107i | −1.58257 | − | 1.22290i | 1.65754 | + | 1.50085i | 0.623338 | + | 1.26943i | 2.58977 | −2.35972 | + | 1.55940i | − | 1.00000i | 2.76598 | − | 1.53277i | |||
109.16 | 0.550383 | − | 1.30272i | −0.707107 | + | 0.707107i | −1.39416 | − | 1.43399i | −2.23019 | − | 0.162008i | 0.531983 | + | 1.31034i | −2.93661 | −2.63541 | + | 1.02695i | − | 1.00000i | −1.43851 | + | 2.81615i | |||
109.17 | 0.750333 | + | 1.19875i | 0.707107 | − | 0.707107i | −0.874002 | + | 1.79892i | 1.95942 | − | 1.07735i | 1.37821 | + | 0.317079i | 1.22137 | −2.81225 | + | 0.302081i | − | 1.00000i | 2.76169 | + | 1.54047i | |||
109.18 | 0.903247 | − | 1.08818i | 0.707107 | − | 0.707107i | −0.368289 | − | 1.96580i | −2.09919 | − | 0.770325i | −0.130770 | − | 1.40815i | 3.05002 | −2.47181 | − | 1.37484i | − | 1.00000i | −2.73434 | + | 1.58851i | |||
109.19 | 1.06224 | + | 0.933621i | −0.707107 | + | 0.707107i | 0.256702 | + | 1.98346i | −1.24079 | + | 1.86022i | −1.41129 | + | 0.0909462i | −1.58988 | −1.57912 | + | 2.34657i | − | 1.00000i | −3.05476 | + | 0.817572i | |||
109.20 | 1.20386 | − | 0.742113i | 0.707107 | − | 0.707107i | 0.898535 | − | 1.78679i | 0.860885 | + | 2.06370i | 0.326501 | − | 1.37601i | −0.707398 | −0.244298 | − | 2.81786i | − | 1.00000i | 2.56788 | + | 1.84553i | |||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
16.e | even | 4 | 1 | inner |
80.q | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 240.2.bl.a | ✓ | 48 |
3.b | odd | 2 | 1 | 720.2.bm.h | 48 | ||
4.b | odd | 2 | 1 | 960.2.bl.a | 48 | ||
5.b | even | 2 | 1 | inner | 240.2.bl.a | ✓ | 48 |
8.b | even | 2 | 1 | 1920.2.bl.a | 48 | ||
8.d | odd | 2 | 1 | 1920.2.bl.b | 48 | ||
15.d | odd | 2 | 1 | 720.2.bm.h | 48 | ||
16.e | even | 4 | 1 | inner | 240.2.bl.a | ✓ | 48 |
16.e | even | 4 | 1 | 1920.2.bl.a | 48 | ||
16.f | odd | 4 | 1 | 960.2.bl.a | 48 | ||
16.f | odd | 4 | 1 | 1920.2.bl.b | 48 | ||
20.d | odd | 2 | 1 | 960.2.bl.a | 48 | ||
40.e | odd | 2 | 1 | 1920.2.bl.b | 48 | ||
40.f | even | 2 | 1 | 1920.2.bl.a | 48 | ||
48.i | odd | 4 | 1 | 720.2.bm.h | 48 | ||
80.k | odd | 4 | 1 | 960.2.bl.a | 48 | ||
80.k | odd | 4 | 1 | 1920.2.bl.b | 48 | ||
80.q | even | 4 | 1 | inner | 240.2.bl.a | ✓ | 48 |
80.q | even | 4 | 1 | 1920.2.bl.a | 48 | ||
240.bm | odd | 4 | 1 | 720.2.bm.h | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
240.2.bl.a | ✓ | 48 | 1.a | even | 1 | 1 | trivial |
240.2.bl.a | ✓ | 48 | 5.b | even | 2 | 1 | inner |
240.2.bl.a | ✓ | 48 | 16.e | even | 4 | 1 | inner |
240.2.bl.a | ✓ | 48 | 80.q | even | 4 | 1 | inner |
720.2.bm.h | 48 | 3.b | odd | 2 | 1 | ||
720.2.bm.h | 48 | 15.d | odd | 2 | 1 | ||
720.2.bm.h | 48 | 48.i | odd | 4 | 1 | ||
720.2.bm.h | 48 | 240.bm | odd | 4 | 1 | ||
960.2.bl.a | 48 | 4.b | odd | 2 | 1 | ||
960.2.bl.a | 48 | 16.f | odd | 4 | 1 | ||
960.2.bl.a | 48 | 20.d | odd | 2 | 1 | ||
960.2.bl.a | 48 | 80.k | odd | 4 | 1 | ||
1920.2.bl.a | 48 | 8.b | even | 2 | 1 | ||
1920.2.bl.a | 48 | 16.e | even | 4 | 1 | ||
1920.2.bl.a | 48 | 40.f | even | 2 | 1 | ||
1920.2.bl.a | 48 | 80.q | even | 4 | 1 | ||
1920.2.bl.b | 48 | 8.d | odd | 2 | 1 | ||
1920.2.bl.b | 48 | 16.f | odd | 4 | 1 | ||
1920.2.bl.b | 48 | 40.e | odd | 2 | 1 | ||
1920.2.bl.b | 48 | 80.k | odd | 4 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(240, [\chi])\).