Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [690,3,Mod(599,690)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(690, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("690.599");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 690.b (of order \(2\), degree \(1\), 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: | \(18.8011382409\) |
Analytic rank: | \(0\) |
Dimension: | \(88\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
599.1 | −1.41421 | −2.99999 | − | 0.00596006i | 2.00000 | 1.21044 | + | 4.85127i | 4.24263 | + | 0.00842880i | − | 6.90074i | −2.82843 | 8.99993 | + | 0.0357603i | −1.71182 | − | 6.86073i | |||||||
599.2 | −1.41421 | −2.99999 | + | 0.00596006i | 2.00000 | 1.21044 | − | 4.85127i | 4.24263 | − | 0.00842880i | 6.90074i | −2.82843 | 8.99993 | − | 0.0357603i | −1.71182 | + | 6.86073i | ||||||||
599.3 | −1.41421 | −2.90508 | − | 0.748659i | 2.00000 | −4.90392 | − | 0.975491i | 4.10841 | + | 1.05876i | − | 3.09484i | −2.82843 | 7.87902 | + | 4.34983i | 6.93519 | + | 1.37955i | |||||||
599.4 | −1.41421 | −2.90508 | + | 0.748659i | 2.00000 | −4.90392 | + | 0.975491i | 4.10841 | − | 1.05876i | 3.09484i | −2.82843 | 7.87902 | − | 4.34983i | 6.93519 | − | 1.37955i | ||||||||
599.5 | −1.41421 | −2.79936 | − | 1.07869i | 2.00000 | 4.72534 | + | 1.63438i | 3.95890 | + | 1.52550i | 7.69411i | −2.82843 | 6.67285 | + | 6.03929i | −6.68264 | − | 2.31136i | ||||||||
599.6 | −1.41421 | −2.79936 | + | 1.07869i | 2.00000 | 4.72534 | − | 1.63438i | 3.95890 | − | 1.52550i | − | 7.69411i | −2.82843 | 6.67285 | − | 6.03929i | −6.68264 | + | 2.31136i | |||||||
599.7 | −1.41421 | −2.76001 | − | 1.17574i | 2.00000 | −2.52270 | − | 4.31694i | 3.90324 | + | 1.66274i | − | 6.80400i | −2.82843 | 6.23528 | + | 6.49009i | 3.56764 | + | 6.10507i | |||||||
599.8 | −1.41421 | −2.76001 | + | 1.17574i | 2.00000 | −2.52270 | + | 4.31694i | 3.90324 | − | 1.66274i | 6.80400i | −2.82843 | 6.23528 | − | 6.49009i | 3.56764 | − | 6.10507i | ||||||||
599.9 | −1.41421 | −2.16793 | − | 2.07367i | 2.00000 | 4.21296 | − | 2.69276i | 3.06591 | + | 2.93261i | − | 3.87132i | −2.82843 | 0.399825 | + | 8.99111i | −5.95803 | + | 3.80814i | |||||||
599.10 | −1.41421 | −2.16793 | + | 2.07367i | 2.00000 | 4.21296 | + | 2.69276i | 3.06591 | − | 2.93261i | 3.87132i | −2.82843 | 0.399825 | − | 8.99111i | −5.95803 | − | 3.80814i | ||||||||
599.11 | −1.41421 | −2.13931 | − | 2.10317i | 2.00000 | −2.73059 | + | 4.18854i | 3.02545 | + | 2.97433i | 4.11953i | −2.82843 | 0.153334 | + | 8.99869i | 3.86164 | − | 5.92349i | ||||||||
599.12 | −1.41421 | −2.13931 | + | 2.10317i | 2.00000 | −2.73059 | − | 4.18854i | 3.02545 | − | 2.97433i | − | 4.11953i | −2.82843 | 0.153334 | − | 8.99869i | 3.86164 | + | 5.92349i | |||||||
599.13 | −1.41421 | −1.52209 | − | 2.58520i | 2.00000 | −4.64244 | + | 1.85681i | 2.15256 | + | 3.65602i | − | 10.7642i | −2.82843 | −4.36648 | + | 7.86981i | 6.56540 | − | 2.62593i | |||||||
599.14 | −1.41421 | −1.52209 | + | 2.58520i | 2.00000 | −4.64244 | − | 1.85681i | 2.15256 | − | 3.65602i | 10.7642i | −2.82843 | −4.36648 | − | 7.86981i | 6.56540 | + | 2.62593i | ||||||||
599.15 | −1.41421 | −1.42652 | − | 2.63914i | 2.00000 | −3.84011 | − | 3.20212i | 2.01740 | + | 3.73230i | 12.7200i | −2.82843 | −4.93010 | + | 7.52955i | 5.43073 | + | 4.52848i | ||||||||
599.16 | −1.41421 | −1.42652 | + | 2.63914i | 2.00000 | −3.84011 | + | 3.20212i | 2.01740 | − | 3.73230i | − | 12.7200i | −2.82843 | −4.93010 | − | 7.52955i | 5.43073 | − | 4.52848i | |||||||
599.17 | −1.41421 | −1.23483 | − | 2.73408i | 2.00000 | 4.77075 | − | 1.49664i | 1.74631 | + | 3.86657i | 2.54772i | −2.82843 | −5.95040 | + | 6.75224i | −6.74686 | + | 2.11656i | ||||||||
599.18 | −1.41421 | −1.23483 | + | 2.73408i | 2.00000 | 4.77075 | + | 1.49664i | 1.74631 | − | 3.86657i | − | 2.54772i | −2.82843 | −5.95040 | − | 6.75224i | −6.74686 | − | 2.11656i | |||||||
599.19 | −1.41421 | −0.996475 | − | 2.82967i | 2.00000 | 3.01455 | + | 3.98905i | 1.40923 | + | 4.00176i | − | 10.6805i | −2.82843 | −7.01408 | + | 5.63939i | −4.26321 | − | 5.64137i | |||||||
599.20 | −1.41421 | −0.996475 | + | 2.82967i | 2.00000 | 3.01455 | − | 3.98905i | 1.40923 | − | 4.00176i | 10.6805i | −2.82843 | −7.01408 | − | 5.63939i | −4.26321 | + | 5.64137i | ||||||||
See all 88 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
5.b | even | 2 | 1 | inner |
15.d | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 690.3.b.a | ✓ | 88 |
3.b | odd | 2 | 1 | inner | 690.3.b.a | ✓ | 88 |
5.b | even | 2 | 1 | inner | 690.3.b.a | ✓ | 88 |
15.d | odd | 2 | 1 | inner | 690.3.b.a | ✓ | 88 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
690.3.b.a | ✓ | 88 | 1.a | even | 1 | 1 | trivial |
690.3.b.a | ✓ | 88 | 3.b | odd | 2 | 1 | inner |
690.3.b.a | ✓ | 88 | 5.b | even | 2 | 1 | inner |
690.3.b.a | ✓ | 88 | 15.d | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(690, [\chi])\).