Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1860,2,Mod(929,1860)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1860, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1860.929");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1860 = 2^{2} \cdot 3 \cdot 5 \cdot 31 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1860.i (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: | \(14.8521747760\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
The algebraic \(q\)-expansion of this newform has not been computed, 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
929.1 | 0 | −1.72938 | − | 0.0961633i | 0 | −1.38738 | + | 1.75362i | 0 | 3.39946i | 0 | 2.98151 | + | 0.332606i | 0 | ||||||||||||
929.2 | 0 | −1.72938 | − | 0.0961633i | 0 | 1.38738 | + | 1.75362i | 0 | − | 3.39946i | 0 | 2.98151 | + | 0.332606i | 0 | |||||||||||
929.3 | 0 | −1.72938 | + | 0.0961633i | 0 | −1.38738 | − | 1.75362i | 0 | − | 3.39946i | 0 | 2.98151 | − | 0.332606i | 0 | |||||||||||
929.4 | 0 | −1.72938 | + | 0.0961633i | 0 | 1.38738 | − | 1.75362i | 0 | 3.39946i | 0 | 2.98151 | − | 0.332606i | 0 | ||||||||||||
929.5 | 0 | −1.62488 | − | 0.599792i | 0 | −2.11867 | + | 0.714996i | 0 | − | 1.63784i | 0 | 2.28050 | + | 1.94918i | 0 | |||||||||||
929.6 | 0 | −1.62488 | − | 0.599792i | 0 | 2.11867 | + | 0.714996i | 0 | 1.63784i | 0 | 2.28050 | + | 1.94918i | 0 | ||||||||||||
929.7 | 0 | −1.62488 | + | 0.599792i | 0 | −2.11867 | − | 0.714996i | 0 | 1.63784i | 0 | 2.28050 | − | 1.94918i | 0 | ||||||||||||
929.8 | 0 | −1.62488 | + | 0.599792i | 0 | 2.11867 | − | 0.714996i | 0 | − | 1.63784i | 0 | 2.28050 | − | 1.94918i | 0 | |||||||||||
929.9 | 0 | −1.37690 | − | 1.05079i | 0 | −2.02481 | − | 0.948759i | 0 | 3.69636i | 0 | 0.791696 | + | 2.89365i | 0 | ||||||||||||
929.10 | 0 | −1.37690 | − | 1.05079i | 0 | 2.02481 | − | 0.948759i | 0 | − | 3.69636i | 0 | 0.791696 | + | 2.89365i | 0 | |||||||||||
929.11 | 0 | −1.37690 | + | 1.05079i | 0 | −2.02481 | + | 0.948759i | 0 | − | 3.69636i | 0 | 0.791696 | − | 2.89365i | 0 | |||||||||||
929.12 | 0 | −1.37690 | + | 1.05079i | 0 | 2.02481 | + | 0.948759i | 0 | 3.69636i | 0 | 0.791696 | − | 2.89365i | 0 | ||||||||||||
929.13 | 0 | −0.910753 | − | 1.47327i | 0 | −1.00177 | + | 1.99911i | 0 | − | 4.87923i | 0 | −1.34106 | + | 2.68357i | 0 | |||||||||||
929.14 | 0 | −0.910753 | − | 1.47327i | 0 | 1.00177 | + | 1.99911i | 0 | 4.87923i | 0 | −1.34106 | + | 2.68357i | 0 | ||||||||||||
929.15 | 0 | −0.910753 | + | 1.47327i | 0 | −1.00177 | − | 1.99911i | 0 | 4.87923i | 0 | −1.34106 | − | 2.68357i | 0 | ||||||||||||
929.16 | 0 | −0.910753 | + | 1.47327i | 0 | 1.00177 | − | 1.99911i | 0 | − | 4.87923i | 0 | −1.34106 | − | 2.68357i | 0 | |||||||||||
929.17 | 0 | −0.577293 | − | 1.63301i | 0 | −2.10556 | − | 0.752745i | 0 | − | 0.552709i | 0 | −2.33347 | + | 1.88545i | 0 | |||||||||||
929.18 | 0 | −0.577293 | − | 1.63301i | 0 | 2.10556 | − | 0.752745i | 0 | 0.552709i | 0 | −2.33347 | + | 1.88545i | 0 | ||||||||||||
929.19 | 0 | −0.577293 | + | 1.63301i | 0 | −2.10556 | + | 0.752745i | 0 | 0.552709i | 0 | −2.33347 | − | 1.88545i | 0 | ||||||||||||
929.20 | 0 | −0.577293 | + | 1.63301i | 0 | 2.10556 | + | 0.752745i | 0 | − | 0.552709i | 0 | −2.33347 | − | 1.88545i | 0 | |||||||||||
See all 48 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 |
31.b | odd | 2 | 1 | inner |
93.c | even | 2 | 1 | inner |
155.c | odd | 2 | 1 | inner |
465.g | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1860.2.i.b | ✓ | 48 |
3.b | odd | 2 | 1 | inner | 1860.2.i.b | ✓ | 48 |
5.b | even | 2 | 1 | inner | 1860.2.i.b | ✓ | 48 |
15.d | odd | 2 | 1 | inner | 1860.2.i.b | ✓ | 48 |
31.b | odd | 2 | 1 | inner | 1860.2.i.b | ✓ | 48 |
93.c | even | 2 | 1 | inner | 1860.2.i.b | ✓ | 48 |
155.c | odd | 2 | 1 | inner | 1860.2.i.b | ✓ | 48 |
465.g | even | 2 | 1 | inner | 1860.2.i.b | ✓ | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1860.2.i.b | ✓ | 48 | 1.a | even | 1 | 1 | trivial |
1860.2.i.b | ✓ | 48 | 3.b | odd | 2 | 1 | inner |
1860.2.i.b | ✓ | 48 | 5.b | even | 2 | 1 | inner |
1860.2.i.b | ✓ | 48 | 15.d | odd | 2 | 1 | inner |
1860.2.i.b | ✓ | 48 | 31.b | odd | 2 | 1 | inner |
1860.2.i.b | ✓ | 48 | 93.c | even | 2 | 1 | inner |
1860.2.i.b | ✓ | 48 | 155.c | odd | 2 | 1 | inner |
1860.2.i.b | ✓ | 48 | 465.g | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{12} + 62T_{7}^{10} + 1425T_{7}^{8} + 15108T_{7}^{6} + 72416T_{7}^{4} + 121444T_{7}^{2} + 30760 \)
acting on \(S_{2}^{\mathrm{new}}(1860, [\chi])\).