Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1155,2,Mod(769,1155)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1155, 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("1155.769");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1155 = 3 \cdot 5 \cdot 7 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1155.k (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: | \(9.22272143346\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
769.1 | −2.66471 | 1.00000 | 5.10065 | −1.18527 | − | 1.89608i | −2.66471 | 2.09935 | + | 1.61019i | −8.26233 | 1.00000 | 3.15840 | + | 5.05250i | ||||||||||||
769.2 | −2.66471 | 1.00000 | 5.10065 | −1.18527 | + | 1.89608i | −2.66471 | 2.09935 | − | 1.61019i | −8.26233 | 1.00000 | 3.15840 | − | 5.05250i | ||||||||||||
769.3 | −2.52930 | 1.00000 | 4.39737 | −1.63381 | − | 1.52665i | −2.52930 | −2.48590 | − | 0.905718i | −6.06368 | 1.00000 | 4.13240 | + | 3.86136i | ||||||||||||
769.4 | −2.52930 | 1.00000 | 4.39737 | −1.63381 | + | 1.52665i | −2.52930 | −2.48590 | + | 0.905718i | −6.06368 | 1.00000 | 4.13240 | − | 3.86136i | ||||||||||||
769.5 | −2.43104 | 1.00000 | 3.90997 | 2.21714 | + | 0.290295i | −2.43104 | 1.43522 | + | 2.22265i | −4.64322 | 1.00000 | −5.38997 | − | 0.705719i | ||||||||||||
769.6 | −2.43104 | 1.00000 | 3.90997 | 2.21714 | − | 0.290295i | −2.43104 | 1.43522 | − | 2.22265i | −4.64322 | 1.00000 | −5.38997 | + | 0.705719i | ||||||||||||
769.7 | −2.15215 | 1.00000 | 2.63174 | 1.03706 | − | 1.98104i | −2.15215 | −2.36665 | − | 1.18277i | −1.35961 | 1.00000 | −2.23191 | + | 4.26348i | ||||||||||||
769.8 | −2.15215 | 1.00000 | 2.63174 | 1.03706 | + | 1.98104i | −2.15215 | −2.36665 | + | 1.18277i | −1.35961 | 1.00000 | −2.23191 | − | 4.26348i | ||||||||||||
769.9 | −1.83264 | 1.00000 | 1.35857 | 0.538446 | − | 2.17027i | −1.83264 | −0.996474 | + | 2.45093i | 1.17551 | 1.00000 | −0.986778 | + | 3.97732i | ||||||||||||
769.10 | −1.83264 | 1.00000 | 1.35857 | 0.538446 | + | 2.17027i | −1.83264 | −0.996474 | − | 2.45093i | 1.17551 | 1.00000 | −0.986778 | − | 3.97732i | ||||||||||||
769.11 | −1.69546 | 1.00000 | 0.874594 | −2.22547 | − | 0.217459i | −1.69546 | 0.698105 | − | 2.55199i | 1.90808 | 1.00000 | 3.77320 | + | 0.368694i | ||||||||||||
769.12 | −1.69546 | 1.00000 | 0.874594 | −2.22547 | + | 0.217459i | −1.69546 | 0.698105 | + | 2.55199i | 1.90808 | 1.00000 | 3.77320 | − | 0.368694i | ||||||||||||
769.13 | −1.63529 | 1.00000 | 0.674169 | −1.84877 | − | 1.25780i | −1.63529 | 1.79271 | − | 1.94582i | 2.16812 | 1.00000 | 3.02327 | + | 2.05687i | ||||||||||||
769.14 | −1.63529 | 1.00000 | 0.674169 | −1.84877 | + | 1.25780i | −1.63529 | 1.79271 | + | 1.94582i | 2.16812 | 1.00000 | 3.02327 | − | 2.05687i | ||||||||||||
769.15 | −1.17317 | 1.00000 | −0.623669 | 2.20931 | − | 0.344884i | −1.17317 | −1.71739 | + | 2.01260i | 3.07801 | 1.00000 | −2.59190 | + | 0.404608i | ||||||||||||
769.16 | −1.17317 | 1.00000 | −0.623669 | 2.20931 | + | 0.344884i | −1.17317 | −1.71739 | − | 2.01260i | 3.07801 | 1.00000 | −2.59190 | − | 0.404608i | ||||||||||||
769.17 | −1.12050 | 1.00000 | −0.744486 | 1.27295 | + | 1.83837i | −1.12050 | 2.54893 | − | 0.709189i | 3.07519 | 1.00000 | −1.42634 | − | 2.05989i | ||||||||||||
769.18 | −1.12050 | 1.00000 | −0.744486 | 1.27295 | − | 1.83837i | −1.12050 | 2.54893 | + | 0.709189i | 3.07519 | 1.00000 | −1.42634 | + | 2.05989i | ||||||||||||
769.19 | −0.567201 | 1.00000 | −1.67828 | −2.10264 | − | 0.760858i | −0.567201 | −2.63586 | + | 0.228555i | 2.08633 | 1.00000 | 1.19262 | + | 0.431559i | ||||||||||||
769.20 | −0.567201 | 1.00000 | −1.67828 | −2.10264 | + | 0.760858i | −0.567201 | −2.63586 | − | 0.228555i | 2.08633 | 1.00000 | 1.19262 | − | 0.431559i | ||||||||||||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.b | odd | 2 | 1 | inner |
35.c | odd | 2 | 1 | inner |
385.h | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1155.2.k.b | yes | 48 |
5.b | even | 2 | 1 | 1155.2.k.a | ✓ | 48 | |
7.b | odd | 2 | 1 | 1155.2.k.a | ✓ | 48 | |
11.b | odd | 2 | 1 | inner | 1155.2.k.b | yes | 48 |
35.c | odd | 2 | 1 | inner | 1155.2.k.b | yes | 48 |
55.d | odd | 2 | 1 | 1155.2.k.a | ✓ | 48 | |
77.b | even | 2 | 1 | 1155.2.k.a | ✓ | 48 | |
385.h | even | 2 | 1 | inner | 1155.2.k.b | yes | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1155.2.k.a | ✓ | 48 | 5.b | even | 2 | 1 | |
1155.2.k.a | ✓ | 48 | 7.b | odd | 2 | 1 | |
1155.2.k.a | ✓ | 48 | 55.d | odd | 2 | 1 | |
1155.2.k.a | ✓ | 48 | 77.b | even | 2 | 1 | |
1155.2.k.b | yes | 48 | 1.a | even | 1 | 1 | trivial |
1155.2.k.b | yes | 48 | 11.b | odd | 2 | 1 | inner |
1155.2.k.b | yes | 48 | 35.c | odd | 2 | 1 | inner |
1155.2.k.b | yes | 48 | 385.h | even | 2 | 1 | inner |