Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [145,2,Mod(17,145)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(145, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([1, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("145.17");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 145 = 5 \cdot 29 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 145.j (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: | \(1.15783082931\) |
Analytic rank: | \(0\) |
Dimension: | \(26\) |
Relative dimension: | \(13\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
17.1 | −2.73482 | − | 1.63186i | 5.47925 | −0.903087 | + | 2.04559i | 4.46285i | 1.05887 | + | 1.05887i | −9.51511 | 0.337028 | 2.46978 | − | 5.59432i | |||||||||||
17.2 | −2.41122 | 1.85973i | 3.81399 | 1.59267 | − | 1.56952i | − | 4.48422i | −0.291676 | − | 0.291676i | −4.37394 | −0.458586 | −3.84028 | + | 3.78447i | |||||||||||
17.3 | −1.82099 | − | 2.59340i | 1.31599 | 1.75450 | − | 1.38626i | 4.72256i | −0.820621 | − | 0.820621i | 1.24557 | −3.72575 | −3.19492 | + | 2.52437i | |||||||||||
17.4 | −1.77873 | 1.38965i | 1.16389 | −2.14366 | + | 0.636182i | − | 2.47181i | −3.41296 | − | 3.41296i | 1.48721 | 1.06888 | 3.81300 | − | 1.13160i | |||||||||||
17.5 | −1.36192 | 0.228160i | −0.145179 | −1.87852 | − | 1.21291i | − | 0.310736i | 3.45046 | + | 3.45046i | 2.92156 | 2.94794 | 2.55840 | + | 1.65188i | |||||||||||
17.6 | −0.839004 | − | 0.711801i | −1.29607 | 1.24503 | + | 1.85739i | 0.597203i | 1.13987 | + | 1.13987i | 2.76542 | 2.49334 | −1.04459 | − | 1.55836i | |||||||||||
17.7 | −0.342532 | 2.64611i | −1.88267 | 1.82632 | + | 1.29018i | − | 0.906377i | −1.55474 | − | 1.55474i | 1.32994 | −4.00189 | −0.625573 | − | 0.441927i | |||||||||||
17.8 | 0.222351 | − | 1.02589i | −1.95056 | −0.621451 | − | 2.14798i | − | 0.228107i | −2.35964 | − | 2.35964i | −0.878412 | 1.94756 | −0.138180 | − | 0.477605i | ||||||||||
17.9 | 0.895351 | 2.11245i | −1.19835 | −1.71183 | + | 1.43862i | 1.89138i | 1.38465 | + | 1.38465i | −2.86364 | −1.46244 | −1.53269 | + | 1.28807i | ||||||||||||
17.10 | 1.26373 | 0.913274i | −0.402981 | 1.78577 | − | 1.34575i | 1.15413i | 2.03055 | + | 2.03055i | −3.03672 | 2.16593 | 2.25673 | − | 1.70067i | ||||||||||||
17.11 | 1.41066 | − | 2.58872i | −0.0100302 | 1.93694 | + | 1.11726i | − | 3.65181i | −0.510628 | − | 0.510628i | −2.83547 | −3.70148 | 2.73237 | + | 1.57607i | ||||||||||
17.12 | 2.23019 | − | 1.25170i | 2.97373 | −2.19887 | + | 0.406147i | − | 2.79153i | 0.483409 | + | 0.483409i | 2.17160 | 1.43324 | −4.90390 | + | 0.905783i | ||||||||||
17.13 | 2.26693 | 2.65401i | 3.13899 | −0.683805 | − | 2.12895i | 6.01647i | −2.59753 | − | 2.59753i | 2.58201 | −4.04378 | −1.55014 | − | 4.82618i | ||||||||||||
128.1 | −2.73482 | 1.63186i | 5.47925 | −0.903087 | − | 2.04559i | − | 4.46285i | 1.05887 | − | 1.05887i | −9.51511 | 0.337028 | 2.46978 | + | 5.59432i | |||||||||||
128.2 | −2.41122 | − | 1.85973i | 3.81399 | 1.59267 | + | 1.56952i | 4.48422i | −0.291676 | + | 0.291676i | −4.37394 | −0.458586 | −3.84028 | − | 3.78447i | |||||||||||
128.3 | −1.82099 | 2.59340i | 1.31599 | 1.75450 | + | 1.38626i | − | 4.72256i | −0.820621 | + | 0.820621i | 1.24557 | −3.72575 | −3.19492 | − | 2.52437i | |||||||||||
128.4 | −1.77873 | − | 1.38965i | 1.16389 | −2.14366 | − | 0.636182i | 2.47181i | −3.41296 | + | 3.41296i | 1.48721 | 1.06888 | 3.81300 | + | 1.13160i | |||||||||||
128.5 | −1.36192 | − | 0.228160i | −0.145179 | −1.87852 | + | 1.21291i | 0.310736i | 3.45046 | − | 3.45046i | 2.92156 | 2.94794 | 2.55840 | − | 1.65188i | |||||||||||
128.6 | −0.839004 | 0.711801i | −1.29607 | 1.24503 | − | 1.85739i | − | 0.597203i | 1.13987 | − | 1.13987i | 2.76542 | 2.49334 | −1.04459 | + | 1.55836i | |||||||||||
128.7 | −0.342532 | − | 2.64611i | −1.88267 | 1.82632 | − | 1.29018i | 0.906377i | −1.55474 | + | 1.55474i | 1.32994 | −4.00189 | −0.625573 | + | 0.441927i | |||||||||||
See all 26 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
145.j | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 145.2.j.a | yes | 26 |
5.b | even | 2 | 1 | 725.2.j.c | 26 | ||
5.c | odd | 4 | 1 | 145.2.e.a | ✓ | 26 | |
5.c | odd | 4 | 1 | 725.2.e.c | 26 | ||
29.c | odd | 4 | 1 | 145.2.e.a | ✓ | 26 | |
145.e | even | 4 | 1 | 725.2.j.c | 26 | ||
145.f | odd | 4 | 1 | 725.2.e.c | 26 | ||
145.j | even | 4 | 1 | inner | 145.2.j.a | yes | 26 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
145.2.e.a | ✓ | 26 | 5.c | odd | 4 | 1 | |
145.2.e.a | ✓ | 26 | 29.c | odd | 4 | 1 | |
145.2.j.a | yes | 26 | 1.a | even | 1 | 1 | trivial |
145.2.j.a | yes | 26 | 145.j | even | 4 | 1 | inner |
725.2.e.c | 26 | 5.c | odd | 4 | 1 | ||
725.2.e.c | 26 | 145.f | odd | 4 | 1 | ||
725.2.j.c | 26 | 5.b | even | 2 | 1 | ||
725.2.j.c | 26 | 145.e | even | 4 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(145, [\chi])\).