Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [155,2,Mod(4,155)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(155, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([5, 6]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("155.4");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 155 = 5 \cdot 31 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 155.n (of order \(10\), degree \(4\), 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: | \(1.23768123133\) |
Analytic rank: | \(0\) |
Dimension: | \(56\) |
Relative dimension: | \(14\) over \(\Q(\zeta_{10})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
4.1 | −2.53334 | + | 0.823132i | 0.851532 | + | 0.276679i | 4.12224 | − | 2.99498i | −1.95835 | − | 1.07930i | −2.38496 | −1.09214 | − | 1.50319i | −4.84638 | + | 6.67047i | −1.77850 | − | 1.29215i | 5.84956 | + | 1.12224i | ||
4.2 | −1.90735 | + | 0.619737i | 0.725535 | + | 0.235741i | 1.63589 | − | 1.18855i | 2.06323 | − | 0.862016i | −1.52995 | −0.105855 | − | 0.145698i | −0.0260217 | + | 0.0358157i | −1.95622 | − | 1.42128i | −3.40109 | + | 2.92283i | ||
4.3 | −1.58285 | + | 0.514300i | −2.21876 | − | 0.720920i | 0.622883 | − | 0.452551i | −2.23096 | + | 0.151008i | 3.88274 | 1.16442 | + | 1.60269i | 1.20333 | − | 1.65624i | 1.97614 | + | 1.43575i | 3.45362 | − | 1.38641i | ||
4.4 | −1.53412 | + | 0.498465i | −0.906662 | − | 0.294592i | 0.487018 | − | 0.353839i | 0.594355 | + | 2.15563i | 1.53777 | −0.381739 | − | 0.525419i | 1.32551 | − | 1.82441i | −1.69180 | − | 1.22916i | −1.98632 | − | 3.01073i | ||
4.5 | −1.03006 | + | 0.334686i | 2.61471 | + | 0.849570i | −0.669031 | + | 0.486079i | −1.31324 | − | 1.80981i | −2.97764 | 2.94133 | + | 4.04839i | 1.79968 | − | 2.47704i | 3.68788 | + | 2.67940i | 1.95843 | + | 1.42468i | ||
4.6 | −0.276467 | + | 0.0898296i | −0.803274 | − | 0.260999i | −1.54967 | + | 1.12590i | −0.0371642 | − | 2.23576i | 0.245524 | −1.44375 | − | 1.98715i | 0.669025 | − | 0.920834i | −1.84992 | − | 1.34405i | 0.211112 | + | 0.614775i | ||
4.7 | −0.124167 | + | 0.0403444i | 2.85207 | + | 0.926694i | −1.60424 | + | 1.16555i | 2.19114 | + | 0.445972i | −0.391521 | −2.49647 | − | 3.43610i | 0.305651 | − | 0.420692i | 4.84849 | + | 3.52264i | −0.290061 | + | 0.0330252i | ||
4.8 | 0.124167 | − | 0.0403444i | −2.85207 | − | 0.926694i | −1.60424 | + | 1.16555i | 2.19114 | − | 0.445972i | −0.391521 | 2.49647 | + | 3.43610i | −0.305651 | + | 0.420692i | 4.84849 | + | 3.52264i | 0.254076 | − | 0.143776i | ||
4.9 | 0.276467 | − | 0.0898296i | 0.803274 | + | 0.260999i | −1.54967 | + | 1.12590i | −0.0371642 | + | 2.23576i | 0.245524 | 1.44375 | + | 1.98715i | −0.669025 | + | 0.920834i | −1.84992 | − | 1.34405i | 0.190563 | + | 0.621452i | ||
4.10 | 1.03006 | − | 0.334686i | −2.61471 | − | 0.849570i | −0.669031 | + | 0.486079i | −1.31324 | + | 1.80981i | −2.97764 | −2.94133 | − | 4.04839i | −1.79968 | + | 2.47704i | 3.68788 | + | 2.67940i | −0.746995 | + | 2.30373i | ||
4.11 | 1.53412 | − | 0.498465i | 0.906662 | + | 0.294592i | 0.487018 | − | 0.353839i | 0.594355 | − | 2.15563i | 1.53777 | 0.381739 | + | 0.525419i | −1.32551 | + | 1.82441i | −1.69180 | − | 1.22916i | −0.162696 | − | 3.60326i | ||
4.12 | 1.58285 | − | 0.514300i | 2.21876 | + | 0.720920i | 0.622883 | − | 0.452551i | −2.23096 | − | 0.151008i | 3.88274 | −1.16442 | − | 1.60269i | −1.20333 | + | 1.65624i | 1.97614 | + | 1.43575i | −3.60895 | + | 0.908360i | ||
4.13 | 1.90735 | − | 0.619737i | −0.725535 | − | 0.235741i | 1.63589 | − | 1.18855i | 2.06323 | + | 0.862016i | −1.52995 | 0.105855 | + | 0.145698i | 0.0260217 | − | 0.0358157i | −1.95622 | − | 1.42128i | 4.46954 | + | 0.365508i | ||
4.14 | 2.53334 | − | 0.823132i | −0.851532 | − | 0.276679i | 4.12224 | − | 2.99498i | −1.95835 | + | 1.07930i | −2.38496 | 1.09214 | + | 1.50319i | 4.84638 | − | 6.67047i | −1.77850 | − | 1.29215i | −4.07276 | + | 4.34620i | ||
39.1 | −2.53334 | − | 0.823132i | 0.851532 | − | 0.276679i | 4.12224 | + | 2.99498i | −1.95835 | + | 1.07930i | −2.38496 | −1.09214 | + | 1.50319i | −4.84638 | − | 6.67047i | −1.77850 | + | 1.29215i | 5.84956 | − | 1.12224i | ||
39.2 | −1.90735 | − | 0.619737i | 0.725535 | − | 0.235741i | 1.63589 | + | 1.18855i | 2.06323 | + | 0.862016i | −1.52995 | −0.105855 | + | 0.145698i | −0.0260217 | − | 0.0358157i | −1.95622 | + | 1.42128i | −3.40109 | − | 2.92283i | ||
39.3 | −1.58285 | − | 0.514300i | −2.21876 | + | 0.720920i | 0.622883 | + | 0.452551i | −2.23096 | − | 0.151008i | 3.88274 | 1.16442 | − | 1.60269i | 1.20333 | + | 1.65624i | 1.97614 | − | 1.43575i | 3.45362 | + | 1.38641i | ||
39.4 | −1.53412 | − | 0.498465i | −0.906662 | + | 0.294592i | 0.487018 | + | 0.353839i | 0.594355 | − | 2.15563i | 1.53777 | −0.381739 | + | 0.525419i | 1.32551 | + | 1.82441i | −1.69180 | + | 1.22916i | −1.98632 | + | 3.01073i | ||
39.5 | −1.03006 | − | 0.334686i | 2.61471 | − | 0.849570i | −0.669031 | − | 0.486079i | −1.31324 | + | 1.80981i | −2.97764 | 2.94133 | − | 4.04839i | 1.79968 | + | 2.47704i | 3.68788 | − | 2.67940i | 1.95843 | − | 1.42468i | ||
39.6 | −0.276467 | − | 0.0898296i | −0.803274 | + | 0.260999i | −1.54967 | − | 1.12590i | −0.0371642 | + | 2.23576i | 0.245524 | −1.44375 | + | 1.98715i | 0.669025 | + | 0.920834i | −1.84992 | + | 1.34405i | 0.211112 | − | 0.614775i | ||
See all 56 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
31.d | even | 5 | 1 | inner |
155.n | even | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 155.2.n.a | ✓ | 56 |
5.b | even | 2 | 1 | inner | 155.2.n.a | ✓ | 56 |
5.c | odd | 4 | 2 | 775.2.k.h | 56 | ||
31.d | even | 5 | 1 | inner | 155.2.n.a | ✓ | 56 |
155.n | even | 10 | 1 | inner | 155.2.n.a | ✓ | 56 |
155.s | odd | 20 | 2 | 775.2.k.h | 56 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
155.2.n.a | ✓ | 56 | 1.a | even | 1 | 1 | trivial |
155.2.n.a | ✓ | 56 | 5.b | even | 2 | 1 | inner |
155.2.n.a | ✓ | 56 | 31.d | even | 5 | 1 | inner |
155.2.n.a | ✓ | 56 | 155.n | even | 10 | 1 | inner |
775.2.k.h | 56 | 5.c | odd | 4 | 2 | ||
775.2.k.h | 56 | 155.s | odd | 20 | 2 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(155, [\chi])\).