Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [403,2,Mod(216,403)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(403, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([1, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("403.216");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 403 = 13 \cdot 31 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 403.i (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: | \(3.21797120146\) |
Analytic rank: | \(0\) |
Dimension: | \(68\) |
Relative dimension: | \(34\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
216.1 | −1.86899 | − | 1.86899i | − | 2.77664i | 4.98628i | 1.10552 | + | 1.10552i | −5.18952 | + | 5.18952i | 3.07074 | − | 3.07074i | 5.58133 | − | 5.58133i | −4.70973 | − | 4.13241i | ||||||
216.2 | −1.86899 | − | 1.86899i | 2.77664i | 4.98628i | 1.10552 | + | 1.10552i | 5.18952 | − | 5.18952i | 3.07074 | − | 3.07074i | 5.58133 | − | 5.58133i | −4.70973 | − | 4.13241i | |||||||
216.3 | −1.78306 | − | 1.78306i | − | 0.329930i | 4.35859i | −1.02687 | − | 1.02687i | −0.588285 | + | 0.588285i | −0.568114 | + | 0.568114i | 4.20550 | − | 4.20550i | 2.89115 | 3.66195i | |||||||
216.4 | −1.78306 | − | 1.78306i | 0.329930i | 4.35859i | −1.02687 | − | 1.02687i | 0.588285 | − | 0.588285i | −0.568114 | + | 0.568114i | 4.20550 | − | 4.20550i | 2.89115 | 3.66195i | ||||||||
216.5 | −1.50749 | − | 1.50749i | − | 1.65703i | 2.54505i | 2.11743 | + | 2.11743i | −2.49795 | + | 2.49795i | −2.06359 | + | 2.06359i | 0.821650 | − | 0.821650i | 0.254264 | − | 6.38401i | ||||||
216.6 | −1.50749 | − | 1.50749i | 1.65703i | 2.54505i | 2.11743 | + | 2.11743i | 2.49795 | − | 2.49795i | −2.06359 | + | 2.06359i | 0.821650 | − | 0.821650i | 0.254264 | − | 6.38401i | |||||||
216.7 | −1.30566 | − | 1.30566i | − | 2.35582i | 1.40948i | −3.07515 | − | 3.07515i | −3.07590 | + | 3.07590i | 1.59752 | − | 1.59752i | −0.771014 | + | 0.771014i | −2.54991 | 8.03019i | |||||||
216.8 | −1.30566 | − | 1.30566i | 2.35582i | 1.40948i | −3.07515 | − | 3.07515i | 3.07590 | − | 3.07590i | 1.59752 | − | 1.59752i | −0.771014 | + | 0.771014i | −2.54991 | 8.03019i | ||||||||
216.9 | −1.12794 | − | 1.12794i | − | 1.37229i | 0.544506i | 1.86354 | + | 1.86354i | −1.54787 | + | 1.54787i | 2.05385 | − | 2.05385i | −1.64171 | + | 1.64171i | 1.11681 | − | 4.20393i | ||||||
216.10 | −1.12794 | − | 1.12794i | 1.37229i | 0.544506i | 1.86354 | + | 1.86354i | 1.54787 | − | 1.54787i | 2.05385 | − | 2.05385i | −1.64171 | + | 1.64171i | 1.11681 | − | 4.20393i | |||||||
216.11 | −0.840661 | − | 0.840661i | − | 1.32208i | − | 0.586577i | −1.04050 | − | 1.04050i | −1.11142 | + | 1.11142i | −1.92533 | + | 1.92533i | −2.17444 | + | 2.17444i | 1.25210 | 1.74942i | ||||||
216.12 | −0.840661 | − | 0.840661i | 1.32208i | − | 0.586577i | −1.04050 | − | 1.04050i | 1.11142 | − | 1.11142i | −1.92533 | + | 1.92533i | −2.17444 | + | 2.17444i | 1.25210 | 1.74942i | |||||||
216.13 | −0.471532 | − | 0.471532i | − | 3.39270i | − | 1.55532i | 1.41859 | + | 1.41859i | −1.59977 | + | 1.59977i | −1.23146 | + | 1.23146i | −1.67644 | + | 1.67644i | −8.51044 | − | 1.33782i | |||||
216.14 | −0.471532 | − | 0.471532i | 3.39270i | − | 1.55532i | 1.41859 | + | 1.41859i | 1.59977 | − | 1.59977i | −1.23146 | + | 1.23146i | −1.67644 | + | 1.67644i | −8.51044 | − | 1.33782i | ||||||
216.15 | −0.459282 | − | 0.459282i | − | 0.325710i | − | 1.57812i | −0.464951 | − | 0.464951i | −0.149593 | + | 0.149593i | 2.03738 | − | 2.03738i | −1.64337 | + | 1.64337i | 2.89391 | 0.427088i | ||||||
216.16 | −0.459282 | − | 0.459282i | 0.325710i | − | 1.57812i | −0.464951 | − | 0.464951i | 0.149593 | − | 0.149593i | 2.03738 | − | 2.03738i | −1.64337 | + | 1.64337i | 2.89391 | 0.427088i | |||||||
216.17 | 0.194750 | + | 0.194750i | − | 2.63371i | − | 1.92414i | −1.36075 | − | 1.36075i | 0.512915 | − | 0.512915i | 2.99770 | − | 2.99770i | 0.764228 | − | 0.764228i | −3.93641 | − | 0.530014i | |||||
216.18 | 0.194750 | + | 0.194750i | 2.63371i | − | 1.92414i | −1.36075 | − | 1.36075i | −0.512915 | + | 0.512915i | 2.99770 | − | 2.99770i | 0.764228 | − | 0.764228i | −3.93641 | − | 0.530014i | ||||||
216.19 | 0.235416 | + | 0.235416i | − | 2.22877i | − | 1.88916i | −2.22025 | − | 2.22025i | 0.524689 | − | 0.524689i | −2.84607 | + | 2.84607i | 0.915572 | − | 0.915572i | −1.96742 | − | 1.04537i | |||||
216.20 | 0.235416 | + | 0.235416i | 2.22877i | − | 1.88916i | −2.22025 | − | 2.22025i | −0.524689 | + | 0.524689i | −2.84607 | + | 2.84607i | 0.915572 | − | 0.915572i | −1.96742 | − | 1.04537i | ||||||
See all 68 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.d | odd | 4 | 1 | inner |
31.b | odd | 2 | 1 | inner |
403.i | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 403.2.i.a | ✓ | 68 |
13.d | odd | 4 | 1 | inner | 403.2.i.a | ✓ | 68 |
31.b | odd | 2 | 1 | inner | 403.2.i.a | ✓ | 68 |
403.i | even | 4 | 1 | inner | 403.2.i.a | ✓ | 68 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
403.2.i.a | ✓ | 68 | 1.a | even | 1 | 1 | trivial |
403.2.i.a | ✓ | 68 | 13.d | odd | 4 | 1 | inner |
403.2.i.a | ✓ | 68 | 31.b | odd | 2 | 1 | inner |
403.2.i.a | ✓ | 68 | 403.i | even | 4 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(403, [\chi])\).