Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [633,2,Mod(632,633)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(633, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("633.632");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 633 = 3 \cdot 211 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 633.d (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: | \(5.05453044795\) |
Analytic rank: | \(0\) |
Dimension: | \(68\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
632.1 | −2.69215 | −0.336216 | − | 1.69911i | 5.24765 | 2.13595i | 0.905142 | + | 4.57424i | 2.13995i | −8.74315 | −2.77392 | + | 1.14253i | − | 5.75028i | |||||||||||
632.2 | −2.69215 | −0.336216 | + | 1.69911i | 5.24765 | − | 2.13595i | 0.905142 | − | 4.57424i | − | 2.13995i | −8.74315 | −2.77392 | − | 1.14253i | 5.75028i | ||||||||||
632.3 | −2.60443 | 1.06993 | − | 1.36207i | 4.78303 | − | 2.83866i | −2.78656 | + | 3.54742i | 2.67377i | −7.24820 | −0.710484 | − | 2.91465i | 7.39308i | |||||||||||
632.4 | −2.60443 | 1.06993 | + | 1.36207i | 4.78303 | 2.83866i | −2.78656 | − | 3.54742i | − | 2.67377i | −7.24820 | −0.710484 | + | 2.91465i | − | 7.39308i | ||||||||||
632.5 | −2.47350 | 1.19250 | − | 1.25616i | 4.11822 | 0.579575i | −2.94966 | + | 3.10712i | − | 4.56706i | −5.23944 | −0.155884 | − | 2.99595i | − | 1.43358i | ||||||||||
632.6 | −2.47350 | 1.19250 | + | 1.25616i | 4.11822 | − | 0.579575i | −2.94966 | − | 3.10712i | 4.56706i | −5.23944 | −0.155884 | + | 2.99595i | 1.43358i | |||||||||||
632.7 | −2.34810 | −1.26767 | − | 1.18026i | 3.51358 | − | 2.68207i | 2.97662 | + | 2.77136i | − | 0.262668i | −3.55404 | 0.213986 | + | 2.99236i | 6.29778i | ||||||||||
632.8 | −2.34810 | −1.26767 | + | 1.18026i | 3.51358 | 2.68207i | 2.97662 | − | 2.77136i | 0.262668i | −3.55404 | 0.213986 | − | 2.99236i | − | 6.29778i | |||||||||||
632.9 | −2.29624 | 1.68029 | − | 0.420273i | 3.27272 | 4.22206i | −3.85835 | + | 0.965047i | 1.75525i | −2.92248 | 2.64674 | − | 1.41236i | − | 9.69486i | |||||||||||
632.10 | −2.29624 | 1.68029 | + | 0.420273i | 3.27272 | − | 4.22206i | −3.85835 | − | 0.965047i | − | 1.75525i | −2.92248 | 2.64674 | + | 1.41236i | 9.69486i | ||||||||||
632.11 | −2.01825 | −1.66864 | − | 0.464362i | 2.07332 | 1.97456i | 3.36773 | + | 0.937196i | 5.19790i | −0.147976 | 2.56874 | + | 1.54971i | − | 3.98515i | |||||||||||
632.12 | −2.01825 | −1.66864 | + | 0.464362i | 2.07332 | − | 1.97456i | 3.36773 | − | 0.937196i | − | 5.19790i | −0.147976 | 2.56874 | − | 1.54971i | 3.98515i | ||||||||||
632.13 | −1.88878 | −0.343145 | − | 1.69772i | 1.56748 | 0.692716i | 0.648125 | + | 3.20661i | − | 1.71180i | 0.816942 | −2.76450 | + | 1.16513i | − | 1.30839i | ||||||||||
632.14 | −1.88878 | −0.343145 | + | 1.69772i | 1.56748 | − | 0.692716i | 0.648125 | − | 3.20661i | 1.71180i | 0.816942 | −2.76450 | − | 1.16513i | 1.30839i | |||||||||||
632.15 | −1.76207 | 1.49484 | − | 0.874902i | 1.10489 | 0.0681843i | −2.63401 | + | 1.54164i | 2.98322i | 1.57725 | 1.46909 | − | 2.61568i | − | 0.120145i | |||||||||||
632.16 | −1.76207 | 1.49484 | + | 0.874902i | 1.10489 | − | 0.0681843i | −2.63401 | − | 1.54164i | − | 2.98322i | 1.57725 | 1.46909 | + | 2.61568i | 0.120145i | ||||||||||
632.17 | −1.55050 | −1.58549 | − | 0.697289i | 0.404037 | 3.05483i | 2.45830 | + | 1.08114i | − | 1.87042i | 2.47453 | 2.02757 | + | 2.21109i | − | 4.73649i | ||||||||||
632.18 | −1.55050 | −1.58549 | + | 0.697289i | 0.404037 | − | 3.05483i | 2.45830 | − | 1.08114i | 1.87042i | 2.47453 | 2.02757 | − | 2.21109i | 4.73649i | |||||||||||
632.19 | −1.34820 | 0.793870 | − | 1.53941i | −0.182353 | − | 1.28346i | −1.07030 | + | 2.07543i | 2.59500i | 2.94225 | −1.73954 | − | 2.44418i | 1.73036i | |||||||||||
632.20 | −1.34820 | 0.793870 | + | 1.53941i | −0.182353 | 1.28346i | −1.07030 | − | 2.07543i | − | 2.59500i | 2.94225 | −1.73954 | + | 2.44418i | − | 1.73036i | ||||||||||
See all 68 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
211.b | odd | 2 | 1 | inner |
633.d | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 633.2.d.a | ✓ | 68 |
3.b | odd | 2 | 1 | inner | 633.2.d.a | ✓ | 68 |
211.b | odd | 2 | 1 | inner | 633.2.d.a | ✓ | 68 |
633.d | even | 2 | 1 | inner | 633.2.d.a | ✓ | 68 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
633.2.d.a | ✓ | 68 | 1.a | even | 1 | 1 | trivial |
633.2.d.a | ✓ | 68 | 3.b | odd | 2 | 1 | inner |
633.2.d.a | ✓ | 68 | 211.b | odd | 2 | 1 | inner |
633.2.d.a | ✓ | 68 | 633.d | even | 2 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(633, [\chi])\).