Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [33,7,Mod(5,33)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(33, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([5, 4]))
N = Newforms(chi, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("33.5");
S:= CuspForms(chi, 7);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 33 = 3 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 7 \) |
Character orbit: | \([\chi]\) | \(=\) | 33.h (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: | \(7.59178475946\) |
Analytic rank: | \(0\) |
Dimension: | \(88\) |
Relative dimension: | \(22\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
5.1 | −14.6057 | + | 4.74569i | 25.8173 | − | 7.90347i | 139.029 | − | 101.011i | −39.7419 | − | 12.9129i | −339.574 | + | 237.957i | −103.376 | + | 75.1073i | −973.541 | + | 1339.96i | 604.070 | − | 408.093i | 641.741 | ||
5.2 | −12.6998 | + | 4.12642i | −26.9979 | + | 0.338512i | 92.4810 | − | 67.1914i | 73.9941 | + | 24.0421i | 341.471 | − | 115.704i | 162.344 | − | 117.950i | −394.901 | + | 543.535i | 728.771 | − | 18.2782i | −1038.92 | ||
5.3 | −11.7448 | + | 3.81612i | −9.61413 | + | 25.2303i | 71.6006 | − | 52.0208i | −165.956 | − | 53.9225i | 16.6343 | − | 333.014i | −453.911 | + | 329.785i | −177.861 | + | 244.804i | −544.137 | − | 485.135i | 2154.90 | ||
5.4 | −11.2768 | + | 3.66405i | 10.7587 | + | 24.7639i | 61.9636 | − | 45.0192i | 101.905 | + | 33.1109i | −212.060 | − | 239.837i | 468.165 | − | 340.142i | −87.7535 | + | 120.782i | −497.500 | + | 532.855i | −1270.48 | ||
5.5 | −10.0445 | + | 3.26364i | −1.25477 | − | 26.9708i | 38.4627 | − | 27.9448i | 135.541 | + | 44.0400i | 100.627 | + | 266.812i | −206.442 | + | 149.989i | 102.165 | − | 140.618i | −725.851 | + | 67.6842i | −1505.17 | ||
5.6 | −9.48700 | + | 3.08251i | −4.13517 | − | 26.6815i | 28.7242 | − | 20.8694i | −210.493 | − | 68.3933i | 121.476 | + | 240.380i | 290.276 | − | 210.898i | 167.074 | − | 229.958i | −694.801 | + | 220.665i | 2207.77 | ||
5.7 | −6.30214 | + | 2.04769i | 25.8420 | − | 7.82243i | −16.2532 | + | 11.8086i | 66.8441 | + | 21.7190i | −146.842 | + | 102.214i | −126.894 | + | 92.1938i | 327.525 | − | 450.800i | 606.619 | − | 404.295i | −465.734 | ||
5.8 | −5.15978 | + | 1.67652i | 21.4926 | + | 16.3422i | −27.9644 | + | 20.3173i | −104.094 | − | 33.8222i | −138.295 | − | 48.2893i | −30.2053 | + | 21.9454i | 314.319 | − | 432.623i | 194.867 | + | 702.473i | 593.806 | ||
5.9 | −3.91805 | + | 1.27305i | −26.2247 | − | 6.42389i | −38.0467 | + | 27.6425i | −40.1977 | − | 13.0610i | 110.927 | − | 8.21623i | −125.751 | + | 91.3637i | 268.853 | − | 370.045i | 646.467 | + | 336.929i | 174.124 | ||
5.10 | −3.46608 | + | 1.12620i | −11.0612 | + | 24.6303i | −41.0317 | + | 29.8113i | 207.457 | + | 67.4069i | 10.6004 | − | 97.8275i | −309.326 | + | 224.739i | 245.744 | − | 338.237i | −484.301 | − | 544.880i | −794.976 | ||
5.11 | −0.658923 | + | 0.214097i | −17.8783 | + | 20.2328i | −51.3887 | + | 37.3361i | −87.3837 | − | 28.3927i | 7.44866 | − | 17.1596i | 484.811 | − | 352.236i | 51.9309 | − | 71.4767i | −89.7314 | − | 723.456i | 63.6579 | ||
5.12 | 0.658923 | − | 0.214097i | 13.7178 | − | 23.2556i | −51.3887 | + | 37.3361i | 87.3837 | + | 28.3927i | 4.06004 | − | 18.2606i | 484.811 | − | 352.236i | −51.9309 | + | 71.4767i | −352.643 | − | 638.031i | 63.6579 | ||
5.13 | 3.46608 | − | 1.12620i | 20.0067 | − | 18.1310i | −41.0317 | + | 29.8113i | −207.457 | − | 67.4069i | 48.9257 | − | 85.3748i | −309.326 | + | 224.739i | −245.744 | + | 338.237i | 71.5354 | − | 725.482i | −794.976 | ||
5.14 | 3.91805 | − | 1.27305i | −14.2134 | − | 22.9561i | −38.0467 | + | 27.6425i | 40.1977 | + | 13.0610i | −84.9128 | − | 71.8486i | −125.751 | + | 91.3637i | −268.853 | + | 370.045i | −324.961 | + | 652.565i | 174.124 | ||
5.15 | 5.15978 | − | 1.67652i | 22.1839 | + | 15.3907i | −27.9644 | + | 20.3173i | 104.094 | + | 33.8222i | 140.267 | + | 42.2210i | −30.2053 | + | 21.9454i | −314.319 | + | 432.623i | 255.253 | + | 682.852i | 593.806 | ||
5.16 | 6.30214 | − | 2.04769i | 0.546048 | + | 26.9945i | −16.2532 | + | 11.8086i | −66.8441 | − | 21.7190i | 58.7175 | + | 169.005i | −126.894 | + | 92.1938i | −327.525 | + | 450.800i | −728.404 | + | 29.4805i | −465.734 | ||
5.17 | 9.48700 | − | 3.08251i | −26.6534 | + | 4.31225i | 28.7242 | − | 20.8694i | 210.493 | + | 68.3933i | −239.568 | + | 123.070i | 290.276 | − | 210.898i | −167.074 | + | 229.958i | 691.809 | − | 229.872i | 2207.77 | ||
5.18 | 10.0445 | − | 3.26364i | −26.0385 | + | 7.14109i | 38.4627 | − | 27.9448i | −135.541 | − | 44.0400i | −238.237 | + | 156.709i | −206.442 | + | 149.989i | −102.165 | + | 140.618i | 627.010 | − | 371.887i | −1505.17 | ||
5.19 | 11.2768 | − | 3.66405i | 26.8765 | + | 2.57968i | 61.9636 | − | 45.0192i | −101.905 | − | 33.1109i | 312.532 | − | 69.3863i | 468.165 | − | 340.142i | 87.7535 | − | 120.782i | 715.690 | + | 138.666i | −1270.48 | ||
5.20 | 11.7448 | − | 3.81612i | 21.0245 | − | 16.9402i | 71.6006 | − | 52.0208i | 165.956 | + | 53.9225i | 182.283 | − | 279.191i | −453.911 | + | 329.785i | 177.861 | − | 244.804i | 155.061 | − | 712.318i | 2154.90 | ||
See all 88 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
11.c | even | 5 | 1 | inner |
33.h | odd | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 33.7.h.a | ✓ | 88 |
3.b | odd | 2 | 1 | inner | 33.7.h.a | ✓ | 88 |
11.c | even | 5 | 1 | inner | 33.7.h.a | ✓ | 88 |
33.h | odd | 10 | 1 | inner | 33.7.h.a | ✓ | 88 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
33.7.h.a | ✓ | 88 | 1.a | even | 1 | 1 | trivial |
33.7.h.a | ✓ | 88 | 3.b | odd | 2 | 1 | inner |
33.7.h.a | ✓ | 88 | 11.c | even | 5 | 1 | inner |
33.7.h.a | ✓ | 88 | 33.h | odd | 10 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(33, [\chi])\).