Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [93,4,Mod(26,93)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(93, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("93.26");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 93 = 3 \cdot 31 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 93.g (of order \(6\), 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: | \(5.48717763053\) |
Analytic rank: | \(0\) |
Dimension: | \(60\) |
Relative dimension: | \(30\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
26.1 | − | 5.32278i | 4.01232 | + | 3.30171i | −20.3320 | 16.2145 | − | 9.36145i | 17.5743 | − | 21.3567i | 10.0069 | − | 17.3325i | 65.6406i | 5.19736 | + | 26.4950i | −49.8290 | − | 86.3063i | |||||
26.2 | − | 5.02024i | −4.97128 | − | 1.51209i | −17.2028 | −9.62201 | + | 5.55527i | −7.59107 | + | 24.9570i | 13.9460 | − | 24.1553i | 46.2002i | 22.4272 | + | 15.0341i | 27.8888 | + | 48.3048i | |||||
26.3 | − | 4.92894i | 1.70332 | − | 4.90904i | −16.2944 | −0.772143 | + | 0.445797i | −24.1964 | − | 8.39553i | −4.86632 | + | 8.42871i | 40.8827i | −21.1974 | − | 16.7233i | 2.19730 | + | 3.80584i | |||||
26.4 | − | 4.88422i | −3.69925 | + | 3.64905i | −15.8556 | 4.82657 | − | 2.78662i | 17.8227 | + | 18.0679i | −14.4254 | + | 24.9855i | 38.3682i | 0.368936 | − | 26.9975i | −13.6104 | − | 23.5740i | |||||
26.5 | − | 4.38211i | 3.16472 | + | 4.12123i | −11.2029 | −15.9571 | + | 9.21282i | 18.0597 | − | 13.8682i | −8.93192 | + | 15.4705i | 14.0356i | −6.96906 | + | 26.0851i | 40.3716 | + | 69.9257i | |||||
26.6 | − | 3.53784i | −1.72942 | + | 4.89991i | −4.51631 | −3.07857 | + | 1.77741i | 17.3351 | + | 6.11840i | 11.5958 | − | 20.0844i | − | 12.3247i | −21.0182 | − | 16.9480i | 6.28821 | + | 10.8915i | ||||
26.7 | − | 3.26617i | −1.81950 | − | 4.86718i | −2.66788 | 15.0188 | − | 8.67113i | −15.8970 | + | 5.94280i | 7.66329 | − | 13.2732i | − | 17.4156i | −20.3788 | + | 17.7117i | −28.3214 | − | 49.0541i | ||||
26.8 | − | 3.17111i | 5.09574 | − | 1.01660i | −2.05595 | 2.46691 | − | 1.42427i | −3.22377 | − | 16.1591i | −1.63919 | + | 2.83916i | − | 18.8492i | 24.9330 | − | 10.3607i | −4.51653 | − | 7.82286i | ||||
26.9 | − | 2.69884i | −3.51955 | − | 3.82267i | 0.716275 | −10.6925 | + | 6.17331i | −10.3168 | + | 9.49869i | −12.8440 | + | 22.2465i | − | 23.5238i | −2.22558 | + | 26.9081i | 16.6608 | + | 28.8573i | ||||
26.10 | − | 2.66165i | −5.12832 | + | 0.836861i | 0.915642 | 10.2342 | − | 5.90875i | 2.22743 | + | 13.6498i | −0.623888 | + | 1.08061i | − | 23.7303i | 25.5993 | − | 8.58339i | −15.7270 | − | 27.2399i | ||||
26.11 | − | 1.71661i | 2.40525 | − | 4.60595i | 5.05324 | −16.7823 | + | 9.68924i | −7.90663 | − | 4.12889i | 15.8688 | − | 27.4856i | − | 22.4074i | −15.4295 | − | 22.1569i | 16.6327 | + | 28.8086i | ||||
26.12 | − | 1.30390i | 3.55936 | + | 3.78563i | 6.29985 | −1.87197 | + | 1.08078i | 4.93607 | − | 4.64103i | 8.53372 | − | 14.7808i | − | 18.6455i | −1.66195 | + | 26.9488i | 1.40923 | + | 2.44085i | ||||
26.13 | − | 1.27199i | 1.79919 | + | 4.87472i | 6.38204 | 15.8415 | − | 9.14608i | 6.20060 | − | 2.28855i | −15.2681 | + | 26.4450i | − | 18.2938i | −20.5258 | + | 17.5411i | −11.6337 | − | 20.1502i | ||||
26.14 | − | 0.445163i | −5.18371 | + | 0.359415i | 7.80183 | −3.94983 | + | 2.28043i | 0.159999 | + | 2.30760i | 3.17077 | − | 5.49193i | − | 7.03440i | 26.7416 | − | 3.72621i | 1.01517 | + | 1.75832i | ||||
26.15 | − | 0.202553i | −2.31599 | + | 4.65147i | 7.95897 | −10.0418 | + | 5.79761i | 0.942168 | + | 0.469111i | −6.18654 | + | 10.7154i | − | 3.23253i | −16.2724 | − | 21.5455i | 1.17432 | + | 2.03399i | ||||
26.16 | 0.202553i | 2.87029 | − | 4.33144i | 7.95897 | 10.0418 | − | 5.79761i | 0.877346 | + | 0.581386i | −6.18654 | + | 10.7154i | 3.23253i | −10.5228 | − | 24.8650i | 1.17432 | + | 2.03399i | ||||||
26.17 | 0.445163i | −2.28059 | − | 4.66893i | 7.80183 | 3.94983 | − | 2.28043i | 2.07844 | − | 1.01524i | 3.17077 | − | 5.49193i | 7.03440i | −16.5978 | + | 21.2958i | 1.01517 | + | 1.75832i | ||||||
26.18 | 1.27199i | 5.12123 | − | 0.879218i | 6.38204 | −15.8415 | + | 9.14608i | 1.11836 | + | 6.51415i | −15.2681 | + | 26.4450i | 18.2938i | 25.4540 | − | 9.00535i | −11.6337 | − | 20.1502i | ||||||
26.19 | 1.30390i | 5.05813 | + | 1.18968i | 6.29985 | 1.87197 | − | 1.08078i | −1.55122 | + | 6.59528i | 8.53372 | − | 14.7808i | 18.6455i | 24.1693 | + | 12.0351i | 1.40923 | + | 2.44085i | ||||||
26.20 | 1.71661i | −2.78624 | + | 4.38598i | 5.05324 | 16.7823 | − | 9.68924i | −7.52904 | − | 4.78290i | 15.8688 | − | 27.4856i | 22.4074i | −11.4737 | − | 24.4408i | 16.6327 | + | 28.8086i | ||||||
See all 60 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
31.e | odd | 6 | 1 | inner |
93.g | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 93.4.g.a | ✓ | 60 |
3.b | odd | 2 | 1 | inner | 93.4.g.a | ✓ | 60 |
31.e | odd | 6 | 1 | inner | 93.4.g.a | ✓ | 60 |
93.g | even | 6 | 1 | inner | 93.4.g.a | ✓ | 60 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
93.4.g.a | ✓ | 60 | 1.a | even | 1 | 1 | trivial |
93.4.g.a | ✓ | 60 | 3.b | odd | 2 | 1 | inner |
93.4.g.a | ✓ | 60 | 31.e | odd | 6 | 1 | inner |
93.4.g.a | ✓ | 60 | 93.g | even | 6 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(93, [\chi])\).