Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [109,4,Mod(46,109)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(109, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([5]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("109.46");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 109 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 109.e (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: | \(6.43120819063\) |
Analytic rank: | \(0\) |
Dimension: | \(52\) |
Relative dimension: | \(26\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
46.1 | − | 5.10799i | 2.61128 | + | 4.52286i | −18.0916 | 3.73827 | + | 6.47487i | 23.1027 | − | 13.3384i | 15.6971 | + | 27.1882i | 51.5478i | −0.137518 | + | 0.238188i | 33.0736 | − | 19.0951i | |||||
46.2 | − | 4.91334i | −4.35312 | − | 7.53982i | −16.1409 | 9.64430 | + | 16.7044i | −37.0457 | + | 21.3883i | 3.49660 | + | 6.05629i | 39.9990i | −24.3993 | + | 42.2608i | 82.0745 | − | 47.3857i | |||||
46.3 | − | 4.82340i | 2.59386 | + | 4.49270i | −15.2652 | −2.46834 | − | 4.27529i | 21.6701 | − | 12.5112i | −11.9912 | − | 20.7694i | 35.0428i | 0.0437764 | − | 0.0758230i | −20.6215 | + | 11.9058i | |||||
46.4 | − | 4.73022i | −2.45405 | − | 4.25053i | −14.3749 | −5.46049 | − | 9.45785i | −20.1059 | + | 11.6082i | 1.27920 | + | 2.21564i | 30.1548i | 1.45530 | − | 2.52066i | −44.7377 | + | 25.8293i | |||||
46.5 | − | 3.81152i | 0.361158 | + | 0.625545i | −6.52767 | 10.5183 | + | 18.2183i | 2.38428 | − | 1.37656i | −11.5289 | − | 19.9686i | − | 5.61183i | 13.2391 | − | 22.9308i | 69.4393 | − | 40.0908i | ||||
46.6 | − | 3.11861i | 3.08348 | + | 5.34074i | −1.72574 | −10.5158 | − | 18.2139i | 16.6557 | − | 9.61617i | 6.20063 | + | 10.7398i | − | 19.5670i | −5.51566 | + | 9.55341i | −56.8020 | + | 32.7947i | ||||
46.7 | − | 2.75335i | −0.992557 | − | 1.71916i | 0.419077 | 0.307591 | + | 0.532763i | −4.73344 | + | 2.73285i | −1.66075 | − | 2.87650i | − | 23.1806i | 11.5297 | − | 19.9700i | 1.46688 | − | 0.846905i | ||||
46.8 | − | 2.63572i | 4.87560 | + | 8.44479i | 1.05299 | 3.33030 | + | 5.76825i | 22.2581 | − | 12.8507i | −3.38317 | − | 5.85982i | − | 23.8611i | −34.0429 | + | 58.9641i | 15.2035 | − | 8.77774i | ||||
46.9 | − | 2.54381i | −0.513075 | − | 0.888672i | 1.52901 | 3.21600 | + | 5.57027i | −2.26062 | + | 1.30517i | 15.2344 | + | 26.3867i | − | 24.2400i | 12.9735 | − | 22.4708i | 14.1697 | − | 8.18091i | ||||
46.10 | − | 2.29881i | −4.53777 | − | 7.85965i | 2.71547 | −0.994942 | − | 1.72329i | −18.0678 | + | 10.4315i | −11.7430 | − | 20.3394i | − | 24.6328i | −27.6827 | + | 47.9478i | −3.96152 | + | 2.28718i | ||||
46.11 | − | 0.570629i | 2.87339 | + | 4.97685i | 7.67438 | 1.27074 | + | 2.20098i | 2.83993 | − | 1.63964i | 8.44184 | + | 14.6217i | − | 8.94425i | −3.01268 | + | 5.21812i | 1.25594 | − | 0.725119i | ||||
46.12 | − | 0.476992i | 1.29609 | + | 2.24489i | 7.77248 | −4.80343 | − | 8.31978i | 1.07079 | − | 0.618223i | −13.4649 | − | 23.3219i | − | 7.52335i | 10.1403 | − | 17.5636i | −3.96847 | + | 2.29120i | ||||
46.13 | − | 0.392313i | −3.20948 | − | 5.55899i | 7.84609 | −10.7878 | − | 18.6850i | −2.18086 | + | 1.25912i | 7.93910 | + | 13.7509i | − | 6.21662i | −7.10158 | + | 12.3003i | −7.33035 | + | 4.23218i | ||||
46.14 | − | 0.00776848i | −3.57258 | − | 6.18789i | 7.99994 | 7.08629 | + | 12.2738i | −0.0480705 | + | 0.0277535i | 8.53919 | + | 14.7903i | − | 0.124295i | −12.0266 | + | 20.8308i | 0.0953489 | − | 0.0550497i | ||||
46.15 | 0.522266i | 2.15477 | + | 3.73217i | 7.72724 | 9.74051 | + | 16.8711i | −1.94919 | + | 1.12536i | −4.20111 | − | 7.27654i | 8.21381i | 4.21393 | − | 7.29875i | −8.81119 | + | 5.08714i | ||||||
46.16 | 1.65255i | −0.440124 | − | 0.762317i | 5.26906 | −3.75307 | − | 6.50051i | 1.25977 | − | 0.727329i | 7.82854 | + | 13.5594i | 21.9278i | 13.1126 | − | 22.7117i | 10.7424 | − | 6.20215i | ||||||
46.17 | 1.70726i | −2.85787 | − | 4.94998i | 5.08526 | 0.376921 | + | 0.652847i | 8.45092 | − | 4.87914i | −11.1601 | − | 19.3298i | 22.3400i | −2.83489 | + | 4.91017i | −1.11458 | + | 0.643504i | ||||||
46.18 | 2.21870i | 3.85635 | + | 6.67940i | 3.07736 | −5.87494 | − | 10.1757i | −14.8196 | + | 8.55610i | 9.25809 | + | 16.0355i | 24.5774i | −16.2429 | + | 28.1335i | 22.5768 | − | 13.0347i | ||||||
46.19 | 2.78760i | −0.912464 | − | 1.58043i | 0.229303 | 5.82500 | + | 10.0892i | 4.40561 | − | 2.54358i | −4.04575 | − | 7.00744i | 22.9400i | 11.8348 | − | 20.4985i | −28.1246 | + | 16.2378i | ||||||
46.20 | 3.35996i | 3.92788 | + | 6.80329i | −3.28930 | 3.56237 | + | 6.17021i | −22.8588 | + | 13.1975i | −7.86115 | − | 13.6159i | 15.8277i | −17.3565 | + | 30.0623i | −20.7316 | + | 11.9694i | ||||||
See all 52 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
109.e | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 109.4.e.a | ✓ | 52 |
109.e | even | 6 | 1 | inner | 109.4.e.a | ✓ | 52 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
109.4.e.a | ✓ | 52 | 1.a | even | 1 | 1 | trivial |
109.4.e.a | ✓ | 52 | 109.e | even | 6 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(109, [\chi])\).