Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [225,3,Mod(13,225)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(225, base_ring=CyclotomicField(60))
chi = DirichletCharacter(H, H._module([20, 57]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("225.13");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 225 = 3^{2} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 225.x (of order \(60\), degree \(16\), 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: | \(6.13080594811\) |
Analytic rank: | \(0\) |
Dimension: | \(928\) |
Relative dimension: | \(58\) over \(\Q(\zeta_{60})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{60}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
13.1 | −2.51131 | − | 3.10121i | −2.37510 | − | 1.83274i | −2.47919 | + | 11.6637i | −4.45894 | − | 2.26226i | 0.280908 | + | 11.9683i | 7.82954 | + | 2.09792i | 28.1752 | − | 14.3560i | 2.28216 | + | 8.70584i | 4.18206 | + | 19.5094i |
13.2 | −2.39531 | − | 2.95797i | 2.99721 | − | 0.129320i | −2.18039 | + | 10.2579i | 0.996446 | − | 4.89970i | −7.56179 | − | 8.55589i | −6.10048 | − | 1.63462i | 21.9999 | − | 11.2095i | 8.96655 | − | 0.775200i | −16.8800 | + | 8.78888i |
13.3 | −2.37770 | − | 2.93621i | 1.06484 | + | 2.80466i | −2.13625 | + | 10.0503i | 1.94075 | + | 4.60798i | 5.70320 | − | 9.79523i | −1.60340 | − | 0.429629i | 21.1234 | − | 10.7629i | −6.73222 | + | 5.97304i | 8.91549 | − | 16.6548i |
13.4 | −2.23416 | − | 2.75896i | 0.755096 | − | 2.90342i | −1.78873 | + | 8.41532i | 0.0648426 | + | 4.99958i | −9.69741 | + | 4.40342i | −8.97463 | − | 2.40474i | 14.5611 | − | 7.41926i | −7.85966 | − | 4.38472i | 13.6488 | − | 11.3488i |
13.5 | −2.06492 | − | 2.54997i | 0.552560 | − | 2.94867i | −1.40678 | + | 6.61838i | 4.95326 | − | 0.682042i | −8.66001 | + | 4.67977i | 8.45676 | + | 2.26598i | 8.08727 | − | 4.12067i | −8.38936 | − | 3.25864i | −11.9673 | − | 11.2223i |
13.6 | −2.03605 | − | 2.51431i | −2.20067 | + | 2.03888i | −1.34462 | + | 6.32594i | −3.36026 | − | 3.70252i | 9.60706 | + | 1.38192i | −13.0248 | − | 3.48998i | 7.11235 | − | 3.62392i | 0.685940 | − | 8.97382i | −2.46763 | + | 15.9873i |
13.7 | −2.01370 | − | 2.48671i | 0.363939 | + | 2.97784i | −1.29711 | + | 6.10241i | 1.07408 | − | 4.88327i | 6.67218 | − | 6.90150i | 9.13770 | + | 2.44844i | 6.38276 | − | 3.25218i | −8.73510 | + | 2.16750i | −14.3062 | + | 7.16254i |
13.8 | −2.00044 | − | 2.47034i | −1.70954 | + | 2.46525i | −1.26917 | + | 5.97096i | −3.88822 | + | 3.14352i | 9.50985 | − | 0.708432i | 4.38514 | + | 1.17500i | 5.96010 | − | 3.03682i | −3.15491 | − | 8.42891i | 15.5437 | + | 3.31679i |
13.9 | −1.95475 | − | 2.41391i | −2.99126 | − | 0.228865i | −1.17428 | + | 5.52456i | 1.21817 | + | 4.84934i | 5.29469 | + | 7.66800i | 0.0185560 | + | 0.00497207i | 4.56092 | − | 2.32390i | 8.89524 | + | 1.36919i | 9.32464 | − | 12.4198i |
13.10 | −1.88803 | − | 2.33152i | −2.53077 | − | 1.61096i | −1.03969 | + | 4.89135i | 4.35459 | − | 2.45714i | 1.02217 | + | 8.94208i | −6.02581 | − | 1.61461i | 2.67477 | − | 1.36287i | 3.80959 | + | 8.15396i | −13.9504 | − | 5.51367i |
13.11 | −1.84346 | − | 2.27648i | 2.55116 | + | 1.57847i | −0.952387 | + | 4.48063i | −4.96038 | + | 0.628232i | −1.10959 | − | 8.71753i | 3.30239 | + | 0.884874i | 1.51571 | − | 0.772291i | 4.01684 | + | 8.05388i | 10.5744 | + | 10.1341i |
13.12 | −1.83098 | − | 2.26107i | 2.04676 | − | 2.19335i | −0.928316 | + | 4.36738i | −4.92462 | − | 0.864912i | −8.70690 | − | 0.611883i | −0.399002 | − | 0.106912i | 1.20532 | − | 0.614139i | −0.621576 | − | 8.97851i | 7.06127 | + | 12.7186i |
13.13 | −1.75378 | − | 2.16573i | 2.99709 | + | 0.131996i | −0.783024 | + | 3.68384i | 4.65885 | + | 1.81526i | −4.97037 | − | 6.72240i | 5.95805 | + | 1.59646i | −0.580693 | + | 0.295878i | 8.96515 | + | 0.791206i | −4.23921 | − | 13.2734i |
13.14 | −1.45990 | − | 1.80282i | 0.453324 | + | 2.96555i | −0.287223 | + | 1.35128i | 4.88955 | − | 1.04512i | 4.68456 | − | 5.14666i | −9.18780 | − | 2.46186i | −5.41239 | + | 2.75775i | −8.58899 | + | 2.68871i | −9.02240 | − | 7.28923i |
13.15 | −1.33618 | − | 1.65005i | −2.89557 | + | 0.784641i | −0.105629 | + | 0.496948i | 0.831842 | − | 4.93032i | 5.16371 | + | 3.72941i | 9.81243 | + | 2.62923i | −6.60607 | + | 3.36596i | 7.76868 | − | 4.54397i | −9.24676 | + | 5.21523i |
13.16 | −1.30275 | − | 1.60876i | −0.852534 | − | 2.87631i | −0.0593127 | + | 0.279044i | −2.09692 | − | 4.53904i | −3.51667 | + | 5.11865i | −1.50124 | − | 0.402256i | −6.85167 | + | 3.49110i | −7.54637 | + | 4.90431i | −4.57048 | + | 9.28670i |
13.17 | −1.21615 | − | 1.50182i | 2.74216 | + | 1.21678i | 0.0552088 | − | 0.259737i | −1.16333 | + | 4.86278i | −1.50749 | − | 5.59800i | −11.2846 | − | 3.02370i | −7.34461 | + | 3.74227i | 6.03889 | + | 6.67321i | 8.71778 | − | 4.16676i |
13.18 | −1.14635 | − | 1.41562i | −1.14916 | − | 2.77118i | 0.141774 | − | 0.666994i | −1.93109 | + | 4.61204i | −2.60560 | + | 4.80352i | 13.0663 | + | 3.50110i | −7.59885 | + | 3.87181i | −6.35886 | + | 6.36906i | 8.74261 | − | 2.55330i |
13.19 | −1.00635 | − | 1.24274i | −1.88326 | + | 2.33524i | 0.299984 | − | 1.41131i | 4.11042 | + | 2.84684i | 4.79732 | − | 0.00967335i | 2.58466 | + | 0.692557i | −7.75506 | + | 3.95140i | −1.90669 | − | 8.79571i | −0.598647 | − | 7.97311i |
13.20 | −0.919820 | − | 1.13588i | 2.48692 | − | 1.67786i | 0.387485 | − | 1.82297i | −0.264765 | + | 4.99299i | −4.19338 | − | 1.28152i | 2.85419 | + | 0.764777i | −7.63631 | + | 3.89090i | 3.36957 | − | 8.34542i | 5.91498 | − | 4.29191i |
See next 80 embeddings (of 928 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.c | even | 3 | 1 | inner |
25.f | odd | 20 | 1 | inner |
225.x | odd | 60 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 225.3.x.a | ✓ | 928 |
9.c | even | 3 | 1 | inner | 225.3.x.a | ✓ | 928 |
25.f | odd | 20 | 1 | inner | 225.3.x.a | ✓ | 928 |
225.x | odd | 60 | 1 | inner | 225.3.x.a | ✓ | 928 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
225.3.x.a | ✓ | 928 | 1.a | even | 1 | 1 | trivial |
225.3.x.a | ✓ | 928 | 9.c | even | 3 | 1 | inner |
225.3.x.a | ✓ | 928 | 25.f | odd | 20 | 1 | inner |
225.3.x.a | ✓ | 928 | 225.x | odd | 60 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(225, [\chi])\).