Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [169,2,Mod(12,169)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(169, base_ring=CyclotomicField(26))
chi = DirichletCharacter(H, H._module([21]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("169.12");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 169 = 13^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 169.h (of order \(26\), degree \(12\), 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: | \(1.34947179416\) |
Analytic rank: | \(0\) |
Dimension: | \(168\) |
Relative dimension: | \(14\) over \(\Q(\zeta_{26})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{26}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
12.1 | −2.11580 | + | 1.46043i | 0.527540 | + | 0.276874i | 1.63454 | − | 4.30994i | 0.267437 | + | 0.301873i | −1.52053 | + | 0.184625i | −1.09769 | − | 4.45349i | 1.60549 | + | 6.51374i | −1.50256 | − | 2.17683i | −1.00671 | − | 0.248131i |
12.2 | −2.08715 | + | 1.44065i | −1.12364 | − | 0.589730i | 1.57149 | − | 4.14367i | −2.16703 | − | 2.44607i | 3.19479 | − | 0.387918i | 1.08188 | + | 4.38937i | 1.47582 | + | 5.98765i | −0.789415 | − | 1.14367i | 8.04684 | + | 1.98337i |
12.3 | −1.71806 | + | 1.18589i | 2.94692 | + | 1.54666i | 0.836178 | − | 2.20482i | −0.781452 | − | 0.882078i | −6.89716 | + | 0.837466i | 0.664295 | + | 2.69515i | 0.178883 | + | 0.725755i | 4.58799 | + | 6.64685i | 2.38863 | + | 0.588744i |
12.4 | −1.24935 | + | 0.862366i | −1.71162 | − | 0.898326i | 0.107998 | − | 0.284768i | 0.371060 | + | 0.418840i | 2.91310 | − | 0.353715i | −0.356102 | − | 1.44476i | −0.615953 | − | 2.49902i | 0.418452 | + | 0.606232i | −0.824778 | − | 0.203290i |
12.5 | −1.19961 | + | 0.828028i | 0.960584 | + | 0.504153i | 0.0442133 | − | 0.116581i | 2.00541 | + | 2.26364i | −1.56977 | + | 0.190605i | 0.228271 | + | 0.926132i | −0.654173 | − | 2.65409i | −1.03564 | − | 1.50039i | −4.28006 | − | 1.05494i |
12.6 | −0.705316 | + | 0.486844i | −1.15919 | − | 0.608391i | −0.448757 | + | 1.18327i | −0.816038 | − | 0.921117i | 1.11379 | − | 0.135238i | 0.328944 | + | 1.33458i | −0.669753 | − | 2.71730i | −0.730609 | − | 1.05847i | 1.02401 | + | 0.252395i |
12.7 | 0.156578 | − | 0.108078i | 2.47375 | + | 1.29833i | −0.696374 | + | 1.83619i | 0.268145 | + | 0.302673i | 0.527655 | − | 0.0640690i | −0.983215 | − | 3.98906i | 0.180477 | + | 0.732225i | 2.72961 | + | 3.95452i | 0.0746977 | + | 0.0184113i |
12.8 | 0.197515 | − | 0.136335i | 1.40158 | + | 0.735608i | −0.688785 | + | 1.81618i | −0.369290 | − | 0.416842i | 0.377124 | − | 0.0457911i | 0.397115 | + | 1.61116i | 0.226434 | + | 0.918678i | −0.280874 | − | 0.406917i | −0.129771 | − | 0.0319856i |
12.9 | 0.323126 | − | 0.223038i | −2.84614 | − | 1.49377i | −0.654545 | + | 1.72589i | 1.31816 | + | 1.48790i | −1.25283 | + | 0.152121i | 0.0500939 | + | 0.203239i | 0.361363 | + | 1.46611i | 4.16496 | + | 6.03399i | 0.757790 | + | 0.186779i |
12.10 | 0.651938 | − | 0.450000i | −1.14686 | − | 0.601918i | −0.486687 | + | 1.28329i | −2.65258 | − | 2.99414i | −1.01854 | + | 0.123674i | −0.815660 | − | 3.30926i | 0.639345 | + | 2.59392i | −0.751216 | − | 1.08832i | −3.07668 | − | 0.758333i |
12.11 | 1.05649 | − | 0.729245i | −0.277577 | − | 0.145684i | −0.124830 | + | 0.329149i | 2.67487 | + | 3.01931i | −0.399497 | + | 0.0485078i | 0.147281 | + | 0.597543i | 0.722584 | + | 2.93164i | −1.64837 | − | 2.38807i | 5.02780 | + | 1.23924i |
12.12 | 1.62113 | − | 1.11899i | 0.214934 | + | 0.112806i | 0.666728 | − | 1.75802i | 0.457146 | + | 0.516011i | 0.474664 | − | 0.0576347i | −0.520249 | − | 2.11073i | 0.0564755 | + | 0.229130i | −1.67072 | − | 2.42046i | 1.31850 | + | 0.324982i |
12.13 | 1.63772 | − | 1.13043i | 1.71106 | + | 0.898032i | 0.695024 | − | 1.83263i | −2.41202 | − | 2.72261i | 3.81739 | − | 0.463516i | 0.903389 | + | 3.66519i | 0.0190509 | + | 0.0772924i | 0.417062 | + | 0.604219i | −7.02794 | − | 1.73223i |
12.14 | 2.07618 | − | 1.43308i | −1.76851 | − | 0.928185i | 1.54757 | − | 4.08062i | −0.134767 | − | 0.152121i | −5.00190 | + | 0.607340i | 0.524392 | + | 2.12754i | −1.42736 | − | 5.79101i | 0.561902 | + | 0.814056i | −0.497802 | − | 0.122697i |
25.1 | −2.49582 | − | 0.946540i | −0.730291 | − | 1.05801i | 3.83616 | + | 3.39854i | −0.453685 | − | 0.0550874i | 0.821227 | + | 3.33185i | −1.38308 | − | 2.63524i | −3.87657 | − | 7.38618i | 0.477757 | − | 1.25974i | 1.08017 | + | 0.566920i |
25.2 | −2.16507 | − | 0.821103i | 1.73033 | + | 2.50681i | 2.51629 | + | 2.22924i | 3.34407 | + | 0.406044i | −1.68793 | − | 6.84821i | −0.858950 | − | 1.63659i | −1.46534 | − | 2.79198i | −2.22627 | + | 5.87018i | −6.90674 | − | 3.62494i |
25.3 | −1.68720 | − | 0.639871i | 0.0407127 | + | 0.0589826i | 0.940192 | + | 0.832938i | 0.654244 | + | 0.0794397i | −0.0309493 | − | 0.125566i | −0.153151 | − | 0.291806i | 0.623830 | + | 1.18861i | 1.06199 | − | 2.80025i | −1.05301 | − | 0.552663i |
25.4 | −1.52590 | − | 0.578696i | −1.45490 | − | 2.10779i | 0.496446 | + | 0.439813i | −3.46589 | − | 0.420835i | 1.00026 | + | 4.05821i | 0.131788 | + | 0.251102i | 1.01380 | + | 1.93163i | −1.26222 | + | 3.32819i | 5.04505 | + | 2.64785i |
25.5 | −1.49527 | − | 0.567082i | 1.18578 | + | 1.71790i | 0.417236 | + | 0.369639i | −2.96326 | − | 0.359805i | −0.798874 | − | 3.24116i | 1.24562 | + | 2.37332i | 1.07210 | + | 2.04271i | −0.481287 | + | 1.26905i | 4.22684 | + | 2.21842i |
25.6 | −0.622921 | − | 0.236243i | −1.59376 | − | 2.30896i | −1.16480 | − | 1.03192i | 3.47190 | + | 0.421565i | 0.447311 | + | 1.81481i | −1.63679 | − | 3.11863i | 1.10100 | + | 2.09779i | −1.72740 | + | 4.55479i | −2.06313 | − | 1.08281i |
See next 80 embeddings (of 168 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
169.h | even | 26 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 169.2.h.a | ✓ | 168 |
169.h | even | 26 | 1 | inner | 169.2.h.a | ✓ | 168 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
169.2.h.a | ✓ | 168 | 1.a | even | 1 | 1 | trivial |
169.2.h.a | ✓ | 168 | 169.h | even | 26 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(169, [\chi])\).