Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [403,2,Mod(15,403)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(403, base_ring=CyclotomicField(60))
chi = DirichletCharacter(H, H._module([5, 42]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("403.15");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 403 = 13 \cdot 31 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 403.cb (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: | \(3.21797120146\) |
Analytic rank: | \(0\) |
Dimension: | \(576\) |
Relative dimension: | \(36\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
15.1 | −2.56739 | + | 0.985528i | −0.915361 | − | 0.0962083i | 4.13393 | − | 3.72220i | 0.706575 | − | 0.706575i | 2.44490 | − | 0.655110i | −0.0270244 | + | 0.515657i | −4.44807 | + | 8.72982i | −2.10581 | − | 0.447604i | −1.11770 | + | 2.51040i |
15.2 | −2.42084 | + | 0.929275i | 2.30609 | + | 0.242379i | 3.51064 | − | 3.16099i | −2.31557 | + | 2.31557i | −5.80791 | + | 1.55622i | 0.246948 | − | 4.71205i | −3.20682 | + | 6.29373i | 2.32484 | + | 0.494160i | 3.45383 | − | 7.75743i |
15.3 | −2.31845 | + | 0.889969i | 1.00099 | + | 0.105208i | 3.09687 | − | 2.78844i | −0.241939 | + | 0.241939i | −2.41437 | + | 0.646928i | −0.164948 | + | 3.14739i | −2.44344 | + | 4.79553i | −1.94354 | − | 0.413111i | 0.345605 | − | 0.776242i |
15.4 | −2.22333 | + | 0.853455i | 2.95877 | + | 0.310980i | 2.72851 | − | 2.45676i | 3.03274 | − | 3.03274i | −6.84373 | + | 1.83377i | 0.0323199 | − | 0.616700i | −1.80727 | + | 3.54696i | 5.72319 | + | 1.21650i | −4.15446 | + | 9.33107i |
15.5 | −2.18231 | + | 0.837709i | −1.82986 | − | 0.192326i | 2.57441 | − | 2.31801i | 2.00072 | − | 2.00072i | 4.15442 | − | 1.11317i | 0.230296 | − | 4.39430i | −1.55387 | + | 3.04965i | 0.376947 | + | 0.0801226i | −2.69016 | + | 6.04221i |
15.6 | −2.13502 | + | 0.819558i | −2.70347 | − | 0.284147i | 2.40035 | − | 2.16129i | −1.50038 | + | 1.50038i | 6.00485 | − | 1.60899i | −0.159342 | + | 3.04042i | −1.27702 | + | 2.50630i | 4.29359 | + | 0.912632i | 1.97370 | − | 4.43299i |
15.7 | −1.74124 | + | 0.668401i | 3.03251 | + | 0.318730i | 1.09888 | − | 0.989437i | −1.90134 | + | 1.90134i | −5.49338 | + | 1.47195i | −0.213728 | + | 4.07817i | 0.441419 | − | 0.866333i | 6.16010 | + | 1.30937i | 2.03984 | − | 4.58157i |
15.8 | −1.73427 | + | 0.665722i | −2.17361 | − | 0.228456i | 1.07820 | − | 0.970817i | 2.39448 | − | 2.39448i | 3.92171 | − | 1.05082i | −0.177207 | + | 3.38131i | 0.463117 | − | 0.908919i | 1.73795 | + | 0.369412i | −2.55860 | + | 5.74672i |
15.9 | −1.67538 | + | 0.643119i | 0.0849006 | + | 0.00892341i | 0.907016 | − | 0.816681i | −1.60006 | + | 1.60006i | −0.147980 | + | 0.0396511i | −0.0146284 | + | 0.279127i | 0.635067 | − | 1.24639i | −2.92731 | − | 0.622220i | 1.65168 | − | 3.70974i |
15.10 | −1.64926 | + | 0.633093i | 0.350555 | + | 0.0368448i | 0.832971 | − | 0.750010i | 0.166326 | − | 0.166326i | −0.601483 | + | 0.161167i | 0.0307024 | − | 0.585837i | 0.705077 | − | 1.38379i | −2.81291 | − | 0.597903i | −0.169015 | + | 0.379614i |
15.11 | −1.50027 | + | 0.575900i | −1.72223 | − | 0.181014i | 0.432864 | − | 0.389753i | −2.07725 | + | 2.07725i | 2.68806 | − | 0.720264i | 0.206300 | − | 3.93644i | 1.03418 | − | 2.02969i | −0.00112424 | 0.000238966i | 1.92015 | − | 4.31273i | |
15.12 | −1.28262 | + | 0.492351i | 1.16310 | + | 0.122247i | −0.0835907 | + | 0.0752654i | 2.24725 | − | 2.24725i | −1.55200 | + | 0.415857i | −0.137685 | + | 2.62719i | 1.31761 | − | 2.58595i | −1.59659 | − | 0.339365i | −1.77593 | + | 3.98880i |
15.13 | −1.00504 | + | 0.385798i | 2.51527 | + | 0.264366i | −0.625026 | + | 0.562776i | 0.441644 | − | 0.441644i | −2.62994 | + | 0.704690i | 0.134189 | − | 2.56048i | 1.38854 | − | 2.72516i | 3.32227 | + | 0.706170i | −0.273484 | + | 0.614254i |
15.14 | −0.857856 | + | 0.329300i | −2.24412 | − | 0.235866i | −0.858811 | + | 0.773277i | 1.02734 | − | 1.02734i | 2.00280 | − | 0.536649i | 0.00452335 | − | 0.0863108i | 1.31643 | − | 2.58364i | 2.04600 | + | 0.434890i | −0.543006 | + | 1.21961i |
15.15 | −0.632627 | + | 0.242843i | −2.36395 | − | 0.248461i | −1.14505 | + | 1.03100i | −1.89446 | + | 1.89446i | 1.55583 | − | 0.416884i | −0.154670 | + | 2.95128i | 1.08929 | − | 2.13786i | 2.59206 | + | 0.550960i | 0.738430 | − | 1.65854i |
15.16 | −0.448407 | + | 0.172127i | 0.358465 | + | 0.0376761i | −1.31485 | + | 1.18390i | 2.11069 | − | 2.11069i | −0.167223 | + | 0.0448073i | 0.209190 | − | 3.99157i | 0.821917 | − | 1.61310i | −2.80737 | − | 0.596724i | −0.583140 | + | 1.30975i |
15.17 | −0.378014 | + | 0.145106i | 1.58507 | + | 0.166597i | −1.36445 | + | 1.22856i | −2.82295 | + | 2.82295i | −0.623351 | + | 0.167027i | 0.00967852 | − | 0.184677i | 0.705159 | − | 1.38395i | −0.449762 | − | 0.0956000i | 0.657488 | − | 1.47674i |
15.18 | −0.0732518 | + | 0.0281187i | −0.541154 | − | 0.0568776i | −1.48171 | + | 1.33414i | 0.197759 | − | 0.197759i | 0.0412399 | − | 0.0110502i | −0.150499 | + | 2.87169i | 0.142267 | − | 0.279215i | −2.64483 | − | 0.562176i | −0.00892549 | + | 0.0200470i |
15.19 | 0.0302615 | − | 0.0116163i | 2.11154 | + | 0.221931i | −1.48551 | + | 1.33756i | −0.761137 | + | 0.761137i | 0.0664762 | − | 0.0178122i | −0.0268833 | + | 0.512964i | −0.0588479 | + | 0.115496i | 1.47489 | + | 0.313497i | −0.0141915 | + | 0.0318747i |
15.20 | 0.185296 | − | 0.0711283i | 3.02116 | + | 0.317537i | −1.45701 | + | 1.31190i | 0.944103 | − | 0.944103i | 0.582394 | − | 0.156052i | −0.137953 | + | 2.63230i | −0.356880 | + | 0.700416i | 6.09216 | + | 1.29493i | 0.107786 | − | 0.242091i |
See next 80 embeddings (of 576 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.f | odd | 12 | 1 | inner |
31.f | odd | 10 | 1 | inner |
403.cb | even | 60 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 403.2.cb.a | ✓ | 576 |
13.f | odd | 12 | 1 | inner | 403.2.cb.a | ✓ | 576 |
31.f | odd | 10 | 1 | inner | 403.2.cb.a | ✓ | 576 |
403.cb | even | 60 | 1 | inner | 403.2.cb.a | ✓ | 576 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
403.2.cb.a | ✓ | 576 | 1.a | even | 1 | 1 | trivial |
403.2.cb.a | ✓ | 576 | 13.f | odd | 12 | 1 | inner |
403.2.cb.a | ✓ | 576 | 31.f | odd | 10 | 1 | inner |
403.2.cb.a | ✓ | 576 | 403.cb | even | 60 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(403, [\chi])\).