Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [209,2,Mod(26,209)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(209, base_ring=CyclotomicField(30))
chi = DirichletCharacter(H, H._module([6, 10]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("209.26");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 209 = 11 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 209.n (of order \(15\), degree \(8\), 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: | \(1.66887340224\) |
Analytic rank: | \(0\) |
Dimension: | \(144\) |
Relative dimension: | \(18\) over \(\Q(\zeta_{15})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{15}]$ |
$q$-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 | −0.271157 | + | 2.57989i | 2.55309 | − | 0.542675i | −4.62600 | − | 0.983287i | −0.178862 | − | 0.0796343i | 0.707754 | + | 6.73383i | 0.955739 | + | 2.94146i | 2.18790 | − | 6.73367i | 3.48311 | − | 1.55078i | 0.253947 | − | 0.439850i |
26.2 | −0.231976 | + | 2.20710i | −0.373871 | + | 0.0794688i | −2.86120 | − | 0.608166i | −2.71951 | − | 1.21080i | −0.0886667 | − | 0.843607i | −0.383306 | − | 1.17970i | 0.634435 | − | 1.95259i | −2.60717 | + | 1.16079i | 3.30323 | − | 5.72135i |
26.3 | −0.202973 | + | 1.93115i | −0.843003 | + | 0.179186i | −1.73186 | − | 0.368119i | 2.72243 | + | 1.21210i | −0.174929 | − | 1.66434i | −0.0464276 | − | 0.142890i | −0.137677 | + | 0.423727i | −2.06209 | + | 0.918101i | −2.89333 | + | 5.01140i |
26.4 | −0.186518 | + | 1.77460i | 1.74061 | − | 0.369977i | −1.15814 | − | 0.246170i | 1.27150 | + | 0.566107i | 0.331908 | + | 3.15790i | −0.603916 | − | 1.85866i | −0.449939 | + | 1.38477i | 0.152194 | − | 0.0677611i | −1.24177 | + | 2.15081i |
26.5 | −0.125995 | + | 1.19876i | −2.49184 | + | 0.529658i | 0.535145 | + | 0.113749i | 1.68116 | + | 0.748502i | −0.320973 | − | 3.05386i | 0.653590 | + | 2.01154i | −0.948738 | + | 2.91991i | 3.18811 | − | 1.41944i | −1.10909 | + | 1.92100i |
26.6 | −0.0795584 | + | 0.756948i | −2.48657 | + | 0.528537i | 1.38965 | + | 0.295380i | −2.02530 | − | 0.901724i | −0.202247 | − | 1.92425i | −1.58873 | − | 4.88961i | −0.804543 | + | 2.47613i | 3.16305 | − | 1.40828i | 0.843688 | − | 1.46131i |
26.7 | −0.0646268 | + | 0.614883i | 1.13244 | − | 0.240707i | 1.58239 | + | 0.336348i | −1.68382 | − | 0.749683i | 0.0748209 | + | 0.711874i | 1.36446 | + | 4.19938i | −0.691191 | + | 2.12727i | −1.51616 | + | 0.675037i | 0.569787 | − | 0.986900i |
26.8 | −0.0551358 | + | 0.524582i | 3.09393 | − | 0.657636i | 1.68415 | + | 0.357977i | −3.18113 | − | 1.41633i | 0.174397 | + | 1.65928i | −0.546190 | − | 1.68100i | −0.606640 | + | 1.86705i | 6.39930 | − | 2.84915i | 0.918374 | − | 1.59067i |
26.9 | −0.00441207 | + | 0.0419780i | 0.931314 | − | 0.197957i | 1.95455 | + | 0.415453i | 1.90793 | + | 0.849463i | 0.00420082 | + | 0.0399681i | −0.431752 | − | 1.32880i | −0.0521503 | + | 0.160502i | −1.91248 | + | 0.851490i | −0.0440767 | + | 0.0763431i |
26.10 | 0.0181387 | − | 0.172578i | −1.78742 | + | 0.379928i | 1.92684 | + | 0.409563i | 0.351649 | + | 0.156564i | 0.0331458 | + | 0.315361i | 0.447117 | + | 1.37609i | 0.212878 | − | 0.655172i | 0.309894 | − | 0.137973i | 0.0333980 | − | 0.0578471i |
26.11 | 0.0990379 | − | 0.942283i | 1.14568 | − | 0.243521i | 1.07821 | + | 0.229180i | −0.566827 | − | 0.252368i | −0.116000 | − | 1.10367i | −0.602230 | − | 1.85347i | 0.908306 | − | 2.79548i | −1.48736 | + | 0.662217i | −0.293939 | + | 0.509117i |
26.12 | 0.128376 | − | 1.22142i | −1.54616 | + | 0.328647i | 0.480911 | + | 0.102221i | −3.68103 | − | 1.63890i | 0.202925 | + | 1.93071i | 0.194103 | + | 0.597387i | 0.945629 | − | 2.91035i | −0.458021 | + | 0.203924i | −2.47434 | + | 4.28568i |
26.13 | 0.176740 | − | 1.68157i | −0.842424 | + | 0.179063i | −0.840131 | − | 0.178575i | 3.33203 | + | 1.48352i | 0.152216 | + | 1.44824i | 1.34243 | + | 4.13157i | 0.596219 | − | 1.83497i | −2.06302 | + | 0.918516i | 3.08353 | − | 5.34083i |
26.14 | 0.186739 | − | 1.77670i | −1.49480 | + | 0.317729i | −1.16551 | − | 0.247736i | 1.77405 | + | 0.789857i | 0.285373 | + | 2.71515i | −1.40791 | − | 4.33309i | 0.446312 | − | 1.37361i | −0.607163 | + | 0.270327i | 1.73463 | − | 3.00446i |
26.15 | 0.200810 | − | 1.91058i | 2.50157 | − | 0.531725i | −1.65369 | − | 0.351503i | −1.25111 | − | 0.557030i | −0.513562 | − | 4.88622i | 1.08189 | + | 3.32973i | 0.183655 | − | 0.565233i | 3.23447 | − | 1.44008i | −1.31548 | + | 2.27849i |
26.16 | 0.244737 | − | 2.32852i | 0.813081 | − | 0.172826i | −3.40581 | − | 0.723927i | −2.63499 | − | 1.17317i | −0.203437 | − | 1.93557i | −0.397027 | − | 1.22192i | −1.07217 | + | 3.29981i | −2.10940 | + | 0.939167i | −3.37663 | + | 5.84850i |
26.17 | 0.263895 | − | 2.51079i | −3.12628 | + | 0.664512i | −4.27814 | − | 0.909348i | −0.536860 | − | 0.239025i | 0.843441 | + | 8.02480i | 0.0749086 | + | 0.230545i | −1.85186 | + | 5.69944i | 6.59142 | − | 2.93469i | −0.741818 | + | 1.28487i |
26.18 | 0.282066 | − | 2.68368i | 2.05882 | − | 0.437617i | −5.16626 | − | 1.09812i | 3.37594 | + | 1.50307i | −0.593697 | − | 5.64865i | −0.679707 | − | 2.09192i | −2.73649 | + | 8.42205i | 1.30661 | − | 0.581741i | 4.98598 | − | 8.63598i |
49.1 | −2.54591 | − | 0.541151i | −0.0636153 | + | 0.0283233i | 4.36175 | + | 1.94197i | 1.21638 | + | 1.35093i | 0.177286 | − | 0.0376833i | 1.30543 | − | 0.948451i | −5.84232 | − | 4.24470i | −2.00415 | + | 2.22583i | −2.36575 | − | 4.09760i |
49.2 | −2.34811 | − | 0.499107i | 1.50416 | − | 0.669694i | 3.43743 | + | 1.53044i | −2.36262 | − | 2.62396i | −3.86618 | + | 0.821782i | −3.20224 | + | 2.32656i | −3.42342 | − | 2.48726i | −0.193390 | + | 0.214781i | 4.23807 | + | 7.34055i |
See next 80 embeddings (of 144 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.c | even | 5 | 1 | inner |
19.c | even | 3 | 1 | inner |
209.n | even | 15 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 209.2.n.a | ✓ | 144 |
11.c | even | 5 | 1 | inner | 209.2.n.a | ✓ | 144 |
19.c | even | 3 | 1 | inner | 209.2.n.a | ✓ | 144 |
209.n | even | 15 | 1 | inner | 209.2.n.a | ✓ | 144 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
209.2.n.a | ✓ | 144 | 1.a | even | 1 | 1 | trivial |
209.2.n.a | ✓ | 144 | 11.c | even | 5 | 1 | inner |
209.2.n.a | ✓ | 144 | 19.c | even | 3 | 1 | inner |
209.2.n.a | ✓ | 144 | 209.n | even | 15 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(209, [\chi])\).