Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [300,2,Mod(11,300)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(300, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([5, 5, 8]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("300.11");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 300.n (of order \(10\), degree \(4\), 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: | \(2.39551206064\) |
Analytic rank: | \(0\) |
Dimension: | \(224\) |
Relative dimension: | \(56\) over \(\Q(\zeta_{10})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
11.1 | −1.40873 | + | 0.124406i | 0.673811 | − | 1.59561i | 1.96905 | − | 0.350510i | −2.20342 | − | 0.380687i | −0.750715 | + | 2.33161i | − | 0.0822219i | −2.73025 | + | 0.738736i | −2.09196 | − | 2.15028i | 3.15139 | + | 0.262166i | |
11.2 | −1.40679 | + | 0.144680i | −1.60959 | + | 0.639710i | 1.95814 | − | 0.407070i | 2.17904 | + | 0.501802i | 2.17180 | − | 1.13282i | − | 2.67645i | −2.69580 | + | 0.855967i | 2.18154 | − | 2.05934i | −3.13805 | − | 0.390668i | |
11.3 | −1.39445 | + | 0.235585i | −1.04145 | − | 1.38397i | 1.88900 | − | 0.657024i | 1.89530 | − | 1.18653i | 1.77830 | + | 1.68454i | 4.72716i | −2.47934 | + | 1.36121i | −0.830762 | + | 2.88268i | −2.36338 | + | 2.10106i | ||
11.4 | −1.39367 | − | 0.240196i | −0.354401 | + | 1.69541i | 1.88461 | + | 0.669506i | −0.834949 | + | 2.07433i | 0.901146 | − | 2.27770i | 4.41884i | −2.46571 | − | 1.38574i | −2.74880 | − | 1.20171i | 1.66189 | − | 2.69038i | ||
11.5 | −1.38528 | − | 0.284622i | 1.64016 | + | 0.556662i | 1.83798 | + | 0.788561i | 1.45398 | + | 1.69881i | −2.11364 | − | 1.23796i | − | 2.78321i | −2.32167 | − | 1.61550i | 2.38025 | + | 1.82603i | −1.53064 | − | 2.76716i | |
11.6 | −1.31876 | − | 0.510760i | −1.38129 | − | 1.04500i | 1.47825 | + | 1.34714i | −1.62369 | + | 1.53741i | 1.28785 | + | 2.08361i | − | 1.57078i | −1.26139 | − | 2.53158i | 0.815942 | + | 2.88691i | 2.92650 | − | 1.19816i | |
11.7 | −1.29913 | − | 0.558792i | 1.06350 | − | 1.36711i | 1.37550 | + | 1.45189i | 1.51671 | − | 1.64305i | −2.14555 | + | 1.18178i | − | 1.88972i | −0.975659 | − | 2.65482i | −0.737954 | − | 2.90782i | −2.88853 | + | 1.28702i | |
11.8 | −1.25692 | + | 0.648193i | 1.55000 | − | 0.772971i | 1.15969 | − | 1.62945i | 0.411823 | + | 2.19782i | −1.44720 | + | 1.97626i | 3.34784i | −0.401439 | + | 2.79979i | 1.80503 | − | 2.39622i | −1.94224 | − | 2.49554i | ||
11.9 | −1.24841 | + | 0.664442i | 1.66858 | + | 0.464581i | 1.11703 | − | 1.65899i | 0.345709 | − | 2.20918i | −2.39175 | + | 0.528689i | − | 0.309856i | −0.292213 | + | 2.81329i | 2.56833 | + | 1.55038i | 1.03629 | + | 2.98766i | |
11.10 | −1.24461 | + | 0.671529i | −1.27306 | + | 1.17444i | 1.09810 | − | 1.67158i | −2.10329 | − | 0.759058i | 0.795787 | − | 2.31662i | − | 0.738888i | −0.244189 | + | 2.81787i | 0.241361 | − | 2.99028i | 3.12750 | − | 0.467689i | |
11.11 | −1.21555 | − | 0.722800i | −1.65104 | + | 0.523516i | 0.955121 | + | 1.75720i | −0.573135 | − | 2.16137i | 2.38532 | + | 0.557011i | 1.39878i | 0.109105 | − | 2.82632i | 2.45186 | − | 1.72869i | −0.865563 | + | 3.04151i | ||
11.12 | −1.03327 | − | 0.965583i | 0.539210 | + | 1.64598i | 0.135297 | + | 1.99542i | −1.86451 | − | 1.23434i | 1.03218 | − | 2.22140i | − | 3.63805i | 1.78694 | − | 2.19245i | −2.41850 | + | 1.77506i | 0.734688 | + | 3.07575i | |
11.13 | −1.02327 | + | 0.976179i | 0.339606 | + | 1.69843i | 0.0941493 | − | 1.99778i | 2.10329 | + | 0.759058i | −2.00548 | − | 1.40643i | 0.738888i | 1.85385 | + | 2.13617i | −2.76934 | + | 1.15359i | −2.89320 | + | 1.27647i | ||
11.14 | −1.01770 | + | 0.981981i | −1.62298 | − | 0.604914i | 0.0714277 | − | 1.99872i | −0.345709 | + | 2.20918i | 2.24573 | − | 0.978119i | 0.309856i | 1.89002 | + | 2.10424i | 2.26816 | + | 1.96353i | −1.81755 | − | 2.58776i | ||
11.15 | −1.00488 | + | 0.995099i | −0.799639 | − | 1.53642i | 0.0195579 | − | 1.99990i | −0.411823 | − | 2.19782i | 2.33243 | + | 0.748191i | − | 3.34784i | 1.97045 | + | 2.02912i | −1.72115 | + | 2.45716i | 2.60088 | + | 1.79873i | |
11.16 | −0.923787 | − | 1.07080i | 1.20611 | + | 1.24310i | −0.293236 | + | 1.97839i | 2.16410 | − | 0.562738i | 0.216925 | − | 2.43987i | 3.72853i | 2.38935 | − | 1.51361i | −0.0905939 | + | 2.99863i | −2.60175 | − | 1.79747i | ||
11.17 | −0.884246 | − | 1.10368i | 0.846655 | − | 1.51102i | −0.436219 | + | 1.95185i | 0.448288 | + | 2.19067i | −2.41633 | + | 0.401675i | 2.41810i | 2.53994 | − | 1.24447i | −1.56635 | − | 2.55862i | 2.02140 | − | 2.43186i | ||
11.18 | −0.776415 | − | 1.18202i | −0.846655 | + | 1.51102i | −0.794359 | + | 1.83548i | 0.448288 | + | 2.19067i | 2.44341 | − | 0.172411i | − | 2.41810i | 2.78634 | − | 0.486145i | −1.56635 | − | 2.55862i | 2.24137 | − | 2.23076i | |
11.19 | −0.732928 | − | 1.20947i | −1.20611 | − | 1.24310i | −0.925633 | + | 1.77291i | 2.16410 | − | 0.562738i | −0.619499 | + | 2.36986i | − | 3.72853i | 2.82270 | − | 0.179888i | −0.0905939 | + | 2.99863i | −2.26674 | − | 2.20497i | |
11.20 | −0.654964 | + | 1.25340i | 1.65603 | − | 0.507508i | −1.14204 | − | 1.64187i | −1.89530 | + | 1.18653i | −0.448527 | + | 2.40807i | − | 4.72716i | 2.80592 | − | 0.356077i | 2.48487 | − | 1.68090i | −0.245847 | − | 3.15271i | |
See next 80 embeddings (of 224 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
4.b | odd | 2 | 1 | inner |
12.b | even | 2 | 1 | inner |
25.d | even | 5 | 1 | inner |
75.j | odd | 10 | 1 | inner |
100.j | odd | 10 | 1 | inner |
300.n | even | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 300.2.n.a | ✓ | 224 |
3.b | odd | 2 | 1 | inner | 300.2.n.a | ✓ | 224 |
4.b | odd | 2 | 1 | inner | 300.2.n.a | ✓ | 224 |
12.b | even | 2 | 1 | inner | 300.2.n.a | ✓ | 224 |
25.d | even | 5 | 1 | inner | 300.2.n.a | ✓ | 224 |
75.j | odd | 10 | 1 | inner | 300.2.n.a | ✓ | 224 |
100.j | odd | 10 | 1 | inner | 300.2.n.a | ✓ | 224 |
300.n | even | 10 | 1 | inner | 300.2.n.a | ✓ | 224 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
300.2.n.a | ✓ | 224 | 1.a | even | 1 | 1 | trivial |
300.2.n.a | ✓ | 224 | 3.b | odd | 2 | 1 | inner |
300.2.n.a | ✓ | 224 | 4.b | odd | 2 | 1 | inner |
300.2.n.a | ✓ | 224 | 12.b | even | 2 | 1 | inner |
300.2.n.a | ✓ | 224 | 25.d | even | 5 | 1 | inner |
300.2.n.a | ✓ | 224 | 75.j | odd | 10 | 1 | inner |
300.2.n.a | ✓ | 224 | 100.j | odd | 10 | 1 | inner |
300.2.n.a | ✓ | 224 | 300.n | even | 10 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(300, [\chi])\).