Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [418,2,Mod(151,418)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(418, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([3, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("418.151");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 418 = 2 \cdot 11 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 418.m (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: | \(3.33774680449\) |
Analytic rank: | \(0\) |
Dimension: | \(40\) |
Relative dimension: | \(10\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
151.1 | −0.809017 | + | 0.587785i | −2.71373 | + | 0.881745i | 0.309017 | − | 0.951057i | −1.18335 | − | 0.859758i | 1.67718 | − | 2.30844i | 3.23095 | + | 1.04980i | 0.309017 | + | 0.951057i | 4.15981 | − | 3.02228i | 1.46271 | ||
151.2 | −0.809017 | + | 0.587785i | −2.42317 | + | 0.787336i | 0.309017 | − | 0.951057i | 1.61580 | + | 1.17395i | 1.49760 | − | 2.06127i | −2.37273 | − | 0.770947i | 0.309017 | + | 0.951057i | 2.82481 | − | 2.05234i | −1.99724 | ||
151.3 | −0.809017 | + | 0.587785i | −2.05378 | + | 0.667313i | 0.309017 | − | 0.951057i | −2.88627 | − | 2.09700i | 1.26930 | − | 1.74705i | −3.81661 | − | 1.24009i | 0.309017 | + | 0.951057i | 1.34564 | − | 0.977666i | 3.56763 | ||
151.4 | −0.809017 | + | 0.587785i | −1.03661 | + | 0.336815i | 0.309017 | − | 0.951057i | −1.83162 | − | 1.33075i | 0.640660 | − | 0.881793i | 3.63997 | + | 1.18270i | 0.309017 | + | 0.951057i | −1.46594 | + | 1.06506i | 2.26401 | ||
151.5 | −0.809017 | + | 0.587785i | −0.809704 | + | 0.263089i | 0.309017 | − | 0.951057i | 2.31139 | + | 1.67932i | 0.500425 | − | 0.688776i | 1.24718 | + | 0.405232i | 0.309017 | + | 0.951057i | −1.84065 | + | 1.33731i | −2.85703 | ||
151.6 | −0.809017 | + | 0.587785i | 0.819072 | − | 0.266133i | 0.309017 | − | 0.951057i | 2.09583 | + | 1.52271i | −0.506214 | + | 0.696744i | 3.24582 | + | 1.05463i | 0.309017 | + | 0.951057i | −1.82700 | + | 1.32739i | −2.59059 | ||
151.7 | −0.809017 | + | 0.587785i | 0.871743 | − | 0.283246i | 0.309017 | − | 0.951057i | −1.71497 | − | 1.24600i | −0.538767 | + | 0.741549i | −1.29565 | − | 0.420982i | 0.309017 | + | 0.951057i | −1.74734 | + | 1.26952i | 2.11982 | ||
151.8 | −0.809017 | + | 0.587785i | 1.02923 | − | 0.334417i | 0.309017 | − | 0.951057i | −0.0854069 | − | 0.0620518i | −0.636098 | + | 0.875514i | −4.39704 | − | 1.42869i | 0.309017 | + | 0.951057i | −1.47957 | + | 1.07497i | 0.105569 | ||
151.9 | −0.809017 | + | 0.587785i | 2.13440 | − | 0.693509i | 0.309017 | − | 0.951057i | −0.363676 | − | 0.264226i | −1.31913 | + | 1.81563i | 1.26293 | + | 0.410350i | 0.309017 | + | 0.951057i | 1.64766 | − | 1.19709i | 0.449528 | ||
151.10 | −0.809017 | + | 0.587785i | 3.06452 | − | 0.995722i | 0.309017 | − | 0.951057i | 2.54228 | + | 1.84707i | −1.89398 | + | 2.60683i | −2.55384 | − | 0.829793i | 0.309017 | + | 0.951057i | 5.97275 | − | 4.33946i | −3.14243 | ||
189.1 | 0.309017 | − | 0.951057i | −1.52806 | − | 2.10319i | −0.809017 | − | 0.587785i | 0.852902 | + | 2.62496i | −2.47245 | + | 0.803349i | −1.95228 | + | 2.68709i | −0.809017 | + | 0.587785i | −1.16141 | + | 3.57445i | 2.76005 | ||
189.2 | 0.309017 | − | 0.951057i | −1.24511 | − | 1.71374i | −0.809017 | − | 0.587785i | −0.810373 | − | 2.49407i | −2.01463 | + | 0.654592i | 2.23258 | − | 3.07288i | −0.809017 | + | 0.587785i | −0.459575 | + | 1.41443i | −2.62242 | ||
189.3 | 0.309017 | − | 0.951057i | −0.943373 | − | 1.29844i | −0.809017 | − | 0.587785i | −0.757060 | − | 2.32999i | −1.52641 | + | 0.495961i | −2.75444 | + | 3.79116i | −0.809017 | + | 0.587785i | 0.131053 | − | 0.403339i | −2.44990 | ||
189.4 | 0.309017 | − | 0.951057i | −0.775778 | − | 1.06777i | −0.809017 | − | 0.587785i | 1.01291 | + | 3.11743i | −1.25524 | + | 0.407851i | 1.17374 | − | 1.61551i | −0.809017 | + | 0.587785i | 0.388756 | − | 1.19647i | 3.27786 | ||
189.5 | 0.309017 | − | 0.951057i | −0.366195 | − | 0.504024i | −0.809017 | − | 0.587785i | −0.493630 | − | 1.51924i | −0.592516 | + | 0.192520i | 0.534358 | − | 0.735481i | −0.809017 | + | 0.587785i | 0.807109 | − | 2.48403i | −1.59742 | ||
189.6 | 0.309017 | − | 0.951057i | 0.238794 | + | 0.328671i | −0.809017 | − | 0.587785i | 0.255338 | + | 0.785850i | 0.386376 | − | 0.125541i | 0.299067 | − | 0.411631i | −0.809017 | + | 0.587785i | 0.876049 | − | 2.69620i | 0.826292 | ||
189.7 | 0.309017 | − | 0.951057i | 0.897714 | + | 1.23560i | −0.809017 | − | 0.587785i | 1.27104 | + | 3.91184i | 1.45253 | − | 0.471956i | −0.757296 | + | 1.04233i | −0.809017 | + | 0.587785i | 0.206241 | − | 0.634743i | 4.11316 | ||
189.8 | 0.309017 | − | 0.951057i | 1.43219 | + | 1.97124i | −0.809017 | − | 0.587785i | −1.36407 | − | 4.19816i | 2.31733 | − | 0.752945i | 0.381062 | − | 0.524486i | −0.809017 | + | 0.587785i | −0.907563 | + | 2.79319i | −4.41421 | ||
189.9 | 0.309017 | − | 0.951057i | 1.43308 | + | 1.97246i | −0.809017 | − | 0.587785i | 0.0275048 | + | 0.0846511i | 2.31877 | − | 0.753413i | −2.13510 | + | 2.93871i | −0.809017 | + | 0.587785i | −0.909840 | + | 2.80020i | 0.0890075 | ||
189.10 | 0.309017 | − | 0.951057i | 1.97478 | + | 2.71805i | −0.809017 | − | 0.587785i | 0.505434 | + | 1.55557i | 3.19526 | − | 1.03820i | 2.28733 | − | 3.14824i | −0.809017 | + | 0.587785i | −2.56099 | + | 7.88192i | 1.63562 | ||
See all 40 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
209.k | even | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 418.2.m.a | ✓ | 40 |
11.d | odd | 10 | 1 | 418.2.m.b | yes | 40 | |
19.b | odd | 2 | 1 | 418.2.m.b | yes | 40 | |
209.k | even | 10 | 1 | inner | 418.2.m.a | ✓ | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
418.2.m.a | ✓ | 40 | 1.a | even | 1 | 1 | trivial |
418.2.m.a | ✓ | 40 | 209.k | even | 10 | 1 | inner |
418.2.m.b | yes | 40 | 11.d | odd | 10 | 1 | |
418.2.m.b | yes | 40 | 19.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{40} - 19 T_{3}^{38} + 25 T_{3}^{37} + 252 T_{3}^{36} - 475 T_{3}^{35} - 2584 T_{3}^{34} + \cdots + 65610000 \) acting on \(S_{2}^{\mathrm{new}}(418, [\chi])\).