Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [525,2,Mod(131,525)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(525, base_ring=CyclotomicField(30))
chi = DirichletCharacter(H, H._module([15, 12, 25]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("525.131");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 525 = 3 \cdot 5^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 525.bm (of order \(30\), 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: | \(4.19214610612\) |
Analytic rank: | \(0\) |
Dimension: | \(608\) |
Relative dimension: | \(76\) over \(\Q(\zeta_{30})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{30}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
131.1 | −0.578671 | + | 2.72243i | 1.50155 | + | 0.863338i | −5.24969 | − | 2.33731i | 2.12367 | + | 0.700010i | −3.21928 | + | 3.58828i | 2.47291 | + | 0.940586i | 6.12911 | − | 8.43599i | 1.50929 | + | 2.59269i | −3.13464 | + | 5.37648i |
131.2 | −0.572271 | + | 2.69232i | −1.59928 | + | 0.665065i | −5.09402 | − | 2.26800i | −1.14863 | + | 1.91850i | −0.875350 | − | 4.68637i | −2.63822 | − | 0.199474i | 5.78563 | − | 7.96324i | 2.11538 | − | 2.12725i | −4.50790 | − | 4.19039i |
131.3 | −0.548566 | + | 2.58080i | 1.24592 | − | 1.20320i | −4.53252 | − | 2.01801i | −1.83369 | + | 1.27969i | 2.42176 | + | 3.87550i | 1.29192 | − | 2.30888i | 4.59278 | − | 6.32141i | 0.104613 | − | 2.99818i | −2.29672 | − | 5.43437i |
131.4 | −0.540793 | + | 2.54423i | −1.70841 | − | 0.285183i | −4.35356 | − | 1.93833i | 0.488675 | − | 2.18202i | 1.64947 | − | 4.19237i | 1.22413 | + | 2.34553i | 4.22820 | − | 5.81962i | 2.83734 | + | 0.974421i | 5.28728 | + | 2.42332i |
131.5 | −0.530626 | + | 2.49640i | −0.257292 | + | 1.71283i | −4.12336 | − | 1.83584i | 1.58898 | − | 1.57326i | −4.13939 | − | 1.55118i | −1.47493 | − | 2.19649i | 3.77069 | − | 5.18991i | −2.86760 | − | 0.881397i | 3.08432 | + | 4.80154i |
131.6 | −0.528347 | + | 2.48568i | 1.52678 | − | 0.817885i | −4.07236 | − | 1.81313i | −0.273491 | − | 2.21928i | 1.22633 | + | 4.22722i | −2.41288 | + | 1.08537i | 3.67110 | − | 5.05284i | 1.66213 | − | 2.49746i | 5.66091 | + | 0.492741i |
131.7 | −0.505077 | + | 2.37620i | 0.0606037 | − | 1.73099i | −3.56413 | − | 1.58685i | 2.07975 | + | 0.821374i | 4.08257 | + | 1.01829i | −2.37564 | + | 1.16461i | 2.71505 | − | 3.73695i | −2.99265 | − | 0.209809i | −3.00218 | + | 4.52704i |
131.8 | −0.489122 | + | 2.30114i | −1.23199 | − | 1.21745i | −3.22892 | − | 1.43761i | 1.02927 | + | 1.98509i | 3.40412 | − | 2.23951i | 1.46353 | − | 2.20410i | 2.12188 | − | 2.92052i | 0.0356207 | + | 2.99979i | −5.07142 | + | 1.39755i |
131.9 | −0.478695 | + | 2.25208i | −0.600734 | − | 1.62454i | −3.01563 | − | 1.34265i | −2.22631 | + | 0.208681i | 3.94616 | − | 0.575244i | 0.695446 | + | 2.55272i | 1.76069 | − | 2.42338i | −2.27824 | + | 1.95183i | 0.595755 | − | 5.11372i |
131.10 | −0.462774 | + | 2.17718i | 1.15009 | + | 1.29510i | −2.69886 | − | 1.20161i | −2.18803 | − | 0.460993i | −3.35190 | + | 1.90461i | −0.462395 | − | 2.60503i | 1.24847 | − | 1.71838i | −0.354583 | + | 2.97897i | 2.01623 | − | 4.55040i |
131.11 | −0.457158 | + | 2.15076i | −0.177450 | + | 1.72294i | −2.58969 | − | 1.15300i | −0.910547 | + | 2.04228i | −3.62450 | − | 1.16931i | 2.32202 | + | 1.26816i | 1.07888 | − | 1.48495i | −2.93702 | − | 0.611470i | −3.97619 | − | 2.89201i |
131.12 | −0.427547 | + | 2.01145i | −1.51386 | + | 0.841555i | −2.03605 | − | 0.906508i | 2.19435 | + | 0.429892i | −1.04550 | − | 3.40487i | 1.64883 | − | 2.06914i | 0.276476 | − | 0.380537i | 1.58357 | − | 2.54800i | −1.80290 | + | 4.23004i |
131.13 | −0.421130 | + | 1.98126i | 0.963172 | + | 1.43955i | −1.92096 | − | 0.855264i | 0.372730 | + | 2.20478i | −3.25774 | + | 1.30206i | −2.59804 | − | 0.500191i | 0.122328 | − | 0.168370i | −1.14460 | + | 2.77306i | −4.52522 | − | 0.190026i |
131.14 | −0.414854 | + | 1.95173i | −1.61597 | − | 0.623422i | −1.81007 | − | 0.805895i | −1.81656 | − | 1.30389i | 1.88714 | − | 2.89531i | −0.995488 | − | 2.45133i | −0.0218510 | + | 0.0300753i | 2.22269 | + | 2.01486i | 3.29845 | − | 3.00451i |
131.15 | −0.406505 | + | 1.91245i | 1.66519 | + | 0.476610i | −1.66514 | − | 0.741369i | −0.898652 | − | 2.04754i | −1.58840 | + | 2.99085i | 2.59781 | + | 0.501390i | −0.203728 | + | 0.280407i | 2.54569 | + | 1.58729i | 4.28113 | − | 0.886296i |
131.16 | −0.392327 | + | 1.84575i | 0.000111749 | 1.73205i | −1.42579 | − | 0.634803i | 0.827237 | − | 2.07742i | −3.19698 | − | 0.679324i | −0.468357 | + | 2.60397i | −0.487221 | + | 0.670602i | −3.00000 | 0.000387111i | 3.50986 | + | 2.34190i | ||
131.17 | −0.352641 | + | 1.65904i | −1.23560 | + | 1.21379i | −0.800979 | − | 0.356619i | −2.12372 | − | 0.699871i | −1.57801 | − | 2.47795i | −1.86600 | + | 1.87565i | −1.11979 | + | 1.54126i | 0.0534131 | − | 2.99952i | 1.91003 | − | 3.27654i |
131.18 | −0.350482 | + | 1.64889i | 1.30612 | − | 1.13757i | −0.768903 | − | 0.342338i | 0.782565 | + | 2.09466i | 1.41795 | + | 2.55234i | 1.46624 | + | 2.20230i | −1.14773 | + | 1.57971i | 0.411886 | − | 2.97159i | −3.72813 | + | 0.556222i |
131.19 | −0.343956 | + | 1.61819i | 1.58851 | − | 0.690379i | −0.673133 | − | 0.299698i | 2.12496 | − | 0.696100i | 0.570782 | + | 2.80797i | 0.539750 | − | 2.59011i | −1.22830 | + | 1.69060i | 2.04676 | − | 2.19335i | 0.395527 | + | 3.67801i |
131.20 | −0.318971 | + | 1.50064i | 0.109335 | − | 1.72860i | −0.323084 | − | 0.143846i | 0.267996 | − | 2.21995i | 2.55912 | + | 0.715443i | −1.46253 | − | 2.20477i | −1.48460 | + | 2.04338i | −2.97609 | − | 0.377991i | 3.24586 | + | 1.11026i |
See next 80 embeddings (of 608 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
7.d | odd | 6 | 1 | inner |
21.g | even | 6 | 1 | inner |
25.d | even | 5 | 1 | inner |
75.j | odd | 10 | 1 | inner |
175.v | odd | 30 | 1 | inner |
525.bm | even | 30 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 525.2.bm.a | ✓ | 608 |
3.b | odd | 2 | 1 | inner | 525.2.bm.a | ✓ | 608 |
7.d | odd | 6 | 1 | inner | 525.2.bm.a | ✓ | 608 |
21.g | even | 6 | 1 | inner | 525.2.bm.a | ✓ | 608 |
25.d | even | 5 | 1 | inner | 525.2.bm.a | ✓ | 608 |
75.j | odd | 10 | 1 | inner | 525.2.bm.a | ✓ | 608 |
175.v | odd | 30 | 1 | inner | 525.2.bm.a | ✓ | 608 |
525.bm | even | 30 | 1 | inner | 525.2.bm.a | ✓ | 608 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
525.2.bm.a | ✓ | 608 | 1.a | even | 1 | 1 | trivial |
525.2.bm.a | ✓ | 608 | 3.b | odd | 2 | 1 | inner |
525.2.bm.a | ✓ | 608 | 7.d | odd | 6 | 1 | inner |
525.2.bm.a | ✓ | 608 | 21.g | even | 6 | 1 | inner |
525.2.bm.a | ✓ | 608 | 25.d | even | 5 | 1 | inner |
525.2.bm.a | ✓ | 608 | 75.j | odd | 10 | 1 | inner |
525.2.bm.a | ✓ | 608 | 175.v | odd | 30 | 1 | inner |
525.2.bm.a | ✓ | 608 | 525.bm | even | 30 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(525, [\chi])\).