Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [504,2,Mod(347,504)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(504, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 3, 5, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("504.347");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 504.cy (of order \(6\), degree \(2\), 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: | \(4.02446026187\) |
Analytic rank: | \(0\) |
Dimension: | \(184\) |
Relative dimension: | \(92\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
347.1 | −1.41373 | + | 0.0369625i | 1.72290 | + | 0.177799i | 1.99727 | − | 0.104510i | 1.53459 | −2.44229 | − | 0.187678i | 2.64302 | + | 0.120259i | −2.81973 | + | 0.221573i | 2.93677 | + | 0.612662i | −2.16949 | + | 0.0567221i | ||
347.2 | −1.41326 | + | 0.0519573i | 0.179393 | − | 1.72274i | 1.99460 | − | 0.146858i | −2.18143 | −0.164020 | + | 2.44399i | 2.31224 | + | 1.28590i | −2.81126 | + | 0.311183i | −2.93564 | − | 0.618093i | 3.08292 | − | 0.113341i | ||
347.3 | −1.40993 | + | 0.110035i | 1.32847 | + | 1.11138i | 1.97578 | − | 0.310283i | −3.42656 | −1.99534 | − | 1.42079i | 1.12525 | − | 2.39454i | −2.75157 | + | 0.654882i | 0.529666 | + | 2.95287i | 4.83120 | − | 0.377042i | ||
347.4 | −1.40917 | − | 0.119296i | −1.11238 | + | 1.32763i | 1.97154 | + | 0.336217i | −4.38485 | 1.72592 | − | 1.73816i | −2.42141 | + | 1.06619i | −2.73813 | − | 0.708985i | −0.525222 | − | 2.95367i | 6.17901 | + | 0.523095i | ||
347.5 | −1.40410 | − | 0.168823i | 1.00207 | + | 1.41275i | 1.94300 | + | 0.474089i | 3.02374 | −1.16851 | − | 2.15281i | −2.60675 | − | 0.452601i | −2.64813 | − | 0.993691i | −0.991703 | + | 2.83135i | −4.24563 | − | 0.510476i | ||
347.6 | −1.39790 | + | 0.214156i | 1.53386 | − | 0.804539i | 1.90827 | − | 0.598739i | 2.18793 | −1.97189 | + | 1.45315i | −2.62554 | − | 0.326387i | −2.53936 | + | 1.24565i | 1.70544 | − | 2.46809i | −3.05852 | + | 0.468558i | ||
347.7 | −1.37793 | − | 0.318287i | −0.613399 | − | 1.61980i | 1.79739 | + | 0.877155i | 3.53729 | 0.329661 | + | 2.42720i | 1.58882 | − | 2.11557i | −2.19749 | − | 1.78074i | −2.24748 | + | 1.98716i | −4.87414 | − | 1.12587i | ||
347.8 | −1.36752 | + | 0.360384i | −0.541502 | + | 1.64523i | 1.74025 | − | 0.985669i | 0.493530 | 0.147602 | − | 2.44504i | 1.32629 | − | 2.28931i | −2.02461 | + | 1.97508i | −2.41355 | − | 1.78179i | −0.674914 | + | 0.177860i | ||
347.9 | −1.36622 | − | 0.365298i | −1.64155 | + | 0.552549i | 1.73312 | + | 0.998154i | 0.188732 | 2.44456 | − | 0.155249i | 2.31931 | + | 1.27310i | −2.00319 | − | 1.99680i | 2.38938 | − | 1.81407i | −0.257849 | − | 0.0689432i | ||
347.10 | −1.36156 | + | 0.382313i | −1.59594 | − | 0.673028i | 1.70767 | − | 1.04108i | −0.632847 | 2.43027 | + | 0.306217i | −1.94512 | + | 1.79346i | −1.92708 | + | 2.07036i | 2.09407 | + | 2.14823i | 0.861657 | − | 0.241945i | ||
347.11 | −1.34051 | + | 0.450600i | −0.774779 | − | 1.54910i | 1.59392 | − | 1.20807i | −0.714950 | 1.73662 | + | 1.72747i | −1.52030 | − | 2.16534i | −1.59230 | + | 2.33764i | −1.79943 | + | 2.40042i | 0.958395 | − | 0.322157i | ||
347.12 | −1.29887 | + | 0.559401i | −0.869153 | + | 1.49819i | 1.37414 | − | 1.45318i | 2.96535 | 0.290830 | − | 2.43216i | −0.557344 | + | 2.58638i | −0.971921 | + | 2.65619i | −1.48915 | − | 2.60431i | −3.85161 | + | 1.65882i | ||
347.13 | −1.29676 | − | 0.564292i | 1.23687 | − | 1.21249i | 1.36315 | + | 1.46350i | −2.35082 | −2.28812 | + | 0.874349i | −1.09033 | − | 2.41064i | −0.941831 | − | 2.66701i | 0.0597186 | − | 2.99941i | 3.04843 | + | 1.32655i | ||
347.14 | −1.28812 | − | 0.583745i | 0.0908862 | + | 1.72966i | 1.31848 | + | 1.50386i | −0.177604 | 0.892611 | − | 2.28106i | 1.51672 | + | 2.16785i | −0.820487 | − | 2.70681i | −2.98348 | + | 0.314405i | 0.228775 | + | 0.103676i | ||
347.15 | −1.27658 | − | 0.608568i | −1.44822 | − | 0.950089i | 1.25929 | + | 1.55377i | 2.40991 | 1.27056 | + | 2.09420i | −1.71811 | + | 2.01199i | −0.662008 | − | 2.74986i | 1.19466 | + | 2.75187i | −3.07644 | − | 1.46660i | ||
347.16 | −1.22414 | + | 0.708150i | 1.63621 | − | 0.568159i | 0.997046 | − | 1.73375i | −2.54780 | −1.60062 | + | 1.85419i | −0.190034 | + | 2.63892i | 0.00723210 | + | 2.82842i | 2.35439 | − | 1.85926i | 3.11887 | − | 1.80422i | ||
347.17 | −1.16532 | − | 0.801263i | −1.44822 | − | 0.950089i | 0.715955 | + | 1.86746i | −2.40991 | 0.926369 | + | 2.26756i | 1.71811 | − | 2.01199i | 0.662008 | − | 2.74986i | 1.19466 | + | 2.75187i | 2.80833 | + | 1.93097i | ||
347.18 | −1.15683 | + | 0.813481i | −1.71844 | + | 0.216724i | 0.676496 | − | 1.88211i | −2.66759 | 1.81163 | − | 1.64863i | 2.37290 | − | 1.17020i | 0.748476 | + | 2.72760i | 2.90606 | − | 0.744855i | 3.08594 | − | 2.17004i | ||
347.19 | −1.14960 | − | 0.823668i | 0.0908862 | + | 1.72966i | 0.643141 | + | 1.89377i | 0.177604 | 1.32019 | − | 2.06328i | −1.51672 | − | 2.16785i | 0.820487 | − | 2.70681i | −2.98348 | + | 0.314405i | −0.204173 | − | 0.146287i | ||
347.20 | −1.13707 | − | 0.840877i | 1.23687 | − | 1.21249i | 0.585851 | + | 1.91227i | 2.35082 | −2.42597 | + | 0.338629i | 1.09033 | + | 2.41064i | 0.941831 | − | 2.66701i | 0.0597186 | − | 2.99941i | −2.67304 | − | 1.97675i | ||
See next 80 embeddings (of 184 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.d | odd | 2 | 1 | inner |
63.n | odd | 6 | 1 | inner |
504.cy | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 504.2.cy.a | yes | 184 |
7.c | even | 3 | 1 | 504.2.bt.a | ✓ | 184 | |
8.d | odd | 2 | 1 | inner | 504.2.cy.a | yes | 184 |
9.d | odd | 6 | 1 | 504.2.bt.a | ✓ | 184 | |
56.k | odd | 6 | 1 | 504.2.bt.a | ✓ | 184 | |
63.n | odd | 6 | 1 | inner | 504.2.cy.a | yes | 184 |
72.l | even | 6 | 1 | 504.2.bt.a | ✓ | 184 | |
504.cy | even | 6 | 1 | inner | 504.2.cy.a | yes | 184 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
504.2.bt.a | ✓ | 184 | 7.c | even | 3 | 1 | |
504.2.bt.a | ✓ | 184 | 9.d | odd | 6 | 1 | |
504.2.bt.a | ✓ | 184 | 56.k | odd | 6 | 1 | |
504.2.bt.a | ✓ | 184 | 72.l | even | 6 | 1 | |
504.2.cy.a | yes | 184 | 1.a | even | 1 | 1 | trivial |
504.2.cy.a | yes | 184 | 8.d | odd | 2 | 1 | inner |
504.2.cy.a | yes | 184 | 63.n | odd | 6 | 1 | inner |
504.2.cy.a | yes | 184 | 504.cy | even | 6 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(504, [\chi])\).