Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [819,2,Mod(176,819)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(819, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([10, 0, 11]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("819.176");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 819 = 3^{2} \cdot 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 819.ew (of order \(12\), 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: | \(6.53974792554\) |
Analytic rank: | \(0\) |
Dimension: | \(336\) |
Relative dimension: | \(84\) over \(\Q(\zeta_{12})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
176.1 | −1.95788 | + | 1.95788i | −1.00105 | + | 1.41347i | − | 5.66657i | −0.576104 | − | 2.15005i | −0.807482 | − | 4.72733i | −0.258819 | − | 0.965926i | 7.17868 | + | 7.17868i | −0.995814 | − | 2.82990i | 5.33747 | + | 3.08159i | |
176.2 | −1.91264 | + | 1.91264i | −1.50144 | + | 0.863523i | − | 5.31637i | 0.990693 | + | 3.69732i | 1.22011 | − | 4.52332i | 0.258819 | + | 0.965926i | 6.34303 | + | 6.34303i | 1.50865 | − | 2.59306i | −8.96647 | − | 5.17679i | |
176.3 | −1.87295 | + | 1.87295i | 0.832889 | − | 1.51865i | − | 5.01588i | 0.0815777 | + | 0.304452i | 1.28440 | + | 4.40431i | 0.258819 | + | 0.965926i | 5.64860 | + | 5.64860i | −1.61259 | − | 2.52973i | −0.723014 | − | 0.417432i | |
176.4 | −1.86338 | + | 1.86338i | 0.258684 | − | 1.71262i | − | 4.94436i | −0.423179 | − | 1.57933i | 2.70924 | + | 3.67329i | −0.258819 | − | 0.965926i | 5.48647 | + | 5.48647i | −2.86617 | − | 0.886057i | 3.73143 | + | 2.15434i | |
176.5 | −1.83201 | + | 1.83201i | −1.42869 | − | 0.979200i | − | 4.71250i | −0.819999 | − | 3.06028i | 4.41128 | − | 0.823478i | 0.258819 | + | 0.965926i | 4.96932 | + | 4.96932i | 1.08234 | + | 2.79795i | 7.10869 | + | 4.10421i | |
176.6 | −1.80918 | + | 1.80918i | −0.649457 | − | 1.60568i | − | 4.54630i | 0.921409 | + | 3.43874i | 4.07996 | + | 1.72998i | −0.258819 | − | 0.965926i | 4.60672 | + | 4.60672i | −2.15641 | + | 2.08564i | −7.88832 | − | 4.55433i | |
176.7 | −1.80571 | + | 1.80571i | 0.527557 | + | 1.64975i | − | 4.52115i | −0.366549 | − | 1.36798i | −3.93158 | − | 2.02636i | 0.258819 | + | 0.965926i | 4.55245 | + | 4.55245i | −2.44337 | + | 1.74068i | 3.13205 | + | 1.80829i | |
176.8 | −1.64739 | + | 1.64739i | 1.05743 | + | 1.37180i | − | 3.42777i | −0.520990 | − | 1.94436i | −4.00189 | − | 0.517890i | −0.258819 | − | 0.965926i | 2.35209 | + | 2.35209i | −0.763679 | + | 2.90117i | 4.06139 | + | 2.34484i | |
176.9 | −1.63956 | + | 1.63956i | 1.73159 | − | 0.0399945i | − | 3.37635i | −1.02893 | − | 3.84001i | −2.77348 | + | 2.90463i | 0.258819 | + | 0.965926i | 2.25661 | + | 2.25661i | 2.99680 | − | 0.138508i | 7.98294 | + | 4.60895i | |
176.10 | −1.59695 | + | 1.59695i | 0.0991579 | + | 1.72921i | − | 3.10050i | 0.947622 | + | 3.53657i | −2.91981 | − | 2.60311i | −0.258819 | − | 0.965926i | 1.75745 | + | 1.75745i | −2.98034 | + | 0.342930i | −7.16104 | − | 4.13443i | |
176.11 | −1.55099 | + | 1.55099i | 1.57977 | − | 0.710165i | − | 2.81113i | 0.719217 | + | 2.68415i | −1.34874 | + | 3.55166i | −0.258819 | − | 0.965926i | 1.25806 | + | 1.25806i | 1.99133 | − | 2.24379i | −5.27859 | − | 3.04759i | |
176.12 | −1.49802 | + | 1.49802i | −0.188264 | − | 1.72179i | − | 2.48814i | 0.487433 | + | 1.81912i | 2.86130 | + | 2.29725i | 0.258819 | + | 0.965926i | 0.731245 | + | 0.731245i | −2.92911 | + | 0.648302i | −3.45527 | − | 1.99490i | |
176.13 | −1.47984 | + | 1.47984i | −1.33313 | − | 1.10579i | − | 2.37986i | 0.507117 | + | 1.89258i | 3.60921 | − | 0.336422i | 0.258819 | + | 0.965926i | 0.562125 | + | 0.562125i | 0.554455 | + | 2.94832i | −3.55117 | − | 2.05027i | |
176.14 | −1.45048 | + | 1.45048i | −0.883457 | + | 1.48980i | − | 2.20781i | 0.260529 | + | 0.972306i | −0.879492 | − | 3.44237i | −0.258819 | − | 0.965926i | 0.301429 | + | 0.301429i | −1.43901 | − | 2.63235i | −1.78821 | − | 1.03242i | |
176.15 | −1.39334 | + | 1.39334i | −1.02425 | − | 1.39675i | − | 1.88279i | −0.512387 | − | 1.91225i | 3.37328 | + | 0.519029i | −0.258819 | − | 0.965926i | −0.163307 | − | 0.163307i | −0.901840 | + | 2.86124i | 3.37835 | + | 1.95049i | |
176.16 | −1.39126 | + | 1.39126i | 1.52938 | + | 0.813025i | − | 1.87123i | −0.451446 | − | 1.68482i | −3.25890 | + | 0.996634i | −0.258819 | − | 0.965926i | −0.179147 | − | 0.179147i | 1.67798 | + | 2.48684i | 2.97211 | + | 1.71595i | |
176.17 | −1.37577 | + | 1.37577i | 1.69723 | + | 0.345561i | − | 1.78551i | 0.367577 | + | 1.37181i | −2.81042 | + | 1.85959i | 0.258819 | + | 0.965926i | −0.295090 | − | 0.295090i | 2.76117 | + | 1.17299i | −2.39301 | − | 1.38160i | |
176.18 | −1.37300 | + | 1.37300i | −1.58085 | + | 0.707758i | − | 1.77025i | 0.189170 | + | 0.705993i | 1.19875 | − | 3.14225i | 0.258819 | + | 0.965926i | −0.315445 | − | 0.315445i | 1.99816 | − | 2.23772i | −1.22906 | − | 0.709597i | |
176.19 | −1.35386 | + | 1.35386i | 1.29330 | − | 1.15212i | − | 1.66586i | 0.0412922 | + | 0.154105i | −0.191139 | + | 3.31075i | −0.258819 | − | 0.965926i | −0.452377 | − | 0.452377i | 0.345247 | − | 2.98007i | −0.264540 | − | 0.152732i | |
176.20 | −1.25977 | + | 1.25977i | −0.923456 | + | 1.46534i | − | 1.17405i | −0.838983 | − | 3.13113i | −0.682653 | − | 3.00934i | 0.258819 | + | 0.965926i | −1.04050 | − | 1.04050i | −1.29446 | − | 2.70636i | 5.00144 | + | 2.88758i | |
See next 80 embeddings (of 336 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
117.x | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 819.2.ew.a | ✓ | 336 |
9.d | odd | 6 | 1 | 819.2.fy.a | yes | 336 | |
13.f | odd | 12 | 1 | 819.2.fy.a | yes | 336 | |
117.x | even | 12 | 1 | inner | 819.2.ew.a | ✓ | 336 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
819.2.ew.a | ✓ | 336 | 1.a | even | 1 | 1 | trivial |
819.2.ew.a | ✓ | 336 | 117.x | even | 12 | 1 | inner |
819.2.fy.a | yes | 336 | 9.d | odd | 6 | 1 | |
819.2.fy.a | yes | 336 | 13.f | odd | 12 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(819, [\chi])\).