Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [819,2,Mod(229,819)]
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([4, 10, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("819.229");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 819 = 3^{2} \cdot 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 819.ep (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: | \(432\) |
Relative dimension: | \(108\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
229.1 | −1.96596 | − | 1.96596i | −1.25489 | + | 1.19384i | 5.72998i | −0.404127 | − | 0.108286i | 4.81410 | + | 0.120037i | −1.02182 | + | 2.44047i | 7.33299 | − | 7.33299i | 0.149514 | − | 2.99627i | 0.581612 | + | 1.00738i | ||
229.2 | −1.93236 | − | 1.93236i | 1.39105 | − | 1.03198i | 5.46804i | −3.14351 | − | 0.842300i | −4.68217 | − | 0.693857i | 0.961386 | − | 2.46490i | 6.70151 | − | 6.70151i | 0.870042 | − | 2.87107i | 4.44676 | + | 7.70202i | ||
229.3 | −1.89917 | − | 1.89917i | −0.785142 | − | 1.54388i | 5.21370i | −2.25967 | − | 0.605476i | −1.44096 | + | 4.42320i | −2.56472 | + | 0.649790i | 6.10335 | − | 6.10335i | −1.76710 | + | 2.42432i | 3.14159 | + | 5.44139i | ||
229.4 | −1.89057 | − | 1.89057i | 1.73151 | + | 0.0432369i | 5.14849i | 3.71009 | + | 0.994115i | −3.19180 | − | 3.35528i | −2.52357 | − | 0.794740i | 5.95244 | − | 5.95244i | 2.99626 | + | 0.149731i | −5.13473 | − | 8.89361i | ||
229.5 | −1.88944 | − | 1.88944i | 0.160590 | + | 1.72459i | 5.13995i | 0.658675 | + | 0.176491i | 2.95508 | − | 3.56193i | −1.14253 | − | 2.38634i | 5.93274 | − | 5.93274i | −2.94842 | + | 0.553903i | −0.911055 | − | 1.57799i | ||
229.6 | −1.88804 | − | 1.88804i | −1.68739 | − | 0.390799i | 5.12938i | 2.57278 | + | 0.689373i | 2.44801 | + | 3.92370i | 1.85236 | − | 1.88912i | 5.90840 | − | 5.90840i | 2.69455 | + | 1.31886i | −3.55594 | − | 6.15907i | ||
229.7 | −1.75840 | − | 1.75840i | 0.504792 | + | 1.65686i | 4.18391i | −3.80271 | − | 1.01893i | 2.02579 | − | 3.80104i | 2.37359 | + | 1.16879i | 3.84019 | − | 3.84019i | −2.49037 | + | 1.67274i | 4.89498 | + | 8.47835i | ||
229.8 | −1.74030 | − | 1.74030i | 1.46132 | + | 0.929814i | 4.05728i | 0.578919 | + | 0.155121i | −0.924973 | − | 4.16128i | 2.60158 | − | 0.481414i | 3.58028 | − | 3.58028i | 1.27089 | + | 2.71751i | −0.737536 | − | 1.27745i | ||
229.9 | −1.66329 | − | 1.66329i | 0.126557 | − | 1.72742i | 3.53307i | 0.103056 | + | 0.0276137i | −3.08370 | + | 2.66270i | 1.55851 | + | 2.13800i | 2.54993 | − | 2.54993i | −2.96797 | − | 0.437234i | −0.125482 | − | 0.217342i | ||
229.10 | −1.64531 | − | 1.64531i | −1.67759 | + | 0.430934i | 3.41408i | −3.69224 | − | 0.989333i | 3.46917 | + | 2.05113i | 0.423123 | − | 2.61170i | 2.32660 | − | 2.32660i | 2.62859 | − | 1.44586i | 4.44712 | + | 7.70264i | ||
229.11 | −1.63578 | − | 1.63578i | −1.32238 | − | 1.11862i | 3.35153i | −1.82513 | − | 0.489041i | 0.333304 | + | 3.99293i | 2.40039 | + | 1.11271i | 2.21080 | − | 2.21080i | 0.497375 | + | 2.95848i | 2.18554 | + | 3.78546i | ||
229.12 | −1.60413 | − | 1.60413i | −1.72570 | + | 0.148152i | 3.14644i | 1.50819 | + | 0.404118i | 3.00590 | + | 2.53059i | 0.912160 | + | 2.48354i | 1.83903 | − | 1.83903i | 2.95610 | − | 0.511333i | −1.77107 | − | 3.06758i | ||
229.13 | −1.60407 | − | 1.60407i | 0.794613 | + | 1.53902i | 3.14605i | 2.37922 | + | 0.637509i | 1.19408 | − | 3.74330i | −0.371203 | + | 2.61958i | 1.83834 | − | 1.83834i | −1.73718 | + | 2.44585i | −2.79381 | − | 4.83902i | ||
229.14 | −1.60074 | − | 1.60074i | 1.07753 | − | 1.35607i | 3.12472i | 0.414662 | + | 0.111108i | −3.89556 | + | 0.445865i | −2.57658 | + | 0.601027i | 1.80038 | − | 1.80038i | −0.677849 | − | 2.92242i | −0.485909 | − | 0.841619i | ||
229.15 | −1.58613 | − | 1.58613i | 1.55738 | − | 0.758001i | 3.03162i | −1.33707 | − | 0.358267i | −3.67250 | − | 1.26792i | −0.831100 | + | 2.51183i | 1.63628 | − | 1.63628i | 1.85087 | − | 2.36099i | 1.55251 | + | 2.68902i | ||
229.16 | −1.56728 | − | 1.56728i | −1.50556 | − | 0.856331i | 2.91271i | 2.16047 | + | 0.578896i | 1.01751 | + | 3.70173i | −2.49862 | − | 0.869986i | 1.43046 | − | 1.43046i | 1.53340 | + | 2.57851i | −2.47876 | − | 4.29334i | ||
229.17 | −1.55336 | − | 1.55336i | −0.812740 | + | 1.52953i | 2.82587i | 3.56579 | + | 0.955449i | 3.63839 | − | 1.11343i | 2.64211 | − | 0.138853i | 1.28288 | − | 1.28288i | −1.67891 | − | 2.48621i | −4.05480 | − | 7.02312i | ||
229.18 | −1.48199 | − | 1.48199i | 1.56721 | + | 0.737461i | 2.39258i | −2.00664 | − | 0.537677i | −1.22968 | − | 3.41550i | −2.24356 | − | 1.40230i | 0.581795 | − | 0.581795i | 1.91230 | + | 2.31151i | 2.17698 | + | 3.77065i | ||
229.19 | −1.43877 | − | 1.43877i | −0.688101 | + | 1.58950i | 2.14012i | −0.107608 | − | 0.0288335i | 3.27695 | − | 1.29691i | 0.792554 | − | 2.52425i | 0.201598 | − | 0.201598i | −2.05303 | − | 2.18748i | 0.113339 | + | 0.196308i | ||
229.20 | −1.43568 | − | 1.43568i | 0.573580 | − | 1.63432i | 2.12234i | 2.32171 | + | 0.622101i | −3.16984 | + | 1.52288i | −0.895150 | − | 2.48972i | 0.175648 | − | 0.175648i | −2.34201 | − | 1.87483i | −2.44009 | − | 4.22637i | ||
See next 80 embeddings (of 432 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.d | odd | 4 | 1 | inner |
63.t | odd | 6 | 1 | inner |
819.ep | 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.ep.a | ✓ | 432 |
7.d | odd | 6 | 1 | 819.2.fk.a | yes | 432 | |
9.c | even | 3 | 1 | 819.2.fk.a | yes | 432 | |
13.d | odd | 4 | 1 | inner | 819.2.ep.a | ✓ | 432 |
63.t | odd | 6 | 1 | inner | 819.2.ep.a | ✓ | 432 |
91.bb | even | 12 | 1 | 819.2.fk.a | yes | 432 | |
117.y | odd | 12 | 1 | 819.2.fk.a | yes | 432 | |
819.ep | even | 12 | 1 | inner | 819.2.ep.a | ✓ | 432 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
819.2.ep.a | ✓ | 432 | 1.a | even | 1 | 1 | trivial |
819.2.ep.a | ✓ | 432 | 13.d | odd | 4 | 1 | inner |
819.2.ep.a | ✓ | 432 | 63.t | odd | 6 | 1 | inner |
819.2.ep.a | ✓ | 432 | 819.ep | even | 12 | 1 | inner |
819.2.fk.a | yes | 432 | 7.d | odd | 6 | 1 | |
819.2.fk.a | yes | 432 | 9.c | even | 3 | 1 | |
819.2.fk.a | yes | 432 | 91.bb | even | 12 | 1 | |
819.2.fk.a | yes | 432 | 117.y | odd | 12 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(819, [\chi])\).