Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [171,3,Mod(20,171)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(171, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("171.20");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 171 = 3^{2} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 171.q (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: | \(4.65941252056\) |
Analytic rank: | \(0\) |
Dimension: | \(72\) |
Relative dimension: | \(36\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
20.1 | −3.35010 | − | 1.93418i | −1.25076 | − | 2.72683i | 5.48210 | + | 9.49527i | −7.85315 | + | 4.53402i | −1.08401 | + | 11.5543i | 0.551401 | − | 0.955054i | − | 26.9400i | −5.87120 | + | 6.82122i | 35.0784 | |||
20.2 | −3.26875 | − | 1.88722i | −2.62151 | + | 1.45865i | 5.12317 | + | 8.87359i | 3.99506 | − | 2.30655i | 11.3219 | + | 0.179376i | −3.48382 | + | 6.03415i | − | 23.5764i | 4.74465 | − | 7.64776i | −17.4118 | |||
20.3 | −3.09705 | − | 1.78808i | 2.66600 | − | 1.37566i | 4.39447 | + | 7.61144i | 4.10114 | − | 2.36780i | −10.7165 | − | 0.506538i | −1.24181 | + | 2.15087i | − | 17.1260i | 5.21511 | − | 7.33502i | −16.9353 | |||
20.4 | −2.99030 | − | 1.72645i | 0.0547441 | + | 2.99950i | 3.96126 | + | 6.86110i | −1.97273 | + | 1.13895i | 5.01478 | − | 9.06391i | 5.96281 | − | 10.3279i | − | 13.5440i | −8.99401 | + | 0.328410i | 7.86538 | |||
20.5 | −2.96952 | − | 1.71445i | 2.47292 | + | 1.69843i | 3.87870 | + | 6.71811i | −2.50796 | + | 1.44797i | −4.43150 | − | 9.28323i | −3.77541 | + | 6.53920i | − | 12.8838i | 3.23066 | + | 8.40017i | 9.92990 | |||
20.6 | −2.40334 | − | 1.38757i | 1.68015 | − | 2.48538i | 1.85070 | + | 3.20551i | −1.08225 | + | 0.624835i | −7.48661 | + | 3.64189i | 3.10197 | − | 5.37276i | 0.828638i | −3.35421 | − | 8.35160i | 3.46801 | ||||
20.7 | −2.32987 | − | 1.34515i | −2.46014 | − | 1.71689i | 1.61888 | + | 2.80398i | 1.31696 | − | 0.760345i | 3.42235 | + | 7.30940i | −2.02468 | + | 3.50685i | 2.05068i | 3.10461 | + | 8.44757i | −4.09112 | ||||
20.8 | −2.31996 | − | 1.33943i | −2.92815 | − | 0.652642i | 1.58813 | + | 2.75073i | 2.50859 | − | 1.44833i | 5.91901 | + | 5.43615i | 6.06415 | − | 10.5034i | 2.20667i | 8.14812 | + | 3.82207i | −7.75975 | ||||
20.9 | −2.19209 | − | 1.26560i | 2.05203 | + | 2.18841i | 1.20350 | + | 2.08453i | 8.14302 | − | 4.70138i | −1.72858 | − | 7.39426i | 1.59509 | − | 2.76277i | 4.03219i | −0.578305 | + | 8.98140i | −23.8003 | ||||
20.10 | −2.05717 | − | 1.18771i | −2.58832 | + | 1.51678i | 0.821296 | + | 1.42253i | −7.91441 | + | 4.56938i | 7.12609 | − | 0.0461048i | −1.26225 | + | 2.18629i | 5.59982i | 4.39877 | − | 7.85180i | 21.7084 | ||||
20.11 | −1.84414 | − | 1.06471i | −0.0845080 | − | 2.99881i | 0.267231 | + | 0.462857i | 0.590019 | − | 0.340648i | −3.03703 | + | 5.62020i | −6.43659 | + | 11.1485i | 7.37961i | −8.98572 | + | 0.506847i | −1.45077 | ||||
20.12 | −1.77501 | − | 1.02480i | 2.96735 | + | 0.441388i | 0.100449 | + | 0.173983i | −5.79916 | + | 3.34815i | −4.81475 | − | 3.82443i | 0.488361 | − | 0.845867i | 7.78667i | 8.61035 | + | 2.61951i | 13.7248 | ||||
20.13 | −0.692274 | − | 0.399685i | 0.0689180 | − | 2.99921i | −1.68050 | − | 2.91072i | 8.02032 | − | 4.63053i | −1.24645 | + | 2.04873i | 3.10019 | − | 5.36969i | 5.88416i | −8.99050 | − | 0.413399i | −7.40301 | ||||
20.14 | −0.658324 | − | 0.380083i | 1.47417 | + | 2.61282i | −1.71107 | − | 2.96367i | −0.162364 | + | 0.0937407i | 0.0226040 | − | 2.28039i | −4.08229 | + | 7.07074i | 5.64207i | −4.65362 | + | 7.70349i | 0.142517 | ||||
20.15 | −0.511116 | − | 0.295093i | −1.58134 | − | 2.54938i | −1.82584 | − | 3.16245i | −4.25975 | + | 2.45937i | 0.0559453 | + | 1.76967i | 3.55508 | − | 6.15757i | 4.51592i | −3.99870 | + | 8.06290i | 2.90297 | ||||
20.16 | −0.506424 | − | 0.292384i | 2.83118 | − | 0.992190i | −1.82902 | − | 3.16796i | 2.36501 | − | 1.36544i | −1.72388 | − | 0.325322i | 0.857721 | − | 1.48562i | 4.47818i | 7.03112 | − | 5.61813i | −1.59693 | ||||
20.17 | −0.406613 | − | 0.234758i | 0.510975 | + | 2.95616i | −1.88978 | − | 3.27319i | −5.03210 | + | 2.90528i | 0.486214 | − | 1.32197i | 5.32202 | − | 9.21801i | 3.65262i | −8.47781 | + | 3.02105i | 2.72815 | ||||
20.18 | −0.00940539 | − | 0.00543021i | −2.99982 | + | 0.0330393i | −1.99994 | − | 3.46400i | −2.13068 | + | 1.23015i | 0.0283939 | + | 0.0159789i | −1.68412 | + | 2.91698i | 0.0868820i | 8.99782 | − | 0.198224i | 0.0267199 | ||||
20.19 | 0.0217615 | + | 0.0125640i | −2.91865 | − | 0.693896i | −1.99968 | − | 3.46355i | 7.22268 | − | 4.17002i | −0.0547961 | − | 0.0517702i | −3.62212 | + | 6.27369i | − | 0.201008i | 8.03702 | + | 4.05048i | 0.209569 | |||
20.20 | 0.133214 | + | 0.0769109i | 1.82791 | − | 2.37881i | −1.98817 | − | 3.44361i | −7.01226 | + | 4.04853i | 0.426459 | − | 0.176304i | −5.54783 | + | 9.60912i | − | 1.22693i | −2.31749 | − | 8.69651i | −1.24550 | |||
See all 72 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.d | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 171.3.q.a | ✓ | 72 |
3.b | odd | 2 | 1 | 513.3.q.a | 72 | ||
9.c | even | 3 | 1 | 513.3.q.a | 72 | ||
9.d | odd | 6 | 1 | inner | 171.3.q.a | ✓ | 72 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
171.3.q.a | ✓ | 72 | 1.a | even | 1 | 1 | trivial |
171.3.q.a | ✓ | 72 | 9.d | odd | 6 | 1 | inner |
513.3.q.a | 72 | 3.b | odd | 2 | 1 | ||
513.3.q.a | 72 | 9.c | even | 3 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(171, [\chi])\).