Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [261,3,Mod(5,261)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(261, base_ring=CyclotomicField(42))
chi = DirichletCharacter(H, H._module([35, 33]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("261.5");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 261 = 3^{2} \cdot 29 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 261.v (of order \(42\), degree \(12\), 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: | \(7.11173489980\) |
Analytic rank: | \(0\) |
Dimension: | \(696\) |
Relative dimension: | \(58\) over \(\Q(\zeta_{42})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{42}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
5.1 | −1.43075 | + | 3.64550i | 0.801822 | − | 2.89086i | −8.31042 | − | 7.71094i | 1.02064 | + | 6.77148i | 9.39143 | + | 7.05916i | 0.703203 | − | 0.652477i | 25.8869 | − | 12.4665i | −7.71416 | − | 4.63591i | −26.1457 | − | 5.96759i |
5.2 | −1.38256 | + | 3.52270i | 2.99971 | + | 0.0418777i | −7.56576 | − | 7.02000i | −0.455445 | − | 3.02168i | −4.29479 | + | 10.5092i | −0.470271 | + | 0.436347i | 21.5513 | − | 10.3786i | 8.99649 | + | 0.251242i | 11.2742 | + | 2.57325i |
5.3 | −1.37031 | + | 3.49151i | −2.56398 | − | 1.55756i | −7.38064 | − | 6.84823i | −0.849898 | − | 5.63871i | 8.95169 | − | 6.81781i | −2.57897 | + | 2.39294i | 20.5071 | − | 9.87570i | 4.14802 | + | 7.98711i | 20.8522 | + | 4.75938i |
5.4 | −1.33390 | + | 3.39871i | −1.10049 | + | 2.79087i | −6.83975 | − | 6.34636i | −0.781471 | − | 5.18472i | −8.01741 | − | 7.46296i | 3.86951 | − | 3.59038i | 17.5348 | − | 8.44433i | −6.57786 | − | 6.14262i | 18.6638 | + | 4.25989i |
5.5 | −1.25942 | + | 3.20895i | −2.38451 | + | 1.82047i | −5.77903 | − | 5.36216i | 0.930464 | + | 6.17322i | −2.83871 | − | 9.94452i | −8.70372 | + | 8.07587i | 12.0617 | − | 5.80861i | 2.37176 | − | 8.68186i | −20.9814 | − | 4.78888i |
5.6 | −1.18777 | + | 3.02639i | −2.93094 | + | 0.640011i | −4.81603 | − | 4.46863i | 0.685670 | + | 4.54912i | 1.54436 | − | 9.63034i | 7.90119 | − | 7.33124i | 7.52748 | − | 3.62505i | 8.18077 | − | 3.75166i | −14.5818 | − | 3.32821i |
5.7 | −1.14410 | + | 2.91513i | 1.50801 | + | 2.59344i | −4.25679 | − | 3.94973i | −0.447770 | − | 2.97076i | −9.28552 | + | 1.42888i | −9.00164 | + | 8.35230i | 5.09826 | − | 2.45519i | −4.45183 | + | 7.82184i | 9.17244 | + | 2.09355i |
5.8 | −1.12715 | + | 2.87193i | 2.45483 | + | 1.72447i | −4.04532 | − | 3.75351i | 0.472918 | + | 3.13760i | −7.71952 | + | 5.10638i | 4.84953 | − | 4.49971i | 4.22082 | − | 2.03264i | 3.05241 | + | 8.46657i | −9.54404 | − | 2.17836i |
5.9 | −1.09182 | + | 2.78191i | −2.10748 | − | 2.13507i | −3.61477 | − | 3.35402i | 0.713024 | + | 4.73060i | 8.24057 | − | 3.53173i | 0.281758 | − | 0.261433i | 2.50709 | − | 1.20735i | −0.117015 | + | 8.99924i | −13.9386 | − | 3.18140i |
5.10 | −1.08159 | + | 2.75585i | 1.92322 | − | 2.30243i | −3.49267 | − | 3.24073i | −1.29663 | − | 8.60259i | 4.26502 | + | 7.79042i | −4.96575 | + | 4.60754i | 2.03933 | − | 0.982090i | −1.60241 | − | 8.85620i | 25.1099 | + | 5.73117i |
5.11 | −1.04968 | + | 2.67453i | 2.14106 | − | 2.10139i | −3.11908 | − | 2.89409i | 0.0180756 | + | 0.119924i | 3.37281 | + | 7.93212i | 5.54916 | − | 5.14887i | 0.659912 | − | 0.317797i | 0.168307 | − | 8.99843i | −0.339713 | − | 0.0775374i |
5.12 | −1.02316 | + | 2.60697i | −0.635014 | − | 2.93202i | −2.81724 | − | 2.61402i | −0.162656 | − | 1.07915i | 8.29343 | + | 1.34447i | −1.81026 | + | 1.67968i | −0.395726 | + | 0.190571i | −8.19351 | + | 3.72375i | 2.97974 | + | 0.680106i |
5.13 | −0.987218 | + | 2.51539i | 2.99149 | − | 0.225795i | −2.42038 | − | 2.24578i | 1.34341 | + | 8.91298i | −2.38529 | + | 7.74767i | −6.98422 | + | 6.48041i | −1.69986 | + | 0.818611i | 8.89803 | − | 1.35093i | −23.7458 | − | 5.41984i |
5.14 | −0.911365 | + | 2.32212i | −0.687966 | + | 2.92005i | −1.62945 | − | 1.51191i | −0.126374 | − | 0.838435i | −6.15372 | − | 4.25877i | 0.932879 | − | 0.865585i | −3.99423 | + | 1.92352i | −8.05341 | − | 4.01779i | 2.06212 | + | 0.470665i |
5.15 | −0.799677 | + | 2.03754i | −2.97515 | − | 0.385309i | −0.579897 | − | 0.538066i | −0.638382 | − | 4.23539i | 3.16425 | − | 5.75389i | 6.14549 | − | 5.70218i | −6.32829 | + | 3.04754i | 8.70307 | + | 2.29271i | 9.14030 | + | 2.08621i |
5.16 | −0.752873 | + | 1.91829i | 2.07135 | + | 2.17014i | −0.180804 | − | 0.167762i | −1.48108 | − | 9.82631i | −5.72242 | + | 2.33960i | 7.62135 | − | 7.07158i | −6.96871 | + | 3.35595i | −0.419050 | + | 8.99024i | 19.9648 | + | 4.55682i |
5.17 | −0.738339 | + | 1.88126i | −2.89765 | + | 0.776937i | −0.0617723 | − | 0.0573163i | −0.901243 | − | 5.97935i | 0.677828 | − | 6.02486i | −6.52464 | + | 6.05398i | −7.12984 | + | 3.43355i | 7.79274 | − | 4.50258i | 11.9141 | + | 2.71932i |
5.18 | −0.686937 | + | 1.75029i | 0.153024 | + | 2.99609i | 0.340585 | + | 0.316017i | 1.19231 | + | 7.91045i | −5.34914 | − | 1.79029i | 2.90380 | − | 2.69434i | −7.56331 | + | 3.64230i | −8.95317 | + | 0.916950i | −14.6646 | − | 3.34710i |
5.19 | −0.593958 | + | 1.51338i | 0.825786 | − | 2.88411i | 0.994673 | + | 0.922922i | 0.542859 | + | 3.60163i | 3.87427 | + | 2.96277i | −6.59851 | + | 6.12253i | −7.84657 | + | 3.77871i | −7.63616 | − | 4.76331i | −5.77307 | − | 1.31767i |
5.20 | −0.524166 | + | 1.33555i | −2.01600 | + | 2.22165i | 1.42325 | + | 1.32059i | 0.215073 | + | 1.42692i | −1.91041 | − | 3.85699i | −1.72570 | + | 1.60122i | −7.68033 | + | 3.69865i | −0.871474 | − | 8.95771i | −2.01846 | − | 0.460699i |
See next 80 embeddings (of 696 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.d | odd | 6 | 1 | inner |
29.e | even | 14 | 1 | inner |
261.v | odd | 42 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 261.3.v.a | ✓ | 696 |
9.d | odd | 6 | 1 | inner | 261.3.v.a | ✓ | 696 |
29.e | even | 14 | 1 | inner | 261.3.v.a | ✓ | 696 |
261.v | odd | 42 | 1 | inner | 261.3.v.a | ✓ | 696 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
261.3.v.a | ✓ | 696 | 1.a | even | 1 | 1 | trivial |
261.3.v.a | ✓ | 696 | 9.d | odd | 6 | 1 | inner |
261.3.v.a | ✓ | 696 | 29.e | even | 14 | 1 | inner |
261.3.v.a | ✓ | 696 | 261.v | odd | 42 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(261, [\chi])\).