Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [403,2,Mod(60,403)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(403, base_ring=CyclotomicField(20))
chi = DirichletCharacter(H, H._module([5, 6]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("403.60");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 403 = 13 \cdot 31 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 403.bn (of order \(20\), degree \(8\), 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: | \(3.21797120146\) |
Analytic rank: | \(0\) |
Dimension: | \(272\) |
Relative dimension: | \(34\) over \(\Q(\zeta_{20})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{20}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
60.1 | −0.399110 | − | 2.51988i | 0.464849 | + | 0.639810i | −4.28839 | + | 1.39338i | −1.89475 | − | 1.89475i | 1.42672 | − | 1.42672i | −1.39495 | − | 0.710764i | 2.90617 | + | 5.70368i | 0.733779 | − | 2.25834i | −4.01833 | + | 5.53075i |
60.2 | −0.393738 | − | 2.48596i | 1.63498 | + | 2.25036i | −4.12287 | + | 1.33960i | 1.33236 | + | 1.33236i | 4.95055 | − | 4.95055i | −0.882842 | − | 0.449831i | 2.66818 | + | 5.23661i | −1.46390 | + | 4.50541i | 2.78760 | − | 3.83680i |
60.3 | −0.390481 | − | 2.46540i | −1.53915 | − | 2.11846i | −4.02361 | + | 1.30735i | −1.32495 | − | 1.32495i | −4.62184 | + | 4.62184i | −2.80514 | − | 1.42929i | 2.52785 | + | 4.96118i | −1.19183 | + | 3.66808i | −2.74916 | + | 3.78390i |
60.4 | −0.388669 | − | 2.45396i | 0.0803757 | + | 0.110628i | −3.96875 | + | 1.28952i | 0.523296 | + | 0.523296i | 0.240236 | − | 0.240236i | 3.99187 | + | 2.03396i | 2.45104 | + | 4.81045i | 0.921273 | − | 2.83539i | 1.08076 | − | 1.48754i |
60.5 | −0.362855 | − | 2.29098i | −0.115498 | − | 0.158970i | −3.21480 | + | 1.04455i | 2.00673 | + | 2.00673i | −0.322287 | + | 0.322287i | −1.28505 | − | 0.654764i | 1.45346 | + | 2.85258i | 0.915119 | − | 2.81645i | 3.86922 | − | 5.32553i |
60.6 | −0.327123 | − | 2.06537i | −1.61302 | − | 2.22013i | −2.25663 | + | 0.733225i | 0.152237 | + | 0.152237i | −4.05774 | + | 4.05774i | 0.794879 | + | 0.405011i | 0.353884 | + | 0.694537i | −1.40010 | + | 4.30907i | 0.264626 | − | 0.364226i |
60.7 | −0.279767 | − | 1.76638i | 1.47487 | + | 2.02999i | −1.13970 | + | 0.370312i | −2.62399 | − | 2.62399i | 3.17310 | − | 3.17310i | 2.14712 | + | 1.09401i | −0.650866 | − | 1.27740i | −1.01855 | + | 3.13478i | −3.90085 | + | 5.36906i |
60.8 | −0.242952 | − | 1.53394i | −0.916843 | − | 1.26193i | −0.391834 | + | 0.127315i | −1.09885 | − | 1.09885i | −1.71297 | + | 1.71297i | −1.58544 | − | 0.807824i | −1.11966 | − | 2.19746i | 0.175195 | − | 0.539194i | −1.41860 | + | 1.95254i |
60.9 | −0.241622 | − | 1.52554i | −1.55970 | − | 2.14675i | −0.366780 | + | 0.119174i | 2.50940 | + | 2.50940i | −2.89809 | + | 2.89809i | 3.71488 | + | 1.89282i | −1.13200 | − | 2.22168i | −1.24880 | + | 3.84340i | 3.22186 | − | 4.43452i |
60.10 | −0.217339 | − | 1.37222i | 0.253284 | + | 0.348615i | 0.0663516 | − | 0.0215589i | −0.524097 | − | 0.524097i | 0.423330 | − | 0.423330i | 0.759489 | + | 0.386979i | −1.30549 | − | 2.56217i | 0.869671 | − | 2.67657i | −0.605272 | + | 0.833086i |
60.11 | −0.197701 | − | 1.24823i | 1.55219 | + | 2.13641i | 0.383111 | − | 0.124480i | 2.03317 | + | 2.03317i | 2.35987 | − | 2.35987i | 0.0423174 | + | 0.0215618i | −1.37862 | − | 2.70570i | −1.22789 | + | 3.77906i | 2.13592 | − | 2.93984i |
60.12 | −0.194801 | − | 1.22992i | 0.902955 | + | 1.24281i | 0.427346 | − | 0.138853i | −1.41853 | − | 1.41853i | 1.35267 | − | 1.35267i | −4.54364 | − | 2.31510i | −1.38469 | − | 2.71762i | 0.197800 | − | 0.608766i | −1.46835 | + | 2.02102i |
60.13 | −0.170304 | − | 1.07526i | 0.498908 | + | 0.686688i | 0.774943 | − | 0.251794i | 2.61824 | + | 2.61824i | 0.653399 | − | 0.653399i | −0.113188 | − | 0.0576723i | −1.39120 | − | 2.73038i | 0.704420 | − | 2.16798i | 2.36938 | − | 3.26117i |
60.14 | −0.159192 | − | 1.00510i | −0.939090 | − | 1.29255i | 0.917230 | − | 0.298026i | −2.17200 | − | 2.17200i | −1.14964 | + | 1.14964i | 3.27405 | + | 1.66821i | −1.36955 | − | 2.68789i | 0.138265 | − | 0.425535i | −1.83731 | + | 2.52884i |
60.15 | −0.0811295 | − | 0.512232i | −1.59762 | − | 2.19894i | 1.64631 | − | 0.534920i | 1.75447 | + | 1.75447i | −0.996753 | + | 0.996753i | −4.04043 | − | 2.05870i | −0.878462 | − | 1.72408i | −1.35589 | + | 4.17300i | 0.756357 | − | 1.04104i |
60.16 | −0.0419805 | − | 0.265054i | 1.85928 | + | 2.55908i | 1.83362 | − | 0.595780i | −0.657016 | − | 0.657016i | 0.600242 | − | 0.600242i | −0.541111 | − | 0.275710i | −0.478554 | − | 0.939216i | −2.16493 | + | 6.66296i | −0.146563 | + | 0.201727i |
60.17 | 0.000221639 | 0.00139937i | −0.216440 | − | 0.297904i | 1.90211 | − | 0.618033i | 1.05381 | + | 1.05381i | 0.000368907 | 0 | 0.000368907i | 1.63845 | + | 0.834831i | 0.00257288 | + | 0.00504957i | 0.885150 | − | 2.72421i | −0.00124111 | + | 0.00170824i | |
60.18 | 0.00544638 | + | 0.0343871i | −1.45104 | − | 1.99718i | 1.90096 | − | 0.617659i | −2.64841 | − | 2.64841i | 0.0607744 | − | 0.0607744i | −1.91884 | − | 0.977699i | 0.0632049 | + | 0.124047i | −0.956174 | + | 2.94280i | 0.0766470 | − | 0.105496i |
60.19 | 0.0137208 | + | 0.0866298i | 1.01677 | + | 1.39947i | 1.89480 | − | 0.615657i | −0.943639 | − | 0.943639i | −0.107284 | + | 0.107284i | 2.98091 | + | 1.51885i | 0.158971 | + | 0.311998i | 0.00237083 | − | 0.00729668i | 0.0687997 | − | 0.0946947i |
60.20 | 0.0381722 | + | 0.241010i | 0.264626 | + | 0.364226i | 1.84548 | − | 0.599634i | −0.0177153 | − | 0.0177153i | −0.0776807 | + | 0.0776807i | −3.14122 | − | 1.60053i | 0.436524 | + | 0.856726i | 0.864417 | − | 2.66040i | 0.00359333 | − | 0.00494579i |
See next 80 embeddings (of 272 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.d | odd | 4 | 1 | inner |
31.f | odd | 10 | 1 | inner |
403.bn | even | 20 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 403.2.bn.a | ✓ | 272 |
13.d | odd | 4 | 1 | inner | 403.2.bn.a | ✓ | 272 |
31.f | odd | 10 | 1 | inner | 403.2.bn.a | ✓ | 272 |
403.bn | even | 20 | 1 | inner | 403.2.bn.a | ✓ | 272 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
403.2.bn.a | ✓ | 272 | 1.a | even | 1 | 1 | trivial |
403.2.bn.a | ✓ | 272 | 13.d | odd | 4 | 1 | inner |
403.2.bn.a | ✓ | 272 | 31.f | odd | 10 | 1 | inner |
403.2.bn.a | ✓ | 272 | 403.bn | even | 20 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(403, [\chi])\).