Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [637,2,Mod(88,637)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(637, base_ring=CyclotomicField(42))
chi = DirichletCharacter(H, H._module([34, 35]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("637.88");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 637 = 7^{2} \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 637.bp (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: | \(5.08647060876\) |
Analytic rank: | \(0\) |
Dimension: | \(756\) |
Relative dimension: | \(63\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
88.1 | −2.55146 | + | 1.00138i | −0.303707 | − | 0.146257i | 4.04110 | − | 3.74960i | −3.66144 | + | 0.274387i | 0.921355 | + | 0.0690460i | −2.63968 | − | 0.179128i | −4.17748 | + | 8.67462i | −1.79962 | − | 2.25666i | 9.06726 | − | 4.36656i |
88.2 | −2.43880 | + | 0.957159i | −2.82876 | − | 1.36226i | 3.56549 | − | 3.30829i | −0.302823 | + | 0.0226935i | 8.20268 | + | 0.614706i | 2.57737 | + | 0.597636i | −3.25549 | + | 6.76010i | 4.27566 | + | 5.36151i | 0.716804 | − | 0.345195i |
88.3 | −2.41692 | + | 0.948570i | 1.89451 | + | 0.912349i | 3.47559 | − | 3.22488i | 1.43933 | − | 0.107862i | −5.44431 | − | 0.407994i | 2.21216 | + | 1.45133i | −3.08813 | + | 6.41257i | 0.886329 | + | 1.11142i | −3.37641 | + | 1.62599i |
88.4 | −2.40235 | + | 0.942851i | −0.390492 | − | 0.188051i | 3.41619 | − | 3.16976i | 1.38128 | − | 0.103512i | 1.11540 | + | 0.0835878i | 0.480491 | − | 2.60175i | −2.97878 | + | 6.18549i | −1.75335 | − | 2.19863i | −3.22071 | + | 1.55101i |
88.5 | −2.39358 | + | 0.939409i | −1.57406 | − | 0.758028i | 3.38061 | − | 3.13675i | 3.68021 | − | 0.275794i | 4.47974 | + | 0.335710i | −1.05923 | + | 2.42447i | −2.91375 | + | 6.05047i | 0.0325956 | + | 0.0408736i | −8.54978 | + | 4.11736i |
88.6 | −2.24631 | + | 0.881613i | 2.07682 | + | 1.00014i | 2.80258 | − | 2.60042i | −3.10203 | + | 0.232465i | −5.54693 | − | 0.415685i | 1.94145 | − | 1.79743i | −1.90889 | + | 3.96385i | 1.44242 | + | 1.80874i | 6.76318 | − | 3.25698i |
88.7 | −2.21641 | + | 0.869875i | 1.15945 | + | 0.558362i | 2.68967 | − | 2.49565i | 1.79264 | − | 0.134340i | −3.05552 | − | 0.228980i | −2.36417 | + | 1.18772i | −1.72435 | + | 3.58064i | −0.837911 | − | 1.05071i | −3.85637 | + | 1.85713i |
88.8 | −2.18175 | + | 0.856274i | 2.75338 | + | 1.32596i | 2.56073 | − | 2.37601i | −1.56425 | + | 0.117224i | −7.14257 | − | 0.535261i | −2.17276 | + | 1.50967i | −1.51852 | + | 3.15323i | 3.95246 | + | 4.95622i | 3.31243 | − | 1.59518i |
88.9 | −2.12012 | + | 0.832087i | −2.53379 | − | 1.22021i | 2.33645 | − | 2.16791i | −1.25010 | + | 0.0936823i | 6.38726 | + | 0.478659i | −2.64572 | − | 0.0132775i | −1.17328 | + | 2.43635i | 3.06070 | + | 3.83799i | 2.57242 | − | 1.23881i |
88.10 | −2.06002 | + | 0.808498i | −1.04079 | − | 0.501218i | 2.12391 | − | 1.97070i | −2.71907 | + | 0.203766i | 2.54928 | + | 0.191042i | 0.772418 | + | 2.53049i | −0.861620 | + | 1.78917i | −1.03845 | − | 1.30217i | 5.43659 | − | 2.61812i |
88.11 | −2.01195 | + | 0.789633i | −0.739417 | − | 0.356085i | 1.95833 | − | 1.81706i | 1.06565 | − | 0.0798593i | 1.76885 | + | 0.132557i | 0.267626 | − | 2.63218i | −0.629693 | + | 1.30757i | −1.45053 | − | 1.81890i | −2.08097 | + | 1.00214i |
88.12 | −1.75260 | + | 0.687843i | 1.74124 | + | 0.838539i | 1.13236 | − | 1.05068i | 3.79954 | − | 0.284736i | −3.62848 | − | 0.271917i | 2.60964 | − | 0.435647i | 0.371913 | − | 0.772285i | 0.458315 | + | 0.574709i | −6.46320 | + | 3.11251i |
88.13 | −1.69388 | + | 0.664798i | 1.46656 | + | 0.706257i | 0.961158 | − | 0.891824i | −0.587608 | + | 0.0440351i | −2.95369 | − | 0.221348i | −1.88488 | − | 1.85668i | 0.543844 | − | 1.12930i | −0.218477 | − | 0.273962i | 0.966061 | − | 0.465230i |
88.14 | −1.58862 | + | 0.623486i | 0.0795416 | + | 0.0383052i | 0.668864 | − | 0.620615i | −3.18924 | + | 0.239001i | −0.150244 | − | 0.0112592i | 2.60068 | − | 0.486269i | 0.805296 | − | 1.67222i | −1.86561 | − | 2.33940i | 4.91747 | − | 2.36813i |
88.15 | −1.56862 | + | 0.615640i | −1.77186 | − | 0.853281i | 0.615466 | − | 0.571069i | 0.139416 | − | 0.0104478i | 3.30469 | + | 0.247652i | 2.63298 | + | 0.259694i | 0.848421 | − | 1.76177i | 0.540916 | + | 0.678287i | −0.212259 | + | 0.102218i |
88.16 | −1.43231 | + | 0.562140i | −1.25604 | − | 0.604875i | 0.269405 | − | 0.249971i | 2.05445 | − | 0.153960i | 2.13905 | + | 0.160300i | −0.494528 | + | 2.59912i | 1.08986 | − | 2.26311i | −0.658718 | − | 0.826007i | −2.85606 | + | 1.37541i |
88.17 | −1.39678 | + | 0.548196i | −2.79574 | − | 1.34636i | 0.184376 | − | 0.171076i | 4.02490 | − | 0.301624i | 4.64310 | + | 0.347952i | 0.853845 | − | 2.50419i | 1.13834 | − | 2.36379i | 4.13300 | + | 5.18262i | −5.45655 | + | 2.62774i |
88.18 | −1.39588 | + | 0.547841i | 2.88642 | + | 1.39003i | 0.182234 | − | 0.169089i | 2.44886 | − | 0.183517i | −4.79059 | − | 0.359005i | −0.897086 | − | 2.48902i | 1.13950 | − | 2.36620i | 4.52878 | + | 5.67890i | −3.31777 | + | 1.59775i |
88.19 | −1.39074 | + | 0.545824i | 1.60197 | + | 0.771470i | 0.170123 | − | 0.157851i | −0.583615 | + | 0.0437359i | −2.64901 | − | 0.198516i | 0.511124 | + | 2.59591i | 1.14602 | − | 2.37973i | 0.100684 | + | 0.126254i | 0.787784 | − | 0.379377i |
88.20 | −1.32370 | + | 0.519514i | −1.03568 | − | 0.498757i | 0.0161854 | − | 0.0150178i | 0.809944 | − | 0.0606969i | 1.63004 | + | 0.122155i | −2.62057 | − | 0.364129i | 1.22034 | − | 2.53406i | −1.04660 | − | 1.31239i | −1.04059 | + | 0.501122i |
See next 80 embeddings (of 756 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
637.bp | even | 42 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 637.2.bp.a | ✓ | 756 |
13.e | even | 6 | 1 | 637.2.bz.a | yes | 756 | |
49.g | even | 21 | 1 | 637.2.bz.a | yes | 756 | |
637.bp | even | 42 | 1 | inner | 637.2.bp.a | ✓ | 756 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
637.2.bp.a | ✓ | 756 | 1.a | even | 1 | 1 | trivial |
637.2.bp.a | ✓ | 756 | 637.bp | even | 42 | 1 | inner |
637.2.bz.a | yes | 756 | 13.e | even | 6 | 1 | |
637.2.bz.a | yes | 756 | 49.g | even | 21 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(637, [\chi])\).