Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [847,2,Mod(4,847)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(847, base_ring=CyclotomicField(330))
chi = DirichletCharacter(H, H._module([220, 6]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("847.4");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 847 = 7 \cdot 11^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 847.bc (of order \(165\), degree \(80\), 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: | \(6.76332905120\) |
Analytic rank: | \(0\) |
Dimension: | \(6880\) |
Relative dimension: | \(86\) over \(\Q(\zeta_{165})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{165}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
4.1 | −1.23061 | − | 2.44389i | −1.47523 | − | 1.63841i | −3.26729 | + | 4.40817i | 0.491572 | − | 0.781153i | −2.18867 | + | 5.62155i | −2.37186 | + | 1.17230i | 9.40152 | + | 1.62700i | −0.194496 | + | 1.85051i | −2.51399 | − | 0.240057i |
4.2 | −1.21978 | − | 2.42238i | 1.80004 | + | 1.99915i | −3.18917 | + | 4.30277i | 0.174931 | − | 0.277981i | 2.64705 | − | 6.79890i | −1.30506 | − | 2.30148i | 8.96818 | + | 1.55200i | −0.442859 | + | 4.21352i | −0.886754 | − | 0.0846747i |
4.3 | −1.21689 | − | 2.41665i | −1.18965 | − | 1.32124i | −3.16845 | + | 4.27482i | 1.15150 | − | 1.82984i | −1.74530 | + | 4.48276i | 2.34736 | − | 1.22061i | 8.85416 | + | 1.53227i | −0.0168215 | + | 0.160046i | −5.82334 | − | 0.556061i |
4.4 | −1.21359 | − | 2.41009i | 0.687247 | + | 0.763265i | −3.14482 | + | 4.24293i | −2.05054 | + | 3.25850i | 1.00550 | − | 2.58261i | 0.677642 | + | 2.55750i | 8.72461 | + | 1.50985i | 0.203320 | − | 1.93446i | 10.3418 | + | 0.987520i |
4.5 | −1.15760 | − | 2.29891i | −1.07578 | − | 1.19477i | −2.75402 | + | 3.71568i | −1.87138 | + | 2.97379i | −1.50135 | + | 3.85618i | 1.30507 | − | 2.30147i | 6.65763 | + | 1.15215i | 0.0434029 | − | 0.412951i | 9.00279 | + | 0.859663i |
4.6 | −1.14692 | − | 2.27770i | 1.16642 | + | 1.29544i | −2.68158 | + | 3.61794i | −0.208637 | + | 0.331544i | 1.61284 | − | 4.14254i | 2.29105 | − | 1.32329i | 6.29053 | + | 1.08862i | −0.00404607 | + | 0.0384958i | 0.994450 | + | 0.0949585i |
4.7 | −1.09197 | − | 2.16858i | 0.240894 | + | 0.267540i | −2.31940 | + | 3.12929i | −0.922980 | + | 1.46670i | 0.317131 | − | 0.814543i | −2.60006 | − | 0.489558i | 4.53398 | + | 0.784636i | 0.300038 | − | 2.85467i | 4.18852 | + | 0.399955i |
4.8 | −1.08702 | − | 2.15874i | 2.06298 | + | 2.29117i | −2.28762 | + | 3.08642i | 0.177774 | − | 0.282499i | 2.70354 | − | 6.94398i | 1.43027 | + | 2.22583i | 4.38632 | + | 0.759082i | −0.679993 | + | 6.46970i | −0.803085 | − | 0.0766853i |
4.9 | −1.07561 | − | 2.13607i | 0.177504 | + | 0.197138i | −2.21495 | + | 2.98837i | 1.97862 | − | 3.14421i | 0.230177 | − | 0.591204i | −1.28890 | − | 2.31057i | 4.05264 | + | 0.701336i | 0.306230 | − | 2.91358i | −8.84447 | − | 0.844544i |
4.10 | −1.02809 | − | 2.04171i | −0.754573 | − | 0.838039i | −1.92068 | + | 2.59135i | 0.372021 | − | 0.591175i | −0.935261 | + | 2.40220i | 0.0111053 | + | 2.64573i | 2.76048 | + | 0.477720i | 0.180658 | − | 1.71884i | −1.58948 | − | 0.151777i |
4.11 | −1.01908 | − | 2.02380i | 0.492008 | + | 0.546430i | −1.86635 | + | 2.51805i | 1.64901 | − | 2.62043i | 0.604475 | − | 1.55258i | 2.29599 | + | 1.31469i | 2.53257 | + | 0.438278i | 0.257071 | − | 2.44587i | −6.98370 | − | 0.666863i |
4.12 | −1.00780 | − | 2.00142i | −1.79161 | − | 1.98979i | −1.79909 | + | 2.42731i | −1.69658 | + | 2.69602i | −2.17681 | + | 5.59109i | −2.62261 | − | 0.349202i | 2.25516 | + | 0.390271i | −0.435794 | + | 4.14630i | 7.10569 | + | 0.678511i |
4.13 | −0.934263 | − | 1.85538i | −2.28606 | − | 2.53893i | −1.37865 | + | 1.86005i | 0.103820 | − | 0.164979i | −2.57488 | + | 6.61352i | 2.41879 | + | 1.07214i | 0.645316 | + | 0.111676i | −0.906493 | + | 8.62471i | −0.403093 | − | 0.0384907i |
4.14 | −0.899031 | − | 1.78541i | 0.525828 | + | 0.583992i | −1.18850 | + | 1.60350i | −1.01707 | + | 1.61622i | 0.569926 | − | 1.46384i | 1.93192 | − | 1.80767i | −0.00801139 | − | 0.00138643i | 0.249035 | − | 2.36941i | 3.79999 | + | 0.362855i |
4.15 | −0.860197 | − | 1.70829i | 1.38309 | + | 1.53608i | −0.987379 | + | 1.33215i | 0.622437 | − | 0.989109i | 1.43433 | − | 3.68405i | −1.91129 | + | 1.82947i | −0.644208 | − | 0.111485i | −0.133012 | + | 1.26553i | −2.22510 | − | 0.212471i |
4.16 | −0.860059 | − | 1.70801i | −1.04358 | − | 1.15901i | −0.986678 | + | 1.33121i | −1.34289 | + | 2.13397i | −1.08207 | + | 2.77927i | 1.52260 | + | 2.16372i | −0.646320 | − | 0.111850i | 0.0593327 | − | 0.564513i | 4.79982 | + | 0.458327i |
4.17 | −0.848040 | − | 1.68414i | −1.50427 | − | 1.67066i | −0.926242 | + | 1.24967i | 2.04012 | − | 3.24193i | −1.53795 | + | 3.95020i | −1.36153 | + | 2.26853i | −0.825864 | − | 0.142921i | −0.214697 | + | 2.04271i | −7.18998 | − | 0.686560i |
4.18 | −0.844935 | − | 1.67798i | −0.215561 | − | 0.239404i | −0.910766 | + | 1.22879i | 0.977338 | − | 1.55308i | −0.219580 | + | 0.563987i | −1.33462 | − | 2.28447i | −0.870953 | − | 0.150724i | 0.302737 | − | 2.88035i | −3.43182 | − | 0.327699i |
4.19 | −0.800529 | − | 1.58979i | 1.78562 | + | 1.98313i | −0.695664 | + | 0.938578i | −0.801920 | + | 1.27432i | 1.72332 | − | 4.42631i | −0.643792 | − | 2.56623i | −1.45875 | − | 0.252447i | −0.430786 | + | 4.09865i | 2.66787 | + | 0.254751i |
4.20 | −0.792818 | − | 1.57448i | −0.885258 | − | 0.983179i | −0.659493 | + | 0.889776i | −0.557106 | + | 0.885292i | −0.846142 | + | 2.17330i | 2.54101 | + | 0.737059i | −1.55021 | − | 0.268275i | 0.130627 | − | 1.24283i | 1.83555 | + | 0.175274i |
See next 80 embeddings (of 6880 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.c | even | 3 | 1 | inner |
121.g | even | 55 | 1 | inner |
847.bc | even | 165 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 847.2.bc.a | ✓ | 6880 |
7.c | even | 3 | 1 | inner | 847.2.bc.a | ✓ | 6880 |
121.g | even | 55 | 1 | inner | 847.2.bc.a | ✓ | 6880 |
847.bc | even | 165 | 1 | inner | 847.2.bc.a | ✓ | 6880 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
847.2.bc.a | ✓ | 6880 | 1.a | even | 1 | 1 | trivial |
847.2.bc.a | ✓ | 6880 | 7.c | even | 3 | 1 | inner |
847.2.bc.a | ✓ | 6880 | 121.g | even | 55 | 1 | inner |
847.2.bc.a | ✓ | 6880 | 847.bc | even | 165 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(847, [\chi])\).