Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [847,2,Mod(23,847)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(847, base_ring=CyclotomicField(66))
chi = DirichletCharacter(H, H._module([22, 42]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("847.23");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 847 = 7 \cdot 11^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 847.u (of order \(33\), degree \(20\), 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: | \(1720\) |
Relative dimension: | \(86\) over \(\Q(\zeta_{33})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{33}]$ |
$q$-expansion
The algebraic \(q\)-expansion of this newform has not been computed, 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
23.1 | −1.58299 | − | 2.22300i | −0.529306 | − | 0.916785i | −1.78173 | + | 5.14797i | −2.73161 | + | 2.60458i | −1.20013 | + | 2.62791i | 0.944173 | − | 2.47155i | 9.02743 | − | 2.65069i | 0.939671 | − | 1.62756i | 10.1141 | + | 1.94933i |
23.2 | −1.54654 | − | 2.17181i | 0.114913 | + | 0.199036i | −1.67084 | + | 4.82758i | 0.314136 | − | 0.299528i | 0.254550 | − | 0.557387i | 2.59071 | + | 0.536841i | 7.95224 | − | 2.33499i | 1.47359 | − | 2.55233i | −1.13634 | − | 0.219012i |
23.3 | −1.54389 | − | 2.16809i | −1.53393 | − | 2.65685i | −1.66287 | + | 4.80455i | 1.34609 | − | 1.28349i | −3.39205 | + | 7.42756i | −1.07048 | − | 2.41952i | 7.87637 | − | 2.31271i | −3.20589 | + | 5.55276i | −4.86092 | − | 0.936866i |
23.4 | −1.54361 | − | 2.16770i | 1.30997 | + | 2.26894i | −1.66206 | + | 4.80220i | 0.519449 | − | 0.495293i | 2.89630 | − | 6.34201i | 2.54979 | − | 0.706084i | 7.86862 | − | 2.31044i | −1.93206 | + | 3.34643i | −1.87548 | − | 0.361469i |
23.5 | −1.53915 | − | 2.16144i | −0.737739 | − | 1.27780i | −1.64868 | + | 4.76356i | 1.21935 | − | 1.16265i | −1.62639 | + | 3.56131i | −2.34511 | + | 1.22493i | 7.74177 | − | 2.27319i | 0.411483 | − | 0.712709i | −4.38976 | − | 0.846058i |
23.6 | −1.41974 | − | 1.99374i | −1.04317 | − | 1.80682i | −1.30522 | + | 3.77118i | −1.72874 | + | 1.64835i | −2.12130 | + | 4.64501i | −0.295823 | + | 2.62916i | 4.67495 | − | 1.37269i | −0.676390 | + | 1.17154i | 5.74074 | + | 1.10644i |
23.7 | −1.41639 | − | 1.98904i | 0.754247 | + | 1.30639i | −1.29599 | + | 3.74452i | 2.88425 | − | 2.75013i | 1.53017 | − | 3.35060i | −0.521292 | − | 2.59389i | 4.59784 | − | 1.35005i | 0.362222 | − | 0.627387i | −9.55536 | − | 1.84164i |
23.8 | −1.39839 | − | 1.96377i | 0.752732 | + | 1.30377i | −1.24675 | + | 3.60224i | −1.42488 | + | 1.35862i | 1.50769 | − | 3.30138i | −2.64184 | − | 0.143811i | 4.19115 | − | 1.23063i | 0.366789 | − | 0.635297i | 4.66055 | + | 0.898247i |
23.9 | −1.30730 | − | 1.83585i | 1.04438 | + | 1.80891i | −1.00717 | + | 2.91001i | −0.634596 | + | 0.605086i | 1.95557 | − | 4.28211i | −0.553085 | + | 2.58730i | 2.33411 | − | 0.685355i | −0.681441 | + | 1.18029i | 1.94046 | + | 0.373992i |
23.10 | −1.28860 | − | 1.80958i | 0.333684 | + | 0.577958i | −0.959972 | + | 2.77366i | −1.44180 | + | 1.37475i | 0.615878 | − | 1.34859i | −0.464073 | − | 2.60473i | 1.99315 | − | 0.585241i | 1.27731 | − | 2.21237i | 4.34563 | + | 0.837552i |
23.11 | −1.28779 | − | 1.80845i | −1.05900 | − | 1.83424i | −0.957945 | + | 2.76780i | 2.85294 | − | 2.72028i | −1.95336 | + | 4.27725i | 2.60448 | − | 0.465469i | 1.97870 | − | 0.580997i | −0.742952 | + | 1.28683i | −8.59347 | − | 1.65626i |
23.12 | −1.26662 | − | 1.77872i | −0.0320353 | − | 0.0554867i | −0.905377 | + | 2.61591i | 0.962361 | − | 0.917609i | −0.0581187 | + | 0.127262i | 1.16758 | + | 2.37418i | 1.60942 | − | 0.472568i | 1.49795 | − | 2.59452i | −2.85111 | − | 0.549507i |
23.13 | −1.26589 | − | 1.77769i | 1.32996 | + | 2.30355i | −0.903582 | + | 2.61073i | −3.13876 | + | 2.99281i | 2.41143 | − | 5.28030i | 1.96223 | + | 1.77473i | 1.59701 | − | 0.468923i | −2.03757 | + | 3.52918i | 9.29362 | + | 1.79120i |
23.14 | −1.24481 | − | 1.74809i | −0.628335 | − | 1.08831i | −0.852131 | + | 2.46207i | −0.293339 | + | 0.279699i | −1.12030 | + | 2.45312i | −2.19146 | − | 1.48239i | 1.24649 | − | 0.366003i | 0.710390 | − | 1.23043i | 0.854089 | + | 0.164612i |
23.15 | −1.23063 | − | 1.72817i | −1.49574 | − | 2.59069i | −0.818005 | + | 2.36347i | −1.74478 | + | 1.66364i | −2.63647 | + | 5.77306i | 2.63832 | − | 0.198173i | 1.01990 | − | 0.299469i | −2.97445 | + | 5.15190i | 5.02223 | + | 0.967955i |
23.16 | −1.16442 | − | 1.63519i | 1.21406 | + | 2.10281i | −0.663855 | + | 1.91808i | 2.54551 | − | 2.42714i | 2.02483 | − | 4.43376i | −2.62336 | + | 0.343509i | 0.0572360 | − | 0.0168060i | −1.44787 | + | 2.50779i | −6.93286 | − | 1.33620i |
23.17 | −1.08714 | − | 1.52668i | 1.52438 | + | 2.64031i | −0.494736 | + | 1.42945i | 0.554663 | − | 0.528870i | 2.37369 | − | 5.19765i | 0.743974 | − | 2.53900i | −0.876412 | + | 0.257338i | −3.14749 | + | 5.45162i | −1.41042 | − | 0.271835i |
23.18 | −1.06888 | − | 1.50103i | −1.66808 | − | 2.88919i | −0.456452 | + | 1.31883i | 1.17953 | − | 1.12468i | −2.55379 | + | 5.59203i | −0.367949 | + | 2.62004i | −1.06864 | + | 0.313782i | −4.06496 | + | 7.04072i | −2.94896 | − | 0.568365i |
23.19 | −1.05144 | − | 1.47654i | −0.136956 | − | 0.237214i | −0.420511 | + | 1.21499i | 1.03151 | − | 0.983547i | −0.206256 | + | 0.451637i | 1.54090 | − | 2.15073i | −1.24233 | + | 0.364781i | 1.46249 | − | 2.53310i | −2.53682 | − | 0.488933i |
23.20 | −1.02678 | − | 1.44191i | 0.607575 | + | 1.05235i | −0.370686 | + | 1.07103i | 2.00114 | − | 1.90808i | 0.893547 | − | 1.95660i | 0.986575 | + | 2.45493i | −1.47193 | + | 0.432196i | 0.761704 | − | 1.31931i | −4.80599 | − | 0.926279i |
See next 80 embeddings (of 1720 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.c | even | 3 | 1 | inner |
121.e | even | 11 | 1 | inner |
847.u | even | 33 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 847.2.u.a | ✓ | 1720 |
7.c | even | 3 | 1 | inner | 847.2.u.a | ✓ | 1720 |
121.e | even | 11 | 1 | inner | 847.2.u.a | ✓ | 1720 |
847.u | even | 33 | 1 | inner | 847.2.u.a | ✓ | 1720 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
847.2.u.a | ✓ | 1720 | 1.a | even | 1 | 1 | trivial |
847.2.u.a | ✓ | 1720 | 7.c | even | 3 | 1 | inner |
847.2.u.a | ✓ | 1720 | 121.e | even | 11 | 1 | inner |
847.2.u.a | ✓ | 1720 | 847.u | even | 33 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(847, [\chi])\).