Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [847,2,Mod(241,847)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(847, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([1, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("847.241");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 847 = 7 \cdot 11^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 847.i (of order \(6\), degree \(2\), 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: | \(24\) |
Relative dimension: | \(12\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
241.1 | −2.13688 | + | 1.23373i | 2.50488 | + | 1.44619i | 2.04417 | − | 3.54061i | 1.33074 | − | 0.768306i | −7.13683 | 2.50945 | + | 0.838250i | 5.15289i | 2.68294 | + | 4.64699i | −1.89576 | + | 3.28355i | ||||
241.2 | −2.03242 | + | 1.17342i | 0.219729 | + | 0.126861i | 1.75383 | − | 3.03772i | 0.452529 | − | 0.261268i | −0.595443 | 2.16845 | + | 1.51586i | 3.53824i | −1.46781 | − | 2.54233i | −0.613154 | + | 1.06201i | ||||
241.3 | −1.38255 | + | 0.798214i | −0.877650 | − | 0.506712i | 0.274290 | − | 0.475085i | −0.282779 | + | 0.163263i | 1.61786 | 1.21424 | − | 2.35066i | − | 2.31709i | −0.986487 | − | 1.70864i | 0.260637 | − | 0.451436i | |||
241.4 | −1.01511 | + | 0.586075i | 1.45404 | + | 0.839493i | −0.313032 | + | 0.542188i | −1.91629 | + | 1.10637i | −1.96802 | −2.45295 | − | 0.991472i | − | 3.07814i | −0.0905026 | − | 0.156755i | 1.29683 | − | 2.24618i | |||
241.5 | −0.589336 | + | 0.340253i | −2.61084 | − | 1.50737i | −0.768455 | + | 1.33100i | −2.80754 | + | 1.62094i | 2.05155 | 2.64379 | − | 0.101846i | − | 2.40689i | 3.04431 | + | 5.27290i | 1.10306 | − | 1.91055i | |||
241.6 | −0.117445 | + | 0.0678066i | −0.690164 | − | 0.398467i | −0.990805 | + | 1.71612i | 3.22334 | − | 1.86100i | 0.108075 | −0.139659 | + | 2.64206i | − | 0.539959i | −1.18245 | − | 2.04806i | −0.252376 | + | 0.437127i | |||
241.7 | 0.117445 | − | 0.0678066i | −0.690164 | − | 0.398467i | −0.990805 | + | 1.71612i | 3.22334 | − | 1.86100i | −0.108075 | 0.139659 | − | 2.64206i | 0.539959i | −1.18245 | − | 2.04806i | 0.252376 | − | 0.437127i | ||||
241.8 | 0.589336 | − | 0.340253i | −2.61084 | − | 1.50737i | −0.768455 | + | 1.33100i | −2.80754 | + | 1.62094i | −2.05155 | −2.64379 | + | 0.101846i | 2.40689i | 3.04431 | + | 5.27290i | −1.10306 | + | 1.91055i | ||||
241.9 | 1.01511 | − | 0.586075i | 1.45404 | + | 0.839493i | −0.313032 | + | 0.542188i | −1.91629 | + | 1.10637i | 1.96802 | 2.45295 | + | 0.991472i | 3.07814i | −0.0905026 | − | 0.156755i | −1.29683 | + | 2.24618i | ||||
241.10 | 1.38255 | − | 0.798214i | −0.877650 | − | 0.506712i | 0.274290 | − | 0.475085i | −0.282779 | + | 0.163263i | −1.61786 | −1.21424 | + | 2.35066i | 2.31709i | −0.986487 | − | 1.70864i | −0.260637 | + | 0.451436i | ||||
241.11 | 2.03242 | − | 1.17342i | 0.219729 | + | 0.126861i | 1.75383 | − | 3.03772i | 0.452529 | − | 0.261268i | 0.595443 | −2.16845 | − | 1.51586i | − | 3.53824i | −1.46781 | − | 2.54233i | 0.613154 | − | 1.06201i | |||
241.12 | 2.13688 | − | 1.23373i | 2.50488 | + | 1.44619i | 2.04417 | − | 3.54061i | 1.33074 | − | 0.768306i | 7.13683 | −2.50945 | − | 0.838250i | − | 5.15289i | 2.68294 | + | 4.64699i | 1.89576 | − | 3.28355i | |||
362.1 | −2.13688 | − | 1.23373i | 2.50488 | − | 1.44619i | 2.04417 | + | 3.54061i | 1.33074 | + | 0.768306i | −7.13683 | 2.50945 | − | 0.838250i | − | 5.15289i | 2.68294 | − | 4.64699i | −1.89576 | − | 3.28355i | |||
362.2 | −2.03242 | − | 1.17342i | 0.219729 | − | 0.126861i | 1.75383 | + | 3.03772i | 0.452529 | + | 0.261268i | −0.595443 | 2.16845 | − | 1.51586i | − | 3.53824i | −1.46781 | + | 2.54233i | −0.613154 | − | 1.06201i | |||
362.3 | −1.38255 | − | 0.798214i | −0.877650 | + | 0.506712i | 0.274290 | + | 0.475085i | −0.282779 | − | 0.163263i | 1.61786 | 1.21424 | + | 2.35066i | 2.31709i | −0.986487 | + | 1.70864i | 0.260637 | + | 0.451436i | ||||
362.4 | −1.01511 | − | 0.586075i | 1.45404 | − | 0.839493i | −0.313032 | − | 0.542188i | −1.91629 | − | 1.10637i | −1.96802 | −2.45295 | + | 0.991472i | 3.07814i | −0.0905026 | + | 0.156755i | 1.29683 | + | 2.24618i | ||||
362.5 | −0.589336 | − | 0.340253i | −2.61084 | + | 1.50737i | −0.768455 | − | 1.33100i | −2.80754 | − | 1.62094i | 2.05155 | 2.64379 | + | 0.101846i | 2.40689i | 3.04431 | − | 5.27290i | 1.10306 | + | 1.91055i | ||||
362.6 | −0.117445 | − | 0.0678066i | −0.690164 | + | 0.398467i | −0.990805 | − | 1.71612i | 3.22334 | + | 1.86100i | 0.108075 | −0.139659 | − | 2.64206i | 0.539959i | −1.18245 | + | 2.04806i | −0.252376 | − | 0.437127i | ||||
362.7 | 0.117445 | + | 0.0678066i | −0.690164 | + | 0.398467i | −0.990805 | − | 1.71612i | 3.22334 | + | 1.86100i | −0.108075 | 0.139659 | + | 2.64206i | − | 0.539959i | −1.18245 | + | 2.04806i | 0.252376 | + | 0.437127i | |||
362.8 | 0.589336 | + | 0.340253i | −2.61084 | + | 1.50737i | −0.768455 | − | 1.33100i | −2.80754 | − | 1.62094i | −2.05155 | −2.64379 | − | 0.101846i | − | 2.40689i | 3.04431 | − | 5.27290i | −1.10306 | − | 1.91055i | |||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.d | odd | 6 | 1 | inner |
11.b | odd | 2 | 1 | inner |
77.i | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 847.2.i.a | ✓ | 24 |
7.d | odd | 6 | 1 | inner | 847.2.i.a | ✓ | 24 |
11.b | odd | 2 | 1 | inner | 847.2.i.a | ✓ | 24 |
11.c | even | 5 | 4 | 847.2.r.e | 96 | ||
11.d | odd | 10 | 4 | 847.2.r.e | 96 | ||
77.i | even | 6 | 1 | inner | 847.2.i.a | ✓ | 24 |
77.n | even | 30 | 4 | 847.2.r.e | 96 | ||
77.p | odd | 30 | 4 | 847.2.r.e | 96 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
847.2.i.a | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
847.2.i.a | ✓ | 24 | 7.d | odd | 6 | 1 | inner |
847.2.i.a | ✓ | 24 | 11.b | odd | 2 | 1 | inner |
847.2.i.a | ✓ | 24 | 77.i | even | 6 | 1 | inner |
847.2.r.e | 96 | 11.c | even | 5 | 4 | ||
847.2.r.e | 96 | 11.d | odd | 10 | 4 | ||
847.2.r.e | 96 | 77.n | even | 30 | 4 | ||
847.2.r.e | 96 | 77.p | odd | 30 | 4 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{24} - 16 T_{2}^{22} + 166 T_{2}^{20} - 1016 T_{2}^{18} + 4507 T_{2}^{16} - 12706 T_{2}^{14} + 25928 T_{2}^{12} - 32188 T_{2}^{10} + 28015 T_{2}^{8} - 11234 T_{2}^{6} + 3163 T_{2}^{4} - 58 T_{2}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(847, [\chi])\).