Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [603,2,Mod(200,603)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(603, 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("603.200");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 603 = 3^{2} \cdot 67 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 603.l (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: | \(4.81497924188\) |
Analytic rank: | \(0\) |
Dimension: | \(132\) |
Relative dimension: | \(66\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
200.1 | −1.39584 | + | 2.41767i | −1.57689 | − | 0.716525i | −2.89677 | − | 5.01735i | 0.979162 | + | 1.69596i | 3.93342 | − | 2.81226i | −0.484838 | − | 0.279921i | 10.5904 | 1.97318 | + | 2.25977i | −5.46704 | ||||
200.2 | −1.34567 | + | 2.33076i | 0.375699 | − | 1.69081i | −2.62164 | − | 4.54082i | −1.69632 | − | 2.93811i | 3.43532 | + | 3.15094i | −2.38340 | − | 1.37606i | 8.72877 | −2.71770 | − | 1.27048i | 9.13072 | ||||
200.3 | −1.33814 | + | 2.31773i | 0.836295 | + | 1.51678i | −2.58126 | − | 4.47087i | −0.663575 | − | 1.14934i | −4.63456 | − | 0.0913554i | 2.68931 | + | 1.55268i | 8.46378 | −1.60122 | + | 2.53695i | 3.55183 | ||||
200.4 | −1.24929 | + | 2.16383i | 1.72470 | − | 0.159387i | −2.12144 | − | 3.67444i | 2.04806 | + | 3.54734i | −1.80976 | + | 3.93108i | 1.26737 | + | 0.731715i | 5.60401 | 2.94919 | − | 0.549790i | −10.2345 | ||||
200.5 | −1.23668 | + | 2.14199i | −1.34361 | + | 1.09302i | −2.05875 | − | 3.56586i | −1.63023 | − | 2.82365i | −0.679631 | − | 4.22973i | −2.34312 | − | 1.35280i | 5.23732 | 0.610593 | − | 2.93721i | 8.06430 | ||||
200.6 | −1.22339 | + | 2.11898i | −0.333349 | + | 1.69967i | −1.99338 | − | 3.45263i | 0.149207 | + | 0.258435i | −3.19375 | − | 2.78572i | 0.0311178 | + | 0.0179659i | 4.86116 | −2.77776 | − | 1.13317i | −0.730156 | ||||
200.7 | −1.18642 | + | 2.05493i | 0.117735 | − | 1.72804i | −1.81517 | − | 3.14397i | 0.408528 | + | 0.707591i | 3.41134 | + | 2.29212i | 4.18823 | + | 2.41807i | 3.86854 | −2.97228 | − | 0.406902i | −1.93874 | ||||
200.8 | −1.18232 | + | 2.04784i | 0.779490 | − | 1.54674i | −1.79576 | − | 3.11035i | 1.18737 | + | 2.05659i | 2.24586 | + | 3.42501i | −3.42944 | − | 1.97999i | 3.76339 | −1.78479 | − | 2.41133i | −5.61541 | ||||
200.9 | −1.16336 | + | 2.01499i | −1.59193 | + | 0.682477i | −1.70679 | − | 2.95625i | 0.719334 | + | 1.24592i | 0.476791 | − | 4.00168i | 1.79079 | + | 1.03391i | 3.28900 | 2.06845 | − | 2.17290i | −3.34736 | ||||
200.10 | −1.13151 | + | 1.95983i | 1.69797 | − | 0.341928i | −1.56063 | − | 2.70309i | −1.14681 | − | 1.98634i | −1.25114 | + | 3.71462i | 0.398259 | + | 0.229935i | 2.53742 | 2.76617 | − | 1.16116i | 5.19052 | ||||
200.11 | −1.00472 | + | 1.74023i | −1.02992 | − | 1.39258i | −1.01894 | − | 1.76486i | −0.0100438 | − | 0.0173963i | 3.45819 | − | 0.393147i | −1.41764 | − | 0.818474i | 0.0761201 | −0.878531 | + | 2.86848i | 0.0403649 | ||||
200.12 | −1.00027 | + | 1.73252i | −1.71459 | − | 0.245331i | −1.00108 | − | 1.73392i | −1.52156 | − | 2.63541i | 2.14009 | − | 2.72516i | 3.85817 | + | 2.22752i | 0.00431800 | 2.87962 | + | 0.841285i | 6.08787 | ||||
200.13 | −0.948352 | + | 1.64259i | 0.853494 | + | 1.50717i | −0.798743 | − | 1.38346i | 1.60208 | + | 2.77488i | −3.28507 | − | 0.0273793i | 0.395721 | + | 0.228469i | −0.763450 | −1.54310 | + | 2.57271i | −6.07734 | ||||
200.14 | −0.930733 | + | 1.61208i | 1.58684 | + | 0.694223i | −0.732528 | − | 1.26878i | −0.136455 | − | 0.236347i | −2.59606 | + | 1.91197i | −2.43107 | − | 1.40358i | −0.995781 | 2.03611 | + | 2.20324i | 0.508012 | ||||
200.15 | −0.838601 | + | 1.45250i | 1.47894 | + | 0.901512i | −0.406503 | − | 0.704084i | −1.89359 | − | 3.27979i | −2.54969 | + | 1.39216i | 2.42055 | + | 1.39750i | −1.99083 | 1.37455 | + | 2.66657i | 6.35186 | ||||
200.16 | −0.803054 | + | 1.39093i | −1.71183 | + | 0.263919i | −0.289792 | − | 0.501935i | 1.80724 | + | 3.13023i | 1.00760 | − | 2.59297i | −4.37008 | − | 2.52307i | −2.28134 | 2.86069 | − | 0.903565i | −5.80525 | ||||
200.17 | −0.797907 | + | 1.38202i | 1.31849 | − | 1.12321i | −0.273312 | − | 0.473390i | 0.456179 | + | 0.790125i | 0.500259 | + | 2.71838i | −2.17126 | − | 1.25358i | −2.31932 | 0.476814 | − | 2.96187i | −1.45595 | ||||
200.18 | −0.762360 | + | 1.32045i | 0.182202 | + | 1.72244i | −0.162384 | − | 0.281258i | −0.908294 | − | 1.57321i | −2.41329 | − | 1.07253i | −2.60798 | − | 1.50572i | −2.55426 | −2.93360 | + | 0.627664i | 2.76979 | ||||
200.19 | −0.746535 | + | 1.29304i | −0.997353 | + | 1.41608i | −0.114628 | − | 0.198542i | 1.00458 | + | 1.73999i | −1.08649 | − | 2.34677i | 1.99677 | + | 1.15284i | −2.64384 | −1.01057 | − | 2.82467i | −2.99982 | ||||
200.20 | −0.716678 | + | 1.24132i | 0.0154551 | − | 1.73198i | −0.0272550 | − | 0.0472070i | −2.10023 | − | 3.63771i | 2.13887 | + | 1.26046i | 0.623932 | + | 0.360227i | −2.78858 | −2.99952 | − | 0.0535358i | 6.02077 | ||||
See next 80 embeddings (of 132 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.d | odd | 6 | 1 | inner |
67.b | odd | 2 | 1 | inner |
603.l | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 603.2.l.a | ✓ | 132 |
9.d | odd | 6 | 1 | inner | 603.2.l.a | ✓ | 132 |
67.b | odd | 2 | 1 | inner | 603.2.l.a | ✓ | 132 |
603.l | even | 6 | 1 | inner | 603.2.l.a | ✓ | 132 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
603.2.l.a | ✓ | 132 | 1.a | even | 1 | 1 | trivial |
603.2.l.a | ✓ | 132 | 9.d | odd | 6 | 1 | inner |
603.2.l.a | ✓ | 132 | 67.b | odd | 2 | 1 | inner |
603.2.l.a | ✓ | 132 | 603.l | even | 6 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(603, [\chi])\).