Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [403,2,Mod(57,403)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(403, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([9, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("403.57");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 403 = 13 \cdot 31 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 403.be (of order \(12\), degree \(4\), 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: | \(3.21797120146\) |
Analytic rank: | \(0\) |
Dimension: | \(136\) |
Relative dimension: | \(34\) over \(\Q(\zeta_{12})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
57.1 | −1.92370 | + | 1.92370i | 2.94175 | + | 1.69842i | − | 5.40127i | −0.883813 | − | 0.236817i | −8.92630 | + | 2.39180i | −2.87039 | + | 0.769118i | 6.54303 | + | 6.54303i | 4.26925 | + | 7.39456i | 2.15576 | − | 1.24463i | |
57.2 | −1.87740 | + | 1.87740i | −2.43895 | − | 1.40813i | − | 5.04929i | 1.13969 | + | 0.305379i | 7.22253 | − | 1.93527i | 0.691082 | − | 0.185175i | 5.72474 | + | 5.72474i | 2.46566 | + | 4.27065i | −2.71297 | + | 1.56634i | |
57.3 | −1.75183 | + | 1.75183i | 0.421253 | + | 0.243210i | − | 4.13779i | 0.481245 | + | 0.128949i | −1.16402 | + | 0.311899i | 0.251958 | − | 0.0675120i | 3.74504 | + | 3.74504i | −1.38170 | − | 2.39317i | −1.06895 | + | 0.617161i | |
57.4 | −1.71373 | + | 1.71373i | 0.673401 | + | 0.388788i | − | 3.87374i | 3.38103 | + | 0.905945i | −1.82031 | + | 0.487750i | −0.0747797 | + | 0.0200372i | 3.21109 | + | 3.21109i | −1.19769 | − | 2.07446i | −7.34672 | + | 4.24163i | |
57.5 | −1.63991 | + | 1.63991i | −1.62343 | − | 0.937289i | − | 3.37862i | −3.90548 | − | 1.04647i | 4.19936 | − | 1.12521i | −3.18409 | + | 0.853175i | 2.26082 | + | 2.26082i | 0.257021 | + | 0.445173i | 8.12076 | − | 4.68852i | |
57.6 | −1.60653 | + | 1.60653i | 1.41629 | + | 0.817697i | − | 3.16189i | −2.93216 | − | 0.785669i | −3.58898 | + | 0.961663i | 4.32754 | − | 1.15956i | 1.86662 | + | 1.86662i | −0.162744 | − | 0.281882i | 5.97281 | − | 3.44840i | |
57.7 | −1.31740 | + | 1.31740i | −1.87836 | − | 1.08447i | − | 1.47108i | 1.82236 | + | 0.488300i | 3.90323 | − | 1.04587i | −2.35057 | + | 0.629834i | −0.696799 | − | 0.696799i | 0.852156 | + | 1.47598i | −3.04406 | + | 1.75749i | |
57.8 | −1.10405 | + | 1.10405i | −1.49319 | − | 0.862095i | − | 0.437851i | −2.41909 | − | 0.648194i | 2.60035 | − | 0.696763i | 2.14125 | − | 0.573746i | −1.72469 | − | 1.72469i | −0.0135854 | − | 0.0235306i | 3.38644 | − | 1.95516i | |
57.9 | −1.06684 | + | 1.06684i | −1.24662 | − | 0.719735i | − | 0.276298i | 3.17907 | + | 0.851829i | 2.09779 | − | 0.562100i | 4.69129 | − | 1.25703i | −1.83892 | − | 1.83892i | −0.463962 | − | 0.803606i | −4.30033 | + | 2.48279i | |
57.10 | −1.05928 | + | 1.05928i | 1.91771 | + | 1.10719i | − | 0.244140i | −0.214757 | − | 0.0575440i | −3.20421 | + | 0.858565i | 2.75305 | − | 0.737677i | −1.85994 | − | 1.85994i | 0.951740 | + | 1.64846i | 0.288442 | − | 0.166532i | |
57.11 | −0.968661 | + | 0.968661i | 0.615455 | + | 0.355333i | 0.123393i | 1.15257 | + | 0.308831i | −0.940364 | + | 0.251970i | −2.39214 | + | 0.640972i | −2.05685 | − | 2.05685i | −1.24748 | − | 2.16069i | −1.41560 | + | 0.817300i | ||
57.12 | −0.890053 | + | 0.890053i | 2.76707 | + | 1.59757i | 0.415612i | 2.13125 | + | 0.571067i | −3.88476 | + | 1.04092i | 1.08500 | − | 0.290724i | −2.15002 | − | 2.15002i | 3.60445 | + | 6.24308i | −2.40520 | + | 1.38865i | ||
57.13 | −0.689883 | + | 0.689883i | −2.73323 | − | 1.57803i | 1.04812i | 0.999543 | + | 0.267827i | 2.97426 | − | 0.796952i | −0.376277 | + | 0.100823i | −2.10285 | − | 2.10285i | 3.48036 | + | 6.02817i | −0.874337 | + | 0.504799i | ||
57.14 | −0.472261 | + | 0.472261i | 2.26659 | + | 1.30862i | 1.55394i | −3.80869 | − | 1.02054i | −1.68843 | + | 0.452414i | −3.01874 | + | 0.808868i | −1.67839 | − | 1.67839i | 1.92496 | + | 3.33413i | 2.28066 | − | 1.31674i | ||
57.15 | −0.456690 | + | 0.456690i | −0.104092 | − | 0.0600978i | 1.58287i | −2.27787 | − | 0.610354i | 0.0749841 | − | 0.0200919i | 0.192857 | − | 0.0516759i | −1.63626 | − | 1.63626i | −1.49278 | − | 2.58556i | 1.31902 | − | 0.761539i | ||
57.16 | −0.308073 | + | 0.308073i | 1.87353 | + | 1.08168i | 1.81018i | 3.38449 | + | 0.906872i | −0.910421 | + | 0.243947i | −3.42053 | + | 0.916529i | −1.17382 | − | 1.17382i | 0.840073 | + | 1.45505i | −1.32205 | + | 0.763289i | ||
57.17 | −0.0974514 | + | 0.0974514i | 0.799719 | + | 0.461718i | 1.98101i | 1.71922 | + | 0.460663i | −0.122929 | + | 0.0329386i | 2.76053 | − | 0.739683i | −0.387954 | − | 0.387954i | −1.07363 | − | 1.85959i | −0.212432 | + | 0.122648i | ||
57.18 | 0.0740210 | − | 0.0740210i | −1.19559 | − | 0.690275i | 1.98904i | 2.69078 | + | 0.720992i | −0.139594 | + | 0.0374041i | −0.0708884 | + | 0.0189945i | 0.295273 | + | 0.295273i | −0.547040 | − | 0.947500i | 0.252543 | − | 0.145806i | ||
57.19 | 0.0798971 | − | 0.0798971i | −1.31416 | − | 0.758731i | 1.98723i | 1.61887 | + | 0.433775i | −0.165618 | + | 0.0443772i | −4.54713 | + | 1.21840i | 0.318568 | + | 0.318568i | −0.348656 | − | 0.603889i | 0.164001 | − | 0.0946857i | ||
57.20 | 0.185549 | − | 0.185549i | −2.28373 | − | 1.31851i | 1.93114i | −3.12819 | − | 0.838196i | −0.668395 | + | 0.179096i | −1.29814 | + | 0.347834i | 0.729421 | + | 0.729421i | 1.97696 | + | 3.42420i | −0.735961 | + | 0.424907i | ||
See next 80 embeddings (of 136 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.d | odd | 4 | 1 | inner |
31.e | odd | 6 | 1 | inner |
403.be | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 403.2.be.c | ✓ | 136 |
13.d | odd | 4 | 1 | inner | 403.2.be.c | ✓ | 136 |
31.e | odd | 6 | 1 | inner | 403.2.be.c | ✓ | 136 |
403.be | even | 12 | 1 | inner | 403.2.be.c | ✓ | 136 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
403.2.be.c | ✓ | 136 | 1.a | even | 1 | 1 | trivial |
403.2.be.c | ✓ | 136 | 13.d | odd | 4 | 1 | inner |
403.2.be.c | ✓ | 136 | 31.e | odd | 6 | 1 | inner |
403.2.be.c | ✓ | 136 | 403.be | even | 12 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(403, [\chi])\):
\( T_{2}^{68} + 2 T_{2}^{67} + 2 T_{2}^{66} + 263 T_{2}^{64} + 524 T_{2}^{63} + 522 T_{2}^{62} + \cdots + 18225 \) |
\( T_{7}^{136} + 2 T_{7}^{135} + 2 T_{7}^{134} - 40 T_{7}^{133} - 1872 T_{7}^{132} - 3112 T_{7}^{131} + \cdots + 63\!\cdots\!41 \) |