Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [624,2,Mod(157,624)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(624, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("624.157");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 624 = 2^{4} \cdot 3 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 624.x (of order \(4\), 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: | \(4.98266508613\) |
Analytic rank: | \(0\) |
Dimension: | \(56\) |
Relative dimension: | \(28\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
157.1 | −1.40239 | − | 0.182491i | 0.707107 | + | 0.707107i | 1.93339 | + | 0.511848i | −1.29029 | + | 1.29029i | −0.862598 | − | 1.12068i | − | 4.74894i | −2.61796 | − | 1.07064i | 1.00000i | 2.04496 | − | 1.57402i | |||
157.2 | −1.36726 | + | 0.361394i | 0.707107 | + | 0.707107i | 1.73879 | − | 0.988237i | 2.24248 | − | 2.24248i | −1.22234 | − | 0.711253i | − | 0.711007i | −2.02023 | + | 1.97956i | 1.00000i | −2.25563 | + | 3.87647i | |||
157.3 | −1.33481 | − | 0.467208i | −0.707107 | − | 0.707107i | 1.56343 | + | 1.24727i | 1.08546 | − | 1.08546i | 0.613487 | + | 1.27422i | − | 3.56874i | −1.50415 | − | 2.39531i | 1.00000i | −1.95602 | + | 0.941749i | |||
157.4 | −1.30024 | − | 0.556219i | −0.707107 | − | 0.707107i | 1.38124 | + | 1.44644i | −2.42862 | + | 2.42862i | 0.526101 | + | 1.31271i | 5.20571i | −0.991405 | − | 2.64898i | 1.00000i | 4.50863 | − | 1.80694i | ||||
157.5 | −1.29421 | + | 0.570097i | −0.707107 | − | 0.707107i | 1.34998 | − | 1.47566i | −2.02025 | + | 2.02025i | 1.31827 | + | 0.512028i | − | 0.634897i | −0.905894 | + | 2.67943i | 1.00000i | 1.46289 | − | 3.76637i | |||
157.6 | −1.14499 | + | 0.830062i | −0.707107 | − | 0.707107i | 0.621993 | − | 1.90082i | 1.63652 | − | 1.63652i | 1.39657 | + | 0.222686i | 2.06925i | 0.865628 | + | 2.69271i | 1.00000i | −0.515380 | + | 3.23220i | ||||
157.7 | −1.01862 | + | 0.981027i | 0.707107 | + | 0.707107i | 0.0751712 | − | 1.99859i | 0.918279 | − | 0.918279i | −1.41396 | − | 0.0265817i | 3.67302i | 1.88410 | + | 2.10954i | 1.00000i | −0.0345202 | + | 1.83623i | ||||
157.8 | −0.913674 | − | 1.07944i | 0.707107 | + | 0.707107i | −0.330401 | + | 1.97252i | −0.470608 | + | 0.470608i | 0.117218 | − | 1.40935i | 2.14920i | 2.43111 | − | 1.44559i | 1.00000i | 0.937977 | + | 0.0780132i | ||||
157.9 | −0.881982 | − | 1.10549i | −0.707107 | − | 0.707107i | −0.444217 | + | 1.95004i | −0.687536 | + | 0.687536i | −0.158044 | + | 1.40535i | − | 2.87123i | 2.54755 | − | 1.22883i | 1.00000i | 1.36646 | + | 0.153670i | |||
157.10 | −0.779416 | + | 1.18005i | 0.707107 | + | 0.707107i | −0.785021 | − | 1.83950i | −0.439534 | + | 0.439534i | −1.38555 | + | 0.283289i | − | 3.49065i | 2.78255 | + | 0.507371i | 1.00000i | −0.176091 | − | 0.861251i | |||
157.11 | −0.546564 | + | 1.30433i | −0.707107 | − | 0.707107i | −1.40253 | − | 1.42580i | −0.741100 | + | 0.741100i | 1.30878 | − | 0.535819i | 3.54681i | 2.62628 | − | 1.05007i | 1.00000i | −0.561578 | − | 1.37170i | ||||
157.12 | −0.327918 | + | 1.37567i | 0.707107 | + | 0.707107i | −1.78494 | − | 0.902215i | −3.01052 | + | 3.01052i | −1.20462 | + | 0.740873i | 4.33033i | 1.82646 | − | 2.15964i | 1.00000i | −3.15428 | − | 5.12869i | ||||
157.13 | −0.326306 | − | 1.37605i | 0.707107 | + | 0.707107i | −1.78705 | + | 0.898028i | −0.534609 | + | 0.534609i | 0.742284 | − | 1.20375i | − | 1.29164i | 1.81886 | + | 2.16604i | 1.00000i | 0.910097 | + | 0.561205i | |||
157.14 | −0.173487 | − | 1.40353i | −0.707107 | − | 0.707107i | −1.93980 | + | 0.486990i | −0.503626 | + | 0.503626i | −0.869773 | + | 1.11512i | 0.824309i | 1.02004 | + | 2.63809i | 1.00000i | 0.794228 | + | 0.619483i | ||||
157.15 | −0.125504 | + | 1.40863i | −0.707107 | − | 0.707107i | −1.96850 | − | 0.353578i | 2.13999 | − | 2.13999i | 1.08480 | − | 0.907310i | − | 2.18307i | 0.745117 | − | 2.72852i | 1.00000i | 2.74588 | + | 3.28304i | |||
157.16 | 0.179347 | − | 1.40280i | 0.707107 | + | 0.707107i | −1.93567 | − | 0.503174i | 2.10708 | − | 2.10708i | 1.11874 | − | 0.865109i | 0.641112i | −1.05301 | + | 2.62511i | 1.00000i | −2.57791 | − | 3.33371i | ||||
157.17 | 0.511838 | + | 1.31834i | −0.707107 | − | 0.707107i | −1.47604 | + | 1.34955i | 1.19874 | − | 1.19874i | 0.570284 | − | 1.29413i | 0.592345i | −2.53467 | − | 1.25518i | 1.00000i | 2.19392 | + | 0.966791i | ||||
157.18 | 0.512178 | + | 1.31821i | 0.707107 | + | 0.707107i | −1.47535 | + | 1.35031i | 2.44843 | − | 2.44843i | −0.569950 | + | 1.29428i | − | 5.13203i | −2.53564 | − | 1.25322i | 1.00000i | 4.48157 | + | 1.97351i | |||
157.19 | 0.617594 | − | 1.27223i | −0.707107 | − | 0.707107i | −1.23715 | − | 1.57145i | 3.01500 | − | 3.01500i | −1.33631 | + | 0.462899i | − | 2.10775i | −2.76331 | + | 0.603431i | 1.00000i | −1.97374 | − | 5.69783i | |||
157.20 | 0.989564 | + | 1.01033i | 0.707107 | + | 0.707107i | −0.0415264 | + | 1.99957i | 0.996563 | − | 0.996563i | −0.0146826 | + | 1.41414i | 3.61917i | −2.06131 | + | 1.93675i | 1.00000i | 1.99302 | + | 0.0206930i | ||||
See all 56 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
16.e | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 624.2.x.b | ✓ | 56 |
4.b | odd | 2 | 1 | 2496.2.x.b | 56 | ||
16.e | even | 4 | 1 | inner | 624.2.x.b | ✓ | 56 |
16.f | odd | 4 | 1 | 2496.2.x.b | 56 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
624.2.x.b | ✓ | 56 | 1.a | even | 1 | 1 | trivial |
624.2.x.b | ✓ | 56 | 16.e | even | 4 | 1 | inner |
2496.2.x.b | 56 | 4.b | odd | 2 | 1 | ||
2496.2.x.b | 56 | 16.f | odd | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{56} + 16 T_{5}^{53} + 1092 T_{5}^{52} + 224 T_{5}^{51} + 128 T_{5}^{50} + 13552 T_{5}^{49} + 478260 T_{5}^{48} + 176096 T_{5}^{47} + 102144 T_{5}^{46} + 4553088 T_{5}^{45} + 109377472 T_{5}^{44} + \cdots + 23\!\cdots\!76 \)
acting on \(S_{2}^{\mathrm{new}}(624, [\chi])\).