Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [420,2,Mod(19,420)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(420, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 0, 3, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("420.19");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 420.bk (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: | \(3.35371688489\) |
Analytic rank: | \(0\) |
Dimension: | \(96\) |
Relative dimension: | \(48\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
19.1 | −1.40539 | − | 0.157734i | −0.866025 | + | 0.500000i | 1.95024 | + | 0.443354i | −2.23392 | − | 0.0978861i | 1.29597 | − | 0.566094i | −2.28831 | − | 1.32803i | −2.67092 | − | 0.930704i | 0.500000 | − | 0.866025i | 3.12409 | + | 0.489933i |
19.2 | −1.40244 | + | 0.182077i | 0.866025 | − | 0.500000i | 1.93370 | − | 0.510706i | −1.21850 | − | 1.87490i | −1.12351 | + | 0.858905i | −1.54335 | + | 2.14897i | −2.61891 | + | 1.06832i | 0.500000 | − | 0.866025i | 2.05025 | + | 2.40758i |
19.3 | −1.39456 | + | 0.234979i | −0.866025 | + | 0.500000i | 1.88957 | − | 0.655384i | 2.22645 | − | 0.207142i | 1.09023 | − | 0.900776i | 1.32770 | + | 2.28850i | −2.48111 | + | 1.35798i | 0.500000 | − | 0.866025i | −3.05624 | + | 0.812041i |
19.4 | −1.34009 | − | 0.451826i | 0.866025 | − | 0.500000i | 1.59171 | + | 1.21098i | 1.35968 | − | 1.77518i | −1.38647 | + | 0.278754i | 2.56678 | − | 0.641578i | −1.58588 | − | 2.34200i | 0.500000 | − | 0.866025i | −2.62417 | + | 1.76458i |
19.5 | −1.33244 | − | 0.473912i | 0.866025 | − | 0.500000i | 1.55081 | + | 1.26292i | −1.63361 | + | 1.52686i | −1.39089 | + | 0.255802i | −0.935228 | − | 2.47494i | −1.46786 | − | 2.41772i | 0.500000 | − | 0.866025i | 2.90030 | − | 1.26027i |
19.6 | −1.27723 | + | 0.607202i | −0.866025 | + | 0.500000i | 1.26261 | − | 1.55107i | −2.23451 | + | 0.0834732i | 0.802509 | − | 1.16447i | 2.60462 | − | 0.464731i | −0.670831 | + | 2.74772i | 0.500000 | − | 0.866025i | 2.80329 | − | 1.46341i |
19.7 | −1.26422 | + | 0.633844i | 0.866025 | − | 0.500000i | 1.19648 | − | 1.60263i | −0.558189 | + | 2.16528i | −0.777921 | + | 1.18103i | 0.242117 | + | 2.63465i | −0.496796 | + | 2.78446i | 0.500000 | − | 0.866025i | −0.666777 | − | 3.09118i |
19.8 | −1.23847 | − | 0.682786i | 0.866025 | − | 0.500000i | 1.06761 | + | 1.69122i | 1.44800 | + | 1.70391i | −1.41394 | + | 0.0279240i | −1.82043 | + | 1.91990i | −0.167457 | − | 2.82347i | 0.500000 | − | 0.866025i | −0.629895 | − | 3.09891i |
19.9 | −1.21054 | − | 0.731152i | −0.866025 | + | 0.500000i | 0.930834 | + | 1.77018i | 1.44800 | + | 1.70391i | 1.41394 | + | 0.0279240i | 1.82043 | − | 1.91990i | 0.167457 | − | 2.82347i | 0.500000 | − | 0.866025i | −0.507051 | − | 3.12136i |
19.10 | −1.13701 | + | 0.840954i | 0.866025 | − | 0.500000i | 0.585592 | − | 1.91235i | 2.13108 | + | 0.677136i | −0.564204 | + | 1.29679i | 2.06024 | − | 1.65994i | 0.942373 | + | 2.66682i | 0.500000 | − | 0.866025i | −2.99250 | + | 1.02223i |
19.11 | −1.12697 | + | 0.854361i | −0.866025 | + | 0.500000i | 0.540133 | − | 1.92568i | 0.789103 | + | 2.09220i | 0.548806 | − | 1.30338i | −1.62304 | − | 2.08943i | 1.03651 | + | 2.63166i | 0.500000 | − | 0.866025i | −2.67680 | − | 1.68368i |
19.12 | −1.07664 | − | 0.916974i | −0.866025 | + | 0.500000i | 0.318317 | + | 1.97451i | −1.63361 | + | 1.52686i | 1.39089 | + | 0.255802i | 0.935228 | + | 2.47494i | 1.46786 | − | 2.41772i | 0.500000 | − | 0.866025i | 3.15891 | − | 0.145905i |
19.13 | −1.06134 | − | 0.934643i | −0.866025 | + | 0.500000i | 0.252886 | + | 1.98395i | 1.35968 | − | 1.77518i | 1.38647 | + | 0.278754i | −2.56678 | + | 0.641578i | 1.58588 | − | 2.34200i | 0.500000 | − | 0.866025i | −3.10224 | + | 0.613260i |
19.14 | −0.987366 | + | 1.01248i | −0.866025 | + | 0.500000i | −0.0502184 | − | 1.99937i | −0.336963 | − | 2.21053i | 0.348845 | − | 1.37051i | −1.65799 | + | 2.06181i | 2.07390 | + | 1.92326i | 0.500000 | − | 0.866025i | 2.57082 | + | 1.84144i |
19.15 | −0.839296 | − | 1.13824i | 0.866025 | − | 0.500000i | −0.591164 | + | 1.91063i | −2.23392 | − | 0.0978861i | −1.29597 | − | 0.566094i | 2.28831 | + | 1.32803i | 2.67092 | − | 0.930704i | 0.500000 | − | 0.866025i | 1.76351 | + | 2.62489i |
19.16 | −0.834650 | + | 1.14165i | 0.866025 | − | 0.500000i | −0.606717 | − | 1.90575i | −1.74710 | + | 1.39558i | −0.152005 | + | 1.40602i | −2.20715 | − | 1.45893i | 2.68209 | + | 0.897981i | 0.500000 | − | 0.866025i | −0.135049 | − | 3.15939i |
19.17 | −0.571370 | + | 1.29365i | 0.866025 | − | 0.500000i | −1.34707 | − | 1.47831i | 0.335062 | − | 2.21082i | 0.152005 | + | 1.40602i | −2.20715 | − | 1.45893i | 2.68209 | − | 0.897981i | 0.500000 | − | 0.866025i | 2.66859 | + | 1.69665i |
19.18 | −0.543538 | − | 1.30559i | −0.866025 | + | 0.500000i | −1.40913 | + | 1.41928i | −1.21850 | − | 1.87490i | 1.12351 | + | 0.858905i | 1.54335 | − | 2.14897i | 2.61891 | + | 1.06832i | 0.500000 | − | 0.866025i | −1.78555 | + | 2.60994i |
19.19 | −0.493780 | − | 1.32521i | 0.866025 | − | 0.500000i | −1.51236 | + | 1.30872i | 2.22645 | − | 0.207142i | −1.09023 | − | 0.900776i | −1.32770 | − | 2.28850i | 2.48111 | + | 1.35798i | 0.500000 | − | 0.866025i | −1.37388 | − | 2.84824i |
19.20 | −0.383148 | + | 1.36132i | −0.866025 | + | 0.500000i | −1.70640 | − | 1.04318i | −2.08286 | + | 0.813448i | −0.348845 | − | 1.37051i | −1.65799 | + | 2.06181i | 2.07390 | − | 1.92326i | 0.500000 | − | 0.866025i | −0.309322 | − | 3.14711i |
See all 96 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
5.b | even | 2 | 1 | inner |
7.d | odd | 6 | 1 | inner |
20.d | odd | 2 | 1 | inner |
28.f | even | 6 | 1 | inner |
35.i | odd | 6 | 1 | inner |
140.s | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 420.2.bk.a | ✓ | 96 |
4.b | odd | 2 | 1 | inner | 420.2.bk.a | ✓ | 96 |
5.b | even | 2 | 1 | inner | 420.2.bk.a | ✓ | 96 |
7.d | odd | 6 | 1 | inner | 420.2.bk.a | ✓ | 96 |
20.d | odd | 2 | 1 | inner | 420.2.bk.a | ✓ | 96 |
28.f | even | 6 | 1 | inner | 420.2.bk.a | ✓ | 96 |
35.i | odd | 6 | 1 | inner | 420.2.bk.a | ✓ | 96 |
140.s | even | 6 | 1 | inner | 420.2.bk.a | ✓ | 96 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
420.2.bk.a | ✓ | 96 | 1.a | even | 1 | 1 | trivial |
420.2.bk.a | ✓ | 96 | 4.b | odd | 2 | 1 | inner |
420.2.bk.a | ✓ | 96 | 5.b | even | 2 | 1 | inner |
420.2.bk.a | ✓ | 96 | 7.d | odd | 6 | 1 | inner |
420.2.bk.a | ✓ | 96 | 20.d | odd | 2 | 1 | inner |
420.2.bk.a | ✓ | 96 | 28.f | even | 6 | 1 | inner |
420.2.bk.a | ✓ | 96 | 35.i | odd | 6 | 1 | inner |
420.2.bk.a | ✓ | 96 | 140.s | even | 6 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(420, [\chi])\).