Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [195,2,Mod(59,195)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(195, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([6, 6, 11]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("195.59");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 195 = 3 \cdot 5 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 195.bh (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: | \(1.55708283941\) |
Analytic rank: | \(0\) |
Dimension: | \(96\) |
Relative dimension: | \(24\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
59.1 | −2.37850 | + | 0.637316i | 0.0570156 | − | 1.73111i | 3.51903 | − | 2.03171i | −1.24285 | + | 1.85885i | 0.967655 | + | 4.15378i | −0.462194 | + | 1.72493i | −3.59279 | + | 3.59279i | −2.99350 | − | 0.197401i | 1.77144 | − | 5.21336i |
59.2 | −2.37850 | + | 0.637316i | 1.47068 | − | 0.914933i | 3.51903 | − | 2.03171i | 1.85885 | − | 1.24285i | −2.91490 | + | 3.11345i | 0.462194 | − | 1.72493i | −3.59279 | + | 3.59279i | 1.32580 | − | 2.69115i | −3.62918 | + | 4.14079i |
59.3 | −1.91529 | + | 0.513200i | −1.72511 | − | 0.154893i | 1.67290 | − | 0.965849i | 1.31191 | + | 1.81077i | 3.38357 | − | 0.588662i | 0.434913 | − | 1.62312i | 0.0957651 | − | 0.0957651i | 2.95202 | + | 0.534415i | −3.44197 | − | 2.79487i |
59.4 | −1.91529 | + | 0.513200i | 0.996697 | + | 1.41654i | 1.67290 | − | 0.965849i | 1.81077 | + | 1.31191i | −2.63593 | − | 2.20158i | −0.434913 | + | 1.62312i | 0.0957651 | − | 0.0957651i | −1.01319 | + | 2.82373i | −4.14142 | − | 1.58340i |
59.5 | −1.66849 | + | 0.447071i | −1.28242 | + | 1.16422i | 0.851938 | − | 0.491866i | −2.11813 | + | 0.716618i | 1.61922 | − | 2.51582i | −0.830623 | + | 3.09993i | 1.24129 | − | 1.24129i | 0.289197 | − | 2.98603i | 3.21369 | − | 2.14262i |
59.6 | −1.66849 | + | 0.447071i | −0.367032 | + | 1.69272i | 0.851938 | − | 0.491866i | 0.716618 | − | 2.11813i | −0.144374 | − | 2.98837i | 0.830623 | − | 3.09993i | 1.24129 | − | 1.24129i | −2.73057 | − | 1.24256i | −0.248718 | + | 3.85445i |
59.7 | −1.25247 | + | 0.335598i | −0.377636 | − | 1.69038i | −0.276000 | + | 0.159349i | −0.638573 | − | 2.14295i | 1.04027 | + | 1.99042i | −0.589843 | + | 2.20132i | 2.12594 | − | 2.12594i | −2.71478 | + | 1.27670i | 1.51896 | + | 2.46967i |
59.8 | −1.25247 | + | 0.335598i | 1.65273 | − | 0.518148i | −0.276000 | + | 0.159349i | −2.14295 | − | 0.638573i | −1.89610 | + | 1.20362i | 0.589843 | − | 2.20132i | 2.12594 | − | 2.12594i | 2.46304 | − | 1.71272i | 2.89828 | + | 0.0806242i |
59.9 | −0.578879 | + | 0.155110i | −0.110304 | − | 1.72853i | −1.42101 | + | 0.820420i | 1.54681 | + | 1.61473i | 0.331966 | + | 0.983504i | 1.15087 | − | 4.29509i | 1.54287 | − | 1.54287i | −2.97567 | + | 0.381329i | −1.14588 | − | 0.694809i |
59.10 | −0.578879 | + | 0.155110i | 1.55211 | − | 0.768741i | −1.42101 | + | 0.820420i | 1.61473 | + | 1.54681i | −0.779243 | + | 0.685756i | −1.15087 | + | 4.29509i | 1.54287 | − | 1.54287i | 1.81807 | − | 2.38634i | −1.17466 | − | 0.644957i |
59.11 | −0.481640 | + | 0.129055i | −1.71178 | − | 0.264216i | −1.51673 | + | 0.875684i | 1.67675 | − | 1.47936i | 0.858561 | − | 0.0936571i | −0.339617 | + | 1.26747i | 1.32268 | − | 1.32268i | 2.86038 | + | 0.904558i | −0.616670 | + | 0.928914i |
59.12 | −0.481640 | + | 0.129055i | 1.08471 | + | 1.35034i | −1.51673 | + | 0.875684i | −1.47936 | + | 1.67675i | −0.696707 | − | 0.510390i | 0.339617 | − | 1.26747i | 1.32268 | − | 1.32268i | −0.646820 | + | 2.92944i | 0.496128 | − | 0.998509i |
59.13 | 0.481640 | − | 0.129055i | −1.08471 | − | 1.35034i | −1.51673 | + | 0.875684i | −1.67675 | + | 1.47936i | −0.696707 | − | 0.510390i | −0.339617 | + | 1.26747i | −1.32268 | + | 1.32268i | −0.646820 | + | 2.92944i | −0.616670 | + | 0.928914i |
59.14 | 0.481640 | − | 0.129055i | 1.71178 | + | 0.264216i | −1.51673 | + | 0.875684i | 1.47936 | − | 1.67675i | 0.858561 | − | 0.0936571i | 0.339617 | − | 1.26747i | −1.32268 | + | 1.32268i | 2.86038 | + | 0.904558i | 0.496128 | − | 0.998509i |
59.15 | 0.578879 | − | 0.155110i | −1.55211 | + | 0.768741i | −1.42101 | + | 0.820420i | −1.54681 | − | 1.61473i | −0.779243 | + | 0.685756i | 1.15087 | − | 4.29509i | −1.54287 | + | 1.54287i | 1.81807 | − | 2.38634i | −1.14588 | − | 0.694809i |
59.16 | 0.578879 | − | 0.155110i | 0.110304 | + | 1.72853i | −1.42101 | + | 0.820420i | −1.61473 | − | 1.54681i | 0.331966 | + | 0.983504i | −1.15087 | + | 4.29509i | −1.54287 | + | 1.54287i | −2.97567 | + | 0.381329i | −1.17466 | − | 0.644957i |
59.17 | 1.25247 | − | 0.335598i | −1.65273 | + | 0.518148i | −0.276000 | + | 0.159349i | 0.638573 | + | 2.14295i | −1.89610 | + | 1.20362i | −0.589843 | + | 2.20132i | −2.12594 | + | 2.12594i | 2.46304 | − | 1.71272i | 1.51896 | + | 2.46967i |
59.18 | 1.25247 | − | 0.335598i | 0.377636 | + | 1.69038i | −0.276000 | + | 0.159349i | 2.14295 | + | 0.638573i | 1.04027 | + | 1.99042i | 0.589843 | − | 2.20132i | −2.12594 | + | 2.12594i | −2.71478 | + | 1.27670i | 2.89828 | + | 0.0806242i |
59.19 | 1.66849 | − | 0.447071i | 0.367032 | − | 1.69272i | 0.851938 | − | 0.491866i | 2.11813 | − | 0.716618i | −0.144374 | − | 2.98837i | −0.830623 | + | 3.09993i | −1.24129 | + | 1.24129i | −2.73057 | − | 1.24256i | 3.21369 | − | 2.14262i |
59.20 | 1.66849 | − | 0.447071i | 1.28242 | − | 1.16422i | 0.851938 | − | 0.491866i | −0.716618 | + | 2.11813i | 1.61922 | − | 2.51582i | 0.830623 | − | 3.09993i | −1.24129 | + | 1.24129i | 0.289197 | − | 2.98603i | −0.248718 | + | 3.85445i |
See all 96 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
5.b | even | 2 | 1 | inner |
13.f | odd | 12 | 1 | inner |
15.d | odd | 2 | 1 | inner |
39.k | even | 12 | 1 | inner |
65.s | odd | 12 | 1 | inner |
195.bh | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 195.2.bh.a | ✓ | 96 |
3.b | odd | 2 | 1 | inner | 195.2.bh.a | ✓ | 96 |
5.b | even | 2 | 1 | inner | 195.2.bh.a | ✓ | 96 |
5.c | odd | 4 | 2 | 975.2.bo.h | 96 | ||
13.f | odd | 12 | 1 | inner | 195.2.bh.a | ✓ | 96 |
15.d | odd | 2 | 1 | inner | 195.2.bh.a | ✓ | 96 |
15.e | even | 4 | 2 | 975.2.bo.h | 96 | ||
39.k | even | 12 | 1 | inner | 195.2.bh.a | ✓ | 96 |
65.o | even | 12 | 1 | 975.2.bo.h | 96 | ||
65.s | odd | 12 | 1 | inner | 195.2.bh.a | ✓ | 96 |
65.t | even | 12 | 1 | 975.2.bo.h | 96 | ||
195.bc | odd | 12 | 1 | 975.2.bo.h | 96 | ||
195.bh | even | 12 | 1 | inner | 195.2.bh.a | ✓ | 96 |
195.bn | odd | 12 | 1 | 975.2.bo.h | 96 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
195.2.bh.a | ✓ | 96 | 1.a | even | 1 | 1 | trivial |
195.2.bh.a | ✓ | 96 | 3.b | odd | 2 | 1 | inner |
195.2.bh.a | ✓ | 96 | 5.b | even | 2 | 1 | inner |
195.2.bh.a | ✓ | 96 | 13.f | odd | 12 | 1 | inner |
195.2.bh.a | ✓ | 96 | 15.d | odd | 2 | 1 | inner |
195.2.bh.a | ✓ | 96 | 39.k | even | 12 | 1 | inner |
195.2.bh.a | ✓ | 96 | 65.s | odd | 12 | 1 | inner |
195.2.bh.a | ✓ | 96 | 195.bh | even | 12 | 1 | inner |
975.2.bo.h | 96 | 5.c | odd | 4 | 2 | ||
975.2.bo.h | 96 | 15.e | even | 4 | 2 | ||
975.2.bo.h | 96 | 65.o | even | 12 | 1 | ||
975.2.bo.h | 96 | 65.t | even | 12 | 1 | ||
975.2.bo.h | 96 | 195.bc | odd | 12 | 1 | ||
975.2.bo.h | 96 | 195.bn | odd | 12 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(195, [\chi])\).