Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [861,2,Mod(419,861)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(861, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 2, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("861.419");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 861 = 3 \cdot 7 \cdot 41 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 861.l (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: | \(6.87511961403\) |
Analytic rank: | \(0\) |
Dimension: | \(216\) |
Relative dimension: | \(108\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
419.1 | −2.76490 | −1.28288 | + | 1.16371i | 5.64469 | 3.67998i | 3.54703 | − | 3.21755i | −1.30191 | − | 2.30327i | −10.0772 | 0.291548 | − | 2.98580i | − | 10.1748i | |||||||||
419.2 | −2.76490 | 1.28288 | − | 1.16371i | 5.64469 | − | 3.67998i | −3.54703 | + | 3.21755i | −2.30327 | − | 1.30191i | −10.0772 | 0.291548 | − | 2.98580i | 10.1748i | |||||||||
419.3 | −2.54686 | −1.08190 | + | 1.35258i | 4.48648 | − | 3.20407i | 2.75545 | − | 3.44484i | 2.56048 | − | 0.666289i | −6.33272 | −0.658969 | − | 2.92673i | 8.16030i | |||||||||
419.4 | −2.54686 | 1.08190 | − | 1.35258i | 4.48648 | 3.20407i | −2.75545 | + | 3.44484i | −0.666289 | + | 2.56048i | −6.33272 | −0.658969 | − | 2.92673i | − | 8.16030i | |||||||||
419.5 | −2.54534 | −1.72582 | − | 0.146743i | 4.47878 | − | 1.45380i | 4.39282 | + | 0.373512i | −2.11962 | + | 1.58342i | −6.30935 | 2.95693 | + | 0.506505i | 3.70042i | |||||||||
419.6 | −2.54534 | 1.72582 | + | 0.146743i | 4.47878 | 1.45380i | −4.39282 | − | 0.373512i | 1.58342 | − | 2.11962i | −6.30935 | 2.95693 | + | 0.506505i | − | 3.70042i | |||||||||
419.7 | −2.51614 | −0.724750 | − | 1.57313i | 4.33098 | 1.53774i | 1.82358 | + | 3.95822i | −2.26225 | − | 1.37194i | −5.86509 | −1.94947 | + | 2.28025i | − | 3.86917i | |||||||||
419.8 | −2.51614 | 0.724750 | + | 1.57313i | 4.33098 | − | 1.53774i | −1.82358 | − | 3.95822i | −1.37194 | − | 2.26225i | −5.86509 | −1.94947 | + | 2.28025i | 3.86917i | |||||||||
419.9 | −2.49211 | −1.73194 | − | 0.0198358i | 4.21059 | 0.195467i | 4.31617 | + | 0.0494328i | 2.64473 | − | 0.0733311i | −5.50903 | 2.99921 | + | 0.0687086i | − | 0.487125i | |||||||||
419.10 | −2.49211 | 1.73194 | + | 0.0198358i | 4.21059 | − | 0.195467i | −4.31617 | − | 0.0494328i | −0.0733311 | + | 2.64473i | −5.50903 | 2.99921 | + | 0.0687086i | 0.487125i | |||||||||
419.11 | −2.39321 | −1.20157 | − | 1.24749i | 3.72747 | 2.99101i | 2.87562 | + | 2.98550i | 1.32481 | + | 2.29017i | −4.13422 | −0.112448 | + | 2.99789i | − | 7.15812i | |||||||||
419.12 | −2.39321 | 1.20157 | + | 1.24749i | 3.72747 | − | 2.99101i | −2.87562 | − | 2.98550i | 2.29017 | + | 1.32481i | −4.13422 | −0.112448 | + | 2.99789i | 7.15812i | |||||||||
419.13 | −2.13419 | −0.283105 | + | 1.70876i | 2.55477 | − | 0.157669i | 0.604199 | − | 3.64682i | −2.21993 | + | 1.43942i | −1.18399 | −2.83970 | − | 0.967514i | 0.336495i | |||||||||
419.14 | −2.13419 | 0.283105 | − | 1.70876i | 2.55477 | 0.157669i | −0.604199 | + | 3.64682i | 1.43942 | − | 2.21993i | −1.18399 | −2.83970 | − | 0.967514i | − | 0.336495i | |||||||||
419.15 | −2.09764 | −0.839690 | + | 1.51490i | 2.40008 | 2.02468i | 1.76136 | − | 3.17771i | 1.67336 | + | 2.04936i | −0.839219 | −1.58984 | − | 2.54409i | − | 4.24705i | |||||||||
419.16 | −2.09764 | 0.839690 | − | 1.51490i | 2.40008 | − | 2.02468i | −1.76136 | + | 3.17771i | 2.04936 | + | 1.67336i | −0.839219 | −1.58984 | − | 2.54409i | 4.24705i | |||||||||
419.17 | −2.06720 | −1.24187 | − | 1.20738i | 2.27332 | − | 4.31335i | 2.56719 | + | 2.49590i | 0.905579 | − | 2.48595i | −0.565003 | 0.0844609 | + | 2.99881i | 8.91656i | |||||||||
419.18 | −2.06720 | 1.24187 | + | 1.20738i | 2.27332 | 4.31335i | −2.56719 | − | 2.49590i | −2.48595 | + | 0.905579i | −0.565003 | 0.0844609 | + | 2.99881i | − | 8.91656i | |||||||||
419.19 | −2.04053 | −0.410008 | − | 1.68282i | 2.16376 | − | 2.10095i | 0.836632 | + | 3.43385i | −2.14224 | + | 1.55268i | −0.334148 | −2.66379 | + | 1.37994i | 4.28704i | |||||||||
419.20 | −2.04053 | 0.410008 | + | 1.68282i | 2.16376 | 2.10095i | −0.836632 | − | 3.43385i | 1.55268 | − | 2.14224i | −0.334148 | −2.66379 | + | 1.37994i | − | 4.28704i | |||||||||
See next 80 embeddings (of 216 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
7.b | odd | 2 | 1 | inner |
21.c | even | 2 | 1 | inner |
41.c | even | 4 | 1 | inner |
123.f | odd | 4 | 1 | inner |
287.g | odd | 4 | 1 | inner |
861.l | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 861.2.l.a | ✓ | 216 |
3.b | odd | 2 | 1 | inner | 861.2.l.a | ✓ | 216 |
7.b | odd | 2 | 1 | inner | 861.2.l.a | ✓ | 216 |
21.c | even | 2 | 1 | inner | 861.2.l.a | ✓ | 216 |
41.c | even | 4 | 1 | inner | 861.2.l.a | ✓ | 216 |
123.f | odd | 4 | 1 | inner | 861.2.l.a | ✓ | 216 |
287.g | odd | 4 | 1 | inner | 861.2.l.a | ✓ | 216 |
861.l | even | 4 | 1 | inner | 861.2.l.a | ✓ | 216 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
861.2.l.a | ✓ | 216 | 1.a | even | 1 | 1 | trivial |
861.2.l.a | ✓ | 216 | 3.b | odd | 2 | 1 | inner |
861.2.l.a | ✓ | 216 | 7.b | odd | 2 | 1 | inner |
861.2.l.a | ✓ | 216 | 21.c | even | 2 | 1 | inner |
861.2.l.a | ✓ | 216 | 41.c | even | 4 | 1 | inner |
861.2.l.a | ✓ | 216 | 123.f | odd | 4 | 1 | inner |
861.2.l.a | ✓ | 216 | 287.g | odd | 4 | 1 | inner |
861.2.l.a | ✓ | 216 | 861.l | even | 4 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(861, [\chi])\).