Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [931,2,Mod(99,931)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(931, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("931.99");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 931 = 7^{2} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 931.w (of order \(9\), degree \(6\), 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: | \(7.43407242818\) |
Analytic rank: | \(0\) |
Dimension: | \(30\) |
Relative dimension: | \(5\) over \(\Q(\zeta_{9})\) |
Twist minimal: | no (minimal twist has level 133) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{9}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
99.1 | −1.80311 | − | 0.656278i | −0.100636 | − | 0.570737i | 1.28842 | + | 1.08111i | 0.131215 | − | 0.110102i | −0.193104 | + | 1.09515i | 0 | 0.305182 | + | 0.528591i | 2.50346 | − | 0.911187i | −0.308853 | + | 0.112413i | ||
99.2 | −1.65002 | − | 0.600557i | 0.488762 | + | 2.77191i | 0.829796 | + | 0.696282i | −2.66117 | + | 2.23299i | 0.858223 | − | 4.86723i | 0 | 0.804890 | + | 1.39411i | −4.62552 | + | 1.68355i | 5.73201 | − | 2.08628i | ||
99.3 | 0.712406 | + | 0.259295i | 0.218901 | + | 1.24145i | −1.09180 | − | 0.916129i | −0.752493 | + | 0.631417i | −0.165955 | + | 0.941177i | 0 | −1.29838 | − | 2.24887i | 1.32580 | − | 0.482551i | −0.699804 | + | 0.254708i | ||
99.4 | 1.29036 | + | 0.469651i | −0.457545 | − | 2.59487i | −0.0876440 | − | 0.0735420i | −0.327974 | + | 0.275202i | 0.628286 | − | 3.56318i | 0 | −1.45172 | − | 2.51445i | −3.70490 | + | 1.34847i | −0.552451 | + | 0.201076i | ||
99.5 | 2.39006 | + | 0.869910i | 0.524166 | + | 2.97269i | 3.42354 | + | 2.87269i | 2.17073 | − | 1.82146i | −1.33319 | + | 7.56088i | 0 | 3.14003 | + | 5.43870i | −5.74307 | + | 2.09031i | 6.77267 | − | 2.46505i | ||
442.1 | −1.80311 | + | 0.656278i | −0.100636 | + | 0.570737i | 1.28842 | − | 1.08111i | 0.131215 | + | 0.110102i | −0.193104 | − | 1.09515i | 0 | 0.305182 | − | 0.528591i | 2.50346 | + | 0.911187i | −0.308853 | − | 0.112413i | ||
442.2 | −1.65002 | + | 0.600557i | 0.488762 | − | 2.77191i | 0.829796 | − | 0.696282i | −2.66117 | − | 2.23299i | 0.858223 | + | 4.86723i | 0 | 0.804890 | − | 1.39411i | −4.62552 | − | 1.68355i | 5.73201 | + | 2.08628i | ||
442.3 | 0.712406 | − | 0.259295i | 0.218901 | − | 1.24145i | −1.09180 | + | 0.916129i | −0.752493 | − | 0.631417i | −0.165955 | − | 0.941177i | 0 | −1.29838 | + | 2.24887i | 1.32580 | + | 0.482551i | −0.699804 | − | 0.254708i | ||
442.4 | 1.29036 | − | 0.469651i | −0.457545 | + | 2.59487i | −0.0876440 | + | 0.0735420i | −0.327974 | − | 0.275202i | 0.628286 | + | 3.56318i | 0 | −1.45172 | + | 2.51445i | −3.70490 | − | 1.34847i | −0.552451 | − | 0.201076i | ||
442.5 | 2.39006 | − | 0.869910i | 0.524166 | − | 2.97269i | 3.42354 | − | 2.87269i | 2.17073 | + | 1.82146i | −1.33319 | − | 7.56088i | 0 | 3.14003 | − | 5.43870i | −5.74307 | − | 2.09031i | 6.77267 | + | 2.46505i | ||
491.1 | −1.59663 | − | 1.33973i | 2.73420 | − | 0.995166i | 0.407048 | + | 2.30848i | −0.0434172 | + | 0.246231i | −5.69875 | − | 2.07417i | 0 | 0.358588 | − | 0.621093i | 4.18734 | − | 3.51360i | 0.399204 | − | 0.334972i | ||
491.2 | −1.30141 | − | 1.09201i | −2.86843 | + | 1.04402i | 0.153879 | + | 0.872693i | −0.224994 | + | 1.27600i | 4.87308 | + | 1.77366i | 0 | −0.946138 | + | 1.63876i | 4.83975 | − | 4.06104i | 1.68622 | − | 1.41491i | ||
491.3 | 0.220093 | + | 0.184680i | −2.07321 | + | 0.754587i | −0.332962 | − | 1.88832i | 0.608956 | − | 3.45356i | −0.595656 | − | 0.216801i | 0 | 0.562764 | − | 0.974735i | 1.43067 | − | 1.20047i | 0.771831 | − | 0.647643i | ||
491.4 | 0.467268 | + | 0.392084i | −0.141115 | + | 0.0513615i | −0.282687 | − | 1.60320i | −0.519693 | + | 2.94733i | −0.0860764 | − | 0.0313292i | 0 | 1.10647 | − | 1.91647i | −2.28086 | + | 1.91387i | −1.39844 | + | 1.17343i | ||
491.5 | 1.44463 | + | 1.21219i | 1.90886 | − | 0.694769i | 0.270260 | + | 1.53272i | 0.445192 | − | 2.52481i | 3.59980 | + | 1.31022i | 0 | 0.418312 | − | 0.724538i | 0.862920 | − | 0.724076i | 3.70369 | − | 3.10776i | ||
785.1 | −1.59663 | + | 1.33973i | 2.73420 | + | 0.995166i | 0.407048 | − | 2.30848i | −0.0434172 | − | 0.246231i | −5.69875 | + | 2.07417i | 0 | 0.358588 | + | 0.621093i | 4.18734 | + | 3.51360i | 0.399204 | + | 0.334972i | ||
785.2 | −1.30141 | + | 1.09201i | −2.86843 | − | 1.04402i | 0.153879 | − | 0.872693i | −0.224994 | − | 1.27600i | 4.87308 | − | 1.77366i | 0 | −0.946138 | − | 1.63876i | 4.83975 | + | 4.06104i | 1.68622 | + | 1.41491i | ||
785.3 | 0.220093 | − | 0.184680i | −2.07321 | − | 0.754587i | −0.332962 | + | 1.88832i | 0.608956 | + | 3.45356i | −0.595656 | + | 0.216801i | 0 | 0.562764 | + | 0.974735i | 1.43067 | + | 1.20047i | 0.771831 | + | 0.647643i | ||
785.4 | 0.467268 | − | 0.392084i | −0.141115 | − | 0.0513615i | −0.282687 | + | 1.60320i | −0.519693 | − | 2.94733i | −0.0860764 | + | 0.0313292i | 0 | 1.10647 | + | 1.91647i | −2.28086 | − | 1.91387i | −1.39844 | − | 1.17343i | ||
785.5 | 1.44463 | − | 1.21219i | 1.90886 | + | 0.694769i | 0.270260 | − | 1.53272i | 0.445192 | + | 2.52481i | 3.59980 | − | 1.31022i | 0 | 0.418312 | + | 0.724538i | 0.862920 | + | 0.724076i | 3.70369 | + | 3.10776i | ||
See all 30 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
19.e | even | 9 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 931.2.w.b | 30 | |
7.b | odd | 2 | 1 | 133.2.v.b | ✓ | 30 | |
7.c | even | 3 | 1 | 931.2.v.e | 30 | ||
7.c | even | 3 | 1 | 931.2.x.d | 30 | ||
7.d | odd | 6 | 1 | 931.2.v.d | 30 | ||
7.d | odd | 6 | 1 | 931.2.x.e | 30 | ||
19.e | even | 9 | 1 | inner | 931.2.w.b | 30 | |
133.u | even | 9 | 1 | 931.2.x.d | 30 | ||
133.w | even | 9 | 1 | 931.2.v.e | 30 | ||
133.x | odd | 18 | 1 | 931.2.x.e | 30 | ||
133.y | odd | 18 | 1 | 133.2.v.b | ✓ | 30 | |
133.y | odd | 18 | 1 | 2527.2.a.r | 15 | ||
133.z | odd | 18 | 1 | 931.2.v.d | 30 | ||
133.ba | even | 18 | 1 | 2527.2.a.s | 15 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
133.2.v.b | ✓ | 30 | 7.b | odd | 2 | 1 | |
133.2.v.b | ✓ | 30 | 133.y | odd | 18 | 1 | |
931.2.v.d | 30 | 7.d | odd | 6 | 1 | ||
931.2.v.d | 30 | 133.z | odd | 18 | 1 | ||
931.2.v.e | 30 | 7.c | even | 3 | 1 | ||
931.2.v.e | 30 | 133.w | even | 9 | 1 | ||
931.2.w.b | 30 | 1.a | even | 1 | 1 | trivial | |
931.2.w.b | 30 | 19.e | even | 9 | 1 | inner | |
931.2.x.d | 30 | 7.c | even | 3 | 1 | ||
931.2.x.d | 30 | 133.u | even | 9 | 1 | ||
931.2.x.e | 30 | 7.d | odd | 6 | 1 | ||
931.2.x.e | 30 | 133.x | odd | 18 | 1 | ||
2527.2.a.r | 15 | 133.y | odd | 18 | 1 | ||
2527.2.a.s | 15 | 133.ba | even | 18 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(931, [\chi])\):
\( T_{2}^{30} + 3 T_{2}^{28} - 7 T_{2}^{27} - 18 T_{2}^{26} - 6 T_{2}^{25} + 136 T_{2}^{24} + 135 T_{2}^{23} + 867 T_{2}^{22} + 877 T_{2}^{21} - 1023 T_{2}^{20} - 117 T_{2}^{19} + 13145 T_{2}^{18} + 5592 T_{2}^{17} + 20736 T_{2}^{16} + \cdots + 11449 \) |
\( T_{3}^{30} - 3 T_{3}^{29} + 6 T_{3}^{28} + 12 T_{3}^{27} - 102 T_{3}^{26} + 258 T_{3}^{25} + 423 T_{3}^{24} - 2601 T_{3}^{23} + 8712 T_{3}^{22} - 14465 T_{3}^{21} - 7476 T_{3}^{20} + 105963 T_{3}^{19} - 34109 T_{3}^{18} + 156342 T_{3}^{17} + \cdots + 26569 \) |