Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [148,2,Mod(15,148)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(148, base_ring=CyclotomicField(36))
chi = DirichletCharacter(H, H._module([18, 13]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("148.15");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 148 = 2^{2} \cdot 37 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 148.q (of order \(36\), degree \(12\), 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: | \(1.18178594991\) |
Analytic rank: | \(0\) |
Dimension: | \(192\) |
Relative dimension: | \(16\) over \(\Q(\zeta_{36})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{36}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
15.1 | −1.35843 | + | 0.393269i | 0.312666 | − | 0.113801i | 1.69068 | − | 1.06846i | −2.13888 | − | 3.05464i | −0.379981 | + | 0.277553i | −2.14447 | − | 0.378127i | −1.87648 | + | 2.11632i | −2.21332 | + | 1.85720i | 4.10682 | + | 3.30837i |
15.2 | −1.35454 | − | 0.406466i | −1.17609 | + | 0.428063i | 1.66957 | + | 1.10115i | 0.506128 | + | 0.722826i | 1.76706 | − | 0.101787i | −3.36523 | − | 0.593381i | −1.81392 | − | 2.17018i | −1.09817 | + | 0.921478i | −0.391767 | − | 1.18482i |
15.3 | −1.28548 | + | 0.589529i | 2.82129 | − | 1.02687i | 1.30491 | − | 1.51565i | 1.88251 | + | 2.68851i | −3.02134 | + | 2.98325i | −2.77218 | − | 0.488810i | −0.783913 | + | 2.71762i | 4.60710 | − | 3.86581i | −4.00488 | − | 2.34622i |
15.4 | −1.25586 | − | 0.650244i | 1.92957 | − | 0.702306i | 1.15437 | + | 1.63323i | −0.635590 | − | 0.907717i | −2.87994 | − | 0.372694i | 2.95389 | + | 0.520851i | −0.387722 | − | 2.80173i | 0.931877 | − | 0.781938i | 0.207974 | + | 1.55325i |
15.5 | −0.987028 | + | 1.01281i | −0.343248 | + | 0.124932i | −0.0515514 | − | 1.99934i | 0.568635 | + | 0.812094i | 0.212263 | − | 0.470955i | 2.90658 | + | 0.512509i | 2.07582 | + | 1.92119i | −2.19592 | + | 1.84260i | −1.38375 | − | 0.225643i |
15.6 | −0.954588 | − | 1.04344i | −2.65144 | + | 0.965046i | −0.177523 | + | 1.99211i | −0.859834 | − | 1.22797i | 3.53800 | + | 1.84539i | 3.28436 | + | 0.579122i | 2.24810 | − | 1.71641i | 3.80070 | − | 3.18917i | −0.460523 | + | 2.06939i |
15.7 | −0.185722 | − | 1.40197i | 2.65144 | − | 0.965046i | −1.93101 | + | 0.520752i | −0.859834 | − | 1.22797i | −1.84539 | − | 3.53800i | −3.28436 | − | 0.579122i | 1.08871 | + | 2.61050i | 3.80070 | − | 3.18917i | −1.56188 | + | 1.43352i |
15.8 | −0.106823 | + | 1.41017i | −1.93332 | + | 0.703672i | −1.97718 | − | 0.301279i | −1.75433 | − | 2.50545i | −0.785775 | − | 2.80149i | 1.00092 | + | 0.176489i | 0.636063 | − | 2.75598i | 0.944446 | − | 0.792484i | 3.72052 | − | 2.20627i |
15.9 | 0.254373 | + | 1.39115i | 1.68945 | − | 0.614911i | −1.87059 | + | 0.707741i | 0.733711 | + | 1.04785i | 1.28518 | + | 2.19386i | 0.596743 | + | 0.105222i | −1.46040 | − | 2.42224i | 0.178005 | − | 0.149364i | −1.27108 | + | 1.28724i |
15.10 | 0.309135 | − | 1.38001i | −1.92957 | + | 0.702306i | −1.80887 | − | 0.853220i | −0.635590 | − | 0.907717i | 0.372694 | + | 2.87994i | −2.95389 | − | 0.520851i | −1.73664 | + | 2.23251i | 0.931877 | − | 0.781938i | −1.44914 | + | 0.596516i |
15.11 | 0.559312 | − | 1.29891i | 1.17609 | − | 0.428063i | −1.37434 | − | 1.45299i | 0.506128 | + | 0.722826i | 0.101787 | − | 1.76706i | 3.36523 | + | 0.593381i | −2.65599 | + | 0.972471i | −1.09817 | + | 0.921478i | 1.22197 | − | 0.253130i |
15.12 | 0.902174 | + | 1.08907i | −1.68945 | + | 0.614911i | −0.372164 | + | 1.96507i | 0.733711 | + | 1.04785i | −2.19386 | − | 1.28518i | −0.596743 | − | 0.105222i | −2.47586 | + | 1.36752i | 0.178005 | − | 0.149364i | −0.479249 | + | 1.74441i |
15.13 | 1.14892 | + | 0.824611i | 1.93332 | − | 0.703672i | 0.640035 | + | 1.89482i | −1.75433 | − | 2.50545i | 2.80149 | + | 0.785775i | −1.00092 | − | 0.176489i | −0.827143 | + | 2.70478i | 0.944446 | − | 0.792484i | 0.0504299 | − | 4.32520i |
15.14 | 1.17445 | − | 0.787832i | −0.312666 | + | 0.113801i | 0.758642 | − | 1.85053i | −2.13888 | − | 3.05464i | −0.277553 | + | 0.379981i | 2.14447 | + | 0.378127i | −0.566923 | − | 2.77103i | −2.21332 | + | 1.85720i | −4.91854 | − | 1.90243i |
15.15 | 1.27790 | − | 0.605792i | −2.82129 | + | 1.02687i | 1.26603 | − | 1.54828i | 1.88251 | + | 2.68851i | −2.98325 | + | 3.02134i | 2.77218 | + | 0.488810i | 0.679923 | − | 2.74549i | 4.60710 | − | 3.86581i | 4.03433 | + | 2.29522i |
15.16 | 1.41030 | − | 0.105088i | 0.343248 | − | 0.124932i | 1.97791 | − | 0.296413i | 0.568635 | + | 0.812094i | 0.470955 | − | 0.212263i | −2.90658 | − | 0.512509i | 2.75831 | − | 0.625887i | −2.19592 | + | 1.84260i | 0.887289 | + | 1.08554i |
19.1 | −1.41220 | − | 0.0753572i | 1.79147 | − | 1.50323i | 1.98864 | + | 0.212840i | −1.25294 | − | 2.68694i | −2.64321 | + | 1.98786i | 0.306442 | − | 0.841942i | −2.79233 | − | 0.450432i | 0.428749 | − | 2.43156i | 1.56693 | + | 3.88892i |
19.2 | −1.37766 | + | 0.319439i | −1.79147 | + | 1.50323i | 1.79592 | − | 0.880160i | −1.25294 | − | 2.68694i | 1.98786 | − | 2.64321i | −0.306442 | + | 0.841942i | −2.19301 | + | 1.78625i | 0.428749 | − | 2.43156i | 2.58444 | + | 3.30146i |
19.3 | −1.35715 | − | 0.397685i | 0.898867 | − | 0.754239i | 1.68369 | + | 1.07943i | 1.24600 | + | 2.67206i | −1.51984 | + | 0.666147i | −1.12931 | + | 3.10275i | −1.85575 | − | 2.13453i | −0.281859 | + | 1.59850i | −0.628371 | − | 4.12190i |
19.4 | −1.26747 | + | 0.627309i | −0.898867 | + | 0.754239i | 1.21297 | − | 1.59019i | 1.24600 | + | 2.67206i | 0.666147 | − | 1.51984i | 1.12931 | − | 3.10275i | −0.539858 | + | 2.77643i | −0.281859 | + | 1.59850i | −3.25548 | − | 2.60513i |
See next 80 embeddings (of 192 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
37.i | odd | 36 | 1 | inner |
148.q | even | 36 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 148.2.q.b | ✓ | 192 |
4.b | odd | 2 | 1 | inner | 148.2.q.b | ✓ | 192 |
37.i | odd | 36 | 1 | inner | 148.2.q.b | ✓ | 192 |
148.q | even | 36 | 1 | inner | 148.2.q.b | ✓ | 192 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
148.2.q.b | ✓ | 192 | 1.a | even | 1 | 1 | trivial |
148.2.q.b | ✓ | 192 | 4.b | odd | 2 | 1 | inner |
148.2.q.b | ✓ | 192 | 37.i | odd | 36 | 1 | inner |
148.2.q.b | ✓ | 192 | 148.q | even | 36 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{192} + 12 T_{3}^{190} + 42 T_{3}^{188} + 3174 T_{3}^{186} + 35427 T_{3}^{184} + 79668 T_{3}^{182} + 5671575 T_{3}^{180} + 62327286 T_{3}^{178} + 103869771 T_{3}^{176} + 6648514522 T_{3}^{174} + \cdots + 10\!\cdots\!84 \)
acting on \(S_{2}^{\mathrm{new}}(148, [\chi])\).