Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [68,2,Mod(3,68)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(68, base_ring=CyclotomicField(16))
chi = DirichletCharacter(H, H._module([8, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("68.3");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 68 = 2^{2} \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 68.i (of order \(16\), degree \(8\), 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: | \(0.542982733745\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Relative dimension: | \(6\) over \(\Q(\zeta_{16})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{16}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
3.1 | −1.17748 | − | 0.783289i | 0.418777 | − | 2.10534i | 0.772915 | + | 1.84461i | −0.561585 | − | 0.840472i | −2.14219 | + | 2.15097i | −1.73093 | − | 1.15657i | 0.534775 | − | 2.77741i | −1.48543 | − | 0.615285i | 0.00292235 | + | 1.42952i |
3.2 | −0.705238 | − | 1.22582i | −0.563072 | + | 2.83076i | −1.00528 | + | 1.72899i | 0.991762 | + | 1.48428i | 3.86710 | − | 1.30613i | −1.11612 | − | 0.745767i | 2.82840 | + | 0.0129412i | −4.92449 | − | 2.03979i | 1.12003 | − | 2.26249i |
3.3 | −0.278734 | + | 1.38647i | −0.418777 | + | 2.10534i | −1.84461 | − | 0.772915i | −0.561585 | − | 0.840472i | −2.80226 | − | 1.16745i | 1.73093 | + | 1.15657i | 1.58578 | − | 2.34207i | −1.48543 | − | 0.615285i | 1.32182 | − | 0.544354i |
3.4 | 0.238486 | − | 1.39396i | 0.0741201 | − | 0.372627i | −1.88625 | − | 0.664880i | −1.26427 | − | 1.89211i | −0.501751 | − | 0.192187i | 3.23087 | + | 2.15880i | −1.37666 | + | 2.47079i | 2.63828 | + | 1.09281i | −2.93903 | + | 1.31109i |
3.5 | 0.368108 | + | 1.36547i | 0.563072 | − | 2.83076i | −1.72899 | + | 1.00528i | 0.991762 | + | 1.48428i | 4.07257 | − | 0.273169i | 1.11612 | + | 0.745767i | −2.00913 | − | 1.99083i | −4.92449 | − | 2.03979i | −1.66165 | + | 1.90059i |
3.6 | 1.15431 | + | 0.817044i | −0.0741201 | + | 0.372627i | 0.664880 | + | 1.88625i | −1.26427 | − | 1.89211i | −0.390010 | + | 0.369569i | −3.23087 | − | 2.15880i | −0.773668 | + | 2.72056i | 2.63828 | + | 1.09281i | 0.0865750 | − | 3.21705i |
7.1 | −1.26407 | − | 0.634144i | 0.968913 | + | 0.647407i | 1.19572 | + | 1.60320i | 0.0404181 | + | 0.203195i | −0.814220 | − | 1.43280i | 2.58907 | + | 0.514998i | −0.494813 | − | 2.78481i | −0.628394 | − | 1.51708i | 0.0777640 | − | 0.282483i |
7.2 | −1.19689 | + | 0.753296i | −1.51958 | − | 1.01535i | 0.865091 | − | 1.80322i | −0.618394 | − | 3.10888i | 2.58363 | + | 0.0705700i | 1.63813 | + | 0.325845i | 0.322943 | + | 2.80993i | 0.130134 | + | 0.314170i | 3.08205 | + | 3.25515i |
7.3 | 0.226060 | − | 1.39603i | 1.80045 | + | 1.20302i | −1.89779 | − | 0.631174i | 0.177432 | + | 0.892011i | 2.08647 | − | 2.24153i | −3.16204 | − | 0.628968i | −1.31015 | + | 2.50669i | 0.646311 | + | 1.56033i | 1.28538 | − | 0.0460519i |
7.4 | 0.313668 | + | 1.37899i | 1.51958 | + | 1.01535i | −1.80322 | + | 0.865091i | −0.618394 | − | 3.10888i | −0.923513 | + | 2.41397i | −1.63813 | − | 0.325845i | −1.75857 | − | 2.21528i | 0.130134 | + | 0.314170i | 4.09314 | − | 1.82791i |
7.5 | 0.827293 | − | 1.14699i | −1.80045 | − | 1.20302i | −0.631174 | − | 1.89779i | 0.177432 | + | 0.892011i | −2.86936 | + | 1.06985i | 3.16204 | + | 0.628968i | −2.69892 | − | 0.846081i | 0.646311 | + | 1.56033i | 1.16992 | + | 0.534441i |
7.6 | 1.34224 | + | 0.445422i | −0.968913 | − | 0.647407i | 1.60320 | + | 1.19572i | 0.0404181 | + | 0.203195i | −1.01214 | − | 1.30055i | −2.58907 | − | 0.514998i | 1.61927 | + | 2.31904i | −0.628394 | − | 1.51708i | −0.0362570 | + | 0.290739i |
11.1 | −1.41399 | − | 0.0249511i | −1.51338 | + | 0.301030i | 1.99875 | + | 0.0705615i | 1.90890 | + | 1.27549i | 2.14742 | − | 0.387894i | 2.53332 | + | 3.79138i | −2.82447 | − | 0.149645i | −0.571942 | + | 0.236906i | −2.66735 | − | 1.85116i |
11.2 | −1.28454 | + | 0.591578i | 2.70865 | − | 0.538783i | 1.30007 | − | 1.51981i | −2.42403 | − | 1.61969i | −3.16062 | + | 2.29446i | 0.747003 | + | 1.11797i | −0.770902 | + | 2.72134i | 4.27484 | − | 1.77070i | 4.07193 | + | 0.646542i |
11.3 | −1.01749 | − | 0.982201i | 1.51338 | − | 0.301030i | 0.0705615 | + | 1.99875i | 1.90890 | + | 1.27549i | −1.83552 | − | 1.18015i | −2.53332 | − | 3.79138i | 1.89138 | − | 2.10301i | −0.571942 | + | 0.236906i | −0.689499 | − | 3.17272i |
11.4 | −0.489996 | − | 1.32661i | −2.70865 | + | 0.538783i | −1.51981 | + | 1.30007i | −2.42403 | − | 1.61969i | 2.04198 | + | 3.32933i | −0.747003 | − | 1.11797i | 2.46939 | + | 1.37917i | 4.27484 | − | 1.77070i | −0.960932 | + | 4.00939i |
11.5 | 0.114599 | + | 1.40956i | 0.935416 | − | 0.186066i | −1.97373 | + | 0.323068i | 0.763429 | + | 0.510107i | 0.369469 | + | 1.29720i | −0.225807 | − | 0.337944i | −0.681573 | − | 2.74508i | −1.93126 | + | 0.799952i | −0.631540 | + | 1.13456i |
11.6 | 1.07774 | − | 0.915678i | −0.935416 | + | 0.186066i | 0.323068 | − | 1.97373i | 0.763429 | + | 0.510107i | −0.837764 | + | 1.05707i | 0.225807 | + | 0.337944i | −1.45912 | − | 2.42301i | −1.93126 | + | 0.799952i | 1.28988 | − | 0.149290i |
23.1 | −1.17748 | + | 0.783289i | 0.418777 | + | 2.10534i | 0.772915 | − | 1.84461i | −0.561585 | + | 0.840472i | −2.14219 | − | 2.15097i | −1.73093 | + | 1.15657i | 0.534775 | + | 2.77741i | −1.48543 | + | 0.615285i | 0.00292235 | − | 1.42952i |
23.2 | −0.705238 | + | 1.22582i | −0.563072 | − | 2.83076i | −1.00528 | − | 1.72899i | 0.991762 | − | 1.48428i | 3.86710 | + | 1.30613i | −1.11612 | + | 0.745767i | 2.82840 | − | 0.0129412i | −4.92449 | + | 2.03979i | 1.12003 | + | 2.26249i |
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
17.e | odd | 16 | 1 | inner |
68.i | even | 16 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 68.2.i.b | ✓ | 48 |
3.b | odd | 2 | 1 | 612.2.bd.d | 48 | ||
4.b | odd | 2 | 1 | inner | 68.2.i.b | ✓ | 48 |
12.b | even | 2 | 1 | 612.2.bd.d | 48 | ||
17.e | odd | 16 | 1 | inner | 68.2.i.b | ✓ | 48 |
51.i | even | 16 | 1 | 612.2.bd.d | 48 | ||
68.i | even | 16 | 1 | inner | 68.2.i.b | ✓ | 48 |
204.t | odd | 16 | 1 | 612.2.bd.d | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
68.2.i.b | ✓ | 48 | 1.a | even | 1 | 1 | trivial |
68.2.i.b | ✓ | 48 | 4.b | odd | 2 | 1 | inner |
68.2.i.b | ✓ | 48 | 17.e | odd | 16 | 1 | inner |
68.2.i.b | ✓ | 48 | 68.i | even | 16 | 1 | inner |
612.2.bd.d | 48 | 3.b | odd | 2 | 1 | ||
612.2.bd.d | 48 | 12.b | even | 2 | 1 | ||
612.2.bd.d | 48 | 51.i | even | 16 | 1 | ||
612.2.bd.d | 48 | 204.t | odd | 16 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{48} + 8 T_{3}^{46} + 16 T_{3}^{44} - 384 T_{3}^{42} - 2944 T_{3}^{40} + 13600 T_{3}^{38} + \cdots + 17866064896 \) acting on \(S_{2}^{\mathrm{new}}(68, [\chi])\).