Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [340,2,Mod(57,340)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(340, base_ring=CyclotomicField(16))
chi = DirichletCharacter(H, H._module([0, 4, 15]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("340.57");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 340 = 2^{2} \cdot 5 \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 340.bd (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: | \(2.71491366872\) |
Analytic rank: | \(0\) |
Dimension: | \(72\) |
Relative dimension: | \(9\) over \(\Q(\zeta_{16})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{16}]$ |
$q$-expansion
The algebraic \(q\)-expansion of this newform has not been computed, 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
57.1 | 0 | −1.50403 | + | 2.25094i | 0 | −2.02389 | − | 0.950727i | 0 | 0.694602 | − | 3.49200i | 0 | −1.65656 | − | 3.99930i | 0 | ||||||||||
57.2 | 0 | −1.35240 | + | 2.02401i | 0 | 2.11145 | − | 0.736056i | 0 | −0.777549 | + | 3.90900i | 0 | −1.11958 | − | 2.70291i | 0 | ||||||||||
57.3 | 0 | −1.25555 | + | 1.87906i | 0 | 0.726829 | + | 2.11464i | 0 | 0.0284166 | − | 0.142860i | 0 | −0.806426 | − | 1.94689i | 0 | ||||||||||
57.4 | 0 | −0.514627 | + | 0.770194i | 0 | −1.14734 | − | 1.91927i | 0 | −0.415329 | + | 2.08800i | 0 | 0.819692 | + | 1.97891i | 0 | ||||||||||
57.5 | 0 | 0.286067 | − | 0.428130i | 0 | 1.53584 | + | 1.62517i | 0 | 0.632123 | − | 3.17790i | 0 | 1.04659 | + | 2.52669i | 0 | ||||||||||
57.6 | 0 | 0.349423 | − | 0.522949i | 0 | 1.09010 | − | 1.95235i | 0 | 0.0348214 | − | 0.175059i | 0 | 0.996671 | + | 2.40618i | 0 | ||||||||||
57.7 | 0 | 0.841077 | − | 1.25876i | 0 | −0.704990 | + | 2.12202i | 0 | −0.993774 | + | 4.99604i | 0 | 0.270982 | + | 0.654209i | 0 | ||||||||||
57.8 | 0 | 1.27530 | − | 1.90862i | 0 | −2.15814 | − | 0.585194i | 0 | 0.311904 | − | 1.56805i | 0 | −0.868385 | − | 2.09647i | 0 | ||||||||||
57.9 | 0 | 1.87474 | − | 2.80575i | 0 | 2.20111 | − | 0.393819i | 0 | −0.0564111 | + | 0.283598i | 0 | −3.20952 | − | 7.74847i | 0 | ||||||||||
73.1 | 0 | −0.529264 | + | 2.66079i | 0 | 0.427916 | + | 2.19474i | 0 | 2.51922 | + | 1.68329i | 0 | −4.02805 | − | 1.66847i | 0 | ||||||||||
73.2 | 0 | −0.441472 | + | 2.21943i | 0 | −2.23590 | − | 0.0274503i | 0 | −2.81028 | − | 1.87777i | 0 | −1.95932 | − | 0.811579i | 0 | ||||||||||
73.3 | 0 | −0.402101 | + | 2.02150i | 0 | −0.456223 | − | 2.18903i | 0 | 2.17863 | + | 1.45572i | 0 | −1.15314 | − | 0.477647i | 0 | ||||||||||
73.4 | 0 | 0.0289619 | − | 0.145601i | 0 | 0.607979 | − | 2.15183i | 0 | −1.82288 | − | 1.21801i | 0 | 2.75128 | + | 1.13962i | 0 | ||||||||||
73.5 | 0 | 0.0352247 | − | 0.177087i | 0 | −0.198315 | + | 2.22726i | 0 | −3.20617 | − | 2.14230i | 0 | 2.74152 | + | 1.13557i | 0 | ||||||||||
73.6 | 0 | 0.0357590 | − | 0.179773i | 0 | 2.03483 | + | 0.927070i | 0 | 0.528939 | + | 0.353426i | 0 | 2.74060 | + | 1.13519i | 0 | ||||||||||
73.7 | 0 | 0.197416 | − | 0.992478i | 0 | −2.21386 | + | 0.314337i | 0 | 4.11814 | + | 2.75165i | 0 | 1.82560 | + | 0.756188i | 0 | ||||||||||
73.8 | 0 | 0.486949 | − | 2.44806i | 0 | 2.17164 | − | 0.532896i | 0 | 1.76999 | + | 1.18267i | 0 | −2.98423 | − | 1.23611i | 0 | ||||||||||
73.9 | 0 | 0.588527 | − | 2.95872i | 0 | −1.22786 | + | 1.86879i | 0 | −1.96902 | − | 1.31565i | 0 | −5.63604 | − | 2.33452i | 0 | ||||||||||
133.1 | 0 | −2.36788 | + | 1.58217i | 0 | 2.10787 | − | 0.746254i | 0 | 1.12893 | − | 0.224559i | 0 | 1.95555 | − | 4.72111i | 0 | ||||||||||
133.2 | 0 | −1.89969 | + | 1.26933i | 0 | −1.28747 | + | 1.82823i | 0 | 2.80650 | − | 0.558249i | 0 | 0.849575 | − | 2.05105i | 0 | ||||||||||
See all 72 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
85.o | even | 16 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 340.2.bd.a | ✓ | 72 |
5.c | odd | 4 | 1 | 340.2.bi.a | yes | 72 | |
17.e | odd | 16 | 1 | 340.2.bi.a | yes | 72 | |
85.o | even | 16 | 1 | inner | 340.2.bd.a | ✓ | 72 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
340.2.bd.a | ✓ | 72 | 1.a | even | 1 | 1 | trivial |
340.2.bd.a | ✓ | 72 | 85.o | even | 16 | 1 | inner |
340.2.bi.a | yes | 72 | 5.c | odd | 4 | 1 | |
340.2.bi.a | yes | 72 | 17.e | odd | 16 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(340, [\chi])\).