Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1008,2,Mod(689,1008)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1008, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 5, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1008.689");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1008.df (of order \(6\), degree \(2\), not 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: | \(8.04892052375\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Relative dimension: | \(24\) over \(\Q(\zeta_{6})\) |
Twist minimal: | no (minimal twist has level 504) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
689.1 | 0 | −1.73204 | + | 0.00732041i | 0 | 2.59337 | 0 | 2.61256 | − | 0.417759i | 0 | 2.99989 | − | 0.0253584i | 0 | ||||||||||||
689.2 | 0 | −1.72872 | − | 0.107304i | 0 | −3.60777 | 0 | −0.415579 | − | 2.61291i | 0 | 2.97697 | + | 0.370999i | 0 | ||||||||||||
689.3 | 0 | −1.70521 | + | 0.303722i | 0 | 2.22830 | 0 | −2.51733 | − | 0.814280i | 0 | 2.81551 | − | 1.03582i | 0 | ||||||||||||
689.4 | 0 | −1.64232 | − | 0.550252i | 0 | −1.28783 | 0 | −1.10056 | + | 2.40599i | 0 | 2.39445 | + | 1.80738i | 0 | ||||||||||||
689.5 | 0 | −1.51239 | + | 0.844203i | 0 | −2.84763 | 0 | 2.64481 | − | 0.0704652i | 0 | 1.57464 | − | 2.55353i | 0 | ||||||||||||
689.6 | 0 | −1.24547 | + | 1.20367i | 0 | 2.04899 | 0 | −1.41312 | + | 2.23676i | 0 | 0.102371 | − | 2.99825i | 0 | ||||||||||||
689.7 | 0 | −0.994080 | − | 1.41838i | 0 | −1.58600 | 0 | 1.06431 | − | 2.42224i | 0 | −1.02361 | + | 2.81997i | 0 | ||||||||||||
689.8 | 0 | −0.958980 | − | 1.44234i | 0 | 4.11484 | 0 | 2.54819 | − | 0.711830i | 0 | −1.16071 | + | 2.76636i | 0 | ||||||||||||
689.9 | 0 | −0.930597 | − | 1.46082i | 0 | 1.36828 | 0 | −2.64451 | − | 0.0810554i | 0 | −1.26798 | + | 2.71887i | 0 | ||||||||||||
689.10 | 0 | −0.549450 | + | 1.64259i | 0 | −1.05582 | 0 | −1.79851 | − | 1.94045i | 0 | −2.39621 | − | 1.80504i | 0 | ||||||||||||
689.11 | 0 | −0.310783 | + | 1.70394i | 0 | 2.76937 | 0 | 1.21939 | + | 2.34800i | 0 | −2.80683 | − | 1.05911i | 0 | ||||||||||||
689.12 | 0 | −0.149059 | + | 1.72562i | 0 | −2.22094 | 0 | 2.45091 | + | 0.996507i | 0 | −2.95556 | − | 0.514438i | 0 | ||||||||||||
689.13 | 0 | 0.210634 | − | 1.71920i | 0 | −1.07485 | 0 | 1.26701 | + | 2.32265i | 0 | −2.91127 | − | 0.724244i | 0 | ||||||||||||
689.14 | 0 | 0.498607 | + | 1.65873i | 0 | −0.623597 | 0 | 0.996837 | − | 2.45078i | 0 | −2.50278 | + | 1.65411i | 0 | ||||||||||||
689.15 | 0 | 0.601162 | − | 1.62438i | 0 | −0.207028 | 0 | 1.37075 | − | 2.26297i | 0 | −2.27721 | − | 1.95303i | 0 | ||||||||||||
689.16 | 0 | 0.859090 | − | 1.50398i | 0 | −4.19811 | 0 | −1.67151 | + | 2.05087i | 0 | −1.52393 | − | 2.58411i | 0 | ||||||||||||
689.17 | 0 | 1.03050 | − | 1.39215i | 0 | 2.20884 | 0 | −2.16520 | − | 1.52049i | 0 | −0.876153 | − | 2.86921i | 0 | ||||||||||||
689.18 | 0 | 1.11231 | + | 1.32769i | 0 | 3.53401 | 0 | −1.30083 | − | 2.30388i | 0 | −0.525517 | + | 2.95361i | 0 | ||||||||||||
689.19 | 0 | 1.14647 | + | 1.29831i | 0 | −0.0525740 | 0 | −2.44149 | + | 1.01937i | 0 | −0.371197 | + | 2.97695i | 0 | ||||||||||||
689.20 | 0 | 1.45594 | + | 0.938214i | 0 | −1.81173 | 0 | −1.69266 | + | 2.03345i | 0 | 1.23951 | + | 2.73196i | 0 | ||||||||||||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
63.s | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1008.2.df.e | 48 | |
3.b | odd | 2 | 1 | 3024.2.df.e | 48 | ||
4.b | odd | 2 | 1 | 504.2.cx.a | yes | 48 | |
7.d | odd | 6 | 1 | 1008.2.ca.e | 48 | ||
9.c | even | 3 | 1 | 3024.2.ca.e | 48 | ||
9.d | odd | 6 | 1 | 1008.2.ca.e | 48 | ||
12.b | even | 2 | 1 | 1512.2.cx.a | 48 | ||
21.g | even | 6 | 1 | 3024.2.ca.e | 48 | ||
28.f | even | 6 | 1 | 504.2.bs.a | ✓ | 48 | |
36.f | odd | 6 | 1 | 1512.2.bs.a | 48 | ||
36.h | even | 6 | 1 | 504.2.bs.a | ✓ | 48 | |
63.k | odd | 6 | 1 | 3024.2.df.e | 48 | ||
63.s | even | 6 | 1 | inner | 1008.2.df.e | 48 | |
84.j | odd | 6 | 1 | 1512.2.bs.a | 48 | ||
252.n | even | 6 | 1 | 1512.2.cx.a | 48 | ||
252.bn | odd | 6 | 1 | 504.2.cx.a | yes | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
504.2.bs.a | ✓ | 48 | 28.f | even | 6 | 1 | |
504.2.bs.a | ✓ | 48 | 36.h | even | 6 | 1 | |
504.2.cx.a | yes | 48 | 4.b | odd | 2 | 1 | |
504.2.cx.a | yes | 48 | 252.bn | odd | 6 | 1 | |
1008.2.ca.e | 48 | 7.d | odd | 6 | 1 | ||
1008.2.ca.e | 48 | 9.d | odd | 6 | 1 | ||
1008.2.df.e | 48 | 1.a | even | 1 | 1 | trivial | |
1008.2.df.e | 48 | 63.s | even | 6 | 1 | inner | |
1512.2.bs.a | 48 | 36.f | odd | 6 | 1 | ||
1512.2.bs.a | 48 | 84.j | odd | 6 | 1 | ||
1512.2.cx.a | 48 | 12.b | even | 2 | 1 | ||
1512.2.cx.a | 48 | 252.n | even | 6 | 1 | ||
3024.2.ca.e | 48 | 9.c | even | 3 | 1 | ||
3024.2.ca.e | 48 | 21.g | even | 6 | 1 | ||
3024.2.df.e | 48 | 3.b | odd | 2 | 1 | ||
3024.2.df.e | 48 | 63.k | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} - 72 T_{5}^{22} - 12 T_{5}^{21} + 2175 T_{5}^{20} + 768 T_{5}^{19} - 36005 T_{5}^{18} + \cdots - 6476 \) acting on \(S_{2}^{\mathrm{new}}(1008, [\chi])\).