Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4009,2,Mod(1,4009)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4009, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4009.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 4009 = 19 \cdot 211 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 4009.a (trivial) |
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: | yes |
Analytic conductor: | \(32.0120261703\) |
Analytic rank: | \(0\) |
Dimension: | \(83\) |
Twist minimal: | yes |
Fricke sign: | \(-1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1.1 | −2.76212 | −1.26441 | 5.62933 | −0.901770 | 3.49245 | 2.01058 | −10.0247 | −1.40127 | 2.49080 | ||||||||||||||||||
1.2 | −2.70959 | 1.36959 | 5.34187 | 3.33123 | −3.71102 | 3.43608 | −9.05510 | −1.12423 | −9.02626 | ||||||||||||||||||
1.3 | −2.67158 | −3.12490 | 5.13731 | 1.64338 | 8.34840 | −0.166545 | −8.38157 | 6.76498 | −4.39041 | ||||||||||||||||||
1.4 | −2.57059 | 1.65554 | 4.60795 | −0.615415 | −4.25572 | −3.21164 | −6.70397 | −0.259188 | 1.58198 | ||||||||||||||||||
1.5 | −2.54651 | 2.15718 | 4.48471 | 0.595796 | −5.49327 | 3.08751 | −6.32733 | 1.65341 | −1.51720 | ||||||||||||||||||
1.6 | −2.52441 | 2.53119 | 4.37265 | −2.23047 | −6.38976 | −2.32994 | −5.98955 | 3.40692 | 5.63062 | ||||||||||||||||||
1.7 | −2.39927 | −1.08640 | 3.75651 | 4.25844 | 2.60656 | 3.20578 | −4.21434 | −1.81974 | −10.2171 | ||||||||||||||||||
1.8 | −2.36270 | −3.38908 | 3.58234 | 0.417258 | 8.00738 | 4.46721 | −3.73860 | 8.48588 | −0.985855 | ||||||||||||||||||
1.9 | −2.35956 | −1.58207 | 3.56753 | −1.28760 | 3.73299 | −1.93582 | −3.69869 | −0.497055 | 3.03817 | ||||||||||||||||||
1.10 | −2.30870 | −1.28180 | 3.33008 | 1.74419 | 2.95929 | −4.60684 | −3.07074 | −1.35699 | −4.02679 | ||||||||||||||||||
1.11 | −2.26307 | −2.97799 | 3.12147 | −3.73061 | 6.73939 | −2.90027 | −2.53796 | 5.86843 | 8.44263 | ||||||||||||||||||
1.12 | −2.15681 | −0.442776 | 2.65181 | −2.84036 | 0.954981 | 0.461477 | −1.40583 | −2.80395 | 6.12610 | ||||||||||||||||||
1.13 | −2.15273 | 1.11488 | 2.63426 | −2.91632 | −2.40003 | 1.19746 | −1.36540 | −1.75705 | 6.27807 | ||||||||||||||||||
1.14 | −1.98014 | 0.389885 | 1.92094 | 2.34447 | −0.772025 | −3.40620 | 0.156544 | −2.84799 | −4.64237 | ||||||||||||||||||
1.15 | −1.90192 | 0.698740 | 1.61729 | 0.858137 | −1.32894 | 1.83968 | 0.727887 | −2.51176 | −1.63210 | ||||||||||||||||||
1.16 | −1.77808 | −1.50294 | 1.16155 | 1.25808 | 2.67234 | 4.62687 | 1.49083 | −0.741168 | −2.23696 | ||||||||||||||||||
1.17 | −1.73978 | 2.70968 | 1.02684 | 3.17028 | −4.71425 | 1.26105 | 1.69309 | 4.34237 | −5.51560 | ||||||||||||||||||
1.18 | −1.71565 | −0.772476 | 0.943456 | −1.16427 | 1.32530 | −0.902518 | 1.81266 | −2.40328 | 1.99748 | ||||||||||||||||||
1.19 | −1.59861 | 3.13645 | 0.555538 | 2.23749 | −5.01395 | 4.84396 | 2.30912 | 6.83733 | −3.57686 | ||||||||||||||||||
1.20 | −1.56314 | 0.463979 | 0.443399 | 3.91464 | −0.725264 | −3.83453 | 2.43318 | −2.78472 | −6.11913 | ||||||||||||||||||
See all 83 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(19\) | \(1\) |
\(211\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 4009.2.a.f | ✓ | 83 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4009.2.a.f | ✓ | 83 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{83} - 11 T_{2}^{82} - 70 T_{2}^{81} + 1189 T_{2}^{80} + 1162 T_{2}^{79} - 60709 T_{2}^{78} + \cdots + 818676 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4009))\).