Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2009,4,Mod(1,2009)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2009, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2009.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 2009 = 7^{2} \cdot 41 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 2009.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: | \(118.534837202\) |
Analytic rank: | \(0\) |
Dimension: | \(60\) |
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 | −5.49786 | −2.52638 | 22.2265 | 1.76575 | 13.8897 | 0 | −78.2152 | −20.6174 | −9.70787 | ||||||||||||||||||
1.2 | −5.33246 | −8.03659 | 20.4351 | −13.4123 | 42.8548 | 0 | −66.3096 | 37.5867 | 71.5204 | ||||||||||||||||||
1.3 | −5.29714 | 8.61912 | 20.0597 | 18.7572 | −45.6567 | 0 | −63.8820 | 47.2892 | −99.3594 | ||||||||||||||||||
1.4 | −5.18542 | 5.95392 | 18.8886 | −18.5974 | −30.8736 | 0 | −56.4620 | 8.44915 | 96.4352 | ||||||||||||||||||
1.5 | −5.08467 | 8.85003 | 17.8539 | −9.49670 | −44.9995 | 0 | −50.1039 | 51.3230 | 48.2876 | ||||||||||||||||||
1.6 | −5.08058 | −0.507104 | 17.8123 | 13.5498 | 2.57638 | 0 | −49.8519 | −26.7428 | −68.8407 | ||||||||||||||||||
1.7 | −4.82815 | 1.12572 | 15.3110 | 7.07953 | −5.43512 | 0 | −35.2986 | −25.7328 | −34.1810 | ||||||||||||||||||
1.8 | −4.66327 | 6.89357 | 13.7460 | −2.31451 | −32.1465 | 0 | −26.7953 | 20.5213 | 10.7932 | ||||||||||||||||||
1.9 | −4.10843 | −6.89455 | 8.87916 | 16.3865 | 28.3257 | 0 | −3.61198 | 20.5348 | −67.3228 | ||||||||||||||||||
1.10 | −4.08931 | −7.07895 | 8.72244 | −1.42150 | 28.9480 | 0 | −2.95427 | 23.1115 | 5.81297 | ||||||||||||||||||
1.11 | −3.99014 | −6.15813 | 7.92125 | −10.2830 | 24.5718 | 0 | 0.314211 | 10.9226 | 41.0305 | ||||||||||||||||||
1.12 | −3.81033 | −2.35464 | 6.51862 | −16.4970 | 8.97195 | 0 | 5.64455 | −21.4557 | 62.8589 | ||||||||||||||||||
1.13 | −3.56350 | 5.16516 | 4.69851 | −4.50228 | −18.4060 | 0 | 11.7649 | −0.321131 | 16.0439 | ||||||||||||||||||
1.14 | −3.49060 | −0.191137 | 4.18427 | 0.318964 | 0.667182 | 0 | 13.3192 | −26.9635 | −1.11337 | ||||||||||||||||||
1.15 | −3.09138 | −0.499673 | 1.55662 | 19.7649 | 1.54468 | 0 | 19.9189 | −26.7503 | −61.1008 | ||||||||||||||||||
1.16 | −2.70807 | −10.0009 | −0.666347 | 16.3679 | 27.0833 | 0 | 23.4691 | 73.0187 | −44.3254 | ||||||||||||||||||
1.17 | −2.67118 | 8.92227 | −0.864788 | 15.7867 | −23.8330 | 0 | 23.6795 | 52.6070 | −42.1690 | ||||||||||||||||||
1.18 | −2.42047 | 3.67223 | −2.14132 | 8.32094 | −8.88852 | 0 | 24.5468 | −13.5147 | −20.1406 | ||||||||||||||||||
1.19 | −2.27328 | 4.10084 | −2.83221 | −16.2838 | −9.32234 | 0 | 24.6246 | −10.1831 | 37.0177 | ||||||||||||||||||
1.20 | −2.18585 | −3.77251 | −3.22204 | 1.48830 | 8.24616 | 0 | 24.5297 | −12.7682 | −3.25321 | ||||||||||||||||||
See all 60 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(7\) | \(1\) |
\(41\) | \(1\) |
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 | 2009.4.a.o | yes | 60 |
7.b | odd | 2 | 1 | 2009.4.a.n | ✓ | 60 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2009.4.a.n | ✓ | 60 | 7.b | odd | 2 | 1 | |
2009.4.a.o | yes | 60 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2009))\):
\( T_{2}^{60} - 2 T_{2}^{59} - 353 T_{2}^{58} + 704 T_{2}^{57} + 58689 T_{2}^{56} - 116674 T_{2}^{55} + \cdots - 13\!\cdots\!36 \) |
\( T_{3}^{60} - 24 T_{3}^{59} - 808 T_{3}^{58} + 23280 T_{3}^{57} + 281662 T_{3}^{56} + \cdots + 18\!\cdots\!12 \) |