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: | \(30\) |
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 | −5.44610 | −3.46823 | 21.6600 | −9.65556 | 18.8883 | 0 | −74.3935 | −14.9714 | 52.5851 | ||||||||||||||||||
1.2 | −5.25966 | −9.95056 | 19.6641 | 0.690716 | 52.3366 | 0 | −61.3490 | 72.0137 | −3.63293 | ||||||||||||||||||
1.3 | −4.59222 | 5.30191 | 13.0885 | −17.8900 | −24.3475 | 0 | −23.3675 | 1.11023 | 82.1547 | ||||||||||||||||||
1.4 | −4.43600 | 5.27473 | 11.6781 | 19.0439 | −23.3987 | 0 | −16.3160 | 0.822745 | −84.4787 | ||||||||||||||||||
1.5 | −4.33219 | 4.25218 | 10.7679 | 6.40888 | −18.4212 | 0 | −11.9908 | −8.91899 | −27.7645 | ||||||||||||||||||
1.6 | −3.69842 | −7.66955 | 5.67831 | 4.73696 | 28.3652 | 0 | 8.58659 | 31.8219 | −17.5193 | ||||||||||||||||||
1.7 | −3.13685 | 0.726413 | 1.83984 | −8.18759 | −2.27865 | 0 | 19.3235 | −26.4723 | 25.6833 | ||||||||||||||||||
1.8 | −3.13019 | 9.57945 | 1.79808 | 15.3925 | −29.9855 | 0 | 19.4132 | 64.7659 | −48.1816 | ||||||||||||||||||
1.9 | −2.61264 | 0.0204702 | −1.17409 | 12.1971 | −0.0534812 | 0 | 23.9686 | −26.9996 | −31.8666 | ||||||||||||||||||
1.10 | −2.48350 | −6.40015 | −1.83221 | 4.02052 | 15.8948 | 0 | 24.4183 | 13.9619 | −9.98498 | ||||||||||||||||||
1.11 | −1.76668 | −6.65681 | −4.87886 | −16.8187 | 11.7604 | 0 | 22.7528 | 17.3131 | 29.7131 | ||||||||||||||||||
1.12 | −1.15126 | 8.67479 | −6.67460 | −8.17766 | −9.98692 | 0 | 16.8943 | 48.2520 | 9.41460 | ||||||||||||||||||
1.13 | −1.00255 | −0.387093 | −6.99489 | 15.3378 | 0.388082 | 0 | 15.0332 | −26.8502 | −15.3769 | ||||||||||||||||||
1.14 | −0.941072 | 8.32984 | −7.11438 | −21.2675 | −7.83898 | 0 | 14.2237 | 42.3862 | 20.0142 | ||||||||||||||||||
1.15 | −0.208351 | 3.07702 | −7.95659 | −2.07481 | −0.641101 | 0 | 3.32458 | −17.5319 | 0.432289 | ||||||||||||||||||
1.16 | −0.116020 | −5.21300 | −7.98654 | −11.3743 | 0.604813 | 0 | 1.85476 | 0.175418 | 1.31965 | ||||||||||||||||||
1.17 | 1.32102 | 10.0678 | −6.25490 | 8.86662 | 13.2998 | 0 | −18.8311 | 74.3601 | 11.7130 | ||||||||||||||||||
1.18 | 1.36514 | −2.26242 | −6.13639 | 19.1582 | −3.08852 | 0 | −19.2982 | −21.8815 | 26.1536 | ||||||||||||||||||
1.19 | 1.40105 | 4.34317 | −6.03706 | −3.78319 | 6.08500 | 0 | −19.6666 | −8.13684 | −5.30043 | ||||||||||||||||||
1.20 | 1.65259 | −8.44588 | −5.26896 | −2.08319 | −13.9575 | 0 | −21.9281 | 44.3328 | −3.44266 | ||||||||||||||||||
See all 30 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.i | yes | 30 |
7.b | odd | 2 | 1 | 2009.4.a.h | ✓ | 30 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2009.4.a.h | ✓ | 30 | 7.b | odd | 2 | 1 | |
2009.4.a.i | yes | 30 | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2009))\):
\( T_{2}^{30} - T_{2}^{29} - 171 T_{2}^{28} + 169 T_{2}^{27} + 12973 T_{2}^{26} - 12569 T_{2}^{25} + \cdots + 82043329536 \) |
\( T_{3}^{30} - 12 T_{3}^{29} - 492 T_{3}^{28} + 6120 T_{3}^{27} + 105467 T_{3}^{26} + \cdots + 84\!\cdots\!04 \) |