Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4011,2,Mod(1,4011)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4011, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("4011.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 4011 = 3 \cdot 7 \cdot 191 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 4011.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: | \(32.0279962507\) |
Analytic rank: | \(0\) |
Dimension: | \(28\) |
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 | −2.77095 | −1.00000 | 5.67816 | 3.12132 | 2.77095 | 1.00000 | −10.1920 | 1.00000 | −8.64902 | ||||||||||||||||||
1.2 | −2.75611 | −1.00000 | 5.59616 | −2.11611 | 2.75611 | 1.00000 | −9.91144 | 1.00000 | 5.83224 | ||||||||||||||||||
1.3 | −2.60224 | −1.00000 | 4.77168 | −2.58506 | 2.60224 | 1.00000 | −7.21258 | 1.00000 | 6.72695 | ||||||||||||||||||
1.4 | −2.55579 | −1.00000 | 4.53209 | 4.15915 | 2.55579 | 1.00000 | −6.47150 | 1.00000 | −10.6299 | ||||||||||||||||||
1.5 | −2.32219 | −1.00000 | 3.39258 | −2.33514 | 2.32219 | 1.00000 | −3.23383 | 1.00000 | 5.42264 | ||||||||||||||||||
1.6 | −2.23874 | −1.00000 | 3.01194 | 1.19635 | 2.23874 | 1.00000 | −2.26547 | 1.00000 | −2.67832 | ||||||||||||||||||
1.7 | −1.81456 | −1.00000 | 1.29264 | 0.616523 | 1.81456 | 1.00000 | 1.28355 | 1.00000 | −1.11872 | ||||||||||||||||||
1.8 | −1.80294 | −1.00000 | 1.25060 | −3.82317 | 1.80294 | 1.00000 | 1.35113 | 1.00000 | 6.89294 | ||||||||||||||||||
1.9 | −1.70611 | −1.00000 | 0.910828 | 3.20896 | 1.70611 | 1.00000 | 1.85825 | 1.00000 | −5.47486 | ||||||||||||||||||
1.10 | −1.53559 | −1.00000 | 0.358037 | −0.986091 | 1.53559 | 1.00000 | 2.52138 | 1.00000 | 1.51423 | ||||||||||||||||||
1.11 | −0.815618 | −1.00000 | −1.33477 | 3.08846 | 0.815618 | 1.00000 | 2.71990 | 1.00000 | −2.51900 | ||||||||||||||||||
1.12 | −0.786373 | −1.00000 | −1.38162 | 2.33863 | 0.786373 | 1.00000 | 2.65921 | 1.00000 | −1.83903 | ||||||||||||||||||
1.13 | −0.482188 | −1.00000 | −1.76749 | −4.13435 | 0.482188 | 1.00000 | 1.81664 | 1.00000 | 1.99354 | ||||||||||||||||||
1.14 | −0.386952 | −1.00000 | −1.85027 | 3.15570 | 0.386952 | 1.00000 | 1.48987 | 1.00000 | −1.22110 | ||||||||||||||||||
1.15 | −0.268740 | −1.00000 | −1.92778 | −1.86903 | 0.268740 | 1.00000 | 1.05555 | 1.00000 | 0.502284 | ||||||||||||||||||
1.16 | −0.0484532 | −1.00000 | −1.99765 | 1.78925 | 0.0484532 | 1.00000 | 0.193699 | 1.00000 | −0.0866947 | ||||||||||||||||||
1.17 | 0.343679 | −1.00000 | −1.88188 | −1.00728 | −0.343679 | 1.00000 | −1.33412 | 1.00000 | −0.346179 | ||||||||||||||||||
1.18 | 0.544491 | −1.00000 | −1.70353 | 2.42913 | −0.544491 | 1.00000 | −2.01654 | 1.00000 | 1.32264 | ||||||||||||||||||
1.19 | 0.602953 | −1.00000 | −1.63645 | −1.53490 | −0.602953 | 1.00000 | −2.19261 | 1.00000 | −0.925472 | ||||||||||||||||||
1.20 | 1.03548 | −1.00000 | −0.927780 | −1.23903 | −1.03548 | 1.00000 | −3.03166 | 1.00000 | −1.28299 | ||||||||||||||||||
See all 28 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(1\) |
\(7\) | \(-1\) |
\(191\) | \(1\) |
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 | 4011.2.a.l | ✓ | 28 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4011.2.a.l | ✓ | 28 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{28} + 6 T_{2}^{27} - 27 T_{2}^{26} - 221 T_{2}^{25} + 212 T_{2}^{24} + 3487 T_{2}^{23} + \cdots + 96 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4011))\).