Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8041,2,Mod(1,8041)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8041, 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("8041.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8041 = 11 \cdot 17 \cdot 43 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8041.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: | \(64.2077082653\) |
Analytic rank: | \(1\) |
Dimension: | \(74\) |
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 | −2.77959 | −0.489198 | 5.72614 | −0.717152 | 1.35977 | −1.78399 | −10.3571 | −2.76069 | 1.99339 | ||||||||||||||||||
1.2 | −2.77350 | −1.80573 | 5.69228 | 3.84925 | 5.00818 | −3.56425 | −10.2405 | 0.260654 | −10.6759 | ||||||||||||||||||
1.3 | −2.72314 | 3.10551 | 5.41549 | −3.61264 | −8.45674 | −4.38477 | −9.30085 | 6.64419 | 9.83772 | ||||||||||||||||||
1.4 | −2.65828 | 2.25599 | 5.06647 | 2.74441 | −5.99706 | 1.84456 | −8.15154 | 2.08950 | −7.29541 | ||||||||||||||||||
1.5 | −2.63295 | 1.21952 | 4.93245 | −0.402998 | −3.21093 | 4.56516 | −7.72100 | −1.51278 | 1.06108 | ||||||||||||||||||
1.6 | −2.60873 | −2.96905 | 4.80548 | −3.25113 | 7.74546 | 1.59414 | −7.31874 | 5.81527 | 8.48133 | ||||||||||||||||||
1.7 | −2.58183 | 0.610196 | 4.66583 | −3.25776 | −1.57542 | −0.634845 | −6.88270 | −2.62766 | 8.41096 | ||||||||||||||||||
1.8 | −2.56683 | −0.396689 | 4.58863 | 1.36693 | 1.01823 | 2.28939 | −6.64457 | −2.84264 | −3.50868 | ||||||||||||||||||
1.9 | −2.38029 | 3.05182 | 3.66578 | 3.97493 | −7.26423 | −3.90026 | −3.96504 | 6.31363 | −9.46150 | ||||||||||||||||||
1.10 | −2.34491 | −2.19404 | 3.49862 | 1.59097 | 5.14484 | −2.37576 | −3.51413 | 1.81381 | −3.73069 | ||||||||||||||||||
1.11 | −2.21900 | −1.42344 | 2.92394 | −3.24388 | 3.15861 | 4.34231 | −2.05022 | −0.973808 | 7.19815 | ||||||||||||||||||
1.12 | −2.15071 | 2.55470 | 2.62555 | −0.0279293 | −5.49442 | −2.62385 | −1.34537 | 3.52651 | 0.0600678 | ||||||||||||||||||
1.13 | −2.14896 | −2.65732 | 2.61805 | 2.21050 | 5.71048 | 0.288836 | −1.32816 | 4.06133 | −4.75028 | ||||||||||||||||||
1.14 | −2.02657 | 2.55915 | 2.10699 | −3.25953 | −5.18630 | 3.04044 | −0.216828 | 3.54925 | 6.60567 | ||||||||||||||||||
1.15 | −1.90535 | −0.951044 | 1.63036 | −1.53040 | 1.81207 | −2.50310 | 0.704287 | −2.09552 | 2.91594 | ||||||||||||||||||
1.16 | −1.89916 | −0.231765 | 1.60683 | 2.27252 | 0.440160 | 0.501161 | 0.746704 | −2.94628 | −4.31590 | ||||||||||||||||||
1.17 | −1.85483 | 0.655235 | 1.44041 | 4.10800 | −1.21535 | −1.24342 | 1.03795 | −2.57067 | −7.61965 | ||||||||||||||||||
1.18 | −1.83407 | 2.68858 | 1.36381 | −0.219056 | −4.93104 | 2.19442 | 1.16682 | 4.22847 | 0.401763 | ||||||||||||||||||
1.19 | −1.66618 | −0.454715 | 0.776164 | 0.422618 | 0.757638 | −4.38423 | 2.03913 | −2.79323 | −0.704158 | ||||||||||||||||||
1.20 | −1.65992 | 0.748300 | 0.755318 | −4.19197 | −1.24211 | −4.84394 | 2.06607 | −2.44005 | 6.95831 | ||||||||||||||||||
See all 74 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(11\) | \(1\) |
\(17\) | \(1\) |
\(43\) | \(1\) |
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 | 8041.2.a.h | ✓ | 74 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8041.2.a.h | ✓ | 74 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{74} + 7 T_{2}^{73} - 89 T_{2}^{72} - 721 T_{2}^{71} + 3587 T_{2}^{70} + 35347 T_{2}^{69} + \cdots + 37888 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8041))\).