Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1045,6,Mod(1,1045)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1045, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1045.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1045 = 5 \cdot 11 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 1045.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: | \(167.601091705\) |
Analytic rank: | \(0\) |
Dimension: | \(40\) |
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 | −10.7623 | −11.3539 | 83.8277 | 25.0000 | 122.194 | −59.3342 | −557.787 | −114.090 | −269.058 | ||||||||||||||||||
1.2 | −10.4622 | 24.7793 | 77.4578 | 25.0000 | −259.246 | 156.281 | −475.589 | 371.012 | −261.555 | ||||||||||||||||||
1.3 | −10.3994 | 4.60548 | 76.1474 | 25.0000 | −47.8942 | 17.5294 | −459.106 | −221.790 | −259.985 | ||||||||||||||||||
1.4 | −10.1541 | −26.5431 | 71.1059 | 25.0000 | 269.522 | 225.388 | −397.085 | 461.539 | −253.853 | ||||||||||||||||||
1.5 | −8.55086 | 12.0717 | 41.1172 | 25.0000 | −103.224 | −150.646 | −77.9596 | −97.2734 | −213.771 | ||||||||||||||||||
1.6 | −8.31261 | 15.2747 | 37.0994 | 25.0000 | −126.972 | 144.510 | −42.3897 | −9.68428 | −207.815 | ||||||||||||||||||
1.7 | −7.39269 | −19.3511 | 22.6518 | 25.0000 | 143.057 | −146.772 | 69.1080 | 131.466 | −184.817 | ||||||||||||||||||
1.8 | −7.16026 | 27.0461 | 19.2693 | 25.0000 | −193.657 | −115.417 | 91.1552 | 488.494 | −179.006 | ||||||||||||||||||
1.9 | −6.98489 | −3.88370 | 16.7886 | 25.0000 | 27.1272 | 123.805 | 106.250 | −227.917 | −174.622 | ||||||||||||||||||
1.10 | −6.34868 | −22.4542 | 8.30576 | 25.0000 | 142.555 | 214.893 | 150.427 | 261.192 | −158.717 | ||||||||||||||||||
1.11 | −5.82679 | −5.33310 | 1.95150 | 25.0000 | 31.0749 | −202.310 | 175.086 | −214.558 | −145.670 | ||||||||||||||||||
1.12 | −5.02898 | 3.76125 | −6.70939 | 25.0000 | −18.9153 | 170.122 | 194.669 | −228.853 | −125.724 | ||||||||||||||||||
1.13 | −3.84457 | 26.7977 | −17.2193 | 25.0000 | −103.025 | 117.853 | 189.227 | 475.114 | −96.1141 | ||||||||||||||||||
1.14 | −2.87435 | 1.31331 | −23.7381 | 25.0000 | −3.77492 | −104.963 | 160.211 | −241.275 | −71.8588 | ||||||||||||||||||
1.15 | −2.77766 | −22.7115 | −24.2846 | 25.0000 | 63.0848 | −147.613 | 156.340 | 272.811 | −69.4415 | ||||||||||||||||||
1.16 | −1.49444 | 1.90416 | −29.7667 | 25.0000 | −2.84565 | 61.7350 | 92.3065 | −239.374 | −37.3610 | ||||||||||||||||||
1.17 | −1.40465 | −17.9315 | −30.0269 | 25.0000 | 25.1875 | 73.4146 | 87.1263 | 78.5376 | −35.1163 | ||||||||||||||||||
1.18 | −0.609213 | 20.7237 | −31.6289 | 25.0000 | −12.6251 | −114.174 | 38.7636 | 186.471 | −15.2303 | ||||||||||||||||||
1.19 | −0.153758 | −4.07857 | −31.9764 | 25.0000 | 0.627114 | −123.189 | 9.83689 | −226.365 | −3.84396 | ||||||||||||||||||
1.20 | 0.237841 | 14.0227 | −31.9434 | 25.0000 | 3.33516 | 13.4367 | −15.2083 | −46.3651 | 5.94602 | ||||||||||||||||||
See all 40 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(5\) | \(-1\) |
\(11\) | \(-1\) |
\(19\) | \(-1\) |
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 | 1045.6.a.h | ✓ | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1045.6.a.h | ✓ | 40 | 1.a | even | 1 | 1 | trivial |