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: | \(1\) |
Dimension: | \(36\) |
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 | −10.8010 | −21.4786 | 84.6626 | 25.0000 | 231.991 | 40.5180 | −568.812 | 218.330 | −270.026 | ||||||||||||||||||
1.2 | −10.6840 | 22.9747 | 82.1485 | 25.0000 | −245.462 | 48.0772 | −535.788 | 284.836 | −267.101 | ||||||||||||||||||
1.3 | −9.59224 | −21.5591 | 60.0111 | 25.0000 | 206.800 | −136.266 | −268.689 | 221.796 | −239.806 | ||||||||||||||||||
1.4 | −9.55197 | −7.71824 | 59.2400 | 25.0000 | 73.7243 | 175.466 | −260.196 | −183.429 | −238.799 | ||||||||||||||||||
1.5 | −9.26988 | 10.9289 | 53.9306 | 25.0000 | −101.309 | 145.096 | −203.294 | −123.560 | −231.747 | ||||||||||||||||||
1.6 | −9.10660 | 19.2419 | 50.9302 | 25.0000 | −175.228 | −172.410 | −172.390 | 127.251 | −227.665 | ||||||||||||||||||
1.7 | −7.64458 | −6.22870 | 26.4397 | 25.0000 | 47.6158 | 101.320 | 42.5064 | −204.203 | −191.115 | ||||||||||||||||||
1.8 | −7.00111 | −3.95422 | 17.0156 | 25.0000 | 27.6840 | −243.106 | 104.908 | −227.364 | −175.028 | ||||||||||||||||||
1.9 | −6.95850 | −10.9341 | 16.4207 | 25.0000 | 76.0849 | −81.2421 | 108.408 | −123.446 | −173.963 | ||||||||||||||||||
1.10 | −6.79783 | −25.2972 | 14.2104 | 25.0000 | 171.966 | 60.0170 | 120.930 | 396.947 | −169.946 | ||||||||||||||||||
1.11 | −5.96881 | 24.3951 | 3.62665 | 25.0000 | −145.610 | 53.9217 | 169.355 | 352.122 | −149.220 | ||||||||||||||||||
1.12 | −4.65971 | 14.3366 | −10.2871 | 25.0000 | −66.8044 | 127.161 | 197.046 | −37.4616 | −116.493 | ||||||||||||||||||
1.13 | −3.77729 | −30.8759 | −17.7321 | 25.0000 | 116.627 | −222.797 | 187.852 | 710.323 | −94.4321 | ||||||||||||||||||
1.14 | −3.50600 | −5.81498 | −19.7080 | 25.0000 | 20.3873 | 29.7384 | 181.288 | −209.186 | −87.6500 | ||||||||||||||||||
1.15 | −3.31390 | −19.0151 | −21.0181 | 25.0000 | 63.0141 | 82.9347 | 175.697 | 118.573 | −82.8475 | ||||||||||||||||||
1.16 | −2.55950 | 0.0390510 | −25.4489 | 25.0000 | −0.0999513 | −217.307 | 147.041 | −242.998 | −63.9876 | ||||||||||||||||||
1.17 | −1.33765 | 19.4352 | −30.2107 | 25.0000 | −25.9975 | −167.518 | 83.2162 | 134.726 | −33.4413 | ||||||||||||||||||
1.18 | −0.114186 | 8.15329 | −31.9870 | 25.0000 | −0.930987 | 179.478 | 7.30638 | −176.524 | −2.85464 | ||||||||||||||||||
1.19 | 0.0328525 | −8.60052 | −31.9989 | 25.0000 | −0.282548 | −22.0929 | −2.10252 | −169.031 | 0.821311 | ||||||||||||||||||
1.20 | 1.07599 | 26.0994 | −30.8422 | 25.0000 | 28.0828 | −166.075 | −67.6179 | 438.177 | 26.8998 | ||||||||||||||||||
See all 36 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.b | ✓ | 36 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1045.6.a.b | ✓ | 36 | 1.a | even | 1 | 1 | trivial |