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: | \(35\) |
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 | −11.2768 | 20.0827 | 95.1666 | −25.0000 | −226.469 | −33.0605 | −712.318 | 160.315 | 281.920 | ||||||||||||||||||
1.2 | −10.0372 | −21.3301 | 68.7460 | −25.0000 | 214.096 | 45.5863 | −368.828 | 211.975 | 250.931 | ||||||||||||||||||
1.3 | −9.69226 | 4.57537 | 61.9400 | −25.0000 | −44.3457 | 180.537 | −290.186 | −222.066 | 242.307 | ||||||||||||||||||
1.4 | −9.31795 | 4.33678 | 54.8242 | −25.0000 | −40.4099 | −224.649 | −212.674 | −224.192 | 232.949 | ||||||||||||||||||
1.5 | −9.20757 | −16.1697 | 52.7793 | −25.0000 | 148.883 | −66.9325 | −191.327 | 18.4588 | 230.189 | ||||||||||||||||||
1.6 | −8.58946 | 14.9980 | 41.7788 | −25.0000 | −128.825 | −101.857 | −83.9948 | −18.0599 | 214.737 | ||||||||||||||||||
1.7 | −7.58272 | 22.1703 | 25.4976 | −25.0000 | −168.111 | 53.6476 | 49.3055 | 248.522 | 189.568 | ||||||||||||||||||
1.8 | −7.13459 | −23.4497 | 18.9024 | −25.0000 | 167.304 | −10.1832 | 93.4458 | 306.890 | 178.365 | ||||||||||||||||||
1.9 | −6.89804 | 25.8209 | 15.5830 | −25.0000 | −178.113 | 221.117 | 113.245 | 423.717 | 172.451 | ||||||||||||||||||
1.10 | −6.12909 | −3.52111 | 5.56569 | −25.0000 | 21.5812 | 66.3590 | 162.018 | −230.602 | 153.227 | ||||||||||||||||||
1.11 | −5.42281 | −30.3382 | −2.59316 | −25.0000 | 164.518 | 20.9364 | 187.592 | 677.408 | 135.570 | ||||||||||||||||||
1.12 | −5.16335 | 0.950399 | −5.33984 | −25.0000 | −4.90724 | −84.8257 | 192.799 | −242.097 | 129.084 | ||||||||||||||||||
1.13 | −3.68873 | −20.7447 | −18.3933 | −25.0000 | 76.5215 | 188.389 | 185.887 | 187.342 | 92.2182 | ||||||||||||||||||
1.14 | −3.41781 | −5.73624 | −20.3185 | −25.0000 | 19.6054 | 236.087 | 178.815 | −210.096 | 85.4453 | ||||||||||||||||||
1.15 | −2.54765 | 26.9200 | −25.5095 | −25.0000 | −68.5826 | 6.38524 | 146.514 | 481.684 | 63.6912 | ||||||||||||||||||
1.16 | −1.44052 | 10.2879 | −29.9249 | −25.0000 | −14.8199 | 14.1538 | 89.2038 | −137.159 | 36.0129 | ||||||||||||||||||
1.17 | −0.635479 | −27.7922 | −31.5962 | −25.0000 | 17.6614 | 19.6289 | 40.4140 | 529.409 | 15.8870 | ||||||||||||||||||
1.18 | −0.259047 | 8.53779 | −31.9329 | −25.0000 | −2.21168 | −211.963 | 16.5616 | −170.106 | 6.47616 | ||||||||||||||||||
1.19 | 1.05998 | −6.05236 | −30.8764 | −25.0000 | −6.41538 | −213.822 | −66.6477 | −206.369 | −26.4995 | ||||||||||||||||||
1.20 | 1.69647 | 12.5434 | −29.1220 | −25.0000 | 21.2794 | 107.740 | −103.691 | −85.6632 | −42.4117 | ||||||||||||||||||
See all 35 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.a | ✓ | 35 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1045.6.a.a | ✓ | 35 | 1.a | even | 1 | 1 | trivial |