Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [547,8,Mod(1,547)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(547, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("547.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 547 \) |
Weight: | \( k \) | \(=\) | \( 8 \) |
Character orbit: | \([\chi]\) | \(=\) | 547.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: | \(170.874608940\) |
Analytic rank: | \(1\) |
Dimension: | \(156\) |
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 | −22.1152 | 83.7935 | 361.080 | −144.371 | −1853.11 | 55.2475 | −5154.61 | 4834.35 | 3192.79 | ||||||||||||||||||
1.2 | −22.0518 | −47.8990 | 358.283 | −388.461 | 1056.26 | −154.038 | −5078.15 | 107.313 | 8566.26 | ||||||||||||||||||
1.3 | −21.6719 | −67.1835 | 341.672 | 441.406 | 1455.99 | −502.268 | −4630.68 | 2326.62 | −9566.11 | ||||||||||||||||||
1.4 | −21.6321 | −15.5597 | 339.946 | 99.8206 | 336.588 | 1402.54 | −4584.82 | −1944.90 | −2159.32 | ||||||||||||||||||
1.5 | −21.5657 | −93.3798 | 337.078 | −365.072 | 2013.80 | −27.9939 | −4508.91 | 6532.79 | 7873.02 | ||||||||||||||||||
1.6 | −21.1866 | −59.2391 | 320.872 | 86.1150 | 1255.07 | −167.825 | −4086.30 | 1322.27 | −1824.48 | ||||||||||||||||||
1.7 | −21.1072 | 70.0639 | 317.514 | 140.672 | −1478.85 | −431.091 | −4000.10 | 2721.94 | −2969.18 | ||||||||||||||||||
1.8 | −20.7147 | 73.8020 | 301.098 | 73.8971 | −1528.79 | 1490.12 | −3585.68 | 3259.74 | −1530.76 | ||||||||||||||||||
1.9 | −20.7120 | 65.0038 | 300.986 | −306.003 | −1346.36 | −1693.80 | −3582.88 | 2038.50 | 6337.92 | ||||||||||||||||||
1.10 | −20.6976 | 32.5628 | 300.390 | −462.112 | −673.972 | −452.526 | −3568.07 | −1126.66 | 9564.62 | ||||||||||||||||||
1.11 | −20.4450 | −5.52261 | 289.998 | 429.047 | 112.910 | 1075.95 | −3312.04 | −2156.50 | −8771.86 | ||||||||||||||||||
1.12 | −20.1873 | 34.9985 | 279.528 | −313.532 | −706.526 | 1440.96 | −3058.93 | −962.103 | 6329.36 | ||||||||||||||||||
1.13 | −19.9440 | 23.7563 | 269.761 | −441.947 | −473.794 | 8.48766 | −2827.28 | −1622.64 | 8814.16 | ||||||||||||||||||
1.14 | −19.8798 | −54.3245 | 267.205 | −433.266 | 1079.96 | 1338.12 | −2767.35 | 764.147 | 8613.23 | ||||||||||||||||||
1.15 | −19.7400 | −43.2832 | 261.668 | −352.760 | 854.411 | −1491.16 | −2638.61 | −313.564 | 6963.48 | ||||||||||||||||||
1.16 | −19.6554 | −60.4425 | 258.337 | 408.742 | 1188.02 | 556.317 | −2561.83 | 1466.29 | −8034.01 | ||||||||||||||||||
1.17 | −19.3499 | 9.03641 | 246.418 | 20.2179 | −174.853 | 193.981 | −2291.37 | −2105.34 | −391.214 | ||||||||||||||||||
1.18 | −18.7781 | −27.5200 | 224.616 | 480.924 | 516.772 | −221.639 | −1814.26 | −1429.65 | −9030.83 | ||||||||||||||||||
1.19 | −18.3687 | −35.4413 | 209.409 | 105.765 | 651.010 | 751.780 | −1495.38 | −930.915 | −1942.76 | ||||||||||||||||||
1.20 | −18.2383 | 47.2008 | 204.635 | 260.548 | −860.861 | −206.356 | −1397.68 | 40.9151 | −4751.95 | ||||||||||||||||||
See next 80 embeddings (of 156 total) |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(547\) | \(-1\) |
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 | 547.8.a.a | ✓ | 156 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
547.8.a.a | ✓ | 156 | 1.a | even | 1 | 1 | trivial |