Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [471,8,Mod(1,471)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(471, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("471.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 471 = 3 \cdot 157 \) |
Weight: | \( k \) | \(=\) | \( 8 \) |
Character orbit: | \([\chi]\) | \(=\) | 471.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: | \(147.133347003\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
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 | −22.3612 | −27.0000 | 372.021 | −432.818 | 603.751 | −1328.71 | −5456.60 | 729.000 | 9678.32 | ||||||||||||||||||
1.2 | −22.2958 | −27.0000 | 369.104 | 234.355 | 601.987 | 1270.63 | −5375.60 | 729.000 | −5225.14 | ||||||||||||||||||
1.3 | −20.6187 | −27.0000 | 297.131 | 165.459 | 556.705 | −1385.76 | −3487.25 | 729.000 | −3411.55 | ||||||||||||||||||
1.4 | −19.9953 | −27.0000 | 271.812 | −231.869 | 539.873 | 342.382 | −2875.57 | 729.000 | 4636.29 | ||||||||||||||||||
1.5 | −18.2609 | −27.0000 | 205.462 | 63.1999 | 493.045 | 499.385 | −1414.52 | 729.000 | −1154.09 | ||||||||||||||||||
1.6 | −17.8012 | −27.0000 | 188.881 | −168.224 | 480.631 | −317.259 | −1083.76 | 729.000 | 2994.59 | ||||||||||||||||||
1.7 | −16.8482 | −27.0000 | 155.860 | 189.080 | 454.900 | 904.411 | −469.396 | 729.000 | −3185.64 | ||||||||||||||||||
1.8 | −16.2410 | −27.0000 | 135.769 | −491.991 | 438.506 | 940.580 | −126.175 | 729.000 | 7990.41 | ||||||||||||||||||
1.9 | −16.0952 | −27.0000 | 131.055 | 414.780 | 434.570 | 1067.44 | −49.1712 | 729.000 | −6675.96 | ||||||||||||||||||
1.10 | −15.7094 | −27.0000 | 118.786 | 126.488 | 424.154 | 1592.00 | 144.749 | 729.000 | −1987.05 | ||||||||||||||||||
1.11 | −14.4638 | −27.0000 | 81.2016 | 346.642 | 390.523 | −1595.50 | 676.883 | 729.000 | −5013.77 | ||||||||||||||||||
1.12 | −13.7948 | −27.0000 | 62.2971 | −136.436 | 372.460 | −1182.74 | 906.360 | 729.000 | 1882.11 | ||||||||||||||||||
1.13 | −12.5583 | −27.0000 | 29.7115 | 553.645 | 339.075 | −406.725 | 1234.34 | 729.000 | −6952.86 | ||||||||||||||||||
1.14 | −10.6994 | −27.0000 | −13.5238 | −456.397 | 288.883 | −1238.76 | 1514.21 | 729.000 | 4883.15 | ||||||||||||||||||
1.15 | −9.95430 | −27.0000 | −28.9118 | −469.613 | 268.766 | −412.443 | 1561.95 | 729.000 | 4674.67 | ||||||||||||||||||
1.16 | −9.40303 | −27.0000 | −39.5830 | 163.955 | 253.882 | 68.7987 | 1575.79 | 729.000 | −1541.67 | ||||||||||||||||||
1.17 | −8.95758 | −27.0000 | −47.7618 | 476.788 | 241.855 | 984.094 | 1574.40 | 729.000 | −4270.87 | ||||||||||||||||||
1.18 | −8.93486 | −27.0000 | −48.1683 | 7.15974 | 241.241 | −664.570 | 1574.04 | 729.000 | −63.9712 | ||||||||||||||||||
1.19 | −6.65005 | −27.0000 | −83.7769 | 355.102 | 179.551 | −1545.65 | 1408.33 | 729.000 | −2361.45 | ||||||||||||||||||
1.20 | −3.69501 | −27.0000 | −114.347 | −349.671 | 99.7651 | 479.777 | 895.473 | 729.000 | 1292.04 | ||||||||||||||||||
See all 48 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(1\) |
\(157\) | \(1\) |
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 | 471.8.a.c | ✓ | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
471.8.a.c | ✓ | 48 | 1.a | even | 1 | 1 | trivial |