Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [29,18,Mod(1,29)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(29, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 18, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("29.1");
S:= CuspForms(chi, 18);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 29 \) |
Weight: | \( k \) | \(=\) | \( 18 \) |
Character orbit: | \([\chi]\) | \(=\) | 29.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: | \(53.1344053299\) |
Analytic rank: | \(0\) |
Dimension: | \(21\) |
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 | −716.767 | 6161.78 | 382683. | −44516.1 | −4.41656e6 | −2.28029e7 | −1.80347e8 | −9.11726e7 | 3.19077e7 | ||||||||||||||||||
1.2 | −687.348 | −21372.9 | 341375. | −1.16997e6 | 1.46906e7 | −9.37855e6 | −1.44551e8 | 3.27660e8 | 8.04176e8 | ||||||||||||||||||
1.3 | −645.655 | 14883.2 | 285798. | 954733. | −9.60943e6 | 1.80737e7 | −9.98999e7 | 9.23704e7 | −6.16428e8 | ||||||||||||||||||
1.4 | −553.336 | −346.147 | 175108. | 641880. | 191536. | −1.77073e7 | −2.43669e7 | −1.29020e8 | −3.55175e8 | ||||||||||||||||||
1.5 | −361.395 | 9478.77 | −465.313 | 215546. | −3.42559e6 | −1.26142e7 | 4.75370e7 | −3.92930e7 | −7.78973e7 | ||||||||||||||||||
1.6 | −332.840 | −18225.8 | −20289.6 | −318485. | 6.06628e6 | 9.82044e6 | 5.03792e7 | 2.03040e8 | 1.06005e8 | ||||||||||||||||||
1.7 | −275.201 | 7065.32 | −55336.6 | −620891. | −1.94438e6 | 2.72618e7 | 5.12998e7 | −7.92215e7 | 1.70870e8 | ||||||||||||||||||
1.8 | −237.844 | −14184.6 | −74502.2 | 1.22166e6 | 3.37373e6 | 1.05564e7 | 4.88946e7 | 7.20633e7 | −2.90566e8 | ||||||||||||||||||
1.9 | −65.9873 | 22515.6 | −126718. | 342266. | −1.48575e6 | 1.63719e7 | 1.70108e7 | 3.77814e8 | −2.25852e7 | ||||||||||||||||||
1.10 | −39.6552 | 13037.0 | −129499. | −1.72282e6 | −516984. | −1.48112e7 | 1.03330e7 | 4.08230e7 | 6.83187e7 | ||||||||||||||||||
1.11 | −3.42944 | −6754.54 | −131060. | 511876. | 23164.3 | −1.86320e7 | 898966. | −8.35164e7 | −1.75545e6 | ||||||||||||||||||
1.12 | 59.1674 | 15524.2 | −127571. | 1.59085e6 | 918527. | −2.66210e7 | −1.53032e7 | 1.11861e8 | 9.41262e7 | ||||||||||||||||||
1.13 | 217.779 | −424.636 | −83644.2 | −874312. | −92476.9 | 3.28654e6 | −4.67607e7 | −1.28960e8 | −1.90407e8 | ||||||||||||||||||
1.14 | 290.453 | −16308.4 | −46709.2 | 805745. | −4.73681e6 | 2.28724e7 | −5.16370e7 | 1.36823e8 | 2.34031e8 | ||||||||||||||||||
1.15 | 327.881 | 6727.77 | −23565.9 | 1.45971e6 | 2.20591e6 | 2.18126e7 | −5.07029e7 | −8.38773e7 | 4.78610e8 | ||||||||||||||||||
1.16 | 361.094 | −2259.76 | −683.052 | −433238. | −815984. | −1.41185e7 | −4.75760e7 | −1.24034e8 | −1.56440e8 | ||||||||||||||||||
1.17 | 460.739 | 20203.5 | 81208.7 | −1.66304e6 | 9.30854e6 | 1.23855e7 | −2.29740e7 | 2.79040e8 | −7.66230e8 | ||||||||||||||||||
1.18 | 500.684 | −16927.2 | 119612. | −1.40156e6 | −8.47517e6 | −1.54199e7 | −5.73773e6 | 1.57390e8 | −7.01740e8 | ||||||||||||||||||
1.19 | 616.905 | 17761.1 | 249500. | 768235. | 1.09569e7 | 1.64486e6 | 7.30585e7 | 1.86316e8 | 4.73928e8 | ||||||||||||||||||
1.20 | 652.859 | 4140.75 | 295153. | 358296. | 2.70332e6 | 1.05214e7 | 1.07122e8 | −1.11994e8 | 2.33917e8 | ||||||||||||||||||
See all 21 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(29\) | \(-1\) |
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 | 29.18.a.b | ✓ | 21 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
29.18.a.b | ✓ | 21 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{21} - 256 T_{2}^{20} - 2069749 T_{2}^{19} + 559124270 T_{2}^{18} + 1763203907568 T_{2}^{17} + \cdots + 35\!\cdots\!00 \) acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(29))\).