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: | \(1\) |
| Dimension: | \(18\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{18} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{18} - 1610997 x^{16} - 28978880 x^{15} + 1054878119348 x^{14} + 33471007935200 x^{13} + \cdots - 13\!\cdots\!24 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{42}\cdot 3^{14}\cdot 17 \) |
| Twist minimal: | yes |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{18} - 1610997 x^{16} - 28978880 x^{15} + 1054878119348 x^{14} + 33471007935200 x^{13} + \cdots - 13\!\cdots\!24 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 16\!\cdots\!43 \nu^{17} + \cdots - 84\!\cdots\!20 ) / 19\!\cdots\!80 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 16\!\cdots\!43 \nu^{17} + \cdots - 16\!\cdots\!40 ) / 98\!\cdots\!40 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 15\!\cdots\!77 \nu^{17} + \cdots - 29\!\cdots\!20 ) / 65\!\cdots\!60 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 77\!\cdots\!79 \nu^{17} + \cdots + 16\!\cdots\!48 ) / 19\!\cdots\!80 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 13\!\cdots\!55 \nu^{17} + \cdots + 22\!\cdots\!96 ) / 10\!\cdots\!60 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 63\!\cdots\!75 \nu^{17} + \cdots + 15\!\cdots\!28 ) / 19\!\cdots\!80 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 31\!\cdots\!08 \nu^{17} + \cdots + 13\!\cdots\!80 ) / 82\!\cdots\!20 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 27\!\cdots\!67 \nu^{17} + \cdots + 16\!\cdots\!04 ) / 19\!\cdots\!80 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 33\!\cdots\!95 \nu^{17} + \cdots + 52\!\cdots\!56 ) / 19\!\cdots\!80 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 53\!\cdots\!81 \nu^{17} + \cdots + 33\!\cdots\!92 ) / 24\!\cdots\!60 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 56\!\cdots\!91 \nu^{17} + \cdots - 10\!\cdots\!36 ) / 19\!\cdots\!80 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 28\!\cdots\!01 \nu^{17} + \cdots - 21\!\cdots\!20 ) / 98\!\cdots\!40 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 60\!\cdots\!87 \nu^{17} + \cdots + 10\!\cdots\!84 ) / 19\!\cdots\!80 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 93\!\cdots\!13 \nu^{17} + \cdots + 14\!\cdots\!88 ) / 28\!\cdots\!40 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 30\!\cdots\!69 \nu^{17} + \cdots - 73\!\cdots\!68 ) / 89\!\cdots\!40 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 18\!\cdots\!97 \nu^{17} + \cdots + 66\!\cdots\!28 ) / 39\!\cdots\!36 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 27\beta _1 + 178999 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{8} - \beta_{7} + \beta_{5} + 3\beta_{4} + 57\beta_{3} + 599\beta_{2} + 301435\beta _1 + 4829561 \)
|
| \(\nu^{4}\) | \(=\) |
\( 22 \beta_{17} - 15 \beta_{16} + 27 \beta_{15} - 22 \beta_{14} - 227 \beta_{13} + 5 \beta_{12} + \cdots + 53950339128 \)
|
| \(\nu^{5}\) | \(=\) |
\( 6502 \beta_{17} + 26222 \beta_{16} + 14142 \beta_{15} + 13460 \beta_{14} - 55452 \beta_{13} + \cdots + 3670371007985 \)
|
| \(\nu^{6}\) | \(=\) |
\( 2741518 \beta_{17} - 16111451 \beta_{16} + 28296471 \beta_{15} - 8988742 \beta_{14} + \cdots + 18\!\cdots\!96 \)
|
| \(\nu^{7}\) | \(=\) |
\( 3257270462 \beta_{17} + 16924258150 \beta_{16} + 14194420086 \beta_{15} + 7254310276 \beta_{14} + \cdots + 21\!\cdots\!13 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 2096037017234 \beta_{17} - 9462019876619 \beta_{16} + 17586320843399 \beta_{15} + \cdots + 69\!\cdots\!36 \)
|
| \(\nu^{9}\) | \(=\) |
\( 746776349354846 \beta_{17} + \cdots + 11\!\cdots\!25 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 22\!\cdots\!30 \beta_{17} + \cdots + 27\!\cdots\!84 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 20\!\cdots\!78 \beta_{17} + \cdots + 63\!\cdots\!53 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 14\!\cdots\!98 \beta_{17} + \cdots + 11\!\cdots\!84 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 38\!\cdots\!06 \beta_{17} + \cdots + 33\!\cdots\!65 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 84\!\cdots\!06 \beta_{17} + \cdots + 48\!\cdots\!08 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 30\!\cdots\!66 \beta_{17} + \cdots + 17\!\cdots\!81 \)
|
| \(\nu^{16}\) | \(=\) |
\( - 45\!\cdots\!02 \beta_{17} + \cdots + 21\!\cdots\!44 \)
|
| \(\nu^{17}\) | \(=\) |
\( - 19\!\cdots\!38 \beta_{17} + \cdots + 87\!\cdots\!57 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−615.069 | −9589.87 | 247238. | 540648. | 5.89843e6 | 1.54300e7 | −7.14501e7 | −3.71746e7 | −3.32536e8 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −608.916 | 12508.7 | 239707. | −1.45732e6 | −7.61677e6 | 1.09811e7 | −6.61493e7 | 2.73284e7 | 8.87386e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −509.707 | −18148.6 | 128729. | 1.42408e6 | 9.25045e6 | −2.65975e7 | 1.19411e6 | 2.00230e8 | −7.25865e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −483.296 | 19868.0 | 102503. | −300011. | −9.60214e6 | −1.05230e7 | 1.38073e7 | 2.65598e8 | 1.44994e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −415.129 | 11060.0 | 41259.9 | 1.55169e6 | −4.59133e6 | 5.04285e6 | 3.72836e7 | −6.81605e6 | −6.44152e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −315.569 | −7907.47 | −31488.2 | −995824. | 2.49535e6 | −2.13865e7 | 5.12990e7 | −6.66120e7 | 3.14251e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −235.315 | 964.844 | −75699.1 | 864499. | −227042. | 6.42601e6 | 4.86562e7 | −1.28209e8 | −2.03429e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −87.1238 | 11891.8 | −123481. | −509857. | −1.03606e6 | −8.46695e6 | 2.21777e7 | 1.22759e7 | 4.44206e7 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −2.09672 | −3304.37 | −131068. | 820444. | 6928.31 | 6.94935e6 | 549632. | −1.18221e8 | −1.72024e6 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 19.4887 | −18855.2 | −130692. | −785915. | −367464. | −8.64265e6 | −5.10144e6 | 2.26380e8 | −1.53164e7 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 54.5616 | −10196.6 | −128095. | −1.33117e6 | −556341. | 2.01292e7 | −1.41406e7 | −2.51701e7 | −7.26309e7 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 229.733 | 12724.1 | −78294.6 | 93032.9 | 2.92314e6 | 1.03405e7 | −4.80985e7 | 3.27614e7 | 2.13728e7 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 235.511 | −19946.7 | −75606.8 | 778691. | −4.69767e6 | −1.72672e7 | −4.86750e7 | 2.68732e8 | 1.83390e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 421.353 | 14930.9 | 46466.2 | −57183.5 | 6.29116e6 | −1.89653e7 | −3.56489e7 | 9.37907e7 | −2.40944e7 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 516.149 | −14037.1 | 135338. | −336022. | −7.24525e6 | 1.38849e7 | 2.20183e6 | 6.79008e7 | −1.73438e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 525.865 | −5290.76 | 145463. | 1.28880e6 | −2.78223e6 | −1.26851e7 | 7.56747e6 | −1.01148e8 | 6.77737e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | 577.016 | 2597.06 | 201875. | −942114. | 1.49854e6 | 2.00475e7 | 4.08544e7 | −1.22395e8 | −5.43614e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.18 | 692.544 | 5331.24 | 348545. | −1.21070e6 | 3.69212e6 | −2.86223e7 | 1.50609e8 | −1.00718e8 | −8.38462e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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.a | ✓ | 18 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 29.18.a.a | ✓ | 18 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{18} - 1610997 T_{2}^{16} - 28978880 T_{2}^{15} + 1054878119348 T_{2}^{14} + 33471007935200 T_{2}^{13} + \cdots - 13\!\cdots\!24 \)
acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(29))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{18} + \cdots - 13\!\cdots\!24 \)
|
| $3$ |
\( T^{18} + \cdots - 10\!\cdots\!00 \)
|
| $5$ |
\( T^{18} + \cdots + 40\!\cdots\!00 \)
|
| $7$ |
\( T^{18} + \cdots - 11\!\cdots\!24 \)
|
| $11$ |
\( T^{18} + \cdots - 41\!\cdots\!16 \)
|
| $13$ |
\( T^{18} + \cdots + 51\!\cdots\!84 \)
|
| $17$ |
\( T^{18} + \cdots - 15\!\cdots\!16 \)
|
| $19$ |
\( T^{18} + \cdots - 17\!\cdots\!00 \)
|
| $23$ |
\( T^{18} + \cdots + 49\!\cdots\!76 \)
|
| $29$ |
\( (T + 500246412961)^{18} \)
|
| $31$ |
\( T^{18} + \cdots + 75\!\cdots\!64 \)
|
| $37$ |
\( T^{18} + \cdots - 10\!\cdots\!00 \)
|
| $41$ |
\( T^{18} + \cdots + 91\!\cdots\!00 \)
|
| $43$ |
\( T^{18} + \cdots - 37\!\cdots\!36 \)
|
| $47$ |
\( T^{18} + \cdots + 13\!\cdots\!84 \)
|
| $53$ |
\( T^{18} + \cdots + 16\!\cdots\!76 \)
|
| $59$ |
\( T^{18} + \cdots - 19\!\cdots\!00 \)
|
| $61$ |
\( T^{18} + \cdots - 15\!\cdots\!64 \)
|
| $67$ |
\( T^{18} + \cdots + 11\!\cdots\!16 \)
|
| $71$ |
\( T^{18} + \cdots + 10\!\cdots\!56 \)
|
| $73$ |
\( T^{18} + \cdots - 75\!\cdots\!00 \)
|
| $79$ |
\( T^{18} + \cdots + 11\!\cdots\!00 \)
|
| $83$ |
\( T^{18} + \cdots + 83\!\cdots\!64 \)
|
| $89$ |
\( T^{18} + \cdots - 53\!\cdots\!00 \)
|
| $97$ |
\( T^{18} + \cdots - 58\!\cdots\!24 \)
|
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