Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [29,16,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, 16, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("29.1");
S:= CuspForms(chi, 16);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 29 \) |
| Weight: | \( k \) | \(=\) | \( 16 \) |
| 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: | \(41.3811164790\) |
| Analytic rank: | \(0\) |
| Dimension: | \(19\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{19} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{19} - 505005 x^{17} - 8736364 x^{16} + 105356631548 x^{15} + 3420215362096 x^{14} + \cdots - 44\!\cdots\!16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | multiple of \( 2^{43}\cdot 3^{6}\cdot 5^{5}\cdot 7 \) |
| 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 \(1,\beta_1,\ldots,\beta_{18}\) 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
\( x^{19} - 505005 x^{17} - 8736364 x^{16} + 105356631548 x^{15} + 3420215362096 x^{14} + \cdots - 44\!\cdots\!16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 59\!\cdots\!61 \nu^{18} + \cdots - 30\!\cdots\!76 ) / 65\!\cdots\!00 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 59\!\cdots\!61 \nu^{18} + \cdots + 13\!\cdots\!76 ) / 32\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 94\!\cdots\!59 \nu^{18} + \cdots - 59\!\cdots\!84 ) / 43\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 35\!\cdots\!07 \nu^{18} + \cdots + 15\!\cdots\!08 ) / 46\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 14\!\cdots\!05 \nu^{18} + \cdots - 18\!\cdots\!12 ) / 13\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 74\!\cdots\!53 \nu^{18} + \cdots + 27\!\cdots\!44 ) / 65\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 27\!\cdots\!13 \nu^{18} + \cdots + 95\!\cdots\!60 ) / 13\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 29\!\cdots\!73 \nu^{18} + \cdots + 14\!\cdots\!68 ) / 13\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 29\!\cdots\!41 \nu^{18} + \cdots + 12\!\cdots\!84 ) / 11\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 24\!\cdots\!75 \nu^{18} + \cdots - 11\!\cdots\!36 ) / 65\!\cdots\!00 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 19\!\cdots\!75 \nu^{18} + \cdots + 11\!\cdots\!48 ) / 32\!\cdots\!00 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 14\!\cdots\!23 \nu^{18} + \cdots + 79\!\cdots\!32 ) / 18\!\cdots\!00 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 10\!\cdots\!83 \nu^{18} + \cdots - 47\!\cdots\!16 ) / 13\!\cdots\!00 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 59\!\cdots\!57 \nu^{18} + \cdots - 28\!\cdots\!04 ) / 65\!\cdots\!00 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 30\!\cdots\!57 \nu^{18} + \cdots + 15\!\cdots\!36 ) / 32\!\cdots\!00 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 67\!\cdots\!63 \nu^{18} + \cdots - 27\!\cdots\!00 ) / 65\!\cdots\!00 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 76\!\cdots\!55 \nu^{18} + \cdots - 38\!\cdots\!28 ) / 65\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 27\beta _1 + 53157 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{10} + \beta_{8} + 3\beta_{5} - 5\beta_{4} + 45\beta_{3} - 218\beta_{2} + 87665\beta _1 + 1379506 \)
|
| \(\nu^{4}\) | \(=\) |
\( 7 \beta_{18} + 31 \beta_{17} + 47 \beta_{16} - 54 \beta_{15} + 26 \beta_{14} - 15 \beta_{13} + \cdots + 4664726863 \)
|
| \(\nu^{5}\) | \(=\) |
\( 16072 \beta_{18} - 2708 \beta_{17} + 11700 \beta_{16} + 8304 \beta_{15} - 5388 \beta_{14} + \cdots + 260984001266 \)
|
| \(\nu^{6}\) | \(=\) |
\( 3915367 \beta_{18} + 4882891 \beta_{17} + 10810295 \beta_{16} - 10942042 \beta_{15} + \cdots + 470860361097227 \)
|
| \(\nu^{7}\) | \(=\) |
\( 3597611948 \beta_{18} - 628009328 \beta_{17} + 2507855464 \beta_{16} + 1845855848 \beta_{15} + \cdots + 39\!\cdots\!06 \)
|
| \(\nu^{8}\) | \(=\) |
\( 832835913943 \beta_{18} + 615782998715 \beta_{17} + 1770712901975 \beta_{16} - 1637457910058 \beta_{15} + \cdots + 50\!\cdots\!67 \)
|
| \(\nu^{9}\) | \(=\) |
\( 583180611734656 \beta_{18} - 94921137385132 \beta_{17} + 410975391117564 \beta_{16} + \cdots + 54\!\cdots\!82 \)
|
| \(\nu^{10}\) | \(=\) |
\( 13\!\cdots\!19 \beta_{18} + \cdots + 55\!\cdots\!91 \)
|
| \(\nu^{11}\) | \(=\) |
\( 83\!\cdots\!16 \beta_{18} + \cdots + 72\!\cdots\!22 \)
|
| \(\nu^{12}\) | \(=\) |
\( 20\!\cdots\!23 \beta_{18} + \cdots + 63\!\cdots\!71 \)
|
| \(\nu^{13}\) | \(=\) |
\( 11\!\cdots\!92 \beta_{18} + \cdots + 93\!\cdots\!54 \)
|
| \(\nu^{14}\) | \(=\) |
\( 28\!\cdots\!23 \beta_{18} + \cdots + 72\!\cdots\!31 \)
|
| \(\nu^{15}\) | \(=\) |
\( 14\!\cdots\!12 \beta_{18} + \cdots + 11\!\cdots\!62 \)
|
| \(\nu^{16}\) | \(=\) |
\( 38\!\cdots\!95 \beta_{18} + \cdots + 84\!\cdots\!59 \)
|
| \(\nu^{17}\) | \(=\) |
\( 18\!\cdots\!48 \beta_{18} + \cdots + 15\!\cdots\!26 \)
|
| \(\nu^{18}\) | \(=\) |
\( 51\!\cdots\!23 \beta_{18} + \cdots + 99\!\cdots\!91 \)
|
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 |
|
−336.813 | 2324.50 | 80675.1 | −106883. | −782922. | 1.25371e6 | −1.61357e7 | −8.94561e6 | 3.59996e7 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −315.699 | 6780.95 | 66897.8 | 237520. | −2.14074e6 | −3.45609e6 | −1.07747e7 | 3.16324e7 | −7.49848e7 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −288.415 | 2292.83 | 50415.0 | −130035. | −661285. | −2.37154e6 | −5.08966e6 | −9.09186e6 | 3.75041e7 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −245.307 | −5730.81 | 27407.6 | 9338.19 | 1.40581e6 | −2.46876e6 | 1.31495e6 | 1.84933e7 | −2.29072e6 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −242.095 | −6397.06 | 25841.9 | −60464.2 | 1.54870e6 | 3.86623e6 | 1.67677e6 | 2.65735e7 | 1.46381e7 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −189.523 | −724.070 | 3150.93 | 55248.6 | 137228. | 3.05347e6 | 5.61311e6 | −1.38246e7 | −1.04709e7 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −107.927 | 5774.54 | −21119.8 | 251167. | −623228. | 1.51116e6 | 5.81594e6 | 1.89964e7 | −2.71077e7 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −101.235 | 2276.05 | −22519.5 | 12349.2 | −230415. | −3.54158e6 | 5.59702e6 | −9.16851e6 | −1.25017e6 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −76.4115 | −1229.26 | −26929.3 | −150576. | 93929.3 | 1.80369e6 | 4.56156e6 | −1.28378e7 | 1.15057e7 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | −56.7174 | −5630.97 | −29551.1 | 330421. | 319374. | 1.54784e6 | 3.53458e6 | 1.73590e7 | −1.87406e7 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 56.0143 | 4540.18 | −29630.4 | −174709. | 254315. | 2.13001e6 | −3.49520e6 | 6.26430e6 | −9.78623e6 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 106.831 | 493.761 | −21355.2 | −265725. | 52748.8 | −3.15726e6 | −5.78202e6 | −1.41051e7 | −2.83876e7 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 123.487 | −2327.40 | −17518.9 | 218629. | −287404. | −1.93890e6 | −6.20979e6 | −8.93214e6 | 2.69980e7 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 160.461 | −4545.76 | −7020.13 | −154867. | −729419. | −1.74373e6 | −6.38446e6 | 6.31503e6 | −2.48502e7 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 207.928 | 6078.50 | 10466.0 | 181747. | 1.26389e6 | 3.03379e6 | −4.63720e6 | 2.25992e7 | 3.77902e7 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 281.041 | 4013.45 | 46215.9 | 150501. | 1.12794e6 | −1.57306e6 | 3.77941e6 | 1.75891e6 | 4.22968e7 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | 323.719 | −4313.95 | 72026.2 | −272706. | −1.39651e6 | 1.00404e6 | 1.27087e7 | 4.26129e6 | −8.82802e7 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.18 | 345.887 | −415.807 | 86869.8 | 271690. | −143822. | 3.51139e6 | 1.87131e7 | −1.41760e7 | 9.39741e7 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.19 | 354.773 | 6648.33 | 93096.1 | −172154. | 2.35865e6 | 417614. | 2.14028e7 | 2.98514e7 | −6.10757e7 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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 | 29.16.a.b | ✓ | 19 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 29.16.a.b | ✓ | 19 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{19} - 505005 T_{2}^{17} - 8736364 T_{2}^{16} + 105356631548 T_{2}^{15} + 3420215362096 T_{2}^{14} + \cdots - 44\!\cdots\!16 \)
acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(29))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{19} + \cdots - 44\!\cdots\!16 \)
|
| $3$ |
\( T^{19} + \cdots + 60\!\cdots\!00 \)
|
| $5$ |
\( T^{19} + \cdots + 87\!\cdots\!00 \)
|
| $7$ |
\( T^{19} + \cdots - 71\!\cdots\!64 \)
|
| $11$ |
\( T^{19} + \cdots - 12\!\cdots\!48 \)
|
| $13$ |
\( T^{19} + \cdots - 94\!\cdots\!72 \)
|
| $17$ |
\( T^{19} + \cdots + 91\!\cdots\!32 \)
|
| $19$ |
\( T^{19} + \cdots - 28\!\cdots\!80 \)
|
| $23$ |
\( T^{19} + \cdots + 14\!\cdots\!24 \)
|
| $29$ |
\( (T + 17249876309)^{19} \)
|
| $31$ |
\( T^{19} + \cdots + 30\!\cdots\!32 \)
|
| $37$ |
\( T^{19} + \cdots - 18\!\cdots\!60 \)
|
| $41$ |
\( T^{19} + \cdots + 31\!\cdots\!00 \)
|
| $43$ |
\( T^{19} + \cdots - 11\!\cdots\!68 \)
|
| $47$ |
\( T^{19} + \cdots - 23\!\cdots\!48 \)
|
| $53$ |
\( T^{19} + \cdots + 21\!\cdots\!44 \)
|
| $59$ |
\( T^{19} + \cdots - 37\!\cdots\!00 \)
|
| $61$ |
\( T^{19} + \cdots - 25\!\cdots\!36 \)
|
| $67$ |
\( T^{19} + \cdots + 13\!\cdots\!96 \)
|
| $71$ |
\( T^{19} + \cdots - 73\!\cdots\!36 \)
|
| $73$ |
\( T^{19} + \cdots + 22\!\cdots\!00 \)
|
| $79$ |
\( T^{19} + \cdots + 27\!\cdots\!00 \)
|
| $83$ |
\( T^{19} + \cdots + 49\!\cdots\!68 \)
|
| $89$ |
\( T^{19} + \cdots - 25\!\cdots\!60 \)
|
| $97$ |
\( T^{19} + \cdots + 80\!\cdots\!96 \)
|
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