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: | \(1\) |
| Dimension: | \(16\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 381613 x^{14} - 3733354 x^{13} + 57580338072 x^{12} + 1053633121552 x^{11} + \cdots + 56\!\cdots\!24 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | multiple of \( 2^{33}\cdot 3^{7}\cdot 5^{4}\cdot 7^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 381613 x^{14} - 3733354 x^{13} + 57580338072 x^{12} + 1053633121552 x^{11} + \cdots + 56\!\cdots\!24 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 16\!\cdots\!55 \nu^{15} + \cdots + 81\!\cdots\!64 ) / 17\!\cdots\!00 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 16\!\cdots\!55 \nu^{15} + \cdots - 34\!\cdots\!36 ) / 89\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 82\!\cdots\!05 \nu^{15} + \cdots + 71\!\cdots\!56 ) / 35\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 99\!\cdots\!77 \nu^{15} + \cdots + 95\!\cdots\!80 ) / 17\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 43\!\cdots\!53 \nu^{15} + \cdots + 23\!\cdots\!36 ) / 30\!\cdots\!20 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 10\!\cdots\!37 \nu^{15} + \cdots - 53\!\cdots\!44 ) / 61\!\cdots\!60 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 36\!\cdots\!37 \nu^{15} + \cdots + 75\!\cdots\!08 ) / 17\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 19\!\cdots\!13 \nu^{15} + \cdots - 30\!\cdots\!84 ) / 89\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 24\!\cdots\!95 \nu^{15} + \cdots - 62\!\cdots\!96 ) / 89\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 29\!\cdots\!99 \nu^{15} + \cdots + 71\!\cdots\!24 ) / 44\!\cdots\!00 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 15\!\cdots\!11 \nu^{15} + \cdots + 12\!\cdots\!56 ) / 17\!\cdots\!00 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 61\!\cdots\!07 \nu^{15} + \cdots - 82\!\cdots\!68 ) / 44\!\cdots\!00 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 20\!\cdots\!21 \nu^{15} + \cdots - 17\!\cdots\!80 ) / 11\!\cdots\!00 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 98\!\cdots\!07 \nu^{15} + \cdots - 35\!\cdots\!44 ) / 35\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 2\beta_{2} + 15\beta _1 + 47702 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{12} + \beta_{7} - 2\beta_{6} - 3\beta_{5} + 10\beta_{4} + 18\beta_{3} - 25\beta_{2} + 79843\beta _1 + 700008 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 26 \beta_{15} - 28 \beta_{14} - 184 \beta_{13} + 190 \beta_{12} + \beta_{11} + 11 \beta_{10} + \cdots + 3808528384 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 4104 \beta_{15} - 43846 \beta_{14} - 32270 \beta_{13} + 122775 \beta_{12} + 17840 \beta_{11} + \cdots + 115959973678 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 4579746 \beta_{15} - 11460904 \beta_{14} - 34431540 \beta_{13} + 30333060 \beta_{12} + \cdots + 347573444024052 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 1121977292 \beta_{15} - 9881493294 \beta_{14} - 6630482574 \beta_{13} + 12644235463 \beta_{12} + \cdots + 16\!\cdots\!42 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 659116655354 \beta_{15} - 2339885330292 \beta_{14} - 5022430018816 \beta_{13} + 3813265240298 \beta_{12} + \cdots + 33\!\cdots\!32 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 190056304510352 \beta_{15} + \cdots + 22\!\cdots\!26 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 87\!\cdots\!38 \beta_{15} + \cdots + 34\!\cdots\!04 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 26\!\cdots\!84 \beta_{15} + \cdots + 29\!\cdots\!54 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 11\!\cdots\!34 \beta_{15} + \cdots + 36\!\cdots\!36 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 35\!\cdots\!44 \beta_{15} + \cdots + 38\!\cdots\!54 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 13\!\cdots\!50 \beta_{15} + \cdots + 39\!\cdots\!72 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 44\!\cdots\!32 \beta_{15} + \cdots + 47\!\cdots\!42 \)
|
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 |
|
−353.626 | −4035.27 | 92283.1 | 246585. | 1.42698e6 | 507417. | −2.10461e7 | 1.93454e6 | −8.71986e7 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −310.732 | −3409.71 | 63786.1 | −228689. | 1.05951e6 | −499515. | −9.63832e6 | −2.72276e6 | 7.10609e7 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −289.475 | 4726.56 | 51028.0 | 42761.9 | −1.36822e6 | 3.84346e6 | −5.28581e6 | 7.99150e6 | −1.23785e7 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −234.508 | −187.442 | 22225.8 | 226297. | 43956.6 | −691945. | 2.47222e6 | −1.43138e7 | −5.30684e7 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −180.238 | 5816.94 | −282.236 | −201198. | −1.04843e6 | 10784.5 | 5.95691e6 | 1.94879e7 | 3.62635e7 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −142.411 | −1743.30 | −12487.2 | −336740. | 248265. | 212194. | 6.44482e6 | −1.13098e7 | 4.79554e7 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −70.3494 | −4845.56 | −27819.0 | 4719.52 | 340882. | −2.30433e6 | 4.26226e6 | 9.13052e6 | −332016. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −9.50673 | 3610.03 | −32677.6 | 115177. | −34319.6 | −703154. | 622174. | −1.31661e6 | −1.09496e6 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 43.1693 | −694.438 | −30904.4 | 52528.3 | −29978.4 | 2.43575e6 | −2.74869e6 | −1.38667e7 | 2.26761e6 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 53.0072 | −6754.31 | −29958.2 | −192618. | −358027. | 1.68404e6 | −3.32494e6 | 3.12717e7 | −1.02101e7 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 124.963 | 7403.83 | −17152.2 | −74071.9 | 925205. | −3.34724e6 | −6.23818e6 | 4.04678e7 | −9.25625e6 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 142.980 | 2566.25 | −12324.6 | 282994. | 366923. | −961224. | −6.44736e6 | −7.76328e6 | 4.04625e7 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 238.149 | 2503.07 | 23947.1 | −210621. | 596104. | 1.58826e6 | −2.10069e6 | −8.08355e6 | −5.01592e7 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 263.654 | −7375.70 | 36745.2 | 191009. | −1.94463e6 | −906747. | 1.04862e6 | 4.00521e7 | 5.03603e7 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 276.899 | 1914.97 | 43905.0 | −57638.4 | 530254. | −1.55025e6 | 3.08382e6 | −1.06818e7 | −1.59600e7 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 320.024 | −2709.91 | 69647.1 | 57494.2 | −867237. | −3.02383e6 | 1.18022e7 | −7.00527e6 | 1.83995e7 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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.a | ✓ | 16 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 29.16.a.a | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{16} + 128 T_{2}^{15} - 373933 T_{2}^{14} - 38720582 T_{2}^{13} + 55753547496 T_{2}^{12} + \cdots + 29\!\cdots\!40 \)
acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(29))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} + \cdots + 29\!\cdots\!40 \)
|
| $3$ |
\( T^{16} + \cdots - 18\!\cdots\!92 \)
|
| $5$ |
\( T^{16} + \cdots - 56\!\cdots\!00 \)
|
| $7$ |
\( T^{16} + \cdots - 22\!\cdots\!28 \)
|
| $11$ |
\( T^{16} + \cdots + 26\!\cdots\!88 \)
|
| $13$ |
\( T^{16} + \cdots + 44\!\cdots\!64 \)
|
| $17$ |
\( T^{16} + \cdots + 15\!\cdots\!48 \)
|
| $19$ |
\( T^{16} + \cdots + 19\!\cdots\!80 \)
|
| $23$ |
\( T^{16} + \cdots - 33\!\cdots\!16 \)
|
| $29$ |
\( (T - 17249876309)^{16} \)
|
| $31$ |
\( T^{16} + \cdots + 13\!\cdots\!96 \)
|
| $37$ |
\( T^{16} + \cdots - 32\!\cdots\!52 \)
|
| $41$ |
\( T^{16} + \cdots - 10\!\cdots\!12 \)
|
| $43$ |
\( T^{16} + \cdots + 27\!\cdots\!80 \)
|
| $47$ |
\( T^{16} + \cdots + 26\!\cdots\!88 \)
|
| $53$ |
\( T^{16} + \cdots + 44\!\cdots\!08 \)
|
| $59$ |
\( T^{16} + \cdots + 11\!\cdots\!00 \)
|
| $61$ |
\( T^{16} + \cdots - 14\!\cdots\!20 \)
|
| $67$ |
\( T^{16} + \cdots - 24\!\cdots\!40 \)
|
| $71$ |
\( T^{16} + \cdots - 21\!\cdots\!36 \)
|
| $73$ |
\( T^{16} + \cdots - 25\!\cdots\!36 \)
|
| $79$ |
\( T^{16} + \cdots - 44\!\cdots\!00 \)
|
| $83$ |
\( T^{16} + \cdots - 98\!\cdots\!84 \)
|
| $89$ |
\( T^{16} + \cdots + 41\!\cdots\!00 \)
|
| $97$ |
\( T^{16} + \cdots + 40\!\cdots\!04 \)
|
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