Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [261,10,Mod(1,261)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(261, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("261.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 261 = 3^{2} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 261.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: | \(134.424353239\) |
| Analytic rank: | \(1\) |
| Dimension: | \(9\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{9} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{9} - 2901 x^{7} - 4592 x^{6} + 2830996 x^{5} + 7409504 x^{4} - 1038861888 x^{3} - 2974719488 x^{2} + \cdots + 456378417152 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{9}\cdot 3^{2}\cdot 7 \) |
| Twist minimal: | no (minimal twist has level 29) |
| 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 \(1,\beta_1,\ldots,\beta_{8}\) 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
\( x^{9} - 2901 x^{7} - 4592 x^{6} + 2830996 x^{5} + 7409504 x^{4} - 1038861888 x^{3} - 2974719488 x^{2} + \cdots + 456378417152 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 645 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 24206118505 \nu^{8} + 3628782688552 \nu^{7} - 51380960778211 \nu^{6} + \cdots - 31\!\cdots\!12 ) / 62\!\cdots\!24 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 30645811893 \nu^{8} + 46114646956 \nu^{7} + 72238283274393 \nu^{6} + \cdots - 56\!\cdots\!84 ) / 31\!\cdots\!12 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 145943679107 \nu^{8} + 3948353615584 \nu^{7} + 511126104077119 \nu^{6} + \cdots - 23\!\cdots\!72 ) / 62\!\cdots\!24 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 206831158817 \nu^{8} + 10132200890616 \nu^{7} + 426846090538933 \nu^{6} + \cdots + 59\!\cdots\!96 ) / 62\!\cdots\!24 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 40920390157 \nu^{8} + 422294149298 \nu^{7} + 97500534227889 \nu^{6} + \cdots - 10\!\cdots\!40 ) / 77\!\cdots\!28 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 346626955361 \nu^{8} + 3949803218648 \nu^{7} - 990199083095925 \nu^{6} + \cdots + 15\!\cdots\!12 ) / 62\!\cdots\!24 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 645 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{8} + 5\beta_{7} - \beta_{6} - \beta_{5} - 27\beta_{4} + \beta_{3} + 5\beta_{2} + 937\beta _1 + 1538 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 20 \beta_{8} + 52 \beta_{7} + 54 \beta_{6} - 66 \beta_{5} - 378 \beta_{4} - 96 \beta_{3} + \cdots + 612497 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 2589 \beta_{8} + 6537 \beta_{7} - 1437 \beta_{6} - 1373 \beta_{5} - 42751 \beta_{4} + 3325 \beta_{3} + \cdots + 3296234 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 52276 \beta_{8} + 84564 \beta_{7} + 95262 \beta_{6} - 94106 \beta_{5} - 779282 \beta_{4} + \cdots + 650555745 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 4158157 \beta_{8} + 7523417 \beta_{7} - 1124589 \beta_{6} - 1517101 \beta_{5} - 58467439 \beta_{4} + \cdots + 5557992858 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 93877892 \beta_{8} + 106963044 \beta_{7} + 132621294 \beta_{6} - 107787978 \beta_{5} + \cdots + 724198092865 \)
|
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 |
|
−33.1179 | 0 | 584.794 | 957.580 | 0 | −5228.90 | −2410.80 | 0 | −31713.0 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −30.2923 | 0 | 405.624 | −2211.65 | 0 | −7440.75 | 3222.36 | 0 | 66995.9 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −24.3559 | 0 | 81.2084 | 2213.18 | 0 | −1575.25 | 10492.3 | 0 | −53903.9 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −12.4485 | 0 | −357.034 | −208.913 | 0 | 10778.0 | 10818.2 | 0 | 2600.65 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −4.12402 | 0 | −494.992 | −522.723 | 0 | −9369.41 | 4152.85 | 0 | 2155.72 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 15.4003 | 0 | −274.831 | −1546.00 | 0 | 3958.22 | −12117.4 | 0 | −23808.9 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 19.7716 | 0 | −121.083 | −339.780 | 0 | 3392.81 | −12517.1 | 0 | −6718.00 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 33.5115 | 0 | 611.021 | 1510.04 | 0 | 2235.57 | 3318.35 | 0 | 50603.9 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 35.6552 | 0 | 759.292 | 886.258 | 0 | −3878.27 | 8817.24 | 0 | 31599.7 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(29\) | \(-1\) |
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 | 261.10.a.b | 9 | |
| 3.b | odd | 2 | 1 | 29.10.a.a | ✓ | 9 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 29.10.a.a | ✓ | 9 | 3.b | odd | 2 | 1 | |
| 261.10.a.b | 9 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{9} - 2901 T_{2}^{7} - 4592 T_{2}^{6} + 2830996 T_{2}^{5} + 7409504 T_{2}^{4} + \cdots + 456378417152 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(261))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} + \cdots + 456378417152 \)
|
| $3$ |
\( T^{9} \)
|
| $5$ |
\( T^{9} + \cdots + 35\!\cdots\!50 \)
|
| $7$ |
\( T^{9} + \cdots + 72\!\cdots\!48 \)
|
| $11$ |
\( T^{9} + \cdots - 40\!\cdots\!78 \)
|
| $13$ |
\( T^{9} + \cdots - 14\!\cdots\!94 \)
|
| $17$ |
\( T^{9} + \cdots + 19\!\cdots\!24 \)
|
| $19$ |
\( T^{9} + \cdots + 11\!\cdots\!60 \)
|
| $23$ |
\( T^{9} + \cdots - 27\!\cdots\!32 \)
|
| $29$ |
\( (T - 707281)^{9} \)
|
| $31$ |
\( T^{9} + \cdots - 54\!\cdots\!58 \)
|
| $37$ |
\( T^{9} + \cdots + 15\!\cdots\!20 \)
|
| $41$ |
\( T^{9} + \cdots - 15\!\cdots\!00 \)
|
| $43$ |
\( T^{9} + \cdots + 15\!\cdots\!26 \)
|
| $47$ |
\( T^{9} + \cdots + 26\!\cdots\!66 \)
|
| $53$ |
\( T^{9} + \cdots + 22\!\cdots\!22 \)
|
| $59$ |
\( T^{9} + \cdots - 74\!\cdots\!00 \)
|
| $61$ |
\( T^{9} + \cdots - 35\!\cdots\!36 \)
|
| $67$ |
\( T^{9} + \cdots - 13\!\cdots\!92 \)
|
| $71$ |
\( T^{9} + \cdots - 12\!\cdots\!84 \)
|
| $73$ |
\( T^{9} + \cdots - 71\!\cdots\!00 \)
|
| $79$ |
\( T^{9} + \cdots - 17\!\cdots\!50 \)
|
| $83$ |
\( T^{9} + \cdots + 96\!\cdots\!16 \)
|
| $89$ |
\( T^{9} + \cdots + 53\!\cdots\!80 \)
|
| $97$ |
\( T^{9} + \cdots - 49\!\cdots\!32 \)
|
| show more | |
| show less |