Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [261,8,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, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("261.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 261 = 3^{2} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| 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: | \(81.5324916514\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 1101 x^{8} - 1540 x^{7} + 405148 x^{6} + 870160 x^{5} - 54569376 x^{4} - 87078400 x^{3} + \cdots - 9372051456 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{11}\cdot 3^{4} \) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 1101 x^{8} - 1540 x^{7} + 405148 x^{6} + 870160 x^{5} - 54569376 x^{4} - 87078400 x^{3} + \cdots - 9372051456 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 220 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 6233754023 \nu^{9} + 106065729814 \nu^{8} - 8345660914107 \nu^{7} - 103076032004426 \nu^{6} + \cdots - 34\!\cdots\!28 ) / 28\!\cdots\!40 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 9928619411 \nu^{9} - 28318524622 \nu^{8} - 9378681547239 \nu^{7} - 3833350436422 \nu^{6} + \cdots + 14\!\cdots\!04 ) / 28\!\cdots\!40 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 10050605791 \nu^{9} - 202929968510 \nu^{8} - 7213453276739 \nu^{7} + 131449898257210 \nu^{6} + \cdots - 12\!\cdots\!60 ) / 28\!\cdots\!40 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 1115090686 \nu^{9} + 2118846149 \nu^{8} + 1130304968382 \nu^{7} + 346512500191 \nu^{6} + \cdots - 87\!\cdots\!36 ) / 28\!\cdots\!44 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 6391461179 \nu^{9} + 33150136896 \nu^{8} + 6170330968231 \nu^{7} - 9695139250004 \nu^{6} + \cdots - 60\!\cdots\!52 ) / 14\!\cdots\!20 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 15515011515 \nu^{9} - 81328131916 \nu^{8} + 17293113920495 \nu^{7} + \cdots - 35\!\cdots\!28 ) / 28\!\cdots\!40 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 9184429821 \nu^{9} + 32671902363 \nu^{8} - 9952790479169 \nu^{7} - 50887218595707 \nu^{6} + \cdots + 21\!\cdots\!64 ) / 14\!\cdots\!20 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 220 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{9} + 3\beta_{8} - 3\beta_{7} - 2\beta_{5} + \beta_{4} + 4\beta_{2} + 366\beta _1 + 463 \)
|
| \(\nu^{4}\) | \(=\) |
\( 26\beta_{9} + 14\beta_{8} - 4\beta_{7} - 6\beta_{5} - 14\beta_{4} - 18\beta_{3} + 457\beta_{2} + 1848\beta _1 + 80296 \)
|
| \(\nu^{5}\) | \(=\) |
\( 761 \beta_{9} + 1831 \beta_{8} - 1515 \beta_{7} - 92 \beta_{6} - 1062 \beta_{5} + 477 \beta_{4} + \cdots + 413091 \)
|
| \(\nu^{6}\) | \(=\) |
\( 15798 \beta_{9} + 11746 \beta_{8} - 4672 \beta_{7} - 2880 \beta_{6} - 6246 \beta_{5} - 6882 \beta_{4} + \cdots + 32704068 \)
|
| \(\nu^{7}\) | \(=\) |
\( 411313 \beta_{9} + 894175 \beta_{8} - 642731 \beta_{7} - 61932 \beta_{6} - 487822 \beta_{5} + \cdots + 260432131 \)
|
| \(\nu^{8}\) | \(=\) |
\( 7942202 \beta_{9} + 7361374 \beta_{8} - 2880588 \beta_{7} - 2351280 \beta_{6} - 4263294 \beta_{5} + \cdots + 14002127024 \)
|
| \(\nu^{9}\) | \(=\) |
\( 200619353 \beta_{9} + 411916503 \beta_{8} - 260418267 \beta_{7} - 39484908 \beta_{6} + \cdots + 146013947267 \)
|
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 |
|
−19.3138 | 0 | 245.021 | 408.086 | 0 | 1746.17 | −2260.12 | 0 | −7881.68 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −19.0438 | 0 | 234.668 | −287.010 | 0 | −124.374 | −2031.37 | 0 | 5465.77 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −14.7228 | 0 | 88.7619 | −246.177 | 0 | −1219.72 | 577.696 | 0 | 3624.43 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −9.91427 | 0 | −29.7073 | −326.241 | 0 | 1247.39 | 1563.55 | 0 | 3234.44 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.69492 | 0 | −125.127 | −194.958 | 0 | 1221.62 | 429.029 | 0 | 330.438 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 3.85204 | 0 | −113.162 | 69.8627 | 0 | −1359.73 | −928.965 | 0 | 269.114 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 4.34965 | 0 | −109.081 | 341.807 | 0 | 956.613 | −1031.22 | 0 | 1486.74 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 13.0110 | 0 | 41.2859 | −124.561 | 0 | −962.222 | −1128.24 | 0 | −1620.67 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 21.4083 | 0 | 330.317 | 555.983 | 0 | −1113.12 | 4331.27 | 0 | 11902.7 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 22.0686 | 0 | 359.023 | −376.792 | 0 | 647.379 | 5098.36 | 0 | −8315.26 | |||||||||||||||||||||||||||||||||||||||||||||||||
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 | 261.8.a.f | 10 | |
| 3.b | odd | 2 | 1 | 29.8.a.b | ✓ | 10 | |
| 12.b | even | 2 | 1 | 464.8.a.g | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 29.8.a.b | ✓ | 10 | 3.b | odd | 2 | 1 | |
| 261.8.a.f | 10 | 1.a | even | 1 | 1 | trivial | |
| 464.8.a.g | 10 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} - 1101 T_{2}^{8} - 1540 T_{2}^{7} + 405148 T_{2}^{6} + 870160 T_{2}^{5} - 54569376 T_{2}^{4} + \cdots - 9372051456 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(261))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + \cdots - 9372051456 \)
|
| $3$ |
\( T^{10} \)
|
| $5$ |
\( T^{10} + \cdots + 11\!\cdots\!00 \)
|
| $7$ |
\( T^{10} + \cdots - 36\!\cdots\!64 \)
|
| $11$ |
\( T^{10} + \cdots - 55\!\cdots\!76 \)
|
| $13$ |
\( T^{10} + \cdots - 23\!\cdots\!44 \)
|
| $17$ |
\( T^{10} + \cdots - 19\!\cdots\!24 \)
|
| $19$ |
\( T^{10} + \cdots - 67\!\cdots\!96 \)
|
| $23$ |
\( T^{10} + \cdots - 91\!\cdots\!16 \)
|
| $29$ |
\( (T - 24389)^{10} \)
|
| $31$ |
\( T^{10} + \cdots + 53\!\cdots\!56 \)
|
| $37$ |
\( T^{10} + \cdots + 44\!\cdots\!36 \)
|
| $41$ |
\( T^{10} + \cdots + 10\!\cdots\!00 \)
|
| $43$ |
\( T^{10} + \cdots - 15\!\cdots\!56 \)
|
| $47$ |
\( T^{10} + \cdots - 51\!\cdots\!76 \)
|
| $53$ |
\( T^{10} + \cdots - 81\!\cdots\!24 \)
|
| $59$ |
\( T^{10} + \cdots - 22\!\cdots\!00 \)
|
| $61$ |
\( T^{10} + \cdots + 11\!\cdots\!56 \)
|
| $67$ |
\( T^{10} + \cdots - 70\!\cdots\!84 \)
|
| $71$ |
\( T^{10} + \cdots - 20\!\cdots\!84 \)
|
| $73$ |
\( T^{10} + \cdots - 10\!\cdots\!76 \)
|
| $79$ |
\( T^{10} + \cdots - 93\!\cdots\!56 \)
|
| $83$ |
\( T^{10} + \cdots - 42\!\cdots\!16 \)
|
| $89$ |
\( T^{10} + \cdots + 71\!\cdots\!84 \)
|
| $97$ |
\( T^{10} + \cdots + 33\!\cdots\!76 \)
|
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