Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1,124,Mod(1,1)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1, base_ring=CyclotomicField(1))
chi = DirichletCharacter(H, H._module([]))
N = Newforms(chi, 124, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1.1");
S:= CuspForms(chi, 124);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1 \) |
| Weight: | \( k \) | \(=\) | \( 124 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1.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: | \(95.8076224914\) |
| 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: |
\( x^{10} - 5 x^{9} + \cdots - 30\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | multiple of \( 2^{178}\cdot 3^{70}\cdot 5^{22}\cdot 7^{9}\cdot 11^{6}\cdot 13^{2}\cdot 17\cdot 31^{2}\cdot 41^{3} \) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 5 x^{9} + \cdots - 30\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( 72\nu - 36 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 72\!\cdots\!23 \nu^{9} + \cdots + 25\!\cdots\!76 ) / 13\!\cdots\!76 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 35\!\cdots\!17 \nu^{9} + \cdots - 88\!\cdots\!36 ) / 13\!\cdots\!76 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 30\!\cdots\!95 \nu^{9} + \cdots - 69\!\cdots\!00 ) / 54\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 18\!\cdots\!65 \nu^{9} + \cdots - 38\!\cdots\!00 ) / 58\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 44\!\cdots\!95 \nu^{9} + \cdots - 35\!\cdots\!00 ) / 54\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 36\!\cdots\!65 \nu^{9} + \cdots + 10\!\cdots\!00 ) / 82\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 20\!\cdots\!65 \nu^{9} + \cdots + 24\!\cdots\!00 ) / 41\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 14\!\cdots\!07 \nu^{9} + \cdots + 13\!\cdots\!00 ) / 41\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 36 ) / 72 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 4944179\beta_{2} + 180506488736968722\beta _1 + 16293782570364898259424186091603158432 ) / 5184 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{7} + 597 \beta_{6} + 2345239 \beta_{5} + 883856762568 \beta_{4} + \cdots + 29\!\cdots\!44 ) / 373248 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 114880745000 \beta_{9} - 76030013716792 \beta_{8} + \cdots + 52\!\cdots\!16 ) / 3359232 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 10\!\cdots\!00 \beta_{9} + \cdots + 25\!\cdots\!72 ) / 15116544 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 19\!\cdots\!00 \beta_{9} + \cdots + 40\!\cdots\!96 ) / 45349632 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 30\!\cdots\!00 \beta_{9} + \cdots - 91\!\cdots\!96 ) / 30233088 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 34\!\cdots\!00 \beta_{9} + \cdots + 49\!\cdots\!92 ) / 90699264 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 25\!\cdots\!00 \beta_{9} + \cdots - 12\!\cdots\!88 ) / 2519424 \)
|
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 |
|
−5.43581e18 | −3.35417e28 | 1.89142e37 | 1.63130e43 | 1.82326e47 | −3.09666e51 | −4.50103e55 | −4.73942e58 | −8.86742e61 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −5.11162e18 | −2.22156e29 | 1.54948e37 | −1.83710e43 | 1.13558e48 | 3.56438e51 | −2.48476e55 | 8.34100e56 | 9.39057e61 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −4.07908e18 | 3.50571e29 | 6.00505e36 | −4.74746e42 | −1.43001e48 | −1.17353e51 | 1.88811e55 | 7.43807e58 | 1.93653e61 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.35000e18 | 1.15541e28 | −8.81132e36 | 1.19036e42 | −1.55981e46 | 4.29022e51 | 2.62510e55 | −4.83858e58 | −1.60698e60 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.20239e18 | −3.80552e29 | −9.18808e36 | 5.95717e42 | 4.57573e47 | −7.49927e51 | 2.38337e55 | 9.63009e58 | −7.16285e60 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.83831e18 | 1.91755e29 | −7.25444e36 | −8.31906e42 | 3.52506e47 | −1.53178e52 | −3.28842e55 | −1.17492e58 | −1.52930e61 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.75898e18 | 3.25623e29 | −3.02185e36 | 1.58514e43 | 8.98387e47 | 1.53160e52 | −3.76757e55 | 5.75108e58 | 4.37336e61 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 3.46200e18 | −2.27154e29 | 1.35165e36 | −7.89778e42 | −7.86409e47 | 1.06526e52 | −3.21349e55 | 3.07977e57 | −2.73421e61 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 5.04095e18 | −1.36074e29 | 1.47773e37 | 1.38517e43 | −6.85941e47 | −1.32080e52 | 2.08873e55 | −3.00032e58 | 6.98256e61 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 6.28101e18 | 2.54101e29 | 2.88173e37 | −1.37149e43 | 1.59601e48 | 5.88359e51 | 1.14211e56 | 1.60483e58 | −8.61436e61 | |||||||||||||||||||||||||||||||||||||||||||||||||
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 | 1.124.a.a | ✓ | 10 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1.124.a.a | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace is the entire newspace \(S_{124}^{\mathrm{new}}(\Gamma_0(1))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + \cdots - 10\!\cdots\!76 \)
|
| $3$ |
\( T^{10} + \cdots - 56\!\cdots\!24 \)
|
| $5$ |
\( T^{10} + \cdots - 19\!\cdots\!00 \)
|
| $7$ |
\( T^{10} + \cdots - 80\!\cdots\!76 \)
|
| $11$ |
\( T^{10} + \cdots + 15\!\cdots\!24 \)
|
| $13$ |
\( T^{10} + \cdots + 17\!\cdots\!76 \)
|
| $17$ |
\( T^{10} + \cdots + 47\!\cdots\!24 \)
|
| $19$ |
\( T^{10} + \cdots - 75\!\cdots\!00 \)
|
| $23$ |
\( T^{10} + \cdots - 41\!\cdots\!24 \)
|
| $29$ |
\( T^{10} + \cdots - 56\!\cdots\!00 \)
|
| $31$ |
\( T^{10} + \cdots + 17\!\cdots\!24 \)
|
| $37$ |
\( T^{10} + \cdots - 30\!\cdots\!76 \)
|
| $41$ |
\( T^{10} + \cdots - 14\!\cdots\!76 \)
|
| $43$ |
\( T^{10} + \cdots + 29\!\cdots\!76 \)
|
| $47$ |
\( T^{10} + \cdots - 96\!\cdots\!76 \)
|
| $53$ |
\( T^{10} + \cdots + 18\!\cdots\!76 \)
|
| $59$ |
\( T^{10} + \cdots - 70\!\cdots\!00 \)
|
| $61$ |
\( T^{10} + \cdots - 29\!\cdots\!76 \)
|
| $67$ |
\( T^{10} + \cdots - 27\!\cdots\!76 \)
|
| $71$ |
\( T^{10} + \cdots + 31\!\cdots\!24 \)
|
| $73$ |
\( T^{10} + \cdots - 15\!\cdots\!24 \)
|
| $79$ |
\( T^{10} + \cdots + 13\!\cdots\!00 \)
|
| $83$ |
\( T^{10} + \cdots + 19\!\cdots\!76 \)
|
| $89$ |
\( T^{10} + \cdots - 31\!\cdots\!00 \)
|
| $97$ |
\( T^{10} + \cdots - 11\!\cdots\!76 \)
|
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