Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1,106,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, 106, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1.1");
S:= CuspForms(chi, 106);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1 \) |
| Weight: | \( k \) | \(=\) | \( 106 \) |
| 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: | \(69.8187388595\) |
| Analytic rank: | \(1\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} + \cdots + 48\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | multiple of \( 2^{111}\cdot 3^{44}\cdot 5^{13}\cdot 7^{7}\cdot 11\cdot 13^{3}\cdot 17 \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - x^{7} + \cdots + 48\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( 144\nu - 18 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 27\!\cdots\!11 \nu^{7} + \cdots + 29\!\cdots\!68 ) / 78\!\cdots\!52 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 27\!\cdots\!25 \nu^{7} + \cdots - 45\!\cdots\!44 ) / 78\!\cdots\!52 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 33\!\cdots\!37 \nu^{7} + \cdots - 11\!\cdots\!80 ) / 89\!\cdots\!80 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 71\!\cdots\!67 \nu^{7} + \cdots + 89\!\cdots\!20 ) / 44\!\cdots\!40 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 11\!\cdots\!43 \nu^{7} + \cdots - 35\!\cdots\!80 ) / 47\!\cdots\!20 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 83\!\cdots\!23 \nu^{7} + \cdots + 36\!\cdots\!80 ) / 17\!\cdots\!96 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 18 ) / 144 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 1007775\beta_{2} + 128696535390092\beta _1 + 54299611946833427723289037367328 ) / 20736 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{7} + 253 \beta_{6} + 920971 \beta_{5} + 3050104560 \beta_{4} + \cdots + 69\!\cdots\!48 ) / 2985984 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 261414756976041 \beta_{7} + \cdots + 30\!\cdots\!48 ) / 26873856 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 59\!\cdots\!93 \beta_{7} + \cdots + 23\!\cdots\!48 ) / 241864704 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 73\!\cdots\!07 \beta_{7} + \cdots + 46\!\cdots\!24 ) / 80621568 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 38\!\cdots\!61 \beta_{7} + \cdots + 25\!\cdots\!08 ) / 241864704 \)
|
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 |
|
−1.15346e16 | 2.00325e25 | 9.24825e31 | −4.02660e36 | −2.31067e41 | −1.49411e43 | −5.98851e47 | 2.76064e50 | 4.64453e52 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −9.55432e15 | −1.37857e25 | 5.07201e31 | −2.80039e36 | 1.31713e41 | −2.78207e44 | −9.70270e46 | 6.48087e49 | 2.67558e52 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −7.36840e15 | −8.33754e23 | 1.37285e31 | 6.83377e36 | 6.14343e39 | 4.41963e44 | 1.97741e47 | −1.24542e50 | −5.03539e52 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.56214e15 | 7.24197e24 | −3.40003e31 | −1.17115e36 | −1.85549e40 | −1.82598e44 | 1.91046e47 | −7.27906e49 | 3.00065e51 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.72418e15 | −1.28608e25 | −3.75920e31 | −7.91382e36 | −2.21743e40 | 2.76433e44 | −1.34757e47 | 4.01631e49 | −1.36449e52 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 3.85781e15 | −1.61266e25 | −2.56821e31 | 9.75211e36 | −6.22135e40 | −2.29314e44 | −2.55568e47 | 1.34832e50 | 3.76218e52 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 5.93573e15 | 1.55541e25 | −5.33194e30 | 1.46253e36 | 9.23248e40 | 8.76445e43 | −2.72431e47 | 1.16692e50 | 8.68120e51 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.03267e16 | −2.77402e24 | 6.60762e31 | −1.38808e36 | −2.86465e40 | −3.04428e43 | 2.63449e47 | −1.17542e50 | −1.43343e52 | |||||||||||||||||||||||||||||||||||||||||||
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 | 1.106.a.a | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1.106.a.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace is the entire newspace \(S_{106}^{\mathrm{new}}(\Gamma_0(1))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + \cdots + 84\!\cdots\!56 \)
|
| $3$ |
\( T^{8} + \cdots - 14\!\cdots\!64 \)
|
| $5$ |
\( T^{8} + \cdots - 14\!\cdots\!00 \)
|
| $7$ |
\( T^{8} + \cdots - 56\!\cdots\!84 \)
|
| $11$ |
\( T^{8} + \cdots + 17\!\cdots\!56 \)
|
| $13$ |
\( T^{8} + \cdots - 19\!\cdots\!44 \)
|
| $17$ |
\( T^{8} + \cdots + 78\!\cdots\!36 \)
|
| $19$ |
\( T^{8} + \cdots + 11\!\cdots\!00 \)
|
| $23$ |
\( T^{8} + \cdots + 33\!\cdots\!76 \)
|
| $29$ |
\( T^{8} + \cdots + 23\!\cdots\!00 \)
|
| $31$ |
\( T^{8} + \cdots + 89\!\cdots\!56 \)
|
| $37$ |
\( T^{8} + \cdots + 20\!\cdots\!76 \)
|
| $41$ |
\( T^{8} + \cdots - 98\!\cdots\!44 \)
|
| $43$ |
\( T^{8} + \cdots - 40\!\cdots\!84 \)
|
| $47$ |
\( T^{8} + \cdots + 73\!\cdots\!96 \)
|
| $53$ |
\( T^{8} + \cdots - 17\!\cdots\!64 \)
|
| $59$ |
\( T^{8} + \cdots - 47\!\cdots\!00 \)
|
| $61$ |
\( T^{8} + \cdots + 32\!\cdots\!56 \)
|
| $67$ |
\( T^{8} + \cdots + 11\!\cdots\!36 \)
|
| $71$ |
\( T^{8} + \cdots + 98\!\cdots\!56 \)
|
| $73$ |
\( T^{8} + \cdots + 18\!\cdots\!76 \)
|
| $79$ |
\( T^{8} + \cdots - 82\!\cdots\!00 \)
|
| $83$ |
\( T^{8} + \cdots - 12\!\cdots\!04 \)
|
| $89$ |
\( T^{8} + \cdots + 15\!\cdots\!00 \)
|
| $97$ |
\( T^{8} + \cdots + 13\!\cdots\!96 \)
|
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