Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1,92,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, 92, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1.1");
S:= CuspForms(chi, 92);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1 \) |
| Weight: | \( k \) | \(=\) | \( 92 \) |
| 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: | \(52.4421558310\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 2 x^{6} + \cdots + 37\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | multiple of \( 2^{83}\cdot 3^{31}\cdot 5^{8}\cdot 7^{6}\cdot 11\cdot 13^{3}\cdot 23 \) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 2 x^{6} + \cdots + 37\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( 24\nu - 7 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 34\!\cdots\!59 \nu^{6} + \cdots + 26\!\cdots\!08 ) / 24\!\cdots\!72 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 90\!\cdots\!31 \nu^{6} + \cdots - 49\!\cdots\!48 ) / 12\!\cdots\!88 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 65\!\cdots\!79 \nu^{6} + \cdots + 55\!\cdots\!00 ) / 30\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 16\!\cdots\!49 \nu^{6} + \cdots + 31\!\cdots\!00 ) / 15\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 15\!\cdots\!61 \nu^{6} + \cdots + 61\!\cdots\!00 ) / 12\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 7 ) / 24 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 49571\beta_{2} - 1270960315153\beta _1 + 3285816439692620103304937381 ) / 576 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 39 \beta_{6} + 12966 \beta_{5} - 12676468 \beta_{4} - 1016261168401 \beta_{3} + \cdots - 52\!\cdots\!96 ) / 1728 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 141481024033993 \beta_{6} + \cdots + 27\!\cdots\!93 ) / 5184 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 27\!\cdots\!43 \beta_{6} + \cdots - 10\!\cdots\!02 ) / 864 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 84\!\cdots\!86 \beta_{6} + \cdots + 93\!\cdots\!55 ) / 1728 \)
|
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 |
|
−8.41697e13 | 2.94143e21 | 4.60866e27 | 2.26211e30 | −2.47580e35 | −5.00335e38 | −1.79515e41 | −1.75319e43 | −1.90401e44 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −5.62150e13 | −7.69269e21 | 6.84242e26 | −3.78165e31 | 4.32444e35 | 1.92974e38 | 1.00717e41 | 3.29936e43 | 2.12585e45 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −3.49886e13 | 3.81854e21 | −1.25168e27 | 4.98385e31 | −1.33605e35 | 4.56727e38 | 1.30422e41 | −1.16026e43 | −1.74378e45 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 9.51137e12 | 5.28795e21 | −2.38541e27 | −1.05575e32 | 5.02957e34 | −1.59557e38 | −4.62376e40 | 1.77856e42 | −1.00417e45 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 2.16242e13 | −4.57928e21 | −2.00827e27 | 4.97855e31 | −9.90233e34 | −2.82571e38 | −9.69662e40 | −5.21406e42 | 1.07657e45 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 7.15766e13 | 9.10839e21 | 2.64733e27 | 1.09575e32 | 6.51948e35 | −1.62560e38 | 1.22718e40 | 5.67789e43 | 7.84298e45 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 7.65028e13 | −2.65748e21 | 3.37680e27 | −4.45639e31 | −2.03305e35 | 2.84237e38 | 6.89228e40 | −1.91217e43 | −3.40927e45 | ||||||||||||||||||||||||||||||||||||||||
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 | 1.92.a.a | ✓ | 7 |
| 3.b | odd | 2 | 1 | 9.92.a.b | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1.92.a.a | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 9.92.a.b | 7 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{92}^{\mathrm{new}}(\Gamma_0(1))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + \cdots + 18\!\cdots\!16 \)
|
| $3$ |
\( T^{7} + \cdots + 50\!\cdots\!88 \)
|
| $5$ |
\( T^{7} + \cdots + 10\!\cdots\!00 \)
|
| $7$ |
\( T^{7} + \cdots - 91\!\cdots\!64 \)
|
| $11$ |
\( T^{7} + \cdots + 17\!\cdots\!92 \)
|
| $13$ |
\( T^{7} + \cdots + 32\!\cdots\!28 \)
|
| $17$ |
\( T^{7} + \cdots + 95\!\cdots\!76 \)
|
| $19$ |
\( T^{7} + \cdots - 27\!\cdots\!00 \)
|
| $23$ |
\( T^{7} + \cdots - 11\!\cdots\!32 \)
|
| $29$ |
\( T^{7} + \cdots + 35\!\cdots\!00 \)
|
| $31$ |
\( T^{7} + \cdots - 10\!\cdots\!68 \)
|
| $37$ |
\( T^{7} + \cdots - 83\!\cdots\!44 \)
|
| $41$ |
\( T^{7} + \cdots - 10\!\cdots\!48 \)
|
| $43$ |
\( T^{7} + \cdots - 47\!\cdots\!52 \)
|
| $47$ |
\( T^{7} + \cdots - 22\!\cdots\!04 \)
|
| $53$ |
\( T^{7} + \cdots - 51\!\cdots\!12 \)
|
| $59$ |
\( T^{7} + \cdots + 36\!\cdots\!00 \)
|
| $61$ |
\( T^{7} + \cdots + 38\!\cdots\!92 \)
|
| $67$ |
\( T^{7} + \cdots + 54\!\cdots\!76 \)
|
| $71$ |
\( T^{7} + \cdots - 75\!\cdots\!88 \)
|
| $73$ |
\( T^{7} + \cdots - 57\!\cdots\!32 \)
|
| $79$ |
\( T^{7} + \cdots - 67\!\cdots\!00 \)
|
| $83$ |
\( T^{7} + \cdots + 12\!\cdots\!08 \)
|
| $89$ |
\( T^{7} + \cdots - 13\!\cdots\!00 \)
|
| $97$ |
\( T^{7} + \cdots + 53\!\cdots\!96 \)
|
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