Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [442,2,Mod(35,442)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(442, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("442.35");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 442 = 2 \cdot 13 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 442.e (of order \(3\), degree \(2\), minimal) |
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: | no |
| Analytic conductor: | \(3.52938776934\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} + 10x^{6} - 5x^{5} + 73x^{4} - 33x^{3} + 184x^{2} + 105x + 225 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{8} - x^{7} + 10x^{6} - 5x^{5} + 73x^{4} - 33x^{3} + 184x^{2} + 105x + 225 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 362\nu^{7} - 902\nu^{6} + 3485\nu^{5} - 5890\nu^{4} + 24026\nu^{3} - 54366\nu^{2} + 49268\nu + 18210 ) / 101235 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -36\nu^{7} - 9\nu^{6} - 272\nu^{5} - 160\nu^{4} - 2828\nu^{3} - 1156\nu^{2} - 1320\nu - 5430 ) / 6749 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 45\nu^{7} - 88\nu^{6} + 340\nu^{5} + 200\nu^{4} + 2344\nu^{3} + 1445\nu^{2} + 1650\nu + 25645 ) / 6749 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 157\nu^{7} + 734\nu^{6} + 2686\nu^{5} + 5947\nu^{4} + 15421\nu^{3} + 35037\nu^{2} + 57499\nu + 49155 ) / 20247 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -227\nu^{7} + 638\nu^{6} - 2465\nu^{5} + 6490\nu^{4} - 16994\nu^{3} + 38454\nu^{2} - 44318\nu + 58725 ) / 20247 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -356\nu^{7} + 308\nu^{6} - 1190\nu^{5} + 3667\nu^{4} - 8204\nu^{3} + 18564\nu^{2} + 38689\nu + 28350 ) / 20247 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{6} + \beta_{4} - 5\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} - \beta_{5} - \beta_{4} - 6\beta_{3} \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{7} + 8\beta_{6} + 30\beta_{2} + \beta _1 - 30 \)
|
| \(\nu^{5}\) | \(=\) |
\( -8\beta_{6} + 10\beta_{5} + 8\beta_{4} + 38\beta_{3} - 5\beta_{2} - 38\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -12\beta_{7} + 12\beta_{5} - 56\beta_{4} - 13\beta_{3} + 190 \)
|
| \(\nu^{7}\) | \(=\) |
\( -80\beta_{7} + 57\beta_{6} + 65\beta_{2} + 246\beta _1 - 65 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/442\mathbb{Z}\right)^\times\).
| \(n\) | \(105\) | \(171\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 35.1 |
|
−0.500000 | + | 0.866025i | −1.31019 | + | 2.26932i | −0.500000 | − | 0.866025i | −3.62039 | −1.31019 | − | 2.26932i | −0.231238 | − | 0.400516i | 1.00000 | −1.93322 | − | 3.34843i | 1.81019 | − | 3.13535i | ||||||||||||||||||||||||||||
| 35.2 | −0.500000 | + | 0.866025i | −0.527802 | + | 0.914180i | −0.500000 | − | 0.866025i | −2.05560 | −0.527802 | − | 0.914180i | 1.80702 | + | 3.12985i | 1.00000 | 0.942850 | + | 1.63306i | 1.02780 | − | 1.78021i | |||||||||||||||||||||||||||||
| 35.3 | −0.500000 | + | 0.866025i | 1.06482 | − | 1.84433i | −0.500000 | − | 0.866025i | 1.12965 | 1.06482 | + | 1.84433i | 2.52416 | + | 4.37198i | 1.00000 | −0.767694 | − | 1.32969i | −0.564823 | + | 0.978301i | |||||||||||||||||||||||||||||
| 35.4 | −0.500000 | + | 0.866025i | 1.27317 | − | 2.20520i | −0.500000 | − | 0.866025i | 1.54635 | 1.27317 | + | 2.20520i | −1.59994 | − | 2.77118i | 1.00000 | −1.74194 | − | 3.01713i | −0.773173 | + | 1.33918i | |||||||||||||||||||||||||||||
| 341.1 | −0.500000 | − | 0.866025i | −1.31019 | − | 2.26932i | −0.500000 | + | 0.866025i | −3.62039 | −1.31019 | + | 2.26932i | −0.231238 | + | 0.400516i | 1.00000 | −1.93322 | + | 3.34843i | 1.81019 | + | 3.13535i | |||||||||||||||||||||||||||||
| 341.2 | −0.500000 | − | 0.866025i | −0.527802 | − | 0.914180i | −0.500000 | + | 0.866025i | −2.05560 | −0.527802 | + | 0.914180i | 1.80702 | − | 3.12985i | 1.00000 | 0.942850 | − | 1.63306i | 1.02780 | + | 1.78021i | |||||||||||||||||||||||||||||
| 341.3 | −0.500000 | − | 0.866025i | 1.06482 | + | 1.84433i | −0.500000 | + | 0.866025i | 1.12965 | 1.06482 | − | 1.84433i | 2.52416 | − | 4.37198i | 1.00000 | −0.767694 | + | 1.32969i | −0.564823 | − | 0.978301i | |||||||||||||||||||||||||||||
| 341.4 | −0.500000 | − | 0.866025i | 1.27317 | + | 2.20520i | −0.500000 | + | 0.866025i | 1.54635 | 1.27317 | − | 2.20520i | −1.59994 | + | 2.77118i | 1.00000 | −1.74194 | + | 3.01713i | −0.773173 | − | 1.33918i | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 442.2.e.d | ✓ | 8 |
| 13.c | even | 3 | 1 | inner | 442.2.e.d | ✓ | 8 |
| 13.c | even | 3 | 1 | 5746.2.a.bi | 4 | ||
| 13.e | even | 6 | 1 | 5746.2.a.be | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 442.2.e.d | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 442.2.e.d | ✓ | 8 | 13.c | even | 3 | 1 | inner |
| 5746.2.a.be | 4 | 13.e | even | 6 | 1 | ||
| 5746.2.a.bi | 4 | 13.c | even | 3 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} - T_{3}^{7} + 10T_{3}^{6} - 5T_{3}^{5} + 73T_{3}^{4} - 33T_{3}^{3} + 184T_{3}^{2} + 105T_{3} + 225 \)
acting on \(S_{2}^{\mathrm{new}}(442, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + T + 1)^{4} \)
|
| $3$ |
\( T^{8} - T^{7} + \cdots + 225 \)
|
| $5$ |
\( (T^{4} + 3 T^{3} - 6 T^{2} + \cdots + 13)^{2} \)
|
| $7$ |
\( T^{8} - 5 T^{7} + \cdots + 729 \)
|
| $11$ |
\( T^{8} - 3 T^{7} + \cdots + 121 \)
|
| $13$ |
\( T^{8} + 11 T^{7} + \cdots + 28561 \)
|
| $17$ |
\( (T^{2} + T + 1)^{4} \)
|
| $19$ |
\( T^{8} + 2 T^{7} + \cdots + 308025 \)
|
| $23$ |
\( T^{8} + 7 T^{7} + \cdots + 9409 \)
|
| $29$ |
\( T^{8} - 3 T^{7} + \cdots + 169 \)
|
| $31$ |
\( (T^{4} + 3 T^{3} - 97 T^{2} + \cdots + 49)^{2} \)
|
| $37$ |
\( T^{8} - 11 T^{7} + \cdots + 33489 \)
|
| $41$ |
\( T^{8} - 35 T^{7} + \cdots + 16621929 \)
|
| $43$ |
\( T^{8} - 12 T^{7} + \cdots + 4012009 \)
|
| $47$ |
\( (T^{4} + 9 T^{3} + \cdots + 4021)^{2} \)
|
| $53$ |
\( (T^{4} - T^{3} - 83 T^{2} + \cdots + 819)^{2} \)
|
| $59$ |
\( T^{8} + T^{7} + \cdots + 10452289 \)
|
| $61$ |
\( T^{8} - 14 T^{7} + \cdots + 169 \)
|
| $67$ |
\( T^{8} - 21 T^{7} + \cdots + 19193161 \)
|
| $71$ |
\( T^{8} + 3 T^{7} + \cdots + 35010889 \)
|
| $73$ |
\( (T^{4} + 18 T^{3} + \cdots - 101)^{2} \)
|
| $79$ |
\( (T^{4} - 6 T^{3} + \cdots + 12571)^{2} \)
|
| $83$ |
\( (T^{4} + 19 T^{3} + \cdots + 375)^{2} \)
|
| $89$ |
\( T^{8} + 13 T^{7} + \cdots + 138368169 \)
|
| $97$ |
\( T^{8} + 3 T^{7} + \cdots + 537289 \)
|
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