Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [353,2,Mod(42,353)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(353, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("353.42");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 353 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 353.c (of order \(4\), 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: | \(2.81871919135\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 2x^{7} + 2x^{6} + 10x^{5} + 47x^{4} - 26x^{3} + 8x^{2} + 4x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{8} - 2x^{7} + 2x^{6} + 10x^{5} + 47x^{4} - 26x^{3} + 8x^{2} + 4x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 103\nu^{7} - 182\nu^{6} + 196\nu^{5} + 784\nu^{4} + 5526\nu^{3} - 1439\nu^{2} - 759\nu - 6214 ) / 1669 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -424\nu^{7} + 733\nu^{6} - 661\nu^{5} - 4313\nu^{4} - 21889\nu^{3} + 5713\nu^{2} - 1996\nu - 962 ) / 5007 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 890\nu^{7} - 2318\nu^{6} + 3395\nu^{5} + 6904\nu^{4} + 37184\nu^{3} - 42152\nu^{2} + 48308\nu - 7853 ) / 5007 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -962\nu^{7} + 2348\nu^{6} - 2657\nu^{5} - 8959\nu^{4} - 40901\nu^{3} + 46901\nu^{2} - 13409\nu - 1852 ) / 5007 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 2462\nu^{7} - 6311\nu^{6} + 7310\nu^{5} + 22564\nu^{4} + 100814\nu^{3} - 129983\nu^{2} + 33224\nu + 4594 ) / 5007 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -4045\nu^{7} + 7666\nu^{6} - 7357\nu^{5} - 41111\nu^{4} - 194428\nu^{3} + 83281\nu^{2} - 26647\nu - 13169 ) / 5007 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + 3\beta_{5} - \beta_{3} + \beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + \beta_{6} + 2\beta_{5} - 9\beta_{3} - \beta_{2} - 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{7} + \beta_{4} - 14\beta_{3} - 9\beta_{2} - 14\beta _1 - 32 \)
|
| \(\nu^{5}\) | \(=\) |
\( -14\beta_{6} - 32\beta_{5} + 9\beta_{4} - 14\beta_{2} - 86\beta _1 - 37 \)
|
| \(\nu^{6}\) | \(=\) |
\( -14\beta_{7} - 86\beta_{6} - 221\beta_{5} + 14\beta_{4} + 165\beta_{3} - 165\beta _1 + 14 \)
|
| \(\nu^{7}\) | \(=\) |
\( -86\beta_{7} - 165\beta_{6} - 395\beta_{5} + 867\beta_{3} + 165\beta_{2} + 560 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/353\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(\beta_{5}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 42.1 |
|
0 | −1.57014 | + | 1.57014i | −2.00000 | −1.25170 | + | 1.25170i | 0 | −1.88858 | − | 1.88858i | 0 | − | 1.93068i | 0 | |||||||||||||||||||||||||||||||||||
| 42.2 | 0 | −0.162533 | + | 0.162533i | −2.00000 | 2.91377 | − | 2.91377i | 0 | −3.23884 | − | 3.23884i | 0 | 2.94717i | 0 | |||||||||||||||||||||||||||||||||||||
| 42.3 | 0 | 0.424399 | − | 0.424399i | −2.00000 | −0.753739 | + | 0.753739i | 0 | 1.60254 | + | 1.60254i | 0 | 2.63977i | 0 | |||||||||||||||||||||||||||||||||||||
| 42.4 | 0 | 2.30827 | − | 2.30827i | −2.00000 | 2.09166 | − | 2.09166i | 0 | 2.52489 | + | 2.52489i | 0 | − | 7.65626i | 0 | ||||||||||||||||||||||||||||||||||||
| 311.1 | 0 | −1.57014 | − | 1.57014i | −2.00000 | −1.25170 | − | 1.25170i | 0 | −1.88858 | + | 1.88858i | 0 | 1.93068i | 0 | |||||||||||||||||||||||||||||||||||||
| 311.2 | 0 | −0.162533 | − | 0.162533i | −2.00000 | 2.91377 | + | 2.91377i | 0 | −3.23884 | + | 3.23884i | 0 | − | 2.94717i | 0 | ||||||||||||||||||||||||||||||||||||
| 311.3 | 0 | 0.424399 | + | 0.424399i | −2.00000 | −0.753739 | − | 0.753739i | 0 | 1.60254 | − | 1.60254i | 0 | − | 2.63977i | 0 | ||||||||||||||||||||||||||||||||||||
| 311.4 | 0 | 2.30827 | + | 2.30827i | −2.00000 | 2.09166 | + | 2.09166i | 0 | 2.52489 | − | 2.52489i | 0 | 7.65626i | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 353.c | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 353.2.c.b | ✓ | 8 |
| 353.c | even | 4 | 1 | inner | 353.2.c.b | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 353.2.c.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 353.2.c.b | ✓ | 8 | 353.c | even | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} \)
acting on \(S_{2}^{\mathrm{new}}(353, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} - 2 T^{7} + \cdots + 1 \)
|
| $5$ |
\( T^{8} - 6 T^{7} + \cdots + 529 \)
|
| $7$ |
\( T^{8} + 2 T^{7} + \cdots + 9801 \)
|
| $11$ |
\( (T^{4} + 6 T^{3} - 13 T^{2} + \cdots + 81)^{2} \)
|
| $13$ |
\( T^{8} + 8 T^{7} + \cdots + 2025 \)
|
| $17$ |
\( (T + 3)^{8} \)
|
| $19$ |
\( T^{8} + 92 T^{6} + \cdots + 149769 \)
|
| $23$ |
\( T^{8} + 78 T^{6} + \cdots + 625 \)
|
| $29$ |
\( (T^{4} + 6 T^{3} - 13 T^{2} + \cdots + 99)^{2} \)
|
| $31$ |
\( T^{8} - 18 T^{7} + \cdots + 289 \)
|
| $37$ |
\( T^{8} - 18 T^{5} + \cdots + 14641 \)
|
| $41$ |
\( T^{8} + 132 T^{6} + \cdots + 2209 \)
|
| $43$ |
\( T^{8} + 140 T^{6} + \cdots + 104976 \)
|
| $47$ |
\( T^{8} + 214 T^{6} + \cdots + 2152089 \)
|
| $53$ |
\( T^{8} + 18 T^{7} + \cdots + 121801 \)
|
| $59$ |
\( T^{8} - 18 T^{7} + \cdots + 25 \)
|
| $61$ |
\( (T^{4} + 10 T^{3} + \cdots - 956)^{2} \)
|
| $67$ |
\( T^{8} - 10 T^{7} + \cdots + 6561 \)
|
| $71$ |
\( T^{8} - 24 T^{7} + \cdots + 841 \)
|
| $73$ |
\( (T^{4} - 6 T^{3} + \cdots + 319)^{2} \)
|
| $79$ |
\( T^{8} - 18 T^{7} + \cdots + 64625521 \)
|
| $83$ |
\( (T^{4} - 18 T^{3} + \cdots + 45)^{2} \)
|
| $89$ |
\( T^{8} - 36 T^{7} + \cdots + 641601 \)
|
| $97$ |
\( (T^{4} + 12 T^{3} + \cdots - 6995)^{2} \)
|
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