Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [323,2,Mod(1,323)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(323, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("323.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 323 = 17 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 323.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: | \(2.57916798529\) |
| 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: |
\( x^{7} - x^{6} - 10x^{5} + 9x^{4} + 26x^{3} - 19x^{2} - 12x + 8 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{7} - x^{6} - 10x^{5} + 9x^{4} + 26x^{3} - 19x^{2} - 12x + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 5\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 6\nu^{2} + 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{6} - \nu^{5} - 10\nu^{4} + 7\nu^{3} + 26\nu^{2} - 7\nu - 10 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{6} - 10\nu^{4} + 26\nu^{2} + \nu - 11 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 6\beta_{2} + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{6} - 2\beta_{5} + 7\beta_{3} + 27\beta _1 + 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{6} + 10\beta_{4} + 34\beta_{2} - \beta _1 + 73 \)
|
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 |
|
−2.45698 | 2.71826 | 4.03673 | 0.0680537 | −6.67869 | 2.54721 | −5.00419 | 4.38892 | −0.167206 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.73498 | −1.88181 | 1.01016 | 3.19376 | 3.26491 | −3.45233 | 1.71735 | 0.541222 | −5.54112 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.757869 | 2.16604 | −1.42563 | 1.06616 | −1.64157 | −3.35405 | 2.59618 | 1.69171 | −0.808008 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.590711 | −2.43336 | −1.65106 | −1.51197 | −1.43741 | 2.74743 | −2.15672 | 2.92126 | −0.893139 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.815735 | 0.448052 | −1.33458 | 3.98351 | 0.365491 | 3.53587 | −2.72013 | −2.79925 | 3.24949 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.12468 | 2.27405 | 2.51428 | −2.29600 | 4.83164 | 1.03200 | 1.09268 | 2.17132 | −4.87828 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.41870 | −0.291220 | 3.85010 | 2.49649 | −0.704373 | −2.05614 | 4.47483 | −2.91519 | 6.03826 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(17\) | \(-1\) |
| \(19\) | \(1\) |
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 | 323.2.a.f | ✓ | 7 |
| 3.b | odd | 2 | 1 | 2907.2.a.p | 7 | ||
| 4.b | odd | 2 | 1 | 5168.2.a.ba | 7 | ||
| 5.b | even | 2 | 1 | 8075.2.a.o | 7 | ||
| 17.b | even | 2 | 1 | 5491.2.a.s | 7 | ||
| 19.b | odd | 2 | 1 | 6137.2.a.g | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 323.2.a.f | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 2907.2.a.p | 7 | 3.b | odd | 2 | 1 | ||
| 5168.2.a.ba | 7 | 4.b | odd | 2 | 1 | ||
| 5491.2.a.s | 7 | 17.b | even | 2 | 1 | ||
| 6137.2.a.g | 7 | 19.b | odd | 2 | 1 | ||
| 8075.2.a.o | 7 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} - T_{2}^{6} - 10T_{2}^{5} + 9T_{2}^{4} + 26T_{2}^{3} - 19T_{2}^{2} - 12T_{2} + 8 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(323))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - T^{6} - 10 T^{5} + \cdots + 8 \)
|
| $3$ |
\( T^{7} - 3 T^{6} + \cdots + 8 \)
|
| $5$ |
\( T^{7} - 7 T^{6} + \cdots - 8 \)
|
| $7$ |
\( T^{7} - T^{6} + \cdots + 608 \)
|
| $11$ |
\( T^{7} - 2 T^{6} + \cdots - 288 \)
|
| $13$ |
\( T^{7} - 20 T^{6} + \cdots - 2344 \)
|
| $17$ |
\( (T - 1)^{7} \)
|
| $19$ |
\( (T + 1)^{7} \)
|
| $23$ |
\( T^{7} + 3 T^{6} + \cdots + 7232 \)
|
| $29$ |
\( T^{7} - 10 T^{6} + \cdots - 10 \)
|
| $31$ |
\( T^{7} - 21 T^{6} + \cdots + 11988 \)
|
| $37$ |
\( T^{7} - 21 T^{6} + \cdots - 54496 \)
|
| $41$ |
\( T^{7} - 11 T^{6} + \cdots + 855808 \)
|
| $43$ |
\( T^{7} - 3 T^{6} + \cdots + 6676 \)
|
| $47$ |
\( T^{7} + 13 T^{6} + \cdots - 29664 \)
|
| $53$ |
\( T^{7} + 15 T^{6} + \cdots - 10504 \)
|
| $59$ |
\( T^{7} + 6 T^{6} + \cdots - 15280 \)
|
| $61$ |
\( T^{7} - 15 T^{6} + \cdots + 457432 \)
|
| $67$ |
\( T^{7} - 21 T^{6} + \cdots - 978192 \)
|
| $71$ |
\( T^{7} + 9 T^{6} + \cdots + 2000 \)
|
| $73$ |
\( T^{7} - 23 T^{6} + \cdots + 1031192 \)
|
| $79$ |
\( T^{7} + 25 T^{6} + \cdots + 4469200 \)
|
| $83$ |
\( T^{7} + 14 T^{6} + \cdots + 90212 \)
|
| $89$ |
\( T^{7} + 17 T^{6} + \cdots - 132520 \)
|
| $97$ |
\( T^{7} - 27 T^{6} + \cdots - 3006 \)
|
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