Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3364,2,Mod(1,3364)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3364, 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("3364.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3364 = 2^{2} \cdot 29^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3364.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: | \(26.8616752400\) |
| Analytic rank: | \(1\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.8.3266578125.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} - 10x^{6} + 6x^{5} + 29x^{4} - 9x^{3} - 25x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - x^{7} - 10x^{6} + 6x^{5} + 29x^{4} - 9x^{3} - 25x^{2} - x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{7} - \nu^{6} - 8\nu^{5} + 5\nu^{4} + 14\nu^{3} - 11\nu^{2} - 4\nu + 7 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} + \nu^{5} - 9\nu^{4} - 13\nu^{3} + 15\nu^{2} + 25\nu + 1 ) / 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{7} + \nu^{6} - 9\nu^{5} - 13\nu^{4} + 15\nu^{3} + 28\nu^{2} + \nu - 6 ) / 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{7} - 2\nu^{6} - 9\nu^{5} + 14\nu^{4} + 24\nu^{3} - 23\nu^{2} - 17\nu + 3 ) / 3 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{7} - \nu^{6} - 11\nu^{5} + 8\nu^{4} + 35\nu^{3} - 17\nu^{2} - 31\nu + 1 ) / 3 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -2\nu^{7} + 20\nu^{5} + 5\nu^{4} - 50\nu^{3} - 8\nu^{2} + 30\nu - 4 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{7} + \beta_{6} + \beta_{4} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 4\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 7\beta_{7} + 8\beta_{6} - 2\beta_{5} + 6\beta_{4} + 2\beta_{2} + 2\beta _1 + 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( 12\beta_{7} + 12\beta_{6} - 9\beta_{5} + 11\beta_{4} - 7\beta_{3} + 10\beta_{2} + 21\beta _1 + 22 \)
|
| \(\nu^{6}\) | \(=\) |
\( 49\beta_{7} + 58\beta_{6} - 22\beta_{5} + 41\beta_{4} - 3\beta_{3} + 21\beta_{2} + 24\beta _1 + 102 \)
|
| \(\nu^{7}\) | \(=\) |
\( 107\beta_{7} + 111\beta_{6} - 70\beta_{5} + 96\beta_{4} - 45\beta_{3} + 80\beta_{2} + 130\beta _1 + 196 \)
|
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 |
|
0 | −3.24575 | 0 | 0.507473 | 0 | 2.09779 | 0 | 7.53487 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.27574 | 0 | −4.41066 | 0 | −3.82663 | 0 | 2.17901 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −2.21825 | 0 | −0.358466 | 0 | 3.12857 | 0 | 1.92063 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −0.271575 | 0 | 0.836327 | 0 | 3.75591 | 0 | −2.92625 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0.243314 | 0 | −3.14638 | 0 | −0.923802 | 0 | −2.94080 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 0.498509 | 0 | 1.93765 | 0 | 0.267484 | 0 | −2.75149 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 0.729419 | 0 | 0.767443 | 0 | −2.02171 | 0 | −2.46795 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 2.54007 | 0 | 2.86661 | 0 | −4.47762 | 0 | 3.45197 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(29\) | \( -1 \) |
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 | 3364.2.a.q | ✓ | 8 |
| 29.b | even | 2 | 1 | 3364.2.a.r | yes | 8 | |
| 29.c | odd | 4 | 2 | 3364.2.c.k | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3364.2.a.q | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 3364.2.a.r | yes | 8 | 29.b | even | 2 | 1 | |
| 3364.2.c.k | 16 | 29.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3364))\):
|
\( T_{3}^{8} + 4T_{3}^{7} - 6T_{3}^{6} - 32T_{3}^{5} - T_{3}^{4} + 41T_{3}^{3} - 14T_{3}^{2} - 3T_{3} + 1 \)
|
|
\( T_{5}^{8} + T_{5}^{7} - 21T_{5}^{6} + 7T_{5}^{5} + 109T_{5}^{4} - 156T_{5}^{3} + 51T_{5}^{2} + 18T_{5} - 9 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + 4 T^{7} + \cdots + 1 \)
|
| $5$ |
\( T^{8} + T^{7} - 21 T^{6} + \cdots - 9 \)
|
| $7$ |
\( T^{8} + 2 T^{7} + \cdots + 211 \)
|
| $11$ |
\( T^{8} + 5 T^{7} + \cdots + 5931 \)
|
| $13$ |
\( T^{8} - 4 T^{7} + \cdots + 20851 \)
|
| $17$ |
\( T^{8} + 15 T^{7} + \cdots + 81 \)
|
| $19$ |
\( T^{8} + 11 T^{7} + \cdots - 549 \)
|
| $23$ |
\( T^{8} + 3 T^{7} + \cdots - 31599 \)
|
| $29$ |
\( T^{8} \)
|
| $31$ |
\( T^{8} - 9 T^{7} + \cdots - 134009 \)
|
| $37$ |
\( T^{8} - 2 T^{7} + \cdots + 16651 \)
|
| $41$ |
\( T^{8} + 38 T^{7} + \cdots - 1341909 \)
|
| $43$ |
\( T^{8} - 3 T^{7} + \cdots - 8009 \)
|
| $47$ |
\( T^{8} - 7 T^{7} + \cdots + 8745291 \)
|
| $53$ |
\( T^{8} + 5 T^{7} + \cdots + 5291775 \)
|
| $59$ |
\( T^{8} + 7 T^{7} + \cdots - 36189 \)
|
| $61$ |
\( T^{8} + 39 T^{7} + \cdots + 15536221 \)
|
| $67$ |
\( T^{8} + 26 T^{7} + \cdots - 4409 \)
|
| $71$ |
\( T^{8} - 52 T^{7} + \cdots - 5139 \)
|
| $73$ |
\( T^{8} - 34 T^{7} + \cdots - 744749 \)
|
| $79$ |
\( T^{8} + 31 T^{7} + \cdots + 1438231 \)
|
| $83$ |
\( T^{8} + 17 T^{7} + \cdots - 6356619 \)
|
| $89$ |
\( T^{8} + 24 T^{7} + \cdots + 1672641 \)
|
| $97$ |
\( T^{8} + 22 T^{7} + \cdots + 343171 \)
|
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