Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3332,2,Mod(1,3332)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3332, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("3332.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3332.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: | \(26.6061539535\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 4x^{7} - 8x^{6} + 36x^{5} + 17x^{4} - 76x^{3} - 20x^{2} + 44x + 17 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{8} - 4x^{7} - 8x^{6} + 36x^{5} + 17x^{4} - 76x^{3} - 20x^{2} + 44x + 17 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{7} + 3\nu^{6} + 10\nu^{5} - 23\nu^{4} - 32\nu^{3} + 23\nu^{2} + 27\nu + 6 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} - 3\nu^{5} - 10\nu^{4} + 25\nu^{3} + 30\nu^{2} - 37\nu - 25 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{7} - 3\nu^{6} - 10\nu^{5} + 23\nu^{4} + 32\nu^{3} - 21\nu^{2} - 31\nu - 14 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( -\nu^{7} + 2\nu^{6} + 14\nu^{5} - 15\nu^{4} - 64\nu^{3} + 2\nu^{2} + 69\nu + 29 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -3\nu^{7} + 8\nu^{6} + 35\nu^{5} - 63\nu^{4} - 137\nu^{3} + 59\nu^{2} + 140\nu + 39 ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( -3\nu^{7} + 8\nu^{6} + 35\nu^{5} - 62\nu^{4} - 138\nu^{3} + 50\nu^{2} + 143\nu + 48 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{2} + 2\beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 2\beta_{2} + 8\beta _1 + 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{7} - 3\beta_{6} + \beta_{5} + 10\beta_{4} + \beta_{3} + 11\beta_{2} + 23\beta _1 + 31 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{7} - 13\beta_{6} + 10\beta_{5} + 18\beta_{4} + 11\beta_{3} + 25\beta_{2} + 79\beta _1 + 56 \)
|
| \(\nu^{6}\) | \(=\) |
\( 16\beta_{7} - 44\beta_{6} + 15\beta_{5} + 99\beta_{4} + 20\beta_{3} + 105\beta_{2} + 244\beta _1 + 283 \)
|
| \(\nu^{7}\) | \(=\) |
\( 45\beta_{7} - 161\beta_{6} + 90\beta_{5} + 238\beta_{4} + 115\beta_{3} + 269\beta_{2} + 810\beta _1 + 666 \)
|
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 | −2.43314 | 0 | −0.700514 | 0 | 0 | 0 | 2.92017 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.33282 | 0 | 2.63441 | 0 | 0 | 0 | −1.22360 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.703456 | 0 | −0.200537 | 0 | 0 | 0 | −2.50515 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −0.453360 | 0 | −3.09710 | 0 | 0 | 0 | −2.79446 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 1.26400 | 0 | −1.77431 | 0 | 0 | 0 | −1.40230 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 1.27952 | 0 | 4.13895 | 0 | 0 | 0 | −1.36282 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 3.08774 | 0 | 3.25463 | 0 | 0 | 0 | 6.53416 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 3.29150 | 0 | −0.255523 | 0 | 0 | 0 | 7.83400 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(1\) |
| \(17\) | \(1\) |
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 | 3332.2.a.u | yes | 8 |
| 7.b | odd | 2 | 1 | 3332.2.a.t | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3332.2.a.t | ✓ | 8 | 7.b | odd | 2 | 1 | |
| 3332.2.a.u | yes | 8 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3332))\):
|
\( T_{3}^{8} - 4T_{3}^{7} - 8T_{3}^{6} + 36T_{3}^{5} + 17T_{3}^{4} - 76T_{3}^{3} - 20T_{3}^{2} + 44T_{3} + 17 \)
|
|
\( T_{5}^{8} - 4T_{5}^{7} - 16T_{5}^{6} + 56T_{5}^{5} + 85T_{5}^{4} - 160T_{5}^{3} - 220T_{5}^{2} - 72T_{5} - 7 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} - 4 T^{7} + \cdots + 17 \)
|
| $5$ |
\( T^{8} - 4 T^{7} + \cdots - 7 \)
|
| $7$ |
\( T^{8} \)
|
| $11$ |
\( T^{8} + 4 T^{7} + \cdots - 368 \)
|
| $13$ |
\( T^{8} - 20 T^{7} + \cdots - 16 \)
|
| $17$ |
\( (T + 1)^{8} \)
|
| $19$ |
\( T^{8} - 8 T^{7} + \cdots - 784 \)
|
| $23$ |
\( T^{8} - 4 T^{7} + \cdots + 13168 \)
|
| $29$ |
\( T^{8} + 16 T^{7} + \cdots - 103024 \)
|
| $31$ |
\( T^{8} + 8 T^{7} + \cdots - 82271 \)
|
| $37$ |
\( T^{8} - 8 T^{7} + \cdots - 19216 \)
|
| $41$ |
\( T^{8} - 12 T^{7} + \cdots + 57193 \)
|
| $43$ |
\( T^{8} + 4 T^{7} + \cdots - 231551 \)
|
| $47$ |
\( T^{8} - 4 T^{7} + \cdots - 4624 \)
|
| $53$ |
\( T^{8} - 242 T^{6} + \cdots + 3468593 \)
|
| $59$ |
\( T^{8} + 16 T^{7} + \cdots - 3944816 \)
|
| $61$ |
\( T^{8} - 32 T^{7} + \cdots + 64057 \)
|
| $67$ |
\( T^{8} - 354 T^{6} + \cdots + 2048081 \)
|
| $71$ |
\( T^{8} - 286 T^{6} + \cdots - 1980944 \)
|
| $73$ |
\( T^{8} - 24 T^{7} + \cdots - 9884543 \)
|
| $79$ |
\( T^{8} - 4 T^{7} + \cdots + 230768 \)
|
| $83$ |
\( T^{8} - 28 T^{7} + \cdots - 53521264 \)
|
| $89$ |
\( T^{8} - 20 T^{7} + \cdots + 3056 \)
|
| $97$ |
\( T^{8} - 56 T^{7} + \cdots - 382687 \)
|
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