Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7942,2,Mod(1,7942)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7942, 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("7942.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7942 = 2 \cdot 11 \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7942.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: | \(63.4171892853\) |
| 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: |
\( x^{8} - 3x^{7} - 14x^{6} + 29x^{5} + 64x^{4} - 50x^{3} - 36x^{2} + 30x - 5 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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
\( x^{8} - 3x^{7} - 14x^{6} + 29x^{5} + 64x^{4} - 50x^{3} - 36x^{2} + 30x - 5 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -3\nu^{7} + 10\nu^{6} + 36\nu^{5} - 83\nu^{4} - 159\nu^{3} + 91\nu^{2} + 91\nu - 35 ) / 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 7\nu^{7} - 18\nu^{6} - 108\nu^{5} + 167\nu^{4} + 531\nu^{3} - 191\nu^{2} - 343\nu + 111 ) / 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -9\nu^{7} + 22\nu^{6} + 140\nu^{5} - 193\nu^{4} - 685\nu^{3} + 129\nu^{2} + 369\nu - 97 ) / 8 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 3\nu^{7} - 8\nu^{6} - 44\nu^{5} + 69\nu^{4} + 213\nu^{3} - 57\nu^{2} - 133\nu + 33 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 3\nu^{7} - 8\nu^{6} - 44\nu^{5} + 69\nu^{4} + 213\nu^{3} - 59\nu^{2} - 129\nu + 39 ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 39\nu^{7} - 106\nu^{6} - 572\nu^{5} + 951\nu^{4} + 2747\nu^{3} - 1047\nu^{2} - 1695\nu + 583 ) / 8 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{6} + \beta_{5} + 2\beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{6} + 2\beta_{5} + \beta_{4} + \beta_{2} + 11\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{7} - 10\beta_{6} + 11\beta_{5} + 2\beta_{4} - 3\beta_{3} + 4\beta_{2} + 32\beta _1 + 24 \)
|
| \(\nu^{5}\) | \(=\) |
\( 4\beta_{7} - 20\beta_{6} + 32\beta_{5} + 14\beta_{4} - 15\beta_{3} + 23\beta_{2} + 135\beta _1 + 49 \)
|
| \(\nu^{6}\) | \(=\) |
\( 23\beta_{7} - 106\beta_{6} + 135\beta_{5} + 43\beta_{4} - 81\beta_{3} + 97\beta_{2} + 454\beta _1 + 233 \)
|
| \(\nu^{7}\) | \(=\) |
\( 97\beta_{7} - 294\beta_{6} + 454\beta_{5} + 203\beta_{4} - 367\beta_{3} + 433\beta_{2} + 1756\beta _1 + 621 \)
|
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 |
|
−1.00000 | −2.74399 | 1.00000 | −2.12596 | 2.74399 | −0.877808 | −1.00000 | 4.52948 | 2.12596 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −2.33202 | 1.00000 | −3.95006 | 2.33202 | −0.471110 | −1.00000 | 2.43832 | 3.95006 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | 0.274535 | 1.00000 | 0.892569 | −0.274535 | 2.55547 | −1.00000 | −2.92463 | −0.892569 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 0.629487 | 1.00000 | 1.24752 | −0.629487 | −2.54316 | −1.00000 | −2.60375 | −1.24752 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 0.695383 | 1.00000 | −0.922651 | −0.695383 | 0.361679 | −1.00000 | −2.51644 | 0.922651 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 1.96012 | 1.00000 | 0.342083 | −1.96012 | 2.97928 | −1.00000 | 0.842060 | −0.342083 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.00000 | 3.22193 | 1.00000 | 3.83997 | −3.22193 | −1.13450 | −1.00000 | 7.38086 | −3.83997 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −1.00000 | 3.29456 | 1.00000 | 1.67652 | −3.29456 | −4.86985 | −1.00000 | 7.85409 | −1.67652 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(11\) | \(1\) |
| \(19\) | \(-1\) |
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 | 7942.2.a.bo | ✓ | 8 |
| 19.b | odd | 2 | 1 | 7942.2.a.bp | yes | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7942.2.a.bo | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 7942.2.a.bp | yes | 8 | 19.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7942))\):
|
\( T_{3}^{8} - 5T_{3}^{7} - 7T_{3}^{6} + 62T_{3}^{5} - 36T_{3}^{4} - 167T_{3}^{3} + 243T_{3}^{2} - 112T_{3} + 16 \)
|
|
\( T_{5}^{8} - T_{5}^{7} - 20T_{5}^{6} + 23T_{5}^{5} + 74T_{5}^{4} - 106T_{5}^{3} - 27T_{5}^{2} + 74T_{5} - 19 \)
|
|
\( T_{13}^{8} - 3T_{13}^{7} - 72T_{13}^{6} + 241T_{13}^{5} + 1342T_{13}^{4} - 4374T_{13}^{3} - 7689T_{13}^{2} + 23004T_{13} + 2911 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{8} \)
|
| $3$ |
\( T^{8} - 5 T^{7} + \cdots + 16 \)
|
| $5$ |
\( T^{8} - T^{7} + \cdots - 19 \)
|
| $7$ |
\( T^{8} + 4 T^{7} + \cdots - 16 \)
|
| $11$ |
\( (T + 1)^{8} \)
|
| $13$ |
\( T^{8} - 3 T^{7} + \cdots + 2911 \)
|
| $17$ |
\( T^{8} + 6 T^{7} + \cdots + 241 \)
|
| $19$ |
\( T^{8} \)
|
| $23$ |
\( T^{8} - 9 T^{7} + \cdots + 27920 \)
|
| $29$ |
\( T^{8} - 4 T^{7} + \cdots - 87169 \)
|
| $31$ |
\( T^{8} - 11 T^{7} + \cdots + 8720 \)
|
| $37$ |
\( T^{8} + 10 T^{7} + \cdots + 12269 \)
|
| $41$ |
\( T^{8} - 29 T^{7} + \cdots - 1301 \)
|
| $43$ |
\( T^{8} - 2 T^{7} + \cdots + 16624 \)
|
| $47$ |
\( T^{8} - 259 T^{6} + \cdots - 15280 \)
|
| $53$ |
\( T^{8} - 59 T^{7} + \cdots + 10843831 \)
|
| $59$ |
\( T^{8} - 23 T^{7} + \cdots - 5134144 \)
|
| $61$ |
\( T^{8} + 2 T^{7} + \cdots + 37902131 \)
|
| $67$ |
\( T^{8} + 3 T^{7} + \cdots + 281264 \)
|
| $71$ |
\( T^{8} - T^{7} + \cdots + 8315920 \)
|
| $73$ |
\( T^{8} + 6 T^{7} + \cdots + 58388219 \)
|
| $79$ |
\( T^{8} - 410 T^{6} + \cdots - 9246464 \)
|
| $83$ |
\( T^{8} - 15 T^{7} + \cdots + 5294656 \)
|
| $89$ |
\( T^{8} - 4 T^{7} + \cdots - 24631 \)
|
| $97$ |
\( T^{8} - 62 T^{7} + \cdots + 53342671 \)
|
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