Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [712,2,Mod(1,712)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(712, 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("712.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 712 = 2^{3} \cdot 89 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 712.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: | \(5.68534862392\) |
| 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: |
\( x^{7} - 3x^{6} - 14x^{5} + 44x^{4} + 37x^{3} - 163x^{2} + 84x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 3x^{6} - 14x^{5} + 44x^{4} + 37x^{3} - 163x^{2} + 84x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -3\nu^{6} + 11\nu^{5} + 10\nu^{4} - 77\nu^{3} + 113\nu^{2} - 18\nu - 55 ) / 37 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -5\nu^{6} + 6\nu^{5} + 66\nu^{4} - 42\nu^{3} - 194\nu^{2} + 7\nu + 7 ) / 37 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{6} + 16\nu^{5} - 46\nu^{4} - 75\nu^{3} + 383\nu^{2} - 339\nu + 105 ) / 37 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -6\nu^{6} + 22\nu^{5} + 57\nu^{4} - 228\nu^{3} - 70\nu^{2} + 482\nu - 110 ) / 37 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 8\nu^{6} - 17\nu^{5} - 113\nu^{4} + 193\nu^{3} + 414\nu^{2} - 507\nu - 137 ) / 37 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + \beta_{5} + \beta_{3} - \beta_{2} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + \beta_{5} + \beta_{4} + 2\beta_{3} - 3\beta_{2} + 8\beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 10\beta_{6} + 11\beta_{5} + 2\beta_{4} + 12\beta_{3} - 16\beta_{2} + 2\beta _1 + 38 \)
|
| \(\nu^{5}\) | \(=\) |
\( 16\beta_{6} + 20\beta_{5} + 15\beta_{4} + 28\beta_{3} - 49\beta_{2} + 67\beta _1 - 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( 104\beta_{6} + 122\beta_{5} + 36\beta_{4} + 129\beta_{3} - 206\beta_{2} + 41\beta _1 + 315 \)
|
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 | −2.78359 | 0 | −3.30155 | 0 | −1.75917 | 0 | 4.74839 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.70405 | 0 | 2.81558 | 0 | 4.08028 | 0 | 4.31186 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.0438523 | 0 | −3.54735 | 0 | −0.459103 | 0 | −2.99808 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0.772908 | 0 | 3.79951 | 0 | 0.162237 | 0 | −2.40261 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 1.81284 | 0 | 0.237463 | 0 | 2.13194 | 0 | 0.286399 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 2.53805 | 0 | 0.356505 | 0 | 4.78031 | 0 | 3.44171 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 3.40769 | 0 | 2.63984 | 0 | −2.93649 | 0 | 8.61233 | 0 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(89\) | \(-1\) |
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 | 712.2.a.e | ✓ | 7 |
| 3.b | odd | 2 | 1 | 6408.2.a.m | 7 | ||
| 4.b | odd | 2 | 1 | 1424.2.a.m | 7 | ||
| 8.b | even | 2 | 1 | 5696.2.a.bi | 7 | ||
| 8.d | odd | 2 | 1 | 5696.2.a.bl | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 712.2.a.e | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 1424.2.a.m | 7 | 4.b | odd | 2 | 1 | ||
| 5696.2.a.bi | 7 | 8.b | even | 2 | 1 | ||
| 5696.2.a.bl | 7 | 8.d | odd | 2 | 1 | ||
| 6408.2.a.m | 7 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{7} - 3T_{3}^{6} - 14T_{3}^{5} + 44T_{3}^{4} + 37T_{3}^{3} - 163T_{3}^{2} + 84T_{3} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(712))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} \)
|
| $3$ |
\( T^{7} - 3 T^{6} + \cdots + 4 \)
|
| $5$ |
\( T^{7} - 3 T^{6} + \cdots - 28 \)
|
| $7$ |
\( T^{7} - 6 T^{6} + \cdots + 16 \)
|
| $11$ |
\( T^{7} + 2 T^{6} + \cdots - 6400 \)
|
| $13$ |
\( T^{7} - 4 T^{6} + \cdots - 448 \)
|
| $17$ |
\( T^{7} - 7 T^{6} + \cdots - 140 \)
|
| $19$ |
\( T^{7} + 19 T^{6} + \cdots + 21364 \)
|
| $23$ |
\( T^{7} - 9 T^{6} + \cdots + 8084 \)
|
| $29$ |
\( T^{7} - 26 T^{6} + \cdots + 19904 \)
|
| $31$ |
\( T^{7} - T^{6} + \cdots + 73252 \)
|
| $37$ |
\( T^{7} - 10 T^{6} + \cdots - 10880 \)
|
| $41$ |
\( T^{7} - 14 T^{6} + \cdots + 1277248 \)
|
| $43$ |
\( T^{7} + 27 T^{6} + \cdots + 940 \)
|
| $47$ |
\( T^{7} - 8 T^{6} + \cdots - 66304 \)
|
| $53$ |
\( T^{7} - 25 T^{6} + \cdots + 24716 \)
|
| $59$ |
\( T^{7} + 10 T^{6} + \cdots - 7984 \)
|
| $61$ |
\( T^{7} - 4 T^{6} + \cdots + 727744 \)
|
| $67$ |
\( T^{7} + 4 T^{6} + \cdots - 412928 \)
|
| $71$ |
\( T^{7} + 2 T^{6} + \cdots + 1403392 \)
|
| $73$ |
\( T^{7} + 5 T^{6} + \cdots - 16507916 \)
|
| $79$ |
\( T^{7} + 18 T^{6} + \cdots + 2716160 \)
|
| $83$ |
\( T^{7} + 10 T^{6} + \cdots + 18800 \)
|
| $89$ |
\( (T - 1)^{7} \)
|
| $97$ |
\( T^{7} + 31 T^{6} + \cdots + 661948 \)
|
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