Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [766,2,Mod(1,766)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(766, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("766.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 766 = 2 \cdot 383 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 766.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: | \(6.11654079483\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 5x^{9} - 14x^{8} + 67x^{7} + 83x^{6} - 258x^{5} - 245x^{4} + 188x^{3} + 137x^{2} + 6x - 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 5x^{9} - 14x^{8} + 67x^{7} + 83x^{6} - 258x^{5} - 245x^{4} + 188x^{3} + 137x^{2} + 6x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 5693 \nu^{9} + 36810 \nu^{8} + 81149 \nu^{7} - 615450 \nu^{6} - 698938 \nu^{5} + 2934605 \nu^{4} + \cdots - 417977 ) / 202061 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 10383 \nu^{9} + 63479 \nu^{8} + 75986 \nu^{7} - 799342 \nu^{6} + 48829 \nu^{5} + 2920215 \nu^{4} + \cdots + 116880 ) / 202061 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 11836 \nu^{9} - 44728 \nu^{8} - 222981 \nu^{7} + 575582 \nu^{6} + 1568688 \nu^{5} - 1871416 \nu^{4} + \cdots + 463664 ) / 202061 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 14895 \nu^{9} - 94392 \nu^{8} - 129440 \nu^{7} + 1303516 \nu^{6} + 359879 \nu^{5} - 5357384 \nu^{4} + \cdots - 805322 ) / 202061 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 18278 \nu^{9} + 90924 \nu^{8} + 249322 \nu^{7} - 1228241 \nu^{6} - 1246170 \nu^{5} + \cdots + 564220 ) / 202061 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 30122 \nu^{9} + 146813 \nu^{8} + 440154 \nu^{7} - 1996030 \nu^{6} - 2622120 \nu^{5} + \cdots + 300255 ) / 202061 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 60650 \nu^{9} - 304379 \nu^{8} - 814719 \nu^{7} + 3997122 \nu^{6} + 4454501 \nu^{5} - 15170117 \nu^{4} + \cdots - 177547 ) / 202061 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 63981 \nu^{9} - 329395 \nu^{8} - 840478 \nu^{7} + 4390020 \nu^{6} + 4572979 \nu^{5} - 17109850 \nu^{4} + \cdots - 962103 ) / 202061 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{9} + \beta_{8} + 2\beta_{5} + 2\beta_{3} + \beta_{2} + 2\beta _1 + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( -5\beta_{9} + 6\beta_{8} + 3\beta_{7} - 2\beta_{6} + 7\beta_{5} + 8\beta_{3} + 2\beta_{2} + 12\beta _1 + 10 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 33 \beta_{9} + 37 \beta_{8} + 11 \beta_{7} - 8 \beta_{6} + 45 \beta_{5} - 2 \beta_{4} + 49 \beta_{3} + \cdots + 68 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 164 \beta_{9} + 192 \beta_{8} + 73 \beta_{7} - 52 \beta_{6} + 204 \beta_{5} - 15 \beta_{4} + \cdots + 254 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 858 \beta_{9} + 986 \beta_{8} + 332 \beta_{7} - 249 \beta_{6} + 1054 \beta_{5} - 86 \beta_{4} + \cdots + 1314 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 4219 \beta_{9} + 4908 \beta_{8} + 1743 \beta_{7} - 1288 \beta_{6} + 5051 \beta_{5} - 473 \beta_{4} + \cdots + 5928 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 20919 \beta_{9} + 24210 \beta_{8} + 8342 \beta_{7} - 6288 \beta_{6} + 24901 \beta_{5} - 2384 \beta_{4} + \cdots + 29073 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 102113 \beta_{9} + 118594 \beta_{8} + 41382 \beta_{7} - 31026 \beta_{6} + 120618 \beta_{5} + \cdots + 138286 \)
|
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 | −3.20458 | 1.00000 | −0.978397 | 3.20458 | 2.51132 | −1.00000 | 7.26935 | 0.978397 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −3.13386 | 1.00000 | 2.68856 | 3.13386 | −3.82743 | −1.00000 | 6.82111 | −2.68856 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −1.46158 | 1.00000 | −3.83127 | 1.46158 | −4.61324 | −1.00000 | −0.863771 | 3.83127 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | −1.45408 | 1.00000 | −0.895422 | 1.45408 | −4.01437 | −1.00000 | −0.885646 | 0.895422 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | −1.29305 | 1.00000 | 2.52786 | 1.29305 | 2.54939 | −1.00000 | −1.32802 | −2.52786 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 0.330181 | 1.00000 | 3.19842 | −0.330181 | −1.93229 | −1.00000 | −2.89098 | −3.19842 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.00000 | 1.00075 | 1.00000 | −1.67672 | −1.00075 | −0.108385 | −1.00000 | −1.99850 | 1.67672 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −1.00000 | 2.46206 | 1.00000 | 1.91260 | −2.46206 | 3.01500 | −1.00000 | 3.06172 | −1.91260 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −1.00000 | 3.32566 | 1.00000 | −3.73068 | −3.32566 | −2.18027 | −1.00000 | 8.05999 | 3.73068 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | −1.00000 | 3.42852 | 1.00000 | 3.78504 | −3.42852 | −3.39973 | −1.00000 | 8.75475 | −3.78504 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(383\) | \( -1 \) |
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 | 766.2.a.f | ✓ | 10 |
| 3.b | odd | 2 | 1 | 6894.2.a.bb | 10 | ||
| 4.b | odd | 2 | 1 | 6128.2.a.k | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 766.2.a.f | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 6128.2.a.k | 10 | 4.b | odd | 2 | 1 | ||
| 6894.2.a.bb | 10 | 3.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(766))\):
|
\( T_{3}^{10} - 28T_{3}^{8} - 7T_{3}^{7} + 264T_{3}^{6} + 133T_{3}^{5} - 926T_{3}^{4} - 676T_{3}^{3} + 920T_{3}^{2} + 576T_{3} - 256 \)
|
|
\( T_{5}^{10} - 3 T_{5}^{9} - 33 T_{5}^{8} + 103 T_{5}^{7} + 344 T_{5}^{6} - 1115 T_{5}^{5} - 1262 T_{5}^{4} + \cdots - 3304 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{10} \)
|
| $3$ |
\( T^{10} - 28 T^{8} + \cdots - 256 \)
|
| $5$ |
\( T^{10} - 3 T^{9} + \cdots - 3304 \)
|
| $7$ |
\( T^{10} + 12 T^{9} + \cdots - 2124 \)
|
| $11$ |
\( T^{10} - 15 T^{9} + \cdots + 6 \)
|
| $13$ |
\( T^{10} + 7 T^{9} + \cdots - 4966 \)
|
| $17$ |
\( T^{10} - T^{9} + \cdots - 131732 \)
|
| $19$ |
\( T^{10} + 2 T^{9} + \cdots - 732416 \)
|
| $23$ |
\( T^{10} - 3 T^{9} + \cdots - 192 \)
|
| $29$ |
\( T^{10} - 26 T^{9} + \cdots + 93431328 \)
|
| $31$ |
\( T^{10} - 7 T^{9} + \cdots + 532288 \)
|
| $37$ |
\( T^{10} - 5 T^{9} + \cdots + 157262 \)
|
| $41$ |
\( T^{10} - 37 T^{9} + \cdots + 1093344 \)
|
| $43$ |
\( T^{10} + 25 T^{9} + \cdots + 27672576 \)
|
| $47$ |
\( T^{10} + 4 T^{9} + \cdots + 321536 \)
|
| $53$ |
\( T^{10} - 14 T^{9} + \cdots + 85891418 \)
|
| $59$ |
\( T^{10} - 37 T^{9} + \cdots - 37258322 \)
|
| $61$ |
\( T^{10} - 11 T^{9} + \cdots + 6302914 \)
|
| $67$ |
\( T^{10} + \cdots - 188572608 \)
|
| $71$ |
\( T^{10} - 35 T^{9} + \cdots + 347584 \)
|
| $73$ |
\( T^{10} - 12 T^{9} + \cdots + 10160376 \)
|
| $79$ |
\( T^{10} + \cdots + 698537728 \)
|
| $83$ |
\( T^{10} + \cdots + 2070377304 \)
|
| $89$ |
\( T^{10} - 70 T^{9} + \cdots - 175392 \)
|
| $97$ |
\( T^{10} + 7 T^{9} + \cdots - 20391744 \)
|
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