Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [706,2,Mod(1,706)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(706, 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("706.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 706 = 2 \cdot 353 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 706.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: | \(5.63743838270\) |
| 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: |
\( x^{10} - 2x^{9} - 23x^{8} + 40x^{7} + 164x^{6} - 186x^{5} - 510x^{4} + 200x^{3} + 579x^{2} + 108x - 27 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 2x^{9} - 23x^{8} + 40x^{7} + 164x^{6} - 186x^{5} - 510x^{4} + 200x^{3} + 579x^{2} + 108x - 27 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 1373 \nu^{9} - 7954 \nu^{8} - 13810 \nu^{7} + 149396 \nu^{6} - 112496 \nu^{5} - 631680 \nu^{4} + \cdots - 518760 ) / 80604 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 1537 \nu^{9} + 5186 \nu^{8} + 28238 \nu^{7} - 105736 \nu^{6} - 113462 \nu^{5} + 541746 \nu^{4} + \cdots + 178614 ) / 80604 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 1183 \nu^{9} - 4280 \nu^{8} - 22652 \nu^{7} + 87427 \nu^{6} + 104924 \nu^{5} - 446877 \nu^{4} + \cdots - 130383 ) / 40302 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 1002 \nu^{9} - 2147 \nu^{8} - 21200 \nu^{7} + 39718 \nu^{6} + 130804 \nu^{5} - 145762 \nu^{4} + \cdots + 107343 ) / 26868 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 1862 \nu^{9} + 5107 \nu^{8} + 39628 \nu^{7} - 106208 \nu^{6} - 233668 \nu^{5} + 560364 \nu^{4} + \cdots + 146673 ) / 40302 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 4925 \nu^{9} - 15337 \nu^{8} - 99232 \nu^{7} + 303734 \nu^{6} + 533986 \nu^{5} - 1431258 \nu^{4} + \cdots + 40707 ) / 80604 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 1736 \nu^{9} - 5923 \nu^{8} - 31492 \nu^{7} + 112556 \nu^{6} + 125434 \nu^{5} - 473658 \nu^{4} + \cdots + 68247 ) / 26868 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 2327 \nu^{9} - 7853 \nu^{8} - 42146 \nu^{7} + 151870 \nu^{6} + 160674 \nu^{5} - 670844 \nu^{4} + \cdots - 78135 ) / 26868 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{8} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{9} - 4\beta_{8} + 4\beta_{7} + 3\beta_{6} - 2\beta_{4} - 2\beta_{3} + 10\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 3 \beta_{9} - 13 \beta_{8} - 2 \beta_{7} + 2 \beta_{6} + 12 \beta_{5} + 17 \beta_{4} + 16 \beta_{3} + \cdots + 65 \)
|
| \(\nu^{5}\) | \(=\) |
\( 29\beta_{9} - 59\beta_{8} + 62\beta_{7} + 50\beta_{6} - 29\beta_{4} - 38\beta_{3} - 3\beta_{2} + 118\beta _1 - 8 \)
|
| \(\nu^{6}\) | \(=\) |
\( 47 \beta_{9} - 156 \beta_{8} - 46 \beta_{7} + 30 \beta_{6} + 147 \beta_{5} + 242 \beta_{4} + 207 \beta_{3} + \cdots + 776 \)
|
| \(\nu^{7}\) | \(=\) |
\( 379 \beta_{9} - 741 \beta_{8} + 812 \beta_{7} + 687 \beta_{6} - 24 \beta_{5} - 401 \beta_{4} + \cdots - 371 \)
|
| \(\nu^{8}\) | \(=\) |
\( 592 \beta_{9} - 1814 \beta_{8} - 848 \beta_{7} + 318 \beta_{6} + 1854 \beta_{5} + 3262 \beta_{4} + \cdots + 9565 \)
|
| \(\nu^{9}\) | \(=\) |
\( 4834 \beta_{9} - 9004 \beta_{8} + 10320 \beta_{7} + 8930 \beta_{6} - 674 \beta_{5} - 5672 \beta_{4} + \cdots - 7848 \)
|
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 | −3.42546 | 1.00000 | 3.97309 | −3.42546 | −2.56329 | 1.00000 | 8.73380 | 3.97309 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −2.76155 | 1.00000 | −3.87788 | −2.76155 | −4.40971 | 1.00000 | 4.62617 | −3.87788 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −2.06228 | 1.00000 | 0.825583 | −2.06228 | 0.208846 | 1.00000 | 1.25301 | 0.825583 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | −1.51381 | 1.00000 | −2.49680 | −1.51381 | 2.67678 | 1.00000 | −0.708394 | −2.49680 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 0.0202539 | 1.00000 | 3.34822 | 0.0202539 | −4.10004 | 1.00000 | −2.99959 | 3.34822 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 1.13476 | 1.00000 | 0.988722 | 1.13476 | 0.510966 | 1.00000 | −1.71232 | 0.988722 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | 2.10467 | 1.00000 | 3.35597 | 2.10467 | −1.14068 | 1.00000 | 1.42963 | 3.35597 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.00000 | 2.56715 | 1.00000 | −0.924826 | 2.56715 | 0.791263 | 1.00000 | 3.59027 | −0.924826 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.00000 | 2.67850 | 1.00000 | −3.60127 | 2.67850 | −0.594295 | 1.00000 | 4.17436 | −3.60127 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.00000 | 3.25777 | 1.00000 | 1.40919 | 3.25777 | −3.37985 | 1.00000 | 7.61305 | 1.40919 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(353\) | \( +1 \) |
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 | 706.2.a.h | ✓ | 10 |
| 3.b | odd | 2 | 1 | 6354.2.a.bk | 10 | ||
| 4.b | odd | 2 | 1 | 5648.2.a.m | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 706.2.a.h | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 5648.2.a.m | 10 | 4.b | odd | 2 | 1 | ||
| 6354.2.a.bk | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(706))\):
|
\( T_{3}^{10} - 2 T_{3}^{9} - 26 T_{3}^{8} + 52 T_{3}^{7} + 230 T_{3}^{6} - 450 T_{3}^{5} - 812 T_{3}^{4} + \cdots + 32 \)
|
|
\( T_{5}^{10} - 3 T_{5}^{9} - 34 T_{5}^{8} + 104 T_{5}^{7} + 357 T_{5}^{6} - 1161 T_{5}^{5} - 1002 T_{5}^{4} + \cdots + 1656 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{10} \)
|
| $3$ |
\( T^{10} - 2 T^{9} + \cdots + 32 \)
|
| $5$ |
\( T^{10} - 3 T^{9} + \cdots + 1656 \)
|
| $7$ |
\( T^{10} + 12 T^{9} + \cdots + 24 \)
|
| $11$ |
\( T^{10} - 15 T^{9} + \cdots - 38208 \)
|
| $13$ |
\( T^{10} - 11 T^{9} + \cdots + 67784 \)
|
| $17$ |
\( T^{10} - 7 T^{9} + \cdots + 149648 \)
|
| $19$ |
\( T^{10} - 4 T^{9} + \cdots - 50944 \)
|
| $23$ |
\( T^{10} - 10 T^{9} + \cdots + 6144 \)
|
| $29$ |
\( T^{10} - 14 T^{9} + \cdots - 7006784 \)
|
| $31$ |
\( T^{10} + T^{9} + \cdots - 8354 \)
|
| $37$ |
\( T^{10} - 9 T^{9} + \cdots + 61362776 \)
|
| $41$ |
\( T^{10} - 7 T^{9} + \cdots + 1989004 \)
|
| $43$ |
\( T^{10} - 19 T^{9} + \cdots - 1906832 \)
|
| $47$ |
\( T^{10} + 22 T^{9} + \cdots + 1718784 \)
|
| $53$ |
\( T^{10} + 2 T^{9} + \cdots - 11193568 \)
|
| $59$ |
\( T^{10} + 6 T^{9} + \cdots - 318112 \)
|
| $61$ |
\( T^{10} - 6 T^{9} + \cdots - 8692928 \)
|
| $67$ |
\( T^{10} + 20 T^{9} + \cdots - 3331264 \)
|
| $71$ |
\( T^{10} + 11 T^{9} + \cdots + 43346 \)
|
| $73$ |
\( T^{10} + \cdots + 412620132 \)
|
| $79$ |
\( T^{10} + 10 T^{9} + \cdots + 599288 \)
|
| $83$ |
\( T^{10} + 7 T^{9} + \cdots + 19263632 \)
|
| $89$ |
\( T^{10} - 18 T^{9} + \cdots + 8676352 \)
|
| $97$ |
\( T^{10} + \cdots + 109331504 \)
|
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