Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8379,2,Mod(1,8379)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8379, 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("8379.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8379 = 3^{2} \cdot 7^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8379.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: | \(66.9066518536\) |
| 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: |
\( x^{10} - 2x^{9} - 13x^{8} + 24x^{7} + 57x^{6} - 98x^{5} - 93x^{4} + 152x^{3} + 39x^{2} - 58x - 7 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 931) |
| 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
\( x^{10} - 2x^{9} - 13x^{8} + 24x^{7} + 57x^{6} - 98x^{5} - 93x^{4} + 152x^{3} + 39x^{2} - 58x - 7 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 2 \nu^{9} - 91 \nu^{8} + 169 \nu^{7} + 995 \nu^{6} - 1285 \nu^{5} - 3248 \nu^{4} + 2542 \nu^{3} + \cdots - 574 ) / 229 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 27 \nu^{9} - 31 \nu^{8} - 335 \nu^{7} + 193 \nu^{6} + 1432 \nu^{5} + 109 \nu^{4} - 2486 \nu^{3} + \cdots + 879 ) / 229 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 47 \nu^{9} - 37 \nu^{8} - 651 \nu^{7} + 319 \nu^{6} + 3061 \nu^{5} - 845 \nu^{4} - 5464 \nu^{3} + \cdots - 22 ) / 229 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 57 \nu^{9} + 40 \nu^{8} + 809 \nu^{7} - 382 \nu^{6} - 3761 \nu^{5} + 1093 \nu^{4} + 6266 \nu^{3} + \cdots - 329 ) / 229 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 57 \nu^{9} - 40 \nu^{8} - 809 \nu^{7} + 382 \nu^{6} + 3761 \nu^{5} - 1093 \nu^{4} - 6037 \nu^{3} + \cdots + 787 ) / 229 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 67 \nu^{9} - 43 \nu^{8} - 967 \nu^{7} + 445 \nu^{6} + 4690 \nu^{5} - 1570 \nu^{4} - 8671 \nu^{3} + \cdots + 222 ) / 229 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 81 \nu^{9} + 93 \nu^{8} + 1005 \nu^{7} - 808 \nu^{6} - 4067 \nu^{5} + 1963 \nu^{4} + 5855 \nu^{3} + \cdots - 576 ) / 229 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + \beta_{6} + \beta_{2} + 4\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{8} + \beta_{7} + \beta_{6} - 2\beta_{5} + \beta_{4} + 7\beta_{2} + \beta _1 + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{8} + 8\beta_{7} + 10\beta_{6} - \beta_{5} + \beta_{4} + 10\beta_{2} + 20\beta _1 + 11 \)
|
| \(\nu^{6}\) | \(=\) |
\( -\beta_{9} + 12\beta_{8} + 11\beta_{7} + 13\beta_{6} - 21\beta_{5} + 8\beta_{4} + 48\beta_{2} + 13\beta _1 + 78 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 4 \beta_{9} + 26 \beta_{8} + 56 \beta_{7} + 81 \beta_{6} - 20 \beta_{5} + 11 \beta_{4} - \beta_{3} + \cdots + 97 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 17 \beta_{9} + 110 \beta_{8} + 95 \beta_{7} + 130 \beta_{6} - 176 \beta_{5} + 56 \beta_{4} + \cdots + 480 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 62 \beta_{9} + 253 \beta_{8} + 389 \beta_{7} + 619 \beta_{6} - 239 \beta_{5} + 95 \beta_{4} + \cdots + 788 \)
|
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 |
|
−2.27289 | 0 | 3.16604 | 1.72209 | 0 | 0 | −2.65028 | 0 | −3.91413 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.94026 | 0 | 1.76460 | −0.683496 | 0 | 0 | 0.456739 | 0 | 1.32616 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.74841 | 0 | 1.05694 | 4.09480 | 0 | 0 | 1.64885 | 0 | −7.15939 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.686476 | 0 | −1.52875 | −0.737311 | 0 | 0 | 2.42240 | 0 | 0.506146 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.115985 | 0 | −1.98655 | 0.455421 | 0 | 0 | 0.462382 | 0 | −0.0528223 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.897685 | 0 | −1.19416 | 1.27247 | 0 | 0 | −2.86735 | 0 | 1.14227 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.31483 | 0 | −0.271216 | 4.02523 | 0 | 0 | −2.98627 | 0 | 5.29250 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.58984 | 0 | 0.527606 | −0.921074 | 0 | 0 | −2.34088 | 0 | −1.46436 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.20114 | 0 | 2.84501 | 4.23731 | 0 | 0 | 1.86000 | 0 | 9.32690 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.76052 | 0 | 5.62048 | 2.53457 | 0 | 0 | 9.99440 | 0 | 6.99673 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(7\) | \(1\) |
| \(19\) | \(1\) |
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 | 8379.2.a.ct | 10 | |
| 3.b | odd | 2 | 1 | 931.2.a.p | ✓ | 10 | |
| 7.b | odd | 2 | 1 | 8379.2.a.cs | 10 | ||
| 21.c | even | 2 | 1 | 931.2.a.q | yes | 10 | |
| 21.g | even | 6 | 2 | 931.2.f.q | 20 | ||
| 21.h | odd | 6 | 2 | 931.2.f.r | 20 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 931.2.a.p | ✓ | 10 | 3.b | odd | 2 | 1 | |
| 931.2.a.q | yes | 10 | 21.c | even | 2 | 1 | |
| 931.2.f.q | 20 | 21.g | even | 6 | 2 | ||
| 931.2.f.r | 20 | 21.h | odd | 6 | 2 | ||
| 8379.2.a.cs | 10 | 7.b | odd | 2 | 1 | ||
| 8379.2.a.ct | 10 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8379))\):
|
\( T_{2}^{10} - 2T_{2}^{9} - 13T_{2}^{8} + 24T_{2}^{7} + 57T_{2}^{6} - 98T_{2}^{5} - 93T_{2}^{4} + 152T_{2}^{3} + 39T_{2}^{2} - 58T_{2} - 7 \)
|
|
\( T_{5}^{10} - 16 T_{5}^{9} + 96 T_{5}^{8} - 248 T_{5}^{7} + 154 T_{5}^{6} + 432 T_{5}^{5} - 578 T_{5}^{4} + \cdots - 82 \)
|
|
\( T_{11}^{10} - 62 T_{11}^{8} + 1279 T_{11}^{6} + 400 T_{11}^{5} - 10770 T_{11}^{4} - 8016 T_{11}^{3} + \cdots + 6368 \)
|
|
\( T_{13}^{10} + 12 T_{13}^{9} - 4 T_{13}^{8} - 512 T_{13}^{7} - 1564 T_{13}^{6} + 2088 T_{13}^{5} + \cdots + 1022 \)
|
|
\( T_{17}^{10} - 16 T_{17}^{9} + 34 T_{17}^{8} + 632 T_{17}^{7} - 3745 T_{17}^{6} + 2376 T_{17}^{5} + \cdots + 37646 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} - 2 T^{9} + \cdots - 7 \)
|
| $3$ |
\( T^{10} \)
|
| $5$ |
\( T^{10} - 16 T^{9} + \cdots - 82 \)
|
| $7$ |
\( T^{10} \)
|
| $11$ |
\( T^{10} - 62 T^{8} + \cdots + 6368 \)
|
| $13$ |
\( T^{10} + 12 T^{9} + \cdots + 1022 \)
|
| $17$ |
\( T^{10} - 16 T^{9} + \cdots + 37646 \)
|
| $19$ |
\( (T + 1)^{10} \)
|
| $23$ |
\( T^{10} + 12 T^{9} + \cdots - 917392 \)
|
| $29$ |
\( T^{10} - 12 T^{9} + \cdots + 168952 \)
|
| $31$ |
\( T^{10} + 8 T^{9} + \cdots - 281848 \)
|
| $37$ |
\( T^{10} - 4 T^{9} + \cdots - 792136 \)
|
| $41$ |
\( T^{10} - 40 T^{9} + \cdots - 10740226 \)
|
| $43$ |
\( T^{10} - 4 T^{9} + \cdots - 1282048 \)
|
| $47$ |
\( T^{10} - 16 T^{9} + \cdots - 4289848 \)
|
| $53$ |
\( T^{10} + \cdots - 288453476 \)
|
| $59$ |
\( T^{10} - 36 T^{9} + \cdots - 1952528 \)
|
| $61$ |
\( T^{10} + \cdots + 238217966 \)
|
| $67$ |
\( T^{10} + 28 T^{9} + \cdots + 1231484 \)
|
| $71$ |
\( T^{10} - 12 T^{9} + \cdots + 7611772 \)
|
| $73$ |
\( T^{10} - 352 T^{8} + \cdots + 30259838 \)
|
| $79$ |
\( T^{10} + \cdots - 358185536 \)
|
| $83$ |
\( T^{10} - 254 T^{8} + \cdots + 328888 \)
|
| $89$ |
\( T^{10} + \cdots + 457135072 \)
|
| $97$ |
\( T^{10} + \cdots - 223920578 \)
|
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