Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6162,2,Mod(1,6162)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6162, 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("6162.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6162 = 2 \cdot 3 \cdot 13 \cdot 79 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6162.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: | \(49.2038177255\) |
| 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: |
\( x^{10} - 4x^{9} - 22x^{8} + 84x^{7} + 149x^{6} - 596x^{5} - 300x^{4} + 1684x^{3} - 228x^{2} - 1520x + 688 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 4x^{9} - 22x^{8} + 84x^{7} + 149x^{6} - 596x^{5} - 300x^{4} + 1684x^{3} - 228x^{2} - 1520x + 688 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 5 \nu^{9} + 39 \nu^{8} - 458 \nu^{7} - 225 \nu^{6} + 6621 \nu^{5} - 3966 \nu^{4} - 28004 \nu^{3} + \cdots - 14416 ) / 3592 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 31 \nu^{9} + 297 \nu^{8} - 483 \nu^{7} - 3993 \nu^{6} + 15434 \nu^{5} + 5372 \nu^{4} + \cdots - 103152 ) / 7184 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 46 \nu^{9} + 180 \nu^{8} + 891 \nu^{7} - 3318 \nu^{6} - 4429 \nu^{5} + 19066 \nu^{4} + \cdots + 15528 ) / 3592 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 109 \nu^{9} + 407 \nu^{8} + 2531 \nu^{7} - 7667 \nu^{6} - 21222 \nu^{5} + 42816 \nu^{4} + \cdots + 32656 ) / 7184 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 28 \nu^{9} - 51 \nu^{8} - 679 \nu^{7} + 536 \nu^{6} + 5019 \nu^{5} - 29 \nu^{4} - 12424 \nu^{3} + \cdots + 8352 ) / 1796 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 423 \nu^{9} + 1011 \nu^{8} + 10819 \nu^{7} - 17783 \nu^{6} - 88956 \nu^{5} + 101864 \nu^{4} + \cdots + 153488 ) / 7184 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 561 \nu^{9} + 1551 \nu^{8} + 13941 \nu^{7} - 28635 \nu^{6} - 113468 \nu^{5} + 174328 \nu^{4} + \cdots + 304240 ) / 7184 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 591 \nu^{9} - 1317 \nu^{8} - 15791 \nu^{7} + 23693 \nu^{6} + 137928 \nu^{5} - 144244 \nu^{4} + \cdots - 304528 ) / 7184 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{9} + \beta_{8} - \beta_{6} - \beta_{4} - \beta_{2} + \beta _1 + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{9} + 2\beta_{8} + 2\beta_{7} - \beta_{6} - 3\beta_{4} - \beta_{3} + 10\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( 21\beta_{9} + 20\beta_{8} + 2\beta_{7} - 19\beta_{6} - 19\beta_{4} - 5\beta_{3} - 12\beta_{2} + 22\beta _1 + 51 \)
|
| \(\nu^{5}\) | \(=\) |
\( 85 \beta_{9} + 58 \beta_{8} + 46 \beta_{7} - 45 \beta_{6} + 6 \beta_{5} - 69 \beta_{4} - 37 \beta_{3} + \cdots + 97 \)
|
| \(\nu^{6}\) | \(=\) |
\( 441 \beta_{9} + 380 \beta_{8} + 98 \beta_{7} - 363 \beta_{6} + 24 \beta_{5} - 367 \beta_{4} - 161 \beta_{3} + \cdots + 767 \)
|
| \(\nu^{7}\) | \(=\) |
\( 1945 \beta_{9} + 1374 \beta_{8} + 910 \beta_{7} - 1209 \beta_{6} + 198 \beta_{5} - 1489 \beta_{4} + \cdots + 2365 \)
|
| \(\nu^{8}\) | \(=\) |
\( 9305 \beta_{9} + 7496 \beta_{8} + 2786 \beta_{7} - 7163 \beta_{6} + 832 \beta_{5} - 7359 \beta_{4} + \cdots + 13903 \)
|
| \(\nu^{9}\) | \(=\) |
\( 42057 \beta_{9} + 30478 \beta_{8} + 17910 \beta_{7} - 28017 \beta_{6} + 4782 \beta_{5} - 31737 \beta_{4} + \cdots + 53029 \)
|
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 | −1.00000 | 1.00000 | −3.58357 | −1.00000 | 0.726343 | 1.00000 | 1.00000 | −3.58357 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −1.00000 | 1.00000 | −1.77603 | −1.00000 | 2.72804 | 1.00000 | 1.00000 | −1.77603 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −1.00000 | 1.00000 | −1.13406 | −1.00000 | −2.86806 | 1.00000 | 1.00000 | −1.13406 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | −1.00000 | 1.00000 | −0.592598 | −1.00000 | 4.36026 | 1.00000 | 1.00000 | −0.592598 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | −1.00000 | 1.00000 | −0.340195 | −1.00000 | −0.901495 | 1.00000 | 1.00000 | −0.340195 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | −1.00000 | 1.00000 | 0.432720 | −1.00000 | −1.69263 | 1.00000 | 1.00000 | 0.432720 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | −1.00000 | 1.00000 | 2.30197 | −1.00000 | −1.48369 | 1.00000 | 1.00000 | 2.30197 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.00000 | −1.00000 | 1.00000 | 3.14136 | −1.00000 | 3.30195 | 1.00000 | 1.00000 | 3.14136 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.00000 | −1.00000 | 1.00000 | 3.33245 | −1.00000 | 3.17797 | 1.00000 | 1.00000 | 3.33245 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.00000 | −1.00000 | 1.00000 | 4.21795 | −1.00000 | −4.34870 | 1.00000 | 1.00000 | 4.21795 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(1\) |
| \(13\) | \(1\) |
| \(79\) | \(1\) |
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 | 6162.2.a.bj | ✓ | 10 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 6162.2.a.bj | ✓ | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6162))\):
|
\( T_{5}^{10} - 6 T_{5}^{9} - 13 T_{5}^{8} + 116 T_{5}^{7} - 5 T_{5}^{6} - 578 T_{5}^{5} + 61 T_{5}^{4} + \cdots - 64 \)
|
|
\( T_{7}^{10} - 3 T_{7}^{9} - 36 T_{7}^{8} + 98 T_{7}^{7} + 429 T_{7}^{6} - 923 T_{7}^{5} - 2290 T_{7}^{4} + \cdots - 2560 \)
|
|
\( T_{11}^{10} - 3 T_{11}^{9} - 59 T_{11}^{8} + 196 T_{11}^{7} + 919 T_{11}^{6} - 3591 T_{11}^{5} + \cdots + 4096 \)
|
|
\( T_{17}^{10} - 12 T_{17}^{9} - 64 T_{17}^{8} + 1135 T_{17}^{7} + 56 T_{17}^{6} - 34168 T_{17}^{5} + \cdots + 1576192 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{10} \)
|
| $3$ |
\( (T + 1)^{10} \)
|
| $5$ |
\( T^{10} - 6 T^{9} + \cdots - 64 \)
|
| $7$ |
\( T^{10} - 3 T^{9} + \cdots - 2560 \)
|
| $11$ |
\( T^{10} - 3 T^{9} + \cdots + 4096 \)
|
| $13$ |
\( (T + 1)^{10} \)
|
| $17$ |
\( T^{10} - 12 T^{9} + \cdots + 1576192 \)
|
| $19$ |
\( T^{10} - 17 T^{9} + \cdots - 584576 \)
|
| $23$ |
\( T^{10} - 3 T^{9} + \cdots + 40 \)
|
| $29$ |
\( T^{10} - 6 T^{9} + \cdots - 95488 \)
|
| $31$ |
\( T^{10} - 9 T^{9} + \cdots + 595008 \)
|
| $37$ |
\( T^{10} - 6 T^{9} + \cdots + 441344 \)
|
| $41$ |
\( T^{10} - 28 T^{9} + \cdots - 20838784 \)
|
| $43$ |
\( T^{10} + \cdots - 140349184 \)
|
| $47$ |
\( T^{10} - 13 T^{9} + \cdots + 75520 \)
|
| $53$ |
\( T^{10} - 18 T^{9} + \cdots + 1441152 \)
|
| $59$ |
\( T^{10} - 9 T^{9} + \cdots + 85000192 \)
|
| $61$ |
\( T^{10} - 16 T^{9} + \cdots - 3034624 \)
|
| $67$ |
\( T^{10} - 19 T^{9} + \cdots - 63670536 \)
|
| $71$ |
\( T^{10} + 11 T^{9} + \cdots + 328688 \)
|
| $73$ |
\( T^{10} - 20 T^{9} + \cdots + 30947584 \)
|
| $79$ |
\( (T + 1)^{10} \)
|
| $83$ |
\( T^{10} + \cdots + 460929600 \)
|
| $89$ |
\( T^{10} - 10 T^{9} + \cdots + 2878264 \)
|
| $97$ |
\( T^{10} - 18 T^{9} + \cdots + 4194304 \)
|
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