Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6042,2,Mod(1,6042)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6042, 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("6042.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6042 = 2 \cdot 3 \cdot 19 \cdot 53 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6042.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: | \(48.2456129013\) |
| Analytic rank: | \(1\) |
| Dimension: | \(9\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{9} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{9} - 3x^{8} - 20x^{7} + 69x^{6} + 27x^{5} - 185x^{4} + 8x^{3} + 109x^{2} - 8x - 14 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{8}\) 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
\( x^{9} - 3x^{8} - 20x^{7} + 69x^{6} + 27x^{5} - 185x^{4} + 8x^{3} + 109x^{2} - 8x - 14 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 18 \nu^{8} + 96 \nu^{7} - 599 \nu^{6} - 2018 \nu^{5} + 6181 \nu^{4} + 7311 \nu^{3} - 11661 \nu^{2} + \cdots + 4020 ) / 2078 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 133 \nu^{8} + 676 \nu^{7} + 2752 \nu^{6} - 16721 \nu^{5} - 3014 \nu^{4} + 77067 \nu^{3} + \cdots + 774 ) / 8312 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 895 \nu^{8} - 2846 \nu^{7} - 18066 \nu^{6} + 64977 \nu^{5} + 27438 \nu^{4} - 173817 \nu^{3} + \cdots + 23946 ) / 8312 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 586 \nu^{8} + 1377 \nu^{7} + 12805 \nu^{6} - 32777 \nu^{5} - 39950 \nu^{4} + 96198 \nu^{3} + \cdots - 2730 ) / 4156 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 695 \nu^{8} + 2181 \nu^{7} + 13373 \nu^{6} - 49418 \nu^{5} - 7016 \nu^{4} + 120673 \nu^{3} + \cdots + 20028 ) / 4156 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 1449 \nu^{8} + 4740 \nu^{7} + 26920 \nu^{6} - 105613 \nu^{5} + 4786 \nu^{4} + 227599 \nu^{3} + \cdots + 25494 ) / 8312 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 1541 \nu^{8} - 2864 \nu^{7} - 35292 \nu^{6} + 69901 \nu^{5} + 144790 \nu^{4} - 206163 \nu^{3} + \cdots + 83714 ) / 8312 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{5} - \beta_{4} + \beta_{3} - 2\beta_{2} + 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{8} - 3\beta_{7} + 4\beta_{6} - 2\beta_{5} - 2\beta_{3} + 2\beta_{2} + 13\beta _1 - 5 \)
|
| \(\nu^{4}\) | \(=\) |
\( 4\beta_{8} - 2\beta_{7} + \beta_{6} - 13\beta_{5} - 20\beta_{4} + 16\beta_{3} - 40\beta_{2} - 12\beta _1 + 86 \)
|
| \(\nu^{5}\) | \(=\) |
\( -14\beta_{8} - 61\beta_{7} + 77\beta_{6} - 25\beta_{5} - 55\beta_{3} + 50\beta_{2} + 212\beta _1 - 151 \)
|
| \(\nu^{6}\) | \(=\) |
\( 97 \beta_{8} - 27 \beta_{7} - 9 \beta_{6} - 184 \beta_{5} - 367 \beta_{4} + 273 \beta_{3} - 722 \beta_{2} + \cdots + 1457 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 231 \beta_{8} - 1080 \beta_{7} + 1338 \beta_{6} - 280 \beta_{5} + 96 \beta_{4} - 1147 \beta_{3} + \cdots - 3426 \)
|
| \(\nu^{8}\) | \(=\) |
\( 1923 \beta_{8} - 72 \beta_{7} - 771 \beta_{6} - 2804 \beta_{5} - 6505 \beta_{4} + 5002 \beta_{3} + \cdots + 25992 \)
|
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 | −3.89305 | −1.00000 | −4.93000 | 1.00000 | 1.00000 | −3.89305 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −1.00000 | 1.00000 | −2.61835 | −1.00000 | 3.95780 | 1.00000 | 1.00000 | −2.61835 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −1.00000 | 1.00000 | −2.08688 | −1.00000 | −3.74967 | 1.00000 | 1.00000 | −2.08688 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | −1.00000 | 1.00000 | −0.801114 | −1.00000 | 1.86180 | 1.00000 | 1.00000 | −0.801114 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | −1.00000 | 1.00000 | −0.487125 | −1.00000 | 4.75176 | 1.00000 | 1.00000 | −0.487125 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | −1.00000 | 1.00000 | 0.369644 | −1.00000 | −1.63265 | 1.00000 | 1.00000 | 0.369644 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | −1.00000 | 1.00000 | 0.744377 | −1.00000 | −0.530367 | 1.00000 | 1.00000 | 0.744377 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.00000 | −1.00000 | 1.00000 | 1.40257 | −1.00000 | −2.54548 | 1.00000 | 1.00000 | 1.40257 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.00000 | −1.00000 | 1.00000 | 4.36993 | −1.00000 | −4.18318 | 1.00000 | 1.00000 | 4.36993 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(1\) |
| \(19\) | \(-1\) |
| \(53\) | \(1\) |
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 | 6042.2.a.bc | ✓ | 9 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 6042.2.a.bc | ✓ | 9 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6042))\):
|
\( T_{5}^{9} + 3T_{5}^{8} - 20T_{5}^{7} - 69T_{5}^{6} + 27T_{5}^{5} + 185T_{5}^{4} + 8T_{5}^{3} - 109T_{5}^{2} - 8T_{5} + 14 \)
|
|
\( T_{7}^{9} + 7T_{7}^{8} - 29T_{7}^{7} - 287T_{7}^{6} - 45T_{7}^{5} + 3203T_{7}^{4} + 4969T_{7}^{3} - 6282T_{7}^{2} - 15512T_{7} - 5968 \)
|
|
\( T_{11}^{9} - 4 T_{11}^{8} - 49 T_{11}^{7} + 149 T_{11}^{6} + 593 T_{11}^{5} - 774 T_{11}^{4} + \cdots + 144 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{9} \)
|
| $3$ |
\( (T + 1)^{9} \)
|
| $5$ |
\( T^{9} + 3 T^{8} + \cdots + 14 \)
|
| $7$ |
\( T^{9} + 7 T^{8} + \cdots - 5968 \)
|
| $11$ |
\( T^{9} - 4 T^{8} + \cdots + 144 \)
|
| $13$ |
\( T^{9} + 5 T^{8} + \cdots - 34832 \)
|
| $17$ |
\( T^{9} + 28 T^{8} + \cdots + 4 \)
|
| $19$ |
\( (T - 1)^{9} \)
|
| $23$ |
\( T^{9} + 10 T^{8} + \cdots - 55072 \)
|
| $29$ |
\( T^{9} - 170 T^{7} + \cdots + 33184 \)
|
| $31$ |
\( T^{9} - 5 T^{8} + \cdots + 988192 \)
|
| $37$ |
\( T^{9} + 25 T^{8} + \cdots - 8064112 \)
|
| $41$ |
\( T^{9} + 7 T^{8} + \cdots - 1333728 \)
|
| $43$ |
\( T^{9} + 16 T^{8} + \cdots + 34496 \)
|
| $47$ |
\( T^{9} + 9 T^{8} + \cdots + 138028 \)
|
| $53$ |
\( (T + 1)^{9} \)
|
| $59$ |
\( T^{9} + 3 T^{8} + \cdots + 283279768 \)
|
| $61$ |
\( T^{9} + 16 T^{8} + \cdots + 15736 \)
|
| $67$ |
\( T^{9} + 13 T^{8} + \cdots - 151658496 \)
|
| $71$ |
\( T^{9} + 4 T^{8} + \cdots - 61750528 \)
|
| $73$ |
\( T^{9} + 29 T^{8} + \cdots - 96511744 \)
|
| $79$ |
\( T^{9} - 13 T^{8} + \cdots + 2564 \)
|
| $83$ |
\( T^{9} + 35 T^{8} + \cdots - 170754112 \)
|
| $89$ |
\( T^{9} + 19 T^{8} + \cdots - 18228832 \)
|
| $97$ |
\( T^{9} + 12 T^{8} + \cdots + 2574592 \)
|
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