Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3024,2,Mod(881,3024)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3024, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 5, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3024.881");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3024.cc (of order \(6\), degree \(2\), not minimal) |
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: | no |
| Analytic conductor: | \(24.1467615712\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(8\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 3x^{14} - 9x^{12} - 9x^{10} + 225x^{8} - 81x^{6} - 729x^{4} - 2187x^{2} + 6561 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 3^{6} \) |
| Twist minimal: | no (minimal twist has level 252) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 3x^{14} - 9x^{12} - 9x^{10} + 225x^{8} - 81x^{6} - 729x^{4} - 2187x^{2} + 6561 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{15} + 15\nu^{13} + 72\nu^{11} + 153\nu^{9} - 423\nu^{7} - 891\nu^{5} + 1944\nu^{3} + 17496\nu ) / 15309 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -5\nu^{14} - 12\nu^{12} + 18\nu^{10} + 369\nu^{8} - 153\nu^{6} - 1782\nu^{4} - 4617\nu^{2} + 9477 ) / 5103 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{15} - 9\nu^{13} - 18\nu^{11} + 72\nu^{9} + 657\nu^{7} - 297\nu^{5} - 3888\nu^{3} - 9477\nu ) / 5103 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{14} + 3\nu^{12} + 9\nu^{10} + 9\nu^{8} - 225\nu^{6} + 81\nu^{4} + 2187 ) / 729 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2\nu^{14} + 3\nu^{12} - 45\nu^{10} - 99\nu^{8} + 369\nu^{6} + 1134\nu^{4} - 729\nu^{2} - 6561 ) / 729 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 32 \nu^{15} + 69 \nu^{14} + 87 \nu^{13} - 99 \nu^{12} + 342 \nu^{11} - 702 \nu^{10} + \cdots - 247131 ) / 30618 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 5\nu^{15} + 12\nu^{13} - 18\nu^{11} - 369\nu^{9} + 153\nu^{7} + 1782\nu^{5} + 4617\nu^{3} - 19683\nu ) / 5103 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 5 \nu^{15} - 18 \nu^{14} + 12 \nu^{13} + 27 \nu^{12} - 72 \nu^{11} + 162 \nu^{10} - 207 \nu^{9} + \cdots + 41553 ) / 4374 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 35 \nu^{15} + 48 \nu^{14} + 84 \nu^{13} - 225 \nu^{12} - 504 \nu^{11} - 1080 \nu^{10} + \cdots - 369603 ) / 30618 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 32 \nu^{15} + 69 \nu^{14} - 87 \nu^{13} - 99 \nu^{12} - 342 \nu^{11} - 702 \nu^{10} - 774 \nu^{9} + \cdots - 247131 ) / 30618 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( -10\nu^{15} - 24\nu^{13} + 36\nu^{11} + 738\nu^{9} - 306\nu^{7} - 3564\nu^{5} - 9234\nu^{3} + 24057\nu ) / 5103 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 11\nu^{15} - 3\nu^{13} - 27\nu^{11} - 207\nu^{9} + 261\nu^{7} + 27\nu^{5} + 2673\nu^{3} + 3645\nu ) / 5103 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 5 \nu^{15} + 18 \nu^{14} + 12 \nu^{13} - 27 \nu^{12} - 72 \nu^{11} - 162 \nu^{10} - 207 \nu^{9} + \cdots - 41553 ) / 4374 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 32 \nu^{15} + 273 \nu^{14} + 87 \nu^{13} - 63 \nu^{12} + 342 \nu^{11} - 3024 \nu^{10} + \cdots - 535815 ) / 30618 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( -61\nu^{15} + 30\nu^{13} + 711\nu^{11} + 2574\nu^{9} - 8217\nu^{7} - 18792\nu^{5} - 1215\nu^{3} + 157464\nu ) / 15309 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{11} - 2\beta_{7} ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{14} + \beta_{13} - \beta_{9} + \beta_{6} + \beta_{5} - \beta_{4} ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{15} - \beta_{13} + \beta_{11} - 2\beta_{10} - \beta_{8} + 2\beta_{6} - \beta_{3} + \beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{10} - \beta_{9} + \beta_{8} + 2\beta_{6} + \beta_{4} + 2\beta_{2} + 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( -4\beta_{12} + 3\beta_{10} - 3\beta_{6} - 5\beta_{3} + 6\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -3\beta_{13} + 3\beta_{10} + 3\beta_{8} + 3\beta_{6} - 6\beta_{4} - 3\beta_{2} + 15 \)
|
| \(\nu^{7}\) | \(=\) |
\( -6\beta_{15} - 3\beta_{13} - 3\beta_{11} - 6\beta_{10} - 3\beta_{8} - 9\beta_{7} + 6\beta_{6} + 6\beta_{3} + 12\beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( 3\beta_{13} + 15\beta_{10} - 15\beta_{9} + 12\beta_{8} + 15\beta_{6} + 3\beta_{5} - 6\beta_{4} + 24\beta_{2} - 24 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 27 \beta_{15} - 9 \beta_{13} - 9 \beta_{12} + 39 \beta_{11} - 18 \beta_{10} - 9 \beta_{8} + \cdots + 54 \beta_1 \)
|
| \(\nu^{10}\) | \(=\) |
\( 21 \beta_{14} - 39 \beta_{13} + 63 \beta_{10} - 6 \beta_{9} + 45 \beta_{8} + 42 \beta_{6} - 39 \beta_{5} + \cdots + 72 \)
|
| \(\nu^{11}\) | \(=\) |
\( 72 \beta_{15} + 9 \beta_{13} - 63 \beta_{12} - 54 \beta_{11} + 126 \beta_{10} + 9 \beta_{8} + \cdots + 180 \beta_1 \)
|
| \(\nu^{12}\) | \(=\) |
\( 36 \beta_{14} - 90 \beta_{13} + 63 \beta_{10} - 36 \beta_{9} + 126 \beta_{8} + 27 \beta_{6} + 18 \beta_{5} + \cdots - 198 \)
|
| \(\nu^{13}\) | \(=\) |
\( 81\beta_{13} + 144\beta_{12} - 189\beta_{10} + 81\beta_{8} + 27\beta_{7} + 189\beta_{6} + 342\beta_{3} + 108\beta_1 \)
|
| \(\nu^{14}\) | \(=\) |
\( 297 \beta_{14} + 81 \beta_{13} + 378 \beta_{10} - 378 \beta_{9} + 297 \beta_{8} + 81 \beta_{6} + \cdots - 1026 \)
|
| \(\nu^{15}\) | \(=\) |
\( 297 \beta_{15} + 189 \beta_{13} + 189 \beta_{12} + 540 \beta_{11} + 540 \beta_{10} + 189 \beta_{8} + \cdots + 945 \beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times\).
| \(n\) | \(757\) | \(785\) | \(1135\) | \(2593\) |
| \(\chi(n)\) | \(1\) | \(1 + \beta_{2}\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 881.1 |
|
0 | 0 | 0 | −2.09336 | + | 3.62580i | 0 | 2.64522 | − | 0.0532130i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 881.2 | 0 | 0 | 0 | −1.21244 | + | 2.10001i | 0 | −1.05649 | − | 2.42566i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 881.3 | 0 | 0 | 0 | −0.276914 | + | 0.479629i | 0 | 1.98718 | + | 1.74675i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 881.4 | 0 | 0 | 0 | −0.266780 | + | 0.462077i | 0 | −2.54716 | + | 0.715531i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 881.5 | 0 | 0 | 0 | 0.266780 | − | 0.462077i | 0 | 1.89325 | − | 1.84814i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 881.6 | 0 | 0 | 0 | 0.276914 | − | 0.479629i | 0 | 0.519138 | + | 2.59432i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 881.7 | 0 | 0 | 0 | 1.21244 | − | 2.10001i | 0 | −1.57244 | − | 2.12778i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 881.8 | 0 | 0 | 0 | 2.09336 | − | 3.62580i | 0 | −1.36869 | + | 2.26422i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2897.1 | 0 | 0 | 0 | −2.09336 | − | 3.62580i | 0 | 2.64522 | + | 0.0532130i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2897.2 | 0 | 0 | 0 | −1.21244 | − | 2.10001i | 0 | −1.05649 | + | 2.42566i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2897.3 | 0 | 0 | 0 | −0.276914 | − | 0.479629i | 0 | 1.98718 | − | 1.74675i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2897.4 | 0 | 0 | 0 | −0.266780 | − | 0.462077i | 0 | −2.54716 | − | 0.715531i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2897.5 | 0 | 0 | 0 | 0.266780 | + | 0.462077i | 0 | 1.89325 | + | 1.84814i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2897.6 | 0 | 0 | 0 | 0.276914 | + | 0.479629i | 0 | 0.519138 | − | 2.59432i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2897.7 | 0 | 0 | 0 | 1.21244 | + | 2.10001i | 0 | −1.57244 | + | 2.12778i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2897.8 | 0 | 0 | 0 | 2.09336 | + | 3.62580i | 0 | −1.36869 | − | 2.26422i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
| 9.d | odd | 6 | 1 | inner |
| 63.o | even | 6 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{16} + 24T_{5}^{14} + 459T_{5}^{12} + 2682T_{5}^{10} + 12168T_{5}^{8} + 6939T_{5}^{6} + 2916T_{5}^{4} + 567T_{5}^{2} + 81 \)
acting on \(S_{2}^{\mathrm{new}}(3024, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} \)
|
| $3$ |
\( T^{16} \)
|
| $5$ |
\( T^{16} + 24 T^{14} + \cdots + 81 \)
|
| $7$ |
\( T^{16} - T^{15} + \cdots + 5764801 \)
|
| $11$ |
\( (T^{8} - 3 T^{7} + \cdots + 3969)^{2} \)
|
| $13$ |
\( T^{16} - 48 T^{14} + \cdots + 81 \)
|
| $17$ |
\( (T^{8} - 78 T^{6} + \cdots + 900)^{2} \)
|
| $19$ |
\( (T^{8} + 75 T^{6} + \cdots + 900)^{2} \)
|
| $23$ |
\( (T^{8} - 3 T^{7} + \cdots + 50625)^{2} \)
|
| $29$ |
\( (T^{8} - 6 T^{7} + \cdots + 245025)^{2} \)
|
| $31$ |
\( T^{16} - 72 T^{14} + \cdots + 531441 \)
|
| $37$ |
\( (T^{4} - T^{3} - 66 T^{2} + \cdots + 610)^{4} \)
|
| $41$ |
\( T^{16} + \cdots + 331869318561 \)
|
| $43$ |
\( (T^{8} + 2 T^{7} + \cdots + 461041)^{2} \)
|
| $47$ |
\( T^{16} + \cdots + 24685405970481 \)
|
| $53$ |
\( (T^{8} + 414 T^{6} + \cdots + 41990400)^{2} \)
|
| $59$ |
\( T^{16} + 96 T^{14} + \cdots + 194481 \)
|
| $61$ |
\( T^{16} + \cdots + 12\!\cdots\!01 \)
|
| $67$ |
\( (T^{8} + 7 T^{7} + \cdots + 3940225)^{2} \)
|
| $71$ |
\( (T^{8} + 207 T^{6} + \cdots + 15876)^{2} \)
|
| $73$ |
\( (T^{8} + 243 T^{6} + \cdots + 76176)^{2} \)
|
| $79$ |
\( (T^{8} + 10 T^{7} + \cdots + 319225)^{2} \)
|
| $83$ |
\( T^{16} + \cdots + 30237384321 \)
|
| $89$ |
\( (T^{8} - 648 T^{6} + \cdots + 211004676)^{2} \)
|
| $97$ |
\( T^{16} + \cdots + 83955602727441 \)
|
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