Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3024,2,Mod(2033,3024)]
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, 1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3024.2033");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3024.ca (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: |
\( x^{16} - 8 x^{15} + 23 x^{14} - 8 x^{13} - 131 x^{12} + 380 x^{11} - 289 x^{10} - 880 x^{9} + \cdots + 6561 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 3^{6} \) |
| Twist minimal: | no (minimal twist has level 126) |
| 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
\( x^{16} - 8 x^{15} + 23 x^{14} - 8 x^{13} - 131 x^{12} + 380 x^{11} - 289 x^{10} - 880 x^{9} + \cdots + 6561 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 169 \nu^{15} + 866 \nu^{14} - 1319 \nu^{13} - 2308 \nu^{12} + 13199 \nu^{11} - 19055 \nu^{10} + \cdots + 244944 ) / 47385 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 154 \nu^{15} + 1325 \nu^{14} - 3608 \nu^{13} + 224 \nu^{12} + 22478 \nu^{11} - 55022 \nu^{10} + \cdots + 1285227 ) / 47385 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 50 \nu^{15} - 1352 \nu^{14} + 6827 \nu^{13} - 7676 \nu^{12} - 27422 \nu^{11} + 107246 \nu^{10} + \cdots - 2825604 ) / 47385 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 964 \nu^{15} + 5336 \nu^{14} - 8294 \nu^{13} - 15778 \nu^{12} + 90644 \nu^{11} - 137225 \nu^{10} + \cdots + 2924019 ) / 142155 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 1445 \nu^{15} + 9836 \nu^{14} - 21081 \nu^{13} - 15627 \nu^{12} + 172766 \nu^{11} + \cdots + 7117227 ) / 47385 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 2858 \nu^{15} - 19265 \nu^{14} + 40866 \nu^{13} + 31392 \nu^{12} - 338066 \nu^{11} + 649239 \nu^{10} + \cdots - 14041269 ) / 47385 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 11192 \nu^{15} - 70123 \nu^{14} + 136087 \nu^{13} + 145859 \nu^{12} - 1215277 \nu^{11} + \cdots - 42998607 ) / 142155 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 4120 \nu^{15} + 25571 \nu^{14} - 48788 \nu^{13} - 55006 \nu^{12} + 441224 \nu^{11} + \cdots + 14963454 ) / 28431 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 7903 \nu^{15} - 48406 \nu^{14} + 90897 \nu^{13} + 107784 \nu^{12} - 834172 \nu^{11} + \cdots - 27526311 ) / 47385 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 26068 \nu^{15} - 165707 \nu^{14} + 325403 \nu^{13} + 334861 \nu^{12} - 2873363 \nu^{11} + \cdots - 102032298 ) / 142155 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 33311 \nu^{15} - 207064 \nu^{14} + 396466 \nu^{13} + 441842 \nu^{12} - 3574291 \nu^{11} + \cdots - 121811526 ) / 142155 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 49417 \nu^{15} + 306332 \nu^{14} - 583301 \nu^{13} - 662377 \nu^{12} + 5287136 \nu^{11} + \cdots + 177999930 ) / 142155 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 16898 \nu^{15} + 108472 \nu^{14} - 217033 \nu^{13} - 208526 \nu^{12} + 1883968 \nu^{11} + \cdots + 69445998 ) / 47385 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 56329 \nu^{15} - 353180 \nu^{14} + 683693 \nu^{13} + 735946 \nu^{12} - 6112403 \nu^{11} + \cdots - 213650217 ) / 142155 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 62668 \nu^{15} + 397070 \nu^{14} - 778856 \nu^{13} - 802747 \nu^{12} + 6878696 \nu^{11} + \cdots + 244346949 ) / 142155 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{12} - 2\beta_{9} + \beta_{8} + 2\beta_{7} + \beta_{5} + \beta_{2} - 2\beta _1 + 2 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{15} - 2 \beta_{14} - \beta_{13} - 3 \beta_{12} + \beta_{11} - \beta_{10} - 3 \beta_{9} - \beta_{8} + \cdots + 3 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 2 \beta_{14} - \beta_{13} - 6 \beta_{12} - \beta_{11} - 3 \beta_{10} - 6 \beta_{9} + 4 \beta_{6} + \cdots - 4 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 3 \beta_{14} - \beta_{13} - 7 \beta_{12} - 4 \beta_{11} - 6 \beta_{10} - 2 \beta_{9} - 6 \beta_{8} + \cdots + 8 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 10 \beta_{15} + 3 \beta_{14} + 4 \beta_{13} - 6 \beta_{12} - 6 \beta_{11} - 5 \beta_{10} - 6 \beta_{9} + \cdots - 1 ) / 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 2 \beta_{15} + 7 \beta_{14} - 4 \beta_{13} - \beta_{12} - 13 \beta_{11} + 2 \beta_{10} - 8 \beta_{9} + \cdots + 20 ) / 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 3 \beta_{15} + 6 \beta_{14} - 11 \beta_{13} + 3 \beta_{12} + 13 \beta_{11} + 21 \beta_{10} - 33 \beta_{9} + \cdots - 3 ) / 3 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 68 \beta_{15} + 10 \beta_{14} - 29 \beta_{13} + 16 \beta_{11} + 25 \beta_{10} - 42 \beta_{9} - 121 \beta_{8} + \cdots + 94 ) / 3 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 32 \beta_{15} + 45 \beta_{13} + 4 \beta_{12} + 144 \beta_{11} - 14 \beta_{10} - 40 \beta_{9} + \cdots + 168 ) / 3 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 18 \beta_{15} + 90 \beta_{14} + 115 \beta_{13} - 4 \beta_{12} + 46 \beta_{11} - 162 \beta_{10} + \cdots + 188 ) / 3 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 351 \beta_{15} - 100 \beta_{14} + 207 \beta_{13} + 78 \beta_{12} + 218 \beta_{11} - 333 \beta_{10} + \cdots + 217 ) / 3 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 418 \beta_{15} - 251 \beta_{14} - 16 \beta_{13} - 92 \beta_{12} - 238 \beta_{11} - 416 \beta_{10} + \cdots - 97 ) / 3 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 609 \beta_{15} - 1440 \beta_{14} - 117 \beta_{13} - 230 \beta_{12} + 588 \beta_{11} - 414 \beta_{10} + \cdots + 1243 ) / 3 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 511 \beta_{15} - 835 \beta_{14} - 260 \beta_{13} - 1254 \beta_{12} - 82 \beta_{11} - 812 \beta_{10} + \cdots - 1641 ) / 3 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 1179 \beta_{15} - 1012 \beta_{14} - 431 \beta_{13} - 96 \beta_{12} + 1306 \beta_{11} - 2838 \beta_{10} + \cdots + 208 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times\).
| \(n\) | \(757\) | \(785\) | \(1135\) | \(2593\) |
| \(\chi(n)\) | \(1\) | \(\beta_{8}\) | \(1\) | \(\beta_{8}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 2033.1 |
|
0 | 0 | 0 | −1.77612 | + | 3.07634i | 0 | −2.63804 | − | 0.201867i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2033.2 | 0 | 0 | 0 | −1.14095 | + | 1.97618i | 0 | −1.42337 | + | 2.23025i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2033.3 | 0 | 0 | 0 | −0.714925 | + | 1.23829i | 0 | −0.327442 | − | 2.62541i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2033.4 | 0 | 0 | 0 | −0.483662 | + | 0.837727i | 0 | 2.16249 | + | 1.52435i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2033.5 | 0 | 0 | 0 | 0.0338034 | − | 0.0585493i | 0 | −1.19767 | − | 2.35915i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2033.6 | 0 | 0 | 0 | 0.450129 | − | 0.779646i | 0 | −1.57151 | + | 2.12847i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2033.7 | 0 | 0 | 0 | 1.80966 | − | 3.13442i | 0 | 2.41308 | − | 1.08492i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2033.8 | 0 | 0 | 0 | 1.82207 | − | 3.15592i | 0 | 1.58246 | + | 2.12034i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2609.1 | 0 | 0 | 0 | −1.77612 | − | 3.07634i | 0 | −2.63804 | + | 0.201867i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2609.2 | 0 | 0 | 0 | −1.14095 | − | 1.97618i | 0 | −1.42337 | − | 2.23025i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2609.3 | 0 | 0 | 0 | −0.714925 | − | 1.23829i | 0 | −0.327442 | + | 2.62541i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2609.4 | 0 | 0 | 0 | −0.483662 | − | 0.837727i | 0 | 2.16249 | − | 1.52435i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2609.5 | 0 | 0 | 0 | 0.0338034 | + | 0.0585493i | 0 | −1.19767 | + | 2.35915i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2609.6 | 0 | 0 | 0 | 0.450129 | + | 0.779646i | 0 | −1.57151 | − | 2.12847i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2609.7 | 0 | 0 | 0 | 1.80966 | + | 3.13442i | 0 | 2.41308 | + | 1.08492i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2609.8 | 0 | 0 | 0 | 1.82207 | + | 3.15592i | 0 | 1.58246 | − | 2.12034i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 63.i | even | 6 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{16} + 24 T_{5}^{14} + 24 T_{5}^{13} + 423 T_{5}^{12} + 450 T_{5}^{11} + 3582 T_{5}^{10} + \cdots + 81 \)
acting on \(S_{2}^{\mathrm{new}}(3024, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} \)
|
| $3$ |
\( T^{16} \)
|
| $5$ |
\( T^{16} + 24 T^{14} + \cdots + 81 \)
|
| $7$ |
\( T^{16} + 2 T^{15} + \cdots + 5764801 \)
|
| $11$ |
\( T^{16} - 12 T^{15} + \cdots + 61732449 \)
|
| $13$ |
\( T^{16} + \cdots + 390971529 \)
|
| $17$ |
\( T^{16} - 18 T^{15} + \cdots + 56070144 \)
|
| $19$ |
\( T^{16} - 72 T^{14} + \cdots + 9199089 \)
|
| $23$ |
\( T^{16} + \cdots + 187388721 \)
|
| $29$ |
\( T^{16} + 6 T^{15} + \cdots + 1108809 \)
|
| $31$ |
\( T^{16} + 204 T^{14} + \cdots + 65610000 \)
|
| $37$ |
\( T^{16} + \cdots + 32746159681 \)
|
| $41$ |
\( T^{16} - 6 T^{15} + \cdots + 81 \)
|
| $43$ |
\( T^{16} + \cdots + 2999643361 \)
|
| $47$ |
\( (T^{8} + 18 T^{7} + \cdots + 766944)^{2} \)
|
| $53$ |
\( T^{16} + \cdots + 36759242529 \)
|
| $59$ |
\( (T^{8} - 30 T^{7} + \cdots + 465300)^{2} \)
|
| $61$ |
\( T^{16} + \cdots + 547560000 \)
|
| $67$ |
\( (T^{8} - 14 T^{7} + \cdots + 51028)^{2} \)
|
| $71$ |
\( T^{16} + 486 T^{14} + \cdots + 65610000 \)
|
| $73$ |
\( T^{16} - 150 T^{14} + \cdots + 71115489 \)
|
| $79$ |
\( (T^{8} + 16 T^{7} + \cdots - 985100)^{2} \)
|
| $83$ |
\( T^{16} + \cdots + 953512641 \)
|
| $89$ |
\( T^{16} + \cdots + 131145120363321 \)
|
| $97$ |
\( T^{16} + \cdots + 9120206721024 \)
|
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