Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2014,2,Mod(1,2014)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2014, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("2014.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2014 = 2 \cdot 19 \cdot 53 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2014.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: | \(16.0818709671\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{9} - 2x^{8} - 19x^{7} + 35x^{6} + 110x^{5} - 177x^{4} - 207x^{3} + 242x^{2} + 96x + 8 \)
|
| 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
\( x^{9} - 2x^{8} - 19x^{7} + 35x^{6} + 110x^{5} - 177x^{4} - 207x^{3} + 242x^{2} + 96x + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 81 \nu^{8} - 122 \nu^{7} + 1811 \nu^{6} + 2605 \nu^{5} - 12302 \nu^{4} - 14451 \nu^{3} + \cdots + 2460 ) / 5668 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 15\nu^{8} - 42\nu^{7} - 295\nu^{6} + 761\nu^{5} + 1826\nu^{4} - 3985\nu^{3} - 3841\nu^{2} + 5482\nu + 1676 ) / 218 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 615 \nu^{8} - 1068 \nu^{7} - 11441 \nu^{6} + 17903 \nu^{5} + 62440 \nu^{4} - 84251 \nu^{3} + \cdots + 27296 ) / 5668 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 615 \nu^{8} - 1068 \nu^{7} - 11441 \nu^{6} + 17903 \nu^{5} + 62440 \nu^{4} - 84251 \nu^{3} + \cdots - 1044 ) / 5668 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 79 \nu^{8} + 134 \nu^{7} + 1481 \nu^{6} - 2293 \nu^{5} - 8338 \nu^{4} + 11323 \nu^{3} + 14445 \nu^{2} + \cdots - 3624 ) / 436 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 729 \nu^{8} + 1736 \nu^{7} + 13465 \nu^{6} - 30401 \nu^{5} - 73876 \nu^{4} + 153341 \nu^{3} + \cdots - 37374 ) / 2834 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 540 \nu^{8} + 1076 \nu^{7} + 10184 \nu^{6} - 19003 \nu^{5} - 57452 \nu^{4} + 97789 \nu^{3} + \cdots - 27527 ) / 1417 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - \beta_{4} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + \beta_{5} + \beta_{4} + \beta_{3} + 2\beta_{2} + 8\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{8} - 3\beta_{6} + 9\beta_{5} - 11\beta_{4} + \beta_{3} + \beta_{2} + 41 \)
|
| \(\nu^{5}\) | \(=\) |
\( -3\beta_{8} + 15\beta_{7} + 13\beta_{5} + 9\beta_{4} + 11\beta_{3} + 30\beta_{2} + 71\beta _1 + 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 14\beta_{8} - \beta_{7} - 48\beta_{6} + 81\beta_{5} - 115\beta_{4} + 6\beta_{3} + 24\beta_{2} + 372 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 56 \beta_{8} + 185 \beta_{7} - 9 \beta_{6} + 143 \beta_{5} + 67 \beta_{4} + 102 \beta_{3} + 363 \beta_{2} + \cdots + 39 \)
|
| \(\nu^{8}\) | \(=\) |
\( 149 \beta_{8} + 3 \beta_{7} - 604 \beta_{6} + 760 \beta_{5} - 1182 \beta_{4} + 4 \beta_{3} + 376 \beta_{2} + \cdots + 3552 \)
|
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 | −3.27182 | 1.00000 | −1.34729 | −3.27182 | −2.00275 | 1.00000 | 7.70479 | −1.34729 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −2.69318 | 1.00000 | 3.23274 | −2.69318 | 3.36755 | 1.00000 | 4.25324 | 3.23274 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −1.95271 | 1.00000 | −1.74049 | −1.95271 | −1.72723 | 1.00000 | 0.813071 | −1.74049 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | −1.48870 | 1.00000 | −0.0491488 | −1.48870 | 3.36592 | 1.00000 | −0.783781 | −0.0491488 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 0.130419 | 1.00000 | 2.98517 | 0.130419 | −3.20060 | 1.00000 | −2.98299 | 2.98517 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 0.208260 | 1.00000 | −3.47129 | 0.208260 | 0.219462 | 1.00000 | −2.95663 | −3.47129 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | 1.51703 | 1.00000 | 3.97321 | 1.51703 | 2.50452 | 1.00000 | −0.698621 | 3.97321 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.00000 | 2.42479 | 1.00000 | −1.18215 | 2.42479 | 4.10451 | 1.00000 | 2.87961 | −1.18215 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.00000 | 3.12591 | 1.00000 | 1.59923 | 3.12591 | −2.63138 | 1.00000 | 6.77131 | 1.59923 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(19\) | \(-1\) |
| \(53\) | \(-1\) |
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 | 2014.2.a.i | ✓ | 9 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2014.2.a.i | ✓ | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2014))\):
|
\( T_{3}^{9} + 2T_{3}^{8} - 19T_{3}^{7} - 35T_{3}^{6} + 110T_{3}^{5} + 177T_{3}^{4} - 207T_{3}^{3} - 242T_{3}^{2} + 96T_{3} - 8 \)
|
|
\( T_{7}^{9} - 4T_{7}^{8} - 27T_{7}^{7} + 97T_{7}^{6} + 283T_{7}^{5} - 778T_{7}^{4} - 1434T_{7}^{3} + 2161T_{7}^{2} + 2997T_{7} - 745 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{9} \)
|
| $3$ |
\( T^{9} + 2 T^{8} + \cdots - 8 \)
|
| $5$ |
\( T^{9} - 4 T^{8} + \cdots + 29 \)
|
| $7$ |
\( T^{9} - 4 T^{8} + \cdots - 745 \)
|
| $11$ |
\( T^{9} - 16 T^{8} + \cdots - 1784 \)
|
| $13$ |
\( T^{9} + 11 T^{8} + \cdots - 195 \)
|
| $17$ |
\( T^{9} + 3 T^{8} + \cdots - 199 \)
|
| $19$ |
\( (T - 1)^{9} \)
|
| $23$ |
\( T^{9} - 11 T^{8} + \cdots + 1352 \)
|
| $29$ |
\( T^{9} - 15 T^{8} + \cdots + 32209 \)
|
| $31$ |
\( T^{9} - 105 T^{7} + \cdots + 1167 \)
|
| $37$ |
\( T^{9} - 5 T^{8} + \cdots + 1259 \)
|
| $41$ |
\( T^{9} + 8 T^{8} + \cdots - 24685 \)
|
| $43$ |
\( T^{9} - 23 T^{8} + \cdots + 9920 \)
|
| $47$ |
\( T^{9} - 7 T^{8} + \cdots + 1516953 \)
|
| $53$ |
\( (T - 1)^{9} \)
|
| $59$ |
\( T^{9} - 9 T^{8} + \cdots - 19255381 \)
|
| $61$ |
\( T^{9} - 12 T^{8} + \cdots + 5675085 \)
|
| $67$ |
\( T^{9} - 27 T^{8} + \cdots + 279352 \)
|
| $71$ |
\( T^{9} - 19 T^{8} + \cdots + 96267 \)
|
| $73$ |
\( T^{9} + 17 T^{8} + \cdots - 505800 \)
|
| $79$ |
\( T^{9} - 4 T^{8} + \cdots - 38615 \)
|
| $83$ |
\( T^{9} - 9 T^{8} + \cdots - 260065 \)
|
| $89$ |
\( T^{9} + 3 T^{8} + \cdots - 120 \)
|
| $97$ |
\( T^{9} + 12 T^{8} + \cdots - 286472 \)
|
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