Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4030,2,Mod(1,4030)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4030, 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("4030.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4030 = 2 \cdot 5 \cdot 13 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4030.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: | \(32.1797120146\) |
| 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: |
\( x^{9} - 3x^{8} - 16x^{7} + 46x^{6} + 80x^{5} - 212x^{4} - 133x^{3} + 294x^{2} + 52x - 112 \)
|
| 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
\( x^{9} - 3x^{8} - 16x^{7} + 46x^{6} + 80x^{5} - 212x^{4} - 133x^{3} + 294x^{2} + 52x - 112 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 95 \nu^{8} + 187 \nu^{7} + 1698 \nu^{6} - 2678 \nu^{5} - 9980 \nu^{4} + 10528 \nu^{3} + \cdots - 11056 ) / 236 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 56 \nu^{8} + 127 \nu^{7} + 1009 \nu^{6} - 1832 \nu^{5} - 6150 \nu^{4} + 7186 \nu^{3} + 14220 \nu^{2} + \cdots - 7850 ) / 118 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 71 \nu^{8} - 141 \nu^{7} - 1274 \nu^{6} + 1994 \nu^{5} + 7614 \nu^{4} - 7718 \nu^{3} - 16811 \nu^{2} + \cdots + 8838 ) / 118 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 161 \nu^{8} + 343 \nu^{7} + 2864 \nu^{6} - 4854 \nu^{5} - 16988 \nu^{4} + 18580 \nu^{3} + \cdots - 19368 ) / 236 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 207 \nu^{8} + 441 \nu^{7} + 3716 \nu^{6} - 6342 \nu^{5} - 22280 \nu^{4} + 24900 \nu^{3} + \cdots - 25812 ) / 236 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 247 \nu^{8} - 557 \nu^{7} - 4344 \nu^{6} + 7954 \nu^{5} + 25476 \nu^{4} - 30960 \nu^{3} - 55535 \nu^{2} + \cdots + 28604 ) / 236 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 198 \nu^{8} - 409 \nu^{7} - 3557 \nu^{6} + 5820 \nu^{5} + 21378 \nu^{4} - 22504 \nu^{3} - 47842 \nu^{2} + \cdots + 25526 ) / 118 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{6} + \beta_{3} + \beta_{2} + \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{8} - 2\beta_{4} + \beta_{3} + 8\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{8} + \beta_{7} - 8\beta_{6} + 3\beta_{5} - \beta_{4} + 10\beta_{3} + 10\beta_{2} + 10\beta _1 + 26 \)
|
| \(\nu^{5}\) | \(=\) |
\( 13\beta_{8} + 2\beta_{7} - 2\beta_{6} + 7\beta_{5} - 23\beta_{4} + 14\beta_{3} + \beta_{2} + 66\beta _1 + 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( 35\beta_{8} + 19\beta_{7} - 58\beta_{6} + 51\beta_{5} - 22\beta_{4} + 97\beta_{3} + 88\beta_{2} + 97\beta _1 + 191 \)
|
| \(\nu^{7}\) | \(=\) |
\( 153 \beta_{8} + 37 \beta_{7} - 30 \beta_{6} + 129 \beta_{5} - 236 \beta_{4} + 173 \beta_{3} + 24 \beta_{2} + \cdots + 33 \)
|
| \(\nu^{8}\) | \(=\) |
\( 461 \beta_{8} + 251 \beta_{7} - 423 \beta_{6} + 653 \beta_{5} - 326 \beta_{4} + 964 \beta_{3} + \cdots + 1471 \)
|
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 | −3.25457 | 1.00000 | −1.00000 | 3.25457 | −1.51784 | −1.00000 | 7.59221 | 1.00000 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −2.72026 | 1.00000 | −1.00000 | 2.72026 | 4.31759 | −1.00000 | 4.39981 | 1.00000 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −2.42233 | 1.00000 | −1.00000 | 2.42233 | −2.05608 | −1.00000 | 2.86766 | 1.00000 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | −0.905402 | 1.00000 | −1.00000 | 0.905402 | −0.957463 | −1.00000 | −2.18025 | 1.00000 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | −0.887962 | 1.00000 | −1.00000 | 0.887962 | −2.77086 | −1.00000 | −2.21152 | 1.00000 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 0.777449 | 1.00000 | −1.00000 | −0.777449 | 1.65571 | −1.00000 | −2.39557 | 1.00000 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.00000 | 1.27856 | 1.00000 | −1.00000 | −1.27856 | −0.354220 | −1.00000 | −1.36528 | 1.00000 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −1.00000 | 2.33128 | 1.00000 | −1.00000 | −2.33128 | −4.98848 | −1.00000 | 2.43487 | 1.00000 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −1.00000 | 2.80323 | 1.00000 | −1.00000 | −2.80323 | 3.67164 | −1.00000 | 4.85808 | 1.00000 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(5\) | \(1\) |
| \(13\) | \(1\) |
| \(31\) | \(-1\) |
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 | 4030.2.a.p | ✓ | 9 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4030.2.a.p | ✓ | 9 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{9} + 3T_{3}^{8} - 16T_{3}^{7} - 46T_{3}^{6} + 80T_{3}^{5} + 212T_{3}^{4} - 133T_{3}^{3} - 294T_{3}^{2} + 52T_{3} + 112 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4030))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{9} \)
|
| $3$ |
\( T^{9} + 3 T^{8} + \cdots + 112 \)
|
| $5$ |
\( (T + 1)^{9} \)
|
| $7$ |
\( T^{9} + 3 T^{8} + \cdots - 384 \)
|
| $11$ |
\( T^{9} - 14 T^{8} + \cdots + 2688 \)
|
| $13$ |
\( (T + 1)^{9} \)
|
| $17$ |
\( T^{9} - T^{8} + \cdots + 14774 \)
|
| $19$ |
\( T^{9} - 6 T^{8} + \cdots - 40608 \)
|
| $23$ |
\( T^{9} + 4 T^{8} + \cdots - 87564 \)
|
| $29$ |
\( T^{9} - 15 T^{8} + \cdots - 490962 \)
|
| $31$ |
\( (T - 1)^{9} \)
|
| $37$ |
\( T^{9} + 9 T^{8} + \cdots + 616334 \)
|
| $41$ |
\( T^{9} - 18 T^{8} + \cdots - 2756488 \)
|
| $43$ |
\( T^{9} + 23 T^{8} + \cdots - 853552 \)
|
| $47$ |
\( T^{9} - 3 T^{8} + \cdots + 39983792 \)
|
| $53$ |
\( T^{9} + 6 T^{8} + \cdots - 19871432 \)
|
| $59$ |
\( T^{9} - 28 T^{8} + \cdots - 80488 \)
|
| $61$ |
\( T^{9} - 471 T^{7} + \cdots + 137935922 \)
|
| $67$ |
\( T^{9} + 16 T^{8} + \cdots - 7972016 \)
|
| $71$ |
\( T^{9} - 32 T^{8} + \cdots + 3705216 \)
|
| $73$ |
\( T^{9} + 11 T^{8} + \cdots - 69612632 \)
|
| $79$ |
\( T^{9} + 8 T^{8} + \cdots + 1097376 \)
|
| $83$ |
\( T^{9} - 15 T^{8} + \cdots - 10448076 \)
|
| $89$ |
\( T^{9} - 51 T^{8} + \cdots + 2671726 \)
|
| $97$ |
\( T^{9} + 26 T^{8} + \cdots - 15539098 \)
|
| show more | |
| show less |