Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1342,2,Mod(1,1342)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1342, 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("1342.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1342 = 2 \cdot 11 \cdot 61 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1342.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: | \(10.7159239513\) |
| 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: |
\( x^{9} - 4x^{8} - 15x^{7} + 55x^{6} + 84x^{5} - 239x^{4} - 184x^{3} + 334x^{2} + 54x - 102 \)
|
| 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
\( x^{9} - 4x^{8} - 15x^{7} + 55x^{6} + 84x^{5} - 239x^{4} - 184x^{3} + 334x^{2} + 54x - 102 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 3 \nu^{8} - 331 \nu^{7} + 1710 \nu^{6} + 2737 \nu^{5} - 17595 \nu^{4} - 7114 \nu^{3} + \cdots - 26638 ) / 1852 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -25\nu^{8} + 174\nu^{7} - 103\nu^{6} - 1422\nu^{5} + 2461\nu^{4} + 2450\nu^{3} - 7282\nu^{2} + 500\nu + 2726 ) / 463 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 101 \nu^{8} - 277 \nu^{7} - 2010 \nu^{6} + 3541 \nu^{5} + 15467 \nu^{4} - 13602 \nu^{3} + \cdots + 19582 ) / 1852 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 57 \nu^{8} + 193 \nu^{7} + 1006 \nu^{6} - 2631 \nu^{5} - 6501 \nu^{4} + 9290 \nu^{3} + 16696 \nu^{2} + \cdots - 7008 ) / 926 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 243 \nu^{8} + 969 \nu^{7} + 3314 \nu^{6} - 11655 \nu^{5} - 17675 \nu^{4} + 40482 \nu^{3} + \cdots - 16766 ) / 1852 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 123 \nu^{8} - 319 \nu^{7} - 2512 \nu^{6} + 4459 \nu^{5} + 17635 \nu^{4} - 16684 \nu^{3} - 44216 \nu^{2} + \cdots + 17072 ) / 926 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 172 \nu^{8} - 623 \nu^{7} - 2662 \nu^{6} + 7598 \nu^{5} + 16571 \nu^{4} - 26116 \nu^{3} - 43574 \nu^{2} + \cdots + 21878 ) / 926 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{7} + \beta_{6} + \beta_{2} + \beta _1 + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{8} + 2\beta_{7} + 3\beta_{6} - \beta_{4} + 2\beta_{2} + 8\beta _1 + 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( 5\beta_{8} + 13\beta_{7} + 17\beta_{6} + 3\beta_{5} - 3\beta_{4} + \beta_{3} + 14\beta_{2} + 18\beta _1 + 46 \)
|
| \(\nu^{5}\) | \(=\) |
\( 27\beta_{8} + 42\beta_{7} + 62\beta_{6} + 14\beta_{5} - 23\beta_{4} + 5\beta_{3} + 45\beta_{2} + 90\beta _1 + 116 \)
|
| \(\nu^{6}\) | \(=\) |
\( 122 \beta_{8} + 199 \beta_{7} + 283 \beta_{6} + 87 \beta_{5} - 90 \beta_{4} + 25 \beta_{3} + 215 \beta_{2} + \cdots + 566 \)
|
| \(\nu^{7}\) | \(=\) |
\( 545 \beta_{8} + 765 \beta_{7} + 1117 \beta_{6} + 389 \beta_{5} - 457 \beta_{4} + 113 \beta_{3} + \cdots + 1922 \)
|
| \(\nu^{8}\) | \(=\) |
\( 2345 \beta_{8} + 3300 \beta_{7} + 4758 \beta_{6} + 1848 \beta_{5} - 1895 \beta_{4} + 479 \beta_{3} + \cdots + 8216 \)
|
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 | −3.10215 | 1.00000 | 1.48335 | 3.10215 | 3.05973 | −1.00000 | 6.62334 | −1.48335 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −1.92965 | 1.00000 | 4.03692 | 1.92965 | −2.17250 | −1.00000 | 0.723568 | −4.03692 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −1.34314 | 1.00000 | −3.04006 | 1.34314 | −0.109279 | −1.00000 | −1.19597 | 3.04006 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 0.154444 | 1.00000 | −2.82974 | −0.154444 | −5.26330 | −1.00000 | −2.97615 | 2.82974 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 0.264709 | 1.00000 | 0.0750700 | −0.264709 | 0.786756 | −1.00000 | −2.92993 | −0.0750700 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 1.62849 | 1.00000 | 2.32737 | −1.62849 | 1.25787 | −1.00000 | −0.348031 | −2.32737 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.00000 | 2.85807 | 1.00000 | 4.40377 | −2.85807 | 0.942585 | −1.00000 | 5.16855 | −4.40377 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −1.00000 | 3.16732 | 1.00000 | 1.06273 | −3.16732 | −4.99374 | −1.00000 | 7.03191 | −1.06273 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −1.00000 | 3.30192 | 1.00000 | −1.51941 | −3.30192 | 4.49187 | −1.00000 | 7.90271 | 1.51941 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(11\) | \(-1\) |
| \(61\) | \(1\) |
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 | 1342.2.a.n | ✓ | 9 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1342.2.a.n | ✓ | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1342))\):
|
\( T_{3}^{9} - 5T_{3}^{8} - 11T_{3}^{7} + 78T_{3}^{6} + T_{3}^{5} - 327T_{3}^{4} + 135T_{3}^{3} + 378T_{3}^{2} - 164T_{3} + 16 \)
|
|
\( T_{5}^{9} - 6T_{5}^{8} - 14T_{5}^{7} + 118T_{5}^{6} + 10T_{5}^{5} - 676T_{5}^{4} + 440T_{5}^{3} + 976T_{5}^{2} - 928T_{5} + 64 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{9} \)
|
| $3$ |
\( T^{9} - 5 T^{8} + \cdots + 16 \)
|
| $5$ |
\( T^{9} - 6 T^{8} + \cdots + 64 \)
|
| $7$ |
\( T^{9} + 2 T^{8} + \cdots - 80 \)
|
| $11$ |
\( (T - 1)^{9} \)
|
| $13$ |
\( T^{9} + T^{8} + \cdots - 186072 \)
|
| $17$ |
\( T^{9} - 11 T^{8} + \cdots - 11160 \)
|
| $19$ |
\( T^{9} + 6 T^{8} + \cdots - 19200 \)
|
| $23$ |
\( T^{9} - 21 T^{8} + \cdots + 216320 \)
|
| $29$ |
\( T^{9} + 5 T^{8} + \cdots + 142896 \)
|
| $31$ |
\( T^{9} + 7 T^{8} + \cdots - 489360 \)
|
| $37$ |
\( T^{9} - 16 T^{8} + \cdots - 1665222 \)
|
| $41$ |
\( T^{9} - 13 T^{8} + \cdots - 2754816 \)
|
| $43$ |
\( T^{9} - 188 T^{7} + \cdots - 565184 \)
|
| $47$ |
\( T^{9} - 9 T^{8} + \cdots + 945024 \)
|
| $53$ |
\( T^{9} - 18 T^{8} + \cdots - 4379926 \)
|
| $59$ |
\( T^{9} - T^{8} + \cdots + 1171584 \)
|
| $61$ |
\( (T + 1)^{9} \)
|
| $67$ |
\( T^{9} - 17 T^{8} + \cdots - 20463616 \)
|
| $71$ |
\( T^{9} - 5 T^{8} + \cdots + 301239104 \)
|
| $73$ |
\( T^{9} + 5 T^{8} + \cdots - 394217808 \)
|
| $79$ |
\( T^{9} + 39 T^{8} + \cdots + 1229056 \)
|
| $83$ |
\( T^{9} - 43 T^{8} + \cdots + 42089472 \)
|
| $89$ |
\( T^{9} - 32 T^{8} + \cdots + 92352 \)
|
| $97$ |
\( T^{9} - 22 T^{8} + \cdots + 296222 \)
|
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