Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6069,2,Mod(1,6069)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6069, 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("6069.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6069 = 3 \cdot 7 \cdot 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6069.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: | \(48.4612089867\) |
| 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: |
\( x^{9} - 12x^{7} - 3x^{6} + 45x^{5} + 21x^{4} - 53x^{3} - 39x^{2} + 3 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| 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
\( x^{9} - 12x^{7} - 3x^{6} + 45x^{5} + 21x^{4} - 53x^{3} - 39x^{2} + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{6} - 9\nu^{4} - 3\nu^{3} + 19\nu^{2} + 10\nu - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{7} + \nu^{6} - 9\nu^{5} - 12\nu^{4} + 16\nu^{3} + 29\nu^{2} + 8\nu - 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{7} - \nu^{6} + 10\nu^{5} + 11\nu^{4} - 23\nu^{3} - 26\nu^{2} + 2\nu + 5 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{8} + \nu^{7} - 10\nu^{6} - 11\nu^{5} + 24\nu^{4} + 25\nu^{3} - 7\nu^{2} - 2\nu + 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( -\nu^{8} + 11\nu^{6} + 2\nu^{5} - 36\nu^{4} - 9\nu^{3} + 37\nu^{2} + 10\nu - 7 \)
|
| \(\beta_{7}\) | \(=\) |
\( -\nu^{8} - 2\nu^{7} + 9\nu^{6} + 21\nu^{5} - 13\nu^{4} - 49\nu^{3} - 18\nu^{2} + 8\nu + 1 \)
|
| \(\beta_{8}\) | \(=\) |
\( -\nu^{8} + \nu^{7} + 12\nu^{6} - 8\nu^{5} - 46\nu^{4} + 13\nu^{3} + 56\nu^{2} + 12\nu - 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + \beta_{5} - \beta_{3} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} + 4\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{8} - \beta_{7} + 7\beta_{6} + 7\beta_{5} + 2\beta_{4} - 8\beta_{3} + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{8} - 8\beta_{7} + 11\beta_{6} + 4\beta_{5} + 10\beta_{4} - 11\beta_{3} + 18\beta _1 + 9 \)
|
| \(\nu^{6}\) | \(=\) |
\( 9\beta_{8} - 12\beta_{7} + 47\beta_{6} + 44\beta_{5} + 21\beta_{4} - 56\beta_{3} + \beta_{2} + 2\beta _1 + 74 \)
|
| \(\nu^{7}\) | \(=\) |
\( 12\beta_{8} - 56\beta_{7} + 91\beta_{6} + 47\beta_{5} + 77\beta_{4} - 93\beta_{3} - \beta_{2} + 88\beta _1 + 74 \)
|
| \(\nu^{8}\) | \(=\) |
\( 65 \beta_{8} - 103 \beta_{7} + 314 \beta_{6} + 277 \beta_{5} + 170 \beta_{4} - 378 \beta_{3} + \cdots + 423 \)
|
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 |
|
−2.17095 | −1.00000 | 2.71304 | −2.62536 | 2.17095 | −1.00000 | −1.54797 | 1.00000 | 5.69953 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.02180 | −1.00000 | 2.08766 | 0.796171 | 2.02180 | −1.00000 | −0.177223 | 1.00000 | −1.60970 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.12893 | −1.00000 | −0.725516 | 3.93352 | 1.12893 | −1.00000 | 3.07692 | 1.00000 | −4.44067 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.665189 | −1.00000 | −1.55752 | −2.61465 | 0.665189 | −1.00000 | 2.36642 | 1.00000 | 1.73924 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.445735 | −1.00000 | −1.80132 | 0.514255 | 0.445735 | −1.00000 | 1.69438 | 1.00000 | −0.229221 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.244764 | −1.00000 | −1.94009 | −2.06843 | −0.244764 | −1.00000 | −0.964391 | 1.00000 | −0.506278 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.77703 | −1.00000 | 1.15784 | 0.740175 | −1.77703 | −1.00000 | −1.49654 | 1.00000 | 1.31531 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.79412 | −1.00000 | 1.21886 | 1.56052 | −1.79412 | −1.00000 | −1.40145 | 1.00000 | 2.79976 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.61669 | −1.00000 | 4.84705 | 2.76381 | −2.61669 | −1.00000 | 7.44985 | 1.00000 | 7.23202 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(7\) | \(1\) |
| \(17\) | \(-1\) |
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 | 6069.2.a.z | ✓ | 9 |
| 17.b | even | 2 | 1 | 6069.2.a.bc | yes | 9 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 6069.2.a.z | ✓ | 9 | 1.a | even | 1 | 1 | trivial |
| 6069.2.a.bc | yes | 9 | 17.b | even | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6069))\):
|
\( T_{2}^{9} - 12T_{2}^{7} - 3T_{2}^{6} + 45T_{2}^{5} + 21T_{2}^{4} - 53T_{2}^{3} - 39T_{2}^{2} + 3 \)
|
|
\( T_{5}^{9} - 3T_{5}^{8} - 18T_{5}^{7} + 49T_{5}^{6} + 93T_{5}^{5} - 261T_{5}^{4} - 60T_{5}^{3} + 456T_{5}^{2} - 333T_{5} + 73 \)
|
|
\( T_{11}^{9} - 18 T_{11}^{8} + 120 T_{11}^{7} - 335 T_{11}^{6} + 171 T_{11}^{5} + 888 T_{11}^{4} + \cdots + 9 \)
|
|
\( T_{23}^{9} - 138 T_{23}^{7} + 106 T_{23}^{6} + 5262 T_{23}^{5} - 7506 T_{23}^{4} - 37764 T_{23}^{3} + \cdots + 3961 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} - 12 T^{7} + \cdots + 3 \)
|
| $3$ |
\( (T + 1)^{9} \)
|
| $5$ |
\( T^{9} - 3 T^{8} + \cdots + 73 \)
|
| $7$ |
\( (T + 1)^{9} \)
|
| $11$ |
\( T^{9} - 18 T^{8} + \cdots + 9 \)
|
| $13$ |
\( T^{9} + 21 T^{8} + \cdots - 321 \)
|
| $17$ |
\( T^{9} \)
|
| $19$ |
\( T^{9} + 9 T^{8} + \cdots - 29103 \)
|
| $23$ |
\( T^{9} - 138 T^{7} + \cdots + 3961 \)
|
| $29$ |
\( T^{9} - 6 T^{8} + \cdots - 7989 \)
|
| $31$ |
\( T^{9} - 30 T^{8} + \cdots + 1619 \)
|
| $37$ |
\( T^{9} - 12 T^{8} + \cdots + 5381 \)
|
| $41$ |
\( T^{9} - 9 T^{8} + \cdots - 31971 \)
|
| $43$ |
\( T^{9} - 249 T^{7} + \cdots + 15984369 \)
|
| $47$ |
\( T^{9} + 9 T^{8} + \cdots - 10232559 \)
|
| $53$ |
\( T^{9} - 324 T^{7} + \cdots - 804383 \)
|
| $59$ |
\( T^{9} + 3 T^{8} + \cdots + 1436957 \)
|
| $61$ |
\( T^{9} - 33 T^{8} + \cdots + 203625 \)
|
| $67$ |
\( T^{9} + 15 T^{8} + \cdots + 36527541 \)
|
| $71$ |
\( T^{9} - 156 T^{7} + \cdots + 61071 \)
|
| $73$ |
\( T^{9} - 21 T^{8} + \cdots + 1478193 \)
|
| $79$ |
\( T^{9} - 30 T^{8} + \cdots + 17 \)
|
| $83$ |
\( T^{9} - 342 T^{7} + \cdots + 604251 \)
|
| $89$ |
\( T^{9} + 12 T^{8} + \cdots + 8233703 \)
|
| $97$ |
\( T^{9} - 438 T^{7} + \cdots - 635977 \)
|
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