Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6525,2,Mod(1,6525)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6525, 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("6525.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6525 = 3^{2} \cdot 5^{2} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6525.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: | \(52.1023873189\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{9} - 2x^{8} - 12x^{7} + 21x^{6} + 48x^{5} - 68x^{4} - 73x^{3} + 66x^{2} + 40x - 10 \)
|
| 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
\( x^{9} - 2x^{8} - 12x^{7} + 21x^{6} + 48x^{5} - 68x^{4} - 73x^{3} + 66x^{2} + 40x - 10 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 4\nu + 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2\nu^{8} - 3\nu^{7} - 19\nu^{6} + 26\nu^{5} + 44\nu^{4} - 75\nu^{3} + 5\nu^{2} + 76\nu - 25 ) / 13 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -2\nu^{8} + 3\nu^{7} + 19\nu^{6} - 13\nu^{5} - 57\nu^{4} - 29\nu^{3} + 47\nu^{2} + 106\nu + 25 ) / 13 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 3\nu^{8} - 11\nu^{7} - 22\nu^{6} + 104\nu^{5} + 27\nu^{4} - 288\nu^{3} + 53\nu^{2} + 192\nu - 31 ) / 13 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -4\nu^{8} + 6\nu^{7} + 51\nu^{6} - 65\nu^{5} - 192\nu^{4} + 189\nu^{3} + 185\nu^{2} - 87\nu - 2 ) / 13 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( \nu^{8} - 8\nu^{7} - 3\nu^{6} + 91\nu^{5} - 43\nu^{4} - 304\nu^{3} + 178\nu^{2} + 259\nu - 71 ) / 13 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 4\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{8} + \beta_{6} + \beta_{5} + \beta_{3} + 7\beta_{2} + \beta _1 + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{8} + \beta_{6} + 2\beta_{5} + \beta_{4} + 9\beta_{3} + 11\beta_{2} + 19\beta _1 + 10 \)
|
| \(\nu^{6}\) | \(=\) |
\( -9\beta_{8} + \beta_{7} + 9\beta_{6} + 10\beta_{5} + 3\beta_{4} + 14\beta_{3} + 49\beta_{2} + 10\beta _1 + 78 \)
|
| \(\nu^{7}\) | \(=\) |
\( -13\beta_{8} + \beta_{7} + 11\beta_{6} + 24\beta_{5} + 16\beta_{4} + 71\beta_{3} + 97\beta_{2} + 98\beta _1 + 89 \)
|
| \(\nu^{8}\) | \(=\) |
\( -70\beta_{8} + 11\beta_{7} + 67\beta_{6} + 83\beta_{5} + 46\beta_{4} + 138\beta_{3} + 349\beta_{2} + 85\beta _1 + 479 \)
|
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 |
|
−2.74387 | 0 | 5.52881 | 0 | 0 | 1.10432 | −9.68257 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.18077 | 0 | 2.75577 | 0 | 0 | −4.62113 | −1.64817 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.77820 | 0 | 1.16198 | 0 | 0 | 2.84862 | 1.49016 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.23642 | 0 | −0.471260 | 0 | 0 | 3.27673 | 3.05552 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.200649 | 0 | −1.95974 | 0 | 0 | −0.775135 | 0.794517 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.747618 | 0 | −1.44107 | 0 | 0 | −3.28783 | −2.57260 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.10241 | 0 | −0.784687 | 0 | 0 | −0.259703 | −3.06987 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.07907 | 0 | 2.32253 | 0 | 0 | −0.602349 | 0.670572 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.21081 | 0 | 2.88766 | 0 | 0 | 1.31646 | 1.96245 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(5\) | \( -1 \) |
| \(29\) | \( -1 \) |
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 | 6525.2.a.ca | ✓ | 9 |
| 3.b | odd | 2 | 1 | 6525.2.a.cc | yes | 9 | |
| 5.b | even | 2 | 1 | 6525.2.a.cd | yes | 9 | |
| 15.d | odd | 2 | 1 | 6525.2.a.cb | yes | 9 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 6525.2.a.ca | ✓ | 9 | 1.a | even | 1 | 1 | trivial |
| 6525.2.a.cb | yes | 9 | 15.d | odd | 2 | 1 | |
| 6525.2.a.cc | yes | 9 | 3.b | odd | 2 | 1 | |
| 6525.2.a.cd | yes | 9 | 5.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6525))\):
|
\( T_{2}^{9} + 2T_{2}^{8} - 12T_{2}^{7} - 21T_{2}^{6} + 48T_{2}^{5} + 68T_{2}^{4} - 73T_{2}^{3} - 66T_{2}^{2} + 40T_{2} + 10 \)
|
|
\( T_{7}^{9} + T_{7}^{8} - 27T_{7}^{7} - 3T_{7}^{6} + 199T_{7}^{5} - 89T_{7}^{4} - 270T_{7}^{3} + 50T_{7}^{2} + 125T_{7} + 25 \)
|
|
\( T_{11}^{9} - 2 T_{11}^{8} - 48 T_{11}^{7} + 124 T_{11}^{6} + 635 T_{11}^{5} - 2064 T_{11}^{4} + \cdots - 600 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} + 2 T^{8} + \cdots + 10 \)
|
| $3$ |
\( T^{9} \)
|
| $5$ |
\( T^{9} \)
|
| $7$ |
\( T^{9} + T^{8} + \cdots + 25 \)
|
| $11$ |
\( T^{9} - 2 T^{8} + \cdots - 600 \)
|
| $13$ |
\( T^{9} + T^{8} + \cdots - 5885 \)
|
| $17$ |
\( T^{9} + 12 T^{8} + \cdots + 1070 \)
|
| $19$ |
\( T^{9} + T^{8} + \cdots - 164800 \)
|
| $23$ |
\( T^{9} + 16 T^{8} + \cdots - 640 \)
|
| $29$ |
\( (T - 1)^{9} \)
|
| $31$ |
\( T^{9} - 5 T^{8} + \cdots + 320 \)
|
| $37$ |
\( T^{9} - 148 T^{7} + \cdots - 11520 \)
|
| $41$ |
\( T^{9} + 10 T^{8} + \cdots + 28800 \)
|
| $43$ |
\( T^{9} + 3 T^{8} + \cdots - 1275840 \)
|
| $47$ |
\( T^{9} + 26 T^{8} + \cdots + 4977120 \)
|
| $53$ |
\( T^{9} + 22 T^{8} + \cdots - 3780480 \)
|
| $59$ |
\( T^{9} - 4 T^{8} + \cdots - 28656000 \)
|
| $61$ |
\( T^{9} - 7 T^{8} + \cdots + 46656 \)
|
| $67$ |
\( T^{9} + 5 T^{8} + \cdots - 2311495 \)
|
| $71$ |
\( T^{9} - 322 T^{7} + \cdots + 73200000 \)
|
| $73$ |
\( T^{9} - 10 T^{8} + \cdots + 29099520 \)
|
| $79$ |
\( T^{9} - 10 T^{8} + \cdots + 18312192 \)
|
| $83$ |
\( T^{9} + 46 T^{8} + \cdots - 1287680 \)
|
| $89$ |
\( T^{9} - 4 T^{8} + \cdots + 1503000 \)
|
| $97$ |
\( T^{9} + 7 T^{8} + \cdots - 88640 \)
|
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