Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9248,2,Mod(1,9248)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9248, 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("9248.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9248 = 2^{5} \cdot 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9248.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: | \(73.8456517893\) |
| 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: |
\( x^{9} - 15x^{7} - 7x^{6} + 69x^{5} + 48x^{4} - 110x^{3} - 87x^{2} + 45x + 37 \)
|
| 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
\( x^{9} - 15x^{7} - 7x^{6} + 69x^{5} + 48x^{4} - 110x^{3} - 87x^{2} + 45x + 37 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} - 3\nu^{6} - 8\nu^{5} + 22\nu^{4} + 19\nu^{3} - 43\nu^{2} - 13\nu + 19 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{8} - 3\nu^{7} - 8\nu^{6} + 22\nu^{5} + 19\nu^{4} - 43\nu^{3} - 13\nu^{2} + 19\nu ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 3\nu^{8} - 8\nu^{7} - 27\nu^{6} + 60\nu^{5} + 75\nu^{4} - 122\nu^{3} - 68\nu^{2} + 54\nu + 17 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 3\nu^{8} - 6\nu^{7} - 33\nu^{6} + 44\nu^{5} + 121\nu^{4} - 88\nu^{3} - 166\nu^{2} + 42\nu + 65 ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 4\nu^{8} - 7\nu^{7} - 46\nu^{6} + 48\nu^{5} + 176\nu^{4} - 83\nu^{3} - 250\nu^{2} + 27\nu + 98 ) / 2 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 5\nu^{8} - 9\nu^{7} - 57\nu^{6} + 64\nu^{5} + 213\nu^{4} - 121\nu^{3} - 290\nu^{2} + 51\nu + 111 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{8} + \beta_{7} + \beta_{5} - 2\beta_{4} + \beta_{2} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -2\beta_{8} + 2\beta_{7} + \beta_{6} + \beta_{5} - 4\beta_{4} - 2\beta_{3} + 8\beta_{2} + 9\beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( -10\beta_{8} + 10\beta_{7} + 2\beta_{6} + 9\beta_{5} - 23\beta_{4} - 5\beta_{3} + 15\beta_{2} + 36\beta _1 + 16 \)
|
| \(\nu^{6}\) | \(=\) |
\( -26\beta_{8} + 28\beta_{7} + 10\beta_{6} + 16\beta_{5} - 60\beta_{4} - 30\beta_{3} + 69\beta_{2} + 87\beta _1 + 102 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 95 \beta_{8} + 101 \beta_{7} + 24 \beta_{6} + 79 \beta_{5} - 238 \beta_{4} - 84 \beta_{3} + \cdots + 195 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 278 \beta_{8} + 312 \beta_{7} + 89 \beta_{6} + 191 \beta_{5} - 696 \beta_{4} - 344 \beta_{3} + \cdots + 846 \)
|
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 |
|
0 | −3.10579 | 0 | −2.03395 | 0 | 0.883659 | 0 | 6.64591 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −3.09095 | 0 | −3.92598 | 0 | −3.26034 | 0 | 6.55400 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −2.62107 | 0 | 0.309248 | 0 | −4.59618 | 0 | 3.87001 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.89421 | 0 | 2.78809 | 0 | −0.147450 | 0 | 0.588025 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −1.77501 | 0 | −2.79665 | 0 | 3.67713 | 0 | 0.150658 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | −0.214835 | 0 | −3.35988 | 0 | −2.68602 | 0 | −2.95385 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 0.453073 | 0 | 0.322684 | 0 | 0.528404 | 0 | −2.79472 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 1.04878 | 0 | 2.71408 | 0 | 1.45113 | 0 | −1.90005 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 2.20001 | 0 | −0.0176497 | 0 | 1.14966 | 0 | 1.84003 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(17\) | \( -1 \) |
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 | 9248.2.a.bo | ✓ | 9 |
| 4.b | odd | 2 | 1 | 9248.2.a.bq | yes | 9 | |
| 17.b | even | 2 | 1 | 9248.2.a.br | yes | 9 | |
| 68.d | odd | 2 | 1 | 9248.2.a.bp | yes | 9 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 9248.2.a.bo | ✓ | 9 | 1.a | even | 1 | 1 | trivial |
| 9248.2.a.bp | yes | 9 | 68.d | odd | 2 | 1 | |
| 9248.2.a.bq | yes | 9 | 4.b | odd | 2 | 1 | |
| 9248.2.a.br | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9248))\):
|
\( T_{3}^{9} + 9T_{3}^{8} + 21T_{3}^{7} - 28T_{3}^{6} - 162T_{3}^{5} - 111T_{3}^{4} + 191T_{3}^{3} + 177T_{3}^{2} - 60T_{3} - 19 \)
|
|
\( T_{5}^{9} + 6T_{5}^{8} - 9T_{5}^{7} - 97T_{5}^{6} - 33T_{5}^{5} + 432T_{5}^{4} + 324T_{5}^{3} - 315T_{5}^{2} + 51T_{5} + 1 \)
|
|
\( T_{7}^{9} + 3T_{7}^{8} - 24T_{7}^{7} - 49T_{7}^{6} + 177T_{7}^{5} + 99T_{7}^{4} - 521T_{7}^{3} + 378T_{7}^{2} - 48T_{7} - 17 \)
|
|
\( T_{19}^{9} - 57T_{19}^{7} - 41T_{19}^{6} + 873T_{19}^{5} + 870T_{19}^{4} - 3862T_{19}^{3} - 1731T_{19}^{2} + 6663T_{19} - 2447 \)
|
|
\( T_{43}^{9} - 3 T_{43}^{8} - 339 T_{43}^{7} + 1222 T_{43}^{6} + 38238 T_{43}^{5} - 144075 T_{43}^{4} + \cdots + 8890633 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} \)
|
| $3$ |
\( T^{9} + 9 T^{8} + \cdots - 19 \)
|
| $5$ |
\( T^{9} + 6 T^{8} + \cdots + 1 \)
|
| $7$ |
\( T^{9} + 3 T^{8} + \cdots - 17 \)
|
| $11$ |
\( T^{9} + 6 T^{8} + \cdots - 4672 \)
|
| $13$ |
\( T^{9} + 12 T^{8} + \cdots - 29179 \)
|
| $17$ |
\( T^{9} \)
|
| $19$ |
\( T^{9} - 57 T^{7} + \cdots - 2447 \)
|
| $23$ |
\( T^{9} - 3 T^{8} + \cdots - 71 \)
|
| $29$ |
\( T^{9} + 21 T^{8} + \cdots - 198917 \)
|
| $31$ |
\( T^{9} + 24 T^{8} + \cdots - 239581 \)
|
| $37$ |
\( T^{9} + 12 T^{8} + \cdots - 8390231 \)
|
| $41$ |
\( T^{9} - 15 T^{8} + \cdots - 1549 \)
|
| $43$ |
\( T^{9} - 3 T^{8} + \cdots + 8890633 \)
|
| $47$ |
\( T^{9} - 12 T^{8} + \cdots + 384067 \)
|
| $53$ |
\( T^{9} - 9 T^{8} + \cdots - 503488 \)
|
| $59$ |
\( T^{9} + 45 T^{8} + \cdots - 1677943 \)
|
| $61$ |
\( T^{9} + 3 T^{8} + \cdots + 358085807 \)
|
| $67$ |
\( T^{9} + 6 T^{8} + \cdots + 4191337 \)
|
| $71$ |
\( T^{9} - 9 T^{8} + \cdots + 41761781 \)
|
| $73$ |
\( T^{9} - 30 T^{8} + \cdots - 9721 \)
|
| $79$ |
\( T^{9} - 18 T^{8} + \cdots - 6839288 \)
|
| $83$ |
\( T^{9} - 9 T^{8} + \cdots + 9609011 \)
|
| $89$ |
\( T^{9} - 504 T^{7} + \cdots + 10434311 \)
|
| $97$ |
\( T^{9} - 33 T^{8} + \cdots - 19741184 \)
|
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