Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2299,2,Mod(1,2299)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2299, 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("2299.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2299 = 11^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2299.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: | \(18.3576074247\) |
| Analytic rank: | \(1\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 2x^{6} - 8x^{5} + 14x^{4} + 17x^{3} - 18x^{2} - 16x - 1 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 2x^{6} - 8x^{5} + 14x^{4} + 17x^{3} - 18x^{2} - 16x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 7\nu^{3} + 5\nu^{2} + 8\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{5} + \nu^{4} + 8\nu^{3} - 5\nu^{2} - 13\nu - 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{6} - 8\nu^{4} - 3\nu^{3} + 13\nu^{2} + 13\nu + 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{6} - 9\nu^{4} - 2\nu^{3} + 19\nu^{2} + 8\nu - 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( -\nu^{6} + 8\nu^{4} + 3\nu^{3} - 12\nu^{2} - 14\nu - 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + \beta_{4} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 6\beta_{6} - \beta_{5} + 7\beta_{4} + \beta_{3} + \beta_{2} + 6\beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{6} - \beta_{5} + 2\beta_{4} + 8\beta_{3} + 9\beta_{2} + 28\beta _1 + 7 \)
|
| \(\nu^{6}\) | \(=\) |
\( 35\beta_{6} - 8\beta_{5} + 44\beta_{4} + 11\beta_{3} + 11\beta_{2} + 37\beta _1 + 82 \)
|
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 |
|
−2.50285 | 1.59648 | 4.26424 | 0.630803 | −3.99574 | −0.249675 | −5.66706 | −0.451261 | −1.57880 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.87135 | −1.09616 | 1.50197 | −4.30775 | 2.05131 | 3.98475 | 0.931995 | −1.79843 | 8.06132 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.79190 | −2.48473 | 1.21090 | 0.260416 | 4.45238 | −3.76880 | 1.41399 | 3.17387 | −0.466638 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.0680200 | −0.141369 | −1.99537 | 3.58517 | −0.00961589 | −3.20234 | −0.271765 | −2.98001 | 0.243863 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.692399 | 3.33751 | −1.52058 | −0.880738 | 2.31089 | −3.64547 | −2.43765 | 8.13895 | −0.609822 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.09476 | −0.248246 | −0.801502 | 1.63186 | −0.271770 | 4.48973 | −3.06697 | −2.93837 | 1.78650 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.31092 | −1.96348 | 3.34035 | −1.91976 | −4.53745 | 0.391807 | 3.09745 | 0.855254 | −4.43642 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(11\) | \( -1 \) |
| \(19\) | \( -1 \) |
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 | 2299.2.a.p | ✓ | 7 |
| 11.b | odd | 2 | 1 | 2299.2.a.r | yes | 7 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2299.2.a.p | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 2299.2.a.r | yes | 7 | 11.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2299))\):
|
\( T_{2}^{7} + 2T_{2}^{6} - 8T_{2}^{5} - 14T_{2}^{4} + 17T_{2}^{3} + 18T_{2}^{2} - 16T_{2} + 1 \)
|
|
\( T_{3}^{7} + T_{3}^{6} - 12T_{3}^{5} - 18T_{3}^{4} + 20T_{3}^{3} + 38T_{3}^{2} + 12T_{3} + 1 \)
|
|
\( T_{7}^{7} + 2T_{7}^{6} - 35T_{7}^{5} - 79T_{7}^{4} + 313T_{7}^{3} + 753T_{7}^{2} - 141T_{7} - 77 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + 2 T^{6} + \cdots + 1 \)
|
| $3$ |
\( T^{7} + T^{6} - 12 T^{5} + \cdots + 1 \)
|
| $5$ |
\( T^{7} + T^{6} - 19 T^{5} + \cdots + 7 \)
|
| $7$ |
\( T^{7} + 2 T^{6} + \cdots - 77 \)
|
| $11$ |
\( T^{7} \)
|
| $13$ |
\( T^{7} + 12 T^{6} + \cdots + 7777 \)
|
| $17$ |
\( T^{7} + 12 T^{6} + \cdots - 7 \)
|
| $19$ |
\( (T - 1)^{7} \)
|
| $23$ |
\( T^{7} + 9 T^{6} + \cdots - 3789 \)
|
| $29$ |
\( T^{7} + 18 T^{6} + \cdots - 1363 \)
|
| $31$ |
\( T^{7} + 5 T^{6} + \cdots - 368903 \)
|
| $37$ |
\( T^{7} - 4 T^{6} + \cdots + 10109 \)
|
| $41$ |
\( T^{7} + 25 T^{6} + \cdots - 61253 \)
|
| $43$ |
\( T^{7} - 10 T^{6} + \cdots - 1963 \)
|
| $47$ |
\( T^{7} + 14 T^{6} + \cdots + 98963 \)
|
| $53$ |
\( T^{7} + 15 T^{6} + \cdots + 16651 \)
|
| $59$ |
\( T^{7} + 9 T^{6} + \cdots + 50353 \)
|
| $61$ |
\( T^{7} + 3 T^{6} + \cdots + 1310837 \)
|
| $67$ |
\( T^{7} + 15 T^{6} + \cdots + 484987 \)
|
| $71$ |
\( T^{7} - 13 T^{6} + \cdots - 3979 \)
|
| $73$ |
\( T^{7} - 28 T^{6} + \cdots + 304013 \)
|
| $79$ |
\( T^{7} + 18 T^{6} + \cdots - 108143 \)
|
| $83$ |
\( T^{7} - 10 T^{6} + \cdots + 158647 \)
|
| $89$ |
\( T^{7} - 2 T^{6} + \cdots + 2211325 \)
|
| $97$ |
\( T^{7} - 29 T^{6} + \cdots + 2132767 \)
|
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