Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5915,2,Mod(1,5915)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5915, 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("5915.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5915 = 5 \cdot 7 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5915.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: | \(47.2315127956\) |
| 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: |
\( x^{9} - 15x^{7} + 70x^{5} - 3x^{4} - 108x^{3} + 3x^{2} + 36x + 8 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 455) |
| 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
\( x^{9} - 15x^{7} + 70x^{5} - 3x^{4} - 108x^{3} + 3x^{2} + 36x + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 3\nu^{8} - \nu^{7} - 44\nu^{6} + 14\nu^{5} + 198\nu^{4} - 71\nu^{3} - 279\nu^{2} + 102\nu + 60 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 4\nu^{8} - \nu^{7} - 59\nu^{6} + 16\nu^{5} + 266\nu^{4} - 92\nu^{3} - 373\nu^{2} + 137\nu + 80 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 5\nu^{8} - \nu^{7} - 74\nu^{6} + 16\nu^{5} + 336\nu^{4} - 97\nu^{3} - 479\nu^{2} + 154\nu + 106 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -5\nu^{8} + 2\nu^{7} + 75\nu^{6} - 30\nu^{5} - 348\nu^{4} + 153\nu^{3} + 514\nu^{2} - 213\nu - 126 ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -8\nu^{8} + 3\nu^{7} + 119\nu^{6} - 44\nu^{5} - 546\nu^{4} + 222\nu^{3} + 793\nu^{2} - 305\nu - 186 ) / 2 \)
|
| \(\beta_{8}\) | \(=\) |
\( -9\nu^{8} + 3\nu^{7} + 134\nu^{6} - 45\nu^{5} - 614\nu^{4} + 235\nu^{3} + 886\nu^{2} - 331\nu - 205 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{7} + \beta_{6} - \beta_{3} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{8} - 2\beta_{7} + \beta_{5} - \beta_{3} + 7\beta_{2} + 17 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{8} - 10\beta_{7} + 8\beta_{6} + 2\beta_{4} - 10\beta_{3} + \beta_{2} + 31\beta _1 + 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( 12\beta_{8} - 24\beta_{7} + \beta_{6} + 10\beta_{5} + 3\beta_{4} - 11\beta_{3} + 47\beta_{2} + 4\beta _1 + 112 \)
|
| \(\nu^{7}\) | \(=\) |
\( 14\beta_{8} - 84\beta_{7} + 57\beta_{6} + 4\beta_{5} + 25\beta_{4} - 85\beta_{3} + 16\beta_{2} + 209\beta _1 + 35 \)
|
| \(\nu^{8}\) | \(=\) |
\( 110 \beta_{8} - 225 \beta_{7} + 20 \beta_{6} + 82 \beta_{5} + 43 \beta_{4} - 100 \beta_{3} + 321 \beta_{2} + \cdots + 782 \)
|
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 |
|
−2.59200 | −0.475581 | 4.71844 | −1.00000 | 1.23270 | −1.00000 | −7.04620 | −2.77382 | 2.59200 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.41652 | 3.17117 | 3.83956 | −1.00000 | −7.66318 | −1.00000 | −4.44533 | 7.05629 | 2.41652 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.47414 | 0.507154 | 0.173103 | −1.00000 | −0.747619 | −1.00000 | 2.69311 | −2.74279 | 1.47414 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.356556 | 2.58545 | −1.87287 | −1.00000 | −0.921856 | −1.00000 | 1.38089 | 3.68455 | 0.356556 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.340927 | −2.81640 | −1.88377 | −1.00000 | 0.960187 | −1.00000 | 1.32408 | 4.93211 | 0.340927 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.838328 | −0.624111 | −1.29721 | −1.00000 | −0.523210 | −1.00000 | −2.76414 | −2.61049 | −0.838328 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.46507 | 2.66524 | 0.146416 | −1.00000 | 3.90476 | −1.00000 | −2.71562 | 4.10353 | −1.46507 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.06090 | −0.320757 | 2.24729 | −1.00000 | −0.661047 | −1.00000 | 0.509645 | −2.89711 | −2.06090 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.81585 | −2.69216 | 5.92903 | −1.00000 | −7.58074 | −1.00000 | 11.0636 | 4.24775 | −2.81585 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
| \(7\) | \(1\) |
| \(13\) | \(-1\) |
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 | 5915.2.a.bc | 9 | |
| 13.b | even | 2 | 1 | 5915.2.a.bd | 9 | ||
| 13.d | odd | 4 | 2 | 455.2.d.b | ✓ | 18 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 455.2.d.b | ✓ | 18 | 13.d | odd | 4 | 2 | |
| 5915.2.a.bc | 9 | 1.a | even | 1 | 1 | trivial | |
| 5915.2.a.bd | 9 | 13.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5915))\):
|
\( T_{2}^{9} - 15T_{2}^{7} + 70T_{2}^{5} - 3T_{2}^{4} - 108T_{2}^{3} + 3T_{2}^{2} + 36T_{2} + 8 \)
|
|
\( T_{3}^{9} - 2T_{3}^{8} - 18T_{3}^{7} + 30T_{3}^{6} + 100T_{3}^{5} - 112T_{3}^{4} - 165T_{3}^{3} - 4T_{3}^{2} + 36T_{3} + 8 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} - 15 T^{7} + \cdots + 8 \)
|
| $3$ |
\( T^{9} - 2 T^{8} + \cdots + 8 \)
|
| $5$ |
\( (T + 1)^{9} \)
|
| $7$ |
\( (T + 1)^{9} \)
|
| $11$ |
\( T^{9} + 8 T^{8} + \cdots + 6528 \)
|
| $13$ |
\( T^{9} \)
|
| $17$ |
\( T^{9} - 6 T^{8} + \cdots + 19904 \)
|
| $19$ |
\( T^{9} + 6 T^{8} + \cdots + 1376 \)
|
| $23$ |
\( T^{9} - 12 T^{8} + \cdots - 363008 \)
|
| $29$ |
\( T^{9} - 22 T^{8} + \cdots + 23787832 \)
|
| $31$ |
\( T^{9} + 20 T^{8} + \cdots - 1191104 \)
|
| $37$ |
\( T^{9} + 4 T^{8} + \cdots - 749728 \)
|
| $41$ |
\( T^{9} + 8 T^{8} + \cdots - 1757504 \)
|
| $43$ |
\( T^{9} - 10 T^{8} + \cdots - 23470592 \)
|
| $47$ |
\( T^{9} - 4 T^{8} + \cdots + 846848 \)
|
| $53$ |
\( T^{9} - 22 T^{8} + \cdots + 187392 \)
|
| $59$ |
\( T^{9} + 8 T^{8} + \cdots - 532032 \)
|
| $61$ |
\( T^{9} - 8 T^{8} + \cdots - 268908544 \)
|
| $67$ |
\( T^{9} - 16 T^{8} + \cdots + 1039872 \)
|
| $71$ |
\( T^{9} + 12 T^{8} + \cdots - 957952 \)
|
| $73$ |
\( T^{9} + 10 T^{8} + \cdots - 512 \)
|
| $79$ |
\( T^{9} + 2 T^{8} + \cdots + 115727936 \)
|
| $83$ |
\( T^{9} + 16 T^{8} + \cdots + 3293184 \)
|
| $89$ |
\( T^{9} + 34 T^{8} + \cdots - 43511808 \)
|
| $97$ |
\( T^{9} + 12 T^{8} + \cdots + 32664448 \)
|
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