Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1815,4,Mod(1,1815)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1815, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1815.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1815 = 3 \cdot 5 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1815.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: | \(107.088466660\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{7} - x^{6} - 52x^{5} + 37x^{4} + 765x^{3} - 296x^{2} - 2962x + 1692 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{7} - x^{6} - 52x^{5} + 37x^{4} + 765x^{3} - 296x^{2} - 2962x + 1692 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 15 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{6} - 9\nu^{5} + 56\nu^{4} + 335\nu^{3} - 893\nu^{2} - 2336\nu + 3384 ) / 94 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{6} - 9\nu^{5} + 56\nu^{4} + 429\nu^{3} - 893\nu^{2} - 4498\nu + 3008 ) / 94 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -4\nu^{6} + 11\nu^{5} + 177\nu^{4} - 399\nu^{3} - 1786\nu^{2} + 2312\nu + 1551 ) / 47 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 15\nu^{6} - 53\nu^{5} - 558\nu^{4} + 2025\nu^{3} + 3243\nu^{2} - 14028\nu + 9588 ) / 94 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 15 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} - \beta_{3} + 23\beta _1 + 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{6} + 2\beta_{5} - \beta_{4} + 32\beta_{2} + 3\beta _1 + 344 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{6} + 3\beta_{5} + 36\beta_{4} - 45\beta_{3} - 6\beta_{2} + 606\beta _1 + 177 \)
|
| \(\nu^{6}\) | \(=\) |
\( 47\beta_{6} + 85\beta_{5} - 45\beta_{4} - 24\beta_{3} + 953\beta_{2} + 83\beta _1 + 9000 \)
|
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 |
|
−5.40561 | −3.00000 | 21.2207 | 5.00000 | 16.2168 | −22.3833 | −71.4658 | 9.00000 | −27.0281 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −3.36428 | −3.00000 | 3.31835 | 5.00000 | 10.0928 | 16.1008 | 15.7504 | 9.00000 | −16.8214 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.97676 | −3.00000 | 0.861071 | 5.00000 | 8.93027 | 1.47808 | 21.2508 | 9.00000 | −14.8838 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.589700 | −3.00000 | −7.65225 | 5.00000 | −1.76910 | −20.2476 | −9.23013 | 9.00000 | 2.94850 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 2.12247 | −3.00000 | −3.49513 | 5.00000 | −6.36740 | 18.5956 | −24.3980 | 9.00000 | 10.6123 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 4.56895 | −3.00000 | 12.8753 | 5.00000 | −13.7069 | −33.6877 | 22.2751 | 9.00000 | 22.8448 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 5.46553 | −3.00000 | 21.8720 | 5.00000 | −16.3966 | 24.1442 | 75.8177 | 9.00000 | 27.3276 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(5\) | \(-1\) |
| \(11\) | \(-1\) |
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 | 1815.4.a.bd | yes | 7 |
| 11.b | odd | 2 | 1 | 1815.4.a.bc | ✓ | 7 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1815.4.a.bc | ✓ | 7 | 11.b | odd | 2 | 1 | |
| 1815.4.a.bd | yes | 7 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1815))\):
|
\( T_{2}^{7} - T_{2}^{6} - 52T_{2}^{5} + 37T_{2}^{4} + 765T_{2}^{3} - 296T_{2}^{2} - 2962T_{2} + 1692 \)
|
|
\( T_{7}^{7} + 16T_{7}^{6} - 1490T_{7}^{5} - 14184T_{7}^{4} + 722529T_{7}^{3} + 2670992T_{7}^{2} - 115840784T_{7} + 163131144 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - T^{6} + \cdots + 1692 \)
|
| $3$ |
\( (T + 3)^{7} \)
|
| $5$ |
\( (T - 5)^{7} \)
|
| $7$ |
\( T^{7} + 16 T^{6} + \cdots + 163131144 \)
|
| $11$ |
\( T^{7} \)
|
| $13$ |
\( T^{7} + \cdots - 609973453952 \)
|
| $17$ |
\( T^{7} + \cdots - 78973270549092 \)
|
| $19$ |
\( T^{7} + \cdots - 26372327985216 \)
|
| $23$ |
\( T^{7} + \cdots + 100765569878688 \)
|
| $29$ |
\( T^{7} + \cdots + 90\!\cdots\!12 \)
|
| $31$ |
\( T^{7} + \cdots + 130509871517520 \)
|
| $37$ |
\( T^{7} + \cdots + 478375739919752 \)
|
| $41$ |
\( T^{7} + \cdots + 26\!\cdots\!08 \)
|
| $43$ |
\( T^{7} + \cdots + 19\!\cdots\!28 \)
|
| $47$ |
\( T^{7} + \cdots + 22\!\cdots\!72 \)
|
| $53$ |
\( T^{7} + \cdots + 17\!\cdots\!24 \)
|
| $59$ |
\( T^{7} + \cdots + 950579470970880 \)
|
| $61$ |
\( T^{7} + \cdots - 29\!\cdots\!52 \)
|
| $67$ |
\( T^{7} + \cdots - 39\!\cdots\!68 \)
|
| $71$ |
\( T^{7} + \cdots + 18\!\cdots\!20 \)
|
| $73$ |
\( T^{7} + \cdots - 55\!\cdots\!44 \)
|
| $79$ |
\( T^{7} + \cdots + 38\!\cdots\!56 \)
|
| $83$ |
\( T^{7} + \cdots + 20\!\cdots\!16 \)
|
| $89$ |
\( T^{7} + \cdots + 30\!\cdots\!12 \)
|
| $97$ |
\( T^{7} + \cdots - 45\!\cdots\!60 \)
|
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