Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1150,4,Mod(1,1150)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1150, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("1150.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1150 = 2 \cdot 5^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1150.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: | \(67.8521965066\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 109x^{4} + 94x^{3} + 2808x^{2} + 81x - 9774 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 109x^{4} + 94x^{3} + 2808x^{2} + 81x - 9774 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -113\nu^{5} + 65\nu^{4} + 11996\nu^{3} - 20168\nu^{2} - 281946\nu + 418401 ) / 39393 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 112\nu^{5} + 749\nu^{4} - 3988\nu^{3} - 29978\nu^{2} - 157242\nu - 298611 ) / 39393 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 16\nu^{5} + 107\nu^{4} - 1195\nu^{3} - 7409\nu^{2} + 15054\nu + 70518 ) / 4377 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 194\nu^{5} - 344\nu^{4} - 17225\nu^{3} + 37646\nu^{2} + 271164\nu - 351927 ) / 39393 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{5} - \beta_{4} + \beta_{3} - 2\beta_{2} + 36 \)
|
| \(\nu^{3}\) | \(=\) |
\( 5\beta_{5} - 2\beta_{4} + 4\beta_{3} + 10\beta_{2} + 60\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -110\beta_{5} - 42\beta_{4} + 71\beta_{3} - 172\beta_{2} - 46\beta _1 + 2059 \)
|
| \(\nu^{5}\) | \(=\) |
\( 646\beta_{5} - 58\beta_{4} + 287\beta_{3} + 971\beta_{2} + 3848\beta _1 - 1432 \)
|
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 |
|
2.00000 | −9.71212 | 4.00000 | 0 | −19.4242 | −33.1288 | 8.00000 | 67.3252 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 2.00000 | −5.90184 | 4.00000 | 0 | −11.8037 | −1.63515 | 8.00000 | 7.83172 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 2.00000 | −3.16288 | 4.00000 | 0 | −6.32576 | 3.52475 | 8.00000 | −16.9962 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 2.00000 | 0.926620 | 4.00000 | 0 | 1.85324 | 28.2323 | 8.00000 | −26.1414 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.00000 | 5.96868 | 4.00000 | 0 | 11.9374 | −15.2484 | 8.00000 | 8.62516 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.00000 | 6.88153 | 4.00000 | 0 | 13.7631 | −23.7447 | 8.00000 | 20.3555 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(1\) |
| \(23\) | \(1\) |
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 | 1150.4.a.x | yes | 6 |
| 5.b | even | 2 | 1 | 1150.4.a.w | ✓ | 6 | |
| 5.c | odd | 4 | 2 | 1150.4.b.s | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1150.4.a.w | ✓ | 6 | 5.b | even | 2 | 1 | |
| 1150.4.a.x | yes | 6 | 1.a | even | 1 | 1 | trivial |
| 1150.4.b.s | 12 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1150))\):
|
\( T_{3}^{6} + 5T_{3}^{5} - 99T_{3}^{4} - 332T_{3}^{3} + 2441T_{3}^{2} + 5544T_{3} - 6900 \)
|
|
\( T_{7}^{6} + 42T_{7}^{5} - 471T_{7}^{4} - 34228T_{7}^{3} - 270876T_{7}^{2} + 839880T_{7} + 1951776 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{6} \)
|
| $3$ |
\( T^{6} + 5 T^{5} + \cdots - 6900 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + 42 T^{5} + \cdots + 1951776 \)
|
| $11$ |
\( T^{6} + 49 T^{5} + \cdots - 17113032 \)
|
| $13$ |
\( T^{6} + \cdots - 6485194206 \)
|
| $17$ |
\( T^{6} + \cdots - 39615096960 \)
|
| $19$ |
\( T^{6} + \cdots + 315942692928 \)
|
| $23$ |
\( (T + 23)^{6} \)
|
| $29$ |
\( T^{6} + \cdots + 194358059568 \)
|
| $31$ |
\( T^{6} + \cdots + 1219003806240 \)
|
| $37$ |
\( T^{6} + \cdots - 6510888786112 \)
|
| $41$ |
\( T^{6} + \cdots - 190890178116489 \)
|
| $43$ |
\( T^{6} + \cdots - 8699897931040 \)
|
| $47$ |
\( T^{6} + \cdots - 126686028693600 \)
|
| $53$ |
\( T^{6} + \cdots + 309174347367936 \)
|
| $59$ |
\( T^{6} + \cdots + 98\!\cdots\!52 \)
|
| $61$ |
\( T^{6} + \cdots - 388743615564288 \)
|
| $67$ |
\( T^{6} + \cdots - 71\!\cdots\!60 \)
|
| $71$ |
\( T^{6} + \cdots - 10\!\cdots\!20 \)
|
| $73$ |
\( T^{6} + \cdots + 930500741801387 \)
|
| $79$ |
\( T^{6} + \cdots + 31\!\cdots\!76 \)
|
| $83$ |
\( T^{6} + \cdots - 57\!\cdots\!64 \)
|
| $89$ |
\( T^{6} + \cdots - 13\!\cdots\!32 \)
|
| $97$ |
\( T^{6} + \cdots - 40\!\cdots\!56 \)
|
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