Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1075,4,Mod(1,1075)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1075, 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("1075.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1075 = 5^{2} \cdot 43 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1075.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: | \(63.4270532562\) |
| 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: |
\( x^{6} - x^{5} - 25x^{4} + 13x^{3} + 144x^{2} + 20x - 32 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 215) |
| 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
\( x^{6} - x^{5} - 25x^{4} + 13x^{3} + 144x^{2} + 20x - 32 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 14\nu + 4 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - 19\nu^{2} - 2\nu + 40 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} - \nu^{4} - 23\nu^{3} + 13\nu^{2} + 114\nu + 8 ) / 8 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 8 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} + \beta_{2} + 15\beta _1 + 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( 4\beta_{4} + 19\beta_{2} + 21\beta _1 + 112 \)
|
| \(\nu^{5}\) | \(=\) |
\( 8\beta_{5} + 4\beta_{4} + 46\beta_{3} + 29\beta_{2} + 239\beta _1 + 92 \)
|
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 |
|
−3.24792 | 3.67520 | 2.54900 | 0 | −11.9368 | 6.14699 | 17.7044 | −13.4929 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.31051 | −2.63579 | −2.66155 | 0 | 6.09000 | 24.0652 | 24.6336 | −20.0526 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0.594297 | 7.85474 | −7.64681 | 0 | 4.66805 | −17.8090 | −9.29885 | 34.6969 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.58363 | −5.21468 | −5.49210 | 0 | −8.25814 | 13.3750 | −21.3666 | 0.192863 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 3.43689 | 3.37907 | 3.81223 | 0 | 11.6135 | 12.3943 | −14.3929 | −15.5819 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 4.94361 | −2.05855 | 16.4392 | 0 | −10.1767 | −4.17251 | 41.7203 | −22.7624 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
| \(43\) | \(-1\) |
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 | 1075.4.a.c | 6 | |
| 5.b | even | 2 | 1 | 215.4.a.a | ✓ | 6 | |
| 15.d | odd | 2 | 1 | 1935.4.a.c | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 215.4.a.a | ✓ | 6 | 5.b | even | 2 | 1 | |
| 1075.4.a.c | 6 | 1.a | even | 1 | 1 | trivial | |
| 1935.4.a.c | 6 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 5T_{2}^{5} - 15T_{2}^{4} + 77T_{2}^{3} + 38T_{2}^{2} - 248T_{2} + 120 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1075))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 5 T^{5} + \cdots + 120 \)
|
| $3$ |
\( T^{6} - 5 T^{5} + \cdots - 2760 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} - 34 T^{5} + \cdots + 1822235 \)
|
| $11$ |
\( T^{6} + 35 T^{5} + \cdots - 5558216 \)
|
| $13$ |
\( T^{6} - 41 T^{5} + \cdots - 74056520 \)
|
| $17$ |
\( T^{6} + \cdots + 1109749632 \)
|
| $19$ |
\( T^{6} + \cdots + 70924813928 \)
|
| $23$ |
\( T^{6} + \cdots + 9295570496 \)
|
| $29$ |
\( T^{6} + \cdots + 17097019832 \)
|
| $31$ |
\( T^{6} + \cdots - 398145639081 \)
|
| $37$ |
\( T^{6} + \cdots - 1355647567000 \)
|
| $41$ |
\( T^{6} + \cdots - 78719999235775 \)
|
| $43$ |
\( (T - 43)^{6} \)
|
| $47$ |
\( T^{6} + \cdots + 180770611219200 \)
|
| $53$ |
\( T^{6} + \cdots + 2307009075200 \)
|
| $59$ |
\( T^{6} + \cdots - 89278005357984 \)
|
| $61$ |
\( T^{6} + \cdots + 66378840012088 \)
|
| $67$ |
\( T^{6} + \cdots - 26\!\cdots\!44 \)
|
| $71$ |
\( T^{6} + \cdots - 127688557613472 \)
|
| $73$ |
\( T^{6} + \cdots - 19\!\cdots\!03 \)
|
| $79$ |
\( T^{6} + \cdots + 15\!\cdots\!95 \)
|
| $83$ |
\( T^{6} + \cdots + 52\!\cdots\!00 \)
|
| $89$ |
\( T^{6} + \cdots - 73\!\cdots\!04 \)
|
| $97$ |
\( T^{6} + \cdots - 10\!\cdots\!72 \)
|
| show more | |
| show less |