Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1925,2,Mod(1,1925)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1925, 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("1925.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1925 = 5^{2} \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1925.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: | \(15.3712023891\) |
| 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: |
\( x^{7} - x^{6} - 13x^{5} + 12x^{4} + 47x^{3} - 37x^{2} - 35x + 6 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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
\( x^{7} - x^{6} - 13x^{5} + 12x^{4} + 47x^{3} - 37x^{2} - 35x + 6 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 6\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{6} - 11\nu^{4} - \nu^{3} + 30\nu^{2} + 3\nu - 10 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} + \nu^{4} - 10\nu^{3} - 9\nu^{2} + 21\nu + 12 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{6} + 2\nu^{5} - 13\nu^{4} - 17\nu^{3} + 40\nu^{2} + 25\nu - 6 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 7\beta_{2} + \beta _1 + 23 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{6} + \beta_{5} - \beta_{4} + 9\beta_{3} + 2\beta_{2} + 38\beta _1 + 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -11\beta_{6} + 11\beta_{5} + 15\beta_{4} + 12\beta_{3} + 47\beta_{2} + 14\beta _1 + 143 \)
|
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 |
|
−2.71503 | 0.525975 | 5.37138 | 0 | −1.42804 | −1.00000 | −9.15341 | −2.72335 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.13935 | −2.65419 | 2.57680 | 0 | 5.67823 | −1.00000 | −1.23398 | 4.04471 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.59433 | 3.08438 | 0.541890 | 0 | −4.91751 | −1.00000 | 2.32471 | 6.51337 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.151896 | −2.21537 | −1.97693 | 0 | 0.336506 | −1.00000 | 0.604080 | 1.90787 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.719607 | 0.234508 | −1.48217 | 0 | 0.168754 | −1.00000 | −2.50579 | −2.94501 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.27174 | −1.44691 | 3.16080 | 0 | −3.28700 | −1.00000 | 2.63702 | −0.906454 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.60926 | 2.47161 | 4.80822 | 0 | 6.44906 | −1.00000 | 7.32737 | 3.10886 | 0 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
| \(7\) | \(1\) |
| \(11\) | \(-1\) |
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 | 1925.2.a.bb | ✓ | 7 |
| 5.b | even | 2 | 1 | 1925.2.a.bd | yes | 7 | |
| 5.c | odd | 4 | 2 | 1925.2.b.r | 14 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1925.2.a.bb | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 1925.2.a.bd | yes | 7 | 5.b | even | 2 | 1 | |
| 1925.2.b.r | 14 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1925))\):
|
\( T_{2}^{7} + T_{2}^{6} - 13T_{2}^{5} - 12T_{2}^{4} + 47T_{2}^{3} + 37T_{2}^{2} - 35T_{2} - 6 \)
|
|
\( T_{3}^{7} - 15T_{3}^{5} - 4T_{3}^{4} + 61T_{3}^{3} + 24T_{3}^{2} - 43T_{3} + 8 \)
|
|
\( T_{13}^{7} - 3T_{13}^{6} - 53T_{13}^{5} + 63T_{13}^{4} + 764T_{13}^{3} + 516T_{13}^{2} - 1268T_{13} - 1172 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + T^{6} - 13 T^{5} + \cdots - 6 \)
|
| $3$ |
\( T^{7} - 15 T^{5} + \cdots + 8 \)
|
| $5$ |
\( T^{7} \)
|
| $7$ |
\( (T + 1)^{7} \)
|
| $11$ |
\( (T - 1)^{7} \)
|
| $13$ |
\( T^{7} - 3 T^{6} + \cdots - 1172 \)
|
| $17$ |
\( T^{7} - 2 T^{6} + \cdots + 2313 \)
|
| $19$ |
\( T^{7} - 20 T^{6} + \cdots - 400 \)
|
| $23$ |
\( T^{7} + 11 T^{6} + \cdots + 576 \)
|
| $29$ |
\( T^{7} - 4 T^{6} + \cdots + 960 \)
|
| $31$ |
\( T^{7} - 6 T^{6} + \cdots + 132280 \)
|
| $37$ |
\( T^{7} + 19 T^{6} + \cdots + 4 \)
|
| $41$ |
\( T^{7} - 24 T^{6} + \cdots + 972 \)
|
| $43$ |
\( T^{7} - 103 T^{5} + \cdots - 31568 \)
|
| $47$ |
\( T^{7} - T^{6} + \cdots - 52740 \)
|
| $53$ |
\( T^{7} + 7 T^{6} + \cdots - 4863 \)
|
| $59$ |
\( T^{7} - 9 T^{6} + \cdots - 107496 \)
|
| $61$ |
\( T^{7} - 18 T^{6} + \cdots - 175300 \)
|
| $67$ |
\( T^{7} - 298 T^{5} + \cdots - 1590352 \)
|
| $71$ |
\( T^{7} - 18 T^{6} + \cdots - 4405392 \)
|
| $73$ |
\( T^{7} + 5 T^{6} + \cdots - 3063100 \)
|
| $79$ |
\( T^{7} - 25 T^{6} + \cdots + 4652962 \)
|
| $83$ |
\( T^{7} - 17 T^{6} + \cdots + 5760 \)
|
| $89$ |
\( T^{7} + 16 T^{6} + \cdots + 38784 \)
|
| $97$ |
\( T^{7} - 4 T^{6} + \cdots - 147200 \)
|
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