Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [825,6,Mod(1,825)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(825, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("825.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 825 = 3 \cdot 5^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 825.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: | \(132.316651346\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{7} - 2x^{6} - 152x^{5} + 358x^{4} + 5771x^{3} - 13444x^{2} - 51316x + 92576 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 5 \) |
| 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
\( x^{7} - 2x^{6} - 152x^{5} + 358x^{4} + 5771x^{3} - 13444x^{2} - 51316x + 92576 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + \nu - 44 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} + \nu^{4} - 123\nu^{3} - 57\nu^{2} + 2642\nu + 464 ) / 48 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{5} + 11\nu^{4} + 171\nu^{3} - 1275\nu^{2} - 5642\nu + 21136 ) / 192 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{6} - 3\nu^{5} + 133\nu^{4} + 351\nu^{3} - 3716\nu^{2} - 9276\nu + 14048 ) / 192 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{6} - 148\nu^{4} + 66\nu^{3} + 5315\nu^{2} - 2754\nu - 37040 ) / 192 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - \beta _1 + 44 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{6} + 2\beta_{5} + 2\beta_{4} + 2\beta_{3} - \beta_{2} + 75\beta _1 - 44 \)
|
| \(\nu^{4}\) | \(=\) |
\( -8\beta_{6} - 8\beta_{5} + 8\beta_{4} - 4\beta_{3} + 115\beta_{2} - 161\beta _1 + 3260 \)
|
| \(\nu^{5}\) | \(=\) |
\( 254\beta_{6} + 254\beta_{5} + 238\beta_{4} + 298\beta_{3} - 181\beta_{2} + 6687\beta _1 - 6628 \)
|
| \(\nu^{6}\) | \(=\) |
\( -1124\beta_{6} - 1316\beta_{5} + 1052\beta_{4} - 724\beta_{3} + 11771\beta_{2} - 20709\beta _1 + 288564 \)
|
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 |
|
−10.4855 | 9.00000 | 77.9456 | 0 | −94.3694 | 152.972 | −481.762 | 81.0000 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −7.35371 | 9.00000 | 22.0770 | 0 | −66.1834 | −251.987 | 72.9708 | 81.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −5.42881 | 9.00000 | −2.52798 | 0 | −48.8593 | 29.0935 | 187.446 | 81.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.60892 | 9.00000 | −25.1935 | 0 | −23.4803 | 174.969 | 149.214 | 81.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 2.33276 | 9.00000 | −26.5582 | 0 | 20.9949 | −146.260 | −136.602 | 81.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 5.33680 | 9.00000 | −3.51852 | 0 | 48.0312 | 73.9998 | −189.555 | 81.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 9.20737 | 9.00000 | 52.7757 | 0 | 82.8663 | −97.7871 | 191.289 | 81.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||
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 | 825.6.a.m | ✓ | 7 |
| 5.b | even | 2 | 1 | 825.6.a.o | yes | 7 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 825.6.a.m | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 825.6.a.o | yes | 7 | 5.b | even | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} + 9T_{2}^{6} - 119T_{2}^{5} - 1053T_{2}^{4} + 2894T_{2}^{3} + 27140T_{2}^{2} - 9288T_{2} - 125184 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(825))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + 9 T^{6} + \cdots - 125184 \)
|
| $3$ |
\( (T - 9)^{7} \)
|
| $5$ |
\( T^{7} \)
|
| $7$ |
\( T^{7} + \cdots + 207675630158484 \)
|
| $11$ |
\( (T - 121)^{7} \)
|
| $13$ |
\( T^{7} + \cdots + 30\!\cdots\!36 \)
|
| $17$ |
\( T^{7} + \cdots + 11\!\cdots\!04 \)
|
| $19$ |
\( T^{7} + \cdots + 29\!\cdots\!25 \)
|
| $23$ |
\( T^{7} + \cdots + 56\!\cdots\!54 \)
|
| $29$ |
\( T^{7} + \cdots - 44\!\cdots\!00 \)
|
| $31$ |
\( T^{7} + \cdots + 17\!\cdots\!76 \)
|
| $37$ |
\( T^{7} + \cdots + 50\!\cdots\!44 \)
|
| $41$ |
\( T^{7} + \cdots - 11\!\cdots\!52 \)
|
| $43$ |
\( T^{7} + \cdots - 15\!\cdots\!00 \)
|
| $47$ |
\( T^{7} + \cdots - 20\!\cdots\!28 \)
|
| $53$ |
\( T^{7} + \cdots - 27\!\cdots\!52 \)
|
| $59$ |
\( T^{7} + \cdots - 47\!\cdots\!00 \)
|
| $61$ |
\( T^{7} + \cdots + 24\!\cdots\!24 \)
|
| $67$ |
\( T^{7} + \cdots + 12\!\cdots\!16 \)
|
| $71$ |
\( T^{7} + \cdots - 27\!\cdots\!20 \)
|
| $73$ |
\( T^{7} + \cdots + 22\!\cdots\!88 \)
|
| $79$ |
\( T^{7} + \cdots + 77\!\cdots\!00 \)
|
| $83$ |
\( T^{7} + \cdots - 36\!\cdots\!52 \)
|
| $89$ |
\( T^{7} + \cdots - 42\!\cdots\!00 \)
|
| $97$ |
\( T^{7} + \cdots - 15\!\cdots\!23 \)
|
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