Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [925,2,Mod(1,925)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(925, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("925.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 925 = 5^{2} \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 925.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: | \(7.38616218697\) |
| Analytic rank: | \(1\) |
| Dimension: | \(9\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{9} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{9} - 4x^{8} - 6x^{7} + 30x^{6} + 15x^{5} - 70x^{4} - 22x^{3} + 44x^{2} + 4x - 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 185) |
| 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 \(1,\beta_1,\ldots,\beta_{8}\) 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
\( x^{9} - 4x^{8} - 6x^{7} + 30x^{6} + 15x^{5} - 70x^{4} - 22x^{3} + 44x^{2} + 4x - 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 3\nu + 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 4\nu^{4} - \nu^{3} + 14\nu^{2} - 4\nu - 6 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} - 7\nu^{3} + 8\nu^{2} + 14\nu - 4 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{7} - 5\nu^{6} + 25\nu^{4} - 11\nu^{3} - 34\nu^{2} + 8\nu + 6 ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{8} - 5\nu^{7} + 26\nu^{5} - 13\nu^{4} - 39\nu^{3} + 14\nu^{2} + 10\nu - 2 ) / 2 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( -\nu^{8} + 5\nu^{7} + \nu^{6} - 29\nu^{5} + 8\nu^{4} + 52\nu^{3} - 4\nu^{2} - 22\nu ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 5\beta _1 + 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} - \beta_{4} + 3\beta_{3} + 9\beta_{2} + 9\beta _1 + 20 \)
|
| \(\nu^{5}\) | \(=\) |
\( 4\beta_{5} - 2\beta_{4} + 13\beta_{3} + 24\beta_{2} + 31\beta _1 + 48 \)
|
| \(\nu^{6}\) | \(=\) |
\( 2\beta_{8} + 2\beta_{7} + 17\beta_{5} - 11\beta_{4} + 41\beta_{3} + 81\beta_{2} + 75\beta _1 + 164 \)
|
| \(\nu^{7}\) | \(=\) |
\( 10\beta_{8} + 10\beta_{7} + 2\beta_{6} + 60\beta_{5} - 30\beta_{4} + 141\beta_{3} + 236\beta_{2} + 231\beta _1 + 460 \)
|
| \(\nu^{8}\) | \(=\) |
\( 50 \beta_{8} + 52 \beta_{7} + 10 \beta_{6} + 209 \beta_{5} - 111 \beta_{4} + 445 \beta_{3} + 737 \beta_{2} + \cdots + 1428 \)
|
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 |
|
−2.68489 | −2.42453 | 5.20864 | 0 | 6.50959 | 3.48285 | −8.61484 | 2.87832 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.58855 | 1.45668 | 4.70058 | 0 | −3.77067 | 0.648627 | −6.99059 | −0.878097 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.15179 | −1.93821 | 2.63020 | 0 | 4.17062 | −4.19034 | −1.35605 | 0.756662 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.35604 | −1.13547 | −0.161165 | 0 | 1.53974 | −0.748140 | 2.93062 | −1.71070 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.703111 | 1.64070 | −1.50564 | 0 | −1.15359 | −0.501390 | 2.46485 | −0.308108 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.246891 | −3.34515 | −1.93904 | 0 | 0.825888 | −2.66723 | 0.972515 | 8.19006 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.974151 | 1.62921 | −1.05103 | 0 | 1.58709 | −4.15169 | −2.97216 | −0.345685 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.70607 | −0.742055 | 0.910675 | 0 | −1.26600 | 0.663740 | −1.85846 | −2.44935 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.05104 | −3.14116 | 2.20678 | 0 | −6.44266 | −0.536422 | 0.424122 | 6.86691 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
| \(37\) | \(-1\) |
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 | 925.2.a.l | 9 | |
| 3.b | odd | 2 | 1 | 8325.2.a.cr | 9 | ||
| 5.b | even | 2 | 1 | 925.2.a.m | 9 | ||
| 5.c | odd | 4 | 2 | 185.2.b.a | ✓ | 18 | |
| 15.d | odd | 2 | 1 | 8325.2.a.cq | 9 | ||
| 15.e | even | 4 | 2 | 1665.2.c.e | 18 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 185.2.b.a | ✓ | 18 | 5.c | odd | 4 | 2 | |
| 925.2.a.l | 9 | 1.a | even | 1 | 1 | trivial | |
| 925.2.a.m | 9 | 5.b | even | 2 | 1 | ||
| 1665.2.c.e | 18 | 15.e | even | 4 | 2 | ||
| 8325.2.a.cq | 9 | 15.d | odd | 2 | 1 | ||
| 8325.2.a.cr | 9 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(925))\):
|
\( T_{2}^{9} + 5T_{2}^{8} - 2T_{2}^{7} - 40T_{2}^{6} - 29T_{2}^{5} + 91T_{2}^{4} + 98T_{2}^{3} - 44T_{2}^{2} - 64T_{2} - 12 \)
|
|
\( T_{3}^{9} + 8T_{3}^{8} + 12T_{3}^{7} - 48T_{3}^{6} - 129T_{3}^{5} + 64T_{3}^{4} + 354T_{3}^{3} + 78T_{3}^{2} - 302T_{3} - 162 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} + 5 T^{8} + \cdots - 12 \)
|
| $3$ |
\( T^{9} + 8 T^{8} + \cdots - 162 \)
|
| $5$ |
\( T^{9} \)
|
| $7$ |
\( T^{9} + 8 T^{8} + \cdots - 14 \)
|
| $11$ |
\( T^{9} - 42 T^{7} + \cdots + 1152 \)
|
| $13$ |
\( T^{9} + 6 T^{8} + \cdots - 16 \)
|
| $17$ |
\( T^{9} + 18 T^{8} + \cdots + 7728 \)
|
| $19$ |
\( T^{9} + 4 T^{8} + \cdots + 63768 \)
|
| $23$ |
\( T^{9} + 16 T^{8} + \cdots - 3904 \)
|
| $29$ |
\( T^{9} + 2 T^{8} + \cdots + 1152 \)
|
| $31$ |
\( T^{9} + 6 T^{8} + \cdots + 2168344 \)
|
| $37$ |
\( (T - 1)^{9} \)
|
| $41$ |
\( T^{9} - 2 T^{8} + \cdots + 8368 \)
|
| $43$ |
\( T^{9} + 14 T^{8} + \cdots + 1852736 \)
|
| $47$ |
\( T^{9} + 34 T^{8} + \cdots - 11786326 \)
|
| $53$ |
\( T^{9} - 4 T^{8} + \cdots - 13034336 \)
|
| $59$ |
\( T^{9} - 10 T^{8} + \cdots + 689664 \)
|
| $61$ |
\( T^{9} - 8 T^{8} + \cdots + 6156288 \)
|
| $67$ |
\( T^{9} + 16 T^{8} + \cdots - 6502304 \)
|
| $71$ |
\( T^{9} - 8 T^{8} + \cdots - 5516928 \)
|
| $73$ |
\( T^{9} + 8 T^{8} + \cdots + 1036576 \)
|
| $79$ |
\( T^{9} + 26 T^{8} + \cdots - 70088 \)
|
| $83$ |
\( T^{9} + 70 T^{8} + \cdots - 37115022 \)
|
| $89$ |
\( T^{9} + 8 T^{8} + \cdots - 27504768 \)
|
| $97$ |
\( T^{9} - 2 T^{8} + \cdots + 106138304 \)
|
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