Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3025,2,Mod(1,3025)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3025, 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("3025.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3025 = 5^{2} \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3025.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: | \(24.1547466114\) |
| Analytic rank: | \(1\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 3x^{7} - 7x^{6} + 18x^{5} + 16x^{4} - 30x^{3} - 12x^{2} + 12x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 275) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 3x^{7} - 7x^{6} + 18x^{5} + 16x^{4} - 30x^{3} - 12x^{2} + 12x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 3\nu^{4} - 5\nu^{3} + 12\nu^{2} + 6\nu - 6 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{6} + 3\nu^{5} + 5\nu^{4} - 12\nu^{3} - 6\nu^{2} + 8\nu ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{6} - 3\nu^{5} - 5\nu^{4} + 12\nu^{3} + 8\nu^{2} - 10\nu - 4 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{6} + 2\nu^{5} + 10\nu^{4} - 13\nu^{3} - 26\nu^{2} + 18\nu + 12 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{7} - 3\nu^{6} - 6\nu^{5} + 15\nu^{4} + 11\nu^{3} - 18\nu^{2} - 4\nu + 2 ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -\nu^{7} + 2\nu^{6} + 10\nu^{5} - 13\nu^{4} - 26\nu^{3} + 18\nu^{2} + 12\nu ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + \beta_{6} + 3\beta_{4} + 2\beta_{3} - \beta_{2} + 6\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 3\beta_{7} + 3\beta_{6} + \beta_{5} + 13\beta_{4} + 9\beta_{3} - 2\beta_{2} + 14\beta _1 + 11 \)
|
| \(\nu^{5}\) | \(=\) |
\( 14\beta_{7} + 14\beta_{6} + 3\beta_{5} + 42\beta_{4} + 25\beta_{3} - 9\beta_{2} + 54\beta _1 + 25 \)
|
| \(\nu^{6}\) | \(=\) |
\( 45\beta_{7} + 45\beta_{6} + 14\beta_{5} + 149\beta_{4} + 88\beta_{3} - 25\beta_{2} + 162\beta _1 + 94 \)
|
| \(\nu^{7}\) | \(=\) |
\( 163\beta_{7} + 165\beta_{6} + 45\beta_{5} + 489\beta_{4} + 275\beta_{3} - 88\beta_{2} + 556\beta _1 + 279 \)
|
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 |
|
−2.76541 | −1.64150 | 5.64751 | 0 | 4.53942 | 1.89792 | −10.0869 | −0.305488 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.56247 | 1.79260 | 4.56626 | 0 | −4.59347 | −3.80088 | −6.57595 | 0.213399 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.67203 | 3.10994 | 0.795692 | 0 | −5.19992 | −3.08998 | 2.01364 | 6.67173 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.32162 | −2.02642 | −0.253315 | 0 | 2.67816 | −0.348258 | 2.97803 | 1.10639 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.139403 | −2.98582 | −1.98057 | 0 | 0.416233 | −3.56959 | 0.554904 | 5.91512 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.208285 | 1.89427 | −1.95662 | 0 | 0.394547 | 1.28881 | −0.824104 | 0.588246 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.958815 | 0.899342 | −1.08067 | 0 | 0.862303 | 0.761645 | −2.95380 | −2.19118 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.29384 | −0.0424059 | 3.26171 | 0 | −0.0972724 | −1.13968 | 2.89416 | −2.99820 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
| \(11\) | \(-1\) |
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 | 3025.2.a.bj | 8 | |
| 5.b | even | 2 | 1 | 3025.2.a.bm | 8 | ||
| 11.b | odd | 2 | 1 | 3025.2.a.bn | 8 | ||
| 11.d | odd | 10 | 2 | 275.2.h.e | yes | 16 | |
| 55.d | odd | 2 | 1 | 3025.2.a.bi | 8 | ||
| 55.h | odd | 10 | 2 | 275.2.h.c | ✓ | 16 | |
| 55.l | even | 20 | 4 | 275.2.z.c | 32 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 275.2.h.c | ✓ | 16 | 55.h | odd | 10 | 2 | |
| 275.2.h.e | yes | 16 | 11.d | odd | 10 | 2 | |
| 275.2.z.c | 32 | 55.l | even | 20 | 4 | ||
| 3025.2.a.bi | 8 | 55.d | odd | 2 | 1 | ||
| 3025.2.a.bj | 8 | 1.a | even | 1 | 1 | trivial | |
| 3025.2.a.bm | 8 | 5.b | even | 2 | 1 | ||
| 3025.2.a.bn | 8 | 11.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3025))\):
|
\( T_{2}^{8} + 5T_{2}^{7} - 31T_{2}^{5} - 34T_{2}^{4} + 25T_{2}^{3} + 34T_{2}^{2} - 3T_{2} - 1 \)
|
|
\( T_{3}^{8} - T_{3}^{7} - 16T_{3}^{6} + 15T_{3}^{5} + 74T_{3}^{4} - 67T_{3}^{3} - 105T_{3}^{2} + 90T_{3} + 4 \)
|
|
\( T_{19}^{8} + T_{19}^{7} - 64T_{19}^{6} - 119T_{19}^{5} + 1016T_{19}^{4} + 1735T_{19}^{3} - 5555T_{19}^{2} - 5520T_{19} + 10480 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 5 T^{7} + \cdots - 1 \)
|
| $3$ |
\( T^{8} - T^{7} - 16 T^{6} + \cdots + 4 \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} + 8 T^{7} + \cdots - 31 \)
|
| $11$ |
\( T^{8} \)
|
| $13$ |
\( T^{8} + 9 T^{7} + \cdots + 3475 \)
|
| $17$ |
\( T^{8} + 19 T^{7} + \cdots + 439 \)
|
| $19$ |
\( T^{8} + T^{7} + \cdots + 10480 \)
|
| $23$ |
\( T^{8} - 2 T^{7} + \cdots - 121 \)
|
| $29$ |
\( T^{8} + 7 T^{7} + \cdots - 35495 \)
|
| $31$ |
\( T^{8} + 5 T^{7} + \cdots + 33569 \)
|
| $37$ |
\( T^{8} + 8 T^{7} + \cdots - 4451 \)
|
| $41$ |
\( T^{8} - 197 T^{6} + \cdots + 442576 \)
|
| $43$ |
\( T^{8} + 14 T^{7} + \cdots - 101 \)
|
| $47$ |
\( T^{8} - 11 T^{7} + \cdots - 459469 \)
|
| $53$ |
\( T^{8} - 11 T^{7} + \cdots - 48896 \)
|
| $59$ |
\( T^{8} - 17 T^{7} + \cdots - 78605 \)
|
| $61$ |
\( T^{8} - 2 T^{7} + \cdots + 3319739 \)
|
| $67$ |
\( T^{8} - 7 T^{7} + \cdots + 23994961 \)
|
| $71$ |
\( T^{8} - 17 T^{7} + \cdots + 960859 \)
|
| $73$ |
\( T^{8} + 34 T^{7} + \cdots - 466721 \)
|
| $79$ |
\( T^{8} + 23 T^{7} + \cdots - 524695 \)
|
| $83$ |
\( T^{8} + 41 T^{7} + \cdots - 8329 \)
|
| $89$ |
\( T^{8} + 11 T^{7} + \cdots - 278125 \)
|
| $97$ |
\( T^{8} - 2 T^{7} + \cdots + 85616 \)
|
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