Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9025,2,Mod(1,9025)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9025, 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("9025.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9025 = 5^{2} \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9025.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: | \(72.0649878242\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 2x^{9} - 12x^{8} + 23x^{7} + 47x^{6} - 86x^{5} - 69x^{4} + 115x^{3} + 34x^{2} - 45x + 5 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 2x^{9} - 12x^{8} + 23x^{7} + 47x^{6} - 86x^{5} - 69x^{4} + 115x^{3} + 34x^{2} - 45x + 5 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{8} - 2\nu^{7} - 11\nu^{6} + 20\nu^{5} + 36\nu^{4} - 56\nu^{3} - 31\nu^{2} + 34\nu - 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( 2\nu^{9} - 2\nu^{8} - 23\nu^{7} + 19\nu^{6} + 80\nu^{5} - 53\nu^{4} - 85\nu^{3} + 34\nu^{2} + 12\nu - 5 \)
|
| \(\beta_{5}\) | \(=\) |
\( -2\nu^{9} + 3\nu^{8} + 22\nu^{7} - 30\nu^{6} - 70\nu^{5} + 87\nu^{4} + 55\nu^{3} - 58\nu^{2} + 7\nu \)
|
| \(\beta_{6}\) | \(=\) |
\( -2\nu^{8} + 3\nu^{7} + 22\nu^{6} - 30\nu^{5} - 70\nu^{4} + 87\nu^{3} + 55\nu^{2} - 58\nu + 8 \)
|
| \(\beta_{7}\) | \(=\) |
\( -\nu^{9} + 4\nu^{8} + 8\nu^{7} - 43\nu^{6} - 6\nu^{5} + 136\nu^{4} - 54\nu^{3} - 113\nu^{2} + 54\nu - 1 \)
|
| \(\beta_{8}\) | \(=\) |
\( \nu^{9} - 4\nu^{8} - 7\nu^{7} + 42\nu^{6} - 5\nu^{5} - 127\nu^{4} + 89\nu^{3} + 90\nu^{2} - 84\nu + 12 \)
|
| \(\beta_{9}\) | \(=\) |
\( 4\nu^{9} - 8\nu^{8} - 41\nu^{7} + 82\nu^{6} + 111\nu^{5} - 245\nu^{4} - 30\nu^{3} + 176\nu^{2} - 64\nu + 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{9} - \beta_{8} + \beta_{7} + 2\beta_{6} + \beta_{5} + \beta_{3} + 6\beta_{2} + \beta _1 + 13 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{9} - \beta_{8} + \beta_{7} + 8\beta_{6} + 10\beta_{5} + 7\beta_{4} + 8\beta_{3} + \beta_{2} + 28\beta _1 + 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 9 \beta_{9} - 11 \beta_{8} + 9 \beta_{7} + 23 \beta_{6} + 11 \beta_{5} + 3 \beta_{4} + 11 \beta_{3} + \cdots + 65 \)
|
| \(\nu^{7}\) | \(=\) |
\( 22 \beta_{9} - 12 \beta_{8} + 12 \beta_{7} + 58 \beta_{6} + 77 \beta_{5} + 45 \beta_{4} + 55 \beta_{3} + \cdots + 17 \)
|
| \(\nu^{8}\) | \(=\) |
\( 67 \beta_{9} - 89 \beta_{8} + 67 \beta_{7} + 193 \beta_{6} + 95 \beta_{5} + 39 \beta_{4} + 92 \beta_{3} + \cdots + 357 \)
|
| \(\nu^{9}\) | \(=\) |
\( 181 \beta_{9} - 109 \beta_{8} + 106 \beta_{7} + 417 \beta_{6} + 545 \beta_{5} + 291 \beta_{4} + \cdots + 191 \)
|
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 |
|
−2.33214 | −1.43275 | 3.43888 | 0 | 3.34137 | 3.71046 | −3.35566 | −0.947236 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.99727 | −0.733917 | 1.98908 | 0 | 1.46583 | −4.08796 | 0.0218157 | −2.46137 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.08508 | −2.80183 | −0.822601 | 0 | 3.04021 | 3.25616 | 3.06275 | 4.85025 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.06941 | 3.33626 | −0.856371 | 0 | −3.56781 | 4.50384 | 3.05462 | 8.13062 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.128542 | 0.935507 | −1.98348 | 0 | 0.120252 | −2.14116 | −0.512046 | −2.12483 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.529524 | 0.522620 | −1.71960 | 0 | 0.276740 | −0.449692 | −1.96962 | −2.72687 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.63264 | 0.767996 | 0.665505 | 0 | 1.25386 | 1.80693 | −2.17875 | −2.41018 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.72264 | −2.48447 | 0.967484 | 0 | −4.27984 | −1.01952 | −1.77865 | 3.17257 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.83019 | 3.26153 | 1.34958 | 0 | 5.96920 | 2.06494 | −1.19039 | 7.63755 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.64037 | −1.37095 | 4.97153 | 0 | −3.61980 | −3.64400 | 7.84594 | −1.12051 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
| \(19\) | \(-1\) |
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 | 9025.2.a.cj | yes | 10 |
| 5.b | even | 2 | 1 | 9025.2.a.cg | ✓ | 10 | |
| 19.b | odd | 2 | 1 | 9025.2.a.ch | yes | 10 | |
| 95.d | odd | 2 | 1 | 9025.2.a.ci | yes | 10 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 9025.2.a.cg | ✓ | 10 | 5.b | even | 2 | 1 | |
| 9025.2.a.ch | yes | 10 | 19.b | odd | 2 | 1 | |
| 9025.2.a.ci | yes | 10 | 95.d | odd | 2 | 1 | |
| 9025.2.a.cj | yes | 10 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9025))\):
|
\( T_{2}^{10} - 2T_{2}^{9} - 12T_{2}^{8} + 23T_{2}^{7} + 47T_{2}^{6} - 86T_{2}^{5} - 69T_{2}^{4} + 115T_{2}^{3} + 34T_{2}^{2} - 45T_{2} + 5 \)
|
|
\( T_{3}^{10} - 21T_{3}^{8} - 10T_{3}^{7} + 134T_{3}^{6} + 109T_{3}^{5} - 230T_{3}^{4} - 157T_{3}^{3} + 165T_{3}^{2} + 55T_{3} - 41 \)
|
|
\( T_{7}^{10} - 4 T_{7}^{9} - 36 T_{7}^{8} + 144 T_{7}^{7} + 407 T_{7}^{6} - 1653 T_{7}^{5} - 1545 T_{7}^{4} + \cdots - 2969 \)
|
|
\( T_{11}^{10} + 3 T_{11}^{9} - 64 T_{11}^{8} - 173 T_{11}^{7} + 1311 T_{11}^{6} + 3251 T_{11}^{5} + \cdots + 35279 \)
|
|
\( T_{29}^{10} + 8 T_{29}^{9} - 72 T_{29}^{8} - 509 T_{29}^{7} + 1785 T_{29}^{6} + 8911 T_{29}^{5} + \cdots - 6445 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} - 2 T^{9} + \cdots + 5 \)
|
| $3$ |
\( T^{10} - 21 T^{8} + \cdots - 41 \)
|
| $5$ |
\( T^{10} \)
|
| $7$ |
\( T^{10} - 4 T^{9} + \cdots - 2969 \)
|
| $11$ |
\( T^{10} + 3 T^{9} + \cdots + 35279 \)
|
| $13$ |
\( T^{10} + 5 T^{9} + \cdots + 881 \)
|
| $17$ |
\( T^{10} - 11 T^{9} + \cdots - 71 \)
|
| $19$ |
\( T^{10} \)
|
| $23$ |
\( T^{10} - 32 T^{9} + \cdots - 1521355 \)
|
| $29$ |
\( T^{10} + 8 T^{9} + \cdots - 6445 \)
|
| $31$ |
\( T^{10} + 11 T^{9} + \cdots - 8188589 \)
|
| $37$ |
\( T^{10} - 35 T^{9} + \cdots + 31455011 \)
|
| $41$ |
\( T^{10} - 16 T^{9} + \cdots + 86759 \)
|
| $43$ |
\( T^{10} - 26 T^{9} + \cdots - 781121 \)
|
| $47$ |
\( T^{10} - 19 T^{9} + \cdots + 47784889 \)
|
| $53$ |
\( T^{10} + \cdots - 380920429 \)
|
| $59$ |
\( T^{10} - 2 T^{9} + \cdots - 45595 \)
|
| $61$ |
\( T^{10} - 8 T^{9} + \cdots + 179849 \)
|
| $67$ |
\( T^{10} - 340 T^{8} + \cdots - 12480451 \)
|
| $71$ |
\( T^{10} + \cdots + 163837621 \)
|
| $73$ |
\( T^{10} + \cdots + 113352881 \)
|
| $79$ |
\( T^{10} - 18 T^{9} + \cdots - 25340080 \)
|
| $83$ |
\( T^{10} - 46 T^{9} + \cdots - 67674791 \)
|
| $89$ |
\( T^{10} + \cdots + 115142555 \)
|
| $97$ |
\( T^{10} + \cdots - 675789104 \)
|
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