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: | \(9\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{9} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{9} - 12x^{7} - 4x^{6} + 48x^{5} + 27x^{4} - 72x^{3} - 51x^{2} + 27x + 19 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 95) |
| 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
\( x^{9} - 12x^{7} - 4x^{6} + 48x^{5} + 27x^{4} - 72x^{3} - 51x^{2} + 27x + 19 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{6} - \nu^{5} - 8\nu^{4} + 4\nu^{3} + 17\nu^{2} - 2\nu - 7 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{7} - \nu^{6} - 8\nu^{5} + 4\nu^{4} + 17\nu^{3} - 2\nu^{2} - 7\nu \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{8} - \nu^{7} - 10\nu^{6} + 6\nu^{5} + 33\nu^{4} - 10\nu^{3} - 41\nu^{2} + 4\nu + 14 \)
|
| \(\beta_{6}\) | \(=\) |
\( 2\nu^{8} - 3\nu^{7} - 19\nu^{6} + 21\nu^{5} + 60\nu^{4} - 42\nu^{3} - 72\nu^{2} + 21\nu + 22 \)
|
| \(\beta_{7}\) | \(=\) |
\( -3\nu^{8} + 5\nu^{7} + 29\nu^{6} - 36\nu^{5} - 96\nu^{4} + 72\nu^{3} + 123\nu^{2} - 32\nu - 38 \)
|
| \(\beta_{8}\) | \(=\) |
\( -4\nu^{8} + 6\nu^{7} + 39\nu^{6} - 42\nu^{5} - 129\nu^{4} + 81\nu^{3} + 165\nu^{2} - 33\nu - 53 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{8} + \beta_{7} - \beta_{5} + \beta_{2} + 4\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{8} + 2\beta_{7} + \beta_{6} - \beta_{4} - \beta_{3} + 6\beta_{2} + 7\beta _1 + 12 \)
|
| \(\nu^{5}\) | \(=\) |
\( -7\beta_{8} + 9\beta_{7} + 3\beta_{6} - 7\beta_{5} - \beta_{4} - 2\beta_{3} + 9\beta_{2} + 20\beta _1 + 16 \)
|
| \(\nu^{6}\) | \(=\) |
\( -11\beta_{8} + 21\beta_{7} + 11\beta_{6} - 3\beta_{5} - 9\beta_{4} - 9\beta_{3} + 36\beta_{2} + 45\beta _1 + 60 \)
|
| \(\nu^{7}\) | \(=\) |
\( -46\beta_{8} + 68\beta_{7} + 31\beta_{6} - 42\beta_{5} - 12\beta_{4} - 21\beta_{3} + 69\beta_{2} + 118\beta _1 + 112 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 91 \beta_{8} + 168 \beta_{7} + 90 \beta_{6} - 39 \beta_{5} - 63 \beta_{4} - 66 \beta_{3} + 228 \beta_{2} + \cdots + 349 \)
|
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 |
|
−2.02162 | −1.90055 | 2.08694 | 0 | 3.84218 | −3.09218 | −0.175759 | 0.612075 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.57047 | 2.28502 | 0.466387 | 0 | −3.58856 | 4.01337 | 2.40850 | 2.22131 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.46231 | 1.51036 | 0.138346 | 0 | −2.20860 | −4.07172 | 2.72231 | −0.718827 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.19408 | 2.27318 | −0.574177 | 0 | −2.71435 | 2.19649 | 3.07377 | 2.16734 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.593847 | −1.93003 | −1.64735 | 0 | 1.14614 | −1.06052 | 2.16596 | 0.725033 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.719457 | −1.23428 | −1.48238 | 0 | −0.888013 | −1.29194 | −2.50542 | −1.47655 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.68361 | −3.25202 | 0.834534 | 0 | −5.47512 | 0.548389 | −1.96219 | 7.57562 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.81702 | 0.177104 | 1.30157 | 0 | 0.321803 | −1.07346 | −1.26906 | −2.96863 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.62224 | −0.928776 | 4.87613 | 0 | −2.43547 | 3.83157 | 7.54188 | −2.13737 | 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.cd | 9 | |
| 5.b | even | 2 | 1 | 1805.2.a.u | 9 | ||
| 19.b | odd | 2 | 1 | 9025.2.a.ce | 9 | ||
| 19.f | odd | 18 | 2 | 475.2.l.b | 18 | ||
| 95.d | odd | 2 | 1 | 1805.2.a.t | 9 | ||
| 95.o | odd | 18 | 2 | 95.2.k.b | ✓ | 18 | |
| 95.r | even | 36 | 4 | 475.2.u.c | 36 | ||
| 285.bf | even | 18 | 2 | 855.2.bs.b | 18 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 95.2.k.b | ✓ | 18 | 95.o | odd | 18 | 2 | |
| 475.2.l.b | 18 | 19.f | odd | 18 | 2 | ||
| 475.2.u.c | 36 | 95.r | even | 36 | 4 | ||
| 855.2.bs.b | 18 | 285.bf | even | 18 | 2 | ||
| 1805.2.a.t | 9 | 95.d | odd | 2 | 1 | ||
| 1805.2.a.u | 9 | 5.b | even | 2 | 1 | ||
| 9025.2.a.cd | 9 | 1.a | even | 1 | 1 | trivial | |
| 9025.2.a.ce | 9 | 19.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9025))\):
|
\( T_{2}^{9} - 12T_{2}^{7} - 4T_{2}^{6} + 48T_{2}^{5} + 27T_{2}^{4} - 72T_{2}^{3} - 51T_{2}^{2} + 27T_{2} + 19 \)
|
|
\( T_{3}^{9} + 3T_{3}^{8} - 12T_{3}^{7} - 37T_{3}^{6} + 39T_{3}^{5} + 147T_{3}^{4} - 6T_{3}^{3} - 186T_{3}^{2} - 75T_{3} + 19 \)
|
|
\( T_{7}^{9} - 33T_{7}^{7} - 10T_{7}^{6} + 327T_{7}^{5} + 228T_{7}^{4} - 911T_{7}^{3} - 1029T_{7}^{2} + 147T_{7} + 343 \)
|
|
\( T_{11}^{9} - 36T_{11}^{7} - 40T_{11}^{6} + 282T_{11}^{5} + 321T_{11}^{4} - 517T_{11}^{3} - 108T_{11}^{2} + 135T_{11} - 19 \)
|
|
\( T_{29}^{9} - 15 T_{29}^{8} - 54 T_{29}^{7} + 1519 T_{29}^{6} - 2274 T_{29}^{5} - 33729 T_{29}^{4} + \cdots - 702001 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} - 12 T^{7} + \cdots + 19 \)
|
| $3$ |
\( T^{9} + 3 T^{8} + \cdots + 19 \)
|
| $5$ |
\( T^{9} \)
|
| $7$ |
\( T^{9} - 33 T^{7} + \cdots + 343 \)
|
| $11$ |
\( T^{9} - 36 T^{7} + \cdots - 19 \)
|
| $13$ |
\( T^{9} - 3 T^{8} + \cdots - 9937 \)
|
| $17$ |
\( T^{9} - 9 T^{8} + \cdots - 1 \)
|
| $19$ |
\( T^{9} \)
|
| $23$ |
\( T^{9} - 114 T^{7} + \cdots + 63197 \)
|
| $29$ |
\( T^{9} - 15 T^{8} + \cdots - 702001 \)
|
| $31$ |
\( T^{9} - 30 T^{8} + \cdots - 996623 \)
|
| $37$ |
\( T^{9} + 30 T^{8} + \cdots + 27721 \)
|
| $41$ |
\( T^{9} - 18 T^{8} + \cdots - 363977 \)
|
| $43$ |
\( T^{9} - 6 T^{8} + \cdots + 10099 \)
|
| $47$ |
\( T^{9} + 21 T^{8} + \cdots - 5721697 \)
|
| $53$ |
\( T^{9} - 9 T^{8} + \cdots + 4387499 \)
|
| $59$ |
\( T^{9} - 27 T^{8} + \cdots - 577711 \)
|
| $61$ |
\( T^{9} - 12 T^{8} + \cdots + 1862369 \)
|
| $67$ |
\( T^{9} + 36 T^{8} + \cdots - 60058259 \)
|
| $71$ |
\( T^{9} + 6 T^{8} + \cdots + 92683 \)
|
| $73$ |
\( T^{9} - 9 T^{8} + \cdots + 1023553 \)
|
| $79$ |
\( T^{9} - 45 T^{8} + \cdots - 17803297 \)
|
| $83$ |
\( T^{9} - 171 T^{7} + \cdots + 9829 \)
|
| $89$ |
\( T^{9} + 9 T^{8} + \cdots - 11971 \)
|
| $97$ |
\( T^{9} + 45 T^{8} + \cdots + 191335897 \)
|
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