Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3525,2,Mod(1,3525)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3525, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("3525.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3525 = 3 \cdot 5^{2} \cdot 47 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3525.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: | \(28.1472667125\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{8} - 3x^{7} - 7x^{6} + 24x^{5} + 8x^{4} - 47x^{3} + 8x^{2} + 13x + 2 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 3x^{7} - 7x^{6} + 24x^{5} + 8x^{4} - 47x^{3} + 8x^{2} + 13x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{7} - 3\nu^{6} - 7\nu^{5} + 24\nu^{4} + 7\nu^{3} - 46\nu^{2} + 13\nu + 9 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{7} - 4\nu^{6} - 5\nu^{5} + 32\nu^{4} - 7\nu^{3} - 61\nu^{2} + 33\nu + 11 \)
|
| \(\beta_{5}\) | \(=\) |
\( 2\nu^{7} - 6\nu^{6} - 13\nu^{5} + 47\nu^{4} + 7\nu^{3} - 87\nu^{2} + 36\nu + 14 \)
|
| \(\beta_{6}\) | \(=\) |
\( 3\nu^{7} - 10\nu^{6} - 18\nu^{5} + 79\nu^{4} + \nu^{3} - 148\nu^{2} + 64\nu + 26 \)
|
| \(\beta_{7}\) | \(=\) |
\( -6\nu^{7} + 19\nu^{6} + 38\nu^{5} - 149\nu^{4} - 16\nu^{3} + 275\nu^{2} - 109\nu - 45 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} - \beta_{5} - \beta_{4} + 5\beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{7} + 2\beta_{6} - \beta_{4} + \beta_{3} + 6\beta_{2} + \beta _1 + 13 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{7} + 9\beta_{6} - 6\beta_{5} - 8\beta_{4} - \beta_{3} + \beta_{2} + 26\beta _1 - 5 \)
|
| \(\nu^{6}\) | \(=\) |
\( 10\beta_{7} + 20\beta_{6} + 2\beta_{5} - 11\beta_{4} + 7\beta_{3} + 35\beta_{2} + 10\beta _1 + 65 \)
|
| \(\nu^{7}\) | \(=\) |
\( 13\beta_{7} + 68\beta_{6} - 29\beta_{5} - 58\beta_{4} - 9\beta_{3} + 14\beta_{2} + 140\beta _1 - 16 \)
|
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 |
|
−2.25864 | −1.00000 | 3.10144 | 0 | 2.25864 | 3.65257 | −2.48774 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.60641 | −1.00000 | 0.580562 | 0 | 1.60641 | 2.35394 | 2.28020 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.267165 | −1.00000 | −1.92862 | 0 | 0.267165 | 0.207232 | 1.04959 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.237165 | −1.00000 | −1.94375 | 0 | 0.237165 | −1.64667 | 0.935320 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.936719 | −1.00000 | −1.12256 | 0 | −0.936719 | −3.65526 | −2.92496 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.60965 | −1.00000 | 0.590961 | 0 | −1.60965 | 2.89141 | −2.26805 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.19791 | −1.00000 | 2.83083 | 0 | −2.19791 | −1.05086 | 1.82609 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.62510 | −1.00000 | 4.89115 | 0 | −2.62510 | 5.24764 | 7.58955 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(5\) | \(1\) |
| \(47\) | \(-1\) |
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 | 3525.2.a.be | yes | 8 |
| 5.b | even | 2 | 1 | 3525.2.a.bd | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3525.2.a.bd | ✓ | 8 | 5.b | even | 2 | 1 | |
| 3525.2.a.be | yes | 8 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3525))\):
|
\( T_{2}^{8} - 3T_{2}^{7} - 7T_{2}^{6} + 24T_{2}^{5} + 8T_{2}^{4} - 47T_{2}^{3} + 8T_{2}^{2} + 13T_{2} + 2 \)
|
|
\( T_{7}^{8} - 8T_{7}^{7} - 4T_{7}^{6} + 144T_{7}^{5} - 170T_{7}^{4} - 550T_{7}^{3} + 613T_{7}^{2} + 723T_{7} - 171 \)
|
|
\( T_{11}^{8} + 8T_{11}^{7} - 27T_{11}^{6} - 334T_{11}^{5} - 341T_{11}^{4} + 2140T_{11}^{3} + 3505T_{11}^{2} - 2423T_{11} - 4420 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 3 T^{7} + \cdots + 2 \)
|
| $3$ |
\( (T + 1)^{8} \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} - 8 T^{7} + \cdots - 171 \)
|
| $11$ |
\( T^{8} + 8 T^{7} + \cdots - 4420 \)
|
| $13$ |
\( T^{8} - 10 T^{7} + \cdots + 6107 \)
|
| $17$ |
\( T^{8} - 6 T^{7} + \cdots + 25308 \)
|
| $19$ |
\( T^{8} + 2 T^{7} + \cdots + 3627 \)
|
| $23$ |
\( T^{8} - 10 T^{7} + \cdots - 365714 \)
|
| $29$ |
\( T^{8} + 13 T^{7} + \cdots - 1138486 \)
|
| $31$ |
\( T^{8} - 168 T^{6} + \cdots + 351587 \)
|
| $37$ |
\( T^{8} - 3 T^{7} + \cdots + 283838 \)
|
| $41$ |
\( T^{8} + 16 T^{7} + \cdots - 98438 \)
|
| $43$ |
\( T^{8} - 25 T^{7} + \cdots + 22599 \)
|
| $47$ |
\( (T - 1)^{8} \)
|
| $53$ |
\( T^{8} - 4 T^{7} + \cdots + 1104692 \)
|
| $59$ |
\( T^{8} + 8 T^{7} + \cdots + 3744478 \)
|
| $61$ |
\( T^{8} - 15 T^{7} + \cdots + 24176727 \)
|
| $67$ |
\( T^{8} - 27 T^{7} + \cdots + 2431981 \)
|
| $71$ |
\( T^{8} - 14 T^{7} + \cdots - 3420630 \)
|
| $73$ |
\( T^{8} - 28 T^{7} + \cdots + 1021770 \)
|
| $79$ |
\( T^{8} - 7 T^{7} + \cdots - 19475608 \)
|
| $83$ |
\( T^{8} - 60 T^{7} + \cdots + 66012500 \)
|
| $89$ |
\( T^{8} + 34 T^{7} + \cdots + 16076470 \)
|
| $97$ |
\( T^{8} - 7 T^{7} + \cdots + 11093 \)
|
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