Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2890,2,Mod(1,2890)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2890, 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("2890.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2890 = 2 \cdot 5 \cdot 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2890.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: | \(23.0767661842\) |
| Analytic rank: | \(1\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.8.32887537664.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 4x^{7} - 6x^{6} + 28x^{5} + 13x^{4} - 56x^{3} - 12x^{2} + 32x + 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 170) |
| 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
\( x^{8} - 4x^{7} - 6x^{6} + 28x^{5} + 13x^{4} - 56x^{3} - 12x^{2} + 32x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -11\nu^{7} - 2\nu^{6} + 198\nu^{5} + 134\nu^{4} - 1074\nu^{3} - 682\nu^{2} + 1333\nu + 450 ) / 193 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 14\nu^{7} - 15\nu^{6} - 252\nu^{5} + 233\nu^{4} + 1016\nu^{3} - 483\nu^{2} - 907\nu + 94 ) / 193 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -18\nu^{7} + 102\nu^{6} - 62\nu^{5} - 465\nu^{4} + 734\nu^{3} + 235\nu^{2} - 1012\nu + 210 ) / 193 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 30\nu^{7} - 170\nu^{6} + 39\nu^{5} + 968\nu^{4} - 773\nu^{3} - 1421\nu^{2} + 979\nu + 422 ) / 193 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 30\nu^{7} - 170\nu^{6} + 39\nu^{5} + 968\nu^{4} - 773\nu^{3} - 1228\nu^{2} + 786\nu - 157 ) / 193 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 62\nu^{7} - 287\nu^{6} - 151\nu^{5} + 1666\nu^{4} - 298\nu^{3} - 2525\nu^{2} + 312\nu + 692 ) / 193 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} - \beta_{5} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + 2\beta_{6} - 3\beta_{5} + \beta_{4} - \beta_{3} + 6\beta _1 + 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( 4\beta_{7} + 9\beta_{6} - 15\beta_{5} + \beta_{4} - 2\beta_{3} + 2\beta_{2} + 15\beta _1 + 21 \)
|
| \(\nu^{5}\) | \(=\) |
\( 19\beta_{7} + 25\beta_{6} - 53\beta_{5} + 5\beta_{4} - 13\beta_{3} + 6\beta_{2} + 60\beta _1 + 55 \)
|
| \(\nu^{6}\) | \(=\) |
\( 72\beta_{7} + 90\beta_{6} - 208\beta_{5} + 3\beta_{4} - 37\beta_{3} + 32\beta_{2} + 193\beta _1 + 210 \)
|
| \(\nu^{7}\) | \(=\) |
\( 280\beta_{7} + 286\beta_{6} - 744\beta_{5} + 4\beta_{4} - 154\beta_{3} + 109\beta_{2} + 701\beta _1 + 672 \)
|
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 |
|
1.00000 | −2.88289 | 1.00000 | −1.00000 | −2.88289 | 0.0509530 | 1.00000 | 5.31107 | −1.00000 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −2.33738 | 1.00000 | −1.00000 | −2.33738 | −3.59714 | 1.00000 | 2.46336 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −2.09612 | 1.00000 | −1.00000 | −2.09612 | −1.13654 | 1.00000 | 1.39373 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | −1.06149 | 1.00000 | −1.00000 | −1.06149 | 1.98179 | 1.00000 | −1.87323 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | −0.0768297 | 1.00000 | −1.00000 | −0.0768297 | 3.77672 | 1.00000 | −2.99410 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 0.468680 | 1.00000 | −1.00000 | 0.468680 | −1.40211 | 1.00000 | −2.78034 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | 1.47571 | 1.00000 | −1.00000 | 1.47571 | −3.54824 | 1.00000 | −0.822287 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.00000 | 2.51034 | 1.00000 | −1.00000 | 2.51034 | −4.12543 | 1.00000 | 3.30179 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(1\) |
| \(17\) | \(-1\) |
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 | 2890.2.a.bi | 8 | |
| 17.b | even | 2 | 1 | 2890.2.a.bj | 8 | ||
| 17.c | even | 4 | 2 | 2890.2.b.r | 16 | ||
| 17.e | odd | 16 | 2 | 170.2.k.b | ✓ | 16 | |
| 85.o | even | 16 | 2 | 850.2.o.g | 16 | ||
| 85.p | odd | 16 | 2 | 850.2.l.e | 16 | ||
| 85.r | even | 16 | 2 | 850.2.o.j | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 170.2.k.b | ✓ | 16 | 17.e | odd | 16 | 2 | |
| 850.2.l.e | 16 | 85.p | odd | 16 | 2 | ||
| 850.2.o.g | 16 | 85.o | even | 16 | 2 | ||
| 850.2.o.j | 16 | 85.r | even | 16 | 2 | ||
| 2890.2.a.bi | 8 | 1.a | even | 1 | 1 | trivial | |
| 2890.2.a.bj | 8 | 17.b | even | 2 | 1 | ||
| 2890.2.b.r | 16 | 17.c | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2890))\):
|
\( T_{3}^{8} + 4T_{3}^{7} - 6T_{3}^{6} - 36T_{3}^{5} - 7T_{3}^{4} + 72T_{3}^{3} + 32T_{3}^{2} - 24T_{3} - 2 \)
|
|
\( T_{7}^{8} + 8T_{7}^{7} - 136T_{7}^{5} - 274T_{7}^{4} + 272T_{7}^{3} + 1008T_{7}^{2} + 576T_{7} - 32 \)
|
|
\( T_{13}^{8} + 4T_{13}^{7} - 58T_{13}^{6} - 228T_{13}^{5} + 993T_{13}^{4} + 3528T_{13}^{3} - 6424T_{13}^{2} - 16352T_{13} + 12784 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{8} \)
|
| $3$ |
\( T^{8} + 4 T^{7} + \cdots - 2 \)
|
| $5$ |
\( (T + 1)^{8} \)
|
| $7$ |
\( T^{8} + 8 T^{7} + \cdots - 32 \)
|
| $11$ |
\( T^{8} + 8 T^{7} + \cdots - 68 \)
|
| $13$ |
\( T^{8} + 4 T^{7} + \cdots + 12784 \)
|
| $17$ |
\( T^{8} \)
|
| $19$ |
\( T^{8} - 12 T^{7} + \cdots + 9712 \)
|
| $23$ |
\( T^{8} + 24 T^{7} + \cdots - 45184 \)
|
| $29$ |
\( T^{8} + 28 T^{7} + \cdots + 100112 \)
|
| $31$ |
\( T^{8} + 4 T^{7} + \cdots - 22544 \)
|
| $37$ |
\( T^{8} + 24 T^{7} + \cdots + 1061344 \)
|
| $41$ |
\( T^{8} + 16 T^{7} + \cdots + 3908 \)
|
| $43$ |
\( T^{8} - 268 T^{6} + \cdots - 169864 \)
|
| $47$ |
\( T^{8} + 12 T^{7} + \cdots + 2742304 \)
|
| $53$ |
\( T^{8} + 12 T^{7} + \cdots + 12709904 \)
|
| $59$ |
\( T^{8} + 4 T^{7} + \cdots + 3004 \)
|
| $61$ |
\( T^{8} + 4 T^{7} + \cdots - 140048 \)
|
| $67$ |
\( T^{8} + 8 T^{7} + \cdots + 1132064 \)
|
| $71$ |
\( T^{8} + 36 T^{7} + \cdots + 30736 \)
|
| $73$ |
\( T^{8} + 28 T^{7} + \cdots + 3525826 \)
|
| $79$ |
\( T^{8} - 204 T^{6} + \cdots + 90176 \)
|
| $83$ |
\( T^{8} + 8 T^{7} + \cdots - 45832 \)
|
| $89$ |
\( T^{8} + 12 T^{7} + \cdots + 4118596 \)
|
| $97$ |
\( T^{8} + 28 T^{7} + \cdots - 1214 \)
|
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