Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [510,2,Mod(137,510)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(510, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("510.137");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 510 = 2 \cdot 3 \cdot 5 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 510.l (of order \(4\), degree \(2\), minimal) |
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: | no |
| Analytic conductor: | \(4.07237050309\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | 8.0.4030726144.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 4x^{7} + 4x^{6} - 16x^{5} + 18x^{4} - 8x^{3} + 172x^{2} + 184x + 274 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{8} - 4x^{7} + 4x^{6} - 16x^{5} + 18x^{4} - 8x^{3} + 172x^{2} + 184x + 274 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2\nu^{7} + 4\nu^{6} - 68\nu^{5} + 135\nu^{4} - 304\nu^{3} + 510\nu^{2} + 204\nu + 2017 ) / 375 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} - 13\nu^{6} + 71\nu^{5} - 180\nu^{4} + 313\nu^{3} - 525\nu^{2} + 297\nu - 1339 ) / 375 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 7\nu^{7} - 36\nu^{6} + 62\nu^{5} - 115\nu^{4} + 86\nu^{3} + 85\nu^{2} + 664\nu + 497 ) / 375 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{7} + 33\nu^{6} - 161\nu^{5} + 335\nu^{4} - 658\nu^{3} + 665\nu^{2} - 357\nu + 2769 ) / 375 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 9\nu^{7} - 47\nu^{6} + 74\nu^{5} - 140\nu^{4} + 172\nu^{3} + 15\nu^{2} + 1088\nu + 579 ) / 375 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 13\nu^{7} - 34\nu^{6} - 72\nu^{5} + 125\nu^{4} - 391\nu^{3} + 970\nu^{2} + 1281\nu + 4483 ) / 375 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{7} - 2\beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + \beta _1 - 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( 4\beta_{7} - 2\beta_{6} - \beta_{5} - 2\beta_{4} - 3\beta_{3} - 9\beta_{2} + 2\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( 8\beta_{7} - 4\beta_{6} + 4\beta_{5} + 2\beta_{4} - 12\beta_{3} - 33\beta_{2} + 12\beta _1 + 13 \)
|
| \(\nu^{5}\) | \(=\) |
\( 24\beta_{7} - 39\beta_{6} + 9\beta_{5} + 36\beta_{4} - 26\beta_{3} - 89\beta_{2} + 45\beta _1 + 45 \)
|
| \(\nu^{6}\) | \(=\) |
\( 108\beta_{7} - 165\beta_{6} + 18\beta_{5} + 105\beta_{4} - 50\beta_{3} - 293\beta_{2} + 140\beta _1 + 89 \)
|
| \(\nu^{7}\) | \(=\) |
\( 413\beta_{7} - 520\beta_{6} + 103\beta_{5} + 320\beta_{4} - 175\beta_{3} - 1138\beta_{2} + 387\beta _1 + 177 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/510\mathbb{Z}\right)^\times\).
| \(n\) | \(241\) | \(307\) | \(341\) |
| \(\chi(n)\) | \(1\) | \(\beta_{2}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 137.1 |
|
−0.707107 | + | 0.707107i | −1.00000 | − | 1.41421i | − | 1.00000i | 0.512677 | + | 2.17650i | 1.70711 | + | 0.292893i | −2.07804 | − | 2.07804i | 0.707107 | + | 0.707107i | −1.00000 | + | 2.82843i | −1.90154 | − | 1.17650i | |||||||||||||||||||||||||
| 137.2 | −0.707107 | + | 0.707107i | −1.00000 | − | 1.41421i | − | 1.00000i | 1.90154 | − | 1.17650i | 1.70711 | + | 0.292893i | 2.66383 | + | 2.66383i | 0.707107 | + | 0.707107i | −1.00000 | + | 2.82843i | −0.512677 | + | 2.17650i | ||||||||||||||||||||||||||
| 137.3 | 0.707107 | − | 0.707107i | −1.00000 | + | 1.41421i | − | 1.00000i | −2.21154 | − | 0.330265i | 0.292893 | + | 1.70711i | 0.532935 | + | 0.532935i | −0.707107 | − | 0.707107i | −1.00000 | − | 2.82843i | −1.79733 | + | 1.33026i | ||||||||||||||||||||||||||
| 137.4 | 0.707107 | − | 0.707107i | −1.00000 | + | 1.41421i | − | 1.00000i | 1.79733 | + | 1.33026i | 0.292893 | + | 1.70711i | 2.88128 | + | 2.88128i | −0.707107 | − | 0.707107i | −1.00000 | − | 2.82843i | 2.21154 | − | 0.330265i | ||||||||||||||||||||||||||
| 443.1 | −0.707107 | − | 0.707107i | −1.00000 | + | 1.41421i | 1.00000i | 0.512677 | − | 2.17650i | 1.70711 | − | 0.292893i | −2.07804 | + | 2.07804i | 0.707107 | − | 0.707107i | −1.00000 | − | 2.82843i | −1.90154 | + | 1.17650i | |||||||||||||||||||||||||||
| 443.2 | −0.707107 | − | 0.707107i | −1.00000 | + | 1.41421i | 1.00000i | 1.90154 | + | 1.17650i | 1.70711 | − | 0.292893i | 2.66383 | − | 2.66383i | 0.707107 | − | 0.707107i | −1.00000 | − | 2.82843i | −0.512677 | − | 2.17650i | |||||||||||||||||||||||||||
| 443.3 | 0.707107 | + | 0.707107i | −1.00000 | − | 1.41421i | 1.00000i | −2.21154 | + | 0.330265i | 0.292893 | − | 1.70711i | 0.532935 | − | 0.532935i | −0.707107 | + | 0.707107i | −1.00000 | + | 2.82843i | −1.79733 | − | 1.33026i | |||||||||||||||||||||||||||
| 443.4 | 0.707107 | + | 0.707107i | −1.00000 | − | 1.41421i | 1.00000i | 1.79733 | − | 1.33026i | 0.292893 | − | 1.70711i | 2.88128 | − | 2.88128i | −0.707107 | + | 0.707107i | −1.00000 | + | 2.82843i | 2.21154 | + | 0.330265i | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 15.e | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 510.2.l.d | ✓ | 8 |
| 3.b | odd | 2 | 1 | 510.2.l.e | yes | 8 | |
| 5.c | odd | 4 | 1 | 510.2.l.e | yes | 8 | |
| 15.e | even | 4 | 1 | inner | 510.2.l.d | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 510.2.l.d | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 510.2.l.d | ✓ | 8 | 15.e | even | 4 | 1 | inner |
| 510.2.l.e | yes | 8 | 3.b | odd | 2 | 1 | |
| 510.2.l.e | yes | 8 | 5.c | odd | 4 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(510, [\chi])\):
|
\( T_{7}^{8} - 8T_{7}^{7} + 32T_{7}^{6} - 40T_{7}^{5} + 84T_{7}^{4} - 560T_{7}^{3} + 2592T_{7}^{2} - 2448T_{7} + 1156 \)
|
|
\( T_{23}^{8} + 16 T_{23}^{7} + 128 T_{23}^{6} + 40 T_{23}^{5} - 76 T_{23}^{4} + 11200 T_{23}^{3} + \cdots + 470596 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 1)^{2} \)
|
| $3$ |
\( (T^{2} + 2 T + 3)^{4} \)
|
| $5$ |
\( T^{8} - 4 T^{7} + \cdots + 625 \)
|
| $7$ |
\( T^{8} - 8 T^{7} + \cdots + 1156 \)
|
| $11$ |
\( (T^{2} + 2)^{4} \)
|
| $13$ |
\( T^{8} - 8 T^{7} + \cdots + 61504 \)
|
| $17$ |
\( (T^{4} + 1)^{2} \)
|
| $19$ |
\( T^{8} + 104 T^{6} + \cdots + 150544 \)
|
| $23$ |
\( T^{8} + 16 T^{7} + \cdots + 470596 \)
|
| $29$ |
\( (T^{4} - 4 T^{3} - 48 T^{2} + \cdots + 98)^{2} \)
|
| $31$ |
\( (T^{4} + 4 T^{3} + \cdots - 882)^{2} \)
|
| $37$ |
\( T^{8} + 8 T^{7} + \cdots + 202500 \)
|
| $41$ |
\( (T^{4} + 112 T^{2} + 1984)^{2} \)
|
| $43$ |
\( T^{8} - 16 T^{7} + \cdots + 1731856 \)
|
| $47$ |
\( (T^{2} - 12 T + 72)^{4} \)
|
| $53$ |
\( T^{8} + 8 T^{7} + \cdots + 5184 \)
|
| $59$ |
\( (T^{4} + 16 T^{3} + \cdots + 112)^{2} \)
|
| $61$ |
\( (T^{4} + 12 T^{3} + \cdots - 4078)^{2} \)
|
| $67$ |
\( T^{8} + 16 T^{7} + \cdots + 5992704 \)
|
| $71$ |
\( T^{8} + 296 T^{6} + \cdots + 23059204 \)
|
| $73$ |
\( T^{8} + 16 T^{7} + \cdots + 1327104 \)
|
| $79$ |
\( T^{8} + 200 T^{6} + \cdots + 470596 \)
|
| $83$ |
\( T^{8} - 16 T^{7} + \cdots + 1296 \)
|
| $89$ |
\( (T^{4} - 8 T^{3} + \cdots + 356)^{2} \)
|
| $97$ |
\( T^{8} - 16 T^{7} + \cdots + 3748096 \)
|
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