Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [510,2,Mod(259,510)]
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([0, 2, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("510.259");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 510 = 2 \cdot 3 \cdot 5 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 510.m (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: | 8.0.110166016.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 10x^{6} + 19x^{4} + 10x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{8} + 10x^{6} + 19x^{4} + 10x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{4} + 9\nu^{2} + 7 ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} + \nu^{5} + 10\nu^{4} + 9\nu^{3} + 18\nu^{2} + 9\nu + 5 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} - \nu^{5} + 10\nu^{4} - 9\nu^{3} + 18\nu^{2} - 9\nu + 5 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 3\nu^{6} + \nu^{5} + 28\nu^{4} + 9\nu^{3} + 40\nu^{2} + 13\nu + 13 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2\nu^{7} + 19\nu^{5} + 29\nu^{3} + 9\nu ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -3\nu^{6} + \nu^{5} - 28\nu^{4} + 9\nu^{3} - 40\nu^{2} + 13\nu - 13 ) / 4 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 3\nu^{7} + 29\nu^{5} + 47\nu^{3} + 14\nu ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{6} + \beta_{4} + \beta_{3} - \beta_{2} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{6} + \beta_{4} - 3\beta_{3} - 3\beta_{2} + 2\beta _1 - 6 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{7} - 2\beta_{6} + 3\beta_{5} - 2\beta_{4} - 3\beta_{3} + 3\beta_{2} \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 9\beta_{6} - 9\beta_{4} + 27\beta_{3} + 27\beta_{2} - 14\beta _1 + 40 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 36\beta_{7} + 27\beta_{6} - 54\beta_{5} + 27\beta_{4} + 41\beta_{3} - 41\beta_{2} ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( -36\beta_{6} + 36\beta_{4} - 106\beta_{3} - 106\beta_{2} + 52\beta _1 - 151 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -284\beta_{7} - 203\beta_{6} + 428\beta_{5} - 203\beta_{4} - 307\beta_{3} + 307\beta_{2} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/510\mathbb{Z}\right)^\times\).
| \(n\) | \(241\) | \(307\) | \(341\) |
| \(\chi(n)\) | \(-\beta_{5}\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 259.1 |
|
−1.00000 | −0.707107 | − | 0.707107i | 1.00000 | −1.70711 | − | 1.44423i | 0.707107 | + | 0.707107i | −1.55123 | + | 1.55123i | −1.00000 | 1.00000i | 1.70711 | + | 1.44423i | ||||||||||||||||||||||||||||||||
| 259.2 | −1.00000 | −0.707107 | − | 0.707107i | 1.00000 | −1.70711 | + | 1.44423i | 0.707107 | + | 0.707107i | 3.37966 | − | 3.37966i | −1.00000 | 1.00000i | 1.70711 | − | 1.44423i | |||||||||||||||||||||||||||||||||
| 259.3 | −1.00000 | 0.707107 | + | 0.707107i | 1.00000 | −0.292893 | − | 2.21680i | −0.707107 | − | 0.707107i | −2.56350 | + | 2.56350i | −1.00000 | 1.00000i | 0.292893 | + | 2.21680i | |||||||||||||||||||||||||||||||||
| 259.4 | −1.00000 | 0.707107 | + | 0.707107i | 1.00000 | −0.292893 | + | 2.21680i | −0.707107 | − | 0.707107i | −1.26493 | + | 1.26493i | −1.00000 | 1.00000i | 0.292893 | − | 2.21680i | |||||||||||||||||||||||||||||||||
| 319.1 | −1.00000 | −0.707107 | + | 0.707107i | 1.00000 | −1.70711 | − | 1.44423i | 0.707107 | − | 0.707107i | 3.37966 | + | 3.37966i | −1.00000 | − | 1.00000i | 1.70711 | + | 1.44423i | ||||||||||||||||||||||||||||||||
| 319.2 | −1.00000 | −0.707107 | + | 0.707107i | 1.00000 | −1.70711 | + | 1.44423i | 0.707107 | − | 0.707107i | −1.55123 | − | 1.55123i | −1.00000 | − | 1.00000i | 1.70711 | − | 1.44423i | ||||||||||||||||||||||||||||||||
| 319.3 | −1.00000 | 0.707107 | − | 0.707107i | 1.00000 | −0.292893 | − | 2.21680i | −0.707107 | + | 0.707107i | −1.26493 | − | 1.26493i | −1.00000 | − | 1.00000i | 0.292893 | + | 2.21680i | ||||||||||||||||||||||||||||||||
| 319.4 | −1.00000 | 0.707107 | − | 0.707107i | 1.00000 | −0.292893 | + | 2.21680i | −0.707107 | + | 0.707107i | −2.56350 | − | 2.56350i | −1.00000 | − | 1.00000i | 0.292893 | − | 2.21680i | ||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 85.j | 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.m.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | 1530.2.n.r | 8 | ||
| 5.b | even | 2 | 1 | 510.2.m.b | yes | 8 | |
| 15.d | odd | 2 | 1 | 1530.2.n.q | 8 | ||
| 17.c | even | 4 | 1 | 510.2.m.b | yes | 8 | |
| 51.f | odd | 4 | 1 | 1530.2.n.q | 8 | ||
| 85.j | even | 4 | 1 | inner | 510.2.m.a | ✓ | 8 |
| 255.i | odd | 4 | 1 | 1530.2.n.r | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 510.2.m.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 510.2.m.a | ✓ | 8 | 85.j | even | 4 | 1 | inner |
| 510.2.m.b | yes | 8 | 5.b | even | 2 | 1 | |
| 510.2.m.b | yes | 8 | 17.c | even | 4 | 1 | |
| 1530.2.n.q | 8 | 15.d | odd | 2 | 1 | ||
| 1530.2.n.q | 8 | 51.f | odd | 4 | 1 | ||
| 1530.2.n.r | 8 | 3.b | odd | 2 | 1 | ||
| 1530.2.n.r | 8 | 255.i | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{8} + 4T_{7}^{7} + 8T_{7}^{6} + 32T_{7}^{5} + 460T_{7}^{4} + 2144T_{7}^{3} + 5408T_{7}^{2} + 7072T_{7} + 4624 \)
acting on \(S_{2}^{\mathrm{new}}(510, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{8} \)
|
| $3$ |
\( (T^{4} + 1)^{2} \)
|
| $5$ |
\( (T^{4} + 4 T^{3} + 12 T^{2} + \cdots + 25)^{2} \)
|
| $7$ |
\( T^{8} + 4 T^{7} + \cdots + 4624 \)
|
| $11$ |
\( (T^{4} + 8 T^{3} + 32 T^{2} + \cdots + 16)^{2} \)
|
| $13$ |
\( T^{8} + 76 T^{6} + \cdots + 35344 \)
|
| $17$ |
\( T^{8} + 4 T^{7} + \cdots + 83521 \)
|
| $19$ |
\( (T^{2} + 8)^{4} \)
|
| $23$ |
\( T^{8} - 16 T^{7} + \cdots + 85264 \)
|
| $29$ |
\( T^{8} - 8 T^{7} + \cdots + 12544 \)
|
| $31$ |
\( T^{8} - 16 T^{7} + \cdots + 85264 \)
|
| $37$ |
\( T^{8} - 512 T^{5} + \cdots + 65536 \)
|
| $41$ |
\( T^{8} + 4 T^{7} + \cdots + 784 \)
|
| $43$ |
\( (T^{4} + 12 T^{3} + \cdots + 68)^{2} \)
|
| $47$ |
\( T^{8} + 264 T^{6} + \cdots + 9535744 \)
|
| $53$ |
\( (T^{4} - 8 T^{3} + \cdots - 1168)^{2} \)
|
| $59$ |
\( T^{8} + 284 T^{6} + \cdots + 1336336 \)
|
| $61$ |
\( T^{8} - 16 T^{7} + \cdots + 73984 \)
|
| $67$ |
\( T^{8} + 148 T^{6} + \cdots + 10000 \)
|
| $71$ |
\( T^{8} + 44 T^{7} + \cdots + 795664 \)
|
| $73$ |
\( T^{8} - 20 T^{7} + \cdots + 59043856 \)
|
| $79$ |
\( T^{8} - 16 T^{7} + \cdots + 10000 \)
|
| $83$ |
\( (T^{4} + 8 T^{3} + \cdots - 1600)^{2} \)
|
| $89$ |
\( (T^{4} - 8 T^{3} + \cdots - 1168)^{2} \)
|
| $97$ |
\( T^{8} - 20 T^{7} + \cdots + 85264 \)
|
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