Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2310,2,Mod(1121,2310)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2310, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2310.1121");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2310 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2310.g (of order \(2\), degree \(1\), 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: | \(18.4454428669\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.619810816.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 2x^{5} + 14x^{4} - 8x^{3} + 2x^{2} + 2x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{8} - 2x^{5} + 14x^{4} - 8x^{3} + 2x^{2} + 2x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 59\nu^{7} - 65\nu^{6} - 96\nu^{5} - 142\nu^{4} + 950\nu^{3} - 1032\nu^{2} - 459\nu + 402 ) / 319 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -63\nu^{7} + 64\nu^{6} + 16\nu^{5} + 130\nu^{4} - 1009\nu^{3} + 1448\nu^{2} - 402\nu - 67 ) / 319 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 64\nu^{7} + 16\nu^{6} + 4\nu^{5} - 127\nu^{4} + 944\nu^{3} - 276\nu^{2} + 378\nu + 63 ) / 319 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -64\nu^{7} - 16\nu^{6} - 4\nu^{5} + 127\nu^{4} - 944\nu^{3} + 276\nu^{2} + 260\nu - 63 ) / 319 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 83\nu^{7} - 59\nu^{6} + 65\nu^{5} - 70\nu^{4} + 1304\nu^{3} - 1614\nu^{2} + 1198\nu + 306 ) / 319 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -129\nu^{7} - 112\nu^{6} - 28\nu^{5} + 251\nu^{4} - 1504\nu^{3} - 301\nu^{2} + 225\nu - 441 ) / 319 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 239\nu^{7} - 20\nu^{6} - 5\nu^{5} - 559\nu^{4} + 3286\nu^{3} - 2207\nu^{2} + 644\nu + 1 ) / 319 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + \beta_{3} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{7} + \beta_{6} + \beta_{5} + 4\beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{7} + \beta_{6} - \beta_{5} - 3\beta_{4} + 3\beta_{3} - 2\beta_{2} - \beta _1 + 2 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -2\beta_{7} - 2\beta_{6} + 2\beta_{5} + \beta_{4} + 2\beta _1 - 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 6\beta_{7} + 6\beta_{6} + 4\beta_{5} - 11\beta_{4} - 11\beta_{3} + 12\beta_{2} - 4\beta _1 + 12 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -17\beta_{7} - 15\beta_{6} - 17\beta_{5} + 14\beta_{3} - 56\beta_{2} - 15\beta_1 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 15\beta_{7} - 15\beta_{6} + 31\beta_{5} + 43\beta_{4} - 43\beta_{3} + 60\beta_{2} + 31\beta _1 - 60 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2310\mathbb{Z}\right)^\times\).
| \(n\) | \(211\) | \(661\) | \(1387\) | \(1541\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1121.1 |
|
1.00000 | −1.55654 | − | 0.759725i | 1.00000 | 1.00000i | −1.55654 | − | 0.759725i | 1.00000i | 1.00000 | 1.84564 | + | 2.36509i | 1.00000i | ||||||||||||||||||||||||||||||||||||
| 1121.2 | 1.00000 | −1.55654 | + | 0.759725i | 1.00000 | − | 1.00000i | −1.55654 | + | 0.759725i | − | 1.00000i | 1.00000 | 1.84564 | − | 2.36509i | − | 1.00000i | ||||||||||||||||||||||||||||||||||
| 1121.3 | 1.00000 | −1.29021 | − | 1.15558i | 1.00000 | − | 1.00000i | −1.29021 | − | 1.15558i | − | 1.00000i | 1.00000 | 0.329281 | + | 2.98187i | − | 1.00000i | ||||||||||||||||||||||||||||||||||
| 1121.4 | 1.00000 | −1.29021 | + | 1.15558i | 1.00000 | 1.00000i | −1.29021 | + | 1.15558i | 1.00000i | 1.00000 | 0.329281 | − | 2.98187i | 1.00000i | |||||||||||||||||||||||||||||||||||||
| 1121.5 | 1.00000 | 0.146426 | − | 1.72585i | 1.00000 | 1.00000i | 0.146426 | − | 1.72585i | 1.00000i | 1.00000 | −2.95712 | − | 0.505418i | 1.00000i | |||||||||||||||||||||||||||||||||||||
| 1121.6 | 1.00000 | 0.146426 | + | 1.72585i | 1.00000 | − | 1.00000i | 0.146426 | + | 1.72585i | − | 1.00000i | 1.00000 | −2.95712 | + | 0.505418i | − | 1.00000i | ||||||||||||||||||||||||||||||||||
| 1121.7 | 1.00000 | 1.70032 | − | 0.329998i | 1.00000 | − | 1.00000i | 1.70032 | − | 0.329998i | − | 1.00000i | 1.00000 | 2.78220 | − | 1.12221i | − | 1.00000i | ||||||||||||||||||||||||||||||||||
| 1121.8 | 1.00000 | 1.70032 | + | 0.329998i | 1.00000 | 1.00000i | 1.70032 | + | 0.329998i | 1.00000i | 1.00000 | 2.78220 | + | 1.12221i | 1.00000i | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 33.d | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2310.2.g.d | yes | 8 |
| 3.b | odd | 2 | 1 | 2310.2.g.c | ✓ | 8 | |
| 11.b | odd | 2 | 1 | 2310.2.g.c | ✓ | 8 | |
| 33.d | even | 2 | 1 | inner | 2310.2.g.d | yes | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2310.2.g.c | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 2310.2.g.c | ✓ | 8 | 11.b | odd | 2 | 1 | |
| 2310.2.g.d | yes | 8 | 1.a | even | 1 | 1 | trivial |
| 2310.2.g.d | yes | 8 | 33.d | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2310, [\chi])\):
|
\( T_{13}^{8} + 80T_{13}^{6} + 1888T_{13}^{4} + 14592T_{13}^{2} + 12544 \)
|
|
\( T_{17}^{4} - 32T_{17}^{2} + 48T_{17} - 16 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{8} \)
|
| $3$ |
\( T^{8} + 2 T^{7} + \cdots + 81 \)
|
| $5$ |
\( (T^{2} + 1)^{4} \)
|
| $7$ |
\( (T^{2} + 1)^{4} \)
|
| $11$ |
\( T^{8} - 12 T^{7} + \cdots + 14641 \)
|
| $13$ |
\( T^{8} + 80 T^{6} + \cdots + 12544 \)
|
| $17$ |
\( (T^{4} - 32 T^{2} + \cdots - 16)^{2} \)
|
| $19$ |
\( T^{8} + 28 T^{6} + \cdots + 64 \)
|
| $23$ |
\( T^{8} + 76 T^{6} + \cdots + 1600 \)
|
| $29$ |
\( (T^{4} - 18 T^{3} + \cdots + 104)^{2} \)
|
| $31$ |
\( (T^{4} - 4 T^{3} - 16 T^{2} + \cdots - 16)^{2} \)
|
| $37$ |
\( (T^{4} - 10 T^{3} + \cdots - 376)^{2} \)
|
| $41$ |
\( (T^{4} - 10 T^{3} + \cdots + 776)^{2} \)
|
| $43$ |
\( T^{8} + 112 T^{6} + \cdots + 256 \)
|
| $47$ |
\( T^{8} + 208 T^{6} + \cdots + 43264 \)
|
| $53$ |
\( T^{8} + 332 T^{6} + \cdots + 28987456 \)
|
| $59$ |
\( T^{8} + 304 T^{6} + \cdots + 409600 \)
|
| $61$ |
\( T^{8} + 320 T^{6} + \cdots + 12845056 \)
|
| $67$ |
\( (T^{4} - 4 T^{3} - 16 T^{2} + \cdots - 16)^{2} \)
|
| $71$ |
\( T^{8} + 256 T^{6} + \cdots + 2262016 \)
|
| $73$ |
\( T^{8} + 368 T^{6} + \cdots + 16128256 \)
|
| $79$ |
\( T^{8} + 180 T^{6} + \cdots + 419904 \)
|
| $83$ |
\( (T^{4} - 8 T^{3} + \cdots - 256)^{2} \)
|
| $89$ |
\( T^{8} + 208 T^{6} + \cdots + 1364224 \)
|
| $97$ |
\( (T^{4} - 26 T^{3} + \cdots - 56)^{2} \)
|
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