Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1716,2,Mod(1585,1716)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1716, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("1716.1585");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1716 = 2^{2} \cdot 3 \cdot 11 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1716.e (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: | \(13.7023289869\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 2x^{9} + 2x^{8} - 2x^{7} + 48x^{6} - 110x^{5} + 126x^{4} + 64x^{2} - 144x + 162 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 2x^{9} + 2x^{8} - 2x^{7} + 48x^{6} - 110x^{5} + 126x^{4} + 64x^{2} - 144x + 162 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 3411 \nu^{9} - 130158 \nu^{8} - 37747 \nu^{7} + 2530 \nu^{6} + 126818 \nu^{5} + \cdots - 115386152 ) / 35805167 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 144530 \nu^{9} + 1381489 \nu^{8} - 171818 \nu^{7} + 285649 \nu^{6} - 9259278 \nu^{5} + \cdots + 160036010 ) / 35805167 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 1370026 \nu^{9} + 1193690 \nu^{8} + 321973 \nu^{7} - 1524094 \nu^{6} - 63008247 \nu^{5} + \cdots + 98347275 ) / 322246503 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 254410 \nu^{9} + 1110837 \nu^{8} + 1471754 \nu^{7} - 356652 \nu^{6} + 4313284 \nu^{5} + \cdots + 98625927 ) / 35805167 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 319759 \nu^{9} + 1591556 \nu^{8} - 683359 \nu^{7} + 594068 \nu^{6} - 17377194 \nu^{5} + \cdots + 69310326 ) / 35805167 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 5480104 \nu^{9} + 4774760 \nu^{8} + 1287892 \nu^{7} - 6096376 \nu^{6} - 252032988 \nu^{5} + \cdots + 393389100 ) / 322246503 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 2856305 \nu^{9} - 3619726 \nu^{8} + 2248576 \nu^{7} - 10264222 \nu^{6} + 132086310 \nu^{5} + \cdots - 278085078 ) / 107415501 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 3578972 \nu^{9} - 2401867 \nu^{8} - 4365989 \nu^{7} - 12626710 \nu^{6} + 147573459 \nu^{5} + \cdots - 208971261 ) / 107415501 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 12502657 \nu^{9} + 12571913 \nu^{8} - 1193690 \nu^{7} + 21943289 \nu^{6} - 574089312 \nu^{5} + \cdots + 1005700122 ) / 322246503 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{9} + \beta_{7} - \beta_{6} + \beta_{5} - \beta_{2} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} - 4\beta_{3} \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 6\beta_{9} + \beta_{8} + 4\beta_{7} - 6\beta_{6} - 4\beta_{5} - \beta_{4} + 3\beta_{3} + 6\beta_{2} + 6\beta _1 + 3 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} - 2\beta_{2} - 9\beta _1 - 22 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 18 \beta_{9} - 4 \beta_{8} - 11 \beta_{7} + 20 \beta_{6} - 11 \beta_{5} - 4 \beta_{4} - 17 \beta_{3} + \cdots + 17 \)
|
| \(\nu^{6}\) | \(=\) |
\( 25\beta_{9} + \beta_{8} + 13\beta_{7} - 69\beta_{6} + 137\beta_{3} \)
|
| \(\nu^{7}\) | \(=\) |
\( - 113 \beta_{9} - 27 \beta_{8} - 68 \beta_{7} + 139 \beta_{6} + 68 \beta_{5} + 27 \beta_{4} + \cdots - 149 \)
|
| \(\nu^{8}\) | \(=\) |
\( -122\beta_{5} - 17\beta_{4} + 228\beta_{2} + 508\beta _1 + 901 \)
|
| \(\nu^{9}\) | \(=\) |
\( 739 \beta_{9} + 176 \beta_{8} + 442 \beta_{7} - 984 \beta_{6} + 442 \beta_{5} + 176 \beta_{4} + \cdots - 1190 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1716\mathbb{Z}\right)^\times\).
| \(n\) | \(859\) | \(925\) | \(937\) | \(1145\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1585.1 |
|
0 | 1.00000 | 0 | − | 3.18368i | 0 | − | 0.345397i | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1585.2 | 0 | 1.00000 | 0 | − | 2.89774i | 0 | 3.88168i | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1585.3 | 0 | 1.00000 | 0 | − | 2.84405i | 0 | 0.568102i | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1585.4 | 0 | 1.00000 | 0 | − | 1.06101i | 0 | 1.69275i | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1585.5 | 0 | 1.00000 | 0 | − | 0.502901i | 0 | − | 3.10244i | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 1585.6 | 0 | 1.00000 | 0 | 0.502901i | 0 | 3.10244i | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1585.7 | 0 | 1.00000 | 0 | 1.06101i | 0 | − | 1.69275i | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1585.8 | 0 | 1.00000 | 0 | 2.84405i | 0 | − | 0.568102i | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1585.9 | 0 | 1.00000 | 0 | 2.89774i | 0 | − | 3.88168i | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1585.10 | 0 | 1.00000 | 0 | 3.18368i | 0 | 0.345397i | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1716.2.e.b | ✓ | 10 |
| 3.b | odd | 2 | 1 | 5148.2.e.d | 10 | ||
| 13.b | even | 2 | 1 | inner | 1716.2.e.b | ✓ | 10 |
| 39.d | odd | 2 | 1 | 5148.2.e.d | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1716.2.e.b | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 1716.2.e.b | ✓ | 10 | 13.b | even | 2 | 1 | inner |
| 5148.2.e.d | 10 | 3.b | odd | 2 | 1 | ||
| 5148.2.e.d | 10 | 39.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{10} + 28T_{5}^{8} + 272T_{5}^{6} + 1020T_{5}^{4} + 1016T_{5}^{2} + 196 \)
acting on \(S_{2}^{\mathrm{new}}(1716, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} \)
|
| $3$ |
\( (T - 1)^{10} \)
|
| $5$ |
\( T^{10} + 28 T^{8} + \cdots + 196 \)
|
| $7$ |
\( T^{10} + 28 T^{8} + \cdots + 16 \)
|
| $11$ |
\( (T^{2} + 1)^{5} \)
|
| $13$ |
\( T^{10} + 4 T^{9} + \cdots + 371293 \)
|
| $17$ |
\( (T^{5} - 4 T^{4} + \cdots - 374)^{2} \)
|
| $19$ |
\( T^{10} + 80 T^{8} + \cdots + 11664 \)
|
| $23$ |
\( (T^{5} - 8 T^{4} + \cdots + 108)^{2} \)
|
| $29$ |
\( (T^{5} + 2 T^{4} + \cdots + 4330)^{2} \)
|
| $31$ |
\( T^{10} + 92 T^{8} + \cdots + 252004 \)
|
| $37$ |
\( T^{10} + 116 T^{8} + \cdots + 3136 \)
|
| $41$ |
\( T^{10} + 164 T^{8} + \cdots + 394384 \)
|
| $43$ |
\( (T^{5} - 2 T^{4} + \cdots + 550)^{2} \)
|
| $47$ |
\( T^{10} + 332 T^{8} + \cdots + 37454400 \)
|
| $53$ |
\( (T^{5} + 10 T^{4} + \cdots + 720)^{2} \)
|
| $59$ |
\( T^{10} + \cdots + 631014400 \)
|
| $61$ |
\( (T^{5} + 6 T^{4} + \cdots + 18568)^{2} \)
|
| $67$ |
\( T^{10} + 280 T^{8} + \cdots + 1562500 \)
|
| $71$ |
\( T^{10} + 144 T^{8} + \cdots + 5184 \)
|
| $73$ |
\( T^{10} + 28 T^{8} + \cdots + 16 \)
|
| $79$ |
\( (T^{5} - 256 T^{3} + \cdots - 9882)^{2} \)
|
| $83$ |
\( T^{10} + \cdots + 1233414400 \)
|
| $89$ |
\( T^{10} + 236 T^{8} + \cdots + 10150596 \)
|
| $97$ |
\( T^{10} + \cdots + 409333824 \)
|
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