Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [266,2,Mod(11,266)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(266, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("266.11");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 266 = 2 \cdot 7 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 266.g (of order \(3\), 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: | \(2.12402069377\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Relative dimension: | \(6\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} + 9x^{10} - 2x^{9} + 65x^{8} - 5x^{7} + 143x^{6} + 88x^{5} + 243x^{4} + 66x^{3} + 32x^{2} - 4x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} + 9x^{10} - 2x^{9} + 65x^{8} - 5x^{7} + 143x^{6} + 88x^{5} + 243x^{4} + 66x^{3} + 32x^{2} - 4x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 4255408 \nu^{11} - 1606511 \nu^{10} - 38600176 \nu^{9} - 1200471 \nu^{8} - 274852022 \nu^{7} + \cdots + 1639841567 ) / 552513861 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 17700016 \nu^{11} + 4255408 \nu^{10} - 157693633 \nu^{9} + 74000208 \nu^{8} - 1149300569 \nu^{7} + \cdots + 70943051 ) / 552513861 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 27266096 \nu^{11} + 17657978 \nu^{10} - 240016319 \nu^{9} + 194180298 \nu^{8} + \cdots - 787604984 ) / 552513861 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 70943051 \nu^{11} - 17700016 \nu^{10} - 634232051 \nu^{9} - 15807531 \nu^{8} + \cdots + 278316325 ) / 552513861 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 71349091 \nu^{11} - 19751018 \nu^{10} - 668162251 \nu^{9} - 29627781 \nu^{8} + \cdots - 343942336 ) / 552513861 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 90694069 \nu^{11} - 43720448 \nu^{10} - 806558014 \nu^{9} - 216843036 \nu^{8} + \cdots + 349665416 ) / 552513861 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 30918496 \nu^{11} - 6556843 \nu^{10} + 275689447 \nu^{9} - 122933523 \nu^{8} + 2007118289 \nu^{7} + \cdots - 664852274 ) / 184171287 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 136024183 \nu^{11} + 33492017 \nu^{10} + 1220152639 \nu^{9} + 32164089 \nu^{8} + \cdots + 534950464 ) / 552513861 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 163742032 \nu^{11} + 47017114 \nu^{10} - 1465315834 \nu^{9} + 739826346 \nu^{8} + \cdots + 1297645907 ) / 552513861 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 70943051 \nu^{11} - 17700016 \nu^{10} - 634232051 \nu^{9} - 15807531 \nu^{8} + \cdots + 278316325 ) / 184171287 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{11} - 3\beta_{5} \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{8} - 5\beta_{3} - \beta_{2} - 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -7\beta_{11} - \beta_{9} + \beta_{7} + 17\beta_{5} + \beta_{4} + 7\beta_{2} + 2\beta _1 - 17 \)
|
| \(\nu^{5}\) | \(=\) |
\( 10\beta_{11} + \beta_{10} + 8\beta_{9} + 8\beta_{8} + \beta_{6} - 15\beta_{5} + 30\beta_{3} + 8 \)
|
| \(\nu^{6}\) | \(=\) |
\( -10\beta_{8} - 9\beta_{4} - 27\beta_{3} - 48\beta_{2} - 27\beta _1 + 97 \)
|
| \(\nu^{7}\) | \(=\) |
\( -85\beta_{11} - 57\beta_{9} + \beta_{7} - 9\beta_{6} + 148\beta_{5} + \beta_{4} + 85\beta_{2} + 193\beta _1 - 148 \)
|
| \(\nu^{8}\) | \(=\) |
\( 335\beta_{11} + \beta_{10} + 86\beta_{9} + 86\beta_{8} - 65\beta_{7} + \beta_{6} - 713\beta_{5} + 261\beta_{3} + 86 \)
|
| \(\nu^{9}\) | \(=\) |
\( -65\beta_{10} - 400\beta_{8} - 22\beta_{4} - 1297\beta_{3} - 682\beta_{2} - 1297\beta _1 + 868 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 2379 \beta_{11} - 704 \beta_{9} + 443 \beta_{7} - 22 \beta_{6} + 4930 \beta_{5} + 443 \beta_{4} + \cdots - 4930 \)
|
| \(\nu^{11}\) | \(=\) |
\( 5315 \beta_{11} + 443 \beta_{10} + 2822 \beta_{9} + 2822 \beta_{8} - 283 \beta_{7} + 443 \beta_{6} + \cdots + 2822 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/266\mathbb{Z}\right)^\times\).
| \(n\) | \(115\) | \(211\) |
| \(\chi(n)\) | \(-1 + \beta_{5}\) | \(-1 + \beta_{5}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 11.1 |
|
−1.00000 | −1.13386 | − | 1.96390i | 1.00000 | 2.46576 | 1.13386 | + | 1.96390i | 2.58723 | + | 0.553408i | −1.00000 | −1.07127 | + | 1.85549i | −2.46576 | ||||||||||||||||||||||||||||||||||||||||||||||
| 11.2 | −1.00000 | −0.972618 | − | 1.68462i | 1.00000 | −1.58157 | 0.972618 | + | 1.68462i | 0.438874 | − | 2.60910i | −1.00000 | −0.391973 | + | 0.678918i | 1.58157 | |||||||||||||||||||||||||||||||||||||||||||||||
| 11.3 | −1.00000 | −0.0773345 | − | 0.133947i | 1.00000 | −3.74572 | 0.0773345 | + | 0.133947i | 1.43718 | + | 2.22138i | −1.00000 | 1.48804 | − | 2.57736i | 3.74572 | |||||||||||||||||||||||||||||||||||||||||||||||
| 11.4 | −1.00000 | 0.225896 | + | 0.391263i | 1.00000 | −0.629145 | −0.225896 | − | 0.391263i | −1.64717 | − | 2.07046i | −1.00000 | 1.39794 | − | 2.42131i | 0.629145 | |||||||||||||||||||||||||||||||||||||||||||||||
| 11.5 | −1.00000 | 0.595126 | + | 1.03079i | 1.00000 | 2.68173 | −0.595126 | − | 1.03079i | 0.325659 | + | 2.62563i | −1.00000 | 0.791651 | − | 1.37118i | −2.68173 | |||||||||||||||||||||||||||||||||||||||||||||||
| 11.6 | −1.00000 | 1.36279 | + | 2.36042i | 1.00000 | −2.19106 | −1.36279 | − | 2.36042i | −2.64177 | + | 0.145170i | −1.00000 | −2.21439 | + | 3.83544i | 2.19106 | |||||||||||||||||||||||||||||||||||||||||||||||
| 121.1 | −1.00000 | −1.13386 | + | 1.96390i | 1.00000 | 2.46576 | 1.13386 | − | 1.96390i | 2.58723 | − | 0.553408i | −1.00000 | −1.07127 | − | 1.85549i | −2.46576 | |||||||||||||||||||||||||||||||||||||||||||||||
| 121.2 | −1.00000 | −0.972618 | + | 1.68462i | 1.00000 | −1.58157 | 0.972618 | − | 1.68462i | 0.438874 | + | 2.60910i | −1.00000 | −0.391973 | − | 0.678918i | 1.58157 | |||||||||||||||||||||||||||||||||||||||||||||||
| 121.3 | −1.00000 | −0.0773345 | + | 0.133947i | 1.00000 | −3.74572 | 0.0773345 | − | 0.133947i | 1.43718 | − | 2.22138i | −1.00000 | 1.48804 | + | 2.57736i | 3.74572 | |||||||||||||||||||||||||||||||||||||||||||||||
| 121.4 | −1.00000 | 0.225896 | − | 0.391263i | 1.00000 | −0.629145 | −0.225896 | + | 0.391263i | −1.64717 | + | 2.07046i | −1.00000 | 1.39794 | + | 2.42131i | 0.629145 | |||||||||||||||||||||||||||||||||||||||||||||||
| 121.5 | −1.00000 | 0.595126 | − | 1.03079i | 1.00000 | 2.68173 | −0.595126 | + | 1.03079i | 0.325659 | − | 2.62563i | −1.00000 | 0.791651 | + | 1.37118i | −2.68173 | |||||||||||||||||||||||||||||||||||||||||||||||
| 121.6 | −1.00000 | 1.36279 | − | 2.36042i | 1.00000 | −2.19106 | −1.36279 | + | 2.36042i | −2.64177 | − | 0.145170i | −1.00000 | −2.21439 | − | 3.83544i | 2.19106 | |||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 133.h | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 266.2.g.c | ✓ | 12 |
| 7.c | even | 3 | 1 | 266.2.h.c | yes | 12 | |
| 19.c | even | 3 | 1 | 266.2.h.c | yes | 12 | |
| 133.h | even | 3 | 1 | inner | 266.2.g.c | ✓ | 12 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 266.2.g.c | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 266.2.g.c | ✓ | 12 | 133.h | even | 3 | 1 | inner |
| 266.2.h.c | yes | 12 | 7.c | even | 3 | 1 | |
| 266.2.h.c | yes | 12 | 19.c | even | 3 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} + 9 T_{3}^{10} + 2 T_{3}^{9} + 65 T_{3}^{8} + 5 T_{3}^{7} + 143 T_{3}^{6} - 88 T_{3}^{5} + \cdots + 1 \)
acting on \(S_{2}^{\mathrm{new}}(266, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{12} \)
|
| $3$ |
\( T^{12} + 9 T^{10} + \cdots + 1 \)
|
| $5$ |
\( (T^{6} + 3 T^{5} - 13 T^{4} + \cdots + 54)^{2} \)
|
| $7$ |
\( T^{12} - T^{11} + \cdots + 117649 \)
|
| $11$ |
\( T^{12} + T^{11} + \cdots + 21609 \)
|
| $13$ |
\( T^{12} + 5 T^{11} + \cdots + 2211169 \)
|
| $17$ |
\( T^{12} + 34 T^{10} + \cdots + 9 \)
|
| $19$ |
\( T^{12} + 5 T^{11} + \cdots + 47045881 \)
|
| $23$ |
\( T^{12} - 10 T^{11} + \cdots + 59397849 \)
|
| $29$ |
\( T^{12} - 3 T^{11} + \cdots + 34292736 \)
|
| $31$ |
\( T^{12} - T^{11} + \cdots + 2401 \)
|
| $37$ |
\( T^{12} - 5 T^{11} + \cdots + 32867289 \)
|
| $41$ |
\( T^{12} + T^{11} + \cdots + 998001 \)
|
| $43$ |
\( T^{12} + 102 T^{10} + \cdots + 16556761 \)
|
| $47$ |
\( T^{12} + \cdots + 2797034769 \)
|
| $53$ |
\( (T^{6} + 6 T^{5} - 105 T^{4} + \cdots - 18)^{2} \)
|
| $59$ |
\( T^{12} + \cdots + 27107317449 \)
|
| $61$ |
\( T^{12} - 4 T^{11} + \cdots + 18983449 \)
|
| $67$ |
\( (T^{6} - 7 T^{5} + \cdots + 419504)^{2} \)
|
| $71$ |
\( T^{12} + \cdots + 9814666761 \)
|
| $73$ |
\( T^{12} + \cdots + 78928969249 \)
|
| $79$ |
\( (T^{6} + 22 T^{5} + \cdots + 208208)^{2} \)
|
| $83$ |
\( (T^{6} + 2 T^{5} + \cdots + 160524)^{2} \)
|
| $89$ |
\( T^{12} - 2 T^{11} + \cdots + 35721 \)
|
| $97$ |
\( T^{12} + \cdots + 572501329 \)
|
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