Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1005,2,Mod(766,1005)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1005, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1005.766");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1005 = 3 \cdot 5 \cdot 67 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1005.i (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: | \(8.02496540314\) |
| 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: |
\( x^{12} - 2x^{11} + 8x^{10} - 2x^{9} + 22x^{8} - 6x^{7} + 35x^{6} + 6x^{5} + 22x^{4} + 2x^{3} + 8x^{2} + 2x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{8} \) |
| 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
\( x^{12} - 2x^{11} + 8x^{10} - 2x^{9} + 22x^{8} - 6x^{7} + 35x^{6} + 6x^{5} + 22x^{4} + 2x^{3} + 8x^{2} + 2x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 10 \nu^{10} + 11 \nu^{9} - 54 \nu^{8} - 64 \nu^{7} - 146 \nu^{6} - 118 \nu^{5} - 144 \nu^{4} + \cdots + 10 ) / 29 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 81 \nu^{11} - 187 \nu^{10} + 261 \nu^{9} - 2924 \nu^{8} + 332 \nu^{7} - 6543 \nu^{6} + 3412 \nu^{5} + \cdots - 388 ) / 754 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 81 \nu^{11} + 187 \nu^{10} - 261 \nu^{9} + 2924 \nu^{8} - 332 \nu^{7} + 6543 \nu^{6} - 3412 \nu^{5} + \cdots + 388 ) / 754 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 107 \nu^{11} + 138 \nu^{10} - 688 \nu^{9} - 389 \nu^{8} - 2177 \nu^{7} - 758 \nu^{6} - 3101 \nu^{5} + \cdots - 986 ) / 377 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 280 \nu^{11} - 608 \nu^{10} + 2277 \nu^{9} - 843 \nu^{8} + 5851 \nu^{7} - 2816 \nu^{6} + 9047 \nu^{5} + \cdots + 329 ) / 754 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 418 \nu^{11} - 950 \nu^{10} + 3609 \nu^{9} - 1981 \nu^{8} + 9871 \nu^{7} - 5882 \nu^{6} + 14881 \nu^{5} + \cdots - 751 ) / 754 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 459 \nu^{11} + 977 \nu^{10} - 3903 \nu^{9} + 1466 \nu^{8} - 10720 \nu^{7} + 2963 \nu^{6} + \cdots - 968 ) / 754 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 459 \nu^{11} + 1081 \nu^{10} - 3942 \nu^{9} + 2103 \nu^{8} - 9979 \nu^{7} + 6291 \nu^{6} + \cdots + 813 ) / 754 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 281 \nu^{11} + 459 \nu^{10} - 1943 \nu^{9} - 500 \nu^{8} - 5099 \nu^{7} - 1042 \nu^{6} + \cdots - 1481 ) / 377 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 318 \nu^{11} - 432 \nu^{10} + 2104 \nu^{9} + 1038 \nu^{8} + 6330 \nu^{7} + 2543 \nu^{6} + 8958 \nu^{5} + \cdots + 1335 ) / 377 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 641 \nu^{11} + 1809 \nu^{10} - 6282 \nu^{9} + 5613 \nu^{8} - 15803 \nu^{7} + 15001 \nu^{6} + \cdots + 1567 ) / 754 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{11} + \beta_{10} - \beta_{9} + 2\beta_{7} - 3\beta_{5} + \beta_{2} - 1 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{10} - \beta_{9} + 3\beta_{7} + \beta_{4} - 3\beta_{3} - 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 7\beta_{11} - 4\beta_{8} + \beta_{6} + 10\beta_{5} - 15\beta_{3} - 8\beta_{2} - 10 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 17 \beta_{11} - 24 \beta_{10} + 17 \beta_{9} - 15 \beta_{8} - 40 \beta_{7} + 2 \beta_{6} + 16 \beta_{5} + \cdots + 24 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( -29\beta_{10} + 24\beta_{9} - 52\beta_{7} - 18\beta_{4} + 52\beta_{3} + 4\beta _1 + 55 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -123\beta_{11} + 104\beta_{8} - 19\beta_{6} - 116\beta_{5} + 274\beta_{3} + 159\beta_{2} + 116 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 330 \beta_{11} + 407 \beta_{10} - 330 \beta_{9} + 265 \beta_{8} + 718 \beta_{7} - 56 \beta_{6} + \cdots - 407 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( 542\beta_{10} - 430\beta_{9} + 944\beta_{7} + 358\beta_{4} - 944\beta_{3} - 71\beta _1 - 950 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 2274\beta_{11} - 1862\beta_{8} + 386\beta_{6} + 2194\beta_{5} - 4957\beta_{3} - 2825\beta_{2} - 2194 ) / 2 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 5959 \beta_{11} - 7455 \beta_{10} + 5959 \beta_{9} - 4934 \beta_{8} - 13028 \beta_{7} + 1002 \beta_{6} + \cdots + 7455 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1005\mathbb{Z}\right)^\times\).
| \(n\) | \(136\) | \(202\) | \(671\) |
| \(\chi(n)\) | \(-1 + \beta_{5}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 766.1 |
|
−1.20818 | + | 2.09262i | −1.00000 | −1.91938 | − | 3.32447i | −1.00000 | 1.20818 | − | 2.09262i | −0.746980 | − | 1.29381i | 4.44311 | 1.00000 | 1.20818 | − | 2.09262i | ||||||||||||||||||||||||||||||||||||||||||||
| 766.2 | −0.962490 | + | 1.66708i | −1.00000 | −0.852773 | − | 1.47705i | −1.00000 | 0.962490 | − | 1.66708i | 0.945042 | + | 1.63686i | −0.566818 | 1.00000 | 0.962490 | − | 1.66708i | |||||||||||||||||||||||||||||||||||||||||||||
| 766.3 | −0.538803 | + | 0.933233i | −1.00000 | 0.419384 | + | 0.726394i | −1.00000 | 0.538803 | − | 0.933233i | −0.746980 | − | 1.29381i | −3.05907 | 1.00000 | 0.538803 | − | 0.933233i | |||||||||||||||||||||||||||||||||||||||||||||
| 766.4 | −0.0207748 | + | 0.0359830i | −1.00000 | 0.999137 | + | 1.73056i | −1.00000 | 0.0207748 | − | 0.0359830i | 2.30194 | + | 3.98707i | −0.166127 | 1.00000 | 0.0207748 | − | 0.0359830i | |||||||||||||||||||||||||||||||||||||||||||||
| 766.5 | 0.907532 | − | 1.57189i | −1.00000 | −0.647227 | − | 1.12103i | −1.00000 | −0.907532 | + | 1.57189i | 0.945042 | + | 1.63686i | 1.28061 | 1.00000 | −0.907532 | + | 1.57189i | |||||||||||||||||||||||||||||||||||||||||||||
| 766.6 | 1.32271 | − | 2.29101i | −1.00000 | −2.49914 | − | 4.32863i | −1.00000 | −1.32271 | + | 2.29101i | 2.30194 | + | 3.98707i | −7.93171 | 1.00000 | −1.32271 | + | 2.29101i | |||||||||||||||||||||||||||||||||||||||||||||
| 841.1 | −1.20818 | − | 2.09262i | −1.00000 | −1.91938 | + | 3.32447i | −1.00000 | 1.20818 | + | 2.09262i | −0.746980 | + | 1.29381i | 4.44311 | 1.00000 | 1.20818 | + | 2.09262i | |||||||||||||||||||||||||||||||||||||||||||||
| 841.2 | −0.962490 | − | 1.66708i | −1.00000 | −0.852773 | + | 1.47705i | −1.00000 | 0.962490 | + | 1.66708i | 0.945042 | − | 1.63686i | −0.566818 | 1.00000 | 0.962490 | + | 1.66708i | |||||||||||||||||||||||||||||||||||||||||||||
| 841.3 | −0.538803 | − | 0.933233i | −1.00000 | 0.419384 | − | 0.726394i | −1.00000 | 0.538803 | + | 0.933233i | −0.746980 | + | 1.29381i | −3.05907 | 1.00000 | 0.538803 | + | 0.933233i | |||||||||||||||||||||||||||||||||||||||||||||
| 841.4 | −0.0207748 | − | 0.0359830i | −1.00000 | 0.999137 | − | 1.73056i | −1.00000 | 0.0207748 | + | 0.0359830i | 2.30194 | − | 3.98707i | −0.166127 | 1.00000 | 0.0207748 | + | 0.0359830i | |||||||||||||||||||||||||||||||||||||||||||||
| 841.5 | 0.907532 | + | 1.57189i | −1.00000 | −0.647227 | + | 1.12103i | −1.00000 | −0.907532 | − | 1.57189i | 0.945042 | − | 1.63686i | 1.28061 | 1.00000 | −0.907532 | − | 1.57189i | |||||||||||||||||||||||||||||||||||||||||||||
| 841.6 | 1.32271 | + | 2.29101i | −1.00000 | −2.49914 | + | 4.32863i | −1.00000 | −1.32271 | − | 2.29101i | 2.30194 | − | 3.98707i | −7.93171 | 1.00000 | −1.32271 | − | 2.29101i | |||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 67.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1005.2.i.c | ✓ | 12 |
| 67.c | even | 3 | 1 | inner | 1005.2.i.c | ✓ | 12 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1005.2.i.c | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 1005.2.i.c | ✓ | 12 | 67.c | even | 3 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} + T_{2}^{11} + 11 T_{2}^{10} + 12 T_{2}^{9} + 89 T_{2}^{8} + 91 T_{2}^{7} + 318 T_{2}^{6} + \cdots + 1 \)
acting on \(S_{2}^{\mathrm{new}}(1005, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + T^{11} + \cdots + 1 \)
|
| $3$ |
\( (T + 1)^{12} \)
|
| $5$ |
\( (T + 1)^{12} \)
|
| $7$ |
\( (T^{6} - 5 T^{5} + \cdots + 169)^{2} \)
|
| $11$ |
\( T^{12} + 43 T^{10} + \cdots + 214369 \)
|
| $13$ |
\( T^{12} + 12 T^{11} + \cdots + 5041 \)
|
| $17$ |
\( T^{12} + 8 T^{11} + \cdots + 841 \)
|
| $19$ |
\( T^{12} - 8 T^{11} + \cdots + 1190281 \)
|
| $23$ |
\( T^{12} + 2 T^{11} + \cdots + 9308601 \)
|
| $29$ |
\( T^{12} + \cdots + 109977169 \)
|
| $31$ |
\( T^{12} + \cdots + 385297641 \)
|
| $37$ |
\( T^{12} + 23 T^{10} + \cdots + 32761 \)
|
| $41$ |
\( T^{12} - 10 T^{11} + \cdots + 169 \)
|
| $43$ |
\( (T^{6} + 16 T^{5} + \cdots + 4096)^{2} \)
|
| $47$ |
\( T^{12} - 2 T^{11} + \cdots + 97673689 \)
|
| $53$ |
\( (T^{6} + 8 T^{5} + \cdots + 8128)^{2} \)
|
| $59$ |
\( (T^{6} + 16 T^{5} + \cdots + 339968)^{2} \)
|
| $61$ |
\( T^{12} + \cdots + 275659609 \)
|
| $67$ |
\( T^{12} + \cdots + 90458382169 \)
|
| $71$ |
\( T^{12} + \cdots + 305235841 \)
|
| $73$ |
\( T^{12} + \cdots + 120802081 \)
|
| $79$ |
\( T^{12} + \cdots + 10073534689 \)
|
| $83$ |
\( T^{12} + 2 T^{11} + \cdots + 121801 \)
|
| $89$ |
\( (T^{6} + 36 T^{5} + \cdots - 47552)^{2} \)
|
| $97$ |
\( T^{12} + \cdots + 33609522241 \)
|
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