Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [287,2,Mod(165,287)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(287, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("287.165");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 287 = 7 \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 287.e (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: | \(2.29170653801\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Relative dimension: | \(5\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - x^{9} + 7x^{8} + 4x^{7} + 32x^{6} + 3x^{5} + 30x^{4} - 7x^{3} + 26x^{2} - 5x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - x^{9} + 7x^{8} + 4x^{7} + 32x^{6} + 3x^{5} + 30x^{4} - 7x^{3} + 26x^{2} - 5x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 911 \nu^{9} + 318 \nu^{8} - 583 \nu^{7} + 16404 \nu^{6} - 265 \nu^{5} + 8533 \nu^{4} - 159430 \nu^{3} + \cdots + 31405 ) / 160271 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 4571 \nu^{9} + 29016 \nu^{8} - 53196 \nu^{7} + 154140 \nu^{6} - 24180 \nu^{5} + 778596 \nu^{4} + \cdots + 319366 ) / 160271 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 4889 \nu^{9} - 36294 \nu^{8} + 66539 \nu^{7} - 199961 \nu^{6} + 30245 \nu^{5} - 973889 \nu^{4} + \cdots - 672224 ) / 160271 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 5191 \nu^{9} + 12966 \nu^{8} - 23771 \nu^{7} + 9622 \nu^{6} - 10805 \nu^{5} + 347921 \nu^{4} + \cdots + 388420 ) / 160271 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 31405 \nu^{9} + 32316 \nu^{8} - 219517 \nu^{7} - 126203 \nu^{6} - 988556 \nu^{5} - 94480 \nu^{4} + \cdots + 155382 ) / 160271 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 62396 \nu^{9} + 33989 \nu^{8} - 409567 \nu^{7} - 433019 \nu^{6} - 2138559 \nu^{5} - 984502 \nu^{4} + \cdots - 15562 ) / 160271 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 91217 \nu^{9} + 91837 \nu^{8} - 622469 \nu^{7} - 394293 \nu^{6} - 2774426 \nu^{5} + \cdots + 373160 ) / 160271 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 128866 \nu^{9} + 127955 \nu^{8} - 902380 \nu^{7} - 514881 \nu^{6} - 4140116 \nu^{5} + \cdots + 645973 ) / 160271 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{9} - \beta_{7} - 2\beta_{6} - \beta_{3} - \beta_{2} + \beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} - 2\beta_{4} - 2\beta_{3} - 5\beta_{2} - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -7\beta_{9} + \beta_{8} + 8\beta_{7} + 9\beta_{6} - \beta_{5} - 7\beta_{4} - 10\beta _1 - 9 \)
|
| \(\nu^{5}\) | \(=\) |
\( -18\beta_{9} + 7\beta_{8} + 19\beta_{7} + 14\beta_{6} + 19\beta_{3} + 34\beta_{2} - 34\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 12\beta_{5} + 53\beta_{4} + 60\beta_{3} + 85\beta_{2} + 57 \)
|
| \(\nu^{7}\) | \(=\) |
\( 145\beta_{9} - 48\beta_{8} - 157\beta_{7} - 129\beta_{6} + 48\beta_{5} + 145\beta_{4} + 255\beta _1 + 129 \)
|
| \(\nu^{8}\) | \(=\) |
\( 412\beta_{9} - 109\beta_{8} - 460\beta_{7} - 413\beta_{6} - 460\beta_{3} - 686\beta_{2} + 686\beta_1 \)
|
| \(\nu^{9}\) | \(=\) |
\( -351\beta_{5} - 1146\beta_{4} - 1255\beta_{3} - 1971\beta_{2} - 1069 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/287\mathbb{Z}\right)^\times\).
| \(n\) | \(206\) | \(211\) |
| \(\chi(n)\) | \(-1 + \beta_{6}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 165.1 |
|
−0.985135 | − | 1.70630i | −0.257781 | + | 0.446490i | −0.940981 | + | 1.62983i | 0.257781 | + | 0.446490i | 1.01580 | 1.91822 | − | 1.82221i | −0.232565 | 1.36710 | + | 2.36788i | 0.507899 | − | 0.879706i | ||||||||||||||||||||||||||||||||||
| 165.2 | −0.678233 | − | 1.17473i | 1.11395 | − | 1.92942i | 0.0800004 | − | 0.138565i | −1.11395 | − | 1.92942i | −3.02207 | 2.60927 | + | 0.437865i | −2.92997 | −0.981768 | − | 1.70047i | −1.51103 | + | 2.61719i | |||||||||||||||||||||||||||||||||||
| 165.3 | 0.564230 | + | 0.977276i | −1.46472 | + | 2.53697i | 0.363288 | − | 0.629233i | 1.46472 | + | 2.53697i | −3.30576 | 1.22536 | + | 2.34489i | 3.07683 | −2.79081 | − | 4.83382i | −1.65288 | + | 2.86287i | |||||||||||||||||||||||||||||||||||
| 165.4 | 0.894706 | + | 1.54968i | 1.16033 | − | 2.00974i | −0.600998 | + | 1.04096i | −1.16033 | − | 2.00974i | 4.15260 | −1.87001 | − | 1.87165i | 1.42796 | −1.19271 | − | 2.06584i | 2.07630 | − | 3.59626i | |||||||||||||||||||||||||||||||||||
| 165.5 | 1.20443 | + | 2.08614i | 0.448226 | − | 0.776350i | −1.90131 | + | 3.29316i | −0.448226 | − | 0.776350i | 2.15943 | 0.117164 | + | 2.64316i | −4.34226 | 1.09819 | + | 1.90212i | 1.07971 | − | 1.87012i | |||||||||||||||||||||||||||||||||||
| 247.1 | −0.985135 | + | 1.70630i | −0.257781 | − | 0.446490i | −0.940981 | − | 1.62983i | 0.257781 | − | 0.446490i | 1.01580 | 1.91822 | + | 1.82221i | −0.232565 | 1.36710 | − | 2.36788i | 0.507899 | + | 0.879706i | |||||||||||||||||||||||||||||||||||
| 247.2 | −0.678233 | + | 1.17473i | 1.11395 | + | 1.92942i | 0.0800004 | + | 0.138565i | −1.11395 | + | 1.92942i | −3.02207 | 2.60927 | − | 0.437865i | −2.92997 | −0.981768 | + | 1.70047i | −1.51103 | − | 2.61719i | |||||||||||||||||||||||||||||||||||
| 247.3 | 0.564230 | − | 0.977276i | −1.46472 | − | 2.53697i | 0.363288 | + | 0.629233i | 1.46472 | − | 2.53697i | −3.30576 | 1.22536 | − | 2.34489i | 3.07683 | −2.79081 | + | 4.83382i | −1.65288 | − | 2.86287i | |||||||||||||||||||||||||||||||||||
| 247.4 | 0.894706 | − | 1.54968i | 1.16033 | + | 2.00974i | −0.600998 | − | 1.04096i | −1.16033 | + | 2.00974i | 4.15260 | −1.87001 | + | 1.87165i | 1.42796 | −1.19271 | + | 2.06584i | 2.07630 | + | 3.59626i | |||||||||||||||||||||||||||||||||||
| 247.5 | 1.20443 | − | 2.08614i | 0.448226 | + | 0.776350i | −1.90131 | − | 3.29316i | −0.448226 | + | 0.776350i | 2.15943 | 0.117164 | − | 2.64316i | −4.34226 | 1.09819 | − | 1.90212i | 1.07971 | + | 1.87012i | |||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 287.2.e.c | ✓ | 10 |
| 7.c | even | 3 | 1 | inner | 287.2.e.c | ✓ | 10 |
| 7.c | even | 3 | 1 | 2009.2.a.l | 5 | ||
| 7.d | odd | 6 | 1 | 2009.2.a.m | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 287.2.e.c | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 287.2.e.c | ✓ | 10 | 7.c | even | 3 | 1 | inner |
| 2009.2.a.l | 5 | 7.c | even | 3 | 1 | ||
| 2009.2.a.m | 5 | 7.d | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} - 2 T_{2}^{9} + 10 T_{2}^{8} - 10 T_{2}^{7} + 50 T_{2}^{6} - 47 T_{2}^{5} + 143 T_{2}^{4} + \cdots + 169 \)
acting on \(S_{2}^{\mathrm{new}}(287, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} - 2 T^{9} + \cdots + 169 \)
|
| $3$ |
\( T^{10} - 2 T^{9} + \cdots + 49 \)
|
| $5$ |
\( T^{10} + 2 T^{9} + \cdots + 49 \)
|
| $7$ |
\( T^{10} - 8 T^{9} + \cdots + 16807 \)
|
| $11$ |
\( T^{10} - 6 T^{9} + \cdots + 2809 \)
|
| $13$ |
\( (T^{5} - 2 T^{4} - 8 T^{3} + \cdots - 7)^{2} \)
|
| $17$ |
\( T^{10} - 3 T^{9} + \cdots + 25921 \)
|
| $19$ |
\( T^{10} + 7 T^{9} + \cdots + 405769 \)
|
| $23$ |
\( T^{10} + 38 T^{8} + \cdots + 36481 \)
|
| $29$ |
\( (T^{5} + 10 T^{4} + \cdots + 49)^{2} \)
|
| $31$ |
\( T^{10} - 6 T^{9} + \cdots + 90601 \)
|
| $37$ |
\( T^{10} - 18 T^{9} + \cdots + 61009 \)
|
| $41$ |
\( (T + 1)^{10} \)
|
| $43$ |
\( (T^{5} + 14 T^{4} + \cdots - 2573)^{2} \)
|
| $47$ |
\( T^{10} + 3 T^{9} + \cdots + 1432809 \)
|
| $53$ |
\( T^{10} - 9 T^{9} + \cdots + 2647129 \)
|
| $59$ |
\( T^{10} + \cdots + 324396121 \)
|
| $61$ |
\( T^{10} + \cdots + 159997201 \)
|
| $67$ |
\( T^{10} + \cdots + 372374209 \)
|
| $71$ |
\( (T^{5} - 150 T^{3} + \cdots - 7241)^{2} \)
|
| $73$ |
\( T^{10} + 13 T^{9} + \cdots + 9186961 \)
|
| $79$ |
\( T^{10} + \cdots + 3108174001 \)
|
| $83$ |
\( (T^{5} + 2 T^{4} + \cdots + 2317)^{2} \)
|
| $89$ |
\( T^{10} + \cdots + 2081093161 \)
|
| $97$ |
\( (T^{5} + 27 T^{4} + \cdots - 56763)^{2} \)
|
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