Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6256,2,Mod(1,6256)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6256, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6256.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6256 = 2^{4} \cdot 17 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6256.a (trivial) |
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: | yes |
| Analytic conductor: | \(49.9544115045\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 28 x^{10} - 2 x^{9} + 283 x^{8} + 30 x^{7} - 1236 x^{6} - 164 x^{5} + 2100 x^{4} + 400 x^{3} + \cdots + 64 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 3128) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 28 x^{10} - 2 x^{9} + 283 x^{8} + 30 x^{7} - 1236 x^{6} - 164 x^{5} + 2100 x^{4} + 400 x^{3} + \cdots + 64 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + \nu^{2} - 8\nu - 6 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - \nu^{3} - 10\nu^{2} + 6\nu + 10 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{9} - \nu^{8} - 21\nu^{7} + 15\nu^{6} + 140\nu^{5} - 62\nu^{4} - 300\nu^{3} + 44\nu^{2} + 72\nu - 16 ) / 8 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{9} + \nu^{8} + 21\nu^{7} - 15\nu^{6} - 140\nu^{5} + 62\nu^{4} + 304\nu^{3} - 48\nu^{2} - 104\nu + 32 ) / 8 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{11} - 3 \nu^{10} - 19 \nu^{9} + 55 \nu^{8} + 114 \nu^{7} - 308 \nu^{6} - 228 \nu^{5} + 468 \nu^{4} + \cdots - 32 ) / 16 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -\nu^{9} - \nu^{8} + 23\nu^{7} + 27\nu^{6} - 166\nu^{5} - 222\nu^{4} + 372\nu^{3} + 584\nu^{2} - 16\nu - 128 ) / 8 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( \nu^{10} - 2 \nu^{9} - 20 \nu^{8} + 36 \nu^{7} + 125 \nu^{6} - 202 \nu^{5} - 242 \nu^{4} + 352 \nu^{3} + \cdots - 8 ) / 8 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - \nu^{11} - \nu^{10} + 27 \nu^{9} + 25 \nu^{8} - 254 \nu^{7} - 200 \nu^{6} + 976 \nu^{5} + 588 \nu^{4} + \cdots + 64 ) / 16 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( \nu^{11} - 3 \nu^{10} - 21 \nu^{9} + 61 \nu^{8} + 152 \nu^{7} - 418 \nu^{6} - 456 \nu^{5} + 1092 \nu^{4} + \cdots + 144 ) / 16 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( \nu^{11} + \nu^{10} - 29 \nu^{9} - 23 \nu^{8} + 296 \nu^{7} + 174 \nu^{6} - 1264 \nu^{5} - 516 \nu^{4} + \cdots - 80 ) / 16 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{5} - \beta_{4} + \beta_{2} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{4} + \beta_{2} + 8\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -9\beta_{5} - 9\beta_{4} + 2\beta_{3} + 11\beta_{2} + 2\beta _1 + 41 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 2 \beta_{11} + 2 \beta_{10} - 2 \beta_{9} + 2 \beta_{7} - 2 \beta_{6} + 9 \beta_{5} + 11 \beta_{4} + \cdots + 19 \)
|
| \(\nu^{6}\) | \(=\) |
\( 4\beta_{10} + 4\beta_{7} - 4\beta_{6} - 81\beta_{5} - 73\beta_{4} + 34\beta_{3} + 115\beta_{2} + 42\beta _1 + 373 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 30 \beta_{11} + 38 \beta_{10} - 26 \beta_{9} + 8 \beta_{8} + 38 \beta_{7} - 34 \beta_{6} + 57 \beta_{5} + \cdots + 303 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 4 \beta_{11} + 96 \beta_{10} + 8 \beta_{8} + 92 \beta_{7} - 92 \beta_{6} - 761 \beta_{5} + \cdots + 3601 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 354 \beta_{11} + 554 \beta_{10} - 266 \beta_{9} + 176 \beta_{8} + 550 \beta_{7} - 466 \beta_{6} + \cdots + 4347 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 112 \beta_{11} + 1564 \beta_{10} + 232 \beta_{8} + 1476 \beta_{7} - 1452 \beta_{6} - 7441 \beta_{5} + \cdots + 36277 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 3878 \beta_{11} + 7294 \beta_{10} - 2546 \beta_{9} + 2688 \beta_{8} + 7174 \beta_{7} - 5946 \beta_{6} + \cdots + 58151 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | −2.90874 | 0 | 1.03110 | 0 | −4.47108 | 0 | 5.46078 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.89242 | 0 | −0.474879 | 0 | −0.880105 | 0 | 5.36609 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −2.72115 | 0 | 3.15620 | 0 | 2.62557 | 0 | 4.40465 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.62290 | 0 | −3.70946 | 0 | −1.86189 | 0 | −0.366201 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −0.683361 | 0 | −2.45073 | 0 | 2.32778 | 0 | −2.53302 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | −0.381544 | 0 | 2.84729 | 0 | −4.83307 | 0 | −2.85442 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 0.323636 | 0 | −0.195911 | 0 | 4.73998 | 0 | −2.89526 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 0.483549 | 0 | 0.555438 | 0 | −2.90872 | 0 | −2.76618 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 2.08729 | 0 | 3.11679 | 0 | −0.295966 | 0 | 1.35679 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 2.15483 | 0 | −2.19934 | 0 | −4.92803 | 0 | 1.64328 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0 | 2.75968 | 0 | 3.53649 | 0 | 2.84954 | 0 | 4.61586 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 0 | 3.40112 | 0 | −1.21300 | 0 | 1.63599 | 0 | 8.56763 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(17\) | \( -1 \) |
| \(23\) | \( +1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 6256.2.a.bj | 12 | |
| 4.b | odd | 2 | 1 | 3128.2.a.j | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3128.2.a.j | ✓ | 12 | 4.b | odd | 2 | 1 | |
| 6256.2.a.bj | 12 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6256))\):
|
\( T_{3}^{12} - 28 T_{3}^{10} - 2 T_{3}^{9} + 283 T_{3}^{8} + 30 T_{3}^{7} - 1236 T_{3}^{6} - 164 T_{3}^{5} + \cdots + 64 \)
|
|
\( T_{5}^{12} - 4 T_{5}^{11} - 26 T_{5}^{10} + 108 T_{5}^{9} + 219 T_{5}^{8} - 940 T_{5}^{7} - 783 T_{5}^{6} + \cdots + 128 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} \)
|
| $3$ |
\( T^{12} - 28 T^{10} + \cdots + 64 \)
|
| $5$ |
\( T^{12} - 4 T^{11} + \cdots + 128 \)
|
| $7$ |
\( T^{12} + 6 T^{11} + \cdots - 20288 \)
|
| $11$ |
\( T^{12} - 3 T^{11} + \cdots + 384 \)
|
| $13$ |
\( T^{12} - 17 T^{11} + \cdots + 1415720 \)
|
| $17$ |
\( (T - 1)^{12} \)
|
| $19$ |
\( T^{12} - T^{11} + \cdots + 3072 \)
|
| $23$ |
\( (T + 1)^{12} \)
|
| $29$ |
\( T^{12} - 21 T^{11} + \cdots + 32791232 \)
|
| $31$ |
\( T^{12} + \cdots - 386911232 \)
|
| $37$ |
\( T^{12} + \cdots - 400543872 \)
|
| $41$ |
\( T^{12} + \cdots + 117913728 \)
|
| $43$ |
\( T^{12} + \cdots + 736157696 \)
|
| $47$ |
\( T^{12} + 7 T^{11} + \cdots + 51337920 \)
|
| $53$ |
\( T^{12} + \cdots + 132157184 \)
|
| $59$ |
\( T^{12} + \cdots + 75869531392 \)
|
| $61$ |
\( T^{12} - 17 T^{11} + \cdots - 8184080 \)
|
| $67$ |
\( T^{12} + \cdots + 1305820160 \)
|
| $71$ |
\( T^{12} + \cdots + 140288000 \)
|
| $73$ |
\( T^{12} - 6 T^{11} + \cdots - 22427200 \)
|
| $79$ |
\( T^{12} + 21 T^{11} + \cdots + 8351104 \)
|
| $83$ |
\( T^{12} + \cdots - 808585216 \)
|
| $89$ |
\( T^{12} + \cdots + 9320420864 \)
|
| $97$ |
\( T^{12} + \cdots + 55678196960 \)
|
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