Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8048,2,Mod(1,8048)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8048, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("8048.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8048 = 2^{4} \cdot 503 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8048.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: | \(64.2636035467\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{12} - 5 x^{11} - 13 x^{10} + 94 x^{9} + 15 x^{8} - 616 x^{7} + 368 x^{6} + 1643 x^{5} - 1463 x^{4} + \cdots - 208 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 1006) |
| 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
\( x^{12} - 5 x^{11} - 13 x^{10} + 94 x^{9} + 15 x^{8} - 616 x^{7} + 368 x^{6} + 1643 x^{5} - 1463 x^{4} + \cdots - 208 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{11} + 543 \nu^{10} - 917 \nu^{9} - 9770 \nu^{8} + 14883 \nu^{7} + 57076 \nu^{6} - 82700 \nu^{5} + \cdots + 1864 ) / 7192 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 115 \nu^{11} - 485 \nu^{10} - 2969 \nu^{9} + 10988 \nu^{8} + 28617 \nu^{7} - 87062 \nu^{6} + \cdots + 56136 ) / 7192 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 577 \nu^{11} - 1339 \nu^{10} - 11285 \nu^{9} + 24612 \nu^{8} + 77557 \nu^{7} - 157540 \nu^{6} + \cdots + 65052 ) / 3596 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 1269 \nu^{11} + 3163 \nu^{10} + 25539 \nu^{9} - 60212 \nu^{8} - 183731 \nu^{7} + 402142 \nu^{6} + \cdots - 193432 ) / 7192 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 1533 \nu^{11} + 3651 \nu^{10} + 30291 \nu^{9} - 66456 \nu^{8} - 214779 \nu^{7} + 419298 \nu^{6} + \cdots - 131744 ) / 7192 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 969 \nu^{11} - 2445 \nu^{10} - 18341 \nu^{9} + 44290 \nu^{8} + 120335 \nu^{7} - 276728 \nu^{6} + \cdots + 72944 ) / 3596 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 1163 \nu^{11} - 3185 \nu^{10} - 21833 \nu^{9} + 58386 \nu^{8} + 141625 \nu^{7} - 369564 \nu^{6} + \cdots + 85748 ) / 3596 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 2572 \nu^{11} - 6743 \nu^{10} - 48993 \nu^{9} + 123571 \nu^{8} + 325094 \nu^{7} - 783065 \nu^{6} + \cdots + 209308 ) / 3596 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 646 \nu^{11} + 1630 \nu^{10} + 12527 \nu^{9} - 30126 \nu^{8} - 85018 \nu^{7} + 192876 \nu^{6} + \cdots - 53424 ) / 899 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + \beta_{5} + \beta_{4} + 6\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{10} + 2\beta_{8} + \beta_{5} + \beta_{4} - \beta_{3} + 8\beta_{2} + 24 \)
|
| \(\nu^{5}\) | \(=\) |
\( -3\beta_{9} + 6\beta_{8} + \beta_{7} + 12\beta_{6} + 9\beta_{5} + 15\beta_{4} + 38\beta _1 + 11 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 2 \beta_{11} - 18 \beta_{10} + 39 \beta_{8} + 6 \beta_{6} + 10 \beta_{5} + 24 \beta_{4} - 12 \beta_{3} + \cdots + 162 \)
|
| \(\nu^{7}\) | \(=\) |
\( - \beta_{11} - 9 \beta_{10} - 42 \beta_{9} + 111 \beta_{8} + 14 \beta_{7} + 124 \beta_{6} + 71 \beta_{5} + \cdots + 116 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 32 \beta_{11} - 220 \beta_{10} - 7 \beta_{9} + 517 \beta_{8} + 7 \beta_{7} + 116 \beta_{6} + \cdots + 1195 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 21 \beta_{11} - 197 \beta_{10} - 440 \beta_{9} + 1468 \beta_{8} + 156 \beta_{7} + 1222 \beta_{6} + \cdots + 1222 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 373 \beta_{11} - 2361 \beta_{10} - 167 \beta_{9} + 5950 \beta_{8} + 155 \beta_{7} + 1609 \beta_{6} + \cdots + 9512 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 323 \beta_{11} - 2948 \beta_{10} - 4203 \beta_{9} + 17093 \beta_{8} + 1615 \beta_{7} + 11846 \beta_{6} + \cdots + 12839 \)
|
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 | −3.16835 | 0 | 0.310718 | 0 | −0.631809 | 0 | 7.03843 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.49390 | 0 | 1.94778 | 0 | 1.87622 | 0 | 3.21954 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −2.44300 | 0 | −1.37594 | 0 | −4.82526 | 0 | 2.96827 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −2.31191 | 0 | 0.313584 | 0 | 2.05708 | 0 | 2.34493 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −1.74625 | 0 | −2.58578 | 0 | −0.635213 | 0 | 0.0494057 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | −1.29494 | 0 | 3.51641 | 0 | −1.29880 | 0 | −1.32313 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | −0.264006 | 0 | 1.70058 | 0 | −2.77338 | 0 | −2.93030 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 0.665271 | 0 | 2.52521 | 0 | 3.86663 | 0 | −2.55741 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 0.867509 | 0 | −3.26212 | 0 | −1.33041 | 0 | −2.24743 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 2.17256 | 0 | 2.95208 | 0 | −2.27467 | 0 | 1.72004 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0 | 2.25121 | 0 | 1.33593 | 0 | −4.39363 | 0 | 2.06796 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 0 | 2.76581 | 0 | −2.37847 | 0 | 2.36325 | 0 | 4.64970 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(503\) | \(-1\) |
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 | 8048.2.a.q | 12 | |
| 4.b | odd | 2 | 1 | 1006.2.a.j | ✓ | 12 | |
| 12.b | even | 2 | 1 | 9054.2.a.bi | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1006.2.a.j | ✓ | 12 | 4.b | odd | 2 | 1 | |
| 8048.2.a.q | 12 | 1.a | even | 1 | 1 | trivial | |
| 9054.2.a.bi | 12 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8048))\):
|
\( T_{3}^{12} + 5 T_{3}^{11} - 13 T_{3}^{10} - 94 T_{3}^{9} + 15 T_{3}^{8} + 616 T_{3}^{7} + 368 T_{3}^{6} + \cdots - 208 \)
|
|
\( T_{5}^{12} - 5 T_{5}^{11} - 18 T_{5}^{10} + 121 T_{5}^{9} + 41 T_{5}^{8} - 982 T_{5}^{7} + 746 T_{5}^{6} + \cdots + 312 \)
|
|
\( T_{7}^{12} + 8 T_{7}^{11} - 12 T_{7}^{10} - 217 T_{7}^{9} - 195 T_{7}^{8} + 1780 T_{7}^{7} + 3070 T_{7}^{6} + \cdots + 3271 \)
|
|
\( T_{13}^{12} - 4 T_{13}^{11} - 61 T_{13}^{10} + 294 T_{13}^{9} + 859 T_{13}^{8} - 5825 T_{13}^{7} + \cdots + 784 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} \)
|
| $3$ |
\( T^{12} + 5 T^{11} + \cdots - 208 \)
|
| $5$ |
\( T^{12} - 5 T^{11} + \cdots + 312 \)
|
| $7$ |
\( T^{12} + 8 T^{11} + \cdots + 3271 \)
|
| $11$ |
\( T^{12} + 18 T^{11} + \cdots - 2736 \)
|
| $13$ |
\( T^{12} - 4 T^{11} + \cdots + 784 \)
|
| $17$ |
\( T^{12} + 2 T^{11} + \cdots + 4992 \)
|
| $19$ |
\( T^{12} + 6 T^{11} + \cdots - 424 \)
|
| $23$ |
\( T^{12} + 13 T^{11} + \cdots + 2974977 \)
|
| $29$ |
\( T^{12} - 20 T^{11} + \cdots - 1632384 \)
|
| $31$ |
\( T^{12} + 7 T^{11} + \cdots + 211328 \)
|
| $37$ |
\( T^{12} + \cdots + 388308680 \)
|
| $41$ |
\( T^{12} - 2 T^{11} + \cdots + 11636352 \)
|
| $43$ |
\( T^{12} + 8 T^{11} + \cdots + 22568624 \)
|
| $47$ |
\( T^{12} + \cdots - 421292919 \)
|
| $53$ |
\( T^{12} - 12 T^{11} + \cdots + 1253640 \)
|
| $59$ |
\( T^{12} + 6 T^{11} + \cdots + 96691200 \)
|
| $61$ |
\( T^{12} + 10 T^{11} + \cdots - 19632496 \)
|
| $67$ |
\( T^{12} + \cdots + 112948624 \)
|
| $71$ |
\( T^{12} + 22 T^{11} + \cdots - 1152 \)
|
| $73$ |
\( T^{12} + 23 T^{11} + \cdots - 248 \)
|
| $79$ |
\( T^{12} + 13 T^{11} + \cdots - 11347520 \)
|
| $83$ |
\( T^{12} + \cdots - 184141104 \)
|
| $89$ |
\( T^{12} + \cdots + 1840063488 \)
|
| $97$ |
\( T^{12} + \cdots + 2828620760 \)
|
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