Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3887,2,Mod(1,3887)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3887, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("3887.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3887 = 13^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3887.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: | \(31.0378512657\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - x^{9} - 19x^{8} + 18x^{7} + 127x^{6} - 109x^{5} - 357x^{4} + 252x^{3} + 400x^{2} - 192x - 128 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 299) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - x^{9} - 19x^{8} + 18x^{7} + 127x^{6} - 109x^{5} - 357x^{4} + 252x^{3} + 400x^{2} - 192x - 128 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 3\nu^{9} + 3\nu^{8} - 47\nu^{7} - 44\nu^{6} + 233\nu^{5} + 195\nu^{4} - 397\nu^{3} - 282\nu^{2} + 184\nu + 112 ) / 16 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{9} - \nu^{8} + 21\nu^{7} + 20\nu^{6} - 155\nu^{5} - 137\nu^{4} + 463\nu^{3} + 358\nu^{2} - 448\nu - 240 ) / 16 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{9} - \nu^{8} + 17\nu^{7} + 16\nu^{6} - 99\nu^{5} - 85\nu^{4} + 235\nu^{3} + 174\nu^{2} - 200\nu - 104 ) / 8 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{9} - \nu^{8} - 19\nu^{7} + 14\nu^{6} + 123\nu^{5} - 53\nu^{4} - 305\nu^{3} + 24\nu^{2} + 224\nu + 48 ) / 8 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 7\nu^{9} + 5\nu^{8} - 113\nu^{7} - 70\nu^{6} + 593\nu^{5} + 281\nu^{4} - 1151\nu^{3} - 312\nu^{2} + 736\nu + 64 ) / 32 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( -5\nu^{9} - 3\nu^{8} + 87\nu^{7} + 46\nu^{6} - 507\nu^{5} - 247\nu^{4} + 1137\nu^{3} + 588\nu^{2} - 832\nu - 448 ) / 32 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 3\nu^{9} + 5\nu^{8} - 49\nu^{7} - 74\nu^{6} + 261\nu^{5} + 337\nu^{4} - 511\nu^{3} - 508\nu^{2} + 304\nu + 192 ) / 16 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{8} - \beta_{7} - \beta_{5} + \beta_{4} + 8\beta_{2} + 22 \)
|
| \(\nu^{5}\) | \(=\) |
\( 10 \beta_{9} + 11 \beta_{8} - 9 \beta_{7} + 10 \beta_{6} - 13 \beta_{5} + 11 \beta_{4} - 2 \beta_{3} + \cdots - 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( 3\beta_{9} - 12\beta_{8} - 10\beta_{7} + \beta_{6} - 14\beta_{5} + 16\beta_{4} - 6\beta_{3} + 56\beta_{2} + 134 \)
|
| \(\nu^{7}\) | \(=\) |
\( 80 \beta_{9} + 96 \beta_{8} - 72 \beta_{7} + 82 \beta_{6} - 126 \beta_{5} + 98 \beta_{4} - 22 \beta_{3} + \cdots - 26 \)
|
| \(\nu^{8}\) | \(=\) |
\( 50 \beta_{9} - 110 \beta_{8} - 82 \beta_{7} + 14 \beta_{6} - 140 \beta_{5} + 170 \beta_{4} - 92 \beta_{3} + \cdots + 862 \)
|
| \(\nu^{9}\) | \(=\) |
\( 603 \beta_{9} + 781 \beta_{8} - 561 \beta_{7} + 641 \beta_{6} - 1097 \beta_{5} + 813 \beta_{4} + \cdots - 240 \)
|
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 |
|
−2.76732 | 0.315862 | 5.65804 | 0.739140 | −0.874091 | 4.71764 | −10.1230 | −2.90023 | −2.04543 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.26436 | 2.37123 | 3.12732 | 3.16568 | −5.36932 | −1.09348 | −2.55266 | 2.62273 | −7.16824 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.20349 | −3.23947 | 2.85538 | −3.16617 | 7.13815 | 0.235798 | −1.88483 | 7.49418 | 6.97663 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.33169 | 1.05176 | −0.226590 | −2.04986 | −1.40062 | −2.29592 | 2.96514 | −1.89380 | 2.72979 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.07778 | 3.08342 | −0.838395 | −2.74187 | −3.32324 | 3.42528 | 3.05916 | 6.50750 | 2.95512 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.434464 | −0.910673 | −1.81124 | 3.11372 | −0.395654 | −3.44119 | −1.65585 | −2.17067 | 1.35280 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.29172 | −2.20386 | −0.331447 | −2.25480 | −2.84678 | 4.62032 | −3.01159 | 1.85700 | −2.91259 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.68420 | 1.25765 | 0.836514 | −4.35359 | 2.11813 | −4.21798 | −1.95954 | −1.41831 | −7.33230 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.49125 | −1.82162 | 4.20631 | 4.09233 | −4.53810 | 3.06673 | 5.49645 | 0.318291 | 10.1950 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.74301 | 3.09569 | 5.52411 | 0.455417 | 8.49152 | −3.01719 | 9.66667 | 6.58332 | 1.24922 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(13\) | \(1\) |
| \(23\) | \(-1\) |
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 | 3887.2.a.p | 10 | |
| 13.b | even | 2 | 1 | 299.2.a.g | ✓ | 10 | |
| 39.d | odd | 2 | 1 | 2691.2.a.bc | 10 | ||
| 52.b | odd | 2 | 1 | 4784.2.a.bh | 10 | ||
| 65.d | even | 2 | 1 | 7475.2.a.w | 10 | ||
| 299.c | odd | 2 | 1 | 6877.2.a.o | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 299.2.a.g | ✓ | 10 | 13.b | even | 2 | 1 | |
| 2691.2.a.bc | 10 | 39.d | odd | 2 | 1 | ||
| 3887.2.a.p | 10 | 1.a | even | 1 | 1 | trivial | |
| 4784.2.a.bh | 10 | 52.b | odd | 2 | 1 | ||
| 6877.2.a.o | 10 | 299.c | odd | 2 | 1 | ||
| 7475.2.a.w | 10 | 65.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3887))\):
|
\( T_{2}^{10} + T_{2}^{9} - 19 T_{2}^{8} - 18 T_{2}^{7} + 127 T_{2}^{6} + 109 T_{2}^{5} - 357 T_{2}^{4} + \cdots - 128 \)
|
|
\( T_{3}^{10} - 3 T_{3}^{9} - 19 T_{3}^{8} + 58 T_{3}^{7} + 107 T_{3}^{6} - 343 T_{3}^{5} - 181 T_{3}^{4} + \cdots + 112 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + T^{9} + \cdots - 128 \)
|
| $3$ |
\( T^{10} - 3 T^{9} + \cdots + 112 \)
|
| $5$ |
\( T^{10} + 3 T^{9} + \cdots - 2372 \)
|
| $7$ |
\( T^{10} - 2 T^{9} + \cdots - 5936 \)
|
| $11$ |
\( T^{10} + 3 T^{9} + \cdots - 190148 \)
|
| $13$ |
\( T^{10} \)
|
| $17$ |
\( T^{10} + 3 T^{9} + \cdots - 13568 \)
|
| $19$ |
\( T^{10} + 2 T^{9} + \cdots + 1230932 \)
|
| $23$ |
\( (T - 1)^{10} \)
|
| $29$ |
\( T^{10} - 17 T^{9} + \cdots - 243712 \)
|
| $31$ |
\( T^{10} + 5 T^{9} + \cdots + 114688 \)
|
| $37$ |
\( T^{10} + 16 T^{9} + \cdots + 1769792 \)
|
| $41$ |
\( T^{10} - 16 T^{9} + \cdots + 5227264 \)
|
| $43$ |
\( T^{10} + 9 T^{9} + \cdots - 14500864 \)
|
| $47$ |
\( T^{10} - 11 T^{9} + \cdots + 12838912 \)
|
| $53$ |
\( T^{10} - 8 T^{9} + \cdots + 283136 \)
|
| $59$ |
\( T^{10} + 2 T^{9} + \cdots + 1183744 \)
|
| $61$ |
\( T^{10} - 48 T^{9} + \cdots - 17088512 \)
|
| $67$ |
\( T^{10} - 6 T^{9} + \cdots + 437516 \)
|
| $71$ |
\( T^{10} + 24 T^{9} + \cdots - 598016 \)
|
| $73$ |
\( T^{10} + \cdots + 200656384 \)
|
| $79$ |
\( T^{10} + \cdots - 431722496 \)
|
| $83$ |
\( T^{10} - 21 T^{9} + \cdots - 2333212 \)
|
| $89$ |
\( T^{10} - 16 T^{9} + \cdots + 35697088 \)
|
| $97$ |
\( T^{10} - 40 T^{9} + \cdots - 1231244 \)
|
| show more | |
| show less |