Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3971,2,Mod(1,3971)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3971, 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("3971.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3971 = 11 \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3971.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: | \(31.7085946427\) |
| Analytic rank: | \(0\) |
| Dimension: | \(9\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{9} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{9} - x^{8} - 13x^{7} + 12x^{6} + 51x^{5} - 38x^{4} - 70x^{3} + 30x^{2} + 27x + 3 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 209) |
| 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 \(1,\beta_1,\ldots,\beta_{8}\) 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
\( x^{9} - x^{8} - 13x^{7} + 12x^{6} + 51x^{5} - 38x^{4} - 70x^{3} + 30x^{2} + 27x + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 10\nu^{8} - 24\nu^{7} - 111\nu^{6} + 290\nu^{5} + 323\nu^{4} - 949\nu^{3} - 335\nu^{2} + 842\nu + 230 ) / 73 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 3\nu^{8} + 22\nu^{7} - 26\nu^{6} - 278\nu^{5} + 2\nu^{4} + 1022\nu^{3} + 301\nu^{2} - 1076\nu - 369 ) / 73 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 14\nu^{8} - 19\nu^{7} - 170\nu^{6} + 260\nu^{5} + 569\nu^{4} - 1022\nu^{3} - 542\nu^{2} + 1135\nu + 176 ) / 73 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -22\nu^{8} + 9\nu^{7} + 288\nu^{6} - 127\nu^{5} - 1134\nu^{4} + 511\nu^{3} + 1467\nu^{2} - 626\nu - 360 ) / 73 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -28\nu^{8} + 38\nu^{7} + 340\nu^{6} - 447\nu^{5} - 1138\nu^{4} + 1387\nu^{3} + 1084\nu^{2} - 1102\nu - 206 ) / 73 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 33\nu^{8} - 50\nu^{7} - 432\nu^{6} + 592\nu^{5} + 1701\nu^{4} - 1825\nu^{3} - 2237\nu^{2} + 1304\nu + 540 ) / 73 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{2} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{7} - \beta_{6} - \beta_{5} + 2\beta_{3} + 7\beta_{2} - \beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( 9\beta_{8} + 10\beta_{7} + 9\beta_{6} + 11\beta_{5} + 9\beta_{4} - 9\beta_{2} + 29\beta _1 - 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( -\beta_{8} + 10\beta_{7} - 10\beta_{6} - 10\beta_{5} + \beta_{4} + 23\beta_{3} + 49\beta_{2} - 12\beta _1 + 90 \)
|
| \(\nu^{7}\) | \(=\) |
\( 68\beta_{8} + 81\beta_{7} + 67\beta_{6} + 92\beta_{5} + 70\beta_{4} - 71\beta_{2} + 184\beta _1 - 25 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 14 \beta_{8} + 78 \beta_{7} - 84 \beta_{6} - 82 \beta_{5} + 13 \beta_{4} + 198 \beta_{3} + 347 \beta_{2} + \cdots + 590 \)
|
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 |
|
−2.69565 | 0.0937671 | 5.26651 | −2.24462 | −0.252763 | 0.215810 | −8.80535 | −2.99121 | 6.05070 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.83739 | −3.12711 | 1.37600 | −3.26983 | 5.74572 | −3.63195 | 1.14653 | 6.77879 | 6.00795 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.23186 | 2.02769 | −0.482521 | 2.63096 | −2.49783 | 3.75846 | 3.05812 | 1.11154 | −3.24098 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.396426 | −1.26801 | −1.84285 | 0.441921 | 0.502672 | −0.667657 | 1.52341 | −1.39215 | −0.175189 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.138860 | 1.49679 | −1.98072 | −3.51650 | −0.207844 | −1.46902 | 0.552764 | −0.759634 | 0.488302 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.05438 | 3.26124 | −0.888282 | 1.30020 | 3.43859 | −1.25919 | −3.04535 | 7.63569 | 1.37091 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.59566 | −1.16701 | 0.546144 | −0.413278 | −1.86216 | 2.20560 | −2.31987 | −1.63808 | −0.659453 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.01377 | −2.58057 | 2.05527 | 3.55226 | −5.19667 | −4.09817 | 0.111292 | 3.65935 | 7.15343 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.63637 | 1.26321 | 4.95045 | 1.51888 | 3.33030 | 1.94612 | 7.77846 | −1.40429 | 4.00432 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(11\) | \( +1 \) |
| \(19\) | \( -1 \) |
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 | 3971.2.a.l | 9 | |
| 19.b | odd | 2 | 1 | 3971.2.a.k | 9 | ||
| 19.d | odd | 6 | 2 | 209.2.e.b | ✓ | 18 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 209.2.e.b | ✓ | 18 | 19.d | odd | 6 | 2 | |
| 3971.2.a.k | 9 | 19.b | odd | 2 | 1 | ||
| 3971.2.a.l | 9 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3971))\):
|
\( T_{2}^{9} - T_{2}^{8} - 13T_{2}^{7} + 12T_{2}^{6} + 51T_{2}^{5} - 38T_{2}^{4} - 70T_{2}^{3} + 30T_{2}^{2} + 27T_{2} + 3 \)
|
|
\( T_{3}^{9} - 19T_{3}^{7} + T_{3}^{6} + 110T_{3}^{5} - 18T_{3}^{4} - 224T_{3}^{3} + 35T_{3}^{2} + 148T_{3} - 14 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} - T^{8} - 13 T^{7} + \cdots + 3 \)
|
| $3$ |
\( T^{9} - 19 T^{7} + \cdots - 14 \)
|
| $5$ |
\( T^{9} - 26 T^{7} + \cdots - 87 \)
|
| $7$ |
\( T^{9} + 3 T^{8} + \cdots + 64 \)
|
| $11$ |
\( (T + 1)^{9} \)
|
| $13$ |
\( T^{9} - 11 T^{8} + \cdots + 4798 \)
|
| $17$ |
\( T^{9} + 6 T^{8} + \cdots - 3684 \)
|
| $19$ |
\( T^{9} \)
|
| $23$ |
\( T^{9} - T^{8} + \cdots + 270 \)
|
| $29$ |
\( T^{9} - 13 T^{8} + \cdots + 680694 \)
|
| $31$ |
\( T^{9} - 16 T^{8} + \cdots + 331264 \)
|
| $37$ |
\( T^{9} - 18 T^{8} + \cdots + 1086460 \)
|
| $41$ |
\( T^{9} - 4 T^{8} + \cdots - 11275008 \)
|
| $43$ |
\( T^{9} + T^{8} + \cdots + 6464 \)
|
| $47$ |
\( T^{9} - 14 T^{8} + \cdots - 318 \)
|
| $53$ |
\( T^{9} + 4 T^{8} + \cdots - 16365 \)
|
| $59$ |
\( T^{9} - 31 T^{8} + \cdots + 8751342 \)
|
| $61$ |
\( T^{9} + 3 T^{8} + \cdots + 4046560 \)
|
| $67$ |
\( T^{9} + 2 T^{8} + \cdots + 66714160 \)
|
| $71$ |
\( T^{9} + 12 T^{8} + \cdots - 3181524 \)
|
| $73$ |
\( T^{9} - 26 T^{8} + \cdots + 347318 \)
|
| $79$ |
\( T^{9} - 31 T^{8} + \cdots - 225935 \)
|
| $83$ |
\( T^{9} - 18 T^{8} + \cdots - 3773520 \)
|
| $89$ |
\( T^{9} + 11 T^{8} + \cdots + 54960 \)
|
| $97$ |
\( T^{9} - 16 T^{8} + \cdots + 391369283 \)
|
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