Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8037,2,Mod(1,8037)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8037, 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("8037.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8037 = 3^{2} \cdot 19 \cdot 47 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8037.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: | \(64.1757681045\) |
| 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: |
\( x^{12} - x^{11} - 15 x^{10} + 14 x^{9} + 84 x^{8} - 76 x^{7} - 213 x^{6} + 196 x^{5} + 225 x^{4} + \cdots - 17 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 893) |
| 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
\( x^{12} - x^{11} - 15 x^{10} + 14 x^{9} + 84 x^{8} - 76 x^{7} - 213 x^{6} + 196 x^{5} + 225 x^{4} + \cdots - 17 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{7} - \nu^{6} - 9\nu^{5} + 7\nu^{4} + 23\nu^{3} - 14\nu^{2} - 14\nu + 7 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{11} + 12\nu^{9} + \nu^{8} - 48\nu^{7} - 5\nu^{6} + 73\nu^{5} + 8\nu^{4} - 33\nu^{3} - 11\nu^{2} + 2\nu + 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( 2\nu^{9} - 24\nu^{7} - \nu^{6} + 95\nu^{5} - 139\nu^{3} + 16\nu^{2} + 53\nu - 13 \)
|
| \(\beta_{6}\) | \(=\) |
\( - \nu^{11} - \nu^{10} + 13 \nu^{9} + 13 \nu^{8} - 59 \nu^{7} - 53 \nu^{6} + 116 \nu^{5} + 76 \nu^{4} + \cdots - 4 \)
|
| \(\beta_{7}\) | \(=\) |
\( - \nu^{11} + \nu^{10} + 13 \nu^{9} - 11 \nu^{8} - 61 \nu^{7} + 43 \nu^{6} + 125 \nu^{5} - 70 \nu^{4} + \cdots - 10 \)
|
| \(\beta_{8}\) | \(=\) |
\( \nu^{11} - 14\nu^{9} + 72\nu^{7} - 5\nu^{6} - 170\nu^{5} + 31\nu^{4} + 182\nu^{3} - 53\nu^{2} - 65\nu + 21 \)
|
| \(\beta_{9}\) | \(=\) |
\( \nu^{11} + \nu^{10} - 14 \nu^{9} - 12 \nu^{8} + 71 \nu^{7} + 43 \nu^{6} - 165 \nu^{5} - 42 \nu^{4} + \cdots + 18 \)
|
| \(\beta_{10}\) | \(=\) |
\( \nu^{11} + \nu^{10} - 14 \nu^{9} - 12 \nu^{8} + 71 \nu^{7} + 43 \nu^{6} - 165 \nu^{5} - 42 \nu^{4} + \cdots + 21 \)
|
| \(\beta_{11}\) | \(=\) |
\( \nu^{11} - 15\nu^{9} + 84\nu^{7} - 5\nu^{6} - 217\nu^{5} + 35\nu^{4} + 249\nu^{3} - 70\nu^{2} - 89\nu + 32 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{10} - \beta_{9} + \beta_{2} + 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{11} + \beta_{10} - \beta_{7} + \beta_{4} + 5\beta_{2} + \beta _1 + 12 \)
|
| \(\nu^{5}\) | \(=\) |
\( 7\beta_{10} - 5\beta_{9} - 2\beta_{8} - \beta_{7} + \beta_{6} + 8\beta_{2} + 17\beta _1 + 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( -10\beta_{11} + 10\beta_{10} - 9\beta_{7} + \beta_{6} - \beta_{5} + 8\beta_{4} + 25\beta_{2} + 10\beta _1 + 54 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 3 \beta_{11} + 43 \beta_{10} - 22 \beta_{9} - 18 \beta_{8} - 11 \beta_{7} + 10 \beta_{6} - \beta_{5} + \cdots + 32 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 71 \beta_{11} + 75 \beta_{10} - 3 \beta_{8} - 62 \beta_{7} + 13 \beta_{6} - 10 \beta_{5} + 50 \beta_{4} + \cdots + 264 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 41 \beta_{11} + 258 \beta_{10} - 96 \beta_{9} - 121 \beta_{8} - 89 \beta_{7} + 73 \beta_{6} + \cdots + 251 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 448 \beta_{11} + 508 \beta_{10} - 3 \beta_{9} - 45 \beta_{8} - 391 \beta_{7} + 113 \beta_{6} + \cdots + 1373 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 377 \beta_{11} + 1543 \beta_{10} - 428 \beta_{9} - 737 \beta_{8} - 638 \beta_{7} + 477 \beta_{6} + \cdots + 1756 \)
|
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 |
|
−2.13808 | 0 | 2.57138 | −1.02698 | 0 | −3.43362 | −1.22166 | 0 | 2.19577 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.93999 | 0 | 1.76355 | −1.45338 | 0 | −4.17438 | 0.458705 | 0 | 2.81953 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.75952 | 0 | 1.09589 | 1.58069 | 0 | 4.03503 | 1.59079 | 0 | −2.78124 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.67181 | 0 | 0.794958 | 2.50226 | 0 | −1.08348 | 2.01460 | 0 | −4.18331 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.792001 | 0 | −1.37273 | 1.55768 | 0 | −2.36074 | 2.67121 | 0 | −1.23369 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.332508 | 0 | −1.88944 | 1.32217 | 0 | 1.64871 | −1.29327 | 0 | 0.439633 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.531564 | 0 | −1.71744 | −0.555438 | 0 | 1.54136 | −1.97606 | 0 | −0.295251 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.821767 | 0 | −1.32470 | 0.409207 | 0 | −1.21348 | −2.73213 | 0 | 0.336273 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.38739 | 0 | −0.0751478 | 4.34713 | 0 | −2.61174 | −2.87904 | 0 | 6.03117 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.87636 | 0 | 1.52073 | −3.86690 | 0 | −2.13974 | −0.899279 | 0 | −7.25570 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 1.88968 | 0 | 1.57091 | 0.784971 | 0 | −0.254031 | −0.810846 | 0 | 1.48335 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 2.46212 | 0 | 4.06203 | 1.39858 | 0 | −2.95389 | 5.07697 | 0 | 3.44347 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(19\) | \(-1\) |
| \(47\) | \(1\) |
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 | 8037.2.a.m | 12 | |
| 3.b | odd | 2 | 1 | 893.2.a.a | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 893.2.a.a | ✓ | 12 | 3.b | odd | 2 | 1 | |
| 8037.2.a.m | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8037))\):
|
\( T_{2}^{12} - T_{2}^{11} - 15 T_{2}^{10} + 14 T_{2}^{9} + 84 T_{2}^{8} - 76 T_{2}^{7} - 213 T_{2}^{6} + \cdots - 17 \)
|
|
\( T_{5}^{12} - 7 T_{5}^{11} - 2 T_{5}^{10} + 112 T_{5}^{9} - 245 T_{5}^{8} - 83 T_{5}^{7} + 771 T_{5}^{6} + \cdots + 51 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} - T^{11} + \cdots - 17 \)
|
| $3$ |
\( T^{12} \)
|
| $5$ |
\( T^{12} - 7 T^{11} + \cdots + 51 \)
|
| $7$ |
\( T^{12} + 13 T^{11} + \cdots - 1913 \)
|
| $11$ |
\( T^{12} - 4 T^{11} + \cdots - 1257 \)
|
| $13$ |
\( T^{12} + 17 T^{11} + \cdots - 3363 \)
|
| $17$ |
\( T^{12} - 6 T^{11} + \cdots - 11163 \)
|
| $19$ |
\( (T - 1)^{12} \)
|
| $23$ |
\( T^{12} - 13 T^{11} + \cdots + 19193 \)
|
| $29$ |
\( T^{12} - 2 T^{11} + \cdots + 38861 \)
|
| $31$ |
\( T^{12} + 14 T^{11} + \cdots + 1905723 \)
|
| $37$ |
\( T^{12} + 2 T^{11} + \cdots - 19501671 \)
|
| $41$ |
\( T^{12} + 8 T^{11} + \cdots + 3821789 \)
|
| $43$ |
\( T^{12} + \cdots + 799148091 \)
|
| $47$ |
\( (T + 1)^{12} \)
|
| $53$ |
\( T^{12} + \cdots - 992611897 \)
|
| $59$ |
\( T^{12} + 8 T^{11} + \cdots - 25751271 \)
|
| $61$ |
\( T^{12} + 6 T^{11} + \cdots - 52438721 \)
|
| $67$ |
\( T^{12} + \cdots - 641357603 \)
|
| $71$ |
\( T^{12} - 7 T^{11} + \cdots + 27600303 \)
|
| $73$ |
\( T^{12} + \cdots - 29351787073 \)
|
| $79$ |
\( T^{12} + \cdots + 8484828139 \)
|
| $83$ |
\( T^{12} + \cdots - 30002859661 \)
|
| $89$ |
\( T^{12} + \cdots + 26837987071 \)
|
| $97$ |
\( T^{12} + \cdots - 518103129 \)
|
| show more | |
| show less |