Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3381,2,Mod(1,3381)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3381, 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("3381.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3381 = 3 \cdot 7^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3381.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: | \(26.9974209234\) |
| 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: |
\( x^{10} - 4x^{9} - 6x^{8} + 36x^{7} + x^{6} - 100x^{5} + 26x^{4} + 108x^{3} - 33x^{2} - 36x + 14 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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
\( x^{10} - 4x^{9} - 6x^{8} + 36x^{7} + x^{6} - 100x^{5} + 26x^{4} + 108x^{3} - 33x^{2} - 36x + 14 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{7} - 3\nu^{6} - 7\nu^{5} + 22\nu^{4} + 12\nu^{3} - 37\nu^{2} - 9\nu + 12 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{9} - 3\nu^{8} - 9\nu^{7} + 28\nu^{6} + 27\nu^{5} - 82\nu^{4} - 40\nu^{3} + 91\nu^{2} + 26\nu - 28 \)
|
| \(\beta_{5}\) | \(=\) |
\( -\nu^{9} + 3\nu^{8} + 10\nu^{7} - 31\nu^{6} - 33\nu^{5} + 103\nu^{4} + 45\nu^{3} - 123\nu^{2} - 26\nu + 37 \)
|
| \(\beta_{6}\) | \(=\) |
\( -2\nu^{9} + 7\nu^{8} + 15\nu^{7} - 63\nu^{6} - 30\nu^{5} + 173\nu^{4} + 30\nu^{3} - 176\nu^{2} - 25\nu + 46 \)
|
| \(\beta_{7}\) | \(=\) |
\( 3\nu^{9} - 11\nu^{8} - 21\nu^{7} + 98\nu^{6} + 34\nu^{5} - 266\nu^{4} - 26\nu^{3} + 271\nu^{2} + 30\nu - 70 \)
|
| \(\beta_{8}\) | \(=\) |
\( -4\nu^{9} + 14\nu^{8} + 31\nu^{7} - 128\nu^{6} - 69\nu^{5} + 362\nu^{4} + 83\nu^{3} - 385\nu^{2} - 67\nu + 108 \)
|
| \(\beta_{9}\) | \(=\) |
\( 5\nu^{9} - 17\nu^{8} - 41\nu^{7} + 159\nu^{6} + 103\nu^{5} - 466\nu^{4} - 136\nu^{3} + 514\nu^{2} + 106\nu - 149 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{9} - \beta_{8} + \beta_{4} - \beta_{3} + \beta_{2} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{9} - \beta_{8} - \beta_{7} - 2\beta_{6} + \beta_{5} + \beta_{4} - 2\beta_{3} + 6\beta_{2} + 7\beta _1 + 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( -8\beta_{9} - 8\beta_{8} - \beta_{7} - 2\beta_{6} + 2\beta_{5} + 9\beta_{4} - 10\beta_{3} + 8\beta_{2} + 28\beta _1 + 8 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 11 \beta_{9} - 10 \beta_{8} - 8 \beta_{7} - 18 \beta_{6} + 10 \beta_{5} + 13 \beta_{4} - 22 \beta_{3} + \cdots + 41 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 55 \beta_{9} - 52 \beta_{8} - 9 \beta_{7} - 24 \beta_{6} + 22 \beta_{5} + 68 \beta_{4} - 79 \beta_{3} + \cdots + 53 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 91 \beta_{9} - 75 \beta_{8} - 50 \beta_{7} - 131 \beta_{6} + 79 \beta_{5} + 122 \beta_{4} + \cdots + 230 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 366 \beta_{9} - 319 \beta_{8} - 62 \beta_{7} - 215 \beta_{6} + 183 \beta_{5} + 494 \beta_{4} + \cdots + 345 \)
|
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 |
|
−2.23864 | 1.00000 | 3.01150 | 2.46287 | −2.23864 | 0 | −2.26439 | 1.00000 | −5.51348 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.28424 | 1.00000 | −0.350734 | −1.91754 | −1.28424 | 0 | 3.01890 | 1.00000 | 2.46257 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.11439 | 1.00000 | −0.758127 | 1.17161 | −1.11439 | 0 | 3.07364 | 1.00000 | −1.30563 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.881528 | 1.00000 | −1.22291 | −2.92694 | −0.881528 | 0 | 2.84108 | 1.00000 | 2.58018 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.507824 | 1.00000 | −1.74212 | 3.36946 | 0.507824 | 0 | −1.90033 | 1.00000 | 1.71109 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.533756 | 1.00000 | −1.71510 | −1.09368 | 0.533756 | 0 | −1.98296 | 1.00000 | −0.583758 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.46930 | 1.00000 | 0.158829 | −2.14842 | 1.46930 | 0 | −2.70522 | 1.00000 | −3.15666 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.94871 | 1.00000 | 1.79746 | 1.40111 | 1.94871 | 0 | −0.394699 | 1.00000 | 2.73035 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.42151 | 1.00000 | 3.86372 | 2.93909 | 2.42151 | 0 | 4.51302 | 1.00000 | 7.11705 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.63770 | 1.00000 | 4.95748 | 0.742420 | 2.63770 | 0 | 7.80096 | 1.00000 | 1.95828 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(7\) | \(1\) |
| \(23\) | \(1\) |
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 | 3381.2.a.bl | yes | 10 |
| 7.b | odd | 2 | 1 | 3381.2.a.bk | ✓ | 10 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3381.2.a.bk | ✓ | 10 | 7.b | odd | 2 | 1 | |
| 3381.2.a.bl | yes | 10 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3381))\):
|
\( T_{2}^{10} - 4T_{2}^{9} - 6T_{2}^{8} + 36T_{2}^{7} + T_{2}^{6} - 100T_{2}^{5} + 26T_{2}^{4} + 108T_{2}^{3} - 33T_{2}^{2} - 36T_{2} + 14 \)
|
|
\( T_{5}^{10} - 4 T_{5}^{9} - 16 T_{5}^{8} + 72 T_{5}^{7} + 71 T_{5}^{6} - 428 T_{5}^{5} - 24 T_{5}^{4} + \cdots + 392 \)
|
|
\( T_{11}^{10} - 2 T_{11}^{9} - 57 T_{11}^{8} + 140 T_{11}^{7} + 1038 T_{11}^{6} - 2936 T_{11}^{5} + \cdots - 12953 \)
|
|
\( T_{13}^{10} - 52 T_{13}^{8} + 96 T_{13}^{7} + 747 T_{13}^{6} - 2584 T_{13}^{5} - 350 T_{13}^{4} + \cdots + 5598 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} - 4 T^{9} + \cdots + 14 \)
|
| $3$ |
\( (T - 1)^{10} \)
|
| $5$ |
\( T^{10} - 4 T^{9} + \cdots + 392 \)
|
| $7$ |
\( T^{10} \)
|
| $11$ |
\( T^{10} - 2 T^{9} + \cdots - 12953 \)
|
| $13$ |
\( T^{10} - 52 T^{8} + \cdots + 5598 \)
|
| $17$ |
\( T^{10} - 12 T^{9} + \cdots - 226 \)
|
| $19$ |
\( T^{10} - 26 T^{9} + \cdots - 24113 \)
|
| $23$ |
\( (T + 1)^{10} \)
|
| $29$ |
\( T^{10} - 16 T^{9} + \cdots - 486482 \)
|
| $31$ |
\( T^{10} - 12 T^{9} + \cdots + 8641252 \)
|
| $37$ |
\( T^{10} + 8 T^{9} + \cdots + 679266 \)
|
| $41$ |
\( T^{10} - 10 T^{9} + \cdots + 1815793 \)
|
| $43$ |
\( T^{10} + 4 T^{9} + \cdots - 14455202 \)
|
| $47$ |
\( T^{10} + \cdots - 112744688 \)
|
| $53$ |
\( T^{10} - 14 T^{9} + \cdots + 5454391 \)
|
| $59$ |
\( T^{10} - 38 T^{9} + \cdots + 42698601 \)
|
| $61$ |
\( T^{10} - 14 T^{9} + \cdots + 141967 \)
|
| $67$ |
\( T^{10} - 268 T^{8} + \cdots - 376888 \)
|
| $71$ |
\( T^{10} + \cdots - 472303438 \)
|
| $73$ |
\( T^{10} - 8 T^{9} + \cdots - 2744 \)
|
| $79$ |
\( T^{10} + 16 T^{9} + \cdots + 537938 \)
|
| $83$ |
\( T^{10} - 28 T^{9} + \cdots + 11428 \)
|
| $89$ |
\( T^{10} - 32 T^{9} + \cdots + 1212644 \)
|
| $97$ |
\( T^{10} + \cdots - 608495324 \)
|
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