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: | \(1\) |
| 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: |
\( x^{10} - 4x^{9} - 6x^{8} + 36x^{7} + x^{6} - 100x^{5} + 26x^{4} + 100x^{3} - 17x^{2} - 28x - 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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
\( x^{10} - 4x^{9} - 6x^{8} + 36x^{7} + x^{6} - 100x^{5} + 26x^{4} + 100x^{3} - 17x^{2} - 28x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{9} - 4\nu^{8} + 35\nu^{7} + 44\nu^{6} - 161\nu^{5} - 202\nu^{4} + 193\nu^{3} + 441\nu^{2} + \nu - 220 ) / 47 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 3\nu^{9} + 6\nu^{8} - 76\nu^{7} - 19\nu^{6} + 453\nu^{5} - 73\nu^{4} - 783\nu^{3} + 114\nu^{2} + 163\nu + 48 ) / 47 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -10\nu^{9} + 27\nu^{8} + 81\nu^{7} - 203\nu^{6} - 194\nu^{5} + 353\nu^{4} + 166\nu^{3} - 4\nu^{2} + 5\nu - 66 ) / 47 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 11 \nu^{9} + 25 \nu^{8} + 122 \nu^{7} - 228 \nu^{6} - 486 \nu^{5} + 581 \nu^{4} + 850 \nu^{3} + \cdots - 35 ) / 47 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 13 \nu^{9} + 21 \nu^{8} + 157 \nu^{7} - 184 \nu^{6} - 647 \nu^{5} + 426 \nu^{4} + 996 \nu^{3} + \cdots - 20 ) / 47 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 15 \nu^{9} - 17 \nu^{8} - 192 \nu^{7} + 140 \nu^{6} + 855 \nu^{5} - 318 \nu^{4} - 1471 \nu^{3} + \cdots - 42 ) / 47 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 26 \nu^{9} - 89 \nu^{8} - 173 \nu^{7} + 744 \nu^{6} + 166 \nu^{5} - 1745 \nu^{4} + 358 \nu^{3} + \cdots - 54 ) / 47 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} - \beta_{5} + \beta_{4} + \beta_{2} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 7\beta_{2} + 7\beta _1 + 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{8} + 11\beta_{7} - 2\beta_{6} - 8\beta_{5} + 8\beta_{4} - \beta_{3} + 10\beta_{2} + 28\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{9} + 2 \beta_{8} + 24 \beta_{7} - 10 \beta_{6} - 9 \beta_{5} + 12 \beta_{4} - 10 \beta_{3} + \cdots + 28 \)
|
| \(\nu^{7}\) | \(=\) |
\( 2 \beta_{9} + 13 \beta_{8} + 94 \beta_{7} - 22 \beta_{6} - 53 \beta_{5} + 57 \beta_{4} - 16 \beta_{3} + \cdots - 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( 13 \beta_{9} + 31 \beta_{8} + 220 \beta_{7} - 79 \beta_{6} - 70 \beta_{5} + 106 \beta_{4} - 85 \beta_{3} + \cdots + 117 \)
|
| \(\nu^{9}\) | \(=\) |
\( 31 \beta_{9} + 129 \beta_{8} + 742 \beta_{7} - 185 \beta_{6} - 337 \beta_{5} + 401 \beta_{4} - 172 \beta_{3} + \cdots - 29 \)
|
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 |
|
−2.72964 | 1.00000 | 5.45092 | −1.75794 | −2.72964 | 0 | −9.41975 | 1.00000 | 4.79853 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.24407 | 1.00000 | 3.03587 | 4.44184 | −2.24407 | 0 | −2.32456 | 1.00000 | −9.96781 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.86909 | 1.00000 | 1.49350 | −3.83063 | −1.86909 | 0 | 0.946688 | 1.00000 | 7.15979 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.69362 | 1.00000 | 0.868357 | 1.90171 | −1.69362 | 0 | 1.91658 | 1.00000 | −3.22078 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.795395 | 1.00000 | −1.36735 | 1.31903 | −0.795395 | 0 | 2.67837 | 1.00000 | −1.04915 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.0765467 | 1.00000 | −1.99414 | −2.75264 | 0.0765467 | 0 | −0.305738 | 1.00000 | −0.210706 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.488009 | 1.00000 | −1.76185 | 0.529828 | 0.488009 | 0 | −1.83582 | 1.00000 | 0.258561 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.01915 | 1.00000 | −0.961343 | −2.85504 | 1.01915 | 0 | −3.01804 | 1.00000 | −2.90970 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.54861 | 1.00000 | 0.398191 | 1.02582 | 1.54861 | 0 | −2.48058 | 1.00000 | 1.58859 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.19951 | 1.00000 | 2.83784 | −2.02197 | 2.19951 | 0 | 1.84284 | 1.00000 | −4.44733 | |||||||||||||||||||||||||||||||||||||||||||||||||
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.bh | yes | 10 |
| 7.b | odd | 2 | 1 | 3381.2.a.bg | ✓ | 10 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3381.2.a.bg | ✓ | 10 | 7.b | odd | 2 | 1 | |
| 3381.2.a.bh | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3381))\):
|
\( T_{2}^{10} + 4T_{2}^{9} - 6T_{2}^{8} - 36T_{2}^{7} + T_{2}^{6} + 100T_{2}^{5} + 26T_{2}^{4} - 100T_{2}^{3} - 17T_{2}^{2} + 28T_{2} - 2 \)
|
|
\( T_{5}^{10} + 4 T_{5}^{9} - 24 T_{5}^{8} - 112 T_{5}^{7} + 95 T_{5}^{6} + 708 T_{5}^{5} - 104 T_{5}^{4} + \cdots - 648 \)
|
|
\( T_{11}^{10} + 2 T_{11}^{9} - 65 T_{11}^{8} - 68 T_{11}^{7} + 1302 T_{11}^{6} + 528 T_{11}^{5} + \cdots - 2441 \)
|
|
\( T_{13}^{10} + 16 T_{13}^{9} + 44 T_{13}^{8} - 464 T_{13}^{7} - 2541 T_{13}^{6} + 1616 T_{13}^{5} + \cdots + 1358 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + 4 T^{9} + \cdots - 2 \)
|
| $3$ |
\( (T - 1)^{10} \)
|
| $5$ |
\( T^{10} + 4 T^{9} + \cdots - 648 \)
|
| $7$ |
\( T^{10} \)
|
| $11$ |
\( T^{10} + 2 T^{9} + \cdots - 2441 \)
|
| $13$ |
\( T^{10} + 16 T^{9} + \cdots + 1358 \)
|
| $17$ |
\( T^{10} + 12 T^{9} + \cdots + 815806 \)
|
| $19$ |
\( T^{10} + 26 T^{9} + \cdots + 22783 \)
|
| $23$ |
\( (T - 1)^{10} \)
|
| $29$ |
\( T^{10} + 16 T^{9} + \cdots - 453058 \)
|
| $31$ |
\( T^{10} + 20 T^{9} + \cdots + 1345636 \)
|
| $37$ |
\( T^{10} - 8 T^{9} + \cdots + 1480658 \)
|
| $41$ |
\( T^{10} + \cdots + 116354273 \)
|
| $43$ |
\( T^{10} + 4 T^{9} + \cdots + 805678 \)
|
| $47$ |
\( T^{10} + 6 T^{9} + \cdots - 11309552 \)
|
| $53$ |
\( T^{10} + \cdots - 238691081 \)
|
| $59$ |
\( T^{10} + 42 T^{9} + \cdots - 1101511 \)
|
| $61$ |
\( T^{10} + 14 T^{9} + \cdots + 22502303 \)
|
| $67$ |
\( T^{10} + \cdots - 2613735112 \)
|
| $71$ |
\( T^{10} - 8 T^{9} + \cdots - 1340942 \)
|
| $73$ |
\( T^{10} + \cdots + 497014664 \)
|
| $79$ |
\( T^{10} + \cdots - 1835233214 \)
|
| $83$ |
\( T^{10} + \cdots + 1949250692 \)
|
| $89$ |
\( T^{10} + \cdots - 4423288444 \)
|
| $97$ |
\( T^{10} - 12 T^{9} + \cdots - 43708 \)
|
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