Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7406,2,Mod(1,7406)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7406, 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("7406.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7406 = 2 \cdot 7 \cdot 23^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7406.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: | \(59.1372077370\) |
| Analytic rank: | \(1\) |
| Dimension: | \(10\) |
| Coefficient field: | 10.10.4074747968929.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - x^{9} - 13x^{8} + 8x^{7} + 57x^{6} - 11x^{5} - 96x^{4} - 15x^{3} + 51x^{2} + 19x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 322) |
| 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
\( x^{10} - x^{9} - 13x^{8} + 8x^{7} + 57x^{6} - 11x^{5} - 96x^{4} - 15x^{3} + 51x^{2} + 19x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{9} - 17\nu^{7} + 4\nu^{6} + 95\nu^{5} - 37\nu^{4} - 194\nu^{3} + 85\nu^{2} + 105\nu - 30 ) / 11 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 12 \nu^{9} - 22 \nu^{8} - 138 \nu^{7} + 213 \nu^{6} + 513 \nu^{5} - 576 \nu^{4} - 711 \nu^{3} + \cdots - 30 ) / 11 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 12 \nu^{9} + 19 \nu^{8} + 145 \nu^{7} - 182 \nu^{6} - 579 \nu^{5} + 478 \nu^{4} + 882 \nu^{3} + \cdots + 1 ) / 11 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 15 \nu^{9} + 26 \nu^{8} + 176 \nu^{7} - 248 \nu^{6} - 677 \nu^{5} + 649 \nu^{4} + 988 \nu^{3} + \cdots + 1 ) / 11 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 20 \nu^{9} - 31 \nu^{8} - 242 \nu^{7} + 294 \nu^{6} + 965 \nu^{5} - 759 \nu^{4} - 1442 \nu^{3} + \cdots + 6 ) / 11 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 23 \nu^{9} + 38 \nu^{8} + 273 \nu^{7} - 360 \nu^{6} - 1063 \nu^{5} + 930 \nu^{4} + 1559 \nu^{3} + \cdots + 16 ) / 11 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 26 \nu^{9} + 44 \nu^{8} + 310 \nu^{7} - 423 \nu^{6} - 1216 \nu^{5} + 1127 \nu^{4} + 1799 \nu^{3} + \cdots + 43 ) / 11 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 43 \nu^{9} - 72 \nu^{8} - 508 \nu^{7} + 685 \nu^{6} + 1962 \nu^{5} - 1787 \nu^{4} - 2830 \nu^{3} + \cdots - 61 ) / 11 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{9} + \beta_{7} - \beta_{6} - \beta_{4} - \beta_{3} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{9} + 2\beta_{7} - \beta_{5} - \beta_{3} + 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 7\beta_{9} + 9\beta_{7} - 6\beta_{6} - \beta_{5} - 8\beta_{4} - 7\beta_{3} - \beta_{2} - \beta _1 + 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( 9\beta_{9} + \beta_{8} + 17\beta_{7} - 2\beta_{6} - 11\beta_{5} - 2\beta_{4} - 10\beta_{3} + \beta_{2} + 21\beta _1 - 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 45 \beta_{9} + \beta_{8} + 64 \beta_{7} - 35 \beta_{6} - 10 \beta_{5} - 53 \beta_{4} - 43 \beta_{3} + \cdots + 41 \)
|
| \(\nu^{7}\) | \(=\) |
\( 67 \beta_{9} + 14 \beta_{8} + 121 \beta_{7} - 21 \beta_{6} - 85 \beta_{5} - 23 \beta_{4} - 73 \beta_{3} + \cdots - 7 \)
|
| \(\nu^{8}\) | \(=\) |
\( 287 \beta_{9} + 21 \beta_{8} + 425 \beta_{7} - 206 \beta_{6} - 84 \beta_{5} - 335 \beta_{4} - 262 \beta_{3} + \cdots + 229 \)
|
| \(\nu^{9}\) | \(=\) |
\( 472 \beta_{9} + 139 \beta_{8} + 822 \beta_{7} - 164 \beta_{6} - 591 \beta_{5} - 200 \beta_{4} + \cdots - 32 \)
|
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 |
|
1.00000 | −3.37885 | 1.00000 | 1.49205 | −3.37885 | 1.00000 | 1.00000 | 8.41663 | 1.49205 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −2.72935 | 1.00000 | 3.61116 | −2.72935 | 1.00000 | 1.00000 | 4.44937 | 3.61116 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −2.15060 | 1.00000 | 0.522140 | −2.15060 | 1.00000 | 1.00000 | 1.62506 | 0.522140 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | −1.70062 | 1.00000 | 2.96270 | −1.70062 | 1.00000 | 1.00000 | −0.107906 | 2.96270 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | −1.66408 | 1.00000 | −0.264223 | −1.66408 | 1.00000 | 1.00000 | −0.230830 | −0.264223 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | −0.953154 | 1.00000 | −0.0978227 | −0.953154 | 1.00000 | 1.00000 | −2.09150 | −0.0978227 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | −0.0147545 | 1.00000 | −4.16631 | −0.0147545 | 1.00000 | 1.00000 | −2.99978 | −4.16631 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.00000 | 0.548020 | 1.00000 | −0.970940 | 0.548020 | 1.00000 | 1.00000 | −2.69967 | −0.970940 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.00000 | 1.46032 | 1.00000 | −1.11649 | 1.46032 | 1.00000 | 1.00000 | −0.867474 | −1.11649 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.00000 | 1.58307 | 1.00000 | 1.02774 | 1.58307 | 1.00000 | 1.00000 | −0.493895 | 1.02774 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(-1\) |
| \(23\) | \(1\) |
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 | 7406.2.a.bp | 10 | |
| 23.b | odd | 2 | 1 | 7406.2.a.bo | 10 | ||
| 23.c | even | 11 | 2 | 322.2.i.c | ✓ | 20 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 322.2.i.c | ✓ | 20 | 23.c | even | 11 | 2 | |
| 7406.2.a.bo | 10 | 23.b | odd | 2 | 1 | ||
| 7406.2.a.bp | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7406))\):
|
\( T_{3}^{10} + 9T_{3}^{9} + 23T_{3}^{8} - 12T_{3}^{7} - 125T_{3}^{6} - 103T_{3}^{5} + 158T_{3}^{4} + 219T_{3}^{3} - 12T_{3}^{2} - 68T_{3} - 1 \)
|
|
\( T_{5}^{10} - 3T_{5}^{9} - 18T_{5}^{8} + 62T_{5}^{7} + 19T_{5}^{6} - 143T_{5}^{5} + 9T_{5}^{4} + 96T_{5}^{3} - 9T_{5}^{2} - 12T_{5} - 1 \)
|
|
\( T_{11}^{10} + 8 T_{11}^{9} - 144 T_{11}^{7} - 300 T_{11}^{6} + 502 T_{11}^{5} + 2119 T_{11}^{4} + \cdots - 439 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{10} \)
|
| $3$ |
\( T^{10} + 9 T^{9} + \cdots - 1 \)
|
| $5$ |
\( T^{10} - 3 T^{9} + \cdots - 1 \)
|
| $7$ |
\( (T - 1)^{10} \)
|
| $11$ |
\( T^{10} + 8 T^{9} + \cdots - 439 \)
|
| $13$ |
\( T^{10} + 15 T^{9} + \cdots + 42481 \)
|
| $17$ |
\( T^{10} - 9 T^{9} + \cdots - 40217 \)
|
| $19$ |
\( T^{10} + T^{9} + \cdots - 69497 \)
|
| $23$ |
\( T^{10} \)
|
| $29$ |
\( T^{10} + 21 T^{9} + \cdots + 178441 \)
|
| $31$ |
\( T^{10} + 30 T^{9} + \cdots + 1103057 \)
|
| $37$ |
\( T^{10} + 25 T^{9} + \cdots - 258103 \)
|
| $41$ |
\( T^{10} + 9 T^{9} + \cdots - 14647313 \)
|
| $43$ |
\( T^{10} + 36 T^{9} + \cdots + 65281667 \)
|
| $47$ |
\( T^{10} + 34 T^{9} + \cdots - 27697 \)
|
| $53$ |
\( T^{10} + 23 T^{9} + \cdots - 36871 \)
|
| $59$ |
\( T^{10} + \cdots - 213245791 \)
|
| $61$ |
\( T^{10} + 2 T^{9} + \cdots - 35604097 \)
|
| $67$ |
\( T^{10} + T^{9} + \cdots - 16320809 \)
|
| $71$ |
\( T^{10} + 36 T^{9} + \cdots - 2671657 \)
|
| $73$ |
\( T^{10} + 15 T^{9} + \cdots + 7126967 \)
|
| $79$ |
\( T^{10} + 9 T^{9} + \cdots + 26960263 \)
|
| $83$ |
\( T^{10} + \cdots - 1236283487 \)
|
| $89$ |
\( T^{10} + \cdots - 157247771 \)
|
| $97$ |
\( T^{10} - 44 T^{9} + \cdots - 44300807 \)
|
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