Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [460,6,Mod(1,460)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(460, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("460.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 460 = 2^{2} \cdot 5 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 460.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: | \(73.7765571140\) |
| 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: |
\( x^{10} - x^{9} - 1704 x^{8} + 1306 x^{7} + 964258 x^{6} - 774352 x^{5} - 223171947 x^{4} + \cdots - 406136160000 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{7} \) |
| 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
\( x^{10} - x^{9} - 1704 x^{8} + 1306 x^{7} + 964258 x^{6} - 774352 x^{5} - 223171947 x^{4} + \cdots - 406136160000 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 341 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 394304351279005 \nu^{9} + \cdots + 26\!\cdots\!76 ) / 99\!\cdots\!04 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 10\!\cdots\!47 \nu^{9} + \cdots - 40\!\cdots\!08 ) / 49\!\cdots\!52 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 23\!\cdots\!67 \nu^{9} + \cdots + 10\!\cdots\!36 ) / 99\!\cdots\!04 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 58\!\cdots\!96 \nu^{9} + \cdots - 26\!\cdots\!52 ) / 14\!\cdots\!56 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 59\!\cdots\!09 \nu^{9} + \cdots + 23\!\cdots\!32 ) / 14\!\cdots\!56 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 590819369010327 \nu^{9} + \cdots - 29\!\cdots\!92 ) / 90\!\cdots\!64 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 68\!\cdots\!79 \nu^{9} + \cdots - 30\!\cdots\!36 ) / 99\!\cdots\!04 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 341 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{9} - \beta_{8} + 4\beta_{7} + 4\beta_{6} + 3\beta_{5} + 3\beta_{4} - \beta_{3} + 573\beta _1 + 65 \)
|
| \(\nu^{4}\) | \(=\) |
\( 46 \beta_{9} - 41 \beta_{8} - 20 \beta_{7} - 8 \beta_{6} + 57 \beta_{5} + \beta_{4} - 92 \beta_{3} + \cdots + 195270 \)
|
| \(\nu^{5}\) | \(=\) |
\( 897 \beta_{9} - 1087 \beta_{8} + 3867 \beta_{7} + 4806 \beta_{6} + 4815 \beta_{5} + 3556 \beta_{4} + \cdots + 207566 \)
|
| \(\nu^{6}\) | \(=\) |
\( 54646 \beta_{9} - 49936 \beta_{8} - 26651 \beta_{7} + 4120 \beta_{6} + 85698 \beta_{5} + \cdots + 138023554 \)
|
| \(\nu^{7}\) | \(=\) |
\( 811109 \beta_{9} - 948750 \beta_{8} + 3330089 \beta_{7} + 4524338 \beta_{6} + 5193732 \beta_{5} + \cdots + 238088103 \)
|
| \(\nu^{8}\) | \(=\) |
\( 51908142 \beta_{9} - 47420246 \beta_{8} - 25581282 \beta_{7} + 12270276 \beta_{6} + 92475654 \beta_{5} + \cdots + 107194578543 \)
|
| \(\nu^{9}\) | \(=\) |
\( 730811835 \beta_{9} - 802130691 \beta_{8} + 2803844322 \beta_{7} + 3986068212 \beta_{6} + \cdots + 233083922019 \)
|
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 |
|
0 | −28.2157 | 0 | 25.0000 | 0 | −5.69155 | 0 | 553.125 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −20.3096 | 0 | 25.0000 | 0 | 9.48682 | 0 | 169.481 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −14.6579 | 0 | 25.0000 | 0 | −225.884 | 0 | −28.1460 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −7.98454 | 0 | 25.0000 | 0 | 164.480 | 0 | −179.247 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −7.79100 | 0 | 25.0000 | 0 | 33.2045 | 0 | −182.300 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 5.05345 | 0 | 25.0000 | 0 | −105.271 | 0 | −217.463 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 14.4714 | 0 | 25.0000 | 0 | 173.785 | 0 | −33.5780 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 18.6788 | 0 | 25.0000 | 0 | −233.132 | 0 | 105.896 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 20.0069 | 0 | 25.0000 | 0 | 138.666 | 0 | 157.276 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 29.7482 | 0 | 25.0000 | 0 | 56.3558 | 0 | 641.956 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(-1\) |
| \(23\) | \(-1\) |
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 | 460.6.a.d | ✓ | 10 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 460.6.a.d | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{10} - 9 T_{3}^{9} - 1668 T_{3}^{8} + 12242 T_{3}^{7} + 925814 T_{3}^{6} - 4943324 T_{3}^{5} + \cdots - 424810484928 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(460))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} \)
|
| $3$ |
\( T^{10} + \cdots - 424810484928 \)
|
| $5$ |
\( (T - 25)^{10} \)
|
| $7$ |
\( T^{10} + \cdots + 22\!\cdots\!24 \)
|
| $11$ |
\( T^{10} + \cdots + 28\!\cdots\!20 \)
|
| $13$ |
\( T^{10} + \cdots - 20\!\cdots\!20 \)
|
| $17$ |
\( T^{10} + \cdots - 17\!\cdots\!76 \)
|
| $19$ |
\( T^{10} + \cdots - 24\!\cdots\!64 \)
|
| $23$ |
\( (T - 529)^{10} \)
|
| $29$ |
\( T^{10} + \cdots - 13\!\cdots\!16 \)
|
| $31$ |
\( T^{10} + \cdots + 17\!\cdots\!00 \)
|
| $37$ |
\( T^{10} + \cdots - 10\!\cdots\!92 \)
|
| $41$ |
\( T^{10} + \cdots - 13\!\cdots\!98 \)
|
| $43$ |
\( T^{10} + \cdots - 12\!\cdots\!00 \)
|
| $47$ |
\( T^{10} + \cdots + 25\!\cdots\!36 \)
|
| $53$ |
\( T^{10} + \cdots - 79\!\cdots\!76 \)
|
| $59$ |
\( T^{10} + \cdots + 31\!\cdots\!60 \)
|
| $61$ |
\( T^{10} + \cdots + 13\!\cdots\!08 \)
|
| $67$ |
\( T^{10} + \cdots + 44\!\cdots\!56 \)
|
| $71$ |
\( T^{10} + \cdots - 42\!\cdots\!20 \)
|
| $73$ |
\( T^{10} + \cdots - 10\!\cdots\!72 \)
|
| $79$ |
\( T^{10} + \cdots + 13\!\cdots\!00 \)
|
| $83$ |
\( T^{10} + \cdots + 94\!\cdots\!44 \)
|
| $89$ |
\( T^{10} + \cdots + 18\!\cdots\!80 \)
|
| $97$ |
\( T^{10} + \cdots - 14\!\cdots\!00 \)
|
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