Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [354,6,Mod(1,354)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(354, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("354.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 354 = 2 \cdot 3 \cdot 59 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 354.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: | \(56.7758722138\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 358x^{4} - 404x^{3} + 26492x^{2} - 11664x - 353376 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3^{4} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 358x^{4} - 404x^{3} + 26492x^{2} - 11664x - 353376 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -34\nu^{5} + 1148\nu^{4} - 6156\nu^{3} - 229231\nu^{2} + 2808606\nu + 4257918 ) / 130266 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -251\nu^{5} - 1742\nu^{4} + 82266\nu^{3} + 616126\nu^{2} - 3340824\nu - 13795668 ) / 260532 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -134\nu^{5} - 584\nu^{4} + 39594\nu^{3} + 299605\nu^{2} - 1456758\nu - 11662758 ) / 130266 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 598\nu^{5} - 6143\nu^{4} - 165030\nu^{3} + 1407292\nu^{2} + 8886672\nu - 58414446 ) / 390798 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -1702\nu^{5} - 2557\nu^{4} + 539844\nu^{3} + 1976846\nu^{2} - 23388948\nu - 58933332 ) / 781596 \)
|
| \(\nu\) | \(=\) |
\( ( 2\beta_{5} + \beta_{4} - 3\beta_{3} + \beta _1 - 1 ) / 18 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 10\beta_{5} + 5\beta_{4} + 9\beta_{3} - 24\beta_{2} - \beta _1 + 1069 ) / 9 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 314\beta_{5} + 121\beta_{4} - 336\beta_{3} - 168\beta_{2} + 34\beta _1 + 1673 ) / 9 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 1276\beta_{5} + 488\beta_{4} + 456\beta_{3} - 2584\beta_{2} - 44\beta _1 + 74592 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 87584\beta_{5} + 35116\beta_{4} - 77562\beta_{3} - 69516\beta_{2} + 2950\beta _1 + 1131320 ) / 9 \)
|
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 |
|
−4.00000 | −9.00000 | 16.0000 | −75.0224 | 36.0000 | 17.7979 | −64.0000 | 81.0000 | 300.090 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −4.00000 | −9.00000 | 16.0000 | −33.6768 | 36.0000 | −116.721 | −64.0000 | 81.0000 | 134.707 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −4.00000 | −9.00000 | 16.0000 | −15.9371 | 36.0000 | 113.338 | −64.0000 | 81.0000 | 63.7485 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −4.00000 | −9.00000 | 16.0000 | −14.2062 | 36.0000 | 60.7262 | −64.0000 | 81.0000 | 56.8247 | |||||||||||||||||||||||||||||||||||||
| 1.5 | −4.00000 | −9.00000 | 16.0000 | 62.9440 | 36.0000 | −185.768 | −64.0000 | 81.0000 | −251.776 | |||||||||||||||||||||||||||||||||||||
| 1.6 | −4.00000 | −9.00000 | 16.0000 | 79.8984 | 36.0000 | 56.6271 | −64.0000 | 81.0000 | −319.594 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(1\) |
| \(59\) | \(-1\) |
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 | 354.6.a.g | ✓ | 6 |
| 3.b | odd | 2 | 1 | 1062.6.a.h | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 354.6.a.g | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 1062.6.a.h | 6 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} - 4T_{5}^{5} - 8774T_{5}^{4} - 62240T_{5}^{3} + 16501109T_{5}^{2} + 425064268T_{5} + 2876741688 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(354))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 4)^{6} \)
|
| $3$ |
\( (T + 9)^{6} \)
|
| $5$ |
\( T^{6} + \cdots + 2876741688 \)
|
| $7$ |
\( T^{6} + \cdots + 150404993856 \)
|
| $11$ |
\( T^{6} + \cdots - 17\!\cdots\!88 \)
|
| $13$ |
\( T^{6} + \cdots + 95\!\cdots\!52 \)
|
| $17$ |
\( T^{6} + \cdots + 43\!\cdots\!92 \)
|
| $19$ |
\( T^{6} + \cdots - 10\!\cdots\!12 \)
|
| $23$ |
\( T^{6} + \cdots + 12\!\cdots\!04 \)
|
| $29$ |
\( T^{6} + \cdots - 72\!\cdots\!72 \)
|
| $31$ |
\( T^{6} + \cdots + 51\!\cdots\!24 \)
|
| $37$ |
\( T^{6} + \cdots + 98\!\cdots\!56 \)
|
| $41$ |
\( T^{6} + \cdots - 88\!\cdots\!36 \)
|
| $43$ |
\( T^{6} + \cdots + 22\!\cdots\!48 \)
|
| $47$ |
\( T^{6} + \cdots - 40\!\cdots\!16 \)
|
| $53$ |
\( T^{6} + \cdots - 80\!\cdots\!36 \)
|
| $59$ |
\( (T - 3481)^{6} \)
|
| $61$ |
\( T^{6} + \cdots + 22\!\cdots\!24 \)
|
| $67$ |
\( T^{6} + \cdots - 45\!\cdots\!56 \)
|
| $71$ |
\( T^{6} + \cdots - 11\!\cdots\!92 \)
|
| $73$ |
\( T^{6} + \cdots + 13\!\cdots\!76 \)
|
| $79$ |
\( T^{6} + \cdots + 57\!\cdots\!28 \)
|
| $83$ |
\( T^{6} + \cdots + 97\!\cdots\!04 \)
|
| $89$ |
\( T^{6} + \cdots - 27\!\cdots\!52 \)
|
| $97$ |
\( T^{6} + \cdots + 43\!\cdots\!56 \)
|
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