Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [363,6,Mod(1,363)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(363, 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("363.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 363 = 3 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 363.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: | \(58.2193265921\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.54016037568.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 42x^{4} + 82x^{3} + 274x^{2} - 408x - 363 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| 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
\( x^{6} - 2x^{5} - 42x^{4} + 82x^{3} + 274x^{2} - 408x - 363 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -21\nu^{5} + 89\nu^{4} + 991\nu^{3} - 3208\nu^{2} - 7974\nu + 10119 ) / 2427 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -32\nu^{5} - 57\nu^{4} + 1356\nu^{3} + 2200\nu^{2} - 8337\nu - 10353 ) / 2427 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 128\nu^{5} + 228\nu^{4} - 5424\nu^{3} - 8800\nu^{2} + 43056\nu + 38985 ) / 2427 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 274\nu^{5} + 33\nu^{4} - 11004\nu^{3} - 635\nu^{2} + 57582\nu + 11742 ) / 2427 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -431\nu^{5} + 16\nu^{4} + 16949\nu^{3} + 2732\nu^{2} - 77325\nu - 75354 ) / 2427 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + 4\beta_{2} + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{5} + 4\beta_{4} - \beta_{3} + 2\beta_{2} + 2\beta _1 + 59 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -3\beta_{5} + 4\beta_{4} + 25\beta_{3} + 153\beta_{2} + 33\beta _1 + 1 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 76\beta_{5} + 144\beta_{4} - 37\beta_{3} - 20\beta_{2} + 124\beta _1 + 1655 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -125\beta_{5} + 188\beta_{4} + 796\beta_{3} + 5311\beta_{2} + 1315\beta _1 - 404 ) / 4 \)
|
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 |
|
−10.8582 | −9.00000 | 85.9010 | −20.0082 | 97.7240 | 176.759 | −585.269 | 81.0000 | 217.253 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −4.59126 | −9.00000 | −10.9203 | 63.3456 | 41.3214 | 112.581 | 197.058 | 81.0000 | −290.836 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.138973 | −9.00000 | −31.9807 | −68.3374 | 1.25076 | −15.6761 | 8.89158 | 81.0000 | 9.49704 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.138973 | −9.00000 | −31.9807 | −68.3374 | −1.25076 | 15.6761 | −8.89158 | 81.0000 | −9.49704 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 4.59126 | −9.00000 | −10.9203 | 63.3456 | −41.3214 | −112.581 | −197.058 | 81.0000 | 290.836 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 10.8582 | −9.00000 | 85.9010 | −20.0082 | −97.7240 | −176.759 | 585.269 | 81.0000 | −217.253 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(11\) | \( +1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 363.6.a.o | ✓ | 6 |
| 3.b | odd | 2 | 1 | 1089.6.a.z | 6 | ||
| 11.b | odd | 2 | 1 | inner | 363.6.a.o | ✓ | 6 |
| 33.d | even | 2 | 1 | 1089.6.a.z | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 363.6.a.o | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 363.6.a.o | ✓ | 6 | 11.b | odd | 2 | 1 | inner |
| 1089.6.a.z | 6 | 3.b | odd | 2 | 1 | ||
| 1089.6.a.z | 6 | 33.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 139T_{2}^{4} + 2488T_{2}^{2} - 48 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(363))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 139 T^{4} + \cdots - 48 \)
|
| $3$ |
\( (T + 9)^{6} \)
|
| $5$ |
\( (T^{3} + 25 T^{2} + \cdots - 86613)^{2} \)
|
| $7$ |
\( T^{6} + \cdots - 97312352448 \)
|
| $11$ |
\( T^{6} \)
|
| $13$ |
\( T^{6} + \cdots - 11\!\cdots\!63 \)
|
| $17$ |
\( T^{6} + \cdots - 37\!\cdots\!03 \)
|
| $19$ |
\( T^{6} + \cdots - 14\!\cdots\!52 \)
|
| $23$ |
\( (T^{3} + 1558 T^{2} + \cdots - 15000846072)^{2} \)
|
| $29$ |
\( T^{6} + \cdots - 10\!\cdots\!23 \)
|
| $31$ |
\( (T^{3} - 4804 T^{2} + \cdots + 315824988480)^{2} \)
|
| $37$ |
\( (T^{3} + \cdots - 1383658969825)^{2} \)
|
| $41$ |
\( T^{6} + \cdots - 37\!\cdots\!43 \)
|
| $43$ |
\( T^{6} + \cdots - 21\!\cdots\!72 \)
|
| $47$ |
\( (T^{3} - 6008 T^{2} + \cdots + 250338029568)^{2} \)
|
| $53$ |
\( (T^{3} - 12189 T^{2} + \cdots + 97006130145)^{2} \)
|
| $59$ |
\( (T^{3} + \cdots - 24683076420120)^{2} \)
|
| $61$ |
\( T^{6} + \cdots - 53\!\cdots\!52 \)
|
| $67$ |
\( (T^{3} + \cdots + 83671658705016)^{2} \)
|
| $71$ |
\( (T^{3} + \cdots - 16884330481656)^{2} \)
|
| $73$ |
\( T^{6} + \cdots - 11\!\cdots\!48 \)
|
| $79$ |
\( T^{6} + \cdots - 48\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots - 57\!\cdots\!72 \)
|
| $89$ |
\( (T^{3} + \cdots + 85151598813819)^{2} \)
|
| $97$ |
\( (T^{3} + \cdots - 775593398644457)^{2} \)
|
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