Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [867,6,Mod(1,867)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(867, 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("867.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 867 = 3 \cdot 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 867.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: | \(139.052771778\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 3x^{7} - 217x^{6} + 561x^{5} + 14182x^{4} - 33552x^{3} - 289744x^{2} + 634992x + 110880 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3 \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 3x^{7} - 217x^{6} + 561x^{5} + 14182x^{4} - 33552x^{3} - 289744x^{2} + 634992x + 110880 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 55 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 113 \nu^{7} + 513 \nu^{6} + 14459 \nu^{5} - 75355 \nu^{4} - 6092 \nu^{3} + 2848824 \nu^{2} + \cdots - 6796144 ) / 232832 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - \nu^{7} + 2065 \nu^{6} - 8629 \nu^{5} - 310251 \nu^{4} + 1239828 \nu^{3} + 7978072 \nu^{2} + \cdots + 70805264 ) / 232832 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 155 \nu^{7} + 69 \nu^{6} - 30393 \nu^{5} - 20007 \nu^{4} + 1571988 \nu^{3} + 668088 \nu^{2} + \cdots + 5308368 ) / 232832 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 197 \nu^{7} + 651 \nu^{6} - 46327 \nu^{5} - 115369 \nu^{4} + 3370716 \nu^{3} + 3486504 \nu^{2} + \cdots + 41306544 ) / 232832 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 337 \nu^{7} - 2591 \nu^{6} + 60635 \nu^{5} + 510853 \nu^{4} - 2497932 \nu^{3} - 22427336 \nu^{2} + \cdots + 110647312 ) / 232832 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 55 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} - 2\beta_{5} - \beta_{3} + 3\beta_{2} + 87\beta _1 + 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{7} + 4\beta_{4} - 6\beta_{3} + 129\beta_{2} + 62\beta _1 + 4753 \)
|
| \(\nu^{5}\) | \(=\) |
\( -2\beta_{7} + 151\beta_{6} - 334\beta_{5} + 8\beta_{4} - 189\beta_{3} + 543\beta_{2} + 8843\beta _1 + 3692 \)
|
| \(\nu^{6}\) | \(=\) |
\( 292\beta_{7} + 40\beta_{6} - 216\beta_{5} + 748\beta_{4} - 1104\beta_{3} + 16025\beta_{2} + 14400\beta _1 + 480743 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 264 \beta_{7} + 19449 \beta_{6} - 43610 \beta_{5} + 1752 \beta_{4} - 27201 \beta_{3} + 81255 \beta_{2} + \cdots + 811560 \)
|
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 |
|
−10.4725 | 9.00000 | 77.6731 | −93.2583 | −94.2524 | 150.382 | −478.312 | 81.0000 | 976.647 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −7.67568 | 9.00000 | 26.9161 | 1.48858 | −69.0811 | −193.858 | 39.0226 | 81.0000 | −11.4259 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −6.43720 | 9.00000 | 9.43749 | 90.3789 | −57.9348 | 182.097 | 145.239 | 81.0000 | −581.787 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.162771 | 9.00000 | −31.9735 | −36.9401 | −1.46494 | −14.6956 | 10.4130 | 81.0000 | 6.01276 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 2.35629 | 9.00000 | −26.4479 | 88.7822 | 21.2066 | −172.001 | −137.720 | 81.0000 | 209.196 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 6.60523 | 9.00000 | 11.6290 | −106.715 | 59.4470 | 3.58814 | −134.555 | 81.0000 | −704.875 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 7.48320 | 9.00000 | 23.9983 | 41.0283 | 67.3488 | 185.102 | −59.8783 | 81.0000 | 307.023 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 11.3034 | 9.00000 | 95.7674 | 15.2351 | 101.731 | −158.615 | 720.790 | 81.0000 | 172.209 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(17\) | \(1\) |
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 | 867.6.a.m | yes | 8 |
| 17.b | even | 2 | 1 | 867.6.a.l | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 867.6.a.l | ✓ | 8 | 17.b | even | 2 | 1 | |
| 867.6.a.m | yes | 8 | 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(867))\):
|
\( T_{2}^{8} - 3T_{2}^{7} - 217T_{2}^{6} + 561T_{2}^{5} + 14182T_{2}^{4} - 33552T_{2}^{3} - 289744T_{2}^{2} + 634992T_{2} + 110880 \)
|
|
\( T_{5}^{8} - 19709 T_{5}^{6} + 188700 T_{5}^{5} + 109490096 T_{5}^{4} - 1859294400 T_{5}^{3} + \cdots - 2744764498944 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 3 T^{7} + \cdots + 110880 \)
|
| $3$ |
\( (T - 9)^{8} \)
|
| $5$ |
\( T^{8} + \cdots - 2744764498944 \)
|
| $7$ |
\( T^{8} + \cdots + 14\!\cdots\!40 \)
|
| $11$ |
\( T^{8} + \cdots - 16\!\cdots\!56 \)
|
| $13$ |
\( T^{8} + \cdots + 53\!\cdots\!28 \)
|
| $17$ |
\( T^{8} \)
|
| $19$ |
\( T^{8} + \cdots - 20\!\cdots\!00 \)
|
| $23$ |
\( T^{8} + \cdots + 78\!\cdots\!00 \)
|
| $29$ |
\( T^{8} + \cdots + 83\!\cdots\!20 \)
|
| $31$ |
\( T^{8} + \cdots - 23\!\cdots\!81 \)
|
| $37$ |
\( T^{8} + \cdots + 70\!\cdots\!88 \)
|
| $41$ |
\( T^{8} + \cdots - 53\!\cdots\!16 \)
|
| $43$ |
\( T^{8} + \cdots + 60\!\cdots\!60 \)
|
| $47$ |
\( T^{8} + \cdots - 30\!\cdots\!80 \)
|
| $53$ |
\( T^{8} + \cdots + 78\!\cdots\!88 \)
|
| $59$ |
\( T^{8} + \cdots - 48\!\cdots\!40 \)
|
| $61$ |
\( T^{8} + \cdots + 31\!\cdots\!00 \)
|
| $67$ |
\( T^{8} + \cdots - 33\!\cdots\!48 \)
|
| $71$ |
\( T^{8} + \cdots - 12\!\cdots\!00 \)
|
| $73$ |
\( T^{8} + \cdots + 82\!\cdots\!92 \)
|
| $79$ |
\( T^{8} + \cdots + 49\!\cdots\!96 \)
|
| $83$ |
\( T^{8} + \cdots - 11\!\cdots\!00 \)
|
| $89$ |
\( T^{8} + \cdots + 64\!\cdots\!88 \)
|
| $97$ |
\( T^{8} + \cdots + 39\!\cdots\!85 \)
|
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