Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1407,2,Mod(1,1407)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1407, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("1407.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1407 = 3 \cdot 7 \cdot 67 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1407.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: | \(11.2349515644\) |
| Analytic rank: | \(0\) |
| Dimension: | \(9\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{9} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{9} - x^{8} - 15x^{7} + 13x^{6} + 68x^{5} - 52x^{4} - 85x^{3} + 75x^{2} - 17x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{8}\) 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
\( x^{9} - x^{8} - 15x^{7} + 13x^{6} + 68x^{5} - 52x^{4} - 85x^{3} + 75x^{2} - 17x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 6\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2\nu^{8} - \nu^{7} - 31\nu^{6} + 12\nu^{5} + 146\nu^{4} - 46\nu^{3} - 194\nu^{2} + 85\nu - 9 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( -\nu^{8} + \nu^{7} + 15\nu^{6} - 13\nu^{5} - 68\nu^{4} + 52\nu^{3} + 86\nu^{2} - 75\nu + 12 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -4\nu^{8} + 3\nu^{7} + 61\nu^{6} - 36\nu^{5} - 284\nu^{4} + 130\nu^{3} + 382\nu^{2} - 193\nu + 13 ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 4\nu^{8} - 3\nu^{7} - 61\nu^{6} + 38\nu^{5} + 284\nu^{4} - 150\nu^{3} - 380\nu^{2} + 233\nu - 25 ) / 2 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 7\nu^{8} - 5\nu^{7} - 106\nu^{6} + 60\nu^{5} + 488\nu^{4} - 218\nu^{3} - 641\nu^{2} + 331\nu - 32 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{7} + \beta_{5} - \beta_{4} + 7\beta_{2} + \beta _1 + 17 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{7} + \beta_{6} + 10\beta_{3} - \beta_{2} + 40\beta _1 + 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( 2\beta_{8} + 11\beta_{7} + 3\beta_{6} + 11\beta_{5} - 12\beta_{4} + 48\beta_{2} + 12\beta _1 + 108 \)
|
| \(\nu^{7}\) | \(=\) |
\( 2\beta_{8} + 15\beta_{7} + 17\beta_{6} + 3\beta_{5} + 82\beta_{3} - 14\beta_{2} + 279\beta _1 + 31 \)
|
| \(\nu^{8}\) | \(=\) |
\( 32\beta_{8} + 99\beta_{7} + 49\beta_{6} + 99\beta_{5} - 112\beta_{4} + 4\beta_{3} + 329\beta_{2} + 108\beta _1 + 726 \)
|
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 |
|
−2.71217 | 1.00000 | 5.35585 | −0.582515 | −2.71217 | −1.00000 | −9.10164 | 1.00000 | 1.57988 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.05601 | 1.00000 | 2.22717 | −3.73085 | −2.05601 | −1.00000 | −0.467074 | 1.00000 | 7.67066 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.73396 | 1.00000 | 1.00661 | 3.77135 | −1.73396 | −1.00000 | 1.72250 | 1.00000 | −6.53936 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.0922446 | 1.00000 | −1.99149 | 3.41870 | 0.0922446 | −1.00000 | −0.368194 | 1.00000 | 0.315357 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.259264 | 1.00000 | −1.93278 | −4.28034 | 0.259264 | −1.00000 | −1.01963 | 1.00000 | −1.10974 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.443512 | 1.00000 | −1.80330 | −0.838701 | 0.443512 | −1.00000 | −1.68681 | 1.00000 | −0.371974 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.37441 | 1.00000 | −0.111009 | 0.144984 | 1.37441 | −1.00000 | −2.90138 | 1.00000 | 0.199267 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.54375 | 1.00000 | 4.47066 | 1.78502 | 2.54375 | −1.00000 | 6.28473 | 1.00000 | 4.54063 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.78896 | 1.00000 | 5.77829 | −3.68765 | 2.78896 | −1.00000 | 10.5375 | 1.00000 | −10.2847 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(7\) | \(1\) |
| \(67\) | \(1\) |
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 | 1407.2.a.m | ✓ | 9 |
| 3.b | odd | 2 | 1 | 4221.2.a.z | 9 | ||
| 7.b | odd | 2 | 1 | 9849.2.a.bf | 9 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1407.2.a.m | ✓ | 9 | 1.a | even | 1 | 1 | trivial |
| 4221.2.a.z | 9 | 3.b | odd | 2 | 1 | ||
| 9849.2.a.bf | 9 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1407))\):
|
\( T_{2}^{9} - T_{2}^{8} - 15T_{2}^{7} + 13T_{2}^{6} + 68T_{2}^{5} - 52T_{2}^{4} - 85T_{2}^{3} + 75T_{2}^{2} - 17T_{2} + 1 \)
|
|
\( T_{5}^{9} + 4T_{5}^{8} - 30T_{5}^{7} - 114T_{5}^{6} + 272T_{5}^{5} + 926T_{5}^{4} - 648T_{5}^{3} - 1624T_{5}^{2} - 416T_{5} + 96 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} - T^{8} - 15 T^{7} + \cdots + 1 \)
|
| $3$ |
\( (T - 1)^{9} \)
|
| $5$ |
\( T^{9} + 4 T^{8} + \cdots + 96 \)
|
| $7$ |
\( (T + 1)^{9} \)
|
| $11$ |
\( T^{9} + 2 T^{8} + \cdots + 924 \)
|
| $13$ |
\( T^{9} - 8 T^{8} + \cdots + 814 \)
|
| $17$ |
\( T^{9} - 20 T^{8} + \cdots + 43008 \)
|
| $19$ |
\( T^{9} + 8 T^{8} + \cdots - 306016 \)
|
| $23$ |
\( T^{9} - 16 T^{8} + \cdots - 117504 \)
|
| $29$ |
\( T^{9} - 20 T^{8} + \cdots + 30144 \)
|
| $31$ |
\( T^{9} + 2 T^{8} + \cdots + 87264 \)
|
| $37$ |
\( T^{9} - 10 T^{8} + \cdots + 10224 \)
|
| $41$ |
\( T^{9} + 10 T^{8} + \cdots - 3277728 \)
|
| $43$ |
\( T^{9} - 20 T^{8} + \cdots - 23789248 \)
|
| $47$ |
\( T^{9} + 18 T^{8} + \cdots + 23108178 \)
|
| $53$ |
\( T^{9} - 46 T^{8} + \cdots - 2794752 \)
|
| $59$ |
\( T^{9} + 20 T^{8} + \cdots - 50666202 \)
|
| $61$ |
\( T^{9} - 38 T^{8} + \cdots - 1333794 \)
|
| $67$ |
\( (T + 1)^{9} \)
|
| $71$ |
\( T^{9} + 16 T^{8} + \cdots + 1057536 \)
|
| $73$ |
\( T^{9} - 6 T^{8} + \cdots - 655872 \)
|
| $79$ |
\( T^{9} - 24 T^{8} + \cdots + 6457024 \)
|
| $83$ |
\( T^{9} + 26 T^{8} + \cdots + 32736 \)
|
| $89$ |
\( T^{9} - 4 T^{8} + \cdots - 81024 \)
|
| $97$ |
\( T^{9} + 10 T^{8} + \cdots - 6414 \)
|
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