Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [207,8,Mod(1,207)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(207, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("207.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 207 = 3^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 207.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: | \(64.6637002752\) |
| Analytic rank: | \(1\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 4 x^{11} - 1070 x^{10} + 4076 x^{9} + 403334 x^{8} - 1518684 x^{7} - 64710184 x^{6} + \cdots + 90709421512 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{9}\cdot 3^{7} \) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 4 x^{11} - 1070 x^{10} + 4076 x^{9} + 403334 x^{8} - 1518684 x^{7} - 64710184 x^{6} + \cdots + 90709421512 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 180 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 316\nu + 408 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 321347843172523 \nu^{11} + \cdots - 23\!\cdots\!44 ) / 14\!\cdots\!40 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 773723770993665 \nu^{11} + \cdots - 70\!\cdots\!24 ) / 14\!\cdots\!40 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 986139303013447 \nu^{11} + \cdots - 37\!\cdots\!76 ) / 14\!\cdots\!40 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 561963621614343 \nu^{11} + \cdots - 19\!\cdots\!64 ) / 71\!\cdots\!20 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 29742526540555 \nu^{11} - 174955570280317 \nu^{10} + \cdots + 70\!\cdots\!08 ) / 25\!\cdots\!40 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 215327141267844 \nu^{11} + \cdots + 28\!\cdots\!40 ) / 89\!\cdots\!40 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 10\!\cdots\!31 \nu^{11} + \cdots + 16\!\cdots\!32 ) / 20\!\cdots\!20 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 86\!\cdots\!51 \nu^{11} + \cdots - 38\!\cdots\!20 ) / 14\!\cdots\!40 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 180 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 316\beta _1 - 48 \)
|
| \(\nu^{4}\) | \(=\) |
\( - \beta_{10} + 7 \beta_{9} + \beta_{8} - 4 \beta_{7} + 3 \beta_{6} + 25 \beta_{5} - 3 \beta_{4} + \cdots + 56737 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 10 \beta_{11} - 56 \beta_{10} + 89 \beta_{9} - 46 \beta_{8} + 50 \beta_{7} + 33 \beta_{6} + \cdots + 14482 \)
|
| \(\nu^{6}\) | \(=\) |
\( 122 \beta_{11} - 357 \beta_{10} + 4974 \beta_{9} + 295 \beta_{8} - 2270 \beta_{7} + 2846 \beta_{6} + \cdots + 21009255 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 8724 \beta_{11} - 45970 \beta_{10} + 74028 \beta_{9} - 35074 \beta_{8} + 43860 \beta_{7} + \cdots + 21246666 \)
|
| \(\nu^{8}\) | \(=\) |
\( 131736 \beta_{11} - 114135 \beta_{10} + 2868469 \beta_{9} + 19455 \beta_{8} - 1018548 \beta_{7} + \cdots + 8344733743 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 5655838 \beta_{11} - 27359160 \beta_{10} + 44958123 \beta_{9} - 20106570 \beta_{8} + \cdots + 13853446806 \)
|
| \(\nu^{10}\) | \(=\) |
\( 90649774 \beta_{11} - 43816941 \beta_{10} + 1513662588 \beta_{9} - 40245629 \beta_{8} + \cdots + 3440473233107 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 3158612072 \beta_{11} - 14364529862 \beta_{10} + 24200442898 \beta_{9} - 10371010490 \beta_{8} + \cdots + 7421113053018 \)
|
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 |
|
−22.2614 | 0 | 367.568 | −366.768 | 0 | −1556.74 | −5333.12 | 0 | 8164.76 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −19.4181 | 0 | 249.062 | −164.146 | 0 | 1184.75 | −2350.80 | 0 | 3187.40 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −11.7226 | 0 | 9.42047 | −342.584 | 0 | 1083.50 | 1390.07 | 0 | 4015.99 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −11.6449 | 0 | 7.60291 | 231.149 | 0 | −1154.85 | 1402.01 | 0 | −2691.70 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −9.40410 | 0 | −39.5629 | 482.766 | 0 | 1331.25 | 1575.78 | 0 | −4539.98 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −3.30114 | 0 | −117.102 | −515.245 | 0 | −725.902 | 809.117 | 0 | 1700.90 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.595405 | 0 | −127.645 | 80.2365 | 0 | −870.573 | 152.213 | 0 | −47.7732 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 4.26584 | 0 | −109.803 | 101.908 | 0 | 827.161 | −1014.43 | 0 | 434.722 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 8.27904 | 0 | −59.4574 | −230.336 | 0 | 754.718 | −1551.97 | 0 | −1906.96 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 14.1737 | 0 | 72.8940 | 214.518 | 0 | −935.728 | −781.057 | 0 | 3040.51 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 15.7762 | 0 | 120.887 | 308.253 | 0 | 70.1815 | −112.213 | 0 | 4863.05 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 19.8529 | 0 | 266.136 | −299.751 | 0 | −235.777 | 2742.40 | 0 | −5950.92 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(23\) | \(-1\) |
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 | 207.8.a.g | ✓ | 12 |
| 3.b | odd | 2 | 1 | 207.8.a.h | yes | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 207.8.a.g | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 207.8.a.h | yes | 12 | 3.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} + 16 T_{2}^{11} - 960 T_{2}^{10} - 14336 T_{2}^{9} + 319655 T_{2}^{8} + 4472332 T_{2}^{7} + \cdots - 171002934784 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(207))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + \cdots - 171002934784 \)
|
| $3$ |
\( T^{12} \)
|
| $5$ |
\( T^{12} + \cdots + 44\!\cdots\!00 \)
|
| $7$ |
\( T^{12} + \cdots + 18\!\cdots\!92 \)
|
| $11$ |
\( T^{12} + \cdots + 16\!\cdots\!92 \)
|
| $13$ |
\( T^{12} + \cdots + 25\!\cdots\!00 \)
|
| $17$ |
\( T^{12} + \cdots + 17\!\cdots\!96 \)
|
| $19$ |
\( T^{12} + \cdots + 48\!\cdots\!96 \)
|
| $23$ |
\( (T - 12167)^{12} \)
|
| $29$ |
\( T^{12} + \cdots + 18\!\cdots\!08 \)
|
| $31$ |
\( T^{12} + \cdots + 64\!\cdots\!12 \)
|
| $37$ |
\( T^{12} + \cdots + 15\!\cdots\!04 \)
|
| $41$ |
\( T^{12} + \cdots + 66\!\cdots\!36 \)
|
| $43$ |
\( T^{12} + \cdots + 77\!\cdots\!88 \)
|
| $47$ |
\( T^{12} + \cdots + 41\!\cdots\!84 \)
|
| $53$ |
\( T^{12} + \cdots - 15\!\cdots\!36 \)
|
| $59$ |
\( T^{12} + \cdots - 74\!\cdots\!48 \)
|
| $61$ |
\( T^{12} + \cdots - 17\!\cdots\!84 \)
|
| $67$ |
\( T^{12} + \cdots + 13\!\cdots\!68 \)
|
| $71$ |
\( T^{12} + \cdots - 20\!\cdots\!88 \)
|
| $73$ |
\( T^{12} + \cdots - 22\!\cdots\!56 \)
|
| $79$ |
\( T^{12} + \cdots + 47\!\cdots\!64 \)
|
| $83$ |
\( T^{12} + \cdots - 53\!\cdots\!48 \)
|
| $89$ |
\( T^{12} + \cdots - 55\!\cdots\!04 \)
|
| $97$ |
\( T^{12} + \cdots - 11\!\cdots\!00 \)
|
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