Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [152,8,Mod(1,152)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(152, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("152.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 152 = 2^{3} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 152.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: | \(47.4825238736\) |
| 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: |
\( x^{9} - 2 x^{8} - 13922 x^{7} - 25112 x^{6} + 57411673 x^{5} + 379057666 x^{4} - 62486804160 x^{3} + \cdots + 69542466153984 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{19}\cdot 3\cdot 5 \) |
| 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
\( x^{9} - 2 x^{8} - 13922 x^{7} - 25112 x^{6} + 57411673 x^{5} + 379057666 x^{4} - 62486804160 x^{3} + \cdots + 69542466153984 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 86369939727737 \nu^{8} + \cdots + 12\!\cdots\!44 ) / 12\!\cdots\!96 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 14\!\cdots\!55 \nu^{8} + \cdots + 20\!\cdots\!12 ) / 19\!\cdots\!36 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 67\!\cdots\!97 \nu^{8} + \cdots + 58\!\cdots\!68 ) / 29\!\cdots\!04 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 424739328970720 \nu^{8} + \cdots - 47\!\cdots\!00 ) / 18\!\cdots\!44 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 55\!\cdots\!31 \nu^{8} + \cdots - 87\!\cdots\!08 ) / 19\!\cdots\!36 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 772928886362595 \nu^{8} + \cdots - 75\!\cdots\!56 ) / 12\!\cdots\!96 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 84\!\cdots\!43 \nu^{8} + \cdots - 41\!\cdots\!64 ) / 48\!\cdots\!84 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + \beta_{3} + 3\beta_{2} + 5\beta _1 + 3094 \)
|
| \(\nu^{3}\) | \(=\) |
\( -3\beta_{8} - 6\beta_{7} - 3\beta_{6} + 14\beta_{5} - 22\beta_{4} + 5\beta_{3} - 170\beta_{2} + 5737\beta _1 + 16306 \)
|
| \(\nu^{4}\) | \(=\) |
\( 246 \beta_{8} - 870 \beta_{7} + 6745 \beta_{6} + 218 \beta_{5} + 164 \beta_{4} + 7065 \beta_{3} + \cdots + 17599490 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 30519 \beta_{8} - 22602 \beta_{7} - 18885 \beta_{6} + 135952 \beta_{5} - 156332 \beta_{4} + \cdots + 126800870 \)
|
| \(\nu^{6}\) | \(=\) |
\( 3257124 \beta_{8} - 8998014 \beta_{7} + 43235341 \beta_{6} + 2992102 \beta_{5} + 938212 \beta_{4} + \cdots + 109685571738 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 223599417 \beta_{8} - 15333714 \beta_{7} - 157559025 \beta_{6} + 1091696124 \beta_{5} + \cdots + 937381890942 \)
|
| \(\nu^{8}\) | \(=\) |
\( 31102975458 \beta_{8} - 74462823846 \beta_{7} + 277014842209 \beta_{6} + 28990339890 \beta_{5} + \cdots + 702834772848850 \)
|
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 |
|
0 | −76.7282 | 0 | 510.168 | 0 | −402.237 | 0 | 3700.22 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −64.2165 | 0 | −1.14422 | 0 | 1813.97 | 0 | 1936.76 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −24.7875 | 0 | 150.360 | 0 | −1661.39 | 0 | −1572.58 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −12.9403 | 0 | −448.356 | 0 | −610.684 | 0 | −2019.55 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −1.53304 | 0 | −60.7705 | 0 | 1098.49 | 0 | −2184.65 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 16.1837 | 0 | −161.818 | 0 | −301.718 | 0 | −1925.09 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 47.7024 | 0 | 432.335 | 0 | 207.442 | 0 | 88.5147 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 83.2992 | 0 | −427.913 | 0 | −991.087 | 0 | 4751.76 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 89.0202 | 0 | 89.1384 | 0 | 1311.21 | 0 | 5737.60 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(19\) | \( -1 \) |
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 | 152.8.a.d | ✓ | 9 |
| 4.b | odd | 2 | 1 | 304.8.a.m | 9 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 152.8.a.d | ✓ | 9 | 1.a | even | 1 | 1 | trivial |
| 304.8.a.m | 9 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{9} - 56 T_{3}^{8} - 12530 T_{3}^{7} + 539452 T_{3}^{6} + 47978161 T_{3}^{5} + \cdots + 13870109203776 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(152))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} \)
|
| $3$ |
\( T^{9} + \cdots + 13870109203776 \)
|
| $5$ |
\( T^{9} + \cdots + 63\!\cdots\!00 \)
|
| $7$ |
\( T^{9} + \cdots + 66\!\cdots\!00 \)
|
| $11$ |
\( T^{9} + \cdots + 58\!\cdots\!00 \)
|
| $13$ |
\( T^{9} + \cdots + 12\!\cdots\!12 \)
|
| $17$ |
\( T^{9} + \cdots - 77\!\cdots\!30 \)
|
| $19$ |
\( (T - 6859)^{9} \)
|
| $23$ |
\( T^{9} + \cdots + 75\!\cdots\!84 \)
|
| $29$ |
\( T^{9} + \cdots + 10\!\cdots\!12 \)
|
| $31$ |
\( T^{9} + \cdots + 48\!\cdots\!56 \)
|
| $37$ |
\( T^{9} + \cdots + 72\!\cdots\!12 \)
|
| $41$ |
\( T^{9} + \cdots - 54\!\cdots\!84 \)
|
| $43$ |
\( T^{9} + \cdots + 11\!\cdots\!52 \)
|
| $47$ |
\( T^{9} + \cdots - 11\!\cdots\!08 \)
|
| $53$ |
\( T^{9} + \cdots - 72\!\cdots\!24 \)
|
| $59$ |
\( T^{9} + \cdots + 71\!\cdots\!44 \)
|
| $61$ |
\( T^{9} + \cdots + 18\!\cdots\!00 \)
|
| $67$ |
\( T^{9} + \cdots + 50\!\cdots\!68 \)
|
| $71$ |
\( T^{9} + \cdots - 10\!\cdots\!20 \)
|
| $73$ |
\( T^{9} + \cdots - 30\!\cdots\!34 \)
|
| $79$ |
\( T^{9} + \cdots - 61\!\cdots\!60 \)
|
| $83$ |
\( T^{9} + \cdots + 29\!\cdots\!40 \)
|
| $89$ |
\( T^{9} + \cdots - 37\!\cdots\!72 \)
|
| $97$ |
\( T^{9} + \cdots + 39\!\cdots\!44 \)
|
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