Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [324,8,Mod(1,324)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(324, 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("324.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 324 = 2^{2} \cdot 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 324.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: | \(101.212748257\) |
| Analytic rank: | \(1\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 3x^{6} - 1289x^{5} + 4994x^{4} + 496633x^{3} - 2291461x^{2} - 56851263x + 373225328 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{10}\cdot 3^{15} \) |
| Twist minimal: | no (minimal twist has level 36) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 3x^{6} - 1289x^{5} + 4994x^{4} + 496633x^{3} - 2291461x^{2} - 56851263x + 373225328 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 12407 \nu^{6} - 375026 \nu^{5} + 16746767 \nu^{4} + 358758001 \nu^{3} - 7371057328 \nu^{2} + \cdots + 960757498036 ) / 228604194 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 11395 \nu^{6} - 406034 \nu^{5} - 9274075 \nu^{4} + 421767655 \nu^{3} - 157810180 \nu^{2} + \cdots + 598479021772 ) / 32657742 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 243601 \nu^{6} - 7082210 \nu^{5} - 196374937 \nu^{4} + 7395255553 \nu^{3} + \cdots + 11442148060819 ) / 114302097 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 524141 \nu^{6} - 15422698 \nu^{5} - 418533653 \nu^{4} + 16267932821 \nu^{3} + \cdots + 22521727355330 ) / 228604194 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 263 \nu^{6} + 3634 \nu^{5} + 267503 \nu^{4} - 3751475 \nu^{3} - 55638172 \nu^{2} + \cdots - 1270904432 ) / 72366 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 263 \nu^{6} + 3634 \nu^{5} + 267503 \nu^{4} - 3751475 \nu^{3} - 55638172 \nu^{2} + \cdots - 1265838812 ) / 72366 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{6} + \beta_{5} + 70 ) / 162 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{6} - 4\beta_{5} + 9\beta_{4} - 9\beta_{3} - 18\beta_{2} + 45\beta _1 + 119752 ) / 324 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -1925\beta_{6} + 2103\beta_{5} + 45\beta_{4} - 180\beta_{3} + 1962\beta_{2} - 4473\beta _1 - 310189 ) / 648 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 2957\beta_{6} - 6064\beta_{5} + 15210\beta_{4} - 10431\beta_{3} - 54801\beta_{2} + 88686\beta _1 + 122501515 ) / 648 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 1077285 \beta_{6} + 1267115 \beta_{5} + 2655 \beta_{4} - 98910 \beta_{3} + 1822266 \beta_{2} + \cdots - 306287181 ) / 648 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 349980 \beta_{6} - 617573 \beta_{5} + 1228587 \beta_{4} - 622311 \beta_{3} - 5658951 \beta_{2} + \cdots + 7978160406 ) / 72 \)
|
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 | 0 | 0 | −510.715 | 0 | −753.430 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | −309.722 | 0 | −84.7846 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | −301.945 | 0 | 1477.45 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0 | 0 | 114.920 | 0 | −1474.10 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0 | 0 | 200.155 | 0 | −288.547 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 0 | 0 | 223.048 | 0 | 1036.23 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 0 | 0 | 263.260 | 0 | 170.188 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(1\) |
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 | 324.8.a.c | 7 | |
| 3.b | odd | 2 | 1 | 324.8.a.d | 7 | ||
| 9.c | even | 3 | 2 | 36.8.e.a | ✓ | 14 | |
| 9.d | odd | 6 | 2 | 108.8.e.a | 14 | ||
| 36.f | odd | 6 | 2 | 144.8.i.d | 14 | ||
| 36.h | even | 6 | 2 | 432.8.i.d | 14 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 36.8.e.a | ✓ | 14 | 9.c | even | 3 | 2 | |
| 108.8.e.a | 14 | 9.d | odd | 6 | 2 | ||
| 144.8.i.d | 14 | 36.f | odd | 6 | 2 | ||
| 324.8.a.c | 7 | 1.a | even | 1 | 1 | trivial | |
| 324.8.a.d | 7 | 3.b | odd | 2 | 1 | ||
| 432.8.i.d | 14 | 36.h | even | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{7} + 321 T_{5}^{6} - 258606 T_{5}^{5} - 43514334 T_{5}^{4} + 25120770165 T_{5}^{3} + \cdots + 64\!\cdots\!00 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(324))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} \)
|
| $3$ |
\( T^{7} \)
|
| $5$ |
\( T^{7} + \cdots + 64\!\cdots\!00 \)
|
| $7$ |
\( T^{7} + \cdots - 70\!\cdots\!68 \)
|
| $11$ |
\( T^{7} + \cdots + 70\!\cdots\!67 \)
|
| $13$ |
\( T^{7} + \cdots + 59\!\cdots\!72 \)
|
| $17$ |
\( T^{7} + \cdots - 99\!\cdots\!88 \)
|
| $19$ |
\( T^{7} + \cdots + 39\!\cdots\!84 \)
|
| $23$ |
\( T^{7} + \cdots + 46\!\cdots\!24 \)
|
| $29$ |
\( T^{7} + \cdots - 70\!\cdots\!48 \)
|
| $31$ |
\( T^{7} + \cdots - 64\!\cdots\!80 \)
|
| $37$ |
\( T^{7} + \cdots + 54\!\cdots\!96 \)
|
| $41$ |
\( T^{7} + \cdots + 30\!\cdots\!53 \)
|
| $43$ |
\( T^{7} + \cdots + 13\!\cdots\!87 \)
|
| $47$ |
\( T^{7} + \cdots - 98\!\cdots\!72 \)
|
| $53$ |
\( T^{7} + \cdots - 25\!\cdots\!32 \)
|
| $59$ |
\( T^{7} + \cdots + 20\!\cdots\!33 \)
|
| $61$ |
\( T^{7} + \cdots - 11\!\cdots\!92 \)
|
| $67$ |
\( T^{7} + \cdots - 98\!\cdots\!51 \)
|
| $71$ |
\( T^{7} + \cdots + 28\!\cdots\!88 \)
|
| $73$ |
\( T^{7} + \cdots - 64\!\cdots\!40 \)
|
| $79$ |
\( T^{7} + \cdots - 46\!\cdots\!56 \)
|
| $83$ |
\( T^{7} + \cdots + 35\!\cdots\!72 \)
|
| $89$ |
\( T^{7} + \cdots + 43\!\cdots\!04 \)
|
| $97$ |
\( T^{7} + \cdots - 17\!\cdots\!87 \)
|
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