Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [232,6,Mod(1,232)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(232, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("232.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 232 = 2^{3} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 232.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: | \(37.2090461966\) |
| 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: |
\( x^{8} - 2 x^{7} - 1016 x^{6} + 6608 x^{5} + 330138 x^{4} - 3478200 x^{3} - 28951517 x^{2} + \cdots - 1682520084 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{13} \) |
| 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
\( x^{8} - 2 x^{7} - 1016 x^{6} + 6608 x^{5} + 330138 x^{4} - 3478200 x^{3} - 28951517 x^{2} + \cdots - 1682520084 \)
:
| \(\beta_{1}\) | \(=\) |
\( 8\nu - 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 35629262 \nu^{7} - 325456665 \nu^{6} + 43025974217 \nu^{5} + 237639218723 \nu^{4} + \cdots - 93\!\cdots\!54 ) / 6888751366920 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 150019919 \nu^{7} + 1055313725 \nu^{6} - 145520693569 \nu^{5} - 399262822211 \nu^{4} + \cdots + 24\!\cdots\!68 ) / 6888751366920 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 1615127666 \nu^{7} - 18452132895 \nu^{6} + 1377336017591 \nu^{5} + 7726662739949 \nu^{4} + \cdots - 19\!\cdots\!02 ) / 6888751366920 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 1871830237 \nu^{7} - 19387286010 \nu^{6} + 1623842930542 \nu^{5} + 7853791412338 \nu^{4} + \cdots - 23\!\cdots\!14 ) / 6888751366920 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 2005185173 \nu^{7} + 24263418000 \nu^{6} - 1706571499268 \nu^{5} - 10486563979892 \nu^{4} + \cdots + 25\!\cdots\!46 ) / 6888751366920 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 504591037 \nu^{7} - 5126806583 \nu^{6} + 450393735795 \nu^{5} + 2111199485305 \nu^{4} + \cdots - 71\!\cdots\!16 ) / 1377750273384 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 2 ) / 8 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{7} + 2\beta_{5} - 4\beta_{4} + 16\beta_{3} + 2\beta_{2} - 5\beta _1 + 2030 ) / 8 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 22\beta_{7} + 52\beta_{6} - 17\beta_{5} + 64\beta_{4} + 173\beta_{3} + 89\beta_{2} + 350\beta _1 - 13750 ) / 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 156\beta_{7} - 948\beta_{6} + 1300\beta_{5} - 2672\beta_{4} + 2748\beta_{3} - 3749\beta _1 + 705786 ) / 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 14166 \beta_{7} + 40012 \beta_{6} - 20256 \beta_{5} + 58700 \beta_{4} + 96942 \beta_{3} + 60128 \beta_{2} + \cdots - 9257674 ) / 8 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 213582 \beta_{7} - 874396 \beta_{6} + 769413 \beta_{5} - 1690696 \beta_{4} - 667161 \beta_{3} + \cdots + 299065370 ) / 8 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 9524100 \beta_{7} + 26084520 \beta_{6} - 15469226 \beta_{5} + 40021872 \beta_{4} + 58274874 \beta_{3} + \cdots - 5334790362 ) / 8 \)
|
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 | −26.6968 | 0 | −9.98967 | 0 | 171.676 | 0 | 469.721 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −18.1002 | 0 | 62.9395 | 0 | 52.4741 | 0 | 84.6162 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −11.3901 | 0 | −55.7643 | 0 | −94.0045 | 0 | −113.266 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −4.52694 | 0 | 5.12253 | 0 | −216.995 | 0 | −222.507 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 3.99700 | 0 | 45.5331 | 0 | 188.457 | 0 | −227.024 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 8.55577 | 0 | −101.468 | 0 | 7.42922 | 0 | −169.799 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 22.2250 | 0 | 82.1724 | 0 | −53.0842 | 0 | 250.951 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 28.9363 | 0 | −3.54531 | 0 | 226.046 | 0 | 594.307 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(29\) | \(1\) |
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 | 232.6.a.c | ✓ | 8 |
| 4.b | odd | 2 | 1 | 464.6.a.l | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 232.6.a.c | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 464.6.a.l | 8 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} - 3 T_{3}^{7} - 1301 T_{3}^{6} + 789 T_{3}^{5} + 459853 T_{3}^{4} + 688095 T_{3}^{3} + \cdots + 547964838 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(232))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} - 3 T^{7} + \cdots + 547964838 \)
|
| $5$ |
\( T^{8} + \cdots + 241742315422 \)
|
| $7$ |
\( T^{8} + \cdots - 30\!\cdots\!00 \)
|
| $11$ |
\( T^{8} + \cdots + 22\!\cdots\!10 \)
|
| $13$ |
\( T^{8} + \cdots + 25\!\cdots\!10 \)
|
| $17$ |
\( T^{8} + \cdots - 59\!\cdots\!20 \)
|
| $19$ |
\( T^{8} + \cdots - 18\!\cdots\!00 \)
|
| $23$ |
\( T^{8} + \cdots + 13\!\cdots\!00 \)
|
| $29$ |
\( (T + 841)^{8} \)
|
| $31$ |
\( T^{8} + \cdots + 29\!\cdots\!10 \)
|
| $37$ |
\( T^{8} + \cdots - 86\!\cdots\!36 \)
|
| $41$ |
\( T^{8} + \cdots + 77\!\cdots\!68 \)
|
| $43$ |
\( T^{8} + \cdots - 48\!\cdots\!66 \)
|
| $47$ |
\( T^{8} + \cdots + 15\!\cdots\!10 \)
|
| $53$ |
\( T^{8} + \cdots - 12\!\cdots\!54 \)
|
| $59$ |
\( T^{8} + \cdots + 32\!\cdots\!84 \)
|
| $61$ |
\( T^{8} + \cdots + 94\!\cdots\!88 \)
|
| $67$ |
\( T^{8} + \cdots + 45\!\cdots\!00 \)
|
| $71$ |
\( T^{8} + \cdots + 34\!\cdots\!08 \)
|
| $73$ |
\( T^{8} + \cdots - 48\!\cdots\!12 \)
|
| $79$ |
\( T^{8} + \cdots - 13\!\cdots\!50 \)
|
| $83$ |
\( T^{8} + \cdots + 24\!\cdots\!28 \)
|
| $89$ |
\( T^{8} + \cdots + 33\!\cdots\!00 \)
|
| $97$ |
\( T^{8} + \cdots + 32\!\cdots\!12 \)
|
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