Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [272,10,Mod(1,272)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(272, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("272.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 272 = 2^{4} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 272.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: | \(140.089747437\) |
| 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: |
\( x^{9} - 4 x^{8} - 31396 x^{7} + 586357 x^{6} + 313675917 x^{5} - 9585814506 x^{4} + \cdots - 25\!\cdots\!80 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 2^{30}\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 136) |
| 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
\( x^{9} - 4 x^{8} - 31396 x^{7} + 586357 x^{6} + 313675917 x^{5} - 9585814506 x^{4} + \cdots - 25\!\cdots\!80 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 26\!\cdots\!96 \nu^{8} + \cdots - 13\!\cdots\!75 ) / 16\!\cdots\!05 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 98\!\cdots\!28 \nu^{8} + \cdots + 61\!\cdots\!80 ) / 10\!\cdots\!85 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 35\!\cdots\!76 \nu^{8} + \cdots + 12\!\cdots\!68 ) / 32\!\cdots\!61 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 44\!\cdots\!92 \nu^{8} + \cdots - 41\!\cdots\!45 ) / 18\!\cdots\!45 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 58\!\cdots\!08 \nu^{8} + \cdots + 25\!\cdots\!80 ) / 18\!\cdots\!45 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 57\!\cdots\!48 \nu^{8} + \cdots + 42\!\cdots\!05 ) / 16\!\cdots\!05 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 34\!\cdots\!56 \nu^{8} + \cdots - 17\!\cdots\!03 ) / 32\!\cdots\!61 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{4} - \beta_{3} + \beta_{2} - 45\beta _1 + 27910 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 46 \beta_{8} + 10 \beta_{7} - 35 \beta_{6} - 9 \beta_{5} - 80 \beta_{4} - 201 \beta_{3} + \cdots - 1223353 ) / 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 1515 \beta_{8} - 5133 \beta_{7} - 429 \beta_{6} - 4764 \beta_{5} + 33949 \beta_{4} - 14422 \beta_{3} + \cdots + 630308591 ) / 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 1812214 \beta_{8} + 1120528 \beta_{7} - 1311779 \beta_{6} + 528576 \beta_{5} - 4354784 \beta_{4} + \cdots - 54452253006 ) / 16 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 101538684 \beta_{8} - 270118629 \beta_{7} + 16612623 \beta_{6} - 224554257 \beta_{5} + \cdots + 17141610389254 ) / 16 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 64040099322 \beta_{8} + 56450686638 \beta_{7} - 45946485987 \beta_{6} + 39098246256 \beta_{5} + \cdots - 21\!\cdots\!10 ) / 32 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 2386422524916 \beta_{8} - 5488353263214 \beta_{7} + 916063657857 \beta_{6} - 4332213508968 \beta_{5} + \cdots + 25\!\cdots\!28 ) / 16 \)
|
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 | −234.849 | 0 | −1564.00 | 0 | −3566.46 | 0 | 35470.8 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −153.169 | 0 | −449.651 | 0 | −606.310 | 0 | 3777.82 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −139.924 | 0 | 1717.33 | 0 | 7962.14 | 0 | −104.160 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −14.5389 | 0 | −1033.54 | 0 | 2595.28 | 0 | −19471.6 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 103.431 | 0 | 927.747 | 0 | −8428.53 | 0 | −8985.10 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 126.820 | 0 | 591.985 | 0 | 1824.54 | 0 | −3599.62 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 147.074 | 0 | −2641.35 | 0 | −5808.64 | 0 | 1947.69 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 222.730 | 0 | −1730.54 | 0 | 11991.0 | 0 | 29925.8 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 256.426 | 0 | 2756.02 | 0 | 4790.93 | 0 | 46071.4 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(17\) | \(-1\) |
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 | 272.10.a.k | 9 | |
| 4.b | odd | 2 | 1 | 136.10.a.b | ✓ | 9 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 136.10.a.b | ✓ | 9 | 4.b | odd | 2 | 1 | |
| 272.10.a.k | 9 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{9} - 314 T_{3}^{8} - 81792 T_{3}^{7} + 31019368 T_{3}^{6} + 1199189392 T_{3}^{5} + \cdots - 80\!\cdots\!40 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(272))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} \)
|
| $3$ |
\( T^{9} + \cdots - 80\!\cdots\!40 \)
|
| $5$ |
\( T^{9} + \cdots + 86\!\cdots\!00 \)
|
| $7$ |
\( T^{9} + \cdots - 22\!\cdots\!40 \)
|
| $11$ |
\( T^{9} + \cdots - 97\!\cdots\!20 \)
|
| $13$ |
\( T^{9} + \cdots - 31\!\cdots\!00 \)
|
| $17$ |
\( (T - 83521)^{9} \)
|
| $19$ |
\( T^{9} + \cdots + 65\!\cdots\!00 \)
|
| $23$ |
\( T^{9} + \cdots + 10\!\cdots\!04 \)
|
| $29$ |
\( T^{9} + \cdots + 27\!\cdots\!00 \)
|
| $31$ |
\( T^{9} + \cdots - 11\!\cdots\!28 \)
|
| $37$ |
\( T^{9} + \cdots - 64\!\cdots\!04 \)
|
| $41$ |
\( T^{9} + \cdots - 25\!\cdots\!32 \)
|
| $43$ |
\( T^{9} + \cdots + 15\!\cdots\!00 \)
|
| $47$ |
\( T^{9} + \cdots + 19\!\cdots\!00 \)
|
| $53$ |
\( T^{9} + \cdots - 20\!\cdots\!00 \)
|
| $59$ |
\( T^{9} + \cdots + 11\!\cdots\!40 \)
|
| $61$ |
\( T^{9} + \cdots - 86\!\cdots\!00 \)
|
| $67$ |
\( T^{9} + \cdots - 14\!\cdots\!80 \)
|
| $71$ |
\( T^{9} + \cdots - 24\!\cdots\!36 \)
|
| $73$ |
\( T^{9} + \cdots - 51\!\cdots\!08 \)
|
| $79$ |
\( T^{9} + \cdots + 20\!\cdots\!00 \)
|
| $83$ |
\( T^{9} + \cdots + 12\!\cdots\!08 \)
|
| $89$ |
\( T^{9} + \cdots + 60\!\cdots\!00 \)
|
| $97$ |
\( T^{9} + \cdots - 20\!\cdots\!32 \)
|
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