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: | \(1\) |
| 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: |
\( x^{7} - x^{6} - 2986x^{5} + 8252x^{4} + 2252056x^{3} - 10388768x^{2} - 243559296x - 675998208 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{19}\cdot 3^{4} \) |
| Twist minimal: | no (minimal twist has level 17) |
| 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
\( x^{7} - x^{6} - 2986x^{5} + 8252x^{4} + 2252056x^{3} - 10388768x^{2} - 243559296x - 675998208 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 41053 \nu^{6} + 292545 \nu^{5} + 124232878 \nu^{4} - 1085670148 \nu^{3} + \cdots + 7076106009216 ) / 6260544000 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 17729 \nu^{6} - 178065 \nu^{5} + 51179204 \nu^{4} + 202357936 \nu^{3} - 36808140928 \nu^{2} + \cdots + 2974683182688 ) / 1956420000 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 332131 \nu^{6} - 7237215 \nu^{5} - 1044625906 \nu^{4} + 19522779196 \nu^{3} + \cdots - 58233632015232 ) / 31302720000 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 843359 \nu^{6} - 2902635 \nu^{5} - 2465651834 \nu^{4} + 12878937644 \nu^{3} + \cdots - 113709524082048 ) / 31302720000 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 420671 \nu^{6} - 3690315 \nu^{5} - 1212277946 \nu^{4} + 12473409836 \nu^{3} + \cdots - 75110133002112 ) / 10434240000 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 875471 \nu^{6} - 4842315 \nu^{5} - 2538432746 \nu^{4} + 19805426636 \nu^{3} + \cdots - 100813095107712 ) / 15651360000 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} - 3\beta _1 + 10 ) / 64 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -16\beta_{6} + 16\beta_{5} - 9\beta_{3} - 7\beta_{2} - 43\beta _1 + 54618 ) / 64 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 304\beta_{6} - 48\beta_{5} - 832\beta_{4} - 1479\beta_{3} + 1143\beta_{2} - 5093\beta _1 - 142586 ) / 64 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -22192\beta_{6} + 32944\beta_{5} - 2240\beta_{4} - 14133\beta_{3} - 14075\beta_{2} + 625\beta _1 + 80467666 ) / 64 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 660720 \beta_{6} - 183536 \beta_{5} - 1816384 \beta_{4} - 2606055 \beta_{3} + 1493079 \beta_{2} + \cdots - 343420858 ) / 64 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 34010544 \beta_{6} + 63178160 \beta_{5} + 2280512 \beta_{4} - 24830397 \beta_{3} + \cdots + 131574874914 ) / 64 \)
|
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 | −243.971 | 0 | 1776.79 | 0 | 9598.61 | 0 | 39838.7 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −169.801 | 0 | 195.287 | 0 | 356.628 | 0 | 9149.54 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −109.740 | 0 | −2498.37 | 0 | −2872.61 | 0 | −7640.20 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −106.475 | 0 | 1303.94 | 0 | −9199.27 | 0 | −8346.12 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 116.887 | 0 | −1103.40 | 0 | 5164.29 | 0 | −6020.47 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 171.025 | 0 | 1536.21 | 0 | −3027.69 | 0 | 9566.70 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 254.074 | 0 | 151.544 | 0 | −9407.97 | 0 | 44870.8 | 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.g | 7 | |
| 4.b | odd | 2 | 1 | 17.10.a.b | ✓ | 7 | |
| 12.b | even | 2 | 1 | 153.10.a.f | 7 | ||
| 68.d | odd | 2 | 1 | 289.10.a.b | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 17.10.a.b | ✓ | 7 | 4.b | odd | 2 | 1 | |
| 153.10.a.f | 7 | 12.b | even | 2 | 1 | ||
| 272.10.a.g | 7 | 1.a | even | 1 | 1 | trivial | |
| 289.10.a.b | 7 | 68.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{7} + 88 T_{3}^{6} - 105728 T_{3}^{5} - 9882840 T_{3}^{4} + 3088987488 T_{3}^{3} + \cdots - 24\!\cdots\!00 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(272))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} \)
|
| $3$ |
\( T^{7} + \cdots - 24\!\cdots\!00 \)
|
| $5$ |
\( T^{7} + \cdots - 29\!\cdots\!00 \)
|
| $7$ |
\( T^{7} + \cdots - 13\!\cdots\!04 \)
|
| $11$ |
\( T^{7} + \cdots + 12\!\cdots\!00 \)
|
| $13$ |
\( T^{7} + \cdots + 51\!\cdots\!32 \)
|
| $17$ |
\( (T - 83521)^{7} \)
|
| $19$ |
\( T^{7} + \cdots - 35\!\cdots\!00 \)
|
| $23$ |
\( T^{7} + \cdots - 31\!\cdots\!72 \)
|
| $29$ |
\( T^{7} + \cdots + 34\!\cdots\!60 \)
|
| $31$ |
\( T^{7} + \cdots - 15\!\cdots\!00 \)
|
| $37$ |
\( T^{7} + \cdots + 26\!\cdots\!04 \)
|
| $41$ |
\( T^{7} + \cdots + 11\!\cdots\!40 \)
|
| $43$ |
\( T^{7} + \cdots - 32\!\cdots\!64 \)
|
| $47$ |
\( T^{7} + \cdots + 18\!\cdots\!00 \)
|
| $53$ |
\( T^{7} + \cdots - 68\!\cdots\!00 \)
|
| $59$ |
\( T^{7} + \cdots - 53\!\cdots\!60 \)
|
| $61$ |
\( T^{7} + \cdots - 16\!\cdots\!00 \)
|
| $67$ |
\( T^{7} + \cdots - 18\!\cdots\!16 \)
|
| $71$ |
\( T^{7} + \cdots + 21\!\cdots\!16 \)
|
| $73$ |
\( T^{7} + \cdots + 86\!\cdots\!04 \)
|
| $79$ |
\( T^{7} + \cdots + 18\!\cdots\!00 \)
|
| $83$ |
\( T^{7} + \cdots + 61\!\cdots\!12 \)
|
| $89$ |
\( T^{7} + \cdots - 45\!\cdots\!00 \)
|
| $97$ |
\( T^{7} + \cdots + 15\!\cdots\!00 \)
|
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