Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [11,14,Mod(1,11)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(11, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 14, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("11.1");
S:= CuspForms(chi, 14);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 11 \) |
| Weight: | \( k \) | \(=\) | \( 14 \) |
| Character orbit: | \([\chi]\) | \(=\) | 11.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: | \(11.7954021847\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{7} - 52218x^{5} - 193468x^{4} + 738251616x^{3} + 3692170944x^{2} - 1726202485760x - 15935676776448 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{8}\cdot 3^{4} \) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 52218x^{5} - 193468x^{4} + 738251616x^{3} + 3692170944x^{2} - 1726202485760x - 15935676776448 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 12973 \nu^{6} - 4604688 \nu^{5} + 341302670 \nu^{4} + 105001617556 \nu^{3} + \cdots + 36\!\cdots\!12 ) / 17081555899392 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 12973 \nu^{6} - 4604688 \nu^{5} + 341302670 \nu^{4} + 105001617556 \nu^{3} + \cdots - 21\!\cdots\!36 ) / 17081555899392 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 616001 \nu^{6} + 19091760 \nu^{5} - 29805187322 \nu^{4} - 1002145432348 \nu^{3} + \cdots - 46\!\cdots\!64 ) / 29892722823936 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 4022421 \nu^{6} + 77937392 \nu^{5} - 185580694786 \nu^{4} - 384785503692 \nu^{3} + \cdots - 27\!\cdots\!64 ) / 119570891295744 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 46155143 \nu^{6} + 5099097264 \nu^{5} + 1735968927062 \nu^{4} - 148010624945852 \nu^{3} + \cdots + 27\!\cdots\!48 ) / 597854456478720 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - \beta_{2} + 5\beta _1 + 14919 \)
|
| \(\nu^{3}\) | \(=\) |
\( 24\beta_{5} - 38\beta_{4} + 4\beta_{3} - 36\beta_{2} + 23934\beta _1 + 82910 \)
|
| \(\nu^{4}\) | \(=\) |
\( 780\beta_{6} + 1836\beta_{5} - 1056\beta_{4} + 28458\beta_{3} - 1842\beta_{2} + 300362\beta _1 + 357196746 \)
|
| \(\nu^{5}\) | \(=\) |
\( 150960 \beta_{6} + 959832 \beta_{5} - 1035756 \beta_{4} + 525524 \beta_{3} + 408004 \beta_{2} + \cdots + 4578871032 \)
|
| \(\nu^{6}\) | \(=\) |
\( 33061560 \beta_{6} + 98131224 \beta_{5} - 32286760 \beta_{4} + 770964884 \beta_{3} + \cdots + 9133734054700 \)
|
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 |
|
−173.552 | −20.9570 | 21928.2 | −56340.0 | 3637.13 | −263430. | −2.38394e6 | −1.59388e6 | 9.77790e6 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −141.702 | 1955.79 | 11887.3 | 69271.0 | −277138. | −238258. | −523634. | 2.23078e6 | −9.81580e6 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −56.0982 | −162.233 | −5044.99 | −6342.85 | 9100.97 | −258834. | 742571. | −1.56800e6 | 355823. | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 9.39625 | −1938.71 | −8103.71 | −30409.3 | −18216.6 | 120450. | −153119. | 2.16426e6 | −285733. | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 50.8028 | 1357.75 | −5611.08 | 20995.6 | 68977.3 | 552698. | −701235. | 249153. | 1.06663e6 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 153.093 | 1416.41 | 15245.5 | 8986.14 | 216843. | −143680. | 1.07985e6 | 411906. | 1.37572e6 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 158.059 | −2229.05 | 16790.7 | 46528.4 | −352322. | 141668. | 1.35911e6 | 3.37435e6 | 7.35425e6 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(11\) | \(-1\) |
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 | 11.14.a.b | ✓ | 7 |
| 3.b | odd | 2 | 1 | 99.14.a.g | 7 | ||
| 4.b | odd | 2 | 1 | 176.14.a.h | 7 | ||
| 11.b | odd | 2 | 1 | 121.14.a.c | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 11.14.a.b | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 99.14.a.g | 7 | 3.b | odd | 2 | 1 | ||
| 121.14.a.c | 7 | 11.b | odd | 2 | 1 | ||
| 176.14.a.h | 7 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} - 52218 T_{2}^{5} + 193468 T_{2}^{4} + 738251616 T_{2}^{3} - 3692170944 T_{2}^{2} + \cdots + 15935676776448 \)
acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(11))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + \cdots + 15935676776448 \)
|
| $3$ |
\( T^{7} + \cdots - 55\!\cdots\!28 \)
|
| $5$ |
\( T^{7} + \cdots + 66\!\cdots\!00 \)
|
| $7$ |
\( T^{7} + \cdots - 22\!\cdots\!52 \)
|
| $11$ |
\( (T - 1771561)^{7} \)
|
| $13$ |
\( T^{7} + \cdots + 70\!\cdots\!68 \)
|
| $17$ |
\( T^{7} + \cdots + 13\!\cdots\!96 \)
|
| $19$ |
\( T^{7} + \cdots - 16\!\cdots\!00 \)
|
| $23$ |
\( T^{7} + \cdots - 41\!\cdots\!68 \)
|
| $29$ |
\( T^{7} + \cdots + 76\!\cdots\!00 \)
|
| $31$ |
\( T^{7} + \cdots - 34\!\cdots\!00 \)
|
| $37$ |
\( T^{7} + \cdots + 17\!\cdots\!64 \)
|
| $41$ |
\( T^{7} + \cdots - 18\!\cdots\!12 \)
|
| $43$ |
\( T^{7} + \cdots - 18\!\cdots\!44 \)
|
| $47$ |
\( T^{7} + \cdots - 18\!\cdots\!36 \)
|
| $53$ |
\( T^{7} + \cdots - 20\!\cdots\!16 \)
|
| $59$ |
\( T^{7} + \cdots + 14\!\cdots\!00 \)
|
| $61$ |
\( T^{7} + \cdots + 60\!\cdots\!28 \)
|
| $67$ |
\( T^{7} + \cdots + 13\!\cdots\!48 \)
|
| $71$ |
\( T^{7} + \cdots - 33\!\cdots\!84 \)
|
| $73$ |
\( T^{7} + \cdots + 53\!\cdots\!88 \)
|
| $79$ |
\( T^{7} + \cdots + 11\!\cdots\!00 \)
|
| $83$ |
\( T^{7} + \cdots + 15\!\cdots\!56 \)
|
| $89$ |
\( T^{7} + \cdots + 10\!\cdots\!00 \)
|
| $97$ |
\( T^{7} + \cdots + 69\!\cdots\!04 \)
|
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