Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [41,6,Mod(1,41)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(41, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("41.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 41 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 41.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: | \(6.57573661233\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 101x^{4} + 49x^{3} + 2072x^{2} - 1340x - 1856 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2}\cdot 5 \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 101x^{4} + 49x^{3} + 2072x^{2} - 1340x - 1856 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + 87\nu^{3} + 94\nu^{2} - 1096\nu - 1408 ) / 112 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -3\nu^{5} + 317\nu^{3} - 110\nu^{2} - 6536\nu + 7872 ) / 448 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -3\nu^{5} + 14\nu^{4} + 261\nu^{3} - 992\nu^{2} - 4212\nu + 10112 ) / 448 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -3\nu^{5} + 317\nu^{3} + 114\nu^{2} - 6760\nu + 480 ) / 224 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - 2\beta_{3} + \beta _1 + 33 \)
|
| \(\nu^{3}\) | \(=\) |
\( 7\beta_{5} - 6\beta_{3} - 6\beta_{2} + 65\beta _1 + 15 \)
|
| \(\nu^{4}\) | \(=\) |
\( 91\beta_{5} + 32\beta_{4} - 182\beta_{3} - 24\beta_{2} + 157\beta _1 + 1979 \)
|
| \(\nu^{5}\) | \(=\) |
\( 703\beta_{5} - 710\beta_{3} - 634\beta_{2} + 4653\beta _1 + 2999 \)
|
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 |
|
−9.06451 | −16.3786 | 50.1653 | 56.5754 | 148.464 | 37.4891 | −164.659 | 25.2597 | −512.828 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −6.52592 | 10.3252 | 10.5876 | −16.4049 | −67.3816 | 62.0721 | 139.735 | −136.390 | 107.057 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.68590 | 26.9213 | −29.1577 | −101.954 | −45.3868 | −200.683 | 103.106 | 481.759 | 171.884 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.378588 | −3.56587 | −31.8567 | 80.4202 | −1.34999 | −141.204 | −24.1754 | −230.285 | 30.4461 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 3.88934 | −1.52542 | −16.8730 | −48.3758 | −5.93288 | −18.0308 | −190.084 | −240.673 | −188.150 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 8.00840 | −23.7766 | 32.1345 | −38.2610 | −190.413 | −83.6433 | 1.07718 | 322.329 | −306.410 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(41\) | \(1\) |
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 | 41.6.a.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | 369.6.a.a | 6 | ||
| 4.b | odd | 2 | 1 | 656.6.a.f | 6 | ||
| 5.b | even | 2 | 1 | 1025.6.a.a | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 41.6.a.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 369.6.a.a | 6 | 3.b | odd | 2 | 1 | ||
| 656.6.a.f | 6 | 4.b | odd | 2 | 1 | ||
| 1025.6.a.a | 6 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 5T_{2}^{5} - 91T_{2}^{4} - 345T_{2}^{3} + 1618T_{2}^{2} + 2548T_{2} - 1176 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(41))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 5 T^{5} + \cdots - 1176 \)
|
| $3$ |
\( T^{6} + 8 T^{5} + \cdots + 588816 \)
|
| $5$ |
\( T^{6} + \cdots + 14084905216 \)
|
| $7$ |
\( T^{6} + \cdots + 99450013184 \)
|
| $11$ |
\( T^{6} + \cdots - 487420750929520 \)
|
| $13$ |
\( T^{6} + \cdots + 6140216737792 \)
|
| $17$ |
\( T^{6} + \cdots - 20\!\cdots\!00 \)
|
| $19$ |
\( T^{6} + \cdots - 16\!\cdots\!00 \)
|
| $23$ |
\( T^{6} + \cdots - 14\!\cdots\!32 \)
|
| $29$ |
\( T^{6} + \cdots + 98\!\cdots\!56 \)
|
| $31$ |
\( T^{6} + \cdots - 14\!\cdots\!40 \)
|
| $37$ |
\( T^{6} + \cdots + 16\!\cdots\!56 \)
|
| $41$ |
\( (T + 1681)^{6} \)
|
| $43$ |
\( T^{6} + \cdots + 16\!\cdots\!48 \)
|
| $47$ |
\( T^{6} + \cdots - 19\!\cdots\!96 \)
|
| $53$ |
\( T^{6} + \cdots + 69\!\cdots\!52 \)
|
| $59$ |
\( T^{6} + \cdots - 67\!\cdots\!24 \)
|
| $61$ |
\( T^{6} + \cdots - 15\!\cdots\!12 \)
|
| $67$ |
\( T^{6} + \cdots - 54\!\cdots\!80 \)
|
| $71$ |
\( T^{6} + \cdots + 67\!\cdots\!96 \)
|
| $73$ |
\( T^{6} + \cdots + 20\!\cdots\!16 \)
|
| $79$ |
\( T^{6} + \cdots - 31\!\cdots\!92 \)
|
| $83$ |
\( T^{6} + \cdots + 17\!\cdots\!96 \)
|
| $89$ |
\( T^{6} + \cdots + 27\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots - 47\!\cdots\!76 \)
|
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