Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6845,2,Mod(1,6845)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6845, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6845.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6845 = 5 \cdot 37^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6845.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: | \(54.6576001836\) |
| 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: |
\( x^{7} - 2x^{6} - 9x^{5} + 17x^{4} + 17x^{3} - 31x^{2} - x + 3 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 185) |
| 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
\( x^{7} - 2x^{6} - 9x^{5} + 17x^{4} + 17x^{3} - 31x^{2} - x + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} - 4\nu^{5} - 8\nu^{4} + 33\nu^{3} + 14\nu^{2} - 52\nu - 2 ) / 7 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{6} - 4\nu^{5} - 8\nu^{4} + 40\nu^{3} + 7\nu^{2} - 87\nu + 19 ) / 7 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2\nu^{6} - \nu^{5} - 23\nu^{4} + 10\nu^{3} + 63\nu^{2} - 27\nu - 18 ) / 7 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 5\nu^{6} - 6\nu^{5} - 47\nu^{4} + 46\nu^{3} + 98\nu^{2} - 71\nu - 24 ) / 7 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} - \beta_{3} + \beta_{2} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{6} - 2\beta_{5} + \beta_{4} - 2\beta_{3} + 7\beta_{2} + 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{6} - \beta_{5} + 9\beta_{4} - 12\beta_{3} + 10\beta_{2} + 29\beta _1 + 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( 12\beta_{6} - 20\beta_{5} + 11\beta_{4} - 24\beta_{3} + 49\beta_{2} + 3\beta _1 + 100 \)
|
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 |
|
−2.48125 | 2.20414 | 4.15659 | 1.00000 | −5.46902 | −3.82714 | −5.35102 | 1.85825 | −2.48125 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.59011 | −1.55860 | 0.528450 | 1.00000 | 2.47835 | −1.60228 | 2.33993 | −0.570766 | −1.59011 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.310249 | 2.06190 | −1.90375 | 1.00000 | −0.639704 | −2.12071 | 1.21113 | 1.25144 | −0.310249 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.334924 | −2.38884 | −1.88783 | 1.00000 | −0.800081 | 4.43035 | −1.30213 | 2.70658 | 0.334924 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.40907 | −0.154338 | −0.0145151 | 1.00000 | −0.217473 | −2.41277 | −2.83860 | −2.97618 | 1.40907 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.89024 | 2.79479 | 1.57300 | 1.00000 | 5.28283 | 2.90632 | −0.807130 | 4.81088 | 1.89024 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.74737 | −0.959059 | 5.54805 | 1.00000 | −2.63489 | 0.626225 | 9.74781 | −2.08021 | 2.74737 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( -1 \) |
| \(37\) | \( +1 \) |
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 | 6845.2.a.m | 7 | |
| 37.b | even | 2 | 1 | 6845.2.a.j | 7 | ||
| 37.e | even | 6 | 2 | 185.2.e.b | ✓ | 14 | |
| 185.l | even | 6 | 2 | 925.2.e.b | 14 | ||
| 185.r | odd | 12 | 4 | 925.2.o.c | 28 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 185.2.e.b | ✓ | 14 | 37.e | even | 6 | 2 | |
| 925.2.e.b | 14 | 185.l | even | 6 | 2 | ||
| 925.2.o.c | 28 | 185.r | odd | 12 | 4 | ||
| 6845.2.a.j | 7 | 37.b | even | 2 | 1 | ||
| 6845.2.a.m | 7 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6845))\):
|
\( T_{2}^{7} - 2T_{2}^{6} - 9T_{2}^{5} + 17T_{2}^{4} + 17T_{2}^{3} - 31T_{2}^{2} - T_{2} + 3 \)
|
|
\( T_{7}^{7} + 2T_{7}^{6} - 26T_{7}^{5} - 64T_{7}^{4} + 135T_{7}^{3} + 432T_{7}^{2} + 100T_{7} - 253 \)
|
|
\( T_{17}^{7} + T_{17}^{6} - 57T_{17}^{5} - 67T_{17}^{4} + 991T_{17}^{3} + 1325T_{17}^{2} - 4768T_{17} - 7005 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 2 T^{6} + \cdots + 3 \)
|
| $3$ |
\( T^{7} - 2 T^{6} + \cdots - 7 \)
|
| $5$ |
\( (T - 1)^{7} \)
|
| $7$ |
\( T^{7} + 2 T^{6} + \cdots - 253 \)
|
| $11$ |
\( T^{7} + 5 T^{6} + \cdots - 333 \)
|
| $13$ |
\( T^{7} - 6 T^{6} + \cdots - 83 \)
|
| $17$ |
\( T^{7} + T^{6} + \cdots - 7005 \)
|
| $19$ |
\( T^{7} - 6 T^{6} + \cdots + 341 \)
|
| $23$ |
\( T^{7} + 6 T^{6} + \cdots - 75 \)
|
| $29$ |
\( T^{7} - 6 T^{6} + \cdots + 2649 \)
|
| $31$ |
\( T^{7} - 4 T^{6} + \cdots + 40581 \)
|
| $37$ |
\( T^{7} \)
|
| $41$ |
\( T^{7} - 3 T^{6} + \cdots - 327 \)
|
| $43$ |
\( T^{7} - 19 T^{6} + \cdots - 863977 \)
|
| $47$ |
\( T^{7} - 2 T^{6} + \cdots + 3 \)
|
| $53$ |
\( T^{7} - 2 T^{6} + \cdots + 60687 \)
|
| $59$ |
\( T^{7} + 18 T^{6} + \cdots - 178281 \)
|
| $61$ |
\( T^{7} + 20 T^{6} + \cdots + 64735 \)
|
| $67$ |
\( T^{7} - 20 T^{6} + \cdots + 941 \)
|
| $71$ |
\( T^{7} - 11 T^{6} + \cdots - 64647 \)
|
| $73$ |
\( T^{7} + 18 T^{6} + \cdots - 23788741 \)
|
| $79$ |
\( T^{7} - 23 T^{6} + \cdots + 40461 \)
|
| $83$ |
\( T^{7} - 9 T^{6} + \cdots + 220383 \)
|
| $89$ |
\( T^{7} + 16 T^{6} + \cdots + 4647489 \)
|
| $97$ |
\( T^{7} - 31 T^{6} + \cdots + 2647169 \)
|
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