Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1911,4,Mod(1,1911)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1911, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1911.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1911 = 3 \cdot 7^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1911.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: | \(112.752650021\) |
| 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: |
\( x^{7} - x^{6} - 31x^{5} + 49x^{4} + 158x^{3} - 152x^{2} - 272x - 80 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | no (minimal twist has level 273) |
| 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
\( x^{7} - x^{6} - 31x^{5} + 49x^{4} + 158x^{3} - 152x^{2} - 272x - 80 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + \nu - 9 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{6} + 29\nu^{4} - 22\nu^{3} - 122\nu^{2} + 44\nu + 60 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{6} - 29\nu^{4} + 24\nu^{3} + 128\nu^{2} - 80\nu - 96 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{6} - 2\nu^{5} - 31\nu^{4} + 78\nu^{3} + 134\nu^{2} - 280\nu - 196 ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{6} - 2\nu^{5} - 30\nu^{4} + 78\nu^{3} + 109\nu^{2} - 254\nu - 148 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - \beta _1 + 9 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{4} + 2\beta_{3} - 3\beta_{2} + 21\beta _1 - 9 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{6} - 4\beta_{5} + 25\beta_{2} - 51\beta _1 + 177 \)
|
| \(\nu^{5}\) | \(=\) |
\( -2\beta_{6} + 2\beta_{5} + 56\beta_{4} + 54\beta_{3} - 103\beta_{2} + 515\beta _1 - 443 \)
|
| \(\nu^{6}\) | \(=\) |
\( 58\beta_{6} - 116\beta_{5} - 44\beta_{4} - 48\beta_{3} + 669\beta_{2} - 1775\beta _1 + 4293 \)
|
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 |
|
−4.34886 | −3.00000 | 10.9125 | −14.0642 | 13.0466 | 0 | −12.6662 | 9.00000 | 61.1631 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −0.799047 | −3.00000 | −7.36152 | −12.4065 | 2.39714 | 0 | 12.2746 | 9.00000 | 9.91338 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0.323122 | −3.00000 | −7.89559 | −8.24678 | −0.969367 | 0 | −5.13622 | 9.00000 | −2.66472 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.546656 | −3.00000 | −7.70117 | 0.201298 | −1.63997 | 0 | −8.58313 | 9.00000 | 0.110041 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 3.39552 | −3.00000 | 3.52958 | −13.5908 | −10.1866 | 0 | −15.1794 | 9.00000 | −46.1480 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 3.71128 | −3.00000 | 5.77360 | 8.59767 | −11.1338 | 0 | −8.26280 | 9.00000 | 31.9084 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 5.17132 | −3.00000 | 18.7426 | 18.5094 | −15.5140 | 0 | 55.5532 | 9.00000 | 95.7178 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(7\) | \(-1\) |
| \(13\) | \(-1\) |
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 | 1911.4.a.t | 7 | |
| 7.b | odd | 2 | 1 | 1911.4.a.u | 7 | ||
| 7.d | odd | 6 | 2 | 273.4.i.c | ✓ | 14 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 273.4.i.c | ✓ | 14 | 7.d | odd | 6 | 2 | |
| 1911.4.a.t | 7 | 1.a | even | 1 | 1 | trivial | |
| 1911.4.a.u | 7 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1911))\):
|
\( T_{2}^{7} - 8T_{2}^{6} - 4T_{2}^{5} + 154T_{2}^{4} - 293T_{2}^{3} - 58T_{2}^{2} + 168T_{2} - 40 \)
|
|
\( T_{5}^{7} + 21T_{5}^{6} - 290T_{5}^{5} - 8915T_{5}^{4} - 24754T_{5}^{3} + 553736T_{5}^{2} + 3001808T_{5} - 626480 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 8 T^{6} + \cdots - 40 \)
|
| $3$ |
\( (T + 3)^{7} \)
|
| $5$ |
\( T^{7} + 21 T^{6} + \cdots - 626480 \)
|
| $7$ |
\( T^{7} \)
|
| $11$ |
\( T^{7} + \cdots + 31008383811 \)
|
| $13$ |
\( (T - 13)^{7} \)
|
| $17$ |
\( T^{7} + \cdots - 2042835640703 \)
|
| $19$ |
\( T^{7} + \cdots - 6513116419611 \)
|
| $23$ |
\( T^{7} + \cdots + 553092105904 \)
|
| $29$ |
\( T^{7} + \cdots - 16\!\cdots\!41 \)
|
| $31$ |
\( T^{7} + \cdots - 25\!\cdots\!72 \)
|
| $37$ |
\( T^{7} + \cdots + 41911148651408 \)
|
| $41$ |
\( T^{7} + \cdots - 42\!\cdots\!12 \)
|
| $43$ |
\( T^{7} + \cdots - 492448628815280 \)
|
| $47$ |
\( T^{7} + \cdots + 14\!\cdots\!80 \)
|
| $53$ |
\( T^{7} + \cdots + 35\!\cdots\!95 \)
|
| $59$ |
\( T^{7} + \cdots + 42\!\cdots\!57 \)
|
| $61$ |
\( T^{7} + \cdots + 32\!\cdots\!55 \)
|
| $67$ |
\( T^{7} + \cdots - 275800882001301 \)
|
| $71$ |
\( T^{7} + \cdots + 11\!\cdots\!63 \)
|
| $73$ |
\( T^{7} + \cdots + 14\!\cdots\!72 \)
|
| $79$ |
\( T^{7} + \cdots - 51\!\cdots\!80 \)
|
| $83$ |
\( T^{7} + \cdots - 53\!\cdots\!56 \)
|
| $89$ |
\( T^{7} + \cdots - 13\!\cdots\!24 \)
|
| $97$ |
\( T^{7} + \cdots - 11\!\cdots\!60 \)
|
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