Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [49,12,Mod(1,49)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(49, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("49.1");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 49 = 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 49.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: | \(37.6488158474\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{6} - x^{5} - 1845x^{4} - 3815x^{3} + 664676x^{2} + 2936988x + 2022336 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{10}\cdot 3\cdot 7^{3} \) |
| Twist minimal: | no (minimal twist has level 7) |
| 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
\( x^{6} - x^{5} - 1845x^{4} - 3815x^{3} + 664676x^{2} + 2936988x + 2022336 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 17\nu^{5} + 49\nu^{4} - 31995\nu^{3} - 140677\nu^{2} + 11840146\nu + 44067804 ) / 48804 \)
|
| \(\beta_{3}\) | \(=\) |
\( 4\nu^{2} - 18\nu - 2458 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 117\nu^{5} - 483\nu^{4} - 236879\nu^{3} + 248767\nu^{2} + 102133386\nu + 168638304 ) / 16268 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -551\nu^{5} + 2513\nu^{4} + 990261\nu^{3} - 1069685\nu^{2} - 342136366\nu - 691122672 ) / 48804 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 9\beta _1 + 2458 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -3\beta_{5} - 5\beta_{4} + 7\beta_{3} + 6\beta_{2} + 4515\beta _1 + 21136 ) / 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 61\beta_{5} - 57\beta_{4} + 2825\beta_{3} + 3154\beta_{2} + 35581\beta _1 + 5550640 ) / 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -2911\beta_{5} - 4623\beta_{4} + 10791\beta_{3} + 12584\beta_{2} + 2878987\beta _1 + 21861490 ) / 4 \)
|
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 |
|
−74.0263 | 164.038 | 3431.90 | −3340.56 | −12143.1 | 0 | −102445. | −150239. | 247289. | ||||||||||||||||||||||||||||||||||||
| 1.2 | −45.9737 | −623.603 | 65.5798 | 10589.3 | 28669.3 | 0 | 91139.2 | 211734. | −486831. | |||||||||||||||||||||||||||||||||||||
| 1.3 | −11.2783 | 46.1731 | −1920.80 | −8609.87 | −520.754 | 0 | 44761.3 | −175015. | 97104.6 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −5.70834 | 732.326 | −2015.41 | 8982.22 | −4180.36 | 0 | 23195.3 | 359154. | −51273.6 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 42.0050 | −410.428 | −283.582 | 82.5370 | −17240.0 | 0 | −97938.0 | −8695.65 | 3466.97 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 72.9817 | 335.494 | 3278.32 | 1078.32 | 24484.9 | 0 | 89790.9 | −64590.5 | 78697.8 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \(1\) |
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 | 49.12.a.g | 6 | |
| 7.b | odd | 2 | 1 | 49.12.a.f | 6 | ||
| 7.c | even | 3 | 2 | 7.12.c.a | ✓ | 12 | |
| 7.d | odd | 6 | 2 | 49.12.c.i | 12 | ||
| 21.h | odd | 6 | 2 | 63.12.e.b | 12 | ||
| 28.g | odd | 6 | 2 | 112.12.i.c | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7.12.c.a | ✓ | 12 | 7.c | even | 3 | 2 | |
| 49.12.a.f | 6 | 7.b | odd | 2 | 1 | ||
| 49.12.a.g | 6 | 1.a | even | 1 | 1 | trivial | |
| 49.12.c.i | 12 | 7.d | odd | 6 | 2 | ||
| 63.12.e.b | 12 | 21.h | odd | 6 | 2 | ||
| 112.12.i.c | 12 | 28.g | odd | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(49))\):
|
\( T_{2}^{6} + 22T_{2}^{5} - 7180T_{2}^{4} - 147640T_{2}^{3} + 9562656T_{2}^{2} + 175711488T_{2} + 671680512 \)
|
|
\( T_{3}^{6} - 244 T_{3}^{5} - 587847 T_{3}^{4} + 107151120 T_{3}^{3} + 62366237955 T_{3}^{2} + \cdots + 476286412264803 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 22 T^{5} + \cdots + 671680512 \)
|
| $3$ |
\( T^{6} + \cdots + 476286412264803 \)
|
| $5$ |
\( T^{6} + \cdots + 24\!\cdots\!25 \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( T^{6} + \cdots + 37\!\cdots\!75 \)
|
| $13$ |
\( T^{6} + \cdots + 32\!\cdots\!44 \)
|
| $17$ |
\( T^{6} + \cdots + 54\!\cdots\!37 \)
|
| $19$ |
\( T^{6} + \cdots + 20\!\cdots\!11 \)
|
| $23$ |
\( T^{6} + \cdots + 88\!\cdots\!47 \)
|
| $29$ |
\( T^{6} + \cdots + 28\!\cdots\!00 \)
|
| $31$ |
\( T^{6} + \cdots + 13\!\cdots\!99 \)
|
| $37$ |
\( T^{6} + \cdots + 38\!\cdots\!25 \)
|
| $41$ |
\( T^{6} + \cdots - 95\!\cdots\!00 \)
|
| $43$ |
\( T^{6} + \cdots - 24\!\cdots\!68 \)
|
| $47$ |
\( T^{6} + \cdots - 24\!\cdots\!25 \)
|
| $53$ |
\( T^{6} + \cdots + 14\!\cdots\!01 \)
|
| $59$ |
\( T^{6} + \cdots - 15\!\cdots\!25 \)
|
| $61$ |
\( T^{6} + \cdots + 13\!\cdots\!13 \)
|
| $67$ |
\( T^{6} + \cdots + 96\!\cdots\!75 \)
|
| $71$ |
\( T^{6} + \cdots + 10\!\cdots\!44 \)
|
| $73$ |
\( T^{6} + \cdots - 11\!\cdots\!83 \)
|
| $79$ |
\( T^{6} + \cdots + 14\!\cdots\!47 \)
|
| $83$ |
\( T^{6} + \cdots - 24\!\cdots\!56 \)
|
| $89$ |
\( T^{6} + \cdots + 97\!\cdots\!21 \)
|
| $97$ |
\( T^{6} + \cdots - 18\!\cdots\!64 \)
|
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