Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [49,12,Mod(18,49)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(49, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4]))
N = Newforms(chi, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("49.18");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 49 = 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 49.c (of order \(3\), degree \(2\), not minimal) |
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: | no |
| Analytic conductor: | \(37.6488158474\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 819x^{4} + 10226x^{3} + 664420x^{2} + 3847872x + 22127616 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 7) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{6} - x^{5} + 819x^{4} + 10226x^{3} + 664420x^{2} + 3847872x + 22127616 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{5} + 819\nu^{4} - 670761\nu^{3} + 664420\nu^{2} + 3847872\nu - 3425411094 ) / 274003926 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -823\nu^{5} + 674037\nu^{4} - 4028451\nu^{3} + 546817660\nu^{2} + 3166798656\nu + 283433123736 ) / 3288047112 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -15951\nu^{5} + 16063\nu^{4} - 13155597\nu^{3} - 87989694\nu^{2} - 10672578460\nu - 61808367936 ) / 61376879424 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 15951\nu^{5} - 16063\nu^{4} + 13155597\nu^{3} + 87989694\nu^{2} + 133426337308\nu + 431488512 ) / 61376879424 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 622049 \nu^{5} - 593697 \nu^{4} + 486237843 \nu^{3} + 7842237682 \nu^{2} + \cdots + 2284470062784 ) / 26304376896 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + \beta_{3} + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 12\beta_{5} + 5\beta_{4} + 1097\beta_{3} - 5\beta_1 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 12\beta_{2} - 823\beta _1 - 11323 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -9828\beta_{5} - 9617\beta_{4} - 913373\beta_{3} + 9828\beta_{2} - 913373 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -76092\beta_{5} - 706351\beta_{4} - 15335875\beta_{3} + 706351\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(-1 - \beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 18.1 |
|
−37.5395 | − | 65.0203i | 120.189 | − | 208.173i | −1794.42 | + | 3108.03i | −6421.02 | − | 11121.5i | −18047.3 | 0 | 115685. | 59682.7 | + | 103374.i | −482083. | + | 834993.i | ||||||||||||||||||||||||
| 18.2 | −27.8625 | − | 48.2593i | −400.451 | + | 693.602i | −528.641 | + | 915.633i | 3533.02 | + | 6119.37i | 44630.3 | 0 | −55207.8 | −232149. | − | 402093.i | 196878. | − | 341002.i | |||||||||||||||||||||||||
| 18.3 | 26.9020 | + | 46.5956i | 350.262 | − | 606.672i | −423.435 | + | 733.411i | 374.999 | + | 649.517i | 37691.0 | 0 | 64625.6 | −156794. | − | 271574.i | −20176.4 | + | 34946.6i | |||||||||||||||||||||||||
| 30.1 | −37.5395 | + | 65.0203i | 120.189 | + | 208.173i | −1794.42 | − | 3108.03i | −6421.02 | + | 11121.5i | −18047.3 | 0 | 115685. | 59682.7 | − | 103374.i | −482083. | − | 834993.i | |||||||||||||||||||||||||
| 30.2 | −27.8625 | + | 48.2593i | −400.451 | − | 693.602i | −528.641 | − | 915.633i | 3533.02 | − | 6119.37i | 44630.3 | 0 | −55207.8 | −232149. | + | 402093.i | 196878. | + | 341002.i | |||||||||||||||||||||||||
| 30.3 | 26.9020 | − | 46.5956i | 350.262 | + | 606.672i | −423.435 | − | 733.411i | 374.999 | − | 649.517i | 37691.0 | 0 | 64625.6 | −156794. | + | 271574.i | −20176.4 | − | 34946.6i | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 49.12.c.g | 6 | |
| 7.b | odd | 2 | 1 | 49.12.c.f | 6 | ||
| 7.c | even | 3 | 1 | 7.12.a.b | ✓ | 3 | |
| 7.c | even | 3 | 1 | inner | 49.12.c.g | 6 | |
| 7.d | odd | 6 | 1 | 49.12.a.d | 3 | ||
| 7.d | odd | 6 | 1 | 49.12.c.f | 6 | ||
| 21.h | odd | 6 | 1 | 63.12.a.d | 3 | ||
| 28.g | odd | 6 | 1 | 112.12.a.h | 3 | ||
| 35.j | even | 6 | 1 | 175.12.a.b | 3 | ||
| 35.l | odd | 12 | 2 | 175.12.b.b | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7.12.a.b | ✓ | 3 | 7.c | even | 3 | 1 | |
| 49.12.a.d | 3 | 7.d | odd | 6 | 1 | ||
| 49.12.c.f | 6 | 7.b | odd | 2 | 1 | ||
| 49.12.c.f | 6 | 7.d | odd | 6 | 1 | ||
| 49.12.c.g | 6 | 1.a | even | 1 | 1 | trivial | |
| 49.12.c.g | 6 | 7.c | even | 3 | 1 | inner | |
| 63.12.a.d | 3 | 21.h | odd | 6 | 1 | ||
| 112.12.a.h | 3 | 28.g | odd | 6 | 1 | ||
| 175.12.a.b | 3 | 35.j | even | 6 | 1 | ||
| 175.12.b.b | 6 | 35.l | odd | 12 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{12}^{\mathrm{new}}(49, [\chi])\):
|
\( T_{2}^{6} + 77T_{2}^{5} + 8783T_{2}^{4} + 230450T_{2}^{3} + 25478324T_{2}^{2} + 642446816T_{2} + 50671810816 \)
|
|
\( T_{3}^{6} - 140 T_{3}^{5} + 604780 T_{3}^{4} - 187803504 T_{3}^{3} + 361316641680 T_{3}^{2} + \cdots + 18\!\cdots\!04 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + \cdots + 50671810816 \)
|
| $3$ |
\( T^{6} + \cdots + 18\!\cdots\!04 \)
|
| $5$ |
\( T^{6} + \cdots + 46\!\cdots\!00 \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( T^{6} + \cdots + 11\!\cdots\!04 \)
|
| $13$ |
\( (T^{3} + \cdots - 91\!\cdots\!44)^{2} \)
|
| $17$ |
\( T^{6} + \cdots + 38\!\cdots\!04 \)
|
| $19$ |
\( T^{6} + \cdots + 18\!\cdots\!00 \)
|
| $23$ |
\( T^{6} + \cdots + 17\!\cdots\!84 \)
|
| $29$ |
\( (T^{3} + \cdots - 25\!\cdots\!60)^{2} \)
|
| $31$ |
\( T^{6} + \cdots + 21\!\cdots\!00 \)
|
| $37$ |
\( T^{6} + \cdots + 18\!\cdots\!56 \)
|
| $41$ |
\( (T^{3} + \cdots + 46\!\cdots\!72)^{2} \)
|
| $43$ |
\( (T^{3} + \cdots + 22\!\cdots\!64)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 45\!\cdots\!04 \)
|
| $53$ |
\( T^{6} + \cdots + 12\!\cdots\!44 \)
|
| $59$ |
\( T^{6} + \cdots + 44\!\cdots\!00 \)
|
| $61$ |
\( T^{6} + \cdots + 12\!\cdots\!24 \)
|
| $67$ |
\( T^{6} + \cdots + 56\!\cdots\!96 \)
|
| $71$ |
\( (T^{3} + \cdots + 61\!\cdots\!92)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 23\!\cdots\!16 \)
|
| $79$ |
\( T^{6} + \cdots + 89\!\cdots\!00 \)
|
| $83$ |
\( (T^{3} + \cdots - 57\!\cdots\!72)^{2} \)
|
| $89$ |
\( T^{6} + \cdots + 80\!\cdots\!00 \)
|
| $97$ |
\( (T^{3} + \cdots - 27\!\cdots\!84)^{2} \)
|
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