Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [49,6,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, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("49.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 49 = 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| 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: | \(7.85880717084\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{57}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 14 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{57})\). We also show the integral \(q\)-expansion of the trace form.
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 |
|
0.725083 | −19.6495 | −31.4743 | 46.7492 | −14.2475 | 0 | −46.0241 | 143.103 | 33.8970 | ||||||||||||||||||||||||
| 1.2 | 8.27492 | 25.6495 | 36.4743 | −28.7492 | 212.248 | 0 | 37.0241 | 414.897 | −237.897 | |||||||||||||||||||||||||
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 | 49.6.a.f | 2 | |
| 3.b | odd | 2 | 1 | 441.6.a.l | 2 | ||
| 4.b | odd | 2 | 1 | 784.6.a.v | 2 | ||
| 7.b | odd | 2 | 1 | 7.6.a.b | ✓ | 2 | |
| 7.c | even | 3 | 2 | 49.6.c.d | 4 | ||
| 7.d | odd | 6 | 2 | 49.6.c.e | 4 | ||
| 21.c | even | 2 | 1 | 63.6.a.f | 2 | ||
| 28.d | even | 2 | 1 | 112.6.a.h | 2 | ||
| 35.c | odd | 2 | 1 | 175.6.a.c | 2 | ||
| 35.f | even | 4 | 2 | 175.6.b.c | 4 | ||
| 56.e | even | 2 | 1 | 448.6.a.u | 2 | ||
| 56.h | odd | 2 | 1 | 448.6.a.w | 2 | ||
| 77.b | even | 2 | 1 | 847.6.a.c | 2 | ||
| 84.h | odd | 2 | 1 | 1008.6.a.bq | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7.6.a.b | ✓ | 2 | 7.b | odd | 2 | 1 | |
| 49.6.a.f | 2 | 1.a | even | 1 | 1 | trivial | |
| 49.6.c.d | 4 | 7.c | even | 3 | 2 | ||
| 49.6.c.e | 4 | 7.d | odd | 6 | 2 | ||
| 63.6.a.f | 2 | 21.c | even | 2 | 1 | ||
| 112.6.a.h | 2 | 28.d | even | 2 | 1 | ||
| 175.6.a.c | 2 | 35.c | odd | 2 | 1 | ||
| 175.6.b.c | 4 | 35.f | even | 4 | 2 | ||
| 441.6.a.l | 2 | 3.b | odd | 2 | 1 | ||
| 448.6.a.u | 2 | 56.e | even | 2 | 1 | ||
| 448.6.a.w | 2 | 56.h | odd | 2 | 1 | ||
| 784.6.a.v | 2 | 4.b | odd | 2 | 1 | ||
| 847.6.a.c | 2 | 77.b | even | 2 | 1 | ||
| 1008.6.a.bq | 2 | 84.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(49))\):
|
\( T_{2}^{2} - 9T_{2} + 6 \)
|
|
\( T_{3}^{2} - 6T_{3} - 504 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} - 9T + 6 \)
|
| $3$ |
\( T^{2} - 6T - 504 \)
|
| $5$ |
\( T^{2} - 18T - 1344 \)
|
| $7$ |
\( T^{2} \)
|
| $11$ |
\( T^{2} - 396T - 179904 \)
|
| $13$ |
\( T^{2} - 350T - 195608 \)
|
| $17$ |
\( T^{2} + 1800 T + 727692 \)
|
| $19$ |
\( T^{2} - 3266 T + 2662072 \)
|
| $23$ |
\( T^{2} - 2088 T - 3507456 \)
|
| $29$ |
\( T^{2} - 6696 T + 10304172 \)
|
| $31$ |
\( T^{2} - 20T - 4155200 \)
|
| $37$ |
\( T^{2} - 6232 T + 5554156 \)
|
| $41$ |
\( T^{2} - 6048 T - 7848036 \)
|
| $43$ |
\( T^{2} + 3020 T - 324400352 \)
|
| $47$ |
\( T^{2} + 11700 T - 165954432 \)
|
| $53$ |
\( T^{2} - 9468 T + 21794244 \)
|
| $59$ |
\( T^{2} - 43938 T + 422751336 \)
|
| $61$ |
\( T^{2} - 64754 T + 719128816 \)
|
| $67$ |
\( T^{2} - 24784 T + 99708976 \)
|
| $71$ |
\( T^{2} + \cdots + 2121099264 \)
|
| $73$ |
\( T^{2} + 17452 T - 317520812 \)
|
| $79$ |
\( T^{2} + \cdots - 2508546944 \)
|
| $83$ |
\( T^{2} + 117558 T - 79919784 \)
|
| $89$ |
\( T^{2} + \cdots - 5252421468 \)
|
| $97$ |
\( T^{2} + \cdots - 1000631156 \)
|
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