Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [176,6,Mod(1,176)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(176, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("176.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 176 = 2^{4} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 176.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: | \(28.2275522871\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{37}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 9 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 88) |
| 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 \(\beta = 2\sqrt{37}\). 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 | −5.16553 | 0 | 9.00000 | 0 | 109.821 | 0 | −216.317 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 19.1655 | 0 | 9.00000 | 0 | −157.821 | 0 | 124.317 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(11\) | \( +1 \) |
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 | 176.6.a.h | 2 | |
| 4.b | odd | 2 | 1 | 88.6.a.a | ✓ | 2 | |
| 8.b | even | 2 | 1 | 704.6.a.l | 2 | ||
| 8.d | odd | 2 | 1 | 704.6.a.o | 2 | ||
| 12.b | even | 2 | 1 | 792.6.a.b | 2 | ||
| 44.c | even | 2 | 1 | 968.6.a.b | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 88.6.a.a | ✓ | 2 | 4.b | odd | 2 | 1 | |
| 176.6.a.h | 2 | 1.a | even | 1 | 1 | trivial | |
| 704.6.a.l | 2 | 8.b | even | 2 | 1 | ||
| 704.6.a.o | 2 | 8.d | odd | 2 | 1 | ||
| 792.6.a.b | 2 | 12.b | even | 2 | 1 | ||
| 968.6.a.b | 2 | 44.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 14T_{3} - 99 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(176))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} - 14T - 99 \)
|
| $5$ |
\( (T - 9)^{2} \)
|
| $7$ |
\( T^{2} + 48T - 17332 \)
|
| $11$ |
\( (T + 121)^{2} \)
|
| $13$ |
\( T^{2} + 708T + 123984 \)
|
| $17$ |
\( T^{2} + 1808 T - 941172 \)
|
| $19$ |
\( T^{2} - 1596 T - 1459024 \)
|
| $23$ |
\( T^{2} - 2454 T + 485957 \)
|
| $29$ |
\( T^{2} + 5320 T + 3920832 \)
|
| $31$ |
\( T^{2} - 2230 T - 33857787 \)
|
| $37$ |
\( T^{2} + 10026 T - 35490631 \)
|
| $41$ |
\( T^{2} + 7820 T + 5664400 \)
|
| $43$ |
\( T^{2} + 11812 T + 31639044 \)
|
| $47$ |
\( T^{2} + 344 T - 243211968 \)
|
| $53$ |
\( T^{2} + 16228 T - 348899852 \)
|
| $59$ |
\( T^{2} + 18878 T - 36444947 \)
|
| $61$ |
\( T^{2} - 12672 T - 431944912 \)
|
| $67$ |
\( T^{2} + 50470 T - 177050475 \)
|
| $71$ |
\( T^{2} + 66326 T + 963103461 \)
|
| $73$ |
\( T^{2} + \cdots - 1013334576 \)
|
| $79$ |
\( T^{2} + 17260 T - 319709852 \)
|
| $83$ |
\( T^{2} + \cdots - 6328465100 \)
|
| $89$ |
\( T^{2} + \cdots + 1274084649 \)
|
| $97$ |
\( T^{2} + \cdots - 2434531183 \)
|
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