Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2304,4,Mod(1,2304)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2304, 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("2304.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2304 = 2^{8} \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2304.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: | \(135.940400653\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{13}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 3 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 384) |
| 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 \(\beta = 4\sqrt{13}\). 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 | 0 | 0 | −18.4222 | 0 | 22.4222 | 0 | 0 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | 10.4222 | 0 | −6.42221 | 0 | 0 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( -1 \) |
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 | 2304.4.a.t | 2 | |
| 3.b | odd | 2 | 1 | 768.4.a.j | 2 | ||
| 4.b | odd | 2 | 1 | 2304.4.a.s | 2 | ||
| 8.b | even | 2 | 1 | 2304.4.a.bq | 2 | ||
| 8.d | odd | 2 | 1 | 2304.4.a.bp | 2 | ||
| 12.b | even | 2 | 1 | 768.4.a.p | 2 | ||
| 16.e | even | 4 | 2 | 1152.4.d.i | 4 | ||
| 16.f | odd | 4 | 2 | 1152.4.d.o | 4 | ||
| 24.f | even | 2 | 1 | 768.4.a.e | 2 | ||
| 24.h | odd | 2 | 1 | 768.4.a.k | 2 | ||
| 48.i | odd | 4 | 2 | 384.4.d.c | ✓ | 4 | |
| 48.k | even | 4 | 2 | 384.4.d.e | yes | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 384.4.d.c | ✓ | 4 | 48.i | odd | 4 | 2 | |
| 384.4.d.e | yes | 4 | 48.k | even | 4 | 2 | |
| 768.4.a.e | 2 | 24.f | even | 2 | 1 | ||
| 768.4.a.j | 2 | 3.b | odd | 2 | 1 | ||
| 768.4.a.k | 2 | 24.h | odd | 2 | 1 | ||
| 768.4.a.p | 2 | 12.b | even | 2 | 1 | ||
| 1152.4.d.i | 4 | 16.e | even | 4 | 2 | ||
| 1152.4.d.o | 4 | 16.f | odd | 4 | 2 | ||
| 2304.4.a.s | 2 | 4.b | odd | 2 | 1 | ||
| 2304.4.a.t | 2 | 1.a | even | 1 | 1 | trivial | |
| 2304.4.a.bp | 2 | 8.d | odd | 2 | 1 | ||
| 2304.4.a.bq | 2 | 8.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2304))\):
|
\( T_{5}^{2} + 8T_{5} - 192 \)
|
|
\( T_{7}^{2} - 16T_{7} - 144 \)
|
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\( T_{11}^{2} - 8T_{11} - 3312 \)
|
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\( T_{13}^{2} + 72T_{13} + 464 \)
|
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\( T_{17}^{2} - 36T_{17} - 3004 \)
|
|
\( T_{19}^{2} - 136T_{19} + 1296 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} + 8T - 192 \)
|
| $7$ |
\( T^{2} - 16T - 144 \)
|
| $11$ |
\( T^{2} - 8T - 3312 \)
|
| $13$ |
\( T^{2} + 72T + 464 \)
|
| $17$ |
\( T^{2} - 36T - 3004 \)
|
| $19$ |
\( T^{2} - 136T + 1296 \)
|
| $23$ |
\( T^{2} + 256T + 15552 \)
|
| $29$ |
\( T^{2} - 152T - 11072 \)
|
| $31$ |
\( T^{2} + 80T - 33552 \)
|
| $37$ |
\( T^{2} + 136T + 1296 \)
|
| $41$ |
\( T^{2} - 436T - 35676 \)
|
| $43$ |
\( T^{2} + 712T + 123408 \)
|
| $47$ |
\( T^{2} - 224T - 28224 \)
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| $53$ |
\( T^{2} - 344T - 17216 \)
|
| $59$ |
\( (T - 324)^{2} \)
|
| $61$ |
\( (T + 324)^{2} \)
|
| $67$ |
\( T^{2} - 456T - 427248 \)
|
| $71$ |
\( T^{2} + 2048 T + 1047744 \)
|
| $73$ |
\( T^{2} - 660T - 370332 \)
|
| $79$ |
\( T^{2} - 496T - 662544 \)
|
| $83$ |
\( T^{2} + 776T - 12528 \)
|
| $89$ |
\( T^{2} + 532T + 57444 \)
|
| $97$ |
\( T^{2} + 1220 T - 1238652 \)
|
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