Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1150,4,Mod(1,1150)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1150, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("1150.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1150 = 2 \cdot 5^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1150.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: | \(67.8521965066\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 107x^{3} - 3x^{2} + 2151x - 2916 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 107x^{3} - 3x^{2} + 2151x - 2916 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - 3\nu^{3} - 53\nu^{2} + 111\nu - 567 ) / 45 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - 3\nu^{3} - 98\nu^{2} + 111\nu + 1323 ) / 45 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2\nu^{4} + 9\nu^{3} - 196\nu^{2} - 753\nu + 2646 ) / 45 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + 42 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{4} - 6\beta_{3} + 65\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 9\beta_{4} - 71\beta_{3} + 98\beta_{2} + 84\beta _1 + 2793 \)
|
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 |
|
−2.00000 | −9.48510 | 4.00000 | 0 | 18.9702 | 8.59355 | −8.00000 | 62.9670 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | −2.00000 | −7.21006 | 4.00000 | 0 | 14.4201 | −19.8879 | −8.00000 | 24.9850 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | −2.00000 | 0.534830 | 4.00000 | 0 | −1.06966 | 28.9380 | −8.00000 | −26.7140 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | −2.00000 | 2.88892 | 4.00000 | 0 | −5.77785 | 8.21839 | −8.00000 | −18.6541 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | −2.00000 | 8.27140 | 4.00000 | 0 | −16.5428 | −22.8621 | −8.00000 | 41.4161 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( -1 \) |
| \(23\) | \( +1 \) |
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 | 1150.4.a.q | ✓ | 5 |
| 5.b | even | 2 | 1 | 1150.4.a.v | yes | 5 | |
| 5.c | odd | 4 | 2 | 1150.4.b.r | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1150.4.a.q | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 1150.4.a.v | yes | 5 | 5.b | even | 2 | 1 | |
| 1150.4.b.r | 10 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1150))\):
|
\( T_{3}^{5} + 5T_{3}^{4} - 97T_{3}^{3} - 314T_{3}^{2} + 1829T_{3} - 874 \)
|
|
\( T_{7}^{5} - 3T_{7}^{4} - 944T_{7}^{3} + 972T_{7}^{2} + 165944T_{7} - 929248 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{5} \)
|
| $3$ |
\( T^{5} + 5 T^{4} + \cdots - 874 \)
|
| $5$ |
\( T^{5} \)
|
| $7$ |
\( T^{5} - 3 T^{4} + \cdots - 929248 \)
|
| $11$ |
\( T^{5} + 26 T^{4} + \cdots + 3242920 \)
|
| $13$ |
\( T^{5} - 61 T^{4} + \cdots - 166878 \)
|
| $17$ |
\( T^{5} + 231 T^{4} + \cdots - 414548320 \)
|
| $19$ |
\( T^{5} - 74 T^{4} + \cdots - 112577688 \)
|
| $23$ |
\( (T + 23)^{5} \)
|
| $29$ |
\( T^{5} + \cdots - 20220755440 \)
|
| $31$ |
\( T^{5} + \cdots - 51046245360 \)
|
| $37$ |
\( T^{5} + \cdots + 5132681056 \)
|
| $41$ |
\( T^{5} + \cdots + 4982446167 \)
|
| $43$ |
\( T^{5} + \cdots + 39661228800 \)
|
| $47$ |
\( T^{5} + \cdots - 1683347184 \)
|
| $53$ |
\( T^{5} + \cdots - 18447971014656 \)
|
| $59$ |
\( T^{5} + \cdots + 1123269020040 \)
|
| $61$ |
\( T^{5} + \cdots + 2333556256768 \)
|
| $67$ |
\( T^{5} + \cdots + 35946409847360 \)
|
| $71$ |
\( T^{5} + \cdots - 5642844385736 \)
|
| $73$ |
\( T^{5} + \cdots - 95962864505643 \)
|
| $79$ |
\( T^{5} + \cdots + 12948718884224 \)
|
| $83$ |
\( T^{5} + \cdots - 31156649817688 \)
|
| $89$ |
\( T^{5} + \cdots + 1214254682896 \)
|
| $97$ |
\( T^{5} + \cdots - 79142464495296 \)
|
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