Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [152,2,Mod(3,152)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(152, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([9, 9, 13]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("152.3");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 152 = 2^{3} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 152.v (of order \(18\), degree \(6\), 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: | \(1.21372611072\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{18})\) |
| Coefficient field: | 12.0.101559956668416.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 8x^{6} + 64 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{18}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 8x^{6} + 64 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{6} ) / 8 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{7} ) / 8 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( \nu^{8} ) / 16 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( \nu^{9} ) / 16 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( \nu^{10} ) / 32 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( \nu^{11} ) / 32 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} \)
|
| \(\nu^{4}\) | \(=\) |
\( 4\beta_{4} \)
|
| \(\nu^{5}\) | \(=\) |
\( 4\beta_{5} \)
|
| \(\nu^{6}\) | \(=\) |
\( 8\beta_{6} \)
|
| \(\nu^{7}\) | \(=\) |
\( 8\beta_{7} \)
|
| \(\nu^{8}\) | \(=\) |
\( 16\beta_{8} \)
|
| \(\nu^{9}\) | \(=\) |
\( 16\beta_{9} \)
|
| \(\nu^{10}\) | \(=\) |
\( 32\beta_{10} \)
|
| \(\nu^{11}\) | \(=\) |
\( 32\beta_{11} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/152\mathbb{Z}\right)^\times\).
| \(n\) | \(39\) | \(77\) | \(97\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(-\beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3.1 |
|
−0.909039 | + | 1.08335i | −0.301363 | − | 0.827987i | −0.347296 | − | 1.96962i | 0 | 1.17095 | + | 0.426191i | 0 | 2.44949 | + | 1.41421i | 1.70339 | − | 1.42931i | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 3.2 | 0.909039 | − | 1.08335i | 1.18075 | + | 3.24408i | −0.347296 | − | 1.96962i | 0 | 4.58782 | + | 1.66983i | 0 | −2.44949 | − | 1.41421i | −6.83175 | + | 5.73252i | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 51.1 | −0.909039 | − | 1.08335i | −0.301363 | + | 0.827987i | −0.347296 | + | 1.96962i | 0 | 1.17095 | − | 0.426191i | 0 | 2.44949 | − | 1.41421i | 1.70339 | + | 1.42931i | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 51.2 | 0.909039 | + | 1.08335i | 1.18075 | − | 3.24408i | −0.347296 | + | 1.96962i | 0 | 4.58782 | − | 1.66983i | 0 | −2.44949 | + | 1.41421i | −6.83175 | − | 5.73252i | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 59.1 | −1.39273 | − | 0.245576i | −0.950339 | + | 1.13257i | 1.87939 | + | 0.684040i | 0 | 1.60169 | − | 1.34398i | 0 | −2.44949 | − | 1.41421i | 0.141375 | + | 0.801775i | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 59.2 | 1.39273 | + | 0.245576i | −1.58175 | + | 1.88506i | 1.87939 | + | 0.684040i | 0 | −2.66587 | + | 2.23693i | 0 | 2.44949 | + | 1.41421i | −0.530560 | − | 3.00895i | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 67.1 | −1.39273 | + | 0.245576i | −0.950339 | − | 1.13257i | 1.87939 | − | 0.684040i | 0 | 1.60169 | + | 1.34398i | 0 | −2.44949 | + | 1.41421i | 0.141375 | − | 0.801775i | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 67.2 | 1.39273 | − | 0.245576i | −1.58175 | − | 1.88506i | 1.87939 | − | 0.684040i | 0 | −2.66587 | − | 2.23693i | 0 | 2.44949 | − | 1.41421i | −0.530560 | + | 3.00895i | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 91.1 | −0.483690 | + | 1.32893i | −3.29112 | + | 0.580314i | −1.53209 | − | 1.28558i | 0 | 0.820687 | − | 4.65435i | 0 | 2.44949 | − | 1.41421i | 7.67564 | − | 2.79370i | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 91.2 | 0.483690 | − | 1.32893i | 1.94383 | − | 0.342749i | −1.53209 | − | 1.28558i | 0 | 0.484720 | − | 2.74898i | 0 | −2.44949 | + | 1.41421i | 0.841902 | − | 0.306427i | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 147.1 | −0.483690 | − | 1.32893i | −3.29112 | − | 0.580314i | −1.53209 | + | 1.28558i | 0 | 0.820687 | + | 4.65435i | 0 | 2.44949 | + | 1.41421i | 7.67564 | + | 2.79370i | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 147.2 | 0.483690 | + | 1.32893i | 1.94383 | + | 0.342749i | −1.53209 | + | 1.28558i | 0 | 0.484720 | + | 2.74898i | 0 | −2.44949 | − | 1.41421i | 0.841902 | + | 0.306427i | 0 | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-2}) \) |
| 19.f | odd | 18 | 1 | inner |
| 152.v | even | 18 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 152.2.v.a | ✓ | 12 |
| 4.b | odd | 2 | 1 | 608.2.bh.a | 12 | ||
| 8.b | even | 2 | 1 | 608.2.bh.a | 12 | ||
| 8.d | odd | 2 | 1 | CM | 152.2.v.a | ✓ | 12 |
| 19.f | odd | 18 | 1 | inner | 152.2.v.a | ✓ | 12 |
| 76.k | even | 18 | 1 | 608.2.bh.a | 12 | ||
| 152.s | odd | 18 | 1 | 608.2.bh.a | 12 | ||
| 152.v | even | 18 | 1 | inner | 152.2.v.a | ✓ | 12 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 152.2.v.a | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 152.2.v.a | ✓ | 12 | 8.d | odd | 2 | 1 | CM |
| 152.2.v.a | ✓ | 12 | 19.f | odd | 18 | 1 | inner |
| 152.2.v.a | ✓ | 12 | 152.v | even | 18 | 1 | inner |
| 608.2.bh.a | 12 | 4.b | odd | 2 | 1 | ||
| 608.2.bh.a | 12 | 8.b | even | 2 | 1 | ||
| 608.2.bh.a | 12 | 76.k | even | 18 | 1 | ||
| 608.2.bh.a | 12 | 152.s | odd | 18 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} + 6 T_{3}^{11} + 15 T_{3}^{10} + 48 T_{3}^{9} + 138 T_{3}^{8} + 48 T_{3}^{7} - 932 T_{3}^{6} + \cdots + 5329 \)
acting on \(S_{2}^{\mathrm{new}}(152, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} - 8T^{6} + 64 \)
|
| $3$ |
\( T^{12} + 6 T^{11} + \cdots + 5329 \)
|
| $5$ |
\( T^{12} \)
|
| $7$ |
\( T^{12} \)
|
| $11$ |
\( T^{12} + 66 T^{10} + \cdots + 13461561 \)
|
| $13$ |
\( T^{12} \)
|
| $17$ |
\( T^{12} + 486 T^{9} + \cdots + 11390625 \)
|
| $19$ |
\( T^{12} + 106 T^{9} + \cdots + 47045881 \)
|
| $23$ |
\( T^{12} \)
|
| $29$ |
\( T^{12} \)
|
| $31$ |
\( T^{12} \)
|
| $37$ |
\( T^{12} \)
|
| $41$ |
\( T^{12} + \cdots + 41437894969 \)
|
| $43$ |
\( T^{12} + \cdots + 594823321 \)
|
| $47$ |
\( T^{12} \)
|
| $53$ |
\( T^{12} \)
|
| $59$ |
\( T^{12} + \cdots + 260443853569 \)
|
| $61$ |
\( T^{12} \)
|
| $67$ |
\( T^{12} + \cdots + 87534914769 \)
|
| $71$ |
\( T^{12} \)
|
| $73$ |
\( T^{12} + \cdots + 964620586801 \)
|
| $79$ |
\( T^{12} \)
|
| $83$ |
\( T^{12} + \cdots + 11478765321 \)
|
| $89$ |
\( T^{12} + \cdots + 168425239515625 \)
|
| $97$ |
\( T^{12} + \cdots + 7482560872329 \)
|
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