Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3332,1,Mod(135,3332)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3332, base_ring=CyclotomicField(42))
chi = DirichletCharacter(H, H._module([21, 32, 21]))
N = Newforms(chi, 1, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3332.135");
S:= CuspForms(chi, 1);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3332.cc (of order \(42\), degree \(12\), 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: | \(1.66288462209\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\Q(\zeta_{21})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Projective image: | \(D_{21}\) |
| Projective field: | Galois closure of \(\mathbb{Q}[x]/(x^{21} + \cdots)\) |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3332\mathbb{Z}\right)^\times\).
| \(n\) | \(785\) | \(885\) | \(1667\) |
| \(\chi(n)\) | \(-1\) | \(-\zeta_{42}^{11}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 135.1 |
|
0.826239 | − | 0.563320i | 1.07473 | − | 0.997204i | 0.365341 | − | 0.930874i | 0 | 0.326239 | − | 1.42935i | −0.222521 | − | 0.974928i | −0.222521 | − | 0.974928i | 0.0858993 | − | 1.14625i | 0 | ||||||||||||||||||||||||||||||||||||||||
| 543.1 | 0.826239 | + | 0.563320i | 1.07473 | + | 0.997204i | 0.365341 | + | 0.930874i | 0 | 0.326239 | + | 1.42935i | −0.222521 | + | 0.974928i | −0.222521 | + | 0.974928i | 0.0858993 | + | 1.14625i | 0 | |||||||||||||||||||||||||||||||||||||||||
| 611.1 | 0.0747301 | − | 0.997204i | 1.82624 | − | 0.563320i | −0.988831 | − | 0.149042i | 0 | −0.425270 | − | 1.86323i | −0.222521 | + | 0.974928i | −0.222521 | + | 0.974928i | 2.19158 | − | 1.49419i | 0 | |||||||||||||||||||||||||||||||||||||||||
| 1019.1 | −0.988831 | − | 0.149042i | 1.36534 | − | 0.930874i | 0.955573 | + | 0.294755i | 0 | −1.48883 | + | 0.716983i | −0.900969 | − | 0.433884i | −0.900969 | − | 0.433884i | 0.632289 | − | 1.61105i | 0 | |||||||||||||||||||||||||||||||||||||||||
| 1087.1 | −0.733052 | − | 0.680173i | 1.95557 | + | 0.294755i | 0.0747301 | + | 0.997204i | 0 | −1.23305 | − | 1.54620i | 0.623490 | − | 0.781831i | 0.623490 | − | 0.781831i | 2.78181 | + | 0.858075i | 0 | |||||||||||||||||||||||||||||||||||||||||
| 1495.1 | 0.955573 | − | 0.294755i | 0.266948 | + | 0.680173i | 0.826239 | − | 0.563320i | 0 | 0.455573 | + | 0.571270i | 0.623490 | − | 0.781831i | 0.623490 | − | 0.781831i | 0.341678 | − | 0.317031i | 0 | |||||||||||||||||||||||||||||||||||||||||
| 1563.1 | −0.988831 | + | 0.149042i | 1.36534 | + | 0.930874i | 0.955573 | − | 0.294755i | 0 | −1.48883 | − | 0.716983i | −0.900969 | + | 0.433884i | −0.900969 | + | 0.433884i | 0.632289 | + | 1.61105i | 0 | |||||||||||||||||||||||||||||||||||||||||
| 1971.1 | −0.733052 | + | 0.680173i | 1.95557 | − | 0.294755i | 0.0747301 | − | 0.997204i | 0 | −1.23305 | + | 1.54620i | 0.623490 | + | 0.781831i | 0.623490 | + | 0.781831i | 2.78181 | − | 0.858075i | 0 | |||||||||||||||||||||||||||||||||||||||||
| 2447.1 | 0.365341 | − | 0.930874i | 0.0111692 | − | 0.149042i | −0.733052 | − | 0.680173i | 0 | −0.134659 | − | 0.0648483i | −0.900969 | + | 0.433884i | −0.900969 | + | 0.433884i | 0.966742 | + | 0.145713i | 0 | |||||||||||||||||||||||||||||||||||||||||
| 2515.1 | 0.365341 | + | 0.930874i | 0.0111692 | + | 0.149042i | −0.733052 | + | 0.680173i | 0 | −0.134659 | + | 0.0648483i | −0.900969 | − | 0.433884i | −0.900969 | − | 0.433884i | 0.966742 | − | 0.145713i | 0 | |||||||||||||||||||||||||||||||||||||||||
| 2923.1 | 0.0747301 | + | 0.997204i | 1.82624 | + | 0.563320i | −0.988831 | + | 0.149042i | 0 | −0.425270 | + | 1.86323i | −0.222521 | − | 0.974928i | −0.222521 | − | 0.974928i | 2.19158 | + | 1.49419i | 0 | |||||||||||||||||||||||||||||||||||||||||
| 2991.1 | 0.955573 | + | 0.294755i | 0.266948 | − | 0.680173i | 0.826239 | + | 0.563320i | 0 | 0.455573 | − | 0.571270i | 0.623490 | + | 0.781831i | 0.623490 | + | 0.781831i | 0.341678 | + | 0.317031i | 0 | |||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 68.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-17}) \) |
| 49.g | even | 21 | 1 | inner |
| 3332.cc | odd | 42 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3332.1.cc.b | yes | 12 |
| 4.b | odd | 2 | 1 | 3332.1.cc.a | ✓ | 12 | |
| 17.b | even | 2 | 1 | 3332.1.cc.a | ✓ | 12 | |
| 49.g | even | 21 | 1 | inner | 3332.1.cc.b | yes | 12 |
| 68.d | odd | 2 | 1 | CM | 3332.1.cc.b | yes | 12 |
| 196.o | odd | 42 | 1 | 3332.1.cc.a | ✓ | 12 | |
| 833.z | even | 42 | 1 | 3332.1.cc.a | ✓ | 12 | |
| 3332.cc | odd | 42 | 1 | inner | 3332.1.cc.b | yes | 12 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3332.1.cc.a | ✓ | 12 | 4.b | odd | 2 | 1 | |
| 3332.1.cc.a | ✓ | 12 | 17.b | even | 2 | 1 | |
| 3332.1.cc.a | ✓ | 12 | 196.o | odd | 42 | 1 | |
| 3332.1.cc.a | ✓ | 12 | 833.z | even | 42 | 1 | |
| 3332.1.cc.b | yes | 12 | 1.a | even | 1 | 1 | trivial |
| 3332.1.cc.b | yes | 12 | 49.g | even | 21 | 1 | inner |
| 3332.1.cc.b | yes | 12 | 68.d | odd | 2 | 1 | CM |
| 3332.1.cc.b | yes | 12 | 3332.cc | odd | 42 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} - 13 T_{3}^{11} + 77 T_{3}^{10} - 274 T_{3}^{9} + 650 T_{3}^{8} - 1078 T_{3}^{7} + 1275 T_{3}^{6} + \cdots + 1 \)
acting on \(S_{1}^{\mathrm{new}}(3332, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} - T^{11} + \cdots + 1 \)
|
| $3$ |
\( T^{12} - 13 T^{11} + \cdots + 1 \)
|
| $5$ |
\( T^{12} \)
|
| $7$ |
\( (T^{6} + T^{5} + T^{4} + \cdots + 1)^{2} \)
|
| $11$ |
\( T^{12} + 8 T^{11} + \cdots + 1 \)
|
| $13$ |
\( T^{12} + 5 T^{11} + \cdots + 1 \)
|
| $17$ |
\( T^{12} - T^{11} + \cdots + 1 \)
|
| $19$ |
\( T^{12} \)
|
| $23$ |
\( T^{12} - 2 T^{11} + \cdots + 1 \)
|
| $29$ |
\( T^{12} \)
|
| $31$ |
\( (T^{6} - T^{5} + 3 T^{4} + \cdots + 1)^{2} \)
|
| $37$ |
\( T^{12} \)
|
| $41$ |
\( T^{12} \)
|
| $43$ |
\( T^{12} \)
|
| $47$ |
\( T^{12} \)
|
| $53$ |
\( T^{12} + T^{11} + \cdots + 1 \)
|
| $59$ |
\( T^{12} \)
|
| $61$ |
\( T^{12} \)
|
| $67$ |
\( T^{12} \)
|
| $71$ |
\( T^{12} - 2 T^{11} + \cdots + 1 \)
|
| $73$ |
\( T^{12} \)
|
| $79$ |
\( T^{12} + T^{11} + \cdots + 1 \)
|
| $83$ |
\( T^{12} \)
|
| $89$ |
\( T^{12} + T^{11} + \cdots + 1 \)
|
| $97$ |
\( T^{12} \)
|
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