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: | \(24\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{42})\) |
| Coefficient field: | \(\Q(\zeta_{84})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{24} + x^{22} - x^{18} - x^{16} + x^{12} - x^{8} - x^{6} + x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Projective image: | \(D_{42}\) |
| Projective field: | Galois closure of \(\mathbb{Q}[x]/(x^{42} - \cdots)\) |
$q$-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 \(\left(\mathbb{Z}/3332\mathbb{Z}\right)^\times\).
| \(n\) | \(785\) | \(885\) | \(1667\) |
| \(\chi(n)\) | \(-1\) | \(\zeta_{84}^{16}\) | \(-1\) |
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 | −0.997204 | + | 0.925270i | 0.365341 | − | 0.930874i | 0 | 0.302705 | − | 1.32624i | 0.974928 | − | 0.222521i | 0.222521 | + | 0.974928i | 0.0635609 | − | 0.848162i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 135.2 | −0.826239 | + | 0.563320i | 0.997204 | − | 0.925270i | 0.365341 | − | 0.930874i | 0 | −0.302705 | + | 1.32624i | −0.974928 | + | 0.222521i | 0.222521 | + | 0.974928i | 0.0635609 | − | 0.848162i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 543.1 | −0.826239 | − | 0.563320i | −0.997204 | − | 0.925270i | 0.365341 | + | 0.930874i | 0 | 0.302705 | + | 1.32624i | 0.974928 | + | 0.222521i | 0.222521 | − | 0.974928i | 0.0635609 | + | 0.848162i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 543.2 | −0.826239 | − | 0.563320i | 0.997204 | + | 0.925270i | 0.365341 | + | 0.930874i | 0 | −0.302705 | − | 1.32624i | −0.974928 | − | 0.222521i | 0.222521 | − | 0.974928i | 0.0635609 | + | 0.848162i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 611.1 | −0.0747301 | + | 0.997204i | −0.563320 | + | 0.173761i | −0.988831 | − | 0.149042i | 0 | −0.131178 | − | 0.574730i | −0.974928 | − | 0.222521i | 0.222521 | − | 0.974928i | −0.539102 | + | 0.367554i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 611.2 | −0.0747301 | + | 0.997204i | 0.563320 | − | 0.173761i | −0.988831 | − | 0.149042i | 0 | 0.131178 | + | 0.574730i | 0.974928 | + | 0.222521i | 0.222521 | − | 0.974928i | −0.539102 | + | 0.367554i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1019.1 | 0.988831 | + | 0.149042i | −0.930874 | + | 0.634659i | 0.955573 | + | 0.294755i | 0 | −1.01507 | + | 0.488831i | 0.433884 | − | 0.900969i | 0.900969 | + | 0.433884i | 0.0983929 | − | 0.250701i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1019.2 | 0.988831 | + | 0.149042i | 0.930874 | − | 0.634659i | 0.955573 | + | 0.294755i | 0 | 1.01507 | − | 0.488831i | −0.433884 | + | 0.900969i | 0.900969 | + | 0.433884i | 0.0983929 | − | 0.250701i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1087.1 | 0.733052 | + | 0.680173i | −0.294755 | − | 0.0444272i | 0.0747301 | + | 0.997204i | 0 | −0.185853 | − | 0.233052i | −0.781831 | − | 0.623490i | −0.623490 | + | 0.781831i | −0.870666 | − | 0.268565i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1087.2 | 0.733052 | + | 0.680173i | 0.294755 | + | 0.0444272i | 0.0747301 | + | 0.997204i | 0 | 0.185853 | + | 0.233052i | 0.781831 | + | 0.623490i | −0.623490 | + | 0.781831i | −0.870666 | − | 0.268565i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1495.1 | −0.955573 | + | 0.294755i | −0.680173 | − | 1.73305i | 0.826239 | − | 0.563320i | 0 | 1.16078 | + | 1.45557i | −0.781831 | − | 0.623490i | −0.623490 | + | 0.781831i | −1.80778 | + | 1.67738i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1495.2 | −0.955573 | + | 0.294755i | 0.680173 | + | 1.73305i | 0.826239 | − | 0.563320i | 0 | −1.16078 | − | 1.45557i | 0.781831 | + | 0.623490i | −0.623490 | + | 0.781831i | −1.80778 | + | 1.67738i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1563.1 | 0.988831 | − | 0.149042i | −0.930874 | − | 0.634659i | 0.955573 | − | 0.294755i | 0 | −1.01507 | − | 0.488831i | 0.433884 | + | 0.900969i | 0.900969 | − | 0.433884i | 0.0983929 | + | 0.250701i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1563.2 | 0.988831 | − | 0.149042i | 0.930874 | + | 0.634659i | 0.955573 | − | 0.294755i | 0 | 1.01507 | + | 0.488831i | −0.433884 | − | 0.900969i | 0.900969 | − | 0.433884i | 0.0983929 | + | 0.250701i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1971.1 | 0.733052 | − | 0.680173i | −0.294755 | + | 0.0444272i | 0.0747301 | − | 0.997204i | 0 | −0.185853 | + | 0.233052i | −0.781831 | + | 0.623490i | −0.623490 | − | 0.781831i | −0.870666 | + | 0.268565i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1971.2 | 0.733052 | − | 0.680173i | 0.294755 | − | 0.0444272i | 0.0747301 | − | 0.997204i | 0 | 0.185853 | − | 0.233052i | 0.781831 | − | 0.623490i | −0.623490 | − | 0.781831i | −0.870666 | + | 0.268565i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2447.1 | −0.365341 | + | 0.930874i | −0.149042 | + | 1.98883i | −0.733052 | − | 0.680173i | 0 | −1.79690 | − | 0.865341i | −0.433884 | − | 0.900969i | 0.900969 | − | 0.433884i | −2.94440 | − | 0.443797i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2447.2 | −0.365341 | + | 0.930874i | 0.149042 | − | 1.98883i | −0.733052 | − | 0.680173i | 0 | 1.79690 | + | 0.865341i | 0.433884 | + | 0.900969i | 0.900969 | − | 0.433884i | −2.94440 | − | 0.443797i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2515.1 | −0.365341 | − | 0.930874i | −0.149042 | − | 1.98883i | −0.733052 | + | 0.680173i | 0 | −1.79690 | + | 0.865341i | −0.433884 | + | 0.900969i | 0.900969 | + | 0.433884i | −2.94440 | + | 0.443797i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2515.2 | −0.365341 | − | 0.930874i | 0.149042 | + | 1.98883i | −0.733052 | + | 0.680173i | 0 | 1.79690 | − | 0.865341i | 0.433884 | − | 0.900969i | 0.900969 | + | 0.433884i | −2.94440 | + | 0.443797i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| See all 24 embeddings | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 68.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-17}) \) |
| 4.b | odd | 2 | 1 | inner |
| 17.b | even | 2 | 1 | inner |
| 49.g | even | 21 | 1 | inner |
| 196.o | odd | 42 | 1 | inner |
| 833.z | even | 42 | 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.c | ✓ | 24 |
| 4.b | odd | 2 | 1 | inner | 3332.1.cc.c | ✓ | 24 |
| 17.b | even | 2 | 1 | inner | 3332.1.cc.c | ✓ | 24 |
| 49.g | even | 21 | 1 | inner | 3332.1.cc.c | ✓ | 24 |
| 68.d | odd | 2 | 1 | CM | 3332.1.cc.c | ✓ | 24 |
| 196.o | odd | 42 | 1 | inner | 3332.1.cc.c | ✓ | 24 |
| 833.z | even | 42 | 1 | inner | 3332.1.cc.c | ✓ | 24 |
| 3332.cc | odd | 42 | 1 | inner | 3332.1.cc.c | ✓ | 24 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3332.1.cc.c | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
| 3332.1.cc.c | ✓ | 24 | 4.b | odd | 2 | 1 | inner |
| 3332.1.cc.c | ✓ | 24 | 17.b | even | 2 | 1 | inner |
| 3332.1.cc.c | ✓ | 24 | 49.g | even | 21 | 1 | inner |
| 3332.1.cc.c | ✓ | 24 | 68.d | odd | 2 | 1 | CM |
| 3332.1.cc.c | ✓ | 24 | 196.o | odd | 42 | 1 | inner |
| 3332.1.cc.c | ✓ | 24 | 833.z | even | 42 | 1 | inner |
| 3332.1.cc.c | ✓ | 24 | 3332.cc | odd | 42 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{24} + 11 T_{3}^{22} + 49 T_{3}^{20} + 118 T_{3}^{18} + 192 T_{3}^{16} + 336 T_{3}^{14} + 421 T_{3}^{12} + \cdots + 1 \)
acting on \(S_{1}^{\mathrm{new}}(3332, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{12} + T^{11} - T^{9} + \cdots + 1)^{2} \)
|
| $3$ |
\( T^{24} + 11 T^{22} + \cdots + 1 \)
|
| $5$ |
\( T^{24} \)
|
| $7$ |
\( (T^{12} - T^{10} + T^{8} + \cdots + 1)^{2} \)
|
| $11$ |
\( T^{24} - 10 T^{22} + \cdots + 1 \)
|
| $13$ |
\( (T^{12} - 5 T^{11} + \cdots + 1)^{2} \)
|
| $17$ |
\( (T^{12} - T^{11} + T^{9} + \cdots + 1)^{2} \)
|
| $19$ |
\( T^{24} \)
|
| $23$ |
\( T^{24} + 70 T^{18} + \cdots + 2401 \)
|
| $29$ |
\( T^{24} \)
|
| $31$ |
\( (T^{12} + 7 T^{10} + \cdots + 49)^{2} \)
|
| $37$ |
\( T^{24} \)
|
| $41$ |
\( T^{24} \)
|
| $43$ |
\( T^{24} \)
|
| $47$ |
\( T^{24} \)
|
| $53$ |
\( (T^{12} + T^{11} - T^{9} + \cdots + 1)^{2} \)
|
| $59$ |
\( T^{24} \)
|
| $61$ |
\( T^{24} \)
|
| $67$ |
\( T^{24} \)
|
| $71$ |
\( T^{24} + 6 T^{22} + \cdots + 1 \)
|
| $73$ |
\( T^{24} \)
|
| $79$ |
\( T^{24} + 11 T^{22} + \cdots + 1 \)
|
| $83$ |
\( T^{24} \)
|
| $89$ |
\( (T^{12} - T^{11} - 6 T^{9} + \cdots + 1)^{2} \)
|
| $97$ |
\( T^{24} \)
|
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