Newspace parameters
Level: | \( N \) | \(=\) | \( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 3332.be (of order \(14\), degree \(6\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.66288462209\) |
Analytic rank: | \(0\) |
Dimension: | \(12\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{14})\) |
Coefficient field: | \(\Q(\zeta_{28})\) |
Defining polynomial: |
\( x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Projective image: | \(D_{14}\) |
Projective field: | Galois closure of \(\mathbb{Q}[x]/(x^{14} - \cdots)\) |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3332\mathbb{Z}\right)^\times\).
\(n\) | \(785\) | \(885\) | \(1667\) |
\(\chi(n)\) | \(-1\) | \(-\zeta_{28}^{6}\) | \(-1\) |
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.
Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
407.1 |
|
0.222521 | − | 0.974928i | −0.974928 | + | 1.22252i | −0.900969 | − | 0.433884i | 0 | 0.974928 | + | 1.22252i | 0.781831 | + | 0.623490i | −0.623490 | + | 0.781831i | −0.321552 | − | 1.40881i | 0 | ||||||||||||||||||||||||||||||||||||||||
407.2 | 0.222521 | − | 0.974928i | 0.974928 | − | 1.22252i | −0.900969 | − | 0.433884i | 0 | −0.974928 | − | 1.22252i | −0.781831 | − | 0.623490i | −0.623490 | + | 0.781831i | −0.321552 | − | 1.40881i | 0 | |||||||||||||||||||||||||||||||||||||||||
1359.1 | 0.222521 | + | 0.974928i | −0.974928 | − | 1.22252i | −0.900969 | + | 0.433884i | 0 | 0.974928 | − | 1.22252i | 0.781831 | − | 0.623490i | −0.623490 | − | 0.781831i | −0.321552 | + | 1.40881i | 0 | |||||||||||||||||||||||||||||||||||||||||
1359.2 | 0.222521 | + | 0.974928i | 0.974928 | + | 1.22252i | −0.900969 | + | 0.433884i | 0 | −0.974928 | + | 1.22252i | −0.781831 | + | 0.623490i | −0.623490 | − | 0.781831i | −0.321552 | + | 1.40881i | 0 | |||||||||||||||||||||||||||||||||||||||||
1835.1 | 0.900969 | − | 0.433884i | −0.433884 | + | 1.90097i | 0.623490 | − | 0.781831i | 0 | 0.433884 | + | 1.90097i | −0.974928 | − | 0.222521i | 0.222521 | − | 0.974928i | −2.52446 | − | 1.21572i | 0 | |||||||||||||||||||||||||||||||||||||||||
1835.2 | 0.900969 | − | 0.433884i | 0.433884 | − | 1.90097i | 0.623490 | − | 0.781831i | 0 | −0.433884 | − | 1.90097i | 0.974928 | + | 0.222521i | 0.222521 | − | 0.974928i | −2.52446 | − | 1.21572i | 0 | |||||||||||||||||||||||||||||||||||||||||
2311.1 | −0.623490 | − | 0.781831i | −0.781831 | + | 0.376510i | −0.222521 | + | 0.974928i | 0 | 0.781831 | + | 0.376510i | −0.433884 | − | 0.900969i | 0.900969 | − | 0.433884i | −0.153989 | + | 0.193096i | 0 | |||||||||||||||||||||||||||||||||||||||||
2311.2 | −0.623490 | − | 0.781831i | 0.781831 | − | 0.376510i | −0.222521 | + | 0.974928i | 0 | −0.781831 | − | 0.376510i | 0.433884 | + | 0.900969i | 0.900969 | − | 0.433884i | −0.153989 | + | 0.193096i | 0 | |||||||||||||||||||||||||||||||||||||||||
2787.1 | −0.623490 | + | 0.781831i | −0.781831 | − | 0.376510i | −0.222521 | − | 0.974928i | 0 | 0.781831 | − | 0.376510i | −0.433884 | + | 0.900969i | 0.900969 | + | 0.433884i | −0.153989 | − | 0.193096i | 0 | |||||||||||||||||||||||||||||||||||||||||
2787.2 | −0.623490 | + | 0.781831i | 0.781831 | + | 0.376510i | −0.222521 | − | 0.974928i | 0 | −0.781831 | + | 0.376510i | 0.433884 | − | 0.900969i | 0.900969 | + | 0.433884i | −0.153989 | − | 0.193096i | 0 | |||||||||||||||||||||||||||||||||||||||||
3263.1 | 0.900969 | + | 0.433884i | −0.433884 | − | 1.90097i | 0.623490 | + | 0.781831i | 0 | 0.433884 | − | 1.90097i | −0.974928 | + | 0.222521i | 0.222521 | + | 0.974928i | −2.52446 | + | 1.21572i | 0 | |||||||||||||||||||||||||||||||||||||||||
3263.2 | 0.900969 | + | 0.433884i | 0.433884 | + | 1.90097i | 0.623490 | + | 0.781831i | 0 | −0.433884 | + | 1.90097i | 0.974928 | − | 0.222521i | 0.222521 | + | 0.974928i | −2.52446 | + | 1.21572i | 0 | |||||||||||||||||||||||||||||||||||||||||
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.e | even | 7 | 1 | inner |
196.k | odd | 14 | 1 | inner |
833.r | even | 14 | 1 | inner |
3332.be | odd | 14 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 3332.1.be.c | ✓ | 12 |
4.b | odd | 2 | 1 | inner | 3332.1.be.c | ✓ | 12 |
17.b | even | 2 | 1 | inner | 3332.1.be.c | ✓ | 12 |
49.e | even | 7 | 1 | inner | 3332.1.be.c | ✓ | 12 |
68.d | odd | 2 | 1 | CM | 3332.1.be.c | ✓ | 12 |
196.k | odd | 14 | 1 | inner | 3332.1.be.c | ✓ | 12 |
833.r | even | 14 | 1 | inner | 3332.1.be.c | ✓ | 12 |
3332.be | odd | 14 | 1 | inner | 3332.1.be.c | ✓ | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
3332.1.be.c | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
3332.1.be.c | ✓ | 12 | 4.b | odd | 2 | 1 | inner |
3332.1.be.c | ✓ | 12 | 17.b | even | 2 | 1 | inner |
3332.1.be.c | ✓ | 12 | 49.e | even | 7 | 1 | inner |
3332.1.be.c | ✓ | 12 | 68.d | odd | 2 | 1 | CM |
3332.1.be.c | ✓ | 12 | 196.k | odd | 14 | 1 | inner |
3332.1.be.c | ✓ | 12 | 833.r | even | 14 | 1 | inner |
3332.1.be.c | ✓ | 12 | 3332.be | odd | 14 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} + 7T_{3}^{10} + 21T_{3}^{8} + 35T_{3}^{6} + 49T_{3}^{4} - 49T_{3}^{2} + 49 \)
acting on \(S_{1}^{\mathrm{new}}(3332, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{6} - T^{5} + T^{4} - T^{3} + T^{2} - T + 1)^{2} \)
$3$
\( T^{12} + 7 T^{10} + 21 T^{8} + 35 T^{6} + \cdots + 49 \)
$5$
\( T^{12} \)
$7$
\( T^{12} - T^{10} + T^{8} - T^{6} + T^{4} - T^{2} + \cdots + 1 \)
$11$
\( T^{12} + 7 T^{10} + 21 T^{8} + 35 T^{6} + \cdots + 49 \)
$13$
\( (T^{6} + 5 T^{5} + 11 T^{4} + 13 T^{3} + \cdots + 1)^{2} \)
$17$
\( (T^{6} + T^{5} + T^{4} + T^{3} + T^{2} + T + 1)^{2} \)
$19$
\( T^{12} \)
$23$
\( T^{12} + 35 T^{6} + 98 T^{4} + 49 T^{2} + \cdots + 49 \)
$29$
\( T^{12} \)
$31$
\( (T^{6} - 7 T^{4} + 14 T^{2} - 7)^{2} \)
$37$
\( T^{12} \)
$41$
\( T^{12} \)
$43$
\( T^{12} \)
$47$
\( T^{12} \)
$53$
\( (T^{6} + 2 T^{5} + 4 T^{4} + 8 T^{3} + 9 T^{2} + \cdots + 1)^{2} \)
$59$
\( T^{12} \)
$61$
\( T^{12} \)
$67$
\( T^{12} \)
$71$
\( T^{12} + 14 T^{8} - 14 T^{6} + 49 T^{4} + \cdots + 49 \)
$73$
\( T^{12} \)
$79$
\( (T^{6} - 7 T^{4} + 14 T^{2} - 7)^{2} \)
$83$
\( T^{12} \)
$89$
\( (T^{6} - 2 T^{5} + 4 T^{4} - T^{3} + 2 T^{2} + \cdots + 1)^{2} \)
$97$
\( T^{12} \)
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