Newspace parameters
Level: | \( N \) | \(=\) | \( 3311 = 7 \cdot 11 \cdot 43 \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 3311.bx (of order \(14\), degree \(6\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.65240425683\) |
Analytic rank: | \(0\) |
Dimension: | \(6\) |
Coefficient field: | \(\Q(\zeta_{14})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
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 \(\left(\mathbb{Z}/3311\mathbb{Z}\right)^\times\).
\(n\) | \(904\) | \(1893\) | \(2927\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(\zeta_{14}^{5}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
538.1 |
|
0.400969 | + | 0.193096i | 0 | −0.500000 | − | 0.626980i | 0 | 0 | 1.00000 | −0.178448 | − | 0.781831i | −0.623490 | + | 0.781831i | 0 | ||||||||||||||||||||||||||||
1077.1 | 0.400969 | − | 0.193096i | 0 | −0.500000 | + | 0.626980i | 0 | 0 | 1.00000 | −0.178448 | + | 0.781831i | −0.623490 | − | 0.781831i | 0 | |||||||||||||||||||||||||||||
1231.1 | −0.277479 | − | 1.21572i | 0 | −0.500000 | + | 0.240787i | 0 | 0 | 1.00000 | −0.346011 | − | 0.433884i | 0.900969 | + | 0.433884i | 0 | |||||||||||||||||||||||||||||
1924.1 | −1.12349 | − | 1.40881i | 0 | −0.500000 | + | 2.19064i | 0 | 0 | 1.00000 | 2.02446 | − | 0.974928i | 0.222521 | + | 0.974928i | 0 | |||||||||||||||||||||||||||||
2232.1 | −1.12349 | + | 1.40881i | 0 | −0.500000 | − | 2.19064i | 0 | 0 | 1.00000 | 2.02446 | + | 0.974928i | 0.222521 | − | 0.974928i | 0 | |||||||||||||||||||||||||||||
3233.1 | −0.277479 | + | 1.21572i | 0 | −0.500000 | − | 0.240787i | 0 | 0 | 1.00000 | −0.346011 | + | 0.433884i | 0.900969 | − | 0.433884i | 0 | |||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-7}) \) |
473.n | even | 14 | 1 | inner |
3311.bx | odd | 14 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 3311.1.bx.a | ✓ | 6 |
7.b | odd | 2 | 1 | CM | 3311.1.bx.a | ✓ | 6 |
11.b | odd | 2 | 1 | 3311.1.bx.b | yes | 6 | |
43.f | odd | 14 | 1 | 3311.1.bx.b | yes | 6 | |
77.b | even | 2 | 1 | 3311.1.bx.b | yes | 6 | |
301.w | even | 14 | 1 | 3311.1.bx.b | yes | 6 | |
473.n | even | 14 | 1 | inner | 3311.1.bx.a | ✓ | 6 |
3311.bx | odd | 14 | 1 | inner | 3311.1.bx.a | ✓ | 6 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
3311.1.bx.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
3311.1.bx.a | ✓ | 6 | 7.b | odd | 2 | 1 | CM |
3311.1.bx.a | ✓ | 6 | 473.n | even | 14 | 1 | inner |
3311.1.bx.a | ✓ | 6 | 3311.bx | odd | 14 | 1 | inner |
3311.1.bx.b | yes | 6 | 11.b | odd | 2 | 1 | |
3311.1.bx.b | yes | 6 | 43.f | odd | 14 | 1 | |
3311.1.bx.b | yes | 6 | 77.b | even | 2 | 1 | |
3311.1.bx.b | yes | 6 | 301.w | even | 14 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 2T_{2}^{5} + 4T_{2}^{4} + T_{2}^{3} + 2T_{2}^{2} - 3T_{2} + 1 \)
acting on \(S_{1}^{\mathrm{new}}(3311, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{6} + 2 T^{5} + 4 T^{4} + T^{3} + 2 T^{2} + \cdots + 1 \)
$3$
\( T^{6} \)
$5$
\( T^{6} \)
$7$
\( (T - 1)^{6} \)
$11$
\( T^{6} - T^{5} + T^{4} - T^{3} + T^{2} - T + 1 \)
$13$
\( T^{6} \)
$17$
\( T^{6} \)
$19$
\( T^{6} \)
$23$
\( T^{6} - 2 T^{5} + 4 T^{4} - 8 T^{3} + \cdots + 1 \)
$29$
\( T^{6} - 2 T^{5} + 4 T^{4} - 8 T^{3} + \cdots + 1 \)
$31$
\( T^{6} \)
$37$
\( T^{6} + 7 T^{4} + 14 T^{2} + 7 \)
$41$
\( T^{6} \)
$43$
\( T^{6} - T^{5} + T^{4} - T^{3} + T^{2} - T + 1 \)
$47$
\( T^{6} \)
$53$
\( T^{6} - 5 T^{5} + 11 T^{4} - 13 T^{3} + \cdots + 1 \)
$59$
\( T^{6} \)
$61$
\( T^{6} \)
$67$
\( T^{6} + 2 T^{5} + 4 T^{4} + T^{3} + 2 T^{2} + \cdots + 1 \)
$71$
\( T^{6} \)
$73$
\( T^{6} \)
$79$
\( T^{6} + 7 T^{4} + 14 T^{2} + 7 \)
$83$
\( T^{6} \)
$89$
\( T^{6} \)
$97$
\( T^{6} \)
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