Newspace parameters
Level: | \( N \) | \(=\) | \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 252.bj (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.01223013094\) |
Analytic rank: | \(1\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
Coefficient field: | \(\Q(\zeta_{12})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} - x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).
\(n\) | \(29\) | \(73\) | \(127\) |
\(\chi(n)\) | \(-\zeta_{12}^{2}\) | \(\zeta_{12}^{2}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
103.1 |
|
−1.00000 | − | 1.00000i | −0.866025 | + | 1.50000i | 2.00000i | −3.00000 | + | 1.73205i | 2.36603 | − | 0.633975i | 2.59808 | − | 0.500000i | 2.00000 | − | 2.00000i | −1.50000 | − | 2.59808i | 4.73205 | + | 1.26795i | ||||||||||||||
103.2 | −1.00000 | + | 1.00000i | 0.866025 | − | 1.50000i | − | 2.00000i | −3.00000 | + | 1.73205i | 0.633975 | + | 2.36603i | −2.59808 | + | 0.500000i | 2.00000 | + | 2.00000i | −1.50000 | − | 2.59808i | 1.26795 | − | 4.73205i | ||||||||||||||
115.1 | −1.00000 | − | 1.00000i | 0.866025 | + | 1.50000i | 2.00000i | −3.00000 | − | 1.73205i | 0.633975 | − | 2.36603i | −2.59808 | − | 0.500000i | 2.00000 | − | 2.00000i | −1.50000 | + | 2.59808i | 1.26795 | + | 4.73205i | |||||||||||||||
115.2 | −1.00000 | + | 1.00000i | −0.866025 | − | 1.50000i | − | 2.00000i | −3.00000 | − | 1.73205i | 2.36603 | + | 0.633975i | 2.59808 | + | 0.500000i | 2.00000 | + | 2.00000i | −1.50000 | + | 2.59808i | 4.73205 | − | 1.26795i | ||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
63.t | odd | 6 | 1 | inner |
252.bj | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 252.2.bj.a | yes | 4 |
3.b | odd | 2 | 1 | 756.2.bj.a | 4 | ||
4.b | odd | 2 | 1 | inner | 252.2.bj.a | yes | 4 |
7.d | odd | 6 | 1 | 252.2.n.a | ✓ | 4 | |
9.c | even | 3 | 1 | 252.2.n.a | ✓ | 4 | |
9.d | odd | 6 | 1 | 756.2.n.a | 4 | ||
12.b | even | 2 | 1 | 756.2.bj.a | 4 | ||
21.g | even | 6 | 1 | 756.2.n.a | 4 | ||
28.f | even | 6 | 1 | 252.2.n.a | ✓ | 4 | |
36.f | odd | 6 | 1 | 252.2.n.a | ✓ | 4 | |
36.h | even | 6 | 1 | 756.2.n.a | 4 | ||
63.i | even | 6 | 1 | 756.2.bj.a | 4 | ||
63.t | odd | 6 | 1 | inner | 252.2.bj.a | yes | 4 |
84.j | odd | 6 | 1 | 756.2.n.a | 4 | ||
252.r | odd | 6 | 1 | 756.2.bj.a | 4 | ||
252.bj | even | 6 | 1 | inner | 252.2.bj.a | yes | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
252.2.n.a | ✓ | 4 | 7.d | odd | 6 | 1 | |
252.2.n.a | ✓ | 4 | 9.c | even | 3 | 1 | |
252.2.n.a | ✓ | 4 | 28.f | even | 6 | 1 | |
252.2.n.a | ✓ | 4 | 36.f | odd | 6 | 1 | |
252.2.bj.a | yes | 4 | 1.a | even | 1 | 1 | trivial |
252.2.bj.a | yes | 4 | 4.b | odd | 2 | 1 | inner |
252.2.bj.a | yes | 4 | 63.t | odd | 6 | 1 | inner |
252.2.bj.a | yes | 4 | 252.bj | even | 6 | 1 | inner |
756.2.n.a | 4 | 9.d | odd | 6 | 1 | ||
756.2.n.a | 4 | 21.g | even | 6 | 1 | ||
756.2.n.a | 4 | 36.h | even | 6 | 1 | ||
756.2.n.a | 4 | 84.j | odd | 6 | 1 | ||
756.2.bj.a | 4 | 3.b | odd | 2 | 1 | ||
756.2.bj.a | 4 | 12.b | even | 2 | 1 | ||
756.2.bj.a | 4 | 63.i | even | 6 | 1 | ||
756.2.bj.a | 4 | 252.r | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 6T_{5} + 12 \)
acting on \(S_{2}^{\mathrm{new}}(252, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} + 2 T + 2)^{2} \)
$3$
\( T^{4} + 3T^{2} + 9 \)
$5$
\( (T^{2} + 6 T + 12)^{2} \)
$7$
\( T^{4} - 13T^{2} + 49 \)
$11$
\( T^{4} - 4T^{2} + 16 \)
$13$
\( (T^{2} + 9 T + 27)^{2} \)
$17$
\( (T^{2} + 9 T + 27)^{2} \)
$19$
\( T^{4} + 3T^{2} + 9 \)
$23$
\( T^{4} - 16T^{2} + 256 \)
$29$
\( (T^{2} + 5 T + 25)^{2} \)
$31$
\( (T^{2} - 27)^{2} \)
$37$
\( (T^{2} + 3 T + 9)^{2} \)
$41$
\( (T^{2} + 3 T + 3)^{2} \)
$43$
\( T^{4} - 121 T^{2} + 14641 \)
$47$
\( (T^{2} - 3)^{2} \)
$53$
\( (T^{2} + T + 1)^{2} \)
$59$
\( (T^{2} - 3)^{2} \)
$61$
\( (T^{2} + 27)^{2} \)
$67$
\( (T^{2} + 81)^{2} \)
$71$
\( (T^{2} + 4)^{2} \)
$73$
\( (T^{2} + 21 T + 147)^{2} \)
$79$
\( (T^{2} + 9)^{2} \)
$83$
\( T^{4} + 27T^{2} + 729 \)
$89$
\( (T^{2} - 15 T + 75)^{2} \)
$97$
\( (T^{2} + 3 T + 3)^{2} \)
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