Newspace parameters
Level: | \( N \) | \(=\) | \( 126 = 2 \cdot 3^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 126.n (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(3.43325133094\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
Coefficient field: | \(\Q(\sqrt{2}, \sqrt{-3})\) |
Defining polynomial: |
\( x^{4} + 2x^{2} + 4 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 14) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{4} + 2x^{2} + 4 \)
:
\(\beta_{1}\) | \(=\) |
\( \nu \)
|
\(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 2 \)
|
\(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 2 \)
|
\(\nu\) | \(=\) |
\( \beta_1 \)
|
\(\nu^{2}\) | \(=\) |
\( 2\beta_{2} \)
|
\(\nu^{3}\) | \(=\) |
\( 2\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/126\mathbb{Z}\right)^\times\).
\(n\) | \(29\) | \(73\) |
\(\chi(n)\) | \(1\) | \(1 + \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
19.1 |
|
−0.707107 | + | 1.22474i | 0 | −1.00000 | − | 1.73205i | −2.74264 | − | 1.58346i | 0 | −2.24264 | − | 6.63103i | 2.82843 | 0 | 3.87868 | − | 2.23936i | ||||||||||||||||||||
19.2 | 0.707107 | − | 1.22474i | 0 | −1.00000 | − | 1.73205i | 5.74264 | + | 3.31552i | 0 | 6.24264 | + | 3.16693i | −2.82843 | 0 | 8.12132 | − | 4.68885i | |||||||||||||||||||||
73.1 | −0.707107 | − | 1.22474i | 0 | −1.00000 | + | 1.73205i | −2.74264 | + | 1.58346i | 0 | −2.24264 | + | 6.63103i | 2.82843 | 0 | 3.87868 | + | 2.23936i | |||||||||||||||||||||
73.2 | 0.707107 | + | 1.22474i | 0 | −1.00000 | + | 1.73205i | 5.74264 | − | 3.31552i | 0 | 6.24264 | − | 3.16693i | −2.82843 | 0 | 8.12132 | + | 4.68885i | |||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.d | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 126.3.n.c | 4 | |
3.b | odd | 2 | 1 | 14.3.d.a | ✓ | 4 | |
4.b | odd | 2 | 1 | 1008.3.cg.l | 4 | ||
7.b | odd | 2 | 1 | 882.3.n.b | 4 | ||
7.c | even | 3 | 1 | 882.3.c.f | 4 | ||
7.c | even | 3 | 1 | 882.3.n.b | 4 | ||
7.d | odd | 6 | 1 | inner | 126.3.n.c | 4 | |
7.d | odd | 6 | 1 | 882.3.c.f | 4 | ||
12.b | even | 2 | 1 | 112.3.s.b | 4 | ||
15.d | odd | 2 | 1 | 350.3.k.a | 4 | ||
15.e | even | 4 | 2 | 350.3.i.a | 8 | ||
21.c | even | 2 | 1 | 98.3.d.a | 4 | ||
21.g | even | 6 | 1 | 14.3.d.a | ✓ | 4 | |
21.g | even | 6 | 1 | 98.3.b.b | 4 | ||
21.h | odd | 6 | 1 | 98.3.b.b | 4 | ||
21.h | odd | 6 | 1 | 98.3.d.a | 4 | ||
24.f | even | 2 | 1 | 448.3.s.c | 4 | ||
24.h | odd | 2 | 1 | 448.3.s.d | 4 | ||
28.f | even | 6 | 1 | 1008.3.cg.l | 4 | ||
84.h | odd | 2 | 1 | 784.3.s.c | 4 | ||
84.j | odd | 6 | 1 | 112.3.s.b | 4 | ||
84.j | odd | 6 | 1 | 784.3.c.e | 4 | ||
84.n | even | 6 | 1 | 784.3.c.e | 4 | ||
84.n | even | 6 | 1 | 784.3.s.c | 4 | ||
105.p | even | 6 | 1 | 350.3.k.a | 4 | ||
105.w | odd | 12 | 2 | 350.3.i.a | 8 | ||
168.ba | even | 6 | 1 | 448.3.s.d | 4 | ||
168.be | odd | 6 | 1 | 448.3.s.c | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
14.3.d.a | ✓ | 4 | 3.b | odd | 2 | 1 | |
14.3.d.a | ✓ | 4 | 21.g | even | 6 | 1 | |
98.3.b.b | 4 | 21.g | even | 6 | 1 | ||
98.3.b.b | 4 | 21.h | odd | 6 | 1 | ||
98.3.d.a | 4 | 21.c | even | 2 | 1 | ||
98.3.d.a | 4 | 21.h | odd | 6 | 1 | ||
112.3.s.b | 4 | 12.b | even | 2 | 1 | ||
112.3.s.b | 4 | 84.j | odd | 6 | 1 | ||
126.3.n.c | 4 | 1.a | even | 1 | 1 | trivial | |
126.3.n.c | 4 | 7.d | odd | 6 | 1 | inner | |
350.3.i.a | 8 | 15.e | even | 4 | 2 | ||
350.3.i.a | 8 | 105.w | odd | 12 | 2 | ||
350.3.k.a | 4 | 15.d | odd | 2 | 1 | ||
350.3.k.a | 4 | 105.p | even | 6 | 1 | ||
448.3.s.c | 4 | 24.f | even | 2 | 1 | ||
448.3.s.c | 4 | 168.be | odd | 6 | 1 | ||
448.3.s.d | 4 | 24.h | odd | 2 | 1 | ||
448.3.s.d | 4 | 168.ba | even | 6 | 1 | ||
784.3.c.e | 4 | 84.j | odd | 6 | 1 | ||
784.3.c.e | 4 | 84.n | even | 6 | 1 | ||
784.3.s.c | 4 | 84.h | odd | 2 | 1 | ||
784.3.s.c | 4 | 84.n | even | 6 | 1 | ||
882.3.c.f | 4 | 7.c | even | 3 | 1 | ||
882.3.c.f | 4 | 7.d | odd | 6 | 1 | ||
882.3.n.b | 4 | 7.b | odd | 2 | 1 | ||
882.3.n.b | 4 | 7.c | even | 3 | 1 | ||
1008.3.cg.l | 4 | 4.b | odd | 2 | 1 | ||
1008.3.cg.l | 4 | 28.f | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 6T_{5}^{3} - 9T_{5}^{2} + 126T_{5} + 441 \)
acting on \(S_{3}^{\mathrm{new}}(126, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} + 2T^{2} + 4 \)
$3$
\( T^{4} \)
$5$
\( T^{4} - 6 T^{3} - 9 T^{2} + 126 T + 441 \)
$7$
\( T^{4} - 8 T^{3} + 42 T^{2} + \cdots + 2401 \)
$11$
\( T^{4} + 18 T^{3} + 261 T^{2} + \cdots + 3969 \)
$13$
\( T^{4} + 264T^{2} + 7056 \)
$17$
\( T^{4} - 30 T^{3} + 351 T^{2} + \cdots + 2601 \)
$19$
\( T^{4} - 6 T^{3} + 9 T^{2} + 18 T + 9 \)
$23$
\( T^{4} + 30 T^{3} + 837 T^{2} + \cdots + 3969 \)
$29$
\( (T^{2} + 24 T + 72)^{2} \)
$31$
\( T^{4} + 42 T^{3} - 615 T^{2} + \cdots + 1447209 \)
$37$
\( T^{4} + 62 T^{3} + 4035 T^{2} + \cdots + 36481 \)
$41$
\( T^{4} + 1224 T^{2} + 345744 \)
$43$
\( (T^{2} + 4 T - 68)^{2} \)
$47$
\( T^{4} + 174 T^{3} + 12609 T^{2} + \cdots + 6335289 \)
$53$
\( T^{4} - 78 T^{3} + 4851 T^{2} + \cdots + 1520289 \)
$59$
\( T^{4} - 78 T^{3} - 1215 T^{2} + \cdots + 10517049 \)
$61$
\( T^{4} + 42 T^{3} - 5409 T^{2} + \cdots + 35964009 \)
$67$
\( T^{4} + 58 T^{3} + 6573 T^{2} + \cdots + 10297681 \)
$71$
\( (T^{2} - 12 T - 1764)^{2} \)
$73$
\( T^{4} - 318 T^{3} + \cdots + 47485881 \)
$79$
\( T^{4} - 110 T^{3} + 9525 T^{2} + \cdots + 6630625 \)
$83$
\( T^{4} + 27936 T^{2} + \cdots + 189778176 \)
$89$
\( T^{4} - 378 T^{3} + \cdots + 71419401 \)
$97$
\( T^{4} + 11016 T^{2} + \cdots + 6780816 \)
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