Newspace parameters
Level: | \( N \) | \(=\) | \( 144 = 2^{4} \cdot 3^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 144.q (of order \(6\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(3.92371580679\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-11})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 3 \) |
Twist minimal: | no (minimal twist has level 36) |
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 \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) :
\(\beta_{1}\) | \(=\) | \( ( \nu^{3} + 2\nu^{2} - 2\nu - 3 ) / 6 \) |
\(\beta_{2}\) | \(=\) | \( ( -\nu^{3} + \nu^{2} + 5\nu ) / 3 \) |
\(\beta_{3}\) | \(=\) | \( ( 2\nu^{3} + \nu^{2} + 2\nu - 9 ) / 3 \) |
\(\nu\) | \(=\) | \( ( \beta_{3} + \beta_{2} - 2\beta _1 + 2 ) / 3 \) |
\(\nu^{2}\) | \(=\) | \( ( -\beta_{3} + 2\beta_{2} + 8\beta _1 + 1 ) / 3 \) |
\(\nu^{3}\) | \(=\) | \( ( 4\beta_{3} - 2\beta_{2} - 2\beta _1 + 11 ) / 3 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).
\(n\) | \(37\) | \(65\) | \(127\) |
\(\chi(n)\) | \(1\) | \(\beta_{1}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
65.1 |
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0 | −2.18614 | − | 2.05446i | 0 | −2.05842 | + | 1.18843i | 0 | −4.05842 | + | 7.02939i | 0 | 0.558422 | + | 8.98266i | 0 | ||||||||||||||||||||||
65.2 | 0 | 0.686141 | + | 2.92048i | 0 | 6.55842 | − | 3.78651i | 0 | 4.55842 | − | 7.89542i | 0 | −8.05842 | + | 4.00772i | 0 | |||||||||||||||||||||||
113.1 | 0 | −2.18614 | + | 2.05446i | 0 | −2.05842 | − | 1.18843i | 0 | −4.05842 | − | 7.02939i | 0 | 0.558422 | − | 8.98266i | 0 | |||||||||||||||||||||||
113.2 | 0 | 0.686141 | − | 2.92048i | 0 | 6.55842 | + | 3.78651i | 0 | 4.55842 | + | 7.89542i | 0 | −8.05842 | − | 4.00772i | 0 | |||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.d | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 144.3.q.b | 4 | |
3.b | odd | 2 | 1 | 432.3.q.b | 4 | ||
4.b | odd | 2 | 1 | 36.3.g.a | ✓ | 4 | |
8.b | even | 2 | 1 | 576.3.q.g | 4 | ||
8.d | odd | 2 | 1 | 576.3.q.d | 4 | ||
9.c | even | 3 | 1 | 432.3.q.b | 4 | ||
9.c | even | 3 | 1 | 1296.3.e.e | 4 | ||
9.d | odd | 6 | 1 | inner | 144.3.q.b | 4 | |
9.d | odd | 6 | 1 | 1296.3.e.e | 4 | ||
12.b | even | 2 | 1 | 108.3.g.a | 4 | ||
20.d | odd | 2 | 1 | 900.3.p.a | 4 | ||
20.e | even | 4 | 2 | 900.3.u.a | 8 | ||
24.f | even | 2 | 1 | 1728.3.q.g | 4 | ||
24.h | odd | 2 | 1 | 1728.3.q.h | 4 | ||
36.f | odd | 6 | 1 | 108.3.g.a | 4 | ||
36.f | odd | 6 | 1 | 324.3.c.b | 4 | ||
36.h | even | 6 | 1 | 36.3.g.a | ✓ | 4 | |
36.h | even | 6 | 1 | 324.3.c.b | 4 | ||
60.h | even | 2 | 1 | 2700.3.p.b | 4 | ||
60.l | odd | 4 | 2 | 2700.3.u.b | 8 | ||
72.j | odd | 6 | 1 | 576.3.q.g | 4 | ||
72.l | even | 6 | 1 | 576.3.q.d | 4 | ||
72.n | even | 6 | 1 | 1728.3.q.h | 4 | ||
72.p | odd | 6 | 1 | 1728.3.q.g | 4 | ||
180.n | even | 6 | 1 | 900.3.p.a | 4 | ||
180.p | odd | 6 | 1 | 2700.3.p.b | 4 | ||
180.v | odd | 12 | 2 | 900.3.u.a | 8 | ||
180.x | even | 12 | 2 | 2700.3.u.b | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
36.3.g.a | ✓ | 4 | 4.b | odd | 2 | 1 | |
36.3.g.a | ✓ | 4 | 36.h | even | 6 | 1 | |
108.3.g.a | 4 | 12.b | even | 2 | 1 | ||
108.3.g.a | 4 | 36.f | odd | 6 | 1 | ||
144.3.q.b | 4 | 1.a | even | 1 | 1 | trivial | |
144.3.q.b | 4 | 9.d | odd | 6 | 1 | inner | |
324.3.c.b | 4 | 36.f | odd | 6 | 1 | ||
324.3.c.b | 4 | 36.h | even | 6 | 1 | ||
432.3.q.b | 4 | 3.b | odd | 2 | 1 | ||
432.3.q.b | 4 | 9.c | even | 3 | 1 | ||
576.3.q.d | 4 | 8.d | odd | 2 | 1 | ||
576.3.q.d | 4 | 72.l | even | 6 | 1 | ||
576.3.q.g | 4 | 8.b | even | 2 | 1 | ||
576.3.q.g | 4 | 72.j | odd | 6 | 1 | ||
900.3.p.a | 4 | 20.d | odd | 2 | 1 | ||
900.3.p.a | 4 | 180.n | even | 6 | 1 | ||
900.3.u.a | 8 | 20.e | even | 4 | 2 | ||
900.3.u.a | 8 | 180.v | odd | 12 | 2 | ||
1296.3.e.e | 4 | 9.c | even | 3 | 1 | ||
1296.3.e.e | 4 | 9.d | odd | 6 | 1 | ||
1728.3.q.g | 4 | 24.f | even | 2 | 1 | ||
1728.3.q.g | 4 | 72.p | odd | 6 | 1 | ||
1728.3.q.h | 4 | 24.h | odd | 2 | 1 | ||
1728.3.q.h | 4 | 72.n | even | 6 | 1 | ||
2700.3.p.b | 4 | 60.h | even | 2 | 1 | ||
2700.3.p.b | 4 | 180.p | odd | 6 | 1 | ||
2700.3.u.b | 8 | 60.l | odd | 4 | 2 | ||
2700.3.u.b | 8 | 180.x | even | 12 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 9T_{5}^{3} + 9T_{5}^{2} + 162T_{5} + 324 \)
acting on \(S_{3}^{\mathrm{new}}(144, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( T^{4} + 3 T^{3} + 12 T^{2} + 27 T + 81 \)
$5$
\( T^{4} - 9 T^{3} + 9 T^{2} + 162 T + 324 \)
$7$
\( T^{4} - T^{3} + 75 T^{2} + 74 T + 5476 \)
$11$
\( T^{4} - 36 T^{3} + 441 T^{2} + \cdots + 81 \)
$13$
\( T^{4} - 5 T^{3} + 93 T^{2} + \cdots + 4624 \)
$17$
\( T^{4} + 387 T^{2} + 20736 \)
$19$
\( (T^{2} + T - 74)^{2} \)
$23$
\( T^{4} + 99 T^{3} + 4059 T^{2} + \cdots + 627264 \)
$29$
\( T^{4} + 63 T^{3} + 441 T^{2} + \cdots + 777924 \)
$31$
\( T^{4} - 7 T^{3} + 705 T^{2} + \cdots + 430336 \)
$37$
\( (T^{2} + 32 T - 932)^{2} \)
$41$
\( T^{4} + 18 T^{3} - 1449 T^{2} + \cdots + 2424249 \)
$43$
\( (T^{2} - 23 T + 529)^{2} \)
$47$
\( T^{4} - 81 T^{3} + 2511 T^{2} + \cdots + 104976 \)
$53$
\( T^{4} + 4032 T^{2} + \cdots + 1327104 \)
$59$
\( T^{4} + 126 T^{3} + 5031 T^{2} + \cdots + 68121 \)
$61$
\( T^{4} - 41 T^{3} + 1929 T^{2} + \cdots + 61504 \)
$67$
\( T^{4} + 116 T^{3} + 12765 T^{2} + \cdots + 477481 \)
$71$
\( T^{4} + 1548 T^{2} + 331776 \)
$73$
\( (T^{2} - 43 T - 206)^{2} \)
$79$
\( T^{4} + 83 T^{3} + 7023 T^{2} + \cdots + 17956 \)
$83$
\( T^{4} - 81 T^{3} + 2511 T^{2} + \cdots + 104976 \)
$89$
\( T^{4} + 24768 T^{2} + \cdots + 84934656 \)
$97$
\( T^{4} + 196 T^{3} + \cdots + 86620249 \)
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