Newspace parameters
Level: | \( N \) | \(=\) | \( 144 = 2^{4} \cdot 3^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 144.i (of order \(3\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.14984578911\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-3}) \) |
Defining polynomial: |
\( x^{2} - x + 1 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 72) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).
\(n\) | \(37\) | \(65\) | \(127\) |
\(\chi(n)\) | \(1\) | \(-\zeta_{6}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
49.1 |
|
0 | − | 1.73205i | 0 | 0.500000 | − | 0.866025i | 0 | −1.50000 | − | 2.59808i | 0 | −3.00000 | 0 | |||||||||||||||||||
97.1 | 0 | 1.73205i | 0 | 0.500000 | + | 0.866025i | 0 | −1.50000 | + | 2.59808i | 0 | −3.00000 | 0 | |||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 144.2.i.b | 2 | |
3.b | odd | 2 | 1 | 432.2.i.a | 2 | ||
4.b | odd | 2 | 1 | 72.2.i.a | ✓ | 2 | |
8.b | even | 2 | 1 | 576.2.i.c | 2 | ||
8.d | odd | 2 | 1 | 576.2.i.d | 2 | ||
9.c | even | 3 | 1 | inner | 144.2.i.b | 2 | |
9.c | even | 3 | 1 | 1296.2.a.e | 1 | ||
9.d | odd | 6 | 1 | 432.2.i.a | 2 | ||
9.d | odd | 6 | 1 | 1296.2.a.i | 1 | ||
12.b | even | 2 | 1 | 216.2.i.a | 2 | ||
24.f | even | 2 | 1 | 1728.2.i.h | 2 | ||
24.h | odd | 2 | 1 | 1728.2.i.g | 2 | ||
36.f | odd | 6 | 1 | 72.2.i.a | ✓ | 2 | |
36.f | odd | 6 | 1 | 648.2.a.a | 1 | ||
36.h | even | 6 | 1 | 216.2.i.a | 2 | ||
36.h | even | 6 | 1 | 648.2.a.c | 1 | ||
72.j | odd | 6 | 1 | 1728.2.i.g | 2 | ||
72.j | odd | 6 | 1 | 5184.2.a.n | 1 | ||
72.l | even | 6 | 1 | 1728.2.i.h | 2 | ||
72.l | even | 6 | 1 | 5184.2.a.i | 1 | ||
72.n | even | 6 | 1 | 576.2.i.c | 2 | ||
72.n | even | 6 | 1 | 5184.2.a.x | 1 | ||
72.p | odd | 6 | 1 | 576.2.i.d | 2 | ||
72.p | odd | 6 | 1 | 5184.2.a.s | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
72.2.i.a | ✓ | 2 | 4.b | odd | 2 | 1 | |
72.2.i.a | ✓ | 2 | 36.f | odd | 6 | 1 | |
144.2.i.b | 2 | 1.a | even | 1 | 1 | trivial | |
144.2.i.b | 2 | 9.c | even | 3 | 1 | inner | |
216.2.i.a | 2 | 12.b | even | 2 | 1 | ||
216.2.i.a | 2 | 36.h | even | 6 | 1 | ||
432.2.i.a | 2 | 3.b | odd | 2 | 1 | ||
432.2.i.a | 2 | 9.d | odd | 6 | 1 | ||
576.2.i.c | 2 | 8.b | even | 2 | 1 | ||
576.2.i.c | 2 | 72.n | even | 6 | 1 | ||
576.2.i.d | 2 | 8.d | odd | 2 | 1 | ||
576.2.i.d | 2 | 72.p | odd | 6 | 1 | ||
648.2.a.a | 1 | 36.f | odd | 6 | 1 | ||
648.2.a.c | 1 | 36.h | even | 6 | 1 | ||
1296.2.a.e | 1 | 9.c | even | 3 | 1 | ||
1296.2.a.i | 1 | 9.d | odd | 6 | 1 | ||
1728.2.i.g | 2 | 24.h | odd | 2 | 1 | ||
1728.2.i.g | 2 | 72.j | odd | 6 | 1 | ||
1728.2.i.h | 2 | 24.f | even | 2 | 1 | ||
1728.2.i.h | 2 | 72.l | even | 6 | 1 | ||
5184.2.a.i | 1 | 72.l | even | 6 | 1 | ||
5184.2.a.n | 1 | 72.j | odd | 6 | 1 | ||
5184.2.a.s | 1 | 72.p | odd | 6 | 1 | ||
5184.2.a.x | 1 | 72.n | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - T_{5} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(144, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} + 3 \)
$5$
\( T^{2} - T + 1 \)
$7$
\( T^{2} + 3T + 9 \)
$11$
\( T^{2} - 5T + 25 \)
$13$
\( T^{2} - 5T + 25 \)
$17$
\( (T + 2)^{2} \)
$19$
\( (T - 4)^{2} \)
$23$
\( T^{2} + T + 1 \)
$29$
\( T^{2} - 9T + 81 \)
$31$
\( T^{2} + T + 1 \)
$37$
\( (T + 6)^{2} \)
$41$
\( T^{2} + 3T + 9 \)
$43$
\( T^{2} - T + 1 \)
$47$
\( T^{2} + 3T + 9 \)
$53$
\( (T - 2)^{2} \)
$59$
\( T^{2} - 11T + 121 \)
$61$
\( T^{2} + 7T + 49 \)
$67$
\( T^{2} + T + 1 \)
$71$
\( (T + 4)^{2} \)
$73$
\( (T + 2)^{2} \)
$79$
\( T^{2} - T + 1 \)
$83$
\( T^{2} - T + 1 \)
$89$
\( (T + 18)^{2} \)
$97$
\( T^{2} - 13T + 169 \)
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