Newspace parameters
Level: | \( N \) | \(=\) | \( 72 = 2^{3} \cdot 3^{2} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 72.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(4.24813752041\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(\sqrt{-10}, \sqrt{22})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: |
\( x^{4} - 6x^{2} + 64 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 2^{2} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{4} - 6x^{2} + 64 \)
:
\(\beta_{1}\) | \(=\) |
\( \nu \)
|
\(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 2\nu ) / 4 \)
|
\(\beta_{3}\) | \(=\) |
\( \nu^{2} - 3 \)
|
\(\nu\) | \(=\) |
\( \beta_1 \)
|
\(\nu^{2}\) | \(=\) |
\( \beta_{3} + 3 \)
|
\(\nu^{3}\) | \(=\) |
\( 4\beta_{2} + 2\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/72\mathbb{Z}\right)^\times\).
\(n\) | \(37\) | \(55\) | \(65\) |
\(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
37.1 |
|
−2.34521 | − | 1.58114i | 0 | 3.00000 | + | 7.41620i | − | 6.32456i | 0 | 10.0000 | 4.69042 | − | 22.1359i | 0 | −10.0000 | + | 14.8324i | |||||||||||||||||||||
37.2 | −2.34521 | + | 1.58114i | 0 | 3.00000 | − | 7.41620i | 6.32456i | 0 | 10.0000 | 4.69042 | + | 22.1359i | 0 | −10.0000 | − | 14.8324i | |||||||||||||||||||||||
37.3 | 2.34521 | − | 1.58114i | 0 | 3.00000 | − | 7.41620i | − | 6.32456i | 0 | 10.0000 | −4.69042 | − | 22.1359i | 0 | −10.0000 | − | 14.8324i | ||||||||||||||||||||||
37.4 | 2.34521 | + | 1.58114i | 0 | 3.00000 | + | 7.41620i | 6.32456i | 0 | 10.0000 | −4.69042 | + | 22.1359i | 0 | −10.0000 | + | 14.8324i | |||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
8.b | even | 2 | 1 | inner |
24.h | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 72.4.d.c | ✓ | 4 |
3.b | odd | 2 | 1 | inner | 72.4.d.c | ✓ | 4 |
4.b | odd | 2 | 1 | 288.4.d.c | 4 | ||
8.b | even | 2 | 1 | inner | 72.4.d.c | ✓ | 4 |
8.d | odd | 2 | 1 | 288.4.d.c | 4 | ||
12.b | even | 2 | 1 | 288.4.d.c | 4 | ||
16.e | even | 4 | 2 | 2304.4.a.bx | 4 | ||
16.f | odd | 4 | 2 | 2304.4.a.cc | 4 | ||
24.f | even | 2 | 1 | 288.4.d.c | 4 | ||
24.h | odd | 2 | 1 | inner | 72.4.d.c | ✓ | 4 |
48.i | odd | 4 | 2 | 2304.4.a.bx | 4 | ||
48.k | even | 4 | 2 | 2304.4.a.cc | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
72.4.d.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
72.4.d.c | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
72.4.d.c | ✓ | 4 | 8.b | even | 2 | 1 | inner |
72.4.d.c | ✓ | 4 | 24.h | odd | 2 | 1 | inner |
288.4.d.c | 4 | 4.b | odd | 2 | 1 | ||
288.4.d.c | 4 | 8.d | odd | 2 | 1 | ||
288.4.d.c | 4 | 12.b | even | 2 | 1 | ||
288.4.d.c | 4 | 24.f | even | 2 | 1 | ||
2304.4.a.bx | 4 | 16.e | even | 4 | 2 | ||
2304.4.a.bx | 4 | 48.i | odd | 4 | 2 | ||
2304.4.a.cc | 4 | 16.f | odd | 4 | 2 | ||
2304.4.a.cc | 4 | 48.k | even | 4 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 40 \)
acting on \(S_{4}^{\mathrm{new}}(72, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - 6T^{2} + 64 \)
$3$
\( T^{4} \)
$5$
\( (T^{2} + 40)^{2} \)
$7$
\( (T - 10)^{4} \)
$11$
\( (T^{2} + 1440)^{2} \)
$13$
\( (T^{2} + 3520)^{2} \)
$17$
\( (T^{2} - 5632)^{2} \)
$19$
\( (T^{2} + 14080)^{2} \)
$23$
\( (T^{2} - 22528)^{2} \)
$29$
\( (T^{2} + 60840)^{2} \)
$31$
\( (T - 62)^{4} \)
$37$
\( (T^{2} + 3520)^{2} \)
$41$
\( (T^{2} - 140800)^{2} \)
$43$
\( (T^{2} + 14080)^{2} \)
$47$
\( (T^{2} - 202752)^{2} \)
$53$
\( (T^{2} + 17640)^{2} \)
$59$
\( (T^{2} + 538240)^{2} \)
$61$
\( (T^{2} + 285120)^{2} \)
$67$
\( (T^{2} + 506880)^{2} \)
$71$
\( T^{4} \)
$73$
\( (T - 30)^{4} \)
$79$
\( (T - 94)^{4} \)
$83$
\( (T^{2} + 449440)^{2} \)
$89$
\( (T^{2} - 563200)^{2} \)
$97$
\( (T - 130)^{4} \)
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