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: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-7}) \) |
Defining polynomial: |
\( x^{2} - x + 2 \)
|
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 2 \) |
Twist minimal: | no (minimal twist has level 8) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-7}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/72\mathbb{Z}\right)^\times\).
\(n\) | \(37\) | \(55\) | \(65\) |
\(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 |
|
1.00000 | − | 2.64575i | 0 | −6.00000 | − | 5.29150i | − | 10.5830i | 0 | −8.00000 | −20.0000 | + | 10.5830i | 0 | −28.0000 | − | 10.5830i | |||||||||||||||
37.2 | 1.00000 | + | 2.64575i | 0 | −6.00000 | + | 5.29150i | 10.5830i | 0 | −8.00000 | −20.0000 | − | 10.5830i | 0 | −28.0000 | + | 10.5830i | |||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.b | even | 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.b | 2 | |
3.b | odd | 2 | 1 | 8.4.b.a | ✓ | 2 | |
4.b | odd | 2 | 1 | 288.4.d.a | 2 | ||
8.b | even | 2 | 1 | inner | 72.4.d.b | 2 | |
8.d | odd | 2 | 1 | 288.4.d.a | 2 | ||
12.b | even | 2 | 1 | 32.4.b.a | 2 | ||
15.d | odd | 2 | 1 | 200.4.d.a | 2 | ||
15.e | even | 4 | 2 | 200.4.f.a | 4 | ||
16.e | even | 4 | 2 | 2304.4.a.bn | 2 | ||
16.f | odd | 4 | 2 | 2304.4.a.v | 2 | ||
24.f | even | 2 | 1 | 32.4.b.a | 2 | ||
24.h | odd | 2 | 1 | 8.4.b.a | ✓ | 2 | |
48.i | odd | 4 | 2 | 256.4.a.l | 2 | ||
48.k | even | 4 | 2 | 256.4.a.j | 2 | ||
60.h | even | 2 | 1 | 800.4.d.a | 2 | ||
60.l | odd | 4 | 2 | 800.4.f.a | 4 | ||
120.i | odd | 2 | 1 | 200.4.d.a | 2 | ||
120.m | even | 2 | 1 | 800.4.d.a | 2 | ||
120.q | odd | 4 | 2 | 800.4.f.a | 4 | ||
120.w | even | 4 | 2 | 200.4.f.a | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8.4.b.a | ✓ | 2 | 3.b | odd | 2 | 1 | |
8.4.b.a | ✓ | 2 | 24.h | odd | 2 | 1 | |
32.4.b.a | 2 | 12.b | even | 2 | 1 | ||
32.4.b.a | 2 | 24.f | even | 2 | 1 | ||
72.4.d.b | 2 | 1.a | even | 1 | 1 | trivial | |
72.4.d.b | 2 | 8.b | even | 2 | 1 | inner | |
200.4.d.a | 2 | 15.d | odd | 2 | 1 | ||
200.4.d.a | 2 | 120.i | odd | 2 | 1 | ||
200.4.f.a | 4 | 15.e | even | 4 | 2 | ||
200.4.f.a | 4 | 120.w | even | 4 | 2 | ||
256.4.a.j | 2 | 48.k | even | 4 | 2 | ||
256.4.a.l | 2 | 48.i | odd | 4 | 2 | ||
288.4.d.a | 2 | 4.b | odd | 2 | 1 | ||
288.4.d.a | 2 | 8.d | odd | 2 | 1 | ||
800.4.d.a | 2 | 60.h | even | 2 | 1 | ||
800.4.d.a | 2 | 120.m | even | 2 | 1 | ||
800.4.f.a | 4 | 60.l | odd | 4 | 2 | ||
800.4.f.a | 4 | 120.q | odd | 4 | 2 | ||
2304.4.a.v | 2 | 16.f | odd | 4 | 2 | ||
2304.4.a.bn | 2 | 16.e | even | 4 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 112 \)
acting on \(S_{4}^{\mathrm{new}}(72, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} - 2T + 8 \)
$3$
\( T^{2} \)
$5$
\( T^{2} + 112 \)
$7$
\( (T + 8)^{2} \)
$11$
\( T^{2} + 252 \)
$13$
\( T^{2} + 2800 \)
$17$
\( (T - 14)^{2} \)
$19$
\( T^{2} + 1372 \)
$23$
\( (T - 152)^{2} \)
$29$
\( T^{2} + 25200 \)
$31$
\( (T - 224)^{2} \)
$37$
\( T^{2} + 59248 \)
$41$
\( (T - 70)^{2} \)
$43$
\( T^{2} + 192892 \)
$47$
\( (T + 336)^{2} \)
$53$
\( T^{2} + 1008 \)
$59$
\( T^{2} + 285628 \)
$61$
\( T^{2} + 9072 \)
$67$
\( T^{2} + 30492 \)
$71$
\( (T - 72)^{2} \)
$73$
\( (T + 294)^{2} \)
$79$
\( (T + 464)^{2} \)
$83$
\( T^{2} + 297052 \)
$89$
\( (T + 266)^{2} \)
$97$
\( (T - 994)^{2} \)
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