Newspace parameters
Level: | \( N \) | \(=\) | \( 72 = 2^{3} \cdot 3^{2} \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 72.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(11.5476350265\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | 4.0.218489.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} - x^{3} - 2x^{2} - 8x + 64 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
Coefficient ring index: | \( 2^{6} \) |
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 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 \( x^{4} - x^{3} - 2x^{2} - 8x + 64 \) :
\(\beta_{1}\) | \(=\) | \( 2\nu \) |
\(\beta_{2}\) | \(=\) | \( ( -\nu^{3} + 5\nu^{2} - 2\nu + 2 ) / 2 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{3} + 3\nu^{2} - 2\nu - 12 ) / 2 \) |
\(\nu\) | \(=\) | \( ( \beta_1 ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{3} + \beta_{2} + \beta _1 + 5 ) / 4 \) |
\(\nu^{3}\) | \(=\) | \( ( 5\beta_{3} - 3\beta_{2} + \beta _1 + 33 ) / 4 \) |
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 |
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−3.77200 | − | 4.21569i | 0 | −3.54400 | + | 31.8031i | 73.9600i | 0 | −112.704 | 147.440 | − | 105.021i | 0 | 311.792 | − | 278.977i | ||||||||||||||||||||||
37.2 | −3.77200 | + | 4.21569i | 0 | −3.54400 | − | 31.8031i | − | 73.9600i | 0 | −112.704 | 147.440 | + | 105.021i | 0 | 311.792 | + | 278.977i | ||||||||||||||||||||||
37.3 | 4.77200 | − | 3.03776i | 0 | 13.5440 | − | 28.9924i | 1.38521i | 0 | 160.704 | −23.4400 | − | 179.495i | 0 | 4.20793 | + | 6.61022i | |||||||||||||||||||||||
37.4 | 4.77200 | + | 3.03776i | 0 | 13.5440 | + | 28.9924i | − | 1.38521i | 0 | 160.704 | −23.4400 | + | 179.495i | 0 | 4.20793 | − | 6.61022i | ||||||||||||||||||||||
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 | 72.6.d.b | 4 | |
3.b | odd | 2 | 1 | 8.6.b.a | ✓ | 4 | |
4.b | odd | 2 | 1 | 288.6.d.b | 4 | ||
8.b | even | 2 | 1 | inner | 72.6.d.b | 4 | |
8.d | odd | 2 | 1 | 288.6.d.b | 4 | ||
12.b | even | 2 | 1 | 32.6.b.a | 4 | ||
15.d | odd | 2 | 1 | 200.6.d.a | 4 | ||
15.e | even | 4 | 2 | 200.6.f.a | 8 | ||
24.f | even | 2 | 1 | 32.6.b.a | 4 | ||
24.h | odd | 2 | 1 | 8.6.b.a | ✓ | 4 | |
48.i | odd | 4 | 2 | 256.6.a.k | 4 | ||
48.k | even | 4 | 2 | 256.6.a.n | 4 | ||
60.h | even | 2 | 1 | 800.6.d.a | 4 | ||
60.l | odd | 4 | 2 | 800.6.f.a | 8 | ||
120.i | odd | 2 | 1 | 200.6.d.a | 4 | ||
120.m | even | 2 | 1 | 800.6.d.a | 4 | ||
120.q | odd | 4 | 2 | 800.6.f.a | 8 | ||
120.w | even | 4 | 2 | 200.6.f.a | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8.6.b.a | ✓ | 4 | 3.b | odd | 2 | 1 | |
8.6.b.a | ✓ | 4 | 24.h | odd | 2 | 1 | |
32.6.b.a | 4 | 12.b | even | 2 | 1 | ||
32.6.b.a | 4 | 24.f | even | 2 | 1 | ||
72.6.d.b | 4 | 1.a | even | 1 | 1 | trivial | |
72.6.d.b | 4 | 8.b | even | 2 | 1 | inner | |
200.6.d.a | 4 | 15.d | odd | 2 | 1 | ||
200.6.d.a | 4 | 120.i | odd | 2 | 1 | ||
200.6.f.a | 8 | 15.e | even | 4 | 2 | ||
200.6.f.a | 8 | 120.w | even | 4 | 2 | ||
256.6.a.k | 4 | 48.i | odd | 4 | 2 | ||
256.6.a.n | 4 | 48.k | even | 4 | 2 | ||
288.6.d.b | 4 | 4.b | odd | 2 | 1 | ||
288.6.d.b | 4 | 8.d | odd | 2 | 1 | ||
800.6.d.a | 4 | 60.h | even | 2 | 1 | ||
800.6.d.a | 4 | 120.m | even | 2 | 1 | ||
800.6.f.a | 8 | 60.l | odd | 4 | 2 | ||
800.6.f.a | 8 | 120.q | odd | 4 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 5472T_{5}^{2} + 10496 \)
acting on \(S_{6}^{\mathrm{new}}(72, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - 2 T^{3} - 8 T^{2} - 64 T + 1024 \)
$3$
\( T^{4} \)
$5$
\( T^{4} + 5472 T^{2} + 10496 \)
$7$
\( (T^{2} - 48 T - 18112)^{2} \)
$11$
\( T^{4} + 347768 T^{2} + \cdots + 5520765456 \)
$13$
\( T^{4} + 590944 T^{2} + \cdots + 7999305984 \)
$17$
\( (T^{2} + 100 T - 72252)^{2} \)
$19$
\( T^{4} + 3109816 T^{2} + \cdots + 120994976016 \)
$23$
\( (T^{2} + 1168 T - 2817216)^{2} \)
$29$
\( T^{4} + \cdots + 535633608132864 \)
$31$
\( (T^{2} + 6464 T + 7754752)^{2} \)
$37$
\( T^{4} + \cdots + 306881230162176 \)
$41$
\( (T^{2} - 2284 T - 85109148)^{2} \)
$43$
\( T^{4} + 121686904 T^{2} + \cdots + 23\!\cdots\!56 \)
$47$
\( (T^{2} - 27360 T + 132648192)^{2} \)
$53$
\( T^{4} + 633629792 T^{2} + \cdots + 76\!\cdots\!44 \)
$59$
\( T^{4} + 1322273016 T^{2} + \cdots + 22\!\cdots\!36 \)
$61$
\( T^{4} + 4119483744 T^{2} + \cdots + 41\!\cdots\!16 \)
$67$
\( T^{4} + 4033664568 T^{2} + \cdots + 32\!\cdots\!84 \)
$71$
\( (T^{2} + 103344 T + 2609278272)^{2} \)
$73$
\( (T^{2} - 19988 T - 1604540316)^{2} \)
$79$
\( (T^{2} + 123936 T + 3701816576)^{2} \)
$83$
\( T^{4} + 5708307384 T^{2} + \cdots + 72\!\cdots\!56 \)
$89$
\( (T^{2} - 42316 T - 6875717724)^{2} \)
$97$
\( (T^{2} + 49788 T - 783309052)^{2} \)
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