Newspace parameters
Level: | \( N \) | \(=\) | \( 288 = 2^{5} \cdot 3^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 288.v (of order \(8\), degree \(4\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.29969157821\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(\zeta_{8})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 32) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{8}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/288\mathbb{Z}\right)^\times\).
\(n\) | \(37\) | \(65\) | \(127\) |
\(\chi(n)\) | \(\zeta_{8}\) | \(1\) | \(1\) |
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 |
|
1.41421i | 0 | −2.00000 | 3.12132 | + | 1.29289i | 0 | 1.00000 | + | 1.00000i | − | 2.82843i | 0 | −1.82843 | + | 4.41421i | |||||||||||||||||||||||
109.1 | − | 1.41421i | 0 | −2.00000 | 3.12132 | − | 1.29289i | 0 | 1.00000 | − | 1.00000i | 2.82843i | 0 | −1.82843 | − | 4.41421i | ||||||||||||||||||||||||
181.1 | − | 1.41421i | 0 | −2.00000 | −1.12132 | + | 2.70711i | 0 | 1.00000 | + | 1.00000i | 2.82843i | 0 | 3.82843 | + | 1.58579i | ||||||||||||||||||||||||
253.1 | 1.41421i | 0 | −2.00000 | −1.12132 | − | 2.70711i | 0 | 1.00000 | − | 1.00000i | − | 2.82843i | 0 | 3.82843 | − | 1.58579i | ||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
32.g | even | 8 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 288.2.v.a | 4 | |
3.b | odd | 2 | 1 | 32.2.g.a | ✓ | 4 | |
4.b | odd | 2 | 1 | 1152.2.v.a | 4 | ||
12.b | even | 2 | 1 | 128.2.g.a | 4 | ||
15.d | odd | 2 | 1 | 800.2.y.a | 4 | ||
15.e | even | 4 | 1 | 800.2.ba.a | 4 | ||
15.e | even | 4 | 1 | 800.2.ba.b | 4 | ||
24.f | even | 2 | 1 | 256.2.g.a | 4 | ||
24.h | odd | 2 | 1 | 256.2.g.b | 4 | ||
32.g | even | 8 | 1 | inner | 288.2.v.a | 4 | |
32.h | odd | 8 | 1 | 1152.2.v.a | 4 | ||
48.i | odd | 4 | 1 | 512.2.g.a | 4 | ||
48.i | odd | 4 | 1 | 512.2.g.d | 4 | ||
48.k | even | 4 | 1 | 512.2.g.b | 4 | ||
48.k | even | 4 | 1 | 512.2.g.c | 4 | ||
96.o | even | 8 | 1 | 128.2.g.a | 4 | ||
96.o | even | 8 | 1 | 256.2.g.a | 4 | ||
96.o | even | 8 | 1 | 512.2.g.b | 4 | ||
96.o | even | 8 | 1 | 512.2.g.c | 4 | ||
96.p | odd | 8 | 1 | 32.2.g.a | ✓ | 4 | |
96.p | odd | 8 | 1 | 256.2.g.b | 4 | ||
96.p | odd | 8 | 1 | 512.2.g.a | 4 | ||
96.p | odd | 8 | 1 | 512.2.g.d | 4 | ||
192.q | odd | 16 | 2 | 4096.2.a.e | 4 | ||
192.s | even | 16 | 2 | 4096.2.a.f | 4 | ||
480.br | even | 8 | 1 | 800.2.ba.b | 4 | ||
480.bu | odd | 8 | 1 | 800.2.y.a | 4 | ||
480.cb | even | 8 | 1 | 800.2.ba.a | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
32.2.g.a | ✓ | 4 | 3.b | odd | 2 | 1 | |
32.2.g.a | ✓ | 4 | 96.p | odd | 8 | 1 | |
128.2.g.a | 4 | 12.b | even | 2 | 1 | ||
128.2.g.a | 4 | 96.o | even | 8 | 1 | ||
256.2.g.a | 4 | 24.f | even | 2 | 1 | ||
256.2.g.a | 4 | 96.o | even | 8 | 1 | ||
256.2.g.b | 4 | 24.h | odd | 2 | 1 | ||
256.2.g.b | 4 | 96.p | odd | 8 | 1 | ||
288.2.v.a | 4 | 1.a | even | 1 | 1 | trivial | |
288.2.v.a | 4 | 32.g | even | 8 | 1 | inner | |
512.2.g.a | 4 | 48.i | odd | 4 | 1 | ||
512.2.g.a | 4 | 96.p | odd | 8 | 1 | ||
512.2.g.b | 4 | 48.k | even | 4 | 1 | ||
512.2.g.b | 4 | 96.o | even | 8 | 1 | ||
512.2.g.c | 4 | 48.k | even | 4 | 1 | ||
512.2.g.c | 4 | 96.o | even | 8 | 1 | ||
512.2.g.d | 4 | 48.i | odd | 4 | 1 | ||
512.2.g.d | 4 | 96.p | odd | 8 | 1 | ||
800.2.y.a | 4 | 15.d | odd | 2 | 1 | ||
800.2.y.a | 4 | 480.bu | odd | 8 | 1 | ||
800.2.ba.a | 4 | 15.e | even | 4 | 1 | ||
800.2.ba.a | 4 | 480.cb | even | 8 | 1 | ||
800.2.ba.b | 4 | 15.e | even | 4 | 1 | ||
800.2.ba.b | 4 | 480.br | even | 8 | 1 | ||
1152.2.v.a | 4 | 4.b | odd | 2 | 1 | ||
1152.2.v.a | 4 | 32.h | odd | 8 | 1 | ||
4096.2.a.e | 4 | 192.q | odd | 16 | 2 | ||
4096.2.a.f | 4 | 192.s | even | 16 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 4T_{5}^{3} + 6T_{5}^{2} - 28T_{5} + 98 \)
acting on \(S_{2}^{\mathrm{new}}(288, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} + 2)^{2} \)
$3$
\( T^{4} \)
$5$
\( T^{4} - 4 T^{3} + 6 T^{2} - 28 T + 98 \)
$7$
\( (T^{2} - 2 T + 2)^{2} \)
$11$
\( T^{4} - 8 T^{3} + 18 T^{2} + 4 T + 2 \)
$13$
\( T^{4} - 4 T^{3} + 6 T^{2} - 4 T + 2 \)
$17$
\( (T^{2} + 8)^{2} \)
$19$
\( T^{4} + 8 T^{3} + 18 T^{2} + 68 T + 578 \)
$23$
\( T^{4} + 12 T^{3} + 72 T^{2} + 24 T + 4 \)
$29$
\( T^{4} - 4 T^{3} + 6 T^{2} - 28 T + 98 \)
$31$
\( (T + 4)^{4} \)
$37$
\( T^{4} - 4 T^{3} + 6 T^{2} - 4 T + 2 \)
$41$
\( T^{4} - 12 T^{3} + 72 T^{2} - 24 T + 4 \)
$43$
\( T^{4} - 16 T^{3} + 162 T^{2} + \cdots + 1922 \)
$47$
\( T^{4} + 136T^{2} + 16 \)
$53$
\( T^{4} + 4 T^{3} + 54 T^{2} - 140 T + 98 \)
$59$
\( T^{4} - 16 T^{3} + 114 T^{2} + \cdots + 1058 \)
$61$
\( T^{4} - 4 T^{3} + 6 T^{2} - 4 T + 2 \)
$67$
\( T^{4} + 8 T^{3} + 18 T^{2} + 68 T + 578 \)
$71$
\( T^{4} - 12 T^{3} + 72 T^{2} - 24 T + 4 \)
$73$
\( (T^{2} - 14 T + 98)^{2} \)
$79$
\( (T^{2} + 36)^{2} \)
$83$
\( T^{4} + 16 T^{3} + 114 T^{2} + \cdots + 1058 \)
$89$
\( T^{4} + 12 T^{3} + 72 T^{2} + \cdots + 2116 \)
$97$
\( (T^{2} + 20 T + 28)^{2} \)
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