Newspace parameters
Level: | \( N \) | \(=\) | \( 960 = 2^{6} \cdot 3 \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 960.s (of order \(4\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(7.66563859404\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(4\) over \(\Q(i)\) |
Coefficient field: | 8.0.18939904.2 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
Coefficient ring index: | \( 2^{4} \) |
Twist minimal: | no (minimal twist has level 240) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) 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 \( x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 \) :
\(\beta_{1}\) | \(=\) | \( -\nu^{6} + 3\nu^{5} - 11\nu^{4} + 17\nu^{3} - 24\nu^{2} + 16\nu - 5 \) |
\(\beta_{2}\) | \(=\) | \( 5\nu^{7} - 17\nu^{6} + 60\nu^{5} - 105\nu^{4} + 155\nu^{3} - 133\nu^{2} + 77\nu - 19 \) |
\(\beta_{3}\) | \(=\) | \( 5\nu^{7} - 18\nu^{6} + 63\nu^{5} - 115\nu^{4} + 170\nu^{3} - 152\nu^{2} + 89\nu - 23 \) |
\(\beta_{4}\) | \(=\) | \( 8\nu^{7} - 28\nu^{6} + 98\nu^{5} - 175\nu^{4} + 256\nu^{3} - 223\nu^{2} + 126\nu - 31 \) |
\(\beta_{5}\) | \(=\) | \( 9\nu^{7} - 31\nu^{6} + 108\nu^{5} - 190\nu^{4} + 275\nu^{3} - 236\nu^{2} + 131\nu - 33 \) |
\(\beta_{6}\) | \(=\) | \( 9\nu^{7} - 32\nu^{6} + 111\nu^{5} - 200\nu^{4} + 290\nu^{3} - 253\nu^{2} + 141\nu - 33 \) |
\(\beta_{7}\) | \(=\) | \( 10\nu^{7} - 35\nu^{6} + 123\nu^{5} - 220\nu^{4} + 325\nu^{3} - 285\nu^{2} + 168\nu - 43 \) |
\(\nu\) | \(=\) | \( ( \beta_{7} - \beta_{3} - \beta_{2} + 1 ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{7} + \beta_{6} - \beta_{5} - 2\beta_{3} - 3 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( ( -2\beta_{7} + 2\beta_{6} - \beta_{5} - 3\beta_{4} + 2\beta_{3} + 5\beta_{2} - 5 ) / 2 \) |
\(\nu^{4}\) | \(=\) | \( ( -5\beta_{7} - \beta_{6} + 3\beta_{5} - 6\beta_{4} + 12\beta_{3} + 4\beta_{2} - 2\beta _1 + 7 ) / 2 \) |
\(\nu^{5}\) | \(=\) | \( ( 3\beta_{7} - 10\beta_{6} + 5\beta_{5} + 10\beta_{4} + 6\beta_{3} - 19\beta_{2} - 5\beta _1 + 26 ) / 2 \) |
\(\nu^{6}\) | \(=\) | \( ( 22\beta_{7} - 9\beta_{6} - 11\beta_{5} + 45\beta_{4} - 48\beta_{3} - 32\beta_{2} + 5\beta _1 - 6 ) / 2 \) |
\(\nu^{7}\) | \(=\) | \( ( 7\beta_{7} + 33\beta_{6} - 30\beta_{5} - 83\beta_{3} + 64\beta_{2} + 35\beta _1 - 118 ) / 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/960\mathbb{Z}\right)^\times\).
\(n\) | \(511\) | \(577\) | \(641\) | \(901\) |
\(\chi(n)\) | \(1\) | \(1\) | \(1\) | \(\beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
241.1 |
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0 | −0.707107 | + | 0.707107i | 0 | −0.707107 | − | 0.707107i | 0 | 1.41421i | 0 | − | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||
241.2 | 0 | −0.707107 | + | 0.707107i | 0 | −0.707107 | − | 0.707107i | 0 | 1.41421i | 0 | − | 1.00000i | 0 | ||||||||||||||||||||||||||||||||||||||
241.3 | 0 | 0.707107 | − | 0.707107i | 0 | 0.707107 | + | 0.707107i | 0 | − | 1.41421i | 0 | − | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||
241.4 | 0 | 0.707107 | − | 0.707107i | 0 | 0.707107 | + | 0.707107i | 0 | − | 1.41421i | 0 | − | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||
721.1 | 0 | −0.707107 | − | 0.707107i | 0 | −0.707107 | + | 0.707107i | 0 | − | 1.41421i | 0 | 1.00000i | 0 | ||||||||||||||||||||||||||||||||||||||
721.2 | 0 | −0.707107 | − | 0.707107i | 0 | −0.707107 | + | 0.707107i | 0 | − | 1.41421i | 0 | 1.00000i | 0 | ||||||||||||||||||||||||||||||||||||||
721.3 | 0 | 0.707107 | + | 0.707107i | 0 | 0.707107 | − | 0.707107i | 0 | 1.41421i | 0 | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||||
721.4 | 0 | 0.707107 | + | 0.707107i | 0 | 0.707107 | − | 0.707107i | 0 | 1.41421i | 0 | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
16.e | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 960.2.s.b | 8 | |
3.b | odd | 2 | 1 | 2880.2.t.b | 8 | ||
4.b | odd | 2 | 1 | 240.2.s.b | ✓ | 8 | |
8.b | even | 2 | 1 | 1920.2.s.d | 8 | ||
8.d | odd | 2 | 1 | 1920.2.s.c | 8 | ||
12.b | even | 2 | 1 | 720.2.t.b | 8 | ||
16.e | even | 4 | 1 | inner | 960.2.s.b | 8 | |
16.e | even | 4 | 1 | 1920.2.s.d | 8 | ||
16.f | odd | 4 | 1 | 240.2.s.b | ✓ | 8 | |
16.f | odd | 4 | 1 | 1920.2.s.c | 8 | ||
48.i | odd | 4 | 1 | 2880.2.t.b | 8 | ||
48.k | even | 4 | 1 | 720.2.t.b | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
240.2.s.b | ✓ | 8 | 4.b | odd | 2 | 1 | |
240.2.s.b | ✓ | 8 | 16.f | odd | 4 | 1 | |
720.2.t.b | 8 | 12.b | even | 2 | 1 | ||
720.2.t.b | 8 | 48.k | even | 4 | 1 | ||
960.2.s.b | 8 | 1.a | even | 1 | 1 | trivial | |
960.2.s.b | 8 | 16.e | even | 4 | 1 | inner | |
1920.2.s.c | 8 | 8.d | odd | 2 | 1 | ||
1920.2.s.c | 8 | 16.f | odd | 4 | 1 | ||
1920.2.s.d | 8 | 8.b | even | 2 | 1 | ||
1920.2.s.d | 8 | 16.e | even | 4 | 1 | ||
2880.2.t.b | 8 | 3.b | odd | 2 | 1 | ||
2880.2.t.b | 8 | 48.i | odd | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{2} + 2 \)
acting on \(S_{2}^{\mathrm{new}}(960, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( (T^{4} + 1)^{2} \)
$5$
\( (T^{4} + 1)^{2} \)
$7$
\( (T^{2} + 2)^{4} \)
$11$
\( T^{8} + 8 T^{7} + 32 T^{6} - 16 T^{5} + \cdots + 64 \)
$13$
\( T^{8} - 8 T^{7} + 32 T^{6} - 16 T^{5} + \cdots + 16 \)
$17$
\( (T^{4} - 4 T^{3} - 12 T^{2} + 8)^{2} \)
$19$
\( T^{8} + 8 T^{7} + 32 T^{6} - 96 T^{5} + \cdots + 256 \)
$23$
\( T^{8} + 128 T^{6} + 5248 T^{4} + \cdots + 200704 \)
$29$
\( T^{8} - 24 T^{7} + 288 T^{6} + \cdots + 135424 \)
$31$
\( (T^{4} + 4 T^{3} - 40 T^{2} - 88 T - 28)^{2} \)
$37$
\( T^{8} + 8 T^{7} + 32 T^{6} + \cdots + 414736 \)
$41$
\( T^{8} + 304 T^{6} + \cdots + 17707264 \)
$43$
\( T^{8} - 256 T^{5} + 5408 T^{4} + \cdots + 12544 \)
$47$
\( (T^{4} - 152 T^{2} + 416 T - 224)^{2} \)
$53$
\( T^{8} + 33312 T^{4} + 73984 \)
$59$
\( T^{8} + 8 T^{7} + 32 T^{6} + \cdots + 7529536 \)
$61$
\( T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 4343056 \)
$67$
\( T^{8} + 4224 T^{4} + \cdots + 1183744 \)
$71$
\( T^{8} + 272 T^{6} + 23168 T^{4} + \cdots + 5161984 \)
$73$
\( T^{8} + 176 T^{6} + 4224 T^{4} + \cdots + 50176 \)
$79$
\( (T^{4} - 20 T^{3} + 120 T^{2} - 264 T + 164)^{2} \)
$83$
\( T^{8} + 32 T^{7} + 512 T^{6} + \cdots + 2166784 \)
$89$
\( T^{8} + 592 T^{6} + \cdots + 118026496 \)
$97$
\( (T^{4} + 24 T^{3} + 120 T^{2} - 32 T - 736)^{2} \)
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