Newspace parameters
Level: | \( N \) | \(=\) | \( 950 = 2 \cdot 5^{2} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 950.f (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(7.58578819202\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(4\) over \(\Q(i)\) |
Coefficient field: | 8.0.3317760000.4 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} + 17x^{4} + 256 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{4} \) |
Twist minimal: | yes |
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 \( x^{8} + 17x^{4} + 256 \) :
\(\beta_{1}\) | \(=\) | \( ( \nu^{5} + 5\nu ) / 28 \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{6} + 33\nu^{2} ) / 112 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{5} + 61\nu ) / 28 \) |
\(\beta_{4}\) | \(=\) | \( ( 2\nu^{4} + 17 ) / 7 \) |
\(\beta_{5}\) | \(=\) | \( ( -3\nu^{7} + 13\nu^{3} ) / 448 \) |
\(\beta_{6}\) | \(=\) | \( ( -\nu^{6} - \nu^{2} ) / 16 \) |
\(\beta_{7}\) | \(=\) | \( ( -11\nu^{7} - 251\nu^{3} ) / 448 \) |
\(\nu\) | \(=\) | \( ( \beta_{3} - \beta_1 ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{6} + 7\beta_{2} ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( ( -3\beta_{7} + 11\beta_{5} ) / 2 \) |
\(\nu^{4}\) | \(=\) | \( ( 7\beta_{4} - 17 ) / 2 \) |
\(\nu^{5}\) | \(=\) | \( ( -5\beta_{3} + 61\beta_1 ) / 2 \) |
\(\nu^{6}\) | \(=\) | \( ( -33\beta_{6} - 7\beta_{2} ) / 2 \) |
\(\nu^{7}\) | \(=\) | \( ( -13\beta_{7} - 251\beta_{5} ) / 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/950\mathbb{Z}\right)^\times\).
\(n\) | \(77\) | \(401\) |
\(\chi(n)\) | \(\beta_{2}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
493.1 |
|
−0.707107 | − | 0.707107i | −0.707107 | + | 0.707107i | 1.00000i | 0 | 1.00000 | −2.73861 | + | 2.73861i | 0.707107 | − | 0.707107i | 2.00000i | 0 | ||||||||||||||||||||||||||||||||||
493.2 | −0.707107 | − | 0.707107i | −0.707107 | + | 0.707107i | 1.00000i | 0 | 1.00000 | 2.73861 | − | 2.73861i | 0.707107 | − | 0.707107i | 2.00000i | 0 | |||||||||||||||||||||||||||||||||||
493.3 | 0.707107 | + | 0.707107i | 0.707107 | − | 0.707107i | 1.00000i | 0 | 1.00000 | −2.73861 | + | 2.73861i | −0.707107 | + | 0.707107i | 2.00000i | 0 | |||||||||||||||||||||||||||||||||||
493.4 | 0.707107 | + | 0.707107i | 0.707107 | − | 0.707107i | 1.00000i | 0 | 1.00000 | 2.73861 | − | 2.73861i | −0.707107 | + | 0.707107i | 2.00000i | 0 | |||||||||||||||||||||||||||||||||||
607.1 | −0.707107 | + | 0.707107i | −0.707107 | − | 0.707107i | − | 1.00000i | 0 | 1.00000 | −2.73861 | − | 2.73861i | 0.707107 | + | 0.707107i | − | 2.00000i | 0 | |||||||||||||||||||||||||||||||||
607.2 | −0.707107 | + | 0.707107i | −0.707107 | − | 0.707107i | − | 1.00000i | 0 | 1.00000 | 2.73861 | + | 2.73861i | 0.707107 | + | 0.707107i | − | 2.00000i | 0 | |||||||||||||||||||||||||||||||||
607.3 | 0.707107 | − | 0.707107i | 0.707107 | + | 0.707107i | − | 1.00000i | 0 | 1.00000 | −2.73861 | − | 2.73861i | −0.707107 | − | 0.707107i | − | 2.00000i | 0 | |||||||||||||||||||||||||||||||||
607.4 | 0.707107 | − | 0.707107i | 0.707107 | + | 0.707107i | − | 1.00000i | 0 | 1.00000 | 2.73861 | + | 2.73861i | −0.707107 | − | 0.707107i | − | 2.00000i | 0 | |||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
5.c | odd | 4 | 2 | inner |
19.b | odd | 2 | 1 | inner |
95.d | odd | 2 | 1 | inner |
95.g | even | 4 | 2 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 950.2.f.b | ✓ | 8 |
5.b | even | 2 | 1 | inner | 950.2.f.b | ✓ | 8 |
5.c | odd | 4 | 2 | inner | 950.2.f.b | ✓ | 8 |
19.b | odd | 2 | 1 | inner | 950.2.f.b | ✓ | 8 |
95.d | odd | 2 | 1 | inner | 950.2.f.b | ✓ | 8 |
95.g | even | 4 | 2 | inner | 950.2.f.b | ✓ | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
950.2.f.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
950.2.f.b | ✓ | 8 | 5.b | even | 2 | 1 | inner |
950.2.f.b | ✓ | 8 | 5.c | odd | 4 | 2 | inner |
950.2.f.b | ✓ | 8 | 19.b | odd | 2 | 1 | inner |
950.2.f.b | ✓ | 8 | 95.d | odd | 2 | 1 | inner |
950.2.f.b | ✓ | 8 | 95.g | even | 4 | 2 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(950, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{4} + 1)^{2} \)
$3$
\( (T^{4} + 1)^{2} \)
$5$
\( T^{8} \)
$7$
\( (T^{4} + 225)^{2} \)
$11$
\( T^{8} \)
$13$
\( (T^{4} + 1)^{2} \)
$17$
\( (T^{4} + 225)^{2} \)
$19$
\( (T^{4} - 22 T^{2} + 361)^{2} \)
$23$
\( (T^{4} + 225)^{2} \)
$29$
\( (T^{2} - 15)^{4} \)
$31$
\( T^{8} \)
$37$
\( (T^{4} + 16)^{2} \)
$41$
\( (T^{2} + 60)^{4} \)
$43$
\( T^{8} \)
$47$
\( (T^{4} + 3600)^{2} \)
$53$
\( (T^{4} + 6561)^{2} \)
$59$
\( (T^{2} - 135)^{4} \)
$61$
\( (T - 10)^{8} \)
$67$
\( (T^{4} + 28561)^{2} \)
$71$
\( (T^{2} + 60)^{4} \)
$73$
\( (T^{4} + 225)^{2} \)
$79$
\( (T^{2} - 240)^{4} \)
$83$
\( (T^{4} + 3600)^{2} \)
$89$
\( (T^{2} - 240)^{4} \)
$97$
\( (T^{4} + 4096)^{2} \)
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