Newspace parameters
Level: | \( N \) | \(=\) | \( 800 = 2^{5} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 800.f (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(47.2015280046\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(i, \sqrt{7})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} - 3x^{2} + 4 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
Coefficient ring index: | \( 2^{5} \) |
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 \( x^{4} - 3x^{2} + 4 \) :
\(\beta_{1}\) | \(=\) | \( \nu^{3} - \nu \) |
\(\beta_{2}\) | \(=\) | \( -\nu^{3} + 5\nu \) |
\(\beta_{3}\) | \(=\) | \( 4\nu^{2} - 6 \) |
\(\nu\) | \(=\) | \( ( \beta_{2} + \beta_1 ) / 4 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{3} + 6 ) / 4 \) |
\(\nu^{3}\) | \(=\) | \( ( \beta_{2} + 5\beta_1 ) / 4 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/800\mathbb{Z}\right)^\times\).
\(n\) | \(101\) | \(351\) | \(577\) |
\(\chi(n)\) | \(-1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
49.1 |
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0 | −5.29150 | 0 | 0 | 0 | − | 8.00000i | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||
49.2 | 0 | −5.29150 | 0 | 0 | 0 | 8.00000i | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||
49.3 | 0 | 5.29150 | 0 | 0 | 0 | − | 8.00000i | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||
49.4 | 0 | 5.29150 | 0 | 0 | 0 | 8.00000i | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
8.b | even | 2 | 1 | inner |
40.f | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 800.4.f.a | 4 | |
4.b | odd | 2 | 1 | 200.4.f.a | 4 | ||
5.b | even | 2 | 1 | inner | 800.4.f.a | 4 | |
5.c | odd | 4 | 1 | 32.4.b.a | 2 | ||
5.c | odd | 4 | 1 | 800.4.d.a | 2 | ||
8.b | even | 2 | 1 | inner | 800.4.f.a | 4 | |
8.d | odd | 2 | 1 | 200.4.f.a | 4 | ||
15.e | even | 4 | 1 | 288.4.d.a | 2 | ||
20.d | odd | 2 | 1 | 200.4.f.a | 4 | ||
20.e | even | 4 | 1 | 8.4.b.a | ✓ | 2 | |
20.e | even | 4 | 1 | 200.4.d.a | 2 | ||
40.e | odd | 2 | 1 | 200.4.f.a | 4 | ||
40.f | even | 2 | 1 | inner | 800.4.f.a | 4 | |
40.i | odd | 4 | 1 | 32.4.b.a | 2 | ||
40.i | odd | 4 | 1 | 800.4.d.a | 2 | ||
40.k | even | 4 | 1 | 8.4.b.a | ✓ | 2 | |
40.k | even | 4 | 1 | 200.4.d.a | 2 | ||
60.l | odd | 4 | 1 | 72.4.d.b | 2 | ||
80.i | odd | 4 | 1 | 256.4.a.j | 2 | ||
80.j | even | 4 | 1 | 256.4.a.l | 2 | ||
80.s | even | 4 | 1 | 256.4.a.l | 2 | ||
80.t | odd | 4 | 1 | 256.4.a.j | 2 | ||
120.q | odd | 4 | 1 | 72.4.d.b | 2 | ||
120.w | even | 4 | 1 | 288.4.d.a | 2 | ||
240.z | odd | 4 | 1 | 2304.4.a.bn | 2 | ||
240.bb | even | 4 | 1 | 2304.4.a.v | 2 | ||
240.bd | odd | 4 | 1 | 2304.4.a.bn | 2 | ||
240.bf | even | 4 | 1 | 2304.4.a.v | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8.4.b.a | ✓ | 2 | 20.e | even | 4 | 1 | |
8.4.b.a | ✓ | 2 | 40.k | even | 4 | 1 | |
32.4.b.a | 2 | 5.c | odd | 4 | 1 | ||
32.4.b.a | 2 | 40.i | odd | 4 | 1 | ||
72.4.d.b | 2 | 60.l | odd | 4 | 1 | ||
72.4.d.b | 2 | 120.q | odd | 4 | 1 | ||
200.4.d.a | 2 | 20.e | even | 4 | 1 | ||
200.4.d.a | 2 | 40.k | even | 4 | 1 | ||
200.4.f.a | 4 | 4.b | odd | 2 | 1 | ||
200.4.f.a | 4 | 8.d | odd | 2 | 1 | ||
200.4.f.a | 4 | 20.d | odd | 2 | 1 | ||
200.4.f.a | 4 | 40.e | odd | 2 | 1 | ||
256.4.a.j | 2 | 80.i | odd | 4 | 1 | ||
256.4.a.j | 2 | 80.t | odd | 4 | 1 | ||
256.4.a.l | 2 | 80.j | even | 4 | 1 | ||
256.4.a.l | 2 | 80.s | even | 4 | 1 | ||
288.4.d.a | 2 | 15.e | even | 4 | 1 | ||
288.4.d.a | 2 | 120.w | even | 4 | 1 | ||
800.4.d.a | 2 | 5.c | odd | 4 | 1 | ||
800.4.d.a | 2 | 40.i | odd | 4 | 1 | ||
800.4.f.a | 4 | 1.a | even | 1 | 1 | trivial | |
800.4.f.a | 4 | 5.b | even | 2 | 1 | inner | |
800.4.f.a | 4 | 8.b | even | 2 | 1 | inner | |
800.4.f.a | 4 | 40.f | even | 2 | 1 | inner | |
2304.4.a.v | 2 | 240.bb | even | 4 | 1 | ||
2304.4.a.v | 2 | 240.bf | even | 4 | 1 | ||
2304.4.a.bn | 2 | 240.z | odd | 4 | 1 | ||
2304.4.a.bn | 2 | 240.bd | odd | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 28 \)
acting on \(S_{4}^{\mathrm{new}}(800, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( (T^{2} - 28)^{2} \)
$5$
\( T^{4} \)
$7$
\( (T^{2} + 64)^{2} \)
$11$
\( (T^{2} + 252)^{2} \)
$13$
\( (T^{2} - 2800)^{2} \)
$17$
\( (T^{2} + 196)^{2} \)
$19$
\( (T^{2} + 1372)^{2} \)
$23$
\( (T^{2} + 23104)^{2} \)
$29$
\( (T^{2} + 25200)^{2} \)
$31$
\( (T + 224)^{4} \)
$37$
\( (T^{2} - 59248)^{2} \)
$41$
\( (T + 70)^{4} \)
$43$
\( (T^{2} - 192892)^{2} \)
$47$
\( (T^{2} + 112896)^{2} \)
$53$
\( (T^{2} - 1008)^{2} \)
$59$
\( (T^{2} + 285628)^{2} \)
$61$
\( (T^{2} + 9072)^{2} \)
$67$
\( (T^{2} - 30492)^{2} \)
$71$
\( (T - 72)^{4} \)
$73$
\( (T^{2} + 86436)^{2} \)
$79$
\( (T + 464)^{4} \)
$83$
\( (T^{2} - 297052)^{2} \)
$89$
\( (T + 266)^{4} \)
$97$
\( (T^{2} + 988036)^{2} \)
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