Newspace parameters
Level: | \( N \) | \(=\) | \( 800 = 2^{5} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 800.g (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(0.399252010106\) |
Analytic rank: | \(0\) |
Dimension: | \(1\) |
Coefficient field: | \(\mathbb{Q}\) |
Coefficient ring: | \(\mathbb{Z}\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 200) |
Projective image: | \(D_{3}\) |
Projective field: | Galois closure of 3.1.200.1 |
Artin image: | $D_6$ |
Artin field: | Galois closure of 6.0.3200000.1 |
$q$-expansion
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
751.1 |
|
0 | −1.00000 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-2}) \) |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 800.1.g.a | 1 | |
4.b | odd | 2 | 1 | 200.1.g.a | ✓ | 1 | |
5.b | even | 2 | 1 | 800.1.g.b | 1 | ||
5.c | odd | 4 | 2 | 800.1.e.a | 2 | ||
8.b | even | 2 | 1 | 200.1.g.a | ✓ | 1 | |
8.d | odd | 2 | 1 | CM | 800.1.g.a | 1 | |
12.b | even | 2 | 1 | 1800.1.g.b | 1 | ||
20.d | odd | 2 | 1 | 200.1.g.b | yes | 1 | |
20.e | even | 4 | 2 | 200.1.e.a | 2 | ||
24.h | odd | 2 | 1 | 1800.1.g.b | 1 | ||
40.e | odd | 2 | 1 | 800.1.g.b | 1 | ||
40.f | even | 2 | 1 | 200.1.g.b | yes | 1 | |
40.i | odd | 4 | 2 | 200.1.e.a | 2 | ||
40.k | even | 4 | 2 | 800.1.e.a | 2 | ||
60.h | even | 2 | 1 | 1800.1.g.a | 1 | ||
60.l | odd | 4 | 2 | 1800.1.p.a | 2 | ||
120.i | odd | 2 | 1 | 1800.1.g.a | 1 | ||
120.w | even | 4 | 2 | 1800.1.p.a | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
200.1.e.a | 2 | 20.e | even | 4 | 2 | ||
200.1.e.a | 2 | 40.i | odd | 4 | 2 | ||
200.1.g.a | ✓ | 1 | 4.b | odd | 2 | 1 | |
200.1.g.a | ✓ | 1 | 8.b | even | 2 | 1 | |
200.1.g.b | yes | 1 | 20.d | odd | 2 | 1 | |
200.1.g.b | yes | 1 | 40.f | even | 2 | 1 | |
800.1.e.a | 2 | 5.c | odd | 4 | 2 | ||
800.1.e.a | 2 | 40.k | even | 4 | 2 | ||
800.1.g.a | 1 | 1.a | even | 1 | 1 | trivial | |
800.1.g.a | 1 | 8.d | odd | 2 | 1 | CM | |
800.1.g.b | 1 | 5.b | even | 2 | 1 | ||
800.1.g.b | 1 | 40.e | odd | 2 | 1 | ||
1800.1.g.a | 1 | 60.h | even | 2 | 1 | ||
1800.1.g.a | 1 | 120.i | odd | 2 | 1 | ||
1800.1.g.b | 1 | 12.b | even | 2 | 1 | ||
1800.1.g.b | 1 | 24.h | odd | 2 | 1 | ||
1800.1.p.a | 2 | 60.l | odd | 4 | 2 | ||
1800.1.p.a | 2 | 120.w | even | 4 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} + 1 \)
acting on \(S_{1}^{\mathrm{new}}(800, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T \)
$3$
\( T + 1 \)
$5$
\( T \)
$7$
\( T \)
$11$
\( T - 1 \)
$13$
\( T \)
$17$
\( T - 1 \)
$19$
\( T - 1 \)
$23$
\( T \)
$29$
\( T \)
$31$
\( T \)
$37$
\( T \)
$41$
\( T + 1 \)
$43$
\( T - 2 \)
$47$
\( T \)
$53$
\( T \)
$59$
\( T + 2 \)
$61$
\( T \)
$67$
\( T + 1 \)
$71$
\( T \)
$73$
\( T - 1 \)
$79$
\( T \)
$83$
\( T + 1 \)
$89$
\( T + 1 \)
$97$
\( T + 2 \)
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