Newspace parameters
Level: | \( N \) | \(=\) | \( 200 = 2^{3} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 200.g (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(5.44960528721\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{6}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{2} - 6 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 2 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/200\mathbb{Z}\right)^\times\).
\(n\) | \(101\) | \(151\) | \(177\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
51.1 |
|
−2.00000 | −3.89898 | 4.00000 | 0 | 7.79796 | 0 | −8.00000 | 6.20204 | 0 | ||||||||||||||||||||||||
51.2 | −2.00000 | 5.89898 | 4.00000 | 0 | −11.7980 | 0 | −8.00000 | 25.7980 | 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 | 200.3.g.b | ✓ | 2 |
4.b | odd | 2 | 1 | 800.3.g.b | 2 | ||
5.b | even | 2 | 1 | 200.3.g.d | yes | 2 | |
5.c | odd | 4 | 2 | 200.3.e.b | 4 | ||
8.b | even | 2 | 1 | 800.3.g.b | 2 | ||
8.d | odd | 2 | 1 | CM | 200.3.g.b | ✓ | 2 |
20.d | odd | 2 | 1 | 800.3.g.d | 2 | ||
20.e | even | 4 | 2 | 800.3.e.b | 4 | ||
40.e | odd | 2 | 1 | 200.3.g.d | yes | 2 | |
40.f | even | 2 | 1 | 800.3.g.d | 2 | ||
40.i | odd | 4 | 2 | 800.3.e.b | 4 | ||
40.k | even | 4 | 2 | 200.3.e.b | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
200.3.e.b | 4 | 5.c | odd | 4 | 2 | ||
200.3.e.b | 4 | 40.k | even | 4 | 2 | ||
200.3.g.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
200.3.g.b | ✓ | 2 | 8.d | odd | 2 | 1 | CM |
200.3.g.d | yes | 2 | 5.b | even | 2 | 1 | |
200.3.g.d | yes | 2 | 40.e | odd | 2 | 1 | |
800.3.e.b | 4 | 20.e | even | 4 | 2 | ||
800.3.e.b | 4 | 40.i | odd | 4 | 2 | ||
800.3.g.b | 2 | 4.b | odd | 2 | 1 | ||
800.3.g.b | 2 | 8.b | even | 2 | 1 | ||
800.3.g.d | 2 | 20.d | odd | 2 | 1 | ||
800.3.g.d | 2 | 40.f | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 2T_{3} - 23 \)
acting on \(S_{3}^{\mathrm{new}}(200, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T + 2)^{2} \)
$3$
\( T^{2} - 2T - 23 \)
$5$
\( T^{2} \)
$7$
\( T^{2} \)
$11$
\( T^{2} + 14T - 167 \)
$13$
\( T^{2} \)
$17$
\( T^{2} + 2T - 863 \)
$19$
\( T^{2} - 34T + 73 \)
$23$
\( T^{2} \)
$29$
\( T^{2} \)
$31$
\( T^{2} \)
$37$
\( T^{2} \)
$41$
\( T^{2} - 46T - 2927 \)
$43$
\( (T - 14)^{2} \)
$47$
\( T^{2} \)
$53$
\( T^{2} \)
$59$
\( (T + 82)^{2} \)
$61$
\( T^{2} \)
$67$
\( T^{2} + 62T - 9623 \)
$71$
\( T^{2} \)
$73$
\( T^{2} - 142T + 4177 \)
$79$
\( T^{2} \)
$83$
\( T^{2} + 158T + 4297 \)
$89$
\( T^{2} + 146T - 2447 \)
$97$
\( (T + 94)^{2} \)
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