Newspace parameters
Level: | \( N \) | \(=\) | \( 200 = 2^{3} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 200.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(11.8003820011\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{6}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{2} - 6 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 2 \) |
Twist minimal: | no (minimal twist has level 40) |
Fricke sign: | \(1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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.
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1.1 |
|
0 | −2.89898 | 0 | 0 | 0 | 16.6969 | 0 | −18.5959 | 0 | ||||||||||||||||||||||||
1.2 | 0 | 6.89898 | 0 | 0 | 0 | −12.6969 | 0 | 20.5959 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(-1\) |
\(5\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 200.4.a.l | 2 | |
3.b | odd | 2 | 1 | 1800.4.a.bp | 2 | ||
4.b | odd | 2 | 1 | 400.4.a.v | 2 | ||
5.b | even | 2 | 1 | 200.4.a.k | 2 | ||
5.c | odd | 4 | 2 | 40.4.c.a | ✓ | 4 | |
8.b | even | 2 | 1 | 1600.4.a.ce | 2 | ||
8.d | odd | 2 | 1 | 1600.4.a.cm | 2 | ||
15.d | odd | 2 | 1 | 1800.4.a.bk | 2 | ||
15.e | even | 4 | 2 | 360.4.f.e | 4 | ||
20.d | odd | 2 | 1 | 400.4.a.x | 2 | ||
20.e | even | 4 | 2 | 80.4.c.c | 4 | ||
40.e | odd | 2 | 1 | 1600.4.a.cf | 2 | ||
40.f | even | 2 | 1 | 1600.4.a.cl | 2 | ||
40.i | odd | 4 | 2 | 320.4.c.g | 4 | ||
40.k | even | 4 | 2 | 320.4.c.h | 4 | ||
60.l | odd | 4 | 2 | 720.4.f.m | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
40.4.c.a | ✓ | 4 | 5.c | odd | 4 | 2 | |
80.4.c.c | 4 | 20.e | even | 4 | 2 | ||
200.4.a.k | 2 | 5.b | even | 2 | 1 | ||
200.4.a.l | 2 | 1.a | even | 1 | 1 | trivial | |
320.4.c.g | 4 | 40.i | odd | 4 | 2 | ||
320.4.c.h | 4 | 40.k | even | 4 | 2 | ||
360.4.f.e | 4 | 15.e | even | 4 | 2 | ||
400.4.a.v | 2 | 4.b | odd | 2 | 1 | ||
400.4.a.x | 2 | 20.d | odd | 2 | 1 | ||
720.4.f.m | 4 | 60.l | odd | 4 | 2 | ||
1600.4.a.ce | 2 | 8.b | even | 2 | 1 | ||
1600.4.a.cf | 2 | 40.e | odd | 2 | 1 | ||
1600.4.a.cl | 2 | 40.f | even | 2 | 1 | ||
1600.4.a.cm | 2 | 8.d | odd | 2 | 1 | ||
1800.4.a.bk | 2 | 15.d | odd | 2 | 1 | ||
1800.4.a.bp | 2 | 3.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 4T_{3} - 20 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(200))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} - 4T - 20 \)
$5$
\( T^{2} \)
$7$
\( T^{2} - 4T - 212 \)
$11$
\( T^{2} - 40T - 1136 \)
$13$
\( T^{2} - 104T + 2608 \)
$17$
\( T^{2} - 96T - 3840 \)
$19$
\( T^{2} - 40T - 1136 \)
$23$
\( T^{2} - 284T + 16108 \)
$29$
\( T^{2} + 140T - 1244 \)
$31$
\( T^{2} - 192T - 46080 \)
$37$
\( T^{2} - 200T - 17744 \)
$41$
\( T^{2} + 524T + 65188 \)
$43$
\( T^{2} - 372T + 24012 \)
$47$
\( T^{2} - 84T - 99636 \)
$53$
\( T^{2} + 296T - 268496 \)
$59$
\( T^{2} - 696T - 64752 \)
$61$
\( T^{2} + 692T + 100900 \)
$67$
\( T^{2} - 316T + 23788 \)
$71$
\( T^{2} - 688T + 93760 \)
$73$
\( T^{2} + 656T - 95552 \)
$79$
\( T^{2} + 736T + 129280 \)
$83$
\( T^{2} + 1628 T + 604972 \)
$89$
\( T^{2} + 660T + 53604 \)
$97$
\( T^{2} - 896T + 162304 \)
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