Newspace parameters
| Level: | \( N \) | \(=\) | \( 200 = 2^{3} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 200.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(32.0767639626\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{129}) \) |
| Defining polynomial: |
\( x^{2} - x - 32 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(\beta = 2\sqrt{129}\). 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 | −16.7156 | 0 | 0 | 0 | −94.1469 | 0 | 36.4124 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 28.7156 | 0 | 0 | 0 | 42.1469 | 0 | 581.588 | 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 | 200.6.a.g | 2 | |
| 4.b | odd | 2 | 1 | 400.6.a.q | 2 | ||
| 5.b | even | 2 | 1 | 40.6.a.d | ✓ | 2 | |
| 5.c | odd | 4 | 2 | 200.6.c.e | 4 | ||
| 15.d | odd | 2 | 1 | 360.6.a.l | 2 | ||
| 20.d | odd | 2 | 1 | 80.6.a.i | 2 | ||
| 20.e | even | 4 | 2 | 400.6.c.l | 4 | ||
| 40.e | odd | 2 | 1 | 320.6.a.q | 2 | ||
| 40.f | even | 2 | 1 | 320.6.a.w | 2 | ||
| 60.h | even | 2 | 1 | 720.6.a.z | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 40.6.a.d | ✓ | 2 | 5.b | even | 2 | 1 | |
| 80.6.a.i | 2 | 20.d | odd | 2 | 1 | ||
| 200.6.a.g | 2 | 1.a | even | 1 | 1 | trivial | |
| 200.6.c.e | 4 | 5.c | odd | 4 | 2 | ||
| 320.6.a.q | 2 | 40.e | odd | 2 | 1 | ||
| 320.6.a.w | 2 | 40.f | even | 2 | 1 | ||
| 360.6.a.l | 2 | 15.d | odd | 2 | 1 | ||
| 400.6.a.q | 2 | 4.b | odd | 2 | 1 | ||
| 400.6.c.l | 4 | 20.e | even | 4 | 2 | ||
| 720.6.a.z | 2 | 60.h | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 12T_{3} - 480 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(200))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} - 12T - 480 \)
$5$
\( T^{2} \)
$7$
\( T^{2} + 52T - 3968 \)
$11$
\( T^{2} - 560T + 59824 \)
$13$
\( T^{2} + 1388 T + 407332 \)
$17$
\( T^{2} + 148 T - 3635420 \)
$19$
\( T^{2} + 1000 T - 418736 \)
$23$
\( T^{2} - 2452 T - 6303488 \)
$29$
\( T^{2} - 1340 T - 49780604 \)
$31$
\( T^{2} + 2248 T - 241280 \)
$37$
\( T^{2} - 5940 T - 15253596 \)
$41$
\( T^{2} - 23076 T + 120568068 \)
$43$
\( T^{2} + 17684 T + 18881728 \)
$47$
\( T^{2} - 2908 T - 402029984 \)
$53$
\( T^{2} - 5412 T - 143143164 \)
$59$
\( T^{2} - 62584 T + 848117008 \)
$61$
\( T^{2} - 14108 T - 579150140 \)
$67$
\( T^{2} - 85412 T + 1671660352 \)
$71$
\( T^{2} - 47208 T + 403320960 \)
$73$
\( T^{2} - 67452 T + 254637252 \)
$79$
\( T^{2} + 65904 T - 2159838720 \)
$83$
\( T^{2} + 108724 T + 2694378208 \)
$89$
\( T^{2} + 55020 T - 349140636 \)
$97$
\( T^{2} + 147668 T - 5562250844 \)
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