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: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{241}) \) |
| Defining polynomial: |
\( x^{2} - x - 60 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{241}\). 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 | −19.5242 | 0 | 0 | 0 | 35.0483 | 0 | 138.193 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 11.5242 | 0 | 0 | 0 | −27.0483 | 0 | −110.193 | 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.e | ✓ | 2 |
| 4.b | odd | 2 | 1 | 400.6.a.u | 2 | ||
| 5.b | even | 2 | 1 | 200.6.a.f | yes | 2 | |
| 5.c | odd | 4 | 2 | 200.6.c.f | 4 | ||
| 20.d | odd | 2 | 1 | 400.6.a.r | 2 | ||
| 20.e | even | 4 | 2 | 400.6.c.o | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 200.6.a.e | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 200.6.a.f | yes | 2 | 5.b | even | 2 | 1 | |
| 200.6.c.f | 4 | 5.c | odd | 4 | 2 | ||
| 400.6.a.r | 2 | 20.d | odd | 2 | 1 | ||
| 400.6.a.u | 2 | 4.b | odd | 2 | 1 | ||
| 400.6.c.o | 4 | 20.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 8T_{3} - 225 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(200))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} + 8T - 225 \)
$5$
\( T^{2} \)
$7$
\( T^{2} - 8T - 948 \)
$11$
\( T^{2} + 200T - 96281 \)
$13$
\( T^{2} - 592T - 564048 \)
$17$
\( T^{2} + 278T - 42375 \)
$19$
\( T^{2} + 840T - 662521 \)
$23$
\( T^{2} - 1952 T - 1361988 \)
$29$
\( T^{2} + 4680 T + 196736 \)
$31$
\( T^{2} + 5008 T - 23931140 \)
$37$
\( T^{2} + 12500 T - 17393196 \)
$41$
\( T^{2} + 5334 T - 204026247 \)
$43$
\( T^{2} + 224 T - 279011472 \)
$47$
\( (T + 13036)^{2} \)
$53$
\( T^{2} - 46812 T + 516607236 \)
$59$
\( T^{2} + 81776 T + 1630122048 \)
$61$
\( T^{2} + 46932 T + 550186580 \)
$67$
\( T^{2} + 68808 T + 1158491927 \)
$71$
\( T^{2} - 7448 T - 14650800 \)
$73$
\( T^{2} - 108822 T + 2959307577 \)
$79$
\( T^{2} + 108104 T + 662040300 \)
$83$
\( T^{2} + 27224 T - 3370254177 \)
$89$
\( T^{2} - 70990 T - 4823083791 \)
$97$
\( T^{2} - 96852 T - 1134962524 \)
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