Newspace parameters
| Level: | \( N \) | \(=\) | \( 224 = 2^{5} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 224.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(35.9259756381\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{61}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 15 \)
|
| 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 \(\beta = \sqrt{61}\). 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 | −0.810250 | 0 | 71.6717 | 0 | 49.0000 | 0 | −242.343 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 14.8102 | 0 | −37.6717 | 0 | 49.0000 | 0 | −23.6565 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(-1\) |
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 | 224.6.a.d | yes | 2 |
| 4.b | odd | 2 | 1 | 224.6.a.c | ✓ | 2 | |
| 8.b | even | 2 | 1 | 448.6.a.s | 2 | ||
| 8.d | odd | 2 | 1 | 448.6.a.y | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 224.6.a.c | ✓ | 2 | 4.b | odd | 2 | 1 | |
| 224.6.a.d | yes | 2 | 1.a | even | 1 | 1 | trivial |
| 448.6.a.s | 2 | 8.b | even | 2 | 1 | ||
| 448.6.a.y | 2 | 8.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 14T_{3} - 12 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(224))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} - 14T - 12 \)
$5$
\( T^{2} - 34T - 2700 \)
$7$
\( (T - 49)^{2} \)
$11$
\( T^{2} + 420T - 84976 \)
$13$
\( T^{2} + 490T - 86436 \)
$17$
\( T^{2} + 1056 T + 266828 \)
$19$
\( T^{2} + 1246 T - 3333420 \)
$23$
\( T^{2} - 504T + 47888 \)
$29$
\( T^{2} + 3904 T - 14374772 \)
$31$
\( T^{2} + 2044 T - 24097520 \)
$37$
\( T^{2} + 7488 T + 14005580 \)
$41$
\( T^{2} - 7832 T - 63108260 \)
$43$
\( T^{2} + 10332 T - 45786544 \)
$47$
\( T^{2} + 41972 T + 396339696 \)
$53$
\( T^{2} - 32812 T - 171589148 \)
$59$
\( T^{2} + 48398 T + 421726020 \)
$61$
\( T^{2} + 718 T - 11734460 \)
$67$
\( T^{2} + 12824 T - 792171632 \)
$71$
\( T^{2} + 103992 T + 2694018240 \)
$73$
\( T^{2} + 54100 T - 1049693676 \)
$79$
\( T^{2} + 64568 T + 967369152 \)
$83$
\( T^{2} - 47810 T + 554507556 \)
$89$
\( T^{2} + 17388 T - 12657841308 \)
$97$
\( T^{2} + 97296 T + 62455628 \)
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