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: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 229x^{3} - 272x^{2} + 7973x - 13998 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{12} \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{5} - 229x^{3} - 272x^{2} + 7973x - 13998 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 56\nu^{4} - 64\nu^{3} - 8772\nu^{2} - 20218\nu - 3528 ) / 633 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 28\nu^{4} - 32\nu^{3} - 5652\nu^{2} - 8210\nu + 114075 ) / 633 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 36\nu^{4} + 200\nu^{3} - 8352\nu^{2} - 46908\nu + 206833 ) / 211 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2\beta_{3} + \beta_{2} + 3\beta _1 + 366 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 7\beta_{4} - 45\beta_{3} + 9\beta_{2} + 630\beta _1 + 1298 ) / 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 4\beta_{4} - 339\beta_{3} + 207\beta_{2} + 1552\beta _1 + 58325 ) / 4 \)
|
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 | −25.7852 | 0 | −34.4729 | 0 | 49.0000 | 0 | 421.876 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −19.9790 | 0 | 106.158 | 0 | 49.0000 | 0 | 156.159 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 2.49210 | 0 | −99.5911 | 0 | 49.0000 | 0 | −236.789 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 5.95336 | 0 | 11.6830 | 0 | 49.0000 | 0 | −207.557 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 27.3187 | 0 | 52.2232 | 0 | 49.0000 | 0 | 503.312 | 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.i | ✓ | 5 |
| 4.b | odd | 2 | 1 | 224.6.a.j | yes | 5 | |
| 8.b | even | 2 | 1 | 448.6.a.bf | 5 | ||
| 8.d | odd | 2 | 1 | 448.6.a.be | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 224.6.a.i | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 224.6.a.j | yes | 5 | 4.b | odd | 2 | 1 | |
| 448.6.a.be | 5 | 8.d | odd | 2 | 1 | ||
| 448.6.a.bf | 5 | 8.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{5} + 10T_{3}^{4} - 876T_{3}^{3} - 7592T_{3}^{2} + 107952T_{3} - 208800 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(224))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{5} \)
$3$
\( T^{5} + 10 T^{4} - 876 T^{3} + \cdots - 208800 \)
$5$
\( T^{5} - 36 T^{4} + \cdots - 222365696 \)
$7$
\( (T - 49)^{5} \)
$11$
\( T^{5} - 116 T^{4} + \cdots + 183286596608 \)
$13$
\( T^{5} - 40 T^{4} + \cdots + 247266196544 \)
$17$
\( T^{5} + \cdots - 353511151373280 \)
$19$
\( T^{5} + 3582 T^{4} + \cdots + 71\!\cdots\!08 \)
$23$
\( T^{5} - 472 T^{4} + \cdots + 11\!\cdots\!52 \)
$29$
\( T^{5} - 4754 T^{4} + \cdots + 38\!\cdots\!20 \)
$31$
\( T^{5} + 10500 T^{4} + \cdots + 34\!\cdots\!20 \)
$37$
\( T^{5} - 19642 T^{4} + \cdots + 11\!\cdots\!00 \)
$41$
\( T^{5} - 23398 T^{4} + \cdots + 13\!\cdots\!00 \)
$43$
\( T^{5} - 22044 T^{4} + \cdots - 77\!\cdots\!32 \)
$47$
\( T^{5} - 16004 T^{4} + \cdots + 62\!\cdots\!40 \)
$53$
\( T^{5} - 54246 T^{4} + \cdots - 57\!\cdots\!00 \)
$59$
\( T^{5} + 74366 T^{4} + \cdots - 14\!\cdots\!20 \)
$61$
\( T^{5} - 68316 T^{4} + \cdots - 30\!\cdots\!16 \)
$67$
\( T^{5} + 26560 T^{4} + \cdots + 16\!\cdots\!64 \)
$71$
\( T^{5} - 93072 T^{4} + \cdots + 10\!\cdots\!36 \)
$73$
\( T^{5} - 136098 T^{4} + \cdots + 81\!\cdots\!00 \)
$79$
\( T^{5} - 96080 T^{4} + \cdots - 26\!\cdots\!92 \)
$83$
\( T^{5} + 145894 T^{4} + \cdots - 86\!\cdots\!00 \)
$89$
\( T^{5} - 188554 T^{4} + \cdots + 64\!\cdots\!92 \)
$97$
\( T^{5} + 88146 T^{4} + \cdots + 93\!\cdots\!00 \)
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