Newspace parameters
| Level: | \( N \) | \(=\) | \( 725 = 5^{2} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 725.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(116.278269364\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.3257317.1 |
| Defining polynomial: |
\( x^{4} - 34x^{2} - 27x + 10 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 29) |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 34x^{2} - 27x + 10 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} + 3\nu - 17 ) / 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - 3\nu - 17 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{3} - \nu^{2} - 67\nu - 25 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{2} + 3\beta _1 + 34 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 3\beta_{3} - 32\beta_{2} + 35\beta _1 + 42 ) / 2 \)
|
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 |
|
−5.91663 | 18.2828 | 3.00648 | 0 | −108.173 | −139.558 | 171.544 | 91.2623 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | −4.16235 | 17.5258 | −14.6748 | 0 | −72.9487 | 220.793 | 194.277 | 64.1549 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0.863638 | −7.07413 | −31.2541 | 0 | −6.10949 | 36.7447 | −54.6287 | −192.957 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 9.21534 | −0.734546 | 52.9225 | 0 | −6.76909 | 90.0205 | 192.808 | −242.460 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
| \(29\) | \(1\) |
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 | 725.6.a.a | 4 | |
| 5.b | even | 2 | 1 | 29.6.a.a | ✓ | 4 | |
| 15.d | odd | 2 | 1 | 261.6.a.a | 4 | ||
| 20.d | odd | 2 | 1 | 464.6.a.i | 4 | ||
| 145.d | even | 2 | 1 | 841.6.a.a | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 29.6.a.a | ✓ | 4 | 5.b | even | 2 | 1 | |
| 261.6.a.a | 4 | 15.d | odd | 2 | 1 | ||
| 464.6.a.i | 4 | 20.d | odd | 2 | 1 | ||
| 725.6.a.a | 4 | 1.a | even | 1 | 1 | trivial | |
| 841.6.a.a | 4 | 145.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 69T_{2}^{2} - 168T_{2} + 196 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(725))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - 69 T^{2} - 168 T + 196 \)
$3$
\( T^{4} - 28 T^{3} + 46 T^{2} + \cdots + 1665 \)
$5$
\( T^{4} \)
$7$
\( T^{4} - 208 T^{3} + \cdots - 101924272 \)
$11$
\( T^{4} + 124 T^{3} + \cdots - 3717303119 \)
$13$
\( T^{4} - 460 T^{3} + \cdots - 116053863479 \)
$17$
\( T^{4} + 184 T^{3} + \cdots + 11464717824 \)
$19$
\( T^{4} + 2392 T^{3} + \cdots + 2620094791680 \)
$23$
\( T^{4} - 1192 T^{3} + \cdots + 2033080361984 \)
$29$
\( (T + 841)^{4} \)
$31$
\( T^{4} + \cdots - 421127233952247 \)
$37$
\( T^{4} - 10928 T^{3} + \cdots + 16079593861120 \)
$41$
\( T^{4} + 1120 T^{3} + \cdots + 10\!\cdots\!56 \)
$43$
\( T^{4} - 21420 T^{3} + \cdots + 47\!\cdots\!37 \)
$47$
\( T^{4} + 23772 T^{3} + \cdots - 35\!\cdots\!87 \)
$53$
\( T^{4} + 8860 T^{3} + \cdots + 16\!\cdots\!53 \)
$59$
\( T^{4} + 10840 T^{3} + \cdots - 43\!\cdots\!80 \)
$61$
\( T^{4} - 49448 T^{3} + \cdots + 90\!\cdots\!24 \)
$67$
\( T^{4} - 7840 T^{3} + \cdots + 20\!\cdots\!92 \)
$71$
\( T^{4} + 48744 T^{3} + \cdots - 18\!\cdots\!16 \)
$73$
\( T^{4} - 74992 T^{3} + \cdots - 40\!\cdots\!00 \)
$79$
\( T^{4} + 106076 T^{3} + \cdots - 53\!\cdots\!75 \)
$83$
\( T^{4} + 62888 T^{3} + \cdots - 18\!\cdots\!88 \)
$89$
\( T^{4} - 107568 T^{3} + \cdots - 18\!\cdots\!60 \)
$97$
\( T^{4} - 49520 T^{3} + \cdots - 12\!\cdots\!28 \)
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