Newspace parameters
| Level: | \( N \) | \(=\) | \( 5 \) |
| Weight: | \( k \) | \(=\) | \( 20 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(11.4408348278\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
| Defining polynomial: |
\( x^{4} - 2x^{3} - 323561x^{2} - 26738538x + 10870990650 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{8}\cdot 3^{2}\cdot 5^{2} \) |
| Twist minimal: | yes |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of 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
\( x^{4} - 2x^{3} - 323561x^{2} - 26738538x + 10870990650 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -50\nu^{3} + 12712\nu^{2} + 10576678\nu - 1034892417 ) / 1360737 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -100\nu^{3} + 25424\nu^{2} + 57439676\nu - 2087927994 ) / 453579 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -340\nu^{3} - 224584\nu^{2} + 120441404\nu + 43256659674 ) / 194391 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 6\beta _1 + 40 ) / 80 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -25\beta_{3} + 78\beta_{2} + 722\beta _1 + 6471260 ) / 40 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -12712\beta_{3} + 251195\beta_{2} - 3079258\beta _1 + 1643139760 ) / 80 \)
|
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 |
|
−1387.10 | 48079.9 | 1.39976e6 | −1.95312e6 | −6.66917e7 | 9.13351e7 | −1.21437e9 | 1.14942e9 | 2.70918e9 | ||||||||||||||||||||||||||||||
| 1.2 | −602.379 | −7268.55 | −161427. | −1.95312e6 | 4.37842e6 | −1.76809e8 | 4.13061e8 | −1.10943e9 | 1.17652e9 | |||||||||||||||||||||||||||||||
| 1.3 | 208.721 | −48249.1 | −480724. | −1.95312e6 | −1.00706e7 | 1.75500e8 | −2.09767e8 | 1.16572e9 | −4.07657e8 | |||||||||||||||||||||||||||||||
| 1.4 | 1360.76 | 10517.7 | 1.32738e6 | −1.95312e6 | 1.43121e7 | 1.23996e8 | 1.09282e9 | −1.05164e9 | −2.65773e9 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
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 | 5.20.a.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | 45.20.a.f | 4 | ||
| 4.b | odd | 2 | 1 | 80.20.a.g | 4 | ||
| 5.b | even | 2 | 1 | 25.20.a.c | 4 | ||
| 5.c | odd | 4 | 2 | 25.20.b.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5.20.a.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 25.20.a.c | 4 | 5.b | even | 2 | 1 | ||
| 25.20.b.c | 8 | 5.c | odd | 4 | 2 | ||
| 45.20.a.f | 4 | 3.b | odd | 2 | 1 | ||
| 80.20.a.g | 4 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 420T_{2}^{3} - 2002872T_{2}^{2} - 746347520T_{2} + 237314973696 \)
acting on \(S_{20}^{\mathrm{new}}(\Gamma_0(5))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} + 420 T^{3} + \cdots + 237314973696 \)
$3$
\( T^{4} - 3080 T^{3} + \cdots + 17\!\cdots\!16 \)
$5$
\( (T + 1953125)^{4} \)
$7$
\( T^{4} - 214021400 T^{3} + \cdots - 35\!\cdots\!04 \)
$11$
\( T^{4} - 11585712768 T^{3} + \cdots - 53\!\cdots\!04 \)
$13$
\( T^{4} - 14812333160 T^{3} + \cdots + 38\!\cdots\!36 \)
$17$
\( T^{4} - 849033742440 T^{3} + \cdots - 65\!\cdots\!04 \)
$19$
\( T^{4} - 1978167708560 T^{3} + \cdots + 33\!\cdots\!00 \)
$23$
\( T^{4} + 26569906952760 T^{3} + \cdots - 42\!\cdots\!44 \)
$29$
\( T^{4} - 116267174339640 T^{3} + \cdots - 13\!\cdots\!00 \)
$31$
\( T^{4} - 251049672388688 T^{3} + \cdots - 30\!\cdots\!44 \)
$37$
\( T^{4} - 53471657716520 T^{3} + \cdots + 35\!\cdots\!96 \)
$41$
\( T^{4} + \cdots + 54\!\cdots\!36 \)
$43$
\( T^{4} + \cdots + 10\!\cdots\!96 \)
$47$
\( T^{4} + \cdots - 43\!\cdots\!04 \)
$53$
\( T^{4} + \cdots + 52\!\cdots\!16 \)
$59$
\( T^{4} + \cdots + 37\!\cdots\!00 \)
$61$
\( T^{4} + \cdots + 59\!\cdots\!96 \)
$67$
\( T^{4} + \cdots + 53\!\cdots\!96 \)
$71$
\( T^{4} + \cdots + 10\!\cdots\!76 \)
$73$
\( T^{4} + \cdots - 98\!\cdots\!44 \)
$79$
\( T^{4} + \cdots - 10\!\cdots\!00 \)
$83$
\( T^{4} + \cdots - 18\!\cdots\!24 \)
$89$
\( T^{4} + \cdots - 22\!\cdots\!00 \)
$97$
\( T^{4} + \cdots + 27\!\cdots\!96 \)
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