Newspace parameters
Level: | \( N \) | \(=\) | \( 80 = 2^{4} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 5 \) |
Character orbit: | \([\chi]\) | \(=\) | 80.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(8.26959704671\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{5})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} - x^{3} + 2x^{2} + x + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 2^{6} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \( x^{4} - x^{3} + 2x^{2} + x + 1 \) :
\(\beta_{1}\) | \(=\) | \( \nu^{3} + 2 \) |
\(\beta_{2}\) | \(=\) | \( -\nu^{3} + 4\nu^{2} + 1 \) |
\(\beta_{3}\) | \(=\) | \( 3\nu^{3} - 4\nu^{2} + 8\nu + 1 \) |
\(\nu\) | \(=\) | \( ( \beta_{3} + \beta_{2} - 2\beta _1 + 2 ) / 8 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{2} + \beta _1 - 3 ) / 4 \) |
\(\nu^{3}\) | \(=\) | \( \beta _1 - 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).
\(n\) | \(17\) | \(21\) | \(31\) |
\(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
31.1 |
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0 | − | 9.06914i | 0 | 11.1803 | 0 | − | 33.1248i | 0 | −1.24922 | 0 | ||||||||||||||||||||||||||||
31.2 | 0 | − | 1.32317i | 0 | −11.1803 | 0 | 36.5889i | 0 | 79.2492 | 0 | ||||||||||||||||||||||||||||||
31.3 | 0 | 1.32317i | 0 | −11.1803 | 0 | − | 36.5889i | 0 | 79.2492 | 0 | ||||||||||||||||||||||||||||||
31.4 | 0 | 9.06914i | 0 | 11.1803 | 0 | 33.1248i | 0 | −1.24922 | 0 | |||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 80.5.b.b | ✓ | 4 |
3.b | odd | 2 | 1 | 720.5.e.b | 4 | ||
4.b | odd | 2 | 1 | inner | 80.5.b.b | ✓ | 4 |
5.b | even | 2 | 1 | 400.5.b.h | 4 | ||
5.c | odd | 4 | 2 | 400.5.h.c | 8 | ||
8.b | even | 2 | 1 | 320.5.b.b | 4 | ||
8.d | odd | 2 | 1 | 320.5.b.b | 4 | ||
12.b | even | 2 | 1 | 720.5.e.b | 4 | ||
20.d | odd | 2 | 1 | 400.5.b.h | 4 | ||
20.e | even | 4 | 2 | 400.5.h.c | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
80.5.b.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
80.5.b.b | ✓ | 4 | 4.b | odd | 2 | 1 | inner |
320.5.b.b | 4 | 8.b | even | 2 | 1 | ||
320.5.b.b | 4 | 8.d | odd | 2 | 1 | ||
400.5.b.h | 4 | 5.b | even | 2 | 1 | ||
400.5.b.h | 4 | 20.d | odd | 2 | 1 | ||
400.5.h.c | 8 | 5.c | odd | 4 | 2 | ||
400.5.h.c | 8 | 20.e | even | 4 | 2 | ||
720.5.e.b | 4 | 3.b | odd | 2 | 1 | ||
720.5.e.b | 4 | 12.b | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 84T_{3}^{2} + 144 \)
acting on \(S_{5}^{\mathrm{new}}(80, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( T^{4} + 84T^{2} + 144 \)
$5$
\( (T^{2} - 125)^{2} \)
$7$
\( T^{4} + 2436 T^{2} + \cdots + 1468944 \)
$11$
\( T^{4} + 36624 T^{2} + \cdots + 267911424 \)
$13$
\( (T^{2} + 136 T - 9956)^{2} \)
$17$
\( (T^{2} + 108 T - 69084)^{2} \)
$19$
\( T^{4} + 354624 T^{2} + \cdots + 29793521664 \)
$23$
\( T^{4} + 444324 T^{2} + \cdots + 1220244624 \)
$29$
\( (T^{2} - 1836 T + 461844)^{2} \)
$31$
\( T^{4} + 245904 T^{2} + \cdots + 14411522304 \)
$37$
\( (T^{2} + 1880 T - 880580)^{2} \)
$41$
\( (T^{2} + 216 T - 290916)^{2} \)
$43$
\( T^{4} + 10590324 T^{2} + \cdots + 18918219842064 \)
$47$
\( T^{4} + 13212516 T^{2} + \cdots + 3006714384144 \)
$53$
\( (T^{2} - 1512 T - 5722884)^{2} \)
$59$
\( T^{4} + \cdots + 321523343069184 \)
$61$
\( (T^{2} - 2752 T - 18066644)^{2} \)
$67$
\( T^{4} + 127875444 T^{2} + \cdots + 40\!\cdots\!64 \)
$71$
\( T^{4} + \cdots + 339683772582144 \)
$73$
\( (T^{2} - 11012 T + 21917956)^{2} \)
$79$
\( T^{4} + 204206400 T^{2} + \cdots + 10\!\cdots\!00 \)
$83$
\( T^{4} + 143312436 T^{2} + \cdots + 29\!\cdots\!44 \)
$89$
\( (T^{2} - 4212 T - 126227484)^{2} \)
$97$
\( (T^{2} + 17764 T + 77431924)^{2} \)
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