Newspace parameters
Level: | \( N \) | \(=\) | \( 320 = 2^{6} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 7 \) |
Character orbit: | \([\chi]\) | \(=\) | 320.h (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(73.6173067584\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(\sqrt{6}, \sqrt{-19})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} - 2x^{3} - x^{2} + 2x + 115 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 2^{7}\cdot 3\cdot 5^{4} \) |
Twist minimal: | no (minimal twist has level 80) |
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 \( x^{4} - 2x^{3} - x^{2} + 2x + 115 \) :
\(\beta_{1}\) | \(=\) | \( ( -20\nu^{3} + 30\nu^{2} + 250\nu - 130 ) / 43 \) |
\(\beta_{2}\) | \(=\) | \( -10\nu^{2} + 10\nu + 10 \) |
\(\beta_{3}\) | \(=\) | \( ( 2400\nu^{3} - 3600\nu^{2} + 21600\nu - 10200 ) / 43 \) |
\(\nu\) | \(=\) | \( ( \beta_{3} + 120\beta _1 + 600 ) / 1200 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{3} - 120\beta_{2} + 120\beta _1 + 1800 ) / 1200 \) |
\(\nu^{3}\) | \(=\) | \( ( 7\beta_{3} - 90\beta_{2} - 450\beta _1 + 1200 ) / 600 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).
\(n\) | \(191\) | \(257\) | \(261\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
319.1 |
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0 | −24.4949 | 0 | 65.0000 | − | 106.771i | 0 | −318.434 | 0 | −129.000 | 0 | ||||||||||||||||||||||||||||
319.2 | 0 | −24.4949 | 0 | 65.0000 | + | 106.771i | 0 | −318.434 | 0 | −129.000 | 0 | |||||||||||||||||||||||||||||
319.3 | 0 | 24.4949 | 0 | 65.0000 | − | 106.771i | 0 | 318.434 | 0 | −129.000 | 0 | |||||||||||||||||||||||||||||
319.4 | 0 | 24.4949 | 0 | 65.0000 | + | 106.771i | 0 | 318.434 | 0 | −129.000 | 0 | |||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
5.b | even | 2 | 1 | inner |
20.d | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 320.7.h.e | 4 | |
4.b | odd | 2 | 1 | inner | 320.7.h.e | 4 | |
5.b | even | 2 | 1 | inner | 320.7.h.e | 4 | |
8.b | even | 2 | 1 | 80.7.h.b | ✓ | 4 | |
8.d | odd | 2 | 1 | 80.7.h.b | ✓ | 4 | |
20.d | odd | 2 | 1 | inner | 320.7.h.e | 4 | |
24.f | even | 2 | 1 | 720.7.j.d | 4 | ||
24.h | odd | 2 | 1 | 720.7.j.d | 4 | ||
40.e | odd | 2 | 1 | 80.7.h.b | ✓ | 4 | |
40.f | even | 2 | 1 | 80.7.h.b | ✓ | 4 | |
40.i | odd | 4 | 2 | 400.7.b.g | 4 | ||
40.k | even | 4 | 2 | 400.7.b.g | 4 | ||
120.i | odd | 2 | 1 | 720.7.j.d | 4 | ||
120.m | even | 2 | 1 | 720.7.j.d | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
80.7.h.b | ✓ | 4 | 8.b | even | 2 | 1 | |
80.7.h.b | ✓ | 4 | 8.d | odd | 2 | 1 | |
80.7.h.b | ✓ | 4 | 40.e | odd | 2 | 1 | |
80.7.h.b | ✓ | 4 | 40.f | even | 2 | 1 | |
320.7.h.e | 4 | 1.a | even | 1 | 1 | trivial | |
320.7.h.e | 4 | 4.b | odd | 2 | 1 | inner | |
320.7.h.e | 4 | 5.b | even | 2 | 1 | inner | |
320.7.h.e | 4 | 20.d | odd | 2 | 1 | inner | |
400.7.b.g | 4 | 40.i | odd | 4 | 2 | ||
400.7.b.g | 4 | 40.k | even | 4 | 2 | ||
720.7.j.d | 4 | 24.f | even | 2 | 1 | ||
720.7.j.d | 4 | 24.h | odd | 2 | 1 | ||
720.7.j.d | 4 | 120.i | odd | 2 | 1 | ||
720.7.j.d | 4 | 120.m | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 600 \)
acting on \(S_{7}^{\mathrm{new}}(320, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( (T^{2} - 600)^{2} \)
$5$
\( (T^{2} - 130 T + 15625)^{2} \)
$7$
\( (T^{2} - 101400)^{2} \)
$11$
\( (T^{2} + 6840000)^{2} \)
$13$
\( (T^{2} + 2234400)^{2} \)
$17$
\( (T^{2} + 30825600)^{2} \)
$19$
\( (T^{2} + 6840000)^{2} \)
$23$
\( (T^{2} - 317117400)^{2} \)
$29$
\( (T - 22022)^{4} \)
$31$
\( (T^{2} + 984960000)^{2} \)
$37$
\( (T^{2} + 4076685600)^{2} \)
$41$
\( (T + 62842)^{4} \)
$43$
\( (T^{2} - 254280600)^{2} \)
$47$
\( (T^{2} - 19011384600)^{2} \)
$53$
\( (T^{2} + 11721434400)^{2} \)
$59$
\( (T^{2} + 42688440000)^{2} \)
$61$
\( (T + 45838)^{4} \)
$67$
\( (T^{2} - 20992335000)^{2} \)
$71$
\( (T^{2} + 29795040000)^{2} \)
$73$
\( (T^{2} + 31565414400)^{2} \)
$79$
\( (T^{2} + 295925760000)^{2} \)
$83$
\( (T^{2} - 176947157400)^{2} \)
$89$
\( (T - 197938)^{4} \)
$97$
\( (T^{2} + 1738193385600)^{2} \)
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