Newspace parameters
Level: | \( N \) | \(=\) | \( 480 = 2^{5} \cdot 3 \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 480.v (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(3.83281929702\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(i)\) |
Coefficient field: | \(\Q(\zeta_{8})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/480\mathbb{Z}\right)^\times\).
\(n\) | \(31\) | \(97\) | \(161\) | \(421\) |
\(\chi(n)\) | \(1\) | \(-\zeta_{8}^{2}\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
257.1 |
|
0 | 0.292893 | − | 1.70711i | 0 | −0.707107 | + | 2.12132i | 0 | 3.00000 | + | 3.00000i | 0 | −2.82843 | − | 1.00000i | 0 | ||||||||||||||||||||||
257.2 | 0 | 1.70711 | − | 0.292893i | 0 | 0.707107 | − | 2.12132i | 0 | 3.00000 | + | 3.00000i | 0 | 2.82843 | − | 1.00000i | 0 | |||||||||||||||||||||||
353.1 | 0 | 0.292893 | + | 1.70711i | 0 | −0.707107 | − | 2.12132i | 0 | 3.00000 | − | 3.00000i | 0 | −2.82843 | + | 1.00000i | 0 | |||||||||||||||||||||||
353.2 | 0 | 1.70711 | + | 0.292893i | 0 | 0.707107 | + | 2.12132i | 0 | 3.00000 | − | 3.00000i | 0 | 2.82843 | + | 1.00000i | 0 | |||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
5.c | odd | 4 | 1 | inner |
15.e | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 480.2.v.b | yes | 4 |
3.b | odd | 2 | 1 | inner | 480.2.v.b | yes | 4 |
4.b | odd | 2 | 1 | 480.2.v.a | ✓ | 4 | |
5.c | odd | 4 | 1 | inner | 480.2.v.b | yes | 4 |
8.b | even | 2 | 1 | 960.2.v.d | 4 | ||
8.d | odd | 2 | 1 | 960.2.v.j | 4 | ||
12.b | even | 2 | 1 | 480.2.v.a | ✓ | 4 | |
15.e | even | 4 | 1 | inner | 480.2.v.b | yes | 4 |
20.e | even | 4 | 1 | 480.2.v.a | ✓ | 4 | |
24.f | even | 2 | 1 | 960.2.v.j | 4 | ||
24.h | odd | 2 | 1 | 960.2.v.d | 4 | ||
40.i | odd | 4 | 1 | 960.2.v.d | 4 | ||
40.k | even | 4 | 1 | 960.2.v.j | 4 | ||
60.l | odd | 4 | 1 | 480.2.v.a | ✓ | 4 | |
120.q | odd | 4 | 1 | 960.2.v.j | 4 | ||
120.w | even | 4 | 1 | 960.2.v.d | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
480.2.v.a | ✓ | 4 | 4.b | odd | 2 | 1 | |
480.2.v.a | ✓ | 4 | 12.b | even | 2 | 1 | |
480.2.v.a | ✓ | 4 | 20.e | even | 4 | 1 | |
480.2.v.a | ✓ | 4 | 60.l | odd | 4 | 1 | |
480.2.v.b | yes | 4 | 1.a | even | 1 | 1 | trivial |
480.2.v.b | yes | 4 | 3.b | odd | 2 | 1 | inner |
480.2.v.b | yes | 4 | 5.c | odd | 4 | 1 | inner |
480.2.v.b | yes | 4 | 15.e | even | 4 | 1 | inner |
960.2.v.d | 4 | 8.b | even | 2 | 1 | ||
960.2.v.d | 4 | 24.h | odd | 2 | 1 | ||
960.2.v.d | 4 | 40.i | odd | 4 | 1 | ||
960.2.v.d | 4 | 120.w | even | 4 | 1 | ||
960.2.v.j | 4 | 8.d | odd | 2 | 1 | ||
960.2.v.j | 4 | 24.f | even | 2 | 1 | ||
960.2.v.j | 4 | 40.k | even | 4 | 1 | ||
960.2.v.j | 4 | 120.q | odd | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{2} - 6T_{7} + 18 \)
acting on \(S_{2}^{\mathrm{new}}(480, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( T^{4} - 4 T^{3} + 8 T^{2} - 12 T + 9 \)
$5$
\( T^{4} + 8T^{2} + 25 \)
$7$
\( (T^{2} - 6 T + 18)^{2} \)
$11$
\( (T^{2} + 18)^{2} \)
$13$
\( (T^{2} + 8 T + 32)^{2} \)
$17$
\( T^{4} + 16 \)
$19$
\( T^{4} \)
$23$
\( T^{4} \)
$29$
\( (T^{2} - 18)^{2} \)
$31$
\( (T - 6)^{4} \)
$37$
\( (T^{2} + 4 T + 8)^{2} \)
$41$
\( T^{4} \)
$43$
\( (T^{2} - 12 T + 72)^{2} \)
$47$
\( T^{4} + 20736 \)
$53$
\( T^{4} + 256 \)
$59$
\( (T^{2} - 18)^{2} \)
$61$
\( (T + 6)^{4} \)
$67$
\( T^{4} \)
$71$
\( (T^{2} + 72)^{2} \)
$73$
\( (T^{2} + 10 T + 50)^{2} \)
$79$
\( (T^{2} + 36)^{2} \)
$83$
\( T^{4} + 20736 \)
$89$
\( (T^{2} - 72)^{2} \)
$97$
\( (T^{2} + 10 T + 50)^{2} \)
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