Newspace parameters
Level: | \( N \) | \(=\) | \( 120 = 2^{3} \cdot 3 \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 120.m (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.958204824255\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(\sqrt{3}, \sqrt{-5})\) |
Defining polynomial: |
\( x^{4} + x^{2} + 4 \)
|
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-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
\( x^{4} + x^{2} + 4 \)
:
\(\beta_{1}\) | \(=\) |
\( \nu \)
|
\(\beta_{2}\) | \(=\) |
\( \nu^{2} \)
|
\(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + \nu ) / 2 \)
|
\(\nu\) | \(=\) |
\( \beta_1 \)
|
\(\nu^{2}\) | \(=\) |
\( \beta_{2} \)
|
\(\nu^{3}\) | \(=\) |
\( 2\beta_{3} - \beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/120\mathbb{Z}\right)^\times\).
\(n\) | \(31\) | \(41\) | \(61\) | \(97\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
59.1 |
|
−0.866025 | − | 1.11803i | 1.73205 | −0.500000 | + | 1.93649i | − | 2.23607i | −1.50000 | − | 1.93649i | 0 | 2.59808 | − | 1.11803i | 3.00000 | −2.50000 | + | 1.93649i | |||||||||||||||||||
59.2 | −0.866025 | + | 1.11803i | 1.73205 | −0.500000 | − | 1.93649i | 2.23607i | −1.50000 | + | 1.93649i | 0 | 2.59808 | + | 1.11803i | 3.00000 | −2.50000 | − | 1.93649i | |||||||||||||||||||||
59.3 | 0.866025 | − | 1.11803i | −1.73205 | −0.500000 | − | 1.93649i | − | 2.23607i | −1.50000 | + | 1.93649i | 0 | −2.59808 | − | 1.11803i | 3.00000 | −2.50000 | − | 1.93649i | ||||||||||||||||||||
59.4 | 0.866025 | + | 1.11803i | −1.73205 | −0.500000 | + | 1.93649i | 2.23607i | −1.50000 | − | 1.93649i | 0 | −2.59808 | + | 1.11803i | 3.00000 | −2.50000 | + | 1.93649i | |||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
15.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-15}) \) |
3.b | odd | 2 | 1 | inner |
5.b | even | 2 | 1 | inner |
8.d | odd | 2 | 1 | inner |
24.f | even | 2 | 1 | inner |
40.e | odd | 2 | 1 | inner |
120.m | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 120.2.m.a | ✓ | 4 |
3.b | odd | 2 | 1 | inner | 120.2.m.a | ✓ | 4 |
4.b | odd | 2 | 1 | 480.2.m.a | 4 | ||
5.b | even | 2 | 1 | inner | 120.2.m.a | ✓ | 4 |
5.c | odd | 4 | 2 | 600.2.b.c | 4 | ||
8.b | even | 2 | 1 | 480.2.m.a | 4 | ||
8.d | odd | 2 | 1 | inner | 120.2.m.a | ✓ | 4 |
12.b | even | 2 | 1 | 480.2.m.a | 4 | ||
15.d | odd | 2 | 1 | CM | 120.2.m.a | ✓ | 4 |
15.e | even | 4 | 2 | 600.2.b.c | 4 | ||
20.d | odd | 2 | 1 | 480.2.m.a | 4 | ||
20.e | even | 4 | 2 | 2400.2.b.c | 4 | ||
24.f | even | 2 | 1 | inner | 120.2.m.a | ✓ | 4 |
24.h | odd | 2 | 1 | 480.2.m.a | 4 | ||
40.e | odd | 2 | 1 | inner | 120.2.m.a | ✓ | 4 |
40.f | even | 2 | 1 | 480.2.m.a | 4 | ||
40.i | odd | 4 | 2 | 2400.2.b.c | 4 | ||
40.k | even | 4 | 2 | 600.2.b.c | 4 | ||
60.h | even | 2 | 1 | 480.2.m.a | 4 | ||
60.l | odd | 4 | 2 | 2400.2.b.c | 4 | ||
120.i | odd | 2 | 1 | 480.2.m.a | 4 | ||
120.m | even | 2 | 1 | inner | 120.2.m.a | ✓ | 4 |
120.q | odd | 4 | 2 | 600.2.b.c | 4 | ||
120.w | even | 4 | 2 | 2400.2.b.c | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
120.2.m.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
120.2.m.a | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
120.2.m.a | ✓ | 4 | 5.b | even | 2 | 1 | inner |
120.2.m.a | ✓ | 4 | 8.d | odd | 2 | 1 | inner |
120.2.m.a | ✓ | 4 | 15.d | odd | 2 | 1 | CM |
120.2.m.a | ✓ | 4 | 24.f | even | 2 | 1 | inner |
120.2.m.a | ✓ | 4 | 40.e | odd | 2 | 1 | inner |
120.2.m.a | ✓ | 4 | 120.m | even | 2 | 1 | inner |
480.2.m.a | 4 | 4.b | odd | 2 | 1 | ||
480.2.m.a | 4 | 8.b | even | 2 | 1 | ||
480.2.m.a | 4 | 12.b | even | 2 | 1 | ||
480.2.m.a | 4 | 20.d | odd | 2 | 1 | ||
480.2.m.a | 4 | 24.h | odd | 2 | 1 | ||
480.2.m.a | 4 | 40.f | even | 2 | 1 | ||
480.2.m.a | 4 | 60.h | even | 2 | 1 | ||
480.2.m.a | 4 | 120.i | odd | 2 | 1 | ||
600.2.b.c | 4 | 5.c | odd | 4 | 2 | ||
600.2.b.c | 4 | 15.e | even | 4 | 2 | ||
600.2.b.c | 4 | 40.k | even | 4 | 2 | ||
600.2.b.c | 4 | 120.q | odd | 4 | 2 | ||
2400.2.b.c | 4 | 20.e | even | 4 | 2 | ||
2400.2.b.c | 4 | 40.i | odd | 4 | 2 | ||
2400.2.b.c | 4 | 60.l | odd | 4 | 2 | ||
2400.2.b.c | 4 | 120.w | even | 4 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7} \)
acting on \(S_{2}^{\mathrm{new}}(120, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} + T^{2} + 4 \)
$3$
\( (T^{2} - 3)^{2} \)
$5$
\( (T^{2} + 5)^{2} \)
$7$
\( T^{4} \)
$11$
\( T^{4} \)
$13$
\( T^{4} \)
$17$
\( (T^{2} - 48)^{2} \)
$19$
\( (T - 4)^{4} \)
$23$
\( (T^{2} + 80)^{2} \)
$29$
\( T^{4} \)
$31$
\( (T^{2} + 60)^{2} \)
$37$
\( T^{4} \)
$41$
\( T^{4} \)
$43$
\( T^{4} \)
$47$
\( (T^{2} + 80)^{2} \)
$53$
\( (T^{2} + 20)^{2} \)
$59$
\( T^{4} \)
$61$
\( (T^{2} + 240)^{2} \)
$67$
\( T^{4} \)
$71$
\( T^{4} \)
$73$
\( T^{4} \)
$79$
\( (T^{2} + 60)^{2} \)
$83$
\( (T^{2} - 12)^{2} \)
$89$
\( T^{4} \)
$97$
\( T^{4} \)
show more
show less