Newspace parameters
Level: | \( N \) | \(=\) | \( 240 = 2^{4} \cdot 3 \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 240.o (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.91640964851\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Coefficient field: | \(\Q(\zeta_{24})\) |
Defining polynomial: |
\( x^{8} - x^{4} + 1 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
Coefficient ring index: | \( 2^{10}\cdot 3 \) |
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring
\(\beta_{1}\) | \(=\) |
\( -\zeta_{24}^{6} - \zeta_{24}^{5} - \zeta_{24}^{3} + 2\zeta_{24}^{2} + \zeta_{24} \)
|
\(\beta_{2}\) | \(=\) |
\( \zeta_{24}^{6} - \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \)
|
\(\beta_{3}\) | \(=\) |
\( 4\zeta_{24}^{4} - 2 \)
|
\(\beta_{4}\) | \(=\) |
\( -\zeta_{24}^{6} + \zeta_{24}^{5} + \zeta_{24}^{3} + 2\zeta_{24}^{2} - \zeta_{24} \)
|
\(\beta_{5}\) | \(=\) |
\( 2\zeta_{24}^{7} - 2\zeta_{24}^{6} - 2\zeta_{24}^{5} + 2\zeta_{24}^{4} - 1 \)
|
\(\beta_{6}\) | \(=\) |
\( 4\zeta_{24}^{7} + 2\zeta_{24}^{5} - 2\zeta_{24}^{3} + 2\zeta_{24} \)
|
\(\beta_{7}\) | \(=\) |
\( -2\zeta_{24}^{7} - 2\zeta_{24}^{6} - 2\zeta_{24}^{4} + 2\zeta_{24}^{3} + 2\zeta_{24} + 1 \)
|
\(\zeta_{24}\) | \(=\) |
\( ( 4\beta_{7} + 3\beta_{6} - 2\beta_{5} - 3\beta_{4} + 3\beta_{3} + 4\beta_{2} + 3\beta_1 ) / 24 \)
|
\(\zeta_{24}^{2}\) | \(=\) |
\( ( -\beta_{7} - \beta_{5} + 3\beta_{4} + 2\beta_{2} + 3\beta_1 ) / 12 \)
|
\(\zeta_{24}^{3}\) | \(=\) |
\( ( \beta_{7} + \beta_{5} + 3\beta_{4} + 4\beta_{2} - 3\beta_1 ) / 12 \)
|
\(\zeta_{24}^{4}\) | \(=\) |
\( ( \beta_{3} + 2 ) / 4 \)
|
\(\zeta_{24}^{5}\) | \(=\) |
\( ( 2\beta_{7} + 3\beta_{6} - 4\beta_{5} + 3\beta_{4} + 3\beta_{3} - 4\beta_{2} - 3\beta_1 ) / 24 \)
|
\(\zeta_{24}^{6}\) | \(=\) |
\( ( -\beta_{7} - \beta_{5} + 2\beta_{2} ) / 6 \)
|
\(\zeta_{24}^{7}\) | \(=\) |
\( ( -2\beta_{7} + 3\beta_{6} + 4\beta_{5} + 3\beta_{4} - 3\beta_{3} + 4\beta_{2} - 3\beta_1 ) / 24 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/240\mathbb{Z}\right)^\times\).
\(n\) | \(31\) | \(97\) | \(161\) | \(181\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
239.1 |
|
0 | −1.41421 | − | 1.00000i | 0 | −1.73205 | + | 1.41421i | 0 | 0 | 0 | 1.00000 | + | 2.82843i | 0 | ||||||||||||||||||||||||||||||||||||
239.2 | 0 | −1.41421 | − | 1.00000i | 0 | 1.73205 | + | 1.41421i | 0 | 0 | 0 | 1.00000 | + | 2.82843i | 0 | |||||||||||||||||||||||||||||||||||||
239.3 | 0 | −1.41421 | + | 1.00000i | 0 | −1.73205 | − | 1.41421i | 0 | 0 | 0 | 1.00000 | − | 2.82843i | 0 | |||||||||||||||||||||||||||||||||||||
239.4 | 0 | −1.41421 | + | 1.00000i | 0 | 1.73205 | − | 1.41421i | 0 | 0 | 0 | 1.00000 | − | 2.82843i | 0 | |||||||||||||||||||||||||||||||||||||
239.5 | 0 | 1.41421 | − | 1.00000i | 0 | −1.73205 | − | 1.41421i | 0 | 0 | 0 | 1.00000 | − | 2.82843i | 0 | |||||||||||||||||||||||||||||||||||||
239.6 | 0 | 1.41421 | − | 1.00000i | 0 | 1.73205 | − | 1.41421i | 0 | 0 | 0 | 1.00000 | − | 2.82843i | 0 | |||||||||||||||||||||||||||||||||||||
239.7 | 0 | 1.41421 | + | 1.00000i | 0 | −1.73205 | + | 1.41421i | 0 | 0 | 0 | 1.00000 | + | 2.82843i | 0 | |||||||||||||||||||||||||||||||||||||
239.8 | 0 | 1.41421 | + | 1.00000i | 0 | 1.73205 | + | 1.41421i | 0 | 0 | 0 | 1.00000 | + | 2.82843i | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
4.b | odd | 2 | 1 | inner |
5.b | even | 2 | 1 | inner |
12.b | even | 2 | 1 | inner |
15.d | odd | 2 | 1 | inner |
20.d | odd | 2 | 1 | inner |
60.h | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 240.2.o.b | ✓ | 8 |
3.b | odd | 2 | 1 | inner | 240.2.o.b | ✓ | 8 |
4.b | odd | 2 | 1 | inner | 240.2.o.b | ✓ | 8 |
5.b | even | 2 | 1 | inner | 240.2.o.b | ✓ | 8 |
5.c | odd | 4 | 1 | 1200.2.h.j | 4 | ||
5.c | odd | 4 | 1 | 1200.2.h.n | 4 | ||
8.b | even | 2 | 1 | 960.2.o.d | 8 | ||
8.d | odd | 2 | 1 | 960.2.o.d | 8 | ||
12.b | even | 2 | 1 | inner | 240.2.o.b | ✓ | 8 |
15.d | odd | 2 | 1 | inner | 240.2.o.b | ✓ | 8 |
15.e | even | 4 | 1 | 1200.2.h.j | 4 | ||
15.e | even | 4 | 1 | 1200.2.h.n | 4 | ||
20.d | odd | 2 | 1 | inner | 240.2.o.b | ✓ | 8 |
20.e | even | 4 | 1 | 1200.2.h.j | 4 | ||
20.e | even | 4 | 1 | 1200.2.h.n | 4 | ||
24.f | even | 2 | 1 | 960.2.o.d | 8 | ||
24.h | odd | 2 | 1 | 960.2.o.d | 8 | ||
40.e | odd | 2 | 1 | 960.2.o.d | 8 | ||
40.f | even | 2 | 1 | 960.2.o.d | 8 | ||
60.h | even | 2 | 1 | inner | 240.2.o.b | ✓ | 8 |
60.l | odd | 4 | 1 | 1200.2.h.j | 4 | ||
60.l | odd | 4 | 1 | 1200.2.h.n | 4 | ||
120.i | odd | 2 | 1 | 960.2.o.d | 8 | ||
120.m | even | 2 | 1 | 960.2.o.d | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
240.2.o.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
240.2.o.b | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
240.2.o.b | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
240.2.o.b | ✓ | 8 | 5.b | even | 2 | 1 | inner |
240.2.o.b | ✓ | 8 | 12.b | even | 2 | 1 | inner |
240.2.o.b | ✓ | 8 | 15.d | odd | 2 | 1 | inner |
240.2.o.b | ✓ | 8 | 20.d | odd | 2 | 1 | inner |
240.2.o.b | ✓ | 8 | 60.h | even | 2 | 1 | inner |
960.2.o.d | 8 | 8.b | even | 2 | 1 | ||
960.2.o.d | 8 | 8.d | odd | 2 | 1 | ||
960.2.o.d | 8 | 24.f | even | 2 | 1 | ||
960.2.o.d | 8 | 24.h | odd | 2 | 1 | ||
960.2.o.d | 8 | 40.e | odd | 2 | 1 | ||
960.2.o.d | 8 | 40.f | even | 2 | 1 | ||
960.2.o.d | 8 | 120.i | odd | 2 | 1 | ||
960.2.o.d | 8 | 120.m | even | 2 | 1 | ||
1200.2.h.j | 4 | 5.c | odd | 4 | 1 | ||
1200.2.h.j | 4 | 15.e | even | 4 | 1 | ||
1200.2.h.j | 4 | 20.e | even | 4 | 1 | ||
1200.2.h.j | 4 | 60.l | odd | 4 | 1 | ||
1200.2.h.n | 4 | 5.c | odd | 4 | 1 | ||
1200.2.h.n | 4 | 15.e | even | 4 | 1 | ||
1200.2.h.n | 4 | 20.e | even | 4 | 1 | ||
1200.2.h.n | 4 | 60.l | odd | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7} \)
acting on \(S_{2}^{\mathrm{new}}(240, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( (T^{4} - 2 T^{2} + 9)^{2} \)
$5$
\( (T^{4} - 2 T^{2} + 25)^{2} \)
$7$
\( T^{8} \)
$11$
\( (T^{2} - 24)^{4} \)
$13$
\( (T^{2} + 24)^{4} \)
$17$
\( (T^{2} - 12)^{4} \)
$19$
\( (T^{2} + 12)^{4} \)
$23$
\( (T^{2} + 36)^{4} \)
$29$
\( (T^{2} + 8)^{4} \)
$31$
\( (T^{2} + 12)^{4} \)
$37$
\( (T^{2} + 24)^{4} \)
$41$
\( (T^{2} + 32)^{4} \)
$43$
\( (T^{2} - 72)^{4} \)
$47$
\( (T^{2} + 36)^{4} \)
$53$
\( (T^{2} - 108)^{4} \)
$59$
\( (T^{2} - 24)^{4} \)
$61$
\( (T + 2)^{8} \)
$67$
\( (T^{2} - 72)^{4} \)
$71$
\( (T^{2} - 96)^{4} \)
$73$
\( (T^{2} + 96)^{4} \)
$79$
\( (T^{2} + 108)^{4} \)
$83$
\( (T^{2} + 36)^{4} \)
$89$
\( (T^{2} + 32)^{4} \)
$97$
\( T^{8} \)
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