Newspace parameters
Level: | \( N \) | \(=\) | \( 3840 = 2^{8} \cdot 3 \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 3840.i (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.91640964851\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
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: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 480) |
Projective image: | \(D_{4}\) |
Projective field: | Galois closure of 4.2.3600.1 |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3840\mathbb{Z}\right)^\times\).
\(n\) | \(511\) | \(1537\) | \(2561\) | \(2821\) |
\(\chi(n)\) | \(1\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1409.1 |
|
0 | −0.707107 | − | 0.707107i | 0 | −1.00000 | 0 | 1.41421i | 0 | 1.00000i | 0 | ||||||||||||||||||||||||||||
1409.2 | 0 | −0.707107 | + | 0.707107i | 0 | −1.00000 | 0 | − | 1.41421i | 0 | − | 1.00000i | 0 | |||||||||||||||||||||||||||
1409.3 | 0 | 0.707107 | − | 0.707107i | 0 | −1.00000 | 0 | 1.41421i | 0 | − | 1.00000i | 0 | ||||||||||||||||||||||||||||
1409.4 | 0 | 0.707107 | + | 0.707107i | 0 | −1.00000 | 0 | − | 1.41421i | 0 | 1.00000i | 0 | ||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
20.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-5}) \) |
4.b | odd | 2 | 1 | inner |
5.b | even | 2 | 1 | inner |
24.f | even | 2 | 1 | inner |
24.h | odd | 2 | 1 | inner |
120.i | 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 | 3840.1.i.a | 4 | |
3.b | odd | 2 | 1 | 3840.1.i.b | 4 | ||
4.b | odd | 2 | 1 | inner | 3840.1.i.a | 4 | |
5.b | even | 2 | 1 | inner | 3840.1.i.a | 4 | |
8.b | even | 2 | 1 | 3840.1.i.b | 4 | ||
8.d | odd | 2 | 1 | 3840.1.i.b | 4 | ||
12.b | even | 2 | 1 | 3840.1.i.b | 4 | ||
15.d | odd | 2 | 1 | 3840.1.i.b | 4 | ||
16.e | even | 4 | 1 | 480.1.c.a | ✓ | 4 | |
16.e | even | 4 | 1 | 960.1.c.a | 4 | ||
16.f | odd | 4 | 1 | 480.1.c.a | ✓ | 4 | |
16.f | odd | 4 | 1 | 960.1.c.a | 4 | ||
20.d | odd | 2 | 1 | CM | 3840.1.i.a | 4 | |
24.f | even | 2 | 1 | inner | 3840.1.i.a | 4 | |
24.h | odd | 2 | 1 | inner | 3840.1.i.a | 4 | |
40.e | odd | 2 | 1 | 3840.1.i.b | 4 | ||
40.f | even | 2 | 1 | 3840.1.i.b | 4 | ||
48.i | odd | 4 | 1 | 480.1.c.a | ✓ | 4 | |
48.i | odd | 4 | 1 | 960.1.c.a | 4 | ||
48.k | even | 4 | 1 | 480.1.c.a | ✓ | 4 | |
48.k | even | 4 | 1 | 960.1.c.a | 4 | ||
60.h | even | 2 | 1 | 3840.1.i.b | 4 | ||
80.i | odd | 4 | 1 | 2400.1.l.a | 4 | ||
80.j | even | 4 | 1 | 2400.1.l.a | 4 | ||
80.k | odd | 4 | 1 | 480.1.c.a | ✓ | 4 | |
80.k | odd | 4 | 1 | 960.1.c.a | 4 | ||
80.q | even | 4 | 1 | 480.1.c.a | ✓ | 4 | |
80.q | even | 4 | 1 | 960.1.c.a | 4 | ||
80.s | even | 4 | 1 | 2400.1.l.a | 4 | ||
80.t | odd | 4 | 1 | 2400.1.l.a | 4 | ||
120.i | odd | 2 | 1 | inner | 3840.1.i.a | 4 | |
120.m | even | 2 | 1 | inner | 3840.1.i.a | 4 | |
240.t | even | 4 | 1 | 480.1.c.a | ✓ | 4 | |
240.t | even | 4 | 1 | 960.1.c.a | 4 | ||
240.z | odd | 4 | 1 | 2400.1.l.a | 4 | ||
240.bb | even | 4 | 1 | 2400.1.l.a | 4 | ||
240.bd | odd | 4 | 1 | 2400.1.l.a | 4 | ||
240.bf | even | 4 | 1 | 2400.1.l.a | 4 | ||
240.bm | odd | 4 | 1 | 480.1.c.a | ✓ | 4 | |
240.bm | odd | 4 | 1 | 960.1.c.a | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
480.1.c.a | ✓ | 4 | 16.e | even | 4 | 1 | |
480.1.c.a | ✓ | 4 | 16.f | odd | 4 | 1 | |
480.1.c.a | ✓ | 4 | 48.i | odd | 4 | 1 | |
480.1.c.a | ✓ | 4 | 48.k | even | 4 | 1 | |
480.1.c.a | ✓ | 4 | 80.k | odd | 4 | 1 | |
480.1.c.a | ✓ | 4 | 80.q | even | 4 | 1 | |
480.1.c.a | ✓ | 4 | 240.t | even | 4 | 1 | |
480.1.c.a | ✓ | 4 | 240.bm | odd | 4 | 1 | |
960.1.c.a | 4 | 16.e | even | 4 | 1 | ||
960.1.c.a | 4 | 16.f | odd | 4 | 1 | ||
960.1.c.a | 4 | 48.i | odd | 4 | 1 | ||
960.1.c.a | 4 | 48.k | even | 4 | 1 | ||
960.1.c.a | 4 | 80.k | odd | 4 | 1 | ||
960.1.c.a | 4 | 80.q | even | 4 | 1 | ||
960.1.c.a | 4 | 240.t | even | 4 | 1 | ||
960.1.c.a | 4 | 240.bm | odd | 4 | 1 | ||
2400.1.l.a | 4 | 80.i | odd | 4 | 1 | ||
2400.1.l.a | 4 | 80.j | even | 4 | 1 | ||
2400.1.l.a | 4 | 80.s | even | 4 | 1 | ||
2400.1.l.a | 4 | 80.t | odd | 4 | 1 | ||
2400.1.l.a | 4 | 240.z | odd | 4 | 1 | ||
2400.1.l.a | 4 | 240.bb | even | 4 | 1 | ||
2400.1.l.a | 4 | 240.bd | odd | 4 | 1 | ||
2400.1.l.a | 4 | 240.bf | even | 4 | 1 | ||
3840.1.i.a | 4 | 1.a | even | 1 | 1 | trivial | |
3840.1.i.a | 4 | 4.b | odd | 2 | 1 | inner | |
3840.1.i.a | 4 | 5.b | even | 2 | 1 | inner | |
3840.1.i.a | 4 | 20.d | odd | 2 | 1 | CM | |
3840.1.i.a | 4 | 24.f | even | 2 | 1 | inner | |
3840.1.i.a | 4 | 24.h | odd | 2 | 1 | inner | |
3840.1.i.a | 4 | 120.i | odd | 2 | 1 | inner | |
3840.1.i.a | 4 | 120.m | even | 2 | 1 | inner | |
3840.1.i.b | 4 | 3.b | odd | 2 | 1 | ||
3840.1.i.b | 4 | 8.b | even | 2 | 1 | ||
3840.1.i.b | 4 | 8.d | odd | 2 | 1 | ||
3840.1.i.b | 4 | 12.b | even | 2 | 1 | ||
3840.1.i.b | 4 | 15.d | odd | 2 | 1 | ||
3840.1.i.b | 4 | 40.e | odd | 2 | 1 | ||
3840.1.i.b | 4 | 40.f | even | 2 | 1 | ||
3840.1.i.b | 4 | 60.h | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{149} + 2 \)
acting on \(S_{1}^{\mathrm{new}}(3840, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( T^{4} + 1 \)
$5$
\( (T + 1)^{4} \)
$7$
\( (T^{2} + 2)^{2} \)
$11$
\( T^{4} \)
$13$
\( T^{4} \)
$17$
\( T^{4} \)
$19$
\( T^{4} \)
$23$
\( (T^{2} - 2)^{2} \)
$29$
\( T^{4} \)
$31$
\( T^{4} \)
$37$
\( T^{4} \)
$41$
\( (T^{2} + 4)^{2} \)
$43$
\( (T^{2} - 2)^{2} \)
$47$
\( (T^{2} - 2)^{2} \)
$53$
\( T^{4} \)
$59$
\( T^{4} \)
$61$
\( T^{4} \)
$67$
\( (T^{2} - 2)^{2} \)
$71$
\( T^{4} \)
$73$
\( T^{4} \)
$79$
\( T^{4} \)
$83$
\( (T^{2} + 2)^{2} \)
$89$
\( T^{4} \)
$97$
\( T^{4} \)
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