Newspace parameters
| Level: | \( N \) | \(=\) | \( 45 = 3^{2} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 45.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(2.65508595026\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{41})\) |
| Defining polynomial: |
\( x^{4} + 21x^{2} + 100 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2\cdot 3^{2} \) |
| Twist minimal: | no (minimal twist has level 15) |
| 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
\( x^{4} + 21x^{2} + 100 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + \nu ) / 10 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 13\nu ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( 3\nu^{2} + 32 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 5\beta_1 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 32 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{2} + 65\beta_1 ) / 6 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/45\mathbb{Z}\right)^\times\).
| \(n\) | \(11\) | \(37\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 19.1 |
|
− | 4.70156i | 0 | −14.1047 | −11.1047 | − | 1.29844i | 0 | − | 16.2094i | 28.7016i | 0 | −6.10469 | + | 52.2094i | ||||||||||||||||||||||||
| 19.2 | − | 1.70156i | 0 | 5.10469 | 8.10469 | + | 7.70156i | 0 | − | 22.2094i | − | 22.2984i | 0 | 13.1047 | − | 13.7906i | ||||||||||||||||||||||||
| 19.3 | 1.70156i | 0 | 5.10469 | 8.10469 | − | 7.70156i | 0 | 22.2094i | 22.2984i | 0 | 13.1047 | + | 13.7906i | |||||||||||||||||||||||||||
| 19.4 | 4.70156i | 0 | −14.1047 | −11.1047 | + | 1.29844i | 0 | 16.2094i | − | 28.7016i | 0 | −6.10469 | − | 52.2094i | ||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 45.4.b.b | 4 | |
| 3.b | odd | 2 | 1 | 15.4.b.a | ✓ | 4 | |
| 4.b | odd | 2 | 1 | 720.4.f.j | 4 | ||
| 5.b | even | 2 | 1 | inner | 45.4.b.b | 4 | |
| 5.c | odd | 4 | 1 | 225.4.a.i | 2 | ||
| 5.c | odd | 4 | 1 | 225.4.a.o | 2 | ||
| 12.b | even | 2 | 1 | 240.4.f.f | 4 | ||
| 15.d | odd | 2 | 1 | 15.4.b.a | ✓ | 4 | |
| 15.e | even | 4 | 1 | 75.4.a.c | 2 | ||
| 15.e | even | 4 | 1 | 75.4.a.f | 2 | ||
| 20.d | odd | 2 | 1 | 720.4.f.j | 4 | ||
| 24.f | even | 2 | 1 | 960.4.f.p | 4 | ||
| 24.h | odd | 2 | 1 | 960.4.f.q | 4 | ||
| 60.h | even | 2 | 1 | 240.4.f.f | 4 | ||
| 60.l | odd | 4 | 1 | 1200.4.a.bn | 2 | ||
| 60.l | odd | 4 | 1 | 1200.4.a.bt | 2 | ||
| 120.i | odd | 2 | 1 | 960.4.f.q | 4 | ||
| 120.m | even | 2 | 1 | 960.4.f.p | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 15.4.b.a | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 15.4.b.a | ✓ | 4 | 15.d | odd | 2 | 1 | |
| 45.4.b.b | 4 | 1.a | even | 1 | 1 | trivial | |
| 45.4.b.b | 4 | 5.b | even | 2 | 1 | inner | |
| 75.4.a.c | 2 | 15.e | even | 4 | 1 | ||
| 75.4.a.f | 2 | 15.e | even | 4 | 1 | ||
| 225.4.a.i | 2 | 5.c | odd | 4 | 1 | ||
| 225.4.a.o | 2 | 5.c | odd | 4 | 1 | ||
| 240.4.f.f | 4 | 12.b | even | 2 | 1 | ||
| 240.4.f.f | 4 | 60.h | even | 2 | 1 | ||
| 720.4.f.j | 4 | 4.b | odd | 2 | 1 | ||
| 720.4.f.j | 4 | 20.d | odd | 2 | 1 | ||
| 960.4.f.p | 4 | 24.f | even | 2 | 1 | ||
| 960.4.f.p | 4 | 120.m | even | 2 | 1 | ||
| 960.4.f.q | 4 | 24.h | odd | 2 | 1 | ||
| 960.4.f.q | 4 | 120.i | odd | 2 | 1 | ||
| 1200.4.a.bn | 2 | 60.l | odd | 4 | 1 | ||
| 1200.4.a.bt | 2 | 60.l | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 25T_{2}^{2} + 64 \)
acting on \(S_{4}^{\mathrm{new}}(45, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} + 25T^{2} + 64 \)
$3$
\( T^{4} \)
$5$
\( T^{4} + 6 T^{3} - 110 T^{2} + \cdots + 15625 \)
$7$
\( T^{4} + 756 T^{2} + 129600 \)
$11$
\( (T^{2} - 42 T + 72)^{2} \)
$13$
\( T^{4} + 3780 T^{2} + \cdots + 1327104 \)
$17$
\( T^{4} + 7252 T^{2} + \cdots + 2483776 \)
$19$
\( (T^{2} + 56 T - 5120)^{2} \)
$23$
\( T^{4} + 2464 T^{2} + 6400 \)
$29$
\( (T^{2} + 318 T + 7200)^{2} \)
$31$
\( (T^{2} - 52 T - 800)^{2} \)
$37$
\( T^{4} + 106596 T^{2} + \cdots + 41990400 \)
$41$
\( (T^{2} - 408 T + 40140)^{2} \)
$43$
\( T^{4} + 196128 T^{2} + \cdots + 8256266496 \)
$47$
\( T^{4} + 189712 T^{2} + \cdots + 6186766336 \)
$53$
\( T^{4} + 218644 T^{2} + \cdots + 813390400 \)
$59$
\( (T^{2} + 186 T + 8280)^{2} \)
$61$
\( (T^{2} - 340 T - 65564)^{2} \)
$67$
\( T^{4} + 341712 T^{2} + \cdots + 9419867136 \)
$71$
\( (T^{2} - 36 T - 331776)^{2} \)
$73$
\( T^{4} + 1126224 T^{2} + \cdots + 104976000000 \)
$79$
\( (T^{2} + 380 T - 886400)^{2} \)
$83$
\( T^{4} + 1371040 T^{2} + \cdots + 40558737664 \)
$89$
\( (T^{2} - 1116 T + 98820)^{2} \)
$97$
\( T^{4} + 1475712 T^{2} + \cdots + 196199387136 \)
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