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: | \(x^{4} + 21 x^{2} + 100\) |
| 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 \(x^{4} + 21 x^{2} + 100\):
| \(\beta_{0}\) | \(=\) | \( 1 \) |
| \(\beta_{1}\) | \(=\) | \((\)\( \nu^{3} + \nu \)\()/10\) |
| \(\beta_{2}\) | \(=\) | \((\)\( \nu^{3} + 13 \nu \)\()/2\) |
| \(\beta_{3}\) | \(=\) | \( 3 \nu^{2} + 32 \) |
| \(1\) | \(=\) | \(\beta_0\) |
| \(\nu\) | \(=\) | \((\)\(\beta_{2} - 5 \beta_{1}\)\()/6\) |
| \(\nu^{2}\) | \(=\) | \((\)\(\beta_{3} - 32\)\()/3\) |
| \(\nu^{3}\) | \(=\) | \((\)\(-\beta_{2} + 65 \beta_{1}\)\()/6\) |
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 \( T_{2}^{4} + 25 T_{2}^{2} + 64 \) acting on \(S_{4}^{\mathrm{new}}(45, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ | \( 64 + 25 T^{2} + T^{4} \) |
| $3$ | \( T^{4} \) |
| $5$ | \( 15625 + 750 T - 110 T^{2} + 6 T^{3} + T^{4} \) |
| $7$ | \( 129600 + 756 T^{2} + T^{4} \) |
| $11$ | \( ( 72 - 42 T + T^{2} )^{2} \) |
| $13$ | \( 1327104 + 3780 T^{2} + T^{4} \) |
| $17$ | \( 2483776 + 7252 T^{2} + T^{4} \) |
| $19$ | \( ( -5120 + 56 T + T^{2} )^{2} \) |
| $23$ | \( 6400 + 2464 T^{2} + T^{4} \) |
| $29$ | \( ( 7200 + 318 T + T^{2} )^{2} \) |
| $31$ | \( ( -800 - 52 T + T^{2} )^{2} \) |
| $37$ | \( 41990400 + 106596 T^{2} + T^{4} \) |
| $41$ | \( ( 40140 - 408 T + T^{2} )^{2} \) |
| $43$ | \( 8256266496 + 196128 T^{2} + T^{4} \) |
| $47$ | \( 6186766336 + 189712 T^{2} + T^{4} \) |
| $53$ | \( 813390400 + 218644 T^{2} + T^{4} \) |
| $59$ | \( ( 8280 + 186 T + T^{2} )^{2} \) |
| $61$ | \( ( -65564 - 340 T + T^{2} )^{2} \) |
| $67$ | \( 9419867136 + 341712 T^{2} + T^{4} \) |
| $71$ | \( ( -331776 - 36 T + T^{2} )^{2} \) |
| $73$ | \( 104976000000 + 1126224 T^{2} + T^{4} \) |
| $79$ | \( ( -886400 + 380 T + T^{2} )^{2} \) |
| $83$ | \( 40558737664 + 1371040 T^{2} + T^{4} \) |
| $89$ | \( ( 98820 - 1116 T + T^{2} )^{2} \) |
| $97$ | \( 196199387136 + 1475712 T^{2} + T^{4} \) |
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