Newspace parameters
| Level: | \( N \) | \(=\) | \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1050.h (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(28.6104277578\) |
| 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: | \( 2^{8} \) |
| Twist minimal: | no (minimal twist has level 42) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(451\) | \(701\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 349.1 |
|
− | 1.41421i | −1.73205 | −2.00000 | 0 | 2.44949i | −6.63103 | + | 2.24264i | 2.82843i | 3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||
| 349.2 | − | 1.41421i | −1.73205 | −2.00000 | 0 | 2.44949i | 3.16693 | + | 6.24264i | 2.82843i | 3.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 349.3 | − | 1.41421i | 1.73205 | −2.00000 | 0 | − | 2.44949i | −3.16693 | + | 6.24264i | 2.82843i | 3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||
| 349.4 | − | 1.41421i | 1.73205 | −2.00000 | 0 | − | 2.44949i | 6.63103 | + | 2.24264i | 2.82843i | 3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||
| 349.5 | 1.41421i | −1.73205 | −2.00000 | 0 | − | 2.44949i | −6.63103 | − | 2.24264i | − | 2.82843i | 3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||
| 349.6 | 1.41421i | −1.73205 | −2.00000 | 0 | − | 2.44949i | 3.16693 | − | 6.24264i | − | 2.82843i | 3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||
| 349.7 | 1.41421i | 1.73205 | −2.00000 | 0 | 2.44949i | −3.16693 | − | 6.24264i | − | 2.82843i | 3.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 349.8 | 1.41421i | 1.73205 | −2.00000 | 0 | 2.44949i | 6.63103 | − | 2.24264i | − | 2.82843i | 3.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
| 7.b | odd | 2 | 1 | inner |
| 35.c | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1050.3.h.a | 8 | |
| 5.b | even | 2 | 1 | inner | 1050.3.h.a | 8 | |
| 5.c | odd | 4 | 1 | 42.3.c.a | ✓ | 4 | |
| 5.c | odd | 4 | 1 | 1050.3.f.a | 4 | ||
| 7.b | odd | 2 | 1 | inner | 1050.3.h.a | 8 | |
| 15.e | even | 4 | 1 | 126.3.c.b | 4 | ||
| 20.e | even | 4 | 1 | 336.3.f.c | 4 | ||
| 35.c | odd | 2 | 1 | inner | 1050.3.h.a | 8 | |
| 35.f | even | 4 | 1 | 42.3.c.a | ✓ | 4 | |
| 35.f | even | 4 | 1 | 1050.3.f.a | 4 | ||
| 35.k | even | 12 | 1 | 294.3.g.b | 4 | ||
| 35.k | even | 12 | 1 | 294.3.g.c | 4 | ||
| 35.l | odd | 12 | 1 | 294.3.g.b | 4 | ||
| 35.l | odd | 12 | 1 | 294.3.g.c | 4 | ||
| 40.i | odd | 4 | 1 | 1344.3.f.f | 4 | ||
| 40.k | even | 4 | 1 | 1344.3.f.e | 4 | ||
| 60.l | odd | 4 | 1 | 1008.3.f.g | 4 | ||
| 105.k | odd | 4 | 1 | 126.3.c.b | 4 | ||
| 105.w | odd | 12 | 1 | 882.3.n.a | 4 | ||
| 105.w | odd | 12 | 1 | 882.3.n.d | 4 | ||
| 105.x | even | 12 | 1 | 882.3.n.a | 4 | ||
| 105.x | even | 12 | 1 | 882.3.n.d | 4 | ||
| 140.j | odd | 4 | 1 | 336.3.f.c | 4 | ||
| 280.s | even | 4 | 1 | 1344.3.f.f | 4 | ||
| 280.y | odd | 4 | 1 | 1344.3.f.e | 4 | ||
| 420.w | even | 4 | 1 | 1008.3.f.g | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 42.3.c.a | ✓ | 4 | 5.c | odd | 4 | 1 | |
| 42.3.c.a | ✓ | 4 | 35.f | even | 4 | 1 | |
| 126.3.c.b | 4 | 15.e | even | 4 | 1 | ||
| 126.3.c.b | 4 | 105.k | odd | 4 | 1 | ||
| 294.3.g.b | 4 | 35.k | even | 12 | 1 | ||
| 294.3.g.b | 4 | 35.l | odd | 12 | 1 | ||
| 294.3.g.c | 4 | 35.k | even | 12 | 1 | ||
| 294.3.g.c | 4 | 35.l | odd | 12 | 1 | ||
| 336.3.f.c | 4 | 20.e | even | 4 | 1 | ||
| 336.3.f.c | 4 | 140.j | odd | 4 | 1 | ||
| 882.3.n.a | 4 | 105.w | odd | 12 | 1 | ||
| 882.3.n.a | 4 | 105.x | even | 12 | 1 | ||
| 882.3.n.d | 4 | 105.w | odd | 12 | 1 | ||
| 882.3.n.d | 4 | 105.x | even | 12 | 1 | ||
| 1008.3.f.g | 4 | 60.l | odd | 4 | 1 | ||
| 1008.3.f.g | 4 | 420.w | even | 4 | 1 | ||
| 1050.3.f.a | 4 | 5.c | odd | 4 | 1 | ||
| 1050.3.f.a | 4 | 35.f | even | 4 | 1 | ||
| 1050.3.h.a | 8 | 1.a | even | 1 | 1 | trivial | |
| 1050.3.h.a | 8 | 5.b | even | 2 | 1 | inner | |
| 1050.3.h.a | 8 | 7.b | odd | 2 | 1 | inner | |
| 1050.3.h.a | 8 | 35.c | odd | 2 | 1 | inner | |
| 1344.3.f.e | 4 | 40.k | even | 4 | 1 | ||
| 1344.3.f.e | 4 | 280.y | odd | 4 | 1 | ||
| 1344.3.f.f | 4 | 40.i | odd | 4 | 1 | ||
| 1344.3.f.f | 4 | 280.s | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{2} + 12 T_{11} + 18 \) acting on \(S_{3}^{\mathrm{new}}(1050, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ | \( ( 2 + T^{2} )^{4} \) |
| $3$ | \( ( -3 + T^{2} )^{4} \) |
| $5$ | \( T^{8} \) |
| $7$ | \( 5764801 - 48020 T^{2} + 294 T^{4} - 20 T^{6} + T^{8} \) |
| $11$ | \( ( 18 + 12 T + T^{2} )^{4} \) |
| $13$ | \( ( 28224 - 432 T^{2} + T^{4} )^{2} \) |
| $17$ | \( ( 509796 - 1476 T^{2} + T^{4} )^{2} \) |
| $19$ | \( ( 138384 + 792 T^{2} + T^{4} )^{2} \) |
| $23$ | \( ( 15876 + 396 T^{2} + T^{4} )^{2} \) |
| $29$ | \( ( 30 + T )^{8} \) |
| $31$ | \( ( 186624 + 2592 T^{2} + T^{4} )^{2} \) |
| $37$ | \( ( 4804864 + 5984 T^{2} + T^{4} )^{2} \) |
| $41$ | \( ( 86436 + 1764 T^{2} + T^{4} )^{2} \) |
| $43$ | \( ( 602176 + 5648 T^{2} + T^{4} )^{2} \) |
| $47$ | \( ( 5089536 - 4896 T^{2} + T^{4} )^{2} \) |
| $53$ | \( ( 6906384 + 6408 T^{2} + T^{4} )^{2} \) |
| $59$ | \( ( 2304 + 9504 T^{2} + T^{4} )^{2} \) |
| $61$ | \( ( 2304 + 288 T^{2} + T^{4} )^{2} \) |
| $67$ | \( ( 2715904 + 4448 T^{2} + T^{4} )^{2} \) |
| $71$ | \( ( -5598 + 60 T + T^{2} )^{4} \) |
| $73$ | \( ( 16064064 - 8208 T^{2} + T^{4} )^{2} \) |
| $79$ | \( ( -9344 + 64 T + T^{2} )^{4} \) |
| $83$ | \( ( 451584 - 10944 T^{2} + T^{4} )^{2} \) |
| $89$ | \( ( 37234404 + 22788 T^{2} + T^{4} )^{2} \) |
| $97$ | \( ( 27123264 - 12816 T^{2} + T^{4} )^{2} \) |
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