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 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
\(\beta_{1}\) | \(=\) |
\( -\zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24} \)
|
\(\beta_{2}\) | \(=\) |
\( -\zeta_{24}^{6} + 2\zeta_{24}^{2} \)
|
\(\beta_{3}\) | \(=\) |
\( -2\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \)
|
\(\beta_{4}\) | \(=\) |
\( 4\zeta_{24}^{4} - 2 \)
|
\(\beta_{5}\) | \(=\) |
\( 2\zeta_{24}^{6} \)
|
\(\beta_{6}\) | \(=\) |
\( -\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \)
|
\(\beta_{7}\) | \(=\) |
\( 2\zeta_{24}^{7} + \zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24} \)
|
\(\zeta_{24}\) | \(=\) |
\( ( \beta_{7} + \beta_{6} + \beta_{3} + \beta_1 ) / 4 \)
|
\(\zeta_{24}^{2}\) | \(=\) |
\( ( \beta_{5} + 2\beta_{2} ) / 4 \)
|
\(\zeta_{24}^{3}\) | \(=\) |
\( ( \beta_{6} - \beta_1 ) / 2 \)
|
\(\zeta_{24}^{4}\) | \(=\) |
\( ( \beta_{4} + 2 ) / 4 \)
|
\(\zeta_{24}^{5}\) | \(=\) |
\( ( \beta_{7} - \beta_{6} + \beta_{3} - \beta_1 ) / 4 \)
|
\(\zeta_{24}^{6}\) | \(=\) |
\( ( \beta_{5} ) / 2 \)
|
\(\zeta_{24}^{7}\) | \(=\) |
\( ( \beta_{7} + \beta_{6} - \beta_{3} - \beta_1 ) / 4 \)
|
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
\( T_{11}^{2} + 12T_{11} + 18 \)
acting on \(S_{3}^{\mathrm{new}}(1050, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} + 2)^{4} \)
$3$
\( (T^{2} - 3)^{4} \)
$5$
\( T^{8} \)
$7$
\( T^{8} - 20 T^{6} + 294 T^{4} + \cdots + 5764801 \)
$11$
\( (T^{2} + 12 T + 18)^{4} \)
$13$
\( (T^{4} - 432 T^{2} + 28224)^{2} \)
$17$
\( (T^{4} - 1476 T^{2} + 509796)^{2} \)
$19$
\( (T^{4} + 792 T^{2} + 138384)^{2} \)
$23$
\( (T^{4} + 396 T^{2} + 15876)^{2} \)
$29$
\( (T + 30)^{8} \)
$31$
\( (T^{4} + 2592 T^{2} + 186624)^{2} \)
$37$
\( (T^{4} + 5984 T^{2} + 4804864)^{2} \)
$41$
\( (T^{4} + 1764 T^{2} + 86436)^{2} \)
$43$
\( (T^{4} + 5648 T^{2} + 602176)^{2} \)
$47$
\( (T^{4} - 4896 T^{2} + 5089536)^{2} \)
$53$
\( (T^{4} + 6408 T^{2} + 6906384)^{2} \)
$59$
\( (T^{4} + 9504 T^{2} + 2304)^{2} \)
$61$
\( (T^{4} + 288 T^{2} + 2304)^{2} \)
$67$
\( (T^{4} + 4448 T^{2} + 2715904)^{2} \)
$71$
\( (T^{2} + 60 T - 5598)^{4} \)
$73$
\( (T^{4} - 8208 T^{2} + 16064064)^{2} \)
$79$
\( (T^{2} + 64 T - 9344)^{4} \)
$83$
\( (T^{4} - 10944 T^{2} + 451584)^{2} \)
$89$
\( (T^{4} + 22788 T^{2} + 37234404)^{2} \)
$97$
\( (T^{4} - 12816 T^{2} + 27123264)^{2} \)
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