Newspace parameters
Level: | \( N \) | \(=\) | \( 224 = 2^{5} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 224.g (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(6.10355792167\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(\sqrt{2}, \sqrt{-7})\) |
Defining polynomial: |
\( x^{4} + 6x^{2} + 16 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{2} \) |
Twist minimal: | no (minimal twist has level 56) |
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
\( x^{4} + 6x^{2} + 16 \)
:
\(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 2\nu ) / 4 \)
|
\(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 10\nu ) / 4 \)
|
\(\beta_{3}\) | \(=\) |
\( \nu^{2} + 3 \)
|
\(\nu\) | \(=\) |
\( ( \beta_{2} - \beta_1 ) / 2 \)
|
\(\nu^{2}\) | \(=\) |
\( \beta_{3} - 3 \)
|
\(\nu^{3}\) | \(=\) |
\( -\beta_{2} + 5\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).
\(n\) | \(127\) | \(129\) | \(197\) |
\(\chi(n)\) | \(-1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
15.1 |
|
0 | −3.41421 | 0 | − | 1.54985i | 0 | 2.64575i | 0 | 2.65685 | 0 | |||||||||||||||||||||||||||||
15.2 | 0 | −3.41421 | 0 | 1.54985i | 0 | − | 2.64575i | 0 | 2.65685 | 0 | ||||||||||||||||||||||||||||||
15.3 | 0 | −0.585786 | 0 | − | 9.03316i | 0 | 2.64575i | 0 | −8.65685 | 0 | ||||||||||||||||||||||||||||||
15.4 | 0 | −0.585786 | 0 | 9.03316i | 0 | − | 2.64575i | 0 | −8.65685 | 0 | ||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.d | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 224.3.g.a | 4 | |
3.b | odd | 2 | 1 | 2016.3.g.a | 4 | ||
4.b | odd | 2 | 1 | 56.3.g.a | ✓ | 4 | |
7.b | odd | 2 | 1 | 1568.3.g.h | 4 | ||
8.b | even | 2 | 1 | 56.3.g.a | ✓ | 4 | |
8.d | odd | 2 | 1 | inner | 224.3.g.a | 4 | |
12.b | even | 2 | 1 | 504.3.g.a | 4 | ||
16.e | even | 4 | 2 | 1792.3.d.g | 8 | ||
16.f | odd | 4 | 2 | 1792.3.d.g | 8 | ||
24.f | even | 2 | 1 | 2016.3.g.a | 4 | ||
24.h | odd | 2 | 1 | 504.3.g.a | 4 | ||
28.d | even | 2 | 1 | 392.3.g.h | 4 | ||
28.f | even | 6 | 2 | 392.3.k.j | 8 | ||
28.g | odd | 6 | 2 | 392.3.k.i | 8 | ||
56.e | even | 2 | 1 | 1568.3.g.h | 4 | ||
56.h | odd | 2 | 1 | 392.3.g.h | 4 | ||
56.j | odd | 6 | 2 | 392.3.k.j | 8 | ||
56.p | even | 6 | 2 | 392.3.k.i | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
56.3.g.a | ✓ | 4 | 4.b | odd | 2 | 1 | |
56.3.g.a | ✓ | 4 | 8.b | even | 2 | 1 | |
224.3.g.a | 4 | 1.a | even | 1 | 1 | trivial | |
224.3.g.a | 4 | 8.d | odd | 2 | 1 | inner | |
392.3.g.h | 4 | 28.d | even | 2 | 1 | ||
392.3.g.h | 4 | 56.h | odd | 2 | 1 | ||
392.3.k.i | 8 | 28.g | odd | 6 | 2 | ||
392.3.k.i | 8 | 56.p | even | 6 | 2 | ||
392.3.k.j | 8 | 28.f | even | 6 | 2 | ||
392.3.k.j | 8 | 56.j | odd | 6 | 2 | ||
504.3.g.a | 4 | 12.b | even | 2 | 1 | ||
504.3.g.a | 4 | 24.h | odd | 2 | 1 | ||
1568.3.g.h | 4 | 7.b | odd | 2 | 1 | ||
1568.3.g.h | 4 | 56.e | even | 2 | 1 | ||
1792.3.d.g | 8 | 16.e | even | 4 | 2 | ||
1792.3.d.g | 8 | 16.f | odd | 4 | 2 | ||
2016.3.g.a | 4 | 3.b | odd | 2 | 1 | ||
2016.3.g.a | 4 | 24.f | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 4T_{3} + 2 \)
acting on \(S_{3}^{\mathrm{new}}(224, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( (T^{2} + 4 T + 2)^{2} \)
$5$
\( T^{4} + 84T^{2} + 196 \)
$7$
\( (T^{2} + 7)^{2} \)
$11$
\( (T^{2} + 8 T - 56)^{2} \)
$13$
\( T^{4} + 84T^{2} + 196 \)
$17$
\( (T^{2} - 36 T + 292)^{2} \)
$19$
\( (T^{2} + 4 T - 718)^{2} \)
$23$
\( T^{4} + 1848 T^{2} + 753424 \)
$29$
\( (T^{2} + 504)^{2} \)
$31$
\( T^{4} + 2464 T^{2} + 614656 \)
$37$
\( T^{4} + 3696 T^{2} + 906304 \)
$41$
\( (T^{2} + 20 T - 188)^{2} \)
$43$
\( (T^{2} - 40 T + 392)^{2} \)
$47$
\( T^{4} + 1344 T^{2} + 50176 \)
$53$
\( T^{4} + 9632 T^{2} + 614656 \)
$59$
\( (T^{2} + 92 T + 1874)^{2} \)
$61$
\( T^{4} + 1652 T^{2} + 329476 \)
$67$
\( (T^{2} - 112 T + 2624)^{2} \)
$71$
\( T^{4} + 10752 T^{2} + \cdots + 3211264 \)
$73$
\( (T^{2} - 116 T + 3236)^{2} \)
$79$
\( T^{4} + 8064 T^{2} + \cdots + 9834496 \)
$83$
\( (T^{2} + 44 T + 146)^{2} \)
$89$
\( (T^{2} - 156 T + 4932)^{2} \)
$97$
\( (T^{2} + 68 T - 15772)^{2} \)
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