Newspace parameters
Level: | \( N \) | \(=\) | \( 224 = 2^{5} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 224.i (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.78864900528\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{7})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} + 7x^{2} + 49 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 \( x^{4} + 7x^{2} + 49 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{2} ) / 7 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{3} ) / 7 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( 7\beta_{2} \) |
\(\nu^{3}\) | \(=\) | \( 7\beta_{3} \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).
\(n\) | \(127\) | \(129\) | \(197\) |
\(\chi(n)\) | \(1\) | \(\beta_{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
65.1 |
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0 | −1.32288 | − | 2.29129i | 0 | 1.50000 | − | 2.59808i | 0 | −2.64575 | 0 | −2.00000 | + | 3.46410i | 0 | ||||||||||||||||||||||||
65.2 | 0 | 1.32288 | + | 2.29129i | 0 | 1.50000 | − | 2.59808i | 0 | 2.64575 | 0 | −2.00000 | + | 3.46410i | 0 | |||||||||||||||||||||||||
193.1 | 0 | −1.32288 | + | 2.29129i | 0 | 1.50000 | + | 2.59808i | 0 | −2.64575 | 0 | −2.00000 | − | 3.46410i | 0 | |||||||||||||||||||||||||
193.2 | 0 | 1.32288 | − | 2.29129i | 0 | 1.50000 | + | 2.59808i | 0 | 2.64575 | 0 | −2.00000 | − | 3.46410i | 0 | |||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
7.c | even | 3 | 1 | inner |
28.g | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 224.2.i.c | ✓ | 4 |
3.b | odd | 2 | 1 | 2016.2.s.o | 4 | ||
4.b | odd | 2 | 1 | inner | 224.2.i.c | ✓ | 4 |
7.b | odd | 2 | 1 | 1568.2.i.p | 4 | ||
7.c | even | 3 | 1 | inner | 224.2.i.c | ✓ | 4 |
7.c | even | 3 | 1 | 1568.2.a.m | 2 | ||
7.d | odd | 6 | 1 | 1568.2.a.t | 2 | ||
7.d | odd | 6 | 1 | 1568.2.i.p | 4 | ||
8.b | even | 2 | 1 | 448.2.i.h | 4 | ||
8.d | odd | 2 | 1 | 448.2.i.h | 4 | ||
12.b | even | 2 | 1 | 2016.2.s.o | 4 | ||
21.h | odd | 6 | 1 | 2016.2.s.o | 4 | ||
28.d | even | 2 | 1 | 1568.2.i.p | 4 | ||
28.f | even | 6 | 1 | 1568.2.a.t | 2 | ||
28.f | even | 6 | 1 | 1568.2.i.p | 4 | ||
28.g | odd | 6 | 1 | inner | 224.2.i.c | ✓ | 4 |
28.g | odd | 6 | 1 | 1568.2.a.m | 2 | ||
56.j | odd | 6 | 1 | 3136.2.a.bg | 2 | ||
56.k | odd | 6 | 1 | 448.2.i.h | 4 | ||
56.k | odd | 6 | 1 | 3136.2.a.bv | 2 | ||
56.m | even | 6 | 1 | 3136.2.a.bg | 2 | ||
56.p | even | 6 | 1 | 448.2.i.h | 4 | ||
56.p | even | 6 | 1 | 3136.2.a.bv | 2 | ||
84.n | even | 6 | 1 | 2016.2.s.o | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
224.2.i.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
224.2.i.c | ✓ | 4 | 4.b | odd | 2 | 1 | inner |
224.2.i.c | ✓ | 4 | 7.c | even | 3 | 1 | inner |
224.2.i.c | ✓ | 4 | 28.g | odd | 6 | 1 | inner |
448.2.i.h | 4 | 8.b | even | 2 | 1 | ||
448.2.i.h | 4 | 8.d | odd | 2 | 1 | ||
448.2.i.h | 4 | 56.k | odd | 6 | 1 | ||
448.2.i.h | 4 | 56.p | even | 6 | 1 | ||
1568.2.a.m | 2 | 7.c | even | 3 | 1 | ||
1568.2.a.m | 2 | 28.g | odd | 6 | 1 | ||
1568.2.a.t | 2 | 7.d | odd | 6 | 1 | ||
1568.2.a.t | 2 | 28.f | even | 6 | 1 | ||
1568.2.i.p | 4 | 7.b | odd | 2 | 1 | ||
1568.2.i.p | 4 | 7.d | odd | 6 | 1 | ||
1568.2.i.p | 4 | 28.d | even | 2 | 1 | ||
1568.2.i.p | 4 | 28.f | even | 6 | 1 | ||
2016.2.s.o | 4 | 3.b | odd | 2 | 1 | ||
2016.2.s.o | 4 | 12.b | even | 2 | 1 | ||
2016.2.s.o | 4 | 21.h | odd | 6 | 1 | ||
2016.2.s.o | 4 | 84.n | even | 6 | 1 | ||
3136.2.a.bg | 2 | 56.j | odd | 6 | 1 | ||
3136.2.a.bg | 2 | 56.m | even | 6 | 1 | ||
3136.2.a.bv | 2 | 56.k | odd | 6 | 1 | ||
3136.2.a.bv | 2 | 56.p | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 7T_{3}^{2} + 49 \)
acting on \(S_{2}^{\mathrm{new}}(224, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( T^{4} + 7T^{2} + 49 \)
$5$
\( (T^{2} - 3 T + 9)^{2} \)
$7$
\( (T^{2} - 7)^{2} \)
$11$
\( T^{4} + 7T^{2} + 49 \)
$13$
\( (T + 4)^{4} \)
$17$
\( (T^{2} + T + 1)^{2} \)
$19$
\( T^{4} + 63T^{2} + 3969 \)
$23$
\( T^{4} + 7T^{2} + 49 \)
$29$
\( (T + 4)^{4} \)
$31$
\( T^{4} + 7T^{2} + 49 \)
$37$
\( (T^{2} - 5 T + 25)^{2} \)
$41$
\( (T - 8)^{4} \)
$43$
\( (T^{2} - 112)^{2} \)
$47$
\( T^{4} + 7T^{2} + 49 \)
$53$
\( (T^{2} + 7 T + 49)^{2} \)
$59$
\( T^{4} + 7T^{2} + 49 \)
$61$
\( (T^{2} - 5 T + 25)^{2} \)
$67$
\( T^{4} + 7T^{2} + 49 \)
$71$
\( T^{4} \)
$73$
\( (T^{2} - 9 T + 81)^{2} \)
$79$
\( T^{4} + 7T^{2} + 49 \)
$83$
\( (T^{2} - 112)^{2} \)
$89$
\( (T^{2} - 9 T + 81)^{2} \)
$97$
\( (T + 8)^{4} \)
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