Newspace parameters
Level: | \( N \) | \(=\) | \( 224 = 2^{5} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 224.r (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(6.10355792167\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
Coefficient field: | \(\Q(\zeta_{12})\) |
Defining polynomial: |
\( x^{4} - x^{2} + 1 \)
|
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).
\(n\) | \(127\) | \(129\) | \(197\) |
\(\chi(n)\) | \(-1\) | \(-\zeta_{12}^{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
95.1 |
|
0 | −0.866025 | − | 0.500000i | 0 | −0.500000 | − | 0.866025i | 0 | 7.00000i | 0 | −4.00000 | − | 6.92820i | 0 | ||||||||||||||||||||||||
95.2 | 0 | 0.866025 | + | 0.500000i | 0 | −0.500000 | − | 0.866025i | 0 | − | 7.00000i | 0 | −4.00000 | − | 6.92820i | 0 | ||||||||||||||||||||||||
191.1 | 0 | −0.866025 | + | 0.500000i | 0 | −0.500000 | + | 0.866025i | 0 | − | 7.00000i | 0 | −4.00000 | + | 6.92820i | 0 | ||||||||||||||||||||||||
191.2 | 0 | 0.866025 | − | 0.500000i | 0 | −0.500000 | + | 0.866025i | 0 | 7.00000i | 0 | −4.00000 | + | 6.92820i | 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 | 224.3.r.b | ✓ | 4 |
4.b | odd | 2 | 1 | inner | 224.3.r.b | ✓ | 4 |
7.c | even | 3 | 1 | inner | 224.3.r.b | ✓ | 4 |
7.c | even | 3 | 1 | 1568.3.d.d | 2 | ||
7.d | odd | 6 | 1 | 1568.3.d.c | 2 | ||
8.b | even | 2 | 1 | 448.3.r.b | 4 | ||
8.d | odd | 2 | 1 | 448.3.r.b | 4 | ||
28.f | even | 6 | 1 | 1568.3.d.c | 2 | ||
28.g | odd | 6 | 1 | inner | 224.3.r.b | ✓ | 4 |
28.g | odd | 6 | 1 | 1568.3.d.d | 2 | ||
56.k | odd | 6 | 1 | 448.3.r.b | 4 | ||
56.p | even | 6 | 1 | 448.3.r.b | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
224.3.r.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
224.3.r.b | ✓ | 4 | 4.b | odd | 2 | 1 | inner |
224.3.r.b | ✓ | 4 | 7.c | even | 3 | 1 | inner |
224.3.r.b | ✓ | 4 | 28.g | odd | 6 | 1 | inner |
448.3.r.b | 4 | 8.b | even | 2 | 1 | ||
448.3.r.b | 4 | 8.d | odd | 2 | 1 | ||
448.3.r.b | 4 | 56.k | odd | 6 | 1 | ||
448.3.r.b | 4 | 56.p | even | 6 | 1 | ||
1568.3.d.c | 2 | 7.d | odd | 6 | 1 | ||
1568.3.d.c | 2 | 28.f | even | 6 | 1 | ||
1568.3.d.d | 2 | 7.c | even | 3 | 1 | ||
1568.3.d.d | 2 | 28.g | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - T_{3}^{2} + 1 \)
acting on \(S_{3}^{\mathrm{new}}(224, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( T^{4} - T^{2} + 1 \)
$5$
\( (T^{2} + T + 1)^{2} \)
$7$
\( (T^{2} + 49)^{2} \)
$11$
\( T^{4} - 289 T^{2} + 83521 \)
$13$
\( (T - 24)^{4} \)
$17$
\( (T^{2} + T + 1)^{2} \)
$19$
\( T^{4} - 49T^{2} + 2401 \)
$23$
\( T^{4} - 49T^{2} + 2401 \)
$29$
\( (T - 24)^{4} \)
$31$
\( T^{4} - 1681 T^{2} + \cdots + 2825761 \)
$37$
\( (T^{2} - 49 T + 2401)^{2} \)
$41$
\( (T + 48)^{4} \)
$43$
\( (T^{2} + 576)^{2} \)
$47$
\( T^{4} - 3025 T^{2} + \cdots + 9150625 \)
$53$
\( (T^{2} - 25 T + 625)^{2} \)
$59$
\( T^{4} - 289 T^{2} + 83521 \)
$61$
\( (T^{2} - T + 1)^{2} \)
$67$
\( T^{4} - 4225 T^{2} + \cdots + 17850625 \)
$71$
\( (T^{2} + 9216)^{2} \)
$73$
\( (T^{2} + 95 T + 9025)^{2} \)
$79$
\( T^{4} - 1681 T^{2} + \cdots + 2825761 \)
$83$
\( (T^{2} + 5184)^{2} \)
$89$
\( (T^{2} + 95 T + 9025)^{2} \)
$97$
\( (T - 144)^{4} \)
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