Newspace parameters
Level: | \( N \) | \(=\) | \( 98 = 2 \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 98.c (of order \(3\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(5.78218718056\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{22})\) |
Defining polynomial: |
\( x^{4} + 22x^{2} + 484 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
Coefficient ring index: | \( 2^{2} \) |
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
\( x^{4} + 22x^{2} + 484 \)
:
\(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
\(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 22 \)
|
\(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 11 \)
|
\(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
\(\nu^{2}\) | \(=\) |
\( 22\beta_{2} \)
|
\(\nu^{3}\) | \(=\) |
\( 11\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/98\mathbb{Z}\right)^\times\).
\(n\) | \(3\) |
\(\chi(n)\) | \(-1 - \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
67.1 |
|
−1.00000 | − | 1.73205i | −4.69042 | + | 8.12404i | −2.00000 | + | 3.46410i | 4.69042 | + | 8.12404i | 18.7617 | 0 | 8.00000 | −30.5000 | − | 52.8275i | 9.38083 | − | 16.2481i | ||||||||||||||||||
67.2 | −1.00000 | − | 1.73205i | 4.69042 | − | 8.12404i | −2.00000 | + | 3.46410i | −4.69042 | − | 8.12404i | −18.7617 | 0 | 8.00000 | −30.5000 | − | 52.8275i | −9.38083 | + | 16.2481i | |||||||||||||||||||
79.1 | −1.00000 | + | 1.73205i | −4.69042 | − | 8.12404i | −2.00000 | − | 3.46410i | 4.69042 | − | 8.12404i | 18.7617 | 0 | 8.00000 | −30.5000 | + | 52.8275i | 9.38083 | + | 16.2481i | |||||||||||||||||||
79.2 | −1.00000 | + | 1.73205i | 4.69042 | + | 8.12404i | −2.00000 | − | 3.46410i | −4.69042 | + | 8.12404i | −18.7617 | 0 | 8.00000 | −30.5000 | + | 52.8275i | −9.38083 | − | 16.2481i | |||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | inner |
7.c | even | 3 | 1 | inner |
7.d | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 98.4.c.g | 4 | |
3.b | odd | 2 | 1 | 882.4.g.bi | 4 | ||
7.b | odd | 2 | 1 | inner | 98.4.c.g | 4 | |
7.c | even | 3 | 1 | 98.4.a.h | ✓ | 2 | |
7.c | even | 3 | 1 | inner | 98.4.c.g | 4 | |
7.d | odd | 6 | 1 | 98.4.a.h | ✓ | 2 | |
7.d | odd | 6 | 1 | inner | 98.4.c.g | 4 | |
21.c | even | 2 | 1 | 882.4.g.bi | 4 | ||
21.g | even | 6 | 1 | 882.4.a.w | 2 | ||
21.g | even | 6 | 1 | 882.4.g.bi | 4 | ||
21.h | odd | 6 | 1 | 882.4.a.w | 2 | ||
21.h | odd | 6 | 1 | 882.4.g.bi | 4 | ||
28.f | even | 6 | 1 | 784.4.a.z | 2 | ||
28.g | odd | 6 | 1 | 784.4.a.z | 2 | ||
35.i | odd | 6 | 1 | 2450.4.a.bs | 2 | ||
35.j | even | 6 | 1 | 2450.4.a.bs | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
98.4.a.h | ✓ | 2 | 7.c | even | 3 | 1 | |
98.4.a.h | ✓ | 2 | 7.d | odd | 6 | 1 | |
98.4.c.g | 4 | 1.a | even | 1 | 1 | trivial | |
98.4.c.g | 4 | 7.b | odd | 2 | 1 | inner | |
98.4.c.g | 4 | 7.c | even | 3 | 1 | inner | |
98.4.c.g | 4 | 7.d | odd | 6 | 1 | inner | |
784.4.a.z | 2 | 28.f | even | 6 | 1 | ||
784.4.a.z | 2 | 28.g | odd | 6 | 1 | ||
882.4.a.w | 2 | 21.g | even | 6 | 1 | ||
882.4.a.w | 2 | 21.h | odd | 6 | 1 | ||
882.4.g.bi | 4 | 3.b | odd | 2 | 1 | ||
882.4.g.bi | 4 | 21.c | even | 2 | 1 | ||
882.4.g.bi | 4 | 21.g | even | 6 | 1 | ||
882.4.g.bi | 4 | 21.h | odd | 6 | 1 | ||
2450.4.a.bs | 2 | 35.i | odd | 6 | 1 | ||
2450.4.a.bs | 2 | 35.j | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 88T_{3}^{2} + 7744 \)
acting on \(S_{4}^{\mathrm{new}}(98, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} + 2 T + 4)^{2} \)
$3$
\( T^{4} + 88T^{2} + 7744 \)
$5$
\( T^{4} + 88T^{2} + 7744 \)
$7$
\( T^{4} \)
$11$
\( (T^{2} + 20 T + 400)^{2} \)
$13$
\( (T^{2} - 4312)^{2} \)
$17$
\( T^{4} + 3168 T^{2} + \cdots + 10036224 \)
$19$
\( T^{4} + 88T^{2} + 7744 \)
$23$
\( (T^{2} + 48 T + 2304)^{2} \)
$29$
\( (T + 166)^{4} \)
$31$
\( T^{4} + 42592 T^{2} + \cdots + 1814078464 \)
$37$
\( (T^{2} - 78 T + 6084)^{2} \)
$41$
\( (T^{2} - 155232)^{2} \)
$43$
\( (T - 436)^{4} \)
$47$
\( T^{4} + 42592 T^{2} + \cdots + 1814078464 \)
$53$
\( (T^{2} + 62 T + 3844)^{2} \)
$59$
\( T^{4} + 443608 T^{2} + \cdots + 196788057664 \)
$61$
\( T^{4} + 74008 T^{2} + \cdots + 5477184064 \)
$67$
\( (T^{2} + 580 T + 336400)^{2} \)
$71$
\( (T + 544)^{4} \)
$73$
\( T^{4} + 360448 T^{2} + \cdots + 129922760704 \)
$79$
\( (T^{2} - 680 T + 462400)^{2} \)
$83$
\( (T^{2} - 38808)^{2} \)
$89$
\( T^{4} + 2252800 T^{2} + \cdots + 5075107840000 \)
$97$
\( (T^{2} - 431200)^{2} \)
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