Newspace parameters
Level: | \( N \) | \(=\) | \( 98 = 2 \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 98.c (of order \(3\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.782533939809\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
Coefficient field: | \(\Q(\sqrt{2}, \sqrt{-3})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} + 2x^{2} + 4 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
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 \( x^{4} + 2x^{2} + 4 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{2} ) / 2 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{3} ) / 2 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( 2\beta_{2} \) |
\(\nu^{3}\) | \(=\) | \( 2\beta_{3} \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/98\mathbb{Z}\right)^\times\).
\(n\) | \(3\) |
\(\chi(n)\) | \(\beta_{2}\) |
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 |
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−0.500000 | − | 0.866025i | −0.707107 | + | 1.22474i | −0.500000 | + | 0.866025i | 1.41421 | + | 2.44949i | 1.41421 | 0 | 1.00000 | 0.500000 | + | 0.866025i | 1.41421 | − | 2.44949i | ||||||||||||||||||
67.2 | −0.500000 | − | 0.866025i | 0.707107 | − | 1.22474i | −0.500000 | + | 0.866025i | −1.41421 | − | 2.44949i | −1.41421 | 0 | 1.00000 | 0.500000 | + | 0.866025i | −1.41421 | + | 2.44949i | |||||||||||||||||||
79.1 | −0.500000 | + | 0.866025i | −0.707107 | − | 1.22474i | −0.500000 | − | 0.866025i | 1.41421 | − | 2.44949i | 1.41421 | 0 | 1.00000 | 0.500000 | − | 0.866025i | 1.41421 | + | 2.44949i | |||||||||||||||||||
79.2 | −0.500000 | + | 0.866025i | 0.707107 | + | 1.22474i | −0.500000 | − | 0.866025i | −1.41421 | + | 2.44949i | −1.41421 | 0 | 1.00000 | 0.500000 | − | 0.866025i | −1.41421 | − | 2.44949i | |||||||||||||||||||
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 | 98.2.c.c | 4 | |
3.b | odd | 2 | 1 | 882.2.g.l | 4 | ||
4.b | odd | 2 | 1 | 784.2.i.m | 4 | ||
7.b | odd | 2 | 1 | inner | 98.2.c.c | 4 | |
7.c | even | 3 | 1 | 98.2.a.b | ✓ | 2 | |
7.c | even | 3 | 1 | inner | 98.2.c.c | 4 | |
7.d | odd | 6 | 1 | 98.2.a.b | ✓ | 2 | |
7.d | odd | 6 | 1 | inner | 98.2.c.c | 4 | |
21.c | even | 2 | 1 | 882.2.g.l | 4 | ||
21.g | even | 6 | 1 | 882.2.a.n | 2 | ||
21.g | even | 6 | 1 | 882.2.g.l | 4 | ||
21.h | odd | 6 | 1 | 882.2.a.n | 2 | ||
21.h | odd | 6 | 1 | 882.2.g.l | 4 | ||
28.d | even | 2 | 1 | 784.2.i.m | 4 | ||
28.f | even | 6 | 1 | 784.2.a.l | 2 | ||
28.f | even | 6 | 1 | 784.2.i.m | 4 | ||
28.g | odd | 6 | 1 | 784.2.a.l | 2 | ||
28.g | odd | 6 | 1 | 784.2.i.m | 4 | ||
35.i | odd | 6 | 1 | 2450.2.a.bj | 2 | ||
35.j | even | 6 | 1 | 2450.2.a.bj | 2 | ||
35.k | even | 12 | 2 | 2450.2.c.v | 4 | ||
35.l | odd | 12 | 2 | 2450.2.c.v | 4 | ||
56.j | odd | 6 | 1 | 3136.2.a.bn | 2 | ||
56.k | odd | 6 | 1 | 3136.2.a.bm | 2 | ||
56.m | even | 6 | 1 | 3136.2.a.bm | 2 | ||
56.p | even | 6 | 1 | 3136.2.a.bn | 2 | ||
84.j | odd | 6 | 1 | 7056.2.a.cl | 2 | ||
84.n | even | 6 | 1 | 7056.2.a.cl | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
98.2.a.b | ✓ | 2 | 7.c | even | 3 | 1 | |
98.2.a.b | ✓ | 2 | 7.d | odd | 6 | 1 | |
98.2.c.c | 4 | 1.a | even | 1 | 1 | trivial | |
98.2.c.c | 4 | 7.b | odd | 2 | 1 | inner | |
98.2.c.c | 4 | 7.c | even | 3 | 1 | inner | |
98.2.c.c | 4 | 7.d | odd | 6 | 1 | inner | |
784.2.a.l | 2 | 28.f | even | 6 | 1 | ||
784.2.a.l | 2 | 28.g | odd | 6 | 1 | ||
784.2.i.m | 4 | 4.b | odd | 2 | 1 | ||
784.2.i.m | 4 | 28.d | even | 2 | 1 | ||
784.2.i.m | 4 | 28.f | even | 6 | 1 | ||
784.2.i.m | 4 | 28.g | odd | 6 | 1 | ||
882.2.a.n | 2 | 21.g | even | 6 | 1 | ||
882.2.a.n | 2 | 21.h | odd | 6 | 1 | ||
882.2.g.l | 4 | 3.b | odd | 2 | 1 | ||
882.2.g.l | 4 | 21.c | even | 2 | 1 | ||
882.2.g.l | 4 | 21.g | even | 6 | 1 | ||
882.2.g.l | 4 | 21.h | odd | 6 | 1 | ||
2450.2.a.bj | 2 | 35.i | odd | 6 | 1 | ||
2450.2.a.bj | 2 | 35.j | even | 6 | 1 | ||
2450.2.c.v | 4 | 35.k | even | 12 | 2 | ||
2450.2.c.v | 4 | 35.l | odd | 12 | 2 | ||
3136.2.a.bm | 2 | 56.k | odd | 6 | 1 | ||
3136.2.a.bm | 2 | 56.m | even | 6 | 1 | ||
3136.2.a.bn | 2 | 56.j | odd | 6 | 1 | ||
3136.2.a.bn | 2 | 56.p | even | 6 | 1 | ||
7056.2.a.cl | 2 | 84.j | odd | 6 | 1 | ||
7056.2.a.cl | 2 | 84.n | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 2T_{3}^{2} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(98, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} + T + 1)^{2} \)
$3$
\( T^{4} + 2T^{2} + 4 \)
$5$
\( T^{4} + 8T^{2} + 64 \)
$7$
\( T^{4} \)
$11$
\( (T^{2} - 2 T + 4)^{2} \)
$13$
\( T^{4} \)
$17$
\( T^{4} + 2T^{2} + 4 \)
$19$
\( T^{4} + 50T^{2} + 2500 \)
$23$
\( (T^{2} - 4 T + 16)^{2} \)
$29$
\( (T - 2)^{4} \)
$31$
\( T^{4} + 72T^{2} + 5184 \)
$37$
\( (T^{2} + 10 T + 100)^{2} \)
$41$
\( (T^{2} - 98)^{2} \)
$43$
\( (T - 2)^{4} \)
$47$
\( T^{4} + 8T^{2} + 64 \)
$53$
\( (T^{2} - 2 T + 4)^{2} \)
$59$
\( T^{4} + 2T^{2} + 4 \)
$61$
\( T^{4} + 8T^{2} + 64 \)
$67$
\( (T^{2} + 12 T + 144)^{2} \)
$71$
\( (T + 12)^{4} \)
$73$
\( T^{4} + 2T^{2} + 4 \)
$79$
\( (T^{2} - 4 T + 16)^{2} \)
$83$
\( (T^{2} - 98)^{2} \)
$89$
\( T^{4} + 50T^{2} + 2500 \)
$97$
\( (T^{2} - 98)^{2} \)
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