Newspace parameters
Level: | \( N \) | \(=\) | \( 196 = 2^{2} \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 196.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.56506787962\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(\zeta_{12})\) |
Defining polynomial: |
\( x^{4} - x^{2} + 1 \)
|
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 2^{2} \) |
Twist minimal: | no (minimal twist has level 28) |
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
\(\beta_{1}\) | \(=\) |
\( \zeta_{12}^{3} \)
|
\(\beta_{2}\) | \(=\) |
\( 2\zeta_{12}^{2} - 1 \)
|
\(\beta_{3}\) | \(=\) |
\( -\zeta_{12}^{3} + 2\zeta_{12} \)
|
\(\zeta_{12}\) | \(=\) |
\( ( \beta_{3} + \beta_1 ) / 2 \)
|
\(\zeta_{12}^{2}\) | \(=\) |
\( ( \beta_{2} + 1 ) / 2 \)
|
\(\zeta_{12}^{3}\) | \(=\) |
\( \beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/196\mathbb{Z}\right)^\times\).
\(n\) | \(99\) | \(101\) |
\(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
195.1 |
|
1.00000 | − | 1.00000i | −1.73205 | − | 2.00000i | − | 1.73205i | −1.73205 | + | 1.73205i | 0 | −2.00000 | − | 2.00000i | 0 | −1.73205 | − | 1.73205i | ||||||||||||||||||||
195.2 | 1.00000 | − | 1.00000i | 1.73205 | − | 2.00000i | 1.73205i | 1.73205 | − | 1.73205i | 0 | −2.00000 | − | 2.00000i | 0 | 1.73205 | + | 1.73205i | ||||||||||||||||||||||
195.3 | 1.00000 | + | 1.00000i | −1.73205 | 2.00000i | 1.73205i | −1.73205 | − | 1.73205i | 0 | −2.00000 | + | 2.00000i | 0 | −1.73205 | + | 1.73205i | |||||||||||||||||||||||
195.4 | 1.00000 | + | 1.00000i | 1.73205 | 2.00000i | − | 1.73205i | 1.73205 | + | 1.73205i | 0 | −2.00000 | + | 2.00000i | 0 | 1.73205 | − | 1.73205i | ||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
7.b | odd | 2 | 1 | inner |
28.d | even | 2 | 1 | inner |
Twists
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
28.2.f.a | ✓ | 4 | 7.c | even | 3 | 1 | |
28.2.f.a | ✓ | 4 | 7.d | odd | 6 | 1 | |
28.2.f.a | ✓ | 4 | 28.f | even | 6 | 1 | |
28.2.f.a | ✓ | 4 | 28.g | odd | 6 | 1 | |
196.2.d.b | 4 | 1.a | even | 1 | 1 | trivial | |
196.2.d.b | 4 | 4.b | odd | 2 | 1 | inner | |
196.2.d.b | 4 | 7.b | odd | 2 | 1 | inner | |
196.2.d.b | 4 | 28.d | even | 2 | 1 | inner | |
196.2.f.a | 4 | 7.c | even | 3 | 1 | ||
196.2.f.a | 4 | 7.d | odd | 6 | 1 | ||
196.2.f.a | 4 | 28.f | even | 6 | 1 | ||
196.2.f.a | 4 | 28.g | odd | 6 | 1 | ||
252.2.bf.e | 4 | 21.g | even | 6 | 1 | ||
252.2.bf.e | 4 | 21.h | odd | 6 | 1 | ||
252.2.bf.e | 4 | 84.j | odd | 6 | 1 | ||
252.2.bf.e | 4 | 84.n | even | 6 | 1 | ||
448.2.p.d | 4 | 56.j | odd | 6 | 1 | ||
448.2.p.d | 4 | 56.k | odd | 6 | 1 | ||
448.2.p.d | 4 | 56.m | even | 6 | 1 | ||
448.2.p.d | 4 | 56.p | even | 6 | 1 | ||
700.2.p.a | 4 | 35.i | odd | 6 | 1 | ||
700.2.p.a | 4 | 35.j | even | 6 | 1 | ||
700.2.p.a | 4 | 140.p | odd | 6 | 1 | ||
700.2.p.a | 4 | 140.s | even | 6 | 1 | ||
700.2.t.a | 4 | 35.k | even | 12 | 1 | ||
700.2.t.a | 4 | 35.l | odd | 12 | 1 | ||
700.2.t.a | 4 | 140.w | even | 12 | 1 | ||
700.2.t.a | 4 | 140.x | odd | 12 | 1 | ||
700.2.t.b | 4 | 35.k | even | 12 | 1 | ||
700.2.t.b | 4 | 35.l | odd | 12 | 1 | ||
700.2.t.b | 4 | 140.w | even | 12 | 1 | ||
700.2.t.b | 4 | 140.x | odd | 12 | 1 | ||
1764.2.b.a | 4 | 3.b | odd | 2 | 1 | ||
1764.2.b.a | 4 | 12.b | even | 2 | 1 | ||
1764.2.b.a | 4 | 21.c | even | 2 | 1 | ||
1764.2.b.a | 4 | 84.h | odd | 2 | 1 | ||
3136.2.f.e | 4 | 8.b | even | 2 | 1 | ||
3136.2.f.e | 4 | 8.d | odd | 2 | 1 | ||
3136.2.f.e | 4 | 56.e | even | 2 | 1 | ||
3136.2.f.e | 4 | 56.h | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 3 \)
acting on \(S_{2}^{\mathrm{new}}(196, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} - 2 T + 2)^{2} \)
$3$
\( (T^{2} - 3)^{2} \)
$5$
\( (T^{2} + 3)^{2} \)
$7$
\( T^{4} \)
$11$
\( (T^{2} + 1)^{2} \)
$13$
\( (T^{2} + 12)^{2} \)
$17$
\( (T^{2} + 3)^{2} \)
$19$
\( (T^{2} - 27)^{2} \)
$23$
\( (T^{2} + 1)^{2} \)
$29$
\( (T - 4)^{4} \)
$31$
\( (T^{2} - 3)^{2} \)
$37$
\( (T - 3)^{4} \)
$41$
\( (T^{2} + 12)^{2} \)
$43$
\( (T^{2} + 4)^{2} \)
$47$
\( (T^{2} - 75)^{2} \)
$53$
\( (T + 1)^{4} \)
$59$
\( (T^{2} - 27)^{2} \)
$61$
\( (T^{2} + 27)^{2} \)
$67$
\( (T^{2} + 9)^{2} \)
$71$
\( (T^{2} + 196)^{2} \)
$73$
\( (T^{2} + 75)^{2} \)
$79$
\( (T^{2} + 81)^{2} \)
$83$
\( (T^{2} - 192)^{2} \)
$89$
\( (T^{2} + 243)^{2} \)
$97$
\( (T^{2} + 300)^{2} \)
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