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: | \(8\) |
Coefficient field: | 8.0.1212153856.10 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} - 4x^{6} + 10x^{4} - 16x^{2} + 16 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) 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 \( x^{8} - 4x^{6} + 10x^{4} - 16x^{2} + 16 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{7} + 2\nu^{3} ) / 8 \) |
\(\beta_{3}\) | \(=\) | \( ( -\nu^{6} + 2\nu^{4} - 2\nu^{2} ) / 4 \) |
\(\beta_{4}\) | \(=\) | \( ( -\nu^{7} + 4\nu^{5} - 2\nu^{3} ) / 8 \) |
\(\beta_{5}\) | \(=\) | \( ( -\nu^{4} + 2\nu^{2} - 2 ) / 2 \) |
\(\beta_{6}\) | \(=\) | \( ( \nu^{6} - 2\nu^{4} + 6\nu^{2} - 4 ) / 4 \) |
\(\beta_{7}\) | \(=\) | \( ( -\nu^{7} + 4\nu^{5} - 10\nu^{3} + 8\nu ) / 8 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{6} + \beta_{3} + 1 \) |
\(\nu^{3}\) | \(=\) | \( -\beta_{7} + \beta_{4} + \beta_1 \) |
\(\nu^{4}\) | \(=\) | \( 2\beta_{6} - 2\beta_{5} + 2\beta_{3} \) |
\(\nu^{5}\) | \(=\) | \( 2\beta_{4} + 2\beta_{2} \) |
\(\nu^{6}\) | \(=\) | \( 2\beta_{6} - 4\beta_{5} - 2\beta_{3} - 2 \) |
\(\nu^{7}\) | \(=\) | \( 2\beta_{7} - 2\beta_{4} + 8\beta_{2} - 2\beta_1 \) |
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 |
|
−1.20711 | − | 0.736813i | −2.72291 | 0.914214 | + | 1.77882i | 1.08239i | 3.28684 | + | 2.00627i | 0 | 0.207107 | − | 2.82083i | 4.41421 | 0.797521 | − | 1.30656i | ||||||||||||||||||||||||||||||||
195.2 | −1.20711 | − | 0.736813i | 2.72291 | 0.914214 | + | 1.77882i | − | 1.08239i | −3.28684 | − | 2.00627i | 0 | 0.207107 | − | 2.82083i | 4.41421 | −0.797521 | + | 1.30656i | ||||||||||||||||||||||||||||||||
195.3 | −1.20711 | + | 0.736813i | −2.72291 | 0.914214 | − | 1.77882i | − | 1.08239i | 3.28684 | − | 2.00627i | 0 | 0.207107 | + | 2.82083i | 4.41421 | 0.797521 | + | 1.30656i | ||||||||||||||||||||||||||||||||
195.4 | −1.20711 | + | 0.736813i | 2.72291 | 0.914214 | − | 1.77882i | 1.08239i | −3.28684 | + | 2.00627i | 0 | 0.207107 | + | 2.82083i | 4.41421 | −0.797521 | − | 1.30656i | |||||||||||||||||||||||||||||||||
195.5 | 0.207107 | − | 1.39897i | −2.14144 | −1.91421 | − | 0.579471i | 2.61313i | −0.443508 | + | 2.99581i | 0 | −1.20711 | + | 2.55791i | 1.58579 | 3.65568 | + | 0.541196i | |||||||||||||||||||||||||||||||||
195.6 | 0.207107 | − | 1.39897i | 2.14144 | −1.91421 | − | 0.579471i | − | 2.61313i | 0.443508 | − | 2.99581i | 0 | −1.20711 | + | 2.55791i | 1.58579 | −3.65568 | − | 0.541196i | ||||||||||||||||||||||||||||||||
195.7 | 0.207107 | + | 1.39897i | −2.14144 | −1.91421 | + | 0.579471i | − | 2.61313i | −0.443508 | − | 2.99581i | 0 | −1.20711 | − | 2.55791i | 1.58579 | 3.65568 | − | 0.541196i | ||||||||||||||||||||||||||||||||
195.8 | 0.207107 | + | 1.39897i | 2.14144 | −1.91421 | + | 0.579471i | 2.61313i | 0.443508 | + | 2.99581i | 0 | −1.20711 | − | 2.55791i | 1.58579 | −3.65568 | + | 0.541196i | |||||||||||||||||||||||||||||||||
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 twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 196.2.d.c | ✓ | 8 |
3.b | odd | 2 | 1 | 1764.2.b.k | 8 | ||
4.b | odd | 2 | 1 | inner | 196.2.d.c | ✓ | 8 |
7.b | odd | 2 | 1 | inner | 196.2.d.c | ✓ | 8 |
7.c | even | 3 | 2 | 196.2.f.d | 16 | ||
7.d | odd | 6 | 2 | 196.2.f.d | 16 | ||
8.b | even | 2 | 1 | 3136.2.f.i | 8 | ||
8.d | odd | 2 | 1 | 3136.2.f.i | 8 | ||
12.b | even | 2 | 1 | 1764.2.b.k | 8 | ||
21.c | even | 2 | 1 | 1764.2.b.k | 8 | ||
28.d | even | 2 | 1 | inner | 196.2.d.c | ✓ | 8 |
28.f | even | 6 | 2 | 196.2.f.d | 16 | ||
28.g | odd | 6 | 2 | 196.2.f.d | 16 | ||
56.e | even | 2 | 1 | 3136.2.f.i | 8 | ||
56.h | odd | 2 | 1 | 3136.2.f.i | 8 | ||
84.h | odd | 2 | 1 | 1764.2.b.k | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
196.2.d.c | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
196.2.d.c | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
196.2.d.c | ✓ | 8 | 7.b | odd | 2 | 1 | inner |
196.2.d.c | ✓ | 8 | 28.d | even | 2 | 1 | inner |
196.2.f.d | 16 | 7.c | even | 3 | 2 | ||
196.2.f.d | 16 | 7.d | odd | 6 | 2 | ||
196.2.f.d | 16 | 28.f | even | 6 | 2 | ||
196.2.f.d | 16 | 28.g | odd | 6 | 2 | ||
1764.2.b.k | 8 | 3.b | odd | 2 | 1 | ||
1764.2.b.k | 8 | 12.b | even | 2 | 1 | ||
1764.2.b.k | 8 | 21.c | even | 2 | 1 | ||
1764.2.b.k | 8 | 84.h | odd | 2 | 1 | ||
3136.2.f.i | 8 | 8.b | even | 2 | 1 | ||
3136.2.f.i | 8 | 8.d | odd | 2 | 1 | ||
3136.2.f.i | 8 | 56.e | even | 2 | 1 | ||
3136.2.f.i | 8 | 56.h | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 12T_{3}^{2} + 34 \)
acting on \(S_{2}^{\mathrm{new}}(196, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{4} + 2 T^{3} + 3 T^{2} + 4 T + 4)^{2} \)
$3$
\( (T^{4} - 12 T^{2} + 34)^{2} \)
$5$
\( (T^{4} + 8 T^{2} + 8)^{2} \)
$7$
\( T^{8} \)
$11$
\( (T^{4} + 20 T^{2} + 68)^{2} \)
$13$
\( (T^{4} + 8 T^{2} + 8)^{2} \)
$17$
\( (T^{4} + 20 T^{2} + 2)^{2} \)
$19$
\( (T^{4} - 28 T^{2} + 34)^{2} \)
$23$
\( (T^{4} + 56 T^{2} + 272)^{2} \)
$29$
\( (T^{2} + 8 T + 8)^{4} \)
$31$
\( (T^{4} - 96 T^{2} + 2176)^{2} \)
$37$
\( (T + 4)^{8} \)
$41$
\( (T^{4} + 36 T^{2} + 162)^{2} \)
$43$
\( (T^{4} + 116 T^{2} + 3332)^{2} \)
$47$
\( (T^{4} - 48 T^{2} + 544)^{2} \)
$53$
\( (T^{2} - 4 T - 68)^{4} \)
$59$
\( (T^{4} - 204 T^{2} + 9826)^{2} \)
$61$
\( (T^{4} + 200 T^{2} + 5000)^{2} \)
$67$
\( (T^{4} + 112 T^{2} + 1088)^{2} \)
$71$
\( T^{8} \)
$73$
\( (T^{4} + 260 T^{2} + 12482)^{2} \)
$79$
\( (T^{4} + 80 T^{2} + 1088)^{2} \)
$83$
\( (T^{4} - 108 T^{2} + 1666)^{2} \)
$89$
\( (T^{4} + 52 T^{2} + 578)^{2} \)
$97$
\( (T^{4} + 100 T^{2} + 1250)^{2} \)
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