Newspace parameters
Level: | \( N \) | \(=\) | \( 169 = 13^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 169.d (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(4.60491646769\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(i)\) |
Coefficient field: | \(\Q(i, \sqrt{22})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: |
\( x^{4} + 121 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{4} + 121 \)
:
\(\beta_{1}\) | \(=\) |
\( \nu \)
|
\(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 11 \)
|
\(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 11 \)
|
\(\nu\) | \(=\) |
\( \beta_1 \)
|
\(\nu^{2}\) | \(=\) |
\( 11\beta_{2} \)
|
\(\nu^{3}\) | \(=\) |
\( 11\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/169\mathbb{Z}\right)^\times\).
\(n\) | \(2\) |
\(\chi(n)\) | \(\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
70.1 |
|
−2.34521 | + | 2.34521i | −3.00000 | − | 7.00000i | 2.34521 | − | 2.34521i | 7.03562 | − | 7.03562i | 7.03562 | + | 7.03562i | 7.03562 | + | 7.03562i | 0 | 11.0000i | |||||||||||||||||||
70.2 | 2.34521 | − | 2.34521i | −3.00000 | − | 7.00000i | −2.34521 | + | 2.34521i | −7.03562 | + | 7.03562i | −7.03562 | − | 7.03562i | −7.03562 | − | 7.03562i | 0 | 11.0000i | ||||||||||||||||||||
99.1 | −2.34521 | − | 2.34521i | −3.00000 | 7.00000i | 2.34521 | + | 2.34521i | 7.03562 | + | 7.03562i | 7.03562 | − | 7.03562i | 7.03562 | − | 7.03562i | 0 | − | 11.0000i | ||||||||||||||||||||
99.2 | 2.34521 | + | 2.34521i | −3.00000 | 7.00000i | −2.34521 | − | 2.34521i | −7.03562 | − | 7.03562i | −7.03562 | + | 7.03562i | −7.03562 | + | 7.03562i | 0 | − | 11.0000i | ||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.b | even | 2 | 1 | inner |
13.d | odd | 4 | 2 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 169.3.d.b | ✓ | 4 |
13.b | even | 2 | 1 | inner | 169.3.d.b | ✓ | 4 |
13.c | even | 3 | 2 | 169.3.f.e | 8 | ||
13.d | odd | 4 | 2 | inner | 169.3.d.b | ✓ | 4 |
13.e | even | 6 | 2 | 169.3.f.e | 8 | ||
13.f | odd | 12 | 4 | 169.3.f.e | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
169.3.d.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
169.3.d.b | ✓ | 4 | 13.b | even | 2 | 1 | inner |
169.3.d.b | ✓ | 4 | 13.d | odd | 4 | 2 | inner |
169.3.f.e | 8 | 13.c | even | 3 | 2 | ||
169.3.f.e | 8 | 13.e | even | 6 | 2 | ||
169.3.f.e | 8 | 13.f | odd | 12 | 4 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 121 \)
acting on \(S_{3}^{\mathrm{new}}(169, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} + 121 \)
$3$
\( (T + 3)^{4} \)
$5$
\( T^{4} + 121 \)
$7$
\( T^{4} + 9801 \)
$11$
\( T^{4} + 1936 \)
$13$
\( T^{4} \)
$17$
\( (T^{2} + 9)^{2} \)
$19$
\( T^{4} + 156816 \)
$23$
\( (T^{2} + 144)^{2} \)
$29$
\( (T + 42)^{4} \)
$31$
\( T^{4} + 2509056 \)
$37$
\( T^{4} + 6125625 \)
$41$
\( T^{4} + 4648336 \)
$43$
\( (T^{2} + 2401)^{2} \)
$47$
\( T^{4} + 121 \)
$53$
\( (T + 24)^{4} \)
$59$
\( T^{4} + 7929856 \)
$61$
\( (T - 30)^{4} \)
$67$
\( T^{4} + 2509056 \)
$71$
\( T^{4} + 290521 \)
$73$
\( T^{4} + 2509056 \)
$79$
\( (T - 54)^{4} \)
$83$
\( T^{4} + 4648336 \)
$89$
\( T^{4} + 756250000 \)
$97$
\( T^{4} + 156816 \)
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