Newspace parameters
Level: | \( N \) | \(=\) | \( 169 = 13^{2} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 169.c (of order \(3\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(9.97132279097\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{17})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} - x^{3} + 5x^{2} + 4x + 16 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 13) |
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 \( x^{4} - x^{3} + 5x^{2} + 4x + 16 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( -\nu^{3} + 5\nu^{2} - 5\nu + 16 ) / 20 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{3} + 4 ) / 5 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{3} + 4\beta_{2} + \beta _1 - 4 \) |
\(\nu^{3}\) | \(=\) | \( 5\beta_{3} - 4 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/169\mathbb{Z}\right)^\times\).
\(n\) | \(2\) |
\(\chi(n)\) | \(-1 + \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
22.1 |
|
−0.780776 | + | 1.35234i | −4.34233 | + | 7.52113i | 2.78078 | + | 4.81645i | 3.56155 | −6.78078 | − | 11.7446i | −13.5885 | − | 23.5360i | −21.1771 | −24.2116 | − | 41.9358i | −2.78078 | + | 4.81645i | ||||||||||||||||
22.2 | 1.28078 | − | 2.21837i | 1.84233 | − | 3.19101i | 0.719224 | + | 1.24573i | −0.561553 | −4.71922 | − | 8.17394i | 9.08854 | + | 15.7418i | 24.1771 | 6.71165 | + | 11.6249i | −0.719224 | + | 1.24573i | |||||||||||||||||
146.1 | −0.780776 | − | 1.35234i | −4.34233 | − | 7.52113i | 2.78078 | − | 4.81645i | 3.56155 | −6.78078 | + | 11.7446i | −13.5885 | + | 23.5360i | −21.1771 | −24.2116 | + | 41.9358i | −2.78078 | − | 4.81645i | |||||||||||||||||
146.2 | 1.28078 | + | 2.21837i | 1.84233 | + | 3.19101i | 0.719224 | − | 1.24573i | −0.561553 | −4.71922 | + | 8.17394i | 9.08854 | − | 15.7418i | 24.1771 | 6.71165 | − | 11.6249i | −0.719224 | − | 1.24573i | |||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 169.4.c.j | 4 | |
13.b | even | 2 | 1 | 169.4.c.g | 4 | ||
13.c | even | 3 | 1 | 169.4.a.g | 2 | ||
13.c | even | 3 | 1 | inner | 169.4.c.j | 4 | |
13.d | odd | 4 | 2 | 169.4.e.f | 8 | ||
13.e | even | 6 | 1 | 13.4.a.b | ✓ | 2 | |
13.e | even | 6 | 1 | 169.4.c.g | 4 | ||
13.f | odd | 12 | 2 | 169.4.b.f | 4 | ||
13.f | odd | 12 | 2 | 169.4.e.f | 8 | ||
39.h | odd | 6 | 1 | 117.4.a.d | 2 | ||
39.i | odd | 6 | 1 | 1521.4.a.r | 2 | ||
52.i | odd | 6 | 1 | 208.4.a.h | 2 | ||
65.l | even | 6 | 1 | 325.4.a.f | 2 | ||
65.r | odd | 12 | 2 | 325.4.b.e | 4 | ||
91.t | odd | 6 | 1 | 637.4.a.b | 2 | ||
104.p | odd | 6 | 1 | 832.4.a.z | 2 | ||
104.s | even | 6 | 1 | 832.4.a.s | 2 | ||
143.i | odd | 6 | 1 | 1573.4.a.b | 2 | ||
156.r | even | 6 | 1 | 1872.4.a.bb | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
13.4.a.b | ✓ | 2 | 13.e | even | 6 | 1 | |
117.4.a.d | 2 | 39.h | odd | 6 | 1 | ||
169.4.a.g | 2 | 13.c | even | 3 | 1 | ||
169.4.b.f | 4 | 13.f | odd | 12 | 2 | ||
169.4.c.g | 4 | 13.b | even | 2 | 1 | ||
169.4.c.g | 4 | 13.e | even | 6 | 1 | ||
169.4.c.j | 4 | 1.a | even | 1 | 1 | trivial | |
169.4.c.j | 4 | 13.c | even | 3 | 1 | inner | |
169.4.e.f | 8 | 13.d | odd | 4 | 2 | ||
169.4.e.f | 8 | 13.f | odd | 12 | 2 | ||
208.4.a.h | 2 | 52.i | odd | 6 | 1 | ||
325.4.a.f | 2 | 65.l | even | 6 | 1 | ||
325.4.b.e | 4 | 65.r | odd | 12 | 2 | ||
637.4.a.b | 2 | 91.t | odd | 6 | 1 | ||
832.4.a.s | 2 | 104.s | even | 6 | 1 | ||
832.4.a.z | 2 | 104.p | odd | 6 | 1 | ||
1521.4.a.r | 2 | 39.i | odd | 6 | 1 | ||
1573.4.a.b | 2 | 143.i | odd | 6 | 1 | ||
1872.4.a.bb | 2 | 156.r | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - T_{2}^{3} + 5T_{2}^{2} + 4T_{2} + 16 \)
acting on \(S_{4}^{\mathrm{new}}(169, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - T^{3} + 5 T^{2} + 4 T + 16 \)
$3$
\( T^{4} + 5 T^{3} + 57 T^{2} + \cdots + 1024 \)
$5$
\( (T^{2} - 3 T - 2)^{2} \)
$7$
\( T^{4} + 9 T^{3} + 575 T^{2} + \cdots + 244036 \)
$11$
\( T^{4} - 80 T^{3} + 5412 T^{2} + \cdots + 976144 \)
$13$
\( T^{4} \)
$17$
\( T^{4} + 19 T^{3} + 1499 T^{2} + \cdots + 1295044 \)
$19$
\( T^{4} + 84 T^{3} + 9644 T^{2} + \cdots + 6697744 \)
$23$
\( T^{4} + 196 T^{3} + \cdots + 80856064 \)
$29$
\( T^{4} - 44 T^{3} + \cdots + 1496451856 \)
$31$
\( (T^{2} - 86 T - 3064)^{2} \)
$37$
\( T^{4} - 209 T^{3} + \cdots + 116942596 \)
$41$
\( T^{4} + 230 T^{3} + \cdots + 124724224 \)
$43$
\( T^{4} + 287 T^{3} + \cdots + 4397811856 \)
$47$
\( (T^{2} + 435 T - 14918)^{2} \)
$53$
\( (T^{2} + 118 T - 344)^{2} \)
$59$
\( T^{4} + 368 T^{3} + \cdots + 991746064 \)
$61$
\( T^{4} - 1058 T^{3} + \cdots + 15981005056 \)
$67$
\( T^{4} - 68 T^{3} + \cdots + 51799939216 \)
$71$
\( T^{4} + 131 T^{3} + \cdots + 49503580036 \)
$73$
\( (T^{2} + 456 T - 235316)^{2} \)
$79$
\( (T^{2} + 1008 T + 247216)^{2} \)
$83$
\( (T^{2} + 1958 T + 817664)^{2} \)
$89$
\( T^{4} + 720 T^{3} + \cdots + 260316284944 \)
$97$
\( T^{4} + 928 T^{3} + \cdots + 776999938576 \)
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