Newspace parameters
Level: | \( N \) | \(=\) | \( 819 = 3^{2} \cdot 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 819.j (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(6.53974792554\) |
Analytic rank: | \(1\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{5})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} - x^{3} + 2x^{2} + x + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 91) |
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 \( x^{4} - x^{3} + 2x^{2} + x + 1 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{3} + 1 ) / 2 \) |
\(\beta_{3}\) | \(=\) | \( ( -\nu^{3} + 2\nu^{2} - 2\nu - 1 ) / 2 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{3} + \beta_{2} + \beta_1 \) |
\(\nu^{3}\) | \(=\) | \( 2\beta_{2} - 1 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/819\mathbb{Z}\right)^\times\).
\(n\) | \(92\) | \(379\) | \(703\) |
\(\chi(n)\) | \(1\) | \(1\) | \(-1 - \beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
235.1 |
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−1.30902 | − | 2.26728i | 0 | −2.42705 | + | 4.20378i | 1.11803 | + | 1.93649i | 0 | −2.00000 | + | 1.73205i | 7.47214 | 0 | 2.92705 | − | 5.06980i | ||||||||||||||||||||
235.2 | −0.190983 | − | 0.330792i | 0 | 0.927051 | − | 1.60570i | −1.11803 | − | 1.93649i | 0 | −2.00000 | + | 1.73205i | −1.47214 | 0 | −0.427051 | + | 0.739674i | |||||||||||||||||||||
352.1 | −1.30902 | + | 2.26728i | 0 | −2.42705 | − | 4.20378i | 1.11803 | − | 1.93649i | 0 | −2.00000 | − | 1.73205i | 7.47214 | 0 | 2.92705 | + | 5.06980i | |||||||||||||||||||||
352.2 | −0.190983 | + | 0.330792i | 0 | 0.927051 | + | 1.60570i | −1.11803 | + | 1.93649i | 0 | −2.00000 | − | 1.73205i | −1.47214 | 0 | −0.427051 | − | 0.739674i | |||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 819.2.j.c | 4 | |
3.b | odd | 2 | 1 | 91.2.e.b | ✓ | 4 | |
7.c | even | 3 | 1 | inner | 819.2.j.c | 4 | |
7.c | even | 3 | 1 | 5733.2.a.v | 2 | ||
7.d | odd | 6 | 1 | 5733.2.a.w | 2 | ||
12.b | even | 2 | 1 | 1456.2.r.j | 4 | ||
21.c | even | 2 | 1 | 637.2.e.h | 4 | ||
21.g | even | 6 | 1 | 637.2.a.e | 2 | ||
21.g | even | 6 | 1 | 637.2.e.h | 4 | ||
21.h | odd | 6 | 1 | 91.2.e.b | ✓ | 4 | |
21.h | odd | 6 | 1 | 637.2.a.f | 2 | ||
39.d | odd | 2 | 1 | 1183.2.e.d | 4 | ||
84.n | even | 6 | 1 | 1456.2.r.j | 4 | ||
273.w | odd | 6 | 1 | 1183.2.e.d | 4 | ||
273.w | odd | 6 | 1 | 8281.2.a.z | 2 | ||
273.ba | even | 6 | 1 | 8281.2.a.ba | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
91.2.e.b | ✓ | 4 | 3.b | odd | 2 | 1 | |
91.2.e.b | ✓ | 4 | 21.h | odd | 6 | 1 | |
637.2.a.e | 2 | 21.g | even | 6 | 1 | ||
637.2.a.f | 2 | 21.h | odd | 6 | 1 | ||
637.2.e.h | 4 | 21.c | even | 2 | 1 | ||
637.2.e.h | 4 | 21.g | even | 6 | 1 | ||
819.2.j.c | 4 | 1.a | even | 1 | 1 | trivial | |
819.2.j.c | 4 | 7.c | even | 3 | 1 | inner | |
1183.2.e.d | 4 | 39.d | odd | 2 | 1 | ||
1183.2.e.d | 4 | 273.w | odd | 6 | 1 | ||
1456.2.r.j | 4 | 12.b | even | 2 | 1 | ||
1456.2.r.j | 4 | 84.n | even | 6 | 1 | ||
5733.2.a.v | 2 | 7.c | even | 3 | 1 | ||
5733.2.a.w | 2 | 7.d | odd | 6 | 1 | ||
8281.2.a.z | 2 | 273.w | odd | 6 | 1 | ||
8281.2.a.ba | 2 | 273.ba | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 3T_{2}^{3} + 8T_{2}^{2} + 3T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(819, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} + 3 T^{3} + 8 T^{2} + 3 T + 1 \)
$3$
\( T^{4} \)
$5$
\( T^{4} + 5T^{2} + 25 \)
$7$
\( (T^{2} + 4 T + 7)^{2} \)
$11$
\( (T^{2} + 3 T + 9)^{2} \)
$13$
\( (T + 1)^{4} \)
$17$
\( T^{4} + 6 T^{3} + 47 T^{2} - 66 T + 121 \)
$19$
\( (T^{2} + 3 T + 9)^{2} \)
$23$
\( T^{4} + 12 T^{3} + 113 T^{2} + \cdots + 961 \)
$29$
\( (T^{2} - 20)^{2} \)
$31$
\( (T^{2} + 5 T + 25)^{2} \)
$37$
\( T^{4} - 4 T^{3} + 57 T^{2} + \cdots + 1681 \)
$41$
\( (T^{2} - 20)^{2} \)
$43$
\( (T + 8)^{4} \)
$47$
\( T^{4} + 6 T^{3} + 47 T^{2} - 66 T + 121 \)
$53$
\( T^{4} + 6 T^{3} + 47 T^{2} - 66 T + 121 \)
$59$
\( T^{4} - 6 T^{3} + 47 T^{2} + 66 T + 121 \)
$61$
\( (T^{2} + 3 T + 9)^{2} \)
$67$
\( (T^{2} - 3 T + 9)^{2} \)
$71$
\( (T^{2} - 80)^{2} \)
$73$
\( T^{4} + 8 T^{3} + 93 T^{2} - 232 T + 841 \)
$79$
\( T^{4} + 8 T^{3} + 93 T^{2} - 232 T + 841 \)
$83$
\( T^{4} \)
$89$
\( T^{4} + 5T^{2} + 25 \)
$97$
\( (T^{2} + 8 T - 164)^{2} \)
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