Newspace parameters
Level: | \( N \) | \(=\) | \( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1300.m (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(10.3805522628\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(i)\) |
Coefficient field: | \(\Q(i, \sqrt{5})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} + 3x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
Coefficient ring index: | \( 2 \) |
Twist minimal: | no (minimal twist has level 260) |
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 \( x^{4} + 3x^{2} + 1 \) :
\(\beta_{1}\) | \(=\) | \( \nu^{2} + \nu + 1 \) |
\(\beta_{2}\) | \(=\) | \( \nu^{3} + 2\nu \) |
\(\beta_{3}\) | \(=\) | \( -\nu^{2} + \nu - 1 \) |
\(\nu\) | \(=\) | \( ( \beta_{3} + \beta_1 ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( ( -\beta_{3} + \beta _1 - 2 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( -\beta_{3} + \beta_{2} - \beta_1 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1300\mathbb{Z}\right)^\times\).
\(n\) | \(301\) | \(651\) | \(677\) |
\(\chi(n)\) | \(\beta_{2}\) | \(1\) | \(-\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
57.1 |
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0 | −1.61803 | + | 1.61803i | 0 | 0 | 0 | 0 | 0 | − | 2.23607i | 0 | |||||||||||||||||||||||||||
57.2 | 0 | 0.618034 | − | 0.618034i | 0 | 0 | 0 | 0 | 0 | 2.23607i | 0 | |||||||||||||||||||||||||||||
593.1 | 0 | −1.61803 | − | 1.61803i | 0 | 0 | 0 | 0 | 0 | 2.23607i | 0 | |||||||||||||||||||||||||||||
593.2 | 0 | 0.618034 | + | 0.618034i | 0 | 0 | 0 | 0 | 0 | − | 2.23607i | 0 | ||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
65.k | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1300.2.m.b | 4 | |
5.b | even | 2 | 1 | 260.2.m.b | ✓ | 4 | |
5.c | odd | 4 | 1 | 260.2.r.b | yes | 4 | |
5.c | odd | 4 | 1 | 1300.2.r.b | 4 | ||
13.d | odd | 4 | 1 | 1300.2.r.b | 4 | ||
15.d | odd | 2 | 1 | 2340.2.u.f | 4 | ||
15.e | even | 4 | 1 | 2340.2.bp.e | 4 | ||
20.d | odd | 2 | 1 | 1040.2.bg.j | 4 | ||
20.e | even | 4 | 1 | 1040.2.cd.j | 4 | ||
65.f | even | 4 | 1 | 260.2.m.b | ✓ | 4 | |
65.g | odd | 4 | 1 | 260.2.r.b | yes | 4 | |
65.k | even | 4 | 1 | inner | 1300.2.m.b | 4 | |
195.n | even | 4 | 1 | 2340.2.bp.e | 4 | ||
195.u | odd | 4 | 1 | 2340.2.u.f | 4 | ||
260.l | odd | 4 | 1 | 1040.2.bg.j | 4 | ||
260.u | even | 4 | 1 | 1040.2.cd.j | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
260.2.m.b | ✓ | 4 | 5.b | even | 2 | 1 | |
260.2.m.b | ✓ | 4 | 65.f | even | 4 | 1 | |
260.2.r.b | yes | 4 | 5.c | odd | 4 | 1 | |
260.2.r.b | yes | 4 | 65.g | odd | 4 | 1 | |
1040.2.bg.j | 4 | 20.d | odd | 2 | 1 | ||
1040.2.bg.j | 4 | 260.l | odd | 4 | 1 | ||
1040.2.cd.j | 4 | 20.e | even | 4 | 1 | ||
1040.2.cd.j | 4 | 260.u | even | 4 | 1 | ||
1300.2.m.b | 4 | 1.a | even | 1 | 1 | trivial | |
1300.2.m.b | 4 | 65.k | even | 4 | 1 | inner | |
1300.2.r.b | 4 | 5.c | odd | 4 | 1 | ||
1300.2.r.b | 4 | 13.d | odd | 4 | 1 | ||
2340.2.u.f | 4 | 15.d | odd | 2 | 1 | ||
2340.2.u.f | 4 | 195.u | odd | 4 | 1 | ||
2340.2.bp.e | 4 | 15.e | even | 4 | 1 | ||
2340.2.bp.e | 4 | 195.n | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 2T_{3}^{3} + 2T_{3}^{2} - 4T_{3} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(1300, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( T^{4} + 2 T^{3} + 2 T^{2} - 4 T + 4 \)
$5$
\( T^{4} \)
$7$
\( T^{4} \)
$11$
\( T^{4} + 6 T^{3} + 18 T^{2} + 12 T + 4 \)
$13$
\( (T^{2} - 4 T + 13)^{2} \)
$17$
\( T^{4} + 100 \)
$19$
\( T^{4} - 10 T^{3} + 50 T^{2} + \cdots + 100 \)
$23$
\( T^{4} + 18 T^{3} + 162 T^{2} + \cdots + 1444 \)
$29$
\( T^{4} + 28T^{2} + 16 \)
$31$
\( T^{4} - 10 T^{3} + 50 T^{2} + \cdots + 100 \)
$37$
\( T^{4} + 108T^{2} + 1296 \)
$41$
\( T^{4} - 24 T^{3} + 288 T^{2} + \cdots + 3844 \)
$43$
\( T^{4} - 2 T^{3} + 2 T^{2} + 44 T + 484 \)
$47$
\( (T^{2} + 80)^{2} \)
$53$
\( T^{4} - 12 T^{3} + 72 T^{2} + \cdots + 484 \)
$59$
\( T^{4} + 18 T^{3} + 162 T^{2} + \cdots + 1444 \)
$61$
\( (T^{2} + 2 T - 44)^{2} \)
$67$
\( (T^{2} - 14 T + 4)^{2} \)
$71$
\( T^{4} - 6 T^{3} + 18 T^{2} - 12 T + 4 \)
$73$
\( (T^{2} + 10 T - 20)^{2} \)
$79$
\( T^{4} + 108T^{2} + 1296 \)
$83$
\( T^{4} + 112T^{2} + 256 \)
$89$
\( T^{4} + 100 \)
$97$
\( (T^{2} + 4 T - 176)^{2} \)
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