Newspace parameters
Level: | \( N \) | \(=\) | \( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1300.ba (of order \(6\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(10.3805522628\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{6})\) |
Coefficient field: | 8.0.22581504.2 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: |
\( x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 \)
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Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{4} \) |
Twist minimal: | no (minimal twist has level 260) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\( x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 \)
:
\(\beta_{1}\) | \(=\) |
\( ( \nu^{7} - 2\nu^{6} + \nu^{5} + 4\nu^{4} - 3\nu^{3} - 2\nu^{2} + 8\nu - 8 ) / 4 \)
|
\(\beta_{2}\) | \(=\) |
\( ( \nu^{7} - 3\nu^{6} + 3\nu^{5} + 3\nu^{4} - 7\nu^{3} - 3\nu^{2} + 14\nu - 16 ) / 4 \)
|
\(\beta_{3}\) | \(=\) |
\( ( 3\nu^{7} - 6\nu^{6} - \nu^{5} + 12\nu^{4} - 5\nu^{3} - 22\nu^{2} + 36\nu - 8 ) / 8 \)
|
\(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} + 3\nu^{6} - 3\nu^{5} - 3\nu^{4} + 7\nu^{3} + 3\nu^{2} - 22\nu + 20 ) / 4 \)
|
\(\beta_{5}\) | \(=\) |
\( ( -3\nu^{7} + 8\nu^{6} - 3\nu^{5} - 10\nu^{4} + 13\nu^{3} + 8\nu^{2} - 32\nu + 32 ) / 8 \)
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\(\beta_{6}\) | \(=\) |
\( ( 3\nu^{7} - 7\nu^{6} + 3\nu^{5} + 11\nu^{4} - 15\nu^{3} - 11\nu^{2} + 40\nu - 32 ) / 4 \)
|
\(\beta_{7}\) | \(=\) |
\( ( 7\nu^{7} - 20\nu^{6} + 11\nu^{5} + 30\nu^{4} - 45\nu^{3} - 28\nu^{2} + 116\nu - 88 ) / 4 \)
|
\(\nu\) | \(=\) |
\( ( -\beta_{4} - \beta_{2} + 1 ) / 2 \)
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\(\nu^{2}\) | \(=\) |
\( ( -\beta_{5} - \beta_{4} - \beta_{3} - 2\beta_{2} + \beta _1 + 2 ) / 2 \)
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\(\nu^{3}\) | \(=\) |
\( ( -2\beta_{6} - \beta_{5} - 2\beta_{4} + \beta_{3} - \beta_{2} + 2\beta _1 - 1 ) / 2 \)
|
\(\nu^{4}\) | \(=\) |
\( ( \beta_{7} - 2\beta_{6} + 2\beta_{5} + \beta_{4} - \beta_{2} + 4\beta _1 - 3 ) / 2 \)
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\(\nu^{5}\) | \(=\) |
\( ( -2\beta_{6} + 3\beta_{5} - \beta_{4} + \beta_{3} + 4\beta_{2} + 4\beta _1 + 2 ) / 2 \)
|
\(\nu^{6}\) | \(=\) |
\( ( -\beta_{7} + 6\beta_{6} + 9\beta_{5} - \beta_{3} + \beta_{2} + 3\beta _1 - 1 ) / 2 \)
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\(\nu^{7}\) | \(=\) |
\( ( -6\beta_{7} + 16\beta_{6} + 2\beta_{5} - 3\beta_{4} - 2\beta_{3} + 3\beta_{2} + 2\beta _1 + 17 ) / 2 \)
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Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1300\mathbb{Z}\right)^\times\).
\(n\) | \(301\) | \(651\) | \(677\) |
\(\chi(n)\) | \(-\beta_{6}\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
49.1 |
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0 | −1.38581 | + | 0.800098i | 0 | 0 | 0 | −2.16612 | + | 3.75184i | 0 | −0.219687 | + | 0.380509i | 0 | ||||||||||||||||||||||||||||||||||||
49.2 | 0 | −0.0820885 | + | 0.0473938i | 0 | 0 | 0 | 0.413419 | − | 0.716063i | 0 | −1.49551 | + | 2.59030i | 0 | |||||||||||||||||||||||||||||||||||||
49.3 | 0 | 2.01978 | − | 1.16612i | 0 | 0 | 0 | −0.199902 | + | 0.346241i | 0 | 1.21969 | − | 2.11256i | 0 | |||||||||||||||||||||||||||||||||||||
49.4 | 0 | 2.44811 | − | 1.41342i | 0 | 0 | 0 | −1.04739 | + | 1.81414i | 0 | 2.49551 | − | 4.32235i | 0 | |||||||||||||||||||||||||||||||||||||
849.1 | 0 | −1.38581 | − | 0.800098i | 0 | 0 | 0 | −2.16612 | − | 3.75184i | 0 | −0.219687 | − | 0.380509i | 0 | |||||||||||||||||||||||||||||||||||||
849.2 | 0 | −0.0820885 | − | 0.0473938i | 0 | 0 | 0 | 0.413419 | + | 0.716063i | 0 | −1.49551 | − | 2.59030i | 0 | |||||||||||||||||||||||||||||||||||||
849.3 | 0 | 2.01978 | + | 1.16612i | 0 | 0 | 0 | −0.199902 | − | 0.346241i | 0 | 1.21969 | + | 2.11256i | 0 | |||||||||||||||||||||||||||||||||||||
849.4 | 0 | 2.44811 | + | 1.41342i | 0 | 0 | 0 | −1.04739 | − | 1.81414i | 0 | 2.49551 | + | 4.32235i | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
65.l | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1300.2.ba.c | 8 | |
5.b | even | 2 | 1 | 1300.2.ba.b | 8 | ||
5.c | odd | 4 | 1 | 260.2.x.a | ✓ | 8 | |
5.c | odd | 4 | 1 | 1300.2.y.b | 8 | ||
13.e | even | 6 | 1 | 1300.2.ba.b | 8 | ||
15.e | even | 4 | 1 | 2340.2.dj.d | 8 | ||
20.e | even | 4 | 1 | 1040.2.da.c | 8 | ||
65.l | even | 6 | 1 | inner | 1300.2.ba.c | 8 | |
65.o | even | 12 | 1 | 3380.2.a.p | 4 | ||
65.q | odd | 12 | 1 | 3380.2.f.i | 8 | ||
65.r | odd | 12 | 1 | 260.2.x.a | ✓ | 8 | |
65.r | odd | 12 | 1 | 1300.2.y.b | 8 | ||
65.r | odd | 12 | 1 | 3380.2.f.i | 8 | ||
65.t | even | 12 | 1 | 3380.2.a.q | 4 | ||
195.bf | even | 12 | 1 | 2340.2.dj.d | 8 | ||
260.bg | even | 12 | 1 | 1040.2.da.c | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
260.2.x.a | ✓ | 8 | 5.c | odd | 4 | 1 | |
260.2.x.a | ✓ | 8 | 65.r | odd | 12 | 1 | |
1040.2.da.c | 8 | 20.e | even | 4 | 1 | ||
1040.2.da.c | 8 | 260.bg | even | 12 | 1 | ||
1300.2.y.b | 8 | 5.c | odd | 4 | 1 | ||
1300.2.y.b | 8 | 65.r | odd | 12 | 1 | ||
1300.2.ba.b | 8 | 5.b | even | 2 | 1 | ||
1300.2.ba.b | 8 | 13.e | even | 6 | 1 | ||
1300.2.ba.c | 8 | 1.a | even | 1 | 1 | trivial | |
1300.2.ba.c | 8 | 65.l | even | 6 | 1 | inner | |
2340.2.dj.d | 8 | 15.e | even | 4 | 1 | ||
2340.2.dj.d | 8 | 195.bf | even | 12 | 1 | ||
3380.2.a.p | 4 | 65.o | even | 12 | 1 | ||
3380.2.a.q | 4 | 65.t | even | 12 | 1 | ||
3380.2.f.i | 8 | 65.q | odd | 12 | 1 | ||
3380.2.f.i | 8 | 65.r | odd | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} - 6T_{3}^{7} + 10T_{3}^{6} + 12T_{3}^{5} - 33T_{3}^{4} - 36T_{3}^{3} + 106T_{3}^{2} + 18T_{3} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(1300, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( T^{8} - 6 T^{7} + 10 T^{6} + 12 T^{5} + \cdots + 1 \)
$5$
\( T^{8} \)
$7$
\( T^{8} + 6 T^{7} + 30 T^{6} + 48 T^{5} + \cdots + 9 \)
$11$
\( (T^{2} - 3 T + 3)^{4} \)
$13$
\( T^{8} + 16 T^{6} - 96 T^{5} + \cdots + 28561 \)
$17$
\( T^{8} - 18 T^{7} + 126 T^{6} + \cdots + 729 \)
$19$
\( T^{8} - 30 T^{6} + 867 T^{4} + \cdots + 1089 \)
$23$
\( T^{8} - 6 T^{7} - 18 T^{6} + 180 T^{5} + \cdots + 9 \)
$29$
\( T^{8} + 42 T^{6} + 192 T^{5} + \cdots + 1521 \)
$31$
\( (T^{4} + 96 T^{2} + 2112)^{2} \)
$37$
\( T^{8} + 18 T^{7} + 258 T^{6} + \cdots + 42849 \)
$41$
\( (T^{4} - 6 T^{3} - 21 T^{2} + 198 T + 1089)^{2} \)
$43$
\( T^{8} + 18 T^{7} + 46 T^{6} + \cdots + 5031049 \)
$47$
\( (T^{2} - 12)^{4} \)
$53$
\( T^{8} + 456 T^{6} + 69168 T^{4} + \cdots + 389376 \)
$59$
\( T^{8} - 24 T^{7} + 174 T^{6} + \cdots + 558009 \)
$61$
\( T^{8} + 4 T^{7} + 118 T^{6} + \cdots + 942841 \)
$67$
\( T^{8} - 18 T^{7} + 330 T^{6} + \cdots + 1083681 \)
$71$
\( T^{8} + 36 T^{7} + 414 T^{6} + \cdots + 45198729 \)
$73$
\( (T^{4} + 24 T^{3} + 156 T^{2} - 1584)^{2} \)
$79$
\( (T^{4} - 8 T^{3} - 180 T^{2} + 1504 T - 368)^{2} \)
$83$
\( (T^{4} + 36 T^{3} + 408 T^{2} + 1440 T + 576)^{2} \)
$89$
\( T^{8} - 24 T^{7} + 174 T^{6} + \cdots + 13689 \)
$97$
\( T^{8} - 6 T^{7} + 222 T^{6} + \cdots + 12981609 \)
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