Newspace parameters
Level: | \( N \) | \(=\) | \( 1040 = 2^{4} \cdot 5 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1040.dh (of order \(6\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(8.30444181021\) |
Analytic rank: | \(0\) |
Dimension: | \(12\) |
Relative dimension: | \(6\) over \(\Q(\zeta_{6})\) |
Coefficient field: | 12.0.513226913958144.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{12} - 6x^{10} + 28x^{8} - 46x^{6} + 58x^{4} - 8x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 2^{8} \) |
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 \(1,\beta_1,\ldots,\beta_{11}\) 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 \( x^{12} - 6x^{10} + 28x^{8} - 46x^{6} + 58x^{4} - 8x^{2} + 1 \) :
\(\beta_{1}\) | \(=\) | \( ( -9\nu^{11} + 42\nu^{9} - 196\nu^{7} + 87\nu^{5} - 12\nu^{3} - 732\nu ) / 394 \) |
\(\beta_{2}\) | \(=\) | \( ( -9\nu^{11} + 42\nu^{9} - 196\nu^{7} + 87\nu^{5} - 12\nu^{3} - 1520\nu ) / 394 \) |
\(\beta_{3}\) | \(=\) | \( ( 24\nu^{10} - 112\nu^{8} + 457\nu^{6} - 232\nu^{4} + 32\nu^{2} + 376 ) / 197 \) |
\(\beta_{4}\) | \(=\) | \( ( -56\nu^{10} + 327\nu^{8} - 1526\nu^{6} + 2380\nu^{4} - 3161\nu^{2} + 42 ) / 394 \) |
\(\beta_{5}\) | \(=\) | \( ( -96\nu^{10} + 645\nu^{8} - 3010\nu^{6} + 5656\nu^{4} - 6235\nu^{2} + 860 ) / 394 \) |
\(\beta_{6}\) | \(=\) | \( ( -103\nu^{11} + 612\nu^{9} - 2856\nu^{7} + 4673\nu^{5} - 5916\nu^{3} + 816\nu ) / 394 \) |
\(\beta_{7}\) | \(=\) | \( ( - 33 \nu^{11} + 103 \nu^{10} + 154 \nu^{9} - 612 \nu^{8} - 653 \nu^{7} + 2856 \nu^{6} + 319 \nu^{5} - 4673 \nu^{4} - 44 \nu^{3} + 5916 \nu^{2} - 1305 \nu - 816 ) / 394 \) |
\(\beta_{8}\) | \(=\) | \( ( - 33 \nu^{11} - 103 \nu^{10} + 154 \nu^{9} + 612 \nu^{8} - 653 \nu^{7} - 2856 \nu^{6} + 319 \nu^{5} + 4673 \nu^{4} - 44 \nu^{3} - 5916 \nu^{2} - 1305 \nu + 816 ) / 394 \) |
\(\beta_{9}\) | \(=\) | \( ( 212 \nu^{11} - 9 \nu^{10} - 1252 \nu^{9} + 42 \nu^{8} + 5777 \nu^{7} - 196 \nu^{6} - 9010 \nu^{5} + 87 \nu^{4} + 10658 \nu^{3} - 12 \nu^{2} - 159 \nu - 732 ) / 394 \) |
\(\beta_{10}\) | \(=\) | \( ( -112\nu^{11} + 654\nu^{9} - 3052\nu^{7} + 4760\nu^{5} - 6125\nu^{3} + 84\nu ) / 197 \) |
\(\beta_{11}\) | \(=\) | \( ( - 179 \nu^{11} - 112 \nu^{10} + 1098 \nu^{9} + 654 \nu^{8} - 5124 \nu^{7} - 3052 \nu^{6} + 8691 \nu^{5} + 4760 \nu^{4} - 10614 \nu^{3} - 5928 \nu^{2} + 1464 \nu + 84 ) / 394 \) |
\(\nu\) | \(=\) | \( ( -\beta_{2} + \beta_1 ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{11} + \beta_{9} + \beta_{8} - 4\beta_{4} ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( -\beta_{10} + 2\beta_{6} + 2\beta_1 \) |
\(\nu^{4}\) | \(=\) | \( ( 4\beta_{8} - 4\beta_{7} - \beta_{5} - 13\beta_{4} - 13 ) / 2 \) |
\(\nu^{5}\) | \(=\) | \( ( -\beta_{11} - 6\beta_{10} + \beta_{9} + \beta_{8} + 16\beta_{6} + 6\beta_{2} ) / 2 \) |
\(\nu^{6}\) | \(=\) | \( -8\beta_{11} - 8\beta_{9} - 8\beta_{7} - 3\beta_{3} - 24 \) |
\(\nu^{7}\) | \(=\) | \( ( 6\beta_{8} + 6\beta_{7} + 21\beta_{2} - 65\beta_1 ) / 2 \) |
\(\nu^{8}\) | \(=\) | \( ( -65\beta_{11} - 65\beta_{9} - 65\beta_{8} + 28\beta_{5} + 188\beta_{4} - 28\beta_{3} ) / 2 \) |
\(\nu^{9}\) | \(=\) | \( 14\beta_{11} + 40\beta_{10} - 14\beta_{9} + 14\beta_{7} - 133\beta_{6} - 133\beta_1 \) |
\(\nu^{10}\) | \(=\) | \( ( -266\beta_{8} + 266\beta_{7} + 121\beta_{5} + 757\beta_{4} + 757 ) / 2 \) |
\(\nu^{11}\) | \(=\) | \( ( 121\beta_{11} + 318\beta_{10} - 121\beta_{9} - 121\beta_{8} - 1092\beta_{6} - 318\beta_{2} ) / 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1040\mathbb{Z}\right)^\times\).
\(n\) | \(261\) | \(417\) | \(561\) | \(911\) |
\(\chi(n)\) | \(1\) | \(-1\) | \(-1 - \beta_{4}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
289.1 |
|
0 | −2.57937 | + | 1.48920i | 0 | 0.254102 | + | 2.22158i | 0 | −0.290769 | − | 0.167875i | 0 | 2.93543 | − | 5.08432i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
289.2 | 0 | −1.24744 | + | 0.720209i | 0 | 1.86081 | − | 1.23992i | 0 | −0.601232 | − | 0.347122i | 0 | −0.462598 | + | 0.801244i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
289.3 | 0 | −0.201864 | + | 0.116546i | 0 | −2.11491 | + | 0.726062i | 0 | −3.71537 | − | 2.14507i | 0 | −1.47283 | + | 2.55102i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
289.4 | 0 | 0.201864 | − | 0.116546i | 0 | −2.11491 | − | 0.726062i | 0 | 3.71537 | + | 2.14507i | 0 | −1.47283 | + | 2.55102i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
289.5 | 0 | 1.24744 | − | 0.720209i | 0 | 1.86081 | + | 1.23992i | 0 | 0.601232 | + | 0.347122i | 0 | −0.462598 | + | 0.801244i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
289.6 | 0 | 2.57937 | − | 1.48920i | 0 | 0.254102 | − | 2.22158i | 0 | 0.290769 | + | 0.167875i | 0 | 2.93543 | − | 5.08432i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
529.1 | 0 | −2.57937 | − | 1.48920i | 0 | 0.254102 | − | 2.22158i | 0 | −0.290769 | + | 0.167875i | 0 | 2.93543 | + | 5.08432i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
529.2 | 0 | −1.24744 | − | 0.720209i | 0 | 1.86081 | + | 1.23992i | 0 | −0.601232 | + | 0.347122i | 0 | −0.462598 | − | 0.801244i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
529.3 | 0 | −0.201864 | − | 0.116546i | 0 | −2.11491 | − | 0.726062i | 0 | −3.71537 | + | 2.14507i | 0 | −1.47283 | − | 2.55102i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
529.4 | 0 | 0.201864 | + | 0.116546i | 0 | −2.11491 | + | 0.726062i | 0 | 3.71537 | − | 2.14507i | 0 | −1.47283 | − | 2.55102i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
529.5 | 0 | 1.24744 | + | 0.720209i | 0 | 1.86081 | − | 1.23992i | 0 | 0.601232 | − | 0.347122i | 0 | −0.462598 | − | 0.801244i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
529.6 | 0 | 2.57937 | + | 1.48920i | 0 | 0.254102 | + | 2.22158i | 0 | 0.290769 | − | 0.167875i | 0 | 2.93543 | + | 5.08432i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
13.c | even | 3 | 1 | inner |
65.n | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1040.2.dh.c | 12 | |
4.b | odd | 2 | 1 | 260.2.ba.a | ✓ | 12 | |
5.b | even | 2 | 1 | inner | 1040.2.dh.c | 12 | |
12.b | even | 2 | 1 | 2340.2.de.a | 12 | ||
13.c | even | 3 | 1 | inner | 1040.2.dh.c | 12 | |
20.d | odd | 2 | 1 | 260.2.ba.a | ✓ | 12 | |
20.e | even | 4 | 2 | 1300.2.i.i | 12 | ||
52.i | odd | 6 | 1 | 3380.2.c.b | 6 | ||
52.j | odd | 6 | 1 | 260.2.ba.a | ✓ | 12 | |
52.j | odd | 6 | 1 | 3380.2.c.a | 6 | ||
52.l | even | 12 | 2 | 3380.2.d.c | 12 | ||
60.h | even | 2 | 1 | 2340.2.de.a | 12 | ||
65.n | even | 6 | 1 | inner | 1040.2.dh.c | 12 | |
156.p | even | 6 | 1 | 2340.2.de.a | 12 | ||
260.v | odd | 6 | 1 | 260.2.ba.a | ✓ | 12 | |
260.v | odd | 6 | 1 | 3380.2.c.a | 6 | ||
260.w | odd | 6 | 1 | 3380.2.c.b | 6 | ||
260.bc | even | 12 | 2 | 3380.2.d.c | 12 | ||
260.bj | even | 12 | 2 | 1300.2.i.i | 12 | ||
780.br | even | 6 | 1 | 2340.2.de.a | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
260.2.ba.a | ✓ | 12 | 4.b | odd | 2 | 1 | |
260.2.ba.a | ✓ | 12 | 20.d | odd | 2 | 1 | |
260.2.ba.a | ✓ | 12 | 52.j | odd | 6 | 1 | |
260.2.ba.a | ✓ | 12 | 260.v | odd | 6 | 1 | |
1040.2.dh.c | 12 | 1.a | even | 1 | 1 | trivial | |
1040.2.dh.c | 12 | 5.b | even | 2 | 1 | inner | |
1040.2.dh.c | 12 | 13.c | even | 3 | 1 | inner | |
1040.2.dh.c | 12 | 65.n | even | 6 | 1 | inner | |
1300.2.i.i | 12 | 20.e | even | 4 | 2 | ||
1300.2.i.i | 12 | 260.bj | even | 12 | 2 | ||
2340.2.de.a | 12 | 12.b | even | 2 | 1 | ||
2340.2.de.a | 12 | 60.h | even | 2 | 1 | ||
2340.2.de.a | 12 | 156.p | even | 6 | 1 | ||
2340.2.de.a | 12 | 780.br | even | 6 | 1 | ||
3380.2.c.a | 6 | 52.j | odd | 6 | 1 | ||
3380.2.c.a | 6 | 260.v | odd | 6 | 1 | ||
3380.2.c.b | 6 | 52.i | odd | 6 | 1 | ||
3380.2.c.b | 6 | 260.w | odd | 6 | 1 | ||
3380.2.d.c | 12 | 52.l | even | 12 | 2 | ||
3380.2.d.c | 12 | 260.bc | even | 12 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} - 11T_{3}^{10} + 102T_{3}^{8} - 207T_{3}^{6} + 350T_{3}^{4} - 19T_{3}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(1040, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{12} \)
$3$
\( T^{12} - 11 T^{10} + 102 T^{8} - 207 T^{6} + \cdots + 1 \)
$5$
\( (T^{6} - T^{4} + 8 T^{3} - 5 T^{2} + 125)^{2} \)
$7$
\( T^{12} - 19 T^{10} + 350 T^{8} - 207 T^{6} + \cdots + 1 \)
$11$
\( (T^{2} + T + 1)^{6} \)
$13$
\( T^{12} + 6 T^{10} - 57 T^{8} + \cdots + 4826809 \)
$17$
\( T^{12} - 51 T^{10} + 1918 T^{8} + \cdots + 4879681 \)
$19$
\( (T^{6} + 5 T^{5} + 42 T^{4} + 21 T^{3} + \cdots + 2809)^{2} \)
$23$
\( T^{12} - 59 T^{10} + 2550 T^{8} + \cdots + 5764801 \)
$29$
\( (T^{6} + 5 T^{5} + 42 T^{4} + 21 T^{3} + \cdots + 2809)^{2} \)
$31$
\( (T^{3} - 6 T^{2} - 4 T + 16)^{4} \)
$37$
\( T^{12} - 107 T^{10} + \cdots + 607573201 \)
$41$
\( (T^{6} - T^{5} + 78 T^{4} + 435 T^{3} + \cdots + 32041)^{2} \)
$43$
\( T^{12} - 99 T^{10} + \cdots + 163047361 \)
$47$
\( (T^{6} + 204 T^{4} + 9008 T^{2} + \cdots + 87616)^{2} \)
$53$
\( (T^{6} + 80 T^{4} + 2032 T^{2} + \cdots + 16384)^{2} \)
$59$
\( (T^{6} - T^{5} + 126 T^{4} - 93 T^{3} + \cdots + 11881)^{2} \)
$61$
\( (T^{6} - 11 T^{5} + 130 T^{4} + \cdots + 51529)^{2} \)
$67$
\( T^{12} - 299 T^{10} + \cdots + 123657019201 \)
$71$
\( (T^{6} - 7 T^{5} + 162 T^{4} + 1065 T^{3} + \cdots + 18769)^{2} \)
$73$
\( (T^{6} + 324 T^{4} + 29808 T^{2} + \cdots + 746496)^{2} \)
$79$
\( (T^{3} + 2 T^{2} - 108 T - 512)^{4} \)
$83$
\( (T^{6} + 128 T^{4} + 2784 T^{2} + \cdots + 3136)^{2} \)
$89$
\( (T^{6} - T^{5} + 126 T^{4} - 93 T^{3} + \cdots + 11881)^{2} \)
$97$
\( T^{12} - 299 T^{10} + \cdots + 4499860561 \)
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