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: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{12} - 8x^{10} + 54x^{8} - 78x^{6} + 92x^{4} - 10x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
Coefficient ring index: | \( 2^{2} \) |
Twist minimal: | no (minimal twist has level 65) |
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 \( x^{12} - 8x^{10} + 54x^{8} - 78x^{6} + 92x^{4} - 10x^{2} + 1 \) :
\(\beta_{1}\) | \(=\) | \( ( 108\nu^{11} - 729\nu^{9} + 4768\nu^{7} - 1242\nu^{5} + 135\nu^{3} + 9934\nu ) / 1222 \) |
\(\beta_{2}\) | \(=\) | \( ( 108\nu^{11} - 729\nu^{9} + 4768\nu^{7} - 1242\nu^{5} + 135\nu^{3} + 12378\nu ) / 1222 \) |
\(\beta_{3}\) | \(=\) | \( ( - 92 \nu^{11} + 293 \nu^{10} + 621 \nu^{9} - 2436 \nu^{8} - 4039 \nu^{7} + 16443 \nu^{6} + 1058 \nu^{5} - 26893 \nu^{4} - 115 \nu^{3} + 28014 \nu^{2} - 5181 \nu - 3045 ) / 2444 \) |
\(\beta_{4}\) | \(=\) | \( ( 135 \nu^{11} - 16 \nu^{10} - 1064 \nu^{9} + 108 \nu^{8} + 7182 \nu^{7} - 729 \nu^{6} - 9801 \nu^{5} + 184 \nu^{4} + 12236 \nu^{3} - 20 \nu^{2} - 108 \nu - 3531 ) / 1222 \) |
\(\beta_{5}\) | \(=\) | \( ( -135\nu^{10} + 1064\nu^{8} - 7182\nu^{6} + 9801\nu^{4} - 12236\nu^{2} + 108 ) / 1222 \) |
\(\beta_{6}\) | \(=\) | \( ( 563 \nu^{11} - 655 \nu^{10} - 4564 \nu^{9} + 5185 \nu^{8} + 30807 \nu^{7} - 34846 \nu^{6} - 46495 \nu^{5} + 47553 \nu^{4} + 52486 \nu^{3} - 52601 \nu^{2} - 5705 \nu + 524 ) / 2444 \) |
\(\beta_{7}\) | \(=\) | \( ( 92 \nu^{11} + 563 \nu^{10} - 621 \nu^{9} - 4564 \nu^{8} + 4039 \nu^{7} + 30807 \nu^{6} - 1058 \nu^{5} - 46495 \nu^{4} + 115 \nu^{3} + 52486 \nu^{2} + 5181 \nu - 5705 ) / 2444 \) |
\(\beta_{8}\) | \(=\) | \( ( - 655 \nu^{11} - 92 \nu^{10} + 5185 \nu^{9} + 621 \nu^{8} - 34846 \nu^{7} - 4039 \nu^{6} + 47553 \nu^{5} + 1058 \nu^{4} - 52601 \nu^{3} - 115 \nu^{2} + 524 \nu - 5181 ) / 2444 \) |
\(\beta_{9}\) | \(=\) | \( ( 389\nu^{10} - 3084\nu^{8} + 20817\nu^{6} - 29219\nu^{4} + 35466\nu^{2} - 1222\nu - 3855 ) / 1222 \) |
\(\beta_{10}\) | \(=\) | \( ( 1195\nu^{11} - 9441\nu^{9} + 63574\nu^{7} - 86757\nu^{5} + 100323\nu^{3} - 956\nu ) / 1222 \) |
\(\beta_{11}\) | \(=\) | \( ( 1357\nu^{11} - 10840\nu^{9} + 73170\nu^{7} - 105117\nu^{5} + 124660\nu^{3} - 13550\nu ) / 1222 \) |
\(\nu\) | \(=\) | \( ( \beta_{2} - \beta_1 ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( ( -\beta_{11} + \beta_{10} - 2\beta_{9} - 6\beta_{5} + 2\beta_{4} - 2\beta_{2} + \beta_1 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( 2\beta_{11} - 3\beta_{10} - \beta_{8} + \beta_{7} + \beta_{6} + 2\beta_{2} \) |
\(\nu^{4}\) | \(=\) | \( ( -14\beta_{9} + 2\beta_{7} - 34\beta_{5} + 2\beta_{3} - 7\beta_{2} + 7\beta _1 - 34 ) / 2 \) |
\(\nu^{5}\) | \(=\) | \( ( 23\beta_{11} - 37\beta_{10} - 16\beta_{8} + 16\beta_{6} - 16\beta_{5} + 16\beta_{3} + 37\beta _1 - 16 ) / 2 \) |
\(\nu^{6}\) | \(=\) | \( 23\beta_{11} - 23\beta_{10} + 8\beta_{8} + 8\beta_{7} + 8\beta_{6} - 46\beta_{4} + 23\beta_{2} - 99 \) |
\(\nu^{7}\) | \(=\) | \( ( -108\beta_{7} - 108\beta_{5} + 108\beta_{3} - 145\beta_{2} + 237\beta _1 - 108 ) / 2 \) |
\(\nu^{8}\) | \(=\) | \( ( 299 \beta_{11} - 299 \beta_{10} + 598 \beta_{9} + 108 \beta_{8} + 108 \beta_{6} + 1378 \beta_{5} - 598 \beta_{4} - 108 \beta_{3} + 598 \beta_{2} - 299 \beta _1 + 108 ) / 2 \) |
\(\nu^{9}\) | \(=\) | \( -467\beta_{11} + 766\beta_{10} + 353\beta_{8} - 353\beta_{7} - 353\beta_{6} - 467\beta_{2} \) |
\(\nu^{10}\) | \(=\) | \( ( 3878\beta_{9} - 706\beta_{7} + 8918\beta_{5} - 706\beta_{3} + 1939\beta_{2} - 1939\beta _1 + 8918 ) / 2 \) |
\(\nu^{11}\) | \(=\) | \( ( - 6045 \beta_{11} + 9923 \beta_{10} + 4584 \beta_{8} - 4584 \beta_{6} + 4584 \beta_{5} - 4584 \beta_{3} - 9923 \beta _1 + 4584 ) / 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1040\mathbb{Z}\right)^\times\).
\(n\) | \(261\) | \(417\) | \(561\) | \(911\) |
\(\chi(n)\) | \(1\) | \(-1\) | \(-1 - \beta_{5}\) | \(1\) |
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 |
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0 | −2.33117 | + | 1.34590i | 0 | −2.12291 | + | 0.702335i | 0 | 2.90420 | + | 1.67674i | 0 | 2.12291 | − | 3.67698i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
289.2 | 0 | −1.86449 | + | 1.07646i | 0 | −0.817544 | − | 2.08125i | 0 | −2.54486 | − | 1.46928i | 0 | 0.817544 | − | 1.41603i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
289.3 | 0 | −0.298874 | + | 0.172555i | 0 | 1.44045 | + | 1.71029i | 0 | −1.75765 | − | 1.01478i | 0 | −1.44045 | + | 2.49493i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
289.4 | 0 | 0.298874 | − | 0.172555i | 0 | 1.44045 | − | 1.71029i | 0 | 1.75765 | + | 1.01478i | 0 | −1.44045 | + | 2.49493i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
289.5 | 0 | 1.86449 | − | 1.07646i | 0 | −0.817544 | + | 2.08125i | 0 | 2.54486 | + | 1.46928i | 0 | 0.817544 | − | 1.41603i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
289.6 | 0 | 2.33117 | − | 1.34590i | 0 | −2.12291 | − | 0.702335i | 0 | −2.90420 | − | 1.67674i | 0 | 2.12291 | − | 3.67698i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
529.1 | 0 | −2.33117 | − | 1.34590i | 0 | −2.12291 | − | 0.702335i | 0 | 2.90420 | − | 1.67674i | 0 | 2.12291 | + | 3.67698i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
529.2 | 0 | −1.86449 | − | 1.07646i | 0 | −0.817544 | + | 2.08125i | 0 | −2.54486 | + | 1.46928i | 0 | 0.817544 | + | 1.41603i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
529.3 | 0 | −0.298874 | − | 0.172555i | 0 | 1.44045 | − | 1.71029i | 0 | −1.75765 | + | 1.01478i | 0 | −1.44045 | − | 2.49493i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
529.4 | 0 | 0.298874 | + | 0.172555i | 0 | 1.44045 | + | 1.71029i | 0 | 1.75765 | − | 1.01478i | 0 | −1.44045 | − | 2.49493i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
529.5 | 0 | 1.86449 | + | 1.07646i | 0 | −0.817544 | − | 2.08125i | 0 | 2.54486 | − | 1.46928i | 0 | 0.817544 | + | 1.41603i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
529.6 | 0 | 2.33117 | + | 1.34590i | 0 | −2.12291 | + | 0.702335i | 0 | −2.90420 | + | 1.67674i | 0 | 2.12291 | + | 3.67698i | 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.a | 12 | |
4.b | odd | 2 | 1 | 65.2.n.a | ✓ | 12 | |
5.b | even | 2 | 1 | inner | 1040.2.dh.a | 12 | |
12.b | even | 2 | 1 | 585.2.bs.a | 12 | ||
13.c | even | 3 | 1 | inner | 1040.2.dh.a | 12 | |
20.d | odd | 2 | 1 | 65.2.n.a | ✓ | 12 | |
20.e | even | 4 | 2 | 325.2.e.e | 12 | ||
52.b | odd | 2 | 1 | 845.2.n.e | 12 | ||
52.f | even | 4 | 2 | 845.2.l.f | 24 | ||
52.i | odd | 6 | 1 | 845.2.b.e | 6 | ||
52.i | odd | 6 | 1 | 845.2.n.e | 12 | ||
52.j | odd | 6 | 1 | 65.2.n.a | ✓ | 12 | |
52.j | odd | 6 | 1 | 845.2.b.d | 6 | ||
52.l | even | 12 | 2 | 845.2.d.d | 12 | ||
52.l | even | 12 | 2 | 845.2.l.f | 24 | ||
60.h | even | 2 | 1 | 585.2.bs.a | 12 | ||
65.n | even | 6 | 1 | inner | 1040.2.dh.a | 12 | |
156.p | even | 6 | 1 | 585.2.bs.a | 12 | ||
260.g | odd | 2 | 1 | 845.2.n.e | 12 | ||
260.u | even | 4 | 2 | 845.2.l.f | 24 | ||
260.v | odd | 6 | 1 | 65.2.n.a | ✓ | 12 | |
260.v | odd | 6 | 1 | 845.2.b.d | 6 | ||
260.w | odd | 6 | 1 | 845.2.b.e | 6 | ||
260.w | odd | 6 | 1 | 845.2.n.e | 12 | ||
260.bc | even | 12 | 2 | 845.2.d.d | 12 | ||
260.bc | even | 12 | 2 | 845.2.l.f | 24 | ||
260.bg | even | 12 | 2 | 4225.2.a.bq | 6 | ||
260.bj | even | 12 | 2 | 325.2.e.e | 12 | ||
260.bj | even | 12 | 2 | 4225.2.a.br | 6 | ||
780.br | even | 6 | 1 | 585.2.bs.a | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
65.2.n.a | ✓ | 12 | 4.b | odd | 2 | 1 | |
65.2.n.a | ✓ | 12 | 20.d | odd | 2 | 1 | |
65.2.n.a | ✓ | 12 | 52.j | odd | 6 | 1 | |
65.2.n.a | ✓ | 12 | 260.v | odd | 6 | 1 | |
325.2.e.e | 12 | 20.e | even | 4 | 2 | ||
325.2.e.e | 12 | 260.bj | even | 12 | 2 | ||
585.2.bs.a | 12 | 12.b | even | 2 | 1 | ||
585.2.bs.a | 12 | 60.h | even | 2 | 1 | ||
585.2.bs.a | 12 | 156.p | even | 6 | 1 | ||
585.2.bs.a | 12 | 780.br | even | 6 | 1 | ||
845.2.b.d | 6 | 52.j | odd | 6 | 1 | ||
845.2.b.d | 6 | 260.v | odd | 6 | 1 | ||
845.2.b.e | 6 | 52.i | odd | 6 | 1 | ||
845.2.b.e | 6 | 260.w | odd | 6 | 1 | ||
845.2.d.d | 12 | 52.l | even | 12 | 2 | ||
845.2.d.d | 12 | 260.bc | even | 12 | 2 | ||
845.2.l.f | 24 | 52.f | even | 4 | 2 | ||
845.2.l.f | 24 | 52.l | even | 12 | 2 | ||
845.2.l.f | 24 | 260.u | even | 4 | 2 | ||
845.2.l.f | 24 | 260.bc | even | 12 | 2 | ||
845.2.n.e | 12 | 52.b | odd | 2 | 1 | ||
845.2.n.e | 12 | 52.i | odd | 6 | 1 | ||
845.2.n.e | 12 | 260.g | odd | 2 | 1 | ||
845.2.n.e | 12 | 260.w | odd | 6 | 1 | ||
1040.2.dh.a | 12 | 1.a | even | 1 | 1 | trivial | |
1040.2.dh.a | 12 | 5.b | even | 2 | 1 | inner | |
1040.2.dh.a | 12 | 13.c | even | 3 | 1 | inner | |
1040.2.dh.a | 12 | 65.n | even | 6 | 1 | inner | |
4225.2.a.bq | 6 | 260.bg | even | 12 | 2 | ||
4225.2.a.br | 6 | 260.bj | even | 12 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} - 12T_{3}^{10} + 109T_{3}^{8} - 412T_{3}^{6} + 1177T_{3}^{4} - 140T_{3}^{2} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(1040, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{12} \)
$3$
\( T^{12} - 12 T^{10} + 109 T^{8} + \cdots + 16 \)
$5$
\( (T^{6} + 3 T^{5} + 5 T^{4} + 10 T^{3} + \cdots + 125)^{2} \)
$7$
\( T^{12} - 24 T^{10} + 397 T^{8} + \cdots + 160000 \)
$11$
\( (T^{6} + 13 T^{4} + 16 T^{3} + 169 T^{2} + \cdots + 64)^{2} \)
$13$
\( T^{12} - 15 T^{10} + 39 T^{8} + \cdots + 4826809 \)
$17$
\( T^{12} - 35 T^{10} + 1062 T^{8} + \cdots + 28561 \)
$19$
\( (T^{6} + 6 T^{5} + 37 T^{4} + 14 T^{3} + \cdots + 100)^{2} \)
$23$
\( T^{12} - 12 T^{10} + 109 T^{8} + \cdots + 16 \)
$29$
\( (T^{2} - 3 T + 9)^{6} \)
$31$
\( (T^{3} - 4 T^{2} - 40 T - 40)^{4} \)
$37$
\( T^{12} - 35 T^{10} + 1062 T^{8} + \cdots + 28561 \)
$41$
\( (T^{6} - 7 T^{5} + 78 T^{4} + 213 T^{3} + \cdots + 25)^{2} \)
$43$
\( T^{12} - 80 T^{10} + 6117 T^{8} + \cdots + 65536 \)
$47$
\( (T^{6} + 236 T^{4} + 14640 T^{2} + \cdots + 270400)^{2} \)
$53$
\( (T^{6} + 171 T^{4} + 1040 T^{2} + \cdots + 400)^{2} \)
$59$
\( (T^{6} - 2 T^{5} + 59 T^{4} - 162 T^{3} + \cdots + 18496)^{2} \)
$61$
\( (T^{6} - 3 T^{5} + 58 T^{4} - 83 T^{3} + \cdots + 13225)^{2} \)
$67$
\( T^{12} - 100 T^{10} + \cdots + 406586896 \)
$71$
\( (T^{6} - 6 T^{5} + 37 T^{4} - 46 T^{3} + \cdots + 676)^{2} \)
$73$
\( (T^{6} + 215 T^{4} + 13900 T^{2} + \cdots + 250000)^{2} \)
$79$
\( (T^{3} - 26 T^{2} + 180 T - 160)^{4} \)
$83$
\( (T^{6} + 276 T^{4} + 23600 T^{2} + \cdots + 640000)^{2} \)
$89$
\( (T^{6} - 10 T^{5} + 257 T^{4} + \cdots + 2515396)^{2} \)
$97$
\( T^{12} - 280 T^{10} + \cdots + 41740124416 \)
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