Newspace parameters
Level: | \( N \) | \(=\) | \( 1248 = 2^{5} \cdot 3 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1248.h (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(9.96533017226\) |
Analytic rank: | \(0\) |
Dimension: | \(12\) |
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: | \( x^{12} - 16x^{9} + 92x^{6} - 68x^{3} + 27 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
Coefficient ring index: | \( 2^{6}\cdot 3^{2} \) |
Twist minimal: | no (minimal twist has level 312) |
Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-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 \( x^{12} - 16x^{9} + 92x^{6} - 68x^{3} + 27 \) :
\(\beta_{1}\) | \(=\) | \( ( 3\nu^{10} - 49\nu^{7} + 339\nu^{4} - 737\nu ) / 210 \) |
\(\beta_{2}\) | \(=\) | \( ( 2\nu^{9} - 21\nu^{6} + 16\nu^{3} + 372 ) / 105 \) |
\(\beta_{3}\) | \(=\) | \( ( -3\nu^{9} + 49\nu^{6} - 269\nu^{3} + 107 ) / 70 \) |
\(\beta_{4}\) | \(=\) | \( ( -4\nu^{11} + 67\nu^{8} - 422\nu^{5} + 521\nu^{2} ) / 90 \) |
\(\beta_{5}\) | \(=\) | \( ( -2\nu^{10} + 31\nu^{7} - 166\nu^{4} + 53\nu ) / 30 \) |
\(\beta_{6}\) | \(=\) | \( ( 2\nu^{11} + \nu^{10} - 31\nu^{8} - 18\nu^{7} + 166\nu^{5} + 113\nu^{4} - 23\nu^{2} - 144\nu ) / 30 \) |
\(\beta_{7}\) | \(=\) | \( ( -12\nu^{11} - 17\nu^{10} + 196\nu^{8} + 266\nu^{7} - 1146\nu^{5} - 1501\nu^{4} + 1058\nu^{2} + 688\nu ) / 210 \) |
\(\beta_{8}\) | \(=\) | \( ( \nu^{9} - 15\nu^{6} + 83\nu^{3} - 33 ) / 6 \) |
\(\beta_{9}\) | \(=\) | \( ( 40\nu^{11} + 51\nu^{10} - 595\nu^{8} - 798\nu^{7} + 3050\nu^{5} + 4503\nu^{4} + 475\nu^{2} - 2064\nu ) / 630 \) |
\(\beta_{10}\) | \(=\) | \( ( 4\nu^{11} - \nu^{10} - 62\nu^{8} + 18\nu^{7} + 332\nu^{5} - 113\nu^{4} - 46\nu^{2} + 144\nu ) / 30 \) |
\(\beta_{11}\) | \(=\) | \( ( -76\nu^{11} + 51\nu^{10} + 1183\nu^{8} - 798\nu^{7} - 6488\nu^{5} + 4503\nu^{4} + 2699\nu^{2} - 2064\nu ) / 630 \) |
\(\nu\) | \(=\) | \( ( \beta_{11} + \beta_{9} - \beta_{7} + 3\beta_{5} - 3\beta_1 ) / 6 \) |
\(\nu^{2}\) | \(=\) | \( ( 3\beta_{11} + 2\beta_{10} + 3\beta_{7} + 2\beta_{6} - 3\beta_{4} ) / 6 \) |
\(\nu^{3}\) | \(=\) | \( ( \beta_{8} + 3\beta_{3} - 2\beta_{2} + 8 ) / 2 \) |
\(\nu^{4}\) | \(=\) | \( ( 9\beta_{11} + \beta_{10} + 9\beta_{9} - 9\beta_{7} - 2\beta_{6} + 30\beta_{5} - 6\beta_1 ) / 6 \) |
\(\nu^{5}\) | \(=\) | \( ( 17\beta_{11} + 6\beta_{10} + 14\beta_{9} + 31\beta_{7} + 6\beta_{6} - 39\beta_{4} ) / 6 \) |
\(\nu^{6}\) | \(=\) | \( 7\beta_{8} + 25\beta_{3} - 5\beta_{2} + 18 \) |
\(\nu^{7}\) | \(=\) | \( ( 61\beta_{11} + 36\beta_{10} + 61\beta_{9} - 61\beta_{7} - 72\beta_{6} + 183\beta_{5} + 69\beta_1 ) / 6 \) |
\(\nu^{8}\) | \(=\) | \( ( 21\beta_{11} - 58\beta_{10} + 252\beta_{9} + 273\beta_{7} - 58\beta_{6} - 309\beta_{4} ) / 6 \) |
\(\nu^{9}\) | \(=\) | \( ( 139\beta_{8} + 501\beta_{3} + 16\beta_{2} - 58 ) / 2 \) |
\(\nu^{10}\) | \(=\) | \( ( 225\beta_{11} + 475\beta_{10} + 225\beta_{9} - 225\beta_{7} - 950\beta_{6} + 336\beta_{5} + 1488\beta_1 ) / 6 \) |
\(\nu^{11}\) | \(=\) | \( ( -1051\beta_{11} - 1344\beta_{10} + 2744\beta_{9} + 1693\beta_{7} - 1344\beta_{6} - 1587\beta_{4} ) / 6 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1248\mathbb{Z}\right)^\times\).
\(n\) | \(703\) | \(769\) | \(833\) | \(1093\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
623.1 |
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0 | −1.44101 | − | 0.960984i | 0 | − | 3.66851i | 0 | −5.27581 | 0 | 1.15302 | + | 2.76957i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
623.2 | 0 | −1.44101 | − | 0.960984i | 0 | 3.66851i | 0 | 5.27581 | 0 | 1.15302 | + | 2.76957i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
623.3 | 0 | −1.44101 | + | 0.960984i | 0 | − | 3.66851i | 0 | 5.27581 | 0 | 1.15302 | − | 2.76957i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||
623.4 | 0 | −1.44101 | + | 0.960984i | 0 | 3.66851i | 0 | −5.27581 | 0 | 1.15302 | − | 2.76957i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
623.5 | 0 | −0.111731 | − | 1.72844i | 0 | − | 4.04932i | 0 | 2.99062 | 0 | −2.97503 | + | 0.386242i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||
623.6 | 0 | −0.111731 | − | 1.72844i | 0 | 4.04932i | 0 | −2.99062 | 0 | −2.97503 | + | 0.386242i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
623.7 | 0 | −0.111731 | + | 1.72844i | 0 | − | 4.04932i | 0 | −2.99062 | 0 | −2.97503 | − | 0.386242i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||
623.8 | 0 | −0.111731 | + | 1.72844i | 0 | 4.04932i | 0 | 2.99062 | 0 | −2.97503 | − | 0.386242i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
623.9 | 0 | 1.55274 | − | 0.767460i | 0 | − | 0.380805i | 0 | 2.28519 | 0 | 1.82201 | − | 2.38333i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||
623.10 | 0 | 1.55274 | − | 0.767460i | 0 | 0.380805i | 0 | −2.28519 | 0 | 1.82201 | − | 2.38333i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
623.11 | 0 | 1.55274 | + | 0.767460i | 0 | − | 0.380805i | 0 | −2.28519 | 0 | 1.82201 | + | 2.38333i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||
623.12 | 0 | 1.55274 | + | 0.767460i | 0 | 0.380805i | 0 | 2.28519 | 0 | 1.82201 | + | 2.38333i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
104.h | odd | 2 | 1 | CM by \(\Q(\sqrt{-26}) \) |
3.b | odd | 2 | 1 | inner |
8.d | odd | 2 | 1 | inner |
13.b | even | 2 | 1 | inner |
24.f | even | 2 | 1 | inner |
39.d | odd | 2 | 1 | inner |
312.h | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1248.2.h.b | 12 | |
3.b | odd | 2 | 1 | inner | 1248.2.h.b | 12 | |
4.b | odd | 2 | 1 | 312.2.h.b | ✓ | 12 | |
8.b | even | 2 | 1 | 312.2.h.b | ✓ | 12 | |
8.d | odd | 2 | 1 | inner | 1248.2.h.b | 12 | |
12.b | even | 2 | 1 | 312.2.h.b | ✓ | 12 | |
13.b | even | 2 | 1 | inner | 1248.2.h.b | 12 | |
24.f | even | 2 | 1 | inner | 1248.2.h.b | 12 | |
24.h | odd | 2 | 1 | 312.2.h.b | ✓ | 12 | |
39.d | odd | 2 | 1 | inner | 1248.2.h.b | 12 | |
52.b | odd | 2 | 1 | 312.2.h.b | ✓ | 12 | |
104.e | even | 2 | 1 | 312.2.h.b | ✓ | 12 | |
104.h | odd | 2 | 1 | CM | 1248.2.h.b | 12 | |
156.h | even | 2 | 1 | 312.2.h.b | ✓ | 12 | |
312.b | odd | 2 | 1 | 312.2.h.b | ✓ | 12 | |
312.h | even | 2 | 1 | inner | 1248.2.h.b | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
312.2.h.b | ✓ | 12 | 4.b | odd | 2 | 1 | |
312.2.h.b | ✓ | 12 | 8.b | even | 2 | 1 | |
312.2.h.b | ✓ | 12 | 12.b | even | 2 | 1 | |
312.2.h.b | ✓ | 12 | 24.h | odd | 2 | 1 | |
312.2.h.b | ✓ | 12 | 52.b | odd | 2 | 1 | |
312.2.h.b | ✓ | 12 | 104.e | even | 2 | 1 | |
312.2.h.b | ✓ | 12 | 156.h | even | 2 | 1 | |
312.2.h.b | ✓ | 12 | 312.b | odd | 2 | 1 | |
1248.2.h.b | 12 | 1.a | even | 1 | 1 | trivial | |
1248.2.h.b | 12 | 3.b | odd | 2 | 1 | inner | |
1248.2.h.b | 12 | 8.d | odd | 2 | 1 | inner | |
1248.2.h.b | 12 | 13.b | even | 2 | 1 | inner | |
1248.2.h.b | 12 | 24.f | even | 2 | 1 | inner | |
1248.2.h.b | 12 | 39.d | odd | 2 | 1 | inner | |
1248.2.h.b | 12 | 104.h | odd | 2 | 1 | CM | |
1248.2.h.b | 12 | 312.h | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} + 30T_{5}^{4} + 225T_{5}^{2} + 32 \)
acting on \(S_{2}^{\mathrm{new}}(1248, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{12} \)
$3$
\( (T^{6} - 2 T^{3} + 27)^{2} \)
$5$
\( (T^{6} + 30 T^{4} + 225 T^{2} + 32)^{2} \)
$7$
\( (T^{6} - 42 T^{4} + 441 T^{2} - 1300)^{2} \)
$11$
\( T^{12} \)
$13$
\( (T^{2} - 13)^{6} \)
$17$
\( (T^{6} + 102 T^{4} + 2601 T^{2} + \cdots + 6656)^{2} \)
$19$
\( T^{12} \)
$23$
\( T^{12} \)
$29$
\( T^{12} \)
$31$
\( (T^{2} - 52)^{6} \)
$37$
\( (T^{6} - 222 T^{4} + 12321 T^{2} + \cdots - 87412)^{2} \)
$41$
\( T^{12} \)
$43$
\( (T^{3} - 129 T + 218)^{4} \)
$47$
\( (T^{6} + 282 T^{4} + 19881 T^{2} + \cdots + 336200)^{2} \)
$53$
\( T^{12} \)
$59$
\( T^{12} \)
$61$
\( T^{12} \)
$67$
\( T^{12} \)
$71$
\( (T^{6} + 426 T^{4} + 45369 T^{2} + \cdots + 397832)^{2} \)
$73$
\( T^{12} \)
$79$
\( T^{12} \)
$83$
\( T^{12} \)
$89$
\( T^{12} \)
$97$
\( T^{12} \)
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