Newspace parameters
Level: | \( N \) | \(=\) | \( 385 = 5 \cdot 7 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 385.o (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(3.07424047782\) |
Analytic rank: | \(0\) |
Dimension: | \(16\) |
Relative dimension: | \(8\) over \(\Q(\zeta_{6})\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{16} + \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{16} + 3x^{12} - 7x^{8} + 48x^{4} + 256 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 7 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{U}(1)[D_{6}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) 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 \( x^{16} + 3x^{12} - 7x^{8} + 48x^{4} + 256 \) :
\(\beta_{1}\) | \(=\) | \( \nu^{2} \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{15} + 35\nu^{11} + 217\nu^{7} - 176\nu^{3} ) / 896 \) |
\(\beta_{3}\) | \(=\) | \( ( 3\nu^{12} - 7\nu^{8} + 91\nu^{4} + 256 ) / 112 \) |
\(\beta_{4}\) | \(=\) | \( ( -3\nu^{13} + 7\nu^{9} - 91\nu^{5} - 256\nu ) / 224 \) |
\(\beta_{5}\) | \(=\) | \( ( -3\nu^{12} + 7\nu^{8} + 21\nu^{4} - 256 ) / 112 \) |
\(\beta_{6}\) | \(=\) | \( ( \nu^{14} + 3\nu^{10} - 7\nu^{6} + 48\nu^{2} ) / 64 \) |
\(\beta_{7}\) | \(=\) | \( ( 9\nu^{14} - 21\nu^{10} + 49\nu^{6} + 768\nu^{2} ) / 448 \) |
\(\beta_{8}\) | \(=\) | \( ( \nu^{15} - 12\nu^{13} - 21\nu^{11} + 28\nu^{9} + 49\nu^{7} + 84\nu^{5} + 216\nu^{3} - 1024\nu ) / 448 \) |
\(\beta_{9}\) | \(=\) | \( ( 3\nu^{14} - 7\nu^{10} - 21\nu^{6} + 256\nu^{2} ) / 112 \) |
\(\beta_{10}\) | \(=\) | \( ( 7\nu^{15} - 24\nu^{13} + 21\nu^{11} + 56\nu^{9} - 49\nu^{7} + 168\nu^{5} + 336\nu^{3} - 1152\nu ) / 896 \) |
\(\beta_{11}\) | \(=\) | \( ( 13\nu^{12} + 7\nu^{8} - 91\nu^{4} + 624 ) / 112 \) |
\(\beta_{12}\) | \(=\) | \( ( 7\nu^{15} + 24\nu^{13} + 21\nu^{11} - 56\nu^{9} - 49\nu^{7} - 168\nu^{5} + 336\nu^{3} + 256\nu ) / 896 \) |
\(\beta_{13}\) | \(=\) | \( ( -13\nu^{13} - 7\nu^{9} + 91\nu^{5} - 624\nu ) / 224 \) |
\(\beta_{14}\) | \(=\) | \( ( -9\nu^{15} - 24\nu^{13} + 21\nu^{11} + 56\nu^{9} - 49\nu^{7} + 168\nu^{5} - 768\nu^{3} - 2048\nu ) / 896 \) |
\(\beta_{15}\) | \(=\) | \( ( 2\nu^{15} + 3\nu^{13} - 7\nu^{9} - 21\nu^{5} + 26\nu^{3} + 144\nu ) / 112 \) |
\(\nu\) | \(=\) | \( ( -2\beta_{14} - 3\beta_{12} + \beta_{10} - 2\beta_{8} ) / 7 \) |
\(\nu^{2}\) | \(=\) | \( \beta_1 \) |
\(\nu^{3}\) | \(=\) | \( ( -7\beta_{15} - 6\beta_{14} + 5\beta_{12} + 3\beta_{10} + \beta_{8} ) / 7 \) |
\(\nu^{4}\) | \(=\) | \( \beta_{5} + \beta_{3} \) |
\(\nu^{5}\) | \(=\) | \( ( 3\beta_{14} + \beta_{12} + 2\beta_{10} + 3\beta_{8} - 14\beta_{4} ) / 7 \) |
\(\nu^{6}\) | \(=\) | \( -3\beta_{9} + 4\beta_{7} \) |
\(\nu^{7}\) | \(=\) | \( ( -5\beta_{14} - 4\beta_{12} - 8\beta_{10} + 9\beta_{8} + 21\beta_{2} ) / 7 \) |
\(\nu^{8}\) | \(=\) | \( 3\beta_{11} + 13\beta_{5} + 13 \) |
\(\nu^{9}\) | \(=\) | \( ( 13\beta_{14} - 42\beta_{13} - 26\beta_{12} + 39\beta_{10} + 13\beta_{8} ) / 7 \) |
\(\nu^{10}\) | \(=\) | \( -7\beta_{9} + 12\beta_{6} + 7\beta_1 \) |
\(\nu^{11}\) | \(=\) | \( ( -49\beta_{15} - 3\beta_{14} + 55\beta_{12} + 61\beta_{10} - 52\beta_{8} + 49\beta_{2} ) / 7 \) |
\(\nu^{12}\) | \(=\) | \( 7\beta_{11} + 7\beta_{3} - 55 \) |
\(\nu^{13}\) | \(=\) | \( ( 110\beta_{14} - 98\beta_{13} + 165\beta_{12} - 55\beta_{10} + 110\beta_{8} - 98\beta_{4} ) / 7 \) |
\(\nu^{14}\) | \(=\) | \( 28\beta_{7} + 28\beta_{6} - 69\beta_1 \) |
\(\nu^{15}\) | \(=\) | \( ( 483\beta_{15} + 134\beta_{14} - 177\beta_{12} + 129\beta_{10} + 43\beta_{8} ) / 7 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/385\mathbb{Z}\right)^\times\).
\(n\) | \(211\) | \(232\) | \(276\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(1 + \beta_{5}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
54.1 |
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−1.37793 | + | 2.38665i | 0 | −2.79740 | − | 4.84524i | 1.93649 | + | 1.11803i | 0 | −2.61087 | + | 0.428223i | 9.90680 | 1.50000 | − | 2.59808i | −5.33671 | + | 3.08115i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
54.2 | −1.03419 | + | 1.79127i | 0 | −1.13909 | − | 1.97296i | 1.93649 | + | 1.11803i | 0 | 0.428223 | − | 2.61087i | 0.575379 | 1.50000 | − | 2.59808i | −4.00539 | + | 2.31251i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
54.3 | −0.964601 | + | 1.67074i | 0 | −0.860910 | − | 1.49114i | −1.93649 | − | 1.11803i | 0 | 2.61087 | + | 0.428223i | −0.536664 | 1.50000 | − | 2.59808i | 3.73588 | − | 2.15691i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
54.4 | −0.318275 | + | 0.551268i | 0 | 0.797402 | + | 1.38114i | −1.93649 | − | 1.11803i | 0 | 0.428223 | + | 2.61087i | −2.28827 | 1.50000 | − | 2.59808i | 1.23267 | − | 0.711685i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
54.5 | 0.318275 | − | 0.551268i | 0 | 0.797402 | + | 1.38114i | −1.93649 | − | 1.11803i | 0 | −0.428223 | − | 2.61087i | 2.28827 | 1.50000 | − | 2.59808i | −1.23267 | + | 0.711685i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
54.6 | 0.964601 | − | 1.67074i | 0 | −0.860910 | − | 1.49114i | −1.93649 | − | 1.11803i | 0 | −2.61087 | − | 0.428223i | 0.536664 | 1.50000 | − | 2.59808i | −3.73588 | + | 2.15691i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
54.7 | 1.03419 | − | 1.79127i | 0 | −1.13909 | − | 1.97296i | 1.93649 | + | 1.11803i | 0 | −0.428223 | + | 2.61087i | −0.575379 | 1.50000 | − | 2.59808i | 4.00539 | − | 2.31251i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
54.8 | 1.37793 | − | 2.38665i | 0 | −2.79740 | − | 4.84524i | 1.93649 | + | 1.11803i | 0 | 2.61087 | − | 0.428223i | −9.90680 | 1.50000 | − | 2.59808i | 5.33671 | − | 3.08115i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
164.1 | −1.37793 | − | 2.38665i | 0 | −2.79740 | + | 4.84524i | 1.93649 | − | 1.11803i | 0 | −2.61087 | − | 0.428223i | 9.90680 | 1.50000 | + | 2.59808i | −5.33671 | − | 3.08115i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
164.2 | −1.03419 | − | 1.79127i | 0 | −1.13909 | + | 1.97296i | 1.93649 | − | 1.11803i | 0 | 0.428223 | + | 2.61087i | 0.575379 | 1.50000 | + | 2.59808i | −4.00539 | − | 2.31251i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
164.3 | −0.964601 | − | 1.67074i | 0 | −0.860910 | + | 1.49114i | −1.93649 | + | 1.11803i | 0 | 2.61087 | − | 0.428223i | −0.536664 | 1.50000 | + | 2.59808i | 3.73588 | + | 2.15691i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
164.4 | −0.318275 | − | 0.551268i | 0 | 0.797402 | − | 1.38114i | −1.93649 | + | 1.11803i | 0 | 0.428223 | − | 2.61087i | −2.28827 | 1.50000 | + | 2.59808i | 1.23267 | + | 0.711685i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
164.5 | 0.318275 | + | 0.551268i | 0 | 0.797402 | − | 1.38114i | −1.93649 | + | 1.11803i | 0 | −0.428223 | + | 2.61087i | 2.28827 | 1.50000 | + | 2.59808i | −1.23267 | − | 0.711685i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
164.6 | 0.964601 | + | 1.67074i | 0 | −0.860910 | + | 1.49114i | −1.93649 | + | 1.11803i | 0 | −2.61087 | + | 0.428223i | 0.536664 | 1.50000 | + | 2.59808i | −3.73588 | − | 2.15691i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
164.7 | 1.03419 | + | 1.79127i | 0 | −1.13909 | + | 1.97296i | 1.93649 | − | 1.11803i | 0 | −0.428223 | − | 2.61087i | −0.575379 | 1.50000 | + | 2.59808i | 4.00539 | + | 2.31251i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
164.8 | 1.37793 | + | 2.38665i | 0 | −2.79740 | + | 4.84524i | 1.93649 | − | 1.11803i | 0 | 2.61087 | + | 0.428223i | −9.90680 | 1.50000 | + | 2.59808i | 5.33671 | + | 3.08115i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
55.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-55}) \) |
5.b | even | 2 | 1 | inner |
7.d | odd | 6 | 1 | inner |
11.b | odd | 2 | 1 | inner |
35.i | odd | 6 | 1 | inner |
77.i | even | 6 | 1 | inner |
385.o | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 385.2.o.a | ✓ | 16 |
5.b | even | 2 | 1 | inner | 385.2.o.a | ✓ | 16 |
7.d | odd | 6 | 1 | inner | 385.2.o.a | ✓ | 16 |
11.b | odd | 2 | 1 | inner | 385.2.o.a | ✓ | 16 |
35.i | odd | 6 | 1 | inner | 385.2.o.a | ✓ | 16 |
55.d | odd | 2 | 1 | CM | 385.2.o.a | ✓ | 16 |
77.i | even | 6 | 1 | inner | 385.2.o.a | ✓ | 16 |
385.o | even | 6 | 1 | inner | 385.2.o.a | ✓ | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
385.2.o.a | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
385.2.o.a | ✓ | 16 | 5.b | even | 2 | 1 | inner |
385.2.o.a | ✓ | 16 | 7.d | odd | 6 | 1 | inner |
385.2.o.a | ✓ | 16 | 11.b | odd | 2 | 1 | inner |
385.2.o.a | ✓ | 16 | 35.i | odd | 6 | 1 | inner |
385.2.o.a | ✓ | 16 | 55.d | odd | 2 | 1 | CM |
385.2.o.a | ✓ | 16 | 77.i | even | 6 | 1 | inner |
385.2.o.a | ✓ | 16 | 385.o | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{16} + 16T_{2}^{14} + 173T_{2}^{12} + 1024T_{2}^{10} + 4408T_{2}^{8} + 11048T_{2}^{6} + 19037T_{2}^{4} + 7448T_{2}^{2} + 2401 \)
acting on \(S_{2}^{\mathrm{new}}(385, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{16} + 16 T^{14} + 173 T^{12} + \cdots + 2401 \)
$3$
\( T^{16} \)
$5$
\( (T^{4} - 5 T^{2} + 25)^{4} \)
$7$
\( (T^{8} - 78 T^{4} + 2401)^{2} \)
$11$
\( (T^{4} + 11 T^{2} + 121)^{4} \)
$13$
\( (T^{8} + 104 T^{6} + 3218 T^{4} + 26728 T^{2} + \cdots + 49)^{2} \)
$17$
\( (T^{8} - 68 T^{6} + 3644 T^{4} + \cdots + 960400)^{2} \)
$19$
\( T^{16} \)
$23$
\( T^{16} \)
$29$
\( T^{16} \)
$31$
\( (T^{8} - 106 T^{6} + 11067 T^{4} + \cdots + 28561)^{2} \)
$37$
\( T^{16} \)
$41$
\( T^{16} \)
$43$
\( (T^{8} - 344 T^{6} + 33458 T^{4} + \cdots + 2798929)^{2} \)
$47$
\( T^{16} \)
$53$
\( T^{16} \)
$59$
\( (T^{4} - 12 T^{3} + 5 T^{2} + 516 T + 1849)^{4} \)
$61$
\( T^{16} \)
$67$
\( T^{16} \)
$71$
\( (T^{2} - 8 T - 149)^{8} \)
$73$
\( T^{16} - 584 T^{14} + \cdots + 64\!\cdots\!21 \)
$79$
\( T^{16} \)
$83$
\( (T^{8} + 664 T^{6} + 124178 T^{4} + \cdots + 45064369)^{2} \)
$89$
\( (T^{8} - 354 T^{6} + 117747 T^{4} + \cdots + 57289761)^{2} \)
$97$
\( T^{16} \)
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