Newspace parameters
Level: | \( N \) | \(=\) | \( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 2016.g (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(54.9320212950\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Coefficient field: | 8.0.292213762624.3 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} - x^{7} - 2x^{6} - 2x^{5} + 24x^{4} - 8x^{3} - 32x^{2} - 64x + 256 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
Coefficient ring index: | \( 2^{10} \) |
Twist minimal: | no (minimal twist has level 56) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \( x^{8} - x^{7} - 2x^{6} - 2x^{5} + 24x^{4} - 8x^{3} - 32x^{2} - 64x + 256 \) :
\(\beta_{1}\) | \(=\) | \( ( -\nu^{7} - 5\nu^{6} + 4\nu^{5} - 6\nu^{4} + 4\nu^{3} - 160\nu^{2} + 96\nu ) / 128 \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{7} - 3\nu^{6} - 12\nu^{5} + 6\nu^{4} + 12\nu^{3} - 96\nu ) / 128 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{6} - \nu^{5} - 2\nu^{4} - 2\nu^{3} + 8\nu^{2} + 8\nu - 16 ) / 16 \) |
\(\beta_{4}\) | \(=\) | \( ( -\nu^{7} - \nu^{6} - 14\nu^{4} - 4\nu^{3} - 128 ) / 64 \) |
\(\beta_{5}\) | \(=\) | \( ( 3\nu^{7} - \nu^{6} - 4\nu^{5} - 22\nu^{4} + 84\nu^{3} + 48\nu^{2} + 128\nu - 384 ) / 64 \) |
\(\beta_{6}\) | \(=\) | \( ( \nu^{7} - 3\nu^{5} - 4\nu^{4} + 22\nu^{3} - 88\nu - 64 ) / 16 \) |
\(\beta_{7}\) | \(=\) | \( ( -3\nu^{7} + \nu^{6} + 18\nu^{4} - 20\nu^{3} - 8\nu^{2} + 16\nu + 192 ) / 32 \) |
\(\nu\) | \(=\) | \( ( -\beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + \beta _1 + 1 ) / 8 \) |
\(\nu^{2}\) | \(=\) | \( ( -\beta_{6} + \beta_{5} + 3\beta_{4} - 3\beta_{3} + \beta_{2} - 7\beta _1 + 5 ) / 8 \) |
\(\nu^{3}\) | \(=\) | \( ( 4\beta_{7} + 3\beta_{6} + 5\beta_{5} - \beta_{4} - 3\beta_{3} - 7\beta_{2} + \beta _1 + 13 ) / 8 \) |
\(\nu^{4}\) | \(=\) | \( ( 4\beta_{7} - \beta_{6} + \beta_{5} - 21\beta_{4} - 7\beta_{3} + 5\beta_{2} - 3\beta _1 - 71 ) / 8 \) |
\(\nu^{5}\) | \(=\) | \( ( -4\beta_{7} + 3\beta_{6} - 3\beta_{5} - \beta_{4} - 35\beta_{3} - 71\beta_{2} - 15\beta _1 - 19 ) / 8 \) |
\(\nu^{6}\) | \(=\) | \( ( 12\beta_{7} + 23\beta_{6} - 7\beta_{5} - 61\beta_{4} + 89\beta_{3} - 91\beta_{2} + 29\beta _1 - 55 ) / 8 \) |
\(\nu^{7}\) | \(=\) | \( ( -84\beta_{7} - 21\beta_{6} - 27\beta_{5} - 153\beta_{4} + 21\beta_{3} + 49\beta_{2} + 9\beta _1 - 27 ) / 8 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2016\mathbb{Z}\right)^\times\).
\(n\) | \(127\) | \(577\) | \(1765\) | \(1793\) |
\(\chi(n)\) | \(-1\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
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1135.1 |
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0 | 0 | 0 | − | 6.26788i | 0 | 2.64575i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
1135.2 | 0 | 0 | 0 | − | 5.73252i | 0 | 2.64575i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
1135.3 | 0 | 0 | 0 | − | 4.88287i | 0 | − | 2.64575i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
1135.4 | 0 | 0 | 0 | − | 3.46547i | 0 | 2.64575i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
1135.5 | 0 | 0 | 0 | 3.46547i | 0 | − | 2.64575i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
1135.6 | 0 | 0 | 0 | 4.88287i | 0 | 2.64575i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
1135.7 | 0 | 0 | 0 | 5.73252i | 0 | − | 2.64575i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
1135.8 | 0 | 0 | 0 | 6.26788i | 0 | − | 2.64575i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.d | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 2016.3.g.b | 8 | |
3.b | odd | 2 | 1 | 224.3.g.b | 8 | ||
4.b | odd | 2 | 1 | 504.3.g.b | 8 | ||
8.b | even | 2 | 1 | 504.3.g.b | 8 | ||
8.d | odd | 2 | 1 | inner | 2016.3.g.b | 8 | |
12.b | even | 2 | 1 | 56.3.g.b | ✓ | 8 | |
21.c | even | 2 | 1 | 1568.3.g.m | 8 | ||
24.f | even | 2 | 1 | 224.3.g.b | 8 | ||
24.h | odd | 2 | 1 | 56.3.g.b | ✓ | 8 | |
48.i | odd | 4 | 2 | 1792.3.d.j | 16 | ||
48.k | even | 4 | 2 | 1792.3.d.j | 16 | ||
84.h | odd | 2 | 1 | 392.3.g.m | 8 | ||
84.j | odd | 6 | 2 | 392.3.k.n | 16 | ||
84.n | even | 6 | 2 | 392.3.k.o | 16 | ||
168.e | odd | 2 | 1 | 1568.3.g.m | 8 | ||
168.i | even | 2 | 1 | 392.3.g.m | 8 | ||
168.s | odd | 6 | 2 | 392.3.k.o | 16 | ||
168.ba | even | 6 | 2 | 392.3.k.n | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
56.3.g.b | ✓ | 8 | 12.b | even | 2 | 1 | |
56.3.g.b | ✓ | 8 | 24.h | odd | 2 | 1 | |
224.3.g.b | 8 | 3.b | odd | 2 | 1 | ||
224.3.g.b | 8 | 24.f | even | 2 | 1 | ||
392.3.g.m | 8 | 84.h | odd | 2 | 1 | ||
392.3.g.m | 8 | 168.i | even | 2 | 1 | ||
392.3.k.n | 16 | 84.j | odd | 6 | 2 | ||
392.3.k.n | 16 | 168.ba | even | 6 | 2 | ||
392.3.k.o | 16 | 84.n | even | 6 | 2 | ||
392.3.k.o | 16 | 168.s | odd | 6 | 2 | ||
504.3.g.b | 8 | 4.b | odd | 2 | 1 | ||
504.3.g.b | 8 | 8.b | even | 2 | 1 | ||
1568.3.g.m | 8 | 21.c | even | 2 | 1 | ||
1568.3.g.m | 8 | 168.e | odd | 2 | 1 | ||
1792.3.d.j | 16 | 48.i | odd | 4 | 2 | ||
1792.3.d.j | 16 | 48.k | even | 4 | 2 | ||
2016.3.g.b | 8 | 1.a | even | 1 | 1 | trivial | |
2016.3.g.b | 8 | 8.d | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} + 108T_{5}^{6} + 4164T_{5}^{4} + 66944T_{5}^{2} + 369664 \)
acting on \(S_{3}^{\mathrm{new}}(2016, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( T^{8} \)
$5$
\( T^{8} + 108 T^{6} + 4164 T^{4} + \cdots + 369664 \)
$7$
\( (T^{2} + 7)^{4} \)
$11$
\( (T^{4} + 16 T^{3} - 156 T^{2} - 864 T - 864)^{2} \)
$13$
\( T^{8} + 908 T^{6} + \cdots + 133448704 \)
$17$
\( (T^{4} - 40 T^{3} + 152 T^{2} + 3168 T - 752)^{2} \)
$19$
\( (T^{4} + 28 T^{3} + 266 T^{2} + 1008 T + 1288)^{2} \)
$23$
\( T^{8} + 2488 T^{6} + \cdots + 15607005184 \)
$29$
\( T^{8} + 3344 T^{6} + \cdots + 13389266944 \)
$31$
\( T^{8} + 3744 T^{6} + \cdots + 61917364224 \)
$37$
\( T^{8} + 7440 T^{6} + \cdots + 554696704 \)
$41$
\( (T^{4} + 64 T^{3} - 1768 T^{2} + \cdots + 837776)^{2} \)
$43$
\( (T^{4} - 2716 T^{2} + 58016 T - 217952)^{2} \)
$47$
\( T^{8} + 9280 T^{6} + \cdots + 7542537191424 \)
$53$
\( T^{8} + 3552 T^{6} + \cdots + 16777216 \)
$59$
\( (T^{4} - 52 T^{3} - 11670 T^{2} + \cdots - 11252856)^{2} \)
$61$
\( T^{8} + 13452 T^{6} + \cdots + 3223777158144 \)
$67$
\( (T^{4} + 152 T^{3} + 4268 T^{2} + \cdots - 791808)^{2} \)
$71$
\( T^{8} + \cdots + 221437256269824 \)
$73$
\( (T^{4} + 56 T^{3} - 2856 T^{2} + \cdots - 1726704)^{2} \)
$79$
\( T^{8} + 24960 T^{6} + \cdots + 89369947930624 \)
$83$
\( (T^{4} - 36 T^{3} - 11078 T^{2} + \cdots + 3180872)^{2} \)
$89$
\( (T^{4} - 256 T^{3} + 16568 T^{2} + \cdots - 618736)^{2} \)
$97$
\( (T^{4} - 32 T^{3} - 18152 T^{2} + \cdots + 9539216)^{2} \)
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