Newspace parameters
Level: | \( N \) | \(=\) | \( 864 = 2^{5} \cdot 3^{3} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 864.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(23.5422948407\) |
Analytic rank: | \(0\) |
Dimension: | \(16\) |
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: | \( x^{16} + x^{14} - 8x^{12} + 4x^{10} + 160x^{8} + 64x^{6} - 2048x^{4} + 4096x^{2} + 65536 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
Coefficient ring index: | \( 2^{30}\cdot 3^{10} \) |
Twist minimal: | no (minimal twist has level 216) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) 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 \( x^{16} + x^{14} - 8x^{12} + 4x^{10} + 160x^{8} + 64x^{6} - 2048x^{4} + 4096x^{2} + 65536 \) :
\(\beta_{1}\) | \(=\) | \( ( -\nu^{15} - 9\nu^{13} - 196\nu^{9} + 576\nu^{7} + 1728\nu^{5} + 512\nu^{3} ) / 32768 \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{12} - 3\nu^{10} + 4\nu^{8} - 12\nu^{6} + 208\nu^{4} + 256\nu^{2} - 1024 ) / 512 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{14} + \nu^{12} + 24\nu^{10} + 164\nu^{8} + 544\nu^{6} - 320\nu^{4} + 3584\nu^{2} + 16384 ) / 8192 \) |
\(\beta_{4}\) | \(=\) | \( ( \nu^{15} - 7\nu^{13} + 48\nu^{11} + 132\nu^{9} - 384\nu^{7} + 3136\nu^{5} + 3584\nu^{3} ) / 32768 \) |
\(\beta_{5}\) | \(=\) | \( ( \nu^{12} + \nu^{10} - 8\nu^{8} + 4\nu^{6} - 96\nu^{4} - 192\nu^{2} - 1024 ) / 256 \) |
\(\beta_{6}\) | \(=\) | \( ( \nu^{14} - 7\nu^{12} + 48\nu^{10} + 132\nu^{8} - 1408\nu^{6} + 64\nu^{4} + 13824\nu^{2} + 12288 ) / 4096 \) |
\(\beta_{7}\) | \(=\) | \( ( 3\nu^{15} - 37\nu^{13} - 192\nu^{11} + 460\nu^{9} + 576\nu^{7} - 1600\nu^{5} + 29184\nu^{3} + 49152\nu ) / 32768 \) |
\(\beta_{8}\) | \(=\) | \( ( -7\nu^{14} - 7\nu^{12} - 40\nu^{10} + 516\nu^{8} - 1248\nu^{6} - 7488\nu^{4} - 14848\nu^{2} + 16384 ) / 8192 \) |
\(\beta_{9}\) | \(=\) | \( ( -5\nu^{15} + 19\nu^{13} - 64\nu^{11} - 84\nu^{9} - 448\nu^{7} + 8128\nu^{5} - 15872\nu^{3} - 49152\nu ) / 32768 \) |
\(\beta_{10}\) | \(=\) | \( ( -\nu^{15} + 3\nu^{13} + 12\nu^{11} + 92\nu^{9} + 496\nu^{7} + 64\nu^{5} + 2816\nu^{3} - 2048\nu ) / 4096 \) |
\(\beta_{11}\) | \(=\) | \( ( -7\nu^{14} + 9\nu^{12} + 168\nu^{10} - 188\nu^{8} - 416\nu^{6} + 9152\nu^{4} - 6656\nu^{2} - 65536 ) / 8192 \) |
\(\beta_{12}\) | \(=\) | \( ( -\nu^{15} + 7\nu^{13} + 16\nu^{11} - 68\nu^{9} - 128\nu^{7} + 1216\nu^{5} + 10752\nu^{3} - 28672\nu ) / 4096 \) |
\(\beta_{13}\) | \(=\) | \( ( 15\nu^{15} + 7\nu^{13} - 128\nu^{11} - 132\nu^{9} + 3136\nu^{7} + 2752\nu^{5} - 32256\nu^{3} - 196608\nu ) / 32768 \) |
\(\beta_{14}\) | \(=\) | \( ( 17\nu^{15} + 9\nu^{13} - 80\nu^{11} + 196\nu^{9} + 2176\nu^{7} + 4160\nu^{5} - 29184\nu^{3} + 327680\nu ) / 32768 \) |
\(\beta_{15}\) | \(=\) | \( ( -9\nu^{14} - \nu^{12} + 16\nu^{10} - 164\nu^{8} + 128\nu^{6} - 576\nu^{4} + 35328\nu^{2} - 36864 ) / 4096 \) |
\(\nu\) | \(=\) | \( ( \beta_{14} - \beta_{13} - \beta_{4} + \beta_1 ) / 16 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{15} - \beta_{11} - \beta_{8} + \beta_{6} + 2\beta_{3} + \beta_{2} - 2 ) / 16 \) |
\(\nu^{3}\) | \(=\) | \( ( \beta_{14} - \beta_{13} + 4\beta_{12} - 4\beta_{9} + 4\beta_{7} - \beta_{4} + \beta_1 ) / 16 \) |
\(\nu^{4}\) | \(=\) | \( ( -\beta_{15} + 5\beta_{11} - 3\beta_{8} - \beta_{6} - 8\beta_{5} - 2\beta_{3} + 11\beta_{2} + 34 ) / 16 \) |
\(\nu^{5}\) | \(=\) | \( ( 7\beta_{14} + \beta_{13} + 4\beta_{12} + 28\beta_{9} + 4\beta_{7} + 57\beta_{4} + 31\beta_1 ) / 16 \) |
\(\nu^{6}\) | \(=\) | \( ( 5\beta_{15} - \beta_{11} - 9\beta_{8} - 27\beta_{6} - 8\beta_{5} + 74\beta_{3} - 15\beta_{2} - 74 ) / 16 \) |
\(\nu^{7}\) | \(=\) | \( ( 13\beta_{14} + 27\beta_{13} - 4\beta_{12} + 64\beta_{10} - 28\beta_{9} - 4\beta_{7} - 77\beta_{4} + 197\beta_1 ) / 16 \) |
\(\nu^{8}\) | \(=\) | \( ( -17\beta_{15} + 13\beta_{11} + 85\beta_{8} + 15\beta_{6} - 56\beta_{5} + 350\beta_{3} + 131\beta_{2} - 926 ) / 16 \) |
\(\nu^{9}\) | \(=\) | \( ( - 57 \beta_{14} + 17 \beta_{13} - 108 \beta_{12} + 192 \beta_{10} + 140 \beta_{9} + 148 \beta_{7} + 633 \beta_{4} - 1009 \beta_1 ) / 16 \) |
\(\nu^{10}\) | \(=\) | \( ( -35\beta_{15} + 311\beta_{11} - 49\beta_{8} + 189\beta_{6} + 280\beta_{5} + 826\beta_{3} - 647\beta_{2} - 122 ) / 16 \) |
\(\nu^{11}\) | \(=\) | \( ( - 91 \beta_{14} - 93 \beta_{13} - 68 \beta_{12} + 320 \beta_{10} - 1372 \beta_{9} - 1348 \beta_{7} + 2843 \beta_{4} + 701 \beta_1 ) / 16 \) |
\(\nu^{12}\) | \(=\) | \( ( - 25 \beta_{15} + 85 \beta_{11} + 285 \beta_{8} + 135 \beta_{6} + 2632 \beta_{5} + 1870 \beta_{3} + 3003 \beta_{2} + 12274 ) / 16 \) |
\(\nu^{13}\) | \(=\) | \( ( 2495 \beta_{14} + 25 \beta_{13} + 2420 \beta_{12} + 960 \beta_{10} + 2476 \beta_{9} - 2444 \beta_{7} - 10623 \beta_{4} - 14585 \beta_1 ) / 16 \) |
\(\nu^{14}\) | \(=\) | \( ( - 2971 \beta_{15} - 3953 \beta_{11} - 5529 \beta_{8} + 3653 \beta_{6} + 1624 \beta_{5} + 3914 \beta_{3} - 863 \beta_{2} - 61322 ) / 16 \) |
\(\nu^{15}\) | \(=\) | \( ( 8813 \beta_{14} + 13211 \beta_{13} + 6044 \beta_{12} - 9408 \beta_{10} - 19516 \beta_{9} - 356 \beta_{7} + 25171 \beta_{4} - 27707 \beta_1 ) / 16 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/864\mathbb{Z}\right)^\times\).
\(n\) | \(325\) | \(353\) | \(703\) |
\(\chi(n)\) | \(-1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
271.1 |
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0 | 0 | 0 | − | 8.60639i | 0 | − | 4.81948i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
271.2 | 0 | 0 | 0 | − | 8.60639i | 0 | 4.81948i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
271.3 | 0 | 0 | 0 | − | 5.14075i | 0 | − | 12.6525i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
271.4 | 0 | 0 | 0 | − | 5.14075i | 0 | 12.6525i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
271.5 | 0 | 0 | 0 | − | 4.41296i | 0 | − | 3.59985i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
271.6 | 0 | 0 | 0 | − | 4.41296i | 0 | 3.59985i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
271.7 | 0 | 0 | 0 | − | 0.169019i | 0 | − | 4.44165i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
271.8 | 0 | 0 | 0 | − | 0.169019i | 0 | 4.44165i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
271.9 | 0 | 0 | 0 | 0.169019i | 0 | − | 4.44165i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
271.10 | 0 | 0 | 0 | 0.169019i | 0 | 4.44165i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
271.11 | 0 | 0 | 0 | 4.41296i | 0 | − | 3.59985i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
271.12 | 0 | 0 | 0 | 4.41296i | 0 | 3.59985i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
271.13 | 0 | 0 | 0 | 5.14075i | 0 | − | 12.6525i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
271.14 | 0 | 0 | 0 | 5.14075i | 0 | 12.6525i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
271.15 | 0 | 0 | 0 | 8.60639i | 0 | − | 4.81948i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
271.16 | 0 | 0 | 0 | 8.60639i | 0 | 4.81948i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
8.d | odd | 2 | 1 | inner |
24.f | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 864.3.b.a | 16 | |
3.b | odd | 2 | 1 | inner | 864.3.b.a | 16 | |
4.b | odd | 2 | 1 | 216.3.b.a | ✓ | 16 | |
8.b | even | 2 | 1 | 216.3.b.a | ✓ | 16 | |
8.d | odd | 2 | 1 | inner | 864.3.b.a | 16 | |
12.b | even | 2 | 1 | 216.3.b.a | ✓ | 16 | |
24.f | even | 2 | 1 | inner | 864.3.b.a | 16 | |
24.h | odd | 2 | 1 | 216.3.b.a | ✓ | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
216.3.b.a | ✓ | 16 | 4.b | odd | 2 | 1 | |
216.3.b.a | ✓ | 16 | 8.b | even | 2 | 1 | |
216.3.b.a | ✓ | 16 | 12.b | even | 2 | 1 | |
216.3.b.a | ✓ | 16 | 24.h | odd | 2 | 1 | |
864.3.b.a | 16 | 1.a | even | 1 | 1 | trivial | |
864.3.b.a | 16 | 3.b | odd | 2 | 1 | inner | |
864.3.b.a | 16 | 8.d | odd | 2 | 1 | inner | |
864.3.b.a | 16 | 24.f | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} + 120T_{5}^{6} + 3918T_{5}^{4} + 38232T_{5}^{2} + 1089 \)
acting on \(S_{3}^{\mathrm{new}}(864, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{16} \)
$3$
\( T^{16} \)
$5$
\( (T^{8} + 120 T^{6} + 3918 T^{4} + \cdots + 1089)^{2} \)
$7$
\( (T^{8} + 216 T^{6} + 9966 T^{4} + \cdots + 950625)^{2} \)
$11$
\( (T^{8} - 508 T^{6} + 60486 T^{4} + \cdots + 4060225)^{2} \)
$13$
\( (T^{8} + 600 T^{6} + 90768 T^{4} + \cdots + 58982400)^{2} \)
$17$
\( (T^{8} - 1336 T^{6} + \cdots + 1435500544)^{2} \)
$19$
\( (T^{4} + 8 T^{3} - 756 T^{2} + \cdots + 129088)^{4} \)
$23$
\( (T^{8} + 2376 T^{6} + \cdots + 49861103616)^{2} \)
$29$
\( (T^{8} + 3768 T^{6} + \cdots + 197136000000)^{2} \)
$31$
\( (T^{8} + 3144 T^{6} + \cdots + 69463346481)^{2} \)
$37$
\( (T^{8} + 7128 T^{6} + \cdots + 4052652134400)^{2} \)
$41$
\( (T^{8} - 5368 T^{6} + \cdots + 84239257600)^{2} \)
$43$
\( (T^{4} + 32 T^{3} - 6096 T^{2} + \cdots + 5478400)^{4} \)
$47$
\( (T^{8} + 10176 T^{6} + \cdots + 7614420811776)^{2} \)
$53$
\( (T^{8} + 10296 T^{6} + \cdots + 269102600001)^{2} \)
$59$
\( (T^{8} - 8416 T^{6} + \cdots + 12687769600)^{2} \)
$61$
\( (T^{8} + 25728 T^{6} + \cdots + 872632396283904)^{2} \)
$67$
\( (T^{4} + 32 T^{3} - 13440 T^{2} + \cdots + 8139520)^{4} \)
$71$
\( (T^{8} + 25416 T^{6} + \cdots + 90326016000000)^{2} \)
$73$
\( (T^{4} + 40 T^{3} - 10962 T^{2} + \cdots + 15407905)^{4} \)
$79$
\( (T^{8} + 39000 T^{6} + \cdots + 62\!\cdots\!44)^{2} \)
$83$
\( (T^{8} - 6940 T^{6} + \cdots + 5715511201)^{2} \)
$89$
\( (T^{8} - 22240 T^{6} + \cdots + 338984068710400)^{2} \)
$97$
\( (T^{4} + 52 T^{3} - 12810 T^{2} + \cdots + 4535185)^{4} \)
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