Newspace parameters
Level: | \( N \) | \(=\) | \( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 2016.k (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(16.0978410475\) |
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
|
|
Defining polynomial: | \( x^{16} + 24x^{14} + 192x^{12} + 672x^{10} + 1092x^{8} + 880x^{6} + 352x^{4} + 64x^{2} + 4 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
Coefficient ring index: | \( 2^{25} \) |
Twist minimal: | yes |
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 \( x^{16} + 24x^{14} + 192x^{12} + 672x^{10} + 1092x^{8} + 880x^{6} + 352x^{4} + 64x^{2} + 4 \) :
\(\beta_{1}\) | \(=\) | \( ( -\nu^{14} - 20\nu^{12} - 100\nu^{10} + 4\nu^{8} + 918\nu^{6} + 1440\nu^{4} + 664\nu^{2} + 72 ) / 4 \) |
\(\beta_{2}\) | \(=\) | \( ( -7\nu^{14} - 164\nu^{12} - 1250\nu^{10} - 3984\nu^{8} - 5334\nu^{6} - 3024\nu^{4} - 596\nu^{2} - 16 ) / 8 \) |
\(\beta_{3}\) | \(=\) | \( ( -5\nu^{15} - 110\nu^{13} - 728\nu^{11} - 1625\nu^{9} - 118\nu^{7} + 2276\nu^{5} + 1648\nu^{3} + 270\nu ) / 8 \) |
\(\beta_{4}\) | \(=\) | \( ( -11\nu^{14} - 261\nu^{12} - 2040\nu^{10} - 6818\nu^{8} - 10038\nu^{6} - 6666\nu^{4} - 1856\nu^{2} - 172 ) / 4 \) |
\(\beta_{5}\) | \(=\) | \( ( -6\nu^{14} - 141\nu^{12} - 1082\nu^{10} - 3502\nu^{8} - 4876\nu^{6} - 3046\nu^{4} - 804\nu^{2} - 68 ) / 2 \) |
\(\beta_{6}\) | \(=\) | \( ( - 11 \nu^{15} - 6 \nu^{14} - 260 \nu^{13} - 138 \nu^{12} - 2018 \nu^{11} - 1012 \nu^{10} - 6672 \nu^{9} - 2976 \nu^{8} - 9702 \nu^{7} - 3276 \nu^{6} - 6544 \nu^{5} - 1188 \nu^{4} - 2004 \nu^{3} + \cdots - 16 ) / 16 \) |
\(\beta_{7}\) | \(=\) | \( ( - 11 \nu^{15} + 6 \nu^{14} - 260 \nu^{13} + 138 \nu^{12} - 2018 \nu^{11} + 1012 \nu^{10} - 6672 \nu^{9} + 2976 \nu^{8} - 9702 \nu^{7} + 3276 \nu^{6} - 6544 \nu^{5} + 1188 \nu^{4} - 2004 \nu^{3} + \cdots + 16 ) / 16 \) |
\(\beta_{8}\) | \(=\) | \( ( 11\nu^{15} + 258\nu^{13} + 1971\nu^{11} + 6311\nu^{9} + 8530\nu^{7} + 4916\nu^{5} + 1046\nu^{3} + 38\nu ) / 8 \) |
\(\beta_{9}\) | \(=\) | \( ( - 11 \nu^{15} - 15 \nu^{14} - 261 \nu^{13} - 352 \nu^{12} - 2040 \nu^{11} - 2692 \nu^{10} - 6817 \nu^{9} - 8638 \nu^{8} - 10018 \nu^{7} - 11726 \nu^{6} - 6554 \nu^{5} - 6808 \nu^{4} - 1640 \nu^{3} + \cdots - 76 ) / 8 \) |
\(\beta_{10}\) | \(=\) | \( ( 11 \nu^{15} - 15 \nu^{14} + 261 \nu^{13} - 352 \nu^{12} + 2040 \nu^{11} - 2692 \nu^{10} + 6817 \nu^{9} - 8638 \nu^{8} + 10018 \nu^{7} - 11726 \nu^{6} + 6554 \nu^{5} - 6808 \nu^{4} + 1640 \nu^{3} + \cdots - 76 ) / 8 \) |
\(\beta_{11}\) | \(=\) | \( ( - 11 \nu^{15} - 27 \nu^{14} - 261 \nu^{13} - 636 \nu^{12} - 2040 \nu^{11} - 4904 \nu^{10} - 6817 \nu^{9} - 16030 \nu^{8} - 10018 \nu^{7} - 22886 \nu^{6} - 6554 \nu^{5} - 15248 \nu^{4} + \cdots - 428 ) / 8 \) |
\(\beta_{12}\) | \(=\) | \( ( -19\nu^{15} - 448\nu^{13} - 3460\nu^{11} - 11326\nu^{9} - 16094\nu^{7} - 10328\nu^{5} - 2904\nu^{3} - 316\nu ) / 8 \) |
\(\beta_{13}\) | \(=\) | \( ( -19\nu^{15} - 454\nu^{13} - 3598\nu^{11} - 12338\nu^{9} - 19070\nu^{7} - 13604\nu^{5} - 4092\nu^{3} - 372\nu ) / 8 \) |
\(\beta_{14}\) | \(=\) | \( ( 11\nu^{15} + 261\nu^{13} + 2042\nu^{11} + 6862\nu^{9} + 10334\nu^{7} + 7410\nu^{5} + 2412\nu^{3} + 252\nu ) / 4 \) |
\(\beta_{15}\) | \(=\) | \( ( 141 \nu^{15} - 6 \nu^{14} + 3324 \nu^{13} - 138 \nu^{12} + 25666 \nu^{11} - 1012 \nu^{10} + 84024 \nu^{9} - 2976 \nu^{8} + 119626 \nu^{7} - 3276 \nu^{6} + 77296 \nu^{5} - 1188 \nu^{4} + \cdots - 16 ) / 16 \) |
\(\nu\) | \(=\) | \( ( \beta_{13} - \beta_{12} + \beta_{10} - \beta_{9} - 2\beta_{8} ) / 4 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{11} + 2\beta_{10} + \beta_{9} + 2\beta_{7} - 2\beta_{6} - 2\beta_{5} - 2\beta_{2} + \beta _1 - 12 ) / 4 \) |
\(\nu^{3}\) | \(=\) | \( ( - 3 \beta_{15} + 3 \beta_{14} - 11 \beta_{13} + 7 \beta_{12} - 8 \beta_{10} + 8 \beta_{9} + 16 \beta_{8} - 6 \beta_{7} - 3 \beta_{6} - 4 \beta_{3} ) / 4 \) |
\(\nu^{4}\) | \(=\) | \( ( - 6 \beta_{11} - 10 \beta_{10} - 4 \beta_{9} - 14 \beta_{7} + 14 \beta_{6} + 13 \beta_{5} - 4 \beta_{4} + 12 \beta_{2} - 10 \beta _1 + 48 ) / 2 \) |
\(\nu^{5}\) | \(=\) | \( ( 50 \beta_{15} - 55 \beta_{14} + 126 \beta_{13} - 64 \beta_{12} + 83 \beta_{10} - 83 \beta_{9} - 188 \beta_{8} + 80 \beta_{7} + 30 \beta_{6} + 58 \beta_{3} ) / 4 \) |
\(\nu^{6}\) | \(=\) | \( ( 72 \beta_{11} + 109 \beta_{10} + 37 \beta_{9} + 181 \beta_{7} - 181 \beta_{6} - 161 \beta_{5} + 67 \beta_{4} - 134 \beta_{2} + 140 \beta _1 - 504 ) / 2 \) |
\(\nu^{7}\) | \(=\) | \( ( - 336 \beta_{15} + 378 \beta_{14} - 752 \beta_{13} + 339 \beta_{12} - 478 \beta_{10} + 478 \beta_{9} + 1142 \beta_{8} - 497 \beta_{7} - 161 \beta_{6} - 372 \beta_{3} ) / 2 \) |
\(\nu^{8}\) | \(=\) | \( - 438 \beta_{11} - 636 \beta_{10} - 198 \beta_{9} - 1130 \beta_{7} + 1130 \beta_{6} + 990 \beta_{5} - 450 \beta_{4} + 784 \beta_{2} - 896 \beta _1 + 2910 \) |
\(\nu^{9}\) | \(=\) | \( ( 4248 \beta_{15} - 4812 \beta_{14} + 9127 \beta_{13} - 3907 \beta_{12} + 5723 \beta_{10} - 5723 \beta_{9} - 13942 \beta_{8} + 6108 \beta_{7} + 1860 \beta_{6} + 4632 \beta_{3} ) / 2 \) |
\(\nu^{10}\) | \(=\) | \( ( 10707 \beta_{11} + 15294 \beta_{10} + 4587 \beta_{9} + 27862 \beta_{7} - 27862 \beta_{6} - 24294 \beta_{5} + 11376 \beta_{4} - 18846 \beta_{2} + 22287 \beta _1 - 69716 ) / 2 \) |
\(\nu^{11}\) | \(=\) | \( ( - 52613 \beta_{15} + 59741 \beta_{14} - 111425 \beta_{13} + 46733 \beta_{12} - 69508 \beta_{10} + 69508 \beta_{9} + 170488 \beta_{8} - 74866 \beta_{7} - 22253 \beta_{6} - 57084 \beta_{3} ) / 2 \) |
\(\nu^{12}\) | \(=\) | \( - 65522 \beta_{11} - 93010 \beta_{10} - 27488 \beta_{9} - 170986 \beta_{7} + 170986 \beta_{6} + 148879 \beta_{5} - 70448 \beta_{4} + 114548 \beta_{2} - 137214 \beta _1 + 423384 \) |
\(\nu^{13}\) | \(=\) | \( ( 646906 \beta_{15} - 735189 \beta_{14} + 1363030 \beta_{13} - 567236 \beta_{12} + 848621 \beta_{10} - 848621 \beta_{9} - 2086356 \beta_{8} + 916968 \beta_{7} + 270062 \beta_{6} + \cdots + 700742 \beta_{3} ) / 2 \) |
\(\nu^{14}\) | \(=\) | \( 802232 \beta_{11} + 1136119 \beta_{10} + 333887 \beta_{9} + 2095431 \beta_{7} - 2095431 \beta_{6} - 1823791 \beta_{5} + 866233 \beta_{4} - 1398722 \beta_{2} + 1683568 \beta _1 - 5168936 \) |
\(\nu^{15}\) | \(=\) | \( - 3966632 \beta_{15} + 4509412 \beta_{14} - 8342408 \beta_{13} + 3461601 \beta_{12} - 5190196 \beta_{10} + 5190196 \beta_{9} + 12770386 \beta_{8} - 5614479 \beta_{7} + \cdots - 4294488 \beta_{3} \) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1889.1 |
|
0 | 0 | 0 | −2.61313 | 0 | −1.25928 | − | 2.32685i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1889.2 | 0 | 0 | 0 | −2.61313 | 0 | −1.25928 | + | 2.32685i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1889.3 | 0 | 0 | 0 | −2.61313 | 0 | 1.25928 | − | 2.32685i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1889.4 | 0 | 0 | 0 | −2.61313 | 0 | 1.25928 | + | 2.32685i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1889.5 | 0 | 0 | 0 | −1.08239 | 0 | −2.10100 | − | 1.60804i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1889.6 | 0 | 0 | 0 | −1.08239 | 0 | −2.10100 | + | 1.60804i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1889.7 | 0 | 0 | 0 | −1.08239 | 0 | 2.10100 | − | 1.60804i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1889.8 | 0 | 0 | 0 | −1.08239 | 0 | 2.10100 | + | 1.60804i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1889.9 | 0 | 0 | 0 | 1.08239 | 0 | −2.10100 | − | 1.60804i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1889.10 | 0 | 0 | 0 | 1.08239 | 0 | −2.10100 | + | 1.60804i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1889.11 | 0 | 0 | 0 | 1.08239 | 0 | 2.10100 | − | 1.60804i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1889.12 | 0 | 0 | 0 | 1.08239 | 0 | 2.10100 | + | 1.60804i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1889.13 | 0 | 0 | 0 | 2.61313 | 0 | −1.25928 | − | 2.32685i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1889.14 | 0 | 0 | 0 | 2.61313 | 0 | −1.25928 | + | 2.32685i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1889.15 | 0 | 0 | 0 | 2.61313 | 0 | 1.25928 | − | 2.32685i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1889.16 | 0 | 0 | 0 | 2.61313 | 0 | 1.25928 | + | 2.32685i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
4.b | odd | 2 | 1 | inner |
7.b | odd | 2 | 1 | inner |
12.b | even | 2 | 1 | inner |
21.c | even | 2 | 1 | inner |
28.d | even | 2 | 1 | inner |
84.h | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 2016.2.k.a | ✓ | 16 |
3.b | odd | 2 | 1 | inner | 2016.2.k.a | ✓ | 16 |
4.b | odd | 2 | 1 | inner | 2016.2.k.a | ✓ | 16 |
7.b | odd | 2 | 1 | inner | 2016.2.k.a | ✓ | 16 |
8.b | even | 2 | 1 | 4032.2.k.g | 16 | ||
8.d | odd | 2 | 1 | 4032.2.k.g | 16 | ||
12.b | even | 2 | 1 | inner | 2016.2.k.a | ✓ | 16 |
21.c | even | 2 | 1 | inner | 2016.2.k.a | ✓ | 16 |
24.f | even | 2 | 1 | 4032.2.k.g | 16 | ||
24.h | odd | 2 | 1 | 4032.2.k.g | 16 | ||
28.d | even | 2 | 1 | inner | 2016.2.k.a | ✓ | 16 |
56.e | even | 2 | 1 | 4032.2.k.g | 16 | ||
56.h | odd | 2 | 1 | 4032.2.k.g | 16 | ||
84.h | odd | 2 | 1 | inner | 2016.2.k.a | ✓ | 16 |
168.e | odd | 2 | 1 | 4032.2.k.g | 16 | ||
168.i | even | 2 | 1 | 4032.2.k.g | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2016.2.k.a | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
2016.2.k.a | ✓ | 16 | 3.b | odd | 2 | 1 | inner |
2016.2.k.a | ✓ | 16 | 4.b | odd | 2 | 1 | inner |
2016.2.k.a | ✓ | 16 | 7.b | odd | 2 | 1 | inner |
2016.2.k.a | ✓ | 16 | 12.b | even | 2 | 1 | inner |
2016.2.k.a | ✓ | 16 | 21.c | even | 2 | 1 | inner |
2016.2.k.a | ✓ | 16 | 28.d | even | 2 | 1 | inner |
2016.2.k.a | ✓ | 16 | 84.h | odd | 2 | 1 | inner |
4032.2.k.g | 16 | 8.b | even | 2 | 1 | ||
4032.2.k.g | 16 | 8.d | odd | 2 | 1 | ||
4032.2.k.g | 16 | 24.f | even | 2 | 1 | ||
4032.2.k.g | 16 | 24.h | odd | 2 | 1 | ||
4032.2.k.g | 16 | 56.e | even | 2 | 1 | ||
4032.2.k.g | 16 | 56.h | odd | 2 | 1 | ||
4032.2.k.g | 16 | 168.e | odd | 2 | 1 | ||
4032.2.k.g | 16 | 168.i | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 8T_{5}^{2} + 8 \)
acting on \(S_{2}^{\mathrm{new}}(2016, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{16} \)
$3$
\( T^{16} \)
$5$
\( (T^{4} - 8 T^{2} + 8)^{4} \)
$7$
\( (T^{8} + 4 T^{6} + 70 T^{4} + 196 T^{2} + \cdots + 2401)^{2} \)
$11$
\( (T^{4} + 20 T^{2} + 28)^{4} \)
$13$
\( (T^{4} + 16 T^{2} + 32)^{4} \)
$17$
\( (T^{4} - 40 T^{2} + 8)^{4} \)
$19$
\( (T^{4} + 64 T^{2} + 224)^{4} \)
$23$
\( (T^{4} + 52 T^{2} + 28)^{4} \)
$29$
\( (T^{2} + 2)^{8} \)
$31$
\( (T^{4} + 64 T^{2} + 896)^{4} \)
$37$
\( (T^{2} - 8)^{8} \)
$41$
\( (T^{4} - 136 T^{2} + 4232)^{4} \)
$43$
\( (T^{4} - 152 T^{2} + 5488)^{4} \)
$47$
\( (T^{4} - 128 T^{2} + 896)^{4} \)
$53$
\( (T^{4} + 36 T^{2} + 196)^{4} \)
$59$
\( (T^{4} - 128 T^{2} + 3584)^{4} \)
$61$
\( (T^{4} + 160 T^{2} + 6272)^{4} \)
$67$
\( (T^{4} - 208 T^{2} + 448)^{4} \)
$71$
\( (T^{4} + 244 T^{2} + 14812)^{4} \)
$73$
\( (T^{4} + 144 T^{2} + 2592)^{4} \)
$79$
\( (T^{4} - 336 T^{2} + 21952)^{4} \)
$83$
\( (T^{4} - 256 T^{2} + 896)^{4} \)
$89$
\( (T^{4} - 328 T^{2} + 7688)^{4} \)
$97$
\( (T^{4} + 80 T^{2} + 1568)^{4} \)
show more
show less