Newspace parameters
| Level: | \( N \) | \(=\) | \( 240 = 2^{4} \cdot 3 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 240.y (of order \(4\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(1.91640964851\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(8\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{16} - 6 x^{15} + 14 x^{14} - 10 x^{13} - 26 x^{12} + 78 x^{11} - 66 x^{10} - 74 x^{9} + 233 x^{8} + \cdots + 256 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{5} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{16} - 6 x^{15} + 14 x^{14} - 10 x^{13} - 26 x^{12} + 78 x^{11} - 66 x^{10} - 74 x^{9} + 233 x^{8} + \cdots + 256 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 15 \nu^{15} + 62 \nu^{14} - 94 \nu^{13} - 26 \nu^{12} + 342 \nu^{11} - 530 \nu^{10} - 2 \nu^{9} + \cdots + 1792 ) / 256 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 13 \nu^{15} + 42 \nu^{14} - 36 \nu^{13} - 70 \nu^{12} + 238 \nu^{11} - 202 \nu^{10} - 250 \nu^{9} + \cdots + 64 ) / 128 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 27 \nu^{15} - 88 \nu^{14} + 78 \nu^{13} + 142 \nu^{12} - 498 \nu^{11} + 438 \nu^{10} + 502 \nu^{9} + \cdots - 256 ) / 256 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 11 \nu^{15} - 43 \nu^{14} + 66 \nu^{13} + 16 \nu^{12} - 232 \nu^{11} + 368 \nu^{10} - 24 \nu^{9} + \cdots - 1664 ) / 128 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 4 \nu^{15} - 21 \nu^{14} + 40 \nu^{13} - 8 \nu^{12} - 114 \nu^{11} + 226 \nu^{10} - 78 \nu^{9} + \cdots - 1120 ) / 32 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 17 \nu^{15} + 98 \nu^{14} - 192 \nu^{13} + 50 \nu^{12} + 526 \nu^{11} - 1098 \nu^{10} + 454 \nu^{9} + \cdots + 5696 ) / 128 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 81 \nu^{15} - 348 \nu^{14} + 538 \nu^{13} + 114 \nu^{12} - 1918 \nu^{11} + 3034 \nu^{10} + \cdots - 12032 ) / 256 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 87 \nu^{15} - 378 \nu^{14} + 582 \nu^{13} + 130 \nu^{12} - 2094 \nu^{11} + 3290 \nu^{10} + \cdots - 12544 ) / 256 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 50 \nu^{15} + 213 \nu^{14} - 322 \nu^{13} - 82 \nu^{12} + 1170 \nu^{11} - 1806 \nu^{10} + \cdots + 7104 ) / 128 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 81 \nu^{15} + 408 \nu^{14} - 718 \nu^{13} + 38 \nu^{12} + 2222 \nu^{11} - 4042 \nu^{10} + \cdots + 18048 ) / 256 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 105 \nu^{15} - 484 \nu^{14} + 790 \nu^{13} + 82 \nu^{12} - 2662 \nu^{11} + 4466 \nu^{10} + \cdots - 18560 ) / 256 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 117 \nu^{15} - 612 \nu^{14} + 1114 \nu^{13} - 134 \nu^{12} - 3318 \nu^{11} + 6306 \nu^{10} + \cdots - 29952 ) / 256 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 103 \nu^{15} + 474 \nu^{14} - 774 \nu^{13} - 66 \nu^{12} + 2590 \nu^{11} - 4378 \nu^{10} + \cdots + 18432 ) / 128 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 223 \nu^{15} + 1030 \nu^{14} - 1694 \nu^{13} - 122 \nu^{12} + 5638 \nu^{11} - 9602 \nu^{10} + \cdots + 40448 ) / 256 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 127 \nu^{15} - 602 \nu^{14} + 1016 \nu^{13} + 26 \nu^{12} - 3290 \nu^{11} + 5742 \nu^{10} + \cdots - 25152 ) / 128 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{15} + \beta_{14} - \beta_{13} - \beta_{12} + \beta_{11} + \beta_{10} - \beta_{8} - \beta_{7} + \cdots - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{15} + \beta_{14} - \beta_{13} - 2 \beta_{12} + 2 \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} + \cdots + 1 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{15} + \beta_{14} - \beta_{13} - \beta_{12} - \beta_{11} - \beta_{10} + 2 \beta_{9} - \beta_{8} + \cdots - 4 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( \beta_{15} + \beta_{14} - \beta_{13} + \beta_{10} + \beta_{9} + \beta_{8} - 2 \beta_{7} + \beta_{6} + \cdots + 3 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - \beta_{15} + \beta_{14} - \beta_{13} + 3 \beta_{12} - \beta_{11} + \beta_{10} + 2 \beta_{9} + \cdots - 3 \beta_1 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 5 \beta_{15} + 3 \beta_{14} - 3 \beta_{13} - 2 \beta_{12} + 8 \beta_{11} + 11 \beta_{10} - 5 \beta_{9} + \cdots + 7 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 3 \beta_{15} - 5 \beta_{14} + 5 \beta_{13} - \beta_{12} + 5 \beta_{11} + 5 \beta_{10} - 6 \beta_{9} + \cdots - 2 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 9 \beta_{15} + 3 \beta_{14} + \beta_{13} + 4 \beta_{12} - 6 \beta_{11} + 11 \beta_{10} - 3 \beta_{9} + \cdots + 1 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 7 \beta_{15} - \beta_{14} - 3 \beta_{13} + 11 \beta_{12} - 3 \beta_{11} + 3 \beta_{10} + 2 \beta_{9} + \cdots - 6 ) / 2 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 23 \beta_{15} + 17 \beta_{14} - 25 \beta_{13} + 22 \beta_{12} + 2 \beta_{11} + 17 \beta_{10} + \cdots - 7 ) / 2 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 27 \beta_{15} - 3 \beta_{14} - 29 \beta_{13} + 3 \beta_{12} + 35 \beta_{11} - 5 \beta_{10} + \cdots - 52 ) / 2 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 19 \beta_{15} - 19 \beta_{14} + 35 \beta_{13} + 40 \beta_{12} + 8 \beta_{11} - 11 \beta_{10} + \cdots - 1 ) / 2 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 27 \beta_{15} - 11 \beta_{14} + 59 \beta_{13} + 47 \beta_{12} - 37 \beta_{11} - 51 \beta_{10} + \cdots - 16 ) / 2 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 49 \beta_{15} + 23 \beta_{14} + 89 \beta_{13} + 38 \beta_{12} + 40 \beta_{11} + 119 \beta_{10} + \cdots + 203 ) / 2 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 15 \beta_{15} + 71 \beta_{14} - 87 \beta_{13} - 77 \beta_{12} + 33 \beta_{11} + 113 \beta_{10} + \cdots - 2 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/240\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) | \(97\) | \(161\) | \(181\) |
| \(\chi(n)\) | \(-1\) | \(\beta_{8}\) | \(1\) | \(\beta_{8}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 163.1 |
|
−1.32675 | + | 0.489639i | 1.00000 | 1.52051 | − | 1.29925i | 2.06823 | + | 0.849960i | −1.32675 | + | 0.489639i | −2.08016 | + | 2.08016i | −1.38116 | + | 2.46828i | 1.00000 | −3.16019 | − | 0.114996i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.2 | −0.677660 | − | 1.24128i | 1.00000 | −1.08155 | + | 1.68233i | 0.311968 | + | 2.21420i | −0.677660 | − | 1.24128i | −1.96597 | + | 1.96597i | 2.82117 | + | 0.202464i | 1.00000 | 2.53703 | − | 1.88771i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.3 | −0.481284 | − | 1.32980i | 1.00000 | −1.53673 | + | 1.28002i | −2.17005 | − | 0.539352i | −0.481284 | − | 1.32980i | 3.00806 | − | 3.00806i | 2.44178 | + | 1.42749i | 1.00000 | 0.327178 | + | 3.14531i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.4 | −0.0770377 | + | 1.41211i | 1.00000 | −1.98813 | − | 0.217572i | −2.13688 | + | 0.658594i | −0.0770377 | + | 1.41211i | −3.54781 | + | 3.54781i | 0.460397 | − | 2.79070i | 1.00000 | −0.765389 | − | 3.06825i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.5 | 0.429059 | − | 1.34756i | 1.00000 | −1.63182 | − | 1.15636i | 2.23531 | + | 0.0583995i | 0.429059 | − | 1.34756i | 0.747384 | − | 0.747384i | −2.25841 | + | 1.70282i | 1.00000 | 1.03777 | − | 2.98714i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.6 | 0.812425 | + | 1.15757i | 1.00000 | −0.679932 | + | 1.88088i | 1.45639 | + | 1.69674i | 0.812425 | + | 1.15757i | 1.12791 | − | 1.12791i | −2.72964 | + | 0.741001i | 1.00000 | −0.780893 | + | 3.06434i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.7 | 1.11192 | − | 0.873858i | 1.00000 | 0.472743 | − | 1.94333i | −1.61356 | − | 1.54804i | 1.11192 | − | 0.873858i | 0.143894 | − | 0.143894i | −1.17254 | − | 2.57394i | 1.00000 | −3.14692 | − | 0.311271i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.8 | 1.20932 | + | 0.733173i | 1.00000 | 0.924916 | + | 1.77328i | −2.15140 | + | 0.609492i | 1.20932 | + | 0.733173i | 0.566689 | − | 0.566689i | −0.181602 | + | 2.82259i | 1.00000 | −3.04860 | − | 0.840275i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 187.1 | −1.32675 | − | 0.489639i | 1.00000 | 1.52051 | + | 1.29925i | 2.06823 | − | 0.849960i | −1.32675 | − | 0.489639i | −2.08016 | − | 2.08016i | −1.38116 | − | 2.46828i | 1.00000 | −3.16019 | + | 0.114996i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 187.2 | −0.677660 | + | 1.24128i | 1.00000 | −1.08155 | − | 1.68233i | 0.311968 | − | 2.21420i | −0.677660 | + | 1.24128i | −1.96597 | − | 1.96597i | 2.82117 | − | 0.202464i | 1.00000 | 2.53703 | + | 1.88771i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 187.3 | −0.481284 | + | 1.32980i | 1.00000 | −1.53673 | − | 1.28002i | −2.17005 | + | 0.539352i | −0.481284 | + | 1.32980i | 3.00806 | + | 3.00806i | 2.44178 | − | 1.42749i | 1.00000 | 0.327178 | − | 3.14531i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 187.4 | −0.0770377 | − | 1.41211i | 1.00000 | −1.98813 | + | 0.217572i | −2.13688 | − | 0.658594i | −0.0770377 | − | 1.41211i | −3.54781 | − | 3.54781i | 0.460397 | + | 2.79070i | 1.00000 | −0.765389 | + | 3.06825i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 187.5 | 0.429059 | + | 1.34756i | 1.00000 | −1.63182 | + | 1.15636i | 2.23531 | − | 0.0583995i | 0.429059 | + | 1.34756i | 0.747384 | + | 0.747384i | −2.25841 | − | 1.70282i | 1.00000 | 1.03777 | + | 2.98714i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 187.6 | 0.812425 | − | 1.15757i | 1.00000 | −0.679932 | − | 1.88088i | 1.45639 | − | 1.69674i | 0.812425 | − | 1.15757i | 1.12791 | + | 1.12791i | −2.72964 | − | 0.741001i | 1.00000 | −0.780893 | − | 3.06434i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 187.7 | 1.11192 | + | 0.873858i | 1.00000 | 0.472743 | + | 1.94333i | −1.61356 | + | 1.54804i | 1.11192 | + | 0.873858i | 0.143894 | + | 0.143894i | −1.17254 | + | 2.57394i | 1.00000 | −3.14692 | + | 0.311271i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 187.8 | 1.20932 | − | 0.733173i | 1.00000 | 0.924916 | − | 1.77328i | −2.15140 | − | 0.609492i | 1.20932 | − | 0.733173i | 0.566689 | + | 0.566689i | −0.181602 | − | 2.82259i | 1.00000 | −3.04860 | + | 0.840275i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 80.s | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 240.2.y.e | ✓ | 16 |
| 3.b | odd | 2 | 1 | 720.2.z.f | 16 | ||
| 4.b | odd | 2 | 1 | 960.2.y.e | 16 | ||
| 5.c | odd | 4 | 1 | 240.2.bc.e | yes | 16 | |
| 8.b | even | 2 | 1 | 1920.2.y.i | 16 | ||
| 8.d | odd | 2 | 1 | 1920.2.y.j | 16 | ||
| 15.e | even | 4 | 1 | 720.2.bd.f | 16 | ||
| 16.e | even | 4 | 1 | 960.2.bc.e | 16 | ||
| 16.e | even | 4 | 1 | 1920.2.bc.j | 16 | ||
| 16.f | odd | 4 | 1 | 240.2.bc.e | yes | 16 | |
| 16.f | odd | 4 | 1 | 1920.2.bc.i | 16 | ||
| 20.e | even | 4 | 1 | 960.2.bc.e | 16 | ||
| 40.i | odd | 4 | 1 | 1920.2.bc.i | 16 | ||
| 40.k | even | 4 | 1 | 1920.2.bc.j | 16 | ||
| 48.k | even | 4 | 1 | 720.2.bd.f | 16 | ||
| 80.i | odd | 4 | 1 | 960.2.y.e | 16 | ||
| 80.j | even | 4 | 1 | 1920.2.y.i | 16 | ||
| 80.s | even | 4 | 1 | inner | 240.2.y.e | ✓ | 16 |
| 80.t | odd | 4 | 1 | 1920.2.y.j | 16 | ||
| 240.z | odd | 4 | 1 | 720.2.z.f | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 240.2.y.e | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 240.2.y.e | ✓ | 16 | 80.s | even | 4 | 1 | inner |
| 240.2.bc.e | yes | 16 | 5.c | odd | 4 | 1 | |
| 240.2.bc.e | yes | 16 | 16.f | odd | 4 | 1 | |
| 720.2.z.f | 16 | 3.b | odd | 2 | 1 | ||
| 720.2.z.f | 16 | 240.z | odd | 4 | 1 | ||
| 720.2.bd.f | 16 | 15.e | even | 4 | 1 | ||
| 720.2.bd.f | 16 | 48.k | even | 4 | 1 | ||
| 960.2.y.e | 16 | 4.b | odd | 2 | 1 | ||
| 960.2.y.e | 16 | 80.i | odd | 4 | 1 | ||
| 960.2.bc.e | 16 | 16.e | even | 4 | 1 | ||
| 960.2.bc.e | 16 | 20.e | even | 4 | 1 | ||
| 1920.2.y.i | 16 | 8.b | even | 2 | 1 | ||
| 1920.2.y.i | 16 | 80.j | even | 4 | 1 | ||
| 1920.2.y.j | 16 | 8.d | odd | 2 | 1 | ||
| 1920.2.y.j | 16 | 80.t | odd | 4 | 1 | ||
| 1920.2.bc.i | 16 | 16.f | odd | 4 | 1 | ||
| 1920.2.bc.i | 16 | 40.i | odd | 4 | 1 | ||
| 1920.2.bc.j | 16 | 16.e | even | 4 | 1 | ||
| 1920.2.bc.j | 16 | 40.k | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(240, [\chi])\):
|
\( T_{7}^{16} + 4 T_{7}^{15} + 8 T_{7}^{14} - 32 T_{7}^{13} + 392 T_{7}^{12} + 1264 T_{7}^{11} + \cdots + 2304 \)
|
|
\( T_{11}^{16} - 8 T_{11}^{13} + 1288 T_{11}^{12} - 400 T_{11}^{11} + 32 T_{11}^{10} + 4864 T_{11}^{9} + \cdots + 952576 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} - 2 T^{15} + \cdots + 256 \)
|
| $3$ |
\( (T - 1)^{16} \)
|
| $5$ |
\( T^{16} + 4 T^{15} + \cdots + 390625 \)
|
| $7$ |
\( T^{16} + 4 T^{15} + \cdots + 2304 \)
|
| $11$ |
\( T^{16} - 8 T^{13} + \cdots + 952576 \)
|
| $13$ |
\( T^{16} + 120 T^{14} + \cdots + 8386816 \)
|
| $17$ |
\( T^{16} + 8 T^{15} + \cdots + 9339136 \)
|
| $19$ |
\( T^{16} - 8 T^{15} + \cdots + 5308416 \)
|
| $23$ |
\( T^{16} + \cdots + 330366976 \)
|
| $29$ |
\( T^{16} - 12 T^{15} + \cdots + 45373696 \)
|
| $31$ |
\( T^{16} + \cdots + 1655277803776 \)
|
| $37$ |
\( T^{16} + \cdots + 793886976 \)
|
| $41$ |
\( T^{16} + \cdots + 177209344 \)
|
| $43$ |
\( T^{16} + \cdots + 37555339264 \)
|
| $47$ |
\( T^{16} + 32 T^{15} + \cdots + 65536 \)
|
| $53$ |
\( (T^{8} - 8 T^{7} + \cdots + 189328)^{2} \)
|
| $59$ |
\( T^{16} + \cdots + 8785580546304 \)
|
| $61$ |
\( T^{16} + \cdots + 55857327698176 \)
|
| $67$ |
\( T^{16} + \cdots + 330366976 \)
|
| $71$ |
\( (T^{8} - 392 T^{6} + \cdots + 3900672)^{2} \)
|
| $73$ |
\( T^{16} + \cdots + 2179567984896 \)
|
| $79$ |
\( (T^{8} + 24 T^{7} + \cdots - 1507376)^{2} \)
|
| $83$ |
\( (T^{8} + 4 T^{7} + \cdots - 5888)^{2} \)
|
| $89$ |
\( (T^{8} - 344 T^{6} + \cdots - 3899136)^{2} \)
|
| $97$ |
\( T^{16} + \cdots + 16602430353664 \)
|
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