Newspace parameters
Level: | \( N \) | \(=\) | \( 312 = 2^{3} \cdot 3 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 312.x (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.49133254306\) |
Analytic rank: | \(0\) |
Dimension: | \(16\) |
Relative dimension: | \(8\) over \(\Q(i)\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{16} - 4 x^{13} + 5 x^{12} - 4 x^{11} + 8 x^{10} - 16 x^{9} + 28 x^{8} - 32 x^{7} + 32 x^{6} - 32 x^{5} + 80 x^{4} - 128 x^{3} + 256 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
Coefficient ring index: | \( 2^{8} \) |
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 \( x^{16} - 4 x^{13} + 5 x^{12} - 4 x^{11} + 8 x^{10} - 16 x^{9} + 28 x^{8} - 32 x^{7} + 32 x^{6} - 32 x^{5} + 80 x^{4} - 128 x^{3} + 256 \) :
\(\beta_{1}\) | \(=\) | \( ( - \nu^{15} - 11 \nu^{14} + 32 \nu^{13} - 52 \nu^{12} + 103 \nu^{11} - 155 \nu^{10} + 276 \nu^{9} - 416 \nu^{8} + 564 \nu^{7} - 700 \nu^{6} + 1040 \nu^{5} - 1312 \nu^{4} + 1264 \nu^{3} - 1200 \nu^{2} + \cdots - 768 ) / 1088 \) |
\(\beta_{2}\) | \(=\) | \( ( - 5 \nu^{15} - 4 \nu^{14} - 10 \nu^{13} - 22 \nu^{12} + 39 \nu^{11} - 44 \nu^{10} + 54 \nu^{9} - 74 \nu^{8} + 236 \nu^{7} - 372 \nu^{6} + 304 \nu^{5} - 168 \nu^{4} + 336 \nu^{3} - 1376 \nu^{2} + \cdots + 1600 ) / 1088 \) |
\(\beta_{3}\) | \(=\) | \( ( 5 \nu^{15} + 4 \nu^{14} - 24 \nu^{13} + 22 \nu^{12} - 39 \nu^{11} + 44 \nu^{10} - 88 \nu^{9} + 210 \nu^{8} - 236 \nu^{7} + 236 \nu^{6} - 32 \nu^{5} + 712 \nu^{4} - 608 \nu^{3} + 288 \nu^{2} + 448 \nu + 576 ) / 1088 \) |
\(\beta_{4}\) | \(=\) | \( ( - \nu^{15} + 2 \nu^{14} + 6 \nu^{12} - 21 \nu^{11} + 30 \nu^{10} - 48 \nu^{9} + 66 \nu^{8} - 140 \nu^{7} + 200 \nu^{6} - 248 \nu^{5} + 272 \nu^{4} - 368 \nu^{3} + 560 \nu^{2} - 576 \nu + 320 ) / 192 \) |
\(\beta_{5}\) | \(=\) | \( ( 29 \nu^{15} + 13 \nu^{14} - 44 \nu^{13} - 56 \nu^{12} + 73 \nu^{11} + 109 \nu^{10} - 320 \nu^{9} + 436 \nu^{8} - 376 \nu^{7} + 172 \nu^{6} - 2008 \nu^{5} + 3232 \nu^{4} - 2656 \nu^{3} + 1072 \nu^{2} + \cdots + 9216 ) / 3264 \) |
\(\beta_{6}\) | \(=\) | \( ( - 15 \nu^{15} + 22 \nu^{14} - 64 \nu^{13} + 155 \nu^{12} - 325 \nu^{11} + 650 \nu^{10} - 892 \nu^{9} + 1223 \nu^{8} - 1978 \nu^{7} + 2896 \nu^{6} - 3712 \nu^{5} + 4120 \nu^{4} - 4840 \nu^{3} + \cdots + 4256 ) / 1632 \) |
\(\beta_{7}\) | \(=\) | \( ( 3 \nu^{15} - 4 \nu^{14} + 4 \nu^{13} - 8 \nu^{12} + 19 \nu^{11} - 20 \nu^{10} + 28 \nu^{9} - 20 \nu^{8} + 40 \nu^{7} - 52 \nu^{6} + 16 \nu^{5} + 56 \nu^{4} - 80 \nu^{3} - 112 \nu^{2} + 64 \nu + 448 ) / 192 \) |
\(\beta_{8}\) | \(=\) | \( ( 47 \nu^{15} - 214 \nu^{14} + 366 \nu^{13} - 480 \nu^{12} + 939 \nu^{11} - 1878 \nu^{10} + 2838 \nu^{9} - 3636 \nu^{8} + 4636 \nu^{7} - 7084 \nu^{6} + 9736 \nu^{5} - 10552 \nu^{4} + \cdots - 8512 ) / 3264 \) |
\(\beta_{9}\) | \(=\) | \( ( 56 \nu^{15} - 47 \nu^{14} + 10 \nu^{13} - 80 \nu^{12} + 352 \nu^{11} - 347 \nu^{10} + 286 \nu^{9} - 368 \nu^{8} + 716 \nu^{7} - 1532 \nu^{6} + 1736 \nu^{5} + 304 \nu^{4} - 64 \nu^{3} - 4880 \nu^{2} + \cdots + 3840 ) / 3264 \) |
\(\beta_{10}\) | \(=\) | \( ( 67 \nu^{15} - 62 \nu^{14} - 36 \nu^{13} - 120 \nu^{12} + 375 \nu^{11} - 138 \nu^{10} - 132 \nu^{9} + 468 \nu^{8} + 20 \nu^{7} - 224 \nu^{6} - 1408 \nu^{5} + 3856 \nu^{4} - 3088 \nu^{3} + \cdots + 14464 ) / 3264 \) |
\(\beta_{11}\) | \(=\) | \( ( - 37 \nu^{15} + 69 \nu^{14} - 40 \nu^{13} + 116 \nu^{12} - 235 \nu^{11} + 317 \nu^{10} - 328 \nu^{9} + 248 \nu^{8} - 450 \nu^{7} + 756 \nu^{6} - 756 \nu^{5} - 264 \nu^{4} + 1752 \nu^{3} + \cdots - 5024 ) / 1632 \) |
\(\beta_{12}\) | \(=\) | \( ( - 79 \nu^{15} + 66 \nu^{14} - 22 \nu^{13} + 176 \nu^{12} - 295 \nu^{11} + 182 \nu^{10} + 146 \nu^{9} - 292 \nu^{8} - 120 \nu^{7} - 288 \nu^{6} + 1512 \nu^{5} - 5184 \nu^{4} + 4656 \nu^{3} + \cdots - 16064 ) / 3264 \) |
\(\beta_{13}\) | \(=\) | \( ( 60 \nu^{15} - 37 \nu^{14} - 50 \nu^{13} - 110 \nu^{12} + 280 \nu^{11} - 101 \nu^{10} - 206 \nu^{9} + 106 \nu^{8} + 160 \nu^{7} - 160 \nu^{6} - 1064 \nu^{5} + 3512 \nu^{4} - 2672 \nu^{3} + \cdots + 12352 ) / 1632 \) |
\(\beta_{14}\) | \(=\) | \( ( 120 \nu^{15} - 23 \nu^{14} - 100 \nu^{13} - 220 \nu^{12} + 356 \nu^{11} + 53 \nu^{10} - 616 \nu^{9} + 620 \nu^{8} - 496 \nu^{7} + 1108 \nu^{6} - 3760 \nu^{5} + 8656 \nu^{4} - 6976 \nu^{3} + \cdots + 24704 ) / 3264 \) |
\(\beta_{15}\) | \(=\) | \( ( - 159 \nu^{15} + 206 \nu^{14} - 80 \nu^{13} + 436 \nu^{12} - 1235 \nu^{11} + 1450 \nu^{10} - 1472 \nu^{9} + 1720 \nu^{8} - 3620 \nu^{7} + 5456 \nu^{6} - 4640 \nu^{5} - 256 \nu^{4} + \cdots - 18752 ) / 3264 \) |
\(\nu\) | \(=\) | \( ( 2\beta_{15} - 2\beta_{14} + 3\beta_{13} + 3\beta_{7} - \beta_{6} + \beta_{3} - 1 ) / 4 \) |
\(\nu^{2}\) | \(=\) | \( ( -\beta_{14} + \beta_{13} - \beta_{9} + \beta_{5} + \beta_{3} - \beta_{2} + \beta_1 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( ( 2 \beta_{15} + 2 \beta_{14} + 4 \beta_{11} + 5 \beta_{10} + 3 \beta_{8} - 4 \beta_{5} - \beta_{4} + \beta_{2} - 4 \beta _1 + 3 ) / 4 \) |
\(\nu^{4}\) | \(=\) | \( ( - \beta_{15} - \beta_{12} + 3 \beta_{11} - \beta_{10} + \beta_{8} + \beta_{7} + \beta_{6} + \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 5 ) / 2 \) |
\(\nu^{5}\) | \(=\) | \( ( - 2 \beta_{15} + 2 \beta_{14} - 5 \beta_{13} - 4 \beta_{12} + 4 \beta_{9} - 9 \beta_{7} + 3 \beta_{6} + 5 \beta_{3} + 8 \beta _1 + 3 ) / 4 \) |
\(\nu^{6}\) | \(=\) | \( ( \beta_{14} + \beta_{13} + 2 \beta_{10} - 3 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} - 2 \beta_{6} - 9 \beta_{5} + 2 \beta_{4} + \beta_{3} - \beta_{2} + 3 \beta_1 ) / 2 \) |
\(\nu^{7}\) | \(=\) | \( ( - 10 \beta_{15} - 10 \beta_{14} - 16 \beta_{12} - 4 \beta_{11} - 13 \beta_{10} - 16 \beta_{9} - 11 \beta_{8} + 4 \beta_{5} - 7 \beta_{4} - 9 \beta_{2} + 12 \beta _1 - 3 ) / 4 \) |
\(\nu^{8}\) | \(=\) | \( ( \beta_{15} - 10 \beta_{13} - 7 \beta_{12} - 3 \beta_{11} + 11 \beta_{10} - 3 \beta_{8} - 3 \beta_{7} - 3 \beta_{6} - 3 \beta_{4} + 1 ) / 2 \) |
\(\nu^{9}\) | \(=\) | \( ( - 22 \beta_{15} + 22 \beta_{14} - 27 \beta_{13} + 28 \beta_{12} + 8 \beta_{11} - 28 \beta_{9} + 25 \beta_{7} + 13 \beta_{6} + 8 \beta_{5} + 27 \beta_{3} + 16 \beta _1 - 11 ) / 4 \) |
\(\nu^{10}\) | \(=\) | \( ( 7 \beta_{14} - 11 \beta_{13} - 4 \beta_{10} + 3 \beta_{9} + 4 \beta_{8} - 4 \beta_{7} + 24 \beta_{6} + 25 \beta_{5} - 24 \beta_{4} + 9 \beta_{3} - 9 \beta_{2} + 29 \beta_1 ) / 2 \) |
\(\nu^{11}\) | \(=\) | \( ( - 14 \beta_{15} - 14 \beta_{14} + 24 \beta_{12} + 12 \beta_{11} + 45 \beta_{10} + 24 \beta_{9} - 21 \beta_{8} - 12 \beta_{5} + 39 \beta_{4} - 7 \beta_{2} + 76 \beta _1 - 69 ) / 4 \) |
\(\nu^{12}\) | \(=\) | \( ( - 73 \beta_{15} - 36 \beta_{13} + 7 \beta_{12} + 19 \beta_{11} - 37 \beta_{10} - 19 \beta_{8} - 19 \beta_{7} + 21 \beta_{6} + 21 \beta_{4} - 18 \beta_{3} - 18 \beta_{2} + 51 ) / 2 \) |
\(\nu^{13}\) | \(=\) | \( ( 30 \beta_{15} - 30 \beta_{14} - 69 \beta_{13} - 148 \beta_{12} + 32 \beta_{11} + 148 \beta_{9} - 9 \beta_{7} + 3 \beta_{6} + 32 \beta_{5} - 91 \beta_{3} - 8 \beta _1 + 83 ) / 4 \) |
\(\nu^{14}\) | \(=\) | \( ( - 47 \beta_{14} + 21 \beta_{13} - 26 \beta_{10} + 13 \beta_{9} - 54 \beta_{8} + 54 \beta_{7} - 62 \beta_{6} + 71 \beta_{5} + 62 \beta_{4} + 45 \beta_{3} - 45 \beta_{2} + 99 \beta_1 ) / 2 \) |
\(\nu^{15}\) | \(=\) | \( ( 70 \beta_{15} + 70 \beta_{14} - 84 \beta_{11} - 45 \beta_{10} + 21 \beta_{8} + 84 \beta_{5} - 231 \beta_{4} - 329 \beta_{2} - 100 \beta _1 + 429 ) / 4 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/312\mathbb{Z}\right)^\times\).
\(n\) | \(79\) | \(145\) | \(157\) | \(209\) |
\(\chi(n)\) | \(1\) | \(\beta_{1}\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
161.1 |
|
0 | −1.71560 | − | 0.238125i | 0 | −0.359033 | − | 0.359033i | 0 | −1.33676 | − | 1.33676i | 0 | 2.88659 | + | 0.817056i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
161.2 | 0 | −1.71560 | + | 0.238125i | 0 | 0.359033 | + | 0.359033i | 0 | −1.33676 | − | 1.33676i | 0 | 2.88659 | − | 0.817056i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
161.3 | 0 | −0.265070 | − | 1.71165i | 0 | 1.20485 | + | 1.20485i | 0 | −3.42064 | − | 3.42064i | 0 | −2.85948 | + | 0.907412i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
161.4 | 0 | −0.265070 | + | 1.71165i | 0 | −1.20485 | − | 1.20485i | 0 | −3.42064 | − | 3.42064i | 0 | −2.85948 | − | 0.907412i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
161.5 | 0 | 1.34162 | − | 1.09547i | 0 | −0.429726 | − | 0.429726i | 0 | 0.549230 | + | 0.549230i | 0 | 0.599886 | − | 2.93941i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
161.6 | 0 | 1.34162 | + | 1.09547i | 0 | 0.429726 | + | 0.429726i | 0 | 0.549230 | + | 0.549230i | 0 | 0.599886 | + | 2.93941i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
161.7 | 0 | 1.63905 | − | 0.559912i | 0 | −2.68975 | − | 2.68975i | 0 | −1.79184 | − | 1.79184i | 0 | 2.37300 | − | 1.83545i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
161.8 | 0 | 1.63905 | + | 0.559912i | 0 | 2.68975 | + | 2.68975i | 0 | −1.79184 | − | 1.79184i | 0 | 2.37300 | + | 1.83545i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
281.1 | 0 | −1.71560 | − | 0.238125i | 0 | 0.359033 | − | 0.359033i | 0 | −1.33676 | + | 1.33676i | 0 | 2.88659 | + | 0.817056i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
281.2 | 0 | −1.71560 | + | 0.238125i | 0 | −0.359033 | + | 0.359033i | 0 | −1.33676 | + | 1.33676i | 0 | 2.88659 | − | 0.817056i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
281.3 | 0 | −0.265070 | − | 1.71165i | 0 | −1.20485 | + | 1.20485i | 0 | −3.42064 | + | 3.42064i | 0 | −2.85948 | + | 0.907412i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
281.4 | 0 | −0.265070 | + | 1.71165i | 0 | 1.20485 | − | 1.20485i | 0 | −3.42064 | + | 3.42064i | 0 | −2.85948 | − | 0.907412i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
281.5 | 0 | 1.34162 | − | 1.09547i | 0 | 0.429726 | − | 0.429726i | 0 | 0.549230 | − | 0.549230i | 0 | 0.599886 | − | 2.93941i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
281.6 | 0 | 1.34162 | + | 1.09547i | 0 | −0.429726 | + | 0.429726i | 0 | 0.549230 | − | 0.549230i | 0 | 0.599886 | + | 2.93941i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
281.7 | 0 | 1.63905 | − | 0.559912i | 0 | 2.68975 | − | 2.68975i | 0 | −1.79184 | + | 1.79184i | 0 | 2.37300 | − | 1.83545i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
281.8 | 0 | 1.63905 | + | 0.559912i | 0 | −2.68975 | + | 2.68975i | 0 | −1.79184 | + | 1.79184i | 0 | 2.37300 | + | 1.83545i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
13.d | odd | 4 | 1 | inner |
39.f | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 312.2.x.c | ✓ | 16 |
3.b | odd | 2 | 1 | inner | 312.2.x.c | ✓ | 16 |
4.b | odd | 2 | 1 | 624.2.bf.g | 16 | ||
12.b | even | 2 | 1 | 624.2.bf.g | 16 | ||
13.d | odd | 4 | 1 | inner | 312.2.x.c | ✓ | 16 |
39.f | even | 4 | 1 | inner | 312.2.x.c | ✓ | 16 |
52.f | even | 4 | 1 | 624.2.bf.g | 16 | ||
156.l | odd | 4 | 1 | 624.2.bf.g | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
312.2.x.c | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
312.2.x.c | ✓ | 16 | 3.b | odd | 2 | 1 | inner |
312.2.x.c | ✓ | 16 | 13.d | odd | 4 | 1 | inner |
312.2.x.c | ✓ | 16 | 39.f | even | 4 | 1 | inner |
624.2.bf.g | 16 | 4.b | odd | 2 | 1 | ||
624.2.bf.g | 16 | 12.b | even | 2 | 1 | ||
624.2.bf.g | 16 | 52.f | even | 4 | 1 | ||
624.2.bf.g | 16 | 156.l | odd | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{16} + 218T_{5}^{12} + 1809T_{5}^{8} + 360T_{5}^{4} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(312, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{16} \)
$3$
\( (T^{8} - 2 T^{7} - T^{6} + 6 T^{5} - 8 T^{4} + \cdots + 81)^{2} \)
$5$
\( T^{16} + 218 T^{12} + 1809 T^{8} + \cdots + 16 \)
$7$
\( (T^{8} + 12 T^{7} + 72 T^{6} + 224 T^{5} + \cdots + 324)^{2} \)
$11$
\( T^{16} + 1040 T^{12} + \cdots + 1679616 \)
$13$
\( (T^{8} + 6 T^{6} + 40 T^{5} - 190 T^{4} + \cdots + 28561)^{2} \)
$17$
\( (T^{8} - 62 T^{6} + 537 T^{4} - 680 T^{2} + \cdots + 16)^{2} \)
$19$
\( (T^{8} + 4 T^{7} + 8 T^{6} + 8 T^{5} + \cdots + 576)^{2} \)
$23$
\( (T^{8} - 60 T^{6} + 1188 T^{4} + \cdots + 4096)^{2} \)
$29$
\( (T^{8} + 124 T^{6} + 5604 T^{4} + \cdots + 746496)^{2} \)
$31$
\( (T^{8} - 4 T^{7} + 8 T^{6} - 24 T^{5} + \cdots + 1024)^{2} \)
$37$
\( (T^{8} - 24 T^{7} + 288 T^{6} + \cdots + 114244)^{2} \)
$41$
\( T^{16} + 25416 T^{12} + \cdots + 7376134537216 \)
$43$
\( (T^{8} + 134 T^{6} + 5049 T^{4} + \cdots + 15376)^{2} \)
$47$
\( T^{16} + 35018 T^{12} + \cdots + 290258027536 \)
$53$
\( (T^{8} + 132 T^{6} + 3780 T^{4} + \cdots + 1024)^{2} \)
$59$
\( T^{16} + 23592 T^{12} + \cdots + 7676563456 \)
$61$
\( (T^{4} + 12 T^{3} - 68 T^{2} - 640 T + 2528)^{4} \)
$67$
\( (T^{8} + 8 T^{7} + 32 T^{6} + \cdots + 6411024)^{2} \)
$71$
\( T^{16} + \cdots + 982045424460816 \)
$73$
\( (T^{8} + 4 T^{7} + 8 T^{6} - 168 T^{5} + \cdots + 7884864)^{2} \)
$79$
\( (T^{4} + 20 T^{3} + 94 T^{2} - 160 T - 1216)^{4} \)
$83$
\( T^{16} + 7248 T^{12} + \cdots + 75391979776 \)
$89$
\( T^{16} + \cdots + 340609761939456 \)
$97$
\( (T^{8} - 24 T^{7} + 288 T^{6} + \cdots + 4717584)^{2} \)
show more
show less