Newspace parameters
Level: | \( N \) | \(=\) | \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 2736.m (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(74.5506003290\) |
Analytic rank: | \(0\) |
Dimension: | \(12\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{12} - 2 x^{11} + 50 x^{10} - 136 x^{9} + 2215 x^{8} - 5020 x^{7} + 18282 x^{6} - 12094 x^{5} + 48457 x^{4} - 30372 x^{3} + 89392 x^{2} + 9344 x + 1024 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
Coefficient ring index: | \( 2^{16} \) |
Twist minimal: | no (minimal twist has level 912) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) 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 \( x^{12} - 2 x^{11} + 50 x^{10} - 136 x^{9} + 2215 x^{8} - 5020 x^{7} + 18282 x^{6} - 12094 x^{5} + 48457 x^{4} - 30372 x^{3} + 89392 x^{2} + 9344 x + 1024 \) :
\(\beta_{1}\) | \(=\) | \( ( 1132830139545 \nu^{11} + 83463703932306 \nu^{10} + 82049035182874 \nu^{9} + \cdots - 63\!\cdots\!20 ) / 52\!\cdots\!92 \) |
\(\beta_{2}\) | \(=\) | \( ( - 38271754888553 \nu^{11} + 9342376003726 \nu^{10} + \cdots - 27\!\cdots\!72 ) / 10\!\cdots\!84 \) |
\(\beta_{3}\) | \(=\) | \( ( 5763155937085 \nu^{11} + 13707431665502 \nu^{10} + 264163078401060 \nu^{9} + 394878866521450 \nu^{8} + \cdots + 11\!\cdots\!52 ) / 13\!\cdots\!48 \) |
\(\beta_{4}\) | \(=\) | \( ( 79024470131873 \nu^{11} + 62863161452290 \nu^{10} + \cdots + 94\!\cdots\!84 ) / 10\!\cdots\!84 \) |
\(\beta_{5}\) | \(=\) | \( ( - 104157093719227 \nu^{11} - 31799241318102 \nu^{10} + \cdots - 19\!\cdots\!44 ) / 10\!\cdots\!84 \) |
\(\beta_{6}\) | \(=\) | \( ( - 2248559103131 \nu^{11} + 4524708578702 \nu^{10} - 112234893734550 \nu^{9} + 307066972550728 \nu^{8} + \cdots - 98\!\cdots\!72 ) / 54\!\cdots\!36 \) |
\(\beta_{7}\) | \(=\) | \( ( 451147977297025 \nu^{11} - 972391254937206 \nu^{10} + \cdots + 15\!\cdots\!20 ) / 10\!\cdots\!84 \) |
\(\beta_{8}\) | \(=\) | \( ( 5084441 \nu^{11} - 10363066 \nu^{10} + 255384002 \nu^{9} - 701038728 \nu^{8} + 11325739791 \nu^{7} - 25986026436 \nu^{6} + \cdots + 25239211968 ) / 10877179584 \) |
\(\beta_{9}\) | \(=\) | \( ( 563617568102453 \nu^{11} + \cdots + 24\!\cdots\!20 ) / 10\!\cdots\!84 \) |
\(\beta_{10}\) | \(=\) | \( ( 8386307706881 \nu^{11} - 16626138955879 \nu^{10} + 416130916219964 \nu^{9} + \cdots + 33\!\cdots\!44 ) / 11\!\cdots\!52 \) |
\(\beta_{11}\) | \(=\) | \( ( - 288508475434596 \nu^{11} + 596180148308589 \nu^{10} + \cdots - 13\!\cdots\!16 ) / 26\!\cdots\!96 \) |
\(\nu\) | \(=\) | \( ( 2\beta_{11} + 2\beta_{10} + 2\beta_{9} + 2\beta_{8} - 2\beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} + 2 ) / 8 \) |
\(\nu^{2}\) | \(=\) | \( ( - 2 \beta_{11} + 4 \beta_{10} - 6 \beta_{9} + 16 \beta_{8} + 8 \beta_{7} + 31 \beta_{6} + 2 \beta_{5} + 5 \beta_{4} + 3 \beta_{3} + 7 \beta_{2} - 4 \beta _1 - 64 ) / 8 \) |
\(\nu^{3}\) | \(=\) | \( ( 12\beta_{5} - 35\beta_{4} + 35\beta_{3} - 63\beta_{2} - 4\beta _1 + 18 ) / 4 \) |
\(\nu^{4}\) | \(=\) | \( ( 90 \beta_{11} - 96 \beta_{10} + 338 \beta_{9} - 672 \beta_{8} - 408 \beta_{7} - 1149 \beta_{6} + 106 \beta_{5} + 247 \beta_{4} + 113 \beta_{3} + 345 \beta_{2} - 168 \beta _1 - 2360 ) / 8 \) |
\(\nu^{5}\) | \(=\) | \( ( - 3850 \beta_{11} - 2554 \beta_{10} - 3762 \beta_{9} - 1858 \beta_{8} + 2338 \beta_{7} + 1135 \beta_{6} - 504 \beta_{5} + 1537 \beta_{4} - 1377 \beta_{3} + 2889 \beta_{2} + 72 \beta _1 - 2702 ) / 8 \) |
\(\nu^{6}\) | \(=\) | \( ( -4258\beta_{5} - 11341\beta_{4} - 3867\beta_{3} - 16079\beta_{2} + 6900\beta _1 + 97664 ) / 4 \) |
\(\nu^{7}\) | \(=\) | \( ( 163514 \beta_{11} + 103394 \beta_{10} + 165862 \beta_{9} + 60614 \beta_{8} - 107558 \beta_{7} - 78445 \beta_{6} - 18652 \beta_{5} + 70235 \beta_{4} - 54587 \beta_{3} + 129927 \beta_{2} + \cdots - 176930 ) / 8 \) |
\(\nu^{8}\) | \(=\) | \( ( - 343514 \beta_{11} + 17776 \beta_{10} - 786242 \beta_{9} + 1080640 \beta_{8} + 840344 \beta_{7} + 2009317 \beta_{6} + 165610 \beta_{5} + 514511 \beta_{4} + 125817 \beta_{3} + \cdots - 4139064 ) / 8 \) |
\(\nu^{9}\) | \(=\) | \( ( 670120\beta_{5} - 3225385\beta_{4} + 2178281\beta_{3} - 5845345\beta_{2} + 238472\beta _1 + 9957246 ) / 4 \) |
\(\nu^{10}\) | \(=\) | \( ( 18600482 \beta_{11} + 1973948 \beta_{10} + 37156070 \beta_{9} - 43542928 \beta_{8} - 37900232 \beta_{7} - 85899087 \beta_{6} + 6433250 \beta_{5} + 23319637 \beta_{4} + \cdots - 177340352 ) / 8 \) |
\(\nu^{11}\) | \(=\) | \( ( - 300468378 \beta_{11} - 177177090 \beta_{10} - 327495670 \beta_{9} - 48227286 \beta_{8} + 229470870 \beta_{7} + 241426709 \beta_{6} - 23665836 \beta_{5} + \cdots - 523950514 ) / 8 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).
\(n\) | \(1009\) | \(1217\) | \(1711\) | \(2053\) |
\(\chi(n)\) | \(1\) | \(1\) | \(-1\) | \(1\) |
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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1711.1 |
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0 | 0 | 0 | −8.90000 | 0 | − | 2.78940i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
1711.2 | 0 | 0 | 0 | −8.90000 | 0 | 2.78940i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
1711.3 | 0 | 0 | 0 | −1.38490 | 0 | − | 7.59081i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
1711.4 | 0 | 0 | 0 | −1.38490 | 0 | 7.59081i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
1711.5 | 0 | 0 | 0 | −0.606930 | 0 | − | 2.85501i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
1711.6 | 0 | 0 | 0 | −0.606930 | 0 | 2.85501i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
1711.7 | 0 | 0 | 0 | 3.59607 | 0 | − | 9.49604i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
1711.8 | 0 | 0 | 0 | 3.59607 | 0 | 9.49604i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
1711.9 | 0 | 0 | 0 | 4.02901 | 0 | − | 7.12527i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
1711.10 | 0 | 0 | 0 | 4.02901 | 0 | 7.12527i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
1711.11 | 0 | 0 | 0 | 8.26675 | 0 | − | 9.51333i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
1711.12 | 0 | 0 | 0 | 8.26675 | 0 | 9.51333i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 2736.3.m.f | 12 | |
3.b | odd | 2 | 1 | 912.3.m.a | ✓ | 12 | |
4.b | odd | 2 | 1 | inner | 2736.3.m.f | 12 | |
12.b | even | 2 | 1 | 912.3.m.a | ✓ | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
912.3.m.a | ✓ | 12 | 3.b | odd | 2 | 1 | |
912.3.m.a | ✓ | 12 | 12.b | even | 2 | 1 | |
2736.3.m.f | 12 | 1.a | even | 1 | 1 | trivial | |
2736.3.m.f | 12 | 4.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} - 5T_{5}^{5} - 77T_{5}^{4} + 437T_{5}^{3} + 16T_{5}^{2} - 1644T_{5} - 896 \)
acting on \(S_{3}^{\mathrm{new}}(2736, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{12} \)
$3$
\( T^{12} \)
$5$
\( (T^{6} - 5 T^{5} - 77 T^{4} + 437 T^{3} + \cdots - 896)^{2} \)
$7$
\( T^{12} + 305 T^{10} + \cdots + 1514143744 \)
$11$
\( T^{12} + 885 T^{10} + \cdots + 187580416 \)
$13$
\( (T^{6} - 18 T^{5} - 416 T^{4} + \cdots + 2630656)^{2} \)
$17$
\( (T^{6} - 5 T^{5} - 915 T^{4} + \cdots + 8199352)^{2} \)
$19$
\( (T^{2} + 19)^{6} \)
$23$
\( T^{12} + 2152 T^{10} + \cdots + 99230924406784 \)
$29$
\( (T^{6} + 6 T^{5} - 2096 T^{4} + \cdots + 76842496)^{2} \)
$31$
\( T^{12} + 1312 T^{10} + \cdots + 1396965376 \)
$37$
\( (T^{6} - 16 T^{5} - 2740 T^{4} + \cdots + 6873088)^{2} \)
$41$
\( (T^{6} + 68 T^{5} - 3012 T^{4} + \cdots + 1137344512)^{2} \)
$43$
\( T^{12} + 15745 T^{10} + \cdots + 14\!\cdots\!36 \)
$47$
\( T^{12} + 12849 T^{10} + \cdots + 12\!\cdots\!76 \)
$53$
\( (T^{6} - 118 T^{5} - 1928 T^{4} + \cdots + 2520564736)^{2} \)
$59$
\( T^{12} + 21104 T^{10} + \cdots + 12\!\cdots\!84 \)
$61$
\( (T^{6} + 105 T^{5} - 6647 T^{4} + \cdots + 826249672)^{2} \)
$67$
\( T^{12} + 15472 T^{10} + \cdots + 23\!\cdots\!76 \)
$71$
\( T^{12} + 19408 T^{10} + \cdots + 80\!\cdots\!04 \)
$73$
\( (T^{6} + 79 T^{5} - 11631 T^{4} + \cdots + 1441081768)^{2} \)
$79$
\( T^{12} + 38988 T^{10} + \cdots + 49\!\cdots\!36 \)
$83$
\( T^{12} + 31516 T^{10} + \cdots + 42\!\cdots\!96 \)
$89$
\( (T^{6} + 222 T^{5} + \cdots + 307161915264)^{2} \)
$97$
\( (T^{6} - 160 T^{5} + \cdots + 100074188608)^{2} \)
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