Newspace parameters
Level: | \( N \) | \(=\) | \( 63 = 3^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 63.e (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(10.1041806482\) |
Analytic rank: | \(0\) |
Dimension: | \(12\) |
Relative dimension: | \(6\) over \(\Q(\zeta_{3})\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{12} + \cdots)\) |
Defining polynomial: |
\( x^{12} + 187x^{10} + 25399x^{8} + 1518438x^{6} + 66232188x^{4} + 1297462320x^{2} + 18380851776 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{4}\cdot 3^{5} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{12} + 187x^{10} + 25399x^{8} + 1518438x^{6} + 66232188x^{4} + 1297462320x^{2} + 18380851776 \)
:
\(\beta_{1}\) | \(=\) |
\( \nu \)
|
\(\beta_{2}\) | \(=\) |
\( ( 334805 \nu^{10} + 56078291 \nu^{8} + 7616751407 \nu^{6} + 387910489914 \nu^{4} + 19861967445084 \nu^{2} + \cdots + 120250595793888 ) / 268837476625872 \)
|
\(\beta_{3}\) | \(=\) |
\( ( - 334805 \nu^{11} - 56078291 \nu^{9} - 7616751407 \nu^{7} - 387910489914 \nu^{5} - 19861967445084 \nu^{3} + \cdots - 389088072419760 \nu ) / 268837476625872 \)
|
\(\beta_{4}\) | \(=\) |
\( ( 4837597 \nu^{11} + 1494767143 \nu^{9} + 203024549011 \nu^{7} + 16589437928838 \nu^{5} + 529420847226732 \nu^{3} + \cdots + 10\!\cdots\!80 \nu ) / 537674953251744 \)
|
\(\beta_{5}\) | \(=\) |
\( ( 289\nu^{10} + 39253\nu^{8} + 5331481\nu^{6} + 102358836\nu^{4} + 2005169040\nu^{2} - 465300162504 ) / 11897569332 \)
|
\(\beta_{6}\) | \(=\) |
\( ( 8147771 \nu^{10} + 247743581 \nu^{8} - 579963031 \nu^{6} - 12528900245394 \nu^{4} - 474402432194820 \nu^{2} + \cdots - 15\!\cdots\!32 ) / 268837476625872 \)
|
\(\beta_{7}\) | \(=\) |
\( ( - 25381 \nu^{11} - 3447337 \nu^{9} - 404606113 \nu^{7} - 8989514244 \nu^{5} - 176101022160 \nu^{3} + 6931671291720 \nu ) / 999395823888 \)
|
\(\beta_{8}\) | \(=\) |
\( ( 7113833 \nu^{10} + 1294946627 \nu^{8} + 175884221279 \nu^{6} + 10868775058206 \nu^{4} + 458647852670748 \nu^{2} + \cdots + 89\!\cdots\!20 ) / 134418738312936 \)
|
\(\beta_{9}\) | \(=\) |
\( ( 901281 \nu^{10} + 185645795 \nu^{8} + 23947312631 \nu^{6} + 1451072814470 \nu^{4} + 44932220248548 \nu^{2} + \cdots + 663388031086704 ) / 9956943578736 \)
|
\(\beta_{10}\) | \(=\) |
\( ( 37519 \nu^{11} + 5095963 \nu^{9} + 628528315 \nu^{7} + 13288585356 \nu^{5} + 260318121840 \nu^{3} - 37967330091600 \nu ) / 499697911944 \)
|
\(\beta_{11}\) | \(=\) |
\( ( - 4460963 \nu^{11} - 895743515 \nu^{9} - 121663045655 \nu^{7} - 7874821552212 \nu^{5} - 317256967300860 \nu^{3} + \cdots - 62\!\cdots\!00 \nu ) / 44806246104312 \)
|
\(\nu\) | \(=\) |
\( \beta_1 \)
|
\(\nu^{2}\) | \(=\) |
\( -\beta_{8} - \beta_{5} + 62\beta_{2} \)
|
\(\nu^{3}\) | \(=\) |
\( \beta_{11} - \beta_{10} - 2\beta_{7} + 2\beta_{4} - 85\beta_{3} - 85\beta_1 \)
|
\(\nu^{4}\) | \(=\) |
\( -14\beta_{9} + 139\beta_{8} + 14\beta_{6} - 5230\beta_{2} - 5230 \)
|
\(\nu^{5}\) | \(=\) |
\( -139\beta_{11} - 362\beta_{4} + 8497\beta_{3} \)
|
\(\nu^{6}\) | \(=\) |
\( -2618\beta_{9} - 5236\beta_{6} + 16423\beta_{5} - 2618\beta_{2} + 522864 \)
|
\(\nu^{7}\) | \(=\) |
\( 16423\beta_{10} + 48554\beta_{7} + 911065\beta_1 \)
|
\(\nu^{8}\) | \(=\) |
\( 711172\beta_{9} - 1876447\beta_{8} + 355586\beta_{6} - 1876447\beta_{5} + 55996200\beta_{2} - 355586 \)
|
\(\nu^{9}\) | \(=\) |
\( 1876447 \beta_{11} - 1876447 \beta_{10} - 5886410 \beta_{7} + 5886410 \beta_{4} - 100576825 \beta_{3} - 100576825 \beta_1 \)
|
\(\nu^{10}\) | \(=\) |
\( -43338386\beta_{9} + 212572543\beta_{8} + 43338386\beta_{6} - 6135103078\beta_{2} - 6135103078 \)
|
\(\nu^{11}\) | \(=\) |
\( -212572543\beta_{11} - 685175402\beta_{4} + 11240963497\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).
\(n\) | \(10\) | \(29\) |
\(\chi(n)\) | \(-1 - \beta_{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
37.1 |
|
−5.31117 | + | 9.19921i | 0 | −40.4170 | − | 70.0043i | 33.7376 | − | 58.4352i | 0 | −120.281 | + | 48.3679i | 518.731 | 0 | 358.372 | + | 620.718i | ||||||||||||||||||||||||||||||||||||||||||||
37.2 | −3.54467 | + | 6.13954i | 0 | −9.12931 | − | 15.8124i | −41.1020 | + | 71.1908i | 0 | 112.556 | + | 64.3292i | −97.4172 | 0 | −291.386 | − | 504.695i | |||||||||||||||||||||||||||||||||||||||||||||
37.3 | −2.44476 | + | 4.23445i | 0 | 4.04630 | + | 7.00840i | 21.3752 | − | 37.0229i | 0 | 43.2256 | − | 122.223i | −196.034 | 0 | 104.514 | + | 181.024i | |||||||||||||||||||||||||||||||||||||||||||||
37.4 | 2.44476 | − | 4.23445i | 0 | 4.04630 | + | 7.00840i | −21.3752 | + | 37.0229i | 0 | 43.2256 | − | 122.223i | 196.034 | 0 | 104.514 | + | 181.024i | |||||||||||||||||||||||||||||||||||||||||||||
37.5 | 3.54467 | − | 6.13954i | 0 | −9.12931 | − | 15.8124i | 41.1020 | − | 71.1908i | 0 | 112.556 | + | 64.3292i | 97.4172 | 0 | −291.386 | − | 504.695i | |||||||||||||||||||||||||||||||||||||||||||||
37.6 | 5.31117 | − | 9.19921i | 0 | −40.4170 | − | 70.0043i | −33.7376 | + | 58.4352i | 0 | −120.281 | + | 48.3679i | −518.731 | 0 | 358.372 | + | 620.718i | |||||||||||||||||||||||||||||||||||||||||||||
46.1 | −5.31117 | − | 9.19921i | 0 | −40.4170 | + | 70.0043i | 33.7376 | + | 58.4352i | 0 | −120.281 | − | 48.3679i | 518.731 | 0 | 358.372 | − | 620.718i | |||||||||||||||||||||||||||||||||||||||||||||
46.2 | −3.54467 | − | 6.13954i | 0 | −9.12931 | + | 15.8124i | −41.1020 | − | 71.1908i | 0 | 112.556 | − | 64.3292i | −97.4172 | 0 | −291.386 | + | 504.695i | |||||||||||||||||||||||||||||||||||||||||||||
46.3 | −2.44476 | − | 4.23445i | 0 | 4.04630 | − | 7.00840i | 21.3752 | + | 37.0229i | 0 | 43.2256 | + | 122.223i | −196.034 | 0 | 104.514 | − | 181.024i | |||||||||||||||||||||||||||||||||||||||||||||
46.4 | 2.44476 | + | 4.23445i | 0 | 4.04630 | − | 7.00840i | −21.3752 | − | 37.0229i | 0 | 43.2256 | + | 122.223i | 196.034 | 0 | 104.514 | − | 181.024i | |||||||||||||||||||||||||||||||||||||||||||||
46.5 | 3.54467 | + | 6.13954i | 0 | −9.12931 | + | 15.8124i | 41.1020 | + | 71.1908i | 0 | 112.556 | − | 64.3292i | 97.4172 | 0 | −291.386 | + | 504.695i | |||||||||||||||||||||||||||||||||||||||||||||
46.6 | 5.31117 | + | 9.19921i | 0 | −40.4170 | + | 70.0043i | −33.7376 | − | 58.4352i | 0 | −120.281 | − | 48.3679i | −518.731 | 0 | 358.372 | − | 620.718i | |||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
7.c | even | 3 | 1 | inner |
21.h | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 63.6.e.f | ✓ | 12 |
3.b | odd | 2 | 1 | inner | 63.6.e.f | ✓ | 12 |
7.c | even | 3 | 1 | inner | 63.6.e.f | ✓ | 12 |
7.c | even | 3 | 1 | 441.6.a.bc | 6 | ||
7.d | odd | 6 | 1 | 441.6.a.bd | 6 | ||
21.g | even | 6 | 1 | 441.6.a.bd | 6 | ||
21.h | odd | 6 | 1 | inner | 63.6.e.f | ✓ | 12 |
21.h | odd | 6 | 1 | 441.6.a.bc | 6 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
63.6.e.f | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
63.6.e.f | ✓ | 12 | 3.b | odd | 2 | 1 | inner |
63.6.e.f | ✓ | 12 | 7.c | even | 3 | 1 | inner |
63.6.e.f | ✓ | 12 | 21.h | odd | 6 | 1 | inner |
441.6.a.bc | 6 | 7.c | even | 3 | 1 | ||
441.6.a.bc | 6 | 21.h | odd | 6 | 1 | ||
441.6.a.bd | 6 | 7.d | odd | 6 | 1 | ||
441.6.a.bd | 6 | 21.g | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} + 187T_{2}^{10} + 25399T_{2}^{8} + 1518438T_{2}^{6} + 66232188T_{2}^{4} + 1297462320T_{2}^{2} + 18380851776 \)
acting on \(S_{6}^{\mathrm{new}}(63, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{12} + 187 T^{10} + \cdots + 18380851776 \)
$3$
\( T^{12} \)
$5$
\( T^{12} + 13138 T^{10} + \cdots + 31\!\cdots\!96 \)
$7$
\( (T^{6} - 71 T^{5} + \cdots + 4747561509943)^{2} \)
$11$
\( T^{12} + 902578 T^{10} + \cdots + 29\!\cdots\!76 \)
$13$
\( (T^{3} - 77 T^{2} - 783976 T + 60080636)^{4} \)
$17$
\( T^{12} + 2284392 T^{10} + \cdots + 20\!\cdots\!76 \)
$19$
\( (T^{6} - 4711 T^{5} + \cdots + 13\!\cdots\!64)^{2} \)
$23$
\( T^{12} + 10191432 T^{10} + \cdots + 12\!\cdots\!56 \)
$29$
\( (T^{6} - 66189634 T^{4} + \cdots - 75\!\cdots\!56)^{2} \)
$31$
\( (T^{6} - 11711 T^{5} + \cdots + 73\!\cdots\!01)^{2} \)
$37$
\( (T^{6} + 9091 T^{5} + \cdots + 10\!\cdots\!16)^{2} \)
$41$
\( (T^{6} - 727900480 T^{4} + \cdots - 13\!\cdots\!64)^{2} \)
$43$
\( (T^{3} + 21843 T^{2} + \cdots - 1643621929904)^{4} \)
$47$
\( T^{12} + 613454880 T^{10} + \cdots + 39\!\cdots\!96 \)
$53$
\( T^{12} + 666461298 T^{10} + \cdots + 26\!\cdots\!96 \)
$59$
\( T^{12} + 4695322386 T^{10} + \cdots + 98\!\cdots\!16 \)
$61$
\( (T^{6} + 8078 T^{5} + \cdots + 28\!\cdots\!04)^{2} \)
$67$
\( (T^{6} - 72325 T^{5} + \cdots + 50\!\cdots\!76)^{2} \)
$71$
\( (T^{6} - 3481391808 T^{4} + \cdots - 30\!\cdots\!44)^{2} \)
$73$
\( (T^{6} + 50029 T^{5} + \cdots + 87\!\cdots\!44)^{2} \)
$79$
\( (T^{6} - 50997 T^{5} + \cdots + 12\!\cdots\!89)^{2} \)
$83$
\( (T^{6} - 7727019426 T^{4} + \cdots - 11\!\cdots\!64)^{2} \)
$89$
\( T^{12} + 2820380296 T^{10} + \cdots + 36\!\cdots\!16 \)
$97$
\( (T^{3} + 216524 T^{2} + \cdots - 546802680855102)^{4} \)
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