Newspace parameters
Level: | \( N \) | \(=\) | \( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 1872.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(110.451575531\) |
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: | \( x^{12} + 180x^{10} + 45168x^{8} + 667388x^{6} + 86783544x^{4} + 12732769404x^{2} + 502441386561 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
Coefficient ring index: | \( 2^{12}\cdot 3^{4} \) |
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 \(1,\beta_1,\ldots,\beta_{11}\) 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 \( x^{12} + 180x^{10} + 45168x^{8} + 667388x^{6} + 86783544x^{4} + 12732769404x^{2} + 502441386561 \) :
\(\beta_{1}\) | \(=\) | \( ( - 1344269861 \nu^{10} - 250606322343 \nu^{8} - 76131758398098 \nu^{6} + \cdots - 18\!\cdots\!52 ) / 23\!\cdots\!61 \) |
\(\beta_{2}\) | \(=\) | \( ( - 6765538 \nu^{10} - 1743804036 \nu^{8} - 474007637436 \nu^{6} - 13733030333780 \nu^{4} + 814747851807606 \nu^{2} + \cdots + 29\!\cdots\!28 ) / 11\!\cdots\!67 \) |
\(\beta_{3}\) | \(=\) | \( ( - 631807564226 \nu^{10} + 3013984529298 \nu^{8} + \cdots + 11\!\cdots\!12 ) / 27\!\cdots\!37 \) |
\(\beta_{4}\) | \(=\) | \( ( - 15964996 \nu^{10} - 3393969246 \nu^{8} - 763080668880 \nu^{6} - 23600413111778 \nu^{4} + \cdots - 15\!\cdots\!96 ) / 38\!\cdots\!89 \) |
\(\beta_{5}\) | \(=\) | \( ( - 2996287599784 \nu^{10} - 409443004063530 \nu^{8} + \cdots - 23\!\cdots\!64 ) / 27\!\cdots\!37 \) |
\(\beta_{6}\) | \(=\) | \( ( - 1148249752 \nu^{11} - 216062553213 \nu^{9} - 53768253413382 \nu^{7} + \cdots - 27\!\cdots\!48 \nu ) / 36\!\cdots\!93 \) |
\(\beta_{7}\) | \(=\) | \( ( - 73109282417633 \nu^{11} + \cdots - 71\!\cdots\!63 \nu ) / 55\!\cdots\!97 \) |
\(\beta_{8}\) | \(=\) | \( ( - 20314422394 \nu^{11} - 3898109401794 \nu^{9} - 960883617950322 \nu^{7} + \cdots - 45\!\cdots\!40 \nu ) / 10\!\cdots\!17 \) |
\(\beta_{9}\) | \(=\) | \( ( 1202226983648 \nu^{11} + 97887638037753 \nu^{9} + \cdots + 96\!\cdots\!80 \nu ) / 39\!\cdots\!63 \) |
\(\beta_{10}\) | \(=\) | \( ( 92968292414 \nu^{11} + 8540893578060 \nu^{9} + \cdots + 75\!\cdots\!50 \nu ) / 14\!\cdots\!69 \) |
\(\beta_{11}\) | \(=\) | \( ( 3579014403746 \nu^{11} + 553966438047651 \nu^{9} + \cdots + 15\!\cdots\!04 \nu ) / 39\!\cdots\!63 \) |
\(\nu\) | \(=\) | \( ( -\beta_{11} + \beta_{9} - 3\beta_{8} ) / 12 \) |
\(\nu^{2}\) | \(=\) | \( ( -7\beta_{5} + 6\beta_{4} + 11\beta_{3} + 9\beta_{2} + 36\beta _1 - 360 ) / 12 \) |
\(\nu^{3}\) | \(=\) | \( ( 20\beta_{11} + 171\beta_{10} - 290\beta_{9} - 84\beta_{8} + 378\beta_{7} + 198\beta_{6} ) / 12 \) |
\(\nu^{4}\) | \(=\) | \( ( 989\beta_{5} - 1023\beta_{4} - 1153\beta_{3} + 3366\beta_{2} - 10206\beta _1 - 115872 ) / 12 \) |
\(\nu^{5}\) | \(=\) | \( ( 2033\beta_{11} - 12798\beta_{10} + 16327\beta_{9} + 98457\beta_{8} - 28080\beta_{7} - 562476\beta_{6} ) / 12 \) |
\(\nu^{6}\) | \(=\) | \( ( 23875\beta_{5} + 92862\beta_{4} - 56522\beta_{3} - 523125\beta_{2} - 366462\beta _1 + 16556556 ) / 6 \) |
\(\nu^{7}\) | \(=\) | \( ( 241277 \beta_{11} - 4695147 \beta_{10} + 9580675 \beta_{9} - 14334051 \beta_{8} + 4899690 \beta_{7} + 77134116 \beta_{6} ) / 12 \) |
\(\nu^{8}\) | \(=\) | \( ( - 43664515 \beta_{5} - 8165433 \beta_{4} + 82455095 \beta_{3} + 31895514 \beta_{2} + 527370786 \beta _1 - 1180662336 ) / 12 \) |
\(\nu^{9}\) | \(=\) | \( ( 41264192 \beta_{11} + 1143031725 \beta_{10} - 2438125094 \beta_{9} - 1848832320 \beta_{8} + 612188982 \beta_{7} + 10103980338 \beta_{6} ) / 12 \) |
\(\nu^{10}\) | \(=\) | \( ( 5058472931 \beta_{5} - 8108490750 \beta_{4} - 9667434835 \beta_{3} + 38868148557 \beta_{2} - 59526603972 \beta _1 - 1301887858200 ) / 12 \) |
\(\nu^{11}\) | \(=\) | \( ( - 1125572761 \beta_{11} - 950125491 \beta_{10} + 4234822777 \beta_{9} + 945942587565 \beta_{8} - 357624354582 \beta_{7} - 5021516691840 \beta_{6} ) / 12 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1872\mathbb{Z}\right)^\times\).
\(n\) | \(145\) | \(209\) | \(469\) | \(703\) |
\(\chi(n)\) | \(1\) | \(-1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
287.1 |
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0 | 0 | 0 | − | 17.5993i | 0 | − | 20.9778i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
287.2 | 0 | 0 | 0 | − | 17.5993i | 0 | 20.9778i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
287.3 | 0 | 0 | 0 | − | 11.5817i | 0 | − | 4.74823i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
287.4 | 0 | 0 | 0 | − | 11.5817i | 0 | 4.74823i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
287.5 | 0 | 0 | 0 | − | 0.360788i | 0 | − | 22.0315i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
287.6 | 0 | 0 | 0 | − | 0.360788i | 0 | 22.0315i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
287.7 | 0 | 0 | 0 | 0.360788i | 0 | − | 22.0315i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
287.8 | 0 | 0 | 0 | 0.360788i | 0 | 22.0315i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
287.9 | 0 | 0 | 0 | 11.5817i | 0 | − | 4.74823i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
287.10 | 0 | 0 | 0 | 11.5817i | 0 | 4.74823i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
287.11 | 0 | 0 | 0 | 17.5993i | 0 | − | 20.9778i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
287.12 | 0 | 0 | 0 | 17.5993i | 0 | 20.9778i | 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 |
12.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1872.4.d.b | ✓ | 12 |
3.b | odd | 2 | 1 | inner | 1872.4.d.b | ✓ | 12 |
4.b | odd | 2 | 1 | inner | 1872.4.d.b | ✓ | 12 |
12.b | even | 2 | 1 | inner | 1872.4.d.b | ✓ | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1872.4.d.b | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
1872.4.d.b | ✓ | 12 | 3.b | odd | 2 | 1 | inner |
1872.4.d.b | ✓ | 12 | 4.b | odd | 2 | 1 | inner |
1872.4.d.b | ✓ | 12 | 12.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} + 444T_{5}^{4} + 41604T_{5}^{2} + 5408 \)
acting on \(S_{4}^{\mathrm{new}}(1872, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{12} \)
$3$
\( T^{12} \)
$5$
\( (T^{6} + 444 T^{4} + 41604 T^{2} + \cdots + 5408)^{2} \)
$7$
\( (T^{6} + 948 T^{4} + 234468 T^{2} + \cdots + 4815824)^{2} \)
$11$
\( (T^{6} - 7086 T^{4} + \cdots - 3187690792)^{2} \)
$13$
\( (T - 13)^{12} \)
$17$
\( (T^{6} + 5358 T^{4} + \cdots + 3122080200)^{2} \)
$19$
\( (T^{6} + 14448 T^{4} + \cdots + 332377344)^{2} \)
$23$
\( (T^{6} - 36936 T^{4} + \cdots - 24079120000)^{2} \)
$29$
\( (T^{6} + 113286 T^{4} + \cdots + 5084822232072)^{2} \)
$31$
\( (T^{6} + 32448 T^{4} + \cdots + 128314182464)^{2} \)
$37$
\( (T^{3} - 54 T^{2} - 64128 T - 5001104)^{4} \)
$41$
\( (T^{6} + 57636 T^{4} + \cdots + 11779283072)^{2} \)
$43$
\( (T^{6} + 208344 T^{4} + \cdots + 241274043518976)^{2} \)
$47$
\( (T^{6} - 627486 T^{4} + \cdots - 49\!\cdots\!28)^{2} \)
$53$
\( (T^{6} + 863934 T^{4} + \cdots + 74541512760968)^{2} \)
$59$
\( (T^{6} - 685590 T^{4} + \cdots - 21\!\cdots\!92)^{2} \)
$61$
\( (T^{3} + 96 T^{2} - 342972 T - 48454848)^{4} \)
$67$
\( (T^{6} + 501348 T^{4} + \cdots + 38\!\cdots\!76)^{2} \)
$71$
\( (T^{6} - 1603686 T^{4} + \cdots - 67\!\cdots\!08)^{2} \)
$73$
\( (T^{3} + 138 T^{2} - 1006608 T - 422788816)^{4} \)
$79$
\( (T^{6} + 1710672 T^{4} + \cdots + 95\!\cdots\!96)^{2} \)
$83$
\( (T^{6} - 1601718 T^{4} + \cdots - 58\!\cdots\!88)^{2} \)
$89$
\( (T^{6} + 235308 T^{4} + \cdots + 303816180020000)^{2} \)
$97$
\( (T^{3} - 1122 T^{2} + 100608 T + 42862608)^{4} \)
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