Newspace parameters
Level: | \( N \) | \(=\) | \( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 2268.f (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(18.1100711784\) |
Analytic rank: | \(0\) |
Dimension: | \(16\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{16} - 3x^{14} - 9x^{12} - 9x^{10} + 225x^{8} - 81x^{6} - 729x^{4} - 2187x^{2} + 6561 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
Coefficient ring index: | \( 2^{8}\cdot 3^{8} \) |
Twist minimal: | no (minimal twist has level 252) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) 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 \( x^{16} - 3x^{14} - 9x^{12} - 9x^{10} + 225x^{8} - 81x^{6} - 729x^{4} - 2187x^{2} + 6561 \) :
\(\beta_{1}\) | \(=\) | \( ( -\nu^{14} + 3\nu^{12} + 9\nu^{10} + 9\nu^{8} - 225\nu^{6} + 81\nu^{4} + 2187 ) / 729 \) |
\(\beta_{2}\) | \(=\) | \( ( 5 \nu^{15} + 72 \nu^{14} + 12 \nu^{13} + 135 \nu^{12} - 18 \nu^{11} - 486 \nu^{10} - 369 \nu^{9} - 5994 \nu^{8} - 1548 \nu^{7} + 8667 \nu^{6} + 1782 \nu^{5} + 44712 \nu^{4} + 25029 \nu^{3} + \cdots - 347733 ) / 30618 \) |
\(\beta_{3}\) | \(=\) | \( ( 5 \nu^{15} - 72 \nu^{14} + 12 \nu^{13} - 135 \nu^{12} - 18 \nu^{11} + 486 \nu^{10} - 369 \nu^{9} + 5994 \nu^{8} - 1548 \nu^{7} - 8667 \nu^{6} + 1782 \nu^{5} - 44712 \nu^{4} + 25029 \nu^{3} + \cdots + 347733 ) / 30618 \) |
\(\beta_{4}\) | \(=\) | \( ( -\nu^{14} - 3\nu^{12} + 27\nu^{10} + 63\nu^{8} - 171\nu^{6} - 783\nu^{4} + 486\nu^{2} + 3888 ) / 243 \) |
\(\beta_{5}\) | \(=\) | \( ( 10\nu^{14} + 24\nu^{12} - 36\nu^{10} - 738\nu^{8} + 306\nu^{6} + 3564\nu^{4} + 9234\nu^{2} - 24057 ) / 1701 \) |
\(\beta_{6}\) | \(=\) | \( ( 47\nu^{14} + 12\nu^{12} - 396\nu^{10} - 1314\nu^{8} + 3366\nu^{6} + 8019\nu^{4} + 2916\nu^{2} - 34992 ) / 5103 \) |
\(\beta_{7}\) | \(=\) | \( ( -2\nu^{15} - 12\nu^{13} + 18\nu^{11} + 99\nu^{9} + 198\nu^{7} - 486\nu^{5} - 486\nu^{3} - 4374\nu ) / 2187 \) |
\(\beta_{8}\) | \(=\) | \( ( -\nu^{14} + \nu^{12} + 12\nu^{10} + 36\nu^{8} - 153\nu^{6} - 261\nu^{4} + 216\nu^{2} + 2430 ) / 81 \) |
\(\beta_{9}\) | \(=\) | \( ( - 5 \nu^{15} - 18 \nu^{14} - 12 \nu^{13} + 27 \nu^{12} + 72 \nu^{11} + 162 \nu^{10} + 207 \nu^{9} + 648 \nu^{8} - 396 \nu^{7} - 2349 \nu^{6} - 2268 \nu^{5} - 3888 \nu^{4} + 243 \nu^{3} + \cdots + 41553 ) / 4374 \) |
\(\beta_{10}\) | \(=\) | \( ( 5 \nu^{15} - 18 \nu^{14} + 12 \nu^{13} + 27 \nu^{12} - 72 \nu^{11} + 162 \nu^{10} - 207 \nu^{9} + 648 \nu^{8} + 396 \nu^{7} - 2349 \nu^{6} + 2268 \nu^{5} - 3888 \nu^{4} - 243 \nu^{3} - 2916 \nu^{2} + \cdots + 41553 ) / 4374 \) |
\(\beta_{11}\) | \(=\) | \( ( \nu^{15} + 3\nu^{13} - 63\nu^{9} - 72\nu^{7} + 297\nu^{5} + 1215\nu^{3} - 1458\nu ) / 729 \) |
\(\beta_{12}\) | \(=\) | \( ( 26\nu^{15} - 51\nu^{13} - 207\nu^{11} - 558\nu^{9} + 3177\nu^{7} + 81\nu^{5} + 9720\nu^{3} - 34992\nu ) / 15309 \) |
\(\beta_{13}\) | \(=\) | \( ( 4\nu^{15} - 12\nu^{13} - 9\nu^{11} - 117\nu^{9} + 657\nu^{7} + 162\nu^{5} + 972\nu^{3} - 8748\nu ) / 2187 \) |
\(\beta_{14}\) | \(=\) | \( ( 55\nu^{15} - 120\nu^{13} - 576\nu^{11} - 90\nu^{9} + 5652\nu^{7} + 3726\nu^{5} - 20655\nu^{3} + 13122\nu ) / 15309 \) |
\(\beta_{15}\) | \(=\) | \( ( 61\nu^{15} - 30\nu^{13} - 711\nu^{11} - 2574\nu^{9} + 8217\nu^{7} + 18792\nu^{5} + 1215\nu^{3} - 157464\nu ) / 15309 \) |
\(\nu\) | \(=\) | \( ( \beta_{13} - \beta_{12} - \beta_{11} - \beta_{7} + \beta_{3} + \beta_{2} ) / 6 \) |
\(\nu^{2}\) | \(=\) | \( ( -2\beta_{10} - 2\beta_{9} + 2\beta_{8} - \beta_{4} - 3\beta _1 + 3 ) / 6 \) |
\(\nu^{3}\) | \(=\) | \( ( -2\beta_{15} - \beta_{14} + 4\beta_{12} + 3\beta_{10} - 3\beta_{9} + 3\beta_{7} + 2\beta_{3} + 2\beta_{2} ) / 6 \) |
\(\nu^{4}\) | \(=\) | \( ( -\beta_{10} - \beta_{9} + \beta_{8} + \beta_{6} - 2\beta_{5} - 2\beta_{3} + 2\beta_{2} + 2\beta _1 + 7 ) / 2 \) |
\(\nu^{5}\) | \(=\) | \( ( - \beta_{15} + \beta_{14} + 2 \beta_{13} - 5 \beta_{12} + \beta_{11} + 3 \beta_{10} - 3 \beta_{9} + 4 \beta_{7} + 2 \beta_{3} + 2 \beta_{2} ) / 2 \) |
\(\nu^{6}\) | \(=\) | \( ( 3\beta_{10} + 3\beta_{9} - \beta_{5} - 3\beta_{3} + 3\beta_{2} - 12\beta _1 + 33 ) / 2 \) |
\(\nu^{7}\) | \(=\) | \( ( - 7 \beta_{15} - 5 \beta_{14} + 15 \beta_{13} + 2 \beta_{12} - 12 \beta_{11} + 12 \beta_{10} - 12 \beta_{9} - 3 \beta_{7} + 4 \beta_{3} + 4 \beta_{2} ) / 2 \) |
\(\nu^{8}\) | \(=\) | \( ( - 6 \beta_{10} - 6 \beta_{9} + 15 \beta_{8} + 15 \beta_{6} - 18 \beta_{5} - 3 \beta_{4} - 15 \beta_{3} + 15 \beta_{2} - 15 \beta _1 - 54 ) / 2 \) |
\(\nu^{9}\) | \(=\) | \( - 21 \beta_{15} + 3 \beta_{14} + 3 \beta_{13} + 12 \beta_{12} + 6 \beta_{11} + 27 \beta_{10} - 27 \beta_{9} + 24 \beta_{7} - 3 \beta_{3} - 3 \beta_{2} \) |
\(\nu^{10}\) | \(=\) | \( ( 42 \beta_{10} + 42 \beta_{9} - 15 \beta_{8} + 27 \beta_{6} - 36 \beta_{5} + 39 \beta_{4} - 63 \beta_{3} + 63 \beta_{2} + 9 \beta _1 + 108 ) / 2 \) |
\(\nu^{11}\) | \(=\) | \( ( - 75 \beta_{15} + 3 \beta_{14} + 171 \beta_{13} - 120 \beta_{12} + 18 \beta_{11} + 72 \beta_{10} - 72 \beta_{9} + 9 \beta_{7} - 6 \beta_{3} - 6 \beta_{2} ) / 2 \) |
\(\nu^{12}\) | \(=\) | \( ( 189 \beta_{10} + 189 \beta_{9} + 72 \beta_{6} + 9 \beta_{5} - 18 \beta_{4} - 63 \beta_{3} + 63 \beta_{2} - 342 \beta _1 - 225 ) / 2 \) |
\(\nu^{13}\) | \(=\) | \( ( - 333 \beta_{15} - 153 \beta_{14} + 252 \beta_{13} + 297 \beta_{12} - 171 \beta_{11} + 459 \beta_{10} - 459 \beta_{9} - 144 \beta_{7} - 108 \beta_{3} - 108 \beta_{2} ) / 2 \) |
\(\nu^{14}\) | \(=\) | \( ( 135 \beta_{10} + 135 \beta_{9} + 81 \beta_{8} + 675 \beta_{6} - 396 \beta_{5} + 270 \beta_{4} - 378 \beta_{3} + 378 \beta_{2} + 324 \beta _1 - 2673 ) / 2 \) |
\(\nu^{15}\) | \(=\) | \( ( - 1044 \beta_{15} + 585 \beta_{14} + 594 \beta_{13} + 144 \beta_{12} + 1080 \beta_{11} + 783 \beta_{10} - 783 \beta_{9} + 351 \beta_{7} - 684 \beta_{3} - 684 \beta_{2} ) / 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times\).
\(n\) | \(325\) | \(1135\) | \(1541\) |
\(\chi(n)\) | \(-1\) | \(1\) | \(-1\) |
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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1133.1 |
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0 | 0 | 0 | −4.18671 | 0 | 1.27652 | − | 2.31743i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1133.2 | 0 | 0 | 0 | −4.18671 | 0 | 1.27652 | + | 2.31743i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1133.3 | 0 | 0 | 0 | −2.42488 | 0 | −2.62893 | − | 0.297883i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1133.4 | 0 | 0 | 0 | −2.42488 | 0 | −2.62893 | + | 0.297883i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1133.5 | 0 | 0 | 0 | −0.553827 | 0 | 2.50632 | − | 0.847573i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1133.6 | 0 | 0 | 0 | −0.553827 | 0 | 2.50632 | + | 0.847573i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1133.7 | 0 | 0 | 0 | −0.533560 | 0 | −0.653912 | − | 2.56367i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1133.8 | 0 | 0 | 0 | −0.533560 | 0 | −0.653912 | + | 2.56367i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1133.9 | 0 | 0 | 0 | 0.533560 | 0 | −0.653912 | − | 2.56367i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1133.10 | 0 | 0 | 0 | 0.533560 | 0 | −0.653912 | + | 2.56367i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1133.11 | 0 | 0 | 0 | 0.553827 | 0 | 2.50632 | − | 0.847573i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1133.12 | 0 | 0 | 0 | 0.553827 | 0 | 2.50632 | + | 0.847573i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1133.13 | 0 | 0 | 0 | 2.42488 | 0 | −2.62893 | − | 0.297883i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1133.14 | 0 | 0 | 0 | 2.42488 | 0 | −2.62893 | + | 0.297883i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1133.15 | 0 | 0 | 0 | 4.18671 | 0 | 1.27652 | − | 2.31743i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1133.16 | 0 | 0 | 0 | 4.18671 | 0 | 1.27652 | + | 2.31743i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
7.b | odd | 2 | 1 | inner |
21.c | even | 2 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} - 24T_{5}^{6} + 117T_{5}^{4} - 63T_{5}^{2} + 9 \)
acting on \(S_{2}^{\mathrm{new}}(2268, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{16} \)
$3$
\( T^{16} \)
$5$
\( (T^{8} - 24 T^{6} + 117 T^{4} - 63 T^{2} + \cdots + 9)^{2} \)
$7$
\( (T^{8} - T^{7} - 2 T^{6} + 11 T^{5} + \cdots + 2401)^{2} \)
$11$
\( (T^{8} + 45 T^{6} + 639 T^{4} + 3078 T^{2} + \cdots + 3969)^{2} \)
$13$
\( (T^{8} + 48 T^{6} + 558 T^{4} + 504 T^{2} + \cdots + 9)^{2} \)
$17$
\( (T^{8} - 78 T^{6} + 1467 T^{4} - 6741 T^{2} + \cdots + 900)^{2} \)
$19$
\( (T^{8} + 75 T^{6} + 1476 T^{4} + 5787 T^{2} + \cdots + 900)^{2} \)
$23$
\( (T^{8} + 81 T^{6} + 2151 T^{4} + \cdots + 50625)^{2} \)
$29$
\( (T^{8} + 162 T^{6} + 8523 T^{4} + \cdots + 245025)^{2} \)
$31$
\( (T^{8} + 72 T^{6} + 1053 T^{4} + 1701 T^{2} + \cdots + 729)^{2} \)
$37$
\( (T^{4} - T^{3} - 66 T^{2} + 23 T + 610)^{4} \)
$41$
\( (T^{8} - 177 T^{6} + 9711 T^{4} + \cdots + 576081)^{2} \)
$43$
\( (T^{4} + 2 T^{3} - 75 T^{2} - 193 T + 679)^{4} \)
$47$
\( (T^{8} - 222 T^{6} + 16785 T^{4} + \cdots + 4968441)^{2} \)
$53$
\( (T^{8} + 414 T^{6} + 56133 T^{4} + \cdots + 41990400)^{2} \)
$59$
\( (T^{8} - 96 T^{6} + 2439 T^{4} - 18225 T^{2} + \cdots + 441)^{2} \)
$61$
\( (T^{8} + 351 T^{6} + 42741 T^{4} + \cdots + 35319249)^{2} \)
$67$
\( (T^{4} + 7 T^{3} - 111 T^{2} - 1004 T - 1985)^{4} \)
$71$
\( (T^{8} + 207 T^{6} + 11250 T^{4} + \cdots + 15876)^{2} \)
$73$
\( (T^{8} + 243 T^{6} + 19620 T^{4} + \cdots + 76176)^{2} \)
$79$
\( (T^{4} + 10 T^{3} - 93 T^{2} - 833 T + 565)^{4} \)
$83$
\( (T^{8} - 267 T^{6} + 9693 T^{4} + \cdots + 173889)^{2} \)
$89$
\( (T^{8} - 648 T^{6} + 133731 T^{4} + \cdots + 211004676)^{2} \)
$97$
\( (T^{8} + 372 T^{6} + 34659 T^{4} + \cdots + 9162729)^{2} \)
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