Newspace parameters
Level: | \( N \) | \(=\) | \( 342 = 2 \cdot 3^{2} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 342.x (of order \(18\), degree \(6\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.73088374913\) |
Analytic rank: | \(0\) |
Dimension: | \(12\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{18})\) |
Coefficient field: | 12.0.1952986685049.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{12} - 6 x^{11} + 27 x^{10} - 80 x^{9} + 186 x^{8} - 330 x^{7} + 463 x^{6} - 504 x^{5} + 420 x^{4} - 258 x^{3} + 108 x^{2} - 27 x + 3 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 3^{2} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{18}]$ |
$q$-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 \( x^{12} - 6 x^{11} + 27 x^{10} - 80 x^{9} + 186 x^{8} - 330 x^{7} + 463 x^{6} - 504 x^{5} + 420 x^{4} - 258 x^{3} + 108 x^{2} - 27 x + 3 \) :
\(\beta_{1}\) | \(=\) | \( 3 \nu^{11} - 15 \nu^{10} + 66 \nu^{9} - 175 \nu^{8} + 387 \nu^{7} - 619 \nu^{6} + 804 \nu^{5} - 770 \nu^{4} + 562 \nu^{3} - 277 \nu^{2} + 82 \nu - 11 \) |
\(\beta_{2}\) | \(=\) | \( 3 \nu^{11} - 18 \nu^{10} + 81 \nu^{9} - 239 \nu^{8} + 553 \nu^{7} - 970 \nu^{6} + 1339 \nu^{5} - 1416 \nu^{4} + 1126 \nu^{3} - 637 \nu^{2} + 226 \nu - 37 \) |
\(\beta_{3}\) | \(=\) | \( 7 \nu^{11} - 38 \nu^{10} + 167 \nu^{9} - 463 \nu^{8} + 1031 \nu^{7} - 1704 \nu^{6} + 2234 \nu^{5} - 2204 \nu^{4} + 1632 \nu^{3} - 841 \nu^{2} + 263 \nu - 38 \) |
\(\beta_{4}\) | \(=\) | \( - 7 \nu^{11} + 39 \nu^{10} - 172 \nu^{9} + 485 \nu^{8} - 1089 \nu^{7} + 1831 \nu^{6} - 2433 \nu^{5} + 2453 \nu^{4} - 1856 \nu^{3} + 985 \nu^{2} - 320 \nu + 46 \) |
\(\beta_{5}\) | \(=\) | \( - 8 \nu^{11} + 46 \nu^{10} - 204 \nu^{9} + 586 \nu^{8} - 1328 \nu^{7} + 2269 \nu^{6} - 3048 \nu^{5} + 3122 \nu^{4} - 2386 \nu^{3} + 1286 \nu^{2} - 419 \nu + 62 \) |
\(\beta_{6}\) | \(=\) | \( 16 \nu^{11} - 87 \nu^{10} + 383 \nu^{9} - 1064 \nu^{8} + 2375 \nu^{7} - 3936 \nu^{6} + 5176 \nu^{5} - 5122 \nu^{4} + 3802 \nu^{3} - 1958 \nu^{2} + 610 \nu - 85 \) |
\(\beta_{7}\) | \(=\) | \( - 16 \nu^{11} + 89 \nu^{10} - 393 \nu^{9} + 1108 \nu^{8} - 2491 \nu^{7} + 4191 \nu^{6} - 5577 \nu^{5} + 5631 \nu^{4} - 4267 \nu^{3} + 2272 \nu^{2} - 742 \nu + 110 \) |
\(\beta_{8}\) | \(=\) | \( - 24 \nu^{11} + 131 \nu^{10} - 577 \nu^{9} + 1608 \nu^{8} - 3595 \nu^{7} + 5982 \nu^{6} - 7891 \nu^{5} + 7858 \nu^{4} - 5865 \nu^{3} + 3051 \nu^{2} - 957 \nu + 133 \) |
\(\beta_{9}\) | \(=\) | \( - 36 \nu^{11} + 198 \nu^{10} - 873 \nu^{9} + 2443 \nu^{8} - 5472 \nu^{7} + 9134 \nu^{6} - 12076 \nu^{5} + 12058 \nu^{4} - 9024 \nu^{3} + 4708 \nu^{2} - 1486 \nu + 209 \) |
\(\beta_{10}\) | \(=\) | \( - 36 \nu^{11} + 198 \nu^{10} - 873 \nu^{9} + 2444 \nu^{8} - 5476 \nu^{7} + 9150 \nu^{6} - 12110 \nu^{5} + 12120 \nu^{4} - 9096 \nu^{3} + 4772 \nu^{2} - 1519 \nu + 217 \) |
\(\beta_{11}\) | \(=\) | \( 42 \nu^{11} - 231 \nu^{10} + 1019 \nu^{9} - 2853 \nu^{8} + 6396 \nu^{7} - 10689 \nu^{6} + 14157 \nu^{5} - 14172 \nu^{4} + 10648 \nu^{3} - 5589 \nu^{2} + 1785 \nu - 256 \) |
\(\nu\) | \(=\) | \( ( -\beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} + \beta_{5} + \beta_{4} - \beta_{3} + 1 ) / 3 \) |
\(\nu^{2}\) | \(=\) | \( ( -\beta_{11} - \beta_{10} - \beta_{9} + 2\beta_{5} - 2\beta_{3} + \beta_{2} - \beta _1 - 4 ) / 3 \) |
\(\nu^{3}\) | \(=\) | \( ( 5 \beta_{11} + 5 \beta_{10} + 5 \beta_{9} - 5 \beta_{8} + 2 \beta_{7} + 3 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - \beta_{3} - 3 \beta _1 - 5 ) / 3 \) |
\(\nu^{4}\) | \(=\) | \( ( 11 \beta_{11} + 17 \beta_{10} + 5 \beta_{9} - 7 \beta_{8} - 5 \beta_{7} - 9 \beta_{5} + 2 \beta_{4} + 6 \beta_{3} - 7 \beta_{2} + \beta _1 + 9 ) / 3 \) |
\(\nu^{5}\) | \(=\) | \( ( - 19 \beta_{11} - \beta_{10} - 31 \beta_{9} + 15 \beta_{8} - 21 \beta_{7} - 24 \beta_{6} + 5 \beta_{5} + 9 \beta_{4} + 16 \beta_{3} - 5 \beta_{2} + 20 \beta _1 + 26 ) / 3 \) |
\(\nu^{6}\) | \(=\) | \( ( - 85 \beta_{11} - 97 \beta_{10} - 55 \beta_{9} + 55 \beta_{8} + 11 \beta_{7} - 18 \beta_{6} + 46 \beta_{5} - 17 \beta_{4} - 7 \beta_{3} + 36 \beta_{2} + 24 \beta _1 - 11 ) / 3 \) |
\(\nu^{7}\) | \(=\) | \( ( 20 \beta_{11} - 118 \beta_{10} + 134 \beta_{9} - 4 \beta_{8} + 142 \beta_{7} + 135 \beta_{6} + 3 \beta_{5} - 76 \beta_{4} - 99 \beta_{3} + 56 \beta_{2} - 77 \beta _1 - 120 ) / 3 \) |
\(\nu^{8}\) | \(=\) | \( ( 503 \beta_{11} + 386 \beta_{10} + 440 \beta_{9} - 279 \beta_{8} + 99 \beta_{7} + 228 \beta_{6} - 235 \beta_{5} + 39 \beta_{4} - 89 \beta_{3} - 152 \beta_{2} - 226 \beta _1 - 73 ) / 3 \) |
\(\nu^{9}\) | \(=\) | \( ( 425 \beta_{11} + 1076 \beta_{10} - 319 \beta_{9} - 305 \beta_{8} - 679 \beta_{7} - 564 \beta_{6} - 197 \beta_{5} + 499 \beta_{4} + 437 \beta_{3} - 438 \beta_{2} + 138 \beta _1 + 472 ) / 3 \) |
\(\nu^{10}\) | \(=\) | \( ( - 2299 \beta_{11} - 862 \beta_{10} - 2725 \beta_{9} + 1061 \beta_{8} - 1277 \beta_{7} - 1824 \beta_{6} + 1074 \beta_{5} + 293 \beta_{4} + 990 \beta_{3} + 434 \beta_{2} + 1351 \beta _1 + 825 ) / 3 \) |
\(\nu^{11}\) | \(=\) | \( ( - 4708 \beta_{11} - 6628 \beta_{10} - 985 \beta_{9} + 2697 \beta_{8} + 2277 \beta_{7} + 1329 \beta_{6} + 1982 \beta_{5} - 2493 \beta_{4} - 1247 \beta_{3} + 2779 \beta_{2} + 788 \beta _1 - 1414 ) / 3 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/342\mathbb{Z}\right)^\times\).
\(n\) | \(191\) | \(325\) |
\(\chi(n)\) | \(1 - \beta_{11}\) | \(-\beta_{10}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
29.1 |
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0.939693 | + | 0.342020i | 0.210069 | − | 1.71926i | 0.766044 | + | 0.642788i | 3.32358 | − | 0.586038i | 0.785424 | − | 1.54373i | 0.883892 | + | 1.53095i | 0.500000 | + | 0.866025i | −2.91174 | − | 0.722330i | 3.32358 | + | 0.586038i | ||||||||||||||||||||||||||||||||||||
29.2 | 0.939693 | + | 0.342020i | 1.72962 | + | 0.0916693i | 0.766044 | + | 0.642788i | 0.995493 | − | 0.175532i | 1.59396 | + | 0.677707i | −1.44420 | − | 2.50143i | 0.500000 | + | 0.866025i | 2.98319 | + | 0.317107i | 0.995493 | + | 0.175532i | |||||||||||||||||||||||||||||||||||||
41.1 | −0.173648 | + | 0.984808i | −0.159815 | − | 1.72466i | −0.939693 | − | 0.342020i | −0.587342 | + | 0.699967i | 1.72621 | + | 0.142098i | −1.91369 | + | 3.31462i | 0.500000 | − | 0.866025i | −2.94892 | + | 0.551252i | −0.587342 | − | 0.699967i | |||||||||||||||||||||||||||||||||||||
41.2 | −0.173648 | + | 0.984808i | 0.986166 | + | 1.42389i | −0.939693 | − | 0.342020i | 1.56640 | − | 1.86676i | −1.57351 | + | 0.723928i | 0.240046 | − | 0.415771i | 0.500000 | − | 0.866025i | −1.05495 | + | 2.80839i | 1.56640 | + | 1.86676i | |||||||||||||||||||||||||||||||||||||
59.1 | 0.939693 | − | 0.342020i | 0.210069 | + | 1.71926i | 0.766044 | − | 0.642788i | 3.32358 | + | 0.586038i | 0.785424 | + | 1.54373i | 0.883892 | − | 1.53095i | 0.500000 | − | 0.866025i | −2.91174 | + | 0.722330i | 3.32358 | − | 0.586038i | |||||||||||||||||||||||||||||||||||||
59.2 | 0.939693 | − | 0.342020i | 1.72962 | − | 0.0916693i | 0.766044 | − | 0.642788i | 0.995493 | + | 0.175532i | 1.59396 | − | 0.677707i | −1.44420 | + | 2.50143i | 0.500000 | − | 0.866025i | 2.98319 | − | 0.317107i | 0.995493 | − | 0.175532i | |||||||||||||||||||||||||||||||||||||
185.1 | −0.766044 | − | 0.642788i | −1.45446 | − | 0.940501i | 0.173648 | + | 0.984808i | 0.146688 | + | 0.403023i | 0.509640 | + | 1.65538i | −0.587267 | + | 1.01718i | 0.500000 | − | 0.866025i | 1.23092 | + | 2.73584i | 0.146688 | − | 0.403023i | |||||||||||||||||||||||||||||||||||||
185.2 | −0.766044 | − | 0.642788i | 1.68842 | − | 0.386327i | 0.173648 | + | 0.984808i | −0.944822 | − | 2.59588i | −1.54173 | − | 0.789350i | −1.67878 | + | 2.90773i | 0.500000 | − | 0.866025i | 2.70150 | − | 1.30456i | −0.944822 | + | 2.59588i | |||||||||||||||||||||||||||||||||||||
281.1 | −0.766044 | + | 0.642788i | −1.45446 | + | 0.940501i | 0.173648 | − | 0.984808i | 0.146688 | − | 0.403023i | 0.509640 | − | 1.65538i | −0.587267 | − | 1.01718i | 0.500000 | + | 0.866025i | 1.23092 | − | 2.73584i | 0.146688 | + | 0.403023i | |||||||||||||||||||||||||||||||||||||
281.2 | −0.766044 | + | 0.642788i | 1.68842 | + | 0.386327i | 0.173648 | − | 0.984808i | −0.944822 | + | 2.59588i | −1.54173 | + | 0.789350i | −1.67878 | − | 2.90773i | 0.500000 | + | 0.866025i | 2.70150 | + | 1.30456i | −0.944822 | − | 2.59588i | |||||||||||||||||||||||||||||||||||||
317.1 | −0.173648 | − | 0.984808i | −0.159815 | + | 1.72466i | −0.939693 | + | 0.342020i | −0.587342 | − | 0.699967i | 1.72621 | − | 0.142098i | −1.91369 | − | 3.31462i | 0.500000 | + | 0.866025i | −2.94892 | − | 0.551252i | −0.587342 | + | 0.699967i | |||||||||||||||||||||||||||||||||||||
317.2 | −0.173648 | − | 0.984808i | 0.986166 | − | 1.42389i | −0.939693 | + | 0.342020i | 1.56640 | + | 1.86676i | −1.57351 | − | 0.723928i | 0.240046 | + | 0.415771i | 0.500000 | + | 0.866025i | −1.05495 | − | 2.80839i | 1.56640 | − | 1.86676i | |||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
171.x | even | 18 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 342.2.x.a | ✓ | 12 |
9.d | odd | 6 | 1 | 342.2.bf.a | yes | 12 | |
19.f | odd | 18 | 1 | 342.2.bf.a | yes | 12 | |
171.x | even | 18 | 1 | inner | 342.2.x.a | ✓ | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
342.2.x.a | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
342.2.x.a | ✓ | 12 | 171.x | even | 18 | 1 | inner |
342.2.bf.a | yes | 12 | 9.d | odd | 6 | 1 | |
342.2.bf.a | yes | 12 | 19.f | odd | 18 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{12} - 9 T_{5}^{11} + 36 T_{5}^{10} - 108 T_{5}^{9} + 306 T_{5}^{8} - 702 T_{5}^{7} + 1062 T_{5}^{6} - 729 T_{5}^{5} + 405 T_{5}^{4} - 729 T_{5}^{3} + 648 T_{5}^{2} - 243 T_{5} + 81 \)
acting on \(S_{2}^{\mathrm{new}}(342, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{6} - T^{3} + 1)^{2} \)
$3$
\( T^{12} - 6 T^{11} + 18 T^{10} - 39 T^{9} + \cdots + 729 \)
$5$
\( T^{12} - 9 T^{11} + 36 T^{10} - 108 T^{9} + \cdots + 81 \)
$7$
\( T^{12} + 9 T^{11} + 60 T^{10} + \cdots + 1369 \)
$11$
\( T^{12} + 81 T^{10} + 2223 T^{8} + \cdots + 29241 \)
$13$
\( T^{12} + 12 T^{11} + 75 T^{10} + \cdots + 567009 \)
$17$
\( T^{12} + 27 T^{10} + 225 T^{9} + \cdots + 3143529 \)
$19$
\( T^{12} - 15 T^{11} + 156 T^{10} + \cdots + 47045881 \)
$23$
\( T^{12} - 18 T^{11} + 153 T^{10} + \cdots + 263169 \)
$29$
\( T^{12} + 45 T^{11} + \cdots + 2503100961 \)
$31$
\( (T^{6} + 42 T^{4} + 477 T^{2} + 867)^{2} \)
$37$
\( T^{12} + 192 T^{10} + 11844 T^{8} + \cdots + 47961 \)
$41$
\( T^{12} + 36 T^{11} + 612 T^{10} + \cdots + 11758041 \)
$43$
\( T^{12} - 12 T^{11} + \cdots + 3825298801 \)
$47$
\( T^{12} - 27 T^{11} + 441 T^{10} + \cdots + 22155849 \)
$53$
\( T^{12} + 171 T^{10} + \cdots + 8176318929 \)
$59$
\( T^{12} + 36 T^{11} + \cdots + 114896961 \)
$61$
\( T^{12} + 3 T^{11} + \cdots + 3449565289 \)
$67$
\( T^{12} - 30 T^{11} + 444 T^{10} + \cdots + 45369 \)
$71$
\( T^{12} + 9 T^{11} + 45 T^{10} + \cdots + 33074001 \)
$73$
\( T^{12} + 24 T^{11} + 273 T^{10} + \cdots + 546121 \)
$79$
\( T^{12} - 6 T^{11} + \cdots + 110109157929 \)
$83$
\( T^{12} - 45 T^{11} + 855 T^{10} + \cdots + 51969681 \)
$89$
\( T^{12} + 9 T^{11} + \cdots + 9358821081 \)
$97$
\( T^{12} - 45 T^{11} + 1197 T^{10} + \cdots + 2595321 \)
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