Newspace parameters
Level: | \( N \) | \(=\) | \( 663 = 3 \cdot 13 \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 663.z (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(5.29408165401\) |
Analytic rank: | \(0\) |
Dimension: | \(16\) |
Relative dimension: | \(8\) over \(\Q(\zeta_{6})\) |
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: | \( x^{16} + 20x^{14} + 156x^{12} + 602x^{10} + 1212x^{8} + 1259x^{6} + 665x^{4} + 168x^{2} + 16 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 \( x^{16} + 20x^{14} + 156x^{12} + 602x^{10} + 1212x^{8} + 1259x^{6} + 665x^{4} + 168x^{2} + 16 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( -\nu^{14} - 20\nu^{12} - 156\nu^{10} - 602\nu^{8} - 1208\nu^{6} - 1219\nu^{4} - 553\nu^{2} + 4\nu - 84 ) / 8 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{15} + 2 \nu^{14} + 20 \nu^{13} + 40 \nu^{12} + 156 \nu^{11} + 312 \nu^{10} + 602 \nu^{9} + 1204 \nu^{8} + 1212 \nu^{7} + 2424 \nu^{6} + 1259 \nu^{5} + 2518 \nu^{4} + 665 \nu^{3} + 1314 \nu^{2} + \cdots + 256 ) / 16 \) |
\(\beta_{4}\) | \(=\) | \( ( \nu^{15} - 2 \nu^{14} + 20 \nu^{13} - 40 \nu^{12} + 156 \nu^{11} - 312 \nu^{10} + 602 \nu^{9} - 1204 \nu^{8} + 1212 \nu^{7} - 2424 \nu^{6} + 1259 \nu^{5} - 2518 \nu^{4} + 665 \nu^{3} - 1314 \nu^{2} + \cdots - 256 ) / 16 \) |
\(\beta_{5}\) | \(=\) | \( ( - 5 \nu^{15} - 22 \nu^{14} - 88 \nu^{13} - 432 \nu^{12} - 556 \nu^{11} - 3272 \nu^{10} - 1426 \nu^{9} - 11996 \nu^{8} - 788 \nu^{7} - 21880 \nu^{6} + 2041 \nu^{5} - 18386 \nu^{4} + 2279 \nu^{3} + \cdots - 552 ) / 64 \) |
\(\beta_{6}\) | \(=\) | \( ( - \nu^{15} - 11 \nu^{14} - 20 \nu^{13} - 216 \nu^{12} - 156 \nu^{11} - 1636 \nu^{10} - 602 \nu^{9} - 6006 \nu^{8} - 1212 \nu^{7} - 11044 \nu^{6} - 1259 \nu^{5} - 9641 \nu^{4} - 657 \nu^{3} + \cdots - 508 ) / 16 \) |
\(\beta_{7}\) | \(=\) | \( ( \nu^{15} - 11 \nu^{14} + 20 \nu^{13} - 216 \nu^{12} + 156 \nu^{11} - 1636 \nu^{10} + 602 \nu^{9} - 6006 \nu^{8} + 1212 \nu^{7} - 11044 \nu^{6} + 1259 \nu^{5} - 9641 \nu^{4} + 657 \nu^{3} - 3727 \nu^{2} + \cdots - 508 ) / 16 \) |
\(\beta_{8}\) | \(=\) | \( ( - 17 \nu^{14} - 336 \nu^{12} - 2572 \nu^{10} - 9610 \nu^{8} - 18204 \nu^{6} - 16667 \nu^{4} - 6765 \nu^{2} - 8 \nu - 956 ) / 16 \) |
\(\beta_{9}\) | \(=\) | \( ( 47 \nu^{15} + 90 \nu^{14} + 952 \nu^{13} + 1744 \nu^{12} + 7556 \nu^{11} + 12952 \nu^{10} + 29846 \nu^{9} + 46084 \nu^{8} + 61756 \nu^{7} + 80232 \nu^{6} + 64981 \nu^{5} + 62974 \nu^{4} + \cdots + 2040 ) / 128 \) |
\(\beta_{10}\) | \(=\) | \( ( 21 \nu^{15} + 416 \nu^{13} + 3196 \nu^{11} + 12018 \nu^{9} + 23044 \nu^{7} + 21607 \nu^{5} + 9089 \nu^{3} + 1316 \nu + 16 ) / 32 \) |
\(\beta_{11}\) | \(=\) | \( ( - 63 \nu^{15} - 1224 \nu^{13} - 9124 \nu^{11} - 32630 \nu^{9} - 57148 \nu^{7} - 44837 \nu^{5} - 13403 \nu^{3} + 32 \nu^{2} - 932 \nu + 96 ) / 64 \) |
\(\beta_{12}\) | \(=\) | \( ( 189 \nu^{15} - 18 \nu^{14} + 3720 \nu^{13} - 336 \nu^{12} + 28300 \nu^{11} - 2360 \nu^{10} + 104738 \nu^{9} - 7668 \nu^{8} + 195444 \nu^{7} - 11272 \nu^{6} + 174799 \nu^{5} - 5990 \nu^{4} + \cdots + 168 ) / 128 \) |
\(\beta_{13}\) | \(=\) | \( ( - 195 \nu^{15} - 94 \nu^{14} - 3800 \nu^{13} - 1840 \nu^{12} - 28468 \nu^{11} - 13896 \nu^{10} - 102750 \nu^{9} - 50924 \nu^{8} - 183500 \nu^{7} - 93752 \nu^{6} - 151345 \nu^{5} + \cdots - 4264 ) / 128 \) |
\(\beta_{14}\) | \(=\) | \( ( 125 \nu^{15} + 46 \nu^{14} + 2456 \nu^{13} + 912 \nu^{12} + 18636 \nu^{11} + 7016 \nu^{10} + 68706 \nu^{9} + 26444 \nu^{8} + 127476 \nu^{7} + 50936 \nu^{6} + 113199 \nu^{5} + 48250 \nu^{4} + \cdots + 3144 ) / 64 \) |
\(\beta_{15}\) | \(=\) | \( ( 125 \nu^{15} - 46 \nu^{14} + 2456 \nu^{13} - 912 \nu^{12} + 18636 \nu^{11} - 7016 \nu^{10} + 68706 \nu^{9} - 26444 \nu^{8} + 127476 \nu^{7} - 50936 \nu^{6} + 113199 \nu^{5} - 48250 \nu^{4} + \cdots - 3144 ) / 64 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( -\beta_{14} - \beta_{13} + \beta_{12} + \beta_{11} - \beta_{4} + \beta _1 - 3 \) |
\(\nu^{3}\) | \(=\) | \( -\beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} - 5\beta_1 \) |
\(\nu^{4}\) | \(=\) | \( \beta_{15} + 5 \beta_{14} + 6 \beta_{13} - 6 \beta_{12} - 6 \beta_{11} - \beta_{8} + 5 \beta_{4} + \beta_{3} - \beta_{2} - 6 \beta _1 + 14 \) |
\(\nu^{5}\) | \(=\) | \( - \beta_{15} - 2 \beta_{14} - \beta_{13} + \beta_{12} - \beta_{11} + 2 \beta_{10} + 10 \beta_{7} - 10 \beta_{6} - 6 \beta_{4} - 5 \beta_{3} + 25 \beta _1 - 1 \) |
\(\nu^{6}\) | \(=\) | \( - 10 \beta_{15} - 24 \beta_{14} - 34 \beta_{13} + 34 \beta_{12} + 34 \beta_{11} + 10 \beta_{8} - 25 \beta_{4} - 9 \beta_{3} + 12 \beta_{2} + 33 \beta _1 - 73 \) |
\(\nu^{7}\) | \(=\) | \( 10 \beta_{15} + 21 \beta_{14} + 11 \beta_{13} - 9 \beta_{12} + 9 \beta_{11} - 26 \beta_{10} + 2 \beta_{9} - 72 \beta_{7} + 74 \beta_{6} + 35 \beta_{4} + 26 \beta_{3} - 132 \beta _1 + 13 \) |
\(\nu^{8}\) | \(=\) | \( 74 \beta_{15} + 119 \beta_{14} + 193 \beta_{13} - 193 \beta_{12} - 193 \beta_{11} - 72 \beta_{8} - \beta_{7} - \beta_{6} + 130 \beta_{4} + 63 \beta_{3} - 102 \beta_{2} - 178 \beta _1 + 397 \) |
\(\nu^{9}\) | \(=\) | \( - 71 \beta_{15} - 159 \beta_{14} - 88 \beta_{13} + 56 \beta_{12} - 56 \beta_{11} + 232 \beta_{10} - 32 \beta_{9} - \beta_{8} + 461 \beta_{7} - 493 \beta_{6} + 2 \beta_{5} - 208 \beta_{4} - 154 \beta_{3} + \beta_{2} + 727 \beta _1 - 116 \) |
\(\nu^{10}\) | \(=\) | \( - 494 \beta_{15} - 609 \beta_{14} - 1103 \beta_{13} + 1101 \beta_{12} + 1101 \beta_{11} + 2 \beta_{9} + 461 \beta_{8} + 20 \beta_{7} + 18 \beta_{6} - 697 \beta_{4} - 404 \beta_{3} + 757 \beta_{2} + 953 \beta _1 - 2211 \) |
\(\nu^{11}\) | \(=\) | \( 443 \beta_{15} + 1070 \beta_{14} + 627 \beta_{13} - 295 \beta_{12} + 295 \beta_{11} - 1780 \beta_{10} + 332 \beta_{9} + 20 \beta_{8} - 2800 \beta_{7} + 3132 \beta_{6} - 40 \beta_{5} + 1247 \beta_{4} + 992 \beta_{3} - 20 \beta_{2} + \cdots + 890 \) |
\(\nu^{12}\) | \(=\) | \( 3152 \beta_{15} + 3197 \beta_{14} + 6349 \beta_{13} - 6309 \beta_{12} - 6309 \beta_{11} - 40 \beta_{9} - 2800 \beta_{8} - 244 \beta_{7} - 204 \beta_{6} + 3824 \beta_{4} + 2485 \beta_{3} - 5244 \beta_{2} - 5087 \beta _1 + 12510 \) |
\(\nu^{13}\) | \(=\) | \( - 2596 \beta_{15} - 6822 \beta_{14} - 4226 \beta_{13} + 1374 \beta_{12} - 1374 \beta_{11} + 12636 \beta_{10} - 2852 \beta_{9} - 244 \beta_{8} + 16567 \beta_{7} - 19419 \beta_{6} + 488 \beta_{5} - 7491 \beta_{4} + \cdots - 6318 \) |
\(\nu^{14}\) | \(=\) | \( - 19663 \beta_{15} - 17124 \beta_{14} - 36787 \beta_{13} + 36299 \beta_{12} + 36299 \beta_{11} + 488 \beta_{9} + 16567 \beta_{8} + 2362 \beta_{7} + 1874 \beta_{6} - 21350 \beta_{4} - 14949 \beta_{3} + \cdots - 71585 \) |
\(\nu^{15}\) | \(=\) | \( 14693 \beta_{15} + 42304 \beta_{14} + 27611 \beta_{13} - 5523 \beta_{12} + 5523 \beta_{11} - 85710 \beta_{10} + 22088 \beta_{9} + 2362 \beta_{8} - 96723 \beta_{7} + 118811 \beta_{6} - 4724 \beta_{5} + \cdots + 42855 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/663\mathbb{Z}\right)^\times\).
\(n\) | \(443\) | \(547\) | \(613\) |
\(\chi(n)\) | \(1\) | \(1\) | \(\beta_{10}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
205.1 |
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−1.99023 | + | 1.14906i | −0.500000 | − | 0.866025i | 1.64068 | − | 2.84174i | 1.20591i | 1.99023 | + | 1.14906i | −0.421031 | − | 0.243082i | 2.94471i | −0.500000 | + | 0.866025i | −1.38566 | − | 2.40004i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
205.2 | −1.70329 | + | 0.983395i | −0.500000 | − | 0.866025i | 0.934132 | − | 1.61796i | − | 0.562071i | 1.70329 | + | 0.983395i | 2.67180 | + | 1.54256i | − | 0.259096i | −0.500000 | + | 0.866025i | 0.552738 | + | 0.957371i | |||||||||||||||||||||||||||||||||||||||||||||||||||
205.3 | −0.631839 | + | 0.364792i | −0.500000 | − | 0.866025i | −0.733853 | + | 1.27107i | 0.945208i | 0.631839 | + | 0.364792i | −0.308178 | − | 0.177927i | − | 2.52998i | −0.500000 | + | 0.866025i | −0.344805 | − | 0.597219i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
205.4 | −0.517981 | + | 0.299056i | −0.500000 | − | 0.866025i | −0.821130 | + | 1.42224i | − | 1.38228i | 0.517981 | + | 0.299056i | −1.15547 | − | 0.667109i | − | 2.17848i | −0.500000 | + | 0.866025i | 0.413380 | + | 0.715996i | |||||||||||||||||||||||||||||||||||||||||||||||||||
205.5 | 0.444391 | − | 0.256569i | −0.500000 | − | 0.866025i | −0.868344 | + | 1.50402i | − | 4.12599i | −0.444391 | − | 0.256569i | 0.528206 | + | 0.304960i | 1.91744i | −0.500000 | + | 0.866025i | −1.05860 | − | 1.83355i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
205.6 | 0.847250 | − | 0.489160i | −0.500000 | − | 0.866025i | −0.521445 | + | 0.903170i | − | 1.22925i | −0.847250 | − | 0.489160i | 4.03243 | + | 2.32812i | 2.97692i | −0.500000 | + | 0.866025i | −0.601301 | − | 1.04148i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
205.7 | 1.42396 | − | 0.822122i | −0.500000 | − | 0.866025i | 0.351768 | − | 0.609281i | 3.13210i | −1.42396 | − | 0.822122i | −1.27469 | − | 0.735944i | 2.13170i | −0.500000 | + | 0.866025i | 2.57496 | + | 4.45997i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
205.8 | 2.12774 | − | 1.22845i | −0.500000 | − | 0.866025i | 2.01819 | − | 3.49561i | 0.284332i | −2.12774 | − | 1.22845i | −4.07306 | − | 2.35158i | − | 5.00321i | −0.500000 | + | 0.866025i | 0.349288 | + | 0.604985i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
511.1 | −1.99023 | − | 1.14906i | −0.500000 | + | 0.866025i | 1.64068 | + | 2.84174i | − | 1.20591i | 1.99023 | − | 1.14906i | −0.421031 | + | 0.243082i | − | 2.94471i | −0.500000 | − | 0.866025i | −1.38566 | + | 2.40004i | |||||||||||||||||||||||||||||||||||||||||||||||||||
511.2 | −1.70329 | − | 0.983395i | −0.500000 | + | 0.866025i | 0.934132 | + | 1.61796i | 0.562071i | 1.70329 | − | 0.983395i | 2.67180 | − | 1.54256i | 0.259096i | −0.500000 | − | 0.866025i | 0.552738 | − | 0.957371i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
511.3 | −0.631839 | − | 0.364792i | −0.500000 | + | 0.866025i | −0.733853 | − | 1.27107i | − | 0.945208i | 0.631839 | − | 0.364792i | −0.308178 | + | 0.177927i | 2.52998i | −0.500000 | − | 0.866025i | −0.344805 | + | 0.597219i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
511.4 | −0.517981 | − | 0.299056i | −0.500000 | + | 0.866025i | −0.821130 | − | 1.42224i | 1.38228i | 0.517981 | − | 0.299056i | −1.15547 | + | 0.667109i | 2.17848i | −0.500000 | − | 0.866025i | 0.413380 | − | 0.715996i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
511.5 | 0.444391 | + | 0.256569i | −0.500000 | + | 0.866025i | −0.868344 | − | 1.50402i | 4.12599i | −0.444391 | + | 0.256569i | 0.528206 | − | 0.304960i | − | 1.91744i | −0.500000 | − | 0.866025i | −1.05860 | + | 1.83355i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
511.6 | 0.847250 | + | 0.489160i | −0.500000 | + | 0.866025i | −0.521445 | − | 0.903170i | 1.22925i | −0.847250 | + | 0.489160i | 4.03243 | − | 2.32812i | − | 2.97692i | −0.500000 | − | 0.866025i | −0.601301 | + | 1.04148i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
511.7 | 1.42396 | + | 0.822122i | −0.500000 | + | 0.866025i | 0.351768 | + | 0.609281i | − | 3.13210i | −1.42396 | + | 0.822122i | −1.27469 | + | 0.735944i | − | 2.13170i | −0.500000 | − | 0.866025i | 2.57496 | − | 4.45997i | |||||||||||||||||||||||||||||||||||||||||||||||||||
511.8 | 2.12774 | + | 1.22845i | −0.500000 | + | 0.866025i | 2.01819 | + | 3.49561i | − | 0.284332i | −2.12774 | + | 1.22845i | −4.07306 | + | 2.35158i | 5.00321i | −0.500000 | − | 0.866025i | 0.349288 | − | 0.604985i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.e | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 663.2.z.d | ✓ | 16 |
13.e | even | 6 | 1 | inner | 663.2.z.d | ✓ | 16 |
13.f | odd | 12 | 2 | 8619.2.a.bn | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
663.2.z.d | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
663.2.z.d | ✓ | 16 | 13.e | even | 6 | 1 | inner |
8619.2.a.bn | 16 | 13.f | odd | 12 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{16} - 10 T_{2}^{14} + 72 T_{2}^{12} + 3 T_{2}^{11} - 238 T_{2}^{10} + 570 T_{2}^{8} - 84 T_{2}^{7} - 505 T_{2}^{6} + 63 T_{2}^{5} + 329 T_{2}^{4} - 84 T_{2}^{2} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(663, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{16} - 10 T^{14} + 72 T^{12} + 3 T^{11} + \cdots + 16 \)
$3$
\( (T^{2} + T + 1)^{8} \)
$5$
\( T^{16} + 33 T^{14} + 347 T^{12} + \cdots + 16 \)
$7$
\( T^{16} - 29 T^{14} + 685 T^{12} + \cdots + 196 \)
$11$
\( T^{16} - 3 T^{15} - 37 T^{14} + \cdots + 43264 \)
$13$
\( T^{16} + 2 T^{15} + 27 T^{14} + \cdots + 815730721 \)
$17$
\( (T^{2} + T + 1)^{8} \)
$19$
\( T^{16} - 92 T^{14} + 5838 T^{12} + \cdots + 20702500 \)
$23$
\( T^{16} + 21 T^{15} + 316 T^{14} + \cdots + 268324 \)
$29$
\( T^{16} - 29 T^{15} + \cdots + 60552413476 \)
$31$
\( T^{16} + 298 T^{14} + \cdots + 10276687876 \)
$37$
\( T^{16} + 18 T^{15} + \cdots + 6305630464 \)
$41$
\( T^{16} - 12 T^{15} + \cdots + 420660100 \)
$43$
\( T^{16} + 3 T^{15} + \cdots + 3207636496 \)
$47$
\( T^{16} + 389 T^{14} + \cdots + 528633693184 \)
$53$
\( (T^{8} + 13 T^{7} - 179 T^{6} + \cdots + 36088)^{2} \)
$59$
\( T^{16} + 3 T^{15} + \cdots + 2428161761536 \)
$61$
\( T^{16} - 29 T^{15} + \cdots + 776118712576 \)
$67$
\( T^{16} - 33 T^{15} + \cdots + 2608383272401 \)
$71$
\( T^{16} - 27 T^{15} + \cdots + 23854784466496 \)
$73$
\( T^{16} + 365 T^{14} + 41407 T^{12} + \cdots + 6724 \)
$79$
\( (T^{8} + 7 T^{7} - 468 T^{6} + \cdots + 17872504)^{2} \)
$83$
\( T^{16} + 315 T^{14} + \cdots + 242861056 \)
$89$
\( T^{16} + 3 T^{15} + \cdots + 17701770304 \)
$97$
\( T^{16} - 6 T^{15} + \cdots + 16072882810000 \)
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