Newspace parameters
Level: | \( N \) | \(=\) | \( 819 = 3^{2} \cdot 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 819.g (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(6.53974792554\) |
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: | \( x^{12} - 4 x^{11} + 14 x^{10} - 36 x^{9} + 78 x^{8} - 140 x^{7} + 208 x^{6} - 196 x^{5} + 40 x^{4} - 64 x^{3} + 236 x^{2} - 264 x + 194 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
Coefficient ring index: | \( 2^{4} \) |
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 \( x^{12} - 4 x^{11} + 14 x^{10} - 36 x^{9} + 78 x^{8} - 140 x^{7} + 208 x^{6} - 196 x^{5} + 40 x^{4} - 64 x^{3} + 236 x^{2} - 264 x + 194 \) :
\(\beta_{1}\) | \(=\) | \( ( - 6258 \nu^{11} + 7900 \nu^{10} - 22162 \nu^{9} + 4215 \nu^{8} + 65136 \nu^{7} - 256186 \nu^{6} + 625792 \nu^{5} - 1479489 \nu^{4} + 1755524 \nu^{3} + \cdots - 771987 ) / 2140125 \) |
\(\beta_{2}\) | \(=\) | \( ( - 8512 \nu^{11} + 36285 \nu^{10} - 149568 \nu^{9} + 383995 \nu^{8} - 926996 \nu^{7} + 1772196 \nu^{6} - 2795742 \nu^{5} + 3302064 \nu^{4} - 1446244 \nu^{3} + \cdots + 3423782 ) / 2140125 \) |
\(\beta_{3}\) | \(=\) | \( ( 9171 \nu^{11} + 11225 \nu^{10} + 15964 \nu^{9} + 144795 \nu^{8} - 221562 \nu^{7} + 859957 \nu^{6} - 1283104 \nu^{5} + 2991258 \nu^{4} - 1657868 \nu^{3} + \cdots + 1525554 ) / 2140125 \) |
\(\beta_{4}\) | \(=\) | \( ( - 2790 \nu^{11} + 8056 \nu^{10} - 29771 \nu^{9} + 65716 \nu^{8} - 132036 \nu^{7} + 213650 \nu^{6} - 271768 \nu^{5} + 111659 \nu^{4} + 201716 \nu^{3} + \cdots + 330332 ) / 428025 \) |
\(\beta_{5}\) | \(=\) | \( ( - 22994 \nu^{11} + 65800 \nu^{10} - 230946 \nu^{9} + 524620 \nu^{8} - 1044232 \nu^{7} + 1702802 \nu^{6} - 2220744 \nu^{5} + 938863 \nu^{4} + 1466652 \nu^{3} + \cdots + 618194 ) / 2140125 \) |
\(\beta_{6}\) | \(=\) | \( ( 25907 \nu^{11} - 46675 \nu^{10} + 224748 \nu^{9} - 375610 \nu^{8} + 887806 \nu^{7} - 1099031 \nu^{6} + 1563432 \nu^{5} + 572906 \nu^{4} - 1368996 \nu^{3} + \cdots - 2004752 ) / 2140125 \) |
\(\beta_{7}\) | \(=\) | \( ( 28463 \nu^{11} - 98400 \nu^{10} + 341782 \nu^{9} - 800190 \nu^{8} + 1675204 \nu^{7} - 2685429 \nu^{6} + 3539138 \nu^{5} - 1591946 \nu^{4} - 2973214 \nu^{3} + \cdots - 4358568 ) / 2140125 \) |
\(\beta_{8}\) | \(=\) | \( ( - 7773 \nu^{11} + 17382 \nu^{10} - 78572 \nu^{9} + 148172 \nu^{8} - 351969 \nu^{7} + 511224 \nu^{6} - 785374 \nu^{5} + 328828 \nu^{4} - 56412 \nu^{3} + \cdots + 1846603 ) / 428025 \) |
\(\beta_{9}\) | \(=\) | \( ( - 56953 \nu^{11} + 137950 \nu^{10} - 557042 \nu^{9} + 1132940 \nu^{8} - 2527949 \nu^{7} + 3825224 \nu^{6} - 5650678 \nu^{5} + 2405276 \nu^{4} + \cdots + 2885833 ) / 2140125 \) |
\(\beta_{10}\) | \(=\) | \( ( - 69229 \nu^{11} + 165670 \nu^{10} - 672406 \nu^{9} + 1341490 \nu^{8} - 2962457 \nu^{7} + 4417157 \nu^{6} - 6197264 \nu^{5} + 2333163 \nu^{4} + \cdots + 7244069 ) / 2140125 \) |
\(\beta_{11}\) | \(=\) | \( ( - 113909 \nu^{11} + 315700 \nu^{10} - 1227336 \nu^{9} + 2602945 \nu^{8} - 5797882 \nu^{7} + 8826197 \nu^{6} - 12964584 \nu^{5} + 6118108 \nu^{4} + \cdots + 16352144 ) / 2140125 \) |
\(\nu\) | \(=\) | \( ( \beta_{6} + \beta_{5} - \beta_{3} - \beta _1 + 1 ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( ( -\beta_{9} + \beta_{8} + 2\beta_{5} - 2\beta_{4} - 2 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( ( 2\beta_{11} - 3\beta_{9} - 3\beta_{8} + 3\beta_{7} - 5\beta_{6} - \beta_{5} + 2\beta_{3} + 3\beta_{2} + 5\beta _1 - 1 ) / 2 \) |
\(\nu^{4}\) | \(=\) | \( 2\beta_{11} + 4\beta_{10} - 3\beta_{9} - 5\beta_{8} + 4\beta_{7} - 3\beta_{5} + 4\beta_{4} - 2\beta _1 + 2 \) |
\(\nu^{5}\) | \(=\) | \( ( - 9 \beta_{11} + 20 \beta_{10} - \beta_{9} - \beta_{8} - 15 \beta_{7} + 9 \beta_{6} - 6 \beta_{5} - 8 \beta_{4} - 3 \beta_{3} - 10 \beta_{2} - 24 \beta _1 + 2 ) / 2 \) |
\(\nu^{6}\) | \(=\) | \( - 15 \beta_{11} - 7 \beta_{10} + 18 \beta_{9} + 19 \beta_{8} - 25 \beta_{7} + 12 \beta_{6} - 2 \beta_{5} - 14 \beta_{4} - 4 \beta_{3} + 12 \beta_{2} + 21 \beta _1 - 5 \) |
\(\nu^{7}\) | \(=\) | \( 5 \beta_{11} - 35 \beta_{10} - \beta_{9} + 27 \beta_{8} + 21 \beta_{7} + 10 \beta_{6} + 26 \beta_{5} + 44 \beta_{4} - 5 \beta_{3} + 21 \beta_{2} + 45 \beta _1 - 38 \) |
\(\nu^{8}\) | \(=\) | \( 36 \beta_{11} + 12 \beta_{10} - 84 \beta_{9} - 48 \beta_{8} + 88 \beta_{7} - 144 \beta_{6} + 69 \beta_{5} + 56 \beta_{4} + 68 \beta_{3} - 64 \beta_{2} - 113 \beta _1 - 1 \) |
\(\nu^{9}\) | \(=\) | \( - 20 \beta_{11} + 72 \beta_{10} + 92 \beta_{9} - 184 \beta_{8} - 48 \beta_{7} - 157 \beta_{6} - 96 \beta_{5} - 145 \beta_{4} + 81 \beta_{3} - 60 \beta_{2} - 119 \beta _1 + 469 \) |
\(\nu^{10}\) | \(=\) | \( - 91 \beta_{11} - 56 \beta_{10} + 428 \beta_{9} + 130 \beta_{8} - 295 \beta_{7} + 1001 \beta_{6} - 137 \beta_{5} - 78 \beta_{4} - 493 \beta_{3} + 101 \beta_{2} + 193 \beta _1 + 443 \) |
\(\nu^{11}\) | \(=\) | \( 4 \beta_{11} - 341 \beta_{10} - 474 \beta_{9} + 1121 \beta_{8} - 176 \beta_{7} + 941 \beta_{6} + 938 \beta_{5} - 108 \beta_{4} - 478 \beta_{3} - 55 \beta_{2} - 27 \beta _1 - 2319 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/819\mathbb{Z}\right)^\times\).
\(n\) | \(92\) | \(379\) | \(703\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(-1\) |
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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818.1 |
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−1.21432 | 0 | −0.525428 | − | 1.25354i | 0 | −0.886386 | + | 2.49285i | 3.06668 | 0 | 1.52220i | |||||||||||||||||||||||||||||||||||||||||||||||||||
818.2 | −1.21432 | 0 | −0.525428 | − | 1.25354i | 0 | 0.886386 | − | 2.49285i | 3.06668 | 0 | 1.52220i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
818.3 | −1.21432 | 0 | −0.525428 | 1.25354i | 0 | −0.886386 | − | 2.49285i | 3.06668 | 0 | − | 1.52220i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
818.4 | −1.21432 | 0 | −0.525428 | 1.25354i | 0 | 0.886386 | + | 2.49285i | 3.06668 | 0 | − | 1.52220i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
818.5 | 1.53919 | 0 | 0.369102 | − | 2.66052i | 0 | −1.88127 | + | 1.86033i | −2.51026 | 0 | − | 4.09505i | |||||||||||||||||||||||||||||||||||||||||||||||||||
818.6 | 1.53919 | 0 | 0.369102 | − | 2.66052i | 0 | 1.88127 | − | 1.86033i | −2.51026 | 0 | − | 4.09505i | |||||||||||||||||||||||||||||||||||||||||||||||||||
818.7 | 1.53919 | 0 | 0.369102 | 2.66052i | 0 | −1.88127 | − | 1.86033i | −2.51026 | 0 | 4.09505i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
818.8 | 1.53919 | 0 | 0.369102 | 2.66052i | 0 | 1.88127 | + | 1.86033i | −2.51026 | 0 | 4.09505i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
818.9 | 2.67513 | 0 | 5.15633 | − | 3.05782i | 0 | −2.16221 | − | 1.52475i | 8.44358 | 0 | − | 8.18007i | |||||||||||||||||||||||||||||||||||||||||||||||||||
818.10 | 2.67513 | 0 | 5.15633 | − | 3.05782i | 0 | 2.16221 | + | 1.52475i | 8.44358 | 0 | − | 8.18007i | |||||||||||||||||||||||||||||||||||||||||||||||||||
818.11 | 2.67513 | 0 | 5.15633 | 3.05782i | 0 | −2.16221 | + | 1.52475i | 8.44358 | 0 | 8.18007i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
818.12 | 2.67513 | 0 | 5.15633 | 3.05782i | 0 | 2.16221 | − | 1.52475i | 8.44358 | 0 | 8.18007i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | inner |
39.d | odd | 2 | 1 | inner |
273.g | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 819.2.g.d | yes | 12 |
3.b | odd | 2 | 1 | 819.2.g.c | ✓ | 12 | |
7.b | odd | 2 | 1 | inner | 819.2.g.d | yes | 12 |
13.b | even | 2 | 1 | 819.2.g.c | ✓ | 12 | |
21.c | even | 2 | 1 | 819.2.g.c | ✓ | 12 | |
39.d | odd | 2 | 1 | inner | 819.2.g.d | yes | 12 |
91.b | odd | 2 | 1 | 819.2.g.c | ✓ | 12 | |
273.g | even | 2 | 1 | inner | 819.2.g.d | yes | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
819.2.g.c | ✓ | 12 | 3.b | odd | 2 | 1 | |
819.2.g.c | ✓ | 12 | 13.b | even | 2 | 1 | |
819.2.g.c | ✓ | 12 | 21.c | even | 2 | 1 | |
819.2.g.c | ✓ | 12 | 91.b | odd | 2 | 1 | |
819.2.g.d | yes | 12 | 1.a | even | 1 | 1 | trivial |
819.2.g.d | yes | 12 | 7.b | odd | 2 | 1 | inner |
819.2.g.d | yes | 12 | 39.d | odd | 2 | 1 | inner |
819.2.g.d | yes | 12 | 273.g | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 3T_{2}^{2} - T_{2} + 5 \)
acting on \(S_{2}^{\mathrm{new}}(819, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{3} - 3 T^{2} - T + 5)^{4} \)
$3$
\( T^{12} \)
$5$
\( (T^{6} + 18 T^{4} + 92 T^{2} + 104)^{2} \)
$7$
\( T^{12} + 6 T^{10} + 95 T^{8} + \cdots + 117649 \)
$11$
\( (T^{3} + 10 T^{2} + 28 T + 20)^{4} \)
$13$
\( T^{12} + 34 T^{10} + 647 T^{8} + \cdots + 4826809 \)
$17$
\( (T^{6} - 116 T^{4} + 3696 T^{2} + \cdots - 20800)^{2} \)
$19$
\( (T^{6} - 80 T^{4} + 1920 T^{2} + \cdots - 13312)^{2} \)
$23$
\( (T^{6} + 54 T^{4} + 716 T^{2} + 200)^{2} \)
$29$
\( (T^{6} + 86 T^{4} + 1100 T^{2} + \cdots + 2888)^{2} \)
$31$
\( (T^{6} - 116 T^{4} + 1264 T^{2} + \cdots - 832)^{2} \)
$37$
\( (T^{6} + 88 T^{4} + 1536 T^{2} + \cdots + 3200)^{2} \)
$41$
\( (T^{6} + 114 T^{4} + 2396 T^{2} + \cdots + 2600)^{2} \)
$43$
\( (T^{3} + 2 T^{2} - 84 T - 104)^{4} \)
$47$
\( (T^{6} + 122 T^{4} + 2876 T^{2} + \cdots + 17576)^{2} \)
$53$
\( (T^{6} + 166 T^{4} + 8844 T^{2} + \cdots + 150152)^{2} \)
$59$
\( (T^{6} + 154 T^{4} + 5404 T^{2} + \cdots + 55016)^{2} \)
$61$
\( (T^{6} + 232 T^{4} + 6304 T^{2} + \cdots + 41600)^{2} \)
$67$
\( T^{12} \)
$71$
\( (T^{3} - 2 T^{2} - 44 T + 20)^{4} \)
$73$
\( (T^{6} - 168 T^{4} + 8064 T^{2} + \cdots - 83200)^{2} \)
$79$
\( (T^{3} + 10 T^{2} - 60 T - 136)^{4} \)
$83$
\( (T^{6} + 202 T^{4} + 9660 T^{2} + \cdots + 87464)^{2} \)
$89$
\( (T^{6} + 178 T^{4} + 5628 T^{2} + \cdots + 17576)^{2} \)
$97$
\( (T^{6} - 172 T^{4} + 5264 T^{2} + \cdots - 20800)^{2} \)
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