Newspace parameters
Level: | \( N \) | \(=\) | \( 927 = 3^{2} \cdot 103 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 927.o (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(7.40213226737\) |
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: | \( x^{16} - 12x^{14} + 97x^{12} - 444x^{10} + 1488x^{8} - 2796x^{6} + 3553x^{4} - 60x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 3^{4} \) |
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 \( x^{16} - 12x^{14} + 97x^{12} - 444x^{10} + 1488x^{8} - 2796x^{6} + 3553x^{4} - 60x^{2} + 1 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( 1551 \nu^{14} - 31123 \nu^{12} + 126720 \nu^{10} - 496980 \nu^{8} - 99276 \nu^{6} - 1405008 \nu^{4} + 23727 \nu^{2} - 1498291 ) / 8150064 \) |
\(\beta_{3}\) | \(=\) | \( ( - 3760 \nu^{15} + 39433 \nu^{13} - 307200 \nu^{11} + 1204800 \nu^{9} - 3772620 \nu^{7} + 3406080 \nu^{5} - 57520 \nu^{3} - 32461655 \nu ) / 8150064 \) |
\(\beta_{4}\) | \(=\) | \( ( - 3760 \nu^{14} + 39433 \nu^{12} - 307200 \nu^{10} + 1204800 \nu^{8} - 3772620 \nu^{6} + 3406080 \nu^{4} - 57520 \nu^{2} - 24311591 ) / 8150064 \) |
\(\beta_{5}\) | \(=\) | \( ( - 141 \nu^{14} + 1642 \nu^{12} - 11520 \nu^{10} + 45180 \nu^{8} - 112086 \nu^{6} + 127728 \nu^{4} - 2157 \nu^{2} - 135698 ) / 78366 \) |
\(\beta_{6}\) | \(=\) | \( ( - 24393 \nu^{15} + 257944 \nu^{13} - 1992960 \nu^{11} + 7816140 \nu^{9} - 23074080 \nu^{7} + 22096944 \nu^{5} - 373161 \nu^{3} - 121145912 \nu ) / 8150064 \) |
\(\beta_{7}\) | \(=\) | \( ( - 29085 \nu^{14} + 315641 \nu^{12} - 2549712 \nu^{10} + 11400444 \nu^{8} - 38999004 \nu^{6} + 72993408 \nu^{4} - 89922237 \nu^{2} + 20633 ) / 8150064 \) |
\(\beta_{8}\) | \(=\) | \( ( - 59690 \nu^{15} + 647223 \nu^{13} - 4876800 \nu^{11} + 19126200 \nu^{9} - 54032484 \nu^{7} + 54071520 \nu^{5} - 913130 \nu^{3} - 232092825 \nu ) / 8150064 \) |
\(\beta_{9}\) | \(=\) | \( ( 138601 \nu^{15} - 1659452 \nu^{13} + 13404864 \nu^{11} - 61231644 \nu^{9} + 205033488 \nu^{7} - 383755776 \nu^{5} + 489043273 \nu^{3} - 108476 \nu ) / 8150064 \) |
\(\beta_{10}\) | \(=\) | \( ( 138601 \nu^{14} - 1659452 \nu^{12} + 13404864 \nu^{10} - 61231644 \nu^{8} + 205033488 \nu^{6} - 383755776 \nu^{4} + 489043273 \nu^{2} - 8258540 ) / 8150064 \) |
\(\beta_{11}\) | \(=\) | \( ( - 173373 \nu^{14} + 2032613 \nu^{12} - 16419216 \nu^{10} + 74454348 \nu^{8} - 251139372 \nu^{6} + 470050944 \nu^{4} - 611652765 \nu^{2} + \cdots + 132869 ) / 8150064 \) |
\(\beta_{12}\) | \(=\) | \( ( - 242430 \nu^{14} + 2945743 \nu^{12} - 23795376 \nu^{10} + 109240584 \nu^{8} - 363961092 \nu^{6} + 681216384 \nu^{4} - 847326990 \nu^{2} + \cdots + 192559 ) / 8150064 \) |
\(\beta_{13}\) | \(=\) | \( ( 138601 \nu^{15} - 1659452 \nu^{13} + 13404864 \nu^{11} - 61231644 \nu^{9} + 205033488 \nu^{7} - 383755776 \nu^{5} + 487005757 \nu^{3} - 108476 \nu ) / 2037516 \) |
\(\beta_{14}\) | \(=\) | \( ( 518332 \nu^{15} - 6208565 \nu^{13} + 50152080 \nu^{11} - 229163100 \nu^{9} + 767098860 \nu^{7} - 1435758720 \nu^{5} + 1817590396 \nu^{3} + \cdots - 405845 \nu ) / 2037516 \) |
\(\beta_{15}\) | \(=\) | \( ( 2454359 \nu^{15} - 29439455 \nu^{13} + 237808560 \nu^{11} - 1087124628 \nu^{9} + 3637390020 \nu^{7} - 6808007040 \nu^{5} + 8606731847 \nu^{3} + \cdots - 1924415 \nu ) / 8150064 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{12} + \beta_{11} + 3\beta_{10} + 3 \) |
\(\nu^{3}\) | \(=\) | \( -\beta_{13} + 4\beta_{9} \) |
\(\nu^{4}\) | \(=\) | \( 4\beta_{12} + 5\beta_{11} + 13\beta_{10} - \beta_{7} + \beta_{5} - 5\beta_{4} + \beta_{2} \) |
\(\nu^{5}\) | \(=\) | \( \beta_{14} - 8\beta_{13} + 17\beta_{9} + \beta_{6} - 8\beta_{3} - 17\beta_1 \) |
\(\nu^{6}\) | \(=\) | \( 7\beta_{5} - 24\beta_{4} + 8\beta_{2} - 58 \) |
\(\nu^{7}\) | \(=\) | \( -\beta_{8} + 10\beta_{6} - 49\beta_{3} - 75\beta_1 \) |
\(\nu^{8}\) | \(=\) | \( -76\beta_{12} - 113\beta_{11} - 264\beta_{10} + 49\beta_{7} - 264 \) |
\(\nu^{9}\) | \(=\) | \( 12\beta_{15} - 61\beta_{14} + 260\beta_{13} - 340\beta_{9} \) |
\(\nu^{10}\) | \(=\) | \( -352\beta_{12} - 527\beta_{11} - 1219\beta_{10} + 272\beta_{7} - 175\beta_{5} + 527\beta_{4} - 272\beta_{2} \) |
\(\nu^{11}\) | \(=\) | \( 97 \beta_{15} - 369 \beta_{14} + 1343 \beta_{13} - 1571 \beta_{9} + 97 \beta_{8} - 466 \beta_{6} + 1440 \beta_{3} + 1571 \beta_1 \) |
\(\nu^{12}\) | \(=\) | \( -780\beta_{5} + 2448\beta_{4} - 1440\beta_{2} + 5687 \) |
\(\nu^{13}\) | \(=\) | \( 660\beta_{8} - 2760\beta_{6} + 7428\beta_{3} + 7355\beta_1 \) |
\(\nu^{14}\) | \(=\) | \( 8015\beta_{12} + 11363\beta_{11} + 26733\beta_{10} - 7428\beta_{7} + 26733 \) |
\(\nu^{15}\) | \(=\) | \( -4080\beta_{15} + 11508\beta_{14} - 33647\beta_{13} + 34748\beta_{9} \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/927\mathbb{Z}\right)^\times\).
\(n\) | \(722\) | \(829\) |
\(\chi(n)\) | \(-1\) | \(-\beta_{10}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
665.1 |
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−1.91212 | − | 1.10397i | 0 | 1.43748 | + | 2.48979i | 0.448288 | + | 0.776457i | 0 | 0.329192 | + | 0.570178i | − | 1.93185i | 0 | − | 1.97958i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
665.2 | −1.78558 | − | 1.03090i | 0 | 1.12553 | + | 1.94948i | 1.67303 | + | 2.89778i | 0 | 0.774131 | + | 1.34083i | − | 0.517638i | 0 | − | 6.89895i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
665.3 | −1.46384 | − | 0.845147i | 0 | 0.428545 | + | 0.742262i | −0.448288 | − | 0.776457i | 0 | −2.42727 | − | 4.20415i | 1.93185i | 0 | 1.51548i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
665.4 | −0.112547 | − | 0.0649791i | 0 | −0.991555 | − | 1.71742i | −1.67303 | − | 2.89778i | 0 | 2.32395 | + | 4.02519i | 0.517638i | 0 | 0.434849i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
665.5 | 0.112547 | + | 0.0649791i | 0 | −0.991555 | − | 1.71742i | 1.67303 | + | 2.89778i | 0 | 2.32395 | + | 4.02519i | − | 0.517638i | 0 | 0.434849i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
665.6 | 1.46384 | + | 0.845147i | 0 | 0.428545 | + | 0.742262i | 0.448288 | + | 0.776457i | 0 | −2.42727 | − | 4.20415i | − | 1.93185i | 0 | 1.51548i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
665.7 | 1.78558 | + | 1.03090i | 0 | 1.12553 | + | 1.94948i | −1.67303 | − | 2.89778i | 0 | 0.774131 | + | 1.34083i | 0.517638i | 0 | − | 6.89895i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
665.8 | 1.91212 | + | 1.10397i | 0 | 1.43748 | + | 2.48979i | −0.448288 | − | 0.776457i | 0 | 0.329192 | + | 0.570178i | 1.93185i | 0 | − | 1.97958i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
881.1 | −1.91212 | + | 1.10397i | 0 | 1.43748 | − | 2.48979i | 0.448288 | − | 0.776457i | 0 | 0.329192 | − | 0.570178i | 1.93185i | 0 | 1.97958i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
881.2 | −1.78558 | + | 1.03090i | 0 | 1.12553 | − | 1.94948i | 1.67303 | − | 2.89778i | 0 | 0.774131 | − | 1.34083i | 0.517638i | 0 | 6.89895i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
881.3 | −1.46384 | + | 0.845147i | 0 | 0.428545 | − | 0.742262i | −0.448288 | + | 0.776457i | 0 | −2.42727 | + | 4.20415i | − | 1.93185i | 0 | − | 1.51548i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
881.4 | −0.112547 | + | 0.0649791i | 0 | −0.991555 | + | 1.71742i | −1.67303 | + | 2.89778i | 0 | 2.32395 | − | 4.02519i | − | 0.517638i | 0 | − | 0.434849i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
881.5 | 0.112547 | − | 0.0649791i | 0 | −0.991555 | + | 1.71742i | 1.67303 | − | 2.89778i | 0 | 2.32395 | − | 4.02519i | 0.517638i | 0 | − | 0.434849i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
881.6 | 1.46384 | − | 0.845147i | 0 | 0.428545 | − | 0.742262i | 0.448288 | − | 0.776457i | 0 | −2.42727 | + | 4.20415i | 1.93185i | 0 | − | 1.51548i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
881.7 | 1.78558 | − | 1.03090i | 0 | 1.12553 | − | 1.94948i | −1.67303 | + | 2.89778i | 0 | 0.774131 | − | 1.34083i | − | 0.517638i | 0 | 6.89895i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
881.8 | 1.91212 | − | 1.10397i | 0 | 1.43748 | − | 2.48979i | −0.448288 | + | 0.776457i | 0 | 0.329192 | − | 0.570178i | − | 1.93185i | 0 | 1.97958i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
103.d | odd | 6 | 1 | inner |
309.g | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 927.2.o.a | ✓ | 16 |
3.b | odd | 2 | 1 | inner | 927.2.o.a | ✓ | 16 |
103.d | odd | 6 | 1 | inner | 927.2.o.a | ✓ | 16 |
309.g | even | 6 | 1 | inner | 927.2.o.a | ✓ | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
927.2.o.a | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
927.2.o.a | ✓ | 16 | 3.b | odd | 2 | 1 | inner |
927.2.o.a | ✓ | 16 | 103.d | odd | 6 | 1 | inner |
927.2.o.a | ✓ | 16 | 309.g | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{16} - 12T_{2}^{14} + 97T_{2}^{12} - 444T_{2}^{10} + 1488T_{2}^{8} - 2796T_{2}^{6} + 3553T_{2}^{4} - 60T_{2}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(927, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{16} - 12 T^{14} + 97 T^{12} - 444 T^{10} + \cdots + 1 \)
$3$
\( T^{16} \)
$5$
\( (T^{8} + 12 T^{6} + 135 T^{4} + 108 T^{2} + \cdots + 81)^{2} \)
$7$
\( (T^{8} - 2 T^{7} + 26 T^{6} - 56 T^{5} + \cdots + 529)^{2} \)
$11$
\( T^{16} + 72 T^{14} + \cdots + 187388721 \)
$13$
\( (T^{4} - 10 T^{3} + 26 T^{2} + 4 T - 44)^{4} \)
$17$
\( T^{16} - 96 T^{14} + \cdots + 27439591201 \)
$19$
\( (T^{8} + 10 T^{7} + 74 T^{6} + 232 T^{5} + \cdots + 1)^{2} \)
$23$
\( (T^{8} + 96 T^{6} + 1592 T^{4} + 1152 T^{2} + \cdots + 16)^{2} \)
$29$
\( T^{16} - 72 T^{14} + 4630 T^{12} + \cdots + 28561 \)
$31$
\( (T^{2} + 12)^{8} \)
$37$
\( (T^{8} + 216 T^{6} + 15408 T^{4} + \cdots + 2509056)^{2} \)
$41$
\( T^{16} - 168 T^{14} + \cdots + 43617904801 \)
$43$
\( (T^{8} - 18 T^{7} + 42 T^{6} + \cdots + 2259009)^{2} \)
$47$
\( T^{16} + 120 T^{14} + \cdots + 96059601 \)
$53$
\( T^{16} + 192 T^{14} + \cdots + 5103121662081 \)
$59$
\( T^{16} - 96 T^{14} + \cdots + 607573201 \)
$61$
\( (T^{4} + 14 T^{3} + 50 T^{2} - 20 T - 236)^{4} \)
$67$
\( (T^{8} - 24 T^{7} + 126 T^{6} + \cdots + 1390041)^{2} \)
$71$
\( (T^{8} + 36 T^{6} + 1215 T^{4} + \cdots + 6561)^{2} \)
$73$
\( (T^{8} + 432 T^{6} + 61416 T^{4} + \cdots + 41525136)^{2} \)
$79$
\( (T^{4} - 28 T^{3} + 200 T^{2} + 160 T - 3776)^{4} \)
$83$
\( T^{16} - 648 T^{14} + \cdots + 91\!\cdots\!41 \)
$89$
\( (T^{4} - 144 T^{2} + 1296)^{4} \)
$97$
\( (T^{8} + 16 T^{7} + 242 T^{6} + 256 T^{5} + \cdots + 1)^{2} \)
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