Newspace parameters
| Level: | \( N \) | \(=\) | \( 72 = 2^{3} \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 72.i (of order \(3\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(11.5476350265\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(8\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} + \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{16} + 943 x^{14} + 354586 x^{12} + 67201893 x^{10} + 6662403684 x^{8} + 324029196504 x^{6} + \cdots + 6387206400 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{26}\cdot 3^{16} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{16} + 943 x^{14} + 354586 x^{12} + 67201893 x^{10} + 6662403684 x^{8} + 324029196504 x^{6} + \cdots + 6387206400 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 52\!\cdots\!89 \nu^{14} + \cdots + 24\!\cdots\!00 ) / 15\!\cdots\!80 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 96\!\cdots\!97 \nu^{14} + \cdots - 68\!\cdots\!76 ) / 92\!\cdots\!28 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 100595975854811 \nu^{15} + \cdots + 26\!\cdots\!00 ) / 53\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 70\!\cdots\!17 \nu^{15} + \cdots - 57\!\cdots\!00 ) / 34\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 14\!\cdots\!56 \nu^{15} + \cdots - 75\!\cdots\!00 ) / 56\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 68\!\cdots\!01 \nu^{15} + \cdots - 18\!\cdots\!00 ) / 22\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 42\!\cdots\!61 \nu^{15} + \cdots + 27\!\cdots\!00 ) / 13\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 10\!\cdots\!20 \nu^{15} + \cdots - 75\!\cdots\!80 ) / 34\!\cdots\!60 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 19\!\cdots\!73 \nu^{15} + \cdots + 50\!\cdots\!00 ) / 13\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 54\!\cdots\!49 \nu^{15} + \cdots - 17\!\cdots\!00 ) / 22\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 38\!\cdots\!33 \nu^{15} + \cdots + 16\!\cdots\!00 ) / 11\!\cdots\!00 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 82\!\cdots\!03 \nu^{15} + \cdots - 16\!\cdots\!00 ) / 22\!\cdots\!00 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 53\!\cdots\!79 \nu^{15} + \cdots + 13\!\cdots\!00 ) / 13\!\cdots\!00 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 16\!\cdots\!83 \nu^{15} + \cdots - 66\!\cdots\!00 ) / 13\!\cdots\!00 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 34\!\cdots\!75 \nu^{15} + \cdots + 21\!\cdots\!80 ) / 27\!\cdots\!80 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{6} + \beta_{5} - 3\beta_{3} + 9\beta _1 - 1 ) / 18 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3 \beta_{15} - 3 \beta_{14} + 3 \beta_{12} + 4 \beta_{10} - 6 \beta_{9} + 3 \beta_{8} - 26 \beta_{6} + \cdots - 4243 ) / 36 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 24 \beta_{15} - 24 \beta_{14} - 9 \beta_{13} + 15 \beta_{12} - 36 \beta_{11} + 90 \beta_{10} + \cdots - 5890 ) / 72 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 1401 \beta_{15} + 1401 \beta_{14} + 213 \beta_{13} - 1620 \beta_{12} + 6 \beta_{11} - 2405 \beta_{10} + \cdots + 1619256 ) / 72 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 3357 \beta_{15} + 2601 \beta_{14} + 564 \beta_{13} - 2415 \beta_{12} + 1878 \beta_{11} + \cdots + 945257 ) / 36 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 298698 \beta_{15} - 298698 \beta_{14} - 51687 \beta_{13} + 373101 \beta_{12} - 22716 \beta_{11} + \cdots - 332912878 ) / 72 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 1582437 \beta_{15} - 972453 \beta_{14} - 37218 \beta_{13} + 1240227 \beta_{12} + 156120 \beta_{11} + \cdots - 546192150 ) / 72 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 62770437 \beta_{15} + 62770437 \beta_{14} + 10270272 \beta_{13} - 83640885 \beta_{12} + \cdots + 70082738614 ) / 72 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 352021971 \beta_{15} + 166878327 \beta_{14} - 34552440 \beta_{13} - 294002589 \beta_{12} + \cdots + 147754833644 ) / 72 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 13199797731 \beta_{15} - 13199797731 \beta_{14} - 1959791403 \beta_{13} + 18639098802 \beta_{12} + \cdots - 14911888389702 ) / 72 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 76342622130 \beta_{15} - 25102033686 \beta_{14} + 16089187701 \beta_{13} + 66811515609 \beta_{12} + \cdots - 38079619770932 ) / 72 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 2787458829303 \beta_{15} + 2787458829303 \beta_{14} + 374766696558 \beta_{13} + \cdots + 31\!\cdots\!52 ) / 72 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 16324660552263 \beta_{15} + 2743832489127 \beta_{14} - 5228025975831 \beta_{13} + \cdots + 94\!\cdots\!94 ) / 72 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 295742863470465 \beta_{15} - 295742863470465 \beta_{14} - 36476716689858 \beta_{13} + \cdots - 34\!\cdots\!86 ) / 36 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 17\!\cdots\!88 \beta_{15} + 24160867844046 \beta_{14} + 744017657089788 \beta_{13} + \cdots - 11\!\cdots\!04 ) / 36 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/72\mathbb{Z}\right)^\times\).
| \(n\) | \(37\) | \(55\) | \(65\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-\beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 25.1 |
|
0 | −15.5066 | + | 1.59554i | 0 | −11.6242 | − | 20.1337i | 0 | −96.0253 | + | 166.321i | 0 | 237.908 | − | 49.4828i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 25.2 | 0 | −11.7287 | + | 10.2683i | 0 | −2.41676 | − | 4.18594i | 0 | 104.276 | − | 180.611i | 0 | 32.1254 | − | 240.867i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 25.3 | 0 | −10.6653 | − | 11.3689i | 0 | 23.8592 | + | 41.3253i | 0 | 82.8535 | − | 143.507i | 0 | −15.5028 | + | 242.505i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 25.4 | 0 | −0.364856 | − | 15.5842i | 0 | −42.2136 | − | 73.1160i | 0 | −29.7517 | + | 51.5315i | 0 | −242.734 | + | 11.3720i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 25.5 | 0 | 1.12435 | + | 15.5479i | 0 | 38.2195 | + | 66.1981i | 0 | −45.4692 | + | 78.7549i | 0 | −240.472 | + | 34.9625i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 25.6 | 0 | 7.01964 | + | 13.9185i | 0 | −35.2589 | − | 61.0702i | 0 | −19.5485 | + | 33.8589i | 0 | −144.449 | + | 195.406i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 25.7 | 0 | 9.05469 | − | 12.6891i | 0 | 46.3431 | + | 80.2687i | 0 | −85.1399 | + | 147.467i | 0 | −79.0250 | − | 229.791i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 25.8 | 0 | 15.5668 | − | 0.822002i | 0 | −4.40840 | − | 7.63556i | 0 | 42.3054 | − | 73.2752i | 0 | 241.649 | − | 25.5918i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.1 | 0 | −15.5066 | − | 1.59554i | 0 | −11.6242 | + | 20.1337i | 0 | −96.0253 | − | 166.321i | 0 | 237.908 | + | 49.4828i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.2 | 0 | −11.7287 | − | 10.2683i | 0 | −2.41676 | + | 4.18594i | 0 | 104.276 | + | 180.611i | 0 | 32.1254 | + | 240.867i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.3 | 0 | −10.6653 | + | 11.3689i | 0 | 23.8592 | − | 41.3253i | 0 | 82.8535 | + | 143.507i | 0 | −15.5028 | − | 242.505i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.4 | 0 | −0.364856 | + | 15.5842i | 0 | −42.2136 | + | 73.1160i | 0 | −29.7517 | − | 51.5315i | 0 | −242.734 | − | 11.3720i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.5 | 0 | 1.12435 | − | 15.5479i | 0 | 38.2195 | − | 66.1981i | 0 | −45.4692 | − | 78.7549i | 0 | −240.472 | − | 34.9625i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.6 | 0 | 7.01964 | − | 13.9185i | 0 | −35.2589 | + | 61.0702i | 0 | −19.5485 | − | 33.8589i | 0 | −144.449 | − | 195.406i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.7 | 0 | 9.05469 | + | 12.6891i | 0 | 46.3431 | − | 80.2687i | 0 | −85.1399 | − | 147.467i | 0 | −79.0250 | + | 229.791i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.8 | 0 | 15.5668 | + | 0.822002i | 0 | −4.40840 | + | 7.63556i | 0 | 42.3054 | + | 73.2752i | 0 | 241.649 | + | 25.5918i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 72.6.i.b | ✓ | 16 |
| 3.b | odd | 2 | 1 | 216.6.i.b | 16 | ||
| 4.b | odd | 2 | 1 | 144.6.i.f | 16 | ||
| 9.c | even | 3 | 1 | inner | 72.6.i.b | ✓ | 16 |
| 9.c | even | 3 | 1 | 648.6.a.h | 8 | ||
| 9.d | odd | 6 | 1 | 216.6.i.b | 16 | ||
| 9.d | odd | 6 | 1 | 648.6.a.i | 8 | ||
| 12.b | even | 2 | 1 | 432.6.i.f | 16 | ||
| 36.f | odd | 6 | 1 | 144.6.i.f | 16 | ||
| 36.h | even | 6 | 1 | 432.6.i.f | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 72.6.i.b | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 72.6.i.b | ✓ | 16 | 9.c | even | 3 | 1 | inner |
| 144.6.i.f | 16 | 4.b | odd | 2 | 1 | ||
| 144.6.i.f | 16 | 36.f | odd | 6 | 1 | ||
| 216.6.i.b | 16 | 3.b | odd | 2 | 1 | ||
| 216.6.i.b | 16 | 9.d | odd | 6 | 1 | ||
| 432.6.i.f | 16 | 12.b | even | 2 | 1 | ||
| 432.6.i.f | 16 | 36.h | even | 6 | 1 | ||
| 648.6.a.h | 8 | 9.c | even | 3 | 1 | ||
| 648.6.a.i | 8 | 9.d | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{16} - 25 T_{5}^{15} + 15039 T_{5}^{14} - 113630 T_{5}^{13} + 153864845 T_{5}^{12} + \cdots + 39\!\cdots\!76 \)
acting on \(S_{6}^{\mathrm{new}}(72, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} \)
|
| $3$ |
\( T^{16} + \cdots + 12\!\cdots\!01 \)
|
| $5$ |
\( T^{16} + \cdots + 39\!\cdots\!76 \)
|
| $7$ |
\( T^{16} + \cdots + 40\!\cdots\!00 \)
|
| $11$ |
\( T^{16} + \cdots + 43\!\cdots\!25 \)
|
| $13$ |
\( T^{16} + \cdots + 14\!\cdots\!00 \)
|
| $17$ |
\( (T^{8} + \cdots + 72\!\cdots\!92)^{2} \)
|
| $19$ |
\( (T^{8} + \cdots - 42\!\cdots\!00)^{2} \)
|
| $23$ |
\( T^{16} + \cdots + 70\!\cdots\!24 \)
|
| $29$ |
\( T^{16} + \cdots + 27\!\cdots\!00 \)
|
| $31$ |
\( T^{16} + \cdots + 51\!\cdots\!16 \)
|
| $37$ |
\( (T^{8} + \cdots + 10\!\cdots\!64)^{2} \)
|
| $41$ |
\( T^{16} + \cdots + 60\!\cdots\!49 \)
|
| $43$ |
\( T^{16} + \cdots + 30\!\cdots\!69 \)
|
| $47$ |
\( T^{16} + \cdots + 27\!\cdots\!84 \)
|
| $53$ |
\( (T^{8} + \cdots - 16\!\cdots\!04)^{2} \)
|
| $59$ |
\( T^{16} + \cdots + 11\!\cdots\!49 \)
|
| $61$ |
\( T^{16} + \cdots + 10\!\cdots\!56 \)
|
| $67$ |
\( T^{16} + \cdots + 29\!\cdots\!09 \)
|
| $71$ |
\( (T^{8} + \cdots + 44\!\cdots\!88)^{2} \)
|
| $73$ |
\( (T^{8} + \cdots + 17\!\cdots\!80)^{2} \)
|
| $79$ |
\( T^{16} + \cdots + 75\!\cdots\!00 \)
|
| $83$ |
\( T^{16} + \cdots + 12\!\cdots\!76 \)
|
| $89$ |
\( (T^{8} + \cdots + 30\!\cdots\!76)^{2} \)
|
| $97$ |
\( T^{16} + \cdots + 21\!\cdots\!69 \)
|
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