Newspace parameters
Level: | \( N \) | \(=\) | \( 243 = 3^{5} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 243.e (of order \(9\), degree \(6\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(14.3374641314\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Relative dimension: | \(8\) over \(\Q(\zeta_{9})\) |
Twist minimal: | no (minimal twist has level 27) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{9}]$ |
$q$-expansion
The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
28.1 | −0.751659 | − | 4.26287i | 0 | −10.0895 | + | 3.67228i | −3.28965 | − | 2.76034i | 0 | 19.2025 | + | 6.98913i | 5.92381 | + | 10.2603i | 0 | −9.29428 | + | 16.0982i | ||||||
28.2 | −0.652567 | − | 3.70089i | 0 | −5.75323 | + | 2.09401i | 4.67730 | + | 3.92472i | 0 | −16.2160 | − | 5.90215i | −3.52788 | − | 6.11047i | 0 | 11.4727 | − | 19.8713i | ||||||
28.3 | −0.336951 | − | 1.91095i | 0 | 3.97937 | − | 1.44837i | −2.07510 | − | 1.74122i | 0 | 11.1989 | + | 4.07607i | −11.8703 | − | 20.5600i | 0 | −2.62817 | + | 4.55212i | ||||||
28.4 | 0.172250 | + | 0.976877i | 0 | 6.59292 | − | 2.39963i | 0.110676 | + | 0.0928685i | 0 | −23.8444 | − | 8.67865i | 7.44756 | + | 12.8996i | 0 | −0.0716572 | + | 0.124114i | ||||||
28.5 | 0.181650 | + | 1.03019i | 0 | 6.48925 | − | 2.36189i | 11.8235 | + | 9.92108i | 0 | 22.1231 | + | 8.05215i | 7.79628 | + | 13.5036i | 0 | −8.07284 | + | 13.9826i | ||||||
28.6 | 0.392076 | + | 2.22357i | 0 | 2.72699 | − | 0.992543i | −14.7308 | − | 12.3606i | 0 | −0.614463 | − | 0.223646i | 12.3077 | + | 21.3175i | 0 | 21.7091 | − | 37.6013i | ||||||
28.7 | 0.748840 | + | 4.24688i | 0 | −9.95772 | + | 3.62432i | 14.2354 | + | 11.9449i | 0 | −17.6749 | − | 6.43313i | −5.59920 | − | 9.69809i | 0 | −40.0687 | + | 69.4011i | ||||||
28.8 | 0.920010 | + | 5.21764i | 0 | −18.8598 | + | 6.86439i | −8.05153 | − | 6.75603i | 0 | 7.26495 | + | 2.64423i | −31.9745 | − | 55.3815i | 0 | 27.8430 | − | 48.2255i | ||||||
55.1 | −4.98740 | + | 1.81526i | 0 | 15.4506 | − | 12.9646i | 0.957594 | + | 5.43079i | 0 | 18.8623 | + | 15.8274i | −32.2942 | + | 55.9352i | 0 | −14.6342 | − | 25.3472i | ||||||
55.2 | −2.99548 | + | 1.09027i | 0 | 1.65587 | − | 1.38944i | −1.24508 | − | 7.06121i | 0 | 3.72090 | + | 3.12220i | 9.30563 | − | 16.1178i | 0 | 11.4282 | + | 19.7942i | ||||||
55.3 | −2.77394 | + | 1.00963i | 0 | 0.547010 | − | 0.458996i | 2.95956 | + | 16.7845i | 0 | −9.82851 | − | 8.24710i | 10.7539 | − | 18.6263i | 0 | −25.1558 | − | 43.5711i | ||||||
55.4 | −0.790355 | + | 0.287666i | 0 | −5.58645 | + | 4.68758i | −1.87827 | − | 10.6522i | 0 | −20.4470 | − | 17.1571i | 6.43113 | − | 11.1390i | 0 | 4.54878 | + | 7.87873i | ||||||
55.5 | 1.08402 | − | 0.394551i | 0 | −5.10893 | + | 4.28690i | 1.33360 | + | 7.56321i | 0 | 9.93710 | + | 8.33822i | −8.46114 | + | 14.6551i | 0 | 4.42972 | + | 7.67249i | ||||||
55.6 | 1.71186 | − | 0.623068i | 0 | −3.58609 | + | 3.00909i | −2.34910 | − | 13.3224i | 0 | 17.5105 | + | 14.6931i | −11.5509 | + | 20.0068i | 0 | −12.3221 | − | 21.3425i | ||||||
55.7 | 3.91280 | − | 1.42414i | 0 | 7.15349 | − | 6.00249i | 2.58756 | + | 14.6748i | 0 | 3.34538 | + | 2.80711i | 2.78612 | − | 4.82571i | 0 | 31.0237 | + | 53.7346i | ||||||
55.8 | 4.39879 | − | 1.60103i | 0 | 10.6577 | − | 8.94287i | −0.544255 | − | 3.08662i | 0 | −23.3667 | − | 19.6070i | 13.8388 | − | 23.9695i | 0 | −7.33583 | − | 12.7060i | ||||||
109.1 | −3.85878 | + | 3.23790i | 0 | 3.01699 | − | 17.1102i | −14.6375 | + | 5.32761i | 0 | 0.978906 | + | 5.55165i | 23.6100 | + | 40.8938i | 0 | 39.2325 | − | 67.9527i | ||||||
109.2 | −2.61428 | + | 2.19364i | 0 | 0.633205 | − | 3.59108i | 10.3200 | − | 3.75618i | 0 | −0.765086 | − | 4.33902i | −7.42861 | − | 12.8667i | 0 | −18.7397 | + | 32.4581i | ||||||
109.3 | −1.78548 | + | 1.49819i | 0 | −0.445841 | + | 2.52849i | 5.86493 | − | 2.13466i | 0 | −0.469354 | − | 2.66184i | −12.3152 | − | 21.3306i | 0 | −7.27356 | + | 12.5982i | ||||||
109.4 | −0.228936 | + | 0.192100i | 0 | −1.37368 | + | 7.79050i | −18.8424 | + | 6.85806i | 0 | 3.73178 | + | 21.1640i | −2.37749 | − | 4.11794i | 0 | 2.99626 | − | 5.18968i | ||||||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
27.e | even | 9 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 243.4.e.c | 48 | |
3.b | odd | 2 | 1 | 243.4.e.b | 48 | ||
9.c | even | 3 | 1 | 27.4.e.a | ✓ | 48 | |
9.c | even | 3 | 1 | 243.4.e.d | 48 | ||
9.d | odd | 6 | 1 | 81.4.e.a | 48 | ||
9.d | odd | 6 | 1 | 243.4.e.a | 48 | ||
27.e | even | 9 | 1 | 27.4.e.a | ✓ | 48 | |
27.e | even | 9 | 1 | inner | 243.4.e.c | 48 | |
27.e | even | 9 | 1 | 243.4.e.d | 48 | ||
27.e | even | 9 | 1 | 729.4.a.d | 24 | ||
27.f | odd | 18 | 1 | 81.4.e.a | 48 | ||
27.f | odd | 18 | 1 | 243.4.e.a | 48 | ||
27.f | odd | 18 | 1 | 243.4.e.b | 48 | ||
27.f | odd | 18 | 1 | 729.4.a.c | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
27.4.e.a | ✓ | 48 | 9.c | even | 3 | 1 | |
27.4.e.a | ✓ | 48 | 27.e | even | 9 | 1 | |
81.4.e.a | 48 | 9.d | odd | 6 | 1 | ||
81.4.e.a | 48 | 27.f | odd | 18 | 1 | ||
243.4.e.a | 48 | 9.d | odd | 6 | 1 | ||
243.4.e.a | 48 | 27.f | odd | 18 | 1 | ||
243.4.e.b | 48 | 3.b | odd | 2 | 1 | ||
243.4.e.b | 48 | 27.f | odd | 18 | 1 | ||
243.4.e.c | 48 | 1.a | even | 1 | 1 | trivial | |
243.4.e.c | 48 | 27.e | even | 9 | 1 | inner | |
243.4.e.d | 48 | 9.c | even | 3 | 1 | ||
243.4.e.d | 48 | 27.e | even | 9 | 1 | ||
729.4.a.c | 24 | 27.f | odd | 18 | 1 | ||
729.4.a.d | 24 | 27.e | even | 9 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{48} - 3 T_{2}^{47} + 3 T_{2}^{46} + 57 T_{2}^{45} - 126 T_{2}^{44} - 702 T_{2}^{43} + 23901 T_{2}^{42} - 48852 T_{2}^{41} - 38196 T_{2}^{40} + 23553 T_{2}^{39} + 2340873 T_{2}^{38} - 16419240 T_{2}^{37} + \cdots + 18\!\cdots\!16 \)
acting on \(S_{4}^{\mathrm{new}}(243, [\chi])\).