Newspace parameters
| Level: | \( N \) | \(=\) | \( 114 = 2 \cdot 3 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 114.i (of order \(9\), degree \(6\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(0.910294583043\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\Q(\zeta_{18})\) |
| Defining polynomial: | \(x^{6} - x^{3} + 1\) |
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{9}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/114\mathbb{Z}\right)^\times\).
| \(n\) | \(77\) | \(97\) |
| \(\chi(n)\) | \(1\) | \(\zeta_{18}^{2}\) |
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 |
|
0.766044 | − | 0.642788i | 0.939693 | + | 0.342020i | 0.173648 | − | 0.984808i | −0.613341 | − | 3.47843i | 0.939693 | − | 0.342020i | −1.85844 | + | 3.21891i | −0.500000 | − | 0.866025i | 0.766044 | + | 0.642788i | −2.70574 | − | 2.27038i | ||||||||||||||||||
| 43.1 | −0.939693 | + | 0.342020i | −0.173648 | + | 0.984808i | 0.766044 | − | 0.642788i | −0.0923963 | − | 0.0775297i | −0.173648 | − | 0.984808i | 2.14543 | + | 3.71599i | −0.500000 | + | 0.866025i | −0.939693 | − | 0.342020i | 0.113341 | + | 0.0412527i | |||||||||||||||||||
| 55.1 | 0.173648 | − | 0.984808i | −0.766044 | − | 0.642788i | −0.939693 | − | 0.342020i | 2.20574 | − | 0.802823i | −0.766044 | + | 0.642788i | −1.78699 | − | 3.09516i | −0.500000 | + | 0.866025i | 0.173648 | + | 0.984808i | −0.407604 | − | 2.31164i | |||||||||||||||||||
| 61.1 | −0.939693 | − | 0.342020i | −0.173648 | − | 0.984808i | 0.766044 | + | 0.642788i | −0.0923963 | + | 0.0775297i | −0.173648 | + | 0.984808i | 2.14543 | − | 3.71599i | −0.500000 | − | 0.866025i | −0.939693 | + | 0.342020i | 0.113341 | − | 0.0412527i | |||||||||||||||||||
| 73.1 | 0.766044 | + | 0.642788i | 0.939693 | − | 0.342020i | 0.173648 | + | 0.984808i | −0.613341 | + | 3.47843i | 0.939693 | + | 0.342020i | −1.85844 | − | 3.21891i | −0.500000 | + | 0.866025i | 0.766044 | − | 0.642788i | −2.70574 | + | 2.27038i | |||||||||||||||||||
| 85.1 | 0.173648 | + | 0.984808i | −0.766044 | + | 0.642788i | −0.939693 | + | 0.342020i | 2.20574 | + | 0.802823i | −0.766044 | − | 0.642788i | −1.78699 | + | 3.09516i | −0.500000 | − | 0.866025i | 0.173648 | − | 0.984808i | −0.407604 | + | 2.31164i | |||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.e | even | 9 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 114.2.i.c | ✓ | 6 |
| 3.b | odd | 2 | 1 | 342.2.u.b | 6 | ||
| 4.b | odd | 2 | 1 | 912.2.bo.d | 6 | ||
| 19.e | even | 9 | 1 | inner | 114.2.i.c | ✓ | 6 |
| 19.e | even | 9 | 1 | 2166.2.a.r | 3 | ||
| 19.f | odd | 18 | 1 | 2166.2.a.p | 3 | ||
| 57.j | even | 18 | 1 | 6498.2.a.bu | 3 | ||
| 57.l | odd | 18 | 1 | 342.2.u.b | 6 | ||
| 57.l | odd | 18 | 1 | 6498.2.a.bp | 3 | ||
| 76.l | odd | 18 | 1 | 912.2.bo.d | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 114.2.i.c | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 114.2.i.c | ✓ | 6 | 19.e | even | 9 | 1 | inner |
| 342.2.u.b | 6 | 3.b | odd | 2 | 1 | ||
| 342.2.u.b | 6 | 57.l | odd | 18 | 1 | ||
| 912.2.bo.d | 6 | 4.b | odd | 2 | 1 | ||
| 912.2.bo.d | 6 | 76.l | odd | 18 | 1 | ||
| 2166.2.a.p | 3 | 19.f | odd | 18 | 1 | ||
| 2166.2.a.r | 3 | 19.e | even | 9 | 1 | ||
| 6498.2.a.bp | 3 | 57.l | odd | 18 | 1 | ||
| 6498.2.a.bu | 3 | 57.j | even | 18 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} - 3 T_{5}^{5} + 12 T_{5}^{4} - 46 T_{5}^{3} + 60 T_{5}^{2} + 12 T_{5} + 1 \) acting on \(S_{2}^{\mathrm{new}}(114, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ | \( 1 + T^{3} + T^{6} \) |
| $3$ | \( 1 - T^{3} + T^{6} \) |
| $5$ | \( 1 + 12 T + 60 T^{2} - 46 T^{3} + 12 T^{4} - 3 T^{5} + T^{6} \) |
| $7$ | \( 3249 + 1026 T + 495 T^{2} + 60 T^{3} + 27 T^{4} + 3 T^{5} + T^{6} \) |
| $11$ | \( 1369 + 777 T + 441 T^{2} + 74 T^{3} + 21 T^{4} + T^{6} \) |
| $13$ | \( 1 - 9 T + 27 T^{2} - 28 T^{3} + 36 T^{4} - 9 T^{5} + T^{6} \) |
| $17$ | \( 9 - 27 T + 9 T^{2} + 24 T^{3} + 18 T^{4} + 3 T^{5} + T^{6} \) |
| $19$ | \( 6859 + 107 T^{3} + T^{6} \) |
| $23$ | \( 157609 - 85752 T + 22626 T^{2} - 3556 T^{3} + 378 T^{4} - 27 T^{5} + T^{6} \) |
| $29$ | \( 1 - 12 T + 60 T^{2} + 46 T^{3} + 12 T^{4} + 3 T^{5} + T^{6} \) |
| $31$ | \( 11881 + 7848 T + 3549 T^{2} + 862 T^{3} + 153 T^{4} + 15 T^{5} + T^{6} \) |
| $37$ | \( ( 17 - 45 T + 3 T^{2} + T^{3} )^{2} \) |
| $41$ | \( 2809 + 2703 T + 1308 T^{2} + 316 T^{3} + 87 T^{4} + 15 T^{5} + T^{6} \) |
| $43$ | \( 63001 + 17319 T + 2406 T^{2} + 280 T^{3} - 3 T^{4} - 3 T^{5} + T^{6} \) |
| $47$ | \( 5041 + 1491 T + 570 T^{2} + 80 T^{3} + 51 T^{4} - 15 T^{5} + T^{6} \) |
| $53$ | \( 395641 + 18870 T + 6186 T^{2} + 1205 T^{3} - 30 T^{4} - 6 T^{5} + T^{6} \) |
| $59$ | \( 81 - 567 T + 1701 T^{2} - 72 T^{3} + 198 T^{4} + 27 T^{5} + T^{6} \) |
| $61$ | \( 289 + 408 T + 375 T^{2} + 215 T^{3} + 84 T^{4} + 15 T^{5} + T^{6} \) |
| $67$ | \( 7017201 + 619866 T + 10890 T^{2} + 834 T^{3} + 72 T^{4} + 3 T^{5} + T^{6} \) |
| $71$ | \( 26569 + 9291 T + 14946 T^{2} + 1304 T^{3} - 87 T^{4} - 3 T^{5} + T^{6} \) |
| $73$ | \( 3249 + 1539 T + 522 T^{2} + 300 T^{3} + 54 T^{4} - 12 T^{5} + T^{6} \) |
| $79$ | \( 32761 - 27693 T + 9918 T^{2} - 2152 T^{3} + 351 T^{4} - 27 T^{5} + T^{6} \) |
| $83$ | \( 2601 + 1836 T + 1143 T^{2} + 210 T^{3} + 45 T^{4} - 3 T^{5} + T^{6} \) |
| $89$ | \( 239121 - 92421 T + 35406 T^{2} - 7104 T^{3} + 756 T^{4} - 42 T^{5} + T^{6} \) |
| $97$ | \( 218089 + 121887 T + 19332 T^{2} + 451 T^{3} + 171 T^{4} - 18 T^{5} + T^{6} \) |
| show more | |
| show less |