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 | −0.0812519 | − | 0.460802i | −0.939693 | + | 0.342020i | 2.20574 | − | 3.82045i | −0.500000 | − | 0.866025i | 0.766044 | + | 0.642788i | −0.358441 | − | 0.300767i | ||||||||||||||||||
| 43.1 | −0.939693 | + | 0.342020i | 0.173648 | − | 0.984808i | 0.766044 | − | 0.642788i | −2.97178 | − | 2.49362i | 0.173648 | + | 0.984808i | −0.613341 | − | 1.06234i | −0.500000 | + | 0.866025i | −0.939693 | − | 0.342020i | 3.64543 | + | 1.32683i | |||||||||||||||||||
| 55.1 | 0.173648 | − | 0.984808i | 0.766044 | + | 0.642788i | −0.939693 | − | 0.342020i | 1.55303 | − | 0.565258i | 0.766044 | − | 0.642788i | −0.0923963 | − | 0.160035i | −0.500000 | + | 0.866025i | 0.173648 | + | 0.984808i | −0.286989 | − | 1.62760i | |||||||||||||||||||
| 61.1 | −0.939693 | − | 0.342020i | 0.173648 | + | 0.984808i | 0.766044 | + | 0.642788i | −2.97178 | + | 2.49362i | 0.173648 | − | 0.984808i | −0.613341 | + | 1.06234i | −0.500000 | − | 0.866025i | −0.939693 | + | 0.342020i | 3.64543 | − | 1.32683i | |||||||||||||||||||
| 73.1 | 0.766044 | + | 0.642788i | −0.939693 | + | 0.342020i | 0.173648 | + | 0.984808i | −0.0812519 | + | 0.460802i | −0.939693 | − | 0.342020i | 2.20574 | + | 3.82045i | −0.500000 | + | 0.866025i | 0.766044 | − | 0.642788i | −0.358441 | + | 0.300767i | |||||||||||||||||||
| 85.1 | 0.173648 | + | 0.984808i | 0.766044 | − | 0.642788i | −0.939693 | + | 0.342020i | 1.55303 | + | 0.565258i | 0.766044 | + | 0.642788i | −0.0923963 | + | 0.160035i | −0.500000 | − | 0.866025i | 0.173648 | − | 0.984808i | −0.286989 | + | 1.62760i | |||||||||||||||||||
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.b | ✓ | 6 |
| 3.b | odd | 2 | 1 | 342.2.u.d | 6 | ||
| 4.b | odd | 2 | 1 | 912.2.bo.c | 6 | ||
| 19.e | even | 9 | 1 | inner | 114.2.i.b | ✓ | 6 |
| 19.e | even | 9 | 1 | 2166.2.a.t | 3 | ||
| 19.f | odd | 18 | 1 | 2166.2.a.n | 3 | ||
| 57.j | even | 18 | 1 | 6498.2.a.bt | 3 | ||
| 57.l | odd | 18 | 1 | 342.2.u.d | 6 | ||
| 57.l | odd | 18 | 1 | 6498.2.a.bo | 3 | ||
| 76.l | odd | 18 | 1 | 912.2.bo.c | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 114.2.i.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 114.2.i.b | ✓ | 6 | 19.e | even | 9 | 1 | inner |
| 342.2.u.d | 6 | 3.b | odd | 2 | 1 | ||
| 342.2.u.d | 6 | 57.l | odd | 18 | 1 | ||
| 912.2.bo.c | 6 | 4.b | odd | 2 | 1 | ||
| 912.2.bo.c | 6 | 76.l | odd | 18 | 1 | ||
| 2166.2.a.n | 3 | 19.f | odd | 18 | 1 | ||
| 2166.2.a.t | 3 | 19.e | even | 9 | 1 | ||
| 6498.2.a.bo | 3 | 57.l | odd | 18 | 1 | ||
| 6498.2.a.bt | 3 | 57.j | even | 18 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 3 T_{5}^{5} - 30 T_{5}^{3} + 36 T_{5}^{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(114, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ | \( 1 + T^{3} + T^{6} \) |
| $3$ | \( 1 + T^{3} + T^{6} \) |
| $5$ | \( 9 + 36 T^{2} - 30 T^{3} + 3 T^{5} + T^{6} \) |
| $7$ | \( 1 + 6 T + 33 T^{2} + 20 T^{3} + 15 T^{4} - 3 T^{5} + T^{6} \) |
| $11$ | \( 2601 - 2295 T + 1413 T^{2} - 438 T^{3} + 99 T^{4} - 12 T^{5} + T^{6} \) |
| $13$ | \( 361 - 57 T + 1407 T^{2} + 800 T^{3} + 186 T^{4} + 21 T^{5} + T^{6} \) |
| $17$ | \( 2601 + 459 T + 495 T^{2} - 24 T^{3} - 18 T^{4} - 3 T^{5} + T^{6} \) |
| $19$ | \( 6859 - 2166 T - 228 T^{2} + 169 T^{3} - 12 T^{4} - 6 T^{5} + T^{6} \) |
| $23$ | \( 9 + 162 T + 846 T^{2} + 246 T^{3} + 108 T^{4} + 15 T^{5} + T^{6} \) |
| $29$ | \( 2601 + 1836 T + 576 T^{2} - 138 T^{3} + 72 T^{4} - 15 T^{5} + T^{6} \) |
| $31$ | \( 289 - 102 T + 87 T^{2} - 16 T^{3} + 15 T^{4} - 3 T^{5} + T^{6} \) |
| $37$ | \( ( 19 - 9 T - 3 T^{2} + T^{3} )^{2} \) |
| $41$ | \( 729 - 729 T + 486 T^{2} - 216 T^{3} + 27 T^{4} + 9 T^{5} + T^{6} \) |
| $43$ | \( 1 + 45 T + 576 T^{2} + 80 T^{3} + 45 T^{4} + 9 T^{5} + T^{6} \) |
| $47$ | \( 25281 + 27189 T + 10836 T^{2} + 1968 T^{3} + 261 T^{4} + 21 T^{5} + T^{6} \) |
| $53$ | \( 47961 - 11826 T + 7218 T^{2} - 2373 T^{3} + 378 T^{4} - 30 T^{5} + T^{6} \) |
| $59$ | \( 23409 - 31671 T + 19197 T^{2} - 3312 T^{3} + 360 T^{4} - 27 T^{5} + T^{6} \) |
| $61$ | \( 1 - 36 T + 549 T^{2} + 323 T^{3} + 72 T^{4} + 9 T^{5} + T^{6} \) |
| $67$ | \( 7921 + 8544 T + 4200 T^{2} + 1070 T^{3} + 156 T^{4} + 15 T^{5} + T^{6} \) |
| $71$ | \( 6561 - 6561 T + 2916 T^{2} - 648 T^{3} + 81 T^{4} - 9 T^{5} + T^{6} \) |
| $73$ | \( 546121 + 281559 T + 32586 T^{2} - 2404 T^{3} + 246 T^{4} - 12 T^{5} + T^{6} \) |
| $79$ | \( 5329 + 7665 T + 3180 T^{2} + 152 T^{3} + 105 T^{4} - 15 T^{5} + T^{6} \) |
| $83$ | \( 3583449 + 511110 T + 78579 T^{2} + 2976 T^{3} + 279 T^{4} + 3 T^{5} + T^{6} \) |
| $89$ | \( 3583449 + 698517 T + 95616 T^{2} + 12174 T^{3} + 1044 T^{4} + 48 T^{5} + T^{6} \) |
| $97$ | \( 1630729 + 34479 T + 3636 T^{2} - 613 T^{3} + 99 T^{4} - 18 T^{5} + T^{6} \) |
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