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 | −0.386659 | − | 2.19285i | 0.939693 | − | 0.342020i | 1.32635 | − | 2.29731i | 0.500000 | + | 0.866025i | 0.766044 | + | 0.642788i | 1.70574 | + | 1.43128i | ||||||||||||||||||
| 43.1 | 0.939693 | − | 0.342020i | 0.173648 | − | 0.984808i | 0.766044 | − | 0.642788i | −0.907604 | − | 0.761570i | −0.173648 | − | 0.984808i | 0.733956 | + | 1.27125i | 0.500000 | − | 0.866025i | −0.939693 | − | 0.342020i | −1.11334 | − | 0.405223i | |||||||||||||||||||
| 55.1 | −0.173648 | + | 0.984808i | 0.766044 | + | 0.642788i | −0.939693 | − | 0.342020i | −3.20574 | + | 1.16679i | −0.766044 | + | 0.642788i | 2.43969 | + | 4.22567i | 0.500000 | − | 0.866025i | 0.173648 | + | 0.984808i | −0.592396 | − | 3.35965i | |||||||||||||||||||
| 61.1 | 0.939693 | + | 0.342020i | 0.173648 | + | 0.984808i | 0.766044 | + | 0.642788i | −0.907604 | + | 0.761570i | −0.173648 | + | 0.984808i | 0.733956 | − | 1.27125i | 0.500000 | + | 0.866025i | −0.939693 | + | 0.342020i | −1.11334 | + | 0.405223i | |||||||||||||||||||
| 73.1 | −0.766044 | − | 0.642788i | −0.939693 | + | 0.342020i | 0.173648 | + | 0.984808i | −0.386659 | + | 2.19285i | 0.939693 | + | 0.342020i | 1.32635 | + | 2.29731i | 0.500000 | − | 0.866025i | 0.766044 | − | 0.642788i | 1.70574 | − | 1.43128i | |||||||||||||||||||
| 85.1 | −0.173648 | − | 0.984808i | 0.766044 | − | 0.642788i | −0.939693 | + | 0.342020i | −3.20574 | − | 1.16679i | −0.766044 | − | 0.642788i | 2.43969 | − | 4.22567i | 0.500000 | + | 0.866025i | 0.173648 | − | 0.984808i | −0.592396 | + | 3.35965i | |||||||||||||||||||
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.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | 342.2.u.e | 6 | ||
| 4.b | odd | 2 | 1 | 912.2.bo.a | 6 | ||
| 19.e | even | 9 | 1 | inner | 114.2.i.a | ✓ | 6 |
| 19.e | even | 9 | 1 | 2166.2.a.q | 3 | ||
| 19.f | odd | 18 | 1 | 2166.2.a.s | 3 | ||
| 57.j | even | 18 | 1 | 6498.2.a.bm | 3 | ||
| 57.l | odd | 18 | 1 | 342.2.u.e | 6 | ||
| 57.l | odd | 18 | 1 | 6498.2.a.br | 3 | ||
| 76.l | odd | 18 | 1 | 912.2.bo.a | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 114.2.i.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 114.2.i.a | ✓ | 6 | 19.e | even | 9 | 1 | inner |
| 342.2.u.e | 6 | 3.b | odd | 2 | 1 | ||
| 342.2.u.e | 6 | 57.l | odd | 18 | 1 | ||
| 912.2.bo.a | 6 | 4.b | odd | 2 | 1 | ||
| 912.2.bo.a | 6 | 76.l | odd | 18 | 1 | ||
| 2166.2.a.q | 3 | 19.e | even | 9 | 1 | ||
| 2166.2.a.s | 3 | 19.f | odd | 18 | 1 | ||
| 6498.2.a.bm | 3 | 57.j | even | 18 | 1 | ||
| 6498.2.a.br | 3 | 57.l | odd | 18 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 9 T_{5}^{5} + 36 T_{5}^{4} + 90 T_{5}^{3} + 162 T_{5}^{2} + 162 T_{5} + 81 \) acting on \(S_{2}^{\mathrm{new}}(114, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ | \( 1 - T^{3} + T^{6} \) |
| $3$ | \( 1 + T^{3} + T^{6} \) |
| $5$ | \( 81 + 162 T + 162 T^{2} + 90 T^{3} + 36 T^{4} + 9 T^{5} + T^{6} \) |
| $7$ | \( 361 - 456 T + 405 T^{2} - 178 T^{3} + 57 T^{4} - 9 T^{5} + T^{6} \) |
| $11$ | \( 81 + 81 T + 81 T^{2} + 18 T^{3} + 9 T^{4} + T^{6} \) |
| $13$ | \( 1369 - 1443 T + 885 T^{2} - 352 T^{3} + 96 T^{4} - 15 T^{5} + T^{6} \) |
| $17$ | \( 81 + 81 T + 81 T^{2} + 72 T^{3} + 36 T^{4} + 9 T^{5} + T^{6} \) |
| $19$ | \( 6859 + 4332 T + 1482 T^{2} + 385 T^{3} + 78 T^{4} + 12 T^{5} + T^{6} \) |
| $23$ | \( 81 - 162 T + 162 T^{2} - 90 T^{3} + 36 T^{4} - 9 T^{5} + T^{6} \) |
| $29$ | \( 29241 - 6156 T + 2106 T^{2} - 72 T^{3} - 18 T^{4} + 9 T^{5} + T^{6} \) |
| $31$ | \( 292681 + 35706 T + 9225 T^{2} + 488 T^{3} + 147 T^{4} + 9 T^{5} + T^{6} \) |
| $37$ | \( ( 1 + 15 T - 9 T^{2} + T^{3} )^{2} \) |
| $41$ | \( 210681 - 111537 T + 27702 T^{2} - 4104 T^{3} + 405 T^{4} - 27 T^{5} + T^{6} \) |
| $43$ | \( 32041 + 23091 T + 7662 T^{2} + 1592 T^{3} + 231 T^{4} + 21 T^{5} + T^{6} \) |
| $47$ | \( 23409 - 15147 T + 6804 T^{2} - 2016 T^{3} + 333 T^{4} - 27 T^{5} + T^{6} \) |
| $53$ | \( 81 + 324 T + 486 T^{2} + 315 T^{3} + 90 T^{4} + T^{6} \) |
| $59$ | \( 6561 + 6561 T + 729 T^{2} - 648 T^{3} + 162 T^{4} - 9 T^{5} + T^{6} \) |
| $61$ | \( 94249 + 12894 T + 1821 T^{2} + 161 T^{3} - 12 T^{4} + 3 T^{5} + T^{6} \) |
| $67$ | \( 2809 - 1272 T + 5502 T^{2} + 224 T^{3} - 66 T^{4} + 3 T^{5} + T^{6} \) |
| $71$ | \( 263169 + 13851 T + 9720 T^{2} + 1728 T^{3} - 27 T^{4} - 9 T^{5} + T^{6} \) |
| $73$ | \( 54289 - 32853 T + 5754 T^{2} - 244 T^{3} + 78 T^{4} + 12 T^{5} + T^{6} \) |
| $79$ | \( 2809 - 14151 T + 30144 T^{2} + 3356 T^{3} + 231 T^{4} + 21 T^{5} + T^{6} \) |
| $83$ | \( 408321 + 92016 T + 26487 T^{2} - 18 T^{3} + 225 T^{4} + 9 T^{5} + T^{6} \) |
| $89$ | \( 431649 - 5913 T - 4374 T^{2} + 900 T^{3} + 90 T^{4} + T^{6} \) |
| $97$ | \( 2679769 + 751383 T + 141156 T^{2} + 17675 T^{3} + 1323 T^{4} + 54 T^{5} + T^{6} \) |
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