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
\( T_{5}^{6} + 9T_{5}^{5} + 36T_{5}^{4} + 90T_{5}^{3} + 162T_{5}^{2} + 162T_{5} + 81 \)
acting on \(S_{2}^{\mathrm{new}}(114, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{6} - T^{3} + 1 \)
$3$
\( T^{6} + T^{3} + 1 \)
$5$
\( T^{6} + 9 T^{5} + 36 T^{4} + 90 T^{3} + \cdots + 81 \)
$7$
\( T^{6} - 9 T^{5} + 57 T^{4} - 178 T^{3} + \cdots + 361 \)
$11$
\( T^{6} + 9 T^{4} + 18 T^{3} + 81 T^{2} + \cdots + 81 \)
$13$
\( T^{6} - 15 T^{5} + 96 T^{4} + \cdots + 1369 \)
$17$
\( T^{6} + 9 T^{5} + 36 T^{4} + 72 T^{3} + \cdots + 81 \)
$19$
\( T^{6} + 12 T^{5} + 78 T^{4} + \cdots + 6859 \)
$23$
\( T^{6} - 9 T^{5} + 36 T^{4} - 90 T^{3} + \cdots + 81 \)
$29$
\( T^{6} + 9 T^{5} - 18 T^{4} + \cdots + 29241 \)
$31$
\( T^{6} + 9 T^{5} + 147 T^{4} + \cdots + 292681 \)
$37$
\( (T^{3} - 9 T^{2} + 15 T + 1)^{2} \)
$41$
\( T^{6} - 27 T^{5} + 405 T^{4} + \cdots + 210681 \)
$43$
\( T^{6} + 21 T^{5} + 231 T^{4} + \cdots + 32041 \)
$47$
\( T^{6} - 27 T^{5} + 333 T^{4} + \cdots + 23409 \)
$53$
\( T^{6} + 90 T^{4} + 315 T^{3} + \cdots + 81 \)
$59$
\( T^{6} - 9 T^{5} + 162 T^{4} + \cdots + 6561 \)
$61$
\( T^{6} + 3 T^{5} - 12 T^{4} + \cdots + 94249 \)
$67$
\( T^{6} + 3 T^{5} - 66 T^{4} + \cdots + 2809 \)
$71$
\( T^{6} - 9 T^{5} - 27 T^{4} + \cdots + 263169 \)
$73$
\( T^{6} + 12 T^{5} + 78 T^{4} + \cdots + 54289 \)
$79$
\( T^{6} + 21 T^{5} + 231 T^{4} + \cdots + 2809 \)
$83$
\( T^{6} + 9 T^{5} + 225 T^{4} + \cdots + 408321 \)
$89$
\( T^{6} + 90 T^{4} + 900 T^{3} + \cdots + 431649 \)
$97$
\( T^{6} + 54 T^{5} + 1323 T^{4} + \cdots + 2679769 \)
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