Newspace parameters
Level: | \( N \) | \(=\) | \( 57 = 3 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 57.i (of order \(9\), degree \(6\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.455147291521\) |
Analytic rank: | \(0\) |
Dimension: | \(6\) |
Coefficient field: | \(\Q(\zeta_{18})\) |
Defining polynomial: |
\( x^{6} - x^{3} + 1 \)
|
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
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 \(\left(\mathbb{Z}/57\mathbb{Z}\right)^\times\).
\(n\) | \(20\) | \(40\) |
\(\chi(n)\) | \(1\) | \(\zeta_{18}^{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
4.1 |
|
1.43969 | + | 0.524005i | −0.173648 | − | 0.984808i | 0.266044 | + | 0.223238i | −1.93969 | + | 1.62760i | 0.266044 | − | 1.50881i | −0.266044 | + | 0.460802i | −1.26604 | − | 2.19285i | −0.939693 | + | 0.342020i | −3.64543 | + | 1.32683i | ||||||||||||||||||
16.1 | −0.266044 | − | 0.223238i | 0.939693 | − | 0.342020i | −0.326352 | − | 1.85083i | −0.233956 | + | 1.32683i | −0.326352 | − | 0.118782i | 0.326352 | + | 0.565258i | −0.673648 | + | 1.16679i | 0.766044 | − | 0.642788i | 0.358441 | − | 0.300767i | |||||||||||||||||||
25.1 | −0.266044 | + | 0.223238i | 0.939693 | + | 0.342020i | −0.326352 | + | 1.85083i | −0.233956 | − | 1.32683i | −0.326352 | + | 0.118782i | 0.326352 | − | 0.565258i | −0.673648 | − | 1.16679i | 0.766044 | + | 0.642788i | 0.358441 | + | 0.300767i | |||||||||||||||||||
28.1 | 0.326352 | + | 1.85083i | −0.766044 | + | 0.642788i | −1.43969 | + | 0.524005i | −0.826352 | − | 0.300767i | −1.43969 | − | 1.20805i | 1.43969 | − | 2.49362i | 0.439693 | + | 0.761570i | 0.173648 | − | 0.984808i | 0.286989 | − | 1.62760i | |||||||||||||||||||
43.1 | 1.43969 | − | 0.524005i | −0.173648 | + | 0.984808i | 0.266044 | − | 0.223238i | −1.93969 | − | 1.62760i | 0.266044 | + | 1.50881i | −0.266044 | − | 0.460802i | −1.26604 | + | 2.19285i | −0.939693 | − | 0.342020i | −3.64543 | − | 1.32683i | |||||||||||||||||||
55.1 | 0.326352 | − | 1.85083i | −0.766044 | − | 0.642788i | −1.43969 | − | 0.524005i | −0.826352 | + | 0.300767i | −1.43969 | + | 1.20805i | 1.43969 | + | 2.49362i | 0.439693 | − | 0.761570i | 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 | 57.2.i.a | ✓ | 6 |
3.b | odd | 2 | 1 | 171.2.u.a | 6 | ||
4.b | odd | 2 | 1 | 912.2.bo.b | 6 | ||
19.e | even | 9 | 1 | inner | 57.2.i.a | ✓ | 6 |
19.e | even | 9 | 1 | 1083.2.a.m | 3 | ||
19.f | odd | 18 | 1 | 1083.2.a.n | 3 | ||
57.j | even | 18 | 1 | 3249.2.a.x | 3 | ||
57.l | odd | 18 | 1 | 171.2.u.a | 6 | ||
57.l | odd | 18 | 1 | 3249.2.a.w | 3 | ||
76.l | odd | 18 | 1 | 912.2.bo.b | 6 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
57.2.i.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
57.2.i.a | ✓ | 6 | 19.e | even | 9 | 1 | inner |
171.2.u.a | 6 | 3.b | odd | 2 | 1 | ||
171.2.u.a | 6 | 57.l | odd | 18 | 1 | ||
912.2.bo.b | 6 | 4.b | odd | 2 | 1 | ||
912.2.bo.b | 6 | 76.l | odd | 18 | 1 | ||
1083.2.a.m | 3 | 19.e | even | 9 | 1 | ||
1083.2.a.n | 3 | 19.f | odd | 18 | 1 | ||
3249.2.a.w | 3 | 57.l | odd | 18 | 1 | ||
3249.2.a.x | 3 | 57.j | even | 18 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 3T_{2}^{5} + 6T_{2}^{4} - 8T_{2}^{3} + 3T_{2}^{2} + 3T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(57, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{6} - 3 T^{5} + 6 T^{4} - 8 T^{3} + \cdots + 1 \)
$3$
\( T^{6} - T^{3} + 1 \)
$5$
\( T^{6} + 6 T^{5} + 18 T^{4} + 30 T^{3} + \cdots + 9 \)
$7$
\( T^{6} - 3 T^{5} + 9 T^{4} - 2 T^{3} + \cdots + 1 \)
$11$
\( T^{6} - 3 T^{5} + 27 T^{4} + \cdots + 1369 \)
$13$
\( T^{6} + 6 T^{5} + 30 T^{4} + 125 T^{3} + \cdots + 361 \)
$17$
\( T^{6} + 12 T^{5} + 114 T^{4} + \cdots + 2809 \)
$19$
\( T^{6} - 18 T^{5} + 162 T^{4} + \cdots + 6859 \)
$23$
\( T^{6} + 9 T^{5} + 18 T^{4} + \cdots + 5329 \)
$29$
\( T^{6} - 12 T^{5} + 60 T^{4} + \cdots + 2809 \)
$31$
\( T^{6} + 24 T^{5} + 387 T^{4} + \cdots + 239121 \)
$37$
\( (T^{3} - 6 T^{2} - 9 T + 51)^{2} \)
$41$
\( T^{6} - 6 T^{5} + 81 T^{4} + \cdots + 2601 \)
$43$
\( T^{6} - 18 T^{5} + 144 T^{4} + \cdots + 361 \)
$47$
\( T^{6} + 33 T^{5} + 477 T^{4} + \cdots + 45369 \)
$53$
\( T^{6} + 24 T^{5} + 375 T^{4} + \cdots + 83521 \)
$59$
\( T^{6} + 54 T^{4} - 224 T^{3} + \cdots + 7921 \)
$61$
\( T^{6} + 21 T^{5} + 204 T^{4} + \cdots + 1369 \)
$67$
\( T^{6} - 30 T^{5} + 372 T^{4} + \cdots + 87616 \)
$71$
\( T^{6} - 18 T^{5} + 144 T^{4} + \cdots + 23409 \)
$73$
\( T^{6} + 27 T^{5} + 324 T^{4} + \cdots + 11881 \)
$79$
\( T^{6} - 12 T^{5} + 168 T^{4} + \cdots + 2809 \)
$83$
\( T^{6} - 15 T^{5} + 189 T^{4} + \cdots + 2601 \)
$89$
\( T^{6} - 45 T^{5} + 900 T^{4} + \cdots + 1265625 \)
$97$
\( T^{6} - 21 T^{5} + 210 T^{4} + \cdots + 72361 \)
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