Newspace parameters
Level: | \( N \) | \(=\) | \( 342 = 2 \cdot 3^{2} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 342.u (of order \(9\), degree \(6\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.73088374913\) |
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: | no (minimal twist has level 114) |
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 \(\left(\mathbb{Z}/342\mathbb{Z}\right)^\times\).
\(n\) | \(191\) | \(325\) |
\(\chi(n)\) | \(1\) | \(\zeta_{18}^{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
55.1 |
|
0.173648 | − | 0.984808i | 0 | −0.939693 | − | 0.342020i | −3.20574 | + | 1.16679i | 0 | 1.43969 | + | 2.49362i | −0.500000 | + | 0.866025i | 0 | 0.592396 | + | 3.35965i | ||||||||||||||||||||||||
73.1 | 0.766044 | + | 0.642788i | 0 | 0.173648 | + | 0.984808i | −0.386659 | + | 2.19285i | 0 | 0.326352 | + | 0.565258i | −0.500000 | + | 0.866025i | 0 | −1.70574 | + | 1.43128i | |||||||||||||||||||||||||
199.1 | 0.173648 | + | 0.984808i | 0 | −0.939693 | + | 0.342020i | −3.20574 | − | 1.16679i | 0 | 1.43969 | − | 2.49362i | −0.500000 | − | 0.866025i | 0 | 0.592396 | − | 3.35965i | |||||||||||||||||||||||||
253.1 | 0.766044 | − | 0.642788i | 0 | 0.173648 | − | 0.984808i | −0.386659 | − | 2.19285i | 0 | 0.326352 | − | 0.565258i | −0.500000 | − | 0.866025i | 0 | −1.70574 | − | 1.43128i | |||||||||||||||||||||||||
271.1 | −0.939693 | + | 0.342020i | 0 | 0.766044 | − | 0.642788i | −0.907604 | − | 0.761570i | 0 | −0.266044 | − | 0.460802i | −0.500000 | + | 0.866025i | 0 | 1.11334 | + | 0.405223i | |||||||||||||||||||||||||
289.1 | −0.939693 | − | 0.342020i | 0 | 0.766044 | + | 0.642788i | −0.907604 | + | 0.761570i | 0 | −0.266044 | + | 0.460802i | −0.500000 | − | 0.866025i | 0 | 1.11334 | − | 0.405223i | |||||||||||||||||||||||||
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 | 342.2.u.a | 6 | |
3.b | odd | 2 | 1 | 114.2.i.d | ✓ | 6 | |
12.b | even | 2 | 1 | 912.2.bo.f | 6 | ||
19.e | even | 9 | 1 | inner | 342.2.u.a | 6 | |
19.e | even | 9 | 1 | 6498.2.a.bs | 3 | ||
19.f | odd | 18 | 1 | 6498.2.a.bn | 3 | ||
57.j | even | 18 | 1 | 2166.2.a.u | 3 | ||
57.l | odd | 18 | 1 | 114.2.i.d | ✓ | 6 | |
57.l | odd | 18 | 1 | 2166.2.a.o | 3 | ||
228.v | even | 18 | 1 | 912.2.bo.f | 6 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
114.2.i.d | ✓ | 6 | 3.b | odd | 2 | 1 | |
114.2.i.d | ✓ | 6 | 57.l | odd | 18 | 1 | |
342.2.u.a | 6 | 1.a | even | 1 | 1 | trivial | |
342.2.u.a | 6 | 19.e | even | 9 | 1 | inner | |
912.2.bo.f | 6 | 12.b | even | 2 | 1 | ||
912.2.bo.f | 6 | 228.v | even | 18 | 1 | ||
2166.2.a.o | 3 | 57.l | odd | 18 | 1 | ||
2166.2.a.u | 3 | 57.j | even | 18 | 1 | ||
6498.2.a.bn | 3 | 19.f | odd | 18 | 1 | ||
6498.2.a.bs | 3 | 19.e | even | 9 | 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 \(S_{2}^{\mathrm{new}}(342, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{6} + T^{3} + 1 \)
$3$
\( T^{6} \)
$5$
\( T^{6} + 9 T^{5} + 36 T^{4} + 90 T^{3} + \cdots + 81 \)
$7$
\( T^{6} - 3 T^{5} + 9 T^{4} - 2 T^{3} + \cdots + 1 \)
$11$
\( T^{6} + 3 T^{4} + 2 T^{3} + 9 T^{2} + \cdots + 1 \)
$13$
\( T^{6} + 3 T^{5} + 18 T^{4} + 24 T^{3} + \cdots + 9 \)
$17$
\( T^{6} - 3 T^{5} + 48 T^{4} - 244 T^{3} + \cdots + 289 \)
$19$
\( T^{6} + 18 T^{5} + 162 T^{4} + \cdots + 6859 \)
$23$
\( T^{6} - 21 T^{5} + 210 T^{4} + \cdots + 72361 \)
$29$
\( T^{6} - 3 T^{5} + 18 T^{4} + \cdots + 45369 \)
$31$
\( T^{6} - 9 T^{5} + 81 T^{4} - 54 T^{3} + \cdots + 729 \)
$37$
\( (T^{3} + 9 T^{2} - 57 T - 361)^{2} \)
$41$
\( T^{6} - 15 T^{5} + 177 T^{4} + \cdots + 11881 \)
$43$
\( T^{6} - 3 T^{5} + 99 T^{4} + \cdots + 3249 \)
$47$
\( T^{6} - 9 T^{5} + 63 T^{4} + \cdots + 2809 \)
$53$
\( T^{6} + 12 T^{5} + 174 T^{4} + \cdots + 94249 \)
$59$
\( T^{6} + 27 T^{5} + 324 T^{4} + \cdots + 289 \)
$61$
\( T^{6} - 3 T^{5} - 60 T^{4} + \cdots + 1682209 \)
$67$
\( T^{6} - 21 T^{5} + 126 T^{4} - 24 T^{3} + \cdots + 9 \)
$71$
\( T^{6} + 39 T^{5} + 561 T^{4} + \cdots + 201601 \)
$73$
\( T^{6} - 36 T^{5} + 558 T^{4} + \cdots + 1369 \)
$79$
\( T^{6} + 45 T^{5} + 1125 T^{4} + \cdots + 4515625 \)
$83$
\( T^{6} - 27 T^{5} + 507 T^{4} + \cdots + 253009 \)
$89$
\( T^{6} - 30 T^{5} + 246 T^{4} + \cdots + 11449 \)
$97$
\( T^{6} + 6 T^{5} + 3 T^{4} + 35 T^{3} + \cdots + 2809 \)
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