Newspace parameters
Level: | \( N \) | \(=\) | \( 378 = 2 \cdot 3^{3} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 378.u (of order \(9\), degree \(6\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(3.01834519640\) |
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 \(\left(\mathbb{Z}/378\mathbb{Z}\right)^\times\).
\(n\) | \(29\) | \(325\) |
\(\chi(n)\) | \(\zeta_{18}^{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
43.1 |
|
0.766044 | + | 0.642788i | 1.11334 | − | 1.32683i | 0.173648 | + | 0.984808i | 2.37939 | + | 0.866025i | 1.70574 | − | 0.300767i | 0.173648 | − | 0.984808i | −0.500000 | + | 0.866025i | −0.520945 | − | 2.95442i | 1.26604 | + | 2.19285i | ||||||||||||||||||
85.1 | −0.939693 | − | 0.342020i | 0.592396 | − | 1.62760i | 0.766044 | + | 0.642788i | 0.152704 | − | 0.866025i | −1.11334 | + | 1.32683i | 0.766044 | − | 0.642788i | −0.500000 | − | 0.866025i | −2.29813 | − | 1.92836i | −0.439693 | + | 0.761570i | |||||||||||||||||||
169.1 | −0.939693 | + | 0.342020i | 0.592396 | + | 1.62760i | 0.766044 | − | 0.642788i | 0.152704 | + | 0.866025i | −1.11334 | − | 1.32683i | 0.766044 | + | 0.642788i | −0.500000 | + | 0.866025i | −2.29813 | + | 1.92836i | −0.439693 | − | 0.761570i | |||||||||||||||||||
211.1 | 0.766044 | − | 0.642788i | 1.11334 | + | 1.32683i | 0.173648 | − | 0.984808i | 2.37939 | − | 0.866025i | 1.70574 | + | 0.300767i | 0.173648 | + | 0.984808i | −0.500000 | − | 0.866025i | −0.520945 | + | 2.95442i | 1.26604 | − | 2.19285i | |||||||||||||||||||
295.1 | 0.173648 | − | 0.984808i | −1.70574 | − | 0.300767i | −0.939693 | − | 0.342020i | −1.03209 | + | 0.866025i | −0.592396 | + | 1.62760i | −0.939693 | + | 0.342020i | −0.500000 | + | 0.866025i | 2.81908 | + | 1.02606i | 0.673648 | + | 1.16679i | |||||||||||||||||||
337.1 | 0.173648 | + | 0.984808i | −1.70574 | + | 0.300767i | −0.939693 | + | 0.342020i | −1.03209 | − | 0.866025i | −0.592396 | − | 1.62760i | −0.939693 | − | 0.342020i | −0.500000 | − | 0.866025i | 2.81908 | − | 1.02606i | 0.673648 | − | 1.16679i | |||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
27.e | even | 9 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 378.2.u.a | ✓ | 6 |
27.e | even | 9 | 1 | inner | 378.2.u.a | ✓ | 6 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
378.2.u.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
378.2.u.a | ✓ | 6 | 27.e | even | 9 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} - 3T_{5}^{5} + 3T_{5}^{3} + 9T_{5}^{2} + 9 \)
acting on \(S_{2}^{\mathrm{new}}(378, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{6} + T^{3} + 1 \)
$3$
\( T^{6} + 9T^{3} + 27 \)
$5$
\( T^{6} - 3 T^{5} + 3 T^{3} + 9 T^{2} + \cdots + 9 \)
$7$
\( T^{6} + T^{3} + 1 \)
$11$
\( T^{6} - 6 T^{5} + 18 T^{4} - 3 T^{3} + \cdots + 9 \)
$13$
\( T^{6} + 3 T^{5} + 6 T^{4} + 8 T^{3} + \cdots + 1 \)
$17$
\( T^{6} + 9 T^{4} + 18 T^{3} + 81 T^{2} + \cdots + 81 \)
$19$
\( T^{6} - 3 T^{5} + 33 T^{4} + \cdots + 2809 \)
$23$
\( T^{6} - 6 T^{5} - 36 T^{4} + \cdots + 3249 \)
$29$
\( T^{6} + 3 T^{5} + 108 T^{4} + \cdots + 3249 \)
$31$
\( T^{6} + 3 T^{5} + 24 T^{4} + 17 T^{3} + \cdots + 361 \)
$37$
\( T^{6} - 12 T^{5} + 132 T^{4} + \cdots + 64 \)
$41$
\( T^{6} + 12 T^{5} + 144 T^{4} + \cdots + 36864 \)
$43$
\( T^{6} + 6 T^{5} - 6 T^{4} - 361 T^{3} + \cdots + 5329 \)
$47$
\( T^{6} - 24 T^{5} + 306 T^{4} + \cdots + 47961 \)
$53$
\( (T^{3} - 117 T + 153)^{2} \)
$59$
\( T^{6} + 24 T^{5} + 252 T^{4} + \cdots + 2601 \)
$61$
\( T^{6} + 3 T^{5} + 114 T^{4} + \cdots + 16129 \)
$67$
\( T^{6} - 3 T^{5} - 96 T^{4} + \cdots + 94249 \)
$71$
\( T^{6} + 12 T^{5} + 189 T^{4} + \cdots + 106929 \)
$73$
\( T^{6} - 3 T^{5} + 123 T^{4} + \cdots + 1369 \)
$79$
\( T^{6} - 33 T^{5} + 492 T^{4} + \cdots + 73441 \)
$83$
\( T^{6} + 144 T^{4} - 720 T^{3} + \cdots + 5184 \)
$89$
\( T^{6} - 21 T^{5} + 405 T^{4} + \cdots + 12321 \)
$97$
\( T^{6} + 15 T^{5} + 120 T^{4} + \cdots + 130321 \)
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