Newspace parameters
Level: | \( N \) | \(=\) | \( 378 = 2 \cdot 3^{3} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 378.k (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(3.01834519640\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{6})\) |
Coefficient field: | \(\Q(\zeta_{24})\) |
Defining polynomial: |
\( x^{8} - x^{4} + 1 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 2^{2}\cdot 3^{2} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring
\(\beta_{1}\) | \(=\) |
\( \zeta_{24}^{2} \)
|
\(\beta_{2}\) | \(=\) |
\( \zeta_{24}^{4} \)
|
\(\beta_{3}\) | \(=\) |
\( \zeta_{24}^{6} \)
|
\(\beta_{4}\) | \(=\) |
\( \zeta_{24}^{7} - \zeta_{24}^{5} + \zeta_{24}^{3} + 2\zeta_{24} \)
|
\(\beta_{5}\) | \(=\) |
\( -\zeta_{24}^{7} + 2\zeta_{24}^{5} + 2\zeta_{24}^{3} - \zeta_{24} \)
|
\(\beta_{6}\) | \(=\) |
\( -2\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \)
|
\(\beta_{7}\) | \(=\) |
\( 2\zeta_{24}^{7} + \zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24} \)
|
\(\zeta_{24}\) | \(=\) |
\( ( \beta_{7} + 2\beta_{6} - \beta_{5} + \beta_{4} ) / 6 \)
|
\(\zeta_{24}^{2}\) | \(=\) |
\( \beta_1 \)
|
\(\zeta_{24}^{3}\) | \(=\) |
\( ( -\beta_{7} - \beta_{6} + 2\beta_{5} + 2\beta_{4} ) / 6 \)
|
\(\zeta_{24}^{4}\) | \(=\) |
\( \beta_{2} \)
|
\(\zeta_{24}^{5}\) | \(=\) |
\( ( 2\beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} ) / 6 \)
|
\(\zeta_{24}^{6}\) | \(=\) |
\( \beta_{3} \)
|
\(\zeta_{24}^{7}\) | \(=\) |
\( ( \beta_{7} - 2\beta_{6} + \beta_{5} + \beta_{4} ) / 6 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/378\mathbb{Z}\right)^\times\).
\(n\) | \(29\) | \(325\) |
\(\chi(n)\) | \(-1\) | \(1 - \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
215.1 |
|
−0.866025 | − | 0.500000i | 0 | 0.500000 | + | 0.866025i | −1.22474 | + | 2.12132i | 0 | −1.00000 | − | 2.44949i | − | 1.00000i | 0 | 2.12132 | − | 1.22474i | |||||||||||||||||||||||||||||||
215.2 | −0.866025 | − | 0.500000i | 0 | 0.500000 | + | 0.866025i | 1.22474 | − | 2.12132i | 0 | −1.00000 | + | 2.44949i | − | 1.00000i | 0 | −2.12132 | + | 1.22474i | ||||||||||||||||||||||||||||||||
215.3 | 0.866025 | + | 0.500000i | 0 | 0.500000 | + | 0.866025i | −1.22474 | + | 2.12132i | 0 | −1.00000 | + | 2.44949i | 1.00000i | 0 | −2.12132 | + | 1.22474i | |||||||||||||||||||||||||||||||||
215.4 | 0.866025 | + | 0.500000i | 0 | 0.500000 | + | 0.866025i | 1.22474 | − | 2.12132i | 0 | −1.00000 | − | 2.44949i | 1.00000i | 0 | 2.12132 | − | 1.22474i | |||||||||||||||||||||||||||||||||
269.1 | −0.866025 | + | 0.500000i | 0 | 0.500000 | − | 0.866025i | −1.22474 | − | 2.12132i | 0 | −1.00000 | + | 2.44949i | 1.00000i | 0 | 2.12132 | + | 1.22474i | |||||||||||||||||||||||||||||||||
269.2 | −0.866025 | + | 0.500000i | 0 | 0.500000 | − | 0.866025i | 1.22474 | + | 2.12132i | 0 | −1.00000 | − | 2.44949i | 1.00000i | 0 | −2.12132 | − | 1.22474i | |||||||||||||||||||||||||||||||||
269.3 | 0.866025 | − | 0.500000i | 0 | 0.500000 | − | 0.866025i | −1.22474 | − | 2.12132i | 0 | −1.00000 | − | 2.44949i | − | 1.00000i | 0 | −2.12132 | − | 1.22474i | ||||||||||||||||||||||||||||||||
269.4 | 0.866025 | − | 0.500000i | 0 | 0.500000 | − | 0.866025i | 1.22474 | + | 2.12132i | 0 | −1.00000 | + | 2.44949i | − | 1.00000i | 0 | 2.12132 | + | 1.22474i | ||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
7.d | odd | 6 | 1 | inner |
21.g | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 378.2.k.d | ✓ | 8 |
3.b | odd | 2 | 1 | inner | 378.2.k.d | ✓ | 8 |
7.c | even | 3 | 1 | 2646.2.d.d | 8 | ||
7.d | odd | 6 | 1 | inner | 378.2.k.d | ✓ | 8 |
7.d | odd | 6 | 1 | 2646.2.d.d | 8 | ||
9.c | even | 3 | 1 | 1134.2.l.e | 8 | ||
9.c | even | 3 | 1 | 1134.2.t.f | 8 | ||
9.d | odd | 6 | 1 | 1134.2.l.e | 8 | ||
9.d | odd | 6 | 1 | 1134.2.t.f | 8 | ||
21.g | even | 6 | 1 | inner | 378.2.k.d | ✓ | 8 |
21.g | even | 6 | 1 | 2646.2.d.d | 8 | ||
21.h | odd | 6 | 1 | 2646.2.d.d | 8 | ||
63.i | even | 6 | 1 | 1134.2.t.f | 8 | ||
63.k | odd | 6 | 1 | 1134.2.l.e | 8 | ||
63.s | even | 6 | 1 | 1134.2.l.e | 8 | ||
63.t | odd | 6 | 1 | 1134.2.t.f | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
378.2.k.d | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
378.2.k.d | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
378.2.k.d | ✓ | 8 | 7.d | odd | 6 | 1 | inner |
378.2.k.d | ✓ | 8 | 21.g | even | 6 | 1 | inner |
1134.2.l.e | 8 | 9.c | even | 3 | 1 | ||
1134.2.l.e | 8 | 9.d | odd | 6 | 1 | ||
1134.2.l.e | 8 | 63.k | odd | 6 | 1 | ||
1134.2.l.e | 8 | 63.s | even | 6 | 1 | ||
1134.2.t.f | 8 | 9.c | even | 3 | 1 | ||
1134.2.t.f | 8 | 9.d | odd | 6 | 1 | ||
1134.2.t.f | 8 | 63.i | even | 6 | 1 | ||
1134.2.t.f | 8 | 63.t | odd | 6 | 1 | ||
2646.2.d.d | 8 | 7.c | even | 3 | 1 | ||
2646.2.d.d | 8 | 7.d | odd | 6 | 1 | ||
2646.2.d.d | 8 | 21.g | even | 6 | 1 | ||
2646.2.d.d | 8 | 21.h | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 6T_{5}^{2} + 36 \)
acting on \(S_{2}^{\mathrm{new}}(378, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{4} - T^{2} + 1)^{2} \)
$3$
\( T^{8} \)
$5$
\( (T^{4} + 6 T^{2} + 36)^{2} \)
$7$
\( (T^{2} + 2 T + 7)^{4} \)
$11$
\( (T^{4} - 18 T^{2} + 324)^{2} \)
$13$
\( (T^{4} + 18 T^{2} + 9)^{2} \)
$17$
\( (T^{4} + 6 T^{2} + 36)^{2} \)
$19$
\( (T^{4} - 24 T^{2} + 576)^{2} \)
$23$
\( (T^{4} - 36 T^{2} + 1296)^{2} \)
$29$
\( (T^{4} + 108 T^{2} + 324)^{2} \)
$31$
\( (T^{4} + 6 T^{3} - 39 T^{2} - 306 T + 2601)^{2} \)
$37$
\( (T^{4} + 2 T^{3} + 21 T^{2} - 34 T + 289)^{2} \)
$41$
\( (T^{2} - 6)^{4} \)
$43$
\( (T - 7)^{8} \)
$47$
\( T^{8} + 228 T^{6} + \cdots + 108243216 \)
$53$
\( T^{8} - 216 T^{6} + 45360 T^{4} + \cdots + 1679616 \)
$59$
\( (T^{4} + 6 T^{2} + 36)^{2} \)
$61$
\( (T^{4} - 6 T^{3} + 9 T^{2} + 18 T + 9)^{2} \)
$67$
\( (T^{4} + 10 T^{3} + 147 T^{2} - 470 T + 2209)^{2} \)
$71$
\( (T^{2} + 162)^{4} \)
$73$
\( (T^{4} + 36 T^{3} + 516 T^{2} + 3024 T + 7056)^{2} \)
$79$
\( (T^{4} + 10 T^{3} + 93 T^{2} + 70 T + 49)^{2} \)
$83$
\( (T^{4} - 264 T^{2} + 7056)^{2} \)
$89$
\( T^{8} + 324 T^{6} + 102060 T^{4} + \cdots + 8503056 \)
$97$
\( (T^{4} + 54 T^{2} + 441)^{2} \)
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