Newspace parameters
Level: | \( N \) | \(=\) | \( 294 = 2 \cdot 3 \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 294.g (of order \(6\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(8.01091977219\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
Coefficient field: | \(\Q(\sqrt{2}, \sqrt{-3})\) |
Defining polynomial: |
\( x^{4} + 2x^{2} + 4 \)
|
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 42) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{4} + 2x^{2} + 4 \)
:
\(\beta_{1}\) | \(=\) |
\( \nu \)
|
\(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 2 \)
|
\(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 2 \)
|
\(\nu\) | \(=\) |
\( \beta_1 \)
|
\(\nu^{2}\) | \(=\) |
\( 2\beta_{2} \)
|
\(\nu^{3}\) | \(=\) |
\( 2\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/294\mathbb{Z}\right)^\times\).
\(n\) | \(197\) | \(199\) |
\(\chi(n)\) | \(1\) | \(1 + \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
19.1 |
|
−0.707107 | + | 1.22474i | −1.50000 | + | 0.866025i | −1.00000 | − | 1.73205i | 0.878680 | + | 0.507306i | − | 2.44949i | 0 | 2.82843 | 1.50000 | − | 2.59808i | −1.24264 | + | 0.717439i | |||||||||||||||||
19.2 | 0.707107 | − | 1.22474i | −1.50000 | + | 0.866025i | −1.00000 | − | 1.73205i | 5.12132 | + | 2.95680i | 2.44949i | 0 | −2.82843 | 1.50000 | − | 2.59808i | 7.24264 | − | 4.18154i | |||||||||||||||||||
31.1 | −0.707107 | − | 1.22474i | −1.50000 | − | 0.866025i | −1.00000 | + | 1.73205i | 0.878680 | − | 0.507306i | 2.44949i | 0 | 2.82843 | 1.50000 | + | 2.59808i | −1.24264 | − | 0.717439i | |||||||||||||||||||
31.2 | 0.707107 | + | 1.22474i | −1.50000 | − | 0.866025i | −1.00000 | + | 1.73205i | 5.12132 | − | 2.95680i | − | 2.44949i | 0 | −2.82843 | 1.50000 | + | 2.59808i | 7.24264 | + | 4.18154i | ||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.d | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 294.3.g.b | 4 | |
3.b | odd | 2 | 1 | 882.3.n.a | 4 | ||
7.b | odd | 2 | 1 | 294.3.g.c | 4 | ||
7.c | even | 3 | 1 | 42.3.c.a | ✓ | 4 | |
7.c | even | 3 | 1 | 294.3.g.c | 4 | ||
7.d | odd | 6 | 1 | 42.3.c.a | ✓ | 4 | |
7.d | odd | 6 | 1 | inner | 294.3.g.b | 4 | |
21.c | even | 2 | 1 | 882.3.n.d | 4 | ||
21.g | even | 6 | 1 | 126.3.c.b | 4 | ||
21.g | even | 6 | 1 | 882.3.n.a | 4 | ||
21.h | odd | 6 | 1 | 126.3.c.b | 4 | ||
21.h | odd | 6 | 1 | 882.3.n.d | 4 | ||
28.f | even | 6 | 1 | 336.3.f.c | 4 | ||
28.g | odd | 6 | 1 | 336.3.f.c | 4 | ||
35.i | odd | 6 | 1 | 1050.3.f.a | 4 | ||
35.j | even | 6 | 1 | 1050.3.f.a | 4 | ||
35.k | even | 12 | 2 | 1050.3.h.a | 8 | ||
35.l | odd | 12 | 2 | 1050.3.h.a | 8 | ||
56.j | odd | 6 | 1 | 1344.3.f.f | 4 | ||
56.k | odd | 6 | 1 | 1344.3.f.e | 4 | ||
56.m | even | 6 | 1 | 1344.3.f.e | 4 | ||
56.p | even | 6 | 1 | 1344.3.f.f | 4 | ||
84.j | odd | 6 | 1 | 1008.3.f.g | 4 | ||
84.n | even | 6 | 1 | 1008.3.f.g | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
42.3.c.a | ✓ | 4 | 7.c | even | 3 | 1 | |
42.3.c.a | ✓ | 4 | 7.d | odd | 6 | 1 | |
126.3.c.b | 4 | 21.g | even | 6 | 1 | ||
126.3.c.b | 4 | 21.h | odd | 6 | 1 | ||
294.3.g.b | 4 | 1.a | even | 1 | 1 | trivial | |
294.3.g.b | 4 | 7.d | odd | 6 | 1 | inner | |
294.3.g.c | 4 | 7.b | odd | 2 | 1 | ||
294.3.g.c | 4 | 7.c | even | 3 | 1 | ||
336.3.f.c | 4 | 28.f | even | 6 | 1 | ||
336.3.f.c | 4 | 28.g | odd | 6 | 1 | ||
882.3.n.a | 4 | 3.b | odd | 2 | 1 | ||
882.3.n.a | 4 | 21.g | even | 6 | 1 | ||
882.3.n.d | 4 | 21.c | even | 2 | 1 | ||
882.3.n.d | 4 | 21.h | odd | 6 | 1 | ||
1008.3.f.g | 4 | 84.j | odd | 6 | 1 | ||
1008.3.f.g | 4 | 84.n | even | 6 | 1 | ||
1050.3.f.a | 4 | 35.i | odd | 6 | 1 | ||
1050.3.f.a | 4 | 35.j | even | 6 | 1 | ||
1050.3.h.a | 8 | 35.k | even | 12 | 2 | ||
1050.3.h.a | 8 | 35.l | odd | 12 | 2 | ||
1344.3.f.e | 4 | 56.k | odd | 6 | 1 | ||
1344.3.f.e | 4 | 56.m | even | 6 | 1 | ||
1344.3.f.f | 4 | 56.j | odd | 6 | 1 | ||
1344.3.f.f | 4 | 56.p | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 12T_{5}^{3} + 54T_{5}^{2} - 72T_{5} + 36 \)
acting on \(S_{3}^{\mathrm{new}}(294, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} + 2T^{2} + 4 \)
$3$
\( (T^{2} + 3 T + 3)^{2} \)
$5$
\( T^{4} - 12 T^{3} + 54 T^{2} - 72 T + 36 \)
$7$
\( T^{4} \)
$11$
\( T^{4} - 12 T^{3} + 126 T^{2} + \cdots + 324 \)
$13$
\( T^{4} + 432 T^{2} + 28224 \)
$17$
\( T^{4} + 12 T^{3} - 666 T^{2} + \cdots + 509796 \)
$19$
\( T^{4} - 12 T^{3} - 324 T^{2} + \cdots + 138384 \)
$23$
\( T^{4} + 12 T^{3} + 270 T^{2} + \cdots + 15876 \)
$29$
\( (T - 30)^{4} \)
$31$
\( T^{4} + 72 T^{3} + 1296 T^{2} + \cdots + 186624 \)
$37$
\( T^{4} - 40 T^{3} + 3792 T^{2} + \cdots + 4804864 \)
$41$
\( T^{4} + 1764 T^{2} + 86436 \)
$43$
\( (T^{2} + 64 T - 776)^{2} \)
$47$
\( T^{4} + 168 T^{3} + 11664 T^{2} + \cdots + 5089536 \)
$53$
\( T^{4} - 108 T^{3} + 9036 T^{2} + \cdots + 6906384 \)
$59$
\( T^{4} + 168 T^{3} + 9360 T^{2} + \cdots + 2304 \)
$61$
\( T^{4} + 24 T^{3} + 144 T^{2} + \cdots + 2304 \)
$67$
\( T^{4} + 88 T^{3} + 6096 T^{2} + \cdots + 2715904 \)
$71$
\( (T^{2} + 60 T - 5598)^{2} \)
$73$
\( T^{4} - 24 T^{3} - 3816 T^{2} + \cdots + 16064064 \)
$79$
\( T^{4} + 64 T^{3} + 13440 T^{2} + \cdots + 87310336 \)
$83$
\( T^{4} + 10944 T^{2} + \cdots + 451584 \)
$89$
\( T^{4} - 324 T^{3} + \cdots + 37234404 \)
$97$
\( T^{4} + 12816 T^{2} + \cdots + 27123264 \)
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