Newspace parameters
Level: | \( N \) | \(=\) | \( 294 = 2 \cdot 3 \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 294.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.34760181943\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Coefficient field: | \(\Q(\zeta_{16})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{8} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{16}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/294\mathbb{Z}\right)^\times\).
\(n\) | \(197\) | \(199\) |
\(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
293.1 |
|
− | 1.00000i | −0.923880 | + | 1.46508i | −1.00000 | −1.53073 | 1.46508 | + | 0.923880i | 0 | 1.00000i | −1.29289 | − | 2.70711i | 1.53073i | |||||||||||||||||||||||||||||||||||
293.2 | − | 1.00000i | −0.382683 | + | 1.68925i | −1.00000 | 3.69552 | 1.68925 | + | 0.382683i | 0 | 1.00000i | −2.70711 | − | 1.29289i | − | 3.69552i | |||||||||||||||||||||||||||||||||||
293.3 | − | 1.00000i | 0.382683 | − | 1.68925i | −1.00000 | −3.69552 | −1.68925 | − | 0.382683i | 0 | 1.00000i | −2.70711 | − | 1.29289i | 3.69552i | ||||||||||||||||||||||||||||||||||||
293.4 | − | 1.00000i | 0.923880 | − | 1.46508i | −1.00000 | 1.53073 | −1.46508 | − | 0.923880i | 0 | 1.00000i | −1.29289 | − | 2.70711i | − | 1.53073i | |||||||||||||||||||||||||||||||||||
293.5 | 1.00000i | −0.923880 | − | 1.46508i | −1.00000 | −1.53073 | 1.46508 | − | 0.923880i | 0 | − | 1.00000i | −1.29289 | + | 2.70711i | − | 1.53073i | |||||||||||||||||||||||||||||||||||
293.6 | 1.00000i | −0.382683 | − | 1.68925i | −1.00000 | 3.69552 | 1.68925 | − | 0.382683i | 0 | − | 1.00000i | −2.70711 | + | 1.29289i | 3.69552i | ||||||||||||||||||||||||||||||||||||
293.7 | 1.00000i | 0.382683 | + | 1.68925i | −1.00000 | −3.69552 | −1.68925 | + | 0.382683i | 0 | − | 1.00000i | −2.70711 | + | 1.29289i | − | 3.69552i | |||||||||||||||||||||||||||||||||||
293.8 | 1.00000i | 0.923880 | + | 1.46508i | −1.00000 | 1.53073 | −1.46508 | + | 0.923880i | 0 | − | 1.00000i | −1.29289 | + | 2.70711i | 1.53073i | ||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
7.b | odd | 2 | 1 | inner |
21.c | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 294.2.d.b | ✓ | 8 |
3.b | odd | 2 | 1 | inner | 294.2.d.b | ✓ | 8 |
4.b | odd | 2 | 1 | 2352.2.k.f | 8 | ||
7.b | odd | 2 | 1 | inner | 294.2.d.b | ✓ | 8 |
7.c | even | 3 | 2 | 294.2.f.c | 16 | ||
7.d | odd | 6 | 2 | 294.2.f.c | 16 | ||
12.b | even | 2 | 1 | 2352.2.k.f | 8 | ||
21.c | even | 2 | 1 | inner | 294.2.d.b | ✓ | 8 |
21.g | even | 6 | 2 | 294.2.f.c | 16 | ||
21.h | odd | 6 | 2 | 294.2.f.c | 16 | ||
28.d | even | 2 | 1 | 2352.2.k.f | 8 | ||
84.h | odd | 2 | 1 | 2352.2.k.f | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
294.2.d.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
294.2.d.b | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
294.2.d.b | ✓ | 8 | 7.b | odd | 2 | 1 | inner |
294.2.d.b | ✓ | 8 | 21.c | even | 2 | 1 | inner |
294.2.f.c | 16 | 7.c | even | 3 | 2 | ||
294.2.f.c | 16 | 7.d | odd | 6 | 2 | ||
294.2.f.c | 16 | 21.g | even | 6 | 2 | ||
294.2.f.c | 16 | 21.h | odd | 6 | 2 | ||
2352.2.k.f | 8 | 4.b | odd | 2 | 1 | ||
2352.2.k.f | 8 | 12.b | even | 2 | 1 | ||
2352.2.k.f | 8 | 28.d | even | 2 | 1 | ||
2352.2.k.f | 8 | 84.h | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 16T_{5}^{2} + 32 \)
acting on \(S_{2}^{\mathrm{new}}(294, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} + 1)^{4} \)
$3$
\( T^{8} + 8 T^{6} + 32 T^{4} + 72 T^{2} + \cdots + 81 \)
$5$
\( (T^{4} - 16 T^{2} + 32)^{2} \)
$7$
\( T^{8} \)
$11$
\( (T^{4} + 44 T^{2} + 196)^{2} \)
$13$
\( (T^{4} + 32 T^{2} + 128)^{2} \)
$17$
\( (T^{4} - 36 T^{2} + 162)^{2} \)
$19$
\( (T^{4} + 4 T^{2} + 2)^{2} \)
$23$
\( (T^{2} + 8)^{4} \)
$29$
\( (T^{4} + 24 T^{2} + 16)^{2} \)
$31$
\( (T^{4} + 80 T^{2} + 1568)^{2} \)
$37$
\( (T + 2)^{8} \)
$41$
\( (T^{4} - 100 T^{2} + 1250)^{2} \)
$43$
\( (T^{2} + 4 T - 46)^{4} \)
$47$
\( (T^{4} - 16 T^{2} + 32)^{2} \)
$53$
\( (T^{4} + 152 T^{2} + 4624)^{2} \)
$59$
\( (T^{4} - 20 T^{2} + 2)^{2} \)
$61$
\( (T^{4} + 80 T^{2} + 1568)^{2} \)
$67$
\( (T^{2} + 4 T - 68)^{4} \)
$71$
\( (T^{4} + 144 T^{2} + 3136)^{2} \)
$73$
\( (T^{4} + 4 T^{2} + 2)^{2} \)
$79$
\( (T^{2} - 24 T + 136)^{4} \)
$83$
\( (T^{4} - 116 T^{2} + 2)^{2} \)
$89$
\( (T^{4} - 164 T^{2} + 4802)^{2} \)
$97$
\( (T^{4} + 100 T^{2} + 578)^{2} \)
show more
show less