Newspace parameters
Level: | \( N \) | \(=\) | \( 378 = 2 \cdot 3^{3} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 378.f (of order \(3\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(3.01834519640\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{-3})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: |
\( x^{4} - 2x^{2} + 4 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 3 \) |
Twist minimal: | no (minimal twist has level 126) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{4} - 2x^{2} + 4 \)
:
\(\beta_{1}\) | \(=\) |
\( ( \nu^{2} ) / 2 \)
|
\(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 2\nu ) / 2 \)
|
\(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 4\nu ) / 2 \)
|
\(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 3 \)
|
\(\nu^{2}\) | \(=\) |
\( 2\beta_1 \)
|
\(\nu^{3}\) | \(=\) |
\( ( -2\beta_{3} + 4\beta_{2} ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/378\mathbb{Z}\right)^\times\).
\(n\) | \(29\) | \(325\) |
\(\chi(n)\) | \(-\beta_{1}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
127.1 |
|
0.500000 | − | 0.866025i | 0 | −0.500000 | − | 0.866025i | −0.724745 | − | 1.25529i | 0 | 0.500000 | − | 0.866025i | −1.00000 | 0 | −1.44949 | ||||||||||||||||||||||
127.2 | 0.500000 | − | 0.866025i | 0 | −0.500000 | − | 0.866025i | 1.72474 | + | 2.98735i | 0 | 0.500000 | − | 0.866025i | −1.00000 | 0 | 3.44949 | |||||||||||||||||||||||
253.1 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | −0.724745 | + | 1.25529i | 0 | 0.500000 | + | 0.866025i | −1.00000 | 0 | −1.44949 | |||||||||||||||||||||||
253.2 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | 1.72474 | − | 2.98735i | 0 | 0.500000 | + | 0.866025i | −1.00000 | 0 | 3.44949 | |||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 378.2.f.d | 4 | |
3.b | odd | 2 | 1 | 126.2.f.c | ✓ | 4 | |
4.b | odd | 2 | 1 | 3024.2.r.e | 4 | ||
7.b | odd | 2 | 1 | 2646.2.f.k | 4 | ||
7.c | even | 3 | 1 | 2646.2.e.l | 4 | ||
7.c | even | 3 | 1 | 2646.2.h.m | 4 | ||
7.d | odd | 6 | 1 | 2646.2.e.k | 4 | ||
7.d | odd | 6 | 1 | 2646.2.h.n | 4 | ||
9.c | even | 3 | 1 | inner | 378.2.f.d | 4 | |
9.c | even | 3 | 1 | 1134.2.a.i | 2 | ||
9.d | odd | 6 | 1 | 126.2.f.c | ✓ | 4 | |
9.d | odd | 6 | 1 | 1134.2.a.p | 2 | ||
12.b | even | 2 | 1 | 1008.2.r.e | 4 | ||
21.c | even | 2 | 1 | 882.2.f.j | 4 | ||
21.g | even | 6 | 1 | 882.2.e.n | 4 | ||
21.g | even | 6 | 1 | 882.2.h.l | 4 | ||
21.h | odd | 6 | 1 | 882.2.e.m | 4 | ||
21.h | odd | 6 | 1 | 882.2.h.k | 4 | ||
36.f | odd | 6 | 1 | 3024.2.r.e | 4 | ||
36.f | odd | 6 | 1 | 9072.2.a.bd | 2 | ||
36.h | even | 6 | 1 | 1008.2.r.e | 4 | ||
36.h | even | 6 | 1 | 9072.2.a.bk | 2 | ||
63.g | even | 3 | 1 | 2646.2.e.l | 4 | ||
63.h | even | 3 | 1 | 2646.2.h.m | 4 | ||
63.i | even | 6 | 1 | 882.2.h.l | 4 | ||
63.j | odd | 6 | 1 | 882.2.h.k | 4 | ||
63.k | odd | 6 | 1 | 2646.2.e.k | 4 | ||
63.l | odd | 6 | 1 | 2646.2.f.k | 4 | ||
63.l | odd | 6 | 1 | 7938.2.a.bm | 2 | ||
63.n | odd | 6 | 1 | 882.2.e.m | 4 | ||
63.o | even | 6 | 1 | 882.2.f.j | 4 | ||
63.o | even | 6 | 1 | 7938.2.a.bn | 2 | ||
63.s | even | 6 | 1 | 882.2.e.n | 4 | ||
63.t | odd | 6 | 1 | 2646.2.h.n | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
126.2.f.c | ✓ | 4 | 3.b | odd | 2 | 1 | |
126.2.f.c | ✓ | 4 | 9.d | odd | 6 | 1 | |
378.2.f.d | 4 | 1.a | even | 1 | 1 | trivial | |
378.2.f.d | 4 | 9.c | even | 3 | 1 | inner | |
882.2.e.m | 4 | 21.h | odd | 6 | 1 | ||
882.2.e.m | 4 | 63.n | odd | 6 | 1 | ||
882.2.e.n | 4 | 21.g | even | 6 | 1 | ||
882.2.e.n | 4 | 63.s | even | 6 | 1 | ||
882.2.f.j | 4 | 21.c | even | 2 | 1 | ||
882.2.f.j | 4 | 63.o | even | 6 | 1 | ||
882.2.h.k | 4 | 21.h | odd | 6 | 1 | ||
882.2.h.k | 4 | 63.j | odd | 6 | 1 | ||
882.2.h.l | 4 | 21.g | even | 6 | 1 | ||
882.2.h.l | 4 | 63.i | even | 6 | 1 | ||
1008.2.r.e | 4 | 12.b | even | 2 | 1 | ||
1008.2.r.e | 4 | 36.h | even | 6 | 1 | ||
1134.2.a.i | 2 | 9.c | even | 3 | 1 | ||
1134.2.a.p | 2 | 9.d | odd | 6 | 1 | ||
2646.2.e.k | 4 | 7.d | odd | 6 | 1 | ||
2646.2.e.k | 4 | 63.k | odd | 6 | 1 | ||
2646.2.e.l | 4 | 7.c | even | 3 | 1 | ||
2646.2.e.l | 4 | 63.g | even | 3 | 1 | ||
2646.2.f.k | 4 | 7.b | odd | 2 | 1 | ||
2646.2.f.k | 4 | 63.l | odd | 6 | 1 | ||
2646.2.h.m | 4 | 7.c | even | 3 | 1 | ||
2646.2.h.m | 4 | 63.h | even | 3 | 1 | ||
2646.2.h.n | 4 | 7.d | odd | 6 | 1 | ||
2646.2.h.n | 4 | 63.t | odd | 6 | 1 | ||
3024.2.r.e | 4 | 4.b | odd | 2 | 1 | ||
3024.2.r.e | 4 | 36.f | odd | 6 | 1 | ||
7938.2.a.bm | 2 | 63.l | odd | 6 | 1 | ||
7938.2.a.bn | 2 | 63.o | even | 6 | 1 | ||
9072.2.a.bd | 2 | 36.f | odd | 6 | 1 | ||
9072.2.a.bk | 2 | 36.h | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 2T_{5}^{3} + 9T_{5}^{2} + 10T_{5} + 25 \)
acting on \(S_{2}^{\mathrm{new}}(378, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} - T + 1)^{2} \)
$3$
\( T^{4} \)
$5$
\( T^{4} - 2 T^{3} + 9 T^{2} + 10 T + 25 \)
$7$
\( (T^{2} - T + 1)^{2} \)
$11$
\( (T^{2} - 2 T + 4)^{2} \)
$13$
\( T^{4} + 24T^{2} + 576 \)
$17$
\( (T + 2)^{4} \)
$19$
\( (T^{2} - 10 T + 19)^{2} \)
$23$
\( (T^{2} + T + 1)^{2} \)
$29$
\( T^{4} + 4 T^{3} + 36 T^{2} - 80 T + 400 \)
$31$
\( (T^{2} + 6 T + 36)^{2} \)
$37$
\( (T^{2} - 4 T - 92)^{2} \)
$41$
\( T^{4} + 96T^{2} + 9216 \)
$43$
\( T^{4} + 4 T^{3} + 36 T^{2} - 80 T + 400 \)
$47$
\( T^{4} + 96T^{2} + 9216 \)
$53$
\( (T^{2} - 12 T + 12)^{2} \)
$59$
\( (T^{2} + 2 T + 4)^{2} \)
$61$
\( T^{4} + 18 T^{3} + 249 T^{2} + \cdots + 5625 \)
$67$
\( T^{4} - 16 T^{3} + 216 T^{2} + \cdots + 1600 \)
$71$
\( (T^{2} + 10 T + 1)^{2} \)
$73$
\( (T^{2} + 4 T - 20)^{2} \)
$79$
\( T^{4} + 6 T^{3} + 51 T^{2} - 90 T + 225 \)
$83$
\( (T^{2} - 2 T + 4)^{2} \)
$89$
\( (T^{2} - 24 T + 120)^{2} \)
$97$
\( T^{4} - 4 T^{3} + 36 T^{2} + 80 T + 400 \)
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