Newspace parameters
Level: | \( N \) | \(=\) | \( 304 = 2^{4} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 304.n (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.42745222145\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
Coefficient field: | \(\Q(\zeta_{12})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
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Defining polynomial: | \( x^{4} - x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/304\mathbb{Z}\right)^\times\).
\(n\) | \(97\) | \(191\) | \(229\) |
\(\chi(n)\) | \(1 - \zeta_{12}^{2}\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
31.1 |
|
0 | −0.866025 | − | 1.50000i | 0 | 0.500000 | + | 0.866025i | 0 | − | 2.00000i | 0 | 0 | 0 | |||||||||||||||||||||||||
31.2 | 0 | 0.866025 | + | 1.50000i | 0 | 0.500000 | + | 0.866025i | 0 | 2.00000i | 0 | 0 | 0 | |||||||||||||||||||||||||||
255.1 | 0 | −0.866025 | + | 1.50000i | 0 | 0.500000 | − | 0.866025i | 0 | 2.00000i | 0 | 0 | 0 | |||||||||||||||||||||||||||
255.2 | 0 | 0.866025 | − | 1.50000i | 0 | 0.500000 | − | 0.866025i | 0 | − | 2.00000i | 0 | 0 | 0 | ||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
19.d | odd | 6 | 1 | inner |
76.f | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 304.2.n.c | ✓ | 4 |
3.b | odd | 2 | 1 | 2736.2.bm.k | 4 | ||
4.b | odd | 2 | 1 | inner | 304.2.n.c | ✓ | 4 |
8.b | even | 2 | 1 | 1216.2.n.c | 4 | ||
8.d | odd | 2 | 1 | 1216.2.n.c | 4 | ||
12.b | even | 2 | 1 | 2736.2.bm.k | 4 | ||
19.d | odd | 6 | 1 | inner | 304.2.n.c | ✓ | 4 |
57.f | even | 6 | 1 | 2736.2.bm.k | 4 | ||
76.f | even | 6 | 1 | inner | 304.2.n.c | ✓ | 4 |
152.l | odd | 6 | 1 | 1216.2.n.c | 4 | ||
152.o | even | 6 | 1 | 1216.2.n.c | 4 | ||
228.n | odd | 6 | 1 | 2736.2.bm.k | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
304.2.n.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
304.2.n.c | ✓ | 4 | 4.b | odd | 2 | 1 | inner |
304.2.n.c | ✓ | 4 | 19.d | odd | 6 | 1 | inner |
304.2.n.c | ✓ | 4 | 76.f | even | 6 | 1 | inner |
1216.2.n.c | 4 | 8.b | even | 2 | 1 | ||
1216.2.n.c | 4 | 8.d | odd | 2 | 1 | ||
1216.2.n.c | 4 | 152.l | odd | 6 | 1 | ||
1216.2.n.c | 4 | 152.o | even | 6 | 1 | ||
2736.2.bm.k | 4 | 3.b | odd | 2 | 1 | ||
2736.2.bm.k | 4 | 12.b | even | 2 | 1 | ||
2736.2.bm.k | 4 | 57.f | even | 6 | 1 | ||
2736.2.bm.k | 4 | 228.n | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 3T_{3}^{2} + 9 \)
acting on \(S_{2}^{\mathrm{new}}(304, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( T^{4} + 3T^{2} + 9 \)
$5$
\( (T^{2} - T + 1)^{2} \)
$7$
\( (T^{2} + 4)^{2} \)
$11$
\( (T^{2} + 4)^{2} \)
$13$
\( (T^{2} + 3 T + 3)^{2} \)
$17$
\( (T^{2} + T + 1)^{2} \)
$19$
\( T^{4} + 26T^{2} + 361 \)
$23$
\( T^{4} - T^{2} + 1 \)
$29$
\( (T^{2} + 3 T + 3)^{2} \)
$31$
\( (T^{2} - 12)^{2} \)
$37$
\( (T^{2} + 12)^{2} \)
$41$
\( (T^{2} - 9 T + 27)^{2} \)
$43$
\( T^{4} - 81T^{2} + 6561 \)
$47$
\( T^{4} - 121 T^{2} + 14641 \)
$53$
\( (T^{2} - 15 T + 75)^{2} \)
$59$
\( T^{4} + 75T^{2} + 5625 \)
$61$
\( (T^{2} - 9 T + 81)^{2} \)
$67$
\( T^{4} + 27T^{2} + 729 \)
$71$
\( T^{4} + 243 T^{2} + 59049 \)
$73$
\( (T^{2} - T + 1)^{2} \)
$79$
\( T^{4} + 3T^{2} + 9 \)
$83$
\( (T^{2} + 64)^{2} \)
$89$
\( (T^{2} + 27 T + 243)^{2} \)
$97$
\( (T^{2} + 27 T + 243)^{2} \)
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