Newspace parameters
| Level: | \( N \) | \(=\) | \( 260 = 2^{2} \cdot 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 260.bk (of order \(12\), degree \(4\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(2.07611045255\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{12})\) |
| Defining polynomial: |
\( x^{4} - x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 \(\left(\mathbb{Z}/260\mathbb{Z}\right)^\times\).
| \(n\) | \(41\) | \(131\) | \(157\) |
| \(\chi(n)\) | \(\zeta_{12}\) | \(1\) | \(\zeta_{12}^{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 33.1 |
|
0 | −0.500000 | + | 0.133975i | 0 | −2.00000 | − | 1.00000i | 0 | −2.13397 | + | 1.23205i | 0 | −2.36603 | + | 1.36603i | 0 | ||||||||||||||||||||||
| 97.1 | 0 | −0.500000 | − | 1.86603i | 0 | −2.00000 | + | 1.00000i | 0 | −3.86603 | + | 2.23205i | 0 | −0.633975 | + | 0.366025i | 0 | |||||||||||||||||||||||
| 193.1 | 0 | −0.500000 | + | 1.86603i | 0 | −2.00000 | − | 1.00000i | 0 | −3.86603 | − | 2.23205i | 0 | −0.633975 | − | 0.366025i | 0 | |||||||||||||||||||||||
| 197.1 | 0 | −0.500000 | − | 0.133975i | 0 | −2.00000 | + | 1.00000i | 0 | −2.13397 | − | 1.23205i | 0 | −2.36603 | − | 1.36603i | 0 | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 65.o | even | 12 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 260.2.bk.a | yes | 4 |
| 5.b | even | 2 | 1 | 1300.2.bs.b | 4 | ||
| 5.c | odd | 4 | 1 | 260.2.bf.b | ✓ | 4 | |
| 5.c | odd | 4 | 1 | 1300.2.bn.a | 4 | ||
| 13.f | odd | 12 | 1 | 260.2.bf.b | ✓ | 4 | |
| 65.o | even | 12 | 1 | inner | 260.2.bk.a | yes | 4 |
| 65.s | odd | 12 | 1 | 1300.2.bn.a | 4 | ||
| 65.t | even | 12 | 1 | 1300.2.bs.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 260.2.bf.b | ✓ | 4 | 5.c | odd | 4 | 1 | |
| 260.2.bf.b | ✓ | 4 | 13.f | odd | 12 | 1 | |
| 260.2.bk.a | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 260.2.bk.a | yes | 4 | 65.o | even | 12 | 1 | inner |
| 1300.2.bn.a | 4 | 5.c | odd | 4 | 1 | ||
| 1300.2.bn.a | 4 | 65.s | odd | 12 | 1 | ||
| 1300.2.bs.b | 4 | 5.b | even | 2 | 1 | ||
| 1300.2.bs.b | 4 | 65.t | even | 12 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 2T_{3}^{3} + 5T_{3}^{2} + 4T_{3} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(260, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( T^{4} + 2 T^{3} + 5 T^{2} + 4 T + 1 \)
$5$
\( (T^{2} + 4 T + 5)^{2} \)
$7$
\( T^{4} + 12 T^{3} + 59 T^{2} + \cdots + 121 \)
$11$
\( T^{4} + 8 T^{3} + 41 T^{2} + 130 T + 169 \)
$13$
\( (T^{2} + 4 T + 13)^{2} \)
$17$
\( T^{4} - 6 T^{3} + 9 T^{2} + 9 \)
$19$
\( T^{4} + 8 T^{3} + 41 T^{2} + 130 T + 169 \)
$23$
\( T^{4} + 14 T^{3} + 53 T^{2} + 4 T + 1 \)
$29$
\( (T^{2} + 3 T + 3)^{2} \)
$31$
\( T^{4} + 2916 \)
$37$
\( T^{4} + 12 T^{3} + 35 T^{2} + \cdots + 169 \)
$41$
\( T^{4} - 20 T^{3} + 125 T^{2} + \cdots + 625 \)
$43$
\( T^{4} + 10 T^{3} + 29 T^{2} + \cdots + 529 \)
$47$
\( T^{4} + 128T^{2} + 1024 \)
$53$
\( T^{4} + 4 T^{3} + 8 T^{2} - 88 T + 484 \)
$59$
\( T^{4} + 8 T^{3} + 65 T^{2} + \cdots + 3481 \)
$61$
\( (T^{2} - 9 T + 81)^{2} \)
$67$
\( T^{4} - 28 T^{3} + 591 T^{2} + \cdots + 37249 \)
$71$
\( T^{4} + 8 T^{3} + 17 T^{2} + 22 T + 121 \)
$73$
\( (T^{2} + 16 T + 16)^{2} \)
$79$
\( T^{4} + 56T^{2} + 16 \)
$83$
\( T^{4} + 128T^{2} + 1024 \)
$89$
\( T^{4} + 28 T^{3} + 197 T^{2} + \cdots + 2209 \)
$97$
\( T^{4} - 12 T^{3} + 111 T^{2} + \cdots + 1089 \)
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