Properties

Label 260.2.bj.a
Level $260$
Weight $2$
Character orbit 260.bj
Analytic conductor $2.076$
Analytic rank $0$
Dimension $4$
CM discriminant -4
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 260 = 2^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 260.bj (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.07611045255\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{12}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12}) q^{2} - 2 \zeta_{12} q^{4} + ( - \zeta_{12}^{3} + 2 \zeta_{12}^{2} + \zeta_{12}) q^{5} + ( - 2 \zeta_{12}^{3} + 2) q^{8} + 3 \zeta_{12} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12}) q^{2} - 2 \zeta_{12} q^{4} + ( - \zeta_{12}^{3} + 2 \zeta_{12}^{2} + \zeta_{12}) q^{5} + ( - 2 \zeta_{12}^{3} + 2) q^{8} + 3 \zeta_{12} q^{9} + (3 \zeta_{12}^{2} - \zeta_{12} - 3) q^{10} + (2 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 2 \zeta_{12}) q^{13} + 4 \zeta_{12}^{2} q^{16} + (5 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - \zeta_{12} - 1) q^{17} + (3 \zeta_{12}^{3} - 3) q^{18} + ( - 4 \zeta_{12}^{3} - 2) q^{20} + (3 \zeta_{12}^{2} + 4 \zeta_{12} - 3) q^{25} + (\zeta_{12}^{2} - 5 \zeta_{12} - 1) q^{26} + (2 \zeta_{12}^{3} - 5 \zeta_{12}^{2} - 2 \zeta_{12} + 10) q^{29} + (4 \zeta_{12}^{2} - 4 \zeta_{12} - 4) q^{32} + (3 \zeta_{12}^{3} - 10 \zeta_{12}^{2} + 5) q^{34} - 6 \zeta_{12}^{2} q^{36} + ( - \zeta_{12}^{3} - \zeta_{12}^{2} + 6 \zeta_{12} - 5) q^{37} + ( - 6 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 6 \zeta_{12}) q^{40} + ( - 5 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 5 \zeta_{12}) q^{41} + (6 \zeta_{12}^{3} + 3) q^{45} + ( - 7 \zeta_{12}^{3} + 7 \zeta_{12}) q^{49} + (\zeta_{12}^{3} - 7) q^{50} + ( - 6 \zeta_{12}^{3} + 4) q^{52} + (2 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + 5 \zeta_{12} - 2) q^{53} + (10 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 7 \zeta_{12} + 7) q^{58} + ( - 10 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 5 \zeta_{12} + 6) q^{61} - 8 \zeta_{12}^{3} q^{64} + (8 \zeta_{12}^{2} - \zeta_{12} - 8) q^{65} + (8 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 2 \zeta_{12} + 10) q^{68} + ( - 6 \zeta_{12}^{2} + 6 \zeta_{12} + 6) q^{72} + ( - 8 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + 5 \zeta_{12} + 8) q^{73} + ( - 5 \zeta_{12}^{2} + 7 \zeta_{12} - 5) q^{74} + (8 \zeta_{12}^{2} + 4 \zeta_{12} - 8) q^{80} + 9 \zeta_{12}^{2} q^{81} + ( - 10 \zeta_{12}^{3} + \zeta_{12}^{2} + 9 \zeta_{12} + 9) q^{82} + (9 \zeta_{12}^{3} - 5 \zeta_{12}^{2} - 15 \zeta_{12} + 7) q^{85} + (16 \zeta_{12}^{3} - 16 \zeta_{12}) q^{89} + (9 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 9 \zeta_{12}) q^{90} + (13 \zeta_{12}^{2} - 13 \zeta_{12} - 13) q^{97} + (7 \zeta_{12}^{2} + 7 \zeta_{12} - 7) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 4 q^{5} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} + 4 q^{5} + 8 q^{8} - 6 q^{10} + 6 q^{13} + 8 q^{16} - 12 q^{17} - 12 q^{18} - 8 q^{20} - 6 q^{25} - 2 q^{26} + 30 q^{29} - 8 q^{32} - 12 q^{36} - 22 q^{37} + 4 q^{40} - 8 q^{41} + 12 q^{45} - 28 q^{50} + 16 q^{52} - 18 q^{53} + 34 q^{58} + 12 q^{61} - 16 q^{65} + 24 q^{68} + 12 q^{72} + 22 q^{73} - 30 q^{74} - 16 q^{80} + 18 q^{81} + 38 q^{82} + 18 q^{85} - 6 q^{90} - 26 q^{97} - 14 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/260\mathbb{Z}\right)^\times\).

\(n\) \(41\) \(131\) \(157\)
\(\chi(n)\) \(-1 + \zeta_{12}^{2}\) \(-1\) \(\zeta_{12}^{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
1.36603 + 0.366025i 0 1.73205 + 1.00000i 0.133975 + 2.23205i 0 0 2.00000 + 2.00000i −2.59808 1.50000i −0.633975 + 3.09808i
87.1 1.36603 0.366025i 0 1.73205 1.00000i 0.133975 2.23205i 0 0 2.00000 2.00000i −2.59808 + 1.50000i −0.633975 3.09808i
107.1 −0.366025 + 1.36603i 0 −1.73205 1.00000i 1.86603 + 1.23205i 0 0 2.00000 2.00000i 2.59808 + 1.50000i −2.36603 + 2.09808i
243.1 −0.366025 1.36603i 0 −1.73205 + 1.00000i 1.86603 1.23205i 0 0 2.00000 + 2.00000i 2.59808 1.50000i −2.36603 2.09808i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
65.q odd 12 1 inner
260.bj 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.bj.a 4
4.b odd 2 1 CM 260.2.bj.a 4
5.c odd 4 1 260.2.bj.b yes 4
13.c even 3 1 260.2.bj.b yes 4
20.e even 4 1 260.2.bj.b yes 4
52.j odd 6 1 260.2.bj.b yes 4
65.q odd 12 1 inner 260.2.bj.a 4
260.bj even 12 1 inner 260.2.bj.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.bj.a 4 1.a even 1 1 trivial
260.2.bj.a 4 4.b odd 2 1 CM
260.2.bj.a 4 65.q odd 12 1 inner
260.2.bj.a 4 260.bj even 12 1 inner
260.2.bj.b yes 4 5.c odd 4 1
260.2.bj.b yes 4 13.c even 3 1
260.2.bj.b yes 4 20.e even 4 1
260.2.bj.b yes 4 52.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(260, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{17}^{4} + 12T_{17}^{3} + 117T_{17}^{2} + 594T_{17} + 1089 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 4 T^{3} + 11 T^{2} - 20 T + 25 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 6 T^{3} + 23 T^{2} - 78 T + 169 \) Copy content Toggle raw display
$17$ \( T^{4} + 12 T^{3} + 117 T^{2} + \cdots + 1089 \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} - 30 T^{3} + 371 T^{2} + \cdots + 5041 \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} + 22 T^{3} + 137 T^{2} + \cdots + 169 \) Copy content Toggle raw display
$41$ \( T^{4} + 8 T^{3} + 123 T^{2} + \cdots + 3481 \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} + 18 T^{3} + 162 T^{2} + 54 T + 9 \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} - 12 T^{3} + 183 T^{2} + \cdots + 1521 \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 22 T^{3} + 242 T^{2} + \cdots + 529 \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} - 256 T^{2} + 65536 \) Copy content Toggle raw display
$97$ \( T^{4} + 26 T^{3} + 338 T^{2} + \cdots + 114244 \) Copy content Toggle raw display
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