Properties

Label 260.2.i.d
Level $260$
Weight $2$
Character orbit 260.i
Analytic conductor $2.076$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 260 = 2^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 260.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.07611045255\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 3 - 3 \zeta_{6} ) q^{3} + q^{5} -3 \zeta_{6} q^{7} -6 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( 3 - 3 \zeta_{6} ) q^{3} + q^{5} -3 \zeta_{6} q^{7} -6 \zeta_{6} q^{9} + ( -3 + 3 \zeta_{6} ) q^{11} + ( -1 + 4 \zeta_{6} ) q^{13} + ( 3 - 3 \zeta_{6} ) q^{15} + 7 \zeta_{6} q^{17} -\zeta_{6} q^{19} -9 q^{21} + ( 7 - 7 \zeta_{6} ) q^{23} + q^{25} -9 q^{27} + ( 5 - 5 \zeta_{6} ) q^{29} -4 q^{31} + 9 \zeta_{6} q^{33} -3 \zeta_{6} q^{35} + ( 3 - 3 \zeta_{6} ) q^{37} + ( 9 + 3 \zeta_{6} ) q^{39} + ( -7 + 7 \zeta_{6} ) q^{41} + 9 \zeta_{6} q^{43} -6 \zeta_{6} q^{45} + 8 q^{47} + ( -2 + 2 \zeta_{6} ) q^{49} + 21 q^{51} -6 q^{53} + ( -3 + 3 \zeta_{6} ) q^{55} -3 q^{57} -5 \zeta_{6} q^{59} + 5 \zeta_{6} q^{61} + ( -18 + 18 \zeta_{6} ) q^{63} + ( -1 + 4 \zeta_{6} ) q^{65} + ( -13 + 13 \zeta_{6} ) q^{67} -21 \zeta_{6} q^{69} + 3 \zeta_{6} q^{71} -14 q^{73} + ( 3 - 3 \zeta_{6} ) q^{75} + 9 q^{77} -8 q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} + 12 q^{83} + 7 \zeta_{6} q^{85} -15 \zeta_{6} q^{87} + ( -7 + 7 \zeta_{6} ) q^{89} + ( 12 - 9 \zeta_{6} ) q^{91} + ( -12 + 12 \zeta_{6} ) q^{93} -\zeta_{6} q^{95} + 11 \zeta_{6} q^{97} + 18 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 3q^{3} + 2q^{5} - 3q^{7} - 6q^{9} + O(q^{10}) \) \( 2q + 3q^{3} + 2q^{5} - 3q^{7} - 6q^{9} - 3q^{11} + 2q^{13} + 3q^{15} + 7q^{17} - q^{19} - 18q^{21} + 7q^{23} + 2q^{25} - 18q^{27} + 5q^{29} - 8q^{31} + 9q^{33} - 3q^{35} + 3q^{37} + 21q^{39} - 7q^{41} + 9q^{43} - 6q^{45} + 16q^{47} - 2q^{49} + 42q^{51} - 12q^{53} - 3q^{55} - 6q^{57} - 5q^{59} + 5q^{61} - 18q^{63} + 2q^{65} - 13q^{67} - 21q^{69} + 3q^{71} - 28q^{73} + 3q^{75} + 18q^{77} - 16q^{79} - 9q^{81} + 24q^{83} + 7q^{85} - 15q^{87} - 7q^{89} + 15q^{91} - 12q^{93} - q^{95} + 11q^{97} + 36q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(41\) \(131\) \(157\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1
0.500000 + 0.866025i
0.500000 0.866025i
0 1.50000 2.59808i 0 1.00000 0 −1.50000 2.59808i 0 −3.00000 5.19615i 0
81.1 0 1.50000 + 2.59808i 0 1.00000 0 −1.50000 + 2.59808i 0 −3.00000 + 5.19615i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 260.2.i.d 2
3.b odd 2 1 2340.2.q.a 2
4.b odd 2 1 1040.2.q.b 2
5.b even 2 1 1300.2.i.a 2
5.c odd 4 2 1300.2.bb.e 4
13.c even 3 1 inner 260.2.i.d 2
13.c even 3 1 3380.2.a.b 1
13.e even 6 1 3380.2.a.a 1
13.f odd 12 2 3380.2.f.a 2
39.i odd 6 1 2340.2.q.a 2
52.j odd 6 1 1040.2.q.b 2
65.n even 6 1 1300.2.i.a 2
65.q odd 12 2 1300.2.bb.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.i.d 2 1.a even 1 1 trivial
260.2.i.d 2 13.c even 3 1 inner
1040.2.q.b 2 4.b odd 2 1
1040.2.q.b 2 52.j odd 6 1
1300.2.i.a 2 5.b even 2 1
1300.2.i.a 2 65.n even 6 1
1300.2.bb.e 4 5.c odd 4 2
1300.2.bb.e 4 65.q odd 12 2
2340.2.q.a 2 3.b odd 2 1
2340.2.q.a 2 39.i odd 6 1
3380.2.a.a 1 13.e even 6 1
3380.2.a.b 1 13.c even 3 1
3380.2.f.a 2 13.f odd 12 2

Hecke kernels

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

\( T_{3}^{2} - 3 T_{3} + 9 \)
\( T_{19}^{2} + T_{19} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 9 - 3 T + T^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( 9 + 3 T + T^{2} \)
$11$ \( 9 + 3 T + T^{2} \)
$13$ \( 13 - 2 T + T^{2} \)
$17$ \( 49 - 7 T + T^{2} \)
$19$ \( 1 + T + T^{2} \)
$23$ \( 49 - 7 T + T^{2} \)
$29$ \( 25 - 5 T + T^{2} \)
$31$ \( ( 4 + T )^{2} \)
$37$ \( 9 - 3 T + T^{2} \)
$41$ \( 49 + 7 T + T^{2} \)
$43$ \( 81 - 9 T + T^{2} \)
$47$ \( ( -8 + T )^{2} \)
$53$ \( ( 6 + T )^{2} \)
$59$ \( 25 + 5 T + T^{2} \)
$61$ \( 25 - 5 T + T^{2} \)
$67$ \( 169 + 13 T + T^{2} \)
$71$ \( 9 - 3 T + T^{2} \)
$73$ \( ( 14 + T )^{2} \)
$79$ \( ( 8 + T )^{2} \)
$83$ \( ( -12 + T )^{2} \)
$89$ \( 49 + 7 T + T^{2} \)
$97$ \( 121 - 11 T + T^{2} \)
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