Properties

Label 260.2.u.a
Level $260$
Weight $2$
Character orbit 260.u
Analytic conductor $2.076$
Analytic rank $1$
Dimension $2$
CM discriminant -4
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(2.07611045255\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (i - 1) q^{2} - 2 i q^{4} + (i - 2) q^{5} + (2 i + 2) q^{8} - 3 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (i - 1) q^{2} - 2 i q^{4} + (i - 2) q^{5} + (2 i + 2) q^{8} - 3 q^{9} + ( - 3 i + 1) q^{10} + ( - 3 i - 2) q^{13} - 4 q^{16} - 8 q^{17} + ( - 3 i + 3) q^{18} + (4 i + 2) q^{20} + ( - 4 i + 3) q^{25} + (i + 5) q^{26} - 4 q^{29} + ( - 4 i + 4) q^{32} + ( - 8 i + 8) q^{34} + 6 i q^{36} + ( - 5 i - 5) q^{37} + ( - 2 i - 6) q^{40} + (9 i + 9) q^{41} + ( - 3 i + 6) q^{45} + 7 i q^{49} + (7 i + 1) q^{50} + (4 i - 6) q^{52} + 4 i q^{53} + ( - 4 i + 4) q^{58} - 10 q^{61} + 8 i q^{64} + (4 i + 7) q^{65} + 16 i q^{68} + ( - 6 i - 6) q^{72} + (11 i + 11) q^{73} + 10 q^{74} + ( - 4 i + 8) q^{80} + 9 q^{81} - 18 q^{82} + ( - 8 i + 16) q^{85} + (13 i - 13) q^{89} + (9 i - 3) q^{90} + ( - 5 i + 5) q^{97} + ( - 7 i - 7) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 4 q^{5} + 4 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 4 q^{5} + 4 q^{8} - 6 q^{9} + 2 q^{10} - 4 q^{13} - 8 q^{16} - 16 q^{17} + 6 q^{18} + 4 q^{20} + 6 q^{25} + 10 q^{26} - 8 q^{29} + 8 q^{32} + 16 q^{34} - 10 q^{37} - 12 q^{40} + 18 q^{41} + 12 q^{45} + 2 q^{50} - 12 q^{52} + 8 q^{58} - 20 q^{61} + 14 q^{65} - 12 q^{72} + 22 q^{73} + 20 q^{74} + 16 q^{80} + 18 q^{81} - 36 q^{82} + 32 q^{85} - 26 q^{89} - 6 q^{90} + 10 q^{97} - 14 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(41\) \(131\) \(157\)
\(\chi(n)\) \(i\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
1.00000i
1.00000i
−1.00000 + 1.00000i 0 2.00000i −2.00000 + 1.00000i 0 0 2.00000 + 2.00000i −3.00000 1.00000 3.00000i
239.1 −1.00000 1.00000i 0 2.00000i −2.00000 1.00000i 0 0 2.00000 2.00000i −3.00000 1.00000 + 3.00000i
\(n\): 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.g odd 4 1 inner
260.u even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 260.2.u.a 2
4.b odd 2 1 CM 260.2.u.a 2
5.b even 2 1 260.2.u.b yes 2
13.d odd 4 1 260.2.u.b yes 2
20.d odd 2 1 260.2.u.b yes 2
52.f even 4 1 260.2.u.b yes 2
65.g odd 4 1 inner 260.2.u.a 2
260.u even 4 1 inner 260.2.u.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.u.a 2 1.a even 1 1 trivial
260.2.u.a 2 4.b odd 2 1 CM
260.2.u.a 2 65.g odd 4 1 inner
260.2.u.a 2 260.u even 4 1 inner
260.2.u.b yes 2 5.b even 2 1
260.2.u.b yes 2 13.d odd 4 1
260.2.u.b yes 2 20.d odd 2 1
260.2.u.b yes 2 52.f even 4 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
\( T_{17} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 4T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 4T + 13 \) Copy content Toggle raw display
$17$ \( (T + 8)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T + 4)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 10T + 50 \) Copy content Toggle raw display
$41$ \( T^{2} - 18T + 162 \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 16 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T + 10)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 22T + 242 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 26T + 338 \) Copy content Toggle raw display
$97$ \( T^{2} - 10T + 50 \) Copy content Toggle raw display
show more
show less