Properties

Label 260.2.r.b
Level $260$
Weight $2$
Character orbit 260.r
Analytic conductor $2.076$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(2.07611045255\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + (\beta_{3} - \beta_1 - 1) q^{5} + (\beta_{3} + \beta_{2} + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} + (\beta_{3} - \beta_1 - 1) q^{5} + (\beta_{3} + \beta_{2} + \beta_1) q^{9} + (\beta_{3} + 2 \beta_{2} - 2) q^{11} + (2 \beta_{2} - 3) q^{13} + (\beta_{3} - 2 \beta_{2} + 2) q^{15} + ( - 2 \beta_{3} - \beta_{2} + 1) q^{17} + (3 \beta_{3} + 4 \beta_{2} - 4) q^{19} + ( - 4 \beta_{2} + \beta_1 - 4) q^{23} + 5 q^{25} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{27} + ( - \beta_{3} - 4 \beta_{2} - \beta_1) q^{29} + (4 \beta_{2} + 3 \beta_1 + 4) q^{31} + ( - \beta_{3} - 2 \beta_{2} - \beta_1) q^{33} + ( - 3 \beta_{3} + 3 \beta_1 + 6) q^{37} + ( - 3 \beta_{3} - 2 \beta_1) q^{39} + (5 \beta_{2} - 2 \beta_1 + 5) q^{41} + (2 \beta_{2} + 3 \beta_1 + 2) q^{43} - 5 \beta_{2} q^{45} + (4 \beta_{3} - 4 \beta_1 - 4) q^{47} - 7 q^{49} + ( - \beta_{3} + 4 \beta_{2} - \beta_1) q^{51} + ( - 4 \beta_{3} + \beta_{2} - 1) q^{53} + ( - 3 \beta_{3} - 4 \beta_{2} + 4) q^{55} + ( - \beta_{3} - 6 \beta_{2} - \beta_1) q^{57} + (4 \beta_{2} - \beta_1 + 4) q^{59} + ( - 3 \beta_{3} + 3 \beta_1 + 2) q^{61} + ( - 5 \beta_{3} - 2 \beta_{2} + \beta_1 + 3) q^{65} + (3 \beta_{3} - 4 \beta_{2} + 3 \beta_1) q^{67} + ( - 5 \beta_{3} + 5 \beta_1 - 2) q^{69} + (2 \beta_{2} + \beta_1 + 2) q^{71} + ( - 3 \beta_{3} - 8 \beta_{2} - 3 \beta_1) q^{73} + 5 \beta_{3} q^{75} + ( - 3 \beta_{3} - 3 \beta_1) q^{79} + (3 \beta_{3} - 3 \beta_1 + 1) q^{81} + (2 \beta_{3} - 2 \beta_1 - 8) q^{83} + (5 \beta_{2} - 5) q^{85} + (2 \beta_{2} + 2 \beta_1 + 2) q^{87} + (\beta_{2} + 2 \beta_1 + 1) q^{89} + (\beta_{3} - \beta_1 - 6) q^{93} + ( - 5 \beta_{3} - 10 \beta_{2} + 10) q^{95} + (6 \beta_{3} + 8 \beta_{2} + 6 \beta_1) q^{97} + ( - 4 \beta_{2} - 3 \beta_1 - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} - 6 q^{11} - 12 q^{13} + 10 q^{15} - 10 q^{19} - 18 q^{23} + 20 q^{25} - 4 q^{27} + 10 q^{31} + 12 q^{37} - 2 q^{39} + 24 q^{41} + 2 q^{43} - 28 q^{49} - 12 q^{53} + 10 q^{55} + 18 q^{59} - 4 q^{61} - 28 q^{69} + 6 q^{71} + 10 q^{75} + 16 q^{81} - 24 q^{83} - 20 q^{85} + 4 q^{87} - 20 q^{93} + 30 q^{95} - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} + \nu + 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} + 2\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{2} + \nu - 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + \beta _1 - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{3} + \beta_{2} - \beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(41\) \(131\) \(157\)
\(\chi(n)\) \(\beta_{2}\) \(1\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
177.1
0.618034i
1.61803i
0.618034i
1.61803i
0 −0.618034 + 0.618034i 0 −2.23607 0 0 0 2.23607i 0
177.2 0 1.61803 1.61803i 0 2.23607 0 0 0 2.23607i 0
213.1 0 −0.618034 0.618034i 0 −2.23607 0 0 0 2.23607i 0
213.2 0 1.61803 + 1.61803i 0 2.23607 0 0 0 2.23607i 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
65.f 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.r.b yes 4
3.b odd 2 1 2340.2.bp.e 4
4.b odd 2 1 1040.2.cd.j 4
5.b even 2 1 1300.2.r.b 4
5.c odd 4 1 260.2.m.b 4
5.c odd 4 1 1300.2.m.b 4
13.d odd 4 1 260.2.m.b 4
15.e even 4 1 2340.2.u.f 4
20.e even 4 1 1040.2.bg.j 4
39.f even 4 1 2340.2.u.f 4
52.f even 4 1 1040.2.bg.j 4
65.f even 4 1 inner 260.2.r.b yes 4
65.g odd 4 1 1300.2.m.b 4
65.k even 4 1 1300.2.r.b 4
195.u odd 4 1 2340.2.bp.e 4
260.l odd 4 1 1040.2.cd.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.m.b 4 5.c odd 4 1
260.2.m.b 4 13.d odd 4 1
260.2.r.b yes 4 1.a even 1 1 trivial
260.2.r.b yes 4 65.f even 4 1 inner
1040.2.bg.j 4 20.e even 4 1
1040.2.bg.j 4 52.f even 4 1
1040.2.cd.j 4 4.b odd 2 1
1040.2.cd.j 4 260.l odd 4 1
1300.2.m.b 4 5.c odd 4 1
1300.2.m.b 4 65.g odd 4 1
1300.2.r.b 4 5.b even 2 1
1300.2.r.b 4 65.k even 4 1
2340.2.u.f 4 15.e even 4 1
2340.2.u.f 4 39.f even 4 1
2340.2.bp.e 4 3.b odd 2 1
2340.2.bp.e 4 195.u odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 2T_{3}^{3} + 2T_{3}^{2} + 4T_{3} + 4 \) acting on \(S_{2}^{\mathrm{new}}(260, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 2 T^{3} + 2 T^{2} + 4 T + 4 \) Copy content Toggle raw display
$5$ \( (T^{2} - 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 6 T^{3} + 18 T^{2} + 12 T + 4 \) Copy content Toggle raw display
$13$ \( (T^{2} + 6 T + 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 100 \) Copy content Toggle raw display
$19$ \( T^{4} + 10 T^{3} + 50 T^{2} + \cdots + 100 \) Copy content Toggle raw display
$23$ \( T^{4} + 18 T^{3} + 162 T^{2} + \cdots + 1444 \) Copy content Toggle raw display
$29$ \( T^{4} + 28T^{2} + 16 \) Copy content Toggle raw display
$31$ \( T^{4} - 10 T^{3} + 50 T^{2} + \cdots + 100 \) Copy content Toggle raw display
$37$ \( (T^{2} - 6 T - 36)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 24 T^{3} + 288 T^{2} + \cdots + 3844 \) Copy content Toggle raw display
$43$ \( T^{4} - 2 T^{3} + 2 T^{2} + 44 T + 484 \) Copy content Toggle raw display
$47$ \( (T^{2} - 80)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 12 T^{3} + 72 T^{2} + \cdots + 484 \) Copy content Toggle raw display
$59$ \( T^{4} - 18 T^{3} + 162 T^{2} + \cdots + 1444 \) Copy content Toggle raw display
$61$ \( (T^{2} + 2 T - 44)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 188T^{2} + 16 \) Copy content Toggle raw display
$71$ \( T^{4} - 6 T^{3} + 18 T^{2} - 12 T + 4 \) Copy content Toggle raw display
$73$ \( T^{4} + 140T^{2} + 400 \) Copy content Toggle raw display
$79$ \( T^{4} + 108T^{2} + 1296 \) Copy content Toggle raw display
$83$ \( (T^{2} + 12 T + 16)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 100 \) Copy content Toggle raw display
$97$ \( T^{4} + 368 T^{2} + 30976 \) Copy content Toggle raw display
show more
show less