Properties

Label 3640.2.a.n
Level $3640$
Weight $2$
Character orbit 3640.a
Self dual yes
Analytic conductor $29.066$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3640.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.0655463357\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−0.618034
1.61803
0 −0.618034 0 1.00000 0 1.00000 0 −2.61803 0
1.2 0 1.61803 0 1.00000 0 1.00000 0 −0.381966 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(7\) \(-1\)
\(13\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3640.2.a.n 2
4.b odd 2 1 7280.2.a.z 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3640.2.a.n 2 1.a even 1 1 trivial
7280.2.a.z 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3640))\):

\( T_{3}^{2} - T_{3} - 1 \) Copy content Toggle raw display
\( T_{11}^{2} + 4T_{11} - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 4T - 16 \) Copy content Toggle raw display
$13$ \( (T - 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 7T + 11 \) Copy content Toggle raw display
$19$ \( T^{2} + 3T + 1 \) Copy content Toggle raw display
$23$ \( T^{2} + 8T - 4 \) Copy content Toggle raw display
$29$ \( T^{2} + 7T + 11 \) Copy content Toggle raw display
$31$ \( T^{2} - 5T - 25 \) Copy content Toggle raw display
$37$ \( T^{2} - 3T - 29 \) Copy content Toggle raw display
$41$ \( T^{2} + 15T + 55 \) Copy content Toggle raw display
$43$ \( T^{2} + 6T + 4 \) Copy content Toggle raw display
$47$ \( T^{2} - 20 \) Copy content Toggle raw display
$53$ \( T^{2} - 20 \) Copy content Toggle raw display
$59$ \( T^{2} - 7T - 89 \) Copy content Toggle raw display
$61$ \( T^{2} + 16T + 44 \) Copy content Toggle raw display
$67$ \( T^{2} + 3T + 1 \) Copy content Toggle raw display
$71$ \( T^{2} + 18T + 36 \) Copy content Toggle raw display
$73$ \( T^{2} + 4T - 16 \) Copy content Toggle raw display
$79$ \( T^{2} - 13T - 59 \) Copy content Toggle raw display
$83$ \( (T + 6)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + T - 61 \) Copy content Toggle raw display
$97$ \( T^{2} + 18T + 36 \) Copy content Toggle raw display
show more
show less