Properties

Label 3920.2.a.bw
Level $3920$
Weight $2$
Character orbit 3920.a
Self dual yes
Analytic conductor $31.301$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3920.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.3013575923\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 245)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + (\beta + 1) q^{3} - q^{5} + 2 \beta q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{3} - q^{5} + 2 \beta q^{9} + (2 \beta + 3) q^{11} + (\beta - 3) q^{13} + ( - \beta - 1) q^{15} + (3 \beta + 1) q^{17} + 6 q^{19} + (\beta - 6) q^{23} + q^{25} + ( - \beta + 1) q^{27} + (4 \beta - 3) q^{29} + ( - 3 \beta + 6) q^{31} + (5 \beta + 7) q^{33} + ( - 3 \beta - 2) q^{37} + ( - 2 \beta - 1) q^{39} + ( - 3 \beta + 2) q^{41} - 2 q^{43} - 2 \beta q^{45} + (3 \beta - 3) q^{47} + (4 \beta + 7) q^{51} - 3 \beta q^{53} + ( - 2 \beta - 3) q^{55} + (6 \beta + 6) q^{57} + (3 \beta + 2) q^{59} - 2 \beta q^{61} + ( - \beta + 3) q^{65} + ( - 3 \beta + 4) q^{67} + ( - 5 \beta - 4) q^{69} + (2 \beta + 6) q^{71} - 6 \beta q^{73} + (\beta + 1) q^{75} + (6 \beta + 7) q^{79} + ( - 6 \beta - 1) q^{81} + ( - 3 \beta - 1) q^{85} + (\beta + 5) q^{87} + 8 q^{89} + 3 \beta q^{93} - 6 q^{95} + (3 \beta - 9) q^{97} + (6 \beta + 8) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 2 q^{5} + 6 q^{11} - 6 q^{13} - 2 q^{15} + 2 q^{17} + 12 q^{19} - 12 q^{23} + 2 q^{25} + 2 q^{27} - 6 q^{29} + 12 q^{31} + 14 q^{33} - 4 q^{37} - 2 q^{39} + 4 q^{41} - 4 q^{43} - 6 q^{47} + 14 q^{51} - 6 q^{55} + 12 q^{57} + 4 q^{59} + 6 q^{65} + 8 q^{67} - 8 q^{69} + 12 q^{71} + 2 q^{75} + 14 q^{79} - 2 q^{81} - 2 q^{85} + 10 q^{87} + 16 q^{89} - 12 q^{95} - 18 q^{97} + 16 q^{99}+O(q^{100}) \) 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
−1.41421
1.41421
0 −0.414214 0 −1.00000 0 0 0 −2.82843 0
1.2 0 2.41421 0 −1.00000 0 0 0 2.82843 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 3920.2.a.bw 2
4.b odd 2 1 245.2.a.e 2
7.b odd 2 1 3920.2.a.br 2
12.b even 2 1 2205.2.a.v 2
20.d odd 2 1 1225.2.a.r 2
20.e even 4 2 1225.2.b.i 4
28.d even 2 1 245.2.a.f yes 2
28.f even 6 2 245.2.e.f 4
28.g odd 6 2 245.2.e.g 4
84.h odd 2 1 2205.2.a.t 2
140.c even 2 1 1225.2.a.p 2
140.j odd 4 2 1225.2.b.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.2.a.e 2 4.b odd 2 1
245.2.a.f yes 2 28.d even 2 1
245.2.e.f 4 28.f even 6 2
245.2.e.g 4 28.g odd 6 2
1225.2.a.p 2 140.c even 2 1
1225.2.a.r 2 20.d odd 2 1
1225.2.b.i 4 20.e even 4 2
1225.2.b.j 4 140.j odd 4 2
2205.2.a.t 2 84.h odd 2 1
2205.2.a.v 2 12.b even 2 1
3920.2.a.br 2 7.b odd 2 1
3920.2.a.bw 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3920))\):

\( T_{3}^{2} - 2T_{3} - 1 \) Copy content Toggle raw display
\( T_{11}^{2} - 6T_{11} + 1 \) Copy content Toggle raw display
\( T_{13}^{2} + 6T_{13} + 7 \) Copy content Toggle raw display
\( T_{17}^{2} - 2T_{17} - 17 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 2T - 1 \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 6T + 1 \) Copy content Toggle raw display
$13$ \( T^{2} + 6T + 7 \) Copy content Toggle raw display
$17$ \( T^{2} - 2T - 17 \) Copy content Toggle raw display
$19$ \( (T - 6)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 12T + 34 \) Copy content Toggle raw display
$29$ \( T^{2} + 6T - 23 \) Copy content Toggle raw display
$31$ \( T^{2} - 12T + 18 \) Copy content Toggle raw display
$37$ \( T^{2} + 4T - 14 \) Copy content Toggle raw display
$41$ \( T^{2} - 4T - 14 \) Copy content Toggle raw display
$43$ \( (T + 2)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 6T - 9 \) Copy content Toggle raw display
$53$ \( T^{2} - 18 \) Copy content Toggle raw display
$59$ \( T^{2} - 4T - 14 \) Copy content Toggle raw display
$61$ \( T^{2} - 8 \) Copy content Toggle raw display
$67$ \( T^{2} - 8T - 2 \) Copy content Toggle raw display
$71$ \( T^{2} - 12T + 28 \) Copy content Toggle raw display
$73$ \( T^{2} - 72 \) Copy content Toggle raw display
$79$ \( T^{2} - 14T - 23 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( (T - 8)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 18T + 63 \) Copy content Toggle raw display
show more
show less