Properties

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

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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: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1960)
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.

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

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.by 2
4.b odd 2 1 1960.2.a.q 2
7.b odd 2 1 3920.2.a.bn 2
20.d odd 2 1 9800.2.a.ca 2
28.d even 2 1 1960.2.a.u yes 2
28.f even 6 2 1960.2.q.p 4
28.g odd 6 2 1960.2.q.v 4
140.c even 2 1 9800.2.a.bs 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1960.2.a.q 2 4.b odd 2 1
1960.2.a.u yes 2 28.d even 2 1
1960.2.q.p 4 28.f even 6 2
1960.2.q.v 4 28.g odd 6 2
3920.2.a.bn 2 7.b odd 2 1
3920.2.a.by 2 1.a even 1 1 trivial
9800.2.a.bs 2 140.c even 2 1
9800.2.a.ca 2 20.d odd 2 1

Hecke kernels

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

\( T_{3}^{2} - 2 T_{3} - 1 \)
\( T_{11} - 1 \)
\( T_{13}^{2} + 2 T_{13} - 1 \)
\( T_{17}^{2} + 2 T_{17} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -1 - 2 T + T^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( T^{2} \)
$11$ \( ( -1 + T )^{2} \)
$13$ \( -1 + 2 T + T^{2} \)
$17$ \( -1 + 2 T + T^{2} \)
$19$ \( ( -2 + T )^{2} \)
$23$ \( -14 + 4 T + T^{2} \)
$29$ \( ( -1 + T )^{2} \)
$31$ \( 18 - 12 T + T^{2} \)
$37$ \( -94 - 4 T + T^{2} \)
$41$ \( 34 + 12 T + T^{2} \)
$43$ \( 4 - 12 T + T^{2} \)
$47$ \( 79 - 18 T + T^{2} \)
$53$ \( 62 - 16 T + T^{2} \)
$59$ \( 18 - 12 T + T^{2} \)
$61$ \( 8 - 8 T + T^{2} \)
$67$ \( -2 + T^{2} \)
$71$ \( -36 + 12 T + T^{2} \)
$73$ \( 56 - 16 T + T^{2} \)
$79$ \( 49 - 18 T + T^{2} \)
$83$ \( -128 + T^{2} \)
$89$ \( -16 + 8 T + T^{2} \)
$97$ \( -1 + 14 T + T^{2} \)
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