Properties

Label 1960.2.a.p
Level $1960$
Weight $2$
Character orbit 1960.a
Self dual yes
Analytic conductor $15.651$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1960 = 2^{3} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1960.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(15.6506787962\)
Analytic rank: \(1\)
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 280)
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 \( q + (\beta - 1) q^{3} + q^{5} - 2 \beta q^{9} + (2 \beta - 2) q^{11} + 2 q^{13} + (\beta - 1) q^{15} + ( - 4 \beta - 2) q^{17} - 4 \beta q^{19} + (\beta - 7) q^{23} + q^{25} + ( - \beta - 1) q^{27} + ( - 2 \beta - 5) q^{29} + (2 \beta - 2) q^{31} + ( - 4 \beta + 6) q^{33} + 4 \beta q^{37} + (2 \beta - 2) q^{39} + (2 \beta + 3) q^{41} + ( - 7 \beta + 3) q^{43} - 2 \beta q^{45} + (4 \beta + 6) q^{47} + (2 \beta - 6) q^{51} - 4 \beta q^{53} + (2 \beta - 2) q^{55} + (4 \beta - 8) q^{57} - 4 q^{59} + (4 \beta + 1) q^{61} + 2 q^{65} + ( - 7 \beta - 3) q^{67} + ( - 8 \beta + 9) q^{69} - 12 q^{71} + (4 \beta - 2) q^{73} + (\beta - 1) q^{75} - 4 q^{79} + (6 \beta - 1) q^{81} + (3 \beta - 9) q^{83} + ( - 4 \beta - 2) q^{85} + ( - 3 \beta + 1) q^{87} + (4 \beta - 11) q^{89} + ( - 4 \beta + 6) q^{93} - 4 \beta q^{95} + 6 q^{97} + (4 \beta - 8) q^{99} +O(q^{100}) \) 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 \( 2 q - 2 q^{3} + 2 q^{5} - 4 q^{11} + 4 q^{13} - 2 q^{15} - 4 q^{17} - 14 q^{23} + 2 q^{25} - 2 q^{27} - 10 q^{29} - 4 q^{31} + 12 q^{33} - 4 q^{39} + 6 q^{41} + 6 q^{43} + 12 q^{47} - 12 q^{51} - 4 q^{55} - 16 q^{57} - 8 q^{59} + 2 q^{61} + 4 q^{65} - 6 q^{67} + 18 q^{69} - 24 q^{71} - 4 q^{73} - 2 q^{75} - 8 q^{79} - 2 q^{81} - 18 q^{83} - 4 q^{85} + 2 q^{87} - 22 q^{89} + 12 q^{93} + 12 q^{97} - 16 q^{99}+O(q^{100}) \) 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 −2.41421 0 1.00000 0 0 0 2.82843 0
1.2 0 0.414214 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 1960.2.a.p 2
4.b odd 2 1 3920.2.a.bz 2
5.b even 2 1 9800.2.a.bz 2
7.b odd 2 1 1960.2.a.t 2
7.c even 3 2 280.2.q.d 4
7.d odd 6 2 1960.2.q.q 4
21.h odd 6 2 2520.2.bi.k 4
28.d even 2 1 3920.2.a.bp 2
28.g odd 6 2 560.2.q.j 4
35.c odd 2 1 9800.2.a.br 2
35.j even 6 2 1400.2.q.h 4
35.l odd 12 4 1400.2.bh.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.q.d 4 7.c even 3 2
560.2.q.j 4 28.g odd 6 2
1400.2.q.h 4 35.j even 6 2
1400.2.bh.g 8 35.l odd 12 4
1960.2.a.p 2 1.a even 1 1 trivial
1960.2.a.t 2 7.b odd 2 1
1960.2.q.q 4 7.d odd 6 2
2520.2.bi.k 4 21.h odd 6 2
3920.2.a.bp 2 28.d even 2 1
3920.2.a.bz 2 4.b odd 2 1
9800.2.a.br 2 35.c odd 2 1
9800.2.a.bz 2 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1960))\):

\( T_{3}^{2} + 2T_{3} - 1 \) Copy content Toggle raw display
\( T_{11}^{2} + 4T_{11} - 4 \) Copy content Toggle raw display
\( T_{13} - 2 \) 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
$11$ \( T^{2} + 4T - 4 \) Copy content Toggle raw display
$13$ \( (T - 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 4T - 28 \) Copy content Toggle raw display
$19$ \( T^{2} - 32 \) Copy content Toggle raw display
$23$ \( T^{2} + 14T + 47 \) Copy content Toggle raw display
$29$ \( T^{2} + 10T + 17 \) Copy content Toggle raw display
$31$ \( T^{2} + 4T - 4 \) Copy content Toggle raw display
$37$ \( T^{2} - 32 \) Copy content Toggle raw display
$41$ \( T^{2} - 6T + 1 \) Copy content Toggle raw display
$43$ \( T^{2} - 6T - 89 \) Copy content Toggle raw display
$47$ \( T^{2} - 12T + 4 \) Copy content Toggle raw display
$53$ \( T^{2} - 32 \) Copy content Toggle raw display
$59$ \( (T + 4)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 2T - 31 \) Copy content Toggle raw display
$67$ \( T^{2} + 6T - 89 \) Copy content Toggle raw display
$71$ \( (T + 12)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 4T - 28 \) Copy content Toggle raw display
$79$ \( (T + 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 18T + 63 \) Copy content Toggle raw display
$89$ \( T^{2} + 22T + 89 \) Copy content Toggle raw display
$97$ \( (T - 6)^{2} \) Copy content Toggle raw display
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