Properties

Label 1960.2.a.t
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: \(x^{2} - 2\)
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.

\(f(q)\) \(=\) \( q + ( 1 + \beta ) q^{3} - q^{5} + 2 \beta q^{9} +O(q^{10})\) \( q + ( 1 + \beta ) q^{3} - q^{5} + 2 \beta q^{9} + ( -2 - 2 \beta ) q^{11} -2 q^{13} + ( -1 - \beta ) q^{15} + ( 2 - 4 \beta ) q^{17} -4 \beta q^{19} + ( -7 - \beta ) q^{23} + q^{25} + ( 1 - \beta ) q^{27} + ( -5 + 2 \beta ) q^{29} + ( 2 + 2 \beta ) q^{31} + ( -6 - 4 \beta ) q^{33} -4 \beta q^{37} + ( -2 - 2 \beta ) q^{39} + ( -3 + 2 \beta ) q^{41} + ( 3 + 7 \beta ) q^{43} -2 \beta q^{45} + ( -6 + 4 \beta ) q^{47} + ( -6 - 2 \beta ) q^{51} + 4 \beta q^{53} + ( 2 + 2 \beta ) q^{55} + ( -8 - 4 \beta ) q^{57} + 4 q^{59} + ( -1 + 4 \beta ) q^{61} + 2 q^{65} + ( -3 + 7 \beta ) q^{67} + ( -9 - 8 \beta ) q^{69} -12 q^{71} + ( 2 + 4 \beta ) q^{73} + ( 1 + \beta ) q^{75} -4 q^{79} + ( -1 - 6 \beta ) q^{81} + ( 9 + 3 \beta ) q^{83} + ( -2 + 4 \beta ) q^{85} + ( -1 - 3 \beta ) q^{87} + ( 11 + 4 \beta ) q^{89} + ( 6 + 4 \beta ) q^{93} + 4 \beta q^{95} -6 q^{97} + ( -8 - 4 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 2q^{5} + O(q^{10}) \) \( 2q + 2q^{3} - 2q^{5} - 4q^{11} - 4q^{13} - 2q^{15} + 4q^{17} - 14q^{23} + 2q^{25} + 2q^{27} - 10q^{29} + 4q^{31} - 12q^{33} - 4q^{39} - 6q^{41} + 6q^{43} - 12q^{47} - 12q^{51} + 4q^{55} - 16q^{57} + 8q^{59} - 2q^{61} + 4q^{65} - 6q^{67} - 18q^{69} - 24q^{71} + 4q^{73} + 2q^{75} - 8q^{79} - 2q^{81} + 18q^{83} - 4q^{85} - 2q^{87} + 22q^{89} + 12q^{93} - 12q^{97} - 16q^{99} + O(q^{100}) \)

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 1960.2.a.t 2
4.b odd 2 1 3920.2.a.bp 2
5.b even 2 1 9800.2.a.br 2
7.b odd 2 1 1960.2.a.p 2
7.c even 3 2 1960.2.q.q 4
7.d odd 6 2 280.2.q.d 4
21.g even 6 2 2520.2.bi.k 4
28.d even 2 1 3920.2.a.bz 2
28.f even 6 2 560.2.q.j 4
35.c odd 2 1 9800.2.a.bz 2
35.i odd 6 2 1400.2.q.h 4
35.k even 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.d odd 6 2
560.2.q.j 4 28.f even 6 2
1400.2.q.h 4 35.i odd 6 2
1400.2.bh.g 8 35.k even 12 4
1960.2.a.p 2 7.b odd 2 1
1960.2.a.t 2 1.a even 1 1 trivial
1960.2.q.q 4 7.c even 3 2
2520.2.bi.k 4 21.g even 6 2
3920.2.a.bp 2 4.b odd 2 1
3920.2.a.bz 2 28.d even 2 1
9800.2.a.br 2 5.b even 2 1
9800.2.a.bz 2 35.c odd 2 1

Hecke kernels

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

\( T_{3}^{2} - 2 T_{3} - 1 \)
\( T_{11}^{2} + 4 T_{11} - 4 \)
\( T_{13} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -1 - 2 T + T^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( T^{2} \)
$11$ \( -4 + 4 T + T^{2} \)
$13$ \( ( 2 + T )^{2} \)
$17$ \( -28 - 4 T + T^{2} \)
$19$ \( -32 + T^{2} \)
$23$ \( 47 + 14 T + T^{2} \)
$29$ \( 17 + 10 T + T^{2} \)
$31$ \( -4 - 4 T + T^{2} \)
$37$ \( -32 + T^{2} \)
$41$ \( 1 + 6 T + T^{2} \)
$43$ \( -89 - 6 T + T^{2} \)
$47$ \( 4 + 12 T + T^{2} \)
$53$ \( -32 + T^{2} \)
$59$ \( ( -4 + T )^{2} \)
$61$ \( -31 + 2 T + T^{2} \)
$67$ \( -89 + 6 T + T^{2} \)
$71$ \( ( 12 + T )^{2} \)
$73$ \( -28 - 4 T + T^{2} \)
$79$ \( ( 4 + T )^{2} \)
$83$ \( 63 - 18 T + T^{2} \)
$89$ \( 89 - 22 T + T^{2} \)
$97$ \( ( 6 + T )^{2} \)
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