Properties

Label 1960.2.a.v
Level $1960$
Weight $2$
Character orbit 1960.a
Self dual yes
Analytic conductor $15.651$
Analytic rank $0$
Dimension $3$
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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1944.1
Defining polynomial: \(x^{3} - 9 x - 6\)
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} - q^{5} + ( 3 + \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} - q^{5} + ( 3 + \beta_{1} + \beta_{2} ) q^{9} + ( 1 + \beta_{1} + \beta_{2} ) q^{11} + ( -1 - \beta_{1} + \beta_{2} ) q^{13} -\beta_{1} q^{15} -2 q^{17} + ( -1 + \beta_{1} - \beta_{2} ) q^{19} + ( 1 - \beta_{2} ) q^{23} + q^{25} + ( 6 + 3 \beta_{1} ) q^{27} + ( 4 - \beta_{1} - \beta_{2} ) q^{29} + ( -4 + 2 \beta_{1} ) q^{31} + ( 6 + 4 \beta_{1} ) q^{33} + ( 3 + \beta_{1} - \beta_{2} ) q^{37} + ( -6 - 2 \beta_{2} ) q^{39} + ( -3 - 2 \beta_{2} ) q^{41} + ( 4 - \beta_{1} ) q^{43} + ( -3 - \beta_{1} - \beta_{2} ) q^{45} + ( 5 + \beta_{1} + 3 \beta_{2} ) q^{47} -2 \beta_{1} q^{51} + ( 3 - 3 \beta_{1} - \beta_{2} ) q^{53} + ( -1 - \beta_{1} - \beta_{2} ) q^{55} + ( 6 - 2 \beta_{1} + 2 \beta_{2} ) q^{57} -8 q^{59} + ( 2 - 3 \beta_{1} - \beta_{2} ) q^{61} + ( 1 + \beta_{1} - \beta_{2} ) q^{65} + ( 2 + 3 \beta_{1} ) q^{67} + ( -\beta_{1} + \beta_{2} ) q^{69} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{73} + \beta_{1} q^{75} + ( -6 - 2 \beta_{1} + 2 \beta_{2} ) q^{79} + ( 9 + 6 \beta_{1} ) q^{81} + ( 10 - \beta_{1} ) q^{83} + 2 q^{85} + ( -6 + \beta_{1} ) q^{87} + ( -\beta_{1} + \beta_{2} ) q^{89} + ( 12 - 2 \beta_{1} + 2 \beta_{2} ) q^{93} + ( 1 - \beta_{1} + \beta_{2} ) q^{95} + 2 q^{97} + ( 21 + 7 \beta_{1} + \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{5} + 9q^{9} + O(q^{10}) \) \( 3q - 3q^{5} + 9q^{9} + 3q^{11} - 3q^{13} - 6q^{17} - 3q^{19} + 3q^{23} + 3q^{25} + 18q^{27} + 12q^{29} - 12q^{31} + 18q^{33} + 9q^{37} - 18q^{39} - 9q^{41} + 12q^{43} - 9q^{45} + 15q^{47} + 9q^{53} - 3q^{55} + 18q^{57} - 24q^{59} + 6q^{61} + 3q^{65} + 6q^{67} - 18q^{79} + 27q^{81} + 30q^{83} + 6q^{85} - 18q^{87} + 36q^{93} + 3q^{95} + 6q^{97} + 63q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 9 x - 6\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 6 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 6\)

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
−2.58423
−0.705720
3.28995
0 −2.58423 0 −1.00000 0 0 0 3.67822 0
1.2 0 −0.705720 0 −1.00000 0 0 0 −2.50196 0
1.3 0 3.28995 0 −1.00000 0 0 0 7.82374 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.v 3
4.b odd 2 1 3920.2.a.cb 3
5.b even 2 1 9800.2.a.cf 3
7.b odd 2 1 1960.2.a.w 3
7.c even 3 2 1960.2.q.w 6
7.d odd 6 2 280.2.q.e 6
21.g even 6 2 2520.2.bi.q 6
28.d even 2 1 3920.2.a.cc 3
28.f even 6 2 560.2.q.l 6
35.c odd 2 1 9800.2.a.ce 3
35.i odd 6 2 1400.2.q.j 6
35.k even 12 4 1400.2.bh.i 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.q.e 6 7.d odd 6 2
560.2.q.l 6 28.f even 6 2
1400.2.q.j 6 35.i odd 6 2
1400.2.bh.i 12 35.k even 12 4
1960.2.a.v 3 1.a even 1 1 trivial
1960.2.a.w 3 7.b odd 2 1
1960.2.q.w 6 7.c even 3 2
2520.2.bi.q 6 21.g even 6 2
3920.2.a.cb 3 4.b odd 2 1
3920.2.a.cc 3 28.d even 2 1
9800.2.a.ce 3 35.c odd 2 1
9800.2.a.cf 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1960))\):

\( T_{3}^{3} - 9 T_{3} - 6 \)
\( T_{11}^{3} - 3 T_{11}^{2} - 24 T_{11} + 44 \)
\( T_{13}^{3} + 3 T_{13}^{2} - 24 T_{13} - 68 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( -6 - 9 T + T^{3} \)
$5$ \( ( 1 + T )^{3} \)
$7$ \( T^{3} \)
$11$ \( 44 - 24 T - 3 T^{2} + T^{3} \)
$13$ \( -68 - 24 T + 3 T^{2} + T^{3} \)
$17$ \( ( 2 + T )^{3} \)
$19$ \( 16 - 24 T + 3 T^{2} + T^{3} \)
$23$ \( -7 - 15 T - 3 T^{2} + T^{3} \)
$29$ \( 26 + 21 T - 12 T^{2} + T^{3} \)
$31$ \( -128 + 12 T + 12 T^{2} + T^{3} \)
$37$ \( 96 - 9 T^{2} + T^{3} \)
$41$ \( -381 - 45 T + 9 T^{2} + T^{3} \)
$43$ \( -22 + 39 T - 12 T^{2} + T^{3} \)
$47$ \( 1588 - 96 T - 15 T^{2} + T^{3} \)
$53$ \( 624 - 72 T - 9 T^{2} + T^{3} \)
$59$ \( ( 8 + T )^{3} \)
$61$ \( 544 - 87 T - 6 T^{2} + T^{3} \)
$67$ \( -8 - 69 T - 6 T^{2} + T^{3} \)
$71$ \( T^{3} \)
$73$ \( -336 - 108 T + T^{3} \)
$79$ \( -768 + 18 T^{2} + T^{3} \)
$83$ \( -904 + 291 T - 30 T^{2} + T^{3} \)
$89$ \( -42 - 27 T + T^{3} \)
$97$ \( ( -2 + T )^{3} \)
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