Properties

Label 1960.2.q.t
Level $1960$
Weight $2$
Character orbit 1960.q
Analytic conductor $15.651$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1960 = 2^{3} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1960.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(15.6506787962\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Defining polynomial: \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 280)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + ( 1 - \beta_{1} ) q^{5} + ( -6 + 5 \beta_{1} - \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{3} q^{3} + ( 1 - \beta_{1} ) q^{5} + ( -6 + 5 \beta_{1} - \beta_{2} + \beta_{3} ) q^{9} + ( -4 \beta_{1} + \beta_{3} ) q^{11} + ( 1 - \beta_{2} ) q^{13} + ( 1 + \beta_{2} ) q^{15} + ( -2 \beta_{1} - \beta_{3} ) q^{17} + ( -2 - 2 \beta_{2} + 2 \beta_{3} ) q^{19} + ( -2 - 2 \beta_{2} + 2 \beta_{3} ) q^{23} -\beta_{1} q^{25} + ( -11 - 3 \beta_{2} ) q^{27} + ( -1 + \beta_{2} ) q^{29} + 8 \beta_{1} q^{31} + ( -5 + 8 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{33} + ( 2 - 2 \beta_{1} ) q^{37} + ( -8 \beta_{1} + \beta_{3} ) q^{39} -2 \beta_{2} q^{41} + ( -2 + 2 \beta_{2} ) q^{43} + ( 5 \beta_{1} + \beta_{3} ) q^{45} + ( -3 - 3 \beta_{2} + 3 \beta_{3} ) q^{47} + ( 11 - 8 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{51} + ( -6 \beta_{1} + 2 \beta_{3} ) q^{53} + ( -3 + \beta_{2} ) q^{55} + ( -18 - 2 \beta_{2} ) q^{57} -8 \beta_{1} q^{59} + ( -4 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{61} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{65} + 4 \beta_{1} q^{67} + ( -18 - 2 \beta_{2} ) q^{69} + 8 q^{71} + 6 \beta_{1} q^{73} + ( 1 + \beta_{2} - \beta_{3} ) q^{75} + ( -5 + 8 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{79} + ( -9 \beta_{1} - 8 \beta_{3} ) q^{81} + ( 4 + 4 \beta_{2} ) q^{83} + ( -3 - \beta_{2} ) q^{85} + ( 8 \beta_{1} - \beta_{3} ) q^{87} + ( -8 + 10 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{89} + ( -8 - 8 \beta_{2} + 8 \beta_{3} ) q^{93} + 2 \beta_{3} q^{95} + ( 7 + 5 \beta_{2} ) q^{97} + ( 10 - 2 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + q^{3} + 2q^{5} - 11q^{9} + O(q^{10}) \) \( 4q + q^{3} + 2q^{5} - 11q^{9} - 7q^{11} + 6q^{13} + 2q^{15} - 5q^{17} - 2q^{19} - 2q^{23} - 2q^{25} - 38q^{27} - 6q^{29} + 16q^{31} - 13q^{33} + 4q^{37} - 15q^{39} + 4q^{41} - 12q^{43} + 11q^{45} - 3q^{47} + 19q^{51} - 10q^{53} - 14q^{55} - 68q^{57} - 16q^{59} - 6q^{61} + 3q^{65} + 8q^{67} - 68q^{69} + 32q^{71} + 12q^{73} + q^{75} - 13q^{79} - 26q^{81} + 8q^{83} - 10q^{85} + 15q^{87} - 18q^{89} - 8q^{93} + 2q^{95} + 18q^{97} + 44q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 2 \nu^{2} - 2 \nu - 3 \)\()/6\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + \nu^{2} + 5 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{3} + \nu^{2} + 2 \nu - 9 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} - 2 \beta_{1} + 2\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + 2 \beta_{2} + 8 \beta_{1} + 1\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(4 \beta_{3} - 2 \beta_{2} - 2 \beta_{1} + 11\)\()/3\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1960\mathbb{Z}\right)^\times\).

\(n\) \(981\) \(1081\) \(1177\) \(1471\)
\(\chi(n)\) \(1\) \(-1 + \beta_{1}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
−1.18614 + 1.26217i
1.68614 0.396143i
−1.18614 1.26217i
1.68614 + 0.396143i
0 −1.18614 + 2.05446i 0 0.500000 + 0.866025i 0 0 0 −1.31386 2.27567i 0
361.2 0 1.68614 2.92048i 0 0.500000 + 0.866025i 0 0 0 −4.18614 7.25061i 0
961.1 0 −1.18614 2.05446i 0 0.500000 0.866025i 0 0 0 −1.31386 + 2.27567i 0
961.2 0 1.68614 + 2.92048i 0 0.500000 0.866025i 0 0 0 −4.18614 + 7.25061i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1960.2.q.t 4
7.b odd 2 1 1960.2.q.r 4
7.c even 3 1 280.2.a.c 2
7.c even 3 1 inner 1960.2.q.t 4
7.d odd 6 1 1960.2.a.s 2
7.d odd 6 1 1960.2.q.r 4
21.h odd 6 1 2520.2.a.x 2
28.f even 6 1 3920.2.a.bt 2
28.g odd 6 1 560.2.a.h 2
35.i odd 6 1 9800.2.a.bu 2
35.j even 6 1 1400.2.a.r 2
35.l odd 12 2 1400.2.g.i 4
56.k odd 6 1 2240.2.a.bg 2
56.p even 6 1 2240.2.a.bk 2
84.n even 6 1 5040.2.a.by 2
140.p odd 6 1 2800.2.a.bk 2
140.w even 12 2 2800.2.g.r 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.a.c 2 7.c even 3 1
560.2.a.h 2 28.g odd 6 1
1400.2.a.r 2 35.j even 6 1
1400.2.g.i 4 35.l odd 12 2
1960.2.a.s 2 7.d odd 6 1
1960.2.q.r 4 7.b odd 2 1
1960.2.q.r 4 7.d odd 6 1
1960.2.q.t 4 1.a even 1 1 trivial
1960.2.q.t 4 7.c even 3 1 inner
2240.2.a.bg 2 56.k odd 6 1
2240.2.a.bk 2 56.p even 6 1
2520.2.a.x 2 21.h odd 6 1
2800.2.a.bk 2 140.p odd 6 1
2800.2.g.r 4 140.w even 12 2
3920.2.a.bt 2 28.f even 6 1
5040.2.a.by 2 84.n even 6 1
9800.2.a.bu 2 35.i odd 6 1

Hecke kernels

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

\( T_{3}^{4} - T_{3}^{3} + 9 T_{3}^{2} + 8 T_{3} + 64 \)
\( T_{11}^{4} + 7 T_{11}^{3} + 45 T_{11}^{2} + 28 T_{11} + 16 \)
\( T_{13}^{2} - 3 T_{13} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 64 + 8 T + 9 T^{2} - T^{3} + T^{4} \)
$5$ \( ( 1 - T + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( 16 + 28 T + 45 T^{2} + 7 T^{3} + T^{4} \)
$13$ \( ( -6 - 3 T + T^{2} )^{2} \)
$17$ \( 4 - 10 T + 27 T^{2} + 5 T^{3} + T^{4} \)
$19$ \( 1024 - 64 T + 36 T^{2} + 2 T^{3} + T^{4} \)
$23$ \( 1024 - 64 T + 36 T^{2} + 2 T^{3} + T^{4} \)
$29$ \( ( -6 + 3 T + T^{2} )^{2} \)
$31$ \( ( 64 - 8 T + T^{2} )^{2} \)
$37$ \( ( 4 - 2 T + T^{2} )^{2} \)
$41$ \( ( -32 - 2 T + T^{2} )^{2} \)
$43$ \( ( -24 + 6 T + T^{2} )^{2} \)
$47$ \( 5184 - 216 T + 81 T^{2} + 3 T^{3} + T^{4} \)
$53$ \( 64 - 80 T + 108 T^{2} + 10 T^{3} + T^{4} \)
$59$ \( ( 64 + 8 T + T^{2} )^{2} \)
$61$ \( 576 - 144 T + 60 T^{2} + 6 T^{3} + T^{4} \)
$67$ \( ( 16 - 4 T + T^{2} )^{2} \)
$71$ \( ( -8 + T )^{4} \)
$73$ \( ( 36 - 6 T + T^{2} )^{2} \)
$79$ \( 1024 - 416 T + 201 T^{2} + 13 T^{3} + T^{4} \)
$83$ \( ( -128 - 4 T + T^{2} )^{2} \)
$89$ \( 2304 + 864 T + 276 T^{2} + 18 T^{3} + T^{4} \)
$97$ \( ( -186 - 9 T + T^{2} )^{2} \)
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