Properties

Label 1960.2.q.q
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: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} - \beta_{2} ) q^{3} -\beta_{2} q^{5} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{1} - \beta_{2} ) q^{3} -\beta_{2} q^{5} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{9} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{11} -2 q^{13} + ( -1 - \beta_{3} ) q^{15} + ( -2 - 4 \beta_{1} - 2 \beta_{2} ) q^{17} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{19} + ( \beta_{1} - 7 \beta_{2} + \beta_{3} ) q^{23} + ( -1 - \beta_{2} ) q^{25} + ( 1 - \beta_{3} ) q^{27} + ( -5 + 2 \beta_{3} ) q^{29} + ( -2 + 2 \beta_{1} - 2 \beta_{2} ) q^{31} + ( 4 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} ) q^{33} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{37} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{39} + ( -3 + 2 \beta_{3} ) q^{41} + ( 3 + 7 \beta_{3} ) q^{43} -2 \beta_{1} q^{45} + ( -4 \beta_{1} - 6 \beta_{2} - 4 \beta_{3} ) q^{47} + ( 2 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} ) q^{51} + 4 \beta_{1} q^{53} + ( 2 + 2 \beta_{3} ) q^{55} + ( -8 - 4 \beta_{3} ) q^{57} + ( -4 - 4 \beta_{2} ) q^{59} + ( -4 \beta_{1} - \beta_{2} - 4 \beta_{3} ) q^{61} + 2 \beta_{2} q^{65} + ( 3 + 7 \beta_{1} + 3 \beta_{2} ) q^{67} + ( -9 - 8 \beta_{3} ) q^{69} -12 q^{71} + ( -2 + 4 \beta_{1} - 2 \beta_{2} ) q^{73} + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{75} -4 \beta_{2} q^{79} + ( 1 - 6 \beta_{1} + \beta_{2} ) q^{81} + ( 9 + 3 \beta_{3} ) q^{83} + ( -2 + 4 \beta_{3} ) q^{85} + ( 1 - 3 \beta_{1} + \beta_{2} ) q^{87} + ( -4 \beta_{1} + 11 \beta_{2} - 4 \beta_{3} ) q^{89} + ( -4 \beta_{1} + 6 \beta_{2} - 4 \beta_{3} ) q^{93} + 4 \beta_{1} q^{95} -6 q^{97} + ( -8 - 4 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{3} + 2q^{5} + O(q^{10}) \) \( 4q - 2q^{3} + 2q^{5} + 4q^{11} - 8q^{13} - 4q^{15} - 4q^{17} + 14q^{23} - 2q^{25} + 4q^{27} - 20q^{29} - 4q^{31} + 12q^{33} + 4q^{39} - 12q^{41} + 12q^{43} + 12q^{47} + 12q^{51} + 8q^{55} - 32q^{57} - 8q^{59} + 2q^{61} - 4q^{65} + 6q^{67} - 36q^{69} - 48q^{71} - 4q^{73} - 2q^{75} + 8q^{79} + 2q^{81} + 36q^{83} - 8q^{85} + 2q^{87} - 22q^{89} - 12q^{93} - 24q^{97} - 32q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)

Character values

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

\(n\) \(981\) \(1081\) \(1177\) \(1471\)
\(\chi(n)\) \(1\) \(\beta_{2}\) \(1\) \(1\)

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
−0.707107 + 1.22474i
0.707107 1.22474i
−0.707107 1.22474i
0.707107 + 1.22474i
0 −1.20711 + 2.09077i 0 0.500000 + 0.866025i 0 0 0 −1.41421 2.44949i 0
361.2 0 0.207107 0.358719i 0 0.500000 + 0.866025i 0 0 0 1.41421 + 2.44949i 0
961.1 0 −1.20711 2.09077i 0 0.500000 0.866025i 0 0 0 −1.41421 + 2.44949i 0
961.2 0 0.207107 + 0.358719i 0 0.500000 0.866025i 0 0 0 1.41421 2.44949i 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.q 4
7.b odd 2 1 280.2.q.d 4
7.c even 3 1 1960.2.a.t 2
7.c even 3 1 inner 1960.2.q.q 4
7.d odd 6 1 280.2.q.d 4
7.d odd 6 1 1960.2.a.p 2
21.c even 2 1 2520.2.bi.k 4
21.g even 6 1 2520.2.bi.k 4
28.d even 2 1 560.2.q.j 4
28.f even 6 1 560.2.q.j 4
28.f even 6 1 3920.2.a.bz 2
28.g odd 6 1 3920.2.a.bp 2
35.c odd 2 1 1400.2.q.h 4
35.f even 4 2 1400.2.bh.g 8
35.i odd 6 1 1400.2.q.h 4
35.i odd 6 1 9800.2.a.bz 2
35.j even 6 1 9800.2.a.br 2
35.k even 12 2 1400.2.bh.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.q.d 4 7.b odd 2 1
280.2.q.d 4 7.d odd 6 1
560.2.q.j 4 28.d even 2 1
560.2.q.j 4 28.f even 6 1
1400.2.q.h 4 35.c odd 2 1
1400.2.q.h 4 35.i odd 6 1
1400.2.bh.g 8 35.f even 4 2
1400.2.bh.g 8 35.k even 12 2
1960.2.a.p 2 7.d odd 6 1
1960.2.a.t 2 7.c even 3 1
1960.2.q.q 4 1.a even 1 1 trivial
1960.2.q.q 4 7.c even 3 1 inner
2520.2.bi.k 4 21.c even 2 1
2520.2.bi.k 4 21.g even 6 1
3920.2.a.bp 2 28.g odd 6 1
3920.2.a.bz 2 28.f even 6 1
9800.2.a.br 2 35.j even 6 1
9800.2.a.bz 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 \(S_{2}^{\mathrm{new}}(1960, [\chi])\):

\( T_{3}^{4} + 2 T_{3}^{3} + 5 T_{3}^{2} - 2 T_{3} + 1 \)
\( T_{11}^{4} - 4 T_{11}^{3} + 20 T_{11}^{2} + 16 T_{11} + 16 \)
\( T_{13} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 1 - 2 T + 5 T^{2} + 2 T^{3} + T^{4} \)
$5$ \( ( 1 - T + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( 16 + 16 T + 20 T^{2} - 4 T^{3} + T^{4} \)
$13$ \( ( 2 + T )^{4} \)
$17$ \( 784 - 112 T + 44 T^{2} + 4 T^{3} + T^{4} \)
$19$ \( 1024 + 32 T^{2} + T^{4} \)
$23$ \( 2209 - 658 T + 149 T^{2} - 14 T^{3} + T^{4} \)
$29$ \( ( 17 + 10 T + T^{2} )^{2} \)
$31$ \( 16 - 16 T + 20 T^{2} + 4 T^{3} + T^{4} \)
$37$ \( 1024 + 32 T^{2} + T^{4} \)
$41$ \( ( 1 + 6 T + T^{2} )^{2} \)
$43$ \( ( -89 - 6 T + T^{2} )^{2} \)
$47$ \( 16 - 48 T + 140 T^{2} - 12 T^{3} + T^{4} \)
$53$ \( 1024 + 32 T^{2} + T^{4} \)
$59$ \( ( 16 + 4 T + T^{2} )^{2} \)
$61$ \( 961 + 62 T + 35 T^{2} - 2 T^{3} + T^{4} \)
$67$ \( 7921 + 534 T + 125 T^{2} - 6 T^{3} + T^{4} \)
$71$ \( ( 12 + T )^{4} \)
$73$ \( 784 - 112 T + 44 T^{2} + 4 T^{3} + T^{4} \)
$79$ \( ( 16 - 4 T + T^{2} )^{2} \)
$83$ \( ( 63 - 18 T + T^{2} )^{2} \)
$89$ \( 7921 + 1958 T + 395 T^{2} + 22 T^{3} + T^{4} \)
$97$ \( ( 6 + T )^{4} \)
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