Properties

Label 1960.2.q.j
Level $1960$
Weight $2$
Character orbit 1960.q
Analytic conductor $15.651$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{6} ) q^{3} -\zeta_{6} q^{5} + 2 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( 1 - \zeta_{6} ) q^{3} -\zeta_{6} q^{5} + 2 \zeta_{6} q^{9} + ( -3 + 3 \zeta_{6} ) q^{11} + q^{13} - q^{15} + ( 5 - 5 \zeta_{6} ) q^{17} + 6 \zeta_{6} q^{19} + ( -1 + \zeta_{6} ) q^{25} + 5 q^{27} -5 q^{29} + ( -2 + 2 \zeta_{6} ) q^{31} + 3 \zeta_{6} q^{33} + 4 \zeta_{6} q^{37} + ( 1 - \zeta_{6} ) q^{39} -2 q^{41} + 10 q^{43} + ( 2 - 2 \zeta_{6} ) q^{45} + 9 \zeta_{6} q^{47} -5 \zeta_{6} q^{51} + ( -6 + 6 \zeta_{6} ) q^{53} + 3 q^{55} + 6 q^{57} + ( 6 - 6 \zeta_{6} ) q^{59} + 12 \zeta_{6} q^{61} -\zeta_{6} q^{65} + ( 2 - 2 \zeta_{6} ) q^{67} + ( 14 - 14 \zeta_{6} ) q^{73} + \zeta_{6} q^{75} -\zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} + 12 q^{83} -5 q^{85} + ( -5 + 5 \zeta_{6} ) q^{87} + 2 \zeta_{6} q^{93} + ( 6 - 6 \zeta_{6} ) q^{95} -9 q^{97} -6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} - q^{5} + 2q^{9} + O(q^{10}) \) \( 2q + q^{3} - q^{5} + 2q^{9} - 3q^{11} + 2q^{13} - 2q^{15} + 5q^{17} + 6q^{19} - q^{25} + 10q^{27} - 10q^{29} - 2q^{31} + 3q^{33} + 4q^{37} + q^{39} - 4q^{41} + 20q^{43} + 2q^{45} + 9q^{47} - 5q^{51} - 6q^{53} + 6q^{55} + 12q^{57} + 6q^{59} + 12q^{61} - q^{65} + 2q^{67} + 14q^{73} + q^{75} - q^{79} - q^{81} + 24q^{83} - 10q^{85} - 5q^{87} + 2q^{93} + 6q^{95} - 18q^{97} - 12q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(981\) \(1081\) \(1177\) \(1471\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(1\) \(1\)

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
0.500000 + 0.866025i
0.500000 0.866025i
0 0.500000 0.866025i 0 −0.500000 0.866025i 0 0 0 1.00000 + 1.73205i 0
961.1 0 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 0 0 1.00000 1.73205i 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.j 2
7.b odd 2 1 1960.2.q.f 2
7.c even 3 1 1960.2.a.f 1
7.c even 3 1 inner 1960.2.q.j 2
7.d odd 6 1 1960.2.a.j yes 1
7.d odd 6 1 1960.2.q.f 2
28.f even 6 1 3920.2.a.l 1
28.g odd 6 1 3920.2.a.x 1
35.i odd 6 1 9800.2.a.t 1
35.j even 6 1 9800.2.a.bd 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1960.2.a.f 1 7.c even 3 1
1960.2.a.j yes 1 7.d odd 6 1
1960.2.q.f 2 7.b odd 2 1
1960.2.q.f 2 7.d odd 6 1
1960.2.q.j 2 1.a even 1 1 trivial
1960.2.q.j 2 7.c even 3 1 inner
3920.2.a.l 1 28.f even 6 1
3920.2.a.x 1 28.g odd 6 1
9800.2.a.t 1 35.i odd 6 1
9800.2.a.bd 1 35.j even 6 1

Hecke kernels

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

\( T_{3}^{2} - T_{3} + 1 \)
\( T_{11}^{2} + 3 T_{11} + 9 \)
\( T_{13} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 - T + T^{2} \)
$5$ \( 1 + T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 9 + 3 T + T^{2} \)
$13$ \( ( -1 + T )^{2} \)
$17$ \( 25 - 5 T + T^{2} \)
$19$ \( 36 - 6 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( ( 5 + T )^{2} \)
$31$ \( 4 + 2 T + T^{2} \)
$37$ \( 16 - 4 T + T^{2} \)
$41$ \( ( 2 + T )^{2} \)
$43$ \( ( -10 + T )^{2} \)
$47$ \( 81 - 9 T + T^{2} \)
$53$ \( 36 + 6 T + T^{2} \)
$59$ \( 36 - 6 T + T^{2} \)
$61$ \( 144 - 12 T + T^{2} \)
$67$ \( 4 - 2 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 196 - 14 T + T^{2} \)
$79$ \( 1 + T + T^{2} \)
$83$ \( ( -12 + T )^{2} \)
$89$ \( T^{2} \)
$97$ \( ( 9 + T )^{2} \)
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