Properties

Label 1960.2.q.o
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: \(\Z[a_1, \ldots, a_{5}]\)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 3 - 3 \zeta_{6} ) q^{3} -\zeta_{6} q^{5} -6 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( 3 - 3 \zeta_{6} ) q^{3} -\zeta_{6} q^{5} -6 \zeta_{6} q^{9} + ( 5 - 5 \zeta_{6} ) q^{11} -5 q^{13} -3 q^{15} + ( 7 - 7 \zeta_{6} ) q^{17} + 2 \zeta_{6} q^{19} + 2 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} -9 q^{27} + 7 q^{29} + ( -4 + 4 \zeta_{6} ) q^{31} -15 \zeta_{6} q^{33} + 6 \zeta_{6} q^{37} + ( -15 + 15 \zeta_{6} ) q^{39} -12 q^{41} -2 q^{43} + ( -6 + 6 \zeta_{6} ) q^{45} -\zeta_{6} q^{47} -21 \zeta_{6} q^{51} -5 q^{55} + 6 q^{57} + ( 4 - 4 \zeta_{6} ) q^{59} -4 \zeta_{6} q^{61} + 5 \zeta_{6} q^{65} + ( -8 + 8 \zeta_{6} ) q^{67} + 6 q^{69} + ( -6 + 6 \zeta_{6} ) q^{73} + 3 \zeta_{6} q^{75} + 3 \zeta_{6} q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} -4 q^{83} -7 q^{85} + ( 21 - 21 \zeta_{6} ) q^{87} + 12 \zeta_{6} q^{93} + ( 2 - 2 \zeta_{6} ) q^{95} + 13 q^{97} -30 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 3q^{3} - q^{5} - 6q^{9} + O(q^{10}) \) \( 2q + 3q^{3} - q^{5} - 6q^{9} + 5q^{11} - 10q^{13} - 6q^{15} + 7q^{17} + 2q^{19} + 2q^{23} - q^{25} - 18q^{27} + 14q^{29} - 4q^{31} - 15q^{33} + 6q^{37} - 15q^{39} - 24q^{41} - 4q^{43} - 6q^{45} - q^{47} - 21q^{51} - 10q^{55} + 12q^{57} + 4q^{59} - 4q^{61} + 5q^{65} - 8q^{67} + 12q^{69} - 6q^{73} + 3q^{75} + 3q^{79} - 9q^{81} - 8q^{83} - 14q^{85} + 21q^{87} + 12q^{93} + 2q^{95} + 26q^{97} - 60q^{99} + O(q^{100}) \)

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 1.50000 2.59808i 0 −0.500000 0.866025i 0 0 0 −3.00000 5.19615i 0
961.1 0 1.50000 + 2.59808i 0 −0.500000 + 0.866025i 0 0 0 −3.00000 + 5.19615i 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.o 2
7.b odd 2 1 1960.2.q.a 2
7.c even 3 1 280.2.a.a 1
7.c even 3 1 inner 1960.2.q.o 2
7.d odd 6 1 1960.2.a.o 1
7.d odd 6 1 1960.2.q.a 2
21.h odd 6 1 2520.2.a.i 1
28.f even 6 1 3920.2.a.c 1
28.g odd 6 1 560.2.a.f 1
35.i odd 6 1 9800.2.a.a 1
35.j even 6 1 1400.2.a.n 1
35.l odd 12 2 1400.2.g.a 2
56.k odd 6 1 2240.2.a.a 1
56.p even 6 1 2240.2.a.z 1
84.n even 6 1 5040.2.a.a 1
140.p odd 6 1 2800.2.a.c 1
140.w even 12 2 2800.2.g.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.a.a 1 7.c even 3 1
560.2.a.f 1 28.g odd 6 1
1400.2.a.n 1 35.j even 6 1
1400.2.g.a 2 35.l odd 12 2
1960.2.a.o 1 7.d odd 6 1
1960.2.q.a 2 7.b odd 2 1
1960.2.q.a 2 7.d odd 6 1
1960.2.q.o 2 1.a even 1 1 trivial
1960.2.q.o 2 7.c even 3 1 inner
2240.2.a.a 1 56.k odd 6 1
2240.2.a.z 1 56.p even 6 1
2520.2.a.i 1 21.h odd 6 1
2800.2.a.c 1 140.p odd 6 1
2800.2.g.b 2 140.w even 12 2
3920.2.a.c 1 28.f even 6 1
5040.2.a.a 1 84.n even 6 1
9800.2.a.a 1 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 \(S_{2}^{\mathrm{new}}(1960, [\chi])\):

\( T_{3}^{2} - 3 T_{3} + 9 \)
\( T_{11}^{2} - 5 T_{11} + 25 \)
\( T_{13} + 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 9 - 3 T + T^{2} \)
$5$ \( 1 + T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 25 - 5 T + T^{2} \)
$13$ \( ( 5 + T )^{2} \)
$17$ \( 49 - 7 T + T^{2} \)
$19$ \( 4 - 2 T + T^{2} \)
$23$ \( 4 - 2 T + T^{2} \)
$29$ \( ( -7 + T )^{2} \)
$31$ \( 16 + 4 T + T^{2} \)
$37$ \( 36 - 6 T + T^{2} \)
$41$ \( ( 12 + T )^{2} \)
$43$ \( ( 2 + T )^{2} \)
$47$ \( 1 + T + T^{2} \)
$53$ \( T^{2} \)
$59$ \( 16 - 4 T + T^{2} \)
$61$ \( 16 + 4 T + T^{2} \)
$67$ \( 64 + 8 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 36 + 6 T + T^{2} \)
$79$ \( 9 - 3 T + T^{2} \)
$83$ \( ( 4 + T )^{2} \)
$89$ \( T^{2} \)
$97$ \( ( -13 + T )^{2} \)
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