Properties

Label 1960.2.q.e
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: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 1) q^{3} - \zeta_{6} q^{5} + 2 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} - 1) q^{3} - \zeta_{6} q^{5} + 2 \zeta_{6} q^{9} + ( - 5 \zeta_{6} + 5) q^{11} - q^{13} + q^{15} + ( - 3 \zeta_{6} + 3) q^{17} - 6 \zeta_{6} q^{19} + 6 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} - 5 q^{27} - 9 q^{29} + 5 \zeta_{6} q^{33} - 6 \zeta_{6} q^{37} + ( - \zeta_{6} + 1) q^{39} - 8 q^{41} + 6 q^{43} + ( - 2 \zeta_{6} + 2) q^{45} + 3 \zeta_{6} q^{47} + 3 \zeta_{6} q^{51} + ( - 12 \zeta_{6} + 12) q^{53} - 5 q^{55} + 6 q^{57} + ( - 8 \zeta_{6} + 8) q^{59} - 4 \zeta_{6} q^{61} + \zeta_{6} q^{65} + ( - 4 \zeta_{6} + 4) q^{67} - 6 q^{69} + 8 q^{71} + ( - 10 \zeta_{6} + 10) q^{73} - \zeta_{6} q^{75} + 3 \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{81} + 12 q^{83} - 3 q^{85} + ( - 9 \zeta_{6} + 9) q^{87} - 16 \zeta_{6} q^{89} + (6 \zeta_{6} - 6) q^{95} - 7 q^{97} + 10 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} - q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} - q^{5} + 2 q^{9} + 5 q^{11} - 2 q^{13} + 2 q^{15} + 3 q^{17} - 6 q^{19} + 6 q^{23} - q^{25} - 10 q^{27} - 18 q^{29} + 5 q^{33} - 6 q^{37} + q^{39} - 16 q^{41} + 12 q^{43} + 2 q^{45} + 3 q^{47} + 3 q^{51} + 12 q^{53} - 10 q^{55} + 12 q^{57} + 8 q^{59} - 4 q^{61} + q^{65} + 4 q^{67} - 12 q^{69} + 16 q^{71} + 10 q^{73} - q^{75} + 3 q^{79} - q^{81} + 24 q^{83} - 6 q^{85} + 9 q^{87} - 16 q^{89} - 6 q^{95} - 14 q^{97} + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.e 2
7.b odd 2 1 1960.2.q.m 2
7.c even 3 1 1960.2.a.k 1
7.c even 3 1 inner 1960.2.q.e 2
7.d odd 6 1 280.2.a.b 1
7.d odd 6 1 1960.2.q.m 2
21.g even 6 1 2520.2.a.p 1
28.f even 6 1 560.2.a.e 1
28.g odd 6 1 3920.2.a.r 1
35.i odd 6 1 1400.2.a.k 1
35.j even 6 1 9800.2.a.n 1
35.k even 12 2 1400.2.g.e 2
56.j odd 6 1 2240.2.a.v 1
56.m even 6 1 2240.2.a.j 1
84.j odd 6 1 5040.2.a.be 1
140.s even 6 1 2800.2.a.i 1
140.x odd 12 2 2800.2.g.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.a.b 1 7.d odd 6 1
560.2.a.e 1 28.f even 6 1
1400.2.a.k 1 35.i odd 6 1
1400.2.g.e 2 35.k even 12 2
1960.2.a.k 1 7.c even 3 1
1960.2.q.e 2 1.a even 1 1 trivial
1960.2.q.e 2 7.c even 3 1 inner
1960.2.q.m 2 7.b odd 2 1
1960.2.q.m 2 7.d odd 6 1
2240.2.a.j 1 56.m even 6 1
2240.2.a.v 1 56.j odd 6 1
2520.2.a.p 1 21.g even 6 1
2800.2.a.i 1 140.s even 6 1
2800.2.g.m 2 140.x odd 12 2
3920.2.a.r 1 28.g odd 6 1
5040.2.a.be 1 84.j odd 6 1
9800.2.a.n 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 \(S_{2}^{\mathrm{new}}(1960, [\chi])\):

\( T_{3}^{2} + T_{3} + 1 \) Copy content Toggle raw display
\( T_{11}^{2} - 5T_{11} + 25 \) Copy content Toggle raw display
\( T_{13} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$19$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$23$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$29$ \( (T + 9)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$41$ \( (T + 8)^{2} \) Copy content Toggle raw display
$43$ \( (T - 6)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$53$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$59$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$61$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$67$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$71$ \( (T - 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$79$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$83$ \( (T - 12)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 16T + 256 \) Copy content Toggle raw display
$97$ \( (T + 7)^{2} \) Copy content Toggle raw display
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