Properties

Label 1960.2.q.h
Level $1960$
Weight $2$
Character orbit 1960.q
Analytic conductor $15.651$
Analytic rank $1$
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: \(1\)
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 40)
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.

\(f(q)\) \(=\) \( q -\zeta_{6} q^{5} + 3 \zeta_{6} q^{9} +O(q^{10})\) \( q -\zeta_{6} q^{5} + 3 \zeta_{6} q^{9} + ( -4 + 4 \zeta_{6} ) q^{11} -2 q^{13} + ( -2 + 2 \zeta_{6} ) q^{17} -4 \zeta_{6} q^{19} -4 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} -2 q^{29} + ( 8 - 8 \zeta_{6} ) q^{31} -6 \zeta_{6} q^{37} -6 q^{41} -8 q^{43} + ( 3 - 3 \zeta_{6} ) q^{45} -4 \zeta_{6} q^{47} + ( -6 + 6 \zeta_{6} ) q^{53} + 4 q^{55} + ( 4 - 4 \zeta_{6} ) q^{59} + 2 \zeta_{6} q^{61} + 2 \zeta_{6} q^{65} + ( -8 + 8 \zeta_{6} ) q^{67} + ( 6 - 6 \zeta_{6} ) q^{73} + ( -9 + 9 \zeta_{6} ) q^{81} -16 q^{83} + 2 q^{85} + 6 \zeta_{6} q^{89} + ( -4 + 4 \zeta_{6} ) q^{95} -14 q^{97} -12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{5} + 3q^{9} + O(q^{10}) \) \( 2q - q^{5} + 3q^{9} - 4q^{11} - 4q^{13} - 2q^{17} - 4q^{19} - 4q^{23} - q^{25} - 4q^{29} + 8q^{31} - 6q^{37} - 12q^{41} - 16q^{43} + 3q^{45} - 4q^{47} - 6q^{53} + 8q^{55} + 4q^{59} + 2q^{61} + 2q^{65} - 8q^{67} + 6q^{73} - 9q^{81} - 32q^{83} + 4q^{85} + 6q^{89} - 4q^{95} - 28q^{97} - 24q^{99} + O(q^{100}) \)

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 0 −0.500000 0.866025i 0 0 0 1.50000 + 2.59808i 0
961.1 0 0 0 −0.500000 + 0.866025i 0 0 0 1.50000 2.59808i 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.h 2
7.b odd 2 1 1960.2.q.i 2
7.c even 3 1 40.2.a.a 1
7.c even 3 1 inner 1960.2.q.h 2
7.d odd 6 1 1960.2.a.g 1
7.d odd 6 1 1960.2.q.i 2
21.h odd 6 1 360.2.a.a 1
28.f even 6 1 3920.2.a.s 1
28.g odd 6 1 80.2.a.a 1
35.i odd 6 1 9800.2.a.x 1
35.j even 6 1 200.2.a.c 1
35.l odd 12 2 200.2.c.b 2
56.k odd 6 1 320.2.a.d 1
56.p even 6 1 320.2.a.c 1
63.g even 3 1 3240.2.q.k 2
63.h even 3 1 3240.2.q.k 2
63.j odd 6 1 3240.2.q.x 2
63.n odd 6 1 3240.2.q.x 2
77.h odd 6 1 4840.2.a.f 1
84.n even 6 1 720.2.a.e 1
91.r even 6 1 6760.2.a.i 1
105.o odd 6 1 1800.2.a.v 1
105.x even 12 2 1800.2.f.a 2
112.u odd 12 2 1280.2.d.a 2
112.w even 12 2 1280.2.d.j 2
140.p odd 6 1 400.2.a.e 1
140.w even 12 2 400.2.c.d 2
168.s odd 6 1 2880.2.a.t 1
168.v even 6 1 2880.2.a.bg 1
280.bf even 6 1 1600.2.a.o 1
280.bi odd 6 1 1600.2.a.k 1
280.br even 12 2 1600.2.c.m 2
280.bt odd 12 2 1600.2.c.k 2
308.n even 6 1 9680.2.a.q 1
420.ba even 6 1 3600.2.a.h 1
420.bp odd 12 2 3600.2.f.t 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.2.a.a 1 7.c even 3 1
80.2.a.a 1 28.g odd 6 1
200.2.a.c 1 35.j even 6 1
200.2.c.b 2 35.l odd 12 2
320.2.a.c 1 56.p even 6 1
320.2.a.d 1 56.k odd 6 1
360.2.a.a 1 21.h odd 6 1
400.2.a.e 1 140.p odd 6 1
400.2.c.d 2 140.w even 12 2
720.2.a.e 1 84.n even 6 1
1280.2.d.a 2 112.u odd 12 2
1280.2.d.j 2 112.w even 12 2
1600.2.a.k 1 280.bi odd 6 1
1600.2.a.o 1 280.bf even 6 1
1600.2.c.k 2 280.bt odd 12 2
1600.2.c.m 2 280.br even 12 2
1800.2.a.v 1 105.o odd 6 1
1800.2.f.a 2 105.x even 12 2
1960.2.a.g 1 7.d odd 6 1
1960.2.q.h 2 1.a even 1 1 trivial
1960.2.q.h 2 7.c even 3 1 inner
1960.2.q.i 2 7.b odd 2 1
1960.2.q.i 2 7.d odd 6 1
2880.2.a.t 1 168.s odd 6 1
2880.2.a.bg 1 168.v even 6 1
3240.2.q.k 2 63.g even 3 1
3240.2.q.k 2 63.h even 3 1
3240.2.q.x 2 63.j odd 6 1
3240.2.q.x 2 63.n odd 6 1
3600.2.a.h 1 420.ba even 6 1
3600.2.f.t 2 420.bp odd 12 2
3920.2.a.s 1 28.f even 6 1
4840.2.a.f 1 77.h odd 6 1
6760.2.a.i 1 91.r even 6 1
9680.2.a.q 1 308.n even 6 1
9800.2.a.x 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 \(S_{2}^{\mathrm{new}}(1960, [\chi])\):

\( T_{3} \)
\( T_{11}^{2} + 4 T_{11} + 16 \)
\( T_{13} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 1 + T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 16 + 4 T + T^{2} \)
$13$ \( ( 2 + T )^{2} \)
$17$ \( 4 + 2 T + T^{2} \)
$19$ \( 16 + 4 T + T^{2} \)
$23$ \( 16 + 4 T + T^{2} \)
$29$ \( ( 2 + T )^{2} \)
$31$ \( 64 - 8 T + T^{2} \)
$37$ \( 36 + 6 T + T^{2} \)
$41$ \( ( 6 + T )^{2} \)
$43$ \( ( 8 + T )^{2} \)
$47$ \( 16 + 4 T + T^{2} \)
$53$ \( 36 + 6 T + T^{2} \)
$59$ \( 16 - 4 T + T^{2} \)
$61$ \( 4 - 2 T + T^{2} \)
$67$ \( 64 + 8 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 36 - 6 T + T^{2} \)
$79$ \( T^{2} \)
$83$ \( ( 16 + T )^{2} \)
$89$ \( 36 - 6 T + T^{2} \)
$97$ \( ( 14 + T )^{2} \)
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