Properties

Label 960.3.j.b
Level $960$
Weight $3$
Character orbit 960.j
Analytic conductor $26.158$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 960.j (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(26.1581053786\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 60)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{3} -5 \zeta_{12}^{3} q^{5} + ( -12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{7} + 3 q^{9} +O(q^{10})\) \( q + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{3} -5 \zeta_{12}^{3} q^{5} + ( -12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{7} + 3 q^{9} + ( -6 + 12 \zeta_{12}^{2} ) q^{11} -18 \zeta_{12}^{3} q^{13} + ( -5 + 10 \zeta_{12}^{2} ) q^{15} -10 \zeta_{12}^{3} q^{17} + ( 8 - 16 \zeta_{12}^{2} ) q^{19} + 18 q^{21} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{23} -25 q^{25} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{27} -36 q^{29} + ( -4 + 8 \zeta_{12}^{2} ) q^{31} -18 \zeta_{12}^{3} q^{33} + ( -30 + 60 \zeta_{12}^{2} ) q^{35} + 54 \zeta_{12}^{3} q^{37} + ( -18 + 36 \zeta_{12}^{2} ) q^{39} + 18 q^{41} + ( -24 \zeta_{12} + 12 \zeta_{12}^{3} ) q^{43} -15 \zeta_{12}^{3} q^{45} + 59 q^{49} + ( -10 + 20 \zeta_{12}^{2} ) q^{51} + 26 \zeta_{12}^{3} q^{53} + ( 60 \zeta_{12} - 30 \zeta_{12}^{3} ) q^{55} + 24 \zeta_{12}^{3} q^{57} + ( -18 + 36 \zeta_{12}^{2} ) q^{59} + 74 q^{61} + ( -36 \zeta_{12} + 18 \zeta_{12}^{3} ) q^{63} -90 q^{65} + ( 48 \zeta_{12} - 24 \zeta_{12}^{3} ) q^{67} -12 q^{69} + ( -60 + 120 \zeta_{12}^{2} ) q^{71} + 36 \zeta_{12}^{3} q^{73} + ( 50 \zeta_{12} - 25 \zeta_{12}^{3} ) q^{75} -108 \zeta_{12}^{3} q^{77} + ( -52 + 104 \zeta_{12}^{2} ) q^{79} + 9 q^{81} + ( 104 \zeta_{12} - 52 \zeta_{12}^{3} ) q^{83} -50 q^{85} + ( 72 \zeta_{12} - 36 \zeta_{12}^{3} ) q^{87} + 18 q^{89} + ( -108 + 216 \zeta_{12}^{2} ) q^{91} -12 \zeta_{12}^{3} q^{93} + ( -80 \zeta_{12} + 40 \zeta_{12}^{3} ) q^{95} + 72 \zeta_{12}^{3} q^{97} + ( -18 + 36 \zeta_{12}^{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 12q^{9} + O(q^{10}) \) \( 4q + 12q^{9} + 72q^{21} - 100q^{25} - 144q^{29} + 72q^{41} + 236q^{49} + 296q^{61} - 360q^{65} - 48q^{69} + 36q^{81} - 200q^{85} + 72q^{89} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/960\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(577\) \(641\) \(901\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
319.1
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
−0.866025 0.500000i
0 −1.73205 0 5.00000i 0 −10.3923 0 3.00000 0
319.2 0 −1.73205 0 5.00000i 0 −10.3923 0 3.00000 0
319.3 0 1.73205 0 5.00000i 0 10.3923 0 3.00000 0
319.4 0 1.73205 0 5.00000i 0 10.3923 0 3.00000 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
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 960.3.j.b 4
4.b odd 2 1 inner 960.3.j.b 4
5.b even 2 1 inner 960.3.j.b 4
8.b even 2 1 60.3.f.a 4
8.d odd 2 1 60.3.f.a 4
20.d odd 2 1 inner 960.3.j.b 4
24.f even 2 1 180.3.f.e 4
24.h odd 2 1 180.3.f.e 4
40.e odd 2 1 60.3.f.a 4
40.f even 2 1 60.3.f.a 4
40.i odd 4 1 300.3.c.a 2
40.i odd 4 1 300.3.c.c 2
40.k even 4 1 300.3.c.a 2
40.k even 4 1 300.3.c.c 2
120.i odd 2 1 180.3.f.e 4
120.m even 2 1 180.3.f.e 4
120.q odd 4 1 900.3.c.f 2
120.q odd 4 1 900.3.c.j 2
120.w even 4 1 900.3.c.f 2
120.w even 4 1 900.3.c.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.f.a 4 8.b even 2 1
60.3.f.a 4 8.d odd 2 1
60.3.f.a 4 40.e odd 2 1
60.3.f.a 4 40.f even 2 1
180.3.f.e 4 24.f even 2 1
180.3.f.e 4 24.h odd 2 1
180.3.f.e 4 120.i odd 2 1
180.3.f.e 4 120.m even 2 1
300.3.c.a 2 40.i odd 4 1
300.3.c.a 2 40.k even 4 1
300.3.c.c 2 40.i odd 4 1
300.3.c.c 2 40.k even 4 1
900.3.c.f 2 120.q odd 4 1
900.3.c.f 2 120.w even 4 1
900.3.c.j 2 120.q odd 4 1
900.3.c.j 2 120.w even 4 1
960.3.j.b 4 1.a even 1 1 trivial
960.3.j.b 4 4.b odd 2 1 inner
960.3.j.b 4 5.b even 2 1 inner
960.3.j.b 4 20.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(960, [\chi])\):

\( T_{7}^{2} - 108 \)
\( T_{11}^{2} + 108 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -3 + T^{2} )^{2} \)
$5$ \( ( 25 + T^{2} )^{2} \)
$7$ \( ( -108 + T^{2} )^{2} \)
$11$ \( ( 108 + T^{2} )^{2} \)
$13$ \( ( 324 + T^{2} )^{2} \)
$17$ \( ( 100 + T^{2} )^{2} \)
$19$ \( ( 192 + T^{2} )^{2} \)
$23$ \( ( -48 + T^{2} )^{2} \)
$29$ \( ( 36 + T )^{4} \)
$31$ \( ( 48 + T^{2} )^{2} \)
$37$ \( ( 2916 + T^{2} )^{2} \)
$41$ \( ( -18 + T )^{4} \)
$43$ \( ( -432 + T^{2} )^{2} \)
$47$ \( T^{4} \)
$53$ \( ( 676 + T^{2} )^{2} \)
$59$ \( ( 972 + T^{2} )^{2} \)
$61$ \( ( -74 + T )^{4} \)
$67$ \( ( -1728 + T^{2} )^{2} \)
$71$ \( ( 10800 + T^{2} )^{2} \)
$73$ \( ( 1296 + T^{2} )^{2} \)
$79$ \( ( 8112 + T^{2} )^{2} \)
$83$ \( ( -8112 + T^{2} )^{2} \)
$89$ \( ( -18 + T )^{4} \)
$97$ \( ( 5184 + T^{2} )^{2} \)
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