Properties

Label 960.3.l.c
Level $960$
Weight $3$
Character orbit 960.l
Analytic conductor $26.158$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(26.1581053786\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-5}) \)
Defining polynomial: \(x^{2} + 5\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 2 + \beta ) q^{3} + \beta q^{5} -6 q^{7} + ( -1 + 4 \beta ) q^{9} +O(q^{10})\) \( q + ( 2 + \beta ) q^{3} + \beta q^{5} -6 q^{7} + ( -1 + 4 \beta ) q^{9} + 2 \beta q^{11} -16 q^{13} + ( -5 + 2 \beta ) q^{15} -2 \beta q^{17} + 2 q^{19} + ( -12 - 6 \beta ) q^{21} -6 \beta q^{23} -5 q^{25} + ( -22 + 7 \beta ) q^{27} -14 \beta q^{29} -18 q^{31} + ( -10 + 4 \beta ) q^{33} -6 \beta q^{35} + 16 q^{37} + ( -32 - 16 \beta ) q^{39} -28 \beta q^{41} -16 q^{43} + ( -20 - \beta ) q^{45} + 22 \beta q^{47} -13 q^{49} + ( 10 - 4 \beta ) q^{51} + 2 \beta q^{53} -10 q^{55} + ( 4 + 2 \beta ) q^{57} + 2 \beta q^{59} -82 q^{61} + ( 6 - 24 \beta ) q^{63} -16 \beta q^{65} -24 q^{67} + ( 30 - 12 \beta ) q^{69} + 56 \beta q^{71} -74 q^{73} + ( -10 - 5 \beta ) q^{75} -12 \beta q^{77} + 138 q^{79} + ( -79 - 8 \beta ) q^{81} -42 \beta q^{83} + 10 q^{85} + ( 70 - 28 \beta ) q^{87} -48 \beta q^{89} + 96 q^{91} + ( -36 - 18 \beta ) q^{93} + 2 \beta q^{95} -166 q^{97} + ( -40 - 2 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{3} - 12q^{7} - 2q^{9} + O(q^{10}) \) \( 2q + 4q^{3} - 12q^{7} - 2q^{9} - 32q^{13} - 10q^{15} + 4q^{19} - 24q^{21} - 10q^{25} - 44q^{27} - 36q^{31} - 20q^{33} + 32q^{37} - 64q^{39} - 32q^{43} - 40q^{45} - 26q^{49} + 20q^{51} - 20q^{55} + 8q^{57} - 164q^{61} + 12q^{63} - 48q^{67} + 60q^{69} - 148q^{73} - 20q^{75} + 276q^{79} - 158q^{81} + 20q^{85} + 140q^{87} + 192q^{91} - 72q^{93} - 332q^{97} - 80q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
641.1
2.23607i
2.23607i
0 2.00000 2.23607i 0 2.23607i 0 −6.00000 0 −1.00000 8.94427i 0
641.2 0 2.00000 + 2.23607i 0 2.23607i 0 −6.00000 0 −1.00000 + 8.94427i 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
3.b 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.l.c 2
3.b odd 2 1 inner 960.3.l.c 2
4.b odd 2 1 960.3.l.b 2
8.b even 2 1 15.3.c.a 2
8.d odd 2 1 240.3.l.b 2
12.b even 2 1 960.3.l.b 2
24.f even 2 1 240.3.l.b 2
24.h odd 2 1 15.3.c.a 2
40.e odd 2 1 1200.3.l.g 2
40.f even 2 1 75.3.c.e 2
40.i odd 4 2 75.3.d.b 4
40.k even 4 2 1200.3.c.f 4
72.j odd 6 2 405.3.i.b 4
72.n even 6 2 405.3.i.b 4
120.i odd 2 1 75.3.c.e 2
120.m even 2 1 1200.3.l.g 2
120.q odd 4 2 1200.3.c.f 4
120.w even 4 2 75.3.d.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.3.c.a 2 8.b even 2 1
15.3.c.a 2 24.h odd 2 1
75.3.c.e 2 40.f even 2 1
75.3.c.e 2 120.i odd 2 1
75.3.d.b 4 40.i odd 4 2
75.3.d.b 4 120.w even 4 2
240.3.l.b 2 8.d odd 2 1
240.3.l.b 2 24.f even 2 1
405.3.i.b 4 72.j odd 6 2
405.3.i.b 4 72.n even 6 2
960.3.l.b 2 4.b odd 2 1
960.3.l.b 2 12.b even 2 1
960.3.l.c 2 1.a even 1 1 trivial
960.3.l.c 2 3.b odd 2 1 inner
1200.3.c.f 4 40.k even 4 2
1200.3.c.f 4 120.q odd 4 2
1200.3.l.g 2 40.e odd 2 1
1200.3.l.g 2 120.m even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} + 6 \) acting on \(S_{3}^{\mathrm{new}}(960, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 9 - 4 T + T^{2} \)
$5$ \( 5 + T^{2} \)
$7$ \( ( 6 + T )^{2} \)
$11$ \( 20 + T^{2} \)
$13$ \( ( 16 + T )^{2} \)
$17$ \( 20 + T^{2} \)
$19$ \( ( -2 + T )^{2} \)
$23$ \( 180 + T^{2} \)
$29$ \( 980 + T^{2} \)
$31$ \( ( 18 + T )^{2} \)
$37$ \( ( -16 + T )^{2} \)
$41$ \( 3920 + T^{2} \)
$43$ \( ( 16 + T )^{2} \)
$47$ \( 2420 + T^{2} \)
$53$ \( 20 + T^{2} \)
$59$ \( 20 + T^{2} \)
$61$ \( ( 82 + T )^{2} \)
$67$ \( ( 24 + T )^{2} \)
$71$ \( 15680 + T^{2} \)
$73$ \( ( 74 + T )^{2} \)
$79$ \( ( -138 + T )^{2} \)
$83$ \( 8820 + T^{2} \)
$89$ \( 11520 + T^{2} \)
$97$ \( ( 166 + T )^{2} \)
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