Properties

Label 960.2.d.c
Level $960$
Weight $2$
Character orbit 960.d
Analytic conductor $7.666$
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: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 960.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.66563859404\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + (\beta - 1) q^{5} + \beta q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + (\beta - 1) q^{5} + \beta q^{7} + q^{9} - \beta q^{11} - 2 q^{13} + (\beta - 1) q^{15} + 2 \beta q^{17} + \beta q^{19} + \beta q^{21} + 3 \beta q^{23} + ( - 2 \beta - 3) q^{25} + q^{27} + 2 \beta q^{29} + 8 q^{31} - \beta q^{33} + ( - \beta - 4) q^{35} - 10 q^{37} - 2 q^{39} - 6 q^{41} - 4 q^{43} + (\beta - 1) q^{45} + 3 \beta q^{47} + 3 q^{49} + 2 \beta q^{51} - 10 q^{53} + (\beta + 4) q^{55} + \beta q^{57} + 3 \beta q^{59} - 4 \beta q^{61} + \beta q^{63} + ( - 2 \beta + 2) q^{65} + 12 q^{67} + 3 \beta q^{69} + ( - 2 \beta - 3) q^{75} + 4 q^{77} + 16 q^{79} + q^{81} - 4 q^{83} + ( - 2 \beta - 8) q^{85} + 2 \beta q^{87} + 10 q^{89} - 2 \beta q^{91} + 8 q^{93} + ( - \beta - 4) q^{95} + 8 \beta q^{97} - \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{9} - 4 q^{13} - 2 q^{15} - 6 q^{25} + 2 q^{27} + 16 q^{31} - 8 q^{35} - 20 q^{37} - 4 q^{39} - 12 q^{41} - 8 q^{43} - 2 q^{45} + 6 q^{49} - 20 q^{53} + 8 q^{55} + 4 q^{65} + 24 q^{67} - 6 q^{75} + 8 q^{77} + 32 q^{79} + 2 q^{81} - 8 q^{83} - 16 q^{85} + 20 q^{89} + 16 q^{93} - 8 q^{95}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
1.00000i
1.00000i
0 1.00000 0 −1.00000 2.00000i 0 2.00000i 0 1.00000 0
289.2 0 1.00000 0 −1.00000 + 2.00000i 0 2.00000i 0 1.00000 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
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 960.2.d.c yes 2
3.b odd 2 1 2880.2.d.d 2
4.b odd 2 1 960.2.d.a 2
5.b even 2 1 960.2.d.b yes 2
5.c odd 4 1 4800.2.k.b 2
5.c odd 4 1 4800.2.k.g 2
8.b even 2 1 960.2.d.b yes 2
8.d odd 2 1 960.2.d.d yes 2
12.b even 2 1 2880.2.d.c 2
15.d odd 2 1 2880.2.d.b 2
16.e even 4 1 3840.2.f.b 2
16.e even 4 1 3840.2.f.c 2
16.f odd 4 1 3840.2.f.a 2
16.f odd 4 1 3840.2.f.d 2
20.d odd 2 1 960.2.d.d yes 2
20.e even 4 1 4800.2.k.c 2
20.e even 4 1 4800.2.k.f 2
24.f even 2 1 2880.2.d.a 2
24.h odd 2 1 2880.2.d.b 2
40.e odd 2 1 960.2.d.a 2
40.f even 2 1 inner 960.2.d.c yes 2
40.i odd 4 1 4800.2.k.b 2
40.i odd 4 1 4800.2.k.g 2
40.k even 4 1 4800.2.k.c 2
40.k even 4 1 4800.2.k.f 2
60.h even 2 1 2880.2.d.a 2
80.k odd 4 1 3840.2.f.a 2
80.k odd 4 1 3840.2.f.d 2
80.q even 4 1 3840.2.f.b 2
80.q even 4 1 3840.2.f.c 2
120.i odd 2 1 2880.2.d.d 2
120.m even 2 1 2880.2.d.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
960.2.d.a 2 4.b odd 2 1
960.2.d.a 2 40.e odd 2 1
960.2.d.b yes 2 5.b even 2 1
960.2.d.b yes 2 8.b even 2 1
960.2.d.c yes 2 1.a even 1 1 trivial
960.2.d.c yes 2 40.f even 2 1 inner
960.2.d.d yes 2 8.d odd 2 1
960.2.d.d yes 2 20.d odd 2 1
2880.2.d.a 2 24.f even 2 1
2880.2.d.a 2 60.h even 2 1
2880.2.d.b 2 15.d odd 2 1
2880.2.d.b 2 24.h odd 2 1
2880.2.d.c 2 12.b even 2 1
2880.2.d.c 2 120.m even 2 1
2880.2.d.d 2 3.b odd 2 1
2880.2.d.d 2 120.i odd 2 1
3840.2.f.a 2 16.f odd 4 1
3840.2.f.a 2 80.k odd 4 1
3840.2.f.b 2 16.e even 4 1
3840.2.f.b 2 80.q even 4 1
3840.2.f.c 2 16.e even 4 1
3840.2.f.c 2 80.q even 4 1
3840.2.f.d 2 16.f odd 4 1
3840.2.f.d 2 80.k odd 4 1
4800.2.k.b 2 5.c odd 4 1
4800.2.k.b 2 40.i odd 4 1
4800.2.k.c 2 20.e even 4 1
4800.2.k.c 2 40.k even 4 1
4800.2.k.f 2 20.e even 4 1
4800.2.k.f 2 40.k even 4 1
4800.2.k.g 2 5.c odd 4 1
4800.2.k.g 2 40.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(960, [\chi])\):

\( T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{13} + 2 \) Copy content Toggle raw display
\( T_{31} - 8 \) Copy content Toggle raw display
\( T_{43} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

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