Properties

Label 960.2.h.d
Level $960$
Weight $2$
Character orbit 960.h
Analytic conductor $7.666$
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: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 960.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.66563859404\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 240)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} - \beta_1 q^{5} - 2 \beta_{2} q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} - \beta_1 q^{5} - 2 \beta_{2} q^{7} + 3 q^{9} - 2 \beta_{3} q^{11} - 4 q^{13} - \beta_{2} q^{15} - 6 \beta_1 q^{17} - 2 \beta_{2} q^{19} - 6 \beta_1 q^{21} + 2 \beta_{3} q^{23} - q^{25} + 3 \beta_{3} q^{27} + 6 \beta_1 q^{29} - 2 \beta_{2} q^{31} - 6 q^{33} - 2 \beta_{3} q^{35} + 4 q^{37} - 4 \beta_{3} q^{39} + 12 \beta_1 q^{41} - 4 \beta_{2} q^{43} - 3 \beta_1 q^{45} - 2 \beta_{3} q^{47} - 5 q^{49} - 6 \beta_{2} q^{51} + 6 \beta_1 q^{53} + 2 \beta_{2} q^{55} - 6 \beta_1 q^{57} - 2 \beta_{3} q^{59} + 10 q^{61} - 6 \beta_{2} q^{63} + 4 \beta_1 q^{65} - 4 \beta_{2} q^{67} + 6 q^{69} + 8 \beta_{3} q^{71} - 2 q^{73} - \beta_{3} q^{75} + 12 \beta_1 q^{77} - 6 \beta_{2} q^{79} + 9 q^{81} + 6 \beta_{3} q^{83} - 6 q^{85} + 6 \beta_{2} q^{87} + 8 \beta_{2} q^{91} - 6 \beta_1 q^{93} - 2 \beta_{3} q^{95} + 10 q^{97} - 6 \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{9} - 16 q^{13} - 4 q^{25} - 24 q^{33} + 16 q^{37} - 20 q^{49} + 40 q^{61} + 24 q^{69} - 8 q^{73} + 36 q^{81} - 24 q^{85} + 40 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{12}^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_1 \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
−0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
0 −1.73205 0 1.00000i 0 3.46410i 0 3.00000 0
191.2 0 −1.73205 0 1.00000i 0 3.46410i 0 3.00000 0
191.3 0 1.73205 0 1.00000i 0 3.46410i 0 3.00000 0
191.4 0 1.73205 0 1.00000i 0 3.46410i 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
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b 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.h.d 4
3.b odd 2 1 inner 960.2.h.d 4
4.b odd 2 1 inner 960.2.h.d 4
8.b even 2 1 240.2.h.b 4
8.d odd 2 1 240.2.h.b 4
12.b even 2 1 inner 960.2.h.d 4
24.f even 2 1 240.2.h.b 4
24.h odd 2 1 240.2.h.b 4
40.e odd 2 1 1200.2.h.m 4
40.f even 2 1 1200.2.h.m 4
40.i odd 4 1 1200.2.o.a 4
40.i odd 4 1 1200.2.o.b 4
40.k even 4 1 1200.2.o.a 4
40.k even 4 1 1200.2.o.b 4
120.i odd 2 1 1200.2.h.m 4
120.m even 2 1 1200.2.h.m 4
120.q odd 4 1 1200.2.o.a 4
120.q odd 4 1 1200.2.o.b 4
120.w even 4 1 1200.2.o.a 4
120.w even 4 1 1200.2.o.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.2.h.b 4 8.b even 2 1
240.2.h.b 4 8.d odd 2 1
240.2.h.b 4 24.f even 2 1
240.2.h.b 4 24.h odd 2 1
960.2.h.d 4 1.a even 1 1 trivial
960.2.h.d 4 3.b odd 2 1 inner
960.2.h.d 4 4.b odd 2 1 inner
960.2.h.d 4 12.b even 2 1 inner
1200.2.h.m 4 40.e odd 2 1
1200.2.h.m 4 40.f even 2 1
1200.2.h.m 4 120.i odd 2 1
1200.2.h.m 4 120.m even 2 1
1200.2.o.a 4 40.i odd 4 1
1200.2.o.a 4 40.k even 4 1
1200.2.o.a 4 120.q odd 4 1
1200.2.o.a 4 120.w even 4 1
1200.2.o.b 4 40.i odd 4 1
1200.2.o.b 4 40.k even 4 1
1200.2.o.b 4 120.q odd 4 1
1200.2.o.b 4 120.w even 4 1

Hecke kernels

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

\( T_{7}^{2} + 12 \) Copy content Toggle raw display
\( T_{11}^{2} - 12 \) Copy content Toggle raw display
\( T_{23}^{2} - 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$13$ \( (T + 4)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$37$ \( (T - 4)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$61$ \( (T - 10)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 192)^{2} \) Copy content Toggle raw display
$73$ \( (T + 2)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 108)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( (T - 10)^{4} \) Copy content Toggle raw display
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