Properties

Label 960.2.b.a
Level $960$
Weight $2$
Character orbit 960.b
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.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.66563859404\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - q^{5} + ( - \beta_{2} + \beta_1) q^{7} + ( - \beta_{3} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - q^{5} + ( - \beta_{2} + \beta_1) q^{7} + ( - \beta_{3} + 1) q^{9} + (\beta_{2} - \beta_1) q^{11} - \beta_{3} q^{13} - \beta_1 q^{15} - \beta_{3} q^{17} + (2 \beta_{2} + 2 \beta_1) q^{19} + ( - \beta_{3} - 2) q^{21} + ( - 3 \beta_{2} - 3 \beta_1) q^{23} + q^{25} + ( - 3 \beta_{2} + 2 \beta_1) q^{27} + 2 q^{29} + ( - 3 \beta_{2} + 3 \beta_1) q^{31} + (\beta_{3} + 2) q^{33} + (\beta_{2} - \beta_1) q^{35} + \beta_{3} q^{37} + ( - 3 \beta_{2} + \beta_1) q^{39} - 4 \beta_{3} q^{41} + ( - 3 \beta_{2} - 3 \beta_1) q^{43} + (\beta_{3} - 1) q^{45} + ( - \beta_{2} - \beta_1) q^{47} + 3 q^{49} + ( - 3 \beta_{2} + \beta_1) q^{51} + 10 q^{53} + ( - \beta_{2} + \beta_1) q^{55} + ( - 2 \beta_{3} + 8) q^{57} + (5 \beta_{2} - 5 \beta_1) q^{59} + ( - 3 \beta_{2} - \beta_1) q^{63} + \beta_{3} q^{65} + ( - \beta_{2} - \beta_1) q^{67} + (3 \beta_{3} - 12) q^{69} + (2 \beta_{2} + 2 \beta_1) q^{71} + 10 q^{73} + \beta_1 q^{75} + 4 q^{77} + (\beta_{2} - \beta_1) q^{79} + ( - 2 \beta_{3} - 7) q^{81} + ( - \beta_{2} + \beta_1) q^{83} + \beta_{3} q^{85} + 2 \beta_1 q^{87} - 2 \beta_{3} q^{89} + ( - 2 \beta_{2} - 2 \beta_1) q^{91} + ( - 3 \beta_{3} - 6) q^{93} + ( - 2 \beta_{2} - 2 \beta_1) q^{95} - 14 q^{97} + (3 \beta_{2} + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{5} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{5} + 4 q^{9} - 8 q^{21} + 4 q^{25} + 8 q^{29} + 8 q^{33} - 4 q^{45} + 12 q^{49} + 40 q^{53} + 32 q^{57} - 48 q^{69} + 40 q^{73} + 16 q^{77} - 28 q^{81} - 24 q^{93} - 56 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( -\zeta_{8}^{3} - \zeta_{8}^{2} + \zeta_{8} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\zeta_{8}^{3} + 2\zeta_{8} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_{3} + \beta_{2} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( \beta_{3} - \beta_{2} - \beta_1 ) / 4 \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
671.1
−0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
0.707107 0.707107i
0 −1.41421 1.00000i 0 −1.00000 0 2.00000i 0 1.00000 + 2.82843i 0
671.2 0 −1.41421 + 1.00000i 0 −1.00000 0 2.00000i 0 1.00000 2.82843i 0
671.3 0 1.41421 1.00000i 0 −1.00000 0 2.00000i 0 1.00000 2.82843i 0
671.4 0 1.41421 + 1.00000i 0 −1.00000 0 2.00000i 0 1.00000 + 2.82843i 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
24.f even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 960.2.b.a 4
3.b odd 2 1 960.2.b.b yes 4
4.b odd 2 1 inner 960.2.b.a 4
8.b even 2 1 960.2.b.b yes 4
8.d odd 2 1 960.2.b.b yes 4
12.b even 2 1 960.2.b.b yes 4
24.f even 2 1 inner 960.2.b.a 4
24.h odd 2 1 inner 960.2.b.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
960.2.b.a 4 1.a even 1 1 trivial
960.2.b.a 4 4.b odd 2 1 inner
960.2.b.a 4 24.f even 2 1 inner
960.2.b.a 4 24.h odd 2 1 inner
960.2.b.b yes 4 3.b odd 2 1
960.2.b.b yes 4 8.b even 2 1
960.2.b.b yes 4 8.d odd 2 1
960.2.b.b yes 4 12.b even 2 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
\( T_{29} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 2T^{2} + 9 \) Copy content Toggle raw display
$5$ \( (T + 1)^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 72)^{2} \) Copy content Toggle raw display
$29$ \( (T - 2)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 128)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 72)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$53$ \( (T - 10)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$73$ \( (T - 10)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$97$ \( (T + 14)^{4} \) Copy content Toggle raw display
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