Properties

Label 1560.2.w.d
Level $1560$
Weight $2$
Character orbit 1560.w
Analytic conductor $12.457$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1560.w (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.4566627153\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} - \beta_{2} q^{3} + ( - \beta_{3} + 2 \beta_{2} - \beta_1) q^{4} + \beta_{2} q^{5} + ( - \beta_{3} - 1) q^{6} + ( - \beta_{3} + \beta_{2} - \beta_1 + 3) q^{7} + (2 \beta_{2} + 2) q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} - \beta_{2} q^{3} + ( - \beta_{3} + 2 \beta_{2} - \beta_1) q^{4} + \beta_{2} q^{5} + ( - \beta_{3} - 1) q^{6} + ( - \beta_{3} + \beta_{2} - \beta_1 + 3) q^{7} + (2 \beta_{2} + 2) q^{8} - q^{9} + (\beta_{3} + 1) q^{10} + (\beta_{3} + 3 \beta_{2} - \beta_1 + 1) q^{11} + ( - \beta_{3} + \beta_1) q^{12} + \beta_{2} q^{13} + ( - \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 4) q^{14} + q^{15} + (2 \beta_{3} - 2 \beta_1 + 4) q^{16} - 4 q^{17} + (\beta_1 - 1) q^{18} + (\beta_{3} - \beta_{2} - \beta_1 + 1) q^{19} + (\beta_{3} - \beta_1) q^{20} + ( - \beta_{3} - 3 \beta_{2} + \beta_1 - 1) q^{21} + (3 \beta_{3} + 2 \beta_{2} - \beta_1 + 2) q^{22} - 2 q^{23} + ( - 2 \beta_{2} + 2) q^{24} - q^{25} + (\beta_{3} + 1) q^{26} + \beta_{2} q^{27} + ( - 2 \beta_{3} + 6 \beta_{2} - 4 \beta_1 + 4) q^{28} + ( - 4 \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 4) q^{29} + ( - \beta_1 + 1) q^{30} + ( - 3 \beta_{3} + 3 \beta_{2} - 3 \beta_1 - 5) q^{31} + (4 \beta_{2} - 4 \beta_1) q^{32} + ( - \beta_{3} + \beta_{2} - \beta_1 + 3) q^{33} + (4 \beta_1 - 4) q^{34} + (\beta_{3} + 3 \beta_{2} - \beta_1 + 1) q^{35} + (\beta_{3} - 2 \beta_{2} + \beta_1) q^{36} + ( - 2 \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 2) q^{37} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 - 2) q^{38} + q^{39} + (2 \beta_{2} - 2) q^{40} + (4 \beta_{3} - 4 \beta_{2} + 4 \beta_1 + 2) q^{41} + ( - 3 \beta_{3} - 2 \beta_{2} + \beta_1 - 2) q^{42} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{43} + (4 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{44} - \beta_{2} q^{45} + (2 \beta_1 - 2) q^{46} + (\beta_{3} - \beta_{2} + \beta_1 - 7) q^{47} + ( - 2 \beta_{3} - 2 \beta_1) q^{48} + ( - 6 \beta_{3} + 6 \beta_{2} - 6 \beta_1 + 5) q^{49} + (\beta_1 - 1) q^{50} + 4 \beta_{2} q^{51} + (\beta_{3} - \beta_1) q^{52} + ( - 6 \beta_{3} - 2 \beta_{2} + 6 \beta_1 - 6) q^{53} + (\beta_{3} + 1) q^{54} + (\beta_{3} - \beta_{2} + \beta_1 - 3) q^{55} + (8 \beta_{2} - 4 \beta_1 + 8) q^{56} + ( - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{57} + (2 \beta_{3} - 8 \beta_{2} + 4 \beta_1 + 6) q^{58} + ( - 5 \beta_{3} + \beta_{2} + 5 \beta_1 - 5) q^{59} + ( - \beta_{3} + 2 \beta_{2} - \beta_1) q^{60} + (4 \beta_{3} - 4 \beta_{2} - 4 \beta_1 + 4) q^{61} + ( - 3 \beta_{3} + 6 \beta_{2} + 5 \beta_1 - 2) q^{62} + (\beta_{3} - \beta_{2} + \beta_1 - 3) q^{63} + 8 \beta_{2} q^{64} - q^{65} + ( - \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 4) q^{66} + ( - 3 \beta_{3} - 7 \beta_{2} + 3 \beta_1 - 3) q^{67} + (4 \beta_{3} - 8 \beta_{2} + 4 \beta_1) q^{68} + 2 \beta_{2} q^{69} + (3 \beta_{3} + 2 \beta_{2} - \beta_1 + 2) q^{70} + (\beta_{3} - \beta_{2} + \beta_1 + 15) q^{71} + ( - 2 \beta_{2} - 2) q^{72} + ( - 8 \beta_{3} + 8 \beta_{2} - 8 \beta_1 + 2) q^{73} + (4 \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 6) q^{74} + \beta_{2} q^{75} + (2 \beta_{2} + 2 \beta_1) q^{76} + (6 \beta_{3} + 12 \beta_{2} - 6 \beta_1 + 6) q^{77} + ( - \beta_1 + 1) q^{78} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 14) q^{79} + (2 \beta_{3} + 2 \beta_1) q^{80} + q^{81} + (4 \beta_{3} - 8 \beta_{2} - 2 \beta_1 - 2) q^{82} + (\beta_{3} - 7 \beta_{2} - \beta_1 + 1) q^{83} + ( - 4 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{84} - 4 \beta_{2} q^{85} + (2 \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 4) q^{86} + (4 \beta_{3} - 4 \beta_{2} + 4 \beta_1 + 2) q^{87} + (4 \beta_{3} + 4 \beta_{2} - 4) q^{88} + ( - 4 \beta_{3} + 4 \beta_{2} - 4 \beta_1 - 6) q^{89} + ( - \beta_{3} - 1) q^{90} + (\beta_{3} + 3 \beta_{2} - \beta_1 + 1) q^{91} + (2 \beta_{3} - 4 \beta_{2} + 2 \beta_1) q^{92} + ( - 3 \beta_{3} + 5 \beta_{2} + 3 \beta_1 - 3) q^{93} + (\beta_{3} - 2 \beta_{2} + 7 \beta_1 - 8) q^{94} + (\beta_{3} - \beta_{2} + \beta_1 + 1) q^{95} + ( - 4 \beta_{3} + 4 \beta_{2}) q^{96} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 4) q^{97} + ( - 6 \beta_{3} + 12 \beta_{2} - 5 \beta_1 + 11) q^{98} + ( - \beta_{3} - 3 \beta_{2} + \beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{6} + 12 q^{7} + 8 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 2 q^{6} + 12 q^{7} + 8 q^{8} - 4 q^{9} + 2 q^{10} + 4 q^{12} + 12 q^{14} + 4 q^{15} + 8 q^{16} - 16 q^{17} - 2 q^{18} - 4 q^{20} - 8 q^{23} + 8 q^{24} - 4 q^{25} + 2 q^{26} + 12 q^{28} + 2 q^{30} - 20 q^{31} - 8 q^{32} + 12 q^{33} - 8 q^{34} - 8 q^{38} + 4 q^{39} - 8 q^{40} + 8 q^{41} - 12 q^{44} - 4 q^{46} - 28 q^{47} + 20 q^{49} - 2 q^{50} - 4 q^{52} + 2 q^{54} - 12 q^{55} + 24 q^{56} - 4 q^{57} + 28 q^{58} + 8 q^{62} - 12 q^{63} - 4 q^{65} + 12 q^{66} + 60 q^{71} - 8 q^{72} + 8 q^{73} + 20 q^{74} + 4 q^{76} + 2 q^{78} + 56 q^{79} + 4 q^{81} - 20 q^{82} + 12 q^{84} + 16 q^{86} + 8 q^{87} - 24 q^{88} - 24 q^{89} - 2 q^{90} - 20 q^{94} + 4 q^{95} + 8 q^{96} - 16 q^{97} + 46 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{2} + \zeta_{12} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{2} + \zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( -\beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1560\mathbb{Z}\right)^\times\).

\(n\) \(391\) \(521\) \(781\) \(937\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
781.1
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i
−0.366025 1.36603i 1.00000i −1.73205 + 1.00000i 1.00000i −1.36603 + 0.366025i 1.26795 2.00000 + 2.00000i −1.00000 1.36603 0.366025i
781.2 −0.366025 + 1.36603i 1.00000i −1.73205 1.00000i 1.00000i −1.36603 0.366025i 1.26795 2.00000 2.00000i −1.00000 1.36603 + 0.366025i
781.3 1.36603 0.366025i 1.00000i 1.73205 1.00000i 1.00000i 0.366025 + 1.36603i 4.73205 2.00000 2.00000i −1.00000 −0.366025 1.36603i
781.4 1.36603 + 0.366025i 1.00000i 1.73205 + 1.00000i 1.00000i 0.366025 1.36603i 4.73205 2.00000 + 2.00000i −1.00000 −0.366025 + 1.36603i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1560.2.w.d 4
4.b odd 2 1 6240.2.w.c 4
8.b even 2 1 inner 1560.2.w.d 4
8.d odd 2 1 6240.2.w.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.w.d 4 1.a even 1 1 trivial
1560.2.w.d 4 8.b even 2 1 inner
6240.2.w.c 4 4.b odd 2 1
6240.2.w.c 4 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1560, [\chi])\):

\( T_{7}^{2} - 6T_{7} + 6 \) Copy content Toggle raw display
\( T_{11}^{4} + 24T_{11}^{2} + 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4 \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 6 T + 6)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 24T^{2} + 36 \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$17$ \( (T + 4)^{4} \) Copy content Toggle raw display
$19$ \( T^{4} + 8T^{2} + 4 \) Copy content Toggle raw display
$23$ \( (T + 2)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} + 104T^{2} + 1936 \) Copy content Toggle raw display
$31$ \( (T^{2} + 10 T - 2)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 56T^{2} + 16 \) Copy content Toggle raw display
$41$ \( (T^{2} - 4 T - 44)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 32T^{2} + 64 \) Copy content Toggle raw display
$47$ \( (T^{2} + 14 T + 46)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 224 T^{2} + 10816 \) Copy content Toggle raw display
$59$ \( T^{4} + 152T^{2} + 5476 \) Copy content Toggle raw display
$61$ \( T^{4} + 128T^{2} + 1024 \) Copy content Toggle raw display
$67$ \( T^{4} + 152T^{2} + 484 \) Copy content Toggle raw display
$71$ \( (T^{2} - 30 T + 222)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 4 T - 188)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 28 T + 184)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 104T^{2} + 2116 \) Copy content Toggle raw display
$89$ \( (T^{2} + 12 T - 12)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 8 T + 4)^{2} \) Copy content Toggle raw display
show more
show less