Properties

Label 1560.2.w.c
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: \(\Q(\zeta_{8})\)
Defining polynomial: \( x^{4} + 1 \) 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.

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

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 2 \) 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
−0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i
0.707107 + 0.707107i
−1.41421 1.00000i 2.00000 1.00000i 1.41421i −3.41421 −2.82843 −1.00000 1.41421i
781.2 −1.41421 1.00000i 2.00000 1.00000i 1.41421i −3.41421 −2.82843 −1.00000 1.41421i
781.3 1.41421 1.00000i 2.00000 1.00000i 1.41421i −0.585786 2.82843 −1.00000 1.41421i
781.4 1.41421 1.00000i 2.00000 1.00000i 1.41421i −0.585786 2.82843 −1.00000 1.41421i
\(n\): 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.c 4
4.b odd 2 1 6240.2.w.d 4
8.b even 2 1 inner 1560.2.w.c 4
8.d odd 2 1 6240.2.w.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.w.c 4 1.a even 1 1 trivial
1560.2.w.c 4 8.b even 2 1 inner
6240.2.w.d 4 4.b odd 2 1
6240.2.w.d 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 \(S_{2}^{\mathrm{new}}(1560, [\chi])\):

\( T_{7}^{2} + 4T_{7} + 2 \) Copy content Toggle raw display
\( T_{11}^{4} + 24T_{11}^{2} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2)^{2} \) 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
$7$ \( (T^{2} + 4 T + 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 24T^{2} + 16 \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 68T^{2} + 4 \) Copy content Toggle raw display
$23$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 108T^{2} + 324 \) Copy content Toggle raw display
$31$ \( (T^{2} - 8 T + 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 152T^{2} + 4624 \) Copy content Toggle raw display
$41$ \( (T^{2} + 16 T + 56)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 24T^{2} + 16 \) Copy content Toggle raw display
$47$ \( (T^{2} + 8 T - 16)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 136T^{2} + 16 \) Copy content Toggle raw display
$59$ \( T^{4} + 88T^{2} + 784 \) Copy content Toggle raw display
$61$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 48T^{2} + 64 \) Copy content Toggle raw display
$71$ \( (T^{2} - 4 T - 4)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 20 T + 98)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 8 T - 16)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 72T^{2} + 784 \) Copy content Toggle raw display
$89$ \( (T^{2} - 4 T - 28)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 12 T - 126)^{2} \) Copy content Toggle raw display
show more
show less