Properties

Label 560.3.f.a
Level $560$
Weight $3$
Character orbit 560.f
Analytic conductor $15.259$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 560.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(15.2588948042\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-5}) \)
Defining polynomial: \( x^{2} + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta q^{3} - \beta q^{5} - 7 q^{7} - 11 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta q^{3} - \beta q^{5} - 7 q^{7} - 11 q^{9} - 2 q^{11} + 6 \beta q^{13} + 10 q^{15} - 12 \beta q^{17} - 6 \beta q^{19} - 14 \beta q^{21} - 26 q^{23} - 5 q^{25} - 4 \beta q^{27} - 22 q^{29} - 24 \beta q^{31} - 4 \beta q^{33} + 7 \beta q^{35} + 14 q^{37} - 60 q^{39} + 12 \beta q^{41} + 34 q^{43} + 11 \beta q^{45} - 12 \beta q^{47} + 49 q^{49} + 120 q^{51} - 34 q^{53} + 2 \beta q^{55} + 60 q^{57} - 18 \beta q^{59} - 42 \beta q^{61} + 77 q^{63} + 30 q^{65} - 14 q^{67} - 52 \beta q^{69} - 62 q^{71} + 24 \beta q^{73} - 10 \beta q^{75} + 14 q^{77} - 38 q^{79} - 59 q^{81} + 18 \beta q^{83} - 60 q^{85} - 44 \beta q^{87} + 12 \beta q^{89} - 42 \beta q^{91} + 240 q^{93} - 30 q^{95} + 12 \beta q^{97} + 22 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 14 q^{7} - 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 14 q^{7} - 22 q^{9} - 4 q^{11} + 20 q^{15} - 52 q^{23} - 10 q^{25} - 44 q^{29} + 28 q^{37} - 120 q^{39} + 68 q^{43} + 98 q^{49} + 240 q^{51} - 68 q^{53} + 120 q^{57} + 154 q^{63} + 60 q^{65} - 28 q^{67} - 124 q^{71} + 28 q^{77} - 76 q^{79} - 118 q^{81} - 120 q^{85} + 480 q^{93} - 60 q^{95} + 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/560\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(351\) \(421\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
321.1
2.23607i
2.23607i
0 4.47214i 0 2.23607i 0 −7.00000 0 −11.0000 0
321.2 0 4.47214i 0 2.23607i 0 −7.00000 0 −11.0000 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
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.3.f.a 2
4.b odd 2 1 35.3.d.a 2
7.b odd 2 1 inner 560.3.f.a 2
12.b even 2 1 315.3.h.b 2
20.d odd 2 1 175.3.d.f 2
20.e even 4 2 175.3.c.d 4
28.d even 2 1 35.3.d.a 2
28.f even 6 2 245.3.h.b 4
28.g odd 6 2 245.3.h.b 4
84.h odd 2 1 315.3.h.b 2
140.c even 2 1 175.3.d.f 2
140.j odd 4 2 175.3.c.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.3.d.a 2 4.b odd 2 1
35.3.d.a 2 28.d even 2 1
175.3.c.d 4 20.e even 4 2
175.3.c.d 4 140.j odd 4 2
175.3.d.f 2 20.d odd 2 1
175.3.d.f 2 140.c even 2 1
245.3.h.b 4 28.f even 6 2
245.3.h.b 4 28.g odd 6 2
315.3.h.b 2 12.b even 2 1
315.3.h.b 2 84.h odd 2 1
560.3.f.a 2 1.a even 1 1 trivial
560.3.f.a 2 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 20 \) acting on \(S_{3}^{\mathrm{new}}(560, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 20 \) Copy content Toggle raw display
$5$ \( T^{2} + 5 \) Copy content Toggle raw display
$7$ \( (T + 7)^{2} \) Copy content Toggle raw display
$11$ \( (T + 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 180 \) Copy content Toggle raw display
$17$ \( T^{2} + 720 \) Copy content Toggle raw display
$19$ \( T^{2} + 180 \) Copy content Toggle raw display
$23$ \( (T + 26)^{2} \) Copy content Toggle raw display
$29$ \( (T + 22)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 2880 \) Copy content Toggle raw display
$37$ \( (T - 14)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 720 \) Copy content Toggle raw display
$43$ \( (T - 34)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 720 \) Copy content Toggle raw display
$53$ \( (T + 34)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 1620 \) Copy content Toggle raw display
$61$ \( T^{2} + 8820 \) Copy content Toggle raw display
$67$ \( (T + 14)^{2} \) Copy content Toggle raw display
$71$ \( (T + 62)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 2880 \) Copy content Toggle raw display
$79$ \( (T + 38)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 1620 \) Copy content Toggle raw display
$89$ \( T^{2} + 720 \) Copy content Toggle raw display
$97$ \( T^{2} + 720 \) Copy content Toggle raw display
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